## Abstract

### The design of ultimate brownie pan based on mathematical model

Author(s): Shujuan Yuan, Jingfei Chen, Aimin Yang, Xin Zhou, Qingpeng Ding

In electric oven, pan shape is rectangular or round in general, when a rectangular pan is used to roast food, heat is concentrated on four corners, as a result, the food will be scorched due to overheating, therefore, the rectangular pan can use the space of oven efficiently, but the food can not be roasted well. On a contrary, if a round pan is used, the heat can be distributed evenly on the outer rim of the pan, so the edge of food will not be overheated. This situation causes a lot of trouble to the user of oven. It has become a worthy issue that how to ensure the quality of roasted food, while both the oven space can be used most efficiently while the food visual effect can be the best. In order to solve the quality problem that is to achieve the largest average thermal distribution H of a pan, we have compared the phenomenon of a temperature field tip thermal effect to that of an electric field tip thermal effect, and we have also built the tip thermal effect model of temperature distribution function, and through theoretical derivation, we have obtained the relation equations between temperature distribution function T , aaverage heat distribution function H and tip distance r , semi-apex angle ï¡ . And their monotonous relations has been analyzed and we have obtained when the semi-apex angle is 2 max ï° ï¡ ï½ , which means that a round pan has a minimum curvature and max H . The model is simulated by using Matlab, and we know through three dimensional simulation temperature profile, the curvature is max, in other word, is max when a round tessellation is used, which is in consistent with the result of theoretical derivation. And through using existing radiation thermal conduction model, we make a further verification of the tip thermal effect model, which approves the rationality of the model