Ind. Eng. Chem. Res. 2007, 46, 4245-4256
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Optimal Distributor Networks in Porous Catalyst Pellets. I. Molecular Diffusion Eivind Johannessen, Gang Wang, and Marc-Olivier Coppens* Physical Chemistry and Molecular Thermodynamics, DelftChemTech, Delft UniVersity of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
The effect of introducing a network of distributor channels in a nanoporous catalyst pellet was studied. The diffusivity in the distributor channels is orders of magnitude larger than in the nanoporous material itself. Introduction of distributor channels leads to an increasing volume-averaged diffusivity but a decreasing amount of catalytically active material per unit volume. This implies that, to maximize yield, there is an optimal volume fraction occupied by the distributor network and an optimal way to distribute and connect this fractional volume in a catalyst pellet of a given geometry and volume. It was found that a distributor network consisting of a large number of straight channels with a position-dependent channel diameter, and without branches, is the global optimum for all 2D and 3D catalyst pellet geometries without corners (spheres, infinite cylinders, infinite sheets, etc.) when there is pure molecular diffusion in the distributor channels and a first-order reaction in the catalyst. The volume fraction occupied by the distributor channels should always be less than 0.5. Moreover, the difference between the global optimum and a network of channels with constant diameter is negligible with respect to the yield. In practice, the latter is preferred because of easier synthesis and an almost identical yield compared to the former. When only the skin of a catalyst pellet contributes to the yield, these results also apply to geometries with corners (squares, cubes, etc.) because effects of corners are insignificant. 1. Introduction There has been tremendous progress in the synthesis of nanoporous materials with controlled porosity for applications like catalysis, sorbents, fuel cells, sensors, and microreactors.1-5 Nanoporous materials are attractive for these applications because of the large surface area in the pores. A drawback is that molecules diffuse slowly through the narrow pores, so that the accessibility to the internal surface is limited. In catalysis, this means that the observed reaction rate may be much lower than the intrinsic rate. In many cases, only the skin of a catalyst pellet, with a diameter varying from micrometers to centimeters, contributes to the chemical conversion. It has been realized for a very long time that the accessibility of the active sites is important in catalysis. Therefore, a lot of work has been devoted to effects of pore network structure and pore shape on diffusion and reaction in porous media.6 Pioneering work on optimal design of catalyst pellets was performed by Hegedus and co-workers.7-9 Keil and Rieckmann10,11 optimized the micropore radius and porosity of porous catalysts. Morbidelli et al.12 searched for the optimal distribution of catalyst sites inside a porous catalyst pellet. Also, Bejan’s13 research on fluid flow and heat conduction in hierarchical, selfsimilar trees of channels bears similarities to catalyst network optimization. One way to reduce diffusion limitations is to introduce a network of large distributor channels through which diffusion is much faster than in the nanoporous material itself. However, the channels consume space that could be occupied by the nanoporous, active material and will thus reduce the intrinsic reaction rate per unit given nanoporous catalyst volume. Therefore, there exists a distributor network that provides the optimal combination of volume-averaged diffusivity and intrinsic reaction rate. At present, most features of the optimal network are unknown. Some relevant questions are as follows: How * Corresponding author. Tel.: +31 15 278 4399. Fax: +31 15 278 5006. E-mail:
[email protected].
large should the volume fraction occupied by channels be? What is the optimal density of the channels? How should the channels be organized? What should the size of the channels beswill one channel size suffice or is a broad size distribution needed? How thick should the walls of nanoporous material be? The objective of this paper is to, for the first time, provide a general answer to some of these questions. Quite similar questions could be asked about monoliths and other structured fixed-bed reactors. Moulijn and co-workers’ work perfectly exemplifies how structuring of catalysts and chemical reactors can increase their performance.14-17 Some aspects of the optimal distributor network were studied earlier.18-20 In those studies, much attention was paid to the distribution of channel sizes. Coppens and Froment18 studied a cube made out of nanoporous material, in which they introduced large channels. By assuming that there are no diffusion limitations in the distributor channels, they showed analytically that a broad channel-size distribution sometimes increases the yield significantly compared to a network consisting of channels of equal size. Later, the numerical results of Gheorghiu and Coppens19 suggested that a hierarchy of channel sizes is beneficial. They numerically optimized the yield on a catalyst square by introducing a finite number of channels and found that the optimal network was very complex and strongly dependent on the type of diffusion and on the intrinsic reaction rate. In some situations, the optimal structure was fractal-like. Recently, Wang et al.20 used a continuum approach to the network optimization on a square. Their findings were in contradiction to earlier results. The main result was that the performances of the optimal bidisperse structure (nanopores plus channels of equal size) and the optimal bimodal structure (nanopores plus a distribution of channel sizes) were not significantly different. Since the bidisperse network is easier to synthesize than the bimodal one, the former is favored in practice. Wang et al. also found that there were only diffusion limitations in the distributor channels in the optimal network, which explains why Coppens and Froment18 found different
