國際數(shù)學建模論文二等獎

1、<p>　　The secret of pans</p><p><b>　　Abstract</b></p><p>　　In this article, we research the influence of pan-shape on the maximum number of pans in the oven and heat distributio

2、n of pans. First of all, we analyse working principle of the oven and learn that the main method of heat exchange in oven is convective heat transfer. To make this simpler, we assume the pan-shape are regular polygon and

3、 internal thermal environment in the oven is same.</p><p>　　In problem one, in order to find out a relationship between the shape and amount of pans that are put into the oven, we make researches on various

4、shapes of pans. The area of each pan is A, we can use the the maximum number of pans to stand for the space utilization of oven while its area is certain. With seamless splicing between baking pans, we will fulfill the m

5、aximum utilization of the oven. According to the multiple relationship between the polygon interior angle and the circumferential ang</p><p>　　In problem two, we make researches on different shapes of pans s

6、o as to find out how the shape of each pan itself influence heat distribution when heated. We built finite element analysis model, and Simulate the heat distribution of each pan in the oven with the help of software ansy

7、s 10.0. Then the functional relationship between the pan-shape and heat distribution is Deduced. Analysing the functional relationship, we found that the pan's heat distribution uniform degree increases with the incr

8、</p><p>　　In problem three, we study the optimization problem in the case of taking into account the quantity and distribution of heat pans. Giving weight p and (1-p) to the number of pans and the heat distr

9、ibution of the pan. We can know the relationship between pan-shape and the maximum number of the pans that can be arranged and the degree of heat distribution of pans by the solving the above problems. List the relation

10、of and ，，, then establish the optimization model. We can know the preferred shape </p><p>　　Keywords:pan,oven,finite element model,optimization model,seamless arrangemen</p><p>　　Introduction&l

11、t;/p><p>　　When baking in a rectangular pan heat is concentrated in the 4 corners and the product gets overcooked at the corners (and to a lesser extent at the edges). In a round </p><p>　　pan the

12、heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges. However, since most ovens are rectangular in shape using round pans is not efficient with respect to using the space i

13、n an oven.We need develop a model to show the distribution of heat across the outer edge of a pan for pans of different shapes - rectangular to circular and other shapes in between.</p><p>　　When baking,the&

14、#160;number of pans that the oven can accommodate and the heat distribution in each pan are problems we usually considering.When the total area of the oven is certain and the area

15、of the pan is the same, different shapes of pan make different oven space utilization. Efficient use of oven space is reflected in the number </p><p>　　of pa

16、n it can accommodate, so we are looking for  the best shape of the pan so that  it </p><p>　　can put more quanti

17、ties of pans in a limited area of the oven.If the pans heat evenly distributed,we could avoid overcooking some parts of the food.So we want to find the best pan-shape so that the heat

18、is more evenly distributed when heated.To consider both the number and the heat distribution of pans,we should give weight and () to the number of pans and the heat distribution in each pan to identify the best pan-shap

19、e in the different weights.</p><p><b>　　Notation</b></p><p>　　Assuptions</p><p>　　A width to length ratio of W/L for the oven which is rectangular in shape. Each p

20、an must have an area of . Initially two racks in the oven, evenly spaced.Each pan must be the regular polygon or round.</p><p>　　The area of oven is certain.</p><p>　　Put the same types o

21、f pans into the oven in the same batch.</p><p>　　The thermal environment of oven is certain.</p><p><b>　　Problem1</b></p><p>　　For the first problem,what we take into co

22、nsideration is the maximum number of pans that can be put into the oven. That’s to say ,we need to choose the best shape of pans so that we can make full use of a oven whose area is certain.</p><p>　　In our

23、opinion ,when the pans are seamlessly connected,we can make the maximum utilization of the oven.</p><p>　　the number of polygon at the seam, the interior angle degree of regular polygon, the number of sides

24、of the polygon are assumed.</p><p>　　According to the polygon interior angles and formulas we could get the following formulae:</p><p><b>　　→ </b></p><p>　　Satisfy the

25、premise of a regular polygon seamless , we got the following data .</p><p>　　That is to say among all regular polygons, only equilateral triangle, square and regular hexagon can be connected seamlessly.So we

26、 take triangle, square and regular hexagon into discussion.</p><p>　　We do an assumption that gets the total number of every shape of pans that can be arranged in a certain area oven.The results are as the f

