AP01_6.vp


1 Introduction
Stirring processes are used in many parts of the chemi-

cal, pharmaceutical, food and bioengineering industries.
In most of them the stirring is a basic operation, e.g.
during homogenization (or blending) of miscible liquids,
i.e. the compensation of temperature and concentration
differences. For such a process it is very important to
know the blending efficiency when designing an industrial
plant. However, the optimal design of stirring plants also
involves better economics of the mixing process. In this
paper we study the problems of the blending efficiency
of pitch blade impellers (PBI), especially the course of
homogenization performed by a three blade (3-b) PBI
with various pitch angles and different vessel/impeller
diameter ratios. There are a few papers dealing with
experimentally determined results of blending times of
the PBI [e.g. 1, 2, 3, 4] but no study describing the blend-
ing efficiency of the above mentioned impellers has yet
appeared up to now. Therefore, additional experimental
data needs to be compiled to supplement our knowledge
of the blending efficiency of the 3-b PBI, with different
geometric configurations.

2 Theoretical
Let us consider a three-blade pitched blade impeller

(3-b PBI) in a blending process of low viscous miscible liq-
uids. We can assume the turbulent regime of flow of the
agitated batch. The dependence of the dimensionless time
of homogenization (mixing time) on the Reynolds number
(Re), modified for rotary impeller

Re � nD2� � , (1)

expressed by the function

� �nt nt� Re (2)

can be distributed in three regions (see Fig. 1):

In the range Re < 10, the liquid around the impeller
moves with the impeller rotation. Therefore, the process of
mixing is negligible and the real mixing time is very long.

When the Re > 10, the flow around the impeller is
turbulent. With an increase of the Reynolds number the
viscous forces in the rest of the agitated batch decrease
and the inertia forces increase.

In the range Re > 104, the negligible effects of the viscous
forces and the forces in the whole agitated batch are only
the inertial forces. The dimensionless blending time does
not depend on the Reynolds number but it depends on the
geometry of the agitated system and on the chosen degree of
homogeneity [5]

� �
� �� �

C t
c t c
c c

�
�

�
k

0 k
(3)

where the average concentration in the volume of the
mixed batch apart from the volume of the concentration
impulse �V

� � � �c t
V

c v t V� �
1

, d . (4)

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Acta Polytechnica Vol. 41  No. 6/2001

Study of the Blending Efficiency of
Pitched Blade Impellers

I. Fořt, T. Jirout, F. Rieger, R. Allner, R. Sperling

This paper presents an analysis of the blending efficiency of pitched blade impellers under a turbulent regime of flow of an agitated
low viscous liquid. The conductivity method is used to determine of the blending (homogenization) time of miscible liquids in pilot
plant mixing equipment with standard radial baffles. For the given homogeneity degree (98%) a three-blade pitched blade impeller is
tested with various off- bottom clearances, vessel/ impeller diameter ratios and various impeller pitch angles.
The experimental results show in accordance with theoretical data from the literature, that the greatest effect on the dimensionless
blending time is exhibited by the vessel/ impeller diameter ratio and the impeller pitch angle. The number of total circulations
necessary for reaching the chosen homogeneity degree depends on the impeller pitch angle and amounts more than three. Finally, the
energetic efficiency of the blending process is calculated.The results of this study show, that the highest energetic efficiency of the
three-blade pitched blade impeller appears for the pitch angle � = 24°, the impeller/vessel diameter ratio T/D = 2 and the impeller
off-bottom clearance h/D = 1.

Keywords: pitched blade impeller, blending of liquids, degree of homogeneity, turbulent flow.

n t = f(Re) n t = const.

n t
[-]

Re [-]

creeping flow turbulent regiontransient region

Fig. 1: Typical dependence of dimensionless blending time on
the Reynolds number for a high-speed rotary impeller



The initial concentration is defined as
� �� �c c t0 0� � (5)

and the final concentration of the dissolved matter in the
agitated liquid is

� �� �c c tk � 	 
 . (6)
Quantity c (t) is the instantaneous concentration of the

dissolved matter averaged over the volume of agitated liquid
volume V, reduced about the volume of concentration impulse
�V. For the sake of simplicity we neglect the volume �V with
respect to value V; the value �V amounts to less than one
thousanth of value V.