10.1021/ie061444s CCC: $37.00 © 2007 American Chemical Society Published on Web 03/01/2007
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results. It has also been found experimentally that the diffusion limitations in industrial fluid catalytic cracking catalysts are in between the zeolite crystals, rather than inside the crystals.21,22 The conclusions of Gheorghiu and Coppens19 and Wang et al.20 differ because of the limited number of channels in the study by Gheorghiu and Coppens, which accentuate finite-size effects, in particular, the role played by corners and edges. The work presented in this paper is different from previous work in two ways. Earlier, pellet geometries with corners (cubes and squares) were studied most extensively. In such geometries, the optimal distributor channels close to a corner will generally differ from those in the central parts of each face (edge in 2D). The truly optimal distributor network for geometries with corners is, therefore, often very complicated, with a hierarchy of channel diameters and a varying channel volume fraction over the pellet.19,20 In this paper, only pellet geometries without corners, like spheres, circular disks, infinite cylinders, slabs, and sheets, are studied. There are three reasons for this. First, the main objective of this paper is to study general properties of the optimal distribution network and not the effect of a particular pellet geometry. Second, in a number of industrially important catalyst pellets, only an external layer or “skin’’ is active. Third, the number of pore channels in an industrial pellet is typically very large, diminishing the effect of corners. The results found here also apply to other pellet geometries when only the skin of the pellet contributes to the yield because of strong diffusion limitations, because the effects of the corners are very small in this case. A second difference between this paper and the work by Gheorghiu and Coppens19 is the way in which the distributor network is constructed. Since there are no corners or edges, it is assumed that the distributor network is a periodic assembly of a single, repeating unit, and the properties of the repeating unit are optimized. The optimal distributor network in a finite pellet, with corners, is generally not as simple as the one studied here, but the yield of the true optimum is not significantly larger than that of the optimal, simplified network structure, even for a catalyst square.20 A network consisting of a single, repeating distributor unit is thus, at least, a very good approximation of the true optimum. For pellet geometries without corners, infinitely many symmetry planes can be defined, so there are reasons to believe that the optimal distributor network is indeed a periodic assembly of a single, repeating unit. In order to limit the scope of this paper, it is assumed that there is pure molecular diffusion in the distributor channels. This means that the diffusivity in a channel does not depend on the size of the channel and that the volume-averaged diffusivity in the pellet depends in good approximation only on the volume fraction occupied by channels. It shall be shown that these properties have a great impact on the nature of the optimal network. The more complicated situation where Knudsen diffusion also plays a role, so that the diffusivity depends on channel size, will be discussed in a separate paper. The paper is organized as follows. In the next section, the details of the optimization problem are presented and the model system is defined. In Section 3, properties of the optimal distributor network are discussed. This discussion culminates with the calculation of the channel diameters and the thickness of the nanoporous walls in near-optimal distributor networks in Section 4. Finally, some concluding remarks are given in Section 5. 2. Optimization Problem Consider a nanoporous catalyst particle in which a first-order, isothermal reaction, A f B, takes place. The particle can either
Figure 1. 2D slab with finite thickness (a) and the circular disk (b) are examples of the pellet geometries studied in this paper. The networks of distributor channels are assumed to be periodic assemblies of repeating units (c or d). Straight distributor channels with (c) and without (a, b, and d) branches are studied. The distributor channels are gray, and the nanoporous material is white.
be a 3D object (sphere, cylinder, cube, sheet, etc.) or a 2D object (circle, square, slab, etc.). The 2D particles could be called catalysts on a chip. The effective diffusivity De for species A and the intrinsic reaction rate constant k in the nanoporous material are fixed and known. The effectiveness factor for the pellet is
η)
∫V kc dV kc0V
(1)
where c is the local concentration within the pellet, V is the volume of the pellet (area in 2D), and c0 is the concentration at the pellet’s external surface. In this paper, distributor channels, going from the pellet’s external surface to its interior, are introduced in order to reduce diffusion limitations. The geometrical optimization problem is, thus, to maximize the effectiveness factor (eq 1) by adjusting the properties of the network of distributor channels. The properties of the nanoporous material, and the size and macroscopic geometry of the pellet, are fixed in the optimization. Because only pellet geometries without corners are studied in this paper, the distributor network is assumed to be a periodic assembly of a single, repeating unit. We start our optimization study with a 2D slab of finite thickness, Figure 1a, since the optimization is computationally more expensive for other 2D pellet geometries, like the circular disk in Figure 1b, and 3D geometries, although results are similar. More specifically, the straight distributor channel with no branches (network in Figure 1a, repeating unit in Figure 1d) and distributor channels with N branches (repeating unit with N ) 4 in Figure 1c) are used to illustrate the general properties of the optimal distributor network. These properties are easily generalized to other 2D and 3D geometries (Section 4).
