There are many equations for determining the flow in natural gas pipes and the pressure drops associated with those flows, or vice versa. Our intent is to determine the validity of each equation with respect to flow rates that might be encountered by a plumbing engineer.

Previous articles in this series suggest that either schedule 40 steel pipe or polyethylene pipe (PE) are the normal piping materials used. The inside diameter of each of these pipes is different. Furthermore, there is no standard for what the inlet pressure to these pipes might be and what the expected pressure drops might be. So, there are no standardized tables for higher pressure conditions that exceed pressures outlined in NFPA 54 National Fuel Gas Code and the ICC International Fuel Gas Code.

As a result, if the natural gas system designer wants to deliver natural gas at over 5 psig, he/she could prepare his/her own tables similar to those in NFPA 54 but based on the higher pressure and higher pressure drops. 

Several sources were used to determine the actual flow equations. (1) Considerations About Equations for Steady State Flow in Natural Gas Pipelines by Paulo M. Coelho and Carlos Pinho in the Journal of the Brazilian Society of Mechanical Sciences and Engineering, July-September, 2007;  (2) Crane Technical Paper 410, 2018; (3) Chapter 22 of ASTM MNL 58 Petroleum Refining and Natural Gas Processing, 2013, concerning Transportation of Crude Oil, Natural Gas, and Petroleum Products. 

These texts all indicate that the Darcy-Weisbach equation appears to be the most accurate method for determining pressure drop, but this method was avoided because of the difficulty of determining the value for “f” (friction coefficient). Most of the alternate gas flow equations predate the availability of present-day computers. Calculating “f” involves an iterative process since the square root of “f” is part of the denominator on both sides of the equation for “f”. The Darcy-Weisbach equation is as follows:

                        h_L = f ( )                                                                                           (Equation 1)

Where:             h_L = gas head loss in feet (meters) of fluid – in this case gas

f = friction flow coefficient - dimensionless

L = length of the pipe in feet (meters)

D = internal diameter of the pipe, same units as “L”

V = gas velocity in feet per second (meters per second)

g = gravitational constant 32.174 feet per second^2 (9.806 meters per second^2)

The basis of the AGA flow equations is an “f” value that is a function of Reynolds number. The classic equation for Reynold number is:

                        Re = σ V D /μ                                                                                        (Equation 2)

Where:             σ = gas density

V = gas velocity

D = internal diameter of the pipe

μ = dynamic viscosity – 7E-06 lbm/ft-sec  (0.010392 centipoise)

To assist in the calculation, when density is broken down to the perfect gas law equation and velocity is broken down as a function of flow and density, and then substituted in the classic Reynolds number equation, the following equation can be derived:

                        Re = 4 Q_st 29 S_g P_st / (μ π D  T_st)                                                            (Equation 3)

Where:             Q_st = Gas Flow rate at Standard Conditions

29 = molecular weight of air, 28.9647 lb/lbmol (28.9647 g/gmol)

S_g = specific gravity of natural gas

Pst = standard gas pressure – 14.696 psia (101.325 kPa)

μ = dynamic viscosity – 7E-06 lbm/ft-sec  (0.010392 centipoise)

π= PI = 3.14159

D = internal diameter of the pipe

 = Universal gas constant, 1545.349 lb_f ft/(lbmol °R) [8314.41 J/(kmol °K)]

T_st = Standard gas temperature, 518.67°R (288.15°K)

(Note:  Reynolds number is “dimensionless”, meaning that all units in the numerator and the denominator must cancel. Equations 2 and 3 have not been corrected to include units. The reader will need to use his/her reference material to provide the needed correction factors.)

Notice also, that the Reynolds number in equation 3 is not dependent on the actual pressure and temperature of the gas. The interesting thing about equation 3 is that, if the high pressure drop equations are used, the “f” value will remain unchanged from the inlet to the outlet of the pipe segment. 

