HEFAT2007 5 International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics

HEFAT2007 5 International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics
HEFAT2007
5th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics
Sun City, South Africa
Paper number: AT1
CONVECTION HEAT TRANSFER IN BAFFLED MIXING TANK
Tuomo Aho and Reijo Karvinen
Tampere University of Technology
Energy and Process Engineering
P.O.Box 589, FIN-33101 Tampere, Finland
[email protected]
ABSTRAC
Single -phase flow field and temperature distribution
in a baffled tank stirred by a three- bladed impeller was
investigated both computationally and experimentally.
The computational model employed a sliding mesh
technique in fully three-dimensional grids for a smallscale unit, in which also velocity and heat transfer
measurements were made with water. Turbulence
effects were simulated using the standard k-ε model.
Two different boundary conditions, namely, constant
heat flux and constant temperature on the wall were
used for heat transfer simulations. Mean velocity and
turbulence were measured using LDA. In temperature
measurements
thermocouples were used. By
comparing experimental and modelled results nondimensional variables of velocity were found, which
gave very similar results to the pilot unit and full- size
reactor of 12 m3, for which some modelled heat transfer
results are given.
NOMENCLATURE
D
[m]
d
[m]
H
[m]
k
[m2/sec3]
LDA
L1, L2, L3 [m]
N
[rpm]
NRe
PIV
q
[W/m2]
R
[m]
[m]
R1, R2
[m]
Rbl
r
[m]
T
[°]
U
[m/s]
[m/s]
Wtip
W
[m/s]
x
[m/s]
tank diameter
impeller diameter
tank height
turbulence kinetic energy
Laser-Doppler anemometry
baffle distances
rotational speed
impeller Reynolds number
Particle Image velocimetry
heat flux
tank radius
bottom radius
impeller blade radius
radial coordinate
mean temperature
axial mean velocity
impeller tip speed
tangential mean velocity
axial coordinate
Special characters
ε
[m2/sec3]
μ
[kg/s/m]
ρ
[kg/m3]
τ
[sec]
φ
[°]
turbulence dissipation
dynamic viscosity
density
time
tangential coordinate
Superscripts
*
dimensionless parameter
INTRODUCTION
Impeller-stirred reactors are widely used in the
chemical industry to provide effective mixing of
chemical reactants to form desired products. The type
of impeller determines flow patterns and therefore the
efficiency of the mixing process. Knowledge of factors
such as mixing efficiency, heat transfer rate, residence
time and concentration levels are critical to the
successful operation of an impeller-stirred tank. These
key operating factors are typically investigated by
conducting measurements using a small pilot- or benchscale equipment. Accurate measurements in
commercial scale units are difficult and often
impossible to carry out. The smaller scale testing
process is usually scaled up to commercial scale
operating conditions. Since the scaling procedure is also
very complex, scale-up models for mixing tanks have
been limited. Computational fluid dynamics provides a
useful method to simulate the performance of both
bench-scale and full-size units. Much work on stirred
tank computations in three dimensions has been carried
out in the last few years using different types of
methods. In the first stage, the impeller was replaced by
a jet and turbulence was handled with the two-equation
models of turbulence [1]. Later on, also the impeller
could be included in predictions employing a rotation
mesh around the impeller [2]. In order to couple
together the impeller and the remainder of the tank
there are many possibilities: steady- state, quasi-static
and transient approaches [3]. As a matter of fact, the
methods are same as in the modelling of impeller
pumps [4]. If use is made of two-equation models
details of turbulence cannot be found. If extended
computational resources are available, more
sophisticated approaches like LES can be adopted [5].
There also exist plenty of experimental papers in the
literature. Mean velocities [6] and power consumption
have been measured [7] and dimensionleess numbers
have been tried to found to descript the performance
[8]. During recent years the structure of turbulence in a
mixing tank has also been measured with PIV and LDA
[9,10].
Published data for the case of a stirred tank with heat
transfer are very limited and there exist only some
papers [11]. The objective of this investigation is to
model the velocity field, to map the temperature
distributions and to model heat transfer in a stirred tank.
To obtain this goal, an impeller-stirred tank is modeled
using computational fluid dynamics simulations with a
sliding grid technique, which has been applied
successfully for the stirred tanks in earlier studies
[2,3,12]. Simulations are carried out with two different
boundary conditions for heat transfer. Data from LDA
measurements of mean velocities in a bench-scale unit
are used to validate calculated velocities. Measurements
of actual temperature distribution are provided as a
basis for checking model reliability.
STIRRED VESSEL CONFIGURATION
X*, U *
D
0.875
0.75
L1
0.625
R
0.50
R1
0.375
0.25
R2
d
The calculations were performed using a nonuniform multiblock grid generation technique. Four
grid densities were used for checking the solution
dependency on the grid density. Especially, at the near
wall region, where heat transfer takes place, the grid
density must be large. The fine grid consisted of
115200 cells. The surface grid for the coarse case and a
side cross-section for the fine grid are illustrated in
Figure 2. The solved velocity field was used for heat
transfer calculations as an initial guess. The fluid was
water, for which density and specific heat can be
assumed constant, but the effect of temperature on
viscosity was taken into acconunt. The no-slip
condition is applied at the surface of the vessel, baffles,
impeller blades and shaft. The free surface is treated as
a plane of symmetry. Heat flux through the free surface
and tank bottom was assumed to be zero.
φ, W*
H
R bl
L2
L3
R *, V *
Figure 1 Gross-section and plan view of bench-scale
stirred tank and blade impeller.
Figure 1 shows a schematic diagram of the
experimental stirred tank and blade impeller. The
cylindrical tank with an inner diameter D of 246 mm
was filled with water at room temperature to a height of
329 mm from tank bottom. The mixing vessel had a
dished bottom,the curvature of which is given by R1 and
R2 equal to 197 mm and 38 mm, respectively. The tank
was equipped with two internal baffles each with a
width L1 of 18.5 mm. Both baffles were mounted at a
distance L2 of 18.5 mm (0.075D) from the tank wall.
The clearance between each baffle and the tank bottom
L3 was 61.5 mm. The curved blade impeller diameter
d and width were 138 mm and 25 mm, respectively.
The radius of the blade curvature, Rbl, was 61.5 mm,
and the blades were tilted 10 degrees past vertical (10 degrees rake).
NUMERICAL MODELLING
The sliding grid method was used for threedimensional flow velocity field modeling around the
complex impeller . Turbulence was modeled using the
standard k-ε model in conjunction with logarithmic
wall
functions.
The
commercially
available
computational package CFX was used in modelling. At
the beginning of the calculation, a time-step size was
chosen such that a rotation of the inner grid zone was
one half the angle increment between the impeller
blades, i.e. 60°. The final steady-cyclic operating
conditions were then achieved by starting from the
calculated velocity field as an initial condition and then
choosing a time-step so that the grid rotates through one
azimuthal cell at the each time step.
Figure 2 Coarse surface grid and side cross-section for
fine grid.
EXPERIMENTAL TECHNIQUES
Velocity Measurements
Laser-Doppler anemometry (LDA) system is shown
in Figure 3. The vessel was placed in a square
plexiglass tank filled with water in order to minimize
laser beam refraction through the curved tranparent
plastic surface of the cylindrical mixing vessel. The
tank bottom was manufactured from opaque, white
plastic which extended to a height of 60 mm from the
tank bottom surface. This opaque region could not be
penetrated by the laser beam, so measurements and
data collection in this region was not possible. A
transparent acrylic lid was fitted on the top of the
mixing tank at height H to prevent the entrainment of
air bubbles from the free surface into the flow.
An Ar-ion laser generated a multicolour laser beam,
which was then split into two green and two blue beams
in a transmitter. These beams were transferred through
fiber optical cables to transmitting and receiving optics,
which converged the beams to intersect and form a
small elliptical control volume. Water in the stirred tank
was seeded with titanium oxide particles of an
approximately mean diameter of 10 μm, which
scattered light as they travelled through the control
volume. The scattered laser light was then collected by
receiving optics and the resulting Doppler shift was
measured with photodetectors. A signal processor
converted electric signals from the photodetectors into
digital information for computer processing. On the
computer, measurements were used to produce both
average and fluctuating velocities on-line.
