Title Page
 Table of Contents
 List of Tables
 List of Figures
 Review of literature and experimental...
 Equipment and apparatus
 Calibration and preliminary...
 Test procedure
 Results and discussion
 Conclusions and recommendation...
 Biographical sketch

Title: Infrared scanning in conjunction with boundary element method to determine convective heat transfer coefficients
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Permanent Link: http://ufdc.ufl.edu/UF00090183/00001
 Material Information
Title: Infrared scanning in conjunction with boundary element method to determine convective heat transfer coefficients
Series Title: Infrared scanning in conjunction with boundary element method to determine convective heat transfer coefficients
Physical Description: Book
Creator: Farid, Mohsen Salah-Eldin,
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Bibliographic ID: UF00090183
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 001693354
oclc - 25222038

Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
    List of Figures
        Page vii
        Page viii
        Page ix
        Page x
        Page xi
        Page xii
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
    Review of literature and experimental techniques
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
    Equipment and apparatus
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
    Calibration and preliminary measurements
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
    Test procedure
        Page 84
        Page 85
        Page 86
        Page 87
    Results and discussion
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
    Conclusions and recommendations
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
    Biographical sketch
        Page 121
        Page 122
        Page 123
Full Text







In the Name of God, the most Gracious,
the most Beneficient

god-there Is no deity Save J m, the Cver-oLvin, the
SelfSLbssten t o Ant of A l eing.
l"either lumber overtaLes Aim, nor seep. Jis i all that
is in the heaven and all that iS on earth. Who iS there that
could intercede with him, uneSd it be by JiS leave?
JAe nowS all that lie open befre men and all that li
hidden from them, whereaS they cannot attain to aught of Hi
hnowled ae S that wich e wil A them to attain].
is eternal power overpreads the heavens and the earth,
and their upolding wearied -im not. nd he alone is truly
exalted, tremendous.
-- Holy Quran, chapter 2, verse 255

To my mother and all my brothers.


I would like to thank Dr. C.K. Hsieh, my committee

chairman, for his support and guidance throughout this work.

He taught me how to be thorough and accurate when it comes to

experimental work. I would like to thank him especially for

his great effort and patience in repetitively revising and

contributing to this manuscript.

I would like to thank Dr. Gaither, Dr. Keesling, Dr.

Gater, and Dr. Hansen for their advice and for serving on my

supervisory committee.

I would like to thank Dr. Choi for his kindness in

providing me with a boundary element computer code which I

adapted for this work.

Finally, I would like to thank Brother Kamel Mettiti for

his valuable assistance in producing this manuscript, and all

my family and friends for their great support and





ACKNOWLEDGEMENTS......................................... ii

LIST OF TABLES...................... ...................... vi

LIST OF FIGURES.................. ...................... Vii

NOMENCLATURE ... .......... ..................*** ix

ABSTRACT................................................ xi


1 INTRODUCTION ..................... ................ 1

TECHNIQUES.................................... 8

2.1 Review of Analytical Solution of
Convection Heat Transfer................. 8
2.2 Review of Techniques for Measurement of
Convective Heat Transfer Coefficients.... 16
2.3 Review of the Current State-of-the-art in
Temperature and Position Measurements
by Infrared Scanning.................... 21

3 ANALYSIS.......................................... 24

3.1 Prediction of Convective Coefficient for
Smooth Cylinders........................ 25
3.2 Radiative Heat Transfer Analysis........... 32
3.3 Boundary Element Analysis.................. 42

4 EQUIPMENT AND APPARATUS........................... 53

4.1 Description of Infrared Scanner and
Auxiliary Equipment ...................... 53
4.2 Construction of the Apparatus.............. 58


5.1 Calibration............................... ... 71



5.1.1 Position Calibration................. 71
5.1.2 Temperature Calibration................ 74
5.2 Preliminary Measurements................ 76
5.2.1 Determination of Cylinder-Surface
Diffusion Characteristics ......... 76
5.2.2 Determination of Mirror Reflectivity. 78
5.2.3 Determination of Cylinder Thermal
Conductivity....................... 79

6 TEST PROCEDURE .................................. 84

7 RESULTS AND DISCUSSION.......................... 88

7.1 Temperature Calibration Curves............. 90
7.2 Surface Temperature........................ 96
7.3 Compilation of Relevant Data............... 99
7.4 Local Convective Coefficient................ 101
7.5 Uncertainty Analysis ....................... 106


REFERENCES............................................. 115

BIOGRAPHICAL SKETCH................................... 121


Table Pace
3-1 Expressions for T and q used in equation
(3-42) for linear elements.................... 52

7-1 Electronic display unit settings................. 89

7-2 Compilation of relevant data..................... 100


Figure Page

3-1 Coordinate system for flow over a horizontal
circular cylinder............... ............... 27

3-2 Layout of cylinder and mirrors................... 33

3-3 The cylinder and its images through the mirrors... 35

3-4 Lengthwise reflection of the cylinder in the
top mirror...................................... 37

3-5 Division of regions for radiation analysis........ 38

3-6 Geometry for the shape factor between an area
element and a strip of finite-length............. 41

3-7 System for analysis in the boundary element
method........................................... 44

3-8 Alternative integration path to resolve the
singularity of q*............................... 47

3-9 Geometries related to linear elements.............. 49

4-1 AGA Thermovision System 680 camera unit........... 54

4-2 Multiple views of the function of the infrared
camera.......................................... 55

4-3 AGA Thermovision System 680 display units......... 57

4-4 Cylinder assembly....... .................... ..... 60

.4-5 Reflectivity of 3M 101-C10 Nextel velvet (black)
paint........................................... 262

4-6 Test section and position markers.................. 64

4-7 System configuration.......................... ... 65


Figures Page

4-8 Thermocouple positions used to measure wall and
ambient temperatures........................... 67

4-9 Cartridge heater circuit diagram.................. 69

5-1 Sample analog outputs for scan line running
across (a) single aluminum marker and (b)
double aluminum markers.......................... 73

5-2 Sample analog output for the calibrator............ 75

5-3 Analog outputs for the determination of the
diffusion characteristics of the cylinder....... 77

5-4 Top views of setups used for measurement of
mirror reflectivity............................ 80

5-5 Calibration curves used to determine the
reflectivity of mirror.......................... 81

6-1 Five camera positions used to scan the cylinder
for temperature measurement..................... 85

6-2 Five positions for temperature calibration......... 86

7-1 Temperature calibration curves for camera unit
in position A........... ................ ......... 91

7-2 Temperature calibration curves for camera unit
in position B.................................... 92

7-3 Temperature calibration curves for camera unit
in position C................................ .... 93

7-4 Temperature calibration curves for camera unit
in position D................................... 94

7-5 Temperature calibration curves for camera unit
in position E........................... ..... ... 95

7-6 Temperature distribution plots.................... 97

7-7 Nu/Gr"025 curves................................... 102

7-8 Analytical comparison of Nu between a smooth
cylinder at nonuniform temperature to a smooth
cylinder at uniform temperature.................. 105

7-9 Uncertainty chart................................. 109



A Surface area

B Buoyancy force

c, Specific heat at constant pressure

Fk-. Shape factor

Gr Grashof number

h Mean convection heat transfer coefficient

h, Local convection heat transfer coefficient

k Thermal conductivity

kp Thermal conductivity of the cylinder material

Nu Nusselt number

Pr Prandtl number

p Pressure

Q Heat transfer rate

q Heat flux

q~ Local heat flux

R Cylinder radius

T Temperature

T* Dimensionless temperature; see equation (3-5)

Ti Temperature of the cylinder inner boundary

Ts Bulk temperature of the fluid

T, Surface temperature of the solid wall

Twx Local wall temperature

Two Stagnation point temperature

t Time

V Velocity; V=(u,v,w)

VH Voltage drop across the cartridge heater

Vs Voltage drop across the standard resistance

p Volumetric thermal expansion coefficient

e Emissivity

v Kinematic viscosity

p Density

pj Reflectivity

a Stefan-Boltzmann constant

r Stream function

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy




August 1991

Chairperson: Chung K. Hsieh
Major Department: Mechanical Engineering

Infrared scanning was used in conjunction with the

boundary element method to determine the convective heat

transfer cofficient for free convection for a rough,

horizontal, circular cylinder in ambient air. The infrared

scanning was used to measure the surface radiosity of the

cylinder, which was in turn used to determine the surface

temperature. This temperature was then used as input in a

real-variable boundary element analysis to determine the

surface heat flux. Finally, the radiation effect was

accounted for and the net heat flux was evaluated at the

surface to determine the convective coefficient in the Nusselt


For the cylinder of roughness 3.429 micrometers with the

surface temperature varying from 328 K to 363 K in a fourth-

degree, even-function polynomial and a Grashof number of

4.05x105, the local heat transfer coefficient, expressed in

terms of the Nusselt number over the Grashof number to the

one-fourth power, is found to be 11 percent higher at the

stagnation point than that for similar cylinder with a smooth

surface. This Nusselt-to-Grashof-number ratio also increases

with angle up to 1200, the upper limit at which the boundary

layer theory is valid in the prediction of the heat transfer.

The study covers the physics of the change of the convection

over the surface of the cylinder. It also provides concrete

evidence that the surface roughness enhances the heat transfer

by free convection for a cylinder at nonuniform temperature.



Convective heat transfer coefficients are the dominant

parameters dictating the rate at which the heat can be

exchanged between a solid surface and its surrounding fluid

under specified conditions. In fact, any study of convective

heat transfer ultimately reduces to the study of the means by

which the convective heat transfer coefficient may be


In 1701, Newton [1-2] first defined the heat transfer

rate Q from the surface of a solid at uniform temperature to

a fluid by the equation

Q = hA(Tw-Ts) (1-1)

where h is the mean heat transfer coefficient from the surface

of area A to the fluid, excluding any radiation, Tw is the

surface temperature of the solid wall, and Ts is the bulk

temperature of the fluid.

