Question: In Prob. I the turbine blade is 15 cm long, and for present purposes it may be assumed that blade dimensions and operating conditions are the same along the entire blade length. It appears feasible to allocate up to 0.03 kg/s of air at 200°C to cool the blade. Assuming a hollow blade of mild-steel construction, a minimum wall thickness of 0.75 mm, and any kind of internal inserts as desired (a solid core leaving a narrow passage just inside the outer wall could be used, for example), make a study of the feasibility of internally cooling this blade. Assume that the cooling air can be introduced at the blade root and discharged through the tip. Treat the blade as a simple heat exchanger.
Problem I: Consider the gas turbine blade on which Prob. II is based. It is desired to maintain the blade surface uniformly at 650°C by internal cooling. Calculate the necessary heat flux around the periphery of the blade.
Problem II: Consider again Prob. III, 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 III: 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. IV for remarks about the state of the air just outside the boundary layer in such a situation).
Problem. IV: In Prob. V 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 V: 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.