Proposed space applications, such as the cooling of infrared and x ray telescopes, have generated substantial interest in the behavior of He II flowing in porous materials. For design purposes, classical porous media correlations and room temperature data are often used to obtain order of magnitude estimates of expected pressure drops, while the attendant temperature differences are either ignored or estimated using smooth tube correlations. A more accurate alternative to this procedure is suggested by an empirical extension of the two fluid model. It is shown that four empirical parameters are necessary to describe the pressure and temperature differences induced by He II flow through a porous sample. The three parameters required to determine pressure differences are measured in counterflow and found to compare favorably with those for isothermal flow. The fourth parameter, the Gorter-Mellink constant, differs substantially from smooth tube values. It is concluded that parameter values determined from counterflow can be used to predict pressure and temperature differences in a variety of flows to an accuracy of about + or - 20 pct

Maddocks, J. R., Jr.

Vansciver, Steven W.

English

NASA Technical Reports Server

paper CD-4: 1991
Cryogenic Engineering
Conference. Huntsville, AL
sj _ey
CHARACTERIZING HE II FLOW THROUGH POROUS MATERIALS
USING COUNTERFLOW DATA
J. R. Maddocks and S. W. Van Sciver,
University of Wisconsin, Madison,
Applied Superconductivity Center,
Madison, Wisconsin
AI T_RAg 
Proposed space applications, such as the cooling of infrared and x-ray
telescopes, have generated substantial interest in the behavior of He H
flowing in porous materials. For design purposes, classicalporous
media correlations and room temperature data are often usedto obtain
order of magnitude estimates of expected pressure drops, while the
attendant temperature differences are either ignored or estimated using
smooth tube correlations. A more accurate alternative to this precedure
is suggested by an empirical extension of the two fluid model. It is shown
that four empirical parameters are necessary to describe the pressure
and temperature differences induced by HeII flow through a porous
sample. The three parameters required to determine pressure
differences are measured in counterflow and found to compare favorably
with those for isothermal flow. The fourth parameter, the Gorter-Mellink
constant, differs substantially from smooth tube values. It is concluded
that parameter values determined from counterflow can be used to
predict pressure and temperature differences in a variety of flows to an
accuracy of about +20%.
INTRODUCTION
A general interest in the behavior of He II flowing through porous
materials stems from recent space based technological applications,
which require the management of He II in a weightless environment.
Specific applications include; fine mesh screens and light weight, high
porosity ceramics for fluid acquisition devices,' sintered met_ or packed
metal powders for use as porous venting plugs 2"3and very fine pore
packed powders or ceramics to be used as superleaks. ('s
For design purposes, the Darcy permeability of a specific porous
sample, measured at room temperature, is often used to obtain order of
magnitude estimates of expected pressure drops. Expected temperature
differences are either neglected or estimated using the Gorter-Mellink
relation and smooth tube values of the Gorter-Mellink parameter.
However, experiments show that the major portion of the pressure drop
https://ntrs.nasa.gov/search.jsp?R=19920002077 2020-03-17T15:07:14+00:00Z
in high porosity ceramics results from kinetic energy losses rather than
viscous drag losses due to laminar flow. In addition, the superfluid
losses are not generally negligable. The result is that room temperature
measurements of the permeability do not provide adequate pressure loss
estimates.
Through a set of experiments discussed in the present paper we show
that a better method of characterizingporous media for He II
applications involves a straightforward measurement of counterflow heat
transport. These results combined with knowledge of the porous media
