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# Gas Expansion Adds Resistance to Fluid Flow

by Jeff Sines, Senior Product Engineer at Engineered Software, Inc.

When sizing equipment for piping systems with compressible gas applications, it is important to account for all factors that would influence the flow rate and pressure drop across the equipment. Under-sizing a pipeline, relief header, control valve, orifice, or safety valve in compressible applications can result in catastrophic failure that damages the equipment and piping system. But more importantly, catastrophic failure due to under-sizing equipment can also jeopardize the safety of employees and the surrounding community, as well as impact the reputation of the manufacturing plant, parent company, engineering consulting firms, and equipment manufacturers. The bottom line is effected by expensive repairs and replacement costs, lost production, compensation for injuries or death, and potentially the loss of operating permits from regulatory agencies.

As compressible gas flows through each device in a piping system, head loss causes a drop in static pressure, a decrease in density (expansion) and temperature, and an increase in velocity. Depending on the amount of expansion, the change in density, temperature, and velocity may be insignificant and can be neglected, or it may be significant enough that it must be accounted for when designing a compressible gas system, sizing equipment, or evaluating plant operation. One way to determine the need to use compressible equations is by evaluating the Pressure Drop Ratio (x = dP/P1). For x < 0.2, the change in gas density is small and the assumption of incompressible flow can be made. For x > 0.4, the density change is large and compressible flow should be assumed.

The process of expansion adds resistance to flow that reduces the flow rate for a given overall pressure drop, or results in a higher pressure drop for a given flow rate. The most common method to account for gas expansion is to determine an Expansion Factor to apply to the incompressible fluid flow equation for a particular device. Various equipment manufacturers approach the determination of the Expansion Factor differently, but each method shows that it is a function of the pressure drop across the device, the fluid properties, and some measure of the device’s hydraulic performance. More complex solution methods can be used including adiabatic flow equations (Fanno Flow) and isothermal equations.

## Gas Expansion in Pipelines, Valves, and Fittings

One form of the Darcy equation found in the Crane Technical Paper N. 410 for calculating the mass flow rate of incompressible fluid flow in a pipeline is given by Equation 1:

 (1) $w=0.525d^2 \sqrt{\frac{\Delta P\rho}{K}}$

Where:

• w = mass flow rate (lb/sec)
• d = inside diameter of the pipe (inches)
• ΔP = pressure drop across the pipe and fittings (psi)
• ρ = constant liquid density (lb/ft3)
• K = total resistance coefficient of the pipe and fittings (dimensionless)

According to the Crane TP-410, “investigation of the complete theoretical analysis of adiabatic flow has led to a basis for establishing correction factors which may be applied to the Darcy equation... These correction factors compensate for the changes in fluid properties due to expansion are identified a Y net expansion factors…” The corresponding equation for the flow of compressible gas is given by Equation 2:

 (2) $w=0.525Yd^2 \sqrt{\frac{\Delta P\rho_1}{K}}$

Where:

• ρ1 = gas density at the inlet of the pipe (lb/ft3)
• 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 (ΔP/P1), 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:

 (3) $W_{incompressible}=63.3 C_v \sqrt{dP \rho_1}$

Where:

• W = mass flow rate (lb/hr)
• Cv = valve flow coefficient
• dP = pressure drop (psi)
• ρ1 = gas density at valve inlet (lb/ft3)

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

 (4) $W_{incompressible}=63.3Y C_v \sqrt{dP \rho_1}$
 (5) $Y=1-\frac{x}{3F_{\gamma}x_{TP}}$

Where:

• Y = Expansion Factor
• x = Pressure Drop Ratio (= dP/P1) (dimensionless)
• = Specific Heat Ratio Factor (= γ/1.4) (dimensionless)
• γ = Ratio of Specific Heats = k in the Darcy method (dimensionless)
• xTP = Critical Pressure Drop Ratio Factor with fittings if installed (dimensionless), determined by air test by the manufacturer

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

The Expansion Factor used in the control valve equations is a function of the pressure drop (in the form of the Pressure Drop Ratio, x), the fluid properties (Ratio of Specific Heats, γ) and a measure of the valve’s performance (Critical Pressure Drop Ratio Factor, xTP).

Choked flow conditions occur when the gas velocity approaches the speed of sound and x = Fγ xTP, resulting in an Expansion Factor = 2/3.

The effect of applying the Expansion Factor to the incompressible equation can be seen in the graph of Mass Flow Rate vs. Pressure Drop Ratio. Accounting for the resistance due to the expansion process pulls the control valve performance curve off of the 2nd order curve predicted by the incompressible flow equation. Because the Expansion Factor accounts for the resistance to flow due to the expansion process, the flow rate predicted by the compressible equation is less than the flow rate predicted by the incompressible equation at any value of Pressure Drop, or Pressure Drop Ratio. At choked flow conditions when Y=2/3, the maximum choked flow rate is obtained and no further increase in the pressure drop will result in an increase of the flow rate through the valve.

## Orifice Flow Meters

The ASME MFC-3M standard for measuring fluid flow with orifice, nozzle, and venturi type flow meters also incorporates an expansion factor in the equations to determine the flow rate of compressible gases, except the term they use is the Expansibility Factor, given by Equation 6 for orifices:

 (6) $Y=1-(0.351+0.256 \beta^4+0.93 \beta^8) \Bigg(1-\bigg(\frac{P_2}{P_1}\bigg)^{1/k} \Bigg)$

Where:

• β = Diameter Ratio (= dorifice /dpipe) (dimensionless)
• P2/P1 = Pressure Ratio (dimensionless)
• P1 = Inlet static pressure (psia)
• P2 = Outlet static pressure (psia)
• k = Ratio of Specific Heats (dimensionless)

The Expansibility Factor accounts for additional resistance due to the expansion process and is a function of the pressure drop (in this case represented as the Pressure Ratio, P2/P1), the fluid properties (Ratio of Specific Heats, k) and a measure of the orifice’s hydraulic performance (the diameter ratio, β ).

## 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 (P2/P1) drops to the Critical Flow Pressure Ratio (Pcf/P1) given by Equation 7:

 (7) $\frac{P_{cf}}{P_1}=\bigg(\frac{2}{k+1}\bigg)^{\big( \frac{k}{k-1} \big)}$

Where:

• Pcf = outlet static pressure at which choked flow occurrs

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:

 (8) $C=520 \sqrt{k \bigg( \frac{2}{k+1}\bigg)^{\big( \frac{k+1}{k-1} \big)}}$

When sizing for subcritical (non-choked) flow, a coefficient F2, 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 = P2 / P1 = Pressure Ratio (dimensionless)

The sizing equations for both critical and subcritical flow contain the Discharge Coefficient (Cd) 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 F2 for sizing relief valves (courtesy of API 520).

## Summary

Under-sizing equipment in compressible gas applications can result if the resistance created by the expansion process isn’t accounted for when designing a piping system. Industrial standards that govern the sizing of equipment account for the expansion of a gas differently, but similarities can be seen by comparing the equations that calculate an Expansion Factor that is applied to the incompressible flow equations. The Expansion Factor is a function of the properties of the gas, a measure of the device’s hydraulic performance, and the pressure drop across the device.

Properly sizing equipment, especially in very hazardous applications, will prevent catastrophic failures, unfortunate injuries or death, costly downtime for repairs, environmental emissions, and the loss of regulatory permits to operate the plant.