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# Derivation of One-Dimensional Heat Equation 3363

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This 2 page Bundle was uploaded by Shahmeer Baweja on Saturday November 8, 2014. The Bundle belongs to 3363 at University of Houston taught by a professor in Fall. Since its upload, it has received 66 views. For similar materials see Intro to PDE in Math at University of Houston.

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Date Created: 11/08/14

DERIVATION OF THE HEAT DIFFUSION EQUATION IN ONE SPACE DIMENSION PHILIP W WALKER A rod of length L units of length insulated except perhaps at its ends lies along the maxis with its left end at coordinate 0 and its right end at coordinate L Suppose that the mass density p units of mass divided by units of length and thermal conductivity K0 energygtltlength time gtlttemperature and speci c heat c energymassgtlttemperature at each point in the rod depend only on the mcoordinate of the point Let e and Q be as follows The thermal energy density energy length at 75 units of time after the time origin at points with rst coordinate x is em t The heat flux energytime to the right at time 75 through the cross section consisting of points with rst coordinate x is m t A negative value for mt indicates heat flow to the left The heat energy per unit length being generated per unit time inside the rod at time t at points with rst coordinate x is Qx t A negative value for Q indicates a heat sink Suppose that 0 g a g b g L Conservation of thermal energy tells us that the timerateofchange in thermal energy in the section of the rod consisting of points with rst coordinate x satisfying a g x g b is the net heat energy owing per unit time across the boundaries of this section plus the net heat energy being generated internally in the section Thus b b E em wdsv lta tgt agtltb t c2ltxtgtdw Assuming that e and have continuous rst order partial derivatives we have b b b mtdm a mtdm Qmtdm 1 Thus ab gems mt Qzt div 0 Since this is true for each choice of a and b with 0 g a g b g L if Q is continuous and e and have continuous rst order partials it follows that 36 amt axmt Qxt for 0 gm g L andt 2 0 By de nition the temperature u is given by emt cmpmumt for 0 g x g L and t 2 0 So cmpmmt mt 1 Qmt for 0 g x g L and t 2 0 1 2 PHILIP W WALKER According to Fourier s law of heat conduction 3 g0mt K0ma umt for 0 g x g L and t 2 0 m Thus we arrive at the heat diffusion equation in one space dimension 8a 3 8n ltgt cp KO 875 8x 856 If each of c p and K 0 is constant and there are no internal sinks or sources so that Q is zero we have Qfor0 m Landt20 8u 82u Z f lt ltL d gt atmt ax2mt or0m an t0 where K0 s 0p is the thermal diffusivity

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