Phenomenological, coefficients relations

Here, Lu is a phenomenological coefficient relating the driving force to the corresponding flux, the subscripts i and k refer to various components in the system. 

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. 

Some of the phenomenological coefficients relating forces and fluxes are already familiar from less general treatments of the subject. For example, the phenomenological coefficient relating a concentration gradient and a mass transfer flux is the diffusion coefficient. Other phenomenological coefficients are related to the ionic mobility, the coefficient of thermal conductivity, and the solvent viscosity. These are discussed in more detail later in this chapter. 

The phenomenological coefficient relating the current density and the macroscopic potential gradient is the conductivity due to the ion, k,. Thus, one has the general relationship... 

Phenomenological coefficients relating flux and force are the diffusion coefficient (D. Pick s law), permeability coefficient (Lp. Darcy s law), thermal diffusivity (X. Fourier . law), kinematic viscosity = tn- p). N ton s law), and electrical conductivity (1/R. Ohm s law). Phenomenological equations are summarised in table 1.7. 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

Thus, the perhaps unfamiliar constitutive relations (2.23)-(2.25) yield familiar results when the fields are time harmonic moreover, because of (2.26) and (2.27), physical meaning can now be attached to the phenomenological coefficients even for arbitrarily time-dependent fields. 

Since the entropy production is positive, the transport coefficients Lik must satisfy the relation TAA-Lhh>LhA-TAh [S.R. de Groot, P. Mazur (1962)]. This restricts the range for the charges of transport to aA-Oh< 1, see Eq. (8.56) ff. We should also add that whereas the Ly are phenomenological coefficients appropriate for the description of the experiments on transport, the ly relate directly to the SE s (Eqn. (8.28)) and can be derived from lattice dynamics based theoretical calculations. 

Although irreversible thermodynamics neatly defines the driving forces behind associated flows, so far it has not told us about the relationship between these two properties. Such relations have been obtained from experiment, and famous empirical laws have been established like those of Fourier for heat conduction, Fick for simple binary material diffusion, and Ohm for electrical conductance. These laws are linear relations between force and associated flow rates that, close to equilibrium, seem to be valid. The heat conductivity, diffusion coefficient, and electrical conductivity, or reciprocal resistance, are well-known proportionality constants and as they have been obtained from experiment, they are called phenomenological coefficients Li /... 

In this equation, just as in Newton s law adapted for friction, the reciprocal of the phenomenological coefficient Ln has been introduced and acts as a friction coefficient, a resistance. Recalling the relations... 

Irreversible processes are driven by generalized forces, X, and are characterized by transport (or Onsager) phenomenological coefficients, L [21,22], where these transport coefficients, Lip are defined by linear relations between the generalized flux densities,./, which are the rates of change with time of state variables, and the corresponding generalized forces X . 

Klier (1972) deduced that for 0.6 < p < 1, which corresponds to 0.13 > F(P) > 0, the Kubelka-Munk absorption coefficient should be nearly proportional to the true absorption coefficient. Deviations from the proportionality up to a factor of two occurred for lower reflectance values. In the range p > 0.6, the Kubelka-Munk function should be nearly proportional to the absorber concentration. Through comparison with the radiative-transfer equation formulated by Chandrashekhar (1960), Klier related the phenomenological coefficients to the true absorption and scattering coefficients a and [Pg.142]

The form of the expressions for the rate of entropy production does not uniquely determine the thermodynamic forces or generalized flows. For an open system, for example, we may define the energy flow in various ways. We may also define the diffusion in several alternative ways depending on the choice of reference average velocity. Thus, we may describe the flows and the forces in various ways. If such forces and flows, which are related by the phenomenological coefficients obeying the Onsager relations, are subjected to a linear transformation, then the dissipation function is not affected by that transformation. 

Onsager s reciprocal relations state that, provided a proper choice is made for the flows and forces, the matrix of phenomenological coefficients is symmetrical. These relations are proved to be an implication of the property of microscopic reversibility , which is the symmetry of all mechanical equations of motion of individual particles with respect to time t. The Onsager reciprocal relations are the results of the global gauge symmetries of the Lagrangian, which is related to the entropy of the system considered. This means that the results in general are valid for an arbitrary process. 

In a two-flow system, there are two degrees of freedom in choosing the phenomenological coefficients. With the linear relations of flows and forces, there is one degree of freedom that is I.]2 = hi, and L22 is proportional to V... 

Therefore, the diffusion coefficient is related to the phenomenological coefficient by... 

For a three-component diffusion system, derive the relations between the diffusion coefficients and the phenomenological coefficients under isothermal conditions. 

Equation (6.171) shows the direct relation between the electrical conductance of the solution and the phenomenological coefficient. Similar relations are obtained by measuring the fraction of the total current that is carried by each ion, also under the conditions V/li, = 0. This fraction is called the Hittorf transference number (t,) and is expressed by... 

The electrical conductance, transference number, and diffusion coefficient provide the three relations from which the phenomenological coefficients can be determined, and for a monomonovalent salt we have... 

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