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problem4-39 - PROBLEM 4.39 KNOWN Nodal point configurations...

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PROBLEM 4.39 KNOWN: Nodal point configurations corresponding to a diagonal surface boundary subjected to a convection process and to the tip of a machine tool subjected to constant heat flux and convection ooling. c F IND: Finite-difference equations for the node m,n in the two situations shown. SCHEMATIC: A SSUMPTIONS: (1) Steady-state, 2-D conduction, (2) Constant properties. ANALYSIS: (a) The control volume about node m,n has triangular shape with sides Δ x and Δ y while the diagonal (surface) length is 2 Δ x. The heat rates associated with the control volume are due to onduction, q 1 and q 2 , and to convection, q c . Performing an energy balance, find c ( ) ( ) ( ) ( ) in out 1 2 c m,n-1 m,n m+1,n m,n m,n E E 0 q q q 0 T T T T k x 1 k y 1 h 2 x 1 T T 0 y x = + + = Δ ⋅ + Δ ⋅ + Δ ⋅ = Δ Δ ± ± . Note that we have considered the solid to have unit depth normal to the page. Recognizing that Δ x = y, dividing each term by k and regrouping, find Δ m,n-1 m+1,n m,n h x h x T T 2 T 2 2 T 0. k k Δ Δ + + + = < (b) The control volume about node m,n has triangular shape with sides
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