T-, D- And C-optimum Designs for BET and GAB Adsorption Isotherms

ory Systems 89 (2007) 36–44www.elsevier.com/locate/chemolab

Chemometrics and Intelligent Laborat

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 2007Available online 18 May 2007

Abstract

Adsorption phenomena are described using the relationship between the equilibrium pressure of the gas and the amount adsorbed at constanttemperature, known as adsorption isotherm. The Brunauer–Emmett–Teller (BET) model and the extension known as Guggenheim–Anderson–deBoer (GAB) model are widely used. The modelling of the adsorption phenomena in many chemical and industrial processes is proved to be ofgreat 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 havebeen 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 anyexperimental 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 physi-cochemical processes: some examples are retentions of chemicalsin soils, adsorption of water by food solids to assure its storagestability, textile dyeing and depollution of industrial liquid ef-fluents or determining the surface area and pore distribution ofsolid materials.

The behaviour of the adsorbate on the adsorbent can lead to amonolayer adsorption, where all the adsorbed molecules are incontact with the surface layer of the adsorbent, or to a multilayeradsorption in which the adsorption space accommodates morethan one layer of the adsorbate. The amount of adsorbate neededto cover the surface with a complete monolayer of molecules isknown as monolayer capacity and the surface area of the ad-sorbent may be calculated from the monolayer capacity.

From the wide variety of models used in the literature todescribe 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 multi-layer adsorption phenomena, Brunauer–Emmett–Teller (BET)model [1] and the extension known as Guggenheim-Anderson-de Boer (GAB) model [2–4] are widely used.

In this work, optimum design theory is firstly used to provideresearchers with optimum designs to discriminate between bothmodels, which is an open question among the international com-munity [5]. Then, optimum designs to perform the best estimationof the parameters followingD- and c-optimality criteria are given.These optimum designs provide a valuable reference tool forresearchers to measure the efficiency of any experimental designused by means of comparing it to the optimum.

We have focused on the water adsorption of food and food-stuffs in order to provide with examples and applications of ourwork. Moisture content is used as a critical criterion for judgingthe quality of foods. Knowledge of water adsorption character-istics is needed for shelf life predictions, so important in drying,packaging and storage.

1.1. Adsorption isotherms

As dealingwith water adsorption the following notationwill beused.Water content,we, is usually expressed in terms of amount of

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37L.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 foodmaterial, and p0 isthe 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 modelsof multilayer adsorption. It expresses the amount of gas or wateradsorbed, we, in terms of water activity aw, following a normaldistribution of mean and variance,

E weð Þ ¼ wmBcBaw1� awð Þ 1þ cB � 1ð Þawð Þ ; var weð Þ ¼ r2: ð1Þ

The parameterwmB is the monolayer capacity and cB is relatedexponentially to the enthalpy of adsorption in the first adsorbedlayer. Monolayer capacity is used to measure the surface area ofthe adsorbent. A report of 1985 of the Commission onColloid andSurface Chemistry [6] recommends the BETmodel for a standardevaluation of monolayer values in adsorbate activity values nothigher than 0.3. Meanwhile in water adsorption by foodstuff thislimit is usually increased up to 0.5 [7].

In recent years GAB gas adsorption isotherm has beenwidely used to especially describe the sorption behaviour offoods and it has been recommended by the European ProjectGroup COST90 [8]. This model introduces the idea that thesorption state of the sorbate molecules beyond the first layers isthe same, but different from the pure liquid state. This differencedemands the introduction of a new parameter. Then, the amountof water adsorbed we is normally distributed with the followingmean and variance,

