Caroline Urmès
email: caroline.urmes@gmail.com

Jean-Marc Schweitzer
email: jean-marc.schweitzer@ifpen.fr

Amandine Cabiac
email: amandine.cabiac@ifpen.fr

Yves Schuurman
email: yves.schuurman@ircelyon.univ-lyon1.fr

Kinetic Study of the Selective Hydrogenation of Acetylene over Supported Palladium under Tail-End Conditions

Keywords: power-law, Langmuir-Hinshelwood, kinetic modeling, Pd/α-Al 2 O 3

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Introduction

Ethylene is the largest of the basic chemical building blocks with a global market estimated at more than 140 million tons per year with an increasing growth rate. It is used mainly as precursor for polymers production, for instance polyethylene, vinyl chloride, ethylbenzene, or even ethylene oxide synthesis. New ways of production of ethylene are emerging, such as ethanol dehydration, but steam cracking of naphtha and gas remains the major producer of alkenes. The C2 fraction at the outlet of a steam cracker contains mainly ethane and ethylene, but also traces of acetylene. These trace amounts need to be removed as acetylene is known to poison the Ziegler-Natta catalyst that is used for the polymerization of ethylene. This important issue is done by selective hydrogenation of acetylene, important process in petrochemical industry. Thus, the initial acetylene content, approximately 0.8-1.6%, needs to be reduced to less than 5 ppmv for chemical grade and less than 1 ppmv for polymer grade ethylene. Depending on plant design, selective hydrogenation is carried in two different ways: front-end and tail-end [START_REF] Derrien | Selective hydrogenation applied to refining of petrochemical raw materials produced by steam cracking[END_REF]. In the tail-end configuration, which corresponds to 70% of all units worldwide, the process is placed after CH 4 and H 2 separation. The hydrogen is added in an amount slightly higher than the acetylene concentration and the majority of the stream is ethylene [START_REF] Derrien | Selective hydrogenation applied to refining of petrochemical raw materials produced by steam cracking[END_REF][START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. In front-end configuration, the selective hydrogenation unit is placed upstream of the demethanizer and a larger amount of hydrogen is available (around 20%).

Alumina-supported palladium or bimetallic palladium-silver catalysts are used for this process, assuring very high activity and selectivity for acetylene hydrogenation. The main goal is to reduce the acetylene content without the hydrogenation of ethylene to ethane. Catalyst deactivation by coke formation is very common under tail-end conditions as is the formation of C 4 byproducts.

The kinetics of this reaction have been the subject of several studies [START_REF] Bos | Mechanism and kinetics of the selective hydrogenation of ethyne and ethane[END_REF][START_REF] Borodzinski | The kinetic model of hydrogenation of acetylene-ethylene mixtures over palladium surface covered by carbonaceous deposits[END_REF][START_REF] Cider | Kinetics of cross-desorption of carbon monoxide by the influence of ethyne over palladium/alumina[END_REF][START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts, Part 2: Steady-State Kinetics and Effects of Palladium Particle Size, Carbon Monoxide, and Promoters[END_REF][START_REF] Pachulski | Kinetics and reactor modeling of a Pd-Ag/Al 2 O 3 catalyst during selective hydrogenation of ethyne[END_REF] and have been analyzed in detail by Borodzi ński and Bond [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts, Part 2: Steady-State Kinetics and Effects of Palladium Particle Size, Carbon Monoxide, and Promoters[END_REF]. The most elaborate models consist of 2 or 3 distinct sites, each Each operating condition was tested three times per catalyst loading. Two catalyst loadings were used. All this data was used to calculate the relative standard deviations of both the acetylene conversion and the ethane exit molar flow rate. The relative standard deviations are given in Table 1. Rather large (10-20%) relative standard deviations were found for the molar exit flows of ethane. The variation of the relative flows of acetylene, ethylene, hydrogen, and argon allowed determination of the apparent reaction orders with respect to acetylene, ethylene, and hydrogen. Apparent reaction orders are based on power law expressions for the rates as follows:

-r C 2 H 2 = k 1 P α 1 C 2 H 2 P β 1 H 2 P γ 1 C 2 H 4 r C 2 H 6 = k 2 P α 2 C 2 H 2 P β 2 H 2 P γ 2 C 2 H 4
As no C4 products were experimentally observed, the kinetic analysis is restricted to the hydrogenation of acetylene to ethylene and ethylene to ethane. Both rate equations were integrated numerically and the reaction orders were determined by regression analysis of the acetylene conversion and the molar exit flow rate of ethane of the data set at each temperature separately. The estimated values of the reaction orders are given in Table 2. A negative reaction order for acetylene was observed, -0.9 with respect to acetylene consumption, and (-1)-(-1.5) with respect to ethane formation. This correspond to a strong adsorption of acetylene on the surface of the catalyst. This order is lower than the values reported in the literature for acetylene consumption, which are between 0-(-0.7) [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF][START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Bond | The hydrogenation of acetylene: II. The reaction of acetylene with hydrogen catalyzed by alumina-supported palladium[END_REF] depending on the conditions.

The order for hydrogen, approximately 1.5, is high and hard to explain mechanistically. However, the same range of magnitude was found by Aduriz and al.: 1.3-1.6 [START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF], but under front-end conditions. Most studies report an order of +1 [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. Molero et al. observed a reaction order of hydrogen between 1-1.25, depending on the temperature for acetylene hydrogenation over a Pd foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]. From a careful analysis of the data, they derived that the hydrogen reaction order can vary between values of 1 and 1.5. Excess hydrogen can remove strongly adsorbed carbonaceous species from the catalyst surface and so creates free surface sites.

Regarding the rate of consumption of acetylene, no strong effect is observed for ethylene. The order is close to zero as shown by numerous studies [START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. However, some ethylene adsorption occurs as indicated by the small negative value of the reaction order. For ethane formation, the reaction order in ethylene is much higher, between 0.3 and 1. This is related to the fact that ethylene is the reactant for ethane production.

Inspection of the reaction orders give valuable insights into the reaction mechanism. However, this needs to be further validated by deriving the corresponding rate equation based on a sequence of elementary steps. Catalytic hydrogenation reactions of unsaturated hydrocarbons often follow a series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Saeys | First-principles based kinetic model for the hydrogenation of toluene[END_REF]. This mechanism is given for acetylene hydrogenation via ethylene to ethane in Table 3. A single site for all surface species has been assumed. This assumption will be discussed later on. A negative reaction order for acetylene was observed, -0.9 with respect to acetylene consumption, and (-1)-(-1.5) with respect to ethane formation. This correspond to a strong adsorption of acetylene on the surface of the catalyst. This order is lower than the values reported in the literature for acetylene consumption, which are between 0-(-0.7) [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF][START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Bond | The hydrogenation of acetylene: II. The reaction of acetylene with hydrogen catalyzed by alumina-supported palladium[END_REF] depending on the conditions.

The order for hydrogen, approximately 1.5, is high and hard to explain mechanistically. However, the same range of magnitude was found by Aduriz and al.: 1.3-1.6 [START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF], but under frontend conditions. Most studies report an order of +1 [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. Molero et al. observed a reaction order of hydrogen between 1-1.25, depending on the temperature for acetylene hydrogenation over a Pd foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]. From a careful analysis of the data, they derived that the hydrogen reaction order can vary between values of 1 and 1.5. Excess hydrogen can remove strongly adsorbed carbonaceous species from the catalyst surface and so creates free surface sites.

Regarding the rate of consumption of acetylene, no strong effect is observed for ethylene. The order is close to zero as shown by numerous studies [START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. However, some ethylene adsorption occurs as indicated by the small negative value of the reaction order. For ethane formation, the reaction order in ethylene is much higher, between 0.3 and 1. This is related to the fact that ethylene is the reactant for ethane production.

Inspection of the reaction orders give valuable insights into the reaction mechanism. However, this needs to be further validated by deriving the corresponding rate equation based on a sequence of elementary steps. Catalytic hydrogenation reactions of unsaturated hydrocarbons often follow a series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Saeys | First-principles based kinetic model for the hydrogenation of toluene[END_REF]. This mechanism is given for acetylene hydrogenation via ethylene to ethane in Table 3. A single site for all surface species has been assumed. This assumption will be discussed later on.