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Figure 1d shows the lower half of one channel and half of the two adjacent nanoporous walls in Figure 1a. It is, thus, a repeating unit for the entire distributor network consisting of straight distributor channels without branches in a 2D slab. This repeating unit has width 2L and depth H, meaning that the slab thickness is 2H. The channel diameter and the thickness of the nanoporous walls are d and w, respectively. The macroporosity, the area (volume in 3D) fraction occupied by the distributor channels, is
)
d d+w
(2)
The equivalent repeating unit for a 2D slab with branched distributor channels is given in Figure 1c. The width and depth are again 2L and H, respectively. The channel consists of a stem with diameter d1 and N branches with diameter d2. The area of the islands of nanoporous material is w1w2 (w1 in x-direction and w2 in y-direction), and the macroporosity is
) 1 + (1 - 1)2
(3)
where 1 ) d1/(d1 + w1) and 2 ) Nd2/H. The branched distributor channel (Figure 1c) and the distributor channel without branches (Figure 1d) coincide when N ) 0 and when 2 ) 0 for any value of N. Over a length 2H of external pellet surface, there are 2H/2L distributor channels. This means that, when the length 2H is used as the basis, the density of distributor channels is
M)
H L
(4)
The density of channels per meter of external surface can be calculated from M once H is specified. In 2D Cartesian coordinates (x and y), conservation of mass gives
De∇2c(x, y) - kc(x, y) ) 0
(5)
for the nanoporous material, in which we assume enough pores to be present in each island to allow for a continuum model to be valid,6,23 and
D0∇2c(x, y) ) 0
(6)
for a distributor channel, where D0 is the molecular diffusivity in a distributor channel. The effect of Knudsen diffusion will be discussed in a separate paper. The boundary conditions are as follows (origin at the center of the external surface):
c(x, 0) ) c0,
(∂x∂c)
x)(L
) 0, and
(∂y∂c)
y)H
)0
(7)
Furthermore, continuity of flux and concentration is imposed on the borders between the distributor channel and the nanoporous material. A dimensionless form of the problem is obtained by introducing c˜ ) c/c0, x˜ ) x/L, y˜ ) y/H, the ratio of the diffusion coefficients in the channel and in the nanoporous material D0/ De, and the pellet Thiele modulus
x
φH ) H
k De
The dimensionless forms of eqs 5-7 are then
(8)
M2
∂2c˜ ∂2c˜ + - φH2c˜ ) 0 ∂x˜ 2 ∂y˜ 2
(9)
D0 ∂2c˜ D0 ∂2c˜ + )0 De ∂x˜ 2 De ∂y˜ 2
(10)
M2 and
c˜ (x˜ , 0) ) 1,
(∂x∂c˜˜)
x˜ )(1
) 0 and
(∂y∂c˜˜)
y)1
)0
(11)
respectively. Furthermore, the effectiveness factor, eq 1, for the 2D geometries in parts d and c of Figure 1 is
η)
∫ c˜ dV˜ nanoporous
(12)
where V ˜ nanoporous is the part of the system that is occupied by the nanoporous material. The contribution from the distributor channel is trivially zero. In the next section, the effectiveness factor is maximized using the density of distributor channels, the number of branches, the position-dependent channel diameter, and the macroporosity as decision variables. On the basis of the results, the design of optimized catalyst pellet will be discussed in Section 4. There are several relevant Thiele moduli in this work. The pellet Thiele modulus, eq 8, is the relevant modulus for the completely nanoporous pellet, which is the natural reference system. In order to make the comparison with this pellet easy, φH will be used in all figures. The local Thiele modulus, which is
φL )
x
w 2
k ) L(1 - ) De
x
k 1- φ ) De M H
(13)
for the distributor channel without branches (Figure 1d), is based on the thickness of the nanoporous walls in pellets with a distributor network. This modulus gives the local effectiveness factor ηL ) tanh(φL)/φL and, thus, measures the extent of diffusion limitations in the nanoporous walls. Finally, the distributor network Thiele modulus,
φ0 ) H
x
k ) φH D0
x
De D0
(14)
characterizes pellets with a distributor network, especially for strong diffusion limitations. It shall be shown that many of the properties of the optimal distributor network can be described using φ0. The optimization problem was solved numerically as follows. The MATLAB 6.5 functions “fmincon’’ and “fminbnd’’ were used for the optimization itself. In each function evaluation, FEMLAB 2.3 was used to draw the geometries (see Figure 1), to solve the differential equations (eqs 9-11), and to calculate the effectiveness factor (eq 12). The number of finite elements was increased until the calculated effectiveness factor became mesh-size independent (5000 elements or more). The effectiveness factor is often insensitive to the decision variables, so accurate calculations of the effectiveness factor and the Jacobian matrix were needed in order to obtain the optimal values of the decision variables. By inspection of the response surface for selected parameter values, it was confirmed that the optima presented later are indeed global optima. 3. Properties of the Optimal Distributor Network 3.1. Density of Distributor Channels. The density of distributor channels, M, is one of the most important properties
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geous because it does not cost anything to reduce the channel diameter when there is pure molecular diffusion in the channels. The molecular diffusivity does not depend on the channel diameter, so the area-averaged (volume-averaged in 3D) diffusivity depends on the macroporosity, but it is roughly independent of M. These arguments are very general and apply to any 2D and 3D geometry, as well as for pellet geometries with corners, for sufficiently high values of M. The corresponding distributor network on a 2D circle is shown in Figure 1b. Mathematically, the optimal network has infinitely narrow channels and infinitely thin walls, but the optimal effectiveness factor is more or less constant when φH/M ≈ φL is smaller than ∼0.1 (M ) 10, 100, and 1000 for φH ) 1, 10, and 100, respectively). The reason is that the local effectiveness factor, ηL ) tanh (φL)/φL, is then very close to unity (0.9967 for φL ) 0.1), and only the macroporosity matters. In practice, the truly optimal distributor network can, therefore, be approximated very well by an array of channels with a finite, nonzero diameter and a finite wall thickness, as discussed in Section 4. When the local effectiveness factor is 1, there are no concentration gradients in the x-direction in Figure 1d, and the 2D slab can be described by the following 1D, effective, model,
D()
d2c(y)
- k(1 - ) c(y) ) 0
dy2
(15)
where the area-averaged (volume-averaged in 3D) diffusivity is
D() ) (D0 - De) + De
(16)
and the boundary conditions are
(dydc)
c(0) ) c0, Figure 2. Optimal effectiveness factor (a) and optimal macroporosity (b) of the pellet with distributor channels without branches as a function of the density of distributor channels: D0/De ) 100.
