Kinetics and reactor modelling of fatty acid epoxidation in the presence of heterogeneous catalyst

Chemical Engineering Journal 375 (2019) 121936

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Kinetics and reactor modelling of fatty acid epoxidation in the presence of heterogeneous catalyst

T

Adriana Freites Aguileraa, Pasi Tolvanena, Johan Wärnåa, Sebastien Leveneura,b, Tapio Salmia Laboratory of Industrial Chemistry & Reaction Engineering, Department of Chemical Engineering, Johan Gadolin Process Chemistry Centre, Åbo Akademi University, FI20500 Åbo-Turku, Finland b Laboratoire de Sécurité des Procédés Chimiques, Institut National des Sciences Appliquées de Rouen, FR-76800 Saint-Étienne-du-Rouvray, France a

H I GH L IG H T S

of oleic acid is essentially intensified by involving a heterogeneous catalyst. • Epoxidation role of heterogeneous catalyst is to enhance the perhydrolysis step. • The kinetic and reactor models were derived for the complex multiphase system of epoxidation. • Detailed • The kinetic parameters for the epoxidation of oleic acid were determined.

A R T I C LE I N FO

A B S T R A C T

Keywords: Oleic acid Fatty acid Epoxidation Kinetic model Heterogeneous catalyst Multiphase reactor

Epoxidized fatty acids and fatty acid esters are used as chemical intermediates and environmentally friendly biolubricants. The epoxidation is carried out by the Prileschajew reaction, which implies the use of hydrogen peroxide as oxidation agent and a percarboxylic acid as the oxygen carrier. The epoxidation process is slow in the two-phase system (aqueous phase and oil phase), but it can be considerably enhanced by incorporating a heterogeneous acid catalyst. The results for the epoxidation of a model compound, oleic acid were presented. A cation-exchange resin (Amberlite IR-120) was used as the heterogeneous catalyst and acetic acid was the reaction carrier. Experimental data on oleic acid epoxidation were obtained at 40–50 °C under conventional heating and microwave irradiation. The experimental set was focused on studying the catalytic effect of the solid resin. The kinetic results revealed that the presence of the heterogeneous catalyst completely masked the enhancing effect of microwave radiation. An extensive set of kinetic data obtained from a laboratory-scale loop reactor was modelled with a realistic heterogeneous model taking into account the co-existence of three phases in the chemically complex reaction system. Rate equations were presented for all the reaction steps involved: the perhydrolysis, epoxidation and ring-opening steps. The rate equations were included in the three-phase reactor model and the kinetic parameters of the system were determined by non-linear regression analysis.

1. Introduction Epoxidation of double bonds is a theoretically and practically very important reaction. The smallest epoxide molecule, ethylene oxide is a real bulk product, which is produced in 25 million [1,2] metric tons per year [1]. This process is carried out in gas phase in the presence of silver catalysts [2,3]. Direct epoxidation of larger molecules is however more complicated and hydrogen peroxide is typically used as the epoxidation agent [4,5]. For instance, the commercial process for the production of propylene oxide is based on the use of hydrogen peroxide and titaniumbased catalysts [5]. Water is formed as a stoichiometric co-product when hydrogen peroxide is used as the epoxidation agent. Epoxidation

of longer hydrocarbons, such as fatty acids and fatty acid esters has been investigated by many groups in recent years [6–34], because epoxidized fatty acids and fatty acid esters are important chemical intermediates and they can be used as bio-lubricants. The approach fulfils the principles of green chemistry [35], since the reagents originate from renewable sources and both the reagents and products are bio-degradable. Moreover, the side product of the Kraft pulping process, tall oil, can be used as the source of the fatty acids and their esters [36]. Thus the production of epoxidized fatty acids and fatty acid esters is not competing with the food chain. The overall reaction of the epoxidation of double bonds with hydrogen peroxide is principally simple,

E-mail address: tapio.salmi@abo.fi (T. Salmi). https://doi.org/10.1016/j.cej.2019.121936 Received 18 February 2019; Received in revised form 12 May 2019; Accepted 9 June 2019 Available online 12 June 2019 1385-8947/ © 2019 Elsevier B.V. All rights reserved.

