VALIDATION OF THE LAGRANGIAN APPROACH FOR PREDICTING TURBULENT DISPERSION
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VALIDATION
OF THE LAGRANGIAN APPROACH FOR PREDICTING TURBULENT DISPERSION AND
EVAPORATION OF DROPLETS WITHIN A SPRAY
Nijdam, J.J.1, Guo, B.2, Fletcher, D.F.1,
Langrish, T.A.G.1
1. Department
of Chemical Engineering, University of Sydney, Australia
2. School
of Materials Science and Engineering, University of New South Wales,
Australia
Keywords and Phrases: spray drying; computational fluid dynamics;
CFD; particle drying
ABSTRACT
The accuracy of the Lagrangian approach for predicting droplet
trajectories and evaporation rates within a simple spray has been
addressed. The turbulent dispersion and overall evaporation rates of
droplets are modelled reasonably well, although the downstream
velocity decay of the larger droplets is under-predicted, which is
attributed to a poor estimate of the radial fluctuating velocity of
these droplets at the inlet boundary. Qualitative agreement is found
between the predicted and experimental evolution of the droplet size
distribution downstream of the nozzle. These results show that
smaller droplets evaporate preferentially to larger droplets, because
they disperse more quickly towards the edge of the jet, where the
entrainment of fresh air from the surroundings produces a significant
evaporative driving force. Droplet dispersion and evaporation rates
are highly influenced by the rate of turbulence generation within the
shear layer. This work demonstrates the potential of the Lagrangian
approach for analysing particle trajectories and drying within the
more complex spray dryer system.
INTRODUCTION
Spray drying is the process of atomising a liquid that contains
dissolved solids within a flow of hot air, where moisture is
progressively evaporated from the droplets until dried particles are
produced. This process is normally carried out within a drying
chamber, which is specially designed to provide sufficient
droplet/particle residence times for a given heat load so that the
powder collected at the exit point of the chamber achieves the
desired moisture content without experiencing any heat degradation.
As an illustration, where low air temperatures are necessary for
heat-sensitive products, moisture evaporation from droplets is slow,
and thus long droplet/particle residence times are required in order
to produce powder with the desired moisture content, which calls for
a drying chamber of great height. The droplet/particle residence time
within the drying chamber also affects the functional properties of
the powder, such as the solubility and bulk density, which are
essentially related to the size and morphology of individual
particles. Long residence times enable particles to agglomerate and
increase in size during their passage through the spray dryer,
provided the particles are sufficiently tacky to stick together after
a collision, and the particle number concentration is sufficiently
high to result in frequent collisions. Clearly, the droplet/particle
residence time is an important consideration when designing spray
dryers, and a method is required to predict this quantity before any
attempt can be made to choose an appropriate dryer configuration and
size for a given application. Masters (1976) has described various
methods for evaluating the droplet/particle residence time, which
involve piecewise methods to predict the droplet/particle
trajectories in different regions of the drying chamber. However,
numerous simplifications and assumptions are often associated with
these methods, since the airflow patterns and the interaction of the
spray and airflow turbulence are very complex.
Recently, the more rigorous Computational Fluid Dynamics (CFD) method
has been employed by various workers (Huang et al., 2004;
Langrish et al., 2004; Oakley, 2004; Verdurmen et al.,
2004) to investigate different aspects of the flow within a spray
dryer, with the ultimate aim of providing a validated CFD code that
accurately predicts droplet/particle trajectories and residence
times, and the extent of particle drying and agglomeration within the
spray dryer, so that information can be gained on the likely
properties of the dried powder. Langrish et al. (2004) and
Harvie et al. (2001) have demonstrated good agreement between
experiment and CFD predictions of the swirling motion and transient
nature of airflow within short-form and tall-form spray dryers,
respectively. In addition, Nijdam et al. (2004) and Rüger
et al. (2000) have shown that the turbulent dispersion of
droplets/particles within a simple spray is predicted accurately
using a Lagrangian droplet/particle tracking model, a feature of many
commercially-available CFD codes. This set of papers has illustrated
the potential of CFD to accurately predict both the airflow patterns
and the interaction of the airflow and the discrete phase within a
spray dryer, which should allow droplet/particle trajectories and
residence times to be estimated with a higher degree of precision in
the near future.
