I Congresso da SBI-Agro


Pump curve combination model

C. de L. T. de Andrade
ASAE Member, Irrigation Engineering Researcher,
EMBRAPA (Brazil), 35701-970 - Sete Lagoas, MG, Brazil,
Tel. (031) 773 5644, Fax (031) 773 9252,

R. G. Allen
Associate Professor, Biological and Irrigation Engineering,
Utah State University, Logan, UT 84322-4105,


Um programa de computador foi desenvolvido para combinar curvas de bomba em série e/ou em paralelo e para determinar o desempenho de estações de bombeamento e de bombas individuais dentro da estação. Polinômios de até quinto grau são ajustados aos dados de curvas de bombas individuais e das estações de bombeamento. Interpolação do tipo "cubic spline" é empregada para combinar curvas de bombas de diferentes tamanhos. Indicadores de desempenho da estação de bombeamento e de bombas dentro da estação são calculados para toda a faixa de vazão da estação. Dessa forma, o desempenho das bombas e da estação de bombeamento pode ser analisado para diferentes situações existentes ou desejáveis em um projeto de distribuição de água ou de irrigação. Uma interface gráfica projetada para WINDOWS 95 e 3.xx permite of usuário desenhar um croquí da estação de bombeamento e visualizar os dados fornecidos pelo usuário juntamente com o polinômio ajustado. Os dados fornecidos pelo usuário, as curvas ajustadas com os coeficientes podem ser imprimidos.


Curva de bomba, combinação, modelo


A computer model was developed to combine pump curves in series and/or in parallel and to determine the performance of the pump station and of individual pumps within the station. Polynomial equations are fit to the individual pump curves and to the combined pump station curve. Cubic spline interpolation is employed to combine curves of pumps with different sizes. Performance indicators of the station and of each pump within the station are calculated for the entire station flow rate range. This way the performance of individual pumps or of the pump station can be analyzed for different situations of an existing or desirable water distribution system or irrigation system. A graphical interface designed for Widows 95 and 3.xx allows the user to draw the pump station layout and to view the discrete data entered along with the equations fit. Data entered, plots of the curves and equation coefficients can be printed out.


Pump curve, combination, model.



This model is part of a larger model, SPRINKMOD, developed to simulate pressure and discharge in pressurized irrigation systems (Andrade, 1997). The model performs all combinations of various sizes of pumps and generates the head-flow rate tables and equations to be used by SPRINKMOD. It runs in WINDOWS 95 and 3.11 environment and provides a user-friendly interface for data entry and results display.

Pumps are selected to match the hydraulic requirements of the distribution or irrigation systems. However, distribution and irrigation systems normally operate over a range of flow conditions due to changes in demand, changes in reservoir elevations or position of laterals, or due to changes in friction or local losses caused by valve operation or by the aging process. In these situations, it may be desirable to use multiple pumps that are combined in series and/or in parallel.

When combining pumps in parallel and/or in series, it is important to analyze the total and individual pump efficiencies, NPSHr and input power over the entire range of system flows. It is desirable to have the operating point as close as possible to the point of best efficiency of the combined and individual pumps. Mismatched pumps can reduce efficiency and may even cause water to be pumped backwards or make a pump function like a turbine. In these situations energy is wasted and the motor or engine of a mismatched pump can be damaged. Conducting the analysis of combined pumps by hand is tedious. In general, computer software or spreadsheets are needed to simplify the analysis. Ormsbee and Walski (1989) developed a graphical method to combine pumps in series and/or parallel which has been used to optimize municipal water distribution pumping operations. The model is not practical for more than two pumps combined together. Tarquin and Dowdy (1989) wrote a model to evaluate efficiency and costs for a specific pumping plant. By analyzing efficiencies and power consumption for different ways of combining the same set of pumps, the best combination could be selected leading to savings in pumping costs. No details were given by the authors about the approach used to combine pumps and about data entry and displaying interface, and whether the model can be used for other situations.

The major objective of this study was to develop a user-friendly computer model for combining pump curves in series and/or parallel.



Pump head-flow rate curve

By measuring the head generated by a pump with fixed speed and impeller diameter for a range of flow rates starting at zero, manufacturers generate what is called a pump characteristic curve, pump rating curve or pump head-flow rate curve.

Most of the pump curves can be described mathematically through use of a polynomial equation. The second degree polynomial is commonly used (Gessler, 1981; Tullis, 1989). Ramalan (1988) used a third degree polynomial to describe pump head-flow rate curves and Wheeler (1993) found a fourth degree to be the best model. Molina (1991) estimated missing pump data by fitting a quadratic function (where the linear component of a second degree polynomial was removed) to a measured set of data. Higher order polynomials might be used to describe some unusual characteristic curves although care must be taken as higher degree polynomials can Awiggle@ between data points (Mathews, 1987).

