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Readers of this book will be able to: utilize the fundamental principles of fluid mechanics and thermodynamics to analyze aircraft engines, understand the common gas turbine aircraft propulsion systems and be able to determine the applicability of each, perform system studies of aircraft engine systems for specified flight conditions, perform preliminary aerothermal design of turbomachinery components, and conceive, analyze, and optimize competing preliminary designs for conventional andunconventional missions.
1.1 Conservation Equations 1 1.1.1 Conservation of mass 2 1.1.2 Conservation of momentum 3 1.1.3 Conservation of energy 3 1.2 Flow Machines with No Heat Addition: The Propeller 3 1.2.1 Zero heat addition with Ve > V0 3 1.2.2 Zero heat addition with Ve < V0 4 1.2.3 Zero heat addition with P = constant > 0 4 1.2.4 Propulsive efficiency 4 1.3 Flow Machines with P = 0 and Q = Constant: The Turbojet, Ramjet, and Scramjet 5 1.3.1 Heat addition, Q > 0 5 1.3.2 Constant heat addition, Q = constant > 0 7 1.3.3 Overall efficiency 8 1.3.4 Fuel efficiency 8 1.4 Flow Machines with P = 0, Q = Constant, and A0 = 0: The Rocket 11 1.5 The Special Case of Combined Heat and Power: The Turbofan 12 1.5.1 Very small bypass ratio, β << 1, the turbojet 15 1.5.2 Very large bypass ratio, β >> 1, the turboprop 15 1.5.3 Finite β, the turbofan 16 1.6 Force Field for Air-Breathing Engines 16 1.7 Conditions for Maximum Thrust 22 1.8 Example: Jet and Rocket Engine Performance 25 1.8.1 Jet engine performance 25 1.8.2 Rocket engine performance 26 1.9 Nomenclature 27 1.9.1 Subscripts 28 1.10 Exercises 29 Reference 34
1.1 CONSERVATION EQUATIONS
A flow machine is one that ingests a stream of fluid, processes it internally in some fashion, and then ejects the processed fluid back into the ambient surroundings. An idealization of such a generalized flow machine is depicted schematically in Figure 1.1
In order to develop the basic features of operation of the idealized flow machine without introducing unnecessary algebraic complexity, we make the following assumptions:
Flow through the streamtube entering and leaving the machine is steady and quasi-one-dimensional.
The entrance and exit stations shown are chosen sufficiently far from the flow machine entrance and exit such that pressures at those stations are in equilibrium with their surroundings, that is, pe = p0.
There is no heat transfer across the boundaries of the streamtube or the flow machine into the ambient surroundings.
Frictional forces on the entering and leaving streamtube surfaces are negligible.
Mass injected into the fluid stream within the flow machine, if any, is negligible compared to mass flow entering the flow machine.
With these restrictions in mind we may assess the consequences of applying the basic conservation principles to the streamtube control volume. Some implications of the assumptions used are important to understand.
The assumption of steady flow implies that V0 is constant, that is, the idealized flow machine may be considered to be flying at speed V0 through a stationary atmosphere with the ambient environmental values of pressure, density, and temperature denoted in Figure 1.1 by p0, r0, and T0, respectively. Alternatively, we may consider our coordinate system to be fixed on the flow machine such that the atmosphere constitutes a free stream flow approaching at speed V0 with static conditions of pressure, density, and temperature denoted in Figure 1.1 by p0, r0, and T0, respectively. This (Galilean) transformation of coordinates is possible because the motion is steady.
Another implication arising from the assumption that the flow machine is moving through the atmosphere at constant speed is that there must be no unbalanced force on the machine. Because there will be resistance to motion due to drag D, there must be another force applied that can maintain the constant motion, which is net thrust Fn. The rate at which work must be done to maintain the motion is DV0, and because D = Fn, this required power may also be written as FV0.
1.1.1 Conservation of mass
The net change in the mass flow passing through the flow machine is zero, which may be written as
-ρ0A0V0 + ρeAeVe = 0. (1.1)
This is equivalent to stating that the mass flow m = ρAV = constant throughout the system:
1.1.2 Conservation of momentum
The net change in momentum of the fluid passing through the streamtube is equal to the force on the fluid, or
(-ρ0A0V0)V0 + (ρeAeVe)Ve = F.
