C catalyst

Applied Catalysis A: General 353 (2009) 166–180

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Applied Catalysis A: General journal homepage: www.elsevier.com/locate/apcata

Kinetics of linoleic acid hydrogenation on Pd/C catalyst Andreas Bernas a, Jukka Myllyoja b, Tapio Salmi a, Dmitry Yu. Murzin a,* a b

A˚bo Akademi University, Process Chemistry Centre, FI-20500 A˚bo/Turku, Finland Neste Oil Oy, Technology Centre, FI-06101 Borga˚/Porvoo, Finland

A R T I C L E I N F O

A B S T R A C T

Article history: Received 1 July 2008 Received in revised form 21 September 2008 Accepted 21 October 2008 Available online 13 November 2008

The hydrogenation of linoleic acid was investigated in semi-batch slurry reactors over 5 wt% palladium on carbon (Pd/C) catalyst in temperature and H2 pressure ranges of 40–100 8C and 0.5–20 bar using ndecane as a solvent. The reaction network involves hydrogenation of linoleic acid to monoenoic acid isomers, predominantly oleic acid, and further hydrogenation of monoenoic acids to stearic acid. The reaction kinetics was established in conditions free from diffusional limitations. The rate was temperature and pressure dependent. A reaction network and mechanisms were proposed and corresponding kinetic equations were derived. The parameters of the mechanistic kinetic models were determined by using non-linear regression analysis. The concentrations of linoleic acid, monoenoic acids, and stearic acid were used in parameter estimation. Data fitting allowed discrimination between rival mechanistic models, more specifically the influence of the hydrogen addition. The kinetic models described the formation of the products with satisfying accuracy. ß 2008 Elsevier B.V. All rights reserved.

Keywords: Palladium catalyst Hydrogenation Linoleic acid Mathematical modeling

1. Introduction Linoleic acid (cis-9,cis-12-octadecadienoic acid) is an unsaturated omega-6 carboxylic acid with an 18-carbon chain and two cis double bonds. The first double bond is located at the sixth carbon from the omega end. Linoleic acid is a chief constituent of tall oil and vegetable oils. Oils and foods that contain linoleic acid include safflower oil (78%), poppy seed oil (70%), walnut oil, grass fed cow milk, olive oil, palm oil, sunflower oil, soybean, lard, coconut oil, egg yolks (16%), spirulina, peanut oil, okra, rice bran oil, wheat germ oil, grape seed oil, macadamia oil, pistachio oil, sesame oil. Linoleic acid is used in making soaps, emulsifiers, and quick-drying oils [1]. Linoleic acid has a potential as a raw material for stearic acid (octadecanoic acid) production. Stearic acid, which is completely saturated 18-carbon carboxylic acid, is conventionally prepared by treating animal fat with water at a high pressure and temperature, leading to the hydrolysis of triglycerides, although it can be obtained from the hydrogenation of some unsaturated vegetable oils. Stearic acid is useful as an ingredient in making candles, soaps, plastics, oil pastels and cosmetics, and for softening rubber to name a few examples. It is used to harden soaps, particularly those made with vegetable oils and esters of stearic acid with ethylene glycol, glycol stearate and glycol distearate are used to produce a pearly

* Corresponding author. Tel.: +358 2 215 4985; fax: +358 2 215 4479. E-mail address: dmurzin@abo.fi (D.Yu. Murzin). 0926-860X/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.apcata.2008.10.059

effect in shampoos, soaps, and other cosmetic products. Moreover, stearic acid has recently been used as a model compound in catalytic deoxygenation for production of so called green diesel [2]. In general, reactions with carboxylic acids over heterogeneous catalysts are difficult to perform. A recent innovation [3] has opened new dimensions for production of the anticarcinogenic and antioxidative cis-9,trans-11- and trans-10,cis-12-conjugated linoleic acid isomers by isomerization of linoleic acid over supported metal catalysts. This isomerization was promoted by chemisorbed hydrogen on the metal surface. At the same time, hydrogenation of linoleic acid was a competing side reaction and it was noticed that hydrogenation of linoleic acid was promoted by metals with high hydrogen adsorption capacity such as Pd, especially when supported by active carbon. As demonstrated in Fig. 1, the network of catalytic steps in linoleic acid hydrogenation involves a rivalry between isomerization to conjugated linoleic acid (CLA) and double bond hydrogenation of linoleic acid. CLA isomers do also undergo isomerization steps as well as hydrogenation to monoenoic species. Further, isomerization of monoenoic acids occurs. As can be seen in the figure, the reaction network involves the following steps: (1) double bond migration of linoleic acid to conjugated linoleic acid isomers, (2) double bond hydrogenation of linoleic acid to monoenoic acid isomers, and (3) double bond hydrogenation of monoenoic acid isomers to stearic acid. Moreover there exists (4) double bond hydrogenation of conjugated linoleic acid isomers to monoenoic acid isomers, (5) positional and geometric isomerization of conjugated linoleic acid isomers, (6)

A. Bernas et al. / Applied Catalysis A: General 353 (2009) 166–180

Nomenclature A A B

frequency factor linoleic acid monoenoic acids (oleic acid, elaidic acid, cis-vaccenic acid, and trans-vaccenic acid) C stearic acid c concentration activation energy Ea Z vacant surface site for chemisorption of organic molecule vacant surface site for chemisorption of hydrogen Z0 H atomic hydrogen I number of intermediates k kinetic constant K equilibrium factor l objective function m total catalyst mass N(i) independent basic route i P number of basic routes r consumption/generation rate the gas constant Rgas degree of explanation R2 s state of the system S number of stages SRS sum of squares of the residuals between the model and data T reaction temperature mean temperature of the experiments Tmean w weight matrix for the observations W number of balance equations x design variables experimental points xjk yijk observations y¯ average of data points response variables yp jjy  y p jj2w weighted norm notation ZA chemisorbed linoleic acid chemisorbed hydrogenated reaction intermediate ZAH2 ZB chemisorbed monoenoic acids ZBH2 chemisorbed hydrogenated reaction intermediate ZX chemisorbed half-hydrogenated reaction intermediate ZY chemisorbed half-hydrogenated reaction intermediate

Greek letters u fractional surface coverage u calculated parameters u0 coverage of vacant sites u0H coverage of hydrogen J chemical equilibrium

positional and geometric isomerization of monoenoic acid isomers. The main CLA compounds are the cis-9,trans-11-, trans-10,cis-12-, cis-9,cis-11-, and trans-9,trans-11-CLA isomers, at the same time several other conjugated dienoic isomers of linoleic acid are generated. The monoenoic compounds are the oleic, elaidic, cisand trans-vaccenic acids.

