T-, D- and c-optimum designs for BET and GAB adsorption isotherms

Chemometrics and Intelligent Laboratory Systems 89 (2007) 36 – 44 www.elsevier.com/locate/chemolab

T-, D- and c-optimum designs for BET and GAB adsorption isotherms Licesio J. Rodríguez-Aragón, Jesús López-Fidalgo ⁎ Departamento de Matemáticas, Instituto de Matemática Aplicada a la Ciencia y a la Ingeniería, E. T. S. Ingenieros Industriales, Universidad de Castilla-La Mancha, Avda. Camilo José Cela 3, E-13071, Ciudad Real, Spain Received 19 February 2007; received in revised form 11 May 2007; accepted 11 May 2007 Available online 18 May 2007

Abstract Adsorption phenomena are described using the relationship between the equilibrium pressure of the gas and the amount adsorbed at constant temperature, known as adsorption isotherm. The Brunauer–Emmett–Teller (BET) model and the extension known as Guggenheim–Anderson–de Boer (GAB) model are widely used. The modelling of the adsorption phenomena in many chemical and industrial processes is proved to be of great interest. Consequently, a correct selection of the isotherm model and a correct estimation of the parameters are crucial tasks. The first objective to characterize the adsorption phenomena is to choose which of the models will best fit the data. T-optimum designs have been obtained in order to discriminate between both models. Once the model has been selected the correct estimation of the parameters is crucial. D- and c-optimality criteria are used in this work. Optimum designs are references, which allow the experimenter to measure the efficiency of any experimental design compared to the optimum. © 2007 Elsevier B.V. All rights reserved. Keywords: BET model; GAB model; Isotherm; T-optimum designs; D-optimum design; c-optimum design

1. Introduction Gas adsorption measurements are important in many physicochemical processes: some examples are retentions of chemicals in soils, adsorption of water by food solids to assure its storage stability, textile dyeing and depollution of industrial liquid effluents or determining the surface area and pore distribution of solid materials. The behaviour of the adsorbate on the adsorbent can lead to a monolayer adsorption, where all the adsorbed molecules are in contact with the surface layer of the adsorbent, or to a multilayer adsorption in which the adsorption space accommodates more than one layer of the adsorbate. The amount of adsorbate needed to cover the surface with a complete monolayer of molecules is known as monolayer capacity and the surface area of the adsorbent may be calculated from the monolayer capacity. From the wide variety of models used in the literature to describe the relationship between the equilibrium pressure of the ⁎ Corresponding author. Tel.: +34 926 295 212; fax: +34 926 295 361. E-mail addresses: [email protected] (L.J. Rodríguez-Aragón), [email protected] (J. López-Fidalgo). URL: http://areaestadistica.uclm.es (J. López-Fidalgo). 0169-7439/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chemolab.2007.05.004

gas and the amount adsorbed at constant temperature for multilayer adsorption phenomena, Brunauer–Emmett–Teller (BET) model [1] and the extension known as Guggenheim-Andersonde Boer (GAB) model [2–4] are widely used. In this work, optimum design theory is firstly used to provide researchers with optimum designs to discriminate between both models, which is an open question among the international community [5]. Then, optimum designs to perform the best estimation of the parameters following D- and c-optimality criteria are given. These optimum designs provide a valuable reference tool for researchers to measure the efficiency of any experimental design used by means of comparing it to the optimum. We have focused on the water adsorption of food and foodstuffs in order to provide with examples and applications of our work. Moisture content is used as a critical criterion for judging the quality of foods. Knowledge of water adsorption characteristics is needed for shelf life predictions, so important in drying, packaging and storage. 1.1. Adsorption isotherms As dealing with water adsorption the following notation will be used. Water content, we, is usually expressed in terms of amount of

