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Flow Equations For Low Pressure Natural Gas

Last updated: 11-27-2020

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Flow Equations For Low Pressure Natural Gas

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Flow Equations For Low Pressure Natural Gas
October 31, 2020
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There are several equations and tables for determining the flow in natural gas pipes and the pressure drops associated with those flows, or vise versa. The purpose of this article is to evaluate the available low pressure natural gas flow equations among themselves and with the tables in the codes.
Previous articles in this series were used to evaluate various equations used for determining the pressure drop in high pressure natural gas lines. High pressure was defined as inlet pressures above 1.5 psig (10.3 kPa) to over 50 psig (345 kPa). Also, earlier articles in this series suggested that either schedule 40 steel pipe or polyethylene pipe (PE) are the normal piping materials used. Type K copper is also suggested in the codes for low pressure natural gas piping.  The inside diameters of each of these pipes is different. Currently, standard tables exist in both NFPA 54 National Fuel Gas Code and the ICC International Fuel Gas Code for low pressure natural gas flow in piping.  
Several references were used to evaluate the baseline equation to make the comparison. These texts all indicate that the Darcy-Weisbach equation appears to be the most accurate method for determining pressure drop. However, this method has been 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 personal 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:
                        hL = f 
Where:             hL = gas head loss in feet (meters) of fluid – in this case natural 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 Reynolds number is:
                        Re = ρ V D /μ                                                                                        (Equation 2)
Where:             ρ = gas density
D = internal diameter of the pipe
μ = gas dynamic viscosity – 6.98311E-06 lbm/ft/sec (0.010392 centipoise)
To assist in the calculation, when density is broken down (into 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 Qst 29 Sg Pst / (μ π D 
 Tst)                                                           (Equation 3)
Where:             Qst = Gas Flow rate at Standard Conditions
29 = molecular weight of air, 28.9647 lb/lbmol (28.9647 g/gmol)
Sg = specific gravity of natural gas
Pst = standard gas pressure – 14.696 psia (101.325 kPa)
μ = gas dynamic viscosity – 6.98311E-06 lbm/ft/sec (0.010392 centipoise)
π= PI = 3.14159
D = internal diameter of the pipe
 = Universal gas constant, 1545.349 lbf ft/(lbmol °R) [8314.41 J/(kmol °K)]
Tst = 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).
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                                 (Equation 4)
AGA Partially Turbulent Flow:       
AGA Fully Turbulent Flow:            
   (Note 2 below)                                     (Equation 6)
                  Note 1:  Formerly, the 2.825 value in equation 5 was 2.51 and is the Colebrook-White equation, 1990.
                  Note 2:  Fully Turbulent Flow is normally not encountered with low pressure gas piping.
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. As will be discussed later, there is also a transition between Partially Turbulent Flow and Laminar Flow; this transition is not closely defined because it occurs between “Re” being 2,000 and 4,000.  With Laminar Flow being dependent on the pipe diameter as well as the velocity, Laminar Flow is more prevalent in smaller pipes than in larger pipes.
Reviewing the Moody Diagram that plots Friction Factor “f” against Reynolds Number “Re”, there is a disconnect between Partially Turbulent and Laminar Flow.  Since the smaller pipes that are the subject of this paper have an “ε/D” ratio of 0.0001or less, the “f” for partially turbulent flow will approach the “Re” being equal to 4,000 along the lower, smooth-pipe line.  “f”will equal approximately 0.0413 at this intersection.  The “f” value drops to 0.032 at “Re” being 2,000, and it quickly rises to 0.064 at “Re” being 1,000.  This has resulted in the smaller flow values predicted by the simplified equations on long and/or small pipes being more than double what the flow capacity might actually be.
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 formula). These points were compared with answers created by using each of the alternate equations and flow tables.  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 = 0.010392 centipoise or 6.98E-06 lbm/ft-sec.