167

Catalytic hydrogenation of vegetable oils has been studied intensively in the literature [4]. The main emphasis in these studies has been on hardening of oils in order to improve their stability and melting behavior [4]. There are few systematic studies, however, of the catalyst properties suitable for production of stearic acid from linoleic acid, showing in particular applicability of supported Pd catalysts [5,6]. The kinetic aspects of liquid-phase hydrogenation of fatty acids, mainly stereoselectivity, were addressed in the literature [7]. The aim of the present paper is to investigate the kinetics of Pd/ carbon catalyzed linoleic acid hydrogenation at varied conditions and to present a kinetic model, which is consistent with mechanistic data and observed kinetic regularities. 2. Experimental 2.1. Semi-batch steel reactor The catalytic hydrogenation was studied using analytical grade (assay 99%) linoleic acid provided by Fluka and 5 wt% Pd/C powder catalyst (Aldrich). This catalyst had a surface area of 936 m2/g according to the BET equation and 1214 m2/g according to the Dubinin method. As measured by H2-TDP, the H2 adsorption capacity was 0.173 mmol H2/g catalyst and 3.46 mmol H2/g metal. The maximum H2 desorption signal occurred at 267 8C. Hydrogenation experiments were carried out at varied pressure in a three-phase 300 ml steel Parr autoclave in order to investigate the dependence of the hydrogenation rate on the hydrogen pressure. The reactor could operate at a maximum pressure of 200 bar, in the temperature range of 10 to 350 8C and was equipped with a cooling coil and a 5 mm filtered reactor inlet line, which avoided the pass of the catalyst particles through it. A propeller improved the gas dispersion and could be regulated to work with a stirring rate in the range 50–2000 rpm. The temperature was measured with a thermocouple and the heating jacket around the reactor was automatically controlled with a Brooks instrument. A Brooks 5866 flow controller regulated the hydrogen pressure in the reactor, but the non-efficient automatic control speed could be improved accessing to the desired pressure with a manual regulation of the hydrogen bottle valve. Temperature and pressure profiles were both registered. A bubbling and preheating unit was installed upstream to the reactor in order to feed the reactant solution. A heating jacket could be warmed up to control the temperature in the unit, and hydrogen or argon was introduced in the reactor through it, in a way that the solution could be flashed with argon to remove the oxygen dissolved, or with hydrogen to reach the saturation state. The sample line was rounded by an electrical heating to avoid its plugging in case stearic acid crystals were formed. In addition, the reactor had two gas flow outlets, one of them with a Brooks flow controller, which was used in the catalyst reduction step, and the other without any controlling device, was applied when a fast removal of the gas in the reactor was needed. In a typical semi-batch experiment, the Pd/C catalyst was sieved to a particle size of less than 63 mm and dried in an oven at 100 8C overnight before introducing in the reactor. The reactor was flushed with argon in order to remove the remaining oxygen. The catalyst was reduced inside the reactor under hydrogen pressure above 5 bar and 200 ml/min outflow. The temperature in the reactor was programmed to increase with a heating rate of 10 8C/ min till 200 8C, kept at 200 8C for two hours and cooled down. 400 mg of analytical grade linoleic acid was dissolved in 140 ml of n-decane as to have a reactant solution concentration of 0.01 M. After the in situ catalyst reduction, the reactant solution was introduced in the bubbling chamber, bubbled with hydrogen for few minutes to prevent catalyst oxidation in the presence of air,

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Fig. 1. Linoleic acid isomerization and hydrogenation network in presence of solid catalyst.

preheated to 50 8C and fed inside the reactor. Temperature and pressure were adjusted to experimental conditions, and the stirring was commenced at 20 8C before reaching the desired reaction temperature, as a fast temperature increase was taking place in the moment the hydrogen was in contact with the catalyst surface, due to the high reaction exothermicity. The stirring rate was regulated to 1000 rpm to avoid external mass transfer limitations. The reactant solution was introduced into the reactor under stirring and the reaction time was initialized to zero as soon as the liquid phase came in contact with the catalyst. The course of the reaction was followed by withdrawing samples smaller than 0.1 ml out of a total of 140 ml from the mixture periodically through the catalyst filter to await analysis by gas chromatography, hence the liquid-to-catalyst ratio could be considered to remain constant. 2.2. Semi-batch glass reactor Linoleic acid hydrogenation experiments were conducted in an isothermal 200-ml completely back-mixed at atmospheric pressure operated semi-batch glass reactor, which was provided with a reflux condenser system and a heating jacket using silicone oil as a heat transfer fluid. The aim was to investigate the temperature dependence only at atmospheric pressure since the reactions conducted in the autoclave were very fast. The system was equipped with a gas outlet lock, two gas inlets and stirring baffles to break the vortex and prevent the solid body rotation of the fluid. The temperature was measured with a Pt-100 thermocouple. Hydrogen and nitrogen gas flows were manipulated with calibrated Fischer Porter rotameters and the gases could be led to a separated bubbling unit to saturate the reactant solution before introducing it inside the reactor. The temperature of the reflux condensers cooling medium was set to 20 8C. In a typical experiment, the catalyst was charged into the reactor. Catalyst reduction in situ took place by heating under hydrogen at atmospheric pressure using the same temperature program as in the steel reactor mentioned above. 400 mg of analytical grade linoleic acid was dissolved in 140 ml of n-decane in the bubbling unit. Air above reaction mixture and oxygen dissolved in the liquid phase were purged out by a nitrogen flow of 100 ml/min through the sinter of the bubbling unit while cooling the reactor to the reaction temperature 40, 60 or 80 8C. The liquid phase containing linoleic acid and the solvent was fed into the

reactor and the reaction time was initialized to zero. For calculation of the concentration versus time dependence, samples smaller than 1 ml were withdrawn from the reactor at certain intervals through a sampling valve equipped with a 7 mm catalyst filter to await analysis by gas chromatography. The system was agitated at a stirring rate of 1000 rpm to keep the catalyst uniformly dispersed in the reaction medium and to eliminate effects of external mass transfer. In previous studies investigating linoleic acid isomerization over Ru/C and Ru/Al2O3 catalysts under similar conditions and with the same glass reactor, the absence of external and internal mass transfer was verified by comparing reaction rates at varied agitation rates, catalyst quantities, as well as catalyst particle sizes [3]. When the reaction was carried out at stirring rates 400, 600, 800, and 1000 rpm, otherwise using the same reaction conditions, it was clear from the results that the diluted system was well mixed and it was obvious that the experiments are performed on the plateau of the initial rate against the stirring rate. The initial rate did not depend on the catalyst mass. Rate limitation from resistance of internal diffusion was investigated by performing isomerization experiments at 165 8C in n-decane over 200 mg of hydrogen preactivated Ru/C catalyst samples with different particle diameter intervals from 0 to 45 mm interval to over 180 mm. Catalyst particles smaller than 100 mm gave the same kinetic results. Since the hydrogenation system using Pd/C with a higher reaction rate cannot be directly compared to the Ru catalyzed isomerizations, hydrogenation experiments were carried out at 100 8C and 20 bar using 0.01 M linoleic acid in n-decane, in order to study the mass transfer. The catalyst was sieved and fractions corresponding to particle dimensions below 63 mm, below 90 mm and above 90 mm, as well as unsieved catalyst were tested. The reactant-to-catalyst ratio was 10:1 since 400 mg of analytical grade linoleic acid and 40 mg of catalyst were used. It was concluded that the linoleic acid consumption rate was retarded only by the use of particle sizes above 90 mm. No change in the reaction rate was observed with a stirring rate above 1000 rpm. 2.3. Analysis The samples from the reactor were silylated by using the reagents N,O-bis(trimethylsilyl)trifluoroacetamide (BSTFA) and trimethylchlorosilane (TMCS), both supplied by Acros Organics,

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Table 1 Experimental data on linoleic acid hydrogenation over 5 wt% Pd/C catalyst. N

Reactor

Temperature (8C)

Pressure (bar)

Catalyst mass (g)

1 2 3 4 5 6 7

Glass Glass Glass Glass Steel Steel Steel

40 80 80 60 100 100 100

atm atm H2/N2 = 1 atm 2 10 20

0.02 0.02 0.02 0.04 0.02 0.02 0.02

Conditions: Reactant, 0.4 g of linoleic acid; solvent, 140 ml of n-decane; initial linoleic acid concentration, 0.010187 mol/dm3; hydrogen flow, 100 ml/min, stirring rate, 1000 rpm, reaction time, 120 min.

and analyzed by a gas chromatograph (GC, Hewlett Packard 6890 Series) equipped a 25 meter HP-5 column (inner diameter: 0.20 mm, film thickness: 0.11 mm), flame ionization detector (FID) unit, and an autosampler injector. Helium was used as a carrier gas. The column temperature was initially 150 8C. 0.5 min after the injection, the temperature was increased from 150 to 230 8C with a rate of 7 8C/min, and from 230 to 290 8C with a rate of 10 8C/min. Thereafter the column was purged at 290 8C for 10 min. The silylation operations were conveniently performed with the entire series of samples from each isomerization reaction in parallel. Peaks identities were verified by a gas chromatograph– mass spectrometer system (GC/MS, Hewlett Packard) applying the same GC conditions.

