What Is the Exhaust Velocity of a Rocket Engine?

[.:Endeavour:.]

I want to calculate the exhaust velocity of a rocket. This is the formula that I found at http://en.wikipedia.org/wiki/Rocket_engine_nozzles which is:
$$V_e = \sqrt{{\frac{T*R}{M}}*{\frac{2*k}{k-1}}*[1-(P_e/P)^(^k^-^2^)^/^k] }$$

V_e = Exhaust velocity at nozzle exit, m/s
T = absolute temperature of inlet gas, K
R = Universal gas law constant = 8314.5 J/(kmol·K)
M = the gas molecular mass, kg/kmol (also known as the molecular weight)
k = c_p / c_v = isentropic expansion factor
c_p = specific heat of the gas at constant pressure
c_v = specific heat of the gas at constant volume
P_e = absolute pressure of exhaust gas at nozzle exit, Pa
P = absolute pressure of inlet gas, Pa

I'll use 1 of the Space Shuttle Main Engine for this equation. The statistics for the SSME are:

1. Thrust at Lift Off = 1.8 MN
2. Chamber pressure = 18.9399 MPa
3. Nozzle Area ratio = 77
4. Exhaust Velocity = 3,560 m/s
5. Chamber Temperature = 3,573.15 Kelvin
6. SSME consume 3,917 liters per second of fuel
7. Exit Pressure = 7.2326 kilopascals

For the molecular mass, do I have to find the mass of it in moles for Oxygen and Hydrogen seperately or when they mix together to form H₂O? I would like an explanation on what is c_p / c_v to further understand the equation.

 

[Astronuc]

One would use the mean molecular mass, which in the case of the SSME is H₂O and H₂. The mean or average molecular mass is weighted by the mole fractions.

The mixing ratio for SSME is 6:1 H₂:O₂, which would yield a 2:1 H₂ to H₂O

Ref: http://www.spaceandtech.com/spacedata/engines/ssme_specs.shtml

http://www.pw.utc.com/vgn-ext-templating/v/index.jsp?vgnextrefresh=1&vgnextoid=75a0184c712de010VgnVCM100000c45a529fRCRD

http://www.braeunig.us/space/comb-OH.htm

http://www.braeunig.us/space/propuls.htm - see Eq. 1.22

Sample problems - http://www.braeunig.us/space/problem.htm#1.5

 

[.:Endeavour:.]

If I remeber from chemistry this how I'll calculate the 0₂ and the H₂ mixture which is:
O₂ + H₂ = H₂O
2O₂ + 4H₂ = 2H₂O and CO₂ with CO (because its a conbustion)

I'll check the problems from the link that you posted, thank you for your help. My last question is the exhaust velocity of the rocket is its initial velocity or final velocity? I did a problem dealing with a rocket and I need it to find the velocity_f of a rocket using the P_i = P_f formula.

 

[.:Endeavour:.]

I've looked over the problem from http://www.braeunig.us/space/problem.htm#1.5 and I'll like to figure out how to find the chamber pressure of a rocket engine without actually having to test an engine.

I'll be using the SSME to find it's exhaust velocity. The chamber pressure of the engine is 18.9399 MPa or 18,939,900 Pa or 186,922.2798 atm. The only problem is that you cannot find the rest of its schematics in the chart that the site provides. Is there a way to do it because I think if I changed 186,922.2798 atm to 186.9222798 atm by moving the decimal 3 spaces to the left then I can find its Adiabatic Flame Temperature when the chamber pressure is 186.9222798 atm the temperature is 3,650 Kelvin. Because I moved the decimal 3spaces to the left, I move the decimal 3 spaces to the right which gives me 3,650,000 Kelvin. Can I use this method in finding other variables from the P_chamber when its off the charts?

 

[Astronuc]

The pressure would be determined by the work of the turbopumps which compress the fluids, LOX and LH2, and force them into the combustion chamber. The fluids mix and combust, so that over a short distance, they must go from cryogenic temperatures to about ~3600 K!

The combustion chamber has the maximum pressure in the system, and the pressure falls through the throat with the pressure gradient increasing beyond the minimum diameter in the throat through the nozzle.

The exit velocity, by convention is measured at the nozzle exit.

See page 33 (and figure 2-1) of this Rocket Propulsion Elements by George P. Sutton, Oscar Biblarz

Because I moved the decimal 3spaces to the left, I move the decimal 3 spaces to the right which gives me 3,650,000 Kelvin.

One cannot simply move the decimal 3 places in this way. This is not correct.

One has to look at the change in density of the combustion mixture as temperature increases. One has to consider the mass (continuity), momentum and energy balance equations.

 

[.:Endeavour:.]

But is there away to calculate the Optimum Mixture Ratio, Adiabatic Flame Temperature, Gas Molecular Weight, and the Specific Heat Ratio when the P_c is more than 250 atm?

 

[Astronuc]

Where does one obtain 250 atm, which is significantly greater than 189 in the SSME, which is pretty much optimized? The higher the pressure, the greater the necessary strength of the throat and nozzle materials at temeprature, which means the greater the mass of the structural material in order to minimize the stress.

 

[.:Endeavour:.]

Yeah, it is greater than the SSME engines and also the F-1 engine. So the more pressure there is in the combustion chamber the heavier the engine is going to be to widstand the high pressure and temperature of the engine so it doesn't burst or melt. There is a limit on how much pressure you can have in an engine before it becomes too heavy to fly, right?

 

[Astronuc]

There are practical mechanical limits on structures that must operate at high temperature, and high temperature creep and stress are limiting at temperatures below melting. IIRC, the practical limit is about 0.7 T_melt. The nozzle structure is cooled by the incoming coolant, but then there are significant thermal gradients on the structure. There is a trade off between thickness (in order to reduce stress for a given load) and thiness for cooling.

 

[.:Endeavour:.]

That's the reason why the F-1 engine operated at 69.085 atm and the SSME operate at 186.922 atm because the difference of their fuel. The SSME can obtain a higher pressure because the Adiabatic Flame Temperature for kerosene is higher than hydrogen at low pressures. What temperature is 0.7 T_melt?