Ind. Eng. Chem. Res. 1988,27, 784-790
784
Table I. Comparison between the Mass-Transfer Parameters, &, and k,,Obtained by Kinetic Runs and by Other Approaches, Respectivelya parameters obtained by physical parameters obtained measurements (&,) in the presence of or by calculations the reaction (12,) speed, rpm PL
k,
700 0.10 0.050
1000 0.28 0.066
2000 0.78 0.076
700 0.11 0.068
1000 0.27 0.080
2000 0.75 0.13
nFor example, pL = kLaL, where kL(50 "C) = 0.018 cm/s, has been evaluated with the transient electrochemical method, while aL is the apparent interfacial area reported in Figure 8. On the contrary, k , has been calculated by the empirical correlation suggested by Sano et al. (1974).
becomes readily operative. The change from liquid-solid to gas-liquid mass transfer, observed by changing the stirring rates from 700 to 2000 rpm in the runs plotted in Figure 10, can now be completely justified by observing on Figure 11 that for l / m = 33 one or the other masstransfer regime prevails in the two cases. In Table I, mass-transfer parameters derived from kinetic runs are compared with those obtained through physical measurements or calculations. As can be seen, this agreement is satisfactory. Another interesting observation is that very fast reactions performed in slurry reactors could be useful to easily determine gas-liquid and liquid-solid mass-transfer coefficients by interpreting curves such as those reported in Figure 11. Acknowledgment Montefluos S.p.A. is kindly acknowledged for the financial support to the present work. Nomenclature u L = specific gas-liquid interface area, cm2/cm3 CHz = concentration of hydrogen in liquid phase, mol/cm3 D = diffusion coefficient in liquid phase, cm2/s dI = impeller diameter, cm d, = average diameter of the catalyst particles, cm d, = reactor diameter, cm F, = shape factor of the catalyst, assumed to be 1 HH2= Henry's law constant for hydrogen, (atm cm3)/mol
i ,,i = current intensity measured at time t or at saturation, nA kL = gas-liquid mass-transfer coefficient, cm/s k, = liquid-solid mass-transfer coefficient, cm/s L = height of slurry, cm m = catalyst holdup, g/cm3 p H 2= partial pressure of hydrogen, atm Pow= 8w3d&/(d,2L) R = reaction rate, mol/(cm3s) T = temperature, K t = time, s Greek Symbols pL = liquid density, g/cm3 p p = catalyst density, g/cm3 IL = liquid viscosity, g/cm/s
+ = correction factor in Pow(see Chaudari et al. (1980)) w
= stirring rate, rps
Registry No. THEAQ, 15547-17-8; Pd, 7440-05-3. Literature Cited Alper, E.; Wichtendahl, B.; Dackwer, W. D. Chem. Eng. Sci. 1980, 35, 217. Berglin, T.; Shoon, N. H. Ind Eng. Chem. Process Des. Deu. 1981, 20, 615. Berglin, T.; Shoon, N. H. Ind. Eng. Chem. Process. Des. Deu. 1983, 22, 150. Charpentier, J. C. Advances in Chemical Engineering; Academic: New York, 1981; Vol. 11. Chaudari, R. V.; Ramachandran, P. A. AIChE J. 1980, 26(2), 177. Clark, L. C. US Patent 2 913 386, 1959. Gruniger, H. R.; Sulzberger, H.; Calzaferri, G . Helv. Chim. Acta 1978, 61(7), 2375. Kirdin, K. K.; Franchuk, V. I.; Balabin, I. Y.; Agafonova, M. I. Sou. Chem. Ind. 1970,9, 5. Kirk, T.; Othmer, K. Kird-Othmer Encyclopedia of Chemical Technology, 3rd ed.; Wiley: New York, 1981; Vol. 13. Linek, V.; Vacek, V. Chem. Eng. Sci. 1981, 36, 1747. Nitta, T.; Akimoto, T.; Matsui, A.; Katayama, T. J.Chem. Eng. Jpn. 1983, 16, 352. Powell, R. Hydrogen Peroxide; Noyes: New York, 1968. Sano, Y.; Yamaguchi, N.; Adachi, T. J. Chem. Eng. Jpn. 1974,7,255. Satterfield, C. W.; Sherwood, R. K. The Role of Diffusion in Catalysis; Addison-Wesley: Reading, MA, 1963. Ulmann, T. Encyclopedie der Technischen Chemie; Wiley: New York, 1969; Vol. 17. Zajcev, M. JAOCS, J . Am. Oil Chem. SOC.1960, 37, 11. Received for review February 2, 1987 Revised manuscript received November 30, 1987 Accepted December 14, 1987
Mathematical Model for High-pressure Tubular Reactor for Ethylene Po1ymerization Adriana Brandolin, Numa J. Capiati, Jorge N. Farber, and Enrique 114. Valles" Planta Piloto de Ingenieria Quimica, UNS-CONICET, 12 de Octubre 1842, 8000 Bahia Blanca, Argentina
