Water hammering can be described by the following PDEs:

$\frac{\partial }{\partial t}V\left(x\,t\right)+\frac{1}{\mathrm{\ρ}}\frac{\partial }{\partial x}P\left(x\,t\right)+\frac{\mathrm{friction}\left(\left|V\left(x\,t\right)\right|\right) V\left(x\,t\right)\left|V\left(x\,t\right)\right|}{2 \mathrm{Dia}}=0$

$\frac{\partial }{\partial x}V\left(x\,t\right)+\frac{1}{\mathrm{Ks}}\frac{\partial }{\partial t}P\left(x\,t\right)=0$

where $V\left(x\,t\right)$ and $P\left(x\,t\right)$ are the velocity and pressure at position $x$ and time $t$, friction ( $\left|V\left(x\,t\right)\right|$) is the friction factor at a given velocity, $\mathrm{\rho }$ is the liquid density, $\mathrm{Dia}$ is the pipe diameter, and $\mathrm{Ks}$ is the effective bulk modulus of the system.

Discretizing the PDEs by replacing the spatial derivatives with a central difference approximation gives these equations:

$\frac{\ⅆ}{\ⅆ t}{V}_i\left(t\right)+\frac{1}{\mathrm{\rho }}\frac{{\mathrm{P}}_{i\+1}\left(t\right)-P_{i-1}\left(t\right)}{2 \mathrm{\Δx}}\+\frac{\mathrm{friction}\left(\left|V_i\left(t\right)\right|\right) V_i\left(t\right) \left|V_i\(t\)\right|}{2 \mathrm{Dia}}=0$

$\frac{{\mathrm{V}}_{i\+1}\left(t\right)-V_{i-1}\left(t\right)}{2 \mathrm{\Δx}}\+\frac{1}{\mathrm{Ks}}\frac{\ⅆ}{\ⅆ t}\mathit{ }{P}_i\left(t\right)\=0$

where $i=1..N$.

This application solves the discretized ODEs numerically.

Calculate the steady state pipeline velocity from the Darcy-Weisbach equation:

Number of nodes:

Length of each node:

Spatially discretized form of each PDE:

Generate the entire set of ODEs:

During the initial two seconds, the velocity at the valve is at a steady state. After that, the velocity decreases exponentially to zero as the valve closes.

Pressure at the start and end of the pipeline:

Initial pressure and velocity distribution along the pipeline:

The velocity at node 0 is equal to the velocity at node 1 (because there are no derivatives involving node 0):

Plot pressure dynamics at the valve: