Experimental and numerical assessment of a methodology for performance prediction of Pumps-as-Turbines (PaTs) operating in off-design conditions

Applied Energy 248 (2019) 555–566

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Experimental and numerical assessment of a methodology for performance prediction of Pumps-as-Turbines (PaTs) operating in off-design conditions Mosè Rossi, Alessandra Nigro, Massimiliano Renzi

T

⁎

Free University of Bozen-Bolzano, Faculty of Science and Technology, Piazza Università 5, 39100 Bolzano, Italy

H I GH L IG H T S

of a predicting model able to forecast the performances of PaTs. • Development operational data of PaTs available in literature were used. • 32 analysis plus the normalization process were applied. • Non-dimensional centrifugal pumps were numerically simulated to validate the model. • Three • Percentage relative differences within a range of ± 7% were obtained close to BEP.

A R T I C LE I N FO

A B S T R A C T

Keywords: Hydraulic machines Pumps-as-Turbines Predicting model Performances forecast Laboratory tests Computational fluid dynamics

In this work, an extensive assessment of a predicting model used to evaluate Pumps-as-Turbines’ (PaTs) characteristic curves is presented, with specific attention to the off-design operating conditions. The novelty of the proposed model consists in the possibility to reconstruct the performance curves of a PaT only by knowing a limited number of operating data in turbine mode at the Best Efficiency Point (BEP), which, in many applications, represents a design constraint. The availability of the off-design performance curves supplies important indications for technical and economic evaluations in those applications where a constant flow rate cannot be granted. The predicting model was derived by re-elaborating a wide experimental data-set based on the most relevant scientific literature related to several PaTs operating in turbine mode. The prediction’s capability of the model was validated with experimental tests and confirmed by numerical simulations. The experimental tests were carried on in both direct and reverse modes by inspecting several flow rates. The model data were compared with the experimental ones in order to validate the Computational Fluid Dynamics (CFD) analyses. Subsequently, the numerical model was used to investigate the performances of other two PaTs operating in turbine mode. The study of the performance obtained with the CFD analyses allowed to evaluate the effectiveness of the predicting model, highlighting its pros together with its possible improvements. In general, it is possible to conclude that the proposed model is able to predict the performances of the studied PaTs with errors included in the range of ± 7% with respect to the Best Efficiency Point (BEP) in turbine mode.

1. Introduction Renewable energy sources are taking the field in the energy production systems in order to decrease the emissions of pollutants in the atmosphere: different agreements were signed in the last decades in order to commit the industrialized countries to achieve determined goals until 2020 and over [1–4]. One of the most used renewable energy sources is the hydropower, being the most spread worldwide. Nowadays, the technology used for the large-scale hydropower is quite mature and, in addition, there is a limited availability of new

⁎

geographical sites since most of them had been already exploited. For this reason, the small-scale hydropower is being used as a valuable alternative to the large-scale one, even though substantial modifications of the usual hydraulic turbines must be applied [5,6]. Among the alternative technologies involved in the small-scale hydropower sector, Pump-as-Turbine (PaT) represents a viable and interesting solution for both energy recovery and power generation purposes [7]. PaTs are common pumps, axial or centrifugal, used as turbines. The advantage of this technology is: (i) lower cost with respect to the conventional hydraulic turbines, (ii) higher availability in the market in terms of sizes

Corresponding author. E-mail address: [email protected] (M. Renzi).

https://doi.org/10.1016/j.apenergy.2019.04.123 Received 11 January 2019; Received in revised form 25 March 2019; Accepted 17 April 2019 Available online 06 May 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

φ ψ η Λ NS ω Q H

g P

flow coefficient head coefficient efficiency power coefficient specific speed rotating speed [rad/s] volumetric flow rate [m3/s] head [m]

gravitational acceleration [m/s2] power [W]

Subscript BEP Exp. CFD Model

and spare parts and (iii) a higher operating range in terms of flow rates [7]. One of the main limiting issues of this technology is that there are few available data regarding the turbine mode operation and it is quite complex to forecast the performances of PaTs in turbine mode, only relying on the pump operating data. Indeed, several researchers tried to develop some methods able to estimate the Best Efficiency Point (BEP) in turbine mode by the means of the BEP values in pump mode [8–12]. However, these methods are based on empirical and statistical approaches that rely on specific tested PaTs, thus inducing to misleading results in the BEP’s forecast of a generic hydraulic machine that is not included in the data-set. In the recent years, some scientists studied different methods to generalize the forecast of PaTs’ BEP values in turbine mode. Some of the authors of this work [13] presented also a methodology that is based on the experimental data of 59 different BEP values of PaTs that are available in literature and involves the correlation between the specific speeds (NS) of the PaTs in pump and turbine modes as well as the specific diameters (DS) . Coupling these two parameters, it is possible to evaluate with a good accuracy the flow rate and the head of the PaT operating at BEP in turbine mode by knowing the operating data at BEP in pump mode. Along the same line, Stefanizzi et al. [14] coupled the relations of the specific speeds (NS) of the PaTs in pump and turbine modes with the head prediction factor h , H defined as BEP _TURBINE , of the PaTs operating in both modes. Huang

