Density variation with temperature

For an incompressible fluid, the density variation with temperature is negligible compared to the viscosity variation. Hence, the viscosity variation is a function of temperature only and can be a cause of radical transformation of flow and transition from stable flow to the oscillatory regime. The critical Reynolds number also depends significantly on the specific heat, Prandtl number and micro-channel radius. For flow of high-viscosity fluids in micro-channels of tq < 10 m the critical Reynolds number is less than 2,300. In this case the oscillatory regime occurs at values of Re < 2,300. 

If I, B, t, and q are known, then n and the sign of q are determined from VH and RH is called the Hall coefficient. The results of Kyser and Thompson are shown in Figure 3. At metal concentrations above five mole % all valence electrons in Li, Na, or Ca solutions are found to be free. It is furthermore found, as noted earlier, that Ra is temperature independent. Figure 4 shows the small variation observed and the change expected when the known density variation with temperature is... 

When the density variation with temperature of water is considered, a maximum is found to oceur near 4°C as discussed in Problem 8.14. Near this point of maximum, density is approximately given by ... 

The proportions of flie LNG components vary widely, depending on the source or gas field. Since flie pure components have widely different densities, the density of a particular LNG can vary with its composition, in addition to having a strong density variation with temperature. The net result is a complex liquid with storage problems associated with internal density differences, particularly on a large scale. 

Hot-Water Process. The hot-water process is the only successflil commercial process to be appHed to bitumen recovery from mined tar sands in North America as of 1997 (2). The process utilizes linear and nonlinear variations of bitumen density and water density, respectively, with temperature so that the bitumen that is heavier than water at room temperature becomes lighter than water at 80°C. Surface-active materials in tar sand also contribute to the process (2). The essentials of the hot-water process involve conditioning, separation, and scavenging (Fig. 9). 

Neglecting the variation of the term, which is negligible compared to the variation with temperature in the exponential term, and recalling that the mobilities are less sensitive to temperature than are the charge carrier densities, Eq. (6.30) can be rewritten as... 

Free convection is fluid flow, induced by density gradients owing, for example, to temperature gradients. In gas extraction the supercritical solvent is subject to density variation with only slight changes in pressure and temperature. Furthermore, flow velocities within the processing equipment are low, so that flow owing to free convection may be important. Therefore, conditions for free convective flow must be considered in such types of systems. For isothermal vertical plates ... 

The apparent density of the coated sand was p = 1394 kg m-3, the specific heat was cp =695 J kg-1 K 1, and the thermal conductivity exhibited the following variation with temperature ... 

We now turn attention to a completely different kind of supercritical fluid supercritical water (SCW). Supercritical states of water provide environments with special properties where many reactive processes with important technological applications take place. Two key aspects combine to make chemical reactivity under these conditions so peculiar the solvent high compressibility, which allows for large density variations with relatively minor changes in the applied pressure and the drastic reduction of bulk polarity, clearly manifested in the drop of the macroscopic dielectric constant from e 80 at room temperature to approximately 6 at near-critical conditions. From a microscopic perspective, the unique features of supercritical fluids as reaction media are associated with density inhomogeneities present in these systems [1,4],... 

The thermal conductivity coefficient has been derived from Browniah motion theory by Irving and Kirkwood33 in terms of the equilibrium singlet and pair distribution functions °/a) and °/Thermal conduction under a macroscopic temperature difference involves a gradient in the mean square molecular velocity rather than in the mean molecular velocity. The steady-state radial distribution function then remains spherically symmetric except for a small correction arising from the number density variation with the temperature. As the analysis introduces no new assumptions and is somewhat lengthy, it will not be reproduced here. The resulting equation for the thermal conductivity coefficient x is... 

Here, cs denotes the concentration in mol/kg (molality scale), and [s] is the concentration in mol/liter (molarity scale). Both units are related in that [s] = pwcs where pw= 1 kg/dm3 is the density of water. Its variation with temperature causes the molarity scale to depend on temperature, whereas the molality scale does not. In the temperature range 0-25°C, however, the density of water differs from unity by less than 0.3%, so that [s] = cs with reasonable accuracy. Most Henry coefficients are less well known. From the definitions in Eqs. (8-7) and (8-8), the coefficients involved are related by... 

Fig. 12.3. Plot of density variations with respect to pressure and temperature for CO2. calculated using the program NIST-14.

