Competitive Adsorption and Diffusion of Gases in a Microporous Solid

The experimental and theoretical study of the co-adsorption and co-diffusion of several gases through a microporous solid and the instantaneous (out of equilibrium) distribution of the adsorbed phases is particularly important in many fields, such as gas separation, heterogeneous catalysis, purification of confined atmospheres, reduction of exhaust emissions contributing to global warming, etc. The original NMR imaging technique used gives a signal characteristic of each adsorbed gas at each instant and at each level of the solid and therefore the distribution of several gases in competitive diffusion and adsorption. But it does not allow to determine separately the inter-and intra-crystallite quantities. A new fast and accurate analytical method for the calculation of the coefficients of co-diffusing gases in the intra-and inter-crystallite spaces of microporous solid (here ZSM 5 zeolite) is developed, using high-performance methods (iterative gradient methods of residual functional minimization and analytical methods of influence functions) and mathematical co-adsorption models, as well as the NMR spectra of each adsorbed gas in the bed. These diffusion coefficients and the gas concentrations in the inter-and intra-crystallite spaces are obtained for each position in the bed and for different adsorption times.


Introduction
Knowledge of the co-diffusion and co-adsorption coefficients of reactants and products is essential when a heterogeneous catalytic reaction is performed by flowing gases through a microporous catalyst bed.But generally the distribution of the various reactants adsorbed on the catalyst is very inhomogeneous and, moreover, very variable from one reactant to another.It is therefore necessary to determine at every moment the diffusion coefficient of each reactant in the presence of the others and its instantaneous distribution along the length of the catalyst bed.
Classical H-MRI should be a good technique for monitoring the co-diffusion and co-adsorption of several gases flowing through a microporous bed.However, since the signal obtained is not specific for each gas, this requires that each experiment be performed several times under identical conditions and each time with only one not deuterated gas.To remedy the drawbacks of classical imaging, we have used the NMR imaging technique, named slice selection procedure, to follow the diffusion and adsorption of a gas in a microporous bed [1][2][3].The sample is displaced vertically, stepby-step, relative to a very thin coil detector during the adsorption of the gas.The bed is assumed to consist of N very thin layers of solid, and the region probed is limited to each layer, so that the variation of the concentration of gas absorbed at the level of each layer is obtained as a function of time.An interesting feature of this technique is its ability to visualize directly the co-diffusion of several gases.Indeed, the NMR signals are quantitatively characteristic of the adsorbed gases.They can therefore provide directly, at every moment and at every level of the bed, the distribution of several gases competing in diffusion and adsorption.We have presented in a previous paper the experimental results of the co-diffusion of benzene and hexane through a silicalite bed [4].In [5,6] we have developed a mathematical methodology for efficient linearization of similar models.Using Heaviside's operational method and Laplace'sintegral transformation method, we have built solutions allowing fast calculations for twocomponent co-adsorption in a heterogeneous zeolite bed and for the dehydration of natural gas [7].In this chapter we have improved the methods previously used to compute the diffusion coefficients against time, increasing the accuracy and speed of calculations by significantly reducing the iteration number.This made it possible to use them for the co-adsorption of several gases diffusing along such a column.

Experimental
The NMR imaging technique, the sample-holder bulb containing the liquid phase in equilibrium with the gas phase, and the narrow zone monitored by the detector have been described in [1][2][3], respectively.
The upper face of the cylindrical bed of zeolite crystallites is exposed to a constant pressure of each gas (Figure 1).The diffusion of the two gases is axial in the macropores of the inter-crystallite space (z direction along the height, l, of the bed) and radial in the micropores of the zeolite.According to the experimental conditions, the zeolite bed consists of a large number, N, of very thin layers of solid, of thickness Δl k ¼ l k À l kÀ1 , perpendicular to the propagation of the gas in the z direction.The corresponding coefficients of inter-and intra-crystallite space are D inter,k and D intra,k , respectively.

