EXPERIMENTAL STUDY OF THE COMBINEDEGRAW
:   AND ROTATION EFFECT ON LAMINAR NATURAL

    

CONVECTION IN A SQUARE HORIZONTAL
" ' ENCLOSURE - _.,

..,
5.

MICHIGAN STATE UNIVERSITY
MOHSEN DIAA SHABANA
1990

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This is to certify that the

dissertation entitled

EXPERIMENTAL STUDY OF THE COMBINED GRAVITY AND
ROTATION EFFECT ON LAMINAR NATURAL CONVECTION
IN A SQUARE HORIZONTAL ENCLOSURE

presented by

Mohsen Diaa Shabana

has been accepted towards fulfillment
of the requirements for

 

Ph.D . degree in Mechanical Engineering

 

 

Major professor

(M2244!
/ ,

Date 5v/V’ 96

MS U is an Affirmative Action/Equal Opportunity {autumn 0- 12771

EXPERIMENTAL STUDY OF THE COMBINED GRAVITY AND
ROTATION EFFECT ON LAMINAR NATURAL CONVECTION
IN A SQUARE HORIZONTAL ENCLOSURE

By

Mohsen Diaa Shabana

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1990

ABSTRACT

EXPERIMENTAL STUDY OF THE COMBINED GRAVITY AND
ROTATION EFFECT ON LAMINAR NATURAL CONVECTION
IN A SQUARE HORIZONTAL ENCLOSURE

By

Mohsen Diaa Shabana

The effect of interacting gravity fields and in-plane rotational body forces on
natural convection heat transfer was experimentally studied in a parallelepiped
enclosure of square cross-section which rotates about its horizontal longitudinal axis
at uniform speeds. The working fluid was air, and buoyant flow was induced by
differential heating of two opposite longitudinal walls of length-to-height ratio ten. A
fiber optics-based laser sheet and smoke flow visualization technique was used for
qualitative assessment of the flow field. A Mach-Zehnder interferometer was used
for real time evaluation of the entire temperature field, which allowed subsequent
calculation of local and mean Nusselt numbers at the heated walls at discrete angular
positions. The angular positions selected for data reduction were 0°, 270°, 180°, and
90°, which during clockwise rotation correspond to the enclosure being: heated from
above, vertical with the hot wall on the right side, heated from below, and vertical

with the hot wall on the left side, respectively.

Rayleigh number was fixed at 1.77x105 throughout the experiments, while the
effect of rotation was studied by varying Taylor number from 105 to 107. The results
showed the existance of an intermediate transitional Taylor number (or rotational
Reynolds number) range (1.3x1065 Ta $2.8x106), which distinctly separates two
characteristically different flow regimes: at low speed rotation where gravity was
dominant, and at high speed rotation where the centrifugal force had increased

influence.

At relatively low rotation rates, gravitational buoyancy dominated the flow, and
heat transfer was shown to be highly dependent on the angular position because of
the periodic nature of the imposed gravity field. Also at low speed rotation, the
Coriolis force induced secondary cross flows, which increased mixing of the core
region fluid, and slightly enhanced the heat transfer. However, heat transfer was
shown to decrease by more than 60% over the transitional region as a result of the
growing thermal boundary layer. At relatively high rotation rates, flow was
characterized by near solid body rotation which stabilized the flow, and heat transfer
was shown to again increase steadily with rotation. An interesting result of the study
was obtained in high speed rotation, where the centrifugal buoyancy resulted in a
unique two cell flow pattern, and temperature fields which resemble the classic case

of a fluid layer heated from below under stable conditions.

In general the rotation effect was to stabilize the flow and relatively enhance the
heat transfer prior to and after the transition. Also at slow rotation the effect of
rotation was to render the Nusselt number distribution more uniform. In addition,
detailed flow and temperature field information as well as local and mean N usselt
number distributions were made available, and experimental Nusselt number

correlation was obtained at high rotation rates.

To my wife, Mona
and my children,
Maie, Rehab and Ramy

ii

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to I my major advisor, Professor
John R. Lloyd for his patient guidance, continued support and motivation, and
valuable advice throughout the course of this work, which helped me develop better
appreciation of independent scientific research. I am also grateful to Professor KT.

Yang for his assistance in providing the numerical work.
I would like to thank my committee members, Professor John McGrath,
Professor Craig Somerton, and Professor Byron Drachman for their assistance and

constructive input. Thanks is also due to Dr. Fakhri Hamady for his assistance, and
to Mr. Leonard Eisele of the machine shop for building the experimental equipment.

iii

Table of Contents

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CHAPTER - - ...................................... .PAGE
LIST OF FIGURES -- - .......................................................... vi
LIST OF TABLES ..... x
NOMENCLATURE - ............................ - ..... xi
1. INTRODUCTION -- - - - ...................... 1
1.1 Rationale for the Present Study 1

1.2 Literature Review 4

1.3 Specific Objectives of the Present Study - 10

2. MATHEMATICAL FORMULATION ..... 12
2.1 Problem Statement ...... 12

2.2 Governing Equations and Boundary Conditions - - ....................................... 14

2.3 Dimensionless Form of Governing Equations 20

3. EXPERIMENTAL APARATUS AND DATA AQUISIT ION PROCEDURES 23
3.1 Experimental Facility ....... 23
3.1.1 Test Section 23

3.1.2 Rotation and Alignment 27

3.1.3 Wall Temperature Control 29

3.1.4 Wall Temperature Monitoring -- ............... 31

3.2 Flow Visualization Technique 36
3.2.1 Fiber Optics Rotating Connection 37

3.2.2 Smoke-Laser Sheet Generation ................. 40

3.3 Mach-Zehnder Interferometer ............... 41

 

 

iv

3.3.1
3.3.2

Interferometer Components ......................................................................................
Allignment and Focusing ..........................................................................................

3.4 Experimental Procedures .......................................................................................................

3.4.1
3.4.2

 

 

Preparation ................. _ - - .............................
Photography ...............................................................................................................

4. RESULTS AND DISCUSSION - - - A - - - ..... - .......................................

 

4.1 Flow Field, Temperature Field and Nusselt Number - - - .............................

4.1.1
4.1.2
4.1.3
4.1.4
4.1.5
4.1.6
4.1.7
4.1.8
4.1.9

4.2 Effect of Rotation on the Mean Nusselt Number

 

 

At Rotational Speed 22.5 rpm - - - ...........................
At Rotational Speed 32.0 rpm
At Rotational Speed 38.0 rpm
At Rotational Speed 47.0 rpm -- --
At Rotational Speed 53.0 rpm .....
At Rotational Speed 62.0 rpm
At Rotational Speed 76.0 rpm --
At Rotational Speed 90.0 rpm -- -
At Rotational Spwd 118.0 rpm _ -- - .........................................

 

 

 

 

 

 

 

 

 

4.3 Effect of Rotation on the Overall Mean Nusselt Number ............ -- -

5. SUMMARY AND CONCLUDING REMARKS ..............
APPENDIX A INTERFEROGRAM ANALYSIS .................
APPENDIX B NUSSELT NUMBER ANALYSIS -

APPENDIX C ERROR ANALYSIS

 

 

 

 

 

APPENDIX D COMPUTER PROGRAM AND WTERFEROMETRIC MEASUREMENTS

 

LIST OF REFERENCES - -- -- ...........

 

41
43

45
45
47

106

112

116

120

123

125

127

Figure 2.1
Figure 2.2
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6
Figure 3.7

Figure 3.8
Figure 3.9

Figure 3.10

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

LIST OF FIGURES

The rotating square enclosure .................................................. 13
Fluid element in a rotational frame of reference ..................... 16
Experimental facility ................................ ~ ................................ 2 4
Schematic diagram of the test section ...................................... 25
Test section and supporting table ........... I ................................. 28
Schematic diagram of four channel rotary water coupler ......... 30
Rotary water coupler ................................................................. 32
Slipring-brush assembly ........................................................... 34
Schematic of slipring-brush assembly ..................................... 35
Schematic diagram of flow visualization system. .................... 38
Scematic diagram of lens cable assembly ................................ 39
Schematic diagram of Mach-Zehnder interferometer ............. 42
Effect of rotational speed of 22.5 rpm,

¢ = 0 deg, Ta = 6.34 x 105 ........................................................ 52
Effect of rotational speed of 22.5 rpm,

¢ = 270 deg. Ta = 6.34 x 105 .................................................... 54
Effect of rotational speed of 22.5 rpm,

41 = 180 deg, Ta = 6.34 x 105 ..................................................... 55
Mean N usselt number vs. angular position

at 22.5 rpm, Ta = 6.34 x 105, Ra = 1.84 x 105 .......................... 57
Effect of rotational speed 32 rpm,

41 = 0 deg, Ta = 1.28 x 106 ........................................................ 58
Effect of rotational speed of 32 rpm,

4» = 270 deg, Ta = 1.28 x 106 .................................................... 60
Effect of rotational speed of 32 rpm,

41 = 90 deg, Ta = 1.28 x 106 ...................................................... 62
Mean Nusselt number vs. angular speed

at 32 rpm, Ta = 1.28 x 105, Ra = 1.72 x 105 ............................ 63
Effect of rotational speed of 38 rpm,

¢=0deg,Ta= 1.81 x 106 ....................................................... 65

vi

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 4.17

Frgure 4.18

Figure 4.19

Figure 4.20

Figure 4.21

Figure 4.22

Figure 4.23

Figure 4.24

Figure 4.25

figure 4.26

Effect of rotational speed of 38 rpm,

¢=270 deg, Ta= 1.81 x 106 .................................................... 66
Effect of rotational speed of 38 rpm,

6=88 deg, Ta= 1.81 x 106 ...................................................... 68
Mean N usselt number vs. angular speed
at38rpm,Ra=1.8471105,Ta=1.81x106 ............................ 70
Effect of rotational speed of 47 rpm,

6=Odeg,ra=2.77 x106 ........................................................ 71
Effect of rotational speed of 47 rpm,

6 = 270 deg, Ta 2.77 x 106 ...................................................... 73
Effect of Rotational speed of 47 rpm,

6 = 180 deg, Ta = 2.77 x 106 ................................................... 74
Effect of rotational speed of 47 rpm,

6 = 90 deg, Ta = 2.77 x 106 ..................................................... 75
Mean Nusselt number vs. angular position

at 47 rpm, Ra = 1.71 x 105, Ta = 2.77 x 106 ............................ 76
Effect of rotational speed of 53 rpm,

6 = 0 deg, Ta = 3.52 x 106 ....................................................... 78
Effect of rotational speed of 53 rpm,

6 = 270 deg, Ta = 3.52 x 106 ................................................... 79
Effect of rotational speed of 53 rpm,

6 = 180 deg, Ta = 3.52 x 106 .......... 80
Effect of rotational speed of 53 rpm,

6 = 92 deg, Ta = 3.52 x 10‘5 ..................................................... 81
Mean N usselt number vs. angular position

at 53 rpm, Ra = 1.83 x 10’, Ta = 3.52 x 106 ............................ 83
Effect of rotational speed of 62 rpm ,

6 = 358 deg, Ta = 4.82 x 10° ................................................... 84
Effect of rotational speed of 62 rpm ,

6 = 269 rpm, Ta = 4.82 x 106 ................................................... 86
Effect of rotational speed of 62 rpm ,

6 = 180 deg, Ta = 4.82 x 106 .................................................... 87
Effect of rotational speed of 62 rpm ,

6 = 87 deg, Ta = 4.82 x 106 ..................................................... 88

vii

Figure 4.27
Figure 4.28
Figure 4.29
Figure 4.30
Figure 4.31
Figure 4.32
Figure 4.33
Figure 4.34
Figure 4.35
Figure 4.36
Figure 4.37
Figure 4.38
Figure 4.39
Figure 4.40
Figure 4.41
Figure 4.42

Figure 4.43

Mean Nusselt number vs. angular position

ar 62 rpm, Ra = 1.7 x 105, Ta = 4.82 x 106 .............................. 89

Effect of rotational speed of 76 rpm ,

6 = 358 deg, Ta = 7.24 x 106 ................................................... 91

Effect of rotational speed of 76 rpm ,

6 = 268 deg, Ta = 7.24 x 106 ................................................... 92

Effect of rotational speed of 76 rpm ,

6 = 178 deg, Ta = 7.24 x 106 ................................................... 93

Effect of rotational speed of 76 rpm ,

6 = 90 deg, Ta = 7.24 x 106 ..................................................... 94

Mean N usselt number vs. angular position

at76rpm,Ra=1.7x105,Ta=7.24x106 ............................ 96

Effect of rotational speed of 90 rpm ,

6=3deg,ra=1.02x107 ...................................................... 97

Effect of rotational speed of 90 rpm ,

6 = 270 deg , Ta = 1.02 x 107 .................................................. 98

Effect of rotational speed of 90 rpm ,

6= 88 deg,Ta=1.02 x107 .................................................... 99

Mean Nusselt number vs. angular position

at 90 rpm , Ra = 1.82 x 105,121 =1.02 x107 ...............

Effect of rotational speed of 118 rpm,

6=ldeg,Ta=1.75x107 ............................................

Effect of rotational speed of 118 rpm ,

6 =268 deg , Ta = 1.75 x 107 .........................................

Effect of rotational speed of 118 rpm ,

¢=9Odeg,Ta=1.75x107 ..........................................

Mean Nusselt number vs. angular position

at 118 rpm,Ra= 1.78 x105,Ta=1.75x107 ..............

Mean Nusselt number vs. Ra x Ta

at angular position 6 = 270 deg .....................................

Mean Nusselt number vs. Ra x Ta

at angular position (I) = 180 deg .....................................

Mean Nusselt number vs. Ra x Ta

at angular position 6 = 90 deg .......................................

viii

........... 100

.......... 102

.......... 103

.......... 104

.......... 105

107

109

110

Figure 4.44

Figure 4.45
Figure 4.46

Figure A.1

Mean Nusselt number vs. Ra x Ta

at angular position 6 = 0 deg .................................................. .111
Overall mean Nusselt number vs. Ra x Ta ............................ 113
Correlation of the overall mean

Nusselt number at high rotation rates .................................... 114
Diagram of Temperature vs. Location ..................................... 122

ix

LIST OF TABLES

Table 1 Interferometric Measurements ................................................ 135

H"m§§§w‘mfigrwmwmou

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NOMENCLATURE

Acceleration vector, m/sec2
Specific heat, J/kg . K
Gravitational acceleration, m/sec2
Convective heat transfer coefficient, W/m2 . K
Enclosure height, m

T‘herma conductivity, W/m . K
Enclosure length, m

Nusselt number

Pressure, N/m2

Prandtl number

Position vector

Universal gas constant, J/kg . K
Rayleigh number

Rotational Rayleigh number
Reynolds number

Volumetric heat generation, W/m3
Time, sec.

Temperature, deg. K

Taylor number

Velocity vector, m/sec

Greek Letters

<‘pc-eag

Thermal diffusivity, m2/sec

Coefficient of volumetric therma expansion, 1/K
Temperature difi'erence

Dynamic viscosity, kg/m . sec

Kimenatic viscisity, mz/sec

xi

p Density, kg/m3

6 Angle between cold wall and horizontal, deg.
co Angular speed, rad/sec
Q Rotational speed, rpm
Subscripts

C Cold wall

H Hot wall

p at constant pressure

in Inertial

ro Rotational

0 Reference condition
Superscripts

* Non-dimensional

xii

CHAPTER 1

INTRODUCTION

1.1 Rationale for the Present Study

Energy transfer by natural convection in rotating cavities is a phenomenon of
significant impact on the design of optimal cooling systems, and accurate temperature
control devices for a wide variety of rotating machinery in both industrial and
laboratory applications, such as rotating heat pipes, thermosyphones, and heat
exchangers. Cooling the rotating field windings of high capacity electric generators,
the large blades of high performance gas turbines, and the rotor of a super
conducting generator using liquid helium are also among the numerous examples that
involve the flow of coolant in rotating channels of different geometries and
orientations relative to the axis of rotation. Natural convection also plays an
important role in crucial manufacturing processes such as crystal growth by chemical
vapor transport, and spin casting of molten metals. Another class of natural
convection problems in rotating systems invlove situations where gravity effects are
relatively or completely negligible as in outer space applications. Examples of such
applications are heat transfer in satellites in orbit where gravity is small compared to
the rotational forces, and deliberate heating or cooling of fuel storage tanks aboard

space vehicles for pressure control.

Natural convection currents are the result of buoyancy forces which are
normally induced by body force fields acting on a fluid which posseses density
gradients. In most enclosure situations the density gradients are the result of thermal
gradients due to temperature differences between the enclosure walls and/or internal
heat generation. The process results in energy transfer at the boundaries of the
enclosure. In most stationary enclosures, gravity induces an upward buoyancy field
in the thin fluid layer adjacent to the heated walls, which in general drives the
contained fluid into primary cellular motion, and secondary boundary layer flows that
either accelerate or retard the main flow and therefore increase or decrease the heat
transfer depending on the orientation of the enclosure walls relative to the gravity

VCCIOI'.

In rotating enclosures however, the centrifugal and Coriolis forces of rotation
interact with gravity to produce more complex regimes of buoyant fluid flow. A fluid
element inside a rotating cavity is acted upon by three force fields, the gravitational
field which exerts a periodic force except when the fluid element is rotating about a
vertical axis, a uniform centrifugal field which acts in the radial direction, and a
Coriolis force which drives the fluid element toward the center of curvature of its
path. Note that the strength of the rotational forces howevere is a function of the
radial distance from the center. Each one of these three forces combined with the
density gradients within the fluid creates its own buoyancy field, however it is the
combined complex buoyancy field that can have profound effects on the heat
transfer process inside the enclosure. Note that while the magnitude of the gravity
force is constant the other two forces increase with the speed of rotation, thereby
creating domains of different flow and heat transfer characteristics based on the
influence of the most dominant force (or combination of forces) in a particular

domain.

As will be shown later in the literature review, even though much is known
regarding fluid flow in simple rotating enclosures, there is not a large body of
experimental data relevant to the effect of rotation on heat transfer, and most of the
existing literature in this regard does not acknowledge the effect of an interacting
gravity field. This indicates that much remains unknown in the more complex
rotating enclosure systems with gravity effects, where basic experiments are scarce
and highly required for the understanding of the fundamental aspects of heat transfer
in crucial applications such as those mentioned in the beginning of this section. A
review of the technical literature relevant to convection heat transfer in rotating
enclosures is presented in the following section to provide more insight into the

physical phenomena involved, and more appreciation of the subject.

1.2 Literature Review

Rotating fluid flows and the coupled phenomena of heat and mass tranfer in
rotating systems have been the focus of increased research attention over the past
three decades. Extensive studies in this area range in interest from understanding the
fundamental aspects of the underlying physical phenomena in simple configurations,
to optimization of the transport process in more complex configurations which arise
in industrial applications. Basic studies are generally aimed at understanding the
effect of rotation on, (a) the stability of heated fluid layers, and (b) how it is
manifested in the transfer of heat at the solid boundaries in contact with the rotating
fluid. The solid boundaries fall under one of three categories, (i) a heated extended
horizontal surface that spins about a vertical axis, (ii) an open channel or annulus
which rotates about its longitudinal axis, with either axial or radial forced flow, and
uniform heating, or (iii) a closed container that rotates about a geometric or parallel

axis with different walls at different temperatures.

A wealth of literature is also found for a broad spectrum of industrial
applications that can be divided into two main groups. The first group involves
cooling of rotating assemblies of high performance power generation devices such as
large gas turbine blades, impelers and rotors, field windings of large generators,
motors and centrifugal pumps etc. (Metzger and Afgan 1984, Yang 1987). The
second group involves such manufacturing processes as, crystal growth from a melt
(Langlois 1984,1985) or by chemical vapor deposition (Yang et al. 1988), and pulp
flow in paper refiners (Kheshgi and Scriven 1985). Studies are also found in relevant
applications such as heat transfer augmentation devices (Bergles 1968), rotating heat

pipes (Marto 1984) and thermosyphones (El-Masri 1984, Eckels et al. 1987).

A literature review on the subject of fluid flow and heat transfer in open and
closed rotating enclosures is presented next. This review is in no way a complete
survey of the all related studies, however it is an attempt to cover the most important

studies that relate to the fundamental aspects of the present study.

Chandrasekhar (1961) presented one of the earliest in depth anlalyses of the
stability of the classical case of a fluid layer above a horizontal heated surface which
is known as the Benard-Rayleigh problem. Since then several comprehensuve studies
(Veronis 1968, Torrest and Hudson 1974, Catton 1978, Yang et al. 1988) have
treated the same problem and showed that cellular convection ensues in the fluid
layer when a critical Rayleigh number Rae is reached or when the buoyancy force
created by gravity in the vertical direction overcomes the viscous stability. When the
fluid layer is rotated it is known that the centrifugal and Coriolis accelerations of
rotation will create additional buoyancy fields which greatly affect stability and the
characteristics of the convection flows in the layer. Rossby (1968) studied the effect
of rotation on thin layers of water and mercury and found that the heat transfer
increased in water as rotation increased up to a certain value of Taylor number, and
that it decreased in mercury with increasing Taylor number. He also concluded that
linear theory was insufficient to fully describe the stability characteristics of the
layers. The same problem has been extensively studied by others like Gillman
(1973), Abell and Hudson (1975), Hunter and Riahi (1975), Clever and Busse (1979),
and Homsy and Hudson (1971 & 1972) who showed that the centrifugal force

srabilizes the horizontal layer everywhere except near vertical boundaries.

Biihler and Oertel (1982) treated the cellular convection problem in a
rectangular box that rotates about a vertical geometrical axis, both theoretically
through the use of linear stability theory, and experimentally by using interferometry
to quantitatively interpret the convection flow. In their experimental investigation

they were able to study the influence of the centrifugal and Coriolis forces

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approximately independently of each other through the use of a high Prandtl number
test fluid (silicon oil) and a low Prandtl number test fluid (nitrogen), respectively.
The experiments in this study were done up to a rotation rate of 150 rpm, however
the centrifugal effects were neglected in the physical model which showed good
agreement with the experiment for the case of low Prandtl number only. The results
showed that the side walls have a stabilizing effect because of the additional viscous
shear, and that rotattion characterised by Taylor number increased the critical
Rayleigh number further and delayed the oncet of cellular convection which indicate

that Coriolis force stabilized the flow.

Emphasis in the studies presented so far has been on the effect of rotation on
flow stability and flow patterns in rotating horizontal layers heated from below where
the gravity vector is orthogonal to the plane of the rotational forces, with little or no
emphasis on the temperature field or the heat transfer. However it is found that fluid
flow and the related problem of convection heat transfer have been more widely
studied in rotating open cyliders, rectangular ducts and annuli due to the profound
impact on the design of efficient coolant circulation channels in high capacity rotating
machinery. The problem in this case is complex and more difficult to analyze
because of the superimposed primary forced flow field that can be either axial or
radially inward or outward. Morris (1981) has presented a rather comprehensive
review of this subject for channels rotating about a parallel or perpendicular axis of

rotation.

