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This is to certify that the

thesis entitled

EXPERIMENTAL INVESTIGATION OF TRANSIENT THERMAL
PROPERTY CHANGES OF ALUMINUM 2024—T35l

presented by

Sami Raouf Al—Araji

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in _MechanicaL Engineering

Major profes r

 

 

Date October 12, 1973

 

 

 

 

 

 

 

 

 

ABSTRACT

EXPERIMENTAL INVESTIGATION OF TRANSIENT THERMAL
PROPERTY CHANGES OF ALUMINUM 2024-T351

BY

Sami Raouf Al-Araji

When as-received aluminum 2024-T351 is heated to
350 °F, changes in the microstructure take place. These
changes are further enhanced by heating to higher temper—
atures and/or longer times.

This heating results in increases in the thermal
conductivity (k) and specific heat (cp) values of the
aluminum. A simplified method proposed by Beck was used to
determine the transient values of k and cp.

It was found that the changes in k can approach 25%
while those for cp are less. The changes in k follow a
certain pattern. This pattern begins with a value of k
when the specimen arrives at a given temperature (in the
range 350 to 425 °F) then k increases to some maximum value
while it is maintained at that temperature. The rate of
increase depends on the temperature level (the closer the
temperature to 425 °F the faster it arrives at the maximum

value). After it stays at the maximum value for a period

 

Sami Raouf Al-Araji

of time (for example, at 425 °F it stays for about 15 to 20
minutes) k starts to drop until it reaches a certain value
where it remains relatively constant with time. At this
stage the aluminum becomes over-aged.

It was found that the over—aged value of k at room
temperature as well as the high temperatures is close to
that obtained by the regular annealing process. In order
to predict the k values of aluminum 2024-T351 for different
temperatures and times, a mathematical model was developed

based on the data obtained in this study.

 

EXPERIMENTAL INVESTIGATION OF TRANSIENT THERMAL

PROPERTY CHANGES OF ALUMINUM 2024-T351

BY

Sami Raouf Al-Araji

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1973

 

ti
2:3 ACKNOWLEDGMENTS
3.5;?
I The author is very grateful for guidance and

encouragement during the period of research and during the

preparation of this thesis by the chairman of his Guidance
Committee, Professor James V. Beck and the members of the
Committee, Professors Norman L. Hills, Mahlon C. Smith, and

Howard L. Womochel. The author also wishes to thank
Professors Robert W. Little and Charles R. St. Clair Jr.

for their help and understanding.

Thanks also go to John W. Hoffman, Don Childs, and

Bob Rose of the Division of Engineering Research.

The author wishes to particularly acknowledge the

help of Don Childs during the construction of the experi-

mental apparatus.

ii

TABLE OF CONTENTS

Page
LIST OF TABLES . . . . . . . . . . . . . v
LIST OF FIGURES. . . . . . . . . . . . . viii
LIST OF SYMBOLS. . . . . . . . . . . . . xii
Chapter
I. INTRODUCTION . . . . . . . . . . . l
1.1 Importance of the problem. . . . . 3
1.2 Literature review . . . . . . . 4
II. EXPERIMENTAL PROCEDURES. . . . . . . . 22
2.1 The new method . . . . . . . . 22
2.2 Experimental set- up. . . 28
2.3 Specimen selection and thermocouple
installation . . . . . . . . 34
2.4 Running a test . . . 48
2.5 The reference material and typical
results . . . . . . . . . . 51
III. EXPERIMENTAL RESULTS FOR ALUMINUM 2024-T351 . 72
3.1 Composition of the specimen . . . . 72
3.2 Metallurgical concepts of specimen. . 72
3.3 Experimental strategy . . . . . . 79
3.4 Experimental results . . . . . . 81
3.5 Corrections for errors. . . . . . 88
IV. MODELING. . . . . . . . . . . . . 114
4.1 The development of a mathematical
model. . . 114
4.2 Obtaining thermal conductivity values
from the mathematical model . . . 123
4.3 Example. . . . . . . . . . . 131

 

Chapter Page

V. SUMMARY AND CONCLUSIONS . . . . . . . . 144
5.1 Recommendations for further research . 146
BIBLIOGRAPHY . . . . . . . . . . . . . . 148

iv

 

LIST OF TABLES

Page

Typical values of k and GP for Armco iron
(specimen #1) at room temperature. A
silicone film, 0.015 inch thick was used
on the interface. . . . . . . . . . 55

Typical values of k and cp for Armco iron
(specimen #1) at room temperature. Three
drops of distilled water were used at the
interface . . . . . . . . . . . . 56

Typical values of k and cp for Armco iron
(specimen #3) a new bar, at room temperature
using 3 drops of distilled water at
interface . . . . . . . . . . . . 56

Typical values of k and c for Armco iron
(specimen #3) at room temperature using 6
drops of water at interface . . . . . . 57

Typical values of k and GP for Armco iron
(specimen #3) at room temperature using
0.015 inch film of silicone grease at
interface . . . . . . . . . . . . 57

Average, variance and standard of deviation
for the values obtained at room temperature
using 0.015 inch silicone film on surface
of Armco iron (specimen #1) . . . . . . 58

Average, variance, and standard of deviation
for values obtained at room temperature
using 3 drops of water on surface of Armco
iron (specimen #1) . . . . . . . . . 58

Average, variance, and standard of deviation
for values obtained at room temperature
using 3 drops of water on surface of Armco
iron (specimen #3) . . . . . . . . . 58

 

2.5.10

2.5.11

2.5.12

2.5.13

2.5.14

Page

Average, variance, and standard of deviation

for values obtained at room temperature

using 6 drops of water on surface of Armco

iron (specimen #3) . . . . . . . . . 59
Average, variance, and standard of deviation

for values obtained at room temperature

using 0.015 inch silicone film on surface

of Armco iron (specimen #3) . . . . . . 59
Typical values of k and c for Armco iron

(specimen #3) at 300 ang 400 °F using 0.015

inch film of silicone grease at the

interface . . . . . . . . . . . . 59
Average, variance, and standard of deviation

of the values obtained at 300 and 400 °F of

Armco iron (specimen #3) using 0.015 inch

silicone film at interface . . . . . . 60
Comparison of the present values of k and cp

of Armco iron with those obtained by TPRC . 61
k and CP values obtained from the IBM 1800

data by utilizing program "SIMPL" . . . . 62
Mechanical properties of aluminum 2024-T351

in the annealed condition. . . . . . . 74
Mechanical properties of aluminum 2024—T351

in the final aged condition . . . . . . 77
k and c values for aluminum 2024-T351

(spec1men SE) at 350 °F . . . . . . . 89
k and c values for aluminum 2024-T351

(spec1men 4H) at 375 °F . . . . . . . 90
k and c values for aluminum 2024-T351

(spec1men 14H) at 375 °F . . . . . . . 91
k and c values for aluminum 2024—T351

(spec1men 6H) at 400 °F . . . . . . . 92
k and c values for aluminum 2024-T351

(spec1men 10H) at 400 °F . . . . . . . 93
k and c values for aluminum 2024—T351

(spec1men 11H) at 400 °F . . . . . . . 93

vi

 

3.4.7 k and c values for aluminum 2024-T351

(spec1men 12H) at 400 °F

3.4.8 k and c values of aluminum 2024-T351

(spec1men 3H) at 425 °F .

3.4.9 k and c values of aluminum 2024—T351

(spec1men 13H) at 425 °F .

3.4.10 Composite (average) values of k

0

aluminum 2024-T351 at 375 °F.

3.4.11 Composite (average) values of k

aluminum 2024-T351 at 400 °F.

3.4.12 Composite (average) values of k

aluminum 2024-T351 at 425 °F.

3.4.13 Values of k at room temperature

aged aluminum 2024-T351 .

and

and

and

for

Cp Of

3.4.14 Composite (average) values of k at room
temperature for over-aged aluminum

2024—T351 . . . . . .

4.3.1 Initial and lst corrected estimates at

zero. . . . . . . .
4.3.2 Temperatures at time zero .
4.3.3 Estimates for 0.5 hour time.

‘ 4.3.4 Temperatures at time 0.5 hour

4.3.5 Estimates for 1.0 hour time.

4.3.6 Temperatures at time 1.0 hour

time

Page

94

94

95

95

96

96

97

97

138
138
141
141
142

142

 

 

 

LIST OF FIGURES

Figure Page
2.1.1 Flat plate geometry of the problem . . . . 23
2.1.2 Heat flux and temperature histories for a

specimen . . . . . . . . . . . . 24
2.1.3 Two plates in imperfect contact . . . . . 28
2.2.1 Schematic diagram of the testing assembly. . 30
2.2.2 Photograph of the testing assembly . . . . 32

2.2.3 A cabinet housing the temperature controllers
and the adjacent hydraulic system. . . . 33

2.2.4 The operational panel, computer signal
conditioner (CSC) and the testing assembly. 35

2.3.1 Thermocouple locations at the heated surface
of the specimen. . . . . . . . . . 38

2.3.2 Thermocouple locations at the insulated
surface (x = E) for an experiment on Armco

iron specimen #3 . . . . . . . . . 41
2.3.3 Diagram of the washer-type thermocouple . . 43
2.3.4 Thermocouple location on the side of the

OFHC copper calorimeter . . . . . . . 44
2.3.5 Wiring diagram for the over-all experimental

system. . . . . . . . . . . . . 46
2.3.6 One millivolt input over sixty seconds. . . 47

2.5.1 Temperature history of the calorimeter and
specimen during a test on Armco iron. . . 64

viii

 

Figure

2.5.2

2.5.3

2.5.4

3.2.1

3.2.2a

3.2.2b

3.2.3

3.2.5
3.2.6
3.4.1

3.4.2

Typical values of k, pcp, h and g'during a
given test on Armco iron . . . . . .

Copper at elevated temperature and Armco iron
at room temperature initially. Brought into
intimate contact, both copper and Armco are
surrounded by Fiberglass insulation . .

Copper at elevated temperature and Armco iron
at room temperature initially. Copper is
surrounded by Fiberglass insulation, and
Armco iron is surrounded by Transite guard
heater (heater was off during test) . . .

The temperature history in the present system
for two cases. . . . . . . . . . .

Partial equilibrium diagram for aluminum-
c0pper alloys. . . . . . . . .

Microstructure of aluminum 2024 in the
annealed condition . . . . . . .

The stages in the formation of an equilibrium
precipitate. (a) Supersaturated solid
solution. (b) Transition lattice coherent
with the solid solution. (c) Equilibrium
precipitate essentially independent of the
solid solution . . . . . . . . . .

Aging and over-aging curves for Al-Cu system,
over-aging drops the strength and hardness
0f the alloy 0 O O O O O O O O O 0

Diagram of lattice structure showing distortion
caused by precipitated constituent. . .

Diagram of precipitation heat treatment. . .
Diagram of over-aging. . . . . . . . .

Thermal conductivity as a function of time at
350 °F for aluminum 2024-T351, specimen 5H .

Specific heat as a function of time at 350 °F
for aluminum 2024-T351, specimen 5H . . .

ix

Page

65

68

69‘

70

73

75

75

76

76
78
78

98

99

Figure

3.4.10

3.4.11

3.4.12

Thermal conductivity as a function of time
at 375 °F for aluminum 2024-T351, specimen
4H and 14“ O I I O . O O O O O 0

Specific heat as a function of time at 375 °F
for aluminum 2024-T351, specimen 4H and
14H 0 I I O C I O O O O O O ‘

Thermal conductivity as a function of time
at 400 °F for aluminum 2024-T351, specimens
6H, loH’ 11H, and 12H I I I C C O I

Specific heat as a function of time at 400 °F
for aluminum 2024- -T351, specimens, 6H, 10H,
11H, 12H. . . . . . . . . . .

Thermal conductivity as a function of time at
425 °F for aluminum 2024-T351, specimens
3H and 13H 0 O I O I O D O O O 0

Specific heat as a function of time at 425 °F
for aluminum 2024-T351, specimens 3H and
13H I I I I O O O O O I I O 0

Thermal conductivity as a function of time for
composite curves at 375°, 400°, and 425 °F
for aluminum 2024-T351 . . . . . .

Specific heat as a function of time for
composite curves at 375°, 400°, and 425 °F
for aluminum 2024-T351 . . . . . . .

Thermal conductivity as a function of
temperature given by TPRC . . . . . .

Specific heat as a function of temperature
given by TPRC . . . . . . . . . .

Values of normalized thermal conductivity of
aluminum 2024-T351 for the temperatures
350°, 375°, 400°, and 425 °F versus
normalized time. Data points are for the
time range t=0 to t=tmax . . . . . .

Graph of thermal conductivity values of

aluminum 2024- T351 at zero time (k min)
versus temperature . . . . . .

Page

100

101

102

103

104

105

106

107

108

109

116

124

 

Figure

4.2.2

Maximum values of thermal conductivities of
aluminum 2024-T351 (kmax) versus their
respective temperatures . . . . . . .

Time at which the thermal conductivity of
aluminum 2024-T351 attains its maximum value
(tmax) versus temperature . . . . . .

Aluminum 2024—T351 specimen maintained at
375 °F for two hours then heated to and
maintained at 400 °F . . . . . . . .

Temperature history of an aluminum 2024-T351
specimen . . . . . . . . . . . .

Aluminum 2024-T351 plate maintained at 420 °F
on one side and at 300 °F on the other . .

Geometry for thermal analysis of an aluminum
2024-T351 plate. 0 o o c o a n o 0

xi

Page

125

127

128

131

132

133

 

 

 

LIST OF SYMBOLS

integrator constant
specific heat
°F
constant (VET?)
length or thickness of specimen

half of the washer thickness (in the
washer-type thermocouple)

contact conductance
thermal conductivity

maximum value of thermal conductivity
with time, at temperature

value of thermal conductivity as soon

as the specimen arrives at the desired

temperature

normalized thermal conductivity, taken
k - kmin .

to be E___—:_F_—-f°r a given temperature.
max min

temperature

time

effective time

final time (end of the test) normally
taken to be about 60 seconds

time at which thermal conductivity
becomes maximum

 

IDIIO

>K1

a
a (phase diagram)
8

0 (phase diagram)

 

 

normalized time, taken to be t
max

integrated heat flux (§£%

ft
heat flux ( Btuz)

hr ft
thermal resistance = A?
k.
3

thermal diffusivity
solution of copper in aluminum
thermal linear expansion of a material

solution of aluminum in the inter-
metallic compound CuAl2

density

xiii

 

CHAPTER I

INTRODUCTION

Aluminum 2024—T351 belongs to a class of materials
that undergo certain changes in microstructure when
subjected to high temperatures (for aluminum 350-400 °F).
These microstructure changes result in changes in the
thermal properties of the alloy, namely an increase in the
thermal conductivity (k) and specific heat (cp) values.
These time as well as temperature dependent thermal proper-
ties have not previously been measured for aluminum 2024—
T351 or any other material that undergoes similar metal-
lurgical changes. Because of the need of predicting
temperature utilizing known thermal properties of materials,
a relatively simple experimental method was developed by
Beck [1] to measure the thermal properties.

This is a multi-property method capable of measuring
the thermal conductivity (k), specific heat (cp), thermal
diffusivity (a), and contact conductance (h) simultaneously
over a short period of time for a wide range of temper-

atures. It can be used to determine the thermal properties

 

 

 

of a class of materials that change while held at a given

temperature. This implies that the method must be transient.
Some of the materials that can be tested using this method
are aluminum 2024—T351, ferrous alloys, biological
materials, plastics, and other alloys that undergo phase
change when heated. Since aluminum 2024-T351 exhibits the
unique behavior described earlier, and its importance from
a machine design aspect, it was chosen to be the principal
material to be investigated.

The problem can be considered to have the following
objectives,

a. Development of the experimental procedures by
using a reference material while simultaneously
adding needed components to complete the building
of the experimental apparatus;

b. Investigating aluminum 2024-T351 by determining its
transient thermal properties, namely the thermal
conductivity (k) and specific heat (cp); and

c. Development of a mathematical model based on the
thermal conductivity data obtained in part b above,
and utilizing this model to determine the k values
of aluminum 2024-T351 for different temperatures

and times.

 

 

 

1.1 Importance of the problem

The problem of developing and applying the new

method of thermal property measurement is considered

important because of a number of reasons.

1.

That the simplified method is a rapid, transient
method is in itself important. This is because the
method can be used to make measurements that cannot
otherwise be made during a single experiment. It
is also because rapid measurements result in reduced
cost.

It is a multi—property determination method which
can be used to measure k, cp, a, and h simultane—
ously.

By knowing the thermal properties of a material,
temperatures and heat fluxes can be predicted in
situations previously not possible. An example is
predicting temperatures of aluminum 2024—T351 after
determining its transient thermal conductivity
behavior.

