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When using an Assumed Temperature for Takeoff, is it common operational practice to also use a Derate?

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Yes, that is allowed.

You can combine selection of a fixed derate and an assumed temperature for the takeoff thrust on the FMC N1 Limit page, but only up to a reduction of 25%.

Takeoff Derate

Fixed derates can be selected on the N1 LIMIT page. [...]
Derated takeoff rating can be further reduced by assumed temperature. [...]

Assumed Temperature Thrust Reduction Takeoff

A takeoff thrust less than the full rated thrust may be achieved by using an assumed temperature that is higher than the actual temperature. The desired thrust level is obtained through entry of a SEL TEMP value on the N1 LIMIT page or TAKEOFF REF page 2. [...]
The maximum thrust reduction authorized is 25 percent below any certified rating. [...]
If the assumed temperature method is applied to a fixed derate, application of additional power should not exceed the fixed derate N1 limit as loss of directional control could occur while on the ground.
When the assumed temperature method is used with full rate, the reduced thrust setting is not considered a limitation. If conditions are encountered where additional thrust is desired, the crew can manually apply full thrust.

(Boeing 737NG FCOMv2 11.32.3 - Flight Management, Navigation - Flight Management Computer)

You can see the current selection on the Upper Display Unit:

737 Upper Display Unit

2 Thrust Mode Display

[...] Thrust mode display annunciations are:

  • TO – takeoff
  • TO 1 – derated takeoff one
  • TO 2 – derated takeoff two
  • D-TO – assumed temperature reduced thrust takeoff
  • D-TO 1 – derate one and assumed temperature reduced thrust takeoff
  • D-TO 2 – derate two and assumed temperature reduced thrust takeoff

(Boeing 737NG FCOMv2 7.10.2 - Engines, APU - Side by Side – Displays)

What would actually be used by an airline depends on the exact situation. The derate and the thrust reduction via assumed temperature are not the same thing because the derate affects Vmcg, but the assumed temperature does not (see also What is the difference between Assumed Temperature and Derate takeoff thrust?). Today, most airlines will use an EFB (or a separate tablet computer) to perform the TO calculations and the computer will optimize the thrust reduction while taking all legal requirements for V1 (which may be affected by Vmcg, which is a lower limit for V1), second/fourth segment climb gradient, obstacle clearance and possibly noise reduction into account.

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  • $\begingroup$ The maximum 25% reduction limit refers to %N1? So if a Max power Takeoff is calculated at 98.0 %N1, the maximum reduction can be as low as 73.5% ? I find this to be an extremely low power setting. Or does the 25% limit refer to Engine Power and not %N1? $\endgroup$ – Stamatis Vellis Nov 17 '20 at 11:13
  • $\begingroup$ Yes, that refers to %N1, but not based on the calculated max. takeoff (98% in your example). It is based on the certified rating thrust for that engine, which you can find in the TCDS (Type Certificate Data Sheet). I don't think it is extremely low, if you consider that the OEW of a 737-800 is ~41t and MTOW is ~79t. With such a large range of possible TO weight, you also want a large range of thrust. $\endgroup$ – Bianfable Nov 17 '20 at 11:33
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    $\begingroup$ @StamatisVellis Just realized I was slightly wrong in my last comment: it is not technically 25% reduction in N1, but rather a 25% reduction in actual thrust. Since thrust and N1 are not perfectly proportional (see e.g. this image), this may not be equal to 25% less N1, but it is somewhat close. $\endgroup$ – Bianfable Nov 17 '20 at 11:56
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Yes, both can be used together, allowing actual thrust for takeoff to be much lower than simply 25%. Mostly, Operators prefer the lowest possible thrust so as to have longer engine life, more time on the wing in regular service than off the wing in maintenance.

A simple visualization of the derate is to pretend there's a smaller, less powerful engine on the wing and the airplane performance certified accordingly.

Typically:

Max Assumed Temp on ‘No derate’ is 100 - 25 = 75% of full thrust

Max Assumed Temp on ’10% derate’ is 90 - 22.5 = 67.5% of full thrust

Max Assumed Temp on ’20% derate’ is 80 - 20 = 60% of full thrust

Even at 60% of full thrust, the N1 remains in the higher 80’s for anything I’ve flown. There’s no simple linear relationship between the reduced N1 and percentage reduction of thrust. Also note that the N1 for 100% thrust - No derate, No Assumed Temp - is normally more than 100 and the red line N1 limit higher than that.

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  • $\begingroup$ Indeed. I just made the calculations for maximum allowed Assumed Temperature (67 degrees) plus maximum Derate (22K) for the 737-800W/CFM56-7B26, under Sea Level, ISA conditions. The minimum %N1 (i.e maximum derate) I came up with was 86.1% N1 $\endgroup$ – Stamatis Vellis Nov 17 '20 at 14:05

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