最新年美赛A题H奖58265390

年美赛A题H奖58265390

Summary

We build two basic models for the two problems respectively: one is to show the distribution of heat across the outer edge of the pan for different shapes, rectangular, circular and the transition shape; another is to select the best shape for the pan under the condition of the optimization of combinations of maximal number of pans in the oven and the maximal even heat distribution of the heat for the pan.

We first use finite-difference method to analyze the heat conduct and radiation problem and derive the heat distribution of the rectangular and the circular. In terms of our isothermal curve of the rectangular pan, we analyze the heat distribution of rounded rectangle thoroughly, using finite-element method. We then use nonlinear integer programming method to solve the maximal number of pans in the oven. In the even heat distribution, we define a function to show the degree of the even heat distribution. We use polynomial fitting with multiple variables to solve the objective function For the last problem, combining the results above, we analyze how results vary with the different values of width to length ratio W/L and the weight factor p. At last, we validate that our method is correct and robust by comparing and analyzing its sensitivity and strengths /weaknesses.

Based on the work above, we ultimately put forward that the rounded rectangular shape is perfect considering optimal number of the pans and even heat distribution. And an advertisement is presented for the Brownie Gourmet Magazine.

Contents

1 Introduction (3)

1.1Brownie pan (3)

1.2Background (3)

1.3Problem Description (3)

2. Model for heat distribution (3)

2.1 Problem analysis (3)

2.2 Assumptions (4)

2.3 Definitions (4)

2.4 The model (4)

3 Results of heat distribution (7)

3.1 Basic results (7)

3.2 Analysis (9)

3.3 Analysis of the transition shape—rounded rectangular (9)

4 Model to select the best shape (11)

4.1 Assumptions (11)

4.2 Definitions (11)

4.3 The model (12)

5 Comparision and Degree of fitting (19)

6 Sensitivity (20)

7 Strengths/weaknesses (21)

8 Conclusions (21)

9 Advertisement for new Brownie Magazine (23)

10 References (24)

1 Introduction

1.1Brownie pan

The Brownie Pan is used to make Brownies which are a kind of popular cakes in America. It usually has many lattices in it and is made of metal or other materials to conduct heat well. It is trivially 9×9 inch or 9×13 inch in size. One example of the concrete shape of Brownie pan is shown in Figure 1

Figure 1 the shape of Brownie Pan (source: Google Image)

1.2Background

Brownies are delicious but the Brownie Pan has a fetal drawback. When baking in a rectangular pan, the food can easily get overcooked in the 4 corners, which is very annoying for the greedy gourmets. In a round pan, the heat is evenly distributed over the entire outer edge but is not efficient with respect to using in the space in an oven, which most cake bakers would not like to see. So our goal is to address this problem.

1.3Problem Description

Firstly, we are asked to develop a model to show the distribution of heat across the outer edge of a pan for different shapes, from rectangular to circular including the transition shapes; then we will build another model to select the best shape of the pan following the condition of the optimization of combinations of maximal number of pans in the oven and maximal even distribution of heat for the pan.

2. Model for heat distribution

2.1 Problem analysis

Here we use a finite difference model to illustrate the distribution of heat, and it has been extensively used in modeling for its characteristic ability to handle irregular

geometries and boundary conditions, spatial and temporal properties variations 1. In literature 1, samples with a rectangular geometric form are difficult to heat uniformly, particularly at the corners and edges. They think microwave radiation in the oven can be crudely thought of as impinging on the sample from all, which we generally acknowledge. But they emphasize the rotation.

Generally, when baking in the oven, the cakes absorb heat by three ways: thermal radiation of the pipes in the oven, heat conduction of the pan, and air convection in the oven. Considering that the influence of convection is small, we assume it negligible. So we only take thermal radiation and conduction into account. The heat is transferred from the outside to the inside while water in the cake is on the contrary. The temperature outside increase more rapidly than that inside. And the contact area between the pan and the outside cake is larger than that between the pan and the inside cakes, which illustrate why cakes in the corner get overcooked easily.

2.2 Assumptions

● We take the pan and cakes as black body, so the absorption of heat in each

area unit and time unit is the same, which drastically simplifies our

calculation.

● We assume the air convection negligible, considering its complexity and the

small influence on the temperature increase .

● We neglect the evaporation of water inside the cake, which may impede the

increase of temperature of cakes.

● We ignore the thickness of cakes and the pan, so the model we build is two-

dimensional.

2.3 Definitions

Φ: heat flows into the node

Q: the heat taken in by cakes or pans from the heat pipes

c E ?: energy increase of each cake unit

p E ?: energy increase of the pan unit

,i m n t : temperature at moment i and point (m,n)

C 1: the specific heat capacity of the cake

C 2: the specific heat capacity of the pan

i

pan t : temperature of the pan at moment i

T 1: temperature in the oven, which we assume is a constant

2.4 The model

Here we use finite-difference method to derive the relationship of temperatures at time i-1 and time i at different place and the relationship of temperatures between the pan and the cake.

First we divide a cake into small units, which can be expressed by a metric. In the following section, we will discuss the cake unit in different places of the pan.

Step 1；temperatures of cakes interior

(m,n+1)

x

△

Figure 2 heat flow According to energy conversation principle, we can get

0up down left right c Q E Φ+Φ+Φ+Φ+-?= (2.4.1) Considering Fourier Law and △x=△y, we get

1,,1,,1,,1,,,1,,1,,1,,1,()()()()

i i m n m n i i left m n m n i i m n m n i i right m n m n i i m n m n i i up m n m n i i m n m n

i i down m n m n t t y t t x

t t y t t x t t x t t y t t x t t y λλλλλλλλ--++++---Φ=-?=-?-Φ=?=-?-Φ=?=-?-Φ=?=-? (2.4.2)

According to Stefan-Boltzman Law,

441,[()]i m n Q Ac T t σ=- (2.4.3)

Where A is the area contacting, c is the heat conductance.σis the Stefan-

Boltzmann constant, and equals 5.73×108 Jm -2s -1k -4.

2

1,,()i i c m n m n E cm t t -?=-

(2.4.4) Substituting (2.4.2)-(2.4.4) into (2.4.1), we get

441,1,,1,1,1,1,,4[()]()0

i i i i i i m n m n m n m n m n m n i i m n m n Ac t t t t t T t cm t t σ

λλ

-++--+++-+---=

This equation demonstrates the relationship of temperature at moment i and moment i-1 as well as the relationship of temperature at (m,n) and its surrounding points.

Step 2: temperature of the cake outer and the pan

● For the 4 corners

cake

Figure 3 the relative position of the cake and the pan in the first corner

Because the contacting area is two times, we get

4411,1,122[()]2()i i i i m n pan pan A c T t t c M t t σλ---=-

● For every edge

cake

Figure 4 the relative position of the cake and the pan at the edge

Similarly, we derive

4411,,2[()]()i i i i m n m n pan pan Ac T t t c M t t σλ---=-

Now that we have derived the express of temperatures of cakes both temporally and spatially, we can use iteration to get the curve of temperature with the variables, time and location.