Team #19076 For office use only 19076 - IIT Gandhinagar 19076.pdf · For office use only T1 ___ T2 _ T3 _ T4 ___ Team Control Number 19076 Problem Chosen A For office

Team #19076 Page 1 of 22

For office use only

T1 ________________

T2 ________________

T3 ________________

T4 ________________

Team Control Number

19076

Problem Chosen

A

For office use only

F1 ________________

F2 ________________

F3 ________________

F4 ________________

2013 Mathematical Contest in Modeling (MCM/ICM) Summary Sheet

In this proposal we present a model for optimizing the cooking of food items in the

electrical oven by optimizing the shape of pans used in the same.

We started with the mathematical modelling for the heat distribution along the

edges of a pan. For the doing so we have assumed exponential decaying model for

estimating the amount of energy absorbed at across the edge. The main reason

behind this is that according to the paper “Combined conduction and radiation

heat transfer with variable thermal conductivity and variable refractive index “,

which is available online at www.sciencedirect.com, temperature is decaying

along the length for 1-D object in exponential way. This is a basis for theoretical

and mathematical component of this method. Some other assumptions are also

taken for simplification of the model.

Using above concepts incorporated with concepts of superposition, we have

modelled the variation of heat along the edges of the food item kept in pans of

different shapes (like: square to circle with other like pentagon, hexagon etc. in

between). All the analysis done by us, are on regular polygonal pan shapes for the

sake of simplicity. Thus for a N side polygon, each side faces a heat distribution due

to combination of flux from n sides (including itself)

From the results obtained above, we got relation between the quality of the

food item (defined by us) and the number of edges of the pan in which it was

baked. We also verified that the heat distribution in case of circular pan has

maximum uniformity.

Once done with the quality, we started with the quantitative analysis. For the

sake of simplification of the model, we assumed a particular methodology of

putting the pans which is based on the general human habits. By keeping the area

of the pan constant but variation in shapes, we have finally obtained a relation

between packing efficiency and number of edged in the pan (at different values of

required parameters). The proposal ends with a combined analysis of above

concepts to obtain optimized values of the number of edges (pan shape) on the

basis of user preference.

However, a weakness in our model could be that a lot of assumptions are made

based on logic but no theoretical, experimental background. But for now, the

model remains accurate and reasonable.


Contents

List of Figures: ....................................................................................................................... 3

List of Symbols, Subscripts and Superscripts.......................................................................... 4

1. Introduction: .................................................................................................................. 5

2. Problem Statement: ....................................................................................................... 5

3. Model 1: Heat distribution across the edges for various pan shapes .............................. 6

3.1 Assumptions: .......................................................................................................... 6

3.2 Model Overview:..................................................................................................... 7

3.3 Model Description: ................................................................................................. 7

3.4 Case Study - Rectangular shape pan: ....................................................................... 9

3.5 Model Application: ................................................................................................ 11

3.6 Result and Conclusion: .......................................................................................... 15

4. Model 2: Optimize solution for pan shape.................................................................... 16

4.1 Assumptions: ........................................................................................................ 16

4.2 Model Overview:................................................................................................... 17

4.3 Model Description: ............................................................................................... 17

4.4 Result and Conclusion: .......................................................................................... 21

Limitations: ......................................................................................................................... 21

References: ......................................................................................................................... 22


List of Figures:

Figure 1: Rectangular Pan

Figure 2: Round Pan .............................................................................................................. 5

Figure 3: Oven with two racks ............................................................................................... 5

Figure 4: Schematic of the Model .......................................................................................... 8

Figure 5: Understanding scaling Factor Effect ........................................................................ 8

Figure 6: Heat distribution for rectangular pan...................................................................... 9

Figure 7: Surface plot showing heat variation on rectangular pan ....................................... 10

Figure 8: Contour plot showing heat variation on rectangular pan ...................................... 10

Figure 9: Square .................................................................................................................. 11

Figure 10: Pentagon ............................................................................................................ 11

Figure 11: heat distribution plot for hexagon ...................................................................... 12

Figure 12: heat distribution plot for heptagon ..................................................................... 12

Figure 13: heat distribution plot for octagon ....................................................................... 13

Figure 14: heat distribution plot for nonagon ...................................................................... 14

Figure 15: Relative Burning wrt number of edges ................................................................ 15

Figure 16: packing in circular pan ........................................................................................ 16

Figure 17: Packing efficency for different values of K (W/L) ................................................. 19


