Volume Conservation
The continuity equation describing the conservation of water volume in pipe networks is given by:
1) |
\dfrac{\partial A}{\partial t} + \dfrac{\partial Q}{\partial x} = q |
where
t is time [T],
x is the lateral distance along a pipe [L],
Q is the flow [L3/T],
A is the cross-sectional area [L2], and
q is source/sink flow per unit length [L2/T].
Momentum Conservation
The momentum equation describing the conservation of momentum in pipe networks is given by:
2) |
\dfrac{\partial V}{\partial t} + V \dfrac{\partial V}{\partial x} = -g \dfrac{\partial H}{\partial x} - \dfrac{\tau_b}{\rho R} - \dfrac{F_{ML}}{\rho A} |
where
V is the cross-sectional average velocity [L/T],
H is the hydraulic or piezometric head [L],
g is gravitational acceleration [L/T2],
\tau_b is the boundary shear stress [M/L/T2],
F_{ML} is a minor loss force term [M/L/T2],
\rho is the water density [M/L3], and
R is the hydraulic radius [L].
The hydraulic head is the sum of the water surface elevation and the pressure head, H = z_s + P / \rho g.
The temporal and convective acceleration terms are given on the left hand side of 2), and the force terms are given on the right. The first force term on the right hand side is the pressure gradient term. The latter two are described below.
Boundary Friction
The boundary shear stress is parameterized as in open channel flow, and is given by:
|
\tau_b = \rho C_D |V| V, \hspace{10pt} C_D = \frac{n^2 g}{R^{1/3}} |
where C_D is the drag coefficient computed using the Manning’s roughness coefficient, n.
Minor Losses
Minor losses account for frictional and turbulent momentum losses at pipe entrances and exits, pipe cross-section contractions and expansions, and bends in the pipe. These terms are typically given as a head loss, H_{\textrm{loss}}, and parameterized using a loss coefficient, K_L, which relates the head loss to the either the velocity head in the pipe, or a change in velocity head. This energy loss is incorporated into the momentum equation as:
|
\dfrac{F_{ML}}{\rho A} = g \dfrac{H_{\textrm{loss}}}{2L} |
where L is the distance over which the minor losses are applied. Additional information on the computation of the head loss, H_{\textrm{loss}}, can be found here Pipe Minor Losses.