Hydraulic

Pumps in series calculation — combined curve and high-pressure discharge

In a series configuration the same flow rate passes through every pump and the heads add up — it is the go-to solution for high-pressure systems (tall vertical lifts, long transmission mains) where a single pump cannot reach the required total head.

When to use

Use pumps in series when the total dynamic head (TDH) required by the system exceeds what a single commercial pump can deliver at good efficiency — tall vertical lifts, long transmission mains with high head loss, boiler feed and booster systems. It is also the natural arrangement of multistage pumps, where each impeller is, in practice, a pump in series inside the same casing. Prefer a series setup (rather than one larger machine) when you want modularity, partial pressure redundancy and spare-part standardization.

What a series association of pumps is

Placing pumps in series means connecting the discharge of one pump directly to the suction of the next, so that the same flow rate passes through all of them and the heads add up. It is the classic solution when the system demands a total dynamic head (TDH) that no single commercial pump can deliver at good efficiency: tall vertical lifts, long transmission mains with high head loss, booster systems and boiler feed.

The physical intuition is straightforward: each pump adds energy to the same stream. If the first one raises the pressure by 88 m and the second repeats the work on the same flow, the outlet of the set sits ~176 m above the inlet. A multistage pump is exactly this — several impellers in series inside a single casing.

Combined curve: adding heads at the same flow rate

The central step of the sizing is building the combined curve. Each pump has an H×Q curve, well approximated by a parabola:

H(Q) = a − b·Q²

where a is the shut-off head (head at zero flow) and b governs the droop of the curve. The coefficients come from a Lagrange interpolation through 3 points of the manufacturer’s catalog.

For the series association, you fix a flow rate and add the heads:

H_comb(Q) = H₁(Q) + H₂(Q) + … + H_N(Q)

For N identical pumps this simply becomes H_comb(Q) = N·H(Q) = N·a − N·b·Q². Note the contrast with the parallel arrangement, where you add the flow rates at the same head. Swapping one sum for the other is the most common conceptual error in this type of calculation.

How the operating point is found

The actual flow rate of the set is not free: it is imposed by the intersection between the combined curve and the system curve:

H_sys(Q) = H_static + k·Q²

Here H_static is the static lift (plus any pressure difference between reservoirs) and k lumps all the head losses — friction (via the friction factor from the Colebrook-White (Serghides) method) and minor losses (ΣK of the fittings). The operating point (Q, H)** solves:

N·a − N·b·Q² = H_static + k·Q²

whose closed-form solution is Q* = √[(N·a − H_static) / (N·b + k)]. In a robust calculator, however, a bisection over the real curves (not just the quadratic fit) is used, because the manufacturer’s curve is not always perfectly parabolic.

Why the gain is in pressure, not flow rate

Because the system curve grows with Q², doubling the available pressure does not double the flow rate: head loss reacts to the square and absorbs most of the increase. The useful effect of series is to raise the lift or reach, keeping the flow rate at a moderate level. Whoever needs more flow should look at parallel; whoever needs more pressure, at series.

Practical design considerations

Downstream pressure class. The pressure at the outlet of the last pump is the sum of the heads. Pipes, flanges, valves and joints downstream must withstand that accumulated pressure — plus the water hammer surge. Rating them by the pressure of a single pump is a serious failure.
NPSH only matters at the first pump. The critical suction is at the first pump; the following ones receive the already-pressurized liquid, with ample available NPSH. Concentrate the NPSHa ≥ NPSHr + margin check at the inlet of the set.
Prefer identical pumps. When identical, they split the pressure symmetrically and operate at the same point of their own curve. When different, the lower-capacity unit limits the set.
Operation near the BEP. Keep the operating point between 70% and 120% of the BEP flow rate of each pump to preserve efficiency and reduce vibration and axial thrust.
Protections. Include a check valve, by-pass and anti-surge devices; the pressurized column of the series makes the transient more severe.

Link with the standards

Pumping-station design in Brazil relies on ABNT NBR 12214 (and on NBR 5626 / NBR 10396 for building installations). Pump curves and testing follow the Hydraulic Institute (HI/ANSI) and ISO 9906, which standardize the tolerance grades of the performance acceptance test — the basis of the catalog points that feed the Lagrange interpolation. The system curve, in turn, depends on the friction factor computed by the Colebrook-White (Serghides) method, ensuring consistency between the estimated head loss and the reported operating point.

Formulas and fundamentals

Combined head (same flow rate) H_comb(Q) = H_1(Q) + H_2(Q) + ... + H_N(Q)

In series the flow rate Q (m³/h) is the same through every pump; the heads H_i (m, mwc) add up at the SAME flow rate. For N identical pumps: H_comb(Q) = N·H_pump(Q).

Individual pump curve (quadratic fit) H(Q) = a − b·Q²

The H×Q curve of a centrifugal pump approximated by a parabola; a (m) is the shut-off head (Q=0) and b (m/(m³/h)²) governs the droop. Coefficients obtained by Lagrange interpolation through 3 catalog points.

System curve H_sys(Q) = H_static + k·Q²

H_static (m) is the static lift plus any pressure difference between reservoirs; k (m/(m³/h)²) lumps the friction and minor head losses (k ∝ f·L/D + ΣK). It is independent of the number of pumps.

Operating point H_comb(Q*) = H_sys(Q*) ⇒ N·a − N·b·Q*² = H_static + k·Q*²

The operating point (Q*, H*) is the intersection of the combined curve with the system curve. Solving: Q* = √[(N·a − H_static)/(N·b + k)]. In practice a bisection over the real curves is used.

