Fluxes, entropy generation

This chapter establishes a direct relation between lost work and the fluxes and driving forces of a process. The Carnot cycle is revisited to investigate how the Carnot efficiency is affected by the irreversibilities in the process. We show to what extent the constraints of finite size and finite time reduce the efficiency of the process, but we also show that these constraints still allow a most favorable operation mode, the thermodynamic optimum, where the entropy generation and thus the lost work are at a minimum. Attention is given to the equipartitioning principle, which seems to be a universal characteristic of optimal operation in both animate and inanimate dynamic systems. 

Step 7. Manipulate the equation of change for specific entropy, via definitions of convective and molecular entropy fluxes, to identify all terms that correspond to entropy generation. These terms appear as products of fluxes and forces. 

Step 8. Postulate Unear relations between these fluxes and forces that obey the Curie restriction, and demonstrate that entropy generation can be expressed as a positive-definite quadratic form. 

Fluxes lead to entropy generation in a chemically reactive mixture of N components. 

The concepts discussed in Section 25-9 are applied to binary mixtures of A and B with chemical reaction. Now, the Curie restriction states that there are two first-rank tensorial fluxes, —(q — linear laws. Notice that the two fluxes are not simply q and Ja, but — (q — aJa) and —Ja, as dictated by the classical expression for the rate of entropy generation, which is given by equation (25-49) in canonical form. In other words, one must exercise caution in identifying fluxes and forces such that their products correspond to specific terms in the final expression for sq- The linear laws are... 

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... 

Application of the entropy generation minimization (EGM) method to (21.7) allows identifying the optimal parameters to be considered in the optimization [6]. Concerning the entropy generated by the evaporated mass flux, that parameter is the fraction x of evaporated mass and its optimal value is found by equating to zero the partial derivative of (21.7) with respect to % The optimal evaporated mass fraction is thus found to be... 

Entropy generation of forced convection heat transfer of liquid fluid over the horizontal surface with embedded open parallel microchannels at constant heat flux boundary conditions may be formulated by an integral of the local entropy generation. Embedded open parallel microchannels within the surface can sufficiently reduce both friction and thermal irreversibilities of liquid fluid through slip-flow conditions (Kandlikar et al., 2006 Yarin et al., 2009). 

Sahin, A. Z. (1999). Effect of variable viscosity on the entropy generation and pumping power in a laminar fluid flow through a duct subjected to constant heat flux. Heat Mass Transfer, 35, pp. 499 506, ISSN 0947-7411. 

The first bracketed (square) term is the conductive energy flux, which is the same as used in the energy Eq. 5.5(c). The other bracketed terms represent the dissipative energy flux, as used also in the energy Eq. 5.5(d). Thus, the volumetric rate of entropy generation can be written as ... 

In order to solve the conservation or transport equations (mass, momentum, energy, and entropy) in terms of the dependent variables n, Vo,U, and , we must further resolve the expressions for the flux vectors— P, q, and s and entropy generation Sg. This resolution is the subject of closure, which will be treated in some detail in the next chapter. However, as a matter of illustration and for future reference, we can resolve the flux vector expression for what is called the local equilibrium approximation, i.e., we assume that the iV-molecule distribution function locally follows the equilibrium form developed in Chap. 4, i.e., we write [cf Eq. (4.34)]... 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

Finally, we turn to the equation of entropy conservation and look at the specific expressions for the entropy flux and entropy generation terms. With entropy defined in the s = 1 space, the entropy flux from Eq. (5.86) in dimensionless terms is ... 

Extended Stefan-Maxwell constitutive laws for diffusion Eq. 4 resolve a number of fundamental problems presented by the Nemst-Planck transport formulation Eq. 1. A thermodynamically proper pair of fluxes and driving forces is used, guaranteeing that all the entropy generated by transport is taken into account. The symmetric formulation of Eq. 4 makes it unnecessary to identify a particular species as a solvent - every species in a solution is a solute on equal footing. Use of velocity differences reflects the physical criterion that the forces driving diffusion of species i relative to species j be invariant with respect to the convective velocity. Finally, all possible binary solute/solute interactions are quantified by distinct transport coefficients each species i in the solution has a diffusivity or mobility relative to every other species j, Djj or up, respectively. 

As mentioned before, nonequilibrium thermodynamics could be used to study the entropy generated by an irreversible process (Prigogine, 1945, 1947). The concept ofhnear nonequilibrium thermodynamics is that when the system is close to equilibrium, the hnear relationship can be obtained between the flux and the driving force (Demirel and Sandler, 2004 Lu et al, 2011). Based on our previous linear nonequihbrium thermodynamic studies on the dissolution and crystallization kinetics of potassium inorganic compounds (Ji et al, 2010 Liu et al, 2009 Lu et al, 2011), the nonequihbrium thermodynamic model of CO2 absorption and desorption kinetics by ILs could be studied. Figure 17 shows the schematic diagram of CO2 absorption kinetic process by ILs. In our work, the surface reaction mass transport rate and diffusion mass transport rate were described using the Hnear nonequihbrium thermodynamic theory. 

Heat production associated with the electrochemical reactions is also assumed to be confined at the electrode-electrolyte surface, thus the resulting thermal energy produces a discontinuity of the heat flux. The heat generated within this surface, in fact, represents a heat source for the electrode and the electrolyte domains. The sum of the inward heat fluxes is equal to the heat generated as a result of the electrochemical reactions. As explained in Section 3.3.2, the heat is generated by the increase in entropy, associated with the electrochemical reaction (reversible heat), and to the activation irreversibilities. Therefore, the boundary conditions for Equation (3.7) are ... 

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. 

We show in Appendix A that the rate of entropy production depends upon the product of fluxes and gradients. Fluxes and gradients are connected in the sense that a gradient is capable of generating a flux. For example, heat flows across temperature gradients and material flows across composition gradients. By manipulation of the fluxes or the gradients we can affect the rate of entropy production. 

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