Adiabatic steady state flow in a nozzle

[jason.bourne]

air enters an adaibatic nozzle steadily at 300 kPa, 200 degree celsius and 30m/s and leaves at 100 kPa and 180 m/s.
find the exit temperature?

when m using energy balance equation: h1 + [ (v1)^2 / 2 ] = h2 + [(v2)^2 / 2 ]
i get the answer 184 degree celsius.

but when i use the other relation i.e, pressure temperature relation

[ T1 / T2 ] = [ P1 / p2 ]^ ((k-1)/k) (i used k = 1.4)

i get very different values of temperature. is this equation not valid?
why m i getting different answers?

is this pressure temperature relation only valid for reversible processes?

 

[lsankar]

The first equation (with h1 and h2 in it) works for inviscid adiabatic flows (with no work done on the fluid by propellers, etc). It does not require flow to be reversible. As a result, it will work even across the shock wave.
The second equation works only for isentropic flows - i.e. reversible, adiabaric flows.

Thus, these two equations will yield the same answers (assuming p1, p2, h1, h2 , T1 are somehow known or given) only for isentropic flows.

 

[stewartcs]

air enters an adaibatic nozzle steadily at 300 kPa, 200 degree celsius and 30m/s and leaves at 100 kPa and 180 m/s.
find the exit temperature?

when m using energy balance equation: h1 + [ (v1)^2 / 2 ] = h2 + [(v2)^2 / 2 ]
i get the answer 184 degree celsius.

but when i use the other relation i.e, pressure temperature relation

[ T1 / T2 ] = [ P1 / p2 ]^ ((k-1)/k) (i used k = 1.4)

i get very different values of temperature. is this equation not valid?
why m i getting different answers?

is this pressure temperature relation only valid for reversible processes?

The first equation is the based on the full energy balance (with some assumptions that drop out some terms). It always works.

The second is for an Ideal gas, isentropic, and constant specific heat assumption.

Based on your problem description, the system does not meet the criteria for the second relation. If your system was given as isentropic, ideal gas, with constant specific heats, then you could use the second relation you listed.

BTW you have the T1/T2 and P1/P2 inverted (T2 and P2 are the numerators).

CS

 

[jason.bourne]

thank you so much