ISSN 1070-4272, Russian Journal of Applied Chemistry, 2015, Vol. 88, No. 11, pp. 1855−1859. © Pleiades Publishing, Ltd., 2015.
Original Russian Text © O.M. Flisyuk, Yu.G. Chesnokov, O.N. Krukovskii, E.I. Borisova, 2015, published in Zhurnal Prikladnoi Khimii, 2015, Vol. 88, No. 11,
A Mathematical Model of Liquid Mixing in an Apparatus
with Annular Cross Section by Gas Bubbling
O. M. Flisyuk, Yu. G. Chesnokov, O. N. Krukovskii, and E. I. Borisova
St. Petersburg State Institute of Technology (Technical University), Moskovskii pr. 26, St. Petersburg, 190013 Russia
e-mail: ﬂ email@example.com
Received June 23, 2015
Abstract—A mathematical model was suggested for calculating the mixing process performed by gas bubbling.
The adequacy of the model for an apparatus of annular cross section with gas feeding through a nozzle of small
diameter was conﬁ rmed experimentally. The parameters of the model in relation to the ﬂ ow rate of the gas fed for
mixing were determined. The suggested approaches associated with mixing intensiﬁ cation can ﬁ nd use in spent
nuclear fuel reprocessing and in other processes occurring in gas–liquid and liquid–solid heterogeneous systems.
In spent nuclear fuel reprocessing, a need arises for
mixing uranium- and plutonium-enriched solutions in
annular vessels of the geometry ensuring nuclear safety
. The process should be performed so as to ensure in-
creased mixing intensity, no precipitate formation on the
bottom of the annular vessel, and reduced entrainment
of radioactive aerosols with the air exhaust.
Naturally, the use of mechanical stirring devices in
a narrow annular gap is difﬁ cult. Therefore, bubbling
systems with feeding compressed air are used .
Gas–liquid systems are widely used in chemical
technology [3, 4]. Numerous papers deal with the rela-
tionships of the gas and liquid ﬂ ow in such systems. The
literature concerning the formation of gas bubbles and
their motion in a liquid layer is surveyed in [5, 6]. A for-
mula suitable in a broad interval of Reynolds numbers
was suggested for calculating the resistance force act-
ing on a spherical bubble . The process of gas bubble
formation is analyzed in [8, 9]. Stirring of the gas and
liquid phases in gas–liquid systems was studied in detail
as applied to vessels with bubbled layer. References to
early papers on this subject can be found in , and a
review of later studies is made in [10, 11]. An analytical
model of liquid circulation in a bubbled layer was sug-
gested in . The liquid mixing in vessels with annular
cross section was not considered previously.
Today, with the progress of the computation means,
computer simulation ﬁ nds increasing use in mathemati-
cal modeling of ﬂ ows in gas–liquid systems. For each
phase (gas and liquid), equations of the conservation of
the mass and momentum are written, and these equa-
tions are solved jointly using numerical methods. Of
course, with such approach, it is necessary to use a
number of simplifying assumptions requiring additional
substantiation. Studies in this ﬁ eld are surveyed in .
Such approach is promising in studying the physical re-
lationships of the ﬂ ow of the gas and liquid phases in
various gas–liquid apparatuses. However, such calcula-
tions cannot be an element of the engineering method
for apparatus calculation. In engineering calculations
of various processes occurring in gas–liquid systems, it
is necessary to have a simple model of the structure of
the ﬂ ows, adequately describing the features of the real
This study is aimed at developing a model convenient
for calculations to describe liquid mixing in an apparatus
with annular cross section and at determining the
parameters of this model using experimental data.