Fig. 2. Semi-batch hydrogenation of linoleic acid over 5 wt% Pd/C catalyst. (&) linoleic acid, (^) oleic acid (~) elaidic acid, cis-vaccenic acid, and trans-vaccenic acid, (*) stearic acid. Conditions: reaction temperature, 40 8C; hydrogen pressure, 1 bar; linoleic acid mass, 0.4 g; catalyst mass, 0.02 g; solvent, 140 ml of n-decane; initial linoleic acid concentration, 0.010187 mol/dm3; hydrogen flow, 100 ml/min, stirring rate, 1000 rpm; reaction time, 120 min.

3. Results and discussion 3.1. Kinetics of linoleic acid hydrogenation Experimental sets for the investigated 5 wt% Pd/C catalyst are presented in Table 1. In a typical run, complete hydrogenation of linoleic to stearic acid took place. The pressure dependence was investigated in the steel reactor and due to the high reaction rate, the temperature dependence was studied at atmospheric pressure in a glass reactor with possibility to get a more accurate temperature control and better pressure control at lower pressures. On entry 3 in Table 1, a special experiment was carried out at atmospheric pressure using 50 vol% of hydrogen and 50 vol% of nitrogen in the feed flow in order to get lower than atmospheric hydrogen pressure. The consecutive hydrogenation kinetics including intermediate products was more pronounced at lower temperatures and pressures. The concentration versus time profile for the experiment at 40 8C, which corresponds to entry 1 in Table 1, is presented in Fig. 2. No other reactions were detected except for hydrogenation of C18:2 linoleic acid to C18:1 fatty acids and further hydrogenation of C18:1 fatty acids to C18:0. The linoleic acid was totally converted after 30 min. The concentrations of the monounsaturated compounds oleic acid, elaidic acid, cis-vaccenic acid, and trans-vaccenic acid initially increased with time and started to decrease after reaching a maximum value, as in a typical consecutive reaction. The concentration of stearic acid increased monotonically and this product started to build up already from the beginning of the reaction. It can be concluded that hydrogenation of linoleic acid proceeds consecutively via monounsaturated acids to stearic acid. Although double bond hydrogenation of linoleic acid is accompanied by conjugation of linoleic acid, no isomerization was detected under these hydrogen rich conditions. In Fig. 2, the monounsaturated fatty acid reached a maximum value at the reaction time of 10 min. At this point the composition of monoenoic acids is as follows: 28 mole% oleic acid, 31% elaidic acid, 38% cis-vaccenic acid, and only 2% trans-vaccenic acid. The

Fig. 3. Selectivity to (a) monoenoic acids and (b) stearic acid versus conversion in linoleic acid hydrogenation on Pd catalyst at 0.5 and 1 bar of hydrogen. Conditions: reaction temperature, 80 8C; linoleic acid mass, 0.4 g; catalyst mass, 0.02 g; solvent, 140 ml of n-decane; initial linoleic acid concentration, 0.010187 mol/dm3; total gas flow, 100 ml/min, stirring rate, 1000 rpm; reaction time, 120 min. Note: 0.5 bar H2 = 50 vol% H2 and 50 vol% N2 at atmospheric pressure.

mass ratio between these monoenes were always constant and conversion independent. Fig. 3 demonstrates the selectivity to monoenoic acids and the selectivity to stearic acid at the hydrogen pressure 1 bar and at the partial hydrogen pressure 0.5 bar. It can be concluded that

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although the pressure is increased by a factor of 2, the selectivity versus conversion dependences are not affected. 3.2. Reaction mechanisms Speaking in terms of molecular mechanisms, the heterogeneously catalytic double bond migration and double bond hydrogenation reactions of linoleic acid to conjugated linoleic acid and hydrogenation products over supported metal catalysts are besides the main pathway over metal sites as well believed to take place through several other routes. If the first step involves a C–H bond cleavage, an allylic intermediate is formed on a supported metal atom or on an acidic site of the support [8,9]. Subsequent hydrogenation at a different carbon atom results in a double bond migration. If linoleic acid adsorbs molecularly, it forms a p complex on the surface, with a C C bond coordinated to a Lewis acid site. If Brønsted acid sites are present, protonation of linoleic acid can occur, resulting in a carbenium ion intermediate where the C C character has been lost. Subsequent loss of a proton from a different carbon results in a double bond migration. The metal-catalyzed hydrogenation is thought to occur predominantly via the Horiuti–Polanyi mechanism, describing hydrogenation and isomerization of olefins. Addition of hydrogen to olefins is a surface reaction involving addition of chemisorbed hydrogen to chemisorbed a olefin. The Horiuti–Polanyi mechanism and the catalytic steps taking place in linoleic acid hydrogenation are demonstrated in Fig. 4. The compound containing one or two double bonds is chemisorbed on the surface of the metal. Thereafter a chemisorbed hydrogen atom is added to the chemisorbed acid to give a chemisorbed half-hydrogenated intermediate. If the hydrogen coverage on the catalyst surface is high (as under hydrogen rich conditions), a second hydrogen atom is added to the chemisorbed half-hydrogenated intermediate, leading to a hydrogenated product. If, on the other hand, the coverage of hydrogen is low, (as under hydrogen poor conditions), hydrogen abstraction by the metal takes place from an adjacent carbon. This step could lead to a double bond migration depending on from which carbon atom the hydrogen abstraction occurs. Thereafter the chemisorbed product undergoes desorption from the metal. Free rotation of the half-hydrogenated intermediate, hydrogen abstraction, and desorption of the olefin result in cis/ trans-isomerizations.

