The Colebrook-White Equation vs Swamee-Jain: Which Friction Factor Formula Should You Use?
Two engineers calculating friction factor for the same pipe can use two different equations and get slightly different answers. One solves the problem exactly but requires iteration. The other gives a direct answer in one step but introduces a small error. This guide explains both equations, shows you exactly where the error lies, and tells you which to use for your situation.
The Problem Both Equations Solve
In turbulent pipe flow, you need the Darcy friction factor (f) to calculate pressure drop with the Darcy-Weisbach equation. Friction factor depends on two things: Reynolds number (Re) and relative roughness (ε/D). But there is no single simple formula that gives you f directly from those two inputs across the full range of turbulent flow.
The Moody Chart is a graphical solution to this problem, and it works well for quick hand estimates. But for precise calculations, programmatic use, or situations where you need a formula rather than a graph, you need an equation. Two equations dominate engineering practice: the Colebrook-White equation and the Swamee-Jain approximation.
Both apply only to turbulent flow in circular pipes. For laminar flow (Re below 2,300), neither applies. You use f = 64/Re instead. For a full picture of where each equation fits in the Moody Chart, read the flow zone breakdown in our chart reading guide.
The Colebrook-White Equation
Cyril Frank Colebrook and Cedric Masey White published this equation in 1939 based on experimental data from rough pipes. It became the definitive expression for friction factor in turbulent flow and forms the mathematical basis of the Moody Chart. You can find the original derivation in the 1937 Colebrook paper in Proceedings of the Royal Society.
The equation blends two limiting cases into one expression.
In the middle of the turbulent range, both terms contribute. The equation transitions smoothly between the two extremes, which is why a single expression covers the entire turbulent flow region.
Valid Range
The Colebrook-White equation applies to fully turbulent flow in circular pipes for Re above 4,000 and relative roughness between 0 and 0.05. It produces the same curves you see on the Moody Chart and is accepted as the standard by virtually every major engineering code including ISO 4006 and the ASHRAE Handbook of Fundamentals.
Why Implicit Equations Need Iteration
The Colebrook-White equation is implicit: f appears on both the left side (as 1/√f) and the right side (as √f inside the log). You cannot rearrange it algebraically to get f = something. Try it and you end up with f in terms of expressions that still contain f.
This is not a flaw. It is a consequence of the physics. The friction factor influences both the pressure gradient and the turbulent boundary layer structure simultaneously. There is no closed-form solution that captures both effects independently.
The practical consequence is that you cannot solve the equation by hand in a single step. You need to iterate: start with a guess for f, calculate the right-hand side, use that result as your new f, and repeat until the answer stops changing. With a good initial guess, this converges in 3 to 6 iterations. The Moody Chart Calculator does this automatically using the Newton-Raphson method.
The Initial Guess
The choice of starting value affects how quickly the iteration converges, but not whether it converges. A common initial guess is f₀ = 0.02, which works for most engineering conditions. A better starting guess uses the fully rough approximation:
This gives you the fully rough friction factor, which is an upper bound for smooth and transitional pipes. Starting from this value typically reduces convergence to 2 to 3 iterations.
How Newton-Raphson Solves It
Newton-Raphson is a root-finding method that finds where a function equals zero by repeatedly improving an estimate using the function's slope (derivative). It converges quadratically, meaning the number of correct digits roughly doubles with each iteration.
To apply it to Colebrook-White, rewrite the equation as F(f) = 0:
The Newton-Raphson update is:
f_(n + 1) = f_n − F(f_n) / F′(f_n)Starting from f₀ = 0.02, this typically reaches 6-digit accuracy in 3 to 5 iterations for any physically realistic input. The calculator shows you the iteration count in the results so you can confirm convergence.
Fixed-Point Iteration: The Simple Alternative
A simpler but slower approach is fixed-point iteration. Rearrange the equation so f appears alone on the left:
Substitute your guess for f on the right, compute the result, use it as the new f, and repeat. This converges linearly rather than quadratically, typically needing 10 to 30 iterations for the same accuracy as 3 to 5 Newton-Raphson steps. It is easier to implement by hand but slower in code. For a spreadsheet calculation, fixed-point iteration is usually the practical choice.
The Swamee-Jain Equation
In 1976, P.K. Swamee and A.K. Jain published an explicit approximation of the Colebrook-White equation that gives f directly without iteration. The original paper appeared in the ASCE Journal of the Hydraulics Division.
This is an explicit equation. You put in Re and ε/D and get f out in a single calculation. No iteration required. No initial guess. This makes it ideal for hand calculations, spreadsheets, and quick estimates.
Where the Approximation Comes From
Swamee and Jain derived their formula by curve-fitting the Colebrook-White equation across its valid range. The smooth pipe term 2.51/(Re × √f) in Colebrook-White contains f, which is what makes it implicit. Swamee-Jain replaces this with 5.74/Re⁰·⁹, which approximates the same behavior but without f on the right-hand side. The trade-off is a small systematic error that varies across the Re and ε/D space.
