Moody Chart Colebrook-White

Overview

The Moody chart (Moody, 1944) is the graphical representation of the Darcy-Weisbach friction factor as a function of Reynolds number and relative pipe roughness. The underlying equation is the Colebrook-White (1939) implicit equation, which is the industry standard for calculating friction losses in turbulent pipe flow. These calculations are fundamental to hydraulic design of pipelines, flowlines, and wellbore tubulars.

Theory

Fluid flow in pipes generates friction losses proportional to velocity squared. The friction factor f captures the combined effects of fluid properties (Reynolds number) and pipe roughness. In laminar flow (Re < 2100), f = 64/Re. In turbulent flow (Re > 4000), f depends on both Re and ε/D (relative roughness).

Formulas

Darcy-Weisbach Equation (Pressure Loss)

ΔP = f * (L/D) * (ρ * v^2) / 2

In oilfield units:

ΔP (psi) = f * L * ρ * v^2 / (25.8 * D)

where L = length (ft), D = diameter (ft), ρ = density (lb/ft³), v = velocity (ft/s).

Head Loss

hf = f * (L/D) * v^2 / (2g)

Reynolds Number

Re = ρ * v * D / μ = v * D / ν

In oilfield units:

Re = 928 * ρ * v * D / μ

where ρ in lb/gal, v in ft/s, D in inches, μ in cp.

Colebrook-White Equation (Turbulent Flow)

1/sqrt(f) = -2 * log10(ε/(3.7*D) + 2.51/(Re*sqrt(f)))

This is implicit in f and must be solved iteratively.

Explicit Approximations

Churchill (1977):

f = 8 * ((8/Re)^12 + (A + B)^(-1.5))^(1/12)
A = (2.457 * ln(1/((7/Re)^0.9 + 0.27*ε/D)))^16
B = (37530/Re)^16

Swamee-Jain (1976):

f = 0.25 / (log10(ε/(3.7*D) + 5.74/Re^0.9))^2

Accurate to within 1% of Colebrook-White for Re > 5000 and 10⁻⁶ < ε/D < 10⁻².

Haaland (1983):

1/sqrt(f) = -1.8 * log10((ε/(3.7*D))^1.11 + 6.9/Re)

Laminar Flow (Re < 2100)

f = 64 / Re

Common Pipe Roughness Values

Worked Example

Given: Water (ρ = 62.4 lb/ft³, μ = 1.0 cp) flowing at 5 ft/s through 4" schedule 40 steel pipe (ID = 4.026"), ε = 0.0018", L = 1,000 ft.

Step 1: Reynolds number:

Re = 62.4 * 5 * (4.026/12) / (1.0 * 6.72e-4)
   = 62.4 * 5 * 0.3355 / 6.72e-4
   = 104.64 / 6.72e-4
   = 155,714 → Turbulent

Step 2: Relative roughness:

ε/D = 0.0018 / 4.026 = 0.000447

Step 3: Swamee-Jain approximation:

f = 0.25 / (log10(0.000447/3.7 + 5.74/155714^0.9))^2
  = 0.25 / (log10(1.208e-4 + 1.27e-4))^2
  = 0.25 / (log10(2.478e-4))^2
  = 0.25 / (-3.606)^2
  = 0.25 / 13.00
  = 0.0192

Step 4: Head loss:

hf = 0.0192 * (1000/0.3355) * 5^2 / (2 * 32.2)
   = 0.0192 * 2981 * 0.388
   = 22.2 ft

Step 5: Pressure drop:

ΔP = 62.4 * 22.2 / 144 = 9.6 psi

Valid Ranges

References

1. Moody, L.F. (1944). "Friction Factors for Pipe Flow." Trans. ASME, 66, 671–684.
2. Colebrook, C.F. (1939). "Turbulent Flow in Pipes, with Particular Reference to the Transition Region Between Smooth and Rough Pipe Laws." J. ICE, 11(4), 133–156.
3. Swamee, P.K. & Jain, A.K. (1976). "Explicit Equations for Pipe-Flow Problems." J. Hydraulics Division, ASCE, 102(5), 657–664.
4. Wikipedia — Moody chart: https://en.wikipedia.org/wiki/Moody_chart
5. Engineering Toolbox — Moody Chart: https://www.engineeringtoolbox.com/moody-diagram-d_618.html