Mass Transfer in Multiphase Mechanically Agitated Systems

109
The effect of upper agitator type on the volumetric mass transfer coefficient value in the
gas–solid–liquid system is presented in Fig. 14. 0
2
4
6
8
10
123456
k
L
a
x 10
2
, 1/s
RT- A 315
RT- HE 3

X = 0.5 mass % X = 2.5 mass%
1.71
3.41
6.82
1.71

6.82


123456
k
L
a
x 10
2
, 1/s
RT- A 315
RT- HE 3

X = 0.5 mass % X = 2.5 mass%
1.71
3.41
6.82
1.71
6.82
3.41

Fig. 14. Comparison of values of k
L
a coefficient for two impeller configurations, working in
gas–solid–liquid systems; X ≠ const; n = 15 1/s; various values of superficial gas velocity w
og

×10
-3
m/s
In the three–phase system that, which agitator ensure better conditions to conduct the
process of gas ingredient transfer between gas and liquid phase, depends significantly on
the quantity of gas introduced into the vessel. Comparison of values of k

of agitators about 20 % higher values of the volumetric mass transfer coefficient were
obtained, comparing with the data characterized the vessel with HE 3 impeller as an upper
one (Fig. 12).
The data obtained for three–phase systems were also described mathematically. On the
strength of 150 experimental points Equation (2) was formulated:

G-L-S
L
2
L
12
1
1
b
c
og
P
ka A w
V
mX mX
⎛⎞
⎛⎞
=
⎜⎟
⎜⎟
⎜⎟
++
⎝⎠
⎝⎠
(6)

3
>; w
og
∈ <1.71×10
-3
; 6.82×10
-3
m/s>; X ∈ <0.5; 5 mass %>.

No. Impeller
A b c m
1
m
2

±Δ, %
Exp.
point
1. Rushton turbine (RT) 0.031 0.43 0.515 -186,67 11.921 6.8 100
2. Smith turbine (CD 6) 0.038 0.563 0.67 -388.62 23.469 6.6 98
3. A 315 0.062 0.522 0.774 209.86 -11.038 10.3 108
Table 6. The values of the coefficient A, m
1
, m
2
and exponents b, c in Eq. (6) for single
impeller systems (Kiełbus-Rąpała et al., 2010)

Configuration of impellers
No.

ka A w
VmX
−−
⎛⎞
=
⎜⎟
+
⎝⎠
(7)
The values of the coefficient A, m, and exponents B, C in this Eq. for both impeller designs
are collected in Table 8. The range of application of Eq. 2 is as follows: Re ∈ <9.7; 16.8×10
4
>;
P
G-L-S
/V
L
∈ <1100; 4950 W/m
3
>; w
og
∈ <1.71×10
-3
; 6.82×10
-3
m/s>; X ∈ <0; 0.025>.

Configuration of impellers
No.
lower upper

k
L
a. Within the range of the low values of the superficial gas velocity w
og
, high agitator
speeds n and low mean concentration X of the solids in the liquid, the value of the
coefficient k
L
a increases even about 20 % (for single impeller) comparing to the data
obtained for gas–liquid system. However, this trend decreases with the increase of both
w
og
and X values. For example, the increase of the k
L
a coefficient is equal to only 10 %
for the superficial gas velocity w
og
= 5.12 x10
-3
m/s. Moreover, within the highest range
of the agitator speeds n value of the k
L
a is even lower than that obtained for gas–liquid
system agitated by means of a single impeller.
In the case of using to agitation two impellers on the common shaft k
L
a coefficient
values were lower compared to a gas–liquid system at all superficial gas velocity
values.
4. The volumetric mass transfer coefficient increases, compared to the system without

h
1
off – bottom clearance of lower agitator m
h
2
off – bottom clearance of upper agitator m
i number of the agitators
J number of baffles
k
L
a volumetric mass transfer coefficient s
-1

n agitator speed s
-1

n
JSG
critical impeller speed for gas – solid – liquid system s
-1

P power consumption W
R curvature radius of the blade m
t time s
V
L
liquid volume m
3

V

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6
Gas-Liquid Mass Transfer in an
Unbaffled Vessel Agitated by Unsteadily
Forward-Reverse Rotating Multiple Impellers
Masanori Yoshida
1
, Kazuaki Yamagiwa
1
,

