Selecting a Pump Four Quadrant Curve (English Units)

Summary

During waterhammer analysis of pumping systems there is potential for the pump to operate outside of its normal operating region where flow, head, and speed are positive. It may be required to use a four quadrant pump curve in order to better predict the pump’s behavior when operating outside of this normal operating region. As manufacturers rarely provide four quadrant data for the pump being used, a published four quadrant data set in dimensionless form is often chosen instead and dimensionalized to approximate the behavior of the actual pump being used in the system. For many cases the choice of four quadrant curve has a low impact on the waterhammer analysis results; however, this will not always be the case. This example is intended to provide the engineer with a deeper understanding of how to select four quadrant data sets for a waterhammer analysis, and how to evaluate the transient results predicted by the four quadrant curves.

This example considers a water transfer system for which several pump trip/valve closure cases are being modeled. There is a high possibility for reverse flow through the pump and reverse rotation of the pump impeller as the water is being pumped uphill, thus four quadrant data sets will be considered to model the behavior of the pump when operating with reverse flow and/or reverse rotation.

For those more interested in a quick how-to discussion and do not need in-depth discussion, consult the Overview of performing a sensitivity study for four quadrant analysis topic in Step 2 below.

Topics Covered

• Selecting a four quadrant data set to analyze cases with reverse flow and/or reverse rotation at the pump

• Performing sensitivity studies to assess four quadrant data selection

• Comparison of the Best Efficiency Point (BEP) and Steady State Operating Point (SSOP) dimensional reference point options

Required Knowledge

This example assumes the user has already worked through the Beginner: Valve Closure example, or has a level of knowledge consistent with that topic. You can also watch the AFT Impulse Quick Start Video (Metric Units) on the AFT website, as it covers the majority of the topics discussed in the Valve Closure example.

This example expands on the topics introduced in the Pump Trip with Backflow - Four Quadrant Modeling example. It is recommended to work that example first as an introduction to basic four quadrant modeling concepts before working this example.

Nomenclature

As this is an advanced topic, a brief list of abbreviations is given here for reference.

BEP	Best Efficiency Point of the standard pump curve
DRP	Dimensional Reference point, reference point used to create a four quadrant curve for a chosen four quadrant data set
Ns	Specific speed for the pump
SPC	Standard pump curve – sometimes called here the Manufacturer Curve, this is the head and efficiency (or power) vs. flow rate curve for the actual pump in your system
SSOP	Initial steady state operating point for the actual pump (from the manufacturer’s curve and system curve)
4QBEP	Four quadrant pump curve created with reference to the BEP
4QSSOP	Four quadrant pump curve created with reference to the SSOP

Model Files

This example uses the following files, which are installed in the Examples folder as part of the AFT Impulse installation:

• Metric - Selecting a Pump Four Quadrant Curve - Initial.imp

• Metric - Selecting a Pump Four Quadrant Curve - Final.imp

• 10 Sec Valve Opening.txt

• 10 Sec Valve Closing.txt

Step 1. Start AFT Impulse

From the Start Menu choose the AFT Impulse 9 folder and select AFT Impulse 9.

To ensure that your results are the same as those presented in this documentation, this example should be run using all default AFT Impulse settings, unless you are specifically instructed to do otherwise.

Step 2. Open the Model

Water is being provided to two different processes, represented by Tank A and Tank B, as can be seen in Figure 1. The system is designed to provide water to one tank at a time.

In this transfer system several pump trip scenarios will need to be analyzed which have potential to experience back flow through the pump before the emergency shut-down (ESD) valve fully closes. Different four quadrant data sets will be compared against the manufacturer pump curve data for each of these cases to help the engineer understand the sensitivity to four quadrant data set selection and the range of potential predictions that result. The cases are as follows:

1. Unplanned pump trip after the flow has been transitioned from Tank A to Tank B

   1. The pump operates far below BEP initially (about 60% BEP). Minimal reverse flow occurs at the pump.

2. Unplanned pump trip after the flow has been transitioned from Tank B to Tank A

   1. The pump operates above BEP initially (about 113% BEP). Sustained reverse flow and reverse rotation occur at the pump.

3. Planned pump trip and Valve B closure after Valve A fails open

   1. The pump operated at BEP initially. Sustained reverse flow and rotation occur at the pump.

Note: No pump startup cases are considered for this example, but the procedure is applicable to those situations as well if reverse flow does not occur during the initial steady state solution.

We will start with a given model file that is fully defined for the transient run using the manufacturer's curve. Navigate to the Examples folder installed with AFT Impulse, and open the model file named Metric - Selecting a Pump Four Quadrant Curve - Initial.imp to the Case 1 - Tank A to Tank B scenario.

Note that we have four junctions with transients defined, as is denoted by the red T in the Workspace. These transients occur as two events. For the first event a planned switch from Tank A to Tank B occurs at time zero. Valve A will begin closing over ten seconds, following the transient shown in Figure 2 below. Valve B will begin opening at the same time over ten seconds following the reverse profile of that shown for Valve A. The second event happens twenty seconds into the transient run (ten seconds after the tank transition finishes), at which time an unplanned pump trip occurs and the ESD valve begins to close over seven seconds. The pump is modeled as a pump trip with inertia.

