Reversible adiabatic

Obviously die first law is not all there is to the structure of themiodynamics, since some adiabatic changes occur spontaneously while the reverse process never occurs. An aspect of the second law is that a state fimction, the entropy S, is found that increases in a spontaneous adiabatic process and remains unchanged in a reversible adiabatic process it caimot decrease in any adiabatic process. 

The Carnot cycle is formulated directly from the second law of thermodynamics. It is a perfectly reversible, adiabatic cycle consisting of two constant entropy processes and two constant temperature processes. It defines the ultimate efficiency for any process operating between two temperatures. The coefficient of performance (COP) of the reverse Carnot cycle (refrigerator) is expressed as... 

These derivatives are of importance for reversible, adiabatic processes (such as in an ideal turbine or compressor), since then the entropy is constant. An example is the Joule-Tnomson coefficient. 

Total Pressure is the pressure that would occur if the fluid were brought to rest in a reversible adiabatic process. Many texts and engineers use the words total and stagnation to describe the flow characteristics interchangeably. To be accurate, the stagnation pressure is the pressure that would occur if the fluid were brought to rest adia-baticaUy or diabatically. 

Total temperature is the temperature that would occur when the fluid is brought to rest in a reversible adiabatic manner. Just like its co mie p2L totalpressure, total eMd stagnation temperatures are used interchangeably by many test engineers. 

Isentropic A reversible adiabatic process, in which there is no change in the entropy of the system. 

Total pre.ssure is the pressure of the gas brought to rest in a reversible adiabatic manner. It can be measured by a pitot tube placed in the flow... 

Clearly, if A is zero (no heat transfer), then the normal polytropic relation holds. A point of interest is that if Tjp = (1 — A) then rj = 1 and the expansion becomes isentropic (but not reversible adiabatic). 

A reversible adiabatic process is known as isentropic. Thus, the two conditions are directly related. In actual practice compressors generate friction heat, give off heat, have valve leakage and have piston ring leakage. These deviations... 

If a system be affected in a reversible adiabatic way, the allowed motions are transformed into allowed motions. 

Corollary. In all reversible adiabatic changes the entropy remains constant such changes are therefore isentropic changes. 

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

Figure 2.11 Graph of empirical temperature 0 against. v, a state variable such as pressure, (a) Reversible adiabatic paths (solid lines 1 — 2 and 1— 2)...

For each state identified on the 9 isothermal as 1, 1, l",..., let us draw paths representing reversible adiabatic processes that intersect a second isotherm at 02. The intersections of the reversible adiabatic paths from states 1,1 and 1" on 9 with those on 02 are denoted by 2, 2 and 2", respectively. Along the three paths, 1-2, l -2, and l"-2", no heat is absorbed or liberated because the processes that connect these points are defined to be adiabatic. 

In addition, we have established that there is a sense of direction to the location of the inaccessible states. State 2, the state reached from 1 by a reversible adiabatic path, represents the division between the states on the second isotherm that are accessible and inaccessible from state 1. We represent this schematically in Figure 2.1 lb, where the reversible adiabatic path separates states that are accessible from state 1 from those that are inaccessible. The observation that the reversible path serves as the boundary between the two sets of states will be useful later when we show the direction of allowed processes in terms of the sign of A5(universe). 

Now we can consider the effect of variations of 9 with a second variable, x2. Since we have been general about the nature of the variable, xj, we can expect to obtain similar behavior for the variable X2. We construct isotherms using the same B and 62, but this time in the direction of x2. Our initial point will be the same state 1 as earlier. The value of x2 in state 1 will fix the location of this state on the isotherm in the new direction. A reversible adiabatic path can be constructed that connects state 1 with a state on the second isothermal in the x2 direction. Irreversible states located on one side of this point will be inaccessible from state 1 by adiabatic paths, while states located on the other side of that point will be accessible. Thus, there exist states located on the plane defined by 9 and X2 that are inaccessible from point 1. Similar conclusions can be drawn by considering isotherms localized on the planes formed by 9 and each of the x,. 

