Journal of Chemical Engineering of Japan, Vol. 42, Supplement 1, pp. s79–s84, 2009 Research PaperJournal of Chemical Engineering of Japan, Vol. 42, Supplement 1, pp. s79–s84, 2009 Research PaperJournal of Chemical Engineering of Japan, Vol. 42, Supplement 1, pp. s79–s84, 2009 Research Paper

Discrimination of the Kinetic Models for Isomerization ofn-Butene to Isobutene

Tai-Shang CHEN and Jia-Ming CHERNDepartment of Chemical Engineering, Tatung University,40 Chungshan N. Rd., Sec. 3, Taipei, 10452, Taiwan, R.O.C.

Keywords: Butene, Isomerization, Kinetics, Model, Reaction Network

There has been a considerable interest for long periods in the production of isobutene from n-butene sinceit can be used with methanol to produce MTBE. In this study, the general rate equation method that is basedon the Bodenstein approximation and network reduction technique is used to discriminate the kinetic modelsfor isomerization of n-butene to isobutene. For illustrating the advantages of using the general rate equationmethod, two proposed mechanisms from literatures were used as examples to analyze the yield ratio and toidentify the correct mechanism. The experimental data published in literature were used as the test data inthis study. The mechanism with the byproduct polyisobutene produced from product isobutene and adsorbedisobutene is identified as reasonable mechanism for the isomerization reaction.

Introduction

Isobutene can be used in the reaction with methanolto produce MTBE (methyl tert-butyl ether), which havebecome a highly demanded material of gasoline (Byg-gningsbacka et al., 1999). There has been a consider-able interest in the production of isobutene from n-buteneover different catalysts (Choudhary and Doraiswamy,1971, 1975; Raghavan and Doraiswamy, 1977; Gayuboet al., 1997; Byggningsbacka et al., 1999). In the pastfifty years, there were some published literatures inves-tigated on the kinetic modeling of isomerization of n-butene to isobutene over catalysts or zeolites (Choudharyand Doraiswamy, 1975; Gayubo et al., 1997; Byggnings-backa et al., 1999). Though the reaction system seemsnot very complex apparently, the real reaction mechanismis somehow complicated due to the heterogeneous cat-alytic reaction. In recent years, continuous studies aboutthe dimerization of isobutene to produce diisobutenes orby further oligomerization to produce triisobutenes andtetraisobutenes have been also investigated by several re-searchers due to environmental concerns (Honkela andKrause, 2004; Ouni et al., 2006; Talwalkar et al., 2007).The Langmuir–Hinshelwood-type kinetic models werederived for the dimerization and trimerization reactionsto describe the reaction networks.

Traditionally, the power-law type or the Langmuir–Hinshelwood-type rate equations are usually adopted todescribe the kinetics of heterogeneous catalytic reactions.Choudhary and Doraiswamy (1975) studied the reactionover fluorinated η-alumina catalyst and compared one

Received on July 2, 2008; accepted on December 19, 2008.Correspondence concerning this article should be addressed toT.-S. Chen (E-mail address: tschen@ttu.edu.tw).Presented at ISCRE 20 in Kyoto, September, 2008.

power-law equation and nine Langmuir-Hinshelwood-type rate equations at different temperatures. However,there was no single suitable rate expression that canbe used at various temperatures. Byggningsbacka et al.(1999) investigated the skeletal isomerization of n-buteneto isobutene over ZSM-22 zeolite. They proposed threedifferent types of mechanisms and solved the system bynumerical method. No explicit rate equation was derivedin the paper. Most types of rate equations published in theliteratures are either empirical in nature or must assumesome rate-determining steps (LH mechanisms) and thuscannot be used confidently in scale-up design.

In this study, we focused on the kinetic modelsfor isomerization of n-butene to isobutene. A method-ology named general rate equation method that is basedon the Bodenstein approximation and network reductiontechnique is used to discriminate the reasonable kineticmodels (Chern and Helfferich, 1990; Chern, 2000; Chenand Chern, 2002a, 2002b). For illustrating the advan-tages of using the general rate equation method, two pro-posed mechanisms modified from the literature (Byggn-ingsbacka et al., 1999) were used as examples to ana-lyze the yield ratio and to identify the more suitable andreasonable mechanism. The experimental data from thepublished literature (Choudhary and Doraiswamy, 1975)were used as the test data in this study.