(8) Diffusion of linoleic acid from the bulk of the solution to the catalyst surface does not influence the consumption rate of linoleic acid. (9) As mentioned above, no mass transfer limitations exist with a catalyst particles size of <63 mm, thus the catalyst effectiveness factor is equal to unity. The models involve several steps: adsorption of the reactant on the catalyst surface, catalytic reaction leading to products and release of the products. To simplify the mechanism, desorption of stearic acid is considered fast. The overall reaction can be written as follows þH2

þH2

A!B!C

(1)

where A, B, and C denote respectively linoleic acid, monoenoic acids (oleic acid, elaidic acid, cis-vaccenic acid, and trans-vaccenic acid), and stearic acid. Note that lumping of monoenic acids could be done, since the ratio between them was constant and conversion independent as mentioned earlier. Involving adsorption steps, the scheme takes the form

(2)

where ‘‘ads’’ denote adsorbed compounds. 3.3.1. Mechanism I In this mechanism it is assumes that double bond hydrogenation takes place through the Horiuti–Polanyi mechanism. The elementary steps can be described by 2 reaction routes, i.e. sets of stoichiometric numbers of steps, and written as Elementary steps

N(1) K1

0

0

The kinetic models utilized in the present study were based on the consecutive hydrogenation of linoleic acid to monounsaturated fatty acids and to stearic acid. Consequently, the following considerations were utilized in the build-up of the kinetic model: (1) The kinetic rates were based on the Langmuir concept of ideal surfaces. (2) Only hydrogenation steps are considered, since isomerization was not detected under the hydrogen rich conditions used. (3) There is no non-catalytic hydrogenation reaction of unsaturated fatty acids. (4) Adsorption of fatty acids compounds is competitive in nature. (5) Fatty acids are chemisorbed either through one or through both olefinic bonds. One active site, denoted with Z, is used for chemisorption of one olefinic acid molecule or one intermediate. (6) The position of the double bonds does not affect the hydrogenation rates. Thus, lumping of the monounsaturated compounds is possible. (7) The catalytic surface reactions are irreversible while adsorption/desorption steps are at quasi-equilibria.

N(2)

1. 2Z þ H2 J2Z H

1

2. Z þ AJZA

1

0

1

0

K2

k3

3. ZA þ Z 0 H ! ZX þ Z 0 k4

3.3. Elementary steps and rate equations

Basic routes

4. ZX þ Z 0 H ! ZB þ Z 0 k5

5. ZB þ Z 0 H ! ZY þ Z 0 k6

1

1

0

0

1

6. ZY þ Z 0 H ! Z 0 þ Z þ C

0

1

7. ZBJZ þ B

0

1

K7

Overall reactions N(1): A + H2 ! B N(2): B + H2 ! C In the catalytic steps, symbols Z, Z0 , and H denote a vacant surface site for chemisorption of organic molecule, a vacant surface site for chemisorption of hydrogen, and atomic hydrogen, respectively. Chemisorbed compounds are described by ZA, ZB, etc. The two types of adsorbed intermediate species on the catalyst surface are grouped together in symbols ZX and ZY, which denote chemisorbed half-hydrogenated intermediates descending from A and B. Symbols J, k, and K denote adsorption quasi-equilibria, kinetic and equilibrium constants. In the first step, hydrogen dissociates on the metal surface. Steps 2 and 7 describe adsorption and desorption of A and B. In steps 3–6, hydrogen addition to the organic compounds takes place. In order to get simple rate expressions and to assist the parameter estimation procedure, steps 3 and 5 are written as irreversible steps since the

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Fig. 4. Horiuti–Polanyi mechanism describing hydrogenation and isomerization of olefins: (a) hydrogenation, (b) cis/trans-isomerization, (c) double bond migration.

hydrogenation is rather fast. Moreover, it is assumed that adsorption of hydrogen and organic molecules is non-competitive in nature, hence sites for chemisorption of them are separated. This mechanism is hereafter referred to as Mechanism I(a). Elementary reactions are grouped in steps, and chemical equations of steps contain reactants and surface species. On the right hand side of the equations of steps, the stoichiometric numbers for the different independent routes (N(1), etc.) of the complex heterogeneous catalytic reaction are given. These numbers must be chosen in a way that the overall equations contain no surface species. The equations describing the overall reaction are obtained by the summation of chemical equations of steps multiplied by the stoichiometric numbers. A set of stoichiometric numbers of steps is defined as a reaction route [10,11]. Routes must be essentially different, and it is impossible to obtain one route through multiplication of another route by a number, although their respective overall equations can be identical. The number of basic routes, P, is determined by the following equation, P ¼SþW I

can be written as (4)

0

(5)

0

(6)

0

(7)

r 4 ¼ k4 uX uH r 5 ¼ k5 uB u H r 6 ¼ k6 uY uH

where r, k, and u denote reaction rate, rate constant, and fractional surface coverage, respectively. The rates for the adsorption– desorption steps are assumed to be high compared to the rate of the complex reaction as a whole. Thus, the equilibrium constants, K, for these steps are given by 2

K1 ¼

u0H 0 2 ð1  uH Þ pH2

(8)

K2 ¼

uA u0 cA

(9)

K7 ¼

u0 cB uB

(10)

(3)

where S is the number of steps, W is the number of balance equations, and I is the number of intermediates. Balance equations determine the relationship between adsorbed intermediates. Such equations can correspond to the total coverage equal to unity. It is assumed that the surface of the catalyst is uniform or quasiuniform meaning that rate constants are coverage independent and that organic compounds form an ideal liquid mixture. Speaking in terms of kinetics, the rate equations for each step

0

r 3 ¼ k3 uA uH

The consumption/generation rates are defined by rA ¼ 

1 dcA 0 ¼ r 3 ¼ k3 uA uH m dt

(11)

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rB ¼

1 dcB 0 0 ¼ r 4  r 5 ¼ k4 uX uH  k5 u B uH m dt

(12)

rC ¼

1 dcC 0 ¼ r 6 ¼ k6 u Y u H m dt

(13)

where m denotes total catalyst mass. Steady state approximations for intermediates uX and uY give respectively the equations r3 ¼ r4

(14)

r5 ¼ r6

(15)

Insertion of Eqs. (4) and (5) in Eq. (14) and of Eqs. (6) and (7) in Eq. (15) yields the expressions 0

0

(16)

0

0

(17)

k3 u A u H ¼ k4 u X u H k5 uB uH ¼ k6 uY uH

Coverage of intermediate species uX can be obtained from Eq. (16)

uX ¼

k3 uA k4

(18)

Accordingly coverage of intermediate uY is obtained from Eq. (17) k u Y ¼ 5 uB k6

uB ¼

u0 cB K7

(20) (21)

The coverage of vacant sites, u0, can be obtained from the balance equation

u A þ uB þ u X þ uY þ u0 ¼ 1

(22)

After combination of Eqs. (18)–(21) for uX, uY, uA, and uB, with Eq. (22) one arrives at the following expression

u0 ¼

1 K 2 cA þ ðcB =K 7 Þ þ ðk3 =k4 ÞK 2 cA þ ðk5 =k6 ÞðcB =K 7 Þ þ 1

(23)

The dissociated hydrogen is still remaining in Eqs. (11)–(13). Regarding the coverage of hydrogen, the solution of Eq. (8) is qffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 pH2 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi uH ¼ (24) 1 þ K 1 pH2 First of all, an essential and necessary simplification of the model must be done to assist the parameter estimation. Since fatty acids are adsorbed with the same adsorption strength, the equilibrium constants K2 and K7 follow the relationship K2 ¼

1 K7

K1

2Z þ H2 J2ZH

(26)

The coverage of dissociated hydrogen atoms is then given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffi u H ¼ u 0 K 1 pH2 (27) while the coverage of available active sites can be written

u0 ¼

1 K 2 cA þ ðcB =K 7 Þ þ ðk3 =k4 ÞK 2 cA þ ðk5 =k6 ÞðcB =K 7 Þ þ

(19)

