Page 75 - Introduction to chemical reaction engineering and kinetics

3.5 Notes on Methodology for Parameter Estimation 57

Substitution of the above equations for (- rA), cA, and q in equation 2.4-4b results in

V
1
-=- fA (1 + f~> dfA = -[fA + 2Wl - 121
qo kAO 1 - fA kA
I
Thus, for given V, T, and P, if q. is varied to obtain several values of fA at the outlet, the
expression - [ fA + 2 ln( 1 - fA)] is a linear function of V/q, with slope kA, from which the
latter can be obtained. (The integration above can be done by the substitution x = 1 - fA.)

3.4.2 Experimental Aspects of Measurement of Arrhenius Parameters A and E.4

So far, we have been considering the effect of concentration on the rate of reaction, on
the assumption that temperature is maintained constant during the time of reaction in
a batch reactor or throughout the reactor in a flow reactor. This has led to the idea of
order of a reaction and the associated rate “constant.” The rate of a chemical reaction
usually depends more strongly on temperature, and measuring and describing the ef-
fect of temperature is very important, both for theories of reaction rates and for reactor
performance. Experimentally, it may be possible to investigate the kinetics of a reacting
system at a given temperature, and then to repeat the work at several other tempera-
tures. If this is done, it is found that the rate constant depends on temperature, and it is
through the rate constant that we examine the dependence of rate on temperature, as
provided by the Arrhenius equation 3.1-6, -7, or -8. If this equation appropriately repre-
sents the. effect of temperature on rate, it becomes a matter of conducting experiments
at several temperatures to determine values of A and EA, the Arrhenius parameters.
Taking T into account implies the ability to operate the reactor at a particular T, and
hence to measure and control T. A thermostat is a device in which T is controlled within
specified and measurable limits; an example is a constant-T water bath.
In the case of a BR, the entire reactor vessel may be immersed in such a device.
However, maintaining constant T in the environment surrounding a reactor may be
more easily achieved than maintaining constant temperature throughout the reacting
system inside the reactor. Significant temperature gradients may be established within
the system, particularly for very exothermic or endothermic reactions, unless steps are
taken to eliminate them, such as by efficient stirring and heat transfer.
In the case of a CSTR, external control of T is usually not necessary because the
reactor naturally operates internally at a stationary value of T, if internal mixing is ef-
ficiently accomplished. If may be necessary, however, to provide heat transfer (heating
or cooling) through the walls of the reactor, to maintain relatively high or low temper-
atures. Another means of controlling or varying the operating T is by controlling or
varying the feed conditions (T,, qo, cAo).
In the case of a PFR, it is usually easier to vary Tin a controllable and measurable
way if it is operated as a differential reactor rather than as an integral reactor. In the
latter case, it may be difficult to eliminate an axial gradient in T over the entire length
of the reactor.

3.5 NOTES ON METHODOLOGY FOR PARAMETER ESTIMATION
In Section 3.4, traditional methods of obtaining values of rate parameters from exper-
imental data are described. These mostly involve identification of linear forms of the
rate expressions (combinations of material balances and rate laws). Such methods are
often useful for relatively easy identification of reaction order and Arrhenius parame-
ters, but may not provide the best parameter estimates. In this section, we note methods
that do not require linearization.