Entropy flux rate

For an optical refrigerator, the photon occupation number n in the entropy flux rate (Eq. 16) has contributions from both the fluorescent photons of the refrigerator, rif v), and the ambient thermal photons, a(v). The latter is given... 

To find entropy production rate, we need to relate flux to chemical potential gradients. To the first-order approximation, flux of each component is linearly related to gradients ... 

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... 

Available energy Mass flow rate Mass flux Pressure Gas Constant Entropy per unit mass Entropy flux Entropy production Time... 

Inequalities (3.2) and (3.3) are generalizations of the principle of the minimal entropy production rate in the course of spontaneous evolution of its system to the stationary state. They are independent of any assump tions on the nature of interrelations of fluxes and forces under the condi tions of the local equilibrium. Expression (3.2), due to its very general nature, is referred to as the Qlansdorf-Prigogine universal criterion of evolution. The criterion implies that in any nonequilibrium system with the fixed boundary conditions, the spontaneous processes lead to a decrease in the rate of changes of the entropy production rate induced by spontaneous variations in thermodynamic forces due to processes inside the system (i.e., due to the changes in internal variables). The equals sign in expres sion (3.2) refers to the stationary state. 

Non-equilibrium thermodynamics (NET) offers a systematic way to derive the local entropy production rate, c, of a system. The total entropy production rate is the integral of the local entropy production rate over the volume, V, of the system, but, in a stationary state, it is also equal to the entropy flux out, J, minus the entropy flux into the system,... 

The entropy flux difference and the integral over a can be calculated independently, and they must give the same answer. The entropy production rate governs the transport processes that take place in the system. We have... 

The entropy production rate and the complete set of equations that follows can be most conveniently written for the liquid film, the interface, and the vapour film in series.Film layers of thicknesses 5 and in the liquid and the vapour are illustrated in Figure 5. With constant fluxes (in stationary states), the integration is easy. The approach was called the integrated interface approach.For the three layers, the integrated overall force is the sum of the integrated force across each layer ... 

With these relations, the combined films and interface is regarded as an effective interface . There is no need to assume phase equilibrium between liquid and vapour. The entropy production rate can alternatively be expressed by the measurable heat flux in the vapour and fluxes of mass. - This set of flux equations was used to explain the entropy production in tray distillation columns. However, it has not yet been used for predictive purposes. Much work remains to be done to include these equations in a software that is useful for industrial purposes. 

The starting point for the rigorous derivation of the diffusive fluxes in terms of the activity is the entropy equation as given by (1.170), wherein the entropy flux vector is defined by (1.171) and the rate of entropy production per unit volume is written (1.172) as discussed in sec. 1.2.4. 

This is the local equation for the entropy density. It is of fundamental importance in what follows. The quantity a, the local entropy production, is the entropy produced irreversibly per unit time per unit volume and is analogous to the property a a defined prior to Eq. (10.3.4). The quantity Js is the entropy flux and V Js represents the rate of change of the local entropy due to an inflow of entropy from neighboring regions of the fluid. 

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 expression for shows, that the entropy flux for open systems consists of two parts the thermal flux associated with the heat transfer, and the flux due to diffusion. The second expression consists of four terms associated with, respectively, the heat transfer, diffusion, viscosity, and chemical reactions. The expression for the dissipative function a has quadratic form. It represents the sum of products of two factors a flux (specifically, the heat flux /, diffusion flux momentum flux n, and the rate of a chemical reaction and a thermodynamic force, proportional to gradient of some intensive variable of state (temperature, chemical potential, or velocity). The second factor can also include external force F]t and chemical affinity Aj. 

For the open system of Fig. 2.16, one must express the entropy-flow rate as written in Eq. (2) of Fig. 2.81. Besides the entropy change due to the heat flux, there is an entropy change due to the flux of matter. The matter flux of component i is expressed by its flow-rate term, d ni/dt. The matter flux can be measured, for example, by thermogravimetry as listed in Fig. 2.16. Each flux of matter across the boundary of the system, deO, needs to be multiplied with the partial molar change in the corresponding entropy, S, [= (dS/dnil p, see Fig. 2.19], to obtain the total contribution to d S/dt. The change in partial molar entropy must be known, or be measured independently by heat capacity measurements, for example, from zero kelvin to T, as outlined in Figs. 2.22 and 2.23. 

The entropy of the system is assumed to be exclusively generated by the heat conduction from the hot to the cold fluids. The entropy production rate, at microscopic level, can be estimated as the product of thermal driving force and heat flux. From a macroscopic stand-point the measurable heat flow is used for this computation. A better approximation can be obtained by introducing phenomenological coefficients (Hasse, 1969 Koeijer, 2002 Meeuse, 2003). For our analysis, however, we adopt an alternative approach. The overall steady state entropy equation of change is applied and the production term is related to the net change of entropy. 

The direct representation of the Fourier equation is not possible in a Formal Graph because heat is energy (of the thermal variety) and not a state variable, and also because the heat flux does not correspond to a variable that can be represented in a Formal Graph. The solution consists of taking recourse to state variables of the thermal variety that are the entropy S (basic quantity), the temperature T (effort), and the entropicflow fs (also called entropy production rate). The variation of entropy is linked to the variation of heat according to the energy-per-entity times the variation of entity number... 

The basic framework of RT was developed by Truesdell and Toupin (1960) and Coleman (1964), which excludes the local-equilibrium hypothesis (see Note 3.3, p. 95). Let us define as the entropy flux and r/T as the entropy supply. RT introduces the rate of entropy production F in a part V of the body as... 

In the model considered below, the role of both grain boundary and bulk diffusion in the transformation front and close to it, respectively, is analyzed within the problem of unambiguous determination of the discontinuous precipitation parameters in the binary Pb-Sn system at room temperature [9]. In order to complete this, we use the principle of maximum rate of free energy release and balance of entropy fluxes for the description of discontinuous precipitation kinetics for binary polycrystaUine alloys and independent determination of three basic parameters interlamellar distance, rate of phase transformation front, and concentration profile close to the transformation front. While solving the problem, we also find the optimal concentration distribution of components both along the precipitation lamella behind the transformation front and close to it, as well as the degree of the components separation. 

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