On the determination of diffusivity and sorption coefficients using different time-lag models

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Abstract

Different models have been proposed to fit experimental permeation data in order to obtain membrane transport parameters (sorption capacity and solute diffusivity) when using the time-lag method. Depending on the assumptions, their complexity can vary from a simple linear equation to a more complex non-linear partial differential equation. The applicability, limitations and inaccuracies of these different models are discussed.

A numerical procedure to solve the time-lag equation in a membrane slab considering permeate pressure build up is presented. This procedure involves the numerical computation of an inverse Laplace transform using a fast Fourier transform. This is applicable to the full range of the experimental permeation data, provided that both sorption and diffusivity coefficients are constant within the experimental pressure range.

Due to its simplicity, the linear time-lag equation is the most commonly used model. However, it does not account for permeate pressure build up. It is shown that this leads to errors in the determination of the permeation parameters which increase with the capacitance parameter (related with the volume available for permeation). For values above a certain threshold (η>0.05) these errors are above 5%. Emphasis is put on providing a simple and straight forward way to minimise them.

Introduction

Usually three approaches are used to determine monocomponent diffusion coefficients of gases in dense membranes: the differential method, the integral method and the sorption method [1]. All these three methods are based on the dynamic response obtained from a sudden change in the membrane boundary conditions.

In the sorption method, a pressure increase is applied to both membrane sides and the sorbed amount is continuously measured (usually performed using a gravimetric technique such as electromagnetic suspension weighing). This method can be used with any membrane shape, even at high pressures, provided that its geometry is well known. However, experiments tend to take longer than when using the other methods [1] and thermal effects are a potential error source, since the sorption heat released by the sample does not dissipate well in a stagnant gas.

In the differential method a constant partial pressure difference (driving force) is applied to both sides of the membrane and the permeate flowrate is measured. This can be done directly, using a very sensitive flowmeter, or indirectly, allowing the permeate flowrate to mix with a carrier stream and analysing the flowrate and composition of the stream leaving the cell. Usually a Wicke and Kallenbach cell [2] is used. The differential method allows for maintaining both sides of the membrane at equal total pressure and minimise leakages due to poor membrane sealing or membrane defects. On the other hand, for low permeate flowrates, very sensitive on-line detection systems are required.

In the integral method a quasi-constant pressure difference is applied to both sides of the membrane and the accumulation of gas on the permeate side is measured. This is usually performed using an accurate pressure sensor, which constitutes a major advantage since calibration requirements are minimal and independent of the gas used. On the other hand the method is very sensitive to membrane leakages, particularly for high pressure measurements. This method, usually known as time-lag method, was first suggested by Daynes in 1920 [3]. Apart from the advantages and disadvantages of the other methods [1], this is undoubtedly the most used one.

For the time-lag method, permeation set-ups usually consist of two chambers (feed and permeate chamber) separated by a dense membrane of known dimensions, thermostatically maintained at a given temperature, T. Initially, both chambers are filled with a gas at the same pressure P0 (that may well be approximately zero). At this point, the sorbed gas in the membrane is in equilibrium with the gas phase, meaning that the initial concentration of the sorbed gas in the membrane, C0, is uniform and hence constant in time.

At a given instant (t=0) a step increase on the feed pressure, Ph, is performed. As the gas permeates the membrane the pressure rises in the permeate chamber of known volume, V. The dynamic response in terms of permeate pressure is continuously monitored. The volume of the feed chamber is usually considered to be large enough to allow feed pressure to remain approximately constant during the experiment [4].

The diffusivity and sorption coefficients are calculated by fitting a model to the experimental data, using the least squares method or a similar technique. Several models have been proposed to fit the experimental permeation data, with different assumptions, complexity and restrictions.

The objective of the present work is to discuss the applicability, limitations and inaccuracies of these different models. Emphasis is put in providing simple and straightforward ways to either avoid model’s misapplication or to minimise the errors associated with such misapplications.

Section snippets

Models

Let us assume that, for the membrane–gas pair under study, diffusion transport follows the Fick law and the sorption isotherm is linear. Under these conditions, a constant sorption coefficient (also known as solubility or Henry coefficient), H, and a constant diffusivity coefficient, D, can be defined:C=HPϕ=−D∂C∂zwhere ϕ is the permeate flux through unit cross-section, z the membrane axial coordinate (perpendicular to the membrane surface), P the pressure and C the membrane sorbed gas

Results and discussion

The three models presented in the previous section aim to describe time-lag experiments. According to the different assumptions behind each model, different applicability conditions must be respected. In order to determine the exact applicability, limitations and inaccuracies of these different models, a comparative study is performed. Model 1 is taken as the base case.

Model 1 is the more generic approach, able to describe all the time-lag experiment. Neither model 2 nor model 3 account for the

Conclusions

Linear time-lag equation (model 3) cannot always be applied to experimental permeation data in order to calculate membrane transport parameters (sorption and diffusion coefficients). Two major factors are identified as potential error sources. The first relates to applying linear regression to a time range in which the membrane concentration profile is not yet been fully developed. The second relates to the linear equation’s inability to account for the pressure difference decrease during the

Acknowledgements

We would like to acknowledge Paulo Cruz, João Santos, Fernão D. Magalhães and Peter Janknecht for reviewing this paper and for technical support with the software application made available for download at the website: http://www.fe.up.pt/lepae/timelag. We would also like to acknowledge the grant ref. BD/21672/99 from FCT and financial support from FCT-Sapiens 34224/99.

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