I n d . Eng. Chem. Res. 1988,27, 691-697
enhance microencapsulation. Four weight percent borax in Glauber's salt is effective in achieving a low degree of subcooling without unduly reducing the crystallization temperature. Both nucleation and crystallization temperatures increase with stronger agitation.
Acknowledgment
69 1
Chen, J.; Nelson, R. Report ORNL/TM-8543, 1983; NTIS, Washington, D.C. Cole, R. L.; Hull, J. R.; Lwin, Y.; Cha, Y . S. Report ANL-82-89,1983;
NTIS. Washindon. D.C.
Fouda, A. E.; Desiault, J. G.; Taylor,J. B.; Capes, C. E. Sol. Energy
1984, 32, 57. Herrick, C. S.; Golibersuch, D. C. Technical Information Series Report 77CRD006, 1977; General Electric Company, Schenectady,
NY.
Financial assistance from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
Nomenclature
D, = inside diameter of capsule X,= distance coordinate measured downward from top inside surface of capsule Greek Symbol
= fraction of theoretical latent heat and sensible heat gained by contents of the capsules during heating cycle Registry No. Glauber's salt, 7727-73-3; borax, 1303-96-4; sodium sulfate, 7757-82-6.
Literature Cited Biswas, D. R. Sol. Energy
1977, 19, 99. Chahroudi, D. In Proc. Workshop on Solar Energy Storage Subsystems for Heating and Cooling of Buildings; NTS: Charlottesville,VA, 1975; p 56.
Herrick, C. S.; Zarnoch, K. P. Technical Information Series Report 79CRD249, 1979; General Electric Company, Schenectady, NY. Hodgins, J. W.; Hoffman, T. W. Can. J. Technol. 1955, 33, 293. Kelly, G. E.; Hill, I. E. NBSIR 74-634,Method of Testing for Rating Thermal Storage Devices Based on Thermal Performance; National Bureau of Standards: Washington, D.C., 1974. Marks, S. B. CHEMTECH 1982, March, 182. Marshall, R. In Proceedings of International Conference on Energy Storage; NTS: Brighton, UK, 1981; Vol. I, p 129. Methods of Testing Thermal Storage Devices Based on Thermal Performance; The American Society of Heating, Refrigeration and Air Conditioning Engineers: New York, 1977, Method ASHRAE 94-77. Sozen, Z. Z. Ph.D. Dissertation, University of British Columbia, Vancouver, 1985. Telkes, M. Ind. Eng. Chem. 1952, 44, 1308. Telkes, M. Proc. Workshop on Solar Energy Storage Subsystems for Heating and Cooling Buildings; NTS: Charlottesville, VA, 1975; p 17. Telkes, M. U.S.Patent 3 986 969, 1976.
Received for review April 2, 1986 Revised manuscript received October 6, 1987 Accepted December 4, 1987
A Critical Evaluation of the Plug-Flow Idealization of Tubular-Flow Reactor Data Andrew Hall Cutler,tt Michael Jerry Antal, Jr.,*tand Maitland Jones, Jr,e Department of Mechanical Engineering, University of Hawaii at Manoa, Honolulu, Hawaii 96822, and Department of Chemistry, Princeton University, Princeton, N e w Jersey 08544
Numerous limitations accompany the use of the plug-flow treatment of tubular-flow reactor data. In this paper we present design criteria, summarizing the findings of earlier workers, which overcome these limitations and assure the legitimacy of the plug-flow idealization. By use of a laminar-flow reactor designed to satisfy these criteria, studies were made of the vapor-phase thermolysis of tert-butyl alcohol, whose rate law is well established. A state-of-the-art nonlinear leasbsquares kinetics algorithm was used to analyze the tubular-flow reactor data. When the plug-flow idealization was employed, calculated values of E and In A were found to enjoy good agreement with literature values obtained by using other techniques. These results lead us to conclude that a well designed tubular-flow reactor can be a source of high-quality rate data when idealized as a plug-flow reactor. Because the magnitude of the renewable biomass resource within the U.S.A. is comparable to the national demand far gasoline (OTA, 1980), there is increasing interest in the thermochemical conversion of biopolymers to high-value chemicals and fluid fuels. High yields of vapor-phase monomers and related species are obtained when biopolymers are rapidly heated in an oxygen-free environment (Diebold and Scahill, 1985; Hopkins et al., 1984). Secondary reactions of these vapor-phase materials form more stable alkanes and alkenes as well as carbon oxides and water (Milne and Soltys, 1983; Antal, 1981, 1983a,b, 1985a,b). Because the hydrocarbon-forming reactions occur primarily in the vapor phase, fundamental studies of the thermolysis of vapor-phase species derived 'University of Hawaii at Manoa.
