Ind. Eng. Chem. Res. 1988, 27, 233-237
cylinders, and rings are almost the same for a given particle size and flow rate. For a more detailed calculation, the effects of LIPID,and variable Pe, on h, can be accounted for by using eq 14 and 15. Both approaches predict a maximum h, at D,/D, N 0.15 in agreement with the data. Acknowledgment The authors appreciate the support given by the Monsanto Company for the last stage of this work. Nomenclature
Bi = Biot number, h $ / k , c = heat capacity of fluid
Ifeq= equivalent particle diameter
D, = particle diameter D, = radial diffusivity D, = tube diameter G = mass velocity hbd = heat-transfer coefficient for packed bed ho = overall bed coefficient h, = wall film coefficient k , = effective thermal conductivity k,O = effective conductivity at zero flow k f = thermal conductivity of fluid L = axial distance Nu, = Nusselt number, h&/kf Peh = Peclet number for radial heat transfer Pe, = Peclet number for radial mass transfer, D,uo/D, Pr = Prandtl number of fluid, c,p/kf r = radial distance R = radius of tube Re, = particle Reynolds number, D,G/w T = temperature T,, = average bed temperature T R = temperature of bed at wall T , = wall temperature uo = superficial velocity x = conversion XF = shape factor in eq 4 z = axial distance Greek Symbols
233
/3 = dimensionless parameter in eq 9 = viscosity
p
Literature Cited Agnew, J. B.; Potter, 0. E. Trans. Znst. Chem. Eng. 1970,48, T15. Bauer, R.; Schlunder, E. U. Znt. Chem. Eng. 1978, 18, 181. Beek, J. Adu. Chem. Eng. 1962, 3, 203. Bernard, R. A.; Wilhelm, R. H. Chem. Eng. Prog. 1950, 46, 233. Bunnell, D. G.; Irvin, H. B.; Olson, R. W.; Smith, J. M. Znd. Eng. Chem. 1949,41, 1977. Coberly, C. A.; Marshall, W. R. Chem. Eng. Prog. 1951, 47, 141. Crider, J. E.; Foss, A. S. AZChE J . 1965, 11, 1012. DeDeken, J. C.; Devos, E. F.; Froment, G. F. ACS Symp. Ser. 1982, 196, 182. DeWasch, A. P.; Froment, G. F. Chem. Eng. Sci. 1972, 27, 567. Dixon, A. G.; Labua, L. A. Znt. J . Heat Mass Transfer 1985,28,879. England, R.; Gunn, D. J. Trans. Znst. Chem. Eng. 1970, 48, T265. Fahien, R. W.; Smith, J. M. AZChE J . 1955, I , 28. Holt, A. D. Perry’s Chemical Engineer’s Handbook, 6th ed.; McGraw-Hill: New York, 1984; pp 10-46. Hofmann, H. Chemical Reactor Design and Technology;deLasa, H., Ed.; Wiley: New York, 1986; p 69. Hyman, M. H. Hydrocarbon Process. 1968,47 (7), 131. Lerou, J. J.; Froment, G. F. Chem. Eng. Sci. 1977, 32, 853. Leva, M. Znd. Eng. Chem. 1947,39, 857. Leva, M.. Znd. Eng. Chem. 1950, 42, 2498. Leva, M.; Weintraub, M.; Grummer, M.; Clark, E. L. Znd. Eng. Chem. 1948, 40, 747. Li, C.; Finlayson, B. A. Chem. Eng. Sci. 1977, 32, 1055. Patterson, W. R.; Carberry, J. J. Chem. Eng. Sci. 1984, 39, 1434. Olbrich, W. E.; Potter, 0. E. Chem. Eng. Sci. 1972,27, 1733. Scheele, G. F.; Hanratty, T. J. J . Fluid Mech. 1962, 14, 244. Scheele, G. F.; Hanratty, T. J. AZChE J. 1963, 9, 183. Schwartz, C. E.; Smith, J. M. Znd. Eng. Chem. 1953, 45, 1209. Schwedock, M. J.; Windes, L. C.; Ray, W. H. Paper presented at Chicago AIChE Meeting, Nov. 1985. Singh, C. P. P.; Saraf, D. N. Znd. Eng. Chem. Process Des. Dev. 1979, 18, 1. Vortmeyer, D.; Winter, R. P. ACS Symp. Ser. 1982, 196, 49. Yagi, S.; Kunii, D. AZChE J. 1957, 3, 373. Yagi, S.; Wakao, N. AZChE J. 1959, 5, 79. Received for review December 9, 1985 Revised manuscript receiued September 21, 1987 Accepted October 8, 1987
Reversible Association and Reaction Rates in Homogeneous Catalysis Chen-Chen
Fu and B e n j a m i n J. McCoy*
Department of Chemical Engineering, University of California,Davis, California 95616
