Development of a Predicting Model for Calculating the Geometry and the Characteristic Curves of Pumps Running as Turbines in Both Operating Modes
Abstract
:1. Introduction
 
 The inability of the distribution networks to reach rural settlements. The latter were therefore forced to produce energy independently, also using inverted flow pumps.
 
 The use of power electronics: the adjustment of the machine using inverters makes it possible to obtain acceptable efficiencies for a range of different flow rates (previously the adjustments were manual and only hydraulic).
 
 Lower costs compared to a normal hydraulic turbine, especially for small units, below 50 kW;
 
 Simplicity of installation and maintenance;
 
 The wide range of models available on the market.
 
 The absence of guide vanes, which excludes the possibility of making hydraulic adjustments;
 
 The lower efficiency compared to a welldesigned hydraulic turbine, especially in the offdesign conditions;
 
 The lack of information about the characteristic curves for the turbine, as the manufacturer of the machine supplies only those for the pump.
State of the Art
 Simple statistical correlations that aim to establish a connection between the point of better efficiency (BEP) in pump operation and that in turbine operation. In more detail, from the examination of the position of the BEP points in the pump and turbine operation, of machines for which these are known, laws are derived which can then be used to predict the location of the BEP of a new machine. For example, Child [13], Sharma [14], Alatorre [15], and Stepanoff [16] combine the best head ratio and the best flow ratio with respect to the total efficiency of the pump; Hancock [17] correlates these reports to the total efficiency of the turbine; Schmield [18] relates these relationships to the hydraulic efficiency of the pump; and Grover [19] and Hergt [20] relate these ratios to the characteristic speed of the turbine.
 The PAT performance prediction method using specific speed, where flow rate and head are expressed as a function of specific speed [21]. Different expressions of specific speeds are used, which are gradually refined and improved to ensure better accuracy of the results. Some examples are as follows: Derakshan applied the dimensionless specific speed to obtain different relations, valid for centrifugal pumps with specific speed n_{s} < 60 [22]; Nautiyal proposed an additional parameter through which it is possible to obtain the trend of the prevalence and the flow rate [23]; Singh proposed a correlation based on experimentation performed on a sample of 13 pumps and subsequently applied it to the pump under examination, thus obtaining the relationship between the specific speed in turbine operation and that in pump operation [24]; Tan, by testing the hydraulic performance of centrifugal pumps, used both in direct and reverse operation, obtained different linear relationships between the pump and turbine parameters [25]; Stefanizzi established a relationship between specific speed under pump and turbine mode, based on data obtained from the performance of 27 pumps, and subsequently it used to predict the performance of 11 new PATs [26].
 Empirical correlations: Derakshan’s [27] methodology proposes head–flow and power–flow polynomial curves, interpolated on the available PAT sample. These polynomials are dimensionless based on the values of the flow rate, head, and power of the PAT at the BEP and can be used in a universal way for predicting the curves of head, power, and efficiency versus flow rate for any machine.
 Onedimensional model: Venturini [28] developed a prediction model based on the physics of the machine and consisting in the use of loss coefficients and specific parameters, through an optimization procedure, which is applied to the machine operating as a pump and subsequently as a turbine.
 Numerical analysis and CFD, for axial flow centrifugal pumps, which allow reconstructing, through a structured stepbystep methodology, the characteristic curve in pump mode, and subsequently in turbine mode, and predicting the behavior of the fluid inside the turbomachinery [29].
 For commercial centrifugal radial flow pumps, through computer numerical simulations, a methodology has been developed that makes it possible to predict the characteristic curves, in both operating modes, with errors of less than 10% compared to the mathematical model [30]. The operating conditions of the site are then obtained, providing a methodology that allows the choice of the most suitable turbomachine to obtain electricity in those areas that do not have access to it, exploiting small hydroelectric resources.
2. Materials and Methods
2.1. Fluid Dynamic Model
2.1.1. Pump Operation
Velocity Triangles
 
 Stodola [37,38] assumes that the motion of the fluid at the exit of the impeller is the sum of a main flow, which is guided by the blades, and of a vortex, having a rotation speed equal in modulus but in the opposite direction to that of the impeller. The diameter to which this vortex refers corresponds to the minimum passage section at the exit of the impeller.
 
