# Optimization of Reaction Selectivity Using CFD-Based Compartmental Modeling and Surrogate-Based Optimization

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Integrating CFD-Based Compartmental Model with Surrogate Based Optimization

#### 2.1. CFD-Based Compartmental Model

#### 2.1.1. A Brief Review of Compartmental Model

#### 2.1.2. Compartmental Model Development

_{i}is the concentration of species i,

**N**

_{i}represents the mass flux of species i, and R

_{i}denotes the source of species i. Compartmental models are obtained by volume averaging Equation (1) over each predefined compartment V, as described in Equation (2).

_{i}with the volume-averaged concentration $\overline{{C}_{\mathit{i},{\mathit{K}}_{j}}}$, Equation (2) is modified to Equation (3), where V

_{j}is the volume of control volume K

_{j}, and S

_{j}is the surface of control volume K

_{j}. The mass flux

**N**consist of convection and diffusion, where the diffusion is modelled by Fick’s law with diffusion coefficient D. The source term R

_{i}models the homogeneous consumption and generation of species i, which include chemical reactions and micro mixing. Due to the assumption of homogeneous compartments, the volume integral of source term in Equation (5) is modified as follows:

_{jk}denotes the flow rate from control volume j to control volume k.

#### 2.1.3. Grid Independence

^{−3}s [11], micro-mixing would dominate chemical reaction. As a result, the rate law of chemical reaction should be replaced with micro-mixing models. Although for different reaction kinetics we can use the same steady state solution from CFD, if new reaction kinetics are used, grid independence test should be conducted with the updated model.

#### 2.2. Surrogate-Based Optimization

#### 2.2.1. A Brief Review of Surrogate-Based Optimization

#### 2.2.2. Problem Formulation

_{s}discrete feeding stages. Feeding rate is kept constant in each stage, but different feeding rate are employed for different stages. Each feeding stage m is specified by its duration t

_{m}and the adopted feeding rate f

_{m}, which are not defined a priori, instead they are determined by solving an optimization problem.

_{R}denotes the price of desired product R while y

_{R}denotes its yield. P

_{A}represents the unit cost of raw material A and its consumption is denoted as y

_{A}. Both y

_{R}and y

_{A}are calculated through the simulation φ based on the compartmental model. Addition point n and addition rate profile which is defined by t

_{1}, f

_{1}, t

_{2}, f

_{2}…t

_{Ns}, f

_{Ns}are parameters of this simulation. The first set of constraints describe the simulation based on the compartmental model. The second constraint represents that the total time span of all stages is pre-defined as T.

_{p}, unless computational expense of solving this optimization problem is taken into consideration. However, this is beyond the scope of this paper. The duration of process T is defined as a parameter, which is usually determined in the production scheduling stage. It is recommended to define T similar to the timescale of mixing in mixing controlled processes to maximize time efficiency of reactors.

## 3. Case Study

#### 3.1. Reactor Setup

#### 3.2. Chemical Kinetics

_{1}= 10

^{−3}s

^{−1}and k

_{2}= 7000 m

^{3}kmol

^{−1}s

^{−1}at a PH = 6.6 [38]. Both reactants are dissolved in aqueous solution.

^{−3}M. diazonium solution is added into the stirred tank in a semi-batch manner, the concentration of which is 7.4 × 10

^{−1}M.

#### 3.3. Flow Field Simulation

^{2}order smaller than that of the bulk flow inside the reactor, influence of reagent injection over the flow pattern is ignored.

#### 3.4. Compartmental Modeling and Grid Independence Test

^{−2}m/s. Based on Equation (7), the upper bound of length scale in each compartment should be 1 m. Furthermore, to justify neglecting diffusive mass transfer, Péclet number (Pe) is analyzed according to Equation (8). Since diffusion coefficient in aqueous solutions are in the order of 10

^{−9}m

^{2}/s, the lower bound of compartment length scale is 10

^{−7}m. It can be concluded that since in single phase turbulent flow convective mass transfer rate is usually several orders of magnitude higher than that of diffusion, compartmental model can be safely adopted in most single phase stirred tank reactors.

#### 3.5. Optimization and Results

^{4}$/mol. The prices have profound influence on the optimal operating policy. Feeding points are defined with 3 integer variables, representing the corresponding radial, axial and angular index.