A mathematical model for ethylene polymerization in a high-pressure tubular reactor is proposed. Kinetic data have been determined through a nonlinear regression analysis based on measured reactor temperatures and molecular weight distribution properties. Experimental information has been obtained from two industrial reactors featuring different configurations and operating with oxygen as the single initiator. The model allows good prediction of conversion, molecular weight, and long-chain branching for different configurations and a wide range of operation conditions. The high-pressure process for ethylene polymerization is typically operated at pressures between 1800 and 3000 atm. Temperatures range from 60 to 320 "C,which is an upper bound given by the thermal decomposition of 0888-5885/88/2627-0784$01.50/0
ethylene. A mixture of ethylene feed, initiators, and chain modifiers is compressed to the reactor pressure, together with a recycle stream. Typical conversions range from 15% to 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 5 , 1988 785 2590, depending on the reactor configuration and initiation system. This process has been leading the manufacture of lowdensity polyethylene for several decades. However, a thorough understanding of the process is not available in the open literature, as can be surveyed in classical references (Chen et al., 1976; Agrawal and Han, 1975; Donati et al., 1982; Goto et al., 1981; Ehrlich and Mortimer, 1970; etc.). These contributions summarize current difficulties being faced in the formulation of a model for the process. The limitations originate not only in the particular complexity of the reacting system but in the severe operation conditions as well. Reproduction of these extreme conditions in a laboratory setup makes any study on the thermodynamic compatibility of the reaction mixture, the elucidation of the kinetic mechanism, or the measurement of kinetic and transport parameters very difficult. The available kinetic information is characterized by the lack of agreement among different sources. It is not surprising that kinetic constants, probably interpreted through oversimplified models, differ by 1or 2 orders of magnitude. Despite serious limitations being faced in a detailed description of the system, this work points to the formulation of a realistic model for the process. This model should serve the engineering goal of good prediction of reactor states and product specifications over a wide range of operation conditions. This purpose is based on some previous progress made in this area by earlier contributors and the availability of experimental information from two industrial reactors featuring different configurations and operation conditions. Experimental temperatures and product properties can be used to estimate the kinetic parameters through a general nonlinear regression. Although several initiators at different injection points are commonly used in industrial operation, the present experimental data were planned for operation with oxygen as a single initiator. This introduces considerable simplification to the kinetic analysis and permits emphasis on the basic kinetic steps, with independence from the initiation mechanisms. This separate problem of complex initiation systems is reserved for later studies. The accomplishment of a useful model is essential for further work on development of optimal design criteria and operation policies.
Model Aspects The complexity of the problem has been previously faced through a variety of simplifying assumptions involving both the free-radical kinetic mechanism and the reactor model itself. Discussion and justification of some of these assumptions are available in classical references (Chen et al., 1976; Agrawal and Han, 1975; Donati et al., 1982; Goto et al., 1981; Ehrlich and Mortimer, 1970; etc.). The proposed model is based on the following basic hypotheses: plug flow, reaction mixture forming a single supercritical phase, nonsteady state of free radicals, and variation of physical properties along the axial coordinate. The typical pulsed flow operation characterizing this type of reactors has been studied by Donati et al. (1982). These authors found no significant differences between average mixing efficiencies, heat-transfer coefficients, and pressure drops, when compared with the corresponding constant flow values. The formalism of generating function techniques has been applied for characterization of the molecular weight distribution of radicals and dead polymer, through their first few moments (Ray and Laurence, 1977; Katz and Saidel, 1967; Saidel and Katz, 1968).
l'*.i
I 14.' I
2anc I ' 3 '
-c
a",
L,
jo
Zone
Zone
4
1
I
I
I
ICi
I
I
~
4 !"l
+I!",
t-2
Figure 1. Reactor scheme.
