Effectiveness Factor and Mass Transfer Coefficient in Wedge and

Article pubs.acs.org/JPCC

Effectiveness Factor and Mass Transfer Coefficient in Wedge and Funnel Pores Using a Generalized Fick−Jacobs Model Aldo Ledesma-Durán, Saúl Iván Hernández Hernández, and Iván Santamaría-Holek* Unidad Multidiscliplinaria de Docencia e Investigación-Juriquilla, Facultad de Ciencias, Universidad Nacional Autónoma de México, Juriquilla CP 76230, Querétaro Mexico S Supporting Information *

ABSTRACT: In this work we study the diffusion−adsorption process in porous media and analyze the effect that the irregular geometry of the pores has on the efficiency of two types of adsorption processes: (a) when there is a net flux along the pore and (b) when the pore is completely saturated. In the first case, we measured the mass transfer coefficient, which is the constant of proportionality between the net flux and the difference of concentration. In the second case, we measure the effectiveness factor, which is the ratio between the actual rate of adsorption and the rate which would be achieved if the entire surface were at the same external concentration. In order to perform this analysis, we use a generalized Fick−Jacobs equation that considers the net effect of diffusion and adsorption along the direction of transport. For this analysis we have used wedge-shaped and conical pores, due to the simplicity of the treatment and their importance in the elaboration of a new brand of artificial materials. We have proved that the enhancement or diminution of the mass transfer coefficient or the effectiveness factor depend upon the specific rate of adsorption; therefore, they can be controlled using our model as a prediction tool in order to build artificial materials with a specific output flux of material. Additionally, our work allows to find how the Thiele modulus locally depends on the geometry of the pore for a linear reaction.

1. INTRODUCTION

relevant in materials where the specific shape of the pores is crucial. We have chosen porous materials composed of conical pores due to two reasons. First, due to the great importance that this pore geometry has in the elaboration of a new class of artificial materials,10−15 it has important applications which range from chemistry to physiology.16−21 Furthermore, the study of this kind of pores represents a current field of research when coupled with processes such as condensation, capillarity, and hysteresis. 22−25 Second, we chose them due to the mathematical simplicity in their treatment. Notwithstanding, we have to emphasize that the model and methodology employed in this work can be used to study more general geometries.9 Since we are interested in the dependence of macroscopic quantities on the geometry of the pore, we will focus our study in a cell containing only one conical pore. In the limit when the convective velocity is negligible (very low Peclet numbers) and when the pore is long enough compared with its width, the transport properties of such pore can be studied using a generalized FJ scheme.9,26−28 This scheme allow us not just to simplify mathematically the description in terms of an effective mass balance along the direction of the pore but also, and more importantly, allows us to incorporate explicitly in the equation

The study of mass diffusion and adsorption in porous media in different pore geometries has deserved a wide attention due to its importance in a great variety of industrial and technological applications.1−6 The most basic aspects of this kind of processes in simplified bed geometries (like slabs, spheres, cylinders, and oblate spheroids) are well-documented in textbooks, for example, refs 7 and 8. However, for more realistic geometries like those of zeolites, the existence of geometric irregularities and obstacles in the shape of the pores introduces additional complications to the solution of the problem and clouds the general quantitative relation between the geometric properties of the pore and the macroscopic transport properties of the chemical process.7 In this work, we present a systematic treatment for studying the macroscopic quantities related to the processes of diffusion and adsorption in materials, which takes into account the microscopic aspects related to the pore geometry. We proceed to study the adsorption of particles of a gas when they diffuse inside a wedge-shaped or conical pores, whose walls can adsorb particles of the bulk (Figure 1a). The scheme that we use is the generalized Fick−Jacobs (FJ) model, which represents a meanfield approach allowing us to study the processes of diffusion and adsorption inside a pore in terms of its specific geometry.9 Therefore, the characteristic scale of length is in the middle between the molecular shape of the particles and the macroscopic shape of a bed particle. Therefore, our work is © 2016 American Chemical Society

Received: September 13, 2016 Revised: November 30, 2016 Published: November 30, 2016 29153

DOI: 10.1021/acs.jpcc.6b09282 J. Phys. Chem. C 2016, 120, 29153−29161

Article

The Journal of Physical Chemistry C

the boundary conditions which corresponds to the case of net flux across the pore in order to find the mass transfer coefficient that we define. In section 4, we defined the effectiveness factor and solve the generalized FJ equation with boundary conditions related to saturation conditions. In section 5, we find a way of expressing the relation between the interplay of diffusion and adsorption and the irregularity of the pore geometry. For doing this, we find a generalized Thiele modulus. Finally, in section 6, we discuss the results as well as their advantages and scopes in the design of materials.