27、ollowing.</p><p><b>　　Table 1.</b></p><p>　　So if S and are given, the number of each shape of pans is determined.</p><p>　　Taking into consideration the actual situati

28、on about covering and aesthetics of oven , we assume that the range of aspect ratio is 0.4 to 1 .Then list the results here.</p><p><b>　　Table 2.</b></p><p>　　Ovens in different aspe

29、ct ratios corresponding to the number of polygons of each shape.</p><p>　　From this model ,we find that it is theoretically true that square can be placed up to the maximum number when the oven’s area is cer

30、tain.</p><p><b>　　Problem2</b></p><p>　　(1）The basic theory of Temperature Analysis:</p><p>　　There are two energy transfer when we heating food by oven,one is convectiv

31、e heat transfer ,the other is radiation heat transfer,convective heat transfer is the most important way of the heat exchanger.we can know the Incompressible fluid continuity equation by the physics knowledge:</p>

32、<p>　　According to the theorey of kinetic energy , we know that resultant force to do work is equal to the amount of change in kinetic energy .For the fluid of arbitrary infinitesimal control-volume, all forces tha

33、t act in the volume are equal to fluid momentum rate in control-volume.The external forces include the surface force, (normal pressures and tangential viscous force) and volume force.We can get the Momentum differential

34、equations in three directions according to the conservation relations.</p><p>　　Symbol description:</p><p>　　--air density，--time,</p><p>　　--component of forces of volumetric force

35、 in the x,y,z direction，</p><p>　　--pressure field ，--viscosity coefficient</p><p>　　Resultant force to do work is equal to the heat which is part of work done by viscous stress to control-volum

36、e in unit time.According to the first law of thermodynamics we can get the following energy differential equations .</p><p>　　Symbol description:</p><p>　　--Specific heat capacity in the air und

37、er a certain pressure.</p><p>　　--The coefficient of thermal conductivity of the air</p><p>　　--temperature field </p><p>　　We know that the temperature is constant in the oven, so

38、 we can solve these 5 quantities()about the fluid.</p><p>　　Build finite element analysis model</p><p>　　1. The pan in the oven heating analysis</p><p>　　We will simplify the differ

39、ent shapes of the pan to the thin polyhedral before we analyse.As an example, we analyse triangle pans' heat distribution in the oven.</p><p><b>　　.</b></p><p>　　Figure 1. Mimic

40、 diagram that heat flows through the triangle</p><p>　　Parameter setting</p><p>　　Assuming a constant temperature inside the oven is 200 . The initial temperature of the pan is set to 0 . The ov

41、en heat transfer coefficient is set to 125 . The specific heat of the pan is set at 500. Density of the pan is set to 7000.The thermal conductivity of pan is set to 50 . The heating time is set to 60s.</p><p&g

42、t;　　3.Software simulation</p><p>　　We can analyse the various shapes of pans by Finite element analysis method and software named ansys10.0 after we know pans' heat distribution in the oven.</p>&

43、lt;p>　　The analytical results are as follows:</p><p>　　Figure 2. Distribution of heat in the different shapes of the pan after heating</p><p>　　Analysis of the results</p><p>　　

44、According to the analysis of the results,it is obvious that with most uniform temperature distribution of the circular pan , and the numbers of sides of the pan shape , temperature of pan is more and more uniform.</p&

45、gt;<p>　　A temperature range R of each shape of the pan from the figure can be seen, and it was summarized as follows :</p><p><b>　　Table 3.</b></p><p>　　Temperature different

46、ials of pans in different shapes</p><p>　　Analysing pan’s variable with temperature range relationship, with software spss19.0 for regression analysis,and the result is following:</p><p><b&

47、gt;　　Table 4.</b></p><p>　　The regression analysis results</p><p>　　So the relationship was</p><p>　　Pan evenly distributs heat and temperature range is linear correlation.,an

48、d is the proportional coefficient.</p><p>　　The smaller , the more uniform heat the pan distributs. When, is the Least. So, the most uniform heat pan is circular.</p><p><b>　　Problem3<

49、;/b></p><p>　　For the third problems, pan-shape’s selection depends on the number of pans() and the Heat distribution ()of the pan in the oven.What we should do is to make as large as possible and let a

