This preview shows page 1. Sign up to view the full content.
Unformatted text preview: lysis: (continued)
(c) To fit the mixture data to the extended Langmuir equation, to determine the best
values of the four coefficients (qC3)m, (qC3=)m, KC3, and KC3=, Eqs. (1) and (2) above must be used
simultaneously. Polymath does not permit more than one equation to be fitted at a time.
Alternatively, the two equations can be added together to give one equation in the total loading,
qt = qC3 + qC3=. However, this does not insure a good fit to the data for the individual loadings.
Nevertheless, this technique was applied with Polymath. It was found that the minimum sum of
squares of the deviations of the fit from the experimental data was not very sensitive to the values
of the four coefficients. One reasonable set that was close to the set determined from the pure
component data was:
(qC3)m = 2.5213, (qC3=)m = 2.8201, KC3 = 0.002116, and KC3= = 0.004752
Unfortunately, this set showed little improvement in the fit of the mixture data as compared to
the fit of the set determined in Part (b) from the pure component data. The plots are not much
different from those on the previous page.
Another procedure that can be applied is to use the program MathCAD with the Minerr
function. The MathCAD program and result is shown on the next page. Minerr searches for the
values of the four constants that provide the best fit to the data. The objective function is the
sum of the squares of the deviations of the data from the predictions (SSE) by the Eqs. (1) and
(2). SSE =
i qC3 i − qC3 K pC3
m C3 2 + qC3= i − i 1 + KC3 pC3 i + KC3= pC3= i qC3= K
m C3= 2
i 1 + KC3 pC3 i + KC3= pC3= i (3)
where the four constants are varied to find the smallest value of SSE. Initial quesses for the four
constants must be provided, which, as shown on the next page for the MathCAD program were:
(qC3)m = 2.5, (qC3=)m = 2.8, KC3 = 0.002, and KC3= = 0.002
The values computed are seen to be:
(qC3)m = 2.153, (qC3=)m = 4.042, KC3 = 0.001642, and KC3= = 0.001997, with SSE = 3.501,
which does not indicate a very good fit.
Unfortunately, different initial guesses give other results, none of which gives much, if any
improvement to the fit.
(d) The procedure outlined in Part (c), although not very satisfactory, was applied to the
extended Langmuir-Freundlich isotherm, Eq. (15-33), which for the total loading is:
qC3 0 k C3 PyC3 C3
qC3= 0 k C3= PyC3= C3=
1 + k C3 PyC3 C3 + k C3= PyC3= C3= 1 + k C3 PyC3 C3 + k C3= PyC3= C3=
When used with the nonlinear regression program of Polymath, the results obtained were not
satisfactory because some coefficients were negative for the best fit.
When MathCAD was used with the Minerr function according to the program and result
is shown on a following page, the following values were obtained for the six constants:
(qC3)m = 1.706, (qC3=)m = 4.507, KC3 = 0.0270, KC3= = 0.03773, nC3 = 1.59, and nC3= = 1.90, with
SSE = 2.583, which is a better fit than for the extended Langmuir equation. However, other
initial guesses for the six constants gave other results, because of the the great sensitivity. E...
View Full Document
This document was uploaded on 02/24/2014 for the course CBE 2124 at NYU Poly.
- Spring '11
- The Land