2. GENERALIZED FICK−JACOBS EQUATION FOR THE DIFFUSION AND ADSORPTION PROCESSES In this section we present a brief summary of the hypotheses and deduction of the mean-field model based on the generalized FJ scheme, originally formulated in ref 9. We also discuss in detail its use and relation with the problem of adsorption in conical pores. Let us start by considering a fluid obeying the Fick diffusion laws summarized in the diffusion equation. In this case, the concentration of particles is given by

Figure 1. Geometry of the pore. (a) Section of a conical pore (top) and of a flat pore of wedge-shaped transversal section (bottom). In the first case, our model is the projection along the angular coordinate, and in the second case the projection is along the z direction. (b) Basic geometry: width of the pore w(x), middle line h(x), function form of the two walls w1(x) and w2(x), length of the walls l1 and l2, and the longitudinal length of the membrane L. The point where the pore would close if the two lines were continued is x0. The entry of material is defined in all cases as x = 0, and the end of the membrane or the output is x = L. All quantities can be expressed in terms of the slopes m1 and m2 and the y-intersections b1 and b2. See eqs 3.

⎛ ∂ 2C ∂C ∂ 2C ⎞ = D0⎜ 2 + 2 ⎟ ∂t ∂y ⎠ ⎝ ∂x

the relevant geometric aspects related to the shape of the pore and the position of active sites of adsorption.9 The concentration profiles obtained for these systems have allowed us to compute the flux at the entry of the pore in a adsorption-diffusion system and to find the dependence that two important coefficients commonly used in the design of reactors have with the shape of the pore. The first coefficient is the mass transfer, which measures the enhancement in the diffusive transport across a membrane which is subjected to a concentration difference due to the chemical reaction at the surface.8,29 Since the material inside the pore is adsorbed, there is a local diminution of the concentration due to the reaction. This, in turn, increases the concentration gradient augmenting the net flux along the material. In contrast, if one is not interested in membranes where there is a net flux but in a saturated system (like bed particles in a tube reactor), then the parameter measuring the efficiency of the walls is the effectiveness factor. This parameter represents the ratio between the actual amount adsorbed by the pore and the amount that would be adsorbed if the entire surface of the pore were subject to the external concentration.30,31 Both parameters measure how one process (diffusion or adsorption) limits or enhances the efficiency of the other. In both cases, our study shows the mathematical dependence of these two quantities with the length ratio between the pore entry and the pore output and, in a remarkable way, how they depend on the rate reaction of adsorption. Therefore, this model allows us to make some conclusive statements with respect to the design of materials composed by conical pores used in adsorption processes. In all our work, we are assuming that the chemical adsorption reaction or the internal diffusion inside the pore are the rate limiting steps. The organization of this work is as follows. In section 2, we establish the hypotheses and scopes of the generalized FJ scheme that we use and explain the relation of the diffusive transport with the width of the pore and its tortuousness, as well as the relation of the adsorption efficiency with the length of the pore walls. In section 3, we solve the FJ equation using

(1)

where D0 is the diffusion coefficient of the particles in the fluid in a nonconfined system. In a confined system like a pore of solid reflecting walls, the diffusion can be studied only along the direction of transport. This can be done by averaging the original diffusion coefficient along the transverse coordinate y and incorporating the effects of the solid walls through an spatial-coordinate dependent coefficient DFJ(x). The particular form of this reduction scheme is given by the FJ equation.32 The value of the effective diffusion coefficient has been deduced in different ways for different pore geometries.26,32−36 Some of these examples are discussed in ref 26. In general terms, this coefficient depends upon the local width of the pore and the position of its middle line; see Figure 1b. However, as in our case, when the pore walls are not entirely reflective but the particles can be attached to the pore walls by adsorption, then the averaging procedure along y leads to a generalized FJ equation9 of the form ∂C 1 ∂ ⎡ ∂C ⎤ (x , t ) = ⎢D(x) ω(x) ⎦⎥ + γ(x) R ⎣ ∂t ω(x) ∂x ∂x

(2)

In this equation, the changes in the average concentration C depend on the effective diffusion given by the FJ expression in the first term at the right-hand side, and the enhanced adsorption reaction given in the second term. The width of the pore is w(x), the density of wall-length is γ(x), and DFJ(x) is the effective diffusion coefficient. R is the volumetric rate of adsorption, which usually depends on the concentration itself in chemical reactors.9 The reduction scheme of the generalized FJ model presented in eq 2 is valid when the walls of the pore are impermeable to diffusion, the length of the pore is much greater than its width,32 and the characteristic times of reaction are much lower than the characteristic time of diffusion.9 In the case of the conical pore considered in this work, the shape of the walls can be represented by the linear equations (see Figure 1) w1(x) = m1x + b1 29154

(3a) DOI: 10.1021/acs.jpcc.6b09282 J. Phys. Chem. C 2016, 120, 29153−29161

Article


In the stationary case, the left-hand term in eq 2 is zero. Substituting the geometric parameters of eqs 4−6 and the reaction term of eq 7 into eq 2, we can obtain the nondimensional equation

and

w2(x) = m2x + b2

(3b)