best efficiency point experimental computational fluid dynamics predicting model

design operating conditions [20]. Venturini et al. [21] developed a physics-based simulation model able to evaluate the performances of centrifugal PaTs through an optimization procedure that identifies specific parameters: these ones, together with the loss coefficients and the performances’ curves in pump mode, supply the main characteristics of a PaT operating in turbine mode. However, the knowledge of several dimensional parameters that characterize the machine are needed to apply this model (e.g. blades’ angle and width, casing clearance, disk friction, etc.) that can be hardly obtained from pumps’ manufacturers. Along the same line, Barbarelli et al. [22] developed a one-dimensional numerical code able to reconstruct the PaTs’ performance: also in this case, some geometrical construction dimensions of the hydraulic machine are necessary to run the model and assess the internal losses. Yang et al. [23] developed a theoretical method to predict the performance of PaTs using theoretical analysis and empirical correlations that require the geometrical characteristics of both impeller and volute of the machine. The pump was also tested on a PaT open test rig in order to verify the accuracy of theoretical and numerical prediction methods; moreover, Computational Fluid Dynamics (CFD) simulations of a single-stage centrifugal pump were also performed in both direct and reverse modes. Indeed, the CFD software are taking the field in the analysis of turbomachinery, especially to study those zones where it is not always easy to investigate experimentally, like the influence of the tip clearances on the vortex characteristics in a mixed flow pump used as turbine [24]. As regards the final applications, PaTs are commonly used in both civil and industrial plants. For instance, Water Distribution Networks (WDNs) present different Pressure Reducing Valves (PRVs) that regulate the pressure inside the water grid without recovering energy: for this reason, PRVs are being replaced with PaTs. Several researchers applied both BEP and off-design forecasting models to select the right PaT according to the WDN’s operating data. Kramer et al. [25] studied the feasibility of installing PaTs in WDNs on both technical and economic points of view, while Lydon et al. [26] showed that it could be recovered up to 40% of the gross power potential of an existing PRV by using a PaT. PaTs are also involved in chemical plants, which are named Hydraulic Power Recovery Turbines (HPRTs) because of their coupling on the same shaft of the feed pump involved in the same chemical process [27]: in this case, the aim of the PaT is to supply part of the mechanical energy, deriving by the one recovered from the processed liquid, to the feed pump [28,29]. The aim of this paper is to extend and cross-validate a model proposed by some of the authors of this work [18] for the performances estimation of PaTs operating in off-design conditions, being the BEP in turbine mode already known. Indeed, in literature, as previously reported, there are several models capable of predicting the BEP in turbine mode, but only few of them allow also to reconstruct the entire performance curves of a PaT. Moreover, it is worth to notice that the desired rated operating conditions, in terms of available flow rate, head, rotating speed and efficiency in turbine mode, are often set as a design constraint for a PaT, being therefore known a priori. On the basis of the BEP operating condition, the definition of the entire operating curve is required in several applications like energy recovery in WDNs, irrigation systems, and some industrial processes, where it is very rare

HBEP _PUMP

et al. [15] carried out an innovative theoretical approach to predict both flow rate and head at BEP in pump and turbine modes by using the principle of the characteristic matching between rotor and volute: moreover, this method uses theoretical formulas based on the Euler equation applied to turbomachinery and velocity relations at the inlet and at the outlet of the rotor. However, the forecast of the BEP is not the main issue related to the performances’ evaluation of PaTs in turbine mode: together with this, also their behaviour when operating in offdesign conditions is being studied in both pump and turbine modes, with particular attention to the cavitation phenomenon [16,17]. Some of the authors of this work [18] tried to define objectively the trend of the main performances’ curves (characteristic, power and efficiency) of PaTs by using data from laboratory tests performed by several researchers on 32 different hydraulic pumps, mainly centrifugal ones, and applying the non-dimensional analysis together with a normalization process in order to carry on a generalized analysis that can be applied to several typologies and designs of machines. The outcomes of this study showed an interesting behaviour of PaTs when they are operating far from the BEP. When PaTs are operating in the low flow rates region, there is a substantial efficiency drop; for this reason, it is not recommended to operate with flow rates that are too low than the BEP one. On the other hand, a different behaviour occurs when PaTs operate with flow rates higher than the BEP one. In this case, the efficiency trend is quite flat and it remains close to the maximum value, resulting in high produced power. However, a higher produced power involves a higher mechanical torque and, thus, higher mechanical stresses on the impeller, bearings and shaft [19]. These two aspects must be considered during the design phase of the PaTs. By using the same data collected from the literature, a useful tool was developed, based on the Artificial Neural Network (ANN) methodology, to forecast both BEP and off556

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to have a constant flow rate. For this reason, the knowledge of the entire characteristic curve of the PaT, as well as the efficiency one, is crucial for a proper technical and economic assessment. At present, as previously described, the models available in literature for the off-design performance evaluation of PaTs [21–23] mainly rely on the knowledge of accurate geometrical characteristics of a pump, which are rarely available from the manufacturers. Therefore, the additional advancement of this work is to supply users of PaTs with a simple tool, capable of giving operational information on the selected machine for a specific application, even in off-design conditions; moreover, it has a strong practical relevance as it is based on performance data that can be easily obtained from the datasheet of pumps’ manufacturers, without requiring specific knowledge of the geometrical design parameters of the pump. Specifically, three different centrifugal PaTs are studied in order to evaluate the accuracy and the range of applicability of the model: their behaviour is evaluated through CFD analyses and compared with the results of the predicting model and the experimental data, which are provided for only one PaT. Section 2 explains the development of the predicting model, the set-up of both test rig used to perform the laboratory tests and the set-up of the CFD simulations. Section 3 shows the results of the laboratory tests of one of the three analysed centrifugal PaTs [30] and the validation of the numerical simulations performed on the same machine. Finally, the performances’ curves (characteristic, power and efficiency) obtained from the experimental campaign, CFD simulations and the predicting model are compared to assess the accuracy of the model. The same procedure is followed to study other two centrifugal PaTs, whose CAD files are available from ANSYS® and SIMSCALE® libraries. In addition, the prediction capability of the presented model is further assessed by comparing the results with the ones of Singh et al. [31] and Novara et al. [32], taking as reference the experimental data available in [31]. It is worth to notice that only these models were selected since they are the only ones that allow to reconstruct the off-design performance with the data available in their works. Finally, Section 4 reports the conclusions of the work.

Fig. 1. Non-dimensional normalized characteristic curve.

Fig. 2. Non-dimensional normalized efficiency curve.

Λ=

2.1. Predicting model The model developed by some of the authors of this work [18] and presented in this paper predicts the performances of PaTs in turbine mode, using formulas obtained by re-elaborating experimental data available in literature. Data obtained by laboratory tests refer to 32 different hydraulic pumps, mainly centrifugal ones, and they are related to both BEP and off-design operating conditions. Regarding these data, different machines and operational ranges are considered in the study. The flow rate of the analysed PaTs ranges between 0.008 m3/s and 0.222 m3/s, the head varies between 1.99 m and 99.52 m, the rotating speed changes from 750 rpm to 2445 rpm, the impeller diameter goes from 0.165 m to 0.300 m, the specific speed is included between 0.17 and 2.39 and, finally, the efficiency assumes values between 0.43 and 0.87. As previously reported in the introduction, a non-dimensional analysis is used and, subsequently, the non-dimensional coefficients are normalized with respect to the BEP values of each machine in turbine mode. In this work the flow coefficient, the head coefficient, specific speed and the power coefficient are defined as follows:

Q nD3

(1)

ψ=

gH (nD)2

(2)

Ns =

Q0.5 (gH)0.75

(4)

The performances of these machines, elaborated from the data-set of experimental data in design and off-design operating conditions, are analysed and compared to depict a possible common trend on the characteristic curves in turbine mode, even in presence of machines with different dimensions and design solutions. Figs. 1 and 2 show all the experimental points, normalized with respect to the BEP values in turbine mode, of the head coefficient and efficiency as a function of the flow coefficient. Figs. 1 and 2 pinpoint that the obtained trends resemble the same behaviour of both characteristic and efficiency curves of a generic hydraulic turbine. Moreover, despite the different characteristics and design of the machines from which the data points are taken, the operating points lay on a well-defined trendline. The graphs supply also an indication of the PaTs’ behaviour when they operate far from the BEP: indeed, the efficiency drop is considerable with flow rates lower than 40% of the BEP one, meaning that it is not suitable to operate in that zone. On the contrary, PaTs have a different behaviour when operating with flow rates higher than the BEP value: in this zone, the efficiency trend is quite flat and it reduces of a small amount with respect to the maximum value. This aspect must be considered during the design phase of the PaTs due to the higher mechanical stresses on the impeller, bearings and shaft. In addition, Fig. 2 also shows that the experimental data related to the PaT’s efficiency, located in the zone where the flow rates are lower than the BEP one, are more spread out compared to the ones that are located in the zone where the flow rates are higher than the BEP value. Given the experimental data points, it is also possible to define polynomial functions able to interpolate the experimental data to describe the normalized performances’ curves of PaTs. Eqs. (1) and (2) represent the correlations obtained for the head coefficient, ψ , and the efficiency, η, as a function of the normalized flow coefficient, φ , respectively.

2. Research and methods

ϕ=

P ρn3D5

ψ/ ψBEP_TURBINE = 0.2394·(φ / φBEP_TURBINE )2 + 0.769·(φ / φBEP_TURBINE) (5)

(3) 557

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inside the pipe and, thus, reliable results. Finally, tests are carried on in steady-state operating conditions at different values of the rotating speed, starting from 450 rpm till 1050 rpm with steps of 200 rpm.

η / ηBEP_TURBINE = −1.9788·(φ / φBEP_TURBINE )6 + 9.0636·(φ / φBEP_TURBINE )5 − 13.148·(φ / φBEP_TURBINE

)4

)3

+ 3.8527·(φ / φBEP_TURBINE

+ 4.5614·(φ / φBEP_TURBINE

)2

− 1.3769·(φ / φBEP_TURBINE) (6)

2.3. CFD analysis

After the evaluation of both head coefficient, ψ , and efficiency, η, also the power coefficient, Λ , can be calculated through Eq. (7).

Λ/ΛBEP_TURBINE = η / ηBEP_TURBINE ·φ / φBEP_TURBINE ·Λ/ΛBEP_TURBINE

The CFD analyses were performed on three different centrifugal pumps: the machine A is the same used in the abovementioned test bench, while machines B [33] and C were provided by ANSYS® and SIMSCALE® libraries, respectively. Both performances and characteristics of these machines in pump mode are reported in Table 3. The software used to set-up the numerical simulations is ANSYS® 19.1, while the flow solver used to carry on the numerical analyses is CFX [34]. An example of the computational domain of the three investigated PaTs is shown in Fig. 5, whereas Fig. 6 shows their impellers. Specifically, Fig. 5 shows the computational domain of the machine A that is composed by four sub-volumes: impeller, volute, inlet and outlet pipes. It is worth to notice that both inlet and outlet pipes were added in all the computational domains of the three analysed PaTs in order to minimize the effect of the fixed Boundary Conditions (BCs) on the flow field of the machines at both inlet and exit sections. The length of both inlet and outlet pipes is about five times the diameter of the respective sections. The used BCs related to the numerical simulations of the three machines in turbine mode are reported in Table 4. Regarding the simulations performed on the machine A operating in pump mode, a volumetric flow rate normal to the boundary and a total pressure of 1 bar are imposed as inlet and outlet boundary conditions, respectively. Having several domains that behave differently, the interfaces related to the interaction between the rotating (impeller) and the stationary (volute) domains are created to enable the Frozen Rotor Model (FRM), which is a steady-state method using rotating reference frame to reduce the computational efforts [35]. In the present work, hybrid unstructured meshes were adopted using the mesh tool available in ANSYS® 19.1. The grid independence was ensured by successive mesh refinements until a maximum difference of 0.7%, in terms of two contiguous results of head, was achieved. For sake of conciseness, only an overview of the refined mesh used for the machine B is reported in Fig. 7a. The discretization of the governing equations is based on the Finite Volume Method (FVM). Both diffusive and turbulence terms are discretized by using High-Resolution schemes. [36] The pressure-velocity coupling uses a non-staggered grid layout similar to [37]. Reynolds Average Navier-Stokes (RANS) equations, augmented with the standard k-ω two-equation model, were used for modelling the turbulence [38]. The k-ω turbulence model is particularly adequate to capture the detachment phenomena and flow separation due to adverse pressure gradients that may occur especially when hydraulic machines operate in off-design conditions [39]. An automatic function is employed for the near wall-treatment, which is a blending between the viscous sublayer and the log-law relation that allows to have a consistent y+ insensitive mesh refinement. The software guidelines [40], based on the paper by Knopp [41], suggest that the wall-adjacent vertices have to be in the log-law layer for a correct definition of the wall function, meaning that the y+ range should be between 20 and 200. As it is shown in Fig. 7b, the y+ values on the impeller walls are below 30, except for the tip gap regions where their values are approximately 250–300. However, this threshold value is

(7)