Available NPSH (first pump only) NPSHa = (P_atm − P_v)/(ρ·g) − H_static,suc − h_f,suc

In series the critical suction is at the FIRST pump; the others receive already-pressurized water. P_atm and P_v in Pa, ρ in kg/m³, heads in m. Check NPSHa ≥ NPSHr + margin.

Standards & methods

ABNT NBR 12214 (pumping station design)
ABNT NBR 10396 / NBR 5626 (building water installations)
Hydraulic Institute (HI/ANSI) — pump curves and testing
ISO 9906 (hydraulic performance acceptance test, grades 1/2/3)
Colebrook-White (Serghides) method (friction factor for the system curve)

Typical reference values

Quantity	Typical range	Note
Velocity in the discharge line	1.5 to 3.0 m/s	Above this, head loss and water hammer rise quickly.
Velocity in the suction line	0.6 to 1.5 m/s	Low suction velocity preserves the available NPSH of the first pump.
NPSH margin (NPSHa − NPSHr)	≥ 0.5 m (preferably ≥ 1.0 m)	HI recommends a larger margin for hot water or volatile liquids.
Efficiency around the BEP	70% to 90%	Keep the operating point between 70% and 120% of the BEP flow rate.
Typical number of pumps in series	2 to 4 stages	Beyond this, prefer a dedicated multistage pump.
Pipe test pressure	1.5 × working pressure	In series the accumulated pressure raises the flange/pipe rating downstream.

Worked example

Two identical pumps in series on a high-pressure transmission main

Inputs

Pumps in series (N): 2 ea
Shut-off head of each pump (a): 130 m
Pump droop coefficient (b): 0.018 m/(m³/h)²
System static head (H_static): 60 m
System loss coefficient (k): 0.05 m/(m³/h)²

Results

Operating flow rate (Q*): 48.2 m³/h
Combined TDH (H*): 176.3 m
Head per pump (H*/2): 88.1 m
Flow rate with a single pump: 32.1 m³/h

With k = 0.05 the system is highly resistive: a single pump (130 − 0.018·Q² = 60 + 0.05·Q²) would only reach 32.1 m³/h at 111.5 m. Pairing two in series, the combined curve becomes 260 − 0.036·Q² and the operating point rises to 48.2 m³/h at 176.3 m — each pump delivers 88.1 m, within its own curve. Note that the flow rate grew only ~50% (it did not double), while the available pressure jumped from 111.5 m to 176.3 m: this is exactly what to expect from a series association, a gain in pressure rather than in flow.

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Common mistakes

Adding the FLOW RATES instead of the heads — in series the heads add up at the same flow rate; flow rates add up in a parallel arrangement.
Rating the pressure class of pipe and flanges by the pressure of ONE pump, ignoring the accumulated pressure that reaches the outlet of the last pump.
Checking the NPSH of every pump equally — only the first one is at the critical suction; ignoring this leads to cavitation at the inlet of the set.
Expecting two pumps in series to double the flow rate: because the system curve is quadratic, the flow gain is modest; the real gain is in pressure.
Forgetting the check valve/by-pass between stages and water hammer protection, made worse by the pressurized column.
Operating outside the BEP range of each pump after the association, dropping efficiency and increasing vibration and axial wear.

Frequently asked questions

Do pumps in series add flow rate or head?

They add HEAD. The same flow rate passes through every pump and the heads accumulate at that same flow rate. Flow rates add up (at the same head) in a parallel arrangement.

When should I use series and when parallel?

Series for HIGH-PRESSURE systems with moderate flow (tall vertical lifts, long mains). Parallel for high FLOW with moderate pressure and demand modulation. The choice depends on the slope of the system curve.

Do pumps in series have to be identical?

It is not mandatory, but it is strongly recommended. Identical pumps operate at the same point of their own curve and share the pressure equally. With different pumps each one works at a distinct point, and the lower-capacity unit can limit the flow rate or run at low efficiency.

Why does the flow rate barely increase when adding pumps in series?

Because the system curve is quadratic (H_sys = H_static + k·Q²): as the available pressure rises, head loss rises with the square of the flow rate and 'consumes' most of the gain. The practical result is more pressure (greater lift or reach), not much more flow rate.

Does NPSH change with the series association?

The critical AVAILABLE NPSH is that of the FIRST pump, which sits on the suction side. The following pumps receive already-pressurized water with ample available NPSH. That is why cavitation protection is concentrated at the inlet of the set.

Do I need to reinforce the downstream piping?

Yes. The pressure at the outlet of the last pump is the sum of the heads of all pumps. Flanges, pipes, valves and joints downstream must have a pressure class compatible with that accumulated pressure (plus the water hammer surge), not with that of a single pump.

Glossary

Series association: Arrangement in which the discharge of one pump feeds the suction of the next; same flow rate, heads added.
Combined curve: The H×Q curve resulting from the association; in series it is obtained by adding the pump heads at each flow-rate value.
Operating point: Flow rate and head (Q*, H*) where the combined pump curve intersects the system curve; found by bisection.
TDH (total dynamic head): Energy per unit weight that the pump(s) deliver to the fluid, in meters of water column (mwc).
Shut-off head: Pump head at zero flow (closed valve); it is the coefficient 'a' of the quadratic fit H = a − b·Q².
Multistage pump: A pump with several impellers in series inside the same casing — physically equivalent to pumps in series, generating high pressure compactly.