AEROSPACE PROPULSION SYSTEMS

Transcription

1 AEROSACE ROULSION SYSTEMS Chapter Fundamentals Chapter Rockets Chapter 3 iston Aerodynamic Engines Chapter 4 Gas Turbine Engines Chapter 5 Ramjets and Scramjets 00 John Wiley & Sons (Asia) te Ltd Courtesy NASA Courtesy USAF

2 Aerospace ropulsion Systems CHATER - FUNDAMENTALS Give me six hours to chop down a tree and I will spend the first four sharpening the axe. US resident Abraham Lincoln

3 WHY STUDY ROULSION? From the earliest days of recorded history, many have dreamed of soaring into the sky. Aerospace propulsion systems are the means for attaining powered flight. 00 John Wiley & Sons (Asia) te Ltd -3

4 WHY STUDY ROULSION? The technology has allowed the peoples of the world to be drawn closer together with commercial air transportation 00 John Wiley & Sons (Asia) te Ltd -4

5 WHY STUDY ROULSION? and has expanded our frontiers into space. 00 John Wiley & Sons (Asia) te Ltd -5

6 ropulsion ropulsion means to drive forward. Therefore a propulsion system is a machine that produces a thrusting force to drive an object forward. 00 John Wiley & Sons (Asia) te Ltd -6

7 Thrust Most aerospace propulsion systems produce thrust (F N ) by applying Newton s Third Law of action/reaction. Thrust is produced by accelerating a working gas (normally by adding heat due to chemical combustion). 00 John Wiley & Sons (Asia) te Ltd -7

8 Types of Aerospace ropulsion Systems Rockets Oldest type, dating back to the Han Dynasty in China (circa,000 BC). Used today in space launchers and missiles. Can operate outside Earth s atmosphere. Capable of very high thrusts. Courtesy National Museum of the USAF 00 John Wiley & Sons (Asia) te Ltd -8

9 Types of Aerospace ropulsion Systems iston Aerodynamic Engines Used in general aviation aircraft. Relatively low cost Capable of low thrusts Limited to low subsonic speeds Limited to low altitudes Courtesy National Museum of the USAF 00 John Wiley & Sons (Asia) te Ltd -9

10 Types of Aerospace ropulsion Systems Turbojet Engines The core of all gas turbine engines. No longer used in aircraft, but still used in missiles. Capable of high thrusts. Capable of supersonic speeds with the use of an afterburner. oor fuel efficiency Courtesy NASA 00 John Wiley & Sons (Asia) te Ltd -0

11 Types of Aerospace ropulsion Systems Turbofan Engines Widely used today in commercial and military aircraft. Capable of high to medium thrusts. Capable of supersonic speeds (typically requires an afterburner). Better fuel efficiency than turbojets. 00 John Wiley & Sons (Asia) te Ltd -

12 Types of Aerospace ropulsion Systems Turboprop engines Used in short-range commercial aircraft and military transports/cargo aircraft. More fuel efficient than turbofans. Limited to medium altitudes and subsonic speeds. Short take-off and landing Noisy, vibration Courtesy National Museum of the USAF 00 John Wiley & Sons (Asia) te Ltd -

13 Types of Aerospace ropulsion Systems Turboshaft Engines Used in helicopters and in auxiliary power units (AUs). Optimized to produce shaft power. Generally short in length. Courtesy USAF 00 John Wiley & Sons (Asia) te Ltd -3

14 Types of Aerospace ropulsion Systems Ramjet Engines Used in long-range supersonic missiles and some specialty aircraft. Mechanically simple (few moving parts). Can operate efficiently at Mach.5 to 5.0. Most cannot operate at subsonic speeds, so require a booster rocket. Courtesy NASA 00 John Wiley & Sons (Asia) te Ltd -4

15 Types of Aerospace ropulsion Systems Scramjet Engines Courtesy NASA Used in experimental hypersonic vehicles. Many difficult technical challenges. No operational models yet. Can operate at hypersonic Mach numbers 5.0 to 5.0. Cannot operate at low supersonic or subsonic speeds, so it requires a booster rocket. 00 John Wiley & Sons (Asia) te Ltd -5

16 .a Review of Terms System An identifiable collection of matter that is under investigation. Types of Systems Isolated Uninfluenced by its surroundings. Closed Contains a fixed mass. Open (or Flow System) Mass can transfer across the boundary of a control volume. 00 John Wiley & Sons (Asia) te Ltd -6

17 .a Review of Terms Working fluid Most aerospace propulsion systems operate in a thermodynamic cycle involving transferring heat to and from a working fluid. Generally this is atmospheric air (or air mixed with combustion gases). The properties of air (e.g. density, temperature, pressure, etc.) change with altitude. This limits the altitude and speed (flight envelope) of air-breathing engines. 00 John Wiley & Sons (Asia) te Ltd -7