As Eqs. (5) and (6) show, different polynomial orders are chosen in order to fit better the experimental data-set. Polynomial equations were selected since they are simple to be adopted and, at the same time, they grant a good accuracy in the fitting of the data for the present problem. The obtained R2-values are used to evaluate the accuracy of the trendlines’ equations related to the normalized characteristic (5) and efficiency (6) curves with respect to the experimental data. Precisely, the obtained R2-values of both characteristic and efficiency trendlines are equal to 0.9171 and 0.7856, respectively. The lower value of the R2value of Eq. (6) is due to the higher dispersion of the data in the low flow rate region where the efficiency of the PaTs is more difficult to be predicted and it depends on the flow-detachments and other phenomena, like secondary flows, that might vary depending on the design of each single machine. However, a different polynomial order or other fitting equations would not be able to capture these values all the same and, therefore, there would not be any benefit on the final R2-value. It follows that the predicting model, regarding the efficiency trendline’s forecast, is more precise when dealing with higher flow rates instead of lower ones with respect to the BEP, thus leading to a lower error percentage in the first case compared with the second one. 2.2. Experimental campaign on a commercial PaT: Pump and turbine operating modes In order to better investigate on the behaviour of a centrifugal pump operating in both direct and reverse modes, laboratory tests on a selected PaT were performed [30]. Tests were also used to validate the presented predicting model on a machine that was not included in the definition of Eqs. (5) and (6), respectively. The test bench embeds: a flow meter, whose measurement range, in terms of flow velocity, goes from 0.3 m/s to 10 m/s; two pressure probes, located close to the inlet and discharge section of the machine, having a range of measurement between 0 bar and 10 bar; a torque meter that links the shaft of the centrifugal pump to the electric motor/brake, whose range of measurement varies between 0 N m and 50 N m. Table 1 resumes the characteristics of the equipment used to set-up the test bench, while Figs. 3 and 4 shows a sketch and a picture of the used test bench, respectively. The overall measurement error considering the error propagation is 0.72% for the hydraulic power, 0.32% for the mechanical power and 0.85% for the mechanical efficiency. Tests are performed in order to reconstruct the steady-state performances of the selected PaT, which is named machine A in this work. The main characteristics of the machine A operating in direct mode at its BEP are listed in Table 2. A 3-way valve is used to regulate the flow rate exploited by the hydraulic machine. A period of minimum five minutes, per each regulation, is employed until the desired flow rate value is achieved and stabilized: this procedure is fundamental to have a fully developed flow Table 1 List and properties of the equipment used to set-up the test bench. Measured quantity

Measurement equipment

Range

Accuracy

Signal

Flow rate Pressure Torque

Endress + Hauser Promag 50 W Keller PA33X Kistler Type 4503A50

0.3–10 m/s 0–10 bar 0–50 N m

0.5% (0.2% optional) 0.1% FS Error band (10–40 °C) 0.1–0.2%

4–20 mA 0–10 V 0–5 V

558

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Fig. 3. Sketch of the used test bench.

Fig. 5. Computational domain.

Furthermore, Table 5 reports the number of elements used to discretize the computational domains. The convergence is checked by monitoring the Root Mean Square (RMS) residuals variations between successive iterations and the calculation process is stopped when a very low percentage variation is reached: indeed, the normalized residuals’ drop, considering all the performed simulations, ranges between 10−5 and 10−10. Finally, it is important to highlight that, in the next section, the efficiency values, obtained by numerical simulations, were multiplied by a constant mechanical efficiency of 0.95 in order to consider the mechanical friction losses. This process is fundamental to compare the results obtained by the CFD analyses with the ones measured in the laboratory tests and available from the predicting model. For sense of clarity, the values related to the mechanical power were obtained through Eq. (8) and applied to the numerical simulations performed on the three machines operating in turbine mode, while Eq. (9) is used when the machine A operates as pump.

Fig. 4. Picture of test bench and the equipment [30]. Table 2 Main characteristics of the machine A operating at its BEP in direct mode. 50 m3/h 10 m 0.76 1450 rpm 0.57 193 mm 7 H20 (ρ ≈ 1000 kg/m3)

Flow rate value at the Best Efficiency Point (BEP) Head at the Best Efficiency Point (BEP) Best efficiency value Fixed rotating speed Specific speed Diameter of the impeller Number of blades Liquid processed

Table 3 Main characteristics of the centrifugal pumps. Machine

A

B

C

P[kW] = (ηρgQH)/1000

(8)

Type-Impeller Number of blades Flow rate [m3/s] (BEP) Head [m] (BEP) Efficiency (BEP) Mechanical power [kW] (BEP) Rotating speed [rpm] Impeller diameter [m]

Double disk 7 0.014 10 0.76 1.79 1450 0.193

Single disk 6 0.077 21.59 0.80 18.67 1450 0.281

Single disk 10 0.12 32 0.66 57.08 1500 0.340

P[kW] = (ρgQH/ η)/1000

(9)

3. Results and comments This Section is devoted to the assessment of the accuracy of the proposed predicting model. A comparison between experimental and numerical values was performed on the machine A, operating in both pump and turbine modes, in order to validate the CFD analyses. Once the CFD simulations had been validated, the effectiveness of the predicting model was evaluated through the comparison between the results obtained with the CFD analyses and the predicted ones related to

considered adequate for the evaluation of the global performance of hydraulic machines, unless a detailed study of the tip gap flow or of the clearance volume is performed [24]. 559

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Fig. 6. Impeller of machine A (a), machine B (b) and machine C (c). Table 4 BCs for the numerical simulations of the three machines in turbine mode. Boundary

Fluid dynamic parameters

Turbulence parameters 3

Inlet Outlet Wall Interface Volute to Impeller Interface Impeller to Exit Tube

3

Volumetric Flow Rate normal to boundary: Machine A [0.016–0.027] [m /s] Machine B [0.077–0.138] [m /s] Machine C [0.180–0.276] [m3/s] Average Static Pressure with Radial Equilibrium of 1 [bar] Heat flow = 0 (adiabatic wall), Velocity = 0 (no-slip condition) Conservative Interface Flux, Pitch Angles: 360° Conservative Interface Flux, Pitch Angles: 360°

Intensity 5% – – Conservative Interface Flux Conservative Interface Flux

Fig. 7. Overview of the refined mesh (a) used for machine B and values of the y+ (b) in correspondence of the impeller walls. Table 5 Spatial discretization of machines A, B and C. Subdomains

Impeller Inlet & Outlet Pipes Volute Total

Table 6 Machine A operating in pump mode: experimental data.

Number of elements Machine A

Machine B

Machine C

1,398,917 195,342 431,669 2,025,928

743,544 84,319 227,643 1,055,506

1,514,004 107,752 182,814 1,804,670

all the three hydraulic machines operating in turbine mode, which were previously described. Furthermore, a comparison between the proposed model and the ones of Singh et al. [31] and Novara et al. [32] is carried out in order to compare the prediction capability, as well as its limitations.

