Abstract

The method of virtual rotodynamic pumps is presented, a simple and practical procedure for the solution of problems of pumps connected in parallel, when the installation conditions and/or characteristics of the coupled pumps are not the same. Virtual pumps are imaginary machines without secondary suction and discharge piping. These pumps consist of the real pump supplied by the manufacturer and all the accessories and piping up to the point of connection with the main delivery pipe. These machines are characterized by always working against the same total load when operating in parallel. The methodology is presented by means of an example of a hydraulic simulation of a pumped well field source system.

Keywords:Aspects; Rotodynamic Pumps; Operation; Parallel

Introduction

In different applications of rotodynamic pumps, it is necessary to analyze different operating options, to meet the needs that are presented. One of these options is to combine pumps of equals or different characteristics in parallel. This operation scheme is mainly done to fraction a total flow in partial flows, when working in a variable spending and constant demand or to increase the flow in a pumping system [1].

It is known that, in general terms, if the pumps placed in parallel are the same, the capacity and power necessary for equal loads will double, triple, etc.; Depending on two, three or more equal pumps placed in parallel, and the efficiency of the set will be equal to that of the original pump. This working condition will only be fulfilled, when secondary pipe systems (suction and impulse of each pump), and the installed pumps are equal [2].

For any other situation, the statement raised in the previous paragraph will not be fulfilled and adjustment for the calculation of the real operating point must be made. As a fundamental characteristic to highlight in a parallel pump coupling, it is that the total flow delivered by the combination is always less than the number of pumps by the flow that supplies a single pump [3]. The existing relationship between the flow that supplies a pump and the one that supplies the combination will depend on the characteristics of each of the coupled pumps, the set in parallel and the characteristic curve of the pipe system [4].

Rotodynamic pumps operating in parallel are influenced by mutually: the flow, the load, the power and efficiency of each of them depend essentially on the individual load regimes of the pumps that work together. The cases of operation for the work of rotodynamic pumps in parallel are [5]:
a. Case 1. Pumps of equal hydraulic characteristics, operating against equal pipe systems to the point where the common impulse pipe (node) begins. In this case, the conditions set out above are met, that is, the bombs will be operating against the same load, therefore, it will be perfectly fulfilled that the expenses for equal loads are added.
b. Case 2. Pumps with the same hydraulic characteristics, operating against different piping systems up to the point where the common discharge piping begins (node). For this particular case, since the piping systems are different up to the point where the secondary discharge pipes of the pumps join, it can be stated that these pumps are not operating against the same load, since, in short, they are operating against different systems.
c. Case 3. Pumps of different hydraulic characteristics, operating against equal piping systems up to the point where the common rising main (node) begins. For this scenario, the piping systems are the same for each pump, but, since they are different pumps, the costs will also be different, and consequently, the losses that occur in each section up to the junction point will be different, therefore, the curves of the individual systems will be different, implying that the pumps are operating against different loads at the outlet of the same.
d. Case 4. Pumps of different hydraulic characteristics, operating against different piping systems up to the point where the common rising main (node) begins. This is the so-called general case, for which it is evident, from all the above mentioned in cases 2 and 3, that the pumps will be operating against different loads.

The hydraulic simulation of pumping systems is particularly useful to evaluate the responses of such systems for certain operating conditions. One of the procedures used to carry out the simulation process is the well-known Virtual Pump Method. It is a simple and practical calculation procedure, recommended for those cases where there are pumps connected in parallel, when the installation conditions and/or hydraulic characteristics of the coupled pumps are not the same (Case 4, general case).

Material and Methods

Analytical treatment of the characteristic curves of parallelcoupled rotodynamic pumps The equations modelling the three design characteristic curves of parallel coupled rotodynamic pumps for the case with equal hydraulic characteristics are [6,7]:
Head-capacity curve (parallel),

where: H_P: load developed by the pumps coupled in parallel, (m); A: coefficient of the polynomial representative of the H-Q curve that defines the value of the load developed by the pump for zero flow or closed valve, (m); B and C: coefficients of the polynomial representative of the H-Q curve that depend on the pressure losses inside the pump, (s/m²), (s²/m⁵) respectively; Q: flow rate driven by the combination of pumps in parallel, (m³/s) and n_b: number of pumps working in parallel.

