How to calculate the temperature of the air coming out of a small hole (Mendeleev - Klaiperon)?

Question

Physics

Hello, a little to solve the problem of calculating the air temperature at the outlet of the nozzle.
The essence of the problem. How will the temperature of the outgoing air change from a hole with a certain cross section, knowing the cross section of the inlet channel and the pressure in it.
* An example of the design (if it is not clear described). T1, P1, S1, S2 are known. Find T2.

My solution: I thought about the need to find the speed of the outlet air flow. I took a textbook where I found the derivation of a similar formula . Further, from the Bernoulli

equation , you can find the pressure of the air flow in the outlet channel. (Considering that the height difference can be neglected, and the air flow velocity on the left side is zero) After knowing the air outlet pressure, you can find the temperature using the formula


mendeleev-clapeyron .

But here the critical section is not taken into account. Thus, it seems that this is not correct.

Help solve the problem. Thank you in advance.