Solving nonlinear simultaneous equations in a process model iterative method

10 Solving nonlinear simultaneous equations in a process model iterative method 

Assuming that the liquid stays at the same temperature, the specific volume, v, will be constant throughout the network at its inlet value t = U, so we may write the flow equations as 

Given that the boundary pressures p, Pi. Pi and pt are either input variables or state variables (or explicitly derivable from the model s states), we have in equations (2.84) to (2.90) a set of seven nonlinear simultaneous equations in the seven unknowns )V 2, ( 23, VV24, W45, W46, p2 and p4. We can in this case reduce the order of the problem easily by substituting for the flows into equations (2.89) and (2.90) to give 

But we are still left with two nonlinear simultaneous equations in the two pressure unknowns p2 and p4. 

It is a characteristic of pumped liquid systems and of the steam or gas flow networks with turbines and compressors that no explicit solution is generally available. It is clear from the form of equations (2.91) to (2.92) that no explicit solution can be expected even for 

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