L.J. Rodríguez-Aragón, J. López-Fidalgo / Chemometrics and Intelligent Laboratory Systems 89 (2007) 36–44

water per amount of adsorbent. Water activity, aw =p/p0, where p is the water vapour pressure exerted by the food material, and p0 is the vapour pressure of pure water at the experiment temperature. Water activity can vary within [0,1]. BET gas adsorption isotherm is one of the classical models of multilayer adsorption. It expresses the amount of gas or water adsorbed, we, in terms of water activity aw , following a normal distribution of mean and variance, E ðwe Þ ¼

wmB cB aw ; varðwe Þ ¼ r2 : ð1  aw Þð1 þ ðcB  1Þaw Þ

ð1Þ

The parameter wmB is the monolayer capacity and cB is related exponentially to the enthalpy of adsorption in the first adsorbed layer. Monolayer capacity is used to measure the surface area of the adsorbent. A report of 1985 of the Commission on Colloid and Surface Chemistry [6] recommends the BET model for a standard evaluation of monolayer values in adsorbate activity values not higher than 0.3. Meanwhile in water adsorption by foodstuff this limit is usually increased up to 0.5 [7]. In recent years GAB gas adsorption isotherm has been widely used to especially describe the sorption behaviour of foods and it has been recommended by the European Project Group COST90 [8]. This model introduces the idea that the sorption state of the sorbate molecules beyond the first layers is the same, but different from the pure liquid state. This difference demands the introduction of a new parameter. Then, the amount of water adsorbed we is normally distributed with the following mean and variance, E ðwe Þ ¼

wmG cG kaw ; varðwe Þ ¼ r2 : ð1  kaw Þð1 þ ðcG  1Þkaw Þ

ð2Þ

The parameters wmG and cG are the analog to those in BET model, i. e., mono-layer capacity and energy constant. The additional parameter introduced, k, is a measure of the free enthalpy of the sorbate molecules in these two states: the liquid and the second sorption state. Measurement are then carried out for upper limits of water activity up to 0.8 or even 0.9 [7]. 1.2. Optimum design background and approach An experimental design consists of a planned collection of points aw1, aw2,…, awN, in a given space X. Some of these N points may be repeated, meaning that several observations are taken at the same value of aw. The total number of observations is N and this number is usually pre-determined by experimental cost constraints. A convenient way to understand designs is to treat them as a collection of different points of X, together with the proportion of the N observations allocated at the different points. This suggests the idea of the so-called approximate design as a probability measure ξ on X. Then ξ(aw) is the proportion of observations to be taken at the point aw. Kiefer [9] pioneered this approach. Its many advantages are well documented in design monographs [10]. This approach has been recently applied to find optimum estimation of the parameters of the Michaelis–Menten model [11] and the Arrhenius equation

37

[12]. In what follows, the approximate design approach will be adopted restricting the attention to designs with a finite set of support points. For convenience, the design will be described using a two row matrix with the support points displayed in the first row and their corresponding proportion of observations in the second row. The paper is organized as follows. The aim of Section 2 is to introduce and solve the problem of discriminating between both adsorption isotherms using T-optimality. Once the adequate model has been selected, D- and c-optimality criteria are introduced in Section 3. D- and c-optimum designs are computed in Sections 4 and 5 for each adsorption isotherms respectively. Advice on designing adsorption experiments can be obtained from the efficiency plots. Comparisons between the optimum designs and conclusions are finally presented in Section 6. 2. Designing to discriminate between rival models, T-optimality The first objective to characterize adsorption phenomena is to choose which of the models will best fit the data. The main difference is the water activity range in which the measurements are taken, for BET model the water range is recommended to be XB = [0.05, 0.3–0.5] while GAB model can be used in a wider range XG = [0.05, 0.8–0.9]. BET model is supposed to be exposed to lack of fit beyond 0.5 [7], which according to literature is said to be caused by the linearization of the model. The simplicity of BET model is preferred to the extension made in GAB model while performing a direct non linear regression [13,14]. GAB model is an extension of BET model in which a new parameter k is introduced, being both models identical for k = 1. It is frequently observed that GAB model is treated as a purely empirical one with values of k unwarranted by the physics behind the equations, with k N 1, as remarked in [15]. This fact leads to prefer BET model being the water range extended up to 0.8 or even 0.9 in some works, while in others it is kept up to 0.5 [16]. The most popular design criterion for model discrimination is T-optimality which was proposed by Atkinson and Fedorov [17]. It has lately been extended to non-normal models [18]. A design for model discrimination should provide a large lack of fit sum of squares for an incorrect model [19]. Given two competing models there may be two T-criteria functions, depending on the model considered true. The aim of this work is to obtain a good design to discriminate between GAB and BET models for the wide range of water content XG. It will be used to say whether it is adequate to apply the simpler BET model to adsorption phenomena in XG or not. The T-optimum design provides the most powerful F-test for lack of fit of the second model when the first is true [19]. Consider the general non linear regression model we ¼ gðaw Þ þ ; aw a X ;