A dynamic, hydrogen gas pulse response method was applied t o the homogeneous hydrogenation of a-methylstyrene with Wilkinson’s catalyst, RhC1(PPh3)3, in a well-stirred bubble reactor. A new mathematical model for the hydrogen concentration in the gas bubbles, dissolved in the liquid, and attached t o the catalyst is proposed. T h e results a t 325 K showed that a n overall rate constant, combining rate constants of reversible association of dissolved molecular hydrogen and the subsequent irreversible reaction, could describe the overall rate. The measured overall intrinsic rate constant for the dissolved-hydrogen, first-order reaction was 0.105 m3/(kgs). T h e catalyst is continuously deactivated, possibly due t o inhibition by a-methylstyrene. Hydrogenation in the liquid phase by homogeneous catalysts has been a subject of active study since 1965 when two groups (Young et al., 1965; Coffey, 1965) reported that
tris(triphenylphosphine)chlororhodium, RhCl(PPh,),, now called Wilkinson’s catalyst, catalyzes the hydrogenation of alkenes and alkynes a t 298 K and 1 atm. A variety of organometallic compounds are known to be hydrogenation catalysts (James, 1973). Kinetic investigations (Osborn et al., 1966; Jardine et al., 1967) showed that hydrogenation of heptene, cyclo0888-5885/88/2627-0233$01.50/0
hexene, and hexyne with Wilkinson’s catalyst in a benzene solution is first-order in hydrogen and between zero- and first-order in substrate. The reaction was also found to be first-order in catalyst concentration up to its solubility limit. Osborn et al. (1966) presented a full rate expression showing that, for large concentrations of olefin, the reaction is first-order in H2and zero-order in olefin. Two reaction paths for the hydrogenation of olefins by Wilkinson’s catalyst were postulated by Osborn et al. (1966). For the first path, called the hydride route, it is 0 1988 American Chemical Society
234 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988
assumed that the hydrogen molecule is reversibly associated with the catalyst to form a dihydro-metal complex. The olefin then attacks the complex to give a transition state in which both hydrogen and olefin are bound to the metal, and the hydride ligands are transferred to the olefinic double bond. In the second proposed reaction mechanism, called the olefin route, the olefin molecule is assumed first to associate reversibly with the catalyst. The hydrogen molecule then attaches to the olefin complex to give the same transition state as in the hydride route. These two reaction mechanisms are outlined as follows: hydride route RhCl(PPh,),
-
+ H 2 + H2RhC1(PPh3), olefin
olefin route RhC1(PPh3), + olefin
RhCl(PPh3), + paraffin
HZ
(olefin)[RhCl(PPh,),] RhCl(PPh,), + paraffin
In the present work, a two-step reaction mechanism, based on the hydride route, was developed. An attempt was made to measure the rates of two processes: reversible association of molecular hydrogen with the catalyst, and the subsequent irreversible reaction to hydrogenate the olefin. Criteria for distinguishing between the alternative mechanisms, based on the measured rate data, are discussed below. The two-step mechanism is familiar in explaining heterogeneous catalysis and has been investigated experimentally by Ahn et al. (1985, 1986) and Chen et al. (1986a,b, 1987). For heterogeneous catalysis, the first step is believed to be a physical, nondissociative adsorption of H2 to the metal catalyst (Madon et al., 1978, Chen et al., 1986a,b). An irreversible dissociative chemisorption follows. For the homogeneous catalyst, the association step for the hydrogen bonding to the metal atom may directly be a reversible dissociative chemisorption. The two-step mechanism is essential for the quantitative description of transient phenomena when neither adsorption nor surface reaction controls the heterogeneous catalytic reaction (McCoy, 1984). Because the dynamic response experiments thus penetrate more deeply into the fundamental processes than do steady-state experiments, such studies can reveal aspects of the underlying kinetics. In steady-state analysis, the rates of adsorption and reaction must be equal. For dynamic systems, on the other hand, measurements of the adsorption and reaction (dissociation) rate constants have been accomplished for systems following linear kinetics. These experimental systems include oxidation of aqueous sulfur