Single Fluid Phase

Extensive measurements of flow in and other properties of beds of particles of various shapes, sizes and compositions are reported by 

A long-established correlation of the friction factor is that of Ergun (Chem. Eng. Prog. 48, 89-94, 1952). The average deviation from his line is said to be 20%. The friction factor is 

The pressure gradients for the liquid and vapor phases are calculated on the assumption of their individual flows through the bed, with the correlations of Eqs. (6.108)-(6.112). 

The fraction of the void space occupied by liquid also is of interest. In Sato s work this is given by 

Additional data are included in the friction correlation of Specchia and Baldi [Chem. Eng. Sci. 32, 5 1 5-523 (1977)], which is represented by 

Scheidegger, Physics of Flow through Porous Media, University of Toronto Press, Toronto, Canada, 1974). 

Operation of packed trickle-bed catalytic reactors is with liquid tind gas flow downward together, and of packed mass transfer equipment with gas flow upward and liquid flow down. 

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Fluid samples may be collected downhole at near-reservoir conditions, or at surface. Subsurface samples are more expensive to collect, since they require downhole sampling tools, but are more likely to capture a representative sample, since they are targeted at collecting a single phase fluid. A surface sample is inevitably a two phase sample which requires recombining to recreate the reservoir fluid. Both sampling techniques face the same problem of trying to capture a representative sample (i.e. the correct proportion of gas to oil) when the pressure falls below the bubble point. 

Coker, A. K., Sizing proeess piping for single-phase fluids, The Chemical Engineer, Oetober 10, 1991. 

Coker, A. K., Program Evaluates Pressure Drop for Single-Phase Fluids, Hydrocarbon Processing, vol. 70, no. 2, p. 53. 

Withers, J. G., Tube-Side Heat Transfer and Pressure Drop for Tubes Having Helical Internal Ridging and Turbulent/Tran-sitional Flow of Single-Phase Fluid Part 1 and Part 2 Single Helix Ridging, Heat Trans. Eng, V. 2, Jufy-Sept., Oct.-Dec. (1980) p. 49. 

A certain amount of information on particle-to-liquid and bed-to-wall heat transfer is available for single-phase fluid flow through a packed bed. It is not clear, however, to what extent this information can be applied to... 

A momentum balance for the flow of a two-phase fluid through a horizontal pipe and an energy balance may be written in an expanded form of that applicable to single-phase fluid flow. These equations for two-phase flow cannot be used in practice since the individual phase velocities and local densities are not known. Some simplification is possible if it... 

For non-settling suspensions The standard equation for a single phase fluid is used with the physical properties of the suspension in place of those of the liquid. 

A micro-channel heat sink can be classified as single-phase or two-phase according to the state of the coolant inside it. For single-phase fluid flow in smooth... 

Table 3.3 Experimental results of single-phase fluid flow in smooth micro-channels... 

For single-phase fluid flow in smooth micro-channels of hydraulic diameter from 15 to 4,010 pm, in the range of the Reynolds numbers Re < Recr, the Poiseuille number, Po, is independent of the Reynolds number. Re. 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

Inasmuch as the nature of pipeline elements sets these networks apart from electrical networks (more commonly referred to as electrical circuits) we shall review briefly the modeling of these elements. We shall, however, limit ourselves to the correlations developed for single-phase fluid flow the modeling of two-phase flow is a subject of sufficient diversity and complexity to merit a separate review. 

Evaluation of each term in Eq. (15-51) is straightforward, except for the friction factor. One approach is to treat the two-phase mixture as a pseudo-single phase fluid, with appropriate properties. The friction factor is then found from the usual Newtonian methods (Moody diagram, Churchill equation, etc.) using an appropriate Reynolds number ... 

Several age-distribution functions may be used (Danckwerts, 1953), but they are all interrelated. Some are residence-time distributions and some are not. In the discussion to follow in this section and in Section 13.4, we assume steady-flow of a Newtonian, single-phase fluid of constant density through a vessel without chemical reaction. Ultimately, we are interested in the effect of a spread of residence times on the performance of a chemical reactor, but we concentrate on the characterization of flow here. 

Harter, 1., Darmander, D., and Renard, P. (2000) Single-phase fluid distributor-mixer-extractor for beds of granular solids. US Patent 6,024,871. 

If the volume change of the gas is neglected, and hence accelerative effects are also neglected, then for single-phase fluid flow. Equation (3) becomes... 

Figure 6.8. Friction factors and void fractions in flow of single phase fluids in granular beds, (a) Correlation of the friction factor, Re = DpG/( - e)p and fp = lgcDpe3/Pu2(l - e)](AP/L = 150/Re + 4.2/(Re)1/f [Sa/o et al., J. Chem. Eng. Jpn. 6, 147-152 (1973)]. (b) Void fraction in granular beds as a function of the ratio of particle and tube diameters [Leva, Weintrauh, Crummer, Pollchik, and Storch, U.S. Bur. Mines Bull. 504 (1951)].

In summary, we refer to Figure 5.5, which may be considered as the projection of the entire equilibrium surface on the entropy-volume plane. All of the equilibrium states of the system when it exists in the single-phase fluid state lie in the area above the curves alevd. All of the equilibrium states of the system when it exists in the single-phase solid state lie in the area bounded by the lines bs and sc. These areas are the projections of the primary surfaces. The two-phase systems are represented by the shaded areas alsb, lev, and csvd. These areas are the projections of the derived surfaces for these states. Finally, the triangular area slv represents the projection of the tangent plane at the triple point, and represents all possible states of the system at the triple point. This area also is a projection of a derived surface. 

Hydrostatic (single-phase fluid) Archimedes buoyancy. Effective stress - established at the boundary. Alters interparticle electrical forces (repulsion, van der Waals attraction, hydration). 

It is suggested that the pressure gradient may be estimated from the Navier-Stokes equation of a single-phase fluid by [Corrsin and Lumley, 1956]... 

For the total mass conservation of a single-phase fluid, / represents the fluid density p. jr represents the diffusional flux of total mass, which is zero. For flow systems without chemical reactions, d> = 0. Therefore, from Eq. (5.12), we have the continuity equation as... 

For the momentum conservation of a single-phase fluid, the momentum per unit volume / is equal to the mass flux pU. The momentum flux is thus expressed by the stress tensor i/r = (pi — t). Here p is the static pressure or equilibrium pressure / is a unit tensor and r is the shear stress tensor. Since <1> = —pf where / is the field force per unit mass, Eq. (5.12) gives rise to the momentum equation as... 

Another approach for estimating am is based on the pseudothermodynamic properties of the mixture, as suggested by Rudinger (1980). The equation for the isentropic changes of state of a gas-solid mixture is given by Eq. (6.53). Note that for a closed system the material density of particles and the mass fraction of particles can be treated as constant. Hence, in terms of the case for a single-phase fluid, the speed of sound in a gas-solid mixture can be expressed as... 

Figure 6.7 Single-phase fluid flow, points E to B.

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