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Chapter 11: Problem 51

A jet engine is to be designed for an altitude of \(12,000 \mathrm{m}\) where the atmospheric pressure is \(19.3 \mathrm{kPa}\). The jet nozzle has a supersonic exit Mach number and is perfectly expanded. The stagnation pressure and temperature of the gas are 100 kpa and \(600^{\circ} \mathrm{C}\) The flow rate of gas is \(45 \mathrm{kg} / \mathrm{s}\). Calculate the throat area, exit area, and exit velocity. Use \(k=1.4\) and \(R=260 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) for the gas.

Short Answer

Expert verified
Using the given data, the throat area, exit area, and exit velocity of the jet engine at an altitude of 12,000 m can be calculated by applying the relevant equations of isentropic flow and the ideal gas law. The specific values, however, will depend on precise numerical solutions of these equations.

Step by step solution

01

Calculation of Throat Area

The mass flow rate, \( \dot{m} \), through the throat can be calculated using the equation \( \dot{m} = p_{0t}*A*\sqrt{ \frac{k}{R*T_{0t}} } * \left[ \frac{(2/(k+1))^{((k+1)/(k-1))}}{\sqrt{k-1}} \right] \). Using the given values of \( p_{0t} = 100 kPa, \dot{m} = 45 kg/s, k = 1.4, R = 260 J/kg.K, T_{0t} = 600°C = 873.15 K \), we can solve this equation for the throat area A.
02

Calculation of Exit Area

The exit area can be calculated using the isentropic flow relation for the exit Mach number (M) combined with the ideal gas law, i.e. \( p_e / p_{0t} = (1 + (k-1)/2 * M^2)^{(-k/(k-1))} \). The perfectly expanded condition at the exit implies that \( P_e = P_a \), i.e., the exit pressure equals the atmospheric pressure (19.3 kPa). We can solve these equations simultaneously for M, and then substitute M back into the isentropic relation that links A, M and A_t (throat area), i.e. \( A / A_t = (1/M) * ((2 + (k-1) * M^2) / (k+1))^{((k+1)/(2*(k-1)))} \), to solve for the exit area.
03

Calculation of Exit Velocity

The exit velocity can be calculated using the formula for velocity in terms of the Mach number and temperature, i.e., \( V = M * \sqrt{k * R * T_{0t}} \). Once the exit Mach number has been determined (as in the previous step), this equation can then be solved for V.

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Most popular questions from this chapter

Air flows adiabatically between two sections in a constant area pipe. At upstream section \((1), p_{0,1}=100\) psia, \(T_{0,1}=600^{\circ} \mathrm{R}\) and \(\mathrm{Ma}_{1}=0.5 .\) At downstream section \((2),\) the flow is choked. Estimate the magnitude of the force per anit cross-sectional area exerted by the inside wall of the pipe on the fluid between sections (1) and (2).

Air enters a 4 -cm-square galvanized steel duct with \(p_{0}=\) \(150 \mathrm{kPa}, T_{0}=400 \mathrm{K},\) and \(V_{1}=120 \mathrm{m} / \mathrm{s},\) (Note: \(\mu=2.2 \times 10^{-5}\) \(\left.\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)\) (a) Compute the maximum possible duct length for these conditions. (b) If the actual duct length is 0.75 times the maximum value, calculate the mass flow rate, the exit pressure, and the stagnation pressure. (c) If the actual duct length is 1.3 times the maximum value for the stated conditions, compute the new mass flow rate and inlet velocity. Assume a low back pressure and use the same value of \(f\) as used in the maximum possible duct length case.

An aircraft cruises at a Mach number of 2.0 at an altitude of \(15 \mathrm{km} .\) Inlet air is decelerated to a Mach number of 0.4 at the engine compressor inlet. A normal shock occurs in the inlet diffuser upstream of the compressor inlet at a section where the Mach number is \(1.2 .\) For isentropic diffusion, except across the shock, and for standard atmosphere, determine the stagnation temperature and pressure of the air entering the engine compressor.

Prove that, in Rayleigh flow, the Mach number at the point of maximum temperature is \(1 / \sqrt{k}\).

A normal shock occurs in a stream of oxygen. The oxygen flows at \(\mathrm{Ma}=1.8\) and the upstream pressure and temperature are 15 psia and \(85^{\circ} \mathrm{F}\) (a) Calculate the following on the downstream side of the shock: static pressure, stagnation pressure, static temperature, stagnation temperature, static density, and velocity. (b) If the Mach number is doubled to \(3.6,\) what will be the resulting values of the parameters listed in part (b)?
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