Brill Academic Publishers
P.O. Box 9000, 2300 PA Leiden
The Netherlands
Lecture Series on Computer
and Computational Sciences
Volume 6, 2006, pp. 1-3
Prediction of Energy and Pore Size Distributions
Via Linear Regularization Theory
A. Shahsavand1 , A. Ahmadpour
Chemical Engineering Dept.,Faculty of Engineering
Ferdowsi University of Mashad,
Mashad, P.O. Box 91775-1111
I.R. IRAN
Received -----------; accepted in revised form -----------Abstract: Reliable estimation of energy or pore size distributions is the key element in the design and operation of
all adsorption processes. Direct measurement of these parameters is not practically feasible. Complex theories and
sophisticated models are required to obtain a faithful estimation of these distributions from a set of measured
isotherms. Once the correct energy distribution(ED) is known, the pore size distribution can be computed (or vice
versa) with high accuracy using reliable analytical techniques.
A novel method based on linear regularization theory is presented in this article to extract the ED from highly
noisy adsorption isotherms with least assumptions. Three synthetic noisy adsorption data sets were employed to
illustrate the remarkable performance of the proposed method for capturing single, double and triple peak energy
distributions.
Keywords: Adsorption, Characterization, Energy distribution, Pore size distribution, Linear regularization
1. INTRODUCTION
Analysis of gas and solid physical adsorption equilibrium is important to design separation and
purification processes as well as heterogeneous chemical reactors. The equilibrium between fluid and
adsorbent phases is expressed by adsorption isotherm which provides the relationship between the gas
pressure and the amount of gas or vapor taken up per unit mass of solid at constant temperature.
The first classification of physical adsorption isotherms was presented by Brunauer et al [1]. Many
theories and models have been presented in the literature to describe different types of isotherms.
Gregg and Sing [2] have given a detailed discussion of the various models used to interpret each type
of the isotherms. Ruthven[3], Rudzinski & Everett[4], Do[5] and Yang [6] discussed further theoretical
aspects of this issue.
Conventional treatment of energetic heterogeneity of a solid surface is based upon the concept of
localized adsorption on independent sites with a spectrum of adsorption energies. To keep the analysis
simple, many researchers assume that there is no lateral interaction between the molecules adsorbed on
neighboring sites. In the absence of lateral interactions the spatial distribution of sites on the surface is
unimportant which simplifies the analysis greatly.
The total adsorption (which is simply a summation of adsorption on various sites) is the only
practically measurable quantity. The local adsorption on particular sites of a given energy is not open to
direct measurement and must be specified based on suitable theoretical assumptions. In the absence of
lateral interactions, the fractional adsorption of an adsorbate on various sites of a solid adsorbent is
described by adsorption isotherms such as Langmuir, DR, DA, Toth and Sips. The total adsorption on
all sites of a solid surface with a continuous spectrum of site energies can be represented as:
emax
C ( pi )  C s ( pi )    pi , eF (e)de
(1)
emin
where  ( pi , e j ) is the local adsorption isotherm evaluated at bulk pressure pi and local site energy ej.
The energy distribution F(e) represents the sites with energy between e and e  de .
The problem considered here is to recover an estimate of the distribution F(e) from a given set of N
noisy data pi , C  ( pi ) . This is clearly an ill-posed inverse problem known as Fredholm integral of


the first kind. Advanced stabilization technique of linear regularization is used to alleviate the illconditioning. Unlike conventional methods, there is no need to assume any a priori model for the
adsorption isotherm.
1
Corresponding author. E-mail: shahsavand@um.ac.ir, ashahsavand@yahoo.com
2__________________________________________________ A. Shahsavand , A. Ahmadpour
House et al [7] employed the penalized least square method to predict the ED of a heterogeneous solid
adsorbent. Recently, many researchers applied the non-zero least square, penalized least square or
various regularization techniques for the same objective [8-11]. The majority of the published works
employed the INTEG program developed by Jaroniec et al [12]. The present article shows that zero
order regularization perform more adequately to recover the single, double and triple peak energy
distributions from various sets of synthetic noisy adsorption data. The leave one out cross validation
(LOOCV) criterion coupled with GSVD can also be employed efficiently to compute the optimum
level of regularization.
