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Team Control Number For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________

21566

Problem Chosen

For office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________

A

2013 Mathematical Contest in Modeling (MCM) Summary Sheet (Attach a copy of this page to your solution paper.)

Summary

Brownies enjoys a great popularity in the United States for its tasty flavor. However, what has troubled housewives and restaurant cooks so many years is that when baking in a rectangular pan heat is concentrated in the four corners and the food gets overcooked at the corners; while baking in a round pan the heat is distributed evenly over the entire outer edge and the food is not overcooked there. Due to lacking efficiency of using the space in an oven, circular pans are not the best choice. We are tasked to solve this dilemma and put forward an optimized design for baking pans. To illustrate how heat is distributed in a specific baking pan, the law of Heat Conduction for a three-dimensional object is firstly deduced to construct our model for heat distribution across the outer edges of pans of different shapes - rectangular, circular and elliptical. Secondly, Finite Difference Method is introduced to solve the proposed model numerically and simulation figures of heat distribution for these shapes are drawn explicitly. Our goal of selecting the best type of pan (shape) under the three conditions presented in the problem is achieved by three steps. Firstly, the approach used in Digital Image Processing to detect the edges of an object in a given image is adopted into our model to look for a surface of a maximum even distribution of heat for the pan. In this approach, a temperature function is defined to deduce the constraints satisfied by the optimal surface of the pan. A numerical algorithm to find the optimal surface with an even heat distribution is further proposed. Secondly, given that an oven has definite space, a model is established to determine the best type of pan (shape) to maximize the number of pans placed in an oven . Finally, the two conditions mentioned above are taken into consideration together to build a comprehensive optimal model with weights p and (1-p) . How the results vary with different values of W/L and p is further illustrated.

Key words: Partial Differential Equation, Finite Difference Method, Monte Carlo Method, Optimized Model.

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A Brownie Fan? You will love the Ultimate Baking Pan

Contents

1 Introduction .................................................................................................................................................. 2 2 Plan of Attack ............................................................................................................................................... 2 3 Assumptions ................................................................................................................................................. 3 4 Conventions.................................................................................................................................................. 3 4.1 Terminologies and Notations ............................................................................................................. 3 4.2 Variables ............................................................................................................................................ 3 5 Model One: Distribution of Heat.................................................................................................................. 4 5.1 Elementary Perception ...................................................................................................................... 4 5.2 Theoretical Verification ..................................................................................................................... 4 5.2.1. Three-Dimensional Equation of Heat Conduction ................................................................ 4 5.2.2 Heat Distribution Model of a Brownie in a Rectangular Pan ................................................. 6 5.2.3 Heat Distribution Model of a Brownie in a Circular Pan ....................................................... 7 5.2.4 Finite Difference Method for a Two-Dimensional Plane ....................................................... 7 5.2.5 Finite Difference Equation for a Cuboid ................................................................................ 8 5.3 Simulation ......................................................................................................................................... 9 5.3.1 Simulation Results for a Brownie Being Baked in a Rectangular Pan ................................... 9 5.3.2 Simulation Results for a Brownie Being Baked in a Circular Pan ....................................... 11 5.3.3 Simulation Results for a Brownie Being Baked in an Elliptical Pan.................................... 11 6 Model Two: the Best Type of Pan .............................................................................................................. 12 6.1 Even Distribution of Heat................................................................................................................ 12 6.1.1 Theoretical Derivation .......................................................................................................... 12 6.1.2 Algorithm ............................................................................................................................. 15 6.1.3 Simulation Results................................................................................................................ 18 6.2 Maximum Number of Pans ............................................................................................................. 20 6.3 A Comprehensive Model for an Optimized Pan .............................................................................. 20 6.3.1 Definition of an Unused Space of an Oven. ......................................................................... 21 6.3.2 Comprehensive Model for an Optimized Pan ...................................................................... 22 6.3.3 Simulation Results for the Optimized Pan ........................................................................... 23 7 Future Work ................................................................................................................................................ 25 8 References .................................................................................................................................................. 25 9 Advertising Sheet ....................................................................................................................................... 26

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1 Introduction

A chocolate brownie is a flat, baked square or bar with a texture between a cake and a cookie. Ingredients of brownies include nuts, frosting, whipped cream, chocolate chips, and so on. Because of its tasty flavor, it has enjoyed widespread popularity in North America and has become common lunchbox food or dessert [1]. Many housewives have acquired the skill to bake chocolate brownies with household ovens(as shown in Figure 1) in the United States. These ovens usually use several methods to work. The most commonly used method is to heat the pans inside the oven from its bottom [2]. This method is typically used for baking and roasting. Some other approaches like heating pans from the top are also available. In order to provide faster and more-even cooking, convection ovens (also known as fan-assist ovens) use a small fan to blow hot air around the cooking chamber [2].

Figure 1. A baking oven

Figure 2. A rectangular pan (a) and a circular pan (b)

However, as no cate come in vain, a delicious chocolate brownie requires suitable baking duration and proper heating temperature. Furthermore, the shape of baking pans also has a great impact on the taste. What troubles housewives and restaurant cooks is that when baking in a rectangular pan (see Figure 2. (a)) heat is concentrated in the 4 corners and the food gets overcooked at the corners (and to a lesser extent at the edges); while baking in a round pan (see Figure 2.(b)) the heat is distributed evenly over the entire outer edge and the food is not overcooked at the edges. However, since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven. Therefore, we are tasked to develop a model to solve this dilemma and apply our model to select the best type of pan shapes.

2 Plan of Attack

Our goal to solve this problem breaks into two steps. Firstly, we are required to develop a model to show the distribution of heat across the outer edge of a pan for pans of different shapes - rectangular to circular and other shapes in between. Secondly, based on the first goal, a model that can be used to select the best type of pan (shapes) under three conditions will be developed. Therefore, we will proceed by: Stating assumptions. Through our assumptions, we will concentrate our attention on our approach towards the problem and provide some insight views about the nature of the heat distribution of baking pans of different shapes. Defining terms. Crucial to developing a model is defining clearly the somewhat ambiguous terms like ovens and baking pans, for there is a myriad of bakeware in totally different styles. Presenting our models. We will illustrate how heat is distributed of a brownie being baked in a pan both by perceptual cognition of elementary analysis and theoretical explanation of mechanism analysis. Simulating model and analyzing results. Necessary to verify our models is to carry on simulation process under proper conditions and discuss the results with the phenomena stated in this problem.

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3 Assumptions

In order to ensure that our models remain valid under various potential constraints, we make a number of assumptions as follows before stepping into details about our approaches. The thickness of side walls and the bottom of the baking pans are negligible. Most baking pans are made of metals, which are good thermal conductors, and the thickness of baking pans is not thick enough to retard the process of heat diffusion. Therefore, the thickness of sidewalls and the bottom of the baking pans are negligible. The preheating duration is very short and is therefore negligible. The temperature of the baking pans increases gradually from room temperature after pans are put into the oven from the outside. However, this gradual increasing process is not one of our major concerns about how the product gets overcooked. So our model works when the baking temperature reaches and stays within a stable level. The baking pans can be treated together as a stable hollow heat source without ceiling. Since the oven has two racks and they are evenly spaced, the height of the baking pans is negligible when compared with the distance between the pan and the bottom of oven. The heat distribution of the outer edge of a baking pan can be approximately viewed as that of the external surface of the brownie. Since the baking pans are hollow with open ceiling and the thickness can be negligible, it is much more convenient to build models on a solid brownie.