List of Symbols, Subscripts and Superscripts

Symbol Description Units

A Cross – Section area of Pan cm2

W Width of the oven cm2

L Length of the oven cm2

Ko Width to Length ratio for oven (W/L) -

Kp Width to Length ratio for pan -

Q0 Heat on the outer side of pan Joules

N Number of pans in the oven -

ΔH Heat difference between corner and center of edge

Joules

Ae Effective area of polygon cm2

N Total number of edges in polygonal Pan -

ηe Packing Efficiency of pan in oven -

a Length of rectangle covering the polygon

cm

b Length of rectangle covering the polygon

cm


1. Introduction:

Ovens are one of the most common kitchen appliances used in every house. Today

various kinds of ovens are available in market which includes conventional ovens

(works on natural convection), modern ovens (works on forced convection) and

microwave ovens. They are generally available in rectangular shape and used for

heating, baking, drying etc.

Lot of study has been done on oven to efficiently cook the food. But very little effort

seems to be provided towards improving the pan design. To study the effect of pan

shape on cooking of the food, we have constructed a mathematical model which will

help to choose user, appropriate shape of pan as per his priorities.

Figure 1: Rectangular Pan Figure 2: Round Pan

2. Problem Statement:

We are given a rectangular oven with

constant width to length ratio and having

two evenly spaced racks. Rectangular pans

are commonly used for baking in the oven.

But the product gets overcooked near the

corner of the pan due to higher heat

distribution around the corner of the pan.

To overcome this problem, pans can be

design in circular shape to provide even

Figure 3: Oven with two racks


distribution of heat across the edge. But circular pans are inefficient for space

optimization. Our model can help to select the optimize pan shape for the

combination of above to factors.

3. Model 1: Heat distribution across the edges for various pan shapes

3.1 Assumptions:

1. Oven is considered rectangular in shape, with width W and Length L.

2. Area (A) of the pan is fixed.

3. Heat flux on the outer surfaces of the pans is uniform.

4. Overcooking of the food is due to excessive heat (as compare to center

portion of the edges) absorbed by food at the point of observation. Hence we

are relating the heat distribution across the edge directly to the heat

absorbed by the food.

5. To describe the heat distribution across the edge we have assumed the

exponential decaying function with maximum value at the corners.

6. Model Selection Justification: For obtaining the rate of heat absorption in the

food item the most basic requirement is a model or the law according to

which the heat distribution can be estimated in a food item. After surfing a

lot on the internet and available literature we were unable to find any model

for the same. But in most of the cases decaying function is found to be

exponential in nature. In many of the heat transfer phenomenon also the

variation of Temperature is exponentially decreasing. Also in case of

radioactive decay, the amount of nuclei left after a certain interval of time is

observed to be exponentially decreasing.

7. Another reason to choose exponential function is that, intuitively one can say

that the effect of any heat generating source becomes zero only at infinity so

functions like �

�� ,�� ,�� (� < 1) are possible choices. But out of all of them

�� is easiest to handle. Hence it has been selected.


8. In our model we have considered heat distribution contours to be straight

lines parallel to heat releasing edges which should be circular in nature in

actual case (shown below).

3.2 Model Overview:

Our model based on the observation that food gets overcooked at the corners than

across the edges. So heat absorbed by the food is max at the corners and decreases

at the center of the edge for the same time period. This variation is assumed to be

same as that of heat distribution across the edge. To begin with we have chosen a

rectangular pan to show the uneven heat distribution over its cross section. We

applied our model to square, pentagonal, hexagonal, heptagonal, octagonal and

nonagonal pan shapes and extrapolated the result to show even heat distribution

across the circular pan.

3.3 Model Description:

The heat absorption variation with distance from the side at which it is supplied:

Q = e�� ∗ SF(x)

Where,

SF(x)= Scaling Factor(x) = Transverse Length at x

Transverse Length at x = 0

From the above model, net Heat distribution due to only one edge can be seen

below:


Figure 4: Schematic of the Model

Scaling factor refers to the ratio of transverse length at given distance (x) from origin

to the transvers length of surface at origin( x=0 from which heat is coming).

Thus the basic assumption that we have used in adopting the definition of Scaling

factor is that, ‘Heat is evenly distributed over entire transverse length at given

distance from origin’.

Consider following example to understand effect of scaling factor in a better way.

Analyzing cross section AB (at some particular distance from the origin) for different

shapes,

Figure 5: Understanding scaling Factor Effect


Scaling Factor at Cross section AB (equal distance from each):

Case 1: d/a Case 2: d/b Case 3: d/c

Where:

a, b, c, d are lengths at different cross sections as shown in Figure 4

Thus SF(1)> SF(2)> SF(3).

Now,

Assuming same heat input per unit length from heat sources

Heat distribution trend is: Case 1> Case 2> Case 3.

