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Flow Rate and Volume Estimates from Variable Frequency Drive Operated Drainage Sump Pumps
Emily P. Nelson1, Thomas F. Scherer1, Xinhua Jia1,*
Published in Applied Engineering in Agriculture 40(1): 51-67 (doi: 10.13031/aea.15790). Copyright 2024 American Society of Agricultural and Biological Engineers.
1 Agricultural and Biosystems Engineering, North Dakota State University, Fargo, North Dakota, USA.
* Correspondence: xinhua.jia@ndsu.edu
The authors have paid for open access for this article. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License https://creative commons.org/licenses/by-nc-nd/4.0/
Submitted for review on 23 August 2023 as manuscript number NRES; approved for publication as a Research Article by Associate Editor Dr. Vivek Sharma and Community Editor Dr. Kati Migliaccio of the Natural Resources & Environmental Systems Community of ASABE on 28 November 2023.
Highlights
Abstract. Subsurface drainage plays a crucial role in excess water removal through perforated pipes buried in the soil. The Red River Valley of eastern North Dakota and west central Minnesota often requires pumped drainage outlets due to its flat topography, with many modern systems incorporating variable frequency drives (VFD) for improved efficiency. VFDs adjust pump speeds to match drainage needs, but this dynamic operation complicates accurate flow rate estimation for edge-of-field monitoring. This project aimed to devise an automated method for calculating flow rate estimates from VFD pumping systems involving tank geometry, operational water levels, and pump duty cycling. Linear regressions with calibration data from a transit-time ultrasonic flowmeter produced mixed outcomes. Some linear regressions revealed errors in calculated versus measured flow rate of =14%, while other regressions showed errors nearing 40%. To address this, future efforts should explore methods incorporating the pump’s electrical characteristics. This approach would provide a more accurate representation of the unstable and intricate pumping operations during transient conditions.
Agricultural drainage systems are very common throughout many regions of the United States and are used to minimize losses to crop yields and reduce field operation delays due to saturated conditions in soils with poor natural drainage, high water tables, or shallow claypan layers. Agricultural drainage alleviates saturated conditions by enhancing the soil’s infiltration capacity and percolation rate (Skaggs et al., 1994; Robinson and Rycroft, 1999) to reduce the water table elevation (Waller and Yitayew, 2015; Ghane and Askar, 2021). This increase in water flow through the soil can also amend crop-damaging salinization of the root zone in some soils (Rhoades, 1974).
While drainage can solve the issues of saturated and salinized soils in the region to improve crop production, it also increases the pollutants being expelled to the environment from arable lands (Baker et al., 1975; Bottcher et al., 1981; Gilliam et al., 1999; Skaggs et al., 2006; Ghane et al., 2016). Nitrogen and phosphorous are of special interest (Castellano et al., 2019; Almen et al., 2021; Maas et al., 2022; Dialameh and Ghane, 2023) due to their impact on harmful eutrophication in downstream waters (Water Resources Mission Area, 2019).
It is important to evaluate the effectiveness of agricultural practices, with the first step to determine the chemical loads leaving these systems through edge-of-field monitoring, while the flow rate and drainage effluent volumes must first be known (Dialameh and Ghane, 2022). Generally, these drainage systems are composed of numerous lateral pipes that route water to the main pipe(s) from which the drainage water moves to the outlet. The outlet of the system can either operate under simple, gravity drainage or incorporate a pumped pump station. In areas with little topographic relief, such as the Red River Basin (RRB), the elevation of the outlet will likely be below the base level of the nearby drainage ditches. In these cases, gravity drainage will not suffice, and a pump station must be constructed (ASABE Standards, 2019a; Madramootoo, 2015).
These pump stations are composed of one to three submersible pumps at the bottom of a tank that remove the water from the tank via a pipe into a nearby drainage ditch or storage pond. The volumetric capacity of the tank and the size of the pump(s) will vary from system to system based on the drainage area, drainage coefficient, and resultant drainage rates. The tank must be large enough to hold the drainage water leaving the field during peak flows, yet small enough to have depth changes that will not lead to more than 10 pumping cycles per hour (ASABE Standards, 2019b).
The pump should be selected based on the type of pump (propeller, mixed flow, centrifugal, etc.) and the power of the pump. For drainage applications, axial flow (propeller) pumps generally work the best. To determine the pump’s required power, the drainage system’s drainage coefficient, the acreage being drained, and the head the water needs to be lifted are used (ASABE Standards, 2019b). A pump performance curve can be used to relate the drainage flow rate and the total dynamic head to the required power of the pump.
In a study conducted by Scherer and Jia (2010), a simple method of calculating the flow rate leaving an agricultural pump station through measures of tank geometry, tank water level, and pump duty cycle duration was developed. However, this study utilized a pump station that operated a single-speed, axial flow pump with simple float controls. When a single-speed pump is used, it can be assumed that it will always be operating at its maximum speed. This site has a much simpler setup than current practices in the RRB that use three-phase motors controlled with variable frequency drives (VFDs) that alter the speed of the pump to match the system’s drainage demands.
Since the power of the selected pump in these pump stations is based on the maximum drainage flows, the pumps’ power is oversized much of the time, during periods of moderate to low flows. This raises an issue with the amount of on/off cycling (duty cycling) the pump’s motor will undergo. When the pump’s power is too great for the ongoing drainage flows, the pump will cycle rapidly, leading to excessive wear on the motor and a decreased lifespan. Since three-phase motors can handle more frequent duty cycling, they are better suited for drainage applications. Additionally, by varying the operating speed of the motor to better match the drainage demands, the motor will undergo less frequent cycling. This can be achieved by directly supplying three-phase input power or by altering a single-phase power supply with a VFD or a rotary of static phase converters (Kaiser et al., 2008; Thiruchelvam and Hong, 2017; ASABE Standards, 2019b). In the RRB region, the use of VFDs in pump stations for agricultural drainage has become a very common practice.
