compressor is designed to have a higher value of the pressure ratio. The ultimate limit is reached when the CET reaches the maximum allowable TIT. At this design point little or no fuel can be injected and the oxygen available is plentiful. In such a design, the stoichiometry (the relative amounts of chemical reactants involved: fuel and oxygen) is fuel-lean, quite lean, in fact. However: no heat, no work. Evidently, there must be a middle ground between compression ratio one and the value that leads to the CET raised to equal the TIT. That is indeed the case. The virtues of an engine are two thermodynamic ones: work delivered per unit mass flow air processed (specific work) and work delivered per unit of heat provided (thermal efficiency). An engine with a high specific work will be compact and light in weight because the machinery involved will be appropriately modest while an engine design with high efficiency will be heavier but sparing of fuel use.

These two design conditions do not occur at the same compressor pressure ratio. In fact, as the pressure ratio is increased (in a design sense) the maximum specific work configuration is reached first, followed at a higher pressure by the maximum efficiency point. Thus, a choice for a preference must be made. The engineer would illustrate this with a cool graph, but we are not doing this here! (He would also have jargonized compressor pressure ratio as CPR!). The optimum pressure ratio has to be examined in the context of the question: how is the engine going to be used? Long distance flights require high efficiency for low fuel consumption while short flights benefit from not having to drag a heavy engine up to flight altitude and then back down, relatively often.

#### **10.3 Compressor Aerodynamics**

So how does the blading in these compressors work? The compressor is a machine that exploits the aerodynamics of wings. After all, the propeller is a compressor of sorts, why not build on that? Actually, the propeller is not a compressor, but it does accelerate a flow and the flow so produced could be slowed to raise pressure. A great advantage of flows in a compressor over similar flows in a freestream is that it has to be an *internal* flow with boundaries. That means that bound vorticity on the rotating and any stationary blading will largely avoid the creation of the trailing vortices that propeller blades must live with. A second advantage is that the flow behind such blading can be made relatively uniform, a virtue the propeller blade does not enjoy to the same degree. Finally, the turning that such blading imparts to the flow can be undone by stationary blades behind rotating ones. In order to do that, the compressor would have to be built with two kinds of blades, some that rotate and others that do not. Such a blading combination is called a *stage* and is illustrated in Fig. 10.2, when one looks closely. This arrangement is not unlike the two elements of a radial compressor with its a rotating impeller and a stationary diffuser. The same can be said about the blading combination of a counterrotating propeller.

Unfortunately, in order to fully understand how an axial flow compressor works, we have to draw diagrams that show the changes in speed and direction of the flow.

According to Bernoulli, speed changes that slow the flow result in the pressure increase we seek. A matter that is slightly complicating is that we will have to look at the situation in the reference frame of the blades themselves. In other words, looking at the rotor blades (the rotating ones) we have to look from a rotating viewpoint whereas the goings-on around the stator blades (the stationary ones) can be safely observed in the laboratory.

The flow through the compressor annulus (the space between the inner and outer cases) is helical in nature for reasons that will become clear further on. Commonly made representations of flow through such a compressor as straight-through parallel to the rotational axis involve an excessive use of artistic license and are inaccurate. Air that enters the compressor at 12 o'clock on the face may exit the compression process at 8 o'clock or thereabouts, depending on the design, specifically the compressor pressure ratio. For that reason, the descriptive word "axial" flow compressor is somewhat misleading, but we will stick to it because it is the accepted convention. To set up a helical flow pattern, the incoming axial flow will have to be turned by what is called an *inlet guide vane* (IGV). The first set of (stationary) blades on the engine in Fig. 10.2 are IGVs. This blade set turns the flow and necessarily accelerates it. The pressure will fall through this set of blades. That drop will be made up at the compressor exit where the last stator redirects the flow in the axial direction (in most design cases). Figure 10.3 shows the inlet guide vane flow velocities6 where the axial direction speed is more or less conserved because mass flow rate has to be conserved. The "more or less" words have to do with flow area and density changes that may also be involved in such designs. In some compressor designs, the inlet guide vane is omitted. This omission will be clearer when we look at the fan in that family of engines.

The fan flow is the outer annulus part of the flow that is directed around most of the engine. The fan as a topic will be addressed again after we explore the basic elements of the engine. In the example engine shown in Fig. 10.2, the fan has a single rotor with an *exit guide vane* (visible at the entrance to the bypass duct) to align the flow direction to the rear without any rotating flow component.

The flow situation for the compressor stages that follow the inlet guide vane is illustrated in Fig. 10.4. The rotor turning as indicated accelerates the flow *away* from

6 Velocity is *speed* with a specified *direction*.

**Fig. 10.4** Flow velocity vectors in the laboratory reference frame through a compressor stage. The axial components of all three vectors (not shown) are the mass flow determining flow velocities which are necessarily conserved. The rotor increases the flow velocity (from 1 to 2). The stator restores the flow direction (3) to approximately that of (1). The rotor details are illustrated in Fig. 10.5

the axial direction as shown. The rotor exit velocity in the laboratory reference frame at (2) is greater than at (1). The stator's function is to restore the angle of the flow (3) to roughly the orientation it had before entering the rotor (1). In its absence, the flow would be turned even further away from the axial by a following rotor, an impossible situation if more than a few rotors were employed. Without stators the result would be unacceptably high flow velocities and a limitation of the air throughput. As with a stator, the lengths of axial velocity vectors would also be conserved as with the inlet guide vane. The desired rise in pressure in the stator is realized because the incoming velocity (2) is larger than the outgoing (3).

How does the rotor increase the velocity as indicated? To visualize the velocities in the rotating reference frame, we must add the rotational velocity (red vectors in Fig. 10.5) and examine them in the rotating frame of reference. The green vectors (L1 and L2) in the laboratory reference frame are the same as those shown in Fig. 10.4 (1 and 2). The solid black arrows (R1 and R2) are the velocities in the rotor's frame. They must align with the geometry of the rotor blade at the leading and trailing edges. Proceeding through the rotor blades reduces the relative velocities and the pressure rises. This statement requires proof and fortunately one is available, but not

**Fig. 10.5** Velocity vectors in the blade row of the rotor in its rotating reference frame. R1 is the relative velocity into the blade row and R2 is relative out. Green vectors (L1 and L2) are the laboratory reference frame velocities shown in Fig. 10.4 and the red lines are the rotational speeds for the change of reference

developed here to keep the narrative simple. In sum, pressure rises in both the rotor and the stator.

#### **10.4 How Well?—Efficiency**

The calculation of the pressure rise through a compressor blade row or stage is a simple application of the same energy equation used for determining the pressure on an airfoil, namely Bernoulli or its compressible equivalent. The velocities in play are those relative to the blades. That statement applies to both rotor and stator blading. The air state variables (temperature and density) will also rise as one proceeds from one stage to the next. The degree to which they change is easily related to the changing pressure through what is called a *stage adiabatic efficiency*. Such an efficiency relates the actual property changes to changes associated with an isentropic process, the accomplishment of which is always a design goal. While no one has ever been successful in devising a flow that perfect, the stage efficiency percentage is in practice quite high, reaching and exceeding 90%. It quantifies the modest frictional losses that occur. Its exact value for the compressor as a whole (the so-called *adiabatic compressor efficiency*) is of great interest to the engine performance evaluator. Its value can be gleaned from experimental testing or from detailed examination of the friction losses associated with the flow through the blading.

There is not much to talk about vortices in the flow through an axial flow compressor except that the boundary layers do contribute to rotational flow that ends up as turbulence. Rest assured, however, that the lifting airfoil behaves very much like the wing with a distributed vorticity distribution along the chords of the blades. The Kutta-Joukowski condition applies to fix the flow orientation at the trailing edge and the leading edges of the blades are rounded7 to accommodate a variation in the in-flow conditions.

As with the wing, one can ask only so much of a blade row before the turning or angles of attack are too large for air to follow your geometric imposition on flow direction. The blading of a compressor can stall. Naturally, the compressor would want to be designed to be operated as close to stall as practical to realize as much compression per stage and to keep the number of stages to a minimum. As a matter of record, we note that early on in the history of axial compressor development, the pressure ratio per stage has grown from 1.15 (J47, 1947) to an estimated average of 1.28 (GE 90, 1995). This is no small feat because it dramatically reduced the size and weight of the compressor.

In practice, the number of stages in any compressor is significantly larger than the number of turbine stages because the compressor air moves into a region of

7 In Germany, the need for rounding the leading edges of blades was recognized (in connection with development of the Junkers Jumo 004) with a patent application made in March 1943 by S. H. Decher. The patent titled "Geschwächte Schaufelprofilspitze," A.1808/E57/43 was granted in May 1944.

higher pressure. This adverse pressure gradient leads to the same flow separation issues associated with the wing aerodynamics. The turning that can be asked of each compressor blade is rather small and contrasts dramatically with the large turning angles in a turbine where the pressure falls. Considering that the power produced by the turbine is almost identical to that absorbed by the compressor, blading aerodynamics obviously plays a large role in determining the configuration of a gas turbine engine.

#### **10.5 The Control Problem**

The axial flow compressor operates with blades at an angle of attack to the air streaming through the compressor. The airstream speed and direction relative to the blades are controlled by the speed of rotation of the rotor blades and the engine's air throughput, i.e., the air axial velocity. If that combination of speeds leads to blade stall, the compressor becomes partially or totally nonfunctional. This possibility can occur when the pilot demands rapid changes in thrust. The pilot might therefore be confronted with the delicate operation of the compressor as well as that of the airplane. That is an untenable situation, especially in the military setting of the mid 1940s. An automatic control mechanism was and is necessary to shield the pilot from concern about the compressor operation.

Axial flow compressor development in Great Britain (byMetrovick) initially failed to manage such control and, in flight testing, the axial flow compressor was judged, at the time, to be unworthy of further development because of the excessive workload on the pilot. On the other side, a control system was devised by the German companies Junkers and BMW.8 It was crude but effective. It consisted of an airflow rate control mechanism in the exhaust nozzle that, by measurements linked to it, always held the flow rate through the engine so that angles of attack on the compressor blades were always in a functional position and the compressor avoided stall.

