Page 385 - Numerical Methods for Chemical Engineering

374 8 Bayesian statistics and parameter estimation

c A and c B are the concentrations of the two reactants in M, r R1 is the reaction rate per
−1
unit volume in M/s, and k 1 (T) is a temperature-dependent rate constant in (Ms) .We
determine the value of k 1 (T) by studying the behavior of well-characterized reactors such as
a perfectly-mixed CSTR or a batch reactor (a well-mixed vessel with no inflow or outflow).

Fitting the rate law to steady-state measurements of a CSTR

For a CSTR, we measure k 1 (T) by operating the reactor at steady state at constant volume
(in)
V, constant volumetric flow rate υ, constant inlet concentrations c , and measuring the
j
outlet concentrations c j . We obtain a measurement of the rate though the balance on C,
dc C (in) υ (in)
= 0 = υ c C − c C + Vr R1 ⇒ r R1 = c C − c C (8.4)
dt V
Measurements of c A and c B are necessary to determine k 1 (T) from (8.3). For each CSTR
experiment, the set of predictor variables is x = [c A c B ]. The parameter that we wish to
estimate is θ = [k 1 ]. The response variable is y (r) = [r R1 ].
In these equations, we have assumed that the reaction is elementary. To relax this assump-
v a v b
tion, we fit the data to the rate law r R1 = k 1 c c , in which case the model parameter vector
A B
T
is now θ = [k 1 v a v b ] .
Alternatively, if we wish to study the temperature dependence of the rate constant, we
repeat the experiments at different temperatures. The predictor vector is now x = [c A c B T ].
Assuming an Arrenhius form of the rate law, k 1 (T ) = A 1 exp[−E 1 /RT ], the vector of
T
model parameters is θ = [v a v b A 1 E 1 ] . In each case above, we have single-response data,
with a nonlinear analytical model relating the predictors to the response.

Fitting the rate law to initial rate measurements in a batch reactor

We can also study the kinetics of the reaction in a batch reactor. We charge the reactor with
known initial concentrations c A (0) and c B (0), and measure the concentrations of the species
as functions of time, governed by the initial value problem:
dc A dc B dc C ν a ν b
=−r R1 =−r R1 = r R1 r R1 = k 1 c c
A B
dt dt dt
(8.5)
c A (t = 0) = c A (0) c B (t = 0) = c B (0) c C (t = 0) = 0
For each run, we keep the temperature constant at T. From the slope of c C (t)at t = 0, we
obtain the initial reaction rate:

dc C
ν a
= r R1 = k 1 (T )[c A (0)] [c B (0)] ν b (8.6)
dt
t=0
Let us say that the results of these experiments are those in Table 8.1. Now, the predictor
variables are x = [c A (0) c B (0)] and we have a single response, y = r R1 . The set of model
T
parameters is θ = [k 1 (T ) v a v b ] . Again, we have single-response data and an analytical,
nonlinear model relating the predictors to the response.