Ind. Eng. Chem. Res. 1987,26, 1132-1140
1132
Fane, A. G.; Sawistowski, H. Inst. Chem. Eng., Symp. Ser. 1969,32,
higher point efficiencies due to the Marangoni surface renewal effects (Ellis and Biddulph, 1967). The comparison of the positive and the negative composition range point efficiencies of the n-propanollwater system indicates that since the system properties are unchanged and the variation in the mixture surface tension, and consequently the stabilizing index, is low in the composition range studied, similar point efficiencies result.
1%
Finch, R.; Van Winkle, M. Ind. Eng. Chem. Process Des. Dev. 1964, 3(2),106. Gemhling, S.; Onken, U. Vapour-liquid Equilibrium Data; Dechema Chemistry Series; Dechema: Frankfurt, 1977; Vol. 1, p 2a. Hart, D. J.;Haselden, G. G. Inst. Chem. Eng., Symp. Ser. 1969,32, 1-19. Haselden, G. G.; Thorogood, R. M. Trans. Inst. Chem. Eng. 1964, 42, T8l. Hellusm, J. D.; Braulich, C. J.; Lyda, C. D.; Van Winkle, M. AIChE J . 1958, 4(4), 465. Hofhuis, P. A. M.; Zuiderweg, F. J. Inst. Chem. Ena., - Symp. . . Ser. 1979, 56, 2.2.1. Jeromin, L.; Holik, H.; Knapp, H. Inst. Chem. Eng., Symp. Ser. 1969, 23, 5.49.5. Kalbassi, M. A. Ph.D. Thesis, University of Nottingham, England, 1987. Kalbassi, M. A.; Dribika, M. M.; Biddulph, M. W.; Kler, S.; Lavin, J. T. Proceedings of the Institute of Chemical Engineers (London) International Symposium on Distillation, Brighton, 1987. Lockett, M. J.; Ahmed, I. S.Chem. Eng. Res. Des. 1983, 61, 110. Lockett, M. J.; Uddin, M. S.Trans. Inst. Chem. Eng. 1980,58, 166. Maripuri, V. 0.; Ratcliff, G. A. J . Chem. Eng. Data 1972, 17, 366. Medina, A. G.; McDermott, C.; Ashton, T. Chem. Eng. Sci. 1979,33, 1489. Pruden, B. B.; Hayduk. W.; Laudic, H. Can. J . Chem. Eng. 1974,52 (Feb), 64. Sargent, R. W. H.; Bernard, J. 0. T.; McMillan, W. P.; Schroter, R. C. Symp. Distill., London 1964. Smirnov, N. A.; Vestin, J. Lenningrad Uniu. USSR, Fiz. Khim, 1959, 81. Stabinkov, V. N.; Matyushev, B. Z.; Protsyak, T. B.; Yushanko, M. Pushch. Prom. Kiev 1972, 15. Standart, G. L. Chem. Eng. 1974, Nou, 716. Thomas, W. J.; Hag, M. A. Ind. Eng. Chem. Process Des. Deu. 1976, 15(4), 509. Umholtz, G. L.; Van Winkle, M. Ind. Eng. Chem. 1957, 4 9 ( 2 ) , 226. Young, G. C.; Weber, J. H. Ind. Eng. Chem. Process Des. Deu. 1972, 11(3), 440. Zuiderweg, F. J.; Harmens, A. Chem. Eng. Sci. 1958, 9, 89. Zuiderweg, F. J. Inst. Chem. Eng., Symp. Ser. 1969, 32, 1:55.
Conclusion The modified column suggested here seems to be suitable for point efficiency measurements of highly surface tension positive and negative as well as any other systems. I t eliminates the surface tension induced wall supported froth and minimizes the wetted wall effects. The column in general encourages steady operating conditions. This development is a useful step to simulate conditions, i.e., mixed froth regime of liquid, froth, and sprays on a large tray and further work is now in progress to improve the gas and the liquid contact time on the modified column tray by incorporating higher outlet weir heights without encouraging the wall effects to occur. As such, a column can be installed without inflicting great costs; it is a useful tool to measure reliable point efficiencies. Literature Cited Ashley, M. T.; Haselden, G. C. Trans. Inst. Chem. Eng. 1972, 50. Bainbridge, G. S.; Sawistowski, H. Chem. Eng. Sci. 1964, 19, 992. Biddulph, M. W. AIChE J . 1975; 21(2), 327. Biddulph, M. W.; Dribika, M. M. AIChE J . 1986, 32(8), 1383. Brown, B. R.; England, B. L. Int. Inst. Ref. Commun., London 1961, Sept, 19. Bubble-Tray Design Manual; AIChE: New York, 1955. Dribika, M. M. PhD Thesis, University of Nottingham, England, 1986. Dribika, M. M.; Biddulph, M. W. AIChE J . 1986, 32(11), 1864. Ellis, S.R. M.; Bennett, R. J. J . Inst. Petr. 1960, 43(433), 19. Ellis, S.R. M.; Catchpole, J. P. Dechem Monogr. 1964, 55, 43. Ellis, S. R. M.; Biddulph, M. W. Trans. Inst. Chem. Eng. 1967,45, T223. Ellis, S.R. M.; Legg, R. J. Can. J . Chem. Eng. 1962, Feb, 6.
