Ind. Eng. Chem. Process Des. Dev. 1981, 20, 397-399
W = mass flow rate
X = Lockhart-Martinelli parameter
V = viscous (=[(hp/hL)s~/(AP/
U ) S G l ‘I2
Y = function =
[ ( h p / h L ) T p - (hp/u)SG]/(vSG)2
Greek Letters a, @ = flow configuration factors a*, @*= function of n’ in eq 26
I’ = function of n’ defined in equation, f, = s2(d/D)0.5/ [R
(d/D)”lr
Ztzner-Reed consistency term [= g&’ 8(”’-l) 6 = function of n‘ [= 1 - $1 I.L = viscosity of fluid p = density $ = Lockhart-Martinelli parameter [ ( A P / A L ) T ~ / ( A P / y =
307
+ =u)sP11’2 function of n’ defined in eq 10 Q,w = constants x = function of n’ (= 0.5 - wr)
Subscripts C = coil G = gas gen = generalized L = liquid S = superficial, single-phase s = straight tube TP = two-phase t = turbulent
Dimensionless Groups ( d / D ) = coil curvature ratio De’ = modified Dean number = Re,,,(d/D)1/2= (CP’V(~-~’)p ) / (gcK%(n’-1))
M = dimensionless number proposed by Mujawar and Rao (1978) Re = Reynolds number (dVp)/p
Literature Cited Akagawa, K.; Sakaguchi, T.; Veda, M. Bull. J S M . 1972, 74, 564. knerjee, S.; Rhodes, E.; Scott, D. S. AfChEJ. 1967, 13, 189. Banerjee, S.; Rhodes, E.; Scott, D. S. Can. J . Chem. Eng. 1969, 47, 445. Chlsholm, D. Trans. ASME 1967, 80, 276. Dayasagar, I. Ph.D. Thesis, Indian Instltute of Technology, Bombay, 1975. Hughmark, G. A. Chem. Eng. Prog. 1962, 58(4), 62. Ito, H. Trans. A S M , J. BasiCEng. 1959, 81, 123. Lockhart, R. W.; Martlnelii, R. C. Chem. Eng. frog. 1949. 45, 39. Mashelkar, R. A.; Devarajan, G. V. Trans. Insf. Chem. Eng. 1977, 55, 29. Mulawar, 8. A. Ph.D. Thesis, Indian Institute of Technology, Bombay, 1978. Mujawar, B. A.; Rao, M. R. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 22. Owhadi, A.; Beii, K. J.; Craln, B., Jr. Int. J. Heat Mass Transfer 1966, 1 7 , 1779. Ramaniah, P.; Satyanarayan, A. Chem. Pet. J. (In&) 1976, 7(6), 3. Reddy, C. J. M.; Satyanarayan, A. J. Insfn. Eng. (India) 1977, 58, Part CH I Rlppel, 0. R.; E&, C. M.; Jordon, H. 8. I&. Eng. Chem. Process Des. Dev. 1966, 5, 32.
Received for review June 3, 1980 Accepted December 8, 1980
COMMUNICATIONS Critical Catalyst Concentration in the Liqufd-Phase Oxidation of Hydrocarbons A method to relate the critical catalyst concentration (CCC)to the length of the induction period, 71,the concentration of the hydroperoxide of the hydrocarbon being oxidized in the liquid phase, and an adsorption type constant, A ,
is presented for the heterogeneously catalyzed liquid-phase oxidation of hydrocarbons involving alkylperoxy (ROO-) free-radical termination. CCC is inversely proportional to A and increases as temperature increases. The method is judiciously applied to liquid-phase cumene and cyclohexene oxidations.
