Thermodynamic optimum

t and tf are the initial and final times of the irreversible process and T0 the environmental temperature. The maximization is carried out with the constraints imposed on the process. Equation (5.48) represents the second law of thermodynamics in equality form by subtracting the work equivalent of the entropy produced, which is the decrease in availability in the process. Availability depends on the variables of the system as well as the variables of the environment 

the temperature, pressure, and chemical potential are estimated at ambient conditions. For an optimal control problem, one must specify (i) control variables, volume, rate, voltage, and limits on the variables, (ii) equations that show the time evolution of the system which are usually differential equations describing heat transfer and chemical reactions, (iii) constraints imposed on the system such as conservation equations, and (iv) objective function, which is usually in integral form for the required quantity to be optimized. The value of process time may be fixed or may be part of the optimization. 

Example 5.4 Minimization of entropy production For a fixed design, the minimization of the rate of entropy production may yield optimal solutions in some economic sense. Such a minimization comes with certain set of constraints. For a single force-flow system, the local rate of volumetric entropy production is 

The entropy production is a function of the temperature field. Then, the minimization problem is to obtain the temperature distribution T x) corresponding to a minimum entropy production t using the following Euler-Lagrange equation  

Minimizing the entropy production function with the constraint expressed in Eq. (5.54), the above equation becomes 

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Based on the above observations, one can conclude that at temperatures below about 10°C, the thermodynamic optimum structure is the target structure a combination of both emulsifiers adsorbed at the surface. At higher temperatures, however, there is no clear thermodynamic difference between an emulsion where the surface is covered with El only or an emulsion where the surface is covered by a combination of El and E2. Since El is available in excess... 

Once the minimum theoretical energy requirements and applicable process options have been determined, a formal facilitated workshop follows to modify the process or facility to bring the design closer to the thermodynamic optimum within project economic constraints. 

This chapter establishes a direct relation between lost work and the fluxes and driving forces of a process. The Carnot cycle is revisited to investigate how the Carnot efficiency is affected by the irreversibilities in the process. We show to what extent the constraints of finite size and finite time reduce the efficiency of the process, but we also show that these constraints still allow a most favorable operation mode, the thermodynamic optimum, where the entropy generation and thus the lost work are at a minimum. Attention is given to the equipartitioning principle, which seems to be a universal characteristic of optimal operation in both animate and inanimate dynamic systems. 

Next we use a similar approach to show further, after De Vos [24], that this thermodynamic optimum is not necessarily the economic optimum. After all, the higher the delivered power V ut/ the faster the investments for building the power plant will be returned, but the higher the efficiency r, the better the expenses for processing the primary fuel will be recovered. So, clearly, the economic optimum for the operation of the plant is in between the point for maximum power and the point for maximum efficiency, or... 

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. 

The optimal Reynolds number defines the operating conditions at which the cylindrical system performs a required heat and mass transport, and generates the minimum entropy. These expressions offer a thermodynamically optimum design. Some expressions for the entropy production in a multicomponent fluid take into account the coupling effects between heat and mass transfers. The resulting diffusion fluxes obey generalized Stefan-Maxwell relations including the effects of ordinary, forced, pressure, and thermal diffusion. 

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... 

Example 4.9 Entropy production in separation process Distillation Distillation columns generally operate far from their thermodynamically optimum conditions. In absorption, desorption, membrane separation, and rectification, the major irreversibility is due to mass transfer. The analysis of a sieve tray distillation column reveals that the irreversibility on the tray is mostly due to bubble-liquid interaction on the tray, and mass transfer is the largest contributor to the irreversibility. 

Distillation columns operating with close to uniform thermodynamic forces are analyzed for separating n-pentane from n-hcptanc (Table 4.14), and ethanol from water (Table 4.15). AsEq. (4.113) shows, chemical separation force (v,V/li,/7) should be uniform throughout the column for thermodynamic optimum. Separation of ethanol from water... 

Configurations that minimize s2(X) and s2 q) also minimize entropy production and lead to thermodynamically optimum configurations. Such thermodynamic analysis will contribute to the study of feasibility and economic analysis after relating the level of entropy production to engineering economics. 

A point of interest is the high water/melt mass ratio calculated in RUN4A this result suggests that the mixing geometry in the pouring mode FITS tests may have resulted in fairly water-rich mixtures, far from the thermodynamic optimum water/melt mass ratio. 

On the other hand, different time-temperature policies are optimal for different classes of complex reactions and these are considered in Chapter 2. Although the reversible reaction is also a complex reaction in the sense that two reactions occur, it is equally true that no additional species are involved in the second (reverse) reaction. Hence, the reversible reaction can also be regarded as a simple reaction. If the reaction is endothermic, its reversible nature makes no difference since both the reaction rate constant and the equilibrium constant increase with tanperature, and the maximum practicable temperature continues to be the optimal tanperature. But if the reaction is exothermic, an increase in tanperatuie has opposite effects it lowers the equilibrium constant but raises the rate constant. Hence, a thermodynamic optimum temperature exists. For any reaction such as A 1 with the rate equation -r = A ([A] - [/ ]/ 0, this optimum can be found by integrating the expression... 

Production of syngas in 3D matrix burners. From the viewpoint of thermodynamics, optimum conditions for syngas production through the partial oxidation of methane are O2/CH4 = 0.5 (a = 0.25), a temperature of 1073—1273 K, and a low pressure, 1 atm. However, such a process is difficult to implement in the absence of a catalyst. Therefore, the main technological challenge in the production of syngas by the partial oxidation of natural gas is to ensme a stable conversion process at low values of the oxidant equivalence ratio, within 0.3—0.5. Since mixtures with such a values are well outside the flammability limits under normal conditions, this is not a trivial task. 

Like all other chemical reactions, polymer syntheses may be subdivided into two categories, namely into kinetically controlled (KC) polymerizations and thermodynamically controlled (TC) polymerizations. KC polymerizations are characterized by irreversible reaction steps, equilibration reactions are absent, and the reaction products may be thermodynamically stable or not. TC polymerizations involve rapid equilibration reactions, above all formation of cyclics by back-biting of a reactive chain end (see Formula 5.1), and the reaction products represent the thermodynamic optimum at any stage of the polymerization process. Borderline cases also exist, which means that a rapid KC polymerization is followed by slow equilibration. This combination is typical for many Ring-opening polymerizations (ROPs). 

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