It may be interesting to note that the German development effort initiated to find and produce a good jet engine involved a number of firms that include the two mentioned above as well as Heinkel (where Hans von Ohain was active) and Daimler Benz. The compressors considered included both radial and axial elements as well as hybrid radial/axial devices that spun the outflow, not in the radial direction, but rearward at roughly 45°. Most interesting is that Daimler Benz proposed a configuration that was to be a fanjet. This part of the history is described in greater detail in the bibliography reference entitled "Turbulent Journey" by the present author. When the war tide shifted against Germany, the scope of the jet engine development program was reduced to the projects by Junkers (Jumo 004) and BMW (BMW 003) with only the Junkers effort leading to a production program and the engine's use in primarily

8 A patent (applied for in March 1942) was issued to S. H. Decher for this device in Germany. It was subsequently also granted in the US (after a postwar application) in September 1954 as no. 2688841 A.

two airplanes, the Me-262 fighter and the Ar-234 bomber. The Daimler recognition of the advantages of the turbofan engine was not singular. Back in the mid 1930s, Frank Whittle in Great Britain had recognized the advantages and patented the idea. The difficulties of developing the simpler turbojet were sufficiently challenging that he let the patent lapse.

That German control technology was shared with Allied victors who quickly recognized the value of the axial flow compressor and its advantages. Rather quickly, engineers learned to adapt more sophisticated control systems on the compressor. These successful steps ultimately led to the dominant use of the axial compressor in a large majority of engines. There were exceptions to be sure, but they were few and for special reasons.

Modern turbojet and fan engines employ electronic engine control systems to avoid stall. These systems manage the rate of power lever changes demanded by the pilot so that they are responsive and safe for compressor operation. A mechanical part of such systems may include variable stator angle control on the compressor so that flow angles are always close to optimal. In Fig. 10.4, the flow angle 2–3 is variable to suit such needs. Additional controls are necessary for starting gas turbine engines and these include bleed valves midway in the compressor.

The most common summary of performance of a compressor is on a so-called *compressor map*. It is somewhat akin to a drag polar of a wing and presents most of the important information regarding the characteristics of a compressor. It consists of a plot of pressure ratio versus air weight flow rate. Lines of constant rotational speed and constant efficiency are noted. The efficiency will have a maximum value at some point. Also shown is the line along which the compressor stalls. Stall is encountered at low air flow rates. The control system's function is to keep the point of operation in the region of highest efficiency and away from stall. Air weight flow rate and rotational speed (usually in RPM) are corrected to reflect inlet conditions to make the map as compact as possible. The corrections reflect inlet conditions that differ from those of a standard day, typically defined for aeronautical purposes as 29.92 in/Hg (14.7 psia or 101 kPa) and 15 °C (59 °F).

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Chapter 11 Bypass and Other Engines**

#### **11.1 The Fan on a Turbofan Engine Has No IGV**

Before leaving the subject of velocity diagrams in the compression parts of the engine, the reader may be curious about the flow through a fan that, in practice with high bypass engines, has no inlet guide vanes. The flow geometry is quite similar to that through a propeller as described in Chap. 8. The caption of Fig. 11.1 describes the diagram with flow that exits the stage in the axial direction imposed by the *exit guide vane*. A sketch of the flow through a fan of a modern turbofan engine is shown the figure. The comments about the helical nature of the flow through an "axial" flow compressor also apply to the fan, even with no inlet guide vane. The sketch shows the angular component of the rotor outflow in the laboratory reference frame (displayed as L2). The angular displacement of the stream tubes is much smaller than in a compressor because there is only one stage.

A couple of things about the engine in Fig. 11.2 are worth observing. First, the engine has some resemblance to a propeller driven system with the engine behind the fan as the power source. This and all high bypass engines have done away with the inlet guide vane and located the rotor to face the stream as the first blading element. Strong differences that set this engine type apart from the propeller with its ICE, however, are apparent. Among these is that the power source is the gas turbine in contrast to the ICE of yesteryear. That allowed power levels more than fifty times larger than what the largest reciprocating ICEs ever could. The fan is shrouded so that it can operate in an airstream that has been slowed by an inlet (not shown here) and there is an exit guide vane. Efficiency levels of this engine type are such that non-stop airliners can operate over distances covering half the globe. The post war ICE powered airliner was challenged to cross the Atlantic!

**Fig. 11.1** Sketch: Flow velocity vectors through a fan without IGV. Flow is from left to right with vectors R1 and R2 relative to the moving blade. The laboratory frame rotor exit velocity (L2) is redirected in the axial direction (L3) by the exit guide vane. Image: Rear view of an engine fan shows the exit guide vanes (GE90 engine display at the Museum of Flight, Seattle)

**Fig. 11.2** A modern turbofan engine: GE90-115 (mid 1990s) contrasted to a radial Wright 9 cylinder Whirlwind on a Ford Trimotor (1929); not to the same scale (Courtesy General Electric and agefotostock: Christopher L. Smith)

#### **11.2 Bypass Ratio**

The two fan jet engines pictured in Figs. 10.2 (1960) and 11.2 (1993) are embodiments of the goal to ever better propulsion efficiency by the reduction of the (average) jet velocity in favor of increased air flow rate. The amount of air bypassing the prime mover engine is expressed as a multiple of the air flow rate processed by the engine, i.e., the compressor, the combustor, and the turbine. Early in the age of turbofans the bypass ratio was around 1 (meaning fan and engine flow rates were about equal) and later 2. The widebody jets introduced bypass ratio of 5 and current airplanes may sport this ratio around 10. These increases in bypass ratio have very large impacts on fuel usage because of the improved propulsive efficiency.

The large fan in high bypass engines makes it imperative to consider its design for aerodynamic performance and, importantly, for weight. The early engines of this type, like the Pratt & Whitney JT9D powering the Boeing 747, had a large number (46) of high aspect ratio blades, meaning they had a relatively short chord. These blades had "clappers" between the blades to stiffen them and prevent harmonic vibrations from causing damage or failure. The blades were made of titanium alloys. Attempts to reduce the weight of the fan include the first design involving composite blades made with a metal leading edge to minimize foreign object damage by such events as bird strikes. These blades had a wider chord and were fewer in number. The General Electric GE90-115B has 22 such blades with composite materials (carbon fiber) at the rear of the blades (see Fig. 11.2). Another approach to weight reduction is the use of hollow all-metal, wide chord, blades. This was the approach taken by P&W in its upgrade of the JT9D (renumbered as PW 4000). The challenge of building a light-weight fan did not always go smoothly or without serious consequences, but it was met and serves well in modern fanjet engines. These blades are a tribute to the art and capabilities of modern aerodynamicists and engineers more generally.

#### **11.3 Turboprops and Turboshaft Engines**

The idea of the bypass turbofan is now fully implemented with fan engines used everywhere for commercial transport. An extension of the turbofan idea is the *turboprop* where nearly 'all' of the power is removed from the primary gas turbine engine flow to drive a gear box and with it a propeller. The effective bypass ratio is very large (larger than that of the ducted fan) but, as with an ICE powered airplane, the propeller sets an airplane speed limit.

Naturally, with an engine delivering its power by means of a rotating shaft (a *turboshaft* engine), all kinds of other applications can be imagined. Among them are helicopters, ships, etc. The success of the turboprop engines was very much due to their greater power output capability than were the piston ICEs. They also provided the user with engines that were much easier to maintain in service and could operate much longer between needs for service.

A few words about differences between gas turbine engines and the ICE will clarify the former's great advantage in aviation. The gas turbine is lighter as measured by power or thrust per pound of engine. Typically, at sea level conditions, 5–6 lbs of thrust or horsepower are produced by each pound of a gas turbine engine weight. This is quite an improvement over the ICE at 1 hp/lb where even that level is achieved only for fairly demanding circumstances such as those set for military aircraft.

The gas turbine uses a lower cost "jet fuel", similar to kerosene, rather than aviation gasoline required by the ICE. The lower volatility of the kerosene-like jet fuel also makes it safer.

Finally, we can say that the gas turbine with very few moving parts, and none of them reciprocating, does not require the mechanical maintenance of the ICE. To wit, the P&W R-4360 engine used in the Boeing Model 377 Stratocruiser airliner was fortunate if it lasted longer that 500 h between major overhauls. Allied bombers in WW II routinely replaced entire engines every 100 flight hours. These overhauls were costly in terms of labor and facilities and demanded an inventory of ready-to-go engines in case of an engine failure on an airplane that just came in. In commercial service, engine shutdowns were not uncommon, especially in the summer months. By contrast, the need for extensive overhauls of modern turbofan engines is in excess of 20 or 30 thousand (!) hours.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Chapter 12 Other Components of the Jet Engine**

#### **12.1 The Turbine**

The turbine is the source of all the mechanical power in the engine. It can also be configured in radial or axial inflow configurations. An example of the former is similar to a child's pinwheel. Only Hans von Ohain used such a turbine in the first jet engine that flew in the Heinkel 178. Von Ohain and everybody else understood that the better way to go was to use an axial turbine, adapting the knowledge from steam turbines for power generation that had been in commercial service for over a decade. The radial turbine is now just a footnote, although one can find it used in automotive turbochargers where operating efficiency requirements are not as strict as they would be for an aircraft application (Fig. 12.1).

The description of the function of an axial turbine parallels that of the compressor. Rotors and stators are involved. The first element is stationary and is called a *nozzle*. This nozzle gives the flow kinetic energy in the azimuthal (rotation) direction which the rotor that follows it can harvest. The gas exiting the nozzle may be supersonic. The relative velocity incoming to the rotor will be lower than the exit velocity from the nozzle because the turbine rotor blades are receding away from it.

In contrast to the compressor where the flow in its stages is adequately describable as locally incompressible, the turbine flow is definitely in the compressible regime. The blading operates with a favorable pressure gradient so that large flow turning angles can be withstood without concern for stall. This aspect of the difference between compressors and turbine explains the physical difference between these components. Figures 10.2 (and 12.3) shows that the power transmitted from the turbine to the compressor is generated with relatively few turbine stages while the compressor absorbs the same power as input but must process the air gently (against an adverse pressure) to avoid the stall issue. To be specific, the P&W JT8D engine in Fig. 10.2 has a 1-stage fan and 13 (6 + 7) stages of compression, all driven by 4 (1 + 3) stages in the turbine. The contrast between compressor and turbine blading is starkly illustrated when we note that a single turbine blade row in the high-pressure turbine can power six stages of the compressor.

