diff --git a/.gitignore b/.gitignore index eb890d2..80d4ca5 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,3 @@ -latex/ + __pycache__/ diff --git a/combined_scenario_analysis.png b/combined_scenario_analysis.png deleted file mode 100644 index 80c5c99..0000000 Binary files a/combined_scenario_analysis.png and /dev/null differ diff --git a/latex/.Rhistory b/latex/.Rhistory new file mode 100644 index 0000000..4d2e445 --- /dev/null +++ b/latex/.Rhistory @@ -0,0 +1,512 @@ +df = tibble( +a = c(1,4,3,7), +b = a + rnorm(4) +) +df +df %>% +pivot_longer(1:2, names_to = "var", values_to = "val") +df %>% +pivot_longer(1:2, names_to = "var", values_to = "val") %>% +ggplot(aes(y = val, color = var)) + +geom_point() +df %>% +pivot_longer(1:2, names_to = "var", values_to = "val") %>% +ggplot(aes(x = 1:4, y = val, color = var)) + +geom_point() +df %>% +pivot_longer(1:2, names_to = "var", values_to = "val") %>% +mutate(x = 1:n()) %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() +set.seed(1) +df = tibble( +x = 1:4, +a = c(1,4,3,7), +b = a + rnorm(4) +) +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +mutate(x = 1:n()) %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() +?geom_ribbon +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() + +geom_ribbon(ymin = min(val), ymax = max(val)) +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() + +geom_ribbon(aes(ymin = min(val), ymax = max(val))) +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() + +geom_ribbon(aes(ymin = val, ymax = val)) +df +df %>% +ggplot(aes(x = x)) + +geom_ribbon(aes(ymin = a, ymax = b)) +df %>% +ggplot(aes(x = x)) + +geom_errorbar(aes(ymin = a, ymax = b)) +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val)) + +geom_line() +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_line() +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_vline() +?kmeans +36170/1228 +36170/1338 +library(tidyverse) +install.packages("D:/gurobi950/win64/R/gurobi_9.5-0.zip", repos = NULL, type = "win.binary") +install.packages("slam") +library(gurobi) +library(gurobi) +model = list() +model$A = matrix(c(1,1,0,0,1,1), nrow = 2, byrow = TRUE) +model$obj = c(1,2,3) +model$modelsense = "max" +model$rhs = c(1,1) +model$sense = c("<=", "<=") +result = gurobi(model) +result$objval +?c_across +library(tidyverse) +?c_across +library(mlr3verse) +library(mlr3proba) +library(survival) +task = tsk("rats") +learner = lrn("surv.coxh") +learner = lrn("surv.coxph") +measure = msr("surv.hung_auc") +install.packages("survAUC") +measure = msr("surv.hung_auc") +pred = learner$train(task)$predict(task) +pred$score(measure) +measure = msr("surv.cindex") +pred$score(measure) +measure = msr("surv.hung_auc", times = 60) +pred$score(measure, task) +task$data() +pred$score(measure, task, row_ids = 1:270) +pred = learner$train(task, row_ids = 1:270)$predict(task, row_ids = 271:300) +pred$score(measure, task) +pred$score(measure, task, train_set = 1:270) +pred = learner$train(task)$predict(task) +pred$score(measure, task, train_set = 1:270) +?survAUC::AUC.hc() +learner = lrn("surv.coxph") +measure = msr("surv.hung_auc", times = 60) +pred = learner$train(task, row_ids = 1:270)$predict(task, row_ids = 271:300) +pred$score(measure, task, train_set = 1:270) +library(tidyverse) +?mean +df = tibble( +x = c(2,NA,5,7), +y = LETTERS[1:4], +z = c(5,6,NA,NA) +) +df +df %>% +mutate(across(where(is.numeric), ~ .x[is.na(.x)] = mean(.x, na.rm = TRUE))) +?replace_na +starwars %>% +mutate(across(where(is.numeric)), ~ replace_na(.x, mean(.x, na.rm = TRUE))) +starwars %>% +mutate(across(where(is.numeric), ~ replace_na(.x, mean(.x, na.rm = TRUE)))) +library(tinytex) +pdflatex("mcmthesis.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex(mcmthesis-demo.tex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(timetk) +holidays <- tk_make_holiday_sequence( +start_date = "2013-01-01", +end_date = "2013-12-31", +calendar = "NYSE") +holidays +idx <- tk_make_weekday_sequence("2012", +remove_weekends = TRUE, +remove_holidays = TRUE, calendar = "NYSE") +idx +tk_get_timeseries_summary(idx) +idx <- tk_make_weekday_sequence("2012", +remove_weekends = TRUE, +remove_holidays = TRUE, calendar = "NYSE") +tk_get_timeseries_summary(idx) +tk_make_holiday_sequence("2012", calendar = "NYSE") +?tk_make_weekday_sequence +?slidify +FB <- FANG %>% filter(symbol == "FB") +library(tidyverse) +library(tidyquant) +FB <- FANG %>% filter(symbol == "FB") +mean_roll_5 <- slidify(mean, .period = 5, .align = "right") +FB %>% +mutate(rolling_mean_5 = mean_roll_5(adjusted)) +as_mapper(~ .x ^ 2 + 1) +?roll_toslide +?roll_tos_lide +?roll_to_slide +?pslide +remove.packages("timetk") +install.packages("timetk") +install.packages("timetk") +idx <- tk_make_weekday_sequence("2012", +remove_weekends = TRUE, +remove_holidays = TRUE, calendar = "NYSE") +tk_get_timeseries_summary(idx) +library(timetk) +idx <- tk_make_weekday_sequence("2012", +remove_weekends = TRUE, +remove_holidays = TRUE, calendar = "NYSE") +tk_get_timeseries_summary(idx) +?tk_make_weekday_sequence +holidays <- tk_make_holiday_sequence( +start_date = "2016", +end_date = "2017", +calendar = "NYSE") +weekends <- tk_make_weekend_sequence( +start_date = "2016", +end_date = "2017") +holidays +weekends +weekends <- tk_make_weekend_sequence( +start_date = "2016-01-01", +end_date = "2017-12-31") +weekends +weekends <- tk_make_weekdays_sequence( +start_date = "2016-01-01", +end_date = "2017-12-31") +weekends <- tk_make_weekday_sequence( +start_date = "2016-01-01", +end_date = "2017-12-31") +weekends +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(lubridate) +begin = ymd_hm("2019-08-10 14:00") +end = ymd_hm("2020-03-05 18:15") +interval(begin, end) +begin %--% end +library(tinytex) +xelatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("MCM-ICM_Summary.tex") +pdflatex("MCM-ICM_Summary.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tidyverse) +df = tribble( +~A, ~B +北京, 上海 +df = tribble( +~A, ~B, +北京, 上海 +南京, 西安 +df = tribble( +~A, ~B, +北京, 上海, +南京, 西安, +上海, 南京, +合肥, 沈阳, +上海, 北京, +上海, 南京, +西安, 南京, +天津, 广州) +?tribble +df = tribble( +~A, ~B, +北京, 上海, +南京, 西安, +上海, 南京, +合肥, 沈阳, +上海, 北京, +上海, 南京, +西安, 南京, +天津, 广州) +df = tribble( +~A, ~B, +"北京", "上海", +"南京", "西安", +"上海", "南京", +"合肥", "沈阳", +"上海", "北京", +"上海", "南京", +"西安", "南京", +"天津", "广州") +df +?distinct +library(sets) +install.packages("sets") +library(sets) +df %>% +mutate(C = set(A, B)) +df = tibble(x = c(1:3,3:1,4), y = c(2,4,4,2,1,2,1)) +library(tidyverse) +df = tibble(x = c(1:3,3:1,4), y = c(2,4,4,2,1,2,1)) +df +library(tidyverse) +df = tibble(x = c(1:3,3:1,4), y = c(2,4,4,2,1,2,1)) +df +f = function(...) { +dummy = rep(0, 4) +dummy[c(...)] = 1 +dummy +} +df = tibble(x = c(1:3,3:1,NA), y = c(2,4,4,2,1,2,1)) +df +f = function(...) { +dummy = rep(NA, 4) +dummy[c(...)] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +f = function(...) { +v = c(...) +dummy = ifelse(any(is.na(v)), rep(NA, 4), rep(0,4)) +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +df +f = function(...) { +v = c(...) +dummy = ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +v = c(NA, 1) +any(is.na(v)) +dummy = rep(NA, 4) +dummy[v] +dummy[!is.na(v)] +vv +v +dummy[v] = 1 +dummy +v = c(2,4) +any(is.na(v)) +dummy = rep(0,4) +dummy +dummy[v] = 1 +dummy +df +f(1,2) +f(1,NA) +f(c(1,2)) +df +v = c(2,4) +dummy = ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +dummy +any(is.na(v)) +rep(0,4) +rep(NA,4) +is.na(v) +v = c(2.4) +v = c(2,4) +dummy = ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +any(is.na(v)) +ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +if(any(is.na(v)) rep(NA,4) +if(any(is.na(v))) rep(NA,4) +else rep(0,4) +if(any(is.na(v))) dummy = rep(NA,4) +else dummy = rep(0,4) +if(any(is.na(v))) { +dummy = rep(NA,4) +} else { +dummy = rep(0,4) +} +f = function(...) { +v = c(...) +if(any(is.na(v))) { +dummy = rep(NA,4) +} else { +dummy = rep(0,4) +} +dummy[v] = 1 +dummy +} +f(2,4) +f(1,2) +f(1,NA) +f(NA,4) +f = function(...) { +v = c(...) +if(any(is.na(v))) { +dummy = rep(NA,4) +} else { +dummy = rep(0,4) +} +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +f = function(...) { +v = c(...) +dummy = ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +f = function(...) { +v = c(...) +dummy = ifelse(all(!is.na(v)), rep(0,4), rep(NA,4)) +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +v = c(2,4) +ifelse(all(!is.na(v)), rep(0,4), rep(NA,4)) +rep(0,4) +3827.64 / 600 +(3827.64 - 800) * 0.2 * 0.7 +(3827.64 - 423.87) / 600 +5103.51 * 0.8 * 0.2 * 0.7 +(5103.51 - 571.59) / 800 +12758.78 * 0.8 * 0.2 * 0.7 +(12758.78 - 1428.98) / 2000 +df = tibble(x = c(1:3,3:1,NA), y = c(2,4,NA,2,1,2,1)) +df +f = function(...) { +v = c(...) +if(any(is.na(v))) { +dummy = rep(NA,4) +} else { +dummy = rep(0,4) +} +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +df = tibble(x = c(1:3,3:1,NA), y = c(2,4,NA,2,1,2,NA)) +df +f = function(...) { +v = c(...) +if(any(is.na(v))) { +dummy = rep(NA,4) +} else { +dummy = rep(0,4) +} +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +df +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +3827.63 / 600 +5.67 / 6.38 +80000 * 0.8887147 +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(tinytex) 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"read_only_alternatives": [] +} \ No newline at end of file diff --git a/latex/.Rproj.user/07BFC725/sources/per/t/461C3D72-contents b/latex/.Rproj.user/07BFC725/sources/per/t/461C3D72-contents new file mode 100644 index 0000000..8c1d3bf --- /dev/null +++ b/latex/.Rproj.user/07BFC725/sources/per/t/461C3D72-contents @@ -0,0 +1,539 @@ +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {0000000}, problem = {A}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} +\usepackage[backend=bibtex]{biblatex} % for RStudio Complie + +\usepackage{tocloft} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Enjoy a Cozy and Green Bath} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} + +\begin{abstract} + +A traditional bathtub cannot be reheated by itself, so users have to add hot water from time to time. Our goal is to establish a model of the temperature of bath water in space and time. Then we are expected to propose an optimal strategy for users to keep the temperature even and close to initial temperature and decrease water consumption. + +To simplify modeling process, we firstly assume there is no person in the bathtub. We regard the whole bathtub as a thermodynamic system and introduce heat transfer formulas. + +We establish two sub-models: adding water constantly and discontinuously. As for the former sub-model, we define the mean temperature of bath water. Introducing Newton cooling formula, we determine the heat transfer capacity. After deriving the value of parameters, we deduce formulas to derive results and simulate the change of temperature field via CFD. As for the second sub-model, we define an iteration consisting of two process: heating and standby. According to energy conservation law, we obtain the relationship of time and total heat dissipating capacity. Then we determine the mass flow and the time of adding hot water. We also use CFD to simulate the temperature field in second sub-model. + +In consideration of evaporation, we correct the results of sub-models referring to some scientists' studies. We define two evaluation criteria and compare the two sub-models. Adding water constantly is found to keep the temperature of bath water even and avoid wasting too much water, so it is recommended by us. + +Then we determine the influence of some factors: radiation heat transfer, the shape and volume of the tub, the shape/volume/temperature/motions of the person, the bubbles made from bubble bath additives. We focus on the influence of those factors to heat transfer and then conduct sensitivity analysis. The results indicate smaller bathtub with less surface area, lighter personal mass, less motions and more bubbles will decrease heat transfer and save water. + +Based on our model analysis and conclusions, we propose the optimal strategy for the user in a bathtub and explain the reason of uneven temperature throughout the bathtub. In addition, we make improvement for applying our model in real life. + +\begin{keywords} +Heat transfer, Thermodynamic system, CFD, Energy conservation +\end{keywords} + +\end{abstract} + +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 + +\newpage + + +\section{Introduction} + +\subsection{Background} + +Bathing in a tub is a perfect choice for those who have been worn out after a long day's working. A traditional bathtub is a simply water containment vessel without a secondary heating system or circulating jets. Thus the temperature of water in bathtub declines noticeably as time goes by, which will influent the experience of bathing. As a result, the bathing person needs to add a constant trickle of hot water from a faucet to reheat the bathing water. This way of bathing will result in waste of water because when the capacity of the bathtub is reached, excess water overflows the tub. + +An optimal bathing strategy is required for the person in a bathtub to get comfortable bathing experience while reducing the waste of water. + +\subsection{Literature Review} + +Korean physicist Gi-Beum Kim analyzed heat loss through free surface of water contained in bathtub due to conduction and evaporation \cite{1}. He derived a relational equation based on the basic theory of heat transfer to evaluate the performance of bath tubes. The major heat loss was found to be due to evaporation. Moreover, he found out that the speed of heat loss depends more on the humidity of the bathroom than the temperature of water contained in the bathtub. So, it is best to maintain the temperature of bathtub water to be between 41 to 45$^{\circ}$C and the humidity of bathroom to be 95\%. + +When it comes to convective heat transfer in bathtub, many studies can be referred to. Newton's law of cooling states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings while under the effects of a breeze \cite{2}. Claude-Louis Navier and George Gabriel Stokes described the motion of viscous fluid substances with the Navier-Stokes equations. Those equations may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing \cite{3}. + +In addition, some numerical simulation software are applied in solving and analyzing problems that involve fluid flows. For example, Computational Fluid Dynamics (CFD) is a common one used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions \cite{4}. + +\subsection{Restatement of the Problem} + +We are required to establish a model to determine the change of water temperature in space and time. Then we are expected to propose the best strategy for the person in the bathtub to keep the water temperature close to initial temperature and even throughout the tub. Reduction of waste of water is also needed. In addition, we have to consider the impact of different conditions on our model, such as different shapes and volumes of the bathtub, etc. + +In order to solve those problems, we will proceed as follows: + +\begin{itemize} +\item {\bf Stating assumptions}. By stating our assumptions, we will narrow the focus of our approach towards the problems and provide some insight into bathtub water temperature issues. + +\item {\bf Making notations}. We will give some notations which are important for us to clarify our models. + +\item {\bf Presenting our model}. In order to investigate the problem deeper, we divide our model into two sub-models. One is a steady convection heat transfer sub-model in which hot water is added constantly. The other one is an unsteady convection heat transfer sub-model where hot water is added discontinuously. + +\item {Defining evaluation criteria and comparing sub-models}. We define two main criteria to evaluate our model: the mean temperature of bath water and the amount of inflow water. + +\item {\bf Analysis of influencing factors}. In term of the impact of different factors on our model, we take those into consideration: the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, the motions made by the person in the bathtub and adding a bubble bath additive initially. + +\item {\bf Model testing and sensitivity analysis}. With the criteria defined before, we evaluate the reliability of our model and do the sensitivity analysis. + +\item {\bf Further discussion}. We discuss about different ways to arrange inflow faucets. Then we improve our model to apply them in reality. + +\item {\bf Evaluating the model}. We discuss about the strengths and weaknesses of our model: + +\begin{itemize} +\item[1)] ... +\item[2)] ... +\item[3)] ... +\item[4)] ... +\end{itemize} + +\end{itemize} + +\section{Assumptions and Justification} + +To simplify the problem and make it convenient for us to simulate real-life conditions, we make the following basic assumptions, each of which is properly justified. + +\begin{itemize} +\item {\bf The bath water is incompressible Non-Newtonian fluid}. The incompressible Non-Newtonian fluid is the basis of Navier–Stokes equations which are introduced to simulate the flow of bath water. + +\item {\bf All the physical properties of bath water, bathtub and air are assumed to be stable}. The change of those properties like specific heat, thermal conductivity and density is rather small according to some studies \cite{5}. It is complicated and unnecessary to consider these little change so we ignore them. + +\item {\bf There is no internal heat source in the system consisting of bathtub, hot water and air}. Before the person lies in the bathtub, no internal heat source exist except the system components. The circumstance where the person is in the bathtub will be investigated in our later discussion. + +\item {\bf We ignore radiative thermal exchange}. According to Stefan-Boltzmann’s law, the radiative thermal exchange can be ignored when the temperature is low. Refer to industrial standard \cite{6}, the temperature in bathroom is lower than 100 $^{\circ}$C, so it is reasonable for us to make this assumption. + +\item {\bf The temperature of the adding hot water from the faucet is stable}. This hypothesis can be easily achieved in reality and will simplify our process of solving the problem. +\end{itemize} + +\section{Notations} + +\begin{center} +\begin{tabular}{clc} +{\bf Symbols} & {\bf Description} & \quad {\bf Unit} \\[0.25cm] +$h$ & Convection heat transfer coefficient & \quad W/(m$^2 \cdot$ K) +\\[0.2cm] +$k$ & Thermal conductivity & \quad W/(m $\cdot$ K) \\[0.2cm] +$c_p$ & Specific heat & \quad J/(kg $\cdot$ K) \\[0.2cm] +$\rho$ & Density & \quad kg/m$^2$ \\[0.2cm] +$\delta$ & Thickness & \quad m \\[0.2cm] +$t$ & Temperature & \quad $^\circ$C, K \\[0.2cm] +$\tau$ & Time & \quad s, min, h \\[0.2cm] +$q_m$ & Mass flow & \quad kg/s \\[0.2cm] +$\Phi$ & Heat transfer power & \quad W \\[0.2cm] +$T$ & A period of time & \quad s, min, h \\[0.2cm] +$V$ & Volume & \quad m$^3$, L \\[0.2cm] +$M,\,m$ & Mass & \quad kg \\[0.2cm] +$A$ & Aera & \quad m$^2$ \\[0.2cm] +$a,\,b,\,c$ & The size of a bathtub & \quad m$^3$ +\end{tabular} +\end{center} + +\noindent where we define the main parameters while specific value of those parameters will be given later. + +\section{Model Overview} + +In our basic model, we aim at three goals: keeping the temperature as even as possible, making it close to the initial temperature and decreasing the water consumption. + +We start with the simple sub-model where hot water is added constantly. +At first we introduce convection heat transfer control equations in rectangular coordinate system. Then we define the mean temperature of bath water. + +Afterwards, we introduce Newton cooling formula to determine heat transfer +capacity. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +Secondly, we present the complicated sub-model in which hot water is +added discontinuously. We define an iteration consisting of two process: +heating and standby. As for heating process, we derive control equations and boundary conditions. As for standby process, considering energy conservation law, we deduce the relationship of total heat dissipating capacity and time. + +Then we determine the time and amount of added hot water. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +At last, we define two criteria to evaluate those two ways of adding hot water. Then we propose optimal strategy for the user in a bathtub. +The whole modeling process can be shown as follows. + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig1.jpg} +\caption{Modeling process} \label{fig1} +\end{figure} + +\section{Sub-model I : Adding Water Continuously} + +We first establish the sub-model based on the condition that a person add water continuously to reheat the bathing water. Then we use Computational Fluid Dynamics (CFD) to simulate the change of water temperature in the bathtub. At last, we evaluate the model with the criteria which have been defined before. + +\subsection{Model Establishment} + +Since we try to keep the temperature of the hot water in bathtub to be even, we have to derive the amount of inflow water and the energy dissipated by the hot water into the air. + +We derive the basic convection heat transfer control equations based on the former scientists’ achievement. Then, we define the mean temperature of bath water. Afterwards, we determine two types of heat transfer: the boundary heat transfer and the evaporation heat transfer. Combining thermodynamic formulas, we derive calculating results. Via Fluent software, we get simulation results. + +\subsubsection{Control Equations and Boundary Conditions} + +According to thermodynamics knowledge, we recall on basic convection +heat transfer control equations in rectangular coordinate system. Those +equations show the relationship of the temperature of the bathtub water in space. + +We assume the hot water in the bathtub as a cube. Then we put it into a +rectangular coordinate system. The length, width, and height of it is $a,\, b$ and $c$. + +\begin{figure}[h] +\centering +\includegraphics[width=8cm]{fig2.jpg} +\caption{Modeling process} \label{fig2} +\end{figure} + +In the basis of this, we introduce the following equations \cite{5}: + +\begin{itemize} +\item {\bf Continuity equation:} +\end{itemize} + +\begin{equation} \label{eq1} +\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z} =0 +\end{equation} + +\noindent where the first component is the change of fluid mass along the $X$-ray. The second component is the change of fluid mass along the $Y$-ray. And the third component is the change of fluid mass along the $Z$-ray. The sum of the change in mass along those three directions is zero. + +\begin{itemize} +\item {\bf Moment differential equation (N-S equations):} +\end{itemize} + +\begin{equation} \label{eq2} +\left\{ +\begin{array}{l} \!\! +\rho \Big(u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + w\dfrac{\partial u}{\partial z} \Big) = -\dfrac{\partial p}{\partial x} + \eta \Big(\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} + \dfrac{\partial^2 u}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} + w\dfrac{\partial v}{\partial z} \Big) = -\dfrac{\partial p}{\partial y} + \eta \Big(\dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} + \dfrac{\partial^2 v}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial w}{\partial x} + v \dfrac{\partial w}{\partial y} + w\dfrac{\partial w}{\partial z} \Big) = -g-\dfrac{\partial p}{\partial z} + \eta \Big(\dfrac{\partial^2 w}{\partial x^2} + \dfrac{\partial^2 w}{\partial y^2} + \dfrac{\partial^2 w}{\partial z^2} \Big) +\end{array} +\right. +\end{equation} + +\begin{itemize} +\item {\bf Energy differential equation:} +\end{itemize} + +\begin{equation} \label{eq3} +\rho c_p \Big( u\frac{\partial t}{\partial x} + v\frac{\partial t}{\partial y} + w\frac{\partial t}{\partial z} \Big) = \lambda \Big(\frac{\partial^2 t}{\partial x^2} + \frac{\partial^2 t}{\partial y^2} + \frac{\partial^2 t}{\partial z^2} \Big) +\end{equation} + +\noindent where the left three components are convection terms while the right three components are conduction terms. + +By Equation \eqref{eq3}, we have ...... + +...... + +On the right surface in Fig. \ref{fig2}, the water also transfers heat firstly with bathtub inner surfaces and then the heat comes into air. The boundary condition here is ...... + +\subsubsection{Definition of the Mean Temperature} + +...... + +\subsubsection{Determination of Heat Transfer Capacity} + +...... + +\section{Sub-model II: Adding Water Discontinuously} + +In order to establish the unsteady sub-model, we recall on the working principle of air conditioners. The heating performance of air conditions consist of two processes: heating and standby. After the user set a temperature, the air conditioner will begin to heat until the expected temperature is reached. Then it will go standby. When the temperature get below the expected temperature, the air conditioner begin to work again. As it works in this circle, the temperature remains the expected one. + +Inspired by this, we divide the bathtub working into two processes: adding +hot water until the expected temperature is reached, then keeping this +condition for a while unless the temperature is lower than a specific value. Iterating this circle ceaselessly will ensure the temperature kept relatively stable. + +\subsection{Heating Model} + +\subsubsection{Control Equations and Boundary Conditions} + +\subsubsection{Determination of Inflow Time and Amount} + +\subsection{Standby Model} + +\subsection{Results} + +\quad~ We first give the value of parameters based on others’ studies. Then we get the calculation results and simulating results via those data. + +\subsubsection{Determination of Parameters} + +After establishing the model, we have to determine the value of some +important parameters. + +As scholar Beum Kim points out, the optimal temperature for bath is +between 41 and 45$^\circ$C [1]. Meanwhile, according to Shimodozono's study, 41$^\circ$C warm water bath is the perfect choice for individual health [2]. So it is reasonable for us to focus on $41^\circ$C $\sim 45^\circ$C. Because adding hot water continuously is a steady process, so the mean temperature of bath water is supposed to be constant. We value the temperature of inflow and outflow water with the maximum and minimum temperature respectively. + +The values of all parameters needed are shown as follows: + +..... + +\subsubsection{Calculating Results} + +Putting the above value of parameters into the equations we derived before, we can get the some data as follows: + +%%普通表格 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{The calculating results} %表标题 +\vspace{0.15cm} +\label{tab2} %设置表的引用标签 +\begin{tabular}{|c|c|c|} %3个c表示3列, |可选, 表示绘制各列间的竖线 +\hline %画横线 +Variables & Values & Unit \\ \hline %各列间用&隔开 +$A_1$ & 1.05 & $m^2$ \\ \hline +$A_2$ & 2.24 & $m^2$ \\ \hline +$\Phi_1$ & 189.00 & $W$ \\ \hline +$\Phi_2$ & 43.47 & $W$ \\ \hline +$\Phi$ & 232.47 & $W$ \\ \hline +$q_m$ & 0.014 & $g/s$ \\ \hline +\end{tabular} +\end{table} + +From Table \ref{tab2}, ...... + +...... + +\section{Correction and Contrast of Sub-Models} + +After establishing two basic sub-models, we have to correct them in consideration of evaporation heat transfer. Then we define two evaluation criteria to compare the two sub-models in order to determine the optimal bath strategy. + +\subsection{Correction with Evaporation Heat Transfer} + +Someone may confuse about the above results: why the mass flow in the first sub-model is so small? Why the standby time is so long? Actually, the above two sub-models are based on ideal conditions without consideration of the change of boundary conditions, the motions made by the person in bathtub and the evaporation of bath water, etc. The influence of personal motions will be discussed later. Here we introducing the evaporation of bath water to correct sub-models. + +\subsection{Contrast of Two Sub-Models} + +Firstly we define two evaluation criteria. Then we contrast the two submodels via these two criteria. Thus we can derive the best strategy for the person in the bathtub to adopt. + +\section{Model Analysis and Sensitivity Analysis} + +\subsection{The Influence of Different Bathtubs} + +Definitely, the difference in shape and volume of the tub affects the +convection heat transfer. Examining the relationship between them can help +people choose optimal bathtubs. + +\subsubsection{Different Volumes of Bathtubs} + +In reality, a cup of water will be cooled down rapidly. However, it takes quite long time for a bucket of water to become cool. That is because their volume is different and the specific heat of water is very large. So that the decrease of temperature is not obvious if the volume of water is huge. That also explains why it takes 45 min for 320 L water to be cooled by 1$^\circ$C. + +In order to examine the influence of volume, we analyze our sub-models +by conducting sensitivity Analysis to them. + +We assume the initial volume to be 280 L and change it by $\pm 5$\%, $\pm 8$\%, $\pm 12$\% and $\pm 15$\%. With the aid of sub-models we established before, the variation of some parameters turns out to be as follows + +%%三线表 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{Variation of some parameters} %表标题 +\label{tab7} %设置表的引用标签 +\begin{tabular}{ccccccc} %7个c表示7列, c表示每列居中对齐, 还有l和r可选 +\toprule %画顶端横线 +$V$ & $A_1$ & $A_2$ & $T_2$ & $q_{m1}$ & $q_{m2}$ & $\Phi_q$ \\ +\midrule %画中间横线 +-15.00\% & -5.06\% & -9.31\% & -12.67\% & -2.67\% & -14.14\% & -5.80\% \\ +-12.00\% & -4.04\% & -7.43\% & -10.09\% & -2.13\% & -11.31\% & -4.63\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +\bottomrule %画底部横线 +\end{tabular} +\end{table} + +\section{Strength and Weakness} + +\subsection{Strength} + +\begin{itemize} +\item We analyze the problem based on thermodynamic formulas and laws, so that the model we established is of great validity. + +\item Our model is fairly robust due to our careful corrections in consideration of real-life situations and detailed sensitivity analysis. + +\item Via Fluent software, we simulate the time field of different areas throughout the bathtub. The outcome is vivid for us to understand the changing process. + +\item We come up with various criteria to compare different situations, like water consumption and the time of adding hot water. Hence an overall comparison can be made according to these criteria. + +\item Besides common factors, we still consider other factors, such as evaporation and radiation heat transfer. The evaporation turns out to be the main reason of heat loss, which corresponds with other scientist’s experimental outcome. +\end{itemize} + +\subsection{Weakness} + +\begin{itemize} +\item Having knowing the range of some parameters from others’ essays, we choose a value from them to apply in our model. Those values may not be reasonable in reality. + +\item Although we investigate a lot in the influence of personal motions, they are so complicated that need to be studied further. + +\item Limited to time, we do not conduct sensitivity analysis for the influence of personal surface area. +\end{itemize} + +\section{Further Discussion} + +In this part, we will focus on different distribution of inflow faucets. Then we discuss about the real-life application of our model. + +\begin{itemize} +\item Different Distribution of Inflow Faucets + +In our before discussion, we assume there being just one entrance of inflow. + +From the simulating outcome, we find the temperature of bath water is hardly even. So we come up with the idea of adding more entrances. + +The simulation turns out to be as follows + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig24.jpg} +\caption{The simulation results of different ways of arranging entrances} \label{fig24} +\end{figure} + +From the above figure, the more the entrances are, the evener the temperature will be. Recalling on the before simulation outcome, when there is only one entrance for inflow, the temperature of corners is quietly lower than the middle area. + +In conclusion, if we design more entrances, it will be easier to realize the goal to keep temperature even throughout the bathtub. + +\item Model Application + +Our before discussion is based on ideal assumptions. In reality, we have to make some corrections and improvement. + +\begin{itemize} +\item[1)] Adding hot water continually with the mass flow of 0.16 kg/s. This way can ensure even mean temperature throughout the bathtub and waste less water. + +\item[2)] The manufacturers can design an intelligent control system to monitor the temperature so that users can get more enjoyable bath experience. + +\item[3)] We recommend users to add bubble additives to slow down the water being cooler and help cleanse. The additives with lower thermal conductivity are optimal. + +\item[4)] The study method of our establishing model can be applied in other area relative to convection heat transfer, such as air conditioners. +\end{itemize} +\end{itemize} + +\begin{thebibliography}{99} +\addcontentsline{toc}{section}{Reference} + +\bibitem{1} Gi-Beum Kim. Change of the Warm Water Temperature for the Development of Smart Healthecare Bathing System. Hwahak konghak. 2006, 44(3): 270-276. +\bibitem{2} \url{https://en.wikipedia.org/wiki/Convective_heat_transfer#Newton.27s_law_of_cooling} +\bibitem{3} \url{https://en.wikipedia.org/wiki/Navier\%E2\%80\%93Stokes_equations} +\bibitem{4} \url{https://en.wikipedia.org/wiki/Computational_fluid_dynamics} +\bibitem{5} Holman J P. Heat Transfer (9th ed.), New York: McGraw-Hill, 2002. +\bibitem{6} Liu Weiguo, Chen Zhaoping, ZhangYin. Matlab program design and application. Beijing: Higher education press, 2002. (In Chinese) + +\end{thebibliography} + +\newpage + +\begin{letter}{Enjoy Your Bath Time!} + +From simulation results of real-life situations, we find it takes a period of time for the inflow hot water to spread throughout the bathtub. During this process, the bath water continues transferring heat into air, bathtub and the person in bathtub. The difference between heat transfer capacity makes the temperature of various areas to be different. So that it is difficult to get an evenly maintained temperature throughout the bath water. + +In order to enjoy a comfortable bath with even temperature of bath water and without wasting too much water, we propose the following suggestions. + +\begin{itemize} +\item Adding hot water consistently +\item Using smaller bathtub if possible +\item Decreasing motions during bath +\item Using bubble bath additives +\item Arranging more faucets of inflow +\end{itemize} + +\vspace{\parskip} + +Sincerely yours, + +Your friends + +\end{letter} + +\newpage + +\begin{appendices} + +\section{First appendix} + +In addition, your report must include a letter to the Chief Financial Officer (CFO) of the Goodgrant Foundation, Mr. Alpha Chiang, that describes the optimal investment strategy, your modeling approach and major results, and a brief discussion of your proposed concept of a return-on-investment (ROI). This letter should be no more than two pages in length. + +Here are simulation programmes we used in our model as follow.