Adiabatic systems entropy

This is frequently stated for an isolated system, but the same statement about an adiabatic system is broader.) A2.1.4.6 IRREVERSIBLE CHANGES AND THE MEASUREMENT OF ENTROPY... 

Real fan total pressure difference is smaller for the same volume flow than that of an isentropic, theoretical fan. This is a result of the fan losses. These arise from the entropy generation in adiabatic systems. We investigate the losses separately, i.e., entropy generation in the impeller and casing. 

Leakage losses when the gas is transported from pressure side to suction side. Since the phenomenon is equalizing pressure in an adiabatic system, this will increase the entropy in the system. 

For an isolated (adiabatic) system, AS > 0 for any natural (spontaneous) process from State a to State b, as was proved in Section 6.8. An alternative and probably simpler proof of this proposition can be obtained if we use a temperature-entropy diagram (Fig. 6.13) instead of Figure 6.8. In Figure 6.13, a reversible adiabatic process is represented as a vertical line because AS = 0 for this process. In terms of Figure 6.13, we can state our proposition as follows For an isolated system, a spontaneous process from a to b must lie to the right of the reversible one, because AS = Sb Sa> 0. 

Entropy is central to stability and equilibrium. In a closed adiabatic system, any spontaneous change should lead to an increase in entropy, but the change may also lead to a change in the temperature. For an isothermal system, we require that, in a spontaneous change, the free energy G = H — TS he minimized. This means that for a spontaneous isothermal change... 

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

Equation (1.126) represents the change of entropy for an irreversible process in an adiabatic system as a function of the internal and external parameters. This may be an important property to quantify the level of irreversibility of a change, and hence yields (i) a starting point to relate the economic implications of irreversibility in real processes, and (ii) an insight into the interference between two processes in a system. 

Therefore, the total entropy produced within the system must be discharged across the boundary at stationary state. For a system at stationary state, boundary conditions do not change with time. Consequently, a nonequilibrium stationary state is not possible for an isolated system for which deS/dt = 0. Also, a steady state cannot be maintained in an adiabatic system in which irreversible processes are occurring, since the entropy produced cannot be discharged, as an adiabatic system cannot exchange heat with its surroundings. In equilibrium, all the terms in Eq. (3.48) vanish because of the absence of both entropy flow across the system boundaries and entropy production due to irreversible processes, and we have dJS/dt = d dt = dS/dt = 0. 

Classical thermodynamics states that the change of entropy production as a result of the irreversible phenomena inside a closed adiabatic system is always positive. This principle allows for the entropy to decrease at some place in the systems as long as a larger increase in the entropy at another place compensates for this loss. 

Equation (3.200) may be useful in describing the state of a system. For example, the state of equilibrium can be achieved for an adiabatic system, since the entropy generated by irreversible processes cannot be exchanged with the surroundings. 

If we consider the change of local entropy of a system at steady state ds/dt = 0, the local entropy density must remain constant because external and internal parameters do not change with time. However, the divergence of entropy flow does not vanish div J, = . Therefore, the entropy produced at any point of a system must be removed or transferred by a flow of entropy taking place at that point. A steady state cannot be maintained in an adiabatic system, since the entropy produced by irreversible processes cannot be removed because no entropy flow is exchanged with the environment. For an adiabatic system, equilibrium state is the only time-invariant state. 

Consider now an irreversible process in a closed system wherein no heat transfer occurs. Such a process is represented on the P V diagram of Fig. 5.6, which shows an irreversible, adiabatic expansion of 1 mol of fluid from an initial equilibrium state at point A to a final equilibrium state at pointB. Now suppose the fluid is restored to its initial state by a reversible process consisting of two steps first, the reversible, adiabatic (constant-entropy) compression of tile fluid to tile initial pressure, and second, a reversible, constant-pressure step that restores tile initial volume. If tlie initial process results in an entropy change of tlie fluid, tlien tliere must be heat transfer during tlie reversible, constant-P second step such tliat ... 

H-function in the statistical mechanics of molecular collision, and the excess entropy (S — Seq) for adiabatic systems in classical thermodynamics. 

It may well be true that the entropy of the universe is increasing (see Chapter 6), but whatever it is doing is quite irrelevant to the study of thermodynamics here on Earth. The difference between the two ways of looking at AS presented above essentially involves two different definitions of the system. In our preferred explanation, the system is the water in the pail, and its entropy decreases spontaneously. In the other view, the system is the universe, by implied hypothesis a closed composite adiabatic system, and the pail a portion of this composite system separated from the rest by diathermal walls. In the overall system, entropy increases. In this view, the choice of system is effectively taken from us—we must choose the universe as our system to preserve the dictum that entropy increases in spontaneous processes. 

