**Abstract**

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 intraand 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 coadsorption 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.

**Keywords:** competitive diffusion of gases, competitive adsorption, modeling, diffusion coefficient, Heaviside's operational method, zeolite bed, gradient methods of identification

### **1. 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 [1–3]. 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's integral 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.

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 Δ*lk* ¼ *lk* � *lk*�1, perpendicular to the propagation of the gas in the z direction. The corresponding coefficients of inter- and intra-crystallite space are

*Competitive Adsorption and Diffusion of Gases in a Microporous Solid*

**3. Experimental results: gaseous benzene and hexane adsorption curves**

indicating quantitatively the relative affinity to the two adsorbates.

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,

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 inter-

crystallite 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.

**4. A mathematical model of competitive co-adsorption and co-diffusion**

*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*

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]

*Dinter,k* and *Dintra,k*, respectively.

*DOI: http://dx.doi.org/10.5772/intechopen.88138*

**in microporous solids**

*with permission from ACS].*

**Figure 2.**

**15**

**4.1 Co-adsorption model in general formulation**
