# Calculating the Economic Level of Friction in Pressurized Water Systems

^{*}

## Abstract

**:**

_{0}) in a pressurized water system. J

_{0}represents the economic level of friction losses (ELF), which is dependent on the network’s behavior as well as other parameters, including energy and the pipe costs. As these have prices changed over time, so has the value of J

_{0}. The network-related parameter was obtained from the total costs function and the sum of the operational and capital expenditures. Because these costs exhibited an opposite trend from J, a minimum total cost exists, specifically, J

_{0}. The algebraic expression, which was derived from the mathematical model of the network, was first calculated for the network’s steady state flow and was later generalized for application to a dynamic one. For a network operating in a given context, J

_{0}was fairly stable in terms of dynamic flow variations, providing valuable information. The first piece of information was the ELF itself, which indicated the energy efficiency of the system from the perspective of friction loss. The second indicated which pipes required renewal from a similar perspective. Thirdly, it provided a simple criterion to calculate the diameter of new pipes. Finally, as J

_{0}can be easily updated, when predictions are performed at the network’s designed time fail (e.g., growing urban trends, demand evolution, etc.), decisions can also be updated.

## 1. Introduction

_{0}), establishing a pragmatic target for these friction losses is possible.

_{0}is a dynamic global metric (a unique value for the whole network) that (1) answers an important energy-related question: what friction losses are reasonable in a water network? (2) J

_{0}sets pipe renovation priorities based on the friction losses criterion; (3) helps to define the right diameter for new pipes from J

_{0}and the average circulating flow; (4) allows for the determination of the impact of leakage and pumping inefficiencies on the optimum design of the network; (5) enables the updating of renewal policies to changing context conditions; and (6) in leaky networks, helps to determine the necessary leakage reduction leading to J ≈ J

_{0}.

_{0}, a summary of the hydraulic gradient concept J, is presented. Thirdly, the total costs function, which is the operational expenditures (OPEX) plus the capital expenditures (CAPEX), is established. OPEX costs increase with J, whereas CAPEX costs decrease. Therefore, a minimum value of the total function exists for J

_{0}. Fourthly, to provide clarity to the process, J

_{0}is calculated for the network’s steady state flow. Lastly, the model is extended to dynamic flows, and, to validate the process, two networks flows, static and dynamic, are analysed.

_{0}. However, a sensitivity analysis has demonstrated that J

_{0}is dependent on the context parameters and consequently, tracking the variation in J

_{0}with time is necessary.

## 2. Energy Balance of a Pressurized Water Network

_{1}[5] specifies whether the total energy, E

_{I}, injected into the system is natural (gravitational), E

_{N}, or supplied by pumping (shaft), E

_{P}. The values range from one, which indicates all energy is natural, to zero, in which all energy is derived from pumping. The natural energy, which is hydraulic in nature, is not affected by any reduction, whereas some of the shaft energy is lost due to inefficiencies in the pumping station. These inefficiencies are quantified by the global efficiency parameter η and are dependent on the operating point of the pumps and complementary devices, such as electric motors.

- Compensation energy, ∆E
_{c}. The energy involved in filling and emptying the compensation tanks. Energy is consumed as the tanks are filled and is released when they are emptied. Depending on the analysis interval, either a net consumption or supply of energy exists. When the energy balance is extended over a long-time period, for example, one year, the net result is zero. - Minimum energy required by users, E
_{uo}, is calculated from the user’s demand and the pressure set by the standards. - Excess energy, E
_{e}, is energy that is over the minimum required by the users. This is avoidable energy. - Topographic energy, E
_{t}. This is the energy required by the system as a result of the topography or the configuration. A portion of this energy is unavoidable unless the design is changed [6]. The rest may be managed with pressure reducing valves (PRV) or with pumps as turbines (PAT). - Energy is embedded in leaks or is lost when water is depressurized in domestic tanks.
- Energy dissipates in pipes through friction E
_{f}; this energy is the focus of this paper.

