Modeling the extraction

The extraction of substances from solid substrates with supercritical solvents can be analyzed and modeled in a simple way by considering only the medium values and by determination of unknown coefficients by fitting to the extraction curve and a mass balance [1]. This approach results in simple equations that can represent parts of the extraction curve sufficiently, but fail for others, especially during the first part of the extraction. If the process is to be modeled more accurately, the analysis is far more complex and beyond the scope of this chapter. Nevertheless, some parameters determining the extraction and influencing the result are listed below together with the description of a simplified model that may provide some insight into the applied methods. 

The extraction system involves a bulk solid phase and a fluid phase. The fluid phase comprises the supercritical solvent and the dissolved extract. The solid phase remains within the extraction vessel and the fluid phase is passed through the extraction vessel. Mass transport occurs between the two phases. Any of the following parameters that can influence the process may be of interest. 

Realistically, this large number of parameters cannot be treated within a practical model. 

The goal of the modeling procedure is therefore to obtain a quantitative representation of the process with a simple system of equations and just a few physically meaningful parameters [1]. In such a model, the following parameters are usually considered sufficient to calculate the course and result of an extraction  

Experimental extraction curves can be represented by this type of model, by fitting the kinetic coefficients (mass transfer coefficient to the fluid, effective transport coefficient in the solid, effective axial dispersion coefficient representing deviations from plug flow) to the experimental curves obtained fi om laboratory experiments. With optimized parameters, it is possible to model the whole extraction curve with reasonable accuracy. These parameters can be used to model the extraction curve for extractions in larger vessels, such as from a pilot plant. Therefore, the model can be used to determine the kinetic parameters from a laboratory experiment and they can be used for scaling up the extraction. 

---

The cyclic structures are retained in solution [16, 61, 64] and such dimers must be considered in modelling the extraction equilibria for the recovery of copper using the commercial extractants shown in Table l.[61, 65]... 

In order to develop a continuous separation process, Kataoka et al. [54] simulated permeation of metal ion in continuous countercurrent column. They developed the material balance equation considering back mixing only in the continuous phase and steady-state diffusion in the dispersed emulsion drops which is similar to the Hquid extraction situation. Bart et al. [55] also modeled the extraction of copper in a continuous countercurrent column. They considered only the continuous phase back mixing in the model and assumed that the reaction between copper ions and carrier is slow, so that the differential mass balance equation for external phase in their model is... 

This section describes a methodical procedure that allows reliability issues to be approached efficiently. MEMS reveal specific reliability aspects, which differ considerably from the reliability issues of integrated circuits and macroscopic devices. A classification of typical MEMS-failure modes is given, as well as an overview of lifetime distribution models. The extraction of reliability parameters is a Tack of failures situation using accelerated aging and suitable models. In a case study, the implementation of the methodology is illustrated with a real-fife example of dynamic mechanical stress on a thin membrane in a hot-film mass-airflow sensor. 

Up to now some studies of modeling the extraction process have been published (for example [10-14]). The kinetic modeling of chemical reactions is a special challenge, because the properties of supercritical fluids, changing with temperature and density, may influence the reaction rate of a selected reaction step. 

Luetge, C., Reiss, I., Schleussinger, A. and Schultz, S. (1994) Modeling the extraction of perylene from spiked soil material by dense carbon dioxide, J. Supercrit. Fluids 7, 265-274... 

---