What would be the minimum number of engines required to climb in a 747?

Question

There was a really good Air Crash Investigation by Nat Geo which showed the British Airways Speedbird 9 issue where volcanic ash swept over all four engines and through the air vents inside the cabin. As a result, all engines stopped working.

After around fifty attempts to restart the engines, engine four roared back to life. Although this did bring good news to all onboard, one engine would still not have given them enough power to clear the mountains that were in their path around Jakarta. Due to this, the Captain turned back to sea.

Nat Geo Flight 9. All engines stopped.

I don't think National Geographic would be super accurate but this looks like one engine could have cleared this mountain.

Anyway, this got me thinking.

A 747 can fly on one engine but can't climb? This does make sense, one engine can make a 747 fly for longer but it cannot climb with just one. What then would be the minimum number of engines required for the jet to clear the mountains? Two?

Answer 1

@Michael Hall said it best, in the first sentence of item 1 of his answer:

There are simply too many variables for a definitive answer.

Here are two extremes to illustrate the range of answers insofar as altitude is concerned:

Let's say you have a 747 at it's optimum altitude in cruise with all four engines operating, say 37,000 just for the heck of it. Now let's say you lose one engine. Can you now climb on three engines? No, in fact you won't even be able to keep the altitude you have. You will start drifting down.

Now let's say that same 747 with the same load (actually heavier because it would still have the fuel on board that it used to get to its 37,000 foot cruise) lost an engine while on the takeoff roll but after V1. Will it climb? Yes, even with the drag from the gear and takeoff flaps you will be able to lift off the runway and climb.

But back to the one-engine scenario. My gut level feeling is that any 747, if carrying no passengers or cargo and only, say, 30,000 lbs of fuel and down low (5,000 ft?) and in a clean configuration could eke out a little climb. Can't prove it though.

But start adding weight or drag and the answer would change. The first model to not be able to do it with weight and drag being added would be the 747-100. The first -100s I flew had a max takeoff weight of 735,000 lbs as I remember. The next model to fail would be the -200. 825,000 lbs was the highest max takeoff for the -200s I flew. The highest max takeoff that I have for a -400 on the DOS weight & balance software I wrote was 910,000 lbs.

The biggest difference performance-wise is due to engine size. The more power the one engine has, the more it can keep aloft.

And there are other variables. What was the nature of the failures of the three engines? Are they windmilling or seized? Even whether the remaining engine is an inboard engine (2 or 3) or an outboard engine (1 or 4) is a factor. When it comes to performance, even how many insects are splattered against the leading engine of the wing can make a difference.

Answer 2

We use the polar of the 747 as given in this answer:

Let's suppose that it took off at its MTOW of 377'000kg and that the accident happened somewhere midway i.e. with half of its 198'000 liters of fuel burned: this corresponds to some 0.5x198'000x0.82=81'000kg and therefore to a total weight at the time of the incident of 377'000-81'000=296'000kg=2'903kN.

According to the captain's website, engines number 4 and 3 restarted somewhere at around 12'000ft ($\rho$=0.75kg/m³) and at a speed of some 300kts=154m/s. With this data we get a lift coefficient of:

$C_l=\frac{W}{½\rho V² S}=\frac{2'903'000}{½0.75\times 154² \times 511}=0.64$

According to the polar, this corresponds to a drag coefficient of about 0.038 which gives a drag i.e. a needed thrust of:

$D=½\rho V² S C_d=T=½0.75 \times 154² \times 511 \times 0.038=173kN$.

Each B747-200 turbofan delivered 243kN of thrust at sea level which became more or less one half at 12'000ft i.e. 120kN. So one engine was definitely not even enough to win drag and keep the speed while two were enough not only to win drag but also to gain some altitude. This result perfectly matches the incident's description: "engine number four finally started, and at 13:56 UTC (20:56 Jakarta time), Moody used its power to reduce the rate of descent. Shortly thereafter, engine three restarted, allowing him to climb slowly".

Answer 3

1. There are simply too many variables for a definitive answer. It depends greatly upon weight, how high the mountains are, starting altitude, and distance available to climb.

2. 1800 feet seems much lower than necessary. Under normal conditions in an unpressurized aircraft with passengers you would want to get below 10,000 feet. However, in an emergency where terrain is a factor people will likely remain conscious at 10,000 to 18,000 feet. Above that useful consciousness decreases rapidly.

P.S. For my understanding, is your comment "this looks like one engine could have cleared this mountain" based on your analysis of the picture you posted?!