1-s2.0-S0009250902002142-main

Chemical Engineering Science 57 (2002) 3439–3451www.elsevier.com/locate/ces

Modelling the catalytic steam reforming of methane: discriminationbetween kinetic expressions using sequentially designed experiments�

K. Jarosch, T. El Solh, H. I. de Lasa ∗

Faculty of Engineering Science, Chemical Reactor Engineering Centre, University of Western Ontario, London, Ont., Canada N6A 5B9

Received 26 March 2001; received in revised form 3 September 2001; accepted 4 September 2001

Abstract

The kinetics of steam reforming of methane on a 5uidizable Ni=�-alumina catalyst is studied in a novel CREC riser simulator reactor.The Box–Hill discrimination function is used to design sequential experiments for discrimination between six candidate models for therate of steam reforming of methane carried out over a 5uidized bed of catalyst. Proposed models include one model in which the rate ofreforming is dependent only on the partial pressure of methane (;rst order), a model in which the rate is dependent on the partial pressuresof the products and of the reactants (power law) and four models with various adsorption e<ects. Models are expressed in a form such thatthe parameters are close-to-linear. Discrimination results indicate that the rate of methane reforming has a ;rst order dependency on thepartial pressure of methane in the numerator and the adsorption of methane in the denominator. Estimates of the close-to-linear parametersin the most probable model are made using 85 observations of the conversion of methane made over a 20wt% Ni=�-alumina catalyst. Allparameters are found to be signi;cant at the 95% con;dence level and correlation between the parameters is found to be moderate (¡ 0:84).? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Discrimination; Fluidization; Kinetics; Modelling; Parameter identi;cation; Statistics

1. Introduction

Steam reforming of hydrocarbons is the primary indus-trial process for the production of hydrogen and synthesisgas. The reforming reaction is both endothermic and equi-librium limited. The feedstock for the reforming process,usually natural gas or light distillates, is mixed with steamand brought into contact with a catalyst consisting primar-ily of nickel dispersed on �-alumina. If the temperature issuBciently high, the hydrocarbons react with the steam pro-ducing a mixture of partially reacted hydrocarbons, carbonmonoxide and hydrogen. Carbon dioxide is also present inthe product gas mixture as the water–gas shift reaction si-multaneously takes place.On the industrial scale, reforming is carried out over a

;xed bed of monolithic catalyst packed into tubes. The large

�Manuscript prepared for the special issue of Chemical EngineeringScience in honour of Dr. J. C. Charpentier.

∗ Corresponding author. Tel.: +1-519-661-2144;fax: +1-519-661-3498.

E-mail address: [email protected] (H. I. de Lasa).

quantity of heat required by the reforming reaction is pro-vided by locating the tubes in a furnace. Operating condi-tions depend on the end use of the synthesis gas but outlettemperatures range from 700–900◦C. Operating pressuresare in the region of 2–3 MPa. Molar steam-to-carbon ratiosemployed are in the range 3:1–5:1.A novel multifunctional reactor concept for steam reform-

ing, called CATFORMING that combines high-temperaturehydrogen-permeable membranes with a circulating 5u-idized bed, (CFB) has been under study in the ChemicalReactor Engineering Centre (CREC) at the University ofWestern Ontario (UWO) (Jarosch, 1995, 2000). In theCATFORMER reactor concept (Fig. 1), the reactantgas meets with catalyst and the resulting gas–solidsuspension enters the riser=downer section of the re-actor. The suspension 5ows down the reactor tubeand the reforming and water–gas shift reactions takeplace. After a certain distance, hydrogen is continu-ously removed from the mixture via di<usion througha membrane. Hydrogen removal shifts the conversionof methane to values above those attainable at equilib-rium as well as favorably a<ecting the selectivity. After

0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S 0009 -2509(02)00214 -2

3440 K. Jarosch et al. / Chemical Engineering Science 57 (2002) 3439–3451

Fig. 1. Schematic 5ow diagram of the CATFORMER concept shown asin the down5ow con;guration.

exiting the reactor, the synthesis gas product is separatedfrom the catalyst and sent for further processing or directuse in the synthesis of alcohols.During steam reforming, coke may be formed by two

mechanisms. When the reactant gas is far from the composi-tion at chemical equilibrium, coke may be formed via kineticprocesses. When the reactant gas composition is close to thecomposition at chemical equilibrium, coke will be formedif graphitic carbon formation is predicted at chemical equi-librium. In conventional tubular reforming, the formationof coke by either mechanism is strictly avoided by ensur-ing the ‘equilibrated gas’ principle applies. This means thatconditions are maintained such that the formation of carbonis not favored at equilibrium and the rate of reforming iscontrolled by the rate of heat transfer such that the reactantgas composition is very close to its equilibrium value at anygiven point in the reformer. In this process, coke formationis expected via kinetic processes and as a result, some frac-tion of the catalyst will need to be regenerated. After regen-eration, hot catalyst is recirculated to the down5ow sectionthus relieving a portion of the heat duty required in the pri-mary section of the reactor.Studies have been carried out on aspects of this process

including:

(a) the development of suitable catalysts (El Solh, Jarosch,& de Lasa, 2001),

(b) the development of suitable membranes (Jarosch & deLasa, 2001),

(c) the collection of data on catalyst coking,(d) the study of the e<ect of time-on-stream on catalyst ac-

tivity,(e) the collection of reaction rate data for kinetic modelling(Jarosch, 2000).

Emphasis was also placed on the evaluation of catalysts andmembranes under reactor conditions of temperature (750–

850◦C) and total pressure (2.0–3:0 MPa) closely approxi-mating industrial conditions.

2. Steam reforming of methane

Natural gas is the primary feedstock for the steam re-forming process. As the main component of natural gasis methane, methane was used as analog for natural gas.The chemical processes involved in the steam reform-ing of methane can be expressed using two independentequilibrium-limited reactions. These are the endothermicreforming reaction (Eq. (1)) and the exothermic water–gasshift reaction (Eq. (2)).