E weð Þ ¼ wmGcGkaw1� kawð Þ 1þ cG � 1ð Þkawð Þ ; var weð Þ ¼ r2: ð2Þ

The parameters wmG and cG are the analog to those in BETmodel, i. e., mono-layer capacity and energy constant. Theadditional parameter introduced, k, is a measure of the freeenthalpy of the sorbate molecules in these two states: the liquidand the second sorption state. Measurement are then carried outfor 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 ofpoints aw1, aw2,…, awN, in a given space X. Some of these Npoints may be repeated, meaning that several observations aretaken at the same value of aw. The total number of observationsis N and this number is usually pre-determined by experimentalcost constraints. A convenient way to understand designs is totreat them as a collection of different points of X, together withthe proportion of the N observations allocated at the differentpoints. This suggests the idea of the so-called approximatedesign as a probability measure ξ on X. Then ξ(aw) is theproportion of observations to be taken at the point aw. Kiefer [9]pioneered this approach. Its many advantages are well docu-mented in design monographs [10]. This approach has beenrecently applied to find optimum estimation of the parameters ofthe Michaelis–Menten model [11] and the Arrhenius equation

[12]. In what follows, the approximate design approach will beadopted restricting the attention to designs with a finite set ofsupport points. For convenience, the design will be describedusing a two row matrix with the support points displayed in thefirst row and their corresponding proportion of observations inthe second row.

The paper is organized as follows. The aim of Section 2 is tointroduce and solve the problem of discriminating between bothadsorption isotherms using T-optimality. Once the adequatemodel has been selected, D- and c-optimality criteria are intro-duced in Section 3. D- and c-optimum designs are computed inSections 4 and 5 for each adsorption isotherms respectively.Advice on designing adsorption experiments can be obtainedfrom the efficiency plots. Comparisons between the optimumdesigns and conclusions are finally presented in Section 6.

2. Designing to discriminate between rival models,T-optimality

The first objective to characterize adsorption phenomena isto choose which of the models will best fit the data. The maindifference is the water activity range in which the measurementsare taken, for BET model the water range is recommended to beXB=[0.05, 0.3–0.5] while GAB model can be used in a widerrange XG=[0.05, 0.8–0.9]. BET model is supposed to be ex-posed to lack of fit beyond 0.5 [7], which according to literatureis said to be caused by the linearization of the model. Thesimplicity of BET model is preferred to the extension made inGAB model while performing a direct non linear regression[13,14].

GAB model is an extension of BET model in which a newparameter k is introduced, being both models identical for k=1. Itis frequently observed that GAB model is treated as a purelyempirical one with values of k unwarranted by the physics behindthe equations, with kN1, as remarked in [15]. This fact leads toprefer BET model being the water range extended up to 0.8 oreven 0.9 in some works, while in others it is kept up to 0.5 [16].

The most popular design criterion for model discrimination isT-optimality which was proposed byAtkinson and Fedorov [17].It has lately been extended to non-normal models [18]. A designfor model discrimination should provide a large lack of fit sum ofsquares for an incorrect model [19]. Given two competingmodels there may be two T-criteria functions, depending on themodel considered true. The aim of this work is to obtain a gooddesign to discriminate between GAB and BET models for thewide range of water content XG.

It will be used to say whether it is adequate to apply thesimpler BET model to adsorption phenomena in XG or not. TheT-optimum design provides the most powerful F-test for lack offit of the second model when the first is true [19].

Consider the general non linear regression model

we ¼ g awð Þ þ �; awaX ; ð3Þ

where the random variable � is independent and normallydistributed with zero mean and constant variance σ2. Thefunction η(aw) is either GAB model, ηG(aw, θG), or BET

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

model, ηB (aw, θB), where htG ¼ wmG; cG; kð ÞaXGoℝ3 andhtB ¼ wmB; cBð ÞaXBoℝ2 are parameter vectors. Henceforth,η(aw)=ηG(aw, θG) will be assumed the “true” model with thecorresponding known parameters θG.