Hydrogen adsorbs dissociatively on palladium, requiring two free neighboring surface sites (step (1)). The adsorbed hydrogen atom can react with adsorbed acetylene to form a vinyl intermediate in step [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. This intermediate can react with a second hydrogen atom to form adsorbed ethylene. Neurock and van Santen studied ethylene adsorption on a Pd(111) surface by DFT and found that ethylene adsorbs at low coverages as a di- species and at high coverage as a -bonded species [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. Ethylene can desorb or react with atomic hydrogen to form an ethyl intermediate. This intermediate can react again with a second hydrogen atom to form ethane, which has little interaction with the Pd surface and therefore desorbs instantaneously (step [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]). Table 3. Elementary steps for the reaction C2H2 + H2  C2H4 and C2H4 + H2  C2H6.

N° Elementary

Step

 1 H2 + 2*  2H* 2 2 C2H2 + *  C2H2* 1 3 C2H2* + H*  C2H3* + * 1 4 C2H3* + H*  C2H4* + * 1 5 C2H4*  C2H4 + * 1 6 C2H4* + H*  C2H5* + * 1 7 C2H5* + H*  C2H6 + 2* 1
Even though the reaction mechanism is still rather simple, the derivation of the corresponding rate-equation requires several assumptions. We assume that the adsorption take place according to the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part 1. Solids[END_REF]. The next assumption is with respect to the rate-determining step for the formation of ethylene and ethane. Since for both steps the reaction order in hydrogen was found to be larger than 1, the additions of the second hydrogen atom to the vinyl and ethyl intermediates are assumed to be rate-determining (steps (4) and [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF], respectively). All other steps are assumed to be in A negative reaction order for acetylene was observed, -0.9 with respect to acetyl consumption, and (-1)-(-1.5) with respect to ethane formation. This correspond to a stro adsorption of acetylene on the surface of the catalyst. This order is lower than the values reported the literature for acetylene consumption, which are between 0-(-0.7) [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF][START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Bond | The hydrogenation of acetylene: II. The reaction of acetylene with hydrogen catalyzed by alumina-supported palladium[END_REF] depending on conditions.

The order for hydrogen, approximately 1.5, is high and hard to explain mechanistica However, the same range of magnitude was found by Aduriz and al.: 1.3-1.6 [START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF], but under fro end conditions. Most studies report an order of +1 [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. Molero et al. observed a reaction or of hydrogen between 1-1.25, depending on the temperature for acetylene hydrogenation over a foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]. From a careful analysis of the data, they derived that the hydrogen reaction order can v between values of 1 and 1.5. Excess hydrogen can remove strongly adsorbed carbonaceous spec from the catalyst surface and so creates free surface sites.

Regarding the rate of consumption of acetylene, no strong effect is observed for ethylene. T order is close to zero as shown by numerous studies [START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. However, some ethylene adsorption occ as indicated by the small negative value of the reaction order. For ethane formation, the reaction or in ethylene is much higher, between 0.3 and 1. This is related to the fact that ethylene is the react for ethane production.

Inspection of the reaction orders give valuable insights into the reaction mechanism. Howev this needs to be further validated by deriving the corresponding rate equation based on a seque of elementary steps. Catalytic hydrogenation reactions of unsaturated hydrocarbons often follo series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Saeys | First-principles based kinetic model for the hydrogenation of toluene[END_REF]. T mechanism is given for acetylene hydrogenation via ethylene to ethane in Table 3. A single site all surface species has been assumed. This assumption will be discussed later on.

Hydrogen adsorbs dissociatively on palladium, requiring two free neighboring surface s (step (1)). The adsorbed hydrogen atom can react with adsorbed acetylene to form a vi intermediate in step [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. This intermediate can react with a second hydrogen atom to form adsorb ethylene. Neurock and van Santen studied ethylene adsorption on a Pd(111) surface by DFT a found that ethylene adsorbs at low coverages as a di- species and at high coverage as a -bond species [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. Ethylene can desorb or react with atomic hydrogen to form an ethyl intermediate. T intermediate can react again with a second hydrogen atom to form ethane, which has little interact with the Pd surface and therefore desorbs instantaneously (step [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]). Even though the reaction mechanism is still rather simple, the derivation of the correspond rate-equation requires several assumptions. We assume that the adsorption take place according the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part 1. Solids[END_REF]. The next assumption is with respect to the rate-determining step for formation of ethylene and ethane. Since for both steps the reaction order in hydrogen was found be larger than 1, the additions of the second hydrogen atom to the vinyl and ethyl intermediates assumed to be rate-determining (steps (4) and [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF] A negative reaction order for acetylene was observed, -0.9 with respect to acetylene consumption, and (-1)-(-1.5) with respect to ethane formation. This correspond to a strong adsorption of acetylene on the surface of the catalyst. This order is lower than the values reported in the literature for acetylene consumption, which are between 0-(-0.7) [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF][START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Bond | The hydrogenation of acetylene: II. The reaction of acetylene with hydrogen catalyzed by alumina-supported palladium[END_REF] depending on the conditions.

The order for hydrogen, approximately 1.5, is high and hard to explain mechanistically. However, the same range of magnitude was found by Aduriz and al.: 1.3-1.6 [START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF], but under frontend conditions. Most studies report an order of +1 [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. Molero et al. observed a reaction order of hydrogen between 1-1.25, depending on the temperature for acetylene hydrogenation over a Pd foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]. From a careful analysis of the data, they derived that the hydrogen reaction order can vary between values of 1 and 1.5. Excess hydrogen can remove strongly adsorbed carbonaceous species from the catalyst surface and so creates free surface sites.

Regarding the rate of consumption of acetylene, no strong effect is observed for ethylene. The order is close to zero as shown by numerous studies [START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. However, some ethylene adsorption occurs as indicated by the small negative value of the reaction order. For ethane formation, the reaction order in ethylene is much higher, between 0.3 and 1. This is related to the fact that ethylene is the reactant for ethane production.

Inspection of the reaction orders give valuable insights into the reaction mechanism. However, this needs to be further validated by deriving the corresponding rate equation based on a sequence of elementary steps. Catalytic hydrogenation reactions of unsaturated hydrocarbons often follow a series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Saeys | First-principles based kinetic model for the hydrogenation of toluene[END_REF]. This mechanism is given for acetylene hydrogenation via ethylene to ethane in Table 3. A single site for all surface species has been assumed. This assumption will be discussed later on.

Hydrogen adsorbs dissociatively on palladium, requiring two free neighboring surface sites (step (1)). The adsorbed hydrogen atom can react with adsorbed acetylene to form a vinyl intermediate in step [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. This intermediate can react with a second hydrogen atom to form adsorbed ethylene. Neurock and van Santen studied ethylene adsorption on a Pd(111) surface by DFT and found that ethylene adsorbs at low coverages as a di- species and at high coverage as a -bonded species [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. Ethylene can desorb or react with atomic hydrogen to form an ethyl intermediate. This intermediate can react again with a second hydrogen atom to form ethane, which has little interaction with the Pd surface and therefore desorbs instantaneously (step (7)). Table 3. Elementary steps for the reaction C2H2 + H2  C2H4 and C2H4 + H2  C2H6.

N° Elementary

Step 

1 H2 + 2*  2H* 2 2 C2H2 + *  C2H2* 1 3 C2H2* + H*  C2H3* + * 1 4 C2H3* + H*  C2H4* + * 1 5 C2H4*  C2H4 + * 1 6 C2H4* + H*  C2H5* + * 1 7 C2H5* + H*  C2H6 + 2* 1
Even though the reaction mechanism is still rather simple, the derivation of the corresponding rate-equation requires several assumptions. We assume that the adsorption take place according to the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part 1. Solids[END_REF]. The next assumption is with respect to the rate-determining step for the formation of ethylene and ethane. Since for both steps the reaction order in hydrogen was found to be larger than 1, the additions of the second hydrogen atom to the vinyl and ethyl intermediates are assumed to be rate-determining (steps ( 4) and [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF], respectively). All other steps are assumed to be in A negative reaction order for acetylene was observed, -0.9 with respect to acetylene consumption, and (-1)-(-1.5) with respect to ethane formation. This correspond to a strong adsorption of acetylene on the surface of the catalyst. This order is lower than the values reported in the literature for acetylene consumption, which are between 0-(-0.7) [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF][START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Bond | The hydrogenation of acetylene: II. The reaction of acetylene with hydrogen catalyzed by alumina-supported palladium[END_REF] depending on the conditions.

