Page 40 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes Continuous System Simulation is written by engineers for engineers, launching the partly
symbolical and partly numerical algorithms that drive the procedure of simulation in terms that
are familiar to simulation practitioners with an engineering background, and yet, the text is
precise in its approach and inclusive in its coverage, providing the reader with a methodical and
detailed understanding of the mechanisms that administer the simulation of dynamical systems.

Continuous System Simulation is a extremely software-oriented text, depending on MATLAB.
Homework problems, suggestions for term project, and open research questions conclude every
chapter to intensify the understanding of the student and enhance his or her inspiration.

2.1.1 A Chemical Reactor

As you all know that in a chemical reaction when two substances A and B are brought together
they produce a third chemical substance C. It means if 1 gram of A combines with 1 gram of B
then it produce 2 grams of C. Moreover, the rate of formation of C is proportional to the product
of the amounts of A and B present. This type of reaction is called as forward reaction. In addition
to this forward reaction there is also a backward reaction, means decomposing C back into A
and B. The rate of decomposition of C is proportional to the amount of C present in the mixer.

In other words, we can say at any time t if a, b, and c are the quantities of the chemicals A, B and
C present, respectively, then their rates of increases are described by the following three
differential equations:
da
 k c k ab, ...(1)

dt 2 1
db
 k c k ab, ...(2)

dt 2 1
dc

 2k ab 2k c, ...(3)
dt 1 2
Where k and k are the rate constants. these constants will vary with temperature and pressure,
1 2
but we do not allow the temperature or pressure of the reaction to vary. Given the values of the
constants k and k and the initial quantities of the chemicals A and B , now we wish to determine
1 2
how much of C has been produced as a function of time. Determination of the rate of such
chemical reactions is important in many industrial processes.
A straightforward method of simulating this system is to start at time zero and increment time
in small steps of t. We assume that the quantities of chemicals remain unaltered during each
step and only change 'instantaneously' at the end of the step. Thus the quantity of A (or B or C)
at the end of one such step is given in terms of the quantity at the beginning of the step as:
da(t)
a(t   t) a(t) . t ...(4)


dt
If  t is sufficiently small Eq. (4) is a reasonable representation. Identical equations can be
written for b(t + t) and c(t + t).
Suppose we wish to simulate the system for a period T. We will divide this period T into a large
number N of small periods t. This is:
T = NT
At time zero, we know a(0), b(0), c(0). From these initial values and the values k and k we
1 2
compute the amounts of chemicals at time t as:
a(t) = a(0) + [k .c(0) – k .a(0).b(0)]t
2 1

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