Received for review May 7 , 1986 Revised manuscript received February 8 , 1987 Accepted February 24, 1987
Heterogenizing Homogeneous Catalyst. 2. Effect of Particle Size and Two-Phase Mixed Kinetic Model Bing Joe Hwang and T s e - C h u a n Chou* Department of Chemical Engineering, National Cheng Kung University, Tainan, Taiwan, Republic of China 70101
The effects of particle size and the degree of cross-linking of the resin-carrier catalyst on the oxidation of acetaldehyde were investigated. A two-phase mixed kinetic model was developed, which includes mass transfer, heterogeneous reactions, and homogeneous chain reactions. Both the experimental results and the theoretical calculations of the model show that the reaction rate for the production of peroxides is significantly affected by the particle size and the degree of cross-linking of the resin catalyst. The results also show that only a fraction of the whole catalyst particle is effectual to initiate free radicals, which then diffuse through the pores t o the bulk solution where the homogeneous free-radical chain reactions occur. T h e free radicals in the bulk solution are heterogeneously and homogeneously terminated. The manufacture of peracetic acid (PAA) from acetaldehyde and oxygen, using homogeneous Co3+ions and heterogeneous Co-type resin as catalyst, has been discussed in previous papers (Chou and Lin, 1980,1982,1983; Chou and Lee, 1982,1985). The mechanism of the generation of peracetic acid from the partial oxidation of acetaldehyde, using a Co-type resin as catalyst, has also been proposed (Chou and Lee, 1982,1985). In the present work, 0888-5885/87/2626-1132$01.50/0
a two-phase kinetic model was developed, which should be helpful in understanding the general system in which both heterogeneous and homogeneous chain reactions occur. Various models have been proposed to explain the interaction of diffusion and reactions in the macro- and micropores of a heterogeneous catalyst (Aris, 1975; Ors and Dogu, 1979; Mingle and Smith, 1961; Frisch, 1962; Car0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1133 berry, 1962; Tartarelli et al., 1970) as well as a resin catalyst (Roucis and Eherdt, 1984; Goto et al., 1983; Ihm et al., 1974; Frisch, 1962; Ruckenstein et al., 1971). Most of these models were assumed either a simple or a consecutive reaction which occurred on the active sites of the microand macropores of the resin particle or heterogeneous catalyst. The products were assumed to be formed inside the catalyst particles and not formed by any homogeneous reaction in the bulk solution. The heterogeneously catalyzed liquid-phase oxidation of hydrocarbons involving either free radicals or homogeneous reactions has been reported by several investigators (Dai and Lunsford, 1980; Prasad et al., 1979; Panayotova et al., 1980; Trimm et al., 1979; Neuberg et al., 1972, 1974, 1975; Walsh and Katzer, 1973; Daniel and Keulks, 1972; Meyer et al., 1965; Taylor, 1970; Mukherjee and Graydon, 1967; Gorokhovatskii and Pyatnitskaya, 1972; Gorokhovatskii, 1973; Ingold, 1961; Sadana, 1979; Sadana and Katzer, 1974a,b; Varma and Graydon, 1973; Roginsky and Andrianova, 1968). In general, these investigators emphasized finding the best operating conditions to obtain the highest yield or selectivity of products by this type of catalyst. Some investigators (Betts, 1971; Sadana, 1979) used the inhibitor method to study the effect of free radicals. Very few models or analyses of this type of reaction have been reported. Sadana (1979) discussed the liquidphase oxidation in which the supported copper oxide was used as the initiator and proposed a model which was evaluated by the method of introducing an inhibitor. However, his model included only the homogeneous reactions in the bulk solution and the initiation and termination reaction$ on the surface of catalyst. The transport phenomena and reactions which occurred inside the particle of catalyst were not considered in his model. According to the above review, in models of reaction systems using heterogeneous catalysts, only the Nernst and pore diffusion and the reactions which occurred inside the particle were considered. No homogeneous reactions were included in this type of model. On the other hand, in models of the reaction systems using homogeneous catalysts, only the reactions which occurred in the bulk solution were considered. The models for heterogeneous catalytic reactions or homogeneous catalytic reactions were good or applicable to these two extreme cases. However, these two types of models may not be suitable for the reaction system which includes pore diffusion and the homogeneous reactions which were initiated or catalyzed by a heterogeneous catalyst or heterogenizing homogeneous catalyst. Based on the proposed mechanism of the synthesis of peracetic acid in our previous paper (Chou and Lee, 1985), the heterogenizing homogeneous catalyst played the dual roles of initiation and termination of free radicals. The free radicals initiated by the catalyst transferred from the surface of the catalyst to the bulk solution, where the homogeneous chain reactions proceeded. The products were formed mainly in the bulk solution. Theoretically, free radicals can be generated within the whole particle of the resin catalyst. However, the generated free radicals might be terminated by the wall of pores of the particle during diffusion and could not reach the bulk solution if the generation of the free radicals occurred in rather deep interior of the particle. Unfortunately, no report was found to analyze these phenomena. In the present work, a two-phase kinetic model was developed by an analysis of the synthesis of peracetic acid from acetaldehyde and oxygen using a heterogenizing homogeneous catalyst.