18 Flight envelopes of different aircraft 00 John Wiley & Sons (Asia) te Ltd -8

19 .a Review of Terms Work (W) is a form of energy generated when a force moves something in the direction it is being applied. The definition of work requires movement to occur. The girl in this figure makes a great point, but she obviously did not see her dad typing on the keyboard, moving his mouse or putting paper in the printer. 00 John Wiley & Sons (Asia) te Ltd -9

20 .a Review of Terms ower ( W ) is the rate of doing work. Energy (E) is simply the capacity to do work and exists in many forms (e.g. kinetic, potential, thermal, mechanical, electrical, chemical, magnetic, or nuclear). 00 John Wiley & Sons (Asia) te Ltd -0

21 .a Review of Terms Heat The energy transferred between molecules of one system to another due to a temperature difference. Heat transfer mechanisms Conduction Takes place between two motionless adjacent substances at different temperatures Convection Takes place between a solid surface and a fluid in motion adjacent to it. Radiation Energy emitted by electromagnetic waves. This can occur across a vacuum (space). 00 John Wiley & Sons (Asia) te Ltd -

22 .a Review of Terms Thermal efficiency (h th) A performance measure of merit of a heat engine The ratio of its work output to the total heat added into the system: The nd Law of Thermodynamics states that η th can never reach 00%. th W Q in out in Q Q in Q Q Q out in 00 John Wiley & Sons (Asia) te Ltd -

23 .a Review of Terms rocess A system undergoes a process when its state changes from one equilibrium condition to another. Cycle If a system undergoes a number of processes so that its final state equals its initial state, then the system has undergone a cycle. 00 John Wiley & Sons (Asia) te Ltd -3

24 .a Review of Terms This figure shows an ideal gas turbine engine cycle. rocess 4 is listed as Not ossible because it occurs outside the engine when the exhaust gases diffuse in the ambient air surroundings. However since this is an open system, with new intake air continuously flowing into the engine (restarting the cycle at point ), it effectively is a cycle. 00 John Wiley & Sons (Asia) te Ltd -4

25 .a Review of Terms Reversible processes The original state of the system can be restored, leaving no change in the surroundings. This is an ideal process which does not occur in nature. Adiabatic A process in which no heat transfer occurs. Isentropic A reversible, adiabatic process. 00 John Wiley & Sons (Asia) te Ltd -5

26 .a Review of Terms Entropy (S) A measure of disorder, chaos, or randomness. For an isolated system: ΔS > 0 ΔS = 0 ΔS < 0 Irreversible processes Reversible (isentropic) processes Impossible Isentropic efficiencies will later be used to compare the actual performance of engine components to their idealized performance. 00 John Wiley & Sons (Asia) te Ltd -6

27 .a Review of Terms The nd Law of Thermodynamics shows that processes will naturally become more disordered. By intentionally adding heat or work to a system we can bring more order to a particular attribute of the system. However, even though one attribute of the system may become more ordered, there will always be a net entropy gain (or increase of disorder) due to other irreversibilities (such as friction or heat loss). 00 John Wiley & Sons (Asia) te Ltd -7

28 .b erfect Gas Law (Equation of State) The properties of a working fluid (like air) are very important in analyzing propulsion systems. Air is closely approximated as an ideal gas. Therefore the equation of state for a perfect gas is a useful analysis tool. v RT 00 John Wiley & Sons (Asia) te Ltd -8

29 .b erfect Gas Law (Equation of State) In the previous equation, R is the gas constant. It is related to the universal gas constant (R o ) by: R R M o w R o M w Universal gas constant [=8.345 kj/(kmoles K)] Molecular weight of gas [kg/kmoles] 00 John Wiley & Sons (Asia) te Ltd -9

30 .c Conservation of Mass t control volume dv control surface V da 0 For steady flow: m V A const ant 00 John Wiley & Sons (Asia) te Ltd -30

31 Example. Liquid oxygen (LOX) and liquid hydrogen are steadily injected into a rocket thrust chamber at 8 kg/s and kg/s respectively and ignited. The combustion products are expelled from the rocket through a nozzle with a diameter of 5 cm. If the density of the combustion gases is 0.75 kg/m 3. Determine the exit velocity of the combustion gases. 00 John Wiley & Sons (Asia) te Ltd -3

32 Example. Solution The control volume of the rocket thrust chamber and nozzle is shown by the dotted line in the figure. Since the propellants are flowing at a steady rate, the conservation of mass equations are reduced to: t control volume dv 0, steady flow control surface V da 0 comb e Ve prod m exit m O m H A 00 John Wiley & Sons (Asia) te Ltd -3 m in

33 Example. V e m 4 O A e m H comb prod 8 kg s m d 4 kg s O m kg s 0.5m 0.75 m 3 H comb prod,047.7 m 00 John Wiley & Sons (Asia) te Ltd -33

34 .d Conservation of Momentum F t control volume V dv control surface V V da 00 John Wiley & Sons (Asia) te Ltd -34