New data of k(t,T) and cp(t,T) of aluminum 2024-
T351 at different times and temperatures is given
for each individual specimen as well as grouped for
each selected temperature (example the individual
curves and the composite curves respectively).

This data is new and has not been reported previ-
ously, because rapid transient methods of measuring

thermal properties were not available.

5. The effectiveness of the experimental procedure of
holding the Specimen at a fixed temperature while
finding its thermal properties is demonstrated.

6. By providing hitherto non-existent data of the time
dependence of k and cp of aluminum 2024-T351, solid
state physicists and material scientists may be

aided in understanding such phenomena as stored

energy, lattice dislocation, etc.

1.2 Literature review

This section gives a review of various transient
methods of thermal property measurement. The difference
between these methods and that used herein are particularly
noted. Let us begin with a general one-dimensional heat

conduction equation which can be written as

.131; .3? (er 3.;.) = pcp 1'3;— (1.2.1)
If
n = 0 r + x rectangular coordinates
n = l r + r cylindrical coordinates
n = 2 r + r spherical coordinates
where

T = temperature
t = time

x = position in rectangular coordinates

r = position in cylindrical or spherical coordinates
p = density of material.
The simplest, and usually the preferred, geometry is the
flat plate (n = 0).
The following are examples of some transient methods
of thermal property measurement.
(1) The line-source method
The line-source method is one of the oldest methods
of measuring thermal conductivity. It is based on the
equation for temperature rise with time of an infinitely
long line heat source which receives energy at a constant
rate. The body of material surrounding the line is

thermally infinite and the initial temperature is uniform.

Suppose energy is liberated at the rate of pcp¢(t)

per unit time per unit length of a line parallel to the z-

. . . , . Btu
ax1s and through the p01nt (x ,y ) where (¢(t) is q£(ff_h?))'
If heating starts at t = 0 when the solid is at zero temper-

ature, then the temperature at time t is, see [2]

l

_ 2 _ . .
T(t) = m ft¢(t')e 1‘ /4a(t t ) Ett
0

(1.2.2)

where r2 = (x-x')2 + (y-y')2-

If ¢(t) is set equal to a constant which is desig—

nated q£(Btu/ft-hr), (1.2.2) becomes

qt w e-udu

T(t) = Kid :2 u
471?

 

2
where u = IETP=EIT' a = thermal diffusivity; it can also be

written as

qt . r2
T(t) — - 4E3 E1 - 43? (1.2.3)
where -Ei (-x) E fx _5_ du is the exponential integral.

For small values of x, Ei(-x) can be approximated by
Ei(—x) = Y + lnx-x + (l/4)x2 + 0(x3), where Y is Euler's

constant, 0.5772156. Thus for large values of time t,

.<

q

qt 4st 1
T(t) =m-ln:2—-'m (1.2.4)

or

T(t) = qla [lnt + 1n :4; - y].

Taylor and Underwood, used this method to determine
thermal conductivity values for plastics; for their work
and the values they obtained see [3].

An extension of the line-source method is the
thermal conductivity probe. It is used to measure k for
granular materials, soils, and rocks. The basic equation

is

qfi (tZ-tc)
AT=m1nW (1.2.5)
1 c

where tC (time constant) is called the calibration factor,

and is determined experimentally for each probe and it is

valid only for use with a specific type of material.

Wechsler and Kritz [4], used this method to determine k of

several materials by utilizing different designs of probes.

The new method is different from the line-source

method and its extensions because

1.

The line-source method is quasi-steady state while
the new method is transient;

The geometries are different; and

The line-source method is not a multi-property

method.

2. The modified Angstrom method of determining thermal

diffusivity.

For the modified Angstrom method of determining

thermal diffusivity the sample is in the form of a semi-

infinite rod, although the method could be applied to other

geometrical shapes. The one-dimensional partial differ-

ential equation that gives the temperature T above the

ambient temperature at any position x and time t is

3 T _ 3T
a(——7) — 3? + UT (1.2.6)

where a is the thermal diffusivity and u is the coefficient
of surface heat loss which takes into account any heat loss
by radiation, conduction, and convection. At high temper-
atures the radiation loss will predominate and u will vary
as the cube of the absolute ambient temperature. Since T
is small, only a few degrees at most, the heat loss by
radiation, as well as the heat loss by conduction and con-
vection to any gas surrounding the rod, may be assumed to
vary linearly with temperature difference (Newton's law of
cooling). If radiation is predominant, use of the correct
temperature dependence (Stefan-Boltzmann law), in which the
radiation varies as the fourth power of the respective
temperatures, would not improve the accuracy of the method
and would enormously increase the labor of computation.

If a heat source whose temperature varies sinu-
soidally with time is located at one end of a semi—infinite

radiating rod, the boundary conditions for T(x,t) will be
T(0,t) = A0 + A1 cos (wt + 5) (1.2.7)
and
T(w,t) = 0 (1.2.8)

Then the solution of (1.2.6-8) is given by Sidles

and Danielson [5] to be

T(x,t) = A0 exp(-aox) + A1 exp(—alx) cos(wt-blx+e)

(1.2.9)

where (A0, A1, and e are arbitrary constants), and

1/2
a0 = (5) (1.2.10)
a1 = {(7%)[(u2+w2)1/2 + u]}1/2 (1.2.11)
bl = {(3%)[(u2+w2)1/2 -u]}1/2 (1.2.12)

The temperature oscillations produced at x = 0 will be

propagated along the rod with a velocity
— 2.
v — b (1.2.13)
and will have an amplitude decrement

q = exp (-a1x1) = exp a L (1 2 14)
exp (-alx2) l ‘ °

 

The quantity L = xZ-x1 is the distance between two thermo-

couples, one placed in the rod at x = x and the other

1
placed in the rod at x = x2.

From equations (1.2.13) and (1.2.14),

albl = £11231 (1.2.15)

and, from equations (1.2.11) and (1.2.12),

1;) =

a1 1 (1.2.16)

3F;

Eliminating alb1 from equations (1.2.15) and (1.2.16) we
get the basic equation for the experimental determination

of thermal diffusivity

10

_ Lv
“”713? (1.2.17)

If the density (0) and Specific heat (cp) of the material
are known, then the thermal conductivity (k) of the

material can be determined from

k = upcp (1.2.18)

For the experimental determination of (a) and the procedure
followed see Sidles and Danielson [5]. The new method is
different from the modified Angstrom method since
1. The modified Angstrom method is quasi-steady state
while the new method is transient;

2. The geometries are different (semi-infinite and

finite); and

3. The modified Angstrom method yields directly only
the thermal diffusivity, while the new method is a

multi-property method.

3. The Flash Method

The flash method was first developed and reported
by Parker, Jenkins, Butler, and Abbott [6]. This method
can be used to measure the thermal diffusivity, heat
capacity, and thermal conductivity of a very small speci-
rnen. The energy of a high-intensity, short-duration light
Pulse is absorbed in the front surface of a specimen
(abated with a few millimeters camphor black. The resulting

temperature history of the insulated rear surface is

11

measured by a thermocouple and recorded with an oscillo-

scope and camera.

Theory of the method:
If the initial temperature distribution in a
thermally insulated plate of uniform thickness (L) is

T(x,0), the temperature distribution at any later time (t)

is [6]
w 2 2 2 L , on' ,
T(x,t) = l-fLT(x,0)dx + 3': e-n H at/L cos on f T(x ,0) COS-ETTGX
L L L o
0 n=1
(1.2.19)

when a pulse of radiant energy Q (Btu) is instantaneously
A ft:

and uniformly absorbed in the small depth 9 at the front

surface x = 0 of a thermally insulated solid of uniform

thickness L, the temperature distribution at that instant

is given by

 

T(x,0) = c g + Ti for 0<x<g (1.2.20)
and
T(x,0) = Ti for g<x<L.

With this initial condition, equation (1.2.19) can be

written as

12

[IO

 

w . 2 2 2
_ A on Sin(nfl L) -n H at/L
T(x,t) — De L [1+2 2 cos ‘f"‘THfi§7§§“e 1 + Ti
p n-l
(1.2.21)
where p is the density in l2%~and c is the heat capacity
ft

. Btu . . .
in IEE:F’ In this application only a few terms are needed.

Since 9 is a very small number for opaque materials, it

follows that sin 2%3 z 2%3. At the rear surface where x =

L, the temperature history can be expressed by

2 2

>10

H

 

w 2
[1+2 1 (_1)n e-n at/L ]

n=1

T(L,t) + Ti (1.2.22)

=cL
p 9

Two dimensionless parameters, V and w, can be defined

 

 

V(L,t) = T _ T (1.2.23)
M i
2
w = H at (1.2.24)
L2

TM represents the maximum temperature at the rear surface,

 

The combination of (1.2.22), (1.2.23), and (1.2.24) yields

m 2
v = 1 + 2 Z (-1)n e"n “) (1.2.25)
n=1

V is plotted versus w in [6].

13

Many ways of determining a have been suggested.
One of these is the "one-half time" method of analysis.

When V is equal to 0.5, w is equal to 1.38, and so

a = (1.38L2/H2t (1.2.26)

1/2)I

where tl/Z is the time required for the back surface to
reach one half of the maximum temperature rise.

In another method of analysis the time axis inter-
cept of the extrapolated straight line portion of the curve
(V versus w in [6]) is approximately w = 0.48, which yields

the relationship,
2 2
a = (0.48L /H tx)’ (1.2.27)

where tx is the time axis intercept of the temperature

versus time curve. The values of diffusivity determined by
equation (1.2.27) are considerably less precise than those
determined by equation (1.2.26). The method given by
(1.2.27) requires the finding, by eye, of the straight
portion of a curve and the extrapolation of this line back
to the baseline. This is a subjective method which is
rather difficult and one in which a small error in the
slope determination results in a relatively large error in
the value of the diffusivity. Its advantage is that it is
independent of the final height of the curve and does not
require that the surface distribution of energy be as

uniform as does the half~time method. It is not necessary

14

to know the amount of energy absorbed in the front surface
in order to determine a. However, this quantity must be
determined if measurements of specific heat or thermal
conductivity are required. The product of the density and
the heat capacity of the material is given by (assuming no

heat losses)

WK)

co = _ (1.2.28)
p L(T Ti)

3

and then the thermal conductivity is given by

k = a c 1.2.29
0 p ( )

The foregoing treatment has neglected the variation
of thermal diffusivity with temperature. The method
produces an effective value of diffusivity for the sample.
The effective value of the corresponding temperature is
arbitrarily picked to be the time average of the mean of
the front and back surface temperatures up to the time that
the rear surface reaches one—half of its maximum value.

The dimensionless parameter V(L,t) at the rear
surface is given by (1.2.25). The dimensionless parameter
V(0,t) at the front surface obtained in a similar manner is

given by

m 2
V(0,t) = 1 + 2 X e““ w), (1.2.30)
n=1

15

which is observed to be infinite at w = O. This is not a

realistic physical value. The mean value of V(L,t) and

 

V(0,t) is
w 2
V‘o't) ;_V(L1t) = 1 + 2 2 e( 4“ “) (1.2.31)
n=1
and the effective value of V is
2 “1/2 ” (-4n2w)
ve — 1 + 6“" f 2 e dw (1.2.32)
1/2 0 n=1
2
where w r H at — 1.38,
1/2
L
v = 1 + 2 [ E .1 (1 _ e-4n21.38)] = 1 6
e 4(I.38) _ 2 '
n—l n
(1.2.33)
Therefore, Te - Ti = Ve(TM - Ti) = 1.6(TM - Ti) (1.2.34)

In this method several simplifying assumptions are

l. One-dimensional heat flow is assumed.

2. The energy pulse is assumed to be absorbed in a
very thin layer of the specimen surface in a time
very short compared with the propagation of the
heat wave through the material.

3. a and k are assumed to be independent of the temper-
ature.

4. There are assumed to be no heat losses by conduction

or radiation from the faces.

16

The new method is different from the flash
diffusivity method for the following reasons,

1. In the new method the thermal properties are
evaluated directly. That is, no curve fitting or
visual inspection of the temperature history is
needed; in the proposed method only 3 DVM measure-
ments are used. In one way of analysis of the
flash method, the thermal properties are evaluated
at tl/2‘ t1/2 is the time required for the back
surface to reach half of the maximum temperature
rise. This time requires visual curves fitting.

2. In the flash method, the surface temperature rise
is very high, so that sometimes it causes vapori-
zation and energy losses. This is not so in the
new method.

3. In the new method, the heat losses by conduction
and radiation are smaller than those of the flash
method since there are no "free" faces.

In short, the flash method determines the thermal
diffusivity by the shape of the temperature versus time
curve at the rear surface, the heat capacity by the maximum
temperature indicated by the thermocouple, and the thermal
conductivity by the product of the heat capacity, thermal
diffusivity, and the density.

The flash method was extended and refined by vari—

ous investigators in the field, and applied to a wide range

17

of materials over a wide range of temperatures. It is
reported to be used in over 80% of the current transient

property measurement according to TPRC.

4. Non—linear estimation method (program PROPERTY).

The non-linear estimation method is based on
theoretical findings by Beck [7] for the estimation of
thermal properties. It is developed into a computer
program called "PROPERTY." This program also can predict
the temperature at various depths of the specimen. A
finite difference method is employed for solving the
transient heat conduction equation.

The properties are found by making the calculated
temperatures match the measured temperatures in a least-
squares sense. The boundary-conditions available in this
program are,

1. Temperature boundary condition: in order to
determine the thermal diffusivity, only measured
temperature boundary conditions are necessary. The
program can use temperature histories at two
boundaries as boundary conditions in the model to
calculate the thermal diffusivity.

2. Heat flux boundary condition: to determine
simultaneously the values of thermal conductivity
(k), and specific heat (cp), a nonzero heat flux

history at one boundary must be known. In one mode

18

the program uses heat flux and insulation boundary

conditions in the model to calculate k and c .

Other input variables include the number of regions,
number of nodes in each region, the length of each region,
and times for the given transient data of temperatures or
heat fluxes.

The program output includes the times, the calcu-
lated temperatures "CALTEMP," measured temperatures
"ETEMP," and their differences called a residual "DTETC."
Printed output also includes the sum of square of residuals
"RMS," and sensitivity coefficients symbolized by "BDB."

The values of root-mean-square are a direct measure
of agreement between the model and the experiment. For
perfect agreement between the calculated and the measured
temperature, the value of "RMS" approaches zero. However,
"RMS" will never reach zero since there are always sources
of errors. The sources of errors may be ascribed to
experimental measurement errors, an imperfect model, and
finite-difference calculations errors since the partial
differential equation model is approximated using differ-
ence equations.

The differences between program PROPERTY and the
new method are that:

1. The new method is simpler analytically.
2. One can obtain the values of k, cp' and a immedi-

ately after each test when using the new method.

 

 

19

This is not possible at present using program
PROPERTY .

3. The new method employs a small electronic inte-
grator to obtain the thermal properties while
program PROPERTY is designed for digital computer

application.

5. The linear finite rod method

Klein, Shanks, and Danielson [8] used the linear
finite rod method to measure the thermal diffusivity (a).
In this method, the sample is in the form of a rod with a
coaxial radiation guard of the same material and with the
heater attached to the sample and guard at one end. Three
thermocouples are attached along the length of the rod.

To obtain a data point at a given ambient temper-

ature, the heater is turned on and the outputs of the
thermocouples are recorded as a function of time. The
curve from the first and third thermocouples and an initial
estimate of the diffusivity are then used to calculate a
curve which is compared with the experimental curve from
the second thermocouple. The estimated diffusivity is
varied until the difference between the experimental and
calculated curves is minimized and the best value for the
thermal diffusivity is obtained. In essence, this is what
program "PROPERTY" does also. An extension of this method

is the radial sample method. In this method, the sample

consists of a stack of disks with an axial heater. Three

20

thermocouples are embedded along the radius of the center
disk at three different radii (inner, middle, and outer).

The data are recorded and analyzed in a manner
similar to that used in the finite rod method. In other
words, the temperature data from the inner and outer radii
are used as empirical boundary conditions, and the temper-
ature data from the middle radius are used to obtain (a).

Carter, Maycock, Klein, and Danielson [9] used this
method to evaluate (a) of Armco iron in the range 26° to
895 °C, utilizing a computer program to make the calcu-
lations by the method of finite differences. The new method
is different from these two methods because of geometry,
experimental technique, and calculations or analysis. It
is similar to the difference between the new method and

PROPERTY .

6. Radial heat flow method

McElroy and Moore [10] classified the radial heat
flow method into five classes:
Class I: Cylindrical methods with a central source (or

sink) of energy in a cylinder that is assumed to be in-
finitely long" and does not employ end guards.
Class II: Cylindrical methods with a central energy source
(or sink) in a cylinder generally composed of stacked disks
and employing end guards to minimize axial flow.

Class III: Spherical and ellipsoidal methods in which the

energy source is completely enclosed by the specimen.

