INVESTIGATION OF A LOW HEAT LOSS HIGH
TEMPERATURE THERMAL ENERGY STORAGE SYSTEM
BY
NORMAN ALAN COPE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1981
Copyright 1981
by
Norman Alan Cope
ACKNOWLEDGEMENTS
I would like to express my deepest gratitude to Dr.
Erich A. Farber, for his inspiring guidance and encourage
ment during my course of this research. I would also like
to express my thanks to the other members of my committee
for their constructive criticism and advice during the
course of this work.
I would like to acknowledge the helpful assistance of
the Shop personnel ably led by Mr. Richard Tomlinson.
I would like to acknowledge the patience and encourage
ment of my wife, Elizabeth, without whose help this work
would not have been possible.
Finally, I would like to thank my typist, Denise Jobb,
for her excellent typing and assistance in preparing this
dissertation and John Murdoch for his assistance in prepar
ing some of the drawings of this text.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ..............................
LIST OF TABLES ................................
LIST OF FIGURES ...............................
LIST OF SYMBOLS ...............................
ABSTRACT ......................................
CHAPTER
I INTRODUCTION ...............................
II LITERATURE REVIEW ..........................
II THEORETICAL CONSIDERATIONS OF POWDER
INSULATION .................................
Introduction ...............................
Theoretical Model ..........................
Thermal Conductivity of a Perfect Gas ......
Thermal Conductivity of a NonContinuum
Gas ........................................
ParticletoParticle Conductivity of a
Porous Powder ..............................
Radiant Transfer in a Porous Medium ........
IV EQUIPMENT DESIGN AND MATERIAL PREPARATION
Introduction ...............................
Conductivity Test Apparatus ................
The Design of High Temperature Thermal
Energy Storage System ......................
Diatomaceous Earth Properties ..............
I
i I
Page
iii
vi
vii
x
xiv
1
10
17
17
18
21
25
32
44
49
49
56
64
Page
V EXPERIMENTAL PROCEDURE AND RESULTS ....... 73
Conductivity Test Equipment ............. 73
Thermal Conductivity of Powder Insulation
Under Vacuum ............................ 76
Data and Results from Thermal
Conductivity Test ........................ 77
Discussion of Results of Conductivity
Test ..................................... 78
Range of values ..................... 78
Radiative transfer .................. 81
Effective gas conductivity .......... 82
Particletoparticle conductivity ... 86
Effect of particle size in a non
continuum ........................ 97
Effect of a vacuum level ............ 98
Effect of carbon and iron in
diatomaceous earth ............... 100
Thermal Energy Storage System ............ 101
Results of Prototype Thermal Energy
Storage System Test ...................... 105
Discussion of Results for the Thermal
Energy Storage System Test ............... 108
Insulation performance .............. 108
Heating of organic oil .............. 111
Steel wool blanket ................. 112
Overall storage vessel performance .. 112
Tank support ........... ............ 114
Cost of high temperature storage .... 114
Uncertainty in the measurement of
K ...... ....................... 117
a
VI CONCLUSIONS AND RECOMMENDATIONS .......... 118
APPENDIX A STORAGE VESSEL TEST DATA ........... 122
APPENDIX B CONVECTION HEAT TRANSFER ANALYSIS
FOR A POROUS POWDER ................ 130
APPENDIX C UNCERTAINTY IN THE MEASUREMENT
OF Ka .................... ........ 135
BIBLIOGRAPHY .................. ............... 140
Page
EXTENDED BIBLIOGRAPHY ........................... 143
BIOGRAPHICAL SKETCH ............................. 144
LIST OF TABLES
Table Page
1 High Temperature Insulators ... ................. 16
2 Molecular Weight, Collision Diameter, and
Specific Heat Ratio of Selected Gases ......... 24
3 Diatomaceous Earth Particle Size and Distri
bution Typical Results ....................... 67
4 Particle Density and Void Fraction of
Diatomaceous Earth ............................ 71
5 Average Thermal Conductivity Values For
Evacuated Diatomaceous Earth ................... 79
6 Theoretical Thermal Conductivity of Air and
Argon For NonContinuum Conditions at
Tb = 860 R ...................................... 84
7 Proportioning of Thermal Conductivity into
Gaseous and SolidtoSolid Conductivity ........ 87
8 Estimated Manufacturer's Cost for Storage
Vessels ....................................... 116
LIST OF FIGURES
Figure Page
1 Schematic of a Residential Energy System
Using High Temperature Thermal Energy
Storage ...................................... 6
2 Mechanistic Description of Heat Transfer
Through Porous Powder ......... .............. 19
3 Idealization of a Contact Region ........... 34
4 Two Particles Forming A Contact Region ...... 39
5 Model for Derivation of Radiant Contribution
to Thermal Conductivity Through a Porous
Medium ....................................... 45
6 Conductivity Test Apparatus ................. 50
7 Prototype High Temperature Thermal Energy
Storage System ............................. 57
8 Electron Microscope Photograph of Diatomaceous
Earth Carbon (1% wt.) Mixture. Particles
from Tyler Sieve #500 ....................... 68
9 Electron Microscope Photograph of Diatomaceous
Earth Carbon (1% wt.) Mixture. Particles
from Tyler Sieve #170 ....................... 68
10 Electron Microscope Photograph of Diatomaceous
Earth (as purchased) ........................ 69
11 Electron Microscope Photograph of Carbon
Black (Monarch 500) ......................... 69
12 Conductivity Test Equipment Arrangement ..... 74
13 Conductivity Test Equipment ................. 75
14 Electron Microscope Photograph of Surface
of Diatomaceous Earth Particle, Tyler
Sieve #500 .................................. 92
viii
Figure Page
15 Electron Microscope Photograph of Surface
of Diatomaceous Earth Particle, Tyler
Sieve #170 ................................... 92
16 Apparent Thermal Conductivity of Selected
Diatomaceous Earth Carbon (1% wt.)
Particles .................................... 96
17 Gas Conduction vs. Gas Pressure in Powders
Under Vacuum (Theoretical) .................... 99
18 Thermocouple Location for Prototype High
Temperature Thermal Energy Storage Vessel ..... 103
19 CoolDown of Prototype High Temperature Thermal
Energy Storage System ......................... 106
20 Temperature Profile Through Powder Insula
tion Under Vacuum ............................. 107
21 Storage System Test Equipment Arrangement ..... 109
LIST OF SYMBOLS
A area, ft2
Aa aperture area, ft2
A gas conduction area, ft2
As solid conduction area, ft2
A1 total area, ft2
An Avogadro's number
a accommodation coefficient dimensionlesss)
a average accommodation coefficient dimensionlesss)
B variable, Ibf/ft
b length, ft
Btu British Thermal Unit
C correction factor dimensionlesss)
C average molecular velocity, ft/sec.
C' correction factor, contact points/particle
C specific heat at constant pressure, Btu/lbm F
C specific heat at constant volume, Btu/lbmF
D diameter, ft
d mean void diameter, ft
Dc equivalent contact diameter, ft
D nominal particle diameter, ft
dx differential length, ft
e correction factor, dimensionless
E Young's mpdulus, psi
F force, lbf
OF degree Fahrenheit
ft feet
ft2 square feet
ft3 cubic feet
G constant, dimensionless
h convective heat transfer coefficient, Btu/hrft20F
Hin rate of energy input, Btu/hr
hr hour
Hst rate of energy storage, Btu/hr
I current, A
I insolation, Btu/hrft2
in. inch
K thermal conductivity, Btu/hrftF
Ka apparent thermal conductivity, Btu/hrftoF
kf thermal conductivity of a fluid, Btu/hrftF
K thermal conductivity coefficient of a gas for
g
continuum, Btu/hrftOF
K thermal conductivity coefficient of a gas for
continuum, transition, and noncontinuum,
Btu/hrftF
Kn Knudsen number, X/d, dimensionless
K radiative conductivity coefficient, Btu/hrftF
krv radiative transfer through particle voids,Btu/hrftF
Ks thermal conductivity coefficient through a particle
bed, Btu/hrftF
K modified thermal conductivity coefficient through
a particle bed, Btu/hrftF
ks thermal conductivity of a particle, Btu/hrftF
L modified mean free path or length, ft
Ibm pounds mass
lbf pounds force
M molecular weight, lbm/ibmmole
mm Hg millimeters of mercury
min minute
n contact regions per particle
NA particles per unit area
Nh particles per unit length
Nu Nusselt number, dimensionless
OD outside diameter, ft
P pressure, lbf/ft2
psi pressure difference, lbf/in2
psig gauge pressure, lbf/in2
Q heat transfer, Btu/hr
Q'" energy generated per unit volume, Btu/hrft3
qc heat transfer by convection, Btu/hrft2
Qelec electrical energy dissipation, Btu/hr
gc heat transfer by gas conduction, Btu/hrft2
q heat transfer by convection, Btu/hrft2
qh heat transfer by convection, Btu/hrft2
Qloss heat loss, Btu/hr
qr radiation transfer, Btu/hrft2
qs heat transfer through solids, Btu/hrft2
QT heat transfer, Btu/hr
qT heat flux, Btu/hrft2
Re resistance to heat flow, hrftF/Btu
R particle radius, ft
OR degree Rankine
r radial coordinate, ft
r. inner tank radius, ft
1
r outer tank radius, ft
R universal gas constant, 1.986 Btu/lb mole R
S average stress, lbf/in2
T temperature, OF, OR
Tb average or bulk powder temperature, OF, R
TC average temperature of hot boundary surface, F
TH average temperature of cold boundary surface, F
T average surface temperature, OF, R
V volume, ft3
wt weight, lbf
x length or Xaxis coordinate
y particle contact radius, ft
z coordinate direction, ft
GREEK SYMBOLS
a thermal diffusivity, ft2/hr
B constant, dimensionless
y specific heat ratio, dimensionless
5 void fraction, dimensionless
A finite change of variable
e emissivity, dimensionless
5 StephanBoltzmann constant, 0.1714 x 108
Btu/hr ft2OR4
q thermal efficiency, dimensionless
e coordinate direction, degrees
x mean free path in a continuum, ft
v Poisson ratio, dimensionless
lf viscosity, Ibm/hrft
vm Hg micrometers of mercury
p density lb /ft3
o collision diameter, ft
T time, hours
Phase angle
xiii
Abstract of Dissertation Presented to the Graduate
Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree
of Doctor of Philosophy
INVESTIGATION OF A LOW HEAT LOSS HIGH
TEMPERATURE THERMAL ENERGY STORAGE VESSEL
By
Norman Alan Cope
May 1982
Chairman: Erich A. Farber
Major Department: Mechanical Engineering
A low heat loss thermal energy storage system capable
of storage in a range of 2501000F is described in this
dissertation. The design criteria called for a storage
system that would be simple to construct, use inexpensive
and readily available materials, utilize an insulating
material with a thermal conductivity between 0.003 0.015
Btu/hrftF for temperatures ranging between 2501000F, and
be suitable for residential or commercial applications.