According to Eqs. (9) and (10), the coverage of A and B, uA and uB, can be expressed via the fraction of vacant sites

u A ¼ K 2 u0 cA

chemisorption of them are separated. This is a usual assumption in kinetic modeling of this type of hydrogenation system, but considering that the studied system is very diluted in unsaturated fatty acids and the hydrogen pressure ranges from 0.2 to 20 bar, very low surface coverage could be expected for the adsorbed fatty acid molecules while the corresponding ones for the adsorbed hydrogen are expected to be changing over this wide range of hydrogen pressure, from low to high hydrogen coverages Consequently, a change in the competition regime between the fatty acids and the hydrogen could also be expected. The competitive adsorption regime should not be left aside. A modification of Mechansim I(a) will let us compare non-competitive and competitive hydrogen adsorption. Considering competitive adsorption of all compounds involved, the first elementary step of the two basic routes above takes the form

(25)

The constants to be calculated are the four frequency factors, A, and the four activation energies, Ea, in the kinetic constants k in Eqs. (4)–(7) and adsorption equilibrium constants K1 and K2 in Eqs. (8) and (9). The mathematical model given by Eqs. (11)–(13) portraying the kinetic model of linoleic acid hydrogenation with non-competitive dissociative hydrogen adsorption, referred to as Mechanism I(a), is then ready for parameter estimation. In Mechansim I(a) it is assumed that the adsorption of hydrogen and organic molecules is non-competitive in nature, hence sites for

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 pH2 þ 1 (28)

This mechanism featuring competitive adsorption of all species is hereby denoted Mechanism I(b). 3.3.2. Mechanism II Although metal-catalyzed hydrogenations is generally thought to follow the Horiuti–Polanyi mechanism involving addition of chemisorbed hydrogen to chemisorbed olefin, there are some drawbacks of this mechanistic approach. The coverage of 0 chemisorbed hydrogen, uH , which is given by Eq. (24) is of order 0.5 with respect to hydrogen pressure. The pressure dependence of the kinetics in linoleic acid hydrogenation was found to be more pronounced at pressures 0.5–2 bar compared to that of 2–20 bar of hydrogen pressure. The reaction rate showed a zero order dependence on the pressure at higher pressure values and this will bring some difficulties in finding a well-defined minimum of the sum of squares when estimating the kinetic parameters. Thus, we present a second mechanism, hereafter referred to as Mechanism II, which can be written as Elementary steps

Basic routes N(1)

K1

1. Z þ AJZA K2

2. ZA þ H2 ! ZAH2 K3

3. ZAH2 ! ZB K4

4. ZB þ H2 ! ZBH2

N(2)

1

0

1

0

1

0

0

1

5. ZBH2 ! Z þ C

0

1

6. ZBJZ þ B

1

1

K5

K6

Overall reactions N (1): A + H2 ! B N(2): B + H2 ! C where ZAH2 and ZBH2 are intermediate complexes. This mechanism follows the one advanced for hydrogenation of olefins [12,13] and explains the zero order with respect to H2 at high pressures. In

A. Bernas et al. / Applied Catalysis A: General 353 (2009) 166–180

step 2, the intermediate ZAH2 will generate slowly at low pressures. At high pressures, the surface will have a rich coverage of this intermediate. In step 3, hydrogenation of ZAH2 to ZB take place. As confirmed by the parameter estimation, this is a slow step that will determine the overall rate at high hydrogen pressures. At lower hydrogen pressures Mechanism II is of first order with respect to the hydrogen pressure. Similar to Mechanism I(a), there is no competitive adsorption of organic molecules and H2. Mechanism II is written in such a way that hydrogen reacts directly from the fluid phase. To be more specific, the hydrogen on the Pd surface is in equilibrium with the hydrogen amount in the liquid phase, which in turn is a function of the H2 pressure. The reaction rate is the same no matter if ZAH2 is generated from H2 adsorbed on the metal surface, H2 dissolved in the liquid phase, or H2 in gas phase. Therefore it is convenient to express the reaction rate as a function of the H2 pressure. The rate equations for each step can be written as r 2 ¼ k 2 u A pH2

(29)

r 3 ¼ k3 uAH2

(30)

r 4 ¼ k4 uB pH2

(31)

r 5 ¼ k5 uBH2

(32)

The equilibrium constants, K, are given by K1 ¼

uA u 0 cA

(33)

K6 ¼

u 0 cB uB

(34)

The consumption/generation rates are defined by rA ¼ 

1 dcA ¼ r 2 ¼ k2 u A pH2 m dt

(35)

rB ¼

1 dcB ¼ r 3  r 4 ¼ k3 uAH2  k4 uB pH2 m dt

(36)

rC ¼

1 dcC ¼ r 5 ¼ k5 uBH2 m dt

(37)

Steady state approximations for intermediates uAH2 and uBH2 give respectively the equations r2 ¼ r3

(38)

r4 ¼ r5

(39)

Insertion of Eqs. (29) and (39) in Eq. (38) and of Eqs. (31) and (32) in Eq. (39) yields the expressions

uB ¼

u 0 cB

(45)

K6

The coverage of vacant sites, u0, is obtained from the balance equation

uA þ uB þ uAH2 þ uBH2 þ u0 ¼ 1

u0 ¼

1 K 1 cA þ ðcB =K 6 Þ þ ðk2 K 1 cA pH2 =k3 Þ þ ðk4 cB pH2 =k5 K 6 Þ þ 1

(47)

The equilibrium factors K1 and K6 then follow the relation K1 ¼

1 K6

(48)

The constants to be calculated are the four frequency factors, A, and the four activation energies, Ea, in the kinetic constants k in Eqs. (29)–(32) and only one adsorption equilibrium constant K1 in Eq. (33). The model given by Eqs. (35)–(37), referred to as Mechanism II, is then ready for parameter estimation. 3.3.3. Mechanism III A detailed reaction mechanism would include adsorption/ desorption of hydrogen and organic molecules, hydrogen addition, double bond migration as well as geometric isomerization. In the present study no double bond isomerization products were observed and due to the constant selectivities to monoenoic acids, one arrives at rather simple kinetics. It was possible to improve the degree of explanation and the parameter identifiability by applying a kinetic model using associative adsorption of hydrogen followed by double bond hydrogenation in one step. Very recently applicability of such a mechanism was discussed in a view of observed through NMR studies para-hydrogen-induced polarization in heterogeneous hydrogenation on supported metal catalysts [14]. Compared to the Horiuti–Polanyi mechanism where hydrogen is dissociated on the metal surface, there will be an essential difference in the adsorption/desorption equilibrium equations and in the rate equations using associative hydrogen adsorption. First of all, the term 2Z0 H in the elementary steps is replaced by Z0 H2 Elementary steps

Basic routes N(1)

K1

1. Z 0 þ H2 JZ 0 H2 K2

2. Z þ AJZA

1

N(2) 1

1

0

1

0

4. ZB þ Z 0 H2 ! Z þ C þ Z 0

0

1

1

1

k3

3. ZA þ Z 0 H2 ! ZB þ Z 0 k4

K5

(40)

5. ZBJZ þ B

k4 u B pH2 ¼ k5 uBH2

(41)

Overall reactions N(1): A + H2 ! B N(2): B + H2 ! C The rate equations become

uAH2 ¼

k2 uA pH2 k3

(42)

uBH2 ¼

k4 uB pH2 k5

(43)

According to Eqs. (33) and (34), the coverage of A and B, uA and uB, is expressed via fraction of vacant sites

uA ¼ K 1 u0 cA

(46)

After combination of Eqs. (44), (45), (42) and (43) for uA, uB, uAH2 , and u BH2 , respectively, with Eq. (46) one arrives at the following expression for the coverage of empty sites

k2 u A pH2 ¼ k3 uAH2

The coverage of intermediate species uAH2 and uBH2 are obtained from Eqs. (40) and (41) correspondingly