2. THEORETICAL BACKGROUND
Consider the problem of finding an unknown and underlying function u~( x) from a set of noisy
exemplars ( xi , yi ; i  1,2,..., N ),
yi   ri ( x) u~( x) dx  i
(2)
The relationship between u~( x) and each measured outputs yi ’s, is defined by its own linear response
kernel ri (x) and  i is the measurement error associated with the i th experiment. Usually, the
measured responses ( yi ’s) might “live” in an entirely different function space from u~ ( xi ) .
Given the yi ’s, the kernels, ri (x) ’s and perhaps some information about the measurement errors  i ’s,
the problem is to devise a procedure to find a good statistical estimator of u~( x) which will be denoted
as uˆ ( x) . This is an inherently ill-posed inverse problem. Depending on the smoothness of the kernel
ri (x) , sharp variations in the underlying function u~( x) are smoothed out by the integration.
Conversely, small variations in the data, yi ’s, may correspond to large variations in u~( x) . The
problem is further compounded by the presence of noise in the data.
In practice, we are not interested in every point of the continuous function uˆ ( x) and a large number M
of evenly spaced discrete points x j , j  1,2,....,M will suffice. For a “sufficiently” dense set of x j ’s,
we may replace the integral with the following set of linear equations [13]:
(3)
( RT R) uˆ  RT y
where y and û are vectors of size N and M respectively and the elements of the N  M matrix R are
defined by Rij  ri ( x j )(x j 1  x j ) .
The direct solution of this equation is hopeless and should be avoided. Since M is much greater than N,
the M  M matrix R T R will be singular and the equation will have a large number of highly
degenerate solutions. Such ill-posed inverse problem can be stabilized by imposing some a priori
information (or belief) about the unknown underlying function u~( x) as a constraint on the original
least square merit function. Based on our a priori information about the final solution (energy
distribution), the previous set of linear equations converts to the following equation:
( RT R   ) uˆ  RT y
(4)
where  is called as the level of regularization. Assuming  and  are chosen such that
( R T R  ) is non-singular, the above system admits a unique solution û  . Evidently, increasing 
pulls the solution û  away from fitting the experimental data (information content of matrix R )
towards our a priori information.
The form of matrix  strongly depends on the nature of the a priori information. For example, if the
solution is practically zero in the entire input domain except for a relatively narrow region, then the
zero order regularization technique provides the optimum distribution by replacing   I M . With a
similar approach, the appropriate set of equations for higher order regularization could be found [14].
The above structure is most favorable for employing the Generalized Singular Value Decomposition
(GSVD) technique. The optimum level of regularization can then be efficiently computed by
minimizing the LOOCV ( CV ( ) ) criterion [14,15].
The evaluation of CV ( ) , at each trial value of  requires the inversion of the M  M matrix and may
prove too time consuming. This can be avoided by resorting to the Generalized Singular Value
Decomposition (GSVD) technique [14]. To the best of our knowledge, this procedure has not been
addressed previously.
The following example applies the above procedure for ED prediction on a heterogeneous solid
adsorbent. Once ED is known, the PSD can be calculated using well known analytic procedures [5].
3
Prediction of ED & PSD Via Linear Regularization Theory ____________________________________________
3. ILLUSTRATIVE EXAMPLE
The performance of the various linear regularization method was initially tested for the assumed true
single peak energy distribution of:

F (e)  1.125 exp  e  2.5 0.52

(5)
The total adsorption data was generated at 100 equispaced points in the range 1  pi  1000 mbar. For
each pi , the total amount adsorbed ( C ( pi ) ) was integrated numerically using the above distribution
and Langmuir local adsorption isotherm over the range 1  e  4 kcal/mole with
C s ( pi )  1 mmole / g . The temperature and constant k0 were taken from House et al [7] as
77.5 K and 3.2 x10 9 (mbar)-1 respectively. The result was then contaminated with 10 percent random
noise drawn from a uniform distribution to simulate experimental data (Figure 1a). The previously
mentioned Cross Validation CV ( ) criterion coupled with GSVD is used to compute the optimum
level of regularization. Figure 1b shows the general behavior of CV ( ) versus log ( ) for various
regularization techniques.
(b)
(a)
Figure 1: a) True and noisy Adsorption data, b) Variation of CV criterion with
regularization
 for different orders of
Figure 2 shows the recovered ED and the corresponding total adsorption isotherms at optimum level of
regularization (* ) . Evidently, zero order regularization does a remarkable job and recovers the
underlying energy distribution closely. First order regularization technique performs less adequately
and the eighth order regularization is hopeless as expected. The damping effect of the integral is
explicitly shown in this figure where the predicted adsorbed amounts at * are virtually the same for
all regularization schemes despite the large differences between their associated energy distributions.