A design selected to fulfill the above criteria consists
of a double tank wherein the annular space between the tanks
is filled with a porous powder (silica) and then evacuated
to a pressure of approximately 1 mm Hg. Thermal conductivity
values of 0.0029 0.110 Btu/hrftoF were achieved experi
mentally for vacuum levels ranging from 0.5 3.0 mm Hg and
hot surface temperatures ranging from 340 7110F.
xiv
The results of an analysis of the modes of heat trans
fer through the porous powder under vacuum indicate several
means of potential improvement in the conductivity of an
insulating powder. Based on theoretical considerations,
the means for reducing the conductivity include
1. selecting powder materials with a low thermal
conductivity, such as quartz particles,
2. selecting a particle size where an average void
dimension is obtained that is less than or equal
to the mean free path of the gas molecules at a
selected operating condition,
3. selecting gases whose properties inhibit free
molecular energy transport, such as gases with
high molecular weights, high specific heat
ratios and large molecular crosssections;
several of the more practical gases being air,
argon and carbon dioxide.
The results of this study indicate for the temperature
range investigated, 25010000F, thermal energy storage is
technically and economically feasible. The availability
of inexpensive thermal energy storage will make it possible
to use current energy resources more effectively, particu
larly those that occur on an intermittent basis, such as
direct solar (thermal). The concept of thermal energy
storage in the temperature range indicated suggests the
possibility of an independent energy system, particularly
for residential and commercial applications.
CHAPTER I
INTRODUCTION
The more successful techniques used to store energy,
electrical or thermal, involve chemical or mechanical tech
niques such as leadacid batteries and thermal energy stor
age, respectively. There has been little practical advance
ment in energy storage technology over the techniques
just mentioned. With the current emphasis on energy
efficiency, there is a need to examine the benefits of
developing and using energy storage devices. In many appli
cations the need for thermal energy is greater than the
need for electrical energy. For example, in a household,
about 85% of the energy needs can be provided by thermal
energy. This quantity of thermal energy is needed for
house heating, house cooling by absorption airconditioning,
domestic water heating, and cooking. The remaining 15% of
the energy requirements, for current lifestyles, is needed
in the form of electricity for lighting, refrigeration
and powering communications systems, fans and pumps.
It should be noted that a gasfired or hot water driven
absorption air conditioner generally requires electricity
to run pumps and controls.
It should be recognized in the above example that
energy, in the form of either heat or work, is most effect
ive when used in a direct manner. For example, when thermal
energy is converted into electrical energy (work), there
follows a substantial loss as a result of the second law
of thermodynamics in the form of the Carnot principle. And
while electric energy (work) can be converted into thermal
energy with no loss, the Carnot loss suffered in creating
work from heat can be substantial. However, from a systems
point of view, one may wish to convert from one form of
energy to another for convenience, to enhance the system
usefulness, or to enhance the lifecycle cost of the system.
Research and development of materials for storing
thermal energy indicate that energy storage on a residential,
commercial and industrial scale is feasible. Phase change
materials, such as nitrate salts, can store large quantities
of latent thermal energy associated with its phase change and
organic oils, such as cottonseed oil, can be used to store
large quantities of sensible thermal energy.
A primary difficulty of storing thermal energy at
temperatures of 25010000F is that one must find an econo
mical means of constructing a storage system with low heat
loss characteristics. Very little literature exists in
this area. Most of the research efforts in the past
have been limited to developing insulation and insulating
techniques in the cryogenic field.
Many of the problems encountered in insulating a
cryogenic vessel are also encountered in insulating a
high temperature system. Therefore, the cryogenic area
seems to be a good starting point for developing a high
temperature thermal energy storage system. Shared problems
in the development of high temperature or cryogenic storage
container are
1) heat leaks due to supports and tank
penetrations,
2) insulation and tank materials,
3) outgassing of materials in the evacuated
space,
4) inner tank support structure,
5) leakfree penetration ports for
piping and electrical wiring,
6) improved evacuating techniques when
fine powders are used as the insulating
material, and
7) reasonable material and system cost.
The solutions to the above problems are different for
cryogenic than for high temperature. For example, in
cryogenic vessels, some plastics can be utilized as a
spacing material; whereas, this same material in moderate
or high temperature environments would cause problems like
loss of structural integrity or outgassing.
In constructing a high temperature storage vessel
there are different concepts that one may employ to achieve
the same results. For example, one could build a storage
vessel using a fibrous insulating material under a vacuum
but, obviously, this material would not support the inner
vessel. A porous powder under vacuum could be used to
support the inner vessel and to insulate the inner tank
as well. Both insulating materials may result in a similar
heat loss characteristic; however, there may be advantages
and disadvantages with using one material in place of the
other.
This study investigated the use of powder insulation
at a typical pressure level of 1 mm Hg for insulating a
high temperature thermal energy storage system. This
technique of insulating was chosen over multilayer
insulation because
1) powder insulation works well at pressures of
110 mm Hg and this level is easier to maintain
than a pressure level of 1 um Hg or less, as
required by multilayer insulation,
2) suitable powders for withstanding high temperatures
are readily available,
3) a need exists for many uses in high temperature
technology,
4) it may be possible to form a load bearing member
of powder to support the inner vessel and this
would avoid the structural complexities of build
ing thick walled vessels.
Powder insulations suitable for high temperatures are
readily available and can be anything from wood ash to
sand. The particular material used in this study was
diatomaceous earth, a material readily available, inexpen
sive, and suitable for high temperature use because of low
particle thermal conductivity and a softening point of
16000F.
Projected applications of high temperature storage systems.
The intention of developing a high temperature thermal
energy storage system was to provide this component in a
selfcontained cooking system for residential use [1]. This
system utilizes a concentrating collector with an automatic
solar powered tracking mechanism [2], a heat storage vessel,
and a cooking range. The concentrating collector coolant
and the heat storage media are organic oil.
Four organic oils (cottonseed, peanut, corn and soy
bean oil) were evaluated for the above use [3]. In opera
tion, the organic oil was heated as it circulated through
a heat exchanger located in the focal point of the collector.
With a minimum temperature of 400F, the heated fluid
contained in the storage vessel would circulate through
the heating coils of the cooking range. The main advantage
of this system is that thermal energy is accumulated and
stored for use at a later time, eliminating the disadvantage
of having to wait for direct radiation and the necessity
to cook outdoors.
The system can also be used as the basis for an integra
ted, residential power system as shown in Figure 1. For
A softening point represents a temperature beyond which
the yield stress of a material drops off rapidly with
increasing temperature.
Vapor Generator 
Legend
P Pump
G Electric Generator
ORCS Organic Rankine
Cycle
Figure 1
Schematic of a Residential Energy System
Using High Temperature Thermal Energy Storage
example, energy accumulated in the storage vessel can be
used to drive a heat engine to produce electricity. The
energy rejected from the continuously operating heat engine
can be used to drive either an absorption air conditioner
or heater, domestic water heater, or similar low tempera
ture devices. The high temperature storage medium, such
as organic oil, can be used to power a cooking range.
Energy for the above system may be collected from
available energy resources. Some of these possible re
sources are
1) direct solar, thermal conversion,
2) fossil or nuclear fuel in the form of off
peak electricity,
3) direct solar, electrical to thermal energy
conversion (photovoltaic cells),
4) wind energy (mechanical to thermal energy
conversion).
As emphasized earlier energy is best used in a direct
manner; that is, work is best used as work and thermal energy
as thermal energy. Processes that convert thermal energy to
work, as in the conversion of solar energy to electric in a
photovoltaic cell (3) or to wind (4), processes that trans
form work into thermal energy, such as in statements 3 and
4 above will have irreversible losses associated with them
and these losses will be substantial. This concept is
embodied in the second law of thermodynamics and may be re
viewed in elementary texts on thermodynamics [4]. However, it
could nevertheless be economically practical to convert work
into thermal energy. For example, if an industrial process
requires high temperature fluids, it might prove advantageous
to purchase electrical energy at reduced rates from a utility
company during periods of low demand and to convert that
power into thermal energy to be stored for later use.
As apparent from the above paragraphs, the listed
energy resources occur on an intermittent basis and may
not coincide with the energy use pattern for which they
were intended. Storing energy for later use transforms an
intermittent source of energy into a potentially continuous
commodity.
In addition to residential applications an efficient
cost effective insulating technique is needed for high
temperature batteries [5]. For example, a lithiumsulfur
battery operates between 707 7970F and a sodiumsulfur
battery operates between 446 5720F. A particular problem
associated with operating these batteries is that energy
must be supplied to bring them to operating temperature
and to maintain standby losses. This energy is not avail
able for work and must be viewed as lost work. If an
efficient and cost effective insulating jacket reduced
these losses, then high temperature batteries would be one
step closer to reality.
Scope of investigation. The goal of this research was to
develop a thermal energy storage system suitable for storing
latent or sensible thermal energy between temperatures of
250 to 10000F. The particular objectives required to ob
tain this goal were to
9
1) find a practical insulating technique capable
of reducing the thermal conductivity to less
than 0.015 Btu/hrftoF @ 10000F,
2) identify and experimentally determine the
necessary thermophysical properties of the
insulating material,
3) construct a high temperature thermal energy
storage system to verify the thermodynamic
analysis and results of the insulating material
and to verify the system design.
CHAPTER II
LITERATURE REVIEW
One of the more important facts to emerge from the
literature review was that no significant references were
found concerning efficient, high temperature insulation
suitable for residential or commercial sized thermal
energy storage systems. This search included a manual
search of the library at the University of Florida
and a computer search that included the National Aeronautics
and Space Administration (NASA) and the National Technical
Information Service (NTIS) engineering files.
In view of the above, the goal of this research was
to develop a thermal energy storage system suitable for
storing latent or sensible thermal energy between tempera
tures 250 1000F. The successful development of a
thermal energy storage system depends upon the type and
quality of the insulation used in constructing the system.
