Irreversible adiabatic processes

In the neighborhood of every equilibrium state of a thermodynamic system, there exist states unattainable from it by any adiabatic process (reversible or irreversible). 

Presumably all points on the same surface can be connected by some solution curve (reversible adiabatic process). Flowever, states on surface S2, for example, cannot be connected to states on either Si or S3 by any reversible adiabatic path. Rather, if they can be connected, it must be through irreversible adiabatic paths for which dS 0. We represent two such paths in Figure 2.12 by dashed lines. 

The Caratheodory analysis has shown that a fundamental aspect of the Second Law is that the allowed entropy changes in irreversible adiabatic processes can occur in only one direction. Whether the allowed direction is increasing or decreasing turns out to be inherent in the conventions we adopt for heat and temperature as we will now show. 

The conditions existing during the adiabatic flow in a pipe may be calculated using the approximate expression Pi/ = a constant to give the relation between the pressure and the specific volume of the fluid. In general, however, the value of the index k may not be known for an irreversible adiabatic process. An alternative approach to the problem is therefore desirable.(2,3)... 

Besides the reversible and irreversible processes, there are other processes. Changes implemented at constant pressure are called isobaric process, while those occurring at constant temperature are known as isothermal processes. When a process is carried out under such conditions that heat can neither leave the system nor enter it, one has what is called an adiabatic process. A vacuum flask provides an excellent example a practical adiabatic wall. When a system, after going through a number of changes, reverts to its initial state, it is said to have passed through a cyclic process. 

In a system undergoing a reversible adiabatic process, there is no change in its entropy. This is so because by definition, no heat is absorbed in such a process. A reversible adiabatic process, therefore, proceeds at constant entropy and may be described as isentropic. The entropy, however, is not constant in an irreversible adiabatic process. 

To determine the entropy change in this irreversible adiabatic process, it is necessary to find a reversible path from a to b. An infinite number of reversible paths are possible, and two are illustrated by the dashed lines in Figure 6.7. 

As in the complete cycle (irreversible adiabatic process from atob followed by the three reversible steps)... 

Furthermore, in the four steps of the cycle (Fig. 6.8) three are adiabatic (one irreversible, two reversible). Hence, Qcycie is identical with Q of the isothermal step, that is, Q of Equation (6.104). If g > 0, then W < 0 that is, work would have been performed by the system. In other words, if Q were positive, we would have carried out a cyclical process in which heat at a constant temperature had been converted completely into work. According to the Kelvin-Planck statement of the second law, such a process cannot be carried out. Hence, Q cannot be a positive number. As Q must be either negative or zero, it follows from Equation (6.104) that... 

The actual finite-time Rankine cycle is shown in Fig. 7.17. The cycle is an external and internal irreversible cycle that consists of two irreversible internal adiabatic processes (pump and turbine) and two irreversible external isobaric heat-transfer processes. The heat source and heat sink are... 

The actual finite-time Brayton cycle as shown in Fig. 7.30 consists of two adiabatic processes and two isobaric heat-transfer processes. The cycle exchanges heat with its surroundings in the two isobaric external irreversible heat-transfer processes. By taking into account the rates of heat transfer associated with the cycle, the upper bound of the power output of the cycle can be found as illustrated in Example 7.19. 

ADIABATIC PROCESS. Any thermodynamic process, reversible or irreversible, which takes place in a system without the exchange of heat with the surroundings. When the process is also reversible, it is called isentropic, because, then the entropy of the system remains constant at every step of the process, fin older usage, isentropic processes were called simply adiabatic, or quasistatic adiabatic the distinction between adiabatic and isentropic processes was not always sharply drawn.)... 

In a third experiment, the initial conditions of Vi, Pi for the ideal gas and its temperature, T are identical to those in the first two experiments. This time no heat is supplied (corresponding to an adiabatic process, q = 0) and a shutter is lowered in front of the moveable (frictionless) piston, after which the pressure on the piston is adjusted to, Pf, the same pressure that was experienced at the end of the two previous experiments. The air in the voidage between shutter and piston is now evacuated creating a vacuum, after which the shutter is withdrawn and the gas expands irreversibly. The work done is now ... 