Power-capacity curve (parallel),

where: P_P: power absorbed by the pumps working in parallel, (kW); D: coefficient of the polynomial representing the P-Q, curve, which defines the value of the power consumed by the pump for zero flow, (kW) and E and F: coefficients of the polynomial representing the P-Q curve dependent on the pump power losses, (s·kW/m³), (s²·kW/m⁶) respectively.

Efficiency-capacity (parallel),

where: η_P: efficiency of the parallel pump combination, (dim.) and G and H: coefficients of the polynomial representing the η-Q, curve, (s/m³) and (s²/m⁶) respectively.

The NPSHr behaviour of pumps does not change when they are placed in series or parallel, since by definition it is a variable that characterizes the suction capacity of a pump and is therefore independent of the coupling type.

Second-degree polynomials are usually proposed for all characteristic equations for rotodynamic pumps and pumps resulting from their coupling in parallel, in series, or in the case of a change in rotational speed and/or impeller diameter. In general, the order of the fitting polynomials based on the flow rate can be greater, thereby achieving improved goodness- of-fit, especially for mixed-type, axial, and specially designed pumps. Given these cases, equations 1-3 take the following form:
Head-capacity curve (parallel),

Hydraulic simulation of pumping systems. Pumped source systems

Hydraulic simulation of pumping systems in steady state (the most appropriate for the hydraulic analysis being performed) consists of obtaining the circulating flow rates through the pipes and the load at the nodes of the system using the system equilibrium equations (mass conservation equation, or continuity equation and the energy conservation equation) based on knowing: punctual consumption at the nodes (if any), the piezometric load at least one node, and the relevant characteristics of the pipes (diameter, roughness and length) and the rest of the system elements (pumps, valves, accessories, etc.) [8].

From the set of relationships above, two systems of nonlinear equations are obtained: one applying the continuity equation at nodes, and the other based on the pressure losses of the network elements. Their solution will yield the circulating flow rates and pressures at the nodes. The nonlinear nature of these systems of equations makes the application of numerical resolution methods essential. One of the most popular methods is the Newton-Raphson method, which finds the simultaneous solution to the system of mass and energy balance equations. The problem is solved by the iterative solution of a system of linear equations equal in size to the number of unknown piezometric loads.

The Newton-Raphson method is one of the many iterative methods available. Martínez y Riaño [9] uses the Node Iteration method to perform the hydraulic analysis of a well field pumping system. In this case, the model is first divided into independent subsystems, and then an iterative solution method similar to that used in the well-known Three-Tank Problem is applied. The methodology presented is superior to that presented by Martínez [7], which applies this iterative method to simpler pumping source systems (fewer subsystems) with a Hazen-Willians friction coefficient, C, for the pipes, which is constant throughout the system.

Another of the most widely used numerical methods for successive iterations, primarily due to its ease of programming and use in simulation software such as EPANET, is the Gradient Method, proposed in 1987 by Todini and Pilati. It combines techniques based on optimization methods with techniques based on the nodal Newton-Raphson method. It begins by applying optimization techniques, which guarantee the existence and uniqueness of the solution by minimizing the objective function. These are essential conditions for subsequent convergence when using the Newton-Raphson method. The problem is finally driven to an algebraic solution through the iterative process known as Incomplete Choleski Factorization/Modified Conjugate Gradient (ICF/MCG).

The Virtual Pump Method, on the other hand, has not been the subject of much research in recent years, despite being a simpler procedure than the iterative methods mentioned above. The concept of virtual pumps simplifies the hydraulic calculation of actual operating points and, in general, the physical understanding of the problem. If, for a given pumping system, the characteristic curve of the virtual pump is taken to include both the suction and discharge pipes up to the discharge point, the length of the system against which pumping would be required would be zero, and the corresponding head losses would be zero. Consequently, the characteristic curve of this system would be given simply by the values of static height versus flow rate [10].

Virtual pump method

In everyday practice, various procedures have been used to determine the operating points of each of the pumps operating, either individually or in parallel, and discharging into the same main drive system. These procedures can be summarized as follows [11]:
i. The pressure losses in the suction and secondary discharge of each pump (excluding the main discharge pipe or system) are ignored when calculating the piping system’s characteristic curve.
ii. Any differences in pressure losses that may exist in the secondary discharge and suction of the pumps are ignored, and any pressure losses considered significant in this part of the installation are included when calculating the main discharge piping system’s characteristic curve.

