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1、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 o
2、f 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
3、 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 th
4、e 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 met
5、hod 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 th
6、e 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 shapero
7、unded 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 use
8、d 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 Br
9、ownie 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 t
10、he 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 pa
11、n 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 p
12、an.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 varia
13、tions 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
14、 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
15、: 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 ins
16、ide 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 h
