Ind. Eng. Chem. Res. 1989, 28, 590-599
590
For the quaternary system, UNIQUAC makes a reasonable prediction for the nitromethane/furfural-rich phase, with mole fractions usually within f0.02 of the experimental values. In the methylcyclohexane-rich phase, much larger differences in mole fractions are present, probably due to the limitations of the predictive method. Also, at the single-phase composition values, UNIQUAC still predicts a separation into two phases. Conclusions Current Gibbs energy models are inadequate in extending experimental VLE data to the prediction of LLE data. This work has presented a technique to measure VLE, and more importantly y m , of partially miscible systems for application to LLE. Present g E expressions were shown to fail in attempting to predict the LLE relationships from VLE data. The continuing measurement of derivative data will lead to better understanding of solution behavior and to improved g E models that will better be able to characterize LLE behavior. Acknowledgment We thank E. I. du Pont de Nemours & Company, Inc., for the financial support of this work. Nomenclature FUR = furfural MCH = methylcyclohexane NTM = nitromethane TOL = toluene Ag,, = NRTL binary interaction parameter, cal/mol AuI2 = UNIQUAC binary interaction parameter, cal/mol
Greek Letters a I 2= NRTL binary parameter y = activity coefficient at infinite dilution Registry No. Toluene, 108-88-3; nitromethane, 75-52-5; methylcyclohexane, 108-87-2; furfural, 98-01-1.
Literature Cited Fredenslund, A.; Gmehling, J.; Michelsen, M. L.; Rasmussen, P.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Deu. 1977a,16,450. Fredenslund, A,; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria Using UNIFAC; Elsevier: New York, 1977b. Hussam, A,; Carr, P. W. Anal. Chem. 1985,57, 793. Nicolaides, G. L.; Eckert, C. A. Ind. Eng. Chem. Fundam. 1978, 17(4),331. Pierotti, G. J.;Deal, C. H.; Derr, E. L. Ind. Eng. Chem. 1959,5I,95. Prausnitz, J. M.; Anderson, T. F.; Grens, E. A.; Eckert, C. A.; Hsieh, R.; O’Connell, J. P. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1980. Reid. R. C.: Prausnitz. J. M.: Poling. B. E. ProDerties of’ Gases and Liquids, 4th ed.; McGraw-Hill: -New York,. 1987. Renon, H.; Fabries, J.; Sannier, H.; Leroi, J.; Masson, J. Znd. Eng. Chem. Process Des. Dev. 1977, 16, 139. Schreiber, L. B.; Eckert, C. A. Ind. Eng. Chem. Process Des. Deu. 1971,10, 572. Scott, L. S . Fluid Phase Equilib. 1986,26, 149. Thomas, E. R.; Eckert, C. A. Ind. Eng. Chem. Process Des. Deu. 1984,23, 1984. Thomas, E. R.; Newman, B. A.; Nicolaides, G. L.; Eckert, C. A. J . Chem. Eng. Data 1982a,27, 233. Thomas, E. R.; Newman, B. A.; Long, T. C.; Wood, D. A.; Eckert, C. A. J . Chem. Eng. Data 1982b,27, 399. Wittrig, T. S. BS Thesis, University of Illinois, Urbana, 1977.
Receiued for review July 25, 1988 Accepted December 12, 1988
Experiments and Model for the Oscillatory Oxidation of Benzaldehyde A n n e M. Reimus, Jean M a r i e Massie, and J o h n L. Hudson* Department of Chemical Engineering, University of Virginia, Charlottesuille, Virginia 22901
The oxidation of benzaldehyde is investigated experimentally in a gas-liquid stirred cell reactor into which both gas and liquid phases are added continuously. The gas-liquid interfacial area is known, and the mass-transfer coefficient can be controlled. Regions of steady-state and oscillatory behavior are studied as a function of two system parameters, the liquid residence time and the liquid-side mass-transfer coefficient for oxygen absorption. An eight-variable mathematical model of the oxidation reaction is used to investigate stability and oscillations in the system. The mathematical simulation results were found t o agree qualitatively with those found experimentally throughout most of the two-parameter space. A simpler three-variable model is then developed and shown to predict behavior similar to that of the more complicated model over most of the range of parameters. Many chemical oscillators, such as the well-known Belousov-Zhabotinskii (Belousov, 1959, Zhabotinskii, 1964) and Bray-Liebhafsky (Bray, 1921, Liebhafsky, 1931a,b) reactions, occur in homogeneous liquid-phase systems. However, chemical systems involving gas-liquid reactions can also exhibit multiple steady states and oscillations. Multiplicity and stability of gas-liquid reactors have been investigated numerically by a number of authors. In these studies, the gaseous component is absorbed into the liquid in which a simple exothermic reaction occurs. Schmitz and Amundson (1963) showed that multiplesteady-state and oscillatory behavior can exist in gas-liquid systems. In later studies, overall material and energy balances demonstrated that multiple steady states are possible in both adiabatic (Hoffman et al., 1975; RSlghuram and Shah, 1977) and nonadiabatic (Huang and Varma, 0888-5885/89/2628-0590$01.50/0
1981a; Raghuram et al., 1979) reactors. Ding et al. (1974) have shown experimentally that two stable steady states can occur in the adiabatic chlorination of n-decane. Huang and Varma (1981b) have determined numerically that oscillatory behavior is possible-but not highly probable-in nonadiabatic reactors in which a fast pseudo-first-order reaction occurs in the liquid. Hancock and Kenney (1977) have reported oscillations in the formation of methyl chloride from hydrochloric acid and methanol in a two-phase reactor in which the reactants enter and the products leave in the gas phase, but reaction occurs in the liquid phase. In this case, the oscillatory behavior resulted mainly from a complex interaction between the product vapor pressure and the reactant partial pressure. Oscillations in some more recently discovered gas-liquid systems are caused by positive feedback in the reaction
CZ 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989 591 mechanism itself and thus can occur under isothermal conditions. Jensen and co-workers (Jensen, 1983; Jensen et al., 1984; Roelofs et al., 1987) have found oscillations during the oxidation of benzaldehyde. The reaction was catalyzed by cobalt and bromide ions and was carried out in an acetic acidfwater solution; air was continuously bubbled through the liquid reactants. The oscillations were visually evident in dramatic color changes of the reaction mixture as the dominant form of the cobalt catalyst varied between Co(I1) (pink) and Co(II1) (brown/ black). Investigations into the mechanism of the oxidation of benzaldehyde and other aldehydes catalyzed by metal ions have proposed steps involving free radicals (Bawn, 1983; Hendriks et al., 1977, 1978; Kuo and Chou, 1988). Several detailed mechanisms which predict oscillations in the oxidation of benzaldehyde have been proposed (Roelofs et al., 1983, 1987; Yuan and Noyes, 1988). Several other gas-liquid reaction systems exhibiting oscillations