Step 3. Understand the Need for Sensitivity Studies for Four Quadrant Analysis

When the Standard Pump Curve is used (i.e., not four quadrant data) and reverse flow occurs at a pump, AFT Impulse uses the zero flow value for head and power. On a graph of head vs flow this would be represented as a horizontal line for negative flows. This may be a good estimate for negative flow that either has a small magnitude or happens for a short duration, but using a four quadrant data set is often better for high magnitude or sustained reverse flow, as this method introduces less uncertainty than just assuming that head and power would remain constant. Different four quadrant data sets can produce widely varied results, so choosing the four quadrant data set which best fits the pump being modeled is important.

Considerations for Choosing a Four Quadrant Curve

During the four quadrant data set selection process, there are several factors that will need to be accounted for. A brief background for each metric is provided below:

1. Pump Specific Speed: It is typically good practice to select a four quadrant data set based on the specific speed of the pump, as pumps with similar specific speeds will often behave similarly. However, the four quadrant data set with the closest reported specific speed is not always the best choice to represent the pump being analyzed for multiple reasons, including uncertainty in the accuracy of the reported specific speed for some four quadrant data sets. Thus, comparing multiple four quadrant data sets at specific speeds close to the specific speed of the pump being analyzed is advised. Note that in Impulse, each four quadrant data set is labeled as Preferred, Average, or Use With Caution, indicating the apparent quality of the four quadrant data set and the accuracy of the reported specific speed. When possible, the Preferred data sets should be selected for use in the model, as these sets provide the most trustworthy data. Average curves may be used if no Preferred data sets are available at similar specific speeds to the pump being used. The Use With Caution data sets should only be used when there are no Preferred or Average data sets that have a similar specific speed to the pump in use, or if the user has previous experience with using that data set which gives them confidence in that data set.

2. Comparison to Manufacturer’s Curve: The manufacturer's curve will always provide the most accurate prediction of the pump’s behavior while the pump is operating in the normal pumping zone of operation (i.e. flow in the positive direction, positive head, positive pump speed), and while the pump is operating within the range of flows that were tested by the manufacturer. Therefore it is reasonable to assume that a four quadrant data set which closely matches the manufacturer's curve while in the normal zone of operation will provide more accurate results than a four quadrant curve that shows larger differences from the manufacturer's curve in the normal pumping zone of operation. This comparison to the manufacturer's curve should be done for both the initial steady state operating point and the final operating point after all transients have died out. For this example we will refer to the final operating point after the transients have died out as the final steady state operating point. The reasoning for this will be discussed further throughout the example.

3. Dimensional Reference Point (DRP): Four quadrant data is usually published as a dimensionless set of data to allow it to be used for analysis with pumps besides the test pump. When dimensionalizing the data set for a pump, either the BEP or the SSOP may be used as the dimensional reference point. Many published four quadrant data sets use the BEP as the DRP to non-dimensionalize the data, so it seems logical that the BEP should be used as the DRP for the reverse transformation. However, if the pump is operating far from the BEP in the steady state, the initial steady state operating point predicted using the BEP as a DRP may be very different from the SSOP predicted from the manufacturer's curve. In that case using the SSOP as the DRP is useful to prevent differences from the steady state results using the manufacturer’s curve. However, using the SSOP as the DRP often causes large differences between the manufacturer's curve and the four quadrant curve. In summary, there is often a trade-off between how well the four quadrant curve matches the manufacturer’s curve for the steady state and final steady state, so it is always advised to test both DRP options. Note that if the simulation begins with a significantly incorrect initial flowrate, this will impact all subsequent results in a way that may be difficult to assess.

Procedure for Comparing Four Quadrant Data Sets

1. To perform the sensitivity study explained above the following steps can be taken:

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

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

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

5. 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.

6. Select OK to accept the changes

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

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

9. 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. This makes the analyst’s job easier.

10. In order to be most conservative, use the worst case predictions from among the various four quadrant scenarios for design purposes.

11. 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 (see discussion below for more on this)

    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 displays these curves for you – review the comparisons)

    3. Give more weight to 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

    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 10 above. One reason would be if the transient results for the different scenarios show only minor differences as noted in step 8 above. 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 required, such as for a nuclear system. 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.

Step 3. Run the Model with the Manufacturer's Curve

ØClick Run Model from the Toolbar or Analysis menu, then proceed to the Output window.

Notice that two warnings are given for this scenario, one indicating that negative head rise was calculated at the pump, and one indicating that reverse flow occurred at the pump. It will be easier to discuss these warnings by reviewing the transient head rise and flow rate at the pump.

ØGo to the Graph Results window. In the Graph List Manager on the Quick Access Panel double click the Head Rise and Vol Flow at Pump Outlet graph list item to load the graph shown in Figure 3. You should see that the flow rate remains positive for the majority of the transient, with minor reverse flow occurring after 27 seconds, never reaching a magnitude higher than 30 gal/min. Graphing the pump head rise shows that the head rise reaches zero at about 21 seconds, and is fixed at zero until around 27 seconds. The low magnitude of reverse flow would suggest that a four quadrant curve may not be needed for use in this model. However, the head rise results introduce a large amount of uncertainty into the results, since AFT Impulse artificially fixes the head rise to zero. Using a four quadrant curve will allow Impulse to estimate the negative head rise which would occur at the actual pump.

