# Application of Machine Learning Coupled with Stochastic Numerical Analyses for Sizing Hybrid Surge Vessels on Low-Head Pumping Mains

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## Abstract

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## 1. Introduction

#### Hybrid Surge Vessels

_{tank}and a diameter of D

_{tank}. The central tube is used for ventilation and is termed, in the following work, as a dipping tube or ventilation tube. The ventilation tube has the height and diameter of H

_{dt}and D

_{dt}, respectively. The total vessel volume, ${\forall}_{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}$, is equal to the initial compressed air volume, ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$, in addition to the water volume below it. The compression chamber is the part of the vessel surrounding the central ventilation tube having the same diameter as the vessel and the height of the ventilation tube, and volume ${\forall}_{\mathrm{c}\mathrm{c}}$ is equal to $0.25\mathsf{\pi}({\mathrm{D}}_{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}^{2}-{\mathrm{D}}_{\mathrm{d}\mathrm{t}}^{2}){\mathrm{H}}_{\mathrm{d}\mathrm{t}}$. Air entry and exit to/from the ventilation tube is conducted through inlet and outlet air valves, as shown in the figure. The vacuum valve is equipped with a check valve to allow air only in the ventilation tube, while the release valve is smaller in diameter to let air out more slowly [30].

## 2. Materials and Methods

#### 2.1. Allowable Safe Pressure Limits

_{max}pressure surge has no generally accepted upper limit, notably in countries including the USA and England. Engineering standards and pipe material rating play a large role in regulating this restriction. However, some scholars proposed that the zero gauge pressure head limits the minimum pressure H

_{min}surge; others reported negative gauge pressures of −7 m, e.g., ref. [22]. Practice, however, showed that compared to the scenario of zero pressure head, a −3 m pressure head led to a significant reduction in vessel size with no impact on pipe integrity. To avoid the worst-case situation for the pipe, the surge vessel size is determined in this work to keep H

_{max}40% above the steady-state hydraulic grade and to keep the pipe’s rated pressure from being exceeded. On the other hand, H

_{min}is restricted to a value of −3 m. These restrictions can be verified by looking at the calculated pressure profiles along the pipeline following a pump failure. The calculation of these transient profiles is briefly explained in the following section.

#### 2.2. Numerical Modelling of Pressure Profile along the Pipeline

^{3}/s); a = the wave speed (m/s); A = cross-sectional area of pipeline (m

^{2}); and g = the gravitational acceleration (m/s

^{2}).

^{3}/s); ${\mathrm{Q}}_{\mathrm{d}\mathrm{t}}$ = water flow inside hybrid vessel to/from dipping tube (m

^{3}/s); and ${\mathrm{Q}}_{\mathrm{s}}$ = water flow to and from the hybrid vessel (m

^{3}/s). The air pressure inside the hybrid vessel can be written as:

_{bar}= atmospheric head (m); $\mathrm{H}$ = the piezometric head inside hybrid vessel (m); and ${\mathrm{h}}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}$ = losses of flow through vessel connecting pipe (m), and defined as ${\mathrm{h}}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}={\mathrm{C}}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}{\mathrm{Q}}_{\mathrm{s}}\left|{\mathrm{Q}}_{\mathrm{s}}\right|$, and C

_{loss}= loss coefficient. The air mass flow rate inside the hybrid vessel is written as [30]:

^{3}); ${\mathrm{A}}_{\mathrm{o}\mathrm{r}}$ = the cross-sectional area of the air in/out flow orifice (m

^{2}); $\mathrm{n}$ = polytropic gas equation exponent; and ${\mathrm{v}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ = the maximum air velocity through air orifice and defined as [30]:

^{3}); and $\mathsf{\Delta}\mathrm{t}$ = solution time step (sec.).

_{s}can be calculated as follows:

_{s}can be calculated as follows:

^{2}); ${\mathrm{A}}_{\mathrm{d}\mathrm{t}}$ = cross-sectional area of the dipping tube (m

^{2}); ${\mathsf{\rho}}_{\mathrm{o}}$ = air density at atmospheric pressure (kg/m

^{3}); m = mass of air inside the hybrid surge vessel (kg); ${\mathrm{m}}_{\mathrm{d}\mathrm{t}}^{\mathrm{t}}$ = mass of air inside the dipping tube (kg); ${\mathrm{Q}}_{\mathrm{m}}^{\mathrm{t}+\mathsf{\Delta}\mathrm{t}}$ = mass flow rate of air inside hybrid tank (kg/s); and $\mathrm{n}$ = polytropic gas equation exponent with value of 1.2 assuming the contained air in the tank satisfies the polytropic equation for a perfect gas.

^{3}/s).

^{2}); N = the speed of the pump rotation (rpm), with an average value of 1500 rpm for wind range of centrifugal pumps; and P = pump power (KW) given by the following:

^{3}/h); ${\mathrm{H}}_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}$ = pump head at operating point (m). Pump operation in different zones is represented using the four-quadrant as follow:

#### 2.3. Hybrid Surge Vessel Optimum Sizing Approach

- A uniform probability distribution is allocated to input parameters for sampling within predetermined realistic ranges.
- According to a predetermined number of Monte Carlo Simulations, random samples are taken from input parameters’ distribution.
- For each stochastic run, which is a different pipeline profile and hydraulic situation, combinations of random variables are inserted into the numerical model, which is then solved deterministically for the pressure change throughout the pipeline. In order to optimize the volume of the hybrid surge vessel, each deterministic run is repeated through numerous iterations.
- The least-square linearization (LSL) method is used to determine the lowest influencing input parameters, which may be eliminated without affecting accuracy, using the result of the Method of Characteristic in the previous phase using the parameter coefficient of sensitivity, ${\mathrm{S}}_{{\mathrm{V}}_{\mathrm{i}}}$:$${\mathrm{S}}_{{\mathrm{V}}_{\mathrm{i}}}=\frac{100\times {\mathrm{w}}_{\mathrm{i}}^{2}{\mathsf{\sigma}}_{\u2206{\mathrm{v}}_{\mathrm{i}}}^{2}}{\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{i}}^{2}{\mathsf{\sigma}}_{\u2206{\mathrm{v}}_{\mathrm{i}}}^{2}}$$
_{i}is the coefficient of regression, $\u2206{\mathrm{v}}_{\mathrm{i}}={\mathrm{v}}_{\mathrm{i}}-{\mathrm{m}}_{\mathrm{v}\mathrm{i}}$, difference between parameter and mean, and ${\mathsf{\sigma}}_{{\mathsf{\delta}}_{{\mathrm{v}}_{\mathrm{i}}}}^{2}$ is the variance of $\u2206{\mathrm{v}}_{\mathrm{i}}$. - Then, an approximation relation can be generated using machine learning utilizing generated matrices with random parameters and the associated optimal vessel volume.

#### 2.4. Sizing Predictive Model Development

## 3. Results and Discussions

#### 3.1. Model Setup—Deterministic Run

_{pump}, is the summation of the pipeline static head H

_{s}, and the pipe friction loss head, H

_{f}. The hybrid surge vessel is placed shortly after the pump. A hypothetical case is considered for a low-head profile and governing down surge, with the following data; a = 750 m/s, v = 1.5 m/s, L = 3000 m, D = 1.0 m, H

_{s}= 5 m, f = 0.015, D

_{con}= 0.75 m, and n =1.2. As discussed before, in low-head profiles, the negative pressure surge governs the hybrid surge vessel’s optimization process is presented with H

_{min}equal to −5.2 m following a sudden pump failure without surge protection.

^{3}); ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}$ = maximum expanded air volume in the hybrid vessel (m

^{3}); ${\mathrm{D}}_{\mathrm{c}\mathrm{o}\mathrm{n}}$ = diameter of hybrid vessel connecting pipe (m); L = pipeline length (m); ${\mathrm{H}}_{\mathrm{s}}$ = static head (m); and v = steady flow velocity in pipelines (m/s).

#### 3.2. Hybrid Surge Vessel Sizing

^{3}per vessel. Larger vessels would be possible; however, they would be custom made and this would affect hybrid vessel choice from an economical point of view. Thus, the number of hybrid vessels, ${\mathrm{N}}_{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{s}}$, for a specific case shall be calculated from the following relation:

^{3}with ${\forall}_{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}/{\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}=1.48$ to an optimum value of 23 m

^{3}with ${\forall}_{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}/{\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}=1.11$ through 10 iterations. For protection against negative pressure surges, water is pushed from the hybrid vessel with high rate to compensate for sub-atmospheric pressure in the pipeline and the atmospheric air enters the vessel when the water level in the vessel drops below the dipping tube bottom. Thus, the final expanded air volume, ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}=20.5\mathrm{m}$

^{3}, which is around 10 times the initial air volume inside the vessel, is ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ =1.8 m

^{3}. The atmospheric pressure compensated for most of this expanded volume through the ventilation tube.

_{s}= 6.5 m, f = 0.015, D

_{con}= 0.65 m, and n =1.2. Case two; a = 750 m/s, v = 2.25 m/s, L = 5000 m, D = 1.0 m, H

_{s}= 10 m, f = 0.014, D

_{con}= 0.85 m, and n =1.2. The tank height is considered 4.3 m constant for one big tank and multiple small tanks, and the ventilation tube bottom is constant for all tanks. It is observed that the final expanded air volume in all tanks is the same, and their summation is equal to the expanded air volume for one big tank, with a minor difference of 1%.

#### 3.3. Hybrid Surge Vessel Versus Standard Surge Vessel

^{3}, while the hybrid vessel volume was smaller by 30%, with ${\forall}_{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}$ = 70 m

^{3}. Figure 4 shows the changes in air volume inside the normal and hybrid surge vessels during a protection cycle following a down surge caused by pump failure. The hybrid tank’s economic size is attributed mainly to the much smaller initial air volume required, which is 4 m

^{3}, compared to the surge tank, which requires an initial air volume of 50 m

^{3}. Accordingly, the expanded air volume reached 40 m

^{3}in the hybrid vessel and a larger value of 91 m

^{3}in the surge vessel. In the surge vessel, compensation for low pressure in the pipeline depends mainly on the initial air volume inside the vessel before the surge event, which would push water from the tank to the pipeline with the required volume. The hybrid tank works by the same principle of pushing water to the pipeline, but the ventilation tube would allow the atmospheric pressure to support releasing water from the tank when the water level drops below the compression chamber level. It is to be noted that the efficiency of the hybrid vessel is shown in the expanded air volume that reached 10 times the initial air volume, while it reached only 1.8 times in the surge vessel. However, the amount of water supplied to the pipeline, Q

_{s}, during the first peak of the negative pressure wave for both tanks are the same. It can be calculated from Figure 4 as the difference between the final expanded air volume and the initial air volume, which has an average value of 38 m

^{3}. This is also shown in Figure 5, which demonstrates the water flow from the vessel to the pipeline during the transient protection case. Fluctuations of water flow from and to the pipeline are lower in the case of a hybrid vessel than the surge vessel, as shown in Figure 5, and as supported by vessel air volume fluctuations in Figure 4.

_{2}O. Initially, both tanks’ water level and air pressure above water is maintained by the steady-state hydraulic grade line of the pump main. Thus, the air pressure inside the two tanks is equal, with a slight difference of 2 m accounting for the difference in water level between the two tanks. After a pump fails and down surge waves are created, both tanks release water into the pipeline, which lowers the air pressure inside the tanks.

