Jet engine performance

Design Point

TS Diagram

Temperature vs. entropy (TS) diagrams (see example RHS) are usually used to illustrate the cycle of gas turbine engines. All the reader really needs to know about entropy is that it represents the degree of disorder of the molecules in the fluid and that it tends to increase!

Apart from stations 0 and 8s, stagnation pressure and stagnation temperature are used. Station 0 is ambient.

The processes depicted are:

Freestream (stations 0 to 1)

In the example, the aircraft is stationary, so stations 0 and 1 are coincident. Station 1 is not depicted on the diagram.

Intake (stations 1 to 2)

In the example, a 100% intake pressure recovery is assumed, so stations 1 and 2 are coincident.

Compression (stations 2 to 3)

The ideal process would appear vertical on a TS diagram. In the real process there is friction, turbulence and, possibly, shock losses, making the exit temperature, for a given pressure ratio, higher than ideal. The shallower the positive slope on the TS diagram, the less efficient the compression process.

Combustion (stations 3 to 4)

Heat (usually by burning fuel) is added, raising the temperature of the fluid. There is an associated pressure loss, some of which is unavoidable

Turbine (stations 4 to 5)

The temperature rise in the compressor dictates that there will be an associated temperature drop across the turbine. Ideally the process would be vertical on a TS diagram. However, in the real process, friction and turbulence cause the pressure drop to be greater than ideal. The shallower the negative slope on the TS diagram, the less efficient the expansion process.

Jetpipe (stations 5 to 8)

In the example the jetpipe is very short, so there is no pressure loss. Consequently, stations 5 and 8 are coincident on the TS diagram.

Nozzle (stations 8 to 8s)

These two stations are both at the throat of the (convergent) nozzle. Station 8s represents static conditions. Not shown on the TS diagram is the expansion process, external to the nozzle, down to ambient pressure.

Design Point Performance Equations

In theory, any combination of flight condition/throttle setting can be nominated as the engine performance Design Point. Usually, however, the Design Point corresponds to the highest corrected flow at inlet to the compression system (e.g. Top-of-Climb, Mach 0.85, 35000ft, ISA) , The design point net thrust of any jet engine can be estimated by working through the engine cycle, step by step. Below are the equations for a single spool turbojet.

Freestream

$T_1 = t_0 \cdot (1 + ({\gamma}_c-1)\cdot M^2/2)$

$P_1 = p_0 \cdot (T_1/t_0)^ (T_4-T_5) = w_2 \cdot C_{\mathrm{pc}}(T_3-T_2)$

A simplyfying assumption sometimes made is for the addition of fuel flow to be exactly offset by an overboard compressor bleed, so mass flow remains constant throughout the cycle.

$P4/P5 = (T4/T5)^ = T_8/(({\gamma}_t+1)/2) \,$

$p_{\mathrm{8s}} = P_8/((T_8/t_{\mathrm{8s}})^)$

${\rho}_{\mathrm{8s}} = p_{\mathrm{8s}}/(R \cdot t_{\mathrm{8s}})$

$A_8 = w_8/({\rho}_{\mathrm{8s}} \cdot V_8)$

Gross Thrust

$F_g = C_\mathrm{x}((w_8 \cdot V_8/g) + A_8(p_{\mathrm{8s}} - p_0))$

Ram Drag

$F_r = w_0 \cdot V_0/g$

Net Thrust

$F_n = F_g - F_r \,$

The calculation of the combustor fuel flow is beyond the scope of this text, but is basically proportional to the combustor entry airflow and a function of the combustor temperature rise.

Note that mass flow is the sizing parameter: doubling the airflow, doubles the thrust and the fuel flow. However, the specific fuel consumption (fuel flow/net thrust) is unaffected, assuming scale effects are neglected.

Similar design point calculations can be done for other types of jet engine e.g. turbofan, turboprop, ramjet, etc.

