Liquid saturation densities, equation

Um et al. also examined a transient using their complex model. They saw that in a matter of tens of seconds the current density response reached steady state after a change in potential. However, their model did not include liquid water. The most complex model to examine transients is that of Natarajan and Nguyen. It should be noted that although the model of Bevers et al. has transient equations, they do not report any transient results. Natarajan and Nguyen included liquid saturation effects and water transport in their model. They clearly showed the flooding of the diffusion media and that it takes on the order of a couple of minutes for the profiles to develop. 

The uncertainties in the equation of state are 0.2% in density, 3% in heat capacities, 1% in speed of sound, and 0.5% in vapor pressure and saturation densities. The estimated uncertainty in the liquid phase along the saturation boundary is approximately 3%, increasing to 10% at pressures to 100 MPa, and is estimated at 10% in the vapor phase. The estimated uncertainty in the liquid phase is approximately 5% and is estimated as 10% in the vapor phase. 

An Equation for Liquid-Vapor Saturation Densities as a Function of Pressure... 

The explicit formula pxr — I = (1 — Pr)0 for reduced saturation density as a function of reduced pressure is proposed for the entire liquid-vapor saturation boundary. The expression A 1 depends on Pr p 0.35 depends weakly on Pr, corresponding at Pr = 1 to the critical exponent pc. The parameters A and ft can be related to the Pitzer factor o>. Special cases include the power law pr — 1 = C(1 — Tr)0c. . . and the low-pressure vapor equation prx0 = p0Pr The function A — Ac = g(Pr) is found from data to be a universal function for nonpolar substances. If Ac is correlated with o>, the formula takes on the corresponding-states form pr = /o,(Pr, to). This form predicted the density of saturated liquid and vapor with 0.4% and 0.9% accuracy, respectively, for 38 substances. 

After correction as above and for tantalum saturation, the two enthalpy data points determined by Deimison et al. (1966a) at 1750 and 1763 K are 6% higher than the selected values. Heat capacity values of Novikov et al. (1978) (1800-2100 K) were again only given in the form heat capacity-density and were corrected to heat capacity using the liquid density equations given by Stankus and Khairulin (1991). With this correction, the values tended from 2.5% higher to 6.5% lower than the selected value. 

Teletzke et al. (1982) used Equation 2.50 and the Peng-Rohinson equation of state (Peng and Robinson, 1976) to calculate density profiles for various values of the temperature T and bulk density n. They found that below the saturation density n ° relatively thick films can form near the solid when the temperature is below but not too far from the critical temperature of the fluid. Under these conditions the gradient energy contribution is relatively small, as would be expected in the neighborhood of the critical point. For densities above b°. where the bulk phase is a liquid-vapor mixture, the liquid completely wets the solid (i.e., there is no equilibrium contact angle). 

The introduction of the third parameter, a>, significantly improved the fit to the saturation properties of hydrocarbons for both equations. The SRK equation gives better predictions for the vapor pressure and saturated vapor volumes for alkanes from Ci up to Cio- For saturated liquid densities, the compression factor in the liquid and densities above the critical temperature, the SRK equation gives better results for Ci and C2 only, while the PR equation is better for hydrocarbons higher than ethane (Yu et al. 1986). Thus the choice of the most suitable equation to use will depend on both the size of the molecule and the part of the surface to be considered. 

As the desnity id) of the saturated vapours is often quite small, it can be neglected in comparison to the density of the liquid, D. Therefore, equation (1) becomes,... 

Equations for vapor pressure, liquid volume, saturated liquid density, liquid viscosity, heat capacity, and saturated Hquid surface tension are described in Refs. 13, 15, and 16. 

The modified Rackett equation, density = (P /fiT.,)/ZRA was used. See Spencer, C. F., and R. P Danner, Improved Equation for Prediction of Saturated Liquid Density, y. Chem. Eng. Data 17, 236... 

The regression constants A, B, and D are determined from the nonlinear regression of available data, while C is usually taken as the critical temperature. The hquid density decreases approximately linearly from the triple point to the normal boiling point and then nonhnearly to the critical density (the reciprocal of the critical volume). A few compounds such as water cannot be fit with this equation over the entire range of temperature. Liquid density data to be regressed should be at atmospheric pressure up to the normal boihng point, above which saturated liquid data should be used. Constants for 1500 compounds are given in the DIPPR compilation. 