have also been investigated. Rastogi and Das (1984) have studied the oscillatory behavior of the benzaldehyde oxidation reaction in both batch and continuous reactors. In addition, they have shown that the oxidation of acetaldehyde carried out in a similar manner can exhibit oscillations. Suresh et al. (1988a,b) have noted oscillations in the dissolved oxygen concentration in the two-phase oxidation of cyclohexane. Burger and Field (1984) have discovered a chemical oscillator based on sulfur chemistry involving the reaction of sulfide ion, sulfite ion, methylene blue, and dissolved oxygen. During the reaction, the methylene blue catalyst oscillates between its oxidized, colored form and its reduced form. Although the oxygen was dissolved in solution in the experimental studies, this reaction could also potentially yield oscillatory behavior in a two-phase system. In this paper, stability and oscillations in the two-phase oxidation of benzaldehyde are investigated. The behavior is studied experimentally in a continuous stirred cell reactor in which the liquid is added and removed at a constant rate and oxygen is absorbed from the gas phase. This appears to be the first study of this reaction in which the liquid and gas are both added continuously. Furthermore, an apparatus is used for which the area of the gas-liquid interface is known and the liquid-side mass-transfer coefficient can be controlled by changing the stirring rate. Such reactors are more useful in laboratory studies than those in which the gas is bubbled into the liquid since the latter type does not offer independent control of the mass-transfer coefficient. The dynamics are investigated as a function of two controllable parameters, the liquid residence time and the liquid-side mass-transfer coefficient. In a parallel mathematical study, the reaction mechanism proposed by Roelofs et al. (1983) is used to determine numerically the regions of stability and oscillations in the same two-parameter space. Their proposed scheme is a free-radical mechanism for which the model consists of eight first-order ordinary differential equations. The experimental and simulation results are seen to agree qualitatively as a function of the two system parameters. Furthermore, a simplified model consisting of three first-order ordinary differential equations is developed and shown to give similar results as the more complete model over much of the parameter range. Experiments McGinnis (1987) has characterized the behavior of the benzaldehyde oxidation system in a continuous stirred cell reactor as a function of the liquid-side mass-transfer coefficient and the liquid residence time. The apparatus used was a scraped surface stirred cell reactor based on
Laboratory
Platinum E l e c t r o
To Constant L e v e l Tank
SOlUIiO"
Intel
Figure 1. Schematic diagram of the experimental apparatus.
a Danckwerts' design (Danckwerts and Gillham, 1966) with requisite feed, temperature control, and measurement systems. This design was chosen because it provided a constant, known surface-area-to-volume ratio and allowed for control and an appropriate range of the liquid-side mass-transfer coefficient, kL. A schematic of the reactor is shown is Figure 1. The reactor was constructed of Teflon and Stainless steel materials which resist corrosion by acetic acid. Experiments were run with a liquid volume of approximately 115 mL and a surface-area-to-volume ratio of a = 0.37 cm-'. Two 4-blade, flat-blade turbine stirrers were submerged in the liquid. The lower stirrer provided bulk mixing of the liquid. The upper stirrer was located approximately 1/16 in. below the gas-liquid interface and controlled the gas-liquid mass-transfer coefficient. The stirrers were connected to the same shaft and thus operated at the same stirring speed. The stirring speed was measured by a magnetic Reid switch circuit connected to a frequency counter. Four static baffles projecting from the bottom of the reactor aided in mixing. The liquid-side mass-transfer coefficient was measured by the physical absorption of pure carbon dioxide into distilled water (Goodridge and Bricknell, 1962; Davies et al., 1964). The stirring speed was varied from 60 to 133 rpm during experimental runs, corresponding to values of kLa ranging from 0.0019 to 0.0035 cm/s. Stirring speeds higher than 133 rpm produced instabilities in the gas-liquid interface, changing the surface-area-to-volume ratio, and thus were not studied. The liquid entered the bottom of the reactor and exited through a side port leading to a constant level tank. The feed solution consisted of 0.02 M Co2+,0.008 M Br-, and 0.75 M benzaldehyde in a 90f10 acetic acid/water (w/w) solvent. Reagent grades of cobaltous acetate tetrahydrate and sodium bromide and 98+% pure benzaldehyde were used without further purification. The feed solution was stored in a well-stirred 4-L glass tank. Nitrogen gas was continuously bubbled through the feed solution to prevent oxidation by air. A Cole-Parmer positive displacement pump regulated by a Master Servodyne controller fed the solution to the reactor. The liquid flow rate was varied from 1.4 to 2.61 mL/min, yielding a range of liquid residence times from 264 to 4998 s. The feed tubing was immersed in the constant-temperature water bath housing the reactor unit to preheat the liquid before it entered the reactor. All experiments were run at a constant temperature of 52 "C. The temperature of the reactor contents was monitored with a Teflon-coated thermistor. The pure oxygen environment in the reactor was maintained with a constant flow of oxygen at a rate of 50 mL/min. The
592
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989
oxygen stream was bubbled through a 90/10 acetic acidlwater solution immersed in the constant-temperature water bath to saturate the gas before it entered the reactor. The gas space was open to the atmosphere through an exit port to maintain the pressure at 1 atm. A platinum wire indicating electrode and a Ag/AgCl porous plug junction reference electrode were used to measure the redox potential of the solution. Digital data were taken with a FLUKE 8840A voltmeter and a DEC PDP 11/73 laboratory computer at a rate of 0.5 Hz. Three to five runs were performed at each set of conditions. For steady-state behavior, a mean potential was determined a t each experimental condition. Oscillatory behavior was characterized by three quantities: the midpoint of the minimum and maximum values of oscillation, the amplitude of oscillation, and the period of oscillation. Mean values of these three quantities were determined from the multiple runs at a given set of conditions. The midpoint had at most a 2% deviation from its mean value. The period and amplitude of oscillations generally deviated by 15% and 20% from their respective means.
