Sorption-diffusion models

Sferrazza RA, Escobosa R, and Gooding CH. Estimation of parameters in a sorption-diffusion model of pervaporation. J. Memb. Sci. 1988 35(2) 125-136. 

The sorption-diffusion model described in the foregoing section is very general. The manifestation of the fundamental principles in the transport depends on whether the membrane polymer is in a rubbery state or in a glassy state. All polymers undergo a transition from rubbery to glassy state when the temperature is lowered below the transition temperature, Tg that is characteristic to the polymer. 

It has to be pointed out that experimental results obtained from GFT s polyvinyl alcohol membranes, particularly the line correlating the water mole fraction in the permeate to that in the feed (see Figure 6.28) [246], cannot be reproduced by Equations 6.135 and 6.145. Presumably, the unique shape observed in the above-correlation is either due to the composite nature of the commercial PVA membrane or due to the membrane swelling occurring at the high water contents, or both. Recently, Hauser et al. explained the shape of the above-correlation curve by using the sorption-diffusion model 1247]. 

Steady-state permeability is defined as the flux of penetrant per unit pressure difference across a sample of unit thickness. Permeation through polymers is generally a three step process absorption of penetrant into the polymer matrix, diffusion of penetrant through the matrix and desorption of penetrant at the other side (Kirwan and Strawbridge, 2003). Thus, permeability is influenced both by the dissolution and the diffusion of the penetrant in the polymer matrix. In this sorption-diffusion model of penetrant transport across the polymer, permeability (P) is given as the product of diffusion coefficient () and solubility coefficient () (Callister and Rethwisch, 2010), that is ... 

Because of their intrinsic characteristics, mostly dense rubbery polymers have been considered for the preparation of polymeric catalytic membranes. The mass-transport mechanism considered has been the well-known sorption-diffusion model.ii Modelling the kinetics of the reaction(s) occurring at the occluded catalyst level is a much more complex task. The reaction may be carried out under special operating conditions, for example in a batch reactor where the catalyst is dispersed in a support - or directly inside the catalytic membrane, -" a reaction-rate model is assumed and the related parameters are determined by fitting the global model to the experimental data. In other cases, the kinetic models determined by other authors are used. - In some theoretical studies, a hypothetical reaction-rate model and the respective model parameters are assumed. 

The model by Yawalkar et was extended by Nagy some years later. The same geometrical distribution of the catalyst in the polymer phase and the same sorption-diffusion model for the mass transport, either in the polymer or in the catalyst, was considered. However, that author considered an irreversible first-order reaction in his model. From the study of the catalytic particles size and distribution, membrane thickness, membrane and catalyst diffusion coefficients, among other variables, the author concluded that the mass-transfer rate depends significantly on the size of the catalytic particles and the thickness of the membrane fraction between the surface and the first particles layer and, also, on the usually low diffusion coefficient through the catalytic particles. 

Reverse osmosis models can be divided into three types irreversible thermodynamics models, such as Kedem-Katchalsky and Spiegler-Kedem models nonporous or homogeneous membrane models, such as the solution—diffusion (SD), solution—diffusion—imperfection, and extended solution—diffusion models and pore models, such as the finely porous, preferential sorption—capillary flow, and surface force—pore flow models. Charged RO membrane theories can be used to describe nanofiltration membranes, which are often negatively charged. Models such as Dorman exclusion and the... 

Sorption can require more than a month to reach equilibrium for highly hydrophobic compounds, but can be adequately described by a radial diffusion model accompanied by the retarding influence of sorption. 

The Diffusion Model. The uptake of a solute by a sorbent can be analyzed by a diffusion model, which has been used successfully to model adsorption rates onto activated carbon (74, 75), ion exchangers (72), heterogeneous catalysts (76), and soil columns (77). For the purpose of illustration, we can consider the diffusion of a compound into a spherical sorbent grain under conditions of linear sorption and no exterior mass transfer limitations (73), which is described by... 

Wu and Gschwend (78) successfully employed a radial diffusion model to describe laboratory observed sorption and desorption kinetics. Their data show that sorption and desorption rates were slower for more hydrophobic compounds and sorbents with a larger grain size in a manner consistent with the radial diffusion model. 

In many early experiments, hysteresis was observed for highly hydrophobic compounds such as PCBs (79, 80). Since the time to reach equilibrium can be quite long for strongly hydrophobic compounds, a solute may have never reached equilibrium during the sorption isotherm experiment. Consequently, Kj would be underestimated, which leads to the discrepancy between the sorption and desorption coefficients that was attributed to hysteresis. The case for hysteresis being an artifact is supported by recent data for tetrachlorobenzene (log K = 4.7), illustrating that sorption and desorption require approximately two days to reach equilibrium with approximately equal time constants (78). Finally, the diffusion model is consistent with the observation that the extent of hysteresis was inversely related to particle size (81). 

While first-order models have been used widely to describe the kinetics of solid phase sorption/desorption processes, a number of other models have been employed. These include various ordered equations such as zero-order, second-order, fractional-order, Elovich, power function or fractional power, and parabolic diffusion models. A brief discussion of these models will be provided the final forms of the equations are given in Table 2. 

