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24 0“†013‘
LIBRARY
Michigan State
University
This is to certify that the
thesis entitled
Temperature and Compaction Ratio (Density) Dependence of
Thermal Conductivity of Ceramic Refractory Blankets
presented by
Mehmet Ali Gulgun
has been accepted towards fulfillment
of the requirements for
M.S. degree in Materials Science
WCM’K
Major professor
Date May 15, 1990
0-7639 MS U is an Affirmative Action/Equal Opportunity Institution
PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or baton an. om.
DATE DUE DATE DUE DATE DUE
MSU I: An Affirmative AcuorVEquol Opportunity Institution
TEMPERATURE AND COMPACTION RATIO (DENSITY) DEPENDENCE OF
THERMAL CONDUCTIVITY OF CERAMIC REFRACTORY BLANKETS
BY.
MEHMET ALI GULGUN
A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Department of Metallurgy, Mechanics,
and Materials Science
April 1990
b05559
ABSTRACT
TEMPERATURE AND COMPACTION RATIO (DENSITY) DEPENDENCE
OF THE THERMAL CONDUCTIVITY OF CERAMIC REFRACTORY
BLANKETS
BY
Mehmet Ali Gulgun
Temperature and compaction ratio (density) dependence of
the thermal conductivity of ceramic refractory blankets were
investigated over a temperature range between room temperature
and 970 degrees Celsius. Specimens were compacted from the
as-received densities to 0.2 times their original thickness by
application of a uni-axial force. Thermal conductivity was
measured using the hot-wire technique.
A model was developed to describe the behavior of thermal
conductivity versus temperature and compaction ratio. The
model accounts for the contributions to overall thermal
conductivity due to radiation, gas conduction, convection, as
well as series and parallel fiber solid conduction.
DEDICATION
This work is dedicated to my families,
here, in TURKEY, and in France.
BISMILLAHIRRAHMANIRRAHIM
iii
ACKNOWLEDGEMENTS
I would like to thank Dr. Eldon Case who has been much
more than a thesis advisor to me for his advice, assistance,
patience, support, and enthusiasm through the duration of this
project.
I would also like to thank my fellow researchers Youngman
Kim, Narendra Bettedupur, Chin Chen Chiu, Carol Ann Gamlen,
Won Jae Lee, and Karl Tebeau for their assistance in the
preparation of this thesis.
I would also like to thank Dr. Kalinath Mukherjee, and
Dr. Parwaiz Khan for their assistance and support, and Dr Tom
Vogel from Geology Department for introducing us the
minerological oil technique for refractive index measurements.
'Acknowledgements are also due to Robert Rose from
Mechanical Engineering Department for his assistance with the
high temperature furnace.
I would like express my special thanks to Catherine
Cassara for being a very good friend and for editing the
thesis.
iv
Rage
List of Tables ...................................... viii
List of Figures ..................................... X
1. Introduction ................................... 1
1.1 A Review of the Hot-wire Technique ....... 2
1.1.1 Theory of the Hot-wire Method .... 4
1.1.1.1 Limitations of the Hot-
wire Technique ......... 9
1.1.1.2 Advantages of the Hot-
wire Technique ......... 14
1.1.2 The Hot-wire Thermal Conductivity
Technique ... ......... .. .......... 15
1.1.3 Evaluation of the Technique ...... 16
1.2.1 Review of the Measurements of
Thermal Conductivity in Ceramic
Fiber Insulating Blankets as a
Function of Temperature and Blanket
Bulk Density ..... .......... ...... 26
1.2.2 Review of Theoretical Models for
Heat Transfer in Fibrous Insulators 31
2. Experimental Procedure . ........................ 44
2.1 Sample Characteristics ................... 44
2.2 Equipment ............... ................. 52
2.2.1 Electrical Equipment ............. 52
2.2.2 Measuring System ................. 52
2.2.2.1 Electronic Devices ..... 53
TABLE OF CONTENTS
V
2.2.2.2 Thermocouple Wires .....
2.2.3 Heating Wires ( Hot wire ) .......
2.2.4 Isolation Box ....................
2.2.5 The Pressure Fixtures ............
2.2.5.1 Steel Pressure Fixture .
2.2.5.2 Refractory Brick
Compression Fixture ....
2.2.6 Electrically Heated Furnace ......
2.3 Specimen Assembly .. ......................
2.4 Calculation of the Effective Voltage Drop
across the Heating Wire ..................
2.5 Refractive Index Measurements ............
Results and Discussion .........................
3.1 The Temperature Dependence of the Thermal
Conductivity of Ceramic Fiber Insulating
Blankets ...... . ...... . ...................
3.2 The Model for the Overall Thermal
Conductivity in Fibrous Refractory
Blankets .................................
3.2.1 Radiation Thermal Conductivity ...
3.2.1.1 The Refractive Index ...
3.2.1.2 The Mean Free Path for
Photon Conduction ......
3.2.2 Gas Conduction Thermal Conductivity
3.2.3 Convection Thermal Conductivity ..
3.2.4 Solid Conduction Thermal
Conductivity ....... ' ..............
3.2.4.1 Solid Conductivity in
Saffil Type Alumina Fiber
Blankets ... ............
vi
6O
62
65
68
7O
74
74
84
89
92
95
98
104
105
106
3.2.4.2 Solid Thermal
Conductivity in
Aluminosilicate Fiber
Blankets ... ...... . .....
3.2.5 The Empirical Term C6 . ...........
3.3 The Compaction Ratio Dependence of the
Thermal Conductivity of Ceramic Fiber
Thermal Insulating Blankets ..............
3.4 Correlation between the Theory and
Experimental Data ... ...... . ..............
3.5 Interface Effects between Thin Blankets of
Similar Refractory Materials .............
3.6 Effects of Fluctuations in the Ambient
Temperature on the Thermal Conductivity
Measured by Transient Hot-wire Method ....
3.7 Comments on the Change of Heat Input to the
Specimen .................................
3.8 Effects of the Gap between the Thermocouple
Junction and the Hot Wire ................
Conclusions ....................................
Appendices .....................................
Appendix A. Derivation of the Transient
Temperature Distribution for the
Hot-wire Technique ...............
Appendix B. Thermal Conductivity Data of
Ceramic Fiber Blankets ...........
Appendix C. Gas Thermal Conductivity data for
air at atmospheric pressure ......
Appendix D. Convertion Table for Thermal
Conductivity Units ...............
Appendix E. Configuration of Sensing
Thermocouple Junction and Placement
with Respect to the Hot Wire .....
References .......... . ..........................
vii
108
113
117
122
140
143
145
158
151
154
154
161
205
206
207
208
10.
11.
12.
LIST OF TABLES
Comparison of Thermal Conductivity Values
Obtained by the Hot-wire and the Other
Techniques
Chemical Composition and Selected Physical
Properties of Ceramic Fiber Blankets as
Listed in Manufacturer's Catalogues [39,40]
Chemical Composition and Selected Physical
Properties of Ceramic Fiber Blankets as
Listed in Manufacturer’s Catalogues
[26,37,38]
Chemical Composition and Selected Physical
Properties of Ceramic Fiber Paper Products
as Listed in Manufacturer’s Catalogues
[26,37,38]
Chemical Composition and Selected Physical
Properties of Ceramic Fiber Blankets as
Listed in Manufacturer’s Catalogues [41]
Chemical Composition and Selected Physical
Properties of Ceramic Fiber Blankets as
Listed in Manufacturer’s Catalogues
[42,43]
Measured and Weighted Average Refractive
Indices of the Fibers
Normalized Error in the Mean Free Path
Calculations due to Truncation in the
Expansion of Probability Function
Parameters for the Solid Thermal
Conductivity in Locon Fibers
Parameters for the Solid Thermal
Conductivity in Kaowool K2300 Fibers
Empirical Constants in C Expression for
Saffil, Locon, and Kaowogl K2300 Blankets
Fiber Physical Properties and Blanket
Density
viii
Page
18
47
48
49
50
51
93
99
109
110
116
118
13.
14.
C1.
01.
Dependence of the Measured Thermal
Conductivity Values on Varying Heat Input
through the Hot Wire
The Onset Time for the Linear Portion of
the Temperature versus the Logarithm of
Time Curve as a Function of the Gap
Distance (the Gap between the Sensing
Thermocouple and the Hot Wire)
Air Gas Thermal conductivity at Atmospheric
Pressure
Conversion Factors for Thermal Conductivity
Units
ix
147
150
205
206
10.
11a.
11b.
11c.
LIST OF FIGURES
Comparison between ASTM C177-76, British Standard
1902, and hot-wire methods for ceramic fiber
blankets 0............OOOOOOOOOOOOOO ........ 0....
Thermal conductivity values of Saffil blanket
(96 kg/m nominal density). A comparison
between hot wire and ASTM guarded hot plate
technique C177-76 ..............................
Comparison between hot plate technique, hot wire-
method, and data obtained in this study ........
Comparison between the experimental data and the
thermal conductivity values for K2300 refractory
blanket at different bulk densities given by
manufacturer ...................................
Experimental apparatus for thermal conductivity
measurements at elevated temperatures ..........
Wiring set-up for elevated temperature
experiments ....................................
Steel fixture for compacted specimen experiments
Fixture for thermal conductivity measurements
‘as a function of compaction at elevated
temperatures ........................ . ..........
Schematic of hot-wire power lines for elevated
temperature experiments ...... . .................
Minerological oil method to determine the
refractive index. a) noil > n
fiber
b) noil < “fiber ...............................
Thermal conductivity versus temperature data for
Kaowool K2300 blanket .......... . ...............
Thermal conductivity versus temperature data for
Kaowool K2600 blanket ..........................
Thermal conductivity versus temperature data for
Kaowool NF blanket .............................
REES
21
22
23
24
45
58
61
63
69
72
76
77
11d. Thermal conductivity versus temperature data for
Kaowool ZR blanket ............................. 79
11e. Thermal conductivity versus temperature data for
Locon blanket ........... ..... . ................. 80
11f. Thermal conductivity versus temperature data for
Durablanket-S ...... ............................ 81
llg. Thermal conductivity versus temperature data for
Fibersil Cloth ......... ........................ 82
11h. Thermal conductivity versus temperature data for
Saffil Blanket at 0.8 compaction ratio ......... 83
12. Contribution by each mode of heat transfer in
glass-fiber insulation at atmoshperic pressure
versus blanket density at 65 C ......... . ....... 100
13a. Thermal conductivity versus compaction ratio
data for Saffil blanket at 25 C, 400 C, and
800 C .......................................... 120
13b. Thermal conductivity versus compaction ratio
data for Saffil blanket at 200 C, 600 C, and
97°C 000...... 00000000 0 000000000000000000 0 00000 121
14a. Thermal conductivity versus compaction ratio
data for Locon blanket at 23 C, 400 C, and 800 C 123
14b. Thermal conductivity versus compaction ratio
data for Locon blanket at 200 C, 600 C, and
970 C .......................................... 124
15a. Thermal conductivity versus compaction ratio
data for Kaowool K2300 blanket at 25 C, 400 C,
and 800 C . ..................................... 125
15b. Thermal conductivity versus compaction ratio
data for Kaowool K2300 blanket at 200 C, 600 C,
and 970 C ........... ........................... 126
16a. Thermal conductivity versus compaction ratio
data and theoy prediction for Saffil blanket at
23 c, 400 c, and 800 c ......................... 127
16b. Thermal conductivity versus compaction ratio
data and theoy prediction for Saffil blanket at
200 C, 600 CI and 970 C 00000 0 0 0 0 0 0 0 000000000000 128
17a. Thermal conductivity versus comopaction ratio
data and theoy prediction for Locon blanket at
25 C, 400 C, and 800 C ......................... 130
X1
17b.
18a.
18b.
18C.
18d.
18e.
18f.
19a.
19b.
20.
21.
Thermal conductivity versus comopaction ratio
data and theoy prediction for Locon blanket at
200 C, 600 C, and 970 C ........................
Thermal conductivity versus temperature data and
theory prediction for Saffil blanket at 0.2, 0.5,
0.7 compaction ratio ..... ........... ...........
Thermal conductivity versus temperature data and
theory prediction for Saffil blanket at 0.3, 0.6,
0.8 compaction ratio .... .......................
Thermal conductivity versus temperature data and
theory prediction for Locon blanket at 0.3, 0.6,
0.8 compaction ratio . ..........................
Thermal conductivity versus temperature data and
theory prediction for Locon blanket at 0.5, 0.7,
1.0 compaction ratio ...........................
Thermal conductivity versus temperature data and
theory prediction for Kaowool K2300 blanket at
0.2, 0.5, 0.7, 1.0 compaction ratio ............
Thermal conductivity versus temperature data and
theory prediction for Kaowool K2300 blanket at
0.3, 0.6, 0.8 compaction ratio .................
Thermal conductivity versus compaction ratio
data and theory prediction for Kaowool K2300
blanket at 25 C, 400 C, and 800 C ..............
Thermal conductivity versus compaction ratio
data and theory prediction for Kaowool K2300
blanket at 200 C, 600 C, and 970 C .............
Effects of interface between stacked refractory
blanket layers on the thermal conductivity
measurements .. .................................
Effects of ambient temperature fluctuations on
the thermal conductivity measurements ..........
xii
131
132
1. INTRODUCTION
Refractory ceramic insulating blankets are more and more
popular in many different applications. The high to very high
temperature, structural, and compositional stability and the
superior thermal insulating properties of ceramic fibers lead
many industries to use ceramic fibrous blankets. The major
concern of the field is to determine the optimum thermal
performance of the refractory insulators at the temperatures
of application. Selection of the best insulation requires the
knowledge of how the extrinsic features, like the density
and/or the thickness, of the insulators must be altered during
the installation.
At temperatures above about 400 C radiation becomes
important in the heat transfer process within the insulating
blankets [1]. Since the amount of heat carried by radiation
depends on the distance a photon can travel without being
scattered by the fibers, the density dependence of thermal
conductivity of fibrous blanket insulators is important along
with the temperature dependent variation of thermal
conductivity. In most applications the blankets have to be
compacted to some extent to satisfy dimensional requirements.
This thesis seeks a clear insight into the temperature
and compaction (i.e. density) dependence of thermal
conductivity of fibrous insulating materials in the
temperature range between room temperature and 970 C. The
fibrous insulators were compacted down to 0.2 times their
original, or as received, thicknesses.
The thermal conductivities of the blanket specimens were
measured under varied densities and temperatures by the hot-
wire technique.
1.1. A Review of the Hot-Wire Technique
The hot-wire technique offers an easy and rapid way of
determining the thermal conductivity and thermal diffusivity
of various gaseous, liquid, and solid materials.
Thermal conductivity and diffusivity of liquids and gases
were successfully measured by hot-wire technique with a
cylindrical line source [2] and a flat-line heat source (hot-
strip method)[3]. The hot-wire technique is widely applied to
solid materials ranging from AgCl salt [4] to natural rubber
[5], and for many types of refractory ceramic materials [6, 7,
8, 9, 10, 11, 12, 13, 14].
The hot-wire technique is recommended for the thermal
conductivity measurements of ceramic fiber insulations up to
1600 C [15, 13, 16] over the guarded hot-plate technique.
Hot-plate techniques are limited to relatively low temperature
regimes (up to 1000 C), while the hot-wire technique can be
used for thermal conductivity measurements to 1600 [15].
Versions of the guarded hot-plate technique differ by the
location where the heat flow is measured, and these various
guarded hot-plate techniques give diverging results at high
temperatures [15]. The experimental difficulties at high
temperatures may stem from the increasing heat losses from the
edges of the plates at high temperatures [15]. Another
possible source of error encountered in the guarded hot-plate
technique is the extrapolation of the low temperature thermal
conductivities to obtain thermal conductivities at
temperatures at which the guarded hot-plate technique cannot
be used.
The hot-wire technique is recognized as a standard in
West Germany (DIN 51046) for thermal conductivity measurements
of refractories up to 1600 C [17, 13]. In the early '703 PRE
(Federation Europeenne des Fabricants de Produits
Refractaires) issued a draft for the hot-wire technique as PRE
Recommendation 32 upon the recommendation of a group of
researchers from Great Britain, France, and West Germany. In
the late ’70s Germany submitted DIN 51046 Hot-Wire Method to
ISO (International Standardization Organization) to be
considered as a standard technique.
The German standard, Deutsche Industrie Norm (DIN 51046),
specifies that the technique can be used for thermal
conductivity measurements of materials with thermal
conductivities of 2 W/mK or less [15]. Jeschke [17]
elaborated on the technique and defined experimental
conditions under which the hot-wire method gives accurate
results for measuring thermal conductivities around 6 W/mK.
Bayreuther et.al. [13] advanced the hot-wire method apparatus
further so that thermal conductivities of refractories ranging
from k-values 0.05 W/mK to 25 W/mK can be measured by the hot-
wire technique.
1.1.1. Theory of the Hot-Wire Method
The idea of using an electrically heated thin wire to
determine the thermal conductivity of liquids was first
suggested in 1888 [18]. The first successful hot-wire
measurements were done on the thermal conductivity of liquids
in the middle of 20th century [19].
The transient heat flow line source method (the hot-wire
technique) is based on the relation between the thermal
conductivity and the temperature rise caused by a constant
heat input through the line heat source in an infinite
homogeneous medium [10].
If a continuous line heat source supplies thermal power
at a constant rate, 0, per unit length of the line source, the
radial temperature gradient created by the source in the
surrounding material, initially at a uniform temperature, is
described by either [17]:
dT/dln(t) =’[Q * exp(-r2/4*a*t)] / 4nk (1)
where, Ts
Q:
k.
r-
ta:
or by [10,
*3
II
where, T8
Q:
R:
r:
t:
temperature
Power input
Thermal conductivity
Radial distance from the hot wire
Time from the beginning of heat input
Thermal diffusivity
15]
[Q/zxkl fm{[eXP(-ï¬z)]/fl}*dfl
I‘D
[Q/ka] * I(rn) (2)
temperature rise above initial temperature
Power input through the hot wire
Thermal conductivity
Distance from the line source
1/2 (a*t)'1/2
Thermal diffusivity
time from start of heat input
p= r/(4*a*t)1/2
The term
I(rn)= -c -ln(rn)2 + [(rn)2/2] - [(rn)‘/8]+...
=1/2 [-c -ln(r2/4at) + (r2/4at) - (r2/4at)2/4 +... (3)
c is a constant.
If rn (or r/(a*t)1/2) is sufficiently small, which means
r is very small (in the limit r=0), and t is sufficiently
large, the higher order terms in I(rn) can be neglected
(Appendix A). Thus
T= [Q/Zflk] * [C ‘ ln(rn)] (4)
Then the temperature rise between the times t1 and t2 is given
by:
TZ-T1= [Q/4nk] * ln(t2/t1) (5)
A graph of temperature versus the natural logarithm of time is
a straight line with the slope of Q/4nk. Thus the slope of
the temperature versus logarithm of the time plot is inversely
proportional to the thermal conductivity of the medium
surrounding the line source. In determining thermal
conductivity by the transient heat flow method, temperature,
time, and heat input are measured, and k is calculated from
equation(5).
Prelovsek et al. [20] and Takegoshi et a1. [21] analyzed
the configuration in which the hot wire and the sensing
thermocouple lay on a horizontal plane separating the two
refractory materials with differing thermal and physical
properties. The analysis yielded the following equation for
the temperature gradient:
Tz-T1= [Q/2«(k1+k2)] * ln(t2/t1) (6)
Thus, if one of the thermal conductivities (say k1) is known,
one can determine the thermal conductivity k2 of the second
material using equation(6). This method is called either the
generalized hot-wire technique [20] or hot-wire method of
comparison [21].
Takegoshi et a1. [21] also employed the hot-wire
technique to determine the thermal conductivity of orthogonal
anisotropic materials, such that:
kx= [Q/2«]*[dlnt/de]-kc (7a)
ky= [Q/2u]*[d1nt/d'ry]-kc (7b)
7
k2: [Q/2«]*[dlnt/de]-kc (7c)
where kc is the thermal conductivity of the isotropic
reference material and kx' k , and k2 are the thermal
Y
conductivities in the three orthogonal directions. The
experimental apparatus is exactly the same as in [20] except
that the material of known thermal conductivity becomes
instead the reference material.
For the case where the thermal resistivity of the
infinite medium is a linear function of temperature, such as
1/p = (1/90) (1+0T) (8a)
where a is the temperature coefficient of resistivity, p and
po are thermal conductivities of the medium at temperatures T
and To (initial temperature). Salin and Salin [22] developed
the relationship (8b) for the constant line source of heat in
an infinite medium.
Assuming the density and the specific heat of the medium
are constant so that k/a = constant = kd/ao, the relationship
between the temperature T and heating time t takes the form
ln(T2/T1) z [aQ / 4xko] * 1n(t2/t1) (8b)
where, Q = constant heat input per unit length per unit time
a = temperature coefficient of thermal resistivity
k,k = thermal conductivity at temperature t and reference
temperature, respectively
a,a a thermal diffusivity at temperature t and reference
temperature, respectively
1.1.1.1 Limitations of the Hot-Wire Technique
Four assumptions that limit the hot—wire technique are
[17]:
1. The heat source is a line (hot wire) of infinitely small
radius (r=0).
2. The length of the wire is infinite.
3. The heating element is embedded in an infinite medium.
4. The coefficient of heat transfer between the heating
element and the surrounding medium is infinite, or the
contact resistance between the hot wire and the specimen
is infinitely small.