* Current address:
Energy Science Laboratories, P.O.Box
85608, San Diego, CA 92138-5608. f Princeton University.
0888-5885/88/2627-0691$01.50/0
from the pyrolysis of biopolymer materials, and related model compounds, are the key to the development of new technologies for utilizing the biopolymer resource. Unfortunately, both the pyrolytic vapors of biopolymer materials and the simpler model compounds which mimic their chemistry are condensable at room temperature and are often difficult to synthesize. Moreover, the hydrocarbon-forming thermolysis reactions typically begin above 600 "C, and some of their products are also condensable at temperatures below 300 "C. Becausde of these ccnstraints, conventional methods (including batch reactors, shock tubes, and turbulent-flow reactors) are not well suited for high-temperature, vapor-phase thermolysis studies of biopolymer-related materials. On the other hand, bench-scale tubular-flow reactors possess various design features which enable them to cope with the constraints imposed by biopolymer materials. For this reason, increasing attention (Antal, 1981, 1983a,b, 1985a,b; Stein et al., 1983; Cutler et al., 1987) is being given to the use 1988 American Chemical Society
692 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988
of tubular-flow reactors for the study of biopolymer-related vapor-phase thermolysis chemistry. Of course, tubularflow reactors remain of deep interest to chemists working in the allied fields of photochemistry (Howard, 1979) and high temperature chemistry (Fontijn and Felder, 1979a,b) as well as chemical engineers (Carberry, 1976). Various questions have been raised concerning the quality of kinetic data derived from tubular-flow reactors. Heat-transfer limitations impose significant constraints on the ability of a flow reactor to achieve isothermality over ita functional length. If species diffusion within the reactor were negligible, the parabolic velocity profile of fully developed laminar flow would be sufficient to invalidate the usual plug-flow analysis of the experimental data. With nonnegligible radial diffusion, the effective radial-velocity profile of species within the reactor lies between the laminar- and plug-flow idealizations. Moreover, radial diffusion introduces the potential effects of wall-catalyzed chemistry into the problem. Finally, axial diffusion can play a significant role in some reactor designs. As reviewed in the following sections, these questions concerning the utility of tubular-flow reactors are wellknown and have motivated many earlier papers. However, earlier studies have necessarily treated idealized situations which are rarely encountered in the laboratory. For example, it is customary to assume that only a single species is involved in the chemical reaction and that its thermophysical properties remain constant throughout the temperature and pressure range of interest. In practice, many species may participate in a thermolysis reaction, and their thermophysical properties, if known, can vary in the temperature and pressure range of interest. Moreover, as noted by Come in his recent review (1983a), both errors in the reactor model and errors in the experimental measurements contribute to errors in kinetic data. Comparisons of the relative contributions of these two sources of error to errors in kinetic measurements are lacking in the literature (Come, 1983a). Finally, apparent activation energy can be used to extrapolate rate laws to higher or lower temperature regimes and are sometimes employed to estimate chemical bond strengths. Unfortunately, little work has been reported on the relative influence of errors in the reactor model and errors in experimental measurements on calculated values of the apparent activation energy obtained from tubular-flow reactors. In this paper we first summarize earlier studies of conditions under which kinetic data, derived through the use of an idealized plug-flow treatment of tubular-flow reactor data, can be considered valid. We also review earlier work concerning the influence of errors in experimental measurements on errors in calculated rate data. Following this, we describe a two-zone, low Reynolds number, tubular-flow reactor which was designed to closely approximate an idealized plug-flow reactor when used to study vaporphase, biopolymer thermolysis reactions. By use of this reactor, rate data for the vapor-phase decomposition of tert-butyl alcohol are presented. The rate law governing the decomposition of tert-butyl alcohol has been studied by a variety of experimental techniques and is well established over a wide range of temperatures. Consequently, vapor-phase thermolysis studies of this model compound permit a realistic critique of the quality of kinetic data (rate constants and apparent activation energies) derived from a plug-flow treatment of tubular-flow reactor data. Influence on Kinetic Data of Errors in the Reactor Model Three sources of error can contribute to errors in
Table I. Characteristic Times and Nondimensional Numbers Pertinent to Tubular Flow Reactors characteristic time formula value,” s R / o 9.93 x 10-4 Tfc,R LIB 7.32 X 7fc.L R2/D 1.7 X lo-’ Tsd,R L2/G 1.5 “nd,L 6.80 x 10-3 Ttd R2/% RZ/v 7.90 x 10-2 7md k-I 3.08 X lo-’ Trk nondimensional no. formula value” Re 7md/Tfc,R 80 PF Ttd/Tmd 0.086 sc T,d,R/Tmd 2.2 pee.d “ed,R/‘fc,R 172 Petd
Ttd/Tfc,R
6.8
Da
Tsd,R/Tck
0.55
Values listed are calculated using data from the experiment a t 1083 K, which represent the worst case for employing the plug-flow
idealization in this work.