Turbulent flow of an agitated liquid is realized by its
circulation. For the given homogeneity degree the blending
process takes place during the given number of the total
liquid circulation nC [5]

t n t� C T, [ C = const.] (7)

where tT is the mean time of total liquid circulation, which
can be calculated from the relation

t
V

Q
T

T
� , (8)

where QT is the total volumetric flow of the agitated liquid.
If we consider a “squared” configuration of the volume of
the agitated liquid, i.e.

V T�
�

4
3 (9)

and introduce the total flowrate number

N
Q

nD
Q T

T� 3 (10)

we can finally rearrange Eq. (2) into the form

n t n
T
D NQ

� �
�


�
�
�C

T

�

4
13

, [C = const.] (11)

Eq. (11) includes impeller speed n, impeller diameter D,
and the mixing vessel diameter T.

The energetic blending criterion E indicates the energy
consumption ratio necessary for the chosen homogeneity
degree [6]:

� �E
P t

T
Po n t

D
T

� � �
�


�
�
�

3

5
3

5

�
. (12)

To calculate the energetic criterion we need to know the
impeller power input P and the mixing time for the given
configuration of the agitated system. Quantity P can be calcu-
lated from the power number

Po = P / � n3 D5 (13)

where � is the density of the agitated liquid. The quantity Po
depends on the geometry of the system agitated by a PBI and
this dependence can be expressed by the equation [7]

� � � � � � � �Po T D h D H T n� � �1 507 0 365 0 165 0 140 2 077. sin. . . .B
0.171

� ,(14)

valid for values of the Reynolds number greater than 104. It
consists among others of quantities nB – the number of
impeller blades, � – the pitch angle of the impeller blades,
and h – the impeller off-bottom clearance.

3 Experimental
After the injection of a small amount of tracer �V (about

1 ccm) into the agitated liquid, the concentration changes
and mixing time were measured in appropriate locations in
the agitated liquid. The conductivity method was used for
measuring the blending time. This method is based on the
principle of monitoring the changes in electrical conductivity
within the mixed liquid. The change in electrical conductivity
was caused by adding a sample of concentrated solution of
sodium chloride into the liquid below its surface along the
impeller shaft. In our experiments the injected sample of
liquid had approximately the same density and viscosity as
the mixed liquid, and thus the effect of the Archimedes
number was eliminated. After adding the tracer the time
change of conductivity was measured and recorded and the
blending time was determined. The conductivity cell con-
sisted of two platinum wire electrodes of 0.8 mm diameter in
the shape of a rectangle 8×10 mm, 8 mm apart. The volume
of the conductivity cell was approximately 0.8 ccm. The cell
was located at of the liquid height under the surface of the
mixed liquid and at 110 of the vessel diameter from the wall
of the vessel (see Fig. 2). We chose this position because

during a test with the decolourizing method a dead space was
detected in this volume for the impellers and vessels under in-
vestigation. The volume of the liquid injected varied in
the range 0.5–1.5 ccm. The process of homogenization was
recorded with a fast chart recorder. The time of homogeniza-
tion (the blending time) was found at the moment when the
fluctuation of the measured electrical voltage U was ±2 %.
A typical time course of the indicated voltage is shown in
Fig. 3.

The principle layout of the pilot plant experimental
equipment is shown in Fig. 2. The blending time was deter-
mined at various geometric configurations; in all of the cases
the impeller was pumping liquid downwards – towards the
bottom of the cylindrical vessel and was located in its axis of
symmetry.

Measurements were carried out with two different series
of 3-b PBIs (Fig. 4 shows the geometry of the impellers used)
with diameters of 100 mm and 67 mm, respectively. For each
impeller we used three different pitch angles:

� = 24°, 35°, and 45°.

