(270b) Analysis of Linear and Nonlinear Systems in Chemical Engineering Using Symbolic Computation Approaches

Introduction

The body of work produced by Luss and his students over the
past 50+ years involves the study of linear and nonlinear dynamics for various chemical
reacting systems, which includes the analysis of limit cycles, quasi-periodic
and chaotic behaviors, time series and phase portraits, power spectra, Hopf
bifurcation, and steady-state multiplicity.  A review of this notable body of
work shows that the mathematical analysis of these systems often involves
manual operations of various degrees of complexity along with supporting
numerical analysis from which illustrations are prepared. 

The main objective of this work is to illustrate the
application of the symboblic computation code Maple™ for linear and nonlinear
analysis of chemically reacting systems through several examples to demonstrate
integration of manual mathematical operations, numerical analysis, and
graphical output using an integrated approach. 

Materials and Methods

The examples given below include
models that describe the dynamic behavior of the three-variable autocatalator
and a two-phase adiabatic tubular reactor with external transport resistances. 
The Maple™ code and intermediate output is not included here for brevity, but
it will be illustrated in the presentation along with several other examples of
increasing complexity.  These are also of interest for teaching as well as
research purposes.

a.  Dynamics of a three-variable autocatalator.  The
three-variable autocatalator is described by the following dimensionless nonlinear
ODE’s:

The dimensionless
time is defined by τ = k_u t.  The concentrations of the
intermediates are a, b, and c.  An example of chaotic
behavior for μ = 0.16 is shown in Fig.1a while the power spectrum
is shown in Fig. 1b.

b. Tubular reactor model with external resistances

The dimensionless
equations for the steady-state operation of a fixed-bed reactor with plug flow
and external heat and mass transfer resistances between the mobile phase and
stagnant catalyst phase for a n^th-order reaction are:

The
dimensionless variables and parameters are:

Figure 1.  Time series and power spectrum for the
three-variable autocatalator showing the chaotic behavior.

Results & Conclusions

Eqs. (4) – (8) are a set of
nonlinear first-order ODE’s with implicit right-hand sides.  For certain values
of parameters, multiple solutions may be produced since the right-hand sides
can be multi-valued.  The above system was solved by constructing a Maple™ code
that combined manual mathematical operations with numerical methods for
parametric sensitivity studies.

Fig. 2a shows the dependence of Da
/ J_D on reactant conversion on the surface of the solid phase
at the reactor inlet, ω (0), for various values of the dimensionless
adiabatic temperature rise parameter, B.  The results suggest that multiple
solutions disappear for 5 < B < 10, although the precise value was not
determined.

Fig. 2b is similar to Fig. 2a
except it shows the dependence of Da / J_D on the
parameter y, which is the gas-phase conversion of the reactant. 
Multiple solutions exist once y reaches a particular threshold value.

The results in Fig. 2a and 2b are
in visual agreement with those given by Vortuba et al. (1974), which
were based upon numerical solutions written in Fortran.  The utility of Maple™ as an analysis tool for
linear and nonlinear dynamic systems has much promise in future work.

Figure 2. Dependence of Da/J_D on w for various values of the parameter B
and y. Other parameters:  n = 1, g = 20, J_D / J_H = 1.

Conclusions

Symbolic
programming languages, such as MAPLE™ or MATHEMATICA™, can be effectively used
for analysis of linear and nonlinear mathematical models that describe chemical
reacting systems.  This approach reduces the lengthy, tedious and error-prone
manual operations utilized in common methods.  Use of a code such as MAPLE™
greatly reduces the time required to produce results and provides an
opportunity for more detailed investigations.

References

Votruba,
J., M. Kubicek, and V. Hlavacek, Modeling of chemical reactors – XXVIII. Steady
state operation of tubular adiabatic fixed bed reactor with piston flow and
external heat and mass transfer. Chem. Eng. Sci. 29, 2333-2338 (1974).