35 Example. A rocket motor burning on a test stand steadily exhausts 0 kg/s of combustion gases at an exit velocity of 750 m/s. The static pressure of the exhaust gases exiting the nozzle is 0 ka. Assume an ambient air pressure of 0.3 ka. Determine the force (or thrust) the rocket produces. 00 John Wiley & Sons (Asia) te Ltd -35

36 Example. Solution The external reaction force (R x ) which holds the rocket in place on the test stand is equal in magnitude but opposite in direction to the thrust force produced by the rocket R x F N The control volume encompassing the rocket and test stand is shown by the dotted line in the figure. Since the exhaust gas is flowing at a steady rate, the conservation of momentum equations reduce to: F N t A V dv V V da e 0 e control volume 00 John Wiley & Sons (Asia) te Ltd -36 0, steady flow control surface

37 Example. F N mv Ae e kg m kn N kn 0 s s, m 0 ka 0.3 ka This example shows how the thrust equation for a rocket is derived from the Law of the Conservation of Momentum. 00 John Wiley & Sons (Asia) te Ltd -37

38 .e Conservation of Energy d dt t Q W e dv u g z V da control volume control surface V 00 John Wiley & Sons (Asia) te Ltd -38

39 .e Conservation of Energy The specific enthalpy (h) is defined as: h u Substituting this into the energy equation gives an alternative form: d dt t Q W ' e dv h g z V da control volume control surface V 00 John Wiley & Sons (Asia) te Ltd -39

40 .e Conservation of Energy An important parameter in this equation is the specific heat, defined as the amount of heat required to raise a unit mass through a ºC temperature rise. When volume is held constant: C v u T V When pressure is held constant: C h T 00 John Wiley & Sons (Asia) te Ltd -40

41 .e Conservation of Energy The gas constant (R) is related to these two specific heats by: R dh dt du dt C C v Also a ratio of specific heats ( ) can be defined as: C C 00 John Wiley & Sons (Asia) te Ltd -4 v

42 .e Conservation of Energy A calorically perfect gas is one that has constant specific heats. If such a gas is in a thermodynamic process between two states, then the definitions for C v and C p can be integrated to obtain: u u u C dt C T T v v 00 John Wiley & Sons (Asia) te Ltd -4

43 .e Conservation of Energy And h h h T C dt C T 00 John Wiley & Sons (Asia) te Ltd -43

44 Example.3 Air enters an adiabatic compressor of a turbojet engine at 70 kg/s and at a temperature (T ) of 30 C. It flows steadily through the compressor with no change in velocity and exits at a temperature (T ) of 350 C. Assume that the constant pressure specific heat (C p ) of air is.005 kj/(kg K). Determine the minimum power that must be generated by a turbine in order to drive the compressor at these conditions. 00 John Wiley & Sons (Asia) te Ltd -44

45 Example.3 Solution The control volume around the compressor section is shown by the dotted line in the figure. The energy equation for steady flow is: t control volume No velocity change 0, steady flow No height change 0, adiabatic e dv control surface V h g z d dt V da Q W ' 00 John Wiley & Sons (Asia) te Ltd -45

46 Example.3 Therefore the power required by the compressor to pressurize the air (and thereby also increase its temperature) is equal to the minimum power required by the turbine to drive it. W turbine W T 30C John Wiley & Sons (Asia) te Ltd -46 K T 350C compressor K h h m C T T kg kj K 303K s,5 kj s m or air kgk kw air p

47 . Isentropic Equations Isentropic equations are used often in the analysis of gas turbine engines. (Standardized tables show the isentropic parameters.) These are derived by first applying the st Law of Thermodynamics to an isentropic process, which gives: dq dw du T ds dv dt 00 John Wiley & Sons (Asia) te Ltd -47 C v dt dv C v 47

48 . Isentropic Equations Recall for a perfect gas: T v R Substituting this into the equation yields: dt R dv v d 00 John Wiley & Sons (Asia) te Ltd -48

49 . Isentropic Equations Differentiating gives: dv Cv R dv v d C v R dv v d dv 0 00 John Wiley & Sons (Asia) te Ltd -49

50 . Isentropic Equations Also recall: R C v Substituting this in the equation gives: dv v d dv 0 00 John Wiley & Sons (Asia) te Ltd -50

51 . Isentropic Equations dv v d dv 0 dv v d dv dv 0 d dv v 0 ln ln( v) 0 00 John Wiley & Sons (Asia) te Ltd -5

52 . Isentropic Equations Applying logarithmic identities transforms these equations into: Therefore: ln ln v ln v 0 v const ant C 00 John Wiley & Sons (Asia) te Ltd -5

53 . Isentropic Equations Again recall that for a perfect gas: v RT Substituting this in gives: v R T T R const ant C 00 John Wiley & Sons (Asia) te Ltd -53

54 . Isentropic Equations T R T This results in the following isentropic equation that relates temperature and pressure: R T T 00 John Wiley & Sons (Asia) te Ltd -54