Since the diameter of the hot wire cannot be zero, some
error is introduced because of the finite diameter of the line
source. Jeschke[17] develops an equation (9) to account for
errors introduced by not fulfilling assumptions 1 and 4:
T= [Q/4tk] * [2h + ln(4F°/c) - (4h-W/2wFo) +
(w-2 / 2wFo) * ln(4Fo / c) +... ] (9)
where, T= Temperature of finite heating wire
F Fourier number = at/r2
o
a-= Thermal diffusivity
t= Time from beginning of heat liberation
r= Radius of heating wire
h= Zak/H
H= Coefficient of heat transfer from the wire to the
surrounding medium
(2 *d*Cp) medium
(d*Cp) wire
d*cp== heat capacity
d= density
Cp= specific heat
c= 1.772 (=exp(1) with 1= Euler constant).
In the limit of r (or diameter, d) approaching zero, the
heat capacity ratio approaches 2. For H infinitely large, h
will tend to zero, and equation (9) could be transformed into
equation (3).
Since the diameter of the heating wire cannot be equal to
10
zero (violation of assumption 1), the time temperature graph
will not look exactly as described by equation (5). Van der
Held [19] observed that the plot of the experimental time-
temperature data and the plot of equation (5) are basically
the same in shape. The two curves differ by a shift in the
time variable. The finite diameter and the thermal mass of
the heating wire cause this shift, which can be corrected by
subtracting a constant to (to=r2/4a) from time, t [15].
Furthermore, to can be determined from experimental
results[19]. If the temporal derivative of
TZ-T1= [Q/4nk] ln(t2/t1) (5)
is evaluated at (t-to) the following is obtained [10]:
dT/dt= [Q/4xk] * (t-to)'1 , or (10a)
dt/dT= [4nk/Q] * (t-to) (10b)
Time shift parameter to is determined by plotting dt/dT as a
function of the true time t. The time-axis intercept of the
linear portion of the dt/dT versus t curve is to. The to
value for a 0.254 mm diameter 13 percent Rhodium/Platinum
alloy wire was determined to be 1 second [15]. For the hot
wires in this work, to was approximately 0.8 second as
determined by the equation (10b) [19] and by the experimental
11
method discussed in section 3.2. of this thesis. The effect
of to was essentially negligible for this study and in other
hot wire studies [23].
The second source of error is the finite length of the
heating element (assumption 2). The source length introduces
two errors [10], one due to the distortion of the radial
temperature field at the ends of the hot wire, and second, due
to the heat losses along the heating element. A very short
heating element will loose a significant amount of heat from
the ends, so that the temperature rise at the thermocouple
would be less than what is predicted by the theory. The first
error (due to the change in the boundary conditions along the
length of the hot wire) is insignificant [15, 10]. The heat
losses along the hot wire are also negligible provided that
(r0/10>'1 0
wire length. In this work r was 0.127 mm and 10 was 152 mm,
0
thus ro/lo is about 1200 and these heat losses can be
> 200, where r is the wire radius and 10 is the hot
neglected.
The finite size of the thermal conductivity specimen
apparently contradicts the assumption of infinite medium
surrounding the heating element. German standard 51046
states that the minimum size for refractory bricks should be
200 mm by 100 mm by 50 mm for a valid thermal conductivity
measurement. DIN 51046 would be a very conservative criterion
if it were applied to fibrous ceramic insulations. The
thermal conductivity of fibrous ceramic insulating blankets
12
seldom reaches 1 W/mK for the maximum temperature at which the
hot-wire technique can be applied.
Another criterion developed by Hayashi for refractory
insulations [24] defines the minimum thickness s (cm) and
length 1 (cm) of thermal conductivity specimens as:
s= 10*(k*t)1/2 (11)
2*(234*ka)b (12)
[.0
II
where a = 0.6, b = 1/(2.2 - 2ro), k (kcal/m hr C) is the
thermal conductivity of the specimen, t (hour) is the elapsed
time, and r (mm) is the radius of the wire. It should be
0
noted that in this criterion the specimen size is not
dependent on the thermal diffusivity of the medium (equation
(11) [15, 24].
The plot of temperature, T, versus ln(t-to) gives an s-
shaped curve where the initial curved portion is due to the
neglected terms in equation (4), which are related to the
finite diameter and the thermal properties of the heating
wire. The upper curved portion is a result of the expanding
temperature field front impinging on specimen boundary [23].
13
1.1.1.2. Advantages of the Hot-Wire Technique
1.
The main advantages of the hot-wire technique are:
It is an easy, rapid, and inexpensive method to
determine the thermal conductivity of refractories.
It is a transient heat flow technique. One does not
need to reach the steady state conditions. Unlike the
steady state parallel plate techniques, no artifacts are
created by exposing the specimen to high temperatures
for extended period of time. The effective thermal
conductivity of the material can be measured in the as-
received condition. The high thermal conductivities
obtained for Saffil blanket at high temperatures by the
hot-plate technique may be due to structural changes
occurring in the fibers due to long time exposure to
high temperatures [15].
It measures thermal conductivity at higher temperatures
(1600 C) than the parallel plate techniques (up to
1000 C). The guarded hot-plate technique gives poor
results below 400 C [23] whereas hot-wire technique
is applicable even for temperatures below room
temperature.
14
1.1.2. The Hot-Wire Thermal Conductivity Technique
Experimental apparatus for the hot-wire technique are
varied. Temperatures are measured by either a thermocouple,
or by the temperature-induced resistance changes in the hot
wire [14]. The thermocouple is either welded to the heating
element [15, 17], or placed very close to it by carving a
groove in the insulating material, if applicable [12, 13, 11].
When one is working with solid materials, sufficient pressure
has to be applied to eliminate any possible air gap between
the halves of the specimen (caused by the finite diameter of
the heating element) [11, 25, 17]. Flat rolled wires can
reduce the problem of thermal contact resistance between the
hot wire and the specimen[24]. In the theory of the hot-wire
technique, it is assumed that there is perfect contact between
the heating wire and the surrounding medium. Thus, the heat
released from the hot wire can be transmitted to the medium
without encountering any resistance. An air gap between the
specimen halves violates this assumption, since the heat
transfer coefficient, H [17], from the hot wire to the
surrounding medium is finite.
The effects of having another medium between the hot wire
and the refractory insulators was investigated [17] by carving
a 20 mm diameter cylindrical hole in the specimen. The hole
was filled by a light-weight powder and magnesia powder or
left unfilled (the light-weight powder was not specified
15
[17]). The error due to the air filling the gap was 9.3
percent. Magnesia powder gave -13 percent and the light-
weight powder gave -12 percent errors.
The hot wire is heated resistively by either alternating
current or direct current. AC and DC heating yielded values
that differed by less than one percent [23]. Changes in the
electrical heat input affect the value of thermal conductivity
only very little [This study (section.3.5) , 24, 23].
1.1.3. Evaluation of the Technique
For isothermal parallel plate geometry, heat transfer
analysis of uni-directional heat flow by combined conduction
and radiation dictates that the heat flux within the material
must be constant in steady state. However, at the boundary
between the hot wire and the diathermanous medium, the hot
surface (hot-wire surface) cannot produce the same amount of
radiant heat flux as the diathermanous substance at the same
temperature. The difference in the produced radiant heat flux
is due to the non-similar absorption coefficients of the two
materials. The difference in heat flux must be transferred by
heat conduction. However the increased rate of heat transfer
by conduction requires a higher temperature differential
between the hot surface and the diathermanous material.
From the point of view of the hot-wire technique, that
means a higher temperature rise would be measured at the hot
16
wire-surrounding medium boundary than would be predicted by
the theory of the hot wire. From equation (5), a higher
temperature rise would give a lower thermal conductivity for
diathermanous materials [1]. Thus, the hot-wire technique
should not be applied to diathermanous materials in a
temperature range where the radiative heat transfer becomes
important [1]. Otherwise, the hot-wire technique would give
lower thermal conductivities for diathermanous materials, such
as low density fiber glass insulations [1]. However, the
results presented in this thesis and the results from several
other researchers [23, 15, 26] clearly show that the thermal
conductivities of ceramic insulator blankets measured using
the hot-wire method are 5 to 15 percent higher than the
corresponding hot-plate values. This leads to two
conclusions: 1) either ceramic fibers cannot be classified as
diathermanous materials for the temperature ranges involved
for the hot-wire technique, or 2) the hot-wire method gives
better approximations to the real thermal conductivity of the
material than the guarded hot-plate techniques.
Several researchers compared the hot-wire method with
other standard techniques (Table 1). Generally, the hot-wire
technique yields thermal conductivities similar to those
obtained by the other techniques. The thermal conductivities
for 2300 and 2800 insulating bricks obtained by the hot-wire
technique in the temperature range between room temperature
and 1000 C are 10 to 20 percent higher than the thermal
17
Table 1. Comparison of Thermal Conductivity Values Obtained
by the Hot Wire and Other Techniques.
Material 1km §kother Reference
[W/mK] [W/mK]
Silicon rubber 0.235 0.250 * [14]
Glass 0.996 1.092 * [14]
Glasswool (k3)*** 0.0382 0.0372 ** [21]
Glasswool (k2) 0.0494 0.0467 ** [21]
Fire-brick 1.63 1.58 1 [25]
Insulation block 0.0928 0.0987 1 [25]
t Thermal conductivity measured using hot wire method
§ Thermal conductivity measured using the indicated
method
* Hot plate technique
** One dimnesional steady state method. (The details of this
technique were not stated [21], but it is assumed that
this refers to the hot plate technique.)
*** For the glasswool specimens, k1 and k2 refer to thermal
conductivity measurements parallel and perpendicular to
the fiber planes [21].
1 ASTM 0 201-47.
18
conductivities measured using calorimetric parallel-plate
methods (C 201-47). The discrepancy between the two methods
is higher for higher grade refractories, which have a higher
thermal conductivity. For lower thermal conductivities, the
results obtained from both techniques converge for lower
temperatures [15]. A similar trend was reported by Haupin
[25] for firebricks and furnace insulation blocks.
The ASTM test for thermal conductivity of refractories (C
201-47 Calorimetric Method) gives lower thermal conductivity
values because of inevitable heat losses [25]. The edge
losses from the parallel-plate equipment cause the effective
thermal conductivity to appear lower. In C 201-47, the heat
flux is measured calorimetrically from the cold face. The
measured heat flux is true heat input, which sets up the
temperature gradient minus the edge losses. Thus, a high
temperature gradient with a low measured-heat flux gives a
[lower thermal conductivity [23].
Davis [23] compared the thermal conductivities for 2300
grade insulating brick obtained by hot wire to thermal
conductivities obtained by other techniques on the same brick.
Hot-wire values were always higher than values obtained by
radial flow method proposed by McElroy and Moore [27], by
Klasse Method, and by British Standard 1902. B.S. 1902 (a
modified calorimeter method) gave results that are 10 to 20
percent lower than the hot-wire results. For heavy-duty
ceramic fiber blanket, the hot-wire results are almost in
19
perfect agreement with the results of the ASTM guarded hot-
plate technique (C 177-76) and B.S. 1902 (Figure 1).
Jackson et al.[15] compare the thermal conductivity
obtained using the hot wire to the conductivities measured by
the ASTM C177 guarded hot-plate technique. Thermal
conductivities from both methods agree fairly well up to 800
C. For temperatures above 800 C, the thermal conductivities
measured using guarded hot plate diverge (Figure 2). The
guarded hot-plate method can give high thermal conductivity
values at high temperatures due to increased heat losses from
the edges of the hot plate [15]. In ASTM C 177 (the guarded
hot plate) heat input is measured at the hot face. The edge
losses would cause the temperature rise set up by the measured
heat input to be too small, thus giving an equivalent
conductivity which is too high [15].
In this study, the thermal conductivity of Saffil Blanket
at 48 kg/m3 as received density is measured under various
compactions using the hot-wire technique. The close agreement
between the values obtained in this study and [15] shows the
consistency of the technique (Figure 3).
The behavior of thermal conductivity in an
aluminosilicate insulating blanket (Kaowool K2300) [this
study] at different compactions was very similar to the
behavior of the thermal conductivity in the same commercial
blanket at corresponding densities [26] (Figure 4). The
thermal conductivities in the manufacturer’s catalogue were
20
Thermal Conductivity (W/mK)
c—a ASTM
8—0 8.3 1902
0 Hot wire 0
0.20~
016-9
0.12—
0.08—
0.041
O
0.00 i r T l
0 200 400 600 800
Temperature (C)
Figure 1. Comparison between ASTM C177-76, British Standard
1902, and the Hot wire methods for ceramic fiber
blankets (after [23])
21
Thermal Conductivity [k] (W/mK)
0.4
o-o Hot wire perpendicular to fiber planes
o—o Hot wire parallel to fiber planes
H ASTM C177 Guarded Hot Plate Technique
8—8 Computed values of k from hot wire data
0.3—
0.2-—
0.1 —
. A
2/4/
0-0 I I l l l
O 200 400 600 800 1000
Temperature (C)
Figure 2. Thermal conductivity values of Saffil blanket (96 kg /m3,
nominal density). A comparison between hot wire and
ASTM guarded hot plate Technique C177-76 (after [15])
22
Thermal Conductivity (W/mK)
0.4
e—o Hot wire perpendicular to fiber planes 3 /
o—o Hot wire parallel to fiber planes 96 kg/m /
a—A ASTM C177 Guarded Hot Plate Technique /
G—O Computed values k from hot wire doto / /
x—x Hot wire parallel to fiber planes for 60 kg/m / /
(this StUdY) / /
P
A /
,Ia
/
0-0 I 7 I 7. I
O 200 400 600 800 1000
Temperature (C)
Figure 3. Comparison between hot plate technique, hot wire
method, and data obtained in this study (x-x).
23
THERMAL conoucrlvm/ (W/mK)
BULK DENSITY (Ib/cuft)
6-0 8-0 10.0 12.0
1 L i L l 1
(3 Experimental data 0
. Manufacturer's data .
0.24 .4
9
0.22 a o
o
0.20 '1 o
o
0.18 —
o
0.16 - o
I I I l I
1.0 0.9 0.8 0.7 0.6 0.5
COMPACTION RATIO
Figure 4. Comparison between the experimental data and the
thermal conductivity values for K2300 refractory blanket
at different bulk densities given by the manufacturer.
24
obtained using the standard test method for thermal
conductivities of refractories, ASTM C-201. This standard
technique is a caloriemetric method applicable to materials
with thermal conductivity factor of not more than 28.18 W/mK
(200 Btu in / hr ftz F), for a thickness of 25 mm (1/2") [28].
ASTM C-201 is usually employed for thermal conductivity
measurements of refractory bricks. The thermal conductivity
of Kaowool K2300 refractory blanket measured in this study
with the hot-wire technique is 5 to 15 percent higher than the
corresponding values obtained with caloriemetric techniques
(ASTM c-201) [26].
The literature includes diverse ideas about the
relationship among hot-wire results and various parallel plate
results. A group of researchers who used the hot-wire
technique on fibrous insulating blankets as well as on
refractory bricks [23, 17, 15] argue that the hot-wire method
gives 10 to 20 percent higher values than the guarded hot-
plate measurements. However, Lentz [10] reports that the hot
wire gives 10 to 15 percent lower thermal conductivity values
than the accepted hot-plate values for rock wool (10 lb per
cu.ft) blankets.
25
1.2.1 Review of the Measurements of Thermal Conductivity in
Ceramic Fiber Insulating Blankets as a Function of
Temperature and Blanket Bulk Density
The literature includes little experimental data, or
theoretical work, on the temperature and compaction dependence
of thermal conductivity of ceramic fiber insulating blankets.
The available hot-wire data were presented in section (1.1).
Several investigators measured the thermal conductivity
of fiber insulator blankets at reduced ambient pressure.
Using a guarded radial heat flow apparatus, Pettyjohn
[29] determined thermal conductivity of silica fiber blanket
in the temperature range between 400 and 800 degrees Kelvin
under various pressures from 0.01 to 760 mm Hg. Measurements
were conducted in an air atmosphere with three different
blanket densities, 55, 98, and 149 kg/m3. The thermal
conductivity of silica fiber blankets increases linearly from
0.05 to 0.15 W/mK, in the temperature range 400-780 K and
atmospheric pressure. For reduced ambient pressure, S-shaped
curves were obtained. The thermal conductivity asymptotically
approaches a constant value as the pressure is reduced to
vacuum levels of 0.01 mm Hg. For the three bulk densities at
which the experiments were conducted, thermal conductivity at
all temperatures decreased as the bulk density of the blankets
increased.
A.H. Striepens[30] measured the thermal conductivity of
26
aluminosilicate and aluminosilicate-chrome oxide fiber
insulating blankets using the guarded hot-plate technique.
Conductivities obtained between room temperature and 930 C for
128 and 384 kg/m3 bulk densities showed an exponential-like
behavior for both blanket types, with k-values ranging from
0.04 to 0.3, and 0.05 to 0.17 W/mK for respective bulk
densities. The ambient pressure dependence of k revealed a
similar S-shaped curve similar to that observed for reduced
ambient pressure experiments in silica blankets [29].
Experiments with aluminosilicate blankets were conducted for
four different blanket densities ranging from 98 to 384 kg/m3,
but no specific data for thermal conductivity versus blanket
density were presented [30].
Verschoor and Greebler [31] measured the thermal
conductivity of glass fiber insulation ranging in density from
3 to 134 kg/m3. Tests were conducted at 65 C at
8 kg/m
standard pressure in helium, air, carbondioxide, and freon-12
atmospheres. Thermal conductivity in air was investigated
over a pressure range of 1 micron to 760 mm Hg. With
decreasing ambient pressure from 760 mm Hg to 1 micron Hg, the
thermal conductivity of the blanket shows a typical S-shaped
curve [30, 29] with an asymptotic approach to a constant value
at low pressures. The constant value at low pressures is
assumed to represent the contributions to the overall thermal
conductivity due to radiation and fiber conduction. The
discrepancy between the experimental data and the theory was
27
assumed to be the contribution due to the convective heat
transfer to overall thermal conductivity. the convective
heat transfer was not accounted for in the theory. The
composite thermal conductivity theory for insulating blankets
will be detailed in section 1.2.2. The radiative thermal
conductivity in glass fiber blankets decreased hyperbolically
from 0.017 to 0.003 W/mK in the bulk density range 8.0 to 134
kg/m3 at 68 C.
Tye and Desjarlais [32] investigated, among many other
things, the effects of temperature, bulk density, and
temperature on the apparent thermal conductivity of
aluminosilicate and zirconia fiber blankets. The conductivity
increased exponentially from 0.03 W/mK to 0.37 W/mK in the
temperature range between room temperature and 1000 C. The
blanket density was 74 kg/m3. At room temperature, the plot
of thermal conductivity versus blanket density is a parabola
in the density range between 74 kg/m3 and 180 kg/m3 [32]. The
minimum of the thermal conductivity in the aluminosilicate
3 blanket density at room
blanket was around 110 kg/m
temperature. The thermal conductivity of the same blanket
showed a decreasing function of density at 1000 C for the
density range covered. The shape of the curve at 1000 C [32]
is very similar to the shape of the thermal conductivity
versus compaction ratio plot obtained in this study for
aluminosilicate blanket (Kaowool K2300) at high temperatures
(800 C and 970 C). Thermal conductivity decreased from 0.4
28
W/mK to 0.2 W/mK with increasing bulk density at 1000 C [32].
The measured thermal conductivities increase up to 10 percent
at 300 C ambient temperature and up to 14 percent at 800 C for
temperature differentials varying from 50 to 400 C..
Klarsfield et.al. [33] used the guarded hot-plate
technique to determine the conductivity of glass fiber
insulation at 8 to 120 kg/m3 blanket densities for
measurements between room temperature and 400 C. Thermal
conductivity decreased from 50 mW/mK to 33 mW/mK in the
density range 8 to 40 kg/m3 at room temperature. The minimum
of thermal conductivity of 32 mW/mK occurred at a blanket mass
density of 60 kg/m3. Then k-values increased to 33 mW/mK at
120 kg/m3.
Fiberglass insulation tested at room temperature with an
unguarded technique [34] revealed a thermal conductivity
change from 0.05 to 0.038 W/mK as the bulk density of the
glass fiber mat increased from 8.8 to 17 kg/m3. Graves et al.
[34] argued that a change in the insulation thickness from
0.0762 to 0.1524 m changed the thermal conductivity by up to 5
percent.
Using the hot-wire technique [10], thermal conductivity
of rock wool specimens at room temperature was measured with
various heater currents. Although the heat input through the
hot wire varied between 0.142 and 0.036 W/m, measured thermal
conductivity varied less than 10 percent. For rockwool
specimens, the mean conductivity was found to be 0.04 W/mK,
29
which was confirmed by Takegoshi [21] ( 0.0399 W/mK).
The thermal conductivity values determined for a given
insulating material can vary from researcher to researcher.
Davis [23] and Jackson et al. [15] separately investigated the
temperature dependence of thermal conductivity in alumina
fiber blankets at 96 kg/m3. For the same temperature range,
Davis [23] measured the change in thermal conductivity as from
0.04 to 0.37 W/mK whereas Jackson et al. [15] obtained values
of conductivity ranging between 0.04 and 0.285 W/mK. However,
the temperature dependence of thermal conductivity for
zirconia fiber blankets and heavy-duty fiber blankets is very
similar to the one for alumina fiber blanket if measured by
the same researcher [24].
Hayashi [6] investigated the mass density dependence of
thermal conductivity for two aluminosilicate fiber blankets
with different chemical compositions. The experiments were
conducted in air and helium atmospheres between room
temperature and 1200 C. Thermal conductivity in air
atmosphere increased from 0.04 W/mK to 0.5 W/mK at 106 kg/m3
blanket mass density in the temperature range between room
temperature and 1200 C. The change in the thermal
conductivity for the same material at 430 kg/m3 blanket
density was only from 0.07 to 0.27 W/mK. One of the
aluminosilicate blanket Hayashi [6] investigated has a very
similar chemical and physical properties to Kaowool K2300
blanket. Nevertheless the thermal conductivities measured [6]
30
are up to 80 percent higher than the thermal conductivities
given for Kaowool K2300 at corresponding blanket densities
[26].