Equation (2.3.10) shows that in closed systems, entropy can be generated in two general ways. First, as already discussed in 2.1.2, the lost work 5Wiogt is the energy needed to overcome dissipative forces that act to oppose a mechanical process. Second, the heat-transfer term in (2.3.10) contributes when a finite temperature difference irreversibly drives heat across system boundaries. This second term is zero in two important special cases (a) for adiabatic processes, = 0, and (b) for processes in which heat is driven by a differential temperature difference, Tg t = T dT. In both of these special cases, (2.3.10) reduces to... 

In 7.1.2 we showed that, for adiabatic processes occurring in closed systems, the combined laws (7.1.11) reduce to a requirement that the system entropy must always increase or remain constant. But if the system can exchange heat with its surroundings, then the entropy may increase, decrease, or remain constant, so for nonadiabatic processes, the entropy no longer serves as an indicator for changes. In this and the... 

As we have seen, the only changes which can take place in an adiabatic system are those in which the entropy either increases or remains constant. The same applies, of course, if the system is completely isolated, that is, if it does no work, as well as being within an adiabatic enclosure, so that the internal energy is constant. Thus whenever a system could change from a state of lower to one of higher entropy, dthin an enclosure of constant volume and energy, it is possible, in a thermodynamic sense, for this change to occur. 

Unless the spontaneous change is purely mechanical, it is irreversible. According to the second law, during an infinitesimal change that is irreversible and adiabatic, the entropy increases. For the isolated system, we can therefore write... 

The mathematical statement of the second law is associated with the definition of entropy S, dS = 8q /T. Entropy is a thermodynamic potential and a quantitative measure of irreversibility. For reversible processes, dS is an exact differential of the state function, and the result of the integration does not depend on the path of change or on how the change is carried out when both the initial and final states are at stable equilibrium. The entropy of a closed adiabatic system remains the same in a reversible process, and increases during an irreversible process. A system and its surrounding create an isolated composite system where the sum of the entropies of all reversible changes remains the same, and increases during irreversible processes. 

Figure 4.9. According to the Clausius inequality, the increase in system entropy dSsy by a process is greater than or equal to 6Q/T)sy for an adiabatic system, where SQ = 0, dSsy is greater than or equal to 0.

Entropy of an adiabatic system (5Q = 0) can never decrease the entropy increases at irreversible processes and remains constant at reversible processes. ... 

I Assume two adiabatic systems (a) and (b) each consisting of a steel block with the mass m = f.OOO kg. The mass-specific heat capacity of the steel is Cp = 0.481 kJ/kg K. In the initial state, the temperatures of the two blocks are 0a = 0°C and Of, = 100°C, respectively. At the time ti, the two steel blocks are brought into mutual thermal contact. At the time t2, the two blocks have reached thermal equilibrium at 0ab = 50°C. Calculate for the described process (1) — (2), the increase in entropy for the two subsystems ASa and ASf, respectively, as well as the increase in entropy AS at of the total system ... 

During the adiabatic process, the two systems did not exchange heat with their surroundings. In the process, the system entropy has been increcised by 11.7 J/K therefore, according to the Clausius inequality(4.12), this is an irreversible process. 

An adiabatic system contains two, thermally separated metal blocks A and B. Both of the blocks have the mass 1000 g and the specific heat capacity of the metal is Cp = 0.38 J/g K. In its initial state, block A is in thermal equihbrium at 0 °C and block B is in thermal equilibrium at 100 °C. The blocks are brought into thermal contact and after some time they have obtained equilibrium at the mutual temperature 50 °C. Calculate the total entropy increase AS. niv in the thermodynamic universe by this process and evaluate the result based on the Clausius inequality I... 

Temperature for Compression of Iron SOLUTION Because this process is reversible and adiabatic, the entropy of the system does not change. Hence, we can solve this problem by writings = s T,P) and setting ds to 0. From Table 5.1 ... 

It is still necessary to consider the role of entropy m irreversible changes. To do this we return to the system considered earlier in section A2.1.4.2. the one composed of two subsystems in themial contact, each coupled with the outside tliroiigh movable adiabatic walls. Earlier this system was described as a function of tliree independent variables, F , and 0 (or 7). Now, instead of the temperature, the entropy S = +. S P will be... 

Equation (A2.1.21) includes, as a special case, the statement dS > 0 for adiabatic processes (for which Dq = 0) and, a fortiori, the same statement about processes that may occur in an isolated system (Dq = T)w = 0). If the universe is an isolated system (an assumption that, however plausible, is not yet subject to experimental verification), the first and second laws lead to the famous statement of Clausius The energy of the universe is constant the entropy of the universe tends always toward a maximum. ... 

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