MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...

The physical meaning of m obs is the observed reaction order in Zone I, which will be
explained in more detail in the next chapter.
The unified correction function models how all of the various vs. M T curves
deviate from the vs. M T curve for first order reactions in the band shown in Figure 4.2.
The correction function is used as a multiplier, f c, preceding the first order curve
expression in the following manner:
= f c
1
M T
⎛ 1
⎝ tanh(3MT ) ⎜
1 ⎞
− ⎟
3MT ⎠
. (4.20)
The (0.5M T -2 + 2MT 2 ) part in the correction function needs to be justified. Consider a
function g(M T) = (0.5M T -2 + 2MT 2 ). The first and second order derivatives of function g
are, respectively,
g'(MT )= − 1
3
MT +4MT , (4.21)
g"(MT ) = 3
4 + 4 . (4.22)
MT By setting g'(M T) to zero and comparing the value of g"(M T) to zero, it is easy to prove
that g(M T) takes the minimum value at M T = 1/2 . As M T departs from 1/2 in both
directions, g(M T) grows rapidly, which decreases the value of f c toward unity. This
guarantees that the correction function would not alter the desirable asymptotic features
of the general asymptotic solution. Further, if M T1 and M T2 satisfy Eq. (4.16), it is easy
to prove that g(M T1) = g(M T2).
The (1 - m s) 2 part in the power index of the correction function f c allows a non-
linear interpolation between the values of f c(M T, 1) and f c(M T, 0). In the first order limit,
f c(M T, 1) = 1, since the curve for first order reactions is exact and does not need any
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