Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 459
Influence of Catalyst Pore Size on Demetallation Rate Kuppuswamy Rajagopalan and Dan Luss" Department of Chemical Engineering, University of Houston, Houston, Texas 77004
A model is developed for predicting the influence of pore size on the rate of demetallation which accounts for the influence of the restricted diffusion of the large reactant species in the pores. Numerical calculations indicate that
the demetallation activity decays with time in an almost linear fashion. Simple algebraic expressions are derived for predicting the pore size yielding either the optimal initial activity or the optimal lifetime activity. It is proven that among all catalysts with the same surface area and porosity, the largest initial activity is attained for pellets with a uniform pore size.
Introduction A rapid development of the capacity and technology of hydrotreating processes has been triggered by the recent, strict demands of removing sulfur and metals from residium oil as well as coal and shale derived fuels. A large fraction of the metals in resid oil is contained in high molecular species such as asphaltenes, which consist of highly condensed heterocyclic and aromatic rings to which sulfur, nitrogen, oxygen, and metals (mainly vanadium and nickel) are bound. Yen and Erdman (1962) have shown that the asphaltenes consist of several layers of condensed aromatic plates. Other investigators, such as Larson and Beuther (1966),believe that the asphaltenes are micellar clusters of molecules. The size of these large species is in the range of lo4 to m. The diffusion of large molecules in pores of comparable size is restricted, due to the exclusion of the molecules from part of the volume and to the increased drag caused by the proximity of the solid wall. Ferry (1936) was the first to predict the exclusion effect, and Renkin (1954) and Anderson and Quinn (1974) determined the magnitude of the enhanced drag. A review of experimental results of restricted diffusion was presented by Bean (1972). Drushel (1972) and Richardson and Alley (1975) noted that asphaltene molecules are excluded from entering into fine pores. Several patents are based on the exclusion of asphaltenes from pores smaller than 3 X to 7 x m (Adams and House, 1970). The rate of demetallation is first order with respect to the metal concentration (Sato et al., 1971; Chang and Silvestri, 1974; Spry and Sawyer, 1975; Dautzenburg, 1978). The intrinsic activity of the catalyst is proportional to its surface area. Thus, a catalyst with small pores has a large surface area and a high intrinsic activity. However, because of the large diffusional limitations in small pores, the observed rate will be rather low. Catalysts with large pores have a small surface area so that their activity is not high. The optimal activity is attained for catalysts with an intermediate pore size. Experimental illustrations of the influence of pore size on the activity were presented by Van Zoonen et al. (1963), Sooter and Crynes (1975), Spry and Sawyer (19751, and Richardson and Alley (1975). Metal sulfide deposits and coke plug the pores of demetallation catalysts. Beuther and Schmid (1963), Sat0 et al. (1971), and Dautzenberg et al. (1978) observed that the coke deposit builds up very rapidly to an equilibrium level. Following this initial period, the coking rate becomes rather low. Newson (1975) developed a model predicting the life of demetallation catalysts assuming uniform pore plugging. Hughes and Mann (1978) presented a model for computing 0019-7882/79/1118-0459$01.00/0
the activity of plugged catalysts neglecting the diffusional limitations and assuming that the deposit is in the form of a wedge. Dautzenberg et al. (1978) developed a model which accounts for the change with time in the deposited metal sulfide profile. The above studies did not account for the change of the restricted diffusion with decreasing pore size. Spry and Sawyer (1975) were the first to examine the influence of pore-size distribution on the initial demetallation activity accounting for the influence of pore size on the diffusivity. Their calculations and experiments indicate the existence of an optimal pore size. The purpose of this work is to develop simple algebraic relations predicting pore sizes which yield the optimal initial activity and the optimal lifetime activity. In addition, we examine the influence of pore-size distribution and pellet geometry on the activity of the catalyst. Demetallation in a Single Pore We develop here a model of the demetallation in a single cylindrical pore of length 21. We assume that the reaction is isothermal and first order and consider only the period following the initial fast coking. Thus, the initial diameter refers to the size after the rapid coke buildup has occurred. The restricted diffusivity in the pore can be described as
