School of Knowledge: Water pumps, fitting curves for water pump sets (graph)

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Source: HC360 Pump and Valve Industry Channel Abstract: This article derives the mathematical model of water pumps and applies it in the computer-based selection process for such pumps. Based on the precision requirements of actual projects, the least squares method was used to derive the fitting characteristic curve equation for the water pump, thereby establishing a database of water pump characteristic curves. On this basis, a general formula for the characteristic curves of the combined operation of series and parallel water pump sets, which is practical for use in actual engineering, was derived. The basic principles and approaches to implementing the characteristic curves of different types of water pumps in computer programs were explored using piecewise function representations. For the widely used variable frequency technology at present, the speed control formula for the water pump set was derived. Keywords: water pump, least squares method, fitting curve, series connection, parallel connection. 1 Introduction In design calculations, especially when using computers to write programs for selecting water pumps, mathematical formulas representing the characteristic curves of these pumps are required as a basis for decision-making. Considering the precision requirements of actual engineering projects, this paper uses the least squares method to determine the fitting curve for individual water pumps. Data were processed through computer programs, resulting in a database of formulae for the characteristic curves of five types of water pumps. Furthermore, a general formula for the characteristic curve of the pump set is provided for the combined operation of pumps in series, parallel, and with speed control. 2 Principle of fitting the characteristic curve of a single water pump: Using the principle of least squares, and based on the shape of the water pump’s characteristic curve, polynomials are considered as the fitting basis functions; the first three terms of these basis functions are selected. The fitting formula (1) for forming the water pump characteristic curve. Different pump models correspond to different formula coefficients a, b, c. Using sample data from the characteristic curves of different pump models, 12 sets of data were obtained for each pump on the flow-head and flow-efficiency curves. By substituting these values into equations (2), (3), and (4), the mathematical formulas for the flow-head and flow-efficiency curves of that particular pump model were derived. The formula for the fitting curve and the formulas for calculating the coefficient values are as follows: On this basis, this paper established a database of some pump characteristic curves. For multi-stage pumps, the established pump characteristic curve database contains the curves for pumps of that model with one stage, while also indicating the minimum and maximum number of stages. When selecting water pumps, this paper handles the number of stages and formula coefficients of the pumps by connecting pumps of the same model in series. To facilitate subsequent analysis of the performance of the pump sets, the fitting coefficients for the flow-head and flow-efficiency curves of individual pumps of different models are set as A0, A1, A2, B0, B1, and B2 respectively. The fitting formulas are as follows: H = A0 + A1×G + A2×G² (5) η = B0 + B1×G + B2×G² (6) Where: H – Pump head; η – Pump efficiency; G – Pump flow rate. Due to space constraints, only some data are listed in the table below: Pump Model, Rated Speed, Flow Rate, Head, Coefficient A0, Coefficient A1, Coefficient A2: ISR50-32-125: 2900, 21.7667, 1.5754, -0.582978; ISR50-32-1251: 1450, 4.9528, 0.9653, -0.539651; 14Sh-61470: 132.0884, 70.0960, 1-0.00034429; 14Sh-91470: 88.9009, 60.02387, -0.00019932; D6-25 (multi-stage pump): 2950, 25.3692, 20.8979, -0.66107458. Coefficients B0, B1, B2 for flow efficiency: Minimum stage number, Maximum stage number: 1.389533, 2.9142, -4.460521; 14.7823, 248.4926, -11.343711; 14.8240, 40.39593, -0.005561; 11-11.013160, 0.53633, -0.000781; 11-0.902886, 241.5298, -8.7872312. Analysis of the combined operation mode of pumps: In practical applications, it is sometimes necessary to operate pumps in parallel or in series within a piping system. The purpose is to increase the flow rate in the system or raise the head pressure. In systems where a significant change in flow rate is required, turning some of the parallel water pumps on or off can be used to adjust the flow rate ； When an increase in flow rate is required during expansion projects, using additional pumps to operate in parallel with the existing ones may be a more economical solution than replacing the existing pumps with a single large pump. 3.1 Parallel operation of pumps When it is necessary to increase the flow rate in the system, parallel operation can be employed. At this point, it can be assumed that the inlet and outlet of the water pump operate under the same head pressure. Moreover, the output flow rate in the main pipe is the sum of the flow rates of each water pump. Based on this principle, the performance curves (G–H curves) for the parallel operation of various water pumps can be drawn. 