775 lines
31 KiB
Plaintext
775 lines
31 KiB
Plaintext
|
%%
|
|||
|
|
%% 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:] <insert the exact wording you input into the AI tool>
|
|||
|
|
\item[Output:] <insert the complete output from the AI tool>
|
|||
|
|
\end{description}
|
|||
|
|
\item OpenAI Ernie (Nov 5, 2023 version, Ernie 4.0)
|
|||
|
|
\begin{description}
|
|||
|
|
\item[Query1:] <insert the exact wording of any subsequent input into the AI tool>
|
|||
|
|
\item[Output:] <insert the complete output from the second query>
|
|||
|
|
\end{description}
|
|||
|
|
\item Github CoPilot (Feb 3, 2024 version)
|
|||
|
|
\begin{description}
|
|||
|
|
\item[Query1:] <insert the exact wording you input into the AI tool>
|
|||
|
|
\item[Output:] <insert the complete output from the AI tool>
|
|||
|
|
\end{description}
|
|||
|
|
\item Google Bard (Feb 2, 2024 version)
|
|||
|
|
\begin{description}
|
|||
|
|
\item[Query1:] <insert the exact wording of your query>
|
|||
|
|
\item[Output:] <insert the complete output from the AI tool>
|
|||
|
|
\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'.
|