Bubble detachment time

Decreasing rate of bubble release The mean bubble detachment frequency Af6-1 directly controls the onset of the gas film. For large enough bubbles, such as the infinite cluster, the bubble detachment time Atb becomes so large that the gas film can be formed. Atb is affected by other parameters such as the wetting of the electrode, viscosity and density of the electrolyte, or the local hydrodynamical fluxes. 

The above equation is applicable when the bubble detaches at s = rF. But if the orifice diameter is large, the authors recommend the use of the relation s = (rF + 2 ) to obtain a better value of the time of bubble formation. Then Eq. (39) is modified to... 

Unlike formation in a liquid the boundary of a fluidised bed bubble can only expand by gas flowing across it to produce the drag force that will cause the particles to move appropriately. During the time that a bubble grows to the size shown in Figure 9 the gas that produced it has advanced to fill the volume indicated by the outer broken line. The annular region above and around the bubble now contains an excess of gas and so the powder void-age must increase. This is unstable and as the bubble detaches and rises through the expanded dense phase the powder relaxes and and returns the excess gas to the bubble. This appears to be completed by the time it has risen about one diameter (of order 1/10 second) and thereafter is of constant volume until it coalesces. 

The most suitable technique for studying adsorption kinetics and dynamic surface tension is the maximum bubble pressure method, which allows measurements to be obtained in the millisecond range, particularly if correction for the so-called dead time, t. The dead time is simply the time required to detach the bubble after it has reached its hemispherical shape. A schematic representation of the principle of maximum bubble pressure is shown in Figure 18.14, which describes the evolution of a bubble at the tip of a capillary. The figure also shows the variation of pressure p in the bubble with time. 

For terminal voltages around U 1.5 (which are, as will be seen in Part 2 of this book, typical for micromachining applications), the gas film formation time is similar to the mean detachment time of the gas bubbles. 

The coefficient takes values between = 2 / 3 (ideal sphere) and = 1 (conditions in the MPTl (Fainerman et al. 1994a). In the moment of bubble detachment this coefficient can reach values > 1. Hence, the effective life time of the bubble during this second stage of its growth (effective dead time) results to... 

The lifetime, the time period from the moment of bubble detachment and formation of a new bubble up to the moment of maximum pressure results as the difference t = tb - tj. From this lifetime t the effective surface age teff can be calculated via the relationship... 

As usual, the subscript V denotes equilibrium values is the age of the interface, which is defined as the period of time between the minimum pressure (bubble formation) and the maximum pressure (bubble detachment) in the case of MBPM A, is a dimensionless parameter A = 1 for immobile interfaces in the case of MBPM, A is an apparatus constant that can be determined by calibration experiments [57] as mentioned earlier, and C2 are the bulk concentrations of surfactant ions and counterions, respectively y is the activity coefficient D f[ is an effective diffusivity that depends on the diffusivities and bulk concentrations of surfactant ions, counterions, and inorganic coions = >eff( i 2> 3> ioo> 2oo> The latter dependence is described by explicit formulas... 

As the bubble detaches from the orifice, the dimensions of the bubble will determine the velocity of the rise. The rise of the bubble through the liquid causes a redistribuhon of surfactant on the bubble surface, with the top having a reduced concentration and the polar base having a higher concentration than the equilibrium value. This unequal distribution of surfactant on the bubble surface has an important role in foam stabilisation (due to the surface tension gradients). When the bubble reaches the interface, a thin liquid film is produced on its top. The fife time of this thin film depends on many factors, e.g. surfactant concentration, rate of drainage, surface tension gradient, surface diffusion and external disturbances. 

In Fig. 21.20c, which corresponds to a higher ALR of 0.1 (liquid volume flow rate of 30 L/h), large deformed bubbles are visible in the injection area. As the gas bubbles detach from the injection holes, they again come into contact with those emerging from downstream injection holes. This leads to bubble coalescence and to the formation of an annular flow pattern. This time, the annular flow remains stable, however. 

These concepts were implemented according to the following scheme the liquid element surrounding the bubble and the bulk are considered as two separate dynamic reactors that operate independent of each other and interact at discrete time intervals. In the beginning of the contact time, the interface is being detached from the bulk. When overcome by the bubble, it returns to the bulk and is mixed with it. Hostomsky and Jones (1995) first used such a framework for crystal precipitation in a flat interface stirred cell. To formulate it for a... 

The bubble is now assumed to detach when its center has covered a distance equal to the sum of the radius of the final bubble arid the radius of the orifice. If the radius of the orifice is R and if it is assumed that V0 = (4tt/ 03/3), then the time of bubble formation can be obtained by plotting on the same axes Eq. (9) and (r + K) as a function of time from Eq. (8). 

Equation (28) permits a direct evaluation of the final bubble volume, VF, without first calculating the time of detachment. Calculations made on the basis of the above equation are presented in the form of a curve in Fig. 11 using Davidson and Schuler s (D9) data. 

These authors have assumed the bubble to be expanding at the orifice, and have used the force balance equation at the time of detachment. The various forces considered by these authors are buoyancy, force due to the addition of mass (P2), excess pressure force, surface tension force, drag force, and force due to the inertia of the liquid. 

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