Line Sizing — Pressure Drop vs Velocity vs Cost

Two lines, same fluid, two different answers

A centrifugal pump on a light-hydrocarbon service. 80 m³/h, ρ = 700 kg/m³. On the P&ID the discharge line is 4″ and the suction line is 6″ — same pump, same fluid, same flow, one size apart, and nobody on the review could say why.

Guess first. Is the bigger suction line (a) a drafting error, (b) a vendor's standard nozzle size, or (c) a deliberate and non-negotiable design decision?

(c). And the reason is not in any velocity table.

The belief worth killing is the one every junior engineer is handed in their first week:

"Size the line off the velocity table. 1–3 m/s for liquid, 15–30 m/s for gas."

The table is not wrong. It is a pre-solved answer to a problem you have not read. Velocity is not a design limit — nobody's pipe has ever failed because a number exceeded 3.0. Velocity is simply the variable that five real limits happen to be written in terms of:

• Erosion / erosion-corrosion — solids or droplets impacting the wall, or protective scale being stripped. Scales with ρv², which is why the limit moves with density.
• Noise and vibration — acoustic power in gas lines climbs steeply with velocity; so does flow-induced vibration at tees and dead legs.
• Surge (water hammer) — the Joukowsky pressure rise is ρ·c·Δv. Halve the velocity and you halve the hammer, for free.
• Pressure drop — the hydraulic circuit has a budget: available pump head, an elevation difference for gravity flow, or an NPSH margin.
• Lifetime cost — steel bought once versus pumping energy bought every hour for twenty years.

A velocity table bundles all five into one number for one typical service, and it is right whenever your case looks like the case it was solved for. Change the density, the length, the energy price or the NPSH margin and the bundle comes apart. That is the whole lesson of the 4″ discharge and the 6″ suction: on the discharge, ΔP is bought with pump head, which is cheap per bar. On the suction, every millibar comes straight off NPSH available, and cavitation is not a cost — it is a wall.

Symbol key — every symbol on this sheet

• D — internal diameter (bore), not nominal size · m, mm
• v — bulk mean velocity = Q/A · m/s
• Q — volumetric flow at flowing conditions · m³/h, m³/s
• ρ (rho) — fluid density at flowing conditions · kg/m³
• μ (mu) — dynamic viscosity · Pa·s (1 cP = 0.001 Pa·s)
• Re — Reynolds number = ρvD/μ: inertia ÷ viscosity, dimensionless
• ε (epsilon) — absolute pipe roughness; 0.046 mm for commercial steel · mm
• ε/D — relative roughness — the form the friction correlations actually use · –
• f — Darcy friction factor (the Fanning factor is f/4 — check which one a formula wants) · –
• ΣK — sum of velocity-head loss coefficients for fittings and valves · –
• ΔP — pressure drop · bar, Pa
• L — straight line length · m
• V_e — API 14E erosional velocity = 1.22·C/√ρ (SI) · m/s
• C — API 14E erosional constant; 100 continuous, 125 intermittent, higher if corrosion-controlled · –
• NPSHa — net positive suction head available at the pump inlet · m of liquid
• η (eta) — pump plus driver efficiency · –
• D_econ — economic diameter: the bore where capex plus lifetime pumping cost is least · mm

The four judges, and the arithmetic each one uses

1 — Pressure drop. Everything starts here. The Darcy–Weisbach equation, with fittings carried as velocity heads:

ΔP = ( f · L/D  +  ΣK ) · ½ρv²

Laminar (Re < 2300):   f = 64 / Re                       — no roughness term at all
Turbulent (Colebrook): 1/√f = −2·log₁₀( ε/(3.7D) + 2.51/(Re·√f) )     — implicit, iterate
Churchill (explicit, all regimes):
    A = [ 2.457 · ln( 1 / ((7/Re)^0.9 + 0.27·ε/D) ) ]^16
    B = ( 37530 / Re )^16
    f = 8 · [ (8/Re)^12 + 1/(A+B)^1.5 ]^(1/12)

Churchill matches Colebrook to well under 1 % in the turbulent range and collapses cleanly to 64/Re in the laminar range, which is why it is the one to code.

The number that matters for sizing is ΔP per 100 m, because it is the only form that can be compared between lines of different length. Typical allowances: 0.2–0.5 bar/100 m for pump discharge, 0.05–0.1 bar/100 m for a pump suction, and whatever the elevation gives you for gravity flow.