Chemical Engineering Journal 375 (2019) 121936

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ω *

Notation

Aeq A C Ea K K K K' AA k k0 k’ m MM N n R Ri S T V V̇

vij w-% α β ρB τ

Pre-exponential factor, – Interfacial area, m2 Concentration, mol/L Activation energy, J/mol Equilibrium constant, – Adsorption constant, – Partition coefficient for the phase equilibrium Acetic acid dissociation constant Rate constant (unit depends on the reaction order) Pre-exponential factor (unit depends on the reaction order) Merged rate constant (unit depends on the reaction order) Mass, g Molar mass, g/mol Interfacial flux, kg/s/m2 Amount of substance, mol General gas constant (8.3143 J/molK) Reaction rate, mol/L/min Speed, rpm Temperature, °C or K Volume, L Volumetric flow L/min Stoichiometric coefficient, – Weight percent, – Phase ratio, – Volumetric flow ratio, – Bulk density, g/L Residence time, s

Objective function Acid site

Subscripts and superscripts

Aq cat decom epox L nocat oil perh RO ∧

Aqueous phase Catalytic Decomposition Epoxidation Liquid Non-catalytic Oil phase Perhydrolysis Ring-opening Estimated quantity form the model

Abbreviations AA CA CH DB EOA HP MCMC MW OA PAA ROP W

Acetic acid Carboxylic acid Conventional heating Double bond Epoxyoleic acid Hydrogen peroxide Markov Chain Monte Carlo Microwave Oleic acid Peracetic acid Ring opening product Water

R =R' + H2 O2 ⇌ ROR' + H2 O and it can be carried out in several ways, as described in literature [6–34]. The principle of Prileschajew implies a stepwise process: hydrogen peroxide forms percarboxylic acid by reacting with a carboxylic acid and the percarboxylic acid epoxidizes the double bond. In the previous work of our group, we have studied the epoxidation of oleic acid with hydrogen peroxide in the absence of added catalysts [33]. Acetic acid was used as the reaction carrier to form peracetic acid, which epoxidized the double bonds. Both conventional heating and microwave heating were applied. The results revealed that the epoxidation time could in best cases been diminished with 50% (from 1100 min to 550 min) compared to conventional heating, which implies considerable process intensification. However, the reaction was still rather slow; about 600–1000 min was needed to complete the epoxidation. Therefore, a heterogeneous catalyst, Amberlite IR-120, which has the potential to enhance the first step in the epoxidation process, namely formation of peracetic acid from acetic acid (perhydrolysis) was tested [34]. The results were very encouraging: the epoxidation time was diminished to approx. 300–350 min. The main part of the kinetic data was published recently [34]. In the present article, we develop a kinetic model for the entire epoxidation process and a reactor model for the three-phase system aqueous phase-oil phase-solid catalyst. The model parameters are determined by non-linear regression analysis and the model predictions are compared with experimental data. The aim was to develop a new, general model for the epoxidation of double bonds in the presence of heterogeneous acid catalysts.

Fig. 1. Schematic view of the epoxidation process.

of the epoxidation process is provided in Fig. 1. The process is initiated by perhydrolysis in the aqueous phase, in which the reaction carrier, the carboxylic acid (RCOOH) reacts with hydrogen peroxide (H2O2) giving percarboxylic acid (RCOOOH) and water (H2O). The perhydrolysis step is an acid-catalyzed nucleophilic substitution. Inorganic and organic acids can be used as homogeneous catalysts, e.g. HCl, H2SO4, formic acid, oxalic acid. In the work of Leveneur et al [37] it was shown that solid cation exchange resins are effective heterogeneous catalysts for the perhydrolysis of carboxylic acids, such as acetic acid and propanoic acid. Therefore a cation exchange resin was used as the catalyst in the current work. By using a heterogeneous catalyst, cumbersome catalyst separation process is avoided. The next step in the reaction mechanism, the epoxidation of the fatty acid proceeds in the oil phase. The percarboxylic acid reacts with the double bond of the fatty acid and the carboxylic acid is regenerated and it returns back to the aqueous phase, where it undergoes

2. Molecular reaction mechanism and rate equations Epoxidation of double bonds in fatty acids and fatty acid esters is a complex multiphase reaction system, consisting of oil and aqueous phases and several reaction steps. In the presence of a heterogeneous catalyst, a three-phase reaction system is formed. A schematic overview 2

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nucleophilic attack are generally considered to be rapid and they can thus be merged. The total site balance for the catalyst surface can be written as

perhydrolysis; The reaction cycle is closed (Fig. 1). Besides perhydrolysis and epoxidation, ring-opening reactions take place in the oil phase, yielding different products, depending on the ring-opening reagent. Water, hydrogen peroxide and carboxylic acids can act as ring opening agents (Fig. 1). The overall reaction stoichiometry is summarized below. Perhydrolysis in aqueous phase

where the asterisk (*) refers to a surface site and c0 is the total concentration of sites available on the surface. The Langmuir adsorption equilibria of each species give the relation

RCOOH(aq) + H2 O2 ⇀ RCOOOH(aq) + H2 O (AA)

↽

(HP )

(W )

(PAA)

Transfer of percarboxylic acid to the oil phase

where Ki denotes the adsorption equilibrium constant. The relation (9) is inserted in the balance Eq. (8) from which the concentration of vacant sites is solved,

