Reduced fluid density

For a reduced temperature (7 j = T/Tc) in the range 0.9-1.2, the reduced fluid density (p/ =p/Pc) can increase fi om gas-like values of 0.1 to liquid-like values of 2.5 as the reduced pressure (P/j = P/P is increased to values greater than 1.0. But as is increased to 1.55, the supercritical fluid becomes more expanded and reduced pressures greater than ten are needed to obtain liquid-like densities. By operating in the critical region, the pressure and temperature can be used to regulate density, which in turn regulates the solvent power of a supercritical fluid. 

Consider first linear polymers composed of identical spherical sites that interact intermolecularly via a pair decomposable site-site hard-core potential of diameter d. The dimensionless reduced fluid density is where p , = Np is the site number density. 

Figure 12. The hard-sphere equation-of-state (d = 3.9 A) as a function of reduced fluid density computed for polyethylene at T = 430 K and N = 6429 by various thermodynamic routes free energy (upper solid), compressibility (lower solid), wall (dashed), and GFD (short/long dash). The inset includes attractions by perturbation theory using the GFD curve as the reference system the points represent experimental results. ...

Figure 17. Athermal spinodal phase diagram obtained using PRISM theory " and the compressibility route for two reduced fluid densities (d =

The reduced fluid density tj is defined by Eq. (98), in which Nav is Avogadro s number. 

Initially glass microspheres were used in the 1970s to overcome severe lost circulation problems in the Ural Mountains. The technology has been used in other sites [1189]. Hollow glass beads reduce the density of a drilling fluid and can be used for underbalanced drilling [1199-1201]. Field applications have been reported [73]. 

Foamed cement slurries have been used to provide a low density cement slurry to reduce permeability damage to highly sensitive formations through reduced fluid loss (29). Glass microspheres have also been used to substantially reduce cement slurry density (30, 31). Other additives which reduce cement slurry density to a lesser extent include bentonite, fly ash, silicates, perlite, gilsonite, diatomaceous earth, and oil emulsions (see citations in reference 29). 

This also applies to a body submerged in a fluid that is subject to any acceleration. For example, a solid particle of volume Vs submerged in a fluid within a centrifuge at a point r where the angular velocity is on is subjected to a net radial force equal to Ap on2rVs. Thus, the effect of buoyancy is to effectively reduce the density of the body by an amount equal to the density of the surrounding fluid. 

Choking is a phenomenon that occurs in high speed compressible flow (e.g. in relief systems). It occurs because, as the pressure falls along a pipe or through a nozzle, the fluid density decreases. This, means that the volumetric flow rate and, hence, the velocity increases (because the mass flow is constant). Choking occurs when the downstream pressure is reduced to the point where the velocity cannot increase any more. This effectively limits the maximum velocity and, hence, flow rate of the fluid. 

After sufficient extraction the dissolved substances have to be separated from the fluid in a following step. Decreasing the pressure at constant temperature reduces the fluid-density, and therefore the solvent-power of the fluid. The extracted substances are collected at the bottom of the separator, as shown in Fig. 6.6-2. To close the solvent cycle, the fluid has to be recompressed to extraction pressure. 

The theorem is easily generalized also to the case when the fluid density is p = p(z) or p = p(r). Indeed, in this case, the continuity equation implies that for vz = 0 we have div v = 0, i.e., the problem reduces to that already considered. We note that in an ideally conducting medium, for arbitrary density, the quantity Hz/p is conserved (instead of Hz in the case of an incompressible fluid), since instead of (4) we have in this case the equation... 

It is possible to compare these results with those reported in the literature for supercritical C02. Hyatt ( 8) reports it values for C02 from -0.52 to -0.0 which compares very favorably with the present value of -0.55 at a reduced density of 0.43. Sigman et al. (9) used a number of solutes to determine n values for C02 at fluid densities of 0.86, 0.68, and 0.46 g/cm3 of -0.12, -0.22, and -0.45, respectively. The present values at similar densities are -0.05, -0.10, and -0.25. The source of this discrepancy is unknown, although Sigman et al. attribute the variation in n values among their various indicators to specific solute effects. 

In Figures 5 and 6, one might expect to see two different solubility regions. At low fluid densities where intermolecular forces are reduced and the surfactant concentration is below the CMC, the solubility should increase gradually as the density increases. At higher densities, above the CMC, the solubility should increase rapidly because the total surfactant solubility is dominated by the saturation concentration of micelles in the fluid. This type of behavior is not apparent in Figures 5 and 6, perhaps because the CMC is below 10 M. 

Reducing the bed length while keeping the space velocity the same will reduce the fluid velocity proportionally. This will affect the fluid dynamics and its related aspects such as pressure drop, hold-ups in case of multiphase flow, interphase mass and heat transfer and dispersion. Table II shows the large variation in fluid velocity and Reynolds number in reactors of different size. The dimensionless Reynolds number (Re = u dp p /rj, where u is the superficial fluid velocity, dp the particle diameter, p the fluid density and t] the dynamic viscosity) generally characterizes the hydrodynamic situation. 

As has been mentioned, the phase stability of these microemulsions is dependent upon the fluid density. The continuous phase solvent must have a sufiSciently high dielectric constant to be able to solvate these nanometer-sized droplets. In near-critical and supercritical solvents having low dielectric constants, we observe strong attractive interactions between the droplets giving rise to a limited size of droplet that can be dispersed. Likewise, the magnitude of the predicted van der Waals type of attractive interactions rises sharply as the dielectric constant of the continuous phase is reduced below a region bounded by supercritical and near-critical... 

Here, p is the fluid density and /, is the fluid velocity in the x, direction. When an index, such as i, is repeated in the same term, it means that the term is a summation over all possible values of the index. The first term on the left-hand side describes the change in fluid density over time, and the second term describes the transport of the fluid. For incompressible fluids, which have a constant density, the continuity equation reduces to the following, simpler form ... 

There have been many analytic and numerical studies of the structure that solids induce in an adjacent fluid. Early studies focussed on layering in planes parallel to a flat solid surface. The sharp cutoff in fluid density at the wall induces density modulations with a period set by oscillations in the pair correlation function for the bulk fluid [169 173]. An initial fluid layer forms at the preferred wall fluid spacing. Additional fluid molecules tend to lie in a second layer, at the preferred fluid fluid spacing. This layer induces a third, and so on. The pair correlation function usually decays over a few molecular diameters, except near a critical point or in other special cases. Simulations of simple spherical fluids show on the order of 5 clear layers [174 176], while the number is typically reduced to 3 or less for chain or branched molecules that have several competing length scales [177 180]. 

The small spheres are fluid molecules, and the large spheres are immobile silica particles. The top visualizations are for a disordered material and the bottom visualizations are for an ordered material of the same porosity. The visualizations on the left are for the saturated vapor state, and those on the right are for the corresponding saturated liquid state, (b) Simulated adsorption and desorption isotherms for Lennard-Jones methane in a silica xerogel at reduced temperature kT/Sfi = 0.7. The reduced adsorbate density p = pa is plotted vs the relative pressure X/Xo for methane silica/methane methane well depth ratios ejf/Sff = 1- (open circles) and 1.8 (filled circles) [44]. (Reproduced with permission from S. Ramalingam,... 

---