Ind. Eng. Chem. Res. 1989,28, 381-386
381
KINETICS AND CATALYSIS Effect of Reactant Size Reductions on Catalytic Reaction Rates Phillip E. Savage* and Minoo Javanmardian Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109
The macromolecular constituents of petroleum residua and heavy oils decrease in size during catalytic hydroprocessing. This enhances their effective diffusivity and, because the catalytic reactions are typically diffusion limited, increases the reaction rate. In this paper we develop mathematical models to probe the influence of reactant size reductions on catalytic reaction rates for conditions representative of hydrodemetalation and hydrodesulfurization reactions. This quantitative analysis demonstrated that reactant size reductions occurring in the bulk fluid phase can increase the reaction rate a t most by a factor equal to the reciprocal of the effectiveness factor a t the reactor inlet. Size reductions occurring within the catalyst pores also increase the apparent reaction rate, and this rate enhancement is a function of the fractional size reduction, the Thiele modulus (a), and the ratio (A) of the reactant diameter to the catalyst pore diameter. For example, when d = 3 and A, = 0.52, an order of magnitude reduction in size increases the rate by 18.8%. This quantitative analysis has also suggested a reactor configuration that can take advantage of the reactant size reductions to achieve even higher rates of reaction. The diffusion of large molecules through catalyst pores of comparable dimensions, variously termed restricted, configurational, or hindered diffusion, is an important feature of processes such as coal liquefaction and heavy oil hydroprocessing. In these instances, the effective diffusivity can be significantly lower than the bulk diffusivity because of steric and hydrodynamic constraints, and the reduction in diffusivity is typically expressed as a function of A, the ratio of the reactant diameter, d,, to pore diameter, d,, as shown in eq 1.
In the regime of configurational diffusion, the effective diffusivity of a reactant molecule of constant size increases with increasing catalyst pore size (i.e., decreasing A) because large pores facilitate reactant transport to the catalytically active surface. However, for a spherical catalyst with a constant pore volume, a large pore diameter is obtained at the expense of a high specific surface area and hence catalytic sites. Thus, the effects of pore size on diffusivity and on surface area are in opposition, and as a result there exists an optimal pore size (or optimal value of A) at which the observed reaction rate is maximized. This has been verified experimentally (Shimura et al., 1986; Spry and Sawyer, 1975) and theoretically (Rajagopalan and Luss, 1979; Ruckenstein and Tsai, 1981; Do, 1984). The literature provides several physical and mathematical models that have incorporated the effects of configurational diffusion on catalytic reaction rates. These models have probed the influence of the catalyst pore size and pore size distribution (Shimura et al., 1986; Rajagopalan and Luss, 1979; Spry and Sawyer, 1975; Do, 1984; Young and Rajagopalan, 1985), catalyst pellet size and geometry (Chang and Crynes, 1986; Rajagopalan and Luss, 1979), changes in pore geometry (Prasher et al., 1978b;
* Corresponding author. 0888-5885/89/2628-0381$01.50/0
Rajagopalan and Luss, 1979; Guin et al., 1986; Chang and Crynes, 1986), reactant concentration (Do, 1984), and reactant size distributions (Spry and Sawyer, 1975) on the apparent reaction rate. However, a feature omitted in previous models is the effect of reactant size reductions on the apparent reaction rate. That reactant size reductions do occur is supported by the reported (Sughrue et al., 1988; Takeuchi et al., 1983; Asaoka et al., 1983; Shiroto et al., 1983; Savage et al., 1988) decrease in the molecular weights of the metals- and heteroatom-containing compounds and the macromolecular asphaltenes in heavy oil during hydroprocessing. For example, Asaoka et al. (1983) reported that the molecular weights of the asphaltene fractions from several heavy feedstocks were reduced 5-fold via reactions over a hydrotreating catalyst. They noted that this molecular weight decrease was consistent with asphaltenes being macromolecules containing several condensed polycyclic cores linked together through aliphatic or heteroatomic bridges. In this paper we examine the effects of reactant size reductions, occurring both outside the catalyst pellet in the bulk fluid phase and within the catalyst pores, on catalytic reaction rates.
