MIT 18 303 - The 1-D Heat Equation (44 pages)

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The 1-D Heat Equation



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The 1-D Heat Equation

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Pages:
44
School:
Massachusetts Institute of Technology
Course:
18 303 - Linear Partial Differential Equations

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The 1 D Heat Equation 18 303 Linear Partial Di erential Equations Matthew J Hancock Fall 2006 1 1 1 The 1 D Heat Equation Physical derivation Reference Guenther Lee 1 3 1 4 Myint U Debnath 2 1 and 2 5 Sept 8 2006 In a metal rod with non uniform temperature heat thermal energy is transferred from regions of higher temperature to regions of lower temperature Three physical principles are used here 1 Heat or thermal energy of a body with uniform properties Heat energy cmu where m is the body mass u is the temperature c is the speci c heat units c L2 T 2 U 1 basic units are M mass L length T time U temperature c is the energy required to raise a unit mass of the substance 1 unit in temperature 2 Fourier s law of heat transfer rate of heat transfer proportional to negative temperature gradient Rate of heat transfer u K0 area x 1 where K0 is the thermal conductivity units K0 M LT 3 U 1 In other words heat is transferred from areas of high temp to low temp 3 Conservation of energy Consider a uniform rod of length l with non uniform temperature lying on the x axis from x 0 to x l By uniform rod we mean the density speci c heat c thermal conductivity K0 cross sectional area A are ALL constant Assume the sides 1 of the rod are insulated and only the ends may be exposed Also assume there is no heat source within the rod Consider an arbitrary thin slice of the rod of width x between x and x x The slice is so thin that the temperature throughout the slice is u x t Thus Heat energy of segment c A x u c A xu x t By conservation of energy change of heat in from heat out from heat energy of left boundary right boundary segment in time t From Fourier s Law 1 u c A xu x t t c A xu x t tA K0 x u tA K0 x x x x Rearranging yields recall c A K0 are constant u u K0 u x t t u x t x x x x x t c x Taking the limit t x 0 gives the Heat Equation u 2u 2 t x where K0 c 2 3 is called the thermal di usivity units L2 T Since the slice was chosen arbi trarily the Heat Equation 2 applies throughout



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