Pump and pipe hydraulics: friction loss, system curves, NPSH and surge

A pumped pipeline is defined by two curves: the head the pipework demands at each flow, and the head the pump delivers. This note derives the system curve from the energy equation and the Darcy–Weisbach friction law, sets out the pump-side relationships, and closes with suction (NPSH) and surge checks. A worked example is carried through every step.

Technical note, RB Solutions Ltd, 22 October 2026

The energy equation between two points

For steady, incompressible flow along a streamline, Bernoulli's equation states that the sum of pressure head, velocity head and elevation is constant. Engineering practice uses the extended energy equation, which adds the head supplied by a pump and the head dissipated by friction and fittings between section 1 and section 2 [1]:

\frac{p_{1}}{ρ g} + \frac{v_{1}^{2}}{2 g} + z_{1} + H_{p} = \frac{p_{2}}{ρ g} + \frac{v_{2}^{2}}{2 g} + z_{2} + h_{f} + h_{m}

where p is static pressure (Pa), ρ is liquid density (kg m⁻³), g is gravitational acceleration (9.80665 m s⁻², standard value), v is mean velocity (m s⁻¹), z is elevation (m), H_p is pump total head (m), h_f is pipe friction loss (m) and h_m is the sum of fitting (minor) losses (m).

All terms carry units of metres of the flowing liquid. Converting to pressure requires multiplication by ρg, and a head stated for water must be converted if the liquid density differs.

Darcy–Weisbach friction loss

The Darcy–Weisbach equation is the dimensionally consistent basis for friction loss in full-bore pipe flow of any Newtonian liquid [1]:

h_{f} = f \frac{L}{D} \frac{v^{2}}{2 g}, Re = \frac{ρ v D}{μ}

where f is the Darcy friction factor (dimensionless), L is pipe length (m), D is internal diameter (m), Re is the Reynolds number (dimensionless) and μ is dynamic viscosity (Pa s).

In laminar flow, f = 64/Re exactly. The laminar regime persists to Re of about 2,100, and a transitional region extends to about 4,000 [1]. Empirical formulae such as Hazen–Williams carry dimensional coefficients, are restricted to water near ambient temperature and fail at low Reynolds number. They are not used here.

Colebrook–White and the Swamee–Jain explicit form

For turbulent flow in commercial pipe, Colebrook combined the smooth-pipe and fully rough laws into a single implicit relationship [2]:

\frac{1}{\sqrt{f}} = - 2 \log_{10} (\frac{ε / D}{3.7} + \frac{2.51}{Re \sqrt{f}})

where ε is the equivalent sand-grain roughness (m) and ε/D is relative roughness (dimensionless).

The equation is solved by fixed-point iteration on 1/√f, which converges in a few steps from any reasonable starting value. Swamee and Jain derived an explicit approximation, valid for 10⁻⁶ ≤ ε/D ≤ 10⁻² and 5,000 ≤ Re ≤ 10⁸, and reported agreement with Colebrook–White to within about 1 % over that range [3]:

f = \frac{0.25}{{[\log_{10} (\frac{ε / D}{3.7} + \frac{5.74}{{Re}^{0.9}})]}^{2}}

where all symbols are as defined above; the result is dimensionless.

The roughness value is an input, not a property of the material in perpetuity. Moody's chart gives 0.046 mm for new commercial steel [4]. Tuberculated, scaled or biofouled mains can be an order of magnitude rougher. UK hydraulic tables based on Colebrook–White accordingly tabulate a range of ε values for new and aged surfaces [5].

Minor losses in fittings and valves

Fittings, valves, entries and exits are represented by a loss coefficient K applied to the local velocity head:

h_{m} = (\sum_{i} K_{i}) \frac{v^{2}}{2 g}

where K_i is the dimensionless loss coefficient of fitting i, referred to velocity v (m s⁻¹) in the connected pipe.

Table 1. Representative loss coefficients for turbulent flow, from Perry's Chemical Engineers' Handbook [1]
Fitting	K (dimensionless)
Sharp-edged entrance from a tank	0.5
Exit into a tank (loss of velocity head)	1.0
Standard 90° elbow	0.75
Gate valve, fully open	0.17
Globe valve, fully open	6.0
Swing check valve, fully open	2.0

Single-value coefficients ignore the dependence of K on size and Reynolds number, which the two-constant and three-constant methods presented in Perry's Handbook capture [1]. For control valves, the manufacturer's flow coefficient (K_v) at the expected opening governs, not a generic K.

System curve, pump curve and operating point

Combining the energy equation with Darcy–Weisbach and the minor losses gives the system curve. Static head is independent of flow, and the dynamic component varies approximately with the square of flow in fully turbulent pipe:

H_{sys} (Q) = H_{st} + (f \frac{L}{D} + \sum K) \frac{8}{π^{2} g D^{4}} Q^{2} \approx H_{st} + k Q^{2}

where H_sys is system head (m), H_st is static head including any terminal pressure difference (m), Q is volumetric flow (m³ s⁻¹) and k is the resistance coefficient (s² m⁻⁵). The factor 8/(π²gD⁴) converts v²/2g to a function of Q.

