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First, we consider the edge of the tray heat distribution at a certain moment during the baking process , so time is a constant in our research model .Secondly , we ignore the heat change in the vertical pan direction , and we simplify the heat distribution of the edge of the pan to a heat distribution problem in a two-dimensional plane which has boundary to describe the heat distribution of pan edge clearly ,projecting heat differences between pan corner and other portions .Which is to select a plan view of the heat distribution of food and pan to visually depict the heat distribution difference on bakeware edge.
According to the different pan shapes,We roughly divide the shape of the baking pan into three categories ,and give out respective description:
Rectangle pan:
When the horizontal cross section of the pan is rectangular shape , as is shown in Figure , we create a Cartesian coordinate system with a rectangular vertex as an origin,and each of the rectangular length and width directions as the shaft in the positive direction. Then the equation of the temperature field is satisfied[2] as follows:
?2t?2t?2t?t ????x2?y2?z2?TIt can be simplified in our description of two - dimensional flat space to be:
?2t?2t?2?0 2?x?yBoundary conditions:
tx?0?tx?l?t1y?0?t?y?l2?t0
?0
?t?T?x?0?t?T?x?l1?t?Ty?0?t?Ty?l2Symmetric polygon pan:
When the horizontal cross section of the pan is between rectangular and circle such as
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symmetrical hexagonal,symmetrical octagon and so on,the method of creating a Cartesian coordinate system are in common.
Circle pan:
When the horizontal cross section of the pan is circle,we can suppose ,as is shown in the Figure ,the disk center of the circle as the origin , and base on the disk plane space to create a cylindrical coordinates .Then the equation of the temperature field is satisfied as:
qv?2t1?t1?2t?2t ??????r2r?rr2??2?z2?It can be simplified in our description of two - dimensional flat space to be :
?2t1?t1?2t??2?0 22?rr?rr??Boundary conditions:
ttr?0?0 ?t0 r?r0?????????2?? 2.4 Model solutions and analysis:
According to the model equation we establish which satisfies the temperature distribution in the two-dimensional plane . We use Matlab [3]software to simulate the distribution of the different shape of the baking pan on the temperature and the temperature gradient . As shown below:
Rectangular
. For the different shape of the baking pans, the distribution solution function image
of temperature and temperature gradient function corresponding to different edge of the pans are shown below :
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1-1 1-2
Figure 1 Temperature distribution function and the temperature gradient function
image on rectangular baking pan edge
Figure 1-1 is the temperature distribution image of the edge of the rectangular baking pan , and we can learn from the figure that bakeware outer edge distribute the higher temperature than the hotplate center region , and the temperature distribution uneven in the extending direction of the side of the rectangle apparently, besides, temperature distributed in the four corners is higher than the area extend to the middle area ; Figure 1-2 is the image of the function of the temperature gradient of the edge of the rectangular baking pan . Overall, the temperature gradient on the outer edge shows a tendency to be higher than the temperature gradient of the bakeware intermediate zone. Specifically analyzing from the image , the entire bakeware regional temperature gradients are unevenly distributed , the temperature gradient of four corners is smaller , along the sides of the rectangle in its direction the temperature gradient from the right angle vertex to the midpoint of the length and width shows a significantly increasing trend .
Considering the distribution of the temperature and its gradient ,the regularity can be seen , the uneven distribution of the phenomenon is more obvious at the edge of the rectangular baking pan in the entire edge , at the same time in the four corners of the rectangle , relatively,the higher the temperature was the smaller gradient became, as a
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result heat in the four corners of the pans is the largest with respect to the entire baking pan ,and inwardly,from the outer edge ,the heat of the situation which act as a certain function is gradually reduced .
Axisymmetric hexagonal:
2-1 2-2
Figure 2 Temperature distribution function and the temperature gradient function
image on axisymmetric hexagonal baking pan edge Figure 2-1 is the temperature distribution function image of the edge of the hexagonal bakeware , from the diagram it can be seen that in this case hotplate edge temperature reduced gradually from outside to inside, and the temperature distribution on the outer edges are not completely uniform because of the uneven blocks of color. Where the angle is sharper in the corner, the temperature tend to be especially higher than the bakeware internal area , similarly the sharp extent is lesser the corner temperature is still higher than bakeware middle area , but relative to the sharp corner , the temperature distribution in unsharp area is more uniform to some extend.
Figure 2-2 hexagonal the bakeware temperature gradient function image can be analyzed : Overall , the bakeware temperature gradient along with a gradually decreasing trend from the outside to the inside , but the temperature gradient in the entire outer edge is uneven . Relatively sharp greater degree near the corners , the