Experimental Results Over the entire range of parameters studied experimentally, the reactor was either at steady state or the reaction exhibited one peak periodic oscillations. Typical bifurcation diagrams of the platinum electrode potential versus the liquid-side mass-transfer coefficient multiplied by the surface-area-to-volume ratio (kLa) for six values of the liquid residence time ( 7 ) are shown in Figure 2. For any value of the liquid residence time, an increase in the mass-transfer coefficient yielded a bifurcation from steady-state to oscillatory behavior. The bifurcation occurred in the range 0.0019 s-l 5 kLa I0.0023 s-l for every residence time except the smallest studied ( 7 = 264 s), where it occurred between kLa values of 0.0023 and 0.0025 s-*. In addition, the steady state at kLa = 0.0019 s-l for the largest two residence times ( 7 = 3600 and 4998 s) appeared to be very close to the bifurcation point, since transient oscillations under these conditions took longer to die out than those a t smaller values of 7. Thus, the bifurcation point seems to occur at smaller values of the mass-transfer coefficient for larger residence times. It also appeared that a second bifurcation point at large kLa (back to steady state) was approached for 7 = 4998 s. The set of experimental conditions 7 = 4998 s and kLa = 0.0035 s-' was run 5 times and twice yielded steady-state behavior. The two states (steady and oscillatory), shown in Figure 2f, are probably not a true example of multiplicity. It is more likely that the conditions at the highest mass-transfer coefficient are near a second (probably Hopf) bifurcation. This is suggested by the decreasing amplitude of the oscillations with increasing kLa. The seemingly multiple states were probably caused by uncontrolled variations in experimental conditions near this bifurcation point. (Bistable states in oscillatory chemical reactors are possible, however, and have been observed in other systems (Lamba and Hudson, 1985).) Unfortunately, the nature of the second bifurcation point could not be studied in detail since higher mass-transfer coefficients were not attainable with the experimental system used. Bifurcation diagrams (not shown) have also been constructed as a function of 7 at constant kLa. These show that the bifurcation from steady to oscillatory behavior occurs at smaller values of 7 for larger values of kLa. Typical time traces of oscillating behavior are shown in Figure 3 for mass-transfer coefficients of 0.0025 and 0.0032 s-l a t two residence times, 264 and 4998 s. Transients ranging from 2 to 4 'Iz h were typically seen before the
N
h
t
t
-
rN e
0 0015
0 0040 kLa (s.c.])
:
2
3
0 0040
0 0015
kLa /set') w
->
I
E
+
?I 3
0 0015
0 0040 kLa (set.')
i
I +
:
I
/ I '
-2 0.0015
0 0040 k L a (set.')
Figure 2. Experimental bifurcation diagrams plotted as platinum electrode potential versus kLa. (+) Stable steady state; (I) oscillatory range. 7 = (a) 264 s, (b) 480 s, ( c ) 1728 s, (d) 2700 s, (e) 3600 s, and (f) 4998 s.
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989 593 300
T I
1
i
0 0.0020
0.0036
kLa
(9-1)
Figure 4. Period of oscillation versus kLa for experimental data. T = (0) 264 s, (A)480 s, (X) 1728 s, ( 0 ) 2700 s, (Y)3600 s, and (*) 4998 S.
experimental range of parameters studied. The period of oscillation generally increases with T at constant kLa. Time jsec)
Mathematical Model In this section, we investigate a mathematical model based on the mechanism of Roelofs et al. (1983) for the oxidation of benzaldehyde. The overall reaction is 2PhCHO + 0 2 2PhCOzH
iC)
-
in which benzaldehyde is oxidized to benzoic acid. The proposed detailed mechanism involves 10 reaction steps plus the mass transfer of oxygen from the gas phase to the liquid phase: ( C O ~ ++) Br~ e (CO~+)~B~(Co3+),Br- + PhCHO PhCO' PhCO' PhCO'
Figure 3. Time series of experimental oscillations of platinum electrode potential. (a) T = 264 s, kLa = 0.0025 8;(b) T = 264 s, kLa = 0.0032 s-l; (c) T = 4998 s, kLa = 0.0025 8; and (d) 7 = 4998 s, kLa = 0.0032 s-l.