Dp = DABG(X)
(1)
where DABis the bulk diffusivity and X = rm/rp
(2)
For X < 0.1, G(X) is very close to unity. Various theoretical analyses (Spry and Sawyer, 1975) indicate that for hard sphere molecules G(X) may be approximated rather closely by G(X) = (1 (3) We shall use a pseudo-steady-state approximation to describe the demetallation and diffusion in the pore, as the characteristic times of these rate processes are much smaller than those associated with the changes in the shape of the pores. In accordance with experimental observations, we assume that the metal sulfide deposit does not affect the intrinsic demetallation activity per unit surface area, but that it reduces the surface area and may affect the restricted diffusion in the pores. The corresponding species conservation balances are (4)
subject to the boundary conditions 0 1979 American Chemical Society
460
Ind. Eng. Chem. Process
Des. Dev., Vol. 18, No. 3, 1979
c=co
z = o
(5)
-dC= o
z=1
(6)
dz Introducing the dimensionless variables {=z/l u = c/co
N = c/(rr;(o)fi)
u = l f = O (9) du -=o {= 1 df The instantaneous demetallation rate in a single pore of half length 1 is
s 0
1
2rrpkCdz = 2rr,(O)1kCO77(t)
(11)
where d t ) = Llu(i',t)f(T,t)di.
(17)
The value of r,(O) can be determined experimentally from the relation r,(O) = 2f/(PbSg) (18)
Equation 4 and the corresponding boundary conditions may be rewritten as
R,(l,t) =
sumption of nonintersecting pores was made so that we can later account for the influence of pore-size distribution. The number of pores per unit external surface area is
where S, is the measured surface area per unit weight of catalyst. The observed rate per unit volume of the slab is Rsi(7) =
R,(GL,t)
rr,Z(O)Lfi
(19)
where R,[(7)1/2L,t] is the demetallation rate a t time t of a single pore of half length (7)1/2L. The demetallation in a spherical pellet can be computed assuming that it consists of many nonintersecting pores of different length, which originate at the exterior surface and terminate a t various positions inside the sphere. Following Rester et al. (1969), we define by n(z) dt the number of pores of length between z and z + dz. At a depth between x and x + dx the pore length is between ( T ) I /and ~ X ( T ) ' / ~ ( X + dx). The surface area bounded by this spherical shell is
(12)
is defined as the instantaneous effectiveness factor. Because of the pore plugging, 77 decreases monotornically with time. The rate of metal deposit per unit length of the pore is dm - = cy2rrpkCM dt where cy is the number of metal sulfide molecules per molecule of organometallic reactant. When the metal deposit thickness is 6 m = pmr[r,2(0)- (r,(O) - a)'] = pmrrP2(O)(l- f " ) (14) Thus, eq 13 may be rewritten as
where r is the radius of the sphere. The total surface area is
&
2rr, (0) yr;rzn ( z )dz Defining by q(x)dx the fraction of surface area a t depth between x and x + dx from the exterior surface we obtain from (20) and (21)
Differentiation of (22) yields 7n(fix)
-q'(x) =
The corresponding boundary condition is (16) f=1; t = O Equations 8 and 15 need to be solved simultaneously to yield the metal deposit profile as a function of time. The pseudo-steady-state equation was solved by using a finite difference scheme with 26 nodal points. The computed reactant profile was substituted into (15) and used to compute new values off. The updated values off and the corresponding values of g (eq 1,3, and 7 ) were substituted into ( 8 ) , which was resolved. The computations are continued until complete plugging of the pore-mouth occurs. Numerical experimentation indicated that f values accurate up to at least four significant digits were attained when the time step used in integrating (15) was 0.00369rP(0)l u . Demetallation in a Catalytic Pellet We examine here the demetallation in catalytic pellets. We consider first a semiinfinite slab catalyst of half thickness L. We assume that the pellet consists of many nonintersecting pores with a radius of r,(O) and a length of 2L(7)'iZ, where 7 is the tortuosity factor. The as-
Lfirzn(z)dz For a homogeneous sphere q(x) =
"( 1 ;)' r -
Differentiation of (24) and comparison with '(23) yields n(z) = K
( Gr) 1- -
The total volume occupied by pores in a fresh catalyst is ~V'rrr,2(0)zn(z)dz = t4ar3/3
(26)
Substitution of (25) in (26) yields
The total reaction rate in a single sphere is Lfirn(z)R,(z,t)dz
(28)
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
461
w 4
8-
z
P
t
I
I 0
8 P O R E RADIUS
0
12
IO-^^]
Figure 1. Dependence of demetallation rate on pore size and time kg mol m-3; k = 5 X m s?; rm = 6.25 on stream: c,, = 2 X X m; u = 9.97 X m s-l; 4 = 3; 6 = 0.5.