Because f varies weakly with Re, a design curve should be computed point by point rather than from one value of k. The approximation is adequate near the duty point. The pump's head–flow characteristic, taken from the manufacturer's test curve and verified on acceptance to BS EN ISO 9906 [6], intersects the system curve at the operating point. Curves are required at both maximum and minimum static head, and for both new and aged pipe roughness, since the operating point moves across that envelope.

Pumps in parallel and in series

Identical pumps in parallel add flow at equal head. The combined curve is obtained by doubling the flow of a single pump at each head. Pumps in series add head at equal flow. The gain from a second parallel pump depends on the steepness of the system curve. On a friction-dominated system the combined flow is well below twice the single-pump flow, and each pump moves to the left of its single-pump duty, where efficiency and radial loading need checking.

Affinity laws for speed change

For a given impeller at dynamically similar operating points, the affinity laws relate flow, head and power to rotational speed [1]:

\frac{Q_{2}}{Q_{1}} = \frac{n_{2}}{n_{1}}, \frac{H_{2}}{H_{1}} = {(\frac{n_{2}}{n_{1}})}^{2}, \frac{P_{2}}{P_{1}} = {(\frac{n_{2}}{n_{1}})}^{3}

where n is rotational speed (min⁻¹), Q is flow (m³ s⁻¹), H is head (m) and P is shaft power (W).

The laws transform the pump curve, not the operating point. On a system with significant static head the duty point does not follow the cubic power reduction. Flow falls faster than speed and reaches zero once the shut-off head, (n₂/n₁)²H₀, drops below H_st. Variable-speed duty must therefore be assessed by re-intersecting the scaled pump curve with the actual system curve.

Net positive suction head

Cavitation is avoided when the net positive suction head available at the pump inlet exceeds that required by the pump. For suction from a vessel:

{NPSH}_{a} = \frac{p_{s} - p_{v}}{ρ g} + z_{s} - h_{f, s}

where p_s is absolute pressure on the liquid surface (Pa), p_v is the liquid vapour pressure at pumping temperature (Pa), z_s is the height of the liquid surface above the impeller eye datum (m, negative for suction lift) and h_f,s is the total suction-side loss at the duty flow (m).

The pump's requirement is defined in BS EN ISO 9906 as NPSH3, the NPSH at which the first-stage head falls by 3 % [6]. Since NPSH3 already corresponds to developed cavitation, a margin above it is applied. The required margin depends on pump type, suction energy and liquid, and is taken from the pump supplier and the project specification rather than a single generic figure. NPSH_a must be checked at minimum suction level, maximum liquid temperature and the highest flow the pump can reach, including run-out when a parallel pump trips.

Surge: the Joukowsky relationship

A rapid change in velocity generates a pressure wave travelling at the acoustic speed of the pipe–liquid system. For a change faster than the pipe period 2L/a, the head rise is given by the Joukowsky equation [7]:

Δ p = ρ a Δ v, a = \sqrt{\frac{K / ρ}{1 + \frac{K D}{E e}}}

where Δp is the pressure change (Pa), a is wave speed (m s⁻¹), Δv is the instantaneous velocity change (m s⁻¹), K is the liquid bulk modulus (Pa), E is the pipe-wall elastic modulus (Pa) and e is wall thickness (m). The wave-speed expression is for a thin-walled pipe free to move axially; restraint factors are given in [7].

Pump trip on a rising main, fast valve closure and column separation at high points are the common cases. Joukowsky provides an upper bound for screening. Where it indicates a significant excursion, a transient analysis by the method of characteristics [7] is used to size protection such as surge vessels, air valves or controlled valve closure. The resulting envelope is compared with the pipe pressure class. For water supply pipelines outside buildings, BS EN 805 sets the relationship between design pressure, maximum design pressure including surge, and system test pressure [8].

Worked example: rising main duty, operating point, NPSH and surge

Inputs. Water at 20 °C with ρ = 998.2 kg m⁻³, μ = 1.002 × 10⁻³ Pa s and p_v = 2.339 kPa [1]. Flow is Q = 0.040 m³ s⁻¹ (144 m³ h⁻¹) through a new steel main of internal diameter 0.200 m and length 800 m, with ε = 0.046 mm [4]. Static head is 25.0 m. Fittings are one entrance, one exit, six standard elbows, two open gate valves and one swing check valve, giving ΣK = 0.5 + 1.0 + 6 × 0.75 + 2 × 0.17 + 2.0 = 8.34 (Table 1). The illustrative pump curve, a quadratic fit to a supplier curve, is H = 48.0 − 4,000 Q² (m, with Q in m³ s⁻¹).

Velocity and Reynolds number. A = π × 0.200²/4 = 0.03142 m², so v = 1.273 m s⁻¹. Re = 998.2 × 1.273 × 0.200 / 1.002 × 10⁻³ = 2.54 × 10⁵, which is turbulent. ε/D = 2.30 × 10⁻⁴.