3.2 Series operation of pumps: When the performance curve of the piping system is steep and a single pump is not capable of providing the required head, the pumps should be operated in series. At this point, the outlet of the first water pump is connected to the inlet of the second water pump. The advantage of operating them in series is that the flow rate through both pumps is equal, while the head or pressure generated is equal to the sum of the heads or pressures produced by both pumps. Based on this principle, through further derivation, a general formula for the series operation of multiple water pumps is obtained. 4 Derivation of the fitting curve formulas for pump sets 4.1 Fitting of the characteristic curves for pumps of the same model The fitting curve formulas for individual pumps of different models have been established earlier; based on the principles of series and parallel connection of such pumps, further derivations yield the following general formulas that are practical and useful for actual applications. For multiple pumps of the same model connected in series, the following formula applies: H=a0+a1×G+a2×G²　　　　　　　 (7) Where: a0=nA0, a1=nA1, a2=nA2; n is the number of pumps of the same model connected in series. For multiple pumps of the same model connected in parallel, the following formula applies: H=a0+a1×G+a2×G² (8) Where: a0=A0, a1=A1/n, a2=A2/n²; n is the number of pumps of the same model connected in parallel. 4.2 Discussion on formulating formulas for groups of pumps of different models Since it is difficult to derive a unified formula for groups of pumps of different models connected either in series or in parallel, this paper considers using piecewise functions to represent the flow-rate–head curve of such pump groups. Based on the basic principles of series and parallel connection of water pumps, the characteristic curves of the water pumps are simply added together for a series-connected pump set, and a program is written to represent the characteristic curve of the resulting pump set. As shown in Figure 1: For a parallel water pump set, a piecewise function approach is used. In the computer program, a database of the characteristic curves for individual pumps of different models is called upon; by determining the extreme values of the head output of each pump, a piecewise representation is created to depict the characteristic curves of the pumps. As shown in Figure 2: The fitting formula for the 5-pump water supply system operating at variable speed. The parameters of the pipeline network are constantly changing due to various factors; when the flow rate and pressure in the pipeline connected to the pumps change, the opening degree of the pump outlet valve can be adjusted to meet the required parameters of the pipeline network. But doing this wastes a large amount of electrical energy, which is not economical. Currently, variable frequency technology is widely used in pipelines both domestically and internationally. By collecting parameter signals from the pipelines, the speed of water pumps is controlled to ensure that their operating conditions meet the requirements of those pipeline parameters. This approach has clear advantages in economic analysis. Based on the formula for the parallel operation of pump sets, the formula for adjusting the speed of pump sets when they operate in parallel is derived: In equation (9): n0 – the rated speed of pumps of the same model; ni – the speed of pumps of the same model when their speed is adjusted; a0 = A0; a1 = A1/n; a2 = A2/n2; n – the number of pumps of the same model operating in parallel. 6 Conclusion With the widespread use of computers today, it is necessary to use computational programs to carry out calculations and analyses for various engineering projects. The foundation of expertise in computer calculation and analysis is mathematical models. By organizing the fitting formulas for individual water pumps and pump sets, a general formula was derived. Through this formula, a relatively comprehensive database of pump characteristic curves was established. Corresponding computer programs were developed based on the mathematical models for various operating modes of pumps, and ideal results were achieved through their application in actual projects. These mathematical models are the foundation and prerequisite for design and operational management.