**Table 2.**

*Reactive distillation studies aimed at the production of biodiesel through transesterification.*


*Distillation Processes - From Solar and Membrane Distillation to Reactive Distillation…*


#### **Table 3.**

*Nomenclature of terms used in mathematical modeling.*

**Figure 3.** *Configuration of each stage j in the reactive distillation column. Source: [53].*

The generic plate scheme adopted by the authors is represented in **Figure 3**.

Eq. (5), which represents the mass balance of component i in stage j of the column as a residual function, is given by:

$$f\_{i,j}^{m} = (\mathbf{R}\_{j} + \mathbf{1})n\_{i,j}^{\mathrm{II}} + (\mathbf{Z}\_{j} + \mathbf{1})n\_{i,j}^{\mathrm{I}} - \left(n\_{i,j+1}^{\mathrm{II}} + n\_{i,j-1}^{\mathrm{I}} + F\_{i,j} + \sum\_{k=1}^{nr} \nu\_{i,j}\mathfrak{k}\_{kj}\right) = \mathbf{0} \tag{5}$$

Assuming that the streams that leave the stage are in phase equilibrium, Eq. (6) relates the mole fractions in the liquid and vapor phases:

$$f\_{i,j}^{eq} = \ln\left(\mathbf{x}\_{i,j}^{I} P\_j\right) - \ln\left(\mathbf{x}\_{i,j}^{II} \mathbf{y}\_{i,j}^{II} P\_{i,j}^{sat}\right) = \mathbf{0} \tag{6}$$

In this expression, the Poynting correction and the fugacity coefficient of the pure saturated compounds are neglected. In addition, and the vapor phase is considered to be an ideal gas mixture as a consequence of the

*Reactive Distillation Applied to Biodiesel Production by Esterification: Simulation… DOI: http://dx.doi.org/10.5772/intechopen.102667*

assumption that the column operates at low pressure, close to atmospheric conditions.

The equation for the rate of reaction k in stage j is represented by Eq. (7), which can be expressed as a residual equation by applying the logarithm function (Eq. (8)).

$$\xi\_{k,j} = k\_{k,j} \prod\_{i=1}^{nc} \mathbf{C}\_{i,j}^{a\_{i,k}} \tag{7}$$

$$f\_{kj}^{r} = \ln k\_{kj} + \sum\_{i}^{nc} a\_{i,k} \ln \left(\frac{\mathbf{x}\_{i,j}^{H}}{\mathbf{v}\_{j}^{H}}\right) - \ln \xi\_{kj} = \mathbf{0} \tag{8}$$

Assuming that the molar volume of the liquid phase is that of an ideal solution and describing Eq. (8) as a function of the activity coefficients of the components in the liquid phase, Eq. (9) is obtained.

$$f\_{k,j}^{\tau} = \ln\left(k\_{k\bar{j}}\right) + \sum\_{i}^{nc} \alpha\_{i,k} \ln\left(\varkappa\_{i,j}^{H} + \eta\_{i,j}^{H}\right) - \ln\left(\xi\_{k,j}\right) = 0 \tag{9}$$

Eq. (10), which describes the energy balance of stage j, is needed to calculate the temperature, which is different at each stage of the reactive distillation column. Positive and negative values of Qj correspond to the heat being supplied to or removed from the column, respectively.

$$f\_j^h = (\mathbf{R}\_j + \mathbf{1})H\_j^{\text{II}} + (\mathbf{Z}\_j + \mathbf{1})H\_j^{\text{I}} - \left(H\_{j+1}^{\text{II}} + H\_{j-1}^{\text{I}} + H\_{F\_j} + Q\_j\right) = \mathbf{0} \tag{10}$$

The ratio between the molar flows of vapor and liquid leaving each stage of the column is represented using Eq. (11). This equation is intended to make the condenser and reboiler specifications more flexible by associating the relationship between the liquid and vapor streams that leave the column stages.

$$E\_j = \frac{\left(Z\_j + \mathbf{1}\right) \sum\_{i=1}^{nc} n\_{i,j}^I}{\left(R\_j + \mathbf{1}\right) \sum\_{i=1}^{nc} n\_{i,j}^{II}} \tag{11}$$

