Ind. Eng. Chem. Res. 1991,30, 1281-1289
1281
Network Model Investigation of Gas Transport in Bidisperse Porous Adsorbents John H. Petropoulos,f John K. Petrou,+and Athanasios I. Liapis*** Physical Chemistry Institute, National Research Center for Physical Sciences "DEMOKRITOS", CR-15310 Aghia Paraskevi, Attiki, Athens, Greece, and Department of Chemical Engineering and Biochemical Processing Institute, University of Missouri-Rolla, Rolla, Missouri 65401 -0249
A capillary network model consisting of a micropore network permeated by one of macropores of randomly varying size has been constructed. Although simplified (to keep computer space and time requirements low) in relation to a real bidisperse porous adsorbent or catalyst, it embodies the d e n t pore structural features likely to determine the gas-transport behavior of such porous solids. Suitable model calculations of Knudsen gas-phase and surface diffusion enabled us to (i) validate useful approximate methods for the more economical evaluation of network permeability and (ii) demonstrate certain important characteristic effects of nonrandom bidispene pore structure on transport behavior and their practical consequences, especially in connection with the experimental determination of surface diffusion coefficients,
Introduction Study of the relation between the gas-transport properties of porous solids and their pore structure is of great importance for the intelligent design of the porous adsorbents and catalysts widely used in the chemical and petrochemical industries. Such porous solids are commonly characterized by their pore size distribution, f ( r ) . The simplest and industrially most widely employed methods for the determination of f ( r )involve analysis of nitrogen sorption and mercury penetration isotherms (Gregg and Sing, 1982; Sing 1990, Dullien, 1979),wherein the porous solid is usually modeled as a bundle of cylindrical capillaries. The gas-transport properties of the porous medium may be described on the basis of this model by introducing a structure factor K , which is physically interpreted mainly in terms of the orientation of the capillaries relative to the direction of flow and the tortuosity of flow paths (Barrer, 1963; Johnson and Stewart, 1965; Satterfield, 1970). In practice, however, K is used essentially as an adjustable constant. Furthermore, this model fails to account for important features of the relative permeability properties of porous media, such as the existence of a percolation threshold below which the permeability of the porous medium vanishes although a significant fraction of the pores therein may still be permeable (e.g., Kirkpatrick, 1973). In model studies of gas-phase or gas-phase and surface transport in porous adsorbents by Nicholson and Petropoulos (1971,1975), the porous medium was represented much more realistically as a network of capillaries of randomly varying radius (cf. also the earlier modeling of liquid permeability and liquid-liquid displacement by Fatt (1956) and Schopper (1966)). The permeability, P, of the capillary network was then studied as a function of f ( r )and network connectivity, np The need to resort to a cumbersome numerical evaluation of P was a serious practical obstacle to wide use of this model, but this difficulty has now been largely eliminated by resort to suitable approximate analytical solutions (Nicholson et al., 1988). The network model studies referred to above are representative of porous adsorbents wherein the variability of pore radius may reasonably be considered to be random.
* Author to whom correspondence should be addressed.
National Research Center for Physical Sciences 'DEMOKRITOS". Univeristy of Missouri-Rolla.
*
This would most likely be the case in porous solids produced by compaction of a collection of irregular nonporous particles. Such porous solids are expected to be monodisperse, i.e., characterized by unimodal f ( r )functions, as illustrated in Figure 1. Although the aforementioned model studies included bimodal f(r) functions, it is important to realize that real bidisperse porous structures are typically the result of pressing ill-fitting microporous particles together, as illustrated in Figure 2. What one has, in effect, is a monodisperse network of relatively narrow intraparticle pores permeated by a more or less well-defined monodisperse network of relatively wide interparticle pores. The transport behavior of such nonrandom bidisperse pore networks is expected to be closer to that of the individual macro- and micropore networks arrayed in parallel rather than to the behavior of the corresponding random network with bimodal f ( r ) (Nicholson and Petropoulos, 1971; Nicholson et al., 1988). Accordingly, studies of model bidisperse networks are needed. However, the only method of evaluating the permeability, P, that would be immediately applicable here is the direct numerical solution (DNS) of the transport equations at each node (see below), and one should bear in mind that the network size required to model adequately a real (three-dimensional) bidisperse porous adsorbent would put heavy demands on computer resources. This difficulty stems principally from the need for a statistidy adequate representation of the variability of pore size in the sparse macropore network and would not arise if this structural feature is ignored, as is done in the "periodic" bidisperse network model of Gavalas and Kim (1981). The possibility of circumventing the problem by the use of the Bethe-lattice-type networks, which are amenable to exact analytical treatment, must be noted (Reyes and Jensen, 1985). However, model porous media of this type (with nT > 2) are nonhomogeneous, because the number of Bethe-lattice nodes generated from a given origin increases with distance from that origin faster than the corresponding volume. Thus, for example, if the aforementioned origin is at the center of a spherical particle, the porosity of the particle will increase indefinitely from the as center outward (and at a rate that increases with b), is well illustrated by Figure 3 of Reyes and Jensen (1985). Although a suitably chosen Bethe-type model porous solid may prove useful as a mathematical (as opposed to a physical) approximation to a particular transport property of a homogeneous model porous solid (cf. Reyes and Jensen, 19851, the correspondence between the two and the
0888-5885/91/2630-1281$02.50/00 1991 American Chemical Society
1282 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
solely through micropores, and a third through maaoporea in series with micropores. The basic simplifying assump-
r(bl
(a)
Figure 1. Schematic representation of (a) pack of nonporous particles (monodisperse porous solid); (b) resulting pore radius distribution.
r(bl
(01
Figure 2. Schematic representation of (a) pack of microporous particles (bidisperse porous solid); (b) resulting pore radius distribution.