## 3. Optimum Hydraulic Gradient of a Static Network Flow

_{0}was calculated assuming that:

- Current prices apply. Using current prices for pipelines that have been operating for decades might be considered incoherent. However, current optimum friction cannot depend on a network’s depreciation, and using current prices appears reasonable to achieve the pursued objective. Moreover, when it is time to renew the main (J
_{0}will indicate the new diameter), it will be replaced with pipes at current prices. - Cost variations over time were not considered; firstly, because the optimal hydraulic gradient J
_{0}is a value that depends on the current costs of both energy and materials and therefore changes over time, as do the costs, and must be updated. Secondly, the cost variations were not considered because its update helps to analyze the impact of new policies related to the increase in water and energy tariffs as well as the implementation of new environmental taxes. - The analysis can be performed at all times with current data as opposed to only completing it at the beginning stage, using design hypotheses.
- Only the most significant economic factors, capital and energy costs that are both J-dependent, were considered. Other costs, such as investments in pumping stations, were not included because their dependence on J is minimally relevant. Other costs should only be included if the investment costs are strongly dependent on friction losses, such as pumping costs in a closed loop, and these costs account for a significant percentage of the overall investment.
- The flow rates for the whole system were obtained from the network mathematical model.
- Inefficiencies were coupled and consequently considered. Leaks were included in the model; therefore, flow rates were higher, as were investment and energy costs, as shown in Equations (4) and (16), respectively. The global efficiency η
_{g}that accounts for losses at pumping stations is a multiplying factor of the dissipated energy, as shown in Equation (11). - The model does not consider local losses because they represent a low percentage of the total friction. If they were to be considered, the corresponding equivalent length would be added only in the energy term to the corresponding main length.
- The friction factor was assumed to be constant and equal to the average value resulting for the entire system. This hypothesis could appear to be an over-simplification; as in a network, the friction factor is highly variable, although, in practice, it is not. An analysis of sensitivity proves this in the Sensitivity Analysis section.

#### 3.1. Cost of Installed Pipes

_{p}is calculated [16] from:

_{0}(p) is the constant pipe cost. This figure depends on the nominal pressure, p, and on the pipe material. Both factors are assumed invariable in the analysed system. The value is calculated from commercial values. D is the pipe diameter and c is an exponent depending on the pipe material and the manufacturing process. The literature reports a wide range of possible values for c. The widest range was considered, which covers all possible values, 1 ≤ c ≤ 2 [17] to other slightly more restrictive ranges of 1 ≤ c ≤ 1.75 [18]. In a later paper, Swamee et al. [19] proposed the lowest value reported in the literature, c = 0.866, which is valid for smaller diameter pipes. In each case, the exponent that best fits the catalogue prices should be adopted. Other documented values include 1.51 [20] and 1.24 [21]

_{0}(p) were determined from the price catalogue. In the two examples provided in this paper (PVC and PE materials with nominal pressures of 6 and 10 bars, respectively), the final value for both cases was c = 2. Nevertheless, for ductile cast pipes with a nominal pressure between 30 and 64 bars and 32 and 85 bars, depending on diameter, the c values that best follow the prices trend were 1.6 and 1.2, similar to previously documented figures.

_{i}, which is sensitive to local costs and to the location where the pipe will be installed (urban or peripheral) as well as to its use (irrigation or urban). In short, the final price cost, C

_{pf}, of the installed pipeline is:

_{i}, range from 1.5 (irrigation use) to 3.5 in city center urban water networks. Therefore, each network requires its own individual analysis.

#### 3.2. Annual Cost of the Network as a Function of the Hydraulic Gradient

_{f}per length L) with the flow rate q, as follows:

_{p}(f, p), is the pipe cost constant, equal to:

^{3}/s) and D (m) must be used.

_{i}, and a generic hydraulic gradient J, the total cost (in today’s values) of the whole network C

_{N}, is:

_{N}, is defined as the product of the cost constant k

_{p}and $\sum {l}_{i}{q}_{i}{}^{\frac{2c}{5}}$, we obtain:

_{N}, the average pipe life span n, and the hydraulic energy grade line J.

#### 3.3. Annual Energy Cost Dissipated Through Friction

_{fy}needed to overcome friction losses in pipes is:

_{g}represents the global pumping station performance (annual average value), the annual dissipated energy will be:

^{3}and h in hours/year, E

_{fyg}results in kWh/year.

_{1}> I

_{3}, where I

_{3}is the ratio between E

_{f}(energy dissipated in pipes through friction) and E

_{I}(total energy injected).

_{p}is, through c and n, a pipe dependent parameter. In short:

_{p}, pipe dependent from Equation (15); λ

_{N}, network cost-dependent from Equation (7); and e, energy cost dependent from Equation (12) prove that J

_{0}is a parameter that also depends on the system and working conditions.

## 4. Refining the ELF Calculation Model

#### 4.1. Networks with Different Materials and Nominal Pressures

_{0}, assumes material and pressure uniformity in the network. If this is not the case, the system has to be divided into as many sectors (k denotes the sub-system) as the materials present, finding the function to optimize each one as follows:

#### 4.2. The Energy Supplied to the System is a Combination of Natural and Shaft Energy

_{1}(previously discussed) and I

_{3}(the ratio between E

_{f}and E

_{I}). This second indicator assesses the weighting of friction losses.

_{1}> I

_{3}), a friction reduction has a direct effect on economic savings because it supposes a reduction in the energy consumed in pumps. This, predictably, is the most common scenario.

_{1}> I

_{3}, the energy saved by the decrease of friction is not transferred directly to the energy bill (or only a part). In this case, from an economic point of view, the price of energy should be affected by a factor that compares the savings that have an impact on the pumped energy versus the total energy savings. This approach should be modified if the natural energy saved is recoverable through turbines.

#### 4.3. Energy Costs Calculated from Real Rates with Environmental Taxes

_{ep}, the power cost (in €/kW and month) p

_{ew}, the energy cost (in €/kWh), and h

_{m}the monthly hours of operation, the bill C

_{em}for that period leads to:

_{em}is monthly dependent, which must be considered when extending this average monthly cost to a year. In any case, a similar procedure should be followed. The average yearly cost $\overline{{p}_{e}}$ is calculated by dividing the yearly energy cost (C

_{ey}) by the total energy (kWh) consumed throughout the year (${h}_{y}\xb7\gamma \xb7\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\eta}_{g}$}\right.\sum J\xb7{l}_{i}{q}_{i}$). On the basis of tariff structures and the economic balances, the average energy prices for short, medium, or long periods of hours, months, and years, respectively, were calculated. Notably, as the excess of required power over the contracted value can be avoided, this additional cost was not included.

_{et}, proportional to energy consumption, can be added to the energy cost. In this environmental context, a comparison of the sustainability of pipeline water transport with conventional means of transportation is important, a simple exercise using MJ/tkm [23] as transport intensity unit I

_{t}instead of the traditional kWh/m

^{3}. The friction energy resulting from transport, as the one required by users does not depend on transport, needs to be considered in MJ; it must then be referred to the displaced load (tkm). In short, I

_{t}is the quotient between the work required to overcome friction and the load factor, both per unit of time, leading to:

_{t}, as works quotient, is dimensionless. Nevertheless, this indicator is singular because energies are in different unit systems. In the numerator of Equation (21), the specific weight of water is expressed in MN/m

^{3}(γ = 9.81 × 10

^{–3}MN/m

^{3}), whereas the denominator is in t/m

^{3}(γ = 1 t/m

^{3}), leading to:

## 5. ELF Calculation in a Dynamic Network

_{Nj}, the network hourly cost factor, depends on the hourly flow rate q

_{ij}through the equation:

_{j}is the annual number of hours that the system operates in the interval period j, with $\overline{{p}_{ej}}$ and $\frac{1}{{\eta}_{gj}}$ being the average energy price and global pumping performance in that period, respectively.

_{p}in Equation (15) remains constant.

