Foam expansion ratio

Recently a new technique has been introduced for the study of foam drainage under pressure drop. The especially constructed apparatus allows automated calculation of foam expansion ratio at any instant of time (see Section 5.3.4). 

A great advantage of mixing foam generators is the possibility to regulate both foam expansion ratio and dispersity, though within a narrow range of alterations. For example, at constant ratio of gas and liquid volumes the dispersity of foam increases when the consumption of liquid and gas rises [45]. 

In a monodisperse foam the deformation of spherical bubbles and formation of films at the places of their contact starts when the gas content in the system reaches - 50% (vol.) for simple cubic bubble packing or 74% for close (face-centred) cubic or hexagonal packing (foam expansion ratio - 4). In a polydisperse foam the transition to polyhedral structure starts at expansion ratio n - 10-20, according to [ 10] but, as reported in [51], this can occur at n < 4, the latter being more probable. The structure which corresponds to the transition of bubbles from spherical to polyhedral shape is called occasionally honeycomb structure. 

If the condition for polyhedricity R/r 1 is not fulfilled, the radius of curvature and the area of the cross-section become dependent on the co-ordinates along the length of the Plateau border. Analytical dependence of the radius of curvature on the co-ordinates (the border profile) at different foam expansion ratio is not found. 

The shape of foam films and border profiles in large interval of foam expansion ratio from 10 to 1500 has been experimentally studied in [83], A regular pentagonal dodecahedron made up of transparent organic glass with an elastic rubber balloon inside it which took the shape of a sphere at inflation (Fig. 1.10) was used as a model of foam cell. 

Both Figs. 1.10 and 1.11 indicate that the curvature of the border along its length becomes smaller compared to cross-section curvature even at foam expansion ratios from 40 -80 and rn/a = 0.5-0.35 (r in centimetres, Table 1.1). Hence, at first approximation, the deviation from the border shape typical for a fully polyhedral bubble can be determined by... 

On the other hand a comparison of the ratio between the volumes of real Plateau border and cylindrical border model, obtained by Pertsov et al. [85], with the ratio (rh /rn)2, evaluated from the dodecahedral model [83] indicates that these ratios correlate well only when n > 640 but when n < 300 they differ by almost 30%. However, it should be noted that the method for calculating border volume in [85,86] at low foam expansion ratio differs from the real volume by 10%. Besides, all values of expansion ratio given in the tables of [86] are twice higher. This means that the interpolation equations for calculating the corrections of the volume and hydro- and electrical conductivities in the cylindrical border model must be corrected. The Plateau border cross-section area is determined by Eq. (1.19) when the contact angle between film and border surface is small 6 (or 1 in radians). 

The foam expansion ratio or its reciprocal value the foam liquid volume fraction, called also relative or volume density of a foam, is used as a basic parameter characterizing the liquid to gas ratio in the foam. 

Applying the relation of parameters n, a, r (Eqs. (4.9) and (4.10)) and the dependence of hydrostatic pressure on foam column height (Eq. (1.39)), it is possible to derive the distribution of local foam expansion ratio along the height H. Assuming a border foam (i.e. neglecting the amount of liquid in films) one obtains from Eq. (4.10) and Eqs. (1.40) and (1.42) which account for the p dependence on r, the following relation... 

If it is assumed that at H = 0 the foam is spherical with closely packed cells then the foam expansion ratio can be expressed by... 

Pertsov et al. [9] have also calculated the average (by volume) ultimate equilibrium foam expansion ratio n, at which there is neither liquid flowout from nor liquid suck in the foam... 

Based on the studies of border and film shape in the dodecahedral model Kruglyakov et al. [18] and Kachalova et al. [19] have proposed an expression for foam expansion ratio, using a cylindrical border model with the same cross-sectional radius of curvature. The volume of excess vertex parts was considered in order to estimate the effect of the longitudinal radius of curvature on the border shape... 

Table 4.1 gives a comparison of the values of the real border foam expansion ratio and these calculated from Eqs. (4.10) and (4.25). It can be seen that choosing appropriate K and accounting for the excess part of vertexes, the calculated values of the expansion ratio are close to the real ones even at n = 20. However, for large expansion ratio ranges the correction (5Knr[ in Eq. (4.25)) depends on r differently. 

The capillary pressure and foam dispersity were determined simultaneously. After drying the reduced pressure under the porous plate sharply changed to lower values leading to an immediate liquid suck in from the porous plate into the foam. As a result the foam expansion ratio decreased to values at which the condition for polyhedricity was not fulfilled. The capillary pressure and the expansion ratio were measured again for the new state of the foam obtained. 

Fig. 4.1. Dependence ra/r (1) and ri,/rn (2,3) versus foam expansion ratio curve 3 is calculate with the...

Determination of Foam Expansion Ratio (Foam Density)... 

Foam expansion ratio, or its reciprocal quantity foam density, chracterises the relative content of gas and liquid in this disperse system. 

The simplest technique for determination of the average by volume expansion ratio (and density) is the direct measurement of the total foam volume and the liquid volume in it (or its mass). Using barbotage methods for foam production the foaming process can be run until the initial solution is completely transformed into foam. Foam expansion ratio and its density are calculated by the formula... 

