§ 3.3  Module 3 — Head Loss: Friction

Colebrook, Swamee-Jain, and Choosing f

The friction factor is not a number you compute. It is the number that makes an equation balance — and it will tell you, if you ask, how nearly it balanced.

§3.2 left you reading f off a chart. The chart is a picture of an equation, the equation carries f on both sides of the equals sign, and no rearrangement frees it. That one fact generates everything here: an iteration to solve it, explicit fits to dodge the iteration, folklore about how good the fits are, and a choice of which to use. The choice turns out to be nearly the least consequential decision you will make about the pipe — but you cannot know that until the alternatives are measured in metres.

3.3.1The friction factor is a root, not a formula

The equation behind the Moody chart is Colebrook-White:

  1/√f  =  −2 · log₁₀( (ε/D)/3.7  +  2.51/(Re·√f) )
                            ↑                ↑
                     rough-wall term    smooth-wall term
                     (no Re in it)      (no roughness in it)

Colebrook-White. Everything is dimensionless: f is the Darcy friction factor, Re the Reynolds number, ε/D the relative roughness.

f is on both sides, and that is not an algebraic accident to be tidied away. The equation is a deliberate sum of two limits, each of which is explicit on its own. Delete the roughness term — set ε/D to zero — and what remains rearranges to 1/√f = 2·log₁₀(Re√f) − 0.79935: Prandtl's smooth-pipe law, published with the constant −0.8. Colebrook's 2.51 is 10^0.4 = 2.51189 rounded, shifting that constant by 0.08% and f by under 0.03% — 0.022% at Re = 4×10³, easing to 0.010% by 10⁸. That is the whole disagreement between the two. Now delete the smooth-wall term instead, letting Re run to infinity: 1/√f = 2·log₁₀(D/ε) + 1.1364, the von Kármán–Nikuradse fully-rough law, published with +1.14 (matching which exactly would need 3.7154 in place of 3.7). Both ends are closed forms. The implicitness lives only in the transition between them.

Which end you are near is settled by which of the two terms is larger, and that is the same question as "which region of the Moody chart am I in". The rough-wall term is a constant — no Re appears in it. The smooth-wall term falls away roughly as 1/Re. For a 250 mm main at ε = 0.26 mm the rough term is fixed at 2.81×10⁻⁴; the two are equal at Re = 5.8×10⁴, and by Re = 6.3×10⁵ the smooth term has fallen to a tenth of it and f is within 2.4% of its plateau. That crossover is the knee of the chart.

3.3.2Solving it, and the residual that proves you did

Substitute x = 1/√f and it becomes an ordinary root find: g(x) = x + 2·log₁₀((ε/D)/3.7 + 2.51x/Re) = 0, with f = 1/x². Two schemes are in common use, and they are not equally good.

Successive substitution. Guess f, evaluate Colebrook's right-hand side, take that as the new f, repeat. It converges, because the right-hand side is very insensitive to f — but slowly. From a blind f₀ = 0.02 on the pipe below the residual runs 3.6×10⁻³, 8.4×10⁻⁵, 2.0×10⁻⁶, 4.6×10⁻⁸, 1.1×10⁻⁹: about 1.6 decimal digits a pass. Swept over Re from 4×10³ to 10⁹ and ε/D from 10⁻⁶ to 2×10⁻², the worst case needs 13 passes to reach 10⁻⁹.

Newton on x. Step x ← x − g(x)/g′(x), with g′(x) = 1 + 2(2.51/Re) / (ln10 · ((ε/D)/3.7 + 2.51x/Re)). This works absurdly well, for a reason worth knowing: written in x, the residual is almost exactly a straight line. On the pipe below its slope is 1.0235 and its curvature, as a central second difference over a whole unit of x, is −6.3×10⁻⁴. Newton on a straight line lands on the root in one step. Over the same grid, seeded from the explicit fit of §3.3.3, the worst case is two steps.

