Simplified relations for natural convection in air are of the formwhere α, β are constants...

Simplified relations for natural convection in air are of the form

where α, β are constants; L is a significant length, in ft; DT is Ts T∞, in °F; and h is the convective heat-transfer coefficient, Btu/h ft2 °F. Determine the values for a and b for the plane vertical wall, using the equation from Problem 19.14.

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/cm3 cells-hr, and from respiration energetics, the energy released by respiration is 468 J/mmol 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, CP = 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 temperature (i.e., surface temperature below 310 K or 37°C).

a. What is the Prandtl number (Pr) for the flowing fluid?

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b. Based on the process energy balance and heat-transfer, what is the required heat transfer coefficient, h?

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c. What is the Nusselt number, Nu?

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d. The mean heat-transfer coefficient, h, is an integral averaged value obtained by integrating the local values of h(x) from x = 0 to x = L. If the O2 flow is assumed to be laminar (for flat plate, Re<2 × 105), what is the needed convective heat-transfer correlation?

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e. What are the required values for Reynolds number (Re) and fluid velocity, v∞?