Entropy production rate

Because the rate of entropy production is negative, the device violates the second law and is therefore impossible. Note that the device would be theoretically possible if the final pressure were specified as 400 psia or less by the inventor. That is, at = 400 psia, T = 40°F, h = 290 Btu/lb, and S, = 1.25 Btu/lb °R, the entropy production rate would be... 

Nonlinear Hamiltonian system, geometric transition state theory, 200-201 Nonlinear thermodynamics coefficients linear limit, 36 entropy production rate, 39 parity, 28-29... 

Even if one of the processes is not chemical but is categorized as a phase change, for example, evaporation, the extended De Donder s equation (Equation 13.17), is known to be valid. Any large magnitude of entropy production rate (diS/dt) due to evaporation might give a correlation such as... 

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 ... 

When Equation Al-14 is substituted into Al-13, the entropy production rate a is... 

Here Dt is a positive proportionality constant ( diffusion constant for Et), Jfz is z-ward flow induced by the gradient, and superscript e denotes eigenmodt character of the associated force or flow. The proportionality (13.25) corresponds to Fick s first law of diffusion when Et is dominated by mass transport or to Fourier s heat theorem when Et is dominated by heat transport, but it applies here more deeply to the metric eigenvalues that control all transport phenomena. In the near-equilibrium limit (13.25), the local entropy production rate (13.24) is evaluated as... 

The local entropy production rate (13.24) is then expressed in terms of eigenforces and eigenfiows as... 

The eigenmode expansion (13.30) for the local entropy production rate can be expressed in terms of usual laboratory variables Rh Rt (13.14a, b) using transformation equations analogous to those of Section 11.6. In the present Abased framework, the expansion of dEt in intensities [cf. (11.89)] becomes... 

NEAR-EQUILIBRIUM IRREVERSIBLE THERMODYNAMICS DIFFUSIONAL GEOMETRY 435 This leads to the final expression for the local entropy production rate... 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

The bilinear formulation of the entropy production rate obtained in a practical form is... 

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... 

Solving Exercise 2.5 shows that the products of the forces and reaction rates in Eq. 2.49 appear in the expression for the entropy production rate for the chemical reactions. The forces and reaction rates are therefore conjugate, as expected. 

Rate of net entropy + Rate of entropy production = Rate of change of entropy... 

The rate of entropy production is obtained from the local value of entropy production or entropy source strength d>... 

Second-law analysis can determine the level of energy dissipation from the rate of entropy production in the system. The entropy production approach is especially important in terms of process optimality since it allows the entropy production of each process to be determined separately. The map of the volumetric entropy production rate identifies the regions within the system where excessive entropy production occurs due to irreversible processes. Minimizing of excessive irreversibilities allows a thermodynamic optimum to be achieved for a required task. Estimation of the trade-offs between the various contributions to the rate of entropy production may be helpful for attaining thermodynamically optimum design and operation. 

Figure 4.4 shows the temperature profile for T2 = 300 K and -2.0 < Br < 8.0. The rise of temperature in the middle part of the Couette device is considerably large for high values of Br. Inserting Eqs. (4.26) and (4.29) into Eq. (4.23) yields an expression for the volumetric entropy production rate for a Couette flow. 

Example 4.4 Thermomechanical coupling in a circular Couette flow For a circular Couette flow (Figure 4.5), the entropy production rate for an incompressible Newtonian fluid held between two coaxial cylinders is... 

Here, the terms on the right side of the above equation show the entropy production due to heat transfer and fluid friction, respectively hence, the entropy production expression has the following basic form , Sprod =, Yprod Ar +, S prod A/,. The volumetric entropy production rate is positive and finite as long as temperature or velocity gradients are present in the medium. 

Substituting Eqs. (4.69)-(4.76) into Eq. (4.68) and performing the integration, the entropy production rate can be obtained for a laminar flow on a flat plate... 

One optimum requires a uniformly distributed entropy production rate in a heat exchanger, mixer, or separator. Consider the example of countercurrent and cocurrent heat exchangers shown in Figure 4.11. Temperature profiles... 

The entropy production rate is determined with quasi-steady-state calculations. The following constant gradients in the gas phase at each stage are used ... 

The chemical driving force on a stage has inlet and outlet concentrations as boundary conditions. In the enriching section below the feed plate, a flowfrom liquid to vapor occurs while in the stripping section, the direction of flow is from vapor to liquid. For the column with specified inlet and outlet compositions, the entropy production rates are... 

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