R , = gasconstant S = specific internal surface area, m2/g S i = Sherwoodnumber St = Stanton number T = temperature, "C T* = dimensionless temperature (=T/Tf,in) X = axial coordinate, m x = dimensionless axial coordinate, (=X/d,) Y = radial coordinate, m y = dimensionless radial coordinate ( =2Y/d,) Greek Letters
= isothermaieffectiveness factor = (-M)R(C/pcpT)f,in p = density CR = porosity of packed bed
Subscripts a = axial c = coolant f = fluid phase H = heat transfer i,j = cell numbers in = inlet j = jacket M = mass transfer p = particle r = radial s = solid phase
t = tube w = wall Literature Cited Agnew, J. B., Ph.D. Thesis, Monash University, 1967. Agnew, J. B.. Potter, 0. E., Trans. lnst. Chem. Eng., 48, T15 (1970). Aris, R., "Introduction to the Analysis of Chemical Reactors", Prentice-Hall, Englewood Cliffs, N.J., 1965 Beek, J., "Advances in Chemical Engineering", Vol. 111, Academic Press, New York. N.Y., 1962. Crider, J. E., Foss, A. S., A.l.Ch.E. J., 12, 514 (1966). Deans, H. R., Lapidus, L., A.l.Ch.E. J., 6, 656 (1960). Gel'bshtein. A. I.,Shchleglova. G. G.. Khomenko, A. A,, Kinet. Katal., 4, 625 (1963a). Gel'bshtein, A. I., Siling, M. I., Serguva, G. A,, Shchleglova, C. G., Kinet. Katal., 4, 149 (1963b). Hansen, K. W., Chem. Eng. Sci., 26, 1555 (1971). Lee, R . S. H., Ph.D. Thesis, Monash University, 1974. Lee, R. S. H., Agnew, J. E., Proc. First Australas. Conf. Heat Mass Transf., Melbourne, 6, 17 (1973). McGuire, M. L., Lapidus, L., A.l.Ch.E. J., 11, 85 (1965). Olbrich, W. E., Agnew, J. E., Potter, 0. E., Trans. lnst. Chem. Eng., 44, T207 (1966). Priestley, A. J., Agnew, J. B., lnd. Eng. Chem., Process Des. Dev., 14, 171 (1975). Priestley, A. J., Webster, J. W. C., Agnew, J. E., Chemeca '70 Proc., 63 (1970). Satterfield, C. N., Cadle, D. J., lnd. Eng. Chem., Prod. Res. Dev., 7, 256 (1968). Satterfield, C. N., "Mass Transfer in Heterogeneous Catalysis", M.I.T. Press, Cambridge, Mass.. 1970. Shankar, S., private communication, Monash University, 1976. Villadsen, J., Stewart, W. E., Chem. Eng. Sci., 22, 1483 (1967). Wesselhoft, R. D., Woods, J. M., Smith, J. M., A.l.Ch.E. J., 5, 361 (1959). Weston. 0. F., Agnew, J. B., lndian Chem. Eng., XV, 37 (1973).
Received f o r reuiew August 16, 1976 Accepted April 5,1977
Model Studies for a Vinyl Chloride Tubular Reactor. 2. Dynamic Behavior Ronald S. H. Lee and John B. Agnew' Department of Chemical Engineering, Monash University, Clayton, Victoria, Australia 3 168
The dynamic behavior of a nonadiabatic single-tube reactor for the catalytic hydrochlorination of acetylene has been investigated both theoretically and experimentally for step changes in feed concentration. The dynamic response was found to be influenced primarily by the bed thermal capacitance, the resistance to heat transfer between gas and catalyst pellet, and the nonlinear coupling between concentration and temperature. The thermal capacitance of the tube wall was examined and found to have a negligible effect. A one-dimensional, nonlinear cell model incorporating interphase heat and mass transfer resistances and intraparticle mass diffusional resistance, with quasi-stationary assumptions for interstitial fluid heat and mass capacitances and intraparticle mass capacitance, was found to give a reasonable representation of actual reactor behavior.