The most common type of VFDs used in industry, and the type used at this research site, are pulse width modulated (PWM) inverter-type VFDs. This type of VFD is composed of three main components: a rectifier, a DC bus, and an inverter (Kaiser et al., 2008; Blair, 2017). The rectifier is used to convert AC power to DC power. Then, this DC power is run through a DC bus containing a capacitor that filters and smooths the signal. Finally, the smoothed DC signal goes through the inverter to convert the signal to a three-phase AC power supply, generally through a series of transistors (Kilowatt Classroom, 2003; Kaiser et al., 2008; Blair, 2017).
The output voltage and frequency of the three-phase AC power supply are controlled by the pulses of the transistors, such that the longer a transistor is closed, the higher the signal amplitude (voltage) and the more rapid the switches the higher the output signal’s frequency (Kilowatt Classroom, 2003; Kaiser et al., 2008). Within an inductor motor, there is a constant relationship between the voltage and the frequency up to a frequency of 60 Hz, meaning that the voltage of the signal will increase linearly as the frequency of the signal increases up to that 60 Hz threshold (Kilowatt Classroom, 2003).
The frequency of the signal leaving the VFD is determined based on the drainage demands at a given time. A pressure transducer measuring the water level in the tank indicates the present pumping demand and controls the output from the VFD to vary the pump’s operating speed accordingly (Scherer, 2022). Essentially, these drainage systems have two distinct VFD modes: continuous operation and on/off cycling. In cases where drainage demand requires a VFD output signal frequency =40 Hz, the pump operates continuously. Conversely, when the VFD delivers a signal frequency <40 Hz, the pump initiates a cyclic pattern of turning on and off (T. F. Scherer, personal communication, 22 June 2023).
There are many benefits from incorporating a VFD in a pumping system with multi-rate applications, including energy savings, improved pumping efficiency, power use efficiency, and increased system longevity. All four benefits stem from the reduction in pumping power under low-flow conditions to avoid partial flow in the pump that can cause cavitation or mechanical failure (Kaiser et al., 2008; Thiruchelvam and Hong, 2017). Another benefit of utilizing VFDs within agricultural drainage pump stations is that a VFD can take either three- or single-phase power and produce a variable three-phase output. This allows cheaper and more readily available, single-phase supply power to be used to operate a three-phase motor which will be more efficient, more affordable, and capable of handling more frequent duty cycling (Scherer, 2022).
It is challenging to determine the flow from a VFD-operated agricultural drainage pump station because many common methods developed for flow rate measurement do not lend themselves easily or accurately to these applications, making the development of new approaches an important objective. Few studies have determined an accurate method of flow rate calculation from these agricultural pump stations (Scherer and Jia, 2010), and, only one has done this for varied frequency systems (Leonow and Mönnigmann, 2013).
The pump motor’s operational speed is directly linked to the flow rate exiting the pump. This flow rate serves as the initial factor in gauging the drainage system’s reaction time to rainfall or irrigation incidents. Furthermore, it aids in evaluating existing drainage system design standards, its effectiveness, ascertaining total drainage effluents volumes and calculating chemical loads that exit crop fields via the constructed drainage pathways.
There are numerous applications for this research, and, as drainage is becoming more important for sustainable agricultural intensification (Castellano et al., 2019), being able to calculate accurate flow rate for edge-of-field monitoring is more important than ever. Through similar methods to Scherer and Jia (2010) and Kanwar et al. (1999), a method of automated calculation of the flow rate from VFD-operated agricultural drainage pump systems will be developed. To achieve this research objective, the following tasks are performed to achieve it: (1) develop a method of flow rate calculation, (2) automate this calculation through MATLAB coding, and (3) calibrate and validate the automated calculation through field experiments and statistical analyses.
Materials and Methods
Site Description
The field site is located in Wyndmere Township, Richland County, North Dakota at 46°15'11.28''N, 97°11'50.28''W. This field has 56.3 ha (139 acres) drained at 18.3 m (60 ft) spacings with a drainage coefficient of 0.95 cm/d (3/8 in./d). The major soils at this site are the Forman-Lanka-Tonka complex and the Lankin-Gardena-Tonka complex which, together, compose 91.1% of the field. These complex soils have an overall loamy texture, with isolated areas ranging from clay loams, silt loams, and loams. The remaining 8.9% of the field lies in Mustinka silty clay loam. The Mustinka silty clay loam is a poorly drained soil, while both complex soils in the field are moderately well to well drained. Several freshwater emergent wetlands cover 2.5 ha (6.16 acres) of the field (Ellingson Companies, 2024). Figure 1 depicts the drainage system underlying the research site. This site has a pump station outlet with a single 230 V, 5.6 kW (7.5 hp) axial flow pump (CP06-0075-233-08, Carry Pumps Inc., Caro, Mich.). The pump is operated with a Cerus Titan P series VFD which takes an incoming 230 V single-phase signal and converts it to a 230 V three-phase signal (CI-007-P4, Franklin Electric Cerus, Fort Wayne, Ind.).
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| Figure 1. Map of the subsurface drainage system installed at the Wyndmere site. |
The pump station tank is a 1.2 m (4 ft) diameter corrugated metal culvert that is 4.4 m (14.3 ft) in total depth, and the outlet is a 20.3 cm (8 in.) diameter PVC pipe with a bell end. A metal door in the tank cover allows for easy access when necessary. The riser pipe from the pump is 20.3 cm in diameter. The invert of the 30.5 cm (12 in.) dual-wall tile is 3.2 m (10.5 ft) below the top of the tank. The base of the elbow of the outlet pipe rises 10.2 cm (4 in.) above the top of the tank. Figure 2 is a schematic of the pump station at this site.
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| Figure 2. Schematic diagram of the pump station located at this site. The red lines indicate the on and off water levels for the pump. This image is made roughly to scale. |
Data Collection
The pump station is outfitted with a SignalFire wireless telemetry system (RANGER) that records data every minute and sends it to a cloud-based data server. These data are available through remote access for up to two weeks. It includes values of the water level in the tank, the pump’s operating voltage, frequency, and current, and the water levels at which the pump will turn on and off among others.
The system is set up so that the pump will be turned on when the water level reaches 0.8 m (2.6 ft) above the top of the pump and turned off when the water level is 0.4 m (1.3 ft) above the pump. The water level is measured by a pressure transducer located in a PVC pipe along the edge of the tank that is cut to correspond to the top of the pump (fig. 2). This is considered the operational range of the pump, and it will be used later in the calculations section.
