Question: You always keep at least some fuel in your auxiliary tanks. Today, as you approach the 2-hour point of a 3-hour flight, you are not planning on using them because 3 days ago you told your FBO to top-off your mains, and the 46 gallons in each main will give you a range of roughly 4.5 hours. Suddenly, however, the rear engine starts sputtering with the tell-tale signs of fuel starvation. Your initial thought is that you've lost the engine's mechanical fuel pump, requiring you to activate the right main tank electric pump, but a quick glance at the fuel guages -- both at or near "E" -- tell you that you're out of fuel. Aside from kicking yourself for not having climbed on the wings to visually check the fuel during the pre-flight and for not monitoring the fuel guages in flight, what do you do?
Answer: Don't waste precious time trying to connect the sputtering engine to its auxiliary, which won't re-start the engine because it doesn't have an electric pump to overcome the voids in the line. Instead, switch immediately the running front engine to its auxiliary tank, monitor it for a minute, then switch the windmilling but dead rear engine to the left main (cross-connect), run the left main auxiliary electric pump until the engine re-starts, monitor it for a minute, and then connect it to its auxiliary tank. (You don't want to run off the left main for more than a minute, because it also is probably very low on fuel and you might run it dry; remember that even though the left main is being replenished by the return fuel coming back from the front engine, the rear engine is drawing from the tank roughly twice as much fuel as is coming in.)
But you are not out of the woods yet. Recognizing that you don't know for sure how much fuel is in the auxiliaries, you head for the nearest airport. You are conscious of the fact that one of the auxiliaries may run dry before you get there, so you're ready (when that happens) to switch the sputtering engine to its main tank (which won't be empty then because it got replenished with returning excess fuel that was drawn by the engine from the auxiliary). Remember that you'll have to turn on the electric pump until it re-starts. How long will the fuel in that main last after you switch to it? Easy: about as long as it lasted when the engine was running on the auxiliary (because roughly half of the fuel which the engine drew from the auxiliary was actually used by the engine and the other half was excess fuel which ended up in the main tank). With this in mind, you jot down on your knee-board the time at which you switched the engines to their auxiliary tanks.
Incidentally, if the airport is too close -- say, a few minutes away -- you have a dilemma. You're supposed to put both engines back on their main tanks for landing, but the mains may not have enough fuel (because the engines didn't run long enough off the auxiliaries for the returning-excess-fuel replenishment to get into the main tanks sufficient fuel to sustain the full landing pattern). If you think that landing on the auxiliaries is an option (after all, how likely is it that you'll lose a mechanical fuel pump during the landing and need the electric pump?), think again. You admitted not knowing how much fuel is in the auxiliaries, so you could run out on short final while running on the auxiliaries. The solution here is to circle near the airport at a safe altitude (say 3,000 feet AGL) for 5 or 10 additional minutes on the auxiliaries to further replenish the mains. Then switch one engine to its main and a couple of minutes later switch the other engine to its main (notice that you won't need the electric pumps for this switch-over if you did it while the engines are running well off the auxiliaries). Now you're ready to land, and should be able to run off the mains about the same amount of time that you ran off the auxiliaries.
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