80
CATALYST AGE
I20 [days]
Figure 2. Dependence of demetallation rate on catalyst age for several pore sizes. Parameters same as in Figure 1.
+f ;
Substitution of (27) into (28) yields the following expression for the reaction rate per unit volume
The above integral can be computed rapidly by a Gaussian-Legendre quadrature. In this method the integral is evaluated as a weighted sum of the integrand a t certain quadrature points (the roots of the appropriate Jacobi polynomial). Numerical experimentation indicates that an accuracy of a t least four significant digits is attained using five quadrature points. Figure 1 describes the demetallation rate per unit volume of a slab catalyst as a function of pore radius for several ages of the catalyst. In this and all other examples reported in this paper G ( h ) was computed by eq 3. For all pore sizes the rate decreases with time on stream due to metal-sulfide deposition, which reduces the area on which the reaction can occur and increases the diffusional resistance. The initial activity vs. pore size graph has a sharp peak, so that small deviations from the optimal pore size strongly reduce the initial rate. However, as the catalyst ages the activity vs. pore size graph becomes flatter and the sensitivity of the rate to deviations from the optimal size diminishes. Exactly the same trend was observed for spherical pellets. Figure 2 compares the demetallation of slab catalysts with different pore radii. The graphs indicate that the activity decays in an almost linear fashion and vanishes when the pore radius is reduced to 2rm. Hence, the total life of a catalyst increases with increasing initial pore radius. The pore radius corresponding to the optimum initial activity is 3.1 X m for this example (see Figure 1). For pores smaller than the optimal radius, increasing the pore radius always improves the catalyst performance as this increases both the initial activity and the total life of the catalyst. This is seen from the graphs corresponding to r,(O) of 2 X lo3 m and 3 X lo3 m. For pores larger than the optimal size an increase in pore size reduces the initial activity but increases the total life. Consequently, the activity graphs for these catalysts intersect, as evidenced m and 11 by the graphs corresponding to r,(O) of 7 X x m. Figure 3 compares the demetallation rate per unit volume of catalyst of two pellet geometries-slab and sphere. The diameter of the sphere was selected as thrice the thickness of the slab so that both geometries have the same external surface area to volume ratio. For catalysts
40
12
-SLAB
"E
qi P
Y
w
5
CATALYST AGE
[days]
Figure 3. A comparison of the demetallation rates in a spherical and slab catalyst. All parameters same as in Figure 1 and
4 = 4 = 3.
with the same pore size, the demetallation rate in a slab catalyst is somewhat larger than that in a sphere. Both the sphere and slab catalyst deactivate a t the same time, as the time required to reduce the pore mouth to 2r, is independent of the shape of the catalyst. Optimal Initial Reactivity The initial reactivity of catalysts with very large or very small pores is low since in the former case the intrinsic activity, which is proportional to the surface area, is low, while in the latter case the diffusional resistance is very high. We present here a simple algebraic relation predicting the pore size for which the optimal initial activity is attained. The initial demetallation rate of a pore can be expressed as
R,(.\/;L,o) = 2rr,(0).\/;LkCoq
(30)
where q = ~ ( 0= ) tanh
4=
@/4
Ld-
Substitution of (30) into (19) yields R d 0 ) = c,xov where
(31) (32) (33) (34) (35)
For a specific reacting system and catalysts with the same porosity C, is a constant, and the optimal pore size satisfies the equation (36) which may be rewritten as
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Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
dlnq dln4 -1 d In 4 d In Xo Substitution of (1) into (32) yields --
(37)
$=L
(39)
where
Differentiation of (31), use of (38), and substitution into (37) yields
It follows from (40) that the optimal value of Xo, or equivalently the optimal pore size (= A$,), depends only on $. For large values of 4,(40) degenerates to d In G(Xo) 1= 2 d In Xo When the restricted diffusion factor G(X) is defined by (3), eq 40 becomes
'
100
a : L J k . 7 . r / 3 : k
Figure 4. Dependence of Xo* on
4 and $ for slab and a spherical
catalyst, respectively.