**Fig. 12.1** A cutaway of an automotive turbocharger. Ambient air is drawn in from the left and exits in the spiral diffuser (blue). The red zone is the radial inflow (exhaust from the reciprocating engine) to the turbine wheel. It is relatively simple and performs adequately well (Photo: Quentin Schwinn (NASA))

The image of turbine nozzles and blades with their cooling air bleed holes (Fig. 12.2) suggests that the compressor feeds more than just the combustor. It also provides the cooling air for the turbine. The word "cooling" is a bit curious because the compressor air is, in fact, hot. It is, however, cooler than the combustor exit air and thus effective as a coolant. It must also be at the highest possible pressure to be able to enter the turbine flow through the many small holes. Identifying an optimum amount of cooling air from the compressor has to consider the positive aspect of operating at effectively higher turbine inlet temperature against the compressor power used to provide the cooling air. In modern engines, the cooling bleed is in the range of 5–15% of the air through the compressor.

Looking back at the ICE that also employs cooling air to maintain the temperatures of the cylinders, we note that the fuel energy transferred to the cooling air in that engine is lost for propulsion purposes. The gas turbine, by contrast, loses very little fuel energy heat as a pure loss. The mass of cooling air stays in the flow so that both heat and mass end up in the propulsion jet and contribute to the thrust produced.

#### **12.2 The Combustor**

We have largely left the topic of vortices to describe how a compressor and turbine work. To first order, vorticity is not a major issue in describing the workings of a jet

**Fig. 12.2** Two blades of an aircraft gas turbine nozzle (one blade is covered by yellow tape) and rotor in the shop. This display is set up for training purposes. The rotor blades on the wheel to the left of the nozzle block are covered by orange plastic to protect the students. To the right of the nozzle element, the rotor blades are naked and turn the flow through a rather large angle. The single block element of the nozzle is held in place with a plastic tie. Note the heavy use of cooling air holes to bathe the nozzle material in cool air and minimize contact with the hot combustion gas. The second turbine stage is also shown as two stator sections (4 blades) (Courtesy General Electric, photo by author)

**Fig. 12.3** A P&W J57 (or JT3C) in sectioned model form showing clearly the flow area expansion at the end of the compressor. The model illustrates the late 1950s technology of combustor design and the two compressor segments: low- and high-pressure elements powered by separate sections of the turbine with a connection by two shafts, inside one another. The artifact is located at the Wings Over the Rockies Air and Space Museum, Denver CO (cropped photo by Ryan Frost, https://com mons.wikimedia.org/wiki/File:Sectioned\_Pratt\_%26\_Whitney\_J57.jpg)

engine except to say that the bound vortices on the blades of both compressor and turbine must be in play. In the combustor, the vortex will be seen to play an important role. First, however, a look at the functionality of this component.

Combustion in a jet engine or better, a gas turbine, takes place at nearly constant pressure. The combustion process there takes place just as it does in a candle, the fireplace, or the burner of a hot air balloon. The consequence of burning fuel at constant pressure is that the air involved is heated and, being a gas, it expands proportionately in volume. The ideal gas law, *pv* = *RT*, is in play here. The resulting lower density (= 1/ν) causes the heated air to rise in the fireplace chimney or, when contained by it, causes the hot air balloon to experience a net buoyancy and rise in the cooler atmosphere.

In reality, the (total and static) pressure in the combustion chamber of a jet engine falls a small amount for two reasons: pressure losses associated with the flow through the hardware and the process of adding heat to a flowing medium always leads to a loss in pressure. Any pressure drop is certainly to be avoided because the compressor had to provide it. From a technical viewpoint, one can show that this loss is minimized if the Mach number of the flow to which heat is added is kept small. This is done in a jet engine design by forcing the flow to slow with an area increase as it enters the chamber (see Figs. 12.3 and 12.4).

In the chamber itself, three major challenges are presented to the designer. These are, first, reducing the volume of the space necessary to carry out combustion to a minimum. Specifically, the length of the combustor must also allow the combustion

**Fig. 12.4** Simplified sketch of a gas turbine combustor showing the swirler and other features and functions. The purpose of the swirler is to form a rotating mass of burning gas with helical vorticity on its outer edges. Simultaneously, the two illustrative rotational cells serve to continuously ignite the fresh air charge by bringing burning fuel forward (https://en.wikipedia.org: file: combustor diagram airflow.png by SidewinderX). The lower sketch is a simplified illustration of the changes in gas flow properties (total temperature in red and total pressure in solid green; static pressure in dashed green) that occur in a typical combustor that, in reality, is quite three-dimensional in nature

process to be complete. This is a problem because flame propagation speeds are very much lower than the flow speed through the chamber even when the Mach number there is held low. Here the vortex comes to the rescue. At the burner inlet one or more vortices are formed so that the swirling motion lengthens the residence time of the burning gas. This usually done by swirling the flow with an array that resembles an inlet guide vane (see Fig. 12.4a). The goal is to avoid forming soot, unburned hydrocarbons and simultaneously avoid forming pollutants such a carbon monoxide and nitrogen oxides. The second challenge is to produce a gas *uniformly* high enough in temperature acceptable to the turbine without itself being at risk of material failure due to overheating material surfaces or chemical oxidation. The uniform aspects have to do with the desire to avoid subjecting the blades to a variation in temperature that is hot and hotter at a high frequency. Lastly, the combustion must be stable in that it is, once lit, continuous and free of needing attention.

In practice, the burner of a gas turbine engine is operated as a combination of two processes: primary combustion where the chemistry is such that most oxygen is burned to achieve a stable and very hot flame and a secondary zone where additional air dilutes the combustion gas to a temperature that the turbine can tolerate. Figure 12.4 is a schematic representation of the process.

The lower part of the figure is a simplified sketch of the variation of the important flow parameters through the combustor. The red line shows a one-dimensional (averaged over the available flow area) variation of the total temperature, noting specifically the very hot primary combustion zone and the subsequent cooling by dilution with cooling air.

Also shown in the sketch is the variation of total and static pressures. The total pressure falls slightly during the process of heat addition to the air. The relative value of the static pressure is a reflection of the local Mach number. In diffusion, the total pressure stays constant while the static pressure rises because of the flow area increase. In the burner, the averaged flow behavior is for the Mach number to increase (toward thermal choking but not coming close to that) so that the net result is a decreasing (average) static pressure. The area decrease in the aft half of the combustor as shown in the upper sketch is an additional reason for the static pressure to decrease in the flow direction. While complicated, the design of the combustor mercifully involves subsonic, primarily low Mach number aerodynamics.

One cannot speak about the combustor without mentioning its pivotal role between the compressor and the turbine. The amount of fuel added to the flow must be just the right amount to raise the air temperature from that exiting the compressor to that acceptable by the turbine. The compressor air is hot from the compression work that went into it. Here is where the compression ratio of the compressor really matters.

In any functioning gas turbine engine, the amount of fuel added for combustion is limited and there is always more air processed than the amount necessary to burn the fuel. The excess oxygen flowing through the turbine has two consequences: the environment for the turbine is an oxidizing one that has the potential for setting the stage for material failure due to oxidation of the hardware in the burner and in the turbine. Good design learned from experience in the fields of metallurgy and cooling technique have largely put this concern to rest. The other consequence is that the

**Fig. 12.5** Left: The voluminous combustion chamber on an early British engine, the Whittle W1 on display at the Smithsonian National Air and Space Museum. At right, the burner cans on a GE J47, an early (1947) single shaft, axial flow compressor engine (its turbine is at right under the plastic), on display at General Electric

left-over oxygen can be burned in an afterburner, i.e., in another combustor behind the turbine to obtain thrust for applications and situations where it is needed. An afterburner was used in the supersonic Concorde airliner to achieve cruising speed although not used during cruise. It is used primarily in military fighter jets.

Considering the progress made starting from the early engines of the 1940s to today's reliable machines, the goals in the combustor have largely been brought to a high degree of perfection. Included in the list of desired goals is the need to minimize undesirable combustion product emissions into the atmosphere. The burners have been so reduced in volume that they are hardly discernable as a component of the engine. Failures in operating engines are seldom attributable to the combustor, except perhaps when maintenance is given a short shrift. The combustion chambers and the region around the turbine entry are, therefore, the subject of scrutiny by the people who maintain jet engines.

Figure 12.5 also illustrates the need to keep the combustion chamber short. A long chamber will involve a long rotating shaft connecting the turbine to the compressor. If the shaft is insufficiently stiff in bending, it will be unstable. In the J47 of Fig. 12.5, engineers dealt with this problem by installing a bearing in the middle of the shaft.

In practice, the combustion chambers of a gas turbine are configured in one of two arrangements: straight-through flow or reverse flow. In most modern engines, the ability to achieve combustion in a small volume has led combustors to the use of the straight variety. The various engine cross-sections in this text illustrate this arrangement. Examples of the reverse flow configuration are shown in Figs. 10.2 (I-16) and 12.5 (Whittle W.1A). The reverse flow combustor allows for a short shaft between turbine and compressor.

#### **12.3 Putting It Together into an Engine**

How do the jet engine and its components work to produce thrust? We mentioned that the compressor and turbine function very efficiently. If the component efficiencies were an impossible 100%, then a compressor and turbine hooked together and could function without any consequence to the universe. They aren't, of course, so we have to pay for what they do. How could we do this? … and the answer is: by giving the turbine a higher *volume flow rate* to process. The only practical way to do that is to heat the gas prior to turbine entry so that it expands in volume—again, like the flue gas in the chimney.

We could start by adding just the right amount of heat to the air entering the turbine so that the losses in both components (compressor and turbine) are overcome and the engine operation is self-sustaining. That achievement probably brought the first reason for celebration to the pioneers Whittle and von Ohain! We could then add more heat. As the engine (of a fixed geometry) runs, the air mass flow rate through the engine must be conserved. It depends only on the product of flow velocity and air density. That means for the turbine that as the density is lowered by heating, the velocity must rise at every locale. In turn, the forces acting on the airfoils must increase because they depend on the *dynamic pressure* which depends on density and velocity squared. Heating the gas increases the forces, torque, and power from the turbine!