Development of the Two-Phase Model The mechanism of formation organic peroxides from
Nernst layer
/ A‘. bulk
\\
Figure 1. Scheme representing mechanism of the heterogenizing homogeneous catalyst for the two-phase mixed kinetic model. Ar’ = R - r’ = effective thickness of generating free radical.
partial oxidation of aldehydes, using a heterogenizing homogeneous catalyst, has been previously proposed (Chou and Lee, 1985; Wang, 1984). A two-phase kinetic model including the initiation and mass transfer of free radicals within the catalyst particle and the homogeneous chain reactions in the bulk solutions was developed as diffusion of aldehyde
kl
RCHO absorption of oxygen
RCHO,,)
(1)
k2
initiation k
RCHO(,, + Co3+& RCHO-Co3+ k-3
(3)
(4) RCHO-Co3+ RCO + Co2++ H+ The free radical RCO generated by the catalyst transfers from the interior of the particle to the bulk solution. propagation
-
+ 02(b) k5
RCO RCO,
RCO,
+ RCHO 5 RC03H + RCO
(5) (6)
homogeneous termination
+ k7
~ R c O ~ inactive species heterogeneous termination RC03
k8
S
inactive species
(7)
(8)
regeneration of catalyst RC03H
+ Co2+2RC02 + OH- + Co3+
(9)
formation of byproduct RC03H + RCHO
k-io
byproducts of peroxides
(10)
where R represents the alkyl or aryl group, S is the surface of the catalyst, and the subscript, g, b, and s, indicate the species in the gas phase, bulk solution and on the surface of the catalyst, respectively. As shown in Figure 1, the free radical RCO, which was generated inside the catalyst particle, would diffuse through the pore of the polymer matrix to the bulk solution. The material balance of the free radical, RCO, within the particle can be expressed as dCA,/dt = D,(-d2CA,/dr2
+ BdCA,/rdr) + f r A p
(11)
1134 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 where CAPis the concentration of the free radical, RCO, inside the resin cawyst particle, rApis the rate of formation of free radical RCO, f is the effectiveness factor, and De is the effective diffusivity of the free radical, RCO. In general, the lifetime of a free radical is very short, the free radical generated inside the catalyst particle may be terminated before it diffuses to the bulk solution. As shown in Figure 1, r’is the radius of the limiting boundary, and the difference of R and r’, R - r’, is effective thickness. The free radical, RCO, generated within the sphere with the radius less than r’is assumed to be inactive to initiate the chain propagation in the bulk solution or unable to reach the bulk solution. The possibility of the generated free radicals, which can reach the bulk solution, increases with the radius. T o modify the phenomena of termination within the particle of catalyst, an effectiveness factor was introduced and defined as f = r( - r~’ ’ )
r’IrlR
(12)
where n is an integer number, R - r’is equal to Ar‘which, in general, depends on the pore size or the degree of cross-linking and is independent of the particle size of resin. The effectiveness factor, f , is zero a t the radius of limiting boundary and is equal to one on the surface of the particle. According to eq 4, the rate of the free radical, RCO, formation is rAp
=
(13)
k4CRCHO-Co3+
At pseudo steady state (Chou and Lee, 1985),the material balances of the other species in the system are k3CRCHO(s)CCo3+ ca(-De
(k-3 + k4)CRCHO...Co3+ = 0
(14)
dCAp/drlr=R) + kGCRCHOCRCO3 - k5CAbCO2 = 0 (15)
where C, is equal to A / V , V is the volume of solution, and A is the total effective active surface area of the heterogenizing homogeneous catalyst for initiation, k 5 C A b C 0 2 - k6CRCHOCRC03
- 2k7CRC0:
where C A b is the concentration of the free radical, RCO, in the bulk solution, and C, is the total surface area of the resin or the total termination area of the solid particles, which includes the heterogenizing homogeneous catalyst and inert resin particles per unit volume. In this sytem, CRCHO(,) can be considered as a constant, K (Chou and Lee, 1985). From eq 14, C R C H()...C 03+ can be expressed as = k3/(k-3
+ k4)CRCHO(s)CCo3+
(17)
If a dimensionless variable, r* = ( r - r ’ ) / ( R - r’), is introduced, eq 12 becomes f
= r*n
(18)
Setting a variable transformation
(22)
(23)
+ k4)] (24)
Az = - [ ( R - r’)2r’k3k4K]/[De(n + l ) ( n + 2)(k-,
+ k4)]
(25) For a given reaction system, De, k,, k-3, k4, and K are constants and A , is a function of R , r’, and n, i.e., A1(R, r’, n), and similarly A, is Az(R,r’, n). If the same crosslinking resin is used, both A, and A2 are constants which are a function of particle size only. From eq 19 q d 21, the concentration distribution of the free radical, RCO, within the catalyst particle is given as
CAP= B,r*”+3/r + B2r*n+2/r+ Glr*/r
+ Gz/r
(26)
where G, and G z can be determined by the boundary conditions (B.C.)
