1、For office use only T1 _ T2 _ T3 _ T4 _ Team Control Number 52888 Problem Chosen A For office use only F1 _ F2 _ F3 _ F4 _ Mathematical Contest in Modeling (MCM/ICM) Summary Sheet Summary Its pleasant to go home to take a bath with the evenly maintained temperature of hot water throughout the bathtu
2、b. 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 close as possible to the initial temperature without wasting too much water. This paper proposes a partial d
3、ifferential equation of the heat conduction of the bath water temperature, and an object programming model. Based 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 ca
4、n 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, th
5、e deviation of temperature with the initial condition, water consumption, and the times of switching faucet as the four objectives. To ensure the top priority objectivehomogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysi
6、s 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
7、 equation model. Second, the weights for the remaining three objectives are determined based 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
8、 model of the synthesis score of the strategy is proposed to determine the best one the person in the bathtub can adopt. For example, keeping the temperature as close as the initial condition results in the fewer number of switching faucet while attention to water consumption gives rise to the more
9、number. Third, the paper conducts the analysis of the diverse 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
10、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 s
11、how 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 increased because of the motions of the person in the bathtub. Therefore, the deterministic model can be improved as a stoc
12、hastic one. With the above evaluation model, this paper present 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
13、 even wasting plentiful hot water in reality. Finally, the paper performs sensitivity 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 pr
14、ovide a one-page non-technical explanation for users of the bathtub. Team #52888 Page 1 of 23 Fall in love with your bathtub Abstract Its pleasant to 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 rea
15、lized by the constantly falling water temperature. Therefore, people should continually add hot water to keep the temperature even and as close as possible to the initial temperature without wasting too much water. This paper proposes a partial differential equation of the heat conduction of the bat
16、h water temperature, and an object programming model. Based 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 spatiotempora
17、l 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,
18、 water consumption, and the times of switching faucet as the four objectives. To ensure the top priority objectivehomogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysis are conducted. The simulation and analytical results
19、 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
20、 three objectives are determined based 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 propo
21、sed to determine the best one the person in the bathtub can adopt. For example, keeping the temperature as close as the initial condition results in the fewer number of switching faucet while attention to water consumption gives rise to the more number. Third, the paper conducts the analysis of the
22、diverse 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 i
23、ndicate 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
24、 the temperature effectively, and require fewer number of switching. The surface area and heat transfer coefficient can be increased because 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 pape
25、r present 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 Team #52888 Page 2 of 23 w
26、ater in reality. Finally, the paper performs sensitivity 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-technica
27、l explanation for users 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 settles into the bathtub to clean and relax.
28、However, the bathtub is not a spa-style tub with a secondary hearing system, as time goes by, the temperature of water will drop. In that conditions, we need to solve several problems:(1) Develop a spatiotemporal model of the temperature of the bathtub water to determine the best strategy to keep th
29、e temperature even throughout the bathtub and as close 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
30、the person in the bathtub.(3)The influence of using bubble to models results.(4)Give a one-page non-technical explanation for users that describes your strategy General Assumptions 1.Considering the safety factors as far as possible to save water, the upper temperature limit is set to 45 ; 2.Conside
31、ring the pleasant of taking a bath, the lower temperature limit is set to 33 ; 3. The initial temperature of the bathtub is 40 . Table 1 Model Inputs and Symbols Symbols Definition Unit 0 TInitial temperature of the Bath water T Outer circumstance temperature TWater temperature of the bathtub at the
