Phenomenology mass flows

It is well known that a flow-equilibrium must be treated by the methods of irreversible thermodynamics. In the case of the PDC-column, principally three flows have to be considered within the transport zone (1) the mass flow of the transported P-mer from the sol into the gel (2) the mass flow of this P-mer from the gel into the sol and (3) the flow of free energy from the column liquid into the gel layer required for the maintenance of the flow-equilibrium. If these flows and the corresponding potentials could be expressed analytically by means of molecular parameters, the flow-equilibrium 18) could be calculated by the usual methods 19). However, such a direct way would doubtless be very cumbersome because the system is very complicated (cf. above). These difficulties can be avoided in a purely phenomenological theory, based on perturbation calculus applied to the integrated transport Eq. (3 b) of the PDC-column in a reversible-thermodynamic equilibrium. 

Another well-known example is the coupling between mass flow and heat flow. As a result, an induced effect known as thermal diffusion (Soret effect) may occur because of the temperature gradient. This indicates that a mass flow of component A may occur without the concentration gradient of component A. Dufour effect is an induced heat flow caused by the concentration gradient. These effects represent examples of couplings between two vectorial flows. The cross-phenomenological coefficients relate the Dufour and Soret effects. In order to describe the coupling effects, the thermal diffusion ratio is introduced besides the transport coefficients of thermal conductivity and dififusivity. 

Here, Jq is the total heat flow, J, the mass flow of component i, and Jrj the reaction rate (flow) of reaction j. For chemical reactions, linear phenomenological equations are... 

For a binary fluid at mechanical equilibrium and for diffusion based on the mass-average velocity, we can now establish a set of phenomenological equations (Eqs. 7.6 and 7.7) with nonvanishing cross coefficients, and hence represent the coupled heat and mass flows... 

After identifying the conjugate forces and flows, for small forces of AT and Afi, the heat flow and mass flows may be represented by the following linear phenomenological equations... 

The linear phenomenological equations help determine the degree of coupling between a pair of flows the degree of coupling between heat and mass flows qSq and between the chemical reaction and the transport process of heat... 

Example 9.10 Chemical reaction velocity coupled to heat flow In this case, LSl and LlS vanish. Still, heat and mass flows are coupled. The new phenomenological equations are... 

Level flow occurs at zero load, which is at Afi = 0. At level flow, the mass flow is induced by AT and the phenomenological equation becomes... 

Tables 7.9 and 7.10 show the phenomenological coefficients for the heat and mass flows for the ternary mixture of toluene (1)—chlorobenzene (2)—bromobenzene (3).

The above phenomenological equations represent coupled heat and mass flows with chemical reaction in an anisotropic medium. The phenomenological equations above in vector form are... 

The constant term of the development is omitted because every flow is cancelled out when the forces are equal to zero. The quantities L, L ,iw... are termed the phenomenological coefficient of first, second order etc. In equation [4.10], the existence of combined effects derived from the union of processes apparently independent of each other is included. Specific information about equation [4.10] has been obtained by historic experience. In 1811, Fourier observed that in the first approximation, a heat flow is hnearly dependent on the temperature gap. Subsequently, Fick emphasised that the longitudinal mass flow is proportional to the difference of concentration. At the same time. Ohm established that an electric flow in a conductor depends linearly on the corresponding forces. All these observations are evidence that in a situation close to equilibrium, flows could be assumed to be proportional to the forces, so equation [4.10] could be approximated to ... 

The surface mobility coefficient ms is a phenomenological parameter that represents the net mass flow per unit gradient in surface chemical potential. The physical mechanisms of mass transport underlying this parameter are not well understood quantitatively, but some dependencies can be surmised. Surely, mobility must depend on temperature, binding energy and concentration of the diffusing species. A qualitative argument that leads to an estimate of the mobility coefficient proceeds as follows. 

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