_{0j}is the optimum J

_{0}. Notably, once a J

_{0}for a given time has been adopted, the costs for the remaining periods depend on the selected gradient. The total cost for each hourly interval J

_{0j}must be calculated using the equation:

_{0j}and costs as the axes from among the 24 possibilities, a minimum value was found. The figure shows the absolute or relative characteristic of the selected minimum value. In very singular cases, such as firewater networks with practically zero energy costs, as J

_{0}increases, the required investment and, accordingly, the capital costs in the absence of OPEX decreased. In this case, pumping costs, which are dependent on J, and other criteria, such as a maximum value for the velocity, should be considered. Otherwise, the optimum would not exist (J

_{0}→∞).

## 6. Case Studies

#### 6.1. Case 1: Static Flow Irrigation Network

#### 6.1.1. Network Annual Capital Cost

_{0}(p) was calculated and the exponent c was adjusted. Results were a

_{0}(p) = 256.29 €/m·m

^{2}and c = 2 (Figure 3).

_{i}= 1.6; and pipe life span, n = 50 years. The network cost parameter λ

_{N}was:

#### 6.1.2. Network Annual Capital Cost

_{g}= 0.7. From these values, and taking into account the network load factor, the annual energy cost e was:

_{0}with the actual value $\overline{{J}_{a}}$, at 1.35 m/km, which was the average head loss weighted with flow and length for the entire pipe network, we can conclude that, for the actual operating conditions at current prices, the friction that occurred in the network was above the optimum value of friction. Therefore, the network was convenient to study from a cost-benefit point as well as the possibility to reduce friction losses in the network.

_{1}was 0.217 (i.e., 78.3% of the energy supplied to the system is shaft energy), whereas I

_{3}was 0.044, meaning only 4.44% was friction loss. Therefore, shaft energy exceeded friction inefficiencies (clearly 1 − C

_{1}> I

_{3}) and, therefore, energy costs were correctly estimated.

_{t}, from a strictly economic point of view, was 0.024 MJ/tkm.

#### 6.2. Case 2: Dynamic Urban Network

_{0}(p) = 583.38 €/m·m

^{2}and c = 2. A mean friction factor f = 0.016 was assumed, whereas the installation factor (higher than in the previous urban network case) was F

_{i}= 2.3. As a final point, the same pipe lifespan was adopted (n = 50 years).

_{0,j}are outlined in Table 1.

_{Ty,j}in Equation (33) was calculated assuming a constant value of J

_{0,j}for the entire day but in consideration of both the daily demand pattern and, accordingly, the flow rate for each hour. Once the total cost for each value of J

_{0,j}was calculated, the final solution J

_{0}was the one with the lowest cost. In this case, J

_{0}was 1.65 m/km, which matches the optimum gradient calculated for 4:00 p.m., with a total annual cost of 6512 €, of which 72% was capital costs and 28% was energy costs. Figure 5 shows the results for each time interval as well as the existence of a total minimum cost (optimum). In this case, the value almost coincided with that corresponding to J

_{0}= 1.65 m/km. Should the costs curve not show an absolute minimum value, an exploration of other J values outside the interval defined by the hourly demand patterns would be necessary; in this case, from 1.242 to 4.081 m/km, but such values must always be within reasonable limits, for instance, from 0.5 to 6 m/km. Only in very rare cases, such as with very short operating times in a fire network, the minimum value will not be within the explored interval.

_{0}in systems with significant time flow variations is convenient, such as for cities with a high seasonality factor. Figure 6 illustrates this evolution, showing the limited impact of these variations. Taking into account the same pattern but with different demand factors, from 0.5 to 2.9 m/km, the variation in J

_{0}was quite moderate, ranging from 1.77 to 1.44 m/km (Figure 6). A variation that has little impact on J

_{0}led to a discrete set of solutions (diameters), and similar J

_{0}values will result in the same commercial diameter. Conversely, the impact of this short interval is, in practice, very low. As the range of commercial diameters is discontinuous, regardless of the J value, the solution is the same. For example, in this case study, we obtained identical solutions (diameter = 400 mm) using the upper value (J = 1.77 m/km, D = 376 mm), the lower value (J = 1.44 m/km, D = 392 mm), or the optimum solution (J

_{0}= 1.65 m/km, D = 380 mm).