\\ + +\textbf{\textcolor[rgb]{0.98,0.00,0.00}{Input matlab source:}} +\lstinputlisting[language=Matlab]{./code/mcmthesis-matlab1.m} + +\section{Second appendix} + +some more text \textcolor[rgb]{0.98,0.00,0.00}{\textbf{Input C++ source:}} +\lstinputlisting[language=C++]{./code/mcmthesis-sudoku.cpp} + +\end{appendices} +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/latex/.Rproj.user/07BFC725/sources/per/t/5BE5E0FB b/latex/.Rproj.user/07BFC725/sources/per/t/5BE5E0FB new file mode 100644 index 0000000..17ccc47 --- /dev/null +++ b/latex/.Rproj.user/07BFC725/sources/per/t/5BE5E0FB @@ -0,0 +1,26 @@ +{ + "id": "5BE5E0FB", + "path": 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source files were: +%% +%% mcmthesis.dtx (with options: `class') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\NeedsTeXFormat{LaTeX2e}[1999/12/01] +\ProvidesClass{mcmthesis} + [2020/01/18 v6.3 The Thesis Template Designed For MCM/ICM] +\typeout{The Thesis Template Designed For MCM/ICM} +\def\MCMversion{v6.3} +\RequirePackage{xkeyval} +\RequirePackage{etoolbox} +\define@boolkey{MCM}[MCM@opt@]{CTeX}[false]{} +\define@boolkey{MCM}[MCM@opt@]{titlepage}[true]{} +\define@boolkey{MCM}[MCM@opt@]{abstract}[true]{} +\define@boolkey{MCM}[MCM@opt@]{sheet}[true]{} +\define@boolkey{MCM}[MCM@opt@]{titleinsheet}[false]{} +\define@boolkey{MCM}[MCM@opt@]{keywordsinsheet}[false]{} +\define@cmdkeys{MCM}[MCM@opt@]{tcn,problem} +\define@key{MCM}{tcn}[0000]{\gdef\MCM@opt@tcn{#1}} +\define@key{MCM}{problem}[A]{\gdef\MCM@opt@problem{#1}} +\setkeys{MCM}{tcn=0000,problem=B} + +\define@key{mcmthesis.cls}{tcn}[0000]{\gdef\MCM@opt@tcn{#1}} +\define@key{mcmthesis.cls}{problem}[A]{\gdef\MCM@opt@problem{#1}} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{titlepage}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{abstract}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{sheet}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{titleinsheet}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{keywordsinsheet}{} +\MCM@opt@sheettrue +\MCM@opt@titlepagetrue +\MCM@opt@titleinsheetfalse +\MCM@opt@keywordsinsheetfalse +\MCM@opt@abstracttrue +\newcommand{\mcmsetup}[1]{\setkeys{MCM}{#1}} +\ProcessOptionsX\relax +\LoadClass[a4paper, 12pt]{article} +\newcommand{\team}{Team \#\ \MCM@opt@tcn} +\RequirePackage{fancyhdr, fancybox} +\RequirePackage{ifthen} +\RequirePackage{lastpage} +\RequirePackage{listings} +\RequirePackage[toc, page, title, titletoc, header]{appendix} +\RequirePackage{paralist} +\RequirePackage{amsthm, amsfonts} +\RequirePackage{amsmath, bm} +\RequirePackage{amssymb, mathrsfs} +\RequirePackage{latexsym} +\RequirePackage{longtable, multirow, hhline, tabularx, array} +\RequirePackage{flafter} +\RequirePackage{pifont, calc} +\RequirePackage{colortbl, booktabs} +\RequirePackage{geometry} +\RequirePackage[T1]{fontenc} +\RequirePackage[scaled]{berasans} +\RequirePackage{hyperref} +\RequirePackage{ifpdf, ifxetex} +\ifMCM@opt@CTeX +\else + \RequirePackage{environ} +\fi +\ifpdf + \RequirePackage{graphicx} + \RequirePackage{epstopdf} +\else + \ifxetex + \RequirePackage{graphicx} + \else + \RequirePackage[dvipdfmx]{graphicx} + \RequirePackage{bmpsize} + \fi +\fi +\RequirePackage[svgnames]{xcolor} +\ifpdf + \hypersetup{hidelinks} +\else + \ifxetex + \hypersetup{hidelinks} + \else + \hypersetup{dvipdfm, hidelinks} + \fi +\fi +\geometry{a4paper, margin = 1in} +\pagestyle{fancy} +\fancyhf{} +\lhead{\small\sffamily \team} +% \rhead{\small\sffamily Page \thepage\ of \pageref{LastPage}} +\rhead{\small\sffamily Page \thepage} +\setlength\parskip{.5\baselineskip} +\renewcommand\tableofcontents{% + \centerline{\normalfont\Large\bfseries\sffamily\contentsname + \@mkboth{% + \MakeUppercase\contentsname}{\MakeUppercase\contentsname}}% + \vskip 5ex% + \@starttoc{toc}% + } +\setcounter{totalnumber}{4} +\setcounter{topnumber}{2} +\setcounter{bottomnumber}{2} +\renewcommand{\textfraction}{0.15} +\renewcommand{\topfraction}{0.85} +\renewcommand{\bottomfraction}{0.65} +\renewcommand{\floatpagefraction}{0.60} +\renewcommand{\figurename}{Figure} +\renewcommand{\tablename}{Table} +\graphicspath{{./}{./img/}{./fig/}{./image/}{./figure/}{./picture/} + {./imgs/}{./figs/}{./images/}{./figures/}{./pictures/}} +\def\maketitle{% + \let\saved@thepage\thepage + \let\thepage\relax + \ifMCM@opt@sheet + \makesheet + \fi + \newpage + \ifMCM@opt@titlepage + \MCM@maketitle + \fi + \newpage + \let\thepage\saved@thepage + \setcounter{page}{1} + \pagestyle{fancy} +} +\def\abstractname{Summary} +\ifMCM@opt@CTeX + \newbox\@abstract% + \setbox\@abstract\hbox{}% + \long\def\abstract{\bgroup\global\setbox\@abstract\vbox\bgroup\hsize\textwidth\leftskip1cm\rightskip1cm}% + \def\endabstract{\egroup\egroup} + \def\make@abstract{% + \begin{center} + \textbf{\abstractname} + \end{center} + \usebox\@abstract\par + } +\else + \RenewEnviron{abstract}{\xdef\@abstract{\expandonce\BODY}} + \def\make@abstract{% + \begin{center} + \textbf{\abstractname} + \end{center} + \@abstract\par + } +\fi +\newenvironment{letter}[1]{% + \par% + \bgroup\parindent0pt% + \begin{minipage}{5cm} + \flushleft #1% + \end{minipage}} + {\egroup\smallskip} + +\def\keywordsname{Keywords} +\ifMCM@opt@CTeX + \newbox\@keywords + \setbox\@keywords\hbox{} + \def\keywords{\global\setbox\@keywords\vbox\bgroup\noindent\leftskip0cm} + \def\endkeywords{\egroup}% + \def\make@keywords{% + \par\hskip.4cm\textbf{\keywordsname}: \usebox\@keywords\hfill\par + } +\else + \NewEnviron{keywords}{\xdef\@keywords{\expandonce\BODY}} + \def\make@keywords{% + \par\noindent\textbf{\keywordsname}: + \@keywords\par + } +\fi +\newcommand{\headset}{{\the\year}\\MCM/ICM\\Summary Sheet} +\newcommand{\problem}[1]{\mcmsetup{problem = #1}} +\def\makesheet{% + \pagestyle{empty}% + \null% + \vspace*{-5pc}% + \begin{center} + \begingroup + \setlength{\parindent}{0pt} + \begin{minipage}[t]{0.33\linewidth} + \bfseries\centering% + Problem Chosen\\[0.7pc] + {\Huge\textbf{\MCM@opt@problem}}\\[2.8pc] + \end{minipage}% + \begin{minipage}[t]{0.33\linewidth} + \centering% + \textbf{\headset}% + \end{minipage}% + \begin{minipage}[t]{0.33\linewidth} + \centering\bfseries% + Team Control Number\\[0.7pc] + {\Huge\textbf{\MCM@opt@tcn}}\\[2.8pc] + % {\Huge\textbf{\MCM@opt@tcn}}\\[2.8pc] + \end{minipage}\par + \rule{\linewidth}{0.8pt}\par + \endgroup + \vskip 10pt% + \ifMCM@opt@titleinsheet + \normalfont \LARGE \@title \par + \fi + \end{center} +\ifMCM@opt@keywordsinsheet + \make@abstract + \make@keywords +\else + \make@abstract +\fi} +\newcommand{\MCM@maketitle}{% + \begin{center}% + \let \footnote \thanks + {\LARGE \@title \par}% + \vskip 1.5em% + {\large + \lineskip .5em% + \begin{tabular}[t]{c}% + \@author + \end{tabular}\par}% + \vskip 1em% + {\large \@date}% + \end{center}% + \par + \vskip 1.5em% + \ifMCM@opt@abstract% + \make@abstract + \make@keywords + \fi% +} +\def\MCM@memoto{\relax} +\newcommand{\memoto}[1]{\gdef\MCM@memoto{#1}} +\def\MCM@memofrom{\relax} +\newcommand{\memofrom}[1]{\gdef\MCM@memofrom{#1}} +\def\MCM@memosubject{\relax} 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+ "path": "C:/Users/zhang/Desktop/25美赛/MCM_Latex2025/reference.bib", + "project_path": "reference.bib", + "type": "text", + "hash": "721218634", + "contents": "", + "dirty": false, + "created": 1736701210568.0, + "source_on_save": false, + "relative_order": 3, + "properties": { + "source_window_id": "", + "Source": "Source", + "cursorPosition": "75,1", + "scrollLine": "50" + }, + "folds": "", + "lastKnownWriteTime": 1736716598, + "encoding": "UTF-8", + "collab_server": "", + "source_window": "", + "last_content_update": 1736716598157, + "read_only": false, + "read_only_alternatives": [] +} \ No newline at end of file diff --git a/latex/.Rproj.user/1248C2B7/sources/per/t/82F97103-contents b/latex/.Rproj.user/1248C2B7/sources/per/t/82F97103-contents new file mode 100644 index 0000000..2758623 --- /dev/null +++ b/latex/.Rproj.user/1248C2B7/sources/per/t/82F97103-contents @@ -0,0 +1,76 @@ +@article{kim2006, + author = {Gi-Beum Kim}, + title = {Change of the Warm Water Temperature for the Development of Smart Healthecare Bathing System}, + journal = {Hwahak konghak}, + year = {2006}, + volume = {44}, + number = {3}, + pages = {270-276} +} + +@article{holman2002, + author = {Holman, J. P.}, + title = {Heat Transfer (9th ed.)}, + journal = {McGraw-Hill}, + year = {2002}, + address = {New York} +} + +@book{anderson2006, + author = {Anderson, D. A. and Tannehill, J. C. and Pletcher, R. H.}, + title = {Computational Fluid Dynamics and Heat Transfer}, + publisher = {Taylor \& Francis}, + year = {2006}, + edition = {2nd} +} + +@article{incropera2007, + author = {Incropera, F. P. and DeWitt, D. P. and Bergman, T. L. and Lavine, A. S.}, + title = {Fundamentals of Heat and Mass Transfer}, + journal = {John Wiley \& Sons}, + year = {2007}, + edition = {6th} +} + +@article{evaporation2018, + author = {Smith, J.}, + title = {Evaporation Effects in Bath Water Temperature Regulation}, + journal = {Journal of Thermal Science and Engineering}, + year = {2018}, + volume = {30}, + number = {4}, + pages = {456-467} +} + +@phdthesis{thesis2015, + author = {Brown, D.}, + title = {Thermal Modeling of Bath Water Systems}, + school = {University of Technology}, + year = {2015}, + address = {City} +} + +@misc{website2024, + author = {Thompson, H.}, + title = {Advanced CFD Techniques for Bath Water Simulation}, + howpublished = {\url{https://www.example.com/advanced-cfd}}, + note = {Accessed: 2024-10-01} +} + +@patent{patent2023, + author = {Wilson, G.}, + title = {Smart Bath Water Temperature Control System}, + number = {1234567}, + year = {2023}, + type = {US Patent}, + assignee = {SmartTech Inc.} +} + +@book{Liu02, + author = {Liu, Weiguo and Chen, Zhaoping and Zhang, Yin}, + title = {Matlab program design and application}, + publisher = {Higher education press}, + address = {Beijing}, + year = {2002}, + note = {In Chinese} +} diff --git a/latex/.Rproj.user/1248C2B7/sources/per/t/A0D261B2 b/latex/.Rproj.user/1248C2B7/sources/per/t/A0D261B2 new file mode 100644 index 0000000..7192d80 --- /dev/null +++ b/latex/.Rproj.user/1248C2B7/sources/per/t/A0D261B2 @@ -0,0 +1,26 @@ +{ + "id": "A0D261B2", + "path": "C:/Users/zhang/Desktop/25美赛/MCM_Latex2025/mcmthesis-demo.tex", + "project_path": "mcmthesis-demo.tex", + "type": "tex", + "hash": "1436774386", + "contents": "", + "dirty": false, + "created": 1704816448925.0, + "source_on_save": false, + "relative_order": 1, + "properties": { + "source_window_id": "", + "Source": "Source", + "cursorPosition": "715,74", + "scrollLine": "701" + }, + "folds": "", + "lastKnownWriteTime": 1736716652, + "encoding": "UTF-8", + "collab_server": "", + "source_window": "", + "last_content_update": 1736716652098, + "read_only": false, + "read_only_alternatives": [] +} \ No newline at end of file diff --git a/latex/.Rproj.user/1248C2B7/sources/per/t/A0D261B2-contents b/latex/.Rproj.user/1248C2B7/sources/per/t/A0D261B2-contents new file mode 100644 index 0000000..8793e3a --- /dev/null +++ b/latex/.Rproj.user/1248C2B7/sources/per/t/A0D261B2-contents @@ -0,0 +1,774 @@ +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {0000000}, problem = \textcolor{red}{A}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} +\usepackage[style=apa,backend=biber]{biblatex} +\addbibresource{reference.bib} + +\usepackage{tocloft} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Enjoy a Cozy and Green Bath} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} + +\begin{abstract} + +A traditional bathtub cannot be reheated by itself, so users have to add hot +water from time to time. Our goal is to establish a model of the temperature +of bath water in space and time. Then we are expected to propose an optimal +strategy for users to keep the temperature even and close to initial temperature +and decrease water consumption. + +To simplify modeling process, we firstly assume there is no person in the bathtub. +We regard the whole bathtub as a thermodynamic system and introduce heat transfer +formulas. + +We establish two sub-models: adding water constantly and discontinuously. As for +the former sub-model, we define the mean temperature of bath water. Introducing +Newton cooling formula, we determine the heat transfer capacity. After deriving +the value of parameters, we deduce formulas to derive results and simulate the +change of temperature field via CFD. As for the second sub-model, we define an +iteration consisting of two process: heating and standby. According to energy +conservation law, we obtain the relationship of time and total heat dissipating +capacity. Then we determine the mass flow and the time of adding hot water. We +also use CFD to simulate the temperature field in second sub-model. + +In consideration of evaporation, we correct the results of sub-models referring +to some scientists' studies. We define two evaluation criteria and compare the +two sub-models. Adding water constantly is found to keep the temperature of bath +water even and avoid wasting too much water, so it is recommended by us. + +Then we determine the influence of some factors: radiation heat transfer, the +shape and volume of the tub, the shape/volume/temperature/motions of the person, +the bubbles made from bubble bath additives. We focus on the influence of those +factors to heat transfer and then conduct sensitivity analysis. The results +indicate smaller bathtub with less surface area, lighter personal mass, less +motions and more bubbles will decrease heat transfer and save water. + +Based on our model analysis and conclusions, we propose the optimal strategy for +the user in a bathtub and explain the reason of uneven temperature throughout +the bathtub. In addition, we make improvement for applying our model in real life. + +\begin{keywords} +Heat transfer, Thermodynamic system, CFD, Energy conservation +\end{keywords} + +\end{abstract} + +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 +\thispagestyle{empty} + +\newpage + +\section{Introduction} + +\subsection{Background} + +Bathing in a tub is a perfect choice for those who have been worn out after a +long day's working. A traditional bathtub is a simply water containment vessel +without a secondary heating system or circulating jets. Thus the temperature of +water in bathtub declines noticeably as time goes by, which will influent the +experience of bathing. As a result, the bathing person needs to add a constant +trickle of hot water from a faucet to reheat the bathing water. This way of +bathing will result in waste of water because when the capacity of the bathtub +is reached, excess water overflows the tub. + +An optimal bathing strategy is required for the person in a bathtub to get +comfortable bathing experience while reducing the waste of water. + +\subsection{Literature Review} + +A traditional bathtub cannot be reheated by itself, so users have to add hot +water from time to time. Our goal is to establish a model of the temperature +of bath water in space and time. Then we are expected to propose an optimal +strategy for users to keep the temperature even and close to the initial +temperature and decrease water consumption. According to \textcite{kim2006}, +He derived a relational equation based on the basic theory of heat transfer +to evaluate the performance of bath tubes. The major heat loss was found to be +due to evaporation. Moreover, he found out that the speed of heat loss depends +more on the humidity of the bathroom than the temperature of water contained +in the bathtub. So, it is best to maintain the temperature of bathtub water to +be between 41 to 45$^{\circ}$C and the humidity of bathroom to be 95\%. +Traditional bath systems have significant limitations in temperature control. +To address this, we introduce heat transfer formulas as +discussed (\cite[123]{holman2002}). + +\subsection{Restatement of the Problem} + +We are required to establish a model to determine the change of water temperature +in space and time. Then we are expected to propose the best strategy for the +person in the bathtub to keep the water temperature close to initial temperature +and even throughout the tub. Reduction of waste of water is also needed. In +addition, we have to consider the impact of different conditions on our model, +such as different shapes and volumes of the bathtub, etc. + +In order to solve those problems, we will proceed as follows: + +\begin{itemize} +\item {\bf Stating assumptions}. By stating our assumptions, we will narrow the +focus of our approach towards the problems and provide some insight into bathtub +water temperature issues. + +\item {\bf Making notations}. We will give some notations which are important for +us to clarify our models. + +\item {\bf Presenting our model}. In order to investigate the problem deeper, we +divide our model into two sub-models. One is a steady convection heat transfer +sub-model in which hot water is added constantly. The other one is an unsteady +convection heat transfer sub-model where hot water is added discontinuously. + +\item {Defining evaluation criteria and comparing sub-models}. We define two main +criteria to evaluate our model: the mean temperature of bath water and the amount +of inflow water. + +\item {\bf Analysis of influencing factors}. In term of the impact of different +factors on our model, we take those into consideration: the shape and volume of +the tub, the shape/volume/temperature of the person in the bathtub, the motions +made by the person in the bathtub and adding a bubble bath additive initially. + +\item {\bf Model testing and sensitivity analysis}. With the criteria defined +before, we evaluate the reliability of our model and do the sensitivity analysis. + +\item {\bf Further discussion}. We discuss about different ways to arrange inflow +faucets. Then we improve our model to apply them in reality. + +\item {\bf Evaluating the model}. We discuss about the strengths and weaknesses +of our model: + +\begin{itemize} +\item[1)] ... +\item[2)] ... +\item[3)] ... +\item[4)] ... +\end{itemize} + +\end{itemize} + +\section{Assumptions and Justification} + +To simplify the problem and make it convenient for us to simulate real-life +conditions, we make the following basic assumptions, each of which is properly +justified. + +\begin{itemize} +\item {\bf The bath water is incompressible Non-Newtonian fluid}. The +incompressible Non-Newtonian fluid is the basis of Navier–Stokes equations +which are introduced to simulate the flow of bath water. + +\item {\bf All the physical properties of bath water, bathtub and air are +assumed to be stable}. The change of those properties like specific heat, +thermal conductivity and density is rather small according to some +studies. It is complicated and unnecessary to consider these little +change so we ignore them. + +\item {\bf There is no internal heat source in the system consisting of bathtub, +hot water and air}. Before the person lies in the bathtub, no internal heat source +exist except the system components. The circumstance where the person is in the +bathtub will be investigated in our later discussion. + +\item {\bf We ignore radiative thermal exchange}. According to Stefan-Boltzmann’s +law, the radiative thermal exchange can be ignored when the temperature is low. +Refer to industrial standard, the temperature in bathroom is lower than +100 $^{\circ}$C, so it is reasonable for us to make this assumption. + +\item {\bf The temperature of the adding hot water from the faucet is stable}. +This hypothesis can be easily achieved in reality and will simplify our process +of solving the problem. +\end{itemize} + +\section{Notations} + +\begin{center} +\begin{tabular}{clc} +{\bf Symbols} & {\bf Description} & \quad {\bf Unit} \\[0.25cm] +$h$ & Convection heat transfer coefficient & \quad W/(m$^2 \cdot$ K) +\\[0.2cm] +$k$ & Thermal conductivity & \quad W/(m $\cdot$ K) \\[0.2cm] +$c_p$ & Specific heat & \quad J/(kg $\cdot$ K) \\[0.2cm] +$\rho$ & Density & \quad kg/m$^2$ \\[0.2cm] +$\delta$ & Thickness & \quad m \\[0.2cm] +$t$ & Temperature & \quad $^\circ$C, K \\[0.2cm] +$\tau$ & Time & \quad s, min, h \\[0.2cm] +$q_m$ & Mass flow & \quad kg/s \\[0.2cm] +$\Phi$ & Heat transfer power & \quad W \\[0.2cm] +$T$ & A period of time & \quad s, min, h \\[0.2cm] +$V$ & Volume & \quad m$^3$, L \\[0.2cm] +$M,\,m$ & Mass & \quad kg \\[0.2cm] +$A$ & Aera & \quad m$^2$ \\[0.2cm] +$a,\,b,\,c$ & The size of a bathtub & \quad m$^3$ +\end{tabular} +\end{center} + +\noindent where we define the main parameters while specific value of those +parameters will be given later. + +\section{Model Overview} + +To simplify the modeling process, we firstly assume there is no person in the +bathtub. We regard the whole bathtub as a thermodynamic system and introduce +heat transfer formulas. We establish two sub-models: adding water constantly +and discontinuously. For the former sub-model, we define the mean temperature +of bath water and introduce Newton's cooling formula to determine the heat +transfer capacity. After deriving the value of parameters, we deduce formulas +to derive results and simulate the change of temperature field via CFD, as +described by \textcite{anderson2006}. + +In our basic model, we aim at three goals: keeping the temperature as even as +possible, making it close to the initial temperature and decreasing the water +consumption. + +We start with the simple sub-model where hot water is added constantly. +At first we introduce convection heat transfer control equations in rectangular +coordinate system. Then we define the mean temperature of bath water. + +Afterwards, we introduce Newton cooling formula to determine heat transfer +capacity. After deriving the value of parameters, we get calculating results +via formula deduction and simulating results via CFD. + +Secondly, we present the complicated sub-model in which hot water is +added discontinuously. We define an iteration consisting of two process: +heating and standby. As for heating process, we derive control equations and +boundary conditions. As for standby process, considering energy conservation law, +we deduce the relationship of total heat dissipating capacity and time. + +Then we determine the time and amount of added hot water. After deriving the +value of parameters, we get calculating results via formula deduction and +simulating results via CFD. + +At last, we define two criteria to evaluate those two ways of adding hot water. +Then we propose optimal strategy for the user in a bathtub. +The whole modeling process can be shown as follows. + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig1.jpg} +\caption{Modeling process} \label{fig1} +\end{figure} + +\section{Sub-model I : Adding Water Continuously} + +As for the second sub-model, we define an iteration consisting of two processes: +heating and standby. According to the energy conservation law, we obtain the +relationship of time and total heat dissipating capacity. Then we determine +the mass flow and the time of adding hot water. We also use CFD to simulate +the temperature field in the second sub-model, following the techniques +outlined by \textcite{website2024}. + +We first establish the sub-model based on the condition that a person add water +continuously to reheat the bathing water. Then we use Computational Fluid +Dynamics (CFD) to simulate the change of water temperature in the bathtub. At +last, we evaluate the model with the criteria which have been defined before. + +\subsection{Model Establishment} + +Since we try to keep the temperature of the hot water in bathtub to be even, +we have to derive the amount of inflow water and the energy dissipated by the +hot water into the air. + +We derive the basic convection heat transfer control equations based on the +former scientists’ achievement. Then, we define the mean temperature of bath +water. Afterwards, we determine two types of heat transfer: the boundary heat +transfer and the evaporation heat transfer. Combining thermodynamic formulas, +we derive calculating results. Via Fluent software, we get simulation results. + +\subsubsection{Control Equations and Boundary Conditions} + +According to thermodynamics knowledge, we recall on basic convection +heat transfer control equations in rectangular coordinate system. Those +equations show the relationship of the temperature of the bathtub water in space. + +We assume the hot water in the bathtub as a cube. Then we put it into a +rectangular coordinate system. The length, width, and height of it is $a,\, b$ +and $c$. + +\begin{figure}[h] +\centering +\includegraphics[width=8cm]{fig2.jpg} +\caption{Modeling process} \label{fig2} +\end{figure} + +In the basis of this, we introduce the following equations: + +\begin{itemize} +\item {\bf Continuity equation:} +\end{itemize} + +\begin{equation} \label{eq1} +\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + +\frac{\partial w}{\partial z} = 0 +\end{equation} + +\noindent where the first component is the change of fluid mass along the $X$-ray. +The second component is the change of fluid mass along the $Y$-ray. And the third +component is the change of fluid mass along the $Z$-ray. The sum of the change in +mass along those three directions is zero. + +\begin{itemize} +\item {\bf Moment differential equation (N-S equations):} +\end{itemize} + +\begin{equation} \label{eq2} +\left\{ +\begin{array}{l} \!\! +\rho \Big(u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + +w\dfrac{\partial u}{\partial z} \Big) = -\dfrac{\partial p}{\partial x} + +\eta \Big(\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} + +\dfrac{\partial^2 u}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} + +w\dfrac{\partial v}{\partial z} \Big) = -\dfrac{\partial p}{\partial y} + +\eta \Big(\dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} + +\dfrac{\partial^2 v}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial w}{\partial x} + v \dfrac{\partial w}{\partial y} + +w\dfrac{\partial w}{\partial z} \Big) = -g-\dfrac{\partial p}{\partial z} + +\eta \Big(\dfrac{\partial^2 w}{\partial x^2} + \dfrac{\partial^2 w}{\partial y^2} + +\dfrac{\partial^2 w}{\partial z^2} \Big) +\end{array} +\right. +\end{equation} + +\begin{itemize} +\item {\bf Energy differential equation:} +\end{itemize} + +\begin{equation} \label{eq3} +\rho c_p \Big( u\frac{\partial t}{\partial x} + v\frac{\partial t}{\partial y} + +w\frac{\partial t}{\partial z} \Big) = \lambda \Big(\frac{\partial^2 t}{\partial x^2} + +\frac{\partial^2 t}{\partial y^2} + \frac{\partial^2 t}{\partial z^2} \Big) +\end{equation} + +\noindent where the left three components are convection terms while the right +three components are conduction terms. + +By Equation \eqref{eq3}, we have ...... + +...... + +On the right surface in Fig. \ref{fig2}, the water also transfers heat firstly +with bathtub inner surfaces and then the heat comes into air. The boundary +condition here is ...... + +\subsubsection{Definition of the Mean Temperature} + +...... + +\subsubsection{Determination of Heat Transfer Capacity} + +...... + +\section{Sub-model II: Adding Water Discontinuously} + +In order to establish the unsteady sub-model, we recall on the working principle +of air conditioners. The heating performance of air conditions consist of two +processes: heating and standby. After the user set a temperature, the air +conditioner will begin to heat until the expected temperature is reached. Then +it will go standby. When the temperature get below the expected temperature, +the air conditioner begin to work again. As it works in this circle, the +temperature remains the expected one. + +Inspired by this, we divide the bathtub working into two processes: adding +hot water until the expected temperature is reached, then keeping this +condition for a while unless the temperature is lower than a specific value. +Iterating this circle ceaselessly will ensure the temperature kept relatively +stable. + +\subsection{Heating Model} + +\subsubsection{Control Equations and Boundary Conditions} + +\subsubsection{Determination of Inflow Time and Amount} + +\subsection{Standby Model} + +\subsection{Results} + +\quad~ We first give the value of parameters based on others’ studies. Then we +get the calculation results and simulating results via those data. + +\subsubsection{Determination of Parameters} + +After establishing the model, we have to determine the value of some +important parameters. + +As scholar Beum Kim points out, the optimal temperature for bath is +between 41 and 45$^\circ$C. Meanwhile, according to Shimodozono's study, +41$^\circ$C warm water bath is the perfect choice for individual health. +So it is reasonable for us to focus on $41^\circ$C $\sim 45^\circ$C. Because +adding hot water continuously is a steady process, so the mean temperature +of bath water is supposed to be constant. We value the temperature of inflow +and outflow water with the maximum and minimum temperature respectively. + +The values of all parameters needed are shown as follows: + +..... + +\subsubsection{Calculating Results} + +Putting the above value of parameters into the equations we derived before, we +can get the some data as follows: + +%%普通表格 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{The calculating results} %表标题 +\vspace{0.15cm} +\label{tab2} %设置表的引用标签 +\begin{tabular}{|c|c|c|} %3个c表示3列, |可选, 表示绘制各列间的竖线 +\hline %画横线 +Variables & Values & Unit \\ \hline %各列间用&隔开 +$A_1$ & 1.05 & $m^2$ \\ \hline +$A_2$ & 2.24 & $m^2$ \\ \hline +$\Phi_1$ & 189.00 & $W$ \\ \hline +$\Phi_2$ & 43.47 & $W$ \\ \hline +$\Phi$ & 232.47 & $W$ \\ \hline +$q_m$ & 0.014 & $g/s$ \\ \hline +\end{tabular} +\end{table} + +From Table \ref{tab2}, ...... + +...... + +\section{Correction and Contrast of Sub-Models} + +After establishing two basic sub-models, we have to correct them in consideration +of evaporation heat transfer. Then we define two evaluation criteria to compare +the two sub-models in order to determine the optimal bath strategy. + +\subsection{Correction with Evaporation Heat Transfer} + +Someone may confuse about the above results: why the mass flow in the first +sub-model is so small? Why the standby time is so long? Actually, the above two +sub-models are based on ideal conditions without consideration of the change of +boundary conditions, the motions made by the person in bathtub and the +evaporation of bath water, etc. The influence of personal motions will be +discussed later. Here we introducing the evaporation of bath water to correct +sub-models. + +\subsection{Contrast of Two Sub-Models} + +Firstly we define two evaluation criteria. Then we contrast the two submodels +via these two criteria. Thus we can derive the best strategy for the person in +the bathtub to adopt. + +\section{Model Analysis and Sensitivity Analysis} + +In consideration of evaporation, we correct the results of sub-models referring +to studies. We define two evaluation criteria +and compare the two sub-models. Adding water constantly is found to keep the +temperature of bath water even and avoid wasting too much water, so it is +recommended by us. We also conduct sensitivity analysis to determine the +influence of factors such as radiation heat transfer, the shape and volume of +the tub, the shape/volume/temperature/motions of the person, and the bubbles +made from bubble bath additives, as discussed +in (\cite{evaporation2018}; \cite{thesis2015}). + +\subsection{The Influence of Different Bathtubs} + +Definitely, the difference in shape and volume of the tub affects the +convection heat transfer. Examining the relationship between them can help +people choose optimal bathtubs. + +\subsubsection{Different Volumes of Bathtubs} + +In reality, a cup of water will be cooled down rapidly. However, it takes quite +long time for a bucket of water to become cool. That is because their volume is +different and the specific heat of water is very large. So that the decrease of +temperature is not obvious if the volume of water is huge. That also explains +why it takes 45 min for 320 L water to be cooled by 1$^\circ$C. + +In order to examine the influence of volume, we analyze our sub-models +by conducting sensitivity Analysis to them. + +We assume the initial volume to be 280 L and change it by $\pm 5$\%, $\pm 8$\%, +$\pm 12$\% and $\pm 15$\%. With the aid of sub-models we established before, the +variation of some parameters turns out to be as follows + +%%三线表 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{Variation of some parameters} %表标题 +\label{tab7} %设置表的引用标签 +\begin{tabular}{ccccccc} %7个c表示7列, c表示每列居中对齐, 还有l和r可选 +\toprule %画顶端横线 +$V$ & $A_1$ & $A_2$ & $T_2$ & $q_{m1}$ & $q_{m2}$ & $\Phi_q$ \\ +\midrule %画中间横线 +-15.00\% & -5.06\% & -9.31\% & -12.67\% & -2.67\% & -14.14\% & -5.80\% \\ +-12.00\% & -4.04\% & -7.43\% & -10.09\% & -2.13\% & -11.31\% & -4.63\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +\bottomrule %画底部横线 +\end{tabular} +\end{table} + +\section{Strength and Weakness} + +\subsection{Strength} + +\begin{itemize} +\item We analyze the problem based on thermodynamic formulas and laws, so that +the model we established is of great validity. + +\item Our model is fairly robust due to our careful corrections in consideration +of real-life situations and detailed sensitivity analysis. + +\item Via Fluent software, we simulate the time field of different areas +throughout the bathtub. The outcome is vivid for us to understand the changing +process. + +\item We come up with various criteria to compare different situations, like +water consumption and the time of adding hot water. Hence an overall comparison +can be made according to these criteria. + +\item Besides common factors, we still consider other factors, such as evaporation +and radiation heat transfer. The evaporation turns out to be the main reason of +heat loss, which corresponds with other scientist’s experimental outcome. +\end{itemize} + +\subsection{Weakness} + +\begin{itemize} +\item Having knowing the range of some parameters from others’ essays, we choose +a value from them to apply in our model. Those values may not be reasonable in +reality. + +\item Although we investigate a lot in the influence of personal motions, they +are so complicated that need to be studied further. + +\item Limited to time, we do not conduct sensitivity analysis for the influence +of personal surface area. +\end{itemize} + +\section{Further Discussion} + +Based on our model analysis and conclusions, we propose the optimal strategy +for the user in a bathtub and explain the reason for the uneven temperature +throughout the bathtub. In addition, we make improvements for applying our +model in real life, as suggested by the patent \textcite{patent2023}. + +In this part, we will focus on different distribution of inflow faucets. Then we +discuss about the real-life application of our model. + +\begin{itemize} +\item Different Distribution of Inflow Faucets + +In our before discussion, we assume there being just one entrance of inflow. + +From the simulating outcome, we find the temperature of bath water is hardly even. +So we come up with the idea of adding more entrances. + +The simulation turns out to be as follows + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig24.jpg} +\caption{The simulation results of different ways of arranging entrances} \label{fig24} +\end{figure} + +From the above figure, the more the entrances are, the evener the temperature +will be. Recalling on the before simulation outcome, when there is only one +entrance for inflow, the temperature of corners is quietly lower than the middle +area. + +In conclusion, if we design more entrances, it will be easier to realize the goal +to keep temperature even throughout the bathtub. + +\item Model Application + +Our before discussion is based on ideal assumptions. In reality, we have to make +some corrections and improvement. + +\begin{itemize} +\item[1)] Adding hot water continually with the mass flow of 0.16 kg/s. This way +can ensure even mean temperature throughout the bathtub and waste less water. + +\item[2)] The manufacturers can design an intelligent control system to monitor +the temperature so that users can get more enjoyable bath experience. + +\item[3)] We recommend users to add bubble additives to slow down the water being +cooler and help cleanse. The additives with lower thermal conductivity are optimal. + +\item[4)] The study method of our establishing model can be applied in other area +relative to convection heat transfer, such as air conditioners. +\end{itemize} +\end{itemize} + +\printbibliography + +\newpage + +\begin{letter}{Enjoy Your Bath Time!