Mass density fluid phase

Phase behaviour describes the phase or phases in which a mass of fluid exists at given conditions of pressure, volume (the inverse of the density) and temperature (PVT). The simplest way to start to understand this relationship is by considering a single component, say water, and looking at just two of the variables, say pressure and temperature. 

When we consider many particles settling, the density of the fluid phase effectively becomes the bulk density of the slurry, i.e., the ratio of the total mass of fluid plus solids divided by the total volume. The viscosity of the slurry is considerably higher than that of the fluid alone because of the interference of boundary layers around interacting solid particles and the increase of form drag caused by particles. The viscosity of a slurry is often a function of the rate of shear of its previous history as it affects clustering of particles, and of the shape and roughness of the particles. Each of these factors contributes to a thicker boundary layer. 

It should be noted that the introduction of fluid/solid interaction has no effect on the macroscopic equations since F2jt exists only at the fluid/solid interface. The relaxation time, tk, is estimated based on the viscosity and mass density representations given by Eqs. (20) and (21) of the Mi component and is detailed in Ref. [37, 43, 44], This model has been shown to satisfy Galilean invariance.44 Furthermore, in this interparticle potential model, the separation of a two-phase fluid into its components is automatic.37... 

Beyond this, the combination of a high pressure optical cell with a magnetic coupling balance provides a possibility to measure the weight of the liquid drop and the related density difference between the drop phase and the surrounding fluid phase with time. Thus, a relation between the mass transfer across the fluid interface and the interfacial tension can be detected. 

The pore diffusivity used in this analysis was determined by the Renkin equation4, the axial dispersion coefficient calculated by assuming a constant Peclet number of 0.2, and the mass transfer coefficient from the bulk to the particle surface calculated by the correlation of Wakao and Kaguei. The product of the heat capacity and density of the solid phase was taken to be the same as that used by Raghavan and Ruthven17. The density of the fluid phase was assumed to be that of pure C02 and was calculated from data provided by the Dionix Corporation in their AI-450 SFC software. Constant pressure heat capacities for the mobile phase were also assumed to be that of pure C02 and were taken from Brunner3. 

Fig. 3-2. P/V/T phase diagram of a pure substance (pure solvent) showing domains in which it exists as solid, liquid, gas (vapour), and/or sc-fluid (CP = critical point TP = triple point p = mass density). The inserted isotherms T2 (T2 > Tc) and Tj, T3 Tc) illustrate the pressure-dependent density p of sc-fluids, which can be adjusted from that of a gas to that of a Hquid. The influence of pressure on density is greatest near the critical point, as shown by the greater slope of isotherm T2 compared to that of T3, which is further away from Tc- Isotherm Ti demonstrates the discontinuity in the density at subcritical conditions due to the phase change. This figure is taken from reference [220].

Re is a Reynolds number g is the acceleration due to gravity /d is the absolute viscosity of fluid pi is the mass density of fluid cr is the surface tension of fluid Pg is the mass density of the gas phase (air) and f is the average radius of the bubbles. To give Gi a name, we called it the Peebles number. If the depth of submergence of the bubble diffuser and the rise velocity computed from one of the above equations are known, the time of contact between the gas phase in the bubbles and the surrounding water can be determined. This is illustrated in the next example. 

Temperature correction factor Absolute viscosity of fluid Mass density of the gas phase (air)... 

This approach makes the velocities in the population balance different from the mass- or phase weighted average velocities obtained solving the two-fluid model. This discrepancy is an argument for the formulation of a mass density population balance instead of a number balance to achieve a consistently integrated population balance within the two-fluid modeling framework. 

In an early attempt to calculate the phase fractions in an approximate implicit volume fraction-velocity-pressure correction procedure, Spalding [176, 177, 178, 180] calculated the phase fractions from the respective phase continuity equations. However, experience did show that it was difficult to conserve mass simultaneously for both phases when the algorithm mentioned above was used. For this reason, Spalding [179] suggested that the volume fraction of the dispersed phase may rather be calculated from a discrete equation that is derived from a combination of the two continuity equations. An alternative form of the latter volume fraction equation, particularly designed for fluids with large density differences, was later proposed by Carver [26]. In this method the continuity equations for each phase were normalized by a reference mass density to balance the weight of the error for each phase. 

Since the total mass of the fluid-particle system is conserved, we can deflne the fluid-phase mass density by... 

In the literature on turbulent two-phase flow (Minier Peirano, 2001 Peirano Minier, 2002 Simonin et al, 1993), the fluid phase is usually treated using a separate distribution function whose integral over phase space leads to the fluid-phase mass density. Here, we use a different approach starting from n(f, x, Vp, p, Vf, f). In this approach, we let the internal coordinate be equal to the fluid mass seen by a particle. The fluid-phase mass density is then given by... 

If the particles are composed of multiple chemical species, then usually the fluid phase will be also. In such cases, it is necessary to introduce a vector of internal coordinates whose components are the mass of each chemical species seen by a particle. Obviously, the sum of these internal coordinates is equal to the fluid mass seen by a particle. By definition, if is the mass of component a, then integration over phase space leads to a component fluid-phase mass density ... 

Microscale fluid turbulence is, by deflnition, present only when the continuous fluid phase is present. The coefficients Bpv describe the interaction of the particle phase with the continuous phase. In contrast, Bpvf models rapid fluctuations in the fluid velocity seen by the particle that are not included in the mesoscale drag term Ap. In the mesoscale particle momentum balance, the term that generates Bpv will depend on the fluid-phase mass density and, hence, will be null when the fluid material density (pf) is null. In any case, Bpv models momentum transfer to/from the particle phase in fluid-particle systems for which the total momentum is conserved (see discussion leading to Eq. (5.17)). 

---