Experimental results: gaseous benzene and hexane adsorption curves
The experimental results have been summarized in [4]: the spectrum of each gas at every instant and every level of the solid and the benzene and hexane concentrations along the sample, for each diffusion time.Here we shall only use the evolution, as a function of time, of the benzene and hexane concentrations at different levels of the sample, on which the calculations of the diffusion coefficients and the instantaneous inter-and intra-crystallite concentrations are based [8]. Figure 2 shows clearly that, under the chosen experimental conditions, benzene hinders the diffusion of hexane, and this at every moment.Moreover, it can be noticed that, at equilibrium, the amount of benzene within the zeolite is twice that of hexane, indicating quantitatively the relative affinity to the two adsorbates.
These curves display modulations as a function of time, which must be averaged for all subsequent mathematical representations.These modulations are weak at the lower layers of the tube and can be due to errors in the measurement of small amounts.Those closer to the arrival of the gas are greater and are similar for the two gases.We suggested that these fluctuations may be due to the fact that intercrystallite adsorption at levels close to the gas phase is fast compared to the liquid-gas equilibrium, which is not as instantaneous for a mixture as for a single component [8].Each slight decrease of the gas pressure could correspond a slight fast desorption.

Co-adsorption model in general formulation
The model presented is similar to the bipolar model [2,3,8,9].By developing the approach described by Ruthven and Kärger [10,11] and Petryk et al. [5]  concerning the elaboration of a complex process of co-adsorption and co-diffusion, it is necessary to specify the most important hypotheses limiting the process.
The general hypothesis adopted to develop the model presented in the most general formulation is that the interaction between the co-adsorbed molecules of several gases and the adsorption centers on the surface in the nanoporous crystallites is determined by the nonlinear competitive equilibrium function of the Langmuir type, taking into account physical assumptions [10]: 1. Co-adsorption is caused by the dispersion forces whose interaction is established by Lennard-Jones and the electrostatic forces of gravity and repulsion described by van der Waals [11].
2. The co-diffusion process involves two types of mass transfer: diffusion in the macropores (inter-crystallite space) and diffusion in the micropores of crystallites (intra-crystallite space).
3. During the evolution of the system toward equilibrium, there is a concentration gradient in the macropores and/or in the micropores.
4. Co-adsorption occurs on active centers distributed over the entire inner surface of the nanopores (intra-crystallite space) [10,11].All crystallites are spherical and have the same radius R; the crystallite bed is uniformly packed.
5. Active adsorption centers adsorb molecules of the ith adsorbate, forming molecular layers of adsorbate on their surfaces.
6. Adsorbed molecules are held by active centers for a certain time, depending on the temperature of the process.
Taking into account these hypotheses, we have developed a nonlinear coadsorption model.The meaning of the symbols is given in the nomenclature: with initial conditions boundary conditions for coordinate X of the crystallite boundary and interface conditions for coordinate Z Tt ,Z ðÞ Z¼1 ¼ T initial , ∂ ∂z Tt ,ZÞ ð j Z¼0 ¼ 0: (8) with Here the activation energy is the heat of adsorption defined asΔH s ¼ ϕ À U g s À U ads s ÀÁ À R g T, where U g s À U ads s -the difference between the kinetic energies of the molecule of the ith component of the adsorbate in the gaseous and adsorbed states is the magnitude of the Lennard-Jones potential, averaged over the pore volume of the adsorbent [11].
The non-isothermal model ( 1)-( 8) can easily be transformed into isothermal model, removing the temperature Eq. ( 2) and condition (8) and replacing the functions K s T ðÞ with the corresponding equilibrium constants K s .The competitive diffusion coefficients D intra s and D inter s can be considered as functions of the time and the position of the particle in the zeolite bed.

The inverse model of co-diffusion coefficient identification: application to the benzene-hexane mixture
On the basis of a developed nonlinear co-adsorption model ( 1)-( 8) , we construct an inverse model for the identification of the competitive diffusion coefficients D intra s and D inter s as a function of time and coordinate in the zeolite bed.
The mathematical model of gas diffusion kinetics in the zeolite bed is defined in domains: by the solutions of the system of differential equations with initial conditions boundary and interface conditions for coordinate Z boundary conditions for coordinate X in the particle Additional condition (NMR-experimental data) The problem of the calculation ( 9)-( 15) is to find unknown functions satisfy the condition (15) for every point h k ⊂ Ω k of the kth layer [8,12]. Here where XdX is the average concentration of adsorbed component s in micropores and M s t, Z ðÞ j h k is the experimental distribution of the mass of the sth component absorbed in macro-and micropores at h k ⊂ Ω k (results of NMR data, Figure 2).