173

(44)

0

(49)

0

(50)

r 3 ¼ k3 uA uH2 r 4 ¼ k4 uB u H2 and the equilibrium constants, K, take the form K1 ¼

u0H2 0 ð1  uH2 Þ pH2

(51)

A. Bernas et al. / Applied Catalysis A: General 353 (2009) 166–180

174

K2 ¼

uA u0 cA

(52)

K5 ¼

u0 cB uB

(53)

K1

The consumption/generation rates are now defined by rA ¼ 

1 dcA 0 ¼ r 3 ¼ k3 uA uH2 m dt

Z þ H2 JZH2

u H ¼ u 0 K 1 pH2

1 dcC 0 ¼ r 4 ¼ k4 u B uH2 m dt

(55)

(56)

The coverage of A and B is given by the expressions

u A ¼ K 2 u0 cA uB ¼

(57)

u0 cB

(58)

K5

The coverage of vacant sites, u0, in Eqs. (57) and (58) in a similar fashion as described above is obtained from the balance equation

u A þ uB þ u 0 ¼ 1

(59)

After a combination of Eqs. (57) and (58) for uA and uB with Eq. (59) one arrives at the expression

u0 ¼

1 K 2 cA þ ðcB =K 5 Þ þ 1

(61)

1 K 2 cA þ ðcB =K 5 Þ þ K 1 pH2 þ 1

1 K5

(62)

The constants to be calculated are now only the two frequency factors, A, and the two activation energies, Ea, in the kinetic constants k in Eqs. (49) and (50) and adsorption equilibrium constants K1 and K2 in Eqs. (51) and (52). The model given by Eqs. (54)–(56) is then ready for parameter estimation. In the expressions above it is assumed that the adsorption of hydrogen and organic molecules is non-competitive in nature, hence sites for chemisorption of them are separated. The model above given by Eqs. (54)–(56) is hereafter referred to as Mechanism III(a).

(65)

This mechanism featuring competitive adsorption of associated hydrogen and organic compounds is hereby denoted Mechanism III(b). In order to get a more realistic description of the fatty acid hydrogenation kinetics, Mechanism III(a) and Mechanism III(b) can be combined to a reaction network which utilize hydrogen addition from sites reserved for hydrogen adsorption, denoted by Z0 , and sites for adsorption of all species, denoted by Z. This approach is referred to as Mechanism III(c) and the elementary steps can be written as follows Elementary steps

Basic routes N(1)

K1

0

N(2)

N(3)

N(4)

1. Z þ H2 JZ H2

1

1

0

0

2. Z þ H2 JZH2

0

0

1

1

1

0

1

0

1

0

0

0

0

0

1

0

0

1

0

0

7. ZB þ ZH2 ! 2Z þ C

0

0

0

1

8. ZBJZ þ B

1

1

1

1

K2

K3

3. Z þ AJZA k4

0

4. ZA þ Z H2 ! ZB þ Z

0

k5

5. ZA þ ZH2 ! ZB þ Z k6

6. ZB þ Z 0 H2 ! Z þ Z 0 þ C

The equilibrium factors K2 and K5 then follow the relation K2 ¼

u0 ¼

0

K 1 pH2 1 þ K 1 pH2

(64)

where the coverage of available active sites can be written

(60)

The expression for the associated hydrogen in Eqs. (54)–(56) does not any longer contain the square root of hydrogen. For the 0 coverage of hydrogen u H2 , the solution of Eq. (51) is

u0H2 ¼

(63)

The coverage of adsorbed molecular hydrogen is then given by

(54)

1 dcB 0 0 ¼ r A  r C ¼ r 3  r 4 ¼ k3 uA uH2  k4 uB uH2 rB ¼ m dt rC ¼

Mechansim III(a) can be modified to describe competitive adsorption of molecular hydrogen and organic molecules. Considering competitive adsorption of all compounds involved, the first elementary step of the two basic routes above takes the form

k7

K8

Overall reactions N(1): A + H2 ! B N(2): B + H2 ! C N(3): A + H2 ! B N(4): B + H2 ! C The rate equations become 0

r 4 ¼ k4 uA uH2

(66)

Table 2 Values of the calculated parameters of Mechanism I(a) and I(b). Parameter

Calculated value Mechanism I(a)

A3 A4 A5 A6 Ea3 Ea4 Ea5 Ea6 K1 K2

1

0.640  10 0.495  100 0.137  102 0.728  102 0.272  105 0.100  105 0.521  105 0.270  105 0.930  101 0.176  102

Estimated standard error Mechanism I(b) 1

0.720  10 0.194  101 0.150  102 0.638  102 0.221  105 0.171  105 0.473  105 0.220  105 0.190  101 0.244  101

Note: Dimensions A = mol g1 dm3 min1, Ea = J mol1, K = bar1. Mechanism I(a): SRS = 0.1255  103, R2 = 95.60%. Mechanism I(b): SRS = 0.1265  103, R2 = 95.56%.

Estimated relative standard error (%)

Mechanism I(a)

Mechanism I(b)

Mechanism I(a)

Mechanism I(b)

NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN

NaN NaN NaN NaN NaN NaN NaN NaN Infinity NaN

NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN

NaN NaN NaN NaN NaN NaN NaN NaN Infinity NaN

A. Bernas et al. / Applied Catalysis A: General 353 (2009) 166–180

r 5 ¼ k 5 u A u H2

The consumption/generation rates are now defined by

(67)

0

r 6 ¼ k 6 u B u H2

(68)

r 7 ¼ k 7 u B u H2

(69)

175

rA ¼ 

rB ¼

and the equilibrium constants, K, take the form

1 dcA 0 ¼ r 4 þ r 5 ¼ k4 u A u H2 þ k5 uA u H2 m dt

(74)

1 dcB ¼ rA  rC ¼ r4 þ r5  r6  r7 m dt 0

0

¼ k4 uA uH2 þ k5 uA uH2  k6 u B uH2  k7 uB uH2

0 H2 0 H2 Þ pH2

(75)

u K1 ¼ ð1  u

(70)

u H2 u 0 pH2

(71)

K3 ¼

uA u 0 cA

(72)

uB ¼

K8 ¼

u 0 cB uB

(73)

The coverage of vacant sites, u0, in Eqs. (77) and (78) is now obtained from the balance equation

K2 ¼

1 dcC 0 ¼ r 6 þ r 7 ¼ k6 uB uH2 þ r 7 ¼ k7 uB uH2 m dt

(76)

The coverage of A and B is given by the expressions

uA ¼ K 3 u 0 cA

Parameter

Calculated value

Estimated standard error

Estimated relative standard error (%)

A2 A3 A4 A5 Ea2 Ea3 Ea4 Ea5 K1

0.285  102 0.182  100 0.627  102 0.371  100 0.414  105 0.662  103 0.663  105 0.118  105 0.424  101

0.287  101 0.979  102 0.779  101 0.379  101 0.313  104 0.874  104 0.138  104 0.238  105 0.338  100

10.1 5.4 12.4 10.2 7.6 1319.5 2.1 202.1 8.0

Note: Dimensions A = mol g1 dm3 min1, 0.1122  103, R2 = 96.06%.

Ea = J mol1,

K = bar1.

