1.Introduction

Water is essential for meeting both present and future human needs. However, its uneven distribution in time and space remains the principal barrier to reliable access [1, 2]. The importance of water supply systems increases as infrastructure becomes more complex. Therefore, a substantial body of research has addressed the challenges of improving these systems. Given the breadth of this literature, no single article can cover it in full, and specialized reviews have consequently emerged in particular subfields. A review of publications on the issues of public water supply and water scarcity is presented in [3]. It is indicated that fresh water availability in certain regions is minimal – in some cases insufficient even to meet basic population needs. The need for sustainable and reliable water supply is emphasized. To do this, a revision of water management system is proposed, particularly in water-scarce regions. General recommendations and possible solutions are discussed, such as advanced technological interventions that enhance water-use efficiency, the use of unconventional water resources, and others.

Water supply networks are essential for delivering high-quality water in sufficient quantities to consumers. A water supply system comprises interconnected hydraulic components that ensure the supply and distribution of water in accordance with demand. Optimization of water distribution networks is carried out in order to minimize resource use and costs or to enhance system efficiency and realize greater performance gains. Genetic algorithms (GA) are widely used to solve problems of nonlinear optimization in water supply systems.

In [5], a review of research on the optimization of water supply systems using GA methods is provided. These studies address the design of new water supply systems and the expansion, strengthening, and rehabilitation of existing infrastructure. A comprehensive analysis of all GA-based optimization studies spanning the period from the 1990s to 2021 is presented.

Determining pipe diameters is the simplest application of GA. A set of pipes is selected from commercially available standard sizes so as to minimize construction costs for a given network. Within the GA framework, the network structure is defined as a “chromosome”, a binary or integer sequence of numbers. The first GA simulation program analyzes each investment project in terms of nodal load. The second GA simulation program analyzes the cost of construction. A reasonable choice of the objective (penalty) function enables GA to converge on an optimal solution.

GA and related methods are based on the classical problem of determining the optimal economical diameter of a simple (unbranched) pipeline [6–9]. This problem was first solved in [10]. An increase in the internal pipeline diameter  on the one hand, raises construction cost through greater material consumption. On the other hand, a larger  reduces the hydraulic pressure loss in the pipeline, thereby lowering the operating cost of the pump motor. The capital cost of water supply is determined by the unit cost per pipe length:

                                                              (1)

where    are the empirical constants, the correction  takes into account valve cost, fittings and pipeline installation.

The operating costs of  are defined as the product of the power consumed by the pumping unit during operation and the cost of 1 kWh of electricity. The power required for pumping is the product of the volumetric flow rate (pump supply) and the pressure drop, divided by the efficiency of the pumping unit. In [10], the coefficient of hydraulic losses along the pipe length was adopted as for new steel pipes under turbulent flow (the resistance region of hydraulically smooth pipes). Total costs reduced to one year are given by:

                                                            (2)

where  is a coefficient that takes depreciation into account. Equating the derivative of the  function to zero yields the diameter  at which this function reaches a minimum. The approach proposed in [10] proved highly fruitful: a large number of subsequent publications used and developed it.

Modern research considers various special cases or accounts for factors that were previously overlooked. Many applications of the method [10] to the calculation of specific water supply systems have been published. In [11], for instance, two special cases of economic optimization of pipe diameter under turbulent flow are considered: hydraulically smooth and rough pipes. Different optimal pipe diameters were obtained for flow rate of 20 l/s: 155 mm for rough pipes and 147 mm for smooth pipes.

The calculation of hydraulic losses in particular cases made it possible to simplify the Genereaux equation. Hydraulic losses in local resistances have been taken into account, but static pressure drop was not. The approach considered in [11] is therefore unsuitable when water must be raised to a certain height.

