Liquid
Argon
Maximum
Convective
Heat
Flux
vs
Liquid
Depth
Tom
Peterson
January
12, 1990
DO
EN
237
1


DEPTH
IN
LIQUID
ARGON
VERSUS
MAXIMUM
CONVECTIVE
HEAT
FLUX
by
Tom
Peterson
January 12, 1990
(report
on
work done
in
February 1988)
Introduction.
In
order
to
help answer questions about the magnitude of heat
flux
to
the liquid argon
in
a liquid argon calorimeter which could
cause boiling (bubbles), calculations estimating the heat flux which
can
be
removed by free convection were made
in
February, 1988.
These calculations are intended
to
be
an
estimate of the heat flux
above which boiling would occur.
No
formal writeup was made of
these calculations, although the graph dated 3
Feb
88 and revised
(adding lowvelocity forced convection lines) 19 Feb 88 was
presented
in
several meetings and widely distributed. With this
description of the calculations, copies of the original graph and
calculations are being added
to
the
DO
Engineering Note files.
Assumptions.
The liquid argon surface is
in
equilibrium with argon vapor at a
pressure of 1.3 bar,
so
the surface
is
at 89.70
K.
The liquid
is
entirely at this surface temperature throughout the bulk of the
volume, except locally where
it
is warmed by a solid surface at a
higher temperature than the bulk liquid. This surface temperature
is
taken
to
be the boiling temperature of argon at the pressure
corresponding
to
1.3 bar plus the liquid head; hence it is a function
of depth below the surface. The free and forced convection
correlations used are
'from
Kreith, "Heat Transfer", for heated flat
plates
in
a large (Le.,
no
other objects nearby enough to disturb the
flow) uniform volume of fluid. Heat flux is a function of plate size,
really length along the flow path (since a boundary layer increases
in
thickness starting from the leading edge of the plate), and
orientation (Le., vertical or horizontal).
M~thod.
A table (on page
1)
was made using deltaT above the surface
temperature of 89.70 K as the independent variable. From this the
saturation (boiling) pressure corresponding
to
89.70 K plus deltaT
and depth below the surface were found. It
can
be seen that the
2
liquid density at a constant (surface) temperature is practically

constant with depth, and this density was used to calculate depth
for a given pressure. A density at the elevated temperature
(saturation temperature) also has to be tabulated since it is the
difference between this and the bulk density that drives the free
convection. Other fluid properties (thermal conductivity. viscosity,
and Prandtl number) are found for the pressure at the depth and
an
average of the saturation (surface) and bulk temperatures. Grashof
number includes a factor L cubed (L is plate length).
so
Grashof
divided by L cubed is calculated first.
The tables on. pages 2 and 3 contain Grashof number, Nusselt
number, and convection coefficients for the various lengths and
orientations, tabulated
as
a function of head. The product of
delta~
T
and the convection coefficient gives the heat flux. tabulated
on
page
4.
Convection coefficients are also calculated for
low~velocity
forced convection for comparison.
The resulting heat fluxes were then plotted as a function of
distance below the liquid surface. Note that depth
is
on
the vertical
axis with zero at the top, and heat flux is
on
the horizontal axis.
Results.
The maximum heat flux which can
be
carried away by free
convection (i.e., the heat flux above which boiling occurs)
is
.001
W/sq.cm. at 4 inches below the surface and
0.1
to
0.2 W/sq.cm. 15
feet below the surface. Forced convection over a 1
cm
plate with a
fluid velocity of 1 cm/sec, or a 10 cm plate at 10 cm/sec,
is
about
like free convection. The line for much higher heat flux
is
10
cm/sec flow over a 1
cm
plate.
Discussion.
The two key assumptions here are that the bulk of the liquid is
at the surface temperature, and that the threshold of boiling is when
the solid surface is at the saturation temperature of the liquid. It is
possible that experiments would give free convection heat fluxes
much lower or much higher than these results if these assumptions
are
in
error.
Much higher heat fluxes via free convection, especially near
the surface where I used very small deltaT's, might be possible if
nucleation
of
boiling occurs at some surface temperature
significantly (like a degree) above the saturation temperature of the
liquid. Nucleation would not occur at a lower deltaT than was used
here, since it was just the deltaT to the saturation temperature.
3
So
this calculation took the most pessimistic possible assumption
(in
terms of avoiding boiling) regarding the onset of nucleation.

Conversely, the limits of free convection might be much lower
than calculated here, again especially near the surface, since the
mechanism of heat dissipation
'from
the bulk involves warming of
the bulk liquid above the surface temperature and transfer of heat
through a boundary layer to the surface where evaporation takes
away the heat (Atkinson,
et.
aI., "Heat and Evaporative Mass Transfer
Correlation at the Liquidvapour Interface of Cryogenic Liquids''.
ICEC
10,
1984). Based
on
that paper, the predicted heat flux to
EC,
and the surface area for evaporation
in
EC,
I calculate that the liquid
would
be
superheated to a depth of about 3 feet. This free
convection calculation would result
in
no
allowable heat flux
to
that
depth since the liquid is already at or above saturation temperature.
A third factor which could cause reality to differ from these
calculations is that the geometry is not a small heated flat plate
suspended
in
fluid. Corners, edges and irregularities will disrupt
the boundary layer
and
enhance free convection heat transfer. But a
hot spot
on
the inner vessel wall will have
no
leading edge,
so
the
velocity profile will be different from what is assumed here. It may
be best approximated
by
the 10
cm
plate
in
these calculations since
that is mostly covered by a thicker boundary layer than the 1
cm
plate, but the total heat added is small enough that flow is still
laminar. The vertically oriented 100
cm
plate
has
a higher heat flux
to the liquid than the 10
cm
plate since for the 100
cm
plate flow is
turbulent rather than laminar.
Conclusjons.
As
I have indicated
in
the above discussion, there
is
considerable uncertainty
in
these results. Near the surface I
can
imagine a difference from these calculations
of
two orders of
magnitude
in
either direction. At 15 feet deep I have much more
confidence
in
these predictions; I would expect them
to
be
within a
factor of two of experimental results for the onset of boiling for
heated objects
in
the argon.
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