21

Class IV: Comparative methods in which concentric cylinders
of materials with known and unknown (k) surround a central
energy source (this method is usually not employed on
solids).
Class V: Self-heating radial methods involving joule
heating of a cylindrical specimen and measurement of the
attendant temperature variation along the radius of the
cylinder.

Other methods for determining (k) alone have been
investigated by
Elygg [11], who used steady—state methods in which the
sample is heated directly by passage of an electric current.
Also Powell [12], developed and improved the so called
"Thermal Comparator methods" to determine k of materials.
Laubitz [13], reviewed two experimental systems for
measuring (k), the guarded linear-heat flow method and the
"generalized Forbes' Bar" method.

Null and Lozier [l4], worked on the measurement of the

 

thermal diffusivity (a), using the phase shift method.
Beck, Mitchel, and Pfahl worked on many aspects of thermal
conductivity using transient measurements (see references

15-30).

CHAPTER II

EXPERIMENTAL PROCEDURES

This chapter deals with experimental aspects of the
problem. The experiment is based on the analysis developed
by Beck [1]. The new method necessitated the development
of an experimental apparatus and strategy to determine the
transient thermal properties of different materials. This
chapter is divided into several sections with each one

dealing with an important part of the experiment.

2.1 The new method

 

For detailed analysis of the method, see reference
[1]. In this section, the new method is described as it
applies to the geometry of the problem, namely the flat
plate.

For a flat plate with temperature variable proper-
ties which describes transient heat conduction, equation

(1.2.1) reduces to

DCP 5?. (2.1.1)

22

 

23

Consider the plate to be heated on one side by
applying a heat flux (g) at x = 0, and to be insulated on

the other which is at x = E, see Figure (2.1.1).

 

Plate

q/A —* Insulated

 

(«e—>1

Figure 2.1.1 Flat plate geometry of the problem.

 

 

 

Then the two boundary conditions can be written as

3T(E,t) = 0
3x
(2.1.2a)
g = -k 3T(0,t)
A 3x
and the initial condition can be given as
The following are known: E, p, %, Ti’ Tf where
E = length (or thickness of specimen) (ft)
p = density of specimen (l2?
ft

24

g = total heat added/area = fm (3) dt (Btu)
A 2
0 ft

Ti = initial temperature (°F)

Tf = final temperature (°F)

The heat flux % and typical temperature histories are shown

in Figure 2.1.2.

 

qlA)

Q/A

 

 

0 1’ time

 

 

 

n’ time

Figure 2.1.2 Heat flux and temperature histories
for a specimen.

 

 

 

 

The objective is to find k and cp using measurements at two

locations, x1 and x2 in the plate.

To proceed first integrate equation (2.1.1) over t

T

f be dT # f(x)
'1'. p

1 (2.1.3)

(I)

3 8T _°° .92 =
é-a-i-(k-a-i) dt-f DC at dt f

25

Notice that the boundary conditions result in the right
hand side of (2.1.3) not being a function of x. Now

integrate equation (2.1.3) over x

00

f [f .33., (k g)dx]dt = I°°Ud (k 3%))... = I°°tk 33-31(1); =
O O O

T
w 3T _ f
f k 5; dt — x IT. pcpdt + c1 (2.1.4)
o 1
Q is defined by the equation,
2. °° s
A — g (A) dt

Use the boundary condition at x = 0 given in Equation

(2.1.2a) in the above equation to obtain

- - f°° [k ——L—3T(gxt)] dt
0

S’IIO
I

Introducing this equation and the boundary condition at

x = E given by Equation (2.1.2a) in Equation (2.1.4) yields

at x = 0
-2:
A c1 (2.1.5)
at x = E
Tf
0 = E ITi pcpdT + c1 (2.1.6)

Substitute Equation (2.1.5) into Equation (2.1.6) to get

T
f _ Q
IT pcpdT — AB
1

(2.1.7)

26

Use Equations (2.1.5) and (2.1.7) in Equation (2.1.4) to

obtain

I” k §§ dt = §.L% - 1] (2.1.8)

Now assume that (k) and (0) are constant but let cp = f(T),
and also define
O(x) 2 f00 T(x,t) dt
0

Then Equation (2.1.8) becomes

d9-9. 22-
k diz— A [E 1] (2.1.9)

Integrating Equation (2.1.9) over x yields

2 2
x1 ‘ x2 _

k [0(x1) - 0(x2)] = [-§§-— x1 + x2]

{VIC

solving for the thermal conductivity,

QE x1 2 x2 2
2A [(E— -l) " (E— -1)]

 

 

k:
0(xl) - 0(x5)
or
QE X1 2 X2 2
-— [(—— -1) - (- -1) 1
k = A m E E (2.1.10)

2 f [T(xl,t) - T(x2,t)] dt
0

Equation (2.1.10) is the thermal conductivity equation. In
Equation (2.1.10), xl can be taken as x = 0 and x2 to be

x = E.
Q

A can be determined from a calorimeter of known

thermal properties. OFHC copper can serve as an example

27

of such material. Then we can derive for a flat plate

calorimeter heated on one side and insulated on the other

Q _ _
A’— (pep)cal Ecal (Ti Tf)ca1 (2'1°ll)
or
QESIE
A = (pep)cal Ecal Esp (Ti — Tf)cal

where cal means calorimeter and sp means specimen.
Introducing (2.1.11) into (2.1.10) gives the

expression

 

k = (pép)cal Ecal Esp (Ti _ Tf)ca1

.. (2.1.12)
26 [T(xl,t) - T(x2,t)] dt

 

 

 

Next assume that p in Equation (2.1.7) to be constant (but

not k), then

01'

(2.1.13a)

S
O
'U
Q.)
*3
ll

Therefore Equation (2.1.13a) can be written as

 

- Q
c = _ (2.1.1313)
p A_‘('—_')'Ep Tf T.

J.

 

1

28

Equation (2.1.13b) is the specific heat equation.
The contact conductance h can be obtained from the

following relation
q(t) = h [Tl(t) - T2(t)]

See Figure 2.1.3.

 

 

 

 

 

11:”

 

Figure 2.1.3 Two plates in imperfect
contact.

 

 

 

Integrate the above equation over t to get

Q = g h [T1(t) - T2(t)] dt

If h is constant, then

 

h = ~33 (2.1.14)

5 [Tl(t) - T2(t)] dt

 

 

 

Equation (2.1.14) is the equation for the contact con-

ductance.

2.2 Experimental set-up
The data acquisition system of the Michigan State

University Thermal Properties Measurement Facility consists

29

of a hydraulic system and main testing unit, thermocouple
sensors, computer signal conditioner (CSC), IBM 1800
computer, an electronic integrator, and temperature
controllers. The computer signal conditioner contains a
DC amplifier for each thermocouple (when the integrator
system is used only one amplifier is needed).

The test specimen and the calorimeter are housed
in the main testing unit (or testing assembly). For details
of the testing assembly see Figure 2.2.1. The specimen and
the calorimeter are assembled in exactly the same manner in
their respective cans in the testing assembly. The testing
assembly shown in Figure 2.2.1 consist of two parts
separated by a heater. One part (bottom part) houses the
specimen which sits on three screws 120° apart from each
other. Each screw has a collar that sits on a Transite
block; also each screw has a hole in it that extends from
the tip that carries the specimen to the collar. The
purpose of the holes in the screws is to minimize the heat
losses from the specimen by conduction. Surrounding the
specimen and sitting directly on the Transite block is an
OFHC guard-heater; its purpose is to maintain a constant
temperature zone around the specimen to minimize heat
losses by convection and radiation from the sides and
bottom surface of the specimen. The space between the
outer surface of the guard-heater and the stainless steel
can is filled with Fiberglass insulation to minimize heat

losses from the guard-heater. This whole assembly with the

30

 

 

C

 

 

\\\\\\\

  

  

   

o|o\vimetev

 

 

l._____

7 1/2

 

 

V

 

Constant temperature zone 7/3»
I
Groove for 3/16" diamete u-

~— 3"———I-

heating coi}.

OFhC copper guard—heaten\
I

1/8' thick Transite

lid-
.018‘ thick sta n-
less steel can.

 

 

 

 

.0,-

1“

    

 

Spednum

\\\\\\\ ‘

 

\ \\ \

 

 

 

I: 1.

       

 

 

 

 

 

 

 

 

  

.035 diameter hole

/

Collar (l/h' th{éé3
4" diameter X 1 3/16" thic
Transite block.

Hydraulic cylinde .

Figure 2.2.1

500

 

with Fiber-
glass insulat-
1 ion.

1 1/
2-

lzj/u-

 

 

 

 

\\ Flathead machine
screw fastens
Transite to heater.

.138“ diameter mounting

screw.

#6-32 nut.

diameter X 1' thick steel

base.

3/8' hex cap screw fastens

Transite to steel base.

Schematic diagram of the testing assembly.

 

31

Transite sits on and is fastened to a steel base which in
turn is attached to the piston of the hydraulic cylinder.
The top part houses the calorimeter, and it is assembled
like the bottom part except it does not sit on a piston;
instead it is fastened to the loading frame. The heater
between the specimen and calorimeter cans is also fixed to
the loading frame and its purpose is to heat the mating
surfaces of both the calorimeter and the specimen; see
Figure 2.2.2.

The following points must be considered during the

assembly process of the specimen and the calorimeter.

1. The specimen and calorimeter must have a flat
surface and each must be parallel to the steel
base in its own can.

2. The specimen and calorimeter must have the same
centerline (no offset).

Figure 2.2.2 shows the testing assembly. The
bottom can houses the specimen; the top can houses the
calorimeter; and the two heaters are between.

Figure 2.2.3 shows the cabinet that houses the
temperature controllers and the adjacent hydraulic system.

The temperature controllers are made by Leeds and Northrup

and are attached (from left to right in Figure 2.2.3) to
the
l. specimen surface heater,

2. calorimeter guard-heater,

 

 

 

 

32

 

Photograph of the testing assembly.

Figure 2.2.2

33

.EoumMm owasmupms
enmeshed on» was muoaaonucoo ousumuomfimu ocu mcamsoc pocflbso d

9

no.-.

m.m.~

ousmflm

 

34

3. specimen guard-heater, and
4. calorimeter surface heater.
The temperature controller attached to the specimen surface
heater is the proportional type while the remainder are the
"on-off" type.
Figure 2.2.3 also shows the specimen and the calori-
meter in intimate contact (face to face) during the test
and how the piston brings the specimen can up for this
purpose. It also shows the position of the two surface
heaters during the test.
Figure 2.2.4 concentrates on the operational panel.
The switches on the panel operate the testing assembly,
hydraulic system and surface heaters. To the left of the
operational panel and sitting on the table is the CSC.
For further details on the data acquisition system
of the Michigan State University Thermal Properties Measure—

ment Facility see [31] and [32].

2.3 Specimen selection and thermocqule installation

Two materials have been investigated, one a
reference material and the other the primary material to
be investigated.

The purpose of using a reference material is to
improve and quantify the accuracy of the experimental
apparatus by using a material which has well-known and

established thermal properties. Armco iron is such a

35

msu. 05m Um: HGGO nu "Ur—00 _wfi.m um HmflcnmEOU _0GMQo _MGO upmhmmo 03;.
A v . . . s M Qfiwmwm OGHHMUH
.

v.m.m onsmflm

 

36

material since many investigators have measured its thermal
properties; see [33], and [34].

The primary material investigated is aluminum 2024-
T351. Its transient thermal properties are determined in
this study for different temperatures. Also a mathematical
model is developed to predict the thermal conductivity
values for different times and temperatures (see Chapters
III and IV).

The thickness of the specimen should be chosen to
minimize heat losses from the sides and to maintain a
reasonable duration of the test. It is found that a
reasonable duration is on the order of 30-60 seconds. An
approximate optimum thickness E of the specimen is found by

utilizing the Fourier modulus.

E;- 3 2 (2.3.1)
B

where t = time, a = thermal diffusivity, and E = thickness
of the specimen. (2.3.1) comes from the solution of the
heat conduction equation for a flat plate (0<x<E), one end
maintained at constant temperature and the other end
insulated. The plate has an initial uniform temperature.
These conditions approximate the experimental conditions of
the tests reported herein. By the time indicated by the
Fourier modulus value given by (2.3.1), the temperature in
the idealized problem just described would be nearly

uniform.

37

The experiment must be short in duration because the
integrator is not accurate after four to five minutes and
with longer duration experiments the heat losses become
relatively greater. Yet, the duration of the experiment
must not be too short because one wishes to visually monitor

the AT (i.e., - T ) during the experiment. This

T5P _ SP _

x~0 x-E

requires that the duration be about 30—60 seconds minimum.
Also, one wants to avoid having the order of the experiment
duration be so short as to be on the order of response

time of the various components of the system.

As mentioned earlier the specimen's surface must be
flat and smooth (as for the calorimeter surface) to insure
the best face to face contact with each other. This is
done by first grinding and then lapping the surface.

To measure the temperature, type "J" thermocouples
were used. They are iron—Constantan, number 30 gage wire
which is about 0.010 inches in diameter.

Three thermocouples were used on the heated surface
of the specimen, 120° apart from each other. They are
embedded in 3 pairs of slots with each slot being 0.010
inch wide, 0.010 inch deep and 0.300 inch long. The
distance between the centerline of the slots in each pair
is 0.100 inch. See Figure 2.3.1. The centerline slot of
0.010 inch wide and 0.010 inch deep on the specimen's sur-
face was used for another thermocouple, but it was later

decided that it was not necessary, and left on all speci-

mens' surface for the sake of consistency.

 

38

 

 

Figure 2.3.1 Thermocouple locations at the
heated surface of the specimen.

 

 

 

A thermocouple is formed by placing an iron wire in
one slot and an Constantan wire is placed in the adjacent
one. After placing a wire in a slot, the side of the slot
is very carefully peened over to pinch the wire and hold it
in place. A junction between the two dissimilar metals of
the thermocouple wire is not formed before the wire is
attached to the Specimen; the metal in the specimen becomes
part of the actual measuring junction. This method elimi-
nates the need for adhesives which add mass and can insulate
the thermocouple from the specimen. An attempt was made to
insulate the Slots electrically except at the tip in order
to let the thermocouples measure at 0.3 inches from the
edge of the specimen surface rather than measure the last
point it sees which is the edge. This was done to gain

greater accuracy in the temperature measurement. This

39

attempt failed because in peening over the edges of the
slot the electrical insulation broke down.

A good deal of time was devoted to make the surface
thermocouples read uniformly. It was found that two things
help make this possible. First, the surface of the specimen
must be quite flat. Next it is necessary to provide a
uniform layer of Silicone grease on the surface which is
achieved by utilizing a Specially constructed comb to
provide a thickness of about 0.015 inch.

To determine the temperature difference across the
specimen the temperature at another location beside the
heated surface must be measured. This temperature at x2
was chosen to be E, the insulated surface, for convenience
and for greatest accuracy. By measuring the other temper—
ature as far away as possible from the heated surface we
can achieve the minimum noise to signal ratio. If the
IBM 1800—system is used for data collection from each test,
then the thermocouples at x2 = E (back surface of the
specimen) can be installed in exactly the same way they are
installed on the heated surface. However, most of our data
was collected by using the electronic integrator system,
which prevented using the above mentioned method of thermo-
couple installation on the back surface of the specimen
(or referred to as the insulated surface) for electrically
conducting materials. This is because the temperature

difference in

40

t

fm [T(x1,t) - T(x2,t)] dt = f f
O 0

required by the integrator was taken by appropriately wiring
the thermocouples assuming that the thermocouples were
electrically insulated. This posed a considerable problem
of selecting the best thermally conducting but electrically
insulating adhesive with which to coat the thermocouple and
which could also withstand the high temperatures. Many
experiments and a great deal of time was devoted to this
problem; electrically insulating adhesives vary from one
another in their thermal response and the objective was to
find the best thermally conducting material to minimize
errors in the evaluation of the integral.

Among the many experiments, one in particular
deserves some description since through that experiment a
suitable adhesive was found. In that experiment a Specimen
of Armco iron (3 inches in diameter and 1 inch thick) was
used. Three thermocouples were installed on the heated
surface (x1 = 0) in the manner described above. For the
insulated surface (x2 = E) five holes of 0.035 inch diameter
and l/16 inch deep were drilled at different locations, in
a circular fashion. The center of each hole is 1.2 inch
from the center of the insulated surface. At one location
one pair of slots were machined in the same manner and with
similar dimensions as those on the heated surface, see

Figure 2.3.2.

 

41

 

  

.010” 3Hdiam.

Figure 2.3.2 Thermocouple locations at the insu-
lated surface (x = E) for an experi-

 

 

 

 

Hole #1 was filled by regular shellac, which is an
electrically insulating material. A thermocouple bead was

tightly inserted in the hole.

