Determination of pore size distributions by regularization and finite element collocation
Introduction
The pore size distribution of a porous medium is perhaps its most central characteristic, because of its influence on the transport and equilibrium of molecules in the structure. Numerous processes of technological and scientific interest such as catalysis, adsorption, membrane separations and noncatalytic gas–solid reactions are strongly affected by the pore size distribution. Characterization of the latter has therefore attracted considerable attention in the literature. Among the methods of characterization of the pore size distribution (PSD) adsorption is perhaps the most popular (Gregg and Sing, 1982; Ruthven, 1984; Yang, 1987), though more recently other alternatives such as small angle X-ray and neutron scattering (Dore and North, 1991; Ramsay and Avery, 1991), and measurement of melting point depression by NMR (Hansen et al., 1996), have also been applied. In all of these cases the measured (or dependent) variable is related to the pore size distribution byin which ν is the independent variable and the theoretical response for a pore of size x. For example, for adsorption ν would represent the pressure P), the local isotherm (i.e. the amount adsorbed per unit volume in a pore of size x at pressure P), and Ψ(ν) the overall amount adsorbed per unit mass. Similarly, for small-angle scattering ν would represent the scattering vector, and a local contribution to the measured intensity Ψ(ν).
Given discrete experimental data , and the theoretical kernel determination of the pore size distribution then requires inversion of Eq. (1). This is a well-known ill-conditioned problem and prone to yield unstable solutions. While general methods for this kind of inversion have been proposed (Tikhonov and Arsenin, 1977; Allison, 1979), most of the techniques hitherto used with the various forms of characterization data have been rather ad hoc, utilizing standard distributions for , or are sensitive to small data errors. Since the bulk of the characterization work in the literature has been related to adsorption, we shall direct our further presentation with reference to this method, while recognizing that our technique and developments to be discussed are equally applicable to the other methods as well. Using the language of adsorption, therefore, we re-write Eq. (1)aswhere is the local effective density of the adsorbate in a pore of size x, and Ca(P) the total amount adsorbed, at pressure P.
Several methods for the inversion of the above adsorption integral equation have been proposed, many of which are specific to particular forms of the kernel, or local isotherm, . These include the BJH method for mesoporous solids with surface layering followed by capillary condensation, and the MP method for microporous solids with slit-shaped pores (Gregg and Sing, 1982). Both of these methods rely on direct numerical differentiation of the data for Ca(P) and are therefore sensitive to errors. It may, of course, be recognized that any procedure for the solution of Eq. (2)involves inversion of the integral operator and is subject to this difficulty, since one is effectively operating on the vector of measurements by a discretized differential operator. A more detailed discussion of this problem is available in the recent work of Gupta and Bhatia (1993), who solved a similar equation for determination of growth rate distributions in crystallization.
To alleviate the above difficulty Tikhonov and Arsenin (1977) suggested the method of regularization, involving minimization of a modified objectivein which α is a small variable parameter, and q a parameter of O(1). Here the first term on the right-hand side is a measure of the fidelity to the data, while the second term represents undesirable characteristics to be minimized in the solution. Further the matrix operator A represents a suitable discretization of the integral operator in Eq. (2). This approach has been recently implemented by Mamleev and Bekturov (1996), with q=1, for the solution of energy distributions. The same approach was also used earlier by Gupta and Bhatia (1993) in solving for growth rate distributions in continuous crystallization, and others (Merz, 1980; Britten et al., 1983) for evaluating adsorption energetic heterogeneity. In other work Brown and Travis (1983) have used the Tikhonov method for evaluating pore size distributions from effective diffusivity data, while Arvind and Bhatia (1995) have used it in their nonlinear problem of determining concentration dependent diffusivities from uptake data.
