Pump operating point calculator

What it computes

The pump is described by a shut-off head and a free delivery, which the sheet turns into a falling parabola of head against flow. The pipeline needs the static lift plus a friction loss that grows with the square of the flow. The operating point is the flow at which the two heads are equal, and the sheet solves that directly. It then reports the flow, the head, the pipe velocity (with a warning above 3 m/s) and a table and plot of both curves.

The formulas

• Pump curve: H_p = H_0 (1 - (Q / Q_max)^2), with a shut-off head H_0 and a free delivery Q_max.
• System curve: H_s = H_st + K Q^2, where H_st is the static lift and K = f (L / D) / (2 g A^2), with a constant friction factor f of 0.018, or 0.030 for an aged pipe.
• The curves cross where H_0 - H_0 Q^2 / Q_max^2 = H_st + K Q^2, so Q^2 = (H_0 - H_st) / (K + H_0 / Q_max^2).
• If H_0 is not above the static lift, the pump cannot lift the water and the sheet says so.

Worked example

Defaults: the medium pump (40 m shut-off head, 75 L/s free delivery), a 150 mm pipe 400 m long, a static lift of 15 m and a new pipe. Engine results:

The pump settles at 40.9 L/s and 28.1 m. The velocity of 2.31 m/s is within 3 m/s.

Move the operating point

Marking the pipe as aged raises the friction factor to 0.030, which increases K to 13057.1 s 2/m⁵ and moves the operating point to 35.21 L/s at 31.19 m. A rougher pipe means less flow from the same pump.

Raising the static lift to 35 m instead drops the flow to 18.29 L/s, with the pump working harder at 37.62 m. The table in the sheet shows the heads at flows from 0 to 60 L/s, and the plot shows the crossing.

What this is, and is not

The sheet source

# Pump and pipeline: operating point

A template with inputs. Choose **Input** in the results bar to fill in the form. A pump lifts water
through a pipeline to a higher tank. The pump delivers the head $H_p = H_0 \left(1 - (Q/Q_{max})^2\right)$
at the flow $Q$ (a shut-off head $H_0$ and a free delivery $Q_{max}$). The pipeline needs the head
$H_s = H_{st} + f \frac{L}{D}\frac{v^2}{2g}$, the static lift plus the friction loss with a constant friction
factor $f$. The pump runs where the two curves cross. The table and the plot follow the form.

## Inputs
pump: ? (Small = 1, *Medium = 2, Large = 3) | pump
D:    ? [100 mm = 100 mm, *150 mm = 150 mm, 200 mm = 200 mm] | pipe inside diameter
L_p:  ? 400 m | pipe length
H_st: ? 15 m | static lift, from the water surface to the tank
aged: ? off | aged pipe (rougher: the friction factor is 0.030, not 0.018)

## Pump curves
The input above picks one entry of each list.

H0_tab:   [30 m, 40 m, 50 m] | shut-off head
Qmax_tab: [60 L/s, 75 L/s, 90 L/s] | free delivery
H_0:   H0_tab.pump -> m
Q_max: Qmax_tab.pump -> L/s
if pump == 1: "The pump is the small one."
if pump == 2: "The pump is the medium one."
if pump == 3: "The pump is the large one."

## Pipeline
g_acc: 9.81 m/s^2 | gravitational acceleration
f: 0.018 + aged * 0.012 | friction factor
A: pi * D^2 / 4 -> m^2 | flow area
K: f * L_p / D / (2 * g_acc * A^2) -> s^2/m^5 | friction head per flow squared

## Operating point
The two curves cross where $H_0 - H_0 Q^2/Q_{max}^2 = H_{st} + K Q^2$, so
$Q^2 = (H_0 - H_{st}) / (K + H_0/Q_{max}^2)$.

H_net: H_0 - H_st -> m | head the pump has above the static lift
if H_net <= 0 m: "The pump cannot lift the water to the tank: choose a larger pump or lower the static lift."
Q_op: sqrt(max(H_net, 0 m) / (K + H_0 / Q_max^2)) -> L/s | flow at the operating point
H_op: H_0 * (1 - (Q_op / Q_max)^2) -> m | head at the operating point
v:    Q_op / A -> m/s | velocity in the pipe
if v > 3 m/s: "The velocity is above 3 m/s: expect noise and wear, use a larger pipe." else: "The velocity is within 3 m/s."

## Table
Q_v:  [0, 10, 20, 30, 40, 50, 60] * 1 L/s | flows in the table
H_pv: H_0 * (1 - (Q_v / Q_max)^2) -> m | pump head
H_sv: H_st + K * Q_v^2 -> m | head the pipeline needs
T: augment(Q_v / (1 L/s), H_pv / (1 m), H_sv / (1 m)) | columns: flow in L/s, pump head in m, pipeline head in m

## Curves
H_p(Q): H_0 * (1 - (Q / Q_max)^2) -> m | pump head at the flow Q
H_s(Q): H_st + K * Q^2 -> m | pipeline head at the flow Q

plot H_p(x * 1 L/s) / (1 m) as "pump", H_s(x * 1 L/s) / (1 m) as "pipeline" for x in 0..60

The same crossing, found numerically from the two curves:

if H_net > 0 m:
    Q_num: solve H_p(Q) = H_s(Q) for Q in 0 L/s..Q_max -> L/s | flow where the pump head equals the pipeline head

How to use it

1. Open the template. Choose Open this template on this page. The sheet opens in a new tab of CivilKit Calcs, with the inputs set to the values used on this page.
2. Switch to Input mode. In the results bar choose Input. The input lines become boxes, lists, buttons and ticks.
3. Enter your values. Change the inputs to your own. Pump size, pipe diameter, length, static lift and the aged-pipe box are the inputs. Every change recalculates the whole page.
4. Read and export the result. Read the results and the OK or not OK lines under them. Use Report, then Print PDF, Word or Excel, to keep the calculation with its workings.