# Improved Planning of Energy Recovery in Water Systems Using a New Analytic Approach to PAT Performance Curves

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}and increased the accuracy of the error ellipse using an experimental database of 181 different PATs.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Definition of the Characteristic Numbers of PATs

^{3}/s when the machine operates as a pump, ${H}_{p,BEP}$ is the head in the best efficiency point in m w.c. when the machine operates as a pump, n is the rotational speed in rpm, ${Q}_{t,BEP}$ is the flow in the best efficiency point in m

^{3}/s when the machine operates as a turbine and ${H}_{p,BEP}$ is the head in the best efficiency point in m w.c. when the machine operates as turbine.

_{0}and E, F, G and I are coefficients of the efficiency curve. The efficiency curve is usually fitted to second-degree polynomials, although a higher degree can be used in order to improve the curve fit [17,18].

- (i)
- Choosing a pump according to pump catalogues through the operation point (${Q}_{t},{H}_{t})$ 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 (${Q}_{p},{H}_{p}$) 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:$$RMSE=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{x}{\left[{O}_{i}-{P}_{i}\right]}^{2}}{x}}$$
- 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:$$MAD={\displaystyle \sum}_{1}^{x}\frac{1}{x}\left|{O}_{i}-{P}_{i}\right|$$
- 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:$$MRD={\displaystyle \sum}_{1}^{x}\frac{\left|{O}_{i}-{P}_{i}\right|/{P}_{i}}{x}$$
- 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]:$$BIAS=\frac{{{\displaystyle \sum}}_{i=1}^{N}{\left[{O}_{i}-{P}_{i}\right]}^{}}{x}$$

#### 2.3. Estimation of the Head-Discharge and Efficiency Curves Considering ${n}_{st}$

^{2}, ${P}_{t}$ is the shaft power in the PAT, ${\eta}_{t}$ is the efficiency of the PAT and n is the rotational speed in rps.

#### 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 m^{2}/s) |

${H}_{p,\text{}BEP}$ | Head in the best efficiency point in pump mode (m w.c.) |

${H}_{t,\text{}BEP}$ | Head in the best efficiency point in turbine mode (m w.c.) |

${H}_{t}$ | Head in turbine mode (m w.c.) |

${n}_{s}$ | Specific rotational speed (m, kw) |

${n}_{sp}$ | Specific rotational speed in pump mode (m, kw) |

${n}_{st}$ | Specific rotational speed in turbine mode (m, kw) |

$n$ | Rotational speed in rpm using in specific speed. The units are rps when it is used to determine the dimensionless numbers ($\phi $,$\psi $ and $\pi $). |

${P}_{t,BEP}$ | Shaft power in the best efficiency point in turbine mode (w) |

${P}_{t}$ | Shaft power in turbine mode (w) |

${Q}_{p,\text{}BEP}$ | Discharge in the best efficiency point in pump mode (m^{3}/s) |

${Q}_{t,\text{}BEP}$ | Discharge in the best efficiency point in turbine mode (m^{3}/s) |

${Q}_{t}$ | Discharge in turbine mode (m^{3}/s) |

Greek symbols | |

${\beta}_{Q}$ | Discharge coefficient (dimensionless) |

${\beta}_{H}$ | Head coefficient (dimensionless) |

${\beta}_{\eta}$ | Efficiency coefficient (dimensionless) |

$\rho $ | Water density (kg/ m^{3}) |

$\psi $ | Head number (dimensionless) |

${\psi}_{t}$ | Head number in turbine mode (dimensionless) |

${\eta}_{p,BEP}$ | Best efficiency in pump mode (dimensionless) |

${\eta}_{t,BEP}$ | Best efficiency in turbine mode (dimensionless) |

${\eta}_{t}$ | Efficiency in turbine mode (dimensionless) |

$\pi $ | Power number (dimensionless) |

$\phi $ | Discharge number (dimensionless) |

${\phi}_{t}$ | Discharge number referred to turbine mode (dimensionless) |

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**Figure 6.**Estimation of the head and efficiency curve as a function of BEP (red line is the estimation of (

**a**) $\frac{{H}_{t}}{H}$ and (

**b**) $\frac{{\eta}_{t}}{{\eta}_{t,BEP}}$). Red line is an equation defined by Equations (18) and (20).

**Figure 7.**(

**a**) Average indexes, (

**b**) standard deviation and (

**c**) variation coefficient in the prediction depending on the applied method. The number which is located in each bar (x) indicates the classified order of the method considering the rest of methods.

