Velocity slip

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

The slip velocity between gas and liquid is v, = Vc Vi. For two-phase gas/liqiiid flow, Ri + Rc = 1. A very common mistake in practice is to assume that in situ phase volume fractious are equal to input volume fraclions. 

As flooding is approached, the slip velocity continues to decrease until at the flood point is zero and the following relationship apphes ... 

For design, the slip velocity is derated to 70-80 percent of the calculated value to give some margin of safety this sets the design value of the continuous phase velocity (V ). The column cross sectional area (and therefore diameter) is set by QJVc- With the diameter set, the other dimensions can be set using the ratios given above. 

Transport Transport units can be scaled up on the principles of pneumatic conveying. Mass and heat transfer can be predicted on both the shp velocity during acceleration and the shp velocity at full acceleration. The slip velocity is increased as the sohds concentration is increased. 

Solid-Liquid Mass Transfer There is potentially a major effect of both shear rate and circulation time in these processes. The sohds can either be fragile or rugged. We are looking at the slip velocity of the particle and also whether we can break up agglomerates of particles which may enhance the mass transfer. When the particles become small enough, they tend to follow the flow pattern, so the slip velocity necessary to affect the mass transfer becomes less and less available. 

What this shows is that, from the definition of off-bottom motion to complete uniformity, the effect of mixer power is much less than from going to on-bottom motion to off-bottom suspension. The initial increase in power causes more and more solids to be in active communication with the liquid and has a much greater mass-transfer rate than that occurring above the power level for off-bottom suspension, in which slip velocity between the particles of fluid is the major contributor (Fig. 18-23). 

This can be readily integrated numerically as long as we use the appropriate nonequilibrium equivalent specific volume in the integration. A reasonably simple form for has been suggested by Chisholm (1983), which makes use of estabhshed correlations for the slip velocity K, which depends on the Lockhart-Martinelh parameter X. Integrating Eq. (26-118) gives ... 

Find the equilibrium, homogeneous specific volume v given by Eq, (26-85) and estimate the slip velocity ratio using the following correlation ... 

In the riser, baffles are placed at intervals to break up bubbles by increasing turbulence and shear. At the top erf the riser the expanded section decreases the upward flow rate of the medium and this, together with the lack of baffles, decreases turbulence and shear, which in turn promotes coalescence of bubbles. Larger bubbles form which have increased slip velocity, so they more easily disengage from the medium. 

It is not possible to calculate the in-line concentrations and slip velocity from purely external measurements on the pipe, i.e. a knowledge of the rates at which the two components are delivered from the end of the pipe provides no evidence for what is happening within the pipe, It is thus necessary to measure one or more of the following variables ... 

The absolute linear velocity of the particles u s The absolute linear velocity of the liquid The slip velocity = u L — u s The hold-up of the solids 65 The hold-up of the liquid e/, = 1 —... 

When an industrial pipeline is to be designed, there will be no a priori way of knowing what the in-line concentration of solids or the slip velocity will be. In general, the rate at which solids are to be transported will be specified and it will be necessary to predict the pressure gradient as a function of the properties of the solid particles, the pipe dimensions and the flow velocity. The main considerations will be to select a pipeline diameter, such that the liquid velocity and concentrations of solids in the discharged mixture will give acceptable pressure drops and power requirements and will not lead to conditions where the pipeline is likely to block. 

A sensitivity analysis showed that the calculated value of pressure gradient was very sensitive to the value of the coefficient of friction pF. but was relatively insensitive to the slip velocity uB and to the bed voidage. 

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.)... 

Thus, if the gas is injected in the form of small dispersed bubbles in order to reduce the slip velocity, coalescence rapidly occurs to give large bubbles and slugs. 

The effects of the addition of surface active agent were investigated because the distribution of air might be affected and the slip velocity reduced. As a result of reducing the surface tension from 0.07 to 0.045 N/m, the maximum efficiency of the pump was increased from 49 to 66 per cent. 

The above results show close agreement between the experimental and theoretical friction factor (solid line) in the limiting case of the continuum flow regime. The Knudsen number was varied to determine the influence of rarefaction on the friction factor with ks/H and Ma kept low. The data shows that for Kn < 0.01, the measured friction factor is accurately predicted by the incompressible value. As Kn increased above 0.01, the friction factor was seen to decrease (up to a 50% X as Kn approached 0.15). The experimental friction factor showed agreement within 5% with the first-order slip velocity model. 

The constant Q is evaluated under the boundary conditions at the pipe wall r = a, u = Us. Then, the slip velocity Ug is determined (Goldstein 1965) from a macroscopic point of view ... 

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