Processes 2023 , 11 , 809
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As the depolymerization reaction proceeds, the time evolution is calculated in a similar fashion as that of dissolution by using a random number, ξ , which lies within the range of ( 0,1 ] : Δ t = − ln ξ R P + R S + R H [ min ] (4) These microscopic events repeat until multiple increments of Δ t accumulate to form the mesoscopic time step, at which point updates to the concentration and temperature aremade. The Kappa number ( κ ), a measure of the lignin content in pulp, is described as follows:
c s 1 + c s 2 0.00153 ∗ ( ∑ 5
(5)
κ =
i = 1 c s i )
where c s i is defined as the concentration of the solid component i in a wood chip. Here, i varies from 1 to 5 and is represented by the five solid phase components: high reactive lignin ( s 1 ), low reactive lignin ( s 2 ), cellulose ( s 3 ), xylan ( s 4 ), and galactoglucomman ( s 5 ). Specifically, it is given by:
Mass o f solid component i in wood chip Wood chip volume
(6)
c s i =
The developed multiscale model is now used to generate data offline for different oper- ating conditions by varying the input profiles of the temperature, cooking time, and NaOH concentration. This is conducted in order to obtain a comprehensive set of input–output data. Specific details about the ranges are provided in the next section. The data is then utilized for training the LSTM-ANN model, which serves as a surrogate model.
3. Surrogate Model Development and MPC Design 3.1. LSTM-ANN Network Training
The proposed multiscale model is utilized to generate high-quality data for training the LSTM-ANN model. Specifically, the operating conditions, such as the cooking time, NaOH concentration, and free-liquor temperature, are systematically varied over a range. The multiscale model is evaluated offline to obtain 750 different datasets. The cooking time ranges between 60 and 120 min with increments of 10 min, the NaOH concentration ranges between 50–92 wt% with increments of 3%, and the free-liquor temperature ranges between 390 and 430 K in a time series manner with increments of 10 K. A total of 750 datasets were selected to ensure that all possible permutations and combinations of different input profiles were covered, limiting the unknown regions where the neural networks might have to extrapolate and make subpar predictions. The output values of the Kappa number and cellulose DP are obtained every 3 s by the model, resulting in a varying number of time-series sequence data containing 1200–2400 sampling points per dataset. It is important to note that, while the cooking time and NaOH concentration remain constant throughout the duration of one such model run, the temperature varies over time in every run. The outputs of the model are the sequence data of the Kappa number and cellulose DP . In order to establish a relationship between the past and present outputs, the output values of the previous time step are also considered in the training input. The window length (i.e., the number of past outputs considered) can be modified; however, for this case, we use only the most recent value at time t to predict the outputs at time t + 1. Once a com- prehensive mix of these datasets that exhaustively covers all possible operating conditions is obtained, the next step is to train the neural network model on this generated data. As the pulping process involves both time-series and time-invariant data characteris- tics, these must be handled differently for improved prediction accuracy. Therefore, in this work, we focus on using LSTM networks to handle the temporal behavior of data and ANN to handle the time-invariant data. This combination of LSTM and ANN is used as they
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