Nijdam et al. (2004) have also demonstrated good qualitative
agreement between experiment and Lagrangian predictions of droplet
coalescence. Moreover, they have shown that the Lagrangian
predictions of coalescence are consistent with a rigorous multi-phase
Eulerian calculation, which lends support to the validity of the
Lagrangian approach for modelling both droplet coalescence and
particle agglomeration. These workers have provided a rough estimate
of the coalescence constant that appears in the Lagrangian model of
droplet coalescence; however, the particle agglomeration constant,
which is a function of the stickiness of particles, must also be
estimated before predictions of both droplet coalescence and particle
agglomeration within a spray dryer are feasible. Rüger et al.
(2000) have also shown good agreement between Lagrangian predictions
and experiments of droplet coalescence in a water spray.
Particle stickiness is related to the surface moisture content and
temperature of individual particles, which implies that moisture
evaporation from the discrete phase must also be modelled before any
particle-agglomeration predictions are possible. Harvie et al.
(2002) have demonstrated that the Lagrangian particle-tracking
approach can be used to model such moisture evaporation from droplets
and particles. The underlying assumption used in their drying model
is that the discrete phase experiences two distinct stages of drying.
During the first drying stage, which is generally called the
constant-rate period, the surface of the droplet/particle is
saturated with water, and only the resistance of the droplet/particle
boundary layer to heat and mass transfer influences the evaporation
rate. The second drying stage, known as the falling-rate period,
begins once sufficient moisture has been removed from the particle
such that surface saturation can no longer be maintained, which
results in the formation of a dry outer layer that progressively
thickens, impeding the transport of moisture from the core to the
surface of the particle, so that the drying rate falls steadily.
Harvie et al. (2002) have shown reasonable agreement between
the predicted and measured moisture content of powder at the exit
point of a spray dryer, although no comparisons were made between
predictions and experiment at any locations within the spray dryer.
The aim of this paper is to continue the validation work of Harvie et
al. (2002) by comparing experimental data with Lagrangian
predictions of moisture evaporation from the discrete phase during
the so-called constant-rate period. The experimental data of Nijdam
et al. (2004) for the evaporation of acetone droplets, which
contain no solids to otherwise impede the drying rate, are used for
this purpose. Rüger et al. (2000) have also modelled
evaporation within a spray using the Lagrangian approach, although
they used water rather than acetone, such that the evaporation of
droplets was negligible compared with the coalescence of these
droplets. We neglect droplet coalescence in this work, since
evaporation is the dominant phenomenon in the dilute acetone spray
investigated here. This work is more rigorous than the earlier study
of Harvie et al. (2002), because experimental and predicted
droplet evaporation rates are compared at various locations
throughout a spray, so that the ability of the model to accurately
predict the influence of airflow turbulence on droplet evaporation
can be ascertained. In addition, the validity of the Lagrangian model
for predicting the turbulent dispersion of droplets within an
evaporating spray is assessed in this paper, which should reveal the
potential of this approach for accurately modelling trajectories and
residence times of evaporating droplets and drying particles within
the more complex spray dryer system.