 
 Busemann [39,40] considered radial impellers having thick blades in an infinitesimal logarithmic spiral, which transforms into a rectilinear array made of infinite foil profile planes of infinitesimal thickness. Furthermore, he assumed that the upstream machine elements downstream of the rotor are sufficiently distant to have no effect on the behavior of the fluid in the mobile blade.
 
 Qiu, Mallikarachchi, and Anderson [41,42], who analyzed the various computation models of the slip factor present in the literature, realized their application limits. Based on this analysis procedure, they obtained a unitary formulation of the slip coefficient, which considers both the geometry of the impeller and the flow conditions. This model was derived from the studies of Eckardt [43], who believed that the rotation speed of the vortex was not equal to the rotation speed of the impeller but depended on the blade load, that is, the difference in relative speed between the face under pressure and that in the blade depression.
 
 
 Wiesner carried out an indepth study on the correlations that existed in the literature in that year (1967) to verify which was the most reliable and which provided results as close as possible to those obtained experimentally. From this study, he found that Busemann’s correlation [39,40] is the most reliable if applied to pumps with centrifugal impellers.
Hydraulic Losses
2.1.2. Turbine Operation
Velocity Triangles
 
 Inlet: For the calculation of the velocities, reference is made to their average value.
 
 Volute: For this component, a free vortex distribution is hypothesized, and this assumption is confirmed in the experimental analyses carried out by some researchers.
 
Hydraulic Losses
2.2. Geometric Model
 
 Flow rate and head relative to the best efficiency conditions (Q, H_{m});
 
 Rotational speed at which the machine must work (n);
 
 Head at the shutoff (H_{mo});
 
 Absorbed power at the point of best efficiency (Pe);
 
 Height of the machine (h_{2});
 
 External diameter of the impeller (d_{2}).
2.2.1. Calculation of the Shaft Diameter
2.2.2. Sizing of the Inlet Section
 
 The blade tip diameter, d_{1p}, is obtained through an interpolation function, of order two, which correlates the blade tip diameter with the specific speed, that is$$\frac{{d}_{1p}}{{d}_{2}}=0.00003{n}_{s}^{2}+0.0106{n}_{s}+0.1219$$
 
 The internal diameter of impeller inlet d_{1m} is assumed with design criteria, considering that the shaft must be housed in the impeller hub. It is determined as follows:$${d}_{\mathit{1}m}=k{d}_{shf}$$
 
 The inlet angle of the relative velocity vector, β_{1p}, was obtained by imposing that, in correspondence with the design conditions, the geometric angle is equal to the real angle. This evaluation of the angle of entry is aimed at minimizing, under design conditions, the losses due to shocks. For a correct evaluation of the meridian speed, both the volumetric efficiency η_{v} and the real transit area must be considered. The real transit area considers the overall dimension factor of the blades which in turn depends on the angle β_{1p}. It is clear that there is a need to resort to a recursive procedure for the evaluation of β_{1p}.
 
 The width of the inlet blade, b_{1}: Indicating with θ the inclination of the blade edge with respect to the radial direction, the length of the incoming blade was obtained as follows:$${b}_{1}=\frac{{d}_{\mathit{1}p}{d}_{\mathit{1}m}}{2}\frac{1}{\mathrm{cos}\left(\theta \right)}$$
2.2.3. Determination of the Geometry of the Seals
2.2.4. Determination of the Number of Blades and the Angle β_{2p}
2.2.5. Calculation of the Blade width at the Outlet
2.2.6. Sizing of the Volute
2.2.7. Sizing of the Final Diffuser
2.2.8. Impeller–Case Distance s_{d}
2.3. Measurement of Geometric Parameters
 
 External diameter (d_{1});
 
 Eye diameter of the impeller (d_{2});
 
 External blade width (b_{1});
 
 Blade width at the eye of the impeller (b_{2});
 
 Outflow angle relative to the external diameter (β_{1});
 
 Outflow angle relative to the eye of the impeller (β_{2}).
 