degree of approximation involved must be established on a heuristic case-by-case basis. Hence, the problem of efficient evaluation of the transport properties of homogeneous bidisperse model networks remains. Our own approach to this problem involves (a) judicious simplification of the model network, while retaining the salient structural features likely to control important aspects of transport behavior, and (b) validation and use of good analytical approximations as computational shortcuts, where possible. Encouraging preliminary results on (b) using the simplest possible model network consistent with (a) have already been reported (Petrou et al., 1990). This work is extended in the present paper by a study of gas transport (Knudsen gas-phase and surface diffusion) in a more realistic bidisperse network, coupled with detailed consideration of the connection between the model studied and a real bidisperse porous medium. The results obtained yield a fuller picture of the practical usefulness of different methods of solution, on one hand, and a deeper insight into the effect of bidisperse pore structure on gas transport and especially on surface transport, on the other hand. Before closing this section, we refer briefly to what is perhaps the simplest and most widely known attempt to model bidisperse porous media, namely that of Wakao and Smith (1962). This model assumes that the flux permeating through the porous solid can be divided into three components, one entirely through macropores, another
tions inherent in the model are (i) the aforementioned flux components are parallel and independent, (ii) the fraction of the cross-sectionalarea of the porous solid occupied by each of these components is determined by a probabilistic argument based on cross sectioning the porous solid and rejoining the cut surfaces at random, and (iii) the macroand micropore diffusion coefficients are calculated on the basis of the relevant mean pore size. These assumptions introduce an undetermined degree of approximation in the evaluation of the flux through the porous medium on one hand and render the model unsuitable for a study of the effect of specific pore structural features on transport behavior on the other hand.
Theory Formulation of Pore Gas Flux. The usual simplified expression for the flux J, of a dilute gas, in the absence of adsorption, through a narrow pore, idealized as a uniform cylindrical capillary of radius r and length h,, is Jm = 2/33.rro,PAC,/h, = rB$AC,/h, = P/,AC,/h, (1)
where ACm is the difference in gas concentration between the ends of the pore, Dg is the mean gas molecular speed; B = 2og/3, and P/, = TB? may be termed the permeance (permeabilitytimes cross-sectional area)of the model pore. In eq 1 pore end effects and the effect of finite pore length are neglected for the sake of simplicity (Nicholson and Petropoulos, 1971, 1973; Reyes and Jensen, 1985; Nicholson et al., 1988). In the presence of adsorption of the gas on the pore walk, an “adsorbed” or ysurface”transport term is added to eq 1 (Barrer, 1963; Nicholson and Petropoulos, 197% which now becomes where k, is the Henry law adsorption coefficient, Do, is the surface diffusion coefficient, and P’ is the permeance of the model pore for the given adsorgable gas. Capillary Network Model for a Porous Medium. A porous medium may be conveniently modeled by a large regular array of nodes joined by uniform capillaries of variable radius. Nicholson and Petropoulos (1971,1975) used a cubic (in three dimensions) or square (in two dimensions) array of nodes that could be linked by capillaries lying along the sides (length h) or face diagonals (length h d 2 ) of the square (or cubic) unit cell (cf. Figure 3). We assume that there are Ne layers of nodes, each containing a total of N, nodes, with a number of capillaries meeting at a typical node. A fixed gas-phase concentration difDIRECTION OF GAS FLUX
1 (a)
(b)
Figure 3. Examples of two-dimensional capillary network configurations (for three-dimensional networks, see Nicholson and Petropoulos (1971, 1975)): (a) square network (% = 4); (b) square network with diagonals (% = 8).
Ind. Eng. Chem. Res., Vol. 30, NO. 6,1991 1283 ference AC, is applied between the first ( k = 1) and last ( k = No) layers, while periodic boundary conditions are in force laterally. Radii rij are assigned from the distribution f ( r ) to all N J N , - l)nT/2 capillaries of the network by going through all nodes systematically from i = 1 to i = N J N , - 1). The distribution f(r) is defined in the range ro 5 r Irb and is subject to the normalizing condition
lorbf(r)dr = 1
(3)
To obtain the (steady-state) flux, J, through the model porous medium, it suffices to calculate the flux between any successive layers of nodes, if the node concentrations, C, are known exactly. If the solution C, is not exact, it is best to use an average value over all layers, i.e.