of the optimal distributor network. The network consisting of straight channels without branches in a 2D slab, parts a and d of Figure 1, shall be used here to illustrate some very general results concerning the density of distributor channels when there is pure molecular diffusion in the channels. Figure 2a shows the optimal effectiveness factor as a function of M when D0/De ) 100 and φH ) 1, 10, and 100. For each value of M, the macroporosity has been optimized. The optimal macroporosity, which gives the optimal channel diameter and the optimal wall thickness once H and L are specified, is shown as a function of M in Figure 2b. The important result in Figure 2a is that the optimal effectiveness factor increases in a monotonous way when the density of distributor channels increases, and it converges for large enough M-values. This means that the optimal distributor network consists of a very large number of narrow distributor channels separated by thin nanoporous walls. The thin walls are advantageous because the diffusion limitations in the walls, characterized by the local Thiele modulus (eq 13), are then essentially removed (φL f 0), and the local effectiveness factor (ηL) approaches unity. There are diffusion limitations in the channels, though, and the effectiveness factor for the whole pellet is 0 would lie in between these limiting cases. For small values of M, introduction of branches leads to a significant gain, but the gain decreases as M increases. It is optimal to have a very high density of distributor channels for all N, and in this limit, the optimal effectiveness factor is independent of N. Moreover, Figure 3b shows that 2 for the N f ∞ distributor channel is zero, and is equal for N ) 0 and N f ∞, when M is sufficiently large. This means that the branches are not needed, and that it is overall optimal to have a very large number of straight distributor channels without branches. In summary, introduction of branches is only beneficial when the density of distributor channels is so low that channels without branches cannot remove the diffusion limitations inside the nanoporous material. When the density of channels is high enough, there are only diffusion limitations in the channels themselves. Introduction of branches can then only increase the overall diffusion limitations in the pellet, since branches increase the tortuosity of the distributor network and thereby reduce the effective diffusivity. These findings are very general and apply to any 2D or 3D pellet geometry where there is pure molecular diffusion in the distributor channels. 3.3. Position-Dependent Channel Diameter. The distributor channels that have been discussed so far have the same channel
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diameter from the external surface to the center of the catalyst pellet. A relevant question is whether channels with a varying channel diameter perform significantly better. Since the optimal distributor network has a very high density of channels (M f ∞) and no branches (N ) 0), the optimal variation of the channel diameter was studied within the context of the 1D model presented in Section 3.1 (eqs 15-19). A varying channel diameter translates into a varying macroporosity, (y), in this model, so the modified version of eq 15 is
[
]
dc(y) d D((y)) - k[1 - (y)]c(y) ) 0 dy dy
(21)
The objective is to find the function (y) that maximizes the effectiveness factor
η)
1 kc0H
∫0H k[1 - (y)]c(y) dy
(22)
which now is a functional of (y). Equation 21 must be taken into account in the optimization, and the macroporosity must be between 0 and 1 everywhere. This is a standard problem in optimal control theory.24 In order to use optimal control theory, eq 21 is rewritten as a set of two first-order differential equations:
dJ(y) ) -k[1 - (y)]c(y) dy
(23)
dc(y) J(y) )dy D((y))
Figure 4. Optimal, straight distributor channel (left) and the optimal distributor channel with position-dependent diameter (right): D0/De ) 100, M f ∞.