In this case, the effective diffusion coefficient given by the expression in ref 35 and computed in ref 26 is DFJ(x) =

∂ 2Z 1 ∂Z + − ϕ2 Z = 0 2 u − u0 ∂u ∂u

3D0 (3 + m12 + m22 + m1m2)

(4)

where we have defined the variables Z = C/C0 and u = x/L as the dimensionless concentration and dimensionless distance from the pore entry, respectively. In this particular form of eq 2, the role of the geometry of the pore in the processes of diffusion and adsorption is shown clearly through the dependence only upon two geometrical parameters. First, we have the generalized Thiele modulus

This effective diffusion coefficient establishes how the diminution in the effective flux depends upon the slopes m1 and m2 of the walls. It should be noted that in the case of conical pores the effective diffusion coefficient does not depend on the coordinate x. The width of the pore w(x) can be calculated through w(x) = w2(x) − w1(x).37 In the particular case of a trapezoidal pore w(x) = (m2 − m1)x + (b2 − b1)

ϕ=

(5)

The factor γ(x) appearing in eq 2 is the length density of the wall.9 This means that when γ(x) is integrated along the longitudinal direction, the result is the total length of the walls, ∫ L0 γ(x) dx = l1 + l2 (see Figure 1). In the particular case of a rectangular pore, this integral has the minimum value of 2L, i.e., two times the longitudinal length of the pore. For the trapezoidal pore defined by eqs 3, the factor γ (x ) =

1 + w1,2 x +

1 + w2,2 x is

m12

m22

γ (x ) =

1+

+

1+

γκL2 DFJ

(9)

appearing in the last term of eq 8. This parameter represents the ratio between the characteristic time of reaction and that of diffusion,8 since its square can be written as 2

ϕ =

L2 /DFJ 1/γκ

=

characteristic time of diffusion characteristic time of adsorption

(10)

A large value of the Thiele modulus comprises systems where the limiting step is the diffusion; therefore, material is not always available in the pore walls for being adsorbed.8 Conversely, for low values of the Thiele modulus the slower process is the adsorption, so we can expect that the reduction scheme we present here would work better in this limit. The second important quantity in eq 8 is given by the geometrical parameter x u0 = 0 (11) L

(6)

Although the approximation in eq 2 represents some loss of information respect to solving directly the diffusion equation (eq 1) since it represents only the dynamic of the average concentration along the transversal coordinate, the reduced model has important advantages. The obvious one is that the result of the projected scheme is an unidimensional equation (eq 2), which can be solved more easily than the original mass balance equation in two dimensions. Notwithstanding, the most important aspect of the projected model is that it introduces explicitly the entropic effects due to the confinement into the equation and therefore makes explicit the dependence of the adsorption and diffusion processes on the geometry of the pore. These geometrical aspects are given in eqs 4−6 and (as we will see later), they are closely related with the constriction and inclination of the pore. Furthermore, another advantage of the reduced scheme is that it introduces naturally the boundary conditions related to the active walls directly into the equation, changing the heterogeneous problem of solving eq 1 (with its particular boundary conditions at the walls) into an homogeneous problem in eq 2, where the only two boundary conditions are related to the ends of the pore.8 Therefore, this scheme has allowed us to focus on the external conditions to the porous material which usually constitute the conditions that can be controlled in the experiment. The validity of this generalized FJ model has been verified for equilibrium and nonequilibrium conditions, see ref.9 In order to understand the interplay between the diffusion and adsorption processes in terms of the pore geometry, we have chosen a simple linear reaction of adsorption given by R = − κC

(8)

In this definition, x0 = −(b2 − b1)/(w2 − w1) is the place where the conical pore would close if both lines in eqs 3 were continued, and L is the length of the pore (see Figure 1). Therefore, if we take the pore entry b2 − b1 as fixed, then u0 determines the specific shape of the pore. For x0 → ±∞, the pore tends to be rectangular. For x0 → 1 to the right or x0 → 0 to the left, the pore tends to be narrower in the respective side than in the other. It is worth stressing that for fixed b1 and b2 all the relevant geometric parameters for the transport are determined exclusively by the slopes m1 and m2. This makes our results very suitable for practical comparisons and for the design of specific materials. When the width of the pore is constant, w(x) = cte and u0 in eq 11 tends to infinite. In this cases, eq 8 has a singular point in u0. Therefore, the treatment of tilted pores and rectangular pores has to be carried out using a limiting procedure on eq 8. This allowed us to find eqs 18 and 25 as detailed in the Supporting Information. Summarizing, in this section we have shown that the generalized FJ scheme described in eq 2 allows us to describe the adsorption and desorption processes inside irregular pores of conical shape described in eqs 3. The interplay between those processes and the geometry of the system is dictated through the three geometrical parameters given in eqs 4−6, which are related to the effective diffusion coefficient, the irregular width of the pore, and the length of the walls, respectively. The generalized FJ model can be solved for the linear reaction of adsorption stated in eq 7. In order to facilitate