ð3Þ

where the random variable  is independent and normally distributed with zero mean and constant variance σ2. The function η(aw) is either GAB model, ηG(aw, θG), or BET

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L.J. Rodríguez-Aragón, J. López-Fidalgo / Chemometrics and Intelligent Laboratory Systems 89 (2007) 36–44

model, ηB (aw , θB), where htG ¼ ðwmG ; cG ; k Þa XG oℝ3 and htB ¼ ðwmB ; cB Þa XB oℝ2 are parameter vectors. Henceforth, η(aw) = ηG(aw , θG) will be assumed the “true” model with the corresponding known parameters θG. The T criterion function is then X TBG ðnÞ ¼ min nðawi Þðgðawi Þ  gB ðawi ; hB ÞÞ2 : ð4Þ hB a XB

i

A design ξT⁎ that maximizes TBG(ξ) is called T-optimum. For regular designs the Equivalence Theorem [17] is applied in such a way that a design ξT⁎ is T-optimal if and only if,  2    w aw ; n⁎ ¼ gðaw Þ  g aw ;hˆB B

T



X

  2 nðawi Þ gðawi Þ  gB awi ;hˆB V 0; aw a X ;

ical for the sequence {αs} are limsYl as ¼ 0; Pl conditions P l 2 a ¼ l; s s¼0 s¼0 as bl: • Step 4: A lower bound for the EffBG (ξ) has been proposed, so that the iterative procedure will stop when 

 maxaw aX wðaw ; ns Þ 1 N d; 1þ TBG ðns Þ

ð7Þ

where 0 b δ b 1 is a suitably chosen value, e.g. δ = 0.998 [18]. To ilustraste the T-optimum design calculation process two examples are presented. The range of water content (design space) considered will be XG = [0.05, 0.8]. To calculate the T-optimum design initial estimates for θG obtained from real experimental works are used.

i

ð5Þ with equality at the support points of ξT⁎. Let ξ be any design, then the ratio EffBG (ξ) = TBG (ξ)/TBG(ξT⁎) is a measure of the efficiency of ξ with respect to the T-optimum design ξT⁎. This efficiency will be considered as a measure of the design goodness with respect to this criterion. The construction of T-optimum designs is numerically demanding due to the need of estimating θB for the competing model (6). The criterion function TBG (ξ) has θˆ B as argument and its value will change at each step the algorithm. In practice, a first order algorithm is used to find the T-optimum design [17]. The iterative procedure is as follows. • Step 1: For a given design ξs supported at points aw1, aw2,..., aws, take X hˆB;s ðns Þ ¼ arg min ns ðawi Þðgðawi Þ  gB ðawi ; hB ÞÞ2 : hB a X B

i

ð6Þ • Step 2: Find the point a ws + 1 = arg max a w ∈ X (η(a w ) − ηB (aw , θ B,s)) 2 . • Step 3: Let ξaws + 1 be a design with measure concentrated at the single point aws + 1. A new design is constructed in the following way: ξs + 1 = (1 − αs + 1) ξs + as + 1ξaws + 1, where typ-

Example 1. In [20] water sorption behaviour of coffee was studied for predicting hygroscopic properties as well as designing units for its optimum preservation, storage, etc. The case of coffee roasted with sugar was considered in that work. The results at 25 °C for the GAB isotherm were wmG = 0.03445 g of H2O adsorbed/g of coffee, cG = 11.70 and k = 0.994. Notice that for k = 1 GAB and BET models are identical. To compare both models the T-optimum design will provide the F-test for lack of fit of BET model. After 182 iterations of the algorithm, with αs = 1/(s + 1), a design supported at three experimental points was obtained with a lower efficiency bound of δ = 0.998. 0

