GASEOUS DIFFUSION AND FLOW IN COMMERCIAL CATALYSTS A T PRESSURE LEVELS ABOVE ATMOSPHERIC C H A R L E S N . S A T T E R F I E L D A N D P. J O H N C A D L E Department of Chemical Engineering, Massachusetts Institute of Technology,Cambridge, Mass. 02139
Counterdiffusion measurements of helium and nitrogen in the absence o f forced flow are reported on five commercial catalysts over the pressure range 1 to 65 atm. and at ambient temperature. Surface diffusion of nitrogen made an increasing contribution to the total nitrogen flux with increase in pressure and a t the highest pressure was of comparable magnitude to the volume diffusion flux. Effective surface diffusion coefficients were estimated to b e (0.3 to 0.9)X 1 0-3 sq. cm. per second. The diffusion data were analyzed in terms of a parallel path pore model. The calculated tortuosity factors were sensibly invariant with pressure; i t was concluded that the model can b e used with confidence to predict diffusion in a particular catalyst under a variety of conditions from a single diffusion measurement. Steady-state simultaneous diffusion and forced flow measurements a t ambient temperature and elevated pressures are reported for a prereduced Harshaw methanol synthesis catalyst. The data were analyzed b y the three methods developed b y Evans et al. In terms o f their “dusty gas model” their empirical parameter, C, was found to be a linear function of pressure and independent o f the pressure difference. Analysis in terms of the capillary tube model showed that the diffusion tortuosity factor was about 1.8 times the flow term tortuosity factor.
QUAXTITATIVE estimation of intraparticle mass transfer by Adiffusion in porous solids with wide pore size distributions is beset with difficulties which stem from the present poor understanding of pore geometry, compounded by the fact that both Knudsen and bulk diffusion modes are frequently important. These two diffusion modes interact and are correlated by the transition region equation (Evans et al., 1961; Rothfield, 1963; Scott and Dullien, 1962). T h e crux of the prediction problem is the different temperature, pressure, and structural dependencies of the two mechanisms. I t is a simple matter to define a n effective diffusion coefficient in terms of either the Knudsen or bulk diffusion equation, or indeed, to define a model with a n adjustable parameter such as the tortuosity factor to describe diffusion in porous solids, and t o evaluate this from a diffusion measurement a t room conditions. However, a knowledge of mass transfer rates a t ambient conditions is seldom required, and so to be of value the effective diffusion coefficient or tortuosity factor must be amenable to extrapolation. Complications arise because with change in conditions there is a change in interaction between the diffusion modes and between the modes and the porous structure. There are no clear directives in the literature on suitable extrapolation procedures and, furthermore, no model has been shown to have general validity over a wide range of conditions and materials. T h e primary objective of this study was t o investigate the behavior of the parallel pore model discussed elsewhere (Johnson and Stewart, 1965; Satterfield and Cadle, 1968), in the moderate to high pressure region. With few exceptions, intraparticle mass transfer in catalysts has been treated as occurring solely by diffusive mechanisms. However, it is widely recognized that small pressure gradients may exist in pellets under reaction conditions when there is a change in the total number of moles on reaction, but a contribution from forced flow is generally neglected because of the dependency of the Poiseuille term on the fourth power of the
202
I&EC FUNDAMENTALS
pore radius. Although this action may be justified in most cases under low pressure conditions, it may not be justified at high pressure levels, because with a n increase in pressure level the forced flow contribution assumes more importance in relation to the diffusion transport. At very high pressures diffusion tends to be completely bulk, so for a fixed mole fraction difference and pressure difference across a porous structure the diffusion flux becomes independent of pressure level, whereas the forced flow contribution is almost proportional to the pressure. A second objective of this study was to evaluate methods of predicting simultaneous diffusion and forced flow a t elevated pressure levels and with small pressure differences. Theory of Simultaneous Steady-State Diffusion and Forced Flow
Diffusion and flow in porous media have been recently reviewed by Barrer (1963). The problem of isothermal binary diffusion with a superimposed pressure gradient has been treated by Evans et al. (1962) in terms of a “dusty gas” concept. They have shown that the flux rate relationship has the same form as the transition region equation for pure diffusion conditions, but the pressure dependency of the bulk diffusion coefficient must be accounted for as well as the fact that the flux ratio no longer obeys the inverse square root molecular weight law. The relationship may be expressed as: -1
+
where y = 1 Nos/NoA. The Knudsen diffusion coefficient, D,, is expressed as KAr in order to isolate the invariant terms from the pore radius. Three approaches for correlation and prediction are con-
sidered here, starting with Equation 1. I n the first, the parallel path pore model is applied to Equation 1, yielding the expression :
The net diffusion flux, N r , is calculated a t the mean pressure conditions which prevail for the diffusion and flow case, and it has been evaluated in this work by Equation 10.
