Ketonization Reactions of Carboxylic Acids and Esters over Ceria

Jun 11, 2010 - Tu N. Pham , Tawan Sooknoi , Steven P. Crossley , and Daniel E. Resasco. ACS Catalysis 2013 3 (11), 2456-2473. Abstract | Full Text HTM...
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Ind. Eng. Chem. Res. 2010, 49, 6027–6033

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Ketonization Reactions of Carboxylic Acids and Esters over Ceria-Zirconia as Biomass-Upgrading Processes Christian A. Gaertner, Juan Carlos Serrano-Ruiz, Drew J. Braden, and James A. Dumesic* Department of Chemical and Biological Engineering, UniVersity of Wisconsin, Madison, Wisconsin 53706

Carboxylic acids and esters are common intermediate products formed in biomass conversion processes. Reaction kinetics studies for upgrading mixtures containing carboxylic acids and esters by ketonization reactions (to achieve C-C coupling) over a ceria-zirconia catalyst have been carried out using butanoic acid, pentanoic acid, and hexanoic acid at temperatures of 548, 573, 598, and 623 K. The results from these experimental studies have been consolidated and described using kinetic models. For the ketonization of hexanoic acid, a kinetic model, previously developed for hexanoic acid, has been modified and extended to describe a wider and more applicable concentration range, between 2.5 and 75 mol %. Furthermore, the homo- and the crossketonization of butanoic acid, pentanoic acid, and hexanoic acid have been studied, showing that the rate of cross-ketonization is faster than the rate of homo-ketonization. Because esters are common side products of biomass conversion processes, formed by the esterification of acids with primary and second alcohols, the reaction kinetics for the direct ketonization of esters has also been studied, showing that ester ketonization is slower than ketonization of the corresponding acid. Finally, a kinetic model has been developed to describe the experimentally observed product distribution obtained by reacting a mixture of different biomass-derived species over a ceria-zirconia catalyst. The model includes the ketonization of acids and the ketonization of esters as well as the esterification of acids with primary and secondary alcohols. Introduction The utilization of biomass to produce fuels is a promising way to sustainably produce clean energy and alleviate our societal and economic dependence on fossil fuels.1 However, the production of fuels from biomass requires the development of new chemical pathways to convert highly oxygenated renewable feedstocks into molecules with the appropriate molecular weight and structure for use as liquid fuels.2,3 One of the recently developed processes for this purpose includes the conversion of sorbitol and glucose, two important biomass derivatives, into a hydrophobic mixture of alcohols, acids, ketones, and heterocyclics. These monofunctional intermediates can subsequently undergo upgrading processes to produce targeted fuels (gasoline, diesel, and jet fuels).4 In this respect, ketonization is an important upgrading reaction because it allows the coupling of two carboxylic acids, or esters, into a larger ketone with the simultaneous removal of oxygen in the form of CO2 and water.5 This reaction is particularly useful because carboxylic acids are common intermediates in biomass conversion processes.6 We have recently reported experimental results and a simplified kinetic model for the ketonization of hexanoic acid over a ceria-zirconia catalyst, including esterification with alcohols as side reactions.7,8 However, the organic liquid of monofunctional intermediates produced from glucose and sorbitol contains a mixture of carboxylic acids, ketones, and alcohols (in the C3-C6 range) in significant concentrations (up to 30% wt),9 and thus, a more realistic kinetic study is necessary. Herein, we have tried to address these issues by developing a new kinetic model that covers mixtures of carboxylic acids over a more applicable range of concentrations. In these mixtures, the ketonization of different acids produces a variety of ketones. Symmetric ketones are formed if two identical acids react (homo-ketonization), whereas ketonization of two different acids produces nonsymmetric ketones by cross-ketonization.10 In * To whom correspondence should be addressed. Tel.: +1 608 262 1095. Fax: +1 608 262 0832. E-mail: [email protected].

