Russian Journal of Applied Chemistry, 2013, Vol. 86, No. 2, pp. 225−233.
Pleiades Publishing, Ltd., 2013.
Original Russian Text © D.V. Elizarov, V.V. Elizarov, T.S. Kamaliev, S.G. D’yakonov, 2013, published in Zhurnal Prikladnoi Khimii, 2013, Vol. 86, No. 2,
PROCESSES AND EQUIPMENT
OF CHEMICAL INDUSTRY
Kinetics of Mass Transfer in the Course
of Liquid Extraction in Stirred Vessels
D. V. Elizarov, V. V. Elizarov, T. S. Kamaliev, and S. G. D’yakonov
Kazan National Research University of Technology, Kazan, Tatarstan, Russia
Received September 10, 2012
Abstract—A mathematical model of liquid extraction in stirred vessels is considered. The calculation results are
compared with experimental data on extraction of various liquid mixtures.
Stirring of liquid media containing dispersed phase
elements in the form of droplets or solid particles
is widely used for intensifying the mass transfer.
Numerous papers have been published on this subject
in Russia and other countries [1–4]. Nevertheless, the
mass exchange mechanism at the phase boundary is still
poorly understood, because it is a very complex physical
phenomenon. This especially concerns elements with
the mobile phase boundary, i.e., liquid droplets.
The motion of dispersed particles with the mobile
phase boundary has certain speciﬁ c features compared
to the motion of solids. At the mobile phase boundary,
the tangential constituent of the velocity differs from
zero. As a result, circulation of the medium arises inside
the droplet, which favors better ﬂ owing-around, and the
ﬂ ow detachment starts at higher Reynolds numbers than
for a solid sphere. Therefore, the velocity of a drop is
higher than the velocity of a solid particle of the same
diameter and density. It should also be noted that, at
certain values of the Reynolds and Weber numbers, the
droplets start to deform and oscillate, which leads to a
drastic increase in the resistance coefﬁ cient as compared
to a solid sphere at equal Reynolds numbers.
Analytical calculation of the mass transfer coefﬁ cients
is possible only in a few cases. Mathematical description
of mass exchange processes in stirred vessels is mainly
based on empirical equations relating the mass transfer
coefﬁ cient to the design and operation parameters and to
physical properties of the phases.
One of the ﬁ rst attempts to describe the mass transfer
controlled by the resistance of the continuous phase is
the Higbie model, according to which the mass transfer
occurs owing to multidimensional diffusion during short
contacts of liquid elements with the droplet surface:
where Ре is the Péclet number.
It seems most promising to describe the mass transfer
in two-phase systems with mobile boundaries on the
basis of the theory of the diffusion boundary layer,
which takes into account the hydrodynamic conditions
at the phase boundary. The theory developed by Levich
 suggests gradual change in the substance transfer
rate from the ﬂ ow core to the phase boundary ﬁ rst in
a turbulent layer, then in a viscous layer in which the
contribution of the convective constituent decreases,
and ﬁ nally in a diffusion sublayer with the prevalence
of molecular diffusion. The models suggested in [l–3]
are based on the theory of the diffusion boundary layer.
For systems with the mobile phase boundary, we
suggested in [6, 7] to use the following dependence:
The parameters of the model are the dynamic velocity