The exponential-like increase in the thermal
conductivity with increasing temperature [6] agrees closely
with the trends of thermal conductivity in similar ceramic
fiber insulating blankets [15, 23, 32].
The plot of thermal conductivity versus bulk density at
room temperature shows a minimum at around 110-130 kg/m3
blanket density for aluminosilicate fiber blankets. The
location of the minimum in thermal conductivity agrees well
with the results obtained by other researchers for similar
material [32].
1.2.2. Review of Theoretical Models for Heat Transfer in
Fibrous Insulators
The possible modes of heat transfer in a fiber insulating
mat are gas conduction, convection, radiation, and solid
conduction in the fibers. The current heat transfer models
include two or more of these heat transfer contributions to
describe the behavior of overall thermal conductivity in
fibrous blankets. Most describe the temperature and density
dependence of thermal conductivity in a semi-empirical manner.
A number of investigators only fit the experimental data to an
empirical expression.
31
Hayashi [6] developed a purely empirical exponential
relationship (13) between thermal conductivity, k (W/mK), and
temperature, T (C):
k = a exp(b*T) (13)
The constants a and b both have been expressed as functions of
the blanket’s mass density, where
a = 0.08 - 0.17 p + 0.2 p2 (14)
3
3 - 2*10' p (15)
b = 1.9*10’
Blanket density, p, is expressed in kg/m3. Although equations
13-15 are determined empirically, the method of fitting the
data to the equation was not specified. No physical
explanation or reasoning for the exponential temperature
dependence of thermal conductivity was given [6].
For the temperature and bulk density dependencies of
thermal conductivity in glass fiber and rock wool mats the
following expressions were determined using the method of
least squares fit to the experimental data [34]:
For glass fiber temperature dependence,
at p=11.28 kg/m3
32
k = o.13093*10'2+0.69073*10'4'rm+0.83407*10"9'rm3 (16a)
3
at p=13.54 kg/m
k = 0.23126*10"2+0.69073*10'4Tm+0.68968*10'9Tm3 (16b)
3
at p=16.92 kg/m
k = 0.31345*10'2+0.69073*10'4Tm+0.55197*10'9'rm3 (16c)
where Tm is the ambient temperature. The empirically
determined density dependence of thermal conductivity in glass
fiber mats is given by [34],
1 4 1
k = 0.23185*10' +0.75743*10' p+0.23039 p- (17)
For temperature dependence of thermal conductivity in rock
wool insulating blankets,
at p=28.30 kg/m3
k = 0.12130*10'2+0.69073*10'4'rm+0.10509*10'3'rm3 (18a)
3
at p334.00 kg/m
k = 0.23313*10'2+0.69073*10'4'rm+0.85647t10'9'rm3 (18b)
3
at p=42.50 kg/m
k = 0.45981*10'2+0.69073*10'4'rm+0.64887*10'9Tm3 (18c)
33
The empirically determined density dependence of the rock wool
insulators was given by [34]
2 3 1
k = 0.81506*10’ +0.27777*10' p+0.94198 p" (19)
2 4
where, k . = 0.54818*10' +0.69073*10' Tm (20)
311?
King [35] proposed that radiation dominates heat transfer
in fibrous insulations. This model's [35] basic assumptions
are: (l) the coefficients of convective and conductive heat
transfer are constant with temperature, and (2) the radiant
heat transfer takes place between equally-spaced,
hypothetical fiber planes. The spacing between the fiber
planes depends on the insulation bulk density, fiber diameter,
and fiber orientation. Solid conduction in fibers is ignored.
The theoretical expression (21) [35] for apparent thermal
conductivity directly employs the temperatures at the hot-face
and cold-face of the blanket specimen, which may be
advantageous for the guarded hot-plate technique where these
temperatures are readily available [35]. The apparent thermal
conductivity, K is given as:
app '
k = k + as (T - T
app eff (2 1)
i (TH-TC)
34
where, ks gas thermal conductivity, [W/mK]
axi= total insulation thickness, [m]
i= number of radiant heat transfers occurring between
parallel planes (the number of hypothetical
insulation layers is equal to i as long as cold
surface boundary effects are neglected)
TH= hot surface temperature, [K]
TC= cold surface temperature, [K]
a= Stefan Boltzman constant, 5.6697 W/mzK4
6‘“? effective layer to layer emissivity (assumed to be
temperature independent).
The foregoing terms except k, i, and a“! can be measured.
Emissivity is calculated from total hemispherical emissivity
of the insulating fibers. Constants k and i may be determined
from the simultaneous solution of two equations with two
unknowns, using data at two different test temperatures.
Klarsfield et a1. [33] and Bhattacharyya [36] modeled the
thermal conductivity of low density glass fiber insulators.
After a review of pertinent literature, Bhattacharyya [36]
argues that the total heat transfer in fibrous blankets is due
to conduction, convection and radiation where the conduction
term includes both fiber (solid state) and air conduction
effects. The model assumes the medium is totally scattering
and the boundaries of the medium (which are identical to the
specimen surfaces) are opaque to infrared radiation.
35
Convective heat transfer is related to conduction heat
transfer by [36]
w = 1 + qconv/qcond (22)
The convective heat flow through the horizontal slab may
be ignored [36] so that the insulation’s apparent thermal
conductivity combines conductive and radiative conductivities:
4 on(Tm2+T02) 0
pN’ D+1/ea+1/ec-l
k = k
eff cond (23)
Parameter kco d is the average of contributions due to the
n
conduction calculated for the following two configurations:
(1) fibers are assumed to be perpendicular to the heat flow,
and (2) fibers are of arbitrary orientation [36]. a is the
Stefan Boltzman constant. Tm and TD are mean temperature and
half of the temperature difference between the cold face and
hot boundaries, respectively. D is the specimen thickness, p
denotes the mass density of air. N', the specific scattering
parameter is calculated from the experimental data using
equation (23). £8 and cc are total hemispherical emissivity
of the hot and the cold face, respectively. The calculated
value is substituted into equation (23) to test the fit of the
theoretical expression to the room temperature thermal
conductivity versus bulk density data.
Klarsfield et al. [33] combine gaseous, kg, solid, ks,
36
and radiative, kr' thermal conductivities to give kapp
(mW/mK), the apparent conductivity as
kapp = kg + ks + kr (24)
3
0.2572T0'81+0.0527p0'91[1+0.13T/100]+ 4°T L (25)
2/6-1+A
where , A= (1.an
1"]
II
ambient temperature
p= density of the blanket
a= Stefan Boltzman constant
L: air layer thickness
e= emissivity
The assumed heat transfer is between two parallel isothermal
plane surfaces, where the medium is assumed to be absorbing as
well as scattering. In equation (25) one is defined by the
authors [33] as the mass extinction coefficient (mz/kg),
expressing the total developed area of solid phase per unit
mass of this solid phase absorbing and scattering medium. The
mass extinction coefficient is assumed to contribute to the
decrease of the radiative energy through the layer. A further
assumption is that the extinction coefficient is a
characteristic material property that depends on the nature of
the glass, specific surface, product texture (fiber
orientation and separation), and the temperature. The
37
coefficient am is determined optically. Klarsfield et
al.[33] calculated the extinction coefficient from equation
(25) using the thermal conductivity data. A similar method
was used by Bhattchararyya [36] to determine the specific
scattering parameter. The mass extinction coefficient
determined by caloriemetric and optical methods yielded
scattered values, and no obvious trend in the behavior of the
extinction coefficient versus temperature was present.
In a very intensive study of heat transfer in insulating
blankets of glass fibers, Verschoor and Greebler [31]
expressed the apparent thermal conductivity of fibrous
materials in terms of conductivities due to radiation, gas
conduction, convection, and solid conduction in fibers. When
the volume fraction of fibers, f, is small, and thermal
conductivity of the fibers is large relative to that of
ambient gas, k p is expressed as:
aP
(kcd+kcv+krd) + k (2 6)
kapp = (1-f) s
where ks accounts for the heat transfer due to irregular
contacts between fibers. The following assumptions for the
gas conduction conductivity and the radiative conductivity
model were made [31]: (1) the fibers lie in planes parallel
to the mat they form (but individual fibers within the planes
are randomly oriented), (2) the heat flow is perpendicular to
38
the fiber planes, (3) the fibers are of uniform diameter, D,
(4) there are no non-fibrous particles in the blankets, and
(5) the radiative heat transfer takes place between the fiber
planes that are separated from each other by an average
spacing 1f. Given these five assumptions, the conductivities
due to gas conduction, kcd' and due to radiative conductivity,
krd’ are:
kcd = kg lf/(lf+lg) (27)
k = 4aT 3 1 / a2 (28)
rd m f
where k9 is the free gas conductivity at the temperature of
interest. The parameter lf, the mean free path for a gas
molecule-fiber collision, is 0.785 D/f. 1g is the mean free
path for gas molecule-gas molecule collision. For reduced
ambient pressures the following equations define lg:
l = 8l/p at 65 C (29a)
101/p at 148 0 (29b)
At high vacuum levels, thermal conductivity does not
change with a further decrease in pressure. This constant
value of thermal conductivity at low pressures is assumed to
39
be due to radiation and fiber conduction only, since air
conduction and convection become negligible at extremely low
pressures. Vershoor and Greebler [31] argued that the
differences between the calculated thermal conductivity and
the experimental data may be due to convective heat transfer.
Based on this argument the convective thermal conductivity was
estimated as approximately one tenth of the gas conductivity
(at atmospheric pressure at 65 C mean temperature) [31]. The
derivation of equation (27) will be given in section 3.1.2 of
this thesis.
Pettyjohn [29] proposed a relation between the gaseous
conduction contribution to pressure and temperature in fibrous
insulations:
(kp/k i = 1 (30)
p0
1+ Boga (T/T )n+0.5
o
p d
d = 1r D/4f
(kp/kp°)= the ratio of the gaseous conduction at the
environmental pressure and temperature to that
of the gaseous medium at standard conditions
p°= standard pressure, 760 mm Hg
p = environmental pressure
L“; standard mean free path length
40
T = standard temperature, 273 K
H
II
environmental temperature
n a property constant of the gas (for air n
is 0.754)
D.
II
calculated pore size
D = fiber diameter (1.3 micron in Pettyjohn’s
study)
f = ratio of bulk density to theoretical density
The contributions from radiation, fiber solid conduction,
and convection to the overall thermal conductivity are
determined from the typical S-shaped curve obtained when the
overall thermal conductivity is plotted as a function of
pressure [29]. The radiative conductivity is determined from
the constant value of thermal conductivity at extremely low
pressures [31]. The radiation conductivity is used to
calculate the back scattering cross-section of the insulation
from the equation (31):
N = (31)
where, N= back scattering cross-section per unit volume,m-1
k = radiation contribution, W/mK
J= index of refraction of fibers
a= Stefan Boltzman constant
Tm: mean temperature
41
Striepens [30] obtained equation (32) for the effective
thermal conductivity of fibrous insulations after scrutinizing
the existing theoretical models.
where , Tn=
a:
3
k = “T" L + II—’—‘*——(—14—li + C9] (32)
eff
2/e-l+N’L (l-f) 1 1
f+ 9
mean temperature
Stefan Boltzman constant
specific scattering cross section
specimen thickness
— effective pore size
mean free path of the gas at the temperature and
pressure of interest
volume fraction of fibers
emissivity of the fiber surface
empirical correction term for solid and solid-to-
solid contact conduction
density of the blanket
Equation (32) combines parts of previous models [36, 31].
Striepens [30] assumed parallel fiber planes separated by
the air layers. Radiation transfer occurs by a series of
scattering reflections at the fiber surfaces with absorption
and re-emission of radiation by the fibers. It is assumed
that at atmospheric pressures the gas entrapped in the blanket
pore volume behaves like the free gas.
42
The specific back-scattering cross section was obtained
from room temperature total infrared transmittance
measurements. The effective thermal conductivity data versus
pressure agreed well within 10 percent of values calculated
from equation (32). The total infrared back scattering cross-
section measurements showed considerable spread in the data.
The discrepancy between the experimental data and the theory
is argued to stem from the spread of the radiation parameters
[30].
43
2. EXPERIMENTAL PROCEDURE
For the thermal conductivity measurements of fibrous
insulating blankets, the hot-wire technique was employed
throughout the temperature and pressure range. A schematic of
the experimental set-up is shown in Figure 5. For thermal
conductivity measurements at elevated temperatures, the sample
assembly was placed in an electrically heated furnace (section
2.2.6) (f in Figure 5). The blanket specimens were compacted
by applying uni-axial load perpendicular to the 150 mm x 150
mm face of the specimens via the pressure fixtures described
in section 2.2.5.
2.1. Sample Characteristics
Samples of 7 blankets, 3 paper products, and one cloth
type product made of ceramic fibers from three different
manufacturers and a graphite fiber insulating blanket from a
foreign producer were available. All the products were cut
using regular household steel scissors as square blocks of 150
x 150 mm (6" x 6") and the blanket specimen thickness was at
least 13 mm (1/2") on both sides of the heating element where
44
.... xx 8
T11 (l 1
s H DCP
hW
fl
TCl
TlC3 ATCZ
., .. b l T12 I
A TC3
a = Ammeter; V = Voltmeter; b = Isolating box; f = Furnace;
s = Specunen; DCP = DC power supply; hw = Hot wire;
TCl = Sensing Thermocouple; TC2 = Far Field Thermocouple;
TC3 = Thermocouple # 3;
T11 = Temperature Indicator. #1; T12 = Temperature Indicator # 2;
T1C3 = Furnace Control Umt
Figure 5. Experimental apparatus for thermal conductivity
measurements at elevated temperatures.
45
applicable. The chemical compositions and some physical
properties of the samples are given in Tables 2 to 6.
The blanket insulation Locon and Fibersil Cloth were of
light brown color as received from the manufacturer, but these
changed in color to creamy white after being fired to elevated
temperatures. During the high temperature experiments on
Locon and Fibersil cloth no unpleasant odor developed. The
observed color change may be due to the burn-out of the trace
materials reported in the manufacturer's catalogues, or due to
the binder burn-out. Whatever the source of the color change
and whatever volatiles may have been produced, there was no
noticeable odor that accompanied the color change.
Paper products Kaowool K2300 paper, Ultrafelt paper, and
Carborundum 970 Paper have organic binders and show mass loss
on ignition. During the firing to 200 C, an intense,
irritating odor developed from these products. The binder
burn-out probably caused the odor.
Specimens of graphite blanket were heated up to 800 C in
a rapid high temperature furnace (C&M Co., Bloomfield,NJ).
Between 770-780 C the graphite fibers burned out completely
without leaving any ash behind, if fired in air atmosphere.
Since this work only involved elevated-temperature thermal
conductivity measurements in air atmosphere no elevated-
temperature experiments with the graphite blanket were
conducted.
46
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49
Table 5. Chemical compositions and selected physical
properties of ceramic fiber blankets as listed
in manufacturer's catalogues [41].
Product manufacturer: Zircar Fibrous Ceramics, Florida, N.Y
Properties Saffil
Chemical composition
Al O % . 95.0
$1523 % 5.0
TiO2 % -
ZrO % -
Fe 0 % -
otï¬ef % -
Specific gravity 3.4
Fiber diameter 2-4
(microns)
Bulk density 48.0 *
(kg/m )
Blanket thiCkness 28
(milimeter)
* Measured in this study.
50
Table 6. Chemical compositions and selected physical
properties of graphite fiber blanket as listed
in manufacturer’s catalogues [42,43].
Product manufacturer: Osaka Gas Co., Ltd., Osaka 541, Japan
Properties LFK-220
Chemical composition
Carbon % 99.0
Ash content % 0.04
other % -
Area wcight 2000.0
(9/111 )
Fiber diameter N.A *
(microns)
Bulk density 0.1
(Q/CC)
Blanket thickness 20
(millimeter)
* N.A - not available
51
2 . 2 . Equipment
2.2.1. Electrical Equipment
A 2 kHz rectified frequency DC power supply built by
Michigan State University Electronics and Computer Services
regulated the power input to the hot wire. The device
operates as a current controlled power source. However,
current supply from the power source was kept constant for all
the experiments. The current supplied to the hot wire was
constant to within 1.5 percent. To limit the power output of
the heating wire a resistance in series with the hot wire was
installed (not shown in Figure 5). The series resistance of
6.5 Ohms (10.2 Ohms for elevated temperature experiments) was
made of Chromel wire of 0.38 mm (0.015") diameter and 100 cm
(170 cm) in length.
2.2.2. Measuring System
A Chromel-Alumel K-type thermocouple sensed the
temperature rise in the vicinity of the hot wire within the
insulator specimen. The output of the sensing thermocouple
was simultaneously fed into a strip chart recorder over an ice
point and into a temperature indicator. A second K-type far
field thermocouple measured the ambient temperature reigning
within the specimen. Two multimeters monitored the current
flowing through the hot-wire circuitry, and the voltage drop
52
across the hot wire.
2.2.2.1. Electronic Devices
2.2.2.1.1. Strip Chart Recorder
A strip chart recorder (Omega RD-145 Mv-min, Omega Eng.
Inc., Stamford, CT) monitored the thermocouple output during
the room temperature experiments. For the elevated
temperature experiments, the millivolt equivalents of the
measured temperatures were beyond the range of the strip chart
recorder. It operated with a reading sensitivity of l mV/cm
and the full range of the device was 10 mV. The experiments
were conducted with 0.127 mm (0.005") diameter heating wire
and thermocouple wire. A chart speed of 4 cm/min gave the
best results in terms of accuracy and ease of reading the
recorded data. For lower chart speeds, the slope of the
temperature versus time curve was too steep to get good
readings. Each experimental run was marked on the recorder
paper and all the data was stored in one file.
2.2.2.1.2. Digital Multimeter
A Fluke-77 type multimeter [John Fluke MFG Co.,
Everett,Wa) monitored the current supply to the hot wire with
a precision of 0.01 amperes. A second multimeter of the same
kind was used: a) to read the output voltage of the
53
thermocouple as a double check to strip chart records at room
temperature experiments; and b) to monitor the voltage drop
across the hot-wire assembly. The resolution of the device
for the voltage readings in 300 mV full range was 0.1 mV, and
for the range 0-30 volts it was 0.01 volts.
2.2.2.1.3. Ice Point for K-type Thermocouple
As the reference (or "cold") junction for the
thermocouples MCJ-K Omega (Omega Eng. Inc., Stamford, CT)
miniature cold junctions were used.
2.2.2.1.4 Temperature Controllers (Indicators)
To observe the temperature change in the vicinity of the
hot wire an Omega CN5001K2 (Omega Eng. Inc., Stamford, CT)
type temperature controller with a sensitivity of 1 C and a
full range of 0 to 1000 C was employed. (Thus, only the
digital temperature indicator function of the controller was
used, and the control function was not utilized. The ambient
temperature was monitored with a second temperature controller
of type CN300K-C. The range of the controller was 0 to 1300 C
with a sensitivity of 1 C. Both temperature controllers
employ K-type thermocouples as probes. [Vendor: Omega
Engineering Inc., Stamford, CT]
54
2.2.2.2. Thermocouple Wires
The thermocouples were produced in this study by twisting
the wires and electrically welding the ends of the wires
together using a high current welder [Select-Amp, CRC the
Chemical Rubber Co., Cleveland, Ohio]. A welder current of
2.5 amps was used for the 0.127 mm wire and 5 amps was used
for the 0.254 and 0.38 mm diameter wires. Three different
sizes of K-type thermocouples were made of commercially
available Chromel and Alumel types wires. The wires were
0.127 mm, 0.254 mm and 0.38 mm in diameter [Vendor: Omega
Eng. Inc., Stamford, CT]. During the initial phase of this
research, 0.127 mm and 0.38 mm diameter thermocouples measured
the temperature rise in the vicinity of the hot wire. The
0.38 mm diameter wire thermocouple, with a bead size slightly
bigger than twice the diameter of the wire, responded slowly
to temperature changes. For example, using the 0.38 mm
diameter thermocouple wires, the temperature rise was first
sensed approximately 3 seconds after the power supply had been
turned on. The observed time lag may be because of the
increased thermal capacity of the heating wire and
thermocouple due to the increased wire diameter. This lag was
seen on the strip chart records for the fourth and fifth runs
of experiments done with k-2300 Kaowool blankets. In order to
minimize the time lag in the response of sensing devices,
subsequent room temperature experiments were conducted with
55
0.127 mm (0.005") diameter thermocouple wires. The
thermocouple bead size was approximately 0.15 to 0.2 mm.
The 0.127 mm diameter thermocouple wires were impractical
for elevated temperature experiments since the rate of
oxidation of the wires was so high that the wires had to be
replaced following every elevated temperature experiment above
800 C. Therefore elevated temperature experiments employed
0.254 mm diameter k-type sensing thermocouples, which had to
be replaced every other high temperature experiment.
Oxidation thinned the thermocouple wires, especially the
alumel wire. An impending failure (breakage due to oxidation)
of a thermocouple could be sensed during an experimental run
by a reduced thermocouple output. For the apparatus used in
this thesis, this error manifest itself as a difference of 4
to 6 C in the outputs of the sensing thermocouple and the far
field thermocouple for stabilized specimen temperatures.
In summary, the thermocouple and heating wire
arrangements employed allowed for optimum thermal response.
