Total entropy production

If no current flows (d /dt) = J2 = 0 and hence the first term is interpreted as the entropy production due to thermal conduction driven by the temperature gradient defined be T. In the absence of a temperature gradient however, T = 0 and hence the second term represents entropy production due to resistive heating, which is proportional to the square of the current. Since the sum of these two terms is known experimentally to account for the total entropy production in the process, it is inferred that L12 = L21. 

For an adiabatic system, the rate of total entropy production 5tot is a functional of the concentration field c(x),... 

The second law requires that the total entropy production in a system of several reactions be positive for a closed system removed from equilibrium. However, in the case of thermodynamic coupling of reactions (26), it is not necessary that individual reaction entropy productions be positive. Apparently such reaction systems have not yet been considered in connection with natural water systems. 

To obtain the total entropy production in the tower, Equation 11 must be integrated over the total diffusion volume... 

Example 3.8 Entropy production in a composite system Consider a composite system consisting of subsystem I enclosed inside subsystem H The whole system containing subsystem I and II is isolated. However, in subsystems I and II some irreversible processes may take place. Discuss the total entropy production in the whole system. 

This phenomenon shows that decrease or absorption of entropy in subsystem II may be compensated by a larger entropy production in subsystem I. This is possible only if subsystems I and n are coupled by some suitable coupling mechanisms leading to dS dS I <7.S II>0. With thermodynamic coupling a process in subsystem n may progress in a direction contrary to that determined by its own thermodynamic force. Some biological reactions represent coupled reactions for which the total entropy production is positive. 

This integration enables one to determine the total entropy production. When phenomena at the interface between two phases are considered, the amount of entropy produced is taken per unit surface area. 

There exist a large number of phenomenological laws for example, Fick s law relates to the flow of a substance and its concentration gradient, and the mass action law explores the reaction rate and chemical concentrations or affinities. When two or more of these phenomena occur simultaneously in a system, they may couple and induce new effects, such as facilitated and active transport in biological systems. In active transport, a substrate can flow against the direction imposed by its thermodynamic force. Without the coupling, such uphill transport would be in violation of the second law of thermodynamics. Therefore, dissipation due to either diffusion or chemical reaction can be negative only if these two processes couple and produce a positive total entropy production. 

An irreversible process causes the entropy production in any local element of a system, and the rate of total entropy production is... 

We can also compare the two processes with the same total entropy productions, the same size and duration, and the same phenomenological coefficients. Process 1 has only equipartitioned forces therefore, the duties of these processes will be different. The total entropy productions for the processes are expressed as... 

Ehminating the constant (average) force A av between Eq. (5.83) and the total entropy production d>av = JAY.lv, we obtain... 

Example 5.9 Distribution of driving forces Consider two identical heat exchangers 1 and 2 operating at steady state with the same total entropy production P. Assume that the distributions of the driving forces are different and are characterized by... 

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. 

Example 5.10 Variance and heat exchangers Consider two heat exchangers with the same heat duty and total entropy production. They have different heat transfer areas and different variances (s ])2 < (v2)2- and hence... 

Very large affinity values may cause instability, and lead to new states that are no larger homogeneous in space. This causes a discontinuous decrease of entropy, and has important consequences in oscillating chemical reactions. Such reactions are far from equilibrium, and present undamped fluctuation on a macroscopic scale. Oscillations around a stationary state are possible as long as the total entropy production is positive. 

The second law requires the total entropy production resulting from all the simultaneous reactions to be positive. This has been verified experimentally. Sometimes, a system has two simultaneous coupled reactions, such that... 

The first term on the left of this equation is the thermodynamic force. The force is not necessarily constant when we have minimum lost work. The optimum force that gives the minimum total entropy production rate is obtained from... 

For systems not far from equilibrium, the total entropy production reaches a minimum value and also assures the stability of the stationary state. However, for systems far from equilibrium, there is no such general criterion. Far from equilibrium, we may have order in time and space, such as, appearance of rhythms, oscillations, and morphological structurization. 

We thus arrive at an interesting conclusion regarding thermodynamics and process control. It is not the steady state irreversibility (inefficiency) that matters for control but the ability to alter the rate of total entropy production in response to the system s departure from steady state. We have previously indicated qualitatively how entropy is produced. To see how the rate of entropy production changes with the system s state, we need to perform a quantitative analysis. This requires a brief introduction to the subject of nonequilibrium thermodynamics (Callen, 1985 Haase, 1990). 

The total entropy production as needed in Equation (A.22) is related to the entropy density as follows ... 

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 question of constrained optimisation is answered in a standard manner by Euler-Lagrange optimisations. By formulating the problem with optimal control theory, Johannessen and Kjelstrup explained that the Hamiltonian of the problem was constant in a study of chemical reactors. The total entropy production for a plug flow reactor was written as a function of a position-dependent state variable vector x(z) and the control variable u(z) ... 

To reduce the lost work in industrial process plants, the minimization of entropy production rates in process equipment is suggested as a strategy for future process design and optimization [81]. The method is based on the hypothesis that the state of operation that has a minimum total entropy production is characterized by equipartition of the local entropy production. In this context we need to quantify the entropy sources of the various irreversible unit operations that occur in the industrial system. 

In an irreversible process the total entropy production as well as the local entropy production must be positive (a > 0). 

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... 

Note that this result is general for any reaction, a/V = jR I>i (Aci)o/Ci. The total entropy production near equilibrium does not depend on the rate of reaction but only on the displacements and the equilibrium concentrations. 

Because of spatial curvature an initially stationary array of non-interacting particles (ideal gas) spontaneously generates relative internal (zero-point) motion. This intrinsic microscopic instability is responsible for the dispersal of energy and the source of entropy. Transportation along the interface inverts, not only the time coordinate, but also the entropy production. Integrated over the entire closed universe the total entropy production is zero... 

Now our object would be to evaluate entropy production dj due to irreversible processes inside the system, which are simply the transport of matter and electricity. The total entropy production d6 due to internal as well as external factors would be given by... 

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