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For office use only

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Team Control Number

48649

Problem Chosen

For office use only

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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.

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 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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