addition, because the organic stream obtained from glucose and sorbitol contains alcohols, the formation of esters is the main side reaction competing with ketonization.8 Accordingly, the direct ketonization of esters11,12 has been also studied, and these reactions are included in the present study. As a result, this publication presents a descriptive and more comprehensive kinetic model for the simultaneous ketonization of mixtures of acids and esters over ceria-zirconia that serves as a kinetic tool for the description of this important biomass upgrading reaction. Experimental Section The experimental setup and the analysis techniques have been described elsewhere.7 The ceria-zirconia catalyst was prepared by coprecipitation.13 Experimental data for the ketonization of hexanoic acid and the ketonization of esters have been taken from previous publications.7,8 To determine the reactivity and controlling kinetics of all acids present in mixtures of functional intermediates,4 three different acids were used in the present study, butanoic acid, pentanoic acid, and hexanoic acid. Reacting a mixture of these acids over a ceria-zirconia oxide catalyst involves homo-ketonization and cross-ketonization reactions. Table 1 provides a synopsis of the expected products. Equimolar mixtures of butanoic acid (Aldrich, 99+ %), pentanoic acid (Fluka, g98.0%), and hexanoic acid (Aldrich, 99%) at varying concentrations (2.5, 5, 10, and 20 mol % of each acid) in 2-butanone (Aldrich, 99%) were converted over 1 g of ceria-zirconia oxide at temperatures of Table 1. Ketonization Reactions of Multiple Acids and the Resulting Products reaction

reactant 1

reactant 2

1 2 3 4 5 6

butanoic acid pentanoic acid hexanoic acid butanoic acid butanoic acid pentanoic acid

butanoic acid pentanoic acid hexanoic acid pentanoic acid hexanoic acid hexanoic acid

10.1021/ie1004338  2010 American Chemical Society Published on Web 06/11/2010

product

reaction type

4-heptanone homo-ketonization 5-nonanone 6-undecanone 4-octanone cross-ketonization 4-nonanone 5-decanone

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548, 573, 598, and 623 K. At these conditions, 2-butanone is inert.7 Because of analysis difficulties, 4-nonanone and 5-nonanone could not be separated in the GC, and only one product formation rate is calculated. Reaction Kinetics Models General Approach. In previous publications, a simple model for the combined esterification and ketonization of acids has been introduced.7,8 However, this simplified model was not able to describe the reaction kinetics obtained in the present study over a wider range of partial pressures of hexanoic acid. Thus, a new rate expression for the ketonization of hexanoic acid needed to be developed. In a previous publication, the development of a model for the ketonization of esters has also been briefly mentioned.8 The scope of the current publication is to develop a more detailed description of the reaction kinetics for the simultaneous ketonization of different acids and esters. We have calculated adsorption constants using two different assumptions. In general, the change in standard entropy upon adsorption is calculated by subtracting the gas-phase entropy from the surface entropy of a molecule as follows ∆Sads ) Ssurface - Sgas

[(

S°trans,3D ) R ln

h

3

) [ ] ] + ln

ratehex.ac._ketonization ) khex.ac._ketoniztionPhex.ac.2

(1)

The three-dimensional translational entropy of a molecule can be calculated using statistical thermodynamics, as shown in eq 2 (2πmkBT)3/2

“nlinfit”, using a Levenberg-Marquardt method. The responses used in the parameter estimation algorithm are the ketone flow rates out of the reactor. Confidence intervals (95%) are determined for the optimized parameters using the “nlparci” function in MATLAB, which uses a statistical method based on the asymptotic normal distribution for the parameter estimates. Specific Cases. The model for the ketonization of hexanoic acid has been changed compared to a previous publication.7 The ketonization rate constant and the adsorption constants for hexanoic acid, carbon dioxide, and water are calculated according to eqs 5 and 6. Furthermore, the rate expression for hexanoic acid ketonization has been modified as shown in eq 7. The denominator of the rate expression now includes two independent terms that correspond to the presence of both acidic and basic sites on the ceria-zerconia catalyst. One of the siteblocking terms accounts for adsorption of the reactant, hexanoic acid, and the other site-blocking term accounts for adsorbed carbon dioxide and water, products of ketonization; the catalyst coverage by the ketone product, 6-undecanone, is considered to be negligible.

kBT 5 + h 2

(2)

The entropy of adsorption is then calculated using the following equation ∆Sads ) -S°trans,3D + ∆Sads,temp-indep.

