Russian Journal of Applied Chemistry, 2013, Vol. 86, No. 8, pp. 1204−1211.
Pleiades Publishing, Ltd., 2013.
Original Russian Text © S.G. D’yakonov, V.I. Elizarov, D.V. Elizarov, T.S. Kamaliev, 2013, published in Zhurnal Prikladnoi Khimii, 2013, Vol. 86, No. 8,
OF SYSTEMS AND PROCESSES
Mathematical Simulation of Liquid Extraction
in a Countercurrent Multistage Apparatus
S. G. D’yakonov, V. I. Elizarov, D. V. Elizarov, and T. S. Kamaliev
Kazan National Research Technological University, Kazan, Russia
Received May 18, 2013
Abstract—A technique of calculating a multistage countercurrent liquid extraction in an apparatus with agitation
of phases was proposed. Kinetic parameters of mass transfer was acquired based on concepts of pseudo-laminar
boundary layer on the elements of the dispersed phase. An approximate method for calculating the installation was
presented. The results of the calculation of the process according to a “stage by stage” scheme and approximate
method are well consistent with each other.
Based on a source of energy used for dispersing
one phase inside another phase and agitation of phases,
extractors with external energy input into interacting
liquids providing a high degree of dispersion, the
developed contact surface of phases, and an increase
in the mass transfer velocity were widely applied in
In view of technical and economic parameters the
multistage countercurrent extractors are more effective
than cross current.
In the design of multistage column (rotary disk,
with mechanical agitator, vibration) and mixer-settler
extractors a diameter and a height are key parameters.
The diameter of the device are determined by the
operating speed of the phase and the height, by a
number of separation stages. The number of separation
stages are found by a number of theoretical stages N
and average efﬁ ciency of a stage η:
It is assumed that the method of the setup calculation
according to a ‘‘stage by stage”’ scheme is more
accurate. It makes possible a separate evaluation of the
efﬁ ciency of each stage, which depends on the surface
and coefﬁ cients of the mass transfer. For calculation of
kinetic mass transfer parameters relations derived from
the experimental data processed and summarized by
criterial equations or hydrodynamic models of boundary
layer and turbulent mass transfer depending on a rate
of energy dissipation, a particle size, and properties of
ﬂ uid were used [1–4].
Experimental studies of a diffusion boundary layer
on ﬁ ne particles by holographic interferometry 
have demonstrated self-similarity of a concentration
proﬁ le in the boundary layer, which combines
features of the laminar and turbulent boundary layer.
This boundary layer is classiﬁ ed as pseudo-laminar.
The mass transfer coefﬁ cient in the continuous phase
in the apparatus with agitation was acquired in the
form  β = 3ν
/(2δ), where δ is a thickness of
the boundary layer, ν
, the kinematic viscosity of the
continuous phase, Sc
, Schmidt number.
For calculating the mass transfer coefﬁ cient in the
continuous phase with agitating liquids well established
equations are used [7, 8]:
β = 0.016nd
, β = 0.301(εν
where n, d
is a speed and diameter of the agitator,