Pressure ranges:  Less than 2.0 psig inlet with 0.3 in w.c. drop, less than 2.0 psig with 0.5 in w.c. drop, less than 2.0 psig inlet with 3.0 in w.c. drop, and less than 2.0 psig with 6.0 in w.c. drop. For this article, the inlet gas pressure was established as 14.79 psia (14.43 psia at 500 feet altitude and 10-inches w.c.).
Distances:  10 feet (3 meters) to 2,000 feet (610 meters) for steel pipe and copper tubing; in increments similar to NFPA 54 and IFGC.  (With partially turbulent flow, the boundary layer between the flowing gas and the edge wall is similar to laminar flow and only determined by diameter.  Since the steel pipe tables use a whole group of sizes from 0.622 inches (15.80 mm) to 11.938-inches (304.37 mm), the need to examine Polyethylene pipe was determined to be non-essential.  Furthermore, only one table (NFPA 6.2.1(h) was examined for copper; this table has pipe sizes down to ¼ inch (DN6) pipe size.
Nominal Pipe Sizes:  0.5” (DN15) through 4” (DN100) or 12” (DN300) for steel and ¼ inch (DN6) through 2 inch (DN50) for copper as established in the NFPA 54 and IFGC.
Pipe Materials:  Sch 40 steel pipe and Type K Copper Tubing.
Equations used:  NFPA/IFGC equation, Low Pressure Mueller Equation, and Low Pressure Spitzglass Equation. The values in the NFPA/IFGC tables were also compared; it should be noted here that the low pressure gas equations and the tables in both the NFPA 54 and IFGC codes are the same.  Note, all equations were rearranged to provide Qh (flow per hour) as a function of H1and H2 (inlet and outlet pressures.)
For all of the following equations, “Qh” is flow in SCFH, “H1“ is the inlet pressure in inches w.c., “H2“ is the outlet pressure in inches w.c., “D” is the pipe inside diameter in inches, “Sg” 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 (0.046 mm) and 0.00006 inches (0.0015 mm) for copper tubing.  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 lower):
            Qh = (D * { 19.17 * [ (H1-H2) / ( Cr * L ) ]0.206 } )(1/0.381)                                            (Equation 7)
Where:             Cr = 0.6094 for natural gas
Mueller Low Pressure Equation:
            Qh = ( 2,971 * D2.725 ) / Sg0.425 * [ (H1-H2) / L ) ]0.575                                                (Equation 8)
Spitzglass-Low Pressure Equation:
            Qh = ( 3,350 / Sg0.5 ) * [ (H1-H2) / L ) ]0.5 * [ D5 / ( 1 + 3.6 / D + 0.03 * D ) ]0.5           (Equation 9)
Table 1 [1]  :  For less than 2.0 psig (13.8 kPa-g) inlet pressure and 0.3 inch w.c. (75 Pa) drop, using Schedule 40 steel pipe, sizes ½-inch (DN-15) to 12-inch (DN-300). (Results compared to Darcy)
    Note:  Reynolds Number Range:  1.01E+02 to 3.64E+04.  Due to the heavy concentration of “Re’s” below 4,000, over 65% of the data was eliminated. Essentially all of the data for pipe sizes 1/4” to 3/4” was deemed to be faulty (between 5 and 100% off). 
Other considerations
As discussed, the major problem is the conversion of the flow from Partially Turbulent at Re = 4,000 and Laminar at Re = 2000.  Since “f” =0.0413 at Re = 4,000 and “f” = 0.32 at Re = 2,000, the safe alternative would be to hold the value of “f” at 0.0413between Re = 4,000 and Re = 1,549 (where 64/Re = 0.0413).  By fixing the “f” between these to Reynolds number values, the result will be a conservative value for expected flow and pressure drop.
Since the NFPA, Mueller, and Spitzglass, formulas and the NFPA tables do not outline the Reynolds Number, the first thing to do would be to determine the appropriate critical flow for each pipe size, associated with the Reynolds numbers 4,000 and 1,549.
