# Pressure Drop and Energy Recovery with a New Centrifugal Micro-Turbine: Fundamentals and Application in a Real WDN

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Tests

## 3. Numerical Model

#### 3.1. Turbine Geometry and Boundary Conditions

#### 3.2. Numerical Results

## 4. Application to the Water Distribution Network (WDN) of Funchal City—Madeira, Portugal

#### 4.1. The Optimization Procedure

#### 4.1.1. The Variables

#### 4.1.2. Non-Linear Constraints

#### 4.1.3. Linear Constraints

#### 4.1.4. The Objective Function

^{3}, as representative of the analyzed case study area. According to the formulation proposed by Araujo et al. (2006) [55], the total leaked discharge in the network ${\mathrm{Q}}_{\mathrm{l}}$ can be evaluated as the sum of the discharge leaked in all single nodes:

#### 4.1.5. The Mathematical Model

#### 4.2. Optimization Results

^{3}. When the literature PAT cost model is, instead, employed to evaluate the cost of the installed turbines (see (II)), the solver selects more turbines than valves (i.e., 6 against 1—Figure 14), since the installation of valves could not be convenient compared to turbines, especially where the pipe diameters are large. However, compared to the results in case (I), the increased value of NPV in case (II) (i.e., EUR 11,942,920) is due to the larger water savings, which is equal to 13,673 m

^{3}per day, and smaller investment costs (EUR 55,786). Regarding the power production, it is quite similar between the two cases, obtaining around 65 kW.

## 5. Conclusions

^{3}per day against 8507 m

^{3}per day) and the investment cost is significantly more confined (EUR 55,786 against EUR 482,298), where the power production has been accounted for at around 65 kW. However, in both analyzed cases, the turbines are installed in the middle part of the network, where the available energy is significant, whereas the valves are mainly located in the peripheral area where valves represent a more viable solution due to the limited available potential energy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Symbols

$\mathsf{\alpha}$ | Runner thickness |

${\mathsf{\alpha}}_{\mathrm{max}}$ | Maximum runner thickness |

${\mathsf{\alpha}}_{\mathrm{min}}$ | Minimum runner thickness |

β | Leakage exponent |

γ | Fluid specific weight |

${\mathsf{\delta}}_{\mathrm{w}}^{+}$ | Wall lift-off |

${\mathsf{\zeta}}_{\mathrm{k}}$ | Binary variable of flow direction through the k-th link |

$\mathsf{\lambda}$ | Discount rate |

$\mathsf{\eta}$ | Turbine efficiency |

μ | Flow viscosity |

ρ | Flow density |

τ | Non-dimensional torque |

φ | Non-dimensional discharge |

ψ | Non-dimensional head drop |

${\mathrm{C}}_{\mathrm{k}}$ | Roughness coefficient of the k-th link |

${\mathrm{c}}_{\mathrm{k}}^{\mathrm{T}}$ | Total cost of turbine |

${\mathrm{c}}_{\mathrm{k}}^{\mathrm{V}}$ | Total cost of valve |

${\mathrm{c}}_{\mathrm{w}}$ | Water unit cost |

$\mathrm{D}$ | Turbine diameter |

${\mathrm{D}}_{\mathrm{k}}$ | Diameter of the k-th pipe |

${\mathrm{D}}_{\mathrm{max}}$ | Maximum turbine diameter |

${\mathrm{D}}_{\mathrm{min}}$ | Minimum turbine diameter |

${\mathrm{E}}_{\mathrm{y}}^{\mathrm{p}}$ | Energy income |

$\mathrm{F}$ | External force |

${\mathrm{f}}_{\mathrm{i}}$ | Leakage coefficient at the i-th node |

$\mathrm{g}$ | Gravity acceleration |

$\mathrm{H}$ | Head drop |

${\mathrm{H}}_{\mathrm{i}}$ | Head at i-th node |

${\mathrm{H}}_{\mathrm{k}}{}_{\mathrm{max}}$ | Maximum head drop within the devices |

${\mathrm{H}}_{\mathrm{k}}^{\mathrm{T}}$ | Head drop within turbine |

${\mathrm{H}}_{\mathrm{k}}^{\mathrm{T}+}$ | Positive component of head drop within turbine |

${\mathrm{H}}_{\mathrm{k}}^{\mathrm{T}-}$ | Negative component of head drop within turbine |

${\mathrm{H}}_{\mathrm{k}}^{\mathrm{V}}$ | Head drop within valve |

${\mathrm{H}}_{\mathrm{k}}^{\mathrm{V}+}$ | Positive component of head drop within valve |