(24)
which means that ∈ [0,1] should everywhere be chosen such that the Hamiltonian is maximized. This maximum is either on one of the boundaries (0 or 1) or at an interior point where
Here, J(y) is the molar flux in the y-direction. The corresponding boundary conditions are
c(0) ) c0 J(H) ) 0
(25)
The Hamiltonian of the optimal control problem is now24
h ) k[1 - (y)]c(y) + λJ(y){-k[1 - (y)]c(y)} + J(y) (26) λc(y) D((y))
[
]
where λJ(y) and λc(y) are multiplier functions. The Hamiltonian for this problem consists of three terms. The first term is the integrand in the effectiveness factor. The other terms are products of multiplier functions and the right-hand sides of eqs 23 and 24. The necessary conditions for the optimum are derived from the Hamiltonian of the optimal control problem.24 They consist of four differential equations, eqs 23, 24, and
λc(y) dλJ(y) ∂h ) )dy ∂J(y) D((y))
(27)
dλc(y) ∂h ) -k[1 - (y)][1 - λJ(y)] )dy ∂c(y)
(28)
with the following boundary conditions
c(0) ) c0 J(H) ) 0 λJ(0) ) 0 λc(H) ) 0
(29)
The two new differential equations give the spatial variations of the multiplier functions. The final necessary condition is
(y) ) argmax h ∈[0,1]
(30)
[
][
]
J(y) ∂D((y)) ∂h )0 ) -kc(y)[1 - λJ(y)] + λc(y) 2 ∂(y) ∂(y) D((y)) (31) The necessary conditions for the optimum from optimal control theory must usually be solved numerically. In the present problem, the optimum has some special properties that simplify the problem tremendously, and semianalytical expressions for c(y) and (y) are obtained (eqs 51 and 54). These details are discussed in the Appendix. The right column of Figure 4 shows the optimal variation of the channel diameter for D0/De ) 100 and φH ) 1, 10, 100. The diameter optimally decreases in a monotonous way from the external surface to the center. The optimal channel actually stops before it reaches the center (not visible for φH ) 10, 100 in Figure 4). This is reasonable since the flux should be zero in the center of the pellet anyway. The figure also shows, as expected, that the width of the optimal channel increases as φH increases. The left column of Figure 4 shows the corresponding distributor channels with optimal constant diameter. The dependence on φH is the same for these types of channels as for the channels with optimal varying diameter. Figure 5 shows the ratio of the effectiveness factors for channels with optimally varying diameter (right column in Figure 4) and channels with optimal constant diameter (left column in Figure 4) for D0/De ) 102, 104, 106 and φH ∈ [10-2, 104]. Three important conclusions can be drawn from this figure. First, the gain achieved by making the channel diameter vary is never more than 4%. The extra complexity introduced by varying the diameter is, thus, not needed in practice. Second, the gain is largest for intermediate values of φH. The two types of channels are equivalent for very low φH-values, because it is
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Figure 5. Ratio of the effectiveness factors for distributor channels with varying diameter and constant diameter: M f ∞. Table 1. Largest Reduction (in %) of the Effectiveness Factor of Pellets with Constant Channel Diameter and Pellets with Constant Macroporosity Compared to the Truly Optimal Slabs, Circular Disk/Infinite Cylinders, and Spheres: OH ∈ [10-2, 104], D0/De ∈ [102, 106]
constant diameter constant macroporosity
slab
circular disk/ infinite cylinder
sphere
3.7 3.7
7.2 1.1
8.2 0.5
then optimal to have no distributor channel at all. They are also equivalent for very large φH-values, because only the outer layers of the pellet contribute to the yield, and there the diameters of the two types of channels are the same when φH is large (see Figure 4). Third, the lines for D0/De ) 1, 10, 100 have approximately the same form. If the distributor network Thiele modulus, eq 14, was used on the abscissa, the lines would almost coincide. The maxima would all be at φ0 ≈ 1. The significance of the distributor network Thiele modulus shall be discussed in more detail in Section 3.4. Similar calculations were done for the position-dependent channel diameter in circular disks/infinite cylinders and spheres using a 1D model like the one used for the slab geometry. The truly optimal pellet, where the channel diameter and the macroporosity vary in the radial direction, was compared to pellets with channels of constant diameter (having a cavity in the center) and pellets with constant macroporosity. In the constant macroporosity pellets, the channel diameter is proportional to the radial position for circular disks/infinite cylinders and spheres; see Figure 1b. The largest differences in effectiveness factor between the three classes of pellets are at φ0 ≈ 1, in the same way as for the slab geometry (see Figure 5). Table 1 shows the largest reduction of the effectiveness factor for constant diameter pellets and constant macroporosity pellets compared to the true optimum when φH ∈ [10-2, 104] and D0/ De ∈ [102, 106]. The largest reduction is never more than 8.2%, meaning that both the constant diameter and constant macroporosity pellets are very good approximations to the true optimum. In practice, pellets with constant channel diameter are probably preferred because they are easiest to synthesize. 3.4. Gain and Macroporosity. A distributor network with a very large number of straight distributor channels without branches is a very good approximation to the global optimum for any 2D or 3D catalyst pellet when there are no significant corner effects and there is pure molecular diffusion in the distributor channels. In this section, the properties of this kind of distributor network are studied in more detail using the 1D
Figure 6. Optimal effectiveness factor of the pellet with distributor channels without branches (a), the ratio of effectiveness factor of the optimal pellet and the completely nanoporous pellet (b), and the macroporosity of the optimal pellet (c).