It is customary in convection studies to use local heat

transfer coefficient h, defined by

h = qx (1-2)
(Tw X-Ts)

where q. is.the local heat flux from the wall to the fluid,

and Tw,z is the local wall temperature.

The convective coefficient hX can be determined

analytically or experimentally. To determine the convective

coefficient analytically, the temperature distribution in the

fluid medium surrounding the solid wall must be found. In

order to find the temperature distribution in the fluid, a

complete fluid mechanics problem in the surrounding fluid must

be solved. Then for the general case of the fluid motion in

three dimensions, a complete solution of the problem requires

the determination of three velocity components, the fluid

pressure, temperature, and density, all as functions of

position and time. There are six unknowns and they require

six equations to solve them; they include three momentum

equations, one continuity equation, one energy equation, and

one equation of state for the fluid. For the general case of

fluid properties that are temperature dependent, a complete

analysis calls for the simultaneous solution of these six

equations subject to pertinent boundary conditions. Finally,

the fluid temperature near the wall is found, from which the

temperature gradient of the fluid is evaluated to find hx [3].

It is noted that, in real flow situations, because of flow

separation and others, it is not always possible to obtain

analytical or numerical solutions. Reliance must therefore be

made heavily on experimental methods to determine the heat

transfer coefficient.


The convective heat transfer coefficient can be

determined experimentally by a variety of methods. As shown

in equation (1-2), it can be determined by simultaneously

measuring three quantities, namely, the local surface

temperature of the wall, Tw,,, the bulk temperature of the

fluid, Ts, and the heat flux from the wall to the fluid, q,.

In this dissertation, a combination of experimental and

numerical techniques will be employed for determining the

convective coefficient of a rough circular cylinder suspended

horizontally in ambient air. The convection is buoyancy

driven and the cylinder is heated internally to a steady

state. Because of the variation of the convective coefficient

on the surface of the cylinder, its temperature is nonuniform.

It is intended to use infrared scanning to measure this

temperature, which is in turn used as input in a boundary

element analysis to evaluate the surface heat flux. This flux

will then be used to determine the convective coefficient as

a function of the position on the surface. The experiment

incorporates several features as highlighted below.

It has been a common practice in heat transfer studies

that the temperature of a body in heat exchange with the

surroundings is measured by using thermocouples. In the

present work, this temperature is measured by means of

infrared scanning. Essentially a nondestructive testing

method, the infrared scanning can be used to cover a broad

field of view without making physical contact with the body


whose temperature is measured. However, because of its

sensing surface radiosity, which is a function of the surface

emissivity, temperature, and the surrounding irradiation on

the surface, the infrared scanning requires use of a lengthy

calibration procedure together with detailed analysis and

elaborate surface preparation in order to retrieve the surface

temperature. Yet the number of data that can be measured in

a temperature field is enormous; it is equal to the product of

the number of pixels along a scan line and the number of lines

that can be scanned on a surface. Better still, the scanner

is equipped with a photon detector of fast response; it can

measure the radiation at great speed which is close to real

time. With the additional advantage of the large view field

as alluded to earlier, the infrared scanning holds a distinct

edge over the thermocouples in thermometry.

The present method differs from the conventional method

in another respect, and that is related to how the heat flux

is measured at the surface. As will be reviewed in the next

chapter, in convection experiments, particularly in free

convection, the heat flux can be determined by optical means

or analogy. In these methods, it is the flux at the fluid

side that is determined. As will be discussed fully later,

the optical method is handicapped in measuring three

dimensional flows, while the analogy cannot be used to

simulate position-variant flux conditions. As for forced

convection situations, one can determine the heat flux in the


solid wall by means of heat flux gauges. Then the number of

the flux values that can be obtained in the wall is limited by

the number of the gauges that are physically located in the

wall. In practice, guard heaters are necessary in the wall to

eliminate heat leak, a further complication to the

construction of the apparatus.

In the present experiment, the surface heat flux will be

determined by using a boundary element method--a numerical

technique, not yet extensively used by the heat transfer

community, that possesses features well suited for the present

work. The fact is that, unlike in the finite difference and

finite element methods in which the entire domain and its

boundary must be discretized in order for a heat transfer

analysis, in the boundary element method only the boundary of

the domain needs discretization. As a result, the number of

the dimensions that is analyzed in the method is reduced by

one. Better yet, in the finite difference and finite element

methods, the temperature field in the solid wall must be fully

determined prior to a numerical differentiation to determine

the heat flux, a source of error unless a very small mesh size

is used; in the boundary element method, the surface heat flux

can be determined directly. This considerably reduces the

error in the computation. With the further advantage of the

need of discretizing only the boundary of the domain in the

boundary element method, a large number of nodal points can be

allocated on the surface where the temperature is supplied by


infrared scanning as described earlier. The heat flux can be

determined with accuracy in the boundary element method.

In the present work, the infrared scanning will be used

in conjunction with the boundary element method to determine

the heat transfer coefficient on a horizontal rough cylinder

in natural convection. Such an experiment is selected

primarily for the consideration that the signal-to-noise ratio

for temperature measurement is usually small for the free

convection from a horizontal cylinder. It thus provides for

an ideal experiment to test the limit of the method in the

measurement of the convective coefficient. Second, there has

been a lack of study on natural convection from a 'rough'

cylinder. While there have been numerous studies documented

for free convection, they are limited almost entirely to

'smooth' surfaces.

Yet rough surfaces are often encountered in engineering

practice. It has been established experimentally that the

roughness on a surface can enhance the convection as verified

by forced convection in cross flow over a cylinder. Such a

trend, however, has not been established for free convection.

It is also noted that the roughness effect cannot be studied

analytically; even experimentation by optical means is

handicapped because of the presence of the roughness. The

proposed method thus becomes a viable alternative to test the

roughness effect under buoyancy driven conditions. The

present research fills the need of suggesting a new method for


the measurement of the convective coefficient. It also

contributes to the knowledge related to the enhancement of the

free convection by the roughness as well.


The literature is reviewed in three subject areas,

namely, (i) the analytical solution of the convective heat

transfer coefficients, including studies of transition from

laminar to turbulent flow and flow separation around a

horizontal circular cylinder, (ii) the experimental techniques

used to determine the convective heat transfer coefficients,

and (iii) the current state-of-the-art in the measurement of

temperature and position by infrared scanning. Due to the

extensive work that has been published to date in the first

area, only those related to the free convection that is

addressed in this dissertation will be reviewed in this

chapter. Forced convection will be mentioned only insofar as

it is related to the free convection.

2.1 Review of Analytical Solution of Convection Heat

The heat transfer characteristics for flow around a

horizontal cylinder have been studied in great detail in the

literature because of its simple geometry and practical

importance. It has been well established that, for a forced

cross flow around the cylinder, the boundary layer changes

from laminar to turbulent flow through a transition region,


and the Reynolds number holds the key to the transition. For

the case of free convection, however, the development of the

boundary layer is more complicated. In the free convection,

the surface generating buoyancy is related to the angle that

is included between the tangent to the surface and the

direction of the gravitational force, and this angle is

constantly changing in the downstream direction because of

surface curvature. The buoyancy force now has two components-

-the tangential component drives the tangential motion,

whereas the normal component generates the cross flow effect.

In this instance, the development of the buoyancy driven flow

depends on the Rayleigh number, a product of the Grashof

number and Prandtl number.

For low and moderate values of the Rayleigh number, the

boundary layer solutions do not adequately describe the flow

and heat transfer characteristics because of the omission of

the surface curvature effect and the breakdown of the boundary

layer assumptions. Under these conditions, the temperature

distribution resembles what is expected for flow near a line

heat source or sink as will be explained later. In this

review, only laminar flow at large Rayleigh number will be

covered because of its possibility of solutions by the

boundary layer theory. Review of the solutions of isothermal

cylinders will be given first; the nonisothermal cases will be

covered later.


Hermann [4] is credited with pioneering the theoretical

analysis for free convection from a horizontal isothermal

cylinder by modifying Pohlhausen's similarity solution for

flow over a vertical isothermal plane wall. The boundary

layer thickness at different angles around the cylinder was

obtained by multiplying the corresponding thickness on the

vertical wall by a parameter that was a function of the angle

on the cylinder. It is noted that his solution is strictly

accurate at the angle theta equal to 900, theta zero being at

the stagnation point on the cylinder. Hermann also summarized

the results of seven papers published prior to the release of

his work.

A more accurate solution of the free convection problem

was obtained by Chiang and Kaye [5] through the use of a

Blasius series. Starting with a general analysis that was

applicable to both variable temperature and flux wall

conditions, they obtained numerical results for the isothermal

case. Their results were compared with the experimental data

of Jodlbauer [6] and the analytical results of Hermann [4] at

theta angle equal to 600 and 1200.

Saville and Churchill [7-9] used a Gortler-type series to

analyze the free convection over horizontal cylinders and

vertical axisymmetric bodies with arbitrary body contours.

Their series solutions were shown to converge faster than the

Blasius series employed in the Chiang and Kaye's analysis [5].

A Merk-type series was attempted by Lin and Chao [10] for the


solution of flows around two-dimensional planer and

axisymmetric bodies. Their analysis was applied to the study

of a number of body configurations and the results compared

well with the analytical and experimental studies published

earlier. The circular cylinder may be taken to be a special

case in their investigation.