An alternative equation to describe pump characteristic curves is the cubic spline. Molina (1991) applied cubic spline interpolation to a pump head-discharge curve and system curve in order to find the operating point. Wheeler (1993) suggested the use of splines in place of the fourth degree polynomial. Kincaid and Cheney (1991) describe the methodology to derive cubic spline functions for a set of data.

Pump efficiency and power consumption

Efficiency, Ep is the ratio between the useful energy transferred from the pump to the water, Wp, and the energy needed to drive the pump Bp (Tullis, 1989; Keller and Bliesner, 1990):




Ep = pump efficiency [%]

Wp = water power output [ML2/T3]

Bp = brake power input [ML2/T3]

The energy transferred to water is (Keller and Bliesner, 1990):



H = total operating head [L]

Q = flow rate [L3/T]

K = conversion constant [L2T2/M]

The energy supplied to the pump shaft is:


where all terms have been described previously.

Starting with zero at a zero flow rate, pump efficiency increases up to a certain maximum and then decreases. For low specific speed centrifugal pumps, the brake horse power typically increases with discharge. For high specific speed pumps, the horsepower required near the shutoff head increases rapidly (Tullis, 1989).

Net Positive Suction Head (NPSH)

The pressure necessary at the suction side of the pump to prevent cavitation is referred to as the net positive suction head requirement (NPSHr). Cavitation of water inside a pump can deteriorate the pump performance and can damage the pump components. NPSHr is determined in laboratories and it is a characteristic of the pump design and operation. It is essential that the net positive suction head available (NPSHa), which depends on the system design, exceed the NPSHr with a reasonable margin of safety to ensure satisfactory operation (Tullis, 1989).

Pumps in series

Pumps in series are used when a Asteep@ type of characteristic curve (large dH/dQ) is desirable, such as in situations where the flow rate needs to be kept almost constant over large changes in head. This may occur in an irrigation system where laterals in operation change from high elevation to low elevation positions in the field and vice versa (Allen, 1992). Also, when the total head generated by one pump is not sufficient to meet the total system pressure requirement, pumps in series are employed.

For pumps operating in series, each pump imparts additional energy to the same stream of water. The combined head is essentially equal to the sum of the individual heads for the same flow rate. The combined efficiency for two pumps in series is given by (Tullis, 1989; Keller and Bliesner, 1990; Krivchenko, 1994):



Eps = combined efficiency [%]

Q = flow rate [L3/T]

hp1 = head of pump 1 [L]

hp2 = head of pump 2 [L]

K = unit conversion constant [L2T2/M]

Bp1 = brake power input of pump 1 [ML2/T3]

Bp2 = brake power input of pump 2 [ML2/T3]

The combined NPSHr curve for pumps in series is the same as the NPSHr curve for the most upstream pump, because the other pumps will be operating with high pressure at their intakes as compared to the first pump.

Pumps in parallel

Pumps in parallel are used whenever a Aflat@ type of characteristic curve (small dH/dQ) is required. That is, the pump discharge head decreases gently with an increase in flow rate. A situation like this might occur when the water demand changes with time, such as during the day in a town, along a season in crop land farm when laterals are turned on and off within an irrigation system. Also, pumps may be connected in parallel for safety or to simplify maintenance. In municipalities, it is common to use three pumps in parallel, each one having the capacity for supplying 50% of the required design flow rate. Another option is to have four pumps in parallel, each one capable of supplying 33% of the normal flow requirement (Tullis, 1989).

When pumps are operated in parallel, they work against a common pressure. The combined characteristic curves are developed in a manner similar to that described for pumps in series, except the flow rates rather than heads are added for a common total head.

The combined efficiency for two pumps in parallel is given by (Tullis, 1989; Keller and Bliesner, 1990; Krivchenko, 1994):



Epp = combined efficiency [%]

H = total head [L]

Q1 = flow rate for pump 1 [L3/T]

Q2 = flow rate for pump 2 [L3/T]

K = unit conversion constant [L2T2/M]

Bp1 = brake power input for pump 1 [ML2/T3]

Bp2 = brake power input for pump 2 [ML2/T3]

The combined NPSHr for pumps in parallel is equal to that for the most limiting pump, the largest NPSHr in this case.

Local head loss

When assembling pumps in series/parallel in a pump station small pipe lengths, valves and fixtures are employed causing the loss of head (local or minor).