Introduction Many theoretical papers have been published over the past 20 years on the dynamics and stability of packed tubular reactors for highly exothermic solid-catalyzed gas reactions. Experimental dynamic data for such systems, however, are singularly lacking despite the fact that this type of reactor is widely used industrially. The reasons for this are principally concerned with the difficulties involved in accurately measuring the temperature and concentration changes occurring in packed tubes of small diameter without significantly altering the physical character of the system. Hoiberg et al. (1971) have published one of the very few detailed studies of experimental dynamics known to the au-
thors. The reaction they considered was between oxygen and hydrogen catalyzed by platinum on silica granules; the ratio of tube diameter to granule diameter was about 50:l. Locally linearized one- and two-dimensional continuum models which neglected intraparticle gradients were found to represent the measured frequency response characteristics, provided that the maximum temperature excursion did not exceed 15 "C. Recently Hansen and Jorgensen (1976) reported the results of dynamic experiments on a packed bed reactor for this same reaction, but using a different catalyst support (alumina) and lower tube to particle diameter ratio; data were obtained at low Reynolds numbers ( 1,intraparticle mass accumulation can fairly safely be neglected. In borderline cases a separate model study would seem to be the most appropriate way to decide whether the intraparticle concentration profile follows the pellet temperature profile quasi-stationarily. An alternative measure is the ratio 7 2 / 7 1 , which equals Le/t,, although $his measure proved to be inconclusive in simulation studies described in the previous section. A further simplification that is often made in transient studies is to lump all the heat transfer resistances a t the pellet/fluid boundary, provided that the major resistance lies in the film surrounding the pellet. Finlayson (1971) has shown that the "lumped" heat transfer coefficient may be established Ind. Eng. Chem., Process D e s . Dev., Vol. 16, No. 4, 1977
497
Table I. Definition of Time Constants and Capacitance Ratios Quantity
Physical meaning
Definition
71
time constant for mass transfer inside catalyst
72
time constant for heat transfer inside catalyst
73
residence time in mixing cell
74
time constant for heat transfer to catalyst
75
residence time in reactor
Value
2
tpd,
9.47 s
40,
3.32 s 3.26 x 10-3 s PsCpsdp 6hc
1.09 s
L -
2.17 s
U
76
A
time constant for mass transfer to catalyst
6kc
HCRl
ratio of heat capacity of fluid to solid particles
HCR2
ratio of heat capacity of solid particles to wall
Table 11. Criteria for Model Assumptions Criterion Assumption
ERPfCpf
(1 - CR)Pscps
3.5 x 10-3 2.56 x 10-3 1.12 x 10-1
Value
r4/r1 > 1 Neglect accumulation term in particle 0.12
mass balance Neglect accumulation term in bulk fluid mass balance HCRl < Neglect accumulation term in bulk 0.004 fluid heat balance Nu < 3 Lump particle energy balance r4/73
>7
333 0.0026 1.0
via a one-term collocation solution as
Rearrangement gives
The procedure is recommended when Nu < 3. A summary of the proposed criteria for the applicability of various quasi-stationary assumptions is given in Table I1 together with appropriate values for the vinyl chloride reactor. While bulk fluid mass and energy accumulation terms may safely be neglected, the small value of 74/71 does not offer clear support for the implementation of the quasi-stationary assumption for ,the intraparticle concentration profile. Dynamic Reactor Model The basic model which will be used is the one-dimensional cell model incorporating inter- and intraparticle mass diffusional resistances and interparticle heat transfer resistance. This model was shown in Part I to give excellent prediction of steady-state behavior for the vinyl chloride reactor under similar conditions to those proposed for this study. A diagrammatic representation is shown in Figure 6. In formulating a dynamic version of this model, due consideration is given to the findings of the previous section, viz. that accumulation terms for the interstitial fluid mass and energy balances may be neglected. Also, because of the greatly increased computation time involved, intraparticle mass accumulation is omitted from the initial formulation. 498
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
Figure 6. Diagrammaticrepresentation of one-dimensionalheterogeneous cell model, showing principal heat and mass transfer resistances.
The relevant dimensionless equations for model DRM- 1 are the following. Balances for Interstitial Fluid. (i) Mass Balance 0 = C*f,?l
- C*f,[ - U/StM(C*f,&- C*,,,,)
(7)
(ii) Energy Balance 0 = T*f,[-l - T*f,& - aStw(T*f,l- T*,) - a'StdT*f,l
- T*,,,[)
Balances for Catalyst Pellet. ( i ) Mass Balance
( i i ) Energy Balance 82
dT*S,, Le 7 - 3Nu(T*f,i- T*so,i 1 dt
(8)
106-
104T ('C)
102-
X (crn)
X (crn)
Figure 7. Effect of wall thermal capacitance on reactor dynamic response. (case 3 conditions, Table 111).
