Nonequilibrium thermodynamics thermodynamic branch

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

Under those conditions P behaves as a Lagrangian in mechanics. Furthermore, as P is a nonnegative function for any positive value of the concentrations X,, by a theorem due to Lyapounov, the asymptotic stability of nonequilibrium steady states is ensured (theorem of minimum entropy production.1-23 These steady states are thus characterized by a minimum level of the dissipation in the linear domain of nonequilibrium thermodynamics the systems tend to states approaching equilibrium as much as their constraints permit. Although entropy may be lower than at equilibrium, the equilibrium type of order still prevails. The steady states belong to what has been called the thermodynamic branch, as it contains the equilibrium state as a particular case. 

Beyond the domain of validity of the minimum entropy production theorem (i.e., far from equilibrium), a new type of order may arise. The stability of the thermodynamic branch is no longer automatically ensured by the relations (8). Nevertheless it can be shown that even then, with fixed boundary conditions, nonequilibrium systems always obey to the inequality1... 

Fig. 2. Stability of the thermodynamic branch as a function of some parameter A that measures the chemical system s distance from equilibrium. In the linear range (i.e., for 0 sAssA ) the steady states belong to the thermodynamic branch (a) and are stable. Beyond this domain there may exist a threshold point Ac at which a new stable nonequilibrium branch of solutions (b) appears while the thermodynamic branch becomes unstable.

Fig. 23. Bifurcation diagram of KU2 in the case of Fig. 22. The lower branch of solutions corresponds to the thermodynamic branch. It tends to the asymptotic value K 12, which separates it from nonequilibrium types of solutions.

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

At this point the need arises to become more explicit about the nature of entropy generation. In the case of the heat exchanger, entropy generation appears to be equal to the product of the heat flow and a factor that can be identified as the thermodynamic driving force, A(l/T). In the next chapter we turn to a branch of thermodynamics, better known as irreversible thermodynamics or nonequilibrium thermodynamics, to convey a much more universal message on entropy generation, flows, and driving forces. 

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. 

If a nonequilibrium system consists of several flows caused by various forces, Eq. (3.174) may be generalized in the linear region of the thermodynamic branch (Figure 2.2), and we obtain... 

In far-ffom-equilibrium systems the loss of stability of the thermodynamic branch and the transition to a dissipative structure follows the same general features shown in our simple example. The parameter such as X corresponds to constraints—e.g. flow rates or concentrations maintained at a nonequilibrium value — that keep the system away from equilibrium. When X reaches a particular value, the thermodynamic branch becomes unstable but at the same time new solutions now become possible driven by fluctuations, the system makes a transition to one of the new states. As we did in section 18.4, let us specify the state of the system hy Xk,k = 1,2,. ..,n which in general may be functions of both position r and time t. Let the equation that describes the spatiotemporal evolution of the system be... 

This problem is related to the question of whether there is a minimum work input required before dispersion begins to take place. Or, in nonequilibrium terms What is the critical parameter at which bifurcation (Fig. 19.47) occurs, and what is the value needed to make the system leave the thermodynamic branch ... 

The statistical polymer method presented above still is convenient to the equilibrium model only. However, since that allows estimation of all additive parameters of branched polymers, we can evaluate thermodynamic functions which characterize not only equilibrium but also nonequilibrium situation. 

The thermodynamic basis of the calculation of the maximum possible work potential or chemical exergy of reversible and irreversible chemical reactions is explained and discussed. Combustion is asserted to be fundamentally irreversible. It is a nonequilibrium uncontrollable chain reaction with hot branches, in a cool milieu, and a limited work output proportional to Carnot efficiency x calorific value (Barclay, 2002). 

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