Steam Injectors

I was thinking this morning about posting on the topic of steam injectors, and their applicability to rocket engines – and then the conversation on Arocket turned to steam injectors. With that confirmation in hand, here are some thoughts/calculations. (Note that Universal Transport Systems is not currently planning on using systems like this.)

What is a steam injector?

Simply put, a steam injector takes steam, accelerates it through a nozzle, mixes the steam with water, and finally decelerates the steam/water mixture. Let's follow that through some example calculations:

1) Acceleration through a nozzle: about 3 atmosphere superheated steam is accelerated through a nozzle to the speed of sound (call it 300 m/s), dropping to 1 atmosphere. The steam cools somewhat to (let's say) 110 C. Note that if you slowed the steam back down at this point, you would end up just short of where you started in temperature and pressure.

2) Mix with water: The steam is mixed with water (let's say 90% by mass) at 20 C. Water has twice the heat capacity of steam, so each degree change of the steam changes the mixture temperature by 0.2 degrees. In addition, condensing the steam takes 500 times the energy of raising the water one degree. So the mixture ends up as all water, at about 80 C. The velocity of the mixture must conserve momentum, so it ends up at 30 m/s. But while the mixture is 90% slower than the steam, it is 1000 times denser!

3) Decelerate the mixture: as you decelerate the mixture, the pressure goes up. Because you have made the steam denser, the pressure goes up higher than the original 3 atmospheres! So you have taken used steam to take water to a higher pressure than the original steam.

According the the Bernoulli equation, the velocity squared divided by two plus the pressure divided by the density does not change. This "pump" works as long as the effect on final pressure caused by the change in density is larger than the effect of the slowing of the mixture. From the example above, here is the calculation of pressure gained from slowing the warm water from 15 m/s to a stop:

30^2/2 + P1/1000 = P2/1000
P2=P1+450000

So the pressure goes up by more than 4 atmospheres!

Note that the effect depends on the water increasing in temperature but not increasing in volume. This is where I believe most past efforts to apply this to rocketry have failed – the liquid to be used really needs to be sub-cooled. I'll follow this post up with an application of this to some of my favorite propellants!

Here is a table of some illustrative options:

Substance	water	water	water	water
Gas Heat Capacity	2	2	2	2
Liquid Heat Capacity	4.19	4.19	4.19	4.19
Latent heat of evaporation	2270	2270	2270	2270
Boiling point (C )	100	100	100	100
Gas Velocity (m/s)	300	300	600	1500
Gas Temperature (C )	110	110	110	110
Liquid Temperature (C )	20	20	20	20
Liquid Density (kg/m3)	1000	1000	1000	1000
Mass Mixture Ratio	0.9	0.86	0.86	0.86
Mixed velocity (m/s)	30	42	84	210
Mixed temperature (C )	77.22	97.05	97.05	97.05
Stopped Pressure (Atm)	4.46	8.73	34.93	218.32