As mentioned above, in conventional agitation vessels, baffles are generally attached to the
vessel wall to avoid the formation of a purely rotational liquid flow, resulting in an
undeveloped vertical liquid flow. In contrast, if a rotation of an impeller and a flow
produced by the impeller are allowed to alternate periodically its direction, a sufficient
mixing of liquid phase would be expected in an unbaffled vessel without having anxiety
about the problems encountered with conventional agitation vessels. We developed an
agitator of a forward-reverse rotating shaft whose unsteady rotation proceeds while
alternating periodically its direction at a constant angle (Yoshida et al., 1996). Additionally,
Mass Transfer in Multiphase Systems and its Applications

118
we designed an impeller with four blades as are longer and narrower and are of triangular
sections. The impellers were attached on the agitator shaft to be multiply arranged in an
unbaffled vessel with a liquid height-to-diameter ratio of 2:1. This unbaffled vessel agitated
by the forward-reverse rotating impellers was applied to an air-water system and then its
performance as a gas-liquid contactor was experimentally assessed, with resolutions for the
above-mentioned problems being provided (Yoshida et al., 1996; Yoshida et al., 2002;
Yoshida et al., 2005).
Liquid phases treated in most chemical processes are mixtures of various substances.
Presence of inorganic electrolytes is known to decrease the rate for gas bubbles to coalesce
because of the electrical effect at the gas-liquid interface (Marrucci and Nicodemo, 1967;
Zieminski and Whittemore, 1971). In many cases, the electrical effect creates different gas-
liquid dispersion characteristics, such as decreased size of gas bubbles dispersed in liquid
phase without practical changes in their density, viscosity and surface tension (Linek et al.,
1970; Robinson and Wilke, 1973; Robinson and Wilke, 1974; Van’t Riet, 1979; Hassan and
Robinson, 1980; Linek et al., 1987). The present work assesses the mass transfer
characteristics in aerated electrolyte solutions, following assessment of those in the air-water
system, for the forward-reverse agitation vessel. In conjunction with the volumetric
coefficient of mass transfer as viewed from change in power input, which is a typical
performance characteristic of gas-liquid contactors, the dependences of mass transfer

o
, is π/4. When such a rotation with sinusoidal angular displacement is
expressed in the form of a cosine function, the angular velocity of impeller,
ω
i
, is given by
the sine function as
ω
i
=2π
θ
o
N
fr
sin(2πN
fr
t) (1)
Gas-Liquid Mass Transfer in an Unbaffled Vessel Agitated
by Unsteadily Forward-Reverse Rotating Multiple Impellers

119

Fig. 1. Schematic flow diagram of experimental apparatus. Dimensions in mm. Fig. 2. Structure and dimensions (in mm) of the impeller used.
where N
fr
is the frequency of forward-reverse rotation and was varied from 1.67 to 6.67 Hz
as an agitation rate. A ring sparger with 24 holes of 1.2 mm diameter (the circle passing

A system measuring unsteady torque of the shaft due to unsteady rotation of the impeller
consisted of the fluid force transducing part, impeller displacement transducing part and
signal processing part. In the fluid force transducing part, the strain generated during
operation in a copper alloy coupling having four strain gauges is recorded continuously. In
the impeller displacement transducing part, a switching circuit composed of a light emitting
diode and a phototransistor, etc. pulses the rest point in cycles of forward-reverse rotation of
impeller, thereby adjusting the frequency of forward-reverse rotation and defining the
trigger point of measurements as the rest point. In the signal processing part, the analog
signals of voltage from the fluid force and impeller displacement transducers are input into
a computer after being digitized to permit calculations of the torque of the forward-reverse
rotating shaft. The fluid force transducer detects the strains caused by different forces such
as fluid forces acting on the impeller and shaft and inertia forces due to the acceleration of
the motions of the impeller and shaft. The fluid force acting on the shaft was found to be
negligibly small in analysis, compared with that acting on the impeller. Hence, the moment
of the fluid force acting on the impeller, i.e., the agitation torque, can be obtained by
subtracting the value measured in air from that in liquid, with the impellers attached.
The time-course curve of instantaneous power consumption, P
m
, was obtained by
multiplying the instantaneous torque, T
m
, measured over one cyclic time of forward-reverse
rotation of impeller by the angular velocity of impeller at the corresponding time [Eq. (1)].
The time-averaged power consumption, P
mav
, that is based on the total energy transmitted in
one cycle was graphically determined from the time-course curve of P
m
.
P

