| dc.description.abstract | Despite the role of drops being recognised in many metallurgical
processes, surprisingly no significant progress has been made in this area.
An understanding of the drop formation phenomenon is expected to give
a better insight of the associated mass transfer and heat transfer aspects
of the relevant processes. Keeping this in view, the present studies on
drop formation at conical tips, which is a first step towards a better
understanding of drop formation at the tips of the melting rods, of direct
relevance in a metallurgical process such as ESR, were taken up. The
studies on drop formation at capillary tips provided a useful analogy
for those at conical tips. Both theoretical and experimental approaches
have been employed for the analysis of drop formation.
The principle of minimisation of free energy has been used to
develop a mathematical model to predict the equilibrium profiles of pendant
drops forming at infinitely slow rates at conical tips, using the varia-
tional approach. The same governing equations have been shown to be
deducible by an alternate approach using the force balance criterion.
The dimensionless profile generating equations were solved using fourth
order Runge–Kutta method for which a computer programme was developed
ensuring an accuracy of better than one part per million in the computed
values. The model permits the calculation of drop volume as well as
to follow the changing sequence of drop profiles until the onset of instability.
The maximum drop volume is shown to increase with cone angle.
Also, it is shown through the model that for a specified cone angle there
is a critical rod diameter beyond which the maximum drop volume forming
at its conical tip is independent of the rod diameter. Drop formation
at conical tips with base diameter less than the critical diameter, referred
to as finite cones, have also been analysed until the onset of instability
and the effect of rod diameter, cone angle and physical properties of
the system on the shape and maximum drop volume are predicted. An
absolute method for the determination of surface tension of liquids using
the pendant drop profiles at conical tips, which is shown to be superior
to other pendant drop profile methods, has been proposed.
A special case of the model is the analysis of drop formation at
a flat surface. A notable feature of this is the possibility of multiple
drop formation when the cross-sectional area of the flat surface is considerably
greater than the critical contact area of the drop. As a result of
the enhanced drop–continuous phase reacting interface, multiple drop
formation can be utilised in processes where refining is of paramount
importance.
Using suitable boundary conditions, the present model can also
describe the drop formation at capillary tips which has been extensively
studied for well over a century, especially by chemical engineers. By
deducing a force balance criterion from the profile generating equations,
it has been shown that in general, the Harkins–Brown’s empirical correction
factors proposed in 1919, in connection with the determination of surface
tension of liquids by drop-weight method, do not represent the true fractional
detachment of drops. Instead, modified correction factors have been
proposed. Utilizing these modified correction factors in conjunction with
the extension of the similarity criterion proposed by Worthington, a semi-
empirical model has been proposed for predicting the detached drop volumes
at conical tips.
The model for describing the drop formation at capillary tips has
been extended for finite flow rate conditions as well. The numerical
results show that the equilibrium drop volume and profile are insensitive
to fluid flow rate over a very wide range. Also, based on this model,
a criterion for the onset of jetting has been proposed. Semiempirical
models have also been proposed for the prediction of detached drop volumes
at capillary tips under finite flow rate conditions. The finite flow
rate model is shown to reduce to the infinitely low flow rate model,
when the flow rate tends to zero. The drop formation at conical tips
under finite flow rate conditions is analysed using a dimensional analysis
approach and a correlation between the detached drop volume and flow
rate has been obtained.
Also, Harkins–Brown’s most reliable drop weight data at capillary
tips has been statistically analysed and utilizing the mathematical description
of the relation between dimensionless detached drop volume and
the dimensionless capillary radius, two alternate methods have been proposed
for the calculation of surface tension of liquids. Using the error
analysis approach, the proposed methods have been shown to be better
than the earlier methods over a wide operating range.
A set of apparatus has been designed and fabricated for studies
on drops forming at metal cone tips of specific cone angle under controlled
flow rates, for quantitative verification of the predictions of the model.
Using these apparatus, experiments have been performed with low vapour
pressure liquids 1,1,2,2-tetrachloroethane and chlorobenzene, to study
the effect of cone angle, rod diameter, flow rate and physical properties
of the liquids on drop size. The trends in the theoretical predictions
mentioned earlier have been shown to be in very good agreement with
the experimental results. The drop formation phenomenon was followed
through movie-photography and the agreement between the drop profiles
and the maximum drop volumes with the theoretical predictions have
been shown to be excellent. However, the predicted detached drop volumes
are shown to be underestimated, with a maximum percentage deviation
of 27%. The reasons for the discrepancy have been discussed. The analysis
of the cine films have also yielded useful information on the mechanism
of detachment and the instability time has been estimated for each of
the conical tips of 60°, 90°, 120°, 150° and 180°, studied. The dimension-
less detached drop volumes at each of the conical tips for both the liquids
studied have been shown to be identical, which supports the assumption
behind the extended Worthington’s similarity criterion. Also, this forms
the basis of a comparative method for the determination of surface tension
of liquids.
Experiments have been performed at a capillary tip to study the
effect of flow rate on drop formation. The experimentally obtained critical
profile and the equilibrium drop volume closely tallies with those predicted
theoretically. Also, the detached drop volumes calculated using the proposed
models are shown to be in good agreement with the experimental
values.
To test the validity of the predictions of the models discussed
above in drop formation at the tips of melting rods, which has greater
relevance to metallurgists, a low temperature experimental set-up was
designed and fabricated for melting studies using paraffin wax material,
the liquid state transparency of which has enabled photographic recording
of the drop formation phenomenon. The experiments consisted of melting
the stationary wax rods having conical tips in a hot acetonitrile/alcohol
bath and collecting the solidified wax drops below. The effect of rod
diameter, cone angle, immersion depth and bath temperature on the drop
size was studied. While the trends of the experimental results were in
conformity with the theoretical predictions, quantification of the experimental
data was not possible owing to uncertainties in the physical properties
of the experimental system, despite efforts to minimise these effects. | |
