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Calculation Theory and Method of Solution
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Figure 89 Pseudo Loop
A series of pressure equations describing the pressure loss in each loop must be generated. The
same pipe may be included in many different loops.
Where a loop includes a number of different fluid zones the elevation differences of the nodes and
the density of the fluid in each pipe along the path in the loop must be used to establish the
difference in pressures due to fluid density at the end points of the loop. This pressure difference is
used together with the fluid surface pressure difference to establish the net pressure difference
around the loop.
A series of mass flow continuity equations must be generated for the mass flow rate balance at each
node in the pipeline system.
Pipe Flow Expert will automatically analyze a system, identify all loops and pseudo-loops, generate
the pressure equations throughout the network and remove redundant loop equations (those which
are already fully described by combining other equations). It will also generate the mass flow balance
equations for each node in a pipeline system.
Solving the unknown values
Pipe Flow Expert generates initial estimates for the flow rate at each outlet point in the pipeline
system. The Out-flows are used to estimate the flow rate in each pipe. The pressure losses within
the system are calculated using friction factors obtained from the Colebrook-White equation, and the
friction pressure loss for each pipe is obtained from the Darcy-Weisbach equation. The initial
estimates are unlikely to give a balanced pressure result over the whole system and must be
adjusted using a variation on the Newton method to converge to a final result where all of the flow
rates and pressures within the system are balanced. Pipe Flow Expert defines the elements of the
pipeline system in a series of non-linear matrix equations. Once an approximate solution has been
obtained, the results are refined using a variation on the Newton method algorithm to ensure the
results converge to a balanced flow and pressure result.