Consider the plane wall of thickness 2L, the infinite cylinder of radius r_o, and the sphere of radius r_o. Each configuration is subjected to a constant surface heat flux q"_s. Using the approximate solutions of Table 5.2b for Fo ≥ 0.2, derive expressions for each of the three geometries for the quantity (T_s,act – T_i)/(T_s,lc – T_i). In this expression, T_s,act is the actual surface temperature as determined by the relations of Table 5.2b, and T_s,lc is the temperature associated with lumped capacitance behavior. Determine criteria associated with (T_s,act – T_i)/ (T_s,lc – T_i) ≤ 1.1, that is, determine when the lumped capacitance approximation is accurate to within 10%.

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Geometry Semi-infinite Interior Cases Plane wall of thickness 2L Infinite cylinder Sphere Exterior Cases Sphere Various shapes (Table 4.1, cases 12-15) Length Scale, Le L (arbitrary) L 1₂ Exact Solutions 1 πT 2√ Fo [Fo+1-25 expl-For]* [2Fo + 1 - 25 exp(-6, F0) -1 -1 [3Fo + 1-2 cxp(-6, FO) - n-1 5². 1 5n = nπ None q" (Fo) J₁(5)=0 [1-exp(Fo)erfc(Fo¹2)]-¹ tan(5₂)=5n Fo<0.2 −2 2 Approximate Solutions Fo≥ 0.2 2√ Fo LR LR LR LR -IN 2√ Fo TT 2√ Fo Use exact solution. i 100 + ㅠ 77 4 N + 0.77 V Fo 113 0.77 VFo + Fo+ −4 -1 +]*¹ -1 +1 Maximum Error (%) + 9% None 5.3 2.1 4.5 (A₂/4π)¹/2 "q* =qL/k(T₁-T) and Fo= at/L, where Le is the length scale given in the table, T, is the object surface temperature, and T; is (a) the initial object temperature for the interior cases and (b) the temperature of the infinite medium for the exterior cases. 3.2 Unknown