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ISSN 0023-1584, Kinetics and Catalysis, 2017, Vol. 58, No. 2, pp. 211–217. © Pleiades Publishing, Ltd., 2017.

Mathematical Modeling of Heterogeneous Catalysis Involving Polymer-Supported Catalysts1 H. A. Silva and L. G. Aguiar* Department of Chemical Engineering, Engineering School of Lorena, University of Sao Paulo, 12602-810, Lorena, SP, Brazil *e-mail; [email protected] Received June 3, 2016

Abstract—This study reports the mathematical modeling of catalytic reaction systems involving polymer-supported catalysts. Differential mass balances were applied to species and the partial differential equations were solved through the method of lines in MATLAB®. Etherification and esterification reactions were studied with the present model and validation was performed with literature data, providing fair agreement. Furthermore, the model proved capable of predicting concentration gradients along the catalyst particles, providing an interesting level of detail to represent catalytic heterogeneous systems. Keywords: mathematical modeling, kinetics, etherification, esterification, Amberlyst DOI: 10.1134/S0023158417020112

INTRODUCTION The search for renewable ecologically friendly sources providing an alternative to petroleum has received an increased attention in many fields of academic research. Many studies are focused on developing compounds to be added to the diesel fuel, aiming to reduce soot and nitrogen oxides emission into the atmosphere. In this context, the addition of organic oxygenates, such as alcohols, ethers and esters, is promising. Among these compounds, ethers show physical-chemical properties favorable for this application [1]. Interesting results in terms of octane number and cold flow properties were reported [1–3] using linear ethers with more than nine carbon atoms, such as di-n-pentyl ether (DNPE). Another challenge is the search for processes, which make use of solvents not derived from petroleum. Lactate esters, such as ethyl lactate, represent a viable option because they can be obtained through biomass processing. They are biodegradable and show interesting chemical properties due to which they can be useful in substitution of halogenated solvents derived from petroleum [4]. Heterogeneous catalysts are used in different industry sectors because they are less susceptible to corrosion, can be easier separated and stored than homogeneous catalysts [4, 5]. Among the different heterogeneous catalysts, the polymer-supported systems present better microenvironment than catalysts carried by silica, alumina, and other materials, due to the flexibility of polymer chains [6]. In this kind of application, the most widely used polymers are sty1 The article is published in the original.

rene-divinylbenzene resins since they show excellent thermal and mechanical stability and resistance to degradation [7]. The incorporation of active sites into resins is conducted through its functionalization with electrophile groups (e.g., sulfonation of aromatic rings from styrene monomeric units). The sulfonated resins can be used as heterogeneous catalysts in a variety of organic reactions such as etherification [1, 8–12], esterification [4, 13–17], and transesterification [18–20]. Significant catalytic activity at low temperatures reduces heat expenditure and inhibits parallel reactions, thus increasing yields of products [7, 12]. Nevertheless, not all active sites formed in the functionalization process have the same accessibility. A concentration gradient is formed, in which the active sites concentration is reduced from the external surface to its internal parts of the catalyst particle. A similar gradient is formed during the catalyzed reaction, where there is transport of reagents from outside to inside of the polymeric material. The transport rate is a function of the polymer properties such as the degree of cross-linking [5, 7]. The development of mathematical models that can comprise both reaction kinetics and diffusion behavior has received increasing attention in recent years. Lilja et al. studied esterification reaction with Smopex-101 and obtained diffusion coefficients by applying WilkeChang equation [21, 22]. Meena et al. adopted Adomian decomposition method to represent concentration of species inside the catalyst during a catalyzed reaction [23]. Paakkonen and Krause used an effectiveness factor proportional to the reagents diffusivity in Amberlyst-16 to study iso-amylene esterification