Whereas our ideal or self-sustaining operation left the outlet pressure behind the turbine at an atmospheric level, a greater heat supplied increases of the pressure at the turbine *outlet* because not all the pressure drop through it is needed to run the compressor. There is pressure left over. The engine becomes a pump: it processes air and delivers it at a higher pressure! With the addition of a nozzle, we can create a jet with (more) momentum to get thrust! Hot dog! More thrust is obtained by adding more heat up to the temperature limit that the turbine can tolerate.

The same physical argument about component entry temperature can be used (in reverse) to talk about the compressor where the work (i.e., the power) required for compression is also temperature dependent. Thus, if the compressor inlet air temperature is lowered, the power required is also lowered. One may conclude that the engine would best operated between temperature extremes that are as wide as possible. This is one reason that airplanes with jet engines operate so well in the cold stratosphere, where the temperature is around - 60 °F. Conversely there might be difficulties associated with operating an engine on very hot days when performance will be suboptimal.

The idea that wide temperature extremes are best for extracting mechanical work from a heat engine is a conclusion that the Frenchman Sadi Carnot (1796–1832) reached when he investigated the convertibility of heat to work generally. His examination of the performance of what we call heat engines laid the foundations for the Second Law of Thermodynamics that involved the identification of energy (through the First Law) and entropy as properties, the concept of reversibility, the notion of temperature, the role of the temperature extremes that apply to heat engines and limit their performance. His studies launched a great deal of understanding of the workings of nature and that knowledge fueled an important second period of the Industrial Revolution. The first having been centered on the use of the steam engine.

A truth that follows from the work of Carnot and others is that a perfect heat engine (a Carnot engine) can be imagined, although building it is not practical and may not be possible. The gas turbine engine, like all engines, is limited by the temperature extremes available. Specifically, the successful generation of mechanical power by means of any *heat engine* depends on heat from a high temperature source (or reservoir, as thermodynamicists like to say) intercepted and converted as it flows to a low temperature reservoir (a heat sink). Without *both* of these reservoirs, nothing useful can be made to happen. Consider, for example, that the heat engine that is planet earth and the life on it operates between input from the sun (~ 5800 K or some large number of degrees F) and radiates the waste energy to the near zero absolute temperature of space (~ 3 K). Consider also that a uniformly hot environment is not a place where much can happen. Is that where the notion of 'hell' being hot originated?

Enough of philosophical wanderings and let's get back to our gas turbine engine! The arguments made above concerning the working of a gas turbine engine can be made much more precise using an expanded version of the conservation of energy statement we have used so far to include mechanical work (compressor and turbine) and heat (combustor). The books referenced in the bibliography should be helpful in the quantification of the performance of the engine. The analysis is rather simple in that the relations are algebraic and don't involve much higher mathematics. The takeaway from such analysis is that the temperature extremes involving the maximum (turbine inlet) and minimum (compressor inlet) play a dominant role in the performance.

Finally, is this new engine any good? It is, apparently, a very successful engine to propel airplanes. As a postscript to our description of the gas turbine, it merits callout to some of the features that make it different from the ICE we know so well. The jet engine is a thrust, rather than a power, producing engine. Both engine types use liquid fuels, albeit slightly different kinds. For flight, using a *liquid* fuel is extremely important because of the ease of storing it onboard and handling it. Just contemplate for a moment the need for shoveling a solid fuel like coal on the ships of yesteryear or the steam locomotive. The difficulties involved in storing gaseous fuel under pressure in an airplane where payload volume is very dear have been considered and are not attractive. Aviation had to wait for the invention of the ICE, in part, for that reason.

There are important technical differences between the engines that say a lot about where and how they are used. The ICE works well and efficiently at power levels less than maximum. The gas turbine does not. The gas turbine works most efficiently at full thrust (or power) which is just what is needed for an airplane in cruise. The ICE works very well in automobiles where it hardly ever runs at full power. The demands of traffic insist on that.

In considering the turboshaft gas turbine for small applications, say under 500 hp, (about 400 kW) the boundary layers on the walls of the various flow components become a larger proportion of the total flow, meaning that friction is proportionally more important. The associated reduction in component efficiencies and the total pressure losses become important contributors to increased fuel consumption (per horsepower). The gas turbine is not easily well adapted in small sizes with high efficiency as a goal.

Finally, an important operational difference is that the turbomachinery, the compressor and turbine, has a large inertia invested in rotation. That will result in a slower response to power demand changes when compared to a similar requirement made of the ICE. The slower temporal response has consequences in the military setting or when emergencies are encountered.

#### **12.4 How Do You Start This Thing?**

When we examined the ideal engine consisting of perfect (thermodynamically reversible) compressor and turbine, we implied that all we had to do is spin it up to operating speed and then let it run on its own. That is indeed what has to be done to a real engine. Gas turbine engines typically have connections to a device that has enough power to raise the speed to the point where the engine operation is selfsustaining. The power for an electric motor may be provided by a battery, an auxiliary power unit (an APU), or from a power cart on the ground. Typically, commercial airliner engines are started by an air turbine supplied by an APU. Another source could be air from a storage source or, as it has been done, generated by burning a charge of solid rocket propellant-like material to generate the gas flow. There are interesting circumstances under which such means were or are used but they are quite uncommon. In flight, an engine may "flame out" or stop working and needs to be restarted. In some circumstances, the airplane's speed and altitude may be exploited to have the flight airstream turn the engine over. It is hard to imagine that the circumstances that call for such action are benign.

#### **12.5 Bleed Valves and Variable Stator Geometry**

There is another aspect of starting a jet engine that is interesting. Consider the operation of an axial compressor. For a running situation, the blade height and flow passage widths are very much smaller at the outlet end of the compressor than at its inlet. The reason is that the increasing density associated increasing pressure as we look further down the compressor path is such that a smaller flow area is required at the high-pressure end to pass the mass flow being processed. During starting, that high density is not yet achieved but the mass flow forced in by the inlet must exit at the outlet. That implies that, initially, the flow velocities near the rear are very high, potentially high enough to "choke" the flow by reaching sonic speeds. If and when that happens the increasing flow demanded by the front of the compressor cannot be accommodated and the process is stuck; unless…

**Fig. 12.6** External view of a small gas turbine compressor with variable geometry stators. The five rings are attached to levers that change the orientation of the stators mounted on shafts through the case. The hydraulic lines and the associated cylinder are used to fix the stator orientation

There are two aspects of the design of a modern gas turbine engine that are used to prevent this situation from being problematic. The first is to allow a bleed from the midsection of a compressor to remove partially compressed air from that location so that flow to the rear blockage is reduced.

A second aspect that deals with starting is that in some compressor designs, the stators can be made to change the angles they present to the flow. Thus, they can be oriented to minimize blockage during startup. These two solutions can be employed in concert when they are designed to do so. The pilot sitting in the cockpit's left seat is ignorant of all this because there is an engine control system that takes the necessary measurements and, with actuators of various kinds, does all that is necessary without burdening the pilot with its doings (Fig. 12.6).

Finally, the starting process is eased by the configuration of the engine when it has more than one shaft. Specifically, the starting process is initiated with the high-pressure components, the high-pressure compressor, burner and turbine, while the low-pressure components wait for things to happen on the inner portion of the engine. After all, the high-pressure section of a multi-spool gas turbine is an engine within an engine! When the high-pressure section is running in a sustained way, the low-pressure components follow suit and the entirety springs to life.

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## **Chapter 13 More Components: Inlets, Mixers, and Nozzles**

#### **13.1 Inlets**

We return to vortices to see where they play important roles in the operation of a jet engine. One aspect of an engine installation design is reflective of the comments made on the wing design. Recall that a nice rounded leading edge on the wing is desirable to allow for operation over a wide range of angles of attack. That same thinking also applies to the inlet that has to handle air nicely in flight and on the ground where the airplane is almost stationary. At low speed, the stagnation point (line) on the inlet air flow is located on the outside of the nacelle. Near a nacelle's maximum width, the curious person will find a red line and words to the effect: "don't stand forward of this line during engine runups on the ground, or else"! The point is to warn persons not to participate in the suction exercised by the engine when it is running on the ground. The roundedness on the inlet parallels the design of a wing at its leading edge. The inlet must manage inlet flow to be uniform in low-speed flight, specifically at high angle of attack or with a cross wind on the runway. The rounded lip allows the entering air flow entry to be relatively uniform. At cruise speed the stagnation line is almost coincident with the foremost *highlight* of the nacelle, its leading edge if you will. The two stagnation streamlines are shown in Fig. 13.1.

#### *13.1.1 An Old Wives' Tale About Inlets*

Descriptions of how a jet engine works are sometimes said to involve the notion that the engine "sucks" itself along to propel the airplane. The reality for an airplane in flight is that nothing could be further from what actually happens. In flight, the purpose of the inlet is to slow the freestream. In this way, the inlet performs part of the compression process. There is no suction involved in its function in a cruise flight condition. The question really has to be centered on the pressure at the engine

**Fig. 13.1** Sketch of the flow pattern in and around a subsonic airplane inlet at low speed on takeoff (green streamlines) and at cruise (red). Dashed lines are the stagnation streamlines. The \* denotes the inlet throat area. Under all conditions, the flow Mach number at the engine face is about 0.5

inlet face. That, in turn, has to do with the flow speed, i.e., the Mach number in that location.

For the sake of argument, one can say that the flow Mach number at the engine face, be that a fanjet or a turbojet is about 0.5. The airflow around the blades requires that under most conditions. Subsonic flight inlet performance is close to ideal, meaning the effect of total pressure losses are minimal. One can say that the flow from the outside world where the pressure is the *static pressure* to the engine face always preserves the total pressure associated with the flight speed.

At takeoff conditions, the static pressure at the engine face (local *M* = 0.5) is about 20% lower than the outside (where the local Mach number is near zero), hence air is indeed sucked in. However, during flight, say at *M* = 0.8, the conservation of total pressure will *increase* the static pressure at the engine face because the inlet *decreases* the flow speed. Under such conditions, the pressure at the engine face is about 30% *higher*—no suction here! At flight conditions like those of a Concorde (*M* = 2) or the SR-71 (*M* = 3.2), the pressures at the engine face are about 7 or 40 (respectively) times the static pressure outside, even though these inlets suffer some (minimized by design) loss in total pressure because of the shock waves in the inlet. Certainly, no suction there either! On a supersonic airplane, the inlet does a significant portion of the engine's necessary compression. On a ramjet engine that is necessarily flown at supersonic speeds, the inlet does *all* the air compression and the use of rotating machinery is dispensed with altogether in such an engine.

