Liquid-phase synthesis of butyl tert-butyl ether catalysed by ion-exchange resins: kinetic modelling through in-depth model discrimination

Abstract

The kinetics of the butyl tert-butyl ether (BTBE) synthesis reaction over Amberlyst™ 35 as the catalyst has been studied at 303–356 K in the liquid phase in two different reactor systems: batch and fixed-bed. Internal mass transfer effects were detected at temperatures above 333 K for catalyst particles larger than 0.25 mm. Particles smaller than 0.08 mm did not show mass transfer limitations under the whole assayed temperature range. The best kinetic model has been searched among a large number of kinetic equations resulting from the systematic combination of all possible elementary reactions, adsorbed species, and rate-determining step based, according to the Langmuir–Hinshelwood–Hougen–Watson and the Eley–Rideal formalisms. The significance of the temperature effect on the kinetic parameters and of the effect of the interaction between the catalyst and the reaction medium on the reaction rate has been checked. All proposed kinetic equations have been fitted to experimental rate data free from mass transfer limitations. The model discrimination procedure has been based on mathematical and physicochemical criteria. The resulting kinetic model is consistent with an Eley–Rideal type mechanism where one 1-butanol molecule adsorbs on one active site of the catalyst, it reacts with one isobutene molecule from the liquid phase to give one adsorbed BTBE molecule, which finally desorbs. The rate-determining step is the surface reaction. The catalyst activity is affected by the resin-medium interaction. 1-Butanol adsorption on the catalyst is more exothermic than BTBE adsorption, and isobutene adsorption is negligible.

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Introduction

Reaction between isobutene (IB) and 1-butanol (BuOH) produces butyl tert-butyl ether (BTBE) and can be catalysed by acidic ion-exchange resins. BTBE is an interesting alternative to methyl and ethyl tert-butyl ethers (MTBE and ETBE, respectively) to be used as oxygenated high-octane component in current gasoline formulations. Since 1-butanol can be obtained through fermentation of non-edible biomass,^1,2 it would allow a reduction of the fossil fuel use and dependence in gasoline production. BTBE can be produced at industrial scale in the same reaction units than MTBE or ETBE, in contrast to the next generation biofuels (stemmed from lignocellulose, non-food materials, algal biomass, and energy crops grown on marginal lands) that these days are receiving more attention, but still being under development.^3,4 Thus, BTBE is a feasible option to find cleaner alternatives to traditional automotive fuels in the short- and midterm.

In the course of the addition of IB to BuOH to form BTBE, two side reactions can take place simultaneously depending on the operating conditions (Fig. 1): isobutene hydration to form tert-butyl alcohol (TBA), and isobutene dimerisation to form 2,4,4-trimethyl-1-pentene (TMP1) and 2,4,4-trimethyl-2-pentene (TMP2). Few literature references focused on BTBE synthesis can be found. The most significant works are devoted to the reaction thermodynamics and to the reactivity of primary alcohols with isobutene.^5–9 No data have been found concerning the BTBE etherification kinetics, which is crucial for the potential production of BTBE in industrial units. Due to the reaction similarities, BTBE synthesis is expected to proceed through a mechanism similar to that of MTBE or ETBE and, therefore, reaction rates could be explained by means of an analogous kinetic model.

Fig. 1 Reaction scheme.

MTBE and ETBE etherifications have been studied extensively throughout the years.^10–16 More recently, the production of propyl tert-butyl ether (PTBE), the next ether in the analogous series, obtained by addition of 1-propanol to isobutene, has also been investigated.¹⁷ According to the literature, all these etherification reactions are reversible and exothermic, and the olefin–alcohol–ether mixtures behave strongly non-ideally. The reported reaction rate expressions are basically derived from Langmuir–Hinshelwood–Hougen–Watson (LHHW) or Eley–Rideal (ER) formalisms. It is expected that these reactions proceed through a similar mechanism. However, there are some discrepancies among the published works concerning mainly the number of active sites involved in the rate-determining step and the compounds that actually adsorb significantly on the resin. Possible reasons of such discrepancies are to have tested a limited number of candidate models, to have chosen a model among others with a similar goodness of fit, to have included non significant effects in the kinetic equation, or to have excluded significant effects. The present in-depth kinetic model discrimination study is motivated to assure that all plausible models are considered in the discrimination procedure, and that all parameters included in the proposed kinetic equation are significant.