chemical rate data (Come, 1983a): errors in the reaction model, errors in the experimental measurements, and errors in the reactor model. Throughout this paper, we assume that the chemical mechanism of the thermolysis reaction is well established; consequently no treatment of the influence of errors in the reaction model (such as errors in the assumed order of the reaction) will be presented. The influence of errors in experimental measurements will be reviewed in the following section. In this section we review the substantial body of literature describing investigations of the influence on kinetic data of errors in the reactor model. Assuming fully developed, isothermal flow of a constant density fluid at constant pressure within a tubular reactor, in which a first-order, homogeneous, irreversible, singlestep, chemical reaction A products takes place, the governing species conservation equation is (see Nomenclature section)
-
.,,a’(
u
$)
-
Tck-’ff
= 0 (1)
Heterogeneous, wall-catalyzed reactions have been treated by Katz (1959), Wissler and Schechter (1962), Dranoff (1962), and Lupta and Dranoff (1966) and will not be discussed here. The explicit emphasis of characteristic times displayed in eq 1 (see Table I) and retained throughout the rest of this paper follows the presentations of Hulburt (1944) and Villermaux (1971). The variety of boundary conditions employed with eq 1 and their influence on solutions to eq 1 have been discussed by Fan and Ahn (1962). Two simplificationsof eq 1are often employed to obtain closed-form solutions to the problem. The first neglects terms related to species diffusion in eq 1 and assumes the velocity profile to be flat (u = l),resulting in the familiar “plug-flow” solution
ap = exp(--k7fc,d
(2)
for the average (“cup mixing”), nondimensional concentration of reactant, aptemerging from the reactor. The second idealization presumes negligible species diffusion (T,d,R a) and a fully developed, laminar (Poiseuille) velocity profile [ u = 2(1 - p 2 ) ] . With this assumption we have
-
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 693 a1 =
exp(-k7fC,~f/u)
(3)
and (with { = l ) , following several changes of variables, we obtain the familiar Y1aminar-flow”solution (Carberry, 1976)
Other mathematically equivalent expressions have been presented (Mulcahy and Pethard, 1963); however, the solution involving E3 given by eq 4 is the most convenient since values of the inverse function, E3-l, are readily available (Abramowitz and Stegun, 1966). A related laminar-flow solution for the case of second-order reactions was derived by Denbigh (1951). Careful analysis employing eq 3 and 4 permits one to calculate the fractional errors 6kfk and 6EfE incurred when the plug-flow idealization is erroneously used to treat an ideal laminar flow reactor. For reactant conversions ranging between 0.1 and 0.7, values of 6kfk range between 0.06 and 0.22 and values of 6EfE between 0.04 and 0.11 (Ramayya, 1986). These results indicate the maximum systematic error which can be incurred through the erroneous use of the plug-flow idealization. If species diffusion within the reactor is not negligible (7,d,R * m), the solution of eq 1 is considerably more complicated (Hoyermann, 1975). Three lines of attack have been described in the literature: analytic solutions, semianalytic solutions, and finite-differencesolutions. The following paragraphs review prior work involving these three approaches and summarize criteria under which the plug-flow idealization leads to small errors in the calculated rate constants. Early attempts to develop analytic solutions of eq 1 indicated the importance of nondimensional analyses (Hulburt, 1944) and the impossibility of realizing idealized laminar flow [T,d,R m, u = 2(1- p 2 ) ] in a vapor-phase, tubular-flow reactor (Bosworth, 1948). The seminal work of Taylor (1953, 1954) and Aris (1956) on the dispersion of a solute in a fluid flowing through a tube identified key insights which have been employed in almost every subsequent analytic treatment of eq 1. Taylor (1953, 1954) and Aris (1956) showed that a finite distribution of solute tends to become normally distributed (with no radial concentration gradients) about a point which moves with the mean velocity of the flow in the tube and undergoes axial dispersion governed by a virtual coefficient of diffusivity. Building on these findings, Mulcahy and Pethard (1963) and Fume and Pacey (1980) have reported detailed criteria for evaluating the legitimacy of the plug-flow idealization. Other analytic studies of the influence of axial diffusion (Dickens et al., 1960; Kaufman, 1961; Azatyan, 1972) and heat transfer (Gilbert, 1958) have also been reported. Semianalytic solutions of eq 