Processes 2023 , 11 , 809
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Table2. A comparison of the cellulose DP and Kappa number between the experimental data and model simulation at the end of the pulping process ( t = t end ) for three different wood species [61–63].
Cellulose DP
Kappa Number
Species
Experiment
Model
Experiment
Model
Eucalyptus
1399 1203 1423
1342 1174 1368
8.7
11.4 52.6 10.8
Date Palm Rachis
54
Pinus Radiata
10.6
Meanerror
4% Mean error
8%
Remark4. The listed network configurations are not a comprehensive set. During the process of hyperparameter tuning, a wide range of structures with a varying number of nodes, layers, types of activation functions, and dropout rates were considered. The trained model was tested on 100 test datasets for accuracy evaluation. The parity plots between the multiscale model and LSTM-ANN model for the Kappa number and cellulose DP at the end of the pulping process are shown in Figure 6. An upper and lower bound of 10% was added to the bisector. The plot shows that the model was well-trained and had an average R 2 value of 0.9953 for the Kappa number and 0.9846 for cellulose DP . In order to compare the accuracy when a deep neural network (DNN) was used instead of an LSTM to train the data with a combination of time-invariant and time-varying operating conditions, parity plots between the multiscale model and DNN model are shown in Figure 7. The R 2 values for the Kappa number and cellulose DP are seen to be very low (less than 0.8) despite the network structure having three layers with 40 hidden nodes each (the same number of total nodes as the LSTM model). This shows that the DNN was unable to capture the temporal dependencies as the LSTM did, and thus the aforementioned LSTM network structure was selected for further study. Next, the controller results are presented.
Table3. List of configurations for the LSTM network.
R 2 Kappa
R 2 DP
SerialNo.
Layers Nodes Batch Size Dropout Rate
1 2 3 4 5 6 7
2 2 3 3 3 3 3
20 40 40 20 40 10 40
16 32 16 32 32 16 16
0.2 0.4 0.5 0.2
0.942 0.957 0.995 0.982 0.998 0.975 0.926
0.884 0.901 0.985 0.944 0.992 0.954 0.895
0 0
0.8
4.3. LSTM-ANN-Based MPC As shown in the parity plots, the LSTM-ANN model effectively establishes a relation- ship between the operating conditions and the output set-point values. This LSTM-ANN model is then integrated into the MPC described in the previous section to obtain the optimal input profiles. These profiles are then used in the multiscale model to calculate the Kappa number and cellulose DP in an iterative manner at each time step. To demon- strate the controller’s capabilities, the desired set-point values of Kappa number = 90 and cellulose DP = 700 were set. Figure 8 shows the optimal input profiles for the cooking time, NaOH concentration, and the free-liquor temperature under MPC operation. As expected, the cooking time and NaOH concentration remained constant throughout the process, following the constraints set in Equations (21)–(22).
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