Autor | ${\mathit{\beta}}_{\mathit{Q}}$ | ${\mathit{\beta}}_{\mathit{H}}$ | ${\mathit{\beta}}_{\mathit{\eta}}$ |
---|---|---|---|

Stepanoff [19] | $\frac{1}{\sqrt{{\text{}\eta}_{p,BEP}}}$ | $\frac{1}{{\text{}\eta}_{p,BEP}}$ | 1 |

Mc. Claskey [20] | $\frac{1}{{\text{}\eta}_{p,BEP}}$ | $\frac{1}{{\text{}\eta}_{p,BEP}}$ | 1 |

Alatorre-Frenk [21] | $\frac{0.85{\eta}_{p,BEP}^{5}+0.385}{2{\eta}_{p,BEP}^{9.5}+0.205}$ | $\frac{1}{0.85{\eta}_{p,BEP}^{5}+0.385}$ | $1-\frac{0.03}{{\text{}\eta}_{p,BEP}}$ |

Sharma-Williams [22] | $\frac{1}{{\eta}_{p,BEP}^{0.8}}$ | $\frac{1}{{\eta}_{p,BEP}^{1.2}}$ | 1 |

MICI [23] | 0.9–1.0 | 1.56–1.78 | 0.75–0.80 |

Yang et al. [24] | $\frac{1.2}{{\eta}_{p,BEP}^{0.55}}$ | $\frac{1.2}{{\eta}_{p,BEP}^{1.1}}$ | - |

Hancock [25] | $\frac{1}{{\eta}_{p,BEP}}$ | $\frac{1}{{\eta}_{p,BEP}}$ | - |

Schmiedl [26] | $-1.5+\frac{2.4}{{\eta}_{p,BEP}^{2}}$ | $-1.4+\frac{2.5}{{\eta}_{p,BEP}}$ | - |

Mijailov [27] | $-$0.078${n}_{sp}$$+$ 3.292 | $-$0.078${n}_{sp}$$+$ 3.112 | $-$0.0014${n}_{sp}$$+$ 0.96 |

Audisio [28] | 1.21${\eta}_{p,BEP}^{-0.25}$ | 1.21${\eta}_{p,BEP}^{-0.8}{\left[1+{(0.6+\mathrm{ln}{n}_{sp})}^{2}\right]}^{0.3}$ | 0.95${\eta}_{p,BEP}^{0.7}{\left[1+{(0.5+\mathrm{ln}{n}_{sp})}^{2}\right]}^{-0.25}$ |

Carvalho [29] | 5·10^{−5}${n}_{sp}^{2}-0.0114\text{}{n}_{sp}^{}+1.2246$ | $-$2·10^{−5}${n}_{sp}^{2}+0.0214\text{}{n}_{sp}^{}+0.7688$ | - |

Nautiyal [30] | 30.303[(${\text{}\eta}_{p,BEP}-$ 0.212)/ln(${n}_{sp}$)]$\text{}-$3.424 | 41.667[(${\text{}\eta}_{p,BEP}-$0.212)/ln(${n}_{sp}$)]$\text{}-$5.042 | - |

Barbarelli [31] | $0.00029{n}_{sp}^{2}-0.02771{n}_{sp}+2.01648$ | $-3{10}^{-5}{n}_{sp}^{3}+4.4{10}^{-3}{n}_{sp}^{2}-0.20882{n}_{sp}+4.64293$ | |

Grover [32] | 2.379 − 0.0264${n}_{st}$ | 2.693 − 0.0229${n}_{st}$ | - |

Hergt [33] | $1.3-\frac{1.6}{{n}_{st}-5}$ | $1.3-\frac{6}{{n}_{st}-3}$ | - |

**Table 2.**Proposed empirical expressions to predict the $\frac{{H}_{t}}{{H}_{t,BEP}}$ and $\frac{{P}_{t}}{{P}_{t,BEP}}$ as a function of $\frac{{Q}_{t}}{{Q}_{t,BEP}}$.

