Thermodynamic incompressible liquids

Hie equation of continuity also finds application in thermodynamics, as the flux density of heat from an enclosed volume must be compensated by a corresponding rate of temperature decrease within. Similarly, in fluid dynamics, if the volume contains an incompressible liquid, the flux density of flow from the volume results in an equivalent rate of decrease in the density within the enclosure. 

Chapter 2 appears on the ESS-RGN of VRB Power Systems, Vancouver, formerly the Regenesys power storage system, both for its own sake and for the thermodynamic points it enables the author to make, reiterate or emphasise in relation to his thermodynamic theories. Here are fuel cells without oxygen, and certainly without combustion, but undoubtedly with charge transfer reactions separated by a potential difference. Moreover, here are fuel cells with incompressible liquid reactants and products. 

Flows are typically considered compressible when the density varies by more than 5 to 10 percent. In practice compressible flows are normally limited to gases, supercritical fluids, and multiphase flows containing gases. Liquid flows are normally considered incompressible, except for certain calculations involved in hydraulic transient analysis (see following) where compressibility effects are important even for nearly incompressible liquids with extremely small density variations. Textbooks on compressible gas flow include Shapiro Dynamics and Thermodynamics of Compressible Fluid Flow, vol. 1 and 11, Ronald Press, New York [1953]) and Zucrow and Hofmann (Gas Dynamics, vol. I and II, Wiley, New York [1976]). 

Comparison of the molecular dynamics calculations with the predictions of classical thermodynamics indicates that the Laplace formula is accurate for droplet diameters of 20 tT j (about 3400 molecules) or larger and predicts a Ap value within 3% of the molecular dynamic.s calculations for droplet diameters of 15 o-y (about 1400 molecules). Interestingly, vapor pres-sures calculated from the molecular dynamics simulations suggested that the Kelvin equation is not consistent with the Laplace formula for small droplets. Possible explanations are the additional assumptions on which the Kelvin relation is ba.sed including ideal vapor, incompressible liquid, and bulk-like liquid phase in the droplet. 

For incompressible liquid the following thermodynamic equalities [4] are obeyed... 

Osmotic pressure is a thermodynamic property of the solution. Thus, n is a state variable that depends upon temperature, pressure, and concentration but does not depend upon the membrane as long as the membrane is semipermeable. Osmotic equilibrium requires that the chemical potentials of the solvent on the two sides of the membrane be equal. Note that the solutes are not in equilibrium since they cannot pass through the membrane. Although osmotic pressure can be measured directly, it is usually estimated from other measurements (e.g., Reid, 1966). For an incompressible liquid osmotic pressure can be estimated from vapor pressure measurements. 

The thermodynamic behavior of fluids near critical points is drastically different from the critical behavior implied by classical equations of state. This difference is caused by long-range fluctuations of the order parameter associated with the critical phase transition. In one-component fluids near the vapor-liquid critical point the order parameter may be identified with the density or in incompressible liquid mixtures near the consolute point with the concentration. To account for the effects of the critical fluctuations in practice, a crossover theory has been developed to bridge the gap between nonclassical critical behavior asymptotically close to the critical point and classical behavior further away from the critical point. We shall demonstrate how this theory can be used to incorporate the effects of critical fluctuations into classical cubic equations of state like the van der Waals equation. Furthermore, we shall show how the crossover theory can be applied to represent the thermodynamic properties of one-component fluids as well as phase-equilibria properties of liquid mixtures including closed solubility loops. We shall also consider crossover critical phenomena in complex fluids, such as solutions of electrolytes and polymer solutions. When the structure of a complex fluid is characterized by a nanoscopic or mesoscopic length scale which is comparable to the size of the critical fluctuations, a specific sharp and even nonmonotonic crossover from classical behavior to asymptotic critical behavior is observed. In polymer solutions the crossover temperature corresponds to a state where the correlation length is equal to the radius of gyration of the polymer molecules. A... 

The theory of crossover critical phenomena has been extended to binary mixtures. This extension is based on a principle of isomorphism of critical phenomena which states that the thermodynamic behavior of fluid mixtures is similar to that of one-component fluids provided that the mixtures are kept at a constant value of a hidden field variable C [80-82]. For mixtures with a simple phase diagram in which the critical points of the two components are connected by a continuous critical locus, this hidden field C may be taken as a function of the difference of the critical potentials of the two components [83-85]. Based on this principle crossover equations have been proposed for the thermodynamic properties of a variety of fluid mixtures near the vapor-liquid critical locus [68,69,79,86-89]. A systematic procedure for extending the application to fluids with more complex phase diagrams has been developed by Anisimov et al. [90-92]. This procedure also incorporates crossover between the one-component vapor-liquid critical limit and the liquid-liquid critical limit of incompressible liquid mixtures [90, 91, 93]. 