55 . Isentropic Equations The gas turbine engine cycle is commonly illustrated by a T-S diagram of the engine cycle. The ideal (or isentropic) processes representing the compressor and turbine are shown by the vertical arrows and labeled with a prime symbol ( ). The actual (or non-isentropic) processes are represented by dashed arrows 00 John Wiley & Sons (Asia) te Ltd -55

56 . Isentropic Equations It is also useful to develop isentropic expressions that relate the specific volume ( v) to pressure () and temperature (T). Starting with the definition of enthalpy (h): dh du d du d v dh du dv v d 00 John Wiley & Sons (Asia) te Ltd -56

57 . Isentropic Equations Recall: du C dt dv v Substituting the definition of enthalpy (h) into this equation gives: 0 ( dh dv v d) dv 00 John Wiley & Sons (Asia) te Ltd -57

58 . Isentropic Equations 0 dh v d dh v d Since: dh C p dt then: v d C p dt 00 John Wiley & Sons (Asia) te Ltd -58

59 . Isentropic Equations Since: dt v d C dv p C v dv vd C C v p d C C v p dv v dv v 00 John Wiley & Sons (Asia) te Ltd -59

60 00 John Wiley & Sons (Asia) te Ltd -60. Isentropic Equations Integrating this equation between points and gives: v v v dv d ln ln v v ln ln v v

61 00 John Wiley & Sons (Asia) te Ltd -6. Isentropic Equations Resulting in an isentropic equation relating pressure and specific volume: v v v v

62 . Isentropic Equations Again recall for a perfect gas: RT v This can be substituted in to give a relationship with temperature. RT v RT v v v 00 John Wiley & Sons (Asia) te Ltd -6

63 00 John Wiley & Sons (Asia) te Ltd -63. Isentropic Equations v v v v T T v v v v T T v v T T Resulting in an isentropic equation relating temperature and specific volume:

64 .3 olytropic rocesses olytropic processes follow laws that form the relation: v n const ant C Where 0 < n < 00 John Wiley & Sons (Asia) te Ltd -64

65 .3 olytropic rocesses Isentropic processes can be thought of simply as a polytropic process with n =. This offers an alternative method from isentropic processes of defining efficiencies. However for simplicity, this presentation will deal only with isentropic efficiencies 00 John Wiley & Sons (Asia) te Ltd -65

66 .4 Total properties When a fluid in motion is isentropically brought to rest a temperature and pressure rise occurs. The fluid properties at this point are known as stagnation properties. 00 John Wiley & Sons (Asia) te Ltd -66

67 .4 Total properties In the absence of hydrostatic pressures (e.g. elevation effects of fluid weight on pressure), stagnation properties are equivalent to total properties. 00 John Wiley & Sons (Asia) te Ltd -67

68 .4 Total properties It can be shown by applying the energy equation that the total temperature (T t ) is: Total temperature = Static temperature + Dynamic temperature T t T V C p 00 John Wiley & Sons (Asia) te Ltd -68

69 Mach Number (M) It is often convenient to describe these values in terms of Mach number, rather than velocity. The Mach number (M) is non-dimensional number defined as the ratio of the velocity (V) over the speed of sound (a): M V a 00 John Wiley & Sons (Asia) te Ltd -69

70 Speed of Sound (a) The speed of sound for a perfect gas (such as air) is: a RT 00 John Wiley & Sons (Asia) te Ltd -70

71 .4 Total properties Substituting these definitions into the equations for total pressure ( t ) and total temperature (T t ) gives: T t T M t M 00 John Wiley & Sons (Asia) te Ltd -7

72 X-Function Like Mach number there are several other nondimensional parameters that are useful for propulsion analysis. One commonly used parameter is simply known as the X-function. It is derived by substituting the Equation of State into the steady flow Conservation of Mass equation: AM m R T 00 John Wiley & Sons (Asia) te Ltd -7

73 X-Function By substituting the definitions just derived for t and T t, this equation becomes: m t AM R T t M A portion of this equation is the X-function. 00 John Wiley & Sons (Asia) te Ltd -73

74 X-Function The X-function (or the non-dimensional mass flow rate) is defined as: X M Substituting this back into the previous mass flow equation, gives: X m 00 John Wiley & Sons (Asia) te Ltd -74 M R T A t t

75 Y and Z-Functions The X-function can be further segmented into the Y and Z-functions: X Y The Y-function (or the non-dimensional specific internal thrust reciprocal) is: Z Y M M M 00 John Wiley & Sons (Asia) te Ltd -75

76 Y and Z-Functions The Z-function (or the non-dimensional internal thrust) is: Z M M The utility of these functions will become evident later 00 John Wiley & Sons (Asia) te Ltd -76

