Ind. Eng. Chem. Process Des. Dev. 1004, 23, 769-772
769
Comments on the Intensity Function and Its Use Gordon R. Youngqulst Department of Chemical Engineering, Clarkson College of Technology, Potsdam, New York 13676
The intensity function IF = TOa has been used empirically by several authors to correlate the combined effects
of residence time and temperature on reactant conversion. Here, IF is shown to originate from an approximation to the Arrhenius temperature dependence for the reaction rate. The implications and marginal usefulness of the intensity function are discussed, and an alternate means of data correlation is presented.
Introduction The concept of an "intensity" or "severity" function, defined as IF = TB", was introduced many years ago (Linden and Peck, 1955) to provide an empirical means of correlating the combined effects of residence time B and temperature T on the extent of hydrocarbon pyrolysis reactions. The concept was revived by Davis and Farrell (1973),also in connection with hydrocarbon pyrolysis, and more recently by Sawaguchi et al. (1980) in the treatment of data for thermal degradation of polymers. In this paper, we elucidate the implications of the intensity function and demonstrate that it is simply related to more conventional kinetic representations. The conversion of a reactant normally may be expressed as a function of temperature (via the temperature dependence of associated rate constants) and reaction time. For example, consider a decomposition reaction of the type A products with arbitrary nth-order kinetics rA = -kCAn. For isothermal batch or plug flow reactors, solution of the reactor material balance for a constant density reaction mixture gives XA = 1 - exp[-kB] (n = 1)
For a CSTR or
It should be apparent that, for any isothermal reactor involving a single reaction having -kinetics of the form rA
=- ~ C A ~ ~ C B ...~ ~ C C ~ ~
the reactor material balances may be solved to obtain
kd = f(X,)
(1)
-
That is, for given conversion and feed conditions (and reactor pressure in the case of gas phase reactions), the product kB is a constant.
or
Implications of the Intensity Function The intensity function IF evolves from the assumption that the temperature dependence of the rate constant k [which usually is of the Arrhenius form k = ko exp(-E/ RT)] may be approximated by k = aTlla
(2)
where a and a are empirical "constants". This may be shown by substitution of this approximation into eq 1
f(xA) = kB = ~ Y p / =~ CY[TO"]'/~= d c~l~'/~ (3)
or Yr
I-n
bAO-
kd = -[l - (1- X,)'-n] 1-n For a continuous stirred tank (backmix) reactor or
Thus, for constant XA, IF (or equivalently, kB) is constant, provided a and a are constant. The nature of a may be deduced directly from eq 2 and the Arrhenius form for k -1 --- =d -l n- k 1 -d=l -n k E a dlnT T d(l/T) R T or
RT E
a=--
Similar relationships also may be written for more complex reactions. For A + B = products, where FA = -kCACB, then for an isothermal batch or plug flow reactor
or
-
1 Arrhenius no.
(4)
[Sawaguchi et al. (1980) incorrectly conclude that
The reasons for this will be shown later.] Thus a formally is dependent on temperature and may be computed directly from eq 4 if the activation energy E is known. More commonly, a has been evaluated from experimental data 0196-4305/84/1123-0769$01.50/0
0 1984 American Chemical Soclety
770
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
,
-2,
I
I
I
I
-41 I
I
d l so that
_ -- -1I F R a
EIF
Thus
/
-121 -14
-161
I
I 2.5
2.4
As shown earlier, a = RT/E, so
2.6
I 2.7 LOG, ,,T( K)
I
I
2.8
2.9
Since kIF = k0, substitution into eq 9 gives
3.3
Figure 1. Display of the approximation k=aT'/" to the Arrhenius dependence of the rate constant (E = 20 kcal).