appear to allow the prediction of temperature and pressure gradients
through the medium to within _+ 20%.
GOVERNING EQUATIONS
The equations most commonly used to analyze the behavior of He II in
simple one dimensional geometries are:
Ps (Dvs/Dt) : "VPs" Fs" Fsn, (1)
and
where
Pn (Dvr_Dt) = "VPn + 11nV2vn "Fn + Fsn,
VP s = (ps/p)VP - PssVT,
(2)
(3)
VP n = (pn/p) VP + pssVT. (4)
The empirical forces F n, F s and Fsn are added to account for the effects of
normal fluid turbulence, superfluid turbulence and mutual friction
respectively.
To formulate the equations of motion in a way that is applicable to
porous materials, the following functional forms are assumed. Based on
a previous experiment 6 it is assumed that,
F n =bnPnVn 2, (5)
F s = bspsvs 2. (6)
It is further assumed that the Gorter-Mellink relationship describes the
mutual friction, so that
Fsn = A(T)pspn(v s - vn) 3. (7)
Finally, the empirical Darcy law,
VP = -_(v/k), (8)
is used to replace the laminar term (_nV2vn) in equation (2), where k is the
Darcy permeability.
Each term in equations (1) and (2) is replaced by its appropriate
functional form. Steady state conditions are assumed, so the time
derivatives are set equal to zero, and the equations then reduce to:
2.
VP s = -bsPsVs 2 - A(T)psPn(V s -Vn)3 (9)
VP n = -(TIn/k)v n - bnPnVn2 + A(T)PsPn(Vs - Vn)3.
The present experiment is designed to test the appropriatness 0£ this
empirical model.
(10)
MATERIALS. APPARATUS AND PROCEDURE
The porous material used in the present experiment is a fibrous
ceramic of the type used for heat shields on the space shuttle. The fibers
consist of 78% silica and 22% aluminum borosilicate. The material is
manufactured by Lockheed, 7 using a process which results in an
inhomogeneous and anisotropic final product. It is available in a number
of packing densities. Samples of 6 and 16 lbs/ft 3 are tested in the present
experiment. The porosity (e) of each sample type, determined as
= 1- (Psample/pfiber), is listed in Table 1.
The test section, shown schematically in figure 1, contains two
symmetrically mounted samples separated by approximately 10 ram. It
is configured as such to allow a broader study including isothermal flow
and combined flow.' Within the test section, the samples are mounted in
thin wall stainless steel tubing, to minimize parallel heat conduction
paths. Heat conduction through the ceramic samples may be neglected,
because the thermal conductivity of the sample material is at least six
orders of magnitude smaller than that of He II. To insure that the
samples fit tightly, they are carefully cut using a sharpened piece of the
same stainless tubing in which they are mounted. The outer surface of
the samples is lightly covered with vacuum grease, to ensure their ridgid
placement duing experimentation. The room temperature permeabilities
of several samples are carefully measured using helium gas, and two
closely matched samples are selected and mounted in the test section.
A 110 £_ metal film resistor serves as a heater and is located between
the samples. Allen-Bradley carbon resistors serve as thermometers. As
indicated in figure 1, there are three thermometers, each located in the
liquid. One between the samples and one approximately 2 mm outside
each sample. They provide an absolute temperature resolution of +0.5
inK. Temperature differences across the samples are determined by
subtracting the absolute temperatures measured on either side of the
samples. Finally, pressure drops across each sample are measured
Table 1. Comparison of the permeability and the coefficient b n
sample
6#
16#
porosity kgas kiso kcf (bn)is O (bn)cf
m2]L_l 1 ii1-1
•96 5.7 22 8.5 8600 7300
.90 2.9 10 3.5 13200 11500
3
H1
T1 J T2 j-,,
vacuum can
Figure 1 Schematic representation of
the experimental apparatus•
using Siemens KPY-33R pressure sensors• These sensors have a
nominal full scale range of 10 kPa and are mounted differentially. Their
resolution is +1 Pa.
All data are taken in the steady state with the aid of a Masscomp
computer and associated periferals. The two pressure sensors, and the
bath temperature are sampled sequentially at a burst rate of 1 MHz. This
sequential sampling is repeated 25 times per second, for 32 seconds• The
