2013年美赛A题H奖论文.doc

summarywe build two basic models for the two problems respectively: one is to show the distribution of heat across the outer edge of the pan for different shapes, rectangular, circular and the transition shape; another is to select the best shape for the pan under the condition of the optimization of combinations of maximal number of pans in the oven and the maximal even heat distribution of the heat for the pan.we first use finite-difference method to analyze the heat conduct and radiation problem and derive the heat distribution of the rectangular and the circular. in terms of our isothermal curve of the rectangular pan, we analyze the heat distribution of rounded rectangle thoroughly, using finite-element method. we then use nonlinear integer programming method to solve the maximal number of pans in the oven. in the even heat distribution, we define a function to show the degree of the even heat distribution. we use polynomial fitting with multiple variables to solve the objective function for the last problem, combining the results above, we analyze how results vary with the different values of width to length ratio w/l and the weight factor p. at last, we validate that our method is correct and robust by comparing and analyzing its sensitivity and strengths /weaknesses. based on the work above, we ultimately put forward that the rounded rectangular shape is perfect considering optimal number of the pans and even heat distribution. and an advertisement is presented for the brownie gourmet magazine.contents1 introduction31.1brownie pan31.2background31.3problem description32. model for heat distribution32.1 problem analysis32.2 assumptions42.3 definitions42.4 the model43 results of heat distribution73.1 basic results73.2 analysis93.3 analysis of the transition shaperounded rectangular94 model to select the best shape114.1 assumptions114.2 definitions114.3 the model125 comparision and degree of fitting196 sensitivity207 strengths/weaknesses218 conclusions219 advertisement for new brownie magazine2310 references241 introduction1.1 brownie panthe brownie pan is used to make brownies which are a kind of popular cakes in america. it usually has many lattices in it and is made of metal or other materials to conduct heat well. it is trivially 99 inch or 913 inch in size. one example of the concrete shape of brownie pan is shown in figure 1figure 1 the shape of brownie pan (source: google image)1.2 background brownies are delicious but the brownie pan has a fetal drawback. when baking in a rectangular pan, the food can easily get overcooked in the 4 corners, which is very annoying for the greedy gourmets. in a round pan, the heat is evenly distributed over the entire outer edge but is not efficient with respect to using in the space in an oven, which most cake bakers would not like to see. so our goal is to address this problem.1.3 problem descriptionfirstly, we are asked to develop a model to show the distribution of heat across the outer edge of a pan for different shapes, from rectangular to circular including the transition shapes; then we will build another model to select the best shape of the pan following the condition of the optimization of combinations of maximal number of pans in the oven and maximal even distribution of heat for the pan.2. model for heat distribution2.1 problem analysishere we use a finite difference model to illustrate the distribution of heat, and it has been extensively used in modeling for its characteristic ability to handle irregular geometries and boundary conditions, spatial and temporal properties variations shixiong liu, mika fukuoka, noboru sakai, a finite element model for simulating temperature distributions in rotating food during microwave heating, journal of food engineering, volume 115, issue 1, march 2013 page 49-62. in literature 1, samples with a rectangular geometric form are difficult to heat uniformly, particularly at the corners and edges. they think microwave radiation in the oven can be crudely thought of as impinging on the sample from all, which we generally acknowledge. but they emphasize the rotation.generally, when baking in the oven, the cakes absorb heat by three ways: thermal radiation of the pipes in the oven, heat conduction of the pan, and air convection in the oven. considering that the influence of convection is small, we assume it negligible. so we only take thermal radiation and conduction into account. the heat is transferred from the outside to the inside while water in the cake is on the contrary. the temperature outside increase more rapidly than that inside. and the contact area between the pan and the outside cake is larger than that between the pan and the inside cakes, which illustrate why cakes in the corner get overcooked easily.2.2 assumptionsl we take the pan and cakes as black body, so the absorption of heat in each area unit and time unit is the same, which drastically simplifies our calculation.l we assume the air convection negligible, considering its complexity and the small influence on the temperature increase .l we neglect the evaporation of water inside the cake, which may impede the increase of temperature of cakes.l we ignore the thickness of cakes and the pan, so the model we build is two-dimensional.2.3 definitions: heat flows into the node q: the heat taken in by cakes or pans from the heat pipes: energy increase of each cake unit: energy increase of the pan unit: temperature at moment i and point (m,n)c1: the specific heat capacity of the cakec2: the specific heat capacity of the pan: temperature of the pan at moment it1: temperature in the oven, which we assume is a constant2.4 the modelhere