Capital recovery factor is used to calculate the amount of each level payment to be made at the end of each  period, to recover the present amount at the end of the period, at a predetermined interest rate of  [12]. The factor is computed as follows:

                                                     (3)

A significant part of [12] focuses on the analysis of the dependence of pipe cost on diameter and on determining the empirical constants  and  in (1). Costs for pipes of various materials (two types of reinforced cement concrete, galvanized iron, and three types of polyvinyl chloride) with a diameter from 40 to 160 mm are curved. The index of  degree ranged from 0.61 to 1.53. In some of the graphs shown, the number of points is small and their scatter is large (e.g., PVC Grey Plain). This scatter is likely due to the fact that the author used price lists from various retail stores. Therefore, the representativeness of the sample is questionable. The solution of the set mathematical problem is postponed. Subsequently, the application of the Generaux method in combination with (3) to determine the economic diameter of the pipeline became known as Life-Cycle Cost Analysis (LCCA).

In [13], twenty different sections of a water pipeline in four regions of Brazil were considered, with length ranging from 533 to 45000 m and flow rate from 0.558 to 800 l/s. The material of the pipes is high-pressure polyethylene (HPPE). The calculated optimal diameters of  ranged from 50 to 1000 mm across various sections. The authors compare three methods for calculating  In fact, only one LCCA method was used (with accepted values of  = 0.12 and  = 20), but with two models for pipe cost:

the linear dependence of pipe cost on its weight; the linear dependence of pipe cost on inner diameter.

The results obtained made it possible to determine the values of the empirical coefficient  in the approximate Bresse formula:

                                                                   (4)

Values of  ranging from 0.6 to 1.7 were selected for different sections of the water pipeline. Clearly, this approach is untenable, since the value of  depends on a large number of factors.

In [14], an optimized selection of pipe diameter for a farm at Hamelmalo Agricultural College (HAC) in the Anseba Region of Eritrea was presented. The LCCA method was used to formulate the problem, and the head loss gradient method was employed to solve it. Another attempt to apply an approximate formula was made to determine the optimal pipeline diameter. In particular, Smith’s empirical formula was used:

                                                                  (5)

where  is the pipe diameter, mm;  is the system discharge, m3/h. The value of  is selected from a dedicated table. It can vary from 25 to 34 depending on the operating time of the pumping unit  (h/y) and friction losses. The value of  affects the annual cost of electricity. The annual pumping time was assumed to be 2920 hours, and  was determined to be 27.92 based on the analysis of hydraulic losses. It is clear that (5) is suitable only for preliminary, approximate estimates.

In [15], the results of calculated values of  under different flow regimes are compared: laminar, transient, and turbulent. The calculations were implemented using media of different viscosities. The volumetric flow rates  were 0.111, 0.167 and 1.667 m3/s, respectively. The pipeline length in all cases was the same  = 4000 m. Depreciation charges were 18 rubles per meter of pipeline length and per meter of diameter per year; the electricity tariff was 3 rubles per kWh. The calculated pipe diameters were 0.311, 0.392, 0.419 m, respectively. Therefore, with high viscosity of the pumped liquid, energy costs increase substantially.

The LCCA method was used in [16] to calculate the optimal diameter of the pipeline network on another farm located in the Anseba Region of Eritrea. The results for  were compared with those obtained using 11 approximate formulas from different authors, including (4) and (5). This comparison was based on calculated total pressure losses and an economic analysis of the pipe diameters of the network in question. The authors claim that the Bresse formula proved the most adequate compared with the LCCA model.

The number of publications using specialized software in the field is increasing [17–19]. The WaterGEMS program is used in [17] to optimize the water supply system of Wukro Town in Ethiopia. Neural networks are used in [18] to determine the inner diameter of a drinking water supply system. These programs rely on the same LCCA method. However, specialized programs are not always used correctly.

