Pump System Curve

Every other pump calculation on this site answers a question about one flow. This one describes the pipework at ALL flows, and it is the piece most people are missing when a pump does not do what they expected.

A piping system demands head for two quite different reasons. The first is static: an elevation to lift the liquid through, plus any pressure difference between the source and the destination vessels. That head is a height. It does not care about flow at all — it is there at zero flow and it is unchanged at full flow. The second is friction, and friction in turbulent flow goes as the square of velocity, hence as the square of flow. Add them and you have $H = H_{st} + kQ^2$: a parabola sitting on a pedestal, where the pedestal height is the static lift and the parabola's steepness is everything the pipe, the fittings, the valves and the strainer do.

The shape of that curve decides how the system behaves under control, which is why the split matters more than the total. A system that is nearly all friction — a closed hydronic loop, where the fluid returns to where it started and the static head cancels — is nearly a pure square law, and slowing the pump follows the affinity laws beautifully into the cube-law power saving that variable-speed pumping is sold on. A system that is nearly all static — a lift into a high tank — barely moves as flow changes, so slowing the pump slides it down its own curve toward a nearly horizontal system curve, the flow collapses quickly, and the pump can end up at shutoff churning and heating. A minimum-speed limit on such a system is not optional.

Getting k is easier than it looks, and there are two honest routes. Build it up from Darcy-Weisbach and a fitting schedule, which gives a design value and is the right thing at the drawing stage; it is also, almost always, optimistic against the system as built. Or measure one duty point — a flow reading and the head at that flow, with the static head known from the geometry — and solve for $k = (H - H_{st})/Q^2$. One point pins the whole curve, because the shape is fixed at $Q^2$ and only the scale is unknown, and the measured value folds every fitting, every partly-shut balancing valve and every fouled metre into a single number that describes what you actually have. Take that point at a HIGH flow if you can: the friction term is what you are measuring, it is largest there, and a point taken near shutoff divides two nearly equal numbers and multiplies the instrument error enormously.

And k is not a property of the pipe for ever. Throttling a balancing valve raises it; opening one lowers it; a fouling heat exchanger raises it slowly across a season; a blocking strainer raises it suddenly. Trending k at constant pump speed is one of the cleanest early warnings a plant will give you — it moves long before anyone notices the flow is down.