, after
starting aeration under a given agitation was measured at the midway point of the liquid
depth, i.e., the distance 0.25 m above the vessel bottom, using a DO electrode. When there
was assumed to be little difference of oxygen concentration between the inlet air and outlet
gas, the overall volumetric coefficient based on the liquid volume, K
L
a
L
, was obtained from
the following relation:
ln[(C
L
*
-C
L
)/(C
L
*
-C
Lo
)]=-K
L
a
L
t (4)
where C
Lo
is the initial concentration, C
L
*

noted that the values of volumetric coefficient evaluated in this work are confined to the
control for comparison and would be required for the reliability to be improved.
Photographs of gas bubbles were taken at the midway point of the liquid depth, i.e., the
distance 0.25 m above the vessel bottom. A square column was set around the vessel section
where the photographs were taken. The space between the square column and vessel was
also filled with water to reduce optical distortion. A point immediately inside the vessel
wall was focused on. When a lamp light was collimated through slits to illuminate the
vertical plane including that point, bodies within 25 mm inside the vessel wall could be
almost in focus. The average value of readings of a scale placed in that space was employed
as a measure for comparison. A spheroid could approximate the bubble shape observed on
the photographs. By measuring the major and minor axes for at least 100 bubbles
photographed, the volume-surface mean diameter, d
vs
, was calculated. The overall gas hold-
up,
φ
gD
, based on the gassed liquid volume was determined using the manometric technique
(Robinson and Wilke, 1974). The manometer reading was corrected for the difference of
dynamic pressure, namely, that of the reading measured in ungassed liquid. When the
dispersion is assumed to comprise spherical gas bubbles of size d
vs
, the gas-liquid interfacial
area per unit volume of gassed liquid, a
D
, is calculated from the following equation:
a
D
=6
φ

φ
gD
)/a
D
(6)
3. Power characteristics of forward-reverse agitation vessel
3.1 Viscous and inertial drag coefficients
The following expression is assumed for the torque of the forward-reverse rotating shaft on
which the impellers were attached, i.e., the agitation torque, T
m
:
T
m
=C
d
ρ
D
i
5
ω
i
⏐
ω
i
⏐+C
m
ρ
D
i
5

d
=(3π/8)[(1/π)∮(T
m
/
ρ
D
i
5
θ
o
2
ω
fr
2
)sin(
ω
fr
t)d(
ω
fr
t)] (8)
C
m
=
θ
o
[(1/π)∮(T
m
/
ρ

=(
ρ
D
i
5
θ
o
2
ω
fr
2
)[(8C
d
/3π)sin(
ω
fr
t)+(C
m
/
θ
o
)cos(
ω
fr
t)] (10)

The data of agitation torque, T
m
, measured in electrolyte solutions of different
concentrations when the gassing rate, the agitation rate and the number of impellers were

. The dependences of
the ratios of gassed coefficients to ungassed ones, C
dg
/C
do
and C
mg
/C
mo
, characterizing the
decrease of the resistance of fluid for the impeller rotation due to aeration, on the agitation
conditions were examined. C
dg
/C
do
and C
mg
/C
mo
decreased with increase of N
fr
, whereas the
coefficient ratios were almost independent of n
i
and C
e
. The drag coefficients with variation
of the aeration and agitation condition in the electrolyte solutions were correlated in the
following form:
C

and
0.52V
s
0.69
N
fr
0.69
and that between C
mg
/C
mo
and 0.31V
s
1.07
N
fr
1.07
, respectively. As can be seen
from the figures, the observed values of respective drag coefficients were satisfactorily
reproduced by Eqs (11) and (12).

Gas-Liquid Mass Transfer in an Unbaffled Vessel Agitated
by Unsteadily Forward-Reverse Rotating Multiple Impellers

123

Fig. 3. Time-course of agitation torque, T
m
.


=(
ρ
D
i
5
θ
o
3
ω
fr
3
)sin(
ω
fr
t)[(8C
d
/3π)sin(
ω
fr
t)+(C
m
/
θ
o
)cos(
ω
fr
t)] (13)
Using Eq. (13), the time-averaged power consumption, P
mav

d
and C
m
, determined
experimentally. Agreements between the observed and calculated values were found to be
good. According to Eq. (2), the value of P
mav
was determined by integrating graphically P
m

with the time. On the other hand, combined use of Eq. (14) with Eq. (11) enables to calculate
P
mav
as a function of the aeration and agitation conditions such as V
s
, N
fr
and n
i
. Figure 7 Fig. 6. Time-course of agitation power, P
m
.