In [19], the optimal pipe diameter for transporting treated water in Karbala in Iraq using WaterCAD V8i software is determined. WaterCAD is used for designing and analyzing the operation of a water supply network. Options with four pipe materials (ductile iron, GRIP, concrete, and plastic) with diameters from 0.8 to 2.0 m are compared. Operating costs for pipes of different diameters and materials were given. GRP pipes had the highest operating costs. They are more than twice higher than those of concrete pipes. This is clearly an error. Operating costs are calculated based on electricity costs. The cost of electricity depends on the hydraulic losses along the pipeline. But GRP pipes have noticeably lower hydraulic losses than concrete pipes. The opposite was found in [19].

These works do not take into account that a decrease in diameter and an increase in hydraulic losses will require a more powerful pump. This means that the cost of the pump depends on the diameter. Moreover, this dependence is inverse. In [15], it was noted that the required pump power depends on the diameter. In particular, for fully developed turbulent flow in the pipeline, the required pump power depends inversely on  However, this fact is not taken into account in the costs.

It was proposed to take into account not only the dependence of pipeline cost on the diameter, but also the cost of pumping station on the required capacity in the investment costs [20]:

                                                                 (6)

where  is the cost of one installed kW pump at its actual capacity  It was proposed to calculate  as hydraulic power divided by efficiency (  = 0.77). In [20], the speed of water movement through pipes (0.8 m/s) is recorded instead of flow rate. The examples given have not been brought to a decision. The analysis of (1) has been performed. The dependence of  on  is not defined in (6). Optimal diameters have not been found.

A similar approach was used in [21]. It was noticed that the cost of a polyethylene pipeline is directly proportional to its weight. The mass was calculated using the formula:

                               (7)

where  is the ratio of the inner diameter of the pipe  to its wall thickness    is the density of polyethylene;  is the cost of 1 kg of HPPE. It is important that plastic pipes are selected from a line with the same ability to withstand a given pressure  which is determined by the  value.

In [21], in order to evaluate investment projects of water supply systems, in addition to discounting (interest rate) according to (2), it was proposed to take inflation into account. But the method of accounting for inflation expectations is not yet generally accepted, unlike (2). The hypothesis about the connection between the pipeline diameter and the capacity of the pumping station has been put forward:

                                                                   (8)

where    are the empirical constants of the regression dependence. They should be selected when processing the results of multivariate hydraulic modeling for each water supply network. There are no specific examples and no solution to the problem in [21].

The approach proposed in [20] was brought to its logical conclusion in [22, 23]. It was found that the dependence of pump cost on pump power is close to a linear. In [22], the power in this function was additionally multiplied by the ratio  (the ratio of suction diameter to discharge diameter). The cost per pipe unit length, directly proportional to its volume  was taken from price lists.

It is known that the volume of a cylindrical pipe can be found by the formula:

                                                       (9)

where  is the wall thickness of the pipe. But in [22], a formula like (9) does not contain a multiplier of 2 before the second term. Therefore, the calculation results cannot be considered reliable.

The approach proposed in [23] formally differs from the one in [20, 22], it is called “a new concept for the economic pipeline design”. In fact, it was a transition from the independent variable  to the “economic design velocity” variable  according to the formula:

                                                           (10)

Using the example of a simple pipeline, a mathematical model was presented that allows relating various types of annual costs to  The relationship between the design velocity and various pipeline parameters was presented. The influence of the pipeline length, static pressure, discount coefficient, and flow rate on the value of  was shown. Three different markets were considered in order to estimate the annual cost of the various pipeline components: those of Lebanon, the USA and India. Two different types of pipe materials were considered: ductile iron pipes and low-pressure polyethylene. The economic velocities for low-pressure polyethylene were 1.75 m/s, 1.00 m/s and 1.95 m/s for Lebanon, the USA and India, respectively. This approach is unlikely to hold much promise for design purposes. Apparently recognizing this, the authors of [23] calculated the values of  using  at the end of their article.

It is known that the power consumed by an electric pump unit  can be found by dividing the hydraulic power  by the efficiency

                                         (11)

where  is the density of water, kg/m3;  is the acceleration of gravity, m/s2;  is the pump head, m;  is the efficiency of the pump;  is the efficiency of the electric motor.

