Nonequilibrium global

For many years, the thermodynamic description of macromolecules lagged behind other materials because of the unique tendency of pol5nneric systems to assume nonequilibrium states. Most standard sources of thermodynamic data are, thus, almost devoid of polymer information (1-7). Much of the aversion to include polymer data in standard reference sources can be traced to their nonequilibrium nature. In the meantime, polymer scientists have learned to recognize equilibrium states and utilize nonequilibrium states to explore the history of samples. For a nonequilibrium sample it is possible, for example, to thermally establish how it was transferred into the solid state (determination of the thermal and mechanical history). More recently, it was discovered with the use of temperature-modulated differential scanning calorimetry (TMDSC) that within the global, nonequilibrium structure of semicrystalline polymers, locally reversible melting and crystallization processes are possible on a nanophase level (8). 

Where p defines the shape of the hole energy spectrum. The relaxation time x in Equation 3 is treated as a function of temperature, nonequilibrium glassy state (5), crosslink density and applied stresses instead of as an experimental constant in the Kohlrausch-Williams-Watts function. The macroscopic (global) relaxation time x is related to that of the local state (A) by x = x = i a which results in (11)... 

The latter condition is commonly known as microscopic reversibility or local detailed balance. This property is equivalent to time reversal invariance in deterministic (e.g., thermostatted) dynamics. Although it can be relaxed by requiring just global (rather than detailed) balance, it is physically natural to think of equilibrium as a local property. Microscopic reversibility, a common assumption in nonequilibrium statistical mechanics, is the crucial ingredient in the present derivation. 

The Taylor method. Krishnamurthy and Taylor (88, 89) present and test a nonequilibrium model which includes rate equations for mass transfer, and sometimes reaction, among the traditional MESH equations. These include individual mass and energy balances in the vapor and the liquid and across the interface. An equilibrium equation exists for the interface only. The solution method for these equations is the same as that of the block-banded matrices of the global Newton methods and the style of the method is similar to the Naphtali-Sandholm (Sec. 4.2.9). 

The number of equations, M5C + 1), for a large number of trays and components, can be excessive. The global Newton method will suffer from the same problem of requiring initial values near the answer. This problem is aggravated with nonequilibrium models because of difficulties due to nonideal if-values and enthalpies then compounded by the addition of mass transfer coefficients to the thermodynamic properties and by the large number of equations. Taylor et al. (80) found that the number of sections of packing does not have to be great to properly model the column, and so the number of equations can be reduced. Also, since a system is seldom mass-transfer-limited in the vapor phase, the rate equations for the vapor can be eliminated. To force a solution, a combination of this technique with a homotopy method may be required. 

Nonequilibrium methods (Sec. 4,2.13) tend to be global Newton methods extended to solve mass-transfer-inhibited systems. Nonequilibrium methods are not yet completely extended to more common systems, but these methods should see the greatest amount of development in distillation modeling. 

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 nonequilibrium systems, spontaneous decaying phenomena toward equilibrium take place. When systems are in the vicinity of global equilibrium, linear relations exist between flows Jt and thermodynamic driving forces A). 

The linear nonequilibrium thermodynamics approach can provide a quantified description of the fully coupled phenomena for systems in the vicinity of global equilibrium. 

For nonequilibrium systems far from global equilibrium, the second law does not impose the sign of entropy variation due to the terms djS and d S, as illustrated in Figure 12.2. Therefore, there is no universal Lyapunov function. For a multicomponent fluid system with n components, entropy production in terms of conjugate forces Xu flows Jj, and / number of chemical reactions is... 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

The problem requires analyses of phase-space structures in systems with many degrees of freedom. In particular, appreciating the global structure of the phase space becomes essential for our understanding of reactions under nonequilibrium conditions. In order to make this point clear, we briefly summarize the present status of the study. 

---