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PROBLEM 4.73 KNOWN:Electrical heating elements with known dissipation rate embedded in a ceramic plate of known thermal conductivity; lower surface is insulated, while upper surface is exposed to a convection rocess. pFIND:(a) Temperature distribution within the plate using prescribed grid spacing, (b) Sketch isotherms to illustrate temperature distribution, (c) Heat loss by convection from exposed surface (compare with element dissipation rate), (d) Advantage, if any, in not setting Δx = Δy, (e) Effect of grid size and convection coefficient on the temperature field. SCHEMATIC:ASSUMPTIONS:(1) Steady-state, two-dimensional conduction in ceramic plate, (2) Constant properties, (3) No internal generation, except for Node 7 (or Node 15 for part (e)), (4) Heating element pproximates a line source of negligible wire diameter. aANALYSIS:(a) The prescribed grid for the symmetry element shown above consists of 12 nodal points. Nodes 1-3 are points on a surface experiencing convection; nodes 4-6 and 8-12 are interior nodes. Node 7 is a special case of the interior node having a generation term; because of symmetry, = 25 W/m. The finite-difference equations are derived as follows: htq′Continued...
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