The initial literature review shows that there are basically
three types of insulation suitable for the indicated
temperature range. Based on physical structure they can
be described as
1) fibrous insulation,
2) laminated insulation,
3) granular insulation.
Fibrous insulation includes such materials as fiberboard
and rockwool, which are suitable for low and moderate
temperature applications. Laminated insulation consists
of alternating layers of low density insulation such as
ceramic paper and low emittance radiation shields. Granu
lar insulation includes such materials as sand, micro
sheres of glass, and diatomaceous earth. Laminated and
granular insulation are suitable for use from cryogenic to
high temperature applications.
As described by Jakob [6] some of the earliest work
on conduction through porous material was done by such men
as Maxwell, Nusselt and Eucker. These investigators included
gross simplifications of the heat transfer mechanisms in
their analyses. Their aim was to make it possible to obtain
reasonable answers to a very complex problem with a minimum
of data. It seems that little advancement has been made in
the area of heat transfer through porous materials since
these early studies.
The basic ideas presented by Reference 6 concerning
Maxwell's, Nusselt's, and Eucker's analyses indicate that
all three viewed a porous body as either a base material
with small particles of another substance distributed
throughout that material, or as alternating layers of solids
and gases with heat flow perpendicular to the layers.
One of the classic works on thermal conductivity of
fine powders at reduced pressures, as described by Jakob [6],
was that of Smoluchowski. The essence of Smoluchowski's
work was that at reduced pressures, gas molecules would
experience slip flow in the vicinity of a solid surface.
This causes a discernible temperature discontinuity to
exist between the solid surface and the gas. When this
theory is applied to fine powders, one correctly concludes
that many temperature discontinuities are created. These
added resistnaces in turn reduce the transfer of heat
through porous powder.
Most of the early works of heat flow through porous
material were developed considering ambient or lower tempera
tures. Few studies were found concerning the flow of heat
from one particle to another. One study, by Strong et al.
[7], analyzed heat flow through glass fibers at a pressure
of 10 pm Hg. Even though this investigation worked with
materials with smooth surface geometry, its experimental
particletoparticle conductivity was off by an order of
magnitude when compared with theoretical values of heat
flow perpendicular to the fiberglas rods.
In the first part of the twentieth century, two facts
important in insulation were discovered and then investiga
ted. First, it was noted by Smoluchowski [8] that soot
had a lower conductance than air. At that time it was
thought that the conductivity of air established a minimum
value for thermal insulation. However, Kistler and Caldwell
[9] in 1934 found that silica aerogel had a thermal con
ductivity less than air and reasoned that the lower conduc
tivity resulted from the fact that the pore size of the
material was less than the mean free path of molecules of
air.
The second important fact, noted by Kistler and
described by Wilkes [10], was that when commercial cork
was placed in an evacuated chamber, thermal conduction and
radiant transfer, through the material, at low temperature,
was found to be a small part of the overall heat transfer.
This indicated that air was responsible for the majority of
the energy transport through the material.
In a more recent study, Peterson [1] showed that
successive thin foil radiation shields of low emissivity
wound with fine steel wool spacers were exceptionally good
insulators, especially when used under vacuum conditions.
Based upon the three developments mentioned above, it
was possible to construct superinsulated vessels, at least
for cryogenic purposes. The requirements for space age
technology placed a premium on lightweight, high efficiency
insulation, and, consequently, great interest was shown in
cellular and laminated or multilayer insulation [12, 13].
Most of the development work on multilayer insulation
was done for cryogenic vessels, but some work was performed
for high temperature applications. For example, the
early design of a thermoelectric nuclear power system
for spacecrafts used multilayer insulation operating
at a hot surface temperature of 1285F [14]. The multi
layer consisted of alternating layers of copper foil with
quartz paper separators, and had an upper temperature limit
of 14000F. The vacuum space was designed and built to
hold a pressure of less than 10 pm Hg for at least five
years.
As evident from much of the literature, multilayer
insulation for spacecraft and for cryogenic work was
expensive. Prior to the need for superinsulators for
spacecraft use, evacuated powders were sufficient for most
low temperature pruposes. As pointed out by Black et al.
[15], although powder insulation does not have as low a
thermal conductivity as multilayer insulation, its advantages
are
1) it can operate effectively at a few millimeters
of mercury or less, thus eliminating the need
for high vacuum,
2) it can be used in vessels of intricate design, and
this eliminates the wrapping and bonding of each
component in a vessel, as is necessary with multi
layer insulation,
3) it costs less than multilayer insulation.
In view of this, it seemed that porous powder under partial
vacuum should be suitable for terrestrial high temperature
insulation applications where weight is not a serious dis
advantage.
There is a growing need for efficient high temperature
insulation. The present technique used to insulate large,
high temperature (2001000F) storage vessels is to use
thick layers of granular insulation at atmospheric pressure
[16,17]. This approach seems reasonable for large thermal
energy storage systems in view of the quantity of energy
stored relative to the amount of energy lost. In smaller
systems, such as a residential sized thermal energy
storage vessel capable of storing 106 Btu with minimal
losses, the storage efficiency is more closely tied to the
overall system size and shape. This results from a smaller
volumetosurface ratio. No references were found con
cerning high efficiency insulation suited to small systems.
Available literature indicates that little consideration
has been given to high temperature thermal energy storage
systems. It seems that more attention has been directed
into developing heat storage media than in developing the
heat storage media containment system.
Table 1 has been compiled to illustrate the types of
materials, and their associated thermal conductivities,
that are available as high temperature insulators. Note
that most high temperature insulating materials listed in
Table 1 have similar thermal conductivities. This result
is not unexpected since these materials are composed of
similar materials which is basically SiO2. All of the
above materials are at atmospheric pressure, with the excep
tion of the diatomaceous earth which is at 1 mm Hg. The
reduced pressure results in a thermal conductivity value
that is approximately one order of magnitude lower than
similar material at atmospheric pressure. Lower thermal
conductivities can be obtained as will be explained in the
next chapter.
TABLE 1
High Temperature Insulators
Material Bulk Density Max Temperature Thermal Conductivity
(Ibm/ft3) (OF) Btu/hrftF
Mean Temperature
@ 500F @ 1000F
Diatomaceous earth,
asbestos and bonding
material [18] 18 1600 0.053 0.065
Glass blocks, average
values [18] 1424 1600 0.053 0.074
Microquartz fiber,
blanket [18] 3 3000 0.042 0.075
Rock woll [18] 812 0.049 0.078
SilOCell [19]
Insulating brick 40 2500 0.142 0.163
Diatomaceous earth,
@ 1 mm Hg pressure in
air. Nominal particle
size 50 pm. 20 1600 0.004 0.010 (est.)
CHAPTER III
THEORETICAL CONSIDERATIONS OF POWDER INSULATION
Introduction
It is relatively easy to measure the overall apparent
conductivity of a powder insulation. However, if one
wishes to alter and optimize apparent conductivity other
than by increasing thickness, it is necessary to have
an acceptable theory with which to proceed. If one ideal
izes the properties of the media which form the powder and
spaces, then one can model its apparent conductivity.
Unfortunately, the true geometry is not well modeled by
a simple approximation, nor are the properties adequately
approximated as ideal. As a result, it is very difficult,
perhaps impossible to evolve an analytically tractable
model that also accurately describes the phenomena as
they occur in the powder. This seems to be the case with
modeling of energy transport through porous powder insula
ting material.
In this study diatomaceous earth (98% Si02) was used
as the insulating powder. In view of the above discussion
the major questions encountered in modeling the irregularly
Apparent conductivity is heat flow from conduction, con
vection, and radiation combined and used in equations as
if it were conduction alone.
shaped particles were how to describe factors that enhance
or impede heat flow through this material. These factors
are
1) gaseous thermal conduction, as limited by the
molecular mean free path and influenced by gas
pocket geometry and boundary conditions,
2) conduction through solids, limited especially
by particle geometry and boundary conditions,
3) radiation resistance determined by particle
void space geometry, particle surface properties
and boundary conditions.
The best understood of the processes applicable to heat
transfer through powders seems to be gaseous conduction,
followed by radiation and particletoparticle conduction.
The relative importance of each process is discussed later.
The remainder of this chapter is devoted to developing a
model for heat flow through porous powder insulation which
addresses the above questions.
Theoretical Model
A mechanistic model for heat flow through a porous
powder is sh6wn in Figure 2. The total energy flow through
the gaspowder mixture is a combination of conductive,
convective, and radiant interaction between the solid and
gaseous materials present. Symbolically, the total heat
flow through the gas powder mixture can be written in func
tional form as
qT = qT (c' qr' qh) (1)
For powders with particle size less than 0.2 inches,
investigators such as Hill and Wilhelm [20] found no
evidence of convection in quiescent, gassolid beds for
conditions of interest to this study. Also, in this work,
the powder material was studied at pressures ranging from
atmospheric to less than 1 mm Hg. Consequently, convection
heat transfer was eliminated from further consideration.
Boundary
3
Figure 2
Mechanistic Description of Heat Transfer Through
Porous Powder
(1) Gaseous conduction, (2) solid conduction,
(3) radiant heat transfer through voids, (4) radiant
heat transfer, solidtosolid, (5) conduction,
point contact, (6) convection.
The justification for eliminating convection heat
transfer under the conditions discussed above can be
shown by estimating the convective heat transfer coef
ficient, h, between particles. This analysis is given
in Appendix B. The results of this analysis reveal that
the Nusselt number,
hb
Nu = h (2)
is of order unity at atmospheric pressure and at tempera
tures between 250 1000F, then the convective heat
transfer coefficient reduces by two orders of magnitude.
Hence, convection heat transfer at high temperature and
low pressure can be eliminated from further consideration.
In view of Equation 2, energy transport through a gassolid
mixture in which convection has been eliminated reduces
to primarily a function of conduction. It will be shown
later that net radiant transfer contributes only a small
fraction of energy transfer under conditions encountered
in this work. Thus,Equation 1 is rewritten as
qT '= qT(qc) (3)
The remainder of this chapter is devoted to examining
in detail conduction through both gas and solids and the
small contribution of radiant transfer through porous pow
ders (silica) under partial vacuum.