560

Q [m3/s]

φ

H [m]

ψ

P [kW]

Λ

η

0.007 0.008 0.010 0.013 0.014 0.016 0.018 0.020 0.021

0.006 0.007 0.009 0.012 0.013 0.015 0.016 0.018 0.019

11.8 11.6 11.2 10.7 10.0 9.0 8.0 6.9 5.9

0.135 0.133 0.128 0.122 0.114 0.103 0.091 0.079 0.067

1.351 1.401 1.570 1.844 1.807 1.909 2.018 2.083 2.026

0.00049 0.00061 0.00081 0.00108 0.00113 0.00114 0.00102 0.00092 0.00076

0.60 0.65 0.70 0.74 0.76 0.74 0.70 0.65 0.60

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Table 7 Machine A operating in pump mode: CFD data.

Table 8 Machine A operating in pump mode: percentage relative differences between experimental and numerical results.

Q [m3/s]

φ

H [m]

ψ

P [kW]

Λ

η

0.007 0.008 0.010 0.013 0.014 0.016 0.018 0.020 0.021

0.006 0.007 0.009 0.012 0.013 0.015 0.016 0.018 0.019

11.9 11.6 11.5 10.8 10.2 9.3 8.2 7.1 5.9

0.136 0.133 0.131 0.123 0.117 0.106 0.094 0.081 0.067

1.381 1.359 1.589 1.861 1.868 1.973 1.984 2.019 2.060

0.00051 0.00062 0.00084 0.00109 0.00114 0.00118 0.00110 0.00101 0.00075

0.62 0.67 0.71 0.74 0.75 0.74 0.73 0.69 0.59

φ

Δψ (%)

Δη (%)

ΔΛ (%)

0.006 0.007 0.009 0.012 0.013 0.015 0.016 0.018 0.019

−0.7 0.0 −2.3 −0.8 −2.6 −2.9 −3.3 −2.5 0.0

−3.3 −3.1 −1.4 0.0 1.3 0.0 −4.3 −6.2 1.7

−4.1 −1.6 −3.7 −0.9 −0.9 −3.5 −7.8 −9.8 1.3

3.1. Experimental tests and numerical simulations Table 9 Machine A operating in turbine mode: experimental results.

Tables 6 and 7 show the results for the machine A, operating in pump mode, obtained by laboratory tests and CFD simulations, respectively. Specifically, the values of volumetric flow rate (Q) , head (H) , mechanical power(P) and their corresponding non-dimensional values φ, ψ and Λ, respectively, are listed in the abovementioned tables. Furthermore, also the efficiency (η) values are reported. For sense of clarity, values referred to the BEP are highlighted in bold. In order to compare these results, Fig. 8 shows the trend of ψ (a), η (b) and Λ (c) as a function of φ, obtained by the experimental tests (green and square markers) and CFD simulations (red and diamond markers), showing the accordance between experimental data and CFD simulations. A quantitative analysis of the errors is reported in Table 8 where the percentage relative differences between experimental and numerical results, evaluated according to Eq. (10), are reported.

Δa(%) = (aExp. − a CFD)/aExp.

φ

H [m]

ψ

P [kW]

Λ

η

0.016 0.018 0.020 0.021 0.022 0.024 0.025 0.027

0.015 0.016 0.018 0.019 0.020 0.022 0.023 0.025

10.5 11.9 13.6 14.7 15.7 18.3 20.5 23.7

0.120 0.136 0.155 0.168 0.179 0.209 0.234 0.271

1.170 1.555 2.001 2.302 2.507 3.145 3.670 4.520

0.00128 0.00161 0.00209 0.00243 0.00265 0.00336 0.00393 0.00488

0.71 0.74 0.75 0.76 0.74 0.73 0.73 0.72

Table 10 Machine A operating in turbine mode: CFD results.

(10)

Results show that there is a general good agreement between experimental and numerical values: in particular, the percentage relative differences of the head coefficients are quite small since the maximum absolute value is equal to 3.3%. The efficiency predicted by the CFD analyses is, in general, in agreement with the one measured experimentally, achieving a maximum absolute percentage relative difference of 6.2%. The higher differences between experimental and numerical results are registered in the evaluation of the power coefficient where a maximum absolute percentage relative difference of 9.8% is obtained: indeed, this higher error is due to the error propagation determined by the use of Eq. (11) for its evaluation.

Λ = η·φ·ψ

Q [m3/s]

Q [m3/s]

φ

H [m]

ψ

P [kW]

Λ

η

0.016 0.018 0.020 0.021 0.022 0.024 0.025 0.027

0.015 0.016 0.018 0.019 0.020 0.022 0.023 0.025

11.0 12.4 14.2 15.3 16.5 19.0 21.1 23.7

0.126 0.142 0.162 0.175 0.188 0.217 0.241 0.271

1.174 1.577 2.062 2.364 2.671 3.355 3.829 4.583

0.00129 0.00164 0.00216 0.00249 0.00282 0.00358 0.00410 0.00495

0.68 0.72 0.74 0.75 0.75 0.75 0.74 0.73

In order to validate also the numerical analyses in turbine mode, a qualitative comparison between the experimental and the numerical data is reported in Fig. 8, which shows the trend of ψ (a), η (b) and Λ (c) as a function of φ. Fig. 9 confirms that the results obtained with the CFD simulations are in accordance with the global trend obtained in the experimental tests, also when the machine A is operating in turbine mode. Table 11 lists the percentage relative differences between experimental and numerical results, showing a good agreement. The absolute

(11)

Tables 9 and 10 show the results obtained by the experimental tests and the numerical analyses, respectively, regarding the machine A operating in turbine mode (PaT).

Fig. 8. Machine A operating in pump mode: comparison between experimental and numerical results regarding head coefficient (a), efficiency (b) and power coefficient (c). 561

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Fig. 9. Machine A operating in turbine mode: comparison between experimental and numerical results regarding head coefficient (a), efficiency (b) and power coefficient (c).