Following a different approach of integral method, Merk

and Prins [11] derived an expression for the average Nusselt

number for the cylinder in free convection. The Nusselt

number variation with angle was also evaluated at Prandtl

number equal to 0.7, 1.0, 10, and co. Merkin [12] used a

finite-difference method to solve the convection problem,

whereas Muntasser and Mulligan [13] used a local nonsimilarity

to obtain the solutions for the cylinder at five Prandtl

numbers: 0.72, 0.733, 1.0, 5, and 10. Peripheral variation of

the local Nusselt number with angle was also compared with

those of five studies [9,13] and good agreement was found

between the local nonsimilarity and finite difference

solutions. Using these as basis for further comparison, it

was found that, of the series solutions tested, the Blasius

results are close; Merk and Prins' integral results, however,

are different [9]. On the other hand, Hermann's classical

solution yields good prediction at theta equal to 900 but

fails elsewhere over the cylinder.

A value of 103 has been suggested to be the moderate

value of the Grashof number contained in the Rayleigh number


quoted earlier in this section [9]. It has been generally

accepted that, in free convection analyses, the boundary layer

assumptions are valid for laminar flow. At moderate and low

values of the Grashof number, the boundary layer solutions may

not be accurate. The problem lies in the fact that, at large

angle theta, the boundary layer equations are inadequate to

model the flow phenomena. At large angles, flows converge to

form a rising plume at the top of the cylinder.

Theoretically, as the boundary layer thickens in the

downstream direction, the curvature, streamwise effects, and

motion pressure can no longer be neglected. The threshold

where the boundary layer solutions are applicable may be at an

angle that is no larger than 1300.

Heat transfer at moderate and low Grashof number is

important in engineering practice. Theoretical analysis in

this range, although difficult, is still possible by inclusion

of the boundary layer curvature in a complete two-dimensional

analysis. Peterka and Richardson [14] made a numerical

analysis with the lower limit of the Grashof number extended

all the way down to 103. A decrease of the Grashof number was

found to increase the heat transfer rate. Heat transfer at

extremely small Grashof number of 10-10 to 10-1 has also been

analyzed; see for example [15]. At these small Grashof

numbers, the temperature was divided into near and far fields

in which different solution techniques were applied. A single

Nusselt number relation was then derived for prediction of the


convective coefficient in close agreement with experiments

[9]. For the Grashof number in the range of 10-1 to 10-3,

numerical results have been obtained by Fujii et al. [16],

Kuehn and Goldstein [17], and Farouk and Guceri [18].

Streamline and isothermal plots produced by Kuehn and

Goldstein show clearly the change from nearly vertical flow of

plume characteristic at small Rayleigh number to boundary

layer type flow at large Rayleigh number. Fujii's analytical

results were tested at three Prandtl numbers of 0.7, 10, and

100 and validated in the Grashof number in the range of 10-4

to 104. Comparison with the experimental data of the others

in references [9] was also satisfactory in the Rayleigh range

of 10-10 to 107.

Because of the flow complication described earlier, the

convective heat transfer coefficients are usually measured

experimentally. In this effort, it is appropriate to quote

Morgan [19], who is credited for a comprehensive review of the

convective coefficient data collected from 64 references

published between 1892 and 1970. In his review, an improved

correlation was recommended in the Rayleigh number of 10-10 to

1012. The correlation is strictly useful for the estimation

of the free convection from smooth cylinders in a horizontal

position. Only qualitative results were given for the

roughness effect, and even that reference was made to forced,

convection and information was given from one bibliographic

citation [20]. Morgan's correlation was critically tested by


Fujii et al. [16] who accounted for the curvature terms in the

energy equation in the numerical solution of the free

convection with thick boundary layer. Fujii's results were

shown to be in good agreement with the experimental data in

the Rayleigh range of 10-10 to 107 noted in the preceding

paragraph. Morgan's correlation, however, somewhat

underpredicts the convective coefficient in the range of

Rayleigh number equal to 10-2 to 104 for liquids.

While extensive research has been conducted in the area

of free convection from isothermal cylinders, only a handful

of research has been directed toward studies of nonisothermal

cylinders. Koh and Price [21] used Blasius series to solve

the nonisothermal problem. Like Chiang and Kay [5], they

assumed the surface temperature variation as an even function

of the angle theta with two arbitrary constants. Special

transformations were devised by them so that the resulting

differential equations and boundary conditions were free of

these constants. The convective coefficients can then be

determined by linear combination of the numerical results

obtained by the solution of the differential equations. It

was found that the dependence of the heat transfer from a

horizontal cylinder on Prandtl number is practically the same

as that from a flat plate. However, the heat transfer itself

is greatly affected by the surface temperature variation. The

heat transfer from a nonisothermal cylinder may not be

reliably computed by using the isothermal cylinder results.


Lin and Chern [22] extended the work of Lin and Chao [10]

to study free convection from nonisothermal cylinders. A

Merk-type series was used in their analysis, and two Prandtl

numbers (0.72 and 100) were tested in their numerical

solution. The series behaved well up to an angle of 1300.

Recently, Sparrow and Kang [23] applied the surface

temperature variation to the studies of the critical radius

for cylinders in free convection. They found that using

Morgan's correlation yields a circumferentially averaged heat

transfer coefficient, which is strictly valid for a one-

dimensional analysis yet produces results that are as good as

those of a two-dimensional analysis.

While flow transition from laminar to turbulent is a

frequent occurrence for forced convection in cross flow, such

transition has rarely been encountered in free convection

because of the small cylinder size that has been normally used

in experimentation. Flow separation, however, is commonplace

in free convection and is difficult to analyze. The

separation of buoyancy driven flow from cylindrical surfaces

arises from direct interaction of attached flows, a mechanism

leading to what is so-called interaction caused separation.

Notice that, in sharp contrast to forced convection in which

the separation is caused by adverse pressure gradient, the

interaction caused separation in free convection does not

backflow or reverse but is realigned in an upward motion as

reported in [24]. Study of heat transfer in the separated


region relies heavily on experimentation. Separation angles

range from 1200 to 1500 as reported in the literature [9].

To close this section, a brief review is made of the

literature on the roughness effect on heat transfer.

Achenbach [25-26] measured the skin friction and heat transfer

for cross flow of forced air over a circular cylinder. The

appearance of roughness on the surface has shown to have

effects of increasing skin friction while enhancing heat

transfer. As expected, the critical Reynolds number where the

flow changes from laminar to turbulent is also lowered as the

roughness is increased. There has been a lack of study of the

roughness effect on free convection. Only one reference [20]

has been found, and even that the study was made for heat

transfer from bare stranded electric conductors, a

configuration bearing little resemblance to the roughness

addressed in this work.

2.2 Review of Techniques for Measurement of Convective Heat
Transfer Coefficients

Measurement of the convective heat transfer coefficient

has been a subject of intense study for decades. Special

symposiums have been organized and monographs and books have

been published to deal with the subject [27-29]. Recently,

several journals have also appeared that are dedicated to the

documentation of the experimental work relating heat transfer

to fluid flow [30-32]. Because of the large number of papers


that have been published to date on the measurement of the

convective coefficient, it will be practically impossible even

for a brief review in this dissertation. Thus in lieu of a

literature review, a review of the experimental methods will

be provided. From this review, the merit of the new method

may be better appreciated.

The measurement of the convective coefficients can be

broadly categorized into two classes, namely, direct methods

and indirect methods. Direct methods consist of those that

measure the local heat flux either at the solid side or the

fluid side. Then, by measuring the temperature difference

between the surface and the surroundings, the convective

coefficient can be determined by using Fourier's conduction

law in conjunction with Newton's cooling law. The method

appears simple and naturally suggests the use of temperature

probes to determine the heat flux. However, such probes, no

matter how small, can disturb the flow and temperature fields,

which is undesirable. Preference is thus given to optical

techniques as described below.

Established optical techniques consist of such well known

approaches as shadowgraph, schlieren, and interferometer

methods. In these methods, the index of refraction of the

fluid medium or the first and second derivatives of this index

are measured; and from these measurements, the temperature or

its gradients are inferred. It is noted that, with the

exception of the interferogram that is able to analyze


axisymmetric flow fields using Abel inversion, the optical

path lengths of these techniques are in a direction that is

perpendicular to the heat flux. As a result, they integrate

the refractive index over the path length of the light beam.

They are best suited to the measurement of one- and two-

dimensional fields in which there is no change in the index of

refraction in the path length over the test section.

In general, shadowgraph and schlieren techniques are good

for qualitative studies in which large temperature and density

gradients are present (e.g., shock waves and flame fronts),

whereas the interferometer systems (usually Mach-Zehnder) are

useful for quantitative studies where the temperature

gradients are small (e.g., free convection). Of particular

interest are the development of (i) grid-schlieren systems,

which can be used for quantitative studies by means of

isophotes, (ii) Schmidt-schlieren systems, which permit use of

shadowgraphs to determine heat fluxes, (iii) stereoscopic

schlieren systems, which can be used to study the temperature

fields in three dimensions, and (iv) shearing interferometers,

which allow for direct determination of the heat fluxes thus

facilitating the computation of the convective coefficient.

It is clear that all the optical techniques are non-

intrusive in the sense that the temperature or its derivatives

can be determined without a probe placed in the flow field.

But for these techniques to work, the fluid medium must be

transparent with respect to the light source. In using


mirrors to save on costs, the fields of view of such

techniques are still limited. Optical distortion

(astigmatism) may also pose a problem if an off-axis single-

pass system is constructed with the use of mirrors in a

schlieren system. As for the interferometers, a minimization

of refraction errors tends to increase the errors due to end

effects and vice versa. It is noted that, in the optical

methods described above, only the temperature or its gradients

at the fluid side are determined.

