Determining the acceleration of an aircraft

[David Lewis]

My basic assumption is that, even if the propulsion unit is 100% efficient, it will still consume some power -- an unavoidable consequence of adding momentum to a fluid.

 

[cjl]

But that power can be arbitrarily low by increasing the mass flow sufficiently. The power required for a given thrust depends on the energy required for a given change in momentum, which will depend both on inflow velocity and on the mass flow rate affected by the propulsion unit.

 

[insightful]

The power (as a function of thrust) formula I gave you gives you induced power...

Perhaps you could provide an example. For instance, the 1903 Wright flyer produced 90 lbf thrust with two 8.5 ft long propellers.

 

[David Lewis]

cjl wrote: "Yes, but without knowing the inflow velocity, you can't calculate any meaningful power number."

David Lewis wrote: Good catch. I forgot that kinetic energy is proportional to the square of velocity. It takes 3 times as much energy to increase the velocity of an air mass from 1.0 to 2.0 as it does from zero to 1.0. Yet delta v in both cases is the same. The power formula only applies to static thrust, if my reasoning is correct.

Acceleration seems to be greatest at the beginning, before rolling friction and air resistance come significantly into play, and initial acceleration is roughly equal to static thrust divided by gross takeoff mass. Acceleration gradually drops off to practically nothing before the airplane leaves the runway.

 

[cjl]

Acceleration gradually drops off to practically nothing before the airplane leaves the runway.

I wouldn't necessarily say that's accurate - at 150mph or so, modern jetliners still have a rather dramatic excess of thrust, and can still accelerate fairly hard. Once they leave the ground though, they will tend to throttle back a bit, and once they reach climb speed, they also climb at a gradient such that the airspeed remains constant (so all the excess thrust goes towards climbing, rather than increased speed).

 

[David Lewis]

I agree totally. My observations are not representative.

 

[insightful]

Perhaps you could provide an example. For instance, the 1903 Wright flyer produced 90 lbf thrust with two 8.5 ft long propellers.

The "induced power" formula actually is dimensionally correct. I don't know where the "2" factor comes from, so leaving that out and applying it to the Wright flyer data gives P = 2240 watts or about 3 hp. Their engine actually produced about 12 hp, so this formula (as pointed out) has little practical value.

 

[David Lewis]

This is my conception of acceleration versus time. Acceleration is zero until the brakes are released. Then acceleration quickly ramps up to the thrust divided by mass value and stays fairly constant until liftoff. After liftoff, the airplane noses up and the airplane sees gravity as an additional acceleration component.

The area under the curve (shaded yellow) is the speed at liftoff.

 

[cjl]

Why would acceleration increase after liftoff? It should drop, since the airplane is now climbing (thus some of the excess thrust goes into climb rate, rather than acceleration) and the drag rises dramatically after liftoff (especially induced drag).

 

[David Lewis]

I assume the maximum rate (or angle) of climb speed is higher than the lift off speed and, to reduce stress on the tires, the pilot postpones some acceleration until after liftoff. Additionally, when the airplane transitions from horizontal motion to an ascent, I counted the component of gravitational force parallel to the flight path the same as an acceleration to simplify performance calculations.

As a quick verification, when you are taking off in an airplane, the force with which the seat presses against your back is proportional to acceleration.

 

[cjl]

The pilot definitely will not postpone acceleration - that would increase runway length required. The pilot will simply rotate when the speed is appropriate, and the plane will take off. As for climb speed, yes it is higher than takeoff speed, but the acceleration will still slow at takeoff, since the thrust from the engines will be the same (or reduced) from the takeoff setting, and the drag will dramatically increase when the aircraft takes off.

 

[mfb]

I counted the component of gravitational force parallel to the flight path the same as an acceleration to simplify performance calculations.

Gravity acts downwards, the aircraft is flying upwards. Gravity is reducing the achievable acceleration.
Acceleration doesn't lead to much stress on the tires - they are passive anyway, unlike in cars where they transmit the accelerating force.

As a quick verification, when you are taking off in an airplane, the force with which the seat presses against your back is proportional to acceleration.

It is not, the orientation of the aircraft plays a role as well. Lift the nose and you get force between your back and the seat without any acceleration.

If aircraft would keep accelerating with the same magnitude after liftoff, they would reach their cruise speed within something like 2-3 minutes. That is not the case, they accelerate slowly while mainly gaining altitude.. The acceleration drops significantly after liftoff.

 

[cjl]

Gravity acts downwards, the aircraft is flying upwards. Gravity is reducing the achievable acceleration.
Acceleration doesn't lead to much stress on the tires - they are passive anyway, unlike in cars where they transmit the accelerating force.
It is not, the orientation of the aircraft plays a role as well. Lift the nose and you get force between your back and the seat without any acceleration.

If aircraft would keep accelerating with the same magnitude after liftoff, they would reach their cruise speed within something like 2-3 minutes. That is not the case, they accelerate slowly while mainly gaining altitude.. The acceleration drops significantly after liftoff.

To be fair, the acceleration is reduced because once best climb speed is reached, the aircraft is pitched up so all the excess thrust is used to climb, and the speed is held fairly constant for a good chunk of the climb (usually at something like 250-300mph - it's wherever the excess power is largest). If the aircraft were trying to accelerate rather than climb, it would reach cruise speed quite fast, but there are a lot of very good reasons why they do not accelerate to cruise speed at low altitude before climbing.

 

[mfb]

Sure, it could accelerate faster by staying lower. That would be very unpleasant for those living below the flight path, and it would increase drag way too fast in the dense atmosphere close to the ground. It wouldn't do that for long, however. Drag at sea level is more than twice the drag at typical cruise altitudes of larger airplanes.

 

[cjl]

Yes, but jet engine thrust at sea level is also more than twice the thrust at typical cruising altitude. The limit to safe jet aircraft speed at low altitude isn't whether it has the power to overcome the drag, it's the risk of structural damage (which, admittedly, is still drag-related).

 

[David Lewis]

mfb wrote: "It (force pressing against your back) is not (proportional to acceleration), the orientation of the aircraft plays a role as well. Lift the nose and you get force between your back and the seat without any acceleration."

David Lewis wrote: You're right. My graph attempts to depict what an accelerometer would show. During most of the climb regime, speed would be constant, and the accelerometer readout would display:
g * sin (angle of climb),
where g stands for acceleration of gravity.

 

[cjl]

Yep. That won't necessarily correlate well to engine excess thrust though, since the flight path angle won't be the same as the cabin deck angle.