Team #20427 第 1 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 7npolygon edge number of the pan( ).nIn 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 napproaches 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 ZLW/nmodel. We can know the preferred shape of pan that should be selected when the and is pLW/given.Team #20427 第 2 页 共 2n 页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,the 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 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 of pan it can accommodate, so we are looking for the best shape of the pan so that it can put more quantities 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 is more evenly distributed when heated.To consider both the number and the heat distribution of pans,we should give weight and (p) to the number of pans and the heat distribution in each pan to identify the best p1pan-shape in the different weights.NotationName descriptionaside length of regular polygon.hheight of equilateral triangle.Team #20427 第 3 页 共 2n 页bthe distance between the regular hexagon parallel sides.Athe area of a pan.mthe number of pans in each line in the oven.nthe number of