b) Show how energy transfer by turbulent fluctuations normal to the flow can lead to the following expression for convective heat transfer in the fluid.
𝑞 = −𝑐P𝜌(𝑎 + 𝜀𝐻) dT/dy
Where q is the heat flux, cp is the specific heat capacity, is the density, is the ̅̅̅̅̅̅̅̅
molecular diffusivity, H is the eddy diffusivity of heat (= (𝑇′𝑣′)), T is the temperature and y is the coordinate normal to the flow.
c) A Laval (converging-diverging) nozzle, of circular cross section, fed from a large high pressure reservoir, is used to accelerate a flow such that the Mach number in a test section downstream of the nozzle is 3.2.
i) If the diameter of the throat of the nozzle is 25 mm, calculate the cross-sectional area of the test section.
ii) If the air pressure and density in the test section are 100 kPa and 1.12 kg/m3, respectively, calculate the actual velocity in the test section.
Assume the air behaves as an ideal gas and take, Specific isobaric heat capacity of air = 1.005 kJ/kg K Specific isochoric heat capacity of air = 0.718 kJ/kg K Specific ideal gas constant of air = 0.287 kJ/kg K