_{0}dynamic variability can be explained by Equation (20). Both the numerator λ

_{N}and the denominator e are sensitive to the circulating flows. However, whereas the exponent for the former is 2c/5, which is 0.8 when c is 2, the dissipated energy e increases linearly with the flow. Therefore, as the denominator increases at a faster rate than the numerator, the final value of J

_{0}decreases (Figure 6). However, because both exponents were similar, the result barely changed. Lower values of c in the pipe cost function reduced the weight of the numerator, whereas the energy factor was maintained. This slightly diminished the value of J

_{0}; however, the 2/5 multiplying factor mitigated this effect.

## 7. Sensitivity Analysis

_{0}for different f values from the lowest to the highest, as found in the network analysis, the plotted results show a quasi-flat curve. The selected J

_{0}(f = 0.014) is the network weighted average.

_{0}was very sensitive to certain pipe cost parameters, such as the pipe lifespan n (Figure 8). By shortening the life of pipes, the capital costs became relevant and J

_{0}increased to mitigate this impact, whereas an increase in life expectancy produced the opposite effect. This behavior was not symmetrical. Although a 60% reduction (20 years) in the lifespan of a pipe increased the value of J

_{0}from 1.75 to 3.38 m/km, a similar extension in its life (60% or 80 years) had a reduced impact on the change of J

_{0}(1.75 to 1.25 m/km). The remaining parameters changed linearly.

_{0}sensitivity to two key OPEX parameters: energy cost and operating hours of the system. A dramatic variation occurred when the system only operated a few hours per year. Fire water networks were the most extreme case, where h→0. In the second example, a similar influence of the energy cost in J

_{0}was found.

_{0}with leakage. The sensitivity was low; even when the flows were duplicated (50% efficiency), the optimum gradient barely changed (1.75–1.59 m/km). This was hardly surprising since, in the same system and with an identical framework, J

_{0}varied little with increased flows, as previously shown for the demand factor (Figure 6).

_{0}(Figure 11).

_{0}. Secondly, if the cost of the pumping station is a small percentage of the total cost of the network, the effect is also minimal. Currently, in urban or irrigation networks, this may account for 5% of the total costs. Therefore, we conclude that including pumping costs in the CAPEX is not necessary. Perhaps this would not be the case with a closed industrial water circuit.

## 8. Results Analysis and Validation

_{0}was based on the validity of the supporting hypotheses, on the quality of the mathematical model of the network, and on the consistency of the context parameters. The process to calculate J

_{0}was deterministic; it did not rely on random factors or on forecasts.

_{0}and the hours of operation was obvious. In this example, as the network only operated during off-peak hours, the energy cost was constant. The second network corresponded to a District Metering Area, a subsystem of an urban network, which served a stable population. This example shows that J

_{0}was only slightly sensitive to dynamic changes. To demonstrate this, large flow variations were introduced through the demand factor (Figure 6). In practice, this low variability had a limited impact (in practice, the solution was essentially the same) and the adopted J

_{0}corresponded to the average demand factor. The same effect was analysed in the irrigation network through the hydraulic network efficiency in this case (Figure 10). In the end, we obtained identical conclusions.

_{0}showed high sensitivity to both kinds of costs: first, to CAPEX, through the pipe cost constant value a

_{0}, the diameter exponent in the cost variation equation exponent c, the installation costs F

_{i}

_{,}and the average life, n (Figure 8); and, second, to OPEX, through the time of activity (hours of operation per year), the price of the energy (Figure 9), and the tariff structure, and, to a lesser extent, the pumping efficiency (Figure 11).

_{0}and simultaneously proves that operational costs had a greater impact on the result than the dynamic behavior of the network. An initial analysis suggested, in an apparent contradiction with the above, that J

_{0}was quite sensitive to dynamic flow changes, as the range of 1.215 to 4.06 range was very wide. However, a deeper analysis showed that these values were greatly influenced by the hourly price of energy and were much less sensitive to the flow rates. This was evidenced during the period when the price of energy was lowest (0.042930 €/kWh), as the values of J

_{0}were surprisingly high, showing a much greater influence than that of the circulating flows.