The proper way to attribute performance contribution by the inlet relative to the engine is to discuss the amount of compression done by the inlet and compare that to the amount done by the compressor. One cannot talk about the relative contributions to thrust because inlet, engine, and nozzle work together as a system. It would be a little like talking about the contribution to a runner's athletic performance by his liver!

In flight, the higher pressures on the inlet do raise hardware design issues. The associated forces have to be taken into account for the mounting hardware that is used to affix the inlet to the engine. Insufficiently strong bolts will result in a forward departure of the inlet and the likely end of proper engine operation. The same pressure

**Fig. 13.2** Forces on ancillary engine components. At the engine faces (front and rear) the pressures are higher than outside, hence the bolts holding the inlet and nozzle onto the engine are in tension

forces also play similar roles at the nozzle. High inside pressure and low outside pressure there also require a strong connection between nozzle and engine. The forces associated with that of the restraining bolts have nothing to say about engine thrust (Fig. 13.2).

The DC-10 inlet in Fig. 13.3 is of a typical design for commercial airliner. The internal surfaces of the inlet handily serve as a sound absorbing liner to reduce noise emissions. The many small holes and the cavities behind them are designed to absorb the noise frequencies associated with the blading behind the inlet. The figure also illustrates the use of a vortex generator on the shoulders of the nacelle to improve the flow over the wing that might have stall issues due to the nacelle 'shadowing' the wing at high angles of attack. This vortex generator (there may be two of them in some applications, humorously referred to as "rabbit ears" but technically called a 'strake') is very effective at low speed and aerodynamically close to absent in cruise when the external flow aligns itself closely with that of the 'ears'. In some instances,

**Fig. 13.3** The inlet of a DC-10 airliner wing mounted engine nacelle. One of the two "rabbit ear" vortex generators is located at the 11 o'clock position of the cowling

**Fig. 13.4** A LAN (Argentina) Boeing 767 taking off from Los Angeles. The white "cloud" just inside the inlet is water vapor condensation at the minimum flow area where static temperatures are lowest and, in this case, below the dew point (Photo courtesy Werner Horvath, Airliners.net)

the strake may also be made effective at improving the flow in the space on either side of the strut and under the wing.

On a subsonic inlet like that shown in Fig. 13.3, the highest local flow velocity is always close to the location where the flow area is a minimum. This area is called the *throat*. The flow velocity there is close to sonic but nominally subsonic. During operation at takeoff, the velocities there are high enough that the local static air temperature can drop below the dew point so that condensation takes place, just as it may on the upper surface of the wing. Figure 13.4 shows an airliner taking off under such conditions. There is little performance impact by this condensation!

#### **13.2 Inlet Geometry and Diverters**

On airplanes capable of supersonic flight, the inlet is more complicated. Total pressure recovery, i.e., minimal loss of the total pressure available, is an important design goal. This is a concern from two aspects: total pressure losses through shock waves and flow uniformity if the inlet is located along the fuselage of the airplane. The inlet to a *M* = 2.2 airplane (Fig. 13.5, a McDonnell F-4) embodies features that deal with these concerns.

**Fig. 13.5** A McDonnell F-4 in flight showing the shock generation ramp on the inlet and the boundary layer diverter between the inlet and the fuselage (Picture credit: USAF—holloman.af.mil, curid = 50920146). At right is the Museum of Flight display showing the diverter head-on. Note the small diverter on the scoop at lower right and the inlet instrument probe

The first order is to minimize the losses associated with shock waves. Total pressure losses are minimized when the flow can be made to slow through a series of oblique shock waves rather than a single normal1 shock. The flow entering the engines may or may not have gone through the shock wave generated by the nose of the airplane. Typically, a wave is created by a ramp that reduces the flow Mach number by turning the air flow slightly outward. On each engine, the F-4 has such a flow deflecting ramp adjacent to the fuselage. In general, the inlet compression process is improved when the flow is turned by more than one ramp. This is usually employed for airplanes where operating efficiency is at a premium. The Concorde is an example, Fig. 13.6. In that example, the shock generated on the underside of the wing is the first wave encountered by the airflow to the engines.

In the F-4 image, the ramp is not on the body but offset by a few inches. That offset is to address the second aspect of the design of a supersonic inlet: avoid ingestion of the air slowed by the boundary layer on the airplane's body.

The inlet of the Concorde supersonic airliner is illustrated in a takeoff configuration (photo) and in cruise flight condition (sketch) of Fig. 13.6. There are three wave generating ramps with two hinged elements for each inlet. In the photograph, these are fully retracted so that maximum airflow is admitted as, for example, during takeoff. The keen observer will note doors on the underside of the inlet to discharge excess air or admit additional air when necessary, notably during takeoff. The sketch shows schematically the array of oblique shocks generated by the three deflection surfaces in a cruise configuration. A normal shock will conclude the diffusion process of the supersonic flow. Further slowing of the air takes place in a diverging duct much like that in a subsonic transport engine.

1 A normal shock reduces a supersonic flow to subsonic in one (irreversible) step, see the shadowgraph in Fig. 7.5.

**Fig. 13.6** The two-dimensional inlet of the Concorde supersonic airliner. The wave generating ramps (there are two hinged elements for each inlet) are fully retracted in this picture so that maximum airflow is admitted as, for example, during takeoff. The ramp has the word "DANGER" noted. Note also the diversion of the boundary layer air between the top of the inlet and the wing. The boundary layer is thicker on the inboard side of the inlet, hence a wider gap. The boundary layer air from the underside of the delta wing is dumped to the side of the inlets

The geometric variability together with the need to control the shock boundary layer interactions required for an inlet designed for supersonic flight is very complicated and involves a costly part of the design process of the propulsion system. This description of the essential features of such inlets is necessarily quite simplified. The reader may wish to consult more detailed descriptions of such inlets, specifically those for supersonic military bombers.

The inlet on a Lockheed F-104 (*M* = 2) employs (half) an axisymmetric (conical) shock wave to slow the flow (Fig. 13.7). The boundary layer air is also diverted on

**Fig. 13.7** Axisymmetric inlets for supersonic aircraft, F-104 and SR-71. In the lower right background of the SR-71 is a display of a P&W R4360 with its 28 cylinders (Photos taken by the author at the Museum of Flight in Seattle)

that airplane. A good example of a whole axisymmetric inlet is that of the SR-71 (*M* = 3.2) where a conical shock wave is generated on the sharply pointed cone. From there the air enters the internal portion of the inlet and proceeds to encounter a number of relatively weak shock waves. This inlet is also designed to remove the boundary layer air on the internal inlet surfaces by suction on these surfaces to minimize the deleterious effects of shock/boundary layer interactions. These can cause flow separation from the wall much like those discussed in connection with transonic drag rise for a wing.

In all cases, inlets will complete the diffusion process with a divergent flow section where the near sonic flow is slowed to near *M* = 0.5 that the engine's compressor blades can handle.

#### **13.3 Mixing**

Internal flow mixing in an engine was carried out in many turbofan engine configurations of modest bypass. The Pratt & Whitney JT8D shown in Fig. 10.2 is a good example. The mixer can be seen at the rear of the engine where fan air and engine core exhaust are mixed. The idea is to create a single, uniform jet stream for expansion by a common nozzle. This design is a follow-on from the earlier P&W J57 enginebased JT3D low bypass engine where the fan flow was directed out of the engine via a separate exit nozzle. The performance of mixed and separate flow configurations is quite similar, except that the mixed flow engine has lower noise emissions. A view of a mixer for a high bypass engine from the rear is shown in Fig. 13.8. The idea is to create a large contact area between the two flows with a wide vortex sheet between them so that mixing takes place rapidly. The differential speeds of the mixing flows generate the vortex sheet that allows the mixing.

Modern high-bypass and very-high-bypass engines may or may not sport fan/primary flow mixers as the flow rates are quite disparate. The performance advantage gained has to be balanced against the weight penalty of the associated hardware. An example of a separate flow high-bypass nozzle configuration is shown in Fig. 13.10.

A substantial amount of mixing of very hot and cooler flows also takes place in the combustor to achieve acceptable turbine inlet temperatures as described in Chap. 12.

#### **13.4 The Nozzle**

The last element of a jet engine is the nozzle where the high-pressure gas produced by the pump that is the engine, expands to atmospheric pressure and in the process, creates the propulsive jet. In its simplest form, a nozzle is a duct of decreasing flow area (a convergent nozzle). This geometry is universally used for turbofan engines where the flow exits the engine with a pressure that is not sufficient to drive the

**Fig. 13.8** An internal mixer for engine core flow emanating from "behind" the round central cone with the fan flow originating in the surrounding darker space (Photo by author at the service education facility of General Electric in Evendale, Ohio)

jet to sonic speed. A ratio of total to static pressure (outside) on the order of 1.9 is required to do that. In engines for commercial airliners, the *nozzle pressure ratio*, as this number is called, is kept to lower values because of the need to avoid the wave generation by supersonic turbulent parts of the flow. This aspect of the turbulent mixing with the freestream is the source of much jet noise in high-speed jets. In fact, in the early *turbojet* transports, the supersonic flow noise was so severe that extraordinary means had to the employed to minimize it. The approach used then was to force rapid mixing with the freestream air by external *daisy petal mixers* (Fig. 13.9) and similar devices. Their use imposed a thrust and fuel consumption penalty that had to be borne. The advent of the turbofan obviated the use of such mixers because the jet total pressure was significantly lower.

The older reader may remember listening to the roar of a 1970s commercial jet taking off. The crackle in the tonal mix was from the supersonic cells interacting with the environment. Some business jets may still be equipped with engines with the higher nozzle pressure ratio and their noise may be enjoyed by those who wish to do so.

External mixing is also done on modern jet engines as, for example, on the Boeing 737 shown in Fig. 13.10. The goal is also to limit the volume of the region where the mixing takes place that is the source of jet noise. This is done by getting the mixing done quickly. The scallops in the trailing edge of the nozzle could also be properly described as vortex generators.