Introduction The critical catalyst concentration (CCC) phenomenon has been observed in the heterogeneously catalyzed liquid-phase free-radical oxidation of hydrocarbons (Meyer et al., 1965; Mukherjee and Graydon, 1967; Bacherikova et al., 1971; Evmenenko et al., 1972; Neuberg and Graydon, 1972; Gorokhovatskii and Pyatnitskaya, 1972; Gorokhovatskii, 1973a,b; Varma and Graydon, 1973; Neuberg et al., 1975; Mikhalovskii et al., 1976). A sudden and significant (almost catastrophic) change in rate for a slight change in catalyst concentration (or other rate determining parameter) represents a critical phenomenon and is a characteristic of branched-chain reactions (Gorokhovatakii and Pyatnitskaya, 1972). It appears that the CCC phenomenon is related not only to either the oxidizable hydrocarbon or the catalyst taken alone but, more appropriately, to the system taken as a whole (Gorokhovatskii, 1973b). Apparently, the “dual function” of free-radical initiation and termination exhibited by the compounds of transition metals generally used as oxidation catalysts is also essential for CCC. The “dual function” prevalent 0196-4305/81/1120-0397$01.25/0
predominantly at high catalyst concentrations in cumene (Gorokhovatskii, 1973a) and phenol (Sadana, 1979) oxidations, increases the length of the induction period rI with increasing catalyst loading in heterogeneously catalyzed cumene (Evmenenko et al., 1972; Gorokhovatskii and Pyatnitskaya, 1972) and phenol (Sadana, 1979) oxidations and in homogeneously catalyzed n-decane oxidation (Knorre et al., 1959). Similar increases in rI with an increase in the “nonactive” catalyst support (tending to decrease the “free” hydroperoxide of the hydrocarbon being oxidized in the liquid phase) has been reported in the oxidation of tetralin (Mukherjee and Graydon, 19671, cumene (Vreugdenhil, 1973), and phenol (Sadana and Katzer, 1974a). Recently, it was shown that the CCC value obtained (Sadana, 1979) from the available T I data for aqueousphase phenol oxidation (Sadana and Katzer, 1974a) (wherein the phenoxy (R.,alkoxy) radical terminates on the catalyst surface) compared very favorably with the CCC value obtained from the expression developed for the kinetic chain length (when it equals 0.5 (Neuberg and 0 1981 American Chemical Society
398
Ind. Eng. Chem. Process Des. Dev., Vol. 20,No. 2, 1981
Table I. Estimated Values of Adsorption Constant, A , in Heterogeneously Catalyzed Liquid-PhaseHydrocarbon Oxidations no.
initial hydrocarbon used, mL
1.
cumene, 1000 mL
2.
cyclohexene, 200 mL
critical catalyst, g
temp, K
l/Aa
A
reference
Co20,,60.0 Co-0,. 21.6 MnO;,’ 13.2 MnO,, 3.2
3 03 296 296 333
0.068 0.024 0.015 0.020
14.70 41.67 66.67 50.00
Evmenenko et al. (1972) Neuberg et al. (1972)
1 / A = catalyst concentration/initial hydrocarbon concentration.
Graydon, 1972)) in the steady-state activity regime. This note develops an expression to relate CCC with an adsorption type constant, A , for liquid-phase hydrocarbon oxidations involving either alkylperoxy [ROO.] radical termination both on the catalyst surface and in the liquid phase or ROO. termination just on the catalyst surface. The usefulness of the method is demonstrated by applying it judiciously to liquid-phase cumene (Evmenenko et al., 1972) and cyclohexene (Neuberg and Graydon, 1972) oxidation data available in the literature. Theory The free-radical mechanism of liquid-phase oxidation of hydrocarbons showing the “dual nature” (prevalent predominantly at high catalyst loadings) of the catalyst in the induction period may, in general, be represented by
-
+ cat. R-+ -H-cat. R. + O2 ROOROO. + R H 2 ROOH + RROO. + cat. inactive products kl
RH
(1)
(2) (3)
k4
-
+
ROO.
