Energy & Fuels 1989, 3, 105-108
105
acetate, 141-78-6;4-tert-butylcyclohexanone,98-53-3;4-tert-butylpyridine, 3978-81-2; 2,6-di-tert-butylpyridine,585-48-8; 4tert-butylphenol, 98-54-4; 3,5-di-tert-butylphenol,1138-52-9; 2-tert-butylphenol,88-18-6; 2,6-di-tert-butylphenol, 128-39-2; phenol, 108-95-2;butyl alcohol, 71-36-3; chloroform-d,865-49-6; chloroform,67-66-3;tert-butyl alcohol, 75650; nitrogen, 7727-37-9; 2,2-dimethylbutane,75-83-2; carbon dioxide, 124-38-9.
partment of Energy, Grant DE-FG22-82PC50809. The FTIR spectral studies and some flow microcalorimetry were done by G.L., Visiting Professor from Zhejiang University, China. Most of the flow microcalorimetric studies are from the Ph.D. Thesis of K.L.J. (1985). Registry No. Ppidine, 110-86-1;butylamine, 109-73-9;ethyl
Evaluation of Systematic Error Incurred in the Plug Flow Idealization of Tubular Flow Reactor Data Sundaresh Venkat Ramayya and Michael Jerry Antal, Jr.* Department of Mechanical Engineering, University of Hawaii at Manoa, Honolulu, Hawaii 96822 Received July 11, 1988. Revised Manuscript Received November 7, 1988
When the plug flow idealization is used to treat tubular flow reactor data, systematic error can be introduced into calculated values of the reaction rate constant K and apparent activation energy E . In the worst case (when the plug flow idealization is (mis)used to evaluate data taken from an ideal, laminar flow reactor), the magnitude of the systematic error in K and E can be evaluated in closed form. In all cases the systematic error reduces the calculated values of K and E below their true values. For single-step, irreversible reactions of order ll2,3/41 1,3/2, and 2 at moderate conversions, the fractional systematic error in K does not exceed 20% of its true value, whereas the fractional systematic error in E remains below 10%. Thus, in the worst case the fractional systematic error in K and E due to a misuse of the plug flow idealization is comparable in magnitude to random errors introduced into K and E from uncertainties in analytic techniques and the measurement of residence time a t reaction conditions.
Introduction Tubular flow reactors are used by many chemists and engineers to make kinetic studies of combustion,' pyrolysis: and photolysis3chemistry. Bench scale, tubular flow reactors usually operate in the laminar flow regime. In spite of the fluid's parabolic velocity profile within the tubular reactor, chemical kinetic parameters are almost always obtained from tubular flow reactor data by use of the plug flow idealization. Many experimental and theoretical examinations of the validity of the plug flow idealization have appeared in the literature."18 The re-
sulb of these examinations were recently su"arizedlg in the form of criteria (based on characteristic times describing the operating conditions of the flow reactor) that ensure the validity of the plug flow idealization. In our experience, it is usually possible to design a tubular flow reactor intended for kinetic studies that satisfies these criteria.20 Such reactors may be used in research concerning homogeneous catalytic, pyrolytic, photolytic, or solvolytic phenomena. Although a tubular flow reactor may be initially designed to operate in the plug flow regime, it is not unusual for researchers to extend its use to regimes beyond those originally envisaged. Under such circumstances, significant departures from the intended plug flow regime may occur. This possibility prompts the question: In the worst possible case, how much systematic error will be introduced into kinetic parameters when the plug flow idealization is used to treat laminar flow reactor data? The purpose of this brief paper is to answer this question for single-step, irreversible chemical reactions of order lI2, 3/4, 1, 3/2,and 2. Methods outlined here can be used to evaluate the magnitudes of this systematic error for any other reaction order that may be of interest.
(1) Fontijin, A.; Felder, W. In Reactive Intermediates in the Gas Phase; Setaer, D. W., Ed.;Academic: New York, 1979. (2) Come, G. M. In Pyrolysis: Theory and Industrial Practice; Albright, L. F., Crynes, B. L., Corcoran,W. H., Eds.; Academic: New York, 1983. .~ (3) Howard, C. J. J. Phys. Chem. 1979,83,3. (4) Cleland, F. A.; Wilhelm, R. H. MChE J. 1956,2, 489. (5) Gilbert, M. Combust. Flame 1958,2, 149. (6) Dickens, P. G.; Could, R. D.; Linnett, J. W.; Richmond, A. Nature il,nndnn\ 1QkO. 1R7. RIlR (London) \__.___._,____, 1960,187,686. __., (7) Walker, R. E. Phys. Fluids 1961, 4, 1211. (8) Vignes, J. P.; Trambouze, P. J. Chem. Eng. Sci. 1962,17,73. (9) Mulcahy, M. F. R.; Pethard, M. R. Aust. J. Chem. 1963,16, 527. (10) Pokier, R. V.; Carr, R. W., Jr. J. Phys. Chem. 1971, 1953. (11) Azatyan, V. V. Dokl. Adad. Nauk. SSSR 1972,203, 177. (12) Ogren, P. J. J. Phys. Chem. 1975, 79, 1749. Villermaux, J. Chem. Phys. 1977, 74, 459. (13) Lede, J.; Villermaux; (14) Lede, J.; Villermaux, J. Chem. Phys. 1977, 74, 468. (15) Dang, V. D.; Steinberg, M. Chem. Eng. Sci. 1977, 32, 326. (16) Brown, R. L. J. Res. Natl. Bur. Stand., Sect. A 1978, A83, 1. (17) Furue, H.; Pacy, P. D. J . Phys. Chem. 1980,84, 3139. ~
(18) Dang, V. D.; Stenberg, M. J. Phys. Chem. 1980,84, 214. (19) Cutler, A. H.; Antal, M. J.; Jones, M. I d . Eng. Chem. Res. 1988, 27, 691-697. (20) Leaney, P. W.; Kershenbaum, L. S. Znd. Eng. Chem. Res. 1987, 26,369-373.