Square
Tank
Transmitter and
Receiver Optics
PC
Signal
Processor
Photodetectors
Transmitter
Traverse
Controller
Ar- ion Laser
Heat Transfer Experiments
The apparatus shown in Figure 4 was used to
experimentally obtain the temperature field in the
stirred tank filled with water in a a room temperature.
The geometry of the tank was the same as in velocity
experiments. However, in order to make wall heat
transfer resistance negligible, copper instead of plastic
was used as the wall material of the cylindrical tank.
The dished bottom, similar to the one used in the flow
measurements, was thermally insulated. A cylindrical
heating element encircled the copper wall and provided
a constant heat flux of 5000 W/m2. The outer surface of
heating element was insulated to minimize ambient heat
loss.
Thermocouple
probe
Figure 3 Laser-Doppler velocimetry system.
Thermocouples
X*
The computer-driven traverse controller was used to
move the laser probe in a predetermined grid. Data for
axial and tangential velocity components were
measured at one vertical plane, ie. φ = 0°. A grid with
six different axial heights in Figure 1(X* equal to 0.25,
0.375, 0.50, 0.625, 0.75 and 0.875) and 36 radial
measuring points at each height was set up to obtain a
total of 216 measuring locations in the tank. Due to the
shaft symmetry, only one half of the tank was mapped.
Since there were two pairs of laser beams and the
traverse controller could move in two directions, it was
possible to measure two velocity components
simultaneously. The axial and tangential velocity
components were chosen for measurement, since the
radial velocity component is insignificant in the region
adjacent to wall where heat transfer takes place. The
criterion for the duration of data acquisition was set at
10 000 validated samples or 3 minutes time period.
However, at the bottom part of the tank, there was
distinct discrepancy between the measured and
calculated tangential velocities in region near the wall.
Due to this discrepancy, the velocity measurements
were repeated at the three lowest measuring heights.
The new criterion for data acquisition duration was
increased to 20 000 validated samples or a 5 minutes
time period.
The impeller Reynolds number NRe based on
rotational speed Ns and diameter d is defined as:
ρN s d 2
Ν Re =
μ
Three measured rotation speeds with respective
impeller tip speeds and Reynolds numbers are given in
Table 1. More details can be found in reference [13].
Table 1 Measured rotation speeds.
Experiment
expt1
expt2
expt3
Ns (r/s)
6.52
9.75
11.38
NRe
1.23*104
1.85*104
2.16*104
Wtip (m/s)
2.83
4.23
4.94
Copper
wall
0.875
0.75
φ
0.625
0.50
0.375
0.25
Cylindrical
heating element
R*
Figure 4 Schematic diagram of heat transfer
measurement.
To measure temperature distribution on the wall
surface, six copper-constant thermocouples of 0.3 mm
in diameter were affixed to the wall with a 41 mm
spacing between the heating element and the copper
wall at two tangential measuring locations, namely φ =
0° and φ = 90° (refer to Figure 4). The temperature field
inside in the tank was measured using a probe
composed of three thermocouples spaced 40 mm apart
from each other. Fluid temperature measurements with
the probe were repeated at six different axial heights
(X* equal to 0.25, 0.375, 0.50, 0.625, 0.75 and 0.875, as
shown in Figure 4).
RESULTS AND DISCUSSION
Velocity Field
The number of nodes was increased as computation
progresses to check that calculated results did not
change too much with different grid sizes. The grid
refinement testing was carried out using four nonuniform grid systems as mentioned ealier. The test
showed that the axial velocity is insensitive to the grid
size, the greatest changes in axial velocity occurring
near the impeller shaft. Since heat transfer from the
wall is considered, the most important velocity
components are the axial and tangential components,
while the radial velocity component clearly becomes
zero at the wall. When comparing the predictions made
with the different meshes, very similar qualitative
trends, especially in the near wall area, were obtained.
