Page 469 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

P1: GNH/GUR P2: GPJ Final Pages
Encyclopedia of Physical Science and Technology EN014A-654 July 28, 2001 16:35

Reactors in Process Engineering 31

and r A r B r R r S r T r i
= = = = = .
1 ν A ν B ν R ν S ν T ν i
0.63 − ε I − 3ε II
2
Y B = .
1 With the normal sign convention (positive for products,
4 − ε I
2
negative for reactants), a rate is negative for a reactant
Similar expressions are prepared for the remaining (−r A ) and positive for a product (r R ).
components. Rates of reactions are functions of the thermodynamic
In general, the reaction coordinate or molar extent of state of the system. For a simple system, fixing tempera-
reaction is a bookkeeping method. Numerical values of the ture and composition fixes the rest of the thermodynamic
reaction coordinate depend on how we write the chemical quantitiesorthestate.Thus,theratecanbewritteninterms
reaction. When the initial moles are unknown or when of a temperature-dependent term called the rate constant
preliminary calculations are done, a basis of 1 mol of feed k (constant at fixed temperature) and a concentration term
is usually assumed. The numerical value of the reaction or terms C i .
coordinate depends on this basis but cancels out when a. Example
mole fractions are calculated.
Another commonly used method for determining the −r A = kC A .
extent of reaction is conversion. Conversion is based on
Rates of reaction vary with changes in temperature or con-
initial and final molar quantities of a reactant. This mo-
centration. All reactions are reversible (i.e., have a forward
lar basis can be written in terms of either total moles of
and a reverse reaction). When the rate of the forward re-
reactant or in terms of molar flow rate. In equation form,
action equals the rate of the reverse reaction, there is no
net change in concentrations of any component, and the
N A0 − N A
X A = , system is said to be at thermodynamic equilibrium. This
N A0
condition represents a minimum free energy of the system
where X A is the conversion of reactant A between 0 and 1,
and determines the limits of conversion. The overall rate
N A0 the initial moles of reactant A or initial molar flow
of reaction equals zero at equilibrium. A relationship can
rate of A, and N A the final number of moles or outlet molar
be derived between the forward and reverse rate constants
flow rate of A.
and the overall thermodynamic equilibrium constant. For
For single reactions, fractional conversion is normally
example, consider the reaction
the preferred measure of the extent of reaction. However,
formultiplereactionsthereactioncoordinateisthemethod k 1
−→
of choice. The relationship that exists between conversion A + B ←− R + S.
k 2
and the reaction coordinate is
ν A ε If the forward rate equals k 1 C A C B , and the reverse rate
X A =− . equals k 2 C R C S , the overall rate of disappearance of com-
N A0
ponent A is −r A = k 1 C A C B − k 2 C R C S . At equilibrium,
−r A 0,
2. Rate Expressions
k 1 C R C S
Before designing a chemical reactor, one must know the = ≡ K c ,
reaction(s) rate. Rates of reaction can be written in intrin- k 2 C A C B
sic form or in terms of a specific reactant of interest. An where K c is defined as the thermodynamic equilibrium
intensive measure, based on a unit volume of fluid, is nor- constant based on concentration.
mally used for homogeneous reacting systems. Thus, the Reactions that have very high values of the equilib-
general definition of reaction rate can be written as rium constant are termed irreversible since the value of
1 dN i k 2 must be very small. Without much loss of accuracy,
r i = , these equations can be modeled as dependent only on the
V t dt
forward rate. In this example, if the reaction is essentially
wherer i is the number of moles of componenti that appear irreversible, −r A = k 1 C A C B .
ordisappearbyreactionperunitvolumeandtimeinkgmol Rate expressions must ultimately come from an anal-
t
−1
liter −1 sec , V the total volume of reacting fluid in liters, ysis of experimental data. We cannot normally write a
N i the number of moles of component i in kg mol, and rate equation by inspection of the stoichiometric reaction
t the time in seconds. The rates of formation of products equation; however, a reaction is termed elementary if the
R, S, T,... are related to the rates of disappearance of rate expression can be written by inspection based on the
reactants A, B, ... , by the stoichiometric numbers, stoichiometric numbers.