The bivariate moments of orders m and n of the number chain length density distribution are defined as
where i is the number of branches in the molecule, x is the number of monomer units, and di(x)is the number density distribution of molecules having i branches and x monomer units. Kinetic Mechanism. The proposed kinetic mechanism is based on widely recognized steps for ethylene polymerization. Related contributions have been reviewed by Ehrlich and Mortimer (1970). Further details on oxygen initiation have been more recently discussed by Tatsukami et al. (1980). The reactions involved in our model are the following: oxygen initiation
O2+ M -% R,(1)
(2)
propagation Ri(x) + M combination termination
k, +
R ~ ( x+ 1)
(3)
(4)
thermal degradation
transfer to polymer
transfer to solvent Ri(x) + S
2P i ( x ) + R,(O)
(7) The oxygen initiation mechanism is not completely understood and probably involves several steps. As a simplification, the rate of the oxygen initiation has been globally described as an nth order reaction rate based on oxygen concentration, the order being adjusted through a regression analysis comprising all kinetic parameters. This permits adequate representation of the retardation observed in the oxygen decomposition (Tatsukami et al., 1980). The thermal decomposition of ethylene as a possible initiation mechanism is known to have negligible influence under current operation conditions (Buback, 1980). Similar considerations apply to chain transfer to monomer, which is not significant in the case of ethylene (Woodbrey and Ehrlich, 1963). The termination mechanism through thermal degradation is equivalent to @-scission,and its consideration is crucial for prediction of degrees of polymerization (Chen et al., 1976).
786 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988
Reactor Balances. Statement of mass and energy balances leads to the following set of coupled nonlinear ordinary differential equations: global mass balance (8)
where dp/& = f(T, m, r , u )
(9)
monomer mass balance dm/dz = -(kpmXoo+ m du/dz)/u
(10)
no. of initiator injection pts injection pts locations, z / L
di/dz = -(kii6m solvent balance ds/dz = -(k,sX,
+ i dv/dz)/v + s dv/dz)/v
Water Temperatures 8wi0
(12)
+ 1O3(-AfOkpmX,) (13)
shell energy balance dTc/dz = rDiu(Tc - T)/(Gc~cCpc)
(14)
The following two characteristic equations correspond to the radical and dead polymer chain length distributions. They allow straightforward generation of bivariate moment balances of any desired order to be solved simultaneously with the mass and energy balances (Ray and Laurence, 1977): moments of chain length distribution for radicals nkpmXm,n-l+ kii6m - k,X&,,,
+ kt,JXm6,0
-
moments of chain length distribution for dead polymer
-
&+m,n+1)
- fimn (du/dz)/u (16)
where m = 0 and 1 and n = 0-2. Basic properties of the polymer chain length distribution can be reproduced as combinations of their first few moments. The following definitions are widely employed in characterizing the polymer products: number-average degree of polymerization
Xn= (101 + FOl)/(~Oo+ Fool
0.07 1.89
0.23 2.16
0.34 1.99
0.45 1.99
0.55 1.90
0.67 1.93
0.83 1.87
(11)
reactor energy balance dT/dz = (l/(pCpu))(-U(T - Tc)4/Di
1 0
Operation Conditions monomer feed rate, kg/h 42 000 solvent/monomer mass ratio 4.29 X O/monomer mass ratio 2.85 x 10-5 operation pressure, kg/cm2 2000
z/L
oxygen balance
+ ktp(h"0l
Design Aspects QlL 4.1 x 10-5 no. of heat exchange zones 8 water inlet locations, z / L 0.07, 0.23, 0.34, 0.45, 0.55, 0.67, 0.83, 1.0
dv - = - - -u dp dz p dz
ktzihnn
Table I. Reactor A
(17)
weight-average degree of polymerization (18) 8, = 0 0 2 + Fo2)/(Xo1 + Fod weight-average branch point number (per molecular weight unit)
Physical Properties. Changes in physical properties of the reaction mixture as functions of internal temperature and pressure are adjusted along the axial coordinate. A reaction enthalpy of A H = -22 300 cal/mol has been adopted in the model, assuming that propagation is the
onlv thermallv relevant step. Formulas previously employed by Chen et al. (1976) have been used for calculation of viscosity, specific heat, and thermal conductivity. Ethylene feed mixtures typically contain impurities of methane, ethane, propane, isobutane, and n-butane in total concentrations ranging from 5% to 7% vol. A program has been developed for simulation of mixture densities. Density values for pure monomer and impure feed have been obtained through a multilinear regression, adjusting density predictions calculated with the BWR model (Benedict et al., 1940). The following polynomial function for density as a function of temperature and pressure has been used: p =a
+ bT + c T 2 + dP + eP2 + fTP
(20)
where the coefficients a, b, c, d , e, and f depend on the mixture composition. The performance of these estimations has been verified by comparison with predicted densities through BWR's equation. Values for pure ethylene were obtained from tables by Goldman (1969), which are based on experimental data by Michels and Geldermans (1942) and Michels et al. (1946). In theoretical situations, a first estimation of heattransfer coefficients can be performed with available correlations, as those employed by Chen et al. (1976). In the present simulation of a real-scale equipment, the above estimations have been improved by resorting to energy balances based on experimentaltemperature of reactor and coolant.
