THE DATA REGRESSION SYSTEM - Teach Yourself the Basics of Aspen Plus - page 77

TABLE 6.1 Objective Functions, Aspen Plus Version 7.0
Objective Function

Formed By:

Maximum likelihood All measured variables

Ordinary least squares Pressure and vapor compositions for isothermal VLE data

Temperature and vapor compositions for isobaric VLE data

Barker's method Pressure only

Modified Barker's method Pressure and vapor compositions

Activity coefficients Activity coefficients

Equilibrium constants Equilibrium constants ( K values)

Relative volatility Relative volatility relative to first component

The choice of objective function depends on the type are data that are available. For
vapor - liquid equilibrium the data may be of the type T - P - x - y or T - P - x ,where T
is temperature, P is pressure, and x and y are liquid- and vapor-phase mole fractions,
respectively. T - P - x - y data can be used with all objective functions except Barker's
method, which is used with T - P - x data only. Selection of an objective function may
be found at Propereties/Regression by choosing New regression and the tab Algorithm.
An example of an objective function suitable for use with T - P - x - y data is Activity
Coefficients:

2

n comp

n data

obs
i

calc
i

γ − γ

ψ = (6.19)

obs
i

γ

j = 1 i = 1

where the experimental activity coefficients are calculated using

y i φ i (y 1 ,y 2 P j ,T j )P j
x i P i (T oj )

obs
i

γ = (6.20)

the experimental values and trial values of the activity coefficient parameters. Note the
scaling of the activity coefficients in the objective function by the denominator,

obs
i

γ :

When VLE data are of the type T - P - x the Barker objective function

n data

(P j − P calc ) 2 ψ = (6.21)

j

j = 1

is indicated. Here P cal j is calculated using the trial values of the activity coefficient
equation parameters and the experimental values

j

1 x 1 Pv 1 + γ

j

2 x 2 Pv 2

P calc

= γ (6.22)

j

Fitting liquid - liquid equilibrium data (i.e., T - x I - x II ,where x I and x II refer to the
composition of the two liquid phases in equilibrium) can be done with the least squares
or the maximum likelihood objective function.