The 0.127 mm diameter wires used for both the heating wire and
the thermocouple wires minimized the response time of the
sensing devices. The parallel arrangement of the heating wire
and the thermocouple minimized the escape of heat from the
area of measurement. The temperature rise in the vicinity of
the hot wire was observed less than 1 second later than the
start of the power input.
56
2.2.3. Heating Wires ( Hot Wire )
The hot wire was commercially available Chromel wire with
70.6 microohm-centimeter resistivity at 20 C (68 F) [44]. The
resistivity values for chromel wires, calculated in this study
using the experimental current and voltage readings, were 5
percent higher than the listed electrical resistivity values
[44]. Two different sizes of heating wire were used. First,
a heating wire of 0.38 mm (0.015") in diameter [12] in
conjunction with 0.38 mm (0.015") diameter thermocouple wires
was employed (section 2.2.2.2). To eliminate the time lag as
much as possible a thinner wire size of 0.127 mm (0.005") was
used for both heating wire and thermocouple. The reduced
heating element diameter coupled with the high electrical
current caused an extremely rapid temperature rise in the
specimen.
For the elevated-temperature experiments, 0.38 mm
diameter chromel wires connected the hot wire (Chromel 0.127
mm diameter) to the power leads. Figure 6 is a schematic of
the hot wire, lead wires, and thermocouple arrangement within
the specimen. The two different sizes of Chromel wires
(heating element and lead wires) were twisted and welded
together by an electrical welding machine (section 2.2.2.2)
using 5 amperes current. Typically, the hot-wire lead-wire
assembly failed at the junction after high temperatures.
Before deciding to use Chromel lead wires, copper wires of
57
Sensing
Thermocou
Far-field
Thermocouple /
. H
Lead-Wire
\
ple
Specimen
>L/i
Temperature
indicator
——1
T
//\
Temperature
indicator
'Voltmeter
DC Power supply
Hot-wire
/
Lead-wire
Furnace not shown
Figure 6. Wiring set-up for elevated temperature experiments.
58
0.38 mm diameter were considered as lead wires. But it is
very difficult to weld two wires of different materials and
different cross sections.
2.2.4. Isolation Box
An isolation box was employed for measurements of thermal
conductivity of as-received (uncompressed) blanket specimens.
The isolation box was a 180 mm x 180 mm x 75 mm (7" x 7" x 3")
box made of Zircar-Alumina Silicate boards. The isolation box
helped to stabilize the ambient temperature by restricting the
air flow around the specimen and providing an insulating
medium between the test specimen and the surroundings. Room
temperature experiments were conducted with and without the
alumina silicate isolation box. The experiments without the
isolation box exhibited small perturbations in the time-
temperature data. Although the scatter in the time—
temperature data increased in the absence of an isolation box,
the slope of the logarithm of time versus temperature (and
hence the computed thermal conductivity value) remained
essentially unchanged by the scatter in the data. A second
isolation box of 180 mm by 180 mm by 75 mm was built from
Kaowool M-12 aluminosilicate boards of thickness 12.7 mm
(1/2") (Thermal Ceramics, Augusta, GA). The subsequent
experiments with.uncompacted specimens were done using these
two isolation boxes.
59
2.2.5. The Pressure Fixtures
2.2.5.1. Steel Pressure Fixture
The first fixture used to compress the fiber blankets
(Figure 7) had upper and lower plattens of high strength-high
temperature steel (CRS 1020). Commercially available M-12
studs of length 127 mm and their matching nuts were used in
the fixture. A downward movement of the steel plate acted to
compress the refractory blanket specimens. The blanket
thickness was measured by the advance of the upper plate. The
mass density of the blankets was calculated from the blankets'
change in thickness.
Two plates of M12-type aluminosilicate boards of 12.7 mm
thickness (Thermal Ceramics, Augusta, Ga.) were placed between
the specimen and the steel plattens to prevent thermal contact
between the highly conductive steel and the low conductivity
ceramic blankets.
The steel fixture oxidized excessively in the first high
temperature experiment (1000 C).
2.2.5.2 Refractory Brick Compression Fixture
Another pressure fixture was constructed from K-26
refractory bricks (Thermal Ceramics, Augusta, Ga) (Figure 8).
The blankets were compressed to the desired thicknesses with
the help of two six inch C—clamps. The compressed blankets
60
A J
I I B I I 12.7
i i
S
D ,6 cl D
S
I B l i i
A 19.0
‘ l
A 152.4
A 203.2 T
METRIC ALL DIMENSIONS IN MlLLIMETERS
A - Steel Plates
B - Aluminosilicate Board Plates
C - Studs
D - Aluminosilicate Board Stops
S - Specimen
Figure 7. Steel ï¬xture for compacted specimen experiments.
61
were then placed in U-shaped bricks of length 203 mm and width
115 mm (Figure 8). Stops prepared from aluminosilicate boards
adjusted the amount of compression (D in Figure 7). The U-
shaped fixture was cut from a standard K-26 refractory brick
of initial dimensions 230 mm by 115 mm by 50 mm (9" x 4.5" x
2") using a hack-saw with alloy steel blades. The corners and
the surfaces were finished with a wood file. The stops were
cut with the same hack-saw and the finishing to dimensions was
done by 240 grade wet emery paper. The refractory brick
compaction fixture turned out to be very simple but effective.
After the placement of the specimen assembly in the fixture,
the thickness of the specimen was again measured using a
Vernier Caliper. The fixture with the specimen assembly was
placed in the center of the furnace cavity. The refractory
brick compression fixture was limited in the pressure that
could be applied to the blanket specimens. For example, the
refractory fixtures fractured routinely during experimentation
with Kaowool K2300 blankets at a compaction ratio of 0.2 and
for Locon balnkets at a compaction ratio of 0.3. Failures in
the refractory brick compression fixtures typically originated
from the corners of the U-shaped refractory.
2.2.6 Electrically Heated Furnace
The elevated-temperature environment for the experiments
was provided by an electrically heated furnace (Lucifer
62
203.2
TC
L
,2
HW To temperature To power
indicator supply
ALL DIMENSIONS IN MILLIMETERS
A = Pressure brick .
B 2 Upper. and lower bI’le plates
C = Alumma; 8111cate board
D 2: Compaction stops
C = Sensing thermocouple
HW 2 Hot mm
Figure 8. Fixture for thermal conductivity measurements
as a function of compaction at elevated temperatures.
63
Furnaces Inc., Neshaminy, PA), which employs eight
siliconcarbide resistively heated furnace elements. Although
the maximum use temperature of the furnace was 1500 C, all
experimentation in this thesis was performed at temperatures
no higher than 1050 C. The temperature in the furnace cavity
is sensed with a R-type thermocouple. A Barber Collman R-type
temperature controller provided a cavity temperature stability
of +/- 2.5 to 5.0 percent depending on the heating rate.
Higher heating rates caused larger fluctuations in the
temperature. The high thermal mass of the furnace aids
temperature stability, but the cooling cycle of the furnace is
very extended for the same reason. I
The furnace cavity is 300 mm by 300 mm by 230 mm. The
furnace has two opennings, one door through which the specimen
assembly was placed, and a 15 mm (9/16") diameter tubular
opening through which the thermocouples and hot wire leads
were taken out. During the high temperature thermal
conductivity measurements, the refractory brick compression
fixture and the specimen containing the hot wire and the
thermocouples in it were assembled outside the furnace. Then
the assembly (specimen plus refractory brick compression
fixture) was placed directly on the floor of the furnace
cavity. After the placement, leads for the hot wire and the
thermocouples were led out through the tubular opening in the
back of the furnace to be connected to the power circuit.
64
2.3. Specimen Assembly
The experiments with the uncompacted refractory blanket
specimens utilized the isolation box, described in section
2.2.4. Square blocks of blankets 152 mm x 152mm (6" x 6")
were placed into the isolation box. The hot-wire and
thermocouple assembly was placed between the layers of
blankets. For elevated-temperature thermal conductivity
measurements, the whole sample assembly, including the
isolation box, was put into the furnace (section 2.2.6) For
all materials the original thickness of the specimen was kept
at least 13 mm (1/2") on both sides of the hot wire. For
Kaowool K2300, Kaowool K2600, Locon, Fibersil cloth,
Durablanket D-S, Ultrafelt, and K2300 paper several layers of
the materials were stacked on top of each other to provide the
reported thicknesses.
Two different thermocouples (section 2.2.2.2) were
employed to get the temperature data within the specimen.
Near the hot wire, the temperature rise was measured with a k-
type sensing thermocouple. A second far field thermocouple
measured the ambient temperature. The positioning of the hot
wire, thermocouple wires and measuring junction relative to
each other and relative to the sample was as shown in
Figure 6. The sensing thermocouple wires ran parallel to the
65
hot wire within the specimen to reduce the heat loss through
the thermocouple wires.
The sensing thermocouple was placed in the center of the
specimen, as close as possible to the hot wire yet not
touching the hot wire. On the average, the gap between the
measuring junction of the sensing thermocouple and the hot
wire varied from 0.5 to 1.2 mm, with the extreme cases of 0.0
mm and 2.5 mm gap distances. A constant gap distance between
the thermocouple and the hot wire could not be assured for two
reasons: (1) a groove to insert the hot wire and the
thermocouples cannot be carved into the blanket products as it
is done with refractory brick specimens [7, 12, 13], and (2)
the thermocouple wires and the hot wire were too thin to apply
enough force to assure that they were straight. Parallelism
between the hot wire and the thermocouple wires throughout the
specimen was also a problem with relative inclination angles
of zero to twenty degrees occuring between the hot wire and
thermocouple. Small spatial shifts of the thermocouple
relative to the hot wire inevitably occurred when the thermal
conductivity apparatus (the refractory blanket, hot wire,
thermocouples and isolation fixture) was assembled. Changes
in the gap distance caused the thermal conductivity to vary up
to 5 percent with respect to a standard assumed gap distance
of 0.8 mm. Errors introduced by non-parallel thermocouple and
. hot wire were neglected since the thermocouple wires used in
this work had very small diameter (0.254 mm), so that the heat
66
losses through the thermocouple wires were negligible.
The gap distance was measured using a vernier-caliper
with the precision of 0.05 mm. The accuracy of the
measurements was not better than +/- 0.20 mm because of the
above mentioned reasons. The gap could not be measured after
the upper blanket layer was placed over the thermocouple and
hot wire. The curvature of the wires was assumed not to
change when the upper layer(s) of the blankets were put in
place.
After thermal equilibrium was reached (as gauged by an
ambient temperature change of less than 1 percent over a four
minute period), heating of the hot wire commenced at time t=0
via a constant electrical current, supplied and regulated by
the DC power supply (section 2.2.1). Temperature-time
readings were then recorded for 3 to 4 minutes during the
heating of the refractory blanket by the hot wire. For the
room temperature experiments, the thermocouple output was
recorded by the strip chart recorder. For the elevated
temperature experiments the millivolts equivalent of the
specimen temperatures were out of range of the strip chart
recorder. The data were recorded manually as time-temperature
pairs. Time was measured using an electronic stopwatch. The
strip chart recorder curves (section 2.2.2.1.1) from the room
temperature experiments very closely resembled a logarithmic
plot. The time-temperature data were plotted manually as
temperature versus natural logarithm of elapsed time which
67
gave an S(sigmoid)-shaped curve for the time range plotted.
The middle portion of this S-shaped curve is a straight
line with the slope proportional to the thermal conductivity
of the material surrounding the hot wire. A non-linear least
squares fitting program was employed to determine the slope of
the straight line region. Thermal conductivity then was
calculated from the slope and the power input using the
relationship reported by Carslaw and Jeager [45].
k = (Q * ln(t1/t2)) / (4 * 1r * 1 * (TZ'Tlll (5)
2.4. Calculation of the Effective Voltage Drop across the
Heating Wire
The voltage across the hot wire and the current through
the hot wire changed with the ambient temperature since
resistivity of hot wire is temperature dependent. The
potential drop across the heating element within the specimen
was calculated based on the measured potential drop across the
entire hot-wire and lead assembly. Changes in the hot wire's
and the lead wires' resistance were assumed to be proportional
to their room temperature resistance (Figure 9), according to
the following calculation:
68
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total resistance of hot wire-lead wire
assembly at room temperature
resistance of the hot wire at room
temperature
resistance of lead wires at room temperature
- resistance of the segment of the lead wires
which are heated by the furnace
Length of the lead-wires in the furnace
Total length of lead-wires
2.5. Refractive Index Measurements
The refractive indices for the three blankets, Saffil,
Locon, and Kaowool K2300 were determined experimentally using
the minerological oil technique of geology [46]. This very
simple technique employs a transmitting optical microscope
70
(Nikon Labophot-pol, magnification 40-400) and a set of
different grades of minerological oils, each of which
correspond to a certain index of refraction value.
In the process of measuring the refractive index, the
fibers are placed into the oil on a slide glass. A filtered,
polarized light (sodium light) is sent through the oil which
now contains the fibers. According to the relation between
the refractive indices of the oil and the fiber, the mixture
appears in a color between blue and red. A red color
indicates that the fiber refractive index is higher than the
refractive index of the oil, and blue color indicates that the
fibers have a lower refractive index than the oil. The
coloring become stronger if the refractive indices of the
fibers and the oil are of similar values. If an orangy color
is observed, the index of refraction of the oil matches that
of the fibers [46].
A way to determine whether the refractive index of the
fiber is greater than the refractive index of the oil is to
observe the displacement of the fiber boundaries as the focus
is raised [46]. If the fiber refractive index exceeds the oil
refractive index, the fiber boundaries (Becke Lines) move
towards the center of the fiber as the focus is raised (Figure
10a). The opposite occurs if the refractive index of the oil
is greater than the fiber refractive index (Figure 10b). At
the point where the two refractive indices match one can
observe that the boundaries of the oil and the fibers move in
71
ï¬ber
oil
raise focus
noil > â€fiber
8) Sodium Light
ï¬ber
. raise focus
noil < "ï¬ber
b) Sodium Light
ï¬gure 10.. Mineralogical oil method to determine the refractive index.
3’ noil > "ï¬ber '3) â€oil < "ï¬ber [451°
72
opposite directions as one changes the focus. With the help
of the minerological oil technique, the refractive index of
minerals can be determined up to five significant figures if a
complete set of oils is available. The most suitable range of
refractive index for the oil technique is below 1.7. The oils
for higher refractive indices degrade over a short time [46].
The set of oils available covered a range of indices of
refraction between 1.40 to 1.71, which was sufficient for the
fibers involved in the blankets.
73
3. RESULTS and DISCUSSION
Measurements of thermal conductivity in ceramic fiber
refractory blankets were conducted using the hot-wire
technique. The hot-wire technique and the experimental
procedure were detailed in Sections 1 and 2 of this thesis,
respectively. A theory on the thermal conductivity in fibrous
refractory blankets was developed in Section 3.2. The theory
accounts for the contributions to overall thermal conductivity
by radiation, gas conduction, convection, as well as series
and parallel fiber solid conduction. The discussions of the
individual thermal conductivity contributions due to
radiation, gas conduction, convection, and solid conduction
were given in Sections 3.2.1, 3.2.2, 3.2.3, and 3.2.4,
respectively.
3.1. The Temperature Dependence of the Thermal Conductivity of
Ceramic Fiber Insulating Blankets
Experiments were conducted with ceramic fiber insulating
blankets to determine the temperature and compaction
dependency of thermal conductivity. Eight commercially
available ceramic fiber insulating blankets were investigated.
The experiments were run with no load applied to the blankets
(the as-received densities). Among these eight insulations
74
Fibersil, a product with metal wire meSh, was only tested to
600 C because of physical characteristics of one of its
constituents, whereas the other blankets were heated to 970 C
during the experiments. Tables B01-BSO in Appendix B list the
thermal conductivities of various blanket products tested in
as-received condition at temperatures from room temperature
to 970 C. The thermal conductivity versus temperature curves
are smooth for each material tested (Figures lla-llh). The
thermal conductivity of the blankets increases with increasing
temperature. It should be noted that the rate of increase in
thermal conductivity, however, is different for each blanket.
The rate increase varies among the different ceramic
insulating blankets due to their chemical composition, initial
density, and probably due to the manufacturing technique
applied, which determines the distribution of the fibers in
the blanket. Thus, the overall thermal conductivity of
insulating blankets is a composite property, which is related
to extrinsic features of the blankets, such as the initial
density and the distribution of the fibers in the structure,
as well as the material properties of the fibers and the gas
filling the pore volume.
The exponential-like rapid increase in thermal
conductivity in these blankets is due to the enhanced
radiative heat transfer and air conduction process at high
temperatures.
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3.2. The Model for the Overall Thermal Conductivity in Fibrous
Refractory Blankets
The following discussion is included to point out the
inadequacy of the existing models to describe the behavior of
the thermal conductivity versus temperature and versus density
for fibrous ceramic refractory blankets. The similarities and
the contrasts among two of the existing models and the theory
developed in this thesis are discussed in this section.
To investigate the temperature dependence of thermal
conductivity in the blankets, the logarithm of thermal
conductivity was analyzed with respect to the logarithm of
temperature. The analysis did not yield a linear relationship
between the logarithm of thermal conductivity and logarithm of
temperature, not even for the high temperature region. Thus,
a power-law dependence on temperature was not supported by the
observations. Neither did examination of the logarithm of
thermal conductivity versus temperature data reveal a
consistent straight-line portion. Hayashi [6] states that the
logarithm of thermal conductivity versus temperature data in
aluminosilicate fiber blankets gives linear relationship for
four different densities. However, Hayashi's data [6] can
only be fit to a straight line for the two blanket mass
densities, 0.243 gr/cm3 and 0.430 gr/cm3 [6].
The K-2300 Kaowool blankets tested in this study had
identical chemical composition to one of the insulator
84
blankets Hayashi [6] investigated. The measured density of
the specimen (this study) was about 0.098 gr/cm3. Only the
data from the experiments (this study) at 0.5 compaction ratio
can be fit by a straight line in the natural logarithm of
thermal conductivity versus temperature plot. The K2300
specimen at 0.5 compaction ratio in this study corresponds to
Hayashi's specimen with 0.243 gr/cm3 bulk density, which was
the only data set that gave a straight line. For higher
compaction ratios, i.e lower densities, the ’nearly' linear
portion of the curve was constrained to the 600-1000 C region
of the graph. The thermal conductivity data for 0.098 gr/cm3
bulk density blanket (this study) and the thermal conductivity
data for 0.106 gr/cm3
bulk density specimen [6] are very
similar, but neither data set is linear in a logarithm of
thermal conductivity versus temperature plot. Since two
different studies (this study and [6]) produced similar
results, it cannot be argued that these deviations from a
straight line in the plot of logarithm of thermal conductivity
versus temperature are due to experimental error.
A. J. Jackson et al. [15] used the German Standard DIN
51046 hot-wire procedure and experimented with Saffil blanket
of 0.096 gr/cm3
(6 lb/cuft) bulk density. Jackson [15]
devised an apparatus to measure the thermal conductivity of
fibrous ceramic insulator blankets. With the hot-wire
technique, the thermal conductivity of Saffil blanket was
measured between room temperature and 1600 C [15]. The same
85
kind of refractory blanket was also investigated in this
thesis. Based on the results, Jackson et a1. [15] suggest
that the data correlates better to an exponential function
than to the T3 expression predicted by radiation heat transfer
theory. But no theoretical work on the issue was done,
neither was an equation describing their data developed.
In the analysis of heat transfer in ceramic fiber
insulating blankets, radiation is an important component of
the entire heat transfer process. However, it is not the only
mode of heat transfer occurring in the insulators, not even at
temperatures exceeding 1000 C. J.D. Verschoor and P. Greebler
[31] reported that air conduction within the pore volume in
the insulators is by far the most important mode of heat
transfer at room temperature. The data on thermal
conductivity of air at various temperatures in Table C1 (which
is taken from the Table Bl in [47]) shows that the heat
conduction in air within the pore volume contributes
significantly to the overall thermal conductivity of the
composite blanket.
For example, at 1200 K (930 C) the thermal conductivity
of air is 0.0782 W/m K. The thermal conductivity due to
radiation in alumina fiber blanket can be calculated using the
expression given by Kingery et al. [48]:
2 3
k =16/3 * a * n * T * 1 (36)
f
86
where n is the refractive index of the fibers. The refractive
index for alumina may be estimated as 1.748, which is the
average of the two values n=1.76 [48] and n=1.736 [49]. The
Stefan Boltzman constant a is 5.6697E-08 W/m2 K4. The mean
free path for photon conduction, 1f, (equivalently the average
pore size) is calculated using the analysis in [31]. With
these values, the radiation thermal conductivity is 0.089 W/m
K for alumina fiber blankets, which is comparable to gas
conduction thermal conductivity for air at that temperature.
Thus, gas conduction and radiation have comparable
contributions to the overall thermal conductivity of the
composite insulating blanket for the temperature range covered
in this work.
The heat transfer model developed in this thesis is based
on the argument that these insulating blankets are composite
structures made of air and ceramic fibers. The air, or the
pore volume, is regarded as the continuous, interconnected
phase. It is assumed that the air and the majority of the
fibers build planes that are stacked on top of each other. The
normal directions to the air and fiber planes are parallel to
the direction of heat flow. Furthermore, a small fraction of
fibers is assumed to cross the majority fiber planes.
Dispersed in a matrix of air, the fibers occasionally touch
each other.
Within the pore volume, heat transfer processes occur by
radiation, gas conduction, and free gas convection. In the
87
model these three modes of heat transfer in the pore volume
and a fraction of solid conduction in the ceramic fibers act
in parallel. Thus, these terms combine to give an effective
thermal conductivity for the composite blankets. The last
contribution comes from the series solid conduction in the
parallel majority fiber planes.
Based on these considerations the expression (37) is
developed to model the overall thermal conductivity in ceramic
fiber refractory blankets.
