Question

a. The exit of a heating chamber is connected to diverging nozzle as shown in Figure 5. The inlet pressure, velocity, and temperature are known to be P, V1, and,

T1, respectively.The heating chamber area is constant and known to be A,. It is known that that Mach number at the exit of the diverging section Ma3 is larger than one. The amount of heat q added in the heating section is also known and the known fluid (an ideal gas) has properties k, R,and c, (symbols have their usual meaning). Assuming the flow to be isentropic in the diverging section, and Rayleigh in the heating section, answer the following with brief justifications: \text { Known quantities: } P_{1}, V_{1}, T_{1}, A_{1}, M a_{3}, q, k, R, \text { and } c_{p} i. In terms of some or all of the known quantities, obtain an expression for the Mach number at the inlet (section 1). ii. What can you say about the Mach number at the in let (section 1), i.e. is it subsonic(<1), sonic (=1), or supersonic (>1)? iii.What can you say about the Mach number at the exit of heating section (section 2),i.e. is it subsonic (<1), sonic (=1), or supersonic (>1)? iv. In terms of some or all of the known quantities, what are the stagnation temperatures and pressures at the inlet (section 1)? Detail a strategy to find the Mach number, stagnation temperature, stagnation pressure, temperature, and pressure at the exit of the heating section (section 2) in terms of all or some known quantities? [Note: statements of the form x is a solution of equation f(x) = 0, where f(x) is shown are acceptable).v. From here onwards assume that T02, P02, T2, P2, and Ma2 have been computed in part(v) and hence known vi. In terms of some or all of the known quantities, what are the stagnation temperatures and pressures at the exit of the diverging section (section 3)? vii. In terms of some or all of the known quantities, what are the temperatures and pressures at the exit of the diverging section (section 3)? viii.In terms of some or all of the known quantities, obtain an expression for the mass flow rate through the assembly. ix. If the fluid could not be approximated as an ideal gas, but as a mixture of gases with the following hypothetical relationship between the pressure P, temperature T,density p, and constants R,a,and b obtain an expression for the speed of sound c in the gas. [Hint: Look at first principles definition of speed of sound] P=\rho e^{-b R T}+\rho^{2} a b. In the context of the converging-diverging nozzles, in your own words, briefly explain the differences between incompressible flow and compressible flow under various flow conditions.

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