It is common to express the local losses, hl, by an equation of the form:



Kr = local loss coefficient [ ]

V = average velocity [L/T]

g = acceleration due to gravity [L/T2]

A extensive description of local loss coefficients for the majority of the fixtures used in municipal water systems and power plants has been done by Ruus (1981) and Flammer et al. (1986). Local loss coefficients for most of the irrigation equipment were presented by Keller and Bliesner (1990).


The general procedure for using the pump combination model consists of: 1C Creating Pump Curve File, 2C Creating Pump Station Files, 3C Viewing Pump Station Performance Data.

Creating pump curve files

The user must create a file for each one of the pump curves that are to be used. Once created, pump files can be re-utilized over and over again. Different pump stations can use the same individual pump files.

A dialog box is employed to enter data for individual pump curves (Figure 1).The box has its own menu bar and table where data can be entered for flow rate, pump head, efficiency and the net positive suction head requirement (NPSHr). At least six sets of data points must be entered so that a polynomial with degree of up to five can be fitted to the data. A maximum of 20 data points can be entered. From within this dialog box the user can save the pump data into a file, print out data and chart, edit existing pump data files and view a plot of the data just entered (Figure 2).

Polynomial equations of degree up to five can be fitted to flow rate versus head, flow rate versus efficiency and flow rate versus NPSHr data. This way pump manufacture's data can be smoothed out. The user is allowed to modify the degree of the polynomials and analyze the effect on the fitting graphically or based on the standard error of the regression.

Finally, the pump combination model allows for files created using one system of units to be converted into the other system and retrieved into the dialog box (Figure 1). Both english and metric systems of units can be used.

Creating pump station files

Pump station files can be created or edited in the same way as for individual pumps. Files for individual pumps need to be created prior to pump station file creation. In a pump station, individual pumps are assembled in parallel or in series. Also, groups of pumps in series can be subsequently assembled in parallel. The program is capable of tracking intricate pump station layouts and of generating a unique pump station curve.

Two dialog boxes are directly involved in the pump combination process. Firstly, a dialog box is displayed from which individual pump curve files, previously created, can be selected to be used in the pump station (Figure 3). Any number of available individual pump curve files can be selected in any order. The order of selection is relevant since letters are assigned to each pump selected as a short name, starting with AA@. After closing this first dialog box, a second one is displayed where the user can draw the layout of the pump station in a grid (Figure 4). The previously selected list of pump curve files is stored in a drop-down box. A pump can be selected from the drop-down box and placed in the grid. In addition, a tool box allows the user to select a pipe fixture and place it in the layout between pumps. Following this procedure, a pump station layout can be drawn as closely as possible to the real pump station. Local (minor) loss coefficients can be associated with each pump to account for losses from the selected pump the next downstream pump or intersection (bifurcation or trifurcation) device. Also, local loss coefficients can be entered for each intersection device to incorporate losses from the selected fixture to the next pump or next intersection downstream. Equation 6 was employed to determine the local losses in pipes and derivations presented in the pump station, provided that proper pipe inside diameter and local loss coefficient was entered by the user.

All calculations for pump combination are accomplished in a code module having many functions and subroutines. The grid is scanned and information are stored in arrays so that smaller and easily maintainable functions could be written for each phase of the pump combination process.

Combination calculations are triggered when the user tries to view the graph or to view data for an individual pump by clicking with the right mouse button over the pump icon. The range of possible flow rates for each pump is subdivided to generate a set of discrete data. The lower limit of the range is set to zero flow rate and the upper limit is the largest flow rate entered by the user for a certain pump. The number of intervals is currently fixed at 20 (21 points). For each flow rate value, the corresponding head, efficiency and NPSHr are determined for every pump by using its polynomial equation.

For pumps in series, the heads of individual pumps are added for a common flow rate, while for pumps in parallel, the flow rates are added for a common head. If there is no common flow rate for pumps in series or if there is no common head for pumps in parallel, as occurs if the pumps are of different size or after upstream combination, interpolation is used to find heads for the common flow rate and vice-versa. The program uses a cubic spline type of interpolation to accomplish this task (Kincaid and Cheney, 1991).

Pump efficiencies are averaged by weighting according to head for pumps in series (equation 4) and by weighting according to flow rate for pumps in parallel (equation 5). This is done after minor losses between the two pumps to be combined are subtracted from the pump heads accordingly.

Required NPSH for the pump station is found by selecting the highest NPSHr of individual pumps for which the upstream intake is open. This is done for every value in the range of flow rates for the station.