Experimental Information Experimental data have been obtained from two industrial reactors with different configurations and operating with oxygen as a single initiator in all runs. The main features and typical operation conditions are described in Tables I and I1 and Figure 1. Internal temperature profiles were measured with thermocouples conveniently located along the reactor tube. Similar procedures were followed with water temperatures. Product samples were analyzed by GPC (gel permeation chromatography)and intrinsic viscosity. This information was used for calculation of molecular weights, polydispersities, and branching parameters according to the method proposed by Foster et al. (1980). Numerical Solution Balance equations for moments of chain length distribution of radicals and dead polymer constitute an unbounded system, as each balance depends on higher order moments. This can be easily handled following the closure
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 787 Table 11. Reactor B Design Aspects DilL 7.9 x 10-3 5 no. of heat exchange zones 0.19, 0.42, 0.60, 0.75, 1.0 water inlet locations, z / L 1 no. of initiator injection pts injection pt locations, z/L 0 Operation Conditions 11910 monomer feed rate, kg/h solvent/monomer mass ratio 6.72 X lo4 O/monomer mass ratio 4.20 X operation pressure, kg/cm2 2100
z/L ewiO
0.19 0.97
Water Temperatures 0.42 0.60 0.75 0.97 0.68 0.68
Table 111. Dimensionless Reactor Balance Eauations Dimensionless Variables axial coordinate reaction mixture temp water temp linear velocity initiator concn (0) monomer concn bivariate momenta of radical chain length distribution bivariate momenta of dead polymer chain length distribution solvent concn
1.0 0.97
procedure proposed by Hulburt and Katz (1964). The polymer moments knand pln can be expressed as functions of precursors won = f(P00, Pol, .**,P0,n-1) (21) Pln = f(Pl0, 1111,
Pl,n-1)
(22)
by using Laguerre polynomials as approximating functions and a truncated gamma distribution as the weighting function. Trials with n = 3-5 did not produce any significant difference in the results; therefore, n = 3 was chosen in all calculations. The resulting system contains 18 highly nonlinear ordinary differential equations. Given the nature of the eigenvalues, there is no possible solution with conventional methods. Gear’s method (Gear, 1971) for rigid systems featuring automatic adjustment of the integration step and the order of the algorithm was implemented in a VAX-780 computer. The dimensionless set of equations is presented in Table 111. CPU times varied from 17 to 40 s, depending on the choice of operation conditions and reactor configuration.
Determination of Kinetic Constants An overview of information available in the open literature indicates a complete lack of agreement among reported values for kinetic constants, even for well-recognized steps in ethylene polymerization (Chen et al., 1976; Donati et al., 1982; Lee and Marano, 1979; Shirodkar and Tsien, 1986; Agrawal and Han, 1975; Goto et al., 1981; etc.). Practically all available kinetic information has been used in order to test the proposed model. However, no combination of values permitted a satisfactory representation of experimental temperature profiles, as measured in real scale operation. Nevertheless, some useful conclusions were drawn from these tests, in relation to the sensitivity of temperature, conversion, degree of polymerization, polydispersity, etc., to the different kinetic constants. This knowledge considerably simplified a general nonlinear regression, given the possibility of decoupling certain insensitive functions, in particular those related to the polymer properties. It was proven that kinetic constants for initiation, propagation, and termination are determinant of temperature and conversion profiles. Although to a lesser degree, they also affect the final (MWD) properties of the product. The other set of kinetic constants for thermal degradation, transfer to solvent, and transfer to polymer has a stronger effect on number-average degree of polymerization, polydispersity, and weight-average branch point number (per molecular weight unit). However, they are unable to modify the temperature and conversion profiles to any significant amount.
