Given a transport profile dataset, the results may be studied and compared in terms of empirical functions that describe the transport process in terms of regression parameters that can be asociated with the performance of the membrane system. The parameters are obtained by non-linear regression and are independent for each solution at both sides of the membrane. This is particularly useful when performing system optimizations since the parameters can be used as response variables depending on the optimization goal.
transTrend(trans, model = "paredes", eccen = 1)Data frame with the complete transport information of
interest species. Must be generated using
conc2frac. This is the only non-optional
parameter.
Model to be used in the regression. Default to
'paredes' but 'rodriguez' also allowed.
See details.
Eccentricity factor (\(\gamma\)) for the model when
model is set to 'paredes'.
A list of 4 or 5 components (depending on the model chosen) with the regression information for each phase, the eccentricity factor (only in Paredes model), the name of the model used, and the sumarized results of the regression: \(G_{feed}\) and \(G_{strip}\) values for the Rodriguez model or summarized \(\alpha\) and \(\beta\) parameters with asocciated uncertainty for the Paredes model.
Two empirical equations have been implemented in the function. In the
'rodriguez' model (Rodriguez de San Miguel et al., 2014), the
fractions (\(\Phi\)) in feed or
strip phases as a function of time (\(t\)) are fitted to
$$\Phi(t)=Ae^{-t/d}+y_0$$ where \(A\), \(d\) and \(y_0\) are the
parameters to be found. In this model, parameter \(d\) determines the
steepness of the species concentration change in time, \(y_0\) reflects
the limiting value to which the profiles tend to at long pertraction
times and \(A\) is not supposed to play an important role in the
transport description. The parameters of each phase are summarized in
the functions \(G_{feed}\) and \(G_{strip}\) for the feed and
strip phases: $$G_{feed}=\frac{1}{y_0d},\qquad G_{strip}=\frac{y_0}{d}$$
The bigger each \(G\) function, the better the transport process.
In the 'paredes' model (Paredes and Rodriguez de San Miguel, 2020),
the transported fractions to the strip solution and from the feed solution
are adjusted to the equations:
$$\Phi_s(t)=\frac{\alpha_s t^\gamma}{\beta_s^{-1}+t^\gamma}$$
$$\Phi_f(t)=1-\frac{\alpha_f t^\gamma}{\beta_f^{-1}+t^\gamma}$$
respectively. In those equations, adjustable parameters \(\alpha\) and
\(\beta\) relates the
maximum fraction transported at long pertraction times and the steepness
of the concentration change, respectively. \(\gamma\) is an excentricity
factor to improve the adjustment and does not need to be changed for
systems under similar conditions. The subscripts \(s\) and \(f\)
means strip and feed phases, respectively.
The later model has the disadvantage over the former that the equation to use depends on the phase to be modeled but has the great advantage that if no significant accumulation is presented in the membrane, the parameters \(\alpha\) and \(\beta\) should be quite similar for both phases and a consensus value can be obtained in various simple ways, while the other model yields quite diferent parameters for each phase. Paredes parameters are combined by using meta-analysis tools that consider the associated uncertainty of each one due to lack of fit to get summarized, lower-uncertainty results. Besides, once the \(\gamma\) parameter has been chosen, the later model uses only two parameters and while comparing models with similar performance, the simpler the better.
E. Rodriguez de San Miguel, X. Vital, J. de Gyves, Cr(vi) transport via a sup ported ionic liquid membrane containing cyphos il101 as carrier: System analysis and optimization through experimental design strategies, Journal of Hazardous Materials 273 (2014) 253 - 262. doi:10.1016/j.jhazmat.2014.03.052.
C. Paredes, E. Rodriguez de San Miguel, Polymer inclusion membrane for the recovery and concentration of lithium from seawater. Master thesis, Universidad Nacional Aut<U+00F3>noma de M<U+00E9>xico, M<U+00E9>xico City, M<U+00E9>xico, 2020.