Epoxidation in the oil phase

C=

(PAA)

C(oil) (OA )

→ COC(oil) + RCOOH(oil) (EOA)

(9)

Ci *=Ki Ci C *

RCOOOH(aq) ⇌ RCOOOH(oil)

RCOOOH(oil) +

(8)

CAA*+CPAA*+CHP *+CW *+C *=C0

C *=

(AA)

Transfer of carboxylic acid to the aqueous phase

C0 1 + ΣKi Ci

(10)

and the concentrations of the adsorbed species are given by

RCOOH(oil) ⇌ RCOOH(aq)

Ci *=

Ring opening

COC + H2 O2 → ROP1

Ki Ci C0 1 + ΣKi Ci

(11)

After inserting the this relation to the rate equation is obtained,

COC + RCOOH → ROP2

R1cat = k1 CAA*CHP − k−1 CPAA *CW

COC + H2 O → ROP3

The expressions for the surface concentrations of acetic acetic (AA) and peracetic acid (PAA) are inserted in the rate expression (12) according to Eq. (11), which gives the rate of heterogeneously catalysed perhydrolysis,

COC + RCOOOH → ROP4 By introducing the abbreviations, the reaction scheme can be summarized as follows,

AA + HP → PAA + W

(

kcat CAA CHP −

(1)

R1cat =

CPAA CW K

(12)

) (13)

1 + ΣKi Ci

PAA + OA → EOA + AA

(2)

EOA + HP → ROP1

(3)

EOA + AA → ROP2

(4)

The rate equation for the perhydrolysis in the absence of an added catalyst is given by Leveneur et al. [38] (ncat in Eq. (14)). The overall rate of the perhydrolysis step becomes

EOA + W → ROP3

(5)

R1 = Rperh

EOA + PAA → ROP4

(6)

⎛ kcat ρB + kncat =⎜ 1 + ΣKi Ciaq ⎝

whereROP1, ROP2 , ROP3 and ROP4 denote the corresponding ringopening products. The perhydrolysis step is mainly catalysed by the solid catalyst, but also the carboxylic acid itself is able to catalyse the process, which has been demonstrated by several authors, for example in ref. [16,38,39]. Consequently, the total rate of perhydrolysis in the presence of a heterogeneous catalyst is

R1 = R1ncat + R1cat . ρB

CPAAaq CW aq K ' AA . CAAaq ⎞ ⎛ ⎞ ⎟ CAAaq CHPaq − K CW aq ⎠ ⎠⎝ ⎜

⎟

(14) The epoxidation step is considered to be elementary and the rate equation is obtained in a straightforward way,

R2 = kepox . COAoil. CPAAoil

(15)

The ring opening reactions are acid catalysed, i.e. a proton transfer takes places to the epoxide ring. The steps are summarized below,

(7)

where ρB = mcat / VL , i.e. mass of catalyst-to-liquid volume ratio, the subscripts cat and ncat refer to the heterogeneously catalysed epoxidation and the epoxidation proceeding spontaneously in the absence of an added catalyst. The surface reaction mechanism in the presence of the cation-exchange resin catalyst is described by the scheme below. The main hypothesis for the heterogeneously catalysed mechanism is that it is completely analogous with the homogeneous one, but the dissolved homogeneous acid is replaced by the sulphonic acid groups on the cation exchange resin in the reaction mechanism. The carboxylic acid adsorbs on the acid site on the solid catalyst (*=−SO3 H ) and H2 O2 from the bulk of the aqueous phase makes a nucleophilic attack, whereas H2 O2 adsorbed on the catalyst surface has the role of a spectator. The process can be summarized as

(I)

(IIa)

Yi + H2 O ⇌ ROPi + H3

O+

(IIb)

where Xi is the ring-opening agent (W, HP, AA, PAA) and Yi is the corresponding reaction intermediate. The ring opening with these agents always leads to a formation of a hydroxyl group, whereas the neighbouring group in the product molecule is OH, OOH, OCOCH3 or OOCOCH3, depending on the ring-opening agent [32]. The protonation step is rapid, while the ring opening step is slower. The steps are denoted by

RCOOH + * ⇌ RCOOH * RCOOH * + H2 O2 ⇌ RCOOOH + *+W

EOA + H+ ⇌ EOA+ + W

The proton transfer step is rapid, whereas the nucleophilic attack is presumed to be the rate limiting step. The steps following the

EOA+ + Xi + W ⇌ ROPi + H+ 3

(I) (II)

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The quasi-equilibrium hypothesis for the rapid steps gives

KI =

CEOA +CWoil CEOA CH +

(16)

from which the concentration CE+ is solved and inserted in the rate equation which becomes

RROi = kROPi CH +CEOA CXi

(17) +

In this case, the protons (H ) originate mainly from acetic acid, which is a much stronger acid than peracetic acid. For weak acids the proton concentration is proportional to the square root of the acid concentration. Thus rate Eq. (17) can in the actual case be written as

RROP = (kROP, AA CAAoil +kROP, PAA CPAAoil +kROP, W CW oil + kROP, HP CHPoil ) .CEOAoil.