77 .5a Isentropic Flow in Ducts Steady, isentropic flow through a frictionless duct is one of the simplest fluid dynamics systems to define and analyze. No work can be extracted from a duct. If the flow is isentropic (adiabatic) then there is no heat transfer (in or out), therefore the energy equation is: Q W 0 00 John Wiley & Sons (Asia) te Ltd -77

78 .5a Isentropic Flow in Ducts Since there is no heat transfer then: T t 0 For steady flow ( m 0) in a frictionless duct then also: 0 t Had friction losses been considered (nonisentropic), there would be a loss in total pressure ( t t ). This is evident by inspection of the following equation: s S m R ln 00 John Wiley & Sons (Asia) te Ltd -78 t t

79 .5a Isentropic Flow in Ducts The continuity equation for one-dimensional steady flow through a varying differential control volume can be written as: da dav dv 0 d da A dv V 0 00 John Wiley & Sons (Asia) te Ltd -79

80 .5a Isentropic Flow in Ducts Applying the linear momentum equation for this same control volume gives the equation: p A d da da da AV dv d V dv 0 Combining these equations: d V d da A 0 00 John Wiley & Sons (Asia) te Ltd -80

81 .5a Isentropic Flow in Ducts But since: Substituting this in gives: a s d V d a da A 0 d M V da A 00 John Wiley & Sons (Asia) te Ltd -8

82 .5a Isentropic Flow in Ducts This equation is significant because it shows the effect that Mach number has on flow inside a varying area duct or channel. Duct Geometry Converging (Decreasing area) Diverging (Increasing area) Converging-Diverging (Decreasing-then-Increasing) Entry Mach No. Static ressure Velocity M < Decreasing Increasing M > Increasing Decreasing M < Increasing Decreasing M > Decreasing Increasing M < Decreasing Increasing 00 John Wiley & Sons (Asia) te Ltd -8

83 .5b Turbomachinery Turbomachinery (such as compressors or turbines) are designed to transfer work (but not heat). Compressors are used to increase the pressure of a flow. Turbines are used to extract work (or energy) from the flow. 00 John Wiley & Sons (Asia) te Ltd -83

84 .5b Turbomachinery The power required to drive a compressor is determined by the energy equation as: W c m C Ideally, the temperature ratio in compressors and turbines would be the minimum associated with pressure changes. So the process is isentropic, and T T t t p 00 John Wiley & Sons (Asia) te Ltd -84 T t t T t t

85 .5b Turbomachinery However, in reality there are irreversibilities due to friction on all of the large wetted surfaces. Entropy does increase but not used as a normal basis for analysis. Instead isentropic efficiencies are used as the ideal against which the actual is rated. 00 John Wiley & Sons (Asia) te Ltd -85

86 .5c Combustion Chambers Combustors are designed as steady flow devices. They are essentially ducts with the capacity for heat addition. No work can be extracted. The heat produced by a combustor is: Q mc p T T t t 00 John Wiley & Sons (Asia) te Ltd -86

87 .5d Nozzles Two types of nozzle shapes are primarily used in aerospace propulsion systems: Convergent nozzles Convergent-divergent (condi) nozzles A 3 rd type, divergent nozzles, are used in scramjets. 00 John Wiley & Sons (Asia) te Ltd -87

88 .5d(i) Convergent Nozzles Isentropic flow through a convergent nozzle can best be understood by examining a nozzle with a constant chamber pressure ( c ) and applying decreasing back pressures (points A D) on it. If the nozzle is exhausting gases into the atmosphere, this back pressure is equal to the ambient pressure ( 0 ). Therefore a continuous decrease in the ambient back pressure is equivalent to climbing in altitude 00 John Wiley & Sons (Asia) te Ltd -88

89 .5d (i) Convergent Nozzles At oint A: 0 = c, so there is no mass flow through the nozzle. As 0 is lowered to point B and beyond, the static pressure through the nozzle decreases and the mass flow through the nozzle increases. The Mach number at the nozzle exit plane also increases. Under these conditions, the exit static pressure ( e ) is equal to 0. This is called fully expanded flow. This is an optimal condition for propulsive convergent nozzles, because it maximizes the net thrust. 00 John Wiley & Sons (Asia) te Ltd -89

90 .5d (i) Convergent Nozzles This trend continues until point C is reached, where the fluid exiting the nozzle is equal to the velocity of sound (M e =.0 ). Since flow through a convergent nozzle cannot be accelerated from subsonic velocities to supersonic velocities; therefore as the back pressure continues to decrease past point C (to point D and beyond) no additional mass can flow through the nozzle. This is called choked flow. 00 John Wiley & Sons (Asia) te Ltd -90

91 .5d (i) Convergent Nozzles For choked flow, the exit static pressure is not equal to the back pressure ( e 0 ). Under these conditions the sonic gases will dissipate through a shock system. If the pressure differences are large enough, these shocks will form outside the nozzle. Choked flow means that the mass flow rate has reached a maximum value. 00 John Wiley & Sons (Asia) te Ltd -9