In hole #2 a Mullite tube (ID 3 .033 inch, OD 3
0.035 inch, and length x 1/16 inch) was inserted with the
thermocouple bead inserted inside the tube so that it
touches the sides of the tube but not the bottom of the
hole.

Hole #3 is exactly like hole #2 except the tube was
filled with silicone grease before the thermocouple bead
was inserted.

Hole #4 was coated using Sauereisen cement number
one. The thermocouple bead was also coated with the same
material and then it was inserted tightly in the hole.

Hole #5 has exactly the same features as hole #4
except that the insulating material used is Astroceram

type A.

 

.llllll'l.)

42

In the pair of Slots (#6) the two thermocouple wires
were installed in a similar manner to those installed on
the heated surface.

Several tests were run on this Specimen using the
IBM 1800-system. The temperature reSponse of all the above
mentioned thermocouples were monitored. Since the thermo-
couple in #6 touches the metal directly, it had the best

thermal response. Next to it was the thermocouple in

 

hole #5 (with the Astroceram insulation). The thermocouples
in the #6 and #5 were very close in their temperature
measurement and much better than the rest of the thermo-
couples. Therefore it was concluded that the Astroceram
type A cement is the best thermal conductor and electrical
insulator. For many Armco iron and aluminum specimens one
thermocouple was used on the insulated surface (x2 = E) in
exactly the same way as that of hole #5 described above.

For the rest of the aluminum specimens a washer
in the form of a hollow disk of 1/2 inch outside diameter,
a little less than 1/4 inch inside diameter, and 1/16 inch
thick was used. Through the side of the washer a hole of
0.035 inch diameter was drilled. This hole was filled with
Astroceram type A and the thermocouple head was inserted
in it, see Figure 2.3.3. This washer was placed on the
insulated surface and fixed firmly to the surface by a small
screw. A small layer of Silicone grease was placed between

the washer and the Specimen to enhance the heat transfer.

43

 

 

.035"

Figure 2.3.3 Diagram of the washer-type
thermocouple.

 

 

 

This washer-type thermocouple introduced a small error in
the determination of (k), and a correction was introduced
as will be seen later. One washer-type thermocouple was
used for several tested specimens. It is important to note
that Astroceram type A can withstand temperatures of up to
2000 °F; this is in addition to the desirable qualities
mentioned earlier.

As mentioned in section (2.1) there are two
boundary conditions used in the integration method, heat
flux boundary condition at x = 0 and insulation boundary
condition at x = E. To measure the heat flux at x = 0, a
standard material with known thermal properties was used.
OFHC copper was chosen to be the standard or calorimeter
because its properties are very well established and it is
often used as a standard material. As mentioned earlier,

the calorimeter was a disk of 3 inches diameter and

44

3/4 inch thick. It was assembled in exactly the same
manner as the specimen and mounted in the loading frame
above the specimen; see Figures 2.2.1 and 2.2.2. To
measure the temperature of the calorimeter, only one
thermocouple was installed in a pair of slots located 3/8
inch from the heated surface and 180° apart; see Figure
2.3.4. The dimensions of the slots are similar to those on
the heated surface of the Specimen, namely 0.300 inch long,

0.010 inch wide, and 0.010 inch deep. The thermocouple was

 

installed in exactly the same way as it was installed on
the heated surface of the specimen with the Constantan wire
in one Slot and the iron wire in the other. Since the
calorimeter's thermal properties are known, the heat flux

can be calculated from

9: -
A [pcpE(Ti Tchalorimeter (2°l'11)

 

calorimeter heated surface

.010"

T (114300” /
3/4” =/ J 3/8
1'

 

l

 

 

L, M
F 3 n

Figure 2.3.4 Thermocouple location on the side of the OPEC
copper calorimeter.

 

 

 

45

To collect the data all thermocouples were connected to a
Specially constructed box (which acts as a reference
junction). The panel in the box was connected to a rotating
switch by plain copper wires that carry the signal or
(output) from each thermocouple via the switch to an
amplifier. The switch allowed each thermocouple output to
be read on a separate channel, see Figure 2.3.5. The
amplifier multiplied each signal by a factor of 1000 before
being sent to the digital voltmeter (DVM). This digital
voltmeter, 10 amplifiers and other auxiliaries comprise
what is called the Computer Signal Conditioner (CSC).

The initial and final temperatures of the calori-
meter and specimen were read directly (in volts) using a
DVM. The AT signal (or channel #3 on the rotating switch)
after being amplified 1000 times left the CSC and was fed
to the electronic integrator. There it was integrated over
time for about one minute and the output was read in volts
on another DVM.

The integrated temperature difference can be

related to the integrated voltage difference by
t t

1” AT dt 3 f f AT dt = cI f
0 O O

fCAth
V

using

AT = CV AV

 

 

46

thermocouples panel

calorimeter R :fi

_—”_- §
specimen R th specimen
interface——-———- - _ 5m°° -
Y

4f:)controller
I. If

. g + v + Sblpf

spec1men E - I I I

insulated -n

 

 

 

 

 

 

 

 

 

 

    
  
 

 

 

 

 

surface ——fiv—1_I - fl -

switch numbers stand for: _ _ controller
#1 = calorimeter.
42 = specimen.
#3 = temp. difference between

top surface and bottom con roller
surface of specimen.

I“ = controller for calorimeter's
guard-heater.

#5 controller for specimen's

guard—heater. input

  
  

rotating
switch +

 

 

 

 

 

digital
oltmeter

 

digital
oltmeter

Figure 2.3.5 wiring diagram for the over-all experimental system.

47

If AT = 100 F° resulting from a temperature rise from 80 to
180-80

180 °F then AV = 4.31 - 1.36 = 2.96 volts and CV =
The integrator constant CI was determined experimentally
in the following manner. A constant voltage AVC of l mv
was applied for 60 seconds to the integrator (see Figure
2.3.6) and the output was read on a digital voltmeter as

some number Kl' i.e.,

t
f _ 60 _ _
CI é AVC dtIintegrator - CI(1 mv X 3600 hour) — K1 mv hr

Then solving for CI gives

K1

C = 1
1mvxF6hr

I

This experiment was repeated several times and the constant

was determined to be

C = 0.00252

 

 

 

 

I
AVCI
I mv
() l .’ tints
60 seconds
Figure 2.3.6 One millivolt input over Sixty
seconds.

 

 

 

48

This constant was used in the main equation of thermal

conductivity to obtain the value of k as described below.

 

The specific heat values depend mainly on the
initial and final temperatures of the calorimeter and
Specimen and were read directly in volts from the DVM in

the CSC.

2.4 Runningga test

 

Running a test involves several steps which must be
put in the following order necessary to produce the optimum :
experiment.

1. Turn on all the equipment and let them warm up for
about 20 minutes. This includes the CSC, electronic
integrator, DVM, control panel of the hydraulic
system and all temperature controllers.

2. Zero all measuring devices this includes the CSC,
electronic integrator and DVM.

3. Clean the heated surface of both the specimen and
calorimeter using a clean rag and acetone.

4. Place a film of silicone 0.015 inch thick on the
specimen surface by utilizing a comb specially made
for this purpose.

5. Heat the specimen and maintain it at the desired
temperature by utilizing the surface heater directly
above it and the temperature controller connected
to it. In the meantime turn on the guard-heater

surrounding the specimen and set its temperature

 

49

controller at the same temperature as that of the
specimen.
Heat the calorimeter to constant temperature about
40 to 50 °F above that of the Specimen by utilizing
the surface heater directly below it and its
temperature controller. In the meantime let the
guard-heater surrounding the calorimeter to reach
the same temperature as that of the calorimeter and
hold it there by utilizing the temperature con—
troller. It is important to note that the same
"on-off" switch on the cabinet that houses the
controllers heats both guard-heaters.
After both the specimen and the calorimeter attain
equilibrium at their respective temperatures, do
the following:
a. Write down the initial temperature of the
calorimeter (channel #1 on the rotating switch).
b. Write down the initial temperature of the
Specimen (channel #2 on the rotating switch).
c. Zero the integrator at (-.104 to -.108
depending on whether the integrator drifts
upward or downward), and put the rotating
Switch on channel #3 (the temperature differ-
ence channel).
d. Turn off the guard-heater for both the calori-

meter and the Specimen.

10.

11.

50

e. Turn the switches on the control panel of the
hydraulic System to the following positions
from left to right: "OFF," "AUTO," UP,"

"insulation,' and "OUT."
To start the test, push the button on the control
panel of the hydraulic system. This causes the
bifurcated heater to move out from between the
specimen and calorimeter and the hydraulic cylinder
to bring the specimen upward into intimate contact
with the calorimeter; see Figure 2.2.3. The
duration of the test is about 60 seconds.
When AT decreases to within about 2% of the maximum
AT (by visually monitoring it on the screen of the
CSC), the test should be ended. Then put the
integrator back on "HOLD" and
a. Record the integrator output from the small
digital voltmeter,
b. Record the final temperature of the calori-
meter, and
c. Record the final temperature of the specimen.
Put the switch on the control panel of the hydraulic
system in the "DOWN" position; this separates the
specimen from the calorimeter by bringing the
hydraulic cylinder down to its normal position; see
Figure 2.2.4.
Turn on the guard-heaters for both specimen and

calorimeter.

51

12. Clean the surfaces (interface) of both specimen and
calorimeter by using a clean rag and acetone.

13. Repeat the tests by starting from step (4). Also
zero the amplifier in the CSC and the integrator
prior to each test.

To evaluate k and cp, substitute in Equations

(2.1.12) and (2.1.13b) which reduce to

[V(Ti) - V(TfH

 

 

k = (constant)1 x t calorimgter
0.00252 f f [V(T(x1,t))-V(T(x2,t))l dt
0
(2.4.1)
_ [V(T-) - V(T )] .
c = (constant)2 x [V(Tl) _ V(Tfjlcalorimgggg (2.4.2)
p f i specimen

Note that the constant Cv (section 2.3) cancels in both
these equations. Hence no correction is needed to allow
for the changing temperature-voltage relation as the
temperature is increased. One must, however, take into

consideration (pep) for different temperatures

calorimeter
and the thermal linear expansion for both specimen and
calorimeter for the different temperatures. These and

other corrections are discussed in Chapter III.

2.5 The reference material and typical results
In order to determine the accuracy of the system
and develop the testing procedure, a reference material,

Armco iron was first chosen to be the specimen.

52

The type used is called Armco magnetic ingot iron
for D-C applications. A great number of tests were con—
ducted at different temperatures using water and silicone
films of different thicknesses at the interface.

For room temperature tests, water as well as
silicone were used at the interface. The films were placed
on the specimen's heated surface just before the test.
Silicone films were also placed on the surface of the
calorimeter instead of that of the Specimen. It was found
that the location of the film did not affect the results.
Several drops of water were used for room temperature tests
with the number of drops varying from 3 to 10 drops. It
was found that the optimum number of drops is between 5 to
6 for best results of k and cp. For the silicone, film
thicknesses from 0.010 inch to 0.030 inch were tried and
best results for k and cp were found when the 0.015 inch
film thickness was used. Water cannot be used on the
interface at the elevated temperatures because it evaporates
immediately (at temperatures of 300 °F and above). Hence a
0.015 inch film of silicone was used for all tests. Tables
2.5.1 and 2.5.2 give the k and cp values of Armco iron
(specimen #1) at room temperature. For the values reported
in Table 2.5.1 Silicone film of 0.015 inch was used on the
specimen's surface, while for those reported in Table 2.5.2
three drops of distilled water were used. Specimen #1 was
prepared from an Armco iron bar that was purchased some ten

years ago.

L_ A

53

k and cp values for room temperature tests of Armco
iron (Specimen #3) are reported in Tables 2.5.3 through
2.5.5. Specimen #3 was prepared from a new bar of "Armco
magnetic ingot iron for D-C applications." For the values
reported in Table 2.5.3, three drops of distilled water were
used on the specimen surface, while for those reported in
Table 2.5.4 six drops of distilled water were used. For the
values reported in Table 2.5.5, 0.015 inch Silicone film was
used on the specimen's surface. For the k and cp values
reported in Tables 2.5.3-5 the guard heaters and controllers
were used.

Tables 2.5.6 through 2.5.10 report the average,
variance and standard deviation of the k and cp values
reported in Tables 2.5.1 through 2.5.5 respectively.

For the old Armco iron Specimens one can see that
the average k and cp values in Table 2.5.6 are close to
those reported in Table 2.5.7. This indicates that using a
silicone film of 0.015 inch thickness on the specimen's
surface is just as accurate as using three drops of
distilled water in room temperature tests. When the
specimen used was prepared from the new bar (Armco magnetic
ingot iron for D-C applications) the agreement of the k
values is not quite as close. As Tables 2.5.8-10 indicate
the average k and cp values are quite close when using
0.015 inch silicone film or six drops of distilled water

on the specimen's surface for room temperature tests, while

54

the tests using three drops of distilled water gave values
about 6% higher for k.

Table 2.5.11 gives the k and cp values of Armco
iron (specimen #3) for the elevated temperatures of 300 and
400 °F. For these elevated temperature tests only 0.015
inch silicone film was used on the specimen surface because
the water evaporates at these temperatures. The average,
variance and standard deviation of the k and cp values
given in Table 2.5.11 are reported in Table 2.5.12, while
Table 2.5.13 gives a comparison between the present values
and those reported by TPRC for room temperature and
elevated temperature tests.

Most of these tests were conducted using the inte-
grator method, but some were conducted using the IBM 1800
method to collect the data. A special program (called
SIMPL) was used to determine k and cp and other parameters
from the 1800 data. k and cp values determined by using
program "SIMPL" are given in Table 2.5.14. Comparing the
value of k obtained by using the integrator method with
that by using the IBM 1800 method (using the temperature
difference thermocouple readings) for 400 °F we can see a
difference of about 5% for the case of heat flow from the
calorimeter to Armco and about 9% for the case of heat flow
from Armco to the calorimeter. For room temperature tests
the difference for the k value is about 7.7% for both
methods of heat flow, while the difference for the cp

values is about 5.4% for the heat flow from the calorimeter

Table 2.5.1

Typical values of k and c

55

for Armco iron

(specimen #1) at room temperature. A silicone
film, 0.015 inch thick was used on the inter-

 

 

 

 

face.
k (FEE-4 c ( Btu )
r- t-F p llfiFfiF
41.143 .1098
38.153 .1080
39.87 .1082
40.325 .1085
38.1 .1078
38.658 .1097
38.92 .1078
38.03 .1081
39.596 .1091
40.148 .1086
39.085 .1078
38.471 .1084
38.85 .1078
38.0 .1142
38.502 .1080
38.523 .1078
39.220 .1076
39.012 .1080

 

 

56

Table 2.5.2 Typical values of k and cp for Armco iron
(Specimen #1) at room temperature. Three
drops of distilled water were used at the
interface.

 

 

k ( Btu ) c ( Btu )
hr—ft—F p le-F
39.365 .1096
38.742 .1093
37.6 .1086
39.231 .1086
40.628 .1103
40.247 .1091
39.992 .1082
39.945 .1078
39.935 .1077

 

 

 

 

Table 2.5.3 Typical values of k and c for Armco iron
(specimen #3) a new bar, at room temperature
using 3 drops of distilled water at inter-

 

 

face.
k ( Btu ) c ( Btu )
hr-ft—F p le-F
45.973 .1085
43.712 .1088
45.389 .1080
47.244 .1074

 

 

 

 

57

Table 2.5.4 Typical values of k and cp

for Armco iron

(specimen #3) at room temperature using 6
drops of water at interface.

 

 

 

k(Bt“) c(Bt°)
hr-ft-F p lEm-F
41.086 .1078
43.889 .1139
43.545 .1083
43.3 .1074
44.118 .1073
42.201 .1075
42.811 .1104
42.5 .1089
43.32 .1085

 

 

 

Table 2.5.5 Typical values of k and cp for Armco iron
(specimen #3) at room temperature using 0.015
inch film of silicone grease at interface.

 

 

 

k( Btu ) c (Btu)
hr-ft-F p IEm-F
42.795 .1087
41.373 .1075
42.493 .1074
42.342 .1070
42.159 .1070
42.915 .1072
41.81 .1072
42.722 .1072
40.4 .1102
40.0 .1085
41.84 .1090
40.2 .1082
43.4 .1080
43.2 .1087

 

 

 

Table 2.5.6

58

Average, variance and standard of deviation
for the values obtained at room temperature
using 0.015 inch silicone film on surface of
Armco iron (Specimen #1).

 

 

E = 39.0337111 6p = .1086222
Variance = .7754886 Variance = .0000024
Standard Standard

Deviation = .8806183 Deviation = .0015338

 

 

Table 2.5.7

Average, variance, and standard of deviation
for values obtained at room temperature using
3 drops of water on surface of Armco iron
(Specimen #1).

