Improved Planning of Energy Recovery in Water Systems Using a New Analytic Approach to PAT Performance Curves
Abstract
:1. Introduction
2. Materials and Methods
2.1. Definition of the Characteristic Numbers of PATs
- (i)
- Choosing a pump according to pump catalogues through the operation point ( as a turbine in the network. Therefore, the use of empirical expressions to predict the BEP location of a PAT with respect to its known BEP as a pump () is necessary.
- (ii)
- Defining empirical expressions which enable to define the head-discharge curve and efficiency curve, as well as the runaway curve, as a function of the discharge.
2.2. Coefficient Proposal to Estimate the Operation Point of PAT Using Pump Manufacture
- Determination of the RMSE. This error index is a standard way to measure the error of a model in predicting quantitative data. If the RMSE is zero, this value indicates a perfect fit. Formally, it is defined as follows:
- Determination of the MAD. This index measures the average magnitude of the errors in a set of predictions without considering their direction. It is the average over the test sample of the absolute differences between prediction and actual observation where all individual differences have equal weight. If the MAD is zero, this value indicates a perfect fit. Formally, it is defined as follows:
- Determination of the MRD. This index considers the weight of the error to the variable value. If the MRD is zero, this value indicates a perfect fit. Formally, it is defined as follows:
- Determination of the bias (BIAS). In this case, the index measures the tendency of the prediction in the variable (Q, H or efficiency), determining if the predicted values are smaller or larger than the experimental values. If the BIAS value is negative, it indicates that the method overestimates the variable, while, if the BIAS value is positive, it indicates that the variable is underestimated. This index is defined by the equation [34]:
2.3. Estimation of the Head-Discharge and Efficiency Curves Considering
2.4. Materials
3. Results
3.1. Comparison between the Proposed and Others Methods to Predict the Flow and Head in Pump Mode. First Approach to Predict Runaway Curve
3.2. Estimation of the Operation and Effciency Curve
4. Conclusions
Supplementary Materials
Author Contributions
Funding
Conflicts of Interest
Nomenclature
D | Impeller diameter (m) |
g | Gravity constant (9.81 m2/s) |
Head in the best efficiency point in pump mode (m w.c.) | |
Head in the best efficiency point in turbine mode (m w.c.) | |
Head in turbine mode (m w.c.) | |
Specific rotational speed (m, kw) | |
Specific rotational speed in pump mode (m, kw) | |
Specific rotational speed in turbine mode (m, kw) | |
Rotational speed in rpm using in specific speed. The units are rps when it is used to determine the dimensionless numbers (, and ). | |
Shaft power in the best efficiency point in turbine mode (w) | |
Shaft power in turbine mode (w) | |
Discharge in the best efficiency point in pump mode (m3/s) | |
Discharge in the best efficiency point in turbine mode (m3/s) | |
Discharge in turbine mode (m3/s) | |
Greek symbols | |
Discharge coefficient (dimensionless) | |
Head coefficient (dimensionless) | |
Efficiency coefficient (dimensionless) | |
Water density (kg/ m3) | |
Head number (dimensionless) | |
Head number in turbine mode (dimensionless) | |
Best efficiency in pump mode (dimensionless) | |
Best efficiency in turbine mode (dimensionless) | |
Efficiency in turbine mode (dimensionless) | |
Power number (dimensionless) | |
Discharge number (dimensionless) | |
Discharge number referred to turbine mode (dimensionless) |
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Autor | |||
---|---|---|---|
Stepanoff [19] | 1 | ||
Mc. Claskey [20] | 1 | ||
Alatorre-Frenk [21] | |||
Sharma-Williams [22] | 1 | ||
MICI [23] | 0.9–1.0 | 1.56–1.78 | 0.75–0.80 |
Yang et al. [24] | - | ||
Hancock [25] | - | ||
Schmiedl [26] | - | ||
Mijailov [27] | 0.078 3.292 | 0.078 3.112 | 0.0014 0.96 |
Audisio [28] | 1.21 | 1.21 | 0.95 |
Carvalho [29] | 5·10−5 | 2·10−5 | - |
Nautiyal [30] | 30.303[( 0.212)/ln()]3.424 | 41.667[(0.212)/ln()]5.042 | - |
Barbarelli [31] | |||
Grover [32] | 2.379 − 0.0264 | 2.693 − 0.0229 | - |
Hergt [33] | - |
Author | Variable | Expression | Range (Experimental Data) | Reference |
---|---|---|---|---|
