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