CH4 + H2O� CO + 3H2; (1)

CO + H2O� CO2 + H2: (2)

A wide variety of rate expressions for the steam reform-ing of methane have been proposed. These models range incomplexity from simple ;rst order dependency on methanesuch as those proposed by Munster and Grabke (1981) andProkopiev, Aristov, Parmon, and Giordano (1992) involvingtwo parameters to complex Langmuir–Hinshelwood mod-els with over ten parameters (Xu & Froment, 1989). It isgenerally agreed, with the exception of Al-Ubaid and Wolf(1987, 1988), that the rate of methane reforming has a ;rstorder dependency on methane. Furthermore, it is also agreedthat the rate-determining step in the reforming process is theformation of adsorbed carbon species (Munster & Grabke,1981):

CH4 + ∗–Metal site↔ CAds– ∗+2H2: (3)

The formation of adsorbed carbon from methane is a step-wise process that requires that a C–H bond be broken whilemethane is in the gas phase. The resultant CH3 species mustthen come into contact with an open site on the surface ofthe metal crystal. After being adsorbed to the surface of themetal crystal, the CH3 is transformed into adsorbed carbonby stepwise dehydrogenation (Rostrup-Nielsen, 1993):

CHgas4 → CHgas3 → CH3–∗ → CH2- ∗→ CH1–∗ → CAds– ∗ : (4)

This mechanism leads to the formulation of rate equationsof the following form (Munster & Grabke, 1981):

rCH4 ˙ −kpCH4p�H2 : (5)

In Eq. (5), the value of � is found to depend on temperature,having a value close to −1 at low temperatures (¡ 700◦C)and approaching 0 at high temperatures (¿ 700◦C).Kinetic rate expressions for the steam reforming of

methane found in the literature use the steady state

K. Jarosch et al. / Chemical Engineering Science 57 (2002) 3439–3451 3441

Fig. 2. Two models each possessing one independent variable and havingmultiple parameters.

approximation and take the form (Rostrup-Nielsen, 1993):

rCH4 =−kpCH4f(pH2O; pH2)

(1 + f(pCH4 ; pH2O; pH2 ; pCO; pCO2))

×[1− pCOp3H2

pCH4pH2OK1

]: (6)

Most kinetic data reported in the literature have been col-lected using packed bed reactors at temperatures and pres-sures signi;cantly lower than those employed in industrialreforming using catalyst designed for ;xed bed use. Giventhat the data CATFORMER is a 5uidized reactor, operatedat both high temperature and pressure and that a number ofcompeting mechanisms are reported in the literature, an ef-;cient experimental strategy needed to be devised for theselection of an appropriate kinetic expression.

3. Sequential design for the model discrimination

It is often desirable to encapsulate the behavior of a sys-tem in a mathematical model and just as often there may bemore than one plausible model available. The question is asto how does one select between plausible kinetic models.This is the motivation for the model discrimination. Thereare two possible approaches. The ;rst approach involves col-lecting experimental data estimating the model parametersand then comparing the models based on such criteria suchas residual sum, lack-of-;t and signi;cance of the parame-ters. The second approach involves the use of a statisticalcriterion to select settings of the independent variables (ex-perimental conditions) such that di<erences between modelsare emphasized.Fig. 2, representing two multiparameter models of one

independent variable, illustrates the hazards involved whenthe ;rst method is employed for model discrimination. Ifall the experimental data were collected with settings of theindependent variable in the shaded region, then it will be

diBcult to judge which is the better model as the predictedvalues are very close. For simple models, it may be easyto select unaided a set of experimental conditions that al-low discrimination, but for systems that are more complexthis may be diBcult or impossible. It is under these circum-stances that the second technique, true model discrimina-tion, becomes necessary. The use of a statistical criterion toguide the selection of experimental conditions ensures thatthe experimental conditions are moved into the unshadedregion of Fig. 2 so that the di<erences between the modelsbecome clear.Model discrimination can be thought of as an iterative

process. Starting with a hypothesis, for example the selectionof a set of candidate models, an initial set of experimentsis designed and performed so that the parameters present ineach model can be estimated. With this information and astatistically based criterion, an experimental condition canbe selected that will improve our knowledge of which modelbest describes the behavior. This process continues until theexperimenter is satis;ed that one model is superior to theothers proposed.

4. Criterion for model discrimination

In the section above, it was suggested that a criterioncould be used to design sequential experiments, the con-ditions of which would allow for maximum discriminationbetween the proposed models. One criterion that could beused would be the experimental conditions (levels of the in-dependent variables) that maximizes the di<erence betweenthe model outputs. This criterion would be satisfactory forthe case depicted in Fig. 2. However, this criterion provesto be less satisfactory if the models are of similar mathe-matical form. In the case of models of similar mathemati-cal form, a desirable criterion would not only maximize thedi<erence in the model outputs but would also employ thevariance of the experimental response to maximize the like-lihood of discrimination. An example of such a criterion isthe Kullback discrimination function (Kullback, 1978).The Kullback discrimination function can be formulated

if we have two models, with outputs, Y 1 and Y 2, each out-put being a random variable, that is to say having statisticalproperties such as a mean and variance. If we wish to deter-mine which of the two models is more likely, we can say thatif model 1 is correct Y 1 will be distributed as the probabil-ity density function p1(y). If model 2 is correct Y 2 will bedistributed as p2(y). For example, imagine an observationof the output lies at y′ as in Fig. 3. One could ask the ques-tion, what are the odds that this observation is from p1(y)as opposed to p2(y)? Kullback suggests that the followingratio can be taken as a measure (Kullback, 1978, p. 4):

ln[p1(y)p2(y)

]: (7)


Fig. 3. If the observation y′ made at x is normally distributed, what arethe odds that it is from either p1(y) or p2(y).

