1 Introduction

Nowadays, the development of mathematical models to analyse the behavior of hydraulic systems requires theoretical and experimental laws. In some cases, the direct application of these equations can lead to erroneous results (Simpson and Marchi 2013), being necessary to correct them in order to consider the necessary simplifications in the initial assumptions (e.g. viscosity effects, friction losses, turbulence, vortex). In water distribution networks, the model simulation with installed hydraulic machines is very common, and the energy analyses have a great significance due to the increasing price of the energy (Corominas 2010) and the need to reduce the energy consumption and the system efficiency to satisfy the European standards requirements (Pasten and Santamarina 2012).

Traditionally, these optimizations of energy consumption in water systems have been focused on the reduction of consumed power by installed pumps. Hence, some authors have worked in the development of pumped systems to adapt the rotational speed of the machine to reduce the energy consumption (Sarbu and Borza 1998; Moreno et al. 2010; Simpson and Marchi 2013; Jiménez-Bello et al. 2015) and the pressure in the system (Kevin 1990; Giustolisi et al. 2008; Cabrera et al. 2014). These reductions are of paramount importance in economic and environmental savings, which have been analyzed through different algorithms and software such as EPANET (Rossman 2000) or WaterGEMS (Nazari and Meisami 2008), providing significant tools for water management in pipe systems.

In the last years, different authors have developed researches using new technologies to leverage the pressure reduction in water distribution networks, increasing the global efficiency in the water system (Abbott and Cohen 2009; Dannier et al. 2015; Pérez-Sánchez et al. 2017). Araujo et al. (2006) and Giugni et al. (2014) enumerate different algorithms to optimize the location the optimal place of the Pressure Reduction Valves (PRVs) in a network for leakages control. The initial study of Pump Working as Turbines (PATs) that are installed in water systems to replace PRVs (Ramos and Borga 1999) allowed modellers to analyze the turbine behaviour according to different aspects such as:

  • The morphology of the machine (Yang et al. 2012; Carravetta et al. 2013; Shi et al. 2015).

  • The design of installation schemes and operation strategies (Carravetta et al. 2012, 2014b; Fontana et al. 2012).

  • The design of machines, adapted to specific work conditions (i.e. low head and high range of flows) in these water drinking network such as the tubular propeller (Ramos et al. 2013; Samora et al. 2016b).

  • The analysis of potential recovered energy according to circulating flow along the time through of simulated annealing techniques (Pérez-Sánchez et al. 2016; Samora et al. 2016a; Pérez Sánchez et al. 2018).

  • The economic and feasibility analyses of these installations showing the sustainability and environmental profit of these solutions (Ramos et al. 2010; McNabola et al. 2014).

Furthermore, the study of the performance behavior of these machines, Singh (2005) and Derakhshan and Nourbakhsh (2008) proposed the efficiency and head curves as function of flow according to the specific rotational speed of the machine. These curves can be used in the simulations of energy analyses with good results when the hydraulic machine operates in its nominal rotational speed. However, if the energy studies consider operation strategies with variation of the rotational speed, the use of affinity laws in the simulations can bring erroneous results of recovered energy in the system (Sarbu and Borza 1998; Gulich 2003) since the turbines do not behave as the similarity described. Therefore, considering the need to know the efficiency of the machine as function of rotational speed, the first aim of this research is to obtain the errors between measured and calculated efficiency through the application of affinity laws. This analysis was developed for two machines (with axial and radial impellers) based on experimentation. The second objective is to develop modified affinity laws to establish the best efficiency line (BEL) and the best efficiency head (BEH) of each machine as function of flow. BEL and BEH establish the rotational speed of the machine when the Variable Operating Strategies (VOS) are used (Carravetta et al. 2013), claiming for each flow the best operation point (BEP). The development of both objectives presents the main novelty of this research that obtains the BEL and BEH. These lines can be used on optimization techniques to maximize the recovered energy in water distribution system when the rotational speed changes, obtaining result more exactly than the values when similarity laws are used.