3.4 Case Study - Rectangular shape pan:

Consider a cross section of rectangular pan.

We are showing here the variation of heat absorbed by the food on given pan cross

section.

From above our model, Scaling Factor = 1.

Heat distribution in the pan due to uniform heat flux on:

Side AB : Q = Q� ∗ e�� Side AD : Q = Q� ∗ e

��

Side CD : Q = Q� ∗ e�(��) Side BC: Q = Q� ∗ e

�(��)

Figure 6: Heat distribution for rectangular pan


The net heat distribution obtained has been shown in contour and surface plots

below (plotted in Matlab):

Figure 7: Surface plot showing heat variation on rectangular pan

Figure 8: Contour plot showing heat variation on rectangular pan

As can be seen above that the variation along the edges is found as per expectations

and the maximum heat concentration is at corners. Thus verifying the applicability of

our Model.


3.5 Model Application:

We used the above model to find out the heat distribution across the edges of

square, pentagonal, hexagonal, heptagonal, octagonal and nonagonal pan shapes.

The equations utilized and the results obtained are shown below:

1 + 2.72�� + 2.72�(� � �.��) + 2.72�� (� � �.��)�

Figure 9: Square

Figure 10: Pentagon

1 + 2.72��∗��

��∗�

0.76

0.76+ 2� ∗sin��10�� + 2.72

�(�.��)∗��

�∗�

0.76

0.76+ 2 ∗(0.76− �)∗sin��10��

+ 2.72��.��∗��

��∗�

0.76

1.23− 2� ∗cos��5�� + 2.72�

��.��∗(�.��)∗��

∗�0.76

1.23− 2 ∗(0.76− �)∗cos��5��


1 + 2.72��.�� +

2.72�(��.��)∗��

�

��∗

�.��

�∗(��.��)∗��

��.� �

+

2.72��.��(��.��)�∗��

�

��∗

�.��

�∗��.��(��.��)��

�� .��

+

2.72��

�

��+

2.72�(�.��.��)∗��

�

��∗

�.��

�.��∗��.��(��.��)�∗��

��+

2.72��∗(�.��)�(��.��)�∗��

�

��∗

�.��

�.��∗(��.��)∗��

��

1 + 2.72��(�)

�� ∗��.��

�.��∗��(�)� +

2.72�(�.�� )∗��(�)

�� ∗

��.��

�.��(�.��)∗��(�)� +

2.72�� ∗��(�)� �.��∗(�.��)��

�� ∗

��.��

�.�∗(�.��)��∗��(�)� +

2.72��(�.��)∗��(�)��.��∗(�.��)�

�� ∗

��.��

�.�(�.��)–�∗(�.��)∗��(�)� +

2.72�� (�)� �.��∗(�.��)�

�� ∗

��.��

�.�∗(�.��)��∗��(�)� +

2.72��(�.��)∗��(�)� �.��∗(�.��)�

�� ∗

��.��

�.�(�.��)��(�.��)��(�)�

Figure 11: heat distribution plot for hexagon

Figure 12: heat distribution plot for heptagon


Figure 13: heat distribution plot for octagon

1 + ��.� + ��(�∗��(�.��)) ∗�0.54

0.54 + 2� ∗sin(0.79��)� + ��(�.��) ∗�

0.54

1.3�

+ ��(�.��∗��(�.��))∗�0.54

1.3 − 2� ∗cos(0.79��)� + ��(�.��)∗��(�.��)�

∗�0.54

0.54 + 2 ∗(0.54 − �)∗sin(0.79��)� + ��(�.��.��) ∗�

0.54

1.3�

+ ��(�.��(�.��)∗��(�.��))∗�0.54

1.3 − 2 ∗(0.54 − �)∗cos(0.79��)�


Figure 14: heat distribution plot for nonagon

1 + 2.72��∗��(�.��) ∗�0.4

0.4 + 2� ∗sin(0.87��)� + 2.72�(�.��∗��(�.��))

∗�0.4

1.02+ 2� ∗sin(0.17��)� + 2.72�(�.��∗��(�.��))∗�

0.4

1.16− 2� ∗cos(1.05��)�

+ 2.72�(��∗��(�.��))∗ �0.4

0.76 − 2� ∗cos(0.35��)� + 2.72�(�.��)∗��(�.��)

∗�0.4

0.4 + 2 ∗(0.4 − �)∗sin(0.87��)� + 2.72�(�.��(�.��)∗��(�.��))

∗�0.4

1.02+ 2 ∗(0.4 − �)∗sin(0.17��)� + 2.72�(�.��(�.��)∗��(�.��))

∗�0.4

1.16− 2 ∗(0.4 − �)∗cos(1.05��)� + 2.72�(��(�.��)∗��(�.��))

∗�0.4

0.76− 2 ∗(0.4 − �)∗cos(0.35��)�

Above equations are derived by assuming the uniform heat input from all the edges

of the pan.