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Acta Polytechnica Vol. 41  No. 6/2001

Fig. 2: Cylindrical vessel with a high-speed rotary impeller, radial
baffles and position of the conductivity probe; (1– conduc-
tivity probe, 2– baffle, 3– vessel, 4– impeller)



For all the impellers the three different vessel/impeller
ratios were investigated: T/D = 2, 3, 4.5. The influence of the
impeller off-bottom clearance was studied at the standard
configuration (T/D = 3, � = 45°). All these configurations are
summarized in Tab. 1. Additionally, the blending time was
determined with a dish shaped bottom of the vessel (for
T/D = 2, 3 and � = 45°). All the vessels were equipped with
four baffles distributed equally around the vessel wall (see
Fig. 2). The ratio between baffle width b and the vessel
diameter b/T = 0.1.

A larger interval of the Reynolds number was covered by
using three different levels of viscosities of the tested liquids:

1. Distilled water [viscosity around 1 mPa�s], 2. 25 % w/w
glycerol water solution [viscosity around 2 mPa�s], and
3. 45 % w/w glycerol water solution [viscosity around 4 mPa�s].
The viscosity was measured with the Hoeppler Viscosimeter
B3 (MLW Prufgeraetewerke, Freital, Germany) at five dif-
ferent temperatures, and the viscosity of the mixed liquid was
calculated by linear interpolation from its known tempe-
rature dependence.

The accuracy of the data obtained depends significantly
on the independent variables, i.e. the viscosity of the mixed
liquid, the impeller speed and on the geometric configura-
tion of the agitated system. The impeller speed was measured
by a photoelectric revolution counter having an accuracy of
±1/min. The diameter of the impeller and the width of its
blade were manufactured with an accuracy of ±0.1 mm and
the pitch angle within an accuracy of ±30’. The impeller
position in the vessel and the height of liquid in the vessel
were measued by a ruler with an accurancy of ±1 mm.
The accuracy of measurement of the dynamic viscosity was
±0.1 mPa�s.

The process of homogenization was recorded with a fast
chart recorder with a scale 1 mm ~ 1 second. The accuracy of
determination of the blending time was considered ±1 s. For
each configuration of the agitated system the blending time
was measured at five different values of impeller speed at
constant viscosity of the mixed liquid. Because of the required
confidence level of the final experimental results [3, 4, 6], the
five courses of blending time for each impeller speed were
determined and the average value for the given experiment
was calculated. The blending time measured and calculated
by this process exhibits a relative deviation of ±5 % from the
calculated average value.

4 Results and discussion
The results of the experiments were evaluated from the

point of view of dependence of the blending time on the
geometric parameters of the agitated system and, further,
according to the formulas in the theoretical part of this study,
evaluations were made to investigate the relation between the
blending time and the mixed liquid circulation. Finally, the
blending efficiency of the 3-b PBI was calculated with respect
to various geometries of the investigated system with the aim
to find the optimal arrangement for the blending process. It
follows from all the results of the experiments (see examples
in Fig. 5, 6) that when the Reynolds number (modified for
a rotary impeller) exceeds ten thousand the dimensionless
blending time does not depend on Re.

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Acta Polytechnica Vol. 41  No. 6/2001

U

voltage

U [mV]

time of homogenization t [s]

� U U� �2 %

Fig. 3: Recorded trace (example of the time course of homo-
genization)

Fig. 4: Three-blade pitched blade impeller (D = 67 mm, 100 mm,
h/D = 0.2)

Vessel/Impeller
diameter ratio

T/D [–]

Impeller off-bottom
clearance/ Impeller

diameter ratio
h/D [–]

Diameter of Vessel
T [mm]

Diameter of Impeller
D [mm]

Off-bottom Clearance
h [mm]

2 1 200 100 100
3 1; 0.75; 200 67 67; 50; 33; 22;

0.50; 0.333; 300 100 100; 75; 50; 33;
4.5 1 300 67 100

Table 1: Pilot plant configurations for investigation for nB = 3; (H = T )



Dependence of the blending time on the geometry of
a mixed system

Taking into account the independence of the quantity n t
from the Reynolds number for the turbulent regime of flow of
an agitated liquid, the dependence of the dimensionless
blending time on the geometric parameters of the mixing sys-
tem was expected in the power form