THEORY
In the Lagrangian droplet-tracking approach, the spray is modelled as
a gas-flow carrying a large number of discrete droplet parcels, with
each parcel consisting of a number of physical droplets of similar
size. The gas flow-field is computed using the Eulerian continuum
equations, while the droplet-parcel trajectories within the gas-flow
are calculated by solving the Lagrangian equations of mass and
momentum. The effect of turbulence on the droplet motion is simulated
by the stochastic approach of Gosman and Ioannides (1983), in which
the fluctuating velocity of the droplet parcels is randomly sampled
from a Gaussian distribution, whose standard deviation is a function
of the local turbulence kinetic energy of the air. The droplet
evaporation rate
is modelled using the following equation,
(1)
where
is the diameter of the droplet,
and
are the vapour molar fractions at the surface of the droplet and in
the gas phase, respectively, and
and
are the molecular weights of the vapour and the gas mixture in the
continuous phase, respectively. The diffusivity of vapour in the gas
is given by
,
while the Sherwood number
is estimated from the empirical correlation of Ranz and Marshall
(1952), as follows:
(2)
which is applicable for Reynolds numbers (
)
of up to 200. The energy balance for an evaporating droplet, assuming
that the temperature throughout the droplet
is uniform, is given by the following expression:
(3)
where
is the temperature of the gas,
and
are respectively the mass of the droplet and heat capacity of the
liquid,
is the thermal conductivity of the gas, and
is the latent heat of vaporisation of the liquid. The Nusselt number
is calculated from a correlation analogous to Equation (2), in which
the Prandtl number (
)
replaces the Schmidt number (
).
The vapour mole fraction at the surface of the droplet
is calculated from the following equation,
(4)
where
,
and
are empirically-determined constants in the Antoine equation, and
is the total pressure. Equations (1) to (4) allow the size and
temperature of droplets within each parcel to be calculated as they
progressively evaporate. Two additional transport equations for
enthalpy and mass fraction are required in addition to the gas
momentum, continuity and any turbulence equations, in order to
calculate the temperature
and vapour mole fraction
of the gas, as described in the CFX4.4 manual (ANSYS).
EXPERIMENTAL
DESCRIPTION AND NUMERICAL ASPECTS
The predictions of the Lagrangian model are compared with droplet
evaporation, velocity and spreading rate measurements of the acetone
round-jet spray investigated experimentally by Nijdam et al.
(2004). The experimental apparatus consisted of a wind tunnel and a
nozzle (a long thin tube, 9.8mm in diameter, located centrally at the
exit plane of wind tunnel, as shown in Figure 1), which generated a
round air jet within a co-flow of air with velocities of
approximately 23 m/s and 2.4 m/s, respectively. The air jet was laden
with a dispersion of acetone droplets in the size range from 1 to 90
m,
which were produced by an ultrasonic nebulizer located upstream of
the nozzle. A phase-Doppler anemometer (PDA) was used to measure the
axial velocity and volume flux of the droplets within the jet at
various locations downstream of the nozzle. These experimental
measurements are presented in detail in the paper of Nijdam et al.
(2004). Note that we compare only the experimental and predicted
axial velocities in this work, because the radial velocity component
was not measured in order to improve the accuracy of the droplet
volume flux measurement, as discussed by Nijdam et al. (2004).
Nevertheless, the dispersion of droplets in the jet downstream of the
nozzle can be adequately described by the decay in the axial velocity
profile and the evolution of the droplet volume flux profile.
Evaporative cooling of the acetone droplets within the nozzle ensured
that the air and droplet temperatures at the exit of the nozzle were
well below the ambient room temperature. Due to the considerable
length of the nozzle (24 nozzle diameters), we assumed that the air
and droplet temperatures reached an equilibrium value of 0 °C at
the exit of the nozzle. This temperature was estimated from the ideal
gas law by determining the density of the air at the exit of the
nozzle from a mass balance over the nozzle, which was based on the
known air flow and temperature entering the nozzle and the
PDA-measured airflow exiting the nozzle. The total acetone droplet
flow at the exit of the nozzle was 1.7 ml/min. The temperature of the
air co-flow was measured at 16 °C.