by determining values of I F for which the conversion is the same. That is, at constant XA T101" = T202"
or
-
The result obtained from eq 5 clearly is an average value for the temperature interval Tl T2. As indicated by Figure 1, eq 2 gives a quite saiisfactory representation for the temperature dependence of k over several orders of magnitude. Further, the magnitude of a is expected to be small (for E = 20 kcal and T = 500 K; a = 0.05)and changes modestly over moderate temperature ranges (for E = 20 kcal and T = 600 K;a = 0.06). Users of the intensity function (Davis and Farrell, 1973; Sawaguchi et al., 1980) set f ( X A )= kIF(see eq 1) and assert that KIF has dependence on the intensity function I F which is of the Arrhenius form KIF
= AIF exp[-EIF/RIF]
(6)
where AIF and Emare an apparent frequency factor and activation energy, respectively. Note that, in fact, kIF = k0. As a result of the approximation for the temperature dependence of k inherent in the use of I F kIF
= (~[T80]'/'
= (Y[~F]'~" = AIF ~XP[-EIF/RIF]
(7) (8)
It follows that 1 In kIF = - In IF + In a
(Y
= In AI, - EIF/RIF
(9)
Thus the Arrhenius plots (Le., In kIF vs. 1 / I F ) given by Sawaguchi et al. (1980)and Davis and Farrell (1973)actually are plots of In I F vs. I F scaled by the constant lla. Furthermore d In kIF
d[ a In
In
EIF AIF- - =
RIF
E I n k + I n 0 = In k o - - + In 0 RT
Using EIFIRIF = l/a = E/RT, In AIF
AIF
= In ko
= lzo0
(11)
Thus internal consistency requires that both AIF and EIF be functions of the residence time 0. That Arrhenius plots of In kIF vs. IFare reasonably linear is a consequence of the fact that a usually is quite small (and hence 80 1 in many instances) and that typically ko >> 0 so that In ko >> In 0. It was noted earlier that Sawaguchi et al. (1980)concluded that -In [l - (RT/E) In 01 a= In 0 This erroneous result evolves directly from the assumption that Ewand AIF are true constants. Clearly they are not, as shown above. Sawaguchi et al. further suggest that the value of a "is determined by the residence time, reaction temperature, activation energy, frequency factor, and conversion". Unequivocally,this is not the case. The value of a depends only on the temperature and the activation energy, as shown by eq 4.
Discussion From the author's perspective, the utility of the intensity function is decidedly limited. Its use forces an approximation to the more readily accepted Arrhenius form for the temperature dependence of the rate constant and, indeed, trades the exponent a for the activation energy E as an unknown to be determined experimentally. Moreover, the goal for using the intensity function-showing the combined effects of temperature and residence time on conversion-can be accomplished equally well by using the Arrhenius form for the temperature dependence. This is apparent upon examining the background to eq 1. For reactions with kinetics involving only one rate constant, conversion depends only on the product k0 for given feed conditions and reactor pressure. That is Oko exp[-E/RT] = k,0, = k2e2 = ... = k.0. I 1 = constant = f ( X A ) (12) for combinations of temperature and residence time which produce a particular conversion. From eq 12,at a given conversion In 0 = In [constant] - In ko + E/RT
From eq 9
+ In 0 or
(13)
The activation energy E may be obtained from eq 13,
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984 1
700
I
I
I
I
600 500
I
I
1 .o
I
300
-
200
-
100 90 80
--
I
I
I
I
I
I
c
X, = 0.7
771
I
4
-
-
E
m
Bexp[-E/RT] x 101'(min)
Figure 4. Dependence of conversion on residence time and temperature for polystyrene degradation (data of Madorsky, 1954).
t
30 ?!8
1.L9
1!60
1/61
1.kP
1.163
1.164
1.k5
1.L6'
0.8
lOOO/T(K)
Figure 2. Residence time vs. 1/T for polystyrene degradation (data of Madorsky, 1954).
t
t 385T A 390°C 0
0
395Y
0 400T
i
0.0.0 A
--
-
20
-
0
I 1.47
1
I
1.48
1.49
I
I
1.50 1.51 1000/T(K)
I
I
1.52
1.53
1.54
Figure 3. Residence time vs. 1/T for polyethylene degradation (data of M.adorsky, 1954).
provided data for conversion vs. T and 6 are available. Note that no specific knowledge of the kinetic details is required. Then the combined effects of temperature and residence time may be shown by plotting XA vs. 6 exp[EIRT]. This is equivalent to the intensity function method of plotting XA vs. 1, and avoids obscuring the importance of the activation energy and other aspects of conventional kinetics. To illustrate, the rather complete data of Madorsky (1954) for thermal degradation of polystyrene and polyethylene will be used. These data consist of fraction of polymer volatilized (i.e., conversion) vs. time for several temperatures and are presented in graphical form by Madorsky. To obtain data suitable for the analysis suggested above, the times required for particular conversions at the various temperatures were read from Figures 5 (for polystyrene) and 10 (for polyethylene) of Madorsky's paper. The results naturally are subject to some reading error. Figures 2 and 3 show In 0 vs. 1/T at constant conversion. For polystyrene, the fit of the data to the form
1.0
3.0
2.0
eexp(-E/RT)
n
4.0
5.0
lOZ3(min)
Figure 5. Dependence of conversion on residence time and temperature for polyethylene degradation (data of Madorsky, 1954).