(3)

where the second term represents the additional changes in entropy, and we have assumed for simplicity that this second term is independent of temperature. The adsorption equilibrium constant for a species, i, at the average reaction temperature (middle of the temperature range tested) used for the experiments, Tave, is calculated in eq 4 as K°ads,i, where the adsorption entropy is calculated via eq 3 K°ads,i ) e[(-∆Hads,i/RTave)+(∆Sads,i/R]

(4)

The adsorption constant at the reaction temperature is then calculated by the following equation Kads,i ) K°ads,i e(-Hads,i/R)[(1/T)-(1/Tave)]

(1 + Kads,hex.ac.Phex.ac.)2(1 + Kads,waterPwater + Kads,CO2PCO2)2

(7)

According to Table 1, six unique ketonization reactions can occur between multiple acids present in the proposed reaction system. The ketonization of multiple acids is modeled in a similar way as the ketonization of hexanoic acid. All acids are included together in one site-blocking term in the denominator, while the products, water and carbon dioxide, are included together in a separate term as shown in eq 8. rateacid_ketonization ) kacid_ketonizationPacid1Pacid2 (1 + Kads,but.ac.Pbut.ac. + Kads,pent.ac.Ppent.ac. + Kads,hex.ac.Phex.ac.)2 × (1 + Kads,waterPwater + Kads,CO2PCO2)2

(8) The reaction rate for the ketonization of primary and secondary esters can be written as eq 9, where the adsorbed reactants and products are accounted for in a single site-blocking term; the catalyst coverage by the ketone product is considered negligible. rateester_ketonization ) kester_ketonizationPester1Pester2 (1 + Kads,prim.esterPprim.ester + Kads,sec.esterPsec.ester + Kads,CO2PCO2)2

(5)

(9)

The rate constants are calculated in a similar way as that described above. A standard rate constant, k°i , is calculated at an average temperature, Tave, and the rate constant at a certain reaction temperature is then calculated by the following equation

Experimental results for the processing of simulated feeds with five unique reactions occurring simultaneously have been introduced in a previous publication.8 Esterification reactions taking place in these simulated feeds are modeled with the equations and parameters from previous publications.7,8 The ketonization of multiple acids in the presence of esters is modeled using eq 10.

ki ) K°i e(-Ea/R)[(1/T)-(1/Tave)]

(6)

The above approach has been shown to be useful to describe the reaction kinetics for a variety of chemical reactions.14,15 The kinetic model was implemented in MATLAB to solve the differential equations that describe the changing gas-phase compositions down the length of a plug-flow reactor. The abovementioned model parameters are optimized to fit the experimental data using the nonlinear parameter estimation function

rateacid_ketonization ) kacid_ketonizationPacid1Pacid2 (1 +

∑K i

ads,acid,iPacid,i)

(1 +

2

∑K

ads,ester,iPester,i

+

i

Kads,waterPwater + Kads,CO2PCO2)2

(10)

Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010

Figure 1. Simulation results (numbered solid lines) and experimental results (symbols) for the ketonization of hexanoic acid at 547 (1, f); 572 (2, 3); 598 (3, b); and 622 K (4, 0).

It has been assumed that for the ketonization of acids in the presence of esters, catalyst site blocking takes place by acids as well as by esters formed from primary and secondary alcohols. Results and Discussion Ketonization of Hexanoic Acid. Experimental results for the ketonization of hexanoic acid along with a kinetic model have been described in a previous publication.7 However, the model parameters reported in the previous publication were tested for higher hexanoic acid concentrations (g10 mol %), and it was determined that the former model did not accurately describe the experimental observations for higher acid concentrations. The previous model overpredicts the ketonization reaction rates and does not properly describe the shift from second- to zeroorder reaction kinetics with respect to the partial pressure of hexanoic acid. Figure 1 shows experimental data along with results from the kinetic model of the present study, and Table 2 shows the optimized model parameters for the above-mentioned reaction. Because the values for the binding energies and entropy corrections for carbon dioxide and water are used in other optimization runs, a common value for those parameters, determined by running independent optimizations and comparing the values, has been fixed for the final optimization run. The pre-exponential factor for the ketonization reaction and the entropy correction for hexanoic acid were determined to be insensitive, and those values were fixed for the final optimization run. The activation energy barrier for the ketonization reaction is predicted with good confidence intervals, and the same is true for the binding energy of hexanoic acid. The site-blocking