2 — Erosional velocity. API RP 14E, the most-quoted and most-argued-about line in the field:

V_e = C / √ρ          (USC: ft/s, lb/ft³)
V_e = 1.22 · C / √ρ   (SI: m/s, kg/m³)     C = 100 continuous · 125 intermittent

Know its limits as well as its value. It is empirical, from 1980s offshore two-phase practice. It contains no sand loading, no droplet size, no material, no geometry — yet erosion happens at bends and tees, not in straight pipe, and a single elbow with sand in it will fail long before V_e. API 14E itself now says C may be raised well above 100 for clean, corrosion-controlled, solids-free service, and that erosive service needs a proper model (DNV-RP-O501). Treat V_e as a screening number you must be able to argue about, not as a code limit.

3 — Noise, vibration and surge. Gas lines above roughly 0.3 Mach get loud; 25–30 m/s is a common working ceiling in process gas, and control-valve and orifice noise adds to it. Two-phase lines have a separate problem — flow regime. A line sized on average velocity can sit unknowingly in the slug-flow region of a Baker or Taitel–Dukler map, and slugs deliver momentum pulses at every elbow (that is a dynamic-load problem: see the DLF topic). Liquid lines that are closed quickly get Joukowsky surge, ΔP = ρ·c·Δv — low velocity is the cheapest mitigation there is.

4 — NPSH. On a pump suction, ΔP is not an operating cost; it is subtracted directly from NPSHa. A suction line is therefore sized short, straight and slow — 1.0–1.2 m/s is normal practice — with an eccentric reducer flat-side-up at the nozzle so no vapour pocket forms. The economic optimum on a suction line is almost always smaller than what you actually install, and you overrule it deliberately.

Slide all four judges at once and watch which one is binding — the same fluid, five services, and a pass/fail matrix for every standard size: ▶ open the interactive: process line sizing calc

And the fifth judge, the one that decides when the other four are silent: money.

Total life cost (D) = installed pipe cost + lifetime pumping energy
                    ≈ a·D^1.2 · L     +     (ΔP(D)·Q/η) · hours · years · price
                      ~ D^1.2                        ~ D⁻⁵

Because one term rises as D^1.2 and the other falls as D⁻⁵, there is a genuine minimum. Its position moves with the energy price, the running hours and the economic life — and with nothing else. A line that runs 500 h/yr wants to be small; the same line at 8760 h/yr wants to be a size or two bigger. This is exactly why an intermittent transfer line and a continuous circulating header, carrying the same fluid at the same rate, are honestly different sizes.

Worked example — 100 m³/h of cooling water, 200 m of pipe

Water, ρ = 1000 kg/m³, μ = 1 cP, ε = 0.046 mm, 200 m straight, ΣK ≈ 8, 8000 h/yr, $0.10/kWh, 20 years, η = 0.70. Try 6″ Sch 40, bore 154.1 mm:

A  = π/4 × 0.1541²            = 0.018650 m²
v  = (100/3600) / 0.018650    = 1.489 m/s
Re = 1000 × 1.489 × 0.1541 / 0.001   = 229 500          (fully turbulent)
ε/D = 0.046 / 154.1           = 2.99 × 10⁻⁴
f  (Churchill)                = 0.01750      (Colebrook gives 0.01744 — 0.3 % apart)
ΔP/100 m = f · (100/D) · ½ρv² = 0.0175 × 648.9 × 1109 = 12 600 Pa = 0.126 bar/100 m
ΔP total = 0.0175 × (200/0.1541) × 1109 + 8 × 1109     = 34 100 Pa = 0.341 bar
Power    = 34 100 × 0.02778 / 0.70                     = 1.35 kW
V_e      = 1.22 × 100 / √1000                          = 3.86 m/s   →  v/V_e = 0.39, fine

Now the economics. Installed pipe on an illustrative basis of $55·(D/25.4)^1.2 per metre gives $479/m → $95 700 of steel. Pumping: 1.35 kW × 8000 h × 20 yr × $0.10 = $21 600 of energy. Sweep the diameter and the total bottoms out at 151 mm — which is 6″ Sch 40 to within 2 %.

