Calculation of Activation Energy from Model B
It was proposed in an earlier study (Dunn, 2008) that kinetic parameters for the oxidation of biodiesel can be obtained from solutions toModel B type equations. The rate of increase in the degree of conversion of a substrate (α) is expressed by:
dα/dt = kTf(α) (17)
where kT is the reaction rate constant, described mathematically in Eq. 5, and f(α) mathematically describes the kinetics of the reaction. Rearranging and integration yields the following relationship:
g(α) = kT∆t (18)
where g(α) is determined from f(α) using Eq. 9 and ∆t is the time interval (t − t0). At the induction period of the reaction, IP = ∆t where g(α) is determined between α = 0 at t0 and α = α* at t = t* (∆t = t* − t0). Substituting Eq. 5 for kT in Eq 18, taking the natural log of both sides and rearranging yields:
ln(∆t) = ln[g(α)/Z0] + (EaRg−1)(T)−1(19)
When ∆t = IPR, Eq 19 is equivalent to Eq. 3, theModel B type equations derived in the present study. Therefore, kinetic parameters Ea and Z0 for the oxidation reaction can be calculated from the slope and intercept coefficients from model equations for each FAME. Earlier, an expression was given for calculating Ea from the slope coefficients (B1) in Eq. 6. The following equation can be used to obtain pre-exponential factor Z0 from the intercept coefficients (B0) assuming the kinetic models are known:
Z0 = g(α)[exp{−B0}] (20)
If the reaction kinetics are not known, then Eq. 20 can be rearranged to yield a ratio of Z0 over g(α), which allows calculation of Z0 (and kT) for varying models:
[Z0/g(α)] = exp{−B0} (21)
The results presented in Table 8 include Eadata and Z0/g(α) ratios calculated from applying Eqs. 6 and 21 to the Model B equation data in Table 4 . Also shown are estimates for Z0 factors determined for two popular kinetic models for the oxidation of fatty derivatives, first order (Kamal-Eldin and Yanishlieva, 2005; Litwinienko, 2005; Litwinienko et al., 2000) and “simple” autocatalysis (Adachi et al., 1995; Ishido et al., 2001; Luna et al., 2007). The g(α) functions were determined at α* = 0.5 at the IPR of the FAME (Dunn, 2020).
Table 8 Activation energy (Ea) and pre-exponential factor (Z0) calculated fromModel B equations