Model Development The reduction in the sizes of the macromolecular constituents of heavy oils and residua can result from both thermal (Reynolds and Biggs, 1985; Savage, 1986; Savage et al., 1988) and catalytic (Sughrue et al., 1988) reactions. Consequently, one can reasonably expect the size reductions to occur in the bulk fluid surrounding a catalyst pellet (Le., external size reductions) as well as within the catalyst pores (i.e., internal size reductions). In this section we develop mathematical models to probe the effects of both external and internal size reductions on catalytic reaction rates. These models consider a single, first-order irreversible reaction occurring in an isothermal, spherical 0 1989 American Chemical Society
382 Ind. Eng. Chem. Res., Vol. 28, No. 4, 1989
catalyst. External mass-transport limitations are assumed - same- size, and , the to be absent, all molecules: are of the reactivity of a molecule is taken to be independent of its size. The last assumption is a reasonable one because the intrinsic kinetics of a catalytic reaction such as removal of metals or heteroatoms from a polycyclic core should not be appreciably altered by the presence (or absence) of peripheral substituents that link such cores together. External Size Reductions. The rate of a catalytic reaction can be written as rate = qklffppSaCp= qklffppSaKpCs (2) where the equilibrium partition coefficient, K,, which can be a function of A, relates the reactant concentration at the pore mouth, C,, to the concentration at the exterior surface, C, (Satterfield et al., 1973; Colton et al., 1975; Prasher et al., 1978a). For a spherical catalyst pellet containing identical cylindrical pores of uniform cross section, the specific surface area, Sa,is related to the pore volume, V,, and pore diameter, d,, by 4v, 4V,X s a = - - -(3) d, ds where the relation d = d,/X has been employed to obain the latter equality. gubstituting this expression for Sainto eq 2 permits the reaction rate to be written as 4‘7klrrPpVp~pCs rate = (4) d, The effectiveness factor (9) and partition coefficient (K,) can be functions of X and hence altered by changes in the reactant size. The other parameters in eq 4 (excepting X and d,) are independent of d,. Therefore, the ratio of reaction rates for two molecules with different sizes is
-rate - rate,
~KpdaoX =-- qKpX 1 lloKpodsX0 O o K p d o P
(5)
where the subscript o denotes parameters evaluated at the initial reactant size d,,, and P is introduced to denote the fractional reduction in reactant size. To determine this ratio of rates, we need to calculate the values of the parameters in eq 5. The effectiveness factor is a function of the Thiele modulus, 9, as given by
where 9 is defined by eq 7 for a spherical catalyst pellet. Equations 1 and 3 have been substituted for De and Sa, respectively, to obtain the latter equality.
Experimental data and theoretical developments have led to several different expressions for F(X) (Satterfield et al., 1973; Colton et al., 1975; Prasher et al., 1978a; Spry and Sawyer, 1975; Renkin, 1954; Chantong and Massoth, 1983; Ternan, 1987), but Figure 1, which displays the variation of F(X) with X for three representative but mathematically distinct expressions, shows that the numerical values of F(X) are generally consistent, particularly at low values of A. In the present model, the empirical correlation of Satterfield et al. (1973), given in eq 8, will be used for demonstrative purposes to relate the ratio of the reactant and pore sizes to the effective diffusivity. F ( h ) = e-4.6x
with Kp = 1
(8)
O
.
l
“‘1 1
Satrerheld
0.2
(1973)
I
I
I
0.6
0.0
e t a1
F ( i )= c - * 6 *
0.4
0.6
0.8
1
o
x Figure 1. Effect of X on effective diffusivity.
By use of eq 8 for F(X),the Thiele modulus can now be expressed as
The parameters d,, X, and DBin eq 9, and consequently the numerical value of a, can change as the reactant size decreases. The bulk diffusivity typically increases as the size of the diffusing molecule decreases, but the available empirical correlations and theoretical equations provide conflicting functional forms (Hines and Maddox, 1985). In the present model, the Stokes-Einstein equation, where DB is inversely proportional to d,, will be used, and therefore the quantity d a B in the denominator of eq 9 is constant and independent of d,. The numerical value of 9 can then be calculated for any reduction in reactant size from eq 10 by choosing initial values for X and the Thiele modulus (Le., A, and 9,).
