Entropy flow

In these systems the converter is producing waste heat, which has to be released to the ambient connected to an entropy flow caused by the irreversibilities within the converter. The discharging process will be a heat pump process, where the entropy has to be taken from the ambient. Therefore it is obvious that these systems have to be coupled to the ambient conditions. Such a storage is not self-sufficient. These systems are called indirect thermal energy storages. 

MSN. 197. I. Prigogine, G. Ordonez and T. Petrosky, Microscopic entropy flow and entropy production in resonance scattering, in Proceedings, XXII Solvay Conf. on Physics, The Physics of Communication, Delphi, 2001, World Scientific, Singapore, 2003, pp. 370-388. 

The coefficients (dU/drii)s vn of the dnt in (6.7) are evidently an important new set of intensive properties that control the chemical flows (analogous to the manner in which T controls the entropy flow and P the volume flow). Following Gibbs, we identify each coefficient (dU/dni)s v n as the chemical potential (/ ) of the corresponding species At ... 

However, for a more general system, the total entropy increase will depend upon how much entropy is produced within it and upon how much entropy flows through its boundaries. 

Same heat Change in entropy flow of surroundings... 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

The dJS may be due to a flow of internal energy, convection entropy flow transported along with the macroscopic flow of the substance as a whole, or the entropy flow caused by diffusion of the individual components. The quantity dJS may be positive, negative, or zero in a special case. For a closed, thermally homogeneous system... 

For an irreversible expansion of a real gas at constant temperature due to a heat reservoir, the change of entropy flow is d,.S = 8q/T, where 8q is the heat flow between the gas and the reservoir to maintain the constant temperature. The increase of entropy during the expansion is... 

The term dJS/dt is the reversible entropy change in time as a result of an entropy flow between the system and its surroundings. On the other hand, djS/dt represents the rate of entropy production inside the system. Equation (3.48) shows that the entropy exchange with the surrounding must be negative at stationary state... 

Therefore, the total entropy produced within the system must be discharged across the boundary at stationary state. For a system at stationary state, boundary conditions do not change with time. Consequently, a nonequilibrium stationary state is not possible for an isolated system for which deS/dt = 0. Also, a steady state cannot be maintained in an adiabatic system in which irreversible processes are occurring, since the entropy produced cannot be discharged, as an adiabatic system cannot exchange heat with its surroundings. In equilibrium, all the terms in Eq. (3.48) vanish because of the absence of both entropy flow across the system boundaries and entropy production due to irreversible processes, and we have dJS/dt = d dt = dS/dt = 0. 

For the total entropy to be constant the entropy flowing out of the system must be equal to the entropy entering the system plus the entropy generated within the system ... 

Equation (3.116) shows that the rate of change of the entropy per unit volume of substance is due to the convection entropy flow p.w. the conduction entropy flow j v. and the entropy source strength The conduction entropy flow is... 

The conduction entropy flow consists of the heat flow j" and the diffusion flow j,-. The j" is reduced heat flow that is the difference between the change in energy and the change in enthalpy due to matter flow. With the... 

The entropy flow v, on the other hand, is the result of the exchange of entropy with the surroundings... 

At stationary state, the local entropy density must remain constant because of the condition dsjdt = 0. However, the divergence of entropy flow does not vanish, and we obtain... 

Consider a system with two parts. The parts are separated by a permeable membrane. The two parts may have different temperatures and pressures. Therefore, two generalized flows of substance and heat occur, while the temperature difference and pressure difference are the two thermodynamic forces. In terms of entropy flow Jv. the dissipation function is... 

The metallic circuit we consider has only electrons flowing, and the dissipation function in terms of entropy flow Js is... 

If we consider the change of local entropy of a system at steady state ds/dt = 0, the local entropy density must remain constant because external and internal parameters do not change with time. However, the divergence of entropy flow does not vanish div J, = . Therefore, the entropy produced at any point of a system must be removed or transferred by a flow of entropy taking place at that point. A steady state cannot be maintained in an adiabatic system, since the entropy produced by irreversible processes cannot be removed because no entropy flow is exchanged with the environment. For an adiabatic system, equilibrium state is the only time-invariant state. 

Equation (A.26) suggests that we can relate the entropy flow vector Js to the energy flow vector J., and the component flow vectors J, according to... 

The entropy flow is based on a heat or material flow, the entropy generation on a heat flow in a temperature field, diffusion by mass forces and differences in the chemical potential, mechanical dissipation and chemical reactions. 

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. 

MkSk = net rate of entropy flow due to the flows of mass into and out of the t=i system (5 = entropy per unit mass)... 

To derive the macroscopic entropy balance equation for mixtures we now %t. d — S, in Eq. 2.1-4 and make the same identification for the entropy flow terms as was made in Sec. 4.1, to obtain... 

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