The T criterion function is then

TBG nð Þ ¼ minhBaXB

Xi

n awið Þ g awið Þ � gB awi; hBð Þð Þ2: ð4Þ

A design ξT⁎ that maximizes TBG(ξ) is called T-optimum. Forregular designs the Equivalence Theorem [17] is applied in sucha way that a design ξT⁎ is T-optimal if and only if,

w aw; n⁎T

� � ¼ g awð Þ � gB aw;hB� �� 2

�Xi

n awið Þ g awið Þ � gB awi;hB� �� 2

V 0; aw aX ;

ð5Þwith 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 theefficiency of ξ with respect to the T-optimum design ξT⁎. Thisefficiency will be considered as a measure of the designgoodness with respect to this criterion.

The construction of T-optimum designs is numerically de-manding due to the need of estimating θB for the competingmodel (6). The criterion function TBG (ξ) has θB as argumentand its value will change at each step the algorithm.

In practice, a first order algorithm is used to find theT-optimum design [17].

The iterative procedure is as follows.

• Step 1: For a given design ξs supported at points aw1, aw2,...,aws, take

hB;s nsð Þ ¼ arg minhB a XB

Xi

ns awið Þ g awið Þ � gB awi; hBð Þð Þ2:

ð6Þ

• Step 2: Find the point aws + 1 = arg maxaw∈ X (η(aw)−ηB(aw, θB ,s))

2.• Step 3: Let ξaws + 1

be a design with measure concentrated atthe single point aws+1. A new design is constructed in thefollowing way: ξs+ 1= (1−αs+ 1) ξs+as+1ξaws + 1

, where typ-

Fig. 1. Plot of the condition given by the Equivalence Theore

ical conditions for the sequence {αs} are limsYlas ¼ 0;Pls¼0 as ¼ l;

Pls¼0 a

2sbl:

• Step 4: A lower bound for the EffBG (ξ) has been proposed,so that the iterative procedure will stop when

1þmaxaw aXw aw; nsð ÞTBG nsð Þ

� ��1

N d; ð7Þ

where 0bδb1 is a suitably chosen value, e.g. δ=0.998 [18].

To ilustraste the T-optimum design calculation process twoexamples are presented. The range of water content (design space)considered will be XG=[0.05, 0.8]. To calculate the T-optimumdesign initial estimates for θG obtained from real experimentalworks are used.

Example 1. In [20] water sorption behaviour of coffee wasstudied for predicting hygroscopic properties as well as de-signing units for its optimum preservation, storage, etc. The caseof coffee roasted with sugar was considered in that work. Theresults at 25 °C for the GAB isotherm were wmG=0.03445 g ofH2O adsorbed/g of coffee, cG=11.70 and k=0.994. Notice thatfor k=1 GAB and BET models are identical. To compare bothmodels the T-optimum design will provide the F-test for lack offit of BET model.

After 182 iterations of the algorithm, with αs=1/(s+1), adesign supported at three experimental points was obtained witha lower efficiency bound of δ=0.998.

n⁎T ¼0:056 0:62 0:8

27182

47

51182

0@

1A ¼ 0:056 0:62 0:8

0:15 0:57 0:28

� :

ð8ÞTo check the optimality of the design the Equivalence

Theorem has to be fulfilled. Function (5) is plotted in Fig. 1showing that ψ(aw, ξT⁎)≤0 and that it achieves a maximum atthe support points of the optimum design.

Example 2. In the same work, [20], the measurements weretaken for ordinary roasted coffee. The results at 25° for GABisotherm were wmG=0.04203 g of H2O adsorbed/g of coffee,cG=4.186 and k=0.941.

m in Eq. (5) for Example 1 (left) and Example 2 (right).

Fig. 2. Values of awD as cB varies in the design space XB and for the extendedXG. In both cases for values cBN20, awD=aw0=0.05.


