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Scientific Background

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Scientific Background

During intensive mixing of two-phase liquid-liquid systems two simultaneous processes take place – the breaking of droplets and their collision and coalescence. The final drop size distribution is determined by the kinetics of these two processes.

Systematic studies of industrial mixing equipment provided sufficient data on the hydrodynamics and the distribution of local intensities of turbulence to allow development of quantitative descriptions of the kinetics of breaking and coalescence of droplets in different mixing devices. It was found, in particular, that both the break-up of droplets and the coalescence occur mainly in an area of maximum local turbulent distribution. This area exists in the close neighborhood of the agitator.

Both the breaking process and the coalescence process are of a stochastic nature and are described in terms of probabilities. An individual act of deformation, and the breaking of a drop occurs under action of an instant pulsation of velocity in the vicinity of the drop, on the condition that the amplitude of the pulsation exceeds some “critical” value  v_{b}^{*}   that depends on the interfacial tension and viscosity of liquids. The mean frequency of breaking of droplets in an area with the local turbulent dissipation E may be estimated as
N_{b} = f_{l} P\big(v' \geq v_{b}^{*} \big) \big(1 - P_{l}(0)\big)
where
f_{l}  is a mean frequency of pulsations of the micro-scale l
P\big(v' \geq v_{b}^{*} \big)  is a relative frequency of pulsations with amplitudes v' \geq v_{b}^{*}
P_{l}(0) is the probability of residue of zero droplets in an area of the scale l
v' is the random pulsation velocity, and its frequency is a function of the local energy dissipation value
v_{b}^{*} is a critical pulsation velocity for breaking the droplet (the value is a function of physical properties of liquids)
l is the scale of pulsations.

Coalescence of droplets is also a random event and occurs if the droplets are in contact at the moment when an “efficient” fluctuation of pressure occurs which pushes them together.
According to the Derjaguin, Landau, Verwey and Overbeek (DLVO theory), the neighboring surfaces are kept from conjunction by the effect of electrostatic repulsing pressure p_{r} The magnitude of this pressure depends on many factors including the chemical composition of the substances.
The act of coalescence of the droplets happens only if the squeezing turbulent pressure is high enough to overcome the repulsing pressure.
So, the mean frequency of coalescence is defined as
N_{c} = f_{\lambda} P\big(v'_{n} \geq v^{*}_{c}\big) \big(1 - P_{\lambda}(0) - P_{\lambda}(1)\big)
where
f_{\lambda} is a mean frequency of pulsations of the micro-scale \lambda
P\big(v'_{n} \geq v^{*}_{c}\big) is a relative frequency of pulsations with amplitudes satisfying the condition p' > p_{r} or v'_{n} \geq v^{*}_{c}
P_{\lambda}(0)  is the probability that zero droplets are located in an area of the scale \lambda
P_{\lambda}(1) is the probability that exactly one droplet is located in an area of the scale \lambda
v'_{n} is the random pulsation velocity, and its frequency is a function of the local energy dissipation value
v^{*}_{c} is a critical pulsation velocity for coalescence. Its the value is a function of repulsing pressure
\lambda is the characteristic scale of pulsations
p' is a random pulsation pressure due to turbulent pulsations of scale \lambda
p_{r} is the repulsing pressure.

Together these equations describe the dynamics of the drop size evolution. These equations are solved numerically with respect to the actual geometry of the mixing devices and the properties of the liquids being mixed. The predictive power of these equations for determining drop size distribution has been extensively verified by experimental testing with a wide variety of liquids and mixing regimes.

Thus by designing for the desired distribution of turbulence in mixing tanks for a given process, we can create dynamic regimes where the frequency of droplet coalescence is much higher than the frequency of their breakage. This will shift the equilibrium towards larger droplets.

This specific mixing regime results in a rapid decrease in the concentration of the smallest droplets in the emulsion due to their coalescence with larger droplets. Therefore the drop size distribution becomes narrower.

The formation rate of the smallest droplets increases along with the intensity of turbulence. The upper limit of intensity corresponds to formation of “secondary” small droplets as a result of break-up. Additionally, special design of the rotors is required in order to create appropriate conditions throughout the tank, and to prevent the creation of small droplets in extremely turbulized micro-vortices which are formed around blades of common agitators.

Turbulent Technologies uses proprietary data and patent pending technology and algorithms to determine the optimal turbulent regime specific to a process, for the desired mixing and separation effects. This is then tested at laboratory and/or pilot scale. Scale up to full size industrial equipment is reliably done mathematically, since we design the whole mixing system for correct turbulence conditions and we are not scaling the geometry. In other words we are “scaling the turbulence” and not the mixer.

Selected Papers

TURBULENT SETTLING (TS) TECHNOLOGY FOR SOLVENT EXTRACTION – ALTA 2010

TURBULENT TECHNOLOGIES MIXING SYSTEM – ISEC 2011

INFLUENCE OF TURBULENCE AND VISCOSITIES ON THE KINETICS OF DROP BREAKING – J Dispersion Science 1993

LIQUID LIQUID TWO PHASE FLOW AND TRANSPORT PHENOMENA – Internation Center for Heat and Mass Transfer 1997

MASS TRANSFER EFFICIENCY IN SX MIXERS – ALTA 2013

Homogenous Turbulent Mixing for Reducing Entrainment in Copper SX – Copper 2013

 

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