1、For office use onlyT1 _T2 _T3 _T4 _Team Control Number50930Problem ChosenAFor office use onlyF1 _F2 _F3 _F4 _2015Mathematical Contest in Modeling (MCM/ICM) Summary SheetSummaryOur goal is a model that can use for control the water temperature through a person take a bath.After a person fills a batht
2、ub with some hot water and then he take a bath,the water will gets cooler,it cause the person body discomfort.We construct models to analyze the temperature distribution in the bathtub space with time changing.Our basic heat transfer differential equation model focuses on the Newton cooling law and
3、Fourier heat conduction law.We assume that the person feels comfortable in a temperature interval,consider with saving water,we decide the temperature of water first inject adopt the upper bound.The water gets more cooler with time goes by,we assume a time period and stipulation it is the temperatur
4、e range,use this model can get the the first inject water volume through the temperature decline from maximum value to minimum value.Then we build a model with a partial differential equation,this model explain the water cooling after the fill bathtub.It shows the temperature distribution and water
5、cool down feature.We can obtain the water temperature change with space and time Team #9530 Page 2 of 10by MATLAB.When the temperature decline to the lower limit,the person adds a constant trickle of hot water.At first the bathtub has a certain volume of minimum temperature of the water，in order to
6、make the temperature after mixed with hot water more closer to the original temperature and adding hot water less,we build a heat accumulation model.In the process of adding hot water,we can calculate the temperature change function by this model until the bathtub is full.After the water fill up,wat
7、er volume is a constant value,some of the water will overflow and take away some heat.Now,temperature rise didnt quickly as fill it up before,it should make the inject heat and the air convection heat difference smallest.For the movement of people, can be seen as a simple mixing movement, It plays a
8、 very good role in promoting the evenly of heat mixture. so we put the human bodys degree of motion as a function, and then establish the function and the heat transfer model of the contact, and draw the relationship between them. For the impact of the size of the bathtub, due to the insulation of t
9、he wall of the bathtub, the heat radiation of the whole body is only related to the area of the water surface, So the shape and size of the bath is just the area of the water surface. Thereby affecting the amount of heat radiation, thereby affecting the amount of water added and the temperature diff
10、erence, So after a long and wide bath to determine the length of the bath. The surface area is also determined, and the heat Team #9530 Page 3 of 10transfer rate can be solved by the heat conduction equation, which can be used to calculate the amount of hot water. Finally, considering the effect of
11、foaming agent, after adding the foam, the foam floats on the liquid surface, which is equivalent to a layer of heat transfer medium, This layer of medium is hindered by the convective heat transfer between water and air, thereby affecting the amount of hot water added. ,ContentTitile.1Summary.1Team
12、#9530 Page 4 of 101 Introduction11.1 Restatement of the Problem 11.2 Background .11.1.1 Common Solving Technique11.1.2 Previous Works 11.3 Example.12 Analysis of the Problem .12.1 Outline of the Approach12.2 Basic Assumptions 12.3 Definitions and Key Terms .13 Calculating and Simplifying the Model14
13、 The Model Results15 Validating the Model16 Strengths and Weaknesses 16.1 Strengths16.2 Weaknesses .17 Food for Thought .18 Conclusion.1References .1Appendices 1Appendix A Source Code1Appendix B 1Team #9530 Page 1 of 101 Introduction1.1 Restatement of the ProblemFrom the topic, we know that the bath
14、 is the first to join the quantitative water, With the bathing process, Water temperature gradually decreased, Until we cant accept. As is known to all, we all hope that the bath time can be a little longer, In order to keep the water temperature in the bath in the appropriate range, we have to add
15、hot water to the bath. So we have to find a way to add hot water? This method can be used to make the amount of hot water at least, and the water temperature in the bath and the original closest l. And the size of the human action and the shape of the bathtub to affect the results, Take a bath befor
16、e joining bubble economy is the result of the impact of it, All these are the problems we need to solve.1.2 Background1.3 MeaningThis problem is related to the comfort of people in the process of bathing. So it is worth our improvement and processing, Now people in the work life of the pressure is v
17、ery great, take a bath can make people get a good relaxation, Solve this problem can improve the degree of comfort of bath of people. People need not worry about the cold water, the Team #9530 Page 2 of 10intelligent control of the whole process, and can be the greatest degree of water saving, low c
18、arbon and environmental protection, the economic benefits of the bathtub reaches a maximum2 Symbol definition：Symbol DefineQ Thermal conductivityS The area of isothermal surface Thermal conductivityb Thickness of bathtubt Temperature difference between two sides of bathtubR Thermal resistanceT Bath
19、water temperatureTw Air temperature Local convective heat transfer coefficientc Specific heat capacity Densitym QualityTeam #9530 Page 3 of 103 analysis of the Problem2.1 Basic Assumption The walls of bathtub is heat insulation,namely between the water and the walls without internal energy exchange.
20、 Ignore the change of the density of water and the coefficient of the heat transfer in consequence of waters temperature changes little in the bathtub. Ignore the change of waters phase state. The water is in the uniform mixed state all the time. After doing some research, we find, for human body, t
21、he appropriate temperature of the bath water is between 36 and 42. 2.2 The Foundation of Model2.2.1 heat transfer model Through the investigation and access to datum,we set the initial water temperature as 42.And we simplify the bathtub as a simple model of cuboid.We injecting a certain amount of wa
22、ter in the cuboid and the temperature of water will reduce over time.To begin with,we established the model of heat transfer of the water in the bathtub: Newtons law of cooling: For convective heat transfer that Team #9530 Page 4 of 10phase state changed uniformity,the equation of convection heat tr
23、ansfers rate indicated as:(2-1)()1wwTdQTdSSIn the engineering calculation,the coefficient of the heat transfer is often used,Newtons law can be indicated as(2-2)1/tQSThis equation applies to convective heat transfer betweenthe water in the bathtub and air. Fouriers law: The equation of heat transfer
24、 between the water and the walls of bathtub,shows that convection heat transfers rate is proportional to the temperatures gradient and heat transfers area.(2-3)tdQSnFor flat walls heat conduction,(2-3) can be simplified as tQSdxCalculating and Simplifying the Model4 The Model ResultsTeam #9530 Page 5 of 105 Validating the Model6 Strengths and Weaknesses6.1Strengths6.2Weaknesses7 Food for Thought8 Conclusion.ReferencesTeam #9530 Page 6 of 10AppendicesAppendix A Source CodeHere are the simulation programmes we used in our model as follow.Input matlab source: .Appendix B .Input C+ source：