Valid Range
Swamee-Jain is valid for:
Outside this range, the approximation error increases. At Re below 5,000 (near the transition zone boundary), the smooth pipe term in Colebrook-White is large and the Re⁰·⁹ approximation is less accurate. For most engineering pipe flow, the valid range covers all practical conditions.
Accuracy Comparison
The table below compares friction factor results from both equations across a range of Reynolds numbers and relative roughness values. All Colebrook-White values are solved iteratively to 6-digit accuracy using Newton-Raphson.
| Re | ε/D | Colebrook-White (f) | Swamee-Jain (f) | Error (%) | Zone |
|---|---|---|---|---|---|
| 10,000 | 0.000015 | 0.03084 | 0.03114 | +1.0% | Smooth turbulent |
| 50,000 | 0.000015 | 0.02098 | 0.02095 | −0.1% | Smooth turbulent |
| 100,000 | 0.00046 | 0.01903 | 0.01908 | +0.3% | Transitional rough |
| 500,000 | 0.00046 | 0.01726 | 0.01724 | −0.1% | Transitional rough |
| 1,000,000 | 0.0026 | 0.02371 | 0.02388 | +0.7% | Transitional rough |
| 5,000,000 | 0.0026 | 0.02337 | 0.02353 | +0.7% | Fully rough |
| 10,000,000 | 0.020 | 0.05018 | 0.05019 | 0.0% | Fully rough |
| 6,000 | 0.000015 | 0.03596 | 0.03668 | +2.0% | Low Re smooth |
| 8,000 | 0.040 | 0.07248 | 0.07339 | +1.3% | High roughness |
The highlighted rows show the largest errors: low Reynolds numbers near the transition zone boundary, and very high relative roughness. In both cases, errors stay below 3%. For the majority of practical engineering conditions, the error is under 1%.
Does 3% Matter?
A 3% error on friction factor produces a 3% error on pressure drop through the Darcy-Weisbach equation. On a pump with 10 m of head loss, that is 0.3 m. For most applications, a 0.3 m error is well within the uncertainty of the roughness value itself, which can vary by 20 to 50% depending on pipe age and condition. The friction factor formula is rarely the dominant source of error in a real pipe flow calculation.
The exception is iterative design calculations where friction factor feeds into a sizing loop. A 3% systematic bias in f can compound through multiple pipe segments or iterations and produce a larger aggregate error. In those cases, use Colebrook-White.
Other Explicit Approximations Worth Knowing
Swamee-Jain is the most widely used explicit approximation, but several others appear in engineering references. Here is a brief overview of the most relevant ones.
Churchill (1977)
Churchill published a single equation that covers laminar flow, the transition zone, and the full turbulent range in one expression. This is useful when you are building a calculation that needs to handle all flow regimes without conditional logic. The equation is more complex than Swamee-Jain but more general. It is accurate to within 0.5% across all regimes.
Where A and B are functions of Re and ε/D. See Churchill (1977) in Chemical Engineering for the full expression.
Haaland (1983)
The Haaland equation is another explicit approximation, slightly more accurate than Swamee-Jain near the smooth pipe regime:
Haaland is accurate to within 2% across its valid range and is more commonly seen in European engineering practice and some chemical engineering texts. The structure is similar to Colebrook-White (it uses 1/√f on the left), so it is not quite as simple to apply as Swamee-Jain but slightly more accurate near the smooth pipe limit.
Blasius (1913) — For Smooth Pipes Only
For smooth pipes (ε/D approaching zero) at moderate Reynolds numbers (4,000 to 100,000), the Blasius correlation is the simplest option:
This applies only to smooth pipes and only within Re = 4,000 to 100,000. It is accurate to within 1% in that range. Outside it, errors grow rapidly. You will see Blasius in textbook examples and heat transfer correlations, but it is not appropriate for rough pipes or high Reynolds numbers.
Summary Comparison
| Equation | Type | Accuracy | Valid Range | Best For |
|---|---|---|---|---|
| Colebrook-White | Implicit (iterative) | <0.1% | Re > 4,000, any ε/D | All precise calculations |
| Swamee-Jain | Explicit | ±3% | Re 5,000–10⁸, ε/D 10⁻⁶–0.05 | Hand calcs, spreadsheets, estimates |
| Haaland | Explicit | ±2% | Re > 4,000, any ε/D | Slightly better near smooth pipe |
| Churchill | Explicit | ±0.5% | All regimes | Code covering all flow regimes |
| Blasius | Explicit | ±1% | Re 4,000–100,000, smooth only | Textbook examples, heat transfer |
Which Equation to Use
The right choice depends on your situation. Here is a direct decision guide.
