problem4-76 - PROBLEM 4.76 KNOWN Nodal network and boundary...

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PROBLEM 4.76 KNOWN: Nodal network and boundary conditions for a water-cooled cold plate. FIND: (a) Steady-state temperature distribution for prescribed conditions, (b) Means by which operation ay be extended to larger heat fluxes. m SCHEMATIC: A SSUMPTIONS: (1) Steady-state conditions, (2) Two-dimensional conduction, (3) Constant properties. ANALYSIS: Finite-difference equations must be obtained for each of the 28 nodes. Applying the nergy balance method to regions 1 and 5, which are similar, it follows that e Node 1: ( ) ( ) ( ) ( ) 2 6 y x T x y T y x x y T 0 Δ Δ + Δ Δ Δ Δ + Δ Δ = 1 Node 5: ( ) ( ) ( ) ( ) 4 10 y x T x y T y x x y T 0 Δ Δ + Δ Δ Δ Δ + Δ Δ = 5 Nodal regions 2, 3 and 4 are similar, and the energy balance method yields a finite-difference equation of he form t Nodes 2,3,4: ( ) ( ) ( ) ( ) ( ) m 1,n m 1,n m,n 1 m,n y x T T 2 x y T 2 y x x y T 0 + Δ Δ + + Δ Δ Δ Δ + Δ Δ = Energy balances applied to the remaining combinations of similar nodes yield the following finite- difference equations.
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