${\mathrm{H}}_{\mathrm{k}}^{\mathrm{V}-}$ | Negative component of head drop within valve |

${\mathrm{I}}_{\mathrm{k}}^{\mathrm{T}}$ | Binary variable for turbine location |

${\mathrm{I}}_{\mathrm{k}}^{\mathrm{V}}$ | Binary variable for valve location |

$\mathrm{i}$,$\mathrm{j}$ | Indices for nodes |

${\mathrm{K}}_{\mathrm{i}}$ | Set of nodes linked to the i-th node |

$\mathrm{k}$ | Index for links |

${\mathrm{L}}_{\mathrm{i},\mathrm{j}}$ | Length of pipe connecting nodes i and j |

${\mathrm{L}}_{\mathrm{k}}$ | Length of the k-th link |

$\mathrm{l}$ | Number of links |

$\mathrm{N}$ | Rotational speed |

$\mathrm{NPV}$ | Net present value |

$\mathrm{n}$ | Number of nodes |

${\mathrm{n}}_{\mathrm{T}}$ | Experimental coefficient |

${\mathrm{P}}_{\mathrm{h}}$ | Hydraulic power |

${\mathrm{P}}_{\mathrm{T}}$ | Produced power |

$\mathrm{p}$ | Pressure |

${\mathrm{p}}_{\mathrm{max}}$ | Maximum allowable pressure |

${\mathrm{p}}_{\mathrm{min}}$ | Minimum allowable pressure |

$\mathrm{Q}$ | Experimental discharge |

${\mathrm{Q}}_{\mathrm{l}}$ | Total leaked discharge |

${\mathrm{Q}}_{\mathrm{T}}$ | Discharge through the turbine |

${\mathrm{Q}}_{\mathrm{k}}$ | Discharge through the k-th link |

${\mathrm{Q}}_{\mathrm{max}}$ | Maximum discharge through the k-th link |

${\mathrm{q}}_{\mathrm{i}}^{\mathrm{d}}$ | Demand of the i-th node |

${\mathrm{q}}_{\mathrm{i}}^{\mathrm{l}}$ | Leaked discharge at the i-th node |

${\mathrm{q}}_{\mathrm{k}}^{+}$ | Positive component of discharge through k-th link |

${\mathrm{q}}_{\mathrm{k}}^{-}$ | Negative component of discharge through k-th link |

${\mathrm{q}}_{\mathrm{T}}$ | Experimental coefficient |

${\mathrm{r}}_{\mathrm{k}}$ | Resistance term at the k-th link |

$\mathrm{T}$ | Torque |

$\mathrm{V}$ | Flow velocity |

${\mathrm{W}}_{\mathrm{y}}^{\mathrm{s}}$ | Water saving |

$\mathrm{Y}$ | Number of years |

$\mathrm{y}$ | Index for years |

$\mathrm{z}$ | Node elevation |

## Appendix A

**Figure A1.**Pressure contour plot when devices (CMT and PRV) are installed (

**a**) and without performing any pressure control strategy (

**b**) for case (II).

**Figure A2.**Produced power of the installed turbines (

**a**) and head drop within the devices (

**b**) resulting from case (II).

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**Figure 3.**Pressure contours for different rotational speeds: (