32、 every moment tTime h x X coordinates of an arbitrary point m y Y coordinates of an arbitrary point m z Z coordinates of an arbitrary point m Total heat transfer coefficient of the system 2 / W m K （ ） Team #52888 Page 3 of 23 1 SThe surrounding-surface area of the bathtub 2 m 2 SThe above-surface a
33、rea of water 2 m 1 H Bathtubs thermal conductivity / W m K （ ） DThe thickness of the bathtub wall m 2 H 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 3 m c Specific heat capacity of water /
34、 ( ) J kg Density of water 3 / kg m () vt Flooding rate of hot water 3 / ms r T The temperature of hot water Temperature Model Basic Model A spatio-temporal temperature model of the bathtub water is proposed in this paper. It is a four dimensional partial differential equation with the generation an
35、d 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 set as the positive direction along the x axis, the width is set as the positive
36、 direction along the y axis, while the height is set as the positive direction along the z axis, as shown in figure 1. Figure 1. The three-dimension coordinate system Team #52888 Page 4 of 23 Temperature variation of each point in space includes three aspects: one is the natural heat dissipation of
37、each point in space; the second is the addition of exogenous thermal energy; and the third is the loss of thermal energy. In this way, we build the Partial Differential Equation model as follows: 222 12 2 2 2 ( , , , ) ( , , , ) () f x y z t f x y z t T T T T t x y z c V (1) Where t refers to time;
38、T is the temperature of any point in the space; 1 f is the addition of exogenous thermal energy; 2 f is the loss of thermal energy. According to the requirements of the subject, as well as the preferences of people, the article proposes these following optimization objective functions. A precedence
39、level exists among these objectives, while keeping the temperature even throughout the bathtub must be ensured. Objective 1( .1 O ): keep the temperature even throughout the bathtub; 2 2 1 00 min ( , , , ) ( , , , ) tt VV F t T x y z t dxdydz dt t T x y z t dxdydz dt (2) Objective 2( .2 O ): keep th
40、e temperature as close as possible to the initial temperature; 2 20 0 min ( , , , ) t V F T x y z t T dxdydz dt (3) Objective 3( .3 O ): do not waste too much water; 3 0 min t F v t dt (4) Objective 4( .4 O ): fewer times of switching. 4 min Fn (5) Since the .1 O is the most crucial, we should give
41、priority to this objective. Therefore, the highest priority strategy is given here, which is homogenization of temperature. Strategy 0 Homogenization of T emperature The following three reasons are provided to prove the importance of this strategy. Reason 1-Simulation In this case, we use grid algor
42、ithm to make discretization of the formula (1), and simulate the distribution of water temperature. (1) Without manual intervention, the distribution of water temperature as shown in Team #52888 Page 5 of 23 figure 2. And the variance of the temperature is 0.4962 . 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2
43、0 0.5Length WidthHeight 42 42.5 43 43.5 44 44.5 45 45.5 Distribution of temperature at the length=1 Distribution of temperature at the width=1 Hot waterCool waterFigure 2. Temperature profiles in three-dimension space without manual intervention (2) Adding manual intervention, the distribution of wa
44、ter temperature as shown in figure 3. And the variance of the temperature is 0.005. 0 0.5 1 0 0.5 1 1.5 2 0 0.5Length WidthHeight 44.7 44.75 44.8 44.85 44.9 44.95 45 Distribution of temperature at the length=1 Distribution of temperature at the width=1 Hot waterCool waterFigure 3. Temperature profil
45、es in three-dimension space with manual intervention Comparing figure 2 with figure 3, it is significant that the temperature of water will be homogeneous if we add some manual intervention. Therefore, we can assumed that 222 2 2 2 ( ) 0 TTT x y z in formula (1). Reason 2-Estimation If the temperatu
46、re of any point in the space is different, then 222 2 2 2 ( ) 0 TTT x y z Thus, we find two points 1 1 1 1 ( , , , ) x y z t and 2 2 2 2 ( , , , ) x y z t with: Team #52888 Page 6 of 23 1 1 1 1 2 2 2 2 ( , , , ) ( , , , ) T x y z t T x y z t Therefore, the objective function 1 F could be estimated a
47、s follows: 2 2 00 2 0 0 0 0 1 1 1 1 ( , , , ) ( , , , ) ( , , , ) ( , , , ) 0 tt VV t T x y z t dxdydz dt t T x y z t dxdydz dt T x y z t T x y z t (6) The formula (6) implies that some motion should be taken to make sure that the temperature can be homogeneous quickly in general and 1 0 F . So we c
48、an assumed that: 222 2 2 2 ( ) 0 TTT x y z . Reason 3-Analytical analysis It is supposed that the temperature varies only on x axis but not on the y-z plane. Then a simplified model is proposed as follows: 2 sin 0 0 0, 0, , 0 0 ,0 0 0 t xx x T a T A x l t l T t T l t t T x x l (7) Then we use two ways, Fourier tra