Using the ratio suggested by Kullback (Eq. (7)), the ex-pected value of the odds in favor of selecting model 1 canbe found by integrating

I(1:2) =∫ ∞

−∞p1(y) ln

[p1(y)p2(y)

]dy: (8)

Similarly, the expectation in favor of selecting model 2 canbe found as follows:

I(2:1) =∫ ∞

−∞p2(y) ln

[p2(y)p1(y)

]dy: (9)

Eqs. (8) and (9) can be regarded as the weight of informationin favor of choosing model 1 or 2. For independent randomevents I(1:2) and I(2:1) are additive by the principle ofdivergence (Kullback, 1978, p. 12) so the integral form ofthe Kullback discrimination function can be written as

J (1; 2) = I(1:2) + I(2:1) =∫ ∞

−∞(p1(y)

−p2(y)) ln[p1(y)p2(y)

]dy: (10)

The Kullback discrimination function (Eq. (10)) is a mea-sure of the total information available when discriminatingbetween two hypotheses. When the function (Eq. (10)) ismaximized, discrimination between the models may becomepossible. To accomplish this, the nature of the distributionsneeds to be known.Given that model r is the correct model, it will have an

expected value (E[Yr(X; x(n+1))]) when evaluated at x(n+1)

and this can be written as Y(n+1)r . When the (n + 1)th ex-

perimental response, Y (n+1), is measured, we ;nd that dueto the experimental error, �EXP, it di<ers from the expectedvalue Y

(n+1)r . We can assume that Y (n+1) is normally dis-

tributed around the expected value of each model, Y(n+1)r ,

with a variance of �2Y and that Y(n+1)r is distributed in a lin-

earized region around the predicted value, Y(n+1)r , with a

variance of �2r . When this is the case we can say that Y(n+1)

is distributed around Y(n+1)r with a variance of �2Y + �2r . Af-

ter taking this into account, the probability density function

of Y (n+1) for the rth model can be written as

pr(Y(n+1)) =

1√2�(�2Y + �2r )

×exp[−12(Y (n+1) − Y

(n+1)r )2

�2Y + �2r

]: (11)

If the appropriate substitution of Eq. (11) into Eqs. (8)and (9) is made and integrated, the followings result:

I(1:2) =12ln(�2Y + �22�2Y + �21

)+12�2Y + �21�2Y + �22

− 12

+12(Y(n+1)1 − Y

(n+1)2 )2

�2Y + �22; (12)

I(2:1) =12ln(�2Y + �21�2Y + �22

)+12�2Y + �22�2Y + �21

− 12

+12(Y(n+1)1 − Y

(n+1)2 )2

�2Y + �21: (13)

Eqs. (12) and (13) can be added as per Eq. (10) to give

J (1; 2) =12(�21 − �22)

(1

�2Y + �22− 1

�2Y + �21

)+12

×(

1�2Y + �22

+1

�2Y + �21

)(Y(n+1)1 − Y

(n+1)2 )2:

(14)

Now we can ;nd the vector of independent variables, x(n+1),that maximizes J (1; 2). The development given above isvalid for two models but it can be extended for any numberof models.As neither �2Y nor �

2r ’s are known they need to be esti-

mated. The variance �2Y can be estimated by using s2e , mea-

sured by doing repeat experiments. The variances of themodels, �2r ’s, can be estimated from the variance of theresiduals when the models are ;t to n observations.

5. The Box–Hill discrimination function

Box and Hill (1967) proposed an improvement to theKullback discrimination function that takes into accountwhat is called the prior probability, P

(n)r of the models or

in other words, the probability that model r of a set of vmodels is the correct model after n experiments. Box andHill suggest that by weighting Kullback’s information func-tions, I(i:j), with the prior probabilities, we will be able toincrease the eBciency of the discrimination. The Box–Hill


discrimination function can be developed as follows:

KV = [P(n)1 · · · P(n)v ]

I(1:1) I(1:2) · · · I(1:v)I(2:1) I(2:2) · · · I(2:v)...

......

I(v:1) I(v:2) · · · I(v:v)

×

P(n)1......

P(n)v

: (15)

If we note that the diagonal of the I matrix is zero, then we;nd

KV = P(n)1 P

(n)2 J (1; 2) + P

(n)1 P

(n)3 J (1; 3) + · · ·

+P(n)1 P

(n)v J (1; v) + P

(n)2 P

(n)v J (2; v)

+ · · ·+ P(n)v−1P

(n)v J (v− 1; v); (16)

KV =12

v∑i=1

v∑j=i+1

P(n)i P

(n)j

[(�2i − �2j )

2

(�2Y + �2i )(�2Y + �2j )

+ (Y(n+1)i

−Y (n+1)j )2(

1�2Y + �2i

+1

�2Y + �2j

)]: (17)

The variances required to calculate KV (Eq. (17)) can beestimated as described above. After n experiments, P

(n)r can

be evaluated using Bayes’ theorem with P(n−1)r , the proba-

bility that any given model is correct prior to evaluating theexperimental data, set to 1=v and pr(Y

(n)) evaluated withEq. (11) (with n replacing n + 1). This is to say that be-fore having any information on the behavior in question, ifv plausible models for the behavior are proposed, the prob-ability that any one of them is the correct model is 1=v. Theresult is as follows:

P(n)r =

P(n−1)r pr(Y

(n))∑vr=1 P

(n−1)r pr(Y (n))

: (18)

This method can be extended to accommodate multipleoutputs. The advantage of discrimination on more than oneoutput is that the rate of discrimination is increased signi;-cantly. The Box–Hill criterion also allows for discriminationbetween models of similar mathematical form as it accountsfor the variance of the prediction, a function of the numberof parameters, as well as the value of the sum of the squaredresiduals.

6. Procedure for sequential experimentation

The procedure for a sequential design using the Box–Hilldiscrimination function can be summarized as follows:

(a) Using the best information possible, design an experi-ment to collect n observations with repeats for the esti-mation of �2Y .

(b) Estimate the parameters of the models in question andcalculate �2r for each model.

(c) Calculate P(n)r .

(d) Find the vector of experimental conditions (x(n+1))which maximizes KV .

(e) Run the conditions x(n+1), return to step (b).