In the 1960’s, the American Gas Association (AGA) proposed the AGA equations, which use the general gas equation with simplified, limiting forms of the Colebrook White equations. There are three flow regimes that are encountered in gas pipes:  Laminar Flow, Partially Turbulent Flow, and Fully Turbulent Flow. The “f” value formulas for these are as follows:

Laminar Flow:                              f = 64 / Re            for Re < 2,000 to 4,000                    (Equation 4)

AGA Partially Turbulent Flow:       1 /  = -2 log₁₀ ( 2.825 / (Re ) )  (Note 1 below)          (Equation 5)

AGA Fully Turbulent Flow:            1 /  = -2 log₁₀ ( ε / (3.7 D) )                                      (Equation 6)

                  Note 1:  Formerly, the 2.825 value in equation 5 was 2.51 and is the Colebrook-White equation, 1990.

Where:             Re = Reynolds Number

f = friction flow coefficient - dimensionless

ε = pipe inside diameter roughness, same units as “D”

D = internal diameter of the pipe

According to both Coelho and Pinho and Petroleum Refining and Natural Gas Processing, the transition between Partially Turbulent Flow and Fully Turbulent Flow occurs where the results of the two equations intersect; the higher value of “f” is used.

When calculating pressure drop, Crane Technical Paper 410 indicates that, where the upstream pressure (P₁) and downstream pressure (P₂) are as follows, these generalizations can be made: (1) If the calculated pressure drop (P₁ – P₂) is less than about 10% of the inlet pressure P₁, reasonable accuracy will be obtained if the specific volume ( V = 1/σ ) used in the formula is based on either P₁ or P₂ conditions, whichever are known. (2) If the calculated pressure drop (P₁ – P₂) is greater than about 10% but less than 40% of the inlet pressure P₁, reasonable accuracy will be obtained if the specific volume used in the formula is based on the average P₁ and P₂ conditions. (3) If the calculated pressure drop (P₁ – P₂) is greater than about 40% of the inlet pressure P₁, then different high pressure drop formulas are provided. An alternate would be to break the length of pipe into several segments that meet the conditions above using the outlet pressure of segment “1” as the inlet pressure to segment “2”, and so forth. Keep in mind that pressures “P₁” and “P₂” are absolute pressures and not gauge pressures.

Procedures Followed

In order to come to some conclusions as to the validity of each of the alternate equations discussed below, a program was set up in Excel and in Visual Basic to calculate the value of “f” to 5 significant digits, for each flow point, and then solve for flow based on available pressure drop using the equations outlined above (via Darcy-Weisbach formula). These points were compared with answers created when using each of the alternate equations. Once a set of results was collected for each alternate equation, the total package of results was compared to the Darcy answers by dividing the alternate results by the Darcy answers; one-by-one. The following statistics were collected:  Minimum ratio, maximum ratio, average ratio, and standard deviation.

The comparisons were set up for each of the following:  given inlet pressure, given ending pressure, distance in feet, pipe diameter (actual), and pipe interior surface roughness (where considered). 

Natural Gas Characteristics:  Where the equations allowed input, the following was included:  Natural Gas Specific Gravity = 0.60. Natural Gas Viscosity = 7E-06 lbm/ft-sec or 0.010392 centipoise.

Pressure ranges:  2 psig inlet with 1 psig drop, 3 psig with a 2 psig drop, 5 psig with a 3.5 psig drop, 20 psig with a 2.0 psig drop; and 40 psig with a 4 psig drop.

Distances:  10 feet (3 meters) to 2,000 feet (610 meters) in increments similar to NFPA 54 and IFGC.

Nominal Pipe Sizes:  0.5” (DN-15) thru 6” (DN150) as available.

Pipe Materials:  Sch 40 steel pipe, SDR 11 PE pipe, SDR 13.5 PE pipe.

Equations used:  NFPA/IFGC equation, Mueller Equation, Weymouth Equation, IGT Distribution Equation, Spitzglass-High Pressure Equation, and AGA Plastic Pipe Manual Equations. For lower pressure piping, the values in the NFPA/IFGC tables were also compared. Only steel was considered for the lower pressure pipes, since they would likely be installed above grade. PE as well as steel piping were considered for 20 psig gas and 40 psig gas. Note, all equations were rearranged to provide Q_h (flow per hour) as a function of P₁ and P₂ (inlet and outlet pressures.)