Experimental and theoretical studies of convection heat transfer in a rotating
radial circular tube which is pertinent to the coolant channels of gas turbine rotor
blades were carried out by Schmidt (1951), Barua (1955), Mori and Nakayama
(1968), Mori et al. (1971), Vidyanidhi et al. (1977), Morris and Ayhan (1979),
Firouzian et al. (1985, 1986), and most recently by Northrop and Owen (1988). It

was shown that the Coriolis force induces secondary flows both in the direction of

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the flow and in the perpendicular direction which creates additional mixing and
increases the heat transfer, while the effect of the centrifugal force as represented by
the rotational Rayleigh number was shown to be dependent on the direction of the
primary flow as it caused an increase in Nusselt number for outward flow and a
decrease for inward flow (Morris and Ayhan 1984). Siegel (1985) also included the
effect of the centrifugal buoyancy in a series expansion analysis for the case of low
Taylor number. His results however correlated reasonably with the experimental data

of Monis and Ayhan (1979).

The emphasis in heat transfer in coolant channels also motivated studies of other
configurations of rotation. The case of fully developed laminar and turbulent forced
flow in a horizontal cylinderical tube rotating about its geometrical axis has been
experimentally studied by Cannon and Kays (1969). They found that rotation delayed
the transition to turbulent flow to higher Reynolds numbers. The corresponding heat

transfer problem was studied by Hudson (1970).

Another rotation configuration which has been extensively investigated is that of
a tube rotating about a parallel axis. Mori and Nakayama (1967), Nakayama (1968),
and Sakamoto and Fukui (1970) studied both laminar and turbulent forced convective
heat transfer in a pipe rotating around a parallel axis. The studies showed that
influential secondary flows cause remarkable skewness in the axial velocity and
temperature profiles. Humphery et al. (1967) investigated the convection heat transfer
at the entrance region of the corresponding problems. Woods and Morris (1980)
have numerically treated the fully developed flows in the parallel revolving horizontal
cylinder with axial flow over a broad range of Rayleigh, Reynolds, and Prandtl
numbers. Their correlation agreed with the results of the asymptptic analysis of
Siegwarth et al. (1969) for high Prandtl number fluids. An experimental study by
Stephenson (1984) on turbulent flow and heat transfer in high rotational speed

(parallel mode) circular duct showed that rotational buoyancy effects are not

generally significant, but may become important in fully developed flow at high

rotation rates and low flow rates.

Of equal importance to the parallel rotating cylinder is the parallel rotating
rectangular duct. Flow and heat transfer studies in the rectangular cross-sectioned
ducts are few despite the fact that it can be found in a broad variety of cooling
applications for the simplicity in manufacturing rectangular channels in long power
transmission shafts. Numerical studies by Neti et al. (1985) and Levy et al. (1986)
demonstrated the significant effect of the buoyant secondary flow induced by the

Coriolis force on the heat transfer.

Forced fluid flow in annuli formed between two rotating cylinders and the
associated convective heat transfer have also been the subject of several studies.
Diprirrra and Swinney (1981), Kataoka et al. (1984), and most recently Yong and
Minkowycz (1989) have treated the flow stability and heat and mass transfer in
concentric rotating cylinders. Kuzay and Scott (1977), Kataoka and Deguchi (1987),
Ball et al. (1989) and others have treated vertical annuli when the inner cylinder is
rotating and the outer is stationary. The more complex problem of two eccentric

rotating cylinders has been treated analytically by Singh and Rajvanshi (1982).

Another class of fluid flow and convection heat transfer is that of closed rotating
enclosures. this class of problems is important for understanding the local and overall
heat flux distribution and the associated thermal stresses in internal cavities of
rotating machinery. The current investigation is categorized under this class of
rotating enclosures in which buoyant flows are driven by the density gradient and the
gravity field as well as the rotational body forces that affect every differential volume
element of the working fluid. While the centrigugal and Coriolis forces of rotation act
uniformly in the radial direction perpendicular to the axis of rotation, the effect of
gravity depends entirely on the angle of inclination between the axis of rotation and

the gravity vector.

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Numerical studies by de Vahl Davis et al. (1984) and Randriamampianina et al.
(1987) on the natural convection heat transfer and fluid flow in a differentially heated
annular cavity of square cross-section formed between two vertical concentric
cylinders and two horizontal planes. The studies showed that the heat transfer
initially decreased with increasing rotation, and started to increase at sufficiently high
rotation rates, where it became independent of the Rayleigh number as the
centrifugally driven motion became dominant. Sobel et al. (1986) have
experimentally studied the heat transfer and flow patterns of rapidly spinning systems
in a rectangular annulus similar to that of de Vahl Davis et al. (1984). The study
showed the existance of weak radially inward plumes that rise from the heated
external cylinder, which reflects the effect of centrifugal buoyancy at high rotation
speeds. They were also able to conclude that spinning resulted in two dimensional
motion, and that the heat transfer coefficient is independent of the heater lengh scale
and fluid viscosity. The radial buoyant flow structures dealt with in these studies are
almost independent of the garvity effect since the axis of rotation is vertical.
However more complex flow regimes can result from inclination of the axis of
rotation relative to the gravity vector because of the interaction between the

gravitational and rotational buoyancy fields.

An experimental as well as numerical investigation has been recently carried out
by Hamady (1987) and Hamady et a1. (1990) on the effect of low speed rotation on
convective heat transfer in a square enclosure rotating about its horizontal axis,
similar to that used in the present study. The study showed that low speed rotation
enhances heat transfer due to the secondary radial flows induced by the Coriolis
force. They also showed that the numerical calculations can accurately predict the
experimental results within their range of Rayleigh and Taylor numbers, and were
able to correlate the Nusselt number results with Ra and Ta. Yang et al. (1988b)

performed a similar numerical study in a rotating horizontal cylider heated

f") _

a.
pa

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10

differentially at both ends. They too concluded that at slow rotation the Coriolis force
has a significant influence on the temperature field as it renders the spatial heat flux
distribution more uniform. They also noted that increasing rotation results in more

uniform and weaker buoyancy field.

The brief review presented above shows that knowledge of heat transfer in
rotating enclosures is rather limited, especially when the effect of the gravity field is
significant, and that more related studies are needed over a broad range of the
controling parameters in order to better asses the influence of rotation and understand

the underlying physical phenomena.

1.3 Specific Objectives of the Present Study

In light of the lack of knowledge addressed above in regard to the combined
effect of gravity and roration on fluid flow and heat transfer in enclosures, the present
study is aimed at experimentally investigating such effects in a simple differentially
heated, air filled square enclosure that rotates about its geometrical axis, which is
oriented in a direction orthogonal to the gravity vector. The benefit from the study is
twofold, first to gain physical understanding both qualitatively and quantitatively of
the effect of the interaction of gravitational and rotational fields on fluid flow and
heat transfer in rotating enclosures, second to validate a numerical model developed
by Yang et a1. (1988), which once validated can be utilized to predict the enclosure
flow and heat transfer in experimentally hard to simulate situations, such as more
complex enclosure configurations and boundary conditions, tilted orientations relative

to the gravity vector, and extreme values of the controlling parameters.

The present study is performed in a square enclosure with two opposite
isothermal walls at different temperatures and two adiabatic walls. Such configuration

is chosen to allow direct comparison of the present results with those that are now

11

known for slowly rotating enclosures (Hamady et al. 1990). The Rayleigh number
which represents a measure of the gravity force relative to the viscous shear forces is
held constant, while the rotational Rayleigh and the Taylor numbers which represent
the Coriolis and centrifugal forces, respectively, are varied as a result of changing the
speed of rotation. The speed of rotation ranges from relatively low rates, where the
gravitational bouyancy is expected to dominate the flow. field, to relatively high rates,

where centrifugal buoyancy is expected to become more influential.

Two types of experiments are performed. In the first experiment flow
visualization using smoke and laser sheet is utilized to qualitatively reveal the nature
of the different flow regimes and their subsequent impact on the temperature field. In
the second experiment a Mach-Zehnder interferometer is employed to show the entire
temperature field and quantify the local Nusselt number distribution at the isothermal
walls at angular positions of 0° (which means at the intant the enclosure reaches the
heated from above position), 270° (vertical position), 180° (heated from below
position), and 90° (the opposite vertical position). Finally the mean and overall

Nusselt numbers are calculated and correlated with the controling parameters.

CHAPTER 2

MATHEMATICAL FORMULATION

2.1 Problem Statement

The present study deals with natural convection in a rotating enclosure. As
shown in figure 2.1 the enclosure is a parallelepiped cavity, with a square cross-
section of side H, i.e. aspect ratio = 1, and of length L. In the enclosure the two
opposite longitudinal walls are isothermal while the other four walls are adiabatic.
One of the isothermal walls is maintained at a low temperature, TC, while the other is
maintained at a relatively higher tempetrature, T H, and they are denoted the cold and
hot walls, respectively. The enclosure is filled with air and rotates with a uniform
angular velocity, 9, about its longitudinal axis, which is oriented perpendicular to the
gravity vector, g. Note that bold face symbols are used to represent vector fields

throughout this chapter.

An angle of rotation, 6, is defined as the angle between the horizontal plane
passing through the axis of the enclosure, and the cold wall. In this notation an angle
of rotation 6 = 0° means that the enclosure is heated from above, 6 = 90° or 270° is

heated from the side, and 6 = 180° is heated from below.

Buoyancy flows in this situation are induced by the combined effect of density
gradients which result from the difference in temperature between the two isothermal
walls, and the driving body forces that consist of a periodic gravity force and

12

l3

Adiabatic Wall
Hot Wall (T1 )

   
     

 

 

Adiabatic Wall

 

 

Cold Wall qr)

Figure 2.1 The Rotating Square Enclosure

14

uniform centrifugal and coriolis forces. External forces such as inertial and viscous
forces also help shape the flow patterns either by retarding or assisting the motion of

the fluid.

2.2 Governing Equations and Boundary Conditions .

The differential equations governing natural convection flows in enclosures are
the mass, momentum, and energy conservation equations. For a laminar, variable
property, compressible Newtonian fluid these governing equations can be written in

vector form as follows:

Mass Conservation

-aa%+V-(pv)=0 (2-1)

Momentum Conservation

_D_v

th=-Vp+uV2v+-I;—V(V°v)+pg (2.2)

Energy Conservation

pc,%f-=v-kvr+s (2.3)

where, p, 1.1, cp, and k are the density, the dynamic viscosity, the specific heat at
constant pressure, and the thermal conductivity of the fluid respectively. Also, t is the
time, p is the static pressure, and T is the temperature. v and g represent the velocity

and the gravity vectors, respectively, and 8 represents a volumetric heat source. V

15

and V2 are the gradient and laplacian operators, respectively, while mm is the
substantial derivative (3 / a: + v - V). It must also be noted that the viscous
dissipation and pressure work terms in the energy equation are neglected because of

the small velocities associated with free convection, as shown by Gebhart (1971).

So far the velocity and acceleration components in the governing equations have
been described in an inertial coordinate system, i.e. relative to a fixed point in space.
However, the equations are often described in a rotating coordinate system in order

to simplify the mathematical analysis.

The velocity and acceleration vectors of a typical fluid element, A, shown in
figure 2.2, can be described by defining the position vector, r, relative to the rotating

coordinate system, xyz, as follows:
r=xi+yj+zk (2.4)

where i, j and k are the unit vectors in the x, y and 2 respectively. The velovity, vm,

and the acceleration, am, can now be defined in the rotating coordinate system as

 

follows:
dr
v,.,- m] (2.5)
dvro d2?
3, = = _ (2.6
o [ d: ]I‘O [dt‘z ]ro )

As shown by Potter and Foss (1975), the velocity and acceleration vectors of

fluid element A can also be related to the inertial coordinate system, XYZ, as follows:

I6

Path Of A \’

 

 

Figure 2.2 Fluid Element in a Rotational

Frame of Reference

17

6:2!- =i£ +(t)><r
”' 4’6. at",

= vm + orxr

and,

 

 

+ mxvin
70

 

 

II
E

Blat- (vm + mxr )] + cux( vm + coxr)
r0

 

Dvm
= + 2 (nxvm + (ox((oxr)
D: ,0

where,

to=-f2k

(2.7)

(2.8)

(2.9)

is the angular velocity vector of the rotating xyz-reference frame, and the subscripts,

ro, and, in, refer to the r0tating and the inertial coordinate systems, respectively.

Substituting equations (2.7) and (2.8) into equations (2.1) and (2.2), the

continuity and momentum equations can be described in the rotating frame of

reference as follow:

2P. . =
a: +V (pvm) 0

Dvm
. p D:

 

+P%V(V'vm)+pg

+ 2p (uxvm + p mx(co><r) = - Vp + pv V2 vm

(2.10)

(2.11)

18

where, v is the kinamatic viscocity, and
V-v=V-(vm+mxr)=V-vm

V2 v = V2 (vm + orxr) = V2 vm

Note that two additional terms appear in the momentum equation as a result of
roration. These terms are, (ox(rnxr) , which represents the normal component of the

centrifugal acceleration, and, 2 toxvm , which represents the Coriolis acceleration.

In order to simplify the governing equations the fluid properties are assumed
constant, except for density variations that drive the enclosure buoyant flows. This
assumption is known as the Boussinesq approximation, which is shown by Zhong,
Yang and Lloyd (1985) to be valid in the range of the temperature differences used
in the present study. The Boussinesq approximation states that the density of the fluid
is constant, except when multiplied by the centrifugal or gravitational accelerations in
the equation of motion, where it can be represented by a linear function of

temperature as follows:

P = 90 [1 - I3 (T "‘ T0)] (2.12)

where, B =(1/p) (Hp/3T) , is the thermal expansion coefficient. For a perfect gas
(p = leT) B is equal to (l/To), where 0 refers to a reference condition. Therefore, in
the absence of a heat source the simplified governing equations can be written as

follows:

Mass Conservation

V ° vm = O (2.13)

 

 

19

Momentum Conservation

 

Dv
r0 -' V V2 vro = — .—1-Vpk — 2 mxva - B (T-To) erl
Dt Po

- I3 (T-To) ] g

Energy Conservation

DT
— = v21
Dt 0‘
where the thermal diffusivity a = k/pc ,

o)><(coxr) = — 92r1

H=fi+fl

(2.14)

(2.15)

(2.16)

(2.17)

and, pk=p -- pog - p0 erl, is defined as the kinetic pressure as it results from

dynamic effects, realizing that gravity and centrifugal acceleration are conservative

vector fields.

The boundary conditions are the no-slip condition at all walls for the velocity

field, and prescribed temperamres at the isothermal walls along with zero heat flux at

the adiabatic walls for the temperature field. With the origin (x = y = z = 0) located

at the geometric center of the enclosure and the xyz coordinates positioned

perpendicular to the enclosure surfaces as shown in figure 2.1, those boundary

conditions can be written as follows:

(2. 18a)

(2. 18b)

20
= 0 (2.19)

, v", = o, — = (2.20)

2.3 Dimensionless Form of Governing Equations

The governing equations are usually normalized before performing an
approximate analytical solution or numerical simulation in order to extract
dimensionless groups of parameters that most likely govern the problem. The choice
of appropriate characteristic scales requires appreciation of their physical effect,
which becomes more difficult in enclosure natural convection since different scales

can operate in different regimes of the flow, as pointed out by Yang (1987).

In the present formulation, the height of the enclosure, H, is used as the length
scale, while a/H, 112/a, pow/H2 and the temperature difference (TH - TC) are used as
the velocity, time, pressure and temperature scales respectively. Conditions at the
temperature of the cold wall are taken as reference conditions, i.e. T 0 = TC and
p0 = pC, which was shown to be valid in enclosure flows for the Rayleigh number
used in the present study (Zhong et al. 1985). This results in the following

dimensionless variables:

 

 

e '1
l' 1 - —
t - v70
v r0 - a,”
e __ t
Hzla
a pk
Pk

 

21

T-TC

e'=——
Tit-Tc

e=£
g g

and also,
v‘ = H v and, v"2 = H2 v2

where, the asterisk (*) is used to identify dimensionless variables, and g is the
magnitude of the gravity vector. Substituting all the above dimensionless variables in
equations (2.13), (2.14) and (2.15) results in the following normalized form of the

governing equations:

V’ - v,,‘ = 0 (2.21)
1 DV,0 42 a t e % a e e
ET-V Vm =—Vpk+Ta (kxv,0)-Ra,0r
- Ra 9‘ g' (222)
D9. = v.2 9‘ (2.23)
D:

The dimensionless groups in the momentum equation (2.22) are defined as follows :

Prandtl Number

V
P = — .
r a (2 24)

22

 

 

Rayleigh Number
3 B (TH - Tc) H3
Ra =
vet
Rotational Rayleigh Number
I3 (TH "’ TC) 92 H4
Ra, =
vet
Taylor Number
2
rd ._. M
v2

(2.25)

(2.26)

(2.27)

Prandtl number is a function of fluid properties and it represents the ratio of the

momentum to thermal difusivities. Raleigh number and Rotational Rayleigh number

are measures of the gravitational and centrifugal buoyancies respectively as compared

to the viscous shear forces, while Taylor number is the square of the ratio of Coriolis

force to the viscous force.

CHAPTER 3

EXPERIMENTAL APARATUS AND DATA AQUISITION PROCEDURES

3.1 Experimental Facility

The experimental facility used in the present study is shown in figure 3.1. The
basic facility consists of an enclosure which is rotated by an electric motor. The basic
facility moves on rails and can be positioned inside the test leg of a Mach-Zehnder
interferometer for temperature field measurements, or outside the interferometer for
flow visualization using smoke and laser light. The active test section surfaces are
also connected to two constant temperature water baths through a rotating coupler in
order to control the temperatures of the isothermal walls, and are also connected to a
digital temperature readout through precision slip rings for accurate wall temperature
measurements. Simultaneous photography and observation of the temperature and the
flow fields are done using a high Speed video camera, and the images are stored on

super VHS video tapes for later analysis.

3.1.1 Test Section

Figuer 3.2 shows a schematic of the test section. It consists of two aluminum
plates and two plexiglas plates that form the four walls of a rectangular cavity. The
aluminun plates and the appropriate plexiglas plates can be positioned to achieve the

desired cross-sectional area and aspect ratio of the cavity.

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Figure 3.2 Schematic Diagram of the Test Section

26

The plexiglass plates function as insulated walls, while the aluminum plates are used
as isothermal walls. Each isothermal wall was machined from a 50.8 cm long by
50.8 cm wide aluminum tooling plate of 2.54 cm thickness. Square channels 1.27 cm
deep and 1.27 cm wide were milled into the back side of the aluminum plate for the
circulation of cooling or heating fluid. The front surfaces of the aluminum plates

were polished to a smooth finish of 25 um flatness.

Each aluminum plate was fitted with 18 Copper-Constantan thermocouples to
monitor the spatial and temporal variations in the actual wall temperature. The test
junction of each thermocouple was cemented in a small hole drilled in the back side
of the plate 2.4-4.0 m away from the front surface. A detailed description of the
design of the aluminum plates and the locations of the thermocouples is given by

Bajorek (1981).

The insulated plates were made of plexiglas which has a thermal conductivity of
approximately 0.5-1 W/mK. The plates are 50.8 cm long and 50.8 cm wide, and were
positioned 5.08 cm apart to form a square cavity of side H= 5.08 cm. The edges of
the back side of the plexiglas plates were milled at an angle of 60 degrees leaving a
very small contact area with the aluminum plates in order to minimize lateral
conduction in the insulated walls which will result from the temperature difference
between the two isothermal plates. Also layers of fiber glass wool are used as

backing for the plexiglas plates in order to decrease heat loss.

All four plates are housed inside a rectangular aluminum box, 51.43 cm long,
40.64 cm wide and 51.43 cm high. The sides of the box are made of 1.27 cm thick
alumium plates, while the front and back plates are made of hard aluminum of 2.54
cm thickness. All void spaces inside the box are filled with fiber glass wool for

insulation.

27

Two hard aluminum wheels of 61 cm diameter and 2.54 cm thickness are bolted
to the front and back sides of the box. A plexiglas frame at the center of each end
wheel holds a circular optical flat window tightly against each of the ends of the
square cavity in order to seal the ends and to allow for longitudinal optical
observations. The optical flats are made of high quality glass disks, 8.89 cm
diameter and 2.54 cm thickness, with accuracy of A 0.1 wavelengh. A detailed

description of the design of the test section is given by Hamady (1987).

3.1.2 Rotation and Alignment

As shown in figure 3.2, each of the front and back side wheels of the enclosure
box is supported on two flanged steel wheels 15.24 cm in diameter. The flanged
wheels are grooved and fitted with rubber O-rings for enhanced traction with the test
section wheels. Each pair of front and back flanged wheels is connected together by a
steel axle 2 cm in diameter. The ends of each axle are mounted in heavy duty
bearings, and the bearings are bolted to aluminum blocks which are in turn bolted to

an aluminum base plate of 88 cm length, 61 cm width and 2 cm thickness.

One axle is directly driven by a 3/4 HP DC motor [Master Electric Co.] through
a connecting rod and two universal joints. The motor is connected to a Variac Speed
Control [Type 1702-A General Radio Co.] for easy adjustment of constant rotation

speeds, and both units are mounted on a moving table.

As shown in figure 3.3 the base plate is carried on a heavy duty table and can
be aligned in all three directions by adjustable screws. The heavy duty table [made
by Economy Engineering] provides height adjustment and high capacity load carrying
capability (approximately 2000 lb). The table is supported by V- grooved casters that
ride on two rail tracks in order to position the test section inside a Mach-Zehnder
interferometer for temperature field evaluation, or outside the interferometer for flow

visualization experiments.

 

Table

mg

Figure 3.3 Test Section and Support

29

The rails are bolted to the floor, with rubber pads placed underneath them to
minimize the transmission of the rotation induced vibrations to the optical

components of the interferometer.

The small contact surface between the test section wheels and the supporting
flanged wheels, in addition to carefully balancing the test section worked to minimize
the unwanted vibration of the test section despite its heavy weight. However, a small
yet deleterious amount of low frequency, high amplitude vibrations still resulted from
the test section rotation at high rotation rates. These vibrations caused disturbances in
the optical components and could smear the fringe patterns in interferometry
experiments. Therefore, proper padding underneath the interferometer frame and high
speed photography were used to solve the vibration problem as discussed later in this

chapter.

3.1.3 Wall Temperature Control

The aluminum plates were maintained at uniform temperature by means of
closed loop water circulation from constant temperature baths. A heating bath and
circulator (model D3-L, HAAKE Co.) was used to heat one plate, while a low
temperature bath and circulator (model 90, Fisher Scientific Co.) was used to cool the
other. Circulating water fiom the constant temperature baths in and out of the
aluminum plates during rotation required the design of a special high speed rotary
water feeder.

Figure 3.4 shows a section in the four channel rotating water coupler used in the
present study. The coupler consists of an aluminun cylinder (12.7 cm long x 6.8 cm
I.D.) inside of which five equally spaced sealed bearings (FAFNIR 9108PP) were
fitted and separated by means of four aluminum spacer rings that have the same outer

diameter as the bearings.