where j'takes values corresponding to nodes linked to i but located in an adjacent layer (the factor of l/z allows for counting each capillary twice in the summation); AC&f = C, - ; , C h represents the spacing between layers and the shortest distance between nodes within a layer; and vilh is the length of capillary ( i j 3 ;thus, uy = 1 and uy = 1 or 4 2 in the networks illustrated in parts a and b of Figure 3, respectively. Given J, the permeance P', and permeability P of the model porous medium, assumed to be in the form of a pellet of thickness 1 and cross-sectional area so, can be calculated form their respective definitions, namely J = P'AC,/l = sOpAC,/l (5) For mathematical convenience, we introduce the reduced gas-phase concentration variable ci = (C, - C,J / AC, where CgIrepresents the constant gas-phase concentration at the nodes of the Neth layer. Hence, the boundary values of c are C k i N , = 0;Ck-1 = 1 (6) Note also that 1 = (Ne- l ) h
(7)
From eqs 4-7 we obtain
P' = sJ' = f/2q(l/~y)CP6iyA~~iy J
(8)
a
In the absence of any dependence of the variability of P'pf, Acif on j l , eq 8 reduces to P'= sJ' = '/2N,(N,- l)(P\iyAcpif)El/uy i'
(8a)
were broken brackets denote (arithmetic) average value. Alternatively, one may express P' or P in terms of the 'effective medium average" value of the pore permeance P ~ NThis . is defined by the operation of replacing the actual capillary network by one of identical size and structure ("effective medium"), except that all elementary conductors (capillaries) have the same permeance P'* This implies, incidentally, that the corresponding AcNf = AC,N = constant, where Ac,N follows from the boundary conditions, namely (cf. eq 6)
effective medium network under the same applied AC, is equal to that through the actual capillary network; Le., eq 4 remains valid if we substitute Pkiy = P',N and Acpiy = Ac,N = l / ( N e- 1 ) (cf. eq 9). Hence, eq 8 becomes (10)
P' = SOP = % N P 6 ~ ? 1 / v y
Porous media are commonly characterized by their porosity e and specific pore surface area (per unit volume of the porous medium) A, which are given in terms of the present model by
where it is assumed that the pore volume of the solid is accounted for by the capillaries (Le., the nodes have no volume or, alternatively, the volume of the nodes is apportioned among the capillaries meeting there); j takes % values appropriate to the nodes linked to node i, and proper allowance has been made for counting each capillary twice in the summations. If rij varies independently of vi, eqs 11 and 12 reduce to
Substitution from eqs 11 or l l a into eqs 8, 8a, or 10, as appropriate, yields an expression for P in terms of e. For example, from eqs 10 and l l a , we obtain
P = Ko@\N/7r(r2)
(13)
where KO
= C(l/vy)/CVj i'
(14)
J
is the "orientation" or "anisotropy" factor equal to 1 / 3 or 1 / 2 for isotropic three- or two-dimensional networks, respectively (Nicholson and Petropoulos, 1971, 1975; Nicholson et al., 1988). As shown below (see eq 181, p',,~ for a network with randomly varying r is a function of f ( r )and only. Hence for a real monodisperse porous medium of f(r) known from suitable porosimetry measurements, the only relevant additional modeling parameter required is the network connectivity, (detailed network configuration is immaterial). The diffusion coefficient, D,, of the porous medium for pure gas-phase transport follows immediately from eq 13, namely Dg = p g / f = K$',Ng/r(1'2)
(15)
If, in accordance with eq 1, we define a pore radius rN by P'pNB = ? r B l ' ~ ~ (16) then eq 15 becomes
D, = K&N3/
(r 2 )
,
(154
(9)
A surface diffusion coefficient, D,, for the porous medium is defined by Ak,D, = P - P, = P, = (P'- P'J/sO = P'JsO (17)
The value of P'fl is so chosen that the flux through the
which upon substitution from eqs 10,12a, and 14 becomes
N,-1
c AC,N = (Ne - ~ ) A c , N=
k=l
Ck-1
- c&=N, = 1
1284 Ind. Eng. Chem. Res., Vol. 30,No. 6, 1991
For a network of uniform radius, eq 17a reduces to D, = @O,. The general outcome of eq 17a can be represented as
D,= K@@”,= K@’,
(17b) where the structure factor E,is greater than, equal to, or less than 1 (Nicholson and Petropoulos, 1973, 1975). As indicated above, % = 1when r = constant. Note also that z8 1as n~ m, in which case the network degenerates into a parallel array of pores (Nicholson and Petropoulos, 1973, 1975). Evaluation of Network Gas Flux. (a) Direct Numerical Solution (DNS): This approach (DNS) involves setting up a specific network of finite size and determining the unknown node concentrations ci, i.e., those with N , I i IN J N , - l),by simultaneous solution of the N,(N, 2) equations which result from application of the law of conservation of mass to each node i, namely CP’,ij(C, - C j ) / V j = c i c ( P ; i j / V j ) - Z ( P ’ , i j C j / V j ) = 0
- -
I
I
I
(18) In eq 18,j’ takes a total of n~ values appropriate to those nodes (in the same or adjacent layers) that are linked to node i. The methods of solution previously used by us (Nicholson and Petropoulos, 1971, 1975; see, e.g., Sahimi et al. (1990) for other methods) involved either (i) inversion of the band matrix of the coefficients of ci, cj (Gauss elimination), or (ii) iteration proceeding from an initial guess of the ci (usually the corresponding values in the effective medium network), through application of eq 18 to each node in turn, each time replacing the old ci value by the new one (Gauss-Seidel iteration), until successive cycles yield the same network permeance P’to the specified precision. The permeance P’is evaluated by means of eq 8 or eq 8a if applicable. Method i makes much higher demands on computer storage space than method ii. Method ii, on the other hand, even when speeded up by overrelaxation, is much slower in producing a reasonably precise ci solution. Fortunately, this difficulty is considerably alleviated here, because the rate of convergence of (PkiyAcpiy)is much higher than that of the individual ci. A hybrid method that strikes a balance between computer space and time requirements has also been successfully used (Petrou et al., 1990). In this, the c{s for a whole layer of the network are computed simultaneously by matrix inversion, treating the cis on adjacent layers as known (using for this purpose the values obtained from the previous iteration cycle or from the original guess where necessary). This is repeated for each layer (k = 2 to k = Ne - 11, and iteration is continued over as many cycles as necessary for satisfactory convergence of P’. In all the above cases, the computer requirements increase steeply with network size. Thus, whereas the DNS method of solution has the advantage of applicability to any type of network, ita utility in model computations has limitations, which (as pointed out in the Introduction) become particularly severe in the case of bidisperse porous media of the type under consideration here. A suitable model for the latter case (bidisperse porous macromedia) can be constructed by setting up a Na1Nel pore network and then converting each node to an Nmi: subnet, all needed additional pores being micropores (Petrou et al., 1990). The radii of both macropores (rmc)
Figure 4. (a) Portion of a two-dimensional square network (h= 4); (b) simple and (c) compound Wheatatone bridge unit cells of original network (a) which become single Conductors in the renormalized network.