92 .5d (i) Convergent Nozzles Choked (or underexpanded) flow occurs when the pressure ratio ( te 0 ) is greater than or equal to a critical value. This places a limit on the air mass flow that can flow through the nozzle The optimum (or maximum) thrust occurs when the exhaust gases fully expand to the ambient pressure ( e = 0 ). This maximizes the momentum thrust. Choked flow results in a loss of momentum thrust, but creates a smaller pressure thrust component since ( e > 0 ). This lost momentum thrust may only be recovered by adding a divergent surface (e.g. condi nozzle). 00 John Wiley & Sons (Asia) te Ltd -9

93 Critical ressure Ratio For isentropic flow, an equation can be derived that defines the critical pressure ratio (R crit ) necessary to just choke the nozzle. This is the maximum pressure ratio ( t /) that can be achieved in the nozzle. This will occur when M e =.0 (for a convergent nozzle). R crit t 00 John Wiley & Sons (Asia) te Ltd -93

94 Choke Test The critical pressure ratio is a function of γ only (e.g. for γ =.333 then R crit =.85 and for γ =.4 then R crit =.893). It provides a test to see if a convergent nozzle is choked or not. If the ideal pressure ratio (achieved by full expansion of the flow through the nozzle) exceeds the critical pressure ratio than this ideal ratio cannot be achieved because flow through the nozzle is choked. Choked if: te 0 R crit 00 John Wiley & Sons (Asia) te Ltd -94

95 Choked Flow If the nozzle is choked, then the exhaust pressure ratio ( te / e ) is equal to the maximum or critical pressure ratio (R crit ). Therefore the static pressure of the exhaust gases is: e te e te R te crit 00 John Wiley & Sons (Asia) te Ltd -95

96 Unchoked Flow If the nozzle is not choked, flow is subsonic throughout the nozzle (M e <). Flow through the nozzle can adjust to changes in ambient back pressure (altitude). Ambient pressure changes will propagate upstream from the nozzle exhaust plane at the speed of sound. So for all unchoked flows in a convergent nozzle, the exit pressure will be equal to the ambient back pressure ( e = 0 ). 00 John Wiley & Sons (Asia) te Ltd -96

97 .5d (i) Convergent Nozzles The modes of operation of a convergent nozzle are summarized in the following table: Modes of Operation Convergent Nozzle A e te 0 te e Under-expanded (choked) Rcrit R crit Just choked Rcrit Rcrit Fully expanded (not choked) Rcrit R M e e e 0 < > John Wiley & Sons (Asia) te Ltd -97 crit

98 Critical Temperature Ratio An equation for the critical temperature ratio for choked flow can also be determined: TR crit The exit static temperature for a choked nozzle can be determined from the critical temperature ratio: T e T TR t crit 00 John Wiley & Sons (Asia) te Ltd -98

99 .5d(ii) Convergent-Divergent Nozzles As was done for convergent nozzles, isentropic flow through a condi nozzle can be understood by examining a nozzle with a constant chamber pressure ( c ) and applying decreasing ambient back pressures ( 0 ) (points A D) on it. Again oint A illustrates a limiting case where 0 = c, so there is no mass flow through the nozzle. The ambient back pressure ( 0 ) is lowered to point B and beyond, the static pressure through the nozzle decreases and the mass flow through the nozzle increases. 00 John Wiley & Sons (Asia) te Ltd -99

100 .5d(ii) Convergent-Divergent Nozzles In this range of ambient back pressures, the flow is fully expanded so e = 0. Flow in both the convergent and divergent portions of the nozzle is subsonic. This trend continues until point C is reached. At this point the flow at the throat travels at the speed of sound (M*=.0). Since the flow through the convergent portion of the nozzle cannot be accelerated from subsonic velocities to supersonic velocities; the condi nozzle becomes choked at all pressure ratios below point C (point D). 00 John Wiley & Sons (Asia) te Ltd -00

101 .5d(ii) Convergent-Divergent Nozzles When M*=.0 at the throat, there are two possible isentropic solutions for a given area ratio (A/A*). The flow can either decelerate to a subsonic exit Mach number or accelerate to a supersonic exit Mach number. oint D (and lower) represents the 0 condition where the flow accelerates to a supersonic Mach number in the diverging section of the nozzle. Therefore for 0 lower than the point D, the pressure will decrease in both the convergent and divergent sections of the condi nozzle resulting in supersonic exhaust flow. 00 John Wiley & Sons (Asia) te Ltd -0

102 .5d(ii) Convergent-Divergent Nozzles This is the objective of condi nozzle designs, because a supersonic exhaust gas velocity greatly increases the thrust of a propulsion system. For 0 in between points C and D, an isentropic solution is not possible because shock waves are formed and this is an irreversible process. In this case, shock equations would have to be used to determine the flow properties. 00 John Wiley & Sons (Asia) te Ltd -0

