# Identification of Water Hammering for Centrifugal Pump Drive Systems
## 2 Methodology  
### Mathematical Representation of Water Hammering  
Water hammering is a pressure surge. The pressure profile is calculated by the Joukowsky equation, Equation (1).  
\[\Delta p_{jou}=\rho.a.\Delta v \tag{1}\]  
where \(\Delta v\) is the flow velocity change in m/s, \(\rho\) is the density of the fluid in kg/m\({}^{3}\), a is the wave propagation velocity m/s through the fluid in the pipeline and \(\Delta p_{jou}\) is the pressure change in N/m\({}^{2}\). If the liquid is water, then \(\rho\) is 1000 kg/m\({}^{3}\). Another equation can be represented for pressure-head change, Equation (2).  
\[\Delta h_{jou}=\frac{a}{g}\Delta v \tag{2}\]  
where \(a\) is wave propagation velocity change in m/s, \(g\) is the acceleration due to gravity (9.81 m/s\({}^{2}\)), and \(\Delta h_{jou}\) is the pressure-head change in meter.  
The wave propagation velocity can be defined by modified Hook's law, which depends on the stiffness of the fluid and pipe wall, Equation (3).  
\[a=\sqrt{\frac{1}{\rho[\frac{1}{K}+\frac{D}{Ee}]}} \tag{3}\]  
where \(K\) is the bulk modulus of the fluids (for water \(K\) is \(2.19\times 10^{9}\) pa) \(D\) is the pipe diameter, \(E\) is Young's module of the pipe material, and \(e\) is the wall thickness of the pipe. In the proposed work, the value is 150 m/s based on the pipe wall and pipe diameter of PVC pipe.  
The figure describes pressure and velocity waves for a frictionless pipe when the valve is suddenly closed. In each step, how the pressure profile is changing has been highlighted. In the first case, when \(t=0\), the pressure profile is steady, which is shown by the pressure-head curve running horizontally because of the assumed lack of friction. Under steady-state conditions, the flow velocity is v\({}_{0}\). In the second case, high-pressure \(\Delta\)h has been created due to the sudden closure of the gate valve. The pressure wave is created in the opposite direction to the steady-state direction of the flow. The process takes place in a period \(0<t<\frac{1}{2}\mathrm{T_{r}}\) where T\({}_{\mathrm{r}}\) the amount of time needed by the pressure wave, and to travel up and down the entire length of the pipeline. The value of T\({}_{\mathrm{r}}\) is \(2\frac{\mathrm{L}}{a}\). In the third case at \(t=\frac{1}{2}\mathrm{T_{r}}\), the reservoir pressure is constant, and the unbalance condition has been created. In the fourth case \(t=\mathrm{T_{r}}\) the head of \(-\Delta\)h travels downstream of the gate valve. In the fifth case, after arrival at the closed gate valve, the velocity changes from \(-\mathrm{v_{0}}\) to v = 0. Here, an adverse change in the pressure head is seen. In the sixth case, when v = 0 the low-pressure wave, \(-\Delta h\) travels to upstream of the tank in time \(T_{r}<t<3/2T_{r}\). In the seventh case, the pressure resumes at the reservoir's pressure head. In the eighth case, the wave of increased pressure originating from the reservoir runs back to the gate valve and wave velocity is denoted as v. In the ninth case, at \(t=2\mathrm{T_{r}}\) conditions are the same as at the instant of closure t = 0, and the whole process starts again [21]. Figure 1 is the mathematical representation of the stages of the water hammering situation.