Thermal Conductivity of a Perfect Gas
Introduction. This section examines the thermal
conductivity of a perfect gas. It is implicit in the use
of the property thermal conductivity that a continuum
exists. A continuum is assumed to exist as long as the
mean free path of a molecule is comparable to the smallest
significant dimension of its surroundings. The ratio of
the mean free path, X, of a gas molecule to the average
dimension of its boundaries, 3, is referred to as the
Knudsen number. That is,
Kn = X/i. (4)
This number is useful in identifying the conditions under
which continuum assumptions apply. The need for identify
ing whether a continuum exists is important in determin
ing the proper relationship required for calculating the
thermal conductivity of a gas. As will be shown in the
following sections, the thermal conductivity of a gas in a
gaspowder mixture is influenced by the magnitude of the
Knudsen number.
Conductivity in a perfect gas. This section examines
thermal conductivity of a continuum. A continuum is said
to exist when the mean free path of a gas molecule is small
(Kn < < 1) compared to the dimensions of its boundaries.
For energy transport in a continuum, the gaseous thermal
conductivity is given by
qgc KVT (5)
The conductivity coefficient, K is given by [21]
Kg = (9y 5) (pC A) (6)
Density, p, is given by
Pbl
PMT (7)
P = ^ (7)
u
where P is the pressure, M is the molecular weight of the
gas, R is the universal gas constant, and T is the tempera
ture of the gas (absolute temperature). The average gaseous
molecular velocity, C, is given by
8 RT
c = (22 (S)
The mean free path, X, for a perfect gas is given by
RT
S u (9)
J/7 2A P
n
where o is the collision diameter of the gaseous molecule,
P is pressure and A is Avogadro's number.
Specific heat at constant volume, Cv, is given by
Cv = R /M(Y1) (10)
where Y is the specific heat ratio.
Combining Equations 7, 8, 9, and 10 into Equation 6
yields
g (9y5) R u u(
g 44yl) 70'I T) Y7
n
Equation 11 indicates that gaseous conductivity is a
function of temperature and that it should rise with an
increase in temperature according to T With an increase
in temperature over the range of 250 1000F, and taking
into account the temperature variation of Cv and y, experi
mental values of thermal conductivity of air at atmospheric
pressure change from 0.0192 to 0.0337 Btu/hrftoF.
From Equation 11 one can identify the gaseous molecular
properties that will cause the conductivity coefficient
to be small. For example, one would choose a gas with high
molecular weight, large collision diameter, and a high
specific heat ratio. Table 2 lists some values of M, o
and y for several of the more common gases.
It is interesting to note in Equation 11 that the
thermal conductivity of a perfect gas is independent of
pressure. Mathematically, the reason for this result is
that the direct proportionality of pressure in the density
term, p, cancels with the inverse proportionality of pres
sure in the mean free path, X, expression. Physically,
this means that if the density, or number of gas molecules
in a given space is decreased, the mean free path or dis
tance between collisions of gas molecules increases so that
the transport of energy is unchanged.
TABLE 2
Molecular Weight, Collision Diameter,
Heat Ratio of Selected Gases
M
(lb /lb mole)
m m
39.948
28.97
28.011
44.011
2.015
4.003
28.013
31.999
20.183
o [22]
(ft x 10 )
0.125
1.1975
1.05
1.10
.886
.689
1.23
1.94+
1.61
and Specific
y [23]
dimensionlesss)
1.667
1.40
1.40
1.285
1.40
1.667
1.40
1.40
1.667
Specific heat ratios for gases at low pressure at 800F.
Corrected value [24].
'Corrected value [24].
Molecule
A
Air
CO
CO2
z2
He
N2
02
Ne
The calculation of the thermal conductivity of a gas,
as calculated with the perfect gas continuum assumptions,
yields results that are adequate for describing real
gases in many situations. However, once the variables of
temperature, pressure, and boundary conditions are such
that a continuum no longer exists, that is Kn > 1, then
Equation 6 must be modified. This is the topic of the
following section.
Thermal Conductivity of a NonContinuum Gas
As mentioned in the previous section, the conductivity
of gases is theoretically independent of pressure in a
continuum. However, both Knudsen [25] and Smoluchowski
[8] developed theories concerning gaseous thermal conduction
for noncontinuum. Noncontinuum and in the limit free
molecular energy transfer occurs when the confining
boundaries of a gas molecule become smaller than the
mean free path of the gas molecule. For this condition
another phenomenon occurs that limits the energy transfer
between a gas molecule and a solid boundary. Knudsen
found that when a gas molecule strikes a solid surface
that the gas molecule does not necessarily come into
thermal equilibrium with the surface. That is, a tempera
ture discontinuity was found between the temperature of the
surface and the temperature of the gas molecule leaving
the surface. Thus, Knudsen introduced a constant surface
property, called the thermal accommodation coefficient, a,
which is a measure of the degree of energy exchange between
a gas molecule and a solid surface. The maximum value of
the accommodation coefficient is unity. This value is
approached by surfaces that are roughened.
The energy transfer between parallel plates a distance
x apart and at moderately low pressure is given by [261
K (THT )
q = x + 2A (12)
where B is
2a 2e
Sa (13)
a y+1
and
e = (9y5) (14)
At very low pressures, in the range of molecular conduction,
it was found that energy transfer becomes independent
of the plate spacing, and proportional to the pressure, P.
Rewriting Equation 9 for the mean free path of a gas
molecule as
A = B/P (15)
where
RT
B = (16)
i/To2A
n
and rewriting Equation 12 in view of Equation 15 for condi
tions where energy transfer is independent of x yields,
q = (THTC (17)
Hence, molecular heat conduction is proportional to pressure
at very low pressures, (Kn >'1).
Equation 17 did not appear suitable for the present
investigation for the following reasons:
1. an equation was needed that would be suitable
for energy transfer for continuum, transitional,
and noncontinuum conditions, and not just the
latter,
2. the use of Equation 17 requires a knowledge of
accommodation coefficients and these values are
generally not available,
3. Equation 17 was derived on the basis of parallel
plates and the material under investigation is
powder material with particles and cavities that
are irregular in size and shape and oriented at
random; that is, the boundary dimensions must be
taken into account.
Strong et al. [7] have found that Equation 6, when modi
fied for noncontinuum conditions, gave experimental results
that were in good agreement with theoretical predictions
over the range of continuum, transitional, and noncontinuum
conditions. The material investigated by Strong et al.
[7] was fiberglassrods which formed small cavities, as does
the powder material of this investigation. The basic
assumptions in using Equation 6 were
1. the accommodation coefficient could be approxi
mated as unity, and
2. that by modifying the definition of the mean free
path, the equation would be suitable over the
range of pressure from continuum to noncontinuum
conditions.
Since the accommodation coefficient is a measure of the
degree of energy transfer between a gas molecule and a
solid surface, then it seemed reasonable to conclude
that for a material with cavities on the order of the mean
free path of the gas, that an effective accommodation
coefficient near unit will result. That is, a gas mole
cule will interact with the boundaries of a given cavity
a sufficient number of times to reach equilibrium before
escaping to another cavity.
In a noncontinuum gas there are two space dimensions
that must be taken into account the mean free path, X,
of the gas molecule and the average space or void dimension
of its confining boundaries, J. When the mean free path
is less than the space dimension, 3, Kn < 1, perfect gas
conductivity is independent of pressure, as expressed
by Equation 6. When the mean free path of the gas molecule
is close to or greater than the space dimension, Kn > 1,
then the conduction equation must be modified to take this
into account. A modified mean free path of a gas molecule,
L, is related to the continuum mean free path, A, and
the gas space dimension, a, by Strong et al. [7] as
L = ( + ~)1 (18)
d
In the continuum region where A < < J, then
L 
In the noncontinuum region where A > > J, then
L 
To relate the modified mean free path, L, to gaseous con
ductivity, recall that
K = (9Y5) ('pCvC ) (6)
Substituting Equation 12 into Equation 6 yields
g C(9y5) pCv (i/A + 1/4)1 (19)
where K is the modified gaseous thermal conductivity coef
g
ficient. Equation 19 can now be used to calculate the
thermal conductivity of a gas for continuum, transitional,
and noncontinuum conditions.
Finally, to show that gaseous thermal conductivity,
g is related to pressure, recall Equations 15 and 16
and with the appropriate substitution, Equation 19
becomes
S2R
Kg = (y (S (1/P + 1/B1 (20)
Thus, as the pressure is lowered, at a fixed void geometry
and temperature, the gaseous thermal conductivity decreases
monotonically. Note that as the product of PT becomes
smaller than B, the former value dominates, and it can be
concluded that K varies essentially linearly with pressure.
g
Conversely, as the product of PT becomes large relative to
B, K is seen to be essentially independent of pressure.
g
Equation 20 also suggests that the gaseous conductivity
will remain the same if, at a given temperature, the modified
mean free path, L, is held at a constant value, that is,
L = (1/PT + 1/B)1 = constant (18)
This means that a given value of K_ can be maintained
by either decreasing 3 and increasing P, or by increasing
T and decreasing P. This suggests that by decreasing the
particle size, which will lower the average void dimension,
J, a higher gas pressure will be acceptable. For example,
by selecting particles such that the pressure level is
raised from 0.10 mm Hg to 5 mm Hg, while maintaining a
constant value of K many of the problems associated with
low pressure, such as the need for diffusion pumps, out
gassing of surfaces, and long term leakage, can be avoided
or minimized. It is the control over the particle size,
and consequently the gas space dimension, that makes a
gaspowder mixture attractive for high temperature insula
tion. Both Equations 17 and 20 indicate that molecular
energy transport can be reduced by selecting gases with
high molecular weights and high specific heat ratios.
This same trend is also exhibited for conduction of heat
by gases in a continuum. Equations 17 and 20 do not show
a dependency on the collision crosssection in the regime
of molecular conduction (Pd < < B) since the mean free
path has been superseded by the confining boundaries of
the gas molecule. Also, for noncontinuum or molecular
conduction, K is found to vary with temperature by T .
Hence, molecular conductivity decreases as the temperature
is elevated.
When the thermal conductivity of a gas has been reduced
below the particletoparticle thermal conductivity in a
gas powder bed, little will be gained by further reducing
conduction through the gas. Under these circumstances,
the particletoparticle thermal conductivity sets the lower
limit of the apparent thermal conductivity through a porous
powder under vacuum. (It will be shown later that net
radiant transfer appears negligible for the conditions
encountered in this study).
The following section examines the contribution of
particletoparticle thermal conductivity to the apparent
thermal conductivity of a powder under vacuum.
ParticletoParticle Conductivity of a Porous Powder
Introduction. Conduction through a region of contact
between particles is not easily analyzed except for ideal
geometries. For an idealized geometry one may express
energy transfer between particles of given size, mathema
tically define the contact area, and express the proper
number of contact points. In most real powders, for example,
see Figure 8, page 68, the particles are both irregular in
shape and size and oriented at random. These factors make
it difficult to determine the parameters necessary to
express energy transport by conduction through the powder.