1. Method

1.1 Network reduction techniqueAs shown in the previous papers (Helfferich, 1989;

Chern and Helfferich, 1990; Chern, 2000; Helfferich,2001), the Bodenstein approximation of quasi-stationarybehavior of intermediates permits any simple network tobe reduced to one with only pseudo-single steps between

Copyright c⃝ 2009 The Society of Chemical Engineers, Japan s79

X0

X2

X1

Xn

…

r

Fig. 1 Single loop reaction network with arbitrary numberof intermediates (co-reactants and co-products notshown) (Chern, 2000)

adjacent nodes and between end members and adjacentnodes. Specifically, the net rate contribution of a multi-step, reversible simple network segment between adja-cent nodes X j and Xk

X j ↔ X j+1 ↔ · · · ↔ X j+n ↔ · · · ↔ Xk (1)

(co-reactants and co-products not shown) is

r j→k = 3 jk[X j ]−3k j [Xk] (2)

where the “segment coefficients” 3 are given by

3 jk =

k−1∏i= j

λi,i+1

D jkand 3k j

k−1∏i= j

λi+1,i

D jk(3)

with

D jk =k∑

i= j+1

i−1∏m= j+1

λm,m−1

k−1∏m=i

λm,m+1

(4)

(products∏ = 1 if lower index exceeds upper). The λ

coefficients are the pseudo-first order rate coefficients ofquasi-single molecular steps and are the products of theactual rate coefficients and the concentrations of any co-reactants of the respective steps. For example, for the stepX0 + A↔ X1 + B, λ01 = k01 [A] and λ10 = k10 [B].1.2 Single cycle system

Consider the single-cycle reaction network for ho-mogeneous catalytic reactions shown in Figure 1. Thesteady-state rate through the cycle can be expressed bythe following equations (Chern, 2000):

r =

( n∏i=0

λi,i+1 −n∏

i=0λi+1,i

)[XT]

n∑i=0

Di i

(5)

(taking index n + 1 = 0 for conveniences)

where the generalized rate coefficient λi,i+1 is the prod-uct of the forward rate constant, ki,i+1 and co-reactant

Xj

Xi

Xk

Xj

Xi

Fig. 2 Reduce the multi-pathway reaction network to equiv-alent single cycle reaction system

concentration, and Di i can be obtained from a square ma-trix of order n by n by tearing the closed network into alinear one with the intermediate Xi on both ends. For ex-ample, the D00 in a 4-intermediate cycle X0–X1–X2–X3–X0 is λ12λ23λ34 + λ10λ23λ34 + λ10λ21λ34 + λ10λ21λ32.

If there exist an arbitrary number of parallel paths inbetween intermediates Xi and X j , the λ coefficient couldbe replaced by the loop coefficients L (Chern, 1988):

L i j =m∑

k=1

3(k)i j and L j i =

m∑k=1

3(k)j i (6)

where m is the number of parallel paths.1.3 Reduction of multi-pathway to single cycle

Many enzymatic or catalytic systems contain mul-tiple sub-reactions and the reaction steps can be con-structed as cyclic networks with multi pathways in topol-ogy. Consider a large cyclic reaction network with two in-dependent parallel pathways, as shown in Figure 2. Theparallel pathways between the two node-intermediatescan be lumped together and the multi-pathway systemcan be reduced to an equivalent pseudo single cycle re-action network (Chen and Chern, 2002a).

2. Results and Discussion

The overall isomerization reaction of n-butene toisobutene over catalysts or zeolites can be described as

Acat.←→ B −→ P (7)

where A, B, and P represent n-butene, isobutene and by-product polyisobutene, respectively. After verifying andanalyzing many mechanisms, we proposed two possiblemechanisms with reaction steps listed in Table 1 and re-action network shown in Figures 3(a) and (b). The nota-tion X0 means the fresh or regenerated catalyst site andX1 to Xn mean the adsorbed reactants or product on cata-lyst sites in different forms. The direction of arrow meansthe addition of reactants or desorption of products fromcatalyst sites. It can be seen in Figure 3(a), the byproductpolyisobutene (P) is produced from a pathway with theaddition of n-butene (A) to intermediate X2, (X–B). Inmechanism Figure 3(b), polyisobutene is produced froma pathway with the addition of isobutene (A) to interme-