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The outer rings of the bearings and the outer spacer rings were compressed together
by means of two aluminum end caps mounted on the ends of the cylinder and held
together by four long compression bolts. Four concentric inner spacer rings, having
the same inner diameter as the bearings and the same width of the outer spacer rings,
were compressed between the inner rings of the bearings by means of two aluminum
disks which were located at the ends of the assembly. Both disks were held together

by an axial compression bolt.

The group of five bearings, four outer spacer rings and four inner spacer rings
formed four sealed cylinderical compartments with one inlet and one outlet hole in
each compartment for water circulation. The design allowed for independent rotation
of the inner and outer cylinderical walls of the fixture. The rear disk of the coupler
was bolted to the back wheel of the test section, and four holes were drilled through
both parts to internally connect the inner four aluminum tubes (in the right hand side
of the schematic) to the inlets and outlets of the isothermal walls by short pieces of
tygon tubing. In the mean time, the outer four tubes (in the top of the schematic)
were connected to the inlets and outlets of the constant temperature baths for closed

loop water circulation.

The rotary water coupler was mounted on the back wheel of the test section as
shown in figure 3.5 . The outer cylinder of the coupler was attached to the stationary
base plate of the test section by means of a crank-rocker mechanism that keeps the
outer part of the coupler in an almost upright position, and exerts a small uniform

torque against the rotation of the enclosure.

3.1.4 Wall Temperature Monitoring

As mentioned previously, each one of the isothermal walls was fitted with 18

copper-constantan thermocouples for monitoring the variations in wall temperature.

 

Figure 3.5 Rotary Water Coupler

33

Details of the wiring and connections of the thermocouples can be found in Hamady
(1987). However it is of interest to briefly describe the system used in the present

work for temperature data acquisition.

Testing the consistency of the measurements was done in two stages. First the
readings of all thermocouples in each wall were compared at steady state without
rotation. Then, one representative thermocouple from each wall was chosen and the
stability of its readings during rotation was monitored. Measurement of temperature
during rotation by means of stationary readouts required a system of rotating sliprings
mounted on the test section and non-rotating contact brushes connected to the
measuring instrument. Figure 3.6 shows a photograph of the slipring-brush assembly

used in the present study as mounted on the front side of the test section.

In the no rotation case extension wires from all functioning thermocouples in
each wall (about 15-16 thermocouples in each) were connected to the input leads of a
rotary switch [made by Minneapolis Honeywell Regulator Co.] , and the switch
output was connected to a microprocessor-based thermocouple meter [Omega
Engineering, Inc., model 680/A] with 0.1 °C resoluion and :l: 0.4 °C accuracy.
Variations in the readings of all termocouples in each wall at steady state were found

to be within :1: 0.1 °C.

For the rotation case, one representative thermocouple with readings close to the
average of all thermocouples in each wall (in the no rotation case) was selected. Each
one of the extension wires from the two representative thermocouples was connected
to a silver plated copper slipring (two sliprings for each thermocouple). Each slipring
is 28.58 cm in diameter with a square cross section of 0.64 cm by 0.64 cm. As
shown in figure 3.7, the sliprings were mounted axially on the front wheel of the test
section by means of five flanged plexiglass rings bolted to the wheel. The plexiglas
flanges held the sliprings and kept them 0.64 cm apart from each other and 3.81 cm

from the aluminun body of the test section.

 

Figure 3.6 Slipring- Brush Assembly

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The four pairs of brushes (one pair for each lead wire) were made of Silver
Graphalloy [type TS-109096-6 grade, Graphite Metallizing Corporation] and were
used to transmit the thermocouple voltage signals from the sliprings to the Omega

meter.

As shown in figure 3.7, each pair of brushes was in contact with a slipring, one
on each side in order to maintain proper continous electrical connection. The wires
connecting pairs of brushes (two for each thermocouple) were connected to the
temperature readout through a two way switch. Each brush was attached to a small
brass sheet spring , and each set of four springs on one side of the slipring assembly
was bolted to a plexiglas holder. The holders were mounted on the base plate of the
test section opposite to each other, and an adjustable spring in between the plexiglas
holders was used to exert enough pressure to keep the brushes in continuous contact
with the sliprings during rotation.

Variations in the steady state wall temperature readings during rotation were
within i 0.1 °C for all rotation rates used in the present experiments. However,
additional external pressure on the contact brushes was necessary for the stability of
the readings during high rotation, which showed the need for improved design of the

brush mounts for future work.

3.2 Flow Visualization Technique

Qualitative observations of the flow field in the rotating enclosure were done
utilizing smoke and laser sheet. The technique is widely used in laboratory
experiments for the visualization of flow movement in stationary frames of reference.
However, utilizing the technique in rotating reference frames is hindered by the
complexity of constructing and alligning a set of mirrors capable of simultaneously

bending, rotating, and focusing a laser beam on a cylinderical mirror which rotates

‘VIV

4.x

37

with the enclosure.

3.2.1 Fiber Optics Rotating Connection

Figure 3.8 shows a schematic diagram of the flow visualization system used in
the present study for making use of fiber optics technology to transmit a laser beam
inside the test section during rotation. The system consists of a flexible incoherent
fiber optic cable [model 77526, ORIEL Corporation], 0.617 cm in diameter and
121.9 cm in length. As shown in figure 3.9, the cable was constructed of a bundle of
glass fibers, 0.32 cm in diameter, which was protected by a non-magnetic
interlocking stainless steel sheathing. The ends of the cable were fitted with hard
steel ferrules, 1.1 cm in diameter. One end of the cable was mounted axially in a
plexiglas holder which was bolted to the back wheel of the test section and was
backed by a black insulating cover to protect the back side optical window. Tire
exposed surface of the cable end was positioned perpendicular to and concentric with
the axis of rotation, while the other end of the cable was inserted inside the

aluminum box of the test section through a small hole in the back wheel.

Inside the test section, the end of the cable was mounted in a Lens-Cable
assembly that focused the transmitted laser beam at the axis of a cylinderical lens for
generating a sheet of laser light perpendicular to the axis of rotation at the midplane
of cavity. The assembly was attached to the body of the test section by aluminum
brackets that restrained it from moving during rotation of the test section. The
assembly was fitted into the space between the isothermal plates and rode on the
back side of the transparent plexiglass wall of the cavity with the cyliderical lens

parallel to the axis of rotation.

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Locking Screw
Plexiglas
Holder

Cylinderical
Lens

Steel
Lockrng Screw

  

 

Glass fiber
Bundle

      

Fiber Optic
Cable

   

 

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Ferrule
Adapter

Beam Probe ~
Focusing
Lens

Collimatin g

Locking Screw

Nylon

Figure 3.9 Schematic Diagram of Lens-Cable Assembly

 

 

40

The lens-cable assembly shown in figure 3.9 was machined out of a plexiglas
block, 5.08x7.62x4.45 cm. The block had a small hole for holding a cylinderical lens
0.64 cm in diameter, and a larger hole perpendicular to the axis of the cylinderical
lens for holding a collimating beam probe 1.5 cm 1D. [model 77644, ORIEL

Corporation].

The probe tube held an f/l.7 lens (19 mm focal lengt, 11 mm clear diameter) by
means of an expanding spring ring. A ferrule attached to the end of the fiber optic
cable fit inside the tube, and this cable could then be positioned to set the distance
between the surface of the fiber bundle and the probe lens greater than the focal
length of the lens, thereby producing a focused image of the laser light at a distance
greater than 7.5 cm from the lens. Finally, by moving the assembeled cable end and
beam probe up and down the plexiglas holder, the focused image of the laser light
was adjusted at the axis of the cylinderical lens, which resulted in the sheet of light

required for the flow visualization.

3.2.2 Smoke-Laser Sheet Generation

As shown in figure 3.8, the smoke necessary for visualising the flow inside the
cavity was supplied through a small injection tube that extended from the rear end of
the cavity to the front wheel of the test section. The injection tube was conneCted to
a smoke reservoir through a rotary connector and a tygon tube. The reservoir
pressure was atmosheric (same as the cavity), and was connected by a tygon tube to
a small bladder type hand pump that was used to manually deliver small volumes of

smoke to the cavity when needed.

An argon ion laser [model 94, LEXEL Corporation] was used to generate a low
power beam of light, which could be targeted on the exposed rotating end of the fiber
optic cable to generate a light sheet inside the cavity. The light sheet combined with

the injected smoke revealed the details of m0tion of the flow during rotation at the

41

midplane of the cavity, and a video camera connected to a monitor was used for
simultaneously observing and recording the flow patterns through the front optical

window (See 3.4.2 Photography).

3.3 Mach-Zehnder Interferometer

Interferometry is widely used for non-invasive temperature field measurements
in convection heat transfer and aerodynamic studies. The technique is based on the
fact that the refractive index of a medium changes as its density changes.
Consequently for gases, if the pressure is held constant the complete temperature
field, which produces the density changes, can be evaluated. A Mach-Zehnder
interferometer was used in the present study because it permits the continous
observation of a two dimensional temperature field, and therefore, provides the

required temperature as well as heat transfer data during rotation.

ps 12 3.3.1 Interferometer Components

A schematic diagram of the MachoZehnder interferometer is shown in figure
3.10 . It basically consists of two beam splitters and two mirrors, which are used to
split a collimated beam of light in two different paths. One part continues
undisturbed and is called the reference beam, while the other part, called the
measuring beam, passes through the test section to be altered by the temperature
variations in the beam path. When the two beams are finally recombined,
interference fringes result fi'om phase differences between the undisturbed reference

beam and the measuring beam.

A 70-Watt high pressure mercury vapor lamp which emits white light was
mounted in a lamp housing with a small aperature (about 2 mm) and was used as the

interferometer’s light source. The white light then passed through a narrow band filter

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43

which was mounted in front of the light source to select only green light (5461A ).
The light then passed through two condensing lenses and was focused on a small
mirror (ml) which acted as a near point source at the focal point of a 10.16 cm

diameter spherical rrrirror (C1).

Mirror (Cl) reflected a collimated beam of light onto beam splitter (81), which
reflected 50% of the original beam to mirror (M1) and transrnited the other 50%
through to mirror (M2). The measuring beam was reflected by M1 and passed
through the test cavity to beam splitter (82), which in turn transrrrited it plus the the
beam reflected by (M2) to the spherical mirror (C2). Two optical window plates
identical to the ones mounted on the test cavity were placed in the pathway of the

reference beam between partial rrrirror (SI) and full mirror (M2) for compensation.

The measuring and reference beams were combined at mirror (C2) and focused
on a small mirror (m2), which in turn focused the reflected image through a large
zoom lens onto the picture element of a high shutter speed video camera connected

in series to a S-VHS recorder and a high resolution TV monitor.

All optical components were carried on a heavy gauge flat horizontal steel
platform supported by rigidly constructed steel tubing legs. Two- layers of thick
rubber pads and felt cushions were used underneath the legs to minimize the

transmision of vibrations from the floor to the optical components.

3.3.2 Alignment and Focusing

Aligning the optical components of the interferometer is a time consuming
process that may seem tedious at the beginning, but becomes less complex as
detailed pocedures are developed for coarse adjustment of relative positions and fine
tuning of individual components. Comprehensive discussion of all important aspects
of the basic theory, design and adjustment of a Mach-Zehnder interferometer can be

found in Goldstein (1976). However, it is appropriate to discuss the basic principles

utilized in the present study.

The main idea is to ensure that (i) the original light beam is bright and
collimated, (ii) the reference and measuring beams are closely parallel and their path
lenghs are approximately the same, and (iii) the recombined parts of the reference
and measuring beams are accurately parallel. In order to do that, all components are
placed on the horizontal platform in the arrangement shown in figure 3.10, and all
distances and angles are adjusted using accurate measuring tools and light projections

and reflections.

The light source, filter and condenser lenses are mounted such that a small
bright sharp spot is focused on a small flat mirror (m1). Mirror (m1) is positioned at
the focal point of sperical mirror (C1) to ensure that the width of the collimated
beam reflected by C1 is constant. This is checked by projecting the beam on a flat
screen positioned perpendicular to the beam at increasing distances. The mirrors and
beam splitters are then arranged at the the four comers of a rectangle and set at 45°

as shown.

The parallelism of the two beams is achieved by, aligning the images of two
objects located before (81) along the original beam and separated by a large distance.
The two objects are chosen to be the bright point light source, which is located at
infinity, and the boundaries of the small mirror (m1) together with the vertical wire
used to suspend it. A small telescope is used to look into beam splitter (82) to see
the images, and the alignment is done by slight rotation of 82 and M2, which are

mounted in gimbal mounts that allow for precise tilt about two orthogonal axes.

When the beams are almost parallel, fringes with finite spacing will appear in
the recombined beam which is projected on mirror (C2). By further rrrinute rotation
of $2 and M2 infinite fringe spacing can be achived, in which case the two beams
will be perfectly parallel and the field of view completely iluminated. This mode of

Operation is called "infinite fringe setting".

45

In the infinite fringe mode used in this work, density gradients initiated by heat
transfer in the test section will result in disturbances in the path lengh of the
measuring beam which in turn produce interference fringes that represent lines of

constant refractive index or temperature inside the test cavity.

Finally, the image reflected by mirror (C2) is focused at the picture element of a
video camera through the small flat mirror (m2). A video camera with high speed
shutter [COHU, model 6415] is used to avoid the distortion of fringes by the small
vibrations transmitted to the optics. The camera shutter can time each frame up to
IBM of a second, which is higher than the frequency of any remaining vibrations

in the interferometer system.

3.4 Experimental Procedures

The nature of experimental work does not provide the levels of accuracy and
precision required for controlling all involved parameters to obtain exact experimental
results. However, several steps were taken before each experimental run to ensure
acceptable and known boundary conditions and accurate, repeatable data aquisition.
Prescription of the boundary conditions required knowledge of the temperature of the
two opposite isothermal walls and elemination of any heat flux to the other two
walls. It also required steady rotation about the central horizontal axis at a known
rate. The data included accurate measurements of the temperature field and a clear

picture of the flow field.

3.4.1 Preparation

Prior to performing an experiment the constant temperature hot water ciculator
was set at a temperature slightly higher than required, while the cold water ciculator

was set at a temperature slightly lower than required. The hot and cold temperatures

46

were chosen such that their average was close to ambient temperature in order to
minimize temperature gradients through the optical windows. The optical windows
were covered with insulating foam covers during start up to minimize convective
currents which result from heat loss to the surroundings, and water circulation would
be started with the heated plates in a vertical position. The stationary enclosure
would be left for 3 to 4 hours and then would be rotated for at least another l’/2 to 2
hours until wall temperatures reached steady state. Steady state temperature was
defined when the variation of the temperature of each wall would not exceed i 0.1°
C over a period of 15 to 20 minutes. The wall temperature was measured using a
representative thermocouple from each wall connected to the Omega digital readout
as explained previously in this chapter. The rotation rate was counted and timed

several times using a stOp watch, and an average was taken.

To perform the flow visualizations during rotation, the laser was triggered, a
steady beam of light was targeted on the exposed end of the fiber optic cable, and
then the smoke reservoir was connected to the injection tube through the rotary
connector. The cover of the front optical window was removed, a small amount of
smoke was injected, and the flow motion inside the rotating cavity was recorded on
video tape for 2 to 3 minutes. The exposed end of the fiber optic cable and the rotary
water coupler were both mounted on the back wheel of the test section, which
resulted in a black field of view in the cavity every time the tubes connected to the
coupler intercepted the laser beam. The blackout of the flow field occured over an

angle of less than 15° while the flow patterns were visible for the rest of the cycle.

In the interferometry experiments, the test section was positioned inside the
interferometer prior to starting water circulation, and the interferometer was adjusted
for the infinite fringe mode. The optical windows were then covered and water
circulation started in the stationary enclosure as described above. After 3 to 4 hours

the window cover would be removed, and the fringe pattern checked. Since the walls

47

were isothermal, fringes near the walls must be parallel to them. However, small
temperature gradients in the optical windows resulted in slight deviations, which
required minor adujstment of mirror (MZ) and/or beam splitter (82) in order to
correct the fringe orientation. Then the windows were covered and steady rotation
was held for at least 1 hour. After the wall temperatures again reached steady state
(typically 02 °C less than the stationary steady state temperature) the window covers
were removed and the fringe patterns were recorded on video tape for 2 to 3 minutes

of rotation.

3.4.2 Photography

Flow visualization and interferometry experiments were recorded on S-VHS
video tapes using a MITSUBISHI twin digital U-70 VCR. The video tapes were
played back in single frame mode and displayed on a Panasonic model WV-5410
high resolution monitor. The best frames at the desired angles of rotation were
photographed off the monitor using a 35 mm still camera loaded with a 400 ASA
color film. The analysis of the interferograms was done on an X-Y traveling
microscope built by GAERTNER SCIENTIFIC Corporation with resolution up to
0.0001 inch.

CHAPTER 4

RESULTS AND DISCUSSION

4.1 Flow Field, Temperature Field, and Nusselt Number

The experimental results of this study are presented in the form of flow
visualizations, isotherms and local and mean Nusselt numbers at instantaneous angles
during rotation of the air filled differentially heated square enclosure. The flow
pictures obtained using smoke and laser light represent the streak lines and are used
for the qualitative analysis of the enclosure convective currents, while the isotherms
obtained using a Mach-Zehnder interferometer show the entire temperature field and
are used to calculate the tempereature gradients at the isothermal walls. The local
Nusselt numbers are derived fi'om the wall temperature gradients as described in
Appendix (B).

The quantitative results in the form of local Nusselt number distributions along
the isothermal walls and mean Nusselt number values are used to determine the
suppression or enhancement of heat transfer as a function of the increasing speed of
rotation. The mean Nusselt numbers were calculated by integrating the local Nusselt
number values over the height of the isothermal wall. Both flow visualization and
interferometry experiments were recorded in real time mode on S-VHS video which
allows the storage of 60 frames every second, i.e. one picture every 2°-12° of the

enclosure r0tation for rotational rates of 20-120 rpm respectively. This makes

48

49

available a large volume of preprocessed information which can provide a
comprehensive data base, however the time consuming method of reading the
interferograms makes it difficult to analyze every single one of them. In addition the
periodic nature of the problem allowed for the selection of fewer angular positions
for data reduction at every rotational speed. Four angular positions were chosen
because of their relevance to a great deal of the existing literature on rectangular
enclosures. These four positions are 0° which represents the case of the enclosure
heated from above, 90° which represents the case of a vertical enclosure heated from
the left side, 180° which represnts the case of the enclosure heated from below, and
270° which again represents the vertical enclosure but heated from the right side. In
some cases however data at the angular position of 180°or 90° could not be obtained
because the field of view was obstructed by the tubes connected to the rotary water
coupler. Note that the overall mean Nusselt number at a certain rotational speed
represents the average of the mean Nusselt number values at these selected angular
positions.

In this study the data is presented as a function of Taylor number
6.34x105 5 Ta s 1.75x107, the rotational Rayleigh number 5.3x103 s Ra, s 1.4x105,
and Rayleigh number Ra = (l.77:t0.07)x105. These parameters correspond to
rotational rates of 22.5 rpm 5 Q S 118.0 rpm and a temperature difference between
the isothermal walls of (TH-TC) = 13.3i0.5 °C.

Although the above parameters can be used to predict the individual effects of
the gravity and rotational forces on the enclosure flows, the combined effect of these
forces plus the inertial and viscous shear forces in a particular enclosure configuration
can not be directly forseen. Therefore the present study was necessary to reveal the
effect of the interaction of these forces on the flow and hence the convection heat
transfer in a rotating enclosure. And the simple model of the differentially heated

square cavity was selected for its wide use in the literature on natural convection in

50

enclosures. Most of the published studies on rotating enclosures however deal with
different configurations and orientations with respect to the gravity vector. In
addition to the experimental results of the prsent study the numerical calculation of
the same problem done by K.T. Yang of University of Notre Dame (1990) is
presented in some cases for the purpose of correlation. This numerical algorithm is
based on the governing equations (2.21)-(2.23) and utilizes the QUICK (quadratic
upstream interpolation for convection kinematics) finite difference scheme to
calculate the velocity field, the temperature field and the Nusselt number among other

variables. See H.Q. Yang ( 1986) for further details on the numerical algorithm.

In the case of a vertical stationary enclosure differentially heated from the sides,
gravitational buoyancy produces circular fluid motion near the boundaries while the
slower fluid in the core region plays a small role in the convection process. As the
enclosure is tilted toward the heated from below position instability of the core
region enhances the mixing of the fluid and thereby the heat transfer. On the other
hand, as the enclosure is tilted toward the heated from above position the fluid
becomes more stratified and the heat transfer drops until the process becomes

dominated by conduction.

In the present study the square enclosure of aspect ratio Ax=l.0 rotates in the
clockwise direction about the horizontal Z-aris at steady rates, which induce uniform
rotational buoyancy forces. However when these forces interact with the periodic
gravity field they result in unsteady complex flow patterns. At slow rotation rates
gravity is expected to dominate the motion of the fluid with primary cellular flow
near the walls of the enclosure. In addition the Coriolis force recirculates the
boundary region fluid toward the core region and therefore improves the mixing of
the fluid and enhances the heat transfer. As the speed of rotation increases the
centrifugal force and the wall shear are expected to have increased influence, where

the viscous shear migrating inward from the wall works to produce solid body

51

rotation and reduce the heat transfer to pure conduction in air, while the centrifugal
buoyancy works to produce symmetric flows relative to the rotating coordinate
system and therefore increase the convective heat transfer. In the following
subsections the flow patterns which result as the speed of rotation increases are
presented, and their influence on the temperature field and the Nusselt number
distribution is discussed. Estimates of the errors associated with data acquisition and

reduction are presented in Appendix (C).

4.1.1 At Rotational Speed 22.5 rpm

Parts (b) and (c) of figures 4.1 through 4.3 show the resulting isorherms and
local Nusselt number distributions at rotational speed (2:225 rpm
(Ta=6.34><105, Ra=l.84x105, and Ra,=5.3x103) for the angular positions of, 2°
(heated from above), 270° (vertical), and 182° (heated from below) respectively.
Flow pictures at the same rotation rate are not available, however, flow pictures at
rotation rates of 17.0 rpm and 28.5 rpm for the corresponding angular positions are
shown in part (a) of the aforementioned figures to help predict the flow patterns at
22.5 rpm. Figure 4.1a shows the flow pattern when the hot wall is on top during
clockwise rotation. The pictures show the counter rotation of the core region fluid
while the fluid near the isothermal walls forms a slower boundary layer and
clockwise rotating vortices at the four corners. It can also be observed that the core
fluid is being well mixed which agrees with the nearly uniform temperature in the
midsection of the enclosure shown in figure 4.1b. The is0therms of figure 4.1b also
show the thickness of the boundary layer and the vortical motion at the lower corner
(L.H.S.) of the hot wall and the opposite corner on the cold wall. It is also clear
from the interferogram the existance of thermal gradients at the plexiglas walls which
shows that they are not perfectly insulated. In figure 4.1c the local Nusselt number is
plotted vs. the non-dimensional height of the isothermal walls [Y‘=(2Y+H)/2H],

22.5 rpm
2 deg.