Even with the difficulties of analyzing particleto
particle conductivity mentioned above, and the fact that
there is little in the literature analyzing this topic, it
is useful to develop an approximate mathematical expression
to describe particletoparticle conductivity. With a mathe
mathical expression one may readily identify those variables
that control the behavior of the material under study.
Theoretical Model. For particletoparticle conductivity
to exist there must be an unbroken path connecting the parti
cles and continuing through them. It is through this path
that energy is transferred by solid conduction. A particu
lar problem in analyzing heat transfer through a powder that
consists of irregularly shaped material, such as the material
under consideration in this study, is the difficulty
of mathematically defining the path through which energy
flows. For example, in Figures 8, 9, and 10 (pages 68
and 69) one sees a variety of shapes ranging from per
forated, hollow and solid cylinders to perforated plates,
and irregularly shaped solid pieces. With the variety of
shapes shown, following the path of heat transfer from
particletoparticle appears to be virtually an impossible
task, thus the need for a simplifying model.
In following the path of energy transfer from one
particle to the next, one finds that the region of greatest
resistance occurs at the smallest area, that is, in a region
of contact. If the dimensions of the contact region are
small relative to the bulk particle size, then one may
approximate particletoparticle energy transfer in a
powder by analyzing the resistance due to the contact
region only. This concept is best illustrated by an
example.
Consider a cylinder as a particle body as shown in
Figure 3. Beside this cylinder is a smaller cylinder which
represents the contact region. The two cylinders are in
intimate contact and they are aligned on the same axis.
Each cylinder has a length to diameter ratio of 10:1 and
the size of the two cylinders relative to each other is
100:1. Assuming the conductivity coefficient to be the same
Contact region
(a)
General Particle Shapes In Contact
Particle interface
Contact region
(b)
Idealization Of Particles In Contact
Figure 3
Idealization Of A Contact Region
for both cylinders, one can easily demonstrate that the
resistance to heat flow in the smaller cylinder is 100
times that of the larger cylinder. This indicates that the
contact resistance, as represented by the smaller cylinder,
predominates and that the body resistance of the particles,
as represented by the larger cylinder, is negligible. In
Chapter V it will also be demonstrated that the relative
dimensions of the particle body and contact region are
such that a resistance ratio greater than 100:1 is not
unusual.
Note carefully that a distinction is made between a
contact area and a region of contact. A contact area is
the physical area or interface between two particles. A
contact region is defined to be the contact area along with
the particle materials adjacent to the contact area that
restrict energy transfer between two particles forming the
contact area. See Figure 3. The contact region is viewed
as a continuous path of solid material through which con
duction heat transfer occurs even though an interface
exist between two particles. This assumption is based upon
the existence of sufficient stresses to maintain the two
particles in intimate contact. That is,
S = F/A (21)
where S is the average stress between the particle, F
is the force or weight supported by a particle, and
A is the contact area between two particles. It will
be demonstrated in Chapter 5 that sufficient stress exists
at a contact region to assume intimate, solid contact
between particles.
The contact region of a solid particle can be analyzed
by any suitable coordinate system. For simplicity, cylindri
cal coordinates will be used. Assuming constant conducti
vity, one can express energy conservation in conduction by
32T + 3T 2T +2T q'" 1T
+ rr2 + + + (22)
a27 rar r 2' i27 T Ft (2"T
Since the contact region is small relative to the particle
body dimensions, with the latter varying from 1 im to
100ml in diameter, the particle material conductivity
coefficient, ks, is assumed to be homogeneous and isotropic.
The following assumptions are also made on Equation 22.
1. steadystate,
2. no internal heat generator, and
3. the temperature variations in the r
and 6 directions are negligible.
A steadystate assumption is made since the material
under investigation is intended for use as an insulator
operating under steadystate conditions.
The temperature variations in the r and e directions
are considered negligible since the contact region is small
relative to the particle body.
With the above assumptions, Equation 22 reduces to a
onedimensional equation and heat flow through a contact
region is viewed as quasionedimensional. Thus,
d2T
S= 0 (23)
A method of solution for onedimensional problems in
which q = constant at every crosssection is [27]
d[qxA(x)] = 0 (24)
which gives
QT = qxA(x) = constant (25)
The x denotes heat transfer in the xdirection and
q k dT (26)
x = dx
Introducting Equation 26 into Equation 25, rearranging
and integrating between limits gives
THTL THT
Q H= L H L (27)
(1/k ) fX2 dx/A(x) Re
S X1
where
Re = (1/k ) If dx/A(x) (28)
and represents the conductive resistance through a contact
region. Equation 28 shows that conductive resistance to
heat flow is directly proportional to the path length, dx,
and inversely proportional to the area, A(x).
An electron microscope photograph of a diatomaceous
earth particle, Figures 14 and 15, page 92, shows that
the surface of these particles, where contact occurs, has
rounded or smooth geometric features as shown in Figure
4(a). It would appear that the regions of contact could
be viewed as hemispherical, cylindrical, conical and so
forth. For the purpose of illustration, a hemispherical
contact region will be assumed.
A contact region formed by two hemispherical pro
trusions is shown in Figure 4(a). Note that the area of
contact of the hemispherical protrusions has been flattened,
as illustrated by the solid line, due to forces imposed
on the particles. An equivalent cylindrical heat flow
path has been superimposed on the hemispherical contacts.
This is to illustrate that the hemispherical contact region
is still considered quasionedimensional. The use of a
hemispherical contact region serves these two purposes
1. it closely resembles the actual contact region
of a particle, and
2. it facilitates the calculations of the contact
area with the use of the Hertz formula [28].
The symbols used in analyzing the resistance to heat
flow through a contact region are shown in Figure 4(b).
Equation 28, written in terms shown in Figure 4(b) is,
Re = x2 dx (28)
s x, iy'4x)
The following variable transformations are made.
y = R sin a
x = R cos 6
dx = R sin 8 de,
w^
y
D x
c
Contact particle body
Contact
Region
(a)
Parameters Used in Analyzing Heat Flow Through A
Hemispheric Contact Region
R
y
y
kRh x
h I
(b)
Two Hemispherical Projections Forming A Contact Region
Figure 4
Two Particles Forming A Contact Region
Substituting into Equation 28 and simplifying,
1 @2 _
Re = i R 72 n de (29)
k.rR e1 sine
Evaluating,
R k/1 0 /2(
Re = In tan l /2 (30)
s 2
and
I @*
Re =  In tan (31)
The term 6* represents the angle at which a mimimum contact
radius, y*, occurs for a given particle. The contact radius
y* is defined below. Since the contact angle 0* is less
than one degree as calculated using the Hertz formula (see
Equation 35), then tan e*/2 can be approximated as
tan y*/2(Rh) (32)
Now, Equation 31 is
Re = 1R n R (33)
Since R >> h, the conductive resistance can be written as
1 2R
Re= 1R In 2 (34)
The value of y* represents the radius of the contact
area between two hemispheres in contact and it is calculated
from the Hertz formula [28] by
y = (3F [(1p)/Ej + (1 p)/E2 1/335
y ]8 (35)
D1 D2
where
F = load per contact
D1 = diameter of particle one
D2 = diameter of particle two
P = Poisson's ratio
E = Young's modulus
In Equation 35, DI and D2 are diameters of spheres.
In this text, the values of D1 and D2 represent the dia
meters of two hemispherical protrusions that form a contact
point. Hence, the parameters Di and D2 will be redefined
as D and DC2, respectively. Generally, these two
diameters are not equal but throughout this text the values
of D and D are assumed to be equal and will therefore
be written as D .
Equation 34 represents the conductive resistance to
heat flow through one contact region. Combining Equation
34 with Equation 27 and rearranging results in an equation
for heat flow through one contact region. If there are
Nh particles in series and NA particles in parallel, then
the rate of heat flow through one foot cube of material
can be written
n ks D NA (TH TC)
QT = 2Nh ln(Dc/*) (36)
where
n = number of contact points per particle
(heat flow in) for particle in series
Dc = estimated contact region diameter
NA = number of particles per unit of area
(parallel heat flow)
Nh = number of particles in a unit of
height (series heat flow)
The equivalent contact region diameter, Dc, is the
estimated diameter of a hemispherical protrusion on a parti
cle. This was shown earlier in Figure 4(a). Also note in
Equation 36 that radius, R, has been converted into terms
of diameter, D .
For a unit temperature difference across a given depth
of insulating material, Equation 36 allows one to define
an effective coefficient of particletoparticle thermal
conductivity as
na ks Dc NA
s 2Nh ln(D/y*) (37)
The above equations were devised on the basis of an
array of uniform particles. However, a material such as
diatomaceous earth consists of particles that are nonuni
form in size and shape and are oriented at random. There
fore, an empirically determined correction factor, C, must
be incorporated into Equation 37. The correction factor
will take into account
1. bridging of particle columns,
2. discontinuities in the heat flow path, such
as terminated columns,
3. nonuniformity of particle size and shape, and
4. changes in material properties, such as the
addition of a foreign substance in a pure
base material.
In view of the above, Equation 37 is now
C T k n D NA
Ks 2Nhln(D/y*) (37)
A sample evaluation of the above equation will be discussed
in Chapter V.
Equation 37 was derived for application in the contact
region connecting two irregularly shaped particles. As a
material more closely approximates a sphere, Dc approaches
the diameter of the sphere. In this case, NA = 1/Dc2 and
Nh = 1/Dc. Equation 37 can then be written as
C'k
I= ln(D /* (38)
where
C' = Crn/2
The particletoparticle conductivity, KS, is seen to
depend primarily on the particle conductivity, ks. The
natural log term is essentially a constant for typical
variations of the material parameters encountered in this
study; that is, for a given powder sample.
The value of C' is a constant for a given material.
Incorporated into C' is the value of n, the number of
contact regions leading into a particle. The value
of Ks is seen to vary directly with n. Thus, one should
choose an insulating powder so that n is minimized.
Radiant Transfer in a Porous Medium
Introduction. The ratio of radianttototal heat
transfer in a porous medium has been modeled extensively.
However, there is little explicit experimental data,
identifying radiation separately, with which to judge
the effectiveness of these models. In a porous medium
in which the particles are opaque and small (<102 ft)
and in which the temperatures are less than 2700F, there
is agreement in the literature that radiant compared to
total heat flow is negligible [29,30].
