Entropy production per unit volume

According to irreversible thermodynamics, the entropy production per unit volume S for an isothermal system can be written... 

The rate of entropy production per unit volume may then be written as... 

From Eqs. (3.118), (3.137), and (3.138), the entropy source strength or the rate of local entropy production per unit volume < > is defined by... 

Researchers and engineers extensively utilized second law analysis in the field of heat and fluid flow. Bejan developed the basic approach, methodology, and applications. In two-dimensional Cartesian coordinates, the local rate of entropy production per unit volume in a convective heat transfer is... 

Example 4.5 Entropy production in a flow through an annular packed bed The introduction of suitable packing into a fluid flow passage considerably enhances wall-to-fluid heat transfer, and hence reduces the entropy production due to heat transfer but increases the entropy production due to fluid-flow friction. Heat transfer to a fluid flowing in an annulus has a technical importance because we can heat or cool either or both of the surfaces independently. Entropy production provides a new criterion in analyzing such processes. In terms of the velocity and temperature profiles, the local rate of entropy production per unit volume of an incompressible Newtonian fluid for a two-dimensional annular flow is... 

Entropy production per unit volume in terms of reaction affinities is... 

The value of A is maintained at a nonzero value, and hence at nonequilibrium conditions. The entropy production per unit volume is... 

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 starting point for our analysis is an expression for the rate of entropy production per unit volume. For isothermal, isobaric processes in the absence of external force fields, the rate of entropy production due to ditfusion is given by (cf. Eq. 2.3.1)... 

The starting point for the rigorous derivation of the diffusive fluxes in terms of the activity is the entropy equation as given by (1.170), wherein the entropy flux vector is defined by (1.171) and the rate of entropy production per unit volume is written (1.172) as discussed in sec. 1.2.4. 

In other words, the matrix giving the phenomenological coefficients for the system is symmetric. Another important part of this theory is that one can estimate the entropy generated in an irreversible process. Onsager showed that the local rate of entropy production per unit volume is... 

From the above two cases it is clear that the rate of internal entropy production per unit volume is the sum of the products of the flux and the conjugate driving force. This may he expressed as... 

Integrate this result from r = 0 to r = 00 to obtain the total entropy production per unit volume. Show that the total entropy production, a/V, for the chemical reaction A -f B C, discussed in Section 32.19, is given by... 

On combining the balance equation for energy assuming the absence of velocity gradients, the first law of thermodynamics and assuming Gibbs equation for entropy production for the case of local equilibrium, a the entropy production per unit volume per unit time due to the occurrence of irreversible processes in the system is given by... 

The entropy production per unit volume is ct = Jti— + 7r2— + Jr3-jr Using the relationship in Eqn (b), the entropy production becomes... 

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