2 Material and Methods

2.1 Type of Hydraulic Machines

A general typification of hydraulic machines refers to action or reaction according to the exchange of energy between fluid and impeller at atmospheric pressure or not. Inside the group of the reaction machines, the impeller shape and the specific speed establish a second classification. This parameter is defined as the rotational speed of a similarity turbine (geometrically) to generate one kW when the head is equal to one. The specific speed establishes the impellers’ typology and it is defined by Eq. (1):

$$ {n}_s=n\frac{P^{1/2}}{H^{5/4}} $$
(1)

where n s is the specific speed of the machine in (m, kW); n is the rotational speed of the machine in rpm; P is the power in shaft, which is measured in (kW); and H is the recovered head in (m w.c.).

Based on the specific speed, the impellers can be: radial, semi-axial, or axial according to Fig. 1a. The radial or centrifugal impeller are those in which the fluid enters in the machine in radial direction and exists in axial direction. This type of machines has a specific speed number between 36 and 93 rpm. For ns between 93 and 176 rpm, the machine is called diagonal or semi-axial. In this sort of machines, the inlet of the fluid is diagonal while the outlet has an axial direction. Finally, the axial machines present the fluid direction both inlet and outlet with axial direction and the specific speed is greater than 176 rpm.

Fig. 1
figure 1

(a) Type of impeller according to specific rotational speed (adapted from Alexander et al. 2009) and (b) scheme of operating points

Full size image

2.2 Theoretical Approximation: Affinity Laws

The study of the behavior of turbomachines with equal specific rotational speed can be tackled if the conditions of similarity (geometrical, kinematic and dynamic) are entailed. The first condition is satisfied, when the study is focused on analyzing the behaviour of a machine with different rotational speed. Kinematic condition establishes that in the inlet and outlet impeller, the triangles are similar. These two conditions are defined by Eqs. (2) to (4) (Mataix 2009):

$$ \frac{Q_1}{Q_0}={\left(\frac{D_1}{D_0}\right)}^3\frac{N_1}{N_0} $$
(2)
$$ \frac{H_1}{H_0}={\left(\frac{D_1}{D_0}\right)}^2{\left(\frac{N_1}{N_0}\right)}^2 $$
(3)
$$ \frac{P_1}{P_0}={\left(\frac{D_1}{D_0}\right)}^2{\left(\frac{N_1}{N_0}\right)}^3 $$
(4)

where Q 1 is the flow in the new conditions of rotational speed in m3/s; Q 0 is the flow in nominal rotational speed in m3/s for the BEP; D 1 is the diameter of the impeller in new situation of rotational speed in m; D 0 is the nominal diameter of the impeller in m; N 1 is the new rotational speed in rpm; N 0 is the nominal rotational speed of the impeller in rpm; H 1 is the head in new condition in m w.c.; H 0 is the head in the nominal conditions in m w.c.; P 1 is the shaft power in new conditions in kW; and P 0 is the shaft power in the nominal condition in kW.

Therefore, if hydraulic parameters of the turbomachine (flow, head, and power) can be related through affinity laws and the efficiency of the machine can be indirectly determined for different rotational speeds, keeping the impeller’s size. Based on this assumption, Eqs. (2) to (4) can be simplified into Eqs. (5) to (7) (Mataix 2009):

$$ \frac{Q_1}{Q_0}=\frac{N_1}{N_0}=\alpha $$
(5)
$$ \frac{H_1}{H_0}={\left(\frac{N_1}{N_0}\right)}^2={\alpha}^2 $$
(6)
$$ \frac{P_1}{P_0}={\left(\frac{N_1}{N_0}\right)}^3={\alpha}^3 $$
(7)

where α is the ratio between N 1 and N 0.

The characteristic curve of a turbomachine (as a second-degree polynomial) can be written by Eq. (8) (Mataix 2009):

$$ {H}_0=A{Q}_0^2+B{Q}_0+C $$
(8)

where A, B, and C are coefficients of the characteristic curve.

The efficiency curve can be also established by a second (Mataix 2009) around the nominal point, or by third-degree polynomial if a discretized range of flows is considered (Ulanicki et al. 2008). The nominal efficiency curve is then defined by Eq. (9):

$$ {\eta}_0=E{Q}_0^3+F{Q}_0^2+G{Q}_0+I $$
(9)

where: η 0 is the efficiency of the machine for a flow equal to Q 0 ; E, F, G, and I are coefficients of efficiency curve.