oscillations settled down to a reasonably constant amplitude such as those seen in the figure. The oscillations are all one peak periodic, but slight variations in their shape can be seen. Oscillations at the smaller value of T (264 s) appear more harmonic in shape, whereas a t 7 = 4998 s (especially at the larger value of kLa), the oscillations more closely resemble relaxation oscillations in which the potential drop from maximum to minimum potential occurs very rapidly. Figure 4 shows the period of oscillation as a function of kLa for the six liquid residence times studied. The lines represent quadratic fits of the data points for each residence time. At the smallest residence times, the period remains fairly constant throughout the range of kLa. For intermediate values of 7 (1728 and 2700 s), the period initially increases with kLa and then levels off. At the largest residence times, the period reaches a maximum value at small kLa and then decreases as kLa is increased further. Thus, the maximum period of oscillation appears to occur at smaller kLa for larger residence times in the
Br-
+ Co3+ + Co2+ + H+ (R2)
+ 02(1)
+ ( C O ~ ++) H~ 2 0 PhCO'
Time ( s e c )
-+
(R1)
-
-
PhCO3'
PhC02'
+ -
+ PhC03'
(R3)
+ 2C02++ 2H+ (R4)
2PhC02'
(R5)
+ PhCHO PhCO3H + PhCO' (R6) PhCO,' + Co2+ H+ PhC03H + Co3+ (R7) PhC02H + PhCO' (R8) PhC02' + PhCHO PhC03H + 2C02+ + 2H+ PhC02H + ( C O ~ ++) H ~ 20 PhC03'
-
-
(R9)
co3+ + C O ~ + ( C O ~ + ) ~
-
020) (R1U Other reaction mechanisms have been proposed to describe the two-phase oxidation of benzaldehyde. In a more recent paper, Roelofs et al. (1987) propose a similar, but more detailed, reaction mechanism involving 20 steps, which accounts for four additional reaction intermediates. In addition, Yuan and Noyes (1988) have recently developed an alternate free-radical mechanism based on nine elementary processes with five independent composition variables. A description of the complete mechanism of the reaction is not yet available. Although the earlier mechanism proposed by Roelofs et al. (1983) may not be an exact description of the reaction, simulations based on it do show reasonable agreement with experiment. Thus, this mechanism will serve as a reasonably simple starting point 02(d
594
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989
for the modeling of the oscillatory behavior. The rates of the reaction steps in the Roelofs mechanism are R1 = -kl[(Co3+)2][Br-]+ ~ - , [ ( C O ~ + ) ~ B ~ - ] R2 = k2[( C O ~ + ) ~ [PhCHO] B~-]
d[PhC03'] = R3 - R5 - R6 - R7 dt d[PhC02'] = R4 dt
+ 2R5 - R8 -
d[PhCO'] = R2 - R3 - R4 - R5 dt
R3 = k,[PhCO'] [02(1)]
[PhCO,'] (14
T
[PhC02']
(le)
T
[PhCO']
+ R6 + R8 - T
(If)
R4 = k,[HaO] [PhCO'][(Co3+)2] d[PhCOsH] [PhCOBH] = R6 + R7 - R9 dt T d[Co3+] [C03+] -- R2 + R7 - 2R10 - dt T with the additional algebraic equations
R5 = k,[PhCO*] [PhCO3'] R6 = k,[PhCO3'] [PhCHO] R7 = k7[H+][PhC03'][Co2+] R8 = k,[PhCOz'] [PhCHO]
(1g) (lh)
R9 = kg[H+I2[PhCO3H][Co2+I2 R10 = ~ , , , [ C O ~ + ] ~ R11 = kii[Oz(g)l - k-11[02(1)1 = k-11([02(1)1*- [ 0 2 ( 1 ) 1 ) = k~a([O,(l)l*- [02(1)1) where [O,(l)]* is the solubility of oxygen in solution. The mass transfer of oxygen from the gas phase to the liquid phase is governed both by the solubility constant of oxygen in the acetic acid/water mixture ([02(1)]*= 6 x M at 70 "C and po, = 1atm) and the product of the liquid-phase mass-transfer coefficient, kL, and the surface area-tovolume ratio, a. It is assumed that there is no enhancement of mass transfer due to reaction. Several assumptions were made prior to the development of the mathematical model of the system. Roelofs et al. (1983) assumed that the concentrations of H 2 0 and H+ are constant and can be incorporated into the rate constants. Furthermore, it is assumed that the first reaction is in equilibrium (R1 = 0). Thus, the concentration of [(Co3+)*Br-]can be calculated from kl
[( C O ~ + ) ~ B = ~- ] [ (Co3+I2][Br-] k-1
and the rate of the second reaction can be written as klk2 R2 = ,-[(CO~+)~][B~-][P~CHO] H-1
The values of the rate constants are given a t 70 "C by Roelofs et al. (1983) as klk2/k-l = 0.25 W2s-', k-1 >> kz, k , = 2 x 108 M-1 k4 = 2 x 104 M-1 k 3 = 1 x 108 M-' s-', k , = 1 X lo4 M-' s - ~ ,k , = 8 X lo4 M-' s-', k , = 1 X lo5 klo = 1 x 106 ~ - s-1 1 , k -11 = ~ - 1 kg = 1 x 109 M-2 s-l, M (atm of 0J1. The s-l, and kll/k-ll = 6 X 1x values of k,, k,, and kg given here include the assumed constant concentrations of [H+]and [H20]. For this study, an equilibrium constant of k,/k-l = 12.35 M was used. The mathematical model for the oxidation of benzaldehyde consists of the following eight ordinary differential equations written for the chemical system in a continuous flow reactor:
(14
d[PhCHO] at
= -R2 - R6 - R8
+
[PhCHOIf - [PhCHO] T
(Ib)
[C02+] = 2([(C03+)2]f- [(C03+),]) + 2([(Co3+),Br-],- [(Co3+),Br-])- [Co3+] (2b) Equations 2a and 2b result from the conservation of total bromide and total cobalt, respectively, in the reactor. The subscript f denotes feed concentrations, with [Br& defiied as the total concentration of bromide in the feed in all forms. The feed contains ( C O ~ +(Co3+),Br-, )~, Br-, PhCHO, and H20. The liquid residence time is denoted by T. The set of eight ordinary differential equations was solved as a function of the same two parameters varied experimentally: the liquid residence time and the liquidside mass-transfer coefficient multiplied by the surfacearea-to-volume ratio. The equations were integrated on the CDC Cyber 855 system using the IMSL subroutine DGEAR (Gear's method). The results of simulations for ranges of conditions of 100 s Ir 5 5000 s and 0.0 s-l I kLa 5 0,005 s-l are reported here. Other parameters were chosen as [(Co3+),lf= 0.009 M, [PhCHOIf = 0.75 M, [Br-l, = 0.0072 M, [(Co3+),Br-If = 0.0008 M, po, = 1 atm, and [O,(l)]* = 6 X M. The results of the simulations shall be compared qualitatively to those of the experiments; exact quantitative comparisons are not possible since the experiments were done at 52 "C, and the reaction rate constants of the model were obtained a t 70 "C. Nevertheless, the comparison will show similar trends in the model and experiments. Typical bifurcation diagrams are presented in Figure 5 , in which log [(CO~')~] is shown as a function of kLa for two values of r. In the parameter range under consideration, there is a bifurcation from steady-state to oscillatory behavior with an increase in either r or kLa. For r = 100 s (Figure 5a), a Hopf bifurcation occurs at kLa = 0.0037 s-' and the system is oscillatory for larger values of the parameter. A bifurcation diagram for a larger residence time, T = 4980 s, is shown in Figure 5b. The Hopf bifurcation occurs at kLa = 0.00056 s-l. These results and those of additional studies not presented here show that the model predicts that the Hopf bifurcation occurs at smaller values of kLa as r is increased and at smaller values of T as k,a is increased. This is consistent with the experimental results. In addition, the eight-variable model predicts a second bifurcation back to steady state a t larger values of kLa than are shown here. This strongly supports the experimental evidence of the suspected second bifurcation point at large kLa for r = 4998 s. Examples of the oscillations of [(Co3+),] for r = 100 s and r = 4980 s are shown in Figure 6. These oscillations
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989 595
' 800
Time
\ SPC
1000
,
=u 7
0.005
0.C
800.