and micropores (rmic)are assumed to vary randomly according to distributions fmc(r) and fmic(r)respectively, which together constitute f ( r )for the bidisperse network. The size of the latter is given by N&Ne - l)Nal(Nel1)Nd,2 and is obviously much greater than would be necessary for the purpose of modeling a monodisperse porous solid. (b) Analytical Approximations for Random Networks: Approximate analytical methods of evaluating P i N for infinite networks with randomly distributed radii (and hence P’,,ij) are available. In particular, a simple “effective medium approximation” (EMA,e.g., Kirkpatrick, 1973; Landauer, 1978) has been shown (Koplik, 1981; Nicholson et al., 1988), by comparison against the DNS, to be very successful over a wide range of radius distributions and network connectivity (nT)values (although it may fail in certain extreme cases). According to EMA
0 (19) where z P f f land cp(P’,,)dP’,= f ( r )dr. Equation 19 must be solved iteratively, but the procedure is much less cumbersome than the DNS. Furthermore, it has proved possible (Nicholson et al., 1988) to obtain the solution of eq 19 explicitly in the form of a series in ascending moments of Another method of immediate interest here is real-space renormalization (RSR). In this, a given network of c a p illaries or other elementary conductors, the permeance of which may be designated by P,,iC1)subject to the distri,is renormalized by dividing it into identical unit bution cells (c usters of capillaries or other elementary conductors, cf. Figure 4). These unit cells become single subject to distribution (p2(P’J,in the conductors Ppif2), renormalized network, which must have the same configuration as the original one (Stinchcombe and Watson, 1976; Stauffer, 1985). In principle, (~2(Pb) may be determined by enumerating the different unit cells resulting from all possible combinations of the P$(l)and establishing their relative weights. The renormalization process should then be repeated until satisfactory convergence to (PkfFm))= P f lis realized (Bemasconi, 1978). The rate of convergence may be enhanced by choosing a larger unit cell. Examples of simple and more complex unit cells that have been used for square networks (Bernasconi, 1978; Sahimi et al., 1983)are flustrated in parts b and c of Figure 4, respectively. In practice, the RSR procedure described above has been applied only to the simple percolation problem where P’Jl) = 0,1, and even in this case the number of different Pkj(n)increases with increasing n at such a rate that some simplifying assumption soon becomes necessary (Bernasconi, 1978). In this respect, EMA
n(PP)
Ind. Eng. Chem. Res., Vol. 30,No. 6,1991 1285
0
H
K CI
(a 1
r-
(b)
Figure 6. (a) Simple and (b) compound unit cells used for the application of REMA to the q u a r e bidisperse network studied here (Na= 7): thick line, macropores (permeance Pk-);, thin line, conductors (permeance P‘ & representing unit cells (size Ne2) of a micropore effective me&m network.
has a decided advantage over RSR. However, a single RSR step has been found to be a very useful preliminary operation for the subsequent use of EMA in situations where direct appplication of EMA gives unsatisfactory results, e.g. near the percolation threshold in the simple percolation problem referred to above (Sahimi et al., 1983). This combined RSR-EMA procedure is conveniently referred to as REMA. Bidisperse Network Model While the nonrandomness of bidisperse networks precludes the direct application of approximate analytical procedures such as EMA, the practical feasibility of DNS depends, as already stated, on reducing computer demands to manageable proportions. To this end, we have restricted ourselves (Petrou et al., 1990) to two-dimensional networks and in particular to the square (+ = 4) network (cf. Figure 3a or 4a). A bidisperse network was generated, as indicated in the theoretical section, by converting each node of an Nal(Nel- 1)macropore network (consisting of uniform cylindrical capillaries of length h, and randomly varying radius r,, and, hence, randomly varying permeance P’,,,,) into an Nd2 subnet, all needed additional elementary conductors being “micropore conductors” (length h, randomly varying permeance Pk,,,,ic). Furthermore, in view of the fact that the variability of Pkfic can reasonably be expected to be only a relatively minor effect, the aforementioned subnets were replaced for purposes of computation by the relevant effective medium network, Le., a single permeance PkN ic was assumed for the subnet conductors. With use of = 15, Ne, = 16 (which, on previous experience with random networks, would be expected to provide a reasonably satisfactory representation of the macropore network (Nicholson and Petropoulos, 1975; Nicholson et al., 1988)), a square bidisperse network with subnet size up to say Na2 = 72 (as illustrated in Figure 5), i.e., a total network size of Nal(Nel- 1>Nd2= 15 X 15 X 7 X 7, could be conveniently handled by the computer available (PRIME 9955), using the hybrid DNS method referred to above. From the modeling point of view, the above subnet size would of course, normally be unrealistically low, if the elementary conductors therein, Plpijfic = PIMdc, were to be regarded as representative of actual micropores. This difficulty can be largely circumvented by regarding the Nd2 subnets as the result of renormalization of large Nmi2= (N12Na)2micropore networks, using a square unit cell of size Naa2. Then, by eq 10
Figure 6. Macropore radius distribution (eq 30)used for the computations reported here (solid line), and corresponding continuous distribution (dashed line).