2
f)
-——— = + -—————— (37)
l (l-v
keff [krd+(1+r)*kgcd](l'vf’+(Vf/°6’*ksidc ksldc
where, keff = overall thermal conductivity in fibrous
refractory blankets
vf = volume fraction of fibers
krd = radiative thermal conductivity
kgcd = thermal conductivity due to gas conduction
ksldc = thermal conductivity due to fiber solid
conduction
r = ratio of the convective heat transfer
contribution to gas conduction contribution
for overall heat transfer process in fibrous
88
blankets
C = empirical term related to the fraction of fibers
that are crossing the majority fiber planes
The model for the overall thermal conductivity of ceramic
fiber refractory blankets (equation (37)) utilizes observable
physical quantities as much as possible. Nevertheless,
equation (37) does include the empirical term C6' In
Sections 3.2.1 through 3.2.5 each term in equation (37) is
discussed in detail.
3.2.1. Radiation Thermal Conductivity
The radiative thermal conductivity of the insulating
blankets is a function of the combined effects of the average
lambient temperature and the material properties (such as the
index of refraction or the absorption coefficient). In
addition, radiative thermal conductivity is also a function of
composite properties such as pore size, which measures the
photon mean free path (the average distance a photon travels
before it scatters from or is absorbed by an object).
According to the classical theory of radiation, Kingery [48]
gives two expressions for radiation thermal conductivity:
89
2
krd =(16/3) * (a * n * T3)/ a (33)
=(16/3) * (a * n2 * T3 * 1f) (35)
where, krd = radiation thermal conductivity, in W/mZK or W/mK
a = Stefan-Boltzman constant, in W/mZK4
n = Index of refraction, dimensionless
T = absolute temperature, in K
a = absorptivity or absorption coefficient,
dimensionless
lf = mean free path, in m
Equation (38) has units of W/mZK and equation (36) has units
of W/mK, which are the more convenient thermal conductivity
units for this study. If the mean free path for photons
within the insulation is expressed in microns, a conversion
factor of 10"6
should be employed since all the other terms
are conveniently expressed in meters.
For the radiative conductivity, Verschoor's [31] analysis
yields a similar expression:
krd = (4 * a * T * 1f)/ a2 (28)
90
where the terms are the same as listed for previous equations.
Among these three expressions (36,38,28), expression (36)
given by Kingery was used in this study since it does not
involve the absorption coefficient of the fibers. The
absorption coefficient presents a difficulty since
absorptivity is a strong function of the surface conditions of
the material used, and in this work the absorptivity of the
specific fibers involved in the blankets was not measured.
FTIR spectroscopy [31] seems very promising for determining
the absorptivity of ceramic fibers as a function of incident
optical wavelength. The listed values of the absorption
coefficient for alumina ranged from 0.22 to 0.595 in different
references [50, 51].
The difficulty in determining an appropriate fiber
absorptivity lead to using:
k= (16/3) * a * n2 * T3 * 1f (36)
for modelling radiation thermal conductivity. The two
parameters in equation (36), refractive index, n, and the mean
free path for photon conduction, 1f, are discussed in Sections
3.2.1.1 and 3.2.1.2 in details.
91
3.2.1.1. The Refractive Index
Theoretical calculations of thermal conductivity make use
of the measured refractive indices of the fibers. The
measurements with the minerological oil technique (Section
2.5) gave refractive index values for the three kinds of
fibers that are appreciably different than the "weighted
average†indices that one would estimate by taking the
weighted averages of refractive indices of individual
constituents (Table 7). These estimated values assumed that
the mixture from which the fibers are made is mainly composed
of crystalline forms of the chemical species in the mixture.
However, the measured refractive indices indicate that this
assumption may not be well suited for Locon and Kaowool K2300
fibers. X-ray analysis for examination of the phases in the
fibers and FTIR Spectrography for direct determination of
refractive indices of fibers offer solutions to these
problems.
In the literature, the data on the temperature dependence.
of refractive index of ceramics and glasses are extremely
inadequate. A. J. Moses [52] measured the refractive index of
corundum (alumina) at 293 and 1773 degrees Kelvin. The
temperature dependence of the refractive index for corundum
obeys the relationship dn/dT = 10"5 K.1 over the temperature
range between 293 and 1973 degrees Kelvin, in the wavelength
region from 0.56 to 4.0 miCrons [52].
92
Table 7. Measured and Weighted Average Refractive Indices of
the Fibers
Fiber Measured refractive weighted average*
Index Refractive Index
Saffil 1.68 1.75
Locon 1.55-1.56 1.68
Kaowool** 1.55-1.56 1.66
* These values assume that the mixture from which the
fibers are made is mainly composed of crystalline
forms of chemical species in the mixture
** Kaowool K2300 Fibers
93
In this thesis it is assumed that the refractive index
has the same kind of dependence on temperature as the linear
thermal expansion coefficient. The refractive index is
closely related to the interplanar spacing of the material.
Thus, the changes in the spacing of atomic planes are assumed
to directly affect the refractive index of the material [53].
The temperature dependence of the linear thermal expansion
coefficient for alumina [48] was used to describe the
temperature dependence of the refractive index for Saffil
blanket. A comparison of the dn/dT value [52] and the
temperature dependent behavior of the linear thermal expansion
coefficient [48] shows that the dn/dT for corundum can be
approximated by the average change of the linear thermal
expansion coefficient for alumina over the temperature range
293 to 1900 degrees Kelvin.
Temperature dependence of the linear thermal expansion
coefficient of mullite was employed in place of temperature
dependence of mullite refractive index in conjunction with the
mullite solid thermal conductivity data during the theoretical
investigations. The temperature dependence of the linear
thermal expansion coefficient for fused silica (another
candidate material to approximate the solid thermal
conductivity in aluminosilicate fibers) is extremely low (=0.5
x 10.6 in/in C over the temperature range 0-1000 C). Thus, in
the calculations the change in the refractive index with
temperature for silica glass was neglected.
94
3.2.1.2 The Mean Free Path for Photon Conduction
The term 1 the mean free path for photon conduction is
f:
calculated from a model describing the fibrous blanket
structure. The photon mean free path is closely related to
the bulk density of the blanket. As the mat is compressed,
the mean free path decreases according to the changes in the
bulk density.
To determine the mean free path for photon conduction,
Verschoor and Greebler [31] used kinetic theory and the
probability of photon-fiber collisions. The fibers were
assumed to be randomly distributed in planes parallel to the
blanket surfaces. The direction of heat flow is perpendicular
to these planes. It is further assumed that the fibers are of
a uniform diameter and there are no particles other than
fibers in the blankets.
Then, in a thin volume element of unit cross-sectional
area with an infinitessimal depth x, the x-direction being
parallel to the heat flow, the volume of fibers is (x*v v
f" f
is the volume fraction of fibers. Fiber volume divided by the
cross-sectional area of an individual fiber gives the total
length of the fibers in the volume element. If the total
length is multiplied by the fiber diameter, the total
projected fiber area, A, is
95
4 * V * x
A = (39)
where vf is the volume fraction of fibers and D is the average
fiber diameter. Since the volume element presents a unit
cross-section area perpendicular to the heat flow, A is the
probability for a photon to collide with a fiber within
distance x. At the same time, the probability that a photon
suffers a collision with a fiber within a distance of x is
given by
P = 1 - exp(- x / l (40)
f)
where lf is the mean free path for such a collision.
Equations (39) and (40) define the same probability. Thus,
4 first equate equations (39) and (40), then expand the
exponential in a power series, keeping only the first order
terms. The resulting equation can be solved for the mean free
path for photon fiber collision
1 = (41)
Thus the mean free path, If, for photon conduction is readily
96
obtained from the bulk density, fiber diameter, and specific
gravity of the fibers.
In equation (41), the practical limits for the depth of
the volume element, x, is about the fiber diameter. If x is
orders of magnitude larger than the fiber diameter, the
analysis of projected fiber area on the unit cross-sectional
area of the volume element looses its validity. Assuming that
x is 3 microns, which is the average fiber diameter in almost
all refractory blanket types, one can estimate the maximum
possible error in the mean free path calculations due to
truncation in the expansion of the exp(-R) term in the
probability equation (41). For an lf around 12 microns the
maximum error introduced would be 1/8 or 0.125 times the
x / lf.
Another limitation to the model is implied by the mean
free path itself. As higher compactions (lower compaction
ratios) are achieved, the spacing between the fibers or
between the fiber planes becomes small so that two other
complications arise. First, a spacing of 5 to 10 microns is
comparable to the wavelength of infrared radiation, and thus
multiple scattering may becomes significant. Equation (37),
which embodies the thermal conductivity model developed in
this thesis, does not account for the effects of multiple
infrared scattering.
Second, if the fiber spacing is less than one fiber
diameter then a photon emitted from one fiber in the direction
97
of heat flow will likely scatter from another fiber and bounce
back in the direction opposite to the heat flow before it can
move forward a significant amount. Refraction and reflection
of the photon beams could dramatically reduce the radiative
heat transfer, a phenomenon that equation (37) does not
anticipate. As compaction decreases the fiber spacing, the
error due to truncation of the expanded exponential term in
the equation (41) can become significant (Table 8). The
maximum error due to truncation would be the first neglected
higher order term in the alternating power series expansion.
In Table 8, the maximum error introduced is listed normalized
with respect to the first order term in the power series
expansion. The Verschoor’s [31] equation (41) should not be
used to calculate the mean free path for the photon conduction
when the mean free path value approaches the single fiber
diameter.
3.2.2. Gas Conduction Thermal Conductivity
Gas conduction thermal conductivity is independent of the
insulation density [31] as long as the ambient pressure
remains at one atmosphere. (The thermal conductivity model in
this thesis assumes a constant ambient gas pressure of one
atmosphere). In Figure 12 the contribution of gas conduction
to the total heat transfer process in the fibrous blankets
with changing insulation bulk density is shown along with the
98
Table 8. Normalized Error in the Mean Free Path Calculations
due to Truncation in the Expansion of Probability
Function.
Mean Free Path, 1f x*/ lf (x / 1f)2 **
[microns]
2 (x / 1f)
170.00 0.0176 0.00882
150.00 0.0200 0.01000
130.00 0.0231 0.01154
120.00 0.0250 0.01250
110.00 0.0273 0.01364
100.00 ,0.0300 0.01500
90.00 0.0333 0.01667
80.00 0.0375 0.01875
70.00 0.0429 0.02143
60.00 0.0500 0.02500
50.00 0.0600 0.03000
45.00 0.0667 0.03333
40.00 0.0750 0.03750
35.00 0.0857 0.04286
30.00 0.1000 0.05000
25.00 0.1200 0.06000
24.00 0.1250 0.06250
23.00 0.1304 0.06522
22.00 0.1364 0.06818
21.00 0.1429 0.07143
20.00 0.1500 0.07500
19.00 0.1579 0.07895
18.00 0.1667 0.08333
17.00 0.1765 0.08824
16.00 0.1875 0.09375
15.00 0.2000 0.10000
14.00 0.2143 0.10714
13.00 0.2308 0.11538
12.00 0.2500 0.12500
11.00 0.2727 0.13636
10.00 0.3000 0.15000
9.00 0.3333 0.16667
8.00 0.3750 0.18750
7.00 0.4286 0.21429
6.00 0.5000 0.25000
5.00 0.6000 0.30000
4.00 0.7500 0.37500
3.00 1.0000 0.50000
* x is taken as 3 microns which is the average fiber
diameter for all blankets.
** The normalized truncation error is obtained by dividing
first neglected higher order term by x/lf.
99
THERMAL CONDUCTIVITY [W/mK] x 102
BLANKET DENSITY [lb/CUft]
2.0 4.0 6.0 8.0
1 J ‘ 1 4.0
5.0 -‘
“3.0
4.0—
TOTAL cououcnwry
3.0--J AIR CONDUCTION
"2.0
—1.0
BLANKET DENSITY [kg /m3]
Figure 12. Contribution by each mode of heat transfer in glass-fiber
THERMAL CONDUCTIVITY [Btu in/ft2 hr F]
insulation at atmospheric pressure versus blanket density
at 65 C. (after [31])
100
contributions of radiation, convection, and solid conduction.
Figure 12 represents the contributions at 65 C mean
temperature.
The free gas conductivity of air changes with
temperature, as does the gas conduction thermal conductivity
within the insulating blanket.
The probability of a free gas molecule to collide with
another gas molecule within the distance x is [31]:
Pg= l - exp (-x / 1g) (42)
where 1g is the mean free path of a gas molecule-gas molecule
collision. If the gas permeates a fibrous insulating
blankets, then gas molecule-fiber collisions should be
accounted for, too. Analogous to the equation (42),
expression (43) gives the probability of a gas molecule
hitting a fiber at low pressures:
Pf = 1 - exp (-x / 1f) (43)
where lf is the mean free path for gas molecule-fiber
collision.
The probability of a gas molecule to travel a distance x
before hitting another gas molecule is:
PX = [8XP(-X/1f)] * [exp(-x/lg)] * [l-exp(-dX/lg)]
101
= [exp(-x(1/lf + l/lg)] dx/lg (44)
The two terms in the bracketts in equation (44) give the
probability that a collision will not happen with either a
fiber or another gas molecule within distance of x. The dx/lg
factor gives the probability that a gas molecule will collide
with another molecule in the following infinitessimal distance
dx. The probability of an intermolecular collision for all
values of x gives the mean free path for such a collision as
[31]:
of†[eXP(-X(1/1f + 1/lgm x dx /1g
L = (45)
of†[expt-xu/lf + 1/lgm dx /lg
= 1g 1f /(1f + 19) (46)
Now, if it is assumed that the molecular velocity
distribution is not significantly affected by the molecule
fiber collisions, the air conduction conductivity in the
blanket can be evaluated in the same manner as that of the
free gas, except L is used as mean free path in place of l
9
[31]. This relationship is given as:
102
k = kfg* 1f/(1f+1g) (47)
gcd
k = gas conduction thermal conductivity in the
gcd
insulating blanket
= free gas thermal conductivity
1 = mean free path for gas molecule-fiber collision
l = mean free path for gas molecule-gas molecule
collision
The free path, lg, is a strong function of ambient
pressure. Since all experiments in this study were done at
atmospheric pressure, 19 is fixed. At atmospheric pressure
and at room temperature 19 is very small with respect to 1f,
even at lower compaction ratios. Thus, the ratio in front of
free gas conductivity in equation (47) approaches unity, which
to k . For the model
reduces the express1on for kgcd fg
. developed in this thesis, the expression for free gas
conductivity is used in place of gas conduction thermal
conductivity.
An expression for the temperature dependence of gas
conductivity was obtained by fitting the air conductivity
versus temperature data (Appendix C) to the following
expression
k =A*'I' (43)
103
via a non-linear least-squares procedure, which gave a 0.9995
best fit correlation coefficient for the temperature range
100-1700 K with the parameters:
A = 2.904722E-04
(I!
ll
0.7907285
For the fit, we used 26 data points in the temperature range
specified. Equation (48) describing the temperature
dependence of gas conductivity is used for the calculations of
effective blanket thermal conductivity.
The expression for gas conductivity contains only values
related to air thermal conductivity, since all the experiments
were done in air atmosphere. If the experiments are conducted
in any other atmosphere the values of the constants in
equation (48) should be modified to describe the specific gas
filling the pore volume in the blankets.
3.2.3. Convective Thermal Conductivity
Gas convection is another mechanism of heat transfer
effective within the blanket pore volume. In this thesis two
basic assumptions are made with regard to the thermal
conductivity due to convection: (1) thermal conductivity due
to gas convection is independent of the blanket bulk density,
and (2) the convective thermal conductivity within the
104
insulation varies with the temperature in the same manner as
the thermal conductivity due to gas conduction, kgcd' since
free gas convection and gas conduction are closely related.
In the blanket bulk density range (48 kg/m3 to 480 kg/m3)
investigated in this thesis, assumption (1) is justified,
since the contribution by convection to overall heat transfer
within the fibrous insulation does not change significantly
for blanket bulk densities larger than 48 kg/m3 (Figure 12).
As a result of the proceeding assumptions, the
contributions for overall thermal conductivity due to gas
conduction and convection are combined. In equation (37), the
gas conductivity term is multiplied by a prefactor which
involves r, the ratio of the convective thermal conductivity
to thermal conductivity due to gas conduction.
The analysis of effective heat transfer mechanisms in
glass-fiber insulation at room temperature [31, 54] gave the
ratio, r, as 0.1. The same value of r is used in the model
developed in this thesis.
3.2.4. Solid Conduction Thermal Conductivity
The model assumes that the fibers in the structure can be
categorized into two groups: (1) a very large fraction of
fibers lay on planes perpendicular to the heat flow, and (2) a
small, but nevertheless significant fraction cross these
planes and are oriented essentially parallel to the heat flow
105
direction.
The fibers crossing the air gap between the majority
fiber planes conduct the heat in a parallel manner relative to
radiation, gas conduction, and convection. Thus, the thermal
conductivities due to these contributions can simply be added
after the volume fractions are taken into consideration. The
fraction of fibers that cross the majority fiber planes are
accounted for by an empirically determined term in the overall
thermal conductivity expression (37).
The majority fiber planes separate the air volume in the
direction of heat flow and act as barriers to radiation and
gas conduction. Conduction in the majority planes occurs by
solid conduction, which acts in series with the terms
discussed previously.
The expressions used to approximate the temperature
dependence of solid thermal conductivity in Saffil type
alumina fibers and in Locon and Kaowool K2300 aluminosilicate
fibers are detailed in Sections 3.2.4.1 and 3.2.4.2,
respectively.
3.2.4.1. Solid Conductivity in Saffil Alumina Fiber Blankets
Phonon conduction is the most important contribution for
thermal conduction in theoretically dense solid ceramics.
Thus, an expression defining the temperature dependence of
phonon conductivity could also dictate the behavior of solid
106
conduction versus temperature in the fibers within the Saffil
blanket. The thermal conductivity data for dense poly-
crystalline alumina (specific gravity 3.28) [55] was fit to an
equation for temperature dependence of phonon conduction [56]
via a non-linear least squares program. For the equation:
k = s / (T2) (49)
sld
using the parameters
0)
II
18492
1.11517
N
II
a correlation coefficient of 0.9987 was obtained.
Saffil blanket is composed of relatively pure alumina
fibers (95 percent alumina) with a fiber specific gravity of
3.4. The high alumina content of the fibers suggests that the
fibers may be in crystalline form in spite of the fact that
they may cooled very rapidly during the production of the
fibers from the melt. Thus, the values and the expression
discussed above likely approximate well the temperature
dependence of the thermal conductivity of the Saffil blanket
fibers.
107
3.2.4.2. Solid Thermal Conductivity in Aluminosilicate Fiber
Blankets
The chemical compositions of Locon and Kaowool K2300
aluminosilicate fibers suggest that the crystallographic phase
mullite solidifies from the melt under equilibrium conditions.
Based on this argument, the thermal conductivity in Locon and
Kaowool K2300 was first approximated by the thermal
conductivity data for mullite with 11 percent porosity.
However, neither for Locon nor for Kaowool the fit between the
theory (equation (37)) and the experimental data was
satisfactory. Based on these results, the thermal
conductivities for alumina-mullite mixtures, for Firebrick
80-D , and for fused silica, along with the combinations of
these were examined to approximate the fiber thermal
conductivity in Locon and Kaowool K2300 Blankets (Tables 9 and
10).
The alumina-mullite mixture (50% AL 03-50% mullite)
2
employed to approximate the thermal conductivity in Locon
fibers had the closest specific gravity (2.68) to the specific
gravity of Locon fibers (2.73). The solid thermal
conductivity data of the mixture was fit to equation (48) via
least squares program. The parameters for equation (48) and
the correlation parameters for the theory (equation (37))are
listed in Table-9.
Based on the two observations, the thermal conductivity
108
Table 9. Parameters for the Solid Thermal Conductivity
in Locon Fibers.
Material(s) Eq. Parameters Maximum*** Maximum Fitted
No. of the discrepancy of aver. parameters
* Eq.** % residual% (if any)
50% Mullite- i A-87.18 38 12.40 u, q §
50% A1203 B--.485
Fused Silica ii C-l.263 9.48 4.97 -
D-.1085
E-2.7E-3
Fused Silica 1,11 A-7o.43 10.80 4.04 if“
and Mullite B--.432 and C6
(11% Porosity): C-1.263
D-.1085
E-2.7E-3
* Equation (1): k - A * TB (11): k - c + 0 * exp(E * T)
sldl sld2
**'A and B are constants in eq. (1), C, D, E are constants in eq. (ii)
*** Maximum discrepancy between the average of the experimental data
and the theory prediction.
§ u and q are empirical constants in the expression for C6.
i The two solid thermal conductivities, k and k , are combined
as- k - k * (l-f) + k * f Sldl Sldz
° sld sldl sld2
3; f is the fraction of ksld2
f for this particular set 0.7676 < f < 1.286
109
Table 10. Parameters for the Solid Thermal Conductivity in
Kaowool K2300 Fibers.
Material(s) Eq. Paramaters Maximum*** Maximum Fitted
§ No. of the discrepancy of aver. parameters
* Eq.** % residual% (if any)
80-d §§ §§ 48 18.45 u, q 0
Fire-brick
Fused Silica i A-1.263 20.8 11.0 -
B-.1085
C-2.7E-3
Fused Silica i,§§ A=1.263 13.30 5.48 fT**
and 80—0 B-.1085 and C6
Fire-brick: C-2.7E-3
§§
§ When the solid thermal conductivity of Mullite was used in the
least squares fit program, there was computational complications
due to overflow
§§ N0 simple equation could be fit to the data. Thus thermal
conductivities given by the manufacturer were used
* Equation (i) ks - A + B * exp(C * T)
ldl
** A, B, and C are constants in equation (1)
*** Maximum discrepancy between the average of the experimental data
and the theory prediction.