} + +From simulation results of real-life situations, we find it takes a period of +time for the inflow hot water to spread throughout the bathtub. During this +process, the bath water continues transferring heat into air, bathtub and the +person in bathtub. The difference between heat transfer capacity makes the +temperature of various areas to be different. So that it is difficult to get +an evenly maintained temperature throughout the bath water. + +In order to enjoy a comfortable bath with even temperature of bath water and +without wasting too much water, we propose the following suggestions. + +\begin{itemize} +\item Adding hot water consistently +\item Using smaller bathtub if possible +\item Decreasing motions during bath +\item Using bubble bath additives +\item Arranging more faucets of inflow +\end{itemize} + +\vspace{\parskip} + +Sincerely yours, + +Your friends + +\end{letter} + +\newpage + +\begin{appendices} + +\section{First appendix} + +In addition, your report must include a letter to the Chief Financial Officer +(CFO) of the Goodgrant Foundation, Mr. Alpha Chiang, that describes the optimal +investment strategy, your modeling approach and major results, and a brief +discussion of your proposed concept of a return-on-investment (ROI). This letter +should be no more than two pages in length. + +Here are simulation programmes we used in our model as follow (\cite{Liu02}).\\ + +\textbf{\textcolor[rgb]{0.98,0.00,0.00}{Input matlab source:}} +\lstinputlisting[language=Matlab]{./code/mcmthesis-matlab1.m} + +\section{Second appendix} + +some more text \textcolor[rgb]{0.98,0.00,0.00}{\textbf{Input C++ source:}} +\lstinputlisting[language=C++]{./code/mcmthesis-sudoku.cpp} + +\end{appendices} + +\newpage +\newcounter{lastpage} +\setcounter{lastpage}{\value{page}} +\thispagestyle{empty} + +\section*{Report on Use of AI} + +\begin{enumerate} +\item OpenAI ChatGPT (Nov 5, 2023 version, ChatGPT-4,) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\item OpenAI Ernie (Nov 5, 2023 version, Ernie 4.0) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\item Github CoPilot (Feb 3, 2024 version) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\item Google Bard (Feb 2, 2024 version) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\end{enumerate} + +% 重置页码 +\clearpage +\setcounter{page}{\value{lastpage}} + +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/latex/.Rproj.user/1248C2B7/sources/prop/12468681 b/latex/.Rproj.user/1248C2B7/sources/prop/12468681 new file mode 100644 index 0000000..dc18769 --- /dev/null +++ b/latex/.Rproj.user/1248C2B7/sources/prop/12468681 @@ -0,0 +1,6 @@ +{ + "source_window_id": "", + "Source": "Source", + "cursorPosition": "80,21", + "scrollLine": "72" +} \ No newline at end of file diff --git a/latex/.Rproj.user/1248C2B7/sources/prop/352D9BE2 b/latex/.Rproj.user/1248C2B7/sources/prop/352D9BE2 new file mode 100644 index 0000000..a007fad --- /dev/null +++ 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b/latex/.Rproj.user/4F8C70FB/sources/per/t/5A8D7EA9-contents @@ -0,0 +1,539 @@ +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {0000000}, problem = \textcolor{red}{A}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} +\usepackage[backend=bibtex]{biblatex} % for RStudio Complie + +\usepackage{tocloft} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Enjoy a Cozy and Green Bath} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} + +\begin{abstract} + +A traditional bathtub cannot be reheated by itself, so users have to add hot water from time to time. Our goal is to establish a model of the temperature of bath water in space and time. Then we are expected to propose an optimal strategy for users to keep the temperature even and close to initial temperature and decrease water consumption. + +To simplify modeling process, we firstly assume there is no person in the bathtub. We regard the whole bathtub as a thermodynamic system and introduce heat transfer formulas. + +We establish two sub-models: adding water constantly and discontinuously. As for the former sub-model, we define the mean temperature of bath water. Introducing Newton cooling formula, we determine the heat transfer capacity. After deriving the value of parameters, we deduce formulas to derive results and simulate the change of temperature field via CFD. As for the second sub-model, we define an iteration consisting of two process: heating and standby. According to energy conservation law, we obtain the relationship of time and total heat dissipating capacity. Then we determine the mass flow and the time of adding hot water. We also use CFD to simulate the temperature field in second sub-model. + +In consideration of evaporation, we correct the results of sub-models referring to some scientists' studies. We define two evaluation criteria and compare the two sub-models. Adding water constantly is found to keep the temperature of bath water even and avoid wasting too much water, so it is recommended by us. + +Then we determine the influence of some factors: radiation heat transfer, the shape and volume of the tub, the shape/volume/temperature/motions of the person, the bubbles made from bubble bath additives. We focus on the influence of those factors to heat transfer and then conduct sensitivity analysis. The results indicate smaller bathtub with less surface area, lighter personal mass, less motions and more bubbles will decrease heat transfer and save water. + +Based on our model analysis and conclusions, we propose the optimal strategy for the user in a bathtub and explain the reason of uneven temperature throughout the bathtub. In addition, we make improvement for applying our model in real life. + +\begin{keywords} +Heat transfer, Thermodynamic system, CFD, Energy conservation +\end{keywords} + +\end{abstract} + +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 +\thispagestyle{empty} + +\newpage + +\section{Introduction} + +\subsection{Background} + +Bathing in a tub is a perfect choice for those who have been worn out after a long day's working. A traditional bathtub is a simply water containment vessel without a secondary heating system or circulating jets. Thus the temperature of water in bathtub declines noticeably as time goes by, which will influent the experience of bathing. As a result, the bathing person needs to add a constant trickle of hot water from a faucet to reheat the bathing water. This way of bathing will result in waste of water because when the capacity of the bathtub is reached, excess water overflows the tub. + +An optimal bathing strategy is required for the person in a bathtub to get comfortable bathing experience while reducing the waste of water. + +\subsection{Literature Review} + +Korean physicist Gi-Beum Kim analyzed heat loss through free surface of water contained in bathtub due to conduction and evaporation \cite{1}. He derived a relational equation based on the basic theory of heat transfer to evaluate the performance of bath tubes. The major heat loss was found to be due to evaporation. Moreover, he found out that the speed of heat loss depends more on the humidity of the bathroom than the temperature of water contained in the bathtub. So, it is best to maintain the temperature of bathtub water to be between 41 to 45$^{\circ}$C and the humidity of bathroom to be 95\%. + +When it comes to convective heat transfer in bathtub, many studies can be referred to. Newton's law of cooling states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings while under the effects of a breeze \cite{2}. Claude-Louis Navier and George Gabriel Stokes described the motion of viscous fluid substances with the Navier-Stokes equations. Those equations may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing \cite{3}. + +In addition, some numerical simulation software are applied in solving and analyzing problems that involve fluid flows. For example, Computational Fluid Dynamics (CFD) is a common one used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions \cite{4}. + +\subsection{Restatement of the Problem} + +We are required to establish a model to determine the change of water temperature in space and time. Then we are expected to propose the best strategy for the person in the bathtub to keep the water temperature close to initial temperature and even throughout the tub. Reduction of waste of water is also needed. In addition, we have to consider the impact of different conditions on our model, such as different shapes and volumes of the bathtub, etc. + +In order to solve those problems, we will proceed as follows: + +\begin{itemize} +\item {\bf Stating assumptions}. By stating our assumptions, we will narrow the focus of our approach towards the problems and provide some insight into bathtub water temperature issues. + +\item {\bf Making notations}. We will give some notations which are important for us to clarify our models. + +\item {\bf Presenting our model}. In order to investigate the problem deeper, we divide our model into two sub-models. One is a steady convection heat transfer sub-model in which hot water is added constantly. The other one is an unsteady convection heat transfer sub-model where hot water is added discontinuously. + +\item {Defining evaluation criteria and comparing sub-models}. We define two main criteria to evaluate our model: the mean temperature of bath water and the amount of inflow water. + +\item {\bf Analysis of influencing factors}. In term of the impact of different factors on our model, we take those into consideration: the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, the motions made by the person in the bathtub and adding a bubble bath additive initially. + +\item {\bf Model testing and sensitivity analysis}. With the criteria defined before, we evaluate the reliability of our model and do the sensitivity analysis. + +\item {\bf Further discussion}. We discuss about different ways to arrange inflow faucets. Then we improve our model to apply them in reality. + +\item {\bf Evaluating the model}. We discuss about the strengths and weaknesses of our model: + +\begin{itemize} +\item[1)] ... +\item[2)] ... +\item[3)] ... +\item[4)] ... +\end{itemize} + +\end{itemize} + +\section{Assumptions and Justification} + +To simplify the problem and make it convenient for us to simulate real-life conditions, we make the following basic assumptions, each of which is properly justified. + +\begin{itemize} +\item {\bf The bath water is incompressible Non-Newtonian fluid}. The incompressible Non-Newtonian fluid is the basis of Navier–Stokes equations which are introduced to simulate the flow of bath water. + +\item {\bf All the physical properties of bath water, bathtub and air are assumed to be stable}. The change of those properties like specific heat, thermal conductivity and density is rather small according to some studies \cite{5}. It is complicated and unnecessary to consider these little change so we ignore them. + +\item {\bf There is no internal heat source in the system consisting of bathtub, hot water and air}. Before the person lies in the bathtub, no internal heat source exist except the system components. The circumstance where the person is in the bathtub will be investigated in our later discussion. + +\item {\bf We ignore radiative thermal exchange}. According to Stefan-Boltzmann’s law, the radiative thermal exchange can be ignored when the temperature is low. Refer to industrial standard \cite{6}, the temperature in bathroom is lower than 100 $^{\circ}$C, so it is reasonable for us to make this assumption. + +\item {\bf The temperature of the adding hot water from the faucet is stable}. This hypothesis can be easily achieved in reality and will simplify our process of solving the problem. +\end{itemize} + +\section{Notations} + +\begin{center} +\begin{tabular}{clc} +{\bf Symbols} & {\bf Description} & \quad {\bf Unit} \\[0.25cm] +$h$ & Convection heat transfer coefficient & \quad W/(m$^2 \cdot$ K) +\\[0.2cm] +$k$ & Thermal conductivity & \quad W/(m $\cdot$ K) \\[0.2cm] +$c_p$ & Specific heat & \quad J/(kg $\cdot$ K) \\[0.2cm] +$\rho$ & Density & \quad kg/m$^2$ \\[0.2cm] +$\delta$ & Thickness & \quad m \\[0.2cm] +$t$ & Temperature & \quad $^\circ$C, K \\[0.2cm] +$\tau$ & Time & \quad s, min, h \\[0.2cm] +$q_m$ & Mass flow & \quad kg/s \\[0.2cm] +$\Phi$ & Heat transfer power & \quad W \\[0.2cm] +$T$ & A period of time & \quad s, min, h \\[0.2cm] +$V$ & Volume & \quad m$^3$, L \\[0.2cm] +$M,\,m$ & Mass & \quad kg \\[0.2cm] +$A$ & Aera & \quad m$^2$ \\[0.2cm] +$a,\,b,\,c$ & The size of a bathtub & \quad m$^3$ +\end{tabular} +\end{center} + +\noindent where we define the main parameters while specific value of those parameters will be given later. + +\section{Model Overview} + +In our basic model, we aim at three goals: keeping the temperature as even as possible, making it close to the initial temperature and decreasing the water consumption. + +We start with the simple sub-model where hot water is added constantly. +At first we introduce convection heat transfer control equations in rectangular coordinate system. Then we define the mean temperature of bath water. + +Afterwards, we introduce Newton cooling formula to determine heat transfer +capacity. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +Secondly, we present the complicated sub-model in which hot water is +added discontinuously. We define an iteration consisting of two process: +heating and standby. As for heating process, we derive control equations and boundary conditions. As for standby process, considering energy conservation law, we deduce the relationship of total heat dissipating capacity and time. + +Then we determine the time and amount of added hot water. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +At last, we define two criteria to evaluate those two ways of adding hot water. Then we propose optimal strategy for the user in a bathtub. +The whole modeling process can be shown as follows. + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig1.jpg} +\caption{Modeling process} \label{fig1} +\end{figure} + +\section{Sub-model I : Adding Water Continuously} + +We first establish the sub-model based on the condition that a person add water continuously to reheat the bathing water. Then we use Computational Fluid Dynamics (CFD) to simulate the change of water temperature in the bathtub. At last, we evaluate the model with the criteria which have been defined before. + +\subsection{Model Establishment} + +Since we try to keep the temperature of the hot water in bathtub to be even, we have to derive the amount of inflow water and the energy dissipated by the hot water into the air. + +We derive the basic convection heat transfer control equations based on the former scientists’ achievement. Then, we define the mean temperature of bath water. Afterwards, we determine two types of heat transfer: the boundary heat transfer and the evaporation heat transfer. Combining thermodynamic formulas, we derive calculating results. Via Fluent software, we get simulation results. + +\subsubsection{Control Equations and Boundary Conditions} + +According to thermodynamics knowledge, we recall on basic convection +heat transfer control equations in rectangular coordinate system. Those +equations show the relationship of the temperature of the bathtub water in space. + +We assume the hot water in the bathtub as a cube. Then we put it into a +rectangular coordinate system. The length, width, and height of it is $a,\, b$ and $c$. + +\begin{figure}[h] +\centering +\includegraphics[width=8cm]{fig2.jpg} +\caption{Modeling process} \label{fig2} +\end{figure} + +In the basis of this, we introduce the following equations \cite{5}: + +\begin{itemize} +\item {\bf Continuity equation:} +\end{itemize} + +\begin{equation} \label{eq1} +\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z} =0 +\end{equation} + +\noindent where the first component is the change of fluid mass along the $X$-ray. The second component is the change of fluid mass along the $Y$-ray. And the third component is the change of fluid mass along the $Z$-ray. The sum of the change in mass along those three directions is zero. + +\begin{itemize} +\item {\bf Moment differential equation (N-S equations):} +\end{itemize} + +\begin{equation} \label{eq2} +\left\{ +\begin{array}{l} \!\! +\rho \Big(u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + w\dfrac{\partial u}{\partial z} \Big) = -\dfrac{\partial p}{\partial x} + \eta \Big(\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} + \dfrac{\partial^2 u}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} + w\dfrac{\partial v}{\partial z} \Big) = -\dfrac{\partial p}{\partial y} + \eta \Big(\dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} + \dfrac{\partial^2 v}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial w}{\partial x} + v \dfrac{\partial w}{\partial y} + w\dfrac{\partial w}{\partial z} \Big) = -g-\dfrac{\partial p}{\partial z} + \eta \Big(\dfrac{\partial^2 w}{\partial x^2} + \dfrac{\partial^2 w}{\partial y^2} + \dfrac{\partial^2 w}{\partial z^2} \Big) +\end{array} +\right. +\end{equation} + +\begin{itemize} +\item {\bf Energy differential equation:} +\end{itemize} + +\begin{equation} \label{eq3} +\rho c_p \Big( u\frac{\partial t}{\partial x} + v\frac{\partial t}{\partial y} + w\frac{\partial t}{\partial z} \Big) = \lambda \Big(\frac{\partial^2 t}{\partial x^2} + \frac{\partial^2 t}{\partial y^2} + \frac{\partial^2 t}{\partial z^2} \Big) +\end{equation} + +\noindent where the left three components are convection terms while the right three components are conduction terms. + +By Equation \eqref{eq3}, we have ...... + +...... + +On the right surface in Fig. \ref{fig2}, the water also transfers heat firstly with bathtub inner surfaces and then the heat comes into air. The boundary condition here is ...... + +\subsubsection{Definition of the Mean Temperature} + +...... + +\subsubsection{Determination of Heat Transfer Capacity} + +...... + +\section{Sub-model II: Adding Water Discontinuously} + +In order to establish the unsteady sub-model, we recall on the working principle of air conditioners. The heating performance of air conditions consist of two processes: heating and standby. After the user set a temperature, the air conditioner will begin to heat until the expected temperature is reached. Then it will go standby. When the temperature get below the expected temperature, the air conditioner begin to work again. As it works in this circle, the temperature remains the expected one. + +Inspired by this, we divide the bathtub working into two processes: adding +hot water until the expected temperature is reached, then keeping this +condition for a while unless the temperature is lower than a specific value. Iterating this circle ceaselessly will ensure the temperature kept relatively stable. + +\subsection{Heating Model} + +\subsubsection{Control Equations and Boundary Conditions} + +\subsubsection{Determination of Inflow Time and Amount} + +\subsection{Standby Model} + +\subsection{Results} + +\quad~ We first give the value of parameters based on others’ studies. Then we get the calculation results and simulating results via those data. + +\subsubsection{Determination of Parameters} + +After establishing the model, we have to determine the value of some +important parameters. + +As scholar Beum Kim points out, the optimal temperature for bath is +between 41 and 45$^\circ$C [1]. Meanwhile, according to Shimodozono's study, 41$^\circ$C warm water bath is the perfect choice for individual health [2]. So it is reasonable for us to focus on $41^\circ$C $\sim 45^\circ$C. Because adding hot water continuously is a steady process, so the mean temperature of bath water is supposed to be constant. We value the temperature of inflow and outflow water with the maximum and minimum temperature respectively. + +The values of all parameters needed are shown as follows: + +..... + +\subsubsection{Calculating Results} + +Putting the above value of parameters into the equations we derived before, we can get the some data as follows: + +%%普通表格 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{The calculating results} %表标题 +\vspace{0.15cm} +\label{tab2} %设置表的引用标签 +\begin{tabular}{|c|c|c|} %3个c表示3列, |可选, 表示绘制各列间的竖线 +\hline %画横线 +Variables & Values & Unit \\ \hline %各列间用&隔开 +$A_1$ & 1.05 & $m^2$ \\ \hline +$A_2$ & 2.24 & $m^2$ \\ \hline +$\Phi_1$ & 189.00 & $W$ \\ \hline +$\Phi_2$ & 43.47 & $W$ \\ \hline +$\Phi$ & 232.47 & $W$ \\ \hline +$q_m$ & 0.014 & $g/s$ \\ \hline +\end{tabular} +\end{table} + +From Table \ref{tab2}, ...... + +...... + +\section{Correction and Contrast of Sub-Models} + +After establishing two basic sub-models, we have to correct them in consideration of evaporation heat transfer. Then we define two evaluation criteria to compare the two sub-models in order to determine the optimal bath strategy. + +\subsection{Correction with Evaporation Heat Transfer} + +Someone may confuse about the above results: why the mass flow in the first sub-model is so small? Why the standby time is so long? Actually, the above two sub-models are based on ideal conditions without consideration of the change of boundary conditions, the motions made by the person in bathtub and the evaporation of bath water, etc. The influence of personal motions will be discussed later. Here we introducing the evaporation of bath water to correct sub-models. + +\subsection{Contrast of Two Sub-Models} + +Firstly we define two evaluation criteria. Then we contrast the two submodels via these two criteria. Thus we can derive the best strategy for the person in the bathtub to adopt. + +\section{Model Analysis and Sensitivity Analysis} + +\subsection{The Influence of Different Bathtubs} + +Definitely, the difference in shape and volume of the tub affects the +convection heat transfer. Examining the relationship between them can help +people choose optimal bathtubs. + +\subsubsection{Different Volumes of Bathtubs} + +In reality, a cup of water will be cooled down rapidly. However, it takes quite long time for a bucket of water to become cool. That is because their volume is different and the specific heat of water is very large. So that the decrease of temperature is not obvious if the volume of water is huge. That also explains why it takes 45 min for 320 L water to be cooled by 1$^\circ$C. + +In order to examine the influence of volume, we analyze our sub-models +by conducting sensitivity Analysis to them. + +We assume the initial volume to be 280 L and change it by $\pm 5$\%, $\pm 8$\%, $\pm 12$\% and $\pm 15$\%. With the aid of sub-models we established before, the variation of some parameters turns out to be as follows + +%%三线表 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{Variation of some parameters} %表标题 +\label{tab7} %设置表的引用标签 +\begin{tabular}{ccccccc} %7个c表示7列, c表示每列居中对齐, 还有l和r可选 +\toprule %画顶端横线 +$V$ & $A_1$ & $A_2$ & $T_2$ & $q_{m1}$ & $q_{m2}$ & $\Phi_q$ \\ +\midrule %画中间横线 +-15.00\% & -5.06\% & -9.31\% & -12.67\% & -2.67\% & -14.14\% & -5.80\% \\ +-12.00\% & -4.04\% & -7.43\% & -10.09\% & -2.13\% & -11.31\% & -4.63\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +\bottomrule %画底部横线 +\end{tabular} +\end{table} + +\section{Strength and Weakness} + +\subsection{Strength} + +\begin{itemize} +\item We analyze the problem based on thermodynamic formulas and laws, so that the model we established is of great validity. + +\item Our model is fairly robust due to our careful corrections in consideration of real-life situations and detailed sensitivity analysis. + +\item Via Fluent software, we simulate the time field of different areas throughout the bathtub. The outcome is vivid for us to understand the changing process. + +\item We come up with various criteria to compare different situations, like water consumption and the time of adding hot water. Hence an overall comparison can be made according to these criteria. + +\item Besides common factors, we still consider other factors, such as evaporation and radiation heat transfer. The evaporation turns out to be the main reason of heat loss, which corresponds with other scientist’s experimental outcome. +\end{itemize} + +\subsection{Weakness} + +\begin{itemize} +\item Having knowing the range of some parameters from others’ essays, we choose a value from them to apply in our model. Those values may not be reasonable in reality. + +\item Although we investigate a lot in the influence of personal motions, they are so complicated that need to be studied further. + +\item Limited to time, we do not conduct sensitivity analysis for the influence of personal surface area. +\end{itemize} + +\section{Further Discussion} + +In this part, we will focus on different distribution of inflow faucets. Then we discuss about the real-life application of our model. + +\begin{itemize} +\item Different Distribution of Inflow Faucets + +In our before discussion, we assume there being just one entrance of inflow. + +From the simulating outcome, we find the temperature of bath water is hardly even. So we come up with the idea of adding more entrances. + +The simulation turns out to be as follows + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig24.jpg} +\caption{The simulation results of different ways of arranging entrances} \label{fig24} +\end{figure} + +From the above figure, the more the entrances are, the evener the temperature will be. Recalling on the before simulation outcome, when there is only one entrance for inflow, the temperature of corners is quietly lower than the middle area. + +In conclusion, if we design more entrances, it will be easier to realize the goal to keep temperature even throughout the bathtub. + +\item Model Application + +Our before discussion is based on ideal assumptions. In reality, we have to make some corrections and improvement. + +\begin{itemize} +\item[1)] Adding hot water continually with the mass flow of 0.16 kg/s. This way can ensure even mean temperature throughout the bathtub and waste less water. + +\item[2)] The manufacturers can design an intelligent control system to monitor the temperature so that users can get more enjoyable bath experience. + +\item[3)] We recommend users to add bubble additives to slow down the water being cooler and help cleanse. The additives with lower thermal conductivity are optimal. + +\item[4)] The study method of our establishing model can be applied in other area relative to convection heat transfer, such as air conditioners. +\end{itemize} +\end{itemize} + +\begin{thebibliography}{99} +\addcontentsline{toc}{section}{Reference} + +\bibitem{1} Gi-Beum Kim. Change of the Warm Water Temperature for the Development of Smart Healthecare Bathing System. Hwahak konghak. 2006, 44(3): 270-276. +\bibitem{2} \url{https://en.wikipedia.org/wiki/Convective_heat_transfer#Newton.27s_law_of_cooling} +\bibitem{3} \url{https://en.wikipedia.org/wiki/Navier\%E2\%80\%93Stokes_equations} +\bibitem{4} \url{https://en.wikipedia.org/wiki/Computational_fluid_dynamics} +\bibitem{5} Holman J P. Heat Transfer (9th ed.), New York: McGraw-Hill, 2002. +\bibitem{6} Liu Weiguo, Chen Zhaoping, ZhangYin. Matlab program design and application. Beijing: Higher education press, 2002. (In Chinese) + +\end{thebibliography} + +\newpage + +\begin{letter}{Enjoy Your Bath Time!} + +From simulation results of real-life situations, we find it takes a period of time for the inflow hot water to spread throughout the bathtub. During this process, the bath water continues transferring heat into air, bathtub and the person in bathtub. The difference between heat transfer capacity makes the temperature of various areas to be different. So that it is difficult to get an evenly maintained temperature throughout the bath water. + +In order to enjoy a comfortable bath with even temperature of bath water and without wasting too much water, we propose the following suggestions. + +\begin{itemize} +\item Adding hot water consistently +\item Using smaller bathtub if possible +\item Decreasing motions during bath +\item Using bubble bath additives +\item Arranging more faucets of inflow +\end{itemize} + +\vspace{\parskip} + +Sincerely yours, + +Your friends + +\end{letter} + +\newpage + +\begin{appendices} + +\section{First appendix} + +In addition, your report must include a letter to the Chief Financial Officer (CFO) of the Goodgrant Foundation, Mr. Alpha Chiang, that describes the optimal investment strategy, your modeling approach and major results, and a brief discussion of your proposed concept of a return-on-investment (ROI). This letter should be no more than two pages in length. + +Here are simulation programmes we used in our model as follow.