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terms in the denominator of the rate expression play a significant role in the prediction of the reaction kinetics, which is in agreement with the reported importance of site-blocking phenomena for this series of reactions.7 Introducing the independent site-blocking terms in the ketonization rate expression improved the model predictions. The model now accurately describes the aforementioned shift in reaction order with respect to the acid partial pressure, from second- to zero-order with increasing acid concentration. Ketonization of Multiple Acids. Different ketone species were detected in the reactor effluent, suggesting that multiple combinations of acids occurred simultaneously. Table 1 shows the proposed reactions that may be occurring and the resulting reaction products. The rate of formation for 4-heptanone and 6-undecanone, two of the homo-ketonization products, is markedly slower than the formation rate of the other ketones. Additionally, the formation rate of 6-undecanone is slightly lower than the formation rate of 4-heptanone. The formation rate of the third product of a homo-ketonization, 5-nonanone, cannot be determined and is therefore included in the overall rate of formation for nonanone. The formation rate of 4-octanone and 5-decanone, cross-ketonization products, is approximately twice as fast as the formation rate of the homo-ketonization products, whereas the formation of the octanone is slightly faster. The formation rate of nonanone includes one homo- and one cross-ketonization product and is thus the highest rate (approximately three times as high as the homo-ketonization rates). It is known from the literature10 that cross-ketonization product formation rates are typically two times greater than the homo-ketonization product formation rates, and this behavior is in agreement with the results shown in Figure 2. It has also been reported in the literature that the reactivity of an acid molecule decreases with increasing chain length.10 In the present experimental study, this trend in decreasing activity is observed as the formation rates for longer ketones (5-decanone for the cross-ketonization reactions and 6-undecanone for the homoketonization reactions) are lower than the formation rates for shorter ketones (4-heptanone and 4-octanone). Because 4- and 5-nonanone cannot be distinguished, the aforementioned effect cannot be observed regarding these molecules. In general, homoand cross-ketonization products follow a binomial distribution;10 thus, for equal concentrations of reactants, twice the amount of substrate is available for cross-ketonization (i.e., A + B and B + A) compared to homo-ketonization (A + A), and the ratio of the rates of formation of the cross-ketonization products compared to the homo-ketonization products is equal to 2/1. Table 3 shows the optimized model parameters for the ketonization of multiple acids. Figure 2 shows the experimental results along with the predictions of the kinetic model. For the lower temperatures, the error in the model prediction is comparatively high, in view of the very low ketone concentrations and analysis uncertainties. Thus, error bars have been included in the graphs. Because of the number of kinetic

Table 2. Optimized Parameter Values for the Ketonization of Hexanoic Acid confidence interval (95%) parameter

reaction/species

optimized value

activation energy pre-exponential factor binding energy

ketonization ketonization hexanoic acid carbon dioxide water hexanoic acid carbon dioxide water

57.7 5.4 × 1012 -58.7 -103.0 -96.0 104.0 86.0 20.0

entropy correction

lower bound

upper bound

53.0

62.4

-62.0

-55.3

units kJ/mol µmol/(min atm2 g cat) kJ/mol J/(mol-K)

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Figure 2. Simulation results (numbered solid lines) and experimental results (symbols) for the homo-ketonization and cross-ketonization reactions of a mixture of butanoic acid, pentanoic acid, and hexanoic acid to form (a) 4-heptanone, (b) 4-octanone, (c) 4- and 5-nonanone, (d) 5-decanone, and (e) 6-undecanone at 548 (1, b); 573 (2, 0); 598 (3, f); and 623 K (4, 3).

parameters (22 total parameters), the entropy correction terms for all of the acids were fixed at the value previously determined for hexanoic acid. The binding energies for all three acids were fixed at the same value. Similar to the run with hexanoic acid, the pre-exponential factors were determined to be the least sensitive parameters. Thus, the pre-exponential factors were assumed to be equal for all six ketonization reactions, a value that is manually fitted and is similar to the value used in the hexanoic acid optimization run. Importantly, the kinetic model

of this present study can be used to distinguish between the different types of ketonization reactions and to describe all of the different kinetic phenomena accurately. Ketonization of Esters. Experimental results for the ketonization of esters have recently been published.8 Although different reaction pathways exist for the ketonization of esters,11,12,16 it is assumed that two acid moieties of an ester react to form a symmetrical ketone liberating one alcohol and

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Table 3. Optimized Parameter Values for the Homo- and Cross-Ketonization Reactions of Butanoic Acid, Pentanoic Acid, and Hexanoic Acid as Listed in Table 1 confidence interval (95%) parameter activation energy

pre-exponential factor

binding energy

entropy correction

reaction/species

optimized value

1 2 3 4 5 6 1 2 3 4 5 6 hexanoic acid pentanoic acid butanoic acid carbon dioxide water hexanoic acid pentanoic acid butanoic acid carbon dioxide water

