I was not completely happy with the last post’s vagueness so I’m adding more formal treatment here. This long post still ends up lazily speculating around the advantages and disadvantages of air breathing propulsion in the end though. 🙂

Effective or apparent exhaust velocity (ISP to some, but that is an old-fashioned troublesome term so I won’t use it) is $v_{exeff}=\frac{F}{\dot{m}_{prop}}$. That means, thrust for a certain propellant flow. Higher is better.

For rockets, the thrust $F_R=\dot{m}_{propR}v_{exR}$ but for air breathers, $F_A=\dot{m}_{totA}v_{exA}$. Air breather is denoted by A and rocket by R.

Thus, for the rocket, the effective exhaust velocity is the real exhaust velocity (it exhausts only propellant products). $v_{exeffR} = v_{exR}$.

For the air breather it’s more complicated. Only fuel is used as propellant flow but a lot more is expelled:
$v_{exeffA} = v_{exA}\frac{\dot{m}_{totA}}{\dot{m}_{propA}}$

So, what is the ratio of fuel flows for a rocket and an air breather of same thrust? The inverse ratio of effective exhaust velocity of course. How much better is an air breather in this case?

$R_{breathe}= \frac{v_{exeffA}}{v_{exeffR}} = \frac{v_{exA}}{v_{exR}}\frac{\dot{m}_{totA}}{\dot{m}_{propA}}$ – the Air Breather’s advantage.

So, the rocket might benefit from a higher real exhaust velocity but the air breather benefits from the ratio of total exhaust mass versus propellant mass.

But that’s not all.

### Practical Questions Affecting Effective Exhaust Velocity

If a jet engine burned hydrocarbon propellant to mostly carbon monoxide, CO, like rockets do, 1 kg of kerosene would need roughly 2 kg of oxygen: C8H10+6½O2 => 8CO + 5H2O. This would already drop the air breather’s propellant flow to one third of the total mass flow, or increase effective exhaust velocity (ISP) to 3X the actual exhaust jet velocity . But the ambient air contains nitrogen too, 80% of it actually by mass. Now the exhaust mass consists of 1 part propellant, 2 parts ambient oxygen and 8 parts ambient nitrogen, meaning 11 times the mass of the original propellant.

And jets can burn lean. Mostly, they actually have to, at least in the main burner to prevent the turbine from melting. So the exhaust mass can rise even more. This of course has the effect of lowering the real exhaust velocity. Nevertheless adding more reaction mass (mass that is not from the propellant flow!) at slower exhaust velocity gives more thrust for same energy, same propellant flow and thus a higher ISP. And modern jets are turbofans, where a big portion of the flow is just pumped by the fan. Although then again, afterburners can change this picture.

It is a slightly deeper question of gas dynamics how diluting the temperature to more reaction mass actually behaves – and the chemistry is different in stoichiometric and lean burning too, as the CO turns to CO2. For one, even though fuel consumption goes down, you need physically bigger engines for the same thrust if you run leaner because of the lower exhaust velocity.

An F-16’s F110-GE-129 turbofan engine might at a particular mid-throttle setting exhaust at 600 m/s (Mach 1.7 to 2), consume 1.0 kg/s of fuel and produce a thrust of 49 kN, with an exhaust mass flow of 88 kg/s. (Source, just converted to sane units.)

A typical hydrocarbon rocket producing 50 kN would exhaust at 2500 m/s and need a 20 kg/s propellant flow.

So the real exhaust velocity in this air breather vs rocket comparison case is 0.6 vs 2.5 km/s while the effective exhaust velocity (obsoletely ISP) is 49 km/s vs 2.5 km/s. The air breather is a whopping 20 times better in this regard.

### Other Issues

The F-110 masses about 1800 kg and has a max dry thrust of about 78 kN and with afterburner 130 kN. That yields thrust to mass numbers of 43 and 72 m/s^2. For rockets numbers like 800-1000 m/s^2 are not rare, so a rocket with such thrust would mass perhaps as little as 50 kg.

Rockets are good at thrust, air breathers are good at fuel economy. Acceleration requires thrust, cruising requires fuel economy. Rockets are the same at all speeds, air breathers usually get lousier and lousier at higher speeds until they stop working completely.

Let’s calculate some examples with a few notional craft.

### Use examples

##### Cruiser

An atmospheric winged long distance cruiser masses 10 t empty and 20 t when fueled and requires a takeoff T/W of 0.4 or roughly 80 kN of thrust.

If we strap on an 1.8 ton F110 turbofan (afterburner not used), mass ratio will be 21.8/11.8 = 1.85 and delta vee 49 km/s*ln(1.85) = 30 km/s. (Takeoff T/W 1.8/21.8=0.37)

With a 80 kg rocket engine, the mass ratio will be 20.08/10.08 = 2.00 and delta vee 2.5 km/s*ln(2.00) = 1.7 km/s.

So the cruiser will go much much farther with the turbofan engine than with the rocket, probably at least 15 times as far.

##### Accelerator

A pop-up vertical first stage booster or launch assist platform masses 2 tons empty and has 10 tons of propellants. It also has 10 tons of mass on top of it (upper stages, payload). The 22+ ton stack requires a T/W of 1.5 or at least 320 kN of thrust.