dioxide by activated carbon (Ahn et al., 1985; Recasens et al., 1984) and hydrogenation of a-methylstyrene by alumina-supported palladium (Ahn et al., 1986; Chen et al., 1986a,b). The dynamic method holds promise not only for revealing fundamental kinetic processes but also for probing the effects of catalyst properties and solution environment on the reaction. Hydrogenation by a homogeneous catalyst requires that gaseous hydrogen first dissolve in the solvent-catalystsubstrate solution. In the transient case, the rate of mass transfer from gas bubble to liquid cannot be ignored. Hence, in dynamic response analysis, we measure not only the chemical kinetics parameters but also the gas-to-liquid mass-transfer coefficient. This is readily accomplished by analyzing temporal moments of the hydrogen in the effluent gas a t different flow rates. Expressions for the moments are easily obtained from the governing differential equations.
The experiments are performed by introducing a pulse of hydrogen in nitrogen carrier gas into the well-stirred solution of a-methylstyrene and Wilkinson’s catalyst. Cumene is the product of the hydrogenation, and its concentration in the reaction solution is always less than 5%. The hydrogen concentration in the effluent gas is monitored as a function of time. From the response peaks, the zero and first moments are calculated by numerical integration for runs at various gas flow rates. The zero moment provides the same information obtainable from steady-state data, Le., the overall intrinsic rate constant and the mass-transfer coefficient. The first moment provides the additional transient information that would allow separate evaluation of the association and reaction rate constants if neither rate is controlling. In the present work, we propose a mathematical theory for the homogeneously catalyzed, two-step hydrogenation reaction process. The model includes gas-to-liquid mass transfer, in addition to reversible association and irreversible reaction. Moment expressions are derived for the model. Our procedure represents an attempt to apply dynamic response methods, successful for heterogeneously catalyzed reactions, toward probing the kinetic details of the important and interesting homogeneous catalytic reactions.
Theory Ahn et al. (1985) showed that either a plug-flow or well-stirred model for the gas bubbles in the agitated liquid is satisfactory for quantitatively describing the time evolution of gas concentration. For the case of well-mixed gas bubbles, with mass transfer from gas to liquid included, the mass balance equation for the hydrogen concentration, c,(t), is VBVLdcg/dt = Q(cgo - cg) - kLa,VL(cg/H - CL) (1) The mass balance equation for the dissolved hydrogen in the liquid includes mass transfer from the gas bubbles to the liquid and reversible association of H2 with catalyst, dcL/dt = kLaB(c,/H - cL) - gka(C13- ca/K) (2) The mass balance equation for the concentration of the hydrogen associated with the catalyst, c,, accounts for the reversible association and the irreversible reaction, dc,/dt = k,(cL - c,/K) - k,c, (3) based on the condition that the reaction is zero-order in methylstyrene. The initial conditions are cg = CL = c, = 0 (4) and the input to the system is an impulse of hydrogen gas cgo = cow) (5) The three ordinary differential equations are easily solved in the Laplace domain. Expressions for zero and first moments are computed from the relations that follow directly from the definition of the Laplace transform, mo = L - c g d t / L m c g od t = lim 5-0 tg
(6)
and
The results are m, = [4Ka + Kr) + PKr]/[e(Ka
+ Kr) + PKr(1 + KL)] (8) m, = ([t(Ka + Kr) + PKrI2 + tKL[(Ka + Kr)2 + KaPI)/([t(Ka + Kr)
+ KrP(1 + KL)I2)[(VBVL)/Q](9)
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 235 in terms of dimensionless groups KL = ~ L ~ B V L / Q H
mensionless quantities, the impulse response is
Kr = k,VBVL/Q
(11)
Ka = k,VBVL/KQ
(12)
P = k,oVBVL/Q
(13)
E
= ~L~BVBVL/Q
xg = clr1e-'l8 + c2r2e-@+ c3r3e-@