0 u and q are empirical constants in the expression for C6°
t The two solid thermal conductivities, k and k , are combined
88' k - k * (l-f) + k * fSIdl sld2
' sld sld80-D sldl
11 f is the fraction of ksldl
f for this particular set 0.2071 < f < 1.78
110
Table 10. continued.
Material(s) Eq. Paramaters Maximum*** Maximum Fitted
§ No. of the discrepancy of aver. parameters
* Eq.** % residua1% (if any)
Fused silica i,ii A-l.263 15 7.28 fT*t
and Mullite B-.1085 and C6
(11% porosity): C-2.7E-3
D-70.43
E--.432
Fused Silica i,ii A-l.263 16.08 8.17 fTT**
and Mullite B-.1085 and C6
(30% porosity): C-2.7E-3
D-63.39
E--.464
§ When the solid thermal conductivity of Mullite was used in the
least squares fit program, there was computational complications
due to overflow
E
* Equation (1) ks - A + B * exp(C * T) (i1): ksld2- D * T
ldl
** A, B, C are constants in eq. (i), D and E are constants in eq. (ii)
*** Maximum discrepancy between the average of the experimental data
and the theory prediction.
t The two solid thermal conductivities, k and k , are combined
aS° k - k * (l—f) + k * f Sldl Sldz
' sld sld2 sldl
it f is the fraction of ksldl
t for this particular set 0.52 < f < 1.34
1f for this particular set 0.04697 < f < 1.479
111
of fused silica was employed to approximate the thermal
conductivity in Locon and Kaowool K2300 fibers: (1) the
thermal conductivity of crystalline phases with similar
compositions did not give satisfactory fit for the theory, and
(2) both Locon and Kaowool are produced by blowing fibers from
a melt where high cooling rates and the high silica content of
the fibers suggest that a considerable amount of glassy phase
may be present, along with some dispersed crystallites [47].
Using the same least squares program the best fit to the fused
silica data in the temperature range of interest was
3
k = 1.2634 + 0.10856 exp(2.707 10' T) (50)
sld
where T is the temperature in degrees Kelvin. The correlation
coefficient for the fit was 0.989.
For the theoretical calculations of overall thermal
conductivity in Locon and Kaowool K2300 blankets, equation
(50) was used to approximate the solid thermal conductivity in
the fibers. Table 10 lists the results of the curve fitting
for Kaowool K2300 blanket using the thermal conductivities of
different materials to approximate the fiber thermal
conductivity. Although the least squares program determined
the optimum values of the indicated parameters in equation
(37) (last column in Table 10), the maximum discrepancy
between the theory and the average of the experimental data
and the maximum of the average residuals was still 10 to 15
112
percent.
A literature search on the thermal conductivity of
aluminosilicate glasses, which probably would provide a better
approximation for Kaowool and Locon fiber thermal
conductivity, yielded only one source of data [57]. However,
the data do not show a consistent trend in the behavior of
thermal conductivity versus temperature.
3.2.5 The Empirical Term C6
The only empirical term C in equation (37) for the
6
overall thermal conductivity may relate to the fraction of
fibers that cross the majority fiber planes. In equation (37)
C6 may define the ratio of the total number of fibers to the
number of fibers that cross the majority fiber planes.
Comparison of the experimental and theoretical data via
least squares fitting showed that C6 increased as the
compaction ratio decreased. An increase in C6 signifies that
the number of fibers crossing the majority fiber planes
decreases. Therefore, the contribution of parallel fiber
solid conduction decreases. If the parallel solid conduction
were only due to contact between the parallel fiber planes,
then C should have decreased with decreasing compaction
6
ratio. Thus, the experimental observations support the
assumption that there may be other solid conduction mechanisms,
effective besides the conduction due to contact between the
113
parallel fiber planes.
The distribution of the fibers in the blanket can be
modeled better with the assumption that a small fraction of
fibers are more parallel oriented to the heat flow rather than
perpendicular to it. Furthermore, these fibers make an angle
with the majority fiber planes rather than being exactly
perpendicular to them. Thus, as the fiber blankets are
compacted, the angle between the parallel fibers and the
majority fiber planes decreases. During this re-arrangement
of the fibers’ orientation, the fibers that were more parallel
to the direction of heat flow become more perpendicular to it
due to decreasing angle with the majority fiber planes. As
the parallel fibers become more and more perpendicular to the
heat flow they will conduct the heat in series rather than in
parallel.
The proceeding assumption fits perfectly into the picture
of the fibrous blankets' construction and explains how the
term C6 can increase with decreasing compaction ratio.
To account for the fiber re-orientation process which
occurs during compaction, two different expressions involving
sine-functions were developed to define C Equation (51) was
6'
developed because it shows a similar behavior to the variation
of C6 with compaction ratio for Saffil blanket in the
compaction ratio range 1.0 to 0.2. The variation of C6 with
respect to compaction ratio was obtained via non-linear least
squares fitting over the entire temperature range by assuming
114
C6 as a constant and varying the compaction. Equation (52)
changes with the compaction in the same manner as the value of
C6 for Locon and Kaowool K2300 blankets.
2
C = u + q * (sin cr*«) (51)
C = u + q * (l-sin(cr*;/2) (52)
where u and q are empirical constants and cr is the compaction
ratio. Equation (52) fits to the experimental data best if
all three materials are considered. Equation (51) fits better
for Saffil thermal conductivity data only.
Table 11 lists the constants u and q for the three
blankets investigated. The values of u and q are determined
via a computer program which increases the u and g values by
one, calculates the sum of the average residuals for the
entire data set, compares the new sum with the residuals from
the proceeding loop, and stores only the smaller sum with it's
corresponding u and g values. Hence, u and g were chosen such
that the sum of the average residuals over the entire
compaction range is minimized.
The application of optical and electron microscopy
methods the relation between C and fiber orientation could be
6
a subject for further study.
115
Table 11. Empirical Constants in C6 Expression for Saffil,
Locon, and Kaowool K2300 Blankets.
Saffil Locon Kaowool
u 130 8 7
q 65 13 19
max. average 5.448 x 10"2 4.971 x 10'2 11.1 x 10'2
difference
max maximum 0.121 0.0948 0.208
difference
116
3.3. The Compaction Ratio Dependence of the Thermal
Conductivity of Ceramic Fiber Thermal Insulating
Blankets
The experiments on compacted specimens were conducted
with blankets of Saffil, Locon and Kaowool K2300. The
compaction ratio is defined as the ratio of the compressed
thickness of the specimen to the original, or as-received
thickness of the blanket. In this thesis, compaction ratio
was changed from 1.0 (uncompacted condition) to 0.2 in six
steps. Locon and Kaowool K2300 are typical aluminosilicate
fiber blankets with slightly different chemical compositions.
Saffil blanket was included in this study because unlike Locon
and Kaowool, Saffil’s chemical composition is approximately 95
percent alumina. Relevant physical properties and chemical
compositions of these three products are listed in Table 12.
From room temperature to 800 C, an optimum bulk density,
or compaction, exists at which the overall thermal
conductivity of the insulator blanket is at a minimum.
In this thesis, two minima of the overall thermal
conductivity at room temperature were observed. The first
minimum, at around 0.9 compaction ratio for the three blankets
investigated, may be due to the reduced convective and
radiative heat transfer. The other two mechanisms of heat
transfer, air conduction and solid conduction are shown not to
change significantly with compaction in the low blanket bulk
117
Table 12 Fiber Physical Properties and Blanket Density
Designation K-2300** LOCON*** SAFFIL****
Chemical A1203 45 % 49.5 % 95 %
Composition
$102 53 % 48.3 % 5 %
Impurities 2 % 2 % ...
Specific gravity 2.56 2.73 3.4
Fiber diameter 2.8 2-3 3
(micron)
Bulk Degsity* 98 91.4 48
(kg/m )
* measured
** Thermal Ceramic Co., Plymouth, MI
*** Carborundum Co., Niagara Falls, NY
**** ICI Ltd., England
118
density region [31]. However, convective and radiative
contributions for overall thermal conductivity change
appreciably at high compaction ratios with changing blanket
bulk density (Figure 12). In the theoretical model developed
in this thesis (equation (37)), the convective heat transfer
is assumed to be independent of the bulk density, thus the
model can not predict the first relative minimum in the
experimental data of the overall thermal conductivity versus
compaction ratio. A study of convective heat transfer in fine
fiber meshes may be needed to model the low temperature
behavior of the thermal conductivity of the insulating ceramic
fiber blankets.
A second minimum in the thermal conductivity versus
compaction ratio data was observed for the compaction ranges
covered, for the ambient temperatures less than 800 C. In the
thermal conductivity versus compaction ratio plots, the
temperature at which minima occurred increased as the initial
blanket densities increased. For Saffil (48 kg/m3 initial
density) the thermal conductivity increased monotonically with
decreased compaction ratio at room temperature. At 200 C the
data showed a minimum at about 0.5 to 0.4 compaction ratio
range. For 400 C, the data decreased slightly until 0.3
compaction ratio after which it started increasing (Figure
13a, b).
For Locon (91 kg/m3 initial density) the thermal
conductivity at room temperature and 200 C increased with
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decreased compaction ratio. At 400 C the experimental data
showed a minimum at about 0.6 compaction ratio. The minimum
at 600 C occurred in the compaction ratio region between 0.5
to 0.3 (Figure 14a, b).
Kaowool K2300 blanket showed an increasing thermal
conductivity at room temperature and 200 C with decreased
compaction ratio. At 400 C the thermal conductivity appeared
to be relatively insensitive to changes in the blanket
compaction. At 600 C, a minimum was observed at around 0.4
compaction ratio which shifted to a compaction ratio of 0.25
at 800 C (Figure 15a, b). At 1000 C for Locon and Kaowool
K2300 and at 600 to 1000 C for Saffil blanket, the thermal
conductivity monotonically decreased as compaction ratio
decreased (no minimum was observed).
3.4 Correlation between the Theory and Experimental Data
Equation (37), developed in this thesis to describe the
behavior of overall thermal conductivity with respect to
temperature and blanket compaction, agrees well with the
experimental data for Saffil and Locon blankets.
For Saffil blanket, the maximum discrepancy between the
theory and the average of the data was 12 percent for the
entire range of temperatures and compaction ratios (Figure
16a, b). The maximum of the average residuals was 5.41
percent for the same material.
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For Locon, the maximum discrepancy between equation (37)
and the average of the experimental data was 9.48 percent, and
the maximum of the average residuals was as low as 4.97 for
the entire Locon thermal conductivity data (Figure 17a,b).
The close match between the theoretical and experimental
trends in overall thermal conductivity versus compaction ratio
is repeated in the behavior of thermal conductivity versus
temperature. The theoretical thermal conductivity increases
with increasing temperature in the same manner as the
experimental data do. However, the rate of increase decreases
as the compaction is increased (compaction ratio is decreased)
for each of the three blankets investigated (Figures 18a -f).
The fit between the theory and the experimental data for
Kaowool K2300 blanket exhibited a maximum difference of 20
percent when the thermal conductivity of fused silica was used
to approximate the solid thermal conductivity in Kaowool K2300
fibers. The maximum of the average residuals for all
compactions was approximately 11 percent (Figure 18e, f,
19a, b).
The maximum discrepancy between the theory and the data
for Locon and Kaowool K2300 blankets occurred at the lowest
compaction ratios and high temperatures which indicated that
the multiple scattering discussed in Section 3.2.1.1 may be
important at this range. A refinement in the theory is
possible if the effects of multiple scattering could be
accounted for by a term in equation (37).
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139
At elevated temperatures, the overall thermal
conductivity can be reduced by using finer diameter fibers to
construct blankets [13] (equation (41)):
1 = —— (41)
where d is the fiber diameter, vf is the volume fraction of
fibers, and If is the mean free path. Thus, finer fiber
diameter will give smaller mean free paths for photon
conduction, which will reduce the radiation heat transfer.
3.5. Interface Effects between Thin Blankets of Similar
Refractory Materials
Thermal conductivity experiments were performed with
specimens consisting of multiple layers of Ultrafelt paper
(where each layer was a 2.54 mm thick aluminosilicate paper
product obtained from Thermal Ceramics, Plymouth, Michigan).
Depending on the nature of the mechanical load applied to the
stacked Ultrafelt layer specimens, an apparent interface
effect on the thermal conductivity was either observed or was
absent.
When no externally applied mechanical load was applied to
the stack of Ultrafelt layers, an interface effect was
140
recorded. For example, in two experimental runs for the
unloaded Ultrafelt stacks, the natural logarithm of time
versus temperature plot showed two distinct slopes (shown
schematically in Figure 20). The slope of the initial linear
segment of the curve (from Ta to Tb in Figure 20) corresponds
to a thermal conductivity of 0.041 W/mK. Tb, the time axis
terminus of the initial linear portion of the curve, ranged
from about 35 to 40 seconds. The slope of the second linear
segment of the curve (from Tb to Tc in Figure 20) corresponds
to a thermal conductivity of 0.033 W/mK. Tc, the time axis
terminus of the second linear portion of the curve, ranged
from about 45 to 105 seconds.
Applying an external mechanical load evidently improved
the thermal contact between the Ultrafelt layers to the point
that an interface effect was no longer observed. For example,
when a 0.8 kg dead weight load was applied over a 150 by 150
mm plate, a single straight line segment was observed in the
temperature versus the natural logarithm of time plot for
times between 20 seconds and 90 seconds, with a corresponding
thermal conductivity of 0.0399 W/mK. Thus, the application of
a relatively slight mechanical load to the layered specimen
was apparently sufficient to obviate the interface effect.
Note that the initial linear portion of the unloaded layered
specimen gave a thermal conductivity of 0.041 W/mK, while the
single straight line portion observed in the loaded layered
specimen gave a thermal conductivity which differed by only
141
TEMPERATURE
l
l
|
Te
TA Ts
NATURAL LOGARITH M OF TIME
Figure 20. Effects of interface between stacked refractory blanket
layers on the thermal conductivity measurements.
142
about 2.8 percent. Thus the conductivities computed from the
initial linear part of the unloaded curve and the entire
linear portion of the loaded curve are essentially the same.
3.6. Effects of Fluctuation in the Ambient Temperature on the
Thermal Conductivity Measured by Transient Hot-wire
Method
As mentioned in section 2.3 of this thesis, the ambient
temperature within the specimen was measured with a far field
thermocouple. Recording one set of temperature-time data
pairs took about three minutes on average. Below about 600 C
the ambient temperature remained stable within 1 percent of
the set temperature. Above 600 C, the stability within 0.5
percent of the set temperature value.
Small variations in the ambient temperature can effect
the thermal conductivity measurements. If the ambient
temperature increased or decreased monotonically during the
experiment, the logarithm of time versus temperature curve did
not exhibit the constant slope region needed for thermal
conductivity calculations.
The effects of unstable ambient temperature were
especially pronounced at high temperatures where thermal
conductivity of the blankets was high. At room temperature,
the time-temperature data had a span of 30 to 50 C where
ambient temperature fluctuations up to 4-5 C were allowable.
143
However, at high temperatures, high thermal conductivities led
to small temperature rises, typically 10 to 20 degrees Celsius
at around 1000 C, during the heating of the specimen by the
hot wire. Thus, changes of +/- 1 to 2 degrees Celsius in the
ambient temperature caused relatively important effects on the
temperature-time data.
The effects of increasing or decreasing ambient
temperature on the thermal conductivity measurements at
elevated temperatures can be divided into two cases:
Case I: Decreasing Ambient Temperature.
Consistent decrease in the ambient temperature during the
heating of the specimen by the hot wire caused the temperature
versus natural logarithm of time curve to appear concave
downward. R.P. Tye [58] reported that a concave downward
curve is an obvious indication of unsuitable set-up response.
Case II: Increasing Ambient Temperature
A steady increase in the ambient temperature may hide
deviations in the temperature versus natural logarithm of time
data. The increase in the ambient temperature may offset the
decrease in the slope of the temperature versus natural
logarithm of time graph to yield a straight line with a larger
slope than that of a successful run. Since a straight line
may be obtained in the logarithm of time versus temperature
plot, it may be difficult to recognize the external influence
144
on such data.
Figure 21 shows the affects of cooling and heating on the
plot of natural logarithm of time versus temperature. Curve A
shows the plot of a successful experiment. Curve B is
obtained from a measurement perturbed by decreasing ambient
temperature. Curves C and D present measurements taken under
the influence of increasing ambient temperature.
Fluctuations in the ambient temperature can be detected
during the experimentation via the far-field thermocouple in
order to avoid erroneous data caused by fluctuating
temperatures
3.7. Comments on the Change of Heat Input to the Specimen
During the experiments with Kaowool K2300 blanket and
Saffil L-D mat, the heat input through the hot wire was
controlled by changing the voltage drop across the hot wire.
Changes in the heat input to the hot wire did not
significantly effect the measured thermal conductivities when
all other parameters, such as temperature and compaction
ratio, were held constant (Table 13). However, in each of
experiments dealing with varying heat input to the specimen,
the total change in heat input was less than a factor of two
(Table 13).
The thermal conductivities obtained from different runs
are within +/- 3 percent of the mean for all sets of
145
TEMPERATURE
xC
NATURAL LOGARITH M OF TIME
Figure 21. Effects of ambient temperature fluctuations on the thermal
conductivity measurements.
146
Table 13. Dependence of the Measured Thermal Conductivity
Values on Varying Heat Input through the Hot Wire
Material Temp Run # Heat Input Thermal Conductivity
[C] [W/m] [W/mK]
K2300 600 1 5.446 0.1148 *
K2300 600 2 5.446 0.1225 *
K2300 600 1 10.376 0.1159 *
K2300 600 2 10.376 0.1218 *
The mean value of thermal conductivity ........ 0.1187
Saffil 400 1 4.639 0.0967 **
Saffil 400 2 4.639 0.0933 **
Saffil 400 1 9.362 0.1000 **
Saffil 400 2 9.362 0.1025 **
The mean value of thermal conductivity ........ 0.0981
Saffil 600 1 4.844 0.1139*
Saffil 600 2 4.844 0.1258*
Saffil 600 1 9.696 0.1278*
Saffil 600 2 9.696 0.1248*
The mean value of thermal conductivity ........ 0.1230
Saffil 600 1 4.702 O.1024***
Saffil 600 2 4.702 0.1031***
Saffil 600 1 9.022 0.1057***
Saffil 600 2 9.022 0.1047***
The mean value of thermal conductivity ....... . 0.1039
* For a blanket compaction ratio 0.6
** For a blanket compaction ratio 0.8
*** For a blanket compaction ratio 0.2
147
measurements with varying heat inputs. These results agree
well with data presented by E. Takegoshi et a1. [21] for
silica glass (which Takegoshi referred to as "quartz" glass
[21])-
3.8. Effects of the Gap between the Thermocouple Junction and
the Hot Wire.
EXperiments done with the Saffil blanket at room
temperature revealed that the gap between the thermocouple
junction and the heating element (hot wire) affected the onset
time for the linear portion of the temperature versus natural
logarithm of time plot. In addition, a larger gap between the
hot wire and sensing junction yielded lower values of thermal
conductivity.
This error can be corrected by subtracting a constant to
(Section 1.1.1.1.), which can be determined from the plot of
dT/dt versus time. For each experiment a separate graph needs
to be drawn to determine to.
This thesis attempted to determine to from the straight
line intercept of the linear portion of the graph of
temperature versus natural logarithm of time with the natural
logarithm of time axis. In this study, 1 second was a typical
value of to, while to ranged from approximately 0.0 second to
about 2.0 seconds.
The experiments done with a 1.7 mm gap distance yielded
148
results which are within 4 percent deviation from those
obtained from experiments with only a 0.8 mm gap if the
correction time, to, is incorporated in the calculations.
Although no theoretical proof has been supplied yet, this
technique of determining the correction time tO seems to work
as a self-correcting mechanism.
An attempt has been made to quantify the shift in the
onset of linear the portion in the plot temperature versus
natural logarithm of time due to the differing gap distances
(Table 14).
Gap distances larger than 1.7 mm yielded deviations from
the straight line regime in the temperature versus natural
logarithm of time plot. As the sensing junction moves away
from the hot wire, the data are more easily affected by the
fluctuations in the ambient temperature, which in turn cause
deviations in the plot of temperature versus natural logarithm
of time.
149
Table 14. The Onset Time for the Linear Portion of the
Temperature versus the Logarithm of Time Curve as a
Function of the Gap Distance (the Gap between the
Sensing Thermocouple and the Hot Wire).
Gap distance Time at which linear
(mm) portion started (sec)
1.7 15
1.7 15
1.7 10
1.7 15
1.2 7 - 10
1.2 10
1.2 6 - 10
1.2 10 - 12
0.8 10
0.8 10
0.8 10
0.8 15
150
4. CONCLUSIONS
Thermal conductivity of ceramic fiber insulating blankets
is not an intrinsic property. It is a composite property due
to the two constituents of the composite, the gas and the
fibers. Thermal conductivity of the blankets strongly depends
on the relative amounts of the fibers and the gas volume.
The total heat transfer in fibrous blankets is made of
the contributions from radiation, gas conduction and
convection, as well as from parallel and series solid
conduction in the fibers. Individual modes of heat transfer
gain prevalence at different temperature and compaction
ranges. In the entire temperature range (296-1250 degrees
Kelvin) covered in this thesis, gas conduction is one of the
most important mechanisms of heat transfer in fibrous
insulations. At elevated temperatures, the radiative
contribution to overall thermal conductivity dominates.
