The effective thermal conductivity k_eff for concentric cylinders and concentric spheres is provided in Equations 9.59 and 9.62, respectively. Derive expressions for the critical Rayleigh numbers associated with the cylindrical and spherical geometries, Ra_c,crit and Ra_s,crit, respectively, below which k_eff is minimized. Evaluate Rac,crit and Ras,crit for air, water, and glycerin at a mean temperature of 300 K. For specified inner and outer surface temperatures and inner cylinder or sphere radii, comment on the heat transfer rate for outer cylinder or sphere radii corresponding to Ra_c,crit and Ra_s,crit, respectively.

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Keff k = = 0.386 Pr 0.861+ Pr, 1/4 1/4