Drainage equations

As discussed in Section 1.2.2 the bubble shapes in fairly dry foams and froths (4 gas > 0.83, approximately) are not spheres or distorted spheres, but polyhedrons. In practice there will be distributions of both gas-cell sizes and shapes. In addition to the gas bubbles, froth contains the floated particles, pulp liquor, and a fraction of (hydrophilic) particles that did not float due to bubble attachment, but which were mechanically entrained in the froth. The pulp liquor and these latter particles all have to be allowed to drain back out of the froth. The rate of this drainage will be greatest at the froth-pulp interface (i.e., the bottom of the froth layer) and slowest near the top of the froth layer. Froth drainage equations are discussed elsewhere [53]. The froth needs to be a stable enough foam that some time can be allowed for these drainage processes, and also so that the upper layer(s) of the froth can be swept out of the flotation cell. On the other hand, the froth should not be too stable as a foam so that it will break easily after collection. In addition to the role of the frother, froth stability is also promoted by increasing liquid viscosity. 

The use of Eqs. (5.39), (5.42) and (5.44) to describe the process of foam drainage is accompanied by a rather sophisticated mathematical procedure and requires knowledge of the border profile. For approximate calculation it is more convenient to apply the drainage equation obtained for a cylindrical border model. 

Consider the drainage equation for a single vertical PB channel ... 

Experimental results obtained by Scheludko and co-workers [2-4] with comparatively thick rigid films, produced from dilute solutions of sodium oleate and isoamyl alcohol, gave reasonably good agreement with the drainage equation. However, deviation from the Reynolds equation was observed in many cases due to tangential surface mobility. Surface viscosity can also slow down the drainage. 

A number of other solutions to the foam drainage equation are available, not only for free drainage but also for steady drainage and the so-called solitary waves formed by injecting liquid into the top of a foam column [23]. These solutions appear to agree well with experimental observation [23]. 

For cakes of high permeabihty, the capillary drain height may be an insignificant fraction of cake thickness, and film drainage becomes the controlling factor in a centrifugal field (7). Under unsteady-state conditions, equation 18 represents the drainable Liquid left in the cake as a function of the centrifugal filtration parameters ... 

This reaction is relatively fast and readily reversible so that in drainage basins in carbonate-dominated terranes the stream water commonly will have near-equihbrium concentrations of hydrogen, bicarbonate, and calcium ions. At equiUbrium, the rates of forward and reverse processes represented in equation 5 are equal. 

It is both convenient and reasonable in continuous filtration, except for precoat filters, to assume that the resistance of the filter cloth plus filtrate drainage is neghgible compared to the resistance of the filter cake and to assume that both pressure drop and specific cake resistance remain constant throughout the filter cycle. Equation (18-51), integrated under these conditions, may then be manipulated to give the following relationships ... 

Equation (24-68a) means that the protection potential, U, must be reached at the end of the pipeline (x = L), and that at the drainage point x = 0), the potential must not fall below because due to hydrogen evolution, according to Eq. (2-19), Eq. (24-67)... 

The preceding equations are reported to predict actual heat transfer coefficients only about 15% lower than experimental values—the difference can be attributed to the rippling of the film and early turbulence and drainage instabilities on the bottom side of the tube. ... 

Fig. 27. Various modes of film drainage and the corresponding equations for the rate of film thinning and drainage time are shown. The criteria for specific modes are also indicated.

Payer80 states that the UNSAT-H model was developed to assess the water dynamics of arid sites and, in particular, estimate recharge fluxes for scenarios pertinent to waste disposal facilities. It addresses soil-water infiltration, redistribution, evaporation, plant transpiration, deep drainage, and soil heat flow as one-dimensional processes. The UNSAT-H model simulates water flow using the Richards equation, water vapor diffusion using Fick s law, and sensible heat flow using the Fourier equation. 

In evaluating drainage and filtration materials, an allowable flow rate is divided by a required flow rate to obtain the DR or FS according to the equations below5 ... 

The SWCR system is built on top of the FMC. The purpose of the SWCR system is to prevent infiltration of surface water into the landfill by containing and systematically removing any liquid that collects within it. Actual design levels of surface water infiltration into the drainage layer can be calculated using the water balance equation or the HELP model.36 37... 

Erosion can seriously effect a landfill closure by disrupting the functioning of drainage layers and surface water and LCRSs. Heavy erosion could lead to the exposure of the waste itself. For this reason, it is important to predict the amount of erosion that will occur at a site and reinforce the facility accordingly. The Universal Soil Loss Equation shown below can be used to determine soil loss from water erosion5 ... 

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