ØReturn to the Output window and go to the Pump Summary tab (Figure 4). The BEP was calculated based on the user-specified power curve to be 1,389 m3/hr. Note that in this case the pump is operating at 854.2 m3/hr during the steady state, which is 61.51% of BEP. This will be an important consideration when choosing a dimensional reference point for the four quadrant data set, as was discussed previously.

Step 4. Choose Four Quadrant Data Sets to Analyze

A. Calculating or Estimating Specific Speed

From the first consideration above, the first step for selecting the four quadrant curve will be to determine the specific speed of the pump and choose a four quadrant data set accordingly. AFT Impulse has a built-in tool to estimate the specific speed for the pump on the Specify Four Quadrant Model window if the specific speed is not available from the manufacturer.

ØCreate a child scenario of Case 1 - Tank A to Tank B named Manufacturer's Curve, this scenario will hold the results that were just obtained. Create another child of the Case 1 - Tank A to Tank B scenario named Four Quadrant Curve. Next, open the Pump Properties window. Select Four Quadrant Curve, choose User Specified for the type, then click Specify Model. By clicking Suggest a Data Set towards the top of the window, we can see that Impulse has estimated a specific speed of 51.5 metric units based on the displayed inputs (Figure 5). Looking at the specification sheet for this pump given by the manufacturer, the specific speed is given as 32.0 metric units. In this case the discrepancy between Impulse’s calculations and the manufacturer’s calculations is due to several assumptions that were made in calculating each value.

The first assumption involves the impeller size for the pump. ANSI/HI 14.3.1.3.1-2019 states that the specific speed for any pump should be calculated for the pump at the maximum impeller size. Impulse will assume by default that the pump is untrimmed. However, for this example the impeller was trimmed. The specific speed estimate in Impulse can be improved by entering the trimmed and untrimmed impeller sizes, allowing Impulse to adjust the head and flow values using the affinity laws.

ØSelect Trimmed under Impeller Diameter, and enter 508 mm and 472 mm as the maximum and Standard Pump Curve diameters, respectively. Click Calculate and the estimated specific speed should now be listed as 47.9 metric units, as shown in Figure 6.

Another assumption contributing to this discrepancy involves the flow rate used for the calculation. ANSI/HI 14.3.1.3.1-2019 states that the specific speed should be calculated using the total flow through the pump. However, the standard does note that there is an alternate definition in industry which defines the specific speed as the flow rate per impeller eye. For double suction pumps (such as the pump in this example) this definition produces a specific speed which is reduced by 1/sqr(2), or approximately 0.707 times the total flow definition. If the specific speed value provided by the manufacturer (32.0) is divided by this factor the resulting specific speed of 45.3 matches much more closely with Impulse’s estimate of 47.9 that was just calculated, which confirms that the specific speed reported by the manufacturer was for the flow per impeller eye. In Impulse it is recommended to use the definition of specific speed based on the total flow, as this is generally more consistent with the specific speed definition used for the four quadrant data sets available in Impulse.

ØChoose Select Suggested Data Set and Close to accept the suggested 41.8 m3/sec units data set, which has the closest specific speed to the calculated specific speed chosen above.

Note: In cases like this where the specific speed has been provided by the manufacturer the value provided by the manufacturer will typically be more accurate. However, the engineer should verify whether the manufacturer has reported the specific speed based on the total flow or flow per impeller eye.

B. Setting up Four Quadrant Scenarios

We just determined that the four quadrant data set with the closest specific speed is the 41.8 m3/sec data set. This data set is Average, so it is an acceptable candidate for use. As discussed above, it would also be prudent to test several other data sets with similar specific speeds. The next closest Preferred or Average data sets are the 37.4 m3/sec and 64 m3/sec data sets, so they will also be considered for use in this model.

Each of the selected data sets should now be compared to the manufacturer's curve with the data set dimensionalized at the BEP and the SSOP. From here on we will refer to four quadrant curves that have been dimensionalized using the BEP as 4QBEP curves, and four quadrant curves that have been dimensionalized using the SSOP as 4QSSOP curves for convenience.

Create the scenarios for the selected four quadrant data sets as follows:

1. In the current Four Quadrant Curve Scenario check that BEP is selected for the DRP (dimensional reference point). Note that the head, efficiency, and power curves versus flow can be seen plotted against the Standard Pump Curve (referred to as the manufacturer’s curve for this example) at the bottom of this window, as shown below for head in Figure 7 and power in Figure 8.

Note: The raw data from the graphs shown in the Specify Four Quadrant Model window (Figure 7 and Figure 8) can be copied using the Copy Data button at the bottom of the window to compare this data for all scenarios in Microsoft Excel.