 To calculate the exit angle β_{2} (angle relative to the highpressure area), it was verified that the profile of the impeller blades in the radial plane was approximated by a logarithmic spiral, using a probe mounted in the spindle of the milling machine to reconstruct its shape. The equation of the logarithmic spiral in polar coordinates r, θ (where r is the generic radius of the profile in a radial plane and θ is the angle that this radius forms with the axis of the machine always in the radial plane) is as follows:
 
 To determine the angle β_{1}, it is required that in correspondence with the design conditions (BEP point) there is a correspondence between the geometric angle and the flow angle.
3. Results
3.1. Results of the Fluid Dynamics Model
 
 Oscillations of the rotation speed;
 
 Instability in the torque applied to the motor shaft;
 
 Instability in the head from the turbine or in the flow rate processed;
 
 Cavitation due to the presence of low suction pressures;
 
 Water hammer, which stresses both the piping system and the mechanical parts of the PAT.
3.2. Results of the Geometric Model
4. Sensitivity Analysis
 
 Hub diameter d_{1m};
 
 Width of the blades entering the impeller b_{1};
 
 Suction diameter d_{0};
 
 Width and height of the volute h_{v}, b;
 
 Impeller exit angle β_{2};
 
 Height of the blades exiting the impeller b_{2}.
 
 Q_{P}_{/T} meas: flow rate to BEP measured on the bench;
 
 Q_{P}_{/T} calc: flow rate to the BEP calculated by the model;
 
 H_{P}_{/T} meas: head at BEP measured on the bench;
 
 H_{P}_{/T} calc: head at BEP calculated by the model.
5. Procedure for Predicting the Performance of a Generic Pump
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
Nomenclature
Symbols  
A  generic area 
A_{b,f}  back/front leakage passage area 
A_{θmj}  inlet area of the jth volute sector 
A_{1}, _{2}  passage area at different points of the impeller 
A_{1r, 2r}  real passage area at different points of impeller 
A_{3}  diffusion region passage area 
A_{4}  volute final section area 
A_{5}  final diffuser inlet passage area 
b_{1, 2}  width at different points of impeller 
b_{3}  vaneless diffuser width 
b_{4}  final section volute width 
b_{5}  final section diffuser width 
c_{1, 2}, _{3}, _{4}  absolute fluid velocities at different points of PAT 
c_{m}_{1, m2, m3}  meridional velocities at different points of PAT 
c_{u1}, _{u2, u3, u4}  peripheral velocities at different points of PAT 
cl  radial clearance of the seal 
d  generic diameter 
d_{o}  impeller eye diameter 
d_{1}, d_{2}, d_{3}  diameter at different points of PAT 
d_{eq}  equivalent hydraulic diameter 
d_{shf}  shaft diameter 
d_{f}  diameter of the front seal 
d_{b}  diameter of the rear seal 
Eff_meas  measured efficiency 
Eff_catal  catalog efficiency 
Eff_calc  calculated efficiency 
h_{4}, _{5}  heights at different points of the final diffuser 
h_{v}  volute throat section height 
H  head 
H_{e}  head at BEP of the pump 
H_{m}  real head 
Head_meas  measured real head 
Head_catal  catalog real head 
Head_calc  calculated real head 
H_{mo}  head at the shutoff 
H_{th}  theoretical head (Euler’s head) 
H_{BEP}  head at BEP of the PAT 
K_{v}  volute velocity coefficient 
L_{d}  diffuser length 
n  rotational speed 
ns  characteristic speed 
sd  clearance between the impeller and the case 
P  power 
P_{e}  maximum pump power 
Q  flow rate 
Q_{e}  flow rate at BEP of the pump 
Qs  leakage flow 
Q_{BEP}  flow rate at BEP of the PAT 
R_{4}  final section volute radius 
t_{1}, _{2}  vane thickness 
u_{1}, _{2}  peripheral velocities at different points of impeller 
w_{u}_{1}, _{u}_{2}  peripheral components of relative velocity 
w_{m}_{1}, _{m}_{2}  meridional components of relative velocity 
w_{∞}  average relative velocity 
z  number of blades 
Greek letters  
α_{2}  absolute flow angle in the vaneless diffuser 
α_{d}  final diffuser opening angle 
β  inclination of relative flow to peripheral direction 
β_{1f}, _{2f}  relative flow direction 
β_{1p}, _{2p}  blades angles at different points of impeller 
ΔS_{cl}  lateral surface area 
ΔS_{inn}  increment of inner wall surface 
ΔS_{cp}  increment of peripheral volute surface 
z  dynamic loss coefficient 
η  efficiency 
η_{calc}  calculated efficiency 
η_{H}  hydraulic efficiency 
η_{D}  disc efficiency 
η_{v}  volumetric efficiency 
η_{tot}  total efficiency 
η_{meas}  measured efficiency 
θ  inclination of blade to radial direction 
λ  friction coefficient 
λ_{j}  friction coefficient of a segment of volute 
μ  leakage flow coefficient 
υ  kinematic viscosity 
ξ_{1}, _{2}  vanes blockage factor 
ξ_{d}  localized drag coefficient 
ρ  density of water 
τ_{a}  torsional stress 
ϕ  capacity coefficient 
ω  angular velocity 
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Impeller 
$${h}_{fg}=\lambda \frac{{w}_{\infty}^{2}}{2g}\left(\frac{l}{{d}_{eq}}\right)$$

