AERO PROPULSION EAS 4300
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Theory of Aerospace Propulsion 10 IDEALIZED FLOW MACHINES 11 Conservation Equations A ow machine is one which ingests a stream of uid processes it internally in some fashion and then ejects the processed uid back into the ambient surroundings An idealization of such a generalized ow machine is schematically depicted in Fig 11 P0 32 p0 pg V0 I Ve A0 A Figure 1 1 Schematic diagram of idealized ow machine and associated streamtube control volume In order to develop the basic features of operation of the idealized ow machine without introducing unnecessary algebraic complexity we make the following assumptions 0 The ow through the streamtube entering and leaving the machine is steady and quasionedimensional o The entrance and exit stations shown are chosen sufficiently far from the ow machine entrance and exit such that the pressures at those stations are in equilibrium with their surroundings that is pfpo 0 There is no heat transfer across the boundaries of the streamtube or the ow machine into the ambient surroundings Frictional forces on the entering and leaving streamtube surfaces are negligible Mass injected into the uid stream within the ow machine if any is negligible compared to the mass ow entering the ow machine With these restrictions in mind we may assess the consequences of applying the basic conservation principles to the streamtube control volume There are some implications of the assumptions used which are important to understand The assumption of steady ow implies that V0 is constant that is the idealized ow machine may be considered to be ying at the speed V0 through a stationary atmosphere with the ambient environmental values of pressure density and temperature denoted in Fig 11 by p0 p0 and To respectively Alternatively we may consider our coordinate system to be fixed on the ow machine such that the atmosphere constitutes a free stream Theory of Aerospace Propulsion ow approaching at speed V0 with static conditions of pressure density and temperature denoted in Fig ll by p0 p0 and To respectively This Galilean transformation of coordinates is possible because the motion is steady Another implication arising from the assumption that the ow machine is moving through the atmosphere at constant speed is that there must be no unbalanced force on the machine Since there will be resistance to the motion due to drag D there must be another force applied which can maintain the constant motion and that is the net thrust F n The rate at which work must be done to maintain the motion is DVO and since DFn this required power may also be written as F Vg 111 Conservation of mass The net change in the mass ow passing through the ow machine is zero which may be written as pvoVo p2AV 0 11 This is equivalent to stating that the mass ow m pAVconstant throughout the system 112 Conservation of momentum The net change in momentum of the uid passing through the streamtube is equal to the force on the uid or poAJ V pAV V F Because the mass ow is constant this equation can be abbreviated to the following mV V0F 12 The force acting on the uid is denoted by F and for equilibrium the force exerted on the control volume by the uid is 7F In general the forces on the streamtube are negligible compared to those on the ow machine proper and are neglected One important case where this is not necessarily true is that of the socalled additive drag of inlets in supersonic ight where the force on the entering streamtube surface may not be negligible 113 Conservation of energy The net change in the total enthalpy of the owing uid is equal to the sum of the rate at which heat and work are added to the uid or mh h0VZ VfmAQP 13 The quantities h AQ 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 ow machines Theory of Aerospace Propulsion 12 Flow Machines with No Heat Addition The Propeller Here we assume that AQ0 so that Pmlth igtltnz Vzgt However if no heat is added to the owing uid it is reasonable to expect that the enthalpy of the uid is essentially unchanged in passing through the machine so that h n he which results in P mV Vf mV VKFV 15 Thus the power supplied to the uid is approximately equal to the product of the force on the uid and the average of the velocities entering and leaving the machine 121 Zero heat addition with VB gtV0 Here Fgt0 and therefore Pgt0 so that work is done on the uid This is the case of the propeller the fan and the compressor where the device does work on the uid and produces a force on the uid in the same sense as the entering velocity Note that this means that the force of the uid on the machine is in the opposite sense that is a thrust is developed 122 Zero heat addition with VeltV0 Here Flt0 and Plt0 so that work is done by the uid This is the