 

 

E = 39.5205556 6p = .1088000
Variance = .8413818 Variance = .0000007
Standard Standard

Deviation = .9172687 Deviation = .0008573

 

 

Table 2.5.8 Average, variance, and standard of deviation

for values obtained at room temperature using
3 drops of water on surface of Armco iron
(specimen #3).

 

 

E = 45.5795000 6p = .1081750
Variance = 2.1497497 Variance = .0000004
Standard Standard

Deviation = 1.4662025 Deviation = .0006131

 

 

 

 

 

 

59

Table 2.5.9 Average, variance, and standard of deviation

for values obtained at room temperature using
6 drops of water on surface of Armco iron
(specimen #3).

 

 

E = 42.9744556 6p = .1088889
Variance = .8889280 Variance = .0000045
Standard Standard

Deviation = .9428298 Deviation =.0021139

 

 

 

Table 2.5.10 Average, variance, and standard of deviation

for values obtained at room temperature using
0.015 inch silicone film on surface of Armco
iron (specimen #3).

 

 

E = 41.9749071 6p = .1079857
Variance = 1.2324266 Variance = .0000009
Standard Standard

Deviation = 1.1101471 Deviation = .0009461

 

 

 

Table 2.5.11 Typical values of k and c for Armco iron

(specimen #3) at 300 and 00 °F using 0.015
inch film of silicone grease at the inter-

 

 

 

face.
Btu Btu . .
k (5?:TE:F) cp (1 m-F) Conditions

37.0 .1427 300 °F

38.711 .1412 0.015" film of Si

38.128 .1407
Calori-
meter

35.90 .1556 400 °F + heat

34.80 .1600 0.015" film of Si Armco flow
Calori-
meter +

33.5 .1190 400 °F heat

35.0 .0953 0.015" film of Si Armco flow

 

 

 

 

Table 2.5.12

60

Average, variance, and standard of deviation

of the values obtained at 300 and 400 °F of
Armco iron (Specimen #3) using 0.015 inch

silicone film at interface.

 

For 300 °F (heat flow from calorimeter to Armco iron):

 

 

 

 

 

E = 37.946 8p = .1415

Variance = 0.7555 Variance = 0.1085 x 10'-5
Standard Standard _3
Deviation = 0.870 Deviation = 1.043 x 10
For 400 °F (heat flow from calorimeter to Armco iron):
E = 35.35 5p = .1578

Variance = 0.6050 Variance = 9.68 x 10'-6
Standard Standard _3
Deviation = 0.779 Deviation = 3.115 x 10

 

For 400 °F (heat flow from Armco iron to calorimeter):

 

E = 34.25
Variance

Standard
Deviation

 

E = 0.1072
P
= 1.125 Variance =
Standard
= 1.062 Deviation =

 

2.78 x 10'

4

0.0167

 

 

 

61

 

 

 

 

 

 

 

 

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62

Table 2.5.14 k and 0p values obtained from the IBM 1800 data by
utilizing program "SIMPL."

 

Temp. of k( Btu 6 ( Btu )
Case Direction of heat flux specimen hr-ft-F p lbm-F

 

4 Armco to calorimeter 136-113 °F 45.5 .1092
(hot)

3 Calorimeter to Armco 87-108 °F 45.35 .1142
(hot)

2A Armco to calorimeter 415-397 °F 38.5 .1210
(u51ng Tx==0 & Tx=L)

28 Armco to calorimeter. " 31.1

(using temp difference
TC readings)

1A Calorimeter to Armco 431-452 °F 39.6 .1331
(u51ng Tx=o & Tx=L readings)
1B Calorimeter to Armco " 33.6

(using temp difference
TC readings)

 

 

 

 

 

 

to Armco and about 1.1% for the heat flow from Armco to the
calorimeter. For both methods, the guard-heaters and
controllers were in use. For a description of the experi-
ment using the IBM 1800, see [31]. Since the standard
testing procedure is for the heat flow during the test to
be from the calorimeter to the Armco, we can see that the
two systems are relatively close in determining k and cp.
As indicated above, Table 2.5.13 compares the
values given by TPRC [33] and the present values. The

present room temperature values of k are about 3.5% higher

using 6 drops of water at the interface and about 1.2%

 

63

higher using the 0.015 inch film of silicone grease at the
interface. The present cp values are about 0.9% higher for
6 drops of water while they are nearly the same as those of
TPRC when using the 0.015 inch of silicone film.

For the 300 °F values, the present k values are
about 3.8% higher while the cp values are about 15% higher.

For 400 °F with the heat flow during the test from
the calorimeter to Armco, the present values are about 3.6%
higher in k and about 17.6% higher in cp. But when the
heat flow is from the Armco iron to the calorimeter the
difference is about 0.7% higher in k and about 17.5% lower
in cp. We used the method of heat flow from the calorimeter
to the specimen (during the test) as our standard testing
procedure because it is more consistent in results than that
of specimen to calorimeter.

Figures 2.5.1 and 2.5.2 Show the results of tests
on Armco iron (specimen #3) from the IBM 1800 data by
using program "SIMPL."

Figure 2.5.1 shows the temperature history of the
calorimeter and specimen during a typical test. It also
depicts the actual behavior of the temperature difference
between the heated surface and insulated surface of the
specimen (AT) as well as fAT dT during the test. Figure
2.5.2 shows the integrated heat flux (%) and contact
conductance (h) history during the test. The k and pcp
values calculated during a test are also shown. The

assymptotic values for "large" times are the desired

64

cal
Measured Temperatures
~470°F

 

 

    
  

 

T(E,t)

 

22 1 n 1 1 11
0 10 20 30 40 50 60
TIME(SEC)

AT.°C

 

 

 

 

L
0 10 20 30 40 50 60

TIME(SEC)

Figure 2.5.1 Temperature history of the calorimeter and
specimen during a test on Armco iron.

 

65

 
  
  

 

 

 

 

 

 

 

 

 

140... .
Q/A 9120
1209.6000 2
w M -0 h 1050..
“510% h' / (L 90
+3 80.. h,Btu/hr-ft2-F )
.3 I--60
25 60.. i
H
>< “0‘2000 350 ..- 30
<3 204-
O
O l l l 0
0 10 20 30 L10 50 60
TIME(SEC)
..40
_ .__._.. 600*".5w
k p
DC
140 P 450---1
1300 u
/ pcpx 10-9 20
w M-K 3 300
P200 Btu/ft_F a
-l
20 pCp X 10 i 3 ~10
_100 J/M'K 150-:
O l 1 l J L O
0 10 20 30 no 50 60
' TIME(SEC)

Figure 2.5.2 Typical values of k, pcp, h and 2 during a

. . A
given test on Armco iron.

o/A x 10-8, Btu/ft2

k, Btu/hr-f‘b—F

66

measurements which should be compared with those of TPRC
also indicated.

To minimize the heat losses from the sides and back
surface of the specimen and calorimeter, several insulating
materials were tested as insulators in the system. Many
experiments were performed to determine the best insulating
material or best type of guard-heater. It was found that
the best method (of those tried) of minimizing the heat
losses in the system is to employ an OFHC copper type
guard-heater, one around the calorimeter and another
around the specimen (see Figure 2.2.1). The guard-heaters
must be at the same initial temperatures as those of the
calorimeter and specimen; that is, the guard-heater around
the calorimeter must be at the same temperature as that of
the calorimeter just before the test and the guard-heater
around the specimen must be at the same temperature as that
of the specimen just before the test. It was found that
this procedure gives the best results (see Figure 2.5.5).

Figures 2.5.3, 2.5.4, and 2.5.5 Show the results of
some of the experiments that were performed to study the
nature and magnitude of the heat losses. Figure 2.5.3
shows the results of an experiment in which the copper
calorimeter was initially at 425 °F and the Armco iron
specimen was at room temperature initially. Both the
copper and Armco were surrounded by Fiberglass insulation
that fills the whole Space in the can. (This test was

performed before the copper guard—heaters were built and

67

installed.) They were brought into intimate contact and
were left in that position for about 50 minutes. The
temperatures of both calorimeter and Specimen were recorded
every minute. The temperature is plotted versus the time
in Figure 2.5.3. The same experiment was performed again
except this time the specimen (Armco iron) was surrounded
by a Transite type guard-heater, this heater was turned off
during the test; the temperature history is plotted versus
time in Figure 2.5.4. In another experiment the specimen
(aluminum 2024-T351) was at 400 °F initially and the copper
guard-heater surrounding it was at the same temperature
while the calorimeter (OFHC copper) was at 445° initially
and the copper guard-heater surrounding it was at 400 °F.
Then the specimen was brought into intimate contact with
the calorimeter and left in contact for 30 minutes. During
this time both guard-heaters were "off," (i.e., the temper-
ature controllers were not used) and the temperature history
of both specimen and calorimeter were recorded. This same
experiment was repeated exactly with exception of the
temperature of the guard—heater around the calorimeter
being 445 °F initially (which is the same as the calori-
meter's initial temperature) rather than 400 °F. The
results of both experiments are plotted in Figure 2.5.5.

It is interesting to note that the rate of temperature drop
for the case of the calorimeter's guard-heater initial
temperature not being the same as that of the calorimeter

is about four times higher than that when both the

68

400_

370~

340-

‘///—'OFHC copper calorimeter

 

 

 

34310—
a:
E Armco iron specimen
4;
CE
3.“
E: 280~
[1]
e)
250—
220%
19 i l L l I

 

o 10 20 3o 40 50
TIME (MINUTES)

Figure 2.5.3 Copper at elevated temperature and Armco iron at room
temperature initially. Brought into intimate contact,
both copper and Armco are surrounded by Fiberglass
insulation.

400

380 —

340 F

320 ~

300 _

280

TEMPERATURE°F

260

240 —

220 h

200 _

180 —

 

160

Figure 2.5.4

69

  
  

calorimeter

Armco iron
specimen

 
    

L l l 1 l l l J- l J
5 10 15 20 25 30 35 1+0 45'50

TIME (MINUTES)

Copper at elevated temperature and Armco iron
at room temperature initially. Copper is
surrounded by Fiberglass insulation, and Armco
iron is surrounded by Transite guard-heater
(heater was off during test).

70

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calorimeter and its guard-heater are at the same temperature

just before the test. This indicates that for a minimum of
heat losses in the system the temperatures of the specimen
and its guard—heater should be equal just before the test
and the temperatures of calorimeter and its guard-heater
should be the same initially also. (The calorimeter
temperature must be 40 to 50 °F higher than the specimen
temperature just before the test.) The larger the rate of
decrease of the temperatures of a Specimen, the greater is
the heat losses. Since the rate of temperature decrease
shown in Figure 2.5.5 is considerably less than those shown
in Figures 2.5.3 and 2.5.4, using an OFHC copper guard-
heater around the calorimeter and the Specimen is shown to
be very effective in minimizing the heat losses in the

system.

 

CHAPTER III

EXPERIMENTAL RESULTS FOR ALUMINUM 2024-T351

Aluminum 2024-T351 is a well-known material which
is used in many fields including aircraft construction. It
was chosen to be the principal material to be investigated
because it undergoes some phase changes with temperature
and time. This results in changes of the thermal properties

of the material, mainly the thermal conductivity (k).

3.1 Composition of the specimen

The specimen is Aluminum 2024-T351. It is composed
of aluminum and 3.8-4.9% copper, 0.50% silicone, 0.50%
iron, 0.30-0.9% manganese, 1.2-1.8% magnesium, 0.10%
chromium, 0.25% zinc, and 0.15% others. It is solution
heat treated and stress-relieved (see Sec. 3.2). Its
mechanical and other properties are very well established

and tabulated, see reference [36].

3.2 Metallurgical concepts of Specimen
The principal alloying element of aluminum 2024-

T351 is copper, about 3.8-4.9%. Figure 3.2.1 presents a

72

73

portion of the aluminum-copper system. The alloy can be

represented on the diagram by the vertical at 4%.

 

Liquid Solution

    

 

 

 

   

Temperature
p— — — -— -— — —

_L

4 56 50
Al Copper % —’ CuAI2
Figure 3.2.1 Partial equilibrium diagram for aluminum-

copper alloys.

 

 

 

The diagram indicates that if the alloy is heated
to 925 °F and maintained at this temperature until an
equilibrium composition is reached it will assume the a
condition with the complete solution of the O-phase. The
a condition is a single phase. The 0 phase consists of
hard particles based on the intermetallic compound CuAl2
(0 is a solution of small amount of Al in CuAlz). If the
alloy, after holding in the a-condition, is cooled very
slowly as in annealing, a precipitation of small spheroids
of the O-phase will occur. See Figure 3.2.2a which is a
photomicrograph. The alloy is in its softest condition
when its annealed. In the annealed condition, the alloy
has the following mechanical properties, as Shown in
Table 3.2.1.

The precipitation of the O-phase (which involves

diffusion) is prevented when the alloy is cooled rapidly by

 

74

Table 3.2.1 Mechanical properties of aluminum 2024-T351 in
the annealed condition.

!_..

 

Ultimate strength = 27,000 psi

Yield strength = 11,000 psi

Brinell hardness = 47 (500 kg load, 10 mm ball)

Endurance limit = 13,000 psi (based on 500 million cycles
of completely reversed stress using R. R.
Moore type of machine and specimen)

Modulus of Elasticity = 10.6 x 106 psi

Ultimate Shearing strength = 18,000 psi

 

 

 

water-quenching from the a-condition. After quenching in
this manner the alloy is said to be in the Solution-treated
condition and does not exhibit its maximum strength and
hardness, see Figure 3.2.2b. Upon holding (after the
solution treatment and quenching) either at room temperature
or at some elevated temperature in the a + 0 region, there
is a progressive change in the hardness and strength in
accordance with the data in Figure 3.2.3. This is because
the copper atoms have an extremely strong tendency to
precipitate out of solid solution to form a particular
crystal pattern of their own with aluminum. This crystal
pattern in its first stage of precipitation is called 0".
When sufficient heat is given for increased atomic mobility,
further precipitation takes place, first along the grain
boundaries then along the Slip planes. In reforming into a

separate crystal pattern the copper constituents, in effect,

 

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Figure 3.2.2a Microstructure of aluminum 2024
in the annealed condition.

6

O .‘ lute atom Solvent atom

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Figure 3 . 2 . 2b The stages in the formation of an equilibrium precipitate. (21) Su-

persuturated solid solution.
solution.
solution

(1)) Transition lattice coherent with the solid

(e) Equilibrium precipitate essentially independent of the solid

76

 

aging
Hardness - _____

 

  
 

over-aging

 

 

O 96 hrs
Time
Figure 3.2.3 Aging and over-aging curves for Al-Cu system,

over-aging drops the strength and hardness
of the alloy.

 

 

 

wedge in between the regular crystal pattern and distort

the normal lattice structure as shown in Figure 3.2.4.

 

 

Figure 3.2.4 Diagram of lattice structure showing

distortion caused by precipitated
constituent.

 

 

 

77

With longer times or higher temperatures more
crystals precipitate in the form of 0'. The alloy reaches
or achieves its maximum hardness and strength at a precipi-
tation stage between 0" and 0'. This is due to the fact
that the distortion of the crystal lattice along the slip
planes interfers with smooth slip and thereby increases the
strength of the alloy. In accordance with the modern
conception of the nature of slip, the precipitated particles
and lattice disturbance interfere with the motion of dis-
location. The mechanical properties attributed to the
alloy in this final aged condition are given in Table
3.2.2.

Table 3.2.2 Mechanical properties of aluminum 2024-T351 in
the final aged condition.

 

Ultimate strength = 68,000 psi
Yield strength = 47,000 psi

Brinell hardness = 120 (based on 500 kg load and 10 mm
ball)

Endurance limit = 20,000 psi (based on 500 million cycles
of completely reversed stress using
R. R. Moore type of machine and
specimen)

Modulus of elasticity = 10.6 x 106 psi

Ultimate shearing strength = 41,000 psi

 

 

 

If aging were to continue due to higher temperatures or
longer times at temperatures or both, then the constituents

would combine into larger sizes (0) and the distortion of

78

the atomic lattice would be reduced or eliminated. If it

is at room temperature it is called natural aging, however,
if it is at higher temperatures then it is called artificial
aging. At this stage the strength and hardness of the alloy
drops appreciably, and the alloy becomes over-aged, see
Figure 3.2.3. The process of precipitation heat treatment
is schematically shown in Figure 3.2.5. Precipitation heat
treatment plus over heating with appreciable time is called
over-aging and is diagramed in Figure 3.2.6. 0' and 0"

represent aging while 0 represent over-aging.

 

natural
quench in (at room
cold water temp)
Solution heat treated + super saturated solution + aging
(alloy in a—phase) artificial
(at high
temp
250-300 °F)
Figure 3.2.5 Diagram of precipitation heat treatment.

