Explanation The flow of nitrogen through the nozzle is isentropic, one dimensional and steady. Write the expression for the thermodynamic relation for pressure. ρ 0 = P 0 R T 0 (I) Here, density of nitrogen at the outlet of nozzle is ρ 0 , pressure of nitrogen at the outlet of nozzle is P 0 , gas constant is R , and temperature of nitrogen at the outlet of nozzle is T 0 . The properties of the fluid are at its critical for a Mach number of 1, Refer Table A-32, “One-dimensional isentropic compressible-flow functions for an ideal gas with k = 1.4 ”, write the following expressions of critical pressure, temperature and density for a Mach number of 1. T ∗ = 0.8333 T 0 (II) P ∗ = 0.5283 P 0 (III) ρ ∗ = 0.6339 ρ 0 (IV) Here, the critical temperature is T ∗ , critical pressure is P ∗ , and critical density is ρ * . Write the expression for the critical velocity of sound in the fluid. c ∗ = k R T ∗ (V) Here, critical velocity of sound in the fluid is c ∗ and ratio of specific heat is k . Conclusion: From Table A-1, “Molar mass, gas constant, and critical2point properties”, the value of R for nitrogen at the given temperature is 0.2968 kPa ⋅ m 3 / kg . Since, the stagnation pressure at inlet is the same as the outlet, substitute 700 kPa for P 0 , 0.2968 kPa ⋅ m 3 / kg for R , and 400 K for T 0 in Equation (I). ρ 0 = 700 kPa 0.2968 kPa ⋅ m 3 / kg × 400 K = 5.896 kg / m 3 Substitute 400 K for T 0 in Equation (II)

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Chapter 17.7, Problem 53P

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