case of the turbine where work is extracted from the uid and the uid experiences a retarding force that is the force on the uid 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 123 Zero heat addition with P constant gt0 In this variation we see that the thrust force drops off with ight speed P l V V01 15 2 V 0 In general the velocity ratio VeVg is not much greater than unity thus Vavg N V0 This is the case of a propeller propulsion system where increases in ight speed are limited by the power available This effect is illustrated in Fig 12 124 Propulsive ef ciency The total power expended is not necessarily converted completely into thrust power F V0 the rate at which the force applied to the uid does work Remember that the ight speed V0 is constant and therefore the drag on the vehicle is equal to the thrust produced DF Then the rate at which work must be done to keep the vehicle at constant speed V0 is DV0FV0 On the other hand it has been Theory of Aerospace Propulsion shown that the power expended is PFVaVg so that the propulsive ef ciency 77p may be de ned as the ratio of useful thrust power to total power delivered to the airstream FV FV V o 2 0 0 16 P FV 77p 1 5m w This equation shows that at a given ight speed the ef ciency drops off with increasing exhaust velocity Ve as shown in Fig 13 N r9010 0 01 FIP lbs thrusthorsepower O 0 200 400 600 800 1000 1200 Vavg fps Figure 1 2 The speci c thrust produced by a propeller as a function of the average speed region B 08 2 2 06 t O a 04 2 1 02 n E g 0 0 1 2 3 4 5 VeVo Figure 1 3 The ef ciency of a propeller as a function of the ratio of exit speed to ight speed Theory of Aerospace Propulsion 13 Flow Machines with P0 and AQC0nstant The Turbojet Ramjet and Scramjet Here we assume that no net power is exchanged with the uid so that h h V VfAQ 17 But we may write the kinetic energy term as VJ VfVVV VoVV5 18 m Substituting this back into the first equation and solving for the thrust yields m FV AQ hZ h0 1 9 avg 131 Heat addition AQgt0 If sufficient heat is added to the uid Eq 19 shows that Fgt0 and thrust is produced on the ow machine This is the basis of operation of the simple jet engine The general internal con guration of the practical jet engine is dependent upon the ight speed For ight in the range of 0ltM0lt3 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 turbocompressor is powered by a gas turbine both being attached to a common driveshaft Such an arrangement is called a turbojet engine and is schematically illustrated in Figure 14 I39l turbine inlet compressor nozzle IJ Fuel injector N 6 7 Figure 1 4 Schematic diagram of a typical turbojet engine showing the required turbomachinery components and usual station numbering scheme The combustor burns the injected fuel supplying heat to the ow passing through the turbojet Theory of Aerospace Propulsion For supersonic ight in the Mach number range of 2ltM0lt5 the ram pressure produced by the inlet in slowing down the incoming air to subsonic speeds obviates the need for the compressor and therefore its driving turbine As a consequence a practical jet engine for this ight regime is called a ramjet and its con guration is very simple as shown in Fig 15 where the turbomachinery no longer needed is indicated by dashed lines This simplicity comes at a price however because the ramjet cannot operate effectively at lower speeds In particular it generates no thrust at zero ight speed so that it cannot provide thrust for takeoff The ramjet must be accelerated to supersonic speeds by some other propulsive means before it can produce enough thrust to sustain ight of the vehicle it powers It is therefore often used to power missiles launched from an aircraft or a rocket booster combustor I I I I 4 I I I l 2 nozzle inlet 11 l 2 3 4 5 6 7 Fuel injector Figure 1 5 Schematic diagram of a typical ramjet engine The dashed lines show the turbomachinery components no longer needed The combustor burns the injected fuel supplying heat to the ow passing through the ramj et For hypersonic ight speeds M0gt5 the temperature increase accompanying the ram compression to subsonic speeds is so high that little or no additional heat can be added in the combustor by burning fuel The only alternative is to use the inlet to slow the ow down from the ight speed to some lower supersonic Mach number thereby not increasing the temperature too much However then fuel must be added to a supersonic stream mixed and combusted Achieving this supersonic combustion is very difficult because the high speed in the combustor gives very little time for the mixing and combustion to take place Such a supersonic combustion ramjet is popularly known as a scramj et The general con guration is like that of the ramjet shown in Fig 15 132 Constant heat addition AQc0nstantgt0 In this case we