Vaneless diffuser 
$${h}_{fc}=\frac{\lambda}{2g}\frac{1}{{D}_{h3}}\frac{{c}_{3}^{2}}{sen\left({\alpha}_{2}^{\prime}\right)}\frac{{d}_{3}}{{d}_{2}}\left(\frac{{d}_{3}{d}_{2}}{2}\right)$$

Volute 
$${h}_{fv}={\displaystyle \sum}_{j=1}^{18}{\lambda}_{j}\frac{{\underset{\_}{c}}_{4}^{2}}{2g}\frac{{\left(\Delta {S}_{cl}+\Delta {S}_{inn}+\Delta {S}_{cp}\right)}_{j}}{{A}_{\theta mj}}\frac{{Q}_{j}}{Q}$$

Final diffuser 
$${h}_{fd}=\frac{\lambda}{8sen\left({\alpha}_{d}\right)}\left[1{\left(\frac{{A}_{4}}{{A}_{5}}\right)}^{2}\right]\frac{{\underset{\_}{c}}_{4}^{2}}{2g}$$

Inlet 
$${h}_{inlet}=0.25{\left(\frac{Q}{\frac{\pi {d}_{0}^{2}}{4}}\right)}^{2}\frac{1}{2g}$$
 
Impeller  Shock losses 
$${h}_{shock}=\frac{{\left[{w}_{1}sen(i)\right]}^{2}}{2g}$$

Wake losses 
$${h}_{dg}={\left({\xi}_{2}1\right)}^{2}\frac{{c}_{m2}^{2}}{2g}$$
 
Vaneless diffuser  Instantaneous expansion losses 
$${h}_{dc}=\frac{{c}_{m2}^{2}}{2g}{\left(1\frac{{A}_{2r}}{{A}_{3}}\right)}^{2}$$

Volute  Mixing losses 
$${h}_{dv}=\frac{{c}_{m3}^{2}}{2g}$$

Final diffuser  Diffusion losses ^{1} 
$${h}_{dd}={\xi}_{d}\frac{{c}_{4}^{2}}{2g}$$

$0.025\le c\le 0.075$  ${\xi}_{d}=0.14$ 
$0.075<c\le 0.15$  ${\xi}_{d}=0.20$ 
$0.15<c\le 0.25$  ${\xi}_{d}=0.47$ 
$0.25<c\le 0.35$  ${\xi}_{d}=0.76$ 
$0.35<c\le 0.45$  ${\xi}_{d}=0.95$ 
$0.45<c\le 0.75$  ${\xi}_{d}=1.05$ 
$0.75<c\le 0.90$  ${\xi}_{d}=1.10$ 
Diffuser  Inlet losses 
$${h}_{dd}={\xi}_{d}\frac{{c}_{4}^{2}}{2g}$$