The order for hydrogen, approximately 1.5, is high and hard to explain mechanistically. However, the same range of magnitude was found by Aduriz and al.: 1.3-1.6 [START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF], but under frontend conditions. Most studies report an order of +1 [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. Molero et al. observed a reaction order of hydrogen between 1-1.25, depending on the temperature for acetylene hydrogenation over a Pd foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]. From a careful analysis of the data, they derived that the hydrogen reaction order can vary between values of 1 and 1.5. Excess hydrogen can remove strongly adsorbed carbonaceous species from the catalyst surface and so creates free surface sites.

Regarding the rate of consumption of acetylene, no strong effect is observed for ethylene. The order is close to zero as shown by numerous studies [START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. However, some ethylene adsorption occurs as indicated by the small negative value of the reaction order. For ethane formation, the reaction order in ethylene is much higher, between 0.3 and 1. This is related to the fact that ethylene is the reactant for ethane production.

Inspection of the reaction orders give valuable insights into the reaction mechanism. However, this needs to be further validated by deriving the corresponding rate equation based on a sequence of elementary steps. Catalytic hydrogenation reactions of unsaturated hydrocarbons often follow a series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Saeys | First-principles based kinetic model for the hydrogenation of toluene[END_REF]. This mechanism is given for acetylene hydrogenation via ethylene to ethane in Table 3. A single site for all surface species has been assumed. This assumption will be discussed later on.

Hydrogen adsorbs dissociatively on palladium, requiring two free neighboring surface sites (step (1)). The adsorbed hydrogen atom can react with adsorbed acetylene to form a vinyl intermediate in step [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. This intermediate can react with a second hydrogen atom to form adsorbed ethylene. Neurock and van Santen studied ethylene adsorption on a Pd(111) surface by DFT and found that ethylene adsorbs at low coverages as a di- species and at high coverage as a -bonded species [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. Ethylene can desorb or react with atomic hydrogen to form an ethyl intermediate. This intermediate can react again with a second hydrogen atom to form ethane, which has little interaction with the Pd surface and therefore desorbs instantaneously (step (7)). Table 3. Elementary steps for the reaction C2H2 + H2  C2H4 and C2H4 + H2  C2H6.

N° Elementary

Step

 1 H2 + 2*  2H* 2 2 C2H2 + *  C2H2* 1 3 C2H2* + H*  C2H3* + * 1 4 C2H3* + H*  C2H4* + * 1 5 C2H4*  C2H4 + * 1 6 C2H4* + H*  C2H5* + * 1 7 C2H5* + H*  C2H6 + 2* 1
Even though the reaction mechanism is still rather simple, the derivation of the corresponding rate-equation requires several assumptions. We assume that the adsorption take place according to the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part 1. Solids[END_REF]. The next assumption is with respect to the rate-determining step for the formation of ethylene and ethane. Since for both steps the reaction order in hydrogen was found to be larger than 1, the additions of the second hydrogen atom to the vinyl and ethyl intermediates are assumed to be rate-determining (steps ( 4) and [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF], respectively). All other steps are assumed to be in A negative reaction order for acetylene was observed, -0.9 with respect to acetylene consumption, and (-1)-(-1.5) with respect to ethane formation. This correspond to a strong adsorption of acetylene on the surface of the catalyst. This order is lower than the values reported in the literature for acetylene consumption, which are between 0-(-0.7) [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF][START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Bond | The hydrogenation of acetylene: II. The reaction of acetylene with hydrogen catalyzed by alumina-supported palladium[END_REF] depending on the conditions.

The order for hydrogen, approximately 1.5, is high and hard to explain mechanistically. However, the same range of magnitude was found by Aduriz and al.: 1.3-1.6 [START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF], but under frontend conditions. Most studies report an order of +1 [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. Molero et al. observed a reaction order of hydrogen between 1-1.25, depending on the temperature for acetylene hydrogenation over a Pd foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]. From a careful analysis of the data, they derived that the hydrogen reaction order can vary between values of 1 and 1.5. Excess hydrogen can remove strongly adsorbed carbonaceous species from the catalyst surface and so creates free surface sites.

Regarding the rate of consumption of acetylene, no strong effect is observed for ethylene. The order is close to zero as shown by numerous studies [START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. However, some ethylene adsorption occurs as indicated by the small negative value of the reaction order. For ethane formation, the reaction order in ethylene is much higher, between 0.3 and 1. This is related to the fact that ethylene is the reactant for ethane production.

Inspection of the reaction orders give valuable insights into the reaction mechanism. However, this needs to be further validated by deriving the corresponding rate equation based on a sequence of elementary steps. Catalytic hydrogenation reactions of unsaturated hydrocarbons often follow a series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Saeys | First-principles based kinetic model for the hydrogenation of toluene[END_REF]. This mechanism is given for acetylene hydrogenation via ethylene to ethane in Table 3. A single site for all surface species has been assumed. This assumption will be discussed later on.

Hydrogen adsorbs dissociatively on palladium, requiring two free neighboring surface sites (step (1)). The adsorbed hydrogen atom can react with adsorbed acetylene to form a vinyl intermediate in step [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. This intermediate can react with a second hydrogen atom to form adsorbed ethylene. Neurock and van Santen studied ethylene adsorption on a Pd(111) surface by DFT and found that ethylene adsorbs at low coverages as a di- species and at high coverage as a -bonded species [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. Ethylene can desorb or react with atomic hydrogen to form an ethyl intermediate. This intermediate can react again with a second hydrogen atom to form ethane, which has little interaction with the Pd surface and therefore desorbs instantaneously (step (7)). Table 3. Elementary steps for the reaction C2H2 + H2  C2H4 and C2H4 + H2  C2H6.

N° Elementary

Step 

1 H2 + 2*  2H* 2 2 C2H2 + *  C2H2* 1 3 C2H2* + H*  C2H3* + * 1 4 C2H3* + H*  C2H4* + * 1 5 C2H4*  C2H4 + * 1 6 C2H4* + H*  C2H5* + * 1 7 C2H5* + H*  C2H6 + 2* 1
Even though the reaction mechanism is still rather simple, the derivation of the corresponding rate-equation requires several assumptions. We assume that the adsorption take place according to the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part 1. Solids[END_REF]. The next assumption is with respect to the rate-determining step for the formation of ethylene and ethane. Since for both steps the reaction order in hydrogen was found to be larger than 1, the additions of the second hydrogen atom to the vinyl and ethyl intermediates are assumed to be rate-determining (steps ( 4) and [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF], respectively). All other steps are assumed to be in A negative reaction order for acetylene was observed, -0.9 with respect to acetylene consumption, and (-1)-(-1.5) with respect to ethane formation. This correspond to a strong adsorption of acetylene on the surface of the catalyst. This order is lower than the values reported in the literature for acetylene consumption, which are between 0-(-0.7) [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF][START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Bond | The hydrogenation of acetylene: II. The reaction of acetylene with hydrogen catalyzed by alumina-supported palladium[END_REF] depending on the conditions.

C 2 H 3 *
The order for hydrogen, approximately 1.5, is high and hard to explain mechanistically. However, the same range of magnitude was found by Aduriz and al.: 1.3-1.6 [START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF], but under frontend conditions. Most studies report an order of +1 [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. Molero et al. observed a reaction order of hydrogen between 1-1.25, depending on the temperature for acetylene hydrogenation over a Pd foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]. From a careful analysis of the data, they derived that the hydrogen reaction order can vary between values of 1 and 1.5. Excess hydrogen can remove strongly adsorbed carbonaceous species from the catalyst surface and so creates free surface sites.

Regarding the rate of consumption of acetylene, no strong effect is observed for ethylene. The order is close to zero as shown by numerous studies [START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. However, some ethylene adsorption occurs as indicated by the small negative value of the reaction order. For ethane formation, the reaction order in ethylene is much higher, between 0.3 and 1. This is related to the fact that ethylene is the reactant for ethane production.

Inspection of the reaction orders give valuable insights into the reaction mechanism. However, this needs to be further validated by deriving the corresponding rate equation based on a sequence of elementary steps. Catalytic hydrogenation reactions of unsaturated hydrocarbons often follow a series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Saeys | First-principles based kinetic model for the hydrogenation of toluene[END_REF]. This mechanism is given for acetylene hydrogenation via ethylene to ethane in Table 3. A single site for all surface species has been assumed. This assumption will be discussed later on.

