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.
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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.
Team #19076 Page 2 of 22
Contents
List of Figures: ....................................................................................................................... 3
List of Symbols, Subscripts and Superscripts.......................................................................... 4
Cuts most cooking time by 50%, Brownie Baking in 2 times faster in larger amount. Make the greatest cooking discovery since fire for yourself. Cooking with this oven is simple and easy. You can cook thing as per your choice with a better quality in more amount and in a quicker way in about half of the normal time. Only the food gets hot your kitchen and stays cool.