Thermodynamic properties are characteristics of a system (e.g., pressure, temperature, density, specific volume, enthalpy, entropy, etc.). Because properties depend only on the state of a system, they are said to be path independent (unlike heat and work). Extensive properties are mass dependent (e.g., total system energy and system mass), whereas intensive properties are independent of mass (e.g., temperature and pressure). Specific properties are intensive properties that represent extensive properties divided by the system mass, for example, specific enthalpy is enthalpy per unit mass, h = H/m. In order to apply thermodynamic balance equations, it is necessary to develop thermodynamic property relationships. Properties of certain idealized substances (incompressible liquids and ideal gases with constant specific heats) can be calculated with simple equations of state however, in general, properties require the use of tabulated data or computer solutions of generalized equations of state. 

In fluid mechanics the principles of conservation of mass, conservation of momentum, the first and second laws of thermodynamics, and empirically developed correlations are used to predict the behavior of gases and liquids at rest or in motion. The field is generally divided into fluid statics and fluid dynamics and further subdivided on the basis of compressibility. Liquids can usually be considered as incompressible, while gases are usually assumed to be compressible. 

Biot and Darcy theory shortcomings have been largely overcome by development of a coupled diffusion-dynamic formalism (de la Cruz et al. 1993, Spanos 2001, Spanos et al. 2(X)3). Porosity is treated as an explicit thermodynamic variable, so that dnumerical model development. Nevertheless, if they are solved subject to the assumption of the incompressibility of a liquid saturant, the existence of a slow wave is predicted. It is called the porosity dilation (PD) wave it is not a strain wav, it is a coupled liquid-solid displacement wave, and it has some interesting properties. 

All these models may be specialized also to incompressible fluids, which practically model liquids (at nonextreme, say atmospheric, pressures). Such fluids may be defined mechanically by / = 1 [10, 83], cf. Rems. 26,35 or thermodynamically [24, 43] and this will be discussed at the end of Sect. 3.7. 

Liquids can be expanded or compressed just as gases can. For technical reasons, the design of expansion and compression units for liquids is different from that of gases, but the thermodynamics analysis is the same. A turbine for liquids is usually called an expander, and a compressor for liquids, a pump. The analysis is based on the same equations as for gases, but we treat the subject separately because the relative incompressibility of liquids allows us to use short-cut approximations for the enthalpy and entropy changes of the fluid. The starting point is eqs. (5.20) and (5.30). 

In general, thermodynamic equihbrium between a sohd and a hquid phase (melt, solntion) depends on temperature, pressure, and composition. However, in process technology, the pressure dependence of solid-liqnid eqnilibrinm may often be neglected, because of the virtual incompressibility of the sohd and the liquid phases at moderate pressures. 

Equation (2.8) indicates that in thermodynamics, the internal energy is used as a function of state to characterize the system at constant volume, and also when no work is being performed on or by the system. But the majority of real processes, especially for polymers, take place at constant pressure, because solids and liquids (the only physical states for polymers) are virtually incompressible. For such processes (i.e., those taking place at constant pressure), Gibbs introduced a new function of state, enthalpy H... 

Although there is no such thing as a truly incompressible fluid, this term is used for liquids. The first law of thermodynamics states that for any given system, the change in energy is equal to the difference between the heat transferred to the sys-... 

Some of the coarse-grained parameters, i e and can be easily measured by experiments or in simulations. The other two parameters, %N and the suppression of density fluctuations, XqN, are thermodynamic characteristics, which are not directly related to the structure (i.e., they cannot be simply expressed as a function of the molecular coordinates). If density fluctuations of the polymeric liquid are small on the length scale of interest (e.g., width of an interface between domains), then the value of the compressibility has only a minor relevance and decreasing it even further will not significantly affect the behavior of the system. Thus, field-theoretic calculations often take the idealized limit of strict incompressibility. In particle-based simulations, however, one often softens the constraint in order to facilitate the motion of the interaction centers and, thereby, reduces the viscosity of the polymer liquid. The Flory-Huggins parameter, in turn, is a crucial coarse-grained parameter and different methods have been devised to extract it from experiments or simulations [16, 20-25]. We shall briefly discuss this important issue in Section 5.2.3, and further refer the reader to the literature, where computer simulations have been quantitatively compared with mean field predictions and where the role of fluctuations on the coarse-grained parameters is discussed [16, 22]. 

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