The Phenomenological Equations

Onsager (1931) used a set of equations that expresses in an explicit manner the linear dependence of the thermodynamic flows on the thermodynamic forces. These equations, known as the phenomenological equations, can be expressed as 

Note that the ath flow can be coupled to the /Jth force if the coupling coefficient Lap 0. Onsager suggested this linear law only for systems sufficiently close to equilibrium, where the thermodynamic forces are small. Such linear laws are well known in physics—for example, Ohm s law, Fourier s law, and Fick s law. 

Equation (13.4.1) is a matrix equation and can be easily inverted to give Xa = RafJf where Raf are coupling coefficients that give the forces as linear function of 

The overall symmetry of the system can be used to show that some coefficients in the L or R matrices are zero. If, for example, the force Xp is a vector quantity but the flow Ja is a scalar flow, the coefficient Lap must be a vector quantity. This is, however, impossible in an isotropic homogeneous system in the absence of external forces. Thus a scalar force cannot induce a vector flow and Lap = 0. An example is that of a mixture in which there are chemical reactions. According to the above, the chemical affinity, a scalar force, cannot induce a flow of matter Jj in any particular direction thus simultaneous diffusion and chemical reaction cannot be coupled. 

A general statement of this argument is that in an isotropic system flows and forces of different tensorial orders are not coupled. This is known as the Curie principle. Systems that are anisotropic often have some elements of symmetry which reduce the number of nonzero coefficients from the maximum of n2. To prove these relations one must apply the arguments of Chapter 11 involving parity, reflection symmetries, rotational symmetries, and time-reversal symmetries. 

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Onsager postulates [4, 5] the phenomenological equations for irreversible processes given by... 

The phenomenological equations proposed by Felix Bloch in 19462 have had a profound effect on the development of magnetic resonance, both ESR and NMR, on the ways in which the experiments are described (particularly in NMR), and on the analysis of line widths and saturation behavior. Here we will describe the phenomenological model, derive the Bloch equations and solve them for steady-state conditions. We will also show how the Bloch equations can be extended to treat inter- and intramolecular exchange phenomena and give examples of applications. 

The linearity of drug release was assessed by fitting the release data to the phenomenological equation [10] ... 

Figure 1.16 Relative complex viscosity ( > / f)ol) versus calculated conversion for polymerization at 120 and 150°C. Linear, straight-line fit to the phenomenological equation >/ / r 0 = exp(19.6 X) is also shown...

The rheokinetics of polycaprolactam polymerizing in the monomer shows that below 50 percent conversion, the relative complex viscosity versus conversion of the nylon 6 homopolymerization is defined by the phenomenological equation ri / t]Q = exp(19.6 X), where // is the complex viscosity of nylon 6 anionically polymerizing in its monomer, 0 is the viscosity of caprolactam monomer, and X is fractional conversion. 

In order to quantify diffiisional effects on curing reactions, kinetic models are proposed in the literature [7,54,88,95,99,127-133]. Special techniques, such as dielectric permittivity, dielectric loss factor, ionic conductivity, and dipole relaxation time, are employed because spectroscopic techniques (e.g., FT i.r. or n.m.r.) are ineffective because of the insolubility of the reaction mixture at high conversions. A simple model, Equation 2.23, is presented by Chem and Poehlein [3], where a diffiisional factor,//, is introduced in the phenomenological equation, Equation 2.1. 

II. The Strain Energy Density Function and the Phenomenologic Equations for Elasticity... 

What has been done so far is to take experimental laws and express them in the form of phenomenological equations, i.e., Eqs. (6.300) and (6.301). Just as the phenomenological equations describing the equilibrium properties of material systems constitute the subject matter of equilibrium thermodynamics, the above phenomenological equations describing the flow properties fall within the purview of nonequilibrium thermodynamics. In this latter subject, the Onsager reciprocity relation occupies a fundamental place (see Section 4.5.7). 

The phenomenological equation that relates the flux of material across the boundary to the concentration gradient at that location is given by Fick s first law. 

A simple model is the diode circuit with two metals with known work functions, as in fig. 16 in VI.9. The whole is in thermal equilibrium at temperature T ( Alkemade s diode ) ). The charge Q on the condenser obeys the phenomenological equation... 

However small the effect, it violates the second law of thermodynamics. An array of 1012 diodes would be able to run a flashlight. The answer to this paradox is that one cannot trust the phenomenological equation to the extent that one may use it for deducing a result that is itself of the order of the fluctuations. This example demonstrates the danger of adding a Langevin term to a nonlinear equation. ... 

The first group of theories was based on the superposition approximation (SA) used for truncation of an infinite hierarchy of equations for a reduced distribution function at the pair distribution level [239,240], It was generalized and applied to the reversible reactions by Lee and Karplus [139] and their successors [145,174,176]. After linearization over deviations from equilibrium [52,63], the theory became simpler and finally was recognized as one identical to IET, provided the reduction to the phenomenological equations is not done [31,175], This is why the linearized superposition approach (LSA) provides exactly the same kernel as IET. 

The phenomenological equation that determines the motion of the magnetic moment of a ferromagnetic sample is known generically as the Landau-Lifshitz-Gilbert equation and has two basic modifications. The first form... 

Figure 1 shows a comparison, published by Mori and Ototake [13], of the experimental dependences of viscosity on concentration of dispersions of solid particles based on the data of Vand [34], Robinson [12], Orr and Blocker [5], Dalla Valle and Orr [17] with the theoretical equations based on the hydrodynamic approach used by Einstein (1), Simha (30), Vand (31), Roscoe (44) and the phenomenological equation of Mori and Ototake (14). A more complicated form of the theoretical dependence, naturally makes it possible to describe experimental results over a wider range, but for concentrated dispersions most of theoretical equations remain inapplicable. 

We tried to compare the phenomenological equations (81) and (82) with the experimental results obtained in studies on the slow (creeping) flow of highly-concentrated disperse systems containing a polyfractional filler [98]. 

Equation (3.141) identifies the following flows and forces to be used in the phenomenological equations. 

These equations are called the phenomenological equations, which are capable of describing multiflow systems and the induced effects of the nonconjugate forces on a flow. Generally, any force Xt can produce any flow./, when the cross coefficients are nonzero. Equation (3.175) assumes that the induced flows are also a linear function of non-conjugated forces. For example, ionic diffusion in an aqueous solution may be related to concentration, temperature, and the imposed electromotive force. 

For a two-flow system, we have the phenomenological equations in terms of the flows... 

Equation (3.247) shows that Onsager s reciprocal relations are satisfied in the phenomenological equations (Wisniewski et al., 1976). 

Transform the phenomenological equations when the flows and forces are linearly dependent and the forces are linearly dependent 0. /, I z/2 and 0 = Xt +yX2. 

The phenomenological equations relating the flows and forces defined by Eq. (6.155) are... 

Using a dissipation function or entropy production equation, the conjugate flows and forces are identified and used in the phenomenological equations for simultaneous heat and mass transfer. Consider the heat and diffusion flows in a fluid at mechanical equilibrium not undergoing a chemical reaction. The dissipation function for such a system is... 

We may consider the phenomenological equations for the n + 1 vector flows of J" and j and n I 1 forces of V(l/7 ) and Vx/u. Assuming linear relations between the forces and the flows, we have the following phenomenological equations... 

The heat of transport can be used in the phenomenological equations to eliminate the coefficients or Ljq. After introducing Eq. (7.54) into Eq. (7.52), we obtain the expression for heat flow in terms of the heat of transport... 

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