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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 ?