In contrast to the optical techniques, heat-flux gauges

measure the temperature gradients at the solid side. There

are, in general, two ways by which the solid wall of the

convection experiment can be constructed. In one design, the

heaters (usually electric strip, foil, or cartridge heaters)

are placed at a small distance from the solid/fluid interface

where the convective coefficient is measured. The heat flux

gauges are installed flush with the interface in order to

measure the heat flux. In another design, the heaters are

mounted flush with the interface; then with the help of guard

heaters placed underneath the strip heaters, the direction of

heat transfer from the strip heaters can be controlled so that

the heat flux at the interface from these heaters can be

computed directly from the electric power input.

It is noted that, depending on the use of the gauges

whether in steady state or unsteady state, modern heat flux

gauges are constructed either in the form of wafers of


thermoelectric materials or in the form of thin or thick films

of noble metals. For the wafer gauges, provisions can be made

to match the conductance of the wafer with that of the base

material and to use fine electric lead wires and insulation

materials so that the lateral heat conduction between the

wafer and the base material can be minimized. Furthermore, by

modifying the surface of the wafer to be radiatively

compatible with that of the base material, the overall heat

transfer in the base material can be controlled so that it is

practically independent of the presence of the wafer. Indeed,

modern heat flux gauges can be designed to meet specific

needs. However, using these gauges often complicates the

design and construction of the overall system. In addition,

the measurement of the heat flux is also limited to those

points where the gauges are physically located. Interpolation

and extrapolation may be necessary at other locations, and

this may lead to an error if the heating rate is changed

rapidly and nonlinearly over the surface of the wall.

Indirect methods are based on use of analogies. Of those

analogy methods developed to study convective heat transfer,

the mass transfer analogy is most popular and is convenient to

implement experimentally. The mass transfer analogy can be

applied in gas and liquid environments. It can also be used

for fluids of variable properties. The mass transfer analogy

is particularly attractive in situations where there are needs

to simulate stepwise changes in the boundary temperature and


to measure the temperature and heat flux on a rotating

boundary. However, the analogy becomes handicapped when

simulating heat flux boundary conditions.

2.3 Review of the Current State-of-the-art in Temperature
and Position Measurements by Infrared Scanning

Using infrared scanners to measure temperature is

relatively new to the heat transfer community. Indeed,

infrared scanners have been developed into commercially

available instruments only in 1960's, and those scanners in

use have been primarily for the purposes of infrared

acquisition or qualitative temperature comparison [33-34].

Their use in quantitative temperature measurements has been

largely ignored, and there is a lack of literature dealing

with the use of these instruments in precision thermometry


Major publication of infrared scanning literature dates

back to 1968 when a special issue of Applied Optics [36] was

published that was dedicated to a forum of infrared scanning

literature. Twenty-three papers were included in this issue

with topics covering diversified areas such as infrared

theory, imaging, reflectography, diagnostics, and most of all,

infrared application to nondestructive testing. Since then,

publications have been sporadically found in optical and

radiation journals except two major efforts of organization of

infrared scanning seminars, one by the industries that

manufacture infrared scanning equipment (e.g., Agema Infrared


Systems) and the other by the technical societies that have a

special interest in infrared applications (e.g., SPIE-The

International Society for Optical Engineering). The former

published Infrared Observer which contained primarily work on

qualitative use of infrared scanning, while the latter

published proceeding volumes of Thermosense and Thermal

Imaging dedicated to infrared remote sensing, diagnostics,

control, and instrumentation.

Back in the early seventies, Hsieh and Thompson [37]

pioneered the use of infrared scanning in measuring the

thermal contact resistance of magnetic substances. A

quantitative determination of the surface temperature by

infrared scanning was described in [38]. A calibration

procedure that is specially designed for adapting the infrared

scanning to precision thermometry has been given in [39].

Blackbody calibrators have been constructed and tested for

continuous temperature calibration [40]. The infrared

scanning technique has been shown to be successful in the

measurement of material thermophysical properties, among

others [37,41-42].

Of particular importance is the application of the

infrared scanning in the development of infrared computerized-

axial-tomography (IR CAT) scan, a nondestructive evaluation

method that is able to detect holes and cavities inside a body

by scanning the surface temperature of the body. Today,

regular- and irregular-shaped holes in regular-shaped bodies


can be detected [43-53]. Extension to irregular-shaped bodies

has also been tested by means of a boundary element method

[49]. It is noted that these works will be used as basis for

the development of the experimentation in this study. A

description of these methods will be given later; they are not

repeated here.


Three analyses will be presented in this chapter and they

are related to (i) the theoretical prediction of the

convective heat transfer coefficient for free convection from

smooth horizontal cylinders, (ii) the evaluation of the

radiative flux from a cylindrical surface in the presence of

mirrors, and (iii) the evaluation of the heat flux in the

solid wall by the boundary element method. Here the first

analysis is important in the establishment of a benchmark so

that the present study for rough surfaces can be compared for

roughness effect. In this regard, Koh and Price's analysis

[21] will be used because the surface temperature in this

analysis can be varied as an even function of the angle theta.

There are two adjustable constants in this function, and as

long as these constants are found by curve fitting the surface

temperature, the convective coefficient can be predicted.

Their analysis is superior to that of Lin and Chern [22] in

which the results are limited to two fixed values of the


The second analysis is motivated by the fact that, in the

present experimentation, the infrared scanner will be used to

measure temperature. The infrared scanner is cooled by liquid


nitrogen and cannot be moved freely. Mirrors are thus used

over places where the scanner cannot view them directly. For

the free convection experiment to be performed in this study,

the flow is buoyancy driven and of low speed. The radiation

effect is important and cannot be ignored. Inclusion of the

mirrors in the analysis will enhance the overall accuracy in

the study.

The third analysis involves the evaluation of the heat

flux based on the scanned surface temperature data. In this

effort, the boundary element method will be used. For the

sake of completeness in this dissertation, this method will be

described briefly and all relevant equations provided so that

the computation can be implemented by means of a computer.

3.1 Prediction of Convective Coefficient for Smooth

Free convection analysis differs from a forced convection

analysis in the inclusion of the body force term in the

momentum equation. Since the flow is buoyancy driven in the

free convection, the momentum equation is coupled with the

energy equation--a characteristic that makes the solution of

the free convection problem more difficult than a forced

convection problem.

For free convection over a horizontal cylinder as the one

of interest in this study, the buoyancy force can be divided

into two components: one in the tangential direction and the

other in the normal direction. For a heated cylinder in a


cold environment, the flow starts at the lower stagnation

point, which is located at the bottommost point on the

cylinder; see Figure 3-1. Then the streamwise direction is

always along the surface, with the point of origin being at

the stagnation point; the streamwise coordinate is marked with

x. As expected, the crosswise direction is perpendicular to

and pointed away from the surface, which is marked with y in

the figure. Because of the continuous change of the inclusion

angle that is located between the direction of the buoyancy

force and the surface tangent (angle y in figure), when the

flow moves downstream over the cylinder, the normal component

of the buoyancy force is first directed toward the surface for

8 from 0 to 900 and later away from the surface for 0 from 900

to 1800. It has been established that the angle y is taken to

be positive if it is located above the surface and negative if

located below the surface for the heated cylinder as shown in

the figure.

It is possible to use the same analysis developed in this

section to study the convection of a cold cylinder in a hot

environment. In this case, the buoyancy force is pointed

downward. The stagnation point where the flow starts is now

moved to the topmost point on the cylinder. Since the

direction of the buoyancy force is now reversed (that is

downward), the angle y is taken to be negative if it is

located above the surface and positive if located below the

surface. Clearly the signs are reversed for the cold


eT< T
*Ts < Tu,








Tw< TS

Figure 3-1.

Coordinate system for flow over a horizontal
circular cylinder.


cylinder. It is noted that, as long as these signs and

stagnation point locations are kept tracked and properly

assigned, flow formulation for one is applicable to the other.

In the present analysis, the hot cylinder in a cold

environment will be analyzed, with the understanding that the

same analysis also applies to a cold cylinder in a hot


For the present problem, the conservation equations for

steady laminar flow in a boundary layer on a horizontal smooth

cylinder can be expressed as

u v = (3-1)

Sau au a2u
u +v = v +g (T-T) sinO (3-2)
x ay2

u y_ (3-3)
x dy Pr dy2

where all notations have their usual meaning; see

NOMENCLATURE. Here, the density has been considered a

variable only in the buoyancy force term gp(T-Ts)sine, all

other properties being treated as constant. The viscous

dissipation, compression, and generation effects in the energy

equation and the normal component of the buoyancy force, and

the corresponding pressure variation across the boundary layer

in the momentum equation have all been neglected. The

boundary layer thickness is small compared to the local radius

of the curvature of the surface. This last assumption


excludes the region of large 0 where the boundary layer

thickness is large to the extent that the curvature and normal

buoyancy effects cannot be ignored in the momentum and energy


The relevant boundary conditions are

y = 0 u = v = 0 T = T,(8)
y-0o u 0 T- T

where Tw is the wall temperature, given by the relation

T*(0) T-T = l+aZ +b04 (3-5)
Two TS

and T,, is that wall temperature at the lower stagnation point

where x=0.

For the two-dimensional problem under investigation in

this study, a stream function 1 can be introduced which is

related to the velocities as

u = and v = -a (3-6)

Then the three governing equations in (3-1) through(3-3) can

be reduced to two equations. With the further introduction of

dimensionless groups

To Ts

S= Y(Gr*)1/4 (3-7)

M = (Gr*) -/4

the governing equations take the following forms

MM,7-M,,M9 = M,, + sin6 (3-8)

M,,-M = 14. (3-9)

where Gr* is the Grashof number defined as

Gr*= (Tw Ts)R3 (3-10)

Here, R is the radius of the cylinder.