1000.
Time (sec)
k,a (wr 1 )
Figure 5. Bifurcation diagrams of log [(Co3+),] versus kLa calculated from the eight-variable model. (-) Stable steady state; (- - -) unstable steady state; (-O-)maxima and minima of oscillations. T = (a) 100 s and (b) 4980 s.
are harmonic over the range of kLa investigated at the smaller residence times and for kLa values near the bifurcation point a t all residence times. They change to relaxation oscillations a t large values of both T and kLa. The time traces also show the increased period and amplitude of oscillation at the larger values of residence time. Again, these variations of the oscillation shape and period are similar to those seen experimentally, although the effect is much more pronounced in the numerical results. Figure 7 shows the variation of the oscillation period with the mass-transfer coefficient for seven values of the residence time. The solid lines indicate the results found with the eight-variable model. The values of these numbers compare quite favorably with those found experimentally. All periods, whether determined experimentally or numerically, fall within the range of 50-300 s. The qualitative trends are also similar. At small values of the residence time (7 5 264 s), the period is small and remains fairly constant with variations of kLa. The period first increases and then levels off with increasing kLa for the intermediate 7 = 720 s. At larger residence times, the length of the oscillatory period reaches a maximum and then decreases as kLa is increased. The maximum is more pronounced a t larger residence times. In addition, the period increases in length.with residence time for any value of the mass-transfer coefficient. All of these trends follow what was seen experimentally.
Simplified Model The eight-variable model described in the previous section can be simplified by making several standard assumptions. First, we assume that the concentration of benzaldehyde in the reactor is constant and equal to its feed concentration. As predicted by the eight-variable model, the benzaldehyde concentration varies very little during an oscillation of the other species. This is thus a good approximation except at large values of both T and kLa where there is significant depletion of benzaldehyde. Under the latter conditions, the concentration of benzaldehyde is still relatively constant but is no longer equal to its value in the feed. Second, it was assumed that the concentration of Co3+is zero since, as predicted by the
-
0
P
19000'
Time (sec)
19000'
Time [sec)
0 7
2Oooo.
1
2Oooo.
Figure 6. Time series of oscillations in log [(Co3*),] from the eight-variable model. (a) T = 100 s, kLa = 0.004 s-l; (b) T = 100 s, kLa = 0.005 s-l; (c) T = 4980 s, kta = 0.001 s-l; and (d) T = 4980 s, kLa = 0.005 s-l. 300
r
-
h
m
-2 .-
E
0.0
0.005 kLa
(5-1)
Figure 7. Period of oscillation versus kLa from mathematical models. (-) Eight-variable model; (- - -) three-variable model (eq 100 s, (A) 264 s, (+) 720 8, (X) 1728 s, ( 0 ) 2700 s, (Y) 5a-c). T = (0) 3600 s, and (*) 4980 s.
eight-variable model, the concentration of eo3+is about 3 orders of magnitude less than that of ( C O ~ +Last, ) ~ the
596 Ind. Eng. Chem. Res., Vol. 28, No. 5 , 1989
pseudo-steady-state approximation was applied to perbenzoic acid (PhC03H) and the free radicals PhC0,' and PhC02', since their concentrations are relatively small. An approximation of this type has been made previously in an analysis of oxidation reactions involving radicals in gas-liquid reactions (Morbidelli et al., 1986). The simplification was performed as a series of steps resulting in a succession of mathematical models, each with one less variable than its predecessor. A linearized stability analysis performed on each subsequent model served as a guide to which variable eliminations were valid. This analysis gives eight nonzero eigenvalues for the eightvariable model and one fewer eigenvalue for each subsequent model. Eigenvalues were calculated by using IMSL subroutine EIGRF. The two negative real eigenvalues with smallest absolute values disappear upon making the assumption of constant benzaldehyde concentration and eliminating the cobalt species Co3+. This result is expected since these two variables change only slightly and slowly during oscillations. The three negative eigenvalues with the largest absolute value are eliminated when the pseuc!o-steady-state approximations are made on PhCO,H, PhCO,', and PhC02'. These latter three components are fast variables in the eight-variable model. They are associated with the large negative eigenvalues which in turn correspond to directions of rapid approach to a steady state or limit cycle. Thus, elimination of these components as variables does not change the stability or behavior of the system greatly. With the above assumptions, the mathematical model is reduced to three differential equations in three variables: 02(1), PhCO', and ( C O ~ + )These ~. three are convenient variables with which to model the system. Oxygen is the species being transferred from the gas phase to the liquid phase, where it reacts. The free radical (PhCO') reacts with the oxygen according to mechanism R3. Finally, some form of the cobalt catalyst should be maintained. ( C O ~ ' ) ~ was chosen because it is present in relatively large amounts according to the eight-variable model and, for this model, is used as a feed variable. The three resulting differential equations are
-
IJOU U
,r,",,
V I O I i 11
Figure 8. Time series of oscillations in (a) A = [O,], (b) B = [ ( C O ~ + )and ~ ] , (c) C = [PhCO'] from the three-variable model (eq 3a-c) with T = 4980 s and kLa = 0.005 s-l.