2. Select OK to accept the current settings.

3. Rename the Four Quadrant Curve scenario in the Scenario Manager as Ns 41.8 4QBEP.

4. Clone the Ns 41.8 4QBEP scenario to create a Ns 41.8 4QSSOP scenario, then change the Dimensional Reference Point in the cloned scenario to the Steady-State Operating Point and change the Sample OP value to 854, as can be seen in Figure 9.

5. Create cloned scenarios for the 37.4 m3/sec and 64 m3/sec four quadrant data sets, changing the four quadrant data set and dimensional reference points as needed. When finished the Scenario Manager should appear similar to Figure 10.

6. Run all of the four quadrant scenarios so that they have results.

Note: A batch run can be used to speed up the process of running multiple scenarios. This can be set up from the File menu by selecting Start Batch Run, then selecting all of the scenarios to be run and clicking Start Run.

C. Comparing Four Quadrant Pump Curves

If the engineer’s goal is simply to perform the most conservative analysis possible in the shortest amount of time, then comparing the pump curves as will be done below would be unnecessary. Instead, the engineer should simply proceed to compare the transient results for all four quadrant scenarios and use the maximum/minimum results from all cases, as was discussed at the beginning of this example.

Additionally, the below comparison may prove to be unnecessary if the transient results show negligible differences. However, we will compare the manufacturer’s curve and four quadrant pump curves here to be thorough.

Finding the actual initial and final steady state flow rates

For the next step in the sensitivity analysis we will compare all of the pump curves to the manufacturer’s curve at both the initial steady state operating point and the final steady state operating point, as was mentioned at the beginning of this example. There are two transient events that occur in this model, so in a sense we have two final operating points, one after each event. The final operating point for Event 1 will be when the flow has fully transitioned to Tank B, but Event 2 has not yet begun. The final operating point for Event 2 will be when the ESD valve and Valve A have fully closed, preventing flow in the system. We need to compare the four quadrant curves for the final operating point after Event 1, since the operating point at this time will affect the results for Event 2. After Event 2 the system will be at zero flow and no transients will occur, so this is not a point of interest for comparison.

The initial and final steady state flow rates can be found by running the manufacturer’s curve scenario, as was done earlier in the example. In this case we already saw that the steady state operating point for the pump was 854 m3/hr (see Figure 4, actual flow rate). To find the actual final steady state after the transient effects have completely dispersed create a new scenario using the manufacturer’s curve for the pump, and adjust the model in the workspace to reflect the final steady state. This scenario can then be run using Time Simulation – Steady Only from the Analysis menu, as we are only interested in determining the steady results. This predicts a flow rate of about 1560 m3/hr for the final steady state operating point.

Checking the dimensionalized curves

Each individual four quadrant curve can be compared to the manufacturer’s curve within the Specify Four Quadrant Model window in Impulse, as was seen above in Figure 7 and Figure 8 for the 41.8 m3/s units data set. Each four quadrant curve will intersect the manufacturer’s curve at the dimensional reference point, which is shown in red (1,389 m3/hr, see Figure 4 BEP). For the head, power, and efficiency curves we will want to check how similar the shapes of the curves are and how close they are to each other. Focus should be given in particular to how well the curves match near the initial and final steady state flow rates, as this will indicate how different those operating points will be.

When the BEP is used as the dimensional reference point, there is only one possible effective head and power curve for each four quadrant data set, as the BEP will not change when the system changes. However, if the steady state operating point is chosen then different effective head and power curves for each four quadrant data set will result depending on where the manufacturer’s curve operates during the initial steady state. Use the slider or the text box on the graph in the Specify Four Quadrant Model window to make sure that the graph is displaying the four quadrant curve for the actual steady state operating point before using the graph for comparison. The 41.8 four quadrant data set dimensionalized at the steady state operating point identified above is shown in Figure 9.

If any four quadrant data set shows a substantial discrepancy for the initial or final steady state operating point, then it may be prudent to consider substituting this data set for another, in particular if the data set you are inspecting is marked as Average or Use With Caution.

Pump vs System Curve Comparison

To more easily compare the curves, the operating points for the head curves are shown using a multi-scenario pump vs. system curve that was created in Fathom 12 (Figure 11). The power curves are graphed in Figure 12 using Microsoft Excel. Table 1 listing the operating points shown in Figure 11 is also provided below for convenience. An engineer would not typically need to do this, but it is shown here to help reinforce the concepts discussed.

Each of the 4QBEP head curves cross the manufacturer's curve at the BEP, which is 1,389 m3/hr for this pump as was seen in the Pump Summary in the Output window (see Figure 4, BEP). The SSOP curves cross the manufacturer's curve at the actual operating point (854 m3/hr, see Figure 4 volumetric flow rate). Thus, the steady state flow rates will be identical between the 4QSSOP scenarios and the original scenario using the manufacturer's curve, unlike the 4QBEP curves. This distinction between the 4QSSOP and 4QBEP curves can be important because if the initial steady-state flow differs significantly from the actual pump you are modeling, the entire simulation will be impacted. Indeed, something like a sudden valve closure which happens early in the transient can result in very different peak pressure predictions. For example, the BEP curve that operates furthest from the manufacturer’s curve for the initial steady-state is the 37.4 4QBEP curve which operates at about 610 m3/hr and 26 meters and is 240 m3/hr and 2 meters below the operating point with the manufacturer’s curve.