The simulations presented in this paper are performed using CFX4.4
(ANSYS), a commercial CFD code, which uses a finite-volume
discretisation method. Turbulence is modelled using the standard
-
model described by Launder and Sharma (1974), with the turbulence
parameter
taking on a value of 1.55 in these simulations, which is different
from the value of 1.6 found by McGuirk and Rodi (1979) for a round
jet. McGuirk and Rodi (1979) have explained that
is an empirical constant, which does not take on a universal value
for every type of flow. They have stated that the value for
of 1.6 is only applicable for round jets well away from any potential
core and for jets issuing into stagnant surroundings. However, the
jet investigated in this work has a top-hat axial mean velocity
profile at the exit of the nozzle, which indicates the presence of a
potential core, and the jet issues into a co-flow. We have found that
a value of 1.55 for
is appropriate in the case of such a jet in order to match the
experimental and predicted decay of the centreline axial mean
velocities of the gas and droplets. Clearly, the
-
turbulence model lacks generality, and more sophisticated gas-flow
turbulence models are needed to resolve this. However, such complex
turbulence models are likely to be computationally very expensive,
and simpler formulations are preferable when other phenomena, such as
droplet evaporation and turbulent dispersion, are the main focus of
an investigation.
The acetone parcels of droplets are tracked using the Lagrangian
particle-tracking feature of CFX4.4 and the turbulent dispersion
model of Gosman and Ioannides (1983). The following values were used
for the properties of acetone: liquid density, 790 kg/m3;
latent heat of vaporisation, 543 kJ/kg; liquid and vapour heat
capacities, 2180 and 1466 J/kg K, respectively; diffusivity of vapour
in air, 1.09×10-5 m2/s. The coefficients
A, B and C in the Antoine Equation (see Equation
4) took the values 21.695, 3021.6, and 240.705, respectively. Note
that a transient three-dimensional simulation is required in order to
properly model the movement of droplet parcels throughout time and
space, although we have observed in practice that the jet was in fact
steady and axisymmetric (two dimensional). The evaporation of the
acetone droplets is simulated by employing the spray drier model
incorporated within CFX4.4, which models heat and mass transfer
between the discrete and gas phases, as described above. The
turbulent Prandtl number is set to 0.7 for both the enthalpy and
vapour mass fraction equations (Antonia and Bilger, 1976). The grid
used was a three-dimensional, non-uniform, block-structured grid,
which was refined in regions of high-velocity gradient. A sensitivity
analysis has been conducted to ensure that the converged solution is
independent of the number of parcel tracks used to represent the
acetone droplet flow (40,000) and the grid mesh size (105
nodes); doubling the number of droplet parcels and grid nodes lead to
a minor change in the droplet volume mean diameter (D30)
at the exit of the flow domain of less than 2%. Further details of
the Lagrangian modelling approach can be found in the CFX4
documentation.
RESULTS
AND DISCUSSION
A typical axial mean velocity profile within a fully-developed jet is
similar in appearance to a Gaussian distribution, with a peak axial
mean velocity at the centreline, and a tapering axial mean velocity
with increasing radial distance from the centreline, down to the
co-flow velocity at the edge of the jet. Downstream of the nozzle,
the centreline axial mean velocity progressively decreases (or
decays) as the jet spreads radially outwards. Figure 2 compares the
experimental and predicted centreline axial mean velocity decay rates
with axial distance from the nozzle exit for different sized
droplets. The decay of the centreline axial mean velocity for the
smallest droplets (5 m)
is predicted reasonably accurately, while the decay of the centreline
axial mean velocity for the largest droplets (85 m)
is predicted rather poorly. The smallest droplets are sufficiently
small to behave as tracers for the airflow. Therefore, the good
agreement between the predicted and experimental decay of the
centreline axial mean velocity for the smallest droplets indicates
that the turbulence constant
of 1.55 is appropriate to properly model the spread of air momentum
in this jet. However, the spread of momentum of the larger droplets
is under-predicted by the Lagrangian model. This discrepancy is
probably due to the incorrect assumption that the radial fluctuating
velocities of these droplets at the inlet boundary of the simulation
region, which is located close to the nozzle exit, are zero. The
initial spread of momentum of the larger droplets is likely to be
sensitive to these inlet radial fluctuating velocities, which were
not measured during the experiments.
