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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 ^{o}F; and h is the convective heat-transfer coefficient, Btulh ft^{2} °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 O_{2} gas to provide O_{2 }for aerobic respiration. At present the specific oxygen consumption of the tissue mass is 0.5 mmol O_{2}/cm^{3}cells-hr, and from respiration energetics, the energy released by respiration is 468 J/rnmol O_{2} consumed. We are interested in using the flowing O_{2} gas at 1 atm to control the temperature at the surface of the tissue scaffold. The properties of O_{2} gas at 300 K are ρ = 1.3 kg/m^{3}, 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 O_{2} 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).

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