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

n⁎T ¼0:099 0:64 0:8

47

270

74

135

518

0@

1A ¼ 0:099 0:64 0:8

0:17 0:55 0:28

� : ð9Þ

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

3. Designing to estimate the parameters

T-optimality provides suitable designs to discriminate betweenGAB and BET models. Once the model has been selected, thecorrect estimation of the parameters is crucial. The design criteriaused in this work are D- and c-optimality. D-optimality mini-mizes the volume of the confidence ellipsoid of the parametersand c-optimality is used to estimate linear combinations of theparameters, 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. Thesenominal values represent the best guesses for the parameters atthe beginning of the experiment. Under the normality assump-tion, the information matrix of a design ξ is given by

M n; hð Þ ¼XawaX

f awð Þf t awð Þn awð Þ; ð10Þ

apart from an unimportant multiplicative constant. When thenumber of observations N is large, the covariance matrix of theestimates of θ is known to be approximately σ2/N times theinverse of this matrix [10].

The design criteria used in this work for estimating the modelparameters are given by two criteria functions. D-optimality isgiven by ΦD [M (ξ, θ)]=det M (ξ, θ)−1/m, where m is thenumber of parameters in the model, while c-optimality is definedby Φc[M (ξ, θ)]=ctM (ξ, θ)−1c, being ctθ the linear combinationof the parameters to be estimated. A design that minimizes oneof these two functions among all the designs onX is called aD- orc-optimum design respectively. An advantage of working withapproximate designs is that their optimality can be easily checked

using the Equivalence Theorem. In addition this result providesmethods for the construction of optimum designs [21,22]. For ageneral criterion function Φ non decreasing, convex anddifferentiable, defined on the information matrices, a design ξΦ⁎

is Φ-optimum if and only if

w aw; n⁎U

� � ¼ f t awð ÞjU n⁎U� �

f awð Þ � trM n; hð ÞjU n⁎U� �

z 0; aw aX ;

ð11Þ

with equality at the support points of ξΦ⁎. Here Φ(ξ) is Φ[M (ξ, θ)]for simplicity of notation and ▿Φ(ξ) denotes the gradient ofΦ(ξ).

The goodness of a design is measured by its efficiency,defined by EffΦ(ξ)=Φ(ξΦ⁎)/Φ(ξ). The efficiency can sometimesbe multiplied by 100 and reported in percentage. If its value is50% it means that the design ξ needs to double the totalnumber of observations to perform as good as the optimumdesign ξΦ⁎.

It is important to remark that both BET and GAB modelsare partially non-linear in terms that the models are linear forthe monolayer capacity, wm, and nonlinear for the other param-eters [23]. Therefore, the D- and c-optimum designs obtainedwill be independent of the initial best guesses of the mono-layer capacity and will only depend on the truly nonlinearparameters.

4. Optimum designs to estimate the BET model

For BET model, being θBt=(wmB, cB) the unknown param-

eter vector let f (aw)=∂E(we)/∂θB be evaluated at the best guessof the parameters θB,

f awð Þ ¼ cBaw1� awð Þ 1þ cB � 1ð Þawð Þ ;

wmBaw1þ cB � 1ð Þawð Þ2

!t

;

aw aXB ¼ aw0; awF½ �:ð12Þ

4.1. D-optimum designs

To estimate both parameters of BET model simultaneously,D-optimality criterion has been used. A 2-point design maxi-mizing the determinant of the information matrix is computed.It is well known that the weights of a D-optimum design with mpoints in its support have to be equal. Then the equivalencetheorem is used to check whether this is actually the D-optimumdesign or not. A D-optimum design ξD⁎ is equally weighted attwo points in XB: (awD , awF) if awD∈XB or (aw0, awF) other-wise. The point awD is the unique solution on XB of the fol-lowing equation (Sturm Theorem):

a3wD � a2wD 2awF þ 1

cB � 1

� þ awD awF þ 2

cB � 1

� � awFcB � 1

¼ 0:

ð13Þ

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 casethe 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 optimumdesigns.

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


Note that “equally weighted” means that half of the obser-vations are taken at one of the support points and the other halfat the other.

