INFRARED SCANNING IN CONJUNCTION WITH BOUNDARY
ELEMENT METHOD TO DETERMINE CONVECTIVE
HEAT TRANSFER COEFFICIENTS
By
MOHSEN SALAHELDIN FARID
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1991
In the Name of God, the most Gracious,
the most Beneficient
godthere Is no deity Save J m, the CveroLvin, 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.
ACKNOWLEDGEMENTS
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
understanding.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS......................................... ii
LIST OF TABLES...................... ...................... vi
LIST OF FIGURES.................. ...................... Vii
NOMENCLATURE ... .......... ..................*** ix
ABSTRACT................................................ xi
CHAPTERS
1 INTRODUCTION ..................... ................ 1
2 REVIEW OF LITERATURE AND EXPERIMENTAL
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 Stateoftheart 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 CALIBRATION AND PRELIMINARY MEASUREMENTS......... 71
5.1 Calibration............................... ... 71
iv
Pare
5.1.1 Position Calibration................. 71
5.1.2 Temperature Calibration................ 74
5.2 Preliminary Measurements................ 76
5.2.1 Determination of CylinderSurface
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
8 CONCLUSIONS AND RECOMMENDATIONS................. 111
REFERENCES............................................. 115
BIOGRAPHICAL SKETCH................................... 121
LIST OF TABLES
Table Pace
A A
31 Expressions for T and q used in equation
(342) for linear elements.................... 52
71 Electronic display unit settings................. 89
72 Compilation of relevant data..................... 100
LIST OF FIGURES
Figure Page
31 Coordinate system for flow over a horizontal
circular cylinder............... ............... 27
32 Layout of cylinder and mirrors................... 33
33 The cylinder and its images through the mirrors... 35
34 Lengthwise reflection of the cylinder in the
top mirror...................................... 37
35 Division of regions for radiation analysis........ 38
36 Geometry for the shape factor between an area
element and a strip of finitelength............. 41
37 System for analysis in the boundary element
method........................................... 44
38 Alternative integration path to resolve the
singularity of q*............................... 47
39 Geometries related to linear elements.............. 49
41 AGA Thermovision System 680 camera unit........... 54
42 Multiple views of the function of the infrared
camera.......................................... 55
43 AGA Thermovision System 680 display units......... 57
44 Cylinder assembly....... .................... ..... 60
.45 Reflectivity of 3M 101C10 Nextel velvet (black)
paint........................................... 262
46 Test section and position markers.................. 64
47 System configuration.......................... ... 65
vii
Figures Page
48 Thermocouple positions used to measure wall and
ambient temperatures........................... 67
49 Cartridge heater circuit diagram.................. 69
51 Sample analog outputs for scan line running
across (a) single aluminum marker and (b)
double aluminum markers.......................... 73
52 Sample analog output for the calibrator............ 75
53 Analog outputs for the determination of the
diffusion characteristics of the cylinder....... 77
54 Top views of setups used for measurement of
mirror reflectivity............................ 80
55 Calibration curves used to determine the
reflectivity of mirror.......................... 81
61 Five camera positions used to scan the cylinder
for temperature measurement..................... 85
62 Five positions for temperature calibration......... 86
71 Temperature calibration curves for camera unit
in position A........... ................ ......... 91
72 Temperature calibration curves for camera unit
in position B.................................... 92
73 Temperature calibration curves for camera unit
in position C................................ .... 93
74 Temperature calibration curves for camera unit
in position D................................... 94
75 Temperature calibration curves for camera unit
in position E........................... ..... ... 95
76 Temperature distribution plots.................... 97
77 Nu/Gr"025 curves................................... 102
78 Analytical comparison of Nu between a smooth
cylinder at nonuniform temperature to a smooth
cylinder at uniform temperature.................. 105
79 Uncertainty chart................................. 109
viii
NOMENCLATURE
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 (35)
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 StefanBoltzmann 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
INFRARED SCANNING IN CONJUNCTION WITH BOUNDARY
ELEMENT METHOD TO DETERMINE CONVECTIVE
HEAT TRANSFER COEFFICIENTS
By
MOHSEN SALAHELDIN FARID
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
realvariable 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
number.
