# Methodology to Solve the Multi-Objective Optimization of Acrylic Acid Production Using Neural Networks as Meta-Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Acrylic Acid Production

#### 2.1. Reactor Model

#### 2.2. Propylene Oxidation

#### 2.3. Acrolein Oxidation

#### 2.4. Flammability Limits

## 3. Methodology for Solving a Multi-Objective Optimization Problem

#### 3.1. Definition of the Optimization Problem

_{2}] in the first reactor, denoted as Sum

_{E}, was implemented as a soft constraint expressed as objective function OF

_{9}. This objective function represents the minimization of the integration of the excess oxygen concentration above the lower flammability limit. The integration for handling the oxygen concentration as a soft constraint allows for an insight into the set of decision variables that could violate this constraint, and provides more flexibility in the optimization process.

#### 3.2. Design of Experiments

_{n}(q

^{s}) [30]. In this design, a complete table of the normalized design points can be obtained given the dimensionality of factor space s (decision variables), the number of levels q of the factors, and the desired number of data points n. The normalized information obtained from the selected UD table is then used along with the minimum and maximum values of the decision variables to determine the actual values of decision variables from which the objective functions are calculated. The following uniform designs were used to generate the initial set of process data, divided as learning and validation data sets, to build the ANN: U

_{50}(5

^{8}) and U

_{20}(5

^{8}), i.e., 50 and 20 design points, which translates to using a ratio of 70/30 for the learning and validation data points.

#### 3.3. Artificial Neural Networks as Meta-Models

#### 3.3.1. ANN Architecture

_{P}), air (F

_{A}) and steam (F

_{S1}) to the first reactor, the flowrate of vapor water (F

_{S2}) to the second reactor, and the operating temperature and pressure of both reactors (T

_{1}, T

_{2}, P

_{1}, and P

_{2}).

#### 3.3.2. Building the ANN

^{2}value was plotted against the number of hidden neurons to determine the proper number of hidden neurons for every ANN. The program used to build the networks was coded in FORTRAN.

#### 3.3.3. Modified Garson Algorithm

#### 3.4. Optimization Algorithm

#### 3.5. Ranking of the Pareto Domain

_{k}), the indifference threshold (Q

_{k}), the preference threshold (P

_{k}), and the veto threshold (V

_{k}) [41,42]. Using these quantitative parameters, the NFM performs a pairwise comparison of all Pareto-optimal solutions and attributes a score to each one, which then allows all of the solutions to be ranked. An interesting feature of this ranking method is its robustness, which means that changes in the weights will not incur in major changes of the optimal zones [41].

## 4. Results

#### 4.1. Construction of the Meta-Model

_{50}(5

^{8}) and U

_{20}(5

^{8}) for the training and validation data sets, respectively. The coefficients of determination (R

^{2}) for each of the nine objective functions (Table 1) are plotted as a function of the number of hidden neurons (Figure 5).

_{1}and OF

_{5}) and the heat recovery of the first reactor (OF

_{2}), are relatively well predicted. However, the ANNs of the other six objectives show poorer predictions with R

^{2}values below 0.90. Furthermore, it is not possible to observe a clear trend for those OFs when one would expect the R

^{2}value to increase as the number of neurons increases. These results suggest that, for these objectives, the number of design data points is insufficient to allow the ANN to capture the underlying relationships that exist between the decision variables and the objectives. The large number of input variables as inputs to the ANNs also points to the necessity to present the neural networks with richer information. Since the available tables of uniform design are limited to a relatively small number of design points, it was decided to use well-distributed random design points, which offer the possibility of using any desired number of design points.

_{1}) and the heat recovery of the second reactor (OF

_{6}) respectively are presented in Figure 6. The total number of design points (training and validation) in the data set varied between 70 and 1430. The predictions for OF

_{1}wwere very good for a relatively low number of hidden neurons and a small number of training data points. Indeed, the very high R

^{2}value indicates that the predictions of the neural network for the compression power of the first compressor were independent of the number of design data points above approximately 140. For the heat recovery of the second reactor (OF

_{6}), the coefficient of determination (R

^{2}) increased with the number of design points, whereas it was not a function of the number of hidden neurons above five neurons. This trend was more significant for some objectives due to their dependency on the input or decision variables. For example, a simple dependency prevailed for OF

_{1}as it was mainly correlated to the air flowrate and the operating pressure of the first reactor. In contrast, OF

_{6}is a much more complex dependency as it is affected by a larger number of inputs, namely the four input flowrates and the operating temperature and pressure of the first reactor, and thereby requires more data points to capture the underlying relationships between the inputs of the ANN to properly predict this output.

