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In-Depth Information
Figure 6.3
Wilson equation parameters.
the values of the slope,
a
, and intercept,
b
, will differ. This problem is solved using
the well-known method of least squares, which works in the folllowing manner. An
objective function,
n
data
(y
calc
−
ax
data
−
b)
2
ψ =
(6.18)
i
is a measure of the goodness of fit and is defined as the sum of squares of errors
between the individual values of the dependent values of each of the
y
data points
observed and the values of
y
calculated from the linear model, with trial values of
a
and
b
. The minimum of this function can be found by setting
∂
ψ
/∂a
and
∂
ψ
/∂b
equal
to zero. This generates a simultaneous set of linear equations which can be solved for
the values of
a
and
b
that best fit the data.
The problem of fitting data to the parameters of a nonlinear activity coefficient model
is similar to the above. In Aspen Plus an objective function must be selected from
several alternatives, given in Table 6.1. The maximum likelihood objective function is
minimized directly subject to the constraints imposed by the applicable thermodynamic
relationships using nonlinear programming methods such as those described by Edgar
et al. (2001). All the remaining objective functions may be minimized by solving the
nonlinear equations that arise by setting the partial derivatives of the objective function
relative to the model parameters equal to zero.