As another example, double and triple peak energy distributions were considered.
First order regularisation
1.0
1.0
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
0.0
-0.2
0.0
1
2.5 e (kcal / mole) 4
*  3 104
Eighth order regularisation
3.0
F(e) (mole / kcal)
1.2
F(e) (mole / kcal)
F(e) (mole / kcal)
Zero order regularisation
1.2
1
2.5 e (kcal / mole) 4
*  5  102
0.0
-3.0
1
2.5 e (kcal / mole) 4
*  1.8 106
Figure 2: Single peak Energy distributions and predicted isotherms at optimum * .
4__________________________________________________ A. Shahsavand , A. Ahmadpour
Using similar procedure as before, two 50 data sets with 10% noise level were generated for each case.
The first data sets were produced with equal pressure increments and the second sets were clustered
with 10 points in the first cluster (1-50 mbar), 15 other points in 50-200 mbar range and remaining data
in the span of 200-1000 mbar. Figure 3 illustrates the calculated distributions using zero order
regularization at * for two data sets on both cases. It seems that the performance of regularization
technique is relatively improved as the data points are properly distributed in the input domain.
Equispaced
Non-Equispaced
Double peak
Double peak
*  105
*  7  106
Triple peak
Triple peak
*  4  10 6
*  7  10 6
Figure 3: Performance of optimal zero order regularization for extraction of double and triple peak
energy distributions with equispaced and properly distributed data sets (Blue: truth, Red: predicted).
Evidently, the new method performs quite adequately in the presence of high level of noise for both
single and multiple peak energy distributions. The application of leave one out cross validation (CV)
criterion coupled with Generalized Singular Value Decomposition (GSVD) is crucial for efficient and
rapid estimation of the optimal level of regularization.
References
[1] S. Brunauer, P.H. Emmett and J. Teller, Adsorption of Gases in Multi-Molecular Layers, Am. Chem. Soc., 60,
309, (1938).
[2] S.J. Gregg and K.S.W. Sing, Adsorption, Surface area and Porosity, Academic Press, NY, (1982).
[3] D.M Ruthven, Principles of Adsorption and Adsorption Processes, John Willey & Sons, NY, (1984).
[4] W. Rudzinski, and D.H. Everett, Adsorption of Gases on Heterogeneous Surfaces, Academic Press, San
Diago, (1992).
[5] D.D. Do, Adsorption Analysis: Equilibria and Kinetics, Imperial College Press, London, (1998).
[6] R.T. Yang, Gas Separation by Adsorption Processes, Imperial College Press, London, (1997).
[7] W.A. House, M. Jaroniec, P. Brauer, and P. Fink, Surface heterogeneity effects in nitrogen adsorption on
chemically modified aerosils. II: Adsorption energy distribution functions evaluated using numerical methods,
Thin solid films, 87(4), 323-335, (1982)
[8] S. K. Bhatia, Determination of pore size distributions by regularization and finite element collocation,
Chemical Engineering Science, 53 (18), 3239-3249, (1998).
[9] F. Dion, and A. Lasia, The use of regularization methods in the deconvolution of underlying distributions in
electrochemical processes, Journal of Electroanalytical Chemistry 475, 28–37, (1999).
[10] C. Herdes, M.A. Santos, S. Abello, F. Medina and L.F. Vega, Search for a reliable methodology for PSD
determination based on a combined molecular simulation –regularization–experimental approach, Applied
Surface Science, 252, 538–547, (2005).
[11] P. Podko´scielny, K. Nieszporek, and P. Szabelski, Adsorption from aqueous phenol solutions on
heterogeneous surfaces of activated carbons—Comparison of experimental data and simulations, Colloids and
Surfaces A: Physicochem. Eng. Aspects, 277, 52–58, (2006).
[12] M. Jaroniec, T.J. Pinnavaia, and M.F. Thorpe, Access in Nanoporous Materials, Plenum Press, New York,
(1995).
[13] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical recipes in FORTRAN: The art of
scientific computing, Cambridge University press (1992).
[14] A. Shahsavand, Optimal and Adaptive Radial Basis Function Neural Networks, PhD. Thesis, University of
Surrey, UK, (2000).
[15] G.H. Golub and C.G. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 3rd edition,
(1996).