# Jet engine off-design and transient performance
## Off-design performance calculation

This section should be read in conjunction with the topic Jet Engine Design Point Performance.

### 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 or intake pressure recovery)
    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. turbine rotor inlet temperature $RIT\,$
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 $A_{\mathrm {8despt} }\,$
3) turbine flow capacity e.g. $w_{\mathrm {4corcalc} }\,$ vs $w_{\mathrm {4cordespt} }\,$
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.
Fairly obviously, when an engine is throttled-back, it will lose net thrust. This drop in thrust is mainly caused by the reduction in air mass flow, but the reduction in turbine rotor inlet temperature and degradations in component performance will also contribute.
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 corrected mass flows, pressure ratios, efficiencies, relative shaft speeds, etc. A further refinement is to allow the component off-design pressure losses to vary with corrected mass flow, or Mach number, 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. $A_{\mathrm {8geometricdesign} }\,$ vs $A_{\mathrm {8calc} }/C_{\mathrm {dcalc} }\,$
3) turbine flow capacity e.g. $w_{\mathrm {4corcalc} }\,$ vs $w_{\mathrm {4corturbchar} }\,$
During the Complex Off-design calculation, the operating point on the compressor map is constantly being guessed (in terms of $N_{\mathrm {cor} }\,$ and ${\beta }\,$) to obtain an estimate of the compressor mass flow, pressure ratio and efficiency. After the combustion calculation is completed, the implied compressor mechanical shaft speed is used to estimate the turbine corrected speed (i.e. $N_{\mathrm {turbcor} }\,$). Typically, the turbine load (power demanded) and entry flow and temperature are used to estimate the turbine enthalpy drop/inlet temperature (i.e. $({\delta }H/T)_{\mathrm {turb} }\,$). The estimated turbine corrected speed and enthalpy drop/inlet temperature parameters are used to obtain, from the turbine map, an estimate of the turbine corrected flow ($w_{\mathrm {4corturbchar} }\,$) and efficiency (i.e. ${\eta }_{\mathrm {pt} }\,$). The calculation then continues, in the usual way, through the turbine, jetpipe and nozzle. If the constraints are not within tolerance, the iteration engine makes another guess at the iteration variables and the iterative loop is restarted.
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 it is now superimposed on the compressor map and gives an indication of corrected shaft speed and compressor surge margin.

### Installation effects

More often than not, the design point calculation is for an uninstalled engine. Installation effects are normally introduced at off-design conditions and will depend on the engine application.
A partially installed engine includes the effect of:
a) the real intake having a pressure recovery of less than 100%
b) air being bled from the compression system for cabin/cockpit conditioning and to cool the avionics
c) oil and fuel pump loads on the HP shaft
In addition, in a fully installed engine, various drags erode the effective net thrust of the engine:
1) an air intake spilling air creates drag
2) exhaust gases, exiting the hot nozzle, can scrub the external part of the nozzle plug (where applicable) and create drag
3) if the jet engine is a civil turbofan, bypass air, exiting the cold nozzle, can scrub the gas generator cowl and the submerged portion of the pylon (where applicable) and create drag
Deducting these throttle-dependent drags (where applicable) from the net thrust calculated above gives the streamtube net thrust.
There is, however, another installation effect: freestream air scrubbing an exposed fan cowl and its associated pylon (where applicable) will create drag. Deducting this term from the streamtube net thrust yields the force applied by the engine to the airframe proper.
In a typical military installation, where the engine is buried within the airframe, only some of the above installation effects apply.

## Transient performance calculation

So far we have examined steady state performance modelling.
A crude transient performance model can be developed by relatively minor adjustments to the off-design calculation. A transient acceleration (or deceleration) is assumed to cover a large number of small time steps of, say, 0.01 s duration. During each time step, the shaft speed is assumed to be momentarily constant. So in the modified off-design iteration, $N\,$ is frozen and a new variable, the excess turbine power ${\delta }P_{w}\,$, allowed to float instead. After the iteration has converged, the excess power is used to estimate the change in shaft speed:
Now:
Acceleration torque = spool inertia * shaft angular acceleration
${\delta }\,{\tau }\,$ = $I\,$ $K\,$ $dN\,$ /$dt\,$
Rearranging:
$dN\,$ = ( ${\delta }\,{\tau }\,$/( $I\,$ $K\,$ )) $dt\,$
But:
${\delta }P_{w}\,$ = $2\,$ ${\pi }\,$ $N\,$ ${\delta }\,{\tau }\,$ /$K_{1}\,$
So:
$dN\,$ = ($K_{1}\,$ ${\delta }P_{w}\,$/ ( $2\,$ ${\pi }\,$ $I\,$ $N\,$ $K\,$)) $dt\,$
Or approximating:
${\delta }N\,$ = ($K_{2}\,$ ${\delta }P_{w}\,$ / ($I\,$ $N\,$)) ${\delta }t\,$
This change in shaft speed is used to calculate a new (frozen) shaft speed for the next time interval:
$N_{\mathrm {new} }\,$ = $N_{\mathrm {old} }\,$ + ${\delta }N\,$
The whole process, described above, is then repeated for the new time:
$t_{\mathrm {new} }\,$ = $t_{\mathrm {old} }\,$ + ${\delta }t\,$
The starting point for the transient is some steady state point (e.g. Ground Idle, Sea Level Static, ISA). A ramp of fuel flow versus time is, for instance, fed into the model to simulate, say, a slam acceleration (or deceleration). The transient calculation is first undertaken for time zero, with the steady state fuel flow as the engine match, which should result in zero excess turbine power. By definition, the first transient calculation should reproduce the datum steady state point. The fuel flow for $tnew\,$ is calculated from the fuel flow ramp and is used as the revised engine match in the next transient iterative calculation. This process is repeated until the transient simulation is completed.
It should be noted that the transient model described above is pretty crude, since it only takes into account inertia effects, other effects being ignored. For instance, under transient conditions the entry mass flow to a volume (e.g. jetpipe) needn't be the same as the exit mass flow; i.e. the volume could be acting as an accumulator, storing or discharging gas. Similarly part of the engine structure (e.g. nozzle wall) could be extracting or adding heat to the gas flow, which would affect that component's discharge temperature.
During a Slam Acceleration on a single spool turbojet, the working line of the compressor tends to deviate from the steady state working line and adopt a curved path, initially going towards surge, but slowly returning to the steady state line, as the fuel flow reaches a new higher steady state value. During the initial overfuelling, the inertia of the spool tends to prevent the shaft speed from accelerating rapidly. Naturally, the extra fuel flow increases the turbine rotor entry temperature, $RIT\,$ . Since the turbine operates between two choked planes (i.e. the turbine and nozzle throats), the turbine pressure ratio and the corresponding temperature drop/entry temperature, $\triangle T_{\mathrm {turb} }/RIT\,$, remain approximately constant. Since $RIT\,$ increases, so must the temperature drop across the turbine and the turbine power output. This extra turbine power, increases the temperature rise across the compressor and, therefore, the compressor pressure ratio. Since the corrected speed of the compressor has hardly changed, the working point tends to move upwards, along a line of roughly constant corrected speed. As time progresses the shaft begins to accelerate and the effect just described diminishes.
During a Slam Deceleration, the opposite trend is observed; the transient compressor working line goes below the steady state line.
The transient behaviour of the high pressure (HP) compressor of a turbofan is similar to that described above for a single spool turbojet.

