Page 305 - Modeling of Chemical Kinetics and Reactor Design

Introduction to Reactor Design Fundamentals for Ideal Systems 275

Solution

An initial guess for the fractional conversion X A,guess on the limiting
reactant A is substituted into the derived Equation 5-45. The time t guess
for the fractional conversion is calculated and compared with the given
holding time (t holding ). If t guess < t holding , assume another guess for the
fractional conversion until t guess ≅ t holding . The fractional conversion
X A,guess that gives t guess ≅ t holding is the actual conversion for the batch
operation at that holding time. Table 5-1 gives the results of the
simulation exercise for the 4-min batch operation (using Excel spread-
sheet in Example 5-1).
In this exercise, the fractional conversion X for the 4-min iso-
A
thermal batch operation is 77.1%. Obviously, the required answer is
obtained through trial-and-error. Equation 5-45 can be incorporated
into a computer program or mathematical package such as Mathcad,
Maple, Polymath, or Mathematica.
Related calculation procedures can be made with different stoichio-
metry, which strongly influences the final form of the design expression.
Also, different rate expressions often result in different relationships
between time and conversion. In complex reactions, analytical integra-
tion of the design equation may be cumbersome and tedious, and
perhaps impossible to solve. In such cases, the designer should resort
to numerical integration methods such as the Euler Simpson’s method
X
or graphical evaluation of ∫ f(X)dX by plotting f(X) versus X and
0
determining the area beneath the curve.
Example 5-3
Gas A decomposes irreversibly to form gas B according to the
reaction A → 2B,

− ( r A )= kC 2 A (5-46)

The reaction is second order and is performed in an isothermal
constant pressure batch reactor. Determine the conversion with time

Table 5-1
X 0.75 0.76 0.77 0.771 0.772
A,guess
t , min 3.598 3.782 3.979 3.999 4.02
guess