The droplet volume flux profile within a fully-developed jet is also
similar to a Gaussian distribution, featuring a peak at the
centreline and a tapering volume flux in the radial direction, which
reduces down to zero towards the edge of the jet where droplets are
relatively sparse. The volume-flux profile also decays and spreads
downstream of the nozzle as individual droplets interact with airflow
turbulence, which drags them radially outwards. The half-radius of
the volume-flux profile, which is defined as the radius of the
Gaussian-shaped volume-flux profile corresponding to half the total
height of this profile, is a measure of the extent of this turbulent
spread or dispersion. A small half-radius indicates that the droplets
have not dispersed outwards significantly and that they are
concentrated near the axis of the jet, whereas a relatively large
half-radius indicates that the droplets have been dispersed by their
own inertia or airflow turbulence away from
the jet axis. Figure 3 shows that the turbulent dispersion rate of
droplets downstream of the nozzle appears to be predicted reasonably
accurately, although some caution is required in this interpretation,
since the turbulent dispersion of the droplets is linked to the
spread of momentum, so that improperly modelling one (as has been
demonstrated by the poor momentum predictions of the larger droplets,
shown in Figure 2) will inevitably affect the validity of the other.
Nevertheless, the overall trends are clearly reasonable. Figure 3
implies that smaller droplets disperse radially more rapidly than
larger droplets downstream of the nozzle. This is physically
reasonable because small droplets have relatively low inertia such
that they readily follow the turbulent fluctuations of the airflow,
which tends to disperse the droplets radially outwards. On the
contrary, larger droplets have relatively high inertia, so that they
are less affected by airflow turbulent fluctuations, being more
inclined to continue along the axially-directed trajectories imparted
to them at the nozzle exit, and thus spreading very little in the
radial direction. Very little spreading of droplets occurs in any
droplet size class within the first 10 cm of the nozzle exit, because
insufficient airflow turbulence has diffused from the shear layer,
near the edge of the jet where this turbulence is generated, towards
the centre of the jet to disperse the droplets radially outwards. It
is this diffusion of airflow turbulence into the core of the jet that
is responsible for the dispersion of droplets beyond 10 cm of the
nozzle exit.
Figure 4 compares the experimental and predicted axial evolution of
the total volume flow of droplets. Two different extremes for the
value of the acetone vapour mass fraction at the nozzle exit are
tested in these simulations - no vapour mass fraction and saturated
vapour mass fraction - since this value was not measured during the
experiments. The true mass fraction of acetone vapour lies between
these extremes. In the simulation that assumes no acetone vapour at
the nozzle exit, the flow of acetone droplets reduces immediately on
emerging from the nozzle, since there is a significant driving force
for evaporation within the core of the jet. In the simulation that
assumes vapour-saturated airflow at the nozzle exit, very little
evaporation of acetone droplets occurs within the first 10 cm from
the nozzle exit, because insufficient fresh air from the co-flow has
been transported (via turbulence) to the core of the jet to reduce
the vapour mass fraction below the saturated value. Acetone droplets
begin to evaporate only when the vapour-imbued airflow within the
core of the jet is diluted with sufficient fresh air, entrained from
the co-flow, to provide a significant driving force for evaporation.
The predicted axial evolution of droplet volume flow assuming
saturated acetone vapour at the nozzle exit gives the best agreement
with the experimental data, which suggests that the airflow was close
to vapour saturation at the nozzle exit during the experiments.
Nevertheless, the predicted droplet evaporation rate with distance
from the nozzle exit is similar irrespective of the acetone vapour
fraction adopted at the nozzle exit, and in good agreement with the
experimental data, which demonstrates the accuracy of the Lagrangian
approach for modelling the overall droplet evaporation rate. These
simulations highlight the significant effect of fresh-air entrainment
on the evaporation rate of droplets within the jet.