Building the kinetic models

The candidate models of the BTBE synthesis have been developed from the LHHW and ER formalisms, because it is a heterogeneously catalysed system. The form of a given kinetic equation is characterised by a set of compounds that adsorb on the catalyst, a set of elementary reactions, and the reaction step being the rate-determining step. Following the approach of our previous works,^17,18 the search of the best kinetic model starts with the proposal of all kinetic equations to be tested. They have been obtained from the systematic combination of all possible rate-determining steps, adsorbed and non-adsorbed species on the catalyst, and significant or non-significant temperature dependence of every parameter, as well as the possible inclusion of a term accounting for the effect on rates of the interaction between the resin and the liquid mixture. The aim of this procedure is to avoid dismissing the true model (i.e., the one that most accurately describes the physicochemical reality of the etherification reaction) from the set of candidate models. The BTBE reacting mixture is assumed to be highly non-ideal, similarly to the analogous etherification systems. Therefore, activities were used instead of concentrations and estimated by means of the UNIFAC-Dortmund method, which is an enhancement of the widely known UNIFAC method.¹⁹

All proposed equations match the same general expression (eqn (1)). Models differ in the developed form of each term in the kinetic equation (Table 1).

As seen in Table 1, the kinetic term is k′, an apparent kinetic coefficient, with the temperature dependence shown in eqn (2), since it consists of a product of the kinetic constant of the rate determining step (assumed to follow the Arrhenius law), adsorption equilibrium constants and the chemical equilibrium constant (both assumed to follow the van't Hoff equation). The particular product depends on the considered mechanism. Parameters and are the parameters to be fitted. The mean experimental temperature, T_m, is included to reduce the correlation between both parameters.

Table 1 Alternative forms of the kinetic equation terms (eqn (1)). a stands for activity; b are binary-type variables (allowed values: 1 or 0), used to show generalised equation expressions; subscripts indices j and i refer to the involved compounds (BuOH, BTBE and IB). When the reaction rate-determining step is the adsorption or desorption of a given compound, its activity in eqn (7) or (8), if included, should be replaced by: a_IB → a_BTBE/a_BuOH, a_BuOH → a_BTBE/a_IB, or a_BTBE → a_IBa_BuOH

Term	Form	Observations	Equation
{Kinetic term}		Arrhenius-type temperature dependence	(2)
{Driving force}		Rate-determining step: surface reaction	(3)
		Rate-determining step: BuOH desorption	(4)
		Rate-determining step: IB adsorption	(5)
		Rate-determining step: BTBE desorption	(6)
{Adsorption term}		j = adsorbed compound	(7)
		i, j = adsorbed compounds (derived from eqn (7) when the fraction of unoccupied sites is not significant)	(8)
{Resin–medium interaction}		b P is equal to 1 (interaction effect) or 0 (no interaction)	(9)

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Regarding the driving force term, four different alternatives have been considered, given the considered rate-determining step for the global reaction process, i.e., surface reaction (eqn (3)), 1-butanol adsorption (eqn (4)), isobutene adsorption (eqn (5)), and BTBE desorption (eqn (6)). The parameter K_eq in eqn (3) to (6) stands for the chemical equilibrium constant for the synthesis of BTBE, whose value had been determined experimentally in a previous work:⁷

In the adsorption term, two different alternatives have been considered: whether the fraction of unoccupied active sites in the catalyst surface affects the reaction rate (eqn (7)) or the number of vacant active sites is non-significant (eqn (8), derived from eqn (7) by removing the unity summand, a required mathematical issue to avoid overparameterisation in the model fit). Parameters K_j in eqn (7) correspond to the actual adsorption equilibrium constant of each species j (K_j = K_a,j, except when the rate-determining step is a compound adsorption–desorption; then, for that compound, K_IB = K_a,IB/K_eq, or K_BuOH = K_a,BuOH/K_eq, or K_BTBE = K_a,BTBE·K_eq). In eqn (8), parameters K_j are quotients of adsorption equilibrium constants (K_j = K_a,j/K_a,i, except when the rate-determining step is a compound adsorption–desorption; then, for that compound, K_IB = K_a,IB/(K_a,i·K_eq), or K_BuOH = K_a,BuOH/(K_a,i·K_eq), or K_BTBE = K_a,BTBE·K_eq/K_a,i). For the sake of clarity, Table 2 lists all alternative forms of the adsorption term.

