Distribution of entropy production

Thermodynamic cost analysis relates the thermodynamic limits of separation systems to finite rate processes, and considers the environmental impact through the depletion of natural resources within the exergy loss concept. Still, economic analysis and thermodynamic analysis approaches may not be parallel. For example, it is estimated that a diabatic column has a lower exergy loss (39%) than that of adiabatic distillation however, this may not lead to a gain in the economic sense, yet it is certainly a gain in the thermodynamic sense. The minimization of entropy production is not always an economic criterion sometimes, existing separation equipment may be modified for an even distribution of forces or an even distribution of entropy production. Thermodynamic analysis requires careful interpretation and application. 

Example 5.5 Equipartition principle in separation processes Extraction Since the minimization of entropy production is not always an economic criterion, it is necessary to relate the overall entropy production and its distribution to the economy of the process. To do this, we may consider various processes with different operating configurations. For example, by modifying an existing design, we may attain an even distribution of forces and hence an even distribution of entropy production. 

The above equation suggests that in heat exchanger 1, for example, the cold fluid would be heated more or the use of a larger cold flow rate is possible. Therefore, the heat exchanger with the smallest s2 would achieve the largest duty and be more economic in practice. This simple analysis suggests that the distribution of entropy production may play a more important role than total entropy production. 

Fluid flow and the wall-to-fluid heat transfer in a packed duct are of interest in fixed bed chemical reactors, packed separation columns, heat exchangers, and some heat storage systems. Estimate the distribution of entropy production in a packed duct flow with asymmetric heat effects. 

Using the velocity and temperature gradients, we obtain the dimensionless entropy production for the empty bed (Figure 4.10). Comparison of Figures 4.9 and 4.10 indicates that outside the wall region, the distribution of the rate of entropy production is uniform in the packed bed, which is the thermodynamic optimality criterion. The profile of entropy production shows a typical S shape in the empty bed. 

Some options for achieving a thermodynamic optimum are to improve an existing design so the operation will be less irreversible and to distribute the irreversibilities uniformly over space and time. This approach relates the distribution of irreversibilities to the minimization of entropy production based on linear nonequilibrium thermodynamics. For a transport of single substance, the local rate of entropy production is... 

Since the minimization of entropy production is not always an economic criterion, it is necessary to relate the overall production and distribution of entropy to the economic analysis by considering various processes with different structures and operating configurations. 

According to Eq. (5.87), the quantities A7 0transfer unit. Generally, operating costs are linearly related to dissipation, while investment costs are linearly related to the size of equipment. The optimum size distribution of the transfer units is obtained when amortization cost is equal to the cost of lost energy due to irreversibility. The cost parameters a and b may be different from one transfer unit to another when a = b, then av/F0pt is a constant, and the optimal size distribution leads to equipartition of the local rate of entropy production. The optimal size of a transfer unit can be obtained from Eq. (5.78)... 

Distributing the entropy production as evenly as possible along space and time would allow for the design and operation of an economic separation process. Dissipation equations show that both the driving forces and flows play the same role in quantifying the rate of entropy production. Therefore, the equipartition of entropy production principle may point out that the uniform distribution of driving forces is identical to the uniform distribution of flows. 

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. 

Equations (89)-(94) govern the behavior of an isotropic liquid in an electric field. For very weak fields the liquid is conducting but immobile. With increasing field a non-uniform spatial distribution of Q and E arises due to an ion drift to electrodes or/and charge injection, and the liquid starts moving in order to satisfy the minimum of entropy production. The corresponding critical field is considered to be the threshold field for the appearance of an EHD instability. Below we discuss the two particular cases mentioned earlier, namely the injection mode and the electrolytic mode. 

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