For the cylinder of roughness 3.429 micrometers with the
surface temperature varying from 328 K to 363 K in a fourth
degree, evenfunction 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
onefourth power, is found to be 11 percent higher at the
stagnation point than that for similar cylinder with a smooth
surface. This NusselttoGrashofnumber 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.
xii
CHAPTER 1
INTRODUCTION
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
determined.
In 1701, Newton [12] first defined the heat transfer
rate Q from the surface of a solid at uniform temperature to
a fluid by the equation
Q = hA(TwTs) (11)
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 (12)
(Tw XTs)
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.
3
The convective heat transfer coefficient can be
determined experimentally by a variety of methods. As shown
in equation (12), 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
4
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 positionvariant flux conditions. As for forced
convection situations, one can determine the heat flux in the
5
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 methoda 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
6
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 signaltonoise 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
7
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.
CHAPTER 2
REVIEW OF LITERATURE AND EXPERIMENTAL TECHNIQUES
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 stateoftheart 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
Transfer
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,
8
9
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.
10
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 [79] used a Gortlertype 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 Merktype series was attempted by Lin and Chao [10] for the
11
solution of flows around twodimensional 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
finitedifference 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
12
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 twodimensional
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 1010 to 101 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
13
convective coefficient in close agreement with experiments
[9]. For the Grashof number in the range of 101 to 103,
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 104
to 104. Comparison with the experimental data of the others
in references [9] was also satisfactory in the Rayleigh range
of 1010 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 1010 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
14
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 1010 to 107 noted in the preceding
paragraph. Morgan's correlation, however, somewhat
underpredicts the convective coefficient in the range of
Rayleigh number equal to 102 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.
15
Lin and Chern [22] extended the work of Lin and Chao [10]
to study free convection from nonisothermal cylinders. A
Merktype 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 twodimensional 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 socalled 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
16
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 [2526] 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 [2729]. Recently,
several journals have also appeared that are dedicated to the
documentation of the experimental work relating heat transfer
to fluid flow [3032]. Because of the large number of papers
17
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
18
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 MachZehnder) are
useful for quantitative studies where the temperature
gradients are small (e.g., free convection). Of particular
interest are the development of (i) gridschlieren systems,
which can be used for quantitative studies by means of
isophotes, (ii) Schmidtschlieren 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
19
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 offaxis 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, heatflux 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
20
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
21
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 Stateoftheart 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 [3334].
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
[35].
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. Twentythree 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
22
Systems) and the other by the technical societies that have a
special interest in infrared applications (e.g., SPIEThe
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 [37,4142].
Of particular importance is the application of the
infrared scanning in the development of infrared computerized
axialtomography (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 irregularshaped holes in regularshaped bodies
23
can be detected [4353]. Extension to irregularshaped 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.
CHAPTER 3
ANALYSIS
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
constants.
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
25
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
Cylinders
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 equationa 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
26
cold environment, the flow starts at the lower stagnation
point, which is located at the bottommost point on the
cylinder; see Figure 31. 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
19
eT< T
*Ts < Tu,
0=0
Bn
BtrB
I
Bt
Bn
g
Tw>Ts
Tw< TS
Figure 31.
Coordinate system for flow over a horizontal
circular cylinder.
28
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
environment.
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 = (31)
Sau au a2u
u +v = v +g (TT) sinO (32)
x ay2
u y_ (33)
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(TTs)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
29
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
equations.
The relevant boundary conditions are
y = 0 u = v = 0 T = T,(8)
(34)
y0o u 0 T T
where Tw is the wall temperature, given by the relation
T*(0) TT = l+aZ +b04 (35)
Two TS
and T,, is that wall temperature at the lower stagnation point
where x=0.
For the twodimensional problem under investigation in
this study, a stream function 1 can be introduced which is
related to the velocities as
u = and v = a (36)
Then the three governing equations in (31) through(33) can
be reduced to two equations. With the further introduction of
dimensionless groups
TTs
To Ts
S= Y(Gr*)1/4 (37)
R
M = (Gr*) /4
V
the governing equations take the following forms
MM,7M,,M9 = M,, + sin6 (38)
M,,M = 14. (39)
where Gr* is the Grashof number defined as
Gr*= (Tw Ts)R3 (310)
Here, R is the radius of the cylinder.
