| transTrend {transmem} | R Documentation |
Fits trend equations that model transport profiles
Description
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.
Usage
transTrend(trans, model = "paredes", eccen = 1)
Arguments
trans |
Data frame with the complete transport information of
interest species. Must be generated using
|
model |
Model to be used in the regression. Default to
|
eccen |
Eccentricity factor ( |
Details
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.
Value
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.
Author(s)
Cristhian Paredes, craparedesca@unal.edu.co
Eduardo Rodriguez de San Miguel, erdsmg@unam.mx
References
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 Autónoma de México, México City, México, 2020.