Correspondingly, the boundary conditions become

7 = 0 M = Me = M7 = 0 0 = T*
?7 00 M, 0 0 0

Koh and Price [21] followed Chiang and Kaye [5] to solve

(3-8), (3-9), and (3-11) with a perturbation technique. In

this effort, M, 4, and sinO are expanded in series as

M = eFo(7) +63F1( ) +65F2() +...

0 = Go0(7) +82G(7) +84G2(7) +... (3-12)

SinO = -- 3 -5
6 120

and they are substituted into (3-8) and (3-9). Terms of like

powers of 8 are then combined and assembled in an ascending

order. Their coefficients are set to zero. Taking terms of

01 with a equal to 0, 1, ...,5 yields six ordinary

differential equations of F0, F1, F2, Go, Gi, and G2. A similar

procedure can be applied to derive the boundary conditions for

these functions; see [21].


In marked deviation from the traditional approach where

the ordinary differential equations are solved directly for M

and 0, a procedure leading to the solution that cannot be

generalized to arbitrary values of a and b appearing in the

wall temperature, Koh and Price [21] introduced


F0 = Xi

F, = aX2+X3

F2 = bX4+a2Xs+aX,+X7
Go = Y1

G, = aY2+Y3

G2 = bY+ a2Y5+aY +Y7

where X and Y are functions of q. Thus, instead of F's and

G's, Xi and Yi, where i=l, 2, ...,7, are taken to be the

unknowns and they are governed by 7 sets of ordinary

differential equations and boundary conditions. The advantage

in this transformation lies in the incorporation of the

coefficients a and b in (3-13); those newly derived equations

and conditions for X's and Y's thus become independent of

those coefficients. Linear combination can then be made as

(3-13) to find the F's and G's once X's and Y's are solved.

Finally, the F's and G's are introduced into (3-12) to find M

and 0 to complete the solution.

It is a formidable task to solve the resulting seven sets

of equations and conditions for the X's and Y's. Koh and


Price [21] were able to show that, of those derived, only one

set containing X, and Yj is nonlinear, the others are all

linear equations. A successive substitution was thus applied

together with the Runge-Kutta method to solve for X, and Y1.

They were then used as input in the other sets of equations

and conditions to find the rest X's and Y's.

The Nusselt number relations were derived as follows:

For Pr = 1.0

Nu 0.42143+(0.852a-0.01861)82+(1.0411b (3-14)

For Pr = 0.7

Nu 0.37023+(0.75688a-0.01609) 2+(0.92847b (3-15)

Extension to other Prandtl numbers is also possible as

reported in [21].

Before closing this section, it must be reiterated that

Koh and Price's analysis is strictly valid for smooth

cylinders. Since the boundary layer thickness is taken to be

thin in the analysis, the results hold up to an angle theta of

about 1200.

3.2 Radiative Heat Transfer Analysis

In the experiment, the heated cylinder was laid out as

shown in Figure 3-2, where two schematic views are given. The

profile view is given on the left where the circle represents

0.051 m

---Hated y


Layout of the cylinder and mirrors.

-0.037 m

-----0.127 m---

Figure 3-2.


the heated cylinder. Two front-surface mirrors are placed

equidistance from the cylinder and tilted at 45 degree angle.

The infrared scanner is placed on the right and the top and

bottom of the cylinder are viewed with the help of the


The right figure is a front view of the setup. All

dimensions are given in meters. Notice that the mirrors are

only a fraction of the length of the cylinder. The center

lines of the mirrors coincide with that of the cylinder; see

the vertical dash-dot line in the right figure.

The position of the mirrors in relation to the cylinder

is important. As shown in the left figure, if a tangent line

is drawn from the cylinder that is perpendicular to the upper

mirror, it intersects the mirror at a small distance from its

lower edge. Thus, from the radiation theory, the energy

leaving the surface of the cylinder is able to reflect back

and incident on the cylinder itself. Furthermore, because of

the relative position of the mirror to the cylinder, only the

energy from a portion of the cylinder may be reflected back,

the rest will be lost due to incidence directly or indirectly

(through the mirror) to the surroundings. Thus a partial view

factor exists as shown diagrammatically in Figure 3-3. In

this figure, the dashed line circle marked 1(2) represents the

mirror image of surface 1 (cylinder) in 2 (upper mirror). The

dashed straight lines are used for ray tracing. The solid

circular arc 1(2)* represents that portion of the image of 1



Figure 3-3.

\ \

.\ *



The cylinder and its images through the


in 2 that is able to see the top of the cylinder. The lower

half of the figure shows the reflection from the lower mirror.

Since the mirrors are placed symmetrical to the cylinder,

similar reflection takes place and no repetition is necessary.

To facilitate further viewing the lengthwise reflection

of the cylinder in the mirrors, Figure 3-4 is provided. Here

only the reflection in the top mirror is shown. Consideration

is given to the fact that, in the experimentation, only the

temperature at the center section of the cylinder will be

measured. Expectedly, the rectangle in dashed line is the

projection of 1(2) in Figure 3-3; similarly that rectangle

marked 1(2)* is the projection of 1(2)* in the same figure.

Because of the length of the mirror being shorter than the

cylinder, 1(2)* is not as long as the cylinder. The slanted

dashed lines help to determine this length geometrically. For

example, a ray leaving the surface after one specular

reflection from the mirror and reaching the top center of the

cylinder must originate from about 45 degree position that is

hidden behind the top of the cylinder as shown in Figure 3-3.

The heavy dots in Figures 3-3 and 3-4 help to locate the

points of origin and destination in this tracing.

It is now possible to formulate the net radiative flux

leaving the surface of the cylinder. For the sake of

analysis, the frontal half of the cylinder is divided into 5

regions marked a through e; see Figure 3-5. For regions a and

b, the radiative energy leaving the surface is incident into

Temperature to be measured

Figure 3-4.

Lengthwise reflection of the cylinder in the top mirror.



Division of regions for radiation analysis.

Figure 3-5.


all directions. However, there are energies reflected back to

these regions due to the presence of the mirror 2. The net

radiative flux for region a can then be expressed as

qa = eCa[T (0) -(1 Fa-1(2)*)TS PFa-1(2)Tb (3-16)

where all notations have their usual meaning; see

NOMENCLATURE. Here all surfaces have been assumed to be

diffuse emitters and gray in an environment at low

temperature. The cylinder is a diffuse reflector, whereas the

mirrors are specular reflectors.

The net radiative flux leaving region b can be formulated

in a similar fashion as

qb, = Eo () -(1- Fb-1(2)*)TS -pFb().T:b] (3-17)

It is noted that dividing the cylinder top into two equal

angular regions a and b is to account for the temperature

variation on the cylinder. The total angle occupied by a and

b is 45 degree according to ray tracing as discussed

previously. A finer division of these regions is certainly

possible but unnecessary in this study. It is expected that

the division will enhance the accuracy of the radiation

correction in the analysis.

Again the energy leaving region c will be incident

directly or indirectly into the surroundings. This time no

energy will be reflected back because this region is not


exposed to any mirrors. The radiative flux in this region is

thus simple, given as

qgo = ea[T (0) -T] (3-18)

The radiative fluxes in regions d and e can be formulated

in accordance with those of a and b as

qrd = ETTd() -l-Fd-1(3)*)T Fd-(3)*Ted] (3-19)


qre O[TT () (-Fe-1(3)*)T4 PF 1(3)*Te+d (3-20)

Clearly (3-18) is the limiting case of the other four

equations above. For example, in the absence of those

mirrors, all reduce to (3-18). Also when the cylinder is in

thermal equilibrium with the surroundings with perfect

mirrors, the net radiative fluxes are zero.

Shape factors in equations (3-16), (3-17), (3-19), and

(3-20) can be determined by using the general formula [54]

FA L3 = [( w 2) 1 tan1w. (sinp2-sini1) (3-21)
AA2 [L2(L2 +w2) 3 L I

All notations are defined pictorially in Figure 3-6. In

practice, dA, is taken to be the point where radiation

corrections are to be made. Values of the shape factors are

to be reported in Chapter 7.

1 X

Figure 3-6.

Geometry for the shape factor between an area
element and a strip of finite-length.

3.3 Boundary Element Analysis

Two kinds of boundary element methods have been developed

as reported in the literature--one is real-variable boundary

element method [55-56], the other is complex-variable boundary

element method [57]. While both can be used to solve the

potential problems, the former is preferred in situations

where higher order elements are necessary in the analysis.

However, the method may be handicapped in solving problems

whose domains contain corners [55-56]. The latter has been

developed only recently. It has the advantage that, in the

complex method, the integration of the potential and stream

function in the complex potential can be carried out exactly,

whereas in the real method the integration of the free space

Green's function and its normal derivative must be performed

numerically. The complex method is thus more accurate than

the real method, particularly in solving potential problems

whose domains are surrounded by piecewise linear boundaries.

Also, corners at the boundary do not pose a problem in the

complex method. However, using complex variables, the method

is naturally limited to the solution of two dimensional

problems. The method has been limited in the past to the use

of linear elements. Such a limitation has been lifted

recently. Bailey [58] has developed quadratic-element

complex-variable boundary element methods for the solution of

heat-transfer potential problems. The method has shown to be


more accurate than the comparable real variable methods but

requires considerable bookkeeping and computational efforts.

In the present work, circular cylinders are used for

experimentation. Real variable method with linear elements is

used for analysis which is selected primarily for its

simplicity. As shown in Figure 3-7, the domain for analysis,

n, is surrounded at the outside by S, and at the inside by Si,

S=S,USi. The angle 8 is measured from the bottom stagnation

point, which is consistent with the 8 position shown in Figure

3-1. Normal vectors at the boundaries are pointed outward

following the convention.

