2.6. Mathematical Modeling
A simplified kinetic mechanism for enzyme-catalyzed hydrolysis reactions
was employed in this work; the mechanism equations and molar balances
are described in a previous work performed by the same research group
(Tavares et al., 2018b). The assumptions for the closed and batch system
were considered the same for the ultrasound system as for the
conventional stirring batch process: Enzymatic hydrolysis reaction
defined using the simplified Ping Pong Bi Bi (PPBB) mechanism;
Elementary reaction rate; Ideal mixture in the liquid phase; and
Isobaric and isothermal process.
To solve the equation system the following initial conditions were used:\(C_{\text{TAG}}\left(0\right)=0.27\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\), \(\ C_{\text{DAG}}\left(0\right)=0\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\),\(C_{\text{MAG}}\left(0\right)=0\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\),\(C_{H_{2}O}\left(0\right)=29.14\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\),\(C_{\text{GL}}\left(0\right)=0\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\mathbf{,\ }C_{\text{FFA}}\left(0\right)=0.08\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\),and\(C_{E}\left(0\right)={C_{E}}^{\text{total}}=3.5x10^{-4}\ \frac{\text{mol}}{\text{Kg}}\)for in natura seeds; and\(C_{\text{TAG}}\left(0\right)=0.34\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\), \(\ C_{\text{DAG}}\left(0\right)=0\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\),\(C_{\text{MAG}}\left(0\right)=0\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\),\(C_{H_{2}O}\left(0\right)=33.72\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\),\(C_{\text{GL}}\left(0\right)=0\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\mathbf{,\ }C_{\text{FFA}}\left(0\right)=0.10\ \frac{\text{mol}}{\text{Kg}}\mathbf{\ }\),and\(C_{E}\left(0\right)={C_{E}}^{\text{total}}=1.6x10^{-4}\ \frac{\text{mol}}{\text{Kg}}\ \)for oil-free seeds. The considered molecular mass of the enzyme was
59kDa (Eastmond, 2004) and the enzymatic concentrations were calculated
considering 17% of proteins (lipases) on the castor bean seeds bodies
(Tully and Beevers, 1976).
Rosenbrock’s numerical method (Rosenbrock, 1963) coded in Maple®
software was used to solve the mathematical model with the initial
conditions.
The enzymatic hydrolysis kinetics experimental data was used to estimate
the parameter values in the mathematical model
(k 1,…, k 12)
employing the minimization of the objective function (Equation
3) through the Simplex Dowhill optimization method (Nelder and Mead,
1964).