We should also compare the final system curve to the pump curves. The final steady state will impact any transients that may occur later on in the simulation or after the end of the simulation, such as a sudden valve closure. In this case the Final System Curve represents the steady state after the first event (Valve A closing and Valve B opening), but before the second event (the unplanned pump tripping while the ESD valve closes). Thus any large differences in the final steady state will impact the subsequent pump trip. It can be seen from Figure 11 below that each of the 4QBEP curves show deviations from the manufacturer's curve at the Final System Curve, but the 4QSSOP curves show a much larger deviation than the 4QBEP curves. The 41.8 4QSSOP curve differs by 430 m3/hr and 5 meters from the predicted final operating point, which could have large consequences on the peak pressures.

In Figure 12 we can see similar trends to those observed for the power curves. At the initial steady state the 4QSSOP curves match the manufacturer’s curve perfectly, while the 4QBEP curves deviate by various amounts. Near the final steady state flow the 4QSSOP curves show significantly more deviation than the 4QBEP curves. This is likely due to the fact that the initial steady state flow is far from the final steady state flow.

The 41.8 4QBEP curve seems to show the least overall deviation from the manufacturer’s curve in terms of both head and power based on these qualitative observations. However, using a 4QBEP curve will produce different steady state results than the manufacturer’s curve, which could be problematic. While the 4QSSOP curves match the steady state results perfectly, the results in the transient would likely provide less trustworthy results based on the large deviations they show from the manufacturer's curve at the final steady state system curve. To test this conclusion let's compare the transient results from all six four quadrant curve scenarios to the transient results using the manufacturer's curve.

Step 5. Compare the Transient Results

We now want to compare the results to see how sensitive the results are to the choice of pump curve used for the simulation.

First go to the Output window for any of the four quadrant scenarios. Note that when any of the four quadrant scenarios are run, several information items will be given in the Warnings tab of the Output window. These messages are meant to inform the user of the potential implications that the four quadrant curves may have for the model, which this example explains in detail.

Now go to the Graph Results window and generate a graph of the pump speed and the velocity at the pump outlet, as can be seen in Figure 13 and Figure 14 below. From Figure 13 we can see that the pump does not reach zero speed during the transient simulation, meaning that the pump does not experience reverse rotation. In Figure 14 the velocity results at the outlet of the pump are compared. Recall from Step 3 that reverse flow occurs at the pump at around 27 seconds, though the magnitude of reverse flow was minor. In addition, during Step 3 we found that negative head rise would occur at the pump starting at 21 seconds. Since the pump operates at positive speed, positive flow and positive head for the first 21 seconds of the transient, the manufacturer’s curve will produce the most accurate results during this time. Therefore, the four quadrant curve results can be compared directly to the manufacturer’s curve results to determine the error in the four quadrant curve results up to 21 seconds into the transient run. After 21 seconds one of the four quadrant curves will likely give more accurate results.

For this case there is a clear difference between the initial steady state velocity (at time zero) using the manufacturer’s curve and the steady-state velocities using the 37.4 4QBEP and 64 4QBEP curves. However, as the tank transition causes the flow to increase at the pump, the velocity results for the 4QBEP curves become closer to the results from the manufacturer’s curve, while the 4QSSOP curves diverge. Notice that at 20 seconds when the pump trip and ESD valve closure transients begin, the 4QSSOP curves now show a large difference from the manufacturer’s curve. The 41.8 4QSSOP curve shows the largest difference from the manufacturer’s curve, with a difference of about 0.7 meters/sec. The initial velocities in Figure 14 should and do match those shown in the Figure 11 (and Table 1) steady-state results.

The static pressure at the ESD valve outlet is shown in Figure 15, with the maximum and minimum pressures detailed in Table 2. We can see that the maximum and minimum pressures at the ESD valve occur once the ESD valve has fully closed after 27 seconds. The 4QSSOP pressure results experience significantly decreased peak pressures and do not experience the low pressure peaks that are seen at about 27 and 28.5 seconds for the other Case 1 scenarios.

For example, consider the 41.8 4QSSOP results, which we noted showed the largest difference in the velocity results when compared to the manufacturer’s curve at time 20 seconds. The 41.8 4QSSOP curve has a maximum pressure of about 12.7 bar, as compared to the 16.2 bar peak pressure for the manufacturer’s curve (See Table 2 below). At about 27 seconds a low pressure of 5.7 bar is calculated using the manufacturer’s curve, while the 41.8 4QSSOP curve remains at a pressure of 10 bar.

This difference in results makes sense based on the large differences from the manufacturer's curve at the final system curve, as were shown in Figure 11. Note for completeness that the Figure 11 final steady state values would compare most precisely to the Impulse results if the transient was run longer and the unplanned pump trip at 20 seconds did not occur. In other words, the waterhammer transients need to all die out for Impulse to find the true, final steady state. We can see in Figure 14 at 20 seconds that the transient velocity has roughly but not completely steadied out.