320 kN means 320 kg of rocket engine, mass ratio 22.3/12.3=1.8 and delta vee of 1.5 km/s. Perhaps staging after a continuous 80 s full throttle burn, this thing could coast to a 50 km apogee, come down and be recovered.

Air breather: How many afterburning turbofans do we need to achieve the takeoff T/W? T/W = (n*130/g)/(22+n*1.8 ) = 1.5 <=> 13n=33+2.7n <=> n = 3.2. So the engine mass will be 5.7 t. And the mass ratio will be 27.7/17.7 = 1.56. Also, we can approximate that the effective exhaust velocity drops to 60% of the dry value, or 30 km/s, when using the afterburner for thrust augmentation (source). Thus, the delta vee will be 30 km/s * ln(1.56) = 13 km/s.

Now, assuming some gravity and drag losses, the rocket booster probably wouldn’t reach a velocity over 1 km/s, or roughly Mach 3. This is still probably outside of the particular engine’s specification (it probably chickens out at around Mach 2.5), although it’s possible something could be done to make it at least operate at higher speeds. So it is likely that the F110 air breather engines can simply not be used for this mission.

### Discussion

The air breather’s 13 km/s figure has a lot of room for losses that are inevitable when the design is changed to fly at higher speeds.

It seems that air breathing engines could theoretically have something to contribute to low Mach and low altitude “launch assist platform” style stages. It is relatively straight forward to construct a rocket engine of the aforementioned performance (2.5 km/s exhaust at sea level), but constructing air breathing mach zero to three capable engines is probably not very easy – even when accepting quite low performance.

The P&W J-58 engine of the SR-71 could operate continuously at Mach 3 and had a thrust of 150 kN and a thrust to weight ratio of 5.2. The movable inlets and some other associated big hardware is probably not accounted in that number. And that thrust could be at sea level and static conditions – I do not know how it varies with altitude and speed. Also specific fuel consumption or effective exhaust velocity numbers are hard to get – there is a number of 8,000 gallons per hour which translates to 30.3 m^3/h or 8.4 liters per second. At JP-7 density of 0.8 kg/l it’s 6.7 kg/s. And that makes the effective exhaust velocity 22 km/s. Half of the F-16:s F110 engine. Still much better than the “average rocket” in that regard.

Existing fighter jet engines are optimized to below Mach 2 speeds – the booster uses of such engines are a bit iffy – they could be useful in increasing payload like the current small solid attachments, but also give some headaches with their delicateness and need for recovery. Here is Dani Eder’s positive take on them.

### Booster Use Speculation

How would a purpose-built air breathing turbine engine booster look like? There are a million ways to approach the problem. One interesting avenue could be Mig 25 style low compression ratio turbojets that could be easy to make. Unfortunately they suffer from low thrust at low speed. Or a J-58 copy with modern materials. And then there are the air turborocket cycles handled in the previous post, providing perhaps intermediate action between rockets and jets but bringing additional problems. The actual performance trade of effective exhaust velocity (ok, I’ve said it so many times, it’s v_exe from now on) and thrust to weight vs velocity, is the subject of another post – or rather maybe too much work for simple blog posts as original material anyway.

Or one could forget the complexities and actually use a subsonic aircraft – they have low operating cost, good safety, and even a pretty good payload fraction because of the well lifting wings. Increasing the speed of a carrier aircraft usually decreases its carrying ability faster than it gives useful staging speed, so that actual total payload to orbit drops! Hence Mach 5 winged craft haven’t tended to actually been very cheap launchers when analyzed – they have had sucky payload fractions and have thus grown to behemoth sizes.

Overall, of course, one is left with the question, why bother with it at all? The air breathing engines can provide high ideal delta vee but can’t cope with high speeds. They are having trouble even with launch assist platform style missions. Mach 3 or 5 is nothing for rockets.

Even if we disregard the fuel use advantage, one can look at air breathing engines and see them often being quite dependable and maintainable compared to rocket engines – it is tempting. But there is no fundamental reason why rocket engines couldn’t be like that as well – there just have not been that many effrts towards that goal.

The last relatively carefree rocket engine before the present newspace ones was with the XLR-99 engine for the X-15 hypersonic rocketplane. A quite highly reusable engine, and the few X-15 craft flew about 200 missions in total with only a few engines and craft. I might do a post just on it some day.

The SSME was a reusable engine too, but it was an overambitious program and resulted in a system that can be called refurbishable but requires too much effort between flights to be really along the evolutionary path to more or less operate-like-a-jet rocket engines.

### Conclusion

So, what to think of air breathers for space? I’d give them a few percents of attention, mostly at the low speed end. There are lots of promises every now and then of making easy high speed air breathing propulsion systems – that then just quietly disappear after failing. (From NASP’s hypersonic scramjets of the eighties to the DARPA MIPCC modded fighter jet engines of the noughties etc etc.) It is a hard problem.

I’d say if you want a comprehensive push for better launcher propulsion, concentrate the majority of effort on the much easier problem on making better rocket engines. Better by being reliable and reusable and requiring little maintenance or operations complexity.

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