(7)

where κ is the constant rate of adsorption with units of 1/s. The reason for choosing this functional form for the reaction term is its ubiquity in chemical systems and, as we will see, the simplicity of its mathematical treatment. 29155


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The Journal of Physical Chemistry C the comprehension of the results, we have taken the nondimensional form of eq 2 for the stationary case resulting in eq 8. This particular form of the generalized FJ scheme is very convenient since it allows to understand the interplay between these two processes in terms of only two geometrical parameters, defined in eqs 9 and 11. The first parameter establishes the specific ratio between the characteristic times of the adsorption and diffusion processes, and the second one establishes the form of the pore. The great advantage of using a reduced scheme like the one presented in this work is that it allows us to include explicitly in the equation the relevant geometric parameters in the description of both processes (adsorption and diffusion). In addition, we focus on the conditions external to the material, which are the conditions that usually can be controlled in the experiments. As we will see in the next section, these external conditions determine the kind of interplay between both processes and the geometry of the pore. The external conditions also suggest the type of parameter which could be used in order to measure the effectiveness of the process. In the case when there is a net flux along the pore in the material, it is convenient to use the mass transfer coefficient. However, if the pore in the material is immerse on the fluid on saturation conditions, it is preferable to use the internal effectiveness factor.

In this expression, i = −1 , is the imaginary number. Returning this solution to the original variables, we can calculate the mass flux at the entry of the pore according to J(x = 0) = −DFJ

DFJ k = ϕ f (ϕ , u 0) k0 D0 D

fD (ϕ , u0) = −

I1[− ϕu0] Y0[iϕ(u0 − 1)] − I0[ϕ(u0 − 1)] Y1[iϕu0] I0[ϕu0] Y0[iϕ(u0 − 1)] − I0[ϕ(u0 − 1)] Y0[iϕu0]

(17)

The subscript D marks the fact that this concentration difference at the entry depends upon Dirichlet boundary conditions in both ends of the pore. A particular case of the mass transfer coefficient k occurs when the pore has constant width, i.e., when its slopes are the same: m2 = m1 = m. This includes inclined pores and the rectangular pore m2 = m1 = 0. In these cases, it can be deduced that the mass transfer coefficient is km = ϕ coth ϕ k0

(18)

Here, the subscript m emphasizes that this expression is valid for pores of constant slope m. The detailed deduction of eq 18 is given in the Supporting Information. The eq 16 specifies how the mass transfer coefficient k depends upon the shape of the pore and the Thiele modulus of the system. As this parameter depends in turn on the geometry of the pore, in the following lines we present a study of this coefficient which focuses on the relation of k with the slopes of the pore, m1 and m2, and with the rate of adsorption κ. Therefore, hereinafter we fix the values of the diffusion coefficient D0 = 1 cm2/s, the size of the entry w(0) = b2−b1 = 1 cm, and the length of the pore L = 2 cm. 3.1. Analysis of the Results: Geometrical Dependence of the Mass Transfer Coefficient. In this section we will discuss how the mass transfer coefficient can be maximized or controlled in practice by means of the walls slopes, as predicted by eq 16. Figure 2 establishes how the mass transfer coefficient is affected by the shape of the pore. In particular, it shows that for a given adsorption rate κ there is a specific value of m1 and m2 in which the mass transfer coefficient is maximum. This point (plotted as a red spot in Figure 2) is located on the line of symmetry m2 = −m1 (blue dashed line). For all cases of κ, the optimal value of m2 is positive, confirming that a pore which is narrower at the entry optimizes the diffusive transport. This is due to the fact that the pores with inverse geometry, i.e., a narrower exit end, tend to accumulate material. This statement is corroborated in Figure 3 where the mass transfer coefficient is plotted as a function of the adsorption rate κ for four symmetrical pores. In this plot it is clear that the mass transfer coefficient increases with κ. If one is interested in control the rate at which a substance is delivered, then each level curve in Figure 2 indicates the possible slopes of the pore that are compatible with the required transfer rate.