The core flow of the engine shown in Fig. 13.10 expands the jet by means of an *external expansion nozzle* that is visible as the central cone of that engine (no

**Fig. 13.9** An early turbofan engine on a Boeing 707 (a Rolls-Royce Conway) with a daisy petal mixer that joins the external and the primary core flow of the engine. In this engine, the fan flow exits through two nozzles on the sides of the nacelle. These nozzles are partially obscured in this view but the visible one exits the nacelle between 2 and 4 o'clock positions (The Boeing Company)

**Fig. 13.10** The (convergent and scalloped) fan flow nozzle exit lip of a commercial airliner, a Boeing 737 MAX. Both core flow and fan flows are partially expanded against the conical, external nozzles (Photo by author at the Boeing assembly plant in Renton, Washington)

scallops on the exit lip). An external expansion nozzle operates pretty much as does the axisymmetric inlet, except in reverse: The nozzle accelerates the flow while the diffuser (or inlet) decelerates it. Both devices handle flows that are fairly close to reversible (free of losses). The fan flow in Fig. 13.10 also expands against an external nozzle wall, but to a lesser degree.

This discussion of the nozzle and that of the inlet allows a revisit to the subject of property variations through the entire engine. Specifically, the sketch of Fig. 9.2 can be revisited to show the variation of temperatures and pressures through the jet engine as a system. Figure 13.11 is the earlier figure augmented with the variation of the gas properties through the inlet and the nozzle. The solid lines are of the total values (in the engine reference frame) with the dashed (red and green) lines are an indication of the static values. The static pressure varies from atmospheric value back to atmospheric value, while the higher static temperature in the jet reflects the need to comply with the Second Law of Thermodynamics that requires a certain amount of heat to be wasted. For purposes of this sketch, we consider a simple turbojet rather than a turbofan where two flows would be involved, making for an overly complicated sketch. The variation of the fan flow can be inferred from that through the core engine, without heat addition and without a turbine. Further, to simplify matters in the sketch, the inlet and nozzle flows are taken to be reversible so that the total pressure losses associated in these components are overlooked. In reality, especially in the inlets of supersonic flight airplanes, some total pressure losses would be experienced.

A textbook on propulsion nozzles will state that thrust from a nozzle consists of two parts: the momentum of the moving gas and a force resulting from pressure mismatch between atmospheric pressure and the pressure at the nozzle exit plane. The pressure mismatch contribution to thrust is small in jet engines and absent when the jet flow is subsonic because the pressures in question necessarily equilibrate.

**Fig. 13.11** Schematic variation of properties through a turbojet with reversible inlet and nozzle flows. The black dotted line establishes the level of static quantities in the air processed by the engine. The solid lines are total quantities in the engine reference frame. The dashed lines describe the variation of static quantities. The letters I … N refer to inlet, compressor, burner, turbine, and nozzle, respectively

Hence, we can speak of thrust being largely the momentum term for such nozzles. For propulsion systems in high performance aircraft, the pressure mismatch term does contribute a correction that can be minimized by proper design, but it is safe to say that the momentum term in thrust dominates the performance so that this aspect of thrust can be relegated to being a detail addressed when performance accuracy is desired. For rocket engines designed to fly into space the matter of pressure mismatch is significant as we shall see. For the present, let us stay with jet engines.

#### **13.5 Choking**

For military jets that need large amounts of thrust and fly much faster than transonic commercial transports, the nozzles are often designed and operated with a sonic flow in the throat of the nozzle and further expansion in a divergent cone. Such nozzles are called *convergent-divergent*. The jet from such nozzles is supersonic with a high speed of sound because the gas is hot and it is consequently noisy. Military operations are not as concerned with noise as flight operations at civilian airports are. Often, the engines of military aircraft are equipped with afterburners. When such an afterburner is in use, the exhaust gas is even hotter and lower in density so that the flow limiting throat has to be enlarged to pass the air volume that the engine provides. For this reason, afterburner engines usually have variable geometry nozzles, specifically the throat opening size than can be altered as conditions require.

When the afterburner is in operation, the Mach number of the hot flow in the duct where combustion takes place tends toward *M* = 1 as heat is released by the burning fuel. Under such conditions a limitation on flow rate may be imposed by *thermal choking*2 rather than *geometric choking* set by the minimum flow area or throat. A geometric throat minimum is hardly discernable in Fig. 13.12 on the B-58 engine nozzle displayed in afterburn mode which means that the process of afterburning drives the flow Mach number quite close to unity.

Flow choking is a very important aspect of internal flows, i.e., flows through engines. When a flow is choked, the mass flow rate through the area where *M* = 1.0 is a maximum that can only be influenced by upstream total pressure and total temperature because these two quantities determine the mass density of the gas and the local speed of sound. Specifically, the mass flow rate cannot be altered by lowering downstream pressure because information about the pressure there cannot be transmitted upstream. In practice, choking is a controlling factor in the ability of internal combustion engine's ability to process air past the valves. It is also a factor in controlling the flow through the gas turbine engine because the turbine nozzle is

2 Choking is the phenomenon where flow rate in a stream tube is governed only by conditions upstream of the location with choked flow, i.e., the total pressure and temperature, and not by the downstream static pressure.

**Fig. 13.12** Left: The propulsion nozzle of a supersonic B-58 bomber. Note the throat and the divergent cone that is variable in geometry (in both throat and exit areas) to accommodate afterburning. The flame-holders for this device are visible at the far end of the afterburner cavity with enough flow length to the nozzle throat to allow for more-or-less complete combustion of the fuel added (Photo by author at the National Museum of the USAF). Right: On the Space Shuttle 'Discovery' at the Udvar-Hazy Center of the Smithsonian National Air and Space Museum, the nozzle throats are near the bulkhead and are roughly one eighth the nozzle exit skirt diameter

always choked whereas the primary propulsion nozzle of a jet engine may or may not be. Finally, choking is a central issue in flow through the rocket nozzle.

#### **13.6 More Extreme Nozzles: The Rocket Engine**

The requirement for a nozzle area minimum in an air-breathing engine nozzle parallels that for a convergent-divergent rocket engine where nozzle pressure ratios (chamber to ambient) are much greater and the exit to throat area relation is much more pronounced. The three hydrogen–oxygen rockets engines ('acronymed' as the SSME, the Space Shuttle Main Engines) on the Space Shuttle illustrate this (Fig. 13.12). The steady pressure in the combustion chamber of these engines is more than two hundred times atmospheric pressure (about 3000 psi). During its ascent flight, the nozzle pressure ratio varies from about 200 at sea level and rises dramatically during ascent. Such high nozzle pressure ratios require a significant divergent nozzle skirt so that as much of the available thrust can be realized by matching the flow pressure at the nozzle exit to the atmospheric pressure. The actual design increased the throat area by a factor of 69 and forced the flow to a jet Mach number of about 5 at the nozzle exit.

**Fig. 13.13** The over-expanded SSME on a test stand running with oxygen and hydrogen. The absence of hydrocarbons makes for a beautiful and clear image of the flow (NASA)

#### *13.6.1 Overexpansion*

At launch or as illustrated on a test stand (Fig. 13.13), the nozzle is *overexpanded*. The nozzle skirt expands the flow too much, i.e., it is too long and should have been taken to only about 15 times the throat area. At this condition, the flow must recompress after leaving the exit to match the high atmospheric pressure. This can be seen in the converging boundary of the jet and in the observation of a normal shock wave to bring the flow closer to the relatively high pressure of the environment. Not visible in the picture (because the jet gas is nearly transparent) is that the first (weak) compression wave is a conical oblique shock wave that connects the exit lip of the nozzle and the perimeter of the central normal shock in the image. The normal shock doing the recompression in the middle of the stream is evidently strong because of the greater luminosity behind the shock resulting from the higher static pressure and static temperature (reconversion of kinetic energy to heat!). These static conditions would be those measured by an unfortunate (theoretical) bug were it to make this trip through the nozzle!!

#### *13.6.2 Under-Expansion*

As the vehicle ascends in the atmosphere, the atmospheric pressure around the vehicle and engine falls, and the need for recompression is diminished. The normal shock seen in Fig. 13.13 will move to the rear and weaken. There will be an altitude where the expansion is just right and matches the flow pressure at the nozzle exit. The nozzle will be perfectly expanded at this point. For the SSME engines, this is estimated to occur at an altitude of about 15,000 feet. As the vehicle ascends further, the (static) pressure at the exit will stay as it was and the flow will have to expand further outside to match the ever lower atmospheric pressure. The nozzle is now *under-expanded*. A rocket nozzle design for ascent to space must be optimized for this aspect of performance and the important consideration is the engine weight, specifically the weight of the nozzle skirt.

The nozzle pressure ratios when the engines approach shutoff during flight are very large. Near the end of burn, the expansion after the flow leaves the nozzles is dramatic as shown in Fig. 13.14. The picture is of the Saturn V launch of Apollo 11. Such a nozzle is under-expanded in this picture. Ideally one would have wanted the nozzle skirt to be as large as the expanded jet in the image, but that is evidently not realistic. Here, the jet is made visible because the combustion is of a hydrocarbon fuel rather than hydrogen. In a similar view of the Space Shuttle at high altitude and after solid rocket booster burnout, the large external jet would be hard to see because the water vapor (and a substantial amount of hydrogen) jet produced is almost transparent to visible light.

**Fig. 13.14** The Saturn V rocket in flight carrying Apollo 11. Note the very large width of the jet from an under-expanded nozzle (NASA)

**Fig. 13.15** Over- and under-expansion waves in an afterburning J58 propelling an SR-71 (NASA)

The conical waves that start at the exit lip (Fig. 13.13), both compression shock or expansion waves, are the beginnings of a train of waves that interact with the free boundary and reflect from it in the opposite sense that they impact the boundary. That is, a compression wave will be reflected as an expansion wave and vice versa ad infinitum. Thus, one can imagine the succession of waves in the jet getting weaker as one proceeds further from the end of the nozzle. While it is hard to see this in the image of the Saturn V flight, it is easily observed in an afterburning turbojet (see Fig. 13.15). The picture is of an afterburning Pratt & Whitney J58 on an SR-71 in flight.