\\ + +\textbf{\textcolor[rgb]{0.98,0.00,0.00}{Input matlab source:}} +\lstinputlisting[language=Matlab]{./code/mcmthesis-matlab1.m} + +\section{Second appendix} + +some more text \textcolor[rgb]{0.98,0.00,0.00}{\textbf{Input C++ source:}} +\lstinputlisting[language=C++]{./code/mcmthesis-sudoku.cpp} + +\end{appendices} +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/latex/.Rproj.user/4F8C70FB/sources/prop/6676DB66 b/latex/.Rproj.user/4F8C70FB/sources/prop/6676DB66 new file mode 100644 index 0000000..31ad8a8 --- /dev/null +++ b/latex/.Rproj.user/4F8C70FB/sources/prop/6676DB66 @@ -0,0 +1,6 @@ +{ + "source_window_id": "", + 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mode 100644 index 0000000..d7cce2f --- /dev/null +++ b/latex/.Rproj.user/F69231EA/sources/per/t/110CA16B-contents @@ -0,0 +1,539 @@ +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {0000000}, problem = \textcolor{red}{A}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} +\usepackage[backend=bibtex]{biblatex} % for RStudio Complie + +\usepackage{tocloft} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Enjoy a Cozy and Green Bath} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} + +\begin{abstract} + +A traditional bathtub cannot be reheated by itself, so users have to add hot water from time to time. Our goal is to establish a model of the temperature of bath water in space and time. Then we are expected to propose an optimal strategy for users to keep the temperature even and close to initial temperature and decrease water consumption. + +To simplify modeling process, we firstly assume there is no person in the bathtub. We regard the whole bathtub as a thermodynamic system and introduce heat transfer formulas. + +We establish two sub-models: adding water constantly and discontinuously. As for the former sub-model, we define the mean temperature of bath water. Introducing Newton cooling formula, we determine the heat transfer capacity. After deriving the value of parameters, we deduce formulas to derive results and simulate the change of temperature field via CFD. As for the second sub-model, we define an iteration consisting of two process: heating and standby. According to energy conservation law, we obtain the relationship of time and total heat dissipating capacity. Then we determine the mass flow and the time of adding hot water. We also use CFD to simulate the temperature field in second sub-model. + +In consideration of evaporation, we correct the results of sub-models referring to some scientists' studies. We define two evaluation criteria and compare the two sub-models. Adding water constantly is found to keep the temperature of bath water even and avoid wasting too much water, so it is recommended by us. + +Then we determine the influence of some factors: radiation heat transfer, the shape and volume of the tub, the shape/volume/temperature/motions of the person, the bubbles made from bubble bath additives. We focus on the influence of those factors to heat transfer and then conduct sensitivity analysis. The results indicate smaller bathtub with less surface area, lighter personal mass, less motions and more bubbles will decrease heat transfer and save water. + +Based on our model analysis and conclusions, we propose the optimal strategy for the user in a bathtub and explain the reason of uneven temperature throughout the bathtub. In addition, we make improvement for applying our model in real life. + +\begin{keywords} +Heat transfer, Thermodynamic system, CFD, Energy conservation +\end{keywords} + +\end{abstract} + +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 +\thispagestyle{empty} + +\newpage + +\section{Introduction} + +\subsection{Background} + +Bathing in a tub is a perfect choice for those who have been worn out after a long day's working. A traditional bathtub is a simply water containment vessel without a secondary heating system or circulating jets. Thus the temperature of water in bathtub declines noticeably as time goes by, which will influent the experience of bathing. As a result, the bathing person needs to add a constant trickle of hot water from a faucet to reheat the bathing water. This way of bathing will result in waste of water because when the capacity of the bathtub is reached, excess water overflows the tub. + +An optimal bathing strategy is required for the person in a bathtub to get comfortable bathing experience while reducing the waste of water. + +\subsection{Literature Review} + +Korean physicist Gi-Beum Kim analyzed heat loss through free surface of water contained in bathtub due to conduction and evaporation \cite{1}. He derived a relational equation based on the basic theory of heat transfer to evaluate the performance of bath tubes. The major heat loss was found to be due to evaporation. Moreover, he found out that the speed of heat loss depends more on the humidity of the bathroom than the temperature of water contained in the bathtub. So, it is best to maintain the temperature of bathtub water to be between 41 to 45$^{\circ}$C and the humidity of bathroom to be 95\%. + +When it comes to convective heat transfer in bathtub, many studies can be referred to. Newton's law of cooling states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings while under the effects of a breeze \cite{2}. Claude-Louis Navier and George Gabriel Stokes described the motion of viscous fluid substances with the Navier-Stokes equations. Those equations may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing \cite{3}. + +In addition, some numerical simulation software are applied in solving and analyzing problems that involve fluid flows. For example, Computational Fluid Dynamics (CFD) is a common one used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions \cite{4}. + +\subsection{Restatement of the Problem} + +We are required to establish a model to determine the change of water temperature in space and time. Then we are expected to propose the best strategy for the person in the bathtub to keep the water temperature close to initial temperature and even throughout the tub. Reduction of waste of water is also needed. In addition, we have to consider the impact of different conditions on our model, such as different shapes and volumes of the bathtub, etc. + +In order to solve those problems, we will proceed as follows: + +\begin{itemize} +\item {\bf Stating assumptions}. By stating our assumptions, we will narrow the focus of our approach towards the problems and provide some insight into bathtub water temperature issues. + +\item {\bf Making notations}. We will give some notations which are important for us to clarify our models. + +\item {\bf Presenting our model}. In order to investigate the problem deeper, we divide our model into two sub-models. One is a steady convection heat transfer sub-model in which hot water is added constantly. The other one is an unsteady convection heat transfer sub-model where hot water is added discontinuously. + +\item {Defining evaluation criteria and comparing sub-models}. We define two main criteria to evaluate our model: the mean temperature of bath water and the amount of inflow water. + +\item {\bf Analysis of influencing factors}. In term of the impact of different factors on our model, we take those into consideration: the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, the motions made by the person in the bathtub and adding a bubble bath additive initially. + +\item {\bf Model testing and sensitivity analysis}. With the criteria defined before, we evaluate the reliability of our model and do the sensitivity analysis. + +\item {\bf Further discussion}. We discuss about different ways to arrange inflow faucets. Then we improve our model to apply them in reality. + +\item {\bf Evaluating the model}. We discuss about the strengths and weaknesses of our model: + +\begin{itemize} +\item[1)] ... +\item[2)] ... +\item[3)] ... +\item[4)] ... +\end{itemize} + +\end{itemize} + +\section{Assumptions and Justification} + +To simplify the problem and make it convenient for us to simulate real-life conditions, we make the following basic assumptions, each of which is properly justified. + +\begin{itemize} +\item {\bf The bath water is incompressible Non-Newtonian fluid}. The incompressible Non-Newtonian fluid is the basis of Navier–Stokes equations which are introduced to simulate the flow of bath water. + +\item {\bf All the physical properties of bath water, bathtub and air are assumed to be stable}. The change of those properties like specific heat, thermal conductivity and density is rather small according to some studies \cite{5}. It is complicated and unnecessary to consider these little change so we ignore them. + +\item {\bf There is no internal heat source in the system consisting of bathtub, hot water and air}. Before the person lies in the bathtub, no internal heat source exist except the system components. The circumstance where the person is in the bathtub will be investigated in our later discussion. + +\item {\bf We ignore radiative thermal exchange}. According to Stefan-Boltzmann’s law, the radiative thermal exchange can be ignored when the temperature is low. Refer to industrial standard \cite{6}, the temperature in bathroom is lower than 100 $^{\circ}$C, so it is reasonable for us to make this assumption. + +\item {\bf The temperature of the adding hot water from the faucet is stable}. This hypothesis can be easily achieved in reality and will simplify our process of solving the problem. +\end{itemize} + +\section{Notations} + +\begin{center} +\begin{tabular}{clc} +{\bf Symbols} & {\bf Description} & \quad {\bf Unit} \\[0.25cm] +$h$ & Convection heat transfer coefficient & \quad W/(m$^2 \cdot$ K) +\\[0.2cm] +$k$ & Thermal conductivity & \quad W/(m $\cdot$ K) \\[0.2cm] +$c_p$ & Specific heat & \quad J/(kg $\cdot$ K) \\[0.2cm] +$\rho$ & Density & \quad kg/m$^2$ \\[0.2cm] +$\delta$ & Thickness & \quad m \\[0.2cm] +$t$ & Temperature & \quad $^\circ$C, K \\[0.2cm] +$\tau$ & Time & \quad s, min, h \\[0.2cm] +$q_m$ & Mass flow & \quad kg/s \\[0.2cm] +$\Phi$ & Heat transfer power & \quad W \\[0.2cm] +$T$ & A period of time & \quad s, min, h \\[0.2cm] +$V$ & Volume & \quad m$^3$, L \\[0.2cm] +$M,\,m$ & Mass & \quad kg \\[0.2cm] +$A$ & Aera & \quad m$^2$ \\[0.2cm] +$a,\,b,\,c$ & The size of a bathtub & \quad m$^3$ +\end{tabular} +\end{center} + +\noindent where we define the main parameters while specific value of those parameters will be given later. + +\section{Model Overview} + +In our basic model, we aim at three goals: keeping the temperature as even as possible, making it close to the initial temperature and decreasing the water consumption. + +We start with the simple sub-model where hot water is added constantly. +At first we introduce convection heat transfer control equations in rectangular coordinate system. Then we define the mean temperature of bath water. + +Afterwards, we introduce Newton cooling formula to determine heat transfer +capacity. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +Secondly, we present the complicated sub-model in which hot water is +added discontinuously. We define an iteration consisting of two process: +heating and standby. As for heating process, we derive control equations and boundary conditions. As for standby process, considering energy conservation law, we deduce the relationship of total heat dissipating capacity and time. + +Then we determine the time and amount of added hot water. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +At last, we define two criteria to evaluate those two ways of adding hot water. Then we propose optimal strategy for the user in a bathtub. +The whole modeling process can be shown as follows. + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig1.jpg} +\caption{Modeling process} \label{fig1} +\end{figure} + +\section{Sub-model I : Adding Water Continuously} + +We first establish the sub-model based on the condition that a person add water continuously to reheat the bathing water. Then we use Computational Fluid Dynamics (CFD) to simulate the change of water temperature in the bathtub. At last, we evaluate the model with the criteria which have been defined before. + +\subsection{Model Establishment} + +Since we try to keep the temperature of the hot water in bathtub to be even, we have to derive the amount of inflow water and the energy dissipated by the hot water into the air. + +We derive the basic convection heat transfer control equations based on the former scientists’ achievement. Then, we define the mean temperature of bath water. Afterwards, we determine two types of heat transfer: the boundary heat transfer and the evaporation heat transfer. Combining thermodynamic formulas, we derive calculating results. Via Fluent software, we get simulation results. + +\subsubsection{Control Equations and Boundary Conditions} + +According to thermodynamics knowledge, we recall on basic convection +heat transfer control equations in rectangular coordinate system. Those +equations show the relationship of the temperature of the bathtub water in space. + +We assume the hot water in the bathtub as a cube. Then we put it into a +rectangular coordinate system. The length, width, and height of it is $a,\, b$ and $c$. + +\begin{figure}[h] +\centering +\includegraphics[width=8cm]{fig2.jpg} +\caption{Modeling process} \label{fig2} +\end{figure} + +In the basis of this, we introduce the following equations \cite{5}: + +\begin{itemize} +\item {\bf Continuity equation:} +\end{itemize} + +\begin{equation} \label{eq1} +\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z} =0 +\end{equation} + +\noindent where the first component is the change of fluid mass along the $X$-ray. The second component is the change of fluid mass along the $Y$-ray. And the third component is the change of fluid mass along the $Z$-ray. The sum of the change in mass along those three directions is zero. + +\begin{itemize} +\item {\bf Moment differential equation (N-S equations):} +\end{itemize} + +\begin{equation} \label{eq2} +\left\{ +\begin{array}{l} \!\! +\rho \Big(u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + w\dfrac{\partial u}{\partial z} \Big) = -\dfrac{\partial p}{\partial x} + \eta \Big(\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} + \dfrac{\partial^2 u}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} + w\dfrac{\partial v}{\partial z} \Big) = -\dfrac{\partial p}{\partial y} + \eta \Big(\dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} + \dfrac{\partial^2 v}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial w}{\partial x} + v \dfrac{\partial w}{\partial y} + w\dfrac{\partial w}{\partial z} \Big) = -g-\dfrac{\partial p}{\partial z} + \eta \Big(\dfrac{\partial^2 w}{\partial x^2} + \dfrac{\partial^2 w}{\partial y^2} + \dfrac{\partial^2 w}{\partial z^2} \Big) +\end{array} +\right. +\end{equation} + +\begin{itemize} +\item {\bf Energy differential equation:} +\end{itemize} + +\begin{equation} \label{eq3} +\rho c_p \Big( u\frac{\partial t}{\partial x} + v\frac{\partial t}{\partial y} + w\frac{\partial t}{\partial z} \Big) = \lambda \Big(\frac{\partial^2 t}{\partial x^2} + \frac{\partial^2 t}{\partial y^2} + \frac{\partial^2 t}{\partial z^2} \Big) +\end{equation} + +\noindent where the left three components are convection terms while the right three components are conduction terms. + +By Equation \eqref{eq3}, we have ...... + +...... + +On the right surface in Fig. \ref{fig2}, the water also transfers heat firstly with bathtub inner surfaces and then the heat comes into air. The boundary condition here is ...... + +\subsubsection{Definition of the Mean Temperature} + +...... + +\subsubsection{Determination of Heat Transfer Capacity} + +...... + +\section{Sub-model II: Adding Water Discontinuously} + +In order to establish the unsteady sub-model, we recall on the working principle of air conditioners. The heating performance of air conditions consist of two processes: heating and standby. After the user set a temperature, the air conditioner will begin to heat until the expected temperature is reached. Then it will go standby. When the temperature get below the expected temperature, the air conditioner begin to work again. As it works in this circle, the temperature remains the expected one. + +Inspired by this, we divide the bathtub working into two processes: adding +hot water until the expected temperature is reached, then keeping this +condition for a while unless the temperature is lower than a specific value. Iterating this circle ceaselessly will ensure the temperature kept relatively stable. + +\subsection{Heating Model} + +\subsubsection{Control Equations and Boundary Conditions} + +\subsubsection{Determination of Inflow Time and Amount} + +\subsection{Standby Model} + +\subsection{Results} + +\quad~ We first give the value of parameters based on others’ studies. Then we get the calculation results and simulating results via those data. + +\subsubsection{Determination of Parameters} + +After establishing the model, we have to determine the value of some +important parameters. + +As scholar Beum Kim points out, the optimal temperature for bath is +between 41 and 45$^\circ$C [1]. Meanwhile, according to Shimodozono's study, 41$^\circ$C warm water bath is the perfect choice for individual health [2]. So it is reasonable for us to focus on $41^\circ$C $\sim 45^\circ$C. Because adding hot water continuously is a steady process, so the mean temperature of bath water is supposed to be constant. We value the temperature of inflow and outflow water with the maximum and minimum temperature respectively. + +The values of all parameters needed are shown as follows: + +..... + +\subsubsection{Calculating Results} + +Putting the above value of parameters into the equations we derived before, we can get the some data as follows: + +%%普通表格 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{The calculating results} %表标题 +\vspace{0.15cm} +\label{tab2} %设置表的引用标签 +\begin{tabular}{|c|c|c|} %3个c表示3列, |可选, 表示绘制各列间的竖线 +\hline %画横线 +Variables & Values & Unit \\ \hline %各列间用&隔开 +$A_1$ & 1.05 & $m^2$ \\ \hline +$A_2$ & 2.24 & $m^2$ \\ \hline +$\Phi_1$ & 189.00 & $W$ \\ \hline +$\Phi_2$ & 43.47 & $W$ \\ \hline +$\Phi$ & 232.47 & $W$ \\ \hline +$q_m$ & 0.014 & $g/s$ \\ \hline +\end{tabular} +\end{table} + +From Table \ref{tab2}, ...... + +...... + +\section{Correction and Contrast of Sub-Models} + +After establishing two basic sub-models, we have to correct them in consideration of evaporation heat transfer. Then we define two evaluation criteria to compare the two sub-models in order to determine the optimal bath strategy. + +\subsection{Correction with Evaporation Heat Transfer} + +Someone may confuse about the above results: why the mass flow in the first sub-model is so small? Why the standby time is so long? Actually, the above two sub-models are based on ideal conditions without consideration of the change of boundary conditions, the motions made by the person in bathtub and the evaporation of bath water, etc. The influence of personal motions will be discussed later. Here we introducing the evaporation of bath water to correct sub-models. + +\subsection{Contrast of Two Sub-Models} + +Firstly we define two evaluation criteria. Then we contrast the two submodels via these two criteria. Thus we can derive the best strategy for the person in the bathtub to adopt. + +\section{Model Analysis and Sensitivity Analysis} + +\subsection{The Influence of Different Bathtubs} + +Definitely, the difference in shape and volume of the tub affects the +convection heat transfer. Examining the relationship between them can help +people choose optimal bathtubs. + +\subsubsection{Different Volumes of Bathtubs} + +In reality, a cup of water will be cooled down rapidly. However, it takes quite long time for a bucket of water to become cool. That is because their volume is different and the specific heat of water is very large. So that the decrease of temperature is not obvious if the volume of water is huge. That also explains why it takes 45 min for 320 L water to be cooled by 1$^\circ$C. + +In order to examine the influence of volume, we analyze our sub-models +by conducting sensitivity Analysis to them. + +We assume the initial volume to be 280 L and change it by $\pm 5$\%, $\pm 8$\%, $\pm 12$\% and $\pm 15$\%. With the aid of sub-models we established before, the variation of some parameters turns out to be as follows + +%%三线表 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{Variation of some parameters} %表标题 +\label{tab7} %设置表的引用标签 +\begin{tabular}{ccccccc} %7个c表示7列, c表示每列居中对齐, 还有l和r可选 +\toprule %画顶端横线 +$V$ & $A_1$ & $A_2$ & $T_2$ & $q_{m1}$ & $q_{m2}$ & $\Phi_q$ \\ +\midrule %画中间横线 +-15.00\% & -5.06\% & -9.31\% & -12.67\% & -2.67\% & -14.14\% & -5.80\% \\ +-12.00\% & -4.04\% & -7.43\% & -10.09\% & -2.13\% & -11.31\% & -4.63\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +\bottomrule %画底部横线 +\end{tabular} +\end{table} + +\section{Strength and Weakness} + +\subsection{Strength} + +\begin{itemize} +\item We analyze the problem based on thermodynamic formulas and laws, so that the model we established is of great validity. + +\item Our model is fairly robust due to our careful corrections in consideration of real-life situations and detailed sensitivity analysis. + +\item Via Fluent software, we simulate the time field of different areas throughout the bathtub. The outcome is vivid for us to understand the changing process. + +\item We come up with various criteria to compare different situations, like water consumption and the time of adding hot water. Hence an overall comparison can be made according to these criteria. + +\item Besides common factors, we still consider other factors, such as evaporation and radiation heat transfer. The evaporation turns out to be the main reason of heat loss, which corresponds with other scientist’s experimental outcome. +\end{itemize} + +\subsection{Weakness} + +\begin{itemize} +\item Having knowing the range of some parameters from others’ essays, we choose a value from them to apply in our model. Those values may not be reasonable in reality. + +\item Although we investigate a lot in the influence of personal motions, they are so complicated that need to be studied further. + +\item Limited to time, we do not conduct sensitivity analysis for the influence of personal surface area. +\end{itemize} + +\section{Further Discussion} + +In this part, we will focus on different distribution of inflow faucets. Then we discuss about the real-life application of our model. + +\begin{itemize} +\item Different Distribution of Inflow Faucets + +In our before discussion, we assume there being just one entrance of inflow. + +From the simulating outcome, we find the temperature of bath water is hardly even. So we come up with the idea of adding more entrances. + +The simulation turns out to be as follows + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig24.jpg} +\caption{The simulation results of different ways of arranging entrances} \label{fig24} +\end{figure} + +From the above figure, the more the entrances are, the evener the temperature will be. Recalling on the before simulation outcome, when there is only one entrance for inflow, the temperature of corners is quietly lower than the middle area. + +In conclusion, if we design more entrances, it will be easier to realize the goal to keep temperature even throughout the bathtub. + +\item Model Application + +Our before discussion is based on ideal assumptions. In reality, we have to make some corrections and improvement. + +\begin{itemize} +\item[1)] Adding hot water continually with the mass flow of 0.16 kg/s. This way can ensure even mean temperature throughout the bathtub and waste less water. + +\item[2)] The manufacturers can design an intelligent control system to monitor the temperature so that users can get more enjoyable bath experience. + +\item[3)] We recommend users to add bubble additives to slow down the water being cooler and help cleanse. The additives with lower thermal conductivity are optimal. + +\item[4)] The study method of our establishing model can be applied in other area relative to convection heat transfer, such as air conditioners. +\end{itemize} +\end{itemize} + +\begin{thebibliography}{99} +\addcontentsline{toc}{section}{Reference} + +\bibitem{1} Gi-Beum Kim. Change of the Warm Water Temperature for the Development of Smart Healthecare Bathing System. Hwahak konghak. 2006, 44(3): 270-276. +\bibitem{2} \url{https://en.wikipedia.org/wiki/Convective_heat_transfer#Newton.27s_law_of_cooling} +\bibitem{3} \url{https://en.wikipedia.org/wiki/Navier\%E2\%80\%93Stokes_equations} +\bibitem{4} \url{https://en.wikipedia.org/wiki/Computational_fluid_dynamics} +\bibitem{5} Holman J P. Heat Transfer (9th ed.), New York: McGraw-Hill, 2002. +\bibitem{6} Liu Weiguo, Chen Zhaoping, ZhangYin. Matlab program design and application. Beijing: Higher education press, 2002. (In Chinese) + +\end{thebibliography} + +\newpage + +\begin{letter}{Enjoy Your Bath Time!} + +From simulation results of real-life situations, we find it takes a period of time for the inflow hot water to spread throughout the bathtub. During this process, the bath water continues transferring heat into air, bathtub and the person in bathtub. The difference between heat transfer capacity makes the temperature of various areas to be different. So that it is difficult to get an evenly maintained temperature throughout the bath water. + +In order to enjoy a comfortable bath with even temperature of bath water and without wasting too much water, we propose the following suggestions. + +\begin{itemize} +\item Adding hot water consistently +\item Using smaller bathtub if possible +\item Decreasing motions during bath +\item Using bubble bath additives +\item Arranging more faucets of inflow +\end{itemize} + +\vspace{\parskip} + +Sincerely yours, + +Your friends + +\end{letter} + +\newpage + +\begin{appendices} + +\section{First appendix} + +In addition, your report must include a letter to the Chief Financial Officer (CFO) of the Goodgrant Foundation, Mr. Alpha Chiang, that describes the optimal investment strategy, your modeling approach and major results, and a brief discussion of your proposed concept of a return-on-investment (ROI). This letter should be no more than two pages in length. + +Here are simulation programmes we used in our model as follow.\\ + +\textbf{\textcolor[rgb]{0.98,0.00,0.00}{Input matlab source:}} +\lstinputlisting[language=Matlab]{./code/mcmthesis-matlab1.m} + +\section{Second appendix} + +some more text \textcolor[rgb]{0.98,0.00,0.00}{\textbf{Input C++ source:}} +\lstinputlisting[language=C++]{./code/mcmthesis-sudoku.cpp} + +\end{appendices} +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/latex/.Rproj.user/F69231EA/sources/per/t/F9FF5BF1 b/latex/.Rproj.user/F69231EA/sources/per/t/F9FF5BF1 new file mode 100644 index 0000000..23e8e26 --- /dev/null +++ b/latex/.Rproj.user/F69231EA/sources/per/t/F9FF5BF1 @@ -0,0 +1,26 @@ +{ + "id": "F9FF5BF1", + "path": 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\ifthenelse{\equal{\MCM@memosubject}{\relax}}{}{\item [{Subject:}] \MCM@memosubject} + \ifthenelse{\equal{\MCM@memodate}{\relax}}{}{\item [{Date:}] \MCM@memodate} + \end{description} + \par\noindent + \rule[0.5ex]{\linewidth}{0.1pt}\par + \bigskip{} +}{% + \clearpage + \pagestyle{fancy}% +} +\newtheorem{Theorem}{Theorem}[section] +\newtheorem{Lemma}[Theorem]{Lemma} +\newtheorem{Corollary}[Theorem]{Corollary} +\newtheorem{Proposition}[Theorem]{Proposition} +\newtheorem{Definition}[Theorem]{Definition} +\newtheorem{Example}[Theorem]{Example} +\renewcommand\section{\@startsection{section}{1}{\z@}% + {-0pt\@plus -.2ex \@minus -.2ex}% + {1pt \@plus .2ex}% + {\rmfamily\Large\bfseries}} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-0pt\@plus -.2ex \@minus -.2ex}% + {1pt \@plus .2ex}% + {\rmfamily\large\bfseries}} +\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% + {-.5ex\@plus -1ex \@minus -.2ex}% + {.25ex \@plus .2ex}% + {\rmfamily\normalsize\bfseries}} 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+c:/Users/zhang/AppData/Roaming/TinyTeX/texmf-dist/tex/latex/xkeyval/xkeyval.sty="E9F50CF8" diff --git a/latex/code/mcmthesis-matlab1.m b/latex/code/mcmthesis-matlab1.m new file mode 100644 index 0000000..829e336 --- /dev/null +++ b/latex/code/mcmthesis-matlab1.m @@ -0,0 +1,15 @@ +function [t,seat,aisle]=OI6Sim(n,target,seated) +pab=rand(1,n); +for i=1:n + if pab(i)<0.4 + aisleTime(i)=0; + else + aisleTime(i)=trirnd(3.2,7.1,38.7); + end +end + + + + + + \ No newline at end of file diff --git a/latex/code/mcmthesis-sudoku.cpp b/latex/code/mcmthesis-sudoku.cpp new file mode 100644 index 0000000..24909a0 --- /dev/null +++ b/latex/code/mcmthesis-sudoku.cpp @@ -0,0 +1,41 @@ +//============================================================================ +// Name : Sudoku.cpp +// Author : wzlf11 +// Version : a.0 +// Copyright : Your copyright notice +// Description : Sudoku in C++. +//============================================================================ + +#include +#include +#include + +using namespace std; + +int table[9][9]; + +int main() { + + for(int i = 0; i < 9; i++){ + table[0][i] = i + 1; + } + + srand((unsigned int)time(NULL)); + + shuffle((int *)&table[0], 9); + + while(!put_line(1)) + { + shuffle((int *)&table[0], 9); + } + + for(int x = 0; x < 9; x++){ + for(int y = 0; y < 9; y++){ + cout << table[x][y] << " "; + } + + cout << endl; + } + + return 0; +} diff --git a/latex/example.tex b/latex/example.tex new file mode 100644 index 0000000..50ceb1e --- /dev/null +++ b/latex/example.tex @@ -0,0 +1,779 @@ +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% + + +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {0000000}, problem = \textcolor{red}{A}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} +\usepackage[style=apa,backend=biber]{biblatex} +\addbibresource{reference.bib} + +\usepackage{biblatex} +\addbibresource{reference.bib} % <--- 添加这一行，注意这里要带 .bib 后缀 + +\usepackage{tocloft} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Enjoy a Cozy and Green Bath} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} + +\begin{abstract} + +A traditional bathtub cannot be reheated by itself, so users have to add hot +water from time to time. Our goal is to establish a model of the temperature +of bath water in space and time. Then we are expected to propose an optimal +strategy for users to keep the temperature even and close to initial temperature +and decrease water consumption. + +To simplify modeling process, we firstly assume there is no person in the bathtub. +We regard the whole bathtub as a thermodynamic system and introduce heat transfer +formulas. + +We establish two sub-models: adding water constantly and discontinuously. As for +the former sub-model, we define the mean temperature of bath water. Introducing +Newton cooling formula, we determine the heat transfer capacity. After deriving +the value of parameters, we deduce formulas to derive results and simulate the +change of temperature field via CFD. As for the second sub-model, we define an +iteration consisting of two process: heating and standby. According to energy +conservation law, we obtain the relationship of time and total heat dissipating +capacity. Then we determine the mass flow and the time of adding hot water. We +also use CFD to simulate the temperature field in second sub-model. + +In consideration of evaporation, we correct the results of sub-models referring +to some scientists' studies. We define two evaluation criteria and compare the +two sub-models. Adding water constantly is found to keep the temperature of bath +water even and avoid wasting too much water, so it is recommended by us. + +Then we determine the influence of some factors: radiation heat transfer, the +shape and volume of the tub, the shape/volume/temperature/motions of the person, +the bubbles made from bubble bath additives. We focus on the influence of those +factors to heat transfer and then conduct sensitivity analysis. The results +indicate smaller bathtub with less surface area, lighter personal mass, less +motions and more bubbles will decrease heat transfer and save water. + +Based on our model analysis and conclusions, we propose the optimal strategy for +the user in a bathtub and explain the reason of uneven temperature throughout +the bathtub. In addition, we make improvement for applying our model in real life. + +\begin{keywords} +Heat transfer, Thermodynamic system, CFD, Energy conservation +\end{keywords} + +\end{abstract} + +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 +\thispagestyle{empty} + +\newpage + +\section{Introduction} + +\subsection{Background} + +Bathing in a tub is a perfect choice for those who have been worn out after a +long day's working. A traditional bathtub is a simply water containment vessel +without a secondary heating system or circulating jets. Thus the temperature of +water in bathtub declines noticeably as time goes by, which will influent the +experience of bathing. As a result, the bathing person needs to add a constant +trickle of hot water from a faucet to reheat the bathing water. This way of +bathing will result in waste of water because when the capacity of the bathtub +is reached, excess water overflows the tub. + +An optimal bathing strategy is required for the person in a bathtub to get +comfortable bathing experience while reducing the waste of water. + +\subsection{Literature Review} + +A traditional bathtub cannot be reheated by itself, so users have to add hot +water from time to time. Our goal is to establish a model of the temperature +of bath water in space and time. Then we are expected to propose an optimal +strategy for users to keep the temperature even and close to the initial +temperature and decrease water consumption. According to \textcite{kim2006}, +He derived a relational equation based on the basic theory of heat transfer +to evaluate the performance of bath tubes. The major heat loss was found to be +due to evaporation. Moreover, he found out that the speed of heat loss depends +more on the humidity of the bathroom than the temperature of water contained +in the bathtub. So, it is best to maintain the temperature of bathtub water to +be between 41 to 45$^{\circ}$C and the humidity of bathroom to be 95\%. +Traditional bath systems have significant limitations in temperature control. +To address this, we introduce heat transfer formulas as +discussed (\cite[123]{holman2002}). + +\subsection{Restatement of the Problem} + +We are required to establish a model to determine the change of water temperature +in space and time. Then we are expected to propose the best strategy for the +person in the bathtub to keep the water temperature close to initial temperature +and even throughout the tub. Reduction of waste of water is also needed. In +addition, we have to consider the impact of different conditions on our model, +such as different shapes and volumes of the bathtub, etc. + +In order to solve those problems, we will proceed as follows: + +\begin{itemize} +\item {\bf Stating assumptions}. By stating our assumptions, we will narrow the +focus of our approach towards the problems and provide some insight into bathtub +water temperature issues. + +\item {\bf Making notations}. We will give some notations which are important for +us to clarify our models. + +\item {\bf Presenting our model}. In order to investigate the problem deeper, we +divide our model into two sub-models. One is a steady convection heat transfer +sub-model in which hot water is added constantly. The other one is an unsteady +convection heat transfer sub-model where hot water is added discontinuously. + +\item {Defining evaluation criteria and comparing sub-models}. We define two main +criteria to evaluate our model: the mean temperature of bath water and the amount +of inflow water. + +\item {\bf Analysis of influencing factors}. In term of the impact of different +factors on our model, we take those into consideration: the shape and volume of +the tub, the shape/volume/temperature of the person in the bathtub, the motions +made by the person in the bathtub and adding a bubble bath additive initially. + +\item {\bf Model testing and sensitivity analysis}. With the criteria defined +before, we evaluate the reliability of our model and do the sensitivity analysis. + +\item {\bf Further discussion}. We discuss about different ways to arrange inflow +faucets. Then we improve our model to apply them in reality. + +\item {\bf Evaluating the model}. We discuss about the strengths and weaknesses +of our model: + +\begin{itemize} +\item[1)] ... +\item[2)] ... +\item[3)] ... +\item[4)] ... +\end{itemize} + +\end{itemize} + +\section{Assumptions and Justification} + +To simplify the problem and make it convenient for us to simulate real-life +conditions, we make the following basic assumptions, each of which is properly +justified. + +\begin{itemize} +\item {\bf The bath water is incompressible Non-Newtonian fluid}. The +incompressible Non-Newtonian fluid is the basis of Navier–Stokes equations +which are introduced to simulate the flow of bath water. + +\item {\bf All the physical properties of bath water, bathtub and air are +assumed to be stable}. The change of those properties like specific heat, +thermal conductivity and density is rather small according to some +studies. It is complicated and unnecessary to consider these little +change so we ignore them. + +\item {\bf There is no internal heat source in the system consisting of bathtub, +hot water and air}. Before the person lies in the bathtub, no internal heat source +exist except the system components. The circumstance where the person is in the +bathtub will be investigated in our later discussion. + +\item {\bf We ignore radiative thermal exchange}. According to Stefan-Boltzmann’s +law, the radiative thermal exchange can be ignored when the temperature is low. +Refer to industrial standard, the temperature in bathroom is lower than +100 $^{\circ}$C, so it is reasonable for us to make this assumption. + +\item {\bf The temperature of the adding hot water from the faucet is stable}. +This hypothesis can be easily achieved in reality and will simplify our process +of solving the problem. +\end{itemize} + +\section{Notations} + +\begin{center} +\begin{tabular}{clc} +{\bf Symbols} & {\bf Description} & \quad {\bf Unit} \\[0.25cm] +$h$ & Convection heat transfer coefficient & \quad W/(m$^2 \cdot$ K) +\\[0.2cm] +$k$ & Thermal conductivity & \quad W/(m $\cdot$ K) \\[0.2cm] +$c_p$ & Specific heat & \quad J/(kg $\cdot$ K) \\[0.2cm] +$\rho$ & Density & \quad kg/m$^2$ \\[0.2cm] +$\delta$ & Thickness & \quad m \\[0.2cm] +$t$ & Temperature & \quad $^\circ$C, K \\[0.2cm] +$\tau$ & Time & \quad s, min, h \\[0.2cm] +$q_m$ & Mass flow & \quad kg/s \\[0.2cm] +$\Phi$ & Heat transfer power & \quad W \\[0.2cm] +$T$ & A period of time & \quad s, min, h \\[0.2cm] +$V$ & Volume & \quad m$^3$, L \\[0.2cm] +$M,\,m$ & Mass & \quad kg \\[0.2cm] +$A$ & Aera & \quad m$^2$ \\[0.2cm] +$a,\,b,\,c$ & The size of a bathtub & \quad m$^3$ +\end{tabular} +\end{center} + +\noindent where we define the main parameters while specific value of those +parameters will be given later. + +\section{Model Overview} + +To simplify the modeling process, we firstly assume there is no person in the +bathtub. We regard the whole bathtub as a thermodynamic system and introduce +heat transfer formulas. We establish two sub-models: adding water constantly +and discontinuously. For the former sub-model, we define the mean temperature +of bath water and introduce Newton's cooling formula to determine the heat +transfer capacity. After deriving the value of parameters, we deduce formulas +to derive results and simulate the change of temperature field via CFD, as +described by \textcite{anderson2006}. + +In our basic model, we aim at three goals: keeping the temperature as even as +possible, making it close to the initial temperature and decreasing the water +consumption. + +We start with the simple sub-model where hot water is added constantly. +At first we introduce convection heat transfer control equations in rectangular +coordinate system. Then we define the mean temperature of bath water. + +Afterwards, we introduce Newton cooling formula to determine heat transfer +capacity. After deriving the value of parameters, we get calculating results +via formula deduction and simulating results via CFD. + +Secondly, we present the complicated sub-model in which hot water is +added discontinuously. We define an iteration consisting of two process: +heating and standby. As for heating process, we derive control equations and +boundary conditions. As for standby process, considering energy conservation law, +we deduce the relationship of total heat dissipating capacity and time. + +Then we determine the time and amount of added hot water. After deriving the +value of parameters, we get calculating results via formula deduction and +simulating results via CFD. + +At last, we define two criteria to evaluate those two ways of adding hot water. +Then we propose optimal strategy for the user in a bathtub. +The whole modeling process can be shown as follows. + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig1.jpg} +\caption{Modeling process} \label{fig1} +\end{figure} + +\section{Sub-model I : Adding Water Continuously} + +As for the second sub-model, we define an iteration consisting of two processes: +heating and standby. According to the energy conservation law, we obtain the +relationship of time and total heat dissipating capacity. Then we determine +the mass flow and the time of adding hot water. We also use CFD to simulate +the temperature field in the second sub-model, following the techniques +outlined by \textcite{website2024}. + +We first establish the sub-model based on the condition that a person add water +continuously to reheat the bathing water. Then we use Computational Fluid +Dynamics (CFD) to simulate the change of water temperature in the bathtub. At +last, we evaluate the model with the criteria which have been defined before. + +\subsection{Model Establishment} + +Since we try to keep the temperature of the hot water in bathtub to be even, +we have to derive the amount of inflow water and the energy dissipated by the +hot water into the air. + +We derive the basic convection heat transfer control equations based on the +former scientists’ achievement. Then, we define the mean temperature of bath +water. Afterwards, we determine two types of heat transfer: the boundary heat +transfer and the evaporation heat transfer. Combining thermodynamic formulas, +we derive calculating results. Via Fluent software, we get simulation results. + +\subsubsection{Control Equations and Boundary Conditions} + +According to thermodynamics knowledge, we recall on basic convection +heat transfer control equations in rectangular coordinate system. Those +equations show the relationship of the temperature of the bathtub water in space. + +We assume the hot water in the bathtub as a cube. Then we put it into a +rectangular coordinate system. The length, width, and height of it is $a,\, b$ +and $c$. + +\begin{figure}[h] +\centering +\includegraphics[width=8cm]{fig2.jpg} +\caption{Modeling process} \label{fig2} +\end{figure} + +In the basis of this, we introduce the following equations: + +\begin{itemize} +\item {\bf Continuity equation:} +\end{itemize} + +\begin{equation} \label{eq1} +\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + +\frac{\partial w}{\partial z} = 0 +\end{equation} + +\noindent where the first component is the change of fluid mass along the $X$-ray. +The second component is the change of fluid mass along the $Y$-ray. And the third +component is the change of fluid mass along the $Z$-ray. The sum of the change in +mass along those three directions is zero. + +\begin{itemize} +\item {\bf Moment differential equation (N-S equations):} +\end{itemize} + +\begin{equation} \label{eq2} +\left\{ +\begin{array}{l} \!