� �n t K
T
D

h
D

a b
c� �

�


�
�
�

�
�


�
�
�1

1 1
1sin � (15)

or

n t K
T
D

h
D

a b
c� �

�


�
�
�

�
�


�
�
�2

2 2
2� . (16)

The values of the exponents ai , bi , ci and the parameter ni,
( i = 1, 2) were calculated from the results of the experiments
(see Tab. 2) by means of multidimensional log-log linear re-
gression and they are listed in Tab. 3 together with the corre-
lation coefficients Ri , ( i = 1, 2) of the corresponding regres-
sion equations. It follows from these two tables that there is no
difference between the agreements of the two proposed equa-
tions [(n t)calc vers. (n t)exp] and that the correlation coefficients
of both equations are practically the same, expressing a suit-
able confidence level of the considered power dependence.
When we look at the values of the exponents in Eqs. (15) and
(16) the highest influence on the blending time is for the ves-
sel/impeller diameter ratio: here the values of exponents a1
and a2 are approx. two. Similar values were found for the stan-
dard Rushton turbine impeller [3] and for four-blade [3] and

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Acta Polytechnica Vol. 41  No. 6/2001

n t = f(Re)

10

100

1000

1000 10000 100000

Re [-]

n t [-]

0.92 mPa s ; = 67 mm� D

1.79 mPa s ; = 67 mmD�

4.08 mPa s ; = 67 mmD�

0.92 mPa s ; = 100 mmD�

3.74 mPa s ; = 100 mmD�

( )av.n t

Fig. 5: Dimensionless blending time as a function of Re, ( D = 67, 100 mm, T/D = 3, � = 35° ) for different viscosities of mixed liquid

n t = f(Re)

10

100

1000 10000 100000

Re [-]

n t [-]

0.93 mPa s ; = 67 mmD�

1.79 mPa s ; = 67 mmD�

4.08 mPa s ; = 67mmD�

0.93 mPa s ; = 100 mmD�

2.11 mPa s ; = 100 mmD�

3.71 mPa s ; = 100 mmD�

( )av.n t�

Fig. 6: Dimensionless blending time as a function of Re, ( D = 67, 100 mm, T/D = 3, � = 45° ) for different viscosities of mixed liquid



six blade PBIs [4]. The values of the exponents c1 and c2 are
moderate and do not differ significantly. The higher the pitch
angle, the lower the blending time.

Table 4 presents a survey of calculated values of the num-
ber of total liquid circulations nC necessary for attaining the
given homogeneity degree. The values of the total flow rate
number NQT were taken from the literature [5] and the cor-
responding values of the dimensionless blending time n t
follow from this study. It follows from this table that 3–4
circulations of the mixed liquid are necessary for attaining
the degree of homogeneity required, and that the impeller
off-bottom clearance does not exhibit a significant influence
on this number. It seems that a decrease in the impeller blade

pitch angle causes a decrease in the number of liquid circu-
lations, but independently of the vessel/impeller diameter
ratio.

Energetic blending criterion
Energetic blending criterion E (see Eq. 12) characterizes

in dimensionless form the energy necessary for attaining
the chosen homogeneity of the agitated liquids. The higher
the energetic blending criterion the lower the blending effi-
ciency at the given geometry of agitated system. Table 5 gives
a survey of the values of quantity E following from our experi-
mental data on blending time. The impeller power number
Po was calculated from Eq. (14).