When written in the residual form, as in Eq. (12), the equation has the following form:

$$f\_{\cdot j}^{vl} = \left(Z\_{\cdot j} + \mathbf{1}\right) \sum\_{i=1}^{nc} n\_{i\cdot j}^{I} - E\_{\cdot} \left(R\_{\cdot j} + \mathbf{1}\right) \sum\_{i=1}^{nc} n\_{i\cdot j}^{II} = \mathbf{0} \tag{12}$$

The values of the Ej parameter for each form of operation of the condenser and reboiler (partial or total) are shown in **Table 4**.

In the study developed by [2, 36, 37, 48], all cases of simulated reactive distillation column configurations used a partial reboiler and total condenser (E1 6¼ 0 and EN = 0).

Solving the set of equations that describe a reactive distillation column is an arduous task, and rigorous mathematical models aimed at a computer simulation of this type of equipment were not developed until the 1970s [54].


**Table 4.**

*Characteristics of column heaters (reboiler and condenser).*

In recent decades, commercial software that has specific models and algorithms for reactive distillation operations has been widely used, as shown previously in **Tables 1** and **2**. The simulations developed in the subsequent section, referring to case studies applied to biodiesel production and co-product valuation, use the RADFRAC module present in the commercial Aspen Plus software, which solves the equations of mass balance, energy balance, phase equilibrium and the sum of molar fractions (MESH) [55] through the "inside-out" algorithm [54].

#### **4. Fatty acid esterification simulation**

#### **4.1 Methodology**

#### *4.1.1 Kinetic parameters estimation*

The kinetic parameters for the esterification of fatty acids (FFA) present in corn distillers oil from DDGS (dried distillers grains with solubles) were estimated by a model fitting of the FFA conversion data (**Table 5**) obtained by our group. The reaction (Eq. (13)) was carried out at the temperatures of 150, 175 and 200°C, with ethanol and NbOPO4 (catalyst), following a molar alcohol:FFA ratio of 10:1 and catalyst load of 10% (FFA mass).

The methodology applied aims to estimate the pre-exponential factor (k0) and the activation energy (Ea) of the reaction. To fit the kinetic parameters, the objective function to be minimized is the squared difference between the experimental values of the FFA conversion and those calculated with a reversible pseudohomogeneous model (Eq. (14)). The reaction rate (rF) equation was applied to Eq. (15), which describes a batch reactor in terms of the FFA conversion (x) in a given time (t).

A Nelder–Mead [56] simplex algorithm and a 4th order Runge–Kutta [57] method were used to perform the objective function minimization and conversion equation integration steps, respectively. The reaction rate (rF) was obtained through the model and the specific reaction rate constants k (L/mol.s) were expressed by the Arrhenius equation (Eq. (16)). Through this methodology, the parameters Ea and k0 for the reaction rate constants of the forward and reverse esterification reactions were estimated.

$$\mathbf{F} + \mathbf{A} \leftrightarrow \mathbf{E} + \mathbf{W} \tag{13}$$

$$-r\_F = k\_1 C\_F C\_A - k\_2 C\_E C\_W \tag{14}$$

$$\frac{d\mathbf{x}}{dt} = \frac{r\_F}{C\_{F(t=0)}}\tag{15}$$

$$k = k\_0 e^{-\frac{E}{RT}} \tag{16}$$


*Reactive Distillation Applied to Biodiesel Production by Esterification: Simulation… DOI: http://dx.doi.org/10.5772/intechopen.102667*

#### **Table 5.**

*FFA conversion of the esterification reaction kinetic tests.*

In these equations,


#### *4.1.2 Process simulation*

The compounds defined for the simulation of the fatty acid esterification with ethanol were specified in the Aspen Plus V.12 process simulator, with the fatty acid and oil fraction represented by oleic acid and triolein, respectively. A similar approach was used by other researchers as a simplification of the numerous components of the acid and oil fraction of the feedstock [58–60]. The NRTL thermodynamic model [61] was selected to evaluate the activity coefficients of the components of the reaction mixture and the NRTL binary interaction parameters missing from the simulator database were estimated directly through the Aspen Plus estimation tool that uses the UNIFAC model [62].