Correspondingly, the boundary conditions become
7 = 0 M = Me = M7 = 0 0 = T*
(311)
?7 00 M, 0 0 0
Koh and Price [21] followed Chiang and Kaye [5] to solve
(38), (39), and (311) 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) +... (312)
SinO =  3 5
6 120
and they are substituted into (38) and (39). 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].
31
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
transformations
F0 = Xi
F, = aX2+X3
F2 = bX4+a2Xs+aX,+X7
(313)
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 (313); 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
(313) 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 (312) 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
32
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 RungeKutta 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+(0.852a0.01861)82+(1.0411b (314)
1/4
Gr*'
+0.10702a20.03305a0.00011))4
For Pr = 0.7
Nu 0.37023+(0.75688a0.01609) 2+(0.92847b (315)
1/4
Gr*
+0.09471a20.02885a0.00009)O4
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 32, where two schematic views are given. The
profile view is given on the left where the circle represents
0.051 m
Ll
Hated y
m
Layout of the cylinder and mirrors.
0.037 m
0.127 m
I
Figure 32.
34
the heated cylinder. Two frontsurface 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
mirrors.
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 dashdot 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 33. 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
1(2)
(3
Figure 33.
\ \
.\ *
a+b45
1
0/
/
The cylinder and its images through the
mirrors.
36
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 34 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 33; 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 33.
The heavy dots in Figures 33 and 34 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 35. For regions a and
b, the radiative energy leaving the surface is incident into
Temperature to be measured
Figure 34.
Lengthwise reflection of the cylinder in the top mirror.
a=b
d=e
Division of regions for radiation analysis.
Figure 35.
39
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 Fa1(2)*)TS PFa1(2)Tb (316)
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 Fb1(2)*)TS pFb().T:b] (317)
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
40
exposed to any mirrors. The radiative flux in this region is
thus simple, given as
qgo = ea[T (0) T] (318)
The radiative fluxes in regions d and e can be formulated
in accordance with those of a and b as
qrd = ETTd() lFd1(3)*)T Fd(3)*Ted] (319)
and
qre O[TT () (Fe1(3)*)T4 PF 1(3)*Te+d (320)
Clearly (318) is the limiting case of the other four
equations above. For example, in the absence of those
mirrors, all reduce to (318). Also when the cylinder is in
thermal equilibrium with the surroundings with perfect
mirrors, the net radiative fluxes are zero.
Shape factors in equations (316), (317), (319), and
(320) can be determined by using the general formula [54]
FA L3 = [( w 2) 1 tan1w. (sinp2sini1) (321)
AA2 [L2(L2 +w2) 3 L I
All notations are defined pictorially in Figure 36. 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 36.
Geometry for the shape factor between an area
element and a strip of finitelength.
3.3 Boundary Element Analysis
Two kinds of boundary element methods have been developed
as reported in the literatureone is realvariable boundary
element method [5556], the other is complexvariable 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 [5556]. 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 quadraticelement
complexvariable boundary element methods for the solution of
heattransfer potential problems. The method has shown to be
43
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 37, 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
31. 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 (322)
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, (323)
T(x) = T,(x), x e S
It is intended to use the realvariable boundary element
method to find the heat flux in the solid wall at the outer boundary.
0=0
Figure 37.
System for analysis in the boundary element
method.
45
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 (324)
where S=SUSi and
kaT" kaT (325)
q* = k q = k (325
Here T* is the freespace Green's function, a solution of the
following equation
2T*+A = 0 (326)
where Ai represents the Dirac delta function.