Iterative gradient method of co-diffusion coefficient identification
The calculation of D intra s k and D inter s k is a complex mathematical problem.In general, it is not possible to obtain a correct formulation of the problem ( 9)-( 15) and to construct a unique analytical solution, because of the complexity of taking into account all the physical parameters (variation of temperature and pressure, crystallite structures, nonlinearity of Langmuir isotherms, etc.), as well as the insufficient number of reliable experimental data, measurement errors, and other factors.
Therefore, according to the principle of Tikhonov and Arsenin [13], later developed by Lions [14] and Sergienko et al. [15], the calculation of diffusion coefficients requires the use of the model for each iteration, by minimizing the difference between the calculated values and the experimental data.
The calculation of the diffusion coefficients ( 9)-( 15) is reduced to the problem of minimizing the functional of error ( 16) between the model solution and the experimental data, the solution being refined incrementally by means of a special calculation procedure which uses fast high-performance gradient methods [6,8,12,15].
According to [12,15], and using the error minimization gradient method for the calculation of D intra s k and D inter s k of the sth diffusing component, we obtain the iteration expression for the n + 1th calculation step: where JD inter s k , D intra s k is the error functional, which describes the deviation of the model solution from the experimental data on h k ∈ Ω k , which is written as

Analytical method of co-diffusion coefficient identification
With the help of iterative gradient methods on the basis of the minimization of the residual functional, very precise and fast analytical methods have been developed making it possible to express the diffusion coefficients in the form of timedependent analytic functions (16).For their efficient use, it is necessary to have an extensive experimental database, with at least two experimental observation conditions for the simultaneous calculation of D intra s k and D inter s k coefficients.Our experimental studies were carried out for five Z positions of the swept zeolite layer for each of the adsorbed components.The data were not sufficient to fully apply this simultaneous identification method to these five sections.We therefore used a combination of the analytical method and the iterative gradient method for determining the co-diffusion coefficients.
Using Eqs. ( 9)- (15), it is possible to calculate D intra s k , D inter sk as a function of time using the experimental data obtained by NMR scanning.In particular, in Eqs. ( 9) and ( 10), the co-diffusion coefficients can be set directly as functions of the time t: D intra s k t ðÞ , D inter sk t ðÞ .In this case, the boundary condition (11) can be given in a more general form-also as a function of time: Experimental NMR scanning conditions are defined simultaneously for all P observation surfaces: For simplicity we design ut , Z We substitute the expression vt ,X,Z ðÞ (20) in the observation conditions (19): Integrating parts (23) and taking into account equality we obtain [16] ut , h i ðÞ ¼ χ i t ðÞÀ Let us first put ut , h P ðÞ ¼ μ sP t ðÞ¼C initial s t ðÞ , where Z ¼ h P is the observation surface, approaching the point of entry into the work area Z = 1.
Then Eq. ( 24) for i = P will be with parallel computing D inter si t ðÞ , i ¼ P À 1, 1.

Numerical simulation and analysis: co-diffusion coefficients: concentration profiles in inter-and intra-crystallite spaces
The variation against time of the benzene and hexane intra-crystallite diffusion coefficients, D intra 1,k аnd D intra 2,k , respectively, is presented in Figure 3 for the five coordinate positions: 6, 8, 10, 12, and 14 mm, defined now from the top of the bed.The curves for benzene D intra 1,k are pseudo exponentials.D intra 1,k decreases from 9.0