The flowsheet developed for the process simulation is shown in **Figure 4** and consists of two columns, the first being responsible for the reactive distillation of the reactants fed to the process (C-EST) and the second for removing approximately 95% of the ethyl esters (FAEE) produced (C-DIST).

The simulated reactive distillation column has 22 total stages, of which 14 compose the reactive zone (5 to 18), while the C-DIST column consists of 10 total stages. The columns operating parameters are presented in **Tables 6** and **7**. Both the distillation columns have kettle-type reboilers, however the C-EST column is equipped with a total condenser, while the C-DIST with a partial condenser to separate the ethyl esters from the remaining oil and excess ethanol. It is noteworthy that the liquid phase composition and temperature profile graphs, as well as the conversions obtained follow the data referring to the process after the optimization described later.

#### **Figure 4.**

*Flowsheet of the FFA esterification process (F = feed, P = product, S = intermediate stream, H = heat exchanger, B = pump, V = valve, C = column).*


#### **Table 6.**

*Reactive distillation column operating parameters (C-EST).*


**Table 7.**

*Distillation column operating parameters (C-DIST).*

*Reactive Distillation Applied to Biodiesel Production by Esterification: Simulation… DOI: http://dx.doi.org/10.5772/intechopen.102667*


#### **Table 8.**

*Properties of the oil feed and ethanol streams (F-OIL and F-ETOH).*

The H-1 and H-2 heat exchangers are responsible for heating the oil (F-OIL) and ethanol (F-ETOH) streams up to 200°C and 50°C, respectively, shown in **Table 8**, while the pumps B-1 and B-2 increase the pressure of the feed streams from 1 bar to 10 bar.

#### *4.1.3 Optimization of the reactive distillation column*

The optimization of the process parameters of the esterification reactive distillation column was performed using the MATLAB® R2020b software by implementing the MEIGO package (Metaheuristics for Bioinformatics Global Optimization) [63], an optimization supplement for global optimal search which can be used to optimize industrial processes [64, 65]. The results obtained through Aspen Plus simulations were provided to MATLAB, where the optimization algorithm was performed, and new obtained values of the variables evaluated were used to carry out new simulations iteratively.

The Particle Swarm Optimization (PSO) method was applied to minimize the objective function that describes the conversion of fatty acids (Eq. (17)), starting from an initial population of 50 particles (solution vectors) defined by the algorithm in the pre-defined search intervals.

For the simulation, the varied parameters were molar reflux ratio, internal pressure, molar ratio between distillate stream and total feed, and oil and ethanol feed stages. As restrictions, the reboiler temperature, the recovery of the desired product (ethyl esters) at the bottom of the column and the feed stages of the reagents were evaluated with Eqs. (18)–(20). The reboiler temperature upper limit was defined as 200°C to avoid degradation of the reagents or products and excessive use of the hot utility. It is observed that the minimization of the negative value of the conversion corresponds to the maximization of its positive value.

$$\min\left(\text{conversion}(\%)\right) = -\left(\mathbf{1} - \frac{\text{Prod}\_{\text{Bot}}}{\text{FFA}\_{\text{in}}}\right) \times \mathbf{100} \tag{17}$$

$$T\_{\text{Re}\,b} \le 200^{\circ}C \tag{18}$$

$$\text{Recp}\_{\text{Prod}} = \frac{\text{Prod}\_{\text{Bot}}}{\text{Prod}\_{\text{Top}} + \text{Prod}\_{\text{Bot}}} \ge 0.99 \tag{19}$$

$$F\mathbb{S}\_{\partial\mathring{\mathsf{U}}} \le F\mathbb{S}\_{\acute{E}\boldsymbol{\varepsilon}\partial\!\!/\!/\!/\!/}\tag{20}$$

In these equations, *FFAin* = Molar flow of fatty acids in the feed stream (kmol/h). *TReb* = Reboiler temperature (°C). *RecProd* = Desired product fraction recovered at the column bottom. *ProdBot* = Molar flow of the desired product at the column bottom (kmol/h). *ProdTop* = Molar flow of the desired product at the column top (kmol/h). *FSOil* = Acid oil (corn distillers oil) feed stage. *FSEtOH* = Ethanol feed stage.