4 Conventions

Some basic terms and variables used in this paper will be defined in this section.

4.1 Terminologies and Notations

Definition 1. Oven: The ovens used in this paper are defined to be heated from the bottom only. Ovens used for baking usually use several heating methods to work. Based on features of ovens produced by many companies, the most commonly used method is to heat the oven from the bottom. As a result, this definition allows us to build our models with a single heat source. Definition 2. Pan: Baking pans discussed in this paper defined to be of the single-chamber types. Many companies have a myriad of designs for baking pans and they usually add cavities into the pans of different kinds. It will be very difficult to put forward our model while taking into consideration the different types of cavities. Therefore, this definition allows us to focus on the shape of the baking pan but rather the inner uncertainties of the cavities.

4.2 Variables

Q

m

ρ

c u k Δv Δs

L W N

Heat of an object Mass of an object Density of an object Specific heat capacity of an object Temperature of an object Coefficient of thermal conductivity The overcooked volume of a brownie The unused area of an oven The length of an oven The width of an oven The number of pans placed in an oven

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5 Model One: Distribution of Heat

5.1 Elementary Perception

Why is heat concentrated in the 4 corners when baking in a rectangular pan?

(a)

(b)

Figure 3. Schematic views of a brownie being baked in a rectangular pan (a) and a circular pan (b)

Suppose the cuboid shown in Figure 3(a) is a brownie being baked in a rectangular pan, the colored part of the cuboid is a narrow space at the lower right corner. Unlike other flat surfaces of the cuboid, which are warmed by heat from only one direction and have enough space to transfer heat to the inner part, the corner is warmed by heat from three directions and the heat accumulated at the corner is hard to diffuse due to a lack of open space. Therefore, the heat is distributed unevenly and the temperature of the four corners is higher than other parts of the brownie, making brownies get overcooked easily at the corners. Why is heat distributed evenly over the entire outer edge when baking in a circular pan? Similar to the above illustration, the disk shown in Figure 3(b) can also be imagined as a brownie being baked in a circular pan. The curved surfaces on the side and the flat surface at the bottom are all heated from only one direction (which is the normal vector of the surface) and they all have enough open space to impart heat into the inner part, keeping the temperature at the edges within a certain level. Therefore, the heat is distributed evenly over the entire outer edge and the temperature is not high enough to get brownies overcooked. The above two points of a brownie being baked in a rectangular or a circular pan remain in perceptual cognition. Although we have acquired the elementary knowledge of their general heat distributions, more explicit and visualized methods are necessarily required to illustrate their exact heat distributions. In order to test our fundamental perceptions, we proceed with our exploration by theoretical verification of mechanism analysis.

5.2 Theoretical Verification

5.2.1. Three-Dimensional Equation of Heat Conduction To deduce the equation of heat distribution without loss of generality, we view the brownie being baked in an oven as a thermal conductor of any shape (as shown in Figure 4).

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r n

Figure 4. A brownie of any shape

Let Ω denote the space occupied by the thermal conductor and ?Ω be its boundary. The temperature of any point ( x, y, z ) ∈ Ω at any time is given by u ( x, y, z , t ) . Select any point ( x, y, z ) ∈ Ω and expand it into a sufficiently small neighborhood G ∈ Ω (whose boundary is

?G ) around this point. During a sufficiently small period of time [t1 , t 2 ] , the changes of heat Q in the space of G satisfy the following relationship:

Q2 (t = t 2 ) ? Q1 (t = t 2 ) = φ (t1 ≤ t ≤ t 2 ) ,

(1)

where φ represents the heat which passes through the boundary ?G during the given period. Based on Fundamentals of Thermology, the heat Q of an object is given by

Q = mcu ,

(2)

where m, c, u represents the mass, the specific heat capacity and the temperature of the object respectively. Due to selection of sufficiently small space and period of time, substituting Eq. (2) into the left-hand side of Eq. (1) and using the Theorem Differential Mean Value yield

Q2 ? Q1 = ρΔvc[u ( x1 , y1 , z1 , t 2 ) ? u ( x1 , y1 , z1 , t1 )] = ρΔvc ∫ ut dt

t1 t2

(3)

= ρΔvcut ( x1 , y1 , z1 , t1 )Δt ,

where ρ , Δv respectively denote the density and volume of space G, ( x1 , y1 , z1 ) ∈ G , t1 ∈ [t1 , t 2 ] and Δt = t 2 ? t1 . To calculate the heat passing through the boundary ?G during the given period, we introduce the Law of Heat Conduction, also known as Fourier's law. It states that the time rate of heat transferring through a material is proportional to the negative gradient in the temperature [3]. Therefore, the heat flux density is defined by

r q = ? k ( x, y, z )?u ,

r

(4)

where q is the amount of energy that flows through a unit area per unit time and k ( x, y, z ) is the coefficient of thermal conductivity, which is related to the characteristics of the medium. The minus in Eq.(4) indicates that heat diffuses from an area of high temperatures to places of low temperatures. Based on the Flux Formula in Calculus, the heat φ passes through the boundary ?G into the space G during the given period is given by

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φ = ∫∫ q ? (? n )dsΔt ,

?G

r

r

(5)

where n is the outward unit normal vector to ?G . Suppose u has second order continuous partial derivative to space and has first order continuous partial derivative to time. Using Gauss formula and the Theorem Differential Mean Value, Eq.(5) can be rewritten as

r

φ = ∫∫ q ? (? n )dsΔt = ∫∫ k?u ? ndsΔt

?G ?G

r

r

r

= ∫∫ k

?G

?u dsΔt = ∫∫∫ ΔudvΔt ?n G

(6)

= kΔu ( x2 , y2 , z 2 , t 2 )ΔvΔt ,

where Δ is Laplace Operator and Δu = u xx + u yy + u zz , ( x2 , y2 , z 2 ) ∈ G and t 2 ∈ [t1 , t 2 ] . Now, combine Eq.(3) with Eq.(6) and substitute them into Eq.(1), we have

ρΔvcut ( x1 , y1 , z1 , t1 )Δt = kΔu ( x2 , y2 , z2 , t 2 )ΔvΔt .

(7)

Divide each of the terms in the above equation by ΔvΔt and let Δv → 0, Δt → 0 , and it yields a simpler form:

ρcut ( x, y, z , t1 ) = kΔu ( x, y, z, t1 ) .

Due to the randomicity of t1 , the final equation of thermal conduction is given by

(8)

ρcut ( x, y, z , t ) = kΔu ( x, y, z , t )

or simply

(9)

u t = a 2 Δu ,

(10)

where a 2 = k /( ρc) > 0 , which is called the thermal diffusivity [4]. To simplify our model, the brownies being baked in an oven can be treated as homogeneous and isotropic thermal conductors, thus a 2 is a constant value. Note: Eq.(10) is derived by the assumption that the brownie can be treated as a homogeneous and isotropic thermal conductor. Thus, the parameters k , ρ , c do not change with u . However, in reality, these parameters do change with u . For the fact that the region where a is influenced by u is quite small, therefore, we still can use Eq.(10) in the overcooked area. 5.2.2 Heat Distribution Model of a Brownie in a Rectangular Pan Since temperature is an external expression or characteristic for heat and the brownie is a homogeneous and isotropic thermal conductor, whose temperature is proportional to heat. Therefore, they can be used as the same parameter in our model. From Eq.(10), the equation of heat diffusion of a brownie being baked in a rectangular pan is given by

u t ( x , y , z , t ) = a 2 Δu ( x , y , z , t ) .