The duration of the pumping cycles will vary, with some as short as 20 s. Since the remote monitoring system can only record values every 60 s, some of these cycles are not captured in the data. To fix this issue, a split-core AC current switch sensor (CTV-A, HOBO Onset, Bourne, Mass.) was installed on the power supply line to the VFD and connected to a data logger (U9-001 State Logger, HOBO Onset, Bourne, Mass.) to record the timestamp of each start-up and shutdown of the pump. This sensor and logger pair is rated for 2-20 A with a response time of 440 milliseconds. The background power drawn from the VFD when the pump is not operating is 0.2 A which is well below the threshold at which the sensor will register the pump being on (2 A). When the pump is running, the VFD draws around 14.1 A, which is within the measurable range of the sensor. This allows each cycle to be isolated with high accuracy (0.44 s). Figure 3 shows the electronic components of the pump station.
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| Figure 3. Image of the VFD (indicated by the rectangle), pump control panel (indicated by the oval), HOBO current sensor and state logger (indicated by the circle), and SignalFire RANGER telemetry device (indicated by the trapezoid and labeled) held within the control box of the site’s pump station. |
Calculations
The tank’s storage requirement is calculated from equation 1 (ASABE Standards, 2019b):
(1)
where
S = storage (m3)
C = 0.90 (SI units) or 2 (English units)
Q = flow rate or pumping capacity (L/s)
N = the number of pump duty cycles per hour.
In most VFD-operated pump stations, a pressure transducer is used to control the frequency of the VFD’s output signal and trigger the pump. The variation in the operating speed of the pump makes it difficult to accurately measure the flow rate leaving the system. However, using measures of the ‘on’ and ‘off’ water levels set for the system, the diameter of the tank, and the volume of water held in the vertical section of the discharge pipe (riser pipe), the change in volume for each pumping cycle can be estimated. Then, this volume change, paired with the duration over which this change occurred, can be used to determine the flow rate leaving the pump. This is the approach tested in this study.
A full duty cycle of the pump is defined as one complete ‘off’ and ‘on’ period of the pump. A pumping cycle can start either when the pump turns on or off; in this study, it was decided to start the pumping cycle when the pump turns off as this portion of the duty cycle has a greater variation in length ranging from less than a minute to days:
(2)
where
TDC = duration of the duty cycle (min)
Toff = duration over which the pump was off (min)
Ton = duration over which the pump was on (min).
There are two separate flow rates that must be determined for this system, the flow rate leaving the drainage system and the flow rate leaving the pump, to develop this method. The flow rate leaving the drainage system occurs over the entire duty cycle, but it is calculated over the portion of the duty cycle where the pump is off. To determine this flow rate, the storage volume of the tank, the volume of water held in the riser pipe, and the duration of time that the pump is off for a given duty cycle must be determined (eqs. 3-6).
(3)
(4)
(5)
(6)
where
Vs = storage volume of the tank (m3)
Won = water level at which the pump will turn on (m)
Woff = water level at which the pump will turn off (m)
Atank = cross-sectional area of the cylindrical tank (m2)
Vpipe = volume of the riser pipe (m3)
Apipe = cross-sectional area of the riser pipe (m2)
Vin = volume entering the sump tank each cycle (m3)
hpipe = height of the riser pipe between its invert and the
‘off’ level for the pump (m).
Then, to determine the flow rate of the drainage waters equation 7 is used:
(7)
where
Qin = flow rate of water leaving the drainage system and
entering the sump tank (m3/min).
Generally, these systems do not incorporate check valves, so the water in the riser pipe will flow back into the tank once the pump shuts off. For this reason, the volume of water held in the riser pipe must be subtracted from the storage volume of the tank.
Now, looking at the flow rate of water leaving the pump, we use the same volumes calculated in equations 3 and 4; however, now the portion of the duty cycle when the pump is ‘on’ is considered:
(8)
To determine the flow rate from the pump, equation 9 is used. Because water does not stop draining from the field while the pump is operating, the drainage flow rate calculated in equation 7 needs to be incorporated into the pumping flow rate:
(9)
where
Qout = flow rate of water leaving the pump (m3/min).
The pump station at the Wyndmere site used for this study has pump ‘on’ and ‘off’ levels set at 0.8 and 0.4 m (2.6 and 1.3 ft), respectively, making the systematic change in water level over each pumping cycle 0.4 m (1.3 ft). The diameter of the tank at this site is 1.2 m (4 ft), so the cross-sectional area of the tank is 1.2 m2 (12.6 ft2). The riser pipe is 20.3 cm (8 in.) in diameter, so the cross-sectional area of the riser pipe is 0.03 m2 (0.4 ft2). The invert is 3.3 m (10.8 ft) above the pump’s ‘off’ level. Using these values, the storage volume of the tank and the volume of water held in the riser pipe can be calculated using equations 3 and 4:
(10)
(11)
The final step in determining the flow rates is to use the timestamp data collected with the current sensor and state logger to isolate the timing of both the ‘on’ and ‘off’ portions of each duty cycle. Using these timestamps, the duration of each cycle can be determined down to the second and then converted back to minutes for use in the flow rate calculations outlined in equations 7 and 9. The duration should be in minutes because the most common unit for flow rate used by industry workers and farmers in this area is cubic meters per minute (gallons per minute, gpm).
Another important metric to calculate from this system is the total volume of drainage water since this value is important for understanding how much water is being removed from the agricultural field. To determine the total volume of water leaving the drainage system, Vin is multiplied by the total number of cycles that occurred over the period of interest and added to the product of the sum of the total time that the pump ran and the flow rate leaving the drainage system (eq. 12). Then, for the volume of water being pumped from the pump station, multiply the flow rate of water being pumped out for each cycle by the duration of the pumping period (Ton) for each corresponding cycle to get volumes for individual pumping cycles and then sum those values (eq. 14). The value for the volume of water being pumped out of the tank should ideally match the value of the volume of water being drained from the field over the period of interest.