Table 11 Machine A operating in turbine mode: percentage relative differences between CFD and experimental results. φ

Δψ (%)

Δη (%)

ΔΛ (%)

0.015 0.016 0.018 0.019 0.020 0.022 0.023 0.025

−5.0 −4.4 −4.5 −4.2 −5.0 −3.8 −3.0 0.0

4.2 2.7 1.3 1.3 −1.4 −2.7 −1.4 −1.4

−0.8 −1.9 −3.3 −2.5 −6.4 −6.5 −4.3 −1.4

Table 13 Machine B operating in turbine mode: CFD results. Q [m3/s]

φ

φ/φBEP

H [m]

ψ

P [kW]

Λ

η

0.077 0.085 0.093 0.101 0.109 0.115 0.123 0.130 0.138

0.023 0.025 0.028 0.030 0.032 0.034 0.037 0.039 0.041

0.72 0.78 0.88 0.94 1.00 1.06 1.16 1.22 1.28

14.4 16.0 17.0 18.7 20.8 22.7 24.7 26.9 29.3

0.078 0.086 0.092 0.101 0.112 0.122 0.133 0.145 0.158

7.18 9.61 11.49 14.45 17.57 20.23 23.55 27.10 31.34

0.00118 0.00155 0.00191 0.00236 0.00283 0.00328 0.00389 0.00447 0.00512

0.66 0.72 0.74 0.78 0.79 0.79 0.79 0.79 0.79

Table 14 Machine B operating in turbine mode: predicting model results.

Table 12 Machine A operating in turbine mode: predicting model results. Q [m3/s]

φ

φ/φBEP

H [m]

ψ

P [kW]

Λ

η

Q [m3/s]

φ

φ/φBEP

H [m]

ψ

P [kW]

Λ

η

0.016 0.018 0.020 0.021 0.022 0.024 0.025 0.027

0.015 0.016 0.018 0.019 0.020 0.022 0.023 0.025

0.79 0.84 0.95 1.00 1.05 1.16 1.21 1.32

11.1 12.0 13.8 14.8 15.8 17.9 18.8 21.0

0.127 0.137 0.158 0.169 0.181 0.204 0.215 0.240

1.115 1.441 1.977 2.256 2.523 3.076 3.320 3.949

0.00122 0.00149 0.00208 0.00238 0.00268 0.00328 0.00356 0.00426

0.64 0.68 0.73 0.74 0.74 0.73 0.72 0.71

0.077 0.085 0.093 0.101 0.109 0.115 0.123 0.130 0.138

0.023 0.025 0.028 0.030 0.032 0.034 0.037 0.039 0.041

0.72 0.78 0.88 0.94 1.00 1.06 1.16 1.22 1.28

14.1 15.6 17.8 19.3 21.0 22.6 25.1 26.9 28.6

0.076 0.084 0.096 0.104 0.113 0.122 0.135 0.145 0.154

6.39 8.59 11.86 14.53 17.29 19.63 23.02 25.73 28.65

0.00105 0.00139 0.00196 0.00237 0.00278 0.00319 0.00380 0.00424 0.00467

0.60 0.66 0.73 0.76 0.77 0.77 0.76 0.75 0.74

maximum values of the percentage relative differences of each variable are even lower than the ones registered in pump mode: in particular, they are equal to 5.0%, 4.2% and 6.5% regarding the head coefficient, the efficiency and the power coefficient, respectively.

Resuming, the CFD tool allows to characterise the global fluid dynamic parameters of the tested centrifugal hydraulic machine with a good accuracy in both pump and turbine modes.

Fig. 10. Machine A operating in turbine mode: comparison between the predicting model results with the experimental and numerical ones regarding head coefficient (a), efficiency (b) and power coefficient (c). 562

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model. Tables 13 and 14 show both CFD and predicted results obtained by analysing the machine B, while Tables 15 and 16 list both CFD and forecasted results that refer to the machine C. Fig. 11 shows the trend of ψ/ψBEP (a), η/ηBEP (b) and Λ/ΛBEP (c) as a function of φ/φBEP, obtained by numerical simulations of both machine B (green square markers) and machine C (red diamond markers) and the predicting model (black circle markers). As it was previously done on the machine A, Fig. 11 shows how the predicting model characterises the global trend of the performances’ parameters with a good accuracy. Finally, a quantitative analysis is reported in Table 17, highlighting the percentage relative differences between numerical and predicted results related to head coefficient, efficiency and power coefficient of the all considered PaTs. In this case the percentage relative difference of a generic variablea was defined through Eq. (12).

Table 15 Machine C in turbine mode: CFD results and non-dimensional values. Q [m3/s]

φ

φ/φBEP

H [m]

ψ

P [kW]

Λ

η

0.180 0.192 0.204 0.216 0.228 0.240 0.252 0.264 0.276

0.029 0.031 0.033 0.035 0.037 0.039 0.041 0.043 0.045

0.78 0.84 0.89 0.95 1.00 1.05 1.11 1.16 1.22

52.8 57.0 61.4 65.8 70.4 75.0 79.7 84.5 89.5

0.182 0.196 0.211 0.226 0.242 0.258 0.274 0.291 0.308

60.60 70.86 81.10 92.02 105.50 116.54 130.04 137.87 150.24

0.00343 0.00401 0.00460 0.00522 0.00600 0.00664 0.00741 0.00788 0.00859

0.65 0.66 0.66 0.66 0.67 0.66 0.66 0.63 0.62

Table 16 Machine C in turbine mode: predicting model results and non-dimensional values. Q [m3/s]

φ

φ/φBEP

H [m]

ψ

P [kW]

Λ

η

0.180 0.192 0.204 0.216 0.228 0.240 0.252 0.264 0.276

0.029 0.031 0.033 0.035 0.037 0.039 0.041 0.043 0.045

0.78 0.84 0.89 0.95 1.00 1.05 1.11 1.16 1.22

52.6 57.3 61.6 66.3 70.9 75.9 80.5 85.8 90.7

0.181 0.197 0.212 0.228 0.244 0.261 0.277 0.295 0.312

52.01 64.76 77.66 89.91 103.08 116.15 129.35 144.44 157.17

0.00294 0.00366 0.00441 0.00511 0.00587 0.00662 0.00738 0.00825 0.00899

0.56 0.60 0.63 0.64 0.65 0.65 0.65 0.65 0.64

Δa(%) = (a CFD − aModel)/aModel

(12)

To better assess the results obtained with the predicting model, the values reported in all the previous tables are graphically displayed in Fig. 12. The analysis of the graphs allows to affirm that, if φ /φBEP varies between 0.88 and 1.16, the relative percentage differences of all the investigated variables related to the three machines are included between a range of ± 7%, while, if the range of φ /φBEP goes from 0.72 to 1.32, the relative percentage differences increase up to ± 17%. The reason of this behaviour is mainly due to the Δψ% and Δη% values. It is evident that, as concerns the head coefficient, very low percentage relative differences between the results of the CFD analyses and the predicting model ones are obtained; on the other hand, higher errors are recorded when the values of efficiencies are obtained considering φ /φBEP ≤ 0.88 and φ /φBEP ≥ 1.16 as Table 17 and Fig. 12 show. The higher uncertainty of the efficiency’s forecast of the model when it is used in these extreme operating conditions is due to the nature of the experimental data-set from which the model was derived. Indeed, Fig. 2 evidences that the experimental values are much more dispersed in the region of low flow rates if compared with the ones close to the PaTs’ BEP.