Cycle improvements

Increasing the overall pressure ratio of the compression system raises the combustor entry temperature. Therefore, at a fixed fuel flow and airflow, there is an increase in turbine inlet temperature. Although the higher temperature rise across the compression system implies a larger temperature drop over the turbine system, the nozzle temperature is unaffected, because the same amount of heat is being added to the total system. There is, however, a rise in nozzle pressure, because turbine expansion ratio increases more slowly than the overall pressure ratio. Consequently, net thrust increases, implying a specific fuel consumption (fuel flow/net thrust) decrease.

So turbojets can be made more fuel efficient by raising overall pressure ratio and turbine inlet temperature in unison. However, better turbine materials and/or improved vane/blade cooling are required to cope with increases in both turbine inlet temperature and compressor delivery temperature. Increasing the latter may also require better compressor materials.

Off-design

General

An engine is said to be running off-design if any of the following apply:

a) change of throttle setting

b) change of altitude

c) change of flight speed

d) change of climate

e) change of installation (e.g. customer bleed or power off-take)

f) change in geometry

Although each off-design point is effectively a design point calculation, the resulting cycle (normally) has the same turbine and nozzle geometry as that at the engine design point. Obviously the final nozzle cannot be over or underfilled with flow. This rule also applies to the turbine nozzle guide vanes, which act like small nozzles.

Simple Off-design Calculation

Design point calculations are normally done by a computer program. By the addition of an iterative loop, such a program can also be used to create a simple off-design model.

In an iteration, a calculation is undertaken using guessed values for the variables. At the end of the calculation, the constraint values are analyzed and an attempt is made to improve the guessed values of the variables. The calculation is then repeated using the new guesses. This procedure is repeated until the constraints are within the desired tolerance (e.g. 0.1%). Iteration Variables

The three variables required for a single spool turbojet iteration are the key design variables:

1) some function of combustor fuel flow e.g. $RIT\,$

2) corrected engine mass flow i.e. $w_{\mathrm{2cor}}\,$

3) compressor pressure ratio i.e. $P_3/P_2\,$

Iteration Constraints (or Matching Quantities)

The three constraints imposed would typically be:

1) engine match e.g. $Fn\,$ or $w_{\mathrm{fe}}\,$ or $T_3\,$ , etc'''

2) nozzle area e.g. $A_{\mathrm{8calc}}\,$ vs $A_{\mathrm{8 des pt}}\,$

3) turbine flow capacity e.g. $w_{\mathrm{4cor calc}}\,$ vs $w_{\mathrm{4cor des pt}}\,$ The latter two are the physical constraints that must be met, whilst the former is some measure of throttle setting.

Note Corrected flow is the flow that would pass through a device, if the entry pressure and temperature corresponded to ambient conditions at sea level on a Standard Day. Results Plotted above are the results of several off-design calculations, showing the effect of throttling a jet engine from its design point condition. This line is known as the compressor steady state (as opposed to transient) working line. Over most of the throttle range, the turbine system on a turbojet operates between choked planes. All the turbine throats are choked, as well as the final nozzle. Consequently the turbine pressure ratio stays essentially constant. This implies a fixed $\triangle T_{\mathrm{turb}}/RIT\,$ . Since turbine rotor entry temperature, $RIT\,$ , usually falls with throttling, the temperature drop across the turbine system, $\triangle T_{\mathrm{turb}}\,$ , must also decrease. However, the temperature rise across the compression system, $\triangle T_{\mathrm{comp}}\,$ , is proportional to $\triangle T_{\mathrm{turb}}\,$ . Consequently, the ratio $\triangle T_{\mathrm{comp}}/T_1\,$ must also fall, implying a decrease in the compression system pressure ratio. The non-dimensional (or corrected flow) at compressor exit tends to stay constant, because it 'sees', beyond the combustor, the constant corrected flow of the choked turbine. Consequently, there must be a decrease in compressor entry corrected flow, as compressor pressure ratio falls. Therefore, the compressor steady state working line has a positive slope, as shown above, on the RHS.