K ' AA . CAAoil CW oil

(18)

Fig. 3. Modelling principle for the reactor system.

3. Reactor model 3.1. Mass balances for multiphase stirred tank reactors The structure of the loop reactor system is displayed schematically in Fig. 2. The loop reactor consists of two basic elements, the vigorously agitated stirred tank and the loop, which includes the microwave part (the chimney) and the heat exchanger. The circulation rate was high compared to the reaction rates. A typical residence time in the loop was 18 s, whereas a typical residence time in the stirred tank was 85 s. The principle of complete recycle was applied, so, from the macroscopic viewpoint, the system could be considered as a batch reactor, because the recycle rate was high compared to the reaction rates. However, the temperatures in the stirred vessel and in the loop were different. Consequently, the system was described as two stirred tanks (T1 and T2). The assumption of stirred tank was fully motivated for the reactor vessel, which was exposed to very vigorous stirring (1200 rpm). For the loop, the hypothesis of stirred tank can be motivated by the short residence time, i.e. low reactant conversion for one cycle. At low reactant conversions, different reactor models give very similar conversions and selectivities. Three phases co-exist in the stirred reactor vessel (oil, liquid, solid catalyst), while the loop is a two-phase system, because the catalyst was fixed in the mixing device in the stirred tank. The modelling principle is illustrated in Fig. 3.

Fig. 4. Inlet and outlet flows to the first tank (T1).

The inlet and outlet flows to the first tank (T1) are illustrated in Fig. 4 The mass balance equation for the aqueous-phase (aq) components in T1 can be written as

Ci2 aq Vaq̇ +

∑ vij Rjaq Vaq = Ni1 A1 +

dni1aq dt

+ Ci1aq Vaq̇

(19)

where N and A denote the interfacial flux and interfacial (oil-aqueous phase) area, n is the amount of substance, V̇ is the volumetric flow rate and V is the volume. All the symbols are explained in Notation.

Fig. 2. Schematic flowsheet of the loop reactor system. 4

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A.F. Aguilera, et al.

dCi1aq

For the components in the oil (oil) phase, an analogous mass balance equation can be written,

Ci2oil Voil̇ + Ni1 A1 +

∑ vij Rjoil Voil =

dni2oil + Ci1oil Voil̇ dt

dt −1 1 − α ⎞ ⎛ (β + = ⎛α + Ki ⎠ ⎜⎜ ⎝ ⎝ ⎜

(20)

In the most general case, both mass balance equations are needed, and a suitable correlation for the mass transfer coefficients in the interfacial fluxes (Ni ) is incorporated. Previous experience however indicates that the interfacial mass transfer is rapid compared to the perhydrolysis kinetics [16]. Therefore it is justified to add balance Eqs. (19) and (20) and to apply the hypothesis of phase equilibria for the oil and aqueous phases. The result of the addition becomes

(Ci2 aq − Ci1aq ) Vaq̇ + (Ci2oil − Ci1oil ) Voil̇ + =

dni1aq dt

+

+ α ∑ vij Rjaq + (1 − α )

⎞ ⎟ ⎠

(32)

dCi1oil dt ⎛ (β + = (αKi + 1 − α )−1 ⎜ ⎜ ⎝

(21)

1−β )(Ci2 aq Ki

τ1

− Ci1aq )

+ α ∑ vij Rjaq + (1 − α )

⎞

∑ vij Rjoil⎟ ⎟ ⎠

The amounts of substance are expressed with concentrations

ni1aq = Ci1aq Vaq = Ci1aq αV1

(22)

ni2oil = Ci2oil Voil = Ci1oil (1 − α ) V1

(23)

(33)

By assuming that the ratio between the volumetric flow rates is equal to the ratio between the corresponding volumes of the oil and aqueous phases, β = α ,

where α = Vaq/ V1 and the hypothesis of phase equilibrium is assumed

Ci1oil = Ki−1 Ci1aq

− Ci1aq )

τ1

∑ vij Rjoil⎟

∑ vij Rjaq Vaq + ∑ vij Rjoil Voil

dni2oil dt

1−β )(Ci2 aq Ki

⎟

Vaq̇ Vaq = →β=α V1 V̇

(34)

Thus the operative forms of the mass balances are written as

(24)

dCi1aq

Differentiation of Eq. (24) gives

dt dCiaq dCioil = Ki−1 dt dt

=

(25)