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Table 1 The reaction steps in two proposed mechanisms ofn-butene (n-C4) to isobutene (i-C4) isomerization

Reaction step Mechanism (a) Mechanism (b)Adsorption of n-C4 S + A⇔ SA S + A⇔ SASurface reaction of SA⇔ SB SA⇔ SBn-C4 to i-C4Desorption of i-C4 SB⇔ S + B SB⇔ S + BAdsorption of n-C4 SB + A⇔ SP —Adsorption of i-C4 — SB + B⇔ SPDesorption of polyisobutene SP⇔ S + P SP⇔ S + P

B

P

X0

X2

X3

A

A

X1

(1)(2)

B

P

X0

X2

X3

B

(1)

(2)

A

X1

Fig. 3 Two proposed mechanisms for the isomerization re-action of n-butene to isobutene

diate X2 Note that the reaction pathway (2), X2–X3–X0,is assumed to be irreversible for both mechanisms.

The systems shown in Figures 3(a) and (b) are clas-sified as multi-cycle or multi-pathway reaction systems(Chen and Chern, 2002b). The left two reaction path-ways, (1) and (2), in both Figures 3(a) and (b) can belumped first by Eq. (6), and then the systems can betreated as single cycle systems. It is needless to assumeany step is a rate-determined step. Taking mechanismshown in Figure 3(a) for example, the reaction rate canbe derived by the method (Chen and Chern, 2002a). Thecyclic reaction rate can be written as

r = (k01k12L24 PA − L42k21k10)[XT]∑Di i

(8)

rB/rP, Experimental [ - ]

0 20 40 60 80 100 120 140 160 180 200

rB

/ rP

, P

red

icte

d [

- ]

0

20

40

60

80

100

120

140

160

180

200

300 oC, r2 = 0.9207

335 oC, r2 = 0.8240

365 oC, r2 = 0.8934

400 oC, r2 = 0.8436

435 oC, r2 = 0.8745

Fig. 4 Parity plot for the yield ratio derived for mechanismin Figure 3(a)

where

L24 = k24 + k23 PA

L42 = k42 PB∑Di i = D00 + D11 + D22 + k23

k34D22 (9)

D00 = (k12 + k10)(k24 + k23 PA)+ k10k21

D11 = (k24 + k23 PA)k01 PA + k21k01 PA + k21k32 PB

D22 = k01k12 PA + (k12 + k10)k32 PB

Note that the index 4 is also expressed as 0 for con-veniences. The individual reaction rates of product B andP are

rB = (k24 D22 − k42 PB D00)[XT]∑Di i

(10)

rP = (k23 PA D22)[XT]∑Di i

(11)

The cyclic reaction rate for the mechanism shown in Fig-ure 3(b) can be derived also by the same procedure andexpressed as

r = (k01k12L24 PA − L42k21k10)[XT]∑Di i

(12)

where

L24 = k24 + k23 PBL42 = k42 PBD00 = (k12 + k10)(k24 + k23 PB)+ k10k21

(13)

Note the Di i terms in Eqs. (8) and (12) are different be-cause of the different reactants in the pathways from X2to X3 in the two mechanisms.

Although the Di i terms in Eqs. (8) and (12) for bothmechanisms are complicated to be expressed, we can dis-criminate the kinetic models without the Di i terms. Forthe purpose of identifying which mechanism is correct,

VOL. 42 Supplement 1 2009 s81

rB/rP, Experimental [ - ]

0 25 50 75 100 125 150 175 200

rB

/ rP

, P

redic

ted [ -

]

0

25

50

75

100

125

150

175

200

300 oC, r2 = 0.9982

335 oC, r2 = 0.9886

365 oC, r2 = 0.9996

400 oC, r2 = 0.9974

435 oC, r2 = 0.9986

Fig. 5 Parity plot for the yield ratio derived for mechanismin Figure 3(b)

the yield ratio rB/rP for the two systems can be employedby dividing Eq. (10) to Eq. (11) as

rB/rP = k24 D22 − k42 PB D00

k23 PA D22(14)

After inserting both D22 and D00 into Eq. (14), we can re-arrange the equations, lump all rate coefficients ki j , andexpress the yield ratio rB/rP as a simpler form. For mech-anism in Figure 3(a) with two constants,

rB/rP = kb (1− PB/K PA)− ka PB

PA + ka PB(15)

where

ka = (k12 + k10)k42

k01k12

kb = k24

k23

(16)