Radiant heat transfer in a porous powder. In view of
the above paragraph, net radiant transfer through powders
for conditions encountered in this study was considered
to be negligible. The powders used were on the order of
104 ft in diameter and upper temperature limit was 10000F.
As will be demonstrated in Chapter V, net radiant transfer
through diatomaceous earth powder under typical operating
conditions accounted for approximately 1%, on a theoretical
basis, of total energy transfer. Even though this is
Contact regions for energy transport out of a particle are
considered as a contact region leading into an adjacent
particle.
considered to be negligible, a theoretical analysis of
radiant heat transfer through powders will be presented
for completeness.
The theoretical model presented is similar to those
of Argo and Smith [29] and Schotte [30]. To model radiant
transport the above authors assumed spherical particles
and considered radiant transfer through parallel planes
located on each side of a particle. See Figure 5.
 ) ,
z d ds
T I T2 1 6 z
I x x x
I 6 T1 > T2
Figure 5
Model for Derivation of Radiant Contribution to
Thermal Conductivity through a Porous Medium
The model considers radiant transfer through the
voids between particles and radiant transfer from surface
to surface in series with the solid particle conduction.
The model also assumes that the particles were opaque
and that the particle size was large compared to the
wavelength of radiation. Considering the last restriction,
the particle size used in this study was 15 125 pm.
For a temperature range of 250 10000F, Wien's displace
ment law gives an average wavelength of 7.35 3.57 pm,
respectively. Thus, the condition is well satisfied
with large particles at all conditions, reasonably
well satisfied with medium size particles at higher
temperatures or small particles at lower temperatures,
but is doubtful for the smallest particles at the lowest
temperatures.
One may view radiant energy as a packet of energy,
or photon, instead of a wave. Of the many photons that
participate in radiation passing through a porous powder
most will strike the surfaces that form a void many times
before they could escape. As a result the probability of
absorption is very high compared to the probability of
escape. Hence, the effective absorptivity and emmissivity
may be approximated as unity. By viewing radiant transfer
in terms of photons, it is easy to visualize radiant energy
transfer as meeting great resistance in flowing through a
porous powder, especially for conditions previously
described.
The results of the analysis by Schotte [30] give an
overall radiative conductivity coefficient of
K + krv (29)
F + F
s rv
where 6 is the void fraction, ks is the solid particle
conductivity, and krv is a particle radiation conductivity
coefficient. The latter variable represents radiant heat
transfer between a particle and its neighbor. This
value will be defined shortly.
The righthand side of Equation 39 consists of two
terms. The first term represents radiant heat transfer to
a particle in series with conduction through the particle.
The second term accounts for radiation through the void
space adjacent to the particle.
The radiative coefficient, krv, which represents
radiative transfer between a particle and its neighbor,
is given by Schotte [30] as,+
k = 4eacT3 (40)
where E is the emissivity of the particles, S is the
average void dimension, is the StefanBoltzmann con
stant, and T is absolute temperature. The temperature
used in this equation, T, is the average temperature of
the particles. Argo and Smith [29] have pointed out that
+This equation is presented by Schotte [30] as krv = 0.692d
T3/108. This equation and the one used above are the same.
48
one investigator [31] used a bulk mean temperature in
evaluating Equation 36 and found it in agreement with more
elaborate methods for determining radiative transfer
through powders.
In summary, the above paragraphs illustrate the
role of radiant to total energy transport through a porous
powder. The parameters that apparently control radiant
heat flow through powders are the particle size, opacity
of the material, particle conductivity, void dimension,
and boundary temperatures.
CHAPTER IV
EQUIPMENT DESIGN
AND MATERIAL PREPARATION
Introduction
The intent of the experimental procedure was two
fold. First, apparent thermal conductivities for various
particle sizes were needed to establish the required
insulation thicknesses for a high temperature storage
system. This required the design and construction of
a conductivity test apparatus capable of measuring con
ductivity of a powder material with a hot face temperature
up to 1000F and for vacuum levels down to 100 Vm Hg.
The second objective of the experimental procedure
was to design a prototype high temperature thermal energy
storage system using a porous powder under a moderate
vacuum (P 1 mm Hg) as the insulating material. The
thermal energy storage vessel was designed to use an
organic oil (corn, cottonseed, peanut, or soybean) as the
storage medium.
Conductivity Test Apparatus
The conductivity test apparatus is shown in Figures
6(a) and 6(b). The central core of the test cell consists
electrical wiring port
wax cup thermocouple port
  OD pipe
aluminum plug,
four threads
per inch
' steel wool (grade 000)
(a)
Sectional View Of Conductivity Test Apparatus
Figure 6
Conductivity Test Apparatus
Top View
Centering Plate
Figure 6(a) (Cont'd.)
52
(b)
Figure 6
Photograph of Conductivity Test Apparatus
of two guard heaters and a center test section heater.
The nichrome heater wires were wound on a threaded
aluminum core which served to thermally dampen small devia
tions in the power supply. The aluminum core was wrapped
with a ceramic paper and inserted into a copper sleeve.
The copper sleeve helped to insure a uniformly heated
surface. Chromelalumel thermocouples were welded to the
copper sleeve so that a temperature difference between
the guard heaters and the center test section heater
could be read. In this manner, the guard heaters could
be adjusted to the same temperature (+ 1/2F) as the
center test section heater, insuring radial heat flow in
that section. By measuring the electrical energy dissi
pated in the central test section, and by measuring the
surface temperatures at the boundaries of the powder
insulation, the apparent thermal conductivity was calcula
ted. For a cylindrical geometry,
2Ka (THTC)L
Qelec In(r /ri) (41)
Solving for conductivity,
Qelec ln(r /ri)
Ka ((T42)
a 2i (THT )L
For this apparatus,
Qelec = rate of electrical energy dissipation
L = 1 ft test section length
r = 0.163 ft outside radius of insulating
o material
ri = 0.089 ft inside radius of insulating
material
TC = average cold surface temperature of insu
lation at boundary
TH = average hot surface temperature of insu
lation at boundary
The electrical energy dissipated was calculated by measuring
the voltage and current of the center test section heater
wires. Thus,
Qelec = EI cos p (43)
where,
cos = 1 (assuming the power factor to be unity
for resistance heating)
E = potential
I = current
The value of TC, the cold wall temperature, was measured
on the outside surface of the conductivity test cell.
The calculated estimate of the temperature drop across
the copper wall was negligible (< 0.002F). The heat
losses through the wires, thermocouples,and electrical
conductors had a negligible effect on the test section
energy balance, since the temperature difference between
that section and the guard heaters was held to + 1/2F
of each other.
Pads of fine steel wool were placed at the top and
bottom of the conductivity test cell to facilitate evacua
ting the fine powder without entrainment. The vacuum
ports at the top and bottom of the test cell permitted
a faster evacuation time than a single port.
A vacuum port was placed above the upper guard
heater and below the lower guard heater so that its
presence would not interfere with the center test section.
Several different types of vacuum gauges were used during
the course of testing.
The first was a thermocouple gauge with a range of one
atmosphere to 1 pm Hg. However, it was difficult to
maintain calibration due to powders infiltrating its
sensing element and, consequently, this gauge was not used.
A swivel type McLeod gauge with a range of 1 pm 5 mm Hg
was the second type used. The gauge worked well after a
small piece of fine steel wool (0000) was put in line with
it to prevent powders from reaching the mercury.
Alignment of the central core of the test apparatus
was maintained by a centering plate as shown in Figure 6(a).
Wire feeds were created by packing asbestos string
around the wires. The wires were sealed as they passed
into the test apparatus by melting high temperature wax
around them. The wax had to be reheated periodically
due to differential expansion between the wire and
wax.
The Design of a High Temperature Thermal Energy Storage
System
A high temperature thermal energy storage system was
designed according to the following criteria.
1. Heat loss per day not to exceel 10 Btu/hrft2.
2. The inner tank has a minimum high pressure
rating of 150 psi at 7000F.
3. Both the inner and outer tank have low pressure
rating at 1.32 x 105 psi (100 pm Hg).
4. The inner tank is capable of holding organic
oils.
5. The vessel construction is simple and durable.
6. The vessel requires little maintenance.
7. The inner storage vessel is sized to be compati
ble with a 6' x 8' concentrating collector.
8. The design potential is intended for mass pro
duction at an acceptable cost.
The tank design is shown in Figure 7.
The inner tank was made from a 14 gauge galvanized
steel, 42 gallon water tank. Note that one end of the
water tank was changed from concave to convex to insure
stability at elevated pressures.
The outer tank was constructed from a 14 gauge
galvanized steel 200 gallon water tank. The cylindrical
steel cable
and turnbuckle
2" flange
Figure 7
Prototype High Temperature Thermal Energy Storage System
5/16" bolts, typical
Figure 7(b) (Cont'd)
End View of Tank Showing Internal
Cable Support and Flange Bolt Pattern
1"
Sx 2" x 2"
angle iron
1"
 x 1" x 1"
angle iron
Figure 7Cc) (Cont'd)
External Support for Inner Tank
(d)
Figure 7
Photograph of Prototype High Temperature
Thermal Energy Storage System
portion of the tank was shortened and the concave end was
changed to convex so that the insulation thickness at
the ends of the tanks would be equal to the thickness
between the cylindrical portions of the tank.
The inner tank was suspended by two 1/16" diameter
stainless steel cables. These cables were selected for
their high breaking strength (500 lb. at room temperature)
and low thermal conductivity (10 Btu/hrftoF at room tempera
ture). The two cables had sufficient strength at elevated
temperatures (1000F) to afford a safety factor of two (2).
The inlet and outlet pipes were designed with a
Ushaped trap which served as an expansion loop and a
heat trap. The inlet pipe at the top had the loop placed
close to the inner tank so that it would prevent volatile
vapors from the organic oil from entering the outlet pipe
when the pipe was not in use. The loop on the bottom
outlet pipe was placed at the opposite end of its point
of attachment and the outlet pipe was sloped downward so
that heat conducted through the metal would induce convective
flow in the opposite direction, thereby helping to minimize
heat flow out of the inner tank.
The rings on the outer tank were needed to meet
safety recommendations on externally pressurized tank
design [32].
The inner tank was designed to contain organic oils.
Copper metal cannot be used with organic oils since these
oils are acidic in nature and react chemically with copper.
The long term effect of high temperature organic oils on
steel is presently unknown. At low temperatures, steel
containers have been used successfully to contain organic
oils.