Estimated standard error 0.527  102 0.105  103 0.232  104 0.297  104 0.148  101 0.740  100

Mechanism III(b) A3 0.865  102 A4 0.700  102 Ea3 0.272  105 Ea4 0.272  105 K1 0.143  101 K2 0.660  102

0.187  101 0.608  100 0.342  104 0.385  104 0.179  102 0.160  101

2.2 0.9 12.6 14.2 12.5 2.4

Mechanism III(c) A4 0.224  102 A5 0.156  101 A6 0.590  102 A7 0.107  103 Ea4 0.308  105 Ea5 0.431  105 Ea6 0.327  105 Ea7 0.296  105 K1 0.725  101 K2 0.109  100 K3 0.787  103

0.137  103 0.119  102 0.314  103 0.470  104 0.482  105 0.741  105 0.472  105 0.428  108 0.237  101 0.549  101 0.241  100

612.7 764.3 532.2 4397.0 156.6 171.8 144.3 144397.7 32.7 50.2 30583.8

119.2 106.8 12.5 7.4 35.1 116.1

1

1

Note: Dimensions A = mol g dm min , Ea = J mol Mechanism III(a): SRS = 0.1306  103, R2 = 95.42%. Mechanism III(b): SRS = 0.3578  103, R2 = 87.45%. Mechanism III(c): SRS = 0.1518  103, R2 = 94.67%.

, K = bar

1

.

(79)

1 K 3 cA þ ðcB =K 8 Þ þ K 2 pH2 þ 1

(80)

0

u0H2 ¼

K 1 pH2 1 þ K 1 pH2

(81)

The equilibrium factors K3 and K8 follow the relation SRS =

K3 ¼

1 K8

(82)

The constants to be calculated are now four frequency factors, A, and four activation energies, Ea, in the kinetic constants k in Eqs. (66)–(69) and adsorption equilibrium constants K1, K2, and K3. in Eqs. (70)–(72). The model described by Eqs. (74)–(76) is then ready for parameter estimation. Here, it is assumed that the adsorption of hydrogen and organic molecules compete to some extent. In Mechanism III(c), there are sites reserved for hydrogen adsorption, Z0 , and hydrogen can also adsorb on sites for organic molecules Z. Table 5 Correlation matrix of the parameters of Mechanism II. A2

3

(78)

K8

The coverage of hydrogen uH2 is obtained from Eq. (70) is

Estimated relative standard error (%)

Mechanism III(a) A3 0.442  102 A4 0.981  102 Ea3 0.185  105 Ea4 0.402  105 K1 0.421  101 K2 0.637  100

1

u 0 cB

After a combination of Eqs. (77) and (78) for uA and uB with Eq. (79) one arrives at

u0 ¼

Table 4 Values of the calculated parameters of Mechanism III(a), III(b), and III(c). Estimated value

(77)

u A þ u B þ u H2 þ u 0 ¼ 1

Table 3 Values of the calculated parameters of Mechanism II.

Parameter

rC ¼

A3

A4

A5

Ea2

Ea3

Ea4

Ea5

K1

A2 1.000 A3 0.488 1.000 A4 0.203 0.580 1.000 A5 0.486 0.662 0.711 1.000 Ea2 0.511 0.505 0.336 0.649 1.000 Ea3 0.185 0.171 0.014 0.076 0.273 1.000 Ea4 0.230 0.470 0.442 0.519 0.594 0.223 1.000 Ea5 0.445 0.134 0.513 0.113 0.215 0.617 0.144 1.000 K1 0.913 0.221 0.200 0.545 0.502 0.120 0.193 0.454 1.000

Table 6 Correlation matrix of the parameters of Mechanism III(a).

A3 A4 Ea3 Ea4 K1 K2

A3

A4

Ea3

Ea4

K1

K2

1.000 0.997 0.435 0.597 0.089 0.998

1.000 0.414 0.561 0.085 0.997

1.000 0.252 0.407 0.424

1.000 0.139 0.593

1.000 0.122

1.000

176

A. Bernas et al. / Applied Catalysis A: General 353 (2009) 166–180

Fig. 5. Comparison of (&) linoleic acid, (^) oleic acid, elaidic acid, cis-vaccenic acid, and trans-vaccenic acid, and (~) stearic acid with response simulations (solid line) of Mechanism II for hydrogenation of linoleic acid over 5 wt% Pd/C catalyst. Conditions: (a) T = 40 8C, p = 1 bar, cat. mass = 0.02 g, (b) T = 80 8C, p = 1 bar, cat. mass = 0.02 g, (c) T = 80 8C, p = 0.5 bar, cat. mass = 0.02 g, (d) T = 60 8C, p = 1 bar, cat. mass = 0.04 g, (e) T = 100 8C, p = 2 bar, cat. mass = 0.02 g, (f) T = 100 8C, p = 10 bar, cat. mass = 0.02 g, (g) T = 100 8C, p = 20 bar, cat. mass = 0.02 g. Other conditions: reactant, 0.4 g of linoleic acid; solvent, 140 ml of n-decane; initial linoleic acid concentration, 0.010187 mol/dm3; hydrogen flow, 100 ml/min, stirring rate, 1000 rpm, reaction time, 120 min.

A. Bernas et al. / Applied Catalysis A: General 353 (2009) 166–180

177

expression 2

R ¼

1

jjy  y p jj2 2 jjy  yjj ¯

! (86)

Hence R2 is typically <1. The closer the value is to unity, the more perfect is the fit. As a rule of thumb one might have that a mechanistic model, with a reasonable amount of noise in the data, should have R2-values clearly exceeding 0.9. The systems of differential equations portraying the model were in the parameter estimations solved numerically with the backward difference method by minimization of the sum of residual squares, SRS, with non-linear regression analysis using the Simplex and Levenberg–Marquardt optimization algorithms implemented in the software Modest [16]. In order to get the values of y and yp as close as possible, the above sum was minimized with respect to u using a step size of 0.1 and a value of 1  106 for both the absolute and relative tolerances of the Simplex and Levenberg–Marquardt optimizer, starting with Simplex and thereafter switching to Levenberg–Marquardt.

Fig. 5. (Continued ).

3.4. Parameter estimation

3.5. Modeling results & experimental versus calculated concentration

The kinetic constants k in the rate equations follows the modified Arrhenius dependence [15]

The values of the calculated frequency factors, activation energies, and adsorption constants A, Ea, and K, the estimated standard errors, as well as the estimated relative standard errors (in %) of the tested reaction mechanisms are presented in Tables 2–4. The degree of explanation of Mechanism I(a), I(b), II, III(a), III(b), and III(c) are R2 = 95.60% (10), R2 = 95.56% (10), R2 = 96.06% (9), R2 = 95.42% (6), R2 = 87.45% (6), and R2 = 94.67% (11), which can be considered as rather high for all models, except for Mechanism III(b). The number in the parenthesis after the R2-value is the number of parameters to be calculated for each model. Surprisingly, Mechanism III(a) has an extraordinary high R2-value considering that this model contains a low number of calculated parameters, since the R-squared value usually decreases when the number of parameters to be calculated decreases. The sum of residual squares are SRS = 0.1255  103, SRS = 0.1265  103, SRS = 0.1122  103, SRS = 0.1306  103, SRS = 0.3578  103, and SRS = 0.1518  103, correspondingly. The low R-squared value and the high SRS-value of Mechanism III(b) indicates that there is no competition in adsorption between hydrogen and organic molecules. As several parameters are calculated simultaneously, the problem might be well identified with respect to some parameters, and quite badly identified with respect to the others. Even more usual is the situation where there are strong correlations between parameters: the values of certain parameters can, in suitable relation with each other, be considerably altered without essentially affecting the fit between data and model. The Horiuti–Polanyi mechanism described by Mechanism I(a) and I(b) involving non-competitive and competitive adsorption of reacting species can be questioned already because of the large values of the standard errors of the parameters as shown in Table 2. The problem might be caused by the expression for the coverage of 0 chemisorbed hydrogen, uH , which is given by Eq. (24) and assumes 0.5 order dependence with respect to hydrogen pressure because of dissociative hydrogen chemisorption. Moreover, the fit of experimental and calculated concentrations at low hydrogen pressure was not satisfactory for the Mechanism I(a) and I(b). Therefore it is rational to develop the modeling to involve associative hydrogen adsorption and hydrogen addition directly from the fluid phase. In addition, Mechanism III(c) showed large values of the standard errors of the parameters, once again confirming that adsorption of hydrogen and organic molecules is non-competitive in nature.

