Assume that a Boeing 747 or Airbus A380 takes off at max takeoff weight. Minutes after takeoff, 3 engines fail, leaving only 1 engine operating.
Can the aircraft still achieve a positive rate of climb?
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I doubt that either aircraft can climb on a single engine (though momentum and airspeed could help, as @RonBeyer already commented). The closest an aircraft came to this condition was the British Airways Flight 9 when it ingested volcanic Ash.
During this incident, the aircraft flew on one engine for around 90 seconds. In this case, a report from the flying magazine says:
On one engine they could maintain height at 13,000 ft; over the next five minutes, the next three engines relit, and they begun a climb.
This seems to indicate that the aircraft was not able to climb with one engine. However, during this incident, the aircraft was not climbing, but was descending at 1800m per minute.
Note: There seems to be some confusion about the altitude. The captain's website gives the altitude as 12,000 ft.
I thought we could just check the thrust and weight, but it's not that easy. The static thrust of one engine looks like it is enough for both aircraft:
However, jet engine thrust changes with speed and altitude:
At low altitude where the engine still produces the rated thrust the best L/D speed will correspond to fairly low Mach number, around M0.3–M0.35, which is just above the minimum thrust point. IIRC the reduction is about 20% (exact profiles for specific engines don't seem to be published anywhere; occasionally some research papers have it for older engines, but not the new ones), and
Plus the L/D is probably worse then I estimated above due to the deflected ruder and ailerons needed to compensate for the asymmetric thrust. So, no, it's not enough (L/D 18 is typical value for clean airliner; exact numbers are hard to get).
At cruise altitude the Mach number would be around M0.7–M0.75, where the engines are above their static thrust already, but at those altitudes the engines are only producing about ¼ of the thrust, so you really need all of them. In fact the cruise altitude is always selected to be just below the operating ceiling, and operating ceiling is where the aircraft can no longer sustain more than 500 ft/min climb with all engines operating. If you lose even one engine in cruise, you always have to descend.
No it cannot climb with only one engine.
If it can climb with one engine the plane would be a twin jet not a quad jet. The number of engines is determine by power require for normal operation plus a margin in case one engine malfunctions during climb out. An engine is a very expensive piece of equipment both in term of up front cost and maintenance so two engines are cheaper than four.
If the engine is powerful enough to allow climbing with only one engine the manufacturer would make it a twin, not a quad. Simple as that.