Consider now adiabatic processes wherein no heat transfer occurs. We represent on the PV diagram of Fig. 5.6 an irreversible, adiabatic expansion of a fluid from an initial equilibrium state at point A to a final equilibrium state at point B. Now suppose the fluid is restored to its initial state by a reversible process. If the initial process results in an entropy change of the fluid, then there must be heat transfer during the reversible restoration process such that... 

Figure 5.6 Cycle containing an irreversible adiabatic process A to B.

A reversible adiabatic process is isentropic, meaning that a substance will have the same entropy values at the beginning and end of the process. Systems such as pumps, turbines, nozzles, and diffusers are nearly adiabatic operations and are more efficient when irreversibilities, such as friction, are reduced, and hence operated under isentropic conditions. 

Let us simplify the notation by setting dQj" 2 = dX and dQj 2 - IdS1" 2 TdS. One must now consider the possibilities (a) dQi - TdS > 0, i.e., TdS < dQi, or (b) dQi - TdS < 0, i.e., TdS > dQA. The present discussion must apply to all cases including the special situation where the irreversible process is carried out adiabatically. Under alternative (b) one then obtains TdS > 0 for the adiabatically isolated system, and no contradictions are uncovered. Under alternative (a) one would require TdS < 0 for an irreversible, adiabatic process, and by the corollary to the Second Law, discussed in Section 1.13, this possibility must be ruled out. We thus claim that... 

One may now distinguish between (i) an adiabatic process wherein 3Q - 0 then the entropy of the system increases according to dS - 38, reflecting the occurrence of irreversible processes within the system and (ii) an isentronic process, in... 

Dhar modelled the stretching of a polymer using the stochastic Rouse model, for which distributions of various definitions of the work can be obtained. Two mechanisms for the stretching were considered one where the force on the end of the polymer was constrained and the other where its end was constrained. Dhar commented that the variable selected for the work was only clearly identified as the entropy production in the latter case. In the former case they argue that the average work is non-zero for an adiabatic process, and therefore should not be considered as an entropy production, however we note that the expression is consistent with a product of flux and field as used in linear irreversible thermodynamics. 

The present volume involves several alterations in the presentation of thermodynamic topics covered in the previous editions. Obviously, it is not a trivial exercise to present in a novel fashion any material that covers a period of more than 160 years. However, as best as I can determine the treatment of irreversible phenomena in Sections 1.13, 1.14, and 1.20 appears not to be widely known. Following much indecision, and with encouragement by the editors, I have dropped the various exercises requiring numerical evaluation of formulae developed in the text. After much thought I have also relegated the Caratheodory formulation of the Second Law of Thermodynamics (and a derivation of the Debye-Hiickel equation) as a separate chapter to the end of the book. This permitted me to concentrate on a simpler exposition that directly links entropy to the reversible transfer of heat. It also provides a neat parallelism with the First Law that directly connects energy to work performance in an adiabatic process. A more careful discussion of the basic mechanism that forces electrochemical phenomena has been provided. I have also added material on the effects of curved interfaces and self assembly, and presented a more systematic formulation of the basics of irreversible processes. A discussion of critical phenomena is now included as a separate chapter. Lastly, the treatment of binary solutions has been expanded to deal with asymmetric properties of such systems. 

A single gas stream enters a process at conditions T, P, and leaves at pressure P2. The process is adiabatic. Prove that the outlet temperahire T2 for the actual (irreversible) adiabatic process is greater than that for a reversible adiabatic process. Assume the gas is ideal with constant heat capacities. 

The HS diagram of Fig. 4-2 compares the path of an actual expansion in a turbine with that of an isentropic expansion for the same intake conditions and the same discharge pressure. The isentropic path is the dashed vertical line from point 1 at intake pressure Pi to point 2 at P2. The irreversible path (solid line) starts at point 1 and terminates at point 2 on the isobar for P2. The process is adiabatic, and irreversibilities cause the path to be directed toward increasing entropy. The greater the irreversiblity, the farther point 2 hes to the right on the P2 isobar, and the lower the value of q. 

For statistically independent systems, the property S is extensive, it is invariant during all reversible adiabatic processes, and it increases during all irreversible adiabatic processes. 

Theorem. For any system in any state, a property S exists that remains invariant in any reversible adiabatic process, that increases in any irreversible adiabatic process, and that is... 

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