103 .5d(ii) Convergent-Divergent Nozzles As just stated, condi nozzles are designed to be choked at the throat (M*=.0) so exhaust gases can be accelerated to a supersonic exit velocity in the diverging section. The same choke test derived for convergent nozzles can also be applied to the throat section of condi nozzles. Optimal thrust occurs when the nozzle is sized so that the exhaust gases are fully (or perfectly) expanded ( e = 0 ). Imperfect nozzle expansion is caused by not having an ideal nozzle expansion ratio (ε = A e /A*) for a particular operating altitude. 00 John Wiley & Sons (Asia) te Ltd -03

104 .5d(ii) Convergent-Divergent Nozzles The flow is underexpanded if e > 0 and overexpanded if e < 0 Underexpansion is caused by a less than optimal nozzle expansion ratio, resulting in a loss in momentum thrust. Overexpansion is caused by having a greater than optimal nozzle expansion ratio, which may result in flow separation, which form shocks inside the nozzle. Nozzle performance losses due to overexpanded flow are generally much larger than losses due to underexpanded flow. Courtesy NASA 00 John Wiley & Sons (Asia) te Ltd SR-7B Showing Shock Diamonds in the Exhaust -04

105 .5d(ii) Convergent-Divergent Nozzles Full expansion of an exhaust jet in a fixedcondi nozzle can only be achieved when it is operating at its design pressure ratio. Consequently, fixed-geometry condi nozzles are typically only used in missiles that spend the majority of flight is at a predictable supersonic cruising velocity. Most aerospace propulsion systems with condi nozzles are designed with variable geometry (VG). This allows the area ratio to be variably optimized over a range of flight conditions, improving the condi nozzle s effectiveness at generating thrust Courtesy National Museum of the USAF VG Nozzle on the F00-W- 00 Engine (F-5 Fighter) 00 John Wiley & Sons (Asia) te Ltd -05

106 .6 Shock Waves A body traveling at M < through a compressible fluid (such as air) creates a disturbance that is propagated throughout the fluid by a wave traveling at the local velocity of sound (relative to the body). This creates gradual changes in the fluid properties as it approaches the body. However, if the body is traveling at M > then the fluid is unable to gradually change ahead of the body. Therefore the supersonic body induces a sudden change in fluid properties due to a shock wave. The first panel shows a situation similar to supersonic flow over a body. Like the driver, the flow has no time to prepare for it. The second case is like subsonic flow over a body. Like the driver, the flow ahead of the body gradually changes to prepare for it. 00 John Wiley & Sons (Asia) te Ltd -06

107 .6 Shock Waves Consideration of shock waves is important in the design of intakes, nozzles and ducts of aerospace propulsion systems capable of supersonic velocities. There are two types of shock waves. The simplest type of shock, the normal shock, occurs normal to the flow direction. An oblique shock occurs at an inclined angle to the flow direction. 00 John Wiley & Sons (Asia) te Ltd -07

108 .6a Normal Shocks Equations have been derived to determine the change in properties across a normal shock (Appendix C). T T t t ~ No change across a shock M M M 00 John Wiley & Sons (Asia) te Ltd -08

109 00 John Wiley & Sons (Asia) te Ltd -09.6a Normal Shocks M M M T T M M M M t t

110 Example.4 A normal shock forms on the intake of an aircraft flying at Mach.6 at 0 km. Assume =.4. Determine the Mach number (M ), total pressure ( t ), static pressure ( ), total temperature (T t ), and static temperature (T ) of air after the shock. 00 John Wiley & Sons (Asia) te Ltd -0

111 Example.4 Solution According to the Standard Atmospheric Table, Appendix A, Table A. for 0 km altitude: 6. 5 ka and T 3. 3 K According to Table C., Appendix C ( ) for M. 6 :.4 t M t.80 T T John Wiley & Sons (Asia) te Ltd -

112 Example.4 Assuming isentropic flow outside the intake: t M a. 6 ka.4 T t T M 00 John Wiley & Sons (Asia) te Ltd K K

113 Example.4 Therefore: t.6 ka ka t t 8 t ka ka T T T 9 T 3.3 K K Finally, since there is no change in total temperature across a shock: Tt Tt K 00 John Wiley & Sons (Asia) te Ltd -3

114 .6b Oblique Shocks The methodology of analyzing flow properties across oblique shocks is very similar to normal shocks. Even though an oblique shock is inclined at an angle to the flow direction, it still creates an abrupt change in fluid properties and is adiabatic. Like a normal shock, there is no change in the total temperature (T t ) across an oblique shock. The difference is that an additional variable must be introduced to account for the oblique shock s inclination to the flow direction. 00 John Wiley & Sons (Asia) te Ltd -4

115 .6b Oblique Shocks It can be shown that there is no change in tangential velocity across an oblique shock. V t Vt 00 John Wiley & Sons (Asia) te Ltd -5