       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

C:
/' .
ColdWall
(b) Temperature Field
ColdWall
(a)FlowField
0 e Hotwu
0 Cold all
:1
z A
I: U
B
:3 ‘0
3 80800006 $1P".°I°e
g 0 O . O O O O O
8 2
3
o—ri . . . . . .
0.0 0.2 0.4 0.6 0.8 1.0

Y.
(c (1:225 tn Ra.=1.84x10'
) 6=z° (hraiied rro above)
Picture 6?

Figure 4.1 Effect of Rotational Speed of 22.5 rpm
at 6 = 0 degrees, Ta = 6.34:th5

53

where Y‘=0 is at the lower end of the walls when the hot wall is on the left and the
cold wall is on the right. In this case the Nusselt number distribution is symmetric
and nearly uniform, however a mean value of 3.1 is significantly higher than that of
the stationary enclosure heated from above (approximately 1.0) ,which can be related
to the rotation induced mixing of the core region, which would otherwise be formed

of nearly stagnant layers of stratified fluid.

As can be seen from figure 4.2a, as the hot wall rotates clockwise toward the
vertical position (6=270°) the boundary layer thickens and the comer vortices move
up the hot wall (toward Y'=0.0) and down the cold wall (toward Y‘=1.0) because of
gravity. The core region is squeezed, and a review of the video tape shows it to
reverse direction and rotate with the enclosure. The isorherms in figure 4.2b also
show the increase in the boundary layer thickness and the increased non uniformity
of the core region temperature. The influence of the gravitational buoyancy is
apparent in the local Nusselt number profiles shown in figure 4.2c, where the local
Nusselt number reaches a minimum near the lower end of the hot wall and a
maximum near the upper end, while the cold wall exhibits a reversed distribution.
The mean Nusselt number however remains almost unchanged from that of the
previous angular position (6=O°).

Figure 4.3a shows the flow pattern at angular position 6=l82° when the
enclosure reaches the heated from below position. The vortices are released from the
opposite comers and driven toward the core by the Coriolis force which greatly
enhances the mixing of the core fluid and therefore enhances the convection heat
transfer. The isotherms in part (b) also show the large thermal gradients at the
isothermal walls and at the core region. Note the inverted bell shape of the local
Nusselt number distributions with a maximum near mid-wall and minimum values
near the comers. This is attributed to the formation of new vortex tubes near the four

comers which replace the ones swept to the core.

Cold Wall

Cold Wall

Local Nusselt Number, Nu

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_. v‘
g a”.
2 é
a E
22.5 rpm
270 deg.
(b) Temperature Field
5.
2
E:
28.5 rpm
270 deg.
a 0
v o I
O
o o . ' '
4 o .
O O
a . . . . . o A 0
a 3 D o ..
0 o O
“0.0 0.2 0.4 0.0 0.8 1.0
Yl
(c) n: r,prn Ra=1.x84 10‘
6:270‘5 ivertlcal enclosure)
Plot. 1]

Figure 4.2 Effect of Rotational Speed of 22.5 rpm
at 6: 270 degrees, Ta = 6.347th5

55

17 rpm

 

H.W.

   

 

 

 

 

 

 

 

 

 

 

 

 

Y'<—I
(b) Temperature
Field at 6 = 182 deg.
22.5 rpm
(a) Flow Field at 6 = 182 deg. ((1) Flow Field at 6 = 90 deg.
13 I Hot VII]
. 0 Cold ‘1.“
O
2 a A
z 0 o "
5 ° 0 o D .
.D g . O .
E A _ e o o
:3 u U
z ‘ ‘
a o -
g 4 c t: I
Z 0 0
ii 0 ° '
.3 2 . 4°
° o
“0.0 ' 0.2 ' 0.4 . 0.6 ' 0.8 ' 1.0
Y‘
(C) r,pm Ra: 1.841: 10 0'
111==182°5 (heiated from below)
[Pi ctu e4]

Figure 4.3 Effect of Rotational Speed of 22.5 rpm
at 6 = 182 degrees, Ta = 6.34x105

56

The mean Nusselt number reaches a high of 4.64 which reflects more than 50%

increase over the previous angular position.

At 6=90° the interferogram and the local Nusselt number distributions are not
available because of reasons explained at the beginning of this chapter, but the flow
pictures of figure 4.3d show the initiation of the counter rotation of the core region
and the growth of the secondary vortical flow at the four comers. This pattern of
fluid motion eventually develops during the rest of the cycle untill the enclosure once
again reaches the heated from above position, then the same sequence of events is

repeated.

Figure 4.4 shows the effect of the angular position (orientation of the hot wall)
on the distribution of the mean Nusselt number in one cycle of the enclosure r0tation.
The figure shows good agreement between the experimental mean Nusselt numbers
and the numerical calculation of K.T. Yang (1990). The largest difference between
both seems to be at 270° where the calculated value at the hot wall is about 16% less
than the experimental value, but it is clear that the trend is the same. The sinusoidal
behavior shown in this figure illustrates the influence of the gravitational buoyancy at
22.5 rpm (T a=6.34x105), where the mean Nusselt number reaches its maximum in
the vicinity of the heated from below position (6=l80°). The figure also shows the
deacrease in the mean Nusselt number near 6=270° or when the relative velocity of
the buoyant fluid near the isothermal walls becomes small. This also explains the
significant increase in the mean N usselt number at the opposite vertical position

(6=90°) where the relative fluid velocity at the wall is expected to be higher.

4.1.2 At Rotational Speed 32.0 rpm

Figures 4.5 through 4.8 illustrate the flow and temperature fields at rotational
speed of 32.0 rpm (Ta=l.28x106 ,Ra=1.72x105, and Ra,=1.0x104). At the heated

from above position (6=0°) the counter clockwise rotation of the core region can still

Mean Nusselt Number. Nu

57

 

6.0
- H]. Nam.
d - (1'. Nam.
A Elflxp.
.V. .
5.0- A C "P

 

 

 

 

0-0 r' l ' ' ' l ' ' ' l ' ' ' l ‘ ' ' I ' ‘ l l ' ' T l l ‘ ' l ' 7 ' I
0 4o 80 1 20 160 200 240 280 320 360
93 (°)

Figure 4.4 Mean Nusselt Number vs. An ar Position
at 22.5 rpm, Ta=6.34x10‘, a=1.84x10'

 

58

 

 

 

 

 

 

 

 

 

 

 

(:1) Flow Field (b) Temperature Field
8
0 Hot In“.
0 Cold VIII
5
z A
h. V
B 0 O .
E 4 o o .
:l o O 0 3 O
z ' o
3 4 o o.
a
z 4 e . Q
73‘ ..
Q a
3
G . . . f
0.0 0.2 0.4- 0.8 0.8
Y.
(c) (2: rpm, Ra=l. 72x10‘
¢= 0° (heated from above)
Picture '7]

Figure 4 .5 ElTect of Rotational Speed of 32 rpm

at 6: 0 degrees, Ta: 1. 28x10‘

 

59

be seen in figure 4.5a, however the velocity of the core fluid relative to the enclosure
seems to be less than that of the slower angular speed of 28.5 rpm (comparing the
real time motion on the video tape). The boundary layer and the corner vortices can
still be seen too, with no notable change in their sizes. Some change can be
recognized in the isotherm structures of figure 4.5b when compared to those of the
corresponding angular position at 22.5 rpm. It can be seen that the thermal boundary
layer is thiner in the middle region and thicker near the edges of both isothermal
walls which leads to increased heat transfer near the center of the wall and decreased
heat transfer near the corners. The figure also shows that the core region is composed
of two counterrotating cells. Figure 4.5c shows the distribution of the local Nusselt
number which conforms with the shape of the thermal boundary layer where the local
Nusselt numbers are significantly higher near mid-wall, with approximately 15%

increase in the mean Nusselt number over that at 22.5 rpm.

At 6=270° (vertical position) the flow picture of figure 4.6a shows slight
increase in the boundary layer thickness, it also indicates less mixing of the core
region fluid which suggests decreased influence of the Coriolis force. It is important
to note here that the viscous shear forces at all four walls stretch the corner vortices
in a clockwise motion despite the gravitational buoyant force that works to impose a
counterclockwise circulation on the boundary layer flow. This flow pattern shows the
increased influence of the shear forces which .work to reduce the relative velocity
between the boundary layer fluid and the wall and therefore is expected to reduce the
convection heat transfer. The motion of the core region also confirms the increase in
the shear forces, where the fluid continues to rotate counterclockwise as it did at
slower angular speeds but at a slower and more uniform fashion. In the mean time
the isotherms of figure 4.6b show a generally more uniform and thicker thermal
boundary layer and a nearly isothermal core region. The predicted drop and

uniformity of the heat transfer is illustrated in figure 4.6c, where the amplitude of the

 

(a) Flow Field at 6 = 270 deg.

C.W.

  

H.W.

 

(d) Flow Field at 6 = 180 deg.

Local Nusselt Number, Nu

(b) Temperature Field
at 6 = 270 deg.

 

 

 

 

 

 

 

 

 

 

 

8
O "dwell.
0 Cold '1"
4
O O O O .
o o o o o o ' o o . . .
' ' o o o o o 3
2
0.0 0.2 0.4 0.6 0.8 1.0
(C) Y'

[Picture 8]

0:32 r m, Ra=l.72x10‘
¢>=2 0° (vertical)

Figure 4.6 Effect of Rotational Speed of 32 rpm
at ¢ = 270 degrees, Ta = 1.28x to“

61

sinusoidal shape of the local Nusselt number distribution is less than that of the
corresponding angular position at 22.5, rpm and the maxima and minima are closer to
the center of the isothermal walls. The mean Nusselt number is approximately 19%
less than that of the previous 6=O° position, and 11% less than that of 6=270° at 22.5
rpm.

Figure 4.7a shows the flow pattern as the enclosure reaches the opposite vertical
position (6=91°). At this position the relative velocity of the flow along the
isothermal walls is enhanced by the aiding gravitational buoyancy, and the wall fluid
is swept toward the core more rapidly, while the core region fluid continues to rotate
counterclockwise thereby enhancing the slushing action of the fluid and consequently
the convection process. Figure 4.7b shows the steep thermal gradients near the walls
and the wraping around of the thermally stratified fluid which agree with the flow
pattern and gives indication to a significant increase in the convection heat transfer.
The local Nusselt number distribution of figure 4.7c shows the remarkably high
values of the bell shaped curve at both the hot and cold walls compared to the
opposite vertical position (6=270°) which illustrate the combined influence of the
shear forces and the assisting gravitational buoyancy. In addition, the shift of the
maximum values toward the center of the walls is a direct result of the swirling
motion induced by the Coriolis force. It is therefore reasonable to conclude that the
heat transfer at that angular position is the maximun during the cycle since the wall
shear, the gravitational buoyancy and the Coriolis force are all in favor of the
convective currents. This is illustrated in figure 4.8 by an increase in the mean
Nusselt number of approximately 91% over that of the 6&270° position and 55%
over that of the 6=0° position. Also note that as shown in figure 4.8 the mean
Nusselt number at the heated from above position (which is usually the lowest value
during the cycle in slow angular speeds) is higher than the mean Nusselt number at

the vertical position 6=270°, which shows a phase shift that result from increasing

62

 

 

 

 

 

 

 

 

 

 

 

 

 

(8) Flow Field at ¢ = 90 deg. (b) Temperature Field at
4) = 91 deg.
10 e not 1.11
o a Cold ml
0 O O C C
D O O -
2 V o o o
. c
'- a
B . o .
E . n '
:5 V " . o
z
12’ ' o e
U
«”4 4, °
0
g o o ' 0 o
'3 o
O .
3 2 - o
e I I V V T
0.0 0.2 0.4 0.6 0.8 1.0
Y'
(C) 0:32 1' m, Ra=1.72x10'

6: 1° vertical)
[Picttgre 8a]

Figure 4.7 Effect of Rotational Speed of 32 rpm
at q» = 90 degrees, Ta = 1.28xro‘

Mean Nusselt Number, Nu

63

 

6.0
— EN. Num.
4 - CI. Num.
l A aw. Exp.
5.04 A CV. ‘1’.

 

 

 

 

0-0 "Tl"'l"'l"'r"'l"'l"‘l"‘l"‘T
0 4O 80 120 160 200 240 280' 320 360
¢ (°)

“8“" 4-3 Mean Nusselt Number vs. Angular Position
at 32 rpm, Ta=1.28x10‘,Ra=1.72x10‘

(,4

the rotation rate. The same behavior is also shown in the numerical results, however

at 6=90° the numerical prediction is not as high as the experimental result.

4.1.3 At Rotational Speed 38.0 rpm

At 38 rpm (Ta=l.8lx106) the picture of the flow in the heated from above
position (6=O°) shown in figure 4.9a reveals a rather indistinct flow pattern. Fluid
rotation can barely be seen near the center of the enclosure while the relative velocity
of the fluid near the walls is very small. This flow pattern suggests a state of quasi-
balance between two or more of the buoyant and the retardant forces, and based on
the observations discussed above for the angular speed of 32 rpm it is reasonable to
say that the dominant forces in this balance are the gravitational buoyancy and the
viscous shear migrating inward from the enclosure walls. Furthermore the low
velocities in the core and the nearly stagnant fluid near the walls suggests a drop in
the heat transfer which is also supported by the temperature field of figure 4.9b. The
isotherms show nearly flat layers of thermally stratified fluid and little mixing in the
core region, however the influence of gravity can still be seen in the nonuniform
shape of the thermal boundary layer. A drop in the heat transfer with the increase in
the angular speed can be seen when comparing the distriution of the local Nusselt
number of figure 4.9c to that of figure 4.5c for the angular speed of 32 rpm. The
minimum values near the corners are slightly lower, while the maxima near the
center of the walls are significantly less than those of 32 rpm. This reflects a

decrease in the mean Nusselt number of approximately 28%.

Review of the real time fluid motion recorded on the video tape shows periodic
motion of the core region relative to the enclosure walls, where the fluid is streched
in a tube like shape between two opposite comers as the hot wall moves from top
to bottom then the tube is bent in an S-shaped curve during the other half of the

cycle. Figure 4.10a shows the flow field as the enclosure reaches the vertical position

Local Nusselt Number, Nu

 

65

 

 

 

 

 

 

 

C.W
(a) Flow Field (b) Temperature Field
3 223:3.
1
6
4
o o o . . ° 0
3 ° 0 o
2 . ._ o O
.
"0.0 0.2 0.4 ' 0.6 0.8 1.0
Y'
(c) r,prn Ra=1.4x8

 

 

6: 0° ( eated fro

 

Pt tcture 13

an above)

 

Figure 4.9 Effect of Rotational Speed of 38 rpm
at 6 = 0 degrees, Ta = 1.8lx10‘

 

 

66

 

 

 

 

 

 

 

 

 

 

 

(a) Flow Field (b) Temperature Field
8 e mum
O Cold'ell
é
. 6
I-
Q)
.o
E
:r
z
3 4
0)
i
z o o 0 o o o o o. . c, o 0 e .
E 9 - ' . o v
E” ' O o e o O
c
0.0 0.2 0.4 0.6 0.8 1.0
Y'
(c) 0:38 2' m, Ra=1.84x10‘

= 0° vertical
4, [Pictu(re 12] )

Figure 4.10 Effect of Rotational Speed of 38 rpm
at 6 = 270 degrees, Ta = 1.813(10‘5

67

(6=270°), where the flow pattern near the enclosure walls remains indistinct and slow
and its shape is almost unchanged from the previous angular position (6=270°) while
the fluid in the core starts to stretch. Moreover, the isotherms and local Nusselt
number distributions of figures 4.10b&c also match those of the previous angular
position of 6=0°, with approximately 12% decrease in the mean Nusselt number. This
suggests that the shear forces during this half of the cycle (the hot wall rotating
clockwise from top to bottom) balance the gravitational buoyancy and suppress the

convection heat transfer.

The effect of gravity however shows in the next figure (4.11) at the opposite
angular position (6=88°) when both the gravitational buoyancy along the isothermal
walls and the shear forces act in the same direction. The mixing of the core region
fluid can be detected in the flow picture of figure 4.11a and its effect can be clearly
demonstrated in the shape of the thermally stratified fluid layers of part (b). Figure
4.11c shows the local Nusselt number distributions which are therefore expected to
be higher than those of the other two positions presented above at the same angular
speed (figures 4.9 and 4.10). As expected the mean Nusselt number at this angle
(6=88°) shows 64% increase over that of the heated from above position (6=O°), and
87% increase over that of the opposite vertical position (6=270°). Also note that
these distributions along the isothermal walls are opposite and identical, with their
minima near the comers as a result of the low relative fluid velocities and the
disappearance of secondary comer flows, and their maxima shifted away from the

centers of the walls indicating a decline in the influence of the Coriolis force.

The observations and discussion in this subsection show that the rotation speed
of 38 rpm (Ta=1.81x106, Ra=1.84x105, and Ra,=1.5x104) marks a transition region
in the interaction of gravitational and rotational forces. 'A transition from well defined
primary and secondary enclosure flows dominated by the gravity effects and Coriolis

acceleration throughout the cycle at lower angular speeds, to another region

68

 

(a) Flow Field at 6 = 90 deg. 00 Temperature Field
at 6 = 88 deg.
D
u 0 Hot Wall
0 Cold 'Al
O O
O
a Q -
0 0
' e
. 0
e . a
4 c . . -
O C
. o
O
2 O
"0.0 0.2 0.4 0.6 0.3 1.0
Y'
9:38 , R =1.84 10'
(c) 6:581? (vear'mcal)x

Local Nusselt Number, Nu

 

 

 

 

 

 

 

 

 

 

[Picture 12a]

 

Figure 4.11 Effect of Rotational Speed of 38 rpm
at 6 = 88 degrees, Ta = 1.8lx106

 

69

dominated by shear forces that tend to induce solid body rotation and suppress the
heat transfer to near pure conduction independent of the angular position. However,
figure 4.12 shows that at 38 rpm gravity still has a significant effect on the mean
Nusselt number, especially at 6:90" where it reaches a relatively higher value.
Nevertheless, the overall mean Nusselt number is 26% less than that of angular speed
32rpm, which shows the increasing effect of the shear forces. The numerical results
shown in figure 4.12 however illustrate more uniformity and earlier independence of

the angular position than the experimental results.

4.1.4 At Rotational Speed 47.0 rpm

The preceding discussion seems to suggest that the boundary layer thickness
will increase with the angular speed until the entire enclosure fluid rotates as a solid
body and the local and mean Nusselt numbers become equal to a value of one which
correspond to pure conduction. But interestingly enough, the influence of the
centrifugal force begins to show in the form of a non-boundary layer, 2-cell type,
buoyant flow that works to enhance the convection heat transfer. Figures 4.13
through 4.17 will illustrate the interaction of the centrifugal buoyancy with the
dominant shear forces at Q=47rpm (Ta=2.77x106, Ra=1.71x105, and Ra,=2.14x104).

An interesting flow pattern is shown in figure 4.13a at the angular position
6=358° (or 6: -2°). The streak lines form two cells, a dominant cell which occupies
more than 3/4 of the enclosure and contains two vortical structures which possess very
little circulation, and a smaller cell located at the lower end of the cold wall which
seems to be so stagnant that the smoke does not penetrate into it. This type of fluid
motion which can be characterized as potential flow in addition to the uniformly
spaced isotherms of figure 4.13b indicate continued decline in the corresponding
Nusselt number values. Part (c) shows the local Nusselt number distributions which

conform with the flow and temperature fields. Note that the distributions along the

Mean Nusselt. Number. Nu

70

 

6.0
— KW. Nam.
. - c1. Nuns.
A rut. Exp.
5 0+ A C.'. EXP.
4.0—. A

 

 

 

 

 

1.0'*
do ..,...,...,...,. ,.,e-er
0 4O 80 120 160 200 240 280 320 360
it (°)
Figure 4-12 Mean Nusselt umber vs. lar Position
at 38 rpm cw rotation) a=1.84x10‘

Ta: 1.81x 106

71

 

(a) Flow Field at 6 = 0 deg. (b) Temperature Field

at 6 = 358 deg.

CD

 

O Bel. Wall
0 Cold 'All

 

 

00°00

 

Local Nusselt Number, Nu

 

 

 

 

 

 

 

2 30003“ .“.u-O'o.o.°‘
. . . e o O . . .
0 . . . .
0.0 0.2 0.4 0.6 0.8 1.0
r'
(c) 47 rpm, Ra=1.71x10‘

n:
6=358° heated from above)
Picture 21]

Figure 4.13 Effect of Rotational Speed of 47 rpm
at e = 0 degrees, Ta = 2.77x10‘

72

isothermal walls are no longer mirror images of each other due to the asymmetric
nature of the flow field. Also note the uniform shape of the curves and the low
Nusselt numbers which approach unity near mid-section of the hot wall. It should be
noted here that the mean Nusselt number of the cold wall is about 19% higher than
that of the hot wall. This is explained by the fact that the conductivity of the
plexiglass walls is higher than that of the contained air which result in a higher heat
flux at the cold wall (heat flux in the plexiglas is estimated to be approximately twice
that in air). Therefore the mean Nusselt numbers of the hot wall will be used for
comparison throughout the text. And as expected the mean Nusselt number for this
angular position (heated from above) is found to be 42% lower than that of the
corresponding angular position at Q=38rpm

A comparison between, parts (a), (b) and (c) of figure 4.13, and those of figures
4.14, 4.15 and 4.16 shows very little change in the flow and isotherm patterns
throughout the cycle, and slight redistribution of the local Nusselt number values
along both isothermal walls. The only difference that can be seen in the flow
pictures (more recognizable on the video tape) is the change in the shape of the
smaller cell at the lower end of the cold wall which becomes elliptic as the hot wall
moves from bottom to top (6=l80° to 0°) and takes a more circular shape during the
second half of the cycle. This small oscillating motion and also the little change in
the local Nusselt number distribution from one angular position to the other indicate
that gravity has little influence on the covection process at this speed. Finally,
comparing figure 4.17 with figure 4.12 it becomes obvious that the mean Nusselt
number values at the current angular speed are almost independent of the angular
position with an overall mean Nusselt number of 1.487 , which is almost 50% lower
than that of 38 rpm but still higher than unity. The largest drop in the mean Nusselt
number however occurs at 6=90° where the traditionally higher value decreases by

about 60%. This indicates that the viscuous shear forces have overcome the

 

73

WW:

(21) Flow Field at 6 = 270 deg. (b) Temperature Field

Local Nusselt Number, Nu

at 6 = 268 deg.

 

CD

 

 

 

 

 

 

 

 

 

 

0 women
0 Cold'ul

3

4

2 r . . o ' e V o o . g e 3

°oo‘800 eon'Ooo

4

c . . . .