58.1 60.0 59.3 55.7 55.2 56.1 5.5 × 1012 5.5 × 1012 5.5 × 1012 5.5 × 1012 5.5 × 1012 5.5 × 1012 -55.7 -55.7 -55.7 -103.0 -96.0 104.0 104.0 104.0 86.0 20.0

lower bound

upper bound

55.3 50.0 56.4 53.1 50.0 53.5

61.0 70.1 62.3 58.3 60.4 58.7

units kJ/mol

µmol/(min atm2 g cat)

-57.8 -57.8 -57.8

-53.7 -53.7 -53.7

kJ/mol

J/(mol-K)

Table 4. Optimized Parameter Values for the Ketonization of 1-Pentylhexanoate and 2-Pentylhexanoate confidence interval (95%) parameter activation energy pre-exponential factor binding energy entropy correction

reaction/species

optimized value

lower bound

upper bound

primary ester ketonization secondary ester ketonization primary ester ketonization secondary ester ketonization 1-pentylhexanoate 2-pentylhexanoate carbon dioxide 1-pentylhexanoate 2-pentylhexanoate carbon dioxide

122.3 101.0 2.4 × 1015 60.2 × 1013 -62.1 -62.1 -103.0 59.0 59.0 86.6

121.2 100.3 2.3 × 1015 60.1 × 1013 -62.7

123.5 101.7 2.4 × 1015 60.3 × 1013 -61.6

olefin molecule plus carbon dioxide, which was detected in the gas phase. Potential products from side reactions were not observed. Importantly, the ketonization of esters of secondary alcohols is faster than the same reaction with esters of primary alcohols. Furthermore, unlike acids, a transition in reaction order dependence from second to zero order, with respect to the ester, is not observed. Accordingly, optimizations of kinetic models have been carried out for both esters separately. However, the optimized values for the binding energy and entropy correction term for carbon dioxide have been fixed at values taken from the optimization results for the single ketonization of hexanoic acid. In the first optimization run for the ketonization of esters, only the ester of the primary alcohol has been included. The pre-exponential factor and activation energy barrier for this reaction have been optimized in addition to the binding energy and entropy correction term for the primary ester. In a second optimization run, the parameters for the secondary ester have been optimized. Importantly, the binding energy and entropy correction term for the ester have been fixed and taken from the run with the primary ester as both molecules are very similar. Table 4 shows the optimized parameters for the ketonization of 1-pentylhexanoate and 2-pentylhexanoate using the abovementioned model. The activation energies determined by the kinetic model (101 versus 122 kJ/mol) reflect the trend that esters of a secondary alcohol are more reactive in a ketonization reaction than the

58.5

59.5

units kJ/mol µmol/(min atm2 g cat) kJ/mol J/(mol-K)

esters of a primary alcohol.8,12 The lower pre-exponential factor of the reaction with the primary ester compared to the value for the secondary species is also in agreement with this trend. The strong adsorption of carbon dioxide causes product inhibition by blocking active sites of the catalyst. Site blocking by the ester reactants plays a smaller role in this model. Regarding the confidence intervals, the model describes the experimental data and important trends well. Figure 3 shows the experimental data and the prediction of the kinetic model for the ketonization of both esters. The difference in reaction rates for the ketonization of acids versus the ketonization of esters can be explained by examining the fitted model parameters. Compared to acids, the adsorption of esters on the catalytic surface is less thermodynamically favorable.8 Thus, it is predicted that for the reaction conditions tested, the surface coverage by esters is smaller than the surface coverage by acids. In addition, the predicted activation energy barriers for the ketonization of acids (55.2-60.0 kJ/mol) are lower than the activation energy barriers predicted for the ketonization of esters (101.0 and 122.3 kJ/mol). The resulting rate constants for the ketonization of acids are approximately 1-3 orders of magnitude greater than the rate constants for the ketonization of esters. Simulated Feeds. Insight into the coupling of all five unique reactions can be gained by processing simulated feeds. The series of reactions included in the comprehensive model are ketonization of acids, esterification of acids with primary and secondary alcohols, and ketonization of both different esters.

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Figure 3. Simulation results (lines) and experimental results (symbols) for the ketonization of 1-pentylhexanoate (solid lines, white symbols) and 2-pentylhexanoate (dashed lines, black symbols) at 573 (1, ]); 598 (2, g); 623 (3, 3 and 6, 1); 648 (4, 0 and 7, 9); and 673 K (5, O and 8, b).