Viewing pump station performance data

Pump combination model goes beyond the process of simply computing the combined flow rate, head, efficiency and NPSHr for a pump station. Detailed performance information for each individual pump in a pump station is determined for each point of the final pump station curve.

Head, efficiency, NPSHr and flow rate for the complete pump station and for individual pumps are retained in arrays, so that when the user clicks on a pump in the grid with the mouse=s right-button, the data for that pump are displayed in a table (Figure 5). This is an important feature of the model. The user can analyze the performance of the complete pump station and of individual pumps, as well, for each flow rate condition at the station outlet. Problems of low performance and cavitation can be detected and rectified, or can be avoided if such analysis is done prior to the pump station installation.

The combined set of data for the pump station can be saved in a file so that it can be linked to the main distribution or irrigation system data file later. For pump stations having different pump sizes combined in series and/or in parallel it is unlikely that a unique polynomial equation can describe the head-flow rate, head-efficiency or head-NPSHr data sets. The main model uses spline interpolation instead of equations, so that any kind of pump combination can be well described.

Pump station data can be viewed in a chart or printed out, and existing files can be edited in the same way it is done for individual pump files as described earlier.


The software presented here is part of a bigger software, SPRINKMOD (Andrade, 1997), designed to simulate pressure and discharge in pressurized irrigation systems. The pump combination part of SPRINKMOD can be set aside of the main model to create a stand-alone program. The software runs in the WINDOWS 95 and 3.xx environment which makes it easy to use.


The financial support from EMBRAPA (Brazil) and Utah Agricultural Experimental Station is gratefully acknowledged.


  • Andrade, C. de L. T. de 1997. Pressure and discharge distribution simulation in pressurized irrigation systems. Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Biological and Irrigation Engineering, Utah State University, Logan, Utah, 278.
  • Allen, R.G. 1992. Sprinkle irrigation design, BIE­607 Sprinkle Irrigation Design Course Notes, Biological and Irrigation Engineering Department, Utah State University, Logan.
  • Flammer, G.H., R.W. Jeppson and H.F. Keedy. 1986. Fundamental principles and applications of fluid mechanics, Department of Civil and Environmental Engineering, Utah State University, Logan.
  • Gessler, J. 1981. Analysis of pipe networks, Closed-conduit flow, M.H. Chaudhry and V. Yevjevich, eds., Water Resources Publication, Chelsea, 61-100.
  • Keller, J. and R. D. Bliesner. 1990. Sprinkle and trickle irrigation, Van Nostrand Reinhold, New York.
  • Kincaid, D. R. and E.W. Cheney. 1991. Numerical analysis - mathematics of scientific computing, Brooks/Cole Publishing, Pacific Grove.
  • Krivchenko, G. 1994. Hydraulic machines: turbines and pumps, Lewis Publishers, Boca Raton.
  • Mathews, J. H. 1987. Numerical methods for computer science, engineering and mathematics, Prentice-Hall, Englewood.
  • Molina, E. 1991. Simulation model to predict operating pressures and flow rates for a sprinkler system in operation, report submitted in partial fulfillment of the requirements for the degree of Master of Science in Agricultural and Irrigation Engineering, Utah State University, Logan, Utah.
  • Ormsbee, L.E. and T.M Walski. 1989. Identifying efficient pump combinations. Journal of Water Works Association, AWWA 81(1):31-34.
  • Ramalan, A.A. 1988. Management strategies for gravity sprinkle irrigation system, dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Agricultural and Irrigation Engineering, Utah State University, Logan, Utah.
  • Ruus, E. 1981. Head loss. p13-38. In: M.H. Chaudhry and V. Yevjevich (eds). Closed-conduit flow. Water Resources Publication, Chelsea,
  • Tarquin, A.J. and J. Dowdy. 1989. Optimal pump operation in water distribution. Journal of Hydraulic Engineering, ASCE 115(2):158-168.
  • Tullis, J.P. 1989. Hydraulics of pipelines: pumps, valves, cavitation, transients, John Wiley & Sons, New York.
  • Wheeler, L.A. 1993. A model for predicting the performance of electrical irrigation pumping plants, report submitted in partial fulfillment of the requirements for the degree of Master of Science in Agricultural and Irrigation Engineering, Utah State University, Logan, Utah.



Figure 1: Dialog box for entering new pump curve data.

Figure 2: Plot of flow rate versus head and polynomial equation coefficients.

Figure 3: Dialog box for pump data file selection.

Figure 4: Example of pump station layout.

Figure 5: Pump station data and corresponding data for one of its pumps.