Title Page
 Table of Contents
 List of Tables
 List of Figures
 List of symbols
 Literature review
 Theoretical considerations of powder...
 Equipment design and material...
 Experimental procedure and...
 Conclusions and recommendation...
 Appendix A: Storage vessel test...
 Appendix B: Convection heat transfer...
 Appendix C: Uncertainty in the...
 Biographical sketch

Title: Investigation of a low heat loss high temperature thermal energy storage system
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00099226/00001
 Material Information
Title: Investigation of a low heat loss high temperature thermal energy storage system
Physical Description: xv, 144 leaves : ill. ; 28 cm.
Language: English
Creator: Cope, Norman Alan, 1947-
Copyright Date: 1982
Subject: Heat storage   ( lcsh )
Heat storage devices   ( lcsh )
Mechanical Engineering thesis Ph. D
Dissertations, Academic -- Mechanical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis (Ph. D.)--University of Florida, 1982.
Bibliography: Bibliography: leaves 140-143.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Norman Alan Cope.
 Record Information
Bibliographic ID: UF00099226
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000317574
oclc - 08835194
notis - ABU4399


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Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
    List of Tables
        Page vii
    List of Figures
        Page viii
        Page ix
    List of symbols
        Page x
        Page xi
        Page xii
        Page xiii
        Page xiv
        Page xv
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
    Literature review
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
    Theoretical considerations of powder insulation
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
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        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
    Equipment design and material preparation
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
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        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
    Experimental procedure and results
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
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        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
    Conclusions and recommendations
        Page 118
        Page 119
        Page 120
    Appendix A: Storage vessel test data
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
    Appendix B: Convection heat transfer analysis for a porous powder
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
    Appendix C: Uncertainty in the measurement of Ka
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
        Page 142
        Page 143
    Biographical sketch
        Page 144
        Page 145
        Page 146
        Page 147
Full Text







Copyright 1981


Norman Alan Cope


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.


ACKNOWLEDGEMENTS ..............................
LIST OF TABLES ................................
LIST OF FIGURES ...............................
LIST OF SYMBOLS ...............................
ABSTRACT ......................................

I INTRODUCTION ...............................

II LITERATURE REVIEW ..........................
INSULATION .................................
Introduction ...............................

Theoretical Model ..........................
Thermal Conductivity of a Perfect Gas ......
Thermal Conductivity of a Non-Continuum
Gas ........................................
Particle-to-Particle Conductivity of a
Porous Powder ..............................

Radiant Transfer in a Porous Medium ........

Introduction ...............................

Conductivity Test Apparatus ................
The Design of High Temperature Thermal
Energy Storage System ......................
Diatomaceous Earth Properties ..............


i I











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
Particle-to-particle 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


FOR A POROUS POWDER ................ 130

OF Ka .................... ........ 135

BIBLIOGRAPHY .................. ............... 140


EXTENDED BIBLIOGRAPHY ........................... 143

BIOGRAPHICAL SKETCH ............................. 144


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 Non-Continuum Conditions at
Tb = 860 R ...................................... 84

7 Proportioning of Thermal Conductivity into
Gaseous and Solid-to-Solid Conductivity ........ 87

8 Estimated Manufacturer's Cost for Storage
Vessels ....................................... 116


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


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 Cool-Down 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


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
continuum, Btu/hrftOF
K thermal conductivity coefficient of a gas for
continuum, transition, and non-continuum,
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/ibm-mole
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
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 X-axis coordinate
y particle contact radius, ft
z coordinate direction, ft


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 Stephan-Boltzmann constant, 0.1714 x 10-8
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


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



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 250-1000F 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 250-1000F, 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.

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 cross-sections;
several of the more practical gases being air,
argon and carbon dioxide.

The results of this study indicate for the temperature

range investigated, 250-10000F, 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.