Table IIL. Operating Conditions for Dynamic Investigation
Wall Energy Balance
+
dT*,i - Hw(T*f,i- T*w,i) Hj(T*, - T* w,i) dt Initial and boundary conditions are given by A -
(11)
a t t* = 0: C*f,i = (C*f,iIss;T*f,i= (T*f,iIss; C*s,i = (C*s,i)ss;T*s,i =
(13) (14) Two variants of this model will be considered in the subsequent discussion. In the first, DRM-2, the dynamics of tube wall are neglected, in which case eq 11 becomes (lla)
In the second, DRM-3, the intraparticle mass accumulation term is included in eq 9, which becomes ac*s,1 = 7 at
[
;
S,l
Y2 a Ca Y*
I
- h2C*ns,ley[l-1/T*,.,1
Initial conditions
Final conditions
1
Re, = 7 8 , p = 170 kPa C,, = 5.50, T , = 100 "C Re, = 78, p = 205 kPa C,, = 6.65, T , = 100 "C Re, = 78, p = 240 kPa C,, = 7.1, T , = 100 "C Re, = 7 8 , p = 170 kPa C,,, = 5.50, T , = 100 "C
Re, = 7 8 , p = 170 kPa C,, = 8.25, T , = 100 "C Re, = 78, p = 205 kPa C,, = 9.95, T , = 100 "C Re, = 78, p = 240 kPa Ci, = 11.7, T , = 100 "C Re, = 7 8 , p = 170 kPa C,,, = 11.0, T , = 100 "C
3
(12)
61
Case
2
(T*s,i)ss;
0 = Hw(T*f,i - T*w,l)+ H,(T*, - T*,,,)
Figure 8. Dynamic response of tube wall temperature to 50% step increase in reactor inlet concentration (case 3 conditions, Table 111).
(ga)
Each of the above sets of model equations may be solved by the method of orthogonal collocation. Three interior collocation points were selected for catalyst profile computations. This has been shown by Ferguson and Finlayson (1970) to give a solution accuracy of 0.02%for a linear slab diffusion problem. For models DRM-1 and DRM-2 the number of iterations required for covergence was found to be two, and a dimensionless time-step increment of 50 was also found to give a sufficiently accurate solution. For model DRM-3, however, which required computation of the dynamic response of the intraparticle concentration profile, an initial dimensionless timestep increment of 1was necessary; this was later increased to 4 as the solution proceeded. The reduced step size was necessary because of the presence of a moderately stiff system of ordinary differential equations and a very steep intraparticle concentration profile. Further details are given elsewhere by Lee (1974). Dynamic Wall Effects A comparison was made between dynamic temperature profiles predicted by models DRM-1 and DRM-2, for a step increase in bulk concentration (case 3 in Table 111) to examine the effect of wall thermal capacitance. The axial bed response shown in Figure 7 indicated little difference between the two sets of predictions-no more than 3 "C, in fact. The predicted
4
instantaneous wall temperatures were likewise very close, as shown in Figure 8. Although the ratio of thermal capacitances of wall and catalyst was high (viz. 9:1), the coolant film heat transfer coefficient was also high (570 W/m2 "C) so that the difference between wall and coolant temperature never exceeded 6 "C. For this reason the wall capacitance had little effect on the reactor dynamics for the conditions examined, and hence it could be neglected in further studies under similar conditions. Had the wall heat transfer coefficient been appreciably lower, however, it is clear that the wall thermal capacitance would have had a highly significant effect on the reactor dynamics. Comparison between Experimental and Predicted Dynamics The validity of the proposed dynamic model DRM-1 was tested experimentally, using the experimental vinyl chloride reactor described in Part I. Only one type of perturbation was employed, viz. a step change in hydrogen chloride concentration a t the reactor inlet. The experimental conditions are specified on Table 111. Measured axial temperature profiles were compared with computed axial profiles determined from the radially averaged predictions, assuming that radial temperature profiles were parabolic (as discussed in Part I). For cases 1to 3, a 50% step increase in inlet concentration was introduced a t successively higher operating pressure and hence progressively greater severity. The magnitude of such a large change precluded the use of linearized models as employed by Hoiberg et al. (1971). Figure 9 toll show experimental and predicted axial temperature profiles for cases 1 to 3. Figure 12 applies to case 4, in which a 100% increase in inlet concentration was applied. The overall agreement was considered to be reasonably satisfactory considering the magnitudes of the changes applied and the experimental difficulties involved in obtaining reproducible results in this type of reaction system. Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
499
1
'
'
"
"
' i
"
160-
180
i
140-
TeC) -
120-
1006
'
lo
'
20
x
'
30
'
io
'
(crn)
Figure 9. Measured and predicted axial temperature responses to step increase in inlet concentration (case 1,Table 111).
l00,v'
"
"
10
20
"
30
"
LO
x (cm) Figure 11. Measured and predicted axial temperature responses to step increase in inlet concentration (case 3, Table 111).