Because the mass flow is constant, this equation can be abbreviated to the following:
m(Ve - V0) = F. (1.2)
The force acting on the fluid is denoted by F, and, for equilibrium, the force exerted on the control volume by the fluid is -F. In general, forces on the streamtube are negligible compared to those on the flow machine proper and are neglected. One important case where this is not necessarily true is that of the so-called additive drag of inlets in supersonic flight, where the force on the entering streamtube surface may not be negligible.
1.1.3 Conservation of energy
The net change in the total enthalpy of the flowing fluid is equal to the sum of the rate at which heat and work are added to the fluid, or
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.3)
The quantities h, Q, and P denote enthalpy, heat addition per unit mass, and power added, respectively. We may consider some extreme cases to illustrate several basic kinds of flow machines.
1.2 FLOW MACHINES WITH NO HEAT ADDITION: THE PROPELLER
Here we assume that Q = 0 so that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.4)
However, if no heat is added to the flowing fluid it is reasonable to expect that the enthalpy of the fluid is essentially unchanged in passing through the machine so that he [approximately equals] h0; which results in
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)
Thus the power supplied to the fluid is approximately equal to the product of the force on the fluid and the average of the velocities entering and leaving the machine.
1.2.1 Zero heat addition with Ve > V0
Here F > 0 and therefore P > 0, so that work is done on the fluid. This is the case of the propeller, the fan, and the compressor, where the device does work on the fluid and produces a force on the fluid in the same sense as the entering velocity. Note that this means that the force of the fluid on the machine is in the opposite sense, that is, a thrust is developed.
1.2.2 Zero heat addition with Ve < V0
Here F < 0 and P < 0, so that work is done by the fluid. This is the case of the turbine, where work is extracted from the fluid and the fluid experiences a retarding force, that is, the force on the fluid is in the opposite sense to that of the incoming velocity. The force on the machine is therefore in the same sense as the entering velocity and is therefore a drag force.
1.2.3 Zero heat addition with P = constant > 0
In this variation, we see that thrust force drops off with flight speed:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In general, the velocity ratio Ve/V0 is not much greater than unity, thus Vavg ~ V0. This is the case of a propeller propulsion system where increases in flight speed are limited by the power available. This effect is illustrated in Figure 1.2.
1.2.4 Propulsive efficiency
The total power expended is not necessarily converted completely into thrust power FV0, the rate at which force applied to the fluid does work. Remember that flight speed V0 is constant and therefore the drag on the vehicle is equal to the thrust produced, D = F. Then the rate at which work must be done to keep the vehicle at constant speed V0 is DV0 = FV0. However, it has been shown that the power expended is P = FVavg so that propulsive efficiency ηp may be defined as the ratio of useful thrust power to total power delivered to the airstream:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.6)
This equation shows that at a given flight speed the efficiency drops off with increasing exhaust velocity Ve, as shown in Figure 1.3.
1.3 FLOW MACHINES WITH P = 0 AND Q = CONSTANT: THE TURBOJET, RAMJET, AND SCRAMJET
Here we assume that no net power is exchanged with the fluid so that
(he - h0) + 1/2(V2e - V20) = Q. (1.7)
But we may write the kinetic energy term as
V2e - V20 = (Ve + V0)(Ve - V0) = (Ve + V0)F/m. (1.8)
Substituting this back into the first equation and solving for the thrust yields
F = m/Vavg [Q - (he - h0). (1.9)
1.3.1 Heat addition, Q > 0
If sufficient heat is added to the fluid, Equation (1.9) shows that F > 0 and thrust is produced on the flow machine. This is the basis of operation of the simple jet engine. The general internal configuration of the practical jet engine is dependent on the flight speed. For flight in the range of 0 < M0 < 3, the jet engine requires a compressor to increase the pressure of the incoming air before fuel is added and burned, particularly in the low end of the speed range. The compressor requires a shaft power source to drive it. Both these functions are best supplied by turbomachinery, or rotating machinery. The air compressor (turbo-compressor) is powered by a gas turbine, with both being attached to a common driveshaft. Such an arrangement is called a turbojet engine and is illustrated schematically in Figure 1.4.
Excerpted from Theory of Aerospace Propulsion by Pasquale M. Sforza Copyright © 2012 by Elsevier Inc. . Excerpted by permission of Butterworth-Heinemann. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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