see that the thrust F2poA0AQ h Theory of Aerospace Propulsion Therefore the thrust is essentially constant with ight speed V0 as shown in Fig 16 This is the basic advantage of the jet engine 7 its thrust is independent of the ight speed and therefore it is not speedlimited as is the propeller 39 035 025 02 ramjet region 015 turboj et region 005 F2p 0A oAQ h8 h0 0 500 1000 1500 2000 2500 3000 V0 fps Figure 1 6 The thrust of a turboj et engine as a function of ight speed 133 Overall ef ciency The overall efficiency in the case of thrust produced by means of heat addition may be defined as the ratio of the required thrust power to the rate of heat addition 770 K 1h2 h0 mAQ V AQ avg But we have already shown in Eq 16 that the propulsive efficiency is V0 77 p Vavg We may then define the thermal efficiency as h h 1 2 0 l l 1 77m A Q Then the overall efficiency is the product of the propulsive and thermal efficiencies 770 77p77m 112 The thermal efficiency accounts for the fact that not all the heat added is converted to useable heat power since some is rejected as increased internal energy in the exhaust gases Thus the thermal efficiency illustrates the extent to which the ow machine puts Theory of Aerospace Propulsion the heat added to good use in increasing the kinetic energy of the exhaust stream ie increasing Ve while the propulsive efficiency illustrates the extent to which that increased kinetic energy provides thrust power to maintain ight at the speed V0 In a jet engine these two efficiencies generally drive in different directions with the higher exhaust velocities sustainable at high thermal efficiency leading to lower propulsive efficiencies at a given ight speed 134 Fuel ef ciency A common measure of fuel efficiency is the specific fuel consumption 0 which is defined as the ratio of the fuel weight ow rate to the thrust produced typically this is measured in pounds of fuel per hour per pound of thrust or simply hr39l The fuel weight ow rate may be related to the heat addition by the following equation mAQ nbwf HV 113 This equation is based on the assumption that the rate of heat addition arises from the energy released in the chemical conversion of the fuel added as characterized by H V the heating value of the fuel The burner efficiency 77 represents the ratio of the heat release actually transferred to the owing combustion gases to the total heat release possible Recall that the set of assumptions made at the outset included the requirement that the amount of uid added to the general stream entering the ow machine is negligible It will be shown subsequently when dealing with combustion chambers that the fuel weight ow rate is indeed much smaller than the air weight ow rate so no loss in generality is incurred here Using Eqs 110 and 113 the specific fuel consumption is Wf Vavg c 114 F UthbHV For hydrocarbon fuel HV18900Btulb and velocity in fts so that the specific fuel consumption in pounds of fuel per hour per pound of thrust this becomes V c 245 gtlt10394 avg 771777m Burner efficiency is generally quite high as will be seen in Chapter 3 and for the present purposes it may be taken as 77b095 However the thermal efficiency bears a bit more consideration Expanding Eq 111 leads to the following representation of thermal efficiency Theory of Aerospace Propulsion T2 W 2 l Under typical ight conditions the fuel to air ratio WfWo N 002 the specific heat ratio cpyocpe N 087 the ratio of atmospheric temperature to exhaust stagnation temperature ToTLe N025 m N 095 and HV18900Btulb so that T2 022 T22 ml c T 39 77m p2 22 359 77m 1 CWTM The ratio of static to stagnation temperature in the exhaust where we may take yN 133 is given by T 1 l 1 2 272 1L22 M 1 6 The exhaust Mach number varies from Me N l for subsonic jet aircraft to perhaps 2 for supersonic jet aircraft so that the range of the temperature ratio is 086 gt TeTte gt 06 Then 77m 1 KCWTM Here the value for K lies in the range 00018gtKgt00011 while the exhaust stagnation temperature lies in the range 1250RltTtveltl750R and therefore the thermal efficiency is in the range 45 lt 7711 lt 49 Then taking an intermediate value say hm47 the specific fuel consumption becomes cj569x10394Vmg The specific fuel consumption varies linearly with average velocity as shown in Fig 17 The notional regime for ramjets is also shown in Fig 17 The only difference between turbojets and ramjets is that the latter needs no compressorturbine machinery the compression process being entirely due to ram pressure Theory of Aerospace Propulsion Cj lbshrlb 0 500 1000 1500 2000 2500 3000 3500 4000 Vavg fts Figure 1 7 Speci c fuel consumption as a function of average velocity for turbojets As shown in Eq 114 the speci c fuel consumption of a jet engine increases with reductions in propulsive ef ciency that is with increases