B.C.l -De dCAp/dr = k,(c~, - CAb)
at r = R
Or
r* = 1 (27)
B.C.2
CAP= 0
at r = r’ or
r* = 0
(28)
and k , is the mass-transfer coefficient of the free radical, RCO, from the particle surface to the bulk solution. By use of eq 26 and 28, G, is found to be zero. Thus, eq 26 becomes
CAP= Blr*n+3/r+ Bzr*n+2/r+ G,r*/r
(29)
If eq 29 is differentiated with respect to r , the result is dCAp/dr = [ ( n+ 3)Blr*n+2/(R- r’)r] - [Blr*n+3/r21 + [ ( n+ 2)Bzr*n+1/(R- r’)r] - [B2r*n+2/r2] [Glr*/r2]+ G l / [ ( R- r’)r] (30) Substituting eq 29 and 30 into eq 27 yields
G I = B3 - B ~ C A ~
- kBCRC03Cs = 0
(16)
CRCHO-Co3+
B1 = A1CCo3+ B2 = AZCco3+ A, = -[(R - r’)3k3k4K]/[De(n + 2)(n + 3)(k-,
(31)
where
B3 = A3CCo3+ (32) B, = [-(R - r’)R%,/D,]/[r‘+ R ( R - r’)ks/De]= A, (33) A3 = [ ( R- r’)(l - k,R/D,)(Al + A,) - R ( ( n + 3)A1 + ( n + 2 ) A z ) l / [ r ’ +k,R(R - r r ) / D e l (34) For a given reaction system and a fixed cross-linkingresin, A3 and A4 are constants and similar to A , and A*, which are a function of particle size only, i.e., A3 is A,@, r’, n ) and A4 is A,(R, r’, n). By eq 29 and 31, the concentration of free radical RCO on the surface of the resin particle is obtained as C A =~ ( B , + B2 + B3)/R - (B,C,b/R) (35)
At pseudo steady state, eq 11 can be rearranged to give
If eq 15,27, and.35 are combined, the concentration of the free radical, RCO, in the bulk solution is expressed as CAb = B5 + B6CRC03 (36)
d2u/dr*2 = ( C l / D e ) ( ( R- ~ - ’ ) ~ r *+ ~+ ( Rl - r’)%’r**) (20)
where
u =
rCAp
(19)
B, = A,5CCo3+
where C, = -(k3k4KCc03+)/(k-3 + k 4 ) . Integrating eq 20 twice gives u =
+
B,r*n+l + BZr*n+2 G,r*
B6
+ G,
where G, and G2 are integrating constants, and
(21)
A5
= ASCRCHO
(37)
(38)
+ A3)/(A4 + R + K&Co,/(KsCa)) (39) (40) A , = k e R / ( k s C J / [ A *+ R + k , R C o , / ( h s C a ) I
= ( A , + A2
Ind. Eng. Chem. Res., Vol. 26, No. 6 , 1987
For a given reaction system, a fixed degree of cross-linking resin, and constant oxygen concentration in the solution, A , and A , are constants which are functions of particle size and the active initiating surface area per unit volume of the solution; Le., A , is A 5 ( R ,r’, n, C,) and A , is A6(R, r’, n, Ca). If eq 36 is substituted into eq 16 and then rearranged, eq 16 becomes 2k7C~cO:
B~CRCO, - k,Co,B, = 0
(41)
where B7 = k,Cs
+ (Caks + C & S A ~ / R ) ~ ~ C R C I - I O / (Ca + k5C0, + CaksA,/R) (42)
The value of (C,k, + C,k,A4/R) may be found by using eq 33, which gives C,k, + Cak,A4/R = [ C , k , / ( R 2 - R r ’ ) ] / [ r ’ / ( R 2- Rr’) + k , / D , ] (43) Combining eq 42 and 43 shows that B7 is positive. Solving eq 41, the concentration of free radical, RC03, in the solution is found to be
C R C=~ (-B7 ~ + (B72+ 8k&7B,C0,)’/~)/(4k7) ( 4 4 ) According to eq 6 , the rate of formation of the peroxide, RC03H, is )“f
= k6CRCHOCRC03
(45)
If eq 44 is substituted into eq 45, the general rate equation of producing peroxide is
rf = ~ & R c H o [ - B+~(B72+ 8 k ~ V ~ C 0 , ) ’ / ~ 1 / ( 4 k 7(46) ) Equation 46 is a general rate equation of a two-phase mixed kinetic model, which includes the interior pore diffusion, mass transfer, and reactions inside the catalyst particles and the homogeneous chain reactions in the bulk solution. Equation 46 can be reduced to a rate equation which does not include the pore diffusion, reactions, and mass transfer inside or nearby the particles. The procedures of reducing eq 46 to the special cases are carried out as follows. From eq 24, 25, 34, and 39, eq 37 may be represented as
1135
rf = ( ~ , C R C H Ox/ ~ ~ ~ ) [-k,C, (kg2CS2+ 8k7A7k3k,KCco3+/(k-3+ k4))1/2](54)
+
If eq 3 is a t equilibrium, k-,
>> k4, then
eq 54 becomes