Figure 5 and 6 show quantitative comparisons of the
measured data and predicted normalized axial and
tangential mean velocity profiles at some axial heights
between two baffles using dimensionless presentation
x
r
k
U
W
X ∗ = , R* = , U * =
,W* =
, k* =
R
H
Wtip
Wtip
0.5Wtip2
It can be observed that the predicted velocities show
good agreement with the LDA data. In the measured
apparatus, there was a lid at a height H. However,
during simulations, the symmetry boundary condition
was used to model the free surface. The lid decreases
the velocity near the wall at the upper part of the tank.
In addition, a hole in the lid at tank centerline caused a
downward flow jet. It can be concluded that the
comparison between experimental data and numerical
predictions match both quantitatively very well. Thus,
the computation gives a sound velocity field prediction.
The axial velocity profiles in Figure 7 agree well with
each other. The comparison of tangential velocities
shows quantitatively similar trends although some
differences exist. The reason for differences in
tangential velocity component profiles is due to the
measuring method. The LDA measurements are more
sensitive to disturbances in this case as rotation speed is
increased. It was found that the flow is characterized by
one large top-to-bottom main circulation loop in both
vertical planes. Furthermore, it was noticed that the
center of the vertical loop lies close to the wall at the
bottom part of the tank. The flow patterns are
dominated by the axial velocity component due to the
shape of the impeller and the tank bottom, with a strong
axial jet flowing upwards at the near wall region and
downwards near the shaft. However, at the top of the
tank, there is a zone that is nearly stagnant at the top of
the tank when examining velocities in the crosssectional planes.
0.4
0.3
U*
*
*
X = 0.25
expt 1
expt 2
expt 3
small-scale unit
full-size unit
X = 0.75
U*
computation
experiment
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
0.0
0.2
0.4
0.6
R*
0.0
1.0
Figure 5 Comparison of computed and measured axial
velocities at the high X* = 0.75 (φ = 0°).
W*
X = 0.75
computation
experiment
0.6
R*
1.0
0.6
R*
1.0
*
X = 0.25
expt 1
expt 2
expt 3
small-scale unit
*
W*
0.4
0.40
0.30
0.30
0.2
0.25
0.20
0.20
0.15
0.15
0.10
0.05
0.10
0.0
0.0
0.05
0.0
0.0
0.2
0.4
0.6
R
*
1.0
Figure 6 Comparison of computed and measured
tangential velocities at the high X *= 0.75 (φ = 0°).
The experimentally determined axial and tangential
velocities at three rotation speeds and the predicted
profiles for the small-scale and commercial size-unit
are reported in Figure 7. These profiles at the bottom
part of the tank prove that, if the results are scaled using
the chosen dimensionless variables, they could be
applied to any rotation speed and any size of the tank.
0.2
0.4
Figure 7 Comparison of small and full size computed
axial and tangential velocity profiles to the three
measured data at the high X* = 0.25 (φ = 0°).
At the different heights, the maximum axial velocity
components upwards and downwards have about the
same values. The axial velocity profile shows that the
flow direction is upwards, when R* is between 0.8 and
1.0, but downwards otherwise. The radial component
was about one-tenth in magnitude when compared to
the axial and tangential components. Axial and
tangential components approximately equal in
magnitude and dominate the velocity field. The axial
velocity component dominates at the tank bottom and
naturally decreases with increasing tank height. At the
top part of the tank, the tangential velocity becomes
more significant. Figure 6 shows the effect of baffles on
the tangential velocity components between the baffles.
The presence of a baffle is seen to cause a decrease in
tangential velocity in the region behind the baffles
between baffle and wall. This can be clearly seen
between the baffles especially at the tank top, where the
axial velocity component is quite small compared to
tangential velocity. The profiles in Figure 6 showed
that the actual tangential velocity is highest in a region
halfway up the tank.
Heat Transfer
Figure 8 shows the measured wall temperature
profiles at 200 seconds time intervals. The predicted
wall temperature field after 1000 seconds heating time
is also given in Figure 8. It corresponds to the thick
profile line on the graph. Figure 8 shows a clearly
defined temperature distribution in the axial direction.
At the tank bottom, the axial and tangential velocities
are highest, therefore the rate of heat transfer is highest,
resulting in lowest wall temperatures in this region. The
lowest heat transfer coefficient observed was at the top
of the tank between the baffles, where there is a
relatively stagnant area in the velocity field.