(12)
(13)
(14)
where
Vin,total = total water leaving the field and entering the sump
tank (m3)
Ncycles = number of cycles occurring over the period of
interest
Vout = volume of water being pumped for a given cycle
(m3)
Vout,total = total volume of water being pumped over the
entire time period (m3).
Calculation Automation
Once the method of calculation was determined, the next objective was to automate the process through MATLAB coding. Kanwar et al. (1999) automated a similar calculation using the Basic program. If small datasets are being measured, manual calculations are easy; however, to determine trends in agricultural drainage flow, long-term studies must be conducted and robust data sets must be collected, processed, and analyzed. This necessitates an automated calculation process.
Code Setup and Initial Processing
The first step in creating the code is to upload the necessary data files. In this case, the data from the state event logger and the SignalFire cloud need to be imported. The values included in these two datasets are the pump state (on or off) and the measured water level in the tank, respectively. With the data files imported into MATLAB, a quick data plot showing the tank level versus time with an overlay of the pump state versus time can be made to check that the two datasets are correlated (fig. 4).
Next, an index for when the pump is on and when the pump is off needs to be created. This was accomplished using MATLAB’s ‘find’ function. Then, using these indices, the timestamp for each start-up and shut-off of the pump can be isolated and stored in an array. These indices should be added to the data plot previously created to ensure that the proper times are being isolated. In figure 4, the timestamps are indicated by the pink and black asterisks.
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| Figure 4. An example of the quick data plot showing the tank’s water level in feet, the pump’s state where ‘1’ is on and ‘0’ is off, and the indices for the timing of pump start-ups and shut-offs determined through the code setup and data processing steps. |
Drainage & Pumping Cycle Determinations
To determine the duration of both portions of the duty cycles, the ‘on’ and ‘off’ indices are used. The duration over which the pump was off (the drainage cycle) is calculated by subtracting the time at which the pump turned off from the succeeding time at which the pump turned on. To calculate the duration over which the pump was on (the pumping cycle), the time of the pump shut-off was taken and the time of the preceding pump start-up was subtracted. In MATLAB, the ‘seconds’ function was used to convert the timestamp data from the event logger to a usable data form to perform the arithmetic; then, the durations were converted from seconds to minutes and stored in an array.
Drainage Flow Rates
Using equations 3 and 4, the storage volumes of both the sump tank and the riser pipe were calculated based on site-specific measurements. The next step is to determine the inflow volume (Vin) and use that volume multiplied by the array of drainage cycle durations calculated from the pump’s state data to determine the flow rate of water leaving the drainage system for each cycle (eq. 7). Once the flow rate of the drainage system is calculated, the average flow rate and total effluent volume for the period of data collection can be determined through equations 12 and 15 where Qin,avg is in cubic meters per minute.
(15)
Pumping Flow Rates
Equation 9, the results from equation 7, and the array of pumping cycle durations can be used to calculate the flow rate of water leaving the pump for each pumping cycle occurring over the period being considered. Then, to determine the average flow rate, the outflow volume for each cycle, and the total outflow volume for the entire period, equations 13, 14, and 16 are used where Qout,avg is in cubic meters per minute. The pumped volumes should match the drained volumes for the system.
(16)
Quality Assurance Check
To check if the code was working properly, a quick check was conducted by calculating the percent change (eq. 17) between the inflow and outflow volumes for individual cycles as well as the total inflow and outflow volumes over the data collection period. The volumes pumped from the pump station should match within 2% or less of the volume drained from the field.
(17)
Calibration and Validation
The last step in the research is to calibrate and validate the method through field experiments. To run the calibration and validation experiments, an ultrasonic flowmeter will be utilized to measure the flow of multiple pumping cycles (RašicAmon et al., 2021). Data collected through SignalFire and with the current sensor and state logger pair for the same pumping cycles will be run through the MATLAB code. The calculated values from the code will be statistically compared with the physical field measurements collected with the flowmeter.
Since agricultural drainage waters generally have low TDS, the flowmeter used in this study was a transit-time type of ultrasonic flowmeter [Transport PT868 Portable Liquid Flowmeter, Panametrics (Baker Hughes), Houston, Tex.]. This flowmeter has an accuracy ranging from 0.5% to 2% of the reading typically; however, this is dependent upon proper installation and optimum conditions. The turn-down ratio for this meter is 400:1 when installed on pipes ranging in size from 12.7 mm to 5 m (0.5 in. to 16 ft) and measuring fluids at temperatures between -20°C and 260°C (-4°F and 500°F) (Panametrics, Inc., 2001a,b).
One complication of using flowmeters on drainage outlets is that the pipe is rarely full-flowing. To try and remedy this issue, an elbow fitting was constructed to slide into the outlet to force a full-flowing pipe by raising the elevation of the outlet above the horizontal section of the pipe. However, there was still a high air content in the water with this fitting, and it caused a small amount of siphoning (Azam et al., 2016; Zak’s Lab, 2021) to occur when the pump would shut down. Next, we tried adding an additional 1.4 m (55 in.) vertical section of pipe in the elbow fitting: this caused significant siphoning that lead to large errors (discussed in-depth in the Discussion section). It was decided to place the flowmeter transducers on the riser pipe instead (fig. 5). This allowed the pump station to operate as normal with no interference from the elbow fitting and avoided air bubbles that were incorporated by turbulent flow induced by the first elbow in the outlet pipe.
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| Figure 5. Images depicting the three different methods used in the calibration experiments. |
As mentioned above, three different methods of flow rate measurement with the ultrasonic flowmeter were tested for the calibration experiments. The first method (Small Elbow Method) inserted the special elbow fitting into the existing bell-end outlet and placed the flowmeter transducers on the horizontal section of the outlet pipe (fig. 5b). The second method (Large Elbow Method) was to use both the elbow outlet fitting and an additional vertical section of pipe to raise the elevation of the outlet even higher with the flowmeter transducers on the horizontal outlet pipe (fig. 5c). Then, the third method (No Elbow Method) placed the flowmeter transducers on the vertical, riser pipe and omitted the use of any outlet fitting (fig. 5a). The results of each data collection method are discussed in the next section.