211

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[24]. Silva and Rodrigues assessed the kinetic constant for the diethylacetal synthesis and the components diffusion through Amberlyst-18, using a mathematical model based on mass balances inside and outside the catalyst [25]. Despite the studies indicated above complex systems involving the correlation between the polymer characteristics and its interactions with the compounds of the catalyzed reaction were still not fully explored in literature. From the theoretical aspect, a mathematical model based on spatial concentration variations is an interesting tool to approach catalytic systems involving polymer particles. Thus, the purpose of this work was to develop a mathematical model for heterogeneous catalysis with styrene-based catalysts. A certain degree of detail, such as representation of concentration gradients along the particle, was intended during the model development. Esterification and etherification reactions were studied and the model was validated with literature data. METHODOLOGY The present mathematical modeling was based on differential mass balance for each reaction constituent. Diffusion of components was considered inside the catalyst particles as the liquid phase was considered homogeneous. Two phases were defined for the present system: phase I—solid (particles), and phase II—liquid (solution). Equations (1) and (2) represent the mass balances for generic component A in phases I and II, respectively [26]:

v ∂C A ∂C A ∂C A ∂C + vr + v θ A + 0/ ∂t ∂r r ∂θ r sin θ ∂θ 2 ∂C A ⎞ 1 1 ∂ ⎛ sin θ ∂ C A ⎞ ∂ ⎡ ⎛ = DA ⎢ 2 ⎜ r ⎟+ 2 ⎜ ⎟ ∂ r ⎠ r sin θ ∂θ ⎝ ∂θ ⎠ ⎣r ∂ r ⎝ 2 ∂ CA ⎤ + 2 1 2 + rA , 2 ⎥ r sin θ ∂θ ⎦ k ⎡ II C AN + 1 ⎤ d C AII = − CII ⎢C A − avV1P N P. dt K P ⎥⎦ V ⎣

(1)

(2)

In order to simplify and tune the model to the studied conditions, the following assumptions were made: 1) spherical particles of constant diameter, 2) homogeneous composition of the liquid phase (phase II), 3) concentration variations occur only along the radial direction,

4) absence of density and diffusivity variations inside the particle, 5) transport inside the particle is affected by diffusion only. Based on these assumptions, eq. (1) can be simplified to give:

∂C A 2 ∂ C A ⎞⎤ (3) = D A ⎡ 12 ∂ ⎛⎜ r ⎟ + rA . ⎢ ⎣r ∂ r ⎝ ∂t ∂ r ⎠⎥⎦ Equation (3) is solved making use of the following boundary conditions: 1) t = 0; C AI = 0, 2) r = 0;

∂C A ∂r I

= 0, r =0

I ⎛ ⎞ CA ∂ C AI II r =R ⎟ = DA r = R; k c ⎜C A − ⎜ ∂r KP ⎟ ⎝ ⎠ 3) C AI + rA r = R , K P = II r = R , CA

r =R

r =R

where C AI and C AII are the concentrations of A in phases I and II, respectively, kc is its mass transfer coefficient in phase II, DA is the diffusivity of A in phase I, rA is the reaction rate and Kp is the partition coefficient of A between phases I and II, t is the reaction time, r represents the radial axis and R is the particle radius. The method of lines was applied, and Eq. (3) was described as a set of ordinary differential equations representing layers of the particle. A constant number of layers (N = 10) was assumed. This approach allows the determination of concentration gradients along the catalyst particle. The differential equations were described as follows:

d C Aj dt ⎡C Aj +1 − 2C Aj + C Aj −1 1 C Aj +1 + C Aj −1 ⎤ = DA ⎢ + ⎥ + rA (4) 2 Δ r r Δ r ( ) j ⎣ ⎦ (2 ≤ j ≤ N ), d C A1 C −C = 6D A A2 2 A1 + rA , dt (Δr )

(5)

⎡C − 2C AN +1 + C AN 2 C AN + 2 − C AN ⎤ d C AN +1 = DA ⎢ AN + 2 + ⎥ + rA , dt rj 2Δ r (Δr )2 ⎣ ⎦ KINETICS AND CATALYSIS

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MATHEMATICAL MODELING OF HETEROGENEOUS CATALYSIS Table 1. Experiments considered in the modeling study TemperaRun tura (K)

Initial concentrations (mol/L)

C EthC LA −

1

433

CP = 9.1082

2

433

CP = 8.7440

3

433

CP = 7.6510

4

463

CP = 9.1082

5

333.11 CLA = 5.2310, CEth = 9.4640, CW = 3.8300

6

353.40 CLA = 5.2310, CEth = 9.4640, CW = 3.8300

[8]

⎛ C C ⎞ A ⎜C P2 − W D ⎟ K1 ⎠ = ⎝ , (C P + BC W ) 2

(1 + K s,EthC Eth

C ELC W K2 . 2 + K s,WC W )

[4]

(7)

