## 2 Materials and Methods
### 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,  
\[Q=A\upsilon \tag{4}\]  
where \(Q\) is the water discharge, \(A\) is the area of the water discharge, and \(\upsilon\) is the velocity of the water. According to the continuity equation, the flow in a closed system is always constant. Mathematically,  
\[Q_{\rm I}=Q_{2} \tag{5}\]  
where \(Q_{I}\) is the flow through point 1, and \(Q_{2}\) is the flow through point 2. Furthermore,  
\[A_{1}\upsilon_{1}=A_{2}\upsilon_{2}=>\upsilon_{2}=A_{1}\upsilon_{1}/A_{2} \tag{6}\]  
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 \"\(\upsilon_{2}\)\", 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, \(A\upsilon\) increases, which increases \(\upsilon_{2}\). 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.