It is assumed that the solid medium is homogeneous and

isotropic. The thermal conductivity is constant. There is no

heat generation in the medium, and the system is in steady

state. Under these conditions, the heat conduction in the

medium is governed by the Laplace equation

V2T(x) = 0, x E R2 (3-22)

For the present study, the inner boundary Si is heated

electrically while the outer boundary S, is exposed to a cold

environment. The temperatures at these boundaries are

measured, thus giving the conditions

T(x) = T,(x), x E S, (3-23)
T(x) = T,(x), x e S

It is intended to use the real-variable boundary element

method to find the heat flux in the solid wall at the outer boundary.


Figure 3-7.

System for analysis in the boundary element


Several methods can be used to derive the boundary

element method. Using Green's third identity leads

immediately to the relation [59]

kT, = fTq*dS qTdS (kVT) T'dn (3-24)

where S=SUSi and
-kaT" kaT (3-25)
q* = -k q = -k (3-25

Here T* is the free-space Green's function, a solution of the

following equation

2T*+A = 0 (3-26)

where Ai represents the Dirac delta function.

For the two-dimensional domain of interest in this study,

T* can be related to the distance r between points xi and x as


T* = 1 (3-27)
27r \r)


q* k Vr.n (3-28)

Introducing (3-22) into (3-24) gives an integral equation for

the temperature at point i as

kTi = ITq'dS qT*dS (3-29)


Equation (3-29), as derived, is strictly valid for

evaluating the temperature at an interior point. In order to

develop a boundary element method for the solution of surface

heat flux, the point i must be moved to the boundary. This

relocation of point causes slight difficulty in the evaluation

of the first term on the left-hand side of (3-29), and this

difficulty is resolved by devising an alternative path for the

contour integral as shown in Figure 3-8. The integral is

expressed as

-Tq*dS = lim f T. (k -dS
o-0 (s-sj s, \ an
= T. k n idS +Tilim' kaT() dS

The limit on the right of the equation can then be

incorporated into the Ti on the left of (3-29) and a general

equation derived as

CikT = Tq'dS- qT'dS (3-31)


ci = 1 for xi E (3-32)

c, = l+lim T(E) dS for xi e S 33
S -o0 on (3-33)

Equation (3-33) can be related to the angles in Figure 3-8 by

introducing (3-26) as
(8,-82) (81-82)
Ci = 1+ for xi S (3-34)
27 27r

if principal values are used.

S -s

Figure 3-8.



Alternative integration path to resolve the
singularity of q .

For the numerical solution of equation (3-31), the

boundary is discretized into elements, S=UNj.,Sj. Following the

convention in which the contour integrals are evaluated in

this equation, nodal points are laid out so that, walking

along the boundary, the domain always lies on the left. A cut

is made at node 1; see Figure 3-7. The nodal points are

numbered so that they advance counterclockwisely along the

outer boundary and clockwisely along the inner boundary.

For the linear elements modeled in this study, nodal

points are placed at the ends of each element as shown in

Figure 3-9, where global and local coordinates of the element

are shown. A general integral is found in (3-31) as

UVdS = E UV*dS (3-35)

where U and V represent T, q, or their normal derivatives.

The variable U can be expressed parametrically as

U(t) = [Wat) ,Wb t)][Uj ,Ul]1 (3-36)

where vector notation has been used. The superscript T refers

to matrix transpose.

In the linear element, the weighting function W can be

related to the parameter t as

Wa(t) t) -15t (3-37)
Wb(t) 2

i+1; t=1

Point i located
on Si where I=j

i; t=-1

j+1; t=1

j; t=-1

( x(t)=- xH 1 + +x t
1; (xi y1) 2 2

Point i located in D',
or on Si where i7j

(change x to y for the
parametric equation of y)

Geometries related to linear elements.

Figure 3-9.


This parameter is, in turn, related to the position S and

length 1 of the element

S = 1(l+t) (3-38)

as shown in Figure 3-9. Using (3-37) and (3-38) collectively

enables U in (3-36) to be reduced to Uj at t=-l (S=0), and to

Uj+, at t=l (S=1). It also follows that

JV*dS = ][U Uj1 (3-39)


iaj = JWaV*dS
j (3-40)
Vj = JWbVdS
A system has been adopted above for the notation of V.
Notice that the first subscript of V gives the nodal point

location where the temperature or heat flux is evaluated. It

comes from the r term in the T* and q* equations in (3-27) and

(3-28). The second subscript refers to the boundary element

over which the contour integral is carried out. The

superscript designates the specific weighting function that is

contained in the integrand. For example, the superscript a

refers to Wa, and so on.

For the boundary S=SUSi discretized into N elements of

which points 1 through m are on S, and m+l through N are on Si,

"'VdS = ,m 1, ",,... V + a,

(lb a.(),+^V+2a ,N(+. (l (3-41)

[U U2 ...,Un]T

This equation can be used in (3-31) to derive a matrix

equation useful for evaluating the heat flux or temperature at

any point on the boundary as

HA = GB (3-42)


h = k(q.j-1+j ,J)-6jCk (3-43)

=b a (3-44)

a = T, (3-45)

bj = q (3-46)
A list of equations useful for evaluating T and q is provided

in Table 3-1 [49]. They are given in forms that are

convenient to compute either exactly or approximately using

Gaussian quadratures. Equations for higher order elements

have also been derived as given in reference [49].

Choi [61] developed a computer code for the solution of

the potential problem using the boundary element method. His

code will be adapted for the solution of the heat flux in this


A A Table 3-1.
Expressions for T and q used in equation (3-42) for linear

For a point i on S,

eqi =0
a(i-1) = 0
Ti = -(1.5-lnli)

b(i-1) '- (0.5-lnli1)

k = [xyj._-x.-yj -xi (yjl.-yj) +yi (xj. -x)]
.1(1 /(x-x+(Y-Yi)2 dt
-a +1
1Ti (1;t Y) n[(x -xi) 2 dt2]

4a +1

-2 (x -xi) .[(y +1 -Yi) (x -xi) (xi+1 -xi) CY -YOD]

(1 t) /[(x-xi) (y-Yy2]2 dt
For a point i in ,+1

,i,j k I -X))212

+2 (y-yi) .[(yj-1-yj) (x-x,) (xi+1-xj) (y-y)]}

S(1Tt)/[(x-xi)2+ (y-yi)2]2 dt

.y,i,j 1
Tai'j = ---8 (X-_X)/ [(X-Xi)I+(y-yi)2]. (lFt) dt

y.i.J "1


An infrared scanner will be used to measure the surface

temperature of the cylinder. A description of the infrared

scanner with its auxiliary equipment and the construction of

the test apparatus will be presented in this chapter.

4.1 Description of Infrared Scanner and Auxiliary

An AGA Thermovision System 680 was used for infrared

scanning. The system consists of two major components: an

infrared camera unit converts the invisible infrared radiation

given off by a scanned object into equivalent electronic

signals which are then transferred via interconnecting cables

to the electronic image and signal display unit where the

signals are amplified and displayed for viewing.

As shown in Figures 4-1 and 4-2, the scanning of the

object is accomplished by two rotating (vertical and

horizontal) prisms, two prism-drive motors, magnetic position-

sensing heads, and a collimating lens. A virtual image is

formed by the 3-element, silicon f/1.8 lens of the camera unit

on a plane within the first prism, which scans the object in

a vertical direction by rotating this prism about its

horizontal axis. This results in a horizontal virtual line-










4 1







(position 2)

(position )I

Figure 4-2.

Multiple views of the function of the infrared
Camera (Courtesy of AGA Thermovision 680).



image, which is then scanned horizontally by rotating the

second prism about its vertical axis. The prisms scan the

field of view at a frequency of 1600 lines per second, with

100 pixels per line, and a picture frequency of 16 per second.

The infrared detector in the camera is an indium

antimonide (InSb) photovoltaic cell sensitive to 2 to 5.6

micrometer wavelength band. The detector cell is mounted on

the surface of a 100 cm3 dewar located at the back of the

camera. Since the detector is cooled by liquid nitrogen

(-196 OC) for sensitivity, and the nitrogen is stored in the

dewar flask mounted in a vertical position, the camera unit

cannot be tilted at a large angle. Mirrors are thus used to

view positions where the camera cannot view them directly as

discussed in chapter 3 and also in the description of the

apparatus later in this chapter.

The black and white monitor shown as A in Figure 4-3 is

primarily an oscilloscope. The electronics in this

oscilloscope are synchronized with the positions of the prisms

in the scanning optics of the camera unit so that each point

in the optical field of view is transformed to a corresponding

point on the oscilloscope screen, the intensity of the

modulated beam in the oscilloscope tube being a function of

the received infrared radiation. A real-time infrared picture

in black and white can thus be viewed on the screen with the

warm parts bright and the cold ones dark.


AGA Thermovision System 680 display units.

Figure 4-3 0

c o
~i ~' '



Item B in Figure 4-3 is a ten-tone color monitor which is

also operated on the output video signal of the infrared

camera. It serves to quantify the signal into ten different

colors. This color monitor is particularly useful in

situations where there is difficulty in differentiating the

minute changes in the gray tones observed in the black-and-

white image during camera adjustment.

The electronic display also consists of a single scan-

line profile adapter (C) which is used to select the scan line

from the infrared images displayed in (A) and (B). It

presents the signal along this line in an analog format on

another cathode-ray-tube screen (D). On this screen the

height of the curve corresponds to the power of the radiation

sensed by the detector. For the present study the temperature

is to be measured quantitatively; the analog output is thus

calibrated for thermometry. Calibration of the infrared

signals will be described later.