Fig. 7. Comparison of average agitation power, P
mav
, values observed with those calculated.
Gas-Liquid Mass Transfer in an Unbaffled Vessel Agitated

unit mass of liquid, P
tw
, with the electrolyte concentration, C
e
, and the superficial gas
velocity, V
s
, as parameters. For any system, k
L
a
L
tended to increase almost linearly with P
tw
.
The rate of increase in k
L
a
L
with P
tw
was practically independent of V
s
but differed
depending on the conditions with and without electrolyte in liquid phase.
The results for the baffled vessel agitated by the unidirectionally rotating multiple DT
impellers examined as a control and those reported by Van’t Riet (1979) and Nocentini et al.
(1993) are also shown in Fig. 8. Although the tendency that the dependence of k
L
a
L

suffers
the two counter influences. In the following sections, the mass transfer parameters such as
a
L
and k
L
are addressed for enhancement of the gas-liquid mass transfer in the forward-
reverse agitation vessel to be assessed.
Mass Transfer in Multiphase Systems and its Applications

126

Fig. 8. Comparison of volumetric coefficient, k
L
a
L
, as viewed from change in specific total
power input, P
tw
.
5. Hydrodynamics of forward-reverse agitation vessel
5.1 Mean bubble diameter
The dependence of the size of gas bubbles on aeration and agitation conditions was
investigated in terms of the power input. Figure 9 shows a typical relationship between the
mean bubble diameter, d
vs
, and the total power input per unit mass of liquid, P
tw
, with the
electrolyte concentration, C

tw
a
(15)
The exponent, a, of P
tw
was obtained from the slope of the straight lines as drawn in Fig. 9.
Its value was independent of the electrolyte concentration, C
e
. The coefficient, A, changed
depending on C
e
and its dependence was expressed for the experimental material of this
work as follows:
A=-1.49C
e
0.096
+2.95 (16)
Gas-Liquid Mass Transfer in an Unbaffled Vessel Agitated
by Unsteadily Forward-Reverse Rotating Multiple Impellers

127

Fig. 9. Relationship between mean bubble diameter, d
vs
, and specific total power input, P
tw
.
As a result, the empirical equation of d
vs
is

gD
increased with P
tw
, its values
differed depending on the electrolyte concentration, C
e
, and the superficial gas velocity, V
s
.
The gas hold-up,
φ
gD
, was then analyzed with the aeration and agitation conditions. Based
on the results shown in Fig. 11, the following functional form was inferred for the empirical
equation of
φ
gD
.
Mass Transfer in Multiphase Systems and its Applications

128
φ
gD
=BP
tw
b1
V
s
b2
(18)

e
0.27
+1.32)P
tw
0.46
V
s
0.70
(20) Fig. 12. Comparison of
φ
gD
values observed with those calculated.
Gas-Liquid Mass Transfer in an Unbaffled Vessel Agitated
by Unsteadily Forward-Reverse Rotating Multiple Impellers

129
5.3 Gas-liquid interfacial area
For electrolyte solutions aerated at the velocities ranged in this work, no significant
difference in the mean bubble diameter and the gas hold-up determining the magnitude of
gas-liquid interfacial area was observed between forward-reverse and conventional
agitation vessel. From this, somewhat larger values of volumetric coefficient as illustrated
in Fig. 8 in the former vessel may be a reflection of the contribution of forward-reverse
rotation of the impeller to increase of the liquid-phase mass transfer coefficient with
occurrence of an intensified liquid turbulence in the vicinity of the gas-liquid interface.
6. Mass transfer consideration
6.1 Analysis of mass transfer coefficient
Calderbank and Moo-Young (1961) first examined the liquid-phase mass transfer coefficient,

1/2
d
b
-1/2
D
L
2/3
(21)
Higbie equation:
k
L
=(2/π
1/2
)[D
L
/(d
b
/V
b
)]
1/2
(22)
In these equations,
ρ
is the liquid density,
μ
is the liquid viscosity, and D
L
is the liquid-phase
diffusivity.

values for
forward-reverse agitation vessel were noticeably different from those for conventional one.
That is, k
L
came to exhibit higher values than those by Calderbank and Moo-Young, with the
maximum difference being about three times, when the agitation rate, N
fr
, was increased.
For such a peculiar difference of k
L
with N
fr
for bubbles of similar sizes, consideration is
Mass Transfer in Multiphase Systems and its Applications