Similar to the case of a slab the pore size, or equivalently Xo, which yields the optimal initial demetallation rate has to satisfy (37). Substitutions of (50) and (49) in (37) yields the following condition for the optimal Xo
Differentiation of (50) yields
For all positive 4 24 05I 1 sinh 2 4 Thus, it follows from (42) that
(43)
(44)
where the equality sign holds for large 4 and the superscript * denotes the optimal size. For a spherical pellet the initial activity can be expressed as R&O) = C1XOVs (45) and the initial effectiveness factor in the sphere is defined as L G r n ( z ) tanh (hz)dz 9s =
(46)
h Lfirzn(z)dz
Substitution of (27) into (46) and defining
(49)
Thus, the optimal X,depends only on s/. For large values of $ (51) degenerates to (41). It can be shown that the right-hand side of (52) is bounded in (0, -1). Hence, for G(X) defined by (31, the pore size yielding the optimal initial activity has to satisfy (44). The above analysis indicates that the optimal pore size for a slab and spherical pellet for which $ = $ will be the same for very small or large values of and will differ only in the intermediate range of $. The analysis also points out the strong influence of the restricted diffusion coefficient, G(X), on the optimal pore size. FiguJe 4 describes the graphs of l / X o * as a function of $ and $ for a slab and spherical pellet. In both cases the limiting value of 5, predicted by (44), is attained for sufficiently large values of the Thiele modulus. For a slab catalyst this limiting value is attained for $ larger that about 10, while for a sphere it is attained for values of $ larger than about 100. The reason that for a sphere the asymptotic value of l / X o * is attained-for such high values of the modified Thiele parameter $ is that the sphere contains short pores for which the optimal l/&*is smaller than 5. Figure 4 is useful for computing the pore size corresponding to the optimal initial activity. Optimal Lifetime Activity A catalyst becomes inactive when the pore mouth is reduced to the size of the organometallic molecule (= 2rm). According to (15) this occurs when
+
It follows that the total life of a catalyst is proportional to the initial pore size (rm/&) and catalysts with large pores have a longer life than those with small pores. In practice, catalysts are usually replaced when their activity reaches a certain limiting value and before they are completely inactive. This limiting value may be expressed as a fraction of the optimal initial activity. For example, for a slab the
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
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75
limiting activity can be defined as RL = FR,L*(O) = FC1Xo*q(Ao*)
(54) The total demetallation activity over the useful lifetime of a catalyst equals the integral of the instantaneous activity over its life. The exact determination of the activity-time graph requires numerical solution of the coupled differential equations (15) and (18). Thus, it is not possible to derive a simple algebraic expression predicting the pore size which yields the maximal total activity. Numerous numerical calculations, some of which are shown in Figures 1 and 2, indicate that the deactivation rate is almost independent of time and that the straight line combining the initial and final activity a t (t = t,) is a very good approximation of the activity-time graph. The total activity as computed by this approximation differs only by a few percent from the exact value and the difference is definitely smaller than that due to uncertainties in the kinetic parameters and the model. We shall use this approximate activity-time dependence to derive a simple prediction of the optimal pore size. According to this simplifying assumption the constant rate of deactivation for a slab is
51
451
$=
L J Z k ^ r /DABr,
Figure 5. Dependence of A,* on
*