^{2}); and (3) the minimum number of neurons. The selected set of nine ANNs, one for each objective function, can now be used as the surrogate model to generate the Pareto domain and find the optimal operating set of decisions variables.

_{8}) and the compression power of the first compressor (OF

_{1}), respectively, as a function of the values calculated using the phenomenological model. The green points represent the learning data, the orange points correspond to the validation data, and the grey points to the testing data. The testing data were generated using the phenomenological model by randomly selecting the decision variables within their allowable ranges, as defined in Table 2. This data set was used as a “second validation” to confirm the good adjustment and precision of the ANNs when exposed to new input data. As previously mentioned, it had no impact on the meta-model training. The predictions of Figure 7a,b correspond to the ANNs with the lowest and highest R

^{2}values for all data presented: 0.912 and 0.999, respectively. Predictions for the other OFs were very good as well, having R

^{2}values between the two previous values. The predicted conversion in the second reactor (R-101) had the majority of the points near the 45o-line, but with the lowest R

^{2}value due to few scattered points with poor predictions. When examining the sensitivity parameters of Table 3, the two objective functions that are influenced by a larger number of input variables are the conversions of the first and second reactors (OF

_{4}and OF

_{8}). As mentioned before, the more an objective function is correlated to a larger number of decision variables, the more learning data points are required to obtain better predictions. As a compromise needs to be made between the R

^{2}value and the number of learning data, a value of R

^{2}above 0.9 was considered a good result.

#### 4.2. Multi-Objective Optimization

_{7}) and the heat recovery of the second reactor (OF

_{6}), while Figure 8c,d present the projection on the plane of the conversion of propylene (OF

_{4}) and heat recovery of the first reactor (OF

_{2}). Based on the NFM ranking, the Pareto domain was divided into four different regions: (i) the best solution in red; (ii) Pareto-optimal solutions ranked in the top 5%; (iii) solutions in the next 45%; and (iv) the remaining 50% of the solutions. The best ranked solution of Figure 8b corresponds to a productivity of 1.179 kmol/m

^{3}h and a heat recovery of 10,755 kW in the second reactor. When the values of the decision variables, associated with the best-ranked Pareto-optimal solution, were used within the first-principle based model for comparison purposes, the values for OF

_{7}and OF

_{6}were 1.202 kmol/m

^{3}h and 11,104 kW, respectively, yielding errors in the vicinity of 2%–3%. This was also the case for the other objective functions, as shown in Table 4. When the phenomenological model is used to circumscribe the Pareto domain and then Pareto-optimal solutions are ranked with NFM, as depicted by Figure 8a, values of 1.2524 kmol/m

^{3}h and 11 291 kW were obtained for OF

_{7}and OF

_{6}, respectively, for differences of approximately 7% and 5%. The corresponding conversion in R-100 of the best ranked solution was 94.97% and 97.27% for the meta-model and the phenomenological model, respectively, as shown in Figure 8c,d. In contrast, a conversion of 96.25% was predicted when the decision variables of the best-ranked solution identified with the ANNs were used in the first-principle based model.

_{3}and OF

_{7}which were assigned a weight of 0.15. The resulting optimal solution corresponded to a solution ranked in the top 5% when using the NFM method, more specifically the solution ranked 232th out of 5000. The values of the objective functions of this solution are presented in Table 5.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