(12)

(13)

The solution of eq 8 with the boundary conditions stated in eq 13 can be obtained with Wolfram Mathematica in terms of modified Bessel functions of the first kind and the Bessel function of the second kind, I0 and Y0, respectively. The functional form of the solution is Z(u) =

(16)

where the function f D(ϕ, u0) is

where J(x = 0) is the flux at the entry. The mass transfer coefficient k has units of velocity (cm/s). If no adsorption occurs inside the pore, then this coefficient has the reference value of k0 = D0/L for rectangular pores.8 Its physical meaning is similar to the concept of conductivity in electric circuits theory, since it measures the proportionality between flux (electric current) and concentration difference (potential difference).29 In order to find this mass transfer coefficient, we first establish the dimensionless boundary conditions of the Dirichlet type:

Z(0) = 1, Z(1) = 0

(15)

Comparing this result with the eq 12, we obtain the following expression for the mass transfer coefficient in terms of the Thiele modulus

3. MASS TRANSFER COEFFICIENT Let us consider a pore that is subjected to a concentration difference between both ends, in x = 0 and x = L (see Figure 1), that is, we will have Dirichlet-type boundary conditions. This corresponds to the case of adsorption with net flow, in which one is usually interested in how the reaction of adsorption increases the transport of fluid through the material. Therefore, under these conditions, the mass transfer coefficient k is usually defined as the constant of proportionality between the flux at the entry and the difference of concentration between the two ends of the pore.8 Therefore, if C(0) = C0 and C(L) = 0 are the boundary conditions at the ends of the pore, then we have J(x = 0) = k(C0 − 0)

∂C (x = 0) ∂x

I0[− ϕ(u0 − u)] Y0[iϕ(u0 − 1)] − I0[ϕ(u0 − 1)] Y0[iϕ(u0 − u)] I0[ϕu0] Y0[iϕ(u0 − 1)] − I0[ϕ(u0 − 1)] Y0[iϕu0]

(14) 29156


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Figure 2. Contour plot of the normalized mass transfer diffusion coefficient k/k0 defined in eq 16 in terms of the geometry of the pore. The slopes of the walls of a pore of length L = 2 are m1 and m2, and the length of the entry is w(0) = 1 (see Figure 1). The values of some contour lines are shown in green. The point in red indicates the pore slopes where the mass transfer coefficient is a maximum, that is, its optimal value. The lower part of the orange line establishes the geometries that are not possible. The dashed line in brown sets the pores that are purely inclined. These contour plots of the mass transfer were made by using two different values of the reaction rate κ = 1.5 and 0.05, at the left and right panels, respectively.

Figure 4. Slope of the symmetric pore that maximizes the mass transfer coefficient, as a function of the adsorption rate κ. In the inset figure, the mass transfer coefficient obtained by using the most efficient pore in eq 16 (red markers) is compared with that of a rectangular pore from eq 18 (solid blue line).

on Figure 4, for all the range of values of κ, the optimization of mass transfer occurs for pores between 45.8 and 55.7°. These results are valid for a mouth/length ratio of pore of w(0)/L = 1/2. However, when we change the longitudinal length of the pore, for example, L = 3, the results (not shown) are entirely similar giving slopes between 46 and 56.5°. The main difference is that the maximum angle is attained at a lower value of κ. The inset in Figure 4 shows the comparison between the mass transfer of a rectangular pore in eq 18, and the mass transfer coefficient which results from using the most efficient pore for each value of κ. As it can be seen, the mass transfer can be enhanced in a significant way by using trapezoidal pores which are narrower at the entry. Summarizing, in this section we have shown how to use the generalized FJ reduction scheme in order to find the mass transfer coefficient of a material whose pores are subject to a concentration difference. This study copes with conical pores of variable slopes (Figure 2). We also have proven that the mass transfer coefficient can be enhanced by choosing pores which are narrower at the entry (Figure 3). In particular, we have proven that the transference of mass can be maximized by choosing pore walls of an specific and predictable angle. This angle depends upon the magnitude of the reaction through the factor κ (Figure 4).

Figure 3. Normalized mass transfer coefficient defined in eq 16 as a function of the adsorption rate κ, for four symmetric pores (m2 = −m1) depicted in the inset figure. The values of the slope m2 in eqs 3 are shown.

Since the pore cannot be closed at the entry, there is a restricted zone delimited by m2 ≥ m1 − w(0)/L depicted in brown color. Pores below this line cannot exist. Finally, the line m2 = m1 corresponds to inclined pores of constant width. A particular case of this line is the origin m2 = m1 = 0 which corresponds to the rectangular pore. This line and its neighborhood correspond to solutions of the model with u0 → ∞. They are studied separately in the Supporting Information. Figure 2 was made by returning the definitions of the dimensionless parameters u0 and ϕ in eqs 11 and 9, respectively, to their expressions in terms of DFJ and γ using eqs 4 and 6. 3.2. Analysis of the Results: Optimization of the Mass Transfer Coefficient. Until now, from comparing the two contour plots in Figure 2, our study establishes that the relation between the geometry of the pore and the mass transfer coefficient depends in turn upon the rate of reaction κ. In order to study this dependence, in Figure 4 we plot the slope of the more efficient geometry as a function of the reaction rate κ. In other words, for each value of κ, we find the optimal pore for mass transfer in terms of the slope m2 = −m1. As it can be seen