Ci2 aq − Ci1aq

−1

1 − α⎞ + ⎛α + (α ∑ vij Rjaq + (1 − α ) Ki ⎠ ⎝ ⎜

τ1

⎟

∑ vij Rjoil) (35)

The ratios of the volumetric flow rates of the aqueous and oil phases are expressed by the factor β ,

Vaq̇ = βV̇

(26)

Voil̇ = (1 − β ) V̇

(27)

dCi1oil dt Ci2oil − Ci1oil = + (αKi + 1 − α )−1 (α ∑ vij Rjaq + (1 − α ) τ1

(36)

The balance equation becomes

(Ci2 aq − Ci1aq ) βV̇ + (Ci2oil − Ci1oil )(1 − β ) V̇ +

A schematic view of Tank 2 is provided by Fig. 5. For the second tank (T2), a new derivation of the mass balances is not needed, but we can write analogously to the first tank (T1) (see the flowsheet in Fig. 2),

∑ vij Rjaq αV1 + ∑ vij Rjoil

1 − α ⎞ dCi1aq (1 − α ) V1 = ⎛α + V1 dt Ki ⎠ ⎝ ⎜

∑ vij Rjoil)

⎟

(28)

dCi2 aq

The residence time is defined by

dt V1 = τ1 V̇

=

(29)

Ci1aq − Ci2 aq

−1

1 − α⎞ + ⎛α + (α ∑ vij R' jaq + (1 − α ) Ki ⎠ ⎝ ⎜

τ2

⎟

(37)

after which we obtain

β (Ci2 aq − Ci1aq ) τ1 Rjoil

(1 − β )(Ci2oil − Ci1oil ) + α ∑ vij Rjaq + (1 − α ) τ1 1 − α ⎞ dCi1aq = ⎛α + Ki ⎠ dt ⎝ +

⎜

∑ vij

⎟

(30)

The partition coefficient for the phase equilibrium is assumed to be temperature independent within the narrow temperature interval applied in the experiments,

Ci1oil Ci2oil = = Ki−1 Ci1aq Ci2 aq

∑ vij R' joil)

(31)

After a final arrangement, the mass balances of the components in the aqueous and oil phases become

Fig. 5. Inlet and outlet flows to Tank 2. 5

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A.F. Aguilera, et al.

dCi2oil dt Ci1oil − Ci2oil = + (αKi + 1 − α )−1 (α ∑ vij R' jaq + (1 − α ) τ2

dCPAA2 aq dt −1 CPAA1aq − CPAA2 aq 1 − α⎞ . = + ⎛α + τ2 KPAA ⎠ ⎝ (α (Rper − Rdecom) − (1 − α ) R epox )

∑ vij R' joil)

⎜

(38)

dCW 2 aq

3.2. Application of the general mass balances to perhydrolysis, epoxidation and ring opening

dt

For the actual cases, all the components do not appear in both liquid phases (oil and water), but hydrogen peroxide and water are present in the aqueous phase only, while the fatty acid, the epoxide and the ring opening products are predominantly in the oil phase. For acetic acid and peracetic acid, a phase equilibrium is presumed. Based on this this reasoning, the equilibrium ratios become

=

dCEOA2 oil CEOA1oil − CEOA2oil = + REpox − RRO dt τ2

(50)

KEOA = 0 KROP = 0 KAA ≠ 0 KPAA ≠ 0 The reaction stoichiometry, Eqs. (1)–(6), gives the relation between the generation rates of the components and the reaction rates. After inserting the rate expressions in the balance equations for the first tank (T1), we obtain

dCAA1aq dt −1

⎜

=

ω=

CHP 2 aq − CHP1aq τ1

− Rperh

(40)

=

CW 2 aq − CW 1aq τ1

An extensive set of kinetic experiments was conducted in the loop reactor. Amberlite IR-120 was used as the heterogeneous catalyst to enhance the perhydrolysis rate. The majority of the experiments were carried out under conventional heating, but for the sake of comparison, some experiments were conducted under microwave radiation. The rotation speed of the mixing device, in which the catalyst was placed, was 1200 rpm to guarantee the operation in the kinetic regime. The liquid volume in the glass reactor was varied between 130 and 390 mL and the total volume of the loop was 91 mL. The catalyst load was varied between 2 and 36w-%. The tailored microwave equipment from Sairem was used in some experiments. In the experiments with microwaves the fixed frequency 2.45 GHz was applied. The experimental procedure is described in detail in a previous article of our group [34], and is therefore not repeated here. The essentials of the equipment and the procedure are summarized in Table 1 and the experimental conditions for each experiment are displayed in Table 2. Samples were withdrawn from the reactor vessel (Fig. 2) and the chemical analyses were made: determination of the iodine value (the amount of double bonds) and the oxirane number (amount of epoxide groups) as well as the determination of the concentrations of hydrogen peroxide, acetic acid and peracetic acid. The Greenspan and McKellar methods [44] were applied for the analysis of hydrogen

