Diffusion frequency response method

A fundamental advantage of the frequency response method is its ability to yield information concerning the distribution of molecular mobilities. For example, a bimodal distribution of diffusivites, which is difficult to detect by conventional sorption measurements, leads to two different resonances [49], Moreover, from an analysis of the frequency response spectrum it is even possible to monitor molecular diffusion in combination with chemical reactions [45]. As in conventional sorption experiments, however, the intrusion of heat effects limits the information provided by this technique for fast adsorption-desorption processes [50]. 

Frequency response methods have been found useful in both theoretical and experimental analysis of gas mixing in fluidized beds. Experiments in a fluidized-bed reactor related to mixing theory were made by Bamstone and Harriott 24). Testin and Stuart have measured diffusion coefficients in gas-solid adsorption studies 25). 

Species formed from acetylene (Ay) adsorbed in zeolite Y, mordenite, beta and ZSM-5 have been studied by IR spectroscopy. The dynamics of Ay physisorption has been characterized by the frequency response method (FR). The rate of micropore diffusion governed the transport in Na-mordenite, while sorption was the rate limiting process step for all the H-zeolites. The equilibrium constants (Ka) of Ay sorption have been determined applying the Langmuir rate equation to describe the pressure dependence of the sorption time constants. The -octane hydroconversion activity of Pt/H-zeolites was found to increase linearly with the Ka of Ay sorption on the H-zeoIites. 

In the frequency response method, first applied to the study of zeolitic diffusion by Yasuda [29] and further developed by Rees and coworkers [2,30-33], the volume of a system containing a widely dispersed sample of adsorbent, under a known pressure of sorbate, is subjected to a periodic (usually sinusoidal) perturbation. If there is no mass transfer or if mass transfer is infinitely rapid so that gas-solid mass-transfer equilibrium is always maintained, the pressure in the system should follow the volume perturbation with no phase difference. The effect of a finite resistance to mass transfer is to cause a phase shift so that the pressure response lags behind the volume perturbation. Measuring the in-phase and out-of-phase responses over a range of frequencies yields the characteristic frequency response spectrum, which may be matched to the spectrum derived from the theoretical model in order to determine the time constant of the mass-transfer process. As with other methods the response may be influenced by heat-transfer resistance, so to obtain reliable results, it is essential to carry out sufficient experimental checks to eliminate such effects or to allow for them in the theoretical model. The form of the frequency response spectrum depends on the nature of the dominant mass-transfer resistance and can therefore be helpful in distinguishing between diffusion-controlled and surface-resistance-controlled processes. 

Yasuda, Y., Determination of vapor diffusion coefficients in zeoUte hy the frequency response method, J. Phys. Chem., 86, 1913-1917, 1982. 

Sun, L.M. and Bourdin, V., Measurement of intracrystalline diffusion by the frequency response method analysis and interpretation of bimodal response curves, Chem. Eng. Sci., 48, 3783-3793, 1993. 

Sun, L.M. and Do, D.D., Dynamic study of a closed diffusion cell by using a frequency response method single resonator, J. Chem. Soc., Faraday Trans., 91, 1695-1705, 1995. 

Bourdin, V.. et al., Application of the thermal frequency response method and of pulsed field gradient NMR to study water diffusion in zeolite NaX. Adsorption. 2(3), 205-216 (1996). 

Fig. 18. Self-diffusion coefficients of benzene in NaX at 458 K PFG NMR, O (97) and (92) (JENS, A (13) deduced from NMR lineshape analysis, (10). Comparison with nonequilibrium measurements T, sorption uptake with piezometric control (93) , zero-length column method (96) o, frequency-response and single-step frequency-response technique (98). The region of the results of gravimetric measurements with different specimens (92) is indicated by the hatched areas. Asterisked symbols represent data obtained by extrapolation from lower temperatures with an activation energy confirmed by NMR measurements.

Regarding adsorption and diffusion without reaction, Jordi and Do (49) simulated the expected results for the frequency response by completely numerical methods, with no need for linearization. In a later study, they used a linearized model coupled with analytic solutions for the diffusion inside the particles, which also took into account transport in both macropores and micropores (50). The mathematical details are clearly presented in these papers. 

There are macroscopic (uptake measurements, liquid chromatography, isotopic-transient experiments, and frequency response techniques), and microscopic techniques (nuclear magnetic resonance, NMR and quasielastic neutron spectrometry, QENS) to measure the gas diffusivities through zeolites. The macroscopic methods are characterized by the fact that diffusion occurs as the result of an applied concentration gradient on the other hand, the microscopic methods render self-diffusion of gases in the absence of a concentration gradient [67]. 

The dynamics of methane, propane, isobutane, neopentane and acetylene transport was studied in zeolites H-ZSM-5 and Na-X by the batch frequency response (FR) method. In the applied temperature range of 273-473 K no catalytic conversion of the hydrocarbons occurred. Texturally homogeneous zeolite samples of close to uniform particle shape and size were used. The rate of diffusion in the zeolitic micropores determined the transport rate of alkanes. In contrast, acetylene is a suitable sorptive for probing the acid sites. The diffusion coefficients and the activation energy of isobutane diffusion in H-ZSM-5 were determined. 

Gangwall et al. [47] were the first to apply Fourier analysis for the evaluation of the transport parameters of the Kubin-Kucera model. Gunn et al. applied the frequency response [80] and the pulse response method [83] in order to determine the coefficients of axial dispersion and internal diffusion in packed beds from experiments performed at various Reynolds numbers. Bashi and Gunn [83] compared the methods based on the analytical properties of the Fourier and the Laplace transforms for the calculation of transport coefficients. MacDonnald et al. [84] discussed the applications of the method of moments to the analysis of the profiles of skewed chromatographic peaks. When more than two parameters have to be determined from one single run, the moment analysis method is less suitable, because only the first and second moments are reliable (see Figure 6.9). Therefore, only two parameters can be determined accurately. 

Abstract Theoretical, experimental principles and the applications of the frequency response (FR) method for determining the diffusivities in microporous and bidispersed porous solid materials have been reviewed. Diffusivities of hydrocarbons and some other sorbates in microporous crystals and related pellets measured using the FR technique are presented, and the FR data are analysed to demonstrate the identification of the FR spectra. These results display the ability of the FR method to discriminate multi-kinetic mechanisms, including a surface resistance or surface barrier occurring simultaneously in the systems, which are difficult to be determined using other microscopic or macroscopic methods. The FR measurements also showed that the diffusivity of a system depends significantly on the subtle differences in molecular shape and size of sorbates in various... 

ABSTRACT. The principle features of the frequency-response i paratus developed at Imperial College is described. The apparatus has been used in both its a) full and b) single-step frequency modes to determine the diffiisivities of various hydrocarbons in silicalite-1. Die effect of temperature and loading of sorbate in the silicalite-1 has been ascertained. The effect of the introduction of A1 atoms into the framework of silicalite-1 on the diffusivity of benzene has been detmnined. The diffusion of benzene in NaX has been studied and diffusion coefficients obtained which agree with NMR pulsed field gradient measurements, n-Butane and 2-butyne hydrocarbons were found to generate out-of-phase response curves by the full FR method which could only be fitted by introducing two diffusion coefficients into the solution of the appropriate diffusion equation. 

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