Viscous cross-flow

Pande, K.K. Effect of gravity and viscous cross-flow on hydrocarbon miscible flood performance in heterogeneous reservoirs. 67th Annual Conference of SPE, Washington D.C., Oct 4-7, 1992 SPE 24935. 

The results from the more complex eight layer cross-section, and the previous Brent Sands model, have confirmed that the same recovery mechanisms operate as were discussed in detail for the simple two layer model above. In particular, the importance of the polymer in changing the viscous forces in the oil displacement process is emphasised and results in viscous cross-flow of fluids between layers in heterogeneous formations of this type. The cross-sectional examples also show the importance of carrying out some simple scoping calculations in order to establish the mechanism... 

Figure 1.11 represents the cross-section through a spherical particle over which an ideal non-viscous fluid flows. The fluid is at rest at points 1 and 3 but the fluid velocity is a maximum at points 2 and 4. There is a corresponding decrease in pressure from point 1 to point 2 and from 1 fo 4. However, fhe pressure rises to a maximum again at point 3. If fhe ideal fluid is replaced wifh a real viscous fluid then, as the pressure increases towards point 3, the boundary layer next to the particle surface becomes fhicker and then separates from the surface as in Figure 1.12. This separation of the boundary layer gives rise to...

Detonation, Rayleigh (or Mikhel son) Line and Transformation in. (Called here Rayleigh-MikheTson Line) The Chapman-Jouguet theory deals with adiabatic transformations in steady, non-viscous, onedimensional flows in stream tubes or ducts of constant cross-section. Such transformations can be called Rayleigh transformations. From the equation of continuity valid for flow of constant cross-section and from the momentum equation (Ref 1, p 117 Ref 2, p 99), with use of the formula c = y-Pv for sonic velocity in an ideal gas, can be derived the relationship ... 

It is necessary to discuss another chemical feature related to water-soluble polymers cross-cross-linking — the component that separates viscous systems from gel systems. Viscous systems flow, and it follows, therefore, that they do not possess the tensile properties of muscles. High-viscosity systems have structural integrity, gels provide the necessary combination of tensile strength and elongation or stretch. 

The driving force F in Fl-FFF is the viscous force exerted on a particle by the cross-flow stream. Application of Stokes law gives for F [17] ... 

Pleated ultrafiltration module. The axial filter is convenient for experiments, in that volumes small relative to ordinary ultrafiltration systems can be studied and in that pumping of viscous solutions is limited to that necessary to replace filtrate or concentrate bled from the chamber, rather than that necessary to maintain desired cross flow velocities. There is no obvious reason it could not be scaled up to moderate sizes for practical separations, but so far as we know, no large-volume axial filters are available. For the operations of interest, any of the commercial ultrafiltration systems would be candidates. We have tested one module, recently developed by Gelman, which incorporates a pleated membrane (Figure 5), with somewhat more open feed passages than those of spiral-wound membranes, and which allows backwashing. Other applications of the module were discussed at this symposium by A. Korin in a paper coauthored by G. B. Tanny, and a written account is presumably in these proceedings. 

Equation 40 calculates the viscous, shell cross-flow pressure drop,, psi. 

Higher cross-flow velocity provides higher shear forces, and thus, a thinner biofilm. However, the shear forces of the turbulent flow do not penetrate the viscous sublayer and affect the bacterial monolayer. Fluid velocities in SW modules have very Uttle effect on impeding the initial rate of microbial (or colloidal) foulant deposition, although they can reduce the thickness of the fouling layer. 

There are a large number of computational studies of the effect of flow and heat transfer development in microchannels. Typically these use commercial software and can include effects such as viscous dissipation and variable properties. In some circumstances, allowing for variable fluid properties can be important, especially in small-diameter ducts. Liu et al. [50] made a detailed study of the effect of having a variable fluid thermal conductivity and viscosity for water in a two-dimensional channel that was 100 j,m wide. They sandwiched a heated section of wall (where a constant wall heat flux was applied) between two adiabatic sections and examined the influence ofthe variable properties on the flow fleld and the Nusselt number. Variable properties were shown to induce a non-negligible cross-flow at low Reynolds numbers, which led to a non-negligible heat transfer enhancement. 

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

The mean free path of molecules in air at atmospheric pressure is /free — 1 /(Niiyg), where Nl 2.69 10 cm is the number density of gas molecules and cTg 10 " cm is the cross section for elastic collisions of molecules. These numbers result in /free — 3.7 10 cm, or 37 nm. The mean pore radius of the GDL is in the order of 10 pm, which means that the flow in the GDL pores occurs in a continuum regime. Thus, pressure-driven oxygen transport in a dry porous GDL can be modeled as a viscous Hagen-Poiseuille flow in an equivalent duct. However, determination of the equivalent duct radius and the dependence of this radius on the GDL porosity is a nontrivial task (Tamayol et al., 2012). Much workhas recently been done to develop statistical models of porous GDLs and to calculate viscous gas flows in these systems using Navier-Stokes equations (Thiedmann et al., 2012). 

As shown in Eqn. (6), the drag coefficient of a cylindrical fiber imder cross flow condition is a function of the Reynolds Number, which is generally expressed as Re = pUp,hd/p (i.e. the ratio of inertial force to viscous force). This definition holds true for Newtonian fluids, where shear stress < shear rate. However, the fluids that are often utilized in fiber sweep applications are non-Newtonian. Hence, the Reynolds Number must be redefined using the apparent viscosity function as Re = pUp>d/papp. The viscosity for Newtonian fluids is independent of the shear rate. However, for non-Newtonian fluids, the apparent viscosity varies with shear rate. Applying the Yield Power Law (YPL) rheology model, the apparent viscosity is expressed as ... 

By injecting a viscous polymer solution, the mobihty ratio was decreased and made favorable. The viscosity of the injected polymer solution was typically 35 to 40 cp. If polymer degradation was not significant, this level of viscosity decreased the mobihty ratio from 9.4 to approximately 0.3. When fluids can freely cross flow between strata, the rate of movement of a polymer front is independent of permeability, so long as the reciprocal of the mobility ratio is greater than the permeability ratio between the strata (Sorbie and Seright 1992, Wang et al. 2008). 

Finally we require a case in which mechanism (lii) above dominates momentum transfer. In flow along a cylindrical tube, mechanism (i) is certainly insignificant compared with mechanism (iii) when the tube diameter is large compared with mean free path lengths, and mechanism (ii) can be eliminated completely by limiting attention to the flow of a pure substance. We then have the classical Poiseuille [13] problem, and for a tube of circular cross-section solution of the viscous flow equations gives 2... 

The cross-sectional area of the wick is deterrnined by the required Hquid flow rate and the specific properties of capillary pressure and viscous drag. The mass flow rate is equal to the desired heat-transfer rate divided by the latent heat of vaporization of the fluid. Thus the transfer of 2260 W requires a Hquid (H2O) flow of 1 cm /s at 100°C. Because of porous character, wicks are relatively poor thermal conductors. Radial heat flow through the wick is often the dominant source of temperature loss in a heat pipe therefore, the wick thickness tends to be constrained and rarely exceeds 3 mm. 

---