  Q = 0.060 m³/s      D = 0.250 m      L = 600 m      ε = 0.26 mm      t = 15 °C

  A  = πD²/4              = 0.049087 m²
  v  = Q/A                 = 1.2223 m/s
  ν  (15 °C)             = 1.1370×10⁻⁶ m²/s
  Re = vD/ν               = 2.688×10⁵        — firmly turbulent
  ε/D                      = 0.26/250 = 1.04×10⁻³

  seed  (Swamee-Jain)  f = 0.021016   x = 6.89799   g = −2.51×10⁻²
  step 1               f = 0.0208675  x = 6.92253   g = −1.91×10⁻⁷
  step 2               f = 0.0208675  x = 6.92253   g = 0  (to double precision)

  v²/2g                    = 0.076175 m
  hf = f · (L/D) · v²/2g   = 0.0208675 × 2400 × 0.076175  =  3.815 m

The worked case this lesson keeps returning to. 60 L/s in 250 mm unlined ductile iron, 600 m, water at 15 °C. Computed with src/core/hydraulics.js.

Notice what the middle block reports. Not "converged" but a residual: the number you get by putting the answer back into the equation and seeing how badly it fails to balance. That is the only honest evidence you solved anything, and it costs one line. Report it beside f.

3.3.3The explicit fit, and the window it was fitted in

Swamee-Jain replaces the iteration with one evaluation:

  f  =  0.25 / [ log₁₀( (ε/D)/3.7  +  5.74/Re^0.9 ) ]²
                            ↑                ↑
                     unchanged from    2.51/(Re·√f) with the
                     Colebrook          f folded into an exponent

Swamee-Jain (1976). Compare it term by term with Colebrook above.

The rough-wall term is Colebrook's, untouched, which is why the two agree exactly in the fully-rough limit. The whole approximation — and the whole error — is in the second term: the f inside 2.51/(Re√f) is absorbed into the exponent 0.9. Swamee and Jain fitted that exponent and coefficient over a stated window, and the window is part of the formula, not a footnote: 5×10³ ≤ Re ≤ 10⁸ and 10⁻⁶ ≤ ε/D ≤ 10⁻². That is the window as it is usually quoted, and the provenance is worth stating plainly: it reaches you through textbooks and software documentation rather than off the 1976 page, and secondary accounts do not all state the rough end identically. The bound is load-bearing in everything below, so name the statement of it you relied on — and then measure the deviation where you actually are, which is the only figure that settles anything.

The roughness bound is the one that catches people, because ε/D depends on diameter and small pipes are rough pipes. The §3.2 roughness table, resolved against a 250 mm main:

Swamee-Jain is not the only explicit fit in service. Haaland's 1983 form is at least as common in software, and comparing them is more useful than picking a favourite: measured against the iterated root over the same window, Haaland's worst deviation is 1.42%, at Re = 8.3×10⁴ with ε/D = 2.5×10⁻⁴ — worse than Swamee-Jain mid-range, and far better at the rough corner where Swamee-Jain is at its worst (0.09% against 2.83%). Neither dominates. State which fit you used and what its error is where you used it, rather than being loyal to one.

3.3.4The error surface, measured

Define the error as (f_fit − f_root) / f_root — signed, so the direction stays visible. Swept densely, the surface has a clear shape: the fit is at its best at high Reynolds number and degrades toward low Re and high roughness, which is to say toward the corners of its own window.

• Across the municipal design range — Re ≥ 10⁵, ε/D from 10⁻⁵ to 10⁻² — the worst deviation is +0.91%, at Re = 10⁵ with ε/D = 2.3×10⁻³. Under one per cent everywhere a force main operates.
• Across the full stated validity — Re from 5×10³ to 10⁸, ε/D from 10⁻⁶ to 10⁻² — the worst is +2.83%, at Re = 5×10³ with ε/D = 10⁻²: both bounds biting simultaneously.
• The error is not one-signed. On a 250 mm main at 60 L/s the fit reads high by 0.58% on cement-lined ductile iron, 0.71% on unlined, 0.42% on aged cast iron — but low by 0.58% on PVC or HDPE, where ε/D is 6.0×10⁻⁶.

That last point kills a convenient argument. "The explicit fit is conservative, so any error is on the safe side" is true on iron and false on plastic: near ε/D = 10⁻⁶ the fit under-predicts f, and so under-predicts head loss, by about half a per cent. Small, but not conservative — and a claim of conservatism that fails in one region is worth less than no claim.