When the affinity laws are applied to Eqs. (8) and (9), the new curves of the turbomachines are defined by Eqs. (10) and (11):

$$ {H}_1=A{Q}_1^2+\alpha {Q}_1+{\alpha}^2C $$
(10)
$$ {\eta}_1=\frac{E}{\alpha^3}{Q}_1^3+\frac{F}{\alpha^2}{Q}_1^2+\frac{G}{\alpha }{Q}_1+I $$
(11)

where: η 1 is the efficiency of the turbine for a flow equal to Q 1 .

2.3 Experimental Approximation: Efficiency Curves

The prediction of head and flow values in turbines working with different rotational speeds has a reasonable approximation. Nevertheless, as the affinity laws do not consider the viscosity effects of the fluid inside of the impeller (third condition of similarity), the use of these laws is limited (Simpson and Marchi 2013). Hence, these equations cannot be applied in all flow range to predict the performance, obtaining good results in turbines for ranges between +/− 20% around of the best efficiency point. As the viscosity effect has to be considered, the dynamic similarity is important in these cases. This effect must be considered together with geometrical and kinematic similarity. To consider the dynamic condition, different researchers (Sarbu and Borza 1998; Gulich 2003; Simpson and Marchi 2013) have proposed equations to define the performance as function of the rotational speed variation. Gulich (2003) proposed the Eq. (12), which predicts the performance according to the variation of Reynolds number between both rotational speeds of the machine.

$$ \frac{1-{\eta}_1}{1-{\eta}_0}=K+\left(1-K\right){\left(\frac{{\mathit{\operatorname{Re}}}_1}{{\mathit{\operatorname{Re}}}_0}\right)}^2 $$
(12)

where Re1 is the Reynolds number for the rotational speed N1; Re0 is the Reynolds number for the rotational speed N0; K is the loss coefficient in the impeller as function of the Reynolds number.

Similar equation was proposed by Sarbu and Borza (1998), who also related the Reynold number as function of the rotational speed, defined in Eq. (13):

$$ \frac{1-{\eta}_1}{1-{\eta}_0}={\left(\frac{N_1}{N_0}\right)}^{0.1} $$
(13)

Equations (12) and (13) were tested by Simpson and Marchi (2013), obtaining good results in the prediction of the efficiency, if the rotational speed is not reduced under to 70% of the nominal speed. This is due to the empirical expression exponent changes with viscosity and friction effects in the impeller.

When variable operating strategy (VOS) is applied, the final objective is to determine the best efficiency line (BEL) as function of the rotational speed for each flow (Fig. 1b). The operation point should be fixed in the available maximum point to maximize the efficiency of the recovery system. The knowledge of BEP for each flow along the time (i.e., Q i , Q i + 1 ) as well as the available net head (H Ti ) will allow researchers to know the best efficiency head line (BEH) of the installed machine (H Ri,N0 , H Ri + 1,N1 ) for the recovery system. Both lines allow to recover the maximum energy, helping to define the VOS in the system for the necessary rotational speed (i.e., N 0 , N 2 , …, N i ) in each time.

Traditionally, these variations have been predicted by affinity laws, but only near the BEP. In this BEL, different authors (Carravetta et al. 2014a, b; Fecarotta et al. 2016) proposed modifications in the affinity laws to improve the prediction of the BEP depending on the rotational speed. This proposal was developed for some semi-axial machines with specific rotational speed between 120 and 162 (m, kW). As proposed, the modification of the affinity laws is developed according to the Eqs. (5) to (7) as well as the use of the Suter Parameters (SP), which were defined by Suter (1966) by Eqs. (14) and (15):

  • $$ \mathrm{Head}\ \mathrm{SP}; WH=\frac{h}{n^2+{q}^2} $$
    (14)
  • $$ \mathrm{Torque}\ \mathrm{SP}; WT=\frac{b}{n^2+{q}^2} $$
    (15)

where h, q, n, and b are the head, discharge, velocity, and torque coefficients defined by Eqs. (16) to (19), respectively.