A number of approaches are available for ;nding the ex-perimental conditions x(n+1) that maximize KV ; however, inmost cases the problem is restated as a minimization prob-lem. This restatement is accomplished by simply searchingfor values of x(n+1) that minimize −KV . In this case, a nu-merical tool for ;nding constrained minima of multivariatefunctions from the MATLABTM optimization toolbox calledCONSTR.M was used. The numerical approach employedin this work to ;nd values of x(n+1) that minimize −KV

is reported in detail by Jarosch (2000). MATLABTM is atrademark of the MathWorks Inc.

7. Assumptions and limitations

No statistical technique should be considered as apanacea. These techniques only provide guidelines and thejudgment and experience of the experimenter are still re-quired. This is particularly evident when model discrimina-tion techniques are employed. The input of the experimenteris critical in determining which models will be considered,the number of models considered and the level at which thediscrimination will be terminated. The experimenter mustalways keep in mind that no discrimination technique willbe able to identify the best model if it is not included inthe set being discriminated. The experimenter must alsokeep in mind that if none of the models are adequate, nodiscrimination may be possible.In addition to the limitations inherent in the technique,

a number of assumptions are used. The most important as-sumptions are that the error associated with the lack-of-;t isadditive and that it follows the probability density functionexpressed in Eq. (11). This will only be strictly correct ifthe model in question is the ‘true’ model and behaves in alinear manner. This deviation from the assumption is com-pounded by the fact that the variances �2Y and �

2r must be

estimated using experimentally determined values.In the work presented here, three steps were taken to en-

sure that the models behaved in a linear or close-to-linearmanner. The ;rst step, which also underlines the judgmentof the experimenter as mentioned above was that, models


Fig. 4. Cross-sectional view of a Riser Simulator unit modi;ed for usein the CATFORMING process.

were selected such that the initial Langmuir–Hinshelwoodmodel forms possessed only low order polynomials in thedenominator. Models with high order polynomials, such asthose suggested by Agnelli, Ponzi, and Yeramian (1987),with orders as high as 7, given their essentially empiri-cal character, were not considered. The second step was toensure that the Langmuir–Hinshelwood models were repa-rameterized such that parameters behave in a close-to-linearmanner (Ratkowsky, 1990). The third step was to employthe technique of temperature centering to reduce the cor-relation between the activation energy and pre-exponentialfactor for each parameter.

8. Experimental setup and materials

Experimental trials were carried out in a modi;ed RiserSimulator (de Lasa, 1991) speci;cally manufactured forthe current project (Jarosch, 2000). The Riser Simulator(Fig. 4) is a bench scale reactor used to simulate the behav-ior of riser=downer reactors operating in the fast 5uidizationregime. This is achieved by trapping a sample of catalystbetween two porous metal disks in a basket that is insertedinto the body of the reactor and, when seated, lies beneath aturbine impeller. When the impeller is spun, reactant gas isdrawn through the basket causing the sample of catalyst tobecome fully 5uidized. As the internal recirculation rate ishigh, the Riser Simulator can be used to follow the progressof a well-mixed plug as it 5ows up a riser or down a down-5ow reactor.The unit used in this study was modi;ed to allow for

the insertion of a high-temperature hydrogen membraneby lengthening the upper shell (to allow the basket to be

Fig. 5. Schematic 5ow diagram of the experimental setup.

inserted) and truncating the lower shell. The membrane(when present) is welded into a plate which is insertedbetween the upper and lower shells. The lower shell has apocket located directly below the membrane. Inlet and out-let ports on the lower shell allow sweep gas to remove thepermeate gas. Runs without the membrane can be carriedout using a blank plate.In addition to the Riser Simulator, several other major

pieces of equipment were required to complete the experi-mental setup; the methane and steam injector, the blow downand GC sampling system, and the data acquisition system(Fig. 5). Injections of methane were made using an auto-mated injection system (AIS). The injector consists of a bar-rel ;tted with a plunger actuated by a double-action pneu-matic piston. The volume injected was controlled by restrict-ing the stroke of the actuator. The 5ow through the injectorwas controlled by a slider valve. In the ‘;ll’ position, feedgas is allowed to 5ow from the supply cylinder into the bar-rel and out to the vent via a needle valve. The pressure in theinjector was controlled by adjusting the needle valve. Whenthe slider valve is in the ‘inject’ position, the inlet and out-let ports are closed and a third port is open such that whenthe plunger moves forward, the contents of the barrel areinjected into the reactor through H2OV and HPV1. The si-multaneous generation and injection of the required amountof steam was also necessary. This was accomplished by us-ing a peristaltic pump to circulate heated distilled water ata rate of 50 ml min−1 from a reservoir and through a loopconnected to the six-port valve (H2OV). To make an injec-tion of steam and methane, H2OV was rotated to allow themethane from the AIS to travel through the loop prior toentering the reactor thus carrying the water along before it.In order to speed the vaporization of the water, the sectionof tubing (which lies outside the reactor’s insulating shell)between HPV1 and the reactor was heated independentlyusing a heating tape.Reaction runs were terminated by opening valves, HPV2

and LPV2, and allowing the pressure to equalize. The ratioof volume between the blow down bottle and the reactor is


approximately 6:1, as the bottle is kept at room temperature.This provided a pressure reduction of about 18:1 (dependingon the initial reactor temperature). Product gas would thenbe allowed to 5ow through a sample loop attached to thevalve GCV. Samples of product gas could then be injectedin the GC for TCD analysis.Pressure, temperature and GC output data from each trial

were collected using a PC-based data acquisition system.The pressures in both the reactor and blow down bottle wererecorded for each trial along with their respective tempera-tures. The system was also used to record the response ofthe GC TCD.