The accuracy of the flowmeter was tested in the lab prior to and after the calibration experiments to ensure that the flowmeter was operational. In both laboratory tests, a 5.1 cm (2 in.) pipe was set up with a garden hose connected to one end and an elbow fitting on the other to force a full-flowing pipe. Then, following the setup and programming of the flowmeter, the hose was turned on. The flow meter measurements were compared with the traditional bucket method, where the time to fill an 18.9 L (5 gal) bucket was recorded with a stopwatch. The flow rate was then determined in liter per minute to compare to the flowmeter. In every instance, the flowmeter matched the rate determined with the bucket within = 2.7 L/min (0.7 gpm).
To start the calibration experiment in each case, the flowmeter must be programmed properly. The thickness of the PVC outlet pipe and its outside diameter were determined as 0.8 and 21.9 cm (0.3 and 8.6 in.), respectively, making the inside diameter 20.3 cm (7.89 in.). These are the values used to program the flowmeter for the system it is measuring. Additionally, we assumed that the water was at a temperature of 10°C (50°F) since it was fresh groundwater. With the inputs listed above, the flowmeter handheld calculated that the transducers should be placed 18.2 cm (72 in.) apart for a 2-crossing measurement (V-mounting). We then clamped the transducers onto the pipe with the metal frame after coating them with a thin layer of coolant to ensure that a good seal was made between the transducer and the pipe.
Once the flowmeter was programmed and installed on the pipe, it was ready to collect measurements. Manual readings were recorded for these calibration experiments. Since the VFD-based systems vary the pump’s operating speed throughout a pumping cycle, multiple readings were recorded for each pumping cycle. A reading was recorded when the flowmeter value would stabilize for 3 s. Then, the recorded flow rate values for a given pumping cycle were averaged, and that average value was used to compare to the code-calculated flow rate estimation.
After recording the flowmeter measurements, the flowmeter was turned off, removed from the pipe, and put away. Then, the outlet fitting, if used, was removed from the outlet, and, finally, the data from the state logger was downloaded. Upon returning to the lab, the water level data for the tank was downloaded from SignalFire. Then, the data was run through the MATLAB algorithm, and the flow rate estimations calculated in MATLAB were compared to the flowmeter measurements recorded for the corresponding pumping cycles.
Statistical Analysis
The statistical analyses run to check the accuracy and validity of the algorithm were percent error, linear regression, and analysis of variance (ANOVA). The percent change calculations are used to determine the accuracy of the algorithm estimations in comparison to the flowmeter measurements. These error percentages were calculated in the main MATLAB algorithm and plotted as part of the results. Then, the linear regression and ANOVA analyses were run in a separate MATLAB script. These results were used to determine if the errors in the algorithm were due to a systematic error that could be corrected for to improve the results. To determine whether a strong correlation exists between the two datasets, an R2 value of 0.75 or greater. Then, in order to reject the null hypothesis and support a true (not random) correlation exists, the ANOVA must yield a p-value of =0.05.
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| Figure 6. Results of the first calibration experiment that took place at the Wyndmere site on 27 April 2023. |
Results
Calibration Experiment 1: Wyndmere 27 April 2023
The first calibration experiment was conducted on 27 April 2023. The “Small Elbow Method” was used to collect the calibration data on this date. The code quality assurance check shows that the code was working properly. The trends in the drainage rates and volumes are matching with the trends in the pumped rates and volumes (figs. 6a and 6b) and there is no difference between the drained and pumped volumes from the system (fig. 6c).
The results of the calibration experiment conducted on this date were promising (fig. 6d). Only two of the recorded cycles were above a 15% difference between the flowmeter measurement and the code-calculated estimate. These larger errors were thought to be the result of a high gas volume fraction in the discharge leading to inaccurate flowmeter readings. The increased outlet elevation from the elbow fitting did also create a minor siphoning effect at pump shutdown. The siphoning effect was not thought to be a major issue at this point; rather, two-phase flow was thought to be the main cause of the higher errors.
Calibration Experiment 2: Wyndmere 8 May 2023
The second calibration experiment was conducted on 8 May 2023. Since it was thought that the high gas volume fraction in the discharge was the cause of the larger errors seen in the first calibration, the “Large Elbow Method” was employed on this date to try and decrease the amount of air in the discharge waters. The vertical section of PVC placed above the elbow was 1.4 m (4.6 ft). See figure 5b for the calibration setup. The vertical section of the pipe was very effective in decreasing air content in the discharge; however, it led to a significant siphoning effect at pump shutdown. The large volume of water that was siphoned back into the sump tank with this setup is likely the cause of the inverse relationship between the drainage flow rates and pumping flow rates seen in figure 7b. Due to the additional backflow into the tank after a pumping cycle, the time required to fill the sump tank to the pump start-up level decreases. This artificially makes the drainage flow rates seem higher than they are and the pumping rates lower.
While the water volumes being drained from the field match the water volumes being pumped, we do see an inconsistency in the code-calculated flow rates for the drainage system and pumping system as aforementioned. The errors between the flow meter measurements and the code estimations were much larger for this experiment than the first. They are consistently around 50% (fig. 7d) for each cycle. This time the code estimations were under-predicting the flow rate, whereas the first experiment’s results showed an over-prediction from the code. This indicated that the siphoning effect arising from the use of outlet fittings played a more significant role in the error between the code estimations and the flowmeter measurements than originally thought. The increase in head requirement from the additional vertical section noticeably decreased the observed flow rates, which also indicates that the “Large Elbow Method” setup was inhibiting normal pumping operations.
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| Figure 7. Results of the second calibration experiment that took place at the Wyndmere site on 8 May 2023. |
Additionally, two pumping cycles had to be removed from the calibration dataset (fig. 7d) because the pump state data showed an additional duty cycle that was not observed in the field. Due to this anomaly in the data, these two cycles were omitted from the analysis.
Calibration Experiment 3: Wyndmere 11 May 2023
The third calibration experiment took place on 11 May 2023. Because the results of the first two calibration experiments showed contradictory results, the third experiment tested all three methods of calibration. The “Small Elbow Method” was the first method tested. The quality assurance check shows that the code is operating as written. The errors on this collection date are consistently higher [20% to 30% (fig. 8d)] than the errors for the data set collected with the “Small Elbow Method” on 27 April 2023 (6% to 21%). The errors are also showing that the code is underestimating the flow rate compared with the flowmeter this time, whereas the code was overestimating the flow rate on 27 April 2023.
