As water cools from a temperature in a tank that is above the boiling point outside in a rocket chamber, some will flash to steam, which drives all the water and steam out of the rocket chamber out through a bell nozzle. The exhaust velocity ranges from 260 m / s for a relatively low pressure tank, to 1,016 m / s for a chamber that is at a pressure of 22,090,000 N / m^2. This can be useful for a second stage rocket that starts in a low pressure atmosphere, for instance a rocket launched from a plane.

Thermodynamics can be used to calculate the exhaust velocity of such a system. Start with liquid water at its boiling point at pressure P. The water is partially flashed to steam and expanded isentropically down to 100,000 N / m^2, through an ideal nozzle. How fast is it going?

This can be solved by finding the entropy of saturated liquid water at P and at 100,000 N / m^2 and vapor at 100,000 N / m^2, then solving for the fraction of water that remains liquid under the assumption entropy is constant. From this, subtract the enthalpies to get the kinetic energy of the exhaust, from which the exhaust velocity can be calculated. This is for sea level, at lower pressures the difference in enthalpy would be greater, so the exhaust velocity would be greater.
 
Boiling Hot Water Rocket
Pressure ( N / m^2 ) Fraction Liquid Exhaust Velocity ( m / s )
1,000,000 0.86 260
3,000,000 0.78 424
5,000,000 0.73 517
9,000,000 0.67 643
11,000,000 0.65 692
13,000,000 0.63 738
15,000,000 0.61 781
17,000,000 0.59 823
20,000,000 0.55 892
22,000,000 0.50 983
22,090,000 0.48 1,016

This can also be used to assist a rocket engine by injecting the water into the nozzle of the engine, adding to the thrust at ground level, permitting it to operate with a large bell nozzle. Once the rocket rises and the outside pressure is lower, the water boost is no longer needed as the rocket gases can exit through the large bell nozzle on their own. This is one of the ways of making an aspirated rocket.
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