PDESolve for transients in gas pipelines

[frederico.cms]

Dears,

I'm stuck on learning how to apply a real PDE problem. It is an application that I can observe in my daily life involving gas, the effect of a pressure wave caused by a transient in a gas pipeline when a valve is abruptly closed at the end of the pipeline.

Reading several technical papers on this matter, many numerical simulations of fast transients have been performed to solve the partial differential equation for transient flows in horizontal gas pipelines. Dorao (2011) [1], like many authors, starts the discussion with this system of PDE:

I was thinking (naive?) that I could simply solve this using the PDE Solver, but after many trials and errors, I came to a halt. I´m stuck setting the initial conditions right for the solver:

T0=-0s is during a steady state. Flow, velocity, and density all are the same for the length of pipe L, from L0 at the (inlet) and L1 (outlet), as this is before the transient.

After this, the flow stops:

T1=+0s is the outlet valve closed. Flow, velocity, and density are zero at L1 (outlet), but remain flowing at L0.

It is considered that no gas mass enters the system, hence this is just a simplification for the pulse wave formed by the transient.

[1] https://www.sciencedirect.com/science/article/abs/pii/S1875510011000059

Any help is highly appreciated.

Best regards,

Fred

[Fred_Kohlhepp]

Your problem resembles the reflection of a shock wave at the end of a shock tube.  The valve closure forces a sudden stoppage of flow in the gas, and a "reflected" shock wave travels back through the moving fluid to force a zero velocity behind it.

Shock tubes (used to study high temperature flows) have been studied extensively and used--there are many papers available that address these devices.  Two I found in a quick search (attached):

• NASA TN-D-5518 
• GALCIT 6-Inch Shock Tube

The first presents the equations that define how the shock wave is established.  The second does much the same thing, but it also addresses the reflected shock wave from the closed end.  I haven't tried to directly apply this to your problem, but I suspect it will be helpful.

Good luck!

[frederico.cms]

Many thanks, Fred.

Excellent reference, it sure will help.

I will have to try harder to solve the PDE, because I´m pretty interested to see the graphical result like the technical papers. Really want to see pdesolve working.

Thanks,

Fred Campos

[LucMeekes]

I see two problems:

- You have not defined x0 and x1.

- You use a constant (2 kg/m^-3) for the boundary conditions for rho, this constant is in terms of kg and m, not kg and m, That is, they are not units. Why not replace the numerical constant by rho0 ?

Hope this helps.

Success!
Luc