(ckicm-j,n-i)
+ Ai3 exp(A14B/(B + I))Qmndmn/Cmn +
Ai5 exp(Alfi@/(e+ l))(QmnQ~ldmn/(Cmnd~l) -
+
QooQ~r,n+ldmn/Coodm,n+l))
exp(Alse/(e
l))tQmndmn/Cmn - Qmn db/de)/dJ
+ (84
On the basis of these observations, kinetic parameters were adjusted through a two-stage nonlinear regression. In the first stage, the following objective function was minimized to adjust the temperature profile:
where N is the number of experimental runs, zi is the axial
788 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 Table IV. Kinetic Constants frequency factor, activation energy, kinetic constant ca1.L.mol.s cal/mol oxygen initiation, ki 3.0 X 1O1O 27 941 propagation, 12, 1.0 x 108 5 245 combination termination, 3.0 X lo* 3 950 ~
ktc
thermal degradation, k~ chain transfer to solvent, k!, chain transfer to polymer,
7.3 X 1.7 X
lo6 lo6
4.4 x 108
11315 9 443
1oF
9 500
J=1
08
IO
Figure 2. Relative reactor temperature versus relative axial distance for reactors A and B: (- - -) experimental data; (-) model predictions.
position for experimental temperature measurement, T,*(z,) is the experimental temperature of run j at z = z,, and T,(z,) is the calculated temperature of run j at z = z,. In the above function (F), the order of the oxygen initiation reaction and kinetic constants for initiation, propagation, and combination termination were included as regression variables. Calculations were performed with Powell's method for the minimum of a function of n variables through successive variations of the independent parameter (Powell, 1977). In the second stage of the regression, the following objective function, based on experimentally determined properties of the MWD over N runs, was considered: N
06
a4 zr
Table V. Range of Operating and Design Variables min max variables 2500 1900 operation pressure, kg/cm 403 333 feed temp, K max reaction temp, K 593 438 493 water temp, K 8000 42000 monomer feed rate, kg/ h 0.2 1.3 oxygen feed rate, kg/h 0.0 solvent feed rate, kg/ h 30.0 400 1300 total reactor length, m 0.025 0.051 internal diameter, m
C((1- Xn/Xn*)p+ (1- X,/X,*),2
~
02
kt,
G=
--1 'A -A
+ (1- X/X*),2) (24)
where 8, and Xw are the number- and weight-average degrees of polymerization, respectively, X is the weightaverage branch point number, and Xn*,X,*,and K* denote experimental values. The search was performed on rate constants for chain transfer to polymer, chain transfer to solvent, and thermal degradation. Values for initiation reaction order and rate constants for initiation, propagation, and termination as established in the previous search were used in the minimization of G. The Levenberg-Marquardt algorithm for nonlinear regression was implemented for the calculations (Brown and Dennis, 1972). Final results from both steps of the regression, over 1 2 experimental runs, are summarized in Table IV. A corresponding value of 6 = 1.1 has been determined for the reaction order of the oxygen initiation. The calculation of kinetic parameters has been based on experimental data obtained in the range of operating and design variables detailed in Table V. They represent realistic industrial conditions. The model permitted excellent representation within this range, as described in subsequent sections.