(10)
(21)
where xg
= cg/cgo
(22)
(14)
The zero moment can be rearranged into a convenient form for analyzing experimental data,
-1 - (1 + l/KL) + (l/Kk, + l/k,)QH/aVL 1 - mo
The dimensionless group KL is independent of the gas flow rate Q since aBis proportional to Q. Therefore, one expects a straight line if the inverse of the conversion, 1/(1- mo), is plotted versus QH/(VLU). The slope of the line provides the overall rate constant, ko = l/(l/k, + l/Kk,) (16) The intercept of the plot gives KL and hence the gas-toliquid mass-transfer coefficient, kL. As the zero moment represents the unconverted fraction of reactant, it is not affected by the dead volume (volume of inlet and exit tubing, gas sparger, and headspace above the liquid). However, the dead volume effect cannot be avoided in the measurement of the first moment, which is the time delay for the response pulse. The ice trap in the line between the reactor and the thermal conductivity cell caused a temperature change and a different flow rate out of the ice trap. In our experiments, the dead time between the trap and the detector cell was subtracted from measured first moments. The remaining first moment (reduced) can be written as mi/mo + v d / Q and with the dead volume, Vd = Vo - VBVL pio =
n = 1-3 (24)
(15)
(17)
4 = COS-^ ( R / ( - T 3 / 2 ) ) T=
3a2- aI2 9
al = 1 + KL + t
+ P + Ka + Kr u2 = P(l + KL + Kr) + t(1 + Ka + Kr) + (1 + KL)(Ka + Kr) u3 = t(Ka + Kr) + PKr(1 + KL) a4 = t(Ka + Kr) + PKr
1 ::
c1 = --r2r3(r2 - r3) - r3[r3 - (1
1 ::
- rl)
{ ::
- r2) - r2[r2- (1
c2 = --rlr3(r3
(25)
(28)
(29) (30) (31)
+ KL)] +
+ r3[r3 - (1 + KL)] -
(18)
combined with eq 8-14 we have
+ vB(1 = vO/ VL + (Q/CVL)2(l/ko2 + o/Kk,2)H(1 - mo)'/mo (19)
pl0(Q/ VL)
As the gas holdup, VB, in the liquid is relatively small, the term VB(l - mo)in eq 19 can be neglected. According to eq 19, a straight line is expected if p ? Q / v ~ is plotted versus ( Q / u V L ) ~ H ( ~mo)2/mo.The slope of this plot yields a value of a/KkI2, from which k, can be evaluated, provided K is known. The intercept gives Vo/ VL, which can be compared with direct measurements of volumes for the apparatus. To evaluate k, and k , separately from the moments of the experimental data requires that the association equilibrium constant, K, be known. Knowledge of K as a function of temperature also allows one to determine the heat of the reversible association step. In principle, K could be determined from second moments, which, unfortunately, are not sufficiently accurate for this purpose. For reversible association (no reaction), eq 19 reduces to the expression (20) [ v o / V ~+ (1 + ~ K ) / H ] V L / Q which shows that the slope of a plot of pio versus VL/Q yields a value of K when H and Vo are known. An attractive feature of this system is that the governing first-order differential equations with constant coefficients can be solved analytically to give c,(t). In terms of dipio =
c3 = --rlr2(rl
+ KL)] +
This result, with parameters determined from moment analysis, can be compared to experimental response curve data to confirm the procedure. Experiments The experimental apparatus for the continuous pulse response measurement is pictured schematically in Figure 1. Nitrogen carrier gas flowed through a dryer of anhydrous silica gel. Hydrogen passed through a Deoxo unit (Engelhard) to eliminate residual oxygen and through a dryer. The water bath maintained the reactor a t sufficiently high temperatures to cause significant reaction. The ice trap prevented evaporated cumene or methylstyrene from going into the detector with the effluent gas. The Pyrex reactor was equipped with an eight-bladed, Teflon-coated impeller and eight stationary baffles made of aluminum. The sample loop in the seven-port valve was 3 X lo4 m3 in volume. The signal from the detector was recorded continuously on a strip chart, and points were
236 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 1.4,
VENT
t
16 In
VENT
I
Temp
K
I
32 9 52 106
Intercept
i
I?