_{2}O following operation of the vented tube, while this drop is gradual in the case of the surge vessel with a longer time span. The fast drop in air pressure in the hybrid vessel is due to the smaller initial air volume [31]. It is observed that the air pressure inside the hybrid vessel vacuum valve maintained an atmospheric value of 10 m H

_{2}O during the whole down surge following the initial drop, except when water level inside the hybrid vessel passed the ventilation tube bottom, as seen in Figure 6. The compression chamber is closed by water below it, the air is pressurized above atmospheric value, and the air release valve in the ventilation tube works to maintain air pressure in the tube as the air pressure in the compression chamber.

^{3}and the surge vessel initial air volume was 50 m

^{3}. However, in some low-head pipelines, the initial air volume required for down surge protection in the normal surge vessel would be close to that required in the hybrid surge vessel. In these cases, the expanded air volumes for the two vessels would be similar, implying similar tank volumes and, in some cases, larger hybrid vessel volumes. This can be observed in Figure 8, which shows the variation in air volume inside a normal and hybrid surge vessel. The pump failure case parameters are: a = 1000 m/s, v = 2.5 m/s, L = 6000 m, D = 1.0 m, H

_{s}= 6 m, f = 0.015, D

_{con}= 0.85 m, and n =1.2. The initial air volumes for the surge and hybrid vessels were close; 30 m

^{3}and 24 m

^{3}, respectively. Both vessels compensated for pressure loss in the pipeline, as shown in Figure 9, and kept H

_{min}above −3 m. The expanded air volume following the first wave of the down surge was higher in the hybrid vessel by 7% compared to the surge vessel, with values of 140 m

^{3}and 130 m

^{3}, respectively. Many trials for different ranges of hydraulic and pipe parameters were attempted; results showed that the hybrid vessel volume was larger than the surge vessel volume by about 50% for some cases. Thus, despite being a low-head pipeline with a down surge governing transient, the hybrid vessel would not be economical for design engineers.

_{s}/L. It is observed from the figure that the hybrid vessel is an economical choice compared to the surge vessel for pipe slope H

_{s}/L ≤ 0.0025. For other values of pipe slope H

_{s}/L > 0.0025, both vessels’ final air volume starts to close until reaching the inflection point at H

_{s}/L = 0.0035, where the hybrid vessel volume becomes more than the surge vessel. Analyses showed that a hybrid vessel would be most economical to use on low-head pipelines with slopes H

_{s}/L ≤ 0.001–0.003. This pipe slope range was found to be highly dependent on pipe diameter and flow velocity.

#### 3.4. Model Setup—Stochastic Runs

^{3}, and the expanded air volume ≤ 150 m

^{3}, and the compression chamber volume ≤ 65 m

^{3}.

#### 3.5. Parameters Selection

_{s}= static head (m); H

_{max, min}= pressure limits (m) and ${\forall}_{\mathrm{c}\mathrm{c}}$ = compression chamber volume (m

^{3}). The maximum allowable pipe pressure, H

_{max,}has been set to 1.4 H

_{pump}, while the minimum pressure, H

_{min,}has been set to a value of −3 m. Other parameters controlling the air volume and, thus, the hybrid vessel volume have varying effects on the volume; however, their contribution to the hybrid vessel initial and expanded air volumes has not been discussed before. Using the least-square realization method, various flow and pipe parameters are ranked according to their contribution to the final tank volume. Table 5 shows the least-square realization regression hybrid vessel sensitivity results, ${\mathrm{S}}_{{\mathrm{V}}_{\mathrm{i}}}$ for various input parameters, as presented in Equation (22). It is found that the compression chamber volume, ${\forall}_{\mathrm{c}\mathrm{c}}$, is largely sensitive to the pipeline’s diameter with the contribution of 94% followed by the system static head, H

_{s}with 2% and the pipe friction coefficient with 3%, other parameters; L, v, and a had negligible impact on the chamber volume. On the other hand, the compression chamber volume contributed to the initial air volume, ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$, as an input parameter with more than 95%; with all other input parameters contributing 5%, the highest impact was for the pipe diameter with 2.8%, followed by the wave speed with 1.7%, and the static head with 0.4%. The friction coefficient and flow velocity had zero contribution to the initial air volume. Results in Table 5 show that the wave velocity had the least contribution to the expanded air volume. Other parameters had a larger contribution; the pipe diameter was the highest with 89%, followed by the velocity with 3%, pipe length with 2.7%, and pipe friction with 3.7%. Calculation showed that the hybrid tank volume is mostly influenced by the pipe diameter as the major contributing parameter, followed by the line static head.

#### 3.6. Developed Genetic Programming Models

^{3}), ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ is the initial gas volume (m

^{3}), ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}$ is the expanded air volume (m

^{3}), v is the steady state velocity (m/s), D is the pipeline diameter (m), L is the pipeline length (m), H

_{s}is the static head (m), and f is the Darcy–Weisbach friction coefficient. The compression chamber volume is an influential predictive input for the initial air volume Genetic Programming model only, as shown in Equation (23).

^{2}, and an average of 0.92 for training and 0.91 for testing datasets. The error measures, root mean square error, RMSE, and root absolute error, RAE, showed close figures of training and testing datasets for compression chamber and expanded air volumes. The root mean square error for ${\forall}_{\mathrm{c}\mathrm{c}}$ is 33 and 44 for the training and testing sets, respectively. The root absolute error is 0.25 and 0.33, respectively, for training and testing sets. The root mean square error for ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}$ is 32 and 40, respectively, for training and testing sets, while the root absolute error is 0.14 for both the training and testing sets. the error measures showed lower values for the initial air volume. The root mean square error for ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ is 8 and 10, respectively, for training and testing sets, while the root absolute error is 0.19 and 0.21, respectively, for training and testing sets. On the other hand, the indices E

_{sn}and D

_{ag}showed satisfying values with averages of 0.92 and 0.89 for the training and testing subset, respectively. Figure 12 shows the Q-Q plot of the Method of Characteristics numerical calculated volumes; ${\forall}_{\mathrm{c}\mathrm{c}}$, ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ and ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}$ versus those predicted by the developed genetic programming models using Equations (23)–(25). There was no substantial over- or under-prediction, and the created models demonstrated extremely good agreement with numerical estimated volumes with minor mistakes. The developed model for ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ showed a slight under-predict volume with about 4%.