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Acta Polytechnica Vol. 41  No. 6/2001

D/T [– ] h/D [– ] � [ ° ] (deg) (n t)exp [– ] (n t)calc [– ]
(Eq. 15)

(n t)calc [– ]
(Eq. 16)

2 1 24 32.8 � 2.0 29.8 29.8
2 1 35 25.3 � 1.3 24.5 24.6
2 1 45 18.6 � 1.0 21.7 21.7
3 1 24 60. 3 � 4. 2 66. 9 66.8
3 1 35 58. 3 � 2. 7 55.0 55.3
3 1 45 49. 2 � 2. 2 48.9 48.8

4.5 1 24 142.0 � 8.1 150.1 150.0
4.5 1 35 118.0 � 11.2 123.6 124.2
4.5 1 45 112.0 � 10.5 109.8 109.5
3 0.33 45 54.5 � 3.2 56.1 56.1
3 0.50 45 55.3 � 4.8 53.3 53.3
3 0.77 45 51.8 � 3.5 50.7 50.6

Table 2: Dependance of dimensionless blending time on impeller pitch angle (three-blade pitched blade impeller), C = 0.02; nB = 3

Eq. Ki ai bi ci Ri
15, i = 1 4.49 1.994 	0.125 	0.566 0.992
16, i = 2 36.63 1.994 	0.128 	0.501 0.992

Table 3: Parameters of the power regression for sin � (Eq. 15) and � (Eq. 16), C = 0.02, nB = 3, Re > 10
4

� [ ° ] (deg) D/T [– ] h/D [– ] n t NQT [– ] nC [– ] nC, av [– ]

45 3 0.30 54.1 1.614 4.116

4.072
45 3 0.50 55.3 1.654 4.313
45 3 0.75 51.8 1.706 4.167
45 3 1.00 47.3 1.654 3.961
24 2 1.00 37.8 0.686 3.584

3.32324 3 1.00 60.3 1.024 2.913
24 4.5 1.00 142.0 1.751 3.474
24 3 0.75 60.3 1.331 3.788

4.16235 3 0.75 58.7 1.637 4.534
45 3 0.75 51.8 1.706 4.167

Table 4: Number of total liquid circulations for the given homogeneity degree (C = 0.02) and the number of impeller blades ( nB = 3)



For the pitch angle � = 45° we can see a permanent
increase in the energetic blending criterion with the in-
creasing ratio T/D. For the pitch angles � = 24° and 35°
a monotonous trend cannot be seen, e.g. the energetic crite-
rion for � = 35° is similar for ratios T/D = 3 and 4.5, while at
the ratios T/D = 2 and 4.5 for pitch angle � = 24° the values
of the energetic criterion have a difference of just 4 %. The
difference between the criteria corresponding to the ratios
T/D = 3 and 4.5 is about 20 %.

The vessel/impeller ratio exhibits an influence on the dif-
ference between the values of the energetic blending criterion
when the flat bottom of the vessel is replaced by a dished
bottom. Here the difference is more significant for the ratio
T/D = 3, where the flow pattern of the agitated liquid follows
better the shape of the bottom, and then the blending process
is faster than in the system with the flat bottom.

The data in Tab. 5 show also that there is a dependence
between the energetic blending criterion and the impeller
off-bottom clearance. The energetic criterion E decreases
with increasing off-bottom clearance. The difference of the
energetic criteria between the shortest and longest impeller
off-bottom clearance is around 85 %. The favourable effect of
decreasing the ratio h/D can be explained by the fact that
the liquid is pumped by the impeller towards the bottom,
where the high level of turbulence contributes to the better
blending process.

The energetic blending criterion E increases with increas-
ing pitch angle at the value of ratio T/D = 4.5 but at the ratio
T/D = 3 we cannot find a clear trend for quantity E as a func-
tion of quantity �. Generally, for the off-bottom clearance
h/D = 1 criterion E decreases with increasing pitch angle, but
with certain exceptions especially around the pitch angle
� = 35°.

5 Conclusions
A three-blade pitched blade impeller seems to be suitable

for blending processes under the turbulent regime of flow of
an agitated liquid. The discovered correlation allows us to
calculate the blending time for different pitch angles, ratios
D/T and H2/D at the homogeneity degree C = 0.02. The most
efficient energetic blending criterion among the geometries
investigated seems to appear at the configuration � = 24°,
T/D = 2 , h/D = 1 and, on the other hand, the most inefficient
configuration was found at the geometry � = 45°, T/D = 4.5,
h/D = 1.