_{0}prevented any kind of comparison, even between similar networks. Results must be compared in consideration of the context of the system. Therefore, such a comparison makes sense within the same network (J

_{0}vs. $\overline{{J}_{a}}$). An analysis of a set of real networks would be interesting, assuming that all the information is available, and comparing J

_{0}with $\overline{{J}_{a}}$ in all networks. These differences would provide relevant information on how, in practice, context information affects the gap ($\overline{{J}_{a}}$ – J

_{0}) and when and where networks have or have not been well designed.

## 9. Conclusions

_{0}and the current value, $\overline{{J}_{a}}$, to determine if friction losses are economically acceptable in a given network and context. This metric plays a similar role to the well-established economic level of leakage (ELL). Both are economic benchmarks of network inefficiencies and both are highly dependent on context parameters and service conditions. In addition, J

_{0}is a reliable substitute for the guide values provided in the literature for the hydraulic gradient J and can update the pipe replacement criteria.

_{0}is well established, available metrics can independently assess the three main energy inefficiencies, friction, leaks, and pumping, in a water network. Although these inefficiencies are dependent on each other, they can be determined independently for a given load condition. The overall energy efficiency of a system is a combination of these three partial efficiencies. The final result will only be maximized if the existing interdependence is adequately addressed. For instance, it is illogical that the most efficient point of working pumps corresponds to a leaky system with excessive friction losses. This paper has clarified this issue and is therefore the starting point for further global assessment.

## Author Contributions

## Conflicts of Interest

## Notation

${a}_{0}\left(p\right)$ | Constant, depending on working pressure and material cost of the pipe |

$c$ | Adjustment exponent of material cost evolution |

${C}_{1}$ | Energy source context indicator |

${C}_{ek,y}$ | Annual energy cost dissipated through friction of sub-system k |

${C}_{em}$ | Monthly energy cost |

${C}_{ey}$ | Annual energy cost dissipated through friction |

${C}_{eyj}$ | Annual cost of energy dissipated through friction for the load status in period j |

${c}_{k}$ | Adjustment exponent of material cost evolution of sub-system k |

${C}_{N}$ | Network cost |

${C}_{Nk,y}$ | Annual repercussion of network cost of sub-system k |

${C}_{Ny}$ | Annual repercussion of network cost |

${C}_{Nyj}$ | Annual repercussion of the network cost for the load status in period j |

${C}_{p}$ | Cost per linear meter pipe. Pipe unitary price |

${C}_{pf}$ | Final price of the installed pipeline |

${C}_{pf,i}$ | Final price of the installed pipeline i |

$D$ | Pipe diameter |

$e$ | Energy cost factor |

${e}_{j}$ | Energy cost factor for period j |

${e}_{m}$ | Monthly energy cost factor |

${E}_{e}$ | Excess of energy |

${E}_{fy}$ | Final annual energy dissipated in pipes through friction |

${E}_{fyg}$ | Final annual energy consumed in pipes through friction |

${E}_{I}$ | Total energy injected into the water pressurized water network |

${E}_{N}$ | Natural energy (gravitational) delivered to the water pressurized water network |

${E}_{P}$ | Pumping energy (shaft energy) supplied into the water pressurized water network |

${E}_{t}$ | Topographic energy |

${E}_{uo}$ | Minimum energy required by users |

$f$ | Friction factor |

${F}_{i}$ | Installation factor |

${f}_{p}$ | Investment–installation–construction factor |

${f}_{pk}$ | Investment–installation–construction parameter of sub-system k |

${G}_{Tk,y}$ | Total annual network cost of sub-system k |

${G}_{Ty}$ | Total annual network cost |

${G}_{Tyj}$ | Total annual cost of the network cost for the load status in period j |