In order to solve it, we then consider its boundary conditions and initial condition. Due to our assumptions that the baking pans can be treated as a stable hollow heat source without ceiling, we set different surfaces with the same temperature. Therefore, the boundary conditions of the sidewalls and the bottom surface are defined by

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?u ( x0 , y, z , t ) = u0 , u ( x1 , y, z , t ) = u0 ? ?u ( x, y0 , z , t ) = u0 , u ( x, y1 , z , t ) = u0 , ?u ( x, y, z , t ) = u 0 0 ?

where u0 is the temperature of different surfaces of a rectangular pan while being baked in an oven, and x0 ≤ x ≤ x1 , y0 ≤ y ≤ y1 , z0 ≤ z ≤ z1 . However, the boundary condition of the top surface is quite different, and it is separately defined by

where ε is a sufficiently small value. Because the top surface has close contact with the air, which is poor conductor, and we assume that the heat exchange between them is very small. Due to our assumption that the surface of the brownie nearly does not change with the inflation of the brownie, therefore, the above equation can be rewritten as

?u =ε , ?n

u z ( x, y , z , t ) = ε .

Its initial condition is defined by

u ( x , y , z ,0 ) = l ,

where l is a limited real value of normal room temperature. Combining all the equations above, we derive our final model of heat distribution of a brownie being baked in a rectangular pan:

?ut ( x, y, z , t ) = a 2 Δu ( x, y, z , t ), ? ?u ( x0 , y, z , t ) = u0 , u ( x1 , y, z , t ) = u0 , ? ?u ( x, y0 , z , t ) = u0 , u ( x, y1 , z , t ) = u0 , ?u ( x, y, z , t ) = u , u ( x, y, z , t ) = ε , 0 0 z 1 ? ?u ( x, y, z ,0) = l. ?

(11)

5.2.3 Heat Distribution Model of a Brownie in a Circular Pan Similarly, by applying the above method, the heat distribution model of a brownie being baked in a circular pan is given by

?ut (r , z , t ) = a 2 Δu (r , z , t ), ? ?u (r0 , z , t ) = l , u (r1 , z , t ) = u0 , ? ?u (r , z0 , t ) = u0 , u z (r , z1 , t ) = ε , ?u (r , z ,0) = 0, ?

(12)

where r0 ≤ r ≤ r1 , z0 ≤ z ≤ z1 , l represents a limited value of normal room temperature and ε is a sufficiently small value as explained in Eq.(11). To solve the above models, we have considered several plausible methods. Commonly used approaches are the method of separation of variables, the Fourier transform method and traveling-wave method. Effective as these methods are, they are not suitable or practical for our three-dimensional model of heat distribution. Therefore, novel method to solve this problem is imperatively demanded. 5.2.4 Finite Difference Method for a Two-Dimensional Plane In mathematics, finite difference methods are numerical methods for approximating the solutions to differential equations using finite difference to approximate derivatives[5]. In order to simplify our

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process of deducing the finite difference equation of three dimensions and illustrate it in an explicit way, we first consider the solution of two dimensions. Similar to Eq.(10), the heat distribution equation of a two-dimensional plane is defined by

ut ( x, y, t ) = a 2 Δu ( x, y, t ) .

(13)

Figure 5. Two-dimensional mesh of a plane

To illustrate how finite-difference method works to Eq.(13), we divide a two-dimensional plane into plenty of squares as shown in Figure 5. Every grid (i, j ) has a specific value of temperature ui,j. If the gird size is as small as possible, the finite-difference method works better and the approximation is much closer to the accurate value. Now, we begin our deducing of the relationship among the near grids shown in Figure 6, the first order partial derivative to x of the temperature in Eq.(13) can be rewritten as

ux =

?u ui +1, j ? ui , j ≈ . ?x Δx

Based on the above equation, the second order partial derivative to x of the temperature is given by

? 2u u ? 2ui , j + ui +1, j . uxx = 2 ≈ i ?1, j ( Δx ) 2 ?x

(k ) (k ) (k )

(14)

On the other hand, the partial derivative to t of the temperature in Eq.(13) is defined by

( k +1) (k ) ?u ui , j ? ui , j ut = . ≈ ?t Δt

(15)

Obviously, by substituting Eq.(14) and Eq.(15) into Eq.(13), the finite difference equation of heat distribution for a two-dimensional plane is given by

ui(,kj+1) ? ui(,kj) Δt

k k ? ui(?1), j ? 2ui(,kj) + ui(+1), j ui(,kj)?1 ? 2ui(,kj) + ui(,kj)+1 ? ? ?. + =a ? ? (Δx) 2 (Δy ) 2 ? ? 2

(16)

Based on the above equation, we can further solve the problem of heat distribution for a two-dimensional plane by iteration if its boundary and initial conditions are given. However, the two-dimensional result is not exactly what we want. Therefore, we proceed with our exploration by employing the same method to analyze our three-dimensional brownie. 5.2.5 Finite Difference Equation for a Cuboid Our model of heat distribution of a three-dimensional brownie being baked in a rectangular pan is defined in Eq.(10) and it can be rewritten as

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2 ?u ( x, y, z , t ) ? 2 u ( x, y , z , t ) ? 2 u ( x, y , z , t ) ? 2 ? ? u ( x, y , z , t ) ?. ? =a ? + + ? ?t ?x 2 ?y 2 ?z 2 ? ?

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By employing the above mentioned finite difference method, the finite difference equation for a three-dimensional object in rectangular coordinates is given by

ui(,kj+l1) ? ui(,kj),l , Δt

k k ? ui(?1), j ,l ? 2ui(,kj),l + ui(+1), j ,l ui(,kj)?1,l ? 2ui(,kj),l + ui(,kj)+1,l ui(,kj),l ?1 ? 2ui(,kj),l + ui(,kj),l +1 ? ? =a ? + + ? ? (Δx) 2 (Δy ) 2 (Δz ) 2 ? ? 2

The iteration formula can be derived by transforming the above equation to

rxδ x (ui(,kj),l ) + ryδ y (ui(,kj),l ) + rzδ z (ui(,kj),l ) + ui(,kj),l = ui(,kj+l1) , ,

where

(17)

rx =

and

a 2 Δt a 2 Δt a 2 Δt , ry = , rz = , ( Δx ) 2 ( Δy ) 2 ( Δz ) 2

k k δ x (ui(,kj),l ) = ui(?1), j ,l ? 2ui(,kj),l + ui(+1), j ,l ,

δ y (ui(,kj),l ) = ui(,kj)?1,l ? 2ui(,kj),l + ui(,kj)+1,l , δ z (ui(,kj),l ) = ui(,kj),l ?1 ? 2ui(,kj),l + ui(,kj),l +1 .

Note that Eq.(17) can also be applied to solve the distribution of heat for a circular pan or other shapes.