AA | Acrylic acid | - |

Ac | Acrolein | - |

Ace | Acetaldehyde | - |

AceA | Acetic acid | - |

ANN | Artificial neural network | - |

C_{i} | Molar concentration of i | kmol/m^{3}cat |

CWR | Catalytic wall reactor | - |

DOE | Design of experiments | - |

F_{A} | Molar flowrate of air | kmol/h |

F_{P} | Molar flowrate of propylene | kmol/h |

F_{S} | Molar flowrate of steam/water vapor | kmol/h |

LFL | Lower flammability limits | vol% |

MOC | Minimum oxygen concentration | vol% |

MOO | Multi-objective optimization | - |

$\dot{\mathrm{n}}$ | olar flowrate | kmol/h |

NFM | Net flow method | - |

OF | Objective function | - |

n_{i} | Order of reaction j | - |

p_{i} | Partial pressure of i | bar^{nj} |

P | Pressure | bar |

PBR | Packed bbed reactor | - |

Prop | Propylene | - |

R | Gas constant | J/mol K |

R^{2} | Coefficient of determination | - |

T | Temperature | ^{o}C |

UD | Uniform design | - |

UFL | Upper flammability limits | vol% |

V | Volume | m^{3} |

W | Catalyst weight and connection weight of the ANN | kg |

## Greek Symbols

$\mathsf{\eta}$ | Efficiency of the compressor | % |

$\mathrm{\xi}$ | Extent of reaction | kmol/h |

## Subscripts

1 | First reactor |

2 | Second reactor |

Feed | Feed stream to first reactor |

i | Index of the input layer of the ANN |

j | Index of the hidden layer of the ANN |

k | Index of the output layer of the ANN |

## Appendix A. Set of Rate Equations for Each Reaction

**Table A1.**Experimental parameters for the rate law of acrolein formation [18].

Constants | Value | Units | |
---|---|---|---|

r_{1} | k_{1Red,o} | 0.0628 | $\left[\frac{\mathrm{k}\mathrm{m}\mathrm{o}\mathrm{l}}{\mathrm{k}\mathrm{g}\xb7\mathrm{s}\xb7\mathrm{b}\mathrm{a}\mathrm{r}}\right]$ |

k_{1Ox,o} | 16,000 | $\left[\frac{\mathrm{k}\mathrm{m}\mathrm{o}\mathrm{l}}{\mathrm{k}\mathrm{g}\xb7\mathrm{s}\xb7\mathrm{b}\mathrm{a}{\mathrm{r}}^{0.75}}\right]$ | |

α_{H2O} | 8.2 | $\left[\frac{1}{\mathrm{b}\mathrm{a}\mathrm{r}}\right]$ | |

r_{2} | k_{2,o} | 2.3200 | $\left[\frac{\mathrm{k}\mathrm{m}\mathrm{o}\mathrm{l}}{\mathrm{k}\mathrm{g}\xb7\mathrm{s}\xb7\mathrm{b}\mathrm{a}{\mathrm{r}}^{0.86+0.3}}\right]$ |

r_{3} | k_{3,o} | 0.0150 | $\left[\frac{\mathrm{k}\mathrm{m}\mathrm{o}\mathrm{l}}{\mathrm{k}\mathrm{g}\xb7\mathrm{s}\xb7\mathrm{b}\mathrm{a}\mathrm{r}}\right]$ |

r_{4} | k_{4,o} | 1.4700 | $\left[\frac{\mathrm{k}\mathrm{m}\mathrm{o}\mathrm{l}}{\mathrm{k}\mathrm{g}\xb7\mathrm{s}\xb7\mathrm{b}\mathrm{a}{\mathrm{r}}^{0.73}}\right]$ |

r_{5} | k_{5,o} | 0.0363 | $\left[\frac{\mathrm{k}\mathrm{m}\mathrm{o}\mathrm{l}}{\mathrm{k}\mathrm{g}\xb7\mathrm{s}\xb7\mathrm{b}\mathrm{a}\mathrm{r}}\right]$ |

K_{H2O} | 1.9 | $\left[\frac{1}{\mathrm{b}\mathrm{a}\mathrm{r}}\right]$ | |

r_{6} | k_{6,o} | 0.00034 | |

K_{H2O} | 1.9 | $\left[\frac{1}{\mathrm{b}\mathrm{a}\mathrm{r}}\right]$ | |

r_{7} | k_{7,o} | 1.3800 | |

K_{H2O,AA} | 55.1 | $\left[\frac{1}{\mathrm{b}\mathrm{a}\mathrm{r}}\right]$ | |

r_{8} | k_{8,o} | 0.00038 | |

r_{9} | k_{9,o} | 4.75 × 10^{9} |

**Table A2.**Experimental parameters for the rate law for acrylic acid formation [20].

Constants | Values | Units | |
---|---|---|---|

r_{10} | k_{10,o} | 19436 | $\left[\frac{1}{\mathrm{s}}\right]$ |

K_{o} | 9.78 × 10^{-6} | $\left[\frac{1}{\mathrm{s}}\right]$ | |

r_{11} | k_{11,o} | 49070 | $\left[\frac{1}{\mathrm{s}}\right]$ |

K_{o} | 9.78 × 10^{-6} | $\left[\frac{1}{\mathrm{s}}\right]$ |

_{2}formation:

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**Figure 2.**Reaction scheme for the propylene oxidation reactor (adapted from [18]).

**Figure 3.**Flowchart of the proposed methodology for solving multi-objective optimization using a three-layer artificial neural network (ANN) as meta-model.

**Figure 5.**R

^{2}values for all the objective functions using 50 learning and 20 validation uniform design (UD) design points vs. Number of neurons in the hidden layer.