Hydrogen adsorbs dissociatively on palladium, requiring two free neighboring surface sites (step (1)). The adsorbed hydrogen atom can react with adsorbed acetylene to form a vinyl intermediate in step [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. This intermediate can react with a second hydrogen atom to form adsorbed ethylene. Neurock and van Santen studied ethylene adsorption on a Pd(111) surface by DFT and found that ethylene adsorbs at low coverages as a di- species and at high coverage as a -bonded species [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. Ethylene can desorb or react with atomic hydrogen to form an ethyl intermediate. This intermediate can react again with a second hydrogen atom to form ethane, which has little interaction with the Pd surface and therefore desorbs instantaneously (step (7)).

Table 3.

Elementary steps for the reaction C2H2 + H2  C2H4 and C2H4 + H2  C2H6.

N° Elementary

Step 

1 H2 + 2*  2H* 2 2 C2H2 + *  C2H2* 1 3 C2H2* + H*  C2H3* + * 1 4 C2H3* + H*  C2H4* + * 1 5 C2H4*  C2H4 + * 1 6 C2H4* + H*  C2H5* + * 1 7 C2H5* + H*  C2H6 + 2* 1
Even though the reaction mechanism is still rather simple, the derivation of the corresponding rate-equation requires several assumptions. We assume that the adsorption take place according to the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part 1. Solids[END_REF]. The next assumption is with respect to the rate-determining step for the formation of ethylene and ethane. Since for both steps the reaction order in hydrogen was found to be larger than 1, the additions of the second hydrogen atom to the vinyl and ethyl intermediates are assumed to be rate-determining (steps ( 4) and ( 7), respectively). All other steps are assumed to be in A negative reaction order for acetylene was observed, -0.9 with respect to acetylene consumption, and (-1)-(-1.5) with respect to ethane formation. This correspond to a strong adsorption of acetylene on the surface of the catalyst. This order is lower than the values reported in the literature for acetylene consumption, which are between 0-(-0.7) [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF][START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Bond | The hydrogenation of acetylene: II. The reaction of acetylene with hydrogen catalyzed by alumina-supported palladium[END_REF] depending on the conditions.

The order for hydrogen, approximately 1.5, is high and hard to explain mechanistically. However, the same range of magnitude was found by Aduriz and al.: 1.3-1.6 [START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF], but under frontend conditions. Most studies report an order of +1 [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. Molero et al. observed a reaction order of hydrogen between 1-1.25, depending on the temperature for acetylene hydrogenation over a Pd foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]. From a careful analysis of the data, they derived that the hydrogen reaction order can vary between values of 1 and 1.5. Excess hydrogen can remove strongly adsorbed carbonaceous species from the catalyst surface and so creates free surface sites.

Regarding the rate of consumption of acetylene, no strong effect is observed for ethylene. The order is close to zero as shown by numerous studies [START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. However, some ethylene adsorption occurs as indicated by the small negative value of the reaction order. For ethane formation, the reaction order in ethylene is much higher, between 0.3 and 1. This is related to the fact that ethylene is the reactant for ethane production.

Inspection of the reaction orders give valuable insights into the reaction mechanism. However, this needs to be further validated by deriving the corresponding rate equation based on a sequence of elementary steps. Catalytic hydrogenation reactions of unsaturated hydrocarbons often follow a series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Saeys | First-principles based kinetic model for the hydrogenation of toluene[END_REF]. This mechanism is given for acetylene hydrogenation via ethylene to ethane in Table 3. A single site for all surface species has been assumed. This assumption will be discussed later on.

Hydrogen adsorbs dissociatively on palladium, requiring two free neighboring surface sites (step (1)). The adsorbed hydrogen atom can react with adsorbed acetylene to form a vinyl intermediate in step [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. This intermediate can react with a second hydrogen atom to form adsorbed ethylene. Neurock and van Santen studied ethylene adsorption on a Pd(111) surface by DFT and found that ethylene adsorbs at low coverages as a di- species and at high coverage as a -bonded species [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. Ethylene can desorb or react with atomic hydrogen to form an ethyl intermediate. This intermediate can react again with a second hydrogen atom to form ethane, which has little interaction with the Pd surface and therefore desorbs instantaneously (step (7)). Even though the reaction mechanism is still rather simple, the derivation of the corresponding rate-equation requires several assumptions. We assume that the adsorption take place according to the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part 1. Solids[END_REF]. The next assumption is with respect to the rate-determining step for the formation of ethylene and ethane. Since for both steps the reaction order in hydrogen was found to be larger than 1, the additions of the second hydrogen atom to the vinyl and ethyl intermediates are assumed to be rate-determining (steps ( 4) and ( 7), respectively). All other steps are assumed to be in A negative reaction order for acetylene was observed, -0.9 with respect to acetylene consumption, and (-1)-(-1.5) with respect to ethane formation. This correspond to a strong adsorption of acetylene on the surface of the catalyst. This order is lower than the values reported in the literature for acetylene consumption, which are between 0-(-0.7) [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF][START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Bond | The hydrogenation of acetylene: II. The reaction of acetylene with hydrogen catalyzed by alumina-supported palladium[END_REF] depending on the conditions.

The order for hydrogen, approximately 1.5, is high and hard to explain mechanistically. However, the same range of magnitude was found by Aduriz and al.: 1.3-1.6 [START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF], but under frontend conditions. Most studies report an order of +1 [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. Molero et al. observed a reaction order of hydrogen between 1-1.25, depending on the temperature for acetylene hydrogenation over a Pd foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]. From a careful analysis of the data, they derived that the hydrogen reaction order can vary between values of 1 and 1.5. Excess hydrogen can remove strongly adsorbed carbonaceous species from the catalyst surface and so creates free surface sites.

Regarding the rate of consumption of acetylene, no strong effect is observed for ethylene. The order is close to zero as shown by numerous studies [START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. However, some ethylene adsorption occurs as indicated by the small negative value of the reaction order. For ethane formation, the reaction order in ethylene is much higher, between 0.3 and 1. This is related to the fact that ethylene is the reactant for ethane production.

Inspection of the reaction orders give valuable insights into the reaction mechanism. However, this needs to be further validated by deriving the corresponding rate equation based on a sequence of elementary steps. Catalytic hydrogenation reactions of unsaturated hydrocarbons often follow a series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Saeys | First-principles based kinetic model for the hydrogenation of toluene[END_REF]. This mechanism is given for acetylene hydrogenation via ethylene to ethane in Table 3. A single site for all surface species has been assumed. This assumption will be discussed later on.

Hydrogen adsorbs dissociatively on palladium, requiring two free neighboring surface sites (step (1)). The adsorbed hydrogen atom can react with adsorbed acetylene to form a vinyl intermediate in step [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. This intermediate can react with a second hydrogen atom to form adsorbed ethylene. Neurock and van Santen studied ethylene adsorption on a Pd(111) surface by DFT and found that ethylene adsorbs at low coverages as a di- species and at high coverage as a -bonded species [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. Ethylene can desorb or react with atomic hydrogen to form an ethyl intermediate. This intermediate can react again with a second hydrogen atom to form ethane, which has little interaction with the Pd surface and therefore desorbs instantaneously (step (7)). Even though the reaction mechanism is still rather simple, the derivation of the corresponding rate-equation requires several assumptions. We assume that the adsorption take place according to the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part 1. Solids[END_REF]. The next assumption is with respect to the rate-determining step for the formation of ethylene and ethane. Since for both steps the reaction order in hydrogen was found to be larger than 1, the additions of the second hydrogen atom to the vinyl and ethyl intermediates are assumed to be rate-determining (steps ( 4) and ( 7), respectively). All other steps are assumed to be in A negative reaction order for acetylene was observed, -0.9 with respect to acetylene consumption, and (-1)-(-1.5) with respect to ethane formation. This correspond to a strong adsorption of acetylene on the surface of the catalyst. This order is lower than the values reported in the literature for acetylene consumption, which are between 0-(-0.7) [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF][START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF][START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Bond | The hydrogenation of acetylene: II. The reaction of acetylene with hydrogen catalyzed by alumina-supported palladium[END_REF] depending on the conditions.