5.3 Simulation

5.3.1 Simulation Results for a Brownie Being Baked in a Rectangular Pan Combining the iteration formula shown in Eq.(17) with the boundary condition shown in Eq.(11), the temperature of every meshgrid of a brownie being baked in a rectangular pan can be derived by many times of iteration. Here, we test our model in a cube with a length of 20, a boundary condition of u0 = 5 , ε = 0 , diffusion duration of t = 1000 , and thermal diffusivity of a 2 = 10 ?4 . Remember the caveat that the values of the constraints we have chosen is definitely not a reproduction of a real world situation, it is only a simulation under our proper prescription. Once the values of these parameters are given, the simulation of the real world situation can also be done. The simulation result of the heat distribution for the cube is shown is Figure 6, where temperatures are colored from blue to red , with redder color representing temperature that is thermally higher up along the color bar.

Temperature 5 4.9 20 4.8 4.7 4.6 z 10 4.5 4.4 4.3 0 20 15 10 5 y 0 0 10 5 x 15 20 4.2 4.1 4

15

5

Figure 6. A simulation of heat distribution for a cube being baked in an oven

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In order to show both the outer and inner distributions of heat, the above figure not only includes the heat distribution of the side walls but also that of three slice maps in the inner part of the cube with one of the three axes fixed respectively each time. To be more explicit, a slice map with z-axis being fixed at z=3 is drawn and shown in Figure 7, which is the plane close the bottom of the cube with a remarkable significance worth attention.

Temperature 5

4.5 4 4 3.5 3.5 z 3 3 2.5 2.5 2 20 15 10 5 y 0 0 5 x 15 10 20 1.5

2

Figure 7. A slice map of heat distribution with z-axis fixed at z=3

It can be observed in Figure 7 that the color gets darker and darker from the heart to the edges, with a significantly dark color at the four corners, so does the temperature. This result is exactly in accordance with the phenomena stated in the introduction part. Because this area is warmed by heat from five directions. While the heat from the bottom contributes evenly to every grid of the plane, the other four directions do not, causing the heat to be concentrated at the four corners. Therefore, we proceed our simulation with confidence to analyze the distribution of heat across the outer edge of the rectangular pan. Due to our assumption that the heat distribution of the outer edge of a rectangular pan can be viewed as that of the external surface of the brownie, we then draw the contours of the heat distribution of the plane with the z-axis being fixed at z=2, as shown in Figure 8.

Temperature 20 18 16 14 12 y 10 8 6 4 2 2 4 6 8 10 x 12 14 16 18 20

Figure 8. Contours of heat distribution of the outer edge of a rectangular pan

It can be observed in Figure 8 that the contours at the four corners is quit different from the others. They are convex curves with a radiant at the corners while the others are elliptical or round-like closed curves expanding from the heart. This result is exactly in accordance with the real heat distribution of the outer edge of a rectangular pan. Thus our elementary perception about the heat distribution of a

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brownie being baked in a rectangular pan has been verified by this robust model. 5.3.2 Simulation Results for a Brownie Being Baked in a Circular Pan The MATLAB toolbox for Partial Differential Equations (PDE) provides a dynamic simulation process for this model, we draw an animation of heat diffusion of the brownie being baked in a circular pan as shown in Figure 9. The three states indicate the dynamic diffusion of heat both from outside to inside and from bottom to top. Note that the right axis in Figure 10 represents the height z (with a higher value on the top) of the brownie, while the left axis represents the radius r (with a lower value in the center) of the brownie and the vertical axis represents temperature.

20

20

20 18

18

18 20 16 20 14

15

20

16 14 12

16 14 12

15

15

12 10 10 5 8 6 0 0 0 -0.5 -0.6 -1 -0.8 -1 0

-0.8 -1 -1 10

10 10 5 8 6 0 0 0 -0.5 -0.6 -1 -0.8 -1 -0.2 -0.4 2 0 4

10 5 8 6 4 0 -0.5 -0.6 0 -0.2 -0.4 2

4 -0.2 -0.4 2

0 0

Figure 9. Three middle states of heat distribution of a brownie being baked in a circular pan

Furthermore, by employing the same method used in the simulation of a rectangular pan to our simulation for a brownie being baked in an circular pan, a slice map and a contour of it can be drawn as shown in Figure 10.

Temperature 5 4.5 11 4 3.5 3 z 2.5 9.5 2

Temperature(z=2) 15 14 13 12 11 y 10 9

1.5

10.5

10

9 20 15 10 5 y 0 0 10 5 x 15 20

8

1 0.5 0

7 6

6

8

10 x

12

14

Figure 10. Contours of heat distribution of the outer edge of a rectangular pan

It can be observed in Figure 10 that the contours distribute evenly from circle to circle with an radiant at the center. Indicating that the temperature at the edges of a circular pan is of the same value. This result is exactly in accordance with the real heat distribution of the outer edge of a circular pan and it explains why brownies will not get overcooked at the edges. Note that the uncontinuous contours shown in the right picture of Figure 10 is caused by interpolation. 5.3.3 Simulation Results for a Brownie Being Baked in an Elliptical Pan By employing the same method used in the simulation of a rectangular pan to our simulation for a brownie being baked in an Elliptical pan, a slice map and a contour of it can be drawn as shown in Figure 11.

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Temperature 5 4.5 11 4 3.5 3 z 10 2.5 9.5 2 9 20 15 10 5 y 0 0 5 x 15 10 0.5 20 1.5 1

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Temperature(z=2) 15 14 13 12 11 y 10 9 8 7 6 6

10.5

8

10 x

12

14

Figure 11. Contours of heat distribution of the outer edge of a rectangular pan

Unlike the results of the the rectangular and the circular pan, the above contours are a middle state between them. Neither the brownie gets overcooked at the corners nor the heat distributes evenly at the edges. Note that the uncontinuous contours shown in the right picture of Figure 11 is caused by interpolation. Based on the above theoretical verification and simulations, conclusion can be drawn that although the rectangular pans save more space of a baking oven than the circular pans, they have brownies overcooked while the circular pans do not. It strikes us to think that there must be a balance in between. Therefore, we proceed our exploration by looking for the best type of pan.

6 Model Two: the Best Type of Pan

Our goal of this model is to put forward an optimal design of a baking pan which satisfies the following conditions: Maximize number of pans that can fit in the oven (N), Maximize even distribution of heat (H) for the pan, Optimize a combination of conditions (1) and (2) where weights p and (1- p) are assigned to illustrate how the results vary with different values of W/L and p. However, when taking all these conditions into consideration, the process would be very complex and the comprehensive model would be very difficult to solve. Therefore, we tackle the above three conditions one by one. We then firstly proceed with our exploration by analyzing the second condition.