0:056 @ 27 ¼ n⁎ T 182

1  0:62 0:8 0:056 0:62 0:8 A ¼ : 4 51 0:15 0:57 0:28 7 182 ð8Þ

To check the optimality of the design the Equivalence Theorem has to be fulfilled. Function (5) is plotted in Fig. 1 showing that ψ(aw , ξT⁎) ≤ 0 and that it achieves a maximum at the support points of the optimum design. Example 2. In the same work, [20], the measurements were taken for ordinary roasted coffee. The results at 25° for GAB isotherm were wmG = 0.04203 g of H2O adsorbed/g of coffee, cG = 4.186 and k = 0.941.

Fig. 1. Plot of the condition given by the Equivalence Theorem in Eq. (5) for Example 1 (left) and Example 2 (right).

L.J. Rodríguez-Aragón, J. López-Fidalgo / Chemometrics and Intelligent Laboratory Systems 89 (2007) 36–44

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using the Equivalence Theorem. In addition this result provides methods for the construction of optimum designs [21,22]. For a general criterion function Φ non decreasing, convex and differentiable, defined on the information matrices, a design ξ⁎Φ is Φ-optimum if and only if    ⁎  ⁎ t w aw ; n⁎ U ¼ f ðaw ÞjU nU f ðaw Þ  trM ðn; hÞjU nU z 0; aw a X ;

ð11Þ

Fig. 2. Values of awD as cB varies in the design space XB and for the extended XG. In both cases for values cB N 20, awD = aw0 = 0.05.

A 3-point T-optimum design was obtained after 270 iterations of the algorithm with a lower efficiency bound of δ = 0.998 0

0:099 @ 47 n⁎ T ¼ 270

0:64 74 135

1  0:8 0:099 0:64 A ¼ 5 0:17 0:55 18

0:8 0:28

: ð9Þ

Once again, Function (5) shows how efficient this design is, (Fig. 1).

with equality at the support points of ξ⁎Φ. Here Φ(ξ) is Φ[M (ξ, θ)] for simplicity of notation and ▿Φ(ξ) denotes the gradient of Φ(ξ). The goodness of a design is measured by its efficiency, defined by EffΦ(ξ) = Φ(ξΦ⁎)/Φ(ξ). The efficiency can sometimes be multiplied by 100 and reported in percentage. If its value is 50% it means that the design ξ needs to double the total number of observations to perform as good as the optimum design ξΦ⁎. It is important to remark that both BET and GAB models are partially non-linear in terms that the models are linear for the monolayer capacity, wm, and nonlinear for the other parameters [23]. Therefore, the D- and c-optimum designs obtained will be independent of the initial best guesses of the monolayer capacity and will only depend on the truly nonlinear parameters.

3. Designing to estimate the parameters 4. Optimum designs to estimate the BET model T-optimality provides suitable designs to discriminate between GAB and BET models. Once the model has been selected, the correct estimation of the parameters is crucial. The design criteria used in this work are D- and c-optimality. D-optimality minimizes the volume of the confidence ellipsoid of the parameters and c-optimality is used to estimate linear combinations of the parameters, in particular to give the best estimates of each of them. Let θ be the unknown parameter vector and let f (aw) = ∂E(we)/∂θ be evaluated at the nominal values of θ. These nominal values represent the best guesses for the parameters at the beginning of the experiment. Under the normality assumption, the information matrix of a design ξ is given by M ðn; hÞ ¼

X

For BET model, being θBt = (wmB, cB) the unknown parameter vector let f (aw) = ∂E(we)/∂θB be evaluated at the best guess of the parameters θB, f ðaw Þ ¼

cB aw wmB aw ; ð1  aw Þð1 þ ðcB  1Þaw Þ ð1 þ ðcB  1Þaw Þ2

!t ;

ð12Þ

aw a XB ¼ ½aw0 ; awF :