The accuracy of the model for predicting the evolution of the droplet
size distribution due to evaporation downstream of the nozzle is
evaluated in Figure 5. It appears that the Lagrangian model predicts
the correct qualitative trends, although a quantitative comparison
shows mixed agreement. Both predictions and experiment imply that
there is a gradual shift in the droplet size distribution towards the
larger droplet size classes, which suggests that smaller droplets
evaporate more quickly than larger droplets. This is expected given
that small droplets disperse outwards more readily than large
droplets, and thus move more quickly into the outer regions of the
spray where fresh air entrained from the co-flow ensures that the
vapour mass fraction is relatively low. Nijdam et al. (2004)
have shown that the shift in droplet size distribution towards the
larger droplet size classes is also due in part to the collision and
subsequent coalescence of droplets, although this effect is likely to
be small in comparison with the evaporative effect for the dilute
sprays investigated here. The discrepancy between the experimental
results and the Lagrangian predictions is difficult to resolve
without information regarding the evolution of the experimental
acetone vapour-fraction field, which would otherwise allow a more
detailed analysis of the spray evaporation simulations.
CONCLUSIONS
The decay of the centreline axial mean velocity of the smallest
droplets is predicted reasonably accurately, while the centreline
axial mean velocity decay for the larger droplets is under-predicted,
which we attribute to a poor estimate of the radial fluctuating
velocity of these droplets at the inlet boundary. The turbulent
dispersion and overall evaporation rates of droplets within the
simple spray are modelled reasonably well, which illustrates the
potential of the Lagrangian approach for accurately modelling
droplet/particle trajectories (and hence residence times) and
evaporation within the more complex spray dryer system. However, the
evolution of the droplet size distribution is not modelled properly,
although there is reasonable qualitative agreement between experiment
and predictions. A more detailed experimental investigation of the
effect of vapour mass fraction on the evaporation rates of droplets
in different size classes is required in order to resolve this
discrepancy. Nevertheless, these simulations show that smaller
droplets evaporate more quickly than larger droplets, because they
disperse more readily towards the outer regions of the jet, where the
evaporative driving force is greatest. The importance of turbulence
generation within the shear layer on the rate of droplet dispersion
and evaporation within the jet has been demonstrated.
ACKNOWLEDGMENTS
This work has been supported by the New Zealand Foundation for
Research, Science and Technology under contract UOSY0001, and an
Australian Research Council Large Grant.
NOMENCLATURE
heat
capacity of liquid (J/kgK)
droplet
diameter (m)
nozzle
diameter (m), or diffusion coefficient (m2/s)
latent
heat of vaporisation (J/kg)
gas
thermal conductivity (W/mK)
droplet
mass (kg)
droplet
evaporation rate
(kg/s)
molecular
weight (kg/mol)
half
the width of the volume-flux profile at half the maximum volume flux
(mm)
total
gas pressure (Pa)
temperature
(K, °C)
mean
axial velocity (m/s)
mole
fraction
density
(kg/m3)
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Figure 1.
Spray nozzle configuration (not to scale).
F
igure
2. Centreline axial velocity
for different size classes at various axial locations from the nozzle
exit: experiment (EX) and Lagrangian model (LM).
F
igure
3. The half-radii
of the radial profiles of droplet volume flux for different droplet
size classes at various axial locations from the nozzle exit:
experiment (EX) and Lagrangian model (LM).
Figure
4. The axial evolution of the total flowrate of droplets: comparison
between experiment and two model predictions; one prediction assumes
no acetone vapour at the nozzle exit, while the other prediction
assumes acetone vapour saturation at the nozzle exit.
F
igure
5. Comparison of experiment and model predictions of the droplet size
distribution at various axial locations from the nozzle exit.
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