Optimal Bioreactor Operational Policies for the Enzymatic Hydrolysis of Sugarcane Bagasse

Abstract

The consolidation of the industrial production of second-generation (2G) bioethanol relies on the improvement of the economics of the process. Within this general scope, this paper addresses one aspect that impacts the costs of the biochemical route for producing 2G bioethanol: defining optimal operational policies for the reactor running the enzymatic hydrolysis of the C6 biomass fraction. The use of fed-batch reactors is one common choice for this process, aiming at maximum yields and productivities. The optimization problem for fed-batch reactors usually consists in determining substrate feeding profiles, in order to maximize some performance index. In the present control problem, the performance index and the system dynamics are both linear with respect to the control variable (the trajectory of substrate feed flow). Simple Michaelis–Menten pseudo-homogeneous kinetic models with product inhibition were used in the dynamic modeling of a fed-bath reactor, and two feeding policies were implemented and validated in bench-scale reactors processing pre-treated sugarcane bagasse. The first approach applied classical optimal control theory. The second policy was defined with the purpose of sustaining high rates of glucose production, adding enzyme (Accellerase® 1500) and substrate simultaneously during the reaction course. A methodology is described, which used economical criteria for comparing the performance of the reactor operating in successive batches and in fed-batch modes. Fed-batch mode was less sensitive to enzyme prices than successive batches. Process intensification in the fed-batch reactor led to glucose final concentrations around 200 g/L.

Appendix

The first step in the development of the mathematical formulation of the optimal control algorithm is the dynamic description of the system to be controlled.

$$ \frac{{\mathrm{d}\underline{x}}}{{\mathrm{d}t}}=\underline{f}\left( {\underline{x}(t),\underline{u}(t),t} \right)=\underline{\dot{x}} $$

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Where $\underline{x}(t)$ is the vector of dimension n of state variables, at time t, $\underline{u}(t)$ is the control vector of dimension m, which specifies the input variables of the system model, $\underline{f}$ is a vector of functions (the system model, in our case mass balances), and $\underline{\dot{x}}$ is the vector of state variables, calculated by the differential equations that constitute the model.

The purpose of the optimal control problem is to determine the control policy that will maximize or minimize a specific performance index, subject to restrictions imposed by the physical nature of the problem. The general performance index (functional) is:

$$ J=h\left( {\underline{x}\left( {{t_f}} \right),{t_f}} \right)+\int\limits_{{{t_0}}}^{{{t_f}}} {F\left( {\underline{x},\underline{u},t} \right)\mathrm{d}t} $$

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The necessary conditions for optimization in the presence of constraints are conveniently expressed in terms of the “Hamiltonian” (H); the performance index is maximized/minimized by maximizing/minimizing H:

$$ \mathop{\min}\limits_{{\underline{u}(t),\;{t_f}}}\quad H=F\left( {\underline{x},\underline{u},t} \right)+{{\underline{\lambda}}^T}\underline{f}\left( {\underline{x},\underline{u},t} \right) $$

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By applying the Maximum Principle of Pontryagin, a set of necessary conditions to maximize or minimize the functional are defined as:

$$ \underline{\dot{\lambda}}=-\frac{{\partial H}}{{\partial \underline{x}}}\ Euler\text{--} Lagrange equation $$

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$$ \underline{\dot{x}}=\underline{f}=\frac{{\partial H}}{{\partial \underline{\lambda}}}\ restrictions, the model $$

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$$ {{\left[ {\frac{{\partial h}}{{\partial \underline{x}}}\left( {{t_f}} \right)-\underline{\lambda}\left( {{t_f}} \right)} \right]}^T}\delta {x_f}+\left( {H\left( {{t_f}} \right)+\frac{{\partial h}}{{\partial t}}\left( {{t_f}} \right)} \right)\delta {t_f}=0\ transversality $$

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$$ \frac{{\partial H}}{{\partial \underline{u}}}=\underline{0}\ singular arc $$

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$$ \mathop{u}\limits^{\bullet }=0\ bang\text{--} bang control $$

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In our case study, the performance index is minimized at a fixed final time:

$$ \begin{array}{*{20}c} {J=-P\left( {tf} \right)-X\left( {tf} \right)=-P\left( {tf} \right)-} \hfill \\ {-\left\{ {{{{\left[ {\left( {{S_0}\cdot {V_0}} \right)+\left( {{S_{\mathrm{feed}}}\cdot \left( {V\left( {tf} \right)-{V_0}} \right)} \right)-\left( {S\left( {tf} \right)\cdot V\left( {tf} \right)} \right)} \right]}} \left/ {{\left[ {\left( {{S_0}\cdot {V_0}} \right)+\left( {{S_{\mathrm{feed}}}\cdot \left( {V\left( {tf} \right)-{V_0}} \right)} \right)} \right]}} \right.}} \right\}} \hfill \\ \end{array} $$

