L.M. Sun

Measurement of intracrystalline diffusion by the frequency response method: analysis and interpretation of bimodal response curves

Abstract In this paper we present a detailed investigation of alternative possible for the bimodal behavior of frequency response curves observed experimentally by Shen and Rees for diffusion of p -xylene in silicalite-1. It is shown that a nonisothermal single-diffusion model can well represent the experimental data using consistent equilibrium data, suggesting that the heat effect provides a possible explanation for the bimodal behavior. On the other hand, it is shown that such a behavior could also be described by a diffusion—rearrangement model which considers the straight and sinusoidal channels of silicate-1 as the transport and storage channels. However, with this model, good agreement with the experimental data could be obtained only when the crystals are assumed to be spherical, which is not consistent with the actual geometry.

Frequency response for nonisothermal adsorption in biporous pellets

The frequency response for nonisothermal adsorption in a biporous pellet is analyzed theoretically, using a mathematical model which includes heat and mass transfer resistances in both micropores and macropores. It is confirmed that, when the heat effect is involved, the out-of-phase component may exhibit a bimodal form. Moreover, it is shown that when both macropore diffusion and micropore diffusion resistances are comparable, macropore diffusion behaves like a surface barrier and leads to an intersection of the in-phase and out-of-phase response functions. When either micropore diffusion or macropore diffusion alone is dominant, the frequency response is essentially the same and, therefore, provides no information concerning the nature of the controlling diffusional resistance. Experimental data for light linear paraffins-5A, reported by Yasuda (1991, J. phys. Chem. 95, 2486–2492), are reanalyzed by the present nonisothermal model. It turns out that the reported experimental response can be equally well represented by a nonisothermal model using several different combinations of mass transfer resistances. It, therefore, appears that the bimodal behavior of the experimental out-of-phase data is caused by the heat effect, thus contradicting the conclusion of Yasuda that there are two adspecies with different mobilities. With the reported data, it is, however, not possible to extract reliable values for the intracrystalline diffusion coefficient, and the nature of the controlling mass transfer resistances cannot be established with certainty.

On the heat effect in measurements of sorption kinetics by the frequency response method

Abstract The effect of heat transfer on the frequency response of sorption kinetics is theoretically investigated using a detailed model incorporating a single diffusion resistance, a surface barrier and heat transfer resistance at the adsorbent surface and at the chamber wall. The steady-state periodic solutions of this model are obtained analytically. Furthermore, a moment analysis is undertaken to establish the time constants of the different transfer mechanisms, which can be quantitatively compared so as to identify the dominant mechanism(s). It is shown that frequency response data can be considerably altered by the heat effect and in certain situations the response assumes a bimodal form. Under the isothermal assumption, this second peak can be erroneously attributed to an additional mass transfer process. Moreover, it is demonstrated that the frequency response of pressure and temperature are independent of variation in the applied volume amplitude. It is also confirmed by the present work that the frequency response is very sensitive to the nature of the transfer mechanisms, making it theoretically feasible to discriminate between surface barrier control and intraparticle diffusion.

Numerical simulation of diffusion-limited PSA process models by finite difference methods

A numerical method based on finite differences is proposed for simulating pressure swing adsorption processes with intraparticle diffusion controlling. The problem has two important spatial coordinates: the axial position in the bed and the location within a particle. The method proposed keeps the two coordinate scales separate. Diffusion equations with variable main- and cross-term diffusivities are solved implicitly for the particles at each time step and the solution is expressed in terms of the bed-scale variables. The material balances on the bed can then be solved with the bed-scale variables as the only unknowns. The method is computationally efficient and validated by comparison with an exact solution derived under conditions of plug flow, constant total pressure, and with a binary Langmuir isotherm. As a full example, we treat the kinetic separation of air to produce nitrogen using a fixed bed of carbon molecular sieve. Characteristics of the specific implementation are incorporation of variable-step grids with identical volumes for the intraparticle diffusion equations and the use of the third-order QUICK scheme to discretize the convection terms in the material balances for the bed.