For the twodimensional domain of interest in this study,
T* can be related to the distance r between points xi and x as
[60]
T* = 1 (327)
27r \r)
Then
q* k Vr.n (328)
27rr
Introducing (322) into (324) gives an integral equation for
the temperature at point i as
kTi = ITq'dS qT*dS (329)
46
Equation (329), 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 lefthand side of (329), and this
difficulty is resolved by devising an alternative path for the
contour integral as shown in Figure 38. The integral is
expressed as
Tq*dS = lim f T. (k dS
o0 (ssj s, \ an
(330)
= 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 (329) and a general
equation derived as
CikT = Tq'dS qT'dS (331)
where
ci = 1 for xi E (332)
c, = l+lim T(E) dS for xi e S 33
S o0 on (333)
Equation (333) can be related to the angles in Figure 38 by
introducing (326) as
(8,82) (8182)
Ci = 1+ for xi S (334)
27 27r
if principal values are used.
S s
Figure 38.
n
S
Alternative integration path to resolve the
singularity of q .
For the numerical solution of equation (331), 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 37. 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 39, where global and local coordinates of the element
are shown. A general integral is found in (331) as
n
UVdS = E UV*dS (335)
j1
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 (336)
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 (337)
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 39.
50
This parameter is, in turn, related to the position S and
length 1 of the element
S = 1(l+t) (338)
2
as shown in Figure 39. Using (337) and (338) collectively
enables U in (336) 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 (339)
where
iaj = JWaV*dS
j (340)
Vj = JWbVdS
j
A
A system has been adopted above for the notation of V.
A
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 (327) and
(328). 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 (341)
[U U2 ...,Un]T
This equation can be used in (331) to derive a matrix
equation useful for evaluating the heat flux or temperature at
any point on the boundary as
HA = GB (342)
where
h = k(q.j1+j ,J)6jCk (343)
=b a (344)
a = T, (345)
bj = q (346)
A A
A list of equations useful for evaluating T and q is provided
in Table 31 [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
work.
A A Table 31.
Expressions for T and q used in equation (342) for linear
elements.
For a point i on S,
eqi =0
a(i1) = 0
Ti = (1.5lnli)
b(i1) ' (0.5lnli1)
4r
k = [xyj._x.yj xi (yjl.yj) +yi (xj. x)]
+1
.1(1 /(xx+(YYi)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) /[(xxi) (yYy2]2 dt
For a point i in ,+1
,i,j k I X))212
+2 (yyi) .[(yj1yj) (xx,) (xi+1xj) (yy)]}
S(1Tt)/[(xxi)2+ (yyi)2]2 dt
S+1
.y,i,j 1
Tai'j = 8 (X_X)/ [(XXi)I+(yyi)2]. (lFt) dt
y.i.J "1
CHAPTER 4
EQUIPMENT AND APPARATUS
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
Equipment
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 41 and 42, the scanning of the
object is accomplished by two rotating (vertical and
horizontal) prisms, two prismdrive motors, magnetic position
sensing heads, and a collimating lens. A virtual image is
formed by the 3element, 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
53
O4
0
>4
P
0)
U
4
co
co
0
u
*rq
0
EO
4 1
Q)
U)
0
to
H
a,
00
rU4
*^l
hr
OPTICALMECHANICAL SCANNER TRANSFER OPTICS DETECTOR
HOUSING
(position 2)
(position )I
Figure 42.
Multiple views of the function of the infrared
Camera (Courtesy of AGA Thermovision 680).
CAMERA LENS
56
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 43 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 realtime infrared picture
in black and white can thus be viewed on the screen with the
warm parts bright and the cold ones dark.
57
AGA Thermovision System 680 display units.
Figure 43 0
c o
~i ~' '
i?
B~9~
58
Item B in Figure 43 is a tentone 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 blackand
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 cathoderaytube 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 37) 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.
59
The cylinder was constructed as shown in Figure 44. 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 (copperconstantan, 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. CIR5118/240, 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 Oring. 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
Thermocouples
Cartridge heater
Copper tubing
Figure 44. Cylinder assembly.
61
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 airconditioned 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 101C10
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 45, 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)
0
a
A
JO
o
I I I I
I I I
I 1 1 *
10 12
14 16
18 20
Wavelength, p (microns)
Figure 45. Reflectivity of 3M 101C10 Nextel velvet (black) paint.
I I I I
** r\
63
was located at the middle section of its length; see Figure
46(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 46(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 47. 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
0180
(a)
00
A B
(b)
Test section and position markers.