Two different initial values of the Thiele modulus are examined in this paper; a0= 12 and 3. These were chosen because the implied effectiveness factors of qo = 0.23 and 0.67 are representative of those typically reported (Ternan, 1983; Newson, 1975; Shah and Paraskos, 1975) for heavy feedstock demetalation and hydrodesulfurization, respectively. The initial values of X selected were the corresponding optimal values of Xqpt = 0.22 and 0.52, respectively. Ruckenstein and Tsai (1981) determined the latter value by using the correlation of Satterfield et al. (1973) for F(X) and an approximate relation for effectiveness factors in the range 0.9 > q > 0.3, and following the same procedure, it can also be shown that A, = 0.22 for 11 C 0.3. Note, however, that these values of A,, are approximate and that Rajagopalan and Luss (1979) have shown that A, is not constant but rather varies continuously with 9 (or q) within the ranges of effectiveness factors above. The value of 9 determined by eq 10 can then be used in eq 6 to calculate the corresponding effectiveness factor. Substituting this value of 7 into eq 5 permits determination of the effects of external reactant size reductions on the relative reaction rates. Internal Size Reductions. Equation 11 describes steady-state diffusion and reaction within the pores of an
Ind. Eng. Chem. Res., Vol. 28, No. 4, 1989 383
71
isothermal, spherical catalyst pellet. The boundary con8.0
1
m, = 1 2
\
\
I
ditions are dCldr = 0 at r = 0 and C = C, at r = R. Note, however, that because we are using the empirical correlation of Satterfield et al. (1973) for F(X),where the equilibrium partition coefficient, K p ,was found to be equal to unity, the reactant concentration at the pore mouth will be equal to its concentration at the pellet surface. Thus, the second boundary condition can also be expressed as C = C, at r = R. The effective diffusivity is a function of the reactant size because both the bulk diffusivity and F(X) increase as the size of a reactant molecule decreases within the catalyst. To account for internal reactant size reductions in this model, we again denote the fractional size reduction as (3 and allow this parameter to be a function of the radial position. Clearly the precise dependence of on r will depend on the particular reactant molecule and the kinetics of the reactions responsible for the size reductions. However, our purpose in this investigation is not to model the reactions of a specific feedstock but rather to assess the general importance of reactant size reductions. Consequently, we used the exponential expression shown in eq 12, where CY is an adjustable parameter related to the (3 = cl,/d,, = exp(-cY(l
-r/R))
r
6.0
Case 2
L
1 ' . = n'22 0 23
\\
i
\
2 0.4
0.0 00
10.0
0.2
06
0 8
10
P
,
\
_1
'\
I
mo = 3
\
A. = 0.52 ' i o= 0
0.0 ! 0.0
I
I
0.2
0.6
0.4
0.8
67
.o
P
(12)
size reduction at the center of the catalyst pellet, to provide a relationship between P and r. This function possesses a convenient mathematical form, and it is suitable for probing the general effects of reactant size reductions on catalytic reaction rates. Performing the differentiation indicated on the first term in eq 11, incorporating eq 1,8, and 12, substituting the rate law for a first-order reaction, and defining a dimensionless concentration C#I and radial position $ lead, upon simplification and rearrangement, to
I
Figure 2. Effect of external reactant size reductions on relative reaction rate: (a, top) a, = 12, A, = 0.22, 9, = 0.23. (b, bottom) a, = 3, A, = 0.52, 9, = 0.67.
Results and Discussion External Size Reductions. The effects of a decreasing reactant size on the relative reaction rate, as calculated from eq 5, are presented here for two different cases. In case 1, the size of the reactant molecules decreases, but all catalyst properties (i.e., pore diameter) remain unchanged, so eq 5 and 10 become
P exp(-4.6X0(1 - P))@:C#I = 0 (13) where
Equation 13 was solved numerically by finite-difference techniques to obtain the concentration profile within the catalyst. The values for X, and 9,used were the same as those examined in the previously discussed case of external size reductions. Having computed the concentration profile within the catalyst, the relative rate could be calculated by integrating numerically over the entire catalyst as indicated by re1 rate =
rate with size reduction rate with no size reduction (I~C#II)~ d$ with P # 1 J1@$2d$
with /3 = 1
The effectiveness factor was determined from
This case would correspond physically to the sizes of the reactant molecules decreasing as they traversed a plug-flow reactor packed with identical spherical catalysts. On the other hand, in case 2 we allow the catalyst pore size to decrease by the same proportion that the reactant size decreases so that X is held constant at its initial optimal value, A,. Consequently, the relevant equations are
-rate rate,
- -17 -1 To
P
_CP -- 1 a 0
A reactor packed with catalysts such that their pore sizes decrease down the reactor by the same fraction that the reactant size decreases (i.e., X = A, at each axial position) would be an idealized physical representation of this case. Figure 2 summarizes the results of the reaction rate calculations and shows that for case 1, in which the catalyst pore size is constant, an order of magnitude decrease in the reactant size (i.e., 0 = 0.1) increases the reaction rate by factors of 3.3 and 1.5 for 9,= 12 and 3, respectively. The rate increases because both the bulk diffusivity and F(X) increase as the reactant size decreases, and this reduces the intraparticle diffusional limitations and increases the effectiveness factor. Figure 3 provides quantitative confirmation of the foregoing qualitative argument by
384 Ind. Eng. Chem. Res., Vol. 28, No. 4, 1989 i o -
\
OR-