1.1
Figure 1. Schematic flow diagram of the apparatus: 1, He cylinder; 2, N2 cylinder; 3,pressure regulator; 4, Deoxo unit; 5, silica gel dryer; 6, shut-off valve; 7, needle valve; 8, flow regulator; 9, rotameter; 10, seven-port valve; 11, constant temperature bath; 12, eight-blade baffle; 13, gas sparger; 14, eight-blade impeller; 15,0.15-m- X 0.1-mdiameter reactor; 16, manometer; 17, soap-bubble flow meter; 18, ice trap; 19, thermal conductivity detector. Table I. Operating Conditions for t h e reactor vol, including tubing and ice trap, m3 sample gas, pulse input carrier gas gas flow rate (1 atm, 298 K), m3/s impeller speed, rev/s pressure catalyst loading, u, kg/m3 vol of a-methylstyrene, m3 dead vol, V,, m3 temp, K
I
325
akgm 3 Slope
11 15
1
I
1
Experiments 1.28 x 10-3 pure hydrogen nitrogen 1.14 X 10”-7.12 8.8 atmospheric 3.2 1.09 x 10-3 0.16 x 10-3 325
1.0
YI
0
I
I
I
5
10
15
QH/(VLu)
x
I 20
I 25
I 30
J 35
IO3, m 3 is)iktl
Figure 2. Zero moments of pulse responses for catalyzed H2 reaction with a-methylstyrene. The overall rate constant determined from the slope is 0.105 m3/(kg.s).
h. Hydrogen response curves for reaction runs were then measured a t different flow rates at 325 K. X
lo4
read off manually for numerical integration to obtain moment values. The technical grade a-methylstyrene and cumene (Eastman Kodak) had purities of 98%; contaminants were predominantly aromatic homologues. The catalyst, RhCl(PPh,), (Aldrich), was 99.99% pure. Operating conditions for the reaction and association runs are given in Table I. Preliminary measurements of the conversion at different hydrogen gas concentrations confirmed that the reaction is first-order in hydrogen. At the very high concentrations of a-methylstyrene, the reaction is zero-order in a-methylstyrene. Catalyst stability was tested by measuring conversion of H2 1 and 1 2 h after adding the catalyst to the methylstyrene. Results indicated that the catalyst was less active for the later runs. Precautions ensured that no oxygen, which deactivates Wilkinson’s catalyst (Osborn et al., 1966), entered the reactor. The definite reason for the deactivation is undetermined, but alkenes are known to inhibit Wilkinson’s catalyst (Wadkar and Chaudhari, 1983). Thus, a possible explanation is inhibition by the reactant, a-methylstyrene. For the determination of rate constants, the experiments are performed in a time period short enough (2-3 h) that the catalyst may be assumed stable. To investigate the cause of catalyst deactivation, a small quantity (5 X lo4 m3) of a-methylstyrene was added to a cumene-catalyst solution. Conversions of hydrogen pulses, measured twice in a 6-day interval, were virtually unchanged for this solution. We infer that the large concentration of methylstyrene of reaction runs probably caused the deactivation. Wadkar and Chaudhari (1983) likewise found that the reaction rate of hydrogenation of allyl alcohol with Wilkinson’s catalyst decreased with increasing allyl alcohol concentration, due to an inhibitory reaction of the allyl alcohol with the catalyst. The procedure for the reaction runs was first to dissolve 0.0034 kg of dry catalyst in the methylstyrene in the reactor. The solution was purged with pure nitrogen for 1