rf = (k&~c~o/4k7)(-k8C +, (ka2CS2 (55) where K 3 is the equilibrium constant of eq 3, and A7 is a constant when the particle size distribution is fixed. Equation 55 is the same as eq 20 of our previous paper (Chou and Lee, 1985). If kg2Cs2> A , + R , eq 47 can be simplified to function of the degree of cross-linking of resin. The parameter, C1*, which is equal to k , k , K / ( k _ , + k 4 ) ,is evalB5 = (A7k3k,KCco3+)/[k,Co,(k-3+ kJ1 (51) uated by a data-fitting method, comparing the experiwhere A , = k,C,(A,* Az* + A 3 * ) / R ,and eq 42 can be mental results with the results calculated from eq 46 using simplified to the parameters as shown in Table I. The data of the parameters, k,, D e , and k5, were given in a range, respec(52) B7 = kaCs + [(A4 + R)~~CRCHOI/[~,CO,R/(~,C~)I tively, in the literature (Smith, 1981; Kataoka et al., 1974; Zaikov et al., 1969). However, fixed values of k,, De, and If k&, >> [(A4 + R)~~CRCH~]/[~,C~,R/(~,C,)], then eq 52 k , within their given range are chosen, respectively, for the is further reduced to evaluation of the parameter, C,*. B7 = kgC, (53) Before evaluation of C1*, the values of Ar’, k,, t,, and Then eq 51 and 52 are substituted into eq 46 which can Co2?e analyzed and described as follows: The free radical, be reduced to CH,CO, generated by the catalyst easily combines with the
+
1136 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 Table I. Parameters of the Theoretical Calculations data in this model parameter in lit. ref k,, dm/min 0.06-0.6 0.1 Smith, 1981 cm2/s 5 x 10-7 Kataoka e t al., 1974 De dm2/min 2.2 x 107 Zaikov et al., 1969 k5, M-' min-' 2.9 X lo63.4 x 107 1.62 X lo6 Zaikov et al., 1969 1.62 x 105 ke, M-' min-' k,, M-' min-' 3.01 x 109 Zaikov et al., 1969 3.01 x 109 this work k,, dm min-' 0.8221 5.0 x 10-4 Boudart, 1968 t,, min 5.0 x 10-4 Seidell, 1953 0.01 0.01 cop M 1.56 x 10-3 this work Cl*, M min" q mequiv-'
Table 11. Evaluation of R.," no. of diameter, particles, (NiD?)1O6, Ni cm2 1030i,cm 0.57 9 2.9 1.23 40 60.5 1.84 88 297.9 2.45 156 936.4 1891.4 3.06 202 3.68 210 2 843.9 4.29 188 3 460.0 4.90 146 3 505.5 5.52 97 2 955.6 6.13 45 1691.0 6.74 7 318.0 7.36 2 108.3 1190 sum 18071.4 sum
(NiDi3)i09, cm3 1.7 74.4 548.2 2 294.2 5 787.8 10 465.6 14 843.3 17 176.8 16315.1 10 365.6 2 143.3 797.4 80 813.4 sum
200
Y
i l-l Slope=-5.7964.IO-'
0:
2
&
100
0P-
0 2
0
5
IO
125
OO
D x I O ' , cm
Figure 2. Typical distribution of particle size of a X8-400 Co-type resin as the heterogenizing homogeneous catalyst.
dissolved oxygen to form free radical CH3C03in the bulk solution or near the outer surface of the catalyst where the dissolved oxygen is plentiful. The concentration of free radical CH&O, would be much higher than that of free radical CH3C0 in the bulk solution. Inside the particle, there is a lack of dissolved oxygen to react with the free radical, CH,CO, which is generated by the. catalyst. Therefore, the concentration of free radical, CH,CO, wo@d be much higher than that of the free radical, CH3C03, inside the catalyst particle. Accordingly, only the pore diffusivity, De, and the mass-transfer coefficient, k,, of the free radical, CH3C0, are considered. The chance of the free radical, CH3C0, inside the particle to diffuse into the bulk solution is related to its lifetime, t,. Based on the data reported in the literature (Boudart, 1968), the lifetime was found to be 5 X min in the acetone solution. The parameters, k6 and k,, were obtained from the literature (Zaikov et al., 1969) as shown in Table I. When pore diffusivity and the lifetime of free radical, CH3C0, are known (Kataoka et al., 1974; Boudart, 1968), the effective thickness, Ar ', is estimated by Einstein's equation, Ar' = (2D,t,)1/2
0.5
(57)
The concentration of dissolved oxygen in the bulk solution is assumed to be saturated and equal to 0.01 M in the acetone solution at 20 " C (Seidell, 1953). Equations 7 and 8 are the main termination steps of the reaction mechanism to form peracetic acid (Chou and Lee, 1985). Equation 8 indicates that the heterogeneous termination of the free radical, CH,C03, occurs mainly on the outer surface of the particle (Boudqrt, 1968). There is little chance for the free radical, CH3C03,to be terminated on the internal surface of the catalyst. The total termination surface area, L,, can only be estimated by considering the outer surface area of the particles. L, is calculated by the
c,
250
, dm'e-'
Figure 3. Effect of total outer surface area of the catalyst on the initial reaction rate. Concentration of acetaldehyde, 5.24 M; temperature, 20 OC; oxygen flow rate, 0.3 L/min; volume of solution, 40.0 mL.
following procedures: Figure 2 is a typical particle size distribution of a catalyst weight, W. The total weight and total outer surface area of these particles are equal to n'
W = 4xp/3CNiR?