## Performance software

Over the years a number of software packages have been developed to estimate the design, off-design and transient performance of various types of gas turbine engine. Most are used in-house by the various aero-engine manufacturers, but several software packages 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, GSP http://www.gspteam.com).

## Engine models

Whatever it's sophistication, the off-design performance model is not only used to predict the off-design performance of the engine, but also assist in the design process (e.g. estimating maximum shaft speeds, pressures, temperatures, etc to support component stressing).
Other models may be constructed to simulate the behaviour of the various individual components (e.g. rotor 2 of the compressor) in much more detail.

## Nomenclature

* $A\,$ flow area
* $A_{\mathrm {8calc} }\,$ calculated nozzle effective throat area
* $A_{\mathrm {8despt} }\,$ design point nozzle effective throat area
* $A_{\mathrm {8geometricdesign} }\,$ nozzle geometric throat area
* ${\alpha }\,$ shaft angular acceleration
* ${\beta }\,$ arbitrary lines which dissect the corrected speed lines on a compressor characteristic
* $C_{\mathrm {pc} }\,$ 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
* ${\delta }\,$ ambient pressure/Sea Level ambient pressure
* $({\delta }H/T)_{\mathrm {turb} }\,$ turbine enthalpy drop/inlet temperature
* ${\delta }N\,$ change in mechanical shaft speed
* ${\delta }P_{w}\,$ excess shaft power
* ${\delta }\,{\tau }\,$ excess shaft torque
* ${\eta }_{\mathrm {pc} }\,$ compressor polytropic efficiency
* ${\eta }_{\mathrm {pt} }\,$ turbine polytropic efficiency
* $g\,$ acceleration of gravity
* $F_{g}\,$ gross thrust
* $F_{n}\,$ net thrust
* $F_{r}\,$ ram drag
* ${\gamma }_{\mathrm {c} }\,$ ratio of specific heats for air
* ${\gamma }_{\mathrm {t} }\,$ ratio of specific heats for combustion products
* $I\,$ spool inertia
* $J\,$ mechanical equivalent of heat
* $K\,$ constant
* $K_{1}\,$ constant
* $K_{2}\,$ constant
* $M\,$ flight Mach number
* $N\,$ compressor mechanical shaft speed
* $N_{\mathrm {cor} }\,$ compressor corrected shaft speed
* $N_{\mathrm {turbcor} }\,$ turbine corrected shaft speed
* $p\,$ static pressure
* $P\,$ stagnation (or total) pressure
* $P_{3}/P_{2}\,$ compressor pressure ratio
* $prf\,$ intake pressure recovery factor
* $R\,$ gas constant
* ${\rho }\,$ density
* $SFC\,$ specific fuel consumption
* $RIT\,$ (turbine) rotor inlet temperature
* $t\,$ static temperature or time
* $T\,$ stagnation (or total) temperature
* $T_{1}\,$ intake stagnation temperature
* $T_{3}\,$ compressor delivery total temperature
* ${\theta }\,$ ambient temperature/Sea Level, Standard Day, ambient temperature
* ${\theta }_{T}\,$ total temperature/Sea Level, Standard Day, ambient temperature
* $V\,$ velocity
* $w\,$ mass flow
* $w_{\mathrm {4corcalc} }\,$ calculated turbine entry corrected flow
* $w_{\mathrm {2cor} }\,$ compressor corrected inlet flow
* $w_{\mathrm {4cordespt} }\,$ design point turbine entry corrected flow
* $w_{\mathrm {4corturbchar} }\,$ corrected entry flow from turbine characteristic (or map)
* $w_{\mathrm {fe} }\,$ combustor fuel flow