The minimum storage vessel size was estimated for
a 6 ft x 8 ft (aperture area) concentrating collector
as follows.
Heat delivered to Heat stored in oil Heat losses (44)
storage system and its container from piping
Assuming the pipe losses equal 10% of the energy stored
in the oil and its container, then
Hin = 1.10 H (45)
in sto
and
IsAal T = l.lpVC AT (46)
where Is is solar insolation, Aa is the collector aperture
area, n is the collector average thermal efficiency, p is
density of organic oil (cottonseed), V is the oil tank
volume, C is the average specific heat of the oil, and
AT is the temperature rise of the oil.
Solving for volume
IsAa nt
V =I C T (47)
p
Assuming the following values,
n = 50%
Is = 300 Btu/hrft2
T = 8 hr/day
p = 46.5 Ibm/ft3 @ 600F (estimated)*
C = 0.66 Btu/lbmF @ 6000F (estimated)*
AT = 400F (200 6000F)
A = 48 ft2
a
and solving for V in Equation 47 yields
V = 4.27 ft3 (maximum)
A 42 gallon tank holds 5.62 ft3 and afforded an excess
capacity to serve as an expansion chamber.
The outer tank size was chosen from commercially
available tanks. The most convenient size was a 220
gallon galvanized water tank. The outer tank diameter
was 2.5 ft and the inner tank outside diameter was 1.33
ft. Thus, there was a 0.585 ft space for insulation. The
ends of the tank were designed to leave an equivalent 0.585
ft space.
A preliminary test on diatomaceous earth powder indi
cated that an apparent conductivity value of Ka = 0.01
aL
Cottonseed oil data [3]
Btu/hrftF could be obtained at a hot surface temperature
of 600F, a cold surface temperature of 70F, and at 1 mm
Hg pressure. A 0.585 ft thick layer of insulating powder
for these conditions would hold the daily energy loss
of the storage tank to approximately 9% per day of the
initial energy stored. This approximation is based on
changes in temperature and average values of C over the
interval of temperature change.
Diatomaceous Earth Properties
Diatomaceous earth is the fossilized remains of micro
scopic marine algae. It consists of approximately 98%
Si02 (silica). The diatomaceous earth used in this study
was purchased commercially. Its intended purpose was for
use in swimming pool filters. This material has been
used industrially as high temperature, loose fill, insula
tion and in compressed brick form as a furnace lining.
The loose fill material has an upper temperature limit of
1600F and the brick has an upper limit of 2500F [19].
The apparent reason for the limitation of the loose fill
material is that the powder particles begin to soften
at that temperature. This would increase the pointto
point solid conductivity and also cause loss of volume of
the material due to settling.
In fine powder insulation it is useful to determine
the effect of particle size on apparent thermal conductivity.
One difficulty of determining particle size is that fine
powders tend to clump. An explanation for this phenomenon
will be discussed later. However, this made it difficult
to separate these particles in a sieve and shaker. Grind
ing these particles in a ball and mill reduced their size
but did not make it easier to separate them into various
particle sizes.
It was found that the addition of small amount of
fine carbon black to the diatomaceous earth, either during
the separation process or the grinding process, permitted
the powder to separate quickly in a sieve and shaker. The
reason for this is that the fine carbon particles act like
a lubricant. Carbon particles are shown in Figure 11.
Note the puffy appearance of the agglomerated carbon
particles. This suggests that the large particles may be
comprised of many smaller carbon particles.
A possible explanation of how carbon acts as a lubri
cant may derive from the fact that carbon is electrically
conductive. When the diatomaceous earth particles are
vibrated, the particles become charged with static electri
city. Hence, if particles have opposing charges, they
attract each other. This prevents the particles from pass
ing through the sieve and, consequently, no separation of
particles occur. When carbon powder is added to diato
maceous earth, it short circuits the static charge between
the particles and this permits the particles to pass through
the sieve.
After the diatomaceous earth was separated into its
various particle sizes, its geometry remained essentially
the same. Figure 8 shows particles trapped in Tyler
Sieve #500. Figure 9 shows particles trapped by Tyler
Sieve #170. Both of these photographs vividly display
the irregular characteristics of these powders.
The various particle size distributions obtained in
a sieve and shaker are given in Table 3. The Tyler sieve
number corresponds to screen weaves of the following opening
size
Tyler Sieve No. 120 170 200 270 325 500 <500
Opening (microns) 125 90 75 53 45 25 < 25
The last category, <500, was simply a catch pan for particles
falling through the #500 screen.
The values shown in Table 3 are representative only
of the particle sizes in the original diatomaceous earth.
These values changed somewhat with each test.
The purpose of generating data for Table 3 was to
find the various nominal particle sizes with their corres
ponding quantities. The variables influencing the separa
tion process were the length of time the shaker was operated
and the amount of carbon present. The objective was to
use as little carbon as possible. The general results
of Table 3 follow.
TABLE 3
Diatomaceous Earth Particle Size and Distribution
Typical Results
Diatom.
earth
(grams)
174.50
175.70
190.50
188.75
172.00
177.75
179.50
172.75
179.00
178.50
171.75
200.25
326.00
334.50
Carbon
(grams)
0
1.0
1.0
1.5
1.5
2.0
2.0
3.0
4.0
4.0
4.0
4.0
5.0
5.0
Tyler Sieve
120
62.25
31.55
35.50
40.25
25.00
30.75
31.50
29.50
35.50
31.75
32.75
32.25
0
18.00
170
81.75
73.00
11.25
12.50
13.75
11.25
10.00
10.75
38.75
10.50
9.25
11.00
0
52.00
200
28.25
45.00
42.00
24.50
29 50
23.50
19.50
18.50
63.25
38.00
31.00
25.50
21.00
270
*
15.25
40.00
47.50
47.25
41.50
37.50
148.00
*
148.00
Run #
1
2
3
4
5
6
7
8
9
10
11
12
13+
14'
Number
325
2.00
8.25
35.50
32.00
26.50
34.75
38.50
40.75
36.50
52.75
27.25
29.25
76.00
500
0.25
2.75
49.50
34.50
30.25
29.50
33.75
33.00
9.00
49.50
49.75
35.25
81.00
Shaker
time
(min.)
30
10
15
10
10
10
10
10
8
16
24
60
10
10
"This size sieve not used during this test.
tin ball mill for 30 minutes, carbon added at beginning.
'Carbon added after powder pulverized for 1 hour.
133.00 111.50
20.00 5.00
<500
0
0.90
17.75
6.50
1.00
2.75
6.75
5.25
0
0
25.75
69.00
5.00

Figure 8
Electron Microscope Photograph of Diatomaceous
EarthCarbon (1% wt) Mixture, Particles from
Tyler Sieve #500 (x 1,000)
..
Figure 9
Electron Microscope Photograph of Diatomaceous
EarthCarbon (1% wt) Mixture, Particles from
Tyler Sieve #170 (x 1,000)
Figure 10
Electron Microscope Photograph of Diatomaceous Earth
(as purchased), x 2,000
Figure 11
Electron Microscope Photograph of Carbon Black,
Monarch 500 (x i0,000)
1. Run #1 illustrates the difficulty of
separating raw* diatomaceous earth.
Even after 30 minutes, very few particles
were separated.
2. Run #2 illustrates that a large percentage
of the particles in the powder are indeed
small.
3. The general trend is that as the amount of
carbon increased, the greater the percentage
of small particles and the faster they
separated.
4. A given type of powder with the same amount
of carbon and the same shake time shows a
general consistency in particle distribution,
but some particles of a given size may vary
appreciably. Compare Run #4 and Run #5.
5. Carbon added prior to pulverzing diatoma
ceous earth permits the separation of
fine particles in less time than when it
was added after pulverizing.
Representative data on density of the various particle
sizes are given in Table 4. Note the discrepancy between
the measured void fractions and the calculated void
fractions. This could be due to air being trapped in and
among the particles and not being displaced during the
void measurement process. Another explanation, in view of
Figures 8 and 9, is that the particles do not separate well
and that the mixture of large and small particles causes the
void fraction to remain relatively constant.
Raw diatomaceous earth refers to the material as purchased.
Void fraction is the ratio of the void space in a con
tainer to the total volume of the container.
TABLE 4
Particle Density and Void Fraction of Diatomaceous Earth
Particle Density (Ibm/ft3)t Void Fraction Void Fraction
Size (measured) (measured) (calculated)
Tyler Sieve #120 15.53 0.89
#170 17.57 0.71 0.88
#200 18.98 0.76 0.87
#270 19.70 0.71 0.86
S #325 19.92 0.77 0.86
#500 20.08 0.76 0.86
less than #500 (catch pan) 22.67 0.75 0.84
raw diatomaceous earth 17.35 0.83 0.88
carbon black 11.77
powder for prototype tank 32.84 0.70 0.77
solid diatomaceous* earth 145.0
(SiO2)
* Calculated from molecular weight of Si02.
1% carbon black in powders.
The method used to measure the void fraction and
density was to fill carefully a graduated cylinder of
known weight with the powder and gently tap the cylinder
to settle the particles. The cylinder with the particles
was then weighted. The cylinder with the powder was then
filled with water to the original level of the powders,
and it was then weighed again. With a knowledge of the
volume and weight of the powders, a density was then cal
culated. The void fraction was found by deducing the volume
of water required to fill the graduated cylinder to the
original volume of powder. The ratio of the latter two
quantities is the void fraction.
CHAPTER V
EXPERIMENTAL PROCEDURE AND RESULTS
Conductivity Test Equipment
The conductivity test equipment arrangement is shown
in Figures 12 and 13. Power was supplied through rheostats
connected to a standard 120 volt AC outlet. Variations in
the AC outlet due to daytime peaks were overcome by obtain
ing a longterm average for the data over a period of
several days to more than a week for each datum point.
These data were recorded manually.
The test section heaters were monitored for current
and voltage. This was accomplished by placing an ammeter on
one lead of the heater wire. Voltage was measured on the
heater side of the power supply. The guard heater power
was monitored only for interest in their power levels
relative to the central heater. The guard heater power
levels were adjusted to give a temperature within +0.5F
of the central heater. This power level was usually slightly
more than the power level of the central heater.
The temperature of the four thermocouples inside the
test cell were measured with a digital voltohm meter.
An ice bath was used as a temperature reference for the
Legend
A Ammeter
Rheo. rheostat
V voltmeter
leak valve
Figure 12
Conductivity Test Equipment Arrangement
Figure 13
Conductivity Test Equipment
chromelalumel (Type K) thermocouples. The temperature
levels were recorded manually.