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Through the Hamiltonian (H), the performance index could be maximized/minimized

$$ H={{\underline{\lambda}}^T}\cdot \left( {\underline{f}\left( {\underline{x}} \right)+\underline{{g\left( {\underline{x}} \right)}}\cdot u} \right)={\lambda_1}\cdot \left[ {-r+\left( {{S_{\mathrm{feed}}}-S} \right)D} \right]+{\lambda_2}\cdot \left( {r-PD} \right)+{\lambda_3}\cdot \left( {VD} \right) $$

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Applying the Euler–Lagrange equation:

$$ \begin{array}{*{20}c} {\begin{array}{*{20}c} {{{\dot{\lambda}}_1}={\lambda_1}\cdot \left\{ {D+{{{\left( {e\cdot k} \right)}} \left/ {{\left[ {S+{k_{\mathrm{m}}}\cdot \left( {{P \left/ {{{k_{\mathrm{i}}}+1}} \right.}} \right)} \right]}} \right.}-{{{\left( {S\cdot e\cdot k} \right)}} \left/ {{{{{\left[ {S+{k_{\mathrm{m}}}\cdot (P/{k_{\mathrm{i}}} + 1)} \right]}}^2}}} \right.}} \right\} -} \hfill \\ {\quad \quad {\lambda_2}\cdot \left\{ {{{{\left( {e\cdot k} \right)}} \left/ {{\left[ {S+{k_{\mathrm{m}}}\left( {{P \left/ {{{k_{\mathrm{i}}}+1}} \right.}} \right)} \right]}} \right.}-{{{\left( {S\cdot e\cdot k} \right)}} \left/ {{{{{\left[ {S+{k_{\mathrm{m}}}\cdot \left( {{P \left/ {{{k_{\mathrm{i}}}+1}} \right.}} \right)} \right]}}^2}}} \right.}} \right\}} \hfill \\ \end{array}} \hfill \\ {\begin{array}{*{20}c} {{{\dot{\lambda}}_2}={{{\left( {-{\lambda_1}\cdot S\cdot e\cdot k\cdot {k_{\mathrm{m}}}} \right)}} \left/ {{\left\{ {{k_{\mathrm{i}}}\cdot {{{\left[ {S+{k_{\mathrm{m}}}\cdot \left( {{P \left/ {{{k_{\mathrm{i}}}+1}} \right.}} \right)} \right]}}^2}} \right\}}} \right.}+} \hfill \\ {\quad \quad\;\;{\lambda_2}\cdot \left\{ {u+{{{\left( {S\cdot e\cdot k\cdot {k_{\mathrm{m}}}} \right)}} \left/ {{\left[ {{k_{\mathrm{i}}}\cdot {{{\left( {S+{k_{\mathrm{m}}}\cdot \left( {{P \left/ {{{k_{\mathrm{i}}}+1}} \right.}} \right)} \right)}}^2}} \right]}} \right.}} \right\}} \hfill \\ \end{array}} \hfill \\ {{{\dot{\lambda}}_3}=-{\lambda_3}\cdot u} \hfill \\ \end{array} $$

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Finally, considering the transversality condition for fixed final time (δt _f  = 0) and free final conversion:

$$ \begin{array}{*{20}c} {{\lambda_1}\left( {{t_f}} \right)\cdot ={V \left/ {{\left[ {{S_{\mathrm{feed}}}\cdot V+{V_0}\cdot \left( {{S_0}-{S_{\mathrm{feed}}}} \right)} \right]}} \right.}} \hfill \\ {{\lambda_2}\left( {{t_f}} \right)\cdot =-1} \hfill \\ {{\lambda_3}\left( {{t_f}} \right)\cdot ={{{\left[ {S\cdot {V_0}\cdot \left( {{S_0}-{S_{\mathrm{feed}}}} \right)} \right]}} \left/ {{{{{\left[ {{S_0}\cdot {V_0}+{S_{\mathrm{feed}}}\cdot V- > {S_{\mathrm{feed}}}\cdot {V_0}} \right]}}^2}}} \right.}} \hfill \\ \end{array} $$

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