0.0 0.2 0.4 0.8 0.8 1.0

Y‘
(c) 0:47 r m, Ra=1.71x10‘

=2 8° (vertical
¢ [Picture 19] >

Figure 4.14 Effect of Rotational Speed of 47 rpm
at it = 270 degrees, Ta = 2.77xro‘

74

   

 

 

 

 

local Nusselt Number, Nu

 

 

 

 

 

 

 

(a) Flow Field (b) Temperature Field
8 e not run
o Cold nn
4
a A . O .
" 0 ° 0 o 0 ° ° ° ° ' '
0 e 2 g o e e 0 ’ 8 0 ° 0 o o o
0 . . . . .
0.0 0.2 0.4 0.8 0.8 1.0
Y‘
(c) 0:47 rpm, Ra=1.71x10‘

¢=180° heated from below)
Picture 18]

Figure 4.15 Effect of Rotational Speed of 47 rpm
at 6 = 180 degrees, Ta = 2.7711106

Local Nusselt Number, Nu

(a) Flow Field at d = 90 deg. 0’) TemPerature Field
at6=9ldeg.
D
" 0 Rule“
0 Cold'all
a
4
o 0 .
6°°°°00000°., e.
.°e.eet'.. 00°00
"0.0 0.2 ' 0.4 0.3 0.8 1.0
Y‘
(c) o: 47 _r m Ra: 1.71x10'

75

 

 

 

 

 

 

 

 

 

 

 

1° (vertical)
¢=[P1ct ure 20]

Figure 4.16 Effect of Rotational Speed of 47 rpm
at 6: 90 degrees, Ta: 2. 77x106

 

'M'D

Mean Nusselt. Number, Nu

76

 

 

 

 

 

 

6.0
— lithium.
. - atrium.
A KW. Exp.
5.0“ A C]. pr.
4.0-
3.0-
2.0-
________ -4—--~——-——-.._~_.___._.-_____.————A
*7 ‘ \r A: *4
1.0-

'r'l"‘l"'lr"lr"l"'
0 4O 80 120 160 200 240 280 320 360
¢(°)

Figure4 17 Mean Nusselt umber vs. Angular Position
at 47 rpm cw rotation) =1.71x10‘

Ta: 2 .77x10‘

77

influence of gravity throughout the cycle and that the periodic nature of the flow field
has become insignificant. This also marks the transition to a new flow regime where
the centrifugal buoyancy works to re-establish the convection heat transfer. Again at
Q=47rpm the numerical prediction are in excellent agreement with the experimental

results as shown in figure 4.17.

4.1.5 At Rotational Speed 53.0 rpm

At Q=53rpm (Ta=3.52x106, Ra=1.83x106, and Ra,=2.92x104) the slow quasi-
steady motion seen at 47 rpm continues to dominate the flow. In figure 4.18a the
flow field at 6=0° looks the same as that of figure 4.13a, however the video tape
shows less oscillation in the shape of the small cell, and a larger area of stagnation at
the center of the larger cell. The temperature field and the local Nusselt number
distributions of parts (b) and (c) are almost identical to those of figure 4.13 too, with
less than 5% decrease in the mean Nusselt number which lies within the experimental
error margin. Note that the isotherms of figure 4.18b show increased outward fluid
motion from the mid region of the hot wall. Figure 4.18 also shows the numerically
predicted temperature field, where the isotherms are almost identical to those of the
experiment. An agreement that is also illustrated in the local Nusselt number
distributions of figure 4.18c, however, the drop of the numerical values near the

middle of the hot wall below one (pure conduction) may indicate some error in them.

At figure 4.19 the flow field is shown to be identical to that of figure 4.18, while
the isotherms are more symmetric and so are the local Nusselt number distributions.
The numerical prediction is again in good agreement with the experiment but the
Nusselt number profile is not as symmetric and the local values are slightly lower
near the lower end of the hot wall. Figures 4.20 and 4.21 also show the stable flow
field which is identical to those of figures 4.18 and 4.19, howevere, the flow fields

and Nusselt number profiles at these positions (6:180°, and92°) show some change

Local Nusselt Number, Nu

 

Experimental

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)FlowField
(experimental)
J Numerical
C.W.
(b) Temperature Field
8 O Edi-1131p.
o Cold'en up.
~m'flll‘ln.
- Con'tnlllfl.
4
“ “ c ° o o o o 9 o Arr-“'1’:
n 0 or": '''' M“ 4 ° :0
" 0 e o oy’fl‘JL”
l \r—
"0.0 ' 0.2 0.4 ' 0.6 0.0 ' 1.0
Y‘
(C) 0=53 rpm, Ra=1.83x10'

6=0° (heated from above)

[Picture 24]

Figure 4.18 Effect of Rotational Speed of 53 rpm
at 6 = 0 degrees, Ta = 3.52x106

 

79

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(experimental)
(a) Flow Field
(experimental) 3
U
(numerical)
(b) Temperature Field
3 0 Eat Well I: .
0 Cold Vent
. — mm: “in.
- Cold VIII Nun.
:
z A
c U
0
.0
E
:1
z
3 4
i
z
'5 n :\ O o O 0 Z
" 0 I'"“ """"" ., -_~ M f
.3 \gfir.’, . . o . . e‘Q‘ ~0~n-9
. \—~—/
"0.0 ' 0.2 0.4 ' 0.6 ' 0.0 1.0
Y‘
((3) 0:53 r m, Ra=1.83x10‘

=2 0° vertl a1
‘1, [Pictu(re 23‘] )

Figure 4.19 Eflect of Rotational Speed of 53 rpm

at (I) = 270 degrees, Ta = 3.52x10‘

 

M'H

Local Nusselt Number, Nu

 

80

 

 

 

 

 

 

 

 

 

 

 

 

 

Experimental
¢ = 177 deg.
(a) Flow Field
6 = 180 deg. Numerical
Experimental 6 = 180 deg.
(b) Temperature Field
8 0 Eat Well I: .
0 Cold Ian I
. —- an m min.
- Cold nn NEIL
4
a \ °\\A - D O O O D A A . .
a ‘9. .0. ..“'. ~~~~~~~~~~ _ 0 0
~ \ ~O
.0000. 0’. “2:3—’
“00 0.2 ' 0.4 ' 0.6 ' 0.8 1.0
Y'
(c) rpm Ra=1. 83x

1'0
¢= 177° heated from below)
Picture 25]

Figure 4.20 Effect of Rotational Speed of 53 rpm
at 6 = 180 degrees, Ta = 3.52x10‘5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) Flow Field
4) = 90 deg. 3' 2
m .
Experimental
/ 6 = 90 deg.
Numerical
(b) Temperature Field
3 e m nu hp.

0 Cold nu lip.

— Int I'll! mu.

- Cold nu Nun.
3
z A
h- V
0
.0
El
5
z
3
i

,
z \ O /:\
gn\‘e--°°o ° ° °o/. .1
3 . .~.: ..... :—--:~“,s(f\ ‘0 O g/lé
"0.0 . 0.2 0.4 0.6 0.0 1.0
Y'
(C) 0:53 r m, Ra=1.83x10'

[Picture 22]
Figure 4.21 Effect of Rotational Speed of 53 rpm

6: 2° (vertical)

at it = 92 degrees, Ta = 352x105

82

which is believed to be due the persistent influence of gravity. The excellent
agreement between the experimental and the numerical temperature fields and local
Nusselt number distributions is also clear in parts (b) and (c) of these figures.
Figures 4.19 through 4.21 also show similar characteristics to figures 4.14 through
4.16 of 47 rpm, respectively, which indicate the stability of the enclosure flow at this
region of the controlling parameters (namely Taylor and Rayleigh numbers). The
increase in heat transfer near the ends of the isothermal walls in all these figures is
attributed to the conduction in the insulated walls, but it can be seen that the values

near the center of the hot wall approach pure conduction.

Figure 4.22 shows that at a Taylor number of 3.52x10‘5 the field is greatly
dominated by the wall shear, which as mentioned before, induces solid body rotation
and limits the transfer of heat to the pure conduction mode. The numerical
calculation also predicts the same uniform profile of the mean Nusselt number. The
profile however is not completely flat which again indicates that the influence of
gravity can still be detected at the current rotational speed. Comparing figure 4.22 to
figure 4.17 it can be seen that the mean Nusselt number retains the same uniform
distribution as a function of the angular position but the overall mean is decreasing

slightly ((6%) with increasing rotation rate.

4.1.6 At Rotational Speed 62.0 rpm

The effect of increasing the speed of rotation to 62 rpm
(Ta=4.82x10°, Ra=1.7x105, andRa,=3.7x104) is shown in figures 4.23 through 4.27.
Figure 4.23a shows the responce of the flow field to the increase in the rotation rate
Q where it forms two equivalent cells that extend between the two isothermal walls.
One of these cells (extends between the lower end of the walls) contains the
stagnation region which was observed near the lower end of the cold wall at slower

speeds of rotation, and the fluid circulates around it unable to penetrate into it.

Mean Nusselt. Number, Nu

83

 

 

 

 

 

6.0
— Elihu.
. - 0.1.Num.
A Elk).
5.0_ A C.Y.lxp.
4.0-
3.0-
2.04
A- -------- A" ------- n- ———————— K ———————— A
_,,....—-—-—"'— ‘ __
A ‘ l ‘
1.0-
0-0 "'l"'l"'l"'l"'l"'l"'l"‘l"‘l
0 4O 80 120 160 200 240 280 320 360

¢(°)

Figure 4.22 Mean Nusselt umber vs. An lar Position
at 53 rpm cw rotation) a=1.83x10°

Ta: -3.52x10‘

84

   

C.W.

 

 

 

 

 

 

 

 

 

 

 

(a) Flow Field (b) Temperature Field

6=0deg. 6:358 deg.

e O Bet'all

0 WV“!
:1
z A
cu
B
5
g 4
8
3
'82: on°°°°°3°o‘.:.°.
3 O 3 . . . e 0 o
. e . . e .
"0.0 ' 0.2 ' 0.4 ' 0.6 ' 0.8 ' 1.0
Y'
(c) - 2 rpm, Ra=1.70x10‘

9-6
¢=358° heated from above)
Picture 28]

Figure 4.23 Effect of Rotational Speed of 62 rpm
at 6 = 358 degrees, Ta = 4.1mm6

85

Though the two cells are not identical they are almost equal in size, and they both
emanate from the central region of the hot wall with outward opposite circulation at
their outer boundaries. Again this interesting flow pattern is independent of the
angular position as can be seen from part (a) of figures 4.24, 4.25 and 4.26, however
some variation can still be seen in the isotherm patterns especially at 6=l80° and 90°
(figures 4.25b and 4.26b) which points out the persistance of the influence of gravity.
The slow counter circulation of the two cells result in an area of stagnant fluid near
the center of the hot wall which in turn suppresses the heat transfer in this region to
pure conduction as can be seen from the local Nusselt numbers of parts (c) of the
aforementioned figures. The bending of the isotherms shown in parts (b) also
indicates circulation in the direction perpendicular to the istherrnal walls which
results in the increase of Nusselt number near the ends of the hot wall because the
cold air moves down the insulated walls and impinges on both ends of the hot wall.
The Nusselt numbers near the ends of the cold wall however are higher than what
they should be because of the conduction in the insulated walls. Figure 4.27
illustrates the characteristics of the mean Nusselt number at the current rotational
speed where the independence of the angular position and the low values indicate the
beginning of the complete domination of the centrifugal buoyancy, which is also

supported by the matching numerical calculations.

Note that the interesting two cell flow configuration encountered here arises
from the domination of the imposed uniform centrifugal force which creates a highly
stable two dimensional cellular flow pattern (may form multi-cellular patterns in a
more shalow enclosure) that resembles the classical type of stable Benard convection
(in a stationary heated from below enclosure) as predicted by Edwards(1967).
Therefore, as the speed of rotation increases the two cells are expected to become
more like a mirror image of each other and the circulation is expected to increase,

thus increasing the heat transfer.

86

KKK

(a) Flow Field (b) Temperature Field
6 = 270 deg. <|> = 269 deg-

   

CD

 

0.
g:
a:

CD

 

 

 

Local Nusselt Number, Nu
:1
I 3

.0

.0

 

 

 

 

 

 

 

0.0 ' 0.2 ' 0.4 ' 0.6 ' 0.6 ' 1.0
r
(C) 0:62 r m, Ra=1.70x10'
6: 9° (vertical)
[Picture 29]
Figure 4.24 Effect of Rotational Speed of 62 rpm
at 6 = 269 degrees, Ta = 4.82x10‘

87

CW. \

I" l A
I'LW
Y°<-]

(b) Temperature Field

 

(a) Flow Field

 

O Bot'nll
0 Cold!“

Cb

 

 

00

O
e°°

 

Local Nusselt Number, Nu

CD
.0
.0

500

 

 

 

 

 

 

 

0.0 ' 0.2 ' 0.4 ' 0.6 ' 0.6 1.0
r
(c) =62 rpm, Ra=1. 70x1 10‘

¢=180° heated from below)
Pict ture '31]

Figure 4.25 Effect of Rotational Speed of 62 r m
at 6 = 180 degrees, Ta = 4.82x10

(a) Flow Field (b) Temperature Field

6 = 90 deg. ¢ = 87 deg.

8 O Hel'dl

O Cold'en
:5
z A
a. V
.0
E
9
Z
3 4
8
8
z 0
a n a 2 o o 0 ° 0 . o
3 " ° . ' ° o
O . .
V0.0 0.2 0.4 0.8
(C) Ra=1.70x10'

88

 

 

 

 

 

 

 

 

 

 

 

 

9:62 1'; ,
¢= 7° (V
[Picture 30]

Figure 4.26 Effect of Rotational Speed of 62 rpm
at 6 = 87 degrees, Ta = 4.82x10‘

 

Mean Nusselt Number, Nu

89

 

 

 

 

 

6.0
- KlNurn.
.. — atrium.
A Klfitp.
5.0“ A C.'.lxp.
4.0-
3.0-
]
2.02_________A ________ .A... _______ A- _______ _A
A
1.03
0-0 'filfi"l"'l"'F"'TT"I"'I“'17"1
0 4O 80 120 160 200 240 280 320 360

¢ (°)

Figure 4.27 Mean Nusselt umber vs. An uiar Position
at 62 rpm cw rotation) a=1.70x10°

Ta= 4. 82x106

90

Also as the speed of rotation increases the temperature field is expected to simulate

the situation of a hot plume rising from a horizontal surface into cooler air.

4.1.7 At Rotational Speed 76.0 rpm

The two identical cell configuration predicted above is clearly illustrated in
figures 4.28 through 4.31, where the effect of increasing the rotational speed to 76
rpm (Ta=7.24x106, Ra=1.7x105, and Ra,=5.56x104) is presented. The flow pictures in
parts (a) of these figures belong to a slower rotational speed of 73 rpm, however they
show the two equal cells to continue to circulate outward from the hot wall as a
result of the centrifugal buoyancy. The difference in the brightness of these two cells
is due to the injection of the smoke near the center of one of the insulated the
insulated walls. The isotherms of parts (b) are also identical and symmetric at all
angular positions thereby confirming the stability of the centrifugally driven buoyuant
flow. However the most interesting feature about this buoyant flow regime is its
resemblance to the stable condition of heated from below. The only difference
between the two is that the heated from below temperature field is affected in the
vertical direction only by gravity, therefore it is expected to stretch or elongate
vertically, while in the present situation it is influenced by the circumferencially
uniform centrifugal force, therefore it is expected to stretch outward unifomrly and
become more like a mushroom stemming from the hot wall as the rotational speed is

further increased.

The local Nusselt number distributions along the isothermal walls at the four
designated angular positions are shown in part (c) of figures 4.28 through 4.31. Since
all four profiles are identical it is no longer necessary to compare them to each Other.
However by comparing them to the corresponding profiles at rotational speed of 62
rpm (figures 4.23c through 4.26c) it can be seen that they are more symmetric about
the median to the isothermal walls which further affirrns that the two flow cells are

Local Nusselt Number, Nu.

91

 

 

 

 

 

 

 

 

 

 

 

 

(a) Flow Field (b) Temperature Field
= 0 deg- 6 = 358 deg.
8 0 Boil-11
0 Cold '6]!
4
6 ° 0 o o
O 0 O O
2 0 a 2 ° 0 0 Z . ‘1
O . . . O
O . .
0 e
"0.0 0.2 0.4 0.6 0.6 1.0
r
(9) rpm, Ra=1.70x10

6: 358° [(Heateds fro above)
t 451’

Figure 4.28 Effect of Rotational Speed of 76 rpm
at 6 = 358 degrees, Ta = 7.241(106

 

 

 

 

 

Local Nusselt Number, Nu

 

 

 

 

 

 

(3) Flow Field (b) Temperature Field
6 = 270 deg. 6 = 268 deg.
91 0 Bot Well
0 Cold 'all
4
2 8 R 9 n 8 .2
"0.0 ' 0.2 0.8
(C)

Figure 4.29 Effect of Rotational Speed of 76 rpm

at 6 = 268 degrees, Ta = 7.24x10‘

 

93

CW. \‘

      

 

 

 

 

 

 

 

 

 

 

 

H.W. 16+]
(6) Flow Field (b) Temperature Field
4): 180 deg. ch: 178 deg.
8 0 Hot Wall
0 Cold nn
3
z A
5 U
i
2
g 4
3
g o o 0 o
’5 '5 A A O O o o O O n A . .
3 O . . o
O . . . Q
”00 ' 0.2 ' 0.4 ' 0.6 ' 0.8 . 1.0
r
(C) 0:76 rpm Ra=1.70x10'

¢=i78° heated from below)
cture 342]

Figure 4.30 Effect of Rotational Speed of 76 rpm
at 6 = 178 degrees, Ta = 7.24x10‘

(a) Flow Field (b) Temperature Field
3
0 Hot Wall
0 Cold nn
:1
z A
k. U
0
i
z
3 4
G)
g 0 0 o
z o o o o o 3 o
8 2 ' o o o 0' . '
O
.3 9 . . o o
O . .l . . g
c ' l I V
0.0 0.2 0.4 0.8 0.8 1.0
Y‘

94

 

 

 

 

 

 

 

 

 

 

 

 

(c) 0:76 r m, Ra=1.70x10‘
6: 0° (vertical)
[Picture 343]

Figure 4.31 Effect of Rotational Speed of 76 rpm

at 6 = 90 degrees, Ta = 7.24x10‘

95

identical in size and flow velocities. It can also be seen that the hot wall profiles are
becoming more concave in the middle region and higher near the ends (almost 100%
higher than the middle) with the opposite profile on the cold wall, which points to
increased circulation in the flow cells. Finally, the mean Nusselt number values
shown in figure 4.32 are as uniform as expected, with an overall mean value of 1.4

which is less than 4% higher than that of the previous rOtational speed (62 rpm).

4.1.8 At Rotational Speed 90.0 rpm

Figures 4.33 through 4.36 show the effect of a rotational speed Q=90rpm
(Ta=1.02x107, Ra=1.82x105, and Ra,=8.35x104) on the flow and temperature fields
and the Nusselt number distributions, the flow visualization pictures however are for
a lower rotational speed of 83 rpm. Again the figures show a continuation of the
trend of no dependance on the angular position, and the flow continues to be steady
and highly stable as can be seen from part (a) of the above figures and from the
continuous motion recorded on video tape. The shape of the isotherms of part (b)
indicates smaller temperature gradients near mid-hot wall and higher gradients on the
sides which means increased circulation. Also note that the pocket of hot air
emanating from the hot wall extends further towards the cold wall as predicted. The
numerical calculation of the flow and temperature fields shown in parts (a) and (b)
again agrees with the experiment, however it should be noted that gravity was

neglected in the numerical model in order to match the flow field.

By comparing the local Nusselt number profiles of figures 4.33c to 4.35c with
those of figures 4.28c to 4.31c it can be seen that the lowest nusselt number values
near the middle of the hot wall are almost the same as those of the lower rotational
speed of 76 rpm (approximately equal 1 which correspongs to pure conduction) while
the maxima near the sides are higher, which amounts to a 13% to 20% increase in

the mean Nusselt number values shown in figure 4.36 over the corresponding

Mean Nusselt Number. Nu

96

 

 

A KW. hp.
A 0.1. Bxp.

 

 

0.0 fin...“
0 4o 80

Figure 4.32 Mean Nusselt umber vs. An

at 76 rpm cw rotation)

Ta=

I ' ‘Ifi F '

"'Iffllfl'ln'ln'l
120 160 200 240 280 320 360'

-7.24x10‘

¢ (°)

lar Position
a=1.70210°

97

if V. H.W. \

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

¢ Odeg ¢=3deg.
H.W.
Numerical
(neglecting
. gravity)
¢=0deg- ¢=0deg.
C-W- — 'C—.w.
(a) Flow Field (b) Temperature Field
8 0 Hot Well
0 Cold In]!
E
. 6
L4
0
.D
E
2
2
F3 4 0 ° 0 o 3 o
g ° 0 o o
z o
'3 8 O o 0 ° 0. o. 0.
3’32 ' ‘ '
° 0
O .1 .
O I
"0.0 ' 0.2 ' 0.4 ' 0.6 ' 0.8 ' 1.0
(e) y.

= rpm, Ra=1. 82x10“
¢= 3° o(hearted from above)
[Picture 445]

Figure 4. 33 Effect of Rotational Speed of 90 rpm
at ¢= 3 degrees, Ta= 1.02x107

  

98

 

3:
c.)

_ Experimental

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1C3), /
0' ‘ A 2 0
Numerical \
neglecting
gravity
(a) Flow Field (b) Temperature Field
8 0 Hot Well
0 Cold 'all
:1
z a
L: U
Q
Q
E
:1
Z
:2. 4
3 o D
0 O . .

32 9 . = ° 0 D e
3 . I . o .

"0.0 0.2 0.4 0.6 0.8 1 0

(e)

0:90 r m, Ra=1.82x10‘

= 0° (vertical)

[Picture 440]

Figure 4.34 Effect of Rotational Speed of 90 rpm
at 4) = 270 degrees, Ta = Lozxio"

 

'M'H

 

 

 

 

 

99

 

 

 

 

 

 

:- 2 a >> .2
© Numerical
o = 90 deg. ) neglecting 4) = 38 deg.
gravity
(a) Flow Field (b) Temperature Field
8 0 Hal. Wall
0 Cold In"
E
- 6
L-
G)
.D
E
.‘J
z
u 4
'6 O O 0 O b
Z O 2 ° 0 o . o o
a . - ' ° n
O 2 I . u
0 . g
G e
0.0 &? 0.4 Y. 0.6 0.8 1.0

 

 

 

 

[Picture 443]

Figure 4.35 Effect of Rotational Speed of 90 rpm
at 4) = 88 degrees, Ta = 1.02x107

0:90 r m, Ra=1.82x10‘
¢= 8° (vertical)

 

 

T~ . Experimental \‘
O u
, . , 3 .0
- .2 :3 ,2
, Y Q) = 90 deg.
A .r .. .

