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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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mach number
The Mach number is a dimensionless quantity representing the ratio of an object's speed to the speed of sound in the surrounding medium. Mach numbers are essential in aerodynamics because they help determine flow characteristics. When Mach number is less than 1, the flow is subsonic. At Mach 1, the flow is sonic, and above 1, it becomes supersonic. In the exercise, the aircraft starts cruising at Mach 2, indicating that it is flying twice the speed of sound. This speed classifies it as supersonic.
  • Mach 0.4 at the engine inlet suggests a significant speed reduction from its cruise speed, resulting in subsonic flow.
  • A Mach number of 1.2 at the shock indicates the flow is slightly supersonic.
Decelerating the air effectively ensures that the engine compressor operates most efficiently, preventing shock waves from adversely affecting its function.
Isentropic process
An isentropic process is a thermodynamic process where the entropy remains constant. This concept is crucial in aerodynamics and thermodynamics, as it describes idealized processes that are both adiabatic (no heat transfer) and reversible. In this scenario, the flow into the engine's compressor is isentropic, except across the shock. In practical terms, this means:
  • No energy loss due to heat or friction happens, ensuring the efficiency of energy transfer.
  • The formulas used鈥攕uch as calculating the total temperature and pressure changes鈥攔ely on this assumption to simplify the equations.
Keep in mind that real-word conditions rarely allow for perfect isentropic processes, but the assumptions make calculations more manageable and serve as close approximations.
Normal shock waves
Normal shock waves occur when a supersonic flow meets an abrupt and drastic pressure and temperature increase, causing the flow to become subsonic. These phenomena are characterized by:
  • An increase in static pressure, temperature, and density.
  • A drop in the Mach number from supersonic to subsonic.
In the exercise, a normal shock occurs with a Mach number of 1.2. This interaction significantly changes the properties of the flow, simplifying conditions for further airflow manipulation. Understanding normal shock waves is vital for designing efficient supersonic inlets and diffusers, and managing the airflow before it enters the engine compressor can maximize performance reliability.
Stagnation temperature and pressure
Stagnation temperature and pressure indicate flow properties where dynamic effects have been removed, representing the maximum values experienced by the flow when brought to rest isentropically. They are vital in analyzing high-speed aerodynamic systems including engines and compressors. For the problem mathematical calculations involve:
  • The total temperature ( T03 ) uses the relation based on Mach number to determine how much the air temperature rises from kinetic energy transformation.
  • The stagnation pressure ( P03 ) considers both isentropic changes and the effect of shock waves to find the conditions after the shock in the compressor.
These calculations provide critical insight into system performance and ensure the engine operates optimally under given conditions, converting the high velocities into usable energy effectively.

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Most popular questions from this chapter

Show that for Rayleigh flow, the maximum amount of heat that may be added to the gas is given by: \\[\frac{q_{\max }}{c_{p} T_{1}}=\frac{\left(\mathrm{Ma}_{1}^{2}-1\right)^{2}}{2(k+1) \mathrm{Ma}_{1}^{2}}\\]

Air is supplied to a convergent-divergent nozzle from a reservoir where the pressure is \(100 \mathrm{kPa}\). The air is then discharged through a short pipe into another reservoir where the pressure can be varied. The cross- sectional area of the pipe is twice the area of the throat of the nozzle. Friction and heat transfer may be neglected throughout the flow. If the discharge pipe anstant cross-sectional area, determine the range of static pressure in the pipe for which a normal shock will stand in the divergent section of the nozzle. If the discharge pipe tapers so that its cross- sectional area is reduced by \(25 \%\), show that a normal shock cannot be drawn to the end of the divergent section of the nozzle. Find the maximum strength of shock (as expressed by the upstream Mach number) that can be formed.

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.

An airplane is flying at a flight (or local) Mach number of 0.70 at \(10,000 \mathrm{m}\) in the Standard Atmosphere. Find the ground speed (a) if the air is not moving relative to the ground and (b) if the air is moving at \(30 \mathrm{km} / \mathrm{hr}\) in the opposite direction from the airplane.

Sound waves are very small-amplitude pressure pulses that travel at the "speed of sound." Do very large-amplitude waves such as a blast wave caused by an explosion (see Video \(\vee 11.8\) ) travel less than, equal to, or greater than the speed of sound? Explain.
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