Flow in pipes and head losses

Push water through a straight, level pipe of constant diameter and the pressure at the far end is lower than at the near end, even though nothing about the geometry demanded it. The energy equation between the two points has no elevation term and no velocity term to absorb the difference — it has to go somewhere, and where it goes is friction against the wall, converted into an equivalent height called head loss. Everything in this topic is about pinning that height down.

Fully developed flow

A pipe fed from a reservoir does not start out with the profile laminar and turbulent flow describes. It enters nearly uniform, and a thin layer grows in from the wall — thickening with distance, exactly the way a boundary layer does on any solid surface — until it has spread all the way to the centreline and the profile stops changing shape from one cross-section to the next. Only beyond that point is the flow fully developed, and only there do the formulas in the rest of this topic strictly apply.

The distance it takes is the entrance length, and the two regimes reach it very differently. For laminar flow, theory gives a clean correlation,

which grows linearly with Reynolds number and can run to hundreds of diameters at anything above a modest Re. Turbulent flow develops far faster, because the same mixing that flattens its velocity profile also redistributes momentum across the pipe much more quickly; a commonly used approximate correlation is \(L_h \approx 1.359\,D\,Re^{1/4}\), and in practice turbulent entrance lengths of ten to sixty diameters are typical — an order of magnitude shorter than laminar flow would need at a comparable Reynolds number. Neither correlation is exact in the way the laminar velocity profile itself is exact; both are useful approximations for deciding how much straight pipe to allow before trusting a friction-factor measurement.

For a horizontal pipe of constant diameter, with the velocity the same at both ends because the area has not changed, the energy equation collapses to the plainest possible statement of what head loss is:

the pressure drop, expressed as a height. Nothing about elevation or velocity is doing any work in that equation — the entire drop is the flow giving up mechanical energy to the wall, in exactly the amount the rest of this topic sets out to compute.

The Darcy-Weisbach equation

The general energy equation for a pipe run, between any two points 1 and 2, keeps every term the same as a pump- or turbine-free case except for one addition:

where \(\alpha\) is the kinetic-energy correction factor — near 2 for a laminar parabola, near 1.05 for a turbulent profile — carrying the same non-uniform-profile honesty that the momentum equation's \(\beta\) carries in the linear momentum equation. The major loss — friction distributed along a straight run of length \(L\) and diameter \(D\) — is given by the Darcy-Weisbach equation:

\(f\) here is the Darcy friction factor, and it is worth being deliberate about that name, because a second, unrelated friction factor is in wide use: the Fanning friction factor, defined as exactly one quarter of the Darcy value. Drop a Fanning \(f\) into the Darcy-Weisbach equation, or the reverse, and every head loss in the calculation comes out wrong by a factor of four. The equation above is written for the Darcy factor throughout; check which one a chart or a piece of software is reporting before trusting either.

The friction factor

For laminar flow, \(f\) is not measured. It falls straight out of the exact Hagen–Poiseuille solution:

Substitute that into Darcy-Weisbach and the length and gravity terms rearrange themselves into the classical Hagen–Poiseuille pressure-drop formula, \(\Delta P = 32\mu LV/D^2\) — the same physics, arrived at from two different starting points, which is a reassuring check rather than a coincidence.

Turbulent flow gets no such gift. There is no closed-form derivation of \(f\) for turbulent pipe flow from first principles — it depends on both Reynolds number and the pipe's relative roughness \(\varepsilon/D\), and the relationship between the three is empirical, fitted to decades of pressure-drop measurements on pipes of known roughness. The Colebrook equation is the standard fit:

implicit in \(f\) on both sides, which is precisely why the Moody chart exists at all — it is nothing more than this equation, solved once by its authors for a grid of \(Re\) and \(\varepsilon/D\) and plotted, so nobody has to iterate it by hand. Where an explicit answer is wanted, Haaland's equation approximates Colebrook to within about 1.5%:

Either way, the honest description of turbulent \(f\) is that it is read off a correlation built from measurement, not derived the way the laminar value is. That is not a weakness of the theory; it reflects a real fact about turbulence, that no tractable exact solution for it exists at all.

Minor losses

Every fitting along a pipe run — an entrance, an elbow, a valve, an exit — dissipates energy of its own, in addition to the distributed friction Darcy-Weisbach already accounts for. Each is given its own loss coefficient, defined the same way regardless of what the fitting actually is:

\(K_L\) is measured, not derived, and tabulated per fitting: a sharp-edged entrance runs around \(K_L \approx 0.5\), a smoothly rounded one as low as 0.03; a fully open gate valve costs perhaps \(K_L \approx 0.2\), a globe valve closer to 10. These numbers vary with the specific fitting and manufacturer, so a real design uses the supplier's own data where precision matters — the values here are typical, not universal constants.

The exit loss deserves its own sentence, because \(K_L = 1.0\) surprises people who expect a smooth exit to cost nothing. It is not the geometry of the exit that costs the head — it is what happens after: the jet's entire kinetic energy is dissipated as it mixes into the still fluid of the reservoir it discharges into, however gently the exit itself is shaped. There is no way to avoid that loss by rounding the opening, because the loss is not at the opening.

Minor losses fold into exactly the same bookkeeping as major losses through an equivalent length, \(L_{eq} = K_L D/f\) — the length of ordinary straight pipe that would wear away the same amount of head. Whether a real design keeps the two terms separate or lumps a fitting into an equivalent length is a matter of convenience; the physics does not care which bookkeeping is used, only that every term is counted once.

Solving a pipe system

Pipes in series share one flow rate, so their losses simply add along the path:

Every segment's velocity follows from continuity once the total flow rate and each diameter are known, and the sum above is a single pass of arithmetic once every \(f\) is in hand.

Pipes in parallel invert the bookkeeping. Two branches that share the same start and end points must lose the same head — there is only one pressure at each shared node, however the flow happens to divide between them:

What makes this genuinely harder than the series case is the same honesty the friction factor already forced: \(f\) depends on \(Re\), which depends on the very velocity the system is being solved for. There is no way to write down \(Q_A\) and \(Q_B\) in closed form when \(f\) is turbulent on both branches — the standard approach guesses a friction factor for each branch, solves the resulting velocities from the equal-head condition, checks the Reynolds numbers those velocities imply, and repeats until the assumed and computed friction factors agree.

None of this is a new idea layered on top of the rest of the topic. It is Darcy-Weisbach applied once per branch, minor losses added the same way they always are, and continuity used to tie the branches together — the entire toolkit built up over the previous four sections, applied more than once and reconciled against itself.