6.1 Even Distribution of Heat

6.1.1 Theoretical Derivation Since temperature is the external expression for heat, even distribution of heat also means even distribution of temperature. Our goal then breaks into finding a specific curve surface that minimizes the nuances of the temperature of every point of the pan. In our exploration of looking for this curve surface, we find that there is a great correlation between our approach and the method used in Digital Image Processing (DIP), one of whose major concerns is to detect the boundary of an object in a given image. Their similarities are listed as follows: They both try to find a specific curve of an unknown edge of an object or surface. In DIP, one has to detect the boundary of an object in a given image while the boundary is unknown. In our model, we have to find a curve surface of even distribution of heat in a thermal space while the curve is also unknown

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They both use Functional Analysis to achieve their goals. In DIP one can establish a function of pixel to detect the boundary while in our model we can establish a function of temperature to determine the curve surface. We then employ the method used in DIP to determine the curve surface of even distribution of heat. Tony F. Chan and Luminita A. Vese (2001) [6] have proposed a model for active contours to detect objects in a given two-dimensional image. Their basic idea in the active contour model is to evolve a curve, subject to constraints from a given image u0 . For instance, starting with a curve of any shape around the object to be detected, the curve moves toward its interior normal and stops on the boundary of the object. Assume that the image i u0 is formed by two regions of approximately piecewise-constant intensities, of distinct values u0 o i and u0 . Assume further that the object to be detected is represented by the region with the value u0 . i o Let C0 denote its boundary. Then we have u0 ≈ u0 inside the object (or inside C0 ), and u0 ≈ u0 outside the object (or outside C0 ), as the pixels inside the object is uniquely different from the pixels i of outside. The curve we are looking for is right on boundary of region u0 . Now let us define the following ‘fitting’ term: [6] F (C ) = F1 (C ) + F2 (C ) (18)

F (C ) = ∫

inside ( C )

u0 ( x, y ) ? c1 dxdy + ∫

2

outside ( C )

u0 ( x, y ) ? c2 dxdy,

2

(19)

where C is any other variable curve, and the constants c1 , c2 depending on C , are the averages of

u0 inside C and outside C respectively. In this case, it is obvious that C0 , the boundary of the

object, is the minimizer of the ‘fitting’ term[6]

inf {F1 (C ) + F2 (C )} ≈ 0 ≈ F1 (C0 ) + F2 (C0 ) .

C

Then, we apply this method to look for our desired surface. Due to our assumption that the thickness of the a baking pan is negligible, the given baking pan can be viewed as a surface denoted by S1 . Starting with S1 we can find our desired surface S 0 . Let u ( x, y, z ) be the temperature of every point within S1 . Similar to Eq.(19), we define the following ‘fitting’ term [6]

F ( S ) = F1 ( S ) + F2 ( S ) + F3 ( S ) F (S ) =

inside ( S )

∫ u ( x, y , z ) ? c

2

1

dxdydz +

outside ( S )

∫ u ( x, y , z ) ? c

2 2

dxdydz +

inside ( S )

∫ ?u

2

dxdydz ,

(20)

where S is any other variable curve, and the constants c1 , c2 depending on S , are the average temperatures of u inside S and outside S respectively. The third term in Eq.(20) is the volume of the overcooked part of a brownie. In this case, it is obvious that S 0 , the boundary of the object, is the minimizer of the fitting term [6]

inf {F1 ( S ) + F2 ( S ) + F3 ( S )} ≈ 0 ≈ F1 ( S 0 ) + F2 ( S 0 ) + F3 ( S 0 ) .

S

In order to solve Eq.(20) to get our desired curve surface, level sets are introduced into our model. In mathematics, a level set of a real-valued function f of n variables is a set of the form [7]

Lc ( f ) = {( x1 , L , xn ) f ( x1 , L , xn ) = c},

(21)

that is, a set where the function takes on a given constant value c. When the number of variables is two, a level set is generically a curve, called a level curve. When n = 3, a level set is called a level surface , and for higher values of n the level set is a level hypersurface. We then therefore, use level sets to represent the surface S of Eq.(20).

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S : {( x, y, z , h) φ ( x, y, z , h) = 0}.

(22)

Using the Heaviside Function H , and the one-dimensional Dirac measure δ 0 , and defined respectively by [6]

?1, if z ≥ 0 d H ( z) = ? , δ ( z) = H ( z) . dz ?0, if z < 0

By substituting Eq.(23) into Fi ( S ) (i = 1,2,3) and they can be rewritten as

(23)

F1 ( S ) =

inside ( S )

∫ u ( x, y , z ) ? c

2

2

1

dxdydz = ∫ u ( x, y, z ) ? c1 H (φ ( x, y, z ) )dxdydz ,

2

Ω

F2 ( S ) =

outside ( S )

∫ u ( x, y , z ) ? c

F3 ( S ) =

2

dxdydz = ∫ u ( x, y, z ) ? c2 [1 ? H (φ ( x, y, z ) )]dxdydz ,

2

Ω

inside ( S )

∫ ?u

2

dxdydz = ∫ ?u H (φ ( x, y, z ) )dxdydz .

2

Ω

By substituting the above three equations into Eq.(20) and it can be rewritten as

F ( S ) = ∫ u ? c1 H (φ )dv + ∫ u ? c2 [1 ? H (φ )]dv + ∫ ?u H (φ )dv .

2 2 2 Ω Ω Ω

(24)

The average temperatures c1 , c2 inside and outside S can be rewritten as

c1 =

∫

Ω

u ( x, y, z ) H (φ ( x, y, z ) )dxdydz

∫ H (φ ( x, y, z ))dxdydz

Ω

, c2 =

∫

Ω

u ( x, y, z )[1 ? H (φ ( x, y, z ) )]dxdydz

∫ H (φ ( x, y, z ))dxdydz

Ω

In order to solve Eq.(24), the mothed of calculus of variations is introduced into our model.

? Assume that S : φ = 0 is the optimal surface of Eq.(24) , λ is a real number and ? ( x, y, z , h) = 0 ? is another curve surface, which has the same initial condition with φ . Let ? L(λ ) = F (φ + λ? ) ,

(25)

? ? ? where F (φ + λ? ) ≥ F (φ ) . When λ = 0 , L(λ ) = F (φ ) , which is the minimal value for Eq.(25).

Therefore, λ = 0 is minimal point and

dL =0. dλ λ = 0

2

By substituting Eq.(25) into Eq.(24) and it can be rewritten as

2 ? ? L(λ ) = ∫ u ? c1 (φ + λ? ) H (φ )dv + ∫ u ? c2 (φ + λ? ) [1 ? H (φ )]dv + ∫ ?u H (φ )dv . 2 Ω Ω Ω

Therefore,

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dL ? ? ′ ? = (?2? )c1 (φ + λ? ) u ? c1 (φ + λ? ) H (φ + λ? )dv dλ ∫Ω 2 ? ? + ? u ? c (φ + λ? ) H ′(φ + λ? )dv

Ω Ω 1 2 2

[ ] ] ∫ [ ? ? ? 1 + ∫ (?2? )c′ (φ + λ? )[u ? c (φ + λ? )][ ? H (φ + λ? )]dv ? ? ? ∫ ? [u ? c (φ + λ? )] H ′(φ + λ? )dv

2 Ω 2

(26)

? + ∫ ? (?u ) 2 H ′(φ + λ? )dv

Ω

? ? ? ? As defined in Eq.(23) that H ′(φ + λ? ) = δ (φ + λ? ) and δ (φ + λ? ) = 0 for φ + λ? ≠ 0 , Eq.(26)

can be simplified to

dL ? ? ′ ? = (?2? ) c1 (φ + λ? ) u ? c1 (φ + λ? ) H (φ + λ? ) dλ ∫Ω ? ? ′ ? + c2 (φ + λ? ) u ? c2 (φ + λ? ) 1 ? H (φ + λ? ) dv

{

[

[

]

][

]}

dL ? ? ? ? ′ ? ′ ? = (?2? ) c1 (φ ) u ? c1 (φ ) H (φ )+ c2 (φ ) u ? c2 (φ ) 1 ? H (φ ) dv dλ λ =0 ∫Ω

Due to the randomicity of the parameter ? , the above equation simplifies to

{

[

]

[

][

]}

? ? ? ? ′ ? ′ ? c1 (φ )[u ? c1 (φ )]H (φ ) + c2 (φ )[u ? c2 (φ )] [1 ? H (φ )] = 0 .