For mechanism in Figure 3(b),

rB/rP = r1 − r2

r2

= kb (PA/PB − 1/K )− ka PB

PA + ka PB− 1

(17)

where ka and kb are the lumped constants with the sameforms as Eq. (16) and K is the equilibrium constant for theisomerization reaction n-butene to isobutene. The valueof K can be evaluated from thermodynamic propertiesand the values for the lumped constants can be deter-mined by nonlinear regression. We used a commercialsoftware Sigma Plot ver 9.0 to perform the nonlinear re-gression and parameter fitting. Both yield ratio equationswere employed to fit the data published by Choudharyand Doraiswamy (1975) at five different reaction temper-atures from 300 to 435◦C. The results of comparing theexperimental and the predicted yield ratios and r -square

Table 2 The fit results of kinetic parame-ters of mechanism (b) for all reac-tion temperatures

Parameter Unit Valueka0 — 1.934 × 102

kb0 — 1.227 × 107

Ea kJ/mol −84.83Eb kJ/mol −15.84

rB/rP, Experimental [ - ]

0 25 50 75 100 125 150 175 200

rB

/ rP

, P

red

icte

d [ -

]

0

25

50

75

100

125

150

175

200

300 oC

335 oC

365 oC

400 oC

435 oC

Fig. 6 Parity plot for the yield ratio derived for mechanism(b) and the kinetic parameters for all temperatures

values are shown in Figures 4 and 5. The r-square val-ues for mechanism in Figure 3(a) range from 0.8240 to0.9207 and those for mechanism in Figure 3(b) rangefrom 0.9886 to 0.9996. From statistical viewpoint, mech-anism (b) is more reasonable for the isomerization reac-tion.

The results shown in Figure 5 were evaluated atfive different temperatures simultaneously. Since we haveidentified mechanism in Figure 3(b) is more reasonable,we can express the two kinetic parameters ka and kb inEq. (17) as the Arrenhius forms,

ka = ka0e−Ea/RT (18)

kb = kb0e−Eb/RT (19)

Then the yield ratio equation was fitted to all the 30 ex-perimental runs with reaction temperatures from 300 to435◦C. The results were listed in Table 2 and the r-squarevalue for the nonlinear regression is 0.972. Note thatthe four parameters in Table 2 are lumped constants thatgroup the six rate constants together as shown in Eq. (16).Therefore, the apparent activation energy for ka gives lit-tle information about the individual steps. However, thenegative value of Eb shown in Table 2, that is the appar-ent activation energy for kb suggests that the isobutene

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Fig. 7 Comparison of the relationships between the partialpressures of n-butene and isobutene with the yieldratio

A

B

P X0

X2

X3

B

X1

(1)(2)

X4

Xn

…

Xn+1B

B

Fig. 8 A revised mechanism in Figure 3(b) for multi-stepisobutene addition to produce polyisobutene

desorption step has a lower energy barrier than the ad-sorption of isobutene onto the adsorbed site:

kb = k24

k23= k240e−E24/RT

k230e−E23/RT (20)

= k240

k230e−(E24−E23)/RT = kb0e−Eb/RT

for Eb < 0, E24 < E23. On the other hand, the apparent ac-tivation energy for ka gives little information about the in-dividual steps because of more complicated relationshipshown in Eq. (16). The parity plot for all the experimentsis shown in Figure 6. Figure 6 shows a reasonable agree-ment between the experimental data and the model pre-diction. The slightly higher deviation as shown in Figure6 was due to the significantly error (r-square = 0.9886as shown in Figure 5) in the experiment for temperature335◦C.

Equations (17)–(19) can be used to analyze the char-acteristics of the isomerization reaction. The yield ratiorB /rP is the function of PA and PB . Figure 7 shows the

Table 3 The fit results of number ofmonomers in polyisobutene fordifferent temperatures of mecha-nism (b)

Temperature [◦C] n r -square300 7 0.9980365 37 0.9784400 49 0.9539435 60 0.9692

comparison of the relationships between the partial pres-sures of n-butene and isobutene and the yield ratio. It canbe seen in Figure 7 that the yield ratio increases sharplywhen the partial pressure of n-butene is greater than 0.8atm. This information is useful to help choose the properoperational parameters for reactor operation conditions.

