Chapter 9, Problem 9.26P

Consider an experiment to investigate the transition to turbulent flow in a tree convection boundary layer that develops along a vertical plate suspended in a large room. The plate is constructed of a thin heater that is sandwiched between two aluminum plates and may be assumed to be isothermal. The heated plate is 1 m high and 2 m wide. The quiescent air and the surroundings are both at 25°C.
(a) The exposed surfaces of the aluminum plate are painted with a very thin coating of high emissivity $\left(\epsilon =0.95\right)$ paint. Determine the electrical power that must be supplied to the heater to sustain the plate at a temperature of $T_s=35°C$ . How much of the plate is exposed to turbulent conditions in the free convection boundary layer?

(b) The experimentalist speculates that the roughness of the paint is affecting the transition to turbulence in the boundary layer and decides to remove the paint and polish the aluminum surface $\left(\epsilon =0.05\right)$ . If the same power is supplied to the plate as in part (a), what is the steady-state plate temperature? How much of the plate is exposed turbulent conditions in the free convection boundary layer?