3.2. Assessment of the predicting model The aim of this paragraph is to investigate on the effectiveness of the predicting model used to evaluate the PaTs’ performances in off-design operating conditions. To this end, Table 12 collects the forecasted values of the machine A obtained using the model described in the Section 2.1, while Fig. 10 shows the trend of ψ/ψBEP (a), η/ηBEP (b) and Λ/ΛBEP (c) as a function of φ/φBEP obtained by the predicting model (black circle markers), experimental tests (green square markers) and CFD simulations (red diamond markers). The normalized values adopted in the model assessment allow to generalize the study and to widen its application to whatever machine, independently of its specific BEP in turbine mode. It is possible to conclude that the predicting model well reproduces the trendlines of both measured and numerical results. The assessment of the CFD analyses, which was performed in the previous paragraph, allows to evaluate the effectiveness of the proposed predicting model also through the study of other two centrifugal pumps operating in turbine mode, namely machine B and machine C, comparing the results of the CFD analyses with the ones predicted by the

3.3. Comparison of the proposed model with other models available in literature Finally, the presented model has been compared to other models available in literature for the prediction of PaTs’ characteristic curve operating at part-load conditions. It is worth to notice that also most of these models rely on the availability of the BEP point value, which corresponds to the rated operating conditions in terms of both flow rate and head. As an example, Barbarelli et al. [22] developed two models: the simpler one is based on the design parameters, while a detailed one that requires specific indications of the geometrical dimensions of the impeller, the blades’ number and pitch angle. These information result

Fig. 11. Machines B and C operating in turbine mode: comparison between predicting model and numerical results regarding head coefficient (a), efficiency (b) and power coefficient (c). 563

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Table 17 Machines A, B and C operating in turbine mode: percentage relative differences between numerical and forecasted results related to head coefficients, efficiency and power coefficients. Machine A

Machine B

Machine C

φ /φBEP

Δψ (%)

Δη (%)

ΔΛ (%)

φ /φBEP

Δψ (%)

Δη (%)

ΔΛ (%)

φ /φBEP

Δψ (%)

Δη (%)

ΔΛ (%)

– 0.79 0.84 0.95 1.00 1.05 1.16 1.21 1.32

– −4.5 −0.7 −1.8 −0.6 0.0 2.4 7.6 8.4

– 7.9 7.5 2.8 2.7 2.7 4.2 4.2 4.3

– 3.2 7.2 0.9 2.0 2.9 6.5 12.0 13.0

0.72 0.78 0.88 0.94 1.00 1.06 1.16 1.22 1.28

2.6 2.4 −4.2 −2.9 −0.9 0.0 −1.5 0.0 2.6

10.0 9.1 1.4 2.6 2.6 2.6 3.9 5.3 6.8

12.4 11.5 −2.6 −0.4 1.8 2.8 2.4 5.4 9.6

0.78 0.84 0.89 0.95 1.00 1.05 1.11 1.16 1.22

0.6 −0.5 −0.5 −0.9 −0.8 −1.1 −1.1 −1.4 −1.3

16.1 10.0 4.8 3.1 3.1 1.5 1.5 −3.1 −3.1

16.7 9.6 4.3 2.2 2.2 0.3 0.4 −4.5 −4.4

Fig. 12. Machines A, B and C operating in turbine mode: graphs of the percentage relative differences between numerical and predicted results related to head coefficient (a), efficiency (b) and power coefficient (c).

Fig. 13. Normalized experimental characteristic curves for three PaTs (18.2 rpm (a), 19.7 rpm (b) and 44.7 rpm (c)) and predicted characteristic curves obtained with the proposed model and the models of [31] and [32].

Table 18 Comparison between the normalized non-dimensional characteristic curves of the PaT with a characteristic speed value equal to 18.2 rpm.

Table 19 Comparison between the normalized non-dimensional characteristic curves of the PaT with a characteristic speed value equal to 19.7 rpm.

φ /φBEP

ψ/ψBEP Exp.

ψ/ψBEP Model

ψ/ψBEP [32]

Δa% Model

Δa% [32]

φ /φBEP

ψ/ψBEP Exp.

ψ/ψBEP Model

ψ/ψBEP [32]

Δa% Model

Δa% [32]

0.52 0.62 0.66 0.82 0.92 1.00

0.50 0.58 0.65 0.82 0.93 1.00

0.46 0.57 0.61 0.79 0.91 1.008

0.51 0.55 0.58 0.73 0.87 1.005

−8.0 −1.7 −6.2 −3.7 −2.2 +0.8

+2.0 −5.2 −10.8 −11.0 −6.5 +0.5

0.43 0.76 0.91 0.97 1.00

0.56 0.75 0.90 0.97 1.00

0.37 0.72 0.89 0.97 1.008

0.50 0.66 0.85 0.95 1.005

−33.9 −4.0 −1.1 0.0 +0.8

−10.7 −12 −5.6 −2.1 +0.5

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accurate numerical comparison is not possible; however, it is worth to notice that the algorithm proposed in [31] requires the data of both shape and size of the pump to be applied; moreover, the application of the algorithm is more time consuming than the other models, since a considerable number of equations have to be solved in order to obtain all the operating data of the PaT. The errors of the proposed model lay in the range of ± 4%, except for two single operating conditions at low flow rate, which is to be considered a very good result for practical uses, given the very simple application of the model over the others proposed in literature. Moreover, the model proposed in this work also supplies a figure of the variation of the mechanical efficiency in turbine mode in off-design, which is not reported and discussed in the other works in literature. An improvement of the proposed model for extremely low flow rate values can be probably achieved if another non-dimensional parameter, again related to the BEP in turbine mode, is introduced in both Eqs. (5) and (6).