suggested by eq 13 is excellent. From the slope, the activation energy for polystyrene operation is 40.2 kcal. This is somewhat lower than reported by Madorsky, who found activation energies of 54 and 56 kcal based on reaction rates and 25 and 50% volatilization. For polyethylene the data fit also is quite good, except for the results at 405 O C . -Madorsky noted this also and he attributed deviations at this temperature to spattering associated with excessively high rates of volatilization. From the slope, the activation energy for polyethylene degradation is 76.3 kcal which agrees well with the data of Madorsky, who reported 71 and 80 kcal at 25% and 50% volatilization. Judging from the data fit obtained here, there appears to be no good reason to assign different activation energies to different conversion levels as Madorsky did. Having determined the activation energies, the combined effects of temperature and residence time can be shown by plotting XAvs. B exp[-EIRT], as suggested above. The results for polystyrene and polyethylene are shown in Figures 4 and 5. As expected, the data fall together around single lines. The fit for polyethylene is somewhat better, although both show agreement within anticipated precision for the data used. One last point merits attention here. From eq 12, f ( X A ) = ko6 exp[-E/RT]. As demonstrated earlier, the function f(xA)takes on a form determined by the kinetics of the reaction and the reactor type. Hence, reduced X A vs. 9 exp[-E/RTJ data of the type shown in Figures 4 and 5 are subject to kinetic interpretation, something which is awkward to do using the intensity function formulation. Standard procedures may be used: a form for the kinetics
772
Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 4, 1984
".d!O
015
110
If5
2!0
2!5
3!0
3!5
4!0
4!5
BexpI-E/RT] n lOZ3(min)
Figure 6. Test of first-order kinetics for polyethylene degradation (data of Madorsky, 1954).
is assumed f ( X A ) is evaluated and then tested against the data. To illustrate, the batch reactor data of Figure 5 for polyethylene are reminiscent of fmt-order kinetics. If this is valid, f(XJ = -In (1- X A ) and a plot of In (1- X A ) VS. 6 exp[-E/RT] should be linear through the origin with slope equal to -&,. Figure 6 shows the result. Linear behavior is observed. The intercept is nonzero, but this possibly may be associated with defining time zero adequately when conducting experiments of this sort. Madorsky suggested that polyethylene degradation is approximately first order, and the above analysis supports this conclusion. Madorsky further suggested that polystyrene degradation is approximately zero order. If this is the case, f ( X A ) = CA&A and a plot of X A vs. 6 exp[E/RT] should be linear. Figure 4 is such a plot for polystyrene. The deviation from linearity is not great, so Madorsky's conclusion again is supported by the present analysis.
Concluding Remarks The intensity function defined by IF = TO" was shown to evolve from the approximation k = aT'J" to the Arrhenius form for the temperature dependence of the rate constant k. The exponent a = RT/E, the reciprocal of the Arrhenius number. Although the combined effects of temperature and residence time on reactant conversion can be correlated using IF,the same goal can be achieved with less data manipulation using 6 exp[-E/RT] as the correlating variable. Nomenclature a = intensity function exponent AIF = apparent frequency factor C A = concentration for reactant A at time 6 CAO= initial concentration for reactant A CB = concentration for reactant B at time 0 C B =~ initial concentration of reactant B E = activation energy EIF = apparent activation energy IF = intensity function = TP k = reaction rate constant kIF
= k0
KO = frequency factor
M = feed concentration ratio CBO/CAO n = power law exponent on concentration R = universal gas constant rA = reaction rate for component A T = absolute temperature X A = conversion for reactant A CY = constant in approximation to temperature dependence for k 0 = residence time Literature Cited Davis. H. G.; Farrell, T. J. Ind. Chem. Process Des. D e v . 1973, 12, 17. Linden, H. R.; Peck, R. E. I n d . Eng. Chem. 1955, 47, 2470. Madorsky, S. J . Polym. Sci. 1954, 9 , 133. Sawaguchi, T.; Inami, T.; Kuroki, T.; Ikemura, T. Ind. Eng. Chem. Process D e s . D e v . 1980, 79, 174.
Received for review May 31, 1983 Accepted December 12, 1983