4.1. D-optimum designs f ðaw Þf t ðaw Þnðaw Þ;

ð10Þ

aw aX

apart from an unimportant multiplicative constant. When the number of observations N is large, the covariance matrix of the estimates of θ is known to be approximately σ2/N times the inverse of this matrix [10]. The design criteria used in this work for estimating the model parameters are given by two criteria functions. D-optimality is given by ΦD [M (ξ, θ)] = det M (ξ, θ)− 1/m, where m is the number of parameters in the model, while c-optimality is defined by Φc[M (ξ, θ)] = ctM (ξ, θ)− 1c, being ctθ the linear combination of the parameters to be estimated. A design that minimizes one of these two functions among all the designs on X is called a D- or c-optimum design respectively. An advantage of working with approximate designs is that their optimality can be easily checked

To estimate both parameters of BET model simultaneously, D-optimality criterion has been used. A 2-point design maximizing the determinant of the information matrix is computed. It is well known that the weights of a D-optimum design with m points in its support have to be equal. Then the equivalence theorem is used to check whether this is actually the D-optimum design or not. A D-optimum design ξD⁎ is equally weighted at two points in XB: (awD , awF) if awD ∈ XB or (aw0, awF) otherwise. The point awD is the unique solution on XB of the following equation (Sturm Theorem):

a3wD



a2wD

 2awF þ

 1 2 awF þ awD awF þ  ¼ 0: cB  1 cB  1 cB  1

ð13Þ

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L.J. Rodríguez-Aragón, J. López-Fidalgo / Chemometrics and Intelligent Laboratory Systems 89 (2007) 36–44

Fig. 3. Efficiencies of the five different experimental designs in comparison to the D-optimum design. The efficiency will provide the experimenter with important information to choose an appropriate distribution of the observations.

Note that “equally weighted” means that half of the observations are taken at one of the support points and the other half at the other. The Equivalence Theorem (11) provides a condition to check the D-optimality of ξD⁎. Thus, the design ξD⁎ is D-optimum if and only if  1 f ðaw ÞV m; f t ðaw ÞM n⁎ D

aw a X B ;

ð14Þ

where m = 2 is the number of parameters. The equality is satisfied at the support points. The values of awD for cB ∈ [1,30] and for two different water ranges are plotted in Fig. 2. Example 3. Most of the times a D-optimum design will not satisfy an experimenter due to its radical form, only supported at two points. In real experiments the need of a larger group of support points is mostly required. The experimental points are usually settled along the design space without any kind of special distribution. Any design is compared to the D-optimum by computing its criterion value ΦD(ξ) = det M(ξ, θB)− 1/2 and the efficiency respect to the D-optimum, EffD (ξ) =ΦD (ξD⁎)/ΦD (ξ). In this example the performance of different types of designs for estimating the parameters will be compared to the D-optimum design. All the designs compared are supported at 6 different points and all of them are supposed to be equally weighted. Therefore, N/6 observations will be taken at each point. The total number of observations taken, N, remains constant in all the designs. This number is usually established by experimental cost constrains. The designs to be compared are: • A uniform design, where the support points are uniformly distributed along the design space with a constant spacing parameter, d, which is the distance between every pair of consecutive support points. • An arithmetic design, where the spacing parameter grows in an arithmetical progression (id) from left to right of the design space, where i = 0, …, 5. • A geometric design (left), where the spacing parameter grows in a geometrical progression di from left to right of the design space.

• A geometric design (right), where the geometrical progression goes from right to left of the design space. • A linear inverse design where the interval [η B (a w0 ), η B (awF)] is uniformly divided. Then the corresponding points in the design space XB through the inverse regression function are taken as support points. In all these cases the rates d are setup in such a way that the extremes are the first and the last design points. Efficiencies of these five different experimental designs have been plotted in Fig. 3 for different values of cB and for the two different design spaces. The experimenter may use this graphs as a reference to choose one of the five designs according to the foreseen. 4.2. c-optimum designs In order to estimate a linear combination of the parameters, say ct θB , the c-optimum criterion should be used, being this method useful when single estimations of the parameters are required.

Fig. 4. Elfving set for BET model. Points f (aw0) = (x0, y0)t and f (awF) = (xF , yF) t are the values of f for the extremes of the design space XB. In this particular case the c-optimum designs are both defined by f (awt) =(xt, yt)t and f (awF) =(xF , yF) t. The points (x⁎, 0)t and (0, y⁎) t are convex combinations of these two points f (awt), f (awF) and the corresponding coefficients give the weights of the optimum designs.

