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).