Friction factor, two routes. Iterating Colebrook–White gives f = 0.01679. Swamee–Jain gives f = 0.01685, which is 0.35 % higher and within the stated accuracy of the explicit form [3]. The Colebrook value is carried forward.

Head loss, two routes. The velocity head is v²/2g = 0.0827 m. Friction gives h_f = 0.01679 × (800/0.200) × 0.0827 = 5.55 m. As a check in pressure units, Δp = f(L/D)ρv²/2 = 54.35 kPa, and 54,350/(998.2 × 9.80665) = 5.55 m. Fittings give h_m = 8.34 × 0.0827 = 0.69 m. The system head at 0.040 m³ s⁻¹ is therefore 25.0 + 5.55 + 0.69 = 31.24 m, and k = 6.24/0.040² = 3,901 s² m⁻⁵.

Operating point. Setting 48.0 − 4,000Q² = 25.0 + 3,901Q² gives Q = √(23.0/7,901) = 0.0540 m³ s⁻¹. Substituting into both curves returns 36.36 m, confirming the intersection. The selected pump therefore delivers 35 % more than the 0.040 m³ s⁻¹ requirement unless throttled or speed-controlled. Two such pumps in parallel would give Q = √(23.0/(3,901 + 1,000)) = 0.0685 m³ s⁻¹, only 27 % more than one pump. At 90 % speed the shut-off head becomes 0.81 × 48.0 = 38.9 m and the duty falls to 0.0419 m³ s⁻¹, a 22 % reduction in flow for a 10 % reduction in speed. This illustrates the effect of static head on speed control.

NPSH available. The suction is from an open tank at standard atmospheric pressure (101.325 kPa), with the liquid surface 2.0 m above the impeller datum and a suction-side loss of 0.6 m. NPSH_a = (101,325 − 2,339)/(998.2 × 9.80665) + 2.0 − 0.6 = 10.11 + 1.40 = 11.5 m. This is compared with the supplier's NPSH3 at 0.0540 m³ s⁻¹ plus the specified margin.

Surge screening. With K = 2.2 GPa for water [1], and a steel wall of E = 200 GPa [1] and e = 6.3 mm, the wave speed is a = 1,278 m s⁻¹. An instantaneous stop from 1.273 m s⁻¹ gives Δp = 998.2 × 1,278 × 1.273 = 1.62 MPa (16.2 bar), or 166 m of head, against a steady duty head of about 31 m. The pipe period 2L/a = 1.25 s means that pump run-down on trip is likely to fall within the Joukowsky regime. A full transient analysis is therefore indicated.

Table 2. Worked example results
Quantity	Value	Check
Reynolds number	2.54 × 10⁵	Turbulent
Friction factor (Colebrook–White)	0.01679	Swamee–Jain 0.01685 (+0.35 %)
Friction loss at 0.040 m³ s⁻¹	5.55 m	54.35 kPa ÷ ρg
Fitting loss	0.69 m	ΣK = 8.34
Operating point, one pump	0.0540 m³ s⁻¹ at 36.36 m	Both curves agree
Operating point, two in parallel	0.0685 m³ s⁻¹ at 43.31 m	Both curves agree
NPSH available	11.5 m	Compare with NPSH3 plus margin
Joukowsky surge (instantaneous stop)	16.2 bar (166 m)	a = 1,278 m s⁻¹, 2L/a = 1.25 s

UK practice references

In the United Kingdom, pipe hydraulic design for water and wastewater is normally carried out with the Colebrook–White law, using the tabulated solutions and roughness data of the HR Wallingford tables [5]. Water supply pipelines outside buildings are designed and tested to BS EN 805 [8]. Pump performance is verified by acceptance test to BS EN ISO 9906 [6]. Within buildings, BS EN 806 and BS 8558 apply to water installations. US codes are not used as the governing basis for UK sites.

References

Green, D.W. and Southard, M.Z. (eds) (2019) Perry's Chemical Engineers' Handbook, 9th ed. New York: McGraw-Hill. Section 6, Fluid and Particle Dynamics; Section 2, Physical and Chemical Data.
Colebrook, C.F. (1939) Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. Journal of the Institution of Civil Engineers, 11(4), 133–156.
Swamee, P.K. and Jain, A.K. (1976) Explicit equations for pipe-flow problems. Journal of the Hydraulics Division, ASCE, 102(HY5), 657–664.
Moody, L.F. (1944) Friction factors for pipe flow. Transactions of the ASME, 66(8), 671–684.
HR Wallingford and Barr, D.I.H. (2006) Tables for the Hydraulic Design of Pipes, Sewers and Channels, 8th ed. London: Thomas Telford.
BS EN ISO 9906:2012 Rotodynamic pumps. Hydraulic performance acceptance tests. Grades 1, 2 and 3. London: BSI.
Wylie, E.B. and Streeter, V.L. (1993) Fluid Transients in Systems. Englewood Cliffs: Prentice Hall.
BS EN 805:2000 Water supply. Requirements for systems and components outside buildings. London: BSI.

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