Chapter 4, Problem 4.7P

Free convection heat transfer is sometimes quantified by writing Equation 4.20 as $q_{\text{conv}}=S{k}_{\text{eff}}\Delta T_{1-2},$ where $k_{\text{eff}}$ is an effective thermal conductivity. The ratio $k_{\text{eff}}/k$ is greater than unity because of fluid motion driven by buoyancy forces, as represented by the dashed streamlines.

An experiment for the configuration shown yields a heat transfer rate per unit length of $q_{\text{conv}}^{\text{'}}=110\text{W/m}$ for surface temperatures of $T_1=53°\text{C}$ and $T_2=15°\text{C}$ respectively. For inner and outer cylinders of diameters $d=20\text{mm}$ and $D=60\text{mm,}$ and an eccentricity factor of $z=10\text{mm,}$ determine the value of $k_{\text{eff}}.$ The actual thermal conductivity of the fluid is $k=0.255\text{W/m}\cdot \text{K}\text{.}$