  • $$ \mathrm{Head}\ \mathrm{coefficient}:h=\frac{H}{H_0} $$
    (16)
  • $$ \mathrm{Discharge}\ \mathrm{coefficient}:q=\frac{Q}{Q_0} $$
    (17)
  • $$ \mathrm{Velocity}\ \mathrm{coefficient}:n=\frac{N}{N_0} $$
    (18)
  • $$ \mathrm{Torque}\ \mathrm{coefficient}:b=\frac{T}{T_0} $$
    (19)

Finally, the performance coefficient (e) (which is also called efficiency ratio) is defined by Eq. (20):

$$ e=\frac{\eta }{\eta_B}=\frac{bn}{qh}=\mathrm{tan}\upvarphi \frac{WT}{WH} $$
(20)

where tanφ is the ratio between q and n, according to the third quadrant (p < φ <3p/2), when the machine is working as turbine (PAT). In this case, h and b are positive, while n and q are negative. According to expert references (Carravetta et al. 2014a, b; Fecarotta et al. 2016), the affinity laws can be modified as:

$$ q=\frac{Q_1}{Q_0}={f}_1\left(\alpha \right) $$
(21)
$$ h=\frac{H_1}{H_0}={f}_2\left(\alpha \right) $$
(22)
$$ p=\frac{P_1}{P_0}={f}_3\left(\alpha \right) $$
(23)

where f 1 , f 2 , and f 3 are fitted functions that depend on the experimental data according to α.

2.4 Theoretical vs. Experimental: Error Definition

The determination of the errors is evaluated by Eqs. (24) and (25). Three comparisons have been done: a) experimental data versus classic affinity laws; b) experimental data versus modified affinity laws; and experimental data versus empirical method. Equation (24) defines the absolute relative error between the experimental data and estimated measurements for the same flow value (head or efficiency) and Eq. (25) defines the mean square error, which is determined according to the number of measured data:

$$ Absolute error\ \left({\varepsilon}_i\right)=\left|\frac{X_{est}-{X}_{exp}}{X_{exp}}\right| $$
(24)
$$ Mean square error\ \left({\sigma}_i\right)=\frac{1}{m}\sqrt{\sum_{i=1}^m{\left(\frac{X_{est}-{X}_{exp}}{X_{exp}}\right)}^2} $$
(25)

where i is the tested parameter, which can be q, h, p, or e; X est is the estimated value through affinity laws or empirical method; X exp is the measured value; and m is the number of measurements.

3 Developed Test

3.1 Tested Impellers

The experimental tests have been carried out in CERIS-Hydraulic Lab of Instituto Superior Técnico from the University of Lisbon with two different machines. In both cases the discharge was measured by an electromagnetic flowmeter, the pressure was registered by pressure transducers, and the power by digital wattmeter FLUKE which was connected to the generator. The digital multimeter is able to measure active power, reactive power and power factor. Finally, the rotational speed was measured by a frequency meter.

Two different machines were tested (Fig. 2). On the one hand, an axial machine with an impeller’s size of 85 mm with five blades. This machine has the best efficiency point for a flow of 4.44 l/s and head of 0.24 m w.c.. when the nominal rotational speed is 750 rpm. For this machine, the specific speed number is 283 (m, kW) (Eq. (1)). On the other hand, a radial machine that has an impeller size of 139 mm. For this machine, when the rotational speed is 1020 rpm, the best efficiency point is located for 3.36 l/s and 4 m w.c. and the specific rotational speed is 51 (m, kW). The generator connected to both machines has a maximum efficiency of 62% when the rotational speed is 1350 rpm. The characteristics of the generator are: the rated frequency 50 Hz, the rated power 550 W, the rated current 1.6/2.8 A, the rated phase voltage 230/400 V, the power factor 0.74 and the rated speed 1020 rpm.

Fig. 2
figure 2

Experimental data (Head and Efficiency) for machines: (a and b: axial; c and d, radial)

Full size image

3.2 Rotational Speed Variation

In Fig. 2a and b, the efficiency and head curve for different rotational speed are shown according to the performed tests in the axial machine. In this case, these figures show the head values and the efficiency as function of the flow for different rotational speeds (500, 750, 1000, 1250, and 1500 rpm) as well as the runaway curve.