NT = (YNA
T h e pressure is assumed to be a linear function of the distance through the pellet-namely,
P = Po
+ (PL - P OZ )
(10)
I n Equation 10 NA is defined by Equation 8. T h e forced flow flux was calculated from Equation 11, which represents the application of the parallel path model to the flow equation developed by Wakao et al. (1965).
(3)
I n terms of the reduced coordinates: Y A * = y A / y A o ; I* = x / L ; and w = ( P L - P o ) / P o ,Equation 2 becomes:
where
P- I -Po
+" PL L
A basic assumption of the parallel path pore model is that the composition is not a function of the pore size-Le., sufficient cross passages exist to cause the composition and pressure a t any value of x t o be uniform.. Equation 4 has no analytical solution, but making use of this assumption, it can be rearranged t o a first-order nonlinear differential equation:
The third method of correlation here uses the dusty gas model which invokes a n empirical parameter, CB, determined by experiment. The model is defined by Equation 13:
Each of the three methods of correlation involves one adjustable constant, the tortuosity factor in Equation 5 or 9,
(5)
or the dusty gas constant, CB, in Equation 13. Each can be evaluated from steady-state simultaneous diffusion and flow experiments, and their behavior with change in conditions can be investigated.
with the boundary conditions:
(7) Equation 5 was solved numerically using the fourth-order Runge-Kutta method. The tortuosity factor is the only unknown and it was eval!uated by a trial and error method, the values of 7 being selected until the boundary condition (Equation 7) was satisfied. Initial values of T were available from the constant pressure case, where Equation 2 can be solved analytically to give the expression:
Evans et al. (1962) developed two further methods of predicting simultaneous diffusion and flow, one in terms of the capillary flow equation and t.he other from the concept of a "dusty gas." I n the capillary tube model the net diffusion and flow flux is defined as the sum of the net pure diffusion flux and the forced flow flux.
N"T =
+F
(9)
Experimental
Steady-state simultaneous diffusion and flow studies were performed on a random sample of 12 pellets, '/d-inch by l/*inch, prereduced Harshaw methanol synthesis catalyst. Experiments were designed according to the two-way classification experimental design with two observations per cell, to investigate the influence of pressure and pressure difference and their interaction upon the tortuosity factor. T h e pressure levels were 0, 250, 500, and 750 p.s.i.g., and the pressure differences ( P x - P H e )applied across the pellets were 0, 2, and 4 p.s.i. Counterdiffusion measurements in the absence of forced flow were made on the Harshaw catalyst and also on random samples of four other catalysts a t each of the above pressure levels and at about 950 p.s.i.g. T h e other catalysts and sample sizes were: Prereduced BASF methanol synthesis catalyst. 1 3 pellets,l5 X 5 mm. Prereduced Haldor Tops@ methanol synthesis catalyst. 12 pellets, 7 X 7 mm. Girdler G52, prereduced 3 3 7 , nickel on refractory oxide support. 6 pellets, X inch. Girdler G58, palladium on alumina catalyst. 15 pellets, a/ls X a/l8 inch. Other information is given in Table I . T h e macrovoid fraction is defined as comprising pores exceeding 100-A. radius, VOL. 7
NO. 2
MAY 1960
203
Table 1.
Av. Diam.,
.
Catalyst
Cm
Harshaw BASF Haldor Topwe - .
0.631 0.494 0.593 0.620 0.392
G5 2 G5 8
Table
11.
Av. Length, Cm , 0.596 0.637 0.591 0.640 0.441
Macropore Distributions for synthesis Catalysts
Physical Properties of Catalysts
G. P p f X
6.30 3.92 5.60 3.12 4.12
Three Methanol
(Pore sizes are values of radius) 7 0 Total Pore Volume In pores In pores