All the articles listed above use (11). A common disadvantage of these articles is the assumption that efficiency can be considered constant (  = const). In fact, efficiency depends on the design and power of the pump. Efficiency is a function of flow rate for a particular pump.

A method for calculating the optimal diameter of the pipeline using the pump performance characteristics was proposed in [24]. This made it possible to take into account the change in efficiency when switching to a different pipeline diameter. In addition, it was shown that the mass and cost of HPPE pipes should be calculated not according to (7), but according to the following formula:

                                     (12)

                   (13)

The use of the dependence of the consumed power  of the CNS-38 sectional pumps on the flow rate required additional recalculation to account for changes in the efficiency of the electric motor. Currently, many pump manufacturers include in the technical documentation not only the dependence of  (often denoted as ) on  but also the dependence of the consumed power of the electric motor    This makes it possible to improve the calculation method proposed in [24].

The aim of this article is to improve the method of calculating the optimal diameter  of the pipeline of a water supply system using the energy characteristics of an electric pump unit (accounting for changes in efficiency).

The tasks to be solved are as follows:

development of an improved method for calculating the optimal diameter verification of the significance of the change in efficiency at the operating point of the pumping unit when calculating investigation of the influence of various factors on

2.Materials and Methods

In addition to the water flow rate  the pipeline length  the static head  and the pipeline material HPPE, the cost of a pipe unit length, the cost and performance characteristics of pumps must be determined.

The investment and operating costs of the pipeline of the water supply system are presented in [22–24]:

                                (14)

where  is the investment cost with discount;  is a part of investment cost that does not depend on the pipeline diameter;  is annual operating cost;  is a part of operating cost that does not depend on the pipeline diameter.

To find the optimal pipeline diameter, we equate the derivative (14) to zero:

                                                        (15)

In order to find  the operating costs, depending on the pipeline diameter, are calculated using the following formula:

                                                        (16)

where  is the cost of electricity, $/kWh;  is the work of the pump’s electric motor per year, kWh;  is the pump’s operating time per day, h;  is the power consumed by the electric motor, kW;  is the number of days per year.

The consumed power of the electric motor was calculated using (11) in previous publications. One of the important improvements of the calculation method is that the value of  in this article is calculated based on the energy characteristics of the pumps used. Here we will use the performance characteristics of a line of vertical multistage pumps CR-64[1]. An example of the characteristics is shown in Fig. 1.

The dependence of the head  and the electric motor consumed power  on the flow rate  can be approximated by second-order polynomials:

                            (17)

where            are the empirical coefficients that are determined by the least squares method based on experimental data for each pump. For example, their values for CR-64-4 are shown in Fig. 1:  = 118.3 m,  = 0.1663 m/(dm3/s),  = –0.0918 m/(dm3/s)2,  = 12.04 kW,  = = 0.6904 kW/(dm3/s),  = –0.00827 kW/(dm3/s)2.

Figure. 1. Dependence of the CR-64-4 head and the electric motor consumed power on the flow rate. Points are experimental data[2]; lines are calculations using (17).

The characteristics of the pipeline will be calculated as in [22–24]:

            (18)

where  is the static head, m;  is the length of the pipeline, m;  is the coefficient of hydraulic friction losses along the length of the pipeline;  is the total coefficient of head losses in local resistances;  is the velocity of water in the pipeline, m/s.

The coefficient of hydraulic friction losses will be calculated using Altschul formula:

                           (19)

where  is the Reynolds number;  is the coefficient of kinematic viscosity of water, m2/s;  is the ratio of the absolute roughness of the pipe wall to the inner diameter. The analysis of experimental data in [24, 25] allowed for obtaining the dependence of this ratio on the diameter. For pipes of small and medium diameter (up to 340 mm) of Russian production, the  relationship is well approximated by a linear relationship:  = 8.09·10–5·  This means that in the specified range, the relative roughness of small and medium HPPE can be considered constant  ≈ 8.09·10–5.