The Equivalence Theorem (11) provides a condition to checkthe D-optimality of ξD⁎. Thus, the design ξD⁎ is D-optimum ifand only if

f t awð ÞM n⁎D� ��1

f awð ÞVm; aw a XB; ð14Þ

where m=2 is the number of parameters. The equality issatisfied 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 notsatisfy an experimenter due to its radical form, only supported attwo points. In real experiments the need of a larger group ofsupport points is mostly required. The experimental points areusually settled along the design space without any kind of specialdistribution. Any design is compared to the D-optimum bycomputing its criterion value ΦD(ξ)=detM(ξ, θB)

−1/2 andthe efficiency respect to theD-optimum, EffD (ξ)=ΦD (ξD⁎)/ΦD (ξ).

In this example the performance of different types of designsfor estimating the parameters will be compared to the D-op-timum design. All the designs compared are supported at 6different points and all of them are supposed to be equallyweighted. Therefore, N/6 observations will be taken at eachpoint. The total number of observations taken, N, remains con-stant in all the designs. This number is usually established byexperimental cost constrains. The designs to be compared are:

• A uniform design, where the support points are uniformlydistributed along the design space with a constant spacingparameter, d, which is the distance between every pair ofconsecutive support points.

• An arithmetic design, where the spacing parameter grows inan arithmetical progression (id) from left to right of thedesign space, where i=0, …, 5.

• A geometric design (left), where the spacing parametergrows in a geometrical progression di from left to right of thedesign space.

• A geometric design (right), where the geometrical progres-sion goes from right to left of the design space.

• A linear inverse design where the interval [ηB (aw0),ηB (awF)] is uniformly divided. Then the corresponding pointsin the design space XB through the inverse regression functionare taken as support points.

In all these cases the rates d are setup in such a way that theextremes are the first and the last design points.

Efficiencies of these five different experimental designs havebeen plotted in Fig. 3 for different values of cB and for the twodifferent design spaces. The experimenter may use this graphsas a reference to choose one of the five designs according to theforeseen.

4.2. c-optimum designs

In order to estimate a linear combination of the parameters,say ctθB , the c-optimum criterion should be used, being thismethod useful when single estimations of the parameters arerequired.

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- andthe 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 obtainingoptimum designs to estimate linear combinations of the param-eters. 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 determinedby c⁎, given by the intersection of the line defined by the vectorc and the boundary of the Elfving set Λ. This intersection canbe expressed as a convex combination of vertices of Λ beingthose vertices the support points of the optimum designs.The coefficients of the convex combination are the weightsof each support point of the optimum design. Furthermore,Φc(ξc⁎) =( ||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, thelinear combination of the parameters should be set to ct=(1, 0)and the resulting optimum design is called wmB-optimum. This

Fig. 6. Efficiencies for the five experimental designs. Left, efficiencies to est

method provides analytical designs in this case. The vertices ofΛ 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

f awFð Þ; aws ¼ 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiawF 1� cBð Þp

.(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 startingat − f (awF). Where awt is the real solution in XB of theequation,

�a4wtawF 1� cBð Þ2þ2a3wta2wF 1�cBð Þ2þa2wtða3wF cB�1ð Þ2

�2cBa2wF cB � 1ð Þ þ 6a2wF cB � 1ð Þ � awF 4cB � 6ð Þ � 1Þ

� 2awtawF þ a2wF ¼ 0: ð15Þ(4) The points of the curve f (XB ) between points 2) and 3).

imate wmB for the two design spaces. Right, efficiencies to estimate cB.


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

n⁎cB ¼ awc awFpwmB 1� pwmBð Þ

� ;

pwmB ¼ awF 1þ awc cB � 1ð Þð Þ2awc þ awF 1þ awc cB � 1ð Þ 4þ awc cB � 1ð Þ þ awF cB � 1ð Þð Þð Þ :

ð16Þ

n⁎cB ¼ awc awFpcB 1� pcBð Þ

� ;

pcB ¼ awF awc � 1ð Þ 1þ awc cB � 1ð Þð Þawc þ awF 1þ awc awc þ awF � 4� c awc þ awF � 2ð Þ þ awF cB � 2ð Þð Þð Þ :

ð17Þwhere awc=awt if awt∈XB and awc=aw0 otherwise.