The order for hydrogen, approximately 1.5, is high and hard to explain mechanistically. However, the same range of magnitude was found by Aduriz and al.: 1.3-1.6 [START_REF] Aduriz | Activity and Selectivity of Pd/Alpha-Al 2 O 3 for Ethyne Hydrogenation in a large Excess of Ethene and Hydrogen[END_REF], but under frontend conditions. Most studies report an order of +1 [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF][START_REF] Mcgowm | Hydrogenation of Acetylene in Excess Ethylene on a Alumina-Supported Palladium Catalyst at Atmospheric Pressure in a Spinning Basket Reactor[END_REF][START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. Molero et al. observed a reaction order of hydrogen between 1-1.25, depending on the temperature for acetylene hydrogenation over a Pd foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]. From a careful analysis of the data, they derived that the hydrogen reaction order can vary between values of 1 and 1.5. Excess hydrogen can remove strongly adsorbed carbonaceous species from the catalyst surface and so creates free surface sites.

Regarding the rate of consumption of acetylene, no strong effect is observed for ethylene. The order is close to zero as shown by numerous studies [START_REF] Vincent | A Langmuir-Hinshelwood model for a hydrogen transfer mechanism in the selective hydrogenation of acetylene over a Pd/γ-Al 2 O 3 catalyst prepared by the sol-gel method[END_REF]. However, some ethylene adsorption occurs as indicated by the small negative value of the reaction order. For ethane formation, the reaction order in ethylene is much higher, between 0.3 and 1. This is related to the fact that ethylene is the reactant for ethane production.

Inspection of the reaction orders give valuable insights into the reaction mechanism. However, this needs to be further validated by deriving the corresponding rate equation based on a sequence of elementary steps. Catalytic hydrogenation reactions of unsaturated hydrocarbons often follow a series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF][START_REF] Saeys | First-principles based kinetic model for the hydrogenation of toluene[END_REF]. This mechanism is given for acetylene hydrogenation via ethylene to ethane in Table 3. A single site for all surface species has been assumed. This assumption will be discussed later on.

Hydrogen adsorbs dissociatively on palladium, requiring two free neighboring surface sites (step (1)). The adsorbed hydrogen atom can react with adsorbed acetylene to form a vinyl intermediate in step [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. This intermediate can react with a second hydrogen atom to form adsorbed ethylene. Neurock and van Santen studied ethylene adsorption on a Pd(111) surface by DFT and found that ethylene adsorbs at low coverages as a di- species and at high coverage as a -bonded species [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. Ethylene can desorb or react with atomic hydrogen to form an ethyl intermediate. This intermediate can react again with a second hydrogen atom to form ethane, which has little interaction with the Pd surface and therefore desorbs instantaneously (step ( 7)). Even though the reaction mechanism is still rather simple, the derivation of the corresponding rate-equation requires several assumptions. We assume that the adsorption take place according to the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part 1. Solids[END_REF]. The next assumption is with respect to the rate-determining step for the formation of ethylene and ethane. Since for both steps the reaction order in hydrogen was found to be larger than 1, the additions of the second hydrogen atom to the vinyl and ethyl intermediates are assumed to be rate-determining (steps ( 4) and ( 7), respectively). All other steps are assumed to be in

C 2 H 6 + 2* 1
Hydrogen adsorbs dissociatively on palladium, requiring two free neighboring surface sites (step (1)). The adsorbed hydrogen atom can react with adsorbed acetylene to form a vinyl intermediate in step [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF]. This intermediate can react with a second hydrogen atom to form adsorbed ethylene. Neurock and van Santen studied ethylene adsorption on a Pd(111) surface by DFT and found that ethylene adsorbs at low coverages as a di-σ species and at high coverage as a π-bonded species [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. Ethylene can desorb or react with atomic hydrogen to form an ethyl intermediate. This intermediate can react again with a second hydrogen atom to form ethane, which has little interaction with the Pd surface and therefore desorbs instantaneously (step [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF]).

Even though the reaction mechanism is still rather simple, the derivation of the corresponding rate-equation requires several assumptions. We assume that the adsorption take place according to the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part 1. Solids[END_REF]. The next assumption is with respect to the rate-determining step for the formation of ethylene and ethane. Since for both steps the reaction order in hydrogen was found to be larger than 1, the additions of the second hydrogen atom to the vinyl and ethyl intermediates are assumed to be rate-determining (steps ( 4) and [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF], respectively). All other steps are assumed to be in quasi-equilibrium. To reduce the number of parameters in the rate equation only the most abundant reaction intermediates are kept in the site balance. The full site balance is given as:

θ * + θ H + θ C 2 H 2 + θ C 2 H 3 + θ C 2 H 4 + θ C 2 H 5 = 1
Here, we assume that the coverages of vinyl and ethyl intermediates are much smaller than those of adsorbed acetylene and ethylene and thus can be left out of the site balance. A combined DFT Monte-Carlo study for acetylene hydrogenation over Pd(111) at a hydrogen to acetylene ratio of 1, showed that this is indeed the case [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF]. This same study indicates that the hydrogen coverage is not negligible and that it is larger than the ethylene coverage. Here, we take into account the ethylene coverage, because a negative reaction order in ethylene was observed. The last model assumption states that the rate-determining steps, ( 4) and [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF], are irreversible (or one-way) under the given reaction conditions.

The rates for the two rate-determining steps can be written as:

r 4 = k 4 θ C 2 H 3 θ H = k 4 K 1 K 2 K 3 P C 2 H 2 P H 2 θ 2 * r 7 = k 7 θ C 2 H 5 θ H = k 7 K 1 K 5 K 6 P C 2 H 2 P H 2 θ 2 *
and the site balance is given by:

1 = θ * + θ H + θ C 2 H 2 + θ C 2 H 4
or:

θ * = 1 1 + K 2 P C 2 H 2 + P C 2 H 4 K 5 + K 1 P H 2
By introducing the number of palladium surface atoms per catalyst mass, N S (mol Pd s /kg cat ), and attributing all temperature effects to the change of the rate constant in the rate-determining steps, thus assuming that the adsorption equilibrium constants do not change between 51 and 62 • C, the following rate equations are obtained for the consumption of acetylene and the production of ethane, respectively:

-r C 2 H 2 = N S k 0 4 exp -E 4 RT K 1 K 2 P C 2 H 2 P H 2 1 + K 1 P H 2 + K 2 P C 2 H 2 + P C 2 H 4 K 5 2 r C 2 H 6 = N S k 0 7 exp -E 7 RT K 1 K 5 P C 2 H 4 P H 2 1 + K 1 P H 2 + K 2 P C 2 H 2 + P C 2 H 4 K 5 2
Notice that k 4 and k 7 in the above equation are actually a combination of k 4 *K 3 and k 7 *K 6 , respectively. Table 4 gives the correspondence between the reaction orders of the rate equations and the surface coverages as well as the range of reaction orders that are covered by the model.

dependence on coverage 1-2θ C2H2 1-θ H2 -2θ C2H4 -2θ C2H2 1-θ H2 1-2θ C2H4 Min-max (-1)-1 0-1 (-2)-0 (-2)-0 0-1 (-1)-1
The reaction order with respect to acetylene is smaller for the ethane production than for the acetylene consumption, whereas the reaction order with respect to ethylene is larger for the ethane production than for the acetylene consumption, in agreement with the experimental results. Taking the experimental reaction orders from Table 3, the mean coverage averaged over all tested reaction conditions of acetylene, hydrogen and ethylene can be estimated as θ C2H2 = 0.65-0.95, θ H2 ≈ 0 and θ C2H4 = 0.06-0.15.