we use finite-difference method to derive the relationship of temperatures at time i-1 and time i at different place and the relationship of temperatures between the pan and the cake.first we divide a cake into small units, which can be expressed by a metric. in the following section, we will discuss the cake unit in different places of the pan.step 1；temperatures of cakes interiorfigure 2 heat flowaccording to energy conversation principle, we can get(2.4.1)considering fourier law and x=y, we get(2.4.2)according to stefan-boltzman law,(2.4.3)where a is the area contacting, c is the heat conductance.is the stefan-boltzmann constant, and equals 5.73108 jm-2s-1k-4. heat transfer theory .nz/unitoperations/httrtheory.htm(2.4.4)substituting (2.4.2)-(2.4.4) into (2.4.1), we getthis equation demonstrates the relationship of temperature at moment i and moment i-1 as well as the relationship of temperature at (m,n) and its surrounding points.step 2: temperature of the cake outer and the panl for the 4 cornersfigure 3 the relative position of the cake and the pan in the first cornerbecause the contacting area is two times, we getl for every edgefigure 4 the relative position of the cake and the pan at the edgesimilarly, we derivenow that we have derived the express of temperatures of cakes both temporally and spatially, we can use iteration to get the curve of temperature with the variables, time and location.3 results of heat distribution3.1 basic resultsl rectangularpreliminarily, we focus on one corner only. after running the programme, we obtain the following figure.figure 5 heat distribution at one cornerfigure 5 demonstrates the temperature at the corner is higher than its surrounding points, thats why food at corners get overcooked easily.then we iterate globally, and get figure 6.figure 6 heat distribution in the rectangular panfigure 6 can intuitively illustrates the temperature at corners is the highest, and temperature on the edge is less higher than that at corners, but is much higher than that at interior points, which successfully explains the problem “products get overcooked at the corners but to a lesser extent on the edge”. after drawing the heat distribution in two dimensions, we sample some points from the inside to the outside in a rectangular and obtain the relationship between temperature and iteration times, which is shown in figure 7figure 7from figure 7, the temperatures go up with time going and then keep nearly parallel to the x-axis. on the other hand, temperature at the center ascends the slowest, then edge and corner, which means given cooking time, food at the center of the pan is cooked just well while food at the corner of the pan has already get overcooked, but a lesser extent to the edge.l roundwe use our model to analyze the heat distribution in a round, just adapting the rectangular units into small annuluses, by running our programme, we get the following figure.figure 8 the heat distribution in the circle panfigure 8 shows heat distribution in circle area is even, the products at the edge are cooked to the same extent approximately.3.2 analysis finally, we draw the isothermal curve of the pan.l rectangularfigure 9 the isothermal curve of the rectangular panfigure 9 demenstrates the isothermal lines are almost concentric circles in the center of the pan and become rounded rectangles outer, which provides theory support for following analysis.l circularfigure 10 the isothermal curve of circularthe isothermal curves of the circular are series of concentric circles, demonstrating that the heat is even distributed.3.3 analysis of the transition shaperounded rectangular from the above analysis, we find that the isothermal curve are nearly rounded rectangulars in the rectangular pan, so we perspective the transition shape between rectangular and circular is rounded rectangular, considering the efficiency of using space of the oven and the even heat distribution. in the following section, we will analyze the heat distribution in rounded rectangular pan using finite element approach.during the cooking process, the temperature goes up gradually. but at a certain moment, the temperature can be assumed a constant. so the boundary condition yields dirichlet boundary condition. and the differential equation is:where t is the temperature, and x, y is the abscissa and the ordinate.and the boundary condition is t=constant.after running the programme, we get the heat distribution in a rounded rectangular pan, the results is in the following.figure 11 the heat distribution in a rounded rectangular panfrom 11, we can see the temperature of the edge and the corner is almost the same, so the food wont get overcooked at corners. we can assume the heat in a rounded rectangular is distributed uniformly.we then draw the isothermal curve of the rounded rectangular pan.figure 12 the isothermal curve of the rounded rectangular panto show the heat distribution more intuitively, we also draw the vertical view of the heat distribution in a rounded rectangular pan.figure 13 the vertical view of the heat distribution on a rounded rectangular platform4 model to select the best shape4.1 assumptionsbesides the assumptions given, we also make several other necessary assumptions.l the area of the even equals the area of the pan with small lattices in it.l there is no space between lattices or small pans on the pan.4.2 definitionss: the area of the oven k : the width of the external rectangular of the rounded rectangular h :the length of the external rectangular of the rounded rectangulara1: the ratio of the width and length of the external rectangular of the rounded rectangular, equals k/ha2: the ratio