An accurate calculation is not possible in extended pipeline. Instead, we increase  by 10 %, as in previous works. Then substituting (19) into (18), we get the function:

                 (20)

Next, for each pump of the CR-64 line, we will find the diameter of the pipeline  at which the flow rate will be equal to the set value  (at the operating point of the pumping unit). These values are determined from the equation:

                                                       (21)

The pump head of the CR-64  line and the electric motor expended power  at the operating point can be calculated by substituting  into (17).

Let us next find an approximation of the dependence of the consumed power on the values found:

                                                            (22)

where      are the empirical constants determined by the least squares method for the pipeline and pump line.

Let us substitute (22) into (16) and differentiate:

                                            (23)

The cost of electricity  does not remain constant over time. It may vary markedly in different countries and regions. Here we will use the tariff values of Rusenergosbyt LLC for Russian regions in January 2024 for consumers of the first price category (power consumption up to 670 kW, voltage up to 0.4 kV)[3].  = 0.04776 $/(kW·h) in the Irkutsk region,  = 0.07794 $/(kW·h) in the Kostroma region,  = = 0.1158 $/(kW·h) in the Leningrad region.

Let us find the first term in (15). The discount coefficient will be calculated using (3). We will find the cost of polyethylene pipes using (11). For certainty, we will take the cost of a unit of HPPE mass from the price list[4] (from the 13th of July 2024)  = 2.1 $.kg.

CR-64 pumps with different number of sections at maximum efficiency have the same nominal capacity (64 m3/h), but different nominal head  It is necessary to install a pump with a high  value while reducing the diameter of the pipeline. This, however, leads to an increase in investment costs. The cost of  pumps of the CR-64 line was taken from the price list[5]. Some of the source data are presented in Table 1. The number of impellers is indicated in the pump model designation after the number 64. If another digit follows, this indicates the number of impellers of reduced size. Note that in some cases reducing the impellers does not affect the cost of the pump: for example, CR-64-4 and CR-64-4-1 cost the same.

Table 1. Price list4 of CR-64 pumps with calculated pipeline diameters, head and power consumption (L = 3000 m, HC = 40 m).

No.

Model

CP, $

d1, m

HWP, m

NWP, kW

1

CR-64-2

7772.9

0.264

43.4

11.6

2

CR-64-3-2

8796.2

0.213

45.5

14.0

3

CR-64-3-1

8796.2

0.170

53.5

15.7

4

CR-64-3

10110.7

0.152

63.4

17.4

5

CR-64-4-2

10589.4

0.147

67.2

19.8

6

CR-64-4-1

11758.0

0.139

75.9

21.5

7

CR-64-4

11758.0

0.133

84.9

23.2

8

CR-64-5

12544.0

0.122

106.3

29.0

9

CR-64-6

15339.3

0.115

127.9

34.8

An approximation of the dependence of the pump cost  on the electric motor consumed power  (at the operating point) is necessary for calculating  This dependence turned out to be linear, as in [22, 23]:  Then the first term in (15) can be expressed as follows:

                                     (24)

Let us substitute (23), (24) into (15) and obtain the equations for determining the optimal diameter:

                            (25)

The algebraic equation (25) was solved numerically for different parameter values.

The efficiency at the working point of the pumping unit will be calculated using the following formula:

                                                       (26)

3.Results and Discussion

The power used and the pump pressure at the operating point depend on the pipeline parameters. Table 1 (column 4) shows the results of calculating such a diameter so the flow rate at the operating point is equal to the specified  The calculation was performed using the following parameters:  = 72 m3/h (20 dm3/s),  = 40 m,  = 3000 m. Moreover,  includes necessary free head from the consumer. The pump head and the expended electric motor power at the operating point of the corresponding pumping unit are placed in the last two columns of Table 1.

A graphical interpretation of the examples of determining the operating point of three pumping units from Table 1 with the specified pipeline parameters is shown in Fig. 2. According to it, in order to ensure the flow rate of  at d = 0.122 m, it is necessary to install the CR-64-5 pump (the upper point in Fig. 2). At  = 0.152 m, it is enough to install CR-64-3 pump (the lower point). The latter is noticeably cheaper.

