Page 470 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology EN014A-654 July 28, 2001 16:35

32 Reactors in Process Engineering

Consider the following reversible reaction nism is hypothesized to obtain a rate expression and a
concentration-versus-time plot is made. The equation is
k 1
−→
A + B ←− 2R. smoothed, and the slopes, which are the rates at each
k 2
composition, are evaluated. These rates are then plot-
If this reaction is elementary, the rate expression can be ted versus concentration; and if we obtain a straight line
written as passing through the origin, the rate equation is consis-
2
−r A = k 1 C A C B − k 2 C . tent with the data. If not, another equation is tested. Ki-
R
netic data can also be taken in flow reactors and eval-
In general, an elementary reaction has the form: uated with the above methods and the reactor design
−r A = k 1 C |ν A | C |ν B | ... − k 2 C |ν R | C |ν S | .... equation.
A B R S
Reactions are classified by their order depending
on the sum of the stoichiometric coefficients of each 4. Temperature Dependence of the Rate Constant
term.
On a microscopic scale, atoms and molecules travel faster
a. Examples
and, therefore, have more collisions as the temperature of
−r A = k zero order a system is increased. Since molecular collisions are the
first-order irreversible driving force for chemical reactions, more collisions give
−r A = kC A
a higher rate of reaction. The kinetic theory of gases sug-
2
−r A = kC second-order irreversible
A gests an exponential increase in the number of collisions
first-order reversible
−r A = k 1 C A − k 2 C R
with a rise in temperature. This model fits an extremely
k 1 C A
−r A = complex large number of chemical reactions and is called an
1 + k 2 C A C R Arrhenius temperature dependency, or Arrhenius’ law.
−r A = kC 0.3 C 0.7 complex The general form of this exponential relationship is
A B
k = k 0 e −E/RT ,
3. Use of Kinetic Data
where k is the rate constant, k 0 the frequency factor or
To design a chemical reactor the rate expression must be
pre-exponential, E the activation energy, R the universal
known. Assuming the reaction is known not to be elemen-
gas constant, and T the absolute temperature. For most re-
tary, we must search for a mechanism that describes the
actions, the activation energy is positive, and the rate con-
reaction taking place or use experimental data directly.
stant k increases with temperature. Some reactions have
Mechanisms can be hypothesized as the sum of a series
very little or no temperature dependence and therefore
of elementary reactions with intermediates. Using meth-
activation energy values close to zero. A few complex re-
ods developed by physical chemists, we can hypothesize
actions have a net negative activation energy and actually
whether the proposed mechanism fits the actual experi-
decrease with temperature. These reactions are extremely
mental evidence. If no inconsistencies are found, the hy-
rare.
pothesized mechanism is possibly the actual mechanism.
The Arrhenius temperature dependency for a reaction
However, agreement of the mechanism with the experi-
can be calculated using experimental data. The procedure
mental data does not necessarily mean that the proposed
is to run a reaction at several different temperatures to get
mechanism is correct, since many mechanisms can be hy-
the rate constant k as a function of absolute temperature.
pothesized to fit such data.
From the previous equations ln k = ln k 0 − E/RT ; the nat-
An interpretation of batch or flow reactor data is used to
ural log of k is then plotted versus the reciprocal of the
fit an empirical rate expression. For example, in a simple
absolute temperature. The slope of this line is then −E/R,
batch reactor, concentration is measured as a function of
and the intercept is the ln k 0 .
time. Once the experimental data are available, two meth-
ods can be used to fit a rate expression.
The first, called the integral method of data analysis, B. Design Equations
consists of hypothesizing rate expressions and then testing
1. General Reactor Design Equation
the data to see if the hypothesized rate expression fits the
experimental data. These types of graphing approaches All chemical reactors have at least one thing in common:
are well covered in most textbooks on kinetics or reactor Chemical species are created or destroyed. In developing a
design. general reactor design equation, we focus on what happens
The differential method of analysis of kinetic data deals to the number of moles of a particular species i. Consider
directly with the differential rate of reaction. A mecha- a region of space where chemical species flow into the