Four Quadrant Waterhammer Theory
The waterhammer theory when using four quadrant data in Suter Suter, P., "Representation of Pump Characteristics for Calculation of Waterhammer", Sultzer Technical Review Research Issue, pp. 45-48, 1966. Suter form follows a similar approach to the Standard Pump Curve Waterhammer Theory.
First, we know from the compatibility equations that the pressure change across the pump is related to the mass flow:
We also know that the dimensionless four quadrant parameters are related to the values used for the Dimensional Reference Point:
From the definition of h we have a relationship between the curve FH (Θ), speed and flow.
Multiplying through by ρg we can attain a similar expression in pressure terms:
Similarly, there is a relationship for volumetric flow that can be converted to mass flow by multiplying by density:
The previous equations can be combined with the compatibility equation for the pump:
This equation contains three potential unknowns - speed (α), flow (ν), and FH (Θ). FH and Θ are related via tabulated data. The value of Θ is related to the speed and flow, bringing the unknown variables down to just speed and flow:
As with the Standard Pump Curve, for a fixed speed there is only one unknown - flow. The equations and tabulated data can be iterated on to find a solution for flow and therefore pressure rise across the pump.
During a pump trip or startup, when neither speed nor flow are known, additional equations are required. The necessary torque balance terms are described in Transient - Inertial Models resulting in the below equation for speed change:
It is helpful to modify this equation so that it is directly applicable in four quadrant terms. The following relations are required:
resulting in the following:
This equation represents the full torque balance with unknown driver torque, pump torque, and pump speed in four quadrant form. For a pump trip, the driver torque is zero leaving only two unknowns. For a startup, the driver torque vs speed must be specified. Like the head term above, the dimensionless torque β can be solved for in terms of speed (α), flow (ν), and FB (Θ).
These equations can then be solved for speed, flow, torque and head.