a subsequent study of the effect of chemical reaction on the absorption and diffusion of oxygen in this reaction system. The form of eq 3 can be further simplified by means of several additional assumptions. The bromide ion concentration can be held constant at its value in the feed, [Br-] = [Br-1, = 0.0072 M. Compared to other species involved in the reaction, the bromide ion concentration does not vary significantly during an oscillation and remains nearly equal to the feed concentration at all conditions studied. The cobaltous ion concentration is also assumed constant at [Co2+]= 0.013 M, which is obtained by assuming [B] = 1/3[(Co3+)2]f= 0.003 M in eq 4d. An additional simplification of the model is possible. In order to aid in presenting this simplification, the variables [OZ],[(Co3+)~], and [PhCO'] (A, B, and C) are shown as functions of time in Figure 8 for conditions which yield relaxation oscillations. These were calculated by using eq 3; almost identical results are obtained with the full , I [B] eight-variable model. Especially note the form of variable - = -k,[B][C] + k7[Co2+][PhC03*]+ rlt __ C. Its value is near zero for much of the cycle (region I), [Blf - [BI (3b) with concentration spikes (in region 11) between the ink,[PhCHO][PhCO3'] + T tervals of near-zero concentration. Attempts to reduce the system to two equations were not successful. For example, d[CI klk, a pseudo-steady-state approximation on variable C will not - = -[PhCHO][B-][B] - k,[A][C] + dt k-, work since, although its value is small for most of the cycle (region I), its time rate of change is large during the in[CI k,[PhCO,'][C] + k,[PhCHO][PhCO,'] - - ( 3 ~ ) tervals of region I1 where, in fact, the value of C becomes T larger than that of A. Nevertheless, the form of eq 3 can be simplified somewhere [AI = [02(1)],[B] = [ ( C O ~ + )[C] ~ ] ,= [PhCO'], and what further. In region I, the sum of the four reaction k3[AI [CI terms on the right-hand side of eq 3c can be set equal to [PhC03'] = (44 zero, and this equation can be rearranged to yield an exkS[C] + k,[PhCHO] + k7[Co2'] pression for [PhCO,']. This expression can be substituted into eq 3b. In addition, however, this substitution can be made in region I1 since the first term on the right-hand side of eq 3b dominates in this region so that any error introduced by the substitution is minimized. The sub[ ( C O ~ + ) ~ B=~( -k]1 / k - , ) [B][Br-1 (4c) stitution and a mathematical rearrangement of the terms - [ ( C O ~ + ) ~ B ~ - ] ) in eq 3c yields the following set of differential equations: [Co2+]= 2([BIf - [B]) + 2([(C0~+)~Br-l, (44 and [PhCHO] = 0.75 M. Equations 3a-c will be used in
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989 597 d[BI dt
- -k,[BI[CI + Ki(1 + Kz)[Bl +
P I , - [BI 7
(a)
?
K,[AI[CI - -[CI 7 dt - K1[B1 - [C] K4
-d[CI -
+
K1-K4 are defined in the following manner: klkZ K1 = -[PhCHO][Br-] = 0.00135 s-l k-1 k,[PhCHO] Kz = = 7.175 k7[Co2+]
k3k7 K3 = -[[Cozf]
= 522.7 s - ~
k5
K4 =
k,[PhCHO]
+ k7[Co2+]
= 4.273 X
k5
which were obtained from [PhCHO] = 0.75 M, [Br-] = 0.0072 M, and [Co2+]= 0.0130 M. Equations 5a-c constitute a simplified, tractable model for the benzaldehyde oxidation reaction. They yield oscillations in all three variables similar to those shown in Figure 8 for the same retictor conditions. Although a two-variable model would, in principle, yield all the simple dynamics seen in this system, viz., steady state, Hopf bifurcations, and single peak periodic oscillations, no such two-variable model could be found based on the original chemical mechanism. Equations 5a-c have two features sufficient to produce oscillations in even a two-variable system, the autocatalytic term K,( 1 + K2)[B] in eq 5b and the term -(K3[A][C])/([C] + K4) in eq 5c. Bifurcation diagrams of [ ( C O ~ + )versus ~ ] kLa obtained from the three-variable model (eq 5) are shown in Figure 9 and typical oscillations in Figure 10 for residence times of 7 = 100 and 4980 s. (Equations 3a-c give very similar results.) The bifurcation diagrams show that the simplified three-variable model and the eight-variable model predict the Hopf bifurcation points a t nearly the same values of kLa (at kLa = 0.0037 for 7 = 100 s and at kLa = 0.00046 for 7 = 4980 s from the three-variable model). The three-variable model gives almost identical results with the eight-variable model in every respect at small residence times. This can be seen from a comparison of the bifurcation diagrams in Figures 5a and 9a and the time-dependent behavior in Figures 6a,b and 10a,b (for 7 = 100 s). At larger residence times, there is some difference between the two models, particularly a t larger values of kLa as seen in Figures 6c,d and 10c,d. The oscillations predicted by the three-variable model are slightly larger and somewhat different in shape than those predicted by the full model. However, the three-variable model still shows a change in the oscillation shape as 7 and kLa are varied (becoming more like relaxation oscillations at large values of both 7 and kLa). The differences between the two models a t large 7 and kLa are mainly caused by the depletion of benzaldehyde not accounted for by the three-variable model. For example, at 7 = 4980 and kLa = 0.0050 s-l, the eight-variable model predicts that the concentration of benzaldehyde oscillates between 0.483 and 0.490 M, whereas the three-variable model assumes that it is equal to the feed value, 0.75 M. Thus, the differences between the two models at these conditions are understandable. The periods of the oscillations predicted by the threevariable model are shown as the dashed lines in Figure 7 . They show the same trends as the eight-variable-model
= v
'
0 005
00
k,a
(YK
'I
Figure 9. Bifurcation diagrams of log [(Co3+),] versus kLa calculated from the three-variable model (eq 5a-c). (-1 Stable steady state; (- - -) unstable steady state; (@-) maxima and minima of oscillations. T = (a) 100 s and (b) 4980 s.
results. The period generally increases with either increasing 7 or kLa. However, the three-variable model exhibits a maximum in the period versus kLa curve only for the largest residence time in the parameter range shown. Maxima in period for several of the other residence times (those closest to 4980 s) are still predicted by the threevariable model, but they are shifted to larger values of kLa than are shown here. Concluding Remarks The experimental study shows that stable steady states and one-peak periodic oscillations are possible in the benzaldehyde oxidation reaction in a two-phase continuous reactor. In the experimental range studied, stable steady-state behavior is found only at the smallest absorption rates of oxygen. Larger values of both experimental parameters, the liquid residence time and the liquid-side mass-transfer coefficient, favor oscillatory behavior. The results predicted by the eight-variable mathematical model agree qualitatively in nearly every respect with those found experimentally. A Hopf bifurcation from steady state to oscillatory behavior is predicted with either increasing 7 or kLa. The bifurcation point shifts to smaller values of kLa when 7 is increased and to smaller values of 7 when kLa is increased. Variations in the period and amplitude of oscillations with the two system parameters also agree with those seen experimentally. The numerical and experimental results cannot be quantitatively compared, since the experiments were run at a temperature of 52 "C and the rate constants of the model are given at 70 "C. However, this study shows that the reaction mechanism proposed by Roelofs et al. (1983) gives a reasonable description of the benzaldehyde oxidation reaction. The three-variable model appears to be a reasonable simplification of the eight-variable one. The simpler model predicts essentially the same steady-state behavior and, more importantly, the same locations of the Hopf bifurcations to oscillations as the eight-variable model. It also gives similar variations in the shape and period of the oscillations over the parameter range studied.