These characteristic curves for a rotational speed of 750 rpm can be fitted by polynomial Eqs. (26) and (27):

  • $$ {\displaystyle \begin{array}{l}\mathrm{H}\mathrm{ead}\ \mathrm{curve}\ \left(\mathrm{m}\ \mathrm{w}.\mathrm{c}.\right):\\ {}\mathrm{H}=0.047{\mathrm{Q}}^2-0.219\mathrm{Q}+0.272\kern0.5em \left({\mathrm{R}}^2=0.998\right).\left[4.40<\mathrm{Q}<13.30\ \mathrm{l}/\mathrm{s}\right]\kern0.5em \end{array}} $$
    (26)
  • $$ {\displaystyle \begin{array}{l}\mathrm{E}\mathrm{fficiency}\left(\%\right):\\ {}\mathrm{E}=-0.058{\mathrm{Q}}^4+1.942{\mathrm{Q}}^3-23.690{\mathrm{Q}}^2+118.381\mathrm{Q}\hbox{--} 143.852\kern0.5em \left({\mathrm{R}}^2=0.995\right).\left[4.40<\mathrm{Q}<13.30\ \mathrm{l}/\mathrm{s}\right]\end{array}} $$
    (27)

When the machine is operating in its nominal rotational speed, the flow range oscillates between 4.40 l/s and 13.30 l/s, varying the recovered head between 0.26 and 5.88 m w.c.. For this range of flows, the efficiency of the machine oscillates between 63.65% and 29.62%, being the maximum efficiency of 63.65%, when the flow is 4.44 l/s.

Figure 2c and d show similar results according to tests carried out with the radial machine. In this case, the machine was tested for 600, 900, 1020, and 1200 rpm. The characteristic curve is presented for a rotational speed of 1020 rpm, being defined by the following Eqs. (28) and (29):

  • $$ {\displaystyle \begin{array}{l}\mathrm{H}\mathrm{ead}\ \mathrm{curve}\ \left(\mathrm{m}\ \mathrm{w}.\mathrm{c}.\right):\\ {}\mathrm{H}=0.431{\mathrm{Q}}^2-1.310\mathrm{Q}+3.553\kern0.5em \left({\mathrm{R}}^2=0.995\right)\ \left[1.80<\mathrm{Q}<4.40\ \mathrm{l}/\mathrm{s}\right]\kern0.5em \end{array}} $$
    (28)
  • $$ {\displaystyle \begin{array}{l}\mathrm{E}\mathrm{fficiency}\left(\%\right):\\ {}\mathrm{E}=-11.145{\mathrm{Q}}^2+80.698\mathrm{Q}-\kern0.62em 84.535\kern0.5em \left({\mathrm{R}}^2=0.995\right)\ \left[4.80<\mathrm{Q}<13.30\ \mathrm{l}/\mathrm{s}\right]\end{array}} $$
    (29)

For this machine, the maximum efficiency is 60.5% for a flow of 3.36 l/s.

The runaway curves are also shown in Fig. 2a and c for both machines (axial and radial). When these curves are analyzed, a significant difference can be determined taking into account the VOS. If a same value of the recovered head is established, a reduction of the flow with a high rotational speed can lead to cause the runaway condition in radial machines. In the case of the axial machine, an increasing of the flow keeping the recovered head can be attained the runaway conditions.

3.3 Analytical Versus Experimental Curves

The need to know the head and efficiency values of each machine can lead to develop new curves with different rotational speeds through affinity laws or Suter parameters, when the water manager uses VOS. The obtained results in analyses of axial and radial machines are shown in Fig. 3. The head values of the affinity laws can be observed, coinciding on a great range of flow conditions. Therefore, the use of these laws to develop new equation of recovered head curve can be used. Otherwise, if water managers want to determine the efficiency, the affinity laws can fail. Figure 4b and d show the predicted efficiencies given by the affinity laws, which are different, especially when the maximum values are considered (i.e., maximum efficiency and flow value).