$h$ | Number of operating hours |

$H$ | Piezometric head |

${h}_{f}$ | Pressure losses |

${h}_{j}$ | Number of operating hours in period j |

${h}_{m}$ | Monthly operating hours |

${I}_{3}$ | Friction energy indicator |

${I}_{t}$ | Transport energy intensity (MJ/tkm) |

$J$ | Hydraulic gradient |

$\overline{{J}_{a}}$ | Average actual head loss weighted with flow and length for the entire pipe network |

${J}_{j}$ | Hydraulic gradient for the load status in period j |

${J}_{0}$ | Optimum hydraulic gradient |

${J}_{0j}$ | Optimum hydraulic gradient in period j |

${J}_{0k}$ | Optimum hydraulic gradient of sub-system k |

${k}_{p}\left(f,p\right)$ | Pipe cost constant |

${l}_{i}$ | Pipe length i |

$n$ | Useful life in years of pipes |

${n}_{k}$ | Useful life in years of pipes of sub-system k |

$p$ | Pressure |

$\overline{{p}_{e}}$ | Average price of energy |

$\overline{{p}_{ej}}$ | Average price of energy in period j |

${p}_{ep}$ | Price of power term |

${p}_{et}$ | Penalty parameter for greenhouse gas emissions |

${p}_{ew}$ | Price of energy term |

${q}_{i}$ | Flow rate through pipe i |

${q}_{ij}$ | Flow rate through pipe i in period j |

$\alpha $ | Pipe cost adjustment coefficient |

$\gamma $ | Specific weight of water (N/m^{3}) |

$\mathsf{\Delta}{E}_{c}$ | Compensation energy |

${\eta}_{g}$ | Overall pumping station performance |

${\eta}_{gj}$ | Overall pumping station performance in period j |

${\lambda}_{N}$ | Installation cost factor |

${\lambda}_{Nj}$ | Installation cost factor for the load status in period j |

${\lambda}_{Nk}$ | Installation cost factor of sub-system k |

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**Figure 8.**J

_{0}sensitivity to pipe’s cost (capital expenditures, or CAPEX) parameters (case 1, steady network).

**Figure 9.**J

_{0}sensitivity to two significant operating expenditure (OPEX) parameters (case 1, steady network).

Hour | 00:00 a.m. | 01:00 a.m. | 2:00 a.m. | 3:00 a.m. | 4:00 a.m. | 5:00 a.m. | 6:00 a.m. | 7:00 a.m. | 8:00 a.m. | 9:00 a.m. | 10:00 a.m. | 11:00 a.m. |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${J}_{0,j}$ (m/km) | 3.842 | 4.061 | 4.081 | 3.723 | 3.969 | 3.760 | 3.636 | 3.489 | 1.630 | 1.614 | 1.606 | 1.624 |

Hour | 12:00 p.m. | 1:00 p.m. | 2:00 p.m. | 3:00 p.m. | 4:00 p.m. | 5:00 p.m. | 6:00 p.m. | 7:00 p.m. | 8:00 p.m. | 9:00 p.m. | 10:00 p.m. | 11:00 p.m. |

${J}_{0,j}$ (m/km) | 1.632 | 1.648 | 1.646 | 1.628 | 1.650 | 1.229 | 11.242 | 1.242 | 1.247 | 1.230 | 1.258 | 1.781 |

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**MDPI and ACS Style**

Cabrera, E.; Gómez, E.; Cabrera, E., Jr.; Soriano, J.
Calculating the Economic Level of Friction in Pressurized Water Systems. *Water* **2018**, *10*, 763.
https://doi.org/10.3390/w10060763

**AMA Style**

Cabrera E, Gómez E, Cabrera E Jr., Soriano J.
Calculating the Economic Level of Friction in Pressurized Water Systems. *Water*. 2018; 10(6):763.
https://doi.org/10.3390/w10060763

**Chicago/Turabian Style**

Cabrera, Enrique, Elena Gómez, Enrique Cabrera, Jr., and Javier Soriano.
2018. "Calculating the Economic Level of Friction in Pressurized Water Systems" *Water* 10, no. 6: 763.
https://doi.org/10.3390/w10060763