Results

For all of the following equations, “Q_h” is flow in SCFH, “P₁“ is the inlet pressure, “P₂“ is the outlet pressure, “D” is the pipe inside diameter in inches, “S_g” is the specific gravity, and “L” is the length of the pipe segment in feet. Pipe inside surface roughness was estimated as 0.0018 inches for steel and 0.00006 inches for PE. Note:  the Reynolds Number was created for each range of values so the reader can look at the portion of the Moody Diagram where these flows exist.

NFPA/IFGC Low pressure equation (for 1.5 psig and higher):

            Q_h = (D * { 18.93 * [ (P₁²-P₂²) * Y / ( Cr * L ) ]^0.206 } )^(1/0.381)                                     (Equation 7)

Where:             Y = 0.9992 for natural gas

Cr = 0.6094 for natural gas

Mueller Equation:

            Q_h = ( 2826 * D^2.725 ) / S_g^0.425 * [ (P₁²-P₂²) / L ) ]^0.575                                                (Equation 8)

Weymouth Equation:

            Q_h = ( 2034 * D^2.667 ) / S_g^0.5 * [ (P₁²-P₂²) / L ) ]^0.5                                                    (Equation 9)

IGT Distribution Equation:

            Q_h = ( 2679 * D^2.667 ) / S_g^0.444 * [ (P₁²-P₂²) / L ]^0.555                                                (Equation 10)

Spitzglass-High Pressure Equation:

            Q_h = ( 3410 / S_g^0.5 ) * [ (P₁²-P₂²) / L ) ]^0.5 * [ D⁵ / ( 1 + 3.6 / D + 0.03 * D ) ]^0.5          (Equation 11)

AGA Plastic Pipe Manual

Additional variables include, “T_b” which is “Standard Temperature” or 518.67°R, “P_b” which is “Standard Pressure” or 14.696 psia, “T” is the gas temperature in degrees R; 60°F or 519.67°R used for this analysis, “Sg” is the specific gravity of natural gas (air = 1.0), 0.60 used for this analysis, “µ” is gas viscosity, 7.0E-06 lb_m/ft-sec used for this analysis, “Z” is the gas compressibility factor, 1.0 for low pressure gas, and “ε” is pipe surface roughness in inches ( 0.0018 for steel and 0.00006 for plastic). 

            For Partially Turbulent flow (flow below critical flow where flow turns Fully Turbulent):

            Q_h = D^2.667 * 664.3 * T_b/P_b * [ (P₁²-P₂²) / ( T * L ) ]^0.555 * 1 / ( S_g^0.444 * µ^0.111 )           (Equation 12)

            For Fully Turbulent flow (for higher flow rates):

            Q_h = D^2.5 * 469.2 * T_b/P_b * [ (P₁²-P₂²) / ( S_g *T * Z * L ) ]^0.5 * log₁₀( 3.7 * D / ε )      (Equation 13)

Table 1:  For 2.0 psig (13.8 kPa-g) inlet pressure and 1.0 psig (6.9 kPa-g) drop, using Schedule 40 steel pipe, sizes ½-inch (DN-15) to 6-inch (DN-150).
Equation	Min Ratio	Max Ratio	Average Ratio	Std Dev.
NFPA/IFGC	0.837	1.020	0.915	0.039
Mueller	0.998	1.686	1.344	0.178
Weymouth	0.836	1.227	1.049	0.087
IGT Distribution	0.983	1.476	1.258	0.130
Spitzglass HP	0.582	0.906	0.777	0.086
NFPA Table	0.800	1.020	0.882	0.044
Note:  Reynolds Number Range:  4.1E+03 to 2.9E+06.

 

Table 2:  For 3.0 psig (20.7 kPa-g) inlet pressure and 2.0 psig (13.8 kPa-g) drop, using Schedule 40 steel pipe, sizes ½-inch (DN-15) to 6-inch (DN-150).
Equation	Min Ratio	Max Ratio	Average Ratio	Std Dev.
NFPA/IFGC	0.826	1.024	0.918	0.043
Mueller	0.981	1.754	1.385	0.202
Weymouth	0.824	1.209	1.023	0.089
IGT Distribution	0.969	1.514	1.277	0.147
Spitzglass HP	0.573	0.855	0.757	0.086
NFPA Table	0.826	1.024	0.914	0.043
Note:  Reynolds Number Range:  6.2E+03 to 4.4E+06.

