5. An adsorption model with IPA Description and parameterisation
5.4. Parameterisation of the adsorption model
5.4.2. Procedure 2 Least Squares
The least squares method is probably the most common procedure used for estimating parameter values of a function or model. For this method, it is assumed that the best set of parameters is the one that produces the smallest deviations, or errors, between the observed data
(0)
and those predicted (P) by the model. Therefore in this procedure the parameter values are varied in the search of a set that minimises an estimator of the deviations. Mathematically the estimator, or objective function(<1» ,
commonly used is the sum of the squared errors:[5.19]
where j is an integer varying between 1 and nD, the number of elements in the
data set.
Fitting a simple model to a data set using this method is a procedure that has been extensively documented and various statistical packages contain this as a standard feature. Although several independent variables can be used, only one dependent variable is commonly allowed. For a simple adsorption model the value of
Qs
is given and the values of C or5
are obtained in response. Only C, or5,
is needed because they are related . . In the case of adsorption with IPA, the model has to be solved for two solutes simultaneously, because SIP is equivalent for both solutes and dependent on their relative concentrations. This means having two independent and two dependent variables. Such a set up is not common in statistical packages. A way used here to overcome this limitation is to sum the two dependent variables, which results in a 3D problem for parameter estimation.For processing the adsorption data of this thesis, C was chosen as the independent variable, because it facilitates better the mathematical representation. To define the independent variable two possibilities were
considered: the sum of
Qs
or the sum ofS.
The equations of the adsorptionmodel using PGA could therefore be written as:
QS,2 + QS,l
=9( Cl + C2 ) + kF,1C�l + kF,2C�2 + ks ( kF,1C�l )( kF,2C�2 )
Using SGA1:
QS,2 + QS,l
=9 (Cl + C2 ) + kF,1C�1 + kF,2C�2 + 9kcC1C2
S2 + Sl
=kF,lC�l + kF,2C�2 + 9kcC1C2
And for SGA2:
[5.20] [5.21] [5.22] [5.23] [5.24] [5.25]
These 3D transformed equations were then entered in the TableCurve™ package (Jandel Scientific) for parameterisation by the least squares method. The parameters found are presented in Table 5.2, and examples of the 3D graphs showing the performance of the fitting procedure are shown in Figure 5.18.
The use of a sum of two variables as the dependent variable is a problem for this procedure, since it may cause bias in the estimation process. Variations in the concentration of one of the solutes may be compensated by variations on the other. This is especially critical if the magnitude of one solute is much bigger than the other. As a consequence, the model's goodness of fit and the values for the estimate errors have to be treated with caution. These values should, however, be useful for a preliminary evaluation, especially for comparisons between the IPA approaches.
The use of
Qs
also smoothes the variations in the independent variable,the values of R2 (Table 5.2). Nonetheless the obtained parameter estimates were quite similar using either Qs or S.
Table 5.2. Parameters of the adsorption model obtained by least squares fitting using the three proposed IPA approaches and two different response variables (as explained in the text).
PGA SGA1 SGA2
QS,1 + QS,2 51 + 52 QS,1 + QS,2 51 + 52 QS,1 + QS,2 51 + 52 kF.l 3.425 (1.32) 3.425 (1.32) 8.670 (2.44) 8.702 (2.46) 5.528 (2.26) 5.601 (2.26) NI 0.3596 (0.04) 0.3596 (0.04) 0.2624 (0.10) 0.2604 (0.10) 0.1989 (0.12) 0.1947 (0.12) kF,2 10.802 (1.07) 10.802 (1.07) 11.058 (2.22) 11.043 (2.23) 10.319 (2.18) 10.248 (2.20) N2 0.4806 (0.04) 0.4806 (0.04) 0.5307 (0.08) 0.5310 (0.08) 0.5119 (0.08) 0.5143 (0.08) ks, kc or kr 0.0716 (0.03) 0.0716 (0.03) 0.0138 (0.002) 0.0138 (0.002) 1 .8477 (0.18) 1 .8503 (0.18) R2 0.9981 0.9697 0.9968 0.9483 0.9980 0.9683 St. Err. 3.1612 3.1612 4.1276 4.1276 3.2301 3.2301
Notes: The subscript index represents the solute, 1 for sulphate and 2 for calcium.
Values in brackets represent the error associated with the parameter estimate.
St. Err. is the model's standard error.
Units: kF.1: L kg-I; ks: kg mmol-1; kc :L mmol-1; k,: L kg-I; St. Err.: mmol kg-I.
Remarkably, all the three approaches resulted in similar values for the parameters of the Freundlich isotherm for calcium. Also the errors are relatively small compared to the other parameters. For sulphate the errors are bigger and the estimates are notably different when using different IPA approaches. This may be a result of a higher variability in the sulphate data. The IPA factor presented more variable responses for both the estimate, and its error. The overall results indicate PGA as the most suitable approach, with smaller estimated errors and higher R2 value.
Visual inspection of the 3D graphs shows very little differences between the approaches if using the sum of Qs (Figure 5.18). On the other hand, using
the sum of 5 reveals a distinct pattern for SGA1 as compared to PGA and
adsorption is apparent for SCAl (Figure 5.l8e). However, as shown in Table 5.2, this is not evident in the overall result, where SCAl showed the highest standard error. This feature illustrates the problem with evaluating the model when using the sum of the two solutes as the dependent variable.
70 60 �; �u "2 �!1,60 40 ], <t: 0 o !lO 30 " � '" � $.40 �o .Ji + ;' _\0 10 ':;' 20 10
(a)
(d)
100 j:' �o � "].QO l�O i -E, 100& +- �o � .!O 10(f)
Figure 5.18. 3D representations of the fitting of the adsorption model with IPA using the least squares method. Results for the two response variables and the three proposed IPA approaches are shown.