RESULTS AND DISCUSSION Two catalytic reactions were studied with the present model: etherification of 1-pentanol to form DNPE using Amberlyst 70® with an average diameter of 0.57 mm as reported by Bringue et al. [8] and esterification of lactic acid (LA) with ethanol (Eth) to form ethyl lactate (EL) catalyzed by Amberlyst 15® with an average diameter of 0.685 mm as reported by Pereira et al. [4]. Table 1 presents the initial conditions of each run simulated in the present work. All experiments were conducted in batch reactors of 70 mL for the etherification of 1-pentanol and 77.4 mL for the esterification of lactic acid with ethanol [4, 8]. The curve fitting study resulted in a set of kinetic and thermodynamic parameters obtained for each run, as described in Tables 2 and 3. The curves in Fig. 1 show the simulation results for consumption of 1-pentanol and production of DNPE and water (W) for the run 1. Fair agreement was obtained from the comparison between predicted and experimental data of DNPE concentration. The present model is able to predict the concentration of species over time for each layer through the particle. Figure 2 shows concentration profiles for the surface and an internal layer of the catalyst particle.

Table 2. Parameters considered in the etherification study Parameter

Run 1

Run 2

Run 3

Run 4

0.09919

0.09919

0.09919

0.8398

0.9316 54.0162

0.9316 54.0162

0.9316 54.0162

1.4491 48.0409

DP (m2/s)

3.57 × 10–10

3.57 × 10–10

3.57 × 10–10

3.57 × 10–10

DD (m2/s)

1.77 × 10–10

1.77 × 10–10

1.77 × 10–10

1.77 × 10–10

DW (m2/s)

6.82 × 10–10

6.82 × 10–10

6.82 × 10–10

6.82 × 10–10

kcP (m/s)

6.25 × 10–3

6.25 × 10–3

6.25 × 10–3

6.25 × 10–3

kcP (m/s)

0.0156

0.0156

0.0156

0.0156

kcW (m/s)

0.1470

0.1470

0.1470

0.1470

KpP (dimensionless)

0.0284

0.0284

0.0284

0.0284

KpD (dimensionless)

0.0105

0.0105

0.0105

0.0105

KpW (dimensionless)

0.0560

0.0560

0.0560

0.0560

A (mol g–1 min–1)* В (dimensionless)* K1 (dimensionless)*

* Data fitted in this study, except for Data reported by Bringue et al. [8] KINETICS AND CATALYSIS

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The system of differential equations was numerically solved through MATLAB®. Kinetic and thermodynamic parameters were fitted as described in the section that follows.

where Δr is the distance between two adjacent layers, i.e., Δr = R ; and C AN + 2 = C AN + N ⎡ ⎛ ⎞ ⎤ C II 2Δ r k C − AN +1 + r ( A ) N +1 ⎥ . c⎜ A ⎟ ⎢ DA ⎣ ⎝ KP ⎠ ⎦ Different catalytic reactions were modeled with the present approach and cases from literature were simulated in order to validate it. 1-Pentanol (P) etherification and lactic acid (LA) esterification were modeled through Langmuir–Hinshelwood–Hougen–Watson (LHHW) approach considering the surface reaction as the rate-limiting step. The production rates of di-npentyl ether (DNPE) and ethyl lactate (EL) are described in Eqs (7) and (8), respectively [8, 12]:

rDNPE

rEL = k

Reference

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Table 3. Parameters considered in the esterification study Parameter Run 5 –1

–1

0.7849 3.5910 15.7480 4.1209 1.16 × 10–9 1.50 × 10–9 1.75 × 10–10 1.54 × 10–9 0.0887 0.1077 0.1250 0.0418 0.02460 0.03480 0.00636 0.03500

Concentration, mol/L

k (mol g min ) Ks, Eth (dimensionless) Ks,W (dimensionless) K2 (dimensionless) DLA (m2/s) DEth (m2/s) DEL (m2/s) DW (m2/s) kcLA (m/s) kcEth (m/s) kcEL (m/s) kcW (m/s) KpLA (dimensionless) KpEth (dimensionless) KpEL (dimensionless) KpW (dimensionless) 10 9 8 7 6 5 4 3 2 1 0

2 3 100 150 200 250 300 350 400 Time, min

Fig. 1. Model molar concentrations of 1-pentanol (1), DNPE (2) and water (3) in phase II over time. e—DNPE (experimental). Simulation and comparison obtained with data from run 1.

