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Mathematical Contest in Modeling (MCM/ICM) Summary Sheet
Summary
It’s pleasant t o go home to take a bath with the evenly maintained temperature of hot water throughout the bathtub. This beautiful idea, however, can not be always realized by the constantly falling water temperature. Therefore, people should continually add hot water to keep the temperature even and as clo as possible to the initial temperature without wasting too much water. This paper propos a partial differential equation of the heat conduction of the bath water temperature, and an object programming model. Bad on the Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), this paper illustrates the best strategy the person in the bathtub can adopt to satisfy his desires. First, a spatiotemporal partial differential equation model of the heat conduction of the temperature of the bath water is built. According to the priority, an object programming model is established, which takes the deviation of temperature throughout the bathtub, the deviation of temperature with the initial condition, water consumption, and the times of switching faucet as the four objectives. To ensure the top priority objective—homogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysis are conducted. The simulation and analytical results all imply that the top priority strategy is: The proper motions of the person making the temper
ature well-distributed throughout the bathtub. Therefore, the Partial Differential Equation model (PDE) can be simplified to the ordinary differential equation model.
Second, the weights for the remaining three objectives are determined bad on the tolerance of temperature and the hobby of the person by applying Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Therefore, the evaluation model of the synthesis score of the strategy is propod to determine the best one the person in the bathtub can adopt. For example, keeping the temperature as clo as the initial condition results in the fewer number of switching faucet while attention to water consumption gives ri to the more number. Third, the paper conducts the analysis of the diver parameters in the model to determine the best strategy, respectively, by controlling the other parameters constantly, and adjusting the parameters of the volume, shape of the bathtub and the shape, volume, temperature and the motions and other parameters of the person in turns. All results indicate that the differential model and the evaluation model developed in this paper depends upon the parameters therein. When considering the usage of a bubble bath additive, it is equal to be the obstruction between water and air. Our results show that this strategy can reduce the dropping rate of the temperature
effectively, and require fewer number of switching.
The surface area and heat transfer coefficient can be incread becau of the motions of the person in the bathtub. Therefore, the deterministic model can be improved as a stochastic one. With the above evaluation model, this paper prent the stochastic optimization model to determine the best strategy. Taking the disparity from the initial temperature as the suboptimum objectives, the result of the model reveals that it is very difficult to keep the temperature constant even wasting plentiful hot water in reality.
Finally, the paper performs nsitivity analysis of parameters. The result shows that the shape and the volume of the tub, different hobbies of people will influence the strategies significantly. Meanwhile, combine with the conclusion of the paper, we provide a one-page non-technical explanation for urs of the bathtub.
Fall in love with your bathtub
Abstract
It’s pleasant t o go home to take a bath with the evenly maintained temperature of hot water throughout the bathtub. This beautiful idea, however, can not be always realized by the constantly falling water temperature. Therefore, people should continually add hot water to keep the temperatur
e even and as clo as possible to the initial temperature without wasting too much water. This paper propos a partial differential equation of the heat conduction of the bath water temperature, and an object programming model. Bad on the Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), this paper illustrates the best strategy the person in the bathtub can adopt to satisfy his desires. First, a spatiotemporal partial differential equation model of the heat conduction of the temperature of the bath water is built. According to the priority, an object programming model is established, which takes the deviation of temperature throughout the bathtub, the deviation of temperature with the initial condition, water consumption, and the times of switching faucet as the four objectives. To ensure the top priority objective—homogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysis are conducted. The simulation and analytical results all imply that the top priority strategy is: The proper motions of the person making the temperature well-distributed throughout the bathtub. Therefore, the Partial Differential Equation model (PDE) can be simplified to the ordinary differential equation model.