Use Colebrook-White when:
You are writing code or using a calculator that handles iteration automatically. Accuracy matters and you do not want a systematic 1 to 3% bias. You are in a design loop where friction factor feeds back into pipe sizing or pump selection. The input is near the edges of Swamee-Jain's valid range (Re close to 5,000, or very smooth pipes). You are producing a deliverable where the calculation method will be audited.
Use Swamee-Jain when:
You are doing a hand calculation and want a direct answer. You are building a spreadsheet that does not have iterative solver capability. You need a quick estimate to check an order of magnitude. Your roughness uncertainty already exceeds 3%, which makes Swamee-Jain's error irrelevant. You are teaching or explaining the calculation and want to avoid iteration complexity.
For Spreadsheet Users
Both Excel and Google Sheets support iterative calculation. In Excel, enable iteration under File → Options → Formulas → Enable iterative calculation. Then you can enter the Colebrook-White equation directly as a circular reference and Excel will iterate to a solution. Set maximum iterations to 100 and maximum change to 0.00001 for sufficient accuracy. Alternatively, use Swamee-Jain in a non-iterative spreadsheet and accept the small approximation error.
For Code
In Python, MATLAB, or any language with a math library, implement Newton-Raphson directly. It is 10 to 15 lines of code and converges in under 10 iterations from any reasonable starting point. Swamee-Jain is a single line but introduces a bias that is easy to avoid. Use Colebrook-White in production code.
Worked Example: Both Methods Side by Side
A 200 mm commercial steel pipe carries water at Re = 250,000. Relative roughness ε/D = 0.00023. Find friction factor using both methods.
Colebrook-White (Iterative)
f = 0.01682
Converged in 3 iterations. Accurate to 6 significant figures.
Swamee-Jain (Direct)
f = 0.01688
Single calculation. Error vs Colebrook-White: +0.35%.
The Swamee-Jain result of 0.01688 is 0.35% above the Colebrook-White result of 0.01682. For a pump sizing exercise with 50 m of this pipe and a flow velocity of 2 m/s, this translates to: Colebrook-White gives ΔP = 7,770 Pa, Swamee-Jain gives ΔP = 7,800 Pa, a difference of 30 Pa (about 0.003 m of head). For most applications, this difference is negligible. Verify both results yourself using the Moody Chart Calculator.
Laminar Flow: f = 64/Re
Neither Colebrook-White nor Swamee-Jain applies when Re is below 2,300. In laminar flow, friction factor follows the exact analytical solution:
This is derived directly from the Hagen-Poiseuille equation for laminar pipe flow. It is exact, not an approximation, and it applies regardless of pipe roughness. In laminar flow, the viscous sublayer is thick enough to completely cover all roughness elements, so surface texture has no effect on friction factor.
For the transition zone (Re 2,300 to 4,000), no reliable equation exists. The Moody Chart leaves this region blank. In practice, design your system to stay outside this zone. For more detail on flow regimes and how to calculate Re, see the Reynolds number guide.
Frequently Asked Questions
What is the Colebrook-White equation?
The Colebrook-White equation is the standard implicit formula for Darcy friction factor in turbulent pipe flow. It covers the full turbulent range from smooth to fully rough pipe by combining both limiting behaviors into a single expression. It requires iteration to solve because f appears on both sides.
What is the Swamee-Jain equation?
The Swamee-Jain equation is an explicit approximation of Colebrook-White. Published in 1976, it gives friction factor directly from Re and ε/D in a single calculation with no iteration. It is accurate to within 3% for Re between 5,000 and 10⁸ and ε/D between 10⁻⁶ and 0.05.
Why is the Colebrook-White equation implicit?
Because f appears on both sides of the equation. The physics of turbulent flow links friction factor to both the Reynolds-number-dependent viscous sublayer and the roughness-dominated outer layer simultaneously. There is no algebraic way to separate them, so f cannot be isolated on one side.
How accurate is the Swamee-Jain equation?
Within its valid range, Swamee-Jain is accurate to within ±3% of the Colebrook-White result. The largest errors occur near the transition zone (Re close to 5,000) and at very high relative roughness. For most engineering pipe flow conditions, the error is under 1%.
Which equation does the Moody Chart use?
The Moody Chart is a graphical solution to the Colebrook-White equation for turbulent flow, combined with f = 64/Re for the laminar region. The Moody Chart Calculator on this site solves the Colebrook-White equation using Newton-Raphson iteration to within 0.1% accuracy.
Can I use Swamee-Jain in a spreadsheet?
Yes. Swamee-Jain is ideal for spreadsheets because it is a single explicit formula with no circular references. Enter Re in one cell, ε/D in another, and the Swamee-Jain formula in a third. You get friction factor instantly. For Colebrook-White in Excel, enable iterative calculation under File → Options → Formulas first.
Solve Colebrook-White Instantly
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