Since in the two previous criteria the losses in the duct system part of each machine are considered equal, it follows that, if the pumps work in parallel and lift from the same level in the suction, they must necessarily work against the same total load. It should be noted that calculation procedures based on the above criteria, particularly in the second one, produce good results for some practical problems, especially when the head losses in the ducts of each pump are small compared to those that occur in the main discharge pipe and therefore do not determine the shape of the characteristic curve of the system against which it is pumped [10].

Conceptually, virtual pumps are imaginary machines that lack secondary suction and discharge piping. These pumps consist of the actual pump supplied by the manufacturer and all accessories and piping up to the point of connection with the main discharge piping. These devices are characterized by always working against the same total load when operating in parallel, without requiring assumptions that might imply only approximate solutions, as would be the case with the previous criteria. This concept of virtual pumps greatly simplifies the hydraulic calculation of actual operating points and, in general, the physical understanding of the problem [10].

If, for a given pumping system, the virtual pump’s characteristic curve is taken to include both the suction and discharge piping up to the discharge point, the length of the system against which pumping would be carried out would be zero, and the corresponding head losses would be zero. Consequently, the characteristic curve for this system would be given simply by the head versus flow rate values [5].

Technically speaking, virtual pumps are analogous in their hydraulic behaviour to the installation of a diaphragm between the discharge flange of a rotodynamic pump and the discharge pipe flange. It should be noted that this constitutes a pure throttling, the losses of which directly affect the pump’s headcapacity characteristic curve. As is known, pressure losses (head losses) due to the diaphragm follow a quadratic or parabolic curve. Therefore, the new pump characteristic curve, when the calibrated diaphragm is installed, differs from the previous curve at all points due to the pressure drop [5].

This method can be used for any of the four parallel pump design schemes; it is especially recommended for Case 4 (the general case). As noted, this is an alternative calculation methodology to numerical methods for simulating pumping systems, specifically for pumped source systems. Hydraulically, it is a procedure for addressing the problem of operating rotodynamic pumps operating in parallel in branched pipelines with independent and common pipe sections.

Approach to the virtual pump method

Using Figure 1 as a reference, Bernoulli’s equation is applied from the water levels in the suction tanks of each pump to point A (the common or central node of the pumping source system), where there is a common head HA (piezometric head at node A) for the connections between pumping stations B1 and B2 and node A. In this case, the flow rates of the individual pumps are summed at this node, the beginning of the common section. This means that the total pump head values must be equal at this point [12].

The total load at node A is obtained from the H vs. Q characteristic curves of each individual pump, reducing them according to the system curves of each individual system (noncommon sections). For system B1-A, the following is obtained:

where: H₁: total head of pump 1, (m); H₂: total head of pump 2, (m); H_A: head at node A, (m); Z₁: height of the water level in the suction tank of pump station 1, with respect to an established reference plane, (m); Z₂: height of the water level in the suction tank of pump station 2, with respect to an established reference plane, (m); hf_B1-A: head losses along branch B₁-A, (m) and hf_B2-A: head losses along line B₂-A (m). and K_B1-A y K_B2-A: characteristic coefficients of secondary discharge pipes 1 and 2, respectively, (s²/m⁵).

their general form and for any number of pumps connected to the same node (node A in this example), these equations transform to obtain the general equation for a virtual pump BnA (Equation 9). This transformation process can be seen graphically in Figure 2.

These virtual or reduced head-capacity curves can be coupled in parallel, as shown in Figure 2 (right image), by summing the flow rates at the same total height. The intersection of this H_Pr-Q curve with the common piping system curve (pipe 3 in the left image of Figure 1) gives the total circulation flow rate.

These virtual or reduced head-capacity curves can be coupled in parallel, as shown in Figure 2 (right image), by summing the flow rates at the same total height. The intersection of this HPr-Q curve with the common piping system curve (pipe 3 in the left image of Figure 1) gives the total circulation flow rate.

The individual pump flow rates are obtained by intersecting the horizontal line (right image in Figure 2) with the individual reduced characteristic curves. These flow rates can then be used to obtain the head (reduced and actual), power, efficiency, and NPSH_r values for each pump, respectively.