4. INTERNAL EFFECTIVENESS FACTOR When one is interested on the efficiency of adsorption inside a porous material which is saturated, there are two ways in which the process can be studied. One way is by surrounding the material by fluid at the two ends of the pore. The other way is by closing one of the ends and let the fluid enter only by one side, until it reaches a maximum of concentration.29 While the first form is more useful in practical applications, the second is easier to understand in terms of geometrical variables and therefore is the one we use in this work. In this last case, we have to solve eq 2 using the boundary conditions C(0) = C0 and J(x = L) = 0. The first condition refers to the fixed concentration at the entry C0, and the second one refers to the zero-flux boundary condition at the right of the pore. In this situation, it is customary to define the effectiveness factor η as a practical measure of the adsorption process. This coefficient is defined as the ratio between the amount of mass adsorbed and the mass that would be adsorbed if the entire pore surface were exposed to the external concentration C0.29 Since the pore is 29157


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The Journal of Physical Chemistry C closed, the actual amount of material adsorbed per unit of time can be estimated from the flux entering to the pore. On the other side, the amount that would be adsorbed if the entire pore is subject to a concentration of saturation can be estimated evaluating the volumetric reaction in eq 7 with constant concentration C0.8,29 Therefore, remembering that we are dealing only with the bidimensional case we have that this ratio is

ηm =

mol reaction per volume ⎡⎣ 2 ⎤⎦ ·volume of the pore [cm 2] scm

(19)

In order to find this effectiveness factor, we first establish the dimensionless boundary conditions of the Neumann type for the right end of the pore: Z(0) = 1, Z′(1) = 0

(25)

where the subscript m emphasizes that this expression is valid for pores of constant slope m. For this deduction, see the Supporting Information. Equation 23 establishes the effectiveness factor for trapezoidal pores in terms of the geometry of the pore and the generalized Thiele modulus. From the comparison between eqs 23 and 25, it can be deduced that the adsorption inside an irregular pore can be reduced or enhanced according to the relative lengths of the pore entry and exit. In order to understand this connection, in we next present a study focused on the geometrical dependence of the effectiveness factor η on the slopes of the pore. This is carried out in similar lines that the study of the mass transfer coefficient in the previous section. The contour plots of Figure 5 show the behavior of the effectiveness factor predicted by eq 23 in terms of the slopes m1

mol Net flux inside the pore ⎡⎣ scm ⎤⎦ · Area of entry [cm]

η=

1 tanh ϕ ϕ

(20)

The solution of eq 8 with the boundary conditions stated in eq 20 can be obtained with Wolfram Mathematica in terms of modified Bessel functions of the first and second kind. The functional form of the solution is Z(u) =

i I0[− ϕτ (u0 − u)] Y1[iϕτ (u0 − 1)] + I1[ϕτ (u0 − 1)] Y0[iϕτ (u0 − u)] i I0[ϕτ u0]Y1 [iϕτ (u0 − 1)] + I1[ϕτ (u0 − 1)] Y0[iϕτ u0]

(21)

In order to calculate the effectiveness factor according to the definition given in eq 19, we have to do two things. First, we calculate the flux of this solution according to eq 15, in the same way we did for the mass transfer coefficient. Second, we have to compute the geometrical quantities related to the area of entry and volume of the pore. In the case of a trapezoidal pore defined in eqs 3, the area of entry is A = w(0) = b2 − b1, and the volume of the pore is V = L L⎡⎣(m2 − m1) 2 + (b1 − b1)⎤⎦. In this case, the ratio between the entrance area and the volume of the pore which appears in the definition of the effectiveness factor, eq 19, can be written in terms of u0 defined in eq 11 as −1 area of entry 1⎡ 1 ⎤ = ⎢1 − ⎥ volume of cell L⎣ 2u0 ⎦

Figure 5. Contour plot of the effectiveness factor η defined in eq 23 in terms of the slopes of the pore m1 and m2. The indications on the geometric relations are the same as that in Figure 2. These contour plots of the effectiveness factor were made by using two different values of the reaction rate κ = 1.0 and 0.1, left and rigth sides, respectively.

and m2 for different values of the reaction rate κ. As it emerges from Figure 5, the effectiveness factor increases as the pore becomes narrower at the exit, i.e., for low values of m2 and m1. In order to make this dependence clearer, in Figure 6 we plot the effectiveness factor for the four symmetrical pores depicted in the inset, by using eq 23. As it can be deduced from this figure, the effectiveness factor in a pore can be diminished or

(22)

Substituting the result of these two operations in eq 19 yields the following expression for the effectiveness factor of a trapezoidal pore −1 ⎡ 1 1 ⎤ η= f (ϕ , u 0) ⎢ 1 − ⎥ ϕ N 2u0 ⎦ ⎣