Table 2 Alternative forms of the adsorption term in a LHHW or ER kinetic model for the BTBE synthesis. When the reaction rate-determining step is the adsorption or desorption of a given compound, its activity should be replaced by: a_IB → a_BTBE/a_BuOH, a_BuOH → a_BTBE/a_IB, or a_BTBE → a_IBa_BuOH

No.	Adsorption term	No.	Adsorption term
1	1	8	1 + KBuOHaBuOH
2	a BuOH	9	1 + KBTBEaBTBE
3	a IB	10	1 + KIBaIB
4	a BuOH + KBTBEaBTBE	11	1 + KBuOHaBuOH + KBTBEaBTBE
5	a BuOH + KIBaIB	12	1 + KBuOHaBuOH + KIBaIB
6	a BTBE + KIBaIB	13	1 + KBTBEaBTBE + KIBaIB
7	a BuOH + KBTBEaBTBE + KIBaIB	14	1 + KBuOHaBuOH + KBTBEaBTBE + KIBaIB

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Since K_j parameters are adsorption equilibrium constants or a quotient thereof, their temperature dependence has been expressed according to the van't Hoff equation, as follows:

In case that the temperature dependence of K_j is not significant, the parameter K_T,j should be taken as equal to zero and, thus, K_1,j is the only parameter to be fitted as the K_j estimate.

The exponent n in the adsorption term is related to the number of active sites, or clusters of active sites, involved in the reaction mechanism. Values of n of 1, 2, and 3 are the most likely, according to previous kinetic studies on similar reaction systems.^{11–13,18,20–22}

The resin–medium interaction term (eqn (9)) accounts for the effect of the reaction medium on the catalyst activity promoted by the difference between the solubility parameters of the reaction medium, δ_M, and the catalyst, δ_P, as observed in analogous reaction systems where the Hildebrand solubility parameter was used.^15,23 The binary parameter b_P is equal to 0 if the effect of the interaction between the reaction medium and the catalyst is not significant, or equal to 1 if the interaction effect is significant; then, δ_P could be constant (k_TP = 0) or linearly temperature dependent (k_TP ≠ 0):^15,23

δ_P = k_P1 + k_PT(T − T_m) (12)

where both k_P1 and k_PT are the fitting parameters, and the mean temperature T_m is included to reduce the correlation between both parameters. The remaining variables in eqn (9) are the molar volume of the liquid mixture V̄_M, estimated from the species concentration and the temperature,^24–26 the catalyst porosity in the swollen-state ϕ_P, the Hildebrand solubility parameter of the liquid mixture δ_M, estimated as described in the literature,²⁶ and the gas constant R.

As a result of all possible combinations of the proposed forms of the general kinetic expression terms, a total of 1404 different kinetic equations are obtained, to be fitted individually to the experimental data with the aim of obtaining the best kinetic model.

Experimental section

Materials

Reactants. 1-Butanol (BuOH, >99.8% GC, Sigma-Aldrich), and isobutene (IB) as pure isobutene (>99.9% GC; Air Liquide) or in a synthetic C4 mixture (25% wt isobutene, 40% wt isobutane, 35% wt trans-2-butene; Abelló-Linde). Safety & hazards: all compounds are flammable, and experiments have been carried out under pressure.

Chemical standards used for chromatographic analysis. 2-Methyl-2-propanol (TBA, >99.7% GC, Panreac), 2,4,4-trimethyl-1-pentene (TMP1, >98% GC, Sigma-Aldrich), 2,4,4-trimethyl-2-pentene (TMP2, >98% GC, Sigma-Aldrich). 1-tert-butoxybutane (butyl tert-butyl ether, BTBE, >98% GC) was synthesized and purified in our laboratory.

Catalyst. Amberlyst™ 35 (A35, The Dow Chemical Company, now DuPont), a sulfonic macroreticular styrene–divinylbenzene resin. Its physical properties can be found elsewhere.¹⁷

Apparatus, procedure, and analysis

Experimental runs were carried out in two different reactor setups. Most of the experiments were run in a batch stirred tank reactor, the rest in a continuously operated fixed-bed catalytic reactor. The purpose of the experiments in the fixed-bed reactor was to validate the results in the batch reactor. The initial reaction mixture in batch experiments and the reactor feed in continuous experiments did not contain BTBE. The batch reactor experiments were carried out at 2.0 MPa and at constant temperature, in the range of 318 to 356 K. Initial alcohol to isobutene molar ratio, , varied between 1.0 and 2.0, and pure isobutene was used as reactant. The catalytic fixed-bed reactor experiments were carried out at 1.5 MPa and at constant temperature, between 303 and 333 K. The isobutene source was either pure isobutene or the synthetic C₄ mixture, with an equimolar alcohol–isobutene mixture at the reactor inlet ( = 1.0).