#### 3.7. Comparison with Classical Design Charts

^{2}value, as shown in Table 6. The chart developed by [15] had the lowest R

^{2}value of 0.17, and 0.25 for [13] chart. Thus, the classical charts can only represent 20% of the data. The other statistical error values showed higher values when compared to the developed genetic programming models. The root mean square error for these classical charts showed values ranging from 98 to 339, which are 3 to 10 times higher than those of the genetic programming models. These errors are confirmed in Figure 13, which shows the scatter between the Method of Characteristics solution and the classical charts predictions. It is observed that the solutions by the classical charts show a spread of values around the line of fit with high over-prediction for initial vessel volume for the [15] chart. The chart in [13] was based on the rigid column theory, while [15]’s chart was based on the incompressible flow equations. An overestimation for the rigid column and incompressible flow theories in the initial vessel air volume computation were reported in [15]. Moreover, results in Table 6 showed that E

_{sn}and D

_{ag}indices for the classical charts yielded $-$89 and 0.5 respectively, indicating the charts’ failure to predict the initial volume. Lower error values and excellent value of E

_{sn}and D

_{ag}indices were calculated using the developed models; these can predict more economical vessel sizes.

#### 3.8. Testing and Validation of the Developed Models

#### 3.9. Parametric Analysis of the Developed MODELS

^{3}compared to 50 m

^{3}for [15,22]. In contrast to earlier design models, the genetic programming model exhibited an opposing pattern in the change of the initial air volume with the pipe static head. The chances that the down surge wave following pump failure will fall below the pipe hydraulic grade is reduced as the static head of the pipeline rises. Thus, less initial air volume would be needed to compensate for the sub-atmospheric pressure in the pipeline in case of a surge vessel (as shown for previous models) and more initial air volume would be needed to act as a cushion inside the vessel against reflected upsurges (as shown in genetic programming model).

#### 3.10. Genetic Programming Models’ Application Range

^{2}of 0.70) to help guide design engineers to down surge cases where hybrid vessels would be most economical, as shown in Figure 15 and as follows:

