Mass transfer model equations system geometry

Abstract Chapter 5 provides an examination of the numerical solutions of the dyeing models that can be applied to different conditions. Numerical simulation of the system involves the use of Matlab software to solve systems of highly non-linear simultaneous coupled partial differential equations. The finite difference and finite element methods are introduced The partition of the fibrous assembly geometry into small units of a simple shape, or mesh, is examined. Polygonal shapes used to define the element are briefly described. The defined geometries, boundary conditions, and mesh of the system enable solutions to the equations of flow or mass transfer models. 

The electrical circuit presented in Figure 10.5 yields the impedance response equivalent to equation (10.49) for a single Faradaic reaction coupled with a mass transfer. This circuit is known as the Randles circuit. Such a circuit may provide a building block for development of circuit models as shown in Chapter 9 for the impedance response of a more complicated system involving, for example, coupled reactions or more complicated 2- or 3-dimensional geometries. 

The hydrodynamics of the experimental system can be described theoretically. Such approach is very important for correct interpretation of the experimental results, and for their extrapolation for the conditions not attainable in the existing experimental system. With the mathematical model the parametric study of the system is also possible, what can reveal the most important factors responsible for the occurrence of the specific transport phenomena. The model was presented in details elsewhere [2]. It was based on the equations of the momentum and mass transfer in the simplified two-dimensional geometry of the air-water-surfactant system. Those basic equations were supplemented with the equation of state for the phopsholipid monolayer. The resultant set of equations with the appropriate initial and boundary conditions was solved numerically and led to temporal profiles of the surface density of the surfactant, T [mol m ], surface tension, a [N m ], and velocity of the interface. Vs [m s ]. The surface tension variation and velocity field obtained from the computations can be compared with the results of experiments conducted with the LFB. 

The adsorber model comprises a system of (i) three parabolic partial differential equations for the mass transport of each single component coupled by both sorption isotherm equations and an expression for the temperature dependence of rate coefficients (ii) two differential equations for chemical reaction and (iii) two parabolic partial differential equations for heat transfer. Beside time, the model contains three spatial coordinates that refer to the interstitial column volume, the macropore volume and the micropore volume and that may be of different geometry. The solution of the problem for which a module-wise algorithm was developed, is described in detail in refs. [103,104]. 

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