Variation of inter-crystallite diffusion coefficients (a.u.) for benzene (left) and hexane (right) against time at different positions in the bed.
Zeolites -New Challenges  EÀ13 to about 1.0 EÀ14 a.u.(equilibrium) depending on the position of the crystallite and the time, as well as on the amount of adsorbed gas.The shapes of the variations of D intra 2,k for hexane are roughly the same, but the diffusion coefficients are higher, from about 9.0 EÀ12 to 3.0 EÀ13 a.u.
Figure 4 presents the variation against time of the benzene and hexane diffusion coefficients in inter-crystallite space, D inter 1,k аnd D inter 2,k , for the same positions.These coefficients decrease with time from 2.0 EÀ6 to 1.0 EÀ7 a.u.(equilibrium) for benzene and from 3 EÀ5 to 1.0 EÀ6 a.u.for hexane, depending on the bed position, and increase adsorbed concentrations.
Figure 5 shows the variation against time of the calculated concentrations C for benzene and hexane in the inter-crystallite space.As can be seen, these concentrations approach the equilibrium values for a diffusion time around 250 min.But the variations of the concentrations with time are rather different for the two gases.
Figure 6 shows the variation of the concentrations Q(t,X,z) of adsorbed benzene (left) and hexane (right) in the micropores of the intra-crystallite space from the surface (abscissa-1) to the center (abscissa-0) of the crystallites located between 6 and 14 mm from the top of the bed and after 25-200 min. of diffusion (a, b, c, and d, respectively).The gradients increase, and the mean concentrations decrease with the increasing distance of the particles from the arrival of the gases.The particles at 6 and 8 mm are saturated with benzene after 100 min, but not yet with hexane.

Conclusion
The main result of this work is the possibility, from a single experiment, of simultaneously distributing several co-diffusing gases in a porous solid and of using the methods of mathematical modeling to analyze for each of them the distribution of their concentrations in the intra-and inter-crystallite spaces.
Using the experimental NMR data and proposed co-adsorption models, the identification procedures for calculating the co-diffusion coefficients for two or more components in intra-and inter-crystallite spaces are developed.These procedures use the iterative gradual identification methods on minimizing of the Lagrange error function and rapid analytic methods based on the influence function.The co-diffusion coefficients were obtained as a function of time for different positions along the catalyst bed.In particular, those in the intra-crystallite space were computed by the analytical method which allowed a calculation with a relatively high degree of discretization over time and to reduce practically twice the volume of iterative calculations.Using these results, the concentrations of codiffusing benzene and hexane in the inter-and intra-crystallite spaces were calculated for each time and each position in the bed.

Nomenclature
layer number, subscript k will be added to all the following symbols to specify that they are characteristic of the kth layer c adsorbate concentration in macropores c ∞ equilibrium adsorbate concentration in macropores layer thickness (all layers have the same thickness) L dimensionless bed length (L = 1) q adsorbate concentration in micropores q ∞ equilibrium adsorbate concentration in micropores

Appendix
A. Iterative gradient method of the identification of co-diffusion coefficients The methodology for solving the direct boundary problem ( 9)-( 15), which describes the diffusion process in a heterogeneous nanoporous bed, is developed in [9,12,15].According to [12] the procedure for determining the diffusion coefficients ( 16) requires a special technique for calculating the gradients ∇J n D intras k t ðÞand ∇J n D inters k t ðÞof the residual functional (17).This leads to the problem of optimizing the extended Lagrange function [12,15]: where I s macro , I s micro are the components given by Eqs.(A.2) and (A.3), corresponding to the macro-and microporosity, respectively J s is the residual functional (17), ϕ s k , ψ s k , s,¼1,2 -unknown factors of Lagrange, to be determined from the stationary condition of the functional Φ D inter sk , D intra sk ÀÁ [9,15]: The calculation of the components in Eq. (A.4) is carried out by assuming that the values D inter sk , D intra sk are incremented by ΔD inter sk , ΔD intra sk .As a result, concentration C s k t, Z ðÞ changes by increment ΔC sk t, Z ðÞ and concentration Q s k t, X, Z ðÞ by increment ΔQ s k t, X, Z ðÞ , s ¼ 1, 2. Conjugate problem.The calculation of the increments ΔJ s , ΔJ s macro , and ΔJ s micro in Eq. (A.4) (using integration by parts and the initial and boundary conditions of the direct problem ( 9)-( 15)) leads to solving the additional conjugate problem to determine the Lagrange factors ϕ s k and ψ s k of the functional (A.1) [15]:  [15].We have obtained the solutions ϕ sk and ψ sk of problem (A.5)-(A.10)using Heaviside operational method in [15].
Substituting in the direct problem ( 9)-( 15 Similarly, we write the system of the basic equations of conjugate boundary problem (A.5)-(A.10) in the operator: , , where L * is the conjugate Lagrange operator of operator L.
The calculated increment of the residual functional (17), neglecting secondorder terms, has the form Reporting in Eq. (A.17) the components X s k t, X, Z ðÞ taking into account Eq. (A.18), we obtain the formula which establishes the relationship between the direct problem ( 9)-( 15) and the conjugate problem (A.6)-(A.10)which makes it possible to obtain the analytical expressions of components of the residual functional gradient: Differentiating expression (A.18), by ΔD intra sk and ΔD inter sk , respectively, and calculating the scalar products according to Eq. (A.15), we obtain the required analytical expressions for the gradient of the residual functional in the intra-and inter-crystallite spaces, respectively:  9)-( 14) and inverse problem (A.5)- Zeolites -New Challenges (A.10).They provide high performance of computing process, avoiding a large number of inner loop iterations by using exact analytical methods [2,15].
B. The linearization schema of the nonlinear co-adsorption model: system of linearized problems and construction of solutions The linearization schema of nonlinear co-adsorption (1)-( 8) is shown in order to demonstrate the simplicity of implementation for the case of two diffusing components (m ¼ 2) and isothermal adsorption.The simplified model ( 1)- (8) Langmuir equilibrium ðÞ , ðÞ is represented by the series of Tailor [5]:

Figure 1 .
Figure 1.Distribution of the layers (left) and corresponding parameters (right).

Figure 2 .
Figure 2. Evolution vs. time of the benzene and hexane concentrations (arbitrary units) at different levels of the sample (continuous, experimental curves; dotted, their approximations used for simulation) [from Ref. [8], reprinted with permission from ACS].

Figure 4 .
Figure 4. Variation of inter-crystallite diffusion coefficients (a.u.) for benzene (left) and hexane (right) against time at different positions in the bed.

Figure 5 .
Figure 5. Variation of the inter-crystallite concentration (a.u.) calculated for benzene (left) and hexane (right) against time and at different positions in the bed.
[16]Þ¼D intra sk t ðÞ =R 2 , χ i t ðÞ¼M sk i t ðÞ , i ¼ 1, P, and considering Eq. (10) in flat form, its solution can be written as[16]: , vt ,X,Z ðÞ ¼ Q sk t,X,ZðÞ,ðÞ cos η m X Á cos η m ξ, η m ¼ mπ, inter si t ðÞ , i ¼ P À 1, 1 calculations.All subsequent coefficients D intra si t ðÞ will be calculated by the formula Using calculated D intra sP t ðÞwith the formula (28) on the observation limit h P ,we define the gradient method D inter sP t ðÞin the same way.With D intra sP t ðÞand D inter sP t ðÞ in h P , we calculate С sk t, h P ðÞ , substituting it in μ sPÀ1 t ðÞ¼С sk t, h P ðÞ for the next coefficient D ðÞ¼C s k D intra sk n , D inter sk n ; t, h k ÀÁ þ Q s k D intra sk n , D inter sk n ; t, h k (15)inter sk , D intra sk , С s k t, Z ðÞ , and Q s k t, X, Z ðÞ by the corresponding values with incrementsD inter sk þ ΔD inter sk , D intra sk þ ΔD intra sk , C s k t, Z ðÞ þ ΔC s k t, zðÞ, andQ s k t, X, Z ðÞ þ ΔQ s k t, X, ZðÞ, subtracting the first equations from the transformed ones and neglecting secondorder terms of smallness, we obtain the basic equations of the problem (9)-(15)in terms of increments ΔC s k t, Z ðÞ and ΔQ s k t, X, Z ðÞ , s ¼ 1, 2 in the operator form Lw s k t, X, Z s k ¼L À1 X s k and L À1 is the inverse operator of operator L. ðÞ and ψ s k t, X, Z ðÞ belong to Ω kt and 0, 1 ½ ∪ Ω kt , respectively, L À1 * is the conjugate operator to inverse operator L À1 , and Ψ s k is the solution of conjugate problem (A.5)-(A.10).
Á E s k t ðÞ δ Z À h k ðÞ XdXdZdt (A.14) wherе w for the case of m ¼ 2 is converted into the form Àθ 2ðÞ, where p 1 , p 2 are the co-adsorption equilibrium constants and partial pressure of the gas phase for 1-th and 2-th component and θ 1 , θ 2 , are the intra-crystallite spaces occupied by the corresponding adsorbed molecules.The expression φ s C 1 , C 2 ðÞ ¼