the handbook of food engineering practice crc press chapter 10 ...

log( k ref
k ) = C 1(T-T ref )
C 2 +(T-T ref ) (36)
where k ref the rate constant at the reference temperature T ref (T ref >T g ) and C 1 , C 2 are
system-dependent coefficients. Williams et al (1955), for T ref =T g , using experimental data
for different polymers, estimated average values of the coefficients: C 1 =-17.44 and
C 2 =51.6. In various studies these are used as universal values to establish the applicability
of WLF equation for different systems. This approach can be misleading (Ferry,1980;
Peleg, 1990; Buera and Karel,1993) and effort should be made to obtain and use system
specific values.
Alternative approaches for accessing the applicability of the WLF model and
calculating the values of C 1 and C 2 have been evaluated (Nelson, 1993; Buera and Karel,
1993). Eq. (36) can be rearranged into an equation of a straight line. Thus the plot of
[log k ref 1
k ]-1 vs. T-T ref
is a straight line with a slope equal to C 2 /C 1 and an intercept of
1/C 1 . If the glass transition temperature, Tg, is known, the WLF constants at Tg can be
calculated (Peleg,1992):
C 1 g = C 1 C 2
C 2 +T g -T ref
and C 2 g =C 2+T g - T ref (37)
These values can be compared to the aforementioned average WLF coefficients.
When Tg and reaction rate data at many higher temperatures are available, k g ,
C 1 and C 2 can be estimated from eq.(36) using non linear regression methodology.
Ferry (1980) proposed an additional approach for verifying the WLF
equation and determining the coefficients. A temperature T ∞ , at which the rate of the
reaction is practically zero, is used. T ∞ can be approximated by the difference between the
reference temperature and C 2 i.e. T ∞ =T ref -C 2 . Rearranging eq. (36)
log( k ref
k ) = C 1(T-T ref )
T-T ∞
(38)
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