Overall for Case 1 where negligible reverse flow occurs at the pump, we can see that the large differences observed between the 4QSSOP curves and the manufacturer’s curve at the final steady state resulted in large variation in the results, especially for the pressures at the ESD valve outlet. The differences in the steady state results for the 4QBEP curves did not result in significant error in the transient results for this case due to the fact that the valve closures during the first transient event were not sensitive to the steady state results. In this case one of the 4QBEP curves would be the best choice out of the four quadrant curves analyzed as Case 1 is much more sensitive to the final steady state conditions (i.e., at 20 seconds before the pump trips) than it is to the initial steady state.

Step 6. Case 2 - Tank B to Tank A

Case 1 operated in the normal pumping zone of operation (positive speed, head and flow) for the majority of the run, so four quadrant data was only required for the analysis of the results when the pump experienced negative head. Case 2 and Case 3 may experience more extensive negative flow, speed, and/or head at the pump requiring further use of four quadrant methods.

A. Create Scenarios for Case 2

For Case 2 the system starts with flow to Tank B during the steady state. The flow then transitions from Tank B to Tank A over ten seconds. At twenty seconds into the transient, the pump trips and the ESD valve closes, as occurred in Case 1.

Let's create scenarios for Case 2 and repeat the analysis above. Do the following:

1. Right-click the Case 1 - Tank A to Tank B scenario and select Clone With Children. Rename the scenario to Case 2 - Tank B to Tank A

2. Open the Valve Properties window for Valve A (J5). We will need to set this valve to begin closed, and open over ten seconds.

3. On the Optional tab set the Special Condition to Closed

4. Go to the Transient tab. Under the Transient Data menu select Import from File from the drop-down menu, then browse to 10 Sec Valve Opening.txt in the Impulse Examples folder

5. Select Tab as the Separator, then click OK. The data should now start with Cv = 0 at time 0, and open to Cv = 6000 at 10 seconds

6. Now open the Valve Properties window for Valve B. This valve will need to begin open, then close over 10 seconds.

7. On the Optional tab set the Special Condition to None

8. Go to the Transient tab. Under the Transient Data menu select Import from File from the drop-down menu, then browse to 10 Sec Valve Closing.txt in the Impulse Examples folder

9. Select Tab as the Separator, then click OK. The data should now start with Cv = 60000 at time 0, and close to Cv = 0 at 10 seconds. The Cv vs. time data should be the same as shown earlier in Figure 2 for Case 1, Valve A.

B. Compare Four Quadrant Curves to Manufacturer's Curve

Now that the scenario is fully defined, let's examine the results for the model using the manufacturer's curve.

ØRun the Case 2 - Tank B to Tank A scenario from the shortcut on the toolbar, or from the Analysis menu, and go to the Output window.

There will be several warnings visible for this scenario. A warning is given for reverse flow at the pump. If we were to graph the flow rate at the pump as was done for Case 1 we would see that a significant magnitude of reverse flow occurs towards the end of the transient run, thus it would be wise to use four quadrant data for this case. This is discussed further in Step 6-C.

Warnings are also given that the static pressure is lower than the vapor pressure at several pipe stations. This warning is given because cavitation modeling was disabled in the Transient Control window in order to decrease the run time for this model. When cavitation modeling is disabled the pressure is allowed to drop below an absolute pressure of zero. Though this is not realistic, we will leave cavitation modeling disabled to simplify the model for the purpose of this example – as well as to better be able to see the impact on low pressures from different pump curve choices. This can be done as the cavitation in this model is minor and has a low impact on the pressure results. When modeling an actual system the cavitation modeling should be turned on in Transient Control in order to obtain more realistic results.

ØGo to the Pump Summary tab. For this case the pump is operating at 1559 m3/hr, which is 112% of the BEP (Figure 16).

As was done in Step 4-C for Case 1, let's compare the manufacturer's pump curve to the dimensionalized four quadrant curves. It will again be useful to compare the pump curves to both the initial steady state, and the final steady state for the system after the tank transition and before the pump trips. Note that the final steady state for Case 2 will be equivalent to the initial steady state for Case 1, which means the final steady state operating point for Case 2 occurs at a flow rate of 854 m3/hr.

A multi-scenario pump vs. system curve graph was generated in AFT Fathom for Case 2, as can be seen in Figure 17, with the operating points from Figure 17 shown in Table 3. The power curves for each of the scenarios can be seen in Figure 18.

In this case the pump operates much closer to BEP, resulting in fewer differences between the 4QSSOP curves and the 4QBEP curves. All of the four quadrant pump head curves intersect the initial steady state system curve close to the manufacturer's curve, though the 4QBEP curves do show minor differences in the steady state operating point. At the final system curve there is not a clear trend when comparing the 4QSSOP and 4QBEP curves. Instead the curves are grouped based on the specific speed, rather than dimensional operating point. Both 41.8 curves appear to be closer to the final steady state operating point than the other curves.

Another point of interest is that the 37.4 4QSSOP curve falls below the final system curve at low flowrates. This means that the 37.4 4QSSOP curve predicts the pump provides insufficient head to overcome the system resistance at this time, even though the manufacturer's curve shows otherwise.