of the width and length of the external rectangular of the rounded rectangular, equals w/ln: the amounts of the rounded rectangular in each rowm: the amounts of the rounded rectangular in each columnr: the radius of the rounded rectangularin order to illustrate more clearly, we draw the following sketch.figure 144.3 the modell problem we use nonlinear integer programming to solve the problem.figure 15 the configuration of the rounded rectangular and the panwhere n, m are integers. both a1 and a2 are variables, we can study the relationship of a1 and n at a given a2. here we set a2=0.8.considering the area of the oven s and the area of the small pan a are unknown, we collect some data online, which is shown in the following table.table 1 (source: / )then we calculate the average of s and a , respectively 169.27 inch2 and 16.25 inch2. after running the programme, we get the following results.figure 16 the relationship of the maximal n and the radius of the rounded rectangular rfrom the above figure, we know as r increases, the optimal number of pans decreases. for the data collected can not represent the whole features, the relationship is not obvious .then we change the area of the oven and the area of each pan, we draw another figure.figure 17this figure intuitively shows the relationship of r and n.appearently, the ratio of the width to length of the rounded rectangular has a big influence on the optimization of n. in the following section, we will study this aspect.figure 18 the relationship of n and a1figure 18 illustrates only when the ratio of the width to the length of the rounded rectangular a1 equals the ratio of the width and length of the oven a2, can n be optimized.finally, we take both a1 and a2 as variables and study the relationship of n and a1, a2. the result is as follows.figure 19 the relationship of n and a1, a2l problem to solve this problem, we first define a function u(r) to show the degree of the even heat distribution for different shapes.where s is the area surrounded by the closed isothermal curve most external of the pan.now we think the rationality of the function. the temperature of the same isothermal curve is equivalent. we assume the temperature of the closed isothermal curve most external of the pan is t0 , for the unclosed isothermal, the temperature is higher than t0, and the temperature inside is lower than t0. so we count the number of the pixel points inside the closed isothermal curve most external of the pan num1 and the number of the whole pixel points num2. consequently, u(r)=num1/num2. to illustrate more clearly, we draw the following figure.figure 20 illustration of sthe following table shows the relationship of u and r. and we set a=1 table 2 the points we countr00.1r0.2r0.3r0.4r0.5ru0.850.860.8840.900.9160.93r0.6r0.7r0.8r0.9rru0.940.9650.980.9971according to table 2, we get the scatter digram of u and r. figure 21 the scatter diagram of u and rfirst, we consider u and r is linear, and by data fitting we derive then, we consider u and r is second -order relationship, and we derive another curve.figure 22 another relationship of u and rtable 3 u r0.1r0.2r0.3r0.4r0.5r000.17860.33620.47280.58840.1u00.18210.34280.4820.59980.2u00.18550.34930.49120.61120.3u00.1890.35570.50030.62250.45u00.19240.36220.50930.63380.6u00.19580.36860.51840.64510.7u00.19920.37150.52730.65630.8u00.20260.38140.63630.66740.9u00.2060.38770.54520.67851.0u00.20930.3940.55410.6896u r0.6r0.7r0.8r0.9r1.0r00.6830.75650.80910.84060.85110.1u0.69620.77120.82420.85690.86760.2u0.70940.78580.8480.87320.88410.3u0.72260.80040.86530.88930.90050.45u0.73570.81490.87150.90550.91680.6u0.74870.82940.8870.92150.9330.7u0.76170.84380.90230.93750.94920.8u0.77470.85810.91770.95340.96530.9u0.78750.87240.93290.96930.98141.0u0.80040.88660.94810.98510.9974according to the points we count, we can get the following figure, demonstrating the relationship u and r, a.figure 23 the relationship of u and r, we afrom the analysis above, we find with r increasing, u increases, which means when the radius of the rounded rectangular r increases, the degree of the even heat distribution. given one extreme circumstance, when r gets its maximal value, the pan becomes a circular, and the degree of the even heat distribution is also the most.finally, we can derive that the degree of the even heat distribution increases from the rectangular, rounded rectangular with smaller r, the rounded rectangular with bigger r and the circular.figure 24 the comparision of degree of even heat distribution for different shapesl problem in this section, we add a weight factor p to analyze the results with the varying values of w/l and p. in this problem, there are three variables , a, r and p .what we need to do is to select the best shape for the pan, namely, to select a and r. firstly, we set a and get the relationship of objective function, y and r, p.figure 25 the relationship of y and r, pfrom the figure, we can see with p decreasing and r increasing, which means the degree of the even heat distribution is larger, y increases.then we set p, and get the relationship of the y and r, a.figure 26 the relationship of y and a, rthis figure shows that the value a has little influence on the objective function.5 comparision and degree of fittingl comparisionwhen solving the problem to select the best shape of the pan, we only take the rounded rectangular into account and ignore other shapes, here, we concentrate on one of the polygonthe regular hexagon as an example to demonstrate our model is correct and retional.we use finite element method to derive the heat distribution, just like analyzing the rounded rectangular.figure 27 the heat distribution o