Results and Discussion (a) Kinetic Parameters. The results of the regression, as summarized in Table IV, yield the values of k, = 9.90
x L'.'/ L1.'/(mol1.'.s) at 180 "C and ki= 6.32 X (mol%) at 250 "C. Considering the temperature dependence, there is good agreement with the experiments by L/(mol.s) at 180 "C Ogo (1984), yielding ki = 1.40 X L/(mol.s) at 250 "C. and ki = 9.84 X The propagation rate constant results in k, = 1.43 X lo3 L/(mol.s) at 130 "C and 2200 atm and a corresponding activation energy for propagation within the wide range for vinyl polymerizations (Laird et al., 1956). The group k /ktc1I2 at 130 "C is in agreement with the previously pubfished values (Ehrlich and Mortimer, 1970; Ogo, 1984). The values of the termination rate constant, k, = 2.16 X lo6 L/(mol.s), are 2 orders of magnitude smaller than those published by Takahashi and Ehrlich (1982) and Luft et al. (1983). The activation energy difference ((E, - Etc)/2)has a value of 3270 cal/mol, somewhat lower than 5000-7000 cal/mol, commonly reported for vinyl polymerization. However, higher values of k,, or (E, - Etc)/2than those yielded by the regression produce temperature peaks much sharper than the ones observed experimentally. As established by Chen et al. (1976) and Ehrlich and Mortimer (19701, the thermal degradation reaction is controlled by a high activation energy, as it becomes relevant only at higher temperatures. The present calculations yield an activation energy of Edt= 11315 cal/mol, in agreement with the above considerations. In relation to transfer to solvent, Ehrlich and Mortimer (1970) reported C, = 5.0 X and E, - E, = 3800 cal/mol for n-butane at 130 "C. Results from the regression, C, = 8.9 X and E, - E, = 4198 cal/mol, are also in good agreement. According to studies by Yamamoto and Sugimoto (1979) for the transfer to polymer reaction, E,, - E, = 3600 cal/mol and AVtp = 1 cm/mol, which determines an effective energy difference of 3651 cal/mol at 2200 atm. The present calculations provide E,, - E, = 4200 cal/mol. (b) Simulation. Numerical simulations have been performed for the two reactor configurations A and B, as detailed in Figure 1, having a single feed and a single initiator (oxygen). Design and operation conditions are detailed in Table I for reactor A and Table I1 for reactor B. Results of the simulation of temperature profiles through the proposed model are compared with experimental values (Figure 2). It is worth noting the good agreement obtained for the two different reactor configurations and operation conditions. It is observed that the peaks are not as pronounced as reported in other simulations (Chen et al., 1976; Agrawal and Han, 1975). This model performance agrees with the experimental findings and indicates a more realistic continuation of the reaction shortly after the peaks. Figure 3 contains the conversion profiles for both reactors. Figures 4, 5, and 6 show the calculated profiles for
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 789 I
16
-
128
G
B
.11
Reactor A
I Reactor A
I
a? 4.32
z
2096IA IY
Y W -
z
032
-
0
, '_-_I_
O
6
; I / /
L
06
0.4
0.2
08
10
zr
Figure 5. Polydispersity versus relative axial distance for reactors
A and B.
Ti7 I Reactor B,------------
80 01
0.2
0.4
0.6
0.8
1.0
zr Figure 4. Number-average degree of polymerization versus relative axial distance for reactors A and B. Table VI. Calculated Values of Polymer Properties (at z =
L) property Xn X W
XwIXn
x
reactor A 713 4773 6.7 3.44 x 10-6
0
I
reactor B 671 4780 7.12 3.88 X 10"
//
02
04
06
. bL 08 10
zr
Figure 6. Weight-average branch point number versus relative axial distance for reactors A and B.
plex initiation systems currently employed in industrial operation. The proposed model was proven useful in the prediction of reactor states and product properties over a wide range of operation conditions. It represents a valuable tool for further studies on optimal reactor design, optimal operation policies, and performance of control systems.
Acknowledgment Table VII. Molecular Weight Distribution Properties: Average Percent E r r o r s av % error property exptl range 6.5 Xn 400-800 9.6 X w 2100-5900 XwIXn 3.4-7.6 5.4 30.0 x 4.0 X 10-6-1.0 X lo4
We thank Adriana Serrani for assistance in modeling of thermodynamic properties of mixtures and computer simulation. This work was supported by POLISUR S.M., CONICET, and SECYT.
