Hi,

I am injecting a variable density fluid in a pipe. The density changes by time as the fluid passes through the pipe (the variation is less than 5%) due to the reduction of temperature. Can anyone tell me how does the pressure change with time in the pipe? I am using a SIMPLE method. the initial condition is uniform pressure everywhere and zero velocity, I fix the outlet pressure and inject the fluid at constant volumetric flow rate. I am looking for the pressure variation till it reaches the steady state, you help is greatly appreciated.

I think we need to know the equation of state you're using for the fluid.

I am using the Peng-Robinson EoS. However the density variation is not substantial.

You didn't specify your fluid.

From a Google search, it appears that the Peng-Robinson EOS simply measures the variation Z of a particular fluid from an ideal gas.

To get an idea of the behavior, first assume a constant density in the EOS to calculate a (constant) Z and use that value in the ideal gas law to get a qualitative feel for the behavior. The ideal gas law is a lot easier to work with.

If you're working with a liquid, just use a constant density instead of a complicated EOS.

Then put a free-slip condition along the top and bottom of your flow regime. Free-slip makes the problem one-dimensional in space (no boundary layers) but time dependent.

Look in classical gas dynamics books (Shapiro comes to mind) for closed-form analyses.

With this in hand, put the free-slip boundary condition and simple EOS into your code and see if you get the same behavior. Then (and only then) turn on the EOS and no-slip condition and try the real problem.

Very qualitatively, you'll see a pressure wave (decay) propagate from the fixed outlet pressure to the inlet - assuming that your inlet is subsonic. I don't know if the wave will reflect back to the outlet or not.