FALLSEM2015-16_CP0667_16-Oct-2015_RM02_53-Applications-of-pde---One-Dimensional-Heat-Equation---Nonh - One dimensional Heat flow in the rods with


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One dimensional Heat flow in the rods with nonzero Boundary Conditions Dr. T. Phaneendra , Professor of Mathematics, VIT University, [email protected] 11/4/2013 1 CV & PDE (MAT201)
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Let an aluminum rod of length 20 cm be initially at the uniform temperature 25 ° C. Suppose that at time t = 0, the end x = 0 is suddenly cooled to 0 ° C while the end x = 20 is heated to 60 ° C, and both are thereafter maintained at those temperatures. Find the temperature distribution in the rod at any time t (Given that the thermal Diffusivity of aluminum is α 2 = 0.86). 11/4/2013 2 CV & PDE (MAT201)
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The temperature distribution in the rod is governed by the boundary value problem: α 2 u xx = u t for 0 < x < 20 and t > 0 (1) u (0, t ) = 0, u (20, t ) = 60 for t > 0 (2) ( boundary conditions ) and u ( x, 0) = 25 for 0 < x < 20 (3) ( initial condition ) 11/4/2013 3 CV & PDE (MAT201)
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square6 Recall that separation of variables will only work if both the partial differential equation and the boundary conditions are linear and homogeneous. square6 In the given problem, boundary conditions are not homogeneous (the temperature at the right end is nonzero) 11/4/2013 4 CV & PDE (MAT201)
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square6 There are no sources to add/subtract heat energy anywhere in the bar. Also our boundary conditions are fixed temperatures and so can not change with time and we are not prescribing a heat flux on the boundaries to continually add/subtract heat energy. Thus there will not be any forcing of heat energy into square6 or out of the bar and so while some heat energy may well naturally flow into our out of the bar at the end points as the temperature changes eventually the temperature distribution in the bar should stabilize out and no longer depend on time.
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