(23)

where f N(ϕ, u0) is fN (ϕ , u0) =

I1[−ϕu0] Y1[iϕ(u0 − 1)] − I1[ϕ(u0 − 1)] Y1[iϕu0] I1[ϕu0] Y1[iϕ(u0 − 1)] + i I1[−ϕ(u0 − 1)] Y0[iϕu0] (24)

The subscript N in this equation emphasizes that the solution of eq 8 is computed by using Neumann-zero-flux boundary conditions at the right end of the pore. A particular case of the effectiveness factor occurs when the pore has constant width. In these cases, it can be deduced that this factor is

Figure 6. Effectiveness factor defined in eq 23 as a function of the adsorption rate κ, for four symmetric pores (m2 = −m1) depicted in the inset figures. The values of the slope m2 in eqs 3 are shown. 29158


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The Journal of Physical Chemistry C enhanced depending on the specific relation between the length of the pore entry, w(0), and the exit, w(L). This confirms that when the mouth of the pore is wider than the exit, the effectiveness is increased. In view of these results, we can conclude that the funnel pores (as the one in red in Figure 6) are the appropriate geometries for maximizing the amount of adsorbed material in a structured porous material composed by conical pores. We justify this claim in terms of two observations. First, as we have proved, the effective rate of adsorption at the interior of the pore is greater in a funnel geometry than in a rectangular pore due to the augment of effective area of adsorption. This is taken into account in the description through the wall density-length γ(x). Second, as it can be seen from the cell-pore systems sketched inside Figure 6, the relation between area/volume of the red pore is greater than, for example, the blue pore; therefore, a series of similar pores can be arranged in more compact way in a membrane. This arrangement optimizes the effective external surface area (proportional to the surface porosity ϵS), as well as the active area in the interior of the pore (proportional to the volumetric porosity ϵV).8,29 In summary, in this section we have provided a reduced model able to describe the general behavior of the effectiveness factor in terms of the pore shape for porous systems under saturation conditions. We have proven that as in the case of the mass transfer coefficient the effectiveness factor depends upon the magnitude of the reaction of adsorption and the geometry of the pore in a way predictable by our model (Figure 5). In general, this effectiveness decreases with the reaction rate, and it is optimized for pores which are wider at the entry (Figure 6). This allows us to advance the hypothesis that a good design for materials under saturation conditions can be made with this kind of pore, due to its efficiency of adsorption and their compact arrangement in a membrane.

The right-hand side of this equation only depends on the slopes m1 and m2 of the pore, since DFJ ∝ D0 (see eq 4) and γ is defined in eq 6. Equation 27 emphasizes the fact that the Thiele modulus of a trapezoidal pore is larger than that of a rectangular pore, since γ ≥ 2 and DFJ ≤ D0. In other words, the effective chemical reaction rate is enhanced by the increase of the effective wall in a trapezoidal pore as well, as the effective diffusive transport is lowered by the irregularity of the pore shape. In consequence, the Thiele modulus is always larger in irregular pores than in rectangular ones: ϕ > ϕs. This statement about trapezoidal pores can be extended to pores with more complex geometries. As any pore geometry can be viewed as a series of trapezoidal sections, we can conclude that the presence of funnel or bottlenecks inside an irregular pore enhances the efficiency of the adsorption process. In order to put this statement in terms of geometric quantities, we consider that any geometry in a small section of the pore can be viewed as a trapezoid. In this case, we substitute eqs 4 and 6 in eq 9, and then we write all not in terms of m1 and m2 but rather in terms of the derivatives of the width of the trapezoidal pore wx(x) ≈ m2 − m1 and the derivative of the 1 middle line of the channel hx(x) ≈ 2 (m2 + m1). This allow us to write the Thiele modulus ratio defined in eq 27 in terms of these two quantities, used in the FJ reduction schemes35,37 as ⎡ ⎤ ϕ 5 2 3 ≃ ⎢1 + hx2(x) + wx (x) + ···⎥ ⎣ ⎦ ϕs 4 48

where we have dropped fourth-order terms in the derivatives. This expression is valid for any pore section where wx and hx do not vary to much from zero. This equation means that the ratio between the characteristic times of reaction and diffusion inside an irregular pore is affected by the local tortuousness or inclination of the pore (measured mainly through the derivative of h(x)) and slightly by the pore constriction (measured by the derivative of w(x)).

