国际数学建模论文二等奖.doc

team #20427 第 12 页 共2n 页the secret of pansabstractin this article, we research the influence of pan-shape on the maximum number of pans in the oven and heat distribution 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 internal thermal environment in the oven is same.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 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 maximum utilization of the oven. according to the multiple relationship between the polygon interior angle and the circumferential angle, we come to a conclusion that only equilateral triangle, square and regular hexagon can fulfill seamless splicing.taking the shape of oven in our daily life into consideration, we set the value of aspect ratio arrange from 0.4 to1. with each value of aspect ratio, we make an arrangement of equilateral triangle, square and regular hexagon pan. finally, we reach the conclusion that it is square pan whose amount is the largest when put into a certain area of oven. further studies also show that the utilization of oven is lower if the number of sides of a regular polygon increase, we also find out that circle pans utilization of oven is the lowest. when , the number of pans can be arranged in the oven is reduced with the decreased of the the polygon edge number of the pan().in problem two, we make researches on different shapes of pans so 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 ansys 10.0. then the functional relationship between the pan-shape and heat distribution is deduced. analysing the functional relationship, we found that the pans heat distribution uniform degree increases with the increase of the polygon edge number of the pan(). when the number of sides approaches infinity, heat distribution is the most uniform. that is to say,the round-pans heat distribution is the most uniform.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 distribution 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 of and ，, then establish the optimization model. we can know the preferred shape of pan that should be selected when the and is given.keywords:pan,oven,finite element model,optimization model,seamless arrangemenintroductionwhen 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 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 rectangular in shape using round pans is not efficient with respect to using the space in 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.when baking,thenumberofpans that theovencanaccommodate and the heat distribution in each pan are problemsweusuallyconsidering.when the total area of the oven is certain and the area of the pan is the same, different shapes of pan make different oven space utilization. efficientuseofovenspaceisreflectedin thenumberofpanitcanaccommodate,sowe arelookingforthebestshapeofthepan sothat itcanputmore quantities of pansinalimitedareaoftheoven.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 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-shape in the different weights.notationnamedescriptionside length of regular polygon.height of equilateral triangle.the distance between the regular hexagon parallel sides.the area of a pan.the number of pans in each line in the oven.the number of pans rows in the oven.the number of pans that can fit in the oven.the width of the oven.the length of the oven.area of the oven.temperature range of pansthe maximum temperature of pansthe minimum temperature of pansassuptionsa width to length ratio ofw/l for the oven which is rectangular in shape. each pan must have an area of. initially two racks in the oven, evenly spaced.each pan must be the regular polygon or round.the area of oven is certain.put the same types of pans into the oven in the same batch.the thermal environment of oven is certain.problem1for the first problem,what we take into consideration is the maximum number of pans that can be put into the oven. thats 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.in our opinion ,when the pans are seamlessly connected,we can make the maximum utilization of the oven.the number of polygon at the seam, the interior angle degree of regular polygon, the number of sides of the polygon are assumed.according to the polygon interior angles and formulas we could get the following formulae: satisfy the premise of a regular polygon seamless , we got the following data .that is to say among all regular polygons, only equilateral triangle, square and regular hexagon can be connected seamlessly.so we take triangle, square and regular hexagon into discussion.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 following.table 1.equilateral trianglesquareregular hexagonassumptionsthe triangle side length and high were and .a square side length is .a hexagon side length is , the distance between the parallel sides is .build constraint conditionresulteventually the number of the pansso if s and are given, the number of each shape of pans is determined.taking into consideration the actual situation about covering and aesthetics of oven , we assume that the range of aspect ratio is 0.4 to 1 .then list the results here.table 2.ovens in different aspect ratios corresponding to the number of polygons of each shape.aspect ratio()trianglesquareregular hexagonround0.4182120140.5162118180.6141817150.7182020150.8182018120.915201616116252016from this model ,we find that it is theoretically true that square can be placed up to the maximum number when the ovens area is certain.problem2(1）the basic theory of temperature analysis:there are two energy transfer when we heating food by oven,one is convective 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: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 that 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 equations in three directions according to the conservation relations.symbol description:-air density，-time,-component of forces of volumetric force in the x,y,z direction，-pressure field ，-viscosity coefficientresultant force to do work is equal to the heat which is part of work done by viscous stress to control-volume in unit time.according to the first law of thermodynamics we can get the following energy differential equations .symbol description:-specific heat capacity in the air under a certain pressure.-the coefficient of thermal conductivity of the air-temperature field we know that the temperature is constant in the oven, so we can solve these 5 quantities()about the fluid.(2) build finite element analysis model 1. the pan in the oven heating analysiswe will simplify the different shapes of the pan to the thin polyhedral before we analyse.as an example, we analyse triangle pans heat distribution in the oven.figure 1. mimic diagram that heat flows through the triangle1. parameter setting assuming a constant temperature inside the oven is 200 . the initial temperature of the pan is set to 0 . the oven 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. 3.software simulationwe 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.the analytical results are as follows: figure 2. distribution of heat in the different shapes of the pan after heating2. analysis of the resultsaccording 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.a temperature range r of each shape of the pan from the figure can be seen, and it was summarized as follows :table 3.temperature differentials of pans in different shapespan shapetemperature range rtriangle89.73676.51313.223square83.08374.7628.321regular pentagon81.88374.7677.116regular hexagon80.55774.3966.161regular octagon79.77474.2535.521round79.03474.1144.92analysing pans variable with temperature range relationship, with software spss19.0 for regression analysis,and the result is following:table 4.the regression analysis results coefficentsamodelunstandardized coefficients standardizedcoefficientstsig.bstd. error beta 1/n3.647.39.9789.356.001(constant)1.272.08115.651.000a.dependent variable: ln(r) so the relationship waspan evenly distributs heat and temperature range is linear correlation.,and is the proportional coefficient.the smaller , the more uniform heat the pan distributs. when, is the least. so, the most uniform heat pan is circular.problem3for the third problems, pan-shapes 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 as small as possible.we assume the following formula: in this case,the problem we need to solve is the maximum value of z.we divide the problem into three cases as the following.case1:when，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.case2:when，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.case3：when,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. based on case 1 we know that when aspect ratiois certain,the maximum number of polygonal pans can be certain.symbol description:-the maximum number of round pan can be put in the oven.-the maximum number of regular hexagon pan can be put in the oven.when aspect ratiois certain, we can solve the value of and by the method that is proved in the first problem.then we get the range of .based on case 2 we can obtain the following model.so when aspect ratiois certain, is a function ofin formula 1.we can get the number of sides of the pan by the above equation,and then get the shape of the pan .evaluating the modelstrengths：the computer simulated the heated situation of the pan in the oven heating, making the results clearly visible.using reasonable assumptions simplified the problem.the model combined with the actual situation, having good practical significance。weaknesses：the accurate mathematical relation between the largest number of pan can be arranged and pan shape cant be established.setting same thermal environment to the oven, and the certain error will be in the model. references1xie benming,ju 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, 2008.2saeedmoaveni. finite element analysis theory and apoplication with ansys. beijing: publishing house of electionics industry.,2003.3zhonggeng han. mathematical modeling methods and applications. beijing: the pla information engineering university,2005.45/undergraduate/contests/mcmteaching you how to choose the panas 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 corners 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 rectangular in shape, using round pans is not efficient with respect to using the space in an oven. in order to choose a best shape of pan so that the amount of food is cooked at one time and heat distribution is more even, we now are faced with