# Determining required pressure drop for choked flow conditions to be met

I have pipework system with an inlet pressure, $$P_0$$, which passes from a pipeline of fixed diameter, $$D_0$$, through a tee with reducer to a smaller diameter pipe, $$D_1$$, before returning to another pipeline (same diameter, $$D_0$$) of lower but unknown pressure, $$P_1$$, through another tee and reducer arrangement.

I am attempting to determine the pressure drop required for natural gas flowing through the system to meet choked flow conditions (Mach 1).

I have used the following equation to calculate a critical pressure ratio:

$$\dfrac{p^*}{p_0} = \left(\dfrac{2}{n+1}\right)^{n/(n-1)}$$

Which works out as 0.542 assuming ideal gas condition using $$n = 1.32$$, natural gas ratio of specific heats ($$k$$). Therefore a local pressure drop of approximately $$P_0/2$$ would be required to meet choked flow conditions.

However, I am concerned that this is overly simplifying the problem.

• Did you check your mean velocity in the different pipe sections to sonic velocity data? – J. Ari May 20 '20 at 16:58

Here's a diagram that I believe depicts your situation. I am concerned that this is overly simplifying the problem.

You're probably fine but it may be useful to review assumptions tied to that equation and your system.

## Ideal gas law.

The equation appears in API 520 Part 1 9th Edition (2014) on page 56 under section 5.6 Sizing for Gas or Vapor Relief which assumes that "the pressure-specific volume relationship along an isoentropic path is well described by the expansion relation,"

$$PV^k={constant}$$

where

$$k$$ : is the ideal gas specific heat ratio at the relieving temperature.

Or, in other words, that the gas is an ideal gas.

The document cautions that this assumption may not be valid if the compressibility factor (a.k.a. "Z-factor") falls outside the $$0.8 range, as may be the case for high pressure conditions, high molecular weight gasses such as $$CO_2$$ or $${H_2}{S}$$, or low temperatures. Compressibility is a measure of the "non-ideality" of a gas and is a function of both pressure and temperature.

Based on your use of the ratio of specific heat $$n=1.32$$, it seems you are assuming the fluid is methane (the primary component in natural gas transmission lines). Use a process simulator / equation of state calculator to calculate the compressibility or calculate it manually such as with the method described in the GPSA Handbook (ex: GPSA 12th edition, Section 23 "Physical Properties", page 23-10, "Z-FACTOR FOR GASES") to confirm the ideal gas assumption is still valid.

If the gas cannot be assumed to ideal, then API 520 Part 1 provides a numerical calculation method in Annex B. The choked flow pressure drop is found as a side effect of identifying the maximum "Mass Flux" from a numerical integration process that uses a set of pressure, temperature, and specific volume values describing the state of gas at different points along an isentropic expansion (you'd use a process simulator / equation of state calculator to get these values).

$$G^2 = \left[ \frac{- 2 \cdot \int^P_{P_1} v \cdot dP}{v^2_t} \right]_{\text{max}}$$

where:

$$G$$ : is the mass flux (mass flow per unit area) through the nozzle, [$$\frac{kg}{s \cdot m^2}$$]

$$v$$ : is the specific volume of the fluid [$$\frac{m^3}{kg}$$]

$$P$$ : is the stagnation pressure of the fluid [$${Pa}$$]

$$1$$ : is the fluid inlet condition to the nozzle

$$t$$ : is the fluid condition at the throat of the nozzle where cross-sectional area is minimized (the integral term corresponding to maximum mass flux)

The pressure drop at choke flow ($$P_1 - P$$) is found when additional terms from the $$\int^P_{P_1} v \cdot dP$$ integral slices don't cause $$G$$ to rise any more.

Or you could buy commercial software and trust that they perform the non-ideal gas calculations correctly.

## One-phase gas flow

The equation assumes one-phase gas flow. If two-phase flow is present then a different set of equations are required to calculate the critical pressure ratio (see Annex C in the document).

## Restriction diameters

I am having a hard time imagining a plausible scenario in which your system would be useful unless it contains a valve of some sort on the $$D_1$$ diameter pipe so the amount of gas flow to the second pipeline can be controlled. If a valve is present to perform this task, make sure to identify the location of the tightest flow restriction when designing the system. For example, if you are using a a ball valve, then make sure you know if it's a restricted port ("RP") or a full port ("FP") valve. If you're installing a pressure relief valve, make sure to reevaluate the sizing equations based on the actual orifice areas ("flow areas") and certified capacity values ("coefficient of discharge") provided by the manufacturer for valves they quote you (see the NB-18 "Redbook"). Beware erosion issues that can alter the pipe inner diameter.

## Summary

If you assume the gas is an ideal single-phase gas, then yes, the calculation of the pressure drop needed to achieve choked flow conditions is simply the equation for critical pressure ratio you provided. Otherwise, things get complicated.