The application of the Virtual Pump Method in a pumped source system is detailed below:
i. Determine (graphically or analytically) the actual load of the n pumps for different reference flow rates.
ii. Perform dynamic correction (reduction of the actual pump load due to pressure losses in the piping of each pump’s individual system) for each pump. This can also be done graphically or analytically.
iii. Calculate the static correction, which is obtained from the difference between the water levels in each suction tank and the virtual level established for the system, Z_sv. The latter could be calculated in several ways, all of them valid: as an average of the water levels in all the suction tanks in the pumping system, by setting a fixed value for Z_sv, which could be the smallest value of all the levels, the largest, or the one decided to establish as the standard for the n pumps. The disadvantage of the first option is that it would be necessary to compute n static corrections, while, for the other three variants, the number of static corrections to be performed would be n-1.
iv. However, from the point of view of ease of calculation, the latter choice would be the best option. Although, for the sake of greater agility in calculations and a better understanding of the conceptual idea of the procedure, it is customary to take a single value for the virtual suction level, several values may well be taken depending on each subsystem. In that case, the notation that would be established for the virtual suction levels of each subsystem would be Z_vNSn. Taking the well field in Figure 1 as an example, we would have Z_vNSA (for subsystem B-1, B-2 and secondary node A) and Z_vNSB (for subsystem B-3, secondary node A and main node B).
v. Compute for each pump, the virtual load corresponding to each reference flow according to the following expression:

where: H_nN: virtual head of pump n, (m); H_n: actual head of pump n, (m); hf_n: head losses in the individual system of pump n, (m); Z_n: height of the water level in the suction tank of pump station n, (m) and Z_sv: height of the virtual level set for the pumping system, (m).
vi. Obtain (graphically or analytically) the load-capacity characteristic curves of the virtual pumps B_nN at the node.
vii. Combine (graphically or analytically) the n virtual pumps in parallel to obtain the virtual equivalent pump, B_1-n, at the node.

Conceptually, once this last step has been completed, there would be a virtual pumping station in the node (example: node A for Figure 1 (image on the left)), where the n virtual pumps will be installed with their virtual equivalent in parallel, whose water level in the suction tank will be the virtual level established for the system, Z_sv.

Using the left image in Figure 1 as an example, the pumping system would be simplified to a virtual pumping station-pipedischarge tank scheme. The system could be further restructured by dynamically correcting the virtual pump combination in parallel, B_1-n, from the common pipe. In theory, this would be equivalent to having a pumping station with a virtual pump, B1- n-T, coupled to the suction tank or reservoir (common pipe length equal to zero).

As for the notation of virtual pumps during the process, for practical reasons, it is advisable to follow the approach proposed by Martínez and Riaño [10], which uses a coding system similar to the numerical hydrological classification of rivers. In this regard, the following general notation is formulated: Hm1-n; where the superscript m represents the number of virtualisations that have been performed up to a specific node, while the subscript 1-n represents the number and specific pumps that have participated in the virtualisation process up to that point. This notation criterion is recommended for large wellfield systems with multiple nodes.

For a better understanding of the methodological procedure for applying the Virtual Pump Method, the simplest scheme that can be proposed will be applied to an example of a pumped source system such as the one illustrated in the image on the left of Figure 1.

Results

Application example for a multi-node pumped source system (well field)

Pumps 1 and 2 used in this study are single-stage, deep-well submersible pumps [13]. High-density polyethylene (HDPE) will be used as the piping material, with an absolute pipe roughness of Є = 0.0000025 m. A kinematic viscosity of water of υ = 10-6 m²/s will be assumed. The water levels in the suction tanks are 10 m and 20 m for pumps 1 and 2, respectively. The internal diameter of both pipes is 200 mm. The equivalent pipe lengths are 600 m, 400 m, and 1,000 m for branches 1 and 2 and the common or collector pipe of the system, respectively. Applying steps 1 and 2 described for the procedure for working with the Virtual Pump Method, the results shown in Table 1 and Figures 3 and 4 are obtained.

In step 3, the static correction for the system is determined. For this, it must first be assumed what the virtual level, Z_sv, to be established for the system will be. Of the possible variants, it is proposed that it be the average of both water levels in the two suction reservoirs, i.e. 15 m. Therefore, the static corrections would be calculated as: (Z1 – Z_sv = 10 – 15 = -5 m) y (Z2 – Z_sv = 20 – 15 = 5 m) for pumps 1 and 2 respectively.