Figure 46.
Wooden chamber
System configuration.
Figure 47.
66
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
47 represents this setup. The specifications of the mirrors
have been given in detail in Figure 32.
In addition to the four thermocouples used to measure the
temperatures at the inner boundary of the cylinder, sixteen
thermocouples (copperconstantan, 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 48, were glued to the walls and
the rest of them, numbered 1 to 8, were suspended in mid air
surrounding the cylinder.
1&9
2 & 10
3 & 11
I .
4 & 12
0.91 m
1.22 m
r
0.51 m
I
(Back)
U
(Side)
Figure 48.
Thermocouple positions used to measure wall and ambient
temperatures.
5 & 13
6&14
& 15
8 & 16
0.91 m
2.13 m
0.81 m
I
W~
68
To measure the heat leak from the two ends of the
cylinder, four copperconstantan thermocouples (same gage as
before) were used. They were placed on each side of the
phenolic disk (A) (refer to Figure 44) 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 49, 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 (HewlettPackard 3490A Multimeter).
For the present study, the data from the analog display
(D in Figure 43) were recorded by using a Nikon F camera.
Kodak PlusX 125 blackandwhite 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 49
Cartridge heater circuit diagram.
70
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.
CHAPTER 5
CALIBRATION AND PRELIMINARY MEASUREMENTS
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
72
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 43) until
the scan line ran across one of those markers. Then the
displayed analog signal would appear as Figure 51(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.
73
Figure 51.
(a)
(b)
Sample analog outputs for scan line running
across (a) single aluminum marker and (b)
double aluminum markers.
74
An analog output for the scan line running across double
markers is shown in Figure 51 (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 5 104 m (thickness) was used. The back of
this plate was affixed with a Minco foil heater driven by a
(Kepco Model SM 3610 AM) DC power supply. The front of this
plate was instrumented with a copperconstantan 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 linespread function of the scanner. Its
scanned analog signal (Figure 52) 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
75
Sample analog output for the calibrator.
Figure 52 0
76
calibration eliminates the need for emissivity correction
which would be necessary if a blackbody was used instead for
calibration.
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 CylinderSurface Diffusion
Characteristics
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 53 provides two views of the analog outputone
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
77
Figure 53.
(a)
(b)
Analog outputs for the determination of the
diffusion characteristics of the cylinder.
78
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 46 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 46 (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 46 (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
follows.
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
79
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
reflectivity.
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 54(a) and
55(a). Next, the calibrator was viewed through the mirror
which was placed at 450 and at the same optical length as
Figure 54(a); see Figure 54(b) where LI+L2=L. A second
calibration curve was generated as shown in Figure 55(b).
Division of the scale divisions in Figure 55(b) by those in
55(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
(a)
Top view
w
Figure 54.
Top views of setups used for measurement of
mirror reflectivity.
1 L
L2
(b)
0 2 46 8 0 2 4 6
Analog display signal division
(a)
Figure 55.
Analog display signal division
(b)
Calibration curves used to determine the reflectivity of mirror.
82
surface heat flux in the boundary element analysis.
Measurement of the thermal conductivity is described in this
section.
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(TT) (51)
0 = kA k (51)
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 44) 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 [6264]) and the geometry of the disks, the heat
leak can be evaluated as
QL = 0.103(ATL+ATR (52)
The heat supplied to the cartridge heater corresponds to
that electrical energy used to drive the heater (see Figure
49)
QH = VHI = V, (53)
Substitution of the resistance of the standard resistor gives
a simple relation
Q VHVs (54)
1.164
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) (55)
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.
CHAPTER 6
TEST PROCEDURE
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 61 was used to cover
points 6 through 12 on the surface of the cylinder as shown in
Figure 46. 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 62. 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 multipleposition
B A
Figure 61.
Five camera positions used to scan the
cylinder for temperature measurement.
 ~t
Ic~
i
DI
Five positions for temperature calibration.
Figure 62.
87
calibration is superior to singleposition 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.
CHAPTER 7
RESULTS AND DISCUSSION
When using the electronic display unit to control the
analog output, the controls were set as listed in Table 71.
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.