180 I
"
"
"
"
I
-A . B 30s OS
160
1
1LO
T ("C)
120 100;
'
'
10
'
'
20
'
x (cm)
'
30
'
'
40
"
x
(cm)
Figure 10. Measured and predicted axial temperature responses to step increase in inlet concentration (case 2, Table 111).
Figure 12. Measured and predicted axial temperature responses to step increase in inlet concentration (case 4, Table 111).
I t was found during the course of experimentation that the introduction of an ideal step change in concentration (keeping all other operating variables constant) proved to be difficult to achieve. Even though a pressure controller had been installed it was not possible to eliminate completely transient disturbances in flow rate and pressure which immediately followed the step change; these unwanted transients took about 10 s to finally disappear. A problem was also experienced in maintaining absolutely steady pressure and flow rate throughout the course of the run; although these variations were small, they could possibly have prolonged the transients. Other possible sources of error which are difficult to avoid completely in heterogeneous reactor studies were concerned with catalyst variability and unavoidable deactivation due to loss of mercuric chloride a t the higher operating temperatures. The final steady-state hot-spot for case 4, for example, was about 8 O C lower than was measured in a previous steady-state run a t the same operating conditions. Closer examination of the profiles, particularly in the region of the hot-spot, indicates that the model predicts a faster rate of profile development than was observed experimentally in the initial stage of transient response, the effect being more pronounced a t higher severity operation. This could possibly have been due to the quasi-stationary assumption which was made for the intraparticle concentration profile. However, attempts to solve the model equations incorporating intraparticle mass accumulation (DRM-3) led to severe numerical difficulties and increased the computation time by an order of magnitude (to approximately 4 h on a Burroughs B6700 computer). This resulted from the occurrence of a moderately stiff system of ordinary differential equations and a very steep intraparticle concentration profile. Although various algorithms have been suggested for numerical integration of similar systems of equations (Lapidus et al., 1971), they have not yet been tried. The use of approximate solutions as sug-
gested by Marek (1975) could also be a way around the problem. Another possible delay of significance could have been associated with chemisorption dynamics. Further investigation of this aspect is proposed. Until more efficient numerical techniques are developed, the mathematical model proposed in this paper may be considered to give a reasonably good representation of the dynamic behavior of the vinyl chloride reactor investigated. As this system represents an extreme case as far as Lewis number is concerned, the model should also be considered for other reactions when linearized models are clearly inappropriate.
500
Ind. Eng. Chern.,Process Des. Dev.,Vol. 16, No. 4, 1977
Stability In view of the low Lewis number, some comment is necessary regarding the nature and stability of the catalyst particle steady state. Aris (1975) has discussed this problem in some detail. Wei (1965) and Hlavacek and Marek (1968) have shown that if the Lewis number is less than 1 the transient temperature may exceed by a considerable amount of maximum possible rise under steady-state conditions. For the present system the thermicity factor /3 is low, being of the order of 0.001 to 0.0025, due largely to the low effective diffusivity, while y is approximately 14. Kehoe and Butt (1972) have presented a uniqueness criterion for n t h order reaction when diffusional resistance in the micropores is large. Applying this to the present system, uniqueness is guaranteed for y < 25. Luss and Lee (1970) have demonstrated that a unique steady state may be unstable a t low Lewis numbers. In a particular example for which (3 = 0.15 and y = 30, limit cycling was determined to occur a t Lewis numbers less than about 0.39 with very high hot-spots developing periodically near the particle surface. No instability of this or any other type has been observed with the present system, however. At the lower values of 6 and
y obtaining it is to be expected that the critical Lewis number should also be substantially lower. As Luss and Lee have pointed out, "the pathological behavior demonstrated by systems with low Lewis number is mainly of academic interest.''
