With pressure-fed engines, is any measurable thrust generated by venting the pressurant out the engine bell after flameout?

Question

A pressure-fed rocket engine supplies fuel and oxidiser to the combustion chamber by pressurising the fuel and oxidiser tanks with an inert gas (typically helium or nitrogen). Presumably, when a pressure-fed engine exhausts its fuel and oxidiser and flames out, the gas that was pushing said fuel and oxidiser is now free to vent out the engine bell; does this belch of pressurant produce any significant amount of thrust?

Answer 1

It would in principle produce a small amount of thrust; compressed-gas systems with no combustion are sometimes used for very small attitude control thrusters. Specific impulse is generally poor -- under 100 seconds as opposed to ~300 for hypergolic bipropellants and ~200 for catalyzed hydrazine monoprops. High-expansion ratio upper stage nozzles in particular can extract a remarkable amount of work from even modest chamber pressure.

However, it's not normal to run a chemical propellant stage to complete depletion. If the flow of propellants "trickles", combustion becomes rough and can damage the chamber; if it goes oxidizer-rich because the fuel ran out first, excess hot oxidizer may rapidly damage the chamber or nozzle. Instead, for stages that "burn to depletion", flow is actually shut off at both the fuel and oxidizer valves simultaneously when a particular low level of remaining propellant is sensed. (h/t Organic Marble.)

Answer 2

I've been able to find one case in which an engine's propellant supply was (eventually) run down to zero, and the pressurant was then used to provide additional ∆v (although this wasn't a launcher engine).

The MESSENGER Mercury orbiter used a LEROS 1b engine for main propulsion; this is a pressure-fed hydrazine/MON engine, with the fuel and oxidiser tanks being pressurised with helium. The last of MESSENGER's propellants were used up in late 2014 fighting orbital decay, after which the helium was then, itself, periodically vented through the engine nozzle to hold the orbiter in orbit for as long as possible. To quote Wikipedia (emphasis added):

After running out of propellant for course adjustments, MESSENGER entered its expected terminal phase of orbital decay in late 2014. The spacecraft's operation was extended by several weeks by exploiting its remaining supply of helium gas, which was used to pressurize its propellant tanks, as reaction mass.^[82]

According to the article linked in the citation at the end of that extract, this was, apparently, the first time this had been done, which lends extra credence to @RussellBorogove's point about rocket engines almost never being burned to depletion, especially since it goes on to mention some disadvantages of using pressurant as propellant, and some reasons why this apparently hadn't been done before (emphasis, once again, added):

“The team continues to find inventive ways to keep MESSENGER going, all while providing an unprecedented vantage point for studying Mercury,” said APL’s Stewart Bushman, lead propulsion engineer for the mission. “To my knowledge, this is the first time that helium pressurant has been intentionally used as a cold-gas propellant through hydrazine thrusters. These engines are not optimized to use pressurized gas as a propellant source. They have flow restrictors and orifices for hydrazine that reduce the feed pressure, hampering performance compared with actual cold-gas engines, which are little more than valves with a nozzle.”

“Propellant, though a consumable, is usually not the limiting life factor on a spacecraft, as generally something else goes wrong first,” he continued. “As such, we had to become creative with what we had available. Helium, with its low atomic weight, is preferred as a pressurant because it's light, but rarely as a cold gas propellant because its low mass doesn't get you much bang for your buck.”