PtRe/C catalysts.4 Also, the model is able to describe the transition in the reaction order dependence on the partial pressure of hexanoic acid from second to zero order with increasing acid concentration. The previously introduced simplified kinetic model has been extended to describe the simultaneous ketonization of different carboxylic acids that are found in a biomassderived mixture of functional intermediates,4 such as butanoic acid, pentanoic acid, and hexanoic acid. The mixture of acids undergoes simultaneous homo- and cross-ketonization reactions. Examining the experimental results and model predictions of the multiple acid ketonization reactions reveals two notable trends. First, when considering the product formation rates, homo-ketonization reactions are approximately half as fast as cross-ketonization reactions. Second, the corresponding reactivity decreases with increasing carbon chain length of the acids. The ketonization of esters is an essential process required to upgrade compounds resulting from the combination of monofunctional intermediates produced from glucose and sorbitol over PtRe/C catalysts.4 A comparison of the determined model parameters for the ketonization of acids and esters shows that acids are more reactive than esters, as reflected in the fitted activation energies. This behavior indicates that in the presence of water, the preferred route for conversion of esters would be hydrolysis to form acids followed by ketonization of the acids. The model introduced in the present study includes the ketonization of acids, ketonization of esters, interconversion between acids and esters by esterification, and hydrolysis reactions involving primary as well as secondary alcohols. The current model describes the experimental results obtained using a simulated biomass-derived feed in which all of the aforementioned reactions take place simultaneously. Because acids and esters are also reaction products in other biomass conversion processes,18 the kinetic model of the present study could be extended to other systems where ketonization reactions could be utilized. Acknowledgment

Figure 4. Simulation results (striped bars) and experimental results (solid bars) for the production of primary esters (blue), secondary esters (red), and ketones (black) for the following simulated feeds and conditions: (1) 0.1 mol % hexanoic acid and 0.05 mol % 1- and 2-pentanol at 648 K; (2) 0.1 mol % hexanoic acid and 0.05 mol % 1- and 2-pentanol at 673 K; (3) 0.3 mol % hexanoic acid and 0.05 mol % 1- and 2-pentanol at 648 K; and (4) 0.3 mol % hexanoic acid and 0.05 mol % 1- and 2-pentanol at 673 K.

This work was supported by the U.S. Department of Energy Office of Basic Energy Sciences and the National Science Foundation Chemical and Transport Systems Division of the Directorate for Engineering. J. C. Serrano-Ruiz thanks the Spanish Ministry of Science and Innovation for postdoctoral support. We also thank D. Wang, R. M. West and J. Q. Bond for valuable discussions and technical assistance. Nomenclature

Experimental results were presented in a previous publication.8 Ketonization of alcohols17 was not observed in the processing of the simulated feed nor in a control experiment in which 1or 2-pentanol was tested using 2-butanone as the solvent. Figure 4 shows the experimental results and model-predicted values for the ketonization of simulated feeds. For all temperatures and concentrations tested, the model sufficiently describes the experimental data. Thus, the model captures the more favorable adsorption of acids compared to that of esters, an observation described in a previous publication,8 and accurately describes the five simultaneous reactions occurring. Conclusions We have developed a kinetic model that describes the ketonization of hexanoic acid over a wide range of hexanoic acid concentrations, such as those encountered in organic oils obtained from the processing of glucose and sorbitol over

PtRe/C ) Platinum-rhenium on carbon C ) Carbon CO2 ) Carbon dioxide ∆Hads ) Adsorption enthalpy ∆Sads ) Adsorption entropy Ssurface ) Entropy of adsorbed species on catalytic surface Sgas ) Entropy in the gas phase S°trans,3D ) Three-dimensional translational entropy m ) molar mass kB ) Boltzmann constant h ) Planck constant Tave ) Average temperature R ) Gas constant Kads,i ) Adsorption equilibrium constant of species i K°ads,i ) Adsorption equilibrium constant of species i at average temperature ki ) Rate constant of species i ki° ) Rate constant of species i

Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010 Ea ) Activation energy Pi ) Partial pressure of species i

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ReceiVed for reView February 26, 2010 ReVised manuscript receiVed May 3, 2010 Accepted May 24, 2010 IE1004338