The Ultimate Brownie Pan

Summary

When we are baking, heat distribution of the pan is not even. It’s related to the shape of the pan. To achieve even heat distribution and to maximize the number of pans in the oven ,we are required to design the best shape of the pan.

In this paper, we develop our model from a thermal law, the inverse square law for thermal radiation. By using it, we obtain the heat distribution of pans with different shapes : rectangular pan, regular dodecagon, 24-polygon and round. We find, the more the number of the polygon sides, the evener the heat distribution is.

In order to redesign the shape of the ultimate Brownie Pan, we develop our model from two aspects. On the one hand, we maximize the number of the pan that can fit the oven. We put forward an algorithm, through which we find the rectangular pan is best. On the other hand, we use “variance” to describe the distribution of heat. We find the variance in the round pan is the minimum. It means the round pan is the best to maximize even distribution of heat.

Hence, mixing the two parts, we build an optimized model with two goals. We get a series of naïve result with the different p and W/L. We find the best pan is a rounded rectangular pan, or its four corners is circular arc.

We also do two works in the improved model. We argue the difference between the outer edge variance and overall variance. Then we explain that how the change of W/L influences the maximal number N. Based on the redesigned pan, we make an advertising sheet for the new Brownie Gourmet Magazine.

Keywords: heat distribution, the inverse square law for thermal radiation, the maximum arrange number