(58)
i-1
n'
L, = 4 a 2 N i R i 2
(59)
i=l
where Ni is the number of particles with radius Ri, p is the density of the particles, and n' is the total number of samples for the analysis of particle size. The data of N , and Ri are obtained by the methods which are described under Experimental Procedures. The total outer surface area per unit weight of catalyst is n'
Sd
n'
= ( 4 ~NiRi2) x / ( 4 ~ p / 3 NiR:) i=l
i=l
(60)
If catalyst loading, W, and the particle size are determined, the total outer surface area of the particles per unit volume can be obtained by
c, = SdW/V
= 3W/(pVR,,)
(61)
where R,, = ( c ~ ~ ~ N i R ~ ) / ( c ~ ~ l N i R i z ) . Table I1 shows the results of a typical analysis of a catalyst loading. The values of R,, and C, are equal to 2.236 X lo4 dm and 70.37 dm2L-l, respectively. The total termination surface area, C,, of the catalyst used for evaluating the parameters of Table I is equal to 70.37 dm2 L-1. The evaluation of the rate constant, k,, can be finished by adding inert solid resin, Le., H-form resin, to change
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1137 Table 111. Evaluation of Parameter C1*
~
7.575 6.125 5.340 5.090 7.575 6.125 5.340 5.090 7.575 6.125 5.340 5.090
-6.204 -6.204 -6.204 -6.204 -6.244 -6.244 -6.244 -6.244 -6.284 -6.284 -6.284 -6.284
-9.306 -9.306 -9.306 -9.306 -9.366 -9.366 -9.366 -9.366 -9.426 -9.426 -9.426 -9.426
1.559 1.559 1.559 1.559 1.569 1.569 1.569 1.569 1.579 1.579 1.579 1.579
-1.86 -1.86 -1.86 -1.86 -1.86 -1.86 -1.86 -1.86 -1.86 -1.86 -1.86 -1.86
8.094 8.094 8.094 8.094 8.146 8.146 8.146 8.146 8.199 8.199 8.199 8.199
5.58 4.51 3.93 3.75 5.58 4.51 3.93 3.75 5.58 4.51 3.93 3.75
the total termination surface area. As shown in Figure 3, the initial reaction rate decreased from 1.10 X to 7.80 X M min-l when the total outer surface of catalyst, C,, increased from 70.37 to 246.29 dm2 L-l. Figure 3 indicates that the plot of the initial reaction rate against the total outer surface area of particles results in a straight line with slope -5.796 x mol min-l dm2. The relationship of the total outer surface area and the initial reaction rate can be expressed as
rf = 0.01492 - 5.796
X
lO-,C,
64.35 63.10 62.47 62.22 64.35 63.10 62.47 62.22 64.35 63.10 62.47 62.22
1.5542 1.2640 1.1052 1.0547 1.5607 1.2692 1.1098 1.0590 1.5672 1.2745 1.1144 1.0634
1.55
0.0173
1.56
0.0166
1.57
0.0173
Experimental data Model calculation
(62)
Comparing eq 56 with eq 62, the following relationship is obtained
5
6
= 5.796 X
(63)
where CRcHois 5.24 M. If the data of k, and k7, as shown in Table I, are substituted into eq 63, k8 is found to be 0.8221 dm min-l. As a certain amount of species transfers through the pore with sinks on the wall, the flux of the species would nonlinearly decrease along the axis of the pore (Bird et al., 1960). So we assumed the "n" value of eq 12 to be 2. The evaluation procedures of Cl* are explained as follows: The initial reaction rate, which is calculated by eq 46, is compared with experimental results. A relative error criterion for each experimental data point is defined as l(rcalcd
- rexptl)/rexptll
(64)
and m
e(C1*) = l / m C e i 1=l
(65)
where m is the number of experimental data. During the curve fitting, e(C,*) is restricted to be equal to, or less than, 0.02. At the beginning of the calculation, all parameters such as k,, k,, k,, k7, It8, and De, except Ar', are given as shown in Table I. All the variables such as R , Cc03+,Co2, V, W, and n, except the concentration of acetaldehyde, are fixed. The effective thickness, Ar', is calculated by eq 57. By the definitions of C1 and C1*, which is the pseudoinitiation rate constant, the following relationship is obtained.
c, = c1*cco3+ (66) where Cl* = k g k 4 K / ( k 3+ k4). By use of the parameter C1*, the given parameters, and the given variables, the constants B,-B7 can be calculated by eq 22,23, 32,33,37, 38, and 42, respectively. Then the reaction rate at different concentrations of acetaldehyde can be calculated from eq 46, using the calculated B, and B7,the given parameters, and the given variables. Based on eq 65, the average relative error, e(C,*),can be obtained from experimental
8
7
Concentration of Acetaldehyde
C,~1.&&8/(4k7)
I
0.81
,M
Figure 4. Effect of acetaldehyde concentration on the calculation and experimental initial reaction rates. Concentration of Co3+,3.58 mequiv/g; temperature, 20 "C; oxygen flow rate, 0.14 L/min; volume of solution, 40.0 mL; loading catalyst, 7.5 g/L.
reaction rates and the calculated ones. Part of the results for the searching of the parameter, C1*, are shown in Table 111. When the parameter, C1*, is equal to 1.56 X the average relative error, e(C1*),is equal to 0.0166, which is less than the preset value, 0.02. Figure 4 also shows that the experimental results are correlated well with the calculated ones W g this C1*. Effect of Particle Size. The effect of particle size on the reaction rate occurs mainly on the initiation steps, i.e., eq 3 and 4, and the termination step, i.e.,.eq 8. The initiation steps generate the free radical, CH3C0, both inside and on the surface of the catalyst particle. In general, the pore size of a polystyrene resin depends on the degree of cross-linking of the prepared resin. For one type of resin, such as X8-400, the degree of cross-linking is fixed and the pore size can be assumed to be constant. The pore diffusivity, De, of a species in a system can be considered to be a function of the pore size only. As shown in eq 57, the effective thickness, Ar', is a function of pore diffusivity and the lifetime of the free radical. Therefore, the effective thickness for generating the free radical, CH3C0, is fixed for a certain type of resin, i.e., a fixed pore size resin. The volume of the catalyst particle between R and r' is defined as the active volume in which the generated free radical, CH3C0, can diffuse to the bulk solution and initiate the chain reactions there. Since effective thickness is fixed for a certain type of resin.used, the active volume to generate the free radical, CH3C0,increases with the decreasing of the particle size for a constant catalyst loading. The fraction of active volume for a catalyst particle is calculated from The percentage of active volume increases from 1.2% to
1138 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987
2
Crosslinking of Resin 4 6 8 1 0 1 2 -_.__ Experimental
1. 5
.
datu
mesh *
- Model
cdculalim
200-400
C
z 1.0 N
0 Lo
05 0
D x / 0 4 , dm
Figure 5. Effect of particle size on the fraction of active volume of a particle.