Conclusions The dynamic behavior of a nonadiabatic packed tubular reactor for the catalytic hydrochlorination of acetylene has been examined both experimentally and theoretically for large step perturbations in inlet hydrogen chloride concentration. Simulation of catalyst pellet dynamics indicated the importance of gadpellet heat transfer resistance and the relatively minor effect of intraparticle heat transfer resistance on the average reaction rate, confirming the results of other investigators. A comparative study of thermal and mass capacitances showed that mass and energy accumulation in the interstitial fluid could safely be neglected in comparison with the thermal capacitance of the catalyst pellet. However, it could not be clearly established from theoretical studies whether intraparticle mass accumulation could also be neglected a t the low value of Lewis number prevailing. Dynamic nonlinear models were developed, using as a basis a one-dimensional heterogeneous cell model incorporating inter- and intraparticle mass diffusional resist,ances and interphase gadpellet heat transfer resistance. Because of the otherwise excessive computation time, it was found necessary to invoke a quasistationary assumption for intraparticle concentration profile by neglecting intraparticle mass accumulation. Although the reaction was carried out in a thickwalled metal tube, the effect of wall thermal capacitance on dynamic behavior was found to be negligible because of the large heat transfer coefficient on the coolant side. It should be noted that if the wall heat transfer coefficient is low, wall thermal capacitance may become a dominant dynamic factor. Experimental dynamic axial temperature profiles were determined for large step changes in inlet hydrogen chloride, viz. 50 to 100%.The maximum temperature excursion measured was 60 "C. Model predictions were in reasonably good agreement, although they indicated a faster rate of profile development in the initial stages than was observed experimentally. An attempt to solve the model equations with intraparticle mass accumulation included resulted in severe numerical difficulties and excessive computation time. Further investigation is underway to try to overcome the problem. Most catalyst-pellet/gas systems are characterized by higher
Lewis numbers than was the case in this study. The dynamic model developed should therefore be of wider applicability in cases where local linearization is not possible. Acknowledgments The authors wish to thank the Austrailian Research Grants Commission and Monash University for their financial support. Nomenclature The main list of symbols is given in Part I. Additional ymbols used are the following. C* = dimensionless volume-average concentration G = mass velocity of gas, kgls m2 h , = solid/gas film heat transfer coefficient, W/m2 "C k , = solidlgas mass transfer coefficient, m/s Le = Lewis number ( = pscpsDe/ke) Nu = Nusselt number ( = h,dP/2h,) p = pressure, kPa Ri = internal radius of tube, m Ro = external radius of tube, m t = time, s t' = dimensionless time ( = 4 D , t / d P 2 ) T* = dimensionless volume-average temperature Up = lumped heat transfer coefficient, W/m2 "C
Greek Symbols 61 = t,d,u/D, 6 2 = 61/t, tp t~ T
= porosity of particle
= porosity of bed = time constant, s
Literature Cited Aris, R., "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts", Vol. 11, Clarendon Press, Oxford, 1975. Ferguson, N. B., Finlayson, B. A,, Chem. Eng. J., I, 327 (1970). Finlayson, B. A., Chem. Eng. Sci., 26, 1081 (1971). Hansen, K. W., Chem. Eng. Sci.. 26, 1555 (1971). Hansen, K. W., Chem. Eng. Sci., 28,723 (1973). Hansen, K. W., Jorgensen, S. B., Chem. Eng. Sci., 31,579, 587 (1976). Hlavacek, V., Marek, M., European Symposium on Chemical Reaction Engineering, Brussels, 1968. Hoiberg, J. A., Lyche, B. C., Foss, A. S., A./.Ch.E. J., 17, 1434(1971). Kehoe, J. P. G., Butt, J. B.. Chem. Eng. Sci., 27, 650 (1972). Kabei, R. L.. Carl, G. L., Yurchak. S., A./.Ch.E. J., 14,627 (1968). Lapidus, L.. Seinfeld, J. H., Academic Press, New York, N.Y., 1971. Lee, R. S. H., Ph.D. Thesis, Monash University, 1974. Luss, D., Lee, J. C. M., A./.Ch.E. J., 16, 620 (1970). Marek. M., Stuchl, I., Chem. Eng. Sci., 30,555 (1975). McGreavy. C., Thornton, J. M., Chem. Eng. J., 1, 296 (1970). Vardi, J., Biller, W. F.. Ind. Eng. Chem., Proc. Des. Dev., 7,83 (1968). Vortmeyer, D., Jahnel, W., Chem. Eflg. Sci., 27, 1485 (1972). Wei, J., Chem. Eng. Sci., 20,729 (1965).
Received f o r review August 16, 1976 Accepted April 5 , 1977
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