Predictions of DNPE concentrations for runs 1–4 are shown in Fig. 3. An average R2 of 0.71 was obtained, indicating fair correlation between experimental data and model predictions. The lowest DNPE concentrations were obtained for run 2, in which a higher amount of water was fed. The presence of water

(a)

0.35

(b) 0.05

0.30

Concentration, mol/L

Concentration, mol/L

2.2120 3.3750 15.7150 4.5035 1.16 × 10–9 1.50 × 10–9 1.75 × 10–10 1.54 × 10–9 0.0887 0.1077 0.1250 0.0418 0.02460 0.03480 0.00636 0.03500

An increase in 1-pentanol profiles at the beginning of the process for both layers is observed as depicted in Fig. 2. This behavior indicates diffusion of this component into the internal part of the catalyst. In both positions, the water concentration profile is above the DNPE profile. This result can be expected because water has higher affinity for the acid environment inside the resin in comparison to DNPE [1, 16, 27]. This phenomenon is corroborated by higher values of KP for water (0.056) compared with those obtained for 1-pentanol (0.0284) and DNPE (0.0105).

1

50

Run 6

0.25 0.20

1

0.15 0.10 3

0.05

0.04 1

0.03 0.02

3

0.01

2

2 0

100

200 Time, min

300

400

0

100

200 Time, min

300

400

Fig. 2. Model concentration profiles of 1-pentanol (1), DNPE (2) and water (3) for different layers of the particle (phase I) obtained for run 1: a—surface, b—internal layer (radius = 0.1425 mm). KINETICS AND CATALYSIS

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MATHEMATICAL MODELING OF HETEROGENEOUS CATALYSIS

compounds for the resin Amberlyst-15. Thus, the phenomena observed in Fig. 4 is corroborated by higher values of KP for water (0.0350), followed by ethanol (0.0348), lactic acid (0.0246) and ethyl lactate (0.00636) [1, 4]. The discrepancy between experimental and predicted conversions might be related to the presence of dimers and trimers, which were not taken into account in the reaction mechanism [4]. Nevertheless, fair agreement was obtained between the model predictions and the experimental data of this esterification reaction.

4

3.0

Concentration, mol/L

2.5 2.0 3 1.5 1

1.0

2 0.5

100

0

200 Time, min

215

300

400

Fig. 3. Results of DNPE concentration obtained with data for runs 1–4 at model predictions (1–4, respectively) and from experimental data (e—run 1, n—run 2, +—run 3, ,—run 4).

causes inhibition in the catalyst activity because water has a higher affinity for the catalytic sites than 1-pentanol [16, 27]. The present approach was applied to the esterification of lactic acid with ethanol (runs 5 and 6). The simulation results are presented in Figs. 4 and 5. Expected concentration and conversion profiles were obtained (Figs. 4, 5). The correlation between model and experimental data from runs 5 and 6 presented R2 = 0.7880 and R2 = 0.9418, respectively. As in the case of etherification the rate of diffusion into the particle in the esterification depends on the affinity of

Figure 6 shows the concentration gradients of 1-pentanol, DNPE, lactic acid and ethyl lactate for different reaction times. It can be observed that 1-pentanol and DNPE profiles increase with increasing reaction time and reach a constant gradient after 300 min. Due to a higher 1-pentanol concentration at layers near the particle surface, the reaction rate is higher, thus increasing the DNPE production in this region. During the initial 60 min concentrations of ethyl lactate at external layers of the particle (radius > 0.3 mm) were lower than those of DNPE. This behavior is explained by a lower diffusivity of ethyl lactate inside the resin (1.16 × 10–9 m2/s) compared to those of ethanol (1.50 × 10–9 m2/s), lactic acid (1.75 × 10–9 m2/s) and water (1.54 × 10–9 m2/s). Figure 6 also shows a trend of reaching constant gradient profiles for reactions times close to 300 min. The concentration profiles obtained in phase I indicate that the external layers of the catalyst constitutes the main reaction region, since the reagents concentration are higher in these layers than in the internal parts of the particle. The parameters presented in Tables 2 and 3 are internally consistent since they demonstrate how polarity and affinity govern the relations between chemical compounds and the resins. The highest diffusion values were obtained for Amberlyst-15, which