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Laiq, S.J. Analyzing Transient Behavior and Cost Effectiveness of a TSE Transmission Line Considering Different Pipeline Materials. In Pipelines 2014: From Underground to the Forefront of Innovation and Sustainability; ASCE Library: Reston, VA, USA, 2014; pp. 2201–2210. [Google Scholar] [CrossRef]
- Triki, A.; Chaker, M.A. Compound technique-based inline design strategy for water-hammer control in steel pressurized-piping systems. Int. J. Press. Vessel. Pip.
**2019**, 169, 188–203. [Google Scholar] [CrossRef] - Chaker, M.A.; Triki, A. Investigating the branching redesign strategy for surge control in pressurized steel piping systems. Int. J. Press. Vessel. Pip.
**2020**, 180, 104044. [Google Scholar] [CrossRef] - Garg, R.K.; Kumar, A. Experimental and numerical investigations of water hammer analysis in pipeline with two different materials and their combined configuration. Int. J. Press. Vessel. Pip.
**2020**, 188, 104219. [Google Scholar] [CrossRef] - Ostfeld, A. (Ed.) Water Supply System Analysis—Selected Topics [Internet]; InTech: London, UK, 2012. [Google Scholar] [CrossRef]
- Zhu, M.L.; Shen, B.; Zhang, Y.H.; Wang, T. Research on protection of water hammer in long-distance pressurized water transfer pipeline project. J. Xi’an Univ. Archit. Technol. (Nat. Sci. Ed.)
**2007**, 39, 40–43. [Google Scholar] - Zhiyong, L. Comparative research on protective measures of pump-stopping water hammer in water intake pumping station along river. Trans. Chin. Soc. Agric. Mach.
**2005**, 7, 61–64. [Google Scholar] - Feng, T.; Jia, Y.; Xie, R. Characteristics and protective measures of water hammer in cascade pumping station for long distance pressure water delivery. China Water Wastewater
**2008**, 14, 16. [Google Scholar] - Calamak, M.; Bozkus, Z. Comparison of performance of two run-of-river plants during transient conditions. J. Perform. Constr. Facil.
**2013**, 27, 624–632. [Google Scholar] [CrossRef] - Wu, Y.; Xu, Y.; Wang, C. Research on air valve of water supply pipelines. Procedia Eng.
**2015**, 119, 884–891. [Google Scholar] [CrossRef] - Arefi, M.H.; Ghaeini-Hessaroeyeh, M.; Memarzadeh, R. Numerical modeling of water hammer in long water transmission pipeline. Appl. Water Sci.
**2021**, 11, 140. [Google Scholar] [CrossRef] - Li, G.; Wu, X.; Li, L.; Qiu, W.; Cui, W. Research on water hammer protection for a long-distance water supply system of a deep well pump group. IOP Conf. Ser. Earth Environ. Sci.
**2021**, 826, 012047. [Google Scholar] [CrossRef] - Graze, H.R.; Horlacher, H.B. Air Chamber Design charts. In Proceedings of the 8th Australian Fluid Mechanics Conference, University of Newcastle, Newcastle, NSW, Australia, 28 November–2 December October 1982; Volume 28, pp. 14–18. [Google Scholar]
- Ruus, E.; Karney, B.W. Applied Hydraulic Transients; Ruus Consulting Ltd., Ken Fench: British Columbia, BC, Canada, 1997. [Google Scholar]
- Stephenson, D. Simple guide for design of air vessels for water hammer protection of pumping lines. J. Hydraul. Eng.
**2002**, 128, 792–797. [Google Scholar] [CrossRef] - Izquierdo, J.; Lopez, P.A.; Lopez, G.; Martinez, F.J.; Perez, R. Encapsulation of air vessel design in a neural network. Appl. Math. Model.
**2006**, 30, 395–405. [Google Scholar] [CrossRef] - De Martino, G.; Fontana, N. Simplified approach for the optimal sizing of throttled air chambers. J. Hydraul. Eng.
**2012**, 138, 1101–1109. [Google Scholar] [CrossRef] - Sun, Q.; Wu, Y.B.; Xu, Y.; Jang, T.U. Optimal sizing of an air vessel in a long-distance water-supply pumping system using the SQP method. J. Pipeline Syst. Eng. Pract.
**2016**, 7, 05016001. [Google Scholar] [CrossRef] - Shi, L.; Zhang, J.; Yu, X.; Chen, S. Water hammer protective performance of a spherical air vessel caused by a pump trip. Water Supply
**2019**, 19, 1862–1869. [Google Scholar] [CrossRef] - Wang, X.; Zhang, J.; Yu, X.; Shi, L.; Zhao, W.; Xu, H. Formula for selecting optimal location of air vessel in long-distance pumping systems. Int. J. Press. Vessel. Pip.
**2019**, 172, 127–133. [Google Scholar] [CrossRef] - Miao, D.; Zhang, J.; Chen, S.; Li, D.Z. An approximate analytical method to size an air vessel in a water supply system. Water Sci. Technol. Water Supply
**2017**, 17, 1016–1021. [Google Scholar] [CrossRef] - Sattar, A.M.; Soliman, M.; El-Ansary, A. Preliminary sizing of surge vessels on pumping mains. Urban Water J.
**2019**, 16, 738–748. [Google Scholar] [CrossRef] - Shi, L.; Zhang, J.; Yu, X.D.; Wang, X.T.; Chen, X.Y.; Zhang, Z.X. Optimal volume selection of air vessels in long-distance water supply systems. AQUA—Water Infrastruct. Ecosyst. Soc.
**2021**, 70, 1053–1065. [Google Scholar] [CrossRef] - Wan, W.; Zhang, B. Investigation of water hammer protection in water supply pipeline systems using an intelligent self-controlled surge tank. Energies
**2018**, 11, 1450. [Google Scholar] [CrossRef] - Charlatte FAYAT Group. A.R.A.A. Dipping Tube Surge Vessel. SPT 140-04-GB. 1978. Available online: https://www.charlattereservoirs.fayat.com (accessed on 17 September 2023).
- Verhoeven, R.; Van Poucke, L.; Huygens, M. Waterhammer Protection with Air Vessels A Comparative Study. WIT Trans. Eng. Sci.
**1998**, 18, 3–14. Available online: https://www.witpress.com/elibrary/wit-transactions-on-engineering-sciences/18/6215 (accessed on 17 September 2023). - Ruus, E.; Karney, B.; El-Fitiany, F.A. Charts for water hammer in low head pump discharge lines resulting from water column separation and check valve closure. Can. J. Civ. Eng.
**1984**, 11, 717–742. [Google Scholar] [CrossRef] - Boulos, P.F.; Karney, B.W.; Wood, D.J.; Lingireddy, S. Hydraulic transient guidelines for protecting water distribution systems. J. Am. Water Work. Assoc.
**2005**, 97, 111–124. [Google Scholar] [CrossRef] - Leruth, P.; Pothof, I. Innovative air vessel design for long distance water transmission pipelines. In Proceeding of the 11th International Conference on Pressure Surges, Lisbon, Portugal, 24–26 October 2012; BHR Group Ltd.: Lisbon, Portugal, 2012. [Google Scholar]
- Wang, R.; Li, P.; Wang, Z.; Zhang, F. Dipping Tube Hydropneumatic Tank Theory and Application in Water Distribution Systems. In Proceedings of the 2012 Second International Conference on Electric Technology and Civil Engineering, Washington, DC, USA, 18–20 May 2012; IEEE Computer Society: Washington, DC, USA, 2012; pp. 958–961. Available online: https://dl.acm.org/doi/proceedings/10.5555/2373291 (accessed on 17 September 2023).
- Wang, R.H.; Wang, Z.X.; Zhang, F.; Sun, J.L.; Wang, X.X.; Luo, J.; Yang, H.B. Hydraulic transient prevention with dipping tube hydropneumatic tank. Appl. Mech. Mater.
**2013**, 316, 762–765. [Google Scholar] [CrossRef] - Moghaddas, S.M.; Samani, H.M.; Haghighi, A. Transient protection optimization of pipelines using air-chamber and air-inlet valves. KSCE J. Civ. Eng.
**2017**, 21, 1991–1997. [Google Scholar] [CrossRef] - Hu, J.; Zhai, X.; Hu, X.; Meng, Z.; Zhang, J.; Yang, G. Water Hammer Protection Characteristics and Hydraulic Performance of a Novel Air Chamber with an Adjustable Central Standpipe in a Pressurized Water Supply System. Sustainability
**2023**, 15, 29730. [Google Scholar] [CrossRef] - Bentley Systems. Bentley HAMMER Connect. 2023. Available online: https://www.bentley.com (accessed on 17 September 2023).
- Zhang, Z. Hydraulic Transients and Computations, 1st ed.; Springer International Publishing: Cham, Switzerland, 2020. [Google Scholar]
- Radi, A.; Poli, R. Genetic programming discovers efficient learning rules for the hidden and output layers of feedforward neural networks. In Genetic Programming: Second European Workshop, EuroGP’99 Göteborg, Sweden, 26–27 May 1999 Proceedings 2; Springer Science & Business Media: New York, NY, USA, 1999; pp. 120–134. [Google Scholar] [CrossRef]
- Ruus, E. Charts for water-hammer in pipelines with air chambers. Can. J. Civ. Eng.
**1977**, 4, 293–313. [Google Scholar] [CrossRef] - Sharif, F.; Siosemarde, M.; Merufinia, E.; Esmatsaatlo, M. Comparative hydraulic simulation of water hammer in transition pipe line systems with different diameter and types. J. Civ. Eng. Urban.
**2014**, 4, 282–286, pii: S225204301400043-4. [Google Scholar]