4.2 Construction of the Apparatus

The heated cylinder is designed so that the temperature

is uniform at the inner boundary (Si in Figure 3-7) and

nonuniform at the outer boundary (Sw in the same figure). The

latter is certainly the result of the variation of the

convective coefficient on the surface. The cylinder must

therefore be designed and constructed so that it provides such

conditions at the boundaries.


The cylinder was constructed as shown in Figure 4-4. The

uniform temperature at the inner boundary was accomplished by

using a copper tubing (0.022 m OD, 0.001 m thick, and 0.33 m

long), which was selected for its high thermal conductivity.

Temperatures on the tubing were measured by using 4

thermocouples (copper-constantan, gage 27) placed 60 degree

apart as shown in the figure. Lengthwise grooves were milled

on the outside surface of the tubing so that half of the

thermocouple wire was embedded and epoxyed inside each groove.

The copper tubing was heated by a cartridge heater (Omega

Engineering, Inc., Catalog No. CIR-5118/240, black oxide

finish, maximum temperature rating: 816 OC). The surface of

the heater was wrapped around with sticky aluminum foil so

that, once the heater was slipped into the copper tubing, the

aluminum foil maintained a good thermal contact between the

tubing and the heater. To keep the heater in place while

minimizing the heat leak from the two ends of the heater,

phenolic plugs (A) were used. They were inserted at each end

of the tubing and kept in place by two set screws installed in

the radial direction. To further cut down on the axial heat

leak, another disk (B) was used. This disk was of the same

diameter as the plug but separated from it by an O-ring. A

small air gap was thus formed between the disk and the plug,

which served as excellent insulation for the plug.

The assembled cylinder was then placed at the center of

a mold for casting the cylinder wall. In the present

Hollow plaster cylinder


Cartridge heater

Copper tubing

Figure 4-4. Cylinder assembly.


experiment, plaster of Paris was used for the wall. Once the

cast was done, the cylinder was removed from the mold and

suspended in a horizontal position in an air-conditioned lab

room for two weeks. The heater was turned on at low heat to

facilitate drying. Once thoroughly dried, the cylinder was

ground to its final size of 0.051 m diameter by rubbing it

against a sand paper laying flat on a table. The phenolic

plugs and disks (0.051 m diameter) served to control the

diameter of the cylinder.

The finished cylinder was painted black using 101-C10

Nextel Velvet black paint, a 3M product that has recently been

discontinued in production. A total of seven layers of the

paint was sprayed on the surface of the cylinder in order to

prevent the radiation from the substrate from penetrating

through the surface. The radiative property of this paint was

obtained from the US National Bureau of Standards (now NIST).

As shown in Figure 4-5, the paint has a reflectivity of about

2% up to about 9 micrometer and a slightly higher value of 4%

above 10 micrometer. A peak of about 9% is found at 9.5

micrometer. The surface is thus essentially black. Painting

the surface black is important in the present experiment

because a photon detector has been used in the scanner. The

analysis of the temperature would be extremely difficult if

the surface was not painted black.

As discussed in Chapter 3, the temperature of the

cylinder was measured only over half of its circumference that

' I I I 1 I I I I I *

"3M" Nextel Velvet Black Paint on Aluminum

A at 30C (Before Heating to 200 0C)
a at 200C
0 at 30C (After Heating to 200C)





I 1 1 *

10 12

14 16

18 20

Wavelength, p (microns)

Figure 4-5. Reflectivity of 3M 101-C10 Nextel velvet (black) paint.


** r\-


was located at the middle section of its length; see Figure

4-6(a). The outer boundary of the cylinder was discretized

into 32 elements of equal arc length. The inclusion angle

between neighboring nodes was 11.250. In order to locate the

points where the temperatures were measured, markers of 0.006

m long and 0.001 m wide were cut from sticky aluminum tapes

and affixed on the cylinder as shown in Figure 4-6(b). The

tapes have a high reflectivity and they showed up on the

infrared images and analog outputs when the scan lines ran

across them. To facilitate viewing, markers corresponding to

odd nodes were placed on the left (row A) and even nodes on

the right (row B); see figure. To provide further marking,

double markers were placed at nodes 1,5,9,13,15, and 17. All

markers were placed sufficiently remote from the center

section of the cylinder so that their presence would not

affect the temperature distribution of the cylinder.

To isolate the cylinder from drafts that might affect the

ambient condition, a wooden chamber was constructed as shown

in Figure 4-7. The dimensions of the chamber measured 0.91 m

(W) X 0.91 m (L) X 2.13 m (H). Vent ports (0.76 m (W) X 0.03

(H)) were cut from the side walls (see dashed lines in the

figure) and they were covered with flaps so that their opening

to the surroundings could be adjusted. These vents are

important to avoid stratification of the ambient air in the

chamber. The chamber was open at the front and a large sheet

of plastic film was used to cover the opening. The infrared





Test section and position markers.

Figure 4-6.

Wooden chamber

System configuration.

Figure 4-7.


scanner was placed inside the plastic tent (see figure) and

the movement of the scanner would not disturb the ambient

condition of the cylinder. To ensure that the radiant energy

leaving the surface of the cylinder would not reflect back via

the chamber wall and no energies from the outside would

transmit through the plastic film, the interior surfaces of

the chamber were covered with black papers and the experiment

was performed in a dark room.

The cylinder was suspended horizontally in the chamber by

four chains; each of them was fitted with a turnbuckle to

adjust its length. There were two ears on each end of the

cylinder for hanging. In a suspended position, the cylinder

was at a height of 1.22 m above the ground. The top and

bottom of the cylinder were viewed by means of the mirrors as

described in Chapter 3. The mirror assembly shown in Figure

4-7 represents this setup. The specifications of the mirrors

have been given in detail in Figure 3-2.

In addition to the four thermocouples used to measure the

temperatures at the inner boundary of the cylinder, sixteen

thermocouples (copper-constantan, gage 27) were used to

measure the temperatures of the walls and the ambient air

temperature inside the wooden chamber. Eight of them,

numbered 9 to 16 in Figure 4-8, were glued to the walls and

the rest of them, numbered 1 to 8, were suspended in mid air

surrounding the cylinder.


2 & 10

3 & 11
I---- ------.

4 & 12

---0.91 m----

1.22 m

0.51 m




Figure 4-8.

Thermocouple positions used to measure wall and ambient

5 & 13


& 15

8 & 16

0.91 m-

2.13 m

0.81 m




To measure the heat leak from the two ends of the

cylinder, four copper-constantan thermocouples (same gage as

before) were used. They were placed on each side of the

phenolic disk (A) (refer to Figure 4-4) to measure the

temperature drop across it. With the value of the thermal

conductivity of the phenolic material, the heat leak through

the two ends of the cylinder could be estimated. This heat

leak is important to the determination of the thermal

conductivity of the cylinder material as will be discussed

later (Chapter 5).

Three Leeds & Northrup rotary thermocouple switches were

used for temperature measurements. DC power supply was used

to drive the cartridge heater. It was constructed by using a

rheostat to adjust voltage and a rectifier for changing the AC

to DC. As shown in Figure 4-9, RH represents the cartridge

heater and Rs is a 1.164 Ohm standard resistor. Measuring the

potential drops across them permits the determination of the

power consumption of the heater. In practice, all

thermocouple EMF and voltage drops were measured by using a

digital voltmeter (Hewlett-Packard 3490A Multimeter).

For the present study, the data from the analog display

(D in Figure 4-3) were recorded by using a Nikon F camera.

Kodak Plus-X 125 black-and-white films were used and the

camera was set with a shutter speed of 1/4 second and an

aperture of f/2. Extension rings were used for closeup

photography. The brightness of the grid lines on the analog

AC Input

Figure 4-9-

Cartridge heater circuit diagram.


display screen was adjusted so that the lines barely showed up

on the photographed pictures. Quantification of the analog

signals could thus be made with precision.


Before the infrared scanner can be used for measuring the

cylinder wall temperature, signal calibration and some

preliminary measurements must be made. The calibration

procedure includes position and temperature calibrations. The

preliminary measurements includes determination of (i) surface

diffusion characteristics, (ii) mirror reflectivity, and (iii)

cylinder thermal conductivity. The calibration procedure and

the preliminary measurements will be covered in the following

two sections.

5.1 Calibration

Since the infrared scanner will be used to measure the

local temperature on the cylinder wall under a steady state

condition, T,(8), both the position and the temperature must

be calibrated.

5.1.1 Position Calibration

The position calibration was accomplished in two steps.

As the first step, an engraved aluminum checker board was used

to correct for image distortion in the electronics of the

infrared scanner. This board has accurately spaced, milled


grooves cut on it and the grooves were painted black using the

same 3M paint mentioned earlier for the cylinder surface.

When the board was inserted at the focal point of the scanner,

the grid lines showed up in the infrared image due to the

contrast between the emissivities of the black paint and the

aluminum surface. The infrared image of the grid lines was

then used to correct for the horizontal and vertical image

distortions in electronics [39].

The second step for the position calibration was to

relate the nodal point locations on the cylinder surface to

the corresponding points on the infrared image. This was done

by using the aluminum markers placed on the cylinder surface

as described earlier in Chapter 4. In experimentation, the

temperature distribution at the center section of the cylinder

was scanned with the scanning direction oriented along the

axis of the cylinder. In the measurement of the temperature

of the cylinder, the position of the scan line was adjusted

with the use of the scan line selector (C in Figure 4-3) until

the scan line ran across one of those markers. Then the

displayed analog signal would appear as Figure 5-1(a). There

is a dip of the curve, a result of the low emissivity of the

marker. The elevation of the rest of the curve corresponds to

the radiosity of the cylinder that was scanned. Then by

referring to the physical position of the marker, the angular

position for that temperature corresponding to the radiosity

can be correctly determined.


Figure 5-1.



Sample analog outputs for scan line running
across (a) single aluminum marker and (b)
double aluminum markers.


