# Multi-Objective Optimisation of Biodiesel Synthesis in Simulated Moving Bed Reactor

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

(Acid) (Alcohol) (Biodiesel) (Water)

## 2. Multi-Objective Optimisation

## 3. Optimisation Methodology

## 4. Mathematical Model for Synthesis of Biodiesel in SMBR

_{p}, the flow rate in section P, is regarded as the reference flow rate, to which all other flow rates are described. If α, β and γ are assumed to be the ratio of feed flow rate (F), raffinate flow rate (R) and desorbent flow rate (D) respectively, to the reference flow rate, Q

_{p}, then flow rates in each section can be defined as shown in Figure 1. Simulation of counter-current movement of the solid is achieved by advancing the inlet and withdrawal ports, column by column, in the same direction of the fluid flow, at a predetermined switching time, t

_{s}. During a switch, these ports move simultaneously by one column, in the direction of the flow of mobile phase. To achieve separation between the components, the internal flow rates of the fluid phases within the four sections, and the switching time (which defines the hypothetical solid phase velocity) must be specified appropriately. By suitable selection of switching time, counter-current or co-current movement of the solid phase with respect to the fluid phase can be achieved in each section. Petroulas and co-workers [35] defined for true counter-current moving bed chromatographic reactor (CMCR) a parameter, σ

_{i}, called relative carrying capacity of the solid relative to the fluid stream for any component i as:

_{i}< 1, V

_{i}> 0 (species move with the fluid phase), and when σ

_{i}> 1, V

_{i}< 0 (species move with the solid phase). When σ = 0, it represents fixed bed. Ray and co-workers [37] re-defined the above parameter, σ, for SMBR by replacing the solid-phase velocity, us, in CMCR by a hypothetical solid phase velocity, ζ, defined as ζ = L/t

_{s}for SMBR. They found, both theoretically [1] and experimentally [2], that simulation of the counter-current movement between two components can be achieved when re-defined σ’s were set such that it is greater than 1 for one and less than 1 for the other component. Hence, in the present study if we set σ appropriately, the more strongly adsorbed component (H2O) will move with the solid (resin) stream and could be collected at the extract port, while at the same time the less strongly adsorbed component (biodiesel) will travel with the fluid stream and could be collected at the raffinate port. It should also be noted that the parameter σ defined by the research group of Carr and Aris [35] is similar to β defined by the research group of Hashimoto [38], γ defined by the research group of Ruthven [39] and m defined by the research group of Morbidelli [40].

_{i}is the concentration of component i in the mobile phase (mol/L), t is the time (s), q

_{i}is the concentration of component i in the polymer phase (mol/L), ε is the column void fraction (-), u is the superficial fluid phase flow rate (m/s), z is the axial coordinate (m), υ

_{i}is the stoichiometric coefficient of the component i, R is the reaction rate and D

_{i}is the apparent dispersion coefficient of the component i (m

^{2}/s). For the component i in the jth column during the Nth switching period, u

_{Ø}denotes superficial flow rate in section ϕ (where ϕ = P, Q, R, S) and the reaction rate expressions and adsorption isotherms are given by

Column 1 Column N

_{col}

Column j Column j − 1, j = 1, 2, 3, ….., N

_{col}

- (1)
- Yield of methyl ester (Y
_{ME})—$${Y}_{ME}=\frac{\mathrm{methyl}\mathrm{oleate}\mathrm{collected}\mathrm{in}\mathrm{raffinate}}{\mathrm{oleic}\mathrm{acid}\mathrm{fed}}=\frac{\beta .\left[{{\displaystyle \int}}_{0}^{{t}_{s}}{{C}_{ME,p}^{\left(N\right)}|}_{z={L}_{col}}dt\right]}{\alpha .{C}_{FA,f.{t}_{s}}}$$ - (2)
- Purity of methyl ester (P
_{ME})—$$\begin{array}{ll}{P}_{ME}& =\frac{\mathrm{methyl}\mathrm{oleate}\mathrm{collected}\mathrm{in}\mathrm{raffinate}}{\mathrm{oleic}\mathrm{acid}+\mathrm{water}+\mathrm{methyl}\mathrm{oleate}\mathrm{collected}\mathrm{in}\mathrm{raffinate}}\\ & =\frac{{{\displaystyle \int}}_{0}^{{t}_{s}}{{C}_{ME,p}^{\left(N\right)}|}_{z={L}_{col}}dt}{{{\displaystyle \int}}_{0}^{{t}_{s}}{({C}_{ME,p}^{\left(N\right)}+{C}_{W,p}^{\left(N\right)}+{C}_{FA,p}^{\left(N\right)})|}_{z={L}_{col}}dt}\end{array}$$

## 5. Optimisation of Biodiesel Production in SMBR

_{s}); throughput parameters (feed flow rate, α and/or raffinate flow rate, β); and operating cost parameters (eluent flow rate, γ and flow rate in section P, QP, which is related to the pressure drop in the system). For the design-stage optimisation, an additional parameter, length of the column L

_{col}, was used. Table 1 represents the optimisation problems studied in this work.