**Figure 1.**A schematic of a typical hybrid surge vessel for protection of low head pipelines (1-Hybrid surge vessel, 2-water main, 3-hybrid vessel inlet pipe, 4-dipping tube, 5-air release valve, 6-vacuum valve, 7-check valve).

**Figure 2.**Stochastic Method of Characteristics/Monte Carlo Simulation framework for hybrid vessel sizing model development.

**Figure 4.**Changes in air volume during down surge protection inside surge vessel and hybrid surge vessel for the same pump failure case.

**Figure 5.**Changes in vessel water flow during surge protection for surge and hybrid vessels for the same pump failure case.

**Figure 6.**Changes in water level inside surge and hybrid vessel during surge protection for the same pump failure case.

**Figure 7.**Variations in air pressure inside surge and hybrid vessel during surge protection for the same pump failure case.

**Figure 8.**Changes in air volume inside normal and hybrid surge vessels for the same pump failure case.

**Figure 11.**Probability distributions of (

**a**) the calculated compression chamber; (

**b**) initial; and (

**c**) expanded air volumes.

**Figure 12.**Method of Characteristic optimized ${\forall}_{\mathrm{c}\mathrm{c}}$, ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ and ${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}$ versus that predicted by the genetic programming (GP) model.

Performance Indicator | Equation | Variables |
---|---|---|

Coefficient of determination | ${{\mathrm{R}}_{\mathrm{i}}}^{2}={\left(\frac{\frac{1}{\mathrm{n}}\sum _{\mathrm{j}=1}^{\mathrm{n}}\left({\mathrm{T}}_{\mathrm{j}}-\overline{\mathrm{T}}\right)\left({\mathrm{P}}_{\left(\mathrm{i}\mathrm{j}\right)}-\overline{\mathrm{P}}\right)}{\sqrt{{\sum}_{\mathrm{j}=1}^{\mathrm{n}}{\left({\mathrm{T}}_{\mathrm{j}}-\overline{\mathrm{T}}\right)}^{2}/\mathrm{n}}\sqrt{{\sum}_{\mathrm{j}=1}^{\mathrm{n}}{\left({\mathrm{P}}_{\left(\mathrm{i}\mathrm{j}\right)}-\overline{\mathrm{P}}\right)}^{2}/\mathrm{n}}}\right)}^{2}$ | P_{(ij)} is the value predicted by the program, T_{j} is the target value, and n is the number of samples, $\overline{\mathrm{P}}=1/\mathrm{n}{\sum}_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{j}}$, ${\mathrm{R}}_{\mathrm{O}}^{2},{\mathrm{R}}_{\mathrm{O}}^{\prime 2}=1-{\sum}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{i}}^{2}{\left(1-{\mathrm{k}}^{\prime},\mathrm{k}\right)}^{2}/{\sum}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{P}}_{\mathrm{i}}-\overline{\mathrm{P}}\right)}^{2}$ |

Root mean square error | $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\frac{\sum _{\mathrm{j}=1}^{\mathrm{n}}{\left({\mathrm{T}}_{\mathrm{i}}-{\mathrm{P}}_{\mathrm{i}}\right)}^{2}}{\mathrm{n}}$ | |

Coefficient of efficiency | E_{sn} = 1 − $\frac{\sum _{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{T}}_{\mathrm{i}}-{\mathrm{P}}_{\mathrm{i}})}^{2}}{\sum _{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{T}}_{\mathrm{i}}-\overline{\mathrm{T}})}^{2}}$ | |

Index of agreement | D_{ag} = 1 − $\frac{\sum _{\mathrm{i}=1}^{\mathrm{n}}{{(\mathrm{T}}_{\mathrm{i}}-{\mathrm{P}}_{\mathrm{i}})}^{2}}{\sum _{\mathrm{i}=1}^{\mathrm{n}}{(\left|{\mathrm{P}}_{\mathrm{i}}-\overline{\mathrm{T}}\right|+\left|{\mathrm{T}}_{\mathrm{i}}-\overline{\mathrm{T}}\right|)}^{2}}$ | |

Gradients of the regression | $\mathrm{k}={\sum}_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{T}}_{\mathrm{i}}\times {\mathrm{P}}_{\mathrm{i}}\right)/{\mathrm{P}}_{\mathrm{i}}^{2}\text{}\mathrm{or}\text{}{\mathrm{k}}^{\prime}={\sum}_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{T}}_{\mathrm{i}}\times {\mathrm{P}}_{\mathrm{i}}\right)/{\mathrm{T}}_{\mathrm{i}}^{2}$ | |

Slope of regression line | $\mathrm{m}\prime =\left({\mathrm{R}}^{2}-{\mathrm{R}}_{\mathrm{O}}^{2}\right)/{\mathrm{R}}^{2}$ and $\mathrm{n}\prime =\left({\mathrm{R}}^{2}-{\mathrm{R}}_{\mathrm{O}}^{\prime 2}\right)/{\mathrm{R}}^{2}$ | |

