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A48649Shanghai Jiao Tong University, China.pdf
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A48649Shanghai Jiao Tong University, China
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For office use only
T1
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T2
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T3
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T4
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Team Control Number
48649
Problem Chosen
A
For office use only
F1
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F2
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F3
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F4
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2016
MCM/ICM
Summary Sheet
Summary
Based on the existing Finite Element Method (FEM) technique, we consider two models
to analyze the temperature distribution in the bathtub, including without and with a person in
it. Then, we build two inflow models, namely, the continuous inflow model and the
discontinuous inflow model. At last, we employ Fuzzy Comprehensive Evaluation (FCE) to
evaluate these two inflow models in order to find an optimal solution.
First, regarding the model without a person in the bathtub, we calculate the heat dissipation
in the water-to-air heat convection and build a hot-water tube model for calculating the heat
addition. What’s more, to find the temperature distribution in the bathtub, we simulate such
model on ANSYS, which is a large-scale common finite element software.
Second, we consider the model with a person in the bathtub from two aspects, namely, the
person in the still and the moving states. Specific to the moving state of person, we find that
there exist two ways of dissipating heat, including the water-to-air heat convection and the
water-to-bathtub heat conduction. Furthermore, we exploit MATLAB to find the temperature
curve in natural cooling process. .
Third, based on the analysis of above two conditions, we propose to simulate on the
continuous and discontinuous inflow models of the hot water. Besides, to find an optimal
strategy, we evaluate these two models using fuzzy comprehensive evaluation (FCE), where
the variables include the amount of needed water, the comfortable degree of user, the operation
complexity and the operative difficulty index.
Fourth, to analyze the sensitivity of two models with and without person, we extensively
change the velocity and the temperature of the hot inflow, the air temperature, the surface area,
volume and material of the bathtub, and the volume of the person body.
At last, we further study the model for cleansing with a bubble bath additive. In this case,
we vary the condition of the bubble to observe this model’s performance.
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Team #48649 Page 1 of 44
Team #48649
January 28,2014
Abstract
Naturally a non-spa-style bathtub without a secondary heating system and circulating jet
will get noticeably cooler after being filled with hot water. The water lose heat both to the air
and to the bathtub. One want to keep warm by adding a constant trickle of hot water from the
faucet. To handle fluid’s overflowing, the bathtub is designed in such a way that when the tub
reached its capacity, excess water escapes through an overflow drain.
We build Bathtub without Person model (BWOP) to simulate the water temperature
distribution without people inside, and Bathtub with Person (BWP) model to research the
natural cooling process and to propose reasonable strategy for user to maintain the temperature.
The simulation of BWOP mainly depends on the finite element method (FEM) using ANSYS,
while the simulation of BWP mainly depends on the solving of the thermodynamic equation
using MATLAB.
Keywords: Heat Convection, Heat Conduction, Finite Element Method, ANSYS
Team #48649 Page 2 of 44
Contents
1 Introduction ........................................................................................................................... 4
1.1 Restatement of the Problem ........................................................................................................ 4
2 Assumptions and Justifications ............................................................................................. 4
3 Notations ............................................................................................................................... 5
4 Model for bathtub without a person in it (BWOP model) .................................................... 6
4.1 The calculation of the heat-transfer coefficient .......................................................................... 8
4.1.1 The heat-transfer coefficient between the water surface and the air .................................... 8
4.1.2 The heat-transfer coefficient between the water and the water tube .................................... 9
4.1.3 The heat-transfer coefficient between the hot water and the cube wall ............................. 10
4.2 The heat dissipation between the water surface and the air ...................................................... 11
4.3 The heat addition between the water and the hot water tube .................................................... 11
4.3.1 The heat exchange between the inside and the outside of the tube .................................... 11
4.3.2 The heat exchange between the water and the water tube ................................................. 12
4.4 Simulation of BWOP model based on ANSYS ........................................................................ 14
5 Model for bathtub with a person in it (BWP model) .......................................................... 17
5.1 Simulation of the temperature distribution if the person is still ................................................ 18
5.2 Heat dissipation in BWP model ................................................................................................ 19
5.2.1 Water-Air heat convection ................................................................................................. 19
5.2.2 Water-Bathtub heat conduction ......................................................................................... 20
5.2.3 The total heat dissipation in the bathtub ............................................................................ 21
5.3 Simulation of BWP model with MATLAB .............................................................................. 21
5.4 Heat addition model .................................................................................................................. 23
5.4.1 Continuous Inflow model (CI model) ................................................................................ 24
5.4.2 Discontinuous Inflow model (DCI model) ........................................................................ 26
5.4.3 FCE of the two models ...................................................................................................... 28
6 Sensitivity analysis.............................................................................................................. 29
6.1 Sensitivity analysis for BWOP ................................................................................................. 29
6.1.1 Inflow Velocity .................................................................................................................. 29
6.1.2 Inflow Temperature ........................................................................................................... 31
6.1.3 Air Temperature ................................................................................................................. 32
Team #48649 Page 3 of 44
6.1.4 Conclusion ......................................................................................................................... 34
6.2 Sensitivity analysis for BWP .................................................................................................... 34
6.2.1 Different surface area of the bathtub ................................................................................. 35
6.2.2 Different volume of the bathtub ......................................................................................... 36
6.2.3 Different material of the bathtub ........................................................................................ 37
6.2.4 Different volume of person in the bathtub ......................................................................... 38
6.2.5 Conclusion ......................................................................................................................... 38
7 Model for Cleansing with a Bubble Bath Additive (CBBA model) ................................... 39
8 Weakness and Strength ....................................................................................................... 40
8.1 Strengths ................................................................................................................................... 40
8.2 Weaknesses ............................................................................................................................... 41
References .............................................................................................................................. 42
Appendix ................................................................................................................................ 43
Team #48649 Page 4 of 44
1 Introduction
1.1 Restatement of the Problem
We are required to model the bathtub water’s temperature in space and time, which enables
an individual to pick the best strategy in order to keep the temperature almost a constant in
the bathtub. Besides, the temperature at any time is required to approach the initial state
without wasting too much water.
By analysis, we proposed to decompose the problem into two sub-problems:
Build a model that can simulate the temperature of the bathtub in space and time.
Propose an optimal strategy. The strategy should satisfy two conditions: waste the least
water and let the user feel comfortable.
First, for generality, we set the velocity and temperature of inflow, the size and shape of
bathtub variables. Then, we try to model the temperature inside. On one hand, the model needs
to show the volume-average temperature. On the other hand, it is required to depict the
distribution under the influence of heat loss and addition. Then we can change the inputs to
do several simulations. Only using many simulation results can we build a relationship
between input and output of the complexed thermodynamical system.
Second we seek to use the relationship getting from our simulation to find the best strategy
for keeping the temperature. In our model, “best” doesn’t not only means “the least water” but
“the most comfortable environment for the people” which put weight on the gradient of
temperature and limit on the degree (to avoid scald).
At last, we try to adjust our model to different initial state such as the bathtub size and shape,
the inflow temperature and the motions made by the people. We also take the bubble bath
additive in to consideration.
2 Assumptions and Justifications
1) Water is stability, incompressible viscous liquid. The compressibility of the water is
small enough to be ignored. The water is steady and viscous in cases that we will discuss
later.
2) The fluid matches the Boussinesq assumptions
[1]
.
The fluid viscous dissipation can be ignored.
All the physical properties except density are constant
For density, we only consider its change in the body force entries in momentum
equation, others can be considered as constant.
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