To find out the heat distribution across the edge of the polygon we consider the

effect of heat coming from all other edges of the polygon.

These values are as per our exponential model (e^x* SF(x)).

The above graphs (U-shape graph in shaded region) show the variation of heat

across the bottom edge for all polygons.

For the corners heat concentration increases very rapidly. So we choose heat value

at distance 10% from each corner.


Then we normalize the values of heat by dividing it by heat value at the centre of the

edge. This value will give you the relative burning of the corners with respect to

centre of the edge.

From the graphs drawn above we have obtained the relative burning at the corners

of pan for different shapes of pan.

Relative Burning: Ratio of heat absorbed at a distance of 0.1 times the total edge

length from the corner to centre of the edge.

Figure 15: Relative Burning wrt number of edges

3.6 Result and Conclusion:

1. Area of the pan (A) is same for all shapes (square to circle). So for fixed area

(A), increasing the number of sides, edge length decreases. This decreases

the heat difference between corners and center of the edge (ΔH).

2. The equation for relative burning is � = 3.73∗�(��.��∗�).

3. We can extrapolate this result to show that for circle which can be

considered as polygon of infinite sides, heat distribution is uniform.

y = 3.73e-0.12n

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

3 4 5 6 7 8 9 10

Rel

ati

ve b

urn

ing

Number of edges in polygon

Variation of Unevenity in heat distribution across edge


4. On increasing the number of side of regular polygon, number of axis of

symmetry increases and hence overall symmetry of polygon is increased.

Thus heat distribution is becoming uniform. As circle has infinite number of

axis of symmetry, heat distribution across the edges (circumference) is

expected to be highest uniformity.

4. Model 2: Optimize solution for pan shape

4.1 Assumptions:

1. Oven is considered rectangular in shape, with width W and Length L.

2. Area (A) of the pan is fixed.

3. Length (L) of the oven is fixed.

4. We chose polygon till 9 edges. As generally pan with more than 9 edges is not

used.

5. For simplicity we neglected the spacing between two pans in oven. This

effect of spacing will be same for all shapes of pan.

6. We considered the pans of regular shapes. Considering the factors of

easiness, simplicity, and habitually one can say that most of the users will

prefer to make the regular arrangement of the pans. The point can be

explained by considering the arrangements of circular pans in a rectangular

oven.

Figure 16: packing in circular pan

(a) (b)


We have assumed that a user will prefer and make arrangements as in (a) instead

of (b) even after knowing the fact that b) is better in terms of space optimization

because,

1. The assumption makes our model quite simple, easy to analyze.

2. Making of arrangement – (a) is quicker and convenient for user.

4.2 Model Overview:

In the model, we started with defining the effective area for polygon. We calculated

packing efficiency for pan by assuming pan cross – section area (A) equals 1cm2 and

oven length (L) to be 20cm. This packing efficiency is plotted with the pan shape.

4.3 Model Description:

The figure below shows the arrangements of pans in the oven. The arrangement is

regular and pans are arranged in regular manner. On the basis of above discussion/

assumption, we found that every pan will occupy a rectangular space of the

dimensions shown below:


The effective area occupied by each pan is equal to the area of rectangle

surrounding the polygon.

Effective Area (Ae): Area of the rectangle covering the polygon.

The geometric construction shown below shows the surrounding rectangle for

different polygon.

Ae for different polygons are listed in table below:

Following table shows dimensions of equivalent rectangle for different pan shapes:

Polygon Edge

length (cm)

Length of rectangle covering

the polygon

(cm)

Width of rectangle covering

the polygon (cm)

Actual Area

occupied by

polygon (cm2)

Area of the rectangle covering polygon

(Ae) (cm2)

Area loss per polygon

(cm2)

Square 1.00∗√� 1.00∗√� 1.00*√� A 1.00*A 0

Pentagon 0.76∗√� 1.23*√� 1.17*√� A 1.44*A 0.44*A

Hexagon 0.62∗√� 1.24 ∗√� 1.07*√� A 1.33*A 0.33*A

Heptagon 0.52∗√� 1.18 ∗√� 1.15*√� A 1.36*A 0.36*A

Octagon 0.45 ∗√� 1.1∗√� 1.1 ∗√� A 1.21*A 0.21*A

Nonagon 0.40*√� 1.16∗√� 1.14 ∗√� A 1.32*A 0.32*A


For the irregular polygonal pan, the packing efficiency is given by ηe = ��∗

�

��∗�

�∗�

For pan of regular polygonal pan,

ηe = ��

�

��∗��

�

��,��

�

��∗��

�

��∗�

�∗� =

��

��∗��

�

��,��

�

��∗��

�

��∗�

�∗(��)

Following table shows the packing efficiency value for different values of K (=W/L)

with A=1 cm2 and L=20 cm.

