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

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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 TT T T kx 1 ky 1 h2 x 1T T 0 yx −= + + = −− Δ⋅ + Δ⋅ + = ΔΔ ±± . 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 hx T T T 2 T 0. kk ⎡⎤ ++ + ⎢⎥ ⎣⎦ = < (b) The control volume about node m,n has triangular shape with sides
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This note was uploaded on 12/07/2010 for the course MAE Heat Trans taught by Professor Lee,j.s. during the Spring '10 term at Seoul National.

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