The solution for Eq.(27) is

(27)

? ?c1 (φ ) u ? c1 (φ ) = 0 ? ′ ? ? ? ? ?c2 (φ ) u ? c2 (φ ) = 0 ? ′ ? Our desired surface is derived when φ = 0 .

[ [

] ]

for φ ≥ 0 for φ < 0

(28)

Taking into account the complexity of both the temperature function of u ( x, y, z ) and the

′ ? ′ ? average temperature function of c1 (φ ), c2 (φ ) , direct deduction for the expression of the desired curve

surface from Eq.(28) will be very difficult and almost hardly practical. Therefore, we decide to employ the numerical solution to get the desired surface. 6.1.2 Algorithm Before introducing our algorithm for solving Eq.(28) we firstly define the temperature classes of our algorithm. The highest temperature in our model is defined by class 5 and the lowest temperature is defined by class 1 with other temperatures being define in between. Based on Eq.(2), the temperature is proportional to the heat, we further define that class 4 is a critical condition to get brownies overcooked. If the temperature is above class 4 the brownies will get overcooked while the temperature within class 4 will never get the brownies overcooked. Our goal then breaks into looking for a surface whose heat is evenly distributed with a minimal overcooked volume where the temperature is between class 4 and class 5. The core idea of our algorithm is to look for a baking pan of even heat distribution by many times

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of iteration starting with a given pan of specific shape. Based on the ready-made contours of every layer of the given pan (say a rectangular pan), we choose a set of contours with the same temperature class (say class 4). Therefore, we can construct a new pan with these contours and this is our new research object. Then we apply the method proposed in Model One to determine the heat distribution of this new object. Our goal can be achieved by many times of iteration and the detailed steps of our algorithm are listed as follows. Step 1. Starting with a rectangular pan, we divide the pan into n layers (we set n=7 in our programs) and draw their contours respectively. The contours of temperature distribution of three layer are drawn for example (see Figure 12).

Figure 12. Three contours for the second, the third and the seventh layer

Step 2. Select the contours of temperature class 4 of every layer of the current pan to be the outlines of the next new pan. Redefine the boundary condition of this new pan and employ the method used in Model One to determine the heat distribution of this new pan. The duration of heat diffusion in our simulation is in accordance with the temperature of the center of layer 7. This definition of duration keeps comparable the heat diffusion in every iteration. The iteration process keeps on going until it reaches the stop condition(*), which will further be illustrated below. The object derived at the stop condition is the shape of the pan with even heat distribution we have long be looking for. However, the above algorithm rises three questions worth concerning in step 2 and the stop condition(*) mentioned above will be illustrated here. Question 1. If the contours of temperature class 4 of every layer of the current pan are used to construct the outlines of the next new pan, then how the boundary conditions will be defined? Question 2. How is the stop condition(*) defined? Question 3. How to solve the problem where the area of the new pan becomes smaller and smaller in every iteration since the area of the pan is required to be a constant A ? To make our algorithm work practically, we have tried our best to find suitable solutions to the above three questions, which are listed as follows. Solution 1. Redefinition of Boundary Condition

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Figure 13. Illustration of redefinition of boundary condition

Figure 13 is one of the contours of the seven layers. We choose a curve Y of temperature class around 4 and divide this curve into 14 segments (as the red segments shown above). The segments and the nodes between two segments are all marked from one to fourteen. Let A and B represent node 6 and node 7 respectively and O represents the center of the contour. Suppose there is another point C, the angle between OC and the inverse x-axis is β . The quadrant of C is determined by tan β . If tan β is between tan α and tan γ , we choose OA and OB as its comparisons. Therefore, its exact position is defined by

1 ? ?outside Y , if OC ≥ 2 (OA + OB) ? . ? ?inside Y , if OC < 1 (OA + OB) ? 2 ?

If C is outside or on Y, we then set its temperature class to class 5, as the boundary condition for the new object. Thus, we solve the boundary condition proposed in Question 1. Solution 2. Definition for Stop Condition(*) The iteration stops when the volume of the space get overcooked in a brownie Δv reaches a minimal value, that is Δv → 0 . But how is Δv defined and calculated?

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Figure 14. A schematic view of our desired object

We use Monte Carlo method to calculate the irregular area of every layer. Then, take the average of the area of two neighboring layers. Multiple the average area by the distance between the two neighboring layers, we get the approximated volume between them. We further derive the entire volume Vi of the object shown in Figure 14. Since the space outside the contours (whose temperature class is class 4) of the object is above temperature class 4, the brownies inside this space get overcooked. Therefore, the volume of the space get overcooked in a brownie Δv is defined by

Δv = vi ? vi ?1 .

Thus, the theoretical iteration stops when Δv → 0 . We set the stop condition to be Δv / V = 5% in our simulation process. Solution 3. Compensation for Area Decay Since the area is required to be a constant A, we define a compensation coefficient r for the decayed area of the new pan after every iteration

ri = A

Si

.

This means that we enlarge the area of the new pan artificially to keep it within a constant A. 6.1.3 Simulation Results Based on our algorithm, simulation results of different layers in some iterations are listed as follows.

Initial

Temperature(z=2) 20 18 16 14 12 y y 10 8 6 4 2 2 4 6 8 10 x 12 14 16 18 20 20 18 16 14 12

First time

Temperature(z=2)

Third time

Temperature(z=2) 20 18 16 14 12 y 10 8 6 4 2

10 8 6 4 2 2 4 6 8 10 x 12 14 16 18 20

2

4

6

8

10 x

12

14

16

18

20

Figure 15. Contours for the second layer

Team # 21566

Temperature(z=3) 20 18 16 14 12

Page 19 of 26

Temperature(z=3) 20 18 16 14 12 y 10 8 6 4 2

y 20 18 16 14 12 10 8 6 4 2 Temperature(z=3)

y

10 8 6 4 2 2 4 6 8 10 x 12 14 16 18 20

2

4

6

8

10 x

12

14

16

18

20

2

4

6

8

10 x

12

14

16

18

20

Figure 16. Contours for the third layer

Temperature(z=7) 20 18 16 14 12 y 10 8 6 4 2 2 4 6 8 10 x 12 14 16 18 20

X= 10.404 Y= 10.596 Level= 0.79281

Temperature(z=7) 20 18 16 14 12 y 10 8 6 4 2 2 4 6 8 10 x 12 14 16 18 20

X= 10.404 Y= 10.596 Level= 0.79991

Temperature(z=7) 20 18 16 14 12 y 10 8 6 4 2 2 4 6 8 10 x 12 14 16 18 20

X= 10.404 Y= 10.596 Level= 0.8094

Figure 17. Contours for the top layer It can be observed from the above figures that after four times of iteration starting from a rectangular pan, the contours of the heat distribution of the new pans get more and more like a circle or at least an oval and they become more and more uniformly spaced. Therefore, our desired pan of even distribution has come into being. The parameter of area of every contour and the volume of the new object in every iteration is listed in Table 1.

