Transitional Velocities

Figure 6-32, taken from Govier and Aziz, schematically indicates four flow pattern regions superimposed on a plot of pressure gradient vs. mixture velocity = Vl -t- V5 = Qj + ( s)/A where and Vs are the superficial liquid and solid velocities, Qi, and ( 5 are liquid and solid volumetric flow rates, and A is the pipe cross-sectional area. is the transition velocity above which a bed exists in the bottom of the pipe, part of which is stationary and part of which moves by saltation, with the upper particles tumbling and bouncing over one another, often with Formation of dunes. With a broad particle-size distribution, the finer particles may be fully suspended. Near V 4, the pressure gra-... 

The most important transition velocity, often regarded as the minimum transport or conveying velocity for settling slurries, is V 9- The Durand equation (Durand, Minnesota Int. Hydraulics Conf., Proc., 89, Int. Assoc, for Hydraulic Research [1953] Durand and Condohos, Proc. Colloq. On the Hyd. Tran.spoti of Solids in Pipes, Not. Cool Boord [UK], Paper IV, 39-35 [1952]) gives the minimum transport velocity as... 

Equation 5.2 is found to hold well for non-Newtonian shear-thinning suspensions as well, provided that the liquid flow is turbulent. However, for laminar flow of the liquid, equation 5.2 considerably overpredicts the liquid hold-up e/,. The extent of overprediction increases as the degree of shear-thinning increases and as the liquid Reynolds number becomes progressively less. A modified parameter X has therefore been defined 16 171 for a power-law fluid (Chapter 3) in such a way that it reduces to X both at the superficial velocity uL equal to the transitional velocity (m )f from streamline to turbulent flow and when the liquid exhibits Newtonian properties. The parameter X is defined by the relation... 

Figure 6 Differences in transit velocities in four subjects, before and after leaving the stomach. (Courtesy of Prof. Dr. W. Weitschies.)...

The onset of turbulent fluidisation appears to be almost independent of bed height, or height at the minimum fluidisation velocity, if this condition is sufficiently well defined. It is, however, strongly influenced by the bed diameter which clearly imposes a maximum on the size of the bubble which can form. The critical fluidising velocity tends to become smaller as the column diameter and gas density, and hence pressure, increase. Particle size distribution appears to assert a strong influence on the transition velocity. With particles of wide size distributions, pressure fluctuations in the bed are smaller and the transition velocity tends to be lower. 

Finally at the highest gas velocities we exceed what is called the choking velocity. Above this the bed is in pneumatic transport. This transition velocity depends on the solid flow rate, and according to Bi and Fan (1991) occurs at... 

The minimum flow velocity for which dispersed solids flowing in a pipe remain suspended and do not form a bed of particles along the bottom of the pipe. See also Transition Velocity. 

We note that uopt is of similar magnitude to the transition velocity ve = DJdp (see Eq. 12.22). Since the reduced velocity v is defined as vlve and the reduced plate height h is Hldp (Eq. 12.25), the above optimum conditions (v near ve and H near dp) correspond roughly to unit values of h and v, a conclusion consistent with Figure 12.3. 

At low velocities between the metal and the solution, the solution flow is laminar, while at high velocities it is turbulent. The transition velocity depends on the geometry, flow rate, liquid viscosity, and surface roughness. The Reynolds number accounts for these effects and predicts the transition from laminar to fluid turbulent flow. The Reynolds number is the ratio of convective to viscous forces in the fluid. For pipes experiencing flow parallel to the centerline of the pipe (4,8) ... 

The three modes of particle pushing are clearly shown in the figure. The area between the broken lines represents mode 2. These transition velocities are clearly particle-sizc dependent. This was to be expected, since dynamic eflects (e.g.. viscous drag, wall friction) retard particle motion more strongly the larger the particles. The critical velocity Vc is defined as the central line through the area representing mode 2. 

To keep the particles in suspension, the flow should be at least 0.15m/sec faster than either 1) the critical deposition velocity of the coarsest particles, or 2) the laminar/turbulent flow transition velocity. The flow rate should also be kept below approximately 3 m/sec to minimize pipe wear. The critical deposition velocity is the fluid flow rate that will just keep the coarsest particles suspended, and is dependent on the particle diameter, the effective slurry density, and the slurry viscosity. It is best determined experimentally by slurry loop testing, and for typical slurries it will lie in the range from 1 m/s to 4.5 m/sec. Many empirical models exist for estimating the value of the deposition velocity, such as the following relations, which are valid over the ranges of slurry characteristics typical for coal slurries ... 

Uc transition velocity between bubbling and turbulent fluidization... 

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