130
necessary in terms of hydrodynamic parameters, taking into account not only the mean
bubble diameter but also other important variables such as the rising velocity of a swarm of
gas bubbles that changes depending on the gas hold-up. Specifically, examination is
required of the mass transfer characteristics as viewed from change in the Reynolds number,
which is a parameter characterizing the liquid flow around gas bubble. Fig. 13. Relationship between mass transfer coefficient, k
L
, and gas bubble size. Fig. 14. Relationship between Sh and Re.
Gas-Liquid Mass Transfer in an Unbaffled Vessel Agitated

ρ
V
b
d
b
/
μ
(25)
Sc=
μ
/
ρ
D
L
(26)
The Froessling and Higbie equations mentioned above are expressed in the forms of Eq.
(23), respectively, as
Froessling equation:
Sh=2+0.55Re
1/2
Sc
1/3
(27)
Higbie equation:
Sh=(2/π
1/2
)Re
1/2
Sc
1/2

is the superficial gas velocity.
φ
gD
is the gas hold-up and the values determined as
the average within the vessel were used for calculation from Eq. (29).
Figure 14 shows the relationship between Sh and Re for the forward-reverse agitation vessel.
The Sh values for the conventional agitation vessel as a control and those by Calderbank and
Moo-Young (1961) are also shown in comparison with the theoretical predictions for single
bubble. For the mean bubble diameter, d
vs
, larger than 2.5 mm, the Sh values for the
conventional agitation vessel more closely resembled the values calculated from the Higbie
equation than those calculated from the Froessling equation. For d
vs
smaller than 2.5 mm,
decrease of Sh with decrease of Re was remarkable, with Sh exhibiting the values that are
Mass Transfer in Multiphase Systems and its Applications

132
intermediary between those from the Higbie and Froessling equations. This fact suggests
that the contribution of liquid flow to the mass transfer is reduced considerably if the
internal circulation within gas bubble is prevented from becoming fully developed through
decreased rising velocity of gas bubble that is attributable to its decreased size, namely,
smaller inertial force of rising motion of gas bubble (Tadaki and Maeda, 1963; Sideman et al.,
1966). The relationship between Sh and Re for the forward-reverse agitation vessel under
the condition of lower agitation rates exhibited the same tendency as that for the
conventional agitation vessel, as shown in the figure. That is, Sh under such conditions
increased approximately in proportion to Re
0.5
, according to the Higbie equation, for d

number (Re), the Strouhal number (St), and the Schmidt number (Sc), which are necessary to
express the unsteady mass transfer phenomena, are all derived based on the equation of
motion that is non-dimensionalized under a condition of unsteady liquid flow, the
dimensionless diffusion equation and the dimensionless mass balance equation. Therefore,
Sh in this gas-liquid agitation system is given as a function of Re, St and Sc.
Sh=func. (Re, St, Sc) (30)
Definitions of Sh, Re and Sc are mentioned above; St is defined for single gas bubble as
St=fd
b
/V
b
(31)
The diameter, d
b
, and the velocity, V
b
, for St were identical to those for Re. The frequency of
forward-reverse rotation of impeller, N
fr
, was taken as a characteristic frequency, f.
The experimental results shown in Fig. 14 suggest that the two coexisting liquid flows affect
Sh in a form of their superposition. Then, the degree of effect would be determined by the
relative magnitude of Re characterizing the steady slip flow of surrounding liquid with the
Gas-Liquid Mass Transfer in an Unbaffled Vessel Agitated
by Unsteadily Forward-Reverse Rotating Multiple Impellers

133
rising motion of gas bubble and St characterizing the unsteadily oscillating flow of liquid
around gas bubble by forward-reverse rotation of the impeller. Furthermore, a value of 1/2,
which many researchers (Calderbank, 1959; Yagi and Yoshida, 1975; Nishikawa et al., 1981;

1/2
-func. (Re) and St was examined.
From the slope of the line and the intercept on the axis in the logarithmic plot, the empirical
constants, c and C, were determined for the respective ranges with d
vs
=2.5 mm as a
boundary, and then the following correlation equations were obtained.
d
vs
<2.5 mm:
Sh=[0.0544Re
0.90
+10.0St
0.10
]Sc
1/2
(35)
d
vs
>2.5 mm:
Sh=[(2/π
1/2
)Re
1/2
+180St
0.79
]Sc
1/2
(36)
The result of correlation of Sh is shown in Fig. 15. As shown in the figure, Sh was correlated