Then, the “No Elbow Method” was tested where the transducers were placed on the riser pipe (vertical section of pipe going upward from the pipe in the tank) without any fitting on the outlet. It was hypothesized that this would allow the system to operate as normal and allow for improved results. The results show errors ranging from roughly 2% to 13.8% with the code overestimating the flow rate compared with the flowmeter measurements (fig. 9d). While collecting the data, the transducers on the riser pipe also produced steadier readings from the flowmeter.
To try to enforce the idea that both the “Small Elbow Method” and the “Large Elbow Method” were inducing larger calibration errors by creating a siphoning effect, a final data set was collected on this date where the elbow fitting was inserted into the outlet again while the transducers stayed on the riser pipe. The results of this dataset show that the errors between the code estimations and the flowmeter measurements were in the 20% to 30% range again (fig. 10d) similar to the results of the data collected with the “Small Elbow Method” on this date. The code does show an overestimation in one data set and an underestimation in the other.
These results indicate that when the fitting(s) are in the outlet, it forces water to be siphoned back into the tank when the pump shuts off and that this volume change causes inconsistent and larger errors between the code and flowmeter flow rates. However, it is hard to compensate for this siphoned volume as it is not a consistent volume for each cycle. This will be discussed in-depth in the Discussion section.
Additionally, in the dataset collected on 8 May 2023 at Wyndmere where we were able to use both the elbow fitting and the vertical section of pipe on the outlet to collect data, we see that the underestimation from the code was even larger (51% to 58%) than when just the elbow fitting is used. This enforces the idea that the siphoning effect caused by the fittings is causing larger errors.
 |
| Figure 8. Results of the third calibration experiment that took place at the Wyndmere site on 11 May 2023. This set of results from the third calibration was collected using the elbow fitting with the transducers situated on the outlet pipe about 45° from the top of the pipe (Small Elbow Method). |
 |
| Figure 9. Results of the third calibration experiment that took place at the Wyndmere site on 11 May 2023. This set of results from the third calibration was collected with the transducers situated on the riser pipe and without using any outlet fitting (No Elbow Method). |
While the results collected by the “No Elbow Method” on 11 May 2023 showed close agreement between the flowmeter measurements and the code-calculated estimations, a linear regression analysis was run to determine if a calibration would further improve the resultant flow rate estimations from the algorithm. The linear regression results show that there is no correlation between the flowmeter-measured flow rate and the code-estimated flow rate with an R-square of 0.03. The p-value determined for the linear regression model was 0.501 indicating that the model is not accurately capturing the variation in the data. The results show that there is no systematic error in the code for which a calibration equation could correct and that all errors are random. Figure 11 summarizes the results of the statistical analysis.
Calibration Experiment 4: Wyndmere 15 May 2023
The fourth calibration experiment was conducted on 15 May 2023, and used the “No Elbow Method” because it produces the most stable flowmeter readings and allows for normal pumping operations. The results of this data set do not agree with the previous results collected with the “No Elbow Method” on 11 May 2023. They show consistent errors between the flowmeter measurements and code estimations of 30%-40% (fig. 12d). These results indicate that there is another issue beyond high-air content in the discharge and the siphoning effect causing errors in the code. At this point, the code is not accurately estimating the flow rate consistently.
 |
| Figure 11. Results of linear regression for the calibration experiment using no outlet fitting and transducers placed on the riser pipe that took place at the Wyndmere site on 11 May 2023. The results of these statistical analyses indicate that a calibration of the code is not necessary and will not improve the results because the errors are not systematic but due to random chance with an R-square value of 0.03 and a p-value of 0.501. |
A linear regression between the measured and code-calculated data for the data collected on 15 May 2023 still shows no statistically significant relationship between the flowmeter measurements and algorithm estimations. With an R-square value of 0.17 and a p-value of 0.156, a calibration of the code will not improve the results being seen (fig. 13).
 |
| Figure 10. Results of the third calibration experiment that took place at the Wyndmere site on 11 May 2023. This set of results from the third calibration was collected using the elbow fitting on the outlet and with the transducers situated on the riser pipe (modified “Small Elbow Method”). |
Volumetric Estimation
The results discussed so far show that the ability of the algorithm to estimate flow rate accurately is not consistent, especially when the active pumping portion of the duty cycle is longer (results on 8 May 2023 and 15 May 2023). Conversely, the volumetric estimations of drained and pumped water are shown to match perfectly with one another from the algorithm (figs. 6c, 7c, 8c, 9c, 10c, and 12c). This is likely due to the method of calculating these values. The volumetric estimations are largely based on the flow rate estimations determined in the algorithm, so, as long as the algorithm is running as written, these values should always align with one another. There was no attempt made to physically measure the volumes of water actually being pumped from the tank, as this would have required the additional construction of a tipping bucket system to capture and measure the water at the outlet. Therefore, is it uncertain if the volumes being calculated in the algorithm are accurate; although, it is likely that they are inconsistent estimations due to their derivation from the inconsistent flow rate estimations.
 |
| Figure 12. Results of the fourth calibration experiment that took place at the Wyndmere site on 15 May 2023. This set of results from the third calibration was collected using the elbow fitting on the outlet and with the transducers situated on the riser pipe. |
Discussions
 |
| Figure 13. Results of linear regression for the calibration experiment using no outlet fitting and transducers placed on the riser pipe that took place at the Wyndmere site on 15 May 2023. The results of these statistical analyses indicate that a calibration of the code is not necessary and will not improve the results because the errors are not systematic but due to random chance with an R-square value of 0.17 and a p-value of 0.156. |
Calibration data were collected on four different dates in an attempt to collect data under different drainage conditions and to show whether or not this method was capable of producing consistent results; however, the results were inconclusive. On 27 April 2023, the results of data collected with the “Small Elbow Method” showed that most of the code-calculated flow rate estimations were in relatively good agreement with the flowmeter measurements (= 10%); however, a couple of cycles had errors that ranged 15% to 20% (fig. 6d). Then, on 8 May 2023, there was a significant change when the “Large Elbow Method” was used for data collection. The results from this data showed that there was an error between the code estimations and flowmeter measurements ranging from 50% to 56% for all recorded cycles (fig. 7d). Two of the three data sets from 11 May 2023 were collected using the “Small Elbow Method” or the modified “Small Elbow Method” where the transducer placement changed. The results of these two datasets showed results with errors generally between 20% and 40% (figs. 8d and 9d). The results of these four data sets show that when the elbow fitting is used, the error between code estimations and flowmeter readings is greater (generally >20%) than when no fitting is used (generally =14%). It is now understood that the use of the fittings alters the fluid mechanics of the pump station and leads to errors in the MATLAB algorithm.