Equation (57) is empirical, except for the case where v = 0.5, then Eq. (57) is similar to the parabolic diffusion model. Equation (57) and various modified forms have been used by a number of researchers to describe the kinetics of solid phase sorption/desorption and chemical transformation processes [25, 121-122]. 

Kinetic models proposed for sorption/desorption mechanisms including first-order, multiple first-order, Langmuir-type second-order, and various diffusion rate laws are shown in Sects. 3.2 and 3.4. All except the diffusion models conceptualize specific sites to or from which molecules may sorb or desorb in a first-order fashion. The following points should be taken into consideration [ 181,198] ... 

Pignatello and Xing [107] used two models, the organic matter diffusion model (OMD) and the sorption-retarded pore diffusion model (SRPD), in order to understand better the meaning of slow sorption/desorption observations and mechanisms and to explore the most likely causes of such slow process in natural solid particles. These authors reported that both OMD and SRPD mechanisms operate in the environment, often probably together in the same particle. OMD may predominate in soils that are high in natural OM and low in aggregation, while SRPD may predominate in soils where the opposite conditions exist. 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

The radial diffusion model and the linear sorption model are compared in Fig. 18.76. Since according to Eq. 19-76 the total mass of the chemical associated with the particle aggregate, (M(r), and the macroscopic solid-water distribution ratio, Kd(t), are linearly related ... 

Figure 19.19 Comparison of the solution of the linear sorption model with the radial diffusion model. Numbers on curves show y defined in Eq. 19-86. Y is the fraction of the chemical taken up by the sphere when equilibrium is reached. After Wu and Gschwend (1988).

Explain the difference between the two types of curves shown in Fig. 19.19, which are labeled radial diffusion model and sorption model, respectively. 

In the performance data of various polyamide and related membranes published to date there should be valuable information for molecular design of more excellent barrier materials. But at present a means for their evaluation and optimization is still not clear. One of the reasons may at least come from the competitive flood of proposals for the detailed mechanisms of reverse osmosis, e.g. the solution-diffusion model, the sieve model, the preferential sorption model and so on. 109)... 

Membrane-penetrant systems, whose sorption and diffusion properties can be described by Eqs. (5)-(7) with N = 2 ( dual mode sorption and diffusion models ) have attracted much interest. The most important examples of such systems are considered in the next two sections. 

Application of the dual mode sorption and diffusion models to homogeneous polymer blend-gas systems 26,65) and filled polymers 66) has also been reported. 

It is particularly interesting and instructive to note that application of Henry + Langmuir dual-mode sorption and diffusion models is not confined to glassy polymer-gas systems. Sorption and transport of high affinity ionic species, exemplified by anionic dyes, in charged polymers, exemplified by polyamides at low pH, has been treated in the same way. These systems are of considerable importance both from the bio-mimetic and from the textile processing point of view, but have received limited atten-... 

Whereas the dual sorption and transport model described above unifies independent dilatometric, sorption and transport experiments characterizing the glassy state, an alternate model offered recently by Raucher and Sefcik provides an empirical and fundamentally contradictory fit of sorption, diffusion and single component permeation data in terms of parameters with ambiguous physical meanings (28), The detailed exposition of the dual mode model and the demonstration of the physical significance and consistency of the various equilibrium and transport parameters in the model in the present paper provide a back drop for several brief comments presented in the Appendix regarding the model of Raucher and Sefcik,... 

The gas-polymer-matrix model for sorption and transport of gases in polymers is consistent with the physical evidence that 1) there is only one population of sorbed gas molecules in polymers at any pressure, 2) the physical properties of polymers are perturbed by the presence of sorbed gas, and 3) the perturbation of the polymer matrix arises from gas-polymer interactions. Rather than treating the gas and polymer separately, as in previous theories, the present model treats sorption and transport as occurring through a gas-polymer matrix whose properties change with composition. Simple expressions for sorption, diffusion, permeation and time lag are developed and used to analyze carbon dioxide sorption and transport in polycarbonate. 

Transient-transport measurements are a powerful tool for evaluating the validity of any sorption-transport model. The ability of a model to predict diffusion time lags is a test for its validity, as all the parameters are fixed by the equilibrium sorption and steady state transport, and because the time lag depends on the specific form of the concentration and diffusion gradients developed during the transient-state experiments. 

The importance of adsorbent non-isothermality during the measurement of sorption kinetics has been recognized in recent years. Several mathematical models to describe the non-isothermal sorption kinetics have been formulated [1-9]. Of particular interest are the models describing the uptake during a differential sorption test because they provide relatively simple analytical solutions for data analysis [6-9]. These models assume that mass transfer can be described by the Fickian diffusion model and heat transfer from the solid is controlled by a film resistance outside the adsorbent particle. Diffusion of adsorbed molecules inside the adsorbent and gas diffusion in the interparticle voids have been considered as the controlling mechanism for mass transfer. 

Rounds, S.A. and J.F. Pankow. 1990. Application of a radial diffusion model to describe gas-particle sorption kinetics. Environ. Sci. Technol. 24 1378-1386. 

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