laval nozzle and kinetic energy

[Mindblast]

I'm trying to simulate an air-breathing engine for atmospheric flight using mainly formulas for isenthropic flow from the NASA pages: http://www.grc.nasa.gov/WWW/K-12/airplane/rktthsum.html

Now i came across a little oddity when trying to simulate expansion through a laval nozzle. The kinetic energy of the exhausted airstream seems to be higher than the energy i put into it.

Heres an example:

Assume we use ambient air at sea level, compress it by a ratio of 5 and then exhaust it through a laval nozzle:

ambient pressure: pa =~ 101000 Pa
ambient temperature: Ta = 288K

pressure after adiabatic compression: pc = 505000 Pa
temperature after adiabatic compression: Tc = Ta * ((pc / pa) ^ ((gamma -1) / gamma)) = 288 * 5 ^ (0.4 / 1.4) =~ 456K
with a gamma value for air of ~1.4

now for simplicity i assume a throat area of the laval nozzle At of 1 m^2, the massflow can then be calculated by
mdot = pc * At / sqrt(Tc) * sqrt(gamma / R) * (((gamma + 1) / 2) ^ (-(gamma + 1)/(2*(gamma - 1))))
= 505000 * 1 / 21.35 * sqrt(1.4 / 286.9) * 1.2 ^ (-2.4 / 0.8)
= 956 (kg/s)

now i determine the optimum exit mach-number so that exit pressure pe = ambient pressure pa:

pe/pa = (1 + (gamma - 1) / 2 * Me^2) ^ (-gamma/(gamma-1))

solving for Me:

Me = sqrt(((pe/pa) ^ (-(gamma-1)/gamma) - 1) * 2 / (gamma - 1))
= sqrt(((101000/505000) ^ (-0.4/1.4) - 1) * 2 / (0.4))
=~ 1.71

then i can calculate exit Temperature:

Te = Tc * (1 + (gamma - 1) / 2 * Me^2) ^ -1
= 456 / (1 + 0.2 * 1.71^2)
=~ 288K (ambient temperature as one would expect)

then i calculate the exit velocity as:

Ve = Me * sqrt(gamma * R * Te)
= 1.71 * sqrt(1.4 * 286.9 * 288)
=~ 581.6 m/s

so far so good..

now i calculate the kinetic energy for 1 second of the exhaust airstream by:

Ekin = 0.5 * mdot * Ve ^ 2
= 0.5 * 956 * 581.6^2
=~ 161.7 MJ

now i calculate the energy used to compress that same airstream:

Ecomp = R * mdot / (gamma - 1) * (Tc - Ta)
= 286.9 * 956 / 0.4 * 168
=~ 115.2 MJ

So i have 46.5MJ more kinetic energy in the exhausted airstream than i put into compressing it which can't be correct. But where is my mistake ?

 

[Urwumpe]

Did you include the combustion chamber in the calculations?

EDIT: ah, ok, now I see the math... the number looked very similar to what you get by burning fuel, that is why I asked.

But I think the compression term is missing something critical...shouldn't you have a change in volume flow during the compression inside the nozzle, but only a little change in temperature?

 

[Mindblast]

Did you include the combustion chamber in the calculations?

EDIT: ah, ok, now I see the math... the number looked very similar to what you get by burning fuel, that is why I asked.

Yes the example is a completely abstract setup with just a magical compressor followed by the laval nozzle. The engine i'm trying to simulate will have electrically powered compressors/fans and an electric heating device downstream of the compressors and then the laval-nozzle. I just reduced the thing as much as possible here to simplify the example.

But I think the compression term is missing something critical...shouldn't you have a change in volume flow during the compression inside the nozzle, but only a little change in temperature?

I'm not sure. The whole idea of the math is a kind of Brayton cycle simulation (adiabatic compression, isobar heat addition, adiabatic expansion) but in this example without the heat addition part.
The temperature rise here is just from work done in the adiabatic compression.


I just noticed that the two energy values i calculated above differ exactly by a factor of 1.4 (= gamma), hardly a coincidence... that would mean if i multiply by gamma in the energy equation for the adiabatic compression everything would be fine... weird...

---------- Post added 08-30-2009 at 06:04 PM ---------- Previous post was 08-29-2009 at 10:27 PM ----------

I think i've found the problem.. the equation i used above for compression energy is wrong because it gives the energy used for energy rise at constant volume. When i use the equation for constant pressure instead:

Ecomp = R * gamma / (gamma - 1) * (Tc - Ta) * mdot

or

Ecomp = cp * (Tc - Ta) * mdot

cp = heat capacity of air at constant pressure (1005 J/(kg*K))
gives me ~161MJ just as i would expect

So Urwumpe it seems your suspicion with the volume flow change was right.

This is a bit counterintuitive to me as in a compressor the pressure is not constant obviously but as it gives me exactly the kinetic energy it seems to be correct.

 

[Urwumpe]

Remember Bernoulli: Total pressure = static pressure + dynamic pressure. Is a bit more complex for supersonic flow, but the general idea remains the same.

A nozzle does not change total pressure, increases exhaust speed at the expense of static pressure.