Standard Pump Curve Waterhammer Theory

A pump head curve is a function of flow and speed.

For any given fixed speed, the head curve is only a function of flow. The head curve can be directly adjusted for speed using the affinity laws.

Considering the case where the head/flow curve follows a polynomial relationship, and including a speed ratio term s:

where

Converting volume flow to mass flow and head to pressure:

Because ρ, g, and the curve constants are constant throughout the simulation, they can be combined and the equation simplified.

Solving for ΔP with Compatibility Equations 1 and 2:

and equating the two:

For a fixed speed, the only unknown is mass flow. For quadratic curve fits (d and e are zero), this equation is solved analytically. For higher order curve fits, it is solved for with Newton-Raphson iteration. Upstream and downstream pressures can then be determined from the compatibility equations.

For variable speed pumps, another parameter must be considered fixed - for example, suction or discharge pressure. In these cases, the above equations can be arranged to solve for the speed required.

If the speed and flow can vary throughout the transient without any parameter being fixed - for example, during a pump trip or startup - further equations and data are required. See Inertial Models.

Flows Outside the Normal Pumping Zone

The Standard Pump Curve is only valid for the Normal Pumping Zone - positive speed, flow, and head. To model such situations without additional four quadrant data, assumptions must be made.

Behavior from negative speed is not considered as the characteristics can be significantly different. This leaves two situations not normally covered - reverse flow and flow past the pump maximum.

• Reverse Flow - It is assumed that the head rise across the pump for all reverse flows is equivalent to the head rise at zero flow for the given curve at the given speed. This is a very rough estimate and will not be accurate for large or sustained flows, and a warning will be generated to this effect.

• Flow Exceeding Maximum - At high flows for a given speed there will be a point where head is actually lost across a pump. The Standard Pump Curve does not account for this - instead, it is assumed that the head rise is zero. 

Submerged Pumps

If a Pump is defined as Submerged, the upstream pressure is known and fixed. The same process as before can be used to solve for flow.

Vapor Cavitation Theory

When the calculated pressure at the upstream side of a Pump junction drops below vapor pressure, a vapor cavity forms on that side of the junction.

Usually vapor cavities form on the upstream side of the pump only, but in cases where there is reverse flow through the pump cavities can also form on the downstream side.

When a vapor cavity forms upstream of a pump, the upstream pressure becomes fixed at the vapor pressure. The calculation then is similar to a submerged pump, which also flows from a fixed pressure. Similar to a pipe interior node, when the vapor volume is negative the cavity collapses and the fluid pressure then rises above the vapor pressure.