Fluxes systems, entropy production

The time derivative of this entropy dS/dt can be split into an entropy flux and entropy production, and the H theorem states that this entropy production is positive for nonequilibrium systems. At stationary state, however, the probabilities become time-independent, dp(a)/dt = 0, and the entropy production becomes... 

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... 

In Chapter 4, we introduced transport equations that apply when there are fluxes other than those of matter that contribute to the entropy production. Assuming that both matter and electrons take part in the transport, Eqns. (4.16)-(4.17) have been derived. In non-isothermal systems, we can use the same set of equations but replace... 

The foundation of irreversible thermodynamics is the concept of entropy production. The consequences of entropy production in a dynamic system lead to a natural and general coupling of the driving forces and corresponding fluxes that are present in a nonequilibrium system. 

In a hypothetical system for modeling kinetics, the microscopic cells must be open systems. It is useful to consider entropy as a fluxlike quantity capable of flowing from one part of a system to another, just like energy, mass, and charge. Entropy flux, denoted by Js, is related to the heat flux. An expression that relates Js to measurable fluxes is derived below. Mass, charge, and energy are conserved quantities and additional restrictions on the flux of conserved quantities apply. However, entropy is not conserved—it can be created or destroyed locally. The consequences of entropy production are developed below. 

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. 

The optimal Reynolds number defines the operating conditions at which the cylindrical system performs a required heat and mass transport, and generates the minimum entropy. These expressions offer a thermodynamically optimum design. Some expressions for the entropy production in a multicomponent fluid take into account the coupling effects between heat and mass transfers. The resulting diffusion fluxes obey generalized Stefan-Maxwell relations including the effects of ordinary, forced, pressure, and thermal diffusion. 

The vividness of our world does not rely on processes that are characterized by linear force-flux relations, rather they rely on the nonlinearity of chemical processes. Let us recapitulate some results for proximity to equilibrium (see also Section VI.2.H.) In equilibrium the entropy production (n) is zero. Out of equilibrium, II = T<5 S/I8f > 0 according to the second law of thermodynamics. In a perturbed system the entropy production decreases while we reestablish equilibrium (II < 0), (Fig. 72). For the cases of interest, the entropy production can be written as a product of fluxes and corresponding forces (see Eq. 108). If some of the external forces are kept constant, equilibrium cannot be achieved, only a steady state occurs. In the linear regime this steady state corresponds to a minimum of entropy production (but nonzero). Again this steady state is stable, since any perturbation corresponds to a higher II-value (<5TI > 0) and n < 0.183 The linear concentration profile in a steady state of a diffusion experiment (described in previous sections) may serve as an example. With... 

We begin here the study of thermodynamics in the proper sense of the word, by exploring a variety of physical situations in a system where one or more intensive variables are rendered nonuniform. So long as the variations in T, P, /x or other intensive quantities are small relative to their average values, one can still apply the machinery of equilibrium thermodynamics in a manner discussed later. It will be seen that the identification of conjugate forces and fluxes, the Onsager reciprocity conditions, and the rate of entropy production play a central role in the analysis provided later in the chapter. 

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. 

As mentioned above, the time-integral of the dissipation function takes on the value of the extensive generalised entropy production, 2, over a period, t under suitable circumstances. The main requirement is that the dynamics satisfies the condition know as the adiabatic incompressibility of phase space . In this case, 2, = —JtFefiV, where is the dissipative flux caused by the field, F, p = IKk T) where T is the temperature of the corresponding initial system and V is the volume of the system. An example where such a relation can be applied is if a molten salt at equilibrium was exposed to a constant electric field. In that case the entropy production would be directly proportional to the current induced, and the FR would describe the probability that it would be observed to flow in the + ve or — ve... 

Steady state conditions obtain when the fluxes and forces giving rise to irreversible phenomena in a system remain time-invariant, whereas the properties of the surroundings change. We now render this idea more precise by introducing Prigogine s Theorem Let irreversible processes take place through imposition of n forces X, X2,. ,X that result in n fluxes J, Ji,---, Jn- Let the first k forces remain fixed at values Xj, X ,..., then it is claimed that the rate of entropy production 0 is minimized when the fluxes Jk+i Jk 2, Jn ah vanish. We first prove the theorem and then discuss its relevance to steady state conditions. As before, we set 9 = Jj Xj, to construct the phenomenological... 

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... 

Entropy production is important for a proper definition of fluxes and forces in non-equilibrium systems that are relevant to the industry. A higher accuracy in the fluxes can be obtained using NET. 

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. 

When a system consisting of n fluxes and n forces is allowed to age, it eventually attains a state of equilibrium. It is necessary, therefore to impose certain constraints on the system to achieve the steady state. Let us impose the constraints in such a manner that forces Xh X2,... XK are fixed at constant values while the remaining forces XK+ h Xk + 2,... Xn are allowed to vary. The entropy production is given by... 

In this problem we explore classical irreversible thermodynamics for a multicomponent system, entropy generation, linear laws, and the molecnlar flux of thermal energy for a ternary system. Consider an N-component system (1 < j < Af) in the presence of external force fields and mnltiple chemical reactions (1 < y < / ). g, is the external force per unit mass that acts specifically on component i in the mixture, and r, is the overall rate of production of the mass of component i per unit volume, which is defined by... 

The book is divided into four parts. Part One, which consists of six chapters, deals with basic principles and concepts of non-equilibrium thermodynamics along with discussion of experimental studies related to test and limitation of formalism. Chapter 2 deals with theoretical foundations involving theoretical estimation of entropy production for open system, identification of fluxes and forces and development of steady-state relations using Onsager reciprocity relation. Steady state in the linear range is characterized by minimum entropy production. Under these circumstances, fluctuations regress exactly as in thermodynamics equilibrium. 

Entropy production for a non-equilibrium close to equilibrium is estimated with the help of Gibbs equation with the objective to estimate internal entropy production a = dj5 /df which is needed for characterization of fluxes J and forces X since as we shall later that cr can be expressed as sum of product of fluxes and forces. To illustrate this point, we consider a discontinuous system involving two chambers separated by a barrier but maintained at different temperatures Tj and T. In the present case, heat flow only occurs on account of force generated due to temperature difference (Fig. 2.4). 

Equation (4.10) enables us to spot out the fluxes and forces in the system by remembering that entropy production is the sum of the product of fluxes 7 and forces X , i.e. 

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. 

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