Background:
Six reactions of a batch reactor are modeled together with one algebraic
balance equation of index 1.
The Mathematical Model:
The differential state variables are denoted by u_{1}(t) , ..., u_{6}(t),
and the algebraic variable by u_{7}(t) . Parameters to be
estimated, are k_{1}, k_{m1}, k_{2}, k_{3},
and k_{m3}. We define
u_{8}(t) =
E_{2} u_{1}(t) /(E_{2} + u_{7}(t)
)
u_{9}(t) = E_{3} u_{3}(t)
/(E_{3} + u_{7}(t) )
u_{10}(t) = E_{1} u_{5}(t) /(E_{1}
+ u_{7}(t) )
The six differential equations and the algebraic equation are
u_{1}(t)_{t}
= -k_{2 }u_{2}(t) u_{8}(t)
u_{2}(t)_{t}
= -k_{1 }u_{2}(t) u_{6}(t)
+ k_{m1} u_{10}(t)
- k_{2} u_{2}(t)
u_{8}(t)
u_{3}(t)_{t}
= k_{2} u_{2}(t)
u_{8}(t)
+ k_{3} u_{4}(t)
u_{6}(t) -
k_{m3} u_{9}(t)
u_{4}(t)_{t}
= -k_{3} u_{4}(t)
u_{6}(t) +
k_{m3} u_{2}(t)
u_{5}(t)_{t}
= k_{1} u_{2}(t)
u_{6}(t)
- k_{m1} u_{10}(t)
u_{6}(t)_{t}
= -k_{1} u_{2}(t)
u_{6}(t) -
k_{3} u_{4}(t)
u_{6}(t) +
k_{m1} u10(t)
+ k_{m3} u_{9}(t)
-u_{7}(t)
+ u_{6}(t)
+ u_{8}(t)
+ u_{9}(t)
+ u_{10}(t) = 0.0131
with initial values
u_{1}(0) = 1.5776, u_{2}(0) = 8.32 u_{3}(0) =
0, u_{4}(0) = 0, u_{5}(0) = 0, and u_{6}(0) =
0.0131. For
u_{7}(0), a consistent initial value can be found. The lower index t
denotes the time derivative of the state variables.
Literature:
1. Schittkowski (2002):
Numerical Data Fitting in Dynamical Systems - A Practical Introduction with
Applications and Software,
Kluwer
Academic Publishers
2. Caracotsios M., Stewart W.E. (1985): Sensitivity analysis of Initial
values: Problems with mixed ODEs and algebraic equations, Computers and
Chemical Engineering, Vol. 9, No. 4, 359-365
Implementation:
The complete solution of a data fitting problem is described
in six
steps:
Results:
Then you would like to take a look at reports and graphs:
- parameter values
- experimental data versus fitting criterion
Model equations (or use your own favorite editor):
Measurement data (or use import function for text files or Excel):
Parameters, tolerances, and start of a data fitting run:
Numerical results (computed by the least squares code DFNLP):
Report on parameter values, residuals, performance, etc. (or export to Word):
Experimental data versus fitting criterion (also available for Gnuplot):