where P’opN,mic represents the effective medium average of actual micropore permeances: P’Opij,mic
= d3rijSai:
+ 2?rrij,mi&ps
(21)
Note, however, that N , should still be kept as large as possible, in order to allow for the fact that, in reality, exchange of gas flux between micropores and macropores can occur throughout the length of the latter. With the variability of r,, in the above bidisperse network represented by a simple two-valued distribution function: fmac(r)
= flak - r1) + f26(r - rz), 0 If1 I1,
f2
= 1 - f1 (22)
(where 6 denotes the delta function) a REMA procedure, so designed as to remove the nonrandomness of the bidisperse network in the RSR step, was feasible (Petrou et al., 1990). The unit cells of Figure 5a or 5b (which correspond to the simple and compound Wheatstone bridge type unit cells of Figure 4b or 4c respectively) were used for this purpose, under the designations of REMAl and REMAB, respectively. The permeances P’p(2)of all different unit cells were computed by matrix inversion (this computation step can be shortened drastically by making use of the rules of combination of network conductances). REMA2 was much more expensive in computer time than REMA1, partly due to the larger unit cell size but mainly because there are 2050 different values of Pk(2)in the former w e as against only 10 in the latter case. However, use of REMA2 in the aforementioned previous study (Petrou et al., 1990) was necessary to establish that REMAl is sufficient for precise results. The REMAl computations were far more economical in computer time than the DNS, but it was recognized that thiswas due in large part to the fact that f,(r) was limited to only two values of r. The DNS is largely (though not fully; cf., e.g., Nicholson and Petropoulos, 1975) indifferent to the nature of fmC(r),whereas the computer demands of REMA increase steeply with the number of values r,, can take. It is, therefore, important to demonstrate that realistic continuous fmc(r) distributions can be handled by REMAl through (i) discretization (cf., Sahimi, 1988) of fmac(r)to a reasonable number of rnuc values or (ii) replacement of complete enumeration of different unit cells by a random sampling technique (REMAS). For this purpose, we have chosen here a five-valued discretized form of the symmetrical triangular radius distribution of Nicholson and Petropoulos (1973), which is illustrated in Figure 6. In accordance with what has been said above, the working equations are as follows:
1286 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 Table I. Results of Model Computatione for Gas-Phase Transport and for Gas-Phase and Surface Transport (with ado= 1) in a Square Bidisperse Network with I-@) Given by Eq 3C (~’pNg)Phg P&g/(P)PNI)(PhL Pp ’N P’pN/(P;N)1.269 3.671 5.577 1.519 3.620 4.595 DNSl (15 min) 1.273 3.616 5.517 1.526 3.566 4.538 DNS2 1.268 3.610 5.483 1.519 3.560 4.513 DNSI 1.268 3.594 5.463 1.520 3.544 4.496 DNSI 1.270 3.623 5.510 1.521 3.513 4.536 mean (60 min) 3.627 4.607 1.270 REMAl (52 min) 3.600 4.577 1.272 3.650 5.558 1.523 REMASl (5 min) 1.269 3.680 5.584 1.518 3.628 4.604 REMAS2 (16 min)
-
Typical computation times on a PRIME 9955 computer lire also indicated.
(a) In the Nd2subnets, the permeance of the individual conductors is given by P\Ng,mic P\N.mic
(23)
= Na3P”pNg,mic = NnBTBrN,mi?
= P\Ng,mic + p\Na,mic
-
P’~N = Na3P’OpN
+ ( y2Na3 )T(rmic)kaDs,mk
NaSTBrN,mi?
(24)
rmic
where r N ~and c D are defined following eqs 16 and 17a, respectively, and = ~ $ ~ following d $ ~eq~17b. All required micropore network parameters, in particular P’OpNpmic, (rmic), and lint can be calculated separa1;ely for any specified micropore network by the methods described above (cf. eqs 17 and 19 and Nicholson and Petropoulos (1975)). For convenience, the co_mputations were performed in terms of permeances (P?normalized with respect to P’,,Ng,mic. Thus (234 P\Ng,mic =
-
P\N,mic
- 1)Nd2bidisperse network. The value of P’ N obtained upon setting Phmic= 0 in the Nal(Nel- 1)ddnetwork 2 will be designated by (P\N)mc.In accordance with our symbolism and eq 10, we have
-
= + p\Na,mic = + amic
macrowhere P’OPNrefers to the Nal(Nel- l)Nd2Na32 pore-micropore network and P\NGac to the Nal(Nel - 1) macropore network.
Results and Discussion Computations have been performed by means of both the DNS and REMA (REMA1, REMASl, REMAS2) methods for the square bidisperse network (Nal= 15, Ne, = 16, N, = 7) and the five-valued f&P) distribution function indicated above (Figure 6). The latter is specifically given by 5
flnac(8
where
P’p.mac
= P‘pg,mac + P L , m a c = TBrma: + 2~r,,kJl~, (27)
Equations 26 and 27 are the equivalents of eq 1 and 2, respectively, and upon nmormalization as above they become P‘,,mac
= P&,mac/p\Ng,mic
= rma:/NaBrN,mi:
=
fma:
(264
where we have defined a normalized macropore radius: ?ma,
= rmac/rN,mi>a1’3
(28)
The results are given in terms of the effective medium average permeance P’pNof the conductors of the Nal(Nel
(29b)
= p\N,mac/NaZ
(pkN)mac
(24a)
(b) In the Nal(Nel- 1) macropore network, the permeance of the individual conductors (macropores) is given by P‘pg,mac = TBrma? (26)
(29a)
=
c
f m W m=l
- Fm)
(30)
where PI = 1.8, f z = 2.4, P3 = 3.0, f 4 = 3.6, f5 = 4.2, and fl = f5 = l/g, f z = f4 = 2/9, and f 3 = 3/g. Results (see Table I) have been obtained both for pure gas-phase transport (amic = 0) and for gas-phase and surface transport with amie = 1. For this value of cymic, the permeance of the conductors of the Nd2subneta is doubled, as shown by eq 24a, whereas that of the macropores increases much less, as indicated by eq 27a. In the latter equation, we chose L ~ ~ /= 1/9. N ~This ~ choice / ~ is consistent with a value of Na3between 10 and 100, since iZ,,m,ic(rmjc)/rNmic 1 (Le., amic ao)in eq 25a. The results of Table I show that the presence of surface flux causes a substantial increase of bidisperse network permeance (P’a)but only a very minor enhancement of macropore network permeance Comparison of Gas Flux Evaluation Methods. In Table I, the results obtained by DNS, based on four different random number sequences for radius assignment (DNSl-DNS41, are compared with those of REMA. Agreement is good, in view of the approximate nature of REMA and the effect of limited network size in the case of DNS. Further material improvement of the DNS average result would require computation times well in excess of that required for REMA1. The result for REMASl, based on a 10% random sample of (simple) unit cells, shows that great economy in computation time is possible at the expense of relatively little loss in accuracy. In accordance with what has been said in the preceding section, REMA2 was not practical, but REMAS2 based on a random sample of 200 (compound) unit cells, gave a result in excellent agreement with REMA1, in conjunction with a large saving in computation time. Table I also shows that agreement between the different methods becomes very close indeed when the values of
-
-
((+)a.
