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The values VR, and A in equation 18 are weighted averages of the conduit upstream, middle and downstream values at time t and/or t+Image Modifiedt. The values at time t+Image Modifiedt are the values for the current iteration. At the first iteration they are equal to the previous time step’s values. Adn, Aup, Hdn, and Hup are respectively the conduit cross-sectional area and conduit depths at the upstream (up) and downstream (dn) nodes.V·(Image ModifiedA/Image Modifiedt) is the conduit average area time derivative and conduit average velocity. Image ModifiedA is the change in cross-sectional area of the conduit between two time steps.

The basic unknowns in equation 12 and 18 are Qt+Image Modifiedt, Hup and Hdn. The variables VR, and A in turn are all related to the center conduit Q and conduit end values of H. The equation relating the conduit flows to the node H is the continuity equation at a node:

Anchor
Equation21
Equation21
Equation 21Image Modified                 

Equations 13 and 21 can be solved iteratively to determine the discharge in each link and the head at each node at the end of a time-step Image Modifiedt. The numerical integration of these two equations is accomplished by using an iterative solution using under-relaxation of the linearized momentum and linear continuity equation.

It should be noted that equation 13 has been linearized by:

  1. Using the product of Qt+Image Modifiedt and Qt in the convective inertia term of the dynamic wave equation.
  2. Using equation 11 for calculating the value of Sf, which does not use the quadratic Q.

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pageEXTRAN Theory - Dynamic Wave Solution