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J.
Fluid
Mech.
(1984),
vol.
149,
pp.
33-58
Printed
in
Great Britain
33
Hydrodynamics and heat transfer
associated with condensation on a moving drop:
solutions for intermediate Reynolds numbers
By
T.
SUNDARARAJAN
AND
P.
S.
AYYASWAMY
University
of
Pennsylvania, Philadelphia,
PA
19104
(Received
17
October 1983 and in revised
form
19 June 1984)
The hydrodynamics and heat/mass transport associated with condensation on a
moving drop have been investigated for the intermediate Reynolds-number range
of drop motion
(Re
=
O(100)).
The drop environment is a mixture of saturated vapour
and a non-condensable. The formulation entails a simultaneous solution
of
the
quasi-steady elliptic partial differential equations that describe the flow field and
transport in the gaseous phase, and the motion inside the liquid drop. The heat
transport inside the drop is treated as a transient process. Results are reported for
the interfacial velocities, drag, external and internal flow structure, heat flux, drop
growth rate and temperature-time history inside the drop. The results obtained here
have been compared with experimental data where available, and these show
excellent agreement.
The results reveal several novel features. The surface-shear stress increases with
condensation. The pressure level in the rear
of
the drop is higher. As
a
consequence,
the friction drag is higher and the pressure drag is lower. The total drag coefficient
increases with condensation rate for small values of drop size
or
temperature
differential, and
it
decreases for large values of these parameters. The volume of the
separated-flow region in the rear
of
the drop decreases with condensation. At very
high rates of condensation, the recirculatory wake
is
completely suppressed.
Condensation also delays the appearance of the weak secondary internal vortex
motion in the drop. The heat and mass fluxes are significantly affected by the presence
of the non-condensable in the gaseous phase and by the circulation inside the drop.
1.
Introduction
Condensation on moving drops occurs in a wide variety
of
physical situations. For
instance, condensation on a spray of drops is encountered in various studies on
raindrops, in the design of an emergency cooling spray system in a nuclear-reactor
containment, etc. However, the condensation phenomenon on translating drops has
not been examined in sufficient detail. In particular, the effects of condensation on
drop motion and on drag phenomena do not seem to have been investigated. The
intent of the present paper
is
to analyse the hydrodynamics and the transport
phenomena associated with condensation on
a
single moving drop,
for
a
wide range
of
condensation rates. Although realistic situations demand the consideration of a
spectrum
of
moving drops of various sizes, there
is
still a great need
for
a fundamental
understanding of the single-drop problem.
There are many studies in the literature that involve heat and mass transfer and/or
hydrodynamics of drop motion (Clift, Grace
&
Weber
1978).
Specifically, with regard
34
T.
Sundararajan
and
P.
S.
Ayyaswamy
to condensation on drops, Nix
&
Fukuta (1972), Lou
&
Yang (1979) and Rakhmatulina
(1981) have studied stationary drops. Ford
&
Lekic (1973) have experimentally
investigated the growth rate of
a
water drop in a pure steam environment. Kulic,
Rhodes
&
Sullivan (1975) and Kulic
&
Rhodes (1978) have predicted condensation
heat-transfer rates for droplets moving in air-steam mixtures using standard
heat-transfer correlations. They have also experimentally recorded the temperature-
time history of a water drop experiencing condensation in
a
forced flow of steam and
air (Kulic
&
Rhodes 1977). Sadhal
&
Ayyaswamy (1983) and Chung, Ayyaswamy
&
Sadhal(1984a,
b)
have theoretically examined slowly moving drops. Condensation
in the vicinity of the front stagnation point of
a
drop in high-Reynolds-number motion
has been investigated by Chung
&
Ayyaswamy (1981
a,
b).
A
boundary-layer
formulation appropriate for intermediate Reynolds numbers has been provided by
Sundararajan
&
Ayyaswamy (1984a).
In this paper
we
examine condensation on a spherical drop that
is
translating with
a
flow Reynolds number
Re
=
O(
100). The drop environment is a saturated mixture
of condensable vapour and a non-condensable gas. Condensation on the entire drop
surface has been considered. The flow problems inside and outside the drop have been
simultaneously treated. A constant-property quasi-steady model describes the
transport in the gaseous phase. The quasi-steady flow solutions and transport rates
to the drop are obtained in terms of the instantaneous values of two non-dimensional
parameters (one,
Re,
representing the flow conditions, and the other,
W,
representing
the thermodynamic conditions). The elliptic partial differential equations for the
gaseous phase are solved by a hybrid finite-difference scheme. The scheme involves
central-differencing near the drop surface and upwind-differencing in the far field.
It
provides accurate results near the drop while guaranteeing numerical stability in the
far field. The drop heating is treated
as
a transient process in view of the relatively
slow diffusion of heat in the liquid phase. The heat equation is suitably formulated
in terms
of
a
stream-function coordinate and is solved by the Crank-Nicolson
procedure. The more general transient treatment of the drop interior (both the flow
and heat-up) is the subject of a separate investigation (Sundararajan
&
Ayyaswamy
1984
b).
Although the formulations developed here are suitable for application to many
liquid-vapour systems in the presence of non-condensables, only results appropriate
for the water-steam system with air
as
the non-condensable have been studied.
2.
Physical
description
Consider the introduction of
a
cold water drop of radius
R
into an environment
consisting of
a
mixture of vapour (steam) and non-condensable (air). The total
pressurep, and temperature
T,
of the saturated mixture are taken
to
be prescribed.
The drop is colder than its environment and condensation occurs on the drop surface.
The shear stress
at
the interface due to drop translation will initiate liquid circulation
inside the drop. The heat deposited on the drop surface will heat the drop liquid.
For
a
short period of time following the introduction of the drop, rapid transients in
velocity, temperature and concentration will occur in both the liquid and gaseous
phases.
Let the instantaneous translational velocity of the drop be
U,.
For
a
particle in
the gaseous phase, the typical residence time adjacent to the drop surface is
O(R/
Urn).
On the other hand, for a particle in the liquid phase, the typical residence time is
O(R/U,),
where
Us
is the maximum circulation velocity
at
the drop surface. Since
Condensation
on
a moving
drop
35
Air-steam mixture
,/A/
Dividing stream surface
Wake recirculation
Secondary internal vortex
Primary internal vortex
Axis of symmetry
FIGURE
1.
Geometry
of
the
problem.
the rapid transients arise due to the sudden contact between the two phases, the
residence-time estimates provided above are the characteristic times for the rapid
transient processes. For a circulating water drop translating in a gas-vapour mixture,
U,/U,
may be shown to be
O(10-l)
or less. Therefore it may be inferred that the
period during which rapid transients occur is
O(R/U,)
and corresponds to
a
few
circulation cycles inside the drop. Subsequent to this initial period of rapid transients,
changes in flow and transport take place at a much slower rate until the drop
thermally equilibrates with the outside. This latter period, during which the bulk of
the condensation occurs, is the subject-matter of the present study.
We consider a coordinate frame that coincides with the drop centre and moves with
the instantaneous bulk velocity
U,
of
the drop (figure
1).
The Reynolds number of
translation (hereinafter referred
as
Re,)
is taken to be
O(lOO),
but
less
than, say,
500.
For
Re,
2
500
flow instabilities such as drop oscillations and vortex shedding are
known to occur. The flow may not be laminar and the deformation from the spherical
shape of the drop may be large (Clift
et
al.
1978).
In
our
analysis the drop deformation
due both to inertial effects (Weber number
We)
and
to
hydrostatic-pressure variation
(Eotvos number
Eo)
are assumed to be small. We consider water drops of size
1
mm
diameter
or
less
(Eo
<
0.4
and
We
<
0.3).
Next consider the circulation inside the