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1287 Table 11. Total (Gas-Phase or Gas-Phase and Surface Flux through the Bidisperse Network Relative to That through the Macropore Network Alone) and Relative Magnitude of Synergistic Macropore-Micropore Effects in the Networks Studied Here and by Petrou et el. (1990)
Petrou et al. (1990)
3.120
1.364
1.274
P' /(pkN)mec are compared, i.e., when assessment of the rective contribution of the macropore and micropore networks to the flux through the bidisperse network is the primary object of interest. Transport Behavior. Table I shows that the gas flux in the presence of micropores is higher than that through the macropores alone by 27 70for pure gas-phase transport and by 47% for gas-phase and surface transport (with amic = 1). These values will, of course, depend on the relative size and number of micropores and macropores, i.e., on the magnitude of ( FmC) but will also be materially affected by the shape and breadth of fmC(F). This is illustrated in Table 11, where the present results are compared with those of Petrou et al. (1990) for the same bidisperse network, except that fmaC(P) is given by eq 22 with f l = f i = 0.5 and P, = 2, Pz = 4. In both cases (Pmac) = 3, but the present macropore network contains a high proportion of medium-size pores, whereas the previous one contained only narrow and wide macropores in equal proportions; the resultin marked constriction effects are reflected in a lower value for this case, as shown in Table 11. Hence the micropore contribution, being unimpeded by macropores, should be relatively higher. If (in addition to the fact that the micropore flux is unaffected by macropores) the presence of micropores does not materially disturb the macropore flux, then PkNfor the bidisperse network can be approximated by P ~ given N ~by
(b,,,),,
0.024 0.070
3.623 3.176
1.521 1.681
1.473 1.540
0.032 0.092
Table 111. Surface Flux and Synergistic Macropore-Micropore Effects Thereon, in the Bidisperse Network Studied Here and by Petrou et al. (1990)
P$Sl this work Petrou et al. (1990)
p'pN8
PpNg
p'plvu
0.974 1.081
0.215 0.254
0.907 0.913
Ph-1
AplpN,I
P'pNga
p'pNm
0.205 0.223
0.074 0.185
transport by the synergistic effect. Following eqs 31 and 32, we have PkNe = P k N - PkNg = PkNa + f l k N - PkNga - *kNg= PkNsa + P p N s (33) Table I11 shows that the observed surface flux is, indeed, enhanced over that given by the additivity assumption by 7.4% and 18.5% for the network studied here and for the one investigated by Petrou et al. (1990), respectively (the corresponding values for the gas-phase flux being 2.4% and 7.0%). This observation is of considerablepractical significance, because it means that, in practice, the surface diffusion coefficient may be substantially overestimated. Following ea 17a, we have
In fact, the above flux component additivity assumption is strictly valid only when Pm, is single valued (as in the model of Gavalas and Kim, 1981). In general, we have PkN
= PkNa
+ hPkN
(32)
The results of Table I1 show that A P f l > 0 (Le., there is synergism between macropore and micropore fluxes) and that APPN/P' N~ is enhanced (i) in the presence of surface transport an$(n) in the network studied by Petrou et al. (1990) by comparison with the present one. It is suggested that the mechanism of the observed synergism consists in the micropore network acting as a conduit for bypassing constrictions in the macropore network. Thus, observation i can be immediately understood as due to the relative enhancement of micropore permeance in the presence of surface transport (to which attention has been drawn above); whereas observation ii is attributable to the fact that the macropore network studied here is less highly constricted (as has already been indicated above). On the same basis, one may predict with some confidence that AP'W will be enhanced (other things being equal) also in the case of lower macropore connectivity (or of constrictions along the length of the individual macropores) and in three-dimensional bidisperse networks by comparison with two-dimensional ones. Hence, such synergistic effects are likely to be important in practice. This, in turn, justifies the introduction of the present model in place of the earlier one of Gavalas and Kim (1981). Observation i further suggests that surface transport is likely to be affected even more markedly than gas-phase
(34) where (35)
p'flsdcis given by eqs 24a and 25a, and P'fl.9mac is given by the corresponding expression (cf. eq 29b) = pkN~,mnc = p;N,mac - pkNg,mac 2 (Fmac )k,Ds,mac
- (Fmac)~,,acao
(36) ~ f l ~ ~ ~ / ~ Na32/3 B r ~ ~ 2 where, following eqs 17b and 25a, we have put Daw, = ~gizepoacD0~.Substituting from eqs 25a, 25b, 35, and 36 into eq 34, we get
-K.
(1
+
(&)(E
- I))(
hP kN8 1
+
pk_) (37)
which for a real bidisperse porous medium (large N&) reduces to W R , , m i c = 1 + W ' ~ N ~ / P ~ N ~ (38) ) Note from eq 38 that iz, z xSmic when AP' = 0, and R, > RSdc (D,> D,&J increasingly as APb,/F$m increases.