Another technique for foam expansion ratio determination involves taking samples and can be applied also for foams produced by foam generators. The precision of these direct volume-mass methods depends to a considerable extent on foam uniformity with respect to its expansion ratio and on the sampling conditions [20,21,22], The relative error in the evaluation of n for low expansion ratio foams (n < 100) is about 1-2%. However, for high expansion ratio foams (n > 1000) these methods become inapplicable. 

One of the most widely used methods for foam expansion ratio determination is based on measuring foam electrical conductivity. The expansion ratio of a foam and its conductivity are related according the expression (see Chapter 8)... 

Thus, the precise value of foam expansion ratio, calculated from Eq. (4.32) can be evaluated only if the condition h r a is fulfilled (see Section 4.1). However, the change in solution composition in the foam during foaming, caused both by the different adsorption of solution components and the increase in surfactant concentration in the foam liquid phase because of foam destruction, restricts the use of Eq. (4.32). The effect of surface conductivity is another restriction. 

This technique involves simultaneous counting of the number of impulses in the foam in an empty container and in a water solution container. The foam expansion ratio is calculated from the following formula... 

An important condition which has to be fulfilled when using this method for foam dispersity determination is the absence of an excess hydrostatic pressure in the foam liquid phase. This pressure is equalized to a considerable extent when an equilibrium distribution the foam expansion ratio and the border pressure along the height of the foam column is established. This can be controlled by measuring the pressure in the Plateau borders at a certain level of the foam column by means of a micromanometer. However, if this condition is overlooked, the hydrostatic pressure can introduce a considerable error in the results of bubble size measurements, especially in low expansion ratio foams. Probably, it is the influence of the unrecorded hydrostatic pressure that can explain the lack of correspondence between the bubble size in the foam and the excess pressure in them, observed by Aleynikov[49]. The... 

A device for foam dispersity determination by measuring the local foam expansion ratio and the pressure in Plateau borders is illustrated in Fig. 4.4. It consists of a glass container equipped with platinum electrodes and a micromanometer. The container bottom is a porous plate (a sintered glass filter). The pressure Ap is measured with a capillary micromanometer and the expansion ratio is determined by the electrical conductivity of the foam. The manometer and the electrodes are positioned so that to ensure a distance of 1.0-1.5 cm between them and the porous plate. When the distance is small the liquid in the porous... 

Fig. 4.4. Schematic of the device for determination of foam dispersity by the local foam expansion ratio ...

Here the coefficient k depends on the foam liquid content which means that Eq. (4.49) is valid only for a narrow interval of foam expansion ratios and dispersities. 

For evaluating the geometric coefficient B it is necessary to determine foam expansion ratio and electrical expansion ratio , which is equal to k[/kF (where kL and kF are the specific electrical conductivities of the solution and the foam, respectively). 

For the study of liquid distribution along the height of the foam column section, foam collectors or a system of electrodes situated on various levels can be used [68], The most correct values of the local foam expansion ratio can be obtained only using horizontal porous electrodes. 

In the middle of cells and in faces that are perpendicular to the flowing direction, the borders are branched, which means that the effective number of borders, equivalent to that in a real system, is different from five. The number of independent borders with constant by height radius and length L can be determined by the electro-hydrodynamic analogy between current intensity and liquid flow rate through borders, both being directly proportional to the cross-sectional areas [6,35]. This analogy indicates that the proportionality coefficients (structural coefficients B = 3) in the dependences border hydroconductivity vs. foam expansion ratio and foam electrical conductivity vs. foam expansion ratio, are identical [10]. From the electrical conductivity data about foam expansion ratio it follows... 

Solving together Eqs. (5.4) - (5.7) and accounting for Eq. (4.10), a function of the liquid volumetric flow rate versus foam expansion ratio is obtained... 

Krotov [9] has considered theoretically another case in which the foam column top is irrigated with the surfactant solution and the foam is in contact with the solution through a filter under which a reduced pressure is created. Here, the constancy of foam expansion ratio and border radii along the height of the foam column is fulfilled only when the volumetric flow rate (qo) corresponds to that derived from Eqs. (5.20) or (5.21). If the value of the volumetric flow rate is higher or lower than q0 (q q0), a distribution of n and r by height is observed. For the cases of q > q0 or q < q0 transcendental equations were derived for the... 

A typical dependence of drainage onset on foam column height at foam expansion ratio n = 70 is given in Fig. 5.14. [6,22], For high foam columns (H > 16 cm) zb is small and does not practically depend on H. It is mainly determined by the hydrodynamic properties of the system (borders size and viscosity), i.e. of the microsyneresis rate. For small foam column heights t0 strongly depends on H and is determined by the rate of internal foam collapse. These dependences indicate that for a quantitative description of drainage detailed... 

Increase in foam expansion ratio under constant dispersity leads to decrease in drainage rate because the radius of Plateau borders diminishes. Indeed, Eq. (5.60) indicates that the change in drainage rate is inversely proportional to the square of expansion ratio and... 

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