2016美赛A题论文.docx-资料库

198eb13e-33f6-4517-b823-83630d2827fa.docx.pdf-第1页.png

第1页 / 共19页

198eb13e-33f6-4517-b823-83630d2827fa.docx.pdf-第2页.png

第2页 / 共19页

198eb13e-33f6-4517-b823-83630d2827fa.docx.pdf-第3页.png

第3页 / 共19页

198eb13e-33f6-4517-b823-83630d2827fa.docx.pdf-第4页.png

第4页 / 共19页

198eb13e-33f6-4517-b823-83630d2827fa.docx.pdf-第5页.png

第5页 / 共19页

198eb13e-33f6-4517-b823-83630d2827fa.docx.pdf-第6页.png

第6页 / 共19页

198eb13e-33f6-4517-b823-83630d2827fa.docx.pdf-第7页.png

第7页 / 共19页

198eb13e-33f6-4517-b823-83630d2827fa.docx.pdf-第8页.png

第8页 / 共19页

For office use only T1 T2 T3 T4 For office use only F1 F2 F3 F4 Team Control Number 46731 Problem Chosen A 2016 MCM/ICM Summary Sheet (Your team's summary should be included as the first page of your electronic submission.) Type a summary of your results on this page. Do not include the name of your school, advisor, or team members on this page. Summary Our experience of everyday life tells us that, bathing quality is relevant to the temperature of the water. Hence the optimality of the water-adding strategy in bathing process has become an issue. In this article, we establish two models. One is water temperature variation and distribution model and the other one is finding best strategy model. We put forward some acceptable hypothesis to simplify the model. Wha’st of the word “noticeably”. more, we clear the meaning The first part has five sections: air’s heat radiation, bathtub wall’s heat radiation, person in, hot water in, bubble existed. We discuss some factors that affect water temperature, such as the shape and the volume of the bathtub and person, especially the motions made by the person in the bathtub because the temperature in the bathtub has a great connection with person. Finally, we get the water temperature variation and distribution model. The second part aims to finding the best water strategy. Different strategies are considered for adding hot water and the amount of the hot water. The key of strategies is how to choose the moment of opening the faucet and the lasting time. By developing an optimization model of the strategy, we manage to determine a strategy for different users and extend the model with a finite volume method. Under the condition of the same bathing time, we study what’s the ideal time to add the hot water to make it to the initial temperature.

Team # 46731 Page 2 of 19 We transform the weight p. With the restrict condition of the temperature in the bath and the amount of the hot water, the question can be transformed to a nonlinear optimization. We get the conclusion and optimal results through the PDE (Partial Differential Equations) toolbox of MATLAB. And the result shows a clear superiority for choosing a suitable strategy for different users. Key words: bathtub; heat-conduction; temperature variation and distribution; water strategy; finite volume method;

Team # 46731 Page 3 of 19 Contents Summary.............................................................................................................1... Introduction ...................................................................................................4... I. Analysis .....................................................................................................4... II. III. Assumptions ..............................................................................................6... IV. Notation .....................................................................................................7... V. Models .......................................................................................................8... Part I Water Temperature Variation and Distribution Model ......................... 8.. Air ’s Heat Radiatio..n......................................................................8.. Bathtub Wall ’s Heat Radia..t.i.o..n.........................................................9. Person In .........................................................................................9... HotWaterIn...................................................................................10 Bubble Existed...............................................................................11 Water Temperature Variation and Distribution Mode.l......................11 Part II Finding the best strategy.................................................................12 VI. Results ....................................................................................................1..2.. Part I..........................................................................................................12. Results of Air ’s Heat Radiation M...o..d..e..l.....................................1. 2 ’s Heat R..a..d.i.a..t.i.o..n............................. 1. 3 Results of Bathtub Wall Results of “Perso.n...I.n.....”..........................................................13 Results of “Hot Wate.r..I.n......”....................................................13 Results of bubble existed model ...................................................15. Results of Water Temperature Variation and Distribution Mode.l... 15 Part II ........................................................................................................16. VII. Strengths and Weaknesses..................................................................1..7... VIII. Explanation for users ...........................................................................18. IX. References...................................................................................................19.

