Compressibility and Standard Flow Rates

Q.  When I convert the standard flow rate of a gas to mass or volumetric, do I need to account for compressibility?


A.  Yes and no. Oftentimes the compressibility factor (Z) is neglected (or really assumed to be unity) when performing such conversions, but you must understand the assumption that warrants its omission. Converting standard flows to mass or volumetric flows is typically done using rearrangements of the ideal gas equation. Therein lies the assumption; can the gas you are working with be approximated as an ideal gas under the operating temperature and pressure of the system? If so, using the ideal gas equation is fine, otherwise Z must be determined and included in the gas density calculation. The following equation gives a basic relation between mass, standard volumetric, and actual volumetric flow rates:

w = q_A \rho_A = q_S \rho_S

Where,
w = mass flow rate
q = volumetric flow rate
ρ = gas density
subscript A = evaluated at actual conditions
subscript S = evaluated at standard reference conditions

The density of a gas can be calculated as:

\rho = \frac{PM_r}{ZRT}

where,
P = pressure
Mr = relative molecular mass
Z = compressibility factor
R = universal gas constant
T = temperature

This is essentially the ideal gas equation, but with the inclusion of the compressibility factor to account for real gas behavior.  If we assume an ideal gas then Z = 1 and the term falls out.

Combining the two equations above so that we may readily make whatever conversions needed gives:

w = q_A\frac{P_AM_r}{Z_ART_A} = q_S\frac{P_SM_r}{Z_SRT_S}

This can be rearranged to give a very common conversion equation to calculate actual volumetric flow rate when given a standard flow rate.

q_A = q_S\bigg(\frac{P_S}{P_A}\bigg)\bigg(\frac{T_A}{T_S}\bigg)\bigg(\frac{Z_A}{Z_S}\bigg)

Oftentimes when this equation is encountered, the compressibility factors (ZS and ZA) are taken to be 1, thereby assuming ideal gas behavior at both the actual and reference conditions.  It is valid to assume the gas behaves ideally at the standard reference conditions (i.e., ZS = 1).  Despite the variety of standard reference conditions used, which tend to vary by country, industry, standards organization, and maybe lunar phase – it turns out they’re not all that “standard”, they typically fall in the range of 14.50 to 14.73 psi a (100.0 to 101.6 kPa) and 32 to 68°F (0 to 20°C).  Under these temperatures and pressures most gases can be approximated as ideal.

It is the actual conditions under which the gas is flowing where you need to be careful about assuming ideal gas behavior.  That is, you cannot always assume that ZA = 1, even though a quick search for conversion equations will turn up many that make this assumption without acknowledging it.  Many gas systems do operate under conditions in which the ideal gas assumption can be made, but it is always worth a quick check to ensure that is the case.  To determine whether a gas at a particular state deviates from ideal behavior, Z can be calculated from a cubic equation of state like Peng-Robinson or read from a generalized compressibility chart.  You might also find software applications or online calculators that will calculate gas density, but once again you need to be aware of the assumptions behind the calculation.