Figure 2.12 A set of parallel, isentropic surfaces ordered so that S, > S2 > S3. The solid curve marked 6 rev = 0 represents a reversible adiabatic path that connects two states that lie on the entropy surface. Si. The dashed curves marked 6qm = 0 are irreversible paths that connect states on different entropy surfaces. Only one of these two paths will be allowed the other will be forbidden.

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. 

In the analysis in Section 2.2b, we showed that for any given initial state there are states that are accessible via irreversible adiabatic paths from the initial state, as well as states that are inaccessible from that initial state by way of irreversible adiabatic paths. Figure 2.11b showed that a reversible adiabatic path containing the initial state marked the division between the states that were accessible or inaccessible from that state, with all accessible states lying on one side of the reversible adiabatic path, and all inaccessible states lying on the other side of it. 

The adiabatic expansion of a gas is an example of (b). In the reversible adiabatic expansion of one mole of an ideal monatomic gas, initially at 298.15 K, from a volume of 25 dm3 to a final volume of 50 dm3, 2343 J of energy are added into the surroundings from the work done in the expansion. Since no heat can be exchanged (in an adiabatic process, q = 0), the internal energy of the gas must decrease by 2343 J. As a result, the temperature of the gas falls to 188 K. 

Example 3.8 Show that for the reversible adiabatic expansion of ideal gas with constant heat capacity... 

Solution In a reversible adiabatic expansion, 6qrev = T dS = 0. Thus, the process is isentropic, or one of constant entropy. To obtain an equation relating p, V and T, we start with... 

It is useful to compare the reversible adiabatic and reversible isothermal expansions of the ideal gas. For an isothermal process, the ideal gas equation can be written... 

Figure 3.2 compares a series of reversible isothermal expansions for the ideal gas starting at different initial conditions. Note that the isotherms are parallel. They cannot intersect since this would give the gas the same pressure and volume at two different temperatures. Figure 3.3 shows a similar comparison for a series of reversible adiabatic expansions. Like the isotherms, the adiabats cannot intersect. To do so would violate the Caratheodory principle and the Second Law of Thermodynamics, since the gas would have two different entropies at the same temperature, pressure, and volume. 

In Chapter 2 (Section 2.2a) we qualitatively described the Carnot cycle, but were not able to quantitatively represent the process on a p— V diagram because we did not know the pressure-volume relationship for a reversible adiabatic process. We now know this relationship (see section 3.3c), and in Figure 3.3, we compare a series of p-V adiabats with different starting temperatures for an... 

Figure 3.3 Reversible adiabats of a monatomic ideal gas with CY. m = 3/2/ , starting with the same initial and final volumes, but with different pressures.

When a process is isentropic, q - F a reversible process is isentropic when q = 0, that is a reversible adiabatic process is isentropic. 

The above relations apply for an ideal gas to a reversible adiabatic process which, as already shown, is isentropic. 

The velocity uw = fkP2v2 is shown to be the velocity of a small pressure wave if the pressure-volume relation is given by Pifi = constant. If the expansion approximates to a reversible adiabatic (isentropic) process k y, the ratio of the specific heats of the gases, as indicated in equation 2.30. 

Response time constant 403 Rkster. S. 6-13,655 Return bends, heat exchanger 505 Reversed flow 668 Reversibility, isothermal flow 143 Reversible adiabatic, isentropic flow 148... 

Annular flow reactors, such as that illustrated in Figure 3.2, are sometimes used for reversible, adiabatic, solid-catalyzed reactions where pressure near the end of the reactor must be minimized to achieve a favorable equilibrium. Ethylbenzene dehydrogenation fits this situation. Repeat Problem 3.7 but substitute an annular reactor for the tube. The inside (inlet) radius of the annulus is 0.1m and the outside (outlet) radius is 1.1m. 

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. 

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