Figure 2. Examples of determining the operating point of pumping units at Q0 = 72 m3/h, L = 3000 m, HС = 40 m: 1) CR-64-3, 2) CR-64-4, 3) CR-64-5; 4) characteristic of the pipeline with d1 = 0.152 m, 5) d1 = 0.133 m, 6) d1 = 0.122 m.

The data of their price list[6] and the approximation of the dependence of the cost CR-64 pumps on the electric motor consumed power at the operating point are shown in Fig. 3. The linear function is in good agreement with the points. The determination index, adjusted due to the small number of points, is 0.95. In this case,  =  = 309.21 $/kW.

Figure 3. Relationship between the cost of CR-64 pumps and the electric motor consumed power at the operating point.

Fig. 4 shows that the calculation results according to (22) are in good agreement with the experimental points. The empirical constants in this case are  = 11.1,  = 0.00274,  = 4.239. The adjusted index of determination is 0.986.

Figure 4. Dependence of the electric motor consumed power (NWP) at the operating point and the derivative (NWP′) of CR-64 units on the pipeline diameter. Points are experimental data; lines are calculations using (22).

The results of (25) solved by numerical method for various parameter values are presented in Tables 2–4 and in Figs. 5 and 6:  = 30,  = 0.12, discount coefficient  = 0.124,  = 0.07794 $/(kWh).

The efficiency decreases from 0.793 to 0.648 with an increase in water consumption  from 60 to 80 m3/h according to Table 2. Efficiency  = 0.692 at  = 1000 m,  = 0.753 when increasing the length of the pipeline to 9000 m according to Table 3. Efficiency  = 0.637 at  = 10 m,  = 0.753 with an increase in static head to 40 m according to Table 4. The efficiency changes very significantly. The change in efficiency is less significant (Figs. 5 and 6). Therefore,  = 0.754 at  = 4 h,  = 0.720 at  = 24 h (Fig. 5, line 1,  = 21),  = 0.742 at  = 4 h,  = 0.737 at  = 24 h (Fig. 5, line 3,  = 13.6). In Fig. 6, the efficiency increases from 0.731 to 0.755 with an increase in  from 0.05 to 0.3 (line 2, the project lifecycle is 15 years). At a constant value of  = 0.1, if  = 15 years, then  = 0.740; if  = 30 years, then  = 0.734.

Table 2. Effect of Q0 on the optimal diameter and efficiency, L = 3000 m, HC = 30 m, T0 = 12 h.

Q0, m3/h

d0

Hwp, m

Nwp, kW

η

60

0.147

47.1

11.6

0.793

64

0.151

48.2

12.5

0.758

68

0.154

49.6

13.4

0.725

72

0.157

51.3

14.4

0.695

76

0.160

53.5

15.7

0.669

80

0.162

56.1

16.9

0.648

Table 3. Effect of L on the optimal diameter and efficiency, Q0 = 68 m3/h, HC = 30 m, T0 = 24 h.

L, m

d0

Hwp, m

Nwp, kW

η

1000

0.175

31.9

9.0

0.692

2000

0.172

32.2

10.2

0.694

3000

0.170

41.2

11.4

0.707

6000

0.167

57.2

15.2

0.739

9000

0.166

71.9

18.7

0.753

Table 4. Effect of Hc on the optimal diameter and efficiency, Q0 = 68 m3/h, L = 3000 m, T0 = 24 h.

Hc, m

d0

Hwp, m

Nwp, kW

η

10

0.155

26.5

8.1

0.637

20

0.154

37.8

10.8

0.688

30

0.154

49.6

13.4

0.725

40

0.153

62.8

16.2

0.757

Tables 2–4 and Figs. 5 and 6 allow for analyzing the influence of various factors on the value of the optimal pipeline d