+ ROO-
k6
inactive products
(4)
(5)
where RH is the hydrocarbon. Step 1 is the classical free-radical initiation step with by the catalyst. the abstraction of the hydrogen atom (H.) Steps 2 and 3 comprise the classical propagation cycle. Nearing CCC, step 4 and steps 4 and 5 have been observed experimentally as the predominant free-radical termination steps occurring during cumene (Evmenenko et al., 1972) and cyclohexene (Neuberg et al., 1975) oxidations, respectively. Using the steady-state approximation applied to freeradical species in the system yields for eq 1-5 the concentration of ROO. as
CROO.= - [(k4/k5)(Mc/Vi) f [(k4(MC/VJ/k5Y + (4k,CRH(M~/Vl)/k5)1”~1/2 (6) where (Mc/VI) is catalyst concentration. For sufficiently high values of [(Mc/Vl) / CRH], eq 6 on applying the binomial expansion and retaining the linear term simplifies to (7)
on taking the positive root. A physical consequence of sufficiently high values of [Mc/Vl)/cRH] is to imply that the termination of free radicals occurs predominantly on the catalyst surface (eq 4). One may then neglect eq 5 when compared to eq 4. Thus, on applying the steadystate approximation to eq 1 and 4 yields eq 7 directly. Thus, the rate of hydrocarbon oxidation nearing CCC may be given by
The second-order disappearance of hydrocarbon has not been explicitly observed experimentally for cumene (Evmenenko et al., 1972) and cyclohexene (Neuberg et al., 1975) oxidations for catalyst loadings nearing CCC (primarily because presumably there has been no effort in this direction). However, it would appear reasonable to assume that such a dependence may be expected to be observed for catalyst loadings nearing CCC on considering the predominant free-radical termination steps observed experimentally nearing CCC along with the reaction scheme presented. Equation 8 integrates to give
CRH,O - CRH- CROOH k&$ =t I 71 (9) C R H ~ R H , OCRHCRH,O k4 is the time required to attain the steady-state concentration of hydroperoxide in the liquid phase (Evmenenko et al., 1972; Sadana and Katzer, 1974a,b; Robertson and Waters, 1948; Sadana, 1979). Assuming that the required steady-state hydroperoxide concentration is directly proportional to the catalyst concentration (Sadana and Katzer, 1974a; Sadana, 1979) CROOH,~ = A M / Vi) (10) then, at t = q since CROOH= CROOH+,eq 9 reduces to
I
-1
I
Note that as the catalyst concentration (MJVl) increases, 71 increases as has been observed previously with heterogeneously catalyzed cumene (Gorokhovatskii and Pyatnitskaya, 1972) and homogeneously catalyzed n-decane oxidation (Knorre et al., 1959). The catalyst concentration reaches a critical value when there is no oxidation and = (Sadana, 1979). Then, from eq 11
Table I lists the experimentally observed critical catalyst/ initial hydrocarbon concentration ratio values for liquid-phase cumene and cyclohexene oxidations along with other reaction conditions and the estimated values of A. Since A is an “adsorption type” constant (Sadana and Katzer, 1974a; Sadana, 19791, it may be represented as A = A. exp(AH/RT) (13) where AH is the heat of adsorption of the hydroperoxide of the hydrocarbon being oxidized on the catalyst surface. There is, however, not enough CCC data available for cumene (or any other reactant) oxidation at different temperatures to perform a consistency test for the complete justification of the validity of eq 13. Nevertheless, an increase in the reaction temperature would decrease A
Ind. Eng. Chem. Process Des. Dev. 1981, 20, 399-401
(leading to a higher level of “free” hydroperoxide in the liquid phase) and thus increase the critical catalyst/initial hydrocarbon concentration ratio. This increase in the critical catalyst/initial hydrocarbon concentration ratio with increasing temperature may be noted for Co203catalyzed cumene oxidation (Evmenenko et al., 1972). Similar increases in the “free” hydroperoxide concentration in the liquid phase have caused critical phenomena to occur at higher catalyst/initial hydrocarbon concentration ratios for other liquid-phase hydrocarbon oxidations (Meyer et al., 1965; Mukherjee and Graydon, 1967; Gorokhovatskii, 1973a,b; Neuberg et al., 1975; Mikhalovskii et al., 1976). Conclusions
An expression is developed to relate CCC with an adsorption type constant, A , for liquid-phase hydrocarbon oxidations involving either ROO. free-radical termination both on the catalyst surface and in the liquid phase or ROO. termination just on the catalyst surface. On the complete justification of the validity of eq 13 and the determination of the values of A,, and AH therein one would be able to predict CCC values for different reaction temperatures. Besides, the method permits a plausible insight into the physics and adds to the basic under-