Though we assumed the pathway for polyisobuteneproduction involves only two steps in Figure 3, the yieldratio equation for multi-step isobutene addition can betreated by the same way with the general rate equationmethod. Figure 8 shows a revised scheme for mechanism(b). The product isobutene, (B), would add into reactionpathway (2) in Figure 8 step by step. Finally the product–(isobutene)n–, (P), was desorbed from catalyst.

All steps in the pathway (2) of Figure 8 are also as-sumed to be irreversible. The reaction rate for the path-way involving arbitrary number of irreversible reactionsteps can be treated by the same procedures describedpreviously. The rate of both pathways in Figure 8 can beexpressed as,

r1 = (k2,n+2 D22 − kn+2,2 D00)[XT]

D00 + D11 + D22 +n+1∑k=3

(32k/3k,n+2)

(21)

r2 = k23 PB D22[XT]

D00 + D11 + D22 +n+1∑k=3

(32k/3k,n+2)

(22)

Note that the index 0 in X0 can also be expressed as n+2for convenience. The yield ratio can be expressed by asimilar form as Eq. (17),

rB/rP = r1 − (n − 1)r2

r2(23)

= kb (PA/PB − 1/K )− ka PB

PA + ka PB− (n − 1)

where n is the number of isobutene for constructing poly-isobutene. Equation (23) was used to fit the same data atfive different reaction temperatures from 300 to 435◦Cfor getting ka , kb and n. The fit results are shown in Ta-ble 3. The result of 330◦C is not shown due to lower r-square value (r2 = 0.93) compared with others. Since thedegree of polymerization will be affected by the tempera-ture, it can be seen in Table 3 the extent of polymerization

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of isobutene increases while the reaction temperature in-creases. The explicit reaction rate for the isomerizationreaction can also be easily obtained by using the method-ology to calculate the Di i terms.

Choudhary and Doraiswamy (1975) and Ragha-van and Doraiswamy (1977) proposed nine differentLangmuir–Hinshelwood-type or empirical kinetic mod-els for the system and concluded that different modelsshould be used at different temperatures. The reactionsfor multi-step isobutene addition were ignored in theirstudies. Though different model equations were able to fittheir kinetic data at different temperatures with satisfac-tory accuracy, there was no clear connection between thereaction mechanism and the rate equations they used. Incomparison with their studies, the kinetic model and rateequations proposed in this study provide a better expla-nation of the effect of temperature. Moreover, the extentsof multi-step isobutene addition could be easily assessed.The reaction kinetics of oligomerization of isobutenes(Honkela and Krause, 2004; Ouni et al. 2006; Talwalkaret al. 2007) can also be analyzed by the general rate equa-tion method if the reaction data are available.

Conclusions

In this study the general rate equation method wassystematically applied as a tool to discriminate two pro-posed kinetic models using the product yield ratio. Mech-anism in Figure 3(b) showing polyisobutene produc-ing from the reaction of isobutene with the adsorbedisobutene is more reasonable for the isomerization re-action. Using the general rate equation approach, no as-sumption of equilibrium steps, rate-determining steps, ormost abundant surface species is required. The explicitreaction rate for the isomerization reaction can also beeasily obtained by using the methodology we proposed.The extent of polymerization for different temperaturescan be evaluated by the model proposed in this study.

AcknowledgmentsThe financial support from Tatung University is gratefully ac-

knowledged.

NomenclatureA = n-butene [—]B = isobutene [—]Di j = denominator of rate equation of segment j → k de-

fined in Eq. (4)Ea = lumped activation energy [kJ/mol]K = equilibrium constant [—]

ka = lumped rate coefficients [—]kb = lumped rate coefficients [—]ki j = rate constant of step i → j for path pL i j = loop coefficient of segment i → j defined in Eq. (6)P = polyisobutene [—]PA = pressure of component [atm]R = gas constant [J/mol K]r = reaction rate [mol/L s]X j = reaction species j or intermediate [mol/L]XT = total catalyst concentration [mol/kg cat]

3 jk = segment coefficient of segment j → k for path de-fined in Eq. (3)

λ jk = pseudo first order rate coefficient of step Xi to X j

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