digitized data are then averaged to give a steady-state value.
During a somewhat longer but overlapping time period, the carbon
resistance thermometers are sampled. The thermometers are sampled
in sequence, each for a period of 10 seconds, at a sampling frequency of 6
Hz. This sampling frequency is determined by an A.C. conductance
bridge, which is used to read the output of the thermometers. The bridge
is a null device that g:i'ves a four-wire measurement of the conductance,
by providing a 300 mV, 24 Hz excitation voltage.
EXPERIMENTAL RESULTS
For the purpose of data analysis, it is assumed that any effects due to
mismatched samples can be neglected, so that half the heat deposited
between the samples flows through each sample. Thus the relation
v n = q/(2psTA£) (11)
is used to determine v n, where q is the total heat deposited, A the cross
sectional area and e the porosity•
While it is expected that this assumption will not cause a great deal of
error in the analysis, it means that small differences in permeability (k),
b n, b s and A(T) will not be measureable, as a result of the fixed boundary
conditions.
A pressure gradient is always observed to accompany the flow of heat
through He II in a narrow channel or porous material. The expected
form of the pressure drop is given by the sum of equations (9), and (10) as
VP = -(TI/k)v n - [b n - (pn/Ps)bs] PnVn 2 (12)
i
a,
4OOO
"i,= = T=I.6 K
"'_i_o_ [] T=l.7 K2OOO
_a_ • T=l.8 K
A T=1.9 K
o
-21300
4000 • , •
-0.10 0.00 0.10
normal fluid velocity (m/s)
Figure 2 Velocity dependence of the pressure gradient, as a function of
the temperature, for the 6# samples.
15OO0
-5000
o T=I.5K
• T=I.6K
" T=I.7K
• T=I.8K
• T=I.9K
-15000 , ,
-0.15 -0.05 0.05 0.15
normal fluid velocity (m/s)
Figure 8 Velocity dependence of the pressure gradient, as a function of
the temperature, for the 16# samples.
where the counterflow condition, v s = - (pn/ps)vn, has been used.
Figures 2 and 3 are plots of pressure gradient versus normal fluid
velocity for the 6# and 16# samples. It is evident in both figures that the
linear relationship between VP and v n breaks down at normal velocities in
excess of approximately 10 mm/s. This shows that the Allen and Reekie
rule does not apply, even if modified to use the Darcy law for porous
materials. In addition, the pressure gradients in figures 2 and 3 exhibit a
small but definite temperature dependence. In order to determine if this
temperature dependence is predictable, equation (12) is rewritten in the
form
VP/v n = -a(T) - _(T)v, (13)
where, u(T) - _I/kand _(T) - {bn- (pn/Ps)bs} Pn"
Theoretically, the permeability can be determined from a plot of a(T)
versus lln. Practically, however, there is a large amount of scatter in a(T)
because the pressure drop at low velocities, where laminar flow
dominates, is not much larger than experimental resolution. Therefore,
the permeability is determined using average values of a(T) and qn" The
permeabilities measured in this fashion are recorded in Table 1 and
agree reasonably well with those measured in room temperature gas flow
experiments. Agreement with isothermal permeabilities is not as good,
but this may be due to the method used to obtain isothermal flow. s
In figure 4, the temperature dependence of the quadratic term, _(T),
is compared to that predicted by equation (12). The figure shows little sign
of the expected temperature dependence, though clearly, _(T) is affected
by temperature. A plot of _(T) versus Pn is shown in figure 5. The
relationship is reasonably linear for both sets of samples, implying that
the pressure gradient should be more accurately given by
"--(T_r_)V n - bnPnVn2. (14)
Equation (14) is the expected normal fluid contribution to the pressure
gradient, and seems to imply that the superfluid contribution is
negligable, if not non-existent. Values ofb n, based on equation (14), are
included in Table 1, as are values of b n determined in isothermal flow. s
The two measurements agree fairly well.
While equation (14) implies that the parameter b s is not measureable
in this experiment, it turns out that it is still possible to estimate it.
Isothermal flow measurements ''s from two separate experiments, using