The alternating nature of reflected waves by the free boundary of a jet contrasts to the reflection of waves in a flow with rigid boundaries. There, waves are reflected in kind, meaning that a compression wave is reflected as another compression wave. To some extent this is exploited in the design of high Mach number supersonic inlets where supersonic flow enters the inlet and the internal shape of the inlet causes a series of reflected compression waves to slow the flow as isentropically as practical. Often, although not necessarily, such inlets use an external cone to reduce the flow Mach number from its flight value to something lower where the internal inlet flow completes the Mach number transition to subsonic values. The inlet on the SR-71 shown in Fig. 13.7 illustrates this design approach.

One might wonder whether the addition of further rocket nozzle hardware to take advantage of the high pressure in the flow could be worthwhile. Indeed, it would, but the mechanical complexity of adding more of a nozzle skirt *while the rocket is in flight* is a challenge that reality might rule out as a solution.

#### *13.6.3 Staging*

In practice, this issue is often resolved by staging, i.e., employing a number of series or parallel stages to build a complete rocket. The nozzles can then be designed to operate in an appropriate range of altitudes with acceptable performance. The staging also addresses the concern that a vehicle may be limited in structural strength to a certain level of acceleration. The level of g-forces acting on the structure increases markedly in any ascent to space because the thrust may or may not be reducible as fuel is burned off and the vehicle becomes ever lighter. Further, there is no point to expending power and fuel to haul excess engine thrust capacity and empty tanks to an altitude where they are not needed. To that end, the (parallel) solid propellant boosters (a stage of sorts) of the Space Shuttle were designed to provide a decreasing amount of thrust as they burn during their 125 s of use when they burn out at about 145,000 feet of altitude. The Apollo rocket system is a set of stages in series, meaning they are fired in a sequence after an earlier stage used up its fuel supply.

#### *13.6.4 Specific Impulse and a Little Chemistry*

In order to circle back to the discussion of flow along a streamline, a few comments involving the conservation of total enthalpy, are particularly interesting in connection with the function of a rocket engine like the SSME. Expansion of the rocket's jet into vacuum leads, ideally, to a complete conversion of the total enthalpy along the streamlines to kinetic energy. The velocity associated with that kinetic energy is the momentum (per unit mass) wanted for thrust. The more the better, because the only other way to increase the thrust is to increase mass flow rate and that is costly in terms of weight that must be launched. Thus, we examine the only quantities that determine a high jet velocity in chemical rocket: a high total temperature and a low molecular weight of the gas. Rocket propulsion people call this velocity by a special name: thrust per unit *weight* flow rate of propellant, or *specific impulse* (*Isp*). It is nothing more than the jet velocity divided by the acceleration of gravity. The definition of *Isp* means that it is measured in seconds.

Hydrocarbon fuels burning with oxygen produce a gas with *Isp* = ~ 350 s. A solid propellant might operate with 280 s. The hydrogen–oxygen combustion in the SSME yields a superior performance with about 450 s (~ 14,500 feet/s!) in the vacuum of space but only about 360 s at takeoff where the over-expansion of the flow reduces the performance. Here is a good place to point out that the airbreathing engines that do not have to carry the air (oxygen) used for combustion can be contrasted to a rocket. The specific impulse of a modern turbofan engine in cruise is about 6500 s!3 No wonder we don't use rockets in airliners!

The hydrogen–oxygen reaction is quite energetic yielding high combustion temperatures. The chemically correct proportion of hydrogen and oxygen is 2 · H2 + 1 · O2 to produce two molecules of water vapor. This translates to a ratio of 1:8 by weight. Oxygen is really heavy compared to hydrogen. The resulting jet consisting of pure water vapor has a molecular weight of 18. A maximum total enthalpy per unit mass can be obtained with a reduced temperature and reduced molecular if an excess of hydrogen is processed. Indeed, the SSME burns the reactants *fuel-rich* in 1:6 (H/O) proportion. The molecular weight of the gas is thereby reduced by about 35%. The lower temperature together with the dearth of (chemically very reactive) oxygen ease the design, construction, and reusability of the engine.

We conclude the discussion of rocket flight to space with a comment on dynamic pressure (*q*) tackled earlier in our story. Recall this pressure involves only speed and air density. Enthusiasts of flights to space might note that launch operations often involve notation of the point along the ascent where the rocket reaches a point of "*max q*". That there is such a maximum should be self-evident as the rocket is flying ever faster into an environment of ever lower atmospheric air density, eventually reaching space with near zero density. The *max q* point is where the aerodynamic heating and aerodynamic loads are at maxima. Later in the flight, these aspects become an ever-smaller concern.

Rocket propulsion involves a lot of chemistry and the identification of "good" fuels and reactions is a part of the challenge. This is part of the work done by rocket scientists! On top of that is the challenge of getting the engine to fire up and run reliably, all without sudden surprises!

One of the interesting dimensions of liquid fuel rocketry is that it employs the same thermodynamic cycle as did the first successful engine: the steam engine. Central to the cycle is compression of a liquid: water in the old and the chemical propellants in the new. Expansion in the old is against a piston or through a turbine. In the new, expansion is via a nozzle to create a propulsive jet. The SSME is a steam engine!

#### **13.7 Airplane Range**

Returning all the way back to the discussion of aerodynamic performance of a wing or airplane, we note the aerodynamicist is largely concerned with maximizing the *M L*/*D* to obtain long range and low fuel costs in an airliner or similar transport vehicle. That quantity is also central in the goal of a long-range airplane design. If either of

3 Airplane propulsion people use a fuel consumption number called the thrust specific fuel consumption (TSFC) measured in pounds per hour per pound of thrust. TSFC is essentially the inverse of *Isp*.

the parameters, *M L*/*D or Isp*, has an important influence on the empty weight of an airplane, then some serious optimization will have to be undertaken and there lies the heart of successfully designing an airplane.

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The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Epilogue**

Friction and the necessary existence of the boundary layer on bodies moving through a fluid make life difficult and challenging for the designer of an airplane. The absence of that reality would make flight impossible.

It is the hope that, in looking over these pages, the subject of flight is better appreciated for the technical achievement that it is. The talent and hard-won insight by many into the complex behavior of air we cannot even see is obviously manifest in the state of flight today. Many theoretical musings and postulates had to be tested by experiments in the laboratory and in flight. The latter often took courage and sometimes ended poorly for the person sitting in the pilot's seat. And yet we, the community of the curious and adventurous, learned from the experiences of those who went before us.

A reality that can never be ignored is that technology can and is always a doubleedged sword. It can be used for good and evil. As was the knife that can cut the meat to eat or stab a person, the gun powder that went from fireworks to propel a bullet, so the airplane can bring help to rescue those that need it, help extinguish a fire in a blazing forest, so can it deliver you or me to a distant place. It can also deliver a bomb to an innocent or to a malevolent enemy.

Developing technology is a thrill at the fundamental level but brings with it a need to examine the associated morality. On July 16, 1945, after witnessing the test of the first atomic bomb, Robert Oppenheimer is said to have pondered a piece of Hindu scripture: "Now I am become Death, the destroyer of worlds". Such is the human condition, and the road of life is not easily travelled. We will, however, journey on.

## **Appendix A Equations for Quantitative Descriptions**

The following is a listing of the equations that may be useful to augment the text discussion. Numerical values of the constant are not provided as these differ between the English and the metric (SI) systems.

#### **A.1 Motion with Circular Streamlines**

(Azimuthal) velocity distributions in a vortex and solid body rotation

*r* is the radial distance from center to streamline, Vortex: *Vazimuthal* = Γ /2π*r*; Γ is the vortex strength, Solid body: *Vazimuthal* = ω*r*; ω is the angular speed of rotation.

#### **A.2 Air as a Medium**

Air is the medium of flight. It consists of about 79% nitrogen, 1% argon and 20% oxygen. The predominance of nitrogen in all air-related processes allows a reasonably accurate description of combustion gas as being like air. While the inclusion of real gas effects is relatively straightforward, it not necessary for the qualitative descriptions of the relevant processes. Further, in an elementary examination of the performance of a gas turbine engine, the fuel mass added to the air stream mass can be neglected.

#### **A.3 The Descriptive Constants**

*R*—ideal gas constant as in the equation of state: *p*v = *RT*

*Cv*—specific heat at constant volume, from the internal energy of air: *u* = *C*v*T*

*Cp*—specific heat at constant pressure, from the enthalpy of air: *h* = *CpT* .

[*more precise definitions for the specific heats apply generally, but for our modelling of air, the relations to internal energy and enthalpy as stated are adequate*].

The specific heats *Cp* and *Cv* are, in reality, functions of temperature because of the internal complexity of the molecules. However, for modeling purposes and reasonably good descriptions of air in the context of wing and engine performance, taking them to be constant is acceptable. The ideal gas equation of state, above, is very much accurate.

The definition of the enthalpy, *h*, allows stating: *Cp* = *C*v + *R* and a definition for the specific heat ratio: γ = *Cp*/*C*v = 1.4 for air.

#### **A.4 The Gas Properties**

Here and in the text, the use of absolute temperatures (*T*) and pressures (*p*) is implicit, unless otherwise noted. The carriage of extra reference constants such as atmospheric pressure and a standard temperature is simply too awkward.

The text refers to *static* values of these quantities that are measured by an observer at rest in the medium. Example: a butterfly holding a thermometer and a pressure gauge while carried along by a flow would measure static pressure and temperature.

Total pressure and temperature are the values measured by a stationary observer at a place where a moving fluid is gently brought to rest relative to that observer. The total temperature and total pressure are measured by the butterfly above just before it impacts the leading edge of a wing or similar obstruction that brings the flow to rest.

The total enthalpy, or equivalently the total temperature, is the measure of all the energy in a moving fluid.

The specific volume (cubic meters per kilogram or cubic feet per pound mass, *v*) and the density (kg/m3, etc., ρ) are related through v = 1/ρ.