A mercuryfilled McLeod gauge with a range of 1 pm 5
mm Hg was used to measure the vacuum level in the conducti
vity test cell for data presented in this study. For the
preliminary testing period a thermocouple vacuum gauge
was used. It was found to lose calibration often and a
standard gauge was needed for recalibration. It was found
easier to use the standard gauge (McLeod) directly in the
experiment.
A common vacuum pump with a blankoff pressure of 15
pm Hg was used for evacuating the system. A variable
leak valve was placed at the bottom of the test cell to
regulate pressure.
The objective of this experiment was to find the
change in thermal conductivity due to changes in
1. particle size,
2. vacuum level, and
3. temperature levels.
Thermal Conductivity of Powder Insulation Under Vacuum
For the determination of the thermal conductivity of
a powder, the thermal conductivity apparatus, as shown in
Figure 6, was filled with the powder to be tested. The
powders were heated prior to evacuation so that the elevated
temperatures would aid moisture removal from the powder
particles.
Once the system was thoroughly heated, it was sealed
and evacuation was initiated. The heater elements were
adjusted so that the main heater and guard heater thermo
couples read within 1/2F of each other. This process
usually required several days. After a uniform temperature
was reached, the system was generally operated under this
condition for at least one day to insure that steadystate
had been achieved. This additional time was necessary
to minimize the effect of the daily changes in the AC
power supply and the daily temperature swing of the test
room.
During the testing procedure, data were taken manually,
usually on hourly intervals, for the three heater tempera
tures (6 points), outside surface of test apparatus, test
heater potential and current, and room temperature.
Data and Results from Thermal Conductivity Test
The experimental values obtained from the thermal
conductivity apparatus are shown in Table 5. The four
measurements were
1. pressure level, measured in um Hg,
2. the hot surface temperature, TH, bounding the
powder material adjacent to the main heater,
3. the cold surface temperature, TC, taken on
the outside surface of the test apparatus,
4. power dissipated from the central test section
heater (potential x current).
From the measured data the apparent thermal conductivity
coefficient, Ka, was calculated with the use of Equation
42.
Discussion of Results of Apparent Conductivity Test
Range of values. The range of apparent conductivity
values, as seen in Table 5, vary as follows
Highest value: Ka = 0.0648 Btu/hrftF
@ TH = 631F, P = 760 mm Hg
Lowest value: K = 0.0029 Btu/hrftF
a
@ TH = 3400F, P = 0.6 mm Hg
To put these values in perspective, air has the following
experimental conductivity values [33]
Air at 631F: K = 0.0272 Btu/hrftF
@ P = 760 mm Hg
Air at 340F: K = 0.0212 Btu/hr ftF
@ P = 760 mm Hg
From this, the highest test data value, taken at 760 mm Hg
pressure, is 2.4 times higher than the conductivity of air,
but the lowest value, with a pressure of 0.6 mm Hg, is 7.3
times less than air at 760 mm Hg. This is significant since
the conductivity of air is essentially independent of
pressure except under the low pressure conditions (Kn > 1)
TABLE 5
Average Thermal Conductivity Values
For Evacuated Diatomaceous Earth*
Powder in Tyler Sieve #500 (loose powder)
TC
(OF)
Ka
(Btu/hrftoF)
.00364
.00648
.00750
.00483
.00722
.00790
.00444
.00407
.00417
.06480
.04600
Powder in Tyler
to 45 psi)
Sieve #500 (powders physically compressed
600
725
650
725
775
800
850
900
1000
1000
.00290
.00380
.00540
.00910
.00338
.00327
.00910
.00930
.01230
.01130
Powders contain approximately 1% carbon by weight.
Pressure
(pm Hg)
TH
(OF)
500
600
800
850
1000
1000
1000
1000
1000
760,000
760,000
TABLE 5, Continued
Powder in #325 pan
Pressure
(vm Hg)
(loose powder)
TH
CF)
22 1400 740
23 1800 462
Powder in #170 pan (loose pow
24 175 456
25 200 699
26 300 588
27 800 647
28 1000 344
29 1000 531
30 1000 710
31 3800 557
Raw Diatomaceous Earth (No ca:
32 600 441
33 1000 725
34 2000 545
35 8000 386
36 8000 583
Pulverized Diatomaceous Earth
37 1000 446
38 8000 511
39 8000 715
Air Only
40 760,000 603
Fiberglass Cloth and Aluminum
41 400 720
TC
(CF)
100
90
der)
81
88
108
110
80
85
98
95
rbon added to lo
93
110
96
110
117
(loose powder)
99
101
116
235
Foil 5 layers
110
Ka
(Btu/hrft F)
.00580
.00475
.00635
.00656
.00590
.00723
.00972
.01198
.01343
.01714
ose powder)
.00920
.01200
.00980
.02900
.02740
.00960
.01560
.01448
.07800
.01440
discussed earlier. This indicates that a powder under
vacuum is effective insulation at elevated temperatures.
The particular area of interest in this study was the
conductivity values of the powders near 1 mm Hg, since
this vacuum level was within the range of common mechanical
vacuum pumps. The conductivities of several of the more
thoroughly investigated powders are shown in Figure 16
(page 96). This graph shows the approximate limits and
the general trend of the apparent conductivities associated
with the large and small grain powders tested.
Radiative transfer. The net contribution due to
radiation through a powder insulation, in the temperature
range of interest in this study, appears to be negligible.
Recall Equations 39 and 40,
Kr 1 36 + v (9)
Kr 1/ks/ + /kr +v 6 kr
kr = 4e 5 T3 (40)
Using typical values in these equations, the resulting
coefficient of radiative conductivity is small compared
to other modes of energy transfer. For example, using the
following values for particles in Tyler sieve #500
5 = 0.76 (Table 4)
E = 1.0 (assumed value)
d = 0.5D = 5.75 x 10 ft.
k = 0.76 Btu/hrfto (pyrex glass)
and calculating for radiant transfer for powder in a
plane adjacent to a hot surface boundary of 1000F
(14600R) and for a plane at 250aR gives a range of
K = 0.001 to 0.0001 Btu/hrftF
r
Comparing the above values to a typical apparent conductivi
ties of Ka = 0.015 and 0.005 Btu/hrftF, shows that radiation
accounts for 7% and 2% of the totals, respectively. These
values represent a worst case for both temperatures since
they were calculated for radiant transfer near the hot
surface boundary. For powder near a cold face boundary at
TC = 900F (550R), then Kr represents about 1% compared
to an average value of K = 0.01 Btu/hrftOF. Thus, the
a
net contribution of radiant transfer through a powder for
the temperature limits of this study appears to be negligible.
Effective gas conductivity. Note that Equation 20,
when rearranged gives
(9 5) 2R (1 + 52 AN _1
K u 1 N 1
g 4= 41) MT p RuT
Evaluating with the following parameters for air alone in
a container of one foot cubed,
P = 760,000 m Hg
T = 4920R (32F)
M = 28.97 lbm/lb mole
a = 1.1975 x 109 ft
d = 1.0 ft (since 3 >> A)
y = 1.39
Cv = 0.18 Btu/lbmF
gives a value of K of
g
K = 0.014 Btu/hrftF
and this is in close agreement with published experimental
data [34]. However, if Equation 12 is evaluated at higher
temperatures, around 600F, its theoretical values are
about 18% below the above cited experimental value.
This discrepancy arises because of the departure of real
gases from the assumptions used in deriving Equation 6.
Table 6 has been prepared to show a relative comparison
for air and argon when used under noncontinuum conditions
and at typical mean temperatures encountered in this
study. Table 6 suggests that the smaller particle size
inhibits gaseous conductivity and that at higher tempera
TABLE 6
Theoretical Thermal Conductivity of Air and Argon
For NonContinuum Conditions at Tb = 860R
Particle Size
Gas
d = 0.5 D (ft)
M (Ib /lb mole)
o (ft)
C (Btu/lbmoR)
Tyler Sieve #170
Air
1.625 x 10 4
28.85
1.1975 x 109
0.18
Tyler Sieve #170
Argon
1.625 x 104
39.95
1.25 x 109
0.0748
Tyler Sieve #500
Air
5.75 x 10'
28.85
1.1975 x 109
0.18
Tyler Sieve #500
Argon
5.75 x 10
39.95
1.25 x 109
0.0748
K
a
(Btu/hrftF)
.00012
.00111
.00211
.00381
.00522
.00640
.00740
.01078
.01398
.01642
.01701
.01923
.01955
.01979
.01983
K
a
(Btu/hrftF)
.0000753
.000712
.00134
.00241
.00328
.00399
.00460
.00661
.00846
.00983
.01016
.01138
.01156
.01169
.01171
K
a
(Btu/hrft F)
.000042
.00041
.00080
.00154
.00222
.00286
.00345
.00588
.00907
.01246
.01346
.01815
.01897
.01963
.01975
K
a
(Btu/hrftoF)
.0000268
.000262
.000513
.000983
.143
.00181
.00218
.00368
.00561
.00759
.00816
.01080
.01124
.01167
.01167
Pressure
(pm Hg)
10
100
200
400
600
800
1,000
2,000
4,000
8,000
10,000
50,000
100,000
380,000
760,000
tures, argon performs significantly better than air,
especially in the pressure range of interest (0.50 pm to
10 mm Hg). Carbon dioxide has a lower thermal conductivity
than air, but its values are not as low as those of argon.
In checking these values with Equation 12, keep in mind
that the theoretical conductivites at ambient pressures
tend to fall as much as 20% below experimental values, as
the temperature is raised from 32 to 6000F. It is not
apparent if this deviation continues at reduced pressures.
This nominal diameters used in evaluating the powder
particles were taken as the average of the screen openings
below and above the trapped powder. For example, if powders
were trapped between Tyler Sieve #500, screen opening
25 pm, but passed through Tyler Sieve #325, screen opening
45 im, then the nominal particle size was taken as the
average of the two values, or 35 pm. It is well to point
out that the nominal diameter is used for convenience in
discussing the particles.
Average void diameters of 50% of the nominal diameter
were assumed in view of Figures 8 and 9 and from observation
of diatomaceous earth particles under an optical microscope.
At the onset of this study it was assumed that powders
separated by sieves would be of fairly uniform size.
However, Figures 8 and 9 indicate that the nominal size, as
defined above, is larger than many particles found in the
powders.
