Density dependence of the inversion temperature

In accord with the mean-field theory developed in Section 5.7.5, the inversion temperature, however, does depend on the density of the fluid. This can be seen from plots of Ta — 1 in Fig. 5.29 based on isostress isostrain ensemble simulations. Regardless of T, Ta /ke — 1 turns out to be a nonmonotonic function of density. It has a maximum that increases and shifts to lower densities with decreasing temperature. In the limit 7 — 0, one exj)ccts all curves to approach zero according to [see Ekj. (5.155)] 

Solid lines in Fig. 5.29 represent fits of the mean-field Eq. (5.183) to the simulation data t2dring Op ( ) and b as fitting parameters. Although these fits represent the simulation data remarkably well, both parameters turn out to 

However, in view of the rather complex variation of ay with temperature and density, the mean-field approach is. still very useful because it permits an estimate of the inversion temperature from an analytic expression [see Eq. (5.184)] at moderate computational expense. The computed inversion temperatures are plotted in Figs. 5.30 for various situations. In accord with the plots in Fig. 5.29, the inversion temperature depends strongly on the density. For example, plots in Fig. 5.30(a) show that, over a density range of 0.1 p 0.5, Tj v changes by about a factor of 3. Over wide ranges of temperature and density the data are again well represented by the mean-field exprassion in Eq. (5.184). 

If the suljstrate separation increases one expects the inversion temperature of the confined fluid to eventually coincide with the bulk inversion temperature irrespective of the density. This notion is supported by plots of Tay — 1 for 10, 20, and bulk in Fig. 5.31(a). The maximum of this curve and its location are shifted toward the bulk curve with increasing substrate separation and so does Ti. Similar plots are obtained for other temperatures, thus permitting one to construct the plot in Fig. 5.30(b) parallel to the one in Fig. 5.30(a). As before for Tjnv (0) one expects ATj v(p, Szo) = Tinv (p, 00) — Tinv (p, S20) oc sj from the plot in Fig. 5.30(b) and the mean-field expressions in Eqs. (4.24) and (5.185). For = 20, for instance, a depression of Ti v of about 4% compared with the bulk value is deduced from Fig. 5.30(b). Thus, even if fluids are confined to spaces of mesoscopic dimension, the confinement-induced depression of the inversion temperature should in principle be accessible experimentally given the accuracy with which the 

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Unfortunately, previous work is almost exclusively concerned with the inversion temperature in the limit of vanishing gas density, Ti y (0). The inversion temperature can be linked to the second virial coefficient, which can be measured [210] or computed from rigorous statistical physical expressions [211] with moderate effort. Currently, only the fairly recent study of Heyes and Llaguno is concerned with the density dependence of the inversion temperature from a molecular (i.c., statistical physical) perspective [212]. These authors compute the inversion temperature from isothermal isobaric molecular dynamics simulations of the LJ (12,6) fluid over a wide range of densities and analyze their results through various equations of state. 

Fig. 2 All the observed temperature dependence of the inverse susceptibility of droplet density fluctuation for the WBB system is shown. The vertical axis indicates the inverse of the renormalized susceptibility and the horizontal the renormalized temperature f = r/Gi. The pure Ising region is characterized by the reduced temperature which is smaller than the Ginzburg number i.e. f < 1...

The relative effects of supercitical carbon dioxide density, temperature, extraction cell dimensions (I.D. Length), and cell dead volume on the supercritical fluid extraction (SFE) recoveries of polycyclic aromatic hydrocarbons and methoxychlor from octadecyl sorbents are quantitatively compared. Recoveries correlate directly with the fluid density at constant temperature whereas, the logarithms of the recoveries correlate with the inverse of the extraction temperature at constant density. Decreasing the extraction vessels internal diameter to length ratio and the incorporation of dead volume in the extraction vessel also resulted in increases in SFE recoveries for the system studied. Gas and supercritical fluid chromatographic data proved to be useful predictors of achievable SFE recoveries, but correlations are dependent on SFE experimental variables, including the cell dimensions and dead volume. 

Using physical reasoning, justify the Tm dependence of the diffusion coefficient as shown by Eq. (II-2). Hint Recall that mean molecular velocity is proportional to F 2 and that the density of an ideal gas is inversely proportional to temperature. 

Figure 4 plots, against suspension density, the heat transfer coefficients measured by Basu (1990) over a wide range of bed temperature for 296 pm sand, by Kobro and Brereton (1986) at a temperature of 850°C for 250 pm sand and by Grace and Lim (1989) at 880°C for 250-300 pm sand. The overall heat transfer coefficient is shown to increase with bed temperature. Before radiation becomes dominant in heat transfer, the observed rise in heat transfer coefficient with bed temperature may be explained as follows. The gas convective component is expected to decrease mainly because of the inverse dependence of gas density on temperature. On the other hand, the particles convective component will increase with temperature, thus leading to an increase in gas conductivity, because the latter is dominant for... 

Finally, in closing this section we notice that the inversion temperature defined by Eq. (5.155) is expected to depend on the presence and chemical nature of the solid substrate even in the limit of vanishing density at least in principle (see Section 5.7.5). This is because Z and Z2 depend on the fluid stibstrate potential [see Eqs. (5.158) and (5.161)], which is, in turn, expected to affect B2 (T) through Elq. (5.150). Moreover, we note that, because the above treatment is valid only in the limit p — 0 [and because 82 (T) / / (p)j, the inversion temperature does not depend on the mean density of the confined fluid. 

Since the total mass of the beam/building does not depend on the temperature, the mass density is inversely proportional to the volume of the beam model p a 1 /AL. By considering the thermal effects to all these quantities with Equation (2.155), the thermal dependence of the squared fundamental frequency can be expressed as follows ... 

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