116 .6b Oblique Shocks Substituting this into the normal momentum equation yields equations that show that the components normal to an oblique shock act just like a normal shock, while the components tangential to an oblique shock do not change. Therefore, the fluid property ratios across an oblique shock can be determined by calculating the components normal to the oblique shock and using the normal shock tables 00 John Wiley & Sons (Asia) te Ltd -6

117 .6b Oblique Shocks Figures for oblique shocks are shown in Appendix C. Note that there are two possible solutions (or none at all). A strong shock has a large value of θ and a large pressure ratio across the shock. It generally occurs when the back pressure of a supersonic flow is extremely high. However, this system is unstable and will normally degenerate into the weaker solution. A strong shock will always slow the supersonic flow velocity to a subsonic speed. 00 John Wiley & Sons (Asia) te Ltd -7

118 .6b Oblique Shocks A weak shock is one that has a relatively small value of θ, a smaller pressure ratio across the shock, and a small back pressure. A weak solution occurs more frequently on aerospace system designs than a strong shock. Normally weak shocks will occur on wings, open inlets and planar surfaces. A weak shock will always slow the flow velocity to a lower but still supersonic speed. This is the type of shock that mostly occurs in propulsion systems. 00 John Wiley & Sons (Asia) te Ltd -8

119 .6b Oblique Shocks A third possibility is that there is no solution at all. This can occur if there is a great enough wedge angle. In this case the shock detaches from the body and may occur in front of it. An example of a detached bow shock is shown in the figure. 00 John Wiley & Sons (Asia) te Ltd -9

120 Example.5 Compare the loss in total pressure ratio incurred by a two-dimensional, two-shock spike diffuser and a threeshock diffuser operating at Mach.0. Assume each oblique shock turns the flow through an angle ( ) of John Wiley & Sons (Asia) te Ltd -0

121 Example.5 Solution (a) Two-shock inlet calculations: From oblique flow charts (Figures C.a and b, Appendix C) for M =.0, =.4, and d = 0 the weak shock solution is: 39.4 M. 64 and Therefore: M n M sin.0 sin John Wiley & Sons (Asia) te Ltd -

122 Example.5 The normal shock tables (Table C., Appendix C) can now be used for M n =.7 : t t For the normal shock M =.64. Then again from the normal shock tables: t3 t John Wiley & Sons (Asia) te Ltd -

123 Example.5 So the total pressure recovery across the two-shock inlet is: t t3 t3 t t shock inlet t 00 John Wiley & Sons (Asia) te Ltd -3

124 Example.5 (b) Three-shock inlet calculations: This is done similar to the one-shock inlet. From the oblique shock tables (Figures C-a and b, Appendix C) again for M =.0, =.4, and d = 0 : and M Therefore, once again: M n sin M.7 00 John Wiley & Sons (Asia) te Ltd -4

125 Example.5 Again using the normal shock tables for M n =.7 : t t For the second oblique shock for M =.64, =.4, and d = 0 (Figures C.a and b, Appendix C), = 49.4 and M 3 = M n.64 sin 00 John Wiley & Sons (Asia) te Ltd -5

126 Example.5 Again using the normal shock tables for M n =.5 : t3 t For the normal shock, using the normal shock tables for M 3 =.8 : t t 3 00 John Wiley & Sons (Asia) te Ltd -6

127 Example.5 Therefore: t 4 t 3 shock inlet t 4 t3 t3 t t t Thus there is about a 0% improvement in total pressure ratio gained by using the three-shock inlet over a two-shock inlet at M =.0. If we were to repeat this calculation for M =4.0, there would be a 6% improvement. Thus the improvement increases with higher speeds. 00 John Wiley & Sons (Asia) te Ltd -7

128 .6c Conical Shocks Supersonic flow about a three-dimensional circular cone is more complex than a simple two-dimensional wedge, because after a conical shock the streamlines curve to satisfy the conservation of mass. Therefore a conical shock will be inclined at a lesser angle to the flow direction than a simple two-dimensional oblique shock. This means that a two-dimensional wedge will create a greater flow disturbance than a three-dimensional cone. 00 John Wiley & Sons (Asia) te Ltd -8

129 .6c Conical Shocks This is because flow cannot pass around the side of a twodimensional wedge, since it extends to infinity in the third dimension. Therefore separate flow relations are necessary to analyze a conical shock shown in Figures C.3, C.4, and C.5 in Appendix C. 00 John Wiley & Sons (Asia) te Ltd -9

130 CHATER - SUMMARY The equations of mass, linear momentum, and energy were presented and applied to basic engine components. Equations for isentropic flow were applied to idealized engine model components. The effect of Mach number on isentropic flow was shown. Two different types of nozzles were introduced: convergent and convergent-divergent (condi) nozzles. A limiting factor of nozzles is choked flow, which means that no additional mass can flow through the nozzle. Lastly the formation of shock waves in compressible, supersonic flow was introduced. 00 John Wiley & Sons (Asia) te Ltd -30