Api 521 fifth edition: isothermal flow temperature determination

[vinalso85]

Hi!

To determine the backpressure in the discharging piping of a safety valve, API 521 recommends the use of isothermal relations. The method consist in calculate the pressure at the piping inlet backward from the pipe outlet. The pressure in the outlet will be the maximum of the atmospheric pressure and the critical pressure.

The critical pressure and Mach numbers for a isothermal flow depend on the flow temperature T (which is constant along the flow by definition). API does not specify (or at list I could not find anywhere) how to estimate the flow temperature.

What should be the flow temperature?

a) The surrounding temperature, since we are assuming isothermal flow due to the heat transfer with the environment.
b) Relieving temperature / design temperature / system temperature (whatever we are using to the determine the valve flow rate). This is clearly not the actual behaviour since the isothermal flow starts somewhere downstream the valve oulet, but maybe a conservative approach.
c) Something else

Thanks!!

 

[PaoloPemi]

normally I adopt the temperature estimated at PSV outlet (see also the thread about discharging temperature) presuming that at high velocity (normally a fraction of speed of sound) the fluid doesn't exchange much heat with ambient

 

[vinalso85]

Thank you!!!

I forgot that I still had the link between the pipe inlet and the reservoir assuming adiabatic. However, I must disagree with your statement that there is not much heat exchange in the pipe discharge, if that was the case adiabatic flow would be assumed in the discharge pipe instead of isothermal.

In addition, PSV outlet temperature and flow temperature will be the same if we assume isothermal flow and they should be based on assuming either critical conditions or atmospheric pressure at the system outlet.

I will put all the equations in order and post the algorithm for adiabatic case and isothermal case. It will take some days (sadly I have lots of things to do) but I think it will be useful for the other thread on determining temperatures. Thanks again.

 

[vinalso85]

Hi again,

perhaps this could solve my question:

Assuming we know stagnation properties of the reservoir we are protecting. We also know the PSV dicharge mass flow rate and the geometry downstream the PSV (pipe diameter and length). Ideal gas is considered.

The flow from the safety valve nozzle to the pipe outlet has two regions:

Region 1: Supersonic flow from the nozzle until shock waves transform the flow in subsonic.

This part of the flow is not possible to analyze. The nature will do what is needed to match the boundary conditions of flows and pressures through shock waves until subsonic flow is reached.

The only we can say about this region is that:

-Mass conservation is valid (of course).
-Flow is adiabatic (region is short enough).
-Entropy will increase.

Much better description of this region can be found in the Brandmaier article “Steam Flow Through Safety Valve Vent Pipes”. In order to increase pressure drop in the pipe, thus, increasing the pressure at the inlet, we will assume that this region ends at the valve outlet. Note that the valve outlet has nothing special compared to any other point of the discharge pipe. The problem is that the length of this region seems unpredictable (as far as I know).

Region 2: From end of region 1 to pipe exit

This part is subsonic. It could be considered either adiabatic (Fanno flow) or isothermal (constant temperature). Anyway, the pipe exit has to be either sonic (Ma=1) or subsonic.

Case 1: Adiabatic flow

Step 1: Assume exit pressure as atmospheric
Step 2: Calculate Mach number at the exit (pressure and temperature is related by considering that total enthalpy is constant, adiabatic flow)
If Mach number is bigger than one, then exit pressure is the critical pressure (solve p for Ma = 1)
If Mach number is less than 1 then the exit pressure will be the atmospheric pressure.
Step 3: Calculate inlet pressure (safety valve outlet) with Fanno flow
Step 4: Calculate temperature or any other property that you need.
Step 5: Check that the backpressure is adequate according to your design code (for instance, less than 10% of set pressure).

Case 2: Isothermal flow

Step 1: Assume atmospheric pressure at pipe outlet
Step 2: Calculate temperature knowing that inlet pressure is function of temperature (because flow is adiabatic until that point), the exit temperature is equal than the inlet (isothermal flow). To get the temperature is necessary to solve isothermal equation.
If Mach number at the exit is less than one, then exit pressure is atmospheric pressure. Note that Mach number for isothermal flow is defined using isothermal speed of sound.
If not, Ma=1 and exit pressure is critical. Solve T again with the isothermal equation knowing that the Ma at exit is 1 and the exit pressure is function of temperature.
Step 3: Calculate inlet pressure (safety valve outlet), again thanks that is function of T.
Step 4: Calculate any property that you need.
Step 5: Check that the backpressure is adequate according to your design code (for instance, less than 10% of set pressure).