In order to estimate the 7 parameters in the two rate equations, a multi-response regression analysis of the two data sets at 51 and 62 • C was carried out simultaneously. Initial results showed that all parameters were strongly correlated and no accurate estimates could be determined. Therefore, the hydrogen equilibrium constant was set at a fixed value, calculated from literature data. The following expression is used to calculate the adsorption equilibrium constant:

K i = σ s A S √ 2πM w RT 1 10 13 e (-E d RT ) Pa -1
where σ s is the sticking coefficient, A S the surface area of Pd (1.26 10 4 m 2 /mol), M w the molecular weight (kg/mol) and E d the desorption activation energy (J/mol). A typical value for the pre-exponential factor for desorption of 10 13 s -1 was assumed. Assuming a sticking coefficient for hydrogen adsorption of 0.16 (0.1-0.2 [START_REF] Conrad | Adsorption of hydrogen on palladium single crystal surfaces[END_REF] and 0.17 [START_REF] Michalakab | Uptake, transport, and release of hydrogen from Pd(100)[END_REF]) and a desorption energy of 69 kJ/mol [START_REF] Chou | Calorimetric heat of adsorption measurements on palladium: I. Influence of crystallite size and support on hydrogen adsorption[END_REF], a value of 611 Pa -1 was estimated for K 1 . Fixing the adsorption equilibrium constant for hydrogen forces the model to account for the hydrogen coverage, else due to the hydrogen reaction order ≥1 the hydrogen coverage would be close to zero. Further regression analysis still showed an unacceptable correlation between the parameters. To get an accurate estimate of the adsorption equilibrium constant for acetylene the adsorption equilibrium constant for ethylene had to be set at a fixed value. Although it is not evident from the two rate equations, the correlation between these two equilibrium constants (K 2 and K 5 ) can be revealed by expressing the ethane selectivity as:

S C 2 H 6 = r C 2 H 6 r C 2 H 2 + r C 2 H 6 ≈ r C 2 H 6 r C 2 H 2 = k 7 k 4 P C 2 H 4 P C 2 H 2 1 K 2 K 5
Apparently, the rate constants k 7 and k 4 can be decoupled by fitting the conversion and ethane production but not the term K 2 *K 5 . However, the absolute values of k 7 and k 4 will depend on the values of K 2 and K 5 .

As stated above, ethylene is adsorbed at low coverages as a di-σ species or at high coverage as a π-bonded species with adsorption enthalpies of -60 and -30 kJ/mol, respectively [START_REF] Neurock | A First Principles Analysis of C-H Bond Formation in Ethylene Hydrogenation[END_REF]. A TPD study gave a value of the adsorption enthalpy of -59 kJ/mol [START_REF] Tysoe | Structural, kinetic, and reactive properties of the palladium(111)-ethylene system[END_REF]. This value was used to estimate the equilibrium constant for ethylene at a value of 0.14 Pa -1 . Table 5 reports the values of the parameter estimates with their 95% confidence intervals from the regression analysis of all data. The parameters can be accurately estimated with 95% confidence intervals at the 10% level for those related the acetylene conversion and 20% for the ethane production, similar as the relative standard deviations on the repeated runs (Table 1). No strong parameter correlation was observed; the highest value of 0.85 was between k 0

4 and E 4 . The value of the equilibrium adsorption constant for acetylene is approximately 3 orders of magnitude higher than that of ethylene. This corresponds to an enthalpy of adsorption for acetylene of (-80)-(-90) kJ/mol. Vattuone et al. [START_REF] Vattuone | Energetics and kinetics of the interaction of acetylene and ethylene with Pd{100} and Ni{100}[END_REF] measured differential heat of adsorptions of acetylene over a single crystal of Pd(100) by calorimetry from 110-40 kJ/mol, decreasing with increasing acetylene coverage. An adequate fit of the data was obtained, as shown in Figures 123456, organized per inlet flow of acetylene, ethane and hydrogen for both the acetylene conversion and ethane production. Although the model corresponds to a hydrogen reaction order of approximately 1, both the acetylene conversion and ethane production are well fitted by the model, especially at 51 • C. The power law model gave a significantly larger value for the hydrogen reaction order (~1.45 at 51 • C, Table 2). The cause of this discrepancy between the two models is not clear. The surface coverages as calculated by the model are in the range of θ C2H2 = 0.62-0.95, θ H2 = 0.002-0.01 and θ C2H4 = 0.04-0.36. This is in good agreement with the surface coverages as calculated from the reaction orders. The apparent activation energy for the conversion of acetylene compares well with the value of 40 kJ/mol reported in the literature for acetylene hydrogenation over a Pd foil [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF].

The proposed reaction mechanism based on a single site can adequately represent the experimental data, as shown in Figures 123456. Neurock and coll. [START_REF] Mei | First-principles-based kinetic Monte Carlo simulation of the selective hydrogenation of acetylene over Pd(111)[END_REF] could describe independent experimental data correctly by the same reaction mechanism as the one proposed in this study using also a single site. Bos et al. [START_REF] Bos | A kinetic study of the hydrogenation of ethyne and ethene on a commercial Pd/Al 2 O 3 catalyst[END_REF] could describe their experimental data by very similar rate equations as used here, in the absence of carbon monoxide. However, the addition of carbon monoxide to the feed resulted in a change of the ethane selectivity, which cannot be explained by the above model. As mentioned in the introduction, it could well be that the second site is related to the presence of carbon monoxide on the surface, which is known to have a strong electronic effect and can cause surface reconstruction.

Numerous other studies [START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts. Part 1. Effect of Changes to the Catalyst During Reaction[END_REF][START_REF] Bos | Mechanism and kinetics of the selective hydrogenation of ethyne and ethane[END_REF][START_REF] Borodzinski | The kinetic model of hydrogenation of acetylene-ethylene mixtures over palladium surface covered by carbonaceous deposits[END_REF][START_REF] Borodzinski | Selective Hydrogenation of Ethyne in Ethene-Rich Streams on Palladium Catalysts, Part 2: Steady-State Kinetics and Effects of Palladium Particle Size, Carbon Monoxide, and Promoters[END_REF][START_REF] Pachulski | Kinetics and reactor modeling of a Pd-Ag/Al 2 O 3 catalyst during selective hydrogenation of ethyne[END_REF] and notably the model proposed by Borodzi ński and Cybulski [START_REF] Borodzinski | The kinetic model of hydrogenation of acetylene-ethylene mixtures over palladium surface covered by carbonaceous deposits[END_REF] are based on two distinct active sites. We therefore performed a regression analysis of the model proposed by Borodzi ński and Cybulski to our data set (we used the equations given in the Appendix A of their article, which differ from the text) [START_REF] Borodzinski | The kinetic model of hydrogenation of acetylene-ethylene mixtures over palladium surface covered by carbonaceous deposits[END_REF]. The regression analysis showed that the parameters associated with the second site were not statistically significant and the model was reduced to a single site model. The arguments of Borodzi ński and Cybulski [START_REF] Borodzinski | The kinetic model of hydrogenation of acetylene-ethylene mixtures over palladium surface covered by carbonaceous deposits[END_REF] for a two-site model was that they observed a lack of the effect of the partial pressure of ethylene on r C2H2 /P H2 , indicating that the coverage of active sites by ethylene is negligible, while they observed an effect of the partial pressure of ethylene on r C2H4 /P H2 . The latter effect is also observed in this study and corresponds to the data in Figure 4. The rate of ethane production increases with the partial pressure of ethylene (the partial pressure of ethylene is proportional to the molar flow of ethylene in Figure 4, where the partial pressure of acetylene and the partial pressure of hydrogen are constant). Contrary to Borodzi ński and Cybulski, we do observe an effect of the partial pressure of ethylene on r C2H2 /P H2 . This is demonstrated in Figure 7 where we trace both the data of Borodzi ński and Cybulski [START_REF] Borodzinski | The kinetic model of hydrogenation of acetylene-ethylene mixtures over palladium surface covered by carbonaceous deposits[END_REF] and our data for r C2H2 /P H2 as a function of the partial pressure of ethylene. Despite the rather different conditions of both studies, the r C2H2 /P H2 values are very similar. However, whereas r C2H2 /P H2 is independent of the ethylene partial pressure in the case of the data of Borodzi ński and Cybulski, in our case, r C2H2 /P H2 decreases with increasing ethylene partial pressure. A single site mechanism is thus validated for the conditions used in this study, or at least if two sites are present on the catalyst, the adsorption behavior of ethylene and acetylene on both sites is not different enough to be distinguished from the experiments under these conditions. Borodzi ński and Cybulski [START_REF] Borodzinski | The kinetic model of hydrogenation of acetylene-ethylene mixtures over palladium surface covered by carbonaceous deposits[END_REF] attributed the different active sites on palladium to the deposit of carbonaceous species, creating pockets of different sizes that induce size dependent reactivity. In our study, the amount of carbonaceous species might be rather low, due to the short time on stream of the catalyst and the much higher stoichiometric ratio of hydrogen to acetylene. This is also consistent with the reaction order of hydrogen greater than 1, which was explained by Molero et al. [START_REF] Molero | The Hydrogenation of Acetylene Catalyzed by Palladium: Hydrogen Pressure Dependence[END_REF] by excess hydrogen that removes adsorbed carbonaceous species from the catalyst surface and creates free palladium sites.