Thermal conductivity continuously increases with
increasing temperature. This increase in the overall thermal
conductivity is mainly due to the increased rate of radiative
heat transfer at elevated temperatures. Yet radiation alone
cannot account for the entire behavior.
The low density (high compaction ratio) behavior of
overall thermal conductivity at room temperature can be better
described if convective heat transfer and the solid conduction
151
in and between the fibers can be approximated better than the
model presented here.
An expression for the overall thermal conductivity in
fibrous insulating blankets developed in this thesis accounts
for all the heat transfer contributions and accurately
predicts the thermal conductivity behavior for the temperature
and compaction ratio ranges studied in this research.
The maximum of the average residuals over the entire
compaction ratio range of Saffil blanket is around 5 percent.
For the Locon blanket the maximum of the average residuals was
less than 5 percent. For Kaowool K2300, the maximum of the
average residuals wave was approximately 7 percent. The lack
of a good fit between the experimental data and the theory
for Kaowool K2300 blanket may be due to the lack of data on
the solid thermal conductivity of fibers.
Reliable solid conduction data for individual fiber
materials may certainly improve the model of the refractory
blanket thermal conductivity, especially at low compaction
ratios (high blanket densities). The solid thermal
conductivity of theoretically dense polycrystalline alumina
was utilized in place of thermal conductivity of fiber in
Saffil blankets. The thermal conductivity of Locon fibers was
approximated by the thermal conductivity of fused quartz. The
thermal conductivity of neither mullite nor fused silica, nor
a combination of the mullite and silica can successfully
describe the thermal conductivity of Kaowool K2300 fibers.
152
At low compaction ratios (high blanket densities) where
the fiber spacing is on the order of 5 to 10 microns, multiple
infrared scattering likely becomes an important phenomenon.
The equation for the overall thermal conductivity does not
account for infrared scattering. Thus at low compaction
ratios, and especially in the high temperature region the
theory predicts the thermal conductivity too high.
Smaller diameter fibers could reduce the radiative
contribution by reducing the mean free path for photons.
If the layers of insulating blanket are not in intimate
mechanical contact during the hot wire experiments, the
temperature versus natural logarithm of time plot has two
regions with different slopes. The second slope, which is
higher than the first slope, corresponds to a lower thermal
conductivity and may be due to air gaps between the blanket
layers. Slight loads applied to the stack of blankets
eliminates the air gaps between the blankets and consequently
a single slope is obtained. The slope of the plot obtained
from a loaded experiment is the same as the slope of the first
region in the plot obtained from the experiments with a loose-
packed structure.
The change in the heat input through the hot wire does
not significantly effect measured thermal conductivities.
The hot wire technique is an easy, feasible, and rapid
method to determine the thermal conductivities of ceramic
fiber insulating blankets.
153
APPENDIX A
Derivation of the transient temperature distribution for the
hot-wire technique.
This appendix summerizes the derivation of equation (2) in
Section 1.1.1.
T = -—E——‘—'rnfm ï¬'l e"?2 dB = Q I(rn) (A1)
Note that [59]
2]“ p’1 (=962 d8 =1/2 J“ 3'1 e-fl d5 =-1/2 Ei(-zz) (A2a)
2
then [60],
xf†5’1 e'flz dp - -Ei(-x) = E1(x) (A2b)
The function defined by equation (A2b) is given the name of
"exponential integral", and it is actually a special form of
"incomplete gamma function, r(a,x)" [60].
Incomplete gamma functions are defined by the variable
limit integrals, generalizing the Euler definition of gamma
function (A3) [60]
1
I‘(z)=ofno et tz- dt (A3)
7(a,x) =ofx e‘t taâ€1 dt ï¬(a)>0 (A4a)
154
r(a,x) =xf†e"t ta'l dt (A4b)
The gamma functions (A4) are related by [60]
7(alx) + P(a.x) = I‘(<'=l) (A5)
The power series expansion of 7(a,x) for small x-values [60]
is
a m n xn
1(37X) = X Z ('1) (A5)
n=0 n! (a+n)
Note that,
XI†5’1 e'ï¬ d3 =XJ" a°’1 e'ï¬ d5 = new (A7)
Thus,
E1(x) = g}? [P(a.x)] = gig [P(a) - 7(a.x)] (A8)
However, note that the integral in (A7) diverges
logarithmically as 890. Thus, it is appropriate to split the
divergent term in the power series expansion for 7(a,x), so
that [60]
0 co
gm [rm — {xa(-1)° x + xa )3 H)“
O 0!(a+0) n=0 n! (a+n)
Elm
w n
E (x) = im [r(a) - xa/a] - _im xa Z (-1)n x
1 ;*° A 0 n=1 n! (a+n)
= 3m [ ar(a) - xa ] _ x0 E (_1)n xn
a a n=1 n! (0+n)
155
a
= 1.31 I: aI‘(a) - X (A9)
a
Apply L'Hospital rule, (because of 0/0), to the limit of the
term in the bracketts in equation (A9), where using the
identity
aF(a) = P(a+1) = a!
the following equality can be written
d(ar(a)) = d(a!) (A10)
d(a) d(a)
Then
d(a!) =. d(eln(a!)) (All)
d(a) d(a)
applying the chain rule to the differentiation, the following
is obtained
= d(eln(a!)) * d(1n(a!)) (A12)
d(ln(a!)) d(a)
= e1"(a’) * F(a) ' (A13)
note that the differentiation of d(ln(a!)) is obtained by
d(a)
expanding the factorial term (a!) [60]
156
z! = 2 F(z) = LE3 n! n2 (A14)
(2+1) (2+2) (2+3) ... (2+n)
taking In of both sides [60],
ln(z!) = Big [ln(n!)+ln(nz)-ln(z+1)-ln(z+2)...-ln(z+n)] (A15)
Differentiating equation (A15) with respect to z,
d(12:2;)) = d: )ï¬ig [1n(n1)+zln(n)-1n(z+1)-ln(z+2)...] (A16)
2 Z
= ii? [1n(n)- 1 - 1 ~-- ‘ 1 1
2+1 2+2 z+n
however, the left hand side of the equation (A16) is equal to
F(z) by definition [60]
= d(ln(z!))
d(z)
F(z)
= d Aim [ln(n!)+zln(n)-ln(z+1)-ln(z+2)...-ln(2+n)]
d(z)
From the definition of Euler constant [60]
F<z> = '7 + 2 z
__ (A17)
n=1 n (n+2)
Thus,
157
F(0) = - 1
where 1 = 0.577215... is the Euler constant.
In equation (A9), the first term yielded,
d
(ar(a)) = a!F(a)
d(a)
the second term in the brackets in equation (A9),
a a
1.1.3,!) x = .3113. = x‘:1 ln(x)
d(a)
y = xa
ln(y) = a ln(x)
d(Y) = d(a) ln(x)
d(a)
a
d(Y) = y ln(x) = xa ln(x) = d(x )
d(a) d(a)
Thus,
a m n xn
E1(x) = A}? a!F(a) - x ln(x) - X (-1)
n=1 n n!
on n
= -1 1 - x° ln(x) - Z (-1)n x
n=1 n n!
w n
= -7 - ln(x) - Z (-1)“ x
n=1 n n!
remember the equalities (A2a) and (A2b),
158
(A18)
(A19)
(A20)
(A21a)
(A21b)
(A21c)
(A21d)
(A22)
(A22a)
(A23)
z)†5'1 e-flz d8 - 1/2 Ei(-22) = E1(x2)
†2 n
21‘0 5-1 e-52 d5 = '7/2 - 1/2 ln(zz) - 1/2 X (_1)n (Z )
n=1 n n!
w 2n
= ' C ' ln(z) ‘ l/2 X (-1)n z (24)
n=1 n n!
Thus,
0 -1 '19? Q
________ 8 (2 d8 = _______ I(rn)
2ak â€J“ zzk
where
2 4
I(rn) = [- C - ln(rn) + rn - rn ...]
2 8
0
Thus, the step from equation (2) to equation (3)(Section
1.1.1) for the temperature distribution for the hot-wire
technique has been shown here.
159
The step from equation (3) to equation (4) in Section 1.1.1
necessitates the truncation of higher order terms in I(rn).
To justify this step, the convergence of the series
(’1)
=1 n n!
5M8
has to be shown. First, apply the ratio test, which gives
(_1)n+1 x2n
(n+1) (n+1)! 2
him? =1}..i.om (_1) x n n!
n x2n (n+1) (n+1)!
(-1)
n n!
2
= ï¬$§ (_1) x n n!
(n+1) (n+1) n!
2
= i? (_1) x n
n2+2n+1
= im = 0 < 1
A 2n+2
The series converges. Furthermore since n 1, 2, 3,...m
for each value of n and for x < 1 each term in the series
is less than the one proceeding it. Therefore, the maximum
error due to truncation cannot be larger than the first
truncated term [60].
160
APPENDIX B.
Thermal Conductivity Data of Ceramic Fiber Blankets
Table 801. Thermal Conductivity versus Temperature for
Saffil Blanket at 0.2 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0460 26.0
0.0474 26.0
0.0470 26.0
0.0469 - 26.0
0.0579 188.0
0.0592 192.0
0.0604 194.0
0.0621 197.0
0.0819 399.0
0.0801 401.0
0.0864 403.0
0.0860 401.0
0.0965 606.0
0.1025 604.0
0.1095 606.0
0.1089 606.0
0.1287 805.0
0.1388 803.0
0.1374 804.0
0.1281 807.0
0.1706 982.0
0.1689 977.0
0.1627 981.0
0.1734 983.0
0.1752 980.0
For SAFFIL Blanket
COMPACTION RATIO = 0.2
161
Appendix B. continued.
Table B02. Thermal Conductivity versus Temperature for
Saffil Blanket at 0.3 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0406 26.0
0.0400 27.0
0.0410 27.0
0.0413 26.0
0.0556 209.0
0.0536 205.0
0.0563 206.0
0.0535 204.0
0.0816 409.0
0.0753 406.0
0.0797 406.0
0.0739 404.0
0.1039 606.0
0.1087 609.0
0.1130 608.0
0.1163 608.0
0.1401 805.0
0.1385 803.0
0.1426 804.0
0.1942 978.0
0.1814 976.0
0.1909 976.0
0.1847 975.0
For SAFFIL Blanket
COMPACTION RATIO = 0.3
162
Appendix B. continued.
Table B03. Thermal Conductivity versus Temperature for
Saffil Blanket at 0.5 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0355 21.0
0.0361 24.0
0.0368 25.0
0.0367 26.0
0.0514 210.0
0.0550 214.0
0.0580 209.0
0.0555 204.0
0.0758 396.0
0.0790 398.0
0.0773 398.0
0.0760 398.0
0.1164 607.0
0.1174 598.0
0.1163 599.0
0.1640 799.0
0.1568 800.0
0.1653 803.0
0.1544 806.0
0.2062 977.0
0.2418 977.0
0.2166 976.0
0.2060 971.0
For SAFFIL Blanket
COMPACTION RATIO = 0.5
163
Appendix B. continued.
Table B04. Thermal Conductivity versus Temperature for
Saffil Blanket at 0.6 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0364 25.0
0.0369 25.0
0.0375 24.0
0.0372 23.0
0.0569 210.0
0.0615 209.0
0.0546 212.0
0.0845 408.0
0.0868 405.0
0.0851 406.0
0.0859 407.0
0.1095 600.0
0.1175 603.0
0.1290 604.0
0.1289 606.0
0.1711 803.0
0.1942 802.0
0.1821 801.0
0.1790 802.0
0.2674 976.0
0.2688 975.0
0.2453 976.0
0.2720 977.0
For SAFFIL Blanket
COMPACTION RATIO = 0.6
164
Appendix B. continued.
Table B05. Thermal Conductivity versus Temperature for
Saffil Blanket at 0.7 Compaction Ratio.
Thermal Conductivity Temperature
[W/EK] [K]
0.0369 25.0
0.0375 25.0
0.0384 25.0
0.0390 25.0
0.0597 201.0
0.0572 201.0
0.0586 201.0
0.0561 203.0
0.0886 401.0
0.0860 399.0
0.0870 401.0
0.0855 401.0
0.1477 599.0
0.1465 599.0
0.1449 599.0
0.1483 600.0
0.1935 798.0
0.2140 800.0
0.1934 802.0
0.2130 802.0
0.3054 979.0
0.3086 976.0
0.2903 976.0
0.3004 977.0
For SAFFIL Blanket
COMPACTION RATIO = 0.7
165
Appendix B. continued.
Table B06. Thermal Conductivity versus Temperature for
Saffil Blanket at 0.8 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0368 27.0
0.0372 28.0
0.0341 27.0
0.0369 27.0
0.0627 206.0
0.0594 203.0
0.0579 206.0
0.0602 206.0
0.0933 395.0
0.0930 397.0
0.0990 398.0
0.1009 401.0
0.1592 597.0
0.1607 597.0
0.1578 602.0
0.1720 602.0
0.2698 808.0
0.2208 803.0
0.1889 804.0
0.2327 806.0
0.3800 975.0
0.3367 977.0
0.3835 980.0
0.3469 984.0
For SAFFIL Blanket
COMPACTION RATIO = 0.8
166
Appendix B. continued.
Table B07. Thermal Conductivity versus Temperature for
Locon Blanket at 1.0 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0361 23.0
0.0362 23.0
0.0373 24.0
0.0356 23.0
0.0360 22.0
0.0360 23.0
0.0501 203.0
0.0547 213.0
0.0572 214.0
0.0803 405.0
0.0812 402.0
0.0791 404.0
0.0840 405.0
0.1143 606.0
0.1162 606.0
0.1224 606.0
0.1091 601.0
0.1598 805.0
0.1620 804.0
0.1647 802.0
0.1601 800.0
0.2320 971.0
0.2299 969.0
0.2333 969.0
0.2189 967.0
For LO-CON Blanket
COMPACTION RATIO = 1.0
167
Appendix B. continued.
Table B08. Thermal Conductivity versus Temperature for
Locon Blanket at 0.8 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0373 22.0
0.0373 24.0
0.0368 23.0
0.0376 22.0
0.0566 199.0
0.0565 197.0
0.0558 196.0
0.0562 202.0
0.0787 397.0
0.0832 395.0
0.0835 400.0
0.0858 403.0
0.1214 607.0
0.1177 607.0
0.1137 608.0
0.1151 604.0
0.1584 797.0
0.1558 798.0
0.1504 796.0
0.1549 793.0
0.2106 973.0
0.2105 971.0
0.2052 969.0
For LO-CON Blanket
COMPACTION RATIO = 0.8
168
Appendix B. continued.
Table B09. Thermal Conductivity versus Temperature for
Locon Blanket at 0.7 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0384 26.0
0.0389 26.0
0.0402 29.0
0.0386 29.0
0.0572 205.0
0.0556 202.0
0.0562 200.0
0.0588 205.0
0.0843 409.0
0.0896 408.0
0.0827 408.0
0.0768 410.0
0.1184 606.0
0.1212 610.0
0.1091 611.0
0.1137 603.0
0.1535 801.0
0.1524 801.0
0.1494 800.0
0.1496 798.0
0.1994 972.0
0.2017 971.0
0.2102 972.0
0.1982 974.0
For LO-CON Blanket
COMPACTION RATIO =
0.7
169
Appendix B. continued.
Table B10. Thermal Conductivity versus Temperature for
Locon Blanket at 0.6 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [R]
0.0401 29.0
0.0397 28.0
0.0397 27.0
0.0407 27.0
0.0569 199.0
0.0576 200.0
0.0585 202.0
0.0600 204.0
0.0871 403.0
0.0775 397.0
0.0860 399.0
0.0798 400.0
0.1011 599.0
0.1144 606.0
0.1141 604.0
0.1102 608.0
0.1493 806.0
0.1479 804.0
0.1472 805.0
0.1462 806.0
0.1977 972.0
0.1930 970.0
0.1994 972.0
0.1984 972.0
For LO-CON Blanket
COMPACTION RATIO =
0.6
170
Appendix B. continued.
Table B11. Thermal Conductivity versus Temperature for
Locon Blanket at 0.5 Compaction Ratio.
Thermal Conductivity Temperature
[W/MK] [K]
0.0451 25.0
0.0452 25.0
0.0437 26.0
0.0469 26.0
0.0624 196.0
0.0623 197.0
0.0627 198.0
0.0605 200.0
0.0852 406.0
0.0889 404.0
0.0862 403.0
0.0809 408.0
0.1115 598.0
0.1073 603.0
0.1149 608.0
0.1132 605.0
0.1427 807.0
0.1396 806.0
0.1397 807.0
0.1459 810.0
0.1810 979.0
0.1891 978.0
0.1857 980.0
0.1760 980.0
For LO-CON Blanket
COMPACTION RATIO = 0.5
171
Appendix B. continued.
Table B12. Thermal Conductivity versus Temperature for
Locon Blanket at 0.3 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0475 22.0
0.0483 23.0
0.0488 23.0
0.0497 23.0
0.0619 206.0
0.0666 209.0
0.0651 209.0
0.0624 204.0
0.0794 411.0
0.0877 412.0
0.0864 410.0
0.0838 410.0
0.1061 609.0
0.1106 610.0
0.1097 610.0
0.1042 609.0
0.1334 803.0
0.1310 806.0
0.1321 807.0
0.1589 975.0
0.1598 975.0
0.1639 972.0
0.1587 975.0
For LO-CON Blanket
COMPACTION RATIO = 0.3
172
Appendix B. continued.
Table B13. Thermal Conductivity versus Temperature for
Kaowool K2300 Blanket at 1.0 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0398 25.0
0.0405 25.0
0.0425 25.0
0.0449 25.0
0.0527 196.0
0.0533 207.0
0.0513 203.0
0.0764 400.0
0.0733 405.0
0.0815 400.0
0.1221 605.0
0.1280 603.0
0.1213 605.0
0.1607 603.0
0.2103 807.0
0.1960 806.0
0.1838 803.0
0.1739 807.0
0.2361 1001.0
0.2716 1005.0
0.2414 1002.0
0.2548 1000.0
For Kaowool K2300 Blanket
COMPACTIO RATIO =
1.0
173
Appendix B. continued.
Table B14. Thermal Conductivity versus Temperature for
Kaowool K2300 Blanket at 0.8 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0386 24.0
0.0384 23.0
0.0412 23.0
0.0403 22.0
0.0586 203.0
0.0624 203.0
0.0612 200.0
0.0611 199.0
0.0891 408.0
0.0818 409.0
0.0791 402.0
0.0751 391.0
0.1176 611.0
0.1239 615.0
0.1250 614.0
0.1177 613.0
0.1688 808.0
0.1582 803.0
0.1823 800.0
0.1556 799.0
0.1970 981.0
0.2108 981.0
0.1932 977.0
For Kaowool K2300 Blanket
COMPACTION RATIO =
0.8
174
Appendix B. continued.
Table B15. Thermal Conductivity versus Temperature for
Kaowool K2300 Blanket at 0.7 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0362 23.0
0.0366 24.0
0.0361 25.0
0.0364 26.0
0.0557 198.0
0.0552 201.0
0.0553 205.0
0.0559 207.0
0.0834 401.0
0.0802 400.0
0.0817 401.0
0.0824 ‘401.0
0.1171 594.0
0.1164 596.0
0.1208 597.0
0.1213 598.0
0.1532 795.0
0.1525 796.0
0.1517 797.0
0.1557 799.0
0.1810 968.0
0.1933 971.0
0.1895 972.0
0.1916 971.0
For Kaowool
COMPACTION RATIO =
0.7
K2300 Blanket
175
Appendix B. continued.
Table B16a. Thermal Conductivity versus Temperature for
Kaowool K2300 Blanket at 0.6 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0388 18.0
0.0387 21.0
0.0390 23.0
0.0395 22.0
0.0574 204.0
0.0612 205.0
0.0605 206.0
0.0577 209.0
0.0860 409.0
0.0861 406.0
0.0813 404.0
0.0896 407.0
0.1194 609.0
0.1312 610.0
0.1145 608.0
0.1199 602.0
0.1620 800.0
0.1549 799.0
0.1640 800.0
0.1610 801.0
0.1904 967.0
0.2070 968.0
0.2094 969.0
0.1919 971.0
For Kaowool K2300 Blanket
COMPACTION RATIO = 0.6
176
Appendix B. continued.
Table B16b. Thermal Conductivity versus Temperature for
Kaowool K2300 Blanket at 0.6 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0374 26.0
0.0388 25.0
0.0359 21.0
0.0367 26.0
0.0588 191.0
0.0564 193.0
0.0585 190.0
0.0560 206.0
0.0768 389.0
0.0794 386.0
0.0814 385.0
0.0812 385.0
0.1184 601.0
0.1108 604.0
0.1178 602.0
0.1572 804.0
0.1650 804.0
0.1602 803.0
0.1565 802.0
0.1939 968.0
0.2003 968.0
0.2144 968.0
For Kaowool K2300 Blanket
COMPACTION RATIO = 0.6
177
Appendix B. continued.
Table B17. Thermal Conductivity versus Temperature for
Kaowool K2300 Blanket at 0.5 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0431 31.0
0.0448 37.0
0.0427 38.0
0.0389 28.0
0.0622 206.0
0.0611 208.0
0.0683 206.0
0.0600 208.0
0.0876 406.0
0.0866 402.0
0.0909 405.0
0.0864 404.0
0.1151 608.0
0.1203 607.0
0.1152 607.0
0.1128 608.0
0.1438 803.0
0.1419 801.0
0.1475 802.0
0.1402 806.0
0.1772 973.0
0.1736 978.0
For Kaowool K2300 Blanket
COMPACTION RATIO =
0.5
178
Appendix B. continued.
