Critical Transport Velocities

The power consumption of the coal-methanol mixture is for about 20 % less than that of coal-water mixture for the same value of transport velocity (except the regime close to the critical velocity Vcr) The reason of it is probably lower viscosity of methanol than that of the water. Similar results were also confirmed experimentally by [17],... 

The decrease in the mean advection velocity is due, at least in part, to a decrease in the mean water velocity. The change could also be a manifestation of a decrease in the maximum tidal velocity and/or an increase in the critical erosion velocity due to decreasing mean grain size. [The maximum tidal velocity does decrease by about 12% (4 cm/sec) along the section.] Both of these phenomena would help to reduce the time that sand grains actually spend in motion and to lower the mean velocity of transport with a consequent decrease in the width of the transition zone. The width of the zone will also be decreased by an increase in the sedimentation rate or a decrease in the diffusion coefficient. 

This concept was tested on the basis of experimental data for a onedimensional system. Figure 4 shows the time-dependent decrease of the particulate concentration in an annular flume observed values and those computed according to Equation 12 show good agreement. In the experiments neither the hydrodynamic system nor the characteristics of the suspended matter were varied. One can see that the transport capacity Ceq/Co is constant (always about 50% of the initial concentration C is sedimenting) and independent of the absolute value of Co. This supports the hypothesis that there exists a critical settling velocity t s.cr, which divides the entire amount of particulates into sedimentous and nonsedimentous parts. 

In situ data from the River Neckar were likewise juxtaposed to model data, assuming that the river can be represented initially as a one-dimensional system. Here again the concept of a critical settling velocity, determined by particle and flow system characteristics (leading to a transport capacity Ceq/Co), leads to a satisfactory reproduction of the observed data. 

A number of kinds of emulsions, foams and suspensions may be made to flow in tubes or pipes, at scales ranging from the laboratory (e.g. capillary viscometers. Section 6.3.1) to full-scale industry (e.g. transportation pipelines. Sections 10.2 and 11.3.4). The pressure drop and pumping requirements are functions of the type of flow and of the rheological properties of the dispersion. If the flow rate in a pipeline falls below the critical deposit velocity (also termed the stationary deposit velocity), then particles or emulsion droplets will either sediment or cream to form a layer on the bottom or top wall, respectively, of the pipe. Some correlations that have been developed for the prediction of critical deposit velocity are discussed by Nasr-El-Din [103] and Shook et /. [104]. 

The occurrence of a critical flow velocity can be explained in two ways by the action of shear stress on corrosion product films covering the pipe walls or by mass transport considerations. We first consider the role of shear stress. 

The data of Table 10.26 can now be interpreted in terms of mass transport. Above a critical flow velocity, which depends on pipe diameter, the rate of mass transport becomes sufficiently fast to carry away dissolved corrosion products without forming a salt film. In other words, their surface concentration remains below saturation. Under these conditions scales that are formed by precipitation of corrosion products at lower flow velocity can not exist and the corrosion rate therefore will be higher. An example in case is carbon dioxide corrosion of carbon steel that occurs in fluids containing... 

The minimum point on the hydraulic characteristic curve for a settling slurry corresponds to the critical deposition velocity. This is the flow velocity when particles begin to settle out. Good slurry transport design dictates that the pipe diameter and/or pump are selected so that the velocity in the pipeline over the... 

Continuing increases in operating velocity beyond that required at turbulent fluidization, a critical velocity, commonly called the transport velocity /tr. wiH be reached where a significant particle entrainment occurs. Beyond this point, continuing operation of the bed will not be possible without recycle of the entrained solids. The bed is now said to be in the fast fluidization regime. The transition velocity has been correlated by Bi et al. (1995) as... 

Thomas A. D. 1979.Therole of laminar/turbulent transition in determining the critical deposit veloc ity and the operating pressttre gradient for long distance sltmy pipelines. Paper read at the 6th International Conference of the Hydraulic Transport of Solids in Pipes. Cranfield, UK BHRA Fluid Engineering, pp. 13-26. 

As shown in Figure 10.7, the mode of transport for a given particle size can be estimated from the ratio of shear velocity to settling velocity. The threshold for initiation of motion shown in Figure 10.7 is computed based on Figures 10.2 and 10.5, with critical shear stress (tor, dyn/cm ) converted to critical shear velocity (m ct, cm/s), defined as... 

Erosion of noncohesive sediments depends on the equilibrium suspended load transport capacity and bed load flux relative to local conditions. The mode of transport for a noncohesive particle class k depends on the magnitude of the local shear velocity relative to the particle settling velocity (Ws ) and Shields critical shear velocity for initiation of motion ( cr, fc7 m/s) ... 

If the shear velocity exceeds the critical shear velocity and the particle settling velocity, eroded sediment will be at least partially transported as suspended load. For suspended load, the direction and magnitude of the water-bed sediment exchange flux (Jo, g/m /s) ultimately depends on the difference between the near-bed actual concentration (5 e, g/m ) and the near-bed equilibrium concentration (5eq, g/m ), as described by Equation 10.13. 

A mass transfer coefficient can be defined as the vertical dispersion coefficient (cm /s or m /h) divided by the mean path length (cm or m) between the water compartments, yielding a transport velocity (cm/s or m/h). This in turn can be multiplied by the area of transfer (cm or m ) to give a hypothetical volumetric exchange flow rate (cm /s or m /h). In the absence of data for a specific system, an initial estimate of 0.02 cm /s or 0.0072 m /h can be assumed and the sensitivity of the model results to this parameter evaluated. If the value proves to be critical, measurements of temperature profiles may be desirable as a basis for estimating the mass transfer rate. 

For transporting foam, the critical capillary pressure is reduced as lamellae thin under the influence of both capillary suction and stretching by the pore walls. For a given gas superficial velocity, foam cannot exist if the capillary pressure and the pore-body to pore-throat radii ratio exceed a critical value. The dynamic foam stability theory introduced here proves to be in good agreement with direct measurements of the critical capillary pressure in high permeability sandpacks. 

Clark, P.E. and Quadir, J.A. "Prop Transport in Hydraulic Fractures A Critical Review of Particle Settling Velocity Equations," SPE/DOE paper 9866, 1981 SPE/DOE Low Permeability Symposium, Denver, May 27-29. 

The PDF codes presented in this chapter can be (and have been) extended to include additional random variables. The most obvious extensions are to include the turbulence frequency, the scalar dissipation rate, or velocity acceleration. However, transported PDF methods can also be applied to treat multi-phase flows such as gas-solid turbulent transport. Regardless of the flow under consideration, the numerical issues involved in the accurate treatment of particle convection and coupling with the FV code are essentially identical to those outlined in this chapter. For non-orthogonal grids, the accurate implementation of the particle-convection algorithm is even more critical in determining the success of the PDF simulation. 

When the voltage is critical, regime b), there is no concentration polarization because the electrophoretic transport is equal to the convective transport. Any build up of species on the membrane will be dissipated due to diffusion driven by the concentration difference. In this regime, increasing the tangential velocity is expected to have no influence on the flux because fluid shear can only improve the transport of particles down a concentration gradient. In this case, there is no concentration gradient. 

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