Irreversibility entropy production rate

We may define the volumetric (total) irreversible entropy production rate by w(t) = fya(r, t)dV/k and an average over all fluctuations in which the time-integrated entropy production is positive by then... 

NEAR-EQUILIBRIUM IRREVERSIBLE THERMODYNAMICS DIFFUSIONAL GEOMETRY 435 This leads to the final expression for the local entropy production rate... 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

Irreversible thermodynamics thus accomplishes two things. Firstly, the entropy production rate EE t allows the appropriate thermodynamic forces X, to be deduced if we start with well defined fluxes (eg., T-VijifT) for the isobaric transport of species i, or (IZT)- VT for heat flow). Secondly, through the Onsager relations, the number of transport coefficients can be reduced in a system of n fluxes to l/2-( - 1 )-n. Finally, it should be pointed out that reacting solids are (due to the... 

Second-law analysis can determine the level of energy dissipation from the rate of entropy production in the system. The entropy production approach is especially important in terms of process optimality since it allows the entropy production of each process to be determined separately. The map of the volumetric entropy production rate identifies the regions within the system where excessive entropy production occurs due to irreversible processes. Minimizing of excessive irreversibilities allows a thermodynamic optimum to be achieved for a required task. Estimation of the trade-offs between the various contributions to the rate of entropy production may be helpful for attaining thermodynamically optimum design and operation. 

To reduce the lost work in industrial process plants, the minimization of entropy production rates in process equipment is suggested as a strategy for future process design and optimization [81]. The method is based on the hypothesis that the state of operation that has a minimum total entropy production is characterized by equipartition of the local entropy production. In this context we need to quantify the entropy sources of the various irreversible unit operations that occur in the industrial system. 

Table I compares the work input and rate of entropy production in the practical process with corresponding quantities in the ideal process. This brings out the fact that the single place where the greatest reduction in power can be effected is in the main exchanger, and also shows the relative importance of the individual pieces of equipment as contributors of the total work input. It also shows that the total work input is the sum of (a) the theoretical work, and (b) the work equivalent of irreversible entropy production, given by the product of AS rr and ambient temperature To.

In the presence of a nonuniformity in concentration a diffusive flux occurs, resulting in the creation of entropy. The rate of irreversible entropy production is, in general, a homogeneous quadratic function of the gradients of... 

The degree of completeness of the analogies between Eqs. (331) and (49) is quite remarkable oll and Jfoll are each symmetric, positive-definite forms, as are their direct submatrices too. The positive-definiteness of o stems from the positivity of the rate of irreversible entropy production. In contrast to the proof of the symmetry of the hydrodynamic resistance matrix (B22), the corresponding proof of the symmetry of the diffusion matrix is trivial. The latter may be taken to be symmetric by definition since its antisymmetric part gives rise to no observable macroscopic physical consequence. 

S Wetted surface area of particle (70) velocity gradient (269a) 5 Time rate of irreversible, entropy production (footnote 55)... 

The search for minimum losses is based on the rate or irreversible entropy production per unit length of tube ... 

Hence expressing the rate of irreversible entropy production in terms of a tube length. 

Note that for confined flow, the appearance of a critical Reynolds number or diameter for ReD < ReD opu the contribution to the rate of entropy production of heat transfer is large in comparison with the irreversibilities due to speed o q > Q y. The situation is reversed for ReD > ReD opt this case, viscosity losses predominate o > b q. This indicates that the rate or irreversible entropy production in the frictional losses (defining the drop in pressure) increase more rapidly with speed than the losses due to heat transfer by convection. 

The entropy production rate is equal to a sum of products of generalized forces by generalized fluxes. The laws of thermodynamics of irreversible processes enable us to express these fluxes as functions of these forces. When we do not stray too far from the state of equilibrium, where the fluxes and forces are null, linear relations appear between these terms. The coefficients of these linear laws are the Onsager phenomenological coefficients they are combinations of the coefficients of diffusion, viscosity, heat conduction, etc. In conductive media, the electrical resistance also appears as an Onsager coefficient. 

When a process is completely reversible, the equahty holds, and the lost work is zero. For irreversible processes the inequality holds, and the lost work, that is, the energy that becomes unavailable for work, is positive. The engineering significance of this result is clear The greater the irreversibility of a process, the greater the rate of entropy production and the greater the amount of energy that becomes unavailable for work. Thus, every irreversibility carries with it a price. 

The rate of entropy production is always positive in the present case, since transport processes are irreversible in nature, i.e. always connected with irreversible losses (dissipation) of energy. 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

The number of typical paths generated by the random process increases as exp(/ifl). In this regard, the Kolmogorov-Sinai entropy per unit time is the rate of production of information by the random process. On the other hand, the time-reversed entropy per unit time is the rate of production of information by the time reversals of the typical paths. The thermodynamic entropy production is the difference between these two rates of information production. With the formula (101), we can recover a result by Landauer [50] and Bennett [51] that erasing information in the memory of a computer is an irreversible process of... 

In this chapter, we first introduce the principles of irreversible or nonequilibrium thermodynamics as opposed to those of equilibrium thermodynamics. Then, we identify important thermodynamic forces X (the cause) and their associated flow rates / (the effect). We show how these factors are responsible for the rate with which the entropy production increases and available work decreases in a process. This gives an excellent insight into the origin of the incurred losses. We pay attention to the relation between flows and forces and the possibility of coupling of processes and its implications. 

In Section 3.3, we have shown that the entropy generation rate in the case of heat transfer in a heat exchanger is simply the product of the thermodynamic driving force X = A(l/T), the natural cause, and its effect, the resultant flow / = Q, a velocity or rate. Selected monographs on irreversible thermodynamics, see, for example, [1], show how entropy generation also has roots in other driving forces such as chemical potential differences or affinities. 

Equation (1.89) determines the rate of entropy production due to irreversibility within a control volume. The concept of entropy production is elaborated further in the next section. 

The loss of energy is directly proportional to the rate of entropy production because of irreversible processes in a system. The loss of energy may be estimated based on the temperature of the surroundings of the system T0, and we have... 

Nonequilibrium thermodynamics estimates the rate of entropy production for a process. This estimation is based on the positive and definite entropy due to irreversible processes and of Gibbs relation... 

An irreversible process causes the entropy production in any local element of a system, and the rate of total entropy production is... 

Some options for achieving a thermodynamic optimum are to improve an existing design so the operation will be less irreversible and to distribute the irreversibilities uniformly over space and time. This approach relates the distribution of irreversibilities to the minimization of entropy production based on linear nonequilibrium thermodynamics. For a transport of single substance, the local rate of entropy production is... 

---