Table 1. Parameter of every iteration

Area Second Layer Third Layer Fourth Layer Fifth Layer Sixth layer Seventh layer Volume

Initial 400 400 400 400 400 400 2000 / /

First Iteration 46.2837 208 236 285 296 296 1196.14 803.86 40.20%

Second Iteration 44.1326 198 210 259 266 272 1092.07 104.07 8.70%

Third Iteration 44.1221 197 206 241 248 253 1040.56 51.51 4.72%

Δv （ Δv / V ）

It can be observed from the above table that the volume of the space get overcooked in a brownie Δv is getting smaller and smaller after three times of iteration. We can foresee that by a sufficient times of iteration, the Δv will approaches to 0. Thus, the object will be a great approximation of our desired pan shape with an even heat distribution. We can imagine that the final shape of the pan should be something like the object shown in Figure 18.

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0

-0.1

-0.2

-0.3

-0.4 1 0.5 0 -0.5 -1 -1 -0.5 0.5 0 1

Figure 18. An anticipated object of the final pan

In the above work, we have only considered the even heat distribution condition. However, the usage efficiency of the space in an oven should also be taken into account. Anyway, the best type of baking pans should always balance the trade-offs of each side. Therefore, we proceed our exploration by maxing the number of pans.

6.2 Maximum Number of Pans

Suppose we have a couple of pans to be arranged into an oven, when the oven can no longer contain any other pans, we define Δs to be the area of gaps. Since the area of the oven is S = W × L , then the ratio of unused space to the total space is given by Δs / S . Since the area of the baking pans has a constant value A , putting a maximum number of pans into the oven means the greatest usage efficiency of space. Because the ratio of unused space to the total space should be nonnegative,

? Δs ? inf ? ? = 0 . ?S ?

The limiting condition is achieved when the unused space Δs is 0. Thus, the shape of the pans is rectangular. The length a and the width b of a rectangular pan are the common divisors of the length L and the width W of the oven respectively

L = na, W = mb , where n and m are integers. Therefore, the maximum number of pans is given by

N = n×m .

In this model, we only considered the a maximum number of pans, the results above required the pans to be similar to the shape of the oven . However, in a real situation, baking pans and ovens are not necessarily produced by the same company. Even if it is true, baking pans still may not be able to satisfy the above requirements. Therefore, we are going to discuss more types of cooking pans in the following comprehensive model.

6.3 A Comprehensive Model for an Optimized Pan

In the above two work of 6.1 and 6.2, we put forward with two optimized design of baking pans to satisfy its maximum condition separately. However, as required in this problem, a combination of condition (1) and (2) where weights p and (1-p) are assigned is necessarily demanded. Therefore, we proceed our optimization by putting forward with a comprehensive model. Two major concerns of this model are listed as follows.

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The distribution of heat of the baking pans should be as even as possible. When brownies are being baked in an oven, the unsatisfactory result where mass parts of brownies are overcooked is never expected by anyone. For the sake of an even distribution of heat, the ratio of the volume of overcooked parts of brownies to the total volume of brownies is defined by Δv / V , which indicates the extend of being overcooked. The number of pans placed in an oven should be as many as possible. For the sake of the greatest usage efficiency of the space within an oven, the baking pans should suit themselves to make the best use of the space. Therefore, we define the ratio of unused space to the total space by Δs / S to indicate the extend of space wasted. 6.3.1 Definition of an Unused Space of an Oven. The definition of Δv has already been illustrated in model 6.1, however, the definition of Δs is not. Therefore, we are going to firstly define Δs and show how it varies in different arrangement of baking pans. Δs is the space unused or wasted when pans have been placed in an oven. For different types of pans of different sizes, Δs is calculated differently. (1). For rectangular pans, they have two basic arrangement methods - vertical and horizontal, as shown in Figure 19

Figure 19. Horizontal (a) and vertical (b) arrangements for a rectangular pan Δs for the two types is given by

? L ? ?W ? ?W ? ? L ? Δs1 = WL ? de ? ? ? ? , and Δs2 = WL ? de ? ? ? ? . ? e ?? d ? ? e ?? d ?

Compare Δs1 with Δs2 , the best method for arranging pans in an oven can be achieved by the minimal one. However, the above two methods are very elementary. In order to reach an optimal arrangement, we put forward a modified algorithm for rectangular pans. The core idea of our modified algorithm is to divide the area of the oven to four rectangles. Every part may not be necessarily the same. Place pans in very part with the basic arrangement methods and calculate the area of unused space Δsi . Then, the total amount of area of unused space is given by

Δs = ∑ Δs i .

i =1

4

The best approach to divide the space of the oven can be derived when Δs reaches a minimal value.

(2).

For circular pans, they have two basic arrangement methods , as shown in Figure 20.

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Figure 20. Two basic arrangement methods for a circular pan Δs for the two types is given by

?W ? ? L ? Δs1 = WL ? π r 2 ? ? ? ? , and Δs2 = WL ? π r 2 n0 . ? 2r ? ? 2r ?

where the number of pans n0 which can be placed in an oven is defined by

? g ?1 ? ? L ? ? ? L ? × ? 2 ? ? ? 1? + ? ? g is odd ? ? ? ? 2 ? ? 2r ? ? ? 2 r ? n0 = ? , ? g × ? 2 ? L ? ? 1? ? ? g is even ? 2 ? ? 2r ? ? ? ? ? ? ? where the layers of the circular pans g placed in an oven is defined by

?W ? 2r ? g=? ? +1. ? 3r ?

(3). For elliptical pans, they have two substitution methods , as shown in Figure 21.

Figure 21. two substitution methods for an elliptical pan When the eccentricity E of the oval is between 0.5 to 1, the oval is more compressed and we substitute it with a rectangle. When the eccentricity E of the oval is between 0 to 0.5, the oval is more like a circle and we substitute it with a circle. Therefore, we can calculate the unused space of the oven in a rectangular or circular way. Based on the analysis above, the best type of arrangement of putting baking in an oven can be realized by the minimal value of Δs . 6.3.2 Comprehensive Model for an Optimized Pan After an all-round consideration for the parameters mentioned in our two concerns, we then put forward our comprehensive model. Combine the above two concerns with the weights, our comprehensive model for an optimized pan is given by

Δv Δs ? ? min ? f = p + (1 ? p ) ? . V S ? ?

(29)

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To illustrate how the results vary with different values of p, we do the following discussion: When p=1, min ? f = distribution of heat. When p=0, min ? f =

? ?

Δv ? ? , this is the exact situation in 6.1 which only considers the even V ? Δs ? ? , this is the exact situation in 6.2 which only considers the maximum S ?

