All of the numerical procedures (model integration, soft sensing and filter tuning) were performed in SCILAB (v.6.0.2, ESO Group, Paris, France, 2019) in a computer with an AMD FXTM-8350 and 15.7 Gb of random access memory, running a 64-bit Linux Mint 18.3 as operating system. For the model integration, a Runge-Kutta 4th order algorithm was used (ode function with “rk” specification).
2.1.2. Assay Conditions
Assays were conducted with three feeding profiles. All of the profiles added up to 600 g
substrate and 2.22 g
protein (approximately 10 FPU.g
substrate−1) of enzymatic complex at the end of each process, regardless of the feeding policy. The first assay, Data Set 1, is a High Solids Batch (HSB). This assay had all of the compounds added at the beginning of the process. The other two assays were operated in fed-batch mode, where the feeding profiles added up to the same substrate mass as the one of the batch processes. Feedings were performed at discrete times, as presented in
Table 1.
Data Set 2 is the Low Solids Fed-Batch (LSF) as the feeding intervals were spaced enough not to generate a visible high solids concentration throughout the hydrolysis. In Data Set 3, additions were closer in time when compared to LSF. In this assay, an initial low solids concentration occurred, but with following substrate additions the medium turned to a high-solids load process. Thus, this assay was named Mixed Profile Fed-Batch (MPF). Duplicates were performed for all assays by conducting the experiment in different days under the same policy.
The reactor used was a 3 L working volume baffled jacketed stirred reactor (reactor’s diameter 0.16 m, height 0.37 m and liquid height 0.21 m, baffle’s height 0.19 m, and thickness 15 mm) with two elephant ear impellers (diameter 0.08 m, New Brunswick Scientific®, Edison, NJ, USA) equidistant from one another, the reactor bottom and the liquid surface. The upper impeller generates a downwards pumping flow and the lower impeller pumps upwards. The stirring motor was placed on top of a plate with bearings, so the motor was free to rotate when opposing forces acted upon it. A digital dynamometer (FG 6005 SD, Lutron Electronics, Coopersburg, PA, USA) was coupled to the bearing assembly in order to measure the opposing force. This allows the agitation power to be measured. The reactor media was kept at 50 °C by a thermostatic bath. The agitation speed was 470 RPM.
2.1.4. Mathematical Modeling
The kinetic model that was used as part of the MHE implementation was developed in a previous work [
22]. This kinetic model was the one that best describes glucose and xylose during enzymatic hydrolysis of a steam-exploded sugarcane bagasse under batch and fed-batch operations. A brief description is given in this subsection, and more information can be obtained in the dedicated paper.
To describe the lignocellulosic material hydrolysis, five reactions are considered:
Reaction 1: Cellulose → γCl-Cb Cellobiose
Reaction 2: Cellobiose → γCb-Gl Glucose
Reaction 3: Hemicellulose → γHe-Xy Xylose
Reaction 4: Lignin → Lignin
Reaction 5: Enzyme → Inactive Enzyme
In the reaction scheme, γ are the pseudo-stoichiometric mass relations between substrates and products for each reaction. The values for these parameters were [
5]: γ
Cl-Cb = 1.056 g
Cellobiose g
Cellulose−1, γ
Cl-Gl = 1.111 g
Glucose g
Cellulose−1, γ
Cb-Gl = 1.056 g
Glucose g
Cellobiose−1, γ
He-Xy = 0.841 g
Xylose g
Hemicellulose−1 [
25].
Reactions 1 and 3 are considered to be heterogeneous, as they represent the breakdown of cellulose and hemicellulose. These are modeled via Modified Michaelis–Menten kinetics with competitive product inhibition, presented in Equation (1).
In this equation, αi (g L−1 min−1) are reaction rates, where the subscripted “i” denotes the reaction used, either 1 or 3. The same nomenclature is used in the subsequent variables, ki are kinetic constants (min−1), Ei are enzyme concentrations (g L−1), [Si] are substrate concentrations (g L−1) and Kmi are the Michaelis–Menten modified constant (gEnz L−1), Kpi are competitive inhibition constants for products (g L−1), and Pi are product concentrations (g L−1).
The hydrolysis of cellobiose into glucose (Reaction 2) is homogeneous. This equation is modeled with a Pseudo-Homogeneous Michaelis–Menten with Competitive Product Inhibition model, as presented in Equation (2).
The same variables are used in this equation; however,
Km2 is the Michaelis–Menten constant (g L
−1) for substrate.
Reaction 5 is included, despite lignin being an inert. Thus, this “reaction” just reflects the accumulation of lignin in the reactor when it operates in fed-batch mode.
Enzymatic inactivation is represented by a generic first order equation, as presented in Equation (3).
where
ke is a first order inactivation parameter (min
−1).
Table 2 presents the parameters of these equations.
Following the formalism that is proposed in reference [
26], Equation (4) represents the mass balance for the components in the reactor. Since substrate feeding was discrete, the fed-batch process was solved as a sequence of batch processes by re-initializing the initial conditions at each feeding with the previous final concentrations.
In Equation (4), the column vector in the left-hand side of the equation represents the concentrations of reactive cellulose ([Cl]), cellobiose ([Cb]), glucose ([Gl]), hemicellulose ([He]), xylose ([Xy]), lignin ([Lg]), and enzymatic complex activity ([E]). The resulting 7 × 5 matrix at the right-hand side of the equation is the pseudo-stoichiometric matrix and the vector (αi) are the reaction rates of reaction 1 to 5, as previously described.
The novelty of this model lies in how it was fitted and how it operates. A fuzzy membership rule interpolates between two different set of parameters for the described kinetic equations, one set was fitted using data from Data Set 1, High Solids Model (HSM), and the other was fitted using Data Set 2, the Low Solids Model (LSM). Thus, when solving for the reaction kinetics, both models generate independent reaction rates, αHSM (High Solids Model reaction rate) and αLSM (Low Solids Model reaction rate) for every equation at a same reaction instant. A Fuzzy Model (FM) then interpolates both standalone reaction rates with an optimized Membership Function (MSF). The MSF calculates the Membership Degree (MD) of each model based on the total amount of Reactive Solids (RS), the addition of the current concentration of cellulose and hemicellulose. The Membership Degree of the HSM (MDHSM) is calculated with the linear fuzzy rule. The upper and lower bounds of the MSF were 75.35 g L−1 and 61.52 g L−1, respectively.
With the HSM and LSM membership degrees, the reaction rate for the FM was calculated with Equation (5). It is the output of the Takagi–Sugeno system.
where
αFUZZY are the reaction rates for each hydrolysis reaction. Thus, a prediction of the entire state vector is obtained. SCILAB’s Ordinary Differential Equation Solver (ODE) was used to integrate the state variables, with the generated model parameters.