model given in Section 3.1 (eqs 15-19). The detailed results presented are for a 2D slab like in parts a and d of Figure 1 and for an infinite 3D catalyst sheet of thickness 2H. The results also apply to the skin of any catalyst pellet when the skin thickness H is much smaller than the size of the pellet. Similar
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results can also be given for other geometries, like disks or cylinders, and spheres. Figure 6a shows the effectiveness factor for the pellet with the optimal distributor network consisting of a very large number of straight distributor channels without branches as a function of φH for D0/De ) 102, 104, and 106. These results are replotted in Figure 6b as the ratio of the effectiveness factors of the optimal pellet and the completely nanoporous pellet. For all values of D0/De, the gain by introducing the distributor network increases as φH increases and is constant when φ0 . 1; see eq 14. The optimal effectiveness factor can be several orders of magnitude larger than the effectiveness factor for the completely nanoporous pellet. These results correspond mathematically to M f ∞, but in reality they also apply to finite values of M as long as φL is ∼0.1 or smaller (cf. discussion in Section 3.1). When M is not sufficiently large, the gain reaches a maximum for φ0 ≈ 1 and then diminishes for larger values of φH. This behavior, which is illustrated by the dotted line in Figure 6b (D0/De ) 102, M ) 10), was found by Gheorghiu and Coppens.19 In addition to having a limited number of channels, Knudsen diffusion played a significant role in some cases studied by Gheorghiu and Coppens. Figure 6c shows the macroporosities corresponding to the optimal effectiveness factors in parts a and b of Figure 6. As expected, the optimal macroporosity increases when φH increases for a given D0/De and decreases when D0/De increases for most values of φH. The exception is for very high φH-values, where the optimal macroporosity slightly increases when D0/De increases. The optimal macroporosity is never larger than 0.5. The limits for the gain and the optimal macroporosities when φH is large can be analyzed analytically. From eq 18, the ratio of the effectiveness factors of the pellet with a distributor network and the corresponding completely nanoporous pellet is
η ηnanoporous
tanh φ* φH tanh φH φ*
) (1 - )
(32)
Using eq 19,
lim
φHf∞
(
) x [( ) ]
η ) ηnanoporous
(1 - )
D0 -1 +1 De
(33)
which means that the optimal macroporosity in this limit is
() ( )
D0 -2 De lim (optimal) ) < 0.5 φHf∞ D0 2 -1 De
(34)
lim
φHf∞
(
)
ηoptimal 1 ) ηnanoporous 2
x
D0 -1 De
≈
x
1 2
1 lim D(optimal) ) D0 2
(36)
φHf∞
Similarly, the area-averaged (volume-averaged in 3D) reaction rate constant is
1 1 lim (k(1 - optimal)) ) kD0/(D0 - De) ≈ k 2 2
φHf∞
(37)
Both area-averaged (volume-averaged in 3D) coefficients are, thus, half of their maximum value. This makes sense because half the space is available for diffusion and the other half is available for reaction. For weaker diffusion limitations (smaller φH), k(1 - optimal) > (1/2)k and D < (1/2)D0, meaning that more than half of the space is available for reaction. It is also interesting to calculate the 1D Thiele modulus, eq 19, when φH is large. By introducing eq 34, the 1D Thiele modulus becomes
lim φ* ) H
φHf∞
x
k ≈H (D0 - De)
x
k ) φ0 D0
(38)
This means that the optimal 1D Thiele modulus goes from φH when φH is small (optimal ) 0) to φ0 when φH is large. The latter corresponds to a pellet with reaction rate constant k/2 and diffusivity D0/2. Even though this pellet has weaker diffusion limitations than the original nanoporous pellet, the diffusion limitations will reduce the effectiveness factor when φ0 is significantly larger than unity, as shown in Figure 6a. As a result of this, the center of the pellet with the optimal distributor network will not contribute significantly to the yield for very strong diffusion limitations and can, therefore, be removed without noticeable consequences on the yield. The optimal pellet is, therefore, hollow as sketched in Figure 7. Without including a penalty for catalyst cost, there is no definite way to optimize the shell thickness, but the results in Figure 6 suggest that a reasonable estimate can be obtained from φ0 ≈ 5. The H in the previous sections should then be identified as the shell thickness, not the pellet “diameter’’, and the results that have been presented so far apply to the shell. The analytical analysis for φH f ∞ and the numerical calculations were also performed for circular disks, infinite cylinders, and spheres. Equations 33-38 and their discussion can directly be extended to these other pellet geometries, and it follows that the numerical results are qualitatively the same as for the slab (Figure 6). 4. Near-Optimal Channel Diameter and Wall Thickness
Since D0/De is generally much larger than 1, this equation shows that the limiting optimal macroporosity is always very close to, but less than, 0.5. By introducing the optimal macroporosity in eq 33, the ratio of the effectiveness factors of the pellet with the optimal distributor network and the corresponding completely nanoporous pellet is
D0 De
by introducing the limiting optimal macroporosity in eq 16, the area-averaged (volume-averaged in 3D) diffusivity is
D0 De
(35)
This is in agreement with the limits in Figure 6b. Furthermore,
In summary, it has been shown that a distributor network consisting of a large number of straight channels with a positiondependent channel diameter, and without branches, is the global optimum for all 2D and 3D catalyst pellet geometries without corners when there is pure molecular diffusion in the distributor channels. The only underlying assumption is that the optimal distributor network is a periodic assembly of a single, repeating unit. Moreover, the global optimum is very well-approximated by a network of channels with constant diameter. Whenever only the skin of any catalyst pellet contributes to the yield, these results also apply to geometries with corners (squares, cubes, etc.) because the effects of corners are insignificant. Mathematically, the optimum has infinitely many narrow channels separated by infinitely thin walls. This is only a
