Simple Process Models


Simple Process Models

Simple processes in chemical engineering concern hydrodynamic, diffusion, heat conduction, adsorption and chemical processes.  These are typical non equilibrium processes and the relevant mathematical descriptions concern quantitatively their kinetics.  This gives a ground to utilize the laws of irreversible thermodynamics as mathematical structures building the models of the simple processes.  The quantitative description of irreversible processes depends on the level of the process description.  From such a point of view, one can define three basic levels of description–thermodynamic, hydrodynamic and boltzmann levels.  These different levels if process descriptions form a natural hierarchy.  Thus, going up from one level to the next, the description becomes richer, i.e., more detailed.  This approach allows the kinetic parameters defined at a lower level to be described through relevant kinetic parameters at an upper level.  The thermodynamic level utilizes quantitative descriptions through extensive variables.  If there is a distributed space, the volume must be represented as a set of unit cells, where the variables are the same but have different values in different cells.  The hydrodynamic is the next level, where a new extensive variable participates in the processes. 

This variable is the momentum.  Therefore, the hydrodynamic level of description can be considered as a generalization of the lower, thermodynamic level.  Here, the extensive variables are mass density, momentum, and energy.  In the isolated systems they are conserved and the conservation laws of mass, momentum and  energy are used.  The Boltzmann level is the next upper level of description and concerns only the mass density as a function of the distribution of the molecules in space and their momenta.  The kinetics of irreversible processes employs mathematical structures following from Onsanger’s linear principle.  According to them, the mean value of the time derivatives of the extensive variables and the mean derivatives of their adjoined intensive variables from the equilibrium are expressed through linear relationships.  The principle is valid close to the equilibrium and the coefficients of the proportionality are the kinetic constants.  When the process takes place far from equilibrium, the kinetic constants become kinetic complexes depending on the corresponding intensive variables (in the case of fusion of two identical systems, the extensive variables double, whereas the intensive variables remain the same).

The hydrodynamic level is widely applicable in the mechanics of continua.  Here, the material point corresponds to a sufficient volume of the medium that is simultaneously sufficiently small with respect to the entire volume under consideration and at the same time sufficiently large with respect to the intermolecular distances of the medium.  Modeling in chemical engineering utilizes mathematical structures provided by the mechanics of the continua.  The principal reason for this is the fact that these structures sufficiently well describe the phenomena in detail.  Moreover, they employ physically well defined models with a low number of experimentally defined parameters.

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