As for the D-optimum design, the values for awc versus cBare plotted in Fig. 5 for the two different water ranges. Theproportion of observations to be taken at this point to obtain thewmB-and the cB-optimum designs are also plotted.

Example 4. The wmB- and cB-optimum designs, ξ⁎wmBand ξ⁎cB,

provide a tool for the experimenter to measure the efficiency ofany experimental design, ξ, when only the estimation of one ofthe parameters is required. Any design is compared to the cor-responding c-optimum by obtaining its criterion value Φc(ξ)=ctM (ξ, θB)

− 1c, then the efficiency Effc(ξ)=Φc(ξc⁎)/Φc(ξ).As in Example 3 the previous five experimental designs are

compared with both c-optimum designs. Efficiencies for bothdesign spaces have been plotted and shown in Fig. 6 fordifferent values of cB.

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

Fig. 8. Efficiencies of the five experimental desig

The differences between the efficiencies to estimate eitherwmB or cB should be remarked. For example, in order to esti-mate wmB with initial nominal values of cBN10, the best ex-perimental design from the five proposed is the geometricdesign (right), being the geometric design (left) the one withlowest efficiency. However, in order to estimate cB for the sameconditions the situation turns out to be the opposite.

5. Optimum designs to estimate the GAB model

For GAB model, being θGt=(wmG, cG, k) the unknown

parameter vector, let f (aw)=∂E(we)/∂θG be evaluated at thebest guesses for parameters θG,

f awð Þ ¼ ð cGkaw1� kawð Þ 1þ cG � 1ð Þkawð Þ ;

wmGkaw

1þ cG � 1ð Þkawð Þ2 ;

wmGcGaw 1þ cG � 1ð Þk2a2w� �

1� kawð Þ2 1þ cG � 1ð Þkawð Þ2Þt

; aw aXG ¼ aw0; awF½ �:

ð18Þ

5.1. D-optimum designs

To estimate the three parameters of GAB model simulta-neously D-optimum designs have been calculated. An analy-tical expression of the designs is not available now. The aimis 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 threepoints: aw1, aw2 and awF; where awF is the upper extreme ofXG, aw2∈ XG and aw1 is either in XG or it is the lower extremeof XG, aw0.

For GAB model, D-optimum designs in the design spaceXG=[0.05, 0.8] have been numerically calculated for differentnominal values of the parameters. The support points aw1 andaw2 are plotted in Fig. 7. As the model is partially nonlinear, D-optimum designs depend only on the parameters cG, k, and thedesign space. A design will be D-optimum if and only if thecondition in [14] is fulfilled.

Example 5. As done for BET model in Example 3, the ef-ficiencies of different experimental designs with 6 supportpoints and with the same proportion of observations at each

ns in comparison to the D-optimum design.

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-optimumdesigns 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 cGb5 at the samethree 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 theefficiencies have been obtained for two different values of theparameter k=0.5, 0.8.

In this case, the arithmetic design gives the highestefficiencies for high values of cG, while the uniform designshould be preferred for low values. Optimum designs give thechance to measure the efficiency of any design that want to beperformed.

5.2. c-optimum designs

Since the dimension of the parameter vector θG is three, theresulting Elfving set for GAB model in ℝ3 makes the task ofobtaining the convex combinations much more difficult than forBET model. Although Elfving's method [24] is valid for anydimension, it is rarely used geometrically for more than twoparameters. A computational procedure, proposed for more thantwo parameters [25], has been followed in this work. When asingle parameter is of particular interest c-optimality becomesDs-optimality and the procedures to compute Ds-optimaldesigns 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 convexcombination of vertices of Λ. Caratheodory's Theorem statesthat for each point on the boundary of Λ there is a convexcombination of m points at most.