Table 20 Comparison between the normalized non-dimensional characteristic curves of the PaT with a characteristic speed value equal to 44.7 rpm. φ /φBEP

ψ/ψBEP Exp.

ψ/ψBEP Model

ψ/ψBEP [32]

Δa% Model

Δa% [32]

0.60 0.67 0.69 0.93 0.96 1.00

0.56 0.61 0.67 0.89 0.95 1.00

0.54 0.62 0.66 0.92 0.96 1.008

0.54 0.58 0.60 0.89 0.94 1.005

0.0 +1.6 −1.5 +3.4 +1.1 +0.8

0.0 −4.9 −10.4 0.0 −1.1 +0.5

to be rarely available from the PaT’s manufacturers. A similar procedure was followed also by Venturini et al. [21] that use almost the same geometrical information of the PaT employed in [22]. Unfortunately, a precise comparison with the aforementioned models is not possible since the authors do not supply the quantitative information required to reconstruct the predicted PaT’s performance curves. However, the error related to the performance prediction, reported in their works, is higher than 5%, especially in the low flow rate region. Singh et al. [31] developed a model for the PaTs’ performance prediction operating in offdesign conditions that requires, as input data, the fixed values of both BEP flow rate and head. The model is based on the experimental tests of 13 PaTs, while three additional PaTs, having a characteristic speed of 18.2 rpm, 19.7 rpm and 44.7 rpm, are used to validate their model. A set of equations allows to define the operating conditions in turbine mode and to obtain other parameters needed to reconstruct the offdesign operating characteristics of a PaT. In the work by Novara et al. [32], a model that requires the knowledge of the BEP values in turbine mode was developed adopting correlations based on the specific speed parameter. In order to assess the accuracy of the presented model with respect to the others, the experimental data of three PaTs analysed in the work of [31] were used as reference. Fig. 13 shows the plots of the characteristic curves of the three PaT, reconstructed using the model presented in this paper and the ones of Singh [31] and [32]. The percentage relative differences between the experimental data and the results obtained using the proposed model and the ones of [32] are listed in Tables 18–20, for the PaTs with characteristic speed of 18.2 rpm, 19.7 rpm and 44.7 rpm, respectively. It is worth to notice that the predicted data obtained by the model of [31] are not reported in the abovementioned tables since, in their work, they used different φ /φBEP values that do not correspond to the experimental ones, except for the BEP. Fig. 13a shows that the model of [31] is more accurate with respect to the proposed model within the range of approximately 0.65 and 0.90 φ /φBEP ; on the other hand, for values lower than approximately 0.65 φ /φBEP , the proposed model has a better forecast capability. In comparison with the model of [32], the proposed model has an evident better forecast capability for values of φ /φBEP greater or equal than 0.62 (see Table 18). In Fig. 13b, the model proposed of [31] is definitively better than the proposed model. However, the predicting capability of the model developed by [32] is worse than the proposed one for values of φ /φBEP greater or equal than 0.76 (see Table 19). Finally, Fig. 13c shows that the proposed model fits better the experimental characteristic curve than the one of [31] for values of φ /φBEP lower than approximately 0.80. On the contrary, the model proposed by [31] shows a slightly better prediction capability with respect to the proposed model for values of φ/φBEP higher than approximately 0.80. The model proposed by [32] is the one that better fits the experimental non-dimensional characteristic curve for values of φ /φBEP greater or equal than 0.93 (see Table 20), whereas, for values of φ /φBEP lower than 0.93, the predicted curve is quite far from the experimental one if compared to the proposed model. As a general comment, it is possible to conclude that all the three models have a relatively good prediction capability, being the model of [31] the one that fits better the experimental data, even though an

4. Conclusions In this work, a model capable of predicting the performances of PaTs was presented and its accuracy was assessed, with particular attention to the behavior in off-design conditions. The validation of the model was performed by studying three different machines, operating in both design and off-design conditions and using experimental tests together with the CFD analyses. The predicting model is built by correlating the normalized non-dimensional parameters of a data-set of PaTs whose experimental data are available in literature. One of the three machines was also tested in a test bench developed by some of the authors of this paper to validate the CFD simulations and the proposed model. In order to assess the forecasting capabilities of the model, other two centrifugal PaTs were studied by the means of the CFD simulations. The characteristic curves predicted by the proposed model showed its accuracy on evaluating the performances of a generic pump operating in reverse mode, achieving low percentage errors with respect to the performances obtained through the CFD analyses. In particular, the relative percentage differences between the PaTs’ performances forecasted by the model and the ones obtained by the CFD analyses were between ± 7%, considering the off-design operating conditions close to the BEP. For extremely low flow rates, the relative percentage differences increase up to ± 17% since the experimental values, on which the predicting model is based, are much more dispersed. In order to further assess the accuracy of the proposed model, a comparison between the studies developed by Singh et al. [31] and Novara et al. [32] was performed, taking the experimental data of [31] as reference. Results showed a general good prediction capability also compared to the others methodologies proposed in literature. However, it is worth to notice that the presented model also supplies a figure of the mechanical efficiency of the PaT, which is not directly given in the other works, and it is based on a very simple procedure that only requires the BEP in turbine mode. A further improvement in the forecast capability of the proposed model consists in increasing the feed-in data from experimental tests on PaTs and in adding additional parameters in the description of the fitting equations. References [1] Hartikainen T, Lehtonen J, Mikkonen R. Reduction of greenhouse-gas emissions by utilization of superconductivity in electric-power generation. Appl Energy 2004;78:151–8. [2] Mori Y, Kikegawa Y, Uchida H. A model for detailed evaluation of fossil-energy saving by utilizing unused but possible energy-sources on a city scale. Appl Energy 2007;84:921–35. [3] Zhang Y-J, Wei Y-M. An overview of current research on EU ETS: evidence from its operating mechanism and economic effect. Appl Energy 2010:1804–14. [4] Kuriyama A, Abe N. Ex-post assessment of the Kyoto protocol – quantification of CO2 mitigation impact in both annex B and non-annex B countries. Appl Energy 2018;220:286–95.

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