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 energetic, 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, 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).

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

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

c. What is the Nusselt number, Nu?

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?

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