GC analyses of samples of the reaction medium in both setups allowed quantifying the reactants and products concentrations. Each GC was equipped with a capillary column (HP-PONA 19091S-001, J&W Scientific, Santa Clara, US; 100% dimethylpolysiloxane, 50 m × 0.20 mm × 0.50 μm), helium was used as the carrier gas, and the oven temperature was set at 333 K.

A detailed description of the experimental setup and procedure can be found elsewhere.¹⁷ Further details on the calculation of experimental reaction rates are shown in section A of the ESI,† and a list of all experimental conditions and calculated reaction rates in section B.

Results and discussion

Experimental results

In order to focus the present study on the BTBE synthesis kinetics, the experimental conditions were chosen to minimize the side reactions extension, based on previous works on similar systems.^27,28 Since the amount of byproducts at the end of the experiments was always below 5% wt, they have been further dismissed from the kinetic analysis. Other side reactions, such as 1-butanol dehydration or further isobutene oligomerisation, were not detected. Obtaining a kinetic model from experimental data requires that the calculated reaction rates are intrinsic kinetics, that is, free from mass and heat transfer effects. The consequence of such effects is to increase the resistance of the global reaction process, so the measured reaction rate would be lower than the reaction rate in the absence of transfer effects. In heterogeneous catalysis, external transfer effects can take place between the fluid bulk phase and the catalyst surface, so they can be avoided by sufficiently reducing the thickness of the film surrounding the catalyst: by increasing the stirring speed in a batch reactor, or by increasing the fluid flow rate in a fixed-bed reactor. In the present reaction system, external mass transfer effects are avoided with stirring speeds of 750 rpm in the batch reactor, and flow rates of 0.031 g s⁻¹ in the fixed-bed setup, as determined in previous works.^21,29,30

Internal mass transfer effects occur inside the catalyst. They become more noticeable at larger catalyst bead size and at higher reaction temperature. Assuming that there are not external transfer effects, internal mass transfer effects can be easily checked by plotting the logarithm of measured reaction rates obtained at the same composition vs. the temperature inverse (Fig. 2). If a straight line is obtained, internal mass transfer effects are negligible. For this purpose, a set of experiments was carried out at different temperatures in both reactor systems, i.e., batch and fixed-bed, with the same reactants composition ( = 1.0) in the absence of product, and with 0.25–0.40 mm catalyst bead size (solid symbols in Fig. 2). Experimental points deviate at temperature higher than 333 K (1/T < 3.0 × 10⁻³ K⁻¹ in Fig. 2), what indicates a significant effect of mass transfer resistances above 333 K. Additional experiments at the same composition and at about 343 K and 353 K were carried out using smaller catalyst particles (<0.08 mm, open symbols in Fig. 2), which resulted well aligned with those obtained at lower temperatures, thus free of mass transfer resistance. In the further kinetic analysis, reaction rates affected by transfer effects were dismissed. From the slope of the solid straight line in Fig. 2, the apparent activation energy, E_ap, for the BTBE formation reaction has been estimated as (67 ± 2) kJ mol⁻¹. This value is in the same range of those quoted in the literature for similar reaction systems using A35.^{14,17,18,22,31,32}

Fig. 2 Arrhenius plot of BTBE formation rates with = 1.0 and pure isobutene. Solid symbols: catalyst bead size 0.25–0.40 mm, batch reactor (●) and fixed-bed reactor (▲). Open symbols: catalyst bead size < 0.08 mm, batch reactor (○).

Fitting kinetic models to experimental rate data

The first step in the search of the kinetic model was to fit each of the 1404 built equations to all experimental reaction rates free from mass transfer limitations at once by optimisation of the parameter values. The Levenberg–Marquardt algorithm was used to minimize the residual sum of squares (RSS), defined as:where r_exp is the experimental reaction rate, r_calc is the calculated value, and N is the number of experimental data (N = 136). Weighted residual sum of squares, with different weighting factors, were also tested, resulting in similar model ranking, distribution of residuals and parameter values. Therefore, further analysis refers to RSS.