similar materials, indicate that a reasonable estimate is given by b n = 2b s.
Even though this relationship has been tested to a very limited extent, it
t " 6# sample
• 16# sample
20000
10000
0
0
|
1
Pn / Ps
Figure 4 Temperature dependence of the quadratic coefficient _(T) as
given by equation (12).
1e+6 [] 6# sample
Ie+6 • 16# sample
_-" 8e+5
6e+5
_" 4e+5 •
2e+5
Oe+O • , , , ,
0 20 40 60 80 100
normal fluid density kg/m3
Figure 5 The temperature dependent, quadratic coefficient, _(T), as a
function of rn only. The solid lines represent least square fits that have
been forced through zero.
nonetheless makes it possible to obtain all the necessary parameters for
calculating pressure drop from simple counterflow measurements.
When considering temperature data, the term F s in equation (14) can,
apparently, be ignored. However, the pressure gradient term is often
less than one order of magnitude smaller than Fsn, so that, in general, it
must be included. Equation (9) can be rearranged, then, to solve for Fsr _
giving
psVT - VP = A(T)pnp(Vs-Vn) 3. (15)
Graphic solutions for A(T) are obtained from equation (15) by plotting
the left hand side versus (Vs-Vn) 3. Values of A(T) as a function of
temperature are plotted in figure 6, for both the 6# and 16# samples. The
uncertainty in those values is on the order of+100 m/s kg. For reference,
• • • • g
a sample of the Gorter-Melhnk coefficient for smooth tubes ,s included,
as well as A(T) calculated for the exit channels of the test section. The
values of A(T) from the exit channels agree fairly well with the smooth
tube results. In contrast, the values of A(T) for the porous samples are 2
to 4 times larger than smooth tube results, though the temperature
dependence remains approximately the same. In addition, there appears
to be some dependence on geometry.
Suprisingly, the 16# material shows less deviation from smooth tube
results than does the 6# material. This result is opposite to what would
be expected and remains unexplained. However, given the dramatic
differences in geometry between porous media and smooth tubes, the
relatively small variation of A(T) from smooth tube values of the Gorter-
Mellink coefficient may imply that any geometry dependence of the
parameter is very w .ak. In porous materials such as those considered in
the present experim _nt, very large surface area to volume ratios and
tortuous flow paths nay give rise to inertial and path length effects.
7
t_0
a_
400O
300O
2OOO
1000
a 6# sample
• 16# sample
o Exit tube
g
• Vinen
Q
[] []
[]
• •
• •
o o o o
[]
0
0
I I " ' I I I
1.1 1.3 1.5 1.7 1.9 2.1
Temperature (K)
Figure 6 A plot of
the Gorter-Mellink
constant as a
function of
temperature, for
both 6# and 16#
samples
These effects may, in turn, account for why the parameter A(T) differs
from measurements in smooth tubes.
CONCLUSIONS
The parameters necessary to estimate pressure and temperature
gradients resulting from the flow of He II in high porosity ceramics can
all be determined in counterflow. This approach offers a more accurate
method of characterizing these materials for design purposes.
Experimental results indicate that gradients can be predicted within 20%.
Since counterflow experiments are relatively straightforward, it seems
reasonable to suggest that they will provide a good and relatively easy
characterization of any porous media.
ACKNOWLEDGEMENTS
Work supported in part by NASA/Goddard Space Flight Center under
grant NAG-5-1031.
RF,EERF,XCY_
1. M.J. DiPirro, Cryogenics, 30:193 (1990).
2. H. Rudiger and M. Wanner, _, 27:38 (1987).
3. H. NakaJ, and M. Murakami,_, 27:442, (1987).
4. R. Srinivasan and A Hofmann, _, 25:641, (1985).
5. G.L. Mills and A.R. Urbach, _, 30:206, (1990).
6. J.R. Maddocks and S.W. Van Sciver, in: "Advances in Cryogenic
engineering," Vol. 35A, Plenum Press, New York, (1990), p. 189
7. Lockheed Missles & Space Co. Inc., Astronautics Div. Sunnyvale, Ca.
8. J.R. Maddocks, "Transport Properties of He II flowing Through
Porous Materials,' PhD. Thesis, University of Wisconsin-Madison,
Madison, Wisconsin (1991).
9. W.F. Vinen, Proc. Roy. Soc. (London), A240:114 (1957).


Characterizing He II flow through porous materials using counterflow data

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