#### **4.2 Results**

#### *4.2.1 Kinetic parameters fitting*

The kinetic constants obtained through the discussed methodology are presented in **Table 9**, with the direct reaction of ethyl esters formation indicated by the subscript "1" and the reverse reaction of fatty acids formation indicated by the subscript "2". **Figure 5** shows the comparison between the experimental and calculated conversions, along with the R2 coefficient of the fit for each temperature.

Observing the results presented, it is noted that the data fitting at 200°C presented a high coefficient of determination, while the data fitting at 175°C obtained a reduced R<sup>2</sup> . However, as the temperature in the reactive section of the esterification column is, on average, close to 195°C, it was concluded that due to the excellent results achieved in the data fitting at 200°C, the use of the estimated


#### **Table 9.**

*Estimated kinetic constants for the esterification reaction.*

**Figure 5.** *Experimental () and calculated FFA conversion at 150°C (Δ), 175°C (*◇*) and 200°C (*◯*).*

*Reactive Distillation Applied to Biodiesel Production by Esterification: Simulation… DOI: http://dx.doi.org/10.5772/intechopen.102667*

kinetic parameters would not hinder the development of a simulation faithful to the real behavior of the reaction.

#### *4.2.2 Esterification of fatty acids*

The composition and temperature profiles along the stages of the reactive distillation column (column C-EST in **Figure 4**) are presented in **Figures 6** and **7**.

The liquid phase composition profile of the C-EST column (**Figure 6**) indicates that the major component for all stages with values higher than 5 (closer to the bottom) is triolein, while in the others there is a predominance of oleic acid, ethyl oleate (FAEE), ethanol and water, since only negligible amounts of triolein are evaporated along the column, as seen in the vapor phase mass composition profile. Additionally, there is a significant increase in the fraction of ethanol and water in the liquid state in the first stage due to the use of a total condenser in the reactive distillation column.

The composition of the streams that characterize the main products of the process (S-FAEE, P-OIL and P-FAEE) are presented in **Table 10** and, based on the simulation results, there is a final fatty acids conversion (mol) of 83.97% inside the

**Figure 6.** *Liquid phase mass composition profile (C-EST).*

**Figure 7.** *Column temperature profile (C-EST).*

#### *Distillation Processes - From Solar and Membrane Distillation to Reactive Distillation…*


#### **Table 10.**

*Composition of the main product streams.*


#### **Table 11.**

*Energy demand of the process equipment.*

column, 94.00% of which is recovered in the P-FAEE stream, while 5.83% is recovered in the P-OIL stream. The remaining 0.17% of FAEE is located at the P-ETOH2 stream. The resulting stream of the desired product (P-FAEE) has a purity (FAEE) greater than 98%, resulting in an ester content superior to the value described in Brazilian and European specifications [66, 67].

In **Table 10**, it is possible to observe that there are still traces of ethyl esters present in the oil stream. However, this amount corresponds to less than 1% of the total mass fraction of the stream. P-OIL, therefore, was considered to be nonsignificant. Furthermore, of the 900 kg/h of FFA fed to the process, only 132.41 kg/ h remain, characterizing a reduction of 85.29% of the total fatty acid mass. Finally, the energy demands for H-1, H-2, condensers and reboilers of columns C-EST and C-DIST are presented in **Table 11**.

#### *4.2.3 Optimization of the reactive distillation column*

**Table 12** shows the limits and initial estimates for the variables evaluated for the optimization of the esterification process. **Table 13** displays the constraints imposed on the reboiler temperature, ethyl ester recovery fraction (FAEE), and conversion. The values chosen as initial estimates were obtained by manually setting different values for the reflux molar ratio, condenser pressure, and distillate feed molar ratio, and adopting the best result obtained.

The results obtained are shown in **Figure 8**, with a maximum conversion of 83.97% and the final values of the variables are added to **Table 14**.