 

 

 

 

quench in

cold water
Solution heat treatment + super-saturated solution
(alloy in a phase)

time or temp + " further temp» + 0' over heating_
or both or time or both with appreciable time

 

 

 

[In this case over-aging had brought the value of the thermal con-
ductivity to approximately the same value obtained by the recommended
annealing procedures]

Figure 3.2.6 Diagram of over—aging

 

 

 

79

If the alloy to be used in applications such as
machine design is in the precipitated-hardened condition,
then it is not in equilibrium-state, and is subject to
continuing change in microstructure and properties at a
rate dependent on service temperature. At room temperature
the rate of change is negligible and the mechanical
properties will be virtually unaltered in service.

However, if the service temperature is elevated,
then changes in structure and properties can become very
significant. Such changes can limit the application of the
aluminum alloys in machine design.

The mechanical properties of the aluminum alloys
after the precipitation hardening heat treatment are
frequently enhanced by superimposing hardening and
strengthening by plastic deformation or work-hardening.
These work—hardening operations also contribute to the
unstable condition of the alloy and the work-hardening
effects are altered by the elevation of temperature.

The minor constituents such as iron, manganese,
zinc, etc. in the composition of the 2024 alloy contribute
to the hardening and strengthening of the alloy by simple
solution hardening, by altering the composition of the

O-phase and by forming independent intermetallics.

3.3 Experimental strategy
The experimental strategy consists of:

1. Testing the specimen at room temperature,

80

2. Heating the specimen to and holding it at the
desired temperature and then performing the tests,
and

3. Cooling the specimen down to room temperature and
test it at that temperature.

After installing the specimen in the experimental
apparatus, it must be tested first at room temperature to
determine the thermal conductivity and specific heat values
of the as-received Specimen. Since the purpose of the
investigation is to determine the transient k and cp values
of the specimen at the elevated temperatures, then the
specimen is heated to the desired temperature and maintained
there.

The tests are performed at intervals from the
moment it arrived at the desired temperature to the time
the alloy is over-aged. (See Section 3.2.)

The time intervals between tests are determined by
the temperature level at which the specimen is held. It is
important to note that the higher the temperature at which
the specimen is maintained, the faster the specimen reaches
its over-aged condition. Therefore at temperature levels
of 400 °F and above it is recommended that the intervals
between tests be as short as possible (on the order of 15
minutes or so) particularly during the first two hours.

The intervals can be on the order of an hour or more
between tests for temperature levels of 350—375 °F. When

the specimen reaches its over-aged condition (experimentally,

81

the specimen is over-aged when its thermal conductivity
values do not change significantly with time anymore) it is
cooled down (in air) to room temperature. Tests on the
specimen at room temperature are again performed. In this
case over-aging brought the thermal conductivity values to
approximately the same values obtained by the recommended
annealing procedure.

This experimental strategy was applied to all
aluminum specimens tested in this investigation. The

measured values are given in section 3.4.

3.4 Experimental results

Fourteen specimens of Aluminum 2024-T351 in the
as—received condition have been tested at different temper-
atures to establish transient k and cp values for this
alloy. The test temperatures were 350°, 375°, 400°, and
425 °F.

For test temperatures about 350 °F the alloy under-
goes such slow changes that several days are needed to
arrive at the over—aged condition. Yet, for temperatures
above 425 °F the phase changes take place so rapidly that
an accurate picture of these changes is difficult to obtain
with the present equipment since the smallest time interval
is about 15 minutes. Reasonable experiment durations are
possible for those indicated above.

All the data was collected by using the integrator

system. As was discussed in Chapter II the insulation

 

 

82

thermocouple was embedded directly into the specimen (after
it was insulated with Astroceram) for some of the specimens,
and for the others the washer-type thermocouple mounting was
used; see Figure 2.3.3.

The results of these tests are given in Tables
3.4.1 to 3.4.13. The thermal conductivity and specific
heat values are also plotted versus time in Figures 3.4.1

to 3.4.12.

350° test

 

Table 3.4.1 gives the results of testing aluminum
2024-T351 (specimen SH) at the nominal temperature of 350
°F. The Specimen was heated to 350 °F and maintained at
that temperature for a total of 74 hours. The first test
was performed fifteen minutes after it had arrived at
350 °F. (Zero time is the instant at which the specimen
arrives at the desired temperature.) As the values of k
and cp indicate, the phase changes at this temperature take
place very slowly and it took over three days to establish
how the phase changes influence the thermal properties. The
R and cp values given in Table 3.4.1 are plotted versus time
in Figures 3.4.1 and 3.4.2. The data points did not provide

as a neat pattern as one would desire; yet one can see that

it started at an average minimum of 88.0 H;§§%:§ for k and
increased to a maximum value of about 103 fi;§%%:§-47 hours

later and, then began to drop slightly.

 

 

83

The specific heat values in Figure 3.4.2 do not
appear to have a regular pattern. They appear randomly
scattered in part because the scale on the vertical-axis
covers the small range of .242 to .262--which is about 8%.
Note, however, that the largest specific heat value is
indicated at an early time.

During a test the temperature of a specimen rises
above its nominal temperature by about 15 to 20 °F, this
is because the calorimeter is maintained initially at about
40 °F above the specimen's temperature. The duration of
the test is about one minute, however. Because the
specimen temperature rise above the nominal temperature is
not large and is brief, it is believed that the effect upon

the properties is not large.

375 °F tests

Tables 3.4.2 and 3.4.3 give the k and cp values for
specimens numbered 4H and 14H which were tested at 375 °F.
Specimen 4H was tested over 11.5 hours period. The k values
became larger with time until reaching a maximum at about
5.75 hours and then started to drop. See Figure 3.4.3.
The specimen then was considered to be over-aged and was
allowed to cool in air to room temperature. The specific
heat value started with a maximum value as is shown in
Figure 3.4.4.

Table 3.4.3 shows data for specimen 14H which was

maintained at 375 °F for the long period of 101 hours; see

 

 

 

 

84

Figure 3.4.3 also, and note the two time axes. The k value
increased with time but at a lower rate than specimen 4H.
(Use the 10 hour axes results for this comparison.) From
the 100 hour plot on Figure 3.4.3 note that there are two
maximums. The first one is 108.4 at 4 hours and the second
is at 111.9 HE§%%=F at 39.75 hours. The latter is the
global maximum. This specimen was the last one tested and
the continued rise after 10 hours was completely unexpected.

The specific heat had a maximum value (.265) at
zero time and it fluctuated in a downward way to about .242
over the 101 hour period. The cp values given in Tables
3.4.2 and 3.4.3 are plotted versus time in Figure 3.4.4.
Again one can see this fluctuation in the specific heat
values to be pronounced because the y—axis scale is
enlarged. Also one can see from Figures 3.4.3 and 3.4.4
that the k values and cp values for the two specimen are
quite close indicating good precision.

In collecting the data in Table 3.4.3, the washer—
type thermocouple mounting (described earlier) was used on

the insulated surface.

400 °F tests

Table 3.4.4 gives the 400 °F values of k and cp for
aluminum 2024-T351 specimen 6H. The specimen was maintained
at 400 °F for over 8 hours. The k value started with

Btu . Btu .
99.4 (EE:TE:F)' went to a max1mum of 118.0 (EE:TE:F) in

3.25 hours, and then leveled off and remained relatively

 

85

constant until the end of the test. See Figure 3.4.5. The
corresponding specific heat values started with a maximum

Btu - - Btu
of 0.269 (TEE:F)' went down With time to 0.251 (lbm—F)'
Btu

went up to a value of 0.258 (135:?) at 3.25 hours, dropped

again to a low of 0.249 (Tgfigf) and then fluctuated between

0.251 and 0.254 (013%) until the end of the test. See

 

Figure 3.4.6. It is interesting to note that a local
maximum of cp occurred at exactly the same time the k value
attained its maximum.

Three additional specimens were tested at 400 °F to
establish the pattern and behavior of the alloy at this
temperature, to see if different time intervals between
tests have any effect on this behavior and to demonstrate
the precision of the method. The k and cp values for the
three additional specimens tested at 400 °F are given in
Tables 3.4.5, 3.4.6, and 3.4.7, and in collecting these
values, the washer-type thermocouple mounting was used on
the insulated surface of the specimens. The k and 0p
values given in Tables 3.4.5 through 3.4.7 are also plotted
in Figures 3.4.5 and 3.4.6. Looking at Figure 3.4.5 one

can see that specimen 10H has a maximum k of 109.7
( Btu
hr—ft-F
. Btu
and 12H have a max1mum k of 110.62 (5?:ft:F) at two hours

Btu
hr-ft-F

The maximum value of k for all four specimens shown

) at two hours from zero time while specimens 11H
and 117.0 (

) at three hours respectively.

in Figure 3.4.5 falls in the range of about 110 to 118

Btu .
(5?:fE:F)' and all four spec1mens have the same general

 

86

behavior, that of starting at a given value and attaining a
maximum then leveling off at some constant value. This
constant value while lower than the maximum value k
attained, is always higher than the initial value at zero
time.

The specific heat values for specimens 10H, 11H,
and 12H shown in Figure 3.4.6 follow pretty much the same

pattern as those of specimen 6H that were described earlier.

425 °F tests

Tables 3.4.8 and 3.4.9 give the k and cp values of
aluminum 2024-T351 (specimens 3H and 13H) at 425 °F and
the results are depicted in Figures 3.4.7 and 3.4.8. The k
values for the two tests have the same general shape as
previously noted for other cases. Again the higher
nominal temperature results in a more rapid change with time
than for the previous tests which were for lower nominal
temperatures. There is a 5 to 8% difference between the
values obtained from the two specimens.

Unlike for the other temperatures investigated the
specific heat at 425 °F has a time dependence that is
similar to that for the thermal conductivity. That is,
the initial value is relatively low, a maximum is attained,
etc. There is maximum difference in values of op in the
two tests of 7.6% but most values are less than 4% differ-

ent.

 

 

 

87

Composite results

Tables 3.4.10, 3.4.11, and 3.4.12 give the composite
values of k and op for the 375°, 400°, and 425 °F temper-
atures. By the composite value we mean an average value
for each data point at a given temperature and time. For
example, the composite value of k 375 °F and one hour is
the average of the k value at one hour from Table 3.4.3
and the interpolated k value at one hour from Table 3.4.2.
The interpolated value is the average of the k values at
0.75 and 1.25 hours. That is, we not only have k and cp

values for each specimen tested at a given temperature and

 

time, but we also have the average k and cp values for all
specimens tested at that temperature and time. These
average values for a given temperature are called the
composite values. The composite values were needed in
developing the mathematical model that is discussed in
Chapter IV.

The k and cp values given in Tables 3.4.10, 3.4.11,
and 3.4.12 are plotted versus time in Figures 3.4.9 and
3.4.10. Although the k values at zero time in Tables
3.4.10, 3.4.11, and 3.4.12 do not all start at 99.0
(Hggégrf), they were taken however to be that in Figure
3.4.9 for the sake of uniformity since the average of the
three values is about 98.4 (H;§%%:§).

The values of k at room temperature for over-aged

aluminum 2024-T351 were obtained after the various speci-

mens were cooled in air from their respective nominal

88

temperatures to room temperature. Table 3.4.13 gives the
over-aged values of k at room temperature for all the
specimens tested in this study. Table 3.4.14 lists the
average room temperature values for over-aged aluminum
2024—T351 as well as a measured value for regular annealing.
Note that all the values given in Table 3.4.14 are quite
close.

Figures 3.4.11 and 3.4.12 show a comparison of the
k and cp values at different temperatures as given by TPRC
and the present study. The lower dashed line in Figure
3.4.11 depicts the initial values (time = zero) of k in the
present study. It starts at room temperature (about 540 °R)
and goes up to about 885 °R. The final values of k (over-
aged values) in the present study are depicted by the upper
dashed line, starting at about 885 °R and going down to
room temperature. A few of the specific heat values
obtained in this study are shown in Figure 3.4.12 by the

small dark circles.

3.5 Corrections for errors

To insure accuracy and precision, all sources of
potentially significant errors in the experimental values
must be considered and corrections made if necessary. The
following is a list of corrections that were applied (or

considered) to the values given in Section 3.4.

89

 

 

 

 

 

Table 3.4.1 k and c values for aluminum 2024-T351
(specimen 5H) at 350 °F.

Time (hrs) at temp k(5;g%%:5) Cp(l§%§F)
.25 89.15 .253
.75 86.7 .251

1.25 88.35 .261
2.25 89.50 .247
3.25 90.40 .249
5.75 88.63 .244
6.75 91.20 .247
7.75 93.74 .252
8.75 91.40 .246
9.75 92.65 .246
10.75 96.45 .249
20.75 94.30 .252
21.75 96.84 .243
22.75 98.91 .244
28.75 97.40 .246
29.75 100.50 .249
30.75 102.0 .245
31.75 102.72 .245
45.0 101.65 .246
46.0 102.65 .247
47.0 103.0 .247
55.0 102.0 .247
73.0 99.82 .243
74.0 101.80 .245

 

 

 

 

90

Table 3.4.2 k and c values for aluminum 2024-T351
(specimen 4H) at 375 °F.

 

 

 

Time (hrs) at temp k(fi;§§§:§) cp(TE%gF)
0.0 98.0
.75 104.52815 .249
1.25 107.111 .239
1.75 106.714 .239
2.25 108.547 .241
3.0 108.940 .244
3.75 107.03 .242
4.25 108.05 .245
5.75 110.52 .242
6.25 109.16 .242
6.75 108.75165 .247
7.5 107.35 .244
9.5 104.80 .240
10.5 105.8 .244
11.5 102.93 .245

 

 

 

 

 

Table 3.4.3

k and c

91

values for aluminum 2024-T351
(specimen 14H) at 375 °F.

 

 

Time (hrs) at temp k(5;%%%:§0 cp(TE%gf)

0.0 99.6 .265

1 102.55 .256

2 105.9 .256

3 106.32 .253

4 108.4 .252

5 107.8 .254

6 106.74 .250

8 107.3 .251

9 106.9 .249
10 107.3 .248
11 107.2 .245
24 110.8 .250
25 111.0 .249
39.75 111.9 .248
54.25 111.7 .244
83.25 109.9 .242
101 111.2 .243

 

 

 

 

 

 

 

92

Table 3.4.4 k and c values for aluminum 2024-T351
(specimen 6H) at 400 °F.

 

 

 

Time (hrs) at temp k(fi?g%%:F) cp(lgfigf)
0.0 99.4 .269
0.5 109.8 .258
1.0 114.5 .255
1.5 112.54 .252
2.0 115.33 .251
2.5 114.35 .251
3.25 118.0 .258
3.75 111.30 .249
4.25 110.70 .251
4.75 110.65 .249
5.25 111.3 .251
5.75 111.21 .254
6.25 112.0 .253
6.75 109.9 .252
7.25 109.7 .252
7.75 110.5 .252
8.25 111.4 .254

 

 

 

 

 

 

93

Table 3.4.5 k and c values for aluminum 2024-T351
(specimen 10H) at 400 °F.

 

 

Time (hrs) at temp k(h%§%E:F) cp(1§%%F)
0.0 97.8 .272
1.0 104.7 .256
1.5 108.6 .262
2.0 109.7 .267
3.5 105.5 .254
4.5 105.9 .257
5.5 105.4 .254
6.5 105.8 .252
8.0 106.2 .253
9.0 105.3 .254

 

 

 

 

 

 

Table 3.4.6 k and cp values for aluminum 2024—T351
(specimen 11H) at 400 °F.

 

 

Time (hrs) at temp k(H?g%%:F) cp(TE%%F)
0.0 98.3 .273
0.5 104.4 .257
1.0 110.55 .254
1.5 110.61 .260
2.0 110.62 .260
3.0 107.8 .256
4.0 107.7 .254
5.0 106.0 .253
6.0 106.7 .253
8.5 106.5 .252

 

 

 

 

 

94

Table 3.4.7 k and c values for aluminum 2024-T351
(specimen 12H) at 400 °F.

 

 

. Btu Btu

Time (hrs) at temp k(H?:ft:F) cp(TEE:F)
0.0 105.3 .278
1.0 114.8 .252
2.0 115.31 .255
3.0 117.0 .258
4.0 115.8 .255
5.0 114.6 .249
6.0 113.34 .248
7.0 113.31 .251
8.0 112.8 .250

 

 

 

 

 

 

Table 3.4.8 k and c values of aluminum 2024—T351
(specimen 3H) at 425 °F.

 

 

Time (hrs) at temp k(H;§%%:F) cp(TE%%§)
0.25 91.690 .260
0.75 104.517 .258
1.25 109.471 .281
2.0 109.095 .266
3.0 105.304 .255
4.0 105.481 .252
6.5 101.875 .254
7.5 102.806 .254
8.5 101.890 .252
9.5 102.791 .252

10.5 102.404 .257

 

 

 

 

 

 

Table 3.4.9

k and c

95

values of aluminum 2024-T351
(specimen 13H) at 425 °F.