1288 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
Conclusion It has been shown that a capillary network model can be devised that may be considered to represent 13 real bidisperse porous solid reasonably well, while requiring only very modest computer resources. Previous results (Petrou et al., 1990) demonstrating the successful use of a REMA method of evaluating the permeability of networks with discrete fmc(r)functions have been confirmed here and extended particularly with regard to validation of a sampling variant of this method (REMAS), which is both time-saving and applicable to continuous fmac(r)distributions. As far as the effect of bidisperse pore structure on transport properties is concerned, the assumption of simple additivity of macropore- and micropore-network fluxes is a useful first approximation, but additional synergistic effects are also found. Our results provide some valuable insight into the pore structural parameters that affect the magnitude of these synergistic effects and show that the surface flux in particular may be markedly enhanced as a result. The practical consequence of this is that the surface diffusion coefficient (D,) measured on the bidisperse porous medium, which consists of packed microporous particles, will tend to be correspondingly higher than that (Ds,mic) measured on a single constituent particle. Nomenclature A
= specific pore surface area (eq 12)
Knudsen transport parameter (=2Dg/3),m s-l Cgi = node gas-phase concentrations, mol m-3 D = diffusion coefficient, m2 s-l f ( r ) pore radius distribution function h = shortest distance between nodes in NJVe network, m h, pore length (=vjh in NJVe network), m k, = Henry law adsorption coefficient, m 1 = thickness of model porous medium, m 1 number of pores meeting at a node of a regular network (network connectivity) N , = number of nodes per layer of network (=N,,N, for the model bidisperse network) Ne = number of layers of nodes in network (=(Ne,- 1)N, + 1 for the model bidisperse network) Na1(Nel) I N,(N,) for macropore network Nd2 size of renormalized micropore subnet in model bidisperse network N&2 = Nd2Nd2= actual micropore subnet in model bidisperse network P(P? gas permeability (permeance), m2 s-l (m4s-l) r = pore radius, m so = cross-sectional area of model porous medium, m2 0, mean gas molecular speed, m s-l (variable),,, value pertaining to bidisperse network when the micropores are nonconducting B
Greek Letters
= (=pP~,,ic) ratio of surface to gas-phase flux in a micropore subnet (-microporous particle) given by eq 25a a. = ratio of surface to gas-phase flux in a single micropore with r = rN,mic,given by eq 25b 6(z) delta function of z AC = gas-phase concentration difference applied between the f h t (k = 1) and last (k = Ne) layers of the model porous medium, mol m-3 Al” = defined in eq 32, m4 s-l c = porosity of model porous medium (eq 11) orientation or anisotropy factor defined in eq 14 KO ii, 1 structure factor in equation 17b ami,
vj
h,ij/h
1
Superscripts 0 = parameter pertaining to real micropores (absence of this
-
superscript implies that the micropore parameter in question pertains to the elementary conductom of the model micropore subneta in the bidisperse network, resulting from renormalization of the real micropore subneta; see text) = normalized parameter (dimensionless)
Subscripts
a = value calculated on the assumption of additivity of ma-
cropore and micropore fluxes in the bidisperse network parameter pertaining to gas-phase flux i a particular node in any model network (including the bidisperse network) i j or ij’ 1 a particular pore or elementary conductor in any model network (including the bidisperse network) mac parameter pertaining to the macropore network
g
(NnlNeJ
mic = parameter pertaining to the micropore subneta of the bidisperse network N p effective medium average value p parameter pertaining to individual pores or elementary conductors of any model network (includingthe bidisperse network); absence of this subscript implies that the parameter in question pertaina to the network or model porous medium as a whole s = parameter pertaining to surface flux; absence of subscript g or s implies that the parameter in question pertains to the total gas flux Literature Cited Barrer, R. M. Diffusion in Porous Media. Appl. Mater. Res. 1963, 2, 129-143. Bernasconi, J. Real-Space Renormalization of Bond-Disordered Conductance Lattices. Phys. Rev. B 1978,18, 2185-2191. Dullien, F. A. L. Porous Media: Fluid Transport and Pore Structure; Academic Press: New York, 1979. Fatt, I. The Network Model of Porous Media. Pet. Trans., AZME 1956,207, 144-181. Gavalas, G. R.; Kim, S. Periodic Capillary Models of Diffusion in Porous Solids. Chem. Eng. Sci. 1981,36, 1111-1122. Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. Johnson, M. F. L.; Stewart, W. E. Pore Structure and Gaseous Diffusion in Solid Catalysts. J. Catal. 1965, 4, 248-252. Kirkpatrick, S. Percolation and Conduction. Rev. Mod. Phys. 1973, 45, 574-588. Koplik, J. On the Effective Medium Theory of Random Linear Networks. J. Phys. C Solid State Phys. 1981, 14,4821-4837. Landauer, R. Electrical Conductivity in Inhomogeneous Media. Am. Znst. Phys. Conf. Proc. 1978,40, 2-45. Nicholson, D.; Petropoulos, J. H. Capillary Models for Porous Media: 111. Two-Phase Flow in a Three-Dimensional Network with Gaussian Radius Distribution. J. Phys. D Appl. Phys. 1971,4, 181-189. Nicholson, D.; Petropoulos, J. H. Capillary Models for Porous Media: IV. Flow Properties of Parallel and Serial Capillary Models with Various Radius Distributions. J. Phys. D: Appl. Phys. 1973, 6, 1737-1744. Nicholson, D.; Petropouloe, J. H. Capillary Models for Porous Media: V. Flow Properties of Random Networks with Various Radius Distributions. J. Phys. D Appl. Phys. 1975, 8, 1430-1440. Nicholson, D.; Petrou, J. K.; Petropoulos, J. H. Relation Between MacroscopicConductance and Microscopic Structural Parametera of Stochastic Networks with Application to Fluid Transport in Porous Materials. Chem. Eng. Sci. 1988,43, 1385-1393. Petrou, J. K.; Petropoulos, J. H.; Kanellopoulos, N. K.; Liapis, A. I. Iterative and Renormalized Effective Medium Approximation Methods for Evaluating Transport in Bidisperse Pore Networks. In Fundamentals of Adsorption; Proceedings of the Third International Conference on Fundamentals of Adsorption, May 7-12, 1989, Sonthofen, Bavaria, Federal Republic of Germany; Mersmann, A. B.. Scholl. S. E., Eds.: Eneineerine Foundation: New York,’1990. . Reyes, S.; Jensen, K. F. Estimation of Effective Transport Coefficients in Porous Solids Based on Percolation Concepts. Chem. Eng. Sci. 1985, 40, 1723-1734. I
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Ind. Eng. Chem. Res. 1991,30,1289-1293 Sahimi, M. On the Determination of Transport Properties of Disordered Systems. Chem. Eng. Commun. 1988,64,177-195. Sahimi, M.; Gavalas, G. R.; Tsotsis, T. T. Statistical and Continuum Models of Fluid-Solid Reactions in Porous Media. Chem. Eng. Sci. 1990,45, 1443-1502. Sahimi, M.; Hughes, B. D.; Scriven, L. E.; Davis, H. T. Real-Space Renormalization and Effective-Medium Approximation to the Percolation Conduction Problem. Phys. Reo. E 1983,28,307-311. Schopper, J. R. A Theoretical Investigation of the Formation Factor/Permeability/PorodlityRelationship Using a Network Model. Geophys. Prospect. 1966,14, 301-341. Sing, K. S. W. The Use of Gas Adsorption for the Characterization of Porous Solids. In Fundamentals of Adsorption; Proceedings of the Third International Conference on Fundamentals of Ad-
sorption, May 7-12,1989, Sonthofen, Bavaria, Federal Republic of Germany; Mersmann, A. B., Scholl, S. E., Ma.; Engineering Foundation: New York, 1990. Stauffer, D. Introduction to Percolation Theory; Taylor & Francis: London, 1985. Stinchcombe,R. B.; Watson, B. P. Renormalization Group Approach for Percolation Conductivity. J.Phys. C: Solid State Phys. 1976, 9,3221-3247.
