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problem4-33 - PROBLEM 4.33 KNOWN Plane surface of...

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PROBLEM 4.33 K NOWN: Plane surface of two-dimensional system. FIND: The finite-difference equation for nodal point on this boundary when (a) insulated; compare esult with Eq. 4.42, and when (b) subjected to a constant heat flux. r SCHEMATIC: ASSUMPTIONS: (1) Two-dimensional, steady-state conduction with no generation, (2) Constant roperties, (3) Boundary is adiabatic. p ANALYSIS: (a) Performing an energy balance on the control volume, ( Δ x/2) ⋅Δ y, and using the onduction rate equation, it follows that c (1,2) in out 1 2 3 E E 0 q q q 0 −= + + = ±± () m-1,n m,n m,n-1 m,n m,n+1 m,n TT xx ky 1 k 1 k 1 0 x2y2y −− ΔΔ ⎡⎤ Δ⋅ + + = ⎢⎥ ΔΔΔ ⎣⎦ . (3) Note that there is no heat rate across the control volume surface at the insulated boundary. ecognizing that Δ x = Δ y, the above expression reduces to the form R (4) < m-1,n m,n-1 m,n+1 m,n 2T T T 4T 0. ++ The Eq. 4.42 of Table 4.2 considers the same configuration but with the boundary subjected to a
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