Ratio $RIT/T_1\,$ is the quantity that determines the throttle setting of the engine. So, for instance, raising intake stagnation temperature by increasing flight speed, at a constant $RIT\,$ , will cause the engine to throttle back to a lower corrected flow/pressure ratio.

The Simple Off-design Calculation outlined above is somewhat crude, since it assumes:

1) no variation in compressor and turbine efficiency with throttle setting

2) no change in pressure losses with component entry flow

3) no variation in turbine flow capacity or nozzle discharge coefficient with throttle setting

Furthermore, there is no indication of relative shaft speed or compressor surge margin.

Complex Off-design Calculation

A more refined off-design model can be created using compressor maps and turbine maps to predict off-design efficiencies, relative shaft speeds, etc.

The iteration scheme is similar to that of the Simple Off-design Calculation.

Iteration Variables

Again three variables are required for a single spool turbojet iteration, typically:

1) some function of combustor fuel flow e.g. $RIT\,$ 2) compressor corrected speed e.g. $N_{\mathrm{cor}}\,$

3) an independent variable indicative of the compressor operating point up a speed line e.g. ${\beta}\,$ .

So compressor corrected speed replaces corrected engine mass flow and Beta replaces compressor pressure ratio.

Iteration Constraints (or Matching Quantities)

The three constraints imposed would typically be similar to before:

1) engine match e.g. $F_n\,$ or $w_{\mathrm{fe}}\,$ or $T_{\mathrm{3}}\,$ , etc

2) nozzle area e.g. $A_{\mathrm{8 geometricdesign}}\,$ vs $A_{\mathrm{8calc}}/C_{\mathrm{dcalc}}\,$

3) turbine flow capacity e.g. $w_{\mathrm{4cor calc}}\,$ vs $w_{\mathrm{4cor turb char}}\,$

Plotted on the LHS are the results of several off-design calculations, showing the effect of throttling a jet engine from its design point condition. The line produced is similar to the working line shown above, but is now plotted on the compressor characteristic and gives an indication of corrected shaft speed and compressor surge margin.

Performance Software

Over the years a number of software packages have been developed to estimate the design and off-design performance of various types of gas turbine engine. Most are used in-house by the various aero-engine manufacturers, but several are available to the general public (e.g. GasTurb http://www.gasturb.de, EngineSim http://www.grc.nasa.gov/WWW/K-12//airplane/ngnsim.html).

Husk Plot

A Husk Plot is a concise way of summarizing the performance of a jet engine. The following sections describe how the plot is generated and can be used.

Thrust/SFC Loops

Specific Fuel Consumption (i.e. SFC), defined as fuel flow/net thrust, is an important parameter reflecting the overall thermal (or fuel) efficiency of an engine.

Selecting a point on the plot, net thrust is calculated as follows:

$Fn = (Fn/{\delta}) \cdot {\delta}$

Clearly, net thrust falls with altitude, because of the decrease in ambient pressure.

The corresponding SFC is calculated as follows:

$SFC = (SFC/\sqrt{\theta}) \cdot \sqrt{\theta}$

At a given point on the Husk Plot, SFC falls with decreasing ambient temperature (e.g. increasing altitude or colder climate).The basic reason why SFC increases with flight speed is the implied increase in ram drag.

Although a Husk Plot is a concise way of summarizing the performance of a jet engine, the predictions obtained at altitude will be slightly optimistic. For instance, because ambient temperature remains constant above 11000m (36089ft) altitude, the Husk Plot would yield no change in SFC with increasing altitude. In reality, there would be a small, steady, increase in SFC, owing to the falling Reynolds Number and Specific Heat effects.

Thrust Lapse

The nominal net thrust quoted for a jet engine usually refers to the Sea Level Static (SLS) condition, either for the International Standard Atmosphere (ISA) or a hot day condition (e.g. ISA+10 °C). As an example, the GE90-76B has a take-off static thrust of 76,000 lbf (360 kN) at SLS, ISA+15 °C.