The comparison of the power curves again shows similar trends. In this case it is difficult to determine qualitatively which four quadrant data set best matches the manufacturer curve, though the 64 curves clearly show the most deviations.

If looking for the most reliable four quadrant curve in this case, one of the 41.8 data sets should be chosen due to the closer match that the 41.8 curves show at the final steady state for the head and power curves. If the system is sensitive to the initial steady state then the 41.8 4QSSOP curve should be chosen.

C. Examine Transient Results for Case 2

ØRun each of the four quadrant scenarios using the batch run feature from the file menu, then proceed to the Graph Results window.

From the pump speed and velocity graphs in Figure 19 and Figure 20 it can be seen that the pump will operate in the normal pumping zone of operation with both forwards flow and positive speed up until about 10 seconds into the transient. Depending on the pump curve being used, the pump experiences some reverse flow between 10 seconds and 27 seconds, though most (but not all) of the pump curve results return to forward flow before reversing again. From the pump speed curve we can see that all of the four quadrant curves predict that the pump will begin to operate at negative speeds (reverse rotation) between 24 - 26 seconds. This means that for the time period between 24 - 27 seconds the pump will likely operate with both reverse flow and rotation. There is a high level of uncertainty when using the manufacturer's curve due to the magnitude and duration of reverse flow and reverse rotation, so the manufacturer’s curve should not be used as a measure of accuracy once the flow becomes negative (at about 10 seconds), and especially during negative rotation.

Note: When the Standard Pump Curve option is used to model a pump in Impulse, the speed will not be allowed to go below 0%, similar to how the head rise was limited to a minimum of 0 in Case 1 when the SPC was used.

Further examining the velocity at the pump we can see that there is less deviation in the initial velocity for each of the curves. In this case this is due to the pump operating closer to the BEP than Case 1. However, the distribution of velocities is much broader once the tank transition has finished at around 10 seconds, reflecting the broader distribution we observed in the pump head curves. A maximum difference of 1.5 meters/second occurs at 20 seconds into the run between the 64 4QBEP and 37.4 4QSSOP curves. If a different transient than a pump trip was modeled at 20 seconds, for example a near instant valve closure of the ESD valve, then wildly different pressure transients would result.

Figure 21 shows the pressure results at the outlet of the ESD valve for the pump trip case, while Table 4 details the maximum and minimum pressures shown in the graph. The range of peak pressures predicted using the four quadrant curves at 27 seconds has been reduced from about 4.2 bar to 4 bar likely as a result of the four quadrant curves being more similar when the pump operates closer to BEP. The highest peak pressure of 27.1 bar is predicted by the 37.4 4QSSOP curve, and the lowest peak pressure of 23.1 bar is predicted by the 64 4QBEP curve.

Zooming in on the minimum pressures that occur at about 28 seconds and 30 seconds into the run (shown in Figure 22) shows that the 41.8 pump curves and the 64.0 pump curves predict a minimum pressure above the vapor pressure (0.018 bar), while the manufacturer's curve and 37.4 pump curves predict that the pressure will drop to vapor pressure (The Figure 22 results actually shows these pressures as negative absolute pressures because transient cavitation is being neglected here for exploratory purposes. If this were a real analysis, other runs with transient cavitation models enabled would be highly recommended.) This range of pressure results highlights another reason why the choice of four quadrant curve to be used is important in waterhammer analysis, as this result will impact the decision of what protective equipment may be required for the system.

Be aware that the ESD valve pressures do not represent the overall maximum/minimum pressures for the system in this case. Many of the Case 2 scenarios actually reach vapor pressure at the outlet of Valve B, even the scenarios which did not drop to vapor pressure at the ESD valve. In an actual analysis the engineer would need to assess the impact of cavitation on the model and consider mitigation such as alternate valve closure times/profiles or installing protective equipment, though this will not be discussed further in this example.

To be conservative the engineer could simply choose the extreme case, which would be the 37.4 4QSSOP scenario. For higher accuracy results further comparison of the four quadrant data sets to the manufacturer’s curve could be done as was described in Step 6-B. From step 6-B the two 41.8 curves were determined to be the best fit. The 41.8 4QSSOP pump curve may be the better option since it gave more extreme results than the 41.8 4QBEP pump curve in this case.

Step 7. Case 3 - Valve A Fails Open

A. Create Case Scenarios for Case 3

For Case 3, consider the scenario where flow is being transitioned to Tank B, like in Case 1. However, Valve A malfunctions and is stuck fully open so that Valve A and Valve B are fully open. In this case the elevation differences between Tank A and Tank B would cause Tank A to begin draining. To prevent Tank A from fully draining both the ESD valve and Valve B will be closed, and the pump will be intentionally tripped at the same time to bring the system to a safe stop so the Valve A malfunction can be investigated.

Make the following changes in the model:

1. In the Scenario Manager right-click on the Case 2 - Tank B to Tank A scenario and choose Clone With Children. Rename the scenario Case 3 - Valve A Fails.

2. Open the Properties window for Pump J2 and go to the Transient Tab. Under Initiation of Transient change the Time Absolute to 0 seconds so that the pump trip will begin at the beginning of the run.