In most of the above applications of regularization the sought after distribution has been discretized at appropriate points, usually determined by a quadrature rule involving the entire domain of x as a whole. Such a discretization is usually unstable when regions of relatively large gradients exist. In such cases it is more appropriate to use finite element collocation, which has the advantage that the order of the polynomial in each element can be arbitrarily chosen and varied for convergence. This also has the advantage that one can readily use different isotherm models, (such as capillary condensation, surface layering etc.) in different ranges of pore size while maintaining accuracy. Most existing algorithms have used a single isotherm model, such as the Dubinin equation or density functional theory and are not designed for use with a combination of piecewise continuous isotherm models. This latter possibility is considered in the recent work of Bhatia and Shethna (1994), who used a convergent expansion around standard distributions, along with finite element quadrature, to solve 1for the PSD. The procedure is, however, computationally intensive as it relies on nonlinear parameter estimation methods for determining the unknown variables. Regularization in the linear problem in Eq. (3)may be anticipated to be more efficient.
One of the problems attending the least-squares solution, as in regularization following Eq. (3), is the existence of negative solution values at some points. This is often tackled by by using the algorithm NNLS of Lawson and Hanson (1974), which starts by taking the solution as the null vector, and iteratively adds the solution variables to the vector of unknowns in the least-squares analysis, while ensuring non-negativity. Since the final non-negative solution is only slightly different from the straightforward full least-squares solution with some negative elements, it would seem appropriate to devise an algorithm using this latter solution as a starting point for arriving at the sought after result. This is another distinctive feature of the approach presented here.
The choice of the parameter α is another important consideration in the regularization method. Too small a value leads to unstable solutions, while too large a value results in excessive fitting error. Generalized cross-validation (Wahba, 1977) is one of the established methods for estimating the optimal value of α, but requires much additional computation. In addition it is developed for situations in which the error norm is an absolute one as in Eq. (3). In practice, since a relative error norm is often appropriate we investigate the latter as well, and the usual cross-validation estimate is not viable. However, as observed by Mamleev and Bekturov (1996), the quality or error of the data fit is relatively insensitive to the value of α over a wide range. It therefore suffices to choose α large enough such that oscillations in are absent and the deviation between predictions and measured values of Ca is suitably consistent with the known measurement errors.
In what follows, we present a technique for obtaining the non-negative regularized solution of Eq. (2), using finite element collocation, which can be used for any choice , including use of different models (capillary condensation, surface layering, etc.) in different ranges of pore size x. The method is tested for convergence and stability to measurement error using synthetic data and also applied to actual experimental data. The package regularized inversion of data using finite element collocation (RIDFEC) implementing our method is very robust and applicable also to a variety of other situations besides adsorption.
Section snippets
Theory
In developing our algorithm for the solution of Eq. (2), for sake of generality, we permit different isotherm models in different ranges of pore size, as depicted in Fig. 1. The pore size range 0⩽x⩽xm corresponds to micropores, in which a suitable pore filling isotherm may be used. In the range xm<x⩽xp(P) capillary condensation occurs, with xp(P) following a modified Kelvin type (having the thickness correction) or other equivalent relationship. Surface layering occurs in pores larger than xp(P
Results and discussion
The above regularization method has been implemented in FORTRAN code as package RIDFEC. All the results to be discussed here were obtained on an IBM compatible PC/Pentium −133 computer, and generally required about 15 s of computational time with six elements and nine collocation points per element. This doubled to about 30 s for Ns=10 and N=9. As indicated above the linear system of equations for the regularized solution was solved by means of singular value decompostion, utilizing published
Summary
A finite element collocation technique with regularization has been presented here for inversion of char1acterization data, to determine pore size distributions. The method has been tested against synthetic adsorption data as well as applied to an experimental isotherm. The main conclusions based our study are as follows:
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The algorithm utilizing regularization with finite element collocation produces converged, stable, solutions over a wide range of values of the regularization parameter α, with
Acknowledgements
This research has been supported by the Australian Research Council under the Large Research Grant Scheme.
NOTATIONA discretised operator objective to be minimised amount adsorbed pore size distribution f0 guess vector value of f at jth point in element i G Matrix of inverse weights for data m index of last element in micropore range M number of data points N number of collocation points in each element number of elements P pressure vapour pressure q constant R vector of residuals t layer thickness
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