 

 

 

 

 

. Btu Btu
Time (hrs) at temp k(K?:ft:F) Cp(TEE=F)
0.0 102.11 .235
0.75 117.12 .271
1.25 113.8 .262
1.75 113.4 .261
2.75 112.6 .259
3.75 112.6 .259
4.75 113.3 .260
6.0 112.0 .258

 

Table 3.4.10

OF.

Composite (average) values of k and CP of
aluminum 2024-T351 at 375

 

 

 

 

 

Time (hrs) at temp k(fi;g%%:§0 cp(15%§F)
0.0 99 .2650
1 104.2 .2500
2 106.5 .2480
3 107.6 .2485
4 108.0 .2478
5 108.6 .2488
6 108.2 .2460
7 107.3 .2483
8 106.7 .2470
9 106.2 .2450

10 106.3 .2450
11 105.8 .2450

 

 

 

Table 3.4.11 Composite (average) values of k and cp
aluminum 2024-T351 at 400

96

°F.

of

 

 

 

 

 

. Btu Btu
Time (hrs) at temp k(5;:ft:?) Cp‘IBfi=F)
0.0 100.2 .2730
1 111.14 .2543
2 112.7 .2583
3 112.2 .2570
4 110.1 .2540
5 109.3 .2520
6 109.3 .2520
7 108.9 .2522
8 109.1 .2520

 

Table 3.4.12 Composite (average) values of k and cp of

 

 

 

aluminum 2024-T351 at 425 °F.
Time (hrs) at temp k(3;§§%:§) Cp(1:;EF)
0.0 96.0 .2475
0.75 110.66 .2644
1.25 111.65 .2720
2 111.15 .2637
3 109.1 .2571
4 109.1 .2554
5 108 .2560
6 107 .2558

 

 

 

 

 

 

 

 

Table 3.4.13 Values of k at room temperature for over-aged

aluminum 2024—T351.

 

Specimen number and
temperature

Over-aged values of k
(at room temperature)

 

3H (425 °F, 10.5 hours)
4H (375 °F, 11.5 hours)
SH (350 °F, 74 hours)
GB (400 °F, 8.25 hours)
10H (400 °F, 9 hours)
11H (400 °F, 8.5 hours)
12H (400 °F, 8 hours)
13H (425 °F, 6 hours)

14H (375 °F, 101 hours)

 

 

97.2
94.2
96.9
99.5
96.0
97.5
105.25
102.3

102.8

 

Table 3.4.14 Composite (average) values of k at room
temperature for over—aged aluminum 2024—T351.

 

Process

R value (F;§%%:f)

 

Regular annealing

350 °F-nomina1 temperature
375 °F—nomina1 temperature
400 °F-nomina1 temperature

425 °F-nominal temperature

 

 

97.0
96.9
98.5
99.56

99.75

 

 

 

 

 

 

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106

 

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110

1. Corrections applied to k:

a. Correction due to the changes in the specimen length
ESp and (pEcp)cal in Equation (2.1.12) because of the high
temperatures. This correction is given by the equation

Esp(EpCp) = (E0) (Eopocp)C

[1 + 8 (T - T ) -
cal sp 1 Sp 5

a p O

_28cal(Tca1 - TO)] (3.5.1)

where the subscript (0) indicates a room temperature value,
and B is thermal linear expansion of the material.

That is, Equation (3.5.1) must be substituted in
Equation (2.1.12) for the high temperatures, or simply

evaluate

[l + Bsp(Tsp - To) _ 28cal‘Tcal

- TO)]
and multiply it by the uncorrected k values.

b. Multiply the uncorrected values of k by 0.5% to
take into account the absorption of energy by the silicone
film on the aluminum surface when heated by the calorimeter
during the test.

c. Multiply the uncorrected values of k by 2% when
using the washer-type thermocouple mounting on the insulated
surface of the specimen. This is because the actual
thickness of the specimen where the insulated surface
thermocouple is located is not 1.5 inches but rather at
1.5" + 1/16". The reason for taking 2% correction is

because the thermocouple is located at half way of the

111

washer's thickness, that is at 1/32 inch from either end

of the washer where

 

2. Corrections applied for 5p:
A correction was applied to the 6p values of

aluminum to take into account changes in (pop) at the

cal
high temperatures.

3. Corrections not required
The following corrections were considered for both
k and 6p but not applied.

1. No correction is necessary for the iron-constantan
thermocouple calibration. Note that the constant CV cancels
in both Equations (2.4.1) and (2.4.2).

2. No correction was applied for the heat losses
because it is not clear how to do this at the present time.
The measured specific heat (which is more affected by the
heat losses) of aluminum was about 8% higher than that
reported by TPRC, while that of Armco iron was about 16%
higher.

As it was shown in Chapter II, Armco iron was
tested in two ways; one way was for the heat flow during
the test to be from the calorimeter to Armco and the other

way was to reverse the heat flow.

112

It was found that cp of Armco at the elevated
temperatures (400 °F for example) is about 16% higher than
that given by TPRC, if the heat flow is from the calori-
meter to the specimen while it is about 16% lower than that
reported by TPRC if the heat flow is reversed. If an
average value is taken for both tests at the same temper-

ature, one can see that the op value becomes almost

 

identical to that given by TPRC for the same temperature.
This process of averaging results for the heat

flow in both directions was not performed for the aluminum
because the thermal conductivity was of primary interest
and the heat flow direction did not significantly affect
the k values. All the aluminum values were obtained with
the heat flow from the calorimeter to the specimen.

3. No correction was needed for the calorimeter to
account for the thermal linear expansion. The specific
heat expression is

c = (DE)cal (cpAT)cal

p (Bolsp (if - Tilép

 

At the elevated temperatures (pE)cal and (pE)Sp should be

modified in the following manner

cal _ T0))(cpAT>cal

(poEo)cal (l ” 28cal (T
c =
— T0)) (Tf "' Ti)Sp

p (p E ) 711 - 285p (Ts

o 0 sp p

113

 

 

However,
(1 — zscal (Teal - TO)) 5 (1 - 285p (TSP - To)
Therefore
E
:Ehical (CpéT)cal : (poEo)cal (C AT)cal
T - T.l ( S 7_—E:"'T‘
9 sp ( f 1 sp poEo sp Tf Ti sp

which means a correction to account for the thermal linear

expansion for the calorimeter is not needed.

 

4. Since the thermocouples on the heated surface do
not measure the temperature at the surface but rather at
0.005 inch below the surface (that is, the measurement is
done at the center of the wire) some form of correction for
k is needed. However, since there is no perfect contact
between the thermocouple wire and the surrounding metal,
the thermocouple then reads somewhat higher than it should
be. Therefore there are two mechanisms in operation, one
lowers the temperature measurement by measuring at 0.005
inch below the surface and the other raises the temperature
measurement because of the imperfect contact between the
thermocouple wires and the surrounding metal. These
effects cancel each other out, and a correction for k is
not applied. For further information on the possible
thermocouple groves effect on the heat transfer see

Henning and Parker [37].

CHAPTER IV

MODELING

In order to utilize for design purposes the data

 

delineated in Chapter III (especially those of the thermal
conductivity) a mathematical model is needed to describe it.
This model would enable one to predict the thermal con-
ductivity values for different temperatures and times.

This chapter deals with (1) formulation of the mathematical
model, (2) predicting the thermal conductivity values for
different times and temperatures by using the model, and

(3) describing an example using the ideas presented in (l)

and (2) above.

4.1. The development of a mathematical model

The thermal conductivity values presented in Section
(3.4), were determined at the four different temperatures
of 350°, 375°, 400°, and 425 °F. To enable the designer
to predict the values of thermal conductivity at temper-
atures other than the ones given above and for transient
conditions, it is necessary to develop a mathematical
model that describes the general behavior of thermal con-

ductivity in the temperature range 350-425 °F. As

114

115

mentioned earlier, for temperatures below 350 °F the thermal
conductivity does not change significantly with time. Yet,
for temperatures higher than 425 °F it changes so rapidly
that accurate measurements were not possible. The figures
presented in Section (3.4) show that thermal conductivity
values have a certain pattern; each starts with a relatively
small value, attain a maximum value at a given time
(depending on the temperature level) and then begin to fall
slightly with time before reaching some constant value. It
appears that most of the significant changes take place
between t = 0 and t = tmax (where tmax is the time at which
the thermal conductivity value becomes maximum for a given
temperature). The thermal conductivity data between t = 0
and tmax for all four temperatures (350°, 375°, 400°, and
425 °F) were normalized and plotted versus their normalized

time; see Figure 4.1.1. Normalized thermal conductivity,

+ . .
k , is defined by
+ _ ki kmin
k
k — k
max min
where
ki = thermal conductivity values at times ti in the
range t = 0 to tmax
kmin = thermal conductivity value at time zero for a

given temperature (time zero means the time at
which the specimen arrives at the desired nominal

temperature).

 

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117

kmax = maximum value of thermal conductivity for a

given nominal temperature. The time at which

k is measured is called t .
max max

Normalized time t+ is defined by

 

max

The data in Figure 4.1.1 can be approximated by a curve.
This curve could be described by a number of functions
including,

1. second degree polynomial with

a. fixed end points (at t = 0 and t = tmax) and
zero slope at tmax’
b. fixed end points (at t = 0 and t = tmax) but

not the slope, and
2. third degree polynomial with the same conditions
as in 1(a) and 1(b).
For model 1(a) the curve between t = 0 and tm is

ax
described by

+ _ + + 2
k — 81 + th + 83(t ) (4.1.la)

where 81' 82, B3 are parameters. The three conditions are

+ +

k = 0 at t = 0 (401011))
k+ = 1 at t+ = 1 (4.1.1c)

d dk+ +
a“ -—1 = 0 at t = 1 (4.1.ld)

118

Substitute (4.1.1b) in (4.1.1a) to get

Substitute (4.1.ld) in (4.1.1a) to get
0 = 82 + 283
or
Then
8 = 2, and B3 = -1
Model 1(a) then becomes
k+ = 2t+ — (t+)2

which has no unknown parameters.

(4.1.2)

For model 1(b) the curve is again given by (4.1.1a)

but with the boundary conditions being only

k+ = 0 at t+ = o

k+ = 1 at t+ = 1

Substitute (4.1.3a) in (4.1.1a) to get

(4.1.3a)

(4.1.3b)

119

and substitute (4.1.3b) in (4.1.1a) to find

Then model 1(b) becomes

k+ = th+ + (1 - 82)(t+)2 (4.1.4)

which has only one unknown parameter, namely 82.

 

To estimate 82 one can minimize the sum of squares (least

squares)
S = {[y1 - k+]2 (4 l 5) i
1 i ' '
where
y: = normalized observed thermal conductivity values
(experimental points)
k: = normalized calculated thermal conductivity values

+ _ + _ + 2
where ki - th + (1 82)(t )
Introduce (4.1.4) in (4.1.5) to find

2

+ - (1 - 82)(t+)2] (4.1.6)

8 = Ely: - th

Taking the derivative of S with respect to 82 and setting
the result equal to zero yields a 82 estimate which

minimizes S; then

2 2 2
= ZZIy: - b2t+ - t+ + b2t+ ][-t+ + t 1 = 0

we.)
(I)
N

5120

or

3 2 3
+ + + + + + + + +
{t (t - 1)b2 - Zt (t - 1)b2 = it (t - 1) - {t (t — 1)yi

(4.1.8)

The estimate of 82 is b2,

2
Zt+(t+ — 1>(t+ - Y1)
2 = 2

Zt+ (t+ — l)2

b (4.1.9)

 

. . . . +
To find b substitute all yi's at their respective ti's

2!
(i = 1, 2, ..., n observed data points) in Equation (4.1.9).
To obtain the calculated values of thermal conductivity,

substitute the value obtained for b2 in Equation (4.1.4).

For model 2(a), the curve can be described as:

2 3
t+ + B4t+ (4.1.10a)

+ +
k — 81 + th + 83

where 81’ 8 B and B4 are parameters to be determined.

2’ 3’

The conditions are

k = 0 at t = o (4.1.10b)
k+ = 1 at t+ = 1 (4.1.10c)
dk+ +
-—T+- =-’ 0 at t = 1 (4.1.100)
dt

Substituting (4.1.10b) in (4.1.10a) yields

 

121

and substitute (4.1.10c) in (4.1.10a) to find

1 = 82 + 83 + 84 (4.1.11)
Differentiating (4.1.10a) and using (4.1.10d) gives
0 = 82 + 283 + 384 (4.1.12)

Then solving (4.1.11) and (4.1.12) simultaneously yields

8 = B - 2 (4.1.13)

 

B3 = 3 " 282 (4.1.1.4)

Therefore (4.1.10a) becomes

2 3

+ + 2 — 2)t+ (4.1.15)

+
k - th + (3 — 282)t + (8
and there is only one parameter to evaluate which is 82.
Substitute (4.1.15) in (4.1.5) and follow in the same
manner described earlier to obtain an estimate of 82,
namely b2, that is

2 3 2 3

Z(3t+ - 2t+ — y:)(-t+ + 2t+ - t+ )

2 2 4
Z(t+) (t+ — 1) (4.1.16)

b

 

and find ki for a given temperature as discussed earlier.
For model 2(b), the curve can be described by
Equation (4.1.10a) and conditions (4.1.10b) and (4.1.10c)

only; then the result will be an equal number of unknown

122

parameters and equations which can be solved simultaneously
to obtain these parameters, see Beck [38].

To avoid tedious and error-prone operations in
finding the parameters in the models described in this
section, a computer program has been used. This program
was written by Nicely and Dye [39] of the Department of
Chemistry at Michigan State University. It is capable of
determining the desired parameters by utilizing a subroutine
that is programmed to give the desired mathematical model.

It was found that the parameters in the second
degree polynomial model (Equation (4.1.4)) and the third
degree polynomial model (Equation (4.1.15)) are nearly the
same. For the second degree model b2 was calculated to
be 2.02371 and for the third degree model b2 = 2.06447.
Both values are nearly 2.0.

Since the two models are nearly similar in describ—
ing the experimental data in Figure 4.1.1, it is easier
from a computational point of view to use the second degree
polynomial model. Hence, by letting 82 = 2 in Equation

(4.1.4) yields

 

 

2
k+ = 2t+ - (t+) (4.1.17)
or
ki - kmin 2( ti ) _ ( t )2
E - k . t t
max filln max max

 

 

123

It is interesting to note that Equation (4.1.17) is
identical to Equation (4.1.2).
4.2 Obtaining thermal cogguctivity values from the
mathematical model

To obtain thermal conductivity values at a given
temperature by using Equation (4.1.17), one must know

k , k . , and tm

max min at that temperature.

ax
The thermal conductivity value at zero time (the

time at which the specimen arrives at the desired temper-

ature) is designated k . . The k . values are determined

min min

from the composite curves given in Figure 3.4.9 and also

from Figure 3.4.1 for 350°, 375°, 400°, and 425°F. These

values are plotted versus their respective temperatures in

Figure 4.2.1. For kmin values between room temperature

(about 80 °F) and 350 °F, values were obtained by linear

Btu

hr-ft-F for 80

 

interpolation; the values are 78 and 88.5
and 350 °F respectively.

The maximum values that the thermal conductivity
attains (kmax) for the temperatures 350°, 375°, 400°, and
425 °F are determined from Figure 3.4.9 and Figure 3.4.1.
They are plotted versus their respective temperatures in
Figure 4.2.2. It was found that a second degree polynomial
curve describes the data. Using the Nicely and Dye program

(KINFIT) the model was found to be

k = -0.00244T2 + 2.006T - 299.835 (4.2.1)
max

 

 

 

124

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126

Equation (4.2.1) is based on four data points in the

temperature range 350° to 425 °F.

The time at which the thermal conductivity attains
its maximum value for a given temperature is designated
tmax' It is also determined from the composite curves in
Figure 3.4.9 and Figure 3.4.1 for the temperatures of 350°,
375°, 400°, and 425 °F. These values are plotted versus

their respective temperatures in Figure 4.2.3.

 

A mathematical model suggested by Frame [40]
describes the data very well in the temperature range '
350°—425 °F; see Figure 4.2.3. The model using KINFIT is

found to be

 

tmax — 1.38 (4'2°2)
T - 350
l + 8.4158 (-—f§--)
Using the equations for kmin’ kmax' and tmax given

above, ki can be obtained by using Equation (4.1.17) for
the temperature range 350 to 425 °F. To use Equation
(4.1.17) the following conditions must be met:

1. The time must be less than t .
max

2. For any specific temperature, one must use kmin and

kmax at that temperature, whether one is heating or
cooling the specimen to arrive at that desired
temperature, regardless of the length of time taken

to reach it.

 

3. It is necessary to use as an expression of
max

percent precipitation of the (0) phase, i.e., 47

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128

hours at 350 °F corresponding to 5 hours at 375 °F,

2 hours at 400 °F, etc.

