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Chapter 11: Problem 31

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.

Short Answer

Expert verified
The stagnation pressure and temperature at the compressor inlet are derived using the step-by-step method above and the applied formulae.

Step by step solution

01

Understand the given conditions

Air travels at Mach number 2.0 (M1) at 15km altitude. The Mach number decreases to 0.4 (M2) at the engine compressor inlet point, with a normal shock at Mach number 1.2 (Ms). The process is isentropic (a thermodynamic process that occurs at constant entropy), except at the shock.
02

Calculate the Total Temperature

To find the total temperature (T02) at the engine, we apply the isentropic relation for total temperature at initial cruise condition (M1 = 2). Using the standard atmospheric temperature (T1) at 15km and the isentropic relation: \(T02=T1*(1+(\gamma-1)/2)*M1^2\) where \(\gamma\) refers to the heat capacity ratio which is 1.4 for air.
03

Calculate the Stagnation Pressure After the Shock

The stagnation pressure (P02) after the shock can be found by using the relation for normal shock and isentropic process, \(P02=P1*(1+(\gamma-1)/2)*M1^2/(1+(\gamma+1)/2*Ms^2)^((\gamma)/(\gamma-1))\) where M1 is the Mach number before the shock (2) and Ms is the Mach number at the shock (1.2). Standard atmospheric pressure (P1) at 15km is to be used.
04

Calculate the Stagnation Pressure and Temperature at the Compressor

To find the stagnation pressure (P03) and temperature (T03) at the compressor, we apply the isentropic relations for M2 = 0.4. Hence, \(P03=P02/(1+(\gamma-1)/2)*M2^2)^(gamma/(gamma-1))\) and \(T03=T02/(1+(\gamma-1)/2)*M2^2)\)

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Most popular questions from this chapter

An ideal gas flows with velocity \(V\), pressure \(p\), temperature \(T,\) and density \(\rho .\) Determine a set of equations for stagnation properties, including entropy, if the stagnation process is defined to be isothermal \((T=\text { constant ) rather than isentropic }(s=\text { constant })\).

Air enters a frictionless, constant area duct with \(\mathrm{Ma}=2.5\) \(T_{0}=20^{\circ} \mathrm{C},\) and \(p_{0}=101 \mathrm{kPa}(\mathrm{abs}) .\) The gas is decelerated by heating until a normal shock occurs where the local Mach number is \(1.3 .\) Downstream of the shock, the subsonic fow is accelerated with heating until it exits with a Mach number of \(0.9 .\) Determine the static temperature and pressure, the stagnation temperature and pressure, and the fluid velocity at the duct entrance, just upstream and downstream of the normal shock, and at the duct exit. Sketch the temperature- entropy diagram for this flow.

An ideal gas is to flow isentropically from a large tank where the air is maintained at a temperature and pressure of \(59^{\circ} \mathrm{F}\) and 80 psia to standard atmospheric discharge conditions. Describe in general terms the kind of duct involved and determine the duct exit Mach number and velocity in \(\mathrm{ft} / \mathrm{s}\) if the gas is air.

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)?

The stagnation pressure and temperature of air flowing past a probe are \(120 \mathrm{kPa}(\mathrm{abs})\) and \(100^{\circ} \mathrm{C},\) respectively. The air pressure is \(80 \mathrm{kPa}(\text { abs }) .\) Determine the airspeed and the Mach number considering the flow to be (a) incompressible, (b) compressible.
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