Ind. Eng. Chem. Res., Vol. 46, No. 12, 2007 4253 ∞ d 0.2 optimal ) H φH 1 - ∞
(41)
optimal
where ) d/(d + w) was used to find d/H. Figure 8 shows d/H calculated with eq 41 and ∞optimal from Figure 6c. The dotted diagonal line is the wall thickness, calculated from eq 40. For each value of D0/De, d/H is more or less constant up to φ0 ≈ 1. For φ0 ≈ 3 and larger, optimal ≈ 0.5, so the walls and the channels are nearly equal in width. Example of Use: Consider a catalyst pellet with H ) 10-5 m in which a reaction with intrinsic reaction rate constant k ) 1000 s-1 takes place. The effective diffusivity in the nanoporous material is De ) 10-9 m2/s, and the molecular diffusivity is D0 ) 10-5 m2/s. Figure 8 gives d/H ≈ 10-3 and w/H ≈ 0.02 when φH ) 10-5 × x103/10-9 ) 10 and D0/De ) 104. This means that the near-optimal distributor network consists of channels with diameter d ) 10 nm and nanoporous walls with thickness w ) 200 nm. The same macroporosity data, Figure 6c, can also be translated into channel diameters and wall thicknesses in a 3D sheet of finite thickness. An example, inspired by novel methods to synthesize structured mesoporous materials, is the hexagonal packing of cylindrical channels; see the upper right corner of Figure 9 for an end view. The macroporosity of this packing is
)
d 2 π d + w 2x3
(
)
(42)
By using φL ) 0.1 as before, the wall thickness is still given by eq 40, but d/H is now
d 0.2 ) H φH
Figure 7. Examples of hollow pellets with a distributor network in the shell.
mathematical limit, because, in practice, Knudsen diffusion (gases) or configurational diffusion (liquids) would be the dominating diffusion mechanism in sufficiently narrow channels. Furthermore, networks with channels and walls of finite sizes perform only slightly worse, as discussed in Section 3.1. Good estimates of how small d and w need to be are obtained from φL ) 0.1 (see eq 13) and the optimal macroporosity for M f ∞:
∞optimal ≡ lim optimal Mf∞
(39)
This assures that the diffusion limitations in the nanoporous walls are negligible (tanh(φL)/φL ) 0.9967 when φL ) 0.1). For the 2D slab geometry, φL ) 0.1 and ) ∞optimal give
w 0.2 ) H φH and
(40)
x x
2x3 ∞ π optimal
1-
2x3 ∞ π optimal
(43)
Figure 9 shows d/H calculated with eq 43 and ∞optimal from Figure 6c. Comparison of Figures 8 and 9 shows that the channel diameters are larger in the 3D geometry. For the numerical example discussed above, d/H ≈ 6.5 × 10-3 and w/H ≈ 0.02 for the hexagonal packing of cylindrical channels (see Figure 9), meaning that the near-optimal channel diameter is 65 nm in the 3D geometry, whereas it is only 10 nm in 2D. The optimal wall thickness is 200 nm in both cases. The fact that the nearoptimal channel diameter is larger in 3D means that possible effects of Knudsen diffusion will be less pronounced in 3D than in 2D. The near-optimal channel diameters and wall thicknesses for other packings of cylindrical channels (like cubic and facecentered cubic) are not very different from the values for hexagonal packing. For some combinations of the relevant parameters (H, k, De, and D0), the near-optimal channel diameter and the wall thickness obtained from Figure 8 or 9 are unrealistic or have no physical significance. First, the nanoporous walls should be thick enough to apply the continuum theory that was used here to solve the diffusion and reaction problem. Second, Knudsen diffusion will play a role for gases when the diameter of the distributor channel is comparable to the mean free path under relevant reaction conditions. It is, therefore, recommended to always compare the diameter obtained from Figure 8 or 9 to the relevant mean free path. When the diameter is so low that Knudsen diffusion plays a role, the optimal effectiveness factor obtained here serves as an upper bound for the optimal
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Ind. Eng. Chem. Res., Vol. 46, No. 12, 2007
Figure 8. Near-optimal channel diameter divided by the pellet characteristic length as a function of the pellet Thiele modulus for distributor channels in a 2D slab. The dotted line gives the near-optimal wall thickness, w/H.
insignificant, even though the detailed results presented here are for 2D slabs and 3D sheets. There are many factors that can influence the properties of the optimal distributor network. The effect of Knudsen diffusion shall be discussed in a separate paper. Nonlinear reaction kinetics, equilibrium reactions, nonstoichiometric reactions, flow, and issues like deactivation and selectivity will also affect the properties of the optimal distributor network. These are interesting directions for future work. Another relevant question is the sensitivity of the optimum and how this relates to the capabilities of synthesis methods. It might not be easy to synthesize the truly optimal network or the approximate networks discussed in this paper. However, the fact that a network of channels with a constant diameter and wall thickness would be very close to optimal significantly simplifies the task. The values obtained from the (semi)analytical formulas presented here may guide the effort to synthesize hierarchically structured nanoporous catalyst with optimal performance. Acknowledgment E.J. would like to thank the Norwegian Research Council for the grant “Transport at the nanoscale, surfaces and contact lines’’. G.W. and M.-O.C. are grateful for the financial support from the Delft Research Centre for Computational Science and Engineering. Appendix
Figure 9. Near-optimal channel diameter divided by the pellet characteristic length as a function of the pellet Thiele modulus for hexagonal packing of cylindrical channels in a 3D sheet of finite thickness. The dotted line gives the near-optimal wall thickness, w/H.
