Relative or slip velocity

Detailed consideration of the interaction between particles and fluids is given in Volume 2 to which reference should be made. Briefly, however, if a particle is introduced into a fluid stream flowing vertically upwards it will be transported by the fluid provided that the fluid velocity exceeds the terminal falling velocity m0 of the particle the relative or slip velocity will be approximately o- As the concentration of particles increases this slip velocity will become progressively less and, for a slug of fairly close packed particles, will approximate to the minimum fluidising velocity of the particles. (See Volume 2, Chapter 6.)... 

Fp/fi. Assuming, after Mertes and Rhodes (1955) and Lapidus and Elgin (1957), that for monodisperse fluid-particle systems, the relative or slip velocity is a unique function of voidage (or vice versa) irrespective of the flow direction of either fluid or particles, it follows that U = — Fp. Thus plots of Fp vs. have frequently been interpreted as equivalent to those of 7 vs. . Some caution should be exercised in assuming this... 

Finkelstein et al. (1971) found that, in both counter-current and cocurrent downward flow, both species of binary-sized solids moved at the same rate, provided that the solids residence time was short (< 1 minute) and the size ratio did not exceed 3. For more common situations, however, in which N solids species have left their initial acceleration zone, Richardson and Shabi (1960) have suggested that the relative or slip velocity, C/ri, of each particle species (e.g., each narrow cut of sieved solids of fixed density) depends on the total surrounding voidage in a manner similar to that of the same speeies acting monodispersely in the same liquid, viz.. 

The volume fraction of the gas-liquid mixture in the vessel which is occupied by the gas is called the gas holdup, tpG. If the superficial gas velocity in the vessel is vG, then vJ< G is the true gas velocity relative to the vessel walls. If the liquid flows upward, cocurrently with the gas, at a velocity relative to the vessel walls vL/( 1 - tpG), the relative velocity of gas and liquid, or slip velocity, is... 

By substituting the relative velocity, or slip velocity, between particles and fluid for the velocity of classical fluidization, Kwauk (1963,1992) extended... 

The Reynolds number includes Voo, the drop velocity relative to its surroundings or slip velocity. If drops move with the surrounding fluid, Voc is negligible, and heat and mass transfer rates depend solely on conduction and diffusion, respectively. If drops are suspended as in fluidization, heat and mass transfer coefficients will increase due to increased slip velocity. 

In reality, the slip velocity may not be neglected (except perhaps in a microgravity environment). A drift flux model has therefore been introduced (Zuber and Findlay, 1965) which is an improvement of the homogeneous model. In the drift flux model for one-dimensional two-phase flow, equations of continuity, momentum, and energy are written for the mixture (in three equations). In addition, another continuity equation for one phase is also written, usually for the gas phase. To allow a slip velocity to take place between the two phases, a drift velocity, uGJ, or a diffusion velocity, uGM (gas velocity relative to the velocity of center of mass), is defined as... 

For example, if the entering solids fraction tp is 0.4, the corresponding values of the local solids fraction theoretical expressions for slip, but practical applications depend on experimental observations and correlations (which will be presented later). In gas-liquid or gas-solid flows, (pm will vary along the pipe, because the gas expands as the pressure drops and speeds up as it expands, which tends to increase the slip, which in turn increases the holdup of the denser phase. 

The model based on terminal and slip velocity approach is rather tenuous. It breaks down as the density difference between liquid and solid approaches zero. Under highly turbulent conditions, an accurate estimation of slip velocity is rather difficult, and there is disagreement on whether or not the relative velocity between the solid and liquid alone is enough to obtain an accurate estimate of the mass-transfer coefficient. 

The concepts of slip velocity and characteristic velocity are useful in defining the flooding point and operational regions of different types of column contactors. The shp (or relative) velocity is given by the equation ... 

As a consequence b is always smaller than the slip velocity for the extreme case of negligible renewal of the wake and equals s when/, = 0 or when the rate of renewal is very rapid for finite/. In the latter example the wake loses its identity relative to the emulsion. 

In conclusion, for a homogeneously bubbling bed, the velocity of bubble rise Mb relative to the bed wall is equal to or lower than the bubble slip velocity relative to the continuous phase. 

The backmixing of the solid phase and solid concentration profile become more important when the solid loading is high (e.g. DCL process) or when the solid particles are large with significant relative slip velocity with the liquid phase. If... 

The presence of the capacitance and our knowledge of the relative magnitudes of the slip velocities on the two branches of the flow curve lead to a straightforward conceptual model for the oscillatory flow. We assume for simplicity that there is a large discontinuity in the slip velocity at a critical wall stress tc, rather like the behavior seen in Figure 12.6. At stresses below Tc there is little or no shp, the flow is slow, and the pressure in the reservoir may build up and compress a small amount of melt. When the pressure builds to a value where the wall stress must exceed Xc to maintain the balance of forces, the melt will begin to slip, the flow rate will increase... 

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