9. Methodology for parameter estimation=discrimination

As mentioned above, the measured experimental responsewas the conversion of methane, that is to say, the extent ofthe reforming reaction (Eq. (1)). As the bench scale CAT-FORMER (Riser Simulator) operates in batch mode, thematerial balance must be integrated in order to calculate thereaction extent. This was done in the following manner, ;rstthe extent of the reforming reaction (�REF) based on the ini-tial number of moles of methane injected (N 0CH4) and thenumber of moles consumed is expressed as a function ofcontact time:

d�REFdt

=dNCH4 =dtN 0CH4

: (19)

The number of moles of methane consumed as a functionof contact time was expressed as

dNCH4dt

=−rCH4w; (20)

where rCH4 is a function of the partial pressure of the reac-tants and has a form similar to Eq. (6), and w the mass ofcatalyst.Based on the initial number of moles of methane and

water (N 0H2O) injected, the partial pressures of each speciespresent were calculated as

pCH4 =

[N 0CH4 − �REFN 0T + 2�REF

]PT ; (21a)

pH2O =

[N 0H2O − �REF − �WGS

N 0T + 2�REF

]PT ; (21b)

pCO =[�REF − �WGSN 0T + 2�REF

]PT ; (22a)

pH2 =[3�REF + �WGSN 0T + 2�REF

]PT ; (22b)

pCO2 =[

�WGSN 0T + 2�REF

]PT ; (23a)

PT =NTRTVR

: (23b)

Under the conditions used, the water–gas shift reaction wasfound to be at chemical equilibriumwith the conditions in thereactor at any given time (Jarosch & de Lasa, 1999). Theextent of thewater–gas shift reaction (�WGS) could, therefore,be estimated using the chemical equilibrium relationship:

K2 =pCO2pH2pCOpH2O

: (24)

If Eqs. (21b), (22a), (22b) and (23a) are substituted into Eq.

(24) the extent �WGS can be calculated from the quadraticform:

�WGS =−b−√

b2 − 4ac2a

(25)

with

a= K2 − 1; (26a)

b=−(K2N 0H2O + 3�REF); (26b)

c = K2�REF(N 0H2O − �REF): (26c)

Once a model form was selected for the rate of methaneconsumption, rCH4 , and given a set of parameter values, thepredicted extent of the reforming reaction could be calcu-lated. The parameter estimation procedure was executedusing the non-linear least-squares regression routine,CURVEFIT.M, available in the Optimization Toolbox (ver-sion 2.0) of MATLABTM release 11 version 5.3.0.10183.The integration required in the parameter estimation wasperformed numerically using the MATLABTM m-;leODE45.M available in the MATLABTM Toolbox version5.3 (Jarosch, 2000).

10. The models

Six models were proposed as candidates to represent therate of consumption of methane. The model forms wereselected such that they would represent a broad range ofpossible mechanisms while at the same time maintaining alow degree of parameterization.MODEL I: Being the ‘First Order’ model, it represents

the rate of consumption of methane as having a ;rst or-der dependence on the partial pressure of methane. This be-havior was reported by Munster and Grabke (1981) whennickel foil was used as a catalyst for the steam reforming ofmethane in the temperature range 700–900◦C. This can berepresented as follows:

rCH41 = %∗1pCH4

[1− pCOp3H2

pCH4pH2OK1

]: (27)


MODEL II: In Model 1, adsorption is assumed to playa minimal role in determining the rate; while in Model II,the ‘First Order Adsorption’ model, adsorption of methaneis assumed to play a role in determining the apparent rateof methane consumption.

rCH42 =pCH4

(%∗1 + %∗2pCH4)

[1− pCOp3H2

pCH4pH2OK1

]: (28)

In addition, the formation of an adsorbed carbon species(Eq. (3)) is assumed to be the direct result of methane ad-sorption to the nickel crystal surface. The products, hydro-gen and carbon monoxide, are not adsorbed. Water reactsdirectly with the adsorbed carbon species.Model III: In the ;rst two models proposed, the rate

was assumed to depend only on the partial pressure ofmethane. In Model III, it is assumed that two adjacent siteson the surface of the nickel crystal are required. One forthe non-dissociate adsorption of water and a second for theadsorbed carbon species. Although not technically of sec-ond order, this model was referred to as the ‘Second OrderAdsorption’ model.

rCH43 =pCH4pH2O

(%∗1 + %∗2pCH4 + %∗3pH2O)2

[1− pCOp3H2

pCH4pH2OK1

]:

(29)

Model IV: It was suggested by Agnelli et al. (1987) asone of the ;ve possible mechanistically based rate equationsfor the steam reforming of methane. Of the ;ve rate mod-els suggested by Agnelli et al. (1987), the models selected(Models IV and V) were chosen as they include the basicadsorption e<ects suggested in the other proposed modelswhile at the same time preserving a low order in the denom-inator.

rCH44 =pCH4

(%∗1 + %∗2pH2O + %∗3pCO)

[1− pCOp3H2

pCH4pH2OK1

]:

(30)

As with Model III, adsorbed water in Model IV (Eq. (30))is assumed not to dissociate into hydrogen and an adsorbedoxygen species. Model IV di<ers from Model III in that thecarbon monoxide product is expected to be adsorbed.Model V: It was also suggested by Agnelli et al. (1987)

as one of the ;ve possible mechanistically based rate equa-tions for the steam reforming of methane. Model V di<ersfromModels III and IV in that, in this case, water is assumedto dissociate and form an adsorbed oxygen species. Reac-tion between adjacent adsorbed carbon and oxygen speciesresults in the production of carbon monoxide.

rCH45 =pCH4

(%∗1 + %∗2 (pH2O=pH2) + %∗3pCO + %∗4pCO2)

×[1− pCOp3H2

pCH4pH2OK1

]: (31)

Signi;cant product adsorption is assumed in Model V(Eq. (31)) and as adsorbed carbon monoxide can react withadsorbed oxygen, carbon dioxide is produced. This leads tothe contribution of both carbon dioxide and carbon monox-ide partial pressures to the apparent reforming rate.Model VI: The sixth model, unlike the others, presup-

poses that the rate of consumption of methane can be corre-lated to the respective partial pressures of the reactants andproducts. Model VI, the ‘Power Law’ model, was selectedas such forms are often able to describe chemical kineticbehaviors without having a ;rm mechanistic foundation.

