problem4-36 - PROBLEM 4.36 KNOWN Conduction in a...

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PROBLEM 4.36 KNOWN: Conduction in a one-dimensional (radial) cylindrical coordinate system with volumetric eneration. g F IND: Finite-difference equation for (a) Interior node, m, and (b) Surface node, n, with convection. SCHEMATIC: (a) Interior node, m (b) Surface node with convection, n ASSUMPTIONS: (1) Steady-state, one-dimensional (radial) conduction in cylindrical coordinates, (2) Constant properties. ANALYSIS: (a) The network has nodes spaced at equal Δ r increments with m = 0 at the center; hence, r = m Δ r (or n Δ r). The control volume is ( ) V 2 r r 2 m r r π π . = ⋅Δ ⋅ = Δ Δ ⋅ A A The energy balance is in g a b E E q q qV 0 + = + + = ± ± ± ( ) m-1 m m+1 m r T T r T T k 2 r k 2 r+ q 2 m r r 0. 2 r 2 r π π π Δ Δ + + Δ Δ Δ ± A A Δ = A Recognizing that r = m Δ r, canceling like terms, and regrouping find 2 m-1 m+1 m 1 1 qm r m T m+ T 2mT 0.
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