? ?

number of pans that can be placed in an oven. The best design in this situation is a rectangular pan which is similar to the oven. When 0.5<p<1, Δv / V takes a higher weight. The method to solve Eq.(29) is to focus on Δv / V and try best to minimize it with fewer considerations for Δs / S . The shape of pan can also be derived by many times of iteration with the above algorithm mentioned in 6.1.2. However, when we choose the isotemperature lines, due to our expectations not to have brownies overcooked as much as possible, we should choose the lines with lower temperatures above 4 as small as we can, say 4.2 if p=0.8. This choice is inversely proportional to p. When being baked in an oven, the smaller the volume between the isotemperature surface of the critical state to get brownies overcooked and the outer curve surface is, the smaller Δv / V is. But in the process of iteration, the pan becomes more like a circle gradually. Thus, the ratio of unused space to the total space Δs / S becomes bigger and bigger. However, due to the fact that Δs / S takes a lower weight and Δv / V takes a higher weight, by proper selection of isotemperature surface, we can derive the optimized design. When 0<p<0.5, Δs / S takes a higher weight. The method to solve Eq.(29) is to focus on Δs / S and try best to minimize it with fewer considerations for Δv / V . The shape of pan can also be derived by many times of iteration with the above algorithm mentioned in 6.1.2. However, when we choose the isotemperature lines, due to our expectations to have a minimum number of pan in an oven, we should choose the lines with higher temperatures above 4 as small as we can, say 4.8 if p=0.2. This choice is inversely proportional to p. When being baked in an oven, the bigger the volume between the isotemperature surface of the critical state to get brownies overcooked and the outer curve surface is, the bigger Δv / V is. But in the process of iteration, the pan becomes more like a rectangle gradually. Thus, the ratio of unused space to the total space Δs / S becomes smaller and smaller. However, due to the fact that Δv / V takes a lower weight and Δs / S takes a higher weight, by proper selection of isotemperature surface, we can derive the optimized design. 6.3.3 Simulation Results for the Optimized Pan Based on the analysis above, we then illustrate how the results vary with different values of W/L

and p.

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Figure 22. Contours for the optimized pan when p=0, p=0.3, p=0.7 and p=1

Figure 22 shows the shapes of the baking pans vary with different values of p. The red curves above are the contours with a temperature class around 4 and they indicate the shapes of the pans we derive by different values of p. Like the above analysis, when p=0, Δs / S takes a higher weight, the shape we derive is a rectangle which maximizes the number the pans which can be placed in oven; while p=1, Δv / V takes a higher weight, the shape we derive is a circle which maximizes the even distribution of the heat. While p equals to some values between 0 and 1, the shape is not rectangular or circular but a shape in between, which takes into consideration both the number of pans and the even distribution of heat. On the other hand, the distance between the red curves and the boundary becomes smaller and smaller with the rise of the value of p. This indicates the overcooked volume of a brownie is also smaller and smaller. The detailed changes of these parameters with different values of p are listed in Table 2.

Table 2. How parameters change with different values of p when W/L=2/3

T p=0 p=0.3 p=0.7 p=1 5 4.7 4.3 4

Δv / V

40.20% 21.81% 10.66% 4.72%

Δs / S

1.23% 3.32% 7.38% 14.31%

f 1.23% 8.90% 9.68% 4.72%

It can be concluded from Table 2 that the overcooked volume becomes lower and lower and space usage efficiency of oven becomes higher and higher with the rise of the value of p.

Table 3. How parameters change with different values of W/L when p=0.7

T W/L=1/3 W/L=2/3 W/L=1 4.3 4.3 4.3

Δv / V

10.66% 10.66% 10.66%

Δs / S

11.85% 7.38% 6.92%

f 11.02% 9.68% 9.54%

Parameters in Table 3 are derived by have p fixed at 0.7. Since the overcooked volume is corelated with p, therefore it remains constant too. It can be concluded that the space usage efficiency

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of oven becomes lower and lower with the rise of the value of W/L.

7 Future Work

In Model One, due to our assumption that the baking pans can be treated as a stable hollow heat source without ceiling and the height of the rectangular pan is negligible when compared with the distance between the pan and the bottom of oven, we set different surfaces with the same temperature. However, this may not always be the case, especially when the distribution of thermal air flow is taken into consideration. We also assume that there is almost no heat exchange between the top surface and the air due to the fact that the top surface has close contact with the air, which is poor conductor. However, since most family-owned ovens are of a regular volume, the space in them is very small. In a closed space, it casts doubt on the argument that there hardly is heat exchange between the air and the top of the surface. We will proceed our exploration and put forward a more robust model to show the distribution of heat in different types of pans if time permits. In Model Two, our algorithm provides a practical approach to get our desired shape of even heat distribution. However, the process of every iteration is very complex, the result shown in Figure.15 ~ Figure. 17 are only the result of 3times of iteration due to a lack of sufficient time. Since we only have seven outlines of our desired pan shape, we know for sure that the approximate shape can be achieved by interpolation with the outlines derived after plenty times of iteration .

8 References

[1] http://en.wikipedia.org/wiki/Chocolate_brownie (1st Feb. 2013) [2] http://en.wikipedia.org/wiki/Oven#Cooking (1st Feb. 2013) [3] http://en.wikipedia.org/wiki/Fourier%27s_law#Fourier.27s_law (2nd Feb. 2013) [4] Jianzhong Shen and Feng Liu, Equation of Mathematical Physics, Xi’an Jiaotong University Press, Xi’an, 2010. [5] http://en.wikipedia.org/wiki/Finite_difference_method (2nd Feb. 2013) [6] Tony F. Chan and Luminita A. Vese. Active Contours Without Edges, IEEE Transactions Image Processing, Vol.10, No.2, February 2001. [7] http://en.wikipedia.org/wiki/Level_sets (3rd Feb. 2013) [8] Dakai Wang, Yuqing Hou, Jinye Peng. The method of Partial Differential Equations for Image Processing. Science Press, Beijing, 2010

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9 Advertising Sheet

To: The editor of the New Brownie Gourmet Magazine From: Team # 21566 Date: February 5, 2013 Subject: An advertisement for the Ultimate Brownie Pan Unsatisfied with the brownies you’ve cooked? Do them get overcooked at the corners when being baked in a rectangular pan? Do you have to repeat again and again to bake enough brownies in a circular pan for family get-together? If you do, you will love our Ultimate Brownie Pan? and it will be a great helper to make you a professional cook praised by your beloved ones. Nothing is more satisfying than a homemade, fresh-from-the-oven brownie. But in a conventional rectangular pan, the product can get overcooked at the corners easily. You may say that you can use a circular pan for instead because the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges. But how can you endure that many times of repetition when you have to cook many brownies for your beloved families or honored guests? Our Ultimate Brownie Pan? provides you an even cooked product with moist centers, a paper-thin crackly crust, lots of fair-cooked chewy edges and a greater usage efficiency of your ovens. Our Ultimate Brownie Pan? is not rectangular or circular but a shape in between. The temperature of the outer edges is distributed evenly with a temperature class around 4, which is defined in our paper. The special shape maximizes the number of pans which can be placed in an oven. Therefore, you can enjoy fair-cooked brownies while doing the least work of repetition. Attracted by our Ultimate Brownie Pan?? Then purchase our products in big retailers in major cities worldwide. Sincerely, Team # 21566

赞助商链接

13 *年美国数学建模 A 题* PROBLEM A: The Ultimate Brownie Pan When baking in a rectangular pan heat is concentrated in the 4 corners and the product ...

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