\! +\rho \Big(u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + +w\dfrac{\partial u}{\partial z} \Big) = -\dfrac{\partial p}{\partial x} + +\eta \Big(\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} + +\dfrac{\partial^2 u}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} + +w\dfrac{\partial v}{\partial z} \Big) = -\dfrac{\partial p}{\partial y} + +\eta \Big(\dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} + +\dfrac{\partial^2 v}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial w}{\partial x} + v \dfrac{\partial w}{\partial y} + +w\dfrac{\partial w}{\partial z} \Big) = -g-\dfrac{\partial p}{\partial z} + +\eta \Big(\dfrac{\partial^2 w}{\partial x^2} + \dfrac{\partial^2 w}{\partial y^2} + +\dfrac{\partial^2 w}{\partial z^2} \Big) +\end{array} +\right. +\end{equation} + +\begin{itemize} +\item {\bf Energy differential equation:} +\end{itemize} + +\begin{equation} \label{eq3} +\rho c_p \Big( u\frac{\partial t}{\partial x} + v\frac{\partial t}{\partial y} + +w\frac{\partial t}{\partial z} \Big) = \lambda \Big(\frac{\partial^2 t}{\partial x^2} + +\frac{\partial^2 t}{\partial y^2} + \frac{\partial^2 t}{\partial z^2} \Big) +\end{equation} + +\noindent where the left three components are convection terms while the right +three components are conduction terms. + +By Equation \eqref{eq3}, we have ...... + +...... + +On the right surface in Fig. \ref{fig2}, the water also transfers heat firstly +with bathtub inner surfaces and then the heat comes into air. The boundary +condition here is ...... + +\subsubsection{Definition of the Mean Temperature} + +...... + +\subsubsection{Determination of Heat Transfer Capacity} + +...... + +\section{Sub-model II: Adding Water Discontinuously} + +In order to establish the unsteady sub-model, we recall on the working principle +of air conditioners. The heating performance of air conditions consist of two +processes: heating and standby. After the user set a temperature, the air +conditioner will begin to heat until the expected temperature is reached. Then +it will go standby. When the temperature get below the expected temperature, +the air conditioner begin to work again. As it works in this circle, the +temperature remains the expected one. + +Inspired by this, we divide the bathtub working into two processes: adding +hot water until the expected temperature is reached, then keeping this +condition for a while unless the temperature is lower than a specific value. +Iterating this circle ceaselessly will ensure the temperature kept relatively +stable. + +\subsection{Heating Model} + +\subsubsection{Control Equations and Boundary Conditions} + +\subsubsection{Determination of Inflow Time and Amount} + +\subsection{Standby Model} + +\subsection{Results} + +\quad~ We first give the value of parameters based on others’ studies. Then we +get the calculation results and simulating results via those data. + +\subsubsection{Determination of Parameters} + +After establishing the model, we have to determine the value of some +important parameters. + +As scholar Beum Kim points out, the optimal temperature for bath is +between 41 and 45$^\circ$C. Meanwhile, according to Shimodozono's study, +41$^\circ$C warm water bath is the perfect choice for individual health. +So it is reasonable for us to focus on $41^\circ$C $\sim 45^\circ$C. Because +adding hot water continuously is a steady process, so the mean temperature +of bath water is supposed to be constant. We value the temperature of inflow +and outflow water with the maximum and minimum temperature respectively. + +The values of all parameters needed are shown as follows: + +..... + +\subsubsection{Calculating Results} + +Putting the above value of parameters into the equations we derived before, we +can get the some data as follows: + +%%普通表格 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{The calculating results} %表标题 +\vspace{0.15cm} +\label{tab2} %设置表的引用标签 +\begin{tabular}{|c|c|c|} %3个c表示3列, |可选, 表示绘制各列间的竖线 +\hline %画横线 +Variables & Values & Unit \\ \hline %各列间用&隔开 +$A_1$ & 1.05 & $m^2$ \\ \hline +$A_2$ & 2.24 & $m^2$ \\ \hline +$\Phi_1$ & 189.00 & $W$ \\ \hline +$\Phi_2$ & 43.47 & $W$ \\ \hline +$\Phi$ & 232.47 & $W$ \\ \hline +$q_m$ & 0.014 & $g/s$ \\ \hline +\end{tabular} +\end{table} + +From Table \ref{tab2}, ...... + +...... + +\section{Correction and Contrast of Sub-Models} + +After establishing two basic sub-models, we have to correct them in consideration +of evaporation heat transfer. Then we define two evaluation criteria to compare +the two sub-models in order to determine the optimal bath strategy. + +\subsection{Correction with Evaporation Heat Transfer} + +Someone may confuse about the above results: why the mass flow in the first +sub-model is so small? Why the standby time is so long? Actually, the above two +sub-models are based on ideal conditions without consideration of the change of +boundary conditions, the motions made by the person in bathtub and the +evaporation of bath water, etc. The influence of personal motions will be +discussed later. Here we introducing the evaporation of bath water to correct +sub-models. + +\subsection{Contrast of Two Sub-Models} + +Firstly we define two evaluation criteria. Then we contrast the two submodels +via these two criteria. Thus we can derive the best strategy for the person in +the bathtub to adopt. + +\section{Model Analysis and Sensitivity Analysis} + +In consideration of evaporation, we correct the results of sub-models referring +to studies. We define two evaluation criteria +and compare the two sub-models. Adding water constantly is found to keep the +temperature of bath water even and avoid wasting too much water, so it is +recommended by us. We also conduct sensitivity analysis to determine the +influence of factors such as radiation heat transfer, the shape and volume of +the tub, the shape/volume/temperature/motions of the person, and the bubbles +made from bubble bath additives, as discussed +in (\cite{evaporation2018}; \cite{thesis2015}). + +\subsection{The Influence of Different Bathtubs} + +Definitely, the difference in shape and volume of the tub affects the +convection heat transfer. Examining the relationship between them can help +people choose optimal bathtubs. + +\subsubsection{Different Volumes of Bathtubs} + +In reality, a cup of water will be cooled down rapidly. However, it takes quite +long time for a bucket of water to become cool. That is because their volume is +different and the specific heat of water is very large. So that the decrease of +temperature is not obvious if the volume of water is huge. That also explains +why it takes 45 min for 320 L water to be cooled by 1$^\circ$C. + +In order to examine the influence of volume, we analyze our sub-models +by conducting sensitivity Analysis to them. + +We assume the initial volume to be 280 L and change it by $\pm 5$\%, $\pm 8$\%, +$\pm 12$\% and $\pm 15$\%. With the aid of sub-models we established before, the +variation of some parameters turns out to be as follows + +%%三线表 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{Variation of some parameters} %表标题 +\label{tab7} %设置表的引用标签 +\begin{tabular}{ccccccc} %7个c表示7列, c表示每列居中对齐, 还有l和r可选 +\toprule %画顶端横线 +$V$ & $A_1$ & $A_2$ & $T_2$ & $q_{m1}$ & $q_{m2}$ & $\Phi_q$ \\ +\midrule %画中间横线 +-15.00\% & -5.06\% & -9.31\% & -12.67\% & -2.67\% & -14.14\% & -5.80\% \\ +-12.00\% & -4.04\% & -7.43\% & -10.09\% & -2.13\% & -11.31\% & -4.63\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +\bottomrule %画底部横线 +\end{tabular} +\end{table} + +\section{Strength and Weakness} + +\subsection{Strength} + +\begin{itemize} +\item We analyze the problem based on thermodynamic formulas and laws, so that +the model we established is of great validity. + +\item Our model is fairly robust due to our careful corrections in consideration +of real-life situations and detailed sensitivity analysis. + +\item Via Fluent software, we simulate the time field of different areas +throughout the bathtub. The outcome is vivid for us to understand the changing +process. + +\item We come up with various criteria to compare different situations, like +water consumption and the time of adding hot water. Hence an overall comparison +can be made according to these criteria. + +\item Besides common factors, we still consider other factors, such as evaporation +and radiation heat transfer. The evaporation turns out to be the main reason of +heat loss, which corresponds with other scientist’s experimental outcome. +\end{itemize} + +\subsection{Weakness} + +\begin{itemize} +\item Having knowing the range of some parameters from others’ essays, we choose +a value from them to apply in our model. Those values may not be reasonable in +reality. + +\item Although we investigate a lot in the influence of personal motions, they +are so complicated that need to be studied further. + +\item Limited to time, we do not conduct sensitivity analysis for the influence +of personal surface area. +\end{itemize} + +\section{Further Discussion} + +Based on our model analysis and conclusions, we propose the optimal strategy +for the user in a bathtub and explain the reason for the uneven temperature +throughout the bathtub. In addition, we make improvements for applying our +model in real life, as suggested by the patent \textcite{patent2023}. + +In this part, we will focus on different distribution of inflow faucets. Then we +discuss about the real-life application of our model. + +\begin{itemize} +\item Different Distribution of Inflow Faucets + +In our before discussion, we assume there being just one entrance of inflow. + +From the simulating outcome, we find the temperature of bath water is hardly even. +So we come up with the idea of adding more entrances. + +The simulation turns out to be as follows + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig24.jpg} +\caption{The simulation results of different ways of arranging entrances} \label{fig24} +\end{figure} + +From the above figure, the more the entrances are, the evener the temperature +will be. Recalling on the before simulation outcome, when there is only one +entrance for inflow, the temperature of corners is quietly lower than the middle +area. + +In conclusion, if we design more entrances, it will be easier to realize the goal +to keep temperature even throughout the bathtub. + +\item Model Application + +Our before discussion is based on ideal assumptions. In reality, we have to make +some corrections and improvement. + +\begin{itemize} +\item[1)] Adding hot water continually with the mass flow of 0.16 kg/s. This way +can ensure even mean temperature throughout the bathtub and waste less water. + +\item[2)] The manufacturers can design an intelligent control system to monitor +the temperature so that users can get more enjoyable bath experience. + +\item[3)] We recommend users to add bubble additives to slow down the water being +cooler and help cleanse. The additives with lower thermal conductivity are optimal. + +\item[4)] The study method of our establishing model can be applied in other area +relative to convection heat transfer, such as air conditioners. +\end{itemize} +\end{itemize} + +\printbibliography + +\newpage + +\begin{letter}{Enjoy Your Bath Time!} + +From simulation results of real-life situations, we find it takes a period of +time for the inflow hot water to spread throughout the bathtub. During this +process, the bath water continues transferring heat into air, bathtub and the +person in bathtub. The difference between heat transfer capacity makes the +temperature of various areas to be different. So that it is difficult to get +an evenly maintained temperature throughout the bath water. + +In order to enjoy a comfortable bath with even temperature of bath water and +without wasting too much water, we propose the following suggestions. + +\begin{itemize} +\item Adding hot water consistently +\item Using smaller bathtub if possible +\item Decreasing motions during bath +\item Using bubble bath additives +\item Arranging more faucets of inflow +\end{itemize} + +\vspace{\parskip} + +Sincerely yours, + +Your friends + +\end{letter} + +\newpage + +\begin{appendices} + +\section{First appendix} + +In addition, your report must include a letter to the Chief Financial Officer +(CFO) of the Goodgrant Foundation, Mr. Alpha Chiang, that describes the optimal +investment strategy, your modeling approach and major results, and a brief +discussion of your proposed concept of a return-on-investment (ROI). This letter +should be no more than two pages in length. + +Here are simulation programmes we used in our model as follow (\cite{Liu02}).\\ + +\textbf{\textcolor[rgb]{0.98,0.00,0.00}{Input matlab source:}} +\lstinputlisting[language=Matlab]{./code/mcmthesis-matlab1.m} + +\section{Second appendix} + +some more text \textcolor[rgb]{0.98,0.00,0.00}{\textbf{Input C++ source:}} +\lstinputlisting[language=C++]{./code/mcmthesis-sudoku.cpp} + +\end{appendices} + +\newpage +\newcounter{lastpage} +\setcounter{lastpage}{\value{page}} +\thispagestyle{empty} + +\section*{Report on Use of AI} + +\begin{enumerate} +\item OpenAI ChatGPT (Nov 5, 2023 version, ChatGPT-4,) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\item OpenAI Ernie (Nov 5, 2023 version, Ernie 4.0) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\item Github CoPilot (Feb 3, 2024 version) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\item Google Bard (Feb 2, 2024 version) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\end{enumerate} + +% 重置页码 +\clearpage +\setcounter{page}{\value{lastpage}} + +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/latex/figures/Flow Chart of Model I.png b/latex/figures/Flow Chart of Model I.png new file mode 100644 index 0000000..dbe6c51 Binary files /dev/null and b/latex/figures/Flow Chart of Model I.png differ diff --git a/latex/figures/Flow Chart of Model II.png b/latex/figures/Flow Chart of Model II.png new file mode 100644 index 0000000..1da8ab2 Binary files /dev/null and b/latex/figures/Flow Chart of Model II.png differ diff --git a/latex/figures/Knee Point Detection.png b/latex/figures/Knee Point Detection.png new file mode 100644 index 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thesis + + author + title + type + institution + + + + unpublished + + author + title + + + + + isbn + + + issn + + + ismn + + + gender + + + + + + + reference.bib + reference.bib + + diff --git a/latex/mcmthesis-demo.blg b/latex/mcmthesis-demo.blg new file mode 100644 index 0000000..0d93993 --- /dev/null +++ b/latex/mcmthesis-demo.blg @@ -0,0 +1,15 @@ +[0] Config.pm:308> INFO - This is Biber 2.20 +[0] Config.pm:311> INFO - Logfile is 'mcmthesis-demo.blg' +[93] biber-MSWIN64:340> INFO - === +[130] Biber.pm:420> INFO - Reading 'mcmthesis-demo.bcf' +[225] Biber.pm:994> INFO - Found 8 citekeys in bib section 0 +[235] Biber.pm:4463> INFO - Processing section 0 +[245] Biber.pm:4654> INFO - Looking for bibtex file 'reference.bib' for section 0 +[296] bibtex.pm:1713> INFO - LaTeX decoding ... +[305] bibtex.pm:1519> INFO - Found BibTeX data source 'reference.bib' +[365] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized' +[365] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable' +[365] Biber.pm:4283> INFO - Sorting list 'nty/global//global/global/global' of type 'entry' with template 'nty' and locale 'en-US' +[365] Biber.pm:4289> INFO - No sort tailoring available for locale 'en-US' +[377] bbl.pm:676> INFO - Writing 'mcmthesis-demo.bbl' with encoding 'UTF-8' +[381] bbl.pm:779> INFO - Output to mcmthesis-demo.bbl diff --git a/latex/mcmthesis-demo.fdb_latexmk b/latex/mcmthesis-demo.fdb_latexmk new file mode 100644 index 0000000..79785d2 --- /dev/null +++ b/latex/mcmthesis-demo.fdb_latexmk @@ -0,0 +1,348 @@ +# Fdb version 4 +["biber mcmthesis-demo"] 1770072992.52744 "mcmthesis-demo.bcf" "mcmthesis-demo.bbl" "mcmthesis-demo" 1770073778.84332 0 + "mcmthesis-demo.bcf" 1770073777 109227 7d0b1f5527dfba9a33b2094ce6109c0f "pdflatex" + "reference.bib" 1770072982 2546 3e9a4c8f13662a0d442dd4f7e671b47d "" + (generated) + "mcmthesis-demo.bbl" + "mcmthesis-demo.blg" 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mcmthesis-demo.bbl + + + mcmthesis-demo.bbl + + + mcmthesis-demo.bcf + + + reference.bib + + + diff --git a/latex/mcmthesis-demo.synctex.gz b/latex/mcmthesis-demo.synctex.gz new file mode 100644 index 0000000..88a79e1 Binary files /dev/null and b/latex/mcmthesis-demo.synctex.gz differ diff --git a/latex/mcmthesis-demo.tex b/latex/mcmthesis-demo.tex new file mode 100644 index 0000000..cf054cc --- /dev/null +++ b/latex/mcmthesis-demo.tex @@ -0,0 +1,990 @@ + +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {2618656}, problem = \textcolor{red}{B}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} + +\usepackage[style=numeric,backend=biber]{biblatex} +\addbibresource{reference.bib} +\bibliography{reference.bib} +\usepackage{tocloft} +\usepackage{float} +\usepackage{booktabs} % 用于绘制专业的三线表 +\usepackage{amsmath} % 用于数学符号和公式 +\usepackage[table]{xcolor} % 必须在导言区引入 +\usepackage{multirow} +\usepackage{tabularx} +\definecolor{tableHead}{RGB}{210, 222, 238} % 浅蓝灰色 (表头) +\definecolor{tableRow}{RGB}{245, 247, 250} % 极浅灰色 (隔行变色) +\usepackage{nicematrix, booktabs, tabularx} + +\usepackage{threeparttable} % 表格脚注支持 + +\usepackage{subcaption} % 支持子图环境 +\usepackage{graphicx} % 表格缩放适配宽度 +\usepackage{makecell} % 单元格换行（可选） +\usepackage{colortbl} % 单元格底色 +% 定义专业配色（浅蓝：推荐方案，浅灰：基线方案，避免刺眼） +\definecolor{recommendColor}{RGB}{217,234,249} +\definecolor{baselineColor}{RGB}{240,240,240} + \usepackage[most]{tcolorbox} +\usepackage{xcolor} +\usepackage{siunitx} % 数值对齐优化（可选） +\usepackage{tikz} +\usepackage{enumitem} + +% 定义备忘录专用的颜色 +\definecolor{fbstblue}{RGB}{0, 51, 102} +\definecolor{fbstgold}{RGB}{218, 165, 32} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Resilience, Efficiency, and Ecology: Optimizing Hybrid Interplanetary Transport for the 2050 Moon Colony} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} + + +\begin{abstract} + +To address the logistical requirements for establishing a 100,000-person Moon Colony by 2050, this study develops a comprehensive evaluation system spanning interplanetary transport decision-making, stochastic risk management, and multi-objective ecological balancing. + +We first construct the \textbf{Universal Energy-Equivalent and Temporal Coordination Model (UETCM)}. By establishing \textbf{Universal Energy-Equivalent} measured in Joules as the primary cost proxy, the model bypasses the volatility of monetary valuations and enables a standardized comparison between chemical rockets and space elevator technologies. Grounded in the Tsiolkovsky equation and gravitational mechanics, the model delineates a linear mapping between energy expenditure and payload mass. To resolve the energy-time trade-off, we formulate a system burden function based on the \textbf{time opportunity cost}, which is calibrated at 504 PJ per year through weight equilibrium and geometric centroid methods. Simulation results demonstrate that hybrid transport can complete the 100-million-metric-ton mission in 100.7 years, significantly outperforming standalone elevator and rocket-only scenarios. Under the calibrated cost parameters, a balanced 139-year strategy is identified as the global deterministic optimum. + +To manage real-world uncertainties, we extend the UETCM into a \textbf{Stochastic Risk and Robustness Framework} utilizing Conditional Value-at-Risk and 10,000 Monte Carlo iterations. This analysis identifies significant long-tail risks stemming from mechanical oscillations and system failures. To achieve a 95 percent completion confidence level, we recommend extending the operational timeline to a range of 155 to 160 years, adjusting the space elevator share to between 65 and 70 percent, and maintaining a 10 percent capacity reserve as a strategic buffer. + +For the subsequent operational phase, we establish the \textbf{Life-Support Logistics and Stochastic Water Balance Model (LSL-SWBM)}. This model integrates multi-tier domestic demand with stochastic medical emergency needs approximated via normal distributions. Our findings indicate that annual water demand at the Luxury Standard is more than 35 times that of the Survival Standard. For the Comfort Standard, we recommend a hybrid scheme utilizing both the elevator and low-latitude rocket sites, which completes the initial filling in 4.57 days with a balanced energy footprint. + +Furthermore, environmental footprints are integrated into a \textbf{tri-objective optimization framework}. While rocket-only delivery incurs massive carbon dioxide and stratospheric water vapor emissions, the space elevator remains nearly carbon-neutral. Consequently, the 186-year standalone elevator scenario is proposed as the ultimate sustainable strategy, emitting only 846 million metric tons of carbon dioxide, which represents less than 6.5 percent of the rocket baseline. + +Finally, we provide the Moon Colony Management Agency with actionable strategic recommendations. By proposing a phased hybrid transport plan with rebalancing triggers based on key performance indicators, this study offers a resilient pathway to lunar colonization that ensures logistical reliability while upholding ecological preservation. + + +\end{abstract} + +\begin{keywords} +Universal Energy-Equivalent; Stochastic Risk Analysis ; Multi-objective Optimization +\end{keywords} +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 +\thispagestyle{empty} + +\newpage + +\section{Introduction} + +\subsection{Background} +As human space exploration shifts from short-term landings to permanent settlement, Earth’s finite resources and fragile ecosystems motivate the development of scalable off-world habitats. Yet traditional chemical-propulsion rockets face persistent bottlenecks—limited payload ratios, high marginal launch costs, and non-negligible environmental impacts—making it difficult to sustain the mass logistics required for constructing and operating a 100,000-person Moon Colony. In this context, the Space Elevator System has been proposed as a transformative infrastructure \cite{penoyre2019spacelinepracticalspaceelevator} that can provide routine, energy-efficient access to space when paired with lunar transfer, but it also introduces new reliability and operational uncertainties. Therefore, the central challenge is to quantify and compare the energy (as a cost proxy), timeline, robustness under failures, and environmental footprint of rockets, elevators, and hybrid portfolios to support an actionable construction and sustainment strategy. + + +\begin{figure}[H] +\centering +\includegraphics[width=12cm]{background.png} +\caption{background} +\label{background} +\end{figure} + + + +\subsection{Restatement of the Problem} + +We develop a quantitative model to estimate the cost proxy (energy) and timeline of transporting materials to build a 100,000-person Moon Colony starting in 2050, and to support a defensible recommendation. Specifically, we: +\begin{itemize} + \item Compare three delivery scenarios: elevators-only, rockets-only (selected sites), and a hybrid portfolio; + \item Test robustness under non-ideal operations (e.g., tether dynamics, failures, downtime, and launch interruptions); + \item Quantify one-year water resupply needs after habitation and map them to the delivery model; + \item Evaluate and mitigate environmental impacts across scenarios; + \item Deliver a one-page recommendation letter to the MCM Agency. +\end{itemize} + +% \subsection{Literature Review} + + + + +\subsection{Our Work} + +\begin{figure}[H] + \centering + \includegraphics[width=12cm]{Our Work.png} + \caption{Our Work} + +\end{figure} + + +% \begin{figure}[H] +% \centering +% \includegraphics[width=12cm]{flow.png} +% \caption{the flow chart of our work} +% \label{Site Distribution Map} +% \end{figure} +\section{Assumptions and Justifications} + +To ensure model tractability while maintaining physical realism, we establish the following assumptions: + +\begin{itemize} + \item \textbf{Assumption 1: }Rocket payload capacity is assumed to be standardized at 100--150 metric tons per mission. \\ + \textbf{Justification: }Constant capacity avoids the volatility of payload metrics, facilitating a streamlined and efficient analysis of the total energy-cost required for construction. + + \item \textbf{Assumption 2: }Rocket propulsion systems are assumed to utilize liquid oxygen and methane (LOX/CH$_4$) as propellant. \\ + \textbf{Justification: }Current reusable heavy-lift standards, such as the SpaceX Starship, utilize LOX/CH$_4$ for its high energy density and minimal carbon buildup, representing future interplanetary transport norms. + + \item \textbf{Assumption 3: }Atmospheric drag and energy losses due to vehicle structural mass are neglected during rocket transit. \\ + \textbf{Justification: }These losses vary significantly per mission. Their exclusion reduces model complexity without compromising the macro-level accuracy of the energy expenditure framework. + + \item \textbf{Assumption 4: }Capital expenditures, including R\&D, manufacturing, and labor costs, are excluded from the assessment.\\ + \textbf{Justification: }This study focuses on the operational application phase of the logistics chain. Furthermore, such costs are often proprietary and difficult to estimate for future technologies. + + \item \textbf{Assumption 5: }Industrial water demand is assumed to be excluded from the water replenishment model.