Question: The gas turbine blade of Prob. I is to be cooled to a uniform surface temperature of 650°C by transpiration of air through a porous surface. If the cooling air is available at 200°C, calculate the necessary cooling-air mass-transfer rate per unit of surface area as a function of position along the blade surface. Discuss the probable surface temperature distribution if it is only mechanically feasible to provide a transpiration rate that is uniform along the surface.

Problem I: Consider again Prob. II, but let the skin be of 3 mm thick stainless steel, insulated on the inner side. Treating the skin as a single element of capacitance, calculate the skin temperature as a function of altitude. [The specific heat of stainless steel is 0.46 kJ/(kg . K)].

Problem II: A particular rocket ascends vertically with a velocity that increases approximately linearly with altitude, reaching 3000mls at 60,000m. Consider a point on the cylindrical shell of the rocket 5 m from the nose. Calculate and plot, as functions of altitude, the adiabatic wall temperature, the local convection conductance, and the internal heat flux necessary to prevent the skin temperature from exceeding 50°C (see Prob. III for remarks about the state of the air just outside the boundary layer in such a situation).

Problem III: In Prob. IV it is desired to cool a particular rectangular section of the aircraft body to 65°C. The section is to be 60 cm wide by 90 cm long (in the flow direction) and is located 3 in from the nose. Estimate the total heat-transfer rate necessary to maintain the desired surface temperature. As an approximation, the boundary layer may be treated as if the free-stream velocity were constant along a flat surface for the preceding 3 m. It may also be assumed that the preceding 3 m of surface is adiabatic. To. obtain the state of the air just outside the boundary layer, it is customary, to assume that the air accelerates from behind the normal shock wave at the nose, is entropically to the free-stream static pressure. In this case the local Mach number then becomes 2.27, and the ratio of local absolute static temperature to free-stream stagnation temperature is 0.49. The local static pressure is the same as the free-stream, that is, the pressure at 17,500m altitude.

Problem IV: Consider an aircraft flying at Mach 3 at an altitude of 17,500m. Suppose the aircraft has a hemispherical nose with a radius of 30 cm. If it is desired to maintain the nose at 80°C, what heat flux must be removed at the stagnation point by internal cooling? As a fair approximation, assume that the air passes through a normal detached shock wave and then decelerates is entropically to zero at the stagnation point; then the flow near the stagnation point is approximated by low-velocity flow about a sphere.