 

Table 3:  For 5.0 psig (34.5 kPa-g) inlet pressure and 3.5 psig (24.1 kPa-g) drop, using Schedule 40 steel pipe, sizes ½-inch (DN-15) to 6-inch (DN-150).
Equation	Min Ratio	Max Ratio	Average Ratio	Std Dev.
NFPA/IFGC	0.824	1.032	0.920	0.047
Mueller	0.962	1.806	1.420	0.217
Weymouth	0.809	1.159	0.999	0.092
IGT Distribution	0.949	1.539	1.292	0.158
Spitzglass HP	0.536	0.836	0.739	0.086
NFPA Table	0.785	1.003	0.885	0.386
Note:  Reynolds Number Range:  8.5E+03 to 6.0E+06.

 

Table 4:  For 20.0 psig (137.9 kPa-g) inlet pressure and 2.0 psig (13.8 kPa-g) drop, using Schedule 40 steel pipe, sizes ½-inch (DN-15) to 6-inch (DN-150).
Equation	Min Ratio	Max Ratio	Average Ratio	Std Dev.
NFPA/IFGC	0.849	1.064	0.960	0.050
Mueller	0.992	1.665	1.253	0.151
Weymouth	0.832	0.934	0.878	0.028
IGT Distribution	0.980	1.363	1.139	0.088
Spitzglass HP	0.437	0.859	0.656	0.101
AGA Plast Pipe Manual	0.990	1.042	0.997	0.003
Note:  Reynolds Number Range:  8.8E+03 to 6.3E+06

 

Table 5:  For 20.0 psig (137.9 kPa-g) inlet pressure and 2.0 psig (13.8 kPa-g) drop, using SDR 13.5 PE pipe, sizes 1 inch (DN-25) to 6-inch (DN-150).
Equation	Min Ratio	Max Ratio	Average Ratio	Std Dev.
NFPA/IFGC	0.807	0.905	0.830	0.020
Mueller	0.996	1.578	1.309	0.151
Weymouth	0.664	1.045	0.910	0.094
IGT Distribution	0.978	1.359	1.177	0.117
Spitzglass HP	0.528	0.832	0.696	0.066
AGA Plast Pipe Manual	0.989	1.033	1.000	0.011
Note:  Reynolds Number Range:  2.2E+04 to 5.4E+06.

 

Table 6:  For 20.0 psig (137.9 kPa-g) inlet pressure and 2.0 psig (13.8 kPa-g) drop, using SDR 11 PE pipe, sizes 3/4 inch (DN-20) to 6-inch (DN-150).
Equation	Min Ratio	Max Ratio	Average Ratio	Std Dev.
NFPA/IFGC	0.690	0.926	0.824	0.054
Mueller	0.909	1.523	1.254	0.143
Weymouth	0.594	1.048	0.890	0.102
IGT Distribution	0.836	1.345	1.151	0.125
Spitzglass HP	0.496	0.827	0.654	0.067
AGA Plast Pipe Manual	0.845	1.026	0.978	0.045
Note:  Reynolds Number Range:  1.5E+04 to 4.6E+06.

 

Table 7:  For 40.0 psig (275.8 kPa-g) inlet pressure and 4.0 psig (27.6 kPa-g) drop, using Schedule 40 steel pipe, sizes ½-inch (DN-15) to 6-inch (DN-150).
Equation	Min Ratio	Max Ratio	Average Ratio	Std Dev.
NFPA/IFGC	0.880	1.107	0.996	0.056
Mueller	0.985	1.802	1.352	0.169
Weymouth	0.826	0.917	0.869	0.028
IGT Distribution	0.987	1.442	1.201	0.098
Spitzglass HP	0.435	0.854	0.649	0.100
AGA Plast Pipe Manual	0.998	1.011	0.990	0.001
Note:  Reynolds Number Range:  1.6E+04 to 1.2E+07.