3. Repeat step 2 for the ESD Valve, junction J3.

4. Open the Valve Properties window for Valve A, junction J5. This valve will need to be set to be open, and remain open throughout the transient.

5. On the Optional Tab, set the Special Condition to None.

6. On the Transient tab set the Transient Special Condition to Ignore Transient Data. This will allow us to keep the transient data entered, but prevent the transient from being used for this scenario.

B. Compare four quadrant curves to manufacturer's curve

Now that the scenario is fully defined, let's examine the results for the model using the manufacturer's curve.

ØRun the Case 3 - Valve A Fails scenario from the shortcut on the toolbar, or from the Analysis menu, and go to the Output window.

Once more we will see a warning indicating that the pump did experience reverse flow, which suggests a four quadrant curve should be used. Warnings will also be given for the transient static pressure, as were discussed for Case 2. Lastly, cautions will be given for reverse flow through junctions 5 and 7. These cautions occur since the loss factors through these junctions were entered assuming the losses were for forwards flow through the junctions, and may be inaccurate when reverse flow occurs. We will assume the difference is negligible in this case, though this should be verified for actual analysis.

In the Pump Summary at the top of the window, we can see that the pump operates at 101.2% of BEP.

In the Pipes section of the Output click the Pipes tab to view the steady state results for the pipes. Here we can see that pipes P4 through P6 experience reverse flow at the beginning of the run when both Valve A and Valve B are open, as is expected.

For this scenario the final steady state after the transient is when the pump has tripped and both Valve A and Valve B are closed. This is not meaningful for comparison as all flows will be zero, so the four quadrant pump head curves were simply compared to the initial steady state for the system, as is shown in Figure 23. The power curves can be seen in Figure 24.

As expected the 4QBEP and 4QSSOP curves for each value of specific speed are identical when operating at BEP, so we essentially only need to compare the manufacturer’s curve to 3 four quadrant curves. Since the BEP stays the same in all scenarios, the BEP curves are the same as in the other cases. Thus our conclusions are similar to the conclusions from Case 2. It is clear that the head and power curves for the 37.4 and 41.8 data sets resemble the manufacturer’s curves better than the 64.0 head and power curve. Hence more weight should be given to the results from the 37.4 and 41.8 curves than the 64.0 curve. Between the 37.4 and 41.8 curve, the 37.4 curve seems closer to the manufacturer curve than the 41.8 curve in head, but the 41.8 curve is a better match for the power curve. Since the 37.4 curve is Preferred while the 41.8 curve is Average, the 37.4 curve may be a better choice, even though the qualitative comparison showed a similar match for the curves.

C. Examine transient results for Case 3

From Figure 25 and Figure 26 below it is clear that the four quadrant results diverge from the manufacturer's curve due to backwards flow beginning at the pump at about 3 seconds, and reverse rotation beginning at around 6 - 7 seconds, depending on the four quadrant curve. In Figure 27 we can see a comparison of the pressure results at the outlet of the ESD valve.

While the pump speed results show a significant difference between the scenarios, the velocity results at the pump and the pressure results at the outlet of the ESD valve show relatively minor differences between the four quadrant scenarios. The 37.5 pump curve produces the highest magnitude of reverse velocity out of the four quadrant curves with a minimum velocity of -1.0 meters/second (i.e. a maximum reverse velocity of positive 1.0 meters/second). This larger reverse velocity translates to a slightly larger peak pressure.

When the manufacturer’s curve results are compared to the four quadrant curves, we can see that the difference in reverse velocity is between 0.15 to 0.2 meters/sec, depending on which four quadrant results are being compared to the manufacturer’s curve results. This difference of velocity is reflected in a 1.2 bar difference from the maximum and minimum pressures predicted by the four quadrant curves.

As with Case 2 it should be noted that the ESD valve does not represent the overall maximum/minimum pressures for this model, which instead occur at Valve B. Cavitation is predicted at Valve B, so the engineer would be required to address the low pressures at Valve B based on the requirements for this system.

Conclusion

In this example three different pump trip scenarios were analyzed for a water transfer system in which there was potential for reverse flow and reverse rotation to occur at the pump. Four quadrant data was considered for use to accurately account for this potential reverse flow/rotation.

In situations such as with Case 1 the reverse flow was negligible, but the pump experienced negative head, requiring the engineer to consider four quadrant data. The manufacturer’s curve was considered most accurate up until 21 seconds into the run when negative head was calculated at the pump.

In Case 2 and Case 3 where the reverse flow and reverse rotation was significant in duration and magnitude, the four quadrant curves allowed for more accurate results to be found. By comparing the pump head and power curves to the curve data from the manufacturer four quadrant data sets could be selected for the analysis based on the initial steady state, and final steady state where applicable.

From the results above it is clear that the choice of four quadrant data set and dimensional reference point can have a large impact on transient results. Note that while a pump operating closer to BEP will typically be less impacted by the choice of dimensional reference point, this will not always be the case. Performing a sensitivity analysis such as what was described in this example will allow the engineer to determine the impact that using different four quadrant data sets and dimensional reference points will have on the results of the transient simulation.