Expansion entropy

The specific internal energy of the fluid in the expanded state U2 can be determined as follows If a thermodynamic graph is used, assume an isentropic expansion (entropy s is constant) to atmospheric pressure po- Therefore, follow the constant-entropy line from the initial state to Po- Read h- and V2 at this point, and calculate the specific internal energy U2-... 

Let us look now at vapor-liquid systems with more than one component. A liquid stream at high temperature and pressure is flashed into a drum, i.e., its pressure is reduced as it flows through a restriction (valve) at the inlet of the drum. This sudden expansion is irreversible and occurs at constant enthalpy. If it were a reversible expansion, entropy (not enthalpy) would be conserved. If the drum pressure is lower than the bubblepoint pressure of the feed at the feed temperature, some of the liquid feed will vaporize. 

Steam turbine integration. Figure 6.32 shows a steam turbine expansion on an enthalpy-entropy plot. In an ideal turbine, steam... 

For example, the expansion of a gas requires the release of a pm holding a piston in place or the opening of a stopcock, while a chemical reaction can be initiated by mixing the reactants or by adding a catalyst. One often finds statements that at equilibrium in an isolated system (constant U, V, n), the entropy is maximized . Wliat does this mean ... 

The Carnot refrigeratiou cycle is reversible and consists of adiabatic (iseutropic due to reversible character) compression (1-2), isothermal rejection of heat (2-3), adiabatic expansion (3-4) and isothermal addition of heat (4-1). The temperature-entropy diagram is shown in Fig. 11-70. The Carnot cycle is an unattainable ideal which serves as a standard of comparison and it provides a convenient guide to the temperatures that should be maintained to achieve maximum effectiveness. 

The fact that shock waves continue to steepen until dissipative mechanisms take over means that entropy is generated by the conversion of mechanical energy to heat, so the process is irreversible. By contrast, in a fluid, rarefactions do not usually involve significant energy dissipation, so they can be regarded as reversible, or isentropic, processes. There are circumstances, however, such as in materials with elastic-plastic response, in which plastic deformation during the release process dissipates energy in an irreversible fashion, and the expansion wave is therefore not isentropic. 

A turboexpander generates the deep, low-temperature refrigeration industrially used for gas separation and liquefaetion, and a number of related purposes. It does so by the meehanism of eonstant entropy expansion, together with the produetion of power (a byproduet). The power is generated from the deerease in enthalpy of the stream itself. A turboexpander is a high effieieney turbine with numerous speeial features. These features make it eonveniently usable and reliable for small volumetrie flows at the low temperatures (and often rather high pressures) usually found in these applieations. 

The overall effieieney of a radial-inflow turbine is a funetion of effieieneies from various eomponents sueh as the nozzle and rotor. A typieal turbine expansion enthalpy/entropy diagram is shown in Figure 8-7. The total enthalpy remains eonstant through the nozzle, sinee neither work nor heat is transferred to or from the fluid. Within the rotor, the total enthalpy ehanges. Downstream of the rotor the total enthalpy remains eonstant. 

The internal friction due to vortices occurs in rapid expansion, diverging, and regulation valves. The entropy generation due to those vortices is taken into consideration in the local resistance. The entropy generation is directly proportional to c- thus. 

Throttling The expansion of a fluid through a constricted passage (across which there is a pressure difference), during which no external work is done. The initial and final velocities of the fluid are equal, and there is no heat exchange with external sources. A change in entropy will, however, take place. 

The transition from a ferromagnetic to a paramagnetic state is normally considered to be a classic second-order phase transition that is, there are no discontinuous changes in volume V or entropy S, but there are discontinuous changes in the volumetric thermal expansion compressibility k, and specific heat Cp. The relation among the variables changing at the transition is given by the Ehrenfest relations. 

In adiabatic compression or expansion, no release or gain of heat by the gas occurs, and no change occurs in entropy. This condition is also known as isentropic and is typical of most compression steps. Actual conditions often cause a realistic deviation, but usually these are not sufficiently great to make the calculations in error. Table 12-4 gives representative average k values for a few common gases and vapors. 

A Mollier Diagram is useful for the expansion of a specific gas/vapor or multicomponent vapor fluid. See Figure 12-91 for comparison of (1) constant enthalpy (Joule-Thompson effect), isenthalpic, and (2) isentropic (constant entropy), which provides the colder temperature. Note that the expander indicated on the figure is somewhere between isenthalpic and isentropic or polytropic. See Figure 12-92. ... 

The theoretical values for enthalpy may be read from a Mollier chart, where hj is the enthalpy of the steam at the turbine inlet, and hg is the enthalpy of the steam at the exhaust pressure and at the inlet entropy. The expansion of steam through the turbine is theoretically at constant entropy. These theoretical rates must be corrected for performance inefficiencies of the particular turbine. The calculations presented here are good for the average design, but exact values for a particular make and model turbine must be quoted by... 

Lead, excess entropy of solution of noble metals in, 133 Lead-thalium, solid solution, 126 Lead-tin, system, energy of solution, 143 solution, enthalpy of formation, 143 Lead-zinc, alloy (Pb8Zn2), calculation of thermodynamic quantities, 136 Legendre expansion in total ground state wave function of helium, 294 Lennard-Jones 6-12 potential, in analy-... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

Expanding a gas increases the disorder. Hence, entropy increases with the expansion. 

The entropy changes ASa and ASB can be calculated from equation (2.69), which applies to the isothermal reversible expansion of ideal gas, since AS is independent of the path and the same result is obtained for the expansion during the spontaneous mixing process as during the controlled reversible expansion. Equation (2.69) gives... 

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... 

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. 

Coefficient of Expansion The change in entropy with pressure is related to the coefficient of expansion by... 

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