Equations polar fluids

In 1971 the Stiel polar factor was incorporated in an equation that accounts for polar fluids (88)... 

This equation was empirically derived from 16 polar fluids and has an average error of 2.9%. A technique for estimating surface tension using nonretarded Hamaker constants (89) has also been presented. 

The couple stress method can be used for modeling a special case of micro-polar fluids, i.e., the two-phase flow, wherein the constitutive equation is given by [22,34-38]... 

In 1977 De Santis et al. (J5) as well as Heidemann et al. ( ) calculated the gas-phase fugacities in the systems HjO-air and H2O-N2-CO2 by equation of state in these calculations the liquid phase was not included. One of the authors (7J showed in 1978 that aqueous systems with some inert gases and alkanes as well as H2S and C02 could be represented by an equation of state if the molecular weight of water was artificially increased. An extension of this method applied to alcohols was found to be only partially successful. Gmehling et al. (8) treated polar fluids such as alcohols, ketones and water as monomer-dimer mixtures using Donohue s equation of state (9) various systems including water-methanol and water-ethanol were succussfully represented. 

However, it was Soave s modification [30] of the temperature dependence of the a parameter, which resulted in accurate vapour pressure predictions (especially above 1 bar) for light hydrocarbons, which led to cubic equations of state becoming important tools for the prediction of vapour-liquid equilibria at moderate and high pressures for non-polar fluids. 

Mathias, P. M., and Copeman, T. W., 1983. Extension of Peng-Robinson equation of state to polar fluids and fluid mixtures. Fluid Phase Eq., 13 91-108. 

The method we describe is analogous to the one used by us [52] in order to study dielectric relaxation of polar fluids in the presence of a DC electric field using the Langevin equation. Our calculations are carried out by interpreting the Cartesian components of Gilbert s equation as a set of stochastic nonlinear differential equations of Stratonovich type [12]. 

The units in the equation are debye for p, bar for p, and K for T. Equation (4.224) applies to a variety of nonhydrogen-bonded polar fluids, including ketones, acetaldehyde, acetonitrile, and ethers. Better results are obtained with specihc equations. For ketones use... 

The coefficient b in Equation (4.223) is zero for nonhydrogen-bonded polar fluids. For hydrogen-bonded fluids. 

In the present chapter, we have reviewed our recent efforts to combine experimental and theoretical efforts to refine our knowledge of interatomic potentials and chemical processes at extreme conditions of pressure and temperature. We have demonstrated using selected molecular systems that our equation of state model can be used to accurately predict properties of non-polar and polar fluids including fluid mixtures. The accuracy of the equation of state of polar fluids is significantly enhanced by using a multi-speeies or cluster representation of the fluid. 

As distinct from pure electrodynamics and pure mechanics, the complex permittivity e(v) of a polar fluid is determined by self-consistent interrelation of Maxwell and Newton equations. Indeed, trajectories of the particles depends on applied electromagnetic field E(t). Since the latter disturbs these trajectories, they are not known a priori. 

Even these extensions of the corresponding-states concept, which are meant to ac- count for molecular structure, cannot be expected to be applicable to fluids with permanent dipoles and quadrupoles. Since molecules with strong permanent dipoles interact differently than molecules without dipoles, or th-an molecules with weak dipoles, one would expect the volumetric equation of state for polar fluids to be a function of the dipole moment. In principle, the corresponding-states concept could be further generalized to include this new parameter, but we will not do so here. Instead, we refer you to the book by Reid, Prausnitz, and Poling for a detailed discussion of the corresponding-states correlations commonly used by engineers. ... 

It must be emphasized that the generalization of the Peng-Robinson equation-of-state parameters given by Eqs. 6.7-2, 6.7-3, and 6.7-4 is useful only for hydrocarbons and inorganic gases (O2, Na, CO2. etc ). For polar fluids (water, organic acids, alcohols, etc.), this simple generalization is not accurate, especially at low temperatures... 

In Chapter 10 we consider vapor-liquid equilibria in mixtures. For such calculations it is important to have the correct pure component vapor pressures if the mixture behavior is to be predicted correctly. Therefore, for equation-of-state calculations involving polar fluids, the PRSV equation will be used. 

The equations of fluid motion inside and outside a circulating drop under viscous flow regime were solved by Hadamard (H2) and Rybczynski (R9) in 1911, and are quoted in hydrodynamics textbooks (L2). The complete derivation is also repeated by Levich (L8). Although Hadamard s stream functions are strictly applicable to the viscous region only, visual observations (GIO, S18) indicated that the function approximates actual flows (E2, H3). Hadamard s stream function inside the drop, as given in polar coordinates with the origin at the center of the drop (K5), is... 

The three Navier-Stokes equations can be put in very compact form by using the shorthand notation of vector calculus [6, p. 66 7 8, p. 80]. Furthermore, it is often convenient to use these equations in polar or spherical coordinates their transformations to those coordinate systems are shown in many texts [6, p. 66 8, p. 80]. The corresponding equations for fluids with variable density are also shown in numerous texts [6, p. 66 7 8, p. 80]. If we set /A = 0 in the Navier-Stokes equations, thus dropping the rightmost term, we find the Euler equation which is often used for three-dimensional flow where viscous effects are negligible. 

WDD requires a non-aqueous solution of an enabling surfactant. The surfactant is retained in the non-polar fluid and the dissolved aqueous salts and colloids are removed with the water. The basis of the phenomenon is illustrated by the cartoon shown in Figure 3. A water-covered hydrophilic solid is immersed in a non-polar solution containing dissolved surfactant molecules (designated as V). The water initially is a film, but increasing adsorption of surfactant at the oil/water and oil/solid interfaces rapidly causes the water to bead up. Then the water rolls off as the contact angle approaches 180. This phenomenon is controlled by the venerable Young Equation ... 

Hirata F, Rossky PJ An extended RISM equation for molecular polar fluids, Chem Phys Lett 83(2) 329-334, 1981. 

The modern cubic equations of state provide reliable predictions for pure-component thermodynamic properties at conditions where the substance is a gas, liquid or supercritical. Walas and Valderrama provided a thorough evaluation and recommendations on the use of cubic equation of state for primary and derivative properties. Vapour pressures for non-polar and slightly polar fluids can be calculated precisely from any of the modem cubic equations of state presented above (Soave-Redlich-Kwong, Peng-Robinson or Patel-Teja). The use of a complex funetion for a (such as those proposed by Twu and co-workers ) results in a significant improvement in uncertainty of the predicted values. For associating fluids (such as water and alcohols), a higher-order equation of state with explicit account for association, such as either the Elliott-Suresh-Donohue or CPA equations of state, are preferred. For saturated liquid volumes, a three-parameter cubic equation of state (such as Patel-Teja) should be used, whereas for saturated vapour volumes any modern cubic equation of state can be used. 

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