Nomenclature = scaling constant for ,,A C, = ratio between chain transfer to solvent and propagation rate constants C, = specific heat of reaction mixture, cal/(kgK) C,, = specific heat of coolant, cal/(kgK) Di = internal diameter, m d,, = scaling constant for c ( ~ , Ei= activation energy for oxygen initiation, cal/mol E , = activation energy for propagation, cal/mol E , = activation energy for combination termination, cal/mol E d = activation energy for thermal decomposition, cal/mol E,, = activation energy for transfer to polymer, cal/mol G, = coolant feed rate, m3/s i = initiator concentration (oxygen), mol/L ki = rate constant for oxygen initiation, L1.l/(moll.l.s) It, = rate constant for combination termination, L/(mol.s) ktd = rate constant for thermal degradation, L/s k,, = rate constant for transfer to polymer, L/(gs) k , = rate constant for transfer to solvent, L/(mol.s) L = total reactor length, m LA = total length for reactor A, m m = monomer concentration, mol/L Pi(x) = concentration of polymer containing i branches and x total monomer units in its structure, mol/L R , ( x ) = concentration of radicals containing i branches and x total monomer units in its structure, mol/L s = solvent concentration, mol/L T = reaction mixture temperature, K T , = relative reactor temperature, T , = T / T o , c,
number-average degree of polymerization, polydispersity, and branch point number, respectively. Table VI details the predicted values of molecular weight distribution properties for reactors A and B, under the operation conditions specified in Tables I and 11. Table VI1 contains average percent errors between experimental data and model predictions. The averages were calculated over 12 different runs corresponding to both reactors A and B.
Conclusions Efforts toward a realistic model for tubular reactors for ethylene polymerization have to deal with the particular complexity of the system and the lack of consistent kinetic information in the open literature. A model has been proposed, based on classical hypotheses of plug flow and homogeneous reaction mixture, with prediction of thermodynamic and transport properties along the axial coordinate. Experimental information on reactor states and product specifications has been obtained from two industrial reactors featuring different configurations. Operation with oxygen as a single initiator introduced considerable simplification to a high-dimensional nonlinear regression for estimation of basic kinetic parameters. This information is available for treatment, as a separate problem, of com-
Ind. Eng. Chem. Res. 1988,27, 790-795
790
TOA= feed temperature for reactor A, K T , = coolant temperature, K U = global heat-transfer coefficient, cal/ (m2.K.s) u = reaction mixture velocity, m/s Xn= number-average degree of polymerization 8,= weight-average degree of polymerization z = axial distance, m 2, = relative axial distance, z, = z / L A Greek Symbols
6 = reaction order for oxygen initiation
AH
= reaction enthalpy, cal/mol
p = reaction mixture density, kg/m3 pc = coolant density, kg/m3
X
= weight-average branch point number (per molecular weight) A, = bivariate moment of orders m and n of the radical chain length distribution R i ( x ) ,mol/L pmn = bivariate moment of orders m and n of the dead polymer chain length distribution Pi(x),mol/L Registry No. Ethylene, 74-85-1; polyethylene, 9002-88-4.
Literature Cited Agrawal, S.; Han, C. D. AZChE J . 1975, 21, 449. Benedict, M.; Webb, G. B.; Rubin, L. C. J . Chem. Phys. 1940,4334. Brown, K. M.; Dennis, J. E. Numerische Mathematik 1972, 18, 289. Buback, M. Macromol. Chem. 1980, 181, 373. Chen, C. H.; Vermeychuk, J. G.; Howell, S. A.; Ehrlich, P. AZChE J . 1976, 22, 463. Donati, L.; Marini, M.; Marziano, G.; Mazzaferri, C.; Spampinato, M.; Langiani, E. In Chemical Reaction Engineering; Wei, J., Georgakis, C., Eds.; ACS Symposium Series 196; American Chemical Society: Washington, D.C., 1982; pp 579-590. Ehrlich, P.; Mortimer, G. A. Adu. Polym. Sci. 1970, 1 , 386.
