Continuous system entropy production

A reaction at steady state is not in equilibrium. Nor is it a closed system, as it is continuously fed by fresh reactants, which keep the entropy lower than it would be at equilibrium. In this case the deviation from equilibrium is described by the rate of entropy increase, dS/dt, also referred to as entropy production. It can be shown that a reaction at steady state possesses a minimum rate of entropy production, and, when perturbed, it will return to this state, which is dictated by the rate at which reactants are fed to the system [R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York]. Hence, steady states settle for the smallest deviation from equilibrium possible under the given conditions. Steady state reactions in industry satisfy these conditions and are operated in a regime where linear non-equilibrium thermodynamics holds. Nonlinear non-equilibrium thermodynamics, however, represents a regime where explosions and uncontrolled oscillations may arise. Obviously, industry wants to avoid such situations ... 

As already mentioned, a continual inflow of energy is necessary to maintain the stationary state of a living system. It is mostly chemical energy which is injected into the system, for example by activated amino acids in protein biosynthesis (see Sect. 5.3) or by nucleoside triphosphates in nucleic acid synthesis. Energy flow is always accompanied by entropy production (dS/dt), which is composed of two contributions ... 

The affinities that define the rate of entropy production in continuous systems are therefore gradients of intensive parameters (in entropy representation) rather than discrete differences. For instance, the affinities associated with the z-components of energy and matter flow for constituent k, in this notation would be... 

In a foregoing section, we mentioned that field forces (e,g., of the electric or elastic field) can cause an interface to move. If they are large enough so that inherent counterforces (such as interface tension or friction) do not bring the boundary to a stop, the interface motion would continue and eventually become uniform. In this section, however, we are primarily concerned with boundary motions caused by chemical potential changes. From irreversible thermodynamics, we know that the dissipated Gibbs energy of the discontinuous system is T-ab, where crb here is the entropy production (see Section 4.2). Since dG/dV = dG/dV = crb- T/ A < ), we have with Eqn. (4.8) at the boundary b... 

Specifically this paper describes an expression for the entropy production due to the mass fluxes in binary mass transfer systems with application to continuous differential contactors. 

This equation shows that the entropy production is a quadratic form in all the forces. In continuous systems, the base of reference for diffusion flow affects the values of transport coefficients and the entropy due to diffusion. Prigogine proved the invariance of entropy for an arbitrary base of reference if the system is in mechanical equilibrium and the divergence of viscous tensors vanishes. 

Example 4.8 Chemical reactions and reacting flows The extension of the theory of linear nonequilibrium thermodynamics to nonlinear systems can describe systems far from equilibrium, such as open chemical reactions. Some chemical reactions may include multiple stationary states, periodic and nonperiodic oscillations, chemical waves, and spatial patterns. The determination of entropy of stationary states in a continuously stirred tank reactor may provide insight into the thermodynamics of open nonlinear systems and the optimum operating conditions of multiphase combustion. These conditions may be achieved by minimizing entropy production and the lost available work, which may lead to the maximum net energy output per unit mass of the flow at the reactor exit. 

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 physical meaning of the terms (or group of terms) in the entropy equation is not always obvious. However, the term on the LHS denotes the rate of accumulation of entropy within the control volume per unit volume. On the RHS the entropy flow terms included in show that for open systems the entropy flow consists of two parts one is the reduced heat flow the other is connected with the diffusion flows of matter jc, Secondly, the entropy production terms included in totai demonstrates that the entropy production contains four different contributions. (The third term on the RHS vanishes by use of the continuity equation, but retained for the purpose of indicating possible contributions from the interfacial mass transfer in multiphase flows, discussed later). The first term in totai arises from heat fluxes as conduction and radiation, the third from diffusion, the fourth is connected to the gradients of the velocity field, giving rise to viscous flow, and the fifth is due to chemical reactions. 

These are systems that exchange both energy and matter with the environment through their boundaries. The simplest chemical reaction engineering example is the continuous stirred tank reactor. These systems do not tend toward their thermodynamic equilibrium, but rather towards a state called stationary non-equilibrium state and is characterized by minimum entropy production. Open systems near equilibrium have unique stationary non-equilibrium state, regardless of the initial conditions. However far from equilibrium these systems may exhibit multiplicity of stationary states and may also exhibit periodic states. 

Components dissolved in water are in a state of continuous motion. Ihis is caused by different forces that affect the magnitude and direction of rates. Any spontaneous mass transfer in ultimately results in an increase of the system s entropy. That is why at the foundation of the description of any processes of spontaneous mass transfer are the thermodynamical laws of irreversible processes. According to these laws the rate at which forms entropy, i.e., entropy production a, is associated with flows of matter dispersion through the following equation... 

In continuous systems, the local increase in entropy can be defined by using the entropy density s(x,t), which is the entropy per unit volume. The total entropy change is ds = dgS + diS and results from the flow of entropy due to exchanges with surroundings (dgs) and from the changes inside the system (dis). Therefore, the local entropy production can be defined by... 

Let us combine now the results of the present and the preceding section for a system which is in contact with a bath system with respect to an exchange of heat and matter and which simultaneously performs internal chemical reactions. Assuming that the temperature T of the system equals that of its bath system, which is a frequent situation in view of biological applications, the continuous entropy production S is then given as... 

In (3.23) we have already introduced the concept of reversible changes of the entropy due to idealized reversible exchanges of heat as infinitely slow processes with negligible irreversible entropy production. This concept is immediately extended to include reversible changes of the entropy which may likewise be caused by infinitely slow exchanges of volume V and chemical components N.. In the course of such processes, the system is not removed from equilibrium but is passing through a continuous series of equilibrium states. This means that in each state of such processes the system is characterized by its entropy S as a function of U, V and... 

Integrate the OM-function on the volume V of the system, while the entropy production is transformed by the surface integral taking into account the continuity equation of the electric current density. Take into account also the boundary conditions prescribing the values of the electric potential and current densities on the boundaries of the system. Than we get the following form of the OM-functional... 

Figure 10.3 An expression for the entropy production in a continuous system can be obtained by considering two adjacent cells separated by a small distance 8. The entropy in the region between x and x + 5 is equal to 5 (x)8. The affinity, which is the difference in the chemical potential, is given by A = p(x) p(x + 8) = p(x) — [p(x)+ (0p/0x)8] = (0 x/0x)8...

The local increase of entropy in continuous systems can be defined by using the entropy density s x, t). As was the case for the total entropy, ds = d s + df,s, with diS > 0. We define local entropy production as... 

As an example of a continuous system, let us look at stationary states in heat conduction using the system we considered in Fig. 17.1. For a one-dimensional system the entropy production is... 

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