k ¼ A exp

   Ea 1 1  Rgas T T mean

(83)

In the expression above, A, Ea, Rgas, T, and Tmean denote frequency factor, activation energy, the gas constant, reaction temperature, and mean temperature of the experiments, correspondingly. The closeness of data and model predicted values were measured with two criteria. If the mechanistic model is formally written as s ¼ f ðx; u; cÞ y p ¼ gðsÞ

(84)

where s, yp, x, u, and c denote state of the system, response variables, design variables, calculated parameters, and constants, the sum of residual squares for the observed variables available, yijk, at experimental points, xjk, is defined by SRS ¼ lðuÞ ¼ jjy  y p jj2w ¼

n setsnobsðkÞ X nydatað Xj;kÞ X

ðyi jk  y pi jk Þ2 wi jk

k¼1

j¼1

(85)

i¼1

where the values yp denote the predictions given by the model with the parameter values u, and w gives the weight matrix for the observations. The weighted norm notation jjy  y p jj2w is an abbreviation for the sum of squares. The terms nsets, nobs, and nydata denote number of experimental sets, number of observations, and number of estimations. In order to get the values of y and yp as close as possible, in an average sense, the above sum is minimized with respect to u. The different terms in the sum can be weighed with the weight factors w. A basic principle is that every data point should be divided by the estimated standard deviation of it. If all the response components are of comparable magnitude, the simple choice w ¼ 1 is often used. The minimization of l can be performed with a number of differential numerical optimization methods. The most common measure for the goodness of the fit is the coefficient of determination, the R2-value. The idea is to compare the residuals y  yp given by the model to the residuals of the simplest model one may think of, the average value y¯ of all data points. The R2-value or degree of explanation is given by the

178

A. Bernas et al. / Applied Catalysis A: General 353 (2009) 166–180

The estimated relative standard errors of the parameters of Mechanisms II and III(a), on the other hand, are reasonably low, as shown in Tables 3 and 4. This solution is well identified. The parameter correlation matrixes of the models describing Mechanisms II and III(a) are presented in Tables 5 and 6. Low positive and negative values in the correlation matrix

indicate that parameter correlation to some extent could be avoided. In Mechanism III(a), the model solution is proposing quite similar magnitude values of the frequency factors (Table 4) which would mean that a double bond hydrogenation occurs with the same rate no matter if linoleic acid or monoenoic acids are being

Fig. 6. Comparison of (&) linoleic acid, (^) oleic acid, elaidic acid, cis-vaccenic acid, and trans-vaccenic acid, and (~) = stearic acid with response simulations (solid line) of Mechanism III(a) for hydrogenation of linoleic acid over 5 wt% Pd/C catalyst. Conditions: (a) T = 40 8C, p = 1 bar, cat. mass = 0.02 g, (b) T = 80 8C, p = 1 bar, cat. mass = 0.02 g, (c) T = 80 8C, p = 0.5 bar, cat. mass = 0.02 g, (d) T = 60 8C, p = 1 bar, cat. mass = 0.04 g, (e) T = 100 8C, p = 2 bar, cat. mass = 0.02 g, (f) T = 100 8C, p = 10 bar, cat. mass = 0.02 g, (g) T = 100 8C, p = 20 bar, cat. mass = 0.02 g. Other conditions: reactant, 0.4 g of linoleic acid; solvent, 140 ml of n-decane; initial linoleic acid concentration, 0.010187 mol/dm3; hydrogen flow, 100 ml/min, stirring rate, 1000 rpm, reaction time, 120 min.

A. Bernas et al. / Applied Catalysis A: General 353 (2009) 166–180

Fig. 6. (Continued ).

hydrogenated. In Mechanisms II and III(a) the activation energies were calculated to 20–40 kJ/mol. Similar magnitudes of activation energies have been reported in kinetic modeling of cotton seed oil (triglycerides) hydrogenation on commercial Ni-Al catalyst [17].

179

On the other hand, activation energies over 60 kJ/mol have recently been obtained in modeling of liquid-phase hydrogenation of methyl oleate on Ni/a-Al2O3 catalyst [18]. Let us take a closer look at the fit of the models. The experimental and calculated concentration versus time dependences of Mechanisms II and III(a) are presented in Figs. 5 and 6. The difference between the mechanisms is as follows: In Mechanisms II a reactions step of ZA and H2 generating an intermediate ZAH2 that continues to ZB is assumed. In Mechanisms III(a), ZA and adsorbed molecular hydrogen generates ZB assuming a non-competitive coverage of hydrogen according to Eq. (61). The predicted concentrations of linoleic acid, the monoenoic acids, and stearic acid follow the experimentally measured concentrations very well in both considered mechanisms. It can be seen in Figs. 5(a) and 6(a) that Mechanism II gives a better prediction at low temperatures than that of Mechanism III(a), resulting in a higher degree of explanation and a lower sum of residual squares for Mechanism II. Mechanism III(a), on the other hand, shows a better prediction at low hydrogen pressure than Mechanism II, as demonstrated in 5(c) and 6(c). It can also be concluded that both models are equally good at high pressure values, as seen in the figures. These two models are compared in parity diagrams in Fig. 7. The model of Mechanism II assuming generation of intermediates ZAH2 and ZBH2 results in a only vaguely better fit of the model. Even if the approach used in Mechanism III(a) can be considered as simple, where reaction intermediates have been neglected, this model can be concluded to be the most feasible for practical purposes, in particular reactor modeling. 4. Conclusions The kinetics of linoleic acid hydrogenation over Pd/C catalysts was studied in semi-batch reactor mode. Intrinsic kinetic experiments were carried out at the temperature and hydrogen pressure ranges 40–100 8C and 0.5–20 bar using pure linoleic acid as reactant with varied amounts of 5 wt% Pd/C powder catalyst and n-decane as a solvent. Over such catalyst and at the conditions used, the reaction scheme involves hydrogenation of linoleic acid to monoenoic acid isomers, predominantly oleic acid, and further hydrogenation of monoenoic acids to stearic acid. Several mechanistic models were advanced. The parameters of these kinetic models describing a perfectly mixed reactor system were determined by using non-linear regression analysis. The concentrations of linoleic acid, monoenoic acids, and stearic acid were used in the parameter estimation. The kinetic models with a degree of explanation of more than 0.95 described the formation of the products with satisfying accuracy. Discrimination of the rival kinetic models was performed based on kinetic regularities as well as statistical analysis. References

Fig. 7. Parity diagram of (a) Mechanism II and (b) Mechanism III(a) comparing observed stearic acid concentration with predicted values in linoleic acid hydrogenation.

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