Minimum entropy production

The value of P will be minimal if its variation is equal to zero 

The variation considered in Eq. (3.329) may be subject to various constraints. For example, the flows /, may vary when the forces Xt remain constant. It is also possible that the thermodynamic force may change while the flow remains the same, or they both may change. 

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. 

For an elementary volume, minimum entropy productions under various constraints are 

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Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

In an important paper (TNC.l), they offered for the first time an extension of nonequilibrium thermodynamics to nonlinear transport laws. As could be expected, the situation was by no means as simple as in the linear domain. The authors were hoping to find a variational principle generalizing the principle of minimum entropy production. It soon became obvious that such a principle cannot exist in the nonlinear domain. They succeeded, however, to derive a half-principle They decomposed the differential of the entropy production (1) as follows ... 

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... 

Horiuti Nakamura [75] have considered the possibility that there may be more than one set of stoicheiometric numbers leading to an overall reaction between the species s s. They find that the number of such reaction routes is the row nullity of the matrix [a ]. Nakamura has also considered the affinity of the overall reaction near equilibrium [79] and with Yamazaki [20] has related this to the theorem of minimum entropy production. 

Two important points need to be mentioned. First, as the mole fraction, y, approaches the equilibrium mole fraction, y g, the integrand approaches zero. Thus the point of minimum entropy production coincides with that of minimum reflux. Second, y > y guarantees that the argument of the logarithm cannot be less than unity, which means that Sp > 0. Finally Equation 24 is only applicable to cases wherein the diffusion processes represent the dominant mechanism for entropy production. 

Figure 2 Isolated system (is) and open subsystem (os) with minimum entropy production.

It should also be noted that the Prigogine theorem on the minimum entropy production is applicable to the circuit as a whole and for its individual branches (open subsystems). Actually, the maximum amount of entropy is formed in the environment owing to heat transfer to it from the hydraulic circuit. In the circuit itself the energy imparted to the fluid is entirely spent on its motion along the branches, i.e., on performance of effective work, and the entropy production at given conditions of interaction with the environment takes its minimal value equal to zero. The minimality of AS/ was shown in (Gorban et al., 2001, 2006). 

Example 3.17 Minimum entropy production in a two-flow system Determine the conditions for minimum entropy production for a two-flow system. 

Equation (3.340) yields a paraboloid-like change of dissipation with respect to forces A, andX2, as seen in Figure 3.4. The system tends to minimize the entropy and eventually reaches zero entropy production if there are no restrictions on the forces. On the other hand, if we externally fix the value of one of the forces, for example, A 2 = X20, then the system will tend toward the stationary state characterized by the minimum entropy production at X2 = X20. The system will move along the parabola of Figure 3.4 and stop at point [Pg.147]

Example 3.18 Minimum entropy production in an elementary chemical reaction system Consider a monomolecular reaction, for example, the following isomerization reaction. 

In this open reaction system, the chemical potentials of reactant R and product B are maintained at a fixed value by an inflow of reactant R and an outflow of product B. The concentration of intermediate X is maintained at a nonequilibrium value, while the temperature is kept constant by the reaction exchanging heat with the environment. Determine the condition for minimum entropy production. 

Example 3.19 Minimum energy dissipation in heat conduction Use the minimum entropy production principle to derive the relation for nonstationary heat conduction in an isotropic solid rod. 

Example 3.20 Minimum entropy production in electrical circuits Determine the conditions that minimize the entropy generation in electrical circuits with n elements connected in series. Assume that the voltage drop across the circuit is kept constant. 

By setting (i.S prod/Reynolds number, and hence the minimum entropy production at known values q and./,. For a laminar flow over a plate, we have... 

Another consequence of Eq. (4.91) is that if we arrange the n subsystems in time instead of in space, then the collection of subsystems constitutes the reaction path of a batch reactor where Vk is the volume of subsystem k. For a specified conversion and time, we should minimize the sum of Jk(AGk/T)Vk. This minimization leads to results similar to Eq. (4.91), and supports the principle of equipartition of forces. Hence, for a given total conversion and reaction time, minimum entropy production results when the driving force A GIT is equal in all n time intervals. Similarly, the conversion is maximum for a given entropy production and reaction time when the driving forces are uniform. 

In general, the methods of modeling, analysis, and optimization in engineering begin with deciding on the system geometry, architecture, and components, and the manner in which the components are connected. Engineering analysis involves the mathematical description of the conceptualized system and its performance. Finally, optimization leads to the most favorable conditions for maximum performance (e.g., minimum entropy production or minimum cost). 

The entropy production is a function of the temperature field. Then, the minimization problem is to obtain the temperature distribution T x) corresponding to a minimum entropy production using the following Euler-Lagrange equation ... 

The above equation shows that the steady state is less dissipative for a specified duty in a finite time. This conclusion is in line with the minimum entropy production principle of Prigogine. 

At stationary states, the minimum entropy production ridVd I, = 0 would lead to Jn = Jm, and the reaction velocities will be equal to each other as in Eq. (8.157). 

Example 8.12 Minimum entropy production Consider the following synthesis reaction... 

Minimum exergy loss or minimum entropy production at stationary state provides a general stability criterion. There are two important steady states identified in the cell static head (sh) and level flow (If). At the static head, where ATP production is zero since. /p = 0, the coupling between the respiratory chain and oxidative phosphorylation maintains a phosphate potentialXp, which can be obtained from Eq. (11.151) as (A p)sh = - qXJZ, and the static head force ratio xsh becomes xsh = q. The oxygen flow./, at the static head is obtained from Eqs. (11.151) and (11.152)... 

The first summation term on the right is the minimum entropy production corresponding to the stationary state. The second sum on the right is zero, according to the Onsager reciprocal relations and the Prigogine principle. Therefore, we have... 

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. 

This equation is the result of minimum entropy production at stable steady state and shows that if the system is disturbed by a small perturbation, it will return to the steady state. 

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. 

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