2013美赛论文

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3.3 Model:

In the process of designing the model,we give priority to the consideration of the usage of the oven,the pans we design should have space practicality .Starting from the ideal rack space occupancy being 100% ,we divide the oven bracket into m?n rectangular grids , and the ratio of the width and length of each rectangular grid iswn/lm, and then consider the edge of the hotplate heat distribution uniform, to make each rectangular grid optimized.

In terms of the isotherm distribution and the its difference in color from the rectangle , we know that the rectangular baking tray corners can accumulate more heat than anywhere else of the rectangle,which formed a temperature difference , while rounded one can distribute the heat evenly well .So our first thought to come up with the optimization ideas is rectangular elliptical arc model design process instead of the rectangle at the right angle .Thinking along this idea , rectangle baking pan model is a idea not been optimized , while heat uniformity get maximum is similar to the runway \and edge distribution homogenization condition, we are asking the best model in between these two shapes .Give full consideration to maximize the number of modeling the baking pan and edge distribution homogenization conditions , the best model we are asking for is between the two shapes .

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Figure 6 Pan’s shape changing process in optimization process

Through the simulation of matlab tool , we obtained temperature distribution and isotherms of the following three baking pan shape edge .According to the figure of isothermal line distribution , we can be sure of the correctness of this line of thought .Though the four right angles of the rectangle being optimized as a quarter circle is able to produce good effect to the temperature uniformity distribution optimization of bakeware edge, however, the rectangular edge being optimized to circle also reducing the efficient use of space of the rectangular grid .

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According to the grid envisaged ,the oven racks can be divided into m?n box , and referring to the length and width proportion in the title , we can come to the following relationship :

nwnW? mlmLWhen conducting the optimization to rectangular Cartesian , we optimize the Cartesian into a quarter circular shown below , which the arc of radius is r, and then compared to the rectangle , the spare area B as a result of the optimization can be expressed as :

2 1 2B?r??r4

WL?4B mnr?(0,wn)2Then the efficient area of each pan is:

s?We hope the size of the baking pan ,on the basis of an extension of model-based , can be conducive to the uniform distribution of temperature field , so we make the aspect ratio of the baking pan size getting close to 1 , that is :

mW?1 nLTaking the requirements inherent in the area of the hotplate in the subject into account , there are the following inequality :

WL?4B?A mnIn the process of the optimization ,we gradually use the circle arc of which gradual increase in the radius instead of rectangular Original at right angles .In the process of gradual change of the shape of the pan , the distribution of heat in the baking pan edge also gradually change . On the basis of the distribution of temperature field model , we continue to introduce temperature field distribution uniformity coefficient . We simulate the radius of the arc and the changes of the temperature field distribution uniformity coefficient in the process of change via Matlab ,as a resule we get the

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relationship between the radius change and temperature field distribution .

In the simulation , we take the specifications as rectangular baking pan model being optimized, then the range of the change on r is?0,2.5? ratio unit.Uniformly select a plurality of r ,according to such a pattern drawn diagram of temperature distribution node,selected temperature of the edge and calculate the degree of heat distribution uniform at the point of such graphics .Results calculated are shown in the following table :

Table 2 Specifications for the 5?7 rectangular edge temperature distribution uniform

degree

The specifications of the Pan

rectangle (5×7)

r=0.5 r=1 r=1.5 r=2 r=2.5

Taking the above specifications rectangle for example ,using spss regression curve estimated ,we can know that temperature field uniform distribution of the degree and the quadratic curve of r fitting the best in a variety of curve types .Thus conclude that in the case of other conditions remain unchanged ,along with the increase of

The degree of uniformity

（%） 90.96 91.66 93.68 94.17 94.34 94.67

r,the better the optimization model evenness is,and uniformity coefficient

proportional to a quadratic function of r.That is:

CU?3.21?10?2r2?0.66?10?2r?90.76

Through the analysis above.

Take the quantity of the pan (M)in the oven can be maximized into account we can obtained objective function :

Max s?WL1???4?r2??r2? mn4??