⎟

+ Rperh

(41)

(42)

dCOA1oil COA2oil − COA1oil = − R epox dt τ1

(43)

dCEOA1oil CEOA2oil − CEOA1oil = + REpox − ΣRROP, j dt τ1

(44)

For the second tank (T2), the balance equations become analogously

dt

dCHP 2 aq dt

(52)

4. Experimental procedure and experimental data

⎜

dCAA2 aq

∑ (Ci − Ci)̂ 2

where Ci is the experimental concentration and Ĉ is the estimated concentration obtained from the model Eqs. (39)–(50). The accuracy of the parameters was checked by standard mathematical analysis and a Markov Chain Monte Carlo (MCMC) method [43].

(39)

dt −1 CPAA2 aq − CPAA1aq 1 − α⎞ . = + ⎛α + τ1 KPAA ⎠ ⎝ (α (Rperh − Rdecom ) − (1 − α )(R epox − RROP, PAA))

dt

1 T0

⎟

dCPAA1aq

dCW 1aq

(51) 1 T

and T0 denotes the reference temperature (typiwhere = − cally close to the average temperature of the experiments). The transformation of the Arrhenius equation was done to suppress the mutual correlation of the pre-exponential factor and the activation energy. Furthermore, the parameter k0 gives the numerical value of the rate constant at the reference temperature. The complete model consists of twelve ordinary differential equations (ODEs), an initial value problem (IVP) (six equations for each tank). The ordinary differential equations describing this system can be stiff due to the presence of rapid and slow reactions, therefore the ODEs were solved numerically by the backward difference method implemented in the code ODESSA [40], which was operated under a parameter estimation routine in the software ModEst (Fortran 90 Compaq Visual FORTRAN 6.0) [41]. The objective function, (ω) was minimized by using a combined Simplex and Levenberg-Marquardt [42] algorithm implemented in ModEst,

K OA = 0

dt

−Ea Rθ

1 θ

1 − α⎞ . + ⎛α + τ1 KAA ⎠ ⎝ (−αRperh + (1 − α )(REpox − RROP, AA))

(48)

The temperature dependences of the rate constants were described by the modified Arrhenius equation,

KW → ∞

CAA2 aq − CAA1aq

+ Rperh

(49)

k = k 0. e

dCHP1aq

τ2

(47)

dCOA2oil COA1oil − COA2oil = − R epox dt τ2

KHP → ∞

=

CW 1aq − CW 2 aq

⎟

=

=

CAA1aq − CAA2 aq

−1

1 − α⎞ . (−αRperh + (1 − α ) REpox ) + ⎛α + KAA ⎠ ⎝ (45) ⎜

τ2

CHP1aq − CHP 2 aq τ2

− Rperh

⎟

(46) 6

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Table 1 Summary of experimental conditions. Carboxylic acid (CA) Catalyst Mass of catalyst Temperature DB:HP:CA

Table 5 Estimated kinetic constants, activation energies and acetic acid distribution coefficient for oleic acid epoxidation with MW from the previous study carried out in the absence of added solid catalyst [47].

Oleic acid Amberlite IR-120 2–36 w-% (dry base, in reference to oils mass) 40–50 °C 1:2.3:2.4 and 1:4.8:2.4 (molar ratios)

DB = double bond, HP = hydrogen peroxide, CA = carboxylic acid. Table 2 Experimental matrix. Experiment

T1 (°C)

T2 (°C)

%cat

ρB (g/L)

OA:HP:AA

Heating

Time (h)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.8 50.4 50.0 50.0 40.0 40.0 40.0 40.0 40.0 40.0

53.5 53.5 53.5 53.5 53.5 53.5 52.3 54.9 63.8 52.6 51.9 64.5 60.0 60.0 60.0 60.0 60.0 60.0

0 6 12 12 6 18 6 6 4 4 12 12 9 9 2 2 36 36

0.0 70.0 69.0 70.0 35.0 104.0 35.0 35.0 23.0 23.0 73.0 73.0 70.0 70.0 17.0 18.0 73.0 73.0

1:4.8:2.4 1:4.7:2.4 1:4.9:2.4 1:4.8:2.4 1:4.7:2.4 1:4.7:2.4 1:4.7:2.4 1:4.7:2.4 1:4.7:2.4 1:4.7:2.4 1:4.8:2.4 1:4.8:2.4 1:2.4:2.4 1:2.3:2.4 1:2.1:2.1 1:2.3:2.4 1:2.6:2.6 1:2.6:2.8

