**2. Materials and Methods**

The details of the adverse effects of the flow fluctuations on the output of the PAT are discussed herein, along with the basic concepts of the PWC theory for a better understanding of the proposed PWC technique.

### *2.1. Effect of Flow Variations on the Output of PAT*

The output of any hydro system is solely determined by the water flow, as given below [14,25]:

$$P\_{\rm out} = \eta \rho \varrho H Q \tag{1}$$

where *Pout* is the output power, *η* is the efficiency, *ρ* is the density of water, *g* is the gravitational acceleration due to gravity (9.81 m/s2), *H* is the head, and *Q* is the water flow.

The fluctuation of the flow "*Q*" varies the output of the hydro system. A smooth output can be achieved while maintaining a constant flow, which can be accomplished using the aforementioned techniques. A smooth output makes the system more efficient. The effects of flow fluctuations on the efficiency of PAT are shown in Figure 1, in which five scenarios were considered for the calculations based on a previous study [15]. Notably, the total reduction in flow is 38.4 L/s, which causes a significant decrease of 6.62% in the efficiency of the system. This decrease in plant efficiency is dangerous for the stability of the system.

The reduction in flow can be eliminated using the PWC technique. The basics of this newly proposed technique are discussed herein.

#### *2.2. Water Column Theory*

The atmospheric pressure (0.1 MPa) is created by a column of air over the surface of the earth. Similarly, water exerts pressure at its bottom, owing to the weight of water acting vertically downward. A water column having a height of 10.3 m creates pressure that is equal to the atmospheric pressure, that is, 101 kPa [26]; pressure is the force acting on an area. Mathematically,

$$P = F/A\tag{2}$$

where *P* is the pressure, *F* is the force, and *A* is the area. In this study, "*A*" indicates the area of the water discharge. If the area of discharge decreases, then the pressure will increase while the force remains constant, which is the weight of water working vertically downward and is indicated by the following:

$$w = mg \tag{3}$$

where *w* is the weight of water, *m* is the mass of water, and *g* is the gravitational acceleration. Here, "*g*" is a constant, and the mass of water should be increased to increase the weight. Traditionally, no arrangement can be used for the increment of the mass of water because the penstock is a fixed entity and functions as a singular body. Therefore, flow fluctuations threaten the stability of the system. Therefore, the PWC technique is presented in this study. Increasing the number of PWCs allows more space for a larger mass of water, which increases the force at the same height. In particular, the PWC technique makes the penstock more flexible and can manage increased flows per the requirements.

**Figure 1.** Effect of flow variations on efficiency.

Alternatively, the pressure is inversely proportional to the area of the discharge area. Therefore, to increase the pressure, the discharge area must be reduced. This reduction was achieved using a nozzle to increase the pressure, which boosted the velocity of the water.

As the PAT is a low-head device, and this study is regarding low heads, only the *Q* parameter, which indicates the discharge/flow of water, in Equation (1) can increase the output of the PAT system. Mathematically,

$$Q = Av\tag{4}$$

where *Q* is the water discharge, *A* is the area of the water discharge, and *v* is the velocity of the water. According to the continuity equation, the flow in a closed system is always constant. Mathematically,

$$Q\mathbf{i} = Q\mathbf{j} \tag{5}$$

where *Qı* is the flow through point 1, and *Q*<sup>2</sup> is the flow through point 2. Furthermore,

$$A\mathbf{u}\cdot\mathbf{u} = A\_2\ \upsilon\_2 \Rightarrow \upsilon\_2 = A\_1\upsilon\_1/A\_2\tag{6}$$

The inlet of the PAT is constant since it is a fixed body, and the only option for increasing the value of *Q* through the PAT while maintaining the same height is by increasing the velocity. The nozzle at the bottom, which has a velocity of "*v*2", as indicated in Equation (6), will increase the velocity of water, which will in turn increase the value of '*Q*'.

In this manner, as the number of water columns increases, *Aı* increases, which increases *v*2. The PWC technique injects the additional flow at the inlet of the PAT in the event of a flow decrease with the existing penstock of a plant; the adverse effects of the flow fluctuations are filtered, which smooths the output of the PAT.

#### *2.3. PWC Technique with the Design of Experiment*

The PWC follows certain rules to produce an output. To understand the behavior of the PWC, an experiment in which five water columns were connected in parallel was conducted. This experimental design is called a "double-nozzle setup", in which one nozzle makes the main outlet, which injects the additional water into the existing penstock of the PAT, while each water column has a separate nozzle. Computational fluid dynamics (CFD) simulations were performed using ANSYS software R1 2021 [27]. The dimensions of the PWC and specifications of the FLUENT solver are listed in Table 1.

**Table 1.** Dimension of the water columns and solver settings.


The geometrical dimensions of the PWC were composed in a design modeler, and the settings were changed from solid to fluid, while meshing of the PWC was performed using ANSYS meshing. A tetrahedral mesh was created using a patch-confirming algorithm, and the span angle center was set to fine. Meshing of the PWC is shown in Figure 2, the details

size, and number of elements per PWC.

**Figure 2.** Meshing of PWCs.

**Table 2.** Mesh report of PWC.


of which are provided in Table 2, including the meshing method, inflation layers, element

A mesh independence test was performed for two PWCs to obtain the optimal element size and number of elements. Element sizes of 20, 30, 40, 50, 60, 70, 80, and 90 mm were applied to create 1,477,215, 655,237, 358,456, 237,308, 174,459, 138,933, 100,099, and 77,206 mesh elements, respectively. The optimal element size was 40 mm, which was selected for all the PWC simulation cases. A size of 40 mm yielded 358,456 elements for the two PWCs, as proven by the grid independence test shown in Figure 3.

**Figure 3.** Grid independence test of the two PWCs.

In the experimental design, the dimensions of the PWCs were kept constant, while the number of PWCs was increased individually. The velocity inlet was assigned as the

319

boundary condition at the inlet of the PWCs, whereas the pressure outlet was assigned at the outlet. The velocity and pressure produced in each case, which were processed in CFD-Post, are illustrated in (a–e) of Figure 4.

**Figure 4.** Velocity and pressure produced by (**a**) one water column, (**b**) two water columns, (**c**) three water columns, (**b**) four water columns, and (**e**) five water columns.

Two types of results were obtained from the experimental design study. First, the output behavior of the PWC was determined, and a mathematical model was developed. If the dimensions and output of the water column are known, the optimal number of PWCs can be derived for any flow, velocity, or pressure profile. Second, the optimal number of PWCs was derived for the aforementioned PAT, the water flow of which was variable at its inlet owing to the variations in the head of the upper reservoir during generation.

The results of the flow, outlet velocity, and pressure were calculated in CFD-Post with a function calculator, while the dimensions of the PWC were maintained to be the same; only the number of PWCs was increased individually according to the requirements of the design flow, as shown in Figure 5.

**Figure 5.** Variations in the pressure, flow, and velocity as the number of PWCs increases.

#### *2.4. Findings from the Design Experiment*

The findings obtained from the design experiment are presented (Figure 5) while developing a mathematical model to determine the output of the PWC.