Foster, G. N.; Hamielec, A. E.; MacRury, T. B. In Size Exclusion Chromatography (GPC);Provder, T., Ed.; ACS Symposium Series 138; American Chemical Society: Washington, D.C., 1980; pp 131-148. Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations; Prentice Hall: Englewood Cliffs, NJ, 1971. Goldman, K. In Ethylene and its Industrial Deriuatiues; Miller, S . A., Ed.; Ernest Benn Limited: London, 1969; pp 150-167. Goto, S.; Yamamoto, K.; Furui, S.; Sugimoto, M. J . Appl. Polym. Sci., Appl. Polym. Symp. 1981, 36, 21. Hulburt, H. M.; Katz, S. Chem. Eng. Sci. 1964, 19, 55. Katz, S.; Saidel, G. M. AIChE J . 1967, 13, 319. Laird, R. K.; Morrell, A. G.; Seed, L. Discuss. Faraday SOC.1956,22, 126. Lee, K. H.; Marano, J. P., Jr. In Polymerization Reactors and Processes; Henderson, J. N., Bouton, T. C., Eds.; ACS Symposium Series 104; American Chemical Society: Washington, D.C., 1979; pp 221-252. Luft, G.; Lim, P.; Yokawa, M. Makromol. Chem. 1983, 184, 207. Michels, A.; Geldermans, M. Physica 1942, 9, 967. Michels, A.; Geldermans, M.; De Groot, S. R. Physica 1946,12, 105. Ogo, Y. J . Macromol. Sci.-Rev. Macromol. Chem. 1984, C24, 1. Powell, M. J. D. Math. Programming 1977, 12, 241. Ray, W. H.; Laurence, R. L. In Chemical Reactor Theory;Lapidus, L., Amundson, N., Eds.; Prentice Hall: Englewood Cliffs, NJ, 1977; p 532. Saidel, G. M.; Katz, S. J . Polym. Sei. 1968, 6, 1149. Shirodkar, P. P.; Tsien, G. 0. Chem. Eng. Sci. 1986, 41, 1031. Takahashi, T.; Ehrlich, P. Macromolecules 1982, 15, 714. Tatsukami, Y.; Takahashi, T.; Yoshioka, H. Makromol. Chem. 1980, 181, 1107. Woodbrey, J. C.; Ehrlich, P. J . Am. Chem. SOC.1963, 85, 1580. Yamamoto, K.; Sugimoto, M. J . Macromol. Sci. (Chem.) 1979, A13, 1067. Receiued for review March 5, 1987 Accepted November 3, 1987
The Activity and Stability of Ni/Si02 Catalysts in Water and Methane Reaction Abdurahman S. Al-Ubaid Chemical Engineering Department, The College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia
The activity and stability of Ni/Si02 catalysts were investigated for steam reforming of methane at 565 "C. Catalysts were prepared by the homogeneous-precipitation-deposition method, and the influence of the following parameters was investigated nickel loading, steaming, pretreatment, and steam-methane ratio during the reaction. The activity and characterization results show t h e formation of two types of nickel hydrosilicate on the surface of prepared Ni/SiOz catalysts, and these catalysts had permanent deactivation when used at high steam-methane ratio, which occurs via surface transformation, forming a nickel hydrosilicate layer. Nickel-based catalysts are widely used in many industrial processes, including hydrogenation, methanation of coal synthesis gas, and steam reforming of hydrocarbons. In these processes, Ni is commonly supported on the surface of an oxide support. The purpose of the support is to facilitate the formation of finely divided metallic particles, thus providing high surface catalytic area. Different supports have been used to achieve these goals such as SiOz,A1,0,, MgO, and TiOz (Martin et al., 1981; Bartholomew, 1976). Several parameters, such as the preparation method, pretreatment, nickel loading, etc., influence the activity of the catalysts. In the preparation of Ni/SiOz catalyst, for example, using a basic ammonia solution (Martin et al., 1981) favors nickel dispersion. The support is not necessarily a passive inert carrier, as was the concept; it may play a more significant role in
affecting the catalysts' activity. Taylor et al. (1964, 1965) found that the activity of Ni/silica-alumina is much lower than the activity of Ni/SiOz for the hydrogenolysis of ethane. Furthermore, this activity varies significantly with nickel loading (1-570 Ni). In recent years, much attention has been devoted to investigate the role of the support (Burch and Flambard, 1984; Cairns et al., 1983; Ozdagan et al., 1983; Vance and Bartholomew, 1983; Duprez et al., 1982) in Ni catalysts. It has been found that the activity for COz hydrogenation (Ozdagan et al., 1983) and CO hydrogenation (Vance and Bartholomew, 1983) increases with increasing metal support interaction in the order Ni/SiOz, Ni/Al2O3, and Ni/Ti02. It has been suggested (Burch and Flambard, 1984) that this behavior is part of an electronic or a structural effect, but it has not been possible to differen-
088S-5SS5/88/2627-0790~0~.50/0 0 1988 American Chemical Society