#### 3.5.1. Optimal Location of Feeding

#### 3.5.2. Optimal Rate of Feeding

#### 3.5.3. Traditional Process Design with Perfect-Mixing Assumptions

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CPP | Critical process parameters |

CFD | Computational fluid dynamics |

QbD | Quality by design |

RTD | Residence time distribution |

GA | Genetic algorithm |

RBF | Radial basis function |

CMC | Chemistry, manufacturing, and controls |

ANN | Artificial neural network |

RANS | Reynolds averaged Navier-Stocks |

MRF | Multi-reference frame |

API | Active pharmaceutical ingredient |

SBO | Surrogate-based opitmization |

List of Symbols: | |

c | Concentration |

N | Mass flux |

D | Diffusivity |

R | Source |

V | Volume |

S | Surface area |

Q | Mass flow rate |

Da | Damköhler number |

Pe | Péclet number |

k | Reaction rate constant |

L | Characteristic length |

u | Characteristic velocity |

P | Price |

y | Yield |

N_{s} | Number of feeding stages |

N_{c} | Number of compartments |

t | Duration |

f | Feed rate |

Subscripts: | |

i | ith species |

j | jth compartment |

m | mth stage of operation |

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**Figure 1.**Geometrical dimension of the two-impeller stirred tank. Where (

**A**) stands for the baffles, (

**B**) represents the pitched blade impeller and (

**C**) denotes the Rushton impeller.

**Figure 3.**Convergence of predicted (

**a**) Diazonium and (

**b**) Pyrazolone distribution with number of compartments.

**Figure 4.**Simulated yield of the desired product when addition location is at the bottom corner and near Rushton impeller.

**Figure 5.**Optimal feeding rate profile solved for constant rate, two-stage and three-stage reagent injection policies.

**Figure 6.**Simulated trajectories of (

**a**) Diazo (

**b**) Pyrazolone and (

**c**) Desired product when optimal feeding policies solved with different number of stages are employed.

**Figure 7.**Optimal feeding rate profile solved with CFD-based compartmental model and perfect mixing model.

**Figure 8.**Simulated trajectories of (

**a**) Diazonium (

**b**) Pyrazolone (

**c**) Desired product when different methodologies are employed.

**Table 1.**Comparison of optimal feeding location for (a) constant rate feeding (b) two-stage dynamic feeding policy and (c) three-stage feeding policy.

Reagent Injection Policy | Optimal Injection Location | |
---|---|---|

Height (m) | Radial Position (m) | |

Constant Rate Feeding | 0.1–0.13 | 0.22–0.25 |

Two-stage Dynamic Feeding | 0.1–0.13 | 0.22–0.25 |

Three-stage Dynamic Feeding | 0.1–0.13 | 0.22–0.25 |

**Table 2.**Comparison of optimal process productivity when (a) constant rate feeding (b) two-stage dynamic feeding policy and (c) three-stage feeding policy are adopted.

Reagent Injection Policy | Optimal Process Productivity ($) |
---|---|

Constant Rate Feeding | 6162.90 |

Two-stage Dynamic Feeding | 6410.43 |

Three-stage Dynamic Feeding | 6411.76 |

**Table 3.**Comparison between optimal operating conditions solved with perfect mixing model and the proposed methodology.

Methodology | Simulated Process Productivity ($) |
---|---|

Perfect-mixing Model | 6162.90 |

CFD-based Compartmental Model | 6410.43 |

**Table 4.**Comparing computational expense between the proposed methodology and direct CFD simulation.

Methodology | Computational Expense for One Simulation (s) |
---|---|

Dynamic CFD simulation | 10^{4} |

CFD-based Compartmental Model | 70 |

**Table 5.**Comparison between optimal operating conditions solved with perfect mixing model and the proposed methodology.

Methodology | Simulated Process Productivity ($) |
---|---|

Perfect-mixing Model | 5713.18 |

Compartmental Model (constant rate) | 6126.90 |

Compartmental Model (dynamic rate) | 6411.76 |

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**MDPI and ACS Style**

Yang, S.; Kiang, S.; Farzan, P.; Ierapetritou, M.
Optimization of Reaction Selectivity Using CFD-Based Compartmental Modeling and Surrogate-Based Optimization. *Processes* **2019**, *7*, 9.
https://doi.org/10.3390/pr7010009

**AMA Style**

Yang S, Kiang S, Farzan P, Ierapetritou M.
Optimization of Reaction Selectivity Using CFD-Based Compartmental Modeling and Surrogate-Based Optimization. *Processes*. 2019; 7(1):9.
https://doi.org/10.3390/pr7010009

**Chicago/Turabian Style**

Yang, Shu, San Kiang, Parham Farzan, and Marianthi Ierapetritou.
2019. "Optimization of Reaction Selectivity Using CFD-Based Compartmental Modeling and Surrogate-Based Optimization" *Processes* 7, no. 1: 9.
https://doi.org/10.3390/pr7010009