The discrimination of kinetic models has been carried out by applying the following mathematical and physicochemical criteria to conservatively discard inadequate models:

1. Fitted kinetic equations presenting large RSS values do not provide a satisfactory description of all experimental kinetic data. Fig. 3 shows the obtained RSS, where kinetic equations are ordered from lowest to highest RSS values. Consequently, equations presenting RSS values larger than 100 have been rejected.

Fig. 3 RSS values of fitted kinetic equations.

2. In a suitable kinetic expression, all fitted parameter values should be statistically significant. Their standard error can be estimated from the covariance matrix of the parameters at the optimum. In this regard, models with at least one parameter with an associated standard uncertainty as large as the fitted value have been discarded.

3. Only kinetic equations producing positive values for the apparent activation energy, E_ap, can be accepted, because of the experimental evidence that the reaction rate increases with temperature (Fig. 2), so models with positive fitted values for (eqn (2)) are directly discarded. In addition, positive E_ap values clearly inconsistent with previously reported E_ap for similar reaction systems (in the range of 69.3 to 84 kJ mol⁻¹ for MTBE, ETBE, PTBE, and TAEE syntheses over A35)^{14,17,18,22,31,32} can also be rejected. In particular, E_ap values well above 100 kJ mol⁻¹ (i.e., < −13 000 K⁻¹) or below 45 kJ mol⁻¹ (i.e., > −5000 K⁻¹) have been rejected.

4. The adsorption process of a given compound on the resin is an exothermic process ( < 0). For models with a significant number of unoccupied active sites (eqn (7)), K_j is the adsorption equilibrium constant of compound j, and K_T,j (eqn (11)) corresponds to , thus it should be positive. Models whose range of any K_T,j and its uncertainty clearly falls in negative values have been discarded. Models where all active sites are considered as occupied (eqn (8)) are not affected by this condition, because K_j is defined as a ratio of adsorption equilibrium constants and, therefore, K_T,j = −()/R, so it can be either positive or negative.

5. From its definition, the resin solubility parameter δ_P must be positive. In models where δ_P is temperature dependent, and it is expected to decrease with temperature,²³ so the range of k_PT and its uncertainty cannot fall completely in positive values. Models that do not meet these conditions have been discarded.

6. The Akaike information criterion (AIC), i.e., the relative likelihood of every model, can be used to rank a number of S kinetic equations from more to less plausible upon the basis of robust multimodel inferences.^33,34 The following expressions apply in the Akaike procedure:

Δ_i = AICc_i − AICc_min (16)

where m is the number of experimental points, k is the number of parameters in the fitted equation, AICc is the bias-corrected reduced AIC for situations with low m/k values (i.e., m/k < 40, like in the present case), Δ_i is an estimator of the level of empirical support for a given model (the lowest Δ_i value corresponds to the most plausible candidate model), AICc_min is the minimum AICc among all models, and w_i accounts for the relative likelihood of model i out of the S candidate models.^33,34 Lower Δ_i and higher w_i values are an indication of a better model.

The consideration of the mathematical and physicochemical criteria applied in a conservative way allowed to reduce the number of candidate equations from 1404 to only a few. The best ranked models, their optimal parameter values and ranking criteria (RSS, Δ_i, and w_i) are listed in Table 3, sorted from best to worst.

Table 3 Parameter values for the best fitting BTBE kinetic equations. A “—” sign indicates that the related effect is not included in the model

