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2016
MCM/ICM
Summary Sheet
The Heat Dissipation Model and Optimization Model for a Hot Bath
Abstract
Considering the time and space factors, shape and volume of the tub, volume/temperature/motions of the person, water addition, effect of bubble bath, we provide three feasible models to optimize the water volume.They are the thermal diffusion model of water, Bi-objective optimization model bad on genetic algorithm, thermal diffusion and bi-objective optimization model with a bubble bath.
Model 1: The heat dissipation model of water, describing water temperature variation curve with time and space.Considering three heat dissipation methods, bad on Fourier law, this model can be created. Further, model simulation shows the slowest heat dissipation rate of three kind of shape bathtubs(cuboid, capsule-like-shape and mi spherical shape) we choo at the same volume. Finally, we modify the value of initial water temperature for nsitivity analysis, finding this model is robust enough.
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Model 2: Bi-objective optimization model bad on genetic algorithm. Firstly, we determine the minimum water addition and the most comfortable temperature as two optimization goals. Secondly, we defined the weight coefficient p as comfort degree when bathing and u genetic algorithm getting different minimum water addition and water temperature at every comfort degree. Then by changing the shape of the tub, the shape and motions of the person, we determine the extent to which this model depends upon the factors. Finally, we modify the value of the minimum water temperature for nsitivity analysis, finding this Bi-objective optimization model is robust enough. Model 3:thermal diffusion and bi-objective optimization model with a bubble bath, looking for the effect of a bubble bath on water temperature. Changing the parameters in model 1 and model 2, we can build corresponding models. Then the model simulation showed a bubble bath has an effect of hindering the heat dissipation.
Finally, we analyze the results of our model and the results show that our model is realistic and nsible.
Key words: thermal diffusion model, optimization model, genetic algorithm
Content
1 Introduction (3)
2 The Analysis of Problem (3)
3 Assumptions (3)
4 Model 1: Heat dissipation model of water (4)
4.1 Symbols and definitions (4)一举两得
4.2 Model overview (4)
4.3 Model establishment (4)
4.4 Model simulation (6)
4.5 Results and analysis (6)
4.6 Sensitivity Analysis (7)
5 Model 2: Bi-objective optimization model bad on genetic algorithm (10)
5.1 Symbols and definitions (10)
5.2 Model overview (10)
5.3 Model establishment (10)
5.4 Results and analysis (12)
5.5 Sensitivity Analysis (13)
6 Model 3: Heat dissipation and optimization model with a bubble bath (13)
6.1 Model overview (13)
6.2 Model establishment (13)
6.3 Results and analysis (13)
7 Conclusions (14)
8 Strengths and Weakness (14)
微信群管理规定8.1 Strengths (14)
8.2 Weakness (14)
9 Non-technical explanation (15)
10 References (15)
1 Introduction
Everyone expects a comfortable water temperature when bathing, spa-style tub has been invented to meet this demand. However, how to get a comfortable temperature by controlling the trickle should be took into account when the bathtub is not a spa-style tub with a condary heating system and circulating jets, but rather a simple water containment vesl.
It also should be took into consideration that how to keep the temperature as clo as possible to the initial temperature without wasting too much water. What’s more, the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, and the motions made by the person in the bathtub have influence on the water temperature.
The mathematical modeling and tools will support in developing a reasonable strategy to control the water temperature in bathtub. In this paper, we wish to create a uful model to describe the water temperature distribution and to create a feasible model to keep the temperature even throughout the bathtub and as clo as possible to the initial temperature without wasting too much water.
2 The Analysis of Problem
The person adds a constant trickle of hot water from the faucet to reheat the bathing water, so our goal is to build a uful model to describe the water temperature variation curve with time and space and optimize the water addition under the condition that the water temperature is as clo as possible to the initial temperature.
In order to provide a uful strategy, we should understand the shape of the tub, the volume and temperature of the person, as well as the motions made by the person in bathtub. So it can be divided into three sub-problems:
●Describing water temperature variation with time and space in bathtub.
●Build a model to determine the minimum water addition and the most
comfortable temperature when bathing. Analyze how the shape of bathtub, the shape, temperature and motions of the person affect the water temperature.
●Build a model to look for the effect of a bubble bath.
3 Assumptions
The following assumptions are made for simplicity.
●We assume that all bathtubs mentioned are single bathtubs.
●Heat radiation from surroundings can be ignored.
●Assume that the position of overflow drain and faucet is known.
●Water is full in tub at the at the beginning of bath.
●The faucet is open at the beginning of bath.
●Suppo that room temperature is steady.
4 Model 1: Heat dissipation model of water
4.1 Symbols and definitions symbols definitions
0T The initial temperature of water in bathtub
a T Room temperature
min T The minimum temperature of water in bathtub
1Q ∆ Heat loss from water in bathtub in a certain time
a Q ∆ Heat transfer from water to the bathtu
b wall
b Q ∆ Heat radiated from water to surroundings
c Q ∆ Heat loss from the evaporation of water surface
ε
Background factor 4.2 Model overview
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Bad on Fourier law, in terms of energy, Heat dissipation model describes the law of water temperature changing with time in bathtub, and the temperature simulation curve is provided also. This model provide theoretical basis for analyzing the time when water temperature change from 0T to min T .For convenience of description, this model is built for the condition when there is no people in bathtub.
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4.3 Model establishment
Bad on Fourier law and law of conrvation of energy, heat loss from water consists of three parts: 1) heat transfer from water to the bathtub wall a Q ∆; 2) heat radiated from water to surroundings b Q ∆; 3) heat loss from the evaporation of water surface c Q ∆, the process can be showed clearly in Figure1[1]
Figure1. Heat loss from water in bathtub ()()''''0t
城市英语怎么读a v v v v Q Adt At βρρρρρ∆=-=-⎰ ()4
min 273b Q T Adt εσ∆=+
0t c u Q k ds dt n Ω∂⎛⎫∆= ⎪∂⎝⎭⎰⎰⎰ 1a b c Q Q Q Q ∆=∆+∆+∆
Where
u n
∂∂ is the directional derivative where u is along the normal direction of region Ω. On ba of consulting relevant data [2], the value of parameters in formulas can be obtained:
parameters value
β
210Pa
αρ[3]
0.0074MPa
''αρ[4] 0.008MPa
σ
()-8245.610/W m k ⨯⋅ ε
0.97 k ()20.19/W m C ⋅︒
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Referring to the literature [5], we know that the most comfortable water temperature for people is approximately 40C ︒,and when water temperature is below 37C ︒, one will feel the cold obviously, so we can assume the value of 0T and min T :