**a**) N = 750 rpm; (

**b**) N = 1050 rpm; (

**c**) N = 1500 rpm; (

**d**) N = 1600 rpm; (

**e**) N = 1800 rpm; (

**f**) N = 1850 rpm.

**Figure 5.**Velocity isosurfaces in the turbine runner for different rotational speeds: (

**a**) N = 750 rpm; (

**b**) N = 1050 rpm; (

**c**) N = 1500 rpm; (

**d**) N = 1600 rpm; (

**e**) N = 1800 rpm; (

**f**) N = 1850 rpm.

**Figure 6.**Velocity isosurface in the runner and draft tube for different rotational speeds with H = 32.8 m: (

**a**) N = 750 rpm; (

**b**) N = 1050 rpm; (

**c**) N = 1500 rpm; (

**d**) N = 1600; (

**e**) N = 1800 rpm; (

**f**) N = 1850 rpm.

**Figure 7.**Velocity magnitude for H = 32.8m and different rotational speeds: (

**a**) N = 750 rpm; (

**b**) N = 1050 rpm; (

**c**) N = 1500 rpm; (

**d**) N = 1600; (

**e**) N = 1800 rpm; (

**f**) N = 1850 rpm; (

**g**) general horizontal plan view.

**Figure 8.**Velocity streamlines near the impeller for different rotational speeds: (

**a**) N = 750 rpm; (

**b**) N = 1050 rpm; (

**c**) N = 1500 rpm; (

**d**) N = 1600; (

**e**) N = 1800 rpm; (

**f**) N = 1850 rpm.

**Figure 9.**Detail of velocity streamlines with reference to the top view (

**a**) and bottom view (

**b**) of the guide vane (N = 1500 rpm).

**Figure 10.**Comparison between experimental measurements and numerical results. (

**a**) Efficiency vs. flow rate coefficient; (

**b**) torque coefficient vs. flow rate coefficient; (

**c**) head coefficient vs flow rate coefficient.

**Figure 12.**Total PRV cost depending on pipe diameter according to Garcia et al. (2019) [49].

**Figure 14.**Device positions within the case study WDN according to the results in (I) (

**a**) and (II) (

**b**).

**Figure 15.**Pressure contour plot when devices are installed (

**a**) and without performing any pressure control strategy (

**b**) for case (I).

**Figure 16.**Produced power of the installed turbines (

**a**) and head drop within the devices (

**b**) resulting from case (I).

D (mm) | 150 | |||||

H (m) | 32.8 | |||||

Q (L/s) | 19.49 | 27.98 | 34.1 | 39.19 | 43.72 | 46.26 |

Q (%) | 42 | 60 | 74 | 85 | 95 | 100 |

η (%) | 33 | 40 | 52 | 55 | 62 | 65 |

N (rpm) | 767 | 1064 | 1489 | 1580 | 1792 | 1843 |

P (W) | 2040 | 3590 | 5680 | 6930 | 8710 | 9690 |

V (m/s) | 0.97 | 1.39 | 1.70 | 1.95 | 2.17 | 2.30 |

T (N m) | 25.40 | 32.22 | 36.43 | 41.88 | 46.41 | 50.21 |

P_{h} (W) | 6265 | 8994 | 10,961 | 12,597 | 14,053 | 14,870 |

_{h}—hydraulic power.

NPV (€) | N° Turbines (-) | N° Valves (-) | Av. Power (kW) | Water Saving (m ^{3}/day) | Invest. Cost (€) | |
---|---|---|---|---|---|---|

(I) ^{1} | 7,169,083 | 5 | 4 | 68 | 8507 | 482,298 |

(II) ^{1} | 11,942,920 | 6 | 1 | 65 | 13,673 | 55,786 |

^{1}(I): traditional turbine + generator cost model; (II) PAT + generator cost model.

D (mm) | H_{T}(m) | Q_{T}(L/s) | N (rpm) | η_{T}(%) | |
---|---|---|---|---|---|

Link 89 | 187 | 52 | 11 | 2748 | 75 |

Link 795 | 540 | 50 | 91 | 933 | 81 |

Link 1358 | 500 | 5 | 25 | 319 | 75 |

Link 1375 | 940 | 13.5 | 144 | 278 | 80 |

Link 2485 | 165 | 161 | 15 | 5480 | 77 |

D (mm) | H_{T}(m) | Q_{T}(L/s) | N (rpm) | η_{T}(%) | |
---|---|---|---|---|---|

Link 8 | 685 | 5.95 | 51 | 254 | 77 |

Link 14 | 70 | 50 | 1.6 | 7200 | 68 |

Link 108 | 170 | 119 | 14 | 4573 | 76 |

Link 1241 | 535 | 41.5 | 82 | 858 | 80 |

Link 1242 | 747 | 11.5 | 84 | 323 | 79 |

Link 2485 | 220 | 122.5 | 24 | 3585 | 78 |

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## Share and Cite

**MDPI and ACS Style**

Morani, M.C.; Simão, M.; Gazur, I.; Santos, R.S.; Carravetta, A.; Fecarotta, O.; Ramos, H.M.
Pressure Drop and Energy Recovery with a New Centrifugal Micro-Turbine: Fundamentals and Application in a Real WDN. *Energies* **2022**, *15*, 1528.
https://doi.org/10.3390/en15041528

**AMA Style**

Morani MC, Simão M, Gazur I, Santos RS, Carravetta A, Fecarotta O, Ramos HM.
Pressure Drop and Energy Recovery with a New Centrifugal Micro-Turbine: Fundamentals and Application in a Real WDN. *Energies*. 2022; 15(4):1528.
https://doi.org/10.3390/en15041528

**Chicago/Turabian Style**

Morani, Maria Cristina, Mariana Simão, Ignac Gazur, Rui S. Santos, Armando Carravetta, Oreste Fecarotta, and Helena M. Ramos.
2022. "Pressure Drop and Energy Recovery with a New Centrifugal Micro-Turbine: Fundamentals and Application in a Real WDN" *Energies* 15, no. 4: 1528.
https://doi.org/10.3390/en15041528