Thermodynamics HW Solutions 466

Thermodynamics HW Solutions 466 - insulation at the...

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Chapter 5 Numerical Methods in Heat Conduction 5-75 A plane wall with variable heat generation and constant thermal conductivity is subjected to uniform heat flux at the left (node 0) and convection at the right boundary (node 4). The explicit transient finite difference formulation of the boundary nodes is to be determined. q 0 Assumptions 1 Heat transfer through the wall is given to be transient, and the thermal conductivity to be constant. 2 Heat transfer is one-dimensional since the plate is large relative to its thickness. 3 Radiation heat transfer is negligible. Analysis Using the energy balance approach and taking the direction of all heat transfers to be towards the node under consideration, the implicit finite difference formulations become Left boundary node: t T T C x A x A g A q x T T kA i i i i i Δ Δ = Δ + + Δ + + + + 0 1 0 1 0 0 1 0 1 1 2 ) 2 / ( ρ Right boundary node: t T T C x A x A g T T hA x T T kA i i i i i i i Δ Δ = Δ + + Δ + + + + + + 4 1 4 1 4 1 4 1 1 4 1 3 2 ) 2 / ( ) ( h, T Δ x q 0 0 1 2 3 4 g ( x, t ) 5-76 A plane wall with variable heat generation and constant thermal conductivity is subjected to
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Unformatted text preview: insulation at the left (node 0) and radiation at the right boundary (node 5). The explicit transient finite difference formulation of the boundary nodes is to be determined. Assumptions 1 Heat transfer through the wall is given to be transient and one-dimensional, and the thermal conductivity to be constant. 2 Convection heat transfer is negligible. Analysis Using the energy balance approach and taking the direction of all heat transfers to be towards the node under consideration, the explicit transient finite difference formulations become Left boundary node: t T T C x A x A g x T T kA i i i i i Δ − Δ = Δ + Δ − + 1 1 2 2 & Right boundary node: t T T C x A x A g x T T kA T T A i i i i i i Δ − Δ = Δ + Δ − + − + 5 1 5 5 5 4 4 5 4 i surr 2 2 ] ) ( ) [( εσ & ε T surr Δ x Insulated • • • • • • 0 1 2 3 4 5 g ( x ) 5-69...
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