Fig. 3
figure 3

Experimental vs affinity laws in the axial (a and b) and radial machine (c and d)

Full size image
Fig. 4
figure 4

Absolute error as a function of the flow and rotational speed in axial (a) and radial (b) machines considering the affinity laws. Absolute error in axial (c) and radial (d) machines when the parameters were determined by affinity laws (AL), empirical method (EM), and Sarbu-Borza method (SB)

Full size image

As shown in Fig. 3a and b, in the axial machine when the rotational speed is 750 rpm, the maximum efficiency is 63.60% for a flow of 4.66 l/s, while the maximum efficiency by affinity laws is 62.19%. For the rotational speed of 1000 rpm, the maximum efficiency is 61.56% when the flow is 6.56 l/s, being the maximum efficiency by the affinity laws 62.27% when the flow is 6.03 l/s. This error raises with rotational speeds of 1250 and 1500 rpm. For the rotational speed of 1250 rpm, the maximum measured efficiency is 59.98% and the maximum estimated efficiency is 62.27%, being the flow values of 8.28 and 7.21 l/s, respectively. Similar tendency can be observed in Fig. 3b when the rotational speed is 1500 rpm.

Figure 3c and d show results between measured efficiencies and estimated values by affinity laws for different rotational speeds of 600, 900, 1020, and 1200 rpm in the radial machine, not being coincident the BEP for each rotational speed (e.g., when the rotational speed is 900 rpm, the maximum measured efficiency is 59.23% and the maximum estimated efficiency is 61.82%, being the flows 3.65 l/s and 3.25 l/s, respectively).

Figure 4a shows the absolute error between experimental data and estimated values by affinity laws used to predict head and efficiency for flows (Q) and rotational speeds (N) in the axial machine. The majority of head errors can attain 10% in both cases (axial and radial machines). If efficiency predictions errors are analyzed, the best fits are located in low value zone of efficiency curve, being these errors below to 1%. Figure 4a also shows the maximum errors are near the maximum efficiency point. In the case of radial machine (Fig. 4b), this trend does not occur as for axial machine. These efficiency errors are important to be considered, because when the mean errors are averaged taken into account for all flow range, the obtained mean error is lower than the absolute relative error near the BEP (e.g., when the radial machine is run to 900 rpm, the absolute relative error is 0.06 and the mean square error is 0.02). Therefore, if the mean error is determined for each speed, this can mislead to erroneous analyses. In the axial machine, the mean square errors are 0.05, 0.04, and 0.06 for rotational speeds of 1000, 1250, and 1500 rpm, respectively. These error values are not representative to analyze the maximum efficiency in the machine when the recovery system is operating by VOS. The mean square errors in radial machine are 0.09, 0.02, and 0.01 for speed of 600, 900, and 1200 rpm, respectively.

3.4 Empirical Method Towards New Affinity Laws

If the shown errors in Fig. 4a and b are taken into account, it is necessary to search for new solutions. These solutions have to allow modelers to determine functions with a less error, to be used in the energy studies based on Suter Parameter (SP). To fit the affinity laws, the described methodology in section 2.3 enables to obtain such functions (f 1 , f 2 , and f 3 ). These functions lead for the determination of the BEH and the BEL in both machines, maximizing the efficiency in all ranges of flow by rotational speed variations.

Figures 5 and 6 show the proposed fitted functions for the axial and radial machines, respectively. To do a better analysis, other maximum values of efficiency were measured in the machine for rotational speeds of 250, 1750, 2000, and 2250 rpm. In the case of the radial machine, the maximum efficiency for the rotational speed of 1500 rpm has also been measured. In both machines, the obtained functions present good regression indexes that are higher to 0.94 to parameters of q, h, p, and efficiency ratio, except the value of efficiency ratio in axial machine. For this machine, the value is 0.719.

Fig. 5
figure 5

Proposed functions for axial machine

Full size image
Fig. 6
figure 6

Proposed functions for radial machine

Full size image

Once the functions for each parameter were adjusted by least squares, the errors can be calculated. The error results are shown in Fig. 4c and d for both machines. The errors of q, h, p, and efficiency ratio (e) are shown when the parameters are determined by affinity laws (AL), the proposed empirical method (EM), and Sarbu-Borza’s Method (SB) (since this method is only used to determine efficiency values according to Eq. (13)). In all cases, the less error was obtained by EM. The obtained errors with SB were between the AL and EM. For the EM, the mean square errors were 0.02, 0.02, 0.004 for q, h, and efficiency ratio in the radial machine. When these values were determined for the axial machine, the errors were 0.02, 0.03, and 0.008, respectively. The mean square error by SB was 0.02 for both machines.