Nonisothermal adsorption of water by synthetic NaX zeolite pellets

Abstract Experimental measurements and theoretical analysis of the uptake and surface temperature are presented for adsorption of water by synthetic NaX zeolite pellets of different sizes. It is shown that, under the present experimental conditions, the heat dissipation effect plays a dominant role in the adsorption kinetics, after a short initial stage which is controlled essentially by both the macropore diffusion and the micropore diffusion. An excellent agreement is obtained between the experimental data and theoretical results given by a nonisothermal model for a bidisperse structure, both for the uptake and for the surface temperature curves. The values of diffusivities determined by the curve-fitting method over an initial stage are of the order of D p ∼ 3 × 10 −5 m 2 /s for the macropore diffusion and D ie / R i 2 ∼ 0.003 s −1 for the micropore diffusion, and these values exhibit an independence of the pellet size for all the tested pellets. Furthermore, it is demonstrated that the use of an effective model for the monodisperse structure may also give a good agreement, but the values of diffusivities determined by the best fit are not very meaningful because of their dependence on the pellet size.

Numerical solution of diffusion equations by the finite difference method: efficiency improvement by iso-volumetric spatial discretization

The traditional finite difference method uses in general the equal-spacing discretization for the solution of diffusion equations in spherical particles. This leads to an unequal density of grids per volume inside the spherical particle with more points in the central region than near the surface. Consequently, the numerical results have poor precision, especially at short diffusion times. The efficiency of the finite difference method can be improved if the grid density in the particles is better distributed. This has been confirmed and the iso-volumetric discretization, which gives a constant grid density per volume in the particle, has been found to be significantly more accurate than the usually use equal-spacing discretization. We have considered the simple case of single component diffusion with a constant diffusion coefficient. It has been found that the iso-volumetric discretization performs better than the equal-spacing discretization in more complicated problems as well

Analysis of the temperature frequency response for diffusion in crystals and biporous pellets

Abstract The frequency response behaviour of the adsorbent temperature is analysed in detail for both diffusion in crystals and in biporous pellets. In the case of crystals, it is shown that the temperature frequency response may exhibit very different behaviour under effects of intracrystalline diffusion or surface barrier, making it easier to discriminate between the diffusion and surface barrier processes. For biporous pellets, the frequency response of the surface temperature proves to be very sensitive to the competition between micropore diffusion and macropore diffusion, due to finite-rate heat conduction inside pellets. This high sensitivity can lead to a much better estimation of the micropore diffusion coefficient in the presence of macropore diffusion than the estimation obtained with the gas pressure frequency response. Finally, it is shown that, in the case of high-pressure experiments, the effect of heating by gas compression may be significant if the gaseous adsorbate has a large specific heat ratio and a large heat capacity, and if the heat exchange at the chamber wall is slow enough. The effect of gas compression heating on the temperature frequency response is confirmed experimentally for diffusion of xenon in zeolite 5A crystals. The present paper shows, however, the effect of gas compression heating on the gas pressure can be efficiently eliminated using the frequency response data corrected with respect to blank experiments, as proposed by Yasuda.

On the Determination of Diffusion Coefficients by Uptake Rate Measurements

Abstract Diffusion coefficients are determined by the measurement of adsorbent surface temperatures using an infra-red detector. The advantages of the optical technique are high temperature sensitivity, fast response and the ability to focus on a precise position. This makes it possible to experiment on a single pellet or very few crystals, thus avoiding intrusion of external transfer resistances. The diffusivities obtained for several systems are in accordance with the n.m.r. data. It has been observed that regeneration conditions could have a large impact on the uptake rate and even probably create a surface barrier effect. The possibility to discriminate between the diffusion and surface barrier by the frequency response technique will be presented.

Interference theory—III. Moment solution for linear isothermal multicomponent adsorption in a pellet

Abstract Multicomponent adsorption in a bidisperse pellet has been studied. The interference between components is included both through the adsorption equilibrium and through diffusion in both macropores and micropores. By the application of moment analysis and matrix operations, the moment solution of the system has been obtained and shown to be essentially the same as the solution for a single-component system but in vector form. The results clearly show the interference between components, and its influence on the adsorbed mass and uptake rate.

Determination of the Kinetic and Thermodynamic Parameters of Adsorption Processes by a Volume Step Thermal Method

A volume step method measuring the pressure and the adsorbent temperature of an adsorbent-adsorbate system has been developped. It is shown that this method allows the determination of all the relevant parameters of an adsorption process, kinetic as well as thermodynamic in case of Linear Driving Force mass transfer. The method for determining the parameters can be extended to the case of diffusive mass transfer if the mass transfer kinetics is faster than the heat transfer kinetics. An example is given, showing the determination of the diffusion coefficient of carbon dioxide in NaX zeolite pellets and the change of the diffusion coefficient and of the isosteric heat of adsorption when the adsorbent is not fully dehydrated.