API Standard 520 Part 1 9th Edition¹ Annex B §B.3.1.2 provides an expression in equation B.13 for the isentropic expansion coefficient in terms of thermodynamic state variables, to be evaluated anywhere along the isentropic path, but typically evaluated at the relief conditions since that is readily available:

We have been asked over the years how that equation was derived and if there is an alternative method to determine the isentropic expansion coefficient. The derivation I used when presenting this to the API Standard 520 Task Force is included at the end of this blog for those interested in the mathematics.

Alternative equation. With respect to an alternative, API Standard 520 Part 1 9th Edition Annex B §B.3.1.4 indicates that “an isentropic expansion coefficient can be used based on an average value between the upstream pressure and the pressure in the throat of the nozzle that, in the case of maximum flow, is the critical flow pressure.” This statement is a bit ambiguous as it could be read to determine the isentropic expansion coefficient at both the relief pressure and the throat pressure, then take the average, or to determine an appropriate average value based on the conditions at the relief pressure and throat pressure. We take the latter approach, and use the following:

Where the subscript ‘1’ represents relief conditions, subscript ‘t’ represents estimated throat conditions, P is pressure, v is specific volume, and ρ is mass density. The throat pressure is estimated (60% of the relief pressure unless additional fluid behavior or back pressure information is known), then an isentropic flash is performed from the relief conditions to that estimated throat pressure to obtain the specific volume at the throat.

The difference between the equations above are best described as the difference between a single-point determination using properties at relief and a two-point fit using isentropic flash data. If the fluid expansion actually follows the isentropic expansion of P·vⁿ = constant, and n is actually constant, then both calculations will yield the same result, and the choice between the two is best made based on availability of data. In general, using a process simulator with an isentropic flash routine (for example, the compressor/expander block in Hysys with 100% isentropic efficiency) and selecting the data based on the relief conditions and the estimated throat pressure is the most convenient as the partial derivative term in equation B.13 can be hard to find in some process simulators.

In reality, both assumptions (P·vⁿ = constant, and n is constant) are simply very good approximations for most cases, and thus one may find slight differences in values calculated between the two methods. When the assumptions are not good, then it is best to employ the numerical integration of API Standard 520 Part 1 §B.1.1.5. Our experience has been that the only times these assumptions are potentially questionable are near the thermodynamic critical point, above the thermodynamic critical pressure (even then, if the temperature is high enough, the fluid behaves pretty well), reactive systems, or crossing phase boundaries. Nonetheless, it is possible to use an algorithm to check for outliers that involves both calculation methods – use API (B.13) to calculate n, use this n to calculate an estimated choke pressure, perform an isentropic flash from the relief conditions to this estimated choke pressure, and calculate an n using the alternative method. If the values deviate significantly, or the fluid enters into the two-phase region on the flash, then direct integration should be used.

Derivation. To derive the constant isentropic expansion coefficient in terms of the thermodynamic state variables, you start with the expansion law

Take the partial derivative of both sides with respect to temperature at constant entropy (since this is an isentropic process). The derivative of a constant is zero.

Expand the derivative and collect the terms

Here’s the tricky part that made my breakthrough – the partial derivatives of pressure and specific volume with respect to temperature need to be expanded out using the chain rule.

I also multiplied by T/T because if I rearrange, I get expressions that are related to the partial derivatives of enthalpy and internal energy, respectively (based on fundamental property relations).

And that is how the specific heat ratio is introduced – the derivatives involving enthalpy and internal energy are the definitions of heat capacities at constant pressure and volume, respectively. Thus, n reduces to the following expression (B.13):

Note that this expression holds true anywhere on the path, just that the relief conditions are most convenient since they are known.

For an ideal gas, one can calculate the partial derivative of pressure with respect to specific volume at constant temperature (it is -P/v, thus canceling the first ratio), and find that the isentropic expansion coefficient for an ideal gas is the ratio of ideal gas specific heats (as described in API Standard 520 Part 1 §B.3.2.2).