Team # 46731 Page 4 of 19 I. Introduction A hot water bath can relax people's body and it is one of the most popular ways to relax ever since. However, people are often troubled by the water becauseit is getting colder and colder. In this case, it is crucial to study the water temperature changes with space and time, and to provide people with the best bath strategy. The interaction and system in the bat is rather complex, include the heat transfer, convection and radiation. The heat transfer satisfies Fourier Diffusion and heat, spread fasted under the direction of gradient. Convection is the interaction of the water in the bath where high energy spreads to lower one. In order to study how the different strategies affect the water temperature, we need first establish an equation to reflect this kind of transformation and heat transfer equation is exactly the one. But the difficult is that the equation is under the ideal situation which has a significant difference to the ordinary condition. By combining the shape/volume/temperature of the person in the bathtub, we get a comprehensive view over the pan designs. We can provide different strategy according to the motions made by the customers. II. Analysis With the purpose of raising water temperature as close as possible to the initial temperature, and wasting less water as far as it could be, so the key point will be the power of losing heat on which controlling the heating time and the flow of hot water depends . However, there are many factors impact the power. Factors of the bathtub Shape If the total amount of water needed to bath is changeless, we can assume that the volume of bathtub is constant, too. Energy lost by evaporation of water positively correlated with the water-gas contact area. So it will be better if surface of tub smaller, satisfying the basic human needs of the bath. the intersecting Volume The amount of water that can set is directly determined by the volume of bathtub. Define a system about the water in the bathtub. With the greater the amount of water, the system is more stable, in the case of equal energy interference. Therefore, it is more reasonable to choose a bigger bathtub in the condition of same intersecting

Team # 46731 surface. Factors of Human Page 5 of 19 Person in bathtub is serious mess to original bathtub system. With the influence of human shape, temperature, behavior, the power of losing heat could be volatile. Thereby the water strategy should fit these. It Feeling is a very vague concept to human about the feeling of hot and cold, considering the difference of person. There are two comprehensions about the word “noticeably: ” (1) the rate of temperature changing reaches a certain value, the man feels "cooler"; (2) the changes of water temperature reachesa certain value than the man feels "cooler". We consider the second condition. Temperature Various parts of person in the water get different temperature, but the human circulation and other physiological system will to keep integral temperature balance. It is reasonable to assume that human body maintains a certain temperature. transfer heat Shape Different shapes in the bathtub directly impact on contact area. People will absorb heat in the system, by the area of contact between man and water, thereby affecting additional cooling power of the system. Behavior It is absurd to consider person as a stationary object. Person will inevitably make a variety of actions, which will stimulate waves on the water surface, increasing the surface area of evaporation and changing power. Based on the research of wave, a coefficient is invented to represent the relation between water surface and the active extent of human.

Team # 46731 Page 6 of 19 Figure 1 models flowchart III. Assumptions The air temperature remains constant. The air above the water will be heated, but as the air temperature rises, the density becomes smaller and the air will rise. In the meanwhile, the fresh air adds to the bottom space. So we assume the air temperature remains constant. Evaporation doesn’t lose water quality. Evaporation would reduce the quality of water, but compare to the whole bathtub, this part is so small that it may be entirely neglected. The bathtub has the double layer and vacuum wall. The wall of bathtub will take heat when contacting with water. Most of the bathtubs have the double layer and vacuum wall. So we just consider the layer which contact with water. The bubble covers the water smoothly.

Team # 46731 Page 7 of 19 IV. Notation Symbol Meaning Water temperature Initial water temperature The temperature range of water and bathtub The temperature range of water and human The height of wave Water-bathtub contact area Water-person contact area Water-air contact area(no person) Water-air contact area(person in) Heat transfer coefficient Evaporation coefficient(no bubble) Evaporation coefficient(bubble in) Water specific heat capacity Water density Water volume Heat source intensity Time Add hot water moment The flow of hot water ?? The dry-bulb temperature of the air The extent of human activity

Team # 46731 Page 8 of 19 V. Models Part I Water Temperature Variation and Distribution Model We see the water in the bathtub as a whole system. The average temperature of the system is affected by following factors. Air ’ s heat radiation Bathtub ′swall Human body Motions Human cross section Add in hot water Bubble effects Increase { Decrease{ { Other dissipative We don’ctonsider other dissipative. Air ’s Heat Radiation Figure 2 bathtub system As to the heat radiation between the water and the air, we can see it as the hot water cool naturally without person in the tub. The three ways to radiation from the water are heat loss through convection, loss through evaporation and loss from cooling. According to the total heat radiation of the unit time [1],we have ????= ??????+ ??????+ ??????= [??( - ??) + ??(?"?- ????) + ????(+ 273) 4]???? ?? Where ??????is the heat loss through convection of the unit time,??????is the heat loss through evaporation of the unit time,??????is the heat loss through convection from cooling of the unit time; ?? is the coefficient of heat transfer, unit ??/(?? ?℃);T is the air, unit℃ ; dF is water temperature, unit ℃;??is the dry-bulb temperature of water-air contact area, unitm2; ??is the evaporation coefficient, unit ??/(?? ??????；) ??" ?? pressure of 5.6×10 -8 ??/(?? ???4). the wet air, unit hPa; ??is the optical density which is 0.97. ??is the surface of the water, unit hPa; ?? is the vapour is the vapour pressure of ?? As a result, we know the heat dissipating power Q1 of the areaSis ?? ??(S) = ∫ [??( - ??) + ??(?"?- ??) + ??(?? + 273) 4]????= ??? ?? ?? Where ??represents[??( - ??) + ??(?"?- ??) + ??(?? + 273) 4] 。 ?? ??

资料库

2016美赛A题论文.docx

相关推荐

考试认证

热门标签

最新资料