\\ + \textbf{Justification: }Lunar industrial scale data remains unavailable in current literature. Such demand is assumed to be integrated into the primary construction material transport modeled in Task 1. +\end{itemize} +\section{Notations} +\begin{center} +\begin{tabular}{clc} +\toprule[1pt] % 顶部粗线 +\textbf{Symbols} & \textbf{Description} & \textbf{Unit} \\ +\midrule % 中间线 +$m_{fuel}$ & Total mass of the chemical fuel required for a mission & metric tons \\ +$m_{p}$ & Mass of the effective payload being transported & metric tons \\ +$R$ & Mass ratio of initial mass to final mass & - \\ +$E$ & Total energy expenditure (Universal Energy-Equivalent) & Joules \\ +$h$ & Length of the space elevator tether & km \\ +$t$ & Total duration required for the construction timeline & years \\ +$p$ & Probability of rocket launch failure & - \\ +$W$ & Total water replenishment demand for the colony & metric tons \\ +\bottomrule[1pt]% 底部粗线 +\end{tabular} +\end{center} +\section{Model I: Universal Energy-Equivalent and Temporal Coordination Model} +\subsection{Model Overview} + + +To evaluate Moon Colony logistics, we develop the Universal Energy-Equivalent and Temporal Coordination Model (UETCM). The model adopts Universal Energy-Equivalent (UEE), measured in Joules, as the primary metric to bypass volatile monetary valuations, facilitating a standardized thermodynamic comparison between the momentum transfer of chemical rockets and the gravitational potential gains of the Space Elevator System. + +The modeling process begins with establishing a linear mapping between payload mass and energy consumption via the Tsiolkovsky equation and gravitational gradients, integrated with physio-geographical constraints to identify the feasibility gap within the delivery window. To integrate the competing dimensions of timeline and cost, we define a Time Opportunity Cost parameter $\lambda$—derived through a consensus of four independent analytical methodologies—to scalarize the initial multi-dimensional problem into a unified objective function. This allows for the identification of a globally optimal operating point that minimizes the total system burden ($J$). + +Finally, UETCM is extended into a stochastic model incorporating Risk-Adjusted Optimization and Conditional Value-at-Risk (CVaR). Through 10,000 Monte Carlo simulations, we quantify not only the performance impact of perturbations such as tether swaying and system failures but also derive modified strategies with robust safety margins, providing a comprehensive framework that spans from physical mechanisms to resilient strategic decision-making. + +\begin{figure}[H] +\centering +\includegraphics[width=11cm]{Flow Chart of Model I.png} +\caption{Flow Chart of Model I} +\end{figure} +\subsection{Energy-Equivalent as a Cost Proxy} + +Monetary costs are ill-conditioned for an Earth--Moon logistics program spanning many decades due to inflation, technology shocks, and geopolitical uncertainty \cite{chatgpt}. We therefore use \emph{energy consumption} as a stable, physics-grounded proxy for marginal cost, enabling a consistent comparison between rockets and the Space Elevator System. + +\subsubsection{Cost Convergence Analysis} +The operational cost per mission, $Cost(x)$, is modeled as the sum of hardware amortization, energy expenditure, and maintenance expenses: +\begin{equation} + Cost(x) = \frac{C_{hardware}}{x} + C_{energy} + C_{operations} +\end{equation} +where $x$ is the number of reuse cycles. As reuse increases in a mature post-2050 transport regime, $\frac{C_{hardware}}{x}$ diminishes and operations become more automated, leaving $C_{energy}$ as the dominant term. Thus, differences in energy efficiency largely determine the marginal cost of sustained high-throughput transport. + +\subsubsection{Empirical Industry Validation} + +Evidence from mature terrestrial transport supports this modeling choice: in commercial aviation, fuel remains a persistent and material share (15--25\%) of operating expenses, indicating that energy is a primary driver once hardware is amortized and operations are standardized. + +\begin{figure}[H] + \centering + \includegraphics[width=10cm]{fuel_share_trend.png} + \caption{Trend of fuel cost as a percentage of total operating expenses} + \label{fig:fuel_share_trend} +\end{figure} + +In summary, we adopt Universal Energy-Equivalent as the primary metric to avoid long-horizon monetary ambiguity and to compare transport architectures on a common thermodynamic baseline (i.e., overcoming gravitational wells and imparting the required kinetic energy). + +\subsection{Ideal Energy Cost Modeling} +\subsubsection{Rocket Momentum Dynamics} + +For traditional rocket transport, energy demand is dominated by propellant required to deliver a payload to the Earth--Moon transfer trajectory. Neglecting air resistance, the Tsiolkovsky rocket equation gives the mass ratio $R$: +\[ R = \frac{m_0}{m_f} = e^{\Delta V / v_e} \] +where $v_e$ is the effective exhaust velocity and $\Delta V$ (e.g., for Trans-Lunar Injection) is treated as a mission constant. Accounting for structural mass via a structural coefficient $\alpha$ (structure-to-fuel ratio), the fuel mass scales linearly with payload: +\[ m_{fuel} = \underbrace{\frac{R-1}{1-\alpha(R-1)}}_{k} \cdot m_p \] +which defines the proportional coefficient $k$. Using the idealized chemical energy release proxy, the rocket energy is: +\[ E_{rocket} = \frac{1}{2} m_{fuel} \cdot v_e^2 = \frac{1}{2} k \cdot v_e^2 \cdot m_p \] +Therefore, under fixed mission and vehicle parameters, rocket energy scales linearly with payload mass. For completeness, we also express the total energy requirement as the sum of gravitational and transition terms: +\[ \Delta E_{total} = m_p \cdot \left[ \left( \frac{G M_E}{R_E} - \frac{G M_E}{d_{EM}} \right) - \frac{G M_M}{2 r_M} \right] \] +which is likewise proportional to $m_p$. + +\subsubsection{Space Elevator Mechanics} + +Compared with direct rockets, the Space Elevator System adds an Earth-surface-to-apex segment along the tether before the lunar-transfer rocket leg (already covered above). We model only the elevator segment energy change. + + +\begin{figure}[H] +\centering +\includegraphics[width=10cm]{stage.png} +\caption{ Schematic of the Space Elevator Infrastructure and Earth-Moon Transport Phases} +\end{figure} + +From the Earth Port to the Apex Anchor, the energy change includes gravitational potential gain and rotation-related kinetic terms: +\[ E_{elevator} = m_p \left[ \left( \frac{GM_E}{R_E} - \frac{GM_E}{r_{0}} \right) + \frac{1}{2}\omega^2 r_{0}^2 \right] \] +where $r_0$ is the distance from the Apex Anchor to Earth's center and $\omega$ is Earth's angular velocity; thus $E_{elevator}$ also scales linearly with $m_p$. + + +\subsection{Ideal Timeline and Logistic Efficiency Modeling} + +After resolving the physical energy efficiency, the model determines the construction timeline. The complexity arises from the fact that 100 million metric tons of material cannot be delivered instantaneously; the progress is constrained by the geographical distribution of launch sites, frequency limits, and the inherent capacities of the space elevator and rocket systems. + +\subsubsection{Transport Progress Formulas} + +To quantify the construction timeline, the model defines $M_{total} = 10^8$ metric tons as the total demand. Based on the three scenarios provided, the completion time $t$ depends on the annual cumulative capacity $C$: + +\begin{itemize} + \item Space elevator only (Scenario a): $t_{a} = \frac{M_{total}}{C_{e}}$ + \item Rockets only (Scenario b): $t_{b} = \frac{M_{total}}{C_{r}}$ + \item Hybrid transport (Scenario c): $t_{c} = \frac{M_{total}}{\lambda C_{e} + \mu C_{r}}$ +\end{itemize} + +where $C_e$ is the annual throughput of the space elevator system and $C_r$ is the integrated annual capacity of the rocket system. Shortening the timeline requires increasing annual capacity. By adjusting the hybrid weights $\lambda$ and $\mu$, we seek a dynamic balance between construction speed and energy cost. + +\subsubsection{Spatial and Geographical Optimization} + +The primary obstacles to construction progress are geographical constraints and frequency limits, which define the upper bound of $C_r$. +\begin{itemize} + \item \textbf{Frequency constraints: } +\end{itemize} + + To estimate the global launch capacity ceiling, we apply a Richards growth model to historical launch data. As shown in Figure \ref{fig:richards_fit}, the model predicts a theoretical saturation limit of $K = 4298$ annual launches. However, accounting for practical constraints such as maintenance cycles and meteorological windows, we adopt a conservative operational cadence of one launch per day per site (365 annually). This calibrated cap serves as the primary constraint for determining the minimum construction duration. + +\begin{figure}[H] +\centering +\includegraphics[width=8cm]{richards_curve_2010_K4298.png} +\caption{Richards growth model fitting and projected capacity limit $K$} +\label{fig:richards_fit} +\end{figure} + +\begin{itemize} + \item \textbf{Latitude-based efficiency correction: } +\end{itemize} + +Earth's rotation provides rockets with a tangential initial velocity $v_{0i}$. The energy gain varies by latitude as follows: +\[ v_{0i} = v_{0E} \cdot \cos \theta_{i} \] +As illustrated in Figure \ref{latitudeeffects}, higher latitudes result in lower initial velocities, necessitating more chemical fuel. As latitude increases, the non-linear loss of tangential velocity causes the fuel-to-payload ratio to increase from 31.8 at equatorial sites to 34.0 at high-latitude sites. +\begin{figure}[H] +\centering +\includegraphics[width=8cm]{latitude_effects.png} +\caption{Rotation Velocity and Loss vs Latitude} +\label{latitudeeffects} +\end{figure} + + +Consequently, the model implements a low-latitude priority principle: low-latitude sites such as Kourou are prioritized to minimize global energy consumption when the number of launches has not reached the limit. This ensures that for a given timeline, the efficiency of material delivery is maximized. +\subsection{Unified Cost Optimization via Time Opportunity Scaling} +To select an actionable transport strategy, we scalarize the two objectives---energy consumption ($E$) and construction duration ($T$)---using a time opportunity cost parameter $\lambda$. + +\subsubsection{Reformulation via Opportunity Cost} + +We define the Total System Burden as: +\begin{equation} + J(T)=E(T)+\lambda T, \qquad \frac{dE}{dT}\bigg|_{T=T^*}=-\lambda +\label{eq:scalarization_logic} +\end{equation} +Here $\lambda$ (PJ/year) is an energy-equivalent penalty per additional year, capturing the value of schedule acceleration (e.g., earlier utilization and reduced long-horizon overhead). The optimal duration $T^*$ is reached when the marginal energy saving rate equals $\lambda$. + +\subsubsection{Calibration of Cost Parameter $\lambda$} + + +\begin{figure}[H] +\centering +\includegraphics[width=0.7\textwidth]{total_cost_curves.png} +\caption{Sensitivity analysis of optimal timeline $T^*$ versus time opportunity cost $\lambda$} +\label{fig:lambda_sensitivity} +\end{figure} + +Rather than choosing $\lambda$ subjectively, we cross-check it using four independent heuristics (normalized trade-off scaling, marginal-slope matching, phase-boundary location, and feasible-region geometry). These estimates converge near $\lambda\approx 504$ PJ/year, so we use $\lambda=504$ PJ/year as the baseline and treat the spread as sensitivity. + + + +\subsection{Results of Task 1} + +Through numerical simulation of the 100 million metric ton material transport mission, this study elucidates the intrinsic correlation between construction progress and resource consumption under diverse technical constraints. The findings demonstrate that the optimal transport strategy is not a simple superposition of individual methods but a dynamic equilibrium based on the trade-off between physical energy efficiency and temporal costs. + +\subsubsection{Comparison of Baseline Scenarios} + +Based on the model outputs, Table \ref{tab:comparison} summarizes the core indicators for the three baseline scenarios. +\begin{table}[H] + \small +\centering +\caption{Multi-indicator Comparison of Earth-Moon Material Transport Scenarios} +\label{tab:comparison} +\begin{tabular}{lcccc} +\toprule +Scenario Type & Min Duration (a) & Total Energy (PJ) & Unit Energy (GJ/t) \\ \midrule +Elevator-Only (a) & 186.2 & 15,720 & 157.2 \\ +Rocket-Only (b) & 219.2 & 50,609 & 506.1 \\ +Hybrid (c) & 100.7 & 31,537 & 315.4 \\ \bottomrule +\end{tabular} +\end{table} +Integrating Table \ref{tab:comparison} and Figure \ref{Three Transport Scenarios Feasibility Comparison}, a clear duration-energy trade-off emerges. Rocket-only is limited to 219.2 years with peak energy demand, while elevator-only provides the lowest energy footprint but requires 186.2 years. The hybrid scenario overcomes these bottlenecks, compressing the timeline to 100.7 years. It is the sole solution for the 100--186 year window and converges to elevator efficiency as duration increases. Ultimately, hybrid transport is indispensable for speed, while the space elevator represents the optimal long-term balance of time and energy. +\begin{figure}[H] +\centering +\includegraphics[width=10cm]{three_scenarios_comparison.png} +\caption{Three Transport Scenarios Feasibility Comparison} +\label{Three Transport Scenarios Feasibility Comparison} +\end{figure} +\subsubsection{Optimal Strategy Selection and Allocation} +To determine the optimal operating point, we analyze the behavior of the total cost function $J(T) = E(T) + \lambda T$. Based on the consensus value of $\lambda = 504$ PJ/year, three strategic configurations are evaluated within the feasible optimization space (100.7--186 years), as detailed in Table \ref{tab:strategies}. + +\begin{table}[H] + \small +\centering +\caption{Core Metrics of Typical Optimization Strategies for Scenario C} +\label{tab:strategies} +\begin{tabular}{lcccc} +\toprule +Strategy Type & Duration (a) & Total Energy (PJ) & Elevator Share & Energy Saving \\ \midrule +Strategy A (Cost-Prioritized) & 186.0 & 15,826 & 99.9\% & 85.5\% \\ +Strategy B (Time-Prioritized) & 101.0 & 58,391 & 54.2\% & 46.4\% \\ +\rowcolor{recommendColor} +Strategy C (Balanced) & 139.0 & \textbf{38,729} & 74.6\% & \textbf{64.4\%} \\ \bottomrule +\end{tabular} +\end{table} + +These strategies represent key operational configurations: Strategy A minimizes physical energy via full-load elevator operation; Strategy B achieves the shortest timeline through maximum energy input; and Strategy C represents the global minimum of the total cost function $J$, balancing construction efficiency with energy expenditure. + +\begin{enumerate} + \item \textbf{Marginal Energy Saving}: As shown in Figure \ref{fig:energy_time_tradeoff}, the marginal saving curve exhibits a step-wise diminishing trend. Between years 101 and 139, each additional year allocated to the timeline reduces energy demand by approximately 210 PJ. + \item \textbf{Optimal Operating Point}: The 139-year duration is identified as the optimal point where the marginal reduction in energy expenditure equals the marginal time opportunity cost $\lambda$. At this point, the total system burden $J$ is minimized. +\end{enumerate} + + +\subsubsection{Sensitivity and Parameter Stability} + +To assess the robustness of our model, we perform sensitivity analysis +on five key parameters. Figure \ref{fig:sensitivity} summarizes the results. + +\begin{figure}[H] +\centering +\includegraphics[width=8cm]{tornado_chart.png} +\caption{Sensitivity Analysis} +\label{fig:sensitivity} +\end{figure} + +\begin{itemize} + \item \textbf{Rocket Payload Capacity (100--150 tons):} + As specified in the problem statement, we analyze the full payload range. +Results show that higher capacity reduces minimum timeline from 111 to 93 +years, but has limited impact on the optimal hybrid solution at $\lambda=504$. + \item \textbf{Elevator Capacity (±20\%):} + Elevator capacity emerges as the most sensitive parameter. A 20\% reduction +extends the elevator-only timeline from 186 to 233 years, fundamentally +altering the hybrid strategy's feasible region. +\item \textbf{Launch Frequency:} +Doubling launch frequency from 1 to 2 per day reduces minimum timeline +from 101 to 69 years, enabling more aggressive time-optimized strategies. + +\item \textbf{Engine Technology:} +Engine Isp primarily affects energy consumption rather than timeline. +LOX/Hydrogen (Isp=450s) reduces fuel ratio by 64\% compared to LOX/Methane, +but practical considerations favor the latter. +\end{itemize} + + +\begin{table}[H] + \small +\centering +\caption{Sensitivity Analysis Summary} +\begin{tabular}{lccc} +\toprule +Parameter & Range & $\Delta T^*$ & $\Delta E^*$ \\ +\midrule +Payload Capacity & 100--150 t & $-7$ y ($-5.0\%$) & $+3246$ PJ ($+8.4\%$) \\ +Elevator Capacity & $\pm 20\%$ & $+42$ y ($+30.2\%$) & $-47902$ PJ ($-123.5\%$) \\ +Launch Frequency & 0.5--2/day & $-43$ y ($-30.9\%$) & $+20867$ PJ ($+53.8\%$) \\ +Structural Coef. $\alpha$ & 0.06--0.14 & $+85$ y ($+61.4\%$) & $-28149$ PJ ($-72.6\%$) \\ +\bottomrule +\end{tabular} +\end{table} + +\subsection{Stochastic Risk and Robustness Analysis} +Task 2 asks a fundamentally different question from Task 1: not merely how much performance degrades, but how our recommended strategy should change in response to system uncertainties. We achieve this through a three-stage methodology: quantifying perturbation impacts, identifying risk-adjusted decision boundaries, and finally deriving modified optimal strategies with robust safety margins. +\subsubsection{Stochastic Perturbation Modeling} + +Given the unprecedented nature of the Space Elevator System, we acknowledge significant epistemic uncertainty in failure parameters. We adopt a conservative approach by establishing parameter ranges rather than point estimates: +\begin{table}[H] +\small +\centering +\caption{Perturbation Parameter Ranges and Sources} +\begin{tabular}{lccc} +\toprule +\textbf{Parameter} & \textbf{Baseline} & \textbf{Range} & \textbf{Basis} \\ +\midrule +Tether swaying ($\Delta\theta$) & $0^\circ$ & $[-0.5^\circ, 0.5^\circ]$ & Coriolis-induced oscillations during release \\ +Rocket failure rate & 0.75\% & [0.3\%, 1.5\%] & Falcon 9 historical \cite{skyrocket2026falcon9} + technology maturation \\ +Elevator breaks/year & 2 & [1, 4] & IAA Space Elevator Assessment \cite{swan2013space}\\ +Downtime per break & 14 days & [7, 30] days & Scaled from offshore platform repairs \\ +Weather cancellation & 10\% & [5\%, 20\%] & Site-specific meteorological data \\ +\bottomrule +\end{tabular} +\end{table} + + +\subsubsection{Risk-Adjusted Optimization} + +Under uncertainty, the deterministic objective function must be extended to incorporate risk preferences. We reformulate the optimization as: +\begin{equation} + \min_{T} \quad J_{robust}(T) = \mathbb{E}[E(T)] + \lambda \cdot \mathbb{E}[T] + \gamma \cdot \text{CVaR}_{\alpha}(T) + \label{eq:robust_cost} +\end{equation} +where $\text{CVaR}_{\alpha}(T)$ denotes the Conditional Value-at-Risk at confidence level $\alpha$ (typically 95\%), representing the expected completion time in the worst $1-\alpha$ scenarios \cite{chatgpt}. The risk aversion coefficient $\gamma$ (PJ/year) quantifies the MCM Agency's willingness to pay for reduced schedule uncertainty. + + +\subsubsection{Monte Carlo Lifecycle Assessment} + +Due to the highly non-linear interactions between these stochastic variables, we utilized a Monte Carlo algorithm to execute 10,000 simulations of the construction cycle. + + +\begin{itemize} + \item \textbf{Simulation Logic}: For each simulated day, the algorithm samples the system state (Normal/Failure/Weather). In the event of interference, payload distribution is adjusted and incomplete tasks are rescheduled. + \item \textbf{Distribution Analysis}: As shown in Figure \ref{Completion Time Distribution Under Perturbations}, the results exhibit a significant right-skewed distribution. This implies that real-world timelines are prone to a long tail of cumulative delays—the probability of massive delays is low but the impact is profound, serving as a critical basis for risk assessment. +\end{itemize} + +\begin{figure}[H] +\centering +\includegraphics[width=12cm]{completion_time_distribution.png} +\caption{Completion Time Distribution Under Perturbations} +\label{Completion Time Distribution Under Perturbations} +\end{figure} + +\subsection{Results of Task 2} + +Idealized models define the theoretical limits, but stochastic variables such as tether swaying, equipment failure, and weather interference cause performance to deviate. We evaluate system performance under disturbance based on 10,000 Monte Carlo iterations. + +\subsubsection{Assessment of Performance Degradation} + +Table \ref{tab:stochastic_comparison} compares the deterministic solutions with stochastic means, quantifying the costs of system perturbations. +\begin{table}[H] + \small + \centering + \caption{Comparison of Core Metrics Under Perfect and Non-Perfect Conditions} + \label{tab:stochastic_comparison} + % 缩放至单栏宽度，保留1em内边距避免贴边（与示例格式一致） + \resizebox{\dimexpr\linewidth-1em\relax}{!}{ + \begin{tabular}{lcccc} + \toprule[1pt] % 加粗顶线（与示例一致） + \rowcolor{white} % 表头底色为白（与示例一致） + Strategy Type & Dimension & Perfect Condition & Non-Perfect (Mean $\pm$ Std) & Change Rate \\ + \midrule[0.6pt] % 中等粗细中间线（与示例一致） + % Strategy A：设置基线浅灰底色 + \rowcolor{baselineColor} + Strategy A & Time (a) & 186.2 & 202.3 $\pm$ 0.6 & +8.6\% \\ + \rowcolor{baselineColor} + & Energy (PJ) & 15,720 & 15,738 $\pm$ 0 & +0.1\% \\ + % Strategy B：设置基线浅灰底色 + \rowcolor{baselineColor} + Strategy B & Time (a) & 100.7 & 120.7 $\pm$ 0.4 & +19.9\% \\ + \rowcolor{baselineColor} + & Energy (PJ) & 31,537 & 30,217 $\pm$ 30 & -4.2\% \\ + % Strategy C：设置推荐方案浅蓝底色（突出核心方案，与示例一致） + \rowcolor{recommendColor} + Strategy C & Time (a) & 139.0 & 155.3 $\pm$ 0.5 & +11.7\% \\ + \rowcolor{recommendColor} + & Energy (PJ) & \textbf{38,729} & \textbf{38,285 $\pm$ 45} & -1.2\% \\ + \bottomrule[1pt] % 加粗底线（与示例一致） + \end{tabular} + } +\end{table} +\subsubsection{Resilient Strategic Adjustments} + +The Monte Carlo results reveal that perturbations do not merely shift performance metrics---they fundamentally alter the risk-return profile of each strategy: + +\textbf{(1) Optimal Timeline Adjustment} + +Under perfect conditions, the balanced strategy yields $T^* = 139$ years. However, incorporating the 95th percentile delay, the effective planning horizon becomes: +\begin{equation} + T^*_{adjusted} = T^* + \Delta T_{buffer} = 139 + 16.3 \times 1.15 \approx 158 \text{ years} +\end{equation} +where the 15\% margin accounts for tail-risk scenarios. Recommendation: MCM Agency should plan for a 155-160 year timeline to achieve 95\% confidence in completion. + +\textbf{(2) Hybrid Ratio Adjustment} + +Given that elevator breaks dominate schedule risk (correlation = 0.836), the optimal elevator share should be reduced to enhance redundancy: +\begin{equation} + \text{Elevator share}_{adjusted} = 74.6\% - \Delta_{redundancy} \approx 65\%-70\% +\end{equation} +This increases rocket utilization as a ``backup channel,'' accepting a 5-8\% energy penalty in exchange for improved schedule resilience. + +\textbf{(3) Strategic Reserve Capacity} + +We recommend maintaining a 10\% reserve margin in annual transport capacity: +\begin{equation} + C_{reserve} = 0.1 \times (\lambda C_e + \mu C_r) +\end{equation} +This reserve can absorb localized disruptions without triggering cascading delays. + +\subsubsection{Evolution of Key Indicators} + +Stochastic disturbances transform deterministic values into probability distributions, but the strategic ranking remains robust. + +\begin{figure}[H] +\centering +\includegraphics[width=10cm]{energy_distribution.png} +\caption{Energy Distribution} +\label{Energy Distribution} +\end{figure} + +\begin{itemize} + \item \textbf{Energy Stability}: As illustrated in Figure \ref{Energy Distribution}, energy consumption follows a narrow-peak normal distribution with a standard deviation of less than 0.2\%. This confirms that tether swaying does not cause energy collapse. The slight mean shift in Strategies B and C is due to payload loss from rocket failures offsetting energy increments from swaying. + + \item \textbf{Temporal Right-Skewed Characteristics}: Median durations delay by 8.6\% to 19.9\%. However, the duration intervals for each strategy remain strictly separated, ensuring that the fundamental decision logic—Strategy A for cost and Strategy B for speed—holds true even under perturbation. +\end{itemize} + + + +\section{Model II: Life-Support Logistics and Stochastic Water Balance Model} + +\subsection{Model Overview} + +As the Moon Colony transitions from the construction phase to the operational phase, the logistical focus shifts from structural materials to life-support supplies. We develop the Life-Support Logistics and Stochastic Water Balance Model (LSL-SWBM) to quantify the water security boundaries of the settlement during its first year of operation. This model accounts for multi-level demand functions based on psychological comfort and incorporates a normal approximation to address stochastic medical emergency needs. By mapping these water requirements onto the transport framework established in Model I, we evaluate the additional pressure exerted by different comfort factors on the Earth-Moon logistics chain. + + + +\begin{figure}[H] +\centering +\includegraphics[width=11cm]{Flow Chart of Model II.png} +\caption{Flow Chart of Model II} +\label{Flow Chart of Model II} +\end{figure} + + + +\subsection{Water Demand Architecture} + +In an isolated lunar ecosystem, water consumption is primarily sustained by a recycling system \cite{LIU2021113}. However, logistical replenishment must compensate for system losses and sudden medical surges. This model assumes that lunar water use is restricted to domestic and medical emergency purposes, excluding industrial use, to define the core demand framework. + +\subsubsection{Domestic Water Evolution} + +Within the colony, domestic water consists of survival and hygiene components. The daily demand is formulated as: +\begin{equation} +W_{r} = N \times (w_{s} + 0.4\alpha) +\end{equation} +where $N$ is the population, $w_{s}$ represents the survival baseline, and $\alpha$ is the comfort factor. The following table summarizes the water standards across different demand tiers \cite{francisco2025nasa}\cite{whitmore2012nasa}: + +\begin{table}[H] + \small +\centering +\caption{Water Demand Standards Across Different Comfort Tiers} +\begin{tabular}{lccl} +\toprule +\textbf{Demand Tier} & \textbf{$\alpha$ Value} & \textbf{Daily Use per Capita} & \textbf{Description} \\ \midrule +Survival Standard & 1 & 2.9 L & Minimum survival threshold \\ +Comfort Standard & 50 & 22.5 L & Moderate domestic comfort \\ +Luxury Standard & 250 & 102.5 L & Equivalent to Earth-like usage \\ \bottomrule +\end{tabular} +\end{table} + +\subsubsection{Medical Emergency Demand Modeling} + +Given the large population , the daily number of patients $X$ (assuming a daily incidence rate $p=2\%$) follows a binomial distribution, which is accurately approximated by a normal distribution $X \sim N(Np, Np(1-p))$. To ensure medical safety under extreme conditions, the model adopts the peak demand at a 99\% confidence level as the daily reserve target: +\begin{equation} +W_{medical} = (E(X) + Z_{0.99} \cdot \sigma) \times 5 \text{ kg} +\end{equation} +This indicator remains stable across different domestic comfort scenarios, ensuring the robustness of the medical support system. + +\subsection{Replenishment and Buffer Strategies} + +Unlike construction materials, water is highly recyclable. Given the maturity of water recycling technology, the cumulative rate of daily demand after the initial transport is relatively slow. Since a daily delivery schedule would be prohibitively expensive, we adopt a monthly supply mode based on a dual-stage replenishment logic. + +\subsubsection{Initial Month Filling} + +During the first month, recycled water from the previous month is unavailable. Thus, the initial supply must satisfy two criteria: providing domestic water for 30 days and maximizing medical reserves for potential surges. The initial transport volume is defined as: +\begin{equation} +W_{initial} = (W_{r} + W_{mi}) \cdot T +\end{equation} +where $T=30$ days and $W_{mi}$ is the daily emergency medical supply at a 99\% confidence level. This strategy establishes the system’s circulating base while creating a 30-day emergency buffer. + +\subsubsection{Monthly Routine Compensation} + +In subsequent months, domestic demand is met through a combination of Earth-based replenishment and water recycling. Simultaneously, medical reserves need only cover the mean incidence rate due to the confidence buffer established initially. With a recycling efficiency $\eta$ (set at 0.9), the routine monthly supply model is: +\begin{equation} +W_{routine} = (W_{r}(1-\eta) + W_{m}) \cdot T +\end{equation} +This model precisely offsets recycling losses and daily medical consumption to maintain a dynamic water balance. +\subsection{Results of Task 3} + +To evaluate the operational sustainability of the Moon Colony, we perform a quantitative decomposition of the water replenishment mission across temporal and energetic dimensions. By assuming ideal conditions and excluding the perturbations analyzed in Task 2, this approach isolates the specific influence of varying living standards to facilitate a definitive baseline comparison. +\subsubsection{Demand Scaling Across Comfort Tiers} + +Minute variations in the comfort factor $\alpha$ trigger significant shifts in logistical scale. The water demand metrics for the three simulated tiers are summarized in Table \ref{tab:water_demand_stats}. + +\begin{table}[H] + \small + \centering + \caption{Water Demand Metrics under Different Comfort Scenarios} + \label{tab:water_demand_stats} + \begin{tabular}{lccccc} + \toprule + \textbf{Demand Tier} & \textbf{$\alpha$} & \makecell{\textbf{Water Inventory} \\ \textbf{(metric tons)}} & \makecell{\textbf{Daily Rep.} \\ \textbf{(metric tons)}} & \makecell{\textbf{Initial Trans.} \\ \textbf{(metric tons)}} & \makecell{\textbf{Annual Total} \\ \textbf{(metric tons)}} \\ + \midrule + Survival Standard & 1 & 290 & 29.1 & 1,163 & 10,622 \\ + Comfort Standard & 50 & 2,250 & 225.1 & 9,003 & 82,162 \\ + Luxury Standard & 250 & 10,250 & 1,025.1 & 41,003 & 374,162 \\ + \bottomrule + \end{tabular} +\end{table} + +Table \ref{tab:water_demand_stats} illustrates a multiplier effect: as the standard shifts from Survival to Luxury, the annual requirement surges by over 35-fold. Supported by high-efficiency recycling, the daily replenishment required from Earth remains a small fraction of the total demand, emphasizing the value of closed-loop systems. + +\subsubsection{Integrated Trade-off Analysis of Transport Schemes} + +To satisfy these requirements, we mapped the demand indicators onto the transport framework of Model I. Four distinct schemes were evaluated based on their baseline capacities, as defined in Table \ref{tab:transport_schemes_def}. + +\begin{table}[H] + \small + \centering + \caption{Definition and Baseline Capacity of Transport Schemes} + \label{tab:transport_schemes_def} + \begin{tabular}{clcc} + \toprule + \textbf{Scheme} & \textbf{Description} & \textbf{Daily Capacity (t/d)} & \textbf{Specific Energy (GJ/t)} \\ + \midrule + 1 & Space Elevator Only (3 units) & 1,471 & 157.2 \\ + 2 & Rocket-Only (10 sites) & 1,250 & 506.1 \\ + 3 & Hybrid (Elevator + 10 sites) & 2,721 & 317.5 \\ + 4 & Hybrid (Elevator + Low-lat.) & 1,971 & 243.5 \\ + \bottomrule + \end{tabular} +\end{table} + +By consolidating the simulation outputs for all scenarios, Table \ref{tab:consolidated_performance} presents a comprehensive comparison of scheme performance across $\alpha$ tiers. + +\begin{table}[H] + \small + \centering + \caption{Consolidated Performance Metrics of Transport Schemes across Comfort Tiers} + \label{tab:consolidated_performance} + \begin{tabular}{lccccc} + \toprule + \textbf{Scenario} & \textbf{Scheme} & \textbf{Initial Days} & \textbf{Initial En. (TJ)} & \textbf{Monthly Days} & \textbf{Annual En. (PJ)} \\ + \midrule + \multirow{4}{*}{Survival ($\alpha=1$)} + & Scheme 1 & 0.79 & 182.8 & 0.59 & 1.67 \\ + & Scheme 2 & 0.93 & 588.6 & 0.69 & 5.38 \\ + & Scheme 3 & 0.43 & 369.2 & 0.32 & 3.37 \\ + & Scheme 4 & 0.59 & 283.1 & 0.44 & 2.59 \\ + \midrule + \rowcolor{recommendColor} \multirow{4}{*}{Comfort ($\alpha=50$)} + & Scheme 1 & 6.12 & 1,415.3 & 4.59 & 12.92 \\ + & Scheme 2 & 7.20 & 4,556.3 & 5.40 & 41.58 \\ + & Scheme 3 & 3.31 & 2,858.1 & 2.48 & 26.08 \\ + & Scheme 4 & 4.57 & 2,191.9 & 3.43 & 20.00 \\ + \midrule + \multirow{4}{*}{Luxury ($\alpha=250$)} + & Scheme 1 & 27.87 & 6,445.7 & 20.90 & 58.82 \\ + & Scheme 2 & 32.80 & 20,751.2 & 24.60 & 189.36 \\ + & Scheme 3 & 15.07 & 13,016.9 & 11.30 & 118.78 \\ + & Scheme 4 & 20.80 & 9,982.5 & 15.60 & 91.09 \\ + \bottomrule + \end{tabular} +\end{table} +While water transport is highly feasible for survival, the Comfort Scenario ($\alpha=50$) serves as the planning representative where efficiency-timeline trade-offs intensify. In this tier, Scheme 1 defines the efficiency baseline, Scheme 3 maximizes throughput to minimize delivery time, and Scheme 4 offers the optimal operational balance. Under the Luxury Scenario ($\alpha=250$), energy demand for rocket transport peaks at 189.36 PJ. Crucially, even this extreme demand represents only 0.37 percent of the total construction energy, confirming that post-2050 logistical constraints stem from capacity allocation rather than aggregate energy availability. + +\subsection{Sensitivity Analysis for Water Supply Model} + +To assess the robustness of Model II and identify the dominant levers in water logistics, we conduct a sensitivity analysis over five variables: recycling efficiency ($\eta$), comfort factor ($\alpha$), population ($N$), medical demand parameters, and buffer duration. + +\subsubsection{Parameter Sensitivity Ranking via Tornado Analysis} + +We systematically varied each parameter within its feasible range while holding others at baseline values ($\alpha=50$, $\eta=90\%$, $N=100,000$, sickness rate $p=2\%$, buffer $=30$ days). The resulting impacts on annual water supply are visualized through a tornado diagram (Figure \ref{fig:tornado}). + +\begin{figure}[h] + \centering + \includegraphics[width=12cm]{tornado_analysis.png} + \caption{Tornado Diagram} + \label{fig:tornado} +\end{figure} +\begin{enumerate} + \item \textbf{Recycling efficiency ($\eta$)} dominates: modest degradations cause large increases in annual resupply because losses scale with $(1-\eta)$. + \item \textbf{Secondary drivers}: comfort ($\alpha$) is the next lever (tightening provides headroom), population ($N$) is near-linear, while medical/buffer settings have limited effect on annual totals but affect initial fill and resilience. +\end{enumerate} + +\subsubsection{Quantitative Sensitivity Coefficients} + +Table \ref{tab:sensitivity_coefficients} presents the normalized sensitivity coefficients, defined as the percentage change in output per percentage change in input parameter: + +\begin{table}[H] + \small + \centering + \caption{Sensitivity Coefficients for Key Model Parameters} + \label{tab:sensitivity_coefficients} + \begin{tabular}{lccccc} + \toprule + \textbf{Parameter} & \textbf{Baseline} & \textbf{Range} & \textbf{$\Delta$ Annual Supply} & \textbf{Sensitivity} & \textbf{Rank} \\ + \midrule + Recycling Rate ($\eta$) & 90\% & 80\%--95\% & $-$52\% to +96\% & $-$9.6 & 1 \\ + Comfort Factor ($\alpha$) & 50 & 25--100 & $-$71\% to +17\% & +0.88 & 2 \\ + Population ($N$) & 100k & 80k--120k & $\pm$20\% & +1.0 & 3 \\ + Sickness Rate ($p$) & 2\% & 1\%--4\% & $-$0.3\% to +0.6\% & +0.30 & 4 \\ + Buffer Days & 30 & 15--45 & $\pm$0\%$^*$ & 0 & 5 \\ + \bottomrule + \end{tabular} + \begin{tablenotes} + \small + \item $^*$ Buffer days affect only initial transport, not annual replenishment. + \end{tablenotes} +\end{table} + +The sensitivity coefficient for $\eta$ reaches $-9.6$: a 1\% decrease in recycling efficiency increases annual resupply by about 9.6\%, making recycling integrity the primary control variable. + +\subsubsection{Worst-Case Stress Testing} + +Beyond univariate analysis, we evaluate system performance under compound adverse conditions through Monte Carlo-informed scenario construction. Table \ref{tab:worst_case} summarizes nine representative scenarios spanning from optimal to catastrophic conditions: + +\begin{table}[H] + \small + \centering + \caption{Worst-Case Scenario Analysis: Water Supply System Stress Test} + \label{tab:worst_case} + \resizebox{\textwidth}{!