Table B18. Thermal Conductivity versus Temperature for
Kaowool K2300 Blanket at 0.3 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [R]
0.0518 21.0
0.0511 21.0
0.0518 21.0
0.0504 20.0
0.0718 224.0
0.0706 221.0
0.0703 223.0
0.0686 218.0
0.0938 424.0
0.0913 423.0
0.0947 422.0
0.0991 420.0
0.1088 613.0
0.1157 614.0
0.1217 613.0
0.1111 613.0
0.1335 810.0
0.1375 813.0
0.1361 814.0
0.1376 813.0
0.1735 981.0
0.1670 980.0
0.1697 975.0
0.1714 975.0
0.1657 977.0
For Kaowool K2300 Blanket
COMPACTION RATIO =
0.3
179
Appendix B. continued.
Table B19. Thermal Conductivity versus Temperature for
Kaowool K2300 Blanket at 0.2 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0613 22.0
0.0633 25.0
0.0672 26.0
0.0668 26.0
0.0797 206.0
0.0749 208.0
0.0762 212.0
0.0796 215.0
0.0929 411.0
0.0932 412.0
0.0887 413.0
0.0988 417.0
0.1119 598.0
0.1142 602.0
0.1192 603.0
0.1147 601.0
0.1326 804.0
0.1347 807.0
0.1358 806.0
0.1376 812.0
0.1608 971.0
0.1612 975.0
0.1634 980.0
0.1674 978.0
For Kaowool K2300 Blanket
COMPACTION RATIO =
0.2
180
Appendix B. continued.
Table B20. Thermal Conductivity versus Compaction Ratio
for Saffil Blanket. Room Temperature Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.0363 0.9
0.0363 0.9
0.0359 0.9
0.0359 0.9
0.0368 0.8
0.0372 0.8
0.0341 0.8
0.0369 0.8
0.0369 0.7
0.0375 0.7
0.0384 0.7
0.0390 0.7
0.0364 0.6
0.0369 0.6
0.0375 0.6
0.0372 0.6
0.0355 0.5
0.0361 0.5
0.0368 0.5
0.0367 0.5
0.0406 0.3
0.0400 0.3
0.0410 0.3
0.0413 0.3
0.0460 0.2
0.0474 0.2
0.0470 0.2
0.0469 0.2
For SAFFIL Blanket
TEMPERATURE = 25 C
181
Appendix B. continued.
Table B21. Thermal Conductivity versus Compaction Ratio
for Saffil Blanket. 200 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.0627
0.0594
0.0579
0.0602
0.0597
0.0572
0.0586
0.0561
0.0569
0.0615
0.0546
0.0514
0.0550
0.0580
0.0555
0.0556
0.0536
0.0563
0.0535
0.0579
0.0592
0.0604
0.0621
c>o<3c>o<3c>o<3c>o<5c>o<3c>o<3c>o<3c>o
A)»run>ucaupucnuamtnasm<m~Jq-q~1m¢naam
\ k'J
“i??117.71'-»“€.‘~L‘ “" (‘73.!
For SAFFIL Blanket
TEMPERATURE = 200 C
182
Appendix B. continued.
Table B22. Thermal Conductivity versus Compaction Ratio
for Saffil Blanket. 400 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.0933 0.8
0.0930 0.8
0.0990 0.8
0.1009 0.8
0.0886 0.7
0.0860 0.7
0.0870 0.7
0.0855 0.7
0.0845 0.6
0.0868 0.6
0.0851 0.6
0.0859 0.6
0.0758 0.5
0.0790 0.5
0.0773 0.5
0.0760 0.5
0.0816 0.3
0.0753 0.3
0.0797 0.3
0.0739 0.3
0.0819 0.2
0.0801 0.2
0.0864 0.2
0.0860 0.2
For SAFFIL Blanket
TEMPERATURE = 400 C
183
Appendix B. continued.
Table B23. Thermal Conductivity versus Compaction Ratio
for Saffil Blanket. 600 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.1592 0.8
0.1607 0.8
0.1578 0.8
0.1720 0.8
0.1477 0.7
0.1465 0.7
0.1449 0.7
0.1483 0.7
0.1095 0.6
0.1175 0.6
0.1290 0.6
0.1289 0.6
0.1164 0.5
0.1174 0.5
0.1163 0.5
0.1039 0.3
0.1087 0.3‘
0.1130 0.3
0.1163 0.3
0.0965 0.2
0.1025 0.2
0.1095 0.2
0.1089 0.2
For SAFFIL Blanket
TEMPERATURE = 600 C
184
Appendix B. continued.
Table B24. Thermal Conductivity versus Compaction Ratio
for Saffil Blanket. 800 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.2698
0.2208
0.1889
0.2327
0.1935
0.2140
0.1934
0.2130
0.1711
0.1942
0.1821
0.1790
0.1640
0.1568
0.1653
0.1544
0.1401
0.1385
0.1426
0.1287
0.1388
0.1374
0.1281
wauuuwmunmmosmmmqqqqmoomm
OOOOOOOOOOOOOOOOOOOOOOO
For SAFFIL Blanket
TEMPERATURE = 800 C
185
Appendix B. continued.
Table B25. Thermal Conductivity versus Compaction Ratio
for Saffil Blanket. 970 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.3800 0.8
0.3367 0.8
0.3835 0.8
0.3469 0.8
0.3054 0.7
0.3086 0.7
0.2903 0.7
0.3004 0.7
0.2674 0.6
0.2688 0.6
0.2453 0.6
0.2720 0.6
0.2062 0.5
0.2418 0.5
0.2166 0.5
0.2060 0.5
0.1942 0.3
0.1814 0.3
0.1909 0.3
0.1847 0.3
0.1706 0.2
0.1689 0.2
0.1627 0.2
0.1734 0.2
0.1752 0.2
For SAFFIL Blanket
TEMPERATURE = 970 C
186
Appendix B. continued.
Table B26. Thermal Conductivity versus Compaction Ratio
for Locon Blanket. Room temperature Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.0361
0.0362
0.0373
0.0356
0.0360
0.0360
0.0361
0.0354
0.0370
0.0358
0.0373
0.0373
0.0368
0.0376
0.0384
0.0389
0.0402
0.0386
0.0401
0.0397
0.0397
0.0407
0.0451
0.0452
0.0437
0.0469
0.0475
0.0483
0.0488
0.0497
uuwummmmaxmosmqqqxloomoommsoxosooooooo
OOCOCOOOOOOOOOOOOOOOOOOOHHHHHH
For LO-CON Blanket
TEMPERATURE = 22 C
187
Appendix B. continued.
Table B27. Thermal Conductivity versus Compaction Ratio
for Locon Blanket. 200 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.0501 1.0
0.0547 1.0
0.0572 1.0
0.0566 0.8
0.0565 0.8
0.0558 0.8
0.0562 0.8
0.0572 0.7
0.0556 0.7
0.0562 0.7
0.0588 0.7
0.0569 0.6
0.0576 0.6
0.0585 0.6
0.0600 0.6
0.0624 0.5
0.0623 0.5
0.0627 0.5
0.0605 0.5
0.0619 0.3
0.0666 0.3
0.0651 0.3
0.0624 0.3
For LO-CON Blanket
TEMPERATURE = 200 C
188
Appendix B. continued.
Table B28. Thermal Conductivity versus Compaction Ratio
for Locon Blanket. 400 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.0803
0.0812
0.0791
0.0840
0.0787
0.0832
0.0835
0.0858
0.0843
0.0896
0.0827
0.0768
0.0871
0.0775
0.0860
0.0798
0.0852
0.0889
0.0862
0.0809
0.0794
0.0877
0.0864
0.0838
ooc>cc:c:ooooooooooooooHHr-u-l
catiutauzmcn01m<nosm-q~Jq~4a>mcnaaococ>o
531mm "'—-
For LO-CON Blanket
TEMPERATURE = 400 C
189
Appendix B. continued.
Table B29. Thermal Conductivity versus Compaction Ratio
for Locon Blanket. 600 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.1143
0.1162
0.1224
0.1091
0.1214
0.1177
0.1137
0.1151
0.1184
0.1212
0.1091
0.1137
0.1011
0.1144
0.1141
0.1102
0.1115
0.1073
0.1149
0.1132
0.1061
0.1106
0.1097
0.1042
UUUUUIU'IUIWGGGOQQQQQQGGOOOO
OOOOOOOOOOOOOOOOOOOOHHHH
For LO-CON Blanket
TEMPERATURE a 600 C
190
Appendix B. continued.
Table B30. Thermal Conductivity versus Compaction Ratio
for Locon Blanket. 800 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.1598 1.0
0.1620 1.0
0.1647 1.0
0.1601 1.0
0.1584 0.8
0.1558 0.8
0.1504 0.8
0.1549 0.8
0.1535 0.7
0.1524 0.7
0.1494 0.7
0.1496 0.7
0.1493 0.6
0.1479 0.6
0.1472 0.6
0.1462 0.6
0.1427 0.5
0.1396 0.5
0.1397 0.5
0.1459 0.5
0.1334 0.3
0.1310 0.3
0.1321 0.3
For LO-CON Blanket
TEMPERATURE = 800 C
191
Appendix B. continued.
Table B31. Thermal Conductivity versus Compaction Ratio
for Locon Blanket. 970 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.2320 1.0
0.2299 1.0
0.2333 1.0
0.2189 1.0
0.2106 0.8
0.2105 0.8
0.2052 0.8
0.1994 0.7
0.2017 0.7
0.2102 0.7
0.1982 0.7
0.1977 0.6
0.1930 0.6
0.1994 0.6
0.1984 0.6
0.1810 0.5
0.1891 0.5
0.1857 0.5
0.1760 0.5
0.1589 0.3
0.1598 0.3
0.1639 0.3
0.1587 0.3
For LO-CON Blanket
TEMPERATURE = 970 C
192
Appendix B. continued.
Table B32. Thermal Conductivity versus Compaction Ratio fOr
Kaowool K2300 Blanket. Room temperature Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.0398
0.0405
0.0425
0.0449
0.0331
0.0346
0.0345
0.0343
0.0386
0.0384
0.0412
0.0403
0.0362
0.0366
0.0361
0.0364
0.0388
0.0387
0.0390
0.0395
0.0374
0.0388
0.0359
0.0367
0.0431
0.0448
0.0427
0.0389
0.0518
0.0511
0.0518
0.0504
0.0613
0.0633
0.0672
0.0668
wwwwuuuummmmmosmasmmo‘mqqdqmmmoomsoxomoooo
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOHI—‘PH
For Kaowool K2300 Blanket
TEMPERATURE = 22 C
193
Appendix B. continued.
Table B33. Thermal Conductivity versus Compaction Ratio for
Kaowool K2300 Blanket. 200 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.0527
0.0533
0.0513
0.0586
0.0624
0.0612
0.0611
0.0557
0.0552
0.0553
0.0559
0.0574
0.0612
0.0605
0.0577
0.0588
0.0564
0.0585
0.0560
0.0622
0.0611
0.0683
0.0600
0.0718
0.0706
0.0703
0.0686
0.0797
0.0749
0.0762
0.0796
OOOOOOOOOOOOOOOOOOOOOOOOOOOOI—‘HH
MNNNuuuummuumasoxmosaxasmasqqqqoooooooaooo
For Kaowool K2300 Blanket
TEMPERATURE = 200 C
194
Appendix B. continued.
Table B34. Thermal Conductivity versus Compaction Ratio for
Kaowool K2300 Blanket. 400 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.0764 1.0
0.0733 1.0
0.0815 1.0
0.0891 0.8
0.0818 0.8
0.0791 0.8
0.0751 0.8
0.0834 0.7
0.0802 0.7
0.0817 0.7
0.0824 0.7
0.0860 0.6
0.0861 0.6
0.0813 0.6
0.0896 0.6
0.0768 0.6
0.0794 0.6
0.0814 0.6
0.0812 0.6
0.0876 0.5
0.0851 0.5
0.0867 0.5
0.0864 0.5
0.0938 0.3
0.0913 0.3
0.0947 0.3
0.0991 0.3
0.0929 0.2
0.0932 0.2
0.0887 0.2
0.0988 0.2
For Kaowool K2300 Blanket
TEMPERATURE = 400 C
195
Appendix B. continued.
Table B35. Thermal Conductivity versus Compaction Ratio for
Kaowool K2300 Blanket. 600 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.1221 1.0
0.1280 1.0
0.1213 1.0
0.1607 1.0
0.1176 0.8
0.1239 0.8
0.1250 0.8
0.1177 0.8
0.1171 0.7
0.1164 0.7
0.1208 0.7
0.1213 '0.7
0.1194 0.6
0.1312 0.6
0.1145 0.6
0.1199 0.6
0.1184 0.6
0.1108 0.6
0.1178 0.6
0.1151 0.5
0.1203 0.5
0.1152 0.5
0.1153 0.5
0.1088 0.3
0.1157 0.3
0.1217 0.3
0.1111 0.3
0.1119 0.2
0.1142 0.2
0.1192 0.2
0.1147 0.2
For Kaowool K2300 Blanket
TEMPERATURE = 600 C
196
Appendix B. continued.
Table B36. Thermal Conductivity versus Compaction Ratio for
Kaowool K2300 Blanket. 800 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.2103 1.0
0.1960 1.0
0.1838 1.0
0.1739 1.0
0.1688 0.8
0.1582 0.8
0.1823 0.8
0.1556 0.8
0.1532 0.7
0.1525 0.7
0.1517 0.7
0.1557 0.7
0.1620 0.6
0.1549 0.6
0.1640 0.6
0.1610 0.6
0.1572 0.6
0.1650 0.6
0.1602 0.6
0.1565 0.6
0.1438 0.5
0.1419 0.5
0.1475 0.5
0.1402 0.5
0.1335 0.3
0.1375 0.3
0.1361 0.3
0.1376 0.3
0.1326 0.2
0.1347 0.2
0.1358 0.2
0.1376 0.2
For Kaowool K2300 Blanket
TEMPERATURE = 800 C
197
Appendix B. continued.
Table B37. Thermal Conductivity versus Compaction Ratio for
Kaowool K2300 Blanket. 970 C Data.
Thermal Conductivity Compaction Ratio
[W/mK]
0.2361 1.0
0.2716 1.0
0.2414 1.0
0.2548 1.0
0.1970 0.8
0.2108 0.8
0.1932 0.8
0.1810 0.7
0.1933 0.7
0.1895 0.7
0.1916 0.7
0.1904 0.6
0.2070 0.6
0.2094 0.6
0.1919 0.6
0.1939 0.6
0.2003 0.6
0.2144 0.6
0.1772 0.5
0.1736 0.5
0.1735 0.5
0.1670 0.3
0.1697 0.3
0.1714 0.3
0.1657 0.3
0.1608 0.2
0.1612 0.2
0.1634 0.2
0.1674 0.2
For Kaowool K2300 Blanket
TEMPERATURE = 1000 C
198
Appendix B. continued.
Table B38. Thermal Conductivity versus Temperature for
Kaowool NF Blanket at 0.7 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0393 22.0
0.0410 24.0
0.0417 23.0
0.0400 22.0
0.0591 197.0
0.0578 195.0
0.0600 198.0
0.0604 199.0
0.0780 394.0
0.0808 396.0
0.0800 396.0
0.0831 397.0
0.1075 596.0
0.1153 599.0
0.1252 599.0
0.1132 600.0
0.1456 805.0
0.1472 803.0
0.1498 805.0
0.1460 802.0
0.1750 975.0
0.1841 978.0
0.1858 978.0
0.1871 980.0
For Kaowool-NF Blanket
COMPACTION RATIO =
0.7
199
Appendix B. continued.
Table B39. Thermal Conductivity versus Temperature for
Kaowool NF Blanket at 1.0 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0366 25.0
0.0360 24.0
0.0352 25.0
0.0369 26.0
0.0575 203.0
0.0454 203.0
0.0483 200.0
0.0575 202.0
0.0674 400.0
0.0738 402.0
0.0665 396.0
0.0759 403.0
0.1126 603.0
0.1205 603.0
0.1184 603.0
0.1343 607.0
0.1458 791.0
0.1525 794.0
0.1532 795.0
0.1559 795.0
0.1721 957.0
0.1562 963.0
0.1729 970.0
0.1973 971.0
For Kaowool-NF Blanket
COMPACTION RATIO =
1.0
200
Appendix B. continued.
Table B40. Thermal Conductivity versus Temperature for
Kaowool ZR Blanket at 1.0 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0447 23.0
0.0441 27.0
0.0358 23.0
0.0280 25.0
0.0584 206.0
0.0585 211.0
0.0581 211.0
0.0626 207.0
0.0829 394.0
0.0957 397.0
0.0848 398.0
0.0872 402.0
0.0998 600.0
0.1055 605.0
0.1049 606.0
0.1071 606.0
0.1234 801.0
0.1477 803.0
0.1419 807.0
0.1420 805.0
0.1848 969.0
0.1624 970.0
0.1805 976.0
For Kaowool-ZR Blanket
COMPACTION RATIO =
1.0
201
Appendix B. continued.
Table B41. Thermal Conductivity versus Temperature for
Durablanket-S at 1.0 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0410 25.0
0.0379 26.0
0.0361 20.0
0.0372 24.0
0.0367 22.0
0.0435 21.0
0.0626 204.0
0.0663 203.0
0.0601 203.0
0.0615 206.0
0.0880 401.0
0.0838 399.0
0.0935 401.0
0.0920 398.0
0.0978 401.0
0.1574 602.0
0.1647 605.0
0.1529 604.0
0.1549 606.0
0.2309 800.0
0.2448 800.0
0.2270 803.0
0.2393 800.0
0.2964 973.0
0.3261 973.0
0.2893 975.0
0.3074 979.0
For Durablanket-S
COMPACTION RATIO = 1.0
202
Appendix B. continued.
Table B42. Thermal Conductivity versus Temperature for
Kaowool K2600 Blanket at 1.0 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0409 23.0
0.0403 23.0
0.0442 23.0
0.0401 23.0
0.0437 202.0
0.0500 206.0
0.0559 205.0
0.0493 202.0
0.0818 393.0
0.0886 402.0
0.0885 398.0
0.0914 402.0
0.1108 605.0
0.1149 599.0
0.1428 605.0
0.1105 598.0
0.1554 805.0
0.1670 798.0
0.1490 797.0
0.2162 798.0
0.2368 983.0
0.2555 984.0
0.2410 984.0
0.2294 983.0
For Kaowool K2600 Blanket
COMPACTION RATIO =
1.0
203
Appendix B. continued.
Table B43. Thermal Conductivity versus Temperature for
Fibersil Cloth at 1.0 Compaction Ratio.
Thermal Conductivity Temperature
[W/mK] [K]
0.0423 23.0
0.0590 26.0
0.0635 23.0
0.0898 197.0
0.1024 200.0
0.1032 198.0
0.0991 199.0
0.1379 404.0
0.1420 406.0
0.1288 407.0
0.1278 410.0
0.1915 600.0
0.1764 600.0
0.1830 604.0
0.1710 595.0
For Fibersil Cloth
COMPACTION RATIO =
1.0
204
APPENDIX C
Table C1. Air gas thermal conductivity at atmospheric
pressure. After [47]
Temperature Thermal Conductivity
[K] [W/mK]
100 0.009246
150 0.013735
200 , 0.01809
250 0.02227
300 0.02624
350 0.03003
400 0.03365
450 0.03707
500 0.04038
550 0.04360
600 0.04659
650 0.04953
700 0.05230
750 0.05509
800 0.05779
850 0.06028
900 0.06279
950 0.06525
1000 0.06752
1100 0.0732
1200 0.0782
1300 0.0837
1400 0.0891
1500 0.0946
1600 0.1000
1700 0.105
1800 0.111
1900 0.117
2000 0.124
2100 0.131
2200 0.139
2300 0.149
2400 0.161
2500 0.175
205
APPENDIX D
Factors for Thermal Conductivity Units
CODVGI‘S 1011
182.;~:
_ ~-os x nnn.o q~_.o 4.0. x m44.m n.o. x ~44.. ~44..o Illuullulnu.
Cm Dun
198..%t
~. . oo4.s n.°. x «ne.4 ~-o. x .n~.. snA.. ......I.....
a. sun
8 gm:
.8.» ~35 . #2 x Es ~-2 x no... no:
... .08.
U Q N8
man 832†a: . Rats at»:
:0 den
a to u a so
n.noo n~.~m so.mo onm~.o . ~o. In... to ~
a cow
8 E U ONE
nno.o shun.o oomn.o n-o. x asn.~ ~-o. . to
a as
to8~cg 1862: use 6 8.8 38 x: S:
5 Bo 3o .08. .3 3 3 3 €263 o»
maï¬a: >uï¬>wuococoo
Hmsuogu sou muouomu coamum>coo .Ho
manna
206
Appendix E
Configuration of Sensing Thermocouple Junction and Placement
with Respect to the Hot Wire
Hot wire
I
/\ aâ€
I -
f‘a—j
Thermocouple
where the dimensions in millimeters are, 2 s a s 6,
2 s b s 5, and 4 s c s 7.
The thermocouples were bent manually to the v-shaped junction
configuration. (See also Figures 5 and 6).
207
REFERENCES
Fine, H. A., "Analysis of the Applicability of the Hot Wire
Technique for Determination of the Thermal Conductivity of
Diathermanous Materials", pp. 147-153 in The:mgl_1;an§m1§§ign
Measuremen;§_9f_1n§ulatign§. ASTM STP 660. Editor R- P-
Tye, American Society for Testing and Materials,
Philadelpia,1978.
Sandberg, 0., Andersson, P., and Backstrom, G., "Heat
Capacity and Thermal Conductivity from Pulsed Wire Probe
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