The Coupled Phenomena

The phenomenological coefficients are important in defining the coupled phenomena. For example, the coupled processes of heat and mass transport give rise to the Soret effect (which is the mass diffusion due to heat transfer), and the Dufour effect (which is the heat transport due to mass diffusion). We can identify the cross coefficients of the coupling between the mass diffusion (vectorial process) and chemical reaction (scalar process) in an anisotropic membrane wall. Therefore, the linear nonequilibrium thermodynamics theory provides a unifying approach to examining various processes usually studied under separate disciplines. 

Simultaneous heat and mass transfer plays an important role in various physical, chemical, and biological processes hence, a vast amount of published research is available in the literature. Heat and mass transfer occurs in absorption, distillation extraction, drying, melting and crystallization, evaporation, and condensation. Mass flow due to the temperature gradient is known as the thermal diffusion or Soret effect. Heat flow due to the isothermal chemical potential gradient is known as the diffusion thermoeffect or the Dufour effect. The Dufour effect is characterized by the heat of transport, which represents the heat flow due to the diffusion of component / under isothermal conditions. Soret effect and Dufour effect represent the coupled phenomena between the vectorial flows of heat and mass. Since many chemical reactions within a biological cell produce or consume heat, local temperature gradients may contribute in the transport of materials across biomembranes. 

These equations obey the Onsager reciprocal relations, which state that the phenomenological coefficient matrix is symmetric. The coefficients Lqq and Lu arc associated with the thermal conductivity k and the mutual diffusivity >, respectively. In contrast, the cross coefficients Llq and Lql define the coupling phenomena, namely the thermal diffusion (Soret effect) and the heat flow due to the diffusion of substance / (Dufour effect). 

These equations display the spatial order with the thermodynamically coupled heat and mass flows. Here, the coupling between chemical reactions and transport processes of heat and mass is excluded. The analysis of reaction-diffusion systems would be more complete if we consider heat effects and coupling among fluxes of mass and heat. The nonequi-librium thermodynamics approach is useful for incorporating the coupling phenomena into reaction-diffusion systems. 

These equations describe the coupled phenomena between diffusion and viscous stresses, existing, for example, in diffusion of small molecules in polymer matrix. Other possible couplings occur in shear-induced diffusion and shear-induced separation. 

The electrodes with bubbles evolving seem to lead in the confined electrochemical cell to a hydrodynamic puffing phenomenon , clearly three dimensional and unsteady in the cell. Then all the coupled phenomena, transports and reactions are affected and should actually be three dimensional and unsteady. 

It is common to treat the semiconductor-electrolyte interface in terms of charge and current density boundary conditions. The total charge held within the electrolytic solution and the interfacial states, which balances the charge held in the semiconductor, is assumed to be constant. This provides a derivative boundary condition for the potential at the interface. The fluxes of electrons and holes are constrained by kinetic expressions at the interface. The assumption that the charge is constant in the space charge region is valid in the absence of kinetic and mass-transfer limitations to the electrochemical reactions. Treatment of the influence of kinetic or mass transfer limitations requires solution of the equations governing the coupled phenomena associated with the semiconductor, the electrolyte, and the semiconductor-electrolyte interface. 

The equations above show that any flow is caused by aU the forces and any force is the result of all the flows present in the system. The coefficients and Kgi are called the phenomenological coefficients. The coefficients L,vt the conductance coefficients and the resistance coefficients. The straight coefficients with the same indices relate the conjugated forces and flows. The cross-coefficients with i k represent the coupling phenomena. 

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