#### **A.5 The Flight Equations**

Dynamic pressure: *q* = 1/2 ρ*V*2

[the symbol *q* is conventional and is not related to the same symbol for heat interactions]

Airplane force coefficients: *CL* or *CD* = Lift or Drag/(½ ρ *V*2) · (Wing Area). Propulsive thrust:

[of a system moving at velocity *V*0, producing a jet of speed *Vj*, neglecting the usually small pressure mismatch term]

$$= (Mass\ flow\ rate) \cdot \left(V\_j - V\_0\right)$$

Energy in jet stream:

$$= (Mass\ flow\ rate) \cdot \left| \left(V\_j^2 - V\_0^2\right) \right|$$

Propulsive efficiency involving jet (j) and flight (0) speeds:

$$=\frac{2}{\left(1+\frac{\nu\_i}{\nu\_0}\right)}$$

#### **A.6 The Equations of Aerothermodynamics**

#### *A.6.1 The Energy Equation Along a (Steady) Streamline*

**Definition** *ht* (total enthalpy) = *h* (static) + ½ *V*2.

From which follows (with the definition of *M* and speed of sound, see below) Total temperature:

$$T\_t = T\left(1 + \frac{\chi - 1}{2}M^2\right)$$

Energy equation for steady, compressible fluid flow:

*ht*2 − *ht*1 = *q*(heat) + w(mechanical work)

For adiabatic flow (no heat interaction) and no work interactions:

 $h\_t = h + 1/2$   $V^2$  is conserved.

The energy equation for an incompressible fluid (density (ρ) must be constant, Bernoulli principle):

 $p/\rho + 1/\_2\,\text{V}^2$  is conserved.

#### *A.6.2 Compressibility*

Static speed of sound:

$$a = \sqrt{\gamma RT}$$

Mach number:

$$M = \frac{V}{a}$$

In a steady, reversible flow pressure changes lead to density changes:

$$\frac{d\rho}{\rho} = -M^2 \frac{dV}{V}$$

[see Appendix B; *dx x* is the fractional change in *x*].

#### *A.6.3 Isentropic Relation Between Two States 1 and 2*

$$\frac{p\_2}{p\_1} = \left(\frac{T\_2}{T\_1}\right)^{\frac{\gamma}{\gamma - 1}}$$

*with* γ/(γ – 1) = 3.5 for air. From which follows, Total pressure:

$$p\_t = p\left(1 + \frac{\gamma - 1}{2}M^2\right)^{\frac{\nu}{\gamma - 1}}$$

#### *A.6.4 Heating Total Pressure Loss*

$$\frac{dp\_t}{p\_t} = -\frac{\mathcal{Y}}{2}M^2 \frac{dq}{C\_p T\_t}$$

where *dq* is the differential amount of heat added.

## **Appendix B Some Quantitative Aspects of Aerodynamics and Thermodynamics**

#### **B.1 Entropy is a Fact**

This appendix, as a discussion of entropy, should properly be the beginning of a study of aerodynamics and thermodynamics, the studies of air and heat in motion. In a text such as this, the result might discourage the reader and the aims of the text would have been missed. It is, however, unavoidable to involve the Second Law of Thermodynamics in an honest discussion of the physics of flight. Further, such a discussion is made much more effective with differential calculus because the second law of thermodynamics is conveniently written in this form.

We have described many fluid motion processes as conserving a quantity called the entropy. The following paragraphs are intended to shed a little more light on that subject using slightly more sophisticated tools.

What is this property called entropy? It is an abstract property whose existence follows directly from the second law of the thermodynamics. The logic parallels the thinking about the first law that is also called an energy conservation principle. The concept of the abstract idea of "energy" as a property originated there. In a similar manner, entropy originates from the second law.

What does the second law of thermodynamics really state? In our everyday experience, we observe that work done, such as dragging a brick across the floor, stirring a liquid, or compressing a gas, is convertible to heat. Thus, heat and work are somehow equivalent. On the other hand, can heat be converted to work? Yes, but within limits. That is at the heart of classical thermodynamics and has to be reconciled with the observation that heat only flows from a hot body to a colder one and not in reverse.

A good book on classical thermodynamics will make the case that heat transferred from whatever we are dealing with (the system) to its external environment (or the reverse) can be done reversibly if the transfer occurs with no fall in temperature between the system and its environment. Without proving it here, this amounts to saying that the differentially small amount of heat transferred divided by some measure of temperature can be made reversible. Briefly, the idea is to show that transferring heat can be made reversible when the temperature of the system being examined is always at the same temperature as the environment to which it gains or loses heat. This involves arguing for the (unrealistic) existence of a large (infinite?) array of 'reservoirs' each at slightly different and fixed temperatures. While this is not practical, it is possible in a thought experiment.

The simplest way to express that mathematically is to make the measure of temperature, the temperature itself. Thus, the process *dq*/*T* can be made reversible.

The general form of the First Law is

$$
\partial q = du \,\star \,\partial \mathcal{W}
$$

the symbols *q* and *w* describe heat and work interaction with a system, which for our purposes, could be a volume of air. In a classical statement, the differential "*ð*"s associated with heat and work are given a modified symbols (to contrast with "*d*") to indicate that a *process* is described and, specifically, that *q* and *w* are not properties. On the other hand, *u*, the internal energy (generally dependent on temperature and sometimes even pressure), is a property.

When the first law is written in a reversible form, work done must be written with *ðw* = *pdv*. This statement is the mechanical equivalent of work expressed as force times distance.

Thus, one can write

$$\left(\frac{dq}{T}\right)\_{rev} = \frac{du}{T} + \frac{p\,dv}{T}$$

The right-hand side of this equation involves only properties, hence the left-hand side must also be a property. It is, in fact, a differential of the entropy, *ds*.

For an ideal (*pv* = *RT*) gas, this reads

$$\left(\frac{dq}{T}\right)\_{rev} = ds = \frac{du}{T} + R\frac{dv}{v} = C\_v \frac{dT}{T} + R\frac{dv}{v}$$

In differential form, the ideal gas law can be written as

$$\frac{dp}{p} + \frac{dv}{v} = \frac{dT}{T}$$

so that the entropy can finally be written in terms of the parameters that really interest us, namely pressure and temperature. Before we do, we will focus on air which is also a perfect gas with constant specific heats. For that case, the definition of enthalpy gives

$$C\_p = C\_v + R$$

and the differential entropy becomes

$$\frac{ds}{C\_v} = \frac{dT}{T} - \frac{\gamma - 1}{\gamma} \frac{dp}{p}$$

using the definition of γ. This is the origin of the isentropic (*ds* = 0) relation between pressure and temperature when integrated for a process between two states.

#### **B.2 Compressibility**

While we are immersed in the differential forms of the equations governing various processes, it is also convenient and fairly straightforward to examine the criterion expressed in Chap. 5 to determine whether a flow process involves a gas that must be described as compressible or whether the incompressible (Bernoulli) formulation is acceptable.

To do this we invoke Newton's second law of motion (simply stated as *F* = *ma*) expressed here in differential form applied to an element of air. This relation (often called a momentum equation) reads, for a unit volume and in the absence of friction forces, as

$$dp + \rho \, V \, dV = 0$$

For an incompressible fluid, this relation leads directly to the Bernoulli principle. More generally, this relation can be rewritten in terms of the very convenient fractional changes as

$$\frac{dp}{P} = -\frac{\rho}{P}\frac{V^2}{V}\frac{dV}{V} = -\gamma M^2 \frac{dV}{V}$$

With the speed of sound stated as

$$a^2 = \frac{\mathcal{Y}\mathcal{P}}{\rho} = \mathcal{Y}\mathcal{R}T$$

Thus, the Mach number appears as the important parameter to state whether velocity changes are sufficient to change the pressure of the air in a significant way. With

$$\frac{p\_2}{p\_1} = \left(\frac{p\_2}{p\_1}\right)^{-\nu} \text{ or } \frac{dp}{p} = -\nu \frac{d\rho}{\rho}$$

valid for isentropic processes, the density changes are easily calculated from the velocity changes:

$$\frac{d\rho}{\rho} = -M^2 \frac{dV}{V}$$

showing the important role that Mach number plays in determining whether a flow may be modeled as incompressible when it might not be.

#### **B.3 Boundary Layers**

The momentum equation as a form of Newton's second law of motion is a vector equation with associated directions. If that equation is applied to the direction normal to a surface along which there is flow, we note that there can be little or no flow acceleration in that direction because the boundary does not allow it. It is therefore a safe conclusion to state that the static pressure normal to the surface, i.e., through the boundary layer is uniform in that direction. That is the observation that allows the interpretation that measurements made on the surfaces are accurate reflection of the pressure in the inviscid flow field outside the boundary layer.

## **Appendix C Induced Drag**

An overview of the airplane designer's concern with lift induced drag (Fig. C.1).

**Fig. C.1** This NASA chart illustrates well the role of wing aspect ratio (*AR*) and its role in determining drag due to lift. *A* is the wing area, *s* the wingspan, and *CL* is the wing lift coefficient. (NASA)

© The Author(s) 2022 R. Decher, *The Vortex and The Jet*, https://doi.org/10.1007/978-981-16-8028-1

## **Appendix D Aerodynamic Performance Summary**

The following graphic is a so-called *High Speed Drag Polar* for a Boeing 727–100. The figure is like Fig. 7.12 except that the axes are switched in orientation. The origin is not shown because only the information of practical interest is displayed. Note the significant drag increase as flight Mach number is increased. On several curves, the points of the maximum *L*/*D* is identified (s. D.1).

From the plot, the values for maximum *L*/*D* are roughly as indicated in Table D.1. Thus, for maximum aerodynamic efficiency, the best flight Mach appears to be near 0.70. The table also notes the values of *M L*/*D* for maximum range. That combination of measure maximizes nearer 0.80. Fortunately, the maxima in these two estimated parameters vary slowly with Mach number so that there is some operational flexibility.

**Fig. D.1** From the Performance Engineer's Manual, a technical document issued by the Boeing Company about the Model 727–100 in 1967, a plot of the airplane drag polar: Airplane lift versus drag coefficient for various flight Mach numbers. The circles are approximate locations of maximum *L*/*D* (The Boeing Company)

## **Glossary of Technical Terms**

**Flow separation** A failure of flow to follow the intended contour of a surface.

**Vorticity** A spatial distribution of vortices.

**Wake** The boundary layer air behind a body moving through a fluid.

## **References**

Alfredsson, P.H., and M. Matsubara. Free-stream turbulence, streaky structures and transition in boundary layer flows. AIAA Paper 2000–2534.

Van Dyke, M. 1983. *An Album of Fluid Motion*. Parabolic Press.