PROBLEM 4.33 KNOWN: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. rSCHEMATIC:ASSUMPTIONS:(1) Two-dimensional, steady-state conduction with no generation, (2) Constant roperties, (3) Boundary is adiabatic. pANALYSIS:(a) Performing an energy balance on the control volume, (Δx/2)⋅Δy, and using the onduction rate equation, it follows that c(1,2) inout123EE0 qqq0−=++=±±()m-1,nm,nm,n-1m,nm,n+1m,nTTxxky1k1k10x2y2y−−ΔΔ⎡⎤Δ⋅+⋅+⋅=⎢⎥ΔΔΔ⎣⎦.−(3) Note that there is no heat rate across the control volume surface at the insulated boundary. ecognizing that Δx =Δy, the above expression reduces to the form R(4) <m-1,nm,n-1m,n+1m,n2TTT4T0.++The Eq. 4.42 of Table 4.2 considers the same configuration but with the boundary subjected to a
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