System Curves, the Operating Point & Pump Sizing

Imagine pumping water from a lower tank up to a higher tank, 50 m of pipe in between. Even before any fluid moves, the pump must overcome the sheer elevation difference between the two tank surfaces — the static lift h_static. That head is owed at every flow, including zero. It is the floor the system curve sits on.

As soon as flow starts, friction fights back. Pushing fluid through 50 m of pipe costs major (friction) loss, and every elbow, valve, and entrance/exit adds minor loss. From Module 8 you know both of these scale with the velocity head V²/(2g) — and since V = Q/A for a fixed-area pipe, the velocity head scales with Q². So the total dynamic loss grows with Q². Adding the static floor and the dynamic loss gives the system curve:

H_sys = h_static + K·Q².

Read it as: 'to push a flow Q through this system, the pump must supply h_static plus a friction penalty that quadruples every time the flow doubles.' At Q = 0 the system needs only h_static; at high flow the K·Q² term dominates and the curve climbs steeply. The system curve is a property of the piping, not the pump.

That coefficient K is not magic — it is the pipe-flow losses of Module 8, repackaged. The Darcy–Weisbach major loss is h_f = f(L/D)(V²/2g), with f the Moody-chart friction factor. The minor losses are h_m = ΣK_m(V²/2g), where ΣK_m sums the loss coefficients of every fitting, valve, entrance, and exit. Both carry the same V²/(2g) factor, so factor it out and convert V to Q/A:

h_loss = [fL/D + ΣK_m] · V²/(2g) = [fL/D + ΣK_m] · Q²/(2gA²).

So K = [fL/D + ΣK_m]/(2gA²), with units s²/m⁵ (so that K·Q² comes out in metres). Longer, narrower, rougher, or more fitting-heavy piping all raise K and steepen the system curve. This is the direct link from the Moody chart to pump selection: a higher f (rougher pipe, or operating in a worse Moody regime) raises K, lifts the system curve, and forces a higher H at any given Q — which moves the operating point and may demand a bigger pump.

Now set the two curves on the same H–Q axes. The pump curve H = H₀ − aQ² falls from its shutoff head. The system curve H_sys = h_static + KQ² rises from its static lift. They cross at exactly one point. That crossing is the operating point: the unique flow Q_op at which the head the pump produces equals the head the system demands. There is no other possibility — at any other flow the two heads disagree and the flow adjusts until they match.

This reframes pump selection entirely. You do not choose the operating flow; the intersection of two curves chooses it for you. Your job is to pick a pump whose curve crosses the system curve near the pump's best-efficiency point (BEP). A pump that crosses far to the right of its BEP runs at high flow and low efficiency; one that crosses far to the left (near shutoff) runs hot and risks damage. Good sizing means engineering the intersection.

With the parabolic models, the intersection is a clean quadratic. Set the heads equal and solve for Q:

H₀ − aQ² = h_static + KQ²  ⟹  H₀ − h_static = (a + K)Q²  ⟹  Q_op = √[(H₀ − h_static)/(a + K)].

Then back-substitute for the operating head: H_op = H₀ − a·Q_op² (equivalently H_op = h_static + K·Q_op² — both give the same value, which is a useful check). If H₀ ≤ h_static the pump cannot even overcome the static lift and there is no positive operating flow — the system sits at shutoff.

Real pump curves are rarely perfect parabolas, and the friction factor f itself drifts with Reynolds number, so in practice the intersection is often found by iteration: guess a Q, compute H_pump from the manufacturer's curve and H_sys from the loss equations, see which is larger, and adjust. The algebraic quadratic is the model that teaches you how the operating point moves when you change a parameter; the iteration is what you do on real data. The interactive below lets you feel both: drag a slider and watch the crossing slide.