**Figure 6.**R

^{2}values for different number of random data points used for training and validation vs. Number of neurons in the hidden layer for (

**a**) compression power in C-100 and (

**b**) heat recovery in R-101.

**Figure 8.**Ranked Pareto domain with net flow method (NFM) obtained with: (

**a**) and (

**c**) the phenomenological model and (

**b**) and (

**d**) the ANNs.

**Figure 9.**Best ranked solution for (

**a**) decision variables and (

**b**) objective functions using NFM with ANN, normalized with respect to the best solution of the phenomenological model.

**Table 1.**Objective functions for the multi-objective optimization (MOO) of the acrylic acid production process.

Objective Function | Variable | Max/Min | Equation |
---|---|---|---|

Compression Power in C-100 [kW] | OF_{1} | Min | $\dot{\mathrm{W}}=\frac{{\dot{\mathrm{n}}\mathrm{RT}}_{1}\left({\left(\frac{{\mathrm{P}}_{1}}{{\mathrm{P}}_{\mathrm{Feed}}}\right)}^{\mathrm{a}}-1\right)}{\mathrm{a}\xb7\mathsf{\eta}}$ |

Heat recovery in R-100 [kW] | OF_{2} | Max | ${\dot{\mathrm{H}}}_{\mathrm{rxn}\mathrm{j}}\left(\mathrm{T}\right)={\displaystyle {\displaystyle \sum}_{\mathrm{j}}}\Delta {\mathrm{H}}_{\mathrm{rnx}\mathrm{j}}\left(\mathrm{T}\right)\xb7{\mathsf{\xi}}_{\mathrm{j}}$ |

Productivity in R-100 [kmol/m^{3}h] | OF_{3} | Max | $\mathrm{Prod}=\frac{{\mathrm{F}}_{\mathrm{Acrolein}}}{{\mathrm{V}}_{1}}$ |

Conversion in R-100 [%] | OF_{4} | Max | $\mathrm{Conv}=\frac{({\mathrm{F}}_{\mathrm{reactant}\mathrm{in}}-{\mathrm{F}}_{\mathrm{reactant}\mathrm{out}}}{{\mathrm{F}}_{\mathrm{reactant}\mathrm{in}}})\times 100$ |

Compression Power in C-101 [kW] | OF_{5} | Min | $\dot{\mathrm{W}}=\frac{{\dot{\mathrm{n}}\mathrm{RT}}_{1}\left({\left(\frac{{\mathrm{P}}_{2}}{{\mathrm{P}}_{1}}\right)}^{\mathrm{a}}-1\right)}{\mathrm{a}\xb7\mathsf{\eta}}$ |

Heat recovery in R-101 [kW] | OF_{6} | Max | ${\dot{\mathrm{H}}}_{\mathrm{rxn}\mathrm{j}}\left(\mathrm{T}\right)={\displaystyle {\displaystyle \sum}_{\mathrm{j}}}\Delta {\mathrm{H}}_{\mathrm{rnx}\mathrm{j}}\left(\mathrm{T}\right)\xb7{\mathsf{\xi}}_{\mathrm{j}}$ |

Productivity in R-101 [kmol/m^{3}h] | OF_{7} | Max | $\mathrm{Prod}=\frac{{\mathrm{F}}_{\mathrm{AcrylicA}}}{{\mathrm{V}}_{2}}$ |

Conversion in R-101 [%] | OF_{8} | Max | $\mathrm{Conv}=\frac{({\mathrm{F}}_{\mathrm{reactant}\mathrm{in}}-{\mathrm{F}}_{\mathrm{reactant}\mathrm{out}}}{{\mathrm{F}}_{\mathrm{reactant}\mathrm{in}}})\times 100$ |

Excess oxygen concentration above LFL | OF_{9} | Min | Sum_{E} = 0If [O _{2}] > 0.07 thenSum _{E}=Sum_{E} + ([O_{2}] − 0.07) ∗ dW |

Decision Variables | x | Min | Max | References |
---|---|---|---|---|

Molar flowrate of propylene [kmol/h] | F_{P} | 91 | 203 | |

Molar flowrate of air [kmol/h] | F_{A} | 433 | 2900 | |

Molar flowrate of steam [kmol/h] | F_{S1} | 91 | 3047 | |

Molar flowrate of water vapor [kmol/h] | F_{S2} | 100 | 4000 | |

Temperature in R-100 [°C] | T_{1} | 330 | 430 | [17,18] |

Temperature in R-101 [°C] | T_{2} | 285 | 315 | [20] |

Pressure in R-100 [bar] | P_{1} | 1.05 | 6 | [23,24,25,26] |

Pressure in R-101 [bar] | P_{2} | 3 | 6 | [23,24,26] |

**Table 3.**Relative importance of the input variables on the objective functions in the selected ANN according to the modified Garson method.

