# Unsteady flow inside a pipe with heat exchange and pressure drop

I am studying the behaviour of a hot air gas inside a circular-area (constant area $$A$$) pipe. The gas exchanges heat through the wall and experiences pressure drop while flowing inside the pipe. I want to study the unsteady phase in 1D.

Initially, the hot gas is at rest and at a hot temperature $$T_0$$ and at pressure $$P_0$$ inside a tank upstream to the pipe, whereas the pipe wall is at $$T_{out} and the air inside the pipe is at rest at $$T_{out}$$ and at $$P_{out}

The hot air is an ideal gas (constant $$c_p$$, $$c_v$$, $$\gamma$$).

At the end of the pipe there is a nozzle expelling to the outer atmosphere $$P_{out}$$, $$T_{out}$$.

Moreover, I know the pressure drop equation along the pipe (thanks to Idelcik book) : $$dP = f(geometry,Reynolds,speed,density)$$

I would like to study the transient phase when we open the tank with hot pressured air. I would like to know the governing equations which takes into account the heat exchange, the loss of temperature and the pressure drop.

I know the governing equations for steady flow, and that for steady flow the mass flow $$\dot{m}$$ is regulated by the choked throat of the nozzle.

However, I don't know the governing equations for unsteady flow, plus I have a problem about the calculation of the mass flow and about the calculation of the pressure drop.

What are the governing equations of unsteady flow with heat exchange inside the pipe ?

Plus how should I consider the pressure drop equation in transient ? Should it be written : $$dP = P(x+dx, t+dt) - P(x,t)$$ ?

And how about the mass flow, how can I calculate it ? I thought that at first, before the first wave pressure gets to the throat, the mass flow is regulated by Vena Contracta phenomena in the inlet of the pipe creating a kind of virtual nozzle. Is it wrong ?

I know it is a lot of questions at once, but basically my main problem is : what are the governing equations of unsteady flow inside a pipe with heat exchange and pressure drop along the pipe for 1D flow ?

The thing is I want to implement a finite difference 1D program to study the transient, and so far it is not running, because I feel like my mathematical model is wrong.

Thank you for your help

• Is this for a PhD thesis? Do you need an equation based solution or can you use something like CFD? If CFD is appropriate, is there any code that combines CFD with heat transfer? Commented Sep 6, 2021 at 19:08
• No it is not for a PhD thesis, and unfortunately I do not have a CFD tool-like. I only have Python to tackle this problem. I have implemented the equations from the model I have followed, but I am not sure that my equations are right because it is like a mix of several courses. Plus, it gives absurd results so it has to be wrong. As I cannot found a clear model of equations, I am asking for your help. Commented Sep 6, 2021 at 19:12
• I was just curious for more details. I'm afraid that I can't help, as this is a very difficult problem that involves transient heat transfer, compressible flow, and possibly turbulence. If turbulence is involved, note that this is one of the main unmodelled problems in physics. Commented Sep 6, 2021 at 19:17
• @DavidWhite If you wish it I can edit my post and write all the continuous equations and the discretized equations that I have implemented. In case your are curious, and, why not, if you have comments or critics to give I would be happy to read them Commented Sep 6, 2021 at 19:22
• Have you considered doing the fluid flow problem first before doing the combined flow and heat transfer? Commented Sep 6, 2021 at 20:59