(124b) Application of Euler-Lagrange Approach to Predict the Multiphase Flow with Simultaneous Heat and Momentum Transfer, and Solidification Predicted by A Shrinkage Unsolidified Core Model

ABSTRACT ? The main objective of this work consists in the mathematical
formulation of the gas-liquid system with simultaneous heat and momentum
exchange and solidification phenomenon, followed by numerical simulations to
predict the main phenomenological characteristics of this process. The
mathematical modeling uses an Euler-Lagrange approach and the models are solved
with volume finite method present in commercials CDF codes. Physical properties
of discrete phase like specific heat are estimated as a function of temperature
through correlations based on contribution groups. The prediction of
solidification consists in applying a shrinkage unsolidified core model that
determines the evolution of the solid thickness formed inside the droplets
during solidification. The solidification phenomena and the physical properties
needs to be numerically implemented through sub-routines, programmed in C
language, and linked in the commercial CFD code. The models and the
sub-routines need to be evaluated, and test cases are performed to prove
physical consistency and the results can be compared to previous works. The
simulation results show physical consistency of the model. Therefore, this
models and sub-routines can be applied to industrial case. Finally, through the
analysis of the results, seeks to consistent data and information that allows a
better understanding about the multiphase flow present in granulation towers
and the influence of solidification phenomena in heat transfer between phases,
and the impact of this in the quality of the final product.

  1          
INTRODUCTION

In granulation process,
the melt is sprayed in the top of a cylindrical tower where droplets are formed
and they fall downward in counter-current with ambient air, which cools and
solidifies the droplets, before they are collected in the bottom of the tower.
It is observed in practice that the main problem is the
deficiency in the solidification process. This can cause operational
problems and bad quality of the
grains and, consequently, lost
productivity and profits. Often,
when the grains are not completely
solidified, it could break either by collision between
particles, but also on the walls
of the tower, creating a kind of "cake". It has been
believed that the incomplete solidification of some droplets is due to
formation of the solid thickness inside the droplets which decrease the heat
transfer between both phases because of an additional conductivity resistance
inside the droplets. Thus, there is a motivation to investigate and analyze these aspects to understand these problems and
solve them.

  2          
MATHEMATICAL MODELING AND NUMERICAL SOLUTION

The mathematical modeling uses an
Euler-Lagrange approach and takes into account some hypotheses:

·        
the air is the continuous phase and its modeled in a
Euler framework;

·        
the droplets are the discrete phase and are modeled in
a Lagrange framework;

·        
the continuous phase under turbulent conditions can be
represented by RANS equations;

·        
there are no chemical reactions;

·        
phases interact only by exchanging heat and momentum;

·        
the interactions between the phases are represented by
a two-way coupling;

·        
the solidification phenomena occurs in constant
temperature;

·        
the droplets are spherical;

·        
only heat transfer by convection and conduction (during
solidification) are considered;

·        
during solidification its considered a shrinkage
unsolidified core model (Yuan, 2007), where the solidification occurs from the
outside surface to inside the droplet until the center and the fusion occurs
from the center of the droplet until the outside surface (Figure 1);

·        
biot number << 0.1 for the liquid droplet (there
is no temperature profile inside the liquid droplet).

Figure 1 - Shrinkage unsolidified core model
(Adapted from Yuan, 2007).

The thermal behavior of droplets passes
through three periods:

·        
cooling of droplets until solidification temperature;

·         solidification
of droplets surface at a constant temperature, progressing until the center of
the droplets until complete solidification, where the resistance to the heat
transfer due to the formation of solid thickness cannot be neglected;

·        
cooling of the solids particles.

Additional equations are necessary to
represent the interphase heat and momentum transfer and the turbulence. For the
interphase momentum transfer it is considered only the drag force which
represents the two-way coupling. The thermal exchange between the phases takes
into account the convective heat transfer from the droplets surface to the gas,
and the conductive heat transfer due to the formation of a solid thickness
during solidification; so this term is represented by an overall heat transfer
coefficient, and these terms represents the two-way coupling. An average
procedure is applied to transform the conservative equations, which are
instantaneous in time and space, in average equations, resulting in the
Reynolds Average Navier-Stokes (RANS) equations (Wilcox, 1994). The turbulence is modeled with standard  model (Wilcox, 1994). The
discretization method used is the Finite Volume Method (FVM) (Maliska,
1995) present in the software FLUENT 13.0 from ANSYS^® (ANSYS, 2010). From Figure 2 to Figure 6, it is presented the synthesis of
the model, which includes the Euler-Lagrange conservative equations (Brodkey
and Hershey (1988), Meier (2010)), the turbulent equations, the auxiliary equations, and the specific heat for the discrete
phase as a function of temperature.

Figure 2 - Discrete phase conservative equations.

Figure 3 - Continuous phase conservative and
turbulent equations.

Figure 4 - Physical properties of discrete phase as
a function of temperature.

Figure 5 - Auxiliary equations for the model - 1.

 Figure 6 - Auxiliary equations for the model - 2.

  3          
RESULTS AND DISCUSSIONS

To analyze the performance of both the mathematical model as its physical consistency (especially during solidification) and the subroutines
developed for the calculation of solidification, numerical cases are performed. One three
dimensional main case is performed to validate the physical consistency of
solidification and fusion. A numerical grid is made for the case (Figure 7) that have two inlets for cold and hot
air and two outlets, and one injection of discrete phase (1.2 millimeters in
diameter) in top of the tower, which has 60 meters in height and 2 by 2 meters
in base. The case is configured so that the droplets are cooled to solidify,
and then heated to fuse where cold air enters in the upper inlet of Figure 7 and directed to the top of the tower,
and the lower inlet has hot air which is directed to the bottom of the tower.

Figure 7 - Numerical grid to test solidification
followed by fusion.

In Figure 8 it can be noted that, initially,
the droplet analyzed is only cooled. Then, the solidification starts, and the
solid thickness starts to grow until a maximum value and then, due to the
heating of the droplet, fusion starts, and solid thickness starts to decrease
until zero, so the droplet it is fully liquid again at the end of the tower.
The Figure 9 shows the liquid
diameter inside the droplet, which starts constant during the cooling, decrease
during solidification, increases during fusion, and ends constant again.

Figure 8 - Solid thickness evolution through tower
height.

Figure 9 - Liquid diameter inside the droplet
through the tower height.

In Figure 10, the coefficient initially
decreases due to the velocity reduction when the droplet is injected in the
tower (during cooling period). In solidification period, it decreases due to
the formation of the solid thickness, which produces a conductivity resistance
to the heat transfer and in fusion period, it stars to increase due to disappearance of the solid thickness. Finally it becomes constant, because in this
period there is only convective resistance, which became constant because the
particle has reached its terminal velocity. Figure 11 shows the
temperature behavior of the droplet/particle.

Figure 10 - Overall Heat Transfer Coefficient through
tower height.

Figure 11 - Particle temperature through tower
height.

  4          
CONCLUSIONS

A three dimensional simulation was carried
out to prove physical consistency of the model. The results showed that the
model is able to predict the solidification and fusion phenomena and has
physical consistency. Now the model can be applied to an industrial case and
compared with experimental data and previous works.

  REFERENCES

ANSYS. Ansys fluent user's guide.
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MALISKA, C. R. Transferência de
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MEIER, H. F.. Introdução à
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