Entropy production integral rate

Using Eq. (3.8), the local value of entropy production is related to the rate of entropy increase within the system by a volume integral... 

If the local entropy production, <, is integrated over the volume, it is called the volumetric rate of entropy production... 

The rate of entropy production over the cross-section S" may be calculated by the following integration ... 

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... 

If the system is nonuniform in temperature and pressure, the derived equations pertain to processes in an elementary volume—that is, to the density of distribution of corresponding quantities. Therefore, determining the system total rates of entropy production and energy dissipation must be done with the integration over the volume. 

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... 

The entropy production rate and the complete set of equations that follows can be most conveniently written for the liquid film, the interface, and the vapour film in series.Film layers of thicknesses 5 and in the liquid and the vapour are illustrated in Figure 5. With constant fluxes (in stationary states), the integration is easy. The approach was called the integrated interface approach.For the three layers, the integrated overall force is the sum of the integrated force across each layer ... 

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... 

The time derivative of entropy production is called the rate of entropy production, and can be calculated from the laws of the conservation of mass, energy, and momentum, and the second law of thermodynamics expressed as equality. If the local entropy production, a, is integrated over the volume, it is called the volumetric rate of entropy production. ... 

Generally, a chemical reaction has a nonlinear relation between the rate and the driving force AIT. Chemical reactors are often designed to operate at the maximum rate of reaction. An alternative is a reactor operated with minimum useful work lost. The lost work per unit time in a chemical reactor is given by the Gouy—Stodola theorem, and is obtained by integrating the entropy production rate over the reactor volume ... 

We may define the volumetric (total) irreversible entropy production rate by w(t) = fya(r, t)dV/k and an average over all fluctuations in which the time-integrated entropy production is positive by then... 

The dependence of the entropy production on the process parameters is complicated, even if the inlet mass flowrates and compositions have been fixed. The outlet temperature T, depends on T,, T2, Q, Wr, and the outlet composition depends on Wr. But ITr is limited by the condition > 0 where depends on Tj as well as on the stream 3 composition. On the other hand, Wr is also limited by the reaction kinetics, taking Tj too low, the reaction rate could become negligibly small (a frozen reaction ) even in the presence of a catalyzer. Relatively simple is the case of fast irreversible reactions , such as combustion reactions. Then, in a broad range of parameters, the integral reaction rate Wr attains its maximum (complete combustion) and the outlet composition is fixed by this condition (see also (6.2.116) below). Thus gs depends on T, 72, and Q, given Tq obeying the conditions (6.2.80). At constant T, and T2, (6.2.74) implies... 

The description will first outline the fundamental rate phenomena and driving forces at the microscopic, continuum scale. Then a spatial integration to a macroscopic scale is carried out to explicitly introduce geometric design variables like volume and surfaces, enabling to define the behavior of the process. The description focuses on equations of change for inventories and on the entropy production rate. 

Previously, for 2-methyl pentane cracking we have used Eley-Rideal kinetic rate expressions to describe the inhibition and poisoning effects of species in the feed, as well as intermediate and product species. In order to utilise such kinetic expressions values for the adsorption equilibrium constants are required. A method for estimating adsorption equilibrium constants has been proposed that uses an integrated form of van t Hoff equation. The heats of adsorption have been calculated using proton affinities and heats of condensation. The entropy of adsorption has been calculated using the Sackur-Tetrode expression. 

The local rate of production of entropy decreases when the Reynolds number increases along the plate, but there is no minimum even if we integrate Q over the whole volume of the limiting layer. 

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