Off-Design Performance

Off-design analysis with fixed throats, a two-spool turbofan solved with fsolve, control limits and the theta break, thrust lapse with altitude and speed, and flat rating

A hot afternoon in Dubai

In August, the afternoon high in Dubai averages about 40 °C (Dubai). An airliner taking off there gets less thrust from its engines than it would from the same runway on a winter morning — same engines, throttles fully forward. And at cruise, eleven kilometres up, this lesson's engine gives about a sixth of its take-off thrust. Nothing about the engines has changed. What changes is the air they breathe and the limits the engine control must respect. Civil engines give constant take-off thrust up to a particular air temperature; above it the turbine is at its temperature limit, and every degree costs thrust (flex temp). This lesson computes why.

At the design point the designer chose every number. Off design, the hardware is fixed — throat areas, blade angles, nozzle sizes — and the operating point is whatever satisfies the matching conditions of Lesson 8-1 at the new flight condition and throttle setting. The simplest useful model keeps the component efficiencies constant and the turbine throats choked. For a single-spool turbojet with a choked nozzle,

\tau_t

is then fixed, and the work balance, divided by its value at the design (reference) point R, gives the compressor directly.

\tau_c - 1 = (\tau_c - 1)_R\;\frac{T_{t4}/T_{t2}}{(T_{t4}/T_{t2})_R}

Off-design turbojet with choked turbine and nozzle: the compressor's work follows the corrected throttle setting Tt4/Tt2.

Derivation

Why Tt4/Tt2 is the throttle

With

\tau_t

fixed, the work balance

c_{pc}T_{t2}(\tau_c - 1) = c_{pt}T_{t4}(1 - \tau_t)

divided by its reference value gives the equation above. The compressor's temperature rise — and with it

\pi_c

, the corrected flow and the corrected speed — depends only on

T_{t4}/T_{t2}

A hot day raises

T_{t2}

; with

T_{t4}

capped by the turbine,

T_{t4}/T_{t2}

falls and the engine slides down its running line. Flying faster raises

T_{t2}

through ram compression and does the same, while raising the inlet pressure — which is why thrust can recover at high flight Mach numbers.

💡The two-spool turbofan

A separate-exhaust turbofan has more freedom. Three quantities float: the fan's temperature ratio

\tau_f

; the bypass ratio

\alpha

, set by how the fan nozzle and the core's choked turbine throat divide the air; and the low-pressure turbine's temperature ratio

\tau_{tL}

, free whenever the core nozzle is unchoked. Three conditions close the problem:

the LP shaft balance: fan work equals LP turbine work;
the LP turbine throat (choked) feeding the core nozzle;
the flow split between the fan nozzle and the HP turbine throat.

They are nonlinear in all three unknowns. Lab 1 hands them to scipy.optimize.fsolve, starting each solution from the previous one.

📐Throttling back, and cruise

The lesson's engine — fan pressure ratio 1.55, core pressure ratio 20, bypass ratio 8, 450 kg/s and 134.4 kN at take-off on a 30 °C day (Lab 1):

Bring $T_{t4}$ down from 1750 to 1350 K: thrust halves to 67.7 kN, the fan pressure ratio falls from 1.55 to 1.28, and the bypass ratio rises from 8.0 to 9.1.
Cruise at 11 km and Mach 0.8 with $T_{t4} = 1350$ K: 20.1 kN at a TSFC of 18.1 mg/(N·s), almost twice the take-off value, because the incoming air already moves at 236 m/s.
At take-off neither nozzle is choked; at cruise both are.

💡Control limits and the theta break

"Full throttle" is a control law, not a fuel valve wide open. This lesson's engine holds the turbine entry at its 1750 K limit unless that would push the compressors past their own limit — here an overall pressure ratio of 31; real controls use corrected or mechanical spool speeds. On cold days, at altitude and at low speed the pressure-ratio limit acts, and

T_{t4}

stays below its maximum. On hot days and at high flight Mach numbers

T_{t4}

reaches its limit. The inlet temperature where both limits meet is the theta break, here

\theta = 303.15/288.15 = 1.052

📐Thrust lapse

Full-throttle thrust as a fraction of take-off thrust (Lab 2):

Sea level: 0.77 at Mach 0.3 and 0.62 at Mach 0.8 — the jet's speed advantage over the incoming air shrinks.
11 km: 0.22 standing still, 0.16 at Mach 0.6, rising again to 0.18 at Mach 0.9 as ram compression helps.
Mach 0.8 at 11 km: 23.1 kN, 17 % of take-off, limited by pressure ratio with $T_{t4} = 1411$ K.

📐Flat rating

Take-off thrust against airport temperature at sea level (Lab 3):

Up to 30 °C: 134.4 kN, flat. On a 0 °C day rated thrust needs only $T_{t4} = 1577$ K, 173 K below the limit — margin that buys engine life.
Above 30 °C the turbine sits at 1750 K and thrust falls by about 0.8 % per degree: 92.1 % on a 40 °C Dubai afternoon, 84.9 % at 50 °C.

Essentially, all models are wrong, but some are useful.

– George E. P. Box and Norman R. Draper, Empirical Model-Building and Response Surfaces (1987) (Wikiquote)

⚠Flex: taking off on an imaginary hot day

Most take-offs do not need full thrust. Telling the engine control to assume a hotter day than the real one — "flex temp", or assumed-temperature thrust reduction — gives just the thrust the runway needs and saves engine life. Errors in calculating or entering the flex temperature have caused incidents and accidents, Emirates Flight 407 among them (flex temp).

💡Where you will use this

Lesson 8-3 follows the engine between these steady points. Aircraft performance manuals, take-off charts and mission analyses all rest on thrust-lapse and flat-rating curves like these, and Unit 10's adaptive-cycle engines exist to bend them.

📚Further reading

Jack D. Mattingly and Keith M. Boyer, Elements of Propulsion: Gas Turbines and Rockets (2nd ed., AIAA, 2016), on engine off-design performance and control limits.
Philip P. Walsh and Paul Fletcher, Gas Turbine Performance (2nd ed., Blackwell, 2004), on off-design and ratings.
Flex temp on Wikipedia.

↓But there is a problem

Every number in this lesson is a steady state. Move the throttle and the engine has to get from one to the next. On the way there is a rotor that cannot change speed instantly, a compressor that surges if pushed too hard, and a flame that blows out if starved. Something has to steer between them, many times a second.