Colebrook – White (Liquid)

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. This equation is limited for use when the Reynolds number exceeds 4000.

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]

H=f\bigg( \frac{L}{\frac{D}{12}} \bigg) \bigg( \frac{V_C^2}{2G} \bigg) \~\ V_C=\frac{Q}{A}=\frac{\frac{\frac{\pi D^2}{4}\sqrt{2GH}}{f(\frac{L}{D})}}{\frac{\pi D^2}{4}}

H=f\bigg( \frac{L}{\frac{D}{12}}  \bigg) \bigg( \frac{V_C^2}{2G}  \bigg)  \\~\\  V_C=\frac{Q}{A}=\frac{\frac{\frac{\pi D^2}{4}\sqrt{2GH}}{f(\frac{L}{D})}}{\frac{\pi D^2}{4}}

𝐻 − Head Loss[ft]
𝐿 − Pipe Length[mi]
𝑉_𝐶 − Average Velocity[ft/sec]
𝑓 − Friction Factor
D – Inside Pipe Diameter[in]
𝐺 − Liquid Specific Gravity
𝐴 − Area[ft2]
𝑄 − Flow Rate[BPD]

HP_1=HP_2-H_1+H+H_2 \~\HP_2=\frac{P_2}{.433514SG} \~\ P_1=.433514(SG)(HP_1) \~\\triangle P=P_1-P_2

HP_1=HP_2-H_1+H+H_2 \\~\\HP_2=\frac{P_2}{.433514SG} \\~\\ P_1=.433514(SG)(HP_1) \\~\\\triangle P=P_1-P_2

𝐻𝑃₁ − Upstream Head Pressure[ft]
𝐻𝑃₂ − Downstream Head Pressure[ft]
𝐻₁ − Upstream Elevation[ft]
𝐻₂ − Downstream Elevation[ft]
𝐻 − Head Loss[ft]
𝑆𝐺 − Specific Gravity
𝑃₁ − Upstream Pressure[psig]
𝑃₂−Downstream Pressure[psig]

Case Guide

Part 1: Create Case

Input Parameters

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

Downstream Pressure

Flow Rate

Internal Pipe Diameter

Upstream Pressure

Part 2: Outputs/Reports

Results

• Friction Factor
• Head Loss [ft]
• Pressure Loss [psi/mi]
• Downstream Pressure(psig)
• Flow Rate(Barrels per Day)
• Internal Pipe Diameter(in)
• Upstream Pressure(psig)
• Average Velocity(ft/sec.)
• Erosional Velocity (ft/sec.)
• Sonic Velocity (ft/sec.)

Downstream Pressure

Flow Rate

Internal Pipe Diameter

Upstream Pressure

References

• McAllister, E. W., “Pipeline Rules of Thumb” Gulf Professional Publishing, Seventh Edition
• Menon, Shahi E., “Gas Pipeline Hydraulics”, Systek Technologies, Inc.
• Carroll, Landon and Hudkins, Weston R., “Advanced Pipeline Design”
• American Gas Association (AGA), “Reference: Eq-17-18, Section 17, GPSA”, Engineering Data Book, Eleventh Edition

FAQ