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CHE3123 Chap17-2BW - Two and Three Dimensional Systems...

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Two and Three Dimensional Systems Analytical Solutions Laplace equation for heat transfer (for conduction) is 2 T = 0 (steady state) 2 T 2 T 2 T 0 x 2 y 2 z 2 Solutions available for simple cases (see text)
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Flux plotting Draw lines of constant temperature (isotherms). Lines perpendicular to these will be energy flow lines (along the direction of maximum gradient). If we keep m = n then the shape factor S is: S N # of flow tubes M # of squares in flow tubes
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Pg 250 table 17.1 has a list of shape factors q = S k (T h -T c ) Note: for convective circular cylinders S 2 π ln(r o /r i ) T + T Energy flow T n m
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Analogy solution: Set up electrical flow lines in geometrically similar shape and record voltage where voltage is the same. Numerical solution: Set up 2-D gradient on object. Look at a volume element i,j+1 x i-1,j y i,j i+1,j i,j-1
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The heat input to the element is: δ Q k y ( T i-1,j – T i,j ) + k y (T i+1,j – T i,j ) dt x x + k x (T i,j-1 – T
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