The co-current heat exchanger model (posted 7/16/2013) showed a vector/matrix solution for a set of linear ODEs. In that solution, all of the eigenvalues were distinct. This posting will show how to deal with repeat eigenvalues.
Belfiore (2003) provides several means for solving the startup dynamics of a train of five CSTRs with equal volumes. A single, first order reaction occurs in all reactors. All reactors are assumed to be at the same constant temperature.
The dynamic material balances for this system can be expressed in vector/matrix form. Belfiore has shown one matrix solution (see the attached file for page references) that involved a series approximation. I have presented an alternative solution using eigenvectors, as well as the series solution. As shown in the linked file, the eigenvector solution avoids some problems associated with the series solution.
One possible application of this model is the treatment of chemical waste. The reduction of C.O.D (chemical oxygen demand) may be modeled by a simple reaction, first order in C.O.D. and with the oxygen concentration assumed constant. The reactors (sometimes ponds) often are operated in a dilute manner so the temperature constraint may be satisfied. For more traditional reactions with high heat of reaction, the heat exchange capacity must be able to control the temperature throughout the startup.
Because of the limitations imposed by the isothermal and first order reaction requirements, direct numerical integration of the ODEs is a more generic method. The main purpose of showing this example was to demonstrate the eigenvector solution of linear ODEs when repeat roots are present.
Belfiore, L.A., Transport Phenomena for Chemical Reactor Design, Wiley (2003)