Materials and Methods

Experimental set-up and testing

Kinetic experiments for acetylene hydrogenation were carried out in an integral fixed bed reactor over 0.05 wt. % Pd/α-Al2O3 catalysts under typical tail-end conditions. During the experiment, 0.1 g of catalyst sieved between 100 and 200 µm was used, diluted with 0.5 g of the α-Al2O3 support, crushed and sieved identically, to improve the isothermicity of the catalytic bed. The fixed-bed reactor consisted of a quartz tube with an inner diameter of 4 mm, inserted in a cylindrical oven. A thermocouple was inserted into the catalyst bed. The reaction was carried out at 51 and 62 °C. A flow consisting of different ratios of acetylene, ethylene, hydrogen, and argon, using mass flow controllers, at a total flow rate of 200 mL/min was used and the reactor was operated at 1.26 bar. The experiment at standard conditions was performed regularly to check the stability of the catalyst and the data repeatability. The reactor was loaded twice with a fresh catalyst sample and the experiments were conducted twice. Before the experiments the catalyst was reduced under a flow of 100 mL/min of 50%

Materials and Methods

Experimental set-up and testing

Kinetic experiments for acetylene hydrogenation were carried out in an integral fixed bed reactor over 0.05 wt. % Pd/α-Al 2 O 3 catalysts under typical tail-end conditions. During the experiment, 0.1 g of catalyst sieved between 100 and 200 µm was used, diluted with 0.5 g of the α-Al 2 O 3 support, crushed and sieved identically, to improve the isothermicity of the catalytic bed. The fixed-bed reactor consisted of a quartz tube with an inner diameter of 4 mm, inserted in a cylindrical oven. A thermocouple was inserted into the catalyst bed. The reaction was carried out at 51 and 62 • C. A flow consisting of different ratios of acetylene, ethylene, hydrogen, and argon, using mass flow controllers, at a total flow rate of 200 mL/min was used and the reactor was operated at 1.26 bar. The experiment at standard conditions was performed regularly to check the stability of the catalyst and the data repeatability. The reactor was loaded twice with a fresh catalyst sample and the experiments were conducted twice. Before the experiments the catalyst was reduced under a flow of 100 mL/min of 50% hydrogen in argon at 150 • C for 2 h. The catalyst activity and selectivity were found to be stable during the kinetic runs.

Online analysis was done by passing the outlet gas flow through a small volume infrared gas cell connected to a FTIR spectrometer. Quantitative FTIR analysis of acetylene, ethylene and ethane was carried out. No C4 products were observed. The carbon balance was closed within 5%. Experimental conditions are given in Table 6. By using appropriate criteria, the kinetic measurements were shown to be free from heat and mass transfer limitations [START_REF] Schuurman | Aspects of kinetic modeling of fixed bed reactors[END_REF]. This can be easily verified using the tool developed by Eurokin [START_REF]Fixed-Bed Web Tool at the Eurokin Website[END_REF]. The maximum observed rate of acetylene consumption of 9 mmol/kg cat /s was used to check the criteria given in Table 7. As shown in Table 7 all criteria were met. 3.3 × 10 -4 < 0.05 internal mass transfer (Weisz-Prater) 1.7 × 10 -2 < 0.33 external heat transfer (film) 0.14 K < 1.1 K external heat transfer (radial) 0.17 K < 1.1 K internal heat transfer 3 × 10 -3 K < 1.1 K

Catalyst

0.05 wt. % Pd supported on α-alumina was used as a catalyst. It was prepared by impregnation of the alumina by an aqueous palladium nitrate solution. The material was then dried at 120 • C under ambient air and calcined at 425 • C for 2 h. The detailed protocol has been previously described together with extensive XRD and HRTEM characterization [START_REF] Ramos-Fernandez | Structural and Morphological Characterization of Alumina Supported Pd Nanoparticles Obtained by Colloidal Synthesis[END_REF][START_REF] Didillon | Studies in Surface Science and Catalysis[END_REF]. The alumina support has a porous volume of 0.54 mL/g and a BET surface area of 10 m 2 /g. Electron Transmission Microscopy analysis showed a mean particle diameter of 2.4 ± 0.7 nm, corresponding to a dispersion of ca. 40%. From this and the wt. % of Pd the number of surface palladium atoms was calculated as N S = 2 × 10 -3 mol/kg. The catalyst was sieved between 100 and 200 µm to improve the heat transfer during the kinetic experiments. Runs under standard conditions using the original catalyst spheres in a single pellet reactor configuration showed similar performance as the crushed sample.

Modeling

An integral reactor operation was used and the rate equations have been integrated numerically using the ODEPACK library [START_REF] Hindmarsh | A Systematized Collection of ODE Solvers[END_REF]. A one-dimensional homogeneous reactor model has been used, in agreement with the absence of mass and heat transfer limitations. A non-linear least-square multi-response regression analysis has been performed by a Levenberg-Marquardt minimization algorithm [START_REF] Levenberg | A Method for the Solution of Certain Non-Linear Problems in Least Squares[END_REF][START_REF] Marquardt | An algorithm for least-squares estimation of nonlinear parameters[END_REF]. The fractional acetylene conversion and the molar exit flow of ethane (µmol/s) have been used for the objective function. No weighing factor has been applied. After regression analysis several statistical tests were performed, including the t-test, the 95% confidence intervals of the parameter estimates, the F-value, and binary correlation coefficients.

Conclusions

The kinetics of the selective hydrogenation of acetylene over a 0.05 wt. % Pd/α-Al 2 O 3 catalyst was studied under intrinsic kinetic conditions. The operating conditions were chosen according to the tail-end process. Analysis of the experimental reaction orders gave good insight into the reaction mechanism. It allowed us to estimate the mean surface coverages of hydrogen, acetylene, and ethylene. The proposed reaction mechanism consists of a series of sequential hydrogen additions according to the Horiuti-Polanyi mechanism. The derived rate equations are based on the addition of the second hydrogen atom as rate-determining step for both acetylene and ethylene hydrogenation. Accurate parameter estimation was only possible after fixing the values of the equilibrium adsorption constant of ethylene. The relatively simple Langmuir-Hinshelwood-type rate-equations describe the experimental data adequately. It is based on physically meaningful parameters. The reaction mechanism involves only one active site, which was carefully verified.

Catalysts 2019, 9 , 13 Figure 1 .

 9131 Figure 1. Evolution of acetylene conversion as a function of the acetylene inlet molar flow. Circles: data at 62 °C, squares: data at 51 °C.

Figure 2 .

 2 Figure 2. Ethane outlet molar flow as a function of the acetylene inlet molar flow. Circles: data at 62 °C, squares: data at 51 °C.

Figure 1 . 13 Figure 1 .

 1131 Figure 1. Evolution of acetylene conversion as a function of the acetylene inlet molar flow. Circles: data at 62 • C, squares: data at 51 • C.

Figure 2 .

 2 Figure 2. Ethane outlet molar flow as a function of the acetylene inlet molar flow. Circles: data at 62 °C, squares: data at 51 °C.

Figure 2 .

 2 Figure 2. Ethane outlet molar flow as a function of the acetylene inlet molar flow. Circles: data at 62 • C, squares: data at 51 • C.

Figure 3 .

 3 Figure 3. Evolution of acetylene conversion as a function of the ethylene inlet molar flow. Circles: data at 62 °C, squares: data at 51 °C.

Figure 4 .

 4 Figure 4. Ethane outlet molar flow as a function of the ethylene inlet molar flow. Circles: data at 62 °C, squares: data at 51 °C.

Figure 3 . 13 Figure 3 .

 3133 Figure 3. Evolution of acetylene conversion as a function of the ethylene inlet molar flow. Circles: data at 62 • C, squares: data at 51 • C.

Figure 4 .

 4 Figure 4. Ethane outlet molar flow as a function of the ethylene inlet molar flow. Circles: data at 62 °C, squares: data at 51 °C.

Figure 4 . 13 Figure 5 .

 4135 Figure 4. Ethane outlet molar flow as a function of the ethylene inlet molar flow. Circles: data at 62 • C, squares: data at 51 • C. Catalysts 2019, 9, x FOR PEER REVIEW 9 of 13

Figure 5 .

