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 ( ) V2 rr 2 m ππ . = ⋅Δ ⋅ = Δ Δ ⋅ AA The energy balance is in g a b E E q q qV 0 += + ±± ± () m-1 m m+1 m rT T T k2 r + q2 mr r 0 . 2r πππ ⎡⎤ Δ− Δ −+ + Δ ⎢⎥ ⎣⎦ ΔΔ ± Δ = A Recognizing that r = m Δ r, canceling like terms, and regrouping find 2 m-1 m+1 m 11 q m r m T m+ T 2mT 0. 22 k Δ + ± = < (b) The control volume for the surface node is ( ) r r / 2 π . = ⋅Δ A The energy balance is
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