Derakhshan and Nourbakhsh | <60 (4) | [37] | ||
+ | <60 (4) | [37] | ||
Plugiese et al. | They use Derakhshan’s equation. | <60 (4) | [38] | |
+ | <45 (2) | [38] | ||
Barbarelli et al. | <55 (12) | [31] | ||
+ 1.185 | <55 (12) | [31] | ||
Fecarotta et al. | 120–165 (4) | [13] | ||
1.85 | 120–165 (4) | [13] | ||
Alberizzi et al. | 3 (1) | [39] | ||
−1.9778 | 3 (1) | [39] | ||
Novara and McNabola | <100 (113) | [12] | ||
<100 (113) | [12] |
Coefficient | Empirical Equation | R2 | Experimental Data |
---|---|---|---|
99.34 | 163 | ||
98.85 | 150 | ||
97.59 | 150 | ||
99.34 | 163 | ||
97.15 | 157 | ||
96.39 | 153 | ||
96.39 | 11 | ||
90.92 | 11 |
Method | Flow (Q) | Head (H) | Efficiency | Q,H | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
RMSE | MAD | MRD | BIAS | RMSE | MAD | MRD | BIAS | RMSE | MAD | MRD | BIAS | % data C<1 | |
This study | 0.181 (1) | 0.135 (1) | 0.091 (1) | −0.037 (2) | 0.294 (2) | 0.211 (2) | 0.129 (2) | −0.02 (3) | 0.078 (1) | 0.059 (1) | 0.068 (1) | −0.004 (1) | 79.20 (1) |
Yang | 0.192 (2) | 0.138 (2) | 0.092 (2) | −0.036 (1) | 0.288 (1) | 0.209 (1) | 0.129 (1) | −0.011 (2) | − | − | − | − | 78.81 (2) |
Mc Claskey | 0.205 (3) | 0.16 (3) | 0.107 (3) | −0.073 (3) | 0.466 (7) | 0.381 (7) | 0.206 (7) | −0.343 (8) | 0.114 (4) | 0.088 (4) | 0.106 (4) | 0.08 (4) | 62.42 (5) |
Sharma-Williams | 0.238 (4) | 0.195 (5) | 0.128 (5) | −0.163 (5) | 0.379 (5) | 0.303 (6) | 0.169 (5) | −0.245 (7) | 0.114 (4) | 0.088 (4) | 0.106 (4) | 0.08 (4) | 71.83 (4) |
Audisio | 0.252 (5) | 0.192 (4) | 0.122 (4) | −0.148 (4) | 0.359 (4) | 0.257 (4) | 0.145 (4) | −0.117 (4) | 0.192 (7) | 0.161 (7) | 0.162 (7) | −0.16 (7) | 74.49 (3) |
Alatorre-Frenk | 0.321 (6) | 0.259 (6) | 0.185 (6) | 0.184 (7) | 0.321 (3) | 0.223 (3) | 0.135 (3) | −0.009 (1) | 0.089 (2) | 0.064 (2) | 0.075 (2) | 0.039 (3) | 60.93 (6) |
Stepanoff | 0.339 (7) | 0.29 (7) | 0.187 (7) | −0.285 (8) | 0.466 (7) | 0.381 (7) | 0.206 (7) | −0.343 (8) | 0.114 (4) | 0.09 (4) | 0.106 (4) | 0.08 (4) | 46.98 (8) |
Carvalo | 0.6 (8) | 0.558 (9) | 0.371 (9) | −0.558 (10) | 0.875 (9) | 0.62 (9) | 0.352 (9) | −0.195 (6) | − | − | − | − | 9.27 (11) |
Barbarelli | 0.878 (9) | 0.312 (8) | 0.244 (8) | 0.177 (6) | 10.717 (12) | 2.087 (12) | 1.668 (12) | −1.798 (12) | − | − | − | − | 60.27 (7) |
Nautiyal | 1.304 (10) | 0.784 (10) | 0.504 (10) | −0.332 (9) | 1.858 (10) | 1.166 (10) | 0.655 (10) | −0.505 (10) | − | − | − | − | 21.85 (9) |
Schimiedl | 2.504 (11) | 1.783 (12) | 1.153 (12) | 1.783 (12) | 0.395 (6) | 0.282 (5) | 0.173 (6) | 0.194 (5) | − | − | − | − | 3.35 (12) |
Mijailov | 2.523 (12) | 1.362 (11) | 1.038 (11) | −1.143 (11) | 2.725 (11) | 1.64 (11) | 1.087 (11) | −1.588 (11) | 0.091 (3) | 0.068 (3) | 0.076 (3) | −0.02 (2) | 13.91 (10) |
Method | Flow (Q) | Head (H) | Efficiency | Q,H | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
RMSE | MAD | MRD | BIAS | RMSE | MAD | MRD | BIAS | RMSE | MAD | MRD | BIAS | % C<1 | |
This study | 0.291 (1) | 0.206 (1) | 0.144 (1) | 0.068 (1) | 0.357 (1) | 0.264 (1) | 0.165 (1) | −0.006 (1) | 0.072 (1) | 0.054 (1) | 0.063 (1) | −0.003 (1) | 69.54 (1) |
Hergt | 1.602 (2) | 0.462 (2) | 0.272 (2) | −0.433 (2) | 1.157 (3) | 0.835 (3) | 0.419 (3) | −0.82 (3) | - | - | - | 31.79 (2) | |
Grover | 1.915 (3) | 1.173 (3) | 0.882 (3) | −1.125 (3) | 0.624 (2) | 0.487 (2) | 0.331 (2) | 0.225 (2) | - | - | - | - | 7.28 (3) |
Curve | RMSE | MAD | MRD | BIAS |
---|---|---|---|---|
Runaway | 0.139 | 0.0946 | 0.0529 | −0.0135 |
Zero-speed | 0.1289 | 0.0855 | 0.1036 | −0.0183 |
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Pérez-Sánchez, M.; Sánchez-Romero, F.J.; Ramos, H.M.; López-Jiménez, P.A. Improved Planning of Energy Recovery in Water Systems Using a New Analytic Approach to PAT Performance Curves. Water 2020, 12, 468. https://doi.org/10.3390/w12020468
Pérez-Sánchez M, Sánchez-Romero FJ, Ramos HM, López-Jiménez PA. Improved Planning of Energy Recovery in Water Systems Using a New Analytic Approach to PAT Performance Curves. Water. 2020; 12(2):468. https://doi.org/10.3390/w12020468
Chicago/Turabian StylePérez-Sánchez, Modesto, Francisco Javier Sánchez-Romero, Helena M. Ramos, and P. Amparo López-Jiménez. 2020. "Improved Planning of Energy Recovery in Water Systems Using a New Analytic Approach to PAT Performance Curves" Water 12, no. 2: 468. https://doi.org/10.3390/w12020468