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1989 American Chemical Societv
106 Energy & Fuels, Vol. 3, No. 1, 1989
n 2
1.5
1
0.75 0.5
F;Yap) [ V a p - 11
2[l/(ap)1/2- 11 -In ap 4[i - (ap)1/41 2[1 - (ap)’/2]
Ramayya and Anta1
Table I. Closed Form Expressions for F,-* and F , FdK@ [K%2/2][0.5(1+ 2/K0)2- 2(1 + 2/KO) + In (1 + 2/KO) + 1.51 [K%z/8][0.5(1+ 4/K6’)2- 3(1 + 4/K@ + 3 In (1 + 4/KO) + 1/(1+ 4/K@ + 1.51 2Es [KO/ 21 [K%2/32][32/K%z - 32/KO + KO12 - 6 In (K0/8) - K%’/128] [K%2/8][8/K%2- 8/KB - In (KO/4) + 1.51
The literature offers some scattered insights into the magnitude of these systematic errors. Cleland and Wilhelm4 presented closed-form expressions for first-order reactions from which the systematic error in the reaction rate constant k can be derived. Representative values of the error in 12 for a first-order reaction can also be calculated from their numerical solutions of the governing equations. Alternatively, the results of later workers”18 can also be used to estimate the magnitude of the systematic error in k for first-order reactions. Estimates of the systematic error in the apparent activation energy E and rate constant k for non-first-order reactions are not available. In the following sections, we show how the fractional systematic error in k and E can be evaluated by relatively straightforward methods. Idealized Concentration Profiles Consider the case of an idealized, isothermal, tubular flow reactor in which a homogeneous, irreversible chemical reaction A -,products occurs. Neglecting axial diffusion and assuming the density of the fluid within the reactor to be constant, the species conservation equation can be written as
[ $ + :(
-
$)]/Pe
(uE)
- k0[Ao]”-lan= 0
with boundary conditions a(0,p) =
1
-
+ KOa” = 0
(3)
If a fully developed, laminar flow velocity profile (u = 2(1 - p 2 ) ) exists within the reactor, eq 3 can be solved to determine the laminar flow concentration field al:
where m = 1- n. The second idealization presumes a flat velocity profile within the reactor, resulting in the familiar plug flow approximation ap: n=1 a,(t,m) = exp(-kO{) = (1 - m K ~ 9 { ) ~ / ~n # 1 (5) This case is often realized7J0J2when Da -,0 or when Pe 0 providing the aspect ratio of the reactor is sufficiently large.4,9
-
al Fl(KO) = 21mexp[(-)ixe)w]w-3 dw 1
n=1
(6) where K = (ImlK[A,Jm)/2. Comparable expressions for the familiar plug flow idealization are given by
aP Fp(KO) = exp(-KO)
n=1 n # 1
= [ l - mK8]1/m
(7)
Evaluation of Systematic Errors 6K/K and 6E/E
The plug flow idealization is often employed to deduce kinetic data from converstion 1 - vs residence time 0 measurements obtained from a tubular flow reactor. Bounds on the systematic error that can be incurred through this approach may be estimated by recognizing that the plug flow and laminar flow idealizationsrepresent limiting cases of real tubular flow reactor performance. In one case (Pe m) species diffusion is negligible (laminar, streamline flow), whereas in the second case (Pe 0 and Da 0) species diffusion is so rapid that the reactant is convected down the tube at the mean velocity of flow. Consequently, we can establish a bound on systematic errors resulting from the plug flow idealization by using it to treat ideal, laminar flow reactor data. Thus we define the fractional, systematic error in the rate constant K to be
-
Two further idealizations permit the derivation of two alternative closed form solutions for a. The first presumes negligible radial diffusion within the reactor (Pe a),in which case eq 1 becomes (u$)
Usually the average (“cup mixing”) concentration of reactant emerging from the reactor is measured during an experiment. Following several changes of variables, the velocity-weighted average concentration a of reactant exiting the reactor ({ = 1)for the laminar flow idealization can be expressed as
-
-
6K - Kactuat - Kapprox _ K
Kactual
(8)
where Kwpmx represents the rate constant obtained through the (mis)use of the plug flow idealization to evaluate la= F;l(al)/0 minar flow reactor data. Employing Kapprox with eq 6 we find
Algebraic representations of F;l and Fl for reactions of various orders are given in Table I, and graphs of 6K/K vs conversion 1- a are displayed in Figure 1. Examining Figure 1, we see that for small values (