Second, the weights for the remaining three objectives are determined bad on the tolerance of temperature and the hobby of the person by applying Analytic Hierarchy Process (AHP) and Techniq
谦逊是什么意思ue for Order Preference by Similarity to Ideal Solution (TOPSIS). Therefore, the evaluation model of the synthesis score of the strategy is propod to determine the best one the person in the bathtub can adopt. For example, keeping the temperature as clo as the initial condition results in the fewer number of switching faucet while attention to water consumption gives ri to the more number. Third, the paper conducts the analysis of the diver parameters in the model to determine the best strategy, respectively, by controlling the other parameters constantly, and adjusting the parameters of the volume, shape of the bathtub and the shape, volume, temperature and the motions and other parameters of the person in turns. All results indicate that the differential model and the evaluation model developed in this paper depends upon the parameters therein. When considering the usage of a bubble bath additive, it is equal to be the obstruction between water and air. Our results show that this strategy can reduce the dropping rate of the temperature effectively, and require fewer number of switching.
The surface area and heat transfer coefficient can be incread becau of the motions of the person in the bathtub. Therefore, the deterministic model can be improved as a stochastic one. With the above evaluation model, this paper prent the stochastic optimization model to determine the best strategy. Taking the disparity from the initial temperature as the suboptimum objectives, the resu
lt of the model reveals that it is very difficult to keep the temperature constant even wasting plentiful hot
water in reality.
Finally, the paper performs nsitivity analysis of parameters. The result shows that the shape and the volume of the tub, different hobbies of people will influence the strategies significantly. Meanwhile, combine with the conclusion of the paper, we provide a one-page non-technical explanation for urs of the bathtub.
Keywords:Heat conduction equation; Partial Differential Equation model (PDE Model); Objective programming; Strategy; Analytical Hierarchy Process (AHP) Problem Statement
A person fills a bathtub with hot water and ttles into the bathtub to clean and relax. However, the bathtub is not a spa-style tub with a condary hearing system, as time goes by, the temperature of water will drop. In that conditions,
we need to solve veral problems:(1) Develop a spatiotemporal model of the temperature of the bathtub water to determine the best strategy to keep the temperature even throughout the bathtub a
nd as clo as possible to the initial temperature without wasting too much water;(2) Determine the extent to which your strategy depends on 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.(3)The influence of using b ubble to model’s results.(4)Give a one-page non-technical explanation for urs that describes your strategy
General Assumptions
1.Considering the safety factors as far as possible to save water, the upper temperature limit is t to 45 ℃;
2.Considering the pleasant of taking a bath, the lower temperature limit is t to 33℃;
3.The initial temperature of the bathtub is 40℃.
Table 1
Model Inputs and Symbols
不管你信不信
Symbols Definition Unit
T Initial temperature of the Bath water ℃
℃T
∞Outer circumstance temperature
T Water temperature of the bathtub at the every moment ℃
t Time h
会计法规x X coordinates of an arbitrary point m
y Y coordinates of an arbitrary point m
z Z coordinates of an arbitrary point m
αTotal heat transfer coefficient of the system 2
（）
⋅
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客厅阳台打通一体装修效果图
W m K
1S
The surrounding-surface area of the bathtub 2m 2S The above-surface area of water
2m 1H Bathtub’s thermal conductivity
古印度/W m K ⋅（） D The thickness of the bathtub wall
m 2H Convection coefficient of water
2/W m K ⋅（） a Length of the bathtub
m b Width of the bathtub
m h Height of the bathtub
m V The volume of the bathtub water
3m c Specific heat capacity of water
/()J kg ⋅℃ ρ Density of water
3/kg m ()v t Flooding rate of hot water
3/m s r T
The temperature of hot water ℃
Temperature Model
Basic Model
A spatio-temporal temperature model of the bathtub water is propod in this paper. It is a four dimensional partial differential equation with the generation and loss of heat. Therefore the model can be described as the Thermal Equation.
The three-dimension coordinate system is established on a corner of the bottom of the bathtub as the original point. The length of the tub is t as the positive direction along the x axis, the width is t as the positive direction along the y axis, while the height is t as the positive direction along the z axis, as shown in figure 1.
Figure 1. The three-dimension coordinate system
