The loss of driving power to a pump can lead to pipe collapse due to buckling or the formation of a vapor cavity and its subsequent collapse. Other transient problems may occur due to slam of a swing check valve, or from a discharge valve closing either too quickly (column separation), or too slowly (surging from reverse flow). The characteristic curves of pumps are usually given graphically defining the relationship between pump head, *H*, as a function of the flow rate, *Q*, of the form:

where *A*, *B*, and *C* are the coefficients of the quadratic curve defined for full speed normal (rated) pump operation and *R* is the ratio of the actual pump speed (*N*) to the rated pump speed (*NR*). For constant speed pump operation *R* is equal to 1. This representation is considered appropriate for normal pump operations with variable speed but is not appropriate for pump operation outside the normal zone (turbining, flow reversal, speed reversal, etc.).

During a hydraulic transient event, a pump may experience either a reversal in flow, rotational speed, or both, depending on the situation. It is also possible that the torque and head may reverse in sign during abnormal zones of performance. In extreme cases, the full four quadrant characteristics representing the eight possible zones of operation (four normal and four abnormal) must be used.

Figure 4 – Complete Head and Torque Characteristics of a Radial-Flow Pump in Suter Diagram

Abnormal Pump (Four Quadrant) Characteristics – InfoSurge utilizes the full four quadrant pump characteristics in addition to using the moment of inertia of the moving pump parts to compute pump rundown speeds. This approach is important for situations where abnormal pump operation occurs such as turbining, flow, and speed reversal. The approach suggested by Suter using data from Hollander’s experiments is utilized (Marchal et al. 1985; Mays 2000). The regions of operation shown on the above plot are termed zones (I through IV) and quadrants (*A* through *H*), the latter definition originating from plots of lines of constant head and constant torque on a flow-speed plane. The four quadrant pump characteristic plots depict values of *h*/(*a*^{2} + v^{2}) and *w*/(*a*^{2} + *v*^{2}) where *h* = *H*/*H _{R}*,

*w*=

*T*/

*T*,

_{R}*a*=

*N*/

*N*and

_{R}*v*=

*Q*/

*Q*. The plots are made utilizing 89 data points at intervals of π/44 radians and the plots shown use a unit of π/44 for the abscissa. A description of the various conditions for each of the eight zones of pump operation is shown in Table 2 below.

_{R}Table 2 – Pump Operating Zones and Quadrants

Since full four zone characteristics are seldom, if ever, available for a particular pump (these characteristics are known and tabulated for only a few pump specific speeds), it is standard in transient analysis to take supplied normal zone curves for head and torque as functions of discharge and fit these to one of the tabulated sets of four zone characteristics. To use this approach in InfoSurge, normalized pump data such as that shown in Figure 4 is stored in files for various pumps and the file to be utilized is specified in the input data along with the following required information:

*H _{R}* - Reference Head, ft (m)

* Q _{R}* - Reference Flow, ft

^{3}/s (m

^{3}/s)

*N _{R}* - Reference Speed, RPM

ε - Pump Efficiency

I - Moment of Inertia of Rotating Mass, lb ft^{2} (*Nm*^{2})

* N _{i}* - Initial Pump Speed, RPM

Three files have been constructed using the classic data from the Hollander experiments on centrifugal pumps tabulated in Streeter and Wylie (1983, 1993). An additional five (5) files have been constructed using data provided by Thorley (1991). The appropriate file to be used to simulate a pump is selected based on the specific speed. The specific speed is given by:

where the Reference Flow *Q _{R}* is in gpm (m

^{3}/s) and the Reference Head

*H*is in ft (m). The available files are listed below. The specific speed shown for the files is given in English units with values in parenthesis for SI units. When a standard Pump File is to be utilized, then the specific speed for each pump to be modeled is first calculated and the file selected for the specific speed that is closest to the computed value. Pumps described by a Pump File can be tripped (lose power) and the resulting rundown calculated based on the pump motor inertia and dynamic characteristics described in the Pump File.

_{R}If four quadrant pump characteristic data is available for a particular pump, then a specific file for that data can be constructed and utilized. Following the format utilized in Streeter and Wylie’s textbooks (1983 and 1993), data is defined over a range of 0 to 2π in intervals of π/44 (a total of 89 points). The data is stored in a file as 89 points of head data (*h*/(*a*^{2}+*v*^{2})) followed by 89 points of torque data (*w*/(*a*^{2}+*v*^{2})).