This research was subsidised by research project of the
Ministry of Education of the Czech Republic No. J04/98:
212200008.

List of Symbols
b baffle width, [m]
C degree of homogeneity
c concentration, [kg�m	3]
D impeller diameter, [m]
E energetic blending criterion
H total liquid depth, [m]
h impeller off-bottom clearance, [m]
n impeller speed, [sec	1]
nB number of impeller blades
nt dimensionless blending time
NQT total flow rate number
P impeller power consumption, [W]
Po power number
R correlation coefficient

12

Acta Polytechnica Vol. 41  No. 6/2001

� [ ° ] (deg) D/T [– ] h/D [– ] n t [– ] Po [– ] E [–]

24 2 1 37.8 0.39 430
24 3 1 60.3 0.34 367
24 4.5 1 142.0 0.29 450
35 2 1 25.3 0.80 403
35 3 1 58.7 0.69 572
35 4.5 1 118.0 0.59 528
45 2 1 18.6 1.23 236
45 2 1 18.3 1.23 247 a)

45 3 1 42.3 1.06 329 a)

45 3 1 48.7 1.06 454
45 3 0.75 51.8 1.11 633
45 3 0.50 55.3 1.19 824
45 3 0.33 54.5 1.29 858
45 4.5 1 112.0 0.92 697

a) Dished bottom

Table 5: Dependence of the energetic blending criterion on the geometry of an agitated system with a three-blade pitched blade impel-
ler ( C = 0.02), flat bottom of cylindrical vessel



Re Reynolds number modified for rotary impeller
T vessel diameter, [m]
t homogenization (blending) time, [sec]
tT time of total liquid circulation, [sec]
QT total volumetric flow ratio of agitated liquid, [m

3
�s	1]

U voltage, [V]
V volume of agitated liquid, [m3]
�V volume of concentration impulse, [m3]
� pitch angle of impeller blades, ° (deg)

 dynamic viscosity, [Pa�s]
� density, [kg�m	3]

Subscripts
calc calculated value
exp experimental value
k final value
0 initial value

References
[1] Nagata, Sh.: Mixing. Principles and applications. Kodansha

Ltd., Tokyo, John Wiley & Sons, N. Y., 1975
[2] GVC-Fachauschuss “Mischvorgaenge”: Mischen and

Ruehren. VDI Gesellschaft Verfahrungstechnik und Che-
mieingenieurwesen, Dusseldorf, 1998

[3] Procházka, J., Landau, J.: Homogenization of Miscible
Liquids by Rotary Impellers. Collect. Czech. Chem.
Commun. 26, 1961, pp. 2961–2974

[4] Kvasnička, J.: Thesis. Research Institute of Chemical
Equipment (VÚCHZ-CHEPOS), Brno, 1967

[5] Fořt, I., Valešová, H., Kudrna, V.: Liquid Circulation in
a System with Axial Mixer and Radial Baffles. Collect. Czech.
Chem. Commun. 36, 1971, pp. 164–185

[6] Rieger, F., Novák, V.: Homogenization Efficiency of Heli-
cal Ribbon and Anchor Agitators. Chem. Eng. Jour. 9,
1975, pp. 63–70

[7] Medek, J.: Power Characteristic of Agitators with Flat In-
clined Blades. Int. Chem. Eng. 20, 1980, pp. 665–672.

Doc. Ing. Ivan Fořt, DrSc.
Ing. Tomáš Jirout
Prof. Ing. František Rieger, DrSc.,

Dept. of Process Engineering
phone: +4202 2435 2713
fax: +4202 24310292
e-mail: fort@fsid.cvut.cz

Czech Technical University in Prague
Faculty of Mechanical Engineering
Technická 4, 166 07 Praha 6, Czech Republic

Ing. Ralf Allner
Prof. Dr. Ing. Reinhard Sperling
e-mail: reinhard.sperling@lbv.hs.anhalt.de

Dept. of Chemical Engineering
Anhalt University of Appl. Sciences
Hochschule Anhalt (FH)
Koethen, Germany

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Acta Polytechnica Vol. 41  No. 6/2001