5. EFFECT OF CONSTRICTION AND TORTUOUSNESS ON THE THIELE MODULUS The local tortuousness and constriction affect, in general, the reaction rate and diffusion inside an irregular pore. As a consequence of this fact, one may expect that the Thiele modulus that measures the relative importance of these two factors could be different from point to point along single pores. Here, we show that this is indeed the case and how to quantify this effect. In the last two sections, we have shown that the interplay between the processes of diffusion and adsorption in conical pores depends upon the Thiele modulus defined in eq 9. This coefficient depends on the values of the effective diffusion DFJ, density of wall-length γ, and the magnitude of reaction rate κ. In order to drop out the dependency of the Thiele modulus on the diffusion and reaction rate coefficients and to write it exclusively in terms of geometrical factors, we may calculate the deviation of the Thiele modulus with respect to that of a rectangular pore: ϕs =

2κL2 D0

6. DISCUSSION AND CONCLUSIONS On the basis of the generalized FJ equation, in this work we have formulated a very general model that provides an accurate description of the interplay between diffusion and adsorption processes in conical pores. Our study makes quantitative predictions on the optimization of mass transfer and adsorption processes in membranes and porous media in terms of two quantities, namely, the mass transfer coefficient k and the effectiveness factor η for a first-order adsorption reaction. In particular, we have demonstrated that the mass transfer across a membrane can be enhanced by using symmetric pores which are narrower at the entry and whose walls make angles between 45 and 55°approximately. The particular angle depends upon the longitudinal length of the pore and the rate of reaction. In contrast, when one is interested in enhancing the adsorption process in saturated systems, our study demonstrates that funnel-shaped pores (those whose entry is wider than its exit) can enhance the value of the effectiveness factor in an appreciable amount, when compared with rectangular pores. The validity of the generalized FJ scheme that we have used in this work has to be tested in a rigorous study whose details surpass the extension of this work. This requires a detailed comparison of the solutions of the generalized FJ scheme and the solution of the mass balance equation. In this work, we can only anticipate some results. In Figure 7, we compare the

(26)

This can be achieved by introducing the Thiele modulus ratio ϕ = ϕs

D0 DFJ

γ 2

(28)

(27) 29159


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of k and η in an arrangement of multiple pores of a structured material. This important fact means that our framework is able to incorporate external and volumetric porosities, and as a consequence of this, it represents a valuable exact theoretical reference for the industrial designing of materials with very specific properties of adsorption and mass transfer in terms of the number of pores, their geometry, and its packing. This predictive ability can be relevant to minimize production costs of membranes and other porous materials or for the design of reactors based on wedge and conical pores.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b09282. Detailed deductions of eqs 18 and 25 (PDF)

Figure 7. Concentration profiles using the generalized FJ equation in eq 2 (lines), compared with those obtained by solving directly the bidimensional equation of mass balance, eq 1 (markers). In the top of this figure, we present the changes in the stationary profile for three different values of κ, for a symmetrical pore of slope m2 = −m1 = 0.5. In the bottom, we present the changes in the stationary profile for four different symmetrical pores with the slope m2 indicated there. In this case, the rate constant is fixed for all cases in κ = 0.1. The left and right panels represent the boundary conditions in eqs 13 and 20, respectively.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Aldo Ledesma-Durán: 0000-0003-3258-5616 Notes

The authors declare no competing financial interest.

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concentration profiles of the reduced average concentration deduced directly by solving the mass balance eq 1 (symbols) and the concentration profile of the approximate scheme of eq 2 (lines). In this figure, different concentration profiles for several rate constants κ and slopes m are plotted, with D0, L, and w(0) fixed as before. As it can be seen, the generalized FJ scheme slightly underestimates the flux at the entry when the chemical reaction has a value over κ ≳ 1.5. This was expected for the FJ scheme since in these systems the hypothesis about the hierarchy of characteristic times of diffusive and chemical transport discussed in section 2 is not fulfilled.9 In the bottom row of Figure 7, we plot the solutions of the projected scheme by changing the geometry of the pore for a low value of κ. In this case, the results of the reduced scheme are in good agreement with the direct solution of the heterogeneous problem. Therefore, our preliminary study shows that the possible flow overestimation (and, therefore overestimation of k and η), do not change in a significant way the geometrical implications of our work, at least for Thiele modulus ϕ ≲ 5. This makes our model very convenient for studying systems such as those in refs 10−15 in the limit of internal-diffusion/ reaction-controlled mechanisms. Remarkably, the utilization of the definitions and the methodology proposed in this work can be extended to more general geometries, as long as the pores forming the porous material fulfill the hypotheses of the FJ scheme.9 For example, we believe that the projection scheme used here can be applied in sinusoidal geometries that resemble zeolites,9,38 where the presence of caves and throats constitutes a key factor in the efficiency of those materials for technological purposes. The advantage of our approach to the problem of diffusion and adsorption across irregular pores under flow and saturation conditions is that the generalized FJ scheme provides analytical solutions which permit us to predict with precision the dependences of k and η on the specific geometry of the pore. Since the model framework we propose gives the study of a single pore inside a cell, it allows one to assess the interrelation

ACKNOWLEDGMENTS We are grateful to UNAM-DGAPA-PAPIIT project IN113415 for financial support. ALD acknowledges CONACyT for financial support under fellowship 221505. S.I.H. is grateful to DGTIC-UNAM project No. SC16-1-IR-113.

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Effectiveness Factor and Mass Transfer Coefficient in Wedge and

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