The more successful techniques used to store energy,

electrical or thermal, involve chemical or mechanical tech-

niques such as lead-acid 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 air-conditioning,

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 gas-fired 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 life-cycle 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 250-10000F 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

2) insulation and tank materials,

3) outgassing of materials in the evacuated

4) inner tank support structure,

5) leak-free 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


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 multi-layer

insulation because

1) powder insulation works well at pressures of
1-10 mm Hg and this level is easier to maintain
than a pressure level of 1 um Hg or less, as
required by multi-layer insulation,

2) suitable powders for withstanding high temperatures
are readily available,

3) a need exists for many uses in high temperature

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


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

self-contained 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 -


P Pump
G Electric Generator
ORCS Organic Rankine

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

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


In addition to residential applications an efficient

cost effective insulating technique is needed for high

temperature batteries [5]. For example, a lithium-sulfur

battery operates between 707 7970F and a sodium-sulfur

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 stand-by 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


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 thermo-physical 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.



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

particle-to-particle 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


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 super-insulated 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 multi-layer insulation [12, 13].

Most of the development work on multi-layer 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 multi-layer 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


As evident from much of the literature, multi-layer

insulation for spacecraft and for cryogenic work was

expensive. Prior to the need for super-insulators 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 multi-l-ayer insulation, its advantages


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 multi-layer 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-


There is a growing need for efficient high temperature

insulation. The present technique used to insulate large,

high temperature (200-1000F) 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

volume-to-surface 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.


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] 14-24 1600 0.053 0.074

Micro-quartz fiber,
blanket [18] 3 3000 0.042 0.075

Rock woll [18] 8-12 0.049 0.078

Sil-O-Cell [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.)




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


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 particle-to-particle 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 gas-powder 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, gas-solid 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.


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, solid-to-solid, (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,

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 gas-solid

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

gas-powder 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

PMT (7)
P = ^ (7)

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

S u (9)
J/7 2A P

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(Y-1) (10)

where Y is the specific heat ratio.

Combining Equations 7, 8, 9, and 10 into Equation 6


g (9y-5) R u u(
g 44y-l) 70'I T) Y7-

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.


Molecular Weight, Collision Diameter,
Heat Ratio of Selected Gases

(lb /lb -mole)
m m

o [22]
(ft x 10 )


and Specific

y [23]



Specific heat ratios for gases at low pressure at 800F.

Corrected value [24].
'Corrected value [24].





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 Non-Continuum 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 non-continuum. Non-continuum 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 (TH-T )
q = x + 2A (12)

where B is

2-a 2e
Sa (13)
a y+1


e = (9y-5) (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)


B = (16)

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 non-continuum conditions, and not just the


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 non-continuum conditions, gave experimental results

that were in good agreement with theoretical predictions

over the range of continuum, transitional, and non-continuum

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 non-continuum


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 non-continuum 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)

In the continuum region where A < < J, then

L -

In the non-continuum region where A > > J, then

L -

To relate the modified mean free path, L, to gaseous con-

ductivity, recall that

K = (9Y-5) ('pCvC ) (6)

Substituting Equation 12 into Equation 6 yields

g C(9y-5) pCv (i/A + 1/4)-1 (19)

where K is the modified gaseous thermal conductivity coef-
ficient. Equation 19 can now be used to calculate the

thermal conductivity of a gas for continuum, transitional,

and non-continuum 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


Kg = (y (S (1/P + 1/B-1 (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.
Conversely, as the product of PT becomes large relative to

B, K is seen to be essentially independent of pressure.
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

gas-powder 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 cross-section 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 non-continuum 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 particle-to-particle thermal conductivity in a

gas powder bed, little will be gained by further reducing

conduction through the gas. Under these circumstances,

the particle-to-particle 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

particle-to-particle thermal conductivity to the apparent

thermal conductivity of a powder under vacuum.

Particle-to-Particle 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 particle-to-

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 particle-to-particle conductivity. With a mathe-

mathical expression one may readily identify those variables

that control the behavior of the material under study.

Theoretical Model. For particle-to-particle 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

particle-to-particle 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 particle-to-particle energy transfer in a

powder by analyzing the resistance due to the contact

region only. This concept is best illustrated by an


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


General Particle Shapes In Contact

Particle interface

Contact region

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


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- -F-t (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. steady-state,

2. no internal heat generator, and

3. the temperature variations in the r
and 6 directions are negligible.