.
____ Expermenla1 _.
xz
-
Model
3
40
coIculotion
3 0
80
-150
Figure 7. Effect of the degree of cross-linking of resin or pore diffusivity on the initial reaction rate. Concentration of acetaldehyde, 5.68 M; concentration of Co3+,3.58 mequiv/g; oxygen flow rate, 0.14 L/min; loading of catalyst, 7.5 g/L.
data
crosslinking
I
-145
En De , dm"m/n-'
I20
D x / 0 4 , dm
Figure 6. Effect of particle size on the initial reaction rate. Concentration of acetaldehyde, 5.68 M; temperature, 20 OC; concentration of Co3+, 3.58 mequiv/g; oxygen flow rate, 0.14 L/min; volume of solution, 40.0 mL; loading of catalyst, 7.5 g/L.
32% as the diameter of the catalyst particle decreases from 8.4 x to 4.48 X dm, as shown in Figure 5. For the larger particles, the effect of particle size on the percentage of active volume becomes less significant. The total outer surface area of a fixed catalyst loading increases with the decreasing of the particle size. As shown in eq 8, the rate of heterogeneous termination will increase with the decreasing of particle size. A decrease in the particle size leads to more active volume and total outer surface area per unit catalyst loading. Therefore, decreasing the particle size of the catalyst increases both the rate of initiation and heterogeneous termination. However, the experimental results and the calculated ones show that the effect of particle size on the rate of initiation reaction is more significant than that of heterogeneous termination reaction. The experimental results indicate that increasing the diameter of catalyst particles from 4.48 x to 8.4 x dm decreases the initial reaction rate of producing peracetic acid M min-' by using a Dowex from 1.18 x to 3.65 X X8 cross-linking resin as carrier, as shown in Figure 6. The results also show that the effects of particle size on the initial reaction rate are less significant for larger particles than for smaller particles. Similar results were also obtained by using a Dowex X2 resin as the carrier of the catalyst. For the same particle size, the initial reaction rate by using a Dowex X2 resin as the carrier is much higher than that by using Dowex X8 resin. Figure 6 also shows the calculated results of eq 46 when the parameter of pore diffusivity De is changed. The results indicate that
the tendency of the theoretical calculation is very similar to the experimental results. The theoretical calculation also shows that increasing the pore diffusivity increases the initial reaction rate a t the same size of the catalyst particle. These calculation results are correlated well with the experimental results in changing the particle size of the catalyst. Effect of the Degree of Cross-Linking of Resin. Increasing the degree of cross-linking of resin from 2% to 12% decreases the initial reaction rate from 1.30 X to 1.16 X M min-' for the particle sizes of 200-400 mesh, as shown in Figure 7. For the larger particles, such as lW200-mesh particles, a similar result is obtained. In general, increasing the degree of cross-linking of resin decreases the pore diffusivity and thus decreases the rate of initiation reaction. The relationship of pore diffusivity and the degree of cross-linkingof resin is not tested in this study. However, Kataoka et al. (1974) pointed out that the relationship of pore diffusivity and the degree of cross-linking of resin could be expressed as (68) where a, and b are constants, X is the degree of crosslinking of resin, and DIois the diffusivity of the free radical, CH,CO, in the bulk solution. Accordingly, the degree of cross-linking of resin is a linear function of the logarithm of the pore diffusivity, i.e., In De = a - bX (69) where a is a constant. The solid lines in Figure 7 are the plotes of the calculated initial reaction rate against the logarithmic pore diffusivity, In De, for different particle sizes of the catalyst. The results show that the effect of the degree of cross-linking of resin on the initial reaction rate is very similar to that of the logarithmic pore diffusivity. Effect of Loading of Catalyst. An increase in the loading of a fixed-size catalyst leads to an increase in both the total active volume and the total outer surface area. As mentioned early, the catalyst plays dual roles of initiation and termination reactions. Increasing the active volume increases the rate of formation of peracetic acid. On the other hand, increasing the total outer surface area decreases the reaction rate. By use of smaller catalyst particles 4.48 X lo4 dm in diameter, both the experimental results and the calculated ones show that increasing the loading of the catalyst from 2.5 to 50 g/L increases the initial reaction rate from 7.0 X lo-, to 1.75 X lo-' M min-'.
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1139
I
/----
2.0[
dotc Experimental data Oxlo',
I
L'+
dm
84.0
O 1 I O 4 , dm
I 2
4.48 84.0
o l ' " ' ' ' " " ' ' ' ' l 0
from 9.47 X lo-' to 3.60 X M when the concentration of the heterogenized Co3+ion decreases from 4.78 to 1.78 mequiv/g, as shown in Figure 9. The results of these calculations are fairly reasonable by comparison with the reaction mechanism.
50 100 Loading of Catalyst , g / l
150
Figure 8. Effect of loading of catalyst on the initial reaction rate. Concentration of acetaldehyde, 5.68 M; temperature, 20 "C; concentration of Co3+, 3.58 mequiv/g; oxygen flow rate, 0.14 L/min; volume of solution, 40 mL.