Results and Discussion Zero moments for reaction runs are plotted as a function of gas flow rate in Figure 2. The Henry’s law constant for hydrogen in a-methylstyrene is 12.44 at 325 K (Herskowitz et al., 1978). According to eq 15, the intercept of the straight line provides the gas-to-liquid mass-transfer coefficient. The value of KL, determined by linear regression, is 16.7, which is 5-fold greater than values determined in earlier studies (Ahn et al., 1985, 1986). The difference can be attributed to the effect of suspended particles on mass-transfer coefficient (Joosten et al., 1977). The combined rate constant of association and surface reaction, given by ko = l / ( l / k a + l/Kk,), was obtained from the slope of the line in Figure 2; its value is 0.105 m3/(kg.s). It seems no reaction rate data for the present reaction system have been reported in the literature. Our rate constant, however, is larger than that for the hydrogenation of styrene (ko = 4.78 X lo-, m3/(kgs)) calculated from the data measured by Jardine et al. (1967). Different molecular structures between a-methylstyrene and styrene can have an effect on rate constants. For example, the rate constant for hydrogenation of cyclohexene was found to be about 4 times lower than that of styrene obtained under the same conditions (Jardine et al., 1967). In most of the hydrogenation studies with RhCl(PPh,), as the catalyst, organic solvents such as benzene or hexane were used (James, 1973; Master, 1981). In this study, however, no solvent was added to the solution, providing another possible reason for the higher rate constant measured in our experiments. The slope of the first-moment plot, Figure 3, provides the value of a/Kk,2 that would allow computation of the separate values of k , and it,. However, there is no obvious way to determine K for this reaction system. Therefore, separate values of k, and k, cannot be accurately evaluated from the data. Although the separated K, k,, and k, values cannot be determined, the values of the slopes of the zeroand first-moment plots yield two relations between the three parameters, 1 ka
1
- + - = 9.52 (kg.s)/m3
(36)
Kkr2= 1.23 m3/(kg.s2)
(37)
Kkr
Since all the quantities in these two equations must be
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 237
0 25
0.15
Temp, K
325
u ,k g / m 3
32 93 3 0 139
Slope Intercept
-
a4 = dimensionless quantity defined by eq 31 c, = concentration of hydrogen associated with the catalyst,
i 3 0
0
n
d
00
0 10
50
25
100
75
125
150
175
0
( Q u V L ) 2 ( 1 - rn0?
Wlmo
x
106 , m6 ( s k g ?
Figure 3. First moments of pulse responses for catalyzed H2 reaction with a-methylstyrene.
positive, the following limit conditions can be derived for the separate coefficients,
k, > 0.105 m3/(kg.s)
(38)
It, < 11.7 l / s
(39)
K > 8.97
X
m3/kg
(40)
According to our results, the transient experiment can be described accurately by the single overall rate constant, k,, independently of k, and k,. This is consistent with the criterion (McCoy, 1984) for determining when an adsorption-reaction sequence can be approximated by a single, fluid-phase rate. The criterion for the approximation in the present case follows from eq 19 and can be stated as
E = uK/(l
kmol/kg of RhC1(PPh3), hydrogen concentration in gas phase, kmol/m3 cBo= concentration in feed cg = Laplace transform of cg, kmol.s/m3 CL = hydrogen concentration in bulk liquid, kmol/m3 c1 = dimensionless quantity defined by eq 32 c2 = dimensionless quantity defined by eq 33 c3 = dimensionless quantity defined by eq 34 E = dimensionless quantity defined by eq 36 F = dimensionless quantity defined by eq 35 H = Henry's law constant for solubility of hydrogen in solvent, cg =
+ K k , / k , ) 2