Fig. 10. Efficiencies of the five experimental design

The c-optimum designs ξwmG

⁎, ξcG⁎ and ξk⁎, which look for the

best possible estimates of each of the parameters have beencalculated through the numerical algorithm for the vectors (1, 0, 0),(0, 1, 0) and (0, 0, 1) respectively. The criterion function isΦc (ξ)=ctM (ξ, θG)

−1c, and from Eq. (11) a lower bound for the efficiencyis obtained providing a stopping rule for the numerical algorithm:

Eff c nð Þz1þ inf aw a XGw aw; nð ÞUc nð Þ

¼ 1þinf awaXG �f t awð ÞM n; hGð Þ�1cctM n; hGð Þ�1f awð Þ þ tr ctM n; hGð Þ�1c

� �h ictM n; hGð Þ�1c

:

ð19Þ

The general form of a c-optimum design is,

n⁎c ¼ aw1 aw2 awFpc qc rc

� ; ð20Þ

where awF is the upper bound of XG, aw2∈XG, and aw1 is eitherin XG or the lower bound of XG, aw0.

The support points for c-optimum designs in the designspace XG=[0.05, 0.8] have been calculated with the methodgiven by [25]. As the model is partially nonlinear, the designsonly depend on the parameters cG, k, as well as on the designspace. The support points aw1 and aw2 are shown in Fig. 9 fordifferent values of cG∈ (1, 30) and for two initial best guessesof parameter k=0.5, 0.8. For the cG-optimum designs and for

s in comparison with the wmG-optimum design.

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

EffΦ(ξT⁎) GAB BET

ξT⁎ D- wmG- cG- k- D- wmB- cB-

Example 1 0.84 0.88 0.25 0.99 0.52 0.44 0.19Example 2 0.81 0.69 0.28 0.88 0.59 0.40 0.27


the initial best guess of the parameter k=0.5 the cG-optimumdesign 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 fiveexperimental designs with 6 support points equallyweighted havebeen calculated. These efficiencies for the wmG-optimal designare shown in Fig. 10 for different values of cG and k. The highestefficiencies are obtained, depending on the value of cG: theuniform and arithmetic designs for k=0.5, while for k=0.8, theone with the highest values of efficiency is the uniform design.

6. Conclusions

Optimum designs to discriminate between GAB and BETmodels have been shown through this work. Once the model hasbeen selected, optimum designs for estimating the parametershave been obtained. Optimum designs obtained may be used asreferences for the experimenters to measure the efficiency oftheir designs according to the criteria of T-, D- and c-optimality.

The prior selection of the model is addressed with T-optimalitycriterion, which allows the experimenter to choose betweenthe two models. The choice of one of these two models has beenwidely studied and justified in adsorption problems [7]. Theneed of nominal values for the parameters as initial guessesbecomes a minor problem here due to the amount of works inliterature which supply values for the parameters to be used asinitial estimations. In practice estimating the adsorption of wateron food stuff needs to take measurements periodically. Therefore,estimates from retrospective studies can be used as nominalvalues to obtain the optimum designs.

The choice of the optimality criterion becomes easier oncethe efficiencies of each optimum design in comparison withother criteria are obtained. As an illustrative example, D- andc-efficiencies of the T-optimum designs obtained in Examples1 and 2 have been calculated and shown in Table 1. A highefficiency was obtained for the k-optimum design in Example1, k=0.994 while for the Example 2, k=0.94, the efficiency ismuch lower. The T-optimality criterion considers the GABmodel as the true one, being both models identical for values ofk=1. For nominal values of k near 1 the T-optimum design isvery close to the k-optimum design.

The optimum designs computed in this paper claim forobservations at 2 or 3 points, some of them are quite extreme.This is frequently disliked by the practitioners, who want someother observations to be more confident on the final results.

In this paper the efficiency of typical sequences of pointshave been analyzed according to the reference given by theactual optimum designs.

Acknowledgements

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

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Documents

T-, D- And C-optimum Designs for BET and GAB Adsorption Isotherms