The calculated values of the virtual loads using equation 10 in step 4 are shown in Table 1. As part of step 5, the graphs of the virtual load-capacity curves for pumps 1 and 2 are illustrated in Figures 3 and 4, along with their representative polynomials. These figures also show the graphs of the actual Hvs. Q curves and the head loss curves for each system independent of each pump, along with their respective modelling equations.

Discussion

For the development of step 6, the following graphic and analytical paths are proposed:

Graphical solution

The graphical solution to the problem of two pumps 1 and 2 operating in parallel is obtained by constructing the curve for the two virtual pumps operating in parallel, as follows:
i. Virtual pump B_1A produces a flow rate Q₁ for the load HA (load at node A). Virtual pump B_2A provides a flow rate Q₂ for the same load HA. Both virtual pumps, working together against the load HA, produce the flow rate Q_A = Q₁ + Q₂ = Q_P. The point on the curve for working together is represented by the coordinates: H = HA and Q = Q_A = Q₁ + Q₂.
ii. For different values of H_A, the H_A vs. Q_A, indicated in Figure 5 as the B_1-2A coordinate, corresponds to a virtual pump that replaces the entire system up to point A.
iii. The intersection point of this curve with the characteristic curve of the piping system for the common section A-T (node-discharge tank), excluding the secondary suction and discharge pipes, is the solution sought to the problem, i.e., the actual operating point for this system. The data for the characteristic curve of the piping system for the A-T section are shown in Table 2.

From the graph in Figure 5, it can be seen that the operating flow rate of the system is 247 m³/h for a head of 20.6 m. For this load, the flow rates of each pump are: Q₁ = 137 m³/h and Q₂ = 110 m³/h.

Analytical solution The equations of the head-capacity curves of the original pumps, their respective virtual pumps, the characteristic curve of the A-T pipeline system (common section), and the curve of the parallel combination of the two virtual pumps operating against the collector pipeline system are shown in Figures 3, 4, and 5, in that order. For the case of this last formulation, it is only applicable within the interval where the flow contributions of the two virtual pumps join the A-T system.

Working with the analytical expressions, the following solutions are obtained: operating point for the A-T system: H_A = 20.624 m and Q_A = Q_P = Q₁ + Q₂ = 247.348 m³/h; for this value of H_A, the values of the flow rates supplied by each of the pumps are obtained: Q_1A = Q₁ = 137,599 m³/h and Q_2A = Q₂ = 109,749 m³/h.

Generalizing this procedure to a set of n pumps coupled in a multi-node parallel interconnected system (e.g., a well field as illustrated in Figure 1 (image on the right)), the procedure would be as follows: if a third pump were coupled, whose secondary drive is connected at node B in the aforementioned figure:
i. Curve B_1,2A is transformed into a second virtual curve by subtracting the pressure losses in section A-B from each load for each flow rate Q_1A + Q_2A, obtaining curve B_1,2B
ii. In a similar manner to how the remaining virtual curves for pumps 1 and 2 were obtained, the curve for virtual pump B_3B is obtained
iii. The two previous curves are combined to obtain curve B_1,2,3B
iv. The intersection of this curve with the characteristic curve of the piping system for section B-T is found, a confluence that defines the operating point for the three pumps 1, 2, and 3 operating together in parallel
v. As a final step, the values of HB and QB are obtained, which define the coordinate of the operating point for the case of three pumps operating in parallel.

Conclusion

The virtual pump method is a practical and simple procedure used to obtain fitted characteristic curves for parallel pump combinations. This concept of virtual pumps greatly simplifies the hydraulic calculation of real operating points and, in general, the physical understanding of the problem. This paper provides an introduction to the method and its fundamental hydraulic principles.

The procedure can be summarized as follows: Using the information from the pump characteristic curves and the piping system curves from the pumps to the piping system junction point, the latter are subtracted from the pumps’ load-capacity curves, thus obtaining the fitted pump characteristic curves for the load corresponding to the piping system junction point. Once this first step has been completed, the conventional method of summing the fitted curves can be applied, since they all refer to the same load.

A practical example of two pumps coupled in parallel is presented to illustrate this method more efficiently, showing its advantages and practical utility in simulating the operation of pumping systems.

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