Page 310 - Modeling of Chemical Kinetics and Reactor Design

280 Modeling of Chemical Kinetics and Reactor Design

methods is the Runge-Kutta fourth order method, which is described
in Appendix D. It is computationally efficient and can be generally
recommended for all but very stiff sets of first order ordinary dif-
ferential equations. The availability and the low cost of personal
computers now allow designers to focus more attention on the physical
insights in the design of reactor systems.
Here, the Runge-Kutta fourth order method is employed to solve a
range of reaction kinetic schemes in reactor systems. In using this
numerical technique, it is assumed that all concentrations are known
at the initial time (i.e., t = 0). This allows the initial rate to be
computed, one for each component. In choosing a small time incre-
ment, ∆t, the concentrations will change very little. These small
changes in concentrations can be computed assuming the reaction
rates are constant. The new concentrations are used to recalculate the
reaction rates. The process is repeated until the sum of ∆t attains the
specified final reaction time. Appendix D reviews other numerical
methods for solving first order differential equations.

SERIES REACTIONS

A series of first order irreversible reactions is one in which an
intermediate is formed that can then further react. A generalized series
reaction is



A →  → C
k 2
k 1
B
(5-61)
)
( Desired Unwanted)
(
in a constant volume batch system and constant temperature T. For
the batch reactor, the following rate equations can be written.
For component A disappearing in the system,
− ( r )=− dC A = kC (5-62)
A 1 A
dt
The rate equation of component B disappearing is:

− ( r ) =− dC B = kC − k C (5-63)
B net 2 B 1 A
dt
The rate equation of component C formed is