So on this service the velocity table, the ΔP allowance and the economics all say 6″, and the table earns its reputation. Now change one thing at a time and watch them separate:

• Triple the energy price to $0.30/kWh → optimum moves to ~182 mm → you buy 8″.
• Make it a pump suction instead → the 1.2 m/s NPSH rule overrules the economics upward.
• Make the fluid 250 cP crude → Re collapses to ~260, f = 64/Re = 0.25, and ΔP rather than velocity decides the size.
• Make it 35 kg/m³ wet gas → V_e drops to 20.6 m/s. Here the ΔP allowance actually bites first (10″ passes erosion at 13.7 m/s but fails on 0.17 bar/100 m), and the economics agree: the continuous optimum is 423 mm ≈ 18″. On low-density service the limits and the money pull the same way — which is why wet-gas lines look extravagantly large.

Watch the same duty in two diameters side by side — flow animated, pressure drawn as a colour ramp along the wall, and the 20-year cost of each: ▶ open the interactive: process line sizing 3d. That model makes one point that a spreadsheet hides: the colour fades continuously along the straight run, not in jumps at the elbows. On a 200 m line the two bends and the gate valve are about 5 % of the loss (ΣK = 1.1 against f·L/D = 22.7). Fittings matter on short, congested runs; on long ones, the wall wins — but note the worked example above carries ΣK = 8, where they are a quarter of it.

Common pitfalls

• Using nominal size as the bore. 6″ pipe has a 154.1 mm bore in Sch 40 and 146.3 mm in Sch 80 — and v ∝ D⁻², so the schedule change moves the velocity by 11 %.
• Mixing Darcy and Fanning friction factors. They differ by exactly 4×, which is a 4× error in ΔP and the most common line-sizing blunder there is.
• Applying a water velocity table to a gas or a viscous liquid. Erosional velocity moves with 1/√ρ, and in laminar flow ΔP depends on μ and not on v² at all.
• Quoting API 14E C = 100 as a code limit. It is an empirical screening value with known weaknesses; say so, and say what your service actually is.
• Sizing a pump suction on economics. ΔP there is NPSH, not dollars.
• Forgetting that ΔP ∝ D⁻⁵. Being one size small is not a small error.
• Sizing a two-phase line on average velocity alone without checking the flow regime — slug flow is a mechanical load case, not a hydraulic inconvenience.
• Assuming incompressible ΔP for gas. Valid only while total ΔP stays under roughly 10 % of the absolute inlet pressure; past that the density changes down the line and you need an isothermal or adiabatic compressible calculation.
• Optimising one line in isolation. The pumping cost belongs to the whole circuit, and the control valve at the end must keep enough authority (see the control-valve topic) — the line cannot eat the entire available ΔP.

Outcome

• Velocity is a proxy, not a limit. The real judges are allowable ΔP for the circuit, erosional velocity, noise / vibration / surge, NPSH on suctions, and lifetime cost.
• ΔP ∝ D⁻⁵ for fixed flow, so one size up cuts pressure drop ~2.5–4× (7.8× only at 4″→6″, where 5″ is skipped). Round up when uncertain.
• Use Churchill (or Colebrook) with relative roughness; watch Darcy vs Fanning; laminar is f = 64/Re with no roughness term.
• API 14E V_e = 1.22·C/√ρ (SI) is a screening number with real limitations — quote it, and know its caveats.
• The economic diameter exists because capex rises as ~D^1.2 while pumping cost falls as ~D⁻⁵. It moves with energy price, running hours and life — and with nothing else.
• The same fluid gets different sizes in different services because a different judge is binding in each. That is not inconsistency; it is the method working.
• Interactive: ▶ open the interactive: process line sizing calc — velocity, Re, Churchill f, ΔP, erosional limit, pass/fail per standard size, and a live economic-diameter curve.
• 3D: ▶ open the interactive: process line sizing 3d — two bores, one duty, pressure as a colour ramp.

Open items

• Add a two-phase module: Lockhart–Martinelli or Beggs–Brill ΔP plus a flow-regime map, so slug risk is visible instead of inferred.
• Add compressible (isothermal) gas ΔP so the >10 % cases stop needing a caveat.
• Add a discounted-cash-flow option to the economic curve (present value, not undiscounted sum) and a real installed-cost basis from a live project.

Educational teaching material — simplified and illustrative. Not for engineering design use.
© EPiC Circle · provided by PipeAxis