TABLE 2-396 The Modified Rockett Equation Input Parameters for Calculating Pure Saturated Liquid Densities... 

The basic equation for water flow through saturated porous media was developed by Flenry Darcy in 1856 to calculate the flow of water through sand filters. This equation has been found to be valid for the flow of liquids through porous media when adjusted for the viscosity and density of the liquid. The original form of the equation, designed to calculate discharge is as follows ... 

The results of Na versus Pa are analyzed by an equation first derived by Brunauer, Emmet, and Teller, and the resultant isotherm is called the BET isotherm. Typically one measures the amount of N2 adsorbed for a particular pressure at 78 K (the boiling point of N2 at a pressure of 1 atm) as sketched in Figure 7-24. There are several regimes of an adsorption isotherm. At low densities the density increases linearly with pressure. When the density approaches one monolayer, the surface saturates. As the pressure approaches the saturation pressure of the gas, bulk condensation of liquid OCCUrs. This condensation can occur preferentially in pores of the solid due to capillary condensation, and the amount of gas and pressure where this occurs can be used to determine the pore volume of the catalyst. 

Liquids with low viscosity or large 3 (high density or efficient momentum transfer across the boundary layer) have a rotational diffusion coefficient close to that of the Debye equation [220], eqn. (110). For viscous liquids, the rotational diffusion coefficient tends to saturate to a viscosity-independent value. Tanabe [235] has found perdeuterobenzene rotational diffusion to be well described by the Hynes et al. theory [221, 222]. 

Peneloux et al. [35] have introduced a clever method of improving the saturated liquid molar volume predictions of a cubic equation of state, by translating the calculated volumes without efffecting the prediction of phase equilibrium. The volume-translation parameter is chosen to give the correct saturated liquid volume at some temperature, usually at a reduced temperature Tr = T/Tc = 0.7, which is near the normal boiling point. It is possible to improve the liquid density predictions further by making the translation parameter temperature dependent. 

The equations given predict vapor behavior to high degrees of accuracy but tend to give poor results near and within the liquid region. The compressibility factor can be used to accurately determine gas volumes when used in conjunction with a vitial expansion or an equation such as equation 53 (77). However, the prediction of saturated liquid volume and density requires another technique. A correlation was found in 1958 between the critical compressibility factor and reduced density, based on inert gases. From this correlation an equation for normal and polar substances was developed (78) ... 

Even if no data are available, there are a number of techniques for estimating specific volumes or densities of pure liquid. An equation to estimate saturated volumes that was developed by Rackett and later modified by Spencer and Danner (15) is as follows ... 

Determination of pure component parameters. In order to use the EOS to model real substances one needs to obtain pure component below its critical point, a technique suggested by Joffe et al. (18) was used. This involves the matching of chemical potentials of each component in the liquid and the vapour phases at the vapour pressure of the substance. Also, the actual and predicted saturated liquid densities were matched. The set of equations so obtained was solved by the use of a standard Newton s method to yield the pure component parameters. Values of exl and v for ethanol and water at several temperatures are shown in Table 1. In this calculation vH and z were set to 9.75 x 10"6 m3 mole"1 and 10, respectively (1 ). The capability of the lattice EOS to fit pure component VLE was found to be quite insensitive to variations in z (6[Pg.90]

The ApBq compound does not decompose during dissolution in the liquid phase. In such a case, equations (5.15) and (5.16) retain their form, but the density of A should be replaced by that of ApBq (kg m 3), while the saturation concentration, cs, should be expressed in kilograms of ApBq, not A, per cubic metre of the solution. 

Equation 4.12 represents the work needed to compress the adsorptive from its equilibrium pressure in the gas phase to its saturation pressure P°. From this equation, since it is assumed that the liquefied adsorbate is incompressible, and knowing the normal density of the liquid at the given adsorption temperature, it is possible to obtain the volume of the filled adsorption space by... 

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