Once the errors were analyzed, the functions (f 1 , f 2 , and f 3 ) were known depending on the rotational speed. These functions enabled to draw the BEL and BEH of each machine. Figure 7 shows these lines for the axial (Fig. 7a) and the radial (Fig. 7b) machines as function of flow, according to VOS described in Fig. 1b. In the particular case of the radial machine, the BEL indicates that recovery system is working upper to 55% in the all range of flows. BEL and BEH are significant information in the optimization methodologies for water managers to develop the VOS in each hydraulic machine, thus maximizing the system efficiency.

Fig. 7
figure 7

BEH and BEL for axial (a) and radial (b) hydraulic machines

Full size image

Finally, different functions as a function of the rotational speed can be drawn (i.e., f 1 , f 2 , f 3 and e), joining the depicted procedure to modify the affinity laws with the analysis developed for semi-axial machines by Fecarotta et al. (2016). This analysis is shown in Fig. 8 which correlates functions (i.e., f 1 , f 2 , f 3 , and e) with the variables: velocity ratio and specific rotational speed.

Fig. 8
figure 8

Fit function to different hydraulic machines

Full size image

These functions (f 1 , f 2 , f 3 , and e) presented relative and mean errors smaller than affinity laws and Sarbu-Borza’s method when they were used to predict the real behavior of hydraulic turbines. When the radial machine was analyzed, the empirical mean square errors were 0.02, 0.02, and 0.004 for q, h, and e parameters and the mean square error of affinity laws were 0.11, 0.04, and 0.03, respectively. Therefore, the application of this empirical method reduced the mean error values in 81.81%, 50%, and 86.67%, respectively. For the axial machine, the empirical mean errors were 0.02, 0.03, and 0.008, for q, h, and e parameters, respectively. Against, these errors were 0.06, 0.18, and 0.03 when the efficiency was determined by affinity laws, reducing them to 66.67%, 83.33%, and 73.33%, respectively.

4 Conclusions

This research presents experimental tests for two different hydraulic turbines (axial and radial) that have a specific rotational speed of 283 and 51 (m, kW) respectively. For these machines, the characteristic curves (flow-head, flow-mechanical power and flow-efficiency) were obtained for different rotational speeds. The variable operating strategies (VOS) in recovery systems of water distribution networks involve the need to predict the efficiency and head curve of each machine for different rotational speeds. These curves will enable to develop viability studies based on energy and economic analyses. These obtained curves are for a specific reaction turbine and not general for all turbines.

The prediction of the efficiency with different rotational speeds by the classic affinity laws can imply significant erroneous values. These errors induce the need to develop a modification in the affinity laws to take into account losses in the impeller of each machine. In this research, as novelty, new modified functions are proposed to know the flow, head, power and efficiency parameters that depend on the rotational speed ratio for radial and axial machines.

These functions (f 1 , f 2 , f 3 , and e) were validated for the tested hydraulic machines, presenting relative and mean errors smaller than affinity laws and the Sarbu-Borza’s method. When the radial machine was analyzed, the mean errors were reduced 81.81%, 50%, and 86.67% for q, h, and e parameters when the errors were compared with the affinity laws. In contrast, the mean errors were reduced 66.67%, 83.33%, and 73.33%, respectively for the axial machine.

The knowledge of these functions enables to develop the BEL and BEH lines as function of the flow of each machine. These lines help managers to choose the best operating rules in order to achieve the best efficiency point for each flow. These rules maximize the real recovered energy under optimization strategies. Regarding the analysis of the rotational speeds, when the flows are reduced, keeping a constant head the rotational speed increases in radial machine. This phenomenon can cause that PAT reaches the runaway conditions when the machine is installed in a line or point of a network in which the flow variation is significant.

Finally, the need to correlate the rotational speed and the specific speed with the different characteristic parameters (q, h, p, and e) is of utmost importance. The development of similar studies for different specific rotational speeds will enable to define the BEL and BEH as well as the selection of the best hydraulic machine for each system characteristics.