in Vavg and therefore with the exhaust velocity Ve High heating value fuels will reduce cj while reduced thermal and burner ef ciencies will increase it Heating values for some typical fuels in the liquid phase are shown in Table 11 along with the heating value per unit volume HV Note that hydrogen has about three times the heat release per unit mass of Jet A while only about onequarter the heat release per unit volume Since the volume of an aircraft in uences its drag the quantity HV is an important parameter to consider as well as HV itself and hydrogen is not generally a good alternative to a hydrocarbon fuel Table 1 1 Heating values for typical liquid fuels 14 Flow Machines with P0 AQC0nstant and A00 The Rocket If no air is taken on board the ow machine that is A00 there is no momentum penalty realized and the momentum equation is simply F mV 1 15 A schematic diagram of the ow eld in this special case is shown in Fig 18 Application of the energy equation leads to the following result 10 Theory of Aerospace Propulsion Fi mAQ h h2pAAQ hZ h 116 Here all the gas ejected from the ow machine was onboard the quantity h represents the enthalpy of that internal propellant supply This is the case of the simple rocket and like the pure jet the thrust produced is independent of the ight speed The propulsive ef ciency is now given by V 77p 0 270 1 17 Figure 1 8 Schematic diagram of the idealized ow eld of a rocket The propulsive ef ciency of a rocket is even smaller than that of a jet which in turn is smaller than that of a propeller This lower propulsive ef ciency means that the overall ef ciency of the rocket is also smaller than the jet and propeller The metric for fuel ef ciency used for rockets is the speci c impulse which is de ned as the ratio of the thrust produced to the weight ow rate of propellant or V V 15pim 2 2 118 mg mg g The speci c impulse is typically measured in pounds of thrust per pound or propellant consumed per second or seconds This is essentially the inverse of the speci c fuel consumption but it is based on the weight ow rate of onboard propellants The speci c impulse for jets is based solely on the onboard fuel ow rate and is given by i 3600 wf sfc 119 ISpZl Therefore the speci c impulse of airbreathing jets is greater than that of rockets A comparison of speci c impulse for jets and rockets is shown in Fig 19 Note that 11 Theory of Aerospace Propulsion according to Eq 118 the specific impulse for rockets doesn t depend on ight speed and that therefore we may consider that at some ight speed the specific impulse of the rocket will be equal to or superior to that of a jet as suggested by Fig 19 14000 12000 Jet liquid A 10000 hydrogen fuel 8000 W Jet s 6000 7HC 2 4000 2000 W i Rocket K Hb 0 i 0 2000 4000 6000 8000 10000 12000 V0 fps Figure 1 9 Specific impulse variation for turboj et and rocket engines as a function of average speed Note that rocket 1W is independent of V0 15 The Special Case of Combined Heat and Power The Turbofan If we consider that the ow through the idealized ow machine is divided into a central hot stream denoted by the subscript h and an outer cold stream denoted by the subscript c the conservation of mass equation may be written as follows m5 m0mhmcmhl Jmhl 120 M The quantity measures the ratio of the cold mass ow to the hot mass ow and is called the bypass ratio This idealized ow field is schematically illustrated in Fig 110 A0 39 quotr Vac Tao 1420 39 I gt Vm Tm Am Va 1 1 P0 p0 T0 PeP0 Figure 1 10 Schematic diagram illustrating an idealized onedimensional ow machine with a hot gas exit ow surrounding a cold gas core ow 12 Theory of Aerospace Propulsion Since we are assuming that the ow outside the machine is inviscid the central hot ow does not mix with the cold outer ow and the force on the uid is given by L F FF 1 V V01 V V0 121 The cold outer ow is assumed to have had power added but not any heat Using the results obtained previously for the case of a ow with power added but no heat added we nd that F 122 5 V avg Similarly assuming the hot core ow has had heat added but has no net power added to it we nd that mh m 77mAQ E VWAQ hm 110 1 123 The total force on the uid may then be written as F P poAn77mAQ Vi 124 Vmg 1 5 Vmg This equation may be expanded to show the dependence on ight speed to yield 2P ZpkomAQ V0 L25 V0 V2 1 5 V0 Vm For high bypass ratio the contribution of the jet to the thrust of the ow machine is reduced particularly at low ight speeds compared to the fan contribution At high speeds however the fan contribution drops off substantially while the jet thrust is approximately constant Thus high bypass ratios now as high as 10 on large modern turbofan engines provide the high thrust needed for acceleration and climb during take off while also maintaining required thrust levels for