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 objective homogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analys
6、is 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 differentia
7、l 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 evaluatio
8、n 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
11、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 increased because of the motions of the person in the bathtub. Therefore, the deterministic model can be improved as a sto
12、chastic 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 constan
13、t 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 p
14、rovide 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 re
15、alized 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 ba
16、th 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 spatiotempor
17、al 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 objective homogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysis are conducted. The simulation and analytical resul
19、ts 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 remaini
20、ng 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 pro
21、posed 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 th
22、e 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
23、 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
24、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 stochastic one. With the above evaluation model, this pa
25、per 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
26、 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 provide a one-page non-techni
27、cal 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
29、the 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 b
30、y 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.Consi
31、dering 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 0T Initial temperature of the Bath water T Outer circumstance temperature T Water temperature of the bathtub at
32、the every moment t Time 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 1S The surrounding-surface area of the bathtub 2m 2S The above-surface a
33、rea of water 2m 1H Bathtubs thermal conductivity /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 D
34、ensity of water 3/kgm ()vt Flooding rate of hot water 3/ms rT 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 and loss of heat. Th
35、erefore 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 direction along t
36、he 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 each point in spac
37、e; 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: 222 122 2 2 ( , , , ) ( , , , )() f x y z t f x y z tT T T Tt x y z c V (1) Where t refers to time; T is the temperature o
38、f any point in the space; 1f is the addition of exogenous thermal energy; 2f 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 level exists among these
39、 objectives, while keeping the temperature even throughout the bathtub must be ensured. Objective 1( .1O ): keep the temperature even throughout the bathtub; 22100m in ( , , , ) ( , , , )ttVVF t T x y z t d x d y d z d t t T x y z t d x d y d z d t (2) Objective 2( .2O ): keep the temperature as clo
40、se as possible to the initial temperature; 2200m in ( , , , )t VF T x y z t T d x d y d z d t (3) Objective 3( .3O ): do not waste too much water; 3 0m in tF v t dt (4) Objective 4( .4O ): fewer times of switching. 4 minFn (5) Since the .1O is the most crucial, we should give priority to this object
41、ive. Therefore, the highest priority strategy is given here, which is homogenization of temperature. Strategy 0 Homogenization of Temperature The following three reasons are provided to prove the importance of this strategy. Reason 1-Simulation In this case, we use grid algorithm to make discretizat
42、ion 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 . 00. 20. 40. 60. 81 00. 511. 5200. 5L e n g t hW i d t hHei
43、ght4242. 54343. 54444. 54545. 5D i s t r i b u t i o n o f t e m p e r a t u r ea t t h e l e n g t h = 1D i s t r i b u t i o n o f t e m p e r a t u r ea t t h e w i d t h = 1H o t w a t e rC o o l w a t e rFigure 2. Temperature profiles in three-dimension space without manual intervention (2) Add
44、ing manual intervention, the distribution of water temperature as shown in figure 3. And the variance of the temperature is 0.005 . 00 . 51 00 . 511 . 5200 . 5L e n g t hW i d t hHeight4 4 . 74 4 . 7 54 4 . 84 4 . 8 54 4 . 94 4 . 9 545D i s t r i b u t i o n o f t e m p e r a t u r ea t t h e l e n
45、g t h = 1D i s t r i b u t i o n o f t e m p e r a t u r ea t t h e w i d t h = 1H o t w a t e rC o o l w a t e rFigure 3. Temperature profiles 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 w
46、e add some manual intervention. Therefore, we can assumed that 2222 2 2( ) 0TTTx y z in formula (1). Reason 2-Estimation If the temperature of any point in the space is different, then 2222 2 2( ) 0TTTx 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 #5288
47、8 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 1F could be estimated as follows: 220020 0 0 0 1 1 1 1( , , , ) ( , , , )( , , , ) ( , , , ) 0ttVVt T x y z t d x d y d z d t t T x y z t d x d y d z d tT 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 0F . So we can assumed that: 2222 2 2( ) 0TTTx 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 simpli