*Reactive Distillation Applied to Biodiesel Production by Esterification: Simulation… DOI: http://dx.doi.org/10.5772/intechopen.102667*

**Table 12.**

*Lower, upper limits and initial estimates for the variables evaluated in the esterification reaction optimization process.*


**Table 13.**

*Initial constraints for the response variables for the variables evaluated in the esterification reaction optimization process.*

An additional simulation performed in a CSTR reactor achieved an FFA conversion of 51.06%, while the maximum average conversion in the kinetic tests (200°C) was 49.55%. The simulated CSTR operated at a constant temperature of 200°C with the residence time of 3 h (same duration of the experimental tests) and was fed with streams following equal mass flows and compositions to the RDC column feed streams. Thus, the optimization results represent a significant improvement of 64.45% and 69.46% compared to the CSTR and experimental tests, respectively, inferring that the use of a reactive distillation column could be beneficial to the process.

**Figure 8.**

*Evolution of the FFA conversion as a function of the number of optimization iterations.*


#### **Table 14.**

*Response vector of input variables for the esterification reaction optimization process.*

### **5. Solketal production simulation**

#### **5.1 Justification**

As biodiesel production increases so do the production of glycerol as for each liter of biodiesel produced, approximately 100 mL of crude glycerol are obtained [68]. Among the transformation processes for glycerol to viable chemical intermediates, glycerol ketalization for the production of solketal has gained prominence. Solketal can be used as an additive to increase the octane and fluid dynamic properties of the fuel. The addition of up to 5% by volume of solketal to gasoline leads to a significant decrease in gum formation [69]. With this motivation, this study aims to simulate the operation of a reactive distillation column for the production of solketal from glycerol with acetone using heterogeneous catalysis, with high conversion of reagents and separation of the components of the reaction.

#### **5.2 Methodology**

The applied methodology considers the ketalization reaction of glycerol (G) with acetone (A), forming solketal (S) and water (W). The reaction is considered reversible and elementary, being described by Eq. (21):

$$\mathbf{G} + \mathbf{A} \overset{k\_1}{\underset{k\_{-1}}{\rightleftharpoons}} \mathbf{S} + \mathbf{W} \tag{21}$$

A pseudo-homogeneous model was used to describe the reaction kinetics through a system of differential equations of concentration over time, at different temperatures, in which the kinetic constants of the direct and inverse reaction are represented, respectively, by k1 and k�1(L/mol.s), while the molar concentrations (mol/L) of the species involved are given by CG, CA, CS and CW (Eq. (22)).

$$-r\_G = k\_1 C\_G C\_A - k\_{-1} C\_S C\_W \tag{22}$$

The solution of the system of differential equations using a 4th order Runge Kutta method [57] and the fitting of the kinetic parameters, k1 and k�1, and subsequent estimation of the Arrhenius equation parameters were performed by a Nelder–Mead simplex algorithm [56]. The experimental data used was retrieved from the study of [70].

The kinetic parameters evaluated were later used to predict the solketal formation reaction in a reactive distillation column, using the rigorous RADFRAC distillation model of the Aspen Plus commercial simulation software. The system considered in this study is shown in **Figure 9**.

Using the estimated kinetic parameters, the glycerol ketalization reaction for the production of solketal was modeled in the Aspen Plus software. For the process simulation, the pressure inside the column was set at 10 atm. The feeding of the 13-stage column, RDC in **Figure 9**, are streams GLI-02 and ACE-02, originated from the heating of the currents GLI-01 and ACE-01 up to 95°C and 55°C, by the heat exchangers H1 and H2, respectively.

The ACE-01 stream is composed only of acetone, while GLI-01 contains 80% glycerol and 20% water by mass, disregarding other components such as methanol or dissolved salts normally present in glycerol from biodiesel production processes [71]. The products of reactive distillation are characterized by TOP-P and BOT-P streams, which correspond, respectively, to the streams rich in the most volatile and least volatile substances in the process.

*Reactive Distillation Applied to Biodiesel Production by Esterification: Simulation… DOI: http://dx.doi.org/10.5772/intechopen.102667*

**Figure 9.** *Flowsheet of the solketal production process used in this study.*
