# transTrend: Fits trend equations that model transport profiles In transmem: Treatment of Membrane-Transport Data

## 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

 1 transTrend(trans, model = "paredes", eccen = 1) 

## Arguments

 trans Data frame with the complete transport information of interest species. Must be generated using conc2frac. This is the only non-optional parameter. model Model to be used in the regression. Default to 'paredes' but 'rodriguez' also allowed. See details. eccen Eccentricity factor (γ) for the model when model is set to 'paredes'.

## Details

Two empirical equations have been implemented in the function. In the 'rodriguez' model (Rodriguez de San Miguel et al., 2014), the fractions (Φ) in feed or strip phases as a function of time (t) are fitted to

Φ(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:

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:

Φ_s(t)=\frac{α_s t^γ}{β_s^{-1}+t^γ}

Φ_f(t)=1-\frac{α_f t^γ}{β_f^{-1}+t^γ}

respectively. In those equations, adjustable parameters α and β relates the maximum fraction transported at long pertraction times and the steepness of the concentration change, respectively. γ 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 α and β 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 γ 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 α and β 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.

transmem documentation built on July 1, 2020, 10:38 p.m.