Mean Nusselt Number. Nu

100

 

 

 

 

 

6.0
— 11le
d - CJqum.
A Klikp.
5.0_ A C.'.pr.
4.0‘
3.04
A A A
i __________________________ e, ________
2.04
l. ‘ ‘ A
1.0-
0.0 f'rl"'l"'lfi"l"‘l'T‘l"'l"'f"'l
0 40 80 120 160 200 240 280 320 360

¢ (°)

Figure 4.36 Mean Nusselt umber vs. An lar Position
at 90 rpm cw rotation), a=1.82x10°

Ta = 1.02xio7

101

values shown in figure 4.32 for Q=76rpm. Figure 4.36 also shows flat experimental
and numerical mean Nusselt number profiles which shows the centrifugal buoyancy
to continue to be the dominant force, and the influence of gravity to disappear
completely. The overall mean Nusselt number is also increased by about 15% which

is proportional to the increase in the angular speed.

4.1.9 At Rotational Speed 118.0 rpm

Because of the size and weight of the test section the highest speed of rotation
reached in the present study was 118.0 rpm which corresponds to a Taylor number
Ta=1.78x107. At this speed the Rayleigh number is Ra=1.78x105 and the rotational
Rayleigh number is Ra,=1.4><105. The isotherm patterns of figures 4.37a through
4.39a show that the steady symmetric, two cell flow regime continues to prevail.
They also show the bulging of the hot air pocket near the cold wall. The numerical
calculation also shows a similar behavior, however they are not completely
symmetric like the experiment, and that is because gravity is considered in the model
in this case. This shows that the numerical model over estimates the influence of
gravity at such a high Taylor number. The local Nusselt number profiles of part (b)
of the same figures show noticable increase in the mid-wall Nusselt number values
which suggest increased motion in this previously stagnant region. The numerical
values do not show that improvement in the mid hot wall local Nusselt numbers, and
the hot wall profiles are skewed toward one side of the enclosure, but interestingly
this skewed orientation is the same in all angular positions which suggests that it may

not be the result of over estimation of the influence of gravity.

Figure 4.40 shows significant increase in the mean Nusselt numbers with an
overall mean value of 1.95, almost 20% higher than that of the lower rotational speed
of 90.0 rpm, and more than 43% higher that the lowest overall mean attained at
Q=62rpm.

102

 

  

 

 

 

 

’ - C.W. C.W.

Experimental Numerical
¢=ldeg. ¢=0deg.

(a) Temperature Field

 

 

 

 

Local Nusselt Number, Nu

 

 

 

 

 

 

 

8

O Hot'nflhp.

O Cold'anEp.

— EetWau FL

- Cold'anNum.
6

O
4 n 9——°—.fi3 O
O,’— ~.\ 0
o ’ \
/ \\ o
o ,’ \ f—\
o ,’ \ o.

*8 I ' .\\ G O
n x \0 a . e
g \z’ . ' \\ f

/
n
U I v v v I
0.0 0.2 0.4 0.6 0.8 1.0
Y‘

(b) 0:118 rpm, Ra=1.78x10‘
¢= 1° (heated from above)
[Picture 41]
Figure 4.37 Effect of Rotational Speed of 118 rpm
at (p = 1 degree, Ta = 1.75x107

103

 

 

KR]

Experimenta Numerical
9" - ¢ = 270 deg.

(a) Temperature Field

 

CD

 

 

 

 

 

 

 

 

 

 

O KotV-l‘l 31p
0 Cold n11 Exp
— EM. Wall Nun
- Cold'nllflum.
é
. 6
l-
cu
.D
E
:1
2
fl 4 ’6’0 0 3‘s:
3: ,cv \°
3 4 o 3’ \\°\ /-\
Z o (x/ Q
0‘ o
'8: 2 8 NJ A - . . \\ o A.
/ fl:
'3 \V \\‘J
n
U r r r r r
0.0 0.2 0.4 0.6 0.8 1.0

y.
0» 0:118 r m, Eta—178x10“
¢=2 8° (vertical)
[Picture 43]
Figure 4.38 Effect of Rotational Speed of 118 rpm
at if = 268 degrees, Ta = 1.75x10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.0 3: 0
f. . .
Experimental Numerical
(a) Temperature Field
8
0 KM. Well Rap.
0 Cold wan Erp.
_ — Hot Wall Nun.
- Cold Van Hula.
é’
.. 6
f—
l)
E o
O O
O
2 0 ° 0
g 4 n , , ‘ \ \ o
3 0 o ’//’ \\\\ c: /\
g . 8 O 9/ I .\ a . O
“a X, ' . ‘ \ O o
O 2 7’ \ Q : \\\ I .
c I Y I r l
0.0 0.2 0.4 0.6 0.8 1.0
Y.
(b) (2:118 r m, Ra=1.78x10‘
°(vert ical)

¢[Picture 40]

Figure 4 .39 Effect of Rotational Speed of 118 rpm
at ‘1’: 90 degrees, Ta= 1.75x107

Mean Nusselt Number, Nu

105

 

 

 

 

 

6.0
— 11.le
q - 0.7.Num.
A Klkp.
5.0_ A C.7.pr.
4.0-
1 A
3.0-4l A
__________________________ A--—-'-——---
.i
2.0 A ‘ ‘ ,
1.0-l
0-0 fir'lH'ITHliHIH'l'H[*Hl'r'lrr'l
O 40 80 120 160 200 240 280 320 360

t5 (°)

Figure 4.40 Mean Nusselt Number vs. An ular Position

at 118 rpm (cw rotation), a=1.78x10‘

Ta = 1.7sxlo7

106

4.2 Effect of Rotation on the Mean Nusselt Number

The mean Nusselt number is the result of the numerical integration of the local
Nusselt numbers over the height of the isothermal wall at an angular position. And as
shown above these values were used to illustrate the variation in the enclosure heat
transfer over one cycle at every rotational speed examined. In this subsection the
mean values are discussed as a function of the rotation rate in order to demonstrate
the influence of rotation on the enclosure convection heat transfer at each previously
selected angular position. At a particular position the mean Nusselt number of the
hot wall is expected to be equal to that of the cold wall. However as mentioned
previously in this chapter, lateral conduction in the insulated (plexiglas) walls results
in significant increase in the cold wall values, especially when the convection
currents are slow. Therefore, only the mean Nusselt numbers at the hot wall are
discussed, while the values at the cold wall are shown for their qualitative value.
Note that the mean Nusselt numbers are plotted vs. the product of Rayleigh and
Taylor numbers (instead of the rotational speed) in order to bring to mind the
influence of the combined gravitational and rotational buoyancies on the enclosure
heat transfer. However the rotational Rayleigh number could have also been used in

conjunction with the other two parameters without loss of the physical phenomena.

Figures 4.41 to 4.44 show the inluence of the rotational speed 0 ranging from
22.5 rpm (Ta=6.34x105) to 118 rpm (Ta=l.75><107) at a Rayleigh number,
Ra==(1.77:l:0.07)x105. At the vertical position ¢=270 figure 4.41 shows a sharp decline
of more than 53% in the mean Nusselt number between Q=22.5-47.0 rpm
(Ta=6.34x105-2.77x106). This region was characterized initially by enhanced mixing
due to the combined influence of the gravitational buoyancy and the Coriolis force,
then by the transition to near solid body rotation due to the viscous shear forces.
Then heat uansfer starts to show a moderate increase due to the growing influence of

the centrifugal buoyancy which resulted in the steady two cell flow configuration.

. . .-..u. -.—t_....v
«.2 ...~..»a..-~JZ ..........-.Z 2

Mean Nusselt Number, Nu

107

 

 

 

 

 

6
o H.W. Exp.
0 CW. Exp.
. —— H.W. Num.
- - C.W. Num.
4..
O I I T I I I T
1E+10 1E+12 2E+12 3E+12 4E+12
Ra.Ta

Figure 4.41 Mean Nusselt Number vs. RaxTa
at Angular Position ¢=270 (vertical)

108

The numerical resuls are also in agreement with the experimental results, and they

show that the heat transfer should peak somewhere between 22.5 and 32 rpm.

In figure 4.42 it can be seen that at the heated from below position (¢=180°) the
maximum value of the mean Nusselt number (near the lower end of the scale)
increases by almot 47% over that of the previous vertical position, and that is mainly
due to increased influence of the gravitational buoyancy. Despite the significant
increase in the maximum value the mean Nusselt number decreases sharply to almost
the same value of the previous position (¢=270°) at 47 rpm and then follows the
same trend. The numerical results also show the large increase at low rotation rates

and the identical behavior at higher rates.

At the Opposite vertical position (¢=90°) figure 4.43 shows a profile similar to
that of o=180°, and while the experiment shows about 15% inrease in the peak value
the numerical predictions are almost the same. The results are also in agreement with
those of Hamady et al. (1990), where the mean Nusselt number is shown to increase
consistantly from slower rotation rates (6.5 rpm to 17 rpm) until it connects with the
results of the present study. Figure 4.44 shows the mean Nusselt number profile at
the heated from above position (¢=O°) where the maximum value at slow rotation
rates is low as expected because of the absence of the gravitational buoyancy at this
angular position.

Figures 4.41 to 4.44 clearly show that three regions can be identified where the
heat transfer exhibits unique characters: a low r0tation region where the heat transfer
is enhanced in a periodic fashion due to the combined influence of gravity and
Coriolis acceleration, a moderate rotation region where the heat transfer is greatly
suppressed to a minimum of approximately 1.4 because of the viscous shear forces,

and a high rotation region where the heat transfer is steady and moderately increasing

with the increasing speed of r0tation.

Mean Nusselt Number, Nu

MD

 

 

 

 

 

6
o H.W. Exp.
0 CW. Exp.
a — H.W. Num.
OWLNunL
O T I’ I l I I fl
1E+10 1E+12 2E+12 3E+12 4E+12
Ra.Ta

Fi ure 4.42 Mean Nusselt Number vs. RaxTa
a Angular Position ¢=180 (htd frm below)

Mean Nusselt Number, Nu

110

 

 

 

 

 

6
o H.W. Exp.
9 o C.W. Exp.
.1 —— H.W. Num.
C.W. Num.
- Hamady(1990)
O I l I I T l I T
1E+10 1E+12 2E+12 3E+12 4E+12
Ra.Ta

Figure 4.43 Mean Nusselt Number vs. RaxTa
at Angular Position 95:90 (vertical)

Mean Nusselt Number, Nu

111

 

o H.W. Exp.
0 C.W. Exp.
. — H.W. Num.
—--(1W.Nunr

 

 

17

 

 

o . r
1E+10 1E+12

l l '
2E+12 3E+12 4E+12

Ra.Ta

Figure 4.44 Mean Nusselt Number vs. RaxTa
at Angular Position 93:0 (heated from above)

112

4.3 Effect of Rotation on the Overall Mean Nusselt Number

In figure 4.45 the overall mean Nusselt number is plotted vs. the product of
Rayleigh and Taylor numbers. The overall mean is the average of the mean Nusselt
numbers at the four selected angular positions, which means that it represents the
average heat transfer to be expected over a cycle at a given rotation rate. In this
figure the three regions suggested above are roughly marked. In region 1
(6.34x105STaSl.28x106) the mean Nusselt number is noted to climb slightly from
3.63 to 3.87, an inrease of almost 7% which is supported by a comparable increase in
the numerical values. And though two data points in this region are not sufficient for
obtaining a numerical correlation, the increase seems proportional to RamxTal’8
which further supports the idea of dominating gravitational buoyancy and Coriolis

forces in this region.

Region 2 was characterized by the domination of the wall shear effects, where
the visual study of the flow indicated a transition from the unicellular slushing
motion to the bicellular steady flow pattern. In this region the overall Nusselt number
droped from a maximum of almost 3.9 to a minimum of less than 1.4 over the Taylor
number range of 1.28x1055Ta52.77x106. In region 3 the steady 2 cell flow pattern
was an evidence of the domination of the centrifugal buoyancy which is shown to
enhance the Nusselt number by almost 39% over a Taylor number range of
2.77xlo6srasl.75xlo7. The data in region 3 can be correlated by:

Nu = 0.067 Rafi-2825 (4.1)

this correlation was obtained using the method of least squares, and it predicts the

experimental data in this region within i5% as shown in figure 4.46.

Overall Mean Nusselt Number, Nu

113

 

    

 

 

I I I H.W. Exp.
. I I D C.W. Exp.
Reglon 1 — H.W. Num.
| . l — - C.W. Num.
Region 2
4- , .. ,1 l .
, \ Reglon3
l
2‘ I
l l
o l I. .
1E+11 5E+11 1E+12 5E+12

Ra.Ta

Figure 4.45 Overall Mean Nusselt Number vs. RaxTa

 

Overall Mean Nusselt Number, Nu

114

 

,rs

— Equ 4.1
I H.W. Exp.

OD
l

Region 3

N
l
I

 

 

 

H

20 30

Ra 0.2825
r

Figure 4.46 Correlation of The Overall Mean
Nusselt Number at High Rotation Rates

115

In general the experimental results presented in this chapter and the good
agreement with the numerical predictions provide a comprehensive description of the
qualitative nature of the flow dynamics and accurate estimates of the heat transfer in
the range of parameters considered. Unique flow patterns and a well defined
transition from gravity dominant to centrifugal force dominant regions explained the
augmentation of heat transfer in some regions of the controlling parameters and its
suppression in other regions. The results showed the transition from boundary layer
flow to potential flow as a result of friction forces which greatly suppressed the heat
transfer to near pure conduction. It showed the transition from highly unsteady to
interesting steady and uniform temperature and flow patterns at high speed rotation.
The flow is shown to be periodic at low rotation rates (region 1) and highly stable at
high speed rotation (region 3), while the effect of rotation throughout the study was
to make the flow two dimensional.

CHAPTER 5

SUMMARY AND CONCLUDING REMARKS

In the present study natural convection heat transfer in a rotating enclosure was
experimentally investigated. A horizontally oriented, differentially heated, air filled
parallelepiped enclosure of cross-sectional aspect ratio of one (square) and
longitudinal aspect ratio of ten was confined to rotate about its longitudinal axis at
uniform angular speeds. Real time visual flow observations were done using smoke
and laser sheet at the mid-section of the enclosure. Flow visualizations were used to
qualitatively understand the interaction of gravitational and rotational buoyant flows
and their effect on the heat transfer. A Mach-Zehnder interferometer was used to
study the 2-D spatial and temporal temperature distributions inside the cavity. The
interferometer was adjusted in the infinite fringe mode to directly reveal the entire
temperature field, which allowed subsequent calculation of the local N usselt number

distribution at the enclosure’s isothermal walls.

The study revealed interesting flow and temperature patterns in the Taylor
number range of 105 to lo7 and rotational Rayleigh number range of 103 to 105 for a
Rayleigh number of approximately 2x105. It also established three regions where the
characteristics of heat transfer are greatly influenced by the domination of one or

more of the controlling parameters.

116

117

At slow rotation rates up to 32 rpm, rotation was seen to enhance the heat
transfer and force gravitationally driven buoyant flows near the isothermal walls into
boundary layer flow regimes. The boundary layer flow induced the formation of
vortical structures at opposite comers on the isothermal walls as the hot wall moved
from top to bottom (¢=O° and 270°). These vorticies were swept inward to the core
region due to the Coriolis force during the second half of the cycle. Gravitational
buoyancy, and Coriolis force induced cross flows, resulted in counter rotation of the
core region during the first half of the cycle, and co—rotation during the second half
in an oscilating, sloshing like type of motion which enhanced the mixing of the core

region fluid, and therefore increased the heat transfer.

The convection process at this first domain of low rotation rates is highly
periodic because of the interaction between the influential gravity field and the
uniform less effective rotational force fields. This periodicity was seen in the flow
and isotherm patterns, as well as in the asymmetric profiles of the local Nusselt
number at the isothermal walls and the sinusoidal distribution of their mean values

over one cycle.

At rotation rates from 32 to 47 rpm the thickness of the boundary layer
increased due to increasing viscous friction, until the entire enclosure fluid attained a
state of near solid body rotation. This second region represented a transition from
gravitationally dominated to centrifugally dominated flow fields, and was
charaCterized by a sharp decline in heat transfer, which dropped from a maximum of
3.9 at 32 rpm to a minimum value slightly higher than that of pure conduction in air
at 47 rpm. The periodic nature of the flow was also seen to fade in this transition
region.

In the third region of 53 to 118 rpm the centifugal force became more
dominant, which resulted in an interesting flow regime of two equal, counter rotating

cells which resembled the classic case of a fluid layer above a horizontal plate heated

118

from below under stable conditions, with the difference that the centrifugal force acts
in the radial direction while gravity acts only in the vertical direction. In this region
the heat transfer increased steadily with the rotational speed. The overall Nusselt
number data in this region was correlated with the rotational Rayleigh number which
represents a measure of the centrifugal buoyancy. At 76 rpm the periodic motion
completely disappeared, and the isotherms and the local Nusselt number distributions

became symmetric.

In conclusion, the experimental results of the present study provide a major
contribution to the present knowledge on enclosure heat transfer by establishing the
existance of a transition region from gravity dominated to centrifugally dominated
natural convection regimes in axially rotating enclosures in general. This transition
region was shown to be associated with substantial suppression in heat transfer to

near pure conduction in the working fluid.

In addition, several other aspects about this study are noteworthy from both

academic and application points of view:

The study contributes to the basic understanding of heat transfer in rotating
enclosures under the effect of gravity, an area for which experimental data is almost
non-existent. Consequently, the experimental results were used to validate the
numerical algorithm of Yang, Yang and Lloyd (1988), which showed excellent

agreement with the experiment in most cases.

Also the experimental results provided test cases for the enhancement or
suppression of local or mean heat transfer due to the combined effect of rotation and
gravity. Such cases are extremely useful in the design of certain rotating devices
where temperature control is vital. And by applying the mass transfer analogy the
results can be of equal importance in chemical deposition processes where control

over the uniformity of the rate of mass transfer is essential.

 

119

Increasing the rotation rate to the point where gravity effects become negligible
makes available a tool for simulating the heat transfer process in a near zero gravity

field, a new area of great importance to space lab research and technical applications.

It is also worth noting that the fiber-optics based flow visualization system
develOped for the present study was shown to be a powerfull tool in studying rotating
enclosures in general, and that the interferometry technique used for collecting the
thermal data proved to be suitable for studying rotating systems despite the sensitivity

to the rotation-induced vibrations.

At this point some suggestions are made for future experimental and numerical
work on heat transfer and buoyant fluid flow in rotating enclosures, with either

interacting or negligible gravity field.

Fore a more complete understanding of the physics of the problem it is
important to establish the effect of, the imposed temperature difference and the
properties of the working fluid for a broad range of Rayleigh and Prandtl numbers,

on the transition from gravitationally driven to centrifugally driven buoyant flows.

Also as a natural extension to the present study it is suggested to increase the

rotational speed and study the delay of transition to turbulent flow re giems.

It is also interesting to study the effect of some orientation and geometric
factors like the aspect ratio (a more shallow enclosure), and the angle of inclination

between the axis of rotation and the gravity vector.

Finally, it is of equal importance to study the effect of introducing a distance of
eccentricity between the geometrical axis of the enclosure and the axis of rotation

which simulates off-axis parallel rotating cavities in a wide range of applications.

Appendices

APPENDIX A

INTERFEROGRAM ANALYSIS

When the Mach-Zehnder interferometer introduced in chapter 3 is adjusted in
the infinite fringe setting, each resulting fringe is the locus of points in a 2-
dimensional field where the refractive index of air is constant. According to the

Gladstone-Dale equation,

1111- = C (A.1)
p

where C is the Gladstone-Dale constant, the refractive index is linearly related to the
density of air. This implies that each fringe represents the locus of points in a 2-
dimensional field where the density is constant. Therefore, by utilizing the ideal gas

equation of state,

P = pRT (A2)

at constant static pressure, each fringe can be calibrated as the locus of points where
the temperature is constant (isotherm). Using this result, the Temperature gradient at
the immediate vicinity of the isothermal walls of the enclosure can be measured at
discrete locations and used for calculating the local Nusselt numbers as discussed in

Appendix B.
The temperature differenc between two isotherms is calculated by,

_ (TH-TC)
- m—l

AT-

1

(A.3)

120

 

121

where, m is the number of the fringes in the field. The distance between the wall and
the center of each fringe is then measured using a high resolution traveling
microscope, and the temperature vs. location data is then plotted as shown in the
example of figure A.1, where x=0.0 is at the hot wall and x=1.0 is at the cold wall.
A polynomial function would then be fitted to the measured points, and the
temperature gradient at the hot and cold walls would be obtained by differentiation of

the function at x=0.0 and x=1.0, respectively.

Two sources of error in the process of determining the temperature gradient
contribute to the relatively low accuracy of this method of analysis based on the
infinite fringe mode of interferometry, first the uncertainty in the location of the
center of a fringe, and second and more importantly the difficulty associated with
choosing the most accurate function that predicts the measured data. Estimates of the

error associated with each one of these sources are discussed in Appendix C.

7.
«I.

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a

o. r ad 0.0 To Nd 0.0

P-pnbbP—hbbPPbPPPbbthb-Pn-hnpnPh-upPPb—Pkbbb-.PFbupbb-Pbbbb-thPbbhthP—pbn-hPhpnpnthphnubhhnanb

 

 

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IIIIIIIIIIIIIIIIIIlIIlllllllllllllllll'IlllllillIUIIIIIIIIIIIIIIIIUIIIIIIUIIIIIIIIIUIIIIIIIIIIIII

 

 

0.
0

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To

6.0

ad

04

APPENDIX B

NUSSELT NUMBER ANALYSIS

Heat transfer in the thin layer of air adjacent to the enclosure’s isothermal walls

is dominantly by condution, where the heat flux transfered per unit surface area is,

4 ._. _ ks [22] (13.1)
8x go

where, k, is the thermal conductivity of air at the temperature of the isothermal wall,
and (HT/ax) is the temperature gradient in the thin layer of air in the x—direction
normal to the wall. An energy balance at the thin layer of air yields,

4 conduction = anvection (3.2)
01',
‘ 3T
- k — = h AT ,
[ 8x L w (B 3)

where, h is the convective heat transfer coefficient, and AT is the temperature

difference between the walls. The local Nusselt is defined as,

hH

Nu = —k- (B~4)

which represents a non-dimensional convective heat transfer coefficient. substituting

123

124

equation (B.3) into equation (BA),

19., 3(T—TC)/(TH-TC)

u=—

kc 6(x/H)

 

(8.5)

(3.6)

where, k” and kc are the thermal conductivity of air at the temperature of the hot and

cold walls, respectively, 9 is the non-dimensional temperature, and § is the non-

dimensional distance normal to the wall. The non-dimensional temperature gradient at

the wall (AG/Ag) is evaluated from the interferograms as explained in Appendix ( 1)

and knowing the properties of air at the temperature of the isothermal walls the local

Nusselt number can be calculated.