CH CH CH CH CH CH CH MW MW CH CH MW CH MW MW CH CH MW

50 6 6 50 6 6 6 6 6 6 6 6 6 6 9 9 6 9

Est. Relative Std Error (%)

kperh

0.317E-03

0.375E-04

11.8

k epox

0.348E-03

0.232E-04

6.7

kRO kperhCat

0.219E-01 0.341E-04

0.116E-01 0.288E-05

52.9 8.4

Ea RO Ea perhCat

0.964E + 05 0.595E + 05

0.234E + 06 0.207E + 05

242.9 34.8

KAA Aeq

0.140E-01 0.281E + 01

0.176E-01 0.441E + 00

125.8 15.7

kperh

0.615E-03

0.374E-04

6.1

k epox

0.331E-02

0.517E-03

15.6

kRO Ea perh

0.807E-01 0.600E + 05

0.576E-01 0.119E + 04

71.4 19.8

Ea epox

0.362E + 05

0.167E + 05

46.3

Ea RO KAA

0.708E + 05 0.439E + 00

0.152E + 06 0.870E-01

214.6 19.8

The overall statistical analysis of parameter estimation is provided in Table 3. The numerical values of the parameters obtained from the nonlinear regression analysis are collected in Table 4, while Table 5 displays the values of the parameters obtained with one of the models from our previous work carried out in the absence of heterogeneous catalysts [47]. The rate constant for the homogeneous part of the perhydrolysis shows to be of the same order of magnitude compared with the ones estimated with one of the models developed in our previous study. The rate constants for epoxidation and ring opening are within one order of magnitude in comparison to the mentioned study. Furthermore, the tendency observed in our previous studies [34,47] is maintained: the perhydrolysis is the slowest reaction, followed by the epoxidation and the ring opening reactions. Regarding the contributions of the homogeneous and heterogeneous parts of the perhydrolysis, the estimated values of the rate constants are in agreement with the k

ρ

physical sense of the system given that and 1 + ΣcatK CB

i iaq

≫ kncat

K ' AA . CAAaq CW aq

(Eq. (14)) which make the contribution of the homogeneous part much smaller than the solid-catalyzed part. Preliminary parameter estimation efforts indicated that the contribution of the adsorption terms in the rate Eq. (13) was minor; therefore the simplification 1 + ΣKi Ciaq ≃ 1 was applied. The estimation of the activation energies for perhydrolysis and epoxidation was cumbersome and very reliable results could not be obtained. This is due to two main factors, the small range of temperatures of the experiments and the rapid tendency of the experimental data for acetic acid and peracetic acid to reach equilibrium and remain constant throughout the reaction (Fig. 6). Given that acetic acid and peracetic acid play key roles in both the perhydrolysis and epoxidation and there is a limited temperature range in the experimental set, the activation energies for these two reactions converged towards zero. In comparison to the previous model, the activation energy for the ring opening is in the same order of magnitude but it presents a high margin of error. On the other hand, it is to be expected that the activation energy of the solid-catalyzed perhydrolysis present lower values compared with the homogeneous perhydrolysis, especially when taking into consideration that the result obtained is the apparent activation energy and it comprises the enthalpies of adsorption into the catalyst surface, which are negative and make the apparent activation energy even smaller. However, the activation energies for the solid-catalyzed perhydrolysis and the homogeneous perhydrolysis seem to be within the same order of magnitude because of the lack of accuracy in the determination of the activation energies of this system is due to the narrow temperature window in which the experiments were performed. Taking into consideration that the activation energies have been successfully determined in our previous study and that there was a strong motivation to focus on the enhancing effect of the addition of the solid catalyst, the experimental program was focused on the variation

Table 4 Estimated kinetic constants, activation energies, distribution coefficients and pre-exponential factor of perhydrolysis equilibrium in oleic acid epoxidation. Estimated Std Error

Est. Relative Std Error (%)

5. Parameter estimation results and discussion

2.733E + 03 1.041E + 02 0.4178E + 00 96.19

Estimated Parameter

Estimated Std Error

The units: kperh , kepox = () L/mol/min; kRO = () 1/min; activation energies (Ea ) = () J/mol, KAA = ().

Table 3 Statistical results for oleic acid epoxidation. Total SS (corrected for means) Residual SS Std. Error of estimate Explained (%):

Estimated Parameter

The units: kperh, kepox , kRO = () L/mol/min; kperhCat = () L2/mol/g/min; activation energies (Ea ) = () J/mol, KAA , Aeq = (). Aeq is the pre-exponential factor of the perhydrolysis equilibrium constant, Eq. (14).

peroxide and peracetic acid in the aqueous phase. For the determination of acetic acid, an automatic titrator with a sodium hydroxide solution was used. The amount of double bonds in the oil phase was determined with Hanus solution [45] and the epoxide content was determined with Jay’s method [46], in which HBr is formed in situ with tetraethylammonium bromide (TEAB) and titrated with a perchloric acid solution. From these primary analytical data, the molar concentrations of the components were calculated. The total amount of the samples withdrawn was small (< 10 wt-%) and its effect on the solid–liquid ratio was considered negligible.

