Linear phenomenological equations

A method is described for fitting the Cole-Cole phenomenological equation to isochronal mechanical relaxation scans. The basic parameters in the equation are the unrelaxed and relaxed moduli, a width parameter and the central relaxation time. The first three are given linear temperature coefficients and the latter can have WLF or Arrhenius behavior. A set of these parameters is determined for each relaxation in the specimen by means of nonlinear least squares optimization of the fit of the equation to the data. An interactive front-end is present in the fitting routine to aid in initial parameter estimation for the iterative fitting process. The use of the determined parameters in assisting in the interpretation of relaxation processes is discussed. 

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

A more rigorous way to generalize Pick s law is to use phenomenological equations based on linear irreversible thermodynamics. In this treatment of an N-component system, the diffusive flux of component i is (De Groot and Mazur,... 

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...

In the absence of gradients of salt concentration and temperature, flows of water and electric current in bentonite clay are coupled through a set of linear phenomenological equations, derived from the theory of irreversible thermodynamics (Katchalsky and Curran, 1967), making use of Onsager s Reciprocal Relations (Groenevelt, 1971) ... 

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. 

States away from global equilibrium are called the thermodynamic branch (Figure 2.2). Systems not far from global equilibrium may be extrapolated around equilibrium state. For systems near equilibrium, linear phenomenological equations may represent the transport and rate processes. The linear nonequilibrium thermodynamics theory determines the dissipation function or the rate of entropy production to describe such systems in the vicinity of equilibrium. This theory is particularly useful to describe coupled phenomena, and quantify the level of coupling in physical, chemical, and biological systems without detailed process mechanisms. 

In some systems, the distance from equilibrium reaches a critical point, after which the states in the thermodynamic branch become metastable or unstable. This region is the nonlinear region where the linear phenomenological equations are not valid. We observe bifurcations and multiple solutions in this region. 

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. 

By introducing the linear phenomenological equations, into the entropy production relation, = %JX, we have... 

If we disregard the higher order terms, these expansions become linear relations, and we have the general type of linear phenomenological equations for irreversible phenomena... 

Example 3.11 Transformation of phenomenological equations dependent flows Transform the thermodynamic forces and flows when the forces are independent, while the flows are linearly dependent in a two-flow system 0 y./t +J2. 

The linear phenomenological equations in terms of the resistance coefficients are... 

On the other hand, we have the following linear phenomenological equation for chemical reaction i... 

We can compare these linear phenomenological equations with Eq. (3.277) to obtain the phenomenological coefficients... 

Based on the entropy production, linear phenomenological equations for an isothermal flow of substance / become... 

If a system is not far from global equilibrium, linear phenomenological equations represent the transport and rate processes involving small thermodynamic driving forces. Consider a simple transport process of heat conduction. The rate of entropy production is... 

Since this condition is satisfied for most systems, the linear phenomenological equations are satisfactory approximations for transport processes. 

For an elementary chemical reaction, the local entropy production and the linear phenomenological equation are... 

There is no definite sign for Eq. (3.317). When the generalized flows are expressed by linear phenomenological equations with constant coefficients obeying to the Onsager relations... 

For a set of linear phenomenological equations, consider the following potentials... 

Equations (3.331) and (3.332) indicate that the first derivatives of the potentials represent linear phenomenological equations, while the second derivatives are the Onsager reciprocal relations. 

Assuming that the linear phenomenological equations hold for a two flow coupled system... 

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. 

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... 

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... 

Since the system is isotropic and assuming locality in velocity space, and using the linear nonequilibrium formulations based on the entropy production relation in Eq. (7.198), we have the linear phenomenological equations... 

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