Impeller  Inlet losses 
$${h}_{inlet}=0.5\cdot \left(1\frac{{d}_{2}{b}_{2}}{{d}_{3}{b}_{3}}\right)\frac{{c}_{m3}^{2}}{2g}$$

Shock losses ^{1} 
$${h}_{shock}=\frac{{\left[{w}_{2}sen(i)\right]}^{2}}{2g}$$
 
Instantaneous expansion losses 
$${h}_{dg}={\left({\xi}_{1}1\right)}^{2}\frac{{c}_{m1}^{2}}{2g}$$
 
Volute  Diffusion losses 
$${h}_{diff}=\frac{{c}_{u3}^{2}{c}_{u3bep}^{2}}{2g}$$

Outlet ^{2} 
$${h}_{inlet}=0.25{\left(\frac{Q}{\frac{\pi {d}_{0}^{2}}{4}}\right)}^{2}\frac{1}{2g}+\frac{{c}_{u1}^{2}}{2g}$$

$1.25\le \frac{{b}_{5}}{b}\le 1.75$  ${\xi}_{d}=0.12$ 
$1.75<\frac{{b}_{5}}{b}\le 3$  ${\xi}_{d}=0.30$ 
$3<\frac{{b}_{5}}{b}\le 5$  ${\xi}_{d}=0.40$ 
Pumps  

40335  40250  80220  50160  80160  100200  
Meas  Calc  Meas  Calc  Meas  Calc  Meas  Calc  Meas  Calc  Meas  Calc  
d_{0} (mm)  72.5  82.4  65  65.6  115  95.6  77.1  73.4  120  87.5  143  114 
d_{1m} (mm)  49.9  40.2  37.74  35.9  50  45.4  34  29.6  42  35  50  51 
d_{1} (mm)  60.9  61.3  51.37  50.8  57  70.5  50  51.5  81  61.2  62  82.4 
d_{f} (mm)  84.75  107.3  84.75  85.5  134.75  124.5  89.75  95.6  135  114  159.75  148.4 
d_{3} (mm)  338  251.2  282  273  230  234.3  185  186.2  185  187.3  224  234.3 
d_{b} (mm)  139.7  108.2  99.76  86.2  150  125.5  90  96.4  135  114.9  150  149.6 
b_{1} (mm)  7.2  27.5  8.89  11.5  50  32.7  32  28.6  62  34.3  61  41.1 
b_{2} (mm)  10  16.1  8  7.3  25  21.2  16  16.5  25  21.1  32.5  28.7 
b_{3} (mm)  16  28.2  8  14.7  42.5  37.2  26  28.8  35  37  61  50.3 
b (mm)  26  24  24  27  49  66  67  42  87  47  55  93.6 
β_{1p} (°)  20.49  43.83  38.58  51.9  17.53  42.75  26.36  41.38  23.7  43  20  38.97 
β_{2p} (°)  24  23  20  23  28.26  27  26  26  23.7  26  26  27 
t_{1} (mm)  4  4  4  4  2  4  4  4  4  4  4  4 
t_{2} (mm)  4  4  4  4  2  4  4  4  5  4  4  4 
b_{5} (mm)  40  64.2  89  52.6  80  89  50  89.9  84  63  89  114.5 
cl (mm)  0.125  0.125  0.25  0.125  0.15  0.125  0.25  0.125  0.25  0.125  0.55  0.25 
sd (mm)  7  9  17  9  9  9  21  9  9  9  15  9 
L_{d} (mm)  97  131.5  98  95  90  170  102  93  137  136.5  91  170 
Q_{P}  H_{P}  Q_{T}  H_{T}  

d_{1m}  *  ***  ***  *** 
b_{1}  *  *  **  **** 
d_{0}  **  ***  ****  **** 
h_{v}, b  ****  ***  ***  **** 
b_{2}  **  ***  *  * 
b_{2}  **  ****  **  ** 
h_{v}, b (m)  Δ% Variation  h_{v}, b (m)  Q_{p} meas  Q_{p} calc  E%  H_{p} meas  H_{p} calc  E%  Q_{T} meas  Q_{T} calc  E%  H_{T} meas  H_{T} calc  E% 

