Colebrook – White (Gas)

Introduction

The Colebrook-White equation is recommended for use by those unfamiliar with pipeline flow equations. It will produce the greatest consistency of accuracy over the widest possible range of variables. The Colebrook Equation has good accurate results for lower flow rates (150-250 MMSCFD); However, larger errors exist above a 250 MMSCFD flow rate.

The friction factor is solved using the Newton-Raphson method.

f=\frac{64}{Re};\,Laminar\,Flow: Re < 2000 \~\ \frac{1}{\sqrt{f}}=2\log_{10}(Re\sqrt{f})-0.8;\,Smooth\,Pipe:Re >4000\,and\,\frac{\varepsilon}{D}\rightarrow0\~\ \frac{1}{\sqrt{f}}=-2\log_{10}\biggr(\frac{\varepsilon}{3.7D}+\frac{2.51}{Re\sqrt{f}}\biggr);\,Transitional,\,Colebrook-White\,Eq,Re>4000\~\ \frac{1}{\sqrt{f}}=1.14-2\log_{10}\biggr(\frac{\varepsilon}{D}\biggr);\,Wholly \,Rough

f=\frac{64}{Re};\,Laminar\,Flow: Re < 2000 \\~\\ \frac{1}{\sqrt{f}}=2\log_{10}(Re\sqrt{f})-0.8;\,Smooth\,Pipe:Re >4000\,and\,\frac{\varepsilon}{D}\rightarrow0\\~\\ \frac{1}{\sqrt{f}}=-2\log_{10}\biggr(\frac{\varepsilon}{3.7D}+\frac{2.51}{Re\sqrt{f}}\biggr);\,Transitional,\,Colebrook-White\,Eq,Re>4000\\~\\ \frac{1}{\sqrt{f}}=1.14-2\log_{10}\biggr(\frac{\varepsilon}{D}\biggr);\,Wholly \,Rough

𝑓 − Friction Factor
D – Inside Pipe Diameter[in]
𝜀 − Pipe Roughness[in]

F=\sqrt{\frac{1}{f}}

F=\sqrt{\frac{1}{f}}

𝐹 − Transmission Factor

Q=C_Q(\frac{T_b}{P_b})D^{2.5}F\biggr[ \frac{P_1^2-e^sP_2^2}{GT_fL_eZ} \biggr]^{0.5}

Q=C_Q(\frac{T_b}{P_b})D^{2.5}F\biggr[    \frac{P_1^2-e^sP_2^2}{GT_fL_eZ}    \biggr]^{0.5}

𝑄 − Flow Rate (FT³/day)
𝐶_𝑄 − 38.774
𝑇_𝑏 −Temperature Base (°R)
𝑃_𝑏 − Pressure Base (psia)
𝑇_𝑓 − Gas Flowing Temperature (°R)
𝐷 − Internal Diameter (in)
𝐹 − Transmission Factor
𝑘 − Pipe Roughness
𝑃₁ − Upstream Pressure (psig)
𝑃₂ − Downstream Pressure (psig)
𝐺 − Gas Specific Gravity
𝑍 − Compressibility Factor
L_e − Pipe Segment Length including Expansion (mi)

s=\frac{C_S\triangle HG}{T_fZ}

s=\frac{C_S\triangle HG}{T_fZ}

𝑠 − Elevation adjustment parameter
𝐶_𝑆 − 0.0375
𝑍 − Compressibility Factor
𝑇_𝑓 − Gas Flowing Temperature (°R)
∆𝐻𝐺 − Change in Elevation (ft)

L_e=\frac{(e^s-1)}{s}

L_e=\frac{(e^s-1)}{s}

𝐿_𝑒 − Pipe Segment Length including Expansion (mi)
𝑠 − Elevation adjustment parameter

V=0.75\frac{Q_h}{D^2P_{avg}}

V=0.75\frac{Q_h}{D^2P_{avg}}

𝑉 − Velocity (ft/sec)
𝑄_ℎ − Volumetric flow rate (scf/hr)
𝐷 − Internal Diameter (in)
𝑃_𝑎𝑣𝑔 − Average Pipeline Pressure (psia)

For Small Pressure Drop P2 > 0.8 P1:

P_{avg}=\frac{(P_1-P_2)}{2}

P_{avg}=\frac{(P_1-P_2)}{2}

For Large Pressure Drop:

P_{avg}=\frac{2}{3}\biggr[ P_1+P_2-\frac{P_1P_1}{P_1+P_2} \biggr]

P_{avg}=\frac{2}{3}\biggr[  P_1+P_2-\frac{P_1P_1}{P_1+P_2}    \biggr]

Case Guide

Part 1: Create Case

Input Parameters

• Temperature base(°F)
• Pressure base(psia)
• Gas Flowing Temperature(°F)
• Gas Specific Gravity
• Compressibility Factor
• Pipeline Efficiency Factor
• Upstream Pressure(psig)
• Downstream Pressure(psig)
• Flow Rate(Barrels per Day)
• Internal Pipe Diameter(in)
• Length of Pipeline(mi)
• Upstream Elevation(ft)
• Downstream Elevation(ft)

Downstream Pressure

Flow Rate

Internal Pipe Diameter

Upstream Pressure

Part 2: Outputs/Reports

Results

• Downstream Pressure(psig)
• Flow Rate(ft/sec.)
• Internal Pipe Diameter(in)
• Upstream Pressure(psig)
• Transmission Factor
• Velocity(ft/sec.)
• Erosional Velocity
• Sonic Velocity

Downstream Pressure

Flow Rate

Internal Pipe Diameter

Upstream Pressure

References

• “Pipeline Rules of Thumb” Gulf Professional Publishing, Seventh Edition, McAllister, E. W.
• “Gas Pipeline Hydraulics”, Systek Technologies, Inc., Menon, Shahi E.
• “Advanced Pipeline Design”, Carroll, Landon and Hudkins, Weston R.
• American Gas Association (AGA), “Reference: Eq-17-18, Section 17, GPSA”, Engineering Data Book, Eleventh Edition
• Hydraulic Transients, McGraw-Hill, New York., Streeter, V.L. and Wylie, E.B. (1967)
• Water Hammer Analysis. Jour. Hyd. Div., ASCE., Vol. 88, HY3, pp79-113 May, Streeter, V.L. (1969)
• Unsteady flow calculations by numerical methods’, Jour. Basic Eng., ASME., 94, pp457-466, June. Streeter, V.L. (1972),
• Hydraulic Pipelines, John Wiley & Sons, J. P Tullis (1989)

FAQ