The following formula approximates the critical flow rates based on pipe size:
            QCr = 0.03586 * ReCr * D   ( QCr = 3.9977E-05 * ReCr * D )                                     (Equation 10)
Where:             QCr = Critical Flow Rate where flow converts from Partially Turbulent to “Indeterminant” and from “Indeterminant” to Laminar – CuFt/Hr  (M3/hr).
ReCr = Critical Reynolds Number:  4,000 or 1,549.
D = internal diameter of the pipe – inches (mm)
At flow rates less than the critical flows, the following equations would be used to determine the capacity of the pipe based on diameter.
The following formula approximates the flow rates based on pipe size and a friction factor “f” value of 0.0413.  This equation would be used to determine the capacity of a pipeline where the flow rates are between the two flows, Qcr, where “ReCr”values are between 4,000 and 1,549:
            Qh = 2,380.2 * D2.5 * (Δh / L)0.5   ( Q = 0.000725636* D2.5 * (Δh / L)0.5 )                 (Equation 11)
Where:             Qh = Flow Rate based on pipe diameter, design Δh and Pipe length – CuFt/Hr  (M3/hr).
D = internal diameter of the pipe – inches (mm)
Δh = pressure drop in pipe ( H1 – H2 ) – inches w.c. (Pa)
L = length of the pipe segment – Feet (meters)
Finally, the following formula approximates the flow rates based on pipe size and a friction factor “f” value of 64/Re (the laminar flow friction factor).  This equation would be used to determine the capacity of a pipeline where the flow rates are below the flow where the critical “ReCr” value is 1,549:
            Qh = 101,990 * D4 * Δh / L   ( Q = 8.50273E-06 * D4 * Δh / L  )                             (Equation 12)
Where:             Qh = Flow Rate based on pipe diameter, design Δh and Pipe length – CuFt/Hr  (M3/hr).
D = internal diameter of the pipe – inches (mm)
Δh = pressure drop in pipe ( H1 – H2 ) – inches w.c. (Pa)
L = length of the pipe segment – Feet (meters)
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 (3rd 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 straightforward 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 1.08to 1.12 times greater for the flow capacity at Sg = 0.65 specific gravity.
(subhead) Conclusions
The equations and tables in the NFPA and IFGC provide disappointing comparable values when compared with using the Darcy Equation and the Colebrook-White formula for “f”.  This is primarily because the pipes with small sizes, long lengths, and low pressure drops have flow regimes that fall in the Laminar Flow range.  This is evident from the tables above when the Laminar Flow data is eliminated from the comparison.  Simplified equation and Table estimated Flow Capacity data is 20% to 100% higher than the comparable Darcy data in the Laminar Flow ranges.  The volume ratios for the comparisons reduce significantly, particularly at the higher pressure drops where Laminar Flow is less likely.
The comparisons also improve significantly when the flows in the Critical transition and Laminar Flow regions where Re < 4,000, are eliminated from the comparison.
Above the critical transition region, the Mueller equation provides slightly higher flow rates and lower pressure drops than might be experienced in actual practice (max ratio ~ 1.2).  As a result, this equation is not recommended for typical plumbing applications.
Above the critical transition region, the NFPA/IFGC calculation, the Sptizglass calculation, and the NFPA/IFGC tables provide reasonable and more conservative estimates of the gas flow capacities of these pipelines in these applications.
However, below the critical regions, all of the equations and tables reviewed provide excessive estimates of pipeline flow capacities.  As outlined in the Other Considerations Section, the recommendations are to first determine at what critical flow rates where Re = 4,000 and Re = 1,549 are encountered using equation 10.  Reasonable estimates of the flow capacities can be determined by using equation 11 between the two critical flow rates and by using equation 12 when the flow rates are below the Re = 1,549 flow rate.
The engineer should consider using a specific gravity for natural gas of 0.65 since the higher specific gravity will reduce the carrying capacity of the pipe system.

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