 5 Figure 5. Evolution of acetylene conversion as a function of the hydrogen inlet molar flow. Circles: data at 62 • C, squares: data at 51 • C.

Figure 5 .

 5 Figure 5. Evolution of acetylene conversion as a function of the hydrogen inlet molar flow. Circles: data at 62 °C, squares: data at 51 °C.

Figure 6 .

 6 Figure 6. Ethane outlet molar flow as a function of the hydrogen inlet molar flow. Circles: data at 62 °C, squares: data at 51 °C.

Figure 6 . 13 Figure 7 .

 6137 Figure 6. Ethane outlet molar flow as a function of the hydrogen inlet molar flow. Circles: data at 62 • C, squares: data at 51 • C. Catalysts 2019, 9, x FOR PEER REVIEW 10 of 13

Figure 7 .

 7 Figure 7. Plots of r C2H2 /P H2 vs. ethylene partial pressure. Open symbols: data from Borodzi ński and Cybulski [4]: T = 70 • C, P C2H2 = 0.02 kPa, P H2 = 0.64 kPa. Closed symbols: data from this study: T = 62 • C, P C2H2 = 0.95 kPa, P H2 = 5.5 kPa. The dotted horizontal line indicates the extrapolated average values of r C2H2 /P H2 for the data from Borodzi ński and Cybulski [4].

Table 1 .

 1 Relative standard deviation (rsd) of 6 repeated experiments (3 for each catalyst loading, 2 loadings) for both conversion and ethane outlet flow at 2 conversion levels.

		Operating	Operating	Operating	Operating
		Conditions 1	Conditions 2	Conditions 1	Conditions 2
	T ( • C)	X C2H2	rsd (%)	X C2H2	rsd (%)	F C2H6	rsd (%)	F C2H6	rsd (%)
	51	0.13	9.3	0.36	6.2	0.07	19	0.13	11
	62	0.23	7.8	0.37	8.5	0.11	16	0.16	15

Table 2 .

 2 Estimated values of the reaction orders with their 95% confidence intervals.

	T ( • C)	r C2H2 α 1 (C 2 H 2 )	β 1 (H 2 )	γ 1 (C 2 H 4 )	r C2H6 α 2 (C 2 H 2 )	β 2 (H 2 )	γ 2 (C 2 H 4 )
	51	-0.88 ± 0.09	1.46 ± 0.16	-0.13 ± 0.07 -1.00 ± 0.20 1.42 ± 0.70 1.09 ± 0.50
	62	-0.93 ± 0.07	1.49 ± 0.34	-0.19 ± 0.04 -1.56 ± 0.13 1.69 ± 0.88 0.30 ± 0.15

Table 3 .

 3 Elementary steps for the reaction C 2 H 2 + H 2

		1 (C2H2)	β1 (H2)	γ1 (C2H4)	2 (C2H2)	β2 (H2)	γ2 (C2H4)
	51	-0.88 ± 0.09 1.46 ± 0.16 -0.13 ± 0.07 -1.00 ± 0.20 1.42 ± 0.70 1.09 ± 0.50
	62	-0.93 ± 0.07 1.49 ± 0.34 -0.19 ± 0.04 -1.56 ± 0.13 1.69 ± 0.88 0.30 ± 0.15

Table 3 .

 3 Elementary steps for the reaction C2H2 + H2  C2H4 and C2H4 + H2  C2H6.

	N°	Elementary Step	
	1	H2 + 2*  2H*	2
	2	C2H2 + *  C2H2*	1
	3	C2H2* + H*  C2H3* + *	1
	4	C2H3* + H*  C2H4* + *	1
	5	C2H4*  C2H4 + *	1
	6	C2H4* + H*  C2H5* + *	1
	7	C2H5* + H*  C2H6 + 2*	1

  ± 0.09 1.46 ± 0.16 -0.13 ± 0.07 -1.00 ± 0.20 1.42 ± 0.70 1.09 ± 0.5062-0.93 ± 0.07 1.49 ± 0.34 -0.19 ± 0.04 -1.56 ± 0.13 1.69 ± 0.88 0.30 ± 0.15

	Catalysts 2019, 9, x FOR PEER REVIEW					3 of 13
		1 (C2H2)	β1 (H2)	γ1 (C2H4)	2 (C2H2)	β2 (H2)	γ2 (C2H4)
	51	-0.88
					2H*	2
			2	C 2 H 2 + *

  ± 0.09 1.46 ± 0.16 -0.13 ± 0.07 -1.00 ± 0.20 1.42 ± 0.70 1.09 ± 0.5062-0.93 ± 0.07 1.49 ± 0.34 -0.19 ± 0.04 -1.56 ± 0.13 1.69 ± 0.88 0.30 ± 0.15

	Catalysts 2019, 9, x FOR PEER REVIEW					3 of 13
		1 (C2H2)	β1 (H2)	γ1 (C2H4)	2 (C2H2)	β2 (H2)	γ2 (C2H4)
	51	-0.88
					C 2 H 4 * + *	1
			5		C 2 H 4 *

Table 3 .

 3 Elementary steps for the reaction C2H2 + H2  C2H4 and C2H4 + H2  C2H6.

	N°	Elementary Step	
	1	H2 + 2*  2H*	2
	2	C2H2 + *  C2H2*	1
	3	C2H2* + H*  C2H3* + *	1
	4	C2H3* + H*  C2H4* + *	1
	5	C2H4*  C2H4 + *	1
	6	C2H4* + H*  C2H5* + *	1
	7	C2H5* + H*  C2H6 + 2*	1

Table 3 .

 3 Elementary steps for the reaction C2H2 + H2  C2H4 and C2H4 + H2  C2H6.

	N°	Elementary Step	
	1	H2 + 2*  2H*	2
	2	C2H2 + *  C2H2*	1
	3	C2H2* + H*  C2H3* + *	1
	4	C2H3* + H*  C2H4* + *	1
	5	C2H4*  C2H4 + *	1
	6	C2H4* + H*  C2H5* + *	1
	7	C2H5* + H*  C2H6 + 2*	1

Table 3 .

 3 Elementary steps for the reaction C2H2 + H2  C2H4 and C2H4 + H2  C2H6.

	N°	Elementary Step	
	1	H2 + 2*  2H*	2
	2	C2H2 + *  C2H2*	1
	3	C2H2* + H*  C2H3* + *	1
	4	C2H3* + H*  C2H4* + *	1
	5	C2H4*  C2H4 + *	1
	6	C2H4* + H*  C2H5* + *	1
	7	C2H5* + H*  C2H6 + 2*	1

Table 4 .

 4 Reaction orders corresponding to the derived rate-equations.

	Reaction Order	C 2 H 2	r C2H2 H 2	C 2 H 4	C 2 H 2	r C2H6 H 2	C 2 H 4

Table 5 .

 5 Parameter estimates with their 95% confidence intervals.

		Parameter	Estimated Value
	k 0	K 1 (Pa -1 ) K 2 (Pa -1 ) 4 (mol/mol Pd s /s)	13 (fixed) 107 ± 7 1.4 ± 0.1 × 10 8
	k 0	K 5 (Pa) 7 (mol/mol Pd s /s)	7.4 (fixed) 3.6 ± 0.6 × 10 9
		E 4 (kJ/mol)	48.5 ± 4.2
		E 7 (kJ/mol) N S (mol/kg)	54.8 ± 11 2 × 10 -3 (fixed)

Table 6 .

 6 Experimental conditions.

	Variable	Values
	T ( • C)	51, 62
	F C 2 H 2 (µmol/s)	0.7-3
	F H 2 (µmol/s)	1.6-10
	F C 2 H 4 (µmol/s)	29-105
	F Ar (µmol/s)	34-115
	P (bar)	1.26
	W cat (g)	0.1
	F tot (µmol/s)	149

Table 7 .

 7 Criteria to assess the conditions for the absence of mass and heat transfer limitations.

	Physical Phenomenon	Criteria
	plug flow (axial)	h/d p = 193 > 6
	plug flow (radial)	d R /d p = 40 > 10
	degree of dilution (vol./vol.)	0.89 < 0.97
	external mass transfer (Carberry number)

Funding: This research received no external funding.

Conflicts of Interest:

The authors declare no conflict of interest.