0.024  −5.0%  0.023  26  25.1  3.6%  35  29.90  14.7%  49.3  45.9  7.0%  99.5  109.0  −9.5% 
0.024  −2.5%  0.023  26  26.0  0.0%  35  30.30  13.5%  49.3  47.3  4.0%  99.5  105.1  −5.6% 
0.024  0.0%  0.024  26  26.0  0.0%  35  31.10  11.1%  49.3  48.8  1.1%  99.5  101.5  −2.0% 
0.024  2.5%  0.025  26  27.2  −4.6%  35  31.20  10.9%  49.3  50.2  −1.8%  99.5  98.2  1.4% 
0.024  5.0%  0.025  26  28.3  −9.0%  35  31.30  10.6%  49.3  51.7  −4.8%  99.5  95.0  4.5% 
h_{v}, b (m)  Δ% Variation  h_{v}, b (m)  Q_{p} meas  Q_{p} calc  E%  H_{p} meas  H_{p} calc  E%  Q_{T} meas  Q_{T} calc  E%  H_{T} meas  H_{T} calc  E% 

0.027  −5.0%  0.026  25  25.0  0.0%  20  19.90  3.8%  38.3  37.6  1.8%  43.7  49.0  −12.2% 
0.027  −2.5%  0.026  25  25.0  0.0%  20  19.70  1.7%  38.3  38.7  −1.0%  43.7  47.3  −8.4% 
0.027  0.0%  0.027  25  25.0  0.0%  20  20.00  0.0%  38.3  39.8  −3.7%  43.7  45.9  −5.0% 
0.027  2.5%  0.028  25  25.4  −1.6%  20  20.20  −0.8%  38.3  41.2  −7.4%  43.7  44.9  −2.9% 
0.027  5.0%  0.028  25  26.4  −5.7%  20  20.10  −0.4%  38.3  42.3  −10.2%  43.7  43.6  0.1% 
h_{v}, b (m)  Δ% Variation  h_{v}, b (m)  Q_{p} meas  Q_{p} calc  E%  H_{p} meas  H_{p} calc  E%  Q_{T} meas  Q_{T} calc  E%  H_{T} meas  H_{T} calc  E% 

0.066  −5.0%  0.063  100  100.0  0.0%  14.4  14.28  0.8%  123  117.68  4.3%  20.0  20.7  −3.3% 
0.066  −2.5%  0.064  100  100.0  0.0%  14.4  14.35  0.3%  123  120.66  1.9%  20.0  20.2  −1.1% 
0.066  0.0%  0.066  100  100.0  0.0%  14.4  14.41  −0.1%  123  122.64  0.3%  20.0  19.6  1.9% 
0.066  2.5%  0.068  100  100.0  0.0%  14.4  14.46  −0.4%  123  124.63  −1.3%  20.0  19.1  4.6% 
0.066  5.0%  0.069  100  100.0  0.0%  14.4  14.51  −0.8%  123  126.61  −2.9%  20.0  18.6  7.2% 
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Barbarelli, S.; Pisano, V.; Amelio, M. Development of a Predicting Model for Calculating the Geometry and the Characteristic Curves of Pumps Running as Turbines in Both Operating Modes. Energies 2022, 15, 2669. https://doi.org/10.3390/en15072669
Barbarelli S, Pisano V, Amelio M. Development of a Predicting Model for Calculating the Geometry and the Characteristic Curves of Pumps Running as Turbines in Both Operating Modes. Energies. 2022; 15(7):2669. https://doi.org/10.3390/en15072669
Chicago/Turabian StyleBarbarelli, Silvio, Vincenzo Pisano, and Mario Amelio. 2022. "Development of a Predicting Model for Calculating the Geometry and the Characteristic Curves of Pumps Running as Turbines in Both Operating Modes" Energies 15, no. 7: 2669. https://doi.org/10.3390/en15072669