Cross testing coefficient | ${\mathrm{R}}_{\mathrm{m}}={\mathrm{R}}^{2}\times \left(1-\sqrt{\left|{\mathrm{R}}^{2}-{\mathrm{R}}_{\mathrm{O}}^{2}\right|}\right)$ | |

Standard deviation errors | ${\mathrm{S}}_{\mathrm{e}}=\sqrt{1/\left(\mathrm{n}-1\right){\sum}_{\mathrm{j}=1}^{\mathrm{n}}{\left({\mathrm{e}}_{\mathrm{j}}-\overline{\mathrm{e}}\right)}^{2}}$ | ${\mathrm{e}}_{\mathrm{j}}={\mathrm{P}}_{\mathrm{j}}-{\mathrm{T}}_{\mathrm{j}}$ |

Iteration Number | ${\mathbf{N}}_{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{k}}$ | ${\forall}_{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{k}}$ (m ^{3}) | ${\mathbf{H}}_{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{k}}$(m) | ${\mathbf{D}}_{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{k}}$(m) | ${\forall}_{\mathbf{a}\mathbf{i}\mathbf{r}}^{\mathbf{o}}$ (m ^{3}) | ${\forall}_{\mathbf{c}\mathbf{c}}$ (m ^{3}) | ${\forall}_{\mathbf{a}\mathbf{i}\mathbf{r}}^{\mathbf{f}}$ (m ^{3}) | ${\forall}_{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{k}}/{\forall}_{\mathbf{a}\mathbf{i}\mathbf{r}}^{\mathbf{f}}$ | ${\mathbf{H}}_{\mathbf{m}\mathbf{a}\mathbf{x}}/{\mathbf{H}}_{\mathbf{p}\mathbf{u}\mathbf{m}\mathbf{p}}$ | ${\mathbf{H}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$ (m) |
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 35 | 5.00 | 3.00 | 1.00 | 7.00 | 21.36 | 1.64 | 1.03 | −5.21 |

2 | 2 | 35 | 5.00 | 3.00 | 1.50 | 7.00 | 21.76 | 1.60 | 1.05 | −4.51 |

3 | 2 | 35 | 5.00 | 3.00 | 2.00 | 7.00 | 22.68 | 1.55 | 1.04 | −3.70 |

4 | 2 | 35 | 5.00 | 3.00 | 2.50 | 7.00 | 23.51 | 1.48 | 1.04 | −3.00 |

5 | 2 | 26 | 5.35 | 2.50 | 1.50 | 5.25 | 21.00 | 1.23 | 1.05 | −3.34 |

6 | 2 | 26 | 5.35 | 2.50 | 2.00 | 5.25 | 21.72 | 1.20 | 1.04 | −2.51 |

7 | 2 | 24 | 4.95 | 2.50 | 1.50 | 4.75 | 20.39 | 1.17 | 1.15 | −3.59 |

8 | 2 | 24 | 4.95 | 2.50 | 1.75 | 4.75 | 20.74 | 1.15 | 1.14 | −3.12 |

9 | 2 | 24 | 4.95 | 2.50 | 1.80 | 4.75 | 20.90 | 1.14 | 1.15 | −2.91 |

10 | 2 | 23 | 4.75 | 2.50 | 1.80 | 4.62 | 20.50 | 1.11 | 1.21 | −3.00 |

Case | ${\mathbf{N}}_{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{k}}$ | ${\mathbf{D}}_{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{k}}$(m) | ${\mathbf{H}}_{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{k}}$(m) | ${\forall}_{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{k}}$(m^{3}) | ${\forall}_{\mathbf{a}\mathbf{i}\mathbf{r}}^{\mathbf{o}}$(m^{3}) | ${\forall}_{\mathbf{a}\mathbf{i}\mathbf{r}}^{\mathbf{f}}$(m^{3}) | ${\sum \forall}_{\mathbf{a}\mathbf{i}\mathbf{r}}^{\mathbf{f}}$(m^{3}) |
---|---|---|---|---|---|---|---|

1 | 1 | 3 | 4.3 | 60 | 7.2 | 53.96 | 53.96 |

2 | 2.95 | 30 | 3.6 | 27.23 | 54.46 | ||

3 | 2.45 | 20 | 2.5 | 18.46 | 55.38 | ||

2 | 1 | * 5.2 | 4.3 | 90 | 18 | 87.24 | 87.24 |

3 | 3 | 30 | 6 | 29.47 | 88.41 | ||

4 | 2.6 | 22.5 | 4.5 | 21.59 | 86.36 |

Parameter | Range | |
---|---|---|

Lower Bound | Upper Bound | |

f | 0.015 | 0.030 |

L (m) | 2500 | 15,000 |

v (m/s) | 0.5 | 2.5 |

D (m) | 0.25 | 2.0 |

H_{s} (m) | 5 | 40 |

a (m/s) | 250 | 1400 |

**Table 5.**Least-square regression-vessel volume sensitivity results, ${\mathrm{S}}_{{\mathrm{V}}_{\mathrm{i}}}$ in percentage.

