I would like to estimate the time needed to open a linear single-acting pneumatic actuator. The actuator's piston is loaded with a spring pushing downwards and the operating fluid (air or nitrogen) pushes upwards; other forces like weight and friction are present.

The operating fluid is provided by a large tank (ideally infinite) with stagnation pressure $P_\infty$ and stagnation temperature $T_\infty$.

If the tank is connected directly to the actuator, balance of forces, with fluid force depending on actuator inlet mass flow rate, leads to an ODE which I can solve numerically.

Since tank and actuator are actually connected by a long pipe, $\frac{length}{diameter}\approx 1000$, I need to estimate the time to reach the equilibrium state in the pipe ("pressurise the pipe"). Initial conditions for pipe and actuator are $P|_{t=0}<P_\infty$ and $T|_{t=0}=T_\infty$.

Does anyone know a way to estimate the time required to reach the equilibrium state in the pipe, maybe based on some dimensional argument or some typical time related to the pipe?

If not, I must determine the evolution of pressure, temperature and mass flow rate inside the pipe.

I have been reading literature for compressible flow in pipes, usually coming from natural gas distribution applications, but the problem of transient compressible flow in a pipe, be it isothermal or non-isothermal, leads to a system of PDEs, requiring some sort of boundary condition also at the exit of the pipe. Since I do not know the pressure / temperature / mass flowrate entering the actuator in advance, I am missing something.

Any help or suggestion would be appreciated.

I would like to estimate the time needed to open a linear single-acting pneumatic actuator. The actuator's piston is loaded with a spring pushing downwards and the operating fluid (air or nitrogen) pushes upwards; other forces like weight and friction are present.

The operating fluid is provided by a large tank (ideally infinite) with stagnation pressure $P_\infty$ and stagnation temperature $T_\infty$.

If the tank is connected directly to the actuator, balance of forces, with fluid force depending on actuator inlet mass flow rate, leads to an ODE which I can solve numerically.

Since tank and actuator are actually connected by a long pipe, $\frac{length}{diameter}\approx 1000$, I need to estimate the time to reach the equilibrium state in the pipe ("pressurise the pipe"). Initial conditions for pipe and actuator are $P|_{t=0}<P_\infty$ and $T|_{t=0}=T_\infty$.

Does anyone know a way to estimate the time required to reach the equilibrium state in the pipe, maybe based on some dimensional argument or some typical time related to the pipe?

If not, I must determine the evolution of pressure, temperature and mass flow rate inside the pipe.

I have been reading literature for compressible flow in pipes, usually coming from natural gas distribution applications, but the problem of transient compressible flow in a pipe, be it isothermal or non-isothermal, leads to a system of PDEs.

The model verification examples I found led me to suppose that the PDE was

requiring some sort of boundary condition also at the exit of the pipe. Since I do not know the pressure / temperature / mass flowrate entering the actuator in advance, I am missing something.

but since the PDE is hyperbolic, it actually requires boundary conditions at the inlet only, besides initial conditions.

Any help or suggestion would be appreciated.