Chapter 5 Numerical Methods in Heat Conduction 5-21 A plane wall with variable heat generation and variable thermal conductivity is subjected to specified heat flux and convection at the left boundary (node 0) and radiation at the right boundary (node 5). The complete finite difference formulation of this problem is to be obtained. q0 Convectio Δ x 1 ε • • •0 2 q 0 T surr Radiation h , T ∞ g ( x ) k ( T ) Assumptions 1 Heat transfer through the wall is given to be steady and one-dimensional, and the thermal conductivity and heat generation to be variable. 2 Convection heat transfer at the right surface is negligible. Analysis Using the energy balance approach and taking the direction of all heat transfers to be towards the node under consideration, the finite difference formulations become Node 0 (at left boundary): 0) 2 / ( ) (00 1000 = Δ + Δ − + − + ∞ x A g x T T A k T T hA A q Node 1 (at the mid plane): 0) 2 / ( 1 1 2 1 10 1 = Δ + Δ − + Δ − x A g x T T A k x T T
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