rCH46 = %∗1pCH4p%2H2Op

%3H2p

%4COp

%5CO2

[1− pCOp3H2

pCH4pH2OK1

]:

(32)

In each model the parameters, %∗i , were allowed to varywith temperature (with the exception of the exponents inEq. (32)). This behavior was represented using the Arrhe-nius relationship centered on temperature:

%∗i = %1; i exp[−%2; i

R

(1T

− 1Tc

)]: (33)

This centered form reduces the correlation between thepre-exponential factor %1; i and the activation energy %2:ithereby improving the statistical properties of the estimatefor the pre-exponential factor.Typical Langmuir–Hinshelwood-type rate expressions, in

which the adsorption constants appear in both the numeratorand the denominator of the expression, lead to mathemati-cal models that are highly non-linear with respect to theirparameters. This non-linearity in the parameters results in ahigh degree of parameter correlation. The rate expressionsused here (Eqs. (28)–(31)) have been reparameterized assuggested by Ratkowsky (1990) to provide models that areclose-to-linear with respect to their parameters.

11. Results and discussion

In order to start the process of model discrimination, ini-tial estimates of the parameters for each of the six candi-date models were made using 29 observations collected over20wt% Ni=�-alumina catalyst (Jarosch, 2000). Settings forthe independent variables likely to allow discrimination be-tween the models were then identi;ed using MATLABTM

via the procedure described above. A trial was then carriedout using the conditions identi;ed and a new set of condi-tions was identi;ed. This sequence was repeated until onemodel was considered to be more probable than the oth-ers. The experimental conditions identi;ed in this mannerare reported in Table 1. A total of six sets of experimentalconditions were identi;ed and carried out.As all six models are reasonable, prior to collecting any

data the probability that any one of the models would best de-scribe the data was presumed to be the same. The probability


Table 1Sequentially designed experimental conditions

Experimenta Temperature Pressure Steam-to-methane ratio Mass of catalyst Contact time(◦C) (MPa g) (g mol g mol−1) (g) (s)

1 773 2.406 1 0.700 1002 730 1.393 2.6 0.700 1003 800 0.352 1 0.304 604 750 0.524 2.1 0.700 905 750 1.227 6.3 0.303 606 750 1.262 6.5 0.275 60

aExperiments 1–6 represent the experimental conditions used to obtain observations 30–35 in Fig. 7.

Fig. 6. Posterior probability of the six candidate models after each of thediscrimination trials. The observation at 32.5 represents the redistributionof probability after dropping Models I and III.

that any one of the models would best describe the data was0.167. After 29 experiments, information is available oncethe parameters have been estimated and therefore the pos-terior probability for each model was to be calculated (Fig.6). Of the six models, Models II and III had accumulatedapproximately half of the probability, 0.23 and 0.26, respec-tively. The other four models shared the remaining proba-bility equally.After the ;rst discrimination experiment (observation 30,

Fig. 6), Model II had accumulated a probability of 0.38 andModel III a probability of 0.19. At this point, discrimina-tion was also made between Models I and V and ModelsIV and VI. The results of the second discrimination experi-ment brought the probabilities of Models II and III together.Discrimination continued between the remaining four othermodels. After the third discrimination experiment, Model IIaccumulated a posterior probability of 0.46. The probability

of Model III fell sharply to 0.006 and further dis-crimination between the four remaining models wasobserved.At this point in the discrimination procedure, two mod-

els were removed from consideration, Models I and III. Ofthe models with signi;cant probability, Model I consistentlyhad the lowest probability and as the remaining models con-tained a ;rst order dependency it was removed in order tospeed the discrimination process. Model III was removedfrom the process as its posterior probability was so low asto have no measurable in5uence on the selection of subse-quent experimental conditions. The posterior probabilitiesof the remaining models (II and IV–VI) were then adjustedsuch that the sum was equal to 1.0. The adjusted prior prob-abilities for each model were plotted as observation 32.5 inFig. 6.The subsequent three discrimination experiments (obser-

vations 33–35, Fig. 6) demonstrated the increasing proba-bility of Model II at the expense of Model VI, the power lawmodel. Models IV and Vmaintained approximately the sameprobabilities. The ;nal probabilities were 0.74 for Model II,the ;rst order adsorption model, 0.15 for Model IV and 0.11for Model V.Once the mathematical form of the model was estab-

lished, the parameters for Model II were estimated using 85observations collected over 20wt% Ni=�-alumina catalystover a wide range of temperatures, molar SMRs and totalreactor pressures. The parameter estimates with their corre-sponding 95% con;dence intervals and correlation matrixare presented in Table 2. A plot of the reconciliation betweenthe experimentally observed conversion of methane andthose predicted using the least-squares parameter estimates(Table 2) can be found in Fig. 7.In addition, the parameters for Model II were es-

timated using 52 observations collected over 4wt%Ni=�-alumina catalyst collected over a range of temper-atures, molar SMRs and total reactor pressures similarto those employed in for the data collected over the20wt% Ni=�-alumina catalyst. The parameter estimateswith their corresponding 95% con;dence intervals andcorrelation matrix are presented in Table 3. A plot ofthe reconciliation between the experimentally observed


Table 2Estimates for the parameters in Model II for data collected over 20wt% Ni=�-alumina catalyst including linearized 95% con;dence intervals and parametercorrelation

%1;1 %2;1 %1;2 %2;2(kPa gcat s mol

−1) (J mol−1) (gcat s mol−1) (J mol−1)

Estimate 7:20× 106 1:25× 105 −5:04× 103 2:50× 105Interval ±1:19× 106 ±0:33× 105 ±3:66× 103 ±1:70× 105

�2r = 0:00658 n= 85 Tc = 1041 K

Parameter correlation matrix

%1;1 %2;1 %1;2 %2;2

%1;1 1%2;1 −0:0348 1%1;2 −0:8362 −0:1826 1%2;2 0.4862 0.6029 −0:8392 1

Fig. 7. Reconciliation plot for model II (Eq. (28)) when ;t to 85 exper-imental observations collected over 20wt% Ni=�-alumina catalyst. Datawere collected over a temperature range of 727–855