An important special case is that of incompressible flow. As discussed in Section 1.2, the term incompressible is something of a misnomer, since what is generally meant in fluid mechanics is constant density. However, a flow in which there are temperature gradients is not quite one of constant density since the density varies with temperature. But the criterion for a constant-density flow is that the flow velocity be small compared with the sound speed in the fluid that is, the Mach number must be small. For a small Mach number the pressure changes are small. Therefore when evaluating the derivatives of thermodynamic quantities for an incompressible flow with an imposed spatial variation in temperature, we must hold the pressure, not the density, constant (Landau Lifshitz 1987), whence... 

The primary emphasis in shock tube interferometric studies of the hydrogen-oxygen reaction has been on induction period phenomena. Recently, however, the entire postshock density profiles of a selection of rich, lean and near stoichiometric Ha-Oa-Ar mixtures have been studied by numerical integration of an assumed reaction mechanism. In this manner it was shown that the characteristic features of the profile prior to the end of the density plateau are essentially independent of the recombination kinetics. Thereafter, however, the shape of the profile is largely accounted for by termolecular reactions (e)-(g). Systematic variation of the termolecular rate coefficient values in experimental regimes where recombination is most sensitive to reactions if) or ig)> respectively, has yielded temperature-dependent expressions of the form kf< = AT for kf and kf believed valid over the range 1400-3000 K. The expression of Jacobs et al. was found satisfactory for kf. In all three cases, variation with temperature is small (1-0 m 0-5). Values at 1700 K, kf = 5-9 x 10 (cited above), kf z= 1-9 X 10 , and kf = 3-6 x 10 cm mole sec, are in excellent accord with those listed in Table 2.2. 

Several attempts to estimate the hole density from a comparison of the mean hole volume with the macroscopic volume are described in the literature. The drawback of such approaches is that assumptions must be made as to the value of or on the thermal expansion and compression of the volume that is not detected by o-Ps. Frequently, it is assumed that that this volume, denoted as occupied or bulk volume, expands Uke an amorphous polymer in the glassy state [Hristov et al., 1996 Dlubek et al., 1998c Band ch et al., 2000 Shantarovich et al., 2007]. Another assumption is that no variation with temperature or pressure is shown [Bohlen and Kirchheim, 2001]. Both assumptions are intuitive but physically not proved. The most successful attempt to estimate hole densities comes from a calculation of the hole free volume with... 

Two points stand out when Tables 59.1 and 59.2 are compared the temperature coefficients for interfacial tension are lower than those for surface tension, and there is no correlation between the interfacial tension of a polymer pair and the difference in their surface tensions. The former effect arises because the variation with temperature is a density effect. 

The fluid viscosity and thermal conductivity experience the largest variation with temperature. Compared with the density and the specific heat variation, their influence on heat transfer is significantly higher, e.g. in the case of water. Therefore, density and thermal conductivity can in most cases be considered to be constant The fluid property variation becomes more important with decreasing diameter, where the axial variation is more pronounced than the variation over the cross-section of the channel. In contrast to the viscous dissipation, the significance of property variations increases with decreasing Br [53]. 

The effect of large changes in pressure at constant temperature on the viscosity of various hydrocarbons is shown in Figure 3. There we see that the logarithm of the viscosity of liquid hydrocarbons and hydrocarbon mixtures increases almost linearly with increasing pressure. Alternatively, viscosity can be considered to be a function of density rather than pressure, and this is used in several of the models discussed later. The kinematic viscosity shows similar trends with respect to these variables mentioned above, however its variation with temperature is significantly more linear than dynamic viscosity so that the former is somewhat easier to correlate than the latter. Consequently, some correlations have been developed exclusively for the kinematic viscosity, as will be discussed later. 

As an additional study, (Cao.8,Bao.2)Zr4(P04)6 ceramics were evaluated for their potential as low thermal expansion matrix materials. When made using a sol-gel technique (Process 1), a bulk density of 76% theoretical was achieved after sintering at 1350°C for 24 h. As shown in Table 3, the effects of sintering temperature and time on the bulk thermal expansions of CBZP and CMZP ceramics were similar. The effect of aging at 1350°C on the bulk thermal expansion of this sample (in Table 3) appears to be negligible. The CTE values for the sample sintered at 1350° for 72 h in different temperature ranges are presented in Table 4, and show considerable variation with temperature. 

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