Relative Importance (%) | |||||||||
---|---|---|---|---|---|---|---|---|---|

Objectives/Decision Variables | F_{P} | F_{A} | F_{S1} | F_{S2} | T_{1} | T_{2} | P_{1} | P_{2} | Bias |

OF_{1} | 1.62 | 17.14 | 2.64 | 1.02 | 1.76 | 1.96 | 41.28 | 1.79 | 30.79 |

OF_{2} | 9.07 | 13.75 | 5.23 | 0.79 | 27.76 | 1.93 | 8.22 | 1.33 | 31.90 |

OF_{3} | 13.42 | 9.03 | 3.69 | 1.24 | 41.58 | 0.83 | 12.25 | 3.01 | 14.95 |

OF_{4} | 5.59 | 18.22 | 7.03 | 4.04 | 36.30 | 1.33 | 19.59 | 2.01 | 5.89 |

OF_{5} | 0.62 | 11.45 | 16.24 | 11.69 | 6.36 | 2.22 | 2.02 | 26.39 | 23.02 |

OF_{6} | 15.24 | 18.19 | 9.69 | 7.66 | 18.37 | 1.83 | 15.42 | 2.61 | 10.99 |

OF_{7} | 17.44 | 34.58 | 5.90 | 6.16 | 10.72 | 1.50 | 8.64 | 2.80 | 12.27 |

OF_{8} | 8.64 | 38.13 | 8.07 | 6.60 | 13.50 | 3.42 | 6.16 | 7.38 | 8.10 |

OF_{9} | 10.91 | 26.29 | 15.98 | 0.86 | 4.55 | 0.30 | 10.09 | 0.77 | 30.26 |

**Table 4.**Objective functions of the best ranked solution using NFM from the Pareto domain obtained with a population of 5000; F

_{P}= 210.0 kmol/h, F

_{A}= 1507.9 kmol/h, F

_{S1}= 206.6 kmol/h, F

_{S2}= 100.0 kmol/h, T

_{1}= 697.65 °C, T

_{2}= 580.72 °C, P

_{1}= 1.05bar and P

_{2}= 4.01 bar.

Objective Function | Meta-Model | Phenomenological Model | % Difference |
---|---|---|---|

OF_{1} | 90.42 | 91.10 | 0.75 |

OF_{2} | 23460 | 23700 | 1.02 |

OF_{3} | 0.8992 | 0.9866 | 9.27 |

OF_{4} | 94.97 | 96.25 | 1.34 |

OF_{5} | 6581 | 6794 | 3.19 |

OF_{6} | 10755 | 11104 | 3.19 |

OF_{7} | 1.179 | 1.202 | 1.93 |

OF_{8} | 81.67 | 82.97 | 1.58 |

OF_{9} | 0.016 | 0.000 | - |

**Table 5.**Objective functions of the solution obtained using the weighted sum method from the Pareto domain; F

_{P}= 210.0 kmol/h, F

_{A}= 1636.5 kmol/h, F

_{S1}= 495.74 kmol/h, F

_{S2}= 100.0 kmol/h, T

_{1}= 628.46 °C, T

_{2}= 570.79 °C, P

_{1}= 3.66 bar, and P

_{2}= 6.00 bar.

Objective Function | Meta-Model |
---|---|

OF_{1} | 2170.50 |

OF_{2} | 25349 |

OF_{3} | 1.1457 |

OF_{4} | 100.00 |

OF_{5} | 9450 |

OF_{6} | 13259 |

OF_{7} | 1.331 |

OF_{8} | 88.50 |

OF_{9} | 0.057 |

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**MDPI and ACS Style**

Sepulveda, G.C.; Ochoa, S.; Thibault, J.
Methodology to Solve the Multi-Objective Optimization of Acrylic Acid Production Using Neural Networks as Meta-Models. *Processes* **2020**, *8*, 1184.
https://doi.org/10.3390/pr8091184

**AMA Style**

Sepulveda GC, Ochoa S, Thibault J.
Methodology to Solve the Multi-Objective Optimization of Acrylic Acid Production Using Neural Networks as Meta-Models. *Processes*. 2020; 8(9):1184.
https://doi.org/10.3390/pr8091184

**Chicago/Turabian Style**

Sepulveda, Geraldine Cáceres, Silvia Ochoa, and Jules Thibault.
2020. "Methodology to Solve the Multi-Objective Optimization of Acrylic Acid Production Using Neural Networks as Meta-Models" *Processes* 8, no. 9: 1184.
https://doi.org/10.3390/pr8091184