The necessary conditions of the optimal control problem discussed in Section 3.3, eqs 23, 24, and 27-31, were solved numerically with the following shooting method: the differential equations were integrated from y ) 0 to y ) H using guessed values of J(0) and λc(0). At every position, (y) was chosen such that it maximized h. The two guessed initial conditions were then adjusted until the boundary conditions at y ) H (see eq 29) were fulfilled. Good initial guesses of J(0) and λc(0) were obtained by first discretizing the optimization problem (eqs 22-25) on a coarse grid and solving it using sequential quadratic programming (the MATLAB function “fmincon’’). The numerical results showed that there are relations between c, J, λc, and λJ:
c0[1 - λJ(y)] ) c(y) and c0λc(y) ) J(y) effectiveness factor when Knudsen diffusion is included in the calculations, as will be shown in a future paper. 5. Concluding Remarks The properties of the optimal distributor network in a nanoporous catalyst pellet, in which a first-order irreversible, isothermal reaction A f B takes place, have been studied. In order to limit the scope of the paper, only pellets without corners or pellets where the effect of the corners are negligible, with pure molecular diffusion in the distributor channels, were studied. It was assumed that the optimal network is a periodic assembly of a single, repeating unit. Given this, the truly optimal distributor network contains a very large number of straight, narrow channels with a position-dependent channel diameter and no branches. This optimum is very well-approximated by a network of channels with a constant and much larger diameter. Typically, the channels are macropores rather than mesopores. Furthermore, the optimal macroporosity, the volume or area fraction occupied by the distributor channels, is never larger than 0.5. These results are very general and apply to all 2D and 3D catalyst pellet geometries when the corner effects are
(44)
These relations simplify the necessary conditions tremendously because λJ(y) and λc(y) can be eliminated. Equations 26 and 31 become
h)
{
[J(y)]2 1 k[1 - (y)][c(y)]2 c0 D((y))
}
(45)
and
{
[
]
}
J(y) 2 ∂h 1 ) - k[c(y)]2 + (D0 - De) ) 0 (46) ∂ c0 D((y)) respectively. Furthermore, the differential equations for the multiplier functions, eqs 27 and 28, are not needed any more. A semianalytical solution of the optimal control problem can now be found. The right column of Figure 4 shows that there are two y-domains in the solution of the necessary conditions. In the first domain, 0 e y < H - δ, is between 0 and 1, and in the second domain, H - δ e y e H, is 0. This means that the optimal is found from the weak optimality condition, eq 31, in the first domain, whereas the optimal is identical to its
Ind. Eng. Chem. Res., Vol. 46, No. 12, 2007 4255
lower bound in the second domain. Both results are contained in the strong necessary condition, eq 30. In the first y-domain, eq 46 gives
-k[c(y)]2 +
[
]
J(y) 2 (D0 - De) ) 0 D((y))
(47)
which means that the molar flux is
J(y) )
x
k c(y)D((y)) D0 - De
(48)
By introducing this into eq 24, the concentration gradient becomes
dc(y) J(y) ))dy D((y))
x
k c(y) D 0 - De
(49)
In the second domain, the concentration profile is given by eq 21 with (y) ) 0. The boundary conditions for these differential equations are
c(0) ) c0, c(H - δ) ) cδ, and J(H) ) 0
(50)
Thus, the concentration profile is
{
c(y) )
d ) channel diameter (m) H ) height of repeating unit (m) h ) Hamiltonian of optimal control problem (mol/(m3 s)) J ) molar flux (mol/(m2 s)) k ) reaction rate constant (1/s) L ) width of repeating unit (m) M ) density of distributor channels N ) number of branches V ) volume (m3) w ) thickness of the nanoporous walls (m) x, y ) spatial coordinates (m) Greek Symbols δ ) thickness of solid core (m) ) macroporosity η ) effectiveness factor λJ ) multiplier function λc ) multiplier function (m/s) Dimensionless Numbers φH ) pellet Thiele modulus; φH ) Hxk/De φL ) local Thiele modulus; φL ) (w/2)xk/De φ0 ) distributor network Thiele modulus; φ0 ) Hxk/D0 Literature Cited
y c0 exp -φ1 H-δ
(
(
(
)
exp φ2
c0 exp(-φ1)
H-y H-y + exp -φ2 δ δ exp(φ2) + exp(-φ2)
)
(
)
)
0ey