(а)

(b)

12

8 6 4

4 2 0

50

100

150 200 Time, min

250

1 2 3 300

Concentration, mol/L

Concentration, mol/L

0.06 10

2

0.05 0.04 0.03 0.02

4 1

0.01 3 0

50

100

150 200 Time, min

250

300

Fig. 4. Model concentration profiles of lactic acid (1), ethanol (2), ethyl lactate (3), and water (4) of run 5 for different phases: a—phase II (liquid), b—inside the particle (radius = 0.1713 mm). e—Lactic acid (experiment). KINETICS AND CATALYSIS

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CONCLUSIONS

0.7 2

0.5

1

0.4 0.3 0.2 0.1 50

0

100

150 200 Time, min

250

300

Fig. 5. Model (1, 2) and experimental (e, +) lactic acid conversion results of runs 5 (1, e) and 6 (2, +).

shows higher acidity (4.81 meq H+/g) in comparison with Amberlyst-70 (2.65 meq H+/g) [1], corroborating results found in previous works [21, 22, 25].

0.12

(b)

1

0.012

2 3 4

0.20

DNPE concentration, mol/L

1-P entanol concentration, mol/L Lactic acid concentration, mol/L

(а) 0.25

5

0.15 0.10 0.05

0

0.05

0.10 0.15 0.20 Radius, mm

0.25

(c)

2 3

0.08

4

0.06 0.04 0.02 0.1

0.2 Radius, mm

0.3

5 0.010

4

0.008 3 0.006 0.004

2

0.002

1

0.30

1

0.10

0

A mathematical model based on LHHW and method of lines was developed and validated with esterification and etherification reactions. The model is capable of predicting the concentration of compounds in solution and gradients inside the catalyst particle. Fair predictions were obtained for simulations conducted with literature data. The concentration profiles suggest a faster diffusion of esterification compounds in comparison to etherification components, indicating higher affinity of the former for the resin Amberlyst-15. The present study also provides an explanation for a low yield obtained when water was involved in the etherification reaction. These observed effects were corroborated by the values of kinetic and thermodynamic parameters obtained in the curve-fitting analysis. Besides the reactions studied herein the present model is also useful for application in transesterification and any other catalytic reaction involving spherical catalyst.

0

Ethyl lactate concentration, mol/L

Conversion, a. u.

0.6

4

0.05

0.10 0.15 0.20 Radius, mm

0.25

0.30

(d)

×10–3

4

3

3

2 2 1 1 0

0.1

0.2 Radius, mm

0.3

Fig. 6. Concentration gradients along the catalyst particle for different compounds: 1-pentanol (a), DNPE (b), lactic acid (c), and ethyl lactate (d). Time, min: (a, b) 50 (1), 100 (2), 200 (3), 300 (4), 360 (5); (c, d) 50 (1), 100 (2), 200 (3), 280 (4). Dots: layers defined in the method of lines. KINETICS AND CATALYSIS

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NOTATION av A, B

Particle’s external surface area/volume ratio, grouped factors for fitting purposes,

C AI

concentration of compound A in phase I,

C AII Dj

concentration of compound A in phase II, diffusion coefficient of compound j on phase I, thermodynamic equilibrium constant of etherification reaction, thermodynamic equilibrium constant of esterification reaction, rate coefficient,

K1 K2 k kcj

mass transfer coefficient of compound j,

Ks,Eth

equilibrium constant for ethanol adsorption,

Ks,W

equilibrium constant for water adsorption,

Kpj

partition coefficient of compound j,

Np

total number of particles, radial axis, radius of the catalyst,

r R rDNPE rEL t T

T V1p V θ

II

reaction rate for DNPE synthesis, reaction rate for ethyl lactate, reaction time, temperature, mean experimental temperature, volume of one catalyst particle, volume of phase II. latitudinal angular coordinates,

longitudinal angular coordinates. 0 Subscripts: D DNPE, di-n-pentyl ether, EL ethyl lactate, Eth ethanol, LA lactic acid, P 1-pentanol, W water.

ACKNOWLEDGMENTS The authors are grateful to FAPESP (Project no. 2014/22080-9) for the financial support. REFERENCES 1. Pérez, M.A., Bringué, R., Iborra, M., Tejero, J., and Cunill, F., Appl. Catal., A, 2014, vol. 482, p. 38.

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