Entropy production equation

Here Sq accounts for the heat flow due to heat transfer as well as mass transfer. The enthalpy also is modified as follows 

Subsystem I and II may both exchange matter and energy, and we have 

Comparing this equation with the first law of thermodynamics dH 8q - VdP (for a closed system and for - dP), we obtain 

Equation (3.132) suggests that the energy flows exchanged between subsystem I and n are equal with opposite signs. 

Assuming that the local thermodynamic equilibrium holds, we can write the Gibbs relation in terms of specific properties 

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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. 

Equations (8.30) and (8.32) constitute 27 relations for the transport coefficients lik. Further relations can be obtained by eliminating the three fluxes j5, j7, and j9 from the entropy production equation (4.10) by using Eqns. (8.29) and (8.31)... 

Using Eq. (3.220) in the local entropy production equation, we have... 

In another operating configuration, we can compare the respective size and durations for specified duty and entropy production. Equations (5.71) and (5.72) are still valid, and we have P > Pav2 and J = J2, which yield... 

Using a dissipation function or entropy production equation, the conjugate flows and forces are identified and used in the phenomenological equations for simultaneous heat and mass transfer. Consider the heat and diffusion flows in a fluid at mechanical equilibrium not undergoing a chemical reaction. The dissipation function for such a system is... 

According to the Curie-Prigogine principle, a scalar flow, such as the rate of reaction, cannot be coupled with a vectorial flow of a transport process in an isotropic medium where an equilibrium-dividing surface is symmetric with respect to rotations around any local normal vector. However, the symmetry properties alone are not sufficient for identifying physical coupling the actual physics considered in deriving the entropy production equation and the specific structure, such as anisotropy, are necessary. 

Equation above shows the three contributions to the rate of entropy production due to heat flow, mass flow, and the chemical reaction, respectively, and excludes the viscous and electrical effects. As the membrane is assumed to be an isotropic medium, there will be no coupling between the vectorial heat and mass flows and scalar chemical reaction, according to the Curie-Prigogine principle. Under these conditions, entropy production equation identifies the conjugate forces and flows, and linear relations for coupled heat and mass flows become... 

The term to the right of the equal sign in Eq. (12.32) is the excess entropy production. Equations (12.31) and (12.32) describe the stability of equilibrium and nonequilibrium stationary states. The term 82S is a Lyapunov functional for a stationary state. 

For constant V and S ys, the external work variation, dw is dw < —dE. For a reversible process, the external work is —dE, the maximum external woric available with dV = 0 and dS y. For an irreversible process, the available work is less than the maximum. In terms of the internal entropy production, equation 26 becomes... 

The first term on the right-hand side of Eq. (19.3) refers to convective transport, the second term with the entropy flow vector s refers to the entropy flow by molecular transport, and term gs is the rate of local entropy production. Equation (19.3) goes back to Jaumann ... 

The matrix of phenomenological coefficients is symmetric provided that the conjugate flows and forces are identified by the entropy production equation or the dissipation function. 

Using a dissipation function or entropy production equation, the conjugate flows and forces are identified and used in the phenomenological equations for simultaneous heat and mass transfer. 

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