L.J. Rodríguez-Aragón, J. López-Fidalgo / Chemometrics and Intelligent Laboratory Systems 89 (2007) 36–44

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Fig. 5. Left, variation of awc against cB for the two different design spaces. Right, variation of the proportion of observations to be taken at awc to obtain the wmB- and the cB-optimum designs. Squares represent the evolution for the space design XB = [0.05, 0.5] and triangles for XB = [0.05, 0.8].

Elfving [24] proposed a nice graphical method of obtaining optimum designs to estimate linear combinations of the parameters. Elfving's set is defined as Λ = Hull{ f (XB) ∪ − f (XB)}. “Hull” means that Λ is the smallest convex set containing { f (XB) ∪ − f (XB)}. Then the c-optimum design ξc⁎ is determined by c⁎, given by the intersection of the line defined by the vector c and the boundary of the Elfving set Λ. This intersection can be expressed as a convex combination of vertices of Λ being those vertices the support points of the optimum designs. The coefficients of the convex combination are the weights of each support point of the optimum design. Furthermore, Φc(ξ⁎c ) =( ||c||/||c⁎|| ) 2 . Elfving's set Λ for BET model is shown in Fig. 4. For example in order to estimate wmB, the monolayer capacity, the linear combination of the parameters should be set to ct = (1, 0) and the resulting optimum design is called wmB-optimum. This

method provides analytical designs in this case. The vertices of Λ are the following points and their symmetric ones, (1) The end point of the curve f (XB), f (awF) = (xF, yF). (2) The tangential point f (aws) = (xs , ys) of the line starting at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðawF Þ; aws ¼ 1= awF ð1  cB Þ. (3) Either the starting point of the curve f (XB), f (aw0) = (x0, y0), or the tangential point f (awt) = (xt , yt) of the line starting at − f (awF). Where awt is the real solution in XB of the equation, a4wt awF ð1  cB Þ2 þ2a3wt a2wF ð1  cB Þ2 þa2wt ða3wF ðcB  1Þ2 2cB a2wF ðcB  1Þ þ 6a2wF ðcB  1Þ  awF ð4cB  6Þ  1Þ  2awt awF þ a2wF ¼ 0:

ð15Þ

(4) The points of the curve f (XB ) between points 2) and 3).

Fig. 6. Efficiencies for the five experimental designs. Left, efficiencies to estimate wmB for the two design spaces. Right, efficiencies to estimate cB.

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L.J. Rodríguez-Aragón, J. López-Fidalgo / Chemometrics and Intelligent Laboratory Systems 89 (2007) 36–44

The differences between the efficiencies to estimate either wmB or cB should be remarked. For example, in order to estimate wmB with initial nominal values of cB N 10, the best experimental design from the five proposed is the geometric design (right), being the geometric design (left) the one with lowest efficiency. However, in order to estimate cB for the same conditions the situation turns out to be the opposite. 5. Optimum designs to estimate the GAB model

Fig. 7. Values of aw1 and aw2 as cG varies and for values of k = 0.5, 0.8.

From the geometry of Fig. 4 and Elfving's argument, the wmB- and cB-optimum designs are: n⁎ cB ¼ pwmB ¼



awc pwmB

awF ; ð1  pwmB Þ

awF ð1 þ awc ðcB  1ÞÞ2 : awc þ awF ð1 þ awc ðcB  1Þð4 þ awc ðcB  1Þ þ awF ðcB  1ÞÞÞ

For GAB model, being θGt = (wmG , cG , k) the unknown parameter vector, let f (aw) = ∂E(we)/∂θG be evaluated at the best guesses for parameters θG, f ða w Þ ¼

ð

cG kaw wmG kaw ; ; ð1  kaw Þð1 þ ðcG  1Þkaw Þ ð1 þ ðcG  1Þkaw Þ2   wmG cG aw 1 þ ðcG  1Þk 2 a2w

ð1  kaw Þ2 ð1 þ ðcG  1Þkaw Þ2

pcB



awF awc ; pcB ð1  pcB Þ awF ðawc  1Þð1 þ awc ðcB  1ÞÞ : ¼ awc þ awF ð1 þ awc ðawc þ awF  4  cðawc þ awF  2Þ þ awF ðcB  2ÞÞÞ