Volume (m^{3}) | f | L (m) | v (m/s) | D (m) | H_{s} (m) | a (m/s) | ${\forall}_{\mathbf{c}\mathbf{c}}$(m^{3}) |
---|---|---|---|---|---|---|---|

${\forall}_{\mathrm{c}\mathrm{c}}$ | 3.38 | 0.001 | 0.05 | 94.23 | 2.32 | 0.01 | -- |

${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ | 0.001 | 0.20 | 0 | 2.80 | 0.38 | 1.75 | 95.11 |

${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}$ | 3.7 | 2.71 | 3.43 | 89.39 | 0.66 | 0.066 | -- |

Model | Data Partitioning | R^{2} | RMSE | RAE | E_{sn} | D_{ag} |
---|---|---|---|---|---|---|

[13] | NA | 0.25 | 98 | 1.82 | −6.48 | −0.06 |

[15] | NA | 0.17 | 339 | 4.92 | −89.35 | −0.67 |

GP-${\forall}_{\mathrm{c}\mathrm{c}}$ | Train | 0.88 | 33 | 0.258 | 0.88 | 0.94 |

Test | 0.81 | 44 | 0.334 | 0.78 | 0.89 | |

GP-${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ | Train | 0.94 | 8 | 0.19 | 0.94 | 0.97 |

Test | 0.94 | 10 | 0.21 | 0.93 | 0.96 | |

GP-${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}$ | Train | 0.96 | 32 | 0.149 | 0.95 | 0.98 |

Test | 0.97 | 40 | 0.144 | 0.97 | 0.98 |

**Table 7.**Range of parameters of classical design charts and the developed genetic programming models.

Parameter | [13] | $\mathbf{GP}-{\forall}_{\mathbf{a}\mathbf{i}\mathbf{r}}^{\mathbf{o}}$ | Parameter | [15] | $\mathbf{GP}-{\forall}_{\mathbf{a}\mathbf{i}\mathbf{r}}^{\mathbf{o}}$ |
---|---|---|---|---|---|

${\mathrm{S}}^{\prime}=\frac{{\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}\mathrm{a}}{\mathrm{n}\mathrm{A}\mathrm{L}\mathrm{v}}$ | 0 to 160 | 0 to 18 | ${\mathrm{S}}^{\prime}=\frac{{\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}\mathrm{g}\left({\mathrm{H}}_{\mathrm{s}}+{\mathrm{H}}_{\mathrm{B}\mathrm{a}\mathrm{r}}\right)}{\mathrm{A}\mathrm{L}{\mathrm{v}}^{2}}$ | 0.5 to 8 | 0.02 to 4.2 |

$\frac{{\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{H}}_{\mathrm{s}}}{\mathrm{a}\mathrm{v}/\mathrm{g}}$ | 0.1 to 1 | 0.03 to 2.6 | $\frac{{\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}+{\mathrm{H}}_{\mathrm{B}\mathrm{a}\mathrm{r}}}{{\mathrm{H}}_{\mathrm{s}}+{\mathrm{H}}_{\mathrm{B}\mathrm{a}\mathrm{r}}}$ | 1 to 2.2 | 1.2 to 17.8 |

$\frac{{\mathrm{H}}_{\mathrm{m}\mathrm{i}\mathrm{n}}-{\mathrm{H}}_{\mathrm{s}}}{\mathrm{a}\mathrm{v}/\mathrm{g}}$ | −0.8 to −0.1 | −0.74 to −0.03 | $\frac{{\mathrm{H}}_{\mathrm{m}\mathrm{i}\mathrm{n}}+{\mathrm{H}}_{\mathrm{B}\mathrm{a}\mathrm{r}}}{{\mathrm{H}}_{\mathrm{s}}+{\mathrm{H}}_{\mathrm{B}\mathrm{a}\mathrm{r}}}$ | 0.2 to 0.7 | 0.15 to 0.48 |

**Table 8.**Testing statistical measures for genetic programming prediction models (based on test dataset).

Model | R (R > 0.8) | K (0.85 < K < 1.15) | K′ (0.85 < K′ < 1.15) | m′ (m′ < 0.1) | n′ (n′ < 0.1) |
---|---|---|---|---|---|

GP-${\forall}_{\mathrm{c}\mathrm{c}}$ | 0.96 | 1.01 | 0.97 | −0.09 | −0.09 |

GP-${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ | 0.98 | 1.02 | 0.97 | −0.03 | −0.03 |

GP-${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}$ | 0.99 | 1.04 | 0.95 | −0.01 | 0.00 |

Model | Mean Prediction Error | Deviation of Prediction Error | Width of Uncertainty Band | 95% Prediction Error Interval |
---|---|---|---|---|

[13]-${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ | +0.31 | 0.51 | ±1.01 | +0.01 to +51 |

[15]-${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ | +0.66 | 0.40 | ±0.78 | +0.01 to +7.88 |

GP-${\forall}_{\mathrm{c}\mathrm{c}}$ | −0.08 | 0.24 | ±0.47 | +0.14 to +10.59 |

GP-${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{o}}$ | 0 | 0.13 | ±0.26 | +0.30 to +3.30 |

GP-${\forall}_{\mathrm{a}\mathrm{i}\mathrm{r}}^{\mathrm{f}}$ | −0.12 | 0.13 | ±0.25 | +0.43 to +4.14 |

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**MDPI and ACS Style**

Sattar, A.M.A.; Ghazal, A.N.; Elhakeem, M.; Elansary, A.S.; Gharabaghi, B.
Application of Machine Learning Coupled with Stochastic Numerical Analyses for Sizing Hybrid Surge Vessels on Low-Head Pumping Mains. *Water* **2023**, *15*, 3525.
https://doi.org/10.3390/w15193525

**AMA Style**

Sattar AMA, Ghazal AN, Elhakeem M, Elansary AS, Gharabaghi B.
Application of Machine Learning Coupled with Stochastic Numerical Analyses for Sizing Hybrid Surge Vessels on Low-Head Pumping Mains. *Water*. 2023; 15(19):3525.
https://doi.org/10.3390/w15193525

**Chicago/Turabian Style**

Sattar, Ahmed M. A., Abedalkareem Nedal Ghazal, Mohamed Elhakeem, Amgad S. Elansary, and Bahram Gharabaghi.
2023. "Application of Machine Learning Coupled with Stochastic Numerical Analyses for Sizing Hybrid Surge Vessels on Low-Head Pumping Mains" *Water* 15, no. 19: 3525.
https://doi.org/10.3390/w15193525