Model no.	k′ (mol g^−1 h^−1)		{Driving force}^a	{Adsorption term}							n	δ P (MPa^1/2)		RSS	Δ i	w i
				1st ads^b	K 1,BuOH	K T,BuOH	K 1,IB	K T,IB	K 1,BTBE	K T,BTBE		k P1	k PT
a Form of the driving force: (a) surface reaction (eqn (3)) and (c) isobutene adsorption (eqn (5)). b First summand of the adsorption term.
49	0.320	−9171	(a)	a BuOH	—	—	—	—	−1.119	−4039	1	24.51	—	6.497	0	0.937
48	0.395	−9031	(a)	a BuOH	—	—	—	—	−0.540	—	1	24.18	—	6.981	8	0.021
290	1.917	−8986	(a)	1	−0.334	—	—	—	−0.810	—	3	24.61	—	6.948	9	0.010
173	2.072	−8994	(a)	1	0.332	—	—	—	−0.144	—	2	24.53	—	6.954	9	0.009
56	3.408	−9019	(a)	1	2.951	—	—	—	2.491	—	1	24.32	—	6.977	10	0.007
174	2.230	−8498	(a)	1	0.495	530	—	—	−0.096	—	2	24.50	—	6.920	11	0.004
978	0.412	−9033	(c)	1	—	—	—	—	−1.178	—	3	24.06	—	7.155	11	0.004
291	1.975	−8707	(a)	1	−0.272	308	—	—	−0.804	—	3	24.60	—	6.933	11	0.004
861	0.428	−9042	(c)	1	—	—	—	—	−0.761	—	2	23.94	—	7.185	11	0.003
166	0.088	−9318	(a)	a BuOH	—	—	—	—	−1.655	−3039	2	22.56	—	7.327	16	<0.001
751	0.121	−9346	(c)	a BuOH	—	—	—	—	−0.981	−3650	1	22.24	—	7.429	18	<0.001
165	0.144	−9196	(a)	a BuOH	—	—	—	—	−1.207	—	2	22.26	—	7.749	22	<0.001
750	0.182	−9218	(c)	a BuOH	—	—	—	—	−0.451	—	1	21.89	—	7.888	24	<0.001
2	1.170	−9770	(a)	1	—	—	—	—	—	—	1	20.06	—	9.566	48	<10^−10

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A model is considered as substantially supported by empirical evidence when its Δ_i value is lower than 3.^33,34 The first ranked model (Model 49) stands out as the most plausible one, because of its high probability (w_Model49 = 93.7%) and Δ_Model49 = 0 value, far from the second ranked model (w_Model48 = 2.1%, Δ_Model48 = 8). In addition, some common features are observed among the first ranked models that support the choice of Model 49: i) none of the equations includes the isobutene contribution in the adsorption term (Σw_i = 0), whereas the 1-butanol and BTBE effect appears in the adsorption term for most of the candidate equations (Σw_i = 0.993 and Σw_i ≈ 1, respectively); ii) there is a high probability that the first summand in the adsorption term is a_BuOH (Σw_i = 0.959) in front of being 1 (Σw_i = 0.041); iii) the exponent in the adsorption term is likely to be n = 1 (Σw_i = 0.966); iv) the resin–medium interaction term is suitable (Σw_i ≈ 1); and v) the most likely rate-determining step is the surface reaction (Σw_i = 0.993; driving force type (a) in Table 3).

Fig. 4 shows the adequacy of the prediction of experimental reaction rates by Model 49, the low sensitivity of the model output due to the uncertainty of the fitted parameters and the randomly distributed residuals for each experimental point. Model 49 predicts simultaneously well the experimental data obtained in both batch and fixed-bed reactor systems.

Fig. 4 Comparison between experimental and calculated BTBE reaction rates for Model 49 (a), and residuals distribution (b). Symbols: experimental rates obtained in the batch reactor (○), in the fixed-bed reactor using the C₄ as isobutene source (●), and in the fixed-bed reactor using pure isobutene (■). Error bars in (a) correspond to the sensitivity (standard deviation) of the model output due to the parameters uncertainty (see section C in ESI†).

Selected kinetic model

The proposed kinetic equation Model 49, its parameter estimates, and their standard error are:where:
This expression derives from an ER reaction mechanism consisting of the following steps: i) one molecule of 1-butanol adsorbs on a resin active site (or cluster of active sites); ii) it reacts with one isobutene molecule from solution to give one molecule of adsorbed BTBE; and iii) the BTBE molecule desorbs. The surface reaction step is the rate-limiting step of the overall reaction process. This mechanism is in agreement with previous works on analogous reaction systems, e.g., syntheses of MTBE,¹⁴ ETBE,¹⁵ or PTBE.¹⁷

Since the exponent of the adsorption term n is 1, the apparent kinetic coefficient k′ is equal to the intrinsic kinetic constant k, as follows:

k′ = k K¹⁻ⁿ_a,BuOH = k (19)

and, therefore, the true activation energy of the reaction E_a is equal to the apparent one E_ap. Since = −(9.17 ± 0.03) × 10³ K, then E_a = (76.3 ± 0.3) kJ mol⁻¹.