ð17Þ where awc = awt if awt ∈ XB and awc = aw0 otherwise. As for the D-optimum design, the values for awc versus cB are plotted in Fig. 5 for the two different water ranges. The proportion of observations to be taken at this point to obtain the wmB-and the cB-optimum designs are also plotted. Example 4. The wmB- and cB-optimum designs, ξ⁎wmB and ξ⁎cB , provide a tool for the experimenter to measure the efficiency of any experimental design, ξ, when only the estimation of one of the parameters is required. Any design is compared to the corresponding c-optimum by obtaining its criterion value Φc(ξ) = ctM (ξ, θB)− 1c, then the efficiency Effc(ξ) = Φc(ξ⁎c )/Φc(ξ). As in Example 3 the previous five experimental designs are compared with both c-optimum designs. Efficiencies for both design spaces have been plotted and shown in Fig. 6 for different values of cB.

;

aw a XG ¼ ½aw0 ; awF :

ð18Þ

ð16Þ n⁎ cB ¼

Þ

t

5.1. D-optimum designs To estimate the three parameters of GAB model simultaneously D-optimum designs have been calculated. An analytical expression of the designs is not available now. The aim is to find a design minimizing the expression ΦD [M (ξ, θG)] = det M (ξ, θG )− 1/m, m = 3. The D-optimum designs are equally weighted at three points: aw1, aw2 and awF; where awF is the upper extreme of XG, aw2 ∈ X˚ G and aw1 is either in X˚ G or it is the lower extreme of XG, aw0. For GAB model, D-optimum designs in the design space XG = [0.05, 0.8] have been numerically calculated for different nominal values of the parameters. The support points aw1 and aw2 are plotted in Fig. 7. As the model is partially nonlinear, Doptimum designs depend only on the parameters cG, k, and the design space. A design will be D-optimum if and only if the condition in [14] is fulfilled. Example 5. As done for BET model in Example 3, the efficiencies of different experimental designs with 6 support points and with the same proportion of observations at each

Fig. 8. Efficiencies of the five experimental designs in comparison to the D-optimum design.

L.J. Rodríguez-Aragón, J. López-Fidalgo / Chemometrics and Intelligent Laboratory Systems 89 (2007) 36–44

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Fig. 9. For k = 0.8, the three c-optimum designs are supported at aw1, aw2, awF . The evolution of aw1, aw2 as cG varies are shown, ■. For k = 0.5, wmG- and k-optimum designs are also supported at aw1, aw2, awF . The evolution of aw1, aw2 as cG varies are shown, □. For k = 0.5, cG-optimum design is supported for cG b 5 at the same three support points, □, but for cG ≥ 5 the design becomes singular and it is supported only at aw1 and aw2, Δ.

support point, N/6, have been calculated and shown in Fig. 8. The design space for GAB model is XG = [0.05, 0.8] and the efficiencies have been obtained for two different values of the parameter k = 0.5, 0.8. In this case, the arithmetic design gives the highest efficiencies for high values of cG, while the uniform design should be preferred for low values. Optimum designs give the chance to measure the efficiency of any design that want to be performed.

The c-optimum designs ξ⁎wmG , ξ⁎cG and ξ⁎k , which look for the best possible estimates of each of the parameters have been calculated through the numerical algorithm for the vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1) respectively. The criterion function is Φc (ξ) = ctM (ξ, θG)− 1c, and from Eq. (11) a lower bound for the efficiency is obtained providing a stopping rule for the numerical algorithm:

¼1þ

5.2. c-optimum designs Since the dimension of the parameter vector θG is three, the resulting Elfving set for GAB model in ℝ3 makes the task of obtaining the convex combinations much more difficult than for BET model. Although Elfving's method [24] is valid for any dimension, it is rarely used geometrically for more than two parameters. A computational procedure, proposed for more than two parameters [25], has been followed in this work. When a single parameter is of particular interest c-optimality becomes Ds-optimality and the procedures to compute Ds-optimal designs may be used here as well. To determine a c-optimum design, the intersection u⁎ of the line determined by the vector c and Elfving's set Λ = Hull{ f (XG) ∪− f (XG )} has to be expressed as a convex combination of vertices of Λ. Caratheodory's Theorem states that for each point on the boundary of Λ there is a convex combination of m points at most.