A steady-state assumption is made since the material

under investigation is intended for use as an insulator

operating under steady-state 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

one-dimensional equation and heat flow through a contact

region is viewed as quasi-one-dimensional. Thus,

S= 0 (23)

A method of solution for one-dimensional problems in

which q = constant at every cross-section is [27]

d[qxA(x)] = 0 (24)

which gives

QT = qxA(x) = constant (25)

The x denotes heat transfer in the x-direction and

q k dT (26)
x = dx

Introducting Equation 26 into Equation 25, rearranging

and integrating between limits gives

Q H= -L H L (27)
(1/k ) fX2 dx/A(x) Re
S X1


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 quasi-one-dimensional. 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,


D x

Contact particle body
Parameters Used in Analyzing Heat Flow Through A
Hemispheric Contact Region


k-R-h x
h I

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


R k/1 0 /2(
Re = In tan l /2 (30)
s 2

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(R-h) (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 [(1-p)/Ej + (1 p)/E2 1/335
y ]8 (35)
D1 D2


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)


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 particle-to-particle 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 non-uni-

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. non-uniformity 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

I= ln(D /* (38)


C' = Crn/2

The particle-to-particle 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 radiant-to-total 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 (<10-2 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

10-4 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

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


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


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 right-hand 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 Stefan-Boltzmann 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.


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.




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)
Sectional View Of Conductivity Test Apparatus

Figure 6
Conductivity Test Apparatus

Top View

Centering Plate

Figure 6(a) (Cont'd.)



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. Chromel-alumel 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 (TH-TC)L
Qelec In(r /ri) (41)

Solving for conductivity,

Qelec ln(r /ri)
Ka ((T42)
a 2i (TH-T )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

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)


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 re-heated periodically

due to differential expansion between the wire and


The Design of a High Temperature Thermal Energy Storage

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 10-5 psi (100 pm Hg).

4. The inner tank is capable of holding organic

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

Sx 2" x 2"
angle iron

- x 1" x 1"
angle iron

Figure 7Cc) (Cont'd)

External Support for Inner Tank


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

U-shaped 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


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


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)

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

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

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 point-to-

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


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.


Diatomaceous Earth Particle Size and Distribution
Typical Results





Tyler Sieve






29 50







Run #









"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




Figure 8

Electron Microscope Photograph of Diatomaceous
Earth-Carbon (1% wt) Mixture, Particles from
Tyler Sieve #500 (x 1,000)

Figure 9

Electron Microscope Photograph of Diatomaceous
Earth-Carbon (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

3. The general trend is that as the amount of
carbon increased, the greater the percentage
of small particles and the faster they

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.


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

* 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.



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 long-term 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 volt-ohm meter.

An ice bath was used as a temperature reference for the


A Ammeter
Rheo. rheostat
V voltmeter

leak valve

Figure 12

Conductivity Test Equipment Arrangement

Figure 13

Conductivity Test Equipment

chromel-alumel (Type K) thermocouples. The temperature

levels were recorded manually.

A mercury-filled 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


A common vacuum pump with a blank-off 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


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 steady-state

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


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


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
@ 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)


Average Thermal Conductivity Values
For Evacuated Diatomaceous Earth*

Powder in Tyler Sieve #500 (loose powder)




Powder in Tyler
to 45 psi)

Sieve #500 (powders physically compressed



Powders contain approximately 1% carbon by weight.

(pm Hg)



TABLE 5, Continued

Powder in #325 pan

(vm Hg)

(loose powder)


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





rbon added to lo


(loose powder)



Foil 5 layers


(Btu/hrft F)



ose powder)





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

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
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= 4-1) 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 10-9 ft

d = 1.0 ft (since 3 >> A)

y = 1.39

Cv = 0.18 Btu/lbmF

gives a value of K of

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 non-continuum 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-


Theoretical Thermal Conductivity of Air and Argon
For Non-Continuum Conditions at Tb = 860R

Particle Size
d = 0.5 D (ft)
M (Ib /lb -mole)
o (ft)
C (Btu/lbmoR)

Tyler Sieve #170
1.625 x 10 4
1.1975 x 10-9

Tyler Sieve #170
1.625 x 104
1.25 x 10-9

Tyler Sieve #500
5.75 x 10-'
1.1975 x 109

Tyler Sieve #500
5.75 x 10
1.25 x 10-9





(Btu/hrft F)




(pm Hg)


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


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