Wakao, N.; Smith, J. M. Diffusion in Catalyst Pellets. Chem. Eng. Sci. 1962, 17, 825-834.
Received for reuiew May 14, 1990 Revised manuscript receioed November 19, 1990 Accepted December 2, 1990
Kinetics of the Conversion of Calcium Sulfate to Ammonium Sulfate Using Ammonium Carbonate Aqueous Solution Elamin M. Elkanzi* and Mohamed
F.Chalabi
Department of Chemical and Petroleum Engineering, U.A.E. University, P.O.Box 17555, Al-Ain, U.A.E.
A two-step procedure is described for measuring the intrinsic rate of the slurry reaction between calcium sulfate and ammonium carbonate. In the first step, experiments were conducted following the shrinking core model concept to get estimates of the kinetic parameters. These estimates were then used in a mathematical model to predict an experimental program under which the conditions of chemical reaction control is likely. The second step is to measure the intrinsic kinetic parameters under these favorable conditions. A simple power law model fitted to the experimental data of the second step predicted an activation energy of 72 kJ/mol and a pseudo-first-order reaction with respect to ammonium carbonate. The procedure may prove useful in measuring the intrinsic kinetics of fluidaolid heterogeneous reactions and may reduce the effort put on experimental work to establish favorable conditions for chemical reaction control. Introduction Ammonium sulfate for use as a fertilizer is generally manufactured from anhydrous ammonia and strong sulfuric acid. An alternative process originally developed in Germany as long ago as 1909 (Higson, 1951)is now in use. It is based on combining ammonia and carbon dioxide to produce ammonium carbonate, which then reacted with gypsum or anhydrite in a three-phase slurry reactor to yield ammonium sulfate and calcium carbonate. However, little work is reported regarding the kinetics of the reactions involved (Ganz et al., 1959; Chalabi et al., 1975). This may be attributed to the complexity resulting from the influence of physical processes that must take place in order for the overall reaction to proceed. In the threephase gas-liquid-solid system involved here, the overall rate of reaction at any instant of time may be controlled by one or more of the following steps: (1) gas-liquid mass transfer; (2) chemical reaction between ammonia solution and carbon dioxide to form ammonium carbonate; (3) transfer of ammonium carbonate solution through the liquid-solid film; (4)diffusion of ammonium carbonate solution through the calcium carbonate product layer; ( 5 ) chemical reaction at the surface of unreacted calcium sulphate core. As these steps take place in series, it is evident that the slowest step would be rate controlling. Andrew (1954),in an investigation on carbon dioxide absorption by partially carbonated ammonia solutions, reported that the rate of carbon dioxide absorption is controlled by the reaction between carbon dioxide and ammonia. Chalabi et al. (1975)reported that ammonium carbonate was present in
* Author to whom correspondence should be addressed. 0888-588519112630- 1289$02.50/ 0
appreciable amounts even at high conversions of calcium sulfate when the overall reaction was carried out in a fluidized bed reactor. These findings suggest that the first two steps may not control the overall reaction. The purpose of this paper is to report a kinetic study for the reaction described by the equation CaS04.2H20(s) + (NH4)&03(aq) CaC03(s) + (NH4)2S04(aq) + 2H20(1) (1) A two-step procedure is to be investigated in order to measure the intrinsic rate of the above reaction. The first step is to establish reaction conditions under which chemical reaction control is likely. These reaction conditions, namely, calcium sulfate particle size, calcium sulfate slurry concentration, ammonium carbonate initial concentration, rate of mixing, and reaction temperatures, are predicted by using a mathematical model based on the relative importance of the resistances offered by steps 3-5 above. The kinetic parameters of the model are estimated experimentally by using the shrinking core model concept (Levenspiel, 1972). The second step is to measure the intrinsic kinetics of the reaction by using an experimental program predicted from the results of the first step.
-
Mathematical Model As far as is known, the effects of mass-transfer limitations on the course of reaction 1 have not yet been investigated. Previous experimental investigations (Ganz et al., 1959; Chalabi et al., 1975) have shown that the calcium sulfate conversion increases with decreasing size of calcium sulfate particles and with decreasing slurry concentrations. Therefore it seems a reasonable assumption that the chemical reaction occurs mainly at the surface of the available calcium sulfate. As the reaction proceeds, 0 1991 American Chemical Society