◦C, a molar SMR

range of 1.3–6.8 and reaction pressure range of 2000–2700 kPa.

conversion of methane and those predicted using theleast-squares parameter estimates (Table 3) can be foundin Fig. 8.When the results of the parameter estimation for the

20wt% Ni/�-alumina catalyst are inspected (Table 2), itcan be seen that the four parameters of Model II are all sig-ni;cant at the 95% con;dence level. In addition, the repa-rameterization and temperature centering were successfulin reducing the overall correlation between the parametersto very moderate levels. Only the pre-exponential factor,%1;2, shows some degree of correlation with its associatedactivation energy, %2;2, and the pre-exponential factor %1;1

both at the level 0.84. From this it can be concluded thatthe model is not overparameterized as would be the case ifone or more parameters were not signi;cant and/or highlycorrelated (i.e. correlation near 1.0).Inspecting the results of the parameter estimation for the

4wt% Ni=�-alumina catalyst (Table 3), it can be seen thatthree of the four parameters of Model II are signi;cant atthe 95% con;dence level. As with the 20wt% Ni=�-aluminacatalyst, the reparameterization and temperature centeringwere successful in reducing the overall correlation be-tween the parameters to very moderate levels. However, thepre-exponential factor, %1;2, once again demonstrated thehighest degree of correlation but in this case the correlationis primarily with the pre-exponential factor %1;1. The levelof the correlation was estimated to be at the level 0.99. It isthis high degree of correlation that leads to the large 95%con;dence interval on parameter %1;2.Previously, it was noted that no statistical di<erence could

be found in the behavior of 4 and 20wt% Ni=�-aluminacatalysts in terms of catalytic activity, stability of activitywith time-on-stream and coking activity (Jarosch, 2000; ElSolh et al., 2001). At the root of this similarity in behaviorwas the fact that the relationship between bulk metal load-ing on the catalyst and metal dispersion led to both cata-lysts having similar available metal surface areas, 0.416 and0:406 m2=gcat for the 4 and 20wt% Ni=�-alumina catalysts,respectively. Once again, this similarity is re5ected in theparameter values.In describing the data collected, Model II was not overpa-

rameterized, however, the possibility exists that the modelis underparameterized. This is to say that the experimentalsystem is displaying a behavior not accounted for in themodel. Speci;cally, Model II did not accumulate all theprobability but Models IV and V retained a relatively stableresidual probability throughout the discrimination process.Such a result indicates one or more of the adsorption terms inModels IV and V may contribute signi;cantly to the model.If this is correct, Model II may be underparameterized


Table 3Estimates for the parameters in Model II for data collected over 4wt% Ni=�-alumina catalyst including linearized 95% con;dence intervals and parametercorrelation

%1;1 %2;1 %1;2 %2;2(kPa gcat s mol

−1) (J mol−1) (gcat s mol−1) (J mol−1)

Estimate 8:06× 106 1:41× 105 −6:61× 103 2:57× 105Interval ±3:66× 106 ±0:51× 105 ±7:75× 103 ±1:33× 105

�2r = 0:00518 n= 52 Tc = 1036 K

Parameter correlation matrix

%1;1 %2;1 %1;2 %2;2

%1;1 1%2;1 −0:3565 1%1;2 −0:9857 0.3229 1%2;2 0.7085 −0:3028 −0:7750 1

Fig. 8. Reconciliation plot for model II (Eq. (28)) when ;t to 52 ex-perimental observations collected over 4wt% Ni=�-alumina catalyst. Datawere collected over a temperature range of 718–850

◦C, a molar SMR

range of 1.2–3.8 and reaction pressure range of 2000–2700 kPa.

resulting in residuals that are not randomly distributed anda residual sum that is in5ated.Lack-of-;t due to underparameterization can be observed

in a number of manners. A qualitative assessment can bemade by inspecting a residual or reconciliation plot suchas that presented in Figs. 7 and 8. If a signi;cant e<ectis not included in the model, data points may cluster inhorizontal bands. This banding is a result of changes in theobserved conversion caused by an independent variable notincluded in the model. Vertical banding can be symptomaticof overparameterization. Neither of these e<ects can be seenin the reconciliation plots (Figs. 7 and 8).A quantitative assessment of underparameterization can

be made by inspecting the information content of the resid-

uals. If the model ;ts the data, that is to say, there is noinformation left in the residuals, they should be normallydistributed with an expected value of zero. This assumptioncan be tested by comparing the sum of the squared valuesof the normalized residual, Q, to the value of the cumula-tive '2 distribution with the same degree of freedom as theresiduals. If the model is underparameterized, the sum of thesquared residual will be in5ated and Q will be greater than'2. If this is the case, the null hypothesis, that the modelhas the correct number of parameters, can be rejected andthe alternative hypothesis, that the residuals are not nor-mally distributed (i.e. the model is underparameterized) canbe accepted. The sum of the squared normalized residualsfor Model II when it was ;t to the data acquired over the20wt% Ni=�-alumina catalyst, Q, (∼ 5:15) was found to beless than the value of the inverse cumulative '2 distribu-tion (∼ 103) evaluated at 0.95 with 81 degrees of freedom(number of observations less than the number of parame-ters). Thus, the null hypothesis can be accepted, the ModelII adequately describes the data acquired over the 20wt%Ni=�-alumina catalyst and the residuals are not in5ated dueto underparameterization.When the same test was performed for the residuals from

Model II ;t to the data acquired over the 4wt% acquiredover the 20wt% Ni=�-alumina catalyst the value of Q wasfound to be ∼ 2:92. The value of the inverse cumulative'2 distribution evaluated at 0.95 with 48 degrees of free-dom was found to be ∼ 65:2. From this it was concludedthat Model II was not under parameterized when ;t to theobservations made over the 4wt% Ni=�-alumina catalyst.Model II was parameterized based on the assumption

that the adsorption of methane would follow a Langmuirisotherm as follows:

rCH4 =−kKApCH41 + KApCH4

=pCH4

−[ 1kKA+ 1

k pCH4]

