Where:

= gas density at the inlet of the pipe (lb/ft*ρ*_{1}^{3})*Y*= Net Expansion Factor (dimensionless)

The Net Expansion Factor (Y) is determined graphically from the value of the gas Ratio of Specific Heats (k), the Pressure Drop Ratio (*Δ*P/P_{1}), and the total Resistance Coefficient (K) of the pipe and fittings (if installed), as shown in Figure 1.

A table is also given that defines the limiting values of the Pressure Drop Ratio and Expansion Factor at which sonic (choked) flow occurs for various values of Resistance Coefficients. For example, air with k=1.4 in a pipe with a total Resistance Coefficient K=1.2, sonic conditions occur when the Pressure Drop Ratio reaches 0.552 at an Expansion Factor of about 0.588. For a pipe with a higher total Resistance Coefficient (K) of about 15, sonic flow is reached at an Expansion Factor of about 0.702 when the Pressure Drop Ratio approaches 0.818.

For a given pipe with a fixed total Resistance Coefficient, K, as the Pressure Drop Ratio increases due to higher flow rates, the Expansion Factor decreases, indicating a greater influence of the expansion process on the overall resistance to flow.

**Figure 1.** Determining the Net Expansion Factor (Y) for the compressible form of the Darcy Equation (courtesy of Crane TP-410).

## Control Valves

The control valve industry approaches compressible fluid flow in a similar fashion but uses equations to calculate the Expansion Factor. One form of the control valve sizing equation presented in the ISA 75.01 Standard for Sizing Control Valves (IEC 60532-2-1 equivalent) for incompressible fluid flow is shown in Equation 3:

Where:

- W = mass flow rate (lb/hr)
*C*= valve flow coefficient_{v}- dP = pressure drop (psi)
*ρ*= gas density at valve inlet (lb/ft_{1}^{3})

A similar equation is presented for compressible fluid flow in Equation 4 that incorporates an Expansion Factor (Y) given by Equation 5:

Where:

*Y*= Expansion Factor*x*= Pressure Drop Ratio (= dP/P_{1}) (dimensionless)*Fγ*= Specific Heat Ratio Factor (= γ/1.4) (dimensionless)*γ*= Ratio of Specific Heats = k in the Darcy method (dimensionless)*x*= Critical Pressure Drop Ratio Factor with fittings if installed (dimensionless), determined by air test by the manufacturer_{TP}

**Figure 2.** Graphs of the incompressible equation, compressible equation, and expansion factor for control valves.

## Safety Relief Valves

The API 520 governs the sizing and selection of pressure relieving devices in the petroleum industry and is used for other industries as well. The resistance created by the expansion process is accounted for in the API 520 sizing equations by two coefficients, one for sizing for critical (or choked) flow and one for subcritical flow.

Critical flow is achieved when the Pressure Ratio (P_{2}/P_{1}) drops to the Critical Flow Pressure Ratio (P_{cf}/P_{1}) given by Equation 7:

Where:

*P*= outlet static pressure at which choked flow occurrs_{cf}

When sizing a safety valve for critical flow conditions, the gas expansion is accounted for using a coefficient, C, that is applied to the sizing equation and determined graphically from Figure 3, by table, or by Equation 8:

When sizing for subcritical (non-choked) flow, a coefficient F_{2}, is determined graphically from Figure 4 or calculated using Equation 9 and applied to the subcritical sizing equation in the standard.

(9) | \[ F_2=\sqrt{\big( \frac{k}{k-1} \big)r^{(2/k)}\Bigg[\frac{1-r^{\big(\frac{k-1}{k}\big)}}{1-r} \Bigg] }\] |

Where:

*k*= Ratio of Specific Heats (dimensionless)*r*=*P*= Pressure Ratio (dimensionless)_{2}/ P_{1}

The sizing equations for both critical and subcritical flow contain the Discharge Coefficient (C_{d}) and Discharge Area (A) which defines the hydraulic performance of the relief valve.

**Figure 3.** Graph of coefficient C for sizing relief valves (courtesy of API 520).

**Figure 4.** Graph of coefficient F_{2} for sizing relief valves (courtesy of API 520).