Polygon a(cm) b(cm) �� = ��

�

�� ∗ ��

�

�� ,��

�

�� ∗ ��

�

�� ∗ �

� ∗ �

K=0.1

L = 20 cm W=2 cm

K=0.2 L = 20 cm W=4 cm

K=0.5 L = 20 cm W=10 cm

K=0.75 L = 20 cm W=15 cm

K=1.0 L = 20 cm W=20 cm

Square 1.0 1.0 40/40 80/80 200/200 300/300 400/400

Pentagon 1.0 1.0 17/40 51/80 136/200 192/300 289/400

Hexagon 1.0 1.0 18/40 54/80 144/200 224/300 324/400

Heptagon 1.0 1.0 17/40 51/80 136/200 208/300 289/400

Octagon 1.0 1.0 18/40 54/80 162/200 234/300 324/400

Nonagon 1.0 1.0 17/40 51/80 136/200 221/300 289/400

Following graph shows the variation of packing efficient with number of edges

Figure 17: Packing efficency for different values of K (W/L)

30

40

50

60

70

80

90

100

4 5 6 7 8 9 10

Pa

ckin

g e

ffic

ien

cy o

f P

an %

Number of edges in Pan/polygon

Variation of Packing efficiency with number of edges

(W/L) = 0.1

(W/L) = 0.2

(W/L) = 0.5

(W/L) = 0.75

(W/L) = 1


The curves obtained above shows variation of space utilization by a polygon with its

number of edges for different values of (W/L).

For given condition curves shows fluctuating behavior hence are to be used as such.

This model requires the maximization of space utilization while minimization in heat

variation across the edge.

We tried to approximate graph between ηe and number edges (W/L = 0.5) for higher

order polynomial. For the 5th degree polynomial the graphs shows minimum

deviation (R2 = 1). So we are approximating the packing efficiency by function

g(n)= − 3.7417 ∗n� + 123.83 ∗n�– 1614.5∗n� + 10359∗n�– 32687∗n +

40766.

We define the function for combine effect of heat distribution and packing efficiency

as follows:

Number of edges = H(p)= (g(n)∗p ∗4 )+ �(1 − p)∗�

�(�)��

�

For p=1 (for optimal packing efficiency), n=4 (square shape pans)

For p= 0 (For even heat distribution), n=infinity (circular shape pans)

Following plot shows the variation of pan shape (type of polygon) for different values

of weights (p).

H(p) = n g(n) y(n) p 4 1.00 2.46 1

5 0.68 2.00 0.467

6 0.72 1.8 0.402

7 0.68 1.4 0.297

8 0.81 1.42 0.33

9 0.68 1.36 0.22


4.4 Result and Conclusion:

1. The above graph can be used to find optimize solution as per the preferences

of the user (which is defined by value p).

2. The graph shows a regular trend between the number of edges in pan and

the preference of the user. The optimize value of number of edges reaches

infinity at p=0, which means user is only interested in good quality product

which is obtained for circular pan. Similarly for p=1, space is the only

preference. Hence square is obtained.

Limitations:

1. Our model is not applicable for polygons of irregular shapes.

2. Interpolated polynomial is taken to be exponential whereas a good number

of other functions satisfying the obtained data points. To get the more

accurate interpolated function we need to obtain higher number of data

points by performing similar analysis on larger number of polygons.

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1

Nu

mb

er o

f ed

ges

in p

oly

gon

Weight (p)

Variation of n with wights p


References:

1. Edible Garden, http://www.cookingandme.com/2010/05/types-of-ovens-

how-to-choose-oven.html

2. http://www.sciencedirect.com/science/article/pii/S096030850700017X

3. Geogebra: http://www.geogebra.org/cms/

4. Blender: http://www.blender.org/

5. http://mathworld.wolfram.com/CirclePacking.html

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Documents

Team #19076 For office use only 19076 - IIT Gandhinagar 19076.pdf · For office use only T1 _____ T2 _____ T3 _____ T4 _____ Team Control Number 19076 Problem Chosen A For office

Team #19076 For office use only 19076 - IIT Gandhinagar 19076.pdf · For office use only T1 ___ T2 _ T3 _ T4 ___ Team Control Number 19076 Problem Chosen A For office