Author | Variable | Expression | Range ${\mathit{n}}_{\mathit{s}\mathit{t}}$ (Experimental Data) | Reference |
---|---|---|---|---|

Derakhshan and Nourbakhsh | $\frac{{H}_{t}}{{H}_{t,BEP}}$ | $1.0283{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}-0.5468\frac{{Q}_{t}}{{Q}_{t,BEP}}+0.5314$ | <60 (4) | [37] |

$\frac{{P}_{t}}{{P}_{t,BEP}}$ | $-0.3092{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{3}$ + $2.1472{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}-0.8865\frac{{Q}_{t}}{{Q}_{t,BEP}}+0.0452$ | <60 (4) | [37] | |

Plugiese et al. | $\frac{{H}_{t}}{{H}_{t,BEP}}$ | They use Derakhshan’s equation. | <60 (4) | [38] |

$\frac{{P}_{t}}{{P}_{t,BEP}}$ | $4\xb7{10}^{-3}{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{3}$ + $1.386{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}-0.390\frac{{Q}_{t}}{{Q}_{t,BEP}}$ | <45 (2) | [38] | |

Barbarelli et al. | $\frac{{H}_{t}}{{H}_{t,BEP}}$ | $0.922{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}-0.406\frac{{Q}_{t}}{{Q}_{t,BEP}}+0.483$ | <55 (12) | [31] |

$\frac{{P}_{t}}{{P}_{t,BEP}}$ | $0.040{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{3}$ + 1.185${\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}-0.043\frac{{Q}_{t}}{{Q}_{t,BEP}}-0.183$ | <55 (12) | [31] | |

Fecarotta et al. | $\frac{{H}_{t}}{{H}_{t,BEP}}$ | $1.61{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}-1.41\frac{{Q}_{t}}{{Q}_{t,BEP}}+0.805$ | 120–165 (4) | [13] |

$\frac{{P}_{t}}{{P}_{t,BEP}}$ | 1.85${\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}-0.858\frac{{Q}_{t}}{{Q}_{t,BEP}}+0.00567$ | 120–165 (4) | [13] | |

Alberizzi et al. | $\frac{{H}_{t}}{{H}_{t,BEP}}$ | $0.2394{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}+0.769\frac{{Q}_{t}}{{Q}_{t,BEP}}$ | 3 (1) | [39] |

$\frac{{\eta}_{t}}{{\eta}_{t,BEP}}$ | −1.9778${\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{6}+9.0636{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{5}-13.148{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{4}+3.8527{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{3}+4.5614{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}-1.3769\frac{{Q}_{t}}{{Q}_{t,BEP}}$ | 3 (1) | [39] | |

Novara and McNabola | $\frac{{H}_{t}}{{H}_{t,BEP}}$ | $1.16{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}+\left(0.0099{n}_{st}-1.0627\right)\frac{{Q}_{t}}{{Q}_{t,BEP}}+\left(0.9027-0.0099{n}_{st}\right)$ | <100 (113) | [12] |

$\frac{{P}_{t}}{{P}_{t,BEP}}$ | $1.248{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}+\left(0.0108{n}_{st}-0.2717\right)\frac{{Q}_{t}}{{Q}_{t,BEP}}+\left(0.0237-0.0108{n}_{st}\right)$ | <100 (113) | [12] |

**Table 3.**Proposed empirical relationship to define the best efficiency point of the machine as a function of the specific speed.

Coefficient | Empirical Equation | R^{2} | Experimental Data |
---|---|---|---|

${n}_{st}$ | $0.844564\times {n}_{sp}$ | 99.34 | 163 |

${\beta}_{Q}$ | ${\beta}_{Q}=\frac{1}{0.825861\times \sqrt{{\eta}_{p,BEP}}}$ | 98.85 | 150 |

${\beta}_{H}$ | ${\beta}_{H}=\frac{1.2337}{{\eta}_{p,BEP}}$ | 97.59 | 150 |

${n}_{sp}$ | $1.17619\times {n}_{st}$ | 99.34 | 163 |

${\beta}_{Q}$ | $\frac{1}{0.210551\times \mathrm{ln}\left({n}_{st}\right)}$ | 97.15 | 157 |

${\beta}_{H}$ | $\frac{1}{0.186314\times \mathrm{ln}\left({n}_{st}\right)}$ | 96.39 | 153 |

${k}_{Runaway}$ | ${k}_{Runaway}={\left(\frac{6.83008}{{n}_{st}}\right)}^{2}$ | 96.39 | 11 |

${k}_{zero-speed}$ | ${k}_{zero-speed}={\left(\frac{4.36583}{{n}_{st}}\right)}^{2}$ | 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) |

**Table 5.**Error indexes depending on applied method to predict BEP in pump mode. MAD: mean absolute deviation, MRD: mean relative deviation, BIAS: bias and RMSE: root mean square error.

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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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Pérez-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