50、s small as possible.</p><p>　　We assume the following formula:</p><p>　　In this case,the problem we need to solve is the maximum value of Z.We divide the problem into three cases as the followin

51、g.</p><p>　　Case1:When，</p><p>　　In this case,only the quantity is considered to select the shape of pans. And it is the same with the first question , so we choose the square pans.</p>&

52、lt;p>　　Case2:When，</p><p>　　In this case,we just consider the factors of the heat distribution to select the shape of pans.And it is the same with the second question , so we choose the round pans.</p&

53、gt;<p>　　Case3：When,</p><p>　　Based on case 1 and case 2 we can know that pan-shape should be between square and round if we want to get maximum of Z. </p><p>　　Based on case 1 we know th

54、at when aspect ratiois certain,the maximum number of Polygonal pans can be certain.</p><p>　　Symbol description:</p><p>　　--The maximum number of round pan can be put in the oven.</p><

55、;p>　　--The maximum number of regular hexagon pan can be put in the oven.</p><p>　　When aspect ratiois certain, we can solve the value of and by the method that is proved in the first problem.Then we ge

56、t the range of .</p><p>　　Based on case 2 we can obtain the following model.</p><p>　　So </p><p>　　When aspect ratiois certain, is a function ofin Formula 1.<

57、;/p><p>　　We can get the number of sides of the pan by the above equation,and then get the shape of the pan .</p><p>　　Evaluating the Model</p><p>　　Strengths：</p><p>　　Th

58、e computer simulated the heated situation of the pan in the oven heating, making the results clearly visible.</p><p>　　Using reasonable assumptions simplified the problem.</p><p>　　The model com

59、bined with the actual situation, having good practical significance。</p><p>　　Weaknesses：</p><p>　　The accurate mathematical relation between the largest number of pan can be arranged and pan

60、shape can’t be established.</p><p>　　Setting same thermal environment to the oven, and the certain error will be in the model. </p><p>　　References</p><p>　　[1]XIE Benming,JU

61、Hongchao.Application of Ansys on Analyzing Temperature Field of Space Above Zinc Plating Bath. Journal of civil aviation university of china,Vol.26,No.1,February, 2008.</p><p>　　[2]SaeedMoaveni. Finite Eleme

62、nt Analysis Theory and Apoplication with Ansys. Beijing: Publishing House of Electionics Industry.,2003.</p><p>　　[3]Zhonggeng Han. Mathematical modeling methods and applications. Beijing: The PLA Informatio

63、n Engineering University,2005.</p><p>　　[4]http://www.madio.net</p><p>　　[5]http://www.comap.com/undergraduate/contests/mcm</p><p>　　Teaching you how to choose the pan</p>&l

64、t;p>　　As the living standard increases, most families have easy access to ovens in our daily life. While we are using the oven, we usually find out that baking in a rectangular pan heat is concentrated on the four cor

65、ners and the product gets overcooked at the corners, but when we use a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges. However, since most ovens are re

66、ctangular in shape, using round pans is not efficient with respect to</p><p>　　To solve the multiobjective programming problem, we build the most compact arranged model, finite element analysis model and opt

67、imization model. We will make a brief introduction to our analysis of the procession the following.</p><p>　　When we take the amount of pans contained at one time into consideration only, our goal is to maxi

68、mize the number of pans in the oven. Finally we find out that only equilateral triangle, square and regular hexagon can fulfill seamless splicing. However, no matter how the value of aspect ratio changes, the amount of s

69、quare pans holded per time is always the largest.</p><p>　　With the help of ansys10.0,we make simulation of each shape in heated case. By analysing the temperature distribution of graphs which come from the

70、simulation, we come to the conclusion that when the number of sides approaches infinity, heat distribution is the most uniform. That is to say, the round-pan’s heat distribution is the most even.</p><p>　　Gi

71、ving weight and to the number of pansand the heat distribution of the pan. When taking into account the quantity and distribution of heat pans,we establish the optimization model. We can know the preferred shape of pan

72、that should be selected when the and are given.</p><p>　　Now we have several ways to choose, if you want to cook as more food as possible, you should choose square pan;if you want to cook food of good quali

73、ty, you should choose round-pan. When you have both requirements to these two aspects, we recommend you to choose the Hexagon and Octagon pan.</p><p>　　*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ^!d

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