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Evaluating Compressible Limits of Darcy in PIPE-FLO® Professional

by Jeff Sines, Senior Product Engineer

Modeling a compressible gas piping system in PIPE-FLO Professional requires special attention to the conditions in which the assumptions of incompressible flow break down. Since PIPE-FLO uses the incompressible Darcy-Weisbach equation to calculate pipeline pressure drop and head loss, reasonably accurate results can be achieved when the density of the gas does not change significantly from the inlet to the outlet of the pipeline.

According to the Crane Technical Paper No. 410 and other widely accepted industrial references and text books, the conditions in which the assumption of incompressibility is valid is indicated by the values of the Mach Number and the Pressure Drop Ratio.

The Mach Number is defined as:

LaTeX Math Block
\[Ma = \frac{v}{c}\]

Where:

  • v = local fluid velocity
  • c = local static speed of sound 

The Pressure Drop Ratio is defined as:

LaTeX Math Block
\[PDR = \frac{dP}{P_{in}}\]

Where:

  • dP = change in absolute static pressure from the inlet to the outlet
  • Pin = absolute inlet static pressure

Figure 1: Variation of static pressure, temperature, and density with Mach Number. Isentropic flow assumes no change in total pressure, temperature, and density.

Figure 1 shows the variation of static density with Mach number when evaluating fluid properties at a given location or if the gas can be considered to be flowing isentropically (no heat transfer so no change in Total Temperature, and no friction so no change in Total Pressure between two points). At Mach = 0.1, the static density is about 99.5% of the total density, at Mach = 0.2 it's 98%, and at Mach = 0.3 it's 95.6% of the total density. With this small change in static density over this range, the assumption of incompressibility is valid if the Mach Number of the flow does not exceed about 0.2 (some references use 0.3 Mach).

In order for the assumption of incompressibility to be valid when using the Darcy-Weisbach equation, the following Pressure Drop Ratio limitations must be applied:

  1. If the PDR is less than 10%, reasonable accuracy will be obtained if the density used in the equation is based upon either the upstream or downstream conditions.
  2. If the PDR is between 10% and 40%, reasonable accuracy is achieved by using an average density between the upstream and downstream conditions.
  3. If the PDR is greater than 40%, high Mach Numbers and significant changes in density will be seen, so the Darcy-Weisbach equation should not be used. 

At high Pressure Drop Ratios, alternative methods like the Adiabatic (Fanno Flow) relations, the Isothermal relations, or the modified Darcy equation can be used. These methods account for the change in fluid properties at high velocities. The PIPE-FLO Compressible program uses the Fanno Flow equations to account for the changing density throughout the system and would be an excellent tool for such systems. The equations and solution methods of PIPE-FLO Compressible are scheduled to be incorporated into PIPE-FLO Professional version 17 as a licensed module.

Compressibility Check Sheet for PIPE-FLO Professional

Because the current and previous versions of PIPE-FLO Professional do not have warnings when the limitations of the Darcy equation are exceeded in gas applications, Engineered Software developed the "Compressiblity Check Sheet", a Microsoft Excel file that uses the data export features of X-Link (in v2009) and DataLink (for v12 and above) to make the evaluation substantially easier. 

The Compressibility Check Sheet, shown in Figure 2, extracts design data and calculated results from the PIPE-FLO model, performs some additional calculations for the Mach Number and Pressure Drop Ratio, then checks to see if the incompressible limitations of the Darcy equation are valid. If the limitations are exceeded, the worksheet will give recommendations for making adjustments to the model to provide more accurate results. For the most accurate results, the fluid properties of the Fluid Zone assigned to the pipe must be calculated using a pressure within a reasonable range of the pipe's calculated inlet and outlet pressure. 

Figure 2: Compressibility Check Sheet in Microsoft Excel used to verify the Mach Number and Pressure Drop Ratio limitations are met for the assumption of incompressibility for a gas to be valid. If the limitations are not met, recommendations are made to split the pipe to reduce the Pressure Drop Ratio.

The Compressiblity Check Sheet files for v2009 to v15.0 are attached below, but starting in v15.0, the Compressibility Check Sheet is also added to the "My Engineering Files / Projects" folder when PIPE-FLO is installed. It is recommended that you use this check sheet any time you model a compressible gas system in PIPE-FLO.

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