Hence holding the specimen for 23 1/2 hours at
350 °F is assumed to be equivalent in the progress of
precipitation of the (0) phase to holding it for 2 1/2 hours
at 375 °F, for 1 hour at 400 °F, etc. Further experimental
work should be done to check this assumption with respect
to a non-constant temperature history of the specimen. To
illustrate the above points consider the case of a specimen
which is initially at room temperature and has a step
increase in temperature to 375 °F. After two hours the
temperature is abruptly increased to 400 °F and held there.
The thermal conductivity history is needed for the temper-

ature history depicted in Figure 4.2.4.

 

 

 

 

 

 

H 400‘ 'F‘r—

“' l

3, 375- I

e) ' '

2 ' l l;

0

a ,+ A i’/

Q l

E ' '

o l I

" R.T ‘ ‘ il-
0 2D 25 1fine
t+= 0 (hours)

Figure 4.2.4 Aluminum 2024-T351 specimen main-
tained at 375 °F for two hours then
heated to and maintained at 400 °F.

 

 

 

 

129

For real times from zero to 2 hours the required constants

are

kmin = kmin,375 =
kmax = kmax,375
tmax = tmax,375

Btu
99-0 (m)

Btu

= 108.6 (E?:?E:f7

= 5.0 hours

The t+ values required are simply found using

+
t :————E———__—

tmax,375

For real times greater than 2 hours the km.

tmax values are evaluated at
kmin = kmin,400 =
kmax = kmax,400
tmax = tmax,400

+
The t range must

now be found

where o<t<2 hours

I k ,
in max

400 °F, i.e.,

Btu
99°° (m)

Btu

= 112.7 (H;:?E:f)

= 2.0 hours

and

associated with 400 °F.

An

. . . +
important assumption that we shall make is that the t value

at the end of 375 °F period is the beginning value for the

400 °F range.

This assumption should be investigated

experimentally in a later study.

The t+ value at the end of the 375 °F period shall

be called t+

by

1 as shown in Figure 4.2.4,

this value is given

130

t
t+___ el =2.o=04
1 tmax,375 5’

 

where te is equal to the effective time based on the
1

nominal temperature of 375 °F. To illustrate for 400 °F,

the k value is to be given for time 2.5 hours. The t:

value is given by

t
e
t;=t:+-€-———2———=0.4+4%=0.65
max,400

where te is equal to the effective time for which the
2
specimen is held at 400 °F only. The k value then is

 

given by
— k .
min,400 _ + _ + 2
max,400 min,400
W = 2(0.65) - (0.65)2 = 0.8775
k = 111.0 (RTE-1‘31?)

If we have a continuous case of heating (or cooling) say
from 350 to 430 °F, then to find k we can approximate the
temperature history by a series of steps as shown in

Figure 4.2.5.

 

 

 

131

 

    

 

420
’\ 430 430
i 360 I l +
Vt

g 350 (1'3 | 4
H 350 I +
E I It+ (42 l l
o i F?) l
g I | I l I
.2 l l I l l

1 I a I I l L L

0 l 2 3 4 'Thne
(hourQ

Figure 4.2.5 Temperature history of an aluminum
2024-T351 specimen.

 

 

 

4.3 Example

To demonstrate the application of the findings
mentioned in sections 4.1 and 4.2 to some practical
engineering problems, the following example is given.

Consider a 1 1/2" thick aluminum 2024—T351 plate
held on one side (x = 0) at 420 °F and on the other side
(x = L) at 300 °F. The objective is to find (a) thermal
conductivity (k) as a function of time, and the (b) temper-
ature history. The problem is illustrated by Figure 4.3.1.

Solution: The general heat conduction equation can

be written as [41]

IO)

(4.3.1)

0.:
ll
‘0
O

we)

(fr-3

0T
x (k'gi) p

 

 

132

 

T: 420° 7 = 300° F

 

 

W

X‘ZO le
() )(
Figure 4.3.1 Aluminum 2024-T351 plate maintained at

420 °F on one side and at 300 °F on the
other.

 

 

 

Since the time changes in temperature in the plate
are very slow, Equation (4.3.1) can be modified for a

quasi-steady state condition and therefore can be written as

8 3T 3

5; (k 3;) 0 (4.3.2)
Integrating (4.3.2) yields

u=~k 3T (4.3.3)

A '53:”
where 9%E1 is the heat flux in the positive x—direction.
Consider now an approximate solution of (4.3.3).
Let the plate be divided into an arbitrary number

of equal regions, say six. See Figure 4.3.2. Then

Equation (4.3.3) can be approximated at time ti by

 

 

 

 

 

 

_ i i _ i i _ i _ i
31 = ki T1 T2 = k1 T2 T3 = xi T3 T4 = k1 T4 T5 _
A 1 Ax 2 Ax 3 Ax 4 Ax ‘
i i i
k1 T5 T6 = k1 T6 T7 '(4 3 4)
5 “"IE"‘ 6 Ax ° °
1' T T' T 1' T

 

 

 

:4

)(
TI 72 73 l4 75 76 T7

 

R] R

Figure 4.3.2 Geometry for thermal analysis of an
aluminum 2024-T351 plate.

 

 

 

01'
i i i i i i i i '
a: z T1 T2 = T2 T3 = T3 T4 = T4 T5 = Ts ' T6 =
l 2 3 4 5
i
T6 T7
1 (4.3.5)

R

134

where R; is a thermal resistance defined by
R. = Ax/k;

Equation (4.3.5) can also be written as the set of equations

 

. i . .
IL=1_1
R3 (A ) T3 T4 (4.3.6)
191'. 11
R4 (A ) = T4 - T5
.3: 1 1
R5 (A ) z T5 ’ T6
.1: 1
R6 (A ) = T6 ' T7

If the elements in Equation (4.3.6) are added then

u v o o I o l
i i i i i i g _ _

or

 

(4.3.7)

 

135

To obtain an initial estimate for time zero evaluate
Rg,2' Rg'fl, ..., Rg'l by first assuming that the temperature
in the plate to be linear, i.e., assuming constant Rj' The
1 superscript is an iteration index and starts at 2 = 0.
Since the plate is divided into six equal regions of Ax =

1/4" then one can take

H. 0
T1 — 420 F
T2'0 = 400 °F
Tg'o = 380 °F
T3'0 = 360 °F
Tg’o = 340 °F

0,0 = 320 °F

 

 

6
.H 0
T7 — 300 F
For region one
0,0 _ Ax
R1 ‘ k0,0
l
0 0 A T + T0'0
Evaluate kl’ at the temperature at ffi or at ———§————— =
420 3 40° = 410 °F. Similarly evaluate kg'0 at 390 °F,
etc.

The model (Equation (4.1.17)) must be used to

evaluate the different k values. One must specify the

 

136

length of time that the plate is held at the above mentioned
conditions.
For simplicity let calculations be performed at the
zero, 0.5 and 1.0 hour times.
1. At time zero the temperature in the plate can be
found in a relatively straightforward manner.

The mathematical model is

 

 

 

2
k+ = 2t+ - (t+)
but since the time is zero :1
t.
+ _ 1 =
t — t 0
max
which results in
k+ _ ki - kmin _ 0
k -*k .
max min
or
k. = k .
1 min

A first approximation for time zero uses the

linear temperature profile given above. The km. curve,

In

Figure 4.2.1, is used in the following typical manner,

 

 

1 in x 1 £3 2
R0,0 = Ax = 4' I2'1n = 2 105x10-4 ft ~hr-F
1 E: 99 Btu ‘ Btu

hr-ft-F

 

137

and the rest of the values are given in Table 4.3.1. For

temperatures below 350 °F kmin is evaluated in the manner

described in Section 4.2. Substitute the initial estimates

 

of R2'0 given in Table 4.3.1 in Equation (4.3.7) to get
0.1
g = 420 — 300 = e 885x10+4 Btu
A 13.502xlo'4 hr ft2

0'1 o o 0 0
Use the values g——— , Rl' , ..., and R6, , in the elements

of Equation (4.3.6) to obtain the more accurate values,

Tg’l, Tg'l, ..., and Tg’l; since T1 is known,

0 1
0,1 g ' _ _ 0,1
Rl A ‘ T1 T2
. 0,1
which can be solved for T2 ,
2.105x8.885 = 420 — Tg'l
Tg’l = 401.28 °F

The other temperatures for time zero and the first and
second iterations are given in Table 4.3.2. There is little
difference between the first and second iterations; hence
the iterations are terminated.

2. Next find the temperature distribution at time 1
which is to be at 1/2 hour. During the time interval
between time zero and 1/2 hour the aluminum alloy is under-
going metallurgical changes causing k to change. These

changes are relatively small, however.

 

Table 4.3.1

Initial and lst corrected estimates at time

zero.

138

 

initial estimates at time 0

lst corrected estimates at

 

 

 

 

 

 

 

 

time 0
No. Temp k9'0 R9'0x104 Temp kg'l Ro’lxlo
i i i i

1 410 99.0 2.105 410.64 99.0 2.105
2 390 99.0 2.105 391.92 99.0 2.105
3 370 96.90 2.157 372.99 98.0 2.126
4 350 88.5 2.355 352.97 88.8 2.346
5 330 87.6 2.378 331.97 87.6 2.378
6 310 86.74 2.402 310.71 86.74 2.402

 

 

 

Table 4.3.2

Temperatures at time zero.

 

 

 

 

 

Estimate lst Corrected 2nd Corrected
N099 NO- Tempo. Tg'o Temp.) Tg'l Temp., T3'2

1 420 420 420

2 400 401.28 401.28
3 380 382.56 382.56
4 360 363.42 363.67
5 340 342.52 342.83
6 320 321.42 321.70
7 300 300 300

 

 

 

 

 

 

139

Using the mathematical model for the thermal

 

 

conductivity
2
k.-k. t. t.
l _min = 2(t1 ) _ ( i )
max min max
or

ki - 99 5 5 2

where km , k , and tmax are obtained from Figures 4.2.1,

in max

4.2.2, and 4.2.3 at 410.64 °F. Then

1,1 _ Btu
k1 - 104-82 m

and the other values are given in Table 4.3.3.

Therefore

1 1 ft
1,1 Ax = 4 in x 12 in

 

2
= 1.99x10-4 EEjgflLfif.

tu

and the other values are given in Table 4.3.3.

Substituting in Equation (4.3.7) yields

1,2

 

g = 420 - 300 Btu
A 13.290x10-4 hr ft2
+4 Btu
= 9.04x10 —————5
hr ft

140

1,2
Since §——— , R1'2, Ré'z, ..., and R1’2 are known, sub-

1 6
stitute in the elements of Equation (4.3.6) to obtain the
more accurate values of Té'z, Té'z, ..., Té'z, i.e.,

l 2
1,2 I _ - 1,2
R1 1“” ‘ T1 T2
1.99x9.04 = 420 - T2
T1'2 = 402.02 °F

2

and the other values are given in Table 4.3.4.

3. The aluminum plate is held for one hour at the
above described conditions. Further metallurgical changes
take place in the time interval 0.5 to 1 hour which causes

k to change more. Again, use the mathematical model for k

 

 

max min max max

or for the temperature at Ax/2,

 

k1 - 99 1 1 2
_' -———- — —_ 0
where, k . , k , and t are obtained from Figures 4.2.1,
min max max

4.2.2, and 4.2.3 at 411.01 °F; then

2,1 _ Btu
kl - 107.61 hf—ft_7§

The other values are given in Table 4.3.5 and the temper-

atures are given in Table 4.3.6.

141

Table 4.3.3 Estimates for 0.5 hour time.

 

 

 

 

 

 

 

lst corrected estimate at time 0.5 hour
No. Temp kl’l R1'l x 104
i i
1 410.64 104.82 1.99
2 391.92 102.16 2.04
3 373.16 100.14 2.08
4 353.25 89.0 2.34
5 332.26 87.60 2.38
6 310.85 86.74 2.40

 

Table 4.3.4

Temperatures at time 0.5 hour.

 

 

 

 

 

Estimate lst corrected
Node No. Temp., Ti’l Temp., T%'2
1 420 420
2 401.28 402.02
3 382.56 383.58
4 363.67 364.78
5 342.83 343.63
6 321.70 322.11
7 300 300

 

 

 

 

 

142

Table 4.3.5 Estimates for 1.0 hour time.

 

 

 

 

 

 

 

 

 

 

 

 

 

lst corrected estimate at time 1.0 hour
No. Temp k?’1 R?’1 x 104
i 1
1 411.01 107.61 1.94
2 392.8 107.06 1.95
3 374.18 102.53 2.03
4 354.21 90.50 2.30
5 332.87 87.60 2.38
6 311.06 86.74 2.40
I
Table 4.3.6 Temperatures at time 1.0 hour. 1
Estimate lst corrected
Node No. Temp., Ti’l Temp., T?’

l 420 420
2 402.02 402.09
3 383.58 384.09
4 364.78 365.35
5 343.63 344.12
6 322.11 322.15
7 300 300

 

 

 

 

 

 

143

As can be seen from this example, the thermal con—
ductivity can be obtained by utilizing the mathematical
model given by Equation (4.1.17) for a wide range of times
and temperatures.

By knowing k one can predict the temperature
history of the material which is very important in machine
design applications.

This example was solved on the basis of the problem
being a quasi steady-state since the changes take place
very slowly.

If the property changes are to be very rapid under
given conditions, then one can extend this example to the

fully transient case by solving the following equation,

3 3T _ 8T
'55 “‘3? —°°pat°

Notice, however, that a key assumption is that the thermal
conductivity is a function of its composition which is

discussed in Section 4.2.

 

 

CHAPTER V

SUMMARY AND CONCLUSIONS

The transient thermal properties (namely thermal
conductivity k and specific heat cp) of aluminum 2024~T351
have been determined for the temperature range of 350 to
425 °F. These values have not previously been measured. 1/
Due to the metallurgy of the alloy, precipitation occurs
above 350 °F. This causes changes in the properties with
time. No existing method and procedure was available for
measurement of the properties at the beginning of the
research. A method had been theoretically derived, however
[1]. Hence one task was the development of experimental
apparatus and procedures for measurement of the transient
thermal properties.

In order to determine the accuracy of the experi—
mental apparatus and testing procedure, a reference
material (Armco iron) was tested. It was found that the
measured k and op values of Armco iron were close to those
given in the literature [33].

It was also found that when aluminum 2024-T351 is

subjected to a high temperature in the range 350 to 425 °F,

144

 

 

145

certain changes in the microstructure take place and these
changes influence the values of k and op. The temperature
level influences k considerably and cp to a lesser degree.

The changes in the k value of aluminum 2024-T351 at
the high temperatures were found to exhibit a pattern.
This pattern begins with the k value of the specimen when
it is quickly brought to its nominal temperature (in the
range 350 to 425 °F). The thermal conductivity then
increases to some maximum value while being maintained at
the nominal temperature. The rate of increase depends on
the temperature level with the higher temperatures causing
more rapid changes in k. The thermal conductivity may
dwell at the maximum for a very short time. After which it
starts to drop until it reaches a certain value where it
remains constant with time. At this stage the aluminum
becomes over-aged.

It was found that the over-aged value of k for
aluminum 2024-T351 at room temperature as well as the high
temperatures (350 to 425 °F) is close to that obtained by
the regular annealing process although it tends to be
slightly higher.

In order to predict the k values of aluminum 2024-
T351 for different temperatures and times, a mathematical
model was developed based on the data obtained in this
study.

An example was given in which the mathematical

model was utilized to predict the k values and consequently

 

146

the temperature histories of an aluminum 2024-T351 plate
while it is undergoing transient changes in its thermal

properties.

5.1 Recommendations for further research
This study is the first step toward determining and
understanding of the transient thermal properties of

aluminum 2024-T351 and other materials that undergo changes

 

in its microstructure at elevated temperatures.

Further research is needed to refine what has
already been accomplished, and to extend the study to other
related areas such as solid state physics and material
science.

Some of the suggestions are:

1. Further study of the heat losses in the present
experimental stystem is needed. Also, aluminum
2024-T351 need to be tested at temperatures other
than the ones in this study (example: 360, 370,
380, 390, 410, 420, 430 °F). The specimens to be
used should be prepared from the same aluminum '
2024-T351 bar that was utilized in this study.

2. Develop techniques to take more rapid measurements;
for example the use of the electric heater may
provide a technique for more rapid measurements.

See [1].

147

Extend the experimental techniques to other
materials that undergo phase changes at the
elevated temperatures.

Investigate the possible relationship between the
above mentioned thermal conductivity pattern of
aluminum 2024—T351 at the elevated temperatures and
lattice dislocation, stored energy, and other

related matters of concern to the solid state

 

physicists and material scientists.
Investigate experimentally the key assumptions of
the use of the mathematical model for transient

temperature histories of the aluminum alloy.

 

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