Measurements and calculations also showed that there
was a temperature difference between tangential
locations, thus the heat transfer coefficients vary with
angular position with respect to baffle location and the
wall temperatures were 2-3 degrees higher between the
baffles than near the baffle location. The wall
temperatures were lower at the baffle location since
flow velocity was highest there, resulting in a higher
convective heat transfer rate.
Figure 9 shows a typical predicted temperature field
contour on a vertical plane slice located midway
between the two internal tank baffles, i.e. φ = 0°, for
the constant heat flux boundary condition at heating
time equal to 1200 seconds. Figure 9 shows how
uniform the temperature field is with a constant heat
flux boundary condition. The temperature profile
develops quickly very near the hot wall, while
elsewhere there is adequate mixing to keep the
temperature of liquid almost uniform and it varies only
by a few degrees in the tank interior. However, the
temperature variation along the wall in the axial
direction can be found in this countour also.
The average temperature of the fluid as a function of
time is illustrated in Figure 9. The fluid temperature
was measured with an instrument made up of three
thermocouples, which gaged the same temperatures at
each of three radial distances. The uniform average
temperature variated linearly with respect to time and
the measured values were the same as derived from
heat balance. The comparison of predicted and
measured average liquid temperature predictions agree
well, given confidence to predicted results when using
the constant wall temperatures as well.
1.0
1000 sec
200 sec
X*
0.8
0.7
0.6
τ
0.5
0.4
0.3
0.2
0.1
0.0
20
25
30
35
40
45
50
55
60
o
T (C) 70
75
Figure 8. The axial wall temperature profiles at 200 sec
time intervals (φ = 90°) and predicted wall temperature
field after 1000 sec heating.
60
Measured
Predicted
Heat balance
o
T (C )
40
30
20
10
0
0
300
600
900
τ
1500
Figure 9 Contour of temperature between two baffles
(φ = 0° and τ = 1200 sec) on the top. Measured,
predicted and heat balance average temperature as
function of time, when heat flux on wall is constant.
o
T (C )
100
90
80
70
60
50
40
30
20
AKNOWLEDGEMENT
The authors gratefully acknowledge the support of
Kemira Fine Chemicals Oy and discussions with Dr. P.
Oinas.
Tavg
Tw
REFERENCES
0
100
200
300
τ(sec)
500
Figure 10 Predicted average temperature as function of
time, when Tw = constant in commercial size reactor.
After scaling up the model to the commercial size
unit, the constant wall temperature boundary condition
was used to model the heat transfer in a commercial
unit, which volume was 12 m3. Figure 10 shows the
development of computed mean temperature for a
constant wall temperature boundary condition.
CONCLUSIONS
In the study, the numerical predictions of the flow
field and temperature distribution of a baffled stirred
tank are presented. Grid dependency test showed that
the tangential and radial velocities are more sensitive to
grid size than the axial component. The results indicate
that the flow was dominated by the axial velocity
component, resulting in a major top-to-bottom
recirculation loop inside the tank. Uniform wall heating
boundary condition showed that the wall temperature
and the local heat transfer coefficient strongly depended
on location. However, the temperature distribution of
the liquid was almost uniform and the average
temperature of the liquid varied linearly with respect to
time. The fluid temperature distribution was clearly
defined in the case of an isothermal wall boundary
condition. The liquid was warmest at the upper part of
the tank and a warm core flowed downwards near the
shaft. The variation of the average liquid temperature as
a function of time was exponential. In general, the
experimental data and the model predictions agree very
well. The present study has shown that computational
prediction of good accuracy can be obtained across the
flow field and temperature distribution in a stirred tank.
Thus, a quick assessment of the influence of rotation
speed and geometrical variables, such as other types of
impellers, can be further explored using only numerical
simulations. It is important to note that the twoequation model of turbulence is enough for the
modelling of velocity field and heat transfer.
Since the need for more accurate boundary conditions
on the jacket side is important, the modelling of the
jacket should be scrutinized in future work. Future
work should also be directed towards obtaining a
detailed technique for rapid heat transfer measurements.
A local Nusselt number correlation for a stirred tank as
a function of the Reynolds and Prandtl number should
be the ultimate goal.
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