high speed cruise Let us consider this case in additional detail by rst rewriting Eq 124 in the following form A W F mh 126 W VW Here we assume that the power for the cold outer stream chC is provided by a turbine in the hot central stream where WC denotes the work done one the uid per unit mass of Theory of Aerospace Propulsion uid processed That turbine also may supply power to a compressor in the hot stream but the net power in the hot stream remains zero This power balance may be written as m M Vn thm 127 The total heat added to the ow by burning fuel is given by Eq 113 and subtracting from that the work added to the cold outer stream yields the heat available for transformation to thrust mfganV AQh m 5W 128 Using Eq 128 in Eq 126 gives the following form for the total thrust m HV F mhAQh th i 129 V V avng avg avg Though this particular form is not easily appreciated it does provide an easy means to check the limiting cases which then may make it easier to interpret A schematic diaram of the internal con guration of a basic turbofan engine is shown in Fig 111 cold fan exhaust turbine compressor nozzle hot jet e aust Fuel injector 1 225 3 45 6 7 Figure 1 11 Schematic diagram of a typical turbofan engine showing the required turbomachinery components and usual station numbering scheme The combustor burns the injected fuel supplying heat to the central ow passing out the nozzle and the fan accelerates the cold outer ow of the turbofan 151 Very Small Bypass Ratio ltlt1 the Turbojet In this case there is essentially no cold outer ow and WCNO so that from Eq 128 Theory of Aerospace Propulsion m HV AQh m L 130 mh Using Eq 130 in Eq 129 gives the thrust as A F m mh Qh77m 131 Vaw This is the equation for thrust of a pure turbojet as suggested by Eq 19 The speci c fuel consumption is then m V c fg avgh F 771777mHV Note that this is the speci c fuel consumption for a turbojet as given in Equation 114 In the limit as the bypass ratio approaches zero we recover the results for a turbojet 152 Very Large Bypass Ratio gtgt1 the Turboprop In this case the cold outer mass ow mC is much larger than the hot central mass ow mh but mh itself is not especially small since it provides the power for driving the cold ow However after driving the outer ow the heat available in the hot central ow for the jet is much smaller than work required for the outer ow AQhltltWC and Eq 128 becomes m HV WE m i 133 mh Thus Eq 129 is approximated by m W F E 134 V avg 5 This is the case of the propeller as given by Eq 15 and in particular it corresponds to the turboprop engine where the gas turbine drives the propeller and little jet thrust is produced The specific fuel consumption in this case is Vavg a C W 17 By comparing Eqs 132 and 135 we see that this case of very high bypass ratio has the fuel efficiency advantage since VmgcltVmgh 7711 153 Finite the Turbofan We may now rewrite Eq 129 in terms of specific thrust that is thrust per unit total mass ow obtaining Theory of Aerospace Propulsion E 1 1 gnthanV 1 77m L36 m l39l a Vavg l39l Vvavgg Vavgm Here a is the fuelair ratio and it and the product gmh 77bH V may be considered for the purposes of this discussion as essentially xed by the generic engine design Thus for ltltl only the rst term on the righthand side of Eq 136 contributes to the speci c thrust and the engine is a turboj et while for gtgtl only the second term on the righthand side of the equation contributes to the speci c thrust and the engine is a turboprop As grows from zero we see an increasing contribution from the propellerlike second term and the engine is a turbofan of increasing bypass ratio The speci c fuel consumption may be obtained from Eq 136 as follows 71 L mfg g a nlhanVL 1 77m 6 I J mhmc l j Vavgm g LIavgc Vavgm m a m J 1 37 It is evident from Eq 137 that as the bypass ratio increases the speci c fuel consumption will decrease and this is the advantage of high bypass turbofan engines in a high fuel cost environment Note that mCmh cannot become arbitrarily large because the central hot mass ow mh provides the power needed to drive the cold ow fan 16 The Force Field for Airbreathing Engines Consider the airbreathing engine shown schematically in Fig 112 Air is taken onboard at approximately the ight speed V0 fuel is added and burned and the products of combustion are exhausted at the exit velocity V7 Further downstream the pressure pg achieves equilibrium with the ambient pressure at the ight altitude pg that is p pg Since experimental measurements are generally taken at the engine exit plane station 7 we con ne our attention to stations up to and including that point 142 A0 A 1 r Ve I V0 quot1 c Wf WeWjTEiVZ p0 p2 P0 T1 Te 1 7 e Figure 1 12 Schematic diagram of the control volume for an airbreathing engine Theory of Aerospace Propulsion We may