As discussed in the Four Quadrant Model Overview, four quadrant data sets are measured data sets intended to approximate the pump behavior outside of the Normal Pumping Zone (i.e. pump operating at positive speed and positive head with forwards rotation). In cases where the pump operates beyond these boundaries, four quadrant data may be used as a tool to predict the head and flow at the pump when measured data is not available. Thus, when the pump is operating in the first quadrant, results generated using the pump performance data provided by the manufacturer (the manufacturer curve) will provide the most accurate results. Once the pump begins operating beyond this region, such as when reverse flow begins, the dimensionalized four quadrant data set (four quadrant curve) will likely provide more accurate results than the manufacturer curve. However, different four quadrant sets can provide widely varied results. Caution should be used to ensure that the chosen four quadrant data set provides reasonable results.

In Impulse, a four quadrant curve can either be selected automatically, or may be selected directly by the user.

Automatically Selected Four Quadrant Curve

When the Four Quadrant Curve option is first selected, a four quadrant curve will automatically be selected by Impulse. Impulse selects a four quadrant curve by doing the following:

1) The specific speed for the pump is estimated using the BEP calculated from the user-specified pump curve, and assuming that the pump includes only a single stage and the impeller is untrimmed.

2) Using the estimated specific speed from step 1, the four quadrant data set with the nearest specific speed is selected. By default Impulse will only choose a Preferred or Average data set.

3) The BEP, which is either specified directly by the user in the Pump Configuration window or calculated based on the user-specified pump curve, is used as the Dimensional Reference Point to dimensionalize the four quadrant curve

This automatically selected four quadrant curve will provide a reasonable estimate for most cases. However, it is advised that the user check the validity of Impulse's selection by using the considerations and procedure described below.

Considerations for Selecting a Four Quadrant Curve

The following recommendations from section 6.2 of Walters, Dahl, and Rogers, 2020 Walters, T., Dahl, T. and Rogers, D.C., "Pump Specific Speed and Four Quadrant Data in Waterhammer Simulation - Taking Another Look", Proceedings of the ASME 2020 Pressure Vessels and Piping Conference, PVP 2020, Minneapolis, MN, USA, July 19-24 2020. should be considered when selecting a four quadrant data set for the waterhammer analysis.

1) Comparison to the Manufacturer Curve

The manufacturer curve is measured data from the actual pump being used in the system, and thus will provide the most accurate prediction for the pump behavior. Though it cannot be used to predict behavior beyond the range of the measured data, it can be used for comparison to the four quadrant curve(s) for the steady state simulation. In some systems, the transient results can be highly sensitive to the steady state of the system, thus accuracy of the pump curve used in the steady state can be an indicator of accuracy during the transient. In addition, the final steady state for the system once the transient response has concluded can be compared as an indicator of how close the four quadrant curve will behave to the manufacturer curve during the transient.

2) Pump Specific Speed

As discussed in a previous topic on Four Quadrant Data Sets, it is typically a good practice to select a four quadrant data set based on the specific speed of the pump, as pumps with similar specific speeds will typically be constructed similarly. However, this does not always hold true. Thus, comparing multiple data sets at specific speeds close to the specific speed specified by the pump manufacturer is advised.

3) Dimensional Reference Point

As was previously discussed, the Four Quadrant Dimensional Reference Point can be defined as either the Best Efficiency Point (BEP) or the steady state operating point (SSOP) from the manufacturer's curve. Using the BEP will result in a different operating point during the steady state, but may provide a closer match for the final steady state using the manufacturer's curve. Using the SSOP will allow the steady state operating point to exactly match the manufacturer's curve, but may diverge widely from the final steady state operating point predicted by the manufacturer's curve due to distortions in the dimensionalized curve.

Procedure for Four Quadrant Data Set Selection

A generic procedure for creating a model and analyzing the results when using four quadrant data is given below:

1. Create a child scenario of the scenario that is defined using the manufacturer's curve

2. Open the Pump Properties window and select Four Quadrant Curve as the Performance Curve Used in Simulation

3. Choose User Selected and click Specify Model to launch the Specify Four Quadrant Model window

4. Choose the desired four quadrant data set from the drop-down list at the top of the window, and select Best Efficiency Point as the Dimensional Reference Point.

5. Select Ok to accept the changes

6. Clone the current scenario and change the Dimensional Reference Point to the Steady-State Operating Point in the cloned scenario

7. Repeat steps 1 - 6 for each data set to be analyzed

8. Run all of the newly created scenarios and compare the transient results. Note that frequently results will not differ greatly among scenarios. In that case, results are not very sensitive to the specific choice of four quadrant curve. In order to be most conservative, use the worst case predictions from among the various four quadrant scenarios for design purposes.

9. If the worst case predictions lead to excessive and/or expensive implications in design and/or operations, take a closer look at results. Compare each four quadrant pump curve to the manufacturer’s curve at the initial steady state and, if applicable, the final steady state operating points to determine which scenario(s) appears to give the most accurate results. Criteria used to decide the most reliable four quadrant curve(s) include:

1. Similarity of specific speed, Ns – when making this comparison, be sure you have an accurate Ns value for your pump

2. Overall agreement between the manufacturer’s curve (Standard Pump Curve) and dimensionalized four quadrant curve for head and power vs. flow rate in the normal zone of pump operation (Impulse compares these curves for you – review the comparisons)

3. Give more weight the four quadrant curves which show closer agreement within the range of flow rates that the manufacturer tested the pump at (i.e. the range of flow rates that the pump curve is graphed for on the specification sheet)

4. Give more weight to four quadrant curves which generate transient results similar to the Standard Pump Curve during time frames when the pump is operating in the normal zone of operation (positive flow, head and rotation)

5. Give more weight to four quadrant curves Impulse categorizes as Preferred

6. Good agreement in initial steady-state flow rates between SPC and four quadrant curve

7. Good agreement in final steady-state flow rates between SPC and four quadrant curve – this not as important as agreement in initial steady-state flow rates, but should be considered

Note that there are several reasons why the engineer may choose to skip step 9 above. One reason would be if the transient results for the different scenarios show only minor differences as noted in Step 8. In that case it would be unnecessary to determine which scenario is most accurate, as any scenario will produce similar results. Another reason would be if the engineer’s analysis involves a project where highly conservative results are desired, such as for a nuclear system, or another such instance where less accurate results would be costly (e.g., force installation of an expensive surge relief system that may not truly be needed). In that case the engineer could choose to simply use the most extreme results for the analysis for their design purposes. Note this means that if different scenarios produce the worst-case pressures in different areas of the model, then the engineer may want to use the most extreme results (e.g., highest or lowest pressure) regardless of scenario, rather than only using the results from one scenario.

Four Quadrant Data Set Selection Example

Several walk-through examples describing the basics of four quadrant modeling can be found in AFT Impulse's walkthrough examples.

For a basic introduction to using four quadrant curves see the "Pump Trip With Backflow - Four Quadrant Modeling" example.

An advanced example on performing a sensitivity study to select a four quadrant curve is also available, titled "Selecting a Pump Four Quadrant Curve".