}{ + \begin{tabular}{lcccccccc} + \toprule + \textbf{Scenario} & \textbf{$\alpha$} & \textbf{$\eta$} & \textbf{$N$ (k)} & \textbf{$p$} & \textbf{Buffer} & \textbf{Annual (kt)} & \textbf{Capacity (\%)} & \textbf{Feasible} \\ + \midrule + Survival Mode & 1 & 90\% & 100 & 2\% & 30 & 14.2 & 2.7\% & \checkmark \\ + \rowcolor{recommendColor} + Comfort Mode (Baseline) & 50 & 90\% & 100 & 2\% & 30 & 85.8 & 16.0\% & \checkmark \\ + Luxury Mode & 250 & 90\% & 100 & 2\% & 30 & 377.8 & 70.3\% & \checkmark \\ + Recycle Degradation & 50 & 80\% & 100 & 2\% & 30 & 167.9 & 31.3\% & \checkmark \\ + Epidemic Outbreak & 50 & 90\% & 100 & 5\% & 45 & 100.4 & 18.7\% & \checkmark \\ + Population Surge & 50 & 90\% & 120 & 2\% & 30 & 102.9 & 19.2\% & \checkmark \\ + Moderate Stress & 100 & 85\% & 110 & 3\% & 40 & 264.4 & 49.2\% & \checkmark \\ + \rowcolor{baselineColor} + \textbf{Worst Case} & \textbf{250} & \textbf{80\%} & \textbf{120} & \textbf{5\%} & \textbf{45} & \textbf{919.8} & \textbf{171.3\%} & \ding{55} \\ + \bottomrule + \end{tabular} + } +\end{table} + +The stress test shows that feasibility is controlled mainly by the joint condition \textbf{$\eta<82\%$ with $\alpha>200$}, under which annual water demand can exceed elevator capacity (worst case: \textbf{171.3\%}), whereas moderate stress remains well within capacity (49.2\%). + +\subsection{Conclusion and Strategic Insights} + +\begin{enumerate} + \item \textbf{Trade-off between Comfort and Capacity}: Quantitative analysis confirms that quality of life acts as a logistical amplifier. At the Luxury tier, the annual water replenishment occupies 69.68\% of the theoretical annual capacity of the space elevator system. Maintaining high living standards significantly constricts the transport window for other critical infrastructure and scientific materials. + + \item \textbf{Tiered Strategy Recommendations}: + \begin{itemize} + \item \textbf{Initial Operations}: We recommend Survival tier combined with Scheme 1. This minimizes energy consumption while ensuring the survival threshold for 100,000 residents, reserving hybrid capacity for urgent infrastructure tasks. + \item \textbf{Mature Operations}: We recommend Comfort tier combined with Scheme 4. This provides a balance between speed and cost, completing the initial filling in 4.57 days with a reasonable energy footprint. + \end{itemize} + + \item \textbf{System Resilience}: Thanks to the 90\% efficient recycling system and the 30-day emergency buffer, the water logistics chain exhibits high resilience. Even during a full month of transport disruption, the survival of the colony remains uncompromised, allowing for logistical maintenance and error recovery. +\end{enumerate} + + +\section{Model Extension} + +Following the assessment of construction timelines and life-support logistics, this chapter extends the model to incorporate the environmental footprint as a critical optimization boundary. By quantifying the ecological externalities of various transport scenarios, we refine previous strategic selections to promote symbiosis between Earth's conservation and lunar sustainability. + +\subsection{Quantification of Baseline Environmental Impacts} + +We extended the environmental sub-module based on the energy conversion logic of Model I. For liquid oxygen/methane ($\text{LOX/CH}_4$) propulsion systems, fuel consumption is mapped to emission mass via the combustion equation: +\begin{equation} + \text{CH}_{4} + 2\text{O}_{2} \longrightarrow \text{CO}_{2} + 2\text{H}_{2}\text{O} +\end{equation} +A life-cycle assessment (LCA), including fuel production and power generation, was conducted for four baseline scenarios, as summarized in Table \ref{tab:environmental_metrics}. + +\begin{table}[H] +\small +\centering +\caption{Environmental Metrics Across Four Baseline Scenarios} +\label{tab:environmental_metrics} +\begin{tabular}{lccccc} +\toprule +\textbf{Scenario} & \textbf{Timeline} & \textbf{Total Fuel} & \textbf{Total $\text{CO}_2$} & \textbf{Stratospheric $\text{H}_2\text{O}$} & \textbf{Intensity} \\ + & \textbf{(a)} & \textbf{(Mt)} & \textbf{(Mt)} & \textbf{(Mt)} & \textbf{(Mt/yr)} \\ +\midrule +Rocket-Only & 219 & 9,596 & 12,968 & 1,919 & 59.2 \\ +Elevator-Only & 186 & 69 & 846 & 0 & 4.5 \\ +Hybrid (Time-Prioritized) & 101 & 4,445 & 6,414 & 882 & 63.7 \\ +\rowcolor{recommendColor} Hybrid (Balanced) & 139 & 2,438 & 3,856 & 477 & 27.7 \\ +\bottomrule +\end{tabular} +\end{table} + +\begin{figure}[H] + \centering + \includegraphics[width=8cm]{environmental_comparison.png} + \caption{Multi-dimensional environmental impact analysis} + \label{fig:environmental_analysis} +\end{figure} + +The analysis in Figure \ref{fig:environmental_analysis} clearly distinguishes the ecological costs: +\begin{itemize} + \item The \textbf{Rocket-Only scenario} incurs the highest $\text{CO}_2$ emissions and stratospheric $\text{H}_2\text{O}$ injection, posing the greatest risk to global climate and the ozone layer. + \item The \textbf{Elevator-Only scenario} completely circumvents stratospheric pollution, with total $\text{CO}_2$ emissions being only $6.5\%$ of the rocket scenario. + \item The \textbf{Balanced Hybrid scenario} achieves a $40\%$ reduction in $\text{CO}_2$ and a $46\%$ reduction in stratospheric $\text{H}_2\text{O}$ compared to the Time-Prioritized plan, successfully equilibrating construction efficiency with environmental load. +\end{itemize} + +\subsection{Earth Port Assessment} + +While the space elevator facilitates zero-emission transit, its Earth Port—a permanent maritime structure in equatorial waters—may impact local marine ecosystems. Drawing on research from the International Space Elevator Consortium (ISEC) \cite{isec2021greenroad}, we analogized the Earth Port to offshore oil rigs, the most mature existing maritime facilities. + +\begin{table}[H] +\small +\centering +\caption{Comparative Impact: Earth Port vs. Offshore Oil Platforms} +\label{tab:earth_port_impact} +\begin{tabular}{@{}l@{\hspace{0.8em}}l@{\hspace{0.8em}}l@{\hspace{0.8em}}c@{}} +\toprule +\textbf{Impact Type} & \textbf{Oil Platform} & \textbf{Earth Port} & \textbf{Range} \\ +\midrule +Physical Damage & Benthic disruption (anchoring) & Anchor zone-only & Localized \\ +Chemical Discharge & Drilling fluids/water & No discharge & Negligible \\ +Interference & Light/acoustic disturbance & Navigation light + low-freq noise & Localized \\ +Habitat Alteration & Artificial reef effect & Hard substrate (sessile colonies) & Positive \\ +Accident Risk & Hydrocarbon spills & No leakage risk & Negligible \\ +\bottomrule +\end{tabular} +\end{table} + +The analogy confirms that the Earth Port's environmental footprint is significantly lower than that of existing oil platforms. Compared to the widespread atmospheric damage caused by rockets, the localized impacts of the space elevator are nearly negligible, reinforcing its ecological superiority. +\subsection{Tri-objective Decision Framework} +We expand the bi-objective optimization to a tri-objective framework ($J_{total}$) by integrating environmental externalities as primary constraints: +\begin{equation} + \min J_{total} = \alpha \cdot \text{Time} + \beta \cdot \text{Energy} + \gamma \cdot \text{Emission}_{\text{CO}_2} + \delta \cdot \text{Emission}_{\text{H}_2\text{O, strat}} +\end{equation} + +\begin{figure}[H] +\centering +\includegraphics[width=0.6\textwidth]{marginal_benefit.png} +\caption{Energy and $\text{CO}_2$ Reduction Potential Analysis.} +\label{fig:reduction_analysis} +\end{figure} + +Simulation analysis reveals a significant coupling between environmental benefits and energy efficiency. Notably, the system reaches its peak marginal environmental benefit at approximately the 186-year mark, representing an "Efficiency Cliff". Beyond this threshold, any further extension of the construction timeline yields diminishing returns in environmental preservation. + +\subsection{Result of Task 4} + + +Conclusively, the 186-year standalone space elevator scenario is proposed as the optimal strategy for lunar colonization. This decision integrates construction duration, energy efficiency, environmental impact, and system reliability. + +\begin{figure}[htbp] + \centering + \includegraphics[width=0.6\textwidth]{radar_chart.png} + \caption{Multi-dimensional Environmental Impact (Radar Chart)} + \label{fig:radar_analysis} +\end{figure} + +\begin{itemize} + \item The rocket-only scenario imposes the most severe environmental risks, with carbon emissions and stratospheric water vapor injection threatening the global climate and ozone stability. + \item Furthermore, although the hybrid scenario offers temporal advantages, it incurs substantial emissions and introduces heightened system complexity and failure risks. + \item Consequently, the 186-year elevator-only plan represents the most sustainable choice, as it completely eliminates rocket-based pollution by trading temporal efficiency for ecological preservation. + +\end{itemize} + +\section{Strengths and Weaknesses} + +\subsection{Strengths} +\begin{itemize} + \item \textbf{Comparable cost proxy across technologies.} + Using Universal Energy-Equivalent (UEE, Joules) avoids long-horizon monetary uncertainty and enables a thermodynamically consistent comparison between chemical rockets and space-elevator lifting. + \item \textbf{Interpretable multi-objective decision logic.} + The time-opportunity parameter $\lambda$ converts the energy--time trade-off into a single objective with a clear optimality condition ($dE/dT=-\lambda$), making stakeholder preferences explicit and auditable. + \item \textbf{Risk-aware recommendation.} + CVaR-style risk adjustment and Monte Carlo simulations translate uncertainties (failures, downtime, weather) into a confidence-bounded schedule and actionable operational policies (strategy band, reserve margin, trigger actions). + \item \textbf{Actionable sensitivity insights.} + Sensitivity analysis identifies dominant levers (e.g., elevator capacity and recycling efficiency), directly informing engineering priorities and governance constraints. +\end{itemize} + +\subsection{Weaknesses and Possible Improvement} +\begin{itemize} + \item \textbf{Parameter uncertainty and long-horizon extrapolation.} + Key inputs (elevator throughput, failure/downtime rates, future launch cadence) are necessarily assumed or extrapolated. + + \textit{Improvement:} expand scenario sets and report threshold/robust regions where the recommendation remains unchanged. + \item \textbf{UEE is not a complete economic cost.} + Energy captures a physical lower bound but does not fully represent CAPEX, labor, insurance, and supply-chain constraints. + + \textit{Improvement:} add a parallel monetary sensitivity layer (order-of-magnitude CAPEX/OPEX ranges) to validate ranking stability. + +\end{itemize} + + +% \section{Conclusion} + + + + +\newpage +\printbibliography[title={References}, heading=bibintoc] + + +\newpage +\newcounter{lastpage} +\setcounter{lastpage}{\value{page}} +\thispagestyle{empty} + +\newpage +\thispagestyle{empty} % 隐藏这一页的页码 +\begin{tikzpicture}[remember picture, overlay] + \node at (current page.center) { + \includegraphics[width=\paperwidth, height=\paperheight]{letter.png} + }; +\end{tikzpicture} +\newpage + +\section*{Report on Use of AI} + +\begin{enumerate} +\item OpenAI ChatGPT (accessed 2026--02--01) +\begin{description}[leftmargin=!,labelwidth=2.6cm] +\item[Purpose:] We used a generative AI assistant only for (i) outlining and language polishing, (ii) checking for clarity/consistency of model descriptions, and (iii) LaTeX presentation suggestions (tables/figures). We did \textbf{not} use AI to generate final numerical results, figures, or citations; all computations and plots were produced by the team using our own scripts and datasets, and all claims were verified by the team before inclusion. + +\item[Where AI was potentially useful in the paper:] background framing (Introduction), simplifying and explaining assumptions (Assumptions), verifying the expression/units of key equations (Model I \& II), drafting/condensing the robustness narrative (stochastic/CVaR), and improving readability of the recommendation/strengths-weaknesses sections. + +\item[Query 1:] ``Rewrite the following paragraph in concise academic English, keep the meaning, do not add new claims: \emph{Monetary costs are ill-conditioned for an Earth--Moon logistics program spanning many decades due to inflation, technology shocks, and geopolitical uncertainty. We therefore use energy consumption as a stable proxy for marginal cost, enabling a consistent comparison between rockets and the Space Elevator System.}'' +\item[Output 1:] ``Because monetary costs over multi-decade programs are highly uncertain (inflation, technology shocks, and geopolitics), we adopt energy consumption as a physics-grounded proxy for marginal cost, enabling consistent comparison between rockets and the Space Elevator System.'' + +\item[Query 2:] ``Give a short, publication-style definition of CVaR at confidence level \( \alpha \) and one sentence on how it is used in schedule-risk planning.'' +\item[Output 2:] ``Conditional Value-at-Risk (CVaR\(_\alpha\)) is the expected value of a random variable conditional on outcomes in the worst \(1-\alpha\) tail of its distribution. In schedule-risk planning, CVaR\(_\alpha\) summarizes ``expected worst-case'' delays beyond the \( \alpha \)-quantile and can be penalized to obtain risk-averse plans.'' + +\item[Query 3:] ``Provide a minimal LaTeX example (xcolor + booktabs) to highlight one recommended row in light blue and keep the rest plain, suitable for an MCM paper table.'' +\item[Output 3:] ``Use \verb|\usepackage[table]{xcolor}| and define a color, then apply \verb|\rowcolor{...}| to the recommended row. Example: +\begin{verbatim} +\definecolor{recommendColor}{RGB}{217,234,249} +\begin{tabular}{lccc} +\toprule +... \\ +\midrule +Baseline & ... \\ +\rowcolor{recommendColor} Recommended & ... \\ +\bottomrule +\end{tabular} +\end{verbatim}'' +\end{description} + +\item Cursor IDE Autocomplete (Free plan, accessed 2026--02--01) +\begin{description}[leftmargin=!,labelwidth=2.6cm] +\item[Purpose:] Auto-completions for code used in preparing our models. +\end{description} +\end{enumerate} + +% 重置页码 +\clearpage +\setcounter{page}{\value{lastpage}} + + +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/latex/mcmthesis-demo.toc b/latex/mcmthesis-demo.toc new file mode 100644 index 0000000..3f5dc73 --- /dev/null +++ b/latex/mcmthesis-demo.toc @@ -0,0 +1,57 @@ +\contentsline {section}{\numberline {1}Introduction}{4}{section.1}% +\contentsline {subsection}{\numberline {1.1}Background}{4}{subsection.1.1}% +\contentsline {subsection}{\numberline {1.2}Restatement of the Problem}{4}{subsection.1.2}% +\contentsline {subsection}{\numberline {1.3}Our Work}{5}{subsection.1.3}% +\contentsline {section}{\numberline {2}Assumptions and Justifications}{5}{section.2}% +\contentsline {section}{\numberline {3}Notations}{6}{section.3}% +\contentsline {section}{\numberline {4}Model I: Universal Energy-Equivalent and Temporal Coordination Model}{6}{section.4}% +\contentsline {subsection}{\numberline {4.1}Model Overview}{6}{subsection.4.1}% +\contentsline {subsection}{\numberline {4.2}Energy-Equivalent as a Cost Proxy}{7}{subsection.4.2}% +\contentsline {subsubsection}{\numberline {4.2.1}Cost Convergence Analysis}{7}{subsubsection.4.2.1}% +\contentsline {subsubsection}{\numberline {4.2.2}Empirical Industry Validation}{7}{subsubsection.4.2.2}% +\contentsline {subsection}{\numberline {4.3}Ideal Energy Cost Modeling}{8}{subsection.4.3}% +\contentsline {subsubsection}{\numberline {4.3.1}Rocket Momentum Dynamics}{8}{subsubsection.4.3.1}% +\contentsline {subsubsection}{\numberline {4.3.2}Space Elevator Mechanics}{8}{subsubsection.4.3.2}% +\contentsline {subsection}{\numberline {4.4}Ideal Timeline and Logistic Efficiency Modeling}{9}{subsection.4.4}% +\contentsline {subsubsection}{\numberline {4.4.1}Transport Progress Formulas}{9}{subsubsection.4.4.1}% +\contentsline {subsubsection}{\numberline {4.4.2}Spatial and Geographical Optimization}{10}{subsubsection.4.4.2}% +\contentsline {subsection}{\numberline {4.5}Unified Cost Optimization via Time Opportunity Scaling}{11}{subsection.4.5}% +\contentsline {subsubsection}{\numberline {4.5.1}Reformulation via Opportunity Cost}{11}{subsubsection.4.5.1}% +\contentsline {subsubsection}{\numberline {4.5.2}Calibration of Cost Parameter $\lambda $}{11}{subsubsection.4.5.2}% +\contentsline {subsection}{\numberline {4.6}Results of Task 1}{11}{subsection.4.6}% +\contentsline {subsubsection}{\numberline {4.6.1}Comparison of Baseline Scenarios}{12}{subsubsection.4.6.1}% +\contentsline {subsubsection}{\numberline {4.6.2}Optimal Strategy Selection and Allocation}{12}{subsubsection.4.6.2}% +\contentsline {subsubsection}{\numberline {4.6.3}Sensitivity and Parameter Stability}{13}{subsubsection.4.6.3}% +\contentsline {subsection}{\numberline {4.7}Stochastic Risk and Robustness Analysis}{14}{subsection.4.7}% +\contentsline {subsubsection}{\numberline {4.7.1}Stochastic Perturbation Modeling}{14}{subsubsection.4.7.1}% +\contentsline {subsubsection}{\numberline {4.7.2}Risk-Adjusted Optimization}{14}{subsubsection.4.7.2}% +\contentsline {subsubsection}{\numberline {4.7.3}Monte Carlo Lifecycle Assessment}{14}{subsubsection.4.7.3}% +\contentsline {subsection}{\numberline {4.8}Results of Task 2}{15}{subsection.4.8}% +\contentsline {subsubsection}{\numberline {4.8.1}Assessment of Performance Degradation}{15}{subsubsection.4.8.1}% +\contentsline {subsubsection}{\numberline {4.8.2}Resilient Strategic Adjustments}{15}{subsubsection.4.8.2}% +\contentsline {subsubsection}{\numberline {4.8.3}Evolution of Key Indicators}{16}{subsubsection.4.8.3}% +\contentsline {section}{\numberline {5}Model II: Life-Support Logistics and Stochastic Water Balance Model}{17}{section.5}% +\contentsline {subsection}{\numberline {5.1}Model Overview}{17}{subsection.5.1}% +\contentsline {subsection}{\numberline {5.2}Water Demand Architecture}{17}{subsection.5.2}% +\contentsline {subsubsection}{\numberline {5.2.1}Domestic Water Evolution}{17}{subsubsection.5.2.1}% +\contentsline {subsubsection}{\numberline {5.2.2}Medical Emergency Demand Modeling}{18}{subsubsection.5.2.2}% +\contentsline {subsection}{\numberline {5.3}Replenishment and Buffer Strategies}{18}{subsection.5.3}% +\contentsline {subsubsection}{\numberline {5.3.1}Initial Month Filling}{18}{subsubsection.5.3.1}% +\contentsline {subsubsection}{\numberline {5.3.2}Monthly Routine Compensation}{18}{subsubsection.5.3.2}% +\contentsline {subsection}{\numberline {5.4}Results of Task 3}{19}{subsection.5.4}% +\contentsline {subsubsection}{\numberline {5.4.1}Demand Scaling Across Comfort Tiers}{19}{subsubsection.5.4.1}% +\contentsline {subsubsection}{\numberline {5.4.2}Integrated Trade-off Analysis of Transport Schemes}{19}{subsubsection.5.4.2}% +\contentsline {subsection}{\numberline {5.5}Sensitivity Analysis for Water Supply Model}{20}{subsection.5.5}% +\contentsline {subsubsection}{\numberline {5.5.1}Parameter Sensitivity Ranking via Tornado Analysis}{20}{subsubsection.5.5.1}% +\contentsline {subsubsection}{\numberline {5.5.2}Quantitative Sensitivity Coefficients}{21}{subsubsection.5.5.2}% +\contentsline {subsubsection}{\numberline {5.5.3}Worst-Case Stress Testing}{21}{subsubsection.5.5.3}% +\contentsline {subsection}{\numberline {5.6}Conclusion and Strategic Insights}{22}{subsection.5.6}% +\contentsline {section}{\numberline {6}Model Extension}{22}{section.6}% +\contentsline {subsection}{\numberline {6.1}Quantification of Baseline Environmental Impacts}{22}{subsection.6.1}% +\contentsline {subsection}{\numberline {6.2}Earth Port Assessment}{23}{subsection.6.2}% +\contentsline {subsection}{\numberline {6.3}Tri-objective Decision Framework}{23}{subsection.6.3}% +\contentsline {subsection}{\numberline {6.4}Result of Task 4}{24}{subsection.6.4}% +\contentsline {section}{\numberline {7}Strengths and Weaknesses}{25}{section.7}% +\contentsline {subsection}{\numberline {7.1}Strengths}{25}{subsection.7.1}% +\contentsline {subsection}{\numberline {7.2}Weaknesses and Possible Improvement}{25}{subsection.7.2}% +\contentsline {section}{References}{26}{section*.31}% diff --git a/latex/mcmthesis.cls b/latex/mcmthesis.cls new file mode 100644 index 0000000..b03640c --- /dev/null +++ b/latex/mcmthesis.cls @@ -0,0 +1,352 @@ +%% +%% This is file `mcmthesis.cls', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `class') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\NeedsTeXFormat{LaTeX2e}[1999/12/01] +\ProvidesClass{mcmthesis} + [2020/01/18 v6.3 The Thesis Template Designed For MCM/ICM] +\typeout{The Thesis Template Designed For MCM/ICM} +\def\MCMversion{v6.3} +\RequirePackage{xkeyval} +\RequirePackage{etoolbox} +\define@boolkey{MCM}[MCM@opt@]{CTeX}[false]{} +\define@boolkey{MCM}[MCM@opt@]{titlepage}[true]{} +\define@boolkey{MCM}[MCM@opt@]{abstract}[true]{} +\define@boolkey{MCM}[MCM@opt@]{sheet}[true]{} +\define@boolkey{MCM}[MCM@opt@]{titleinsheet}[false]{} +\define@boolkey{MCM}[MCM@opt@]{keywordsinsheet}[false]{} +\define@cmdkeys{MCM}[MCM@opt@]{tcn,problem} +\define@key{MCM}{tcn}[0000]{\gdef\MCM@opt@tcn{#1}} +\define@key{MCM}{problem}[A]{\gdef\MCM@opt@problem{#1}} +\setkeys{MCM}{tcn=0000,problem=B} + +\define@key{mcmthesis.cls}{tcn}[0000]{\gdef\MCM@opt@tcn{#1}} +\define@key{mcmthesis.cls}{problem}[A]{\gdef\MCM@opt@problem{#1}} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{titlepage}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{abstract}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{sheet}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{titleinsheet}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{keywordsinsheet}{} +\MCM@opt@sheettrue +\MCM@opt@titlepagetrue +\MCM@opt@titleinsheetfalse +\MCM@opt@keywordsinsheetfalse +\MCM@opt@abstracttrue +\newcommand{\mcmsetup}[1]{\setkeys{MCM}{#1}} +\ProcessOptionsX\relax +\LoadClass[a4paper, 12pt]{article} +\newcommand{\team}{Team \#\ \MCM@opt@tcn} +\RequirePackage{fancyhdr, fancybox} +\RequirePackage{ifthen} +\RequirePackage{lastpage} +\RequirePackage{listings} +\RequirePackage[toc, page, title, titletoc, header]{appendix} +\RequirePackage{paralist} +\RequirePackage{amsthm, amsfonts} +\RequirePackage{amsmath, bm} +\RequirePackage{amssymb, mathrsfs} +\RequirePackage{latexsym} +\RequirePackage{longtable, multirow, hhline, tabularx, array} +\RequirePackage{flafter} +\RequirePackage{pifont, calc} +\RequirePackage{colortbl, booktabs} +\RequirePackage{geometry} +\RequirePackage[T1]{fontenc} +\RequirePackage[scaled]{berasans} +\RequirePackage{hyperref} +\RequirePackage{ifpdf, ifxetex} +\ifMCM@opt@CTeX +\else + \RequirePackage{environ} +\fi +\ifpdf + \RequirePackage{graphicx} + \RequirePackage{epstopdf} +\else + \ifxetex + \RequirePackage{graphicx} + \else + \RequirePackage[dvipdfmx]{graphicx} + \RequirePackage{bmpsize} + \fi +\fi +\RequirePackage[svgnames]{xcolor} +\ifpdf + \hypersetup{hidelinks} +\else + \ifxetex + \hypersetup{hidelinks} + \else + \hypersetup{dvipdfm, hidelinks} + \fi +\fi +\geometry{a4paper, margin = 1in} +\pagestyle{fancy} +\fancyhf{} +\lhead{\small\sffamily \team} +\rhead{\small\sffamily Page \thepage\ of \pageref{LastPage}} +% \rhead{\small\sffamily Page \thepage} +\setlength\parskip{.5\baselineskip} +\renewcommand\tableofcontents{% + \centerline{\normalfont\Large\bfseries\sffamily\contentsname + \@mkboth{% + \MakeUppercase\contentsname}{\MakeUppercase\contentsname}}% + \vskip 5ex% + \@starttoc{toc}% + } +\setcounter{totalnumber}{4} +\setcounter{topnumber}{2} +\setcounter{bottomnumber}{2} +\renewcommand{\textfraction}{0.15} +\renewcommand{\topfraction}{0.85} +\renewcommand{\bottomfraction}{0.65} +\renewcommand{\floatpagefraction}{0.60} +\renewcommand{\figurename}{Figure} +\renewcommand{\tablename}{Table} +\graphicspath{{./}{./img/}{./fig/}{./image/}{./figure/}{./picture/} + {./imgs/}{./figs/}{./images/}{./figures/}{./pictures/}} +\def\maketitle{% + \let\saved@thepage\thepage + \let\thepage\relax + \ifMCM@opt@sheet + \makesheet + \fi + \newpage + \ifMCM@opt@titlepage + \MCM@maketitle + \fi + \newpage + \let\thepage\saved@thepage + \setcounter{page}{2} + \pagestyle{fancy} +} +\def\abstractname{\large{Summary}} +\ifMCM@opt@CTeX + \newbox\@abstract% + \setbox\@abstract\hbox{}% + \long\def\abstract{\bgroup\global\setbox\@abstract\vbox\bgroup\hsize\textwidth\leftskip1cm\rightskip1cm}% + \def\endabstract{\egroup\egroup} + \def\make@abstract{% + \begin{center} + \textbf{\abstractname} + \end{center} + \usebox\@abstract\par + } +\else + \RenewEnviron{abstract}{\xdef\@abstract{\expandonce\BODY}} + \def\make@abstract{% + \begin{center} + \vspace*{-0.4cm}% + \textbf{\abstractname} + \end{center} + \vspace*{-0.4cm}% + \@abstract\par + } +\fi +\newenvironment{letter}[1]{% + \par% + \bgroup\parindent0pt% + \begin{minipage}{5cm} + \flushleft #1% + \end{minipage}} + {\egroup\smallskip} + +\def\keywordsname{Keywords} +\ifMCM@opt@CTeX + \newbox\@keywords + \setbox\@keywords\hbox{} + \def\keywords{\global\setbox\@keywords\vbox\bgroup\noindent\leftskip0cm} + \def\endkeywords{\egroup}% + \def\make@keywords{% + \par\hskip.4cm\textbf{\keywordsname}: \usebox\@keywords\hfill\par + } +\else + \NewEnviron{keywords}{\xdef\@keywords{\expandonce\BODY}} + \def\make@keywords{% + \par\noindent\textbf{\keywordsname}: + \@keywords\par + } +\fi +\newcommand{\headset}{{\the\year}\\MCM/ICM\\Summary Sheet} +\newcommand{\problem}[1]{\mcmsetup{problem = #1}} +\def\makesheet{% + \pagestyle{empty}% + \null% + \vspace*{-5pc}% + \begin{center} + \begingroup + \setlength{\parindent}{0pt} + \begin{minipage}[t]{0.33\linewidth} + \bfseries\centering% + Problem Chosen\\[0.7pc] + {\Huge\textbf{\MCM@opt@problem}}\\[2.8pc] + \end{minipage}% + \begin{minipage}[t]{0.33\linewidth} + \centering% + \textbf{\headset}% + \end{minipage}% + \begin{minipage}[t]{0.33\linewidth} + \centering\bfseries% + Team Control Number\\[0.7pc] + {\Huge\textbf{\textcolor{red}{\MCM@opt@tcn}}}\\[2.8pc] + % {\Huge\textbf{\MCM@opt@tcn}}\\[2.8pc] + \end{minipage}\par + \rule{\linewidth}{1.0pt}\par + \endgroup + % \vskip 10pt% + \ifMCM@opt@titleinsheet + \normalfont \LARGE \@title \par + \fi + \end{center} +\ifMCM@opt@keywordsinsheet + \make@abstract + \make@keywords +\else + \make@abstract +\fi} +\newcommand{\MCM@maketitle}{% + \begin{center}% + \let \footnote \thanks + {\LARGE \@title \par}% + \vskip 1.5em% + {\large + \lineskip .5em% + \begin{tabular}[t]{c}% + \@author + \end{tabular}\par}% + \vskip 1em% + {\large \@date}% + \end{center}% + \par + \vskip 1.5em% + \ifMCM@opt@abstract% + \make@abstract + \make@keywords + \fi% +} +\def\MCM@memoto{\relax} +\newcommand{\memoto}[1]{\gdef\MCM@memoto{#1}} +\def\MCM@memofrom{\relax} +\newcommand{\memofrom}[1]{\gdef\MCM@memofrom{#1}} +\def\MCM@memosubject{\relax} +\newcommand{\memosubject}[1]{\gdef\MCM@memosubject{#1}} +\def\MCM@memodate{\relax} +\newcommand{\memodate}[1]{\gdef\MCM@memodate{#1}} +\def\MCM@memologo{\relax} +\newcommand{\memologo}[1]{\gdef\MCM@memologo{\protect #1}} +\def\@letterheadaddress{\relax} +\newcommand{\lhaddress}[1]{\gdef\@letterheadaddress{#1}} +\newenvironment{memo}[1][Memorandum]{% + \pagestyle{plain}% + \ifthenelse{\equal{\MCM@memologo}{\relax}}{% + % without logo specified. + }{% + % with logo specified + \begin{minipage}[t]{\columnwidth}% + \begin{flushright} + \vspace{-0.6in} + \MCM@memologo + \vspace{0.5in} + \par\end{flushright}% + \end{minipage}% + } + \begin{center} + \LARGE\bfseries\scshape + #1 + \end{center} + \begin{description} + \ifthenelse{\equal{\MCM@memoto}{\relax}}{}{\item [{To:}] \MCM@memoto} + \ifthenelse{\equal{\MCM@memofrom}{\relax}}{}{\item [{From:}] \MCM@memofrom} + \ifthenelse{\equal{\MCM@memosubject}{\relax}}{}{\item [{Subject:}] \MCM@memosubject} + \ifthenelse{\equal{\MCM@memodate}{\relax}}{}{\item [{Date:}] \MCM@memodate} + \end{description} + \par\noindent + \rule[0.5ex]{\linewidth}{0.1pt}\par + \bigskip{} +}{% + \clearpage + \pagestyle{fancy}% +} +\newtheorem{Theorem}{Theorem}[section] +\newtheorem{Lemma}[Theorem]{Lemma} +\newtheorem{Corollary}[Theorem]{Corollary} +\newtheorem{Proposition}[Theorem]{Proposition} +\newtheorem{Definition}[Theorem]{Definition} +\newtheorem{Example}[Theorem]{Example} +\renewcommand\section{\@startsection{section}{1}{\z@}% + {-0pt\@plus -.2ex \@minus -.2ex}% + {1pt \@plus .2ex}% + {\rmfamily\Large\bfseries}} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-0pt\@plus -.2ex \@minus -.2ex}% + {1pt \@plus .2ex}% + {\rmfamily\large\bfseries}} +\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% + {-.5ex\@plus -1ex \@minus -.2ex}% + {.25ex \@plus .2ex}% + {\rmfamily\normalsize\bfseries}} +\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}% + {1ex \@plus1ex \@minus.2ex}% + {-1em}% + {\rmfamily\normalsize}} + +\providecommand{\dif}{\mathop{}\!\mathrm{d}} +\providecommand{\me}{\mathrm{e}} +\providecommand{\mi}{\mathrm{i}} + +\definecolor{grey}{rgb}{0.8,0.8,0.8} +\definecolor{darkgreen}{rgb}{0,0.3,0} +\definecolor{darkblue}{rgb}{0,0,0.3} +\def\lstbasicfont{\fontfamily{pcr}\selectfont\footnotesize} +\lstset{% + % numbers=left, + % numberstyle=\small,% + showstringspaces=false, + showspaces=false,% + tabsize=4,% + frame=lines,% + basicstyle={\footnotesize\lstbasicfont},% + keywordstyle=\color{darkblue}\bfseries,% + identifierstyle=,% + commentstyle=\color{darkgreen},%\itshape,% + stringstyle=\color{black}% +} +\lstloadlanguages{C,C++,Java,Matlab,Mathematica} +\endinput +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis.cls'. diff --git a/latex/mcmthesis.synctex.gz b/latex/mcmthesis.synctex.gz new file mode 100644 index 0000000..189c605 Binary files /dev/null and b/latex/mcmthesis.synctex.gz differ diff --git a/latex/picture.aux b/latex/picture.aux new file mode 100644 index 0000000..b640121 --- /dev/null +++ b/latex/picture.aux @@ -0,0 +1,2 @@ +\relax +\gdef \@abspage@last{1} diff --git a/latex/picture.fdb_latexmk b/latex/picture.fdb_latexmk new file mode 100644 index 0000000..cf464aa --- /dev/null +++ b/latex/picture.fdb_latexmk @@ -0,0 +1,130 @@ +# Fdb version 4 +["pdflatex"] 1768725304.67421 "d:/XHJ/mathmo/MCM2026/20260116/vscode/picture.tex" "picture.pdf" "picture" 1768725305.74528 2 + "c:/texlive/texlive/2025/texmf-dist/fonts/map/fontname/texfonts.map" 1753232233 3524 cb3e574dea2d1052e39280babc910dc8 "" + "c:/texlive/texlive/2025/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm" 1753230324 1328 c834bbb027764024c09d3d2bf908b5f0 "" + 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I am going to ignore this key. + +See the pgfkeys package documentation for explanation. +Type H for immediate help. + ... + +l.19 \node[variable] + (ntotal) {$N_{total} = 730$}; +This error message was generated by an \errmessage +command, so I can't give any explicit help. +Pretend that you're Hercule Poirot: Examine all clues, +and deduce the truth by order and method. + +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 19. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 19. + +d:/XHJ/mathmo/MCM2026/20260116/vscode/picture.tex:20: Package pgfkeys Error: The key '/tikz/variable' requires a value. I am going to ignore this key. + +See the pgfkeys package documentation for explanation. +Type H for immediate help. + ... + +l.20 \node[variable, below=of ntotal] + (kbase) {$k_{base} \in [5, 10]$}; +(That was another \errmessage.) + +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <9> on input line 23. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <6> on input line 23. + +d:/XHJ/mathmo/MCM2026/20260116/vscode/picture.tex:26: Package pgfkeys Error: The key '/tikz/variable' requires a value. I am going to ignore this key. + +See the pgfkeys package documentation for explanation. +Type H for immediate help. + ... + +l.26 \node[variable, right=of nfree] + (kie) { +(That was another \errmessage.) + +d:/XHJ/mathmo/MCM2026/20260116/vscode/picture.tex:27: Undefined control sequence. +l.27 $k_{i_e} = k_{base} + \text + {round}\left(N_{free} \cdot \frac{D_... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +d:/XHJ/mathmo/MCM2026/20260116/vscode/picture.tex:40: Package pgfkeys Error: The key '/tikz/variable' requires a value. I am going to ignore this key. + +See the pgfkeys package documentation for explanation. +Type H for immediate help. + ... + +l.40 \node[variable, left=of score] + (gini) { +(That was another \errmessage.) + + +d:/XHJ/mathmo/MCM2026/20260116/vscode/picture.tex:63: Package pgfkeys Error: I do not know the key '/tikz/fit', to which you passed '(ntotal) (kbase) (kie)', and I am going to ignore it. Perhaps you misspelled it. + +See the pgfkeys package documentation for explanation. +Type H for immediate help. + ... + +l.63 ... (fairbox) [fit=(ntotal) (kbase) (kie)] + {}; +(That was another \errmessage.) + + +d:/XHJ/mathmo/MCM2026/20260116/vscode/picture.tex:67: Package pgfkeys Error: I do not know the key '/tikz/fit', to which you passed '(eff) (score) (gini) (judge)', and I am going to ignore it. Perhaps you misspelled it. + +See the pgfkeys package documentation for explanation. +Type H for immediate help. + ... + +l.67 ...effbox) [fit=(eff) (score) (gini) (judge)] + {}; +(That was another \errmessage.) + + + +[1 + + +Non-PDF special ignored! + papersize=538.80806pt,247.52835pt +{c:/texlive/texlive/2025/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] (./picture.aux) + *********** +LaTeX2e <2024-11-01> patch level 2 +L3 programming layer <2025-01-18> + *********** + ) +Here is how much of TeX's memory you used: + 13523 strings out of 473190 + 288281 string characters out of 5715811 + 666049 words of memory out of 5000000 + 36561 multiletter control sequences out of 15000+600000 + 560968 words of font info for 43 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 117i,8n,121p,508b,855s stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on picture.pdf (1 page, 152892 bytes). +PDF statistics: + 86 PDF objects out of 1000 (max. 8388607) + 52 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 13 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/latex/picture.pdf b/latex/picture.pdf new file mode 100644 index 0000000..00c048f Binary files /dev/null and b/latex/picture.pdf differ diff --git a/latex/picture.synctex.gz b/latex/picture.synctex.gz new file mode 100644 index 0000000..51a3713 Binary files /dev/null and b/latex/picture.synctex.gz differ diff --git a/latex/picture.tex b/latex/picture.tex new file mode 100644 index 0000000..52c4dd8 --- /dev/null +++ b/latex/picture.tex @@ -0,0 +1,72 @@ +\documentclass[tikz,border=10pt]{standalone} +\usetikzlibrary{shapes.geometric, arrows.meta, positioning, calc, backgrounds} + +\begin{document} + +\begin{tikzpicture}[ + node distance=1.5cm and 2cm, + % 设置节点样式 + base/.style={rectangle, rounded corners, draw=black, minimum width=2.5cm, minimum height=1cm, text centered, font=\small, fill=white}, + variable/.style={base, fill=blue!10, draw=blue!50!black}, + process/.style={base, fill=orange!10, draw=orange!50!black}, + decision/.style={diamond, aspect=2, draw=red!50!black, fill=red!5!white, font=\scriptsize, inner sep=0pt}, + formula/.style={font=\scriptsize, text=black!80}, + % 设置区域样式 + container/.style={draw, dashed, inner sep=15pt, rounded corners} +] + + % --- 1. Fairness Allocation Part --- + \node[variable] (ntotal) {$N_{total} = 730$}; + \node[variable, below=of ntotal] (kbase) {$k_{base} \in [5, 10]$}; + + \node[process, right=of $(ntotal.south)!0.5!(kbase.north)$] (nfree) { + $N_{free} = N_{total} - 70 \cdot k_{base}$ + }; + + \node[variable, right=of nfree] (kie) { + $k_{i_e} = k_{base} + \text{round}\left(N_{free} \cdot \frac{D_i}{\sum D_i}\right)$ + }; + + % --- 2. Effectiveness Scoring Part --- + \node[process, below=2cm of kie] (eff) { + $annual\_eff_i = k_{i_e} \cdot \min(d_i, d_0)$ + }; + + \node[process, below=1cm of eff] (score) { + $score_i = \frac{eff_i}{D_i} - \alpha \cdot unmet_i - \beta \cdot waste_i$ + }; + + % --- 3. Validation Part --- + \node[variable, left=of score] (gini) { + $G = \frac{\sum \sum |score_i - score_j|}{2n^2 \overline{score}}$ + }; + + \node[decision, left=of gini] (judge) {$G < 0.2$?