1 rely on the well-known separation of variables method to identify eigenfunctions and associated eigenvalues which can be used to study the behavior of solutions to eq 1as a function of the operating conditions of the reactor. When axial diffusion is neglected relative to radial diffusion (RIL 0), eq 1 and the associated boundary conditions prescribe a Sturm-Liouvilletype problem, which has been the subject of numerous studies (Lauwerier, 1959; Wissler and Schechter, 1962; Hsu, 1965; Solomon and Hudson, 1967; Huggins and Cahn, 1967; Ferguson et al., 1969; Bolden et al., 1970; Villermaux, 1971; Ogren, 1975; Lede and Villermaux, 1977a,b). When axial diffusion is not neglected, the separation of variables treatment of eq 1 gives rise to nonorthogonal eigenfunc-
-
-
tions which introduce additional complexities to the problem (Dang and Steinberg, 1977,1980). Nevertheless, extremely insightful studies of this difficult problem were described over 20 years ago (Walker, 1961; Brown, 1978), and complete solutions for both homogeneous and heterogeneous reactions have been recently reported (Dang and Steinberg, 1977, 1980; Judelkis, 1980). Because in many cases the first (and largest) eigenvalue describes the behavior of the reactant over much of the reactor’s length, semianalytic solutions have offered special insights into regimes where the plug-flow idealization is valid. Assuming negligible axial diffusion, several authors have described fully computational, finite-difference solutions of eq 1and the related equation for second-orderreactions (Cleland and Wilhelm, 1956; Krongelb and Strandberg, 1959; Vignes and Trambouze, 1962; Fan and Ahn, 1962; Pokier and Carr, 1971). These solutions have offered both points of verification for the analytic and semianalytic solutions, and additional criteria for the range of validity of the plug-flow idealization. Dispersion in chemical reactors, which includes the influences of diffusion and nonideal flow patterns, has been treated by Danckwerts (1953) and Bischoff and Levenspiel (1962). A particularly interesting study by Batten (1961) revealed the importance of channeling in tubular-flow reactors equipped with narrow inlet and outlet tubes. A chemical reactor departs from the isothermal, plugflow idealization due to the influences of heat-transfer, axial diffusion, Poiseuille flow of the fluid mitigated by radial diffusion, wall effects, dispersion, and other miscellaneous phenomena (such as axial pressure gradients). Table I1 summarizes criteria for judging to what extent these influences cause a functioning reactor to depart from the isothermal, plug-flow idealization. In most cases the authors cited in Table I1 published studies of the influence of one or more nondimensional groups on concentration profiles or measured rate constants. The summary given in Table I1 indicates those values of the nondimensional groups which were found to ensure that the plug-flow value of species concentration or measured rate constant did not depart from the true value by more than a few percent. The interested reader should refer to the original literature for details. Experience with the criteria which assure negligible axial diffusion and negligible radial nonisothermalities indicates that these criteria can be easily satisfied during the design phase of a chemical reactor. Criterion 2 can be troublesome, but its derivation lacked the rigor of Dang and Steinberg (1980) and therefore should not be given excessive emphasis. Often in the literature the magnitude of the Peclet number (Pe) is used to assert the validity of the plug-flow treatment of tubular-flow reactor data. Physical reasoning alone shows that criteria based solely on Pe cannot be adequate. When reaction rates are sufficiently high, reaction will occur before radial diffusion has the opportunity to “smear” the parabolic velocity profile of laminar flow. Thus (as indicated in Table 11),values of both Pe and the Damkohler number, Da,are needed to ascertain the validity of the plug-flow idealization. In Table 11, criteria 5,7, and 8 due to Walker (1961) and Brown (1978) merit close attention, since they examined both the influences of Pe and Da on values of the rate constant obtained using the plug-flow idealization. Criterion 5 also includes some effects of a slow wall reaction and need not be absolutely satisfied. Criterion 6 given by Cleland and Wilhelm (1956) reflects calculated departures of true concentration profiles from idealized profiles. Consequently, it should not be
694 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 m
Table 11. Summary of Criteria for the Validity of the Plug-Flow Idealization criteria" author
S ( P ) = Cs(tj)TDjS(tj) j= 1
Negligible Axial Diffusion
