GOING BACK TO BASICS
The aircraft initially has a certain quantity of stored usable energy made up of:
a. Chemical energy by virtue of its fuel load.
b. Potential energy by virtue of its height.
c. Kinetic energy by virtue of its speed.
To achieve maximum endurable we must expend the stored energy as slowly as possible.
MINIMIZING CHEMICAL ENERGY EXPENDITURE RATE
The rate of expenditure of chemical energy is essentially:
Fuel consumption rate = Thrust required x Specific Fuel Consumption.
(SFC is the mass of fuel that is required per hour to generate each unit of thrust).
Looking at the above equation we can see that to minimize the fuel consumption rate a good starting point would be to minimize the thrust required and to minimize the SFC.
Thrust required is equal to drag, so our ideal speed should be VMD. The drag at any given EAS does not vary significantly with altitude until compressibility effects kick in, so provided we do not go too high, the altitude should not affect the drag, thrust required, or fuel consumption.
SFC is more problematic because it varies with a number of factors including air temperature, air pressure and RPM. The most significant of these factors is RPM.
At very low RPM the majority of the fuel energy is used simply in overcoming friction and keeping the engine running, so little thrust is produced for each unit of fuel burned. This means that the SFC is very high at low RPM. As RPM increases, the basic running costs (in terms of energy) represent a decreasing proportion of the total fuel flow. The aerodynamic and thermal processes also become more efficient. This means that the SFC gradually decrease as RPM increases. Most text books quote a between 90% to 95% RPM as the RPM range within which SFC is lowest. The precise value of the optimum RPM will of course vary from engine to engine, but the figure of 90% to 95% is a reasonable starting point for the purposes of this thread.
Combining the above factors shows that for our maximum endurance we need to fly at VMD with our engines (or engine) running at 90% to 95% RPM.
At low altitude the thrust of both engines running at 90% to 95% RPM will be too great to balance the drag at VMD. Increasing drag using spoilers or flaps would simply waste fuel so this is not an option. But the thrust at any given RPM decreases as altitude increases, so if we climb to suitably high altitude we will achieve the required balance between drag at VMD and thrust at 90% to 95% RPM. But even at this altitude, the fuel flow with both engines running at 90% to 95% RPM will be far greater than that with one engine shut down and the other running at 90% to 95%. So a better solution would be to fly at the altitude where single engine thrust at 90% to 95% RPM is balances the drag at VMD.
MINIMIZING POTENTIAL ENERGY AND KINETIC ENERGY EXPENDITURE RATE
The power required is the rate at which an aircraft expends energy. So to minimize the expenditure rate of stored potential end kinetic energy during the descent to cruise altitude, the best option would be to shut down both engines and fly at VMP. So ideally we would descend with all engine out at VMP then star one engine and run it up to 90%-95% RPM as we level off at cruise altitude at VMD.
Unfortunately this contravenes the requirements to “fly at a single speed” and to “reach the cruise altitude asap”. Reaching the cruise altitude in minimum time would require a maximum speed descent, but then maintaining max sped in the cruise would not achieve maximum endurance.
ADJUSTING FOR REAL-WORLD FACTORS
All of the above does not of course take into account the actual performance characteristics of the specified aircraft and engines. To do this we would need to examine the appropriate data and adjust our airspeed, RPM and altitude accordingly.