An analog output for the scan line running across double

markers is shown in Figure 5-1 (b). Use of the double markers

further clarifies the position on the cylinder. As shown in

the images, the elevation of the curve in (b) is higher than

that in (a), giving indication of the higher temperature along

the scan line in (b). Quantitative determination of this

temperature is provided in the next section.

5.1.2 Temperature Calibration

For temperature calibration, a copper plate 0.045 m (W)

X 0.035 m (H) X 5 10-4 m (thickness) was used. The back of

this plate was affixed with a Minco foil heater driven by a

(Kepco Model SM 36-10 AM) DC power supply. The front of this

plate was instrumented with a copper-constantan thermocouple

placed right at the center to measure temperature. This side

of the plate was also painted black using the same 3M paint

used earlier. The copper plate had a width that was greater

than that of the line-spread function of the scanner. Its

scanned analog signal (Figure 5-2) was thus free from the

amplitude distortions described in [39]. The temperature

calibration was accomplished by varying the DC power input to

the foil heater and scanning the front surface of the

calibrator. By relating the height of the plateau in the

analog display with the true surface temperature indicated by

the thermocouple, calibration curves can be constructed as

will be shown later. It is noted that this temperature


Sample analog output for the calibrator.

Figure 5-2 0


calibration eliminates the need for emissivity correction

which would be necessary if a blackbody was used instead for


5.2 Preliminary Measurements

Before formal tests the cylinder surface diffusion

characteristic, the mirror reflectivity, and the thermal

conductivity of the cylinder must be accurately determined.

5.2.1 Determination of Cylinder-Surface Diffusion

For the present experimentation, the infrared scanner was

used to measure the temperature from a curved surface. It is

necessary to determine the largest radiation angle beyond

which the energy deviates from a diffuse radiator.

Determination of this radiation angle was accomplished by

setting the cylinder in an upright position and waiting for

the cylinder temperature to reach a steady state. Since the

boundary layer thickness over the cylinder now varies in the

axial direction in this position, the temperature of the

cylinder is uniform circumferentially. Scanning along the

cross section of the cylinder permits the determination of

this radiation angle.

Figure 5-3 provides two views of the analog output--one

for the scan line running across the midsection of the

cylinder [see (a)], the other for the scan line running across

the band of the aluminum markers [see (b)]. Then by using


Figure 5-3.



Analog outputs for the determination of the
diffusion characteristics of the cylinder.


these two views collectively together with the physical

positions of the markers, the largest angle for diffuse

radiation can be calculated to be &=460 (see Figure 4-6 for

definitions of angles). Here the angle 0 is taken to be that

point on the curve in (a) where the detected signal starts to

drop from that level at the center. Figure 4-6 (a) shows the

relation between the scan angle, 6, the radiation angle, w,

and the polar angle, 0. Geometrically relating the radiation

angle to the polar angle gives a maximum 0 angle of 450. It

is thus safe to say that, within a limit of 0 equal to 340,

the radiation from the cylinder surface is diffuse. This

latter angle corresponds to seven nodal positions (#6 through

#12 nodes) shown in Figure 4-6 (a).

5.2.2 Determination of Mirror Reflectivity

In the analysis given in section 3.2, the reflection from

the mirror was accounted for by using the reflectivity ps.

The mirror was placed at 450 along the optical path. This

reflectivity was measured by using the method described as


Usually the determination of the reflectivity is a

complicated process. It depends not only on the angle at

which the energy impinges on the surface, but also on the

direction at which the reflected energy is measured. The

reflectivity depends not only on the conditions of the surface

that reflects the energy but also on the characteristics of


the source that irradiates the surface. The latter encompass

the diffusivity and the spectral distribution of the source.

Fortunately, for the present experimentation the mirrors were

used primarily for the deflection of energy that was radiated

by the cylinder which was nearly black and diffuse. Only a

small polar angle of the cylinder was viewed through the

mirror. A simple experiment was thus devised to measure this


The infrared scanner was used to view the calibrator in

two settings. With the calibrator in a vertical position, the

scanner viewed the calibrator directly at a path length L. A

calibration curve was generated that relates the temperatures

of the calibrator with the scale divisions of the detected

energy displayed in the analog output; see Figures 5-4(a) and

5-5(a). Next, the calibrator was viewed through the mirror

which was placed at 450 and at the same optical length as

Figure 5-4(a); see Figure 5-4(b) where LI+L2=L. A second

calibration curve was generated as shown in Figure 5-5(b).

Division of the scale divisions in Figure 5-5(b) by those in

5-5(a) at the same temperature gives the reflectivity of the

mirror as ps = 0.94. The same experiment was repeated for the

other mirror. Similar result was obtained.

5.2.3 Determination of Cylinder Thermal Conductivity

The thermal conductivity of the cylinder material

(plaster of Paris) is important in the determination of the

Top view


Top view


Figure 5-4.

Top views of setups used for measurement of
mirror reflectivity.

1 L



0 2 46 8 0 2 4 6

Analog display signal division


Figure 5-5.

Analog display signal division


Calibration curves used to determine the reflectivity of mirror.


surface heat flux in the boundary element analysis.

Measurement of the thermal conductivity is described in this


In the determination of the thermal conductivity, the

cylinder stood in a vertical position. Since the boundary

layer thickness now varies in the axial direction of the

cylinder, the cylinder temperature is uniform

circumferentially. The scanner can thus be used to determine

the thermal conductivity

The analysis of the thermal conductivity is simple.

Under a steady state condition, the heat conduction through

the cylinder wall is given by the relation

dT 27L(T-T) (5-1)
0 = -kA k (5-1)
Qc dr -PA- ln (r,/r) (51)

where rw is the outer radius of the cylinder, ri is the inner

radius of the cylinder, and L is its length. In this

equation, the Qc can be determined by subtracting the heat

leak through the two ends of the cylinder (Q,) from the total

Joule heat that is supplied to the cartridge heater (Qg).

Here the QL can be determined by measuring the temperature

drop across the phenolic disk A (Figure 4-4) at both ends of

the cylinder, AT, and ATR. Then with the additional knowledge

of the thermal conductivity of the phenolic material (0.33

W/mk from [62-64]) and the geometry of the disks, the heat

leak can be evaluated as

QL = 0.103(ATL+ATR (5-2)

The heat supplied to the cartridge heater corresponds to

that electrical energy used to drive the heater (see Figure


QH = VHI = V, (5-3)

Substitution of the resistance of the standard resistor gives

a simple relation

Q VHVs (5-4)

Further introducing the geometry of the cylinder, the thermal

conductivity of the plaster can be calculated as

k = 0.411 VVs -0.103(ATL+ATR) (5-5)

In practice, the thermal conductivity of the cylinder was

determined after the temperature calibration and the surface

diffusion characterization were over. They provide input as

to where and how the scanned signal can be used to determine

the thermal conductivity.


Because of the radiating surface of the cylinder being

diffuse over a 0 angle of 340 and the dewar used for cooling

the scanner in the camera unit being installed in an upright

position, infrared scanning of the cylinder was accomplished

in five steps.

The camera position A in Figure 6-1 was used to cover

points 6 through 12 on the surface of the cylinder as shown in

Figure 4-6. The camera was then raised to position B to cover

points 14 through 17 through the reflection of the upper

mirror. It was subsequently lowered to position C to cover

points 1 through 4. Finally the camera was tilted at 22.50

from the horizontal plane of the cylinder to cover points 8

through 14 (position D) and 4 through 10 (position E).

Prior to the measurement of the cylinder surface

temperature, calibration of the temperature was also performed

in five steps with the calibration plate (section 5.1) placed

at five positions normal to the axis of the camera yet tangent

to the intended surface of the cylinder; see Figure 6-2. The

voltage to the heater was adjusted so that the plate was

heated to different temperatures that correspond to those of

the cylinder. It is expected that this multiple-position

-B-- A

Figure 6-1.

Five camera positions used to scan the
cylinder for temperature measurement.

--- -~t----




Five positions for temperature calibration.

Figure 6-2.


calibration is superior to single-position calibration in

which the calibrator is set up in a fixed position (usually

vertical) and the calibration results so obtained are applied

to retrieve the temperature on all parts of the cylinder as

will be demonstrated later.

All measurements were made inside the wooden chamber

placed in a darkened laboratory room. The analog outputs were

photographed by using a Nikon camera. Some sample outputs

have been shown in Chapter 5. Preliminary measurements

covering surface diffusion, mirror reflectivity, and thermal

conductivity tests were all made prior to the measurement of

the cylinder temperature. In the course of the experiment,

special attention was given to maintaining the cylinder in a

steady state. This usually takes about 24 hours for the

cylinder. Such a steady state, however, is unnecessary for

the calibrator plate because of its small thermal capacitance

and the uniform heat supplied by the heater.


When using the electronic display unit to control the

analog output, the controls were set as listed in Table 7-1.

Special attention was given to two settings on the unit: the

sensitivity setting and the picture black level setting. The

formal controls the temperature span that can be covered on

the analog display unit, while the latter controls the lowest

temperature level that can be reached on the screen. Once set

for a particular power input to the cartridge heater, these

controls must not be disturbed and they should be set

identically for both temperature calibration and measurement.

The power input to the cartridge heater was adjusted to

give two temperature readings at the inner boundary. Eight

readings was taken from four thermocouples shown in Figure 4-

4. Here at each station, one reading was taken before and one

after the experiment. For the low power setting, the inner

boundary was at 390.72 K, which was taken to be the average of

eight readings ranging from 390.21 K to 391.29 K. At the high

power setting, the inner boundary was at 448.34 K, which was

taken to be the average of eight readings ranging from 446.85

K to 449.79 K.

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