With the use of the outlet fittings, the head pressure acting on the water at the outlet is greater than the head pressure acting on the water directly above the pump, which is greater than the head pressure acting on the water at the top of the riser pipe. This creates a siphoning effect which acts to pull the water at the outlet pipe back into the sump tank until air bubbles reach the highest point in the outlet pipe (breakpoint). Air bubbles reaching the highest point cause a break in the vacuum seal within the pipe (Zhang et al., 2022) that is creating the siphoning effect. Once the seal is broken, water will flow according to gravity once again (Azam et al., 2016). Equations 18-20 explain how this phenomenon occurs and how the water flows against gravity back into the sump tank of the pump station. Using equation 21 and 1 for both the SG (specific gravity) and ? (density) of water at 10°C (50°F), the pressure, in PSI, at points A, B, and C in figure 14 can be calculated as 16.7, 8.5, and 14.7, respectively. Point C is under atmospheric pressure because the water level outside the pump at the same level (Point 1) is under atmospheric pressure (Mott and Untener, 2015; Zak’s Lab, 2021). Essentially, the pressure gradient is PA > PC > PB, and, because water flows from high to low pressure, this leads to a siphoning effect in the pipe.
 |
| Figure 14. Schematic showing the setup that leads to water siphoning back into the tank at the end of a pumping cycle. Point A is located at the base of the elbow attachment and is under a pressure head of 16.7 psi. Point B is located at the elbow located directly above the pump and is under a pressure head of 8.5 psi. Point C and Point 1 are located at the water surface inside the pipe and outside the pipe, respectively, and are both under atmospheric pressure due to the lack of a check valve immediately upstream of the pump. |
(18)
(19)
(20)
Converting Head (ft) to Pressure (psi):
(21)
The volume being siphoned back into the tank would need to be accounted for in the code to avoid the errors induced by the effect; however, it is difficult to determine the exact volume being siphoned since it changes with different siphon geometries and flow rates. Each pump station constructed by Ellingson is unique, so the design characteristics affecting the siphoned volume for a given pump station would need to be investigated. Regarding the process of priming the siphon, Chen et al. (2023) conducted physical studies showing that siphon priming is a dynamic process where air is removed in two stages: air compression and air entrainment. Since the pump station design in this research is not a true siphon (the outlet is not submerged), the air compression stage does not really occur at the pump start-up. Instead, the air in the pipe is simply pushed through the outlet; however, depending upon the flow rate, all of the air in the pipe may not be exhausted and some bubbles may remain in the pipe. This is the air entrainment stage, where bubbles may continue to be removed and exhausted throughout the duration of the pumping cycle. This means that the location of bubbles within the pipe will vary from cycle to cycle, and, subsequently, the volume of water siphoned back into the sump tank prior to an air bubble reaching the breakpoint in the pipe will also vary by cycle (Chen et al., 2023).
As we know, the flow rate in drainage pump stations is highly variable, so the water volume siphoned will also be highly variable and difficult to predict mathematically. There is the potential to measure the volume being siphoned into the sump following the pump shut-off by using a pressure transducer. The pressure transducer would need to collect measurements at a sub-second frequency to accurately capture the volume change, however, which would fill the logger’s memory rapidly. For example, in the study conducted by Chen et al. (2021), they used pressure transducers in the elbow of a siphon to monitor air pocket pressure during pumping operations and collected measurements at a frequency of 128 Hz or every 0.001 s. The water level loggers used in this study [13-Foot Water Level Data Logger (U20-001-04), HOBO Onset, Bourne, Mass.] only have 64 kB of memory which can hold around 21,700 measurements. This means that these loggers would fill up after only 3.6 min. This will not work for continuous drainage flow rate monitoring unless automatic or remote data download is available for the logger.
Instead, it is better to mitigate the significance of the siphoning effect in the calibration by situating the flowmeter transducers on the riser pipe to avoid the need for a fitting on the outlet. This also minimizes the issue of air bubble entrainment in the water, as the flow has not passed through an elbow prior to this point which decreases the turbulence in the flow and the aeration of the water.
Data was collected with the “No Elbow Method” on two separate occasions to test the accuracy of the code estimations in comparison to flowmeter readings. On 11 May 2023, the results from the data set collected in this manner showed small errors between the code estimations and flowmeter measurements (=14%) for all 20 cycles recorded. While this was promising, the final calibration data set collected on 15 May 2023 showed a consistent error between the measured and code-estimated flow rates of 30% to 37%, which is unacceptable.
A possible explanation for these inconsistent results is that the method of calculation in the code cannot accurately account for the rate of the fluid flow through the pump when the pump runs for long periods of time. Because the only consistent volume change that we can utilize in the flow rate calculation process is the storage volume of the tank between pumping cycles, that volume is used as the base to determine the rate of fluid flow through the pump. However, there is no way to isolate the exact time over which this volume is removed by the pump as the water leaving the field through the drainage system continues to enter the sump tank throughout the pumping process. This artificially decreases the pumping rate (first term) in equation 9. While the addition of the drainage rates (second term) in equation 9 should compensate for this phenomenon, it may do so adequately.
There is no easy way to compensate for this issue without the use of a flowmeter in the outlet; however, an alternative approach to determine the flow rate through the use of the electrical characteristics of the pump, a pump curve, and the pump affinity equation relating flow rate to rotor and impeller speed (eq. 22) may be worth investigating.