399
standing of the chemistry involved in the CCC phenomenon. L i t e r a t u r e Cited bcherkwa, I. V.; Gwokhovatskii, Ye. 8.; Evmenenko, N. P. Klmt. Ketal. 1971, 12, 1437. Evmenenko, N. P.; Gorokhovatskii, Ya. B.; Pylenko, Yu. I. Do&/.Akad. mu& SSSR 1972, 202, 1117. Gorokhovatskii, Ya. 8. Cafal. Proc. Inf. Congr. 5th 1972 1973a, 879. Gorokhovatskli, Ya. B. Klmt. Ketal. 1973b, 14, 83. Gorokhovatskii, Ya. B.; Pyatnitskaya, A. I. Kinet. Ketal. 1972, 13, 1527. Knorre, D. 0.; Chugukina, L. 0.; Emanuel, N. M. 2%. F k . Khlm. 1959, 33, 877. Meyer, C.; Clement. G.; Balaceanu, J. C. Proc. Int. Congr. Catal. 3rd 1984 1965, 1 , 134. Mikhalovskii, S. V.; Gorokhovatskii, Ya. 8.; Evmenenko, N. P. Klmt. Ketal. 1976, 17. 1058. MukherJee, A.; Graydon, W. F. J . fbys. C b m . 1967, 71, 4232. Neuberg, H. J.; Graydon. W. F. J . Catal. 1972, 25, 425. Neubeq, H. J.; Phillips, M. J.; Graydon, W. F. J. Catal. 1976, 38. 33. Robertson, A.; Waters, W. A. J . Chem. Soc. 1948, 1535. Sadana, A.; Katzer, J. R. J . Catal. 19741%3 5 , 140. Sadana, A.; Katzer, J. R. I d . Eng. Ghem. Fundam. 1974b, 13, 127. Sadana, A. Ind. Eng. Chem. proces4Des. Dev. 1979, 18, 50. Varma, G. R.; Graydon, W. F. J. Cats/. 1973, 28, 236. Vreugdenhil, A. D. J. Catal. 1973, 2 8 , 493.
Chemical Engineering Division National Chemical Laboratory Poona 41 1 008,India
Ajit Sadana
Received for review May 29, 1980 Accepted October 28, 1980
Effect of Inhibitors on Hydrate Formation A method is proposed to predict the effect of inhibitors on natural gas hydrate formation conditions. Initiil experimental results indicate that the proposed method is much more accurate than the method which forms the basis for current industrial practive of inhibitor injection into natural gas streams.
Introduction
Gas hydrates are members of a group of compounds called clathrates, which are comprised of guest molecules and host molecules. The host, or water molecules, form a lattice which is stabilized by the inclusion of the guest molecules. Conditions favoring hydrate formation are those of high pressures and low temperatures. One of the two structures (I or 11) is formed by the water and gas, depending on the size of the gas molecule. Gas hydrates can be formed by a pure gas or by a mixture of gases. Davidson (1973) gives an excellent review of the physicochemical aspects of hydrates. The formation of natural gas hydrates was first brought to the attention of the natural gas industry by HammerSchmidt (1934), when he determined that freezing in gas pipelines was not due to ice but to gas hydrates. Recently, hydrates have become of interest due to estimates of large deposita of gas in hydrate form by Makogon (1974) and Katz (1971). In his initial effort, Hammerschmidt (1939) presented a method for determining the effect of inhibitors such as calcium chloride, ethanol, and methanol on hydrate formation conditions. Unfortunately, the compositions of the gases used by Hammerschmidt were not reported. Nevertheless the initial study was the basis for the current industrial standard, as in the GPSA Data Book, for methanol injection to inhibit hydrate formation. This work was undertaken to determine a theoretically sound, yet accurate method for determining hydrate inhibition. A modification is made to the model of Parrish 019&4305/81/1120-0399$01.25/0
and Prausnitz (1972) to incorporate the effects of inhibitors. Thermodynamic M o d e l
Shortly after the crystalline structures of hydrates were determined, van der Waals and Platteuw (1959) developed the basic statistical thermodynamics model used in current three-phase (hydrate-gas-water rich liquid) formation predictions as kwL= pwH - RT In y ~ =, pWB + RT&, In (1 -) ,6 + RT In yg, (1) m
The basic equation may be considered as a modification of Raoult’s law with k@ as the chemical potential of water in the unoccupied hydrate lattice, the summation term resulting from the guest molecule occupation of the lattice, and the final term resulting from the normally small solubility of the guest molecule in liquid water. The activity coefficient of water, yw,is normally taken as 1.0 due to the fact that the water concentration is almost pure when hydrocarbons are the hydrate gases. The number of type m cavities per water molecule in the lattice, v, is a constant. The fraction of type m cavities occupied by a type j gas molecule is given by the formula
where the Langmuir constant, Cmj,is a function of tem0 1981 American Chemlcal Soclety