The mean Nusselt number is then calculated by fitting a cubic spline function to

the local Nusselt number values, and numerically integrating over the entire length of

the wall.

APPENDIX C

ERROR ANALYSIS

Three types of errors can affect the accuracy of the experimental results of the
present study, optical measurement errors, data aquisition errors and calculation
errors. A qualitative assessment of the uncertainties resulting from these sources of

enor is discussed here to ensure the acceptability of the resuls.

Optical measurement errors are taken care of in the adjustment of the infinite
field fringe setting before the experiment, where the first and the last fringes in the
field must follow the sufaces of the hot and cold walls, respectively, bad
experimennts were rejected based upon this criterion. However, it is estimated that
about 5% error can result from this source as judged by few repeatability

measurements performed on a single representative experiment.

Kline and McClintock (1953) presented a method for estimating uncertainties in
sin gle-sample experiments, where uncertainty was defined as what one thinks the
error might be, and it may vary considerably depending upon the particular
circumstances of the single observation. As shown in Appendix B, the local Nusselt

number is a function of a number of independent variables,

3(T-T
Nu = Nu (H, k”, kg, in mg (C.1)

or namely, the height of the enclosure, the thermal conducdvities of air at the
temperature of each of the ismhennal walls, the distance from the wall to the center

of the fringe, and the calculated temperature gradient at the wall. Accuracy of better

125

126

than 1% could be achieved in measuring wall heights and temperatures, however
higher uncertainty in the last two variables was possible because of the difficulties
associated with, accurately locating the center (the darkest spot) of a fringe, and the
choice of the best-fit polynomial that predicts the measured data ( see Appendix A)

for calculating the temperature gradient at each wall.

The uncertainty in locating the fringe is estimated from repeated measurements
of the same interferogram to be less than 5%. The best-fit polynomial function was
selected based on the statistic R2 discussed by Draper and Smith (1931, pp. 101-102).
A third degree polynomial gave the best statistical fit, while fits obtained from fourth
and fifth degree polynomials appeared statistically reasonabe too, where they gave
temperature gradient values within 12% of the values of the best fit. Therefore the
third degree polynomial was used in the study with an estimate of uncertainty of 12%

in the temperature gradient alone.

Kline and McClintock (1953) suggested a second-power equation for an upper bound
on the uncertainty in single measurements, which for the variables of the present

study becomes,

 

 

' (3(T-TC) “’2
AN“ _ fl 2 fl 2 A(X/H) 2 Bx/H 2
[Nu L..- (kfl) +< kc) +(—x/H >+< 304C) > ((3.2)
L any J

 

 

therefore, substituting the individual uncertainties of all independent variables in this
equation yields an overall estimated uncertainty of about 14%, which was considered

reasonable regarding the nature of the experimental technique.

APPENDIX D

COMPUTER PROGRAM
AND
INT ERFEROMETRIC NIEASUREMENTS

127

 

 

0000000000000

This program reads the interferometric measurements in data files
with "filename, shown in Table 1,

and then uses IMSL Library subroutines to fit a

shape preserving polynomial function of a known order to the
measurements at each line.

The program then derives the local Nusselt number at the isothermal
walls then integrates over the wall height to give the mean Nusselt
number.

It outputs the mean Nusselt numbers to the tenniral, and generates
three output data files; one for the local values at me hot wall

with the name "PIIfiIenamedat", and one for the cold wall

with the name "PCfilename.dat", and the third for statistical

data with the name ”STATfiIename.dat".

 

DIMENSION THETA(100),X(100),YH(100),B(30),SSPOLY(30),STAT( 10)
& ,CNU(100),YC(100),XD( 100)

REAL VALUEI,CSITG,BREAK(22),CSCOEF(4,22),TH,TC,CHC,DELT,DELTF
& ,HWHT,CWHI‘,XW,HNU(100),BSUM,VALUE2,XF,XP,YP,RSPNS ,XO

INTEGER NDEG,NLINE,NFRG,MLINE(100),NRSPNS

EXTERNAL CSITG

CI'IARACI'ER‘lZ FILNAM

CHARACTER‘M CFILNHFILN

CIIARACFER*16 STFILN,DFNAM,PFNAM

WRITE(5,*) ’ENTER INPUT FILENAME :’

READ(5,100) FILNAM

129

WRITE(5,*) ’ENT ER DEGREE OF POLYNOMIAL (integer):’
READ(5,*) NDEG

100 FORMAT(A)

109 FORMAT(A,A)

110 FORMAT(A,12)
OPEN(UNIT=10,NAME=FILNAM,TYPE=’OLD’)
READ(10,101) NLINE

C 902WRITE(5,*) NLINE
READ(10,101) NFRG
RBAD(10,*) TH
READ(10,*) TC
DELT=TH~TC
DELTF=DELT/(NFRG+1)
READ(10,*) CHC
READ(10,*) HWHT
READ(10,*) CWHT
DO 710 L=l,(NFRG+2)

THETA(L)= (TH - (L-l)*DELTF-TC)/DELT

C 905 WRITE(5,*) THETA(L)

710 CONTINUE

101 FORMATOZ)

WRITE(CFILN,109) ’PC’fILNAM
WRITE(HFILN,109) ’PH’,FILNAM
WRITE(STFILN,109) ’STAT',FILNAM
OPENCUNIT =1 1,NAME=HFILN,TYPE= ’NEW’)
OPEN(UNIT=12,NAME=CFILN,TYPE=’NEW’)
OPEN (UNIT =13,N AME=STFILN,TYPE=’NEW’)

C

130

 

700

WRIT E(5,*) ’If you want the temperature profile(s) along

+ one or more lines in the interferogram, enter [the number

+ of lines]. If no temperature profiles are needed enter [0]:’

READ(5,*) RSPNS

IF(RSPNS.EQ.O) GO TO 600

NRSPNS=RSPNS

IF(NRSPNS.GT.NLINE) THEN

WRITE(5,*) ’This number of lines exceeds the number of lines

+ in the data file.’

751

750

GO TO 700

ELSE

DO 750 J=1,NRSPNS

IF(J.NE.1) GOTO 751

WRITE(5,*) ’Enter the first line number:’
READ(5,*) MLINEU)

GO TO 750

WRITE(5,*) ’Enter the next line numbert’
READ(5,*) MLINE(J)

CONTINUE

ENDIF

 

DO 300 I=1,NLINE
READ(10,*) YD
YH(I)=YD/HWHT
YC(I)=YD/CWHT
READ(10,*) (XD(J),J=1,NFRG+2)

 

320

330

400

340

131

XW=XD(NFRG+2)-XD(1)
DO 320 J=1,NFRG+2
X(J)=(XD(D-XD(1))/XW
CONTINUE
CALL RCURV ((NFRG+2),X,THETA,NDEG,B,SSPOLY,STAT)
WRIT E(13,*) I I
DO 330 J=1,NDEG+1
WRITE(13,*) B(J),SSPOLY(J)
CONTINUE
DO 400 K=1,10
WRITE(13,*) STAT(K)
CONTINUE
HNU(I)=ABS(CHC*B(2))
BSUM=0.0
DO 340 J=1,NDEG
BSUM=BSUM+J*B(J+1)
CONTINUE
CNU(I)=ABS(CHC*BSUM)
WRITE(11,*) YH(I),HNU(I)
WRITE(12,*) YC(I),CNU(I)
DO 720 L=1,NRSPNS
IF(INE.MLINE(L)) GOTO 720
WRITE(5,110) ’ Enter a filename for the fitted curve

+ of line number ’,MLINE(L)

READ(5,100) PFNAM
WRIT'E(5,110) ’ Enter a filename for the actual data

+ of line nember ’,MLINE(L)

 

620

610

640

720
300

C

132

READ(5,100) DFNAM
OPEN(UNIT=14,NAME=PFNAM,TYPE=’NEW’)
OPEN (UNIT:15,NAME=DFNAM,TYPE=’NEW’)
XF=1.0/ 100.0
XO=0.0
WRITE(14,*) XO,B(1)
DO 610 J=l,100
XP=XF*J
=B(1)
DO 620 K=2,NDEG+1
YP=YP+B (K)*(XP**(K- 1))
CONTINUE
WRITE(14,*) XP,YP
CONTINUE
CLOSE(14)
DO 640 J=1,NFRG+2
WRITE(15,*) X(J),THETA(J)
CONTINUE
CLOSE(15)
CONTINUE
CONTINUE
CLOSE(11)
CLOSE(12)
CLOSE(13)
GOTO 800

 

600

D0 302 I=1,NLINE

 

322

332

402

342

302

133

READ(10,*) YD
YH(I)=YD/HW HT
YC(I)=YD/CWHT
READ(10,*) (XD(J),J=1,NFRG+2)

XW=XD(NFRG+2)-XD(1)

DO 322 J=1,NFRG+2

X(J)=(XD(J)-XD(1))/XW

CONTINUE
CALL RCURV ((NFRG+2),X,THETA,NDEG,B,SSPOLY,STAT)
WRITE(13,*) I

DO 332 J=1,NDEG+1

WRITE(13,*) B(J),SSPOLY(J)
CONTINUE
DO 402 K=1,10
WRITE(13,*) STAT(K)

CONTINUE
HNU(I)=ABS(CHC*B(2))

BSUM=0.0

DO 342 J=1,NDEG

BSUM=BSUM+J*B(J+1)

CONTINUE
CNU(I)=ABS(CHC*BSUM)
WRITE(11,*) YI-I(I),HNU(I)
WRITE(12,*) YC(I),CNU(I)

CONTINUE
CLOSE( 1 l)
CLOS E( 1 2)

800

134

CLOSE(13)
CALL CSAKM (NLINE,YH,HNU,BREAK,CSCOEF)
NINTV=NLINE-l
VALUE]=CSITG(YH(1),YH(NLINE),NINTV,BREAK,CSCOEF)
WRITE(5,*) ’The Mean Nusselt Number For The Hot Wall: ’,VALUE1
CALL CSAKM (NLINE,YC,CNU,BREAK,CSCOEF)
NINTV=NLINE~1
VALUE2=CSITG(YC(1),YC(NLINE),NINTV,BREAK,CSCOEF)
WRITE(5,*) ’The Mean Nusselt Number For The Cold Wall = ’,VALUE2
CLOSE(IO)

END

Table 1: lnterferometric Measurements

picture 6

17 ; number of lines across isothermal walls
9 ; number of fringes, excluding wall fringe
33.2 ; hot wall temperature

19.4 ; cold wall temperature

1.0415 ; ratio of thermal conductivities (hot/cold)
1.827 ; hot wall height

1.865 ; cold wall hel'

 

ight

0.10115 ; non-dim. Y coord., followed by non-dim. X’s

0.090 .114
0.203
0.084 .112
0.3045
0.091 .121
0.406
0.089 .130
0.5075
0.091 .144
0.609 -
0.096 .157
0.7105
0.097 .150
0.812
0.098 .138
0.9135
0.091 .128
1.015
0.082 .121
1.1165
0.086 .116
1.218
0.088 .114
1.3195
0.080 .111
1.421
0.082 .111
1.5225
0.084 .117
1.624

.161
.158
.166
.185
.218
.250
.234
.212
.189
.173
.165
.164
.167
.121
.184 .

0.081 .129 .205

1.7255

.217
.204
.220
.268
.370
.374
.328
.287
.255
.232
.220
.220
.219
.237

.303
0.081 .120 .209 .

314

.333 1.035 1.448 1.594 1.708 1.794 1.856
.283 1.137 1.476 1.597 1.701 1.785 1.849
.367 1.223 1.509 1.630 1.724 1.795 1.846
.626 1.269 1.546 1.660 1.740 1.800 1.850
.579 1.288 1.578 1.683 1.749 1.804 1.841
.525 1.257 1.594 1.693 1.750 1.803 1.843
.446 1.206 1.597 1.693 1.747 1.800 1.835
.391 1.114 1.591 1.685 1.744 1.795 1.830
.347 .919 1.567 1.665 1.731 1.788 1.832
.312 .738 1.526 1.638 1.709 1.779 1.833
.295 .611 1.466 1.595 1.683 1.765 1.833
.286 .542 1.348 1.545 1.650 1.753 1.825
.296 .485 1.252 1.501 1.651 1.758 1.828
.320 .487 1.207 1.522 1.702 1.778 1.823
.361 .513 1.167 1.664 1.741 1.795 1.825
.401 .545 1.193 1.701 1.757 1.808 1.828
.427 .593 1.559 1.701 1.760 1.808 1.830

135

Table 1 (cont’d)

picture 1

17
10

0.056 .098
0.051 .112
0.053 .113
0.054 .119
0.056 .118
0.055 .119
0.056 .110
0.053 .104
0.055 .101
0.057 .100
0.056 .096
0.058 .095
0.059 .091
0.058 .088
0.059 .088

0.059 .092

.189
.231
.241
.254
.246
.224
.199
.180
.167
.156
.146
.141
.133
.128
.128
.131
.139

.312
.362
.394
.432

.345
.296
.258
.231
.212
.196
.186
.175
.170
.171
.177
.198

.465

.502
.413
.354
.308
.276
.252
.236
.222
.217
.219
.232
.283

136

.848 1.470 1.675 1.767 1.829 1.876 1.912
.550 1.256 1.635 1.732 1.791 1.837 1.878 1.912
.666 1.439 1.671 1.740 1.793 1.834 1.876 1.912
.721 1.507 1.675 1.740 1.789 1.832 1.873 1.912
.626 1.055 1.660 1.731 1.780 1.825 1.869 1.912
.801 1.640 1.716 1.772 1.819 1.865 1.914
.601 1.602 1.694 1.756 1.807 1.857 1.914
.480 1.550 1.662 1.736 1.796 1.852 1.912
.403 1.480 1.616 1.707 1.777 1.844 1.914
.355 1.361 1.556 1.666 1.755 1.835 1.914
.320 .442 1.457 1.606 1.722 1.821 1.917

.296 .405 1.322 1.529 1.684 1.803 1.922
.278 .384 1.153 1.424 1.635 1.791 1.926
.274 .377 1.007 1.364 1.616 1.791 1.930
.279 .401 .922 1.419 1.654 1.803 1.934

.314 .486 .978 1.492 1.685 1.819 1.943

.419 .704 1.186 1.553 1.717 1.837 1.946

 

1'T

137

Table l (cont’d)
picture 4

17
10

33.2

19.4

1.0415

2.563

2.596

0.142389

0.0 .25 .547 1.013 1.338 1.6041973 2.258 2.4212514 2.578 2.604
0.284778

0.0 .246 .4861.363 1.6491909 2.266 2.405 2.494 2.5412585 2.605
0.427166

0.0 .156 .285 1.109 1.849 2.173 2.382 2.465 2.524 2.568 2.597 2.609
0.569555

0.0 .102 .186 .261 .3901.7892.4212.488 2.533 2.566 2.582 2.603
0.711944

0.0 .085 .143 .211 .299 .969 2.427 2.493 2.525 2.558 2.577 2.603
0.854333

0.0 .067 .122 .180 .245 1.117 2.415 2.487 2.517 2.549 2.577 2.601
0.996722

0.0 .060 .108 .162 .215 .293 .398 2.477 2.506 2.546 2.575 2.599
1.139110 - .

0.0 .048 .093 .147 .187 .264 .348 2.462 2.495 2.539 2.5712598
1.281499

0.0 .039 .091 .130 .171 .236 .317 2.435 2.484 2.526 2.564 2.599
1.423888

0.0 .039 .087 .124 .161 .215 .288 2.419 2.471 2.517 2.556 2.601
1.566277 .

”0% .091 .127 .156 .200 .269 .969 2.452 2.505 2.554 2.601
1.7

0.3 .337 .090 .127 .152 .192 2631.180 2.428 2.4912548 2.599
1. 51 54 ~

0.0 .039 .088 .124 .155 .195 .273 .436 .854 2.463 2.5312599
1.993443

3.03.3? .089 .124 .157 .214 .289 .491 .784 2.426 2.504 2.597
.1 5 2

0.0 .05 .089 .134 .174 .247 .365 .598 .865 2.342 2.46 2.598
2.278221

(21.14) 2(£518 .103 .158 .224 .323 .516 .785 1.019 2.175 2.388 2.597
00 .058 .117 .191 .298 .482 .7481.0081.3361.78 2.339 2.596

 

Table l (cont’d)

picture 7

17

9

32.4

19.5
1.0388
1.925
1.946
0.10694
0.057 .097
0.21389
0.056 .100
0.32083 ,
0.059 .110
0.42778
0.065 .135
0.53472
0.065 .147
0.64167
0.063 .144
0.74861
0.062 .129
0.85556
0.059 .108
0.9625
0.057 .101
1.06944
0.057 .097
1.17638
0.057 .095
1.28333
0.058 .096
1.39028
0.059 .099
1.49722
0.060 .107
1.60417
0.062 .126
1.71111

1.81805
0.070 .164

.157
.159
.181
.221
.251
.227
.190
.164
.148
.139
.137
.140
.147
.165 .
.197
0.072 .155 .
.286

.246
.240
.285
.360
.348
.305
.260
.223
. 198
.184
. 185
. 191
.206

.301
.356
.397

.421
.485
.482
.439
.386
.328
.288
.260

138

.475 1.000 1.375 1.568 1.697 1.810 1.895
.820 1.474 1.605 1.711 1.814 1.896
.728 1.532 1.649 1.759 1.834 1.899
.645 1.598 1.717 1.796 1.852 1.900
.576 1.648 1.750 1.812 1.859 1.897
.525 1.693 1.771 1.822 1.865 1.897
.479 1.710 1.769 1.824 1.864 1.897
.425 1.714 1.771 1.818 1.862 1.897
.721 1.711 1.767 1.808 1.860 1.894
.249 1.583 1.684 1.751 1.808 1.853 1.894
.251 1.540 1.656 1.723 1.786 1.845 1.893
.269 1.506 1.608 1.682 1.754 1.833 1.892
.301 1.446 1.553 1.640 1.727 1.823 1.892
.357 1.387 1.515 1.626 1.738 1.833 1.894
.416 1.333 1.504 1.684 1.786 1.863 1.904
.472 1.309 1.584 1.742 1.821 1.879 1.912
.551 1.141 1.592 1.751 1.827 1.884 1.912

Table 1 (cont’d)

picture 8

17

10

32.4

19.5
1.0388
1.861
1.864
0.10339
0.038 .086
0.20678
0.037 .108
0.31017
0.041 .128
0.41356
0.045 .140
0.51694
0.047 .141
0.62033
0.053 .131
0.72372
0.054 .119
0.82711
0.058 .108
0.9305
0.057 .109
1.03389
0.053 .107
1.13728
0.058 .111
1.24067
0.073 .124
1.34405
0.073 .126
1.44744
0.074 .129
1.55083
0.077 .131
1.65422
0.079 .133
1.75761
0.085 .138

.184
.215
.245
.272
.261
.228
.201
.180
.173
.164
.165
.175
.176
.178
.182
.188
.200

.302
.336
.410
.452
.395
.335
.290
.257
.242
.225
.219
.229
.230
.232
.239
.251
.283

.484

.412
.366
.330
.306
.297
.303
.299
.307
.321
.350
.425

139

.935 1.409 1.692 1.806 1.878 1.934 1.981
.564 1.069 1.562 1.742 1.821 1.880 1.931 1.981
.689 1.146 1.632 1.759 1.830 1.884 1.934 1.982
.674 1.163 1.647 1.771 1.834 1.885 1.934 1.983
.573 1.124 1.653 1.772 1.835 1.885 1.933 1.984
.484 1.075 1.659 1.772 1.834 1.883 1.933 1.984
.911 1.633 1.761 1.828 1.881 1.932 1.980
.765 1.595 1.748 1.821 1.878 1.931 1.983
.591 1.559 1.724 1.806 1.871 1.929 1.985
.514 1.489 1.690 1.787 1.859 1.926 1.985
.470 1.417 1.647 1.762 1.849 1.925 1.986
.472 1.314 1.599 1.736 1.841 1.931 2.005
.467 1.159 1.511 1.685 1.816 1.925 2.010
.469 1.056 1.417 1.626 1.797 1.918 2.013
.503 .975 1.384 1.650 1.817 1.932 2.018
.569 1.011 1.471 1.708 1.847 1.944 2.022
.707 1.122 1.559 1.741 1.869 1.962 2.028

Table 1 (cont’d)

picture 8a

17

8

32.4

19.5
1.0388
1.873
1.884
0.10406
0.036 .109
0.20811
0.045 .155
0.31217
0.049 .161
0.41622
0.052 .142
0.52028
0.054 .120
0.62433
0.053 .105
0.72839
0.052 .096
0.83244
0.053 .085
0.9365
0.056 .084
1.04056
0.057 .082
1.14461
0.055 .083
1.24867
0.055 .082
1.35272
0.054 .082
1.45678
0.054 .083
1.56083
0.052 .087
1.66489
0.052 .093
1.76894
0.048 .105

.253
.312
.263
.222
.187
.160
.141
.126
.116
.114
.110
.109
.112
.117
.132
.154
.199

.295 .385
.247 .319
.214 .269
.187 .235

140

.922 1.373 1.562 1.737 1.861 1.946 1.991
.498 1.474 1.622 1.789 1.880 1.949 1.991
.364 1.576 1.767 1.861 1.911 1.957 1.991
.547 1.888 1.926 1.960 1.990
.415 1.898 1.932 1.961 1.991
.338 1.902 1.932 1.961 1.990
.288 1.902 1.932 1.958 1.984
.161 .205 1.862 1.893 1.924 1.951 1.975
.152 .188 1.848 1.883 1.915 1.946 1.972
.142 1.778 1.828 1.867 1.903 1.937 1.968
.141 1.742 1.800 1.846 1.890 1.929 1.962
.139 1.696 1.766 1.816 1.867 1.913 1.956
.144 1.626 1.717 1.782 1.837 1.897 1.951
.153 1.514 1.662 1.739 1.804 1.872 1.947
.175 .249 .399 1.677 1.759 1.841 1.935
.232 .370 .516 1.574 1.703 1.816 1.929

.326 .471 .617 .7901.618 1.8041931

17 .