(22)
where
RPM1 = a known pump operational speed (rotations/min)
GPM1 = a resultant flow rate (gal/min)
RPM2 = an adjusted operational speed to be determined
(rotations/min)
GPM2 = an adjusted flow rate to be determined (gal/min).
Using the above equation paired with a pump curve developed for the pump in use and the pump’s rating, the flow rate of fluid being pumped under different operational speeds can be estimated. This method has been utilized in agricultural subsurface drainage systems that are running continuously where constant or narrowly variable operational speeds are assumed (fig. 15).
 |
| Figure 15. Flow rate estimation using the pump affinity equation, pump curve, and pump rating method for high flows and continuous pumping from a different field site in Shenford Township, Ransom County, North Dakota. |
Leonow and Mönnigmann (2013) found that this method is only sufficient when single-speed pumps or a narrow range of pump speeds can be assumed for prolonged periods of time. This is largely due to the fact that the transitional stages in pumping (start-up, speed change, and shut-off) lead to instability in pumping operations and complex, two-phase flow (Zhang et al., 2021; Tong et al., 2022) largely due to lags and delays in the pump motor and stator current draw leading to differences between theoretical and actual flow rates (Leonow and Mönnigmann, 2013).
In this paper, a different method of estimating flow rate from variable speed, centrifugal pumps is developed. The method developed utilizes two Q-i (flow rate vs. effective stator current) curves measured for a given pump to determine what they call a “stator current envelope” (Leonow and Mönnigmann, 2013). One curve (the lower boundary curve) is developed by varying the operational speed and measuring the effective stator current and the flow rate while keeping the discharge valve closed (Leonow and Mönnigmann, 2013). When the discharge valve is closed, it creates what is called a “deadhead” condition where the water in the pump is recycled, which can lead to temperature increases and cause pump failure (Leishear, 2018). Care should be taken while developing the lower boundary curve to avoid damage to the pump and safety issues. Then, a second curve (the upper boundary curve) is developed by repeating the same process with the discharge valve wide open. The flow rate is then estimated based on the stator current envelope and the following equations:
(23)
(24)
where
Qest = estimated flow rate
Qhigh = flow rate determined from the upper boundary
curve
= normalized, inverse flow rate as a function of the
stator current envelope (Leonow and
Mönnigmann, 2013).
However, this static model still struggled to capture and model the complex and unstable pumping operations during transient stages. To improve the model, they developed a dynamic model that allowed for the delay and lags in the pump motor and stator current draw to be better captured. Equation 25 is the dynamic model developed in this study. The results when utilizing the dynamic model and equations 23 and 24 showed that the model could closely capture pumping conditions and accurately estimate the flow rate from a variable speed pumping system.
(25)
where
n = speed of the pump
nadj = adjusted pump speed
TL = time lag
s = time
Td = time delay that can be taken from the step responses
while developing the lower boundary curve (Leonow
and Mönnigmann, 2013).
The difference between the systems studied by Leonow and Mönnigmann (2013) and the systems constructed by Ellingson is the pump type. In this study, they used centrifugal pumps whereas Ellingson uses axial flow pumps. However, the main difference between these types of pumps is the direction of flow compared to the impeller. This method should work for axial flow pumps as well, but axial flow pumps do have higher deadhead pressures (Leishear, 2018), so pumping conditions while measuring the lower boundary Q-i curve should be monitored for safety purposes.
One final thought regarding potential causes of the issues seen in the results of the MATLAB algorithm developed in this study is that some degree of siphoning is still occurring beyond the first 90° elbow in the outlet pipe even when the outlet fittings are omitted from the calibration procedure. However, Ellingson does drill an air release into the underside of the first 90° elbow at every pump station to mitigate this phenomenon.
Conclusion
The objectives of this research were to develop a simple method of calculating flow rates for drainage waters leaving agricultural subsurface drainage pump systems operated with a VFD, to automate this process, and to test the validity of this method. The approach taken was to use measures of the sump tank’s geometry, the water levels that operate the pump, and the duty cycling of the pump to calculate the flow rates and water volumes leaving the field and being pumped from the outlet. This method was selected as it is simple and requires only one additional sensor and data logger supplementary to what is already remotely available from the system at minimal additional cost ($260).
This method showed mixed results when tested by a field experiment using an ultrasonic flowmeter to measure the flow rate leaving the outlet and compare to the estimated flow rate calculated by the code. Some test dates showed good agreement between the calculated and measured flow rates with errors =14% while other test dates showed errors approaching 60%. Some of the error is attributed to siphoning that occurs at pump shutdown and to turbulent and two-phase flow in the outlet pipe, which decrease the accuracy in the flowmeter measurements. However, it is also likely that this method is too simple to accurately calculate flow with the complex pumping operations and flow dynamics brought upon by transient pumping conditions as a result of the VFD incorporation. With the simple design of the sump and the available data, this method is not capable of producing consistent, accurate estimations of flow rate, especially under higher drainage rates that lead to prolonged pumping cycles.
Future research should investigate the applicability of utilizing electrical characteristics of the pump to estimate the flow rate from variable speed pumping systems utilizing axial flow pumps. Leonow and Mönnigmann (2013) successfully employed a method using Q-i curves on a system operating a centrifugal pump with a VFD. Additionally, it is possible that a better method of monitoring the tank level to determine the time over which the pump removes the tank’s inflow volume and the volume that is being siphoned back into the tank after each cycle would improve the accuracy of the algorithm estimations. Since the volume of water leaving the field is the most important aspect for environmental and legal purposes, this should be the first goal of future projects, with the flow rate being secondary.
Acknowledgments
The authors would like to thank Alex Anderson and two other producers, who wished to remain anonymous, for letting us work in their fields and for being helpful and patient cooperators. This work was supported by Ellingson Companies, without whose support we would not have been able to complete this project. The authors also wish to thank Dr. Dongqing Lin, Dr. Sulaymon Eshkabilov, and Tayler Johnston for their technical assistance. This work was supported by the USDA National Institute of Food and Agriculture, Hatch project number ND01482. We also want to thank the reviewers and the associate editor for their valuable comments.
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