Naturally, net thrust will decrease with altitude, because of the lower air density. There is also, however, a flight speed effect.

Initially as the aircraft gains speed down the runway, there will be little increase in nozzle pressure and temperature, because the ram rise in the intake is very small. There will also be little change in mass flow. Consequently, nozzle gross thrust initially only increases marginally with flight speed. However, being an air breathing engine (unlike a conventional rocket) there is a penalty for taking on-board air from the atmosphere. This is known as ram drag. Although the penalty is zero at static conditions, it rapidly increases with flight speed causing the net thrust to be eroded.

As flight speed builds up after take-off, the ram rise in the intake starts to have a significant effect upon nozzle pressure/temperature and intake airflow, causing nozzle gross thrust to climb more rapidly. This term now starts to offset the still increasing ram drag, eventually causing net thrust to start to increase. In some engines, the net thrust at say Mach 1.0, sea level can even be slightly greater than the static thrust. Above Mach 1.0, with a subsonic inlet design, shock losses tend to decrease net thrust, however a suitably designed supersonic inlet can give a lower reduction in intake pressure recovery, allowing net thrust to continue to climb in the supersonic regime.

The thrust lapse described above depends on the design specific thrust and, to a certain extent, on how the engine is rated with intake temperature. Three possible ways of rating an engine are depicted on the above Husk Plot. The engine could be rated at constant turbine entry temperature, shown on the plot as $SOT/{\theta}\,$ . Alternatively, a constant mechanical shaft speed could be assumed, depicted as $N_F/\sqrt{\theta}\,$ . A further alternative is a constant compressor corrected speed, shown as $N_F/\sqrt{\theta}_T\,$ . The variation of net thrust with flight Mach number can be clearly seen on the Husk Plot.

Other Trends

The Husk Plot can also be used to indicate trends in the following parameters:

1) turbine entry temperature

$SOT = (SOT/{\theta}) \cdot {\theta}\,$

So as ambient temperature falls (through altitude or climate), turbine entry temperature must also fall to stay at the same non-dimensional point on the Husk Plot. All the other non-dimensional groups (e.g. corrected flow, axial and peripheral Mach numbers, pressure ratios, efficiencies, etc will also stay constant.

2) mechanical shaft speed

$N_F = (N_F/\sqrt{\theta}) \cdot \sqrt{\theta}\,$

Again as ambient temperature falls (through altitude or climate), mechanical shaft speed must also decrease to remain at the same non-dimemsional point.

By definition, compressor corrected speed must remain constant at a given non-dimensional point.

Rated Performance

Civil

Nowadays, civil engines are usually flat-rated on net thrust up to a 'kink-point' climate. So at a given flight condition, net thrust is held approximately constant over a very wide range of ambient temperature, by increasing (HP) turbine rotor inlet temperature (RIT or SOT). However, beyond the kink-point, SOT is held constant and net thrust starts to fall for further increases in ambient temperature. Consequently, aircraft fuel load and/or payload must be decreased.

Usually, for a given rating, the kink-point SOT is held constant, regardless of altitude or flight speed.

Some engines have a special rating, known as the 'Denver Bump'. This invokes a higher RIT than normal, to enable fully laden aircraft to Take-off safely from Denver, CO in the summer months. Denver Airport is extremely hot in the summer and the runways are over a mile above sea level. Both of these factors affect engine thrust.

Military

The rating systems used on military engines vary from engine to engine. A typical military rating structure is shown on the left. At low intake temperatures, the engine tends to operate at maximum corrected speed or corrected flow. As intake temperature rises, a limit on (HP) turbine rotor inlet temperature (SOT) takes effect, progressively reducing corrected flow. At even higher intake temperatures, a limit on compressor delivery temperature (T₃) is invoked, which decreases both SOT and corrected flow.