=pCH4

%∗1 + %∗2pCH4: (34)


Based on this form (Eq. (34)), the sign of %1;2, the inverseof the rate constant for methane consumption, is expected tobe negative. The sign of %1;1 is also expected to be negative.If Tables 2 and 3 are consulted, it can be seen that the signof %1;2 was found to be negative but that the sign of %1;1 wasfound to be positive.A negative apparent adsorption constant can be explained

if the following isotherm is considered:

)=KApCH4

1 + (KA − *)pCH4: (35)

In Eq. (35), when the value of KA is much larger than thevalue *, a typical concave Langmuir chemisorption isothermis the result. When the value of KA is close to the value of* the isotherm approaches a straight line. When the valueof KA is much less than the value of * a convex isotherm oftype III, is the result. Type III isotherms are found in caseswhere physisorption, the adsorption of multiple species persite, occurs below the saturation vapor pressure. Given theparameterization used here (Eq. (34)), a negative apparentadsorption constant would result if KA¡*.In general, to have physical meaning, the value of * should

be taken as the inverse of the saturation vapor pressure forthe species under consideration. In this case the value of *appears to be substantially larger than that if calculated fromthe saturation vapor pressure. This is likely due to the factthat, under the conditions of temperature and pressure usedin this study, methane readily reacts with the surface of thenickel crystal (Eqs. (3) and (4)) producing active carbon.This active carbon is free to participate in the steam reform-ing reaction but may also form reactive coke. Clustering ofactive carbon species as coke can explain high surface cov-erage at pressures removed from saturation.

12. Conclusions

Based on the results gathered during the present study,the following conclusions can be drawn:

(a) The use of a statistical criterion such as the Box–Hillfunction is a powerful tool for the design of experimentsthat optimizes e<ort expended to information acquired,in this case allowing for the discrimination between sixdi<erent plausible models with as few as six sequentialexperiments in addition to those used to make initial theparameter estimates.

(b) Results of the model discrimination indicate that the ad-sorption of methane plays an important role in determin-ing the observed rate of methane consumption duringthe steam reforming of methane in the CATFORMERreactor.

(c) Previous observations (Jarosch, 2000; El Solh etal., 2001) on the behavior of the 4 and 20wt%Ni=�-alumina catalysts led to the conclusion that they

should possess similar kinetic parameters. This was con-;rmed when kinetic parameters of Model II were eval-uated using large data sets collected over both catalysts.

(d) Based on the estimated value of the parameters, theadsorption behavior of methane appears to follow anisotherm of type III. This is indicative of physisorptionstemming from the presence of more than one activecarbon species per site at pressures well below the sat-uration vapor pressure for methane.

Notation

I(i:j) expected value of odds in favor of i over jJ (i; j) I(i:j)+ I(j:i), Kullback’s discrimination functionk rate constant for the reforming reaction,

mol g−1cat kPa−1 s−1

ka; kd adsorption/desorption rate constants for hydrogenke; kdis emergence/dissolution rate constants for atomic

hydrogenKA adsorption constant for methane, kPa−1

KV value of the Box–Hill discrimination function, di-mensionless

K1 equilibrium constant for steam reforming, kPa2

K2 equilibrium constant for water gas shift, dimen-sionless

N number of moles, molpi partial pressure of species i, kPap(y) probability density function for yP pressure, kPaPSat saturation vapor pressure, kPaP∗ relative pressure, pP−1

Sat

P(n)r prior probability of model r after n experiments

Q sum of squared normalized residualsr rate of consumption, mol g−1cat s

−1

R universal gas constant, J mol−1

or kPa cm3 mol−1 K−1

s2e experimental variance measured by performing re-peat experiments

SMR molar steam-to-methane ratio, dimensionlessT temperature, Kv total number of models proposed, dimensionlessVR reactor volume, mlw mass of catalyst, gx(n) matrix containing settings of the independent vari-

ables for model/experimentx(n+1) vector containing the settings of the independent

variables of model Yr(X; x) selected by discrimi-nation criterion after n observations

y′ value of an observed responseY (n) value of experimental observation made using

settings of the independent variables in x(n)

Y(n)r expected value of nth response model Yr(X; x(n))

Y(n)r predicted value of nth response ofmodel Yr(X; x(n))


Greek letters

* Asymptote in generalized isotherm, kPa−1

�REF Extent of the methane reforming reaction, mol�WGS extent of the water gas shift reaction, mol

�EXP experimental error, Y (n)− Y(n), di<erence between

the observed value (experimental observation) andthe expected value of the model

X vector of parameters for model estimated using nobservations

%i parameter in power law rate model, dimensionless%∗i generalized parameter following an Arrhenius tem-

perature relationship%1; i pre-exponential factor for generalized parameter %∗i%2; i activation energy for generalized parameter

%∗i ; J mol−1

�2r variance of model prediction Y(n)r

�2Y variance of observed response, Y (n), estimated us-ing s2e as measured by performing experimental re-peats

) surface coverage, dimensionless'2 Chi-squared value of cumulative distribution

Subscripts/superscripts

C center value(n) evaluated after n number of experimentsr model numberT totalv total number of modelsCH4 methaneH2O water/steamH2 hydrogenCO carbon monoxideCO2 carbon dioxide0 initial

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Al-Ubaid, A., & Wolf, E. (1987). Activity, FTIR studies and catalystcharacterization during methane steam reforming on Ni/Y–zeolitecatalysts. Applied Catalysis, 34, 119–134.

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Jarosch, K., & de Lasa, H. I. (1999). Novel Riser Simulator for methanereforming using high temperature membranes. Chemical EngineeringScience, 54(10), 1455–1460.

Jarosch, K., & de Lasa, H. (2001). Permeability, selectivity and testingof hydrogen di<usion membranes suitable for use in steam reforming.Industrial and Engineering Chemistry Research, 40(23), 5391–5397.

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