The parameter K_BTBE in the adsorption term of eqn (18) corresponds to the ratio between the adsorption equilibrium constants of BTBE and 1-butanol, i.e., K_a,BTBE/K_a,BuOH. It is related to adsorption thermodynamic properties as follows:

where K_a,BTBE is the adsorption equilibrium constant of BTBE, and , and are the adsorption Gibbs free energy, enthalpy and entropy changes of compound j, respectively. The adsorption enthalpy and entropy changes of individual compounds cannot be obtained from the parameter estimates of the kinetic model. Instead, parameters provide differences between BTBE and 1-butanol adsorption enthalpy and entropy changes: = (34 ± 2) kJ mol⁻¹ and = (93 ± 7) J mol⁻¹ K⁻¹. Consequently, 1-butanol adsorption on the resin is more exothermic than BTBE adsorption, and the entropic loss due to adsorption for 1-butanol is larger than for BTBE. A positive difference between the adsorption enthalpies is consistent with values reported in a previous work on adsorption equilibrium of BTBE and 1-butanol on A35, but in the gas phase: = −21.8 kJ mol⁻¹ and = −41.8 kJ mol⁻¹.³⁵

The estimate of the Hildebrand solubility parameter of A35, δ_P (eqn (12)), can be considered as constant in the assayed temperature range and equal to (24.51 ± 0.18) MPa^1/2. This value is slightly larger than the published δ_P values for this catalyst in similar reaction systems, also being constant: (20.5 ± 0.3) MPa^1/2 in the PTBE synthesis,¹⁷ (20.9 ± 2.0) MPa^1/2 in the ETBE synthesis,³¹ and (21.16 ± 0.12) MPa^1/2 in the simultaneous synthesis of ETBE and TAEE.¹⁸ However, it fully agrees with the Hildebrand solubility parameter estimated by means of the Hoftyzer and van Krevelen group contribution method,³⁶ resulting in δ_P = 25.45 MPa^1/2 at 298 K. The Hansen solubility parameter, which is considered as a refinement of the Hildebrand parameter, has also been estimated from group contribution,³⁶ resulting in δ_P = 25.54 MPa^1/2 at 298 K. With the aim of checking whether the choice of the selected model, and thus the reaction mechanism, would be affected in case that δ_P were roughly estimated, all kinetic equations have been fitted again to experimental rate data, but taking δ_P as a fixed value equal to 20.85 MPa^1/2 (the average of the estimated values quoted in literature)^17,18,29 if appearing in the model. The best ranked models obtained with this constraint are listed in Table S2 (section D in ESI†). Globally, the residual sum of squares (RSS) is now slightly higher due to the reduction of the degrees of freedom of the fit, but the group of best ranked models is coincident with the non-restricted case, and Model 49 is again the most plausible one (w_Model49 = 44.7%, followed by w_Model166 = 8.9%). This result reinforces the choice of the selected kinetic equation Model 49 as the best one.

Conclusions

The kinetics of the liquid-phase etherification reaction of isobutene with 1-butanol to produce butyl tert-butyl ether using Amberlyst™ 35 as the catalyst has been studied at 303–356 K in the liquid phase. Significant internal mass transfer limitations have been detected at temperatures above 333 K using 0.25–0.40 mm catalyst bead size, but this effect was not noticeable for bead size smaller than 0.08 mm. Reaction rates free from mass transfer effects obtained in two reaction setups (a batch reactor and a differential tubular reactor) have been used to fit a large number of candidate kinetic equations, each derived from a different mechanism and rate-determining step, based on the Langmuir–Hinshelwood–Hougen–Watson and Eley–Rideal formalisms. The possible effects of the reaction medium on the catalyst activity, and of the temperature on the parameters of the kinetic equations have also been considered. Discrimination of models has been accomplished by applying mathematical and physicochemical criteria in a conservative manner. The Akaike information criterion has been used to rank the reliability of the kinetic models. The best model is derived from an Eley–Rideal type mechanism, where one molecule of 1-butanol adsorbed on one active site reacts with one isobutene molecule from the liquid phase to form one BTBE molecule that, finally, desorbs. The rate-determining step is the surface reaction. The adsorption of 1-butanol on the resin is more exothermic than BTBE adsorption. The interaction between the reaction medium and the resin has a significant effect on the catalytic activity.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors thank The Dow Chemical Company, now DuPont, for providing the ion exchange resin used as the catalyst in this work.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0re00318b

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