Foam drainage equations

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]. 

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. 

Eqs. (1.41) and (1.43) for the capillary pressure are derived assuming that foam is in equilibrium with the surrounding medium (air) under constant pressure. If an isolated foam with constant volume is submitted to drying , then in the calculation of pa the decrease in gas pressure should be considered. This pressure decrease results from the increase in gas volume caused by drainage of liquid from the foam. As far as changes in pressure and liquid volume usually are not large (Ap p0), the decrease in gas pressure can be derived from the equation of Boyle-Mariot. Then... 

The dependence of the radius of border curvature on the co-ordinate of the direction of flow involved in the calculation of flow rate through a foam is determined from Leonard-Lemlich s equation (5.2) as well from that of Laplace while for the drainage process an independent equation is proposed (for example, parabolic equation). 

These equations can be transformed into expressions about both the rate of liquid drainage w and the volume of liquid AVi T released from the foam at the moment x... 

The drainage kinetics can be formally described using the equations of chemical kinetics. This yields expressions for the dependence of the volume of the liquid outflow on the time with respect to the volume of liquid in the foam [7,14,72], So Eq. (5.50) about the liquid volume in a foam can be derived from the following first order differential equation... 

A semi-quantitative estimation of the influence of the structural parameters and physicochemical properties of the foaming solution on the initial drainage rate can be obtained from the equation describing the drainage in a homogenous polyhedral foam, the liquid of which flows out only through the borders [7]... 

There are several equation proposed for the description of drainage of dynamic (moving) foams. The following empirical expression is presented in [70]... 

In the general case it is difficult to predict quantitatively the liquid carry-away with a foam. As mentioned in Chapter 5, analytical equations, describing the liquid distribution along the height of the foam column, have been derived only for a foam that is at hydrostatic equilibrium [57-59], Calculation of the liquid content in a non-equilibrium static foam, performed on the basis of the drainage model of Desai and Kumar [60], are given in [61]. 

Britten and Lavoie 1992 Patino 1995). Liquid drainage from foams has been described in the literature by empirical equations (Hailing 1981 Elizalde et al. 1991 Patino 1995) ... 

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