It is worth asking where the bad corner physically is. Two litres per second in a 250 mm rough concrete main gives Re = 9.0×10³ and ε/D = 1.2×10⁻², and the fit is 2.25% off — but the velocity there is 0.041 m/s, fifteen times below the 0.6 m/s self-cleansing floor of §1.3. That is not a design condition; it is a main that has stopped working. In normal service the corner is reachable only in small bore: a 50 mm sample or chemical-feed line at 0.5 L/s runs at Re = 1.1×10⁴, and on an aged cast iron wall its ε/D is 3×10⁻² — three times the roughness bound, where the quoted error covers nothing and the only figure you have is the one you measure. Measured: the fit reads high by 1.80% there. Past a fit's window the answer is not "invalid", it is unquantified until you sweep it yourself.

So do not quote "Swamee-Jain is within 1% of Colebrook" as a general fact, which is how it is usually passed on. It is true where force mains live and false at the corner, and the gap between those two claims is the gap between a number you measured and one you inherited.

3.3.5Choosing a method for the accuracy the design needs

Per cents on f are not decisions. Metres are. Because f enters Darcy-Weisbach linearly, a 0.71% error in f is a 0.71% error in head loss: on the worked pipe the choice of method is worth 3.842 m against 3.815 m, 27 mm. On a station with 20 m of static lift the total dynamic head is 23.8 m, of which 27 mm is 0.11% — at 60 L/s, 16 W of water power out of 14.0 kW. Now price the other assumptions in the same units.

The bottom row is the design decision; the second row is not. Choosing ε moves this answer by 2.677 m — 82%, or 1.57 kW of water power — roughly a hundred times what choosing between an exact root and a good fit moves it. Temperature is worth three times the method choice, and Darcy-Weisbach carries it free through ν in the Reynolds number. Given that ranking, the answer to "which method" is almost embarrassingly relaxed:

1. Design calculations and checks: iterate Colebrook and report the residual. It is two Newton steps. There is no case for approximating something that costs two lines of arithmetic, and the residual is free evidence.
2. Inside a loop — a system curve evaluated at 200 flows, a network solve, an optimiser — the explicit fit is fine, provided you write down the window you actually swept. Re ≥ 10⁵ with 10⁻⁵ ≤ ε/D ≤ 10⁻² buys you under 1%; carry the lower bound down to the fit's own ε/D = 10⁻⁶ and the worst case becomes 1.14%, at Re = 9.5×10⁷. Two different sentences, and only the one you swept is yours to quote — but ask §3.3.4's question of that corner before you worry about it: ε/D = 10⁻⁶ on new plastic means a 1.5 m main, and Re = 9.5×10⁷ in a 1.5 m main is 72 m/s. Swept across every PVC duty point a station actually runs — 0.6 to 3 m/s, 50 to 2000 mm — the worst is 0.68%. State the window, then check whether its bad corner is anywhere you can stand.
3. By hand, or checking someone else's number: the Moody chart, two significant figures. More precision than the roughness deserves, and it catches order-of-magnitude errors faster than anything else.
4. Fully rough, past the knee of Figure 3.3.1: the closed form 1/√f = 2·log₁₀(D/ε) + 1.1364, which is Colebrook's own fully-rough limit, 2log₁₀(3.7). No iteration and no window — Reynolds number has dropped out of the physics, not been approximated away. The additive constant has not: Nikuradse's published +1.14 is the sand-grain fit of the history note above, and it puts f about 0.1% below every other number in this lesson. Pick one and say which.
5. Below Re 4000: none of the above. Laminar flow is f = 64/Re exactly, from Hagen-Poiseuille. Between 2000 and 4000 nothing is predictable and the honest output is a range, as §3.1 argued. Both explicit fits are silent there and will still return a number if you ask.

One habit carries all of this: in the calculation, write the ε you assumed, the temperature you assumed, and the residual you achieved. Two are judgements and the third is a fact, and keeping them visibly separate is how you avoid this subject's characteristic artefact — a friction factor reported to six figures from a roughness known to one.

What this lesson has not accounted for: not one fitting. Every number above is straight pipe, and inside a station the elbows, valves and header routinely lose more head than the pipe does — Module 4. Nor is ε a constant; it moves over the design life (§3.5), which is why the 98× row is a range and not a number. Hazen-Williams, still the water-industry default and carrying its own unit trap, is §3.4. And the pump remains what it has been since Module 1: a black box supplying whatever head the system demands. Module 6 asks whether one exists that will.