598
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989 3 %
-
Subscripts f = concentration i n the feed T = total concentration of component in all forms Registry No. Benzaldehyde, 100-52-7; cobaltous acetate, 7646-79-9; sodium bromide, 7647-15-6.
Literature Cited
-_
= , 190~10
rlnlc
2l)000 iii(
Figure 10. Time series of oscillations in log [(Co3+),] from the three-trariable model (eq 5a-c). (a) T = 100 s, kLa = 0.004 s-l; (b) T = 100 s, kLa = 0.005 s-l; (c) 7 = 4980 s, kLa = 0.001 s-l; and (d) T = 4980 S, k,a = 0.005 s-'.
Acknowledgment This work was supported in part by Grant CBT-8713070 from the National Science Foundation.
Nomenclature a = reactor surface-area-to-volume ratio ki = rate constant of ith reaction step in Roelofs' mechanism k , = liquid-side mass-transfer coefficient K i= ith constant i n three-variable model PO, = partial pressure of oxygen R i = reaction rate expression of the ith reaction step t = time Greek Symbol T = liquid residence t i m e Superscript * = saturation concentration (solubility)
Bawn, C. E. H. Free radical reactions in solution initiated by heavy metal ions. Discuss Faraday SOC. 1953, 14, 181-190. Belousov, B. P. Sbornik Referatov PO Radiatsionnoi Meditsine, 1958 (Collections of Abstracts on Radiation Medicine). A Periodic Reaction and its Mechanism; Medgiz: Moscow, 1959. Bray, W. C. A periodic reaction in homogeneous solution and its 1921, 43, 1262-1267. relation to catalysis. J . Am. Chem. SOC. Burger, M.; Field, R. J. A new chemical oscillator containing neither metal nor oxyhalogen ions. Nature 1984, 307, 720-721. Danckwerts, P. V.; Gillham, A. J. The design of gas absorbers: I. Methods for predicting rates of absorption with chemical reaction in packed columns, and tests with 1-1/2 in. Raschig rings. Trans. Inst. Chem. Eng. 1966, 44, T42-T54. Davies, J. T.; Kilner, A. A.; Ratcliff, G. A. Effect of diffusivities and surface films on rates of gas absorption. Chem. Eng. Sei. 1964, 19, 583-590. Ding, J. S. Y.; Sharma, S.; Luss, D. Steady-State Multiplicity and Control of the Chlorination of Liquid n-Decane in an Adiabatic Continuously Stirred Tank Reactor. Znd. Eng. Chem. Fundam. 1974, 13, 76-82. Goodrige, F.; Bricknell, D. J. Interfacial resistance in the carbon dioxide-water system. Trans. Znst. Chem. Eng. 1962,40, 54-60. Hancock, M. D.; Kenney, C. N. The stability and dynamics of a gas-liquid reactor. Chem. Eng. Sei. 1977, 32, 629-636. Hendriks, C. F.; VanBeek, H. C. A,; Heertjes, P. M. Autoxidation of Aldehydes in Acetic Acid Solution. Ind. Eng. Chem. Prod. Res. Dei;. 1977, 16, 270-275. Hendriks. C. F.: VanBeek. H. C. A,: Heerties. P. M. The Kinetics of the Autoxidation of Aldehydes in the Presence of Cobalt(I1) and Cobalt(II1) Acetate in Acetic Acid Solution. Znd. Eng. Chem. Prod. Des. Deu. 1978, 17, 260-264. Hoffman, L. A,; Sharma, S.; Luss, D. Steady state multiplicity of adiabatic gas-liquid reactions: I. The single reaction case. AIChE J . 1975, 21, 318-326. Huang, D. T . J . ; Varma, A. Steady-state and dynamic behavior of fast gas-liquid reactions in non-adiabatic continuous stirred tank reactors. Chem. Eng. J. 1981a, 21, 47-57. Huang, D. T.-J.;Varma, A. Steady-state uniqueness and multiplicity of nonadiabatic gas-liquid CSTRs. Part 1: Second-order reaction model. AIChE J . 1981b, 27, 481-489. Jensen, J. H. A New Type of Oscillating Reaction: The Air Oxidation of Benzaldehyde. J . Am. Chem. SOC.1983,105, 2639-2641. Jensen, J. H.; Roelofs, M. G.; Wasserman, E. The mechanism of the oscillating air oxidation of benzaldehyde. In Non-Equilibrium Dynamics in Chemical Systems. Proceedings of the International Symposium, Bordeaux, France, 1984. Kuo, M.-C.: Chou, T.-C. Benzaldehyde oxidation catalyzed by the wall of a tubular bubble column reactor. AZChE J . 1988, 34, 1034-1038. Lamba, P.; Hudson, J. L. Experimental evidence of multiple oscillatory states in a continuous reactor. Chem. Eng. Commun. 1985, 32, 369-375. Liebhafsky, H. A. Reactions involving hydrogen peroxide, iodine and iodate ion. 111. The reduction of iodate ion by hydrogen peroxide. J . Am. Chem. Soc. 1931a, 53, 896-911. Liebhafsky, H. A. Reactions Involving Hydrogen Peroxide, Iodine and Iodate Ion. IV. The Reduction of Iodine to Iodate Ion by Hydrogen Peroxide. J . Am. Chem. SOC.1931b, 53, 2071-2090. McGinnis (Massie), J. M. Oscillations in the oxidation of benzaldehyde in a gas-liquid reactor. Doctoral dissertation, University of Virginia, Charlottesville, 1987. Morbidelli, M.; Paludetto, R.; Carra, S. Gas-liquid autoxidation reactors. Chem. Eng. Sci. 1986, 41, 2299-2307. Raghuram, S.; Shah, Y. T. Criteria for unique and multiple steady states for a gas-liquid reaction in an adiabatic CSTR. Chem. Eng. J . 1977, 13r81-92. Razhuram. S.: Shah. k'.T.: Tiernev. J. W. Multide steadv states in gas-liquid reactor. Chem. Eng. J . 1979, 1?, 63-75. Rastogi, R. P.; Das, I. Chemical oscillations in aerial oxidation of benzaldehyde and acetaldehyde. Indian J . Chem. 1984, 23A,
a
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Roelofs, M. G.; Wasserman, E.; Jensen, J. H.; Nader, A. E. Mechanism of an oscillating organic reaction: Oxidation of benzaldehyde
Ind. Eng. Chem. Res. 1989, 28, 599-608 with O2 catalyzed by Co/Br. J . Am. Chem. SOC. 1983, 105, 6329-6330. Roelofs, M.G.; Wasserman, E.; Jensen, J. H. Oscillations and complex mechanisms: O2 oxidation of benzaldehyde. J.Am. Chem. SOC.1987,209, 4207-4217. Schmitz, R. A.; Amundson, N. R. An analysis of chemical reactor stability and control-Vb (Two-phase gas-liquid and concentrated liquid-liquid systems in physical equilibrium-2). Chem. Eng. Sci. 1963,18, 391-414. Suresh, A. K.; Sridhar, T.; Potter, 0. E. Autocatalytic oxidation of cyclohexane-modeling reaction kinetics. AIChE J . 1988a,34, 69-80.