apply the conservation of momentum principle to a control volume that extends from station 0 to station 7 as shown in Fig 113 Here the control volume is the streamtube which enters the airframe structure that houses the engine For simplicity we will consider a nacelle or engine pod bounded by the station 0 and 7 The force F1 is the force exerted on the uid by the external ow and the force F 2 is the force exerted on the uid by the walls of the nacelle The momentum theorem states that in the absence of body forces the resultant force on the boundary of a control volume xed in inertial space is equal to the momentum ux passing through that boundary Therefore assuming there are no body forces and that the ow is quasionedimensional we have V V FiF2pvop7A7 w7g7 w g 138 E4 P7147 EW7V7 l gt pe pl pa Te 0 l 7 e Figure 1 13 Control volume for onedimensional analysis of an airbreathing engine Now we may consider the forces acting on the nacelle enclosing the engine by examining the freebody diagram in Fig 114 Here F is the force on the nacelle exerted by the aircraft through the pylon F 2 is the force exerted by the uid on the walls of the nacelle and F 4 is the force exerted on the nacelle by the external stream Equilibrium of these forces requires that FF FZ0 139 Then using the results for F2 from Eq 138 in Eq 139 we obtain w0w W F fK jm pvop7A7 E E 140 Adding and subtracting pgA 7 to Eq 140 and rearranging it yields Theory of Aerospace Propulsion F Vo A7 p7 p0 pawn A7F1 F4 141 F4 Figure 1 14 Schematic diagram of the structure containing the ow machine Now Eq 141 is in the form F net thrust 7 nacelle or airframe drag where the rst term in square brackets is the engine contribution to the resultant force F and the second term in square brackets is the airframe contribution to the resultant force F The net thrust provided by the engine is de ned as W W W F 147 p7 po Vo 142 The net thrust written in this form is de ned as F gross thrust 7 ram drag Consider now the drag of the airframe housing the engine which is expressed as DpoA0 A7F1F4 143 Here F1 is the force in the streamwise direction due to the pressure of the external ow on the capture streamtube extending from station 0 to station 1 and it may be expressed as Al E I pdA 144 The quantityA is the crosssectional area of the stream tube The force F4 is the total drag force acting on the nacelle due to the external ow It is comprised of both pressure and friction forces and may be represented by F4 pdADf 145 When the ight velocity Vg0 and the power is off the pressure drag should be zero the pressure and Viscous drag must be zero and therefore the force F10 This means we Theory of Aerospace Propulsion must de ne these forces somewhat more carefully and consistently so that they are accounted for accurately under static test conditions Therefore introduce F39Ip pocMD 146 A7 EFL pp0dADfDp4Df 1 47 Using Eqs 146 and 147 in Eq 143 the nacelle drag may be rewritten as Al A7 DJA p p0dALp p0dADfDpJDND 148 The term DH is called the additive drag due to the inlet it will be seen to be an important factor in the design of inlets for supersonic ight The nacelle drag has interesting ramifications particularly for wind tunnel testing and engineairframe integration Consider the differences between the ow field under poweron and poweroff conditions as illustrated in Figs 115 and 116 The disruption of the ow field in the poweroff case where the nacelle acts like a bluff body leads to the result that the nacelle drag is greater than in the poweron case For the purposes of comparing various jet engines we may use Ve to determine the thrust as done previously n F WV Vo 149 g The net thrust given by Eq 142 is equal to the net thrust given by Eq 149 and this permits us to calculate Ve which is o en called the effective exhaust velocity as follows amp WO V V7 p7 po 150 Figure 1 15 Schematic diagram of ow during powered operation The ow proceeds smoothly over the nacelle and the exit station pressure may be adjusted to match the ambient pressure Theory of Aerospace Propulsion v Figure 1 16 Schematic diagram of ow with power off The ow at the front face is a stagnation point with high pressure The ow may separate from the cowl lip and cause irregular turbulent ow over the nacelle while the exit station pressure is that of a recirculation region and generally lower than the ambient pressure In obtaining this result we have assumed that Wf ltlt wu and since Wf 2 as will be shown in a subsequent chapter the approximation is quite a good one It should be noted that for subsonic exit velocities that is V7 lt 617 the pressure is always equilibrated that is p7 p0 hence the exhaust velocity increases until V7 617 For supersonic ows p7 is not necessarily equal to p0 this situation will be considered subsequently We now turn to