inf aw a XG wðaw ; nÞ Uc ðnÞ h  i inf awaXG f t ðaw ÞM ðn; hG Þ1 cct M ðn; hG Þ1 f ðaw Þ þ tr ct M ðn; hG Þ1 c

Eff c ðnÞz1 þ

ct M ðn; hG Þ1 c

:

ð19Þ The general form of a c-optimum design is, n⁎ c ¼



aw1 pc

aw2 awF qc rc

;

ð20Þ

where awF is the upper bound of XG, aw2 ∈ XG, and aw1 is either in X˚ G or the lower bound of XG, aw0. The support points for c-optimum designs in the design space XG = [0.05, 0.8] have been calculated with the method given by [25]. As the model is partially nonlinear, the designs only depend on the parameters cG, k, as well as on the design space. The support points aw1 and aw2 are shown in Fig. 9 for different values of cG ∈ (1, 30) and for two initial best guesses of parameter k = 0.5, 0.8. For the cG-optimum designs and for

Fig. 10. Efficiencies of the five experimental designs in comparison with the wmG-optimum design.

44

L.J. Rodríguez-Aragón, J. López-Fidalgo / Chemometrics and Intelligent Laboratory Systems 89 (2007) 36–44

Table 1 D- and c- efficiencies of T-optimum designs of Examples 1 and 2

Acknowledgements

EffΦ(ξ⁎T) ξ⁎

GAB D-

wmG-

cG-

k-

D-

wmB-

cB-

Example 1 Example 2

0.84 0.81

0.88 0.69

0.25 0.28

0.99 0.88

0.52 0.59

0.44 0.40

0.19 0.27

T

BET

The Authors would like to thank the Referees for their helpful comments. This work was sponsored by Junta de Comunidades de Castilla-La Mancha PAI07-0019-2036. References

the initial best guess of the parameter k = 0.5 the cG-optimum design changes from three support points to two (aw1, aw2), becoming a singular design, as observed in Fig. 9. Example 6. As in previous examples, the efficiencies for the five experimental designs with 6 support points equally weighted have been calculated. These efficiencies for the wmG-optimal design are shown in Fig. 10 for different values of cG and k. The highest efficiencies are obtained, depending on the value of cG: the uniform and arithmetic designs for k = 0.5, while for k = 0.8, the one with the highest values of efficiency is the uniform design. 6. Conclusions Optimum designs to discriminate between GAB and BET models have been shown through this work. Once the model has been selected, optimum designs for estimating the parameters have been obtained. Optimum designs obtained may be used as references for the experimenters to measure the efficiency of their designs according to the criteria of T-, D- and c-optimality. The prior selection of the model is addressed with T-optimality criterion, which allows the experimenter to choose between the two models. The choice of one of these two models has been widely studied and justified in adsorption problems [7]. The need of nominal values for the parameters as initial guesses becomes a minor problem here due to the amount of works in literature which supply values for the parameters to be used as initial estimations. In practice estimating the adsorption of water on food stuff needs to take measurements periodically. Therefore, estimates from retrospective studies can be used as nominal values to obtain the optimum designs. The choice of the optimality criterion becomes easier once the efficiencies of each optimum design in comparison with other criteria are obtained. As an illustrative example, D- and c-efficiencies of the T-optimum designs obtained in Examples 1 and 2 have been calculated and shown in Table 1. A high efficiency was obtained for the k-optimum design in Example 1, k = 0.994 while for the Example 2, k = 0.94, the efficiency is much lower. The T-optimality criterion considers the GAB model as the true one, being both models identical for values of k = 1. For nominal values of k near 1 the T-optimum design is very close to the k-optimum design. The optimum designs computed in this paper claim for observations at 2 or 3 points, some of them are quite extreme. This is frequently disliked by the practitioners, who want some other observations to be more confident on the final results. In this paper the efficiency of typical sequences of points have been analyzed according to the reference given by the actual optimum designs.

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