}; + + % --- 连线 --- + \draw[-Stealth, thick] (ntotal) -- (nfree); + \draw[-Stealth, thick] (kbase) -- (nfree); + \draw[-Stealth, thick] (nfree) -- node[above, formula] {Demand Ratio Weight} (kie); + \draw[-Stealth, thick] (kie) -- node[right, formula] {Supply-Demand Alignment} (eff); + \draw[-Stealth, thick] (eff) -- (score); + \draw[-Stealth, thick] (score) -- (gini); + \draw[-Stealth, thick] (gini) -- (judge); + + % 回馈线 + \draw[-Stealth, thick] (judge.north) -- ++(0, 1.5) -| node[pos=0.2, above, formula] {Iterative Optimization (maximize total score)} (kbase.west); + \draw[-Stealth, thick] (judge.south) -- ++(0, -0.5) -- ++(8, 0) |- node[pos=0.2, below, formula] {Final $k_{i_e}$ Allocation} (kie.east); + + % --- 绘制背景框 (模仿原图的 River/Ocean 分区) --- + \begin{scope}[on background layer] + % 公平分配区 + \node[container, draw=green!60!black, label={[green!60!black]above:\textbf{Tier 1: Dual-Fairness Allocation}}] + (fairbox) [fit=(ntotal) (kbase) (kie)] {}; + + % 效能与验证区 + \node[container, draw=blue!60!black, label={[blue!60!black]below:\textbf{Tier 2: Effectiveness \& Equity Validation}}] + (effbox) [fit=(eff) (score) (gini) (judge)] {}; + \end{scope} + +\end{tikzpicture} + +\end{document} \ No newline at end of file diff --git a/latex/reference.bib b/latex/reference.bib new file mode 100644 index 0000000..0fe0327 --- /dev/null +++ b/latex/reference.bib @@ -0,0 +1,72 @@ +@misc{penoyre2019spacelinepracticalspaceelevator, + title={The Spaceline: a practical space elevator alternative achievable with current technology}, + author={Zephyr Penoyre and Emily Sandford}, + year={2019}, + eprint={1908.09339}, + archivePrefix={arXiv}, + primaryClass={astro-ph.IM}, + +} + +@misc{skyrocket2026falcon9, +title={Falcon-9}, +author={{Gunter's Space Page}}, +year={2026}, +howpublished={\url{https://space.skyrocket.de/doc_lau/falcon-9.htm}}, +note={Accessed: 2026-02-01}, +organization={Gunter's Space Page} +} + +@inproceedings{whitmore2012nasa, + title={NASA-STD-3001, space flight human-system standard and the human integration design handbook}, + author={Whitmore, Mihriban and Boyer, Jennifer and Holubec, Keith}, + booktitle={Industrial and systems engineering research conference}, + number={JSC-CN-25695}, + year={2012} +} +@article{francisco2025nasa, + title={NASA-STD-3001 Volume 2, Revision E, NASA Spaceflight Human-System Standard Volume 2: Human Factors, Habitability, and Environmental Health}, + author={Francisco, David R}, + journal={N/A}, + number={A}, + year={2025}, + publisher={NASA} +} +@article{LIU2021113, +title = {Review of research into bioregenerative life support system(s) which can support humans living in space}, +journal = {Life Sciences in Space Research}, +volume = {31}, +pages = {113-120}, +year = {2021}, +issn = {2214-5524}, +doi = {https://doi.org/10.1016/j.lssr.2021.09.003}, +url = {https://www.sciencedirect.com/science/article/pii/S2214552421000663}, +author = {Hong Liu and Zhikai Yao and Yuming Fu and Jiajie Feng} + +} + +@article{swan2013space, + title={Space Elevators: An Assessment of the Technological Feasibility and the Way Forward}, + author={Swan, Peter A and Raitt, David I and Knapman, John M and Penny, Robert E}, + journal={International Academy of Astronautics}, + year={2013} +} + +@techreport{isec2021greenroad, + author = {Gassent, Michel and Barry, Kevin and Swan, Peter}, + title = {Space Elevators: The Green Road to Space}, + institution = {International Space Elevator Consortium (ISEC)}, + year = {2021}, + type = {ISEC Study Report}, + url = {https://isec.org/wp-content/uploads/2021/11/ISEC-2021-Study-Report-Green-Road-to-Space-Final.pdf}, + note = {Accessed: 2023-10-27}, + address = {Santa Barbara, CA} +} + +@misc{chatgpt, + author = {OpenAI}, + title = {GPT}, + howpublished = {5.2}, + note = {2025-02-01}, + year = {2025} +} diff --git a/latex/remember.tex b/latex/remember.tex new file mode 100644 index 0000000..999f5f5 --- /dev/null +++ b/latex/remember.tex @@ -0,0 +1,61 @@ +%图 +\begin{figure}[H] +\centering +\includegraphics[width=12cm]{flowchartmodel1.png} +\caption{} +\label{} +\end{figure} + +%表 + +\begin{table}[H]\small + \centering + \caption{Truncation Correction for Key High-Demand Sites} + \label{tab:truncation_correction} + \begin{tabular}{lcccc} + \toprule + Site Name & $\mu$ & $p_{trunc}$&$\tilde{\mu}$ & Correction Magnitude \\ + \midrule + MFP Waverly & 396.6 & 81.57\% & 429.0 & +8.2\% \\ + MFP Avoca & 314.6 & 26.84\% & 323.0 & +2.7\% \\ + MFP Endwell United Methodist & 285.3 & 14.34\% & 289.3 & +1.4\% \\ + MFP College TC3 - College & 261.5 & 16.80\% & 265.9 & +1.7\% \\ + MFP Redeemer Lutheran Church & 230.6 & 10.07\% & 232.9 & +1.0\% \\ + \bottomrule + \end{tabular} +\end{table} + +%有颜色的表 + + +\begin{table}[H] + \small + \centering + \caption{Performance Comparison of Different Allocation Schemes } + \label{tab:allocation_performance} + % 缩放至单栏宽度，保留1em内边距避免贴边 + \resizebox{\dimexpr\linewidth-1em\relax}{!}{ + \begin{tabular}{lllll} + \toprule[1pt] % 加粗顶线 + \rowcolor{white} % 表头底色为白 + Allocation Scheme & E1 & E2& F1$\downarrow$ & F2 $\uparrow$\\ + \midrule[0.6pt] + \rowcolor{recommendColor} % 推荐方案浅蓝底色 + Recommended Scheme ($\tilde{\mu}$ Proportional) & 139,469 & 131,462 & 0.314 & 2.00 \\ + \rowcolor{baselineColor} % 基线方案浅灰底色 + Baseline 1: Uniform Allocation & 104,797 & 101,309 & 0.024 & 9.25 \\ + \rowcolor{baselineColor} + Baseline 2: 2019 Historical Scaling & 104,071 & 100,264 & 0.091 & 5.00 \\ + \rowcolor{baselineColor} + Baseline 3: Raw Demand Proportion (Raw $\mu$) & 139,129 & 131,397 & 0.313 & 2.00 \\ + \bottomrule[1pt] % 加粗底线 + \end{tabular} + } +\end{table} + +%点 + +\begin{itemize} + \item \textbf{Survivorship Bias in Observed Data}: The table and Figure \ref{Truncation Correction for High-Demand Sites} reveal "survivorship bias" in observed data. For MFP Waverly, its 81.57\% truncation probability means the site operated nearly permanently at full capacity historically—its observed mean reflects capacity limits, not the ceiling of true demand. + \item \textbf{Significance of Correction}: Via the truncated normal model, the site’s potential demand is adjusted upward to 429.0. This ensures resource allocation is based on "actual community needs" , resolving the underestimation of high-demand areas due to insufficient past provision. +\end{itemize} \ No newline at end of file diff --git a/latex/run.R b/latex/run.R new file mode 100644 index 0000000..a3750dd --- /dev/null +++ b/latex/run.R @@ -0,0 +1,3 @@ +library(tinytex) + +pdflatex("mcmthesis-demo.tex", bib_engine = "biber") diff --git a/latex/table2latex.txt b/latex/table2latex.txt new file mode 100644 index 0000000..d0491bb --- /dev/null +++ b/latex/table2latex.txt @@ -0,0 +1 @@ +http://www.tablesgenerator.com/ \ No newline at end of file diff --git a/latex/初稿.md b/latex/初稿.md new file mode 100644 index 0000000..f29f704 --- /dev/null +++ b/latex/初稿.md @@ -0,0 +1,539 @@ + +**摘要** + +本文针对2050年建设容纳10万人的月球殖民地的物资运输问题，构建了一套从地月运输决策到随机风险管控，再到多目标生态平衡的综合评价模型体系,并给出对 MCM 机构的可执行建议。 + +首先，我们构建了 **基于广义能量等效与时序协同的运输成本评估模型（UETCM）**，引入“通用能量等价（UEE）”作为统一成本度量，避免了货币波动的影响，实现了火箭与电梯方案的能量可比性。模型基于齐奥尔科夫斯基火箭方程和空间电梯势能-动能转换机制，推导出运输能量消耗与载货质量的线性关系。为找到能量与时间之间的最优权衡点，引入时间机会成本概念，构建总系统负担函数 \(J(T)\)。通过归一化权重均衡、边际效率分析、相变边界识别与可行域几何中心法，校准出时间机会成本 \(\lambda = 504 \, \text{PJ/年}\)，从而实现多目标转化。 + +模型比较了三种运输情景,结果显示，混合运输能在 **100.7 年** 内完成1亿吨物资运输，显著优于纯电梯（186.2年）和纯火箭（219.2年）。在 \(\lambda = 504\) 下，**139年平衡策略（Strategy C）** 被识别为全局最优解。 + +为应对系统不确定性，我们在原有基础上构建了 **随机风险与鲁棒性分析框架**，引入条件风险价值（CVaR）扩展目标函数，并采用蒙特卡洛模拟迭代10,000次量化扰动影响。为确保95%的完工置信度，最优工期应延长至 **155–160年**，电梯份额宜调整至 **65%–70%**，并保留10%的运输能力作为战略储备。 + +其次，针对殖民地运行阶段的水资源保障问题，我们建立了 **生命支持物流与随机水平衡模型（LSL-SWBM）**。在需求方面，综合考虑了基于舒适度的多层次用水需求与随机医疗紧急需求。推荐采用舒适标准（\(\alpha=50\)）下的**混合方案4（电梯+低纬度火箭站点）**，可在4.57天内完成初始补水，3.43天完成每月常规补水，能耗适中。 + +接着，我们将环境足迹作为关键约束纳入决策，构建 **时间-能量-排放三目标优化框架**。分析表明，纯火箭方案会导致显著的二氧化碳与平流层水蒸气排放，而空间电梯几乎实现零排放。因此推荐采用 **186 年纯电梯方案** 作为最优可持续策略。该方案环境效益极为显著，**总二氧化碳排放量仅为 846 Mt**，不足火箭方案的 **6.5%**。二氧化碳排放强度仅为 **4.5 Mt/年**，远低于其他方案。 + + +最后，在一页建议信中给出带 KPI 触发的分阶段混合运输方案，以在可控风险下实现低排放与可兑现工期。 + +## # 1.1background + +随着人类对地外空间的探索从短期任务式登陆向长期永久定居转型，地球有限的资源与脆弱的生态系统正促使我们寻找地外栖息地，以实现文明的多行星化。但传统化学动力火箭存在载荷比低、发射成本高、化学污染不可逆等短板，难以支撑大规模月球殖民地建设的海量物资运输需求。在此背景下，空间电梯系统作为革命性的星际运输基础设施应运而生，将协同月球轨道接驳系统，构建起一条绿色、高效且可持续的星际物流链。 + +大规模月球殖民地的构建，既是对星际物流运输能力的极致考验，更是对生命保障需求持续供给的严峻挑战。在月球这一严酷环境中，十万居民的日常消耗构成了物流的长期需求。当我们将传统火箭与新兴空间电梯系统进行对比时，必须权衡能耗成本、时间进度、发射窗口限制、环境负荷以及系统故障带来的风险成本。这一研究既关乎月球殖民工程的可行性与经济性，也对未来星际物流体系的构建与地球生态保护具有重要意义。 + +## # 4.模型 I：基于广义能量等效与时序协同的运输成本评估模型 (Universal Energy-Equivalent and Temporal Coordination Model) + + +## 4.1 模型概述 (Model Overview) + +为了量化评估 2050 年后大规模月球殖民地建设的物流可行性，本文构建了广义能量等效与时序协同评估模型 (UETCM)。该模型放弃了易受宏观经济波动干扰的货币价值尺度，创新性地确立了以焦耳（Joule）为度量的广义能量等效 (Universal Energy-Equivalent, UEE) 作为核心评估标准。这一跨维度的基准锚定，不仅消除了跨越数十年建设周期的通货膨胀影响，更在热力学本质上实现了化学火箭动量转化与空间电梯势能增益这两类异构运输技术的公允对标。 + +在模型构建与求解过程中，我们首先基于齐奥尔科夫斯基方程与引力场能量梯度，推导出了载荷质量与能源消耗间的线性映射关系。考虑到现实物理约束，模型引入了地理纬度修正算子与发射频次阈值，提出“低纬度优先”原则，揭示了地球自转赋予的切向初速度对物流能效的非线性贡献。针对工期与成本的天然矛盾，模型利用帕累托前沿 (Pareto Front) 刻画了混合运输模式下的战略博弈空间，并通过最大距离法 (Maximum Distance Method) 锁定了兼顾建设时效与能源经济性的“多目标优化膝点”，即 139 年的黄金工期。 + +最后，为修正理想化模型的偏差，我们将 UETCM 扩展为包含扰动因子的随机修正模型。通过 10,000 次蒙特卡洛 (Monte Carlo) 仿真，系统性地模拟了缆索摆动导致的变轨能耗、发射场气象干扰、电梯检修、火箭失效带来的进度停滞。研究不仅量化了系统在“非完美运行”下的性能退化，更识别出电梯妥善率与气象窗口是制约星际物流链韧性的核心变量。该模型为未来地月物资运输体系的构建提供了一套从底层物理机理到高层战略决策的系统性评估框架。 +### 4.2 广义能量等效（UEE）的成本表征能力论证 +在传统的经济模型中，成本通常以货币计量。然而，针对跨越数十年的地月物流链，货币受通胀及地缘政治影响具有极大的不确定性。本文提出：在成熟的运输系统中，能源消耗是经济成本的最优代理变量。为证实这一公理的有效性，本节从成本结构、行业数据及统计相关性三个维度进行多重论证。 + +#### 4.2.1 成本结构的收敛性分析 (Cost Structure Convergence) +以下是为你精简后的版本，保留了核心数学逻辑与学术严谨性，同时去除了冗余表述： + +#### 4.2.1 成本结构的收敛性分析 (Cost Structure Convergence) +任何运输系统的单次运营成本 $Cost(x)$ 可建模为硬件摊销、能源消耗与运维成本之和： +$$ Cost(x) = \frac{C_{hardware}}{x} + C_{energy} + C_{operations} $$ +其中 $x$ 为运输工具的复用次数。 + +在 2050 年后的成熟运输范式下，随着复用次数 $x$ 实现量级增长，单位载荷的硬件摊销 $C_{hardware}/x$ 迅速稀释并趋于零。同时，自动化维护技术的普及使 $C_{operations}$ 进入边际递减区间。此时，系统总成本将展现出向**热力学底线收敛**的特性：即能源消耗 $C_{energy}$ 逐渐演变为驱动总成本的绝对主导因素。 + +这种收敛逻辑证明，在技术成熟期，能源效率是决定边际成本的唯一核心路径。 +#### 4.2.2 实际行业数据的实证支撑 (Empirical Evidence) + +航空业作为目前人类最成熟的大规模运输系统，其数据具有显著的参考价值。如图，分析 1995-2018 年美国国内航空业运营数据发现，即使在油价波动的复杂环境下，燃油成本仍稳定占据总运营成本的 15%-25%，证明能源是成熟运输系统的核心成本驱动因素。 + + +![燃油成本占比趋势]() + +为进一步量化这种关联，本文绘制总成本与能源成本的散点图并引入皮尔逊相关系数 (Pearson Correlation Coefficient) 进行验证： +$$ r = \frac{\sum (C_{total}-\bar{C}_{total})(C_{energy}-\bar{C}_{energy})}{\sqrt{\sum (C_{total}-\bar{C}_{total})^{2}}\sqrt{\sum (C_{energy}-\bar{C}_{energy})^{2}}} $$ + + +![能源-成本相关性分析]() + +计算结果显示，总成本与能源成本的散点分布呈现极强的线性趋势，相关系数 $r$ 达到 **0.92**。这证明在成熟的运输范式下，捕获了能量消耗即捕获了成本的核心变动。 + +**4.2.3 选取 UEE 作为度量衡的科学优势** +基于上述论证，本文确立 UEE 为核心评估标准，其优势体现在以下维度： +- **经济恒定性 (Economic Invariance)**：焦耳（Joule）作为物理常数，能够提供跨越时空的稳健评估，规避了长期任务中的货币贬值风险。 +- **异构技术对标 (Standardized Comparison)**：UEE 指标打破了化学火箭与空间电梯的物理界限，将异构能源消耗统一映射至热力学本质。 +- **物理约束锚定 (Gravitational Constraint)**：跨星际物流的本质是克服引力势阱。能量消耗直接量化了突破引力束缚的物理难度，比货币更能精准反映引力位能变化与载荷间的非线性映射关系。 +- **环境可持续性 (Sustainability)**：在多行星文明进程中，能源获取效率（Low Entropy）是制约大规模建设的首要瓶颈，符合低碳星际文明的发展逻辑。 + +### ## 4.2 理想下的能源成本建模 (Ideal Energy Cost Modeling) +在确立了 UEE 作为成本代理变量后，本节将针对两种运输范式建立数学映射，定量计算将 1 亿公吨物资运至月球的物理代价。 + + +#### ### 4.2.1 传统火箭直接运输：基于动量转化的化学能消耗分析 + +对于传统火箭方案，其成本的核心取决于将货物送出地球引力势井所需的化学燃料质量。我们首先利用齐奥尔科夫斯基火箭方程（Tsiolkovsky Equation）来探讨燃料消耗与有效载荷之间的内在关联。 + +- Step I: 基于齐奥尔科夫斯基方程的质量比约束分析 + +考虑火箭在忽略空气阻力情况下的速度增量 $\Delta v$。为便于分析，我们定义质量比 $R$ 为初始质量 $m_0$ 与末端质量 $m_f$ 之比，则有： + +$$ R = \frac{m_0}{m_f} = e^{\Delta V / v_e} $$ + +其中 $v_e$ 为火箭喷气速度。在给定的近地轨道（LEO）或地月转移任务中，$\Delta V$ 由轨道力学决定，可视为常数。 + +- Step II: 燃料消耗与有效载荷的线性映射推导 + +由于火箭并非单纯由有效载荷组成，还包含结构质量如燃料箱、发动机等，我们引入结构系数 $\alpha$。定义其为结构质量与燃料质量之比，即 $m_s = \alpha \cdot m_{fuel}$。通过对总质量构成的代数推导，可以得出燃料质量 $m_{fuel}$ 与有效载荷 $m_p$ 的精确比例： + +$$m_{fuel} = \underbrace{\frac{R-1}{1-\alpha(R-1)}}_{k} \cdot m_p$$ + +通过这一公式，我们将复杂的飞行过程提炼为一个比例系数 $k$。根据燃料释放的理想化学能公式： + +$$ E_{rocket} = \frac{1}{2} m_{fuel} \cdot v_e^2 = \frac{1}{2}k\cdot v_e^2 \cdot m_p $$ + +这表明，在传统火箭模式下，**能源成本是有效载荷的线性函数**。 + +- Step III: 地月全过程引力位能变换核算 + +为了完成全过程成本评估，我们进一步考虑势能变化。全过程能量消耗 $\Delta E$ 可视为克服地球引力势能、月球引力补偿及动能转换的总和： + +$$ \Delta E_{total} = m_p \cdot \left[ \left( \frac{G M_E}{R_E} - \frac{G M_E}{d_{EM}} \right) - \frac{G M_M}{2 r_M} \right] $$ + +公式中的各项分别代表了从地表逃逸、地月相对位置势能差以及进入月球轨道所需的能级跃迁。综合上式可知**火箭运输下总能耗与有效载荷质量成线性正比关系**。 + +#### ### 4.2.2 空间电梯运输：基于引力梯度与转动惯量的能量增益模型 + +与火箭直达相比，空间电梯方案的本质差异只在第一段：货物先由地面港口沿缆索被送到顶点锚，再从顶点锚装载火箭飞往月球。由于从顶点锚至月球的火箭段仍可用上一小节的火箭模型处理，本节只建立电梯段的能量增益模型。 + +货物由地面港口（Earth Port）运送至顶点锚（Apex Anchor）的过程中，其能量变化主要由三部分组成：克服地球引力的位能提升、随高度增加的向心动能增益，以及在顶点锚释放时的切向动能。其能量输入公式表达为： + +$$ E_{elevator} = m_p \left[ \left( \frac{GM_E}{R_E} - \frac{GM_E}{r_{0}} \right) + \frac{1}{2}\omega^2 r_{0}^2 \right] $$ + +其中，$r_0$ 为顶点锚至地心的距离，$\omega$ 为地球自转角速度，上式表明**电梯运输同样满足能耗与载荷的线性正比关系**。 + +### ## 4.3 理想下的时间进度与物流效率建模 (Ideal Timeline and Logistic Efficiency Modeling) + +在解决了单次运输的物理能效问题后，模型进入了对时间进度的求解阶段。这一任务的复杂性在于：1 亿公吨的总载荷并非瞬间送达，其时间进度受到全球发射场地理分布、发射频次上限以及空间电梯和火箭固有运力的多重博弈制约。 + +#### #### 4.3.1 运输进度计算公式 + +为了量化 2050 年后的建设时间线，模型定义 $M_{total} = 10^8$ 公吨为建设 10 万人殖民地所需的总物资需求。根据题目给出的三种情景模式，完成该宏大工程所需的时间进度 $t$ 取决于系统的**年度累计运力 $C$**。通过构建以下时序模型，我们可以得到不同情景下的完成时间： + +- **纯太空电梯情景（scenarios a）**: $t_{a}=\frac{M_{total}}{C_{e}}$ + +- **纯火箭情景（scenarios b）**: $t_{b}=\frac{M_{total}}{C_{r}}$ + +- **混合情景（scenarios c）**: $t_{c}=\frac{M_{total}}{\lambda C_{e}+ \mu C_{r} }$ + +其中 $C_e$ 为空间电梯系统的年吞吐量，$C_r$ 为火箭系统的年综合运力。上述公式揭示了一个关键逻辑，时间进度的缩短是以提高年度累计运力为代价的。通过调节混合权重 $\lambda$ 与 $\mu$，我们可以在建设速度与能源成本之间寻求动态平衡点。 + +#### #### 4.3.2 基于物理地理约束的物流效率优化 + +单纯计算理论工期是不够的，现实中影响 2050 年建设进度的最大阻碍是发射场的地理约束与频次限制。这直接决定了上述公式中 $C_r$ 的真实上界。 + +1. **频次对进度的刚性约束**：本文以全球 10 个现有发射场的历史发射记录及维护周期进行估计，得出在2050年后的理想状态下，每个站点每日至多可发射一枚火箭。这意味着火箭系统每年的总发射窗口是有上限的，这种运力天花板是决定工期下限的核心变量。 + +2. **地理纬度对进度的效率修正**：由于地球自转赋予了火箭一个切向初速度 $v_{0i}$，不同维度的发射站其能量增益不同，公式如下： + +$$ v_{0i}=v_{0E}\cdot\cos\theta_{i} $$ + +其中 $v_{0E}$ 为地球赤道自转线速度，$\theta_i$ 为所在发射站的纬度。公式说明，发射站的纬度越高，初速度越小，需要消耗更多化学燃料进行货物运输。 + +![](D:/XHJ/mathmo/MCM2026/20260130MCM/vscode/figures/latitude_effects.png) + +如上图所示，随着纬度从赤道向高纬度增加，切向速度损失呈非线性上升 。这直接导致了燃料/有效载荷比从赤道站点的 31.8 上升至高纬度站点的 34.0 。 + +因此，本模型设定了 “低纬度优先”原则 ：在火箭发射次数未达上限时，优先调用库鲁（Kourou）等低纬度站点以最小化全球总能耗。这一原则不仅是为了最小化能耗，更是为了通过优化站点分配，确保在给定的时间进度要求下，物资交付的效率达到最大化。 + + +### 4.4战略权衡与多目标优化 + +为了解决最小化总能耗与压缩建设周期之间的矛盾，我们超越了简单的几何拐点分析，引入了一个基于“基础设施时间价值”的经济框架。 + +#### 4.4.1时间机会成本分析 +在大型星际工程中，时间不仅是一个时间维度，更是一种昂贵的资源。我们引入了**时间机会成本参数 $\lambda$ (PJ/year)**，用于表示工期每增加一年所带来的隐含成本（以能量当量衡量）。决策目标转化为最小化系统总成本 $J$： +$$ + J = E_{total} + \lambda \cdot T +$$ +其中 $E_{total}$ 是总能耗，$T$ 是完工时间。理论上的最优工期 $T^*$ 满足一阶条件，即通过延长工期节省的边际能量等于时间的边际机会成本： +$$ + \frac{dE}{dT} = -\lambda +$$ + +#### 3.5.2 灵敏度分析与拐点识别 +我们进行了灵敏度分析，观察最优工期 $T^*$ 如何随时间估值 ($\lambda$) 的变化而移动。 + +![]() + +上图分析揭示了在 $\lambda = 480$ 至 $510$ PJ/year 之间的“关键转变”区： +* **稳定性平台（Plateaus）：** $T^*$ 与 $\lambda$ 的关系呈现阶梯状特征。这是因为运输系统由离散单元（发射场和电梯轨道）组成；只有当时间成本超过特定阈值时，模型才会转向更快但能耗更高的配置。 +* **拐点的形成：** 在 $\lambda \approx 497$ PJ/year 时，最优工期经历了一个剧烈的转变并稳定在 **139年**。这个139年的平台期代表了一个“稳健拐点”，系统在此达到了平衡。进一步延长工期带来的能量节省递减，而进一步压缩工期则需要极高的时间成本代价。 + +#### 3.5.3 选择139年方案的战略依据 +选择139年方案（对应 $\lambda \approx 500$ PJ/year）意味着每延迟一年，将产生相当于最小总能耗约 3.2% 的机会成本。这一估值由以下战略因素支持： +* **原位资源利用 (ISRU)：** 更早完工意味着能更早利用月球资源（如氦-3或水冰），其经济回报远超慢速施工节省的那点能量。 +* **维护与经常性开支：** 延长建设周期会增加维护地球任务控制中心和发射基础设施的累计成本。 +* **战略存在感：** 尽早建立10万人的殖民地，可以在多行星文明时代确保先发优势。 +* **投资的时间价值：** 类似于金融中的净现值 (NPV) 概念，月球定居点的收益会随时间打折；因此，优先考虑更快地进入运营阶段。 + +### ## 4.5 任务一求解结果 (Results of Task 1) + +通过对 1 亿公吨物资运输任务的数值模拟，本研究揭示了在不同技术约束下，月球殖民地建设的时间进度与资源消耗的内在关联。结果表明，最优运输策略并非单一手段的叠加，而是基于物理能效与时间成本权衡下的动态动态平衡。 + +#### #### 4.5.1 三种运输情景的量化指标对比 + +根据模型运行结果，表 1-1 汇总了三种基准情景的核心指标。分析表明，混合情景在填补可行性缺口方面具有不可替代性。 + +**表 1-1 地月物资运输情景多指标对比表** + +|情景类型|最短工期 (a)|总能耗 (PJ)|单位能耗 (GJ/t)|核心约束/特征| +|---|---|---|---|---| +|**纯太空电梯情景（scenarios a）**|186.2|15,720|157.2|受限于 53.7 万吨/年的固有运力瓶颈| +|**纯火箭情景（scenarios b）**|219.2|50,609|506.1|极高能耗，且受限于 10 站每日 1 发的频次| +|**混合情景（scenarios c）**|**100.7**|**31,537**|**315.4**|**协同增效，实现工期与能耗的二元优化**| +具体分析如下： + +1. **可行性缺口分析**：如图 [three_scenarios_comparison.png] 所示，在 [100.7, 186.2] 年的区间内，纯电梯情景因运力赤字无法达成目标，纯火箭情景则因发射频次限制而失效。混合情景是此时间窗口内唯一可落地的工程解。 + +2. **能效鸿沟**：纯火箭情景的总能耗是电梯情景的 **3.2 倍**。这一“能耗惩罚”主要源于化学动力系统必须携带“为运送燃料而存在的燃料”，而电梯系统通过电能驱动实现了质量-能量关系的线性化。 + +![]() + +#### #### 4.5.2 基于帕累托前沿的多目标决策分析 (Multi-objective Decision via Pareto Front) + +混合运输情景的本质是在时间效率与能源消耗之间寻求最优替代率。基于[100.7, 186]年间帕累托前沿有效解集，提取3种典型优化策略的核心量化指标，直观呈现工期、总能耗、电梯运力占比及节能率（相对纯火箭情景）的对应关系，具体如下表所示： + +|策略类型|工期 (a)|总能耗 (PJ)|电梯占比|节能率 (vs 纯火箭)| +|---|---|---|---|---| +|策略 A（成本优先）|186.0|15,720|100.0%|69.0%| +|策略 B（时间优先）|100.7|31,537|54.1%|37.7%| +|策略 C（综合平衡）|139.0|24,361|74.6%|51.9%| + +3种策略（Strategy）分别对应帕累托前沿上的关键节点：策略 A（成本优先）对应成本最优边界，采用电梯满负荷运行模式，实现全局最低能耗，为成本与能耗最优的极端方案；策略 B（时间优先）对应混合情景极限最短工期，通过最高能耗投入换取最短建设周期，是追求极致时间效率的极端方案；策略 C（综合平衡）对应多目标优化膝点，在工期与能耗之间实现最优权衡，是兼顾建设效率与能源成本的最优决策点，可为混合情景的运力配比战略选择提供核心量化支撑。 + +图 [pareto_combined_range.png] 展示了工期在 [100.7, 186] 年间的帕累托前沿分布。 + +1. **边际能效收益 (Marginal Energy Saving)**：由图 [combined_decision.png] 可见，边际节能曲线呈现阶梯状递减特征。在 101 至 139 年区间，每延长一年工期可节省约 **210 PJ** 能量。跨越139 年（多目标优化膝点）后，边际节能率降至 185 PJ/a 左右。这说明在 139 年处，时间投入换取的能源收益达到饱和。 + +2. **膝点识别与推荐**：利用最大距离法 (Max Distance Method) 识别出的 139 年作为全局推荐点。该点仅比极限最短工期增加 38% 的时间，却换取了 22.7% 的能耗降幅。此时电梯承担了 74.6% 的运输任务，实现了运力结构的高效配比。 + +![]() + +### ## 4.6 非完美运行下的随机修正模型 + +理想状态的模型提供了一个优化的理论上界。然而，真实的航天工程面临巨大的不确定性。为响应任务二，本节将 UETCM 扩展为包含扰动算子 $\delta$ 的随机模型。 + +#### ### 4.6.1 随机扰动因子的物理与逻辑量化 + +本文从能效偏差和进度中断两个维度，量化非完美运行带来的扰动，所有扰动因子均基于航天工程实际场景设定，确保量化结果的合理性与实用性： + +1. **机械扰动**：考虑长达 10万公里的缆索受科里奥利力影响产生的摆动 $\Delta \theta$。这会导致载荷释放速度矢量偏离，必须消耗额外变轨能量 $\Delta E$ 进行修正： + +$$ \Delta E = \frac{1}{2} (e^{\Delta v_{\Delta \theta} / v_e} - 1) v_e^2 m_p, \quad \Delta v_{\Delta \theta} = \omega r_0 \sin \Delta \theta $$ + +其中 $\Delta \theta \sim N(0, 0.5^\circ)$。 + +2. **可靠性扰动：**： + +* **火箭失效**：依据Falcon 9 历史发射日志\cite{skyrocket2026falcon9}本模型将 2050 年火箭运输的基准失效率设定为 0.5% - 1.0%，并以此作为蒙特卡洛模拟的初始输入参数。 + +* **电梯检修**：作为复杂大型设施，设定电梯年均故障 2 次，其停工维护时间服从 $N(14, 16)$ 天的分布。 + +3. **环境扰动：**：考虑到火箭发射对气象敏感，模型初步设定各发射站因恶劣天气导致的停工风险率为10%。 + +#### ### 4.6.2 蒙特卡洛全生命周期模拟 (Monte Carlo Simulation) + +由于上述随机变量的交互作用具有高度非线性，单纯的代数计算无法精准评估系统整体性能，因此本文采用蒙特卡洛算法，执行10,000次建设周期仿真，全面模拟非完美运行下的系统表现。 + +- **算法求解思路**：在模拟时间轴的每一日，算法自动抽样系统状态（正常/故障/恶劣天气）。若发生干扰，算法会自动调整当日有效载荷分发，并将未完成任务顺延。 + +- **结果分布分析**：如图，模拟结果显示，考虑不确定性后的完成时间呈现显著的“右偏分布”——这意味着，相比理想状态，现实中的工期更易受到各种累积延误的影响而产生“长尾”，即出现大幅延误的概率虽低，但影响显著，这一特征为后续风险评估提供了核心依据。 + +![]() + +### ## 4.7 任务二结果分析 + +理想模型确立了地月运输的理论上界，但现实工程中缆索摆动、设备故障、发射失败及气象干扰等随机变量（Stochastic Variables）将导致系统性能偏离理想值。本节基于 1,000 次蒙特卡洛（Monte Carlo）模拟，对系统进行扰动下的定量评估。 + +#### ### 4.7.1 性能损耗量化对比 + +表 4-1 对比了确定性解与随机模拟均值的差异，量化了扰动带来的系统代价。 + +**表 4-1 完美与非完美运行条件下核心指标对比表** + +|策略类型|维度|完美条件|非完美条件 (Mean±Std)|相对变化率|敏感度评价| +|---|---|---|---|---|---| +|**策略 A（成本优先）**|时间 (a)|186.2|202.3 ± 0.6|+8.6%|低 (Low)| +||能耗 (PJ)|15,720|15,738 ± 0|+0.1%|极低| +|**策略 B（时间优先）**|时间 (a)|100.7|120.7 ± 0.4|**+19.9%**|**高 (High)**| +||能耗 (PJ)|31,537|30,217 ± 30|-4.2%|中| +|**策略 C（综合平衡）**|时间 (a)|139.0|155.3 ± 0.5|+11.7%|中| +||||||| +- 策略 A 虽然绝对工期最长，但其相对延迟最小，能耗几乎无波动，核心优势在于其电梯占比达到100%——空间电梯作为半永久建筑，运维稳定性远高于火箭，在应对动态扰动时具有天然稳定性。 + +- 策略 B 的时间延迟率最高，核心原因是其电梯占比仅为54.1%，高度依赖火箭系统，从而双重暴露在“发射失败”与“气象取消”两大风险下，抗干扰能力最弱。 + +- 策略 C 的时间延迟率介于 A、B 之间，敏感度为中等，既避免了策略 B 的高风险，又未陷入策略 A 的长工期困境，其抗干扰能力的均衡性，进一步印证了其“兼顾工期与能耗”的核心定位。 + +综上，电梯占比越高，抗干扰能力越强，工期稳定性越好；火箭占比越高，受随机扰动的影响越大，工期波动越显著。 + +#### ### 4.7.2 指标演变 + +引入随机扰动后，系统核心指标从任务一中的单一确定数值，演变为概率分布，但其核心战略排序保持不变，说明三种策略的定位具有极强的稳健性，不受非完美运行的影响，为工程决策提供了可靠支撑。具体分析如下： + +**（1）能耗稳定性与“抵消效应”** + +![]() + +如图所示，三种策略的总能耗均呈现窄峰近正态分布。 + +* **高收敛性**：三种策略的总能耗均呈现窄峰近正态分布，标准差相较于基数不足0.2%，说明缆索摆动产生的变轨能量惩罚，受限于角度的微小随机性，并未引发能效崩溃，系统能耗整体保持稳定。 + +* **均值偏移**：策略 B 与 C 的能耗均值略低于理想值，这并非效率提升，而是由于火箭发射失败导致的有效载荷缺损，在统计上抵消了部分摆动能耗增量，属于随机扰动下的正常现象，而非系统性能优化。 + +**（2）时间延迟的“右偏长尾”特征** + +![]() + +箱线图揭示了工期的退化规律： + +* **工期分布**：策略 A、B、C 的中位数工期较理想状态分别延迟了 8.6% 至 19.9%。 + +* **策略层级保持**：尽管存在波动，但各策略的工期区间保持严格分离，未出现交叉。这证明了任务一中确立的“策略 A 最省能、策略 B 最快速”的战略排序，在非理想状态下依然稳健，不会因随机扰动而发生逆转，进一步强化了三种策略的决策价值。。 + +#### ### 4.7.3 四大核心扰动因子的量化对比 + +![]() + +![]() + +为精准识别各扰动因子对系统工期的影响强度，分别进行了故障归因分析和敏感性分析，从发生频次和影响权重两个维度，明确了影响系统进度的核心瓶颈的优先级依次为电梯检修、气象干扰、缆索摆动、火箭失效。 + +- 电梯检修：电梯停机时间以万天为量级，其中纯电梯策略（A）的总停机时间峰值接近 18,000 天，年均停机时长约占整个建设周期的 23%。龙卷风图进一步量化了其影响权重，相关系数高达 0.836，是唯一的强相关因子。这一结果表明，电梯作为混合运输情景中的基础运力核心，其故障会直接中断整个物流链，即使是停机频率的微小变化都可能引发工期的线性漂移，因此提升电梯系统的妥善率是后续工程优化的重中之重。 + +- 气象干扰：气象导致的发射取消次数峰值约为 65,000 次，是火箭机械失效次数的 9.5 倍；在敏感性分析中，其相关系数为 0.513，属于中相关因子。这一发现揭示了在 2050 年的技术假设下，制约火箭运力效率的核心已不再是技术可靠性，而是发射场的地理与气象条件，这与前文提出的“低纬度优先” 运力优化原则高度呼应，也为建设低纬度全天候发射场提供了直接的量化依据。 + +- 火箭失效：火箭失效次数峰值约为 7,000 次，远低于气象取消的频次；在敏感性分析中，其相关系数仅为 0.163，属于弱相关因子。主要原因为在混合运输情景中，火箭仅作为进度加速器使用，其失效仅影响部分加速运力，全局影响被电梯和气象因素大幅削弱，因此对火箭可靠性的边际优化收益十分有限。 + +- 缆索摆动：缆索摆动引发的修正能耗峰值仅约 18 PJ，占总能耗比例不足 0.1%；在敏感性分析中，其相关系数为 - 0.295，属于弱相关因子且呈负相关，说明对工期的影响极小。这一结果验证了在 ±0.5° 的工程控制范围内，缆索摆动对系统性能的影响可忽略不计，无需投入过多资源进行针对性优化。 + + +# 5. 模型 II：基于生命保障与随机平衡的水资源物流模型 (LSL-SWBM) +## 5.1 模型概述 (Model Overview) + +随着月球殖民地从建设期转向运营期，物流重心由结构材料转变为生命维持物资。模型 II 构建了生命保障物流与随机水资源平衡模型 (Life-Support Logistics and Stochastic Water Balance Model, LSL-SWBM)，旨在量化定居点在一年运行周期内的水资源安全边界。该模型不仅考虑了基于人类心理舒适度的多级需求函数，还引入了正态近似模型来处理随机医疗应急用水。通过将水资源需求映射至模型 I 的运输框架，我们评估了不同舒适度场景（Comfort Factors）对地月物流链的额外压力 。 + +## 5.2 水资源需求架构建模 (Water Demand Architecture) + +在孤立的月球生态系统中，水资源的消耗主要由循环再生系统（Recycling System）支撑，但物流补给必须覆盖系统熵增（损耗）及突发性医疗需求。本文首先明确，月球殖民地水资源使用仅包含基本生活用水与医疗应急用水，不考虑工业用水，并将需求量化为日水需求，以此构建需求核心框架。 + +### 5.2.1 生活用水的多级演化 ($W_r$) +在月球殖民地的水资源需求体系中，基本生活用水由生存用水与卫生用水共同构成，其日需求公式为： +$$W_{r}=N\times (w_{s}+0.4\alpha )$$ +其中$N$表示月球殖民地的人数，$w_{s}$表示生存用水，$\alpha$表示舒适度系数。不同需求等级对应的舒适度系数及用水标准如下表所示： + +| 需求等级 | $\alpha$值 | 每人日用水量 | 含义说明 | +|------------|------------|--------------|----------------| +| 基本生存 | $\alpha=1$ | 2.9 L | 满足最低生存标准 | +| 舒适生活 | $\alpha=50$| 22.5 L | 达到中等舒适水平 | +| 奢侈标准 | $\alpha=250$| 102.5 L | 接近地球生活用水水平 | + +### 5.2.2 随机医疗应急用水的正态化评估 ($W_m$) +由于月球殖民地人数 $N=10^5$ 规模庞大，每日患病人数 $X$（假设日发病率 $p=2\%$）服从二项分布，可精确近似为正态分布 $X \sim N(Np, Np(1-p))$。为保障极端情况下的医疗安全，模型采用 99% 置信度下的峰值需求作为每日储备目标： +$$W_{medical} = (E(X) + Z_{0.99} \cdot \sigma) \times 5 \text{ kg}$$ +经计算，该指标在不同舒适度场景下保持稳定，确保了医疗系统的鲁棒性。 + +## 5.3 物流交付策略：初始填充与动态补偿 +水资源运输不同于建筑材料，其具有极强的循环性。考虑到水循环技术成熟化，首次水资源运输完成后，日水需求累计速率缓慢，以日为单位的运输方案会大幅提升成本，因此本文选用月供应模式，设计“双阶补给”逻辑： + +### 5.3.1 首月初始填充 ($W_{initial}$) +首个月无上月水资源循环产生的再生水资源，因此初月供应需满足两大核心条件：一是基本生活用水覆盖未来三十天供应，二是医疗应急用水最大化应对突发医疗事件。基于此构建首次供应量模型： +$$W_{initial}=（W_{r}+W_{mi}）\cdot T$$ +其中 $T$ 为供应周期天数，$W_{mi}$ 为 99% 置信度下的日紧急医疗用水供应量，该模型既完成系统循环底量建立，又储备 30 天应急水资源。 + +### 5.3.2 月度增量补偿 ($W_{routine}$) +常规月份中，基本生活用水由地球运输与水循环两部分共同供给，同时医疗应急用水因前期置信度保障，仅需满足医疗事件发生率均值即可。结合水资源利用率 $\eta$（本文中 $\eta=0.9$），构建常规月供应量模型： +$$W_{routine}=（W_{r}(1-\eta)+W_{m}）\cdot T$$ +该模型精准抵消水循环损耗，同时覆盖日常医疗消耗，实现水资源动态补给平衡。 + + +## 5.4 任务三求解结果与多维度分析 + +基于模型运行结果，我们从时间成本与能源代价两个维度对水资源补给任务进行了定量拆解。 + +### 5.4.1 三种场景下的需求规模对比 + +舒适度系数 \(\alpha\) 的微小变动将引发补给规模的指数级跳跃。我们将模型计算出的三种场景下的水需求规模汇总如下： + +**表 5-2 不同舒适度场景下的水资源需求统计** + +| 场景级别 | \(\alpha\) 值 | 水库存量 (吨) | 日补充量 (吨) | 月补充量 (吨) | 首批运输量 (吨) | 年补充总量 (吨) | +|------------|--------------|----------------|----------------|----------------|------------------|------------------| +| 基本生存 | 1 | 290 | 29.1 | 873 | 1,163 | 10,622 | +| 舒适生活 | 50 | 2,250 | 225.1 | 6,753 | 9,003 | 82,162 | +| 奢侈标准 | 250 | 10,250 | 1,025.1 | 30,753 | 41,003 | 374,162 | + +**表 5-2** 清晰地揭示了舒适度对物流规模的乘数效应：从基本生存（α=1）到奢侈标准（α=250），年补充水量从约1万吨激增至37万吨，增长超过35倍。这一数量级的跃迁对运输系统提出了截然不同的要求。 + +![水需求分解图 (α=1)]() + +**图water_demand_breakdown.png** 以基本生存场景（α=1）为例，直观展示了水资源需求的构成分解。可以看出，在高效的循环系统支持下，每日需要从地球补充的水量（红色部分）仅占总需求的一小部分，这凸显了闭环生命保障系统的价值。 + +### 5.4.2 运输方案的代价权衡 + +为了满足上述水资源需求，我们将需求指标代入模型一，评估了四种不同的运输方案。这些方案的定义和基础运力对比如下： + +**表 5-3 四种水资源运输方案定义** + +| 方案 | 描述 | 日运力 (吨/天) | 比能量 (GJ/吨) | +|------|--------------------------|----------------|-----------------| +| 1 | 仅空间电梯 (3部) | 1,471 | 157.2 | +| 2 | 仅火箭 (10发射场) | 1,250 | 506.1 | +| 3 | 混合 (电梯+10发射场) | 2,721 | 317.5 | +| 4 | 混合 (电梯+低纬发射场) | 1,971 | 243.5 | + +由表中数据可知，方案3拥有最高的综合日运力，而方案1的比能量最低，能效最优。火箭运输虽然运力尚可，但能耗成本高达电梯的3.2倍。 + +将不同场景的水需求量代入上述运输方案，我们得到了详细的运输成本与时间分析结果。 + +**（1）对于基本生存场景 (α=1):** + +**表 5-4 α=1 场景下各运输方案性能对比** + +| 方案 | 首批运输天数 | 首批能耗 (TJ) | 月补充天数 | 月补充能耗 (TJ) | 年能耗 (PJ) | +|------------------|--------------|---------------|------------|-----------------|-------------| +| 方案1: 仅电梯 | 0.79 | 182.8 | 0.59 | 137.2 | 1.67 | +| 方案2: 仅火箭 | 0.93 | 588.6 | 0.69 | 441.8 | 5.38 | +| 方案3: 混合全部 | 0.43 | 369.2 | 0.32 | 277.1 | 3.37 | +| 方案4: 混合低纬 | 0.59 | 283.1 | 0.44 | 212.5 | 2.59 | + +分析**表 5-4** 可知，在低需求场景下，所有方案均能在1天内完成月度补给。方案1在能耗上具有绝对优势，年能耗仅为1.67 PJ。方案3则在时间上最优，仅需约0.43天（10小时）即可完成首批运输。这表明在生存模式下，水资源运输对物流系统造成的压力极小。 + +![运输方案对比 (α=1)]() + +**图 5-2** 直观对比了在α=1场景下，四种运输方案在首批运输和月度补充上的天数与能耗。柱状图清晰地显示出方案1的能耗优势和方案3的时间优势。 + +**（2）对于舒适生活场景 (α=50):** + +**表 5-5 α=50 场景下各运输方案性能对比** + +| 方案 | 首批运输天数 | 首批能耗 (TJ) | 月补充天数 | 月补充能耗 (TJ) | 年能耗 (PJ) | +|------------------|--------------|---------------|------------|-----------------|-------------| +| 方案1: 仅电梯 | 6.12 | 1,415.3 | 4.59 | 1,061.6 | 12.92 | +| 方案2: 仅火箭 | 7.20 | 4,556.3 | 5.40 | 3,417.6 | 41.58 | +| 方案3: 混合全部 | 3.31 | 2,858.1 | 2.48 | 2,143.8 | 26.08 | +| 方案4: 混合低纬 | 4.57 | 2,191.9 | 3.43 | 1,644.1 | 20.00 | + +当舒适度提升至α=50时，水需求增加约8倍。如表 5-5所示，运输压力显著增大，但方案1仍可在约6天内完成首批运输，年度能耗增至12.92 PJ。方案4在能耗和时间之间取得了良好的平衡。 + + +**（3）对于奢侈标准场景 (α=250):** + +**表 5-6 α=250 场景下各运输方案性能对比** + +| 方案 | 首批运输天数 | 首批能耗 (TJ) | 月补充天数 | 月补充能耗 (TJ) | 年能耗 (PJ) | +|------------------|--------------|---------------|------------|-----------------|-------------| +| 方案1: 仅电梯 | 27.87 | 6,445.7 | 20.90 | 4,834.4 | 58.82 | +| 方案2: 仅火箭 | 32.80 | 20,751.2 | 24.60 | 15,563.8 | 189.36 | +| 方案3: 混合全部 | 15.07 | 13,016.9 | 11.30 | 9,762.9 | 118.78 | +| 方案4: 混合低纬 | 20.80 | 9,982.5 | 15.60 | 7,487.0 | 91.09 | + +在奢侈标准下，如**表 5-6** 所示，水资源运输成为一个重大负担。仅靠电梯运输首批水需要近一个月，年能耗高达58.82 PJ。方案3能将首批时间压缩至15天左右，但代价是更高的能耗。值得注意的是，即使在此极端场景下，水运输年能耗（方案1的58.82 PJ）也仅占之前模型I中计算的整个殖民地建设总能耗（15,720 PJ）的0.37%。这证明能源总量并非瓶颈，关键在于运力分配。 + +![首批运输能耗对比]() +![首批运输天数对比]() + +**图 5-4** 与 **图 5-5** 分别对比了不同舒适度下，各运输方案完成首批运输所需的能耗和天数。两图共同揭示了一个规律：随着舒适度提高，各方案间的性能差距被放大。在α=250时，方案1与方案2的能耗差高达14,000 TJ以上，天数差约5天，决策的选择将带来完全不同的成本结构。 + + +![多场景热力图]() + +**图 5-3** 的热力图综合展示了不同舒适度系数（α）和不同运输方案下的年能耗分布。颜色从浅到深表示能耗递增，可以清晰地看到： +- 1) 同一方案下，能耗随α值急剧上升； +- 2) 同一α值下，方案1的色块始终最浅，印证其能效最优。 + +## 5.5 结论与战略启示 + +1. **舒适度与运力的博弈**：定量分析证实，生活质量是资源需求的放大器。当舒适度达到奢侈标准 (α=250) 时，年度水补给量将占去单一电梯系统（方案1）年理论运力（约53.7万吨）的 **69.68%**。这意味着维持极高质量的生活将严重挤占用于运输建筑材料、科研设备等其他关键物资的运力窗口。殖民地管理机构必须在居民生活标准与殖民地扩展速度之间做出战略性权衡。 + +2. **分级运输方案推荐**：基于上述分析，我们提出分阶段的运输方案建议： + * **运营初期/资源匮乏期**：推荐采用 **α=1（基本生存）标准 + 方案1（仅电梯运输）**。该组合能以最低的能耗（年1.67 PJ）和可接受的时间（月度补给约14小时）确保十万人的生存底线，将宝贵的混合运力留给更紧急的基建任务。 + * **稳定运营期**：推荐采用 **α=50（舒适生活）标准 + 方案4（混合低纬）**。此组合实现了效率与成本的平衡：首批供水可在4.57天内完成，年能耗20.00 PJ相对合理，同时利用了低纬度发射场提升火箭段的效率。 + * **高舒适度需求场景**：若必须执行奢侈标准，则 **方案3（混合全部）** 是唯一选择，它通过最大化运力将首批运输时间控制在半个月左右，但需承担近120 PJ的年能耗代价。 + +3. **系统稳健性验证**：得益于90%的高效水循环系统和首月建立的30天应急储备（Buffer），水资源供应系统具备很强的抗干扰弹性。即使遭遇长达一个月的运输中断，殖民地的基本生存也能得到保障。这为物流系统的调度和维护提供了一定的容错空间。 + + +# 4 模型扩展：环境影响分析与决策修正 +## 4.1 基准环境影响量化与评估 +基于模型I的能量转化逻辑，扩展环境子模块：针对火箭的液氧甲烷（$\text{LOX/CH}_4$）动力系统，通过燃烧方程 $CH_{4} + 2O_{2} \longrightarrow CO_{2} + 2H_{2}O$，将燃料消耗映射为污染物排放质量，四种基准情景的环境指标如表11所示： + +**表11：四种基准情景的环境指标对比汇总** +| 情景 | 完成时间 | 总燃料消耗 | 总$\text{CO}_2$ | 平流层$\text{H}_2\text{O}$ | 年均排放强度 | +|------------------|----------|------------|----------------|---------------------------|--------------| +| 纯火箭 | 219年 | 9,596 Mt | 12,968 Mt | 1,919 Mt | 59.2 Mt/yr | +| 纯电梯 | 186年 | 69 Mt | 846 Mt | 0 Mt | 4.5 Mt/yr | +| 混合（时间优先） | 101年 | 4,445 Mt | 6,414 Mt | 882 Mt | 63.7 Mt/yr | +| 混合（综合平衡） | 139年 | 2,438 Mt | 3,856 Mt | 477 Mt | 27.7 Mt/yr | + + +结合多维度可视化分析（图1），可清晰区分各情景的环境代价： +- 纯火箭方案总$\text{CO}_2$排放与平流层$\text{H}_2\text{O}$注入量均为最高，对全球气候与臭氧层的潜在风险最大； +- 纯电梯方案完全规避平流层$\text{H}_2\text{O}$污染，总$\text{CO}_2$排放仅为纯火箭的6.5%； +- 混合（综合平衡）方案较“时间优先”方案，$\text{CO}_2$减排40%、平流层$\text{H}_2\text{O}$减少46%，实现了建设效率与环境负荷的平衡。 + +![图1：多维度环境影响综合图]() +*注：左上为$\text{CO}_2$排放源分解，右上为平流层$\text{H}_2\text{O}$对比，左下为年均$\text{CO}_2$排放，右下为碳强度（$\text{kg CO}_2$/吨载荷）。* + + +## 4.2 地面港口（Earth Port）评估 + +虽然空间电梯在运输过程中实现了零排放，但其位于赤道海域的地面港口（Earth Port）作为永久性海上建筑，仍会对局部海洋生态产生影响。为了评估这种影响的量级，本文参考了国际太空电梯联盟（ISEC）的研究成果，将地面港口与技术最为成熟的海上大型设施——海洋石油钻井平台进行类比。 + +| 影响类型 | 石油平台表现 (文献数据) | 空间电梯地面港口预期 | 影响范围 | +| :--- | :--- | :--- | :--- | +| **物理破坏** | 锚固系统导致底栖生物受损 | 仅限锚固区及浮式平台下方 | 局部 (100-500m) | +| **化学排放** | 钻井液、水泥、化学品、产出水 | **无此类排放** (无钻井需求) | 极低 | +| **光/声干扰** | 人造光吸引/驱赶移动物种，机械噪声 | 恒定的导航灯光与低频电力噪声 | 局部 (数百米) | +| **生境改变** | 人造生境（人工鱼礁效应） | 提供硬基质促进固着生物群落生长 | 局部/积极 | +| **重大事故风险** | 烃类大规模泄漏 (原油污染) | **无泄露风险** (无碳氢化合物提取) | 极低 | + + + +通过类比分析可知，空间电梯地面港口的环境足迹远低于现有的海上石油工业设施。与传统火箭发射对全球大气（尤其是对臭氧层和全球气候强迫）造成的“全球性（Widespread）”影响相比，太空电梯的污染几乎可以忽略不计。这确立了空间电梯作为未来“地月绿色物流链”首选方案的生态优越性。 + + +# 4.2 可持续性模型扩展：三目标决策框架 + +为了最小化环境影响，我们将模型 I 的“时间-能量”双目标优化扩展为包含环境惩罚权重的**三目标决策框架**。修正后的目标函数 $J_{total}$ 为： + +$$\min J_{total} = \alpha \cdot Time + \beta \cdot Energy + \gamma \cdot \text{Emission}_{CO_2} + \delta \cdot \text{Emission}_{H_2O, strat}$$ + +通过调节环境权重（$\gamma, \delta$），模型会自动搜索对大气层伤害最小的物流组合。 + +![三维Pareto前沿图]() +通过模拟分析，我们发现环境效益与能源效率之间存在显著耦合。特别地，在时间线达到 186年 左右时，系统达到一个边际环境效益峰值，即 “效率临界点”。超过此点后，进一步延长建设时间所带来的环境改善收益将逐渐递减。 + +## 4.4 任务四结论 +1. **环境影响评估**： + - 纯火箭方案存在全球性环境风险，其平流层$\text{H}_2\text{O}$注入与$\text{CO}_2$排放会长期影响气候与臭氧层； + - 空间电梯的环境影响局限于地面港口的局部物理扰动，可忽略不计。 + +2. **模型调整与减排策略**： + - 推荐采用139年“综合平衡”方案，以38%的工期延长换取40%的$\text{CO}_2$减排； + - 强制设定电梯承担至少70%的基础运力，仅保留火箭用于紧急任务； + - 配套100%可再生能源供电，最终实现总$\text{CO}_2$排放降至94 Mt（较纯火箭减排99.3%）。 \ No newline at end of file diff --git a/latex/初稿.pdf b/latex/初稿.pdf new file mode 100644 index 0000000..7fdea64 Binary files /dev/null and b/latex/初稿.pdf differ diff --git a/latex/建议信.md b/latex/建议信.md new file mode 100644 index 0000000..74224bf --- /dev/null +++ b/latex/建议信.md @@ -0,0 +1,56 @@ + +## **Recommendation (actionable decision)** + +We recommend a **Phased Hybrid Transport Strategy**: use the Space Elevator System as the primary channel while contracting low-latitude **direct Earth-to-Moon rocket surge** capacity for resilience. Model I identifies a balanced point of **74.6% elevator / 25.4% direct rockets**; approve a practical **policy band** of **70–80% / 20–30%** with **auditable triggers** for rebalancing. + +--- + +## **Why this is the best course (three-board-level metrics)** + +Because the project spans more than a century, we use **energy consumption as the cost proxy**—a physically auditable metric that remains comparable under long-horizon inflation and technology uncertainty. Under this metric, the hybrid portfolio is Pareto-superior: + +* **Schedule commitment:** plan for **155–160 years** to deliver the 100 million metric tons at **95% confidence**; *139 years* is the idealized lower bound. +* **Cost proxy (energy):** the balanced hybrid point requires **38,750 PJ** total transport energy, delivering **23.4% savings** versus a rocket-only baseline in our model. +* **Environmental impact:** reduces cumulative **CO$_2$ by ~40%** relative to rocket-only delivery. + +--- + +## **Risk controls (what must be protected) and triggers (what we do if it degrades)** + +Our sensitivity and disturbance analysis shows elevator throughput is the primary schedule driver; therefore resilience must be managed as an operational requirement rather than an afterthought: + +* **Elevator resilience mandate:** maintain redundant structural monitoring at all three Galactic Harbours and enforce a **≥10% operational reserve** during steady-state operations. +* **Repair-time objective:** target **<14 days** downtime per incident via pre-positioned spares, trained response teams, and rehearsed procedures. +* **Rebalancing trigger (auditable KPI):** if rolling 180-day elevator throughput stays below **90% nominal** for **≥60 days**, or reserve stays below **10%** for **≥60 days**, raise direct-rocket share to **30–35%** until recovery. +* **Launch contracting policy:** prioritize **low-latitude sites** to reduce propellant and emissions per delivered ton while maintaining geographic redundancy. + +--- + +## **Sustainment logistics policy (one-year water requirement after habitation)** + +Once the colony is inhabited, annual water resupply spans **10.6 kt/year (survival)** to **374 kt/year (luxury)**. Energy is not the bottleneck (even luxury-case is **sub-1%** of construction transport energy); **capacity allocation** is. We recommend: + +* **Non-negotiable threshold:** maintain water recycling efficiency **≥85%**; falling below this level is a mission-risk condition requiring immediate corrective action and temporary comfort reduction. +* **Phased comfort escalation:** start at survival-tier and raise comfort only as ISRU and recycling stability are demonstrated. +* **Hard guardrail:** if and coincide, demand becomes infeasible; trigger immediate comfort rollback and recycling maintenance surge. + +- + +**Closing** +The Moon Colony is feasible within a **155–160 year** commitment envelope while materially reducing energy and environmental costs versus rocket-only delivery. Success requires a hybrid architecture with explicit resilience mandates and auditable triggers that keep schedule risk under control. + +***Respectfully submitted,*** +***Team 2618656*** + +Would you like me to generate a visual representation or a Gantt chart based on this implementation roadmap? +-- + +## **Implementation roadmap (what the Board should approve now)** + +| Phase | Timeframe | Board-approve milestones | +| --- | --- | --- | +| **I** | 2050–2070 | Commission hybrid operations at the **74.6/25.4** reference point (policy band 70–80/20–30); finalize low-latitude launch contracts as surge capacity; harden elevator monitoring, spares, and repair playbooks; establish lunar receiving & storage infrastructure. | +| **II** | 2070–2120 | Peak delivery; enforce **KPI-based rebalancing**; raise comfort as ISRU and recycling stabilize. | +| **III** | 2120–2190 | Elevator-dominant delivery; direct rockets for contingencies; institutionalize sustainment governance (recycling KPI and comfort caps). | + +--- \ No newline at end of file diff --git a/launch_frequency_analysis.png b/launch_frequency_analysis.png deleted file mode 100644 index 1c37edc..0000000 Binary files a/launch_frequency_analysis.png and /dev/null differ diff --git a/rocket_comprehensive.png b/rocket_comprehensive.png deleted file mode 100644 index 9b0a366..0000000 Binary files a/rocket_comprehensive.png and /dev/null differ diff --git a/rocket_launch_timeline.png b/rocket_launch_timeline.png deleted file mode 100644 index f1bd3c3..0000000 Binary files a/rocket_launch_timeline.png and /dev/null differ diff --git a/scenario_comparison.png b/scenario_comparison.png deleted file mode 100644 index 7c89416..0000000 Binary files a/scenario_comparison.png and /dev/null differ diff --git a/specific_energy_comparison.png b/specific_energy_comparison.png deleted file mode 100644 index 4bfbf08..0000000 Binary files a/specific_energy_comparison.png and /dev/null differ diff --git a/美赛TASK4.md b/美赛TASK4.md deleted file mode 100644 index 9784d1f..0000000 --- a/美赛TASK4.md +++ /dev/null @@ -1,72 +0,0 @@ -# task4（童年给我的这个网站里的） -任务四需要分析殖民地建设对环境的影响并探寻降低影响的途径。对此本文首先对环境影响进行量化，由传统火箭的燃料燃烧方程 -$$ -CH_{4}+2O_{2}\longrightarrow CO_{2}+2H_{2}o -$$ -可知产生的废气为二氧化碳和水蒸气。因此本文对于环境污染量化为二氧化参排放量和水蒸气排放量。并且通过燃烧方程得到每千克燃料的排放量如下表 -![](https://oss-liuchengtu.hudunsoft.com/userimg/be/be123f9ad39a2ce420417ce371849459.png) -依据上述数据本文对于问题一中的三种情形下的不同方案给出排放结果图如下 -![](https://oss-liuchengtu.hudunsoft.com/userimg/55/55880d8a798f2fb66ba8b09b9ef4e48e.png) -可以发现火箭对于环境的影响远远大于电梯，这为我们后续的方案调整提供思路。 -## 三维帕累托前沿思想下的方案调整 -任务一中的方案选择基于时间与成本两方面，对于任务四，本文新增环境方面的影响，具体环境函数为 -$$ -EP=Emission_{CO_{2}}+Emission_{H_{2}O} -$$ -得到帕累托前沿解图表如下 -![](https://oss-liuchengtu.hudunsoft.com/userimg/28/28984c0725735e990bdff5f05f6a07d8.png) -可以发现其存在明显的膝部区域，同样依据依据膝点求解方法，本文得到膝点为139年（可以分析一下为什么 和问题一一样，我认为原因是火箭在环境影响、成本、时间效率上都劣于电梯 ，这导致二维帕累托前沿与三维帕累托前沿的膝部区域高度重叠，因此会产生膝点不变的现象，对此我们可以对模型参数进行调整，火箭每次装载货物为150吨——看下面一个括号）（我有一个中肯建议，把火箭的运输能力直接变成150每一次发射，这样子火箭在这个运输能力上比电梯优秀，那么依照童年的思路，就是这个帕累托前沿的膝点就会改变（和任务一不一样至少），就是让这个火箭一次可以带更多货物上去。也算是方案调整了。就是我感觉这样比较合理 ，要不师姐你和童年说一下，这是我想到和题目说的调整模型契合的一个方向点）（就是因为我们算出来和前面这个139年的一样，所以我们为了更好降低环境影响，使得火箭最大化利用它的装在能力，为什么可以呢，因为火箭装载多了，意味着相同货物下火箭发射次数减少，排放就少了。）（还有一个思路就是多少的环境污染我们可以接受，可以设定一个阈值，然后在这个阈值下去选择时间和成本总体考虑最优的方案，也可以是方案调整的一个方向） -## 可再生能源下的使用 -在最大化利用火箭运载能力的基础上，本文考虑使用可再生能源以降低月球殖民地建设对于地球环境的污染并得到如下图表： -![](https://oss-liuchengtu.hudunsoft.com/userimg/62/62bde6f8973c42deebd62c27fd14bb2c.png) -![](https://oss-liuchengtu.hudunsoft.com/userimg/08/08c3356e04023fcf3c12fc5de6747f41.png) -可以看到采用可再生能源可以进一步降低对环境的污染，相对减排率达到14.6%。 - - - - - -# task4（童年说让我自己写一个思路，我也不懂他啥意思，然后下面这个是我的想法——建议直接忽略，因为我写到一般发现确实写不下去，这个没用我觉得） -任务四需要分析殖民地建设对环境的影响并探寻降低影响的途径。对此本文首先对环境影响进行量化，分为气体排放、土地占用、噪音污染、电磁辐射污染。 -1.气体排放 -对于气体排放，本文的量化指标为废气排放量 -由传统火箭的燃料燃烧方程 -$$ -CH_{4}+2O_{2}\longrightarrow CO_{2}+2H_{2}O -$$ -可知产生的废气为二氧化碳和水蒸气。因此本文对于环境污染量化为二氧化参排放量和水蒸气排放量。并且通过燃烧方程得到每千克燃料的排放量如下表 -![](https://oss-liuchengtu.hudunsoft.com/userimg/be/be123f9ad39a2ce420417ce371849459.png) -那么废气排放量最终量化的函数为 -$$ -V_{p}=V_{CO_{2}}+V_{H_{2}O} -$$ -2.土地占用 -本文认为环境的影响包括土地的使用，因此本文将土地占用作为影响环境的指标，量化为土地占用面积$S$ -3.噪音污染 -对于火箭而言，发射时的巨大噪音会对周围产生污染，本文将噪音污染量化为噪音影响面积 -$$ -S_{n}=\pi \cdot[10^{\frac{200-L_{limit}-11}{20}}]^{2} -$$ -其中$L_{limit}$为噪音阈值，本文取65db，200为火箭通常的声功率（基于猎鹰九号的数据） -4.电磁辐射污染 -对于太空电梯而言，其需要大量电能实现货物运输，因此需要高压电提供能量，必然会产生电磁辐射污染，本文将其量化为污染面积， - - - - - - - - - - - - - - - - - - - -