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Suresh, A. K.; Sridhar, T.; Potter, 0. E. Autocatalytic oxidation of cyclohexane-mass transfer and chemical reaction. AZChE J . 1988b,34, 81-93. Yuan, Z.; Noyes, R. M. An alternative mechanistic explanation for the oscillatory catalyzed oxidation of benzaldehyde by air. Preprint, 1988. Zhabotinskii, A. M. Periodic course of oxidation of malonic acid in solution (Investigation of the kinetics of the reaction of Belousov). Biofitika 1964,9, 306-311.
Received for review J u n e 23, 1988 Accepted December 19, 1988
Thermodynamic Analysis for Rapid Measurements of Equilibrium Adsorption from Binary Gas Mixtures James A. Ritter and Ralph T. Yang* Chemical Engineering Department, State University of New York at Buffalo, Buffalo,New York 14260
Simple numerical and graphical procedures, based on an exact thermodynamic analysis (ETA), were developed t o determine the experimental adsorbed-phase mole fractions of binary gas systems adsorbed a t elevated pressures. Well-defined pure-gas isotherms were required along with the total amount adsorbed of a few data points from mixed-gas constant gas-phase composition isotherms. T h e procedures were tested with binary data consisting of only two or three mixed-gas isotherms containing between three and five data points each. The results from the numerical ETA compared very well with the binary data. The average absolute difference in the adsorbed-phase mole fractions for 69 mixed-gas data points was 0.02. Features of these ETA procedures are as follows: (1) Low-pressure mixed-gas data are not required. (2) The effects of gas-phase fugacity can be determined a priori with little effort. (3) For ideal gas and also for real gas, an iterative procedure is not necessary; a rapid graphical procedure can be used. (4) T h e binary data can be checked for thermodynamic consistency in three ways. It has been shown that classical thermodynamics can be used to determine the adsorbed-phase composition for a binary mixture when only the total amount adsorbed is measured (Van Ness, 1969). However, as Van Ness (1969) noted, the exact thermodynamic analysis (ETA) presented the experimentalist with a formidable problem because it required mixed-gas isotherms measured at constant gasphase composition. As a result, this thermodynamically consistent method has received little attention in the literature (Friederich and Mullins, 1972; Gravelle and Lu, 1978; Myers et al., 1982). Also, in addition to the experimental difficulty mentioned by Van Ness (1969), Sloan and Mullins (1975) and Myers (1986) pointed out that numerous pure- and mixed-gas data points are required to accurately evaluate the adsorbed-phase composition by the ETA. Gravelle and Lu (1978) and Myers et al. (1982) applied the ETA, as presented by Van Ness (1969) for ideal gas, to binary adsorption data measured by a gravimetric method. Friederich and Mullins (1972) designed this gravimetric method exclusively for the ETA, and they also modified the ETA for application to real gas. In all cases, however, adsorption experiments were carried out below atmospheric pressure where the ideal-gas assumption was valid, and the ETA developed by Friederich and Mullins (1972) was derived with an unusual standard state, which unnecessarily complicated their analysis. Sloan and Mullins (1975) and Myers (1986) both presented gravimetric thermodynamic analyses that required far less mixed-gas data for the accurate evaluation of binary adsorption equilibria. The former study actually obtained the experimental mixed-gas information by a chromatographic technique. Although these methods are 0888-58851891 2628-0599$01.50/0
thermodynamically consistent and seem quite plausible, they are not exact in the sense that assumptions are involved. For the former study, the chromatographic information is used to correct an equation of state (EOS), and an empirical activity coefficient model is assumed in the latter study. This raises a question of whether the adsorbed-phase mole fractions determined from these techniques are experimental quantities or predictions from thermodynamically consistent models. Furthermore, the analyses are quite complicated and Myers (1986) points out that his method can require an excessive amount of computer time. This paper introduces relatively simple procedures based on the ETA that require few mixed-gas data points and that obviate the need to measure low-pressure binary adsorption equilibria if the two pure-gas isotherms are well defined at low pressures. Moreover, these ETA procedures involve no assumptions; thus, the adsorbed-phase mole fractions, in addition to being thermodynamically consistent, are true experimental quantities. A volumetric flow-desorption (VFD) method (Reich et al., 1980) is also described in this paper that can be applied to the ETA at elevated pressures. Hence, the equations formulated by Van Ness (1969) for ideal gas are modified here for real gas in a way similar to those developed by Friederich and Mullins (1972). However, when a standard state is selected that is different from that employed by Friederich and Mullins (1972), the analysis is greatly simplified. Numerical and graphical ETA procedures are presented which are valid for both ideal- and real-gas systems. They are applied to three binary gas systems adsorbed at elevated pressures on BPL activated carbon a t 212.7 and 301.4 K (Reich et al., 1980). The results from the ETA 0 1989 American Chemical Society