 

Table 8:  For 40.0 psig (275.8 kPa-g) inlet pressure and 4.0 psig (27.6 kPa-g) drop, using SDR 13.5 PE pipe, sizes 1-inch (DN-25) to 6-inch (DN-150)
Equation	Min Ratio	Max Ratio	Average Ratio	Std Dev.
NFPA/IFGC	0.800	0.872	0.815	0.015
Mueller	0.998	1.621	1.337	0.158
Weymouth	0.623	0.985	0.853	0.087
IGT Distribution	0.970	1.365	1.174	0.118
Spitzglass HP	0.496	0.779	0.652	0.062
AGA Plast Pipe Manual	0.981	1.032	0.996	0.013
Note:  Reynolds Number Range:  4.2E+04 to 1.0E+07.

 

Table 9:  For 40.0 psig (275.8 kPa-g) inlet pressure and 4.0 psig (27.6 kPa-g) drop, using SDR 11 PE pipe, sizes 3/4 inch (DN-20) to 6-inch (DN-150).
Equation	Min Ratio	Max Ratio	Average Ratio	Std Dev.
NFPA/IFGC	0.800	0.888	0.821	0.018
Mueller	0.989	1.613	1.303	0.153
Weymouth	0.618	0.983	0.847	0.088
IGT Distribution	0.969	1.361	1.166	0.116
Spitzglass HP	0.464	0.778	0.623	0.068
AGA Plast Pipe Manual	0.982	1.032	0.993	0.012
Note:  Reynolds Number Range:  2.7E+04 to 9.4E+06.

Other Considerations

The other consideration in this discussion is maximum velocity. This has been reviewed in some detail in the ASPE Plumbing Engineering Design Handbook Volume 3, Chapter 11, that will be published in the Spring 2020. Considerations are noise and erosion. A maximum actual gas velocity of 100 feet per second (30.5 meters per second) should be considered for gas pressures over 10 psig (69.0 kPa-g).

All computations performed used 0.6 as the specific gravity. This was because the tables in NFPA 54 and the IFGC are all based on 0.6 specific gravity. The internet places the specific gravity of natural gas between 0.6 and 0.7. The North American Combustion Handbook (3^rd Edition – 1986) places the specific gravity of natural gas between 0.59 and 0.64. Higher specific gravity means higher viscosity, lower Reynolds number, and higher value for “f”. This means the pressure drop will be higher or the pipe carrying capacity with a specific pressure drop will be lower. The straight forward capacity factor for gas is (0.65/0.60)^0.5; this equates to 1.04 (and approximately 1.06 when “f” is considered). Therefore, the pressure drop will be 1.08 to 1.12 times greater for the flow capacity at Sg = 0.65 specific gravity.

Conclusions

The equations and tables in the NFPA and IFGC provide very comparable values with the ASTM / AGA equations using the Darcy Equation and the Colebrook-White formula for “f”. This analysis is for new clean pipe. Clean pipe will not affect the Partially Turbulent flow regime, since the flow at the pipe surface is laminar. The engineer should consider multiplying any flow in the fully turbulent range by an efficiency factor of 0.90 to 0.97.

The Weymouth equation provides conservative values for flow and higher pressure drops than might be experienced in actual practice. The Spitzglass-High Pressure equation is even more conservative than the Weymouth equation.

The Mueller and IGT Distribution equations provide higher flow rates and lower pressure drops than might be experienced in actual practice. As a result, these equations are not recommended for typical plumbing applications where higher pressures might be used.

The AGA equations in the AGA Plastic Pipe Manual provide very comparable values with the AGA equations using the Darcy Equation and the Colebrook-White formula for “f”.

There are two final considerations when applying these formulas: Maximum velocity and actual specific gravity of the natural gas. A maximum gas velocity of 100 feet per second (30.5 meters per second) to minimize noise and erosion. The specific gravity of the natural gas should be considered as higher specific gravity will result in higher pressure drops or lower pipe carrying capacity at a given pressure drop.

Although these equations have been used to create a carrying capacity (Q_h) in this article, the equations can be used to create a table like NFPA/IFGC for design purposes. The equations can then be rearranged to create a pressure drop calculation, which the engineer can use to backcheck the analysis.