Simplified relations for natural convection in air are of the form h = (T/L) where ,

Simplified relations for natural convection in air are of the form h = α(ΔT/L)^β

where α, β are constants; L is a significant length, in ft; ΔT is T_s, – T_∞, in ^oF; and h is the convective heat-transfer coefficient, Btulh ft² °F. Determine the values for α and β for the plane vertical wall, using the equation from Problem 19.14.

Data From Problem 19.14

An engineered tissue system consists of a flat plate of cell mass immobilized on a scaffold measuring 5 cm in length, and is 0.5 cm thick. The bottom face of the scaffold is exposed to water and organic nutrients. The top face is exposed to flowing O₂ gas to provide O₂ for aerobic respiration. At present the specific oxygen consumption of the tissue mass is 0.5 mmol O₂/cm³cells-hr, and from respiration energetics, the energy released by respiration is 468 J/rnmol O₂ consumed. We are interested in using the flowing O₂ gas at 1 atm to control the temperature at the surface of the tissue scaffold. The properties of O₂ gas at 300 K are ρ = 1.3 kg/m³, C_p = 920 J/kg · K, µ = 2.06 × 10^-5 kg/m sec, and k = 0.027 W/m K. We are interested in determining the O₂ flow rate necessary to keep the surface temperature within 10°C of the flowing gas temper ature (i.e., surface temperature below 310 K or 37°C).