Entropy production due to diffusion

Clearly this result can be generalized to many reactions. We thus have the general result the affinity of a net reaction is equal to the sum of the affinities of the individual reactions. 

When several chemical reactions are taking place, an interesting possibility arises. Consider two reactions that are coupled by one or more reactants that take part in both reactions. For the total entropy production we have 

For this inequality to be satisfied, it is not necessary that each term be positive. It may well be that 

In this case the decrease in entropy due to one reaction is compensated by the increase in entropy due to the other. Such coupled reactions are common in biological systems. The affinity of a reaction is driven away from zero at the expense of another reaction whose affinity tends to zero. 

The concepts of chemical potential and affinity not only describe chemical reactions but also the flow of matter from one region of spaee to another. With the concept of chemical potential, we are now in a position to obtain an expression for the entropy change due to diffusion, an example of an irreversible process we saw in the previous chapter (Fig. 3.8). The idea of chemical potential 

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With the help of this postulate, it is possible to obtain an explicit expression for a, the rate of entropy production per unit volume due to various irreversible processes taking place within the system (see, e.g., Slattery, 1981). The rate of entropy production due to diffusion is... 

The positive definiteness requirement of allows us to derive certain restrictions on the values of the If we substitute the GMS relation for into Eq. 2.3.15 we obtain a very neat and compact expression for the rate of entropy production due to diffusion... 

Rate of entropy production due to diffusion (Chapter 2) [J/m s K] Characteristic diameter of molecule (Section 4.1) [A]... 

This is an elementary example of how the expression for entropy production can be used to obtain linear relations between thermodynamic forces and flows, which often turn out to be empirically discovered laws such as Ohm s law. In section 10.3 we shall see that similar consideration of entropy production due to diffusion leads to another empirically discovered law called the Pick s law of diffusion. Modern thermodynamics enables us incorporate many such phenomenological laws into one unified formalism. 

The interaction between heat and matter flows produces two effects, the Soret effect and the Dufour effect. In the Soret effect, heat flow drives a flow of matter. In the Dufour effect, concentration gradients drive a heat flow. The reciprocal relations in this context can be obtained by writing the entropy production due to diffusion and heat flow ... 

Equation 9 relates the basic extensive properties, the respective driving potentials, and the entropy production in the diffusion process. The first term, s TT, arises from heat transfer effects while the second term is due to mass transfer. For processes wherein the entropy production due to column heat transfer is small relative to the mass-transfer, the s TE term is negligible and Equation 9 simplifies to... 

What we have achieved so far is to express the rate of entropy production due to mass diffusion in terms of a convenient driving force c RTd per unit volume of mixture. Equation 2.3.8 shows that the rate of entropy production is a sum of the products of two quantities the force acting on /, per unit volume, tending to move i relative to the mixture and the relative velocity of the movement of i with respect to the mixture Ojiif is, therefore, the dissipation due to diffusion. 

Finally, Example 8 is more complex in that it comprises entropy production due to chemical reaction in combination with heat transfer, and also diffusion the role of the latter appears as marginal. The example can also be regarded as an example of complex single-node balancing, a kind of thermodynamic analysis included. Concerning the entropy production (or loss of exergy), it turns out that the chemical portion, thus the term -(Gr / Tj ) Wr in (6.2.109) can represent an enormous item in the exergy balance it can be computed, but this is usually all that we can do in practice. In other terms, what kind of work has been actually lost, is a matter of theoretical speculations only. 

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. 

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. 

It should be noted that Q is not the actual entropy production (the latter would also include terms due to diffusion of the gas components etc.), and Ql cannot be generally put equal to the lost exergy of the exit gas , thus to m(H -TQS ) by (6.1.6). The latter value will depend on the zero levels adopted for enthalpy and entropy in the whole set of production units for instance in the combustion chamber, one will take zero levels rather for chemical elements such as C (carbon), thus the final CO2 will be assigned negative enthalpy value at its standard state. 

This entropy production rate contains three contributions. The first one is due to diffusion in u-space and is proportional to the product of the mesoscopic probability current /V and the derivative with respect to particle velocity u of the nonequilibrium chemical potential p (r, u, f) per unit mass... 

That is, the entropy production in the volume consists of three terms, each of which is due to an irreversible process. The first term is the heat conduction term, the second is the mass diffusion term, and the third is the chemical reaction term. The above equation is known as the entropy production equation. 

There exist a large number of phenomenological laws for example, Fick s law relates to the flow of a substance and its concentration gradient, and the mass action law explores the reaction rate and chemical concentrations or affinities. When two or more of these phenomena occur simultaneously in a system, they may couple and induce new effects, such as facilitated and active transport in biological systems. In active transport, a substrate can flow against the direction imposed by its thermodynamic force. Without the coupling, such uphill transport would be in violation of the second law of thermodynamics. Therefore, dissipation due to either diffusion or chemical reaction can be negative only if these two processes couple and produce a positive total entropy production. 

At first sight it might appear that the second law of thermodynamics is violated for reverse diffusion to occur. This is not so. One process may depart from equilibrium in such a sense as to consume entropy provided it is coupled to another process that produces entropy even faster. This is, of course, the basic principle of any pump, whether it moves water uphill or moves heat towards a higher temperature region. For the second law requirement <7 > 0 to hold it is allowable for to be less than zero, corresponding to reverse diffusion for 1, provided <72 and 0-3, due to species 2 and 3 diffusion, be such that the overall entropy production rate is positive (a + 0-2 + <73 > 0). 

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. 

With the aim of expressing the rate of entropy production totai due to mass diffusion in terms of a convenient driving force, we add some... 

The entropy production rate at macroscopic level, (equation 8.5) can be obtained after integration of the local entropy production rate density over the volume element in phase (a). For the sake of simplification the following assumptions are adopted (i) entropy contributions of all involving phases in the GLRDVE are combined in a single variable (E = /(E )) (m) entropy is produced due to mass and heat diffusion in... 

Calculation of the Entropy Production Rate due to Crain Boundary Diffusion... 

FIGURE 1.4 Illustration of basic fuel ceU processes and their relation to the thermodynamic properties of a cell. The electrical work performed by the cell, corresponds to the reaction enthalpy, — A//, minus the reversible heat due to entropy production, —TAS, and minus the sum of irreversible heat losses at finite load, Qi. These losses are caused by kinetic processes at electrochemical interfaces as well as by transport processes in diffusion and conduction media. 

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