The impact of design intake temperature is shown on the right hand side.

An engine with a low design T₁ combines high corrected flow with high rotor turbine temperature (SOT), maximizing net thrust at low T₁ conditions (e.g. Mach 0.9, 30000 ft, ISA). However, although turbine rotor inlet temperature stays constant as T₁ increases, there is a steady decrease in corrected flow, resulting in poor net thrust at high T₁ conditions (e.g. Mach 0.9, sea level, ISA).

Although an engine with a high design T₁ has a high corrected flow at low T₁ conditions, the SOT is low, resulting in a poor net thrust. Only at high T₁ conditions is there the combination of a high corrected flow and a high SOT, to give good thrust characteristics.

A compromise between these two extremes would be to design for a medium intake temperature (say 290 K).

As T₁ increases along the SOT plateau, the engines will throttle back, causing both a decrease in corrected airflow and overall pressure ratio. As shown, the chart implies a common T₃ limit for both the low and high design T₁ cycles. Roughly speaking, the T₃ limit will correspond to a common overall pressure ratio at the T₃ breakpoint. Although both cycles will increase throttle setting as T₁ decreases, the low design T₁ cycle has a greater 'spool-up' before hitting the corrected speed limit. Consequently, the low design T₁ cycle has a higher design overall pressure ratio.

Nomenclature

$A\,$ flow area
$A_{\mathrm{8calc}}\,$ calculated nozzle effective throat area
$A_{\mathrm{8 des pt}}\,$ design point nozzle effective throat area
$A_{\mathrm{8 geometricdesign}}\,$ nozzle geometric throat area
$C_{\mathrm{pc}}\,$ specific heat at constant pressure for air
$C_{\mathrm{pt}}\,$ specific heat at constant pressure for combustion products
$C_{\mathrm{dcalc}}\,$ calculated nozzle discharge coefficient
$C_x\,$ thrust coefficient
$g\,$ acceleration of gravity
$F_g\,$ gross thrust
$F_n\,$ net thrust
$F_r\,$ ram drag
$J\,$ mechanical equivalent of heat
$M\,$ flight Mach number
$N_{\mathrm{cor}}\,$ compressor corrected shaft speed
$t\,$ static pressure
$P\,$ stagnation (or total) pressure
$P_3/P_2\,$ compressor pressure ratio
$prf\,$ intake pressure recovery factor
$R\,$ gas constant
$SFC\,$ specific fuel consumption
$RIT\,$ (turbine) rotor inlet temperature
$t\,$ static temperature
$T\,$ stagnation (or total) temperature
$T_1\,$ intake stagnation temperature
$T_3\,$ compressor delivery total temperature
$V\,$ velocity
$w\,$ mass flow
$w_{\mathrm{4cor calc}}\,$ calculated turbine entry corrected flow
$w_{\mathrm{2cor}}\,$ compressor corrected inlet flow
$w_{\mathrm{4cor des pt}}\,$ design point turbine entry corrected flow
$w_{\mathrm{4cor turb char}}\,$ corrected entry flow from turbine characteristic (or map)
$w_{\mathrm{fe}}\,$ combustor fuel flow
${\beta}\,$ arbitrary lines which disect the corrected speed lines on a compressor characteristic
${\theta}\,$ ambient temperature/Sea Level,Standard Day,ambient temperature
${\theta}_T\,$ total temperature/Sea Level,Standard Day,ambient temperature
${\delta}\,$ ambient pressure/Sea Level ambient pressure
${\rho}\,$ density
${\gamma}_{\mathrm{c}}\,$ ratio of specific heats for air
${\gamma}_{\mathrm{t}}\,$ ratio of specific heats for combustion products
${\eta}_{\mathrm{pc}}\,$ compressor polytropic efficiency
${\eta}_{\mathrm{pt}}\,$ turbine polytropic efficiency

Jet engines

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