considering some of the equations relating the ow properties for jet engines Continuing with our assumption that Wf ltlt wo for simplicity we note that the gross thrust of Eq 142 is now given by W nggoV7A7p7po 13951 Recalling that within the current approximation the weight ow wo W7 and using the equation for sound speed as 612 p the momentum ux in Eq 151 may be written as w gel7 p7 47V72 77P7A7M72 13952 Substituting this result into Eq 151 leads to the following equation for gross thrust F A7p71y7MZ po 153 Similarly we may write the ram drag as W E Vo 70pvoM 154 20 Theory of Aerospace Propulsion If the ow in the exhaust nozzle is assumed to be adiabatic with constant speci c heats then the stagnation temperature is constant through it If it is further assumed that the ow is also reversible that is isentropic then the stagnation pressure is also constant through it The quasionedimensional energy equation under these assumptions may be written as l cpJT7 EV72 cpJTm 155 This equation shows that the sum of the internal or thermal energy is a constant so that converting the thermal energy into kinetic energy permits the nozzle to accelerate the ow and produce the high exhaust velocity necessary for the production of high levels of thrust Solving Eql55 for the exit plane velocity yields 156 The isentropic relation between pressure and temperature is 77 1 i i h 157 158 We see here that the nozzle acts as a mechanical accelerator for the ow converting the thermal energy of the ow to kinetic energy The static to stagnation pressure ratio in Eq 158 may be written as follows amp amp 159 pt p0 pt0 pt The first ratio on the righthand side of Eql590 is that of the exit pressure of the nozzle to the ambient pressure at the ight altitude while the second is a free stream ratio that is a function of the ight Mach number and the third is the ratio of stagnation pressure across the entire control volume This stagnation pressure ratio is dependent upon the operation of the engine and it is indicative of the effectiveness of the engine since the larger the ratio that is the greater the stagnation pressure loss through the engine the lower the exhaust velocity everything else being equal Note that the 21 Theory of Aerospace Propulsion stagnation pressure ratio across the control volume is less than unity because the entropy must increase through the engine and this is re ected in a reduction in stagnation pressure across the engine Although many factors in uence the stagnation pressure loss in an airbreathing engine it will be shown that the greatest losses occur in the inlets of engines in supersonic ight Therefore good inlet design is crucial for the effective operation of engines for supersonic ight 17 Conditions for Maximum Thrust Let us consider the conditions which must be met for the isentropic nozzle of the engine in order that maximum thrust is produced for a given value of total energy in the ow entering the nozzle that is a given value of stagnation temperature T177 which is constant inside the nozzle At the same time we are assuming a given value of the stagnation pressure which in our ideal case of isentropic ow means that the stagnation pressure p17 is also constant throughout the nozzle We are therefore investigating what conditions must be met by the nozzle so as to make the best use of the acceleration capabilities of the nozzle First we note that the gross thrust given in Eq 151 for given engine geometry and ight conditions depends only upon the exit pressure of the nozzle We may search for an extremum in the gross thrust by taking the partial derivative of the gross thrust with respect to the pressure and setting it to zero leading to the following equation 6i 3 an 6A7 p p 0 1601 6177 g 5P7 6P7 7 0 6107 Here we have used the exact form of the equation in that the weight ow shown is that at station 7 In a subsequent chapter we will show that the quasionedimensional momentum equation requires that de Adp 161 Incorporating this result into Eq 160 yields 6F 6A p p 0 162 6107 7 0 6p 7 The gross thrust is an extremum when the exit pressure matches the ambient atmospheric 6A pressure p7pg or when 0 To determ1ne whether or not th1s extremum 1s a P 7 maximum we evaluate the second derivative of the gross thrust with respect to pressure evaluated at the extreme point Thus we find ang 62A 6A lax l 7 gtla7lflal 3963 22 Theory of Aerospace Propulsion We will show in a subsequent chapter dealing solely with the equations of motion that the variation of pressure with area for a quasionedimensional ow has the behavior shown in Fig 117 AK p Subsonic branc E t t M1 f f p E A 5 A6 A7 Supersonic E branch E A A7 A5 A 39 Figure 1 17 General variation of pressure with crosssectional area for a quasi onedimensional ow It is clear that the conditions for an extremum as