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Fig. 6. Fitting of the model to the experimental data for oleic acid epoxidation. In black: epoxyoleic acid, : peracetic acid, : oleic acid, : hydrogen peroxide and : acetic acid. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

concentrations of these components remain virtually constant throughout the reaction (Fig. 6), as there is consumption and generation of both components at similar rates. After preliminary efforts in parameter estimation, the value of the partition coefficient of peracetic acid was fixed to 1, because it turned out that the effect of this parameter was very minor. In principle, the partition coefficients could be measured by separate experiments under non-reactive conditions. However, it is difficult to reach completely non-reactive conditions for the actual system, because the reactions proceed to some extent in the absence of added catalyst, too [33].

of the catalyst amount and the ratios of the initial reactant concentrations. Moreover, it is important to emphasize that we have been bounded to conduct experiments under a narrow temperature window in the presence of the heterogeneous catalyst, because high concentrations of peracetic acid (> 15%) exhibit some degree of explosiveness and instability [48] and the system presents a high risk of a thermal runaway due to the exothermic nature of the reactions involved [49]. The partition coefficients for the phase equilibrium for acetic acid and peracetic acid were not determined very precisely because the 8

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A.F. Aguilera, et al.

Table 6 Correlation matrix of the parameters. kperh

k epox

kRO

kperhCat

Ea RO

Ea perhCat

kperh

1.000

k epox

−0.224

1.000

kRO kperhCat

−0.027 −0.073

0.096 −0.036

1.000 −0.001

1.000

EaRO Ea perhCat

−0.001 0.067

0.003 −0.033

−0.190 −0.010

−0.001 0.116

1.000 0.028

1.000

KAA Aeq

0.019 −0.036

−0.006 −0.086

0.034 0.019

0.015 −0.550

0.006 0.008

−0.046 −0.048

KAA

Aeq

1.000 0.188

1.000

Fig. 7. MCMC sensitivity plot. Predictive distribution for the estimated parameters.

different solutions in this kind of calculations. Finally, a parity plot is displayed in Fig. 9 where generally a good agreement with experimental and calculated values can be observed.

Table 6 displays the correlation matrix for the estimated parameters. In general, the correlation between the parameters is rather low, which supports the model. The fit of the model to the experimental data is compared in Fig. 6, which indicates that the model fits with the experimental data well. The main features of the experimental data are well reproduced by the model: the decrease of the double bond and hydrogen peroxide concentrations, the formation of epoxide as well as the close to steady state behavior of acetic and peracetic acid. Fig. 7 displays the parameter set plot for the sensitivity analysis, demonstrating rather well-defined minima for most of the parameters. Fig. 8 displays the probability distribution of the estimated parameters. The distributions of the values of the rate constants and the acetic acid distribution constant confirm to be normal, whereas the distributions for the activation energies and the pre-exponential factor for perhydrolysis equilibrium Aeq seem to be multimodal. Moreover, it is important to consider the high complexity of the system can lead to

6. Conclusions A new mathematical model was developed for a loop reactor system consisting of a perfectly stirred tank reactor and a loop with a conventional heat exchanger and a microwave device. The model was applied to the epoxidation of a fatty acid model component, oleic acid in the presence of Amberlite IR-120 as a solid catalyst. The model incorporates rate equations for the essential steps of fatty acid epoxidation according to the Prileschajew principle, namely perhydrolysis, epoxidation and ring opening. The reactor model takes into account the co-existence of the oil phase, the aqueous phase and the solid catalyst phase. Non-linear regression analysis revealed that the model gave a very satisfactory description of oleic acid epoxidation in the presence of 9

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A.F. Aguilera, et al.

Fig. 8. MCMC probability distributions of the model parameters.

separately from the kinetic experiments. However, it was illustrated that the approach is principally applicable for the very demanding three-phase system and complex kinetics present in the epoxidation fatty acids. The general principles of the model can be used for epoxidation of other components, too and it has a potential in the process design and optimization. Acknowledgements Financial support from Åbo Akademi Forskningsinstitut is gratefully acknowledged. This work is part of the activities financed by Academy of Finland, the Academy Professor grants 319002 (T. Salmi) and 320115 (A. Freites Aguilera and P. Tolvanen). Institut National des Sciences Appliquées de Rouen (INSA) is gratefully acknowledged for fruitful scientific collaboration. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.cej.2019.121936.

Fig. 9. MCMC parity plot between experimental and calculated values (concentrations).

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