Conclusions The effects of the particle size and the degree of cross-linking of the resin catalyst on the formation rate of peroxides are well explained by the two-phase mixed kinetic model. The developed model includes the masstransfer, heterogeneous reactions and the homogeneous chain reactions. Experimental results as well as the calculating ones show that the initiated free radicals within the particle diffuse through the pore to the bulk solution where the homogeneous chain reactions occur. Only a fraction of the whole heterogenized homogeneous catalyst particle is effective to initiate the free radicals. The active volume of the catalyst increases from 1.2% to 32% as the diameter of the particle decreases from 8.4 X to 4.48X dm. The pseudo rate constant of the initiation steps to generate the free radical, RCO, was found to be 1.56 x M min-' g mequiv-'. The free radicals in the bulk solution are heterogeneously and homogeneously terminated. The rate constant of heterogeneous termination reaction was found to be 0.8221 dm min-'. The two-phase mixed kinetic model can be applied to the systems of homogeneous chain reactions which are initiated by a heterogenizing homogeneous catalyst or a heterogeneous catalyst. Acknowledgment
rf Figure 9. Distribution of concentration of free radical CH&O within the particle. Concentration of acetaldehyde, 5.68 M temperature, 20 OC; concentration of dissolved oxygen, 0.01 M; diameter of particle, 4.48 X lo4 dm; density of particle, 1430 g/dm3.
When the catalyst loading further increases from 50 to 75 g/L, the reaction rate will still remain at 1.75 X M m i d . This is because the initial reaction rate affected by the total active volume is more significant than that by the total outer surface area in the catalyst loading range 2.5-50 g/L. On the other hand, the effects of the total active volume and the total outer surface area on the initial reaction rate are about the same when the catalyst loading is in the range 50-75 g/L, as shown in Figure 8. Similar results were also obtained by using larger catalyst particles 8.40 X dm in diameter. In this case, the calculation results show that the initial reaction rate slightly increases when the loading of the catalyst increases from 50 to 75 g/L, while the experimental results show the difference is insignificant. Distribution of Concentration of Free Radical, CH3C0, within the Particle. The distribution of concentration of free radical, CH3C0, within the catalyst particle can be obtained by calculating eq 22, 23, 29, and 31. Figure 9 shows the typical concentration distribution of free radical, CH,CO, CAP,inside the particle. It shows that CA is zero at radius, r', and a maximum is found near the Tachs, R. The mass transfer of the free radical, CH3C0, diffusing from the surface of the particle to the bulk solution may make CAPbecome a little smaller at the point, r* = 1.0, or radius, R , as shown in Figure 9. According to eq 3 and 4 mentioned before, increasing the concentration of the heterogenized Co3+ion on .the wall of the pore will generate more free radical, CH3C0. The maximum concentration of free radical, CH3C0, decreases
The support of the National Science Council and National Cheng Kung University of the Republic of China is acknowledged.
Nomenclature A = total effective surface area, dm2 a = constant defined in eq 69 Al-A6 = variables defined in eq 24, 25, 34, 33, 39, 40, respectively a, = constant defined in eq 68 A1*-A3* = variables defined in eq 48, 49, 50, respectively b = constant defined in eq 68 Bl-B7 = variables defined in eq 22, 23, 32, 33, 31, 38, respectively C1 = variable defined in eq 20, M min-' C1* = k 3 k 4 K / ( k 3+ k 4 ) ,M g min-' mequiv-' C, = A / V , the total active surface area per unit volume of solution C A=~ concentration of free radical RCO in the bulk solution, M CAP = concentration of free radical RCO in the particle, M C, = total outer surface area of the catalyst particles per volume, dm2 L-l D = diameter of catalyst particle, dm De = pore diffusivity, dm2 min-' DI* = diffusivity in the bulk solution, dm2 min-l e, = relative error at point i, defined in eq 64 e(C1*) = average relative error, defined in eq 65 f = effectiveness factor defined in eq 12 G1, G, = integrating constant of eq 2 1 K = constant, M K3 = equilibrium constant of eq 3 kl, k2, etc. = forward rate constant of eq 1 , 2 , etc., respectively k+ h-,,, etc. = backward rate constants of eq 3, 10, etc., respectively k , = mass-transfer coefficient between catalyst and bulk solution, dm min-' L, = total outer surface area of the catalyst particles, dm2
I n d . Eng. C h e m . Res. JLYBII,
1140
n = integer number defined in eq 12 p = fraction of active volume R = radius of catalyst particle, dm r = distance defined in Figure 1, dm rf = limiting radius to generate free radical, defined in Figure 1
rf = formation rate of peroxides, M min-I R,, = variable defined in eq 61 r* = variable defined as ( r - r f ) / ( R-- r’) Sd = variable defined by eq 60
t , = lifetime of free radical, min variable defined in eq 19 V = volume of solution, L W = weight of loading of catalyst, g X = degree of cross-linking of resin u =
Greek Symbols = density of resin, g dm-3 Ar’ = variable defined in eq 57
p
Literature Cited Aris, R. The Mathematical Theor)! of Diffusion and Reaction in Permeable Catalysts; Oxford University Press: London, 1975; Vols. I and 11. Betts, J. Q. Reo. 1971, 25, 265. Bird, R. B.; Warren, E. S.; Edwin, N. L. Transport Phenomena; Wiley: New York, 1960; pp 466-467. Boudart, M. Kinetics of Chemical Processes; Prentice-Hall: Englewood Cliffs, NJ, 1968; pp 148-149. Caloyannis, A . G.; Graydon, W. F. J . Catal. 1971, 22. 287. Carberry, J.