## 6. Optimisation of Existing Setup

#### Case 1.2: Maximisation of Purity and Minimisation of Desorbent Consumption

## 7. Design Stage Optimisation

_{col}). Two optimisation problems were once again considered for design-stage optimisation:

#### 7.1. Case 2.1 Simultaneous Maximisation of Yield and Purity

_{col}≤ 0.5 (m)]. The Pareto optimal solution is shown in Figure 5. Once again, it is observed that they act in conflicting manner. But a much higher value of purity (97%) can be obtained as compared to case 1.1 where the highest purity value obtained was 87%. Moreover, the highest yield value obtained in case 1.1 was 79% against a purity value of 76%. The yield in this case is 90% corresponding to value of purity being marginally more than 90%. Hence a drastic improvement is achieved when column length is introduced as a decision variable. The purity also acts in a conflicting manner against raffinate flow rate, as is evident from Figure 6. A very low value of β (≈ 0.1) is required to achieve 97% purity, indicating the requirement of a higher residence time in section P. Figure 6 represents that a high value of desorbent flow rate (γ ≈ 3.5) is required to achieve a purity in the range of 94% to 97%. Just as in case 1.1, γ must be kept above a threshold value; further increase in γ will not improve purity. An increase in column length also improves purity, as represented by Figure 6. Larger column length means that the reactants will have more residence time, hence improving the conversion, purity and yield. As far as switch time is concerned, it has increased to about 11 min (see Figure 6) as compared to 5 min in Case 1.1. This is due to the introduction of column length as a decision variable. A higher L

_{col}value means indicates requirement of a higher residence time before a switch is made.

#### 7.2. Case 2.2 Maximisation of Purity and Minimisation of Desorbent Consumption

_{col}≤ 0.5 (m)]. The Pareto optimal solutions are shown in Figure 7. Figure 7 shows the relation between γ and purity. At lower values of γ, a linear relation exists with purity. However, after that, the graph becomes exponential; indicating that a slight increase in purity would require a very high desorbent consumption, just as in case 1.2. Hence, γ should be just high enough above a threshold value (≈2 in this case). Further increase is not necessary.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic flow diagram of the SMBR. The inlets and outlets divide the entire system into four section, P, Q, R and S with respectively p, q, r and s number of columns. The flow retes in each section is given by Qq = (1 − β)Qq, Qr = (1 − β + γ)Qp, Qs = (1 − α)Qp, where α, β and γ are given by F/Qp, Ra/Qp, E/Qp.

**Figure 2.**Existing set-up (Case 1.1 Maximization of Yield and Purity): Pareto optimal solutions and corresponding operating variables.

**Figure 3.**Steady state concentration profiles of methyl oleate-water-olenic acid system; (

**a**) corresponding to point 1 & (

**b**) corresponding to point 2 of Figure 2.

**Figure 4.**Case 1.2 Existing set-up. Maximization of Purity and minimization of Desorbent flow rate: Pareto optimal solutions and corresponding operating variables.

**Figure 6.**Case 2.1 Design-stage Optimization. Maximization of Yield and Purity. Corresponding operating variables for Pareto solutions shown in Figure 5.

**Figure 7.**Case 2.2 Design-stage Optimization. Maximization of Purity and minimization of desorbent consumption. Pareto optimal solutions and corresponding operating variables.

**Table 1.**Optimisation problems along with their objective functions, constraints, decision variables and fixed parameters.

Case | Objective Functions | Constraints | Decision Variables | Fixed Parameters |
---|---|---|---|---|

1.1 Existing setup | Maximum Y_{ME}Maximum P _{ME} | Y_{ME} ≥ 50%P _{ME} ≥ 50% | 1 ≤ t_{s} ≤ 17 (min)0.1 ≤ β ≤ 1 1 ≤ γ ≤ 5 | Q_{P} = 1.66 ml/min α = 0.1 [F] = 0.21 mol/lit L _{col} = 25 cm, N_{col} = 4 |

1.2 Existing setup | Maximum P_{ME}Minimum γ | Y_{ME} ≥ 50%P _{ME} ≥ 50% | Same as Case 1.1 | Same as Case 1.1 |

2.1 Design stage | Maximum Y_{ME}Maximum P _{ME} | Y_{ME} ≥ 50%P _{ME} ≥ 50% | Same as Case 1.1 0.2 ≤ L _{col} ≤ 0.5 (m) | Same as Case 1.1 except L_{col} is not fixed |

2.2 Design stage | Maximum P_{ME}Minimum γ | Y_{ME} ≥ 50%P _{ME} ≥ 50% | Same as Case 2.1 | Same as Case 2.1 |

Number of generations, N_{gen} | 50 |

Population size, P_{pop} | 50 |

Probability of crossover, P_{cross} | 0.65 |

Probability of mutation, P_{mute} | 0.002 |

Spreading parameter, σ | 0.075 |

Sharing function exponent, α | 2.0 |

Random number generator seed, S_{r} | 0.455 |

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Ray, N.M.; Ray, A.K.
Multi-Objective Optimisation of Biodiesel Synthesis in Simulated Moving Bed Reactor. *Separations* **2021**, *8*, 127.
https://doi.org/10.3390/separations8080127

**AMA Style**

Ray NM, Ray AK.
Multi-Objective Optimisation of Biodiesel Synthesis in Simulated Moving Bed Reactor. *Separations*. 2021; 8(8):127.
https://doi.org/10.3390/separations8080127

**Chicago/Turabian Style**

Ray, Nillohit Mitra, and Ajay K. Ray.
2021. "Multi-Objective Optimisation of Biodiesel Synthesis in Simulated Moving Bed Reactor" *Separations* 8, no. 8: 127.
https://doi.org/10.3390/separations8080127