given by Eq 162 are satis ed at either the matched exit case where p 7pg or at the sonic point where p7p the pressure for Ml For the case p7pg Eq 163 reduces to 62F g E 164 opz 6p 7 F7FU From the shape of the curve in Figll3 it is clear that if the exit ow is supersonic the righthand side of Eq 164 is negative and the thrust is a maximum If the exit ow for this case is sonic the righthand side of Eql64 is zero and the extremum is a saddle point It is tempting to say further that if the exit ow is subsonic then the righthand side of Eql64 is positive and the thrust is a minimum However that conclusion doesn t hold up under close scrutiny Recall that we started this section with a consideration of the gross thrust as given by EqlSl which includes the pressure imbalance term A subsonic ow exiting as a free jet from the exit of the nozzle cannot support a pressure difference so the pressure is always matched and the gross thrust for a subsonic jet is just 165 23 Theory of Aerospace Propulsion 6F As a result the first derivative of F g becomes A7 and there is no extremum The P 7 gross thrust may then be illustrated as in Fig 118 Fgm F g P7P0 M 7gt l ltI Subsonic Supersonic exit exit V0 V7617 V7 Figure 1 18 Variation of gross thrust with nozzle exit velocity We may consider the result that the gross thrust is a maximum for matched operation in a somewhat more physical fashion using the illustration in Fig 119 The solid black line represents the nozzle wall and the shaded areas above and below it represents the ambient pressure level If there were no ow the ambient pressure force on one side of the nozzle wall would cancel that on the other side With supersonic ow in the nozzle the pressure distribution is shown as the dashed line and the point where this line crosses into the shaded region indicates the length of nozzle required for matched operation If the nozzle is lengthened by an amount shown as Along the pressure on the lengthened portion of the nozzle will be below p0 and the thrust will drop from the matched value Similarly if the nozzle is shortened by an amount Ashort the portion of the nozzle represented by this length and the positive thrust associated with it will be lost and again the thrust will drop from the matched value Therefore the maximum thrust must occur when the nozzle is matched that is when p7p0 The general behavior of the net thrust is shown in Fig 120 with a notional illustration of the types of nozzle geometries typical of different exit conditions Subsonic jet aircraft generally operate with a choked exit station while supersonic operation requires opening the exit area to produce a supersonic ow and perhaps to attempt to match the exit pressure to the ambient atmospheric pressure 24 Theory of Aerospace Propulsion K pressure Figure 1 19 Illustration of effect of making nozzle longer or shorter than required for matched operation that is p7 30 Net thrust as a function of exit velocity 7 7 F E Saddle point 0 i V0 V7a7 V7 Exit flow Exit flow subsonic supersonic Figure 1 20 Net thrust as a function of exit velocity and illustration of the nozzle geometries appropriate to the indicated conditions 25 T heory of Aerospace Propulsion 18 Example Problems 181 The Pratt amp Whitney J58 jet engine for the SR71 has a fuel ow rate of 8000galhr of JP7 With a fuel to air ratio FA0034 The takeoff gross thrust Fg340001bs at a speed Vo200kts For a nozzle eXit Mach number M71 and a turbine eXit temperature TET1580FT7 Find a the effective eXhaust velocity b the net thrust c the specific fuel consumption d the free stream capture area e the useful power f the nozzle eXit temperature g the nozzle eXit velocity h the nozzle eXit area and i the nozzle eXit pressure a VefnggW lJrAF 2400fps b Net thrust FnFgm0 V0294001bs c CjWfFn15360029400184lbfuellbtththr d A0m0p0V0174 112 D147ft e PusefulFnV014750 hp 11MW f T7Tt71y7 1M722391 2040R086 1750R use m133 g V7a7y7RT7122000fps R1716ft2s2R h Since the gross thrust F gA7p71y7M72p0 and m7pVA7 then A7m1yRTg127p0 7 Fgpo and A7736ft2 D731ft i p7FgA7p01y7M72137p0 182 The German V2 rocket burned about 8000kg of propellant LOX and alcohol With an oxidizer to fuel ratio OF13 in 65s With a nozzle eXit velocity Ve 2000ms While the nozzle was operating in the matched mode 26 Theory ofAerospace Propulsion 1 Find a the thrust produced b the power produced on the launch pad and at a ight speed of 2kms c the speci c impulse and d the overall ef ciency a The thrust for a rocket is given by FmVe so that F8000kg65s2000ms246kN or 553klbs b The power P produced on the pad and at Vo2kms is given by PV0F 0 on the pad and 2250kNs 225MW or 660000hp at 2kms c ISPFmg 246kN8000kg65s98lms2203s d The overall ef ciency 170FV0mAQFV0Wf77bHV quot or 170 FVOmg1OF17bHV Then 170492MNmsl23kgsl272MJkg1l3338 Theory of Aerospace Propulsion 28