Table l (cont’d)

picture 13

17

9

33.2

19.4
1.0415
1.808
1.848
0.10044
0.035 .066
0.20089
0.035 .079
0.30133
0.034 .097
0.40178
0.034 .114
0.50222
0.035 .122
0.60267
0.034 .123
0.70311
0.034 .113
0.80356
0.032 .100

1.00444
0.035 .081
1.10489
0.034 .073
1.20533
0.035 .069
1.30578
0.033 .062
1.40622
0.033 .062
1.50667
0.042 .077
1.60711
0.049 .091
1.70755
0.054 .116

136

.166
.209
.235
.236 .
.223
.201
.177
.154
.134
.118
.109
.106
.107
.121
.153
.210

.268
.358
.460
.450

.356
.308
.263
.228
.201
.177
. 165
. 156
. 162
.195
.262
.341

.581

.297
.265
.256
.264
.33 1
.414
.494

141

.927 1.243 1.420 1.580 1.688 1.767
.684 1.014 1.302 1.487 1.630 1.713 1.774
.751 1.086 1.385 1.576 1.674 1.730 1.774
.773 1.143 1.475 1.624 1.689 1.736 1.773
.763 1.132 1.517 1.631 1.690 1.737 1.772
.694 1.120 1.515 1.630 1.689 1.736 1.773
.606 1.120 1.490 1.618 1.683 1.730 1.772
.529 1.087 1.451 1.596 1.669 1.721 1.771
.434 1.066 1.410 1.567 1.651 1.710 1.763
.360 1.020 1.372 1.544 1.631 1.701 1.761
.931 1.322 1.508 1.608 1.690 1.761
.844 1.270 1.463 1.586 1.678 1.762
.757 1.189 1.424 1.567 1.671 1.760
.630 1.121 1.383 1.559 1.668 1.762
.615 1.061 1.352 1.579 1.691 1.770
.643 1.054 1.380 1.608 1.717 1.780
.700 1.104 1.465 1.645 1.734 1.782

Table 1 (cont’d)

picture 12

17

11

33.2

19.4
1.0415
1.748
1.775
0.09711
0.025 .097
0.19422
0.026 .114
0.29133
0.026 .123
0.38844
0.025 .135
0.48556
0.025 .142
0.58267
0.024 .136
0.67978
0.023 .122
0.77689
0.027 .101
0.874
0.017 .092
0.97111
0.017 .088
1.06822
0.019 .084
1.16533
0.019 .084
1.26244
0.021 .082
1.35956
0.021 1 .080
1.45667
0.020 .076
1.55378
0.013 .072
1.65089
0.010 .060

.196
.222
.241
.265
.274
.255
.227
. 196
. 173
. 157
.151
.145
. 142
.140
.135
.133

120

.359

.426
.429
.408
.376
.336
.302
.267
.247
.231
.224
.219
.218
.215
.211

142

.639 .942 1.198 1.427 1.586 1.692 1.763 1.817 1.850
.739 1.035 1.268 1.457 1.595 1.688 1.756 1.812 1.849
.746 1.071 1.282 1.460 1.598 1.685 1.752 1.809 1.845
.690 1.060 1.281 1.461 1.593 1.683 1.749 1.805 1.843
.628 1.025 1.272 1.447 1.586 1.680 1.746 1.805 1.840
.572 .957 1.250 1.431 1.574 1.672 1.739 1.799 1.835
.510 .883 1.217 1.415 1.557 1.657 1.731 1.794 1.835
.452 .780 1.180 1.391 1.535 1.644 1.723 1.792 1.835
.404 .688 1.125 1.370 1.516 1.625 1.712 1.784 1.833
.374 .622 1.066 1.331 1.490 1.602 1.695 1.777 1.832
.351 .567 .967 1.292 1.458 1.575 1.676 1.768 1.828

.337 .527 .826 1.229 1.428 1.550 1.655 1.760 1.832
.323 .482 .735 1.118 1.392 1.532 1.643 1.757 1.829
.317 .462 .675 .981 1.353 1.515 1.645 1.757 1.833

.314 .443 .635 .889 1.278 1.512 1.665 1.766 1.838

.311 .448 .630 .836 1.169 1.493 1.673 1.773 1.842

.316 .466 .672 .886 1.163 1.468 1.677 1.780 1.847

 

Table 1 (cont’d)

picture 12a

17

9

33.2

19.4
1.0415
1.790
1.790
0.09794
0.032 .054
0.19589
0.034 .076
0.29383
0.039 .097
0.39178
0.046 .101
0.48972
0.046 .097
0.58767
0.048 .093
0.68561
0.053 .090
0.78356
0.051 .085
0.8815
0.055 .087
0.97944
0.058 .088
1.07739
0.059 .086
1.17533
0.059 .085
1.27328
0.058 .085
1.37122
0.060 .087
1.46917
0.059 .086
1.56711
0.057 .087
1.66505
0.058 .093

.131
.203
.238
.223
.196
.174
.155
.138
.131
.125
.120
.116
.114
.114
.117
.127
.154

.239
.219
.202
.188
. 181
.182
.194
.233
.321

143

.316 .800 1.343 1.570 1.697 1.782 1.847 1.892
.451 1.054 1.515 1.668 1.753 1.808 1.856 1.892
.408 1.282 1.653 1.738 1.7901.829 1.865 1.892
.343
.293
.252
.224
.198
.181
.169
.159
.150
.146
.145
.152
.173
.227

.549 1.710 1.765 1.802 1.835 1.864 1.892
.424 1.719 1.767 1.804 1.835 1.865 1.892
.352 1.710 1.759 1.797 1.832 1.863 1.891
.300 1.685 1.742 1.786 1.823 1.858 1.889
.263 1.652 1.721 1.769 1.810 1.850 1.887
.318 1.694 1.749 1.795 1.840 1.885
.283 1.659 1.726 1.778 1.830 1.882
.257 1.618 1.700 1.759 1.819 1.880
.237 1.571 1.669 1.739 1.806 1.876
.226 1.486 1.627 1.709 1.789 1.872
.228 .328 1.577 1.678 1.768 1.866
.253 .383 1.494 1.633 1.745 1.860
.330 .500 .966 1.583 1.732 1.859

.454 .642 .995 1.552 1.767 1.859

. —._- _-— ‘w- I. 111.51“!

 

1" Fl:- nus-m

Table 1 (cont’d)

picture 21

17

9

32.2

19.4
1.0386
1.91

1.952
0.10611
0.043 .113
0.21222
0.049 .137
0.31833
0.048 .171
0.42444
0.059 .203
0.53056
0.061 .218
0.63667
0.058 .235
0.74278
0.054 .233
0.84889
0.052 .231
0.955
0.049 .231
1.06111
0.052 .217
1.16722
0.050 .203
1.27333
0.046 .184
1.37944
0.039 .163
1.48556
0.033 .143
1.59167
0.028 .131
1.69778
0.025 .125
1.80389
0.013 .106

.23 1
.286
.360
.405
.428
.448
.454
.452
.441
.424
.380
.338
.294
.263
.252
.251
.255

.448
.534
.592
.639
.673
.692
.701
.700
.682
.638
.591
.522
.445
.396
.376
.385
.392

144

.733 1.028 1.310 1.508 1.676 1.816 1.887
.794 1.044 1.282 1.466 1.632 1.785 1.876
.851 1.099 1.313 1.484 1.640 1.786 1.871
.901 1.149 1.346 1.510 1.652 1.784 1.871
.948 1.175 1.373 1.529 1.658 1.782 1.876
.967 1.197 1.393 1.545 1.668 1.780 1.872
.955 1.209 1.411 1.559 1.678 1.780 1.872
.976 1.239 1.428 1.567 1.682 1.784 1.864
.972 1.241 1.435 1.574 1.687 1.789 1.862
.936 1.234 1.428 1.573 1.680 1.789 1.862
.859 1.173 1.406 1.559 1.676 1.788 1.863
.771
.663
.584
.548
.567
.591

1.065 1.354 1.535 1.665 1.784 1.861
.941 1.255 1.491 1.643 1.771 1.854
.828 1.151 1.426 1.625 1.768 1.854
.770 1.068 1.366 1.606 1.770 1.861
.786 1.087 1.384 1.630 1.786 1.861
.845 1.159 1.449 1.681 1.808 1.864

 

Table 1 (cont’d)

picture 19

17

9

32.2

19.4
1.0386
1.863
1.863
0.1035
0.026 .097
0.207
0.027 .117
0.3105
0.027 .118
0.414
0.027 .127
0.5175
0.027 .137
0.621
0.028 .151
0.7245
0.031 .156
0.828
0.032 .159
0.9315
0.035 .173
1.035
0.038 .186
1.1385
0.040 .200
1.242
0.043 .204
1.3455
0.047 .201
1.449
0.041 .187
1.5525
0.041 .181
1.656
0.041 .159
1.7595
0.036 .123

.218
.254
.257
.276
.302
.323
.336
.350
.368
.381 .
.380
.377
.363

.334
.316

.372
.405
.430
.465
.494
.529
.550
.578
.590

.585
.559
.527
.495
.496
.475
.421

.607
.640
.663
.710
.761
.812 1.089 l.3361.5361.697 1.8291957
.839
.865
.870
.864 1.145 1.4061574 1.7061.8301.959
.829 1.119 1.3841562 1.7041.8321.960
.784 1.062 1.3361547 1.7001.828 1.963
.726
.689
.671
.659
.627

145

.892 1.167 1.440 1.680 1.850 1.958
.898 1.146 1.391 1.628 1.821 1.958
.923 1.166 1.395 1.629 1.808 1.958
.972 1.217 1.448 1.661 1.816 1.957
1.039 1.285 1.504 1.680 1.825 1.958

1.125 1.372 1.559 1.700 1.829 1.957
1.142 1.403 1.570 1.710 1.829 1.956
1.159 1.415 1.576 1.709 1.831 1.960

.991 1.274 1.520 1.690 1.830 1.968
.923 1.223 1.493 1.682 1.834 1.969
.888 1.184 1.474 1.693 1.842 1.976
.881 1.174 1.470 1.695 1.851 1.980
8741.189 1.497 1.731 1.874 1.986

 

Table 1 (cont’d)

picture 18

17

9

32.2

19.4
1.0386
1.933
1.933
0.10739
0.031 .146
0.21478
0.056 .232
0.32217
0.060 .231
0.42956
0.058 .228
0.53694
0.051 .226
0.64433
0.055 .231
0.75172
0.055 .223
0.85911
0.054 .217
0.9665
0.050 .211
1.07389
0.050 .206
1.18128
0.050 .182
1.28867
0.042 .179
1.39605
0.043 .158
1.50344
0.038 .139
1.61083
0.035 .121
1.71822
0.029 .101
1.82561
0.029 .088

.326
.427
.425
.417
.427
.427
.418 .
.621
.390 .
.377
.351
.327
.295
.268
.223
.186
.167

.412

.532
.639
.631
.636
.641
.635

.569
.535
.500
.459
.415
.352
.308
.280

146

.793 1.058 1.369 1.612 1.758 1.851 1.907
.879 1.107 1.364 1.573 1.731 1.831 1.902
.860 1.084 1.321 1.526 1.693 1.807 1.896
.855 1.069 1.293 1.496 1.661 1.785 1.895
.870 1.0901309 1.497 1.657 1.777 1.889
.871 1.097 1.332 1.514 1.657 1.781 1.882
8761.109 1.341 1.530 1.660 1.781 1.882
8701.107 1.350 1.527 1.659 1.775 1.880
.8461.0901.341 1.519 1.655 1.775 1.876
.801 1.055 1.309 1.502 1.641 1.7641875
.754 1.001 1.254 1.471 1.623 1.754 1.874
.701 .9341.2001.436 1.608 1.751 1.874
.646 .868 1.127 1.388 1.587 1.751 1.875
.591 79410501323 1.561 1.742 1.874
.531 .731 .9821.2691.522 1.726 1.872
.474 .686 .9341.2301.4881.7011.873
.462 .705 .995 1.3201571 1.7701887

 

Table 1 (cont’d)

picture 20

17

9

32.2

19.4
1.0386
1.859
1.884
0.103278
0.079 .103
0.206555
0.084 .162
0.30983
0.090 .212
0.41311
0.093 .234
0.516389
0.096 .244
0.619667
0.097 .240
0.722944

0.099 .239 .
.435
.421
.391
.363
.327
.293
.259
.231
.217
.219

0.826222
0.099 .233
0.9295
0.096 .227
1.03278
0.093 .218
1.13605
0.090 .210
1.23933
0.087 .198
1.34261
0.086 .184
1.44589
0.083 .165
1.54917
0.080 .153
1.65244
0.073 .141
1.75572
0.072 .136

.220
.346
.448
.478
.475
.451

.462

.663
.61 1
.556
.492
.432
.376
.332
.308
.323

147

.756 1.151 1.485 1.748 1.883 1.977 2.016
.634 .977 1.288 1.542 1.728 1.852 1.946 2.018
.745 1.068 1.353 1.554 1.720 1.842 1.935 2.017
.763 1.086 1.347 1.533 1.697 1.823 1.920 2.010
.750 1.059 1.324 1.511 1.675 1.804 1.909 2.010
.726 1.030 1.299 1.494 1.659 1.786 1.894 2.002
.711 1.026 1.295 1.495 1.653 1.777 1.889 1.998
.697 1.010 1.286 1.495 1.650 1.773 1.883 1.994
.964 1.251 1.479 1.644 1.773 1.880 1.984
.875 1.199 1.451 1.631 1.760 1.868 1.987
.799 1.128 1.412 1.604 1.746 1.863 1.984
.718 1.019 1.340 1.571 1.727 1.856 1.987
.619 .884 1.212 1.515 1.704 1.844 1.989

.532 .770 1.077 1.421 1.671 1.832 1.988

.472 .673 .935 1.285 1.598 1.811 1.987

.439 .621 .862 1.173 1.517 1.791 1.987

.471 .673 .928 1.242 1.543 1.819 1.992

 

1F:

Table 1 (cont’d)

picture 24

17

10 ,

32.9

19.2
1.0413
1.825
1.841
0.10139
0.032 .082
0.20278
0.032 .102
0.30417
0.041 .133
0.40556
0.042 .163
0.50694
0.043 .185
0.60833
0.045 .198
0.70972
0.046 .211
0.81111
0.046 .224
0.9125
0.050 .227
1.01389
0.055 .221
1.11528
0.064 .205
1.21667
0.065 .186
1.31805
0.058 .167
1.41944
0.054 .154
1.52083
0.054 .145
1.62222
0.054 .141
1.72361
0.049 .134

. 166
.216 .
.280
.335
.366
.392
.415
.427
.431

.356
.317
.274 .
.249
.233
.239 .
.241

.324

.484
.547
.585
.616
.642
.650
.648
.602
.594
.476

.358
.333

.355

148

.584 .842 1.073 1.273 1.436 1.566 1.687 1.784
.646 .874 1.084 1.260 1.410 1.550 1.670 1.777
.715 .932 1.131 1.295 1.439 1.566 1.678 1.773
.773 .994 1.172 1.332 1.465 1.581 1.682 1.771
.827 1.029 1.207 1.357 1.487 1.594 1.687 1.770
.860 1.064 1.243 1.385 1.505 1.604 1.695 1.770
.873 1.097 1.276 1.408 1.522 1.615 1.695 1.770
.888 1.120 1.289 1.422 1.528 1.617 1.697 1.768
.891 1.126 1.294 1.428 1.532 1.619 1.695 1.767
.847 1.095 1.288 1.422 1.531 1.618 1.696 1.767
.764 1.024 1.249 1.399 1.516 1.608 1.691 1.766
.681 .920 1.173 1.351 1.495 1.595 1.682 1.766
.587 .808 1.067 1.286 1.459 1.579 1.679 1.766
.513 .702 .946 1.202 1.408 1.555 1.673 1.772
.467 .646 .872 1.120 1.358 1.544 1.679 1.777
.482 .649 .872 1.119 1.369 1.562 1.695 1.778
.514 .696 .929 1.196 1.430 1.612 1.720 1.779

 

1F

Table 1 (cont’d)

picture 23

17

10

32.9

19.2
1.0413
1.749
1.787
0.09717
0.057 .131
0.19433
0.058 .151
0.2915
0.052 .157
0.38867
0.048 .168
0.48583
0.048 .179
0.583
0.046 .187
0.68017
0.043 .186
0.77733
0.042 .192
0.8745
0.040 .192
0.97167
0.040 .190
1.06883
0.041 .191
1.166
0.045 .183
1.26317
0.041 .169
1.36033
0.040 .156
1.4575
0.040 .142
1.55467
0.040 .126
1.65183
0.037 .095

.243 .
.268
.286
.3 14
.337
.358
.368
.389
.392
.382
.369
.342
.312
.290
.268

.200

.421
.452
.494
.535
.580
.598
.622
.627
.610
.574
.517
.470
.433

.383
.345

.649
.644
.668
.720
.771

.726
.652
.601
.570
.553
.508

149

.892 1.133 1.359 1.581 1.744 1.841 1.871
.879 1.085 1.282 1.495 1.681 1.819 1.872
.887 1.084 1.278 1.490 1.667 1.803 1.870
.934 1.136 1.328 1.526 1.679 1.797 1.872
.987 1.192 1.384 1.559 1.691 1.793 1.871
.813 1.029 1.246 1.430 1.578 1.698 1.793 1.866
.843 1.067 1.279 1.453 1.588 1.698 1.790 1.860
.870 1.097 1.322 1.468 1.592 1.701 1.792 1.862
.867 1.113 1.328 1.474 1.592 1.695 1.788 1.859
.857 1.097 1.310 1.465 1.587 1.693 1.786 1.863
.797 1.062 1.273 1.444 1.574 1.688 1.783 1.870
.973 1.211 1.409 1.562 1.678 1.780 1.872
.881 1.122 1.353 1.534 1.669 1.775 1.872
.813 1.051 1.3 1.509 1.655 1.772 1.874

.764 1.001 1.242 1.474 1.647 1.772 1.875
.751 .979 1.216 1.457 1.646 1.769 1.872

.718 .956 1.211 1.466 1.652 1.779 1.888

 

Table 1 (cont’d)

picture 25

17

9

32.9

19.2
1.0413
1.834
1.834
0.10189
0.070 .189
0.20378
0.088 .265
0.30567
0.085 .281
0.40756
0.082 .292
0.50944
0.077 .303
0.61133
0.074 .316
0.71322
0.072 .305
0.81511
0.072 .301
0.917
0.080 .283
1.0189
0.082 .260
1.12078
0.080 .235
1.22267
0.075 .208
1.32455
0.069 .182
1.42644
0.059 .158
1.52833

1.63022
0.052 .120
1.73211
0.048 .103

.386
.463
.481
.498
.515
.534
.531
.524
.488
.448
.408
.362
.323
.279
0.055 .137 .
.204
. 179

.61 1
.693
.717
.728
.747
.766
.773

.602
.540
.479
.424
.366
.316
.283

150

.887 1.131 1.397 1.573 1.687 1.761 1.810
.941 1.146 1.363 1.532 1.656 1.746 1.804
.938 1.142 1.336 1.498 1.626 1.731 1.802
.947 1.144 1.322 1.494 1.613 1.721 1.804
.976 1.170 1.352 1.506 1.619 1.718 1.803
.995 1.194 1.382 1.520 1.626 1.719 1.801
.999 1.209 1.399 1.528 1.629 1.721 1.801
.773 1.009 1.214 1.398 1.527 1.627 1.717 1.798
.742 .991 1.196 1.390 1.519 1.623 1.714 1.797
.673 .924 1.158 1.359 1.500 1.607 1.707 1.803
.851 1.081 1.319 1.473 1.590 1.696 1.798
.759
.677
.606
.538
.477
.439

.998 1.246 1.436 1.569 1.688 1.794
.911 1.169 1.389 1.544 1.681 1.797
.817 1.076 1.330 1.511 1.670 1.798
.743 1.003 1.262 1.479 1.658 1.797
.682 .935 1.203 1.440 1.633 1.798
.678 .981 1.275 1.515 1.701 1.820

 

 

Table 1 (cont’d)

picture 22

0.8845
0.086 .202
0.98278

' 0.091 .200

1.08105
0.089 .188
1.17933
0.092 .178
1.27761
0.087 .164
1.37589
0.087 .151
1.47417
0.089 .142
1.57244
0.082 .134
1.67072
0.085 .133

. 164
.259
.353
.386
.398
.41 1
.413
.403
.394
.376
.341
.307
.272
.237
.212
.202
.204

.319
.482
.577
.614
.624
.642
.663
.663
.638
.591
.524
.464
.401
.348
.305
.286
.294

.590

.684
.583
.497
.434
.408
.437

151

.942 1.280 1.523 1.700 1.802 1.877 1.913
.766 1.057 1.323 1.537 1.684 1.787 1.869 1.914
.857 1.124 1.357 1.525 1.674 1.780 1.863 1.914
.884 1.127 1.337 1.509 1.651 1.763 1.858 1.913
.894 1.125 1.331 1.500 1.642 1.758 1.851 1.912
.912 1.150 1.342 1.515 1.644 1.751 1.847 1.908
.932 1.169 1.371 1.531 1.651 1.755 1.845 1.905
.938 1.185 1.390 1.539 1.656 1.755 1.838 1.903
.914 1.153 1.382 1.536 1.659 1.756 1.837 1.902
.851 1.114 1.353 1.527 1.649 1.753 1.832 1.901
.777 1.049 1.307 1.493 1.632 1.741 1.827 1.898
.959 1.226 1.448 1.606 1.728 1.817 1.897
.836 1.113 1.369 1.571 1.707 1.815 1.895
.728 .995 1.259 1.500 1.683 1.804 1.897
.629 .879 1.145 1.409 1.643 1.800 1.897
.585 .819 1.072 1.349 1.607 1.790 1.895
.629 .875 1.147 1.422 1.661 1.828 1.899

 

Table 1 (cont’d)

picture 28

17

10

32.1

19.4
1.0383
1.914
1.938
0.10633
0.073 .107
0.21267
0.073 .122
0.319
0.073 .142
0.42533
0.074 .169
0.53167
0.077 .193
0.638
0.081 .219
0.74433
0.080 .247
0.85066
0.078 .289
0.957
0.082 .292
1.06333
0.086 .282
1.1697
0.084 .254
1.276
0.080 .222
1.38233
0.084 .199
1.48867
0.083 .181
1.595
0.080 .163
1.70133
0.075 .161
1.80767
0.070 .148

. 188
.220
.269
.324
.384
.449
.517
.580
.586
.536

.386
.335
.294
.273
.265
.261

.321
.382
.452
.534
.629
.733
.807
.852
.881
.834
.744
.622
.516

.397
.382
.395

152

.555 .841 1.110 1.366 1.563 1.716 1.841 1.920
.616 .884 1.113 1.333 1.524 1.682 1.825 1.914
.689 .948 1.172 1.369 1.549 1.695 1.829 1.909
.773 1.024 1.236 1.425 1.584 1.719 1.837 1.909
.880 1.110 1.302 1.480 1.624 1.736 1.841 1.909
.971 1.188 1.374 1.526 1.651 1.747 1.844 1.905
1.038 1.251 1.423 1.561 1.671 1.759 1.850 1.902
1.088 1.296 1.449 1.574 1.679 1.765 1.848 1.907
1.108 1.316 1.461 1.577 1.678 1.766 1.849 1.903
1.099 1.305 1.452 1.577 1.680 1.764 1.851 1.899
1.005 1.261 1.422 1.561 1.669 1.760 1.848 1.905
.895 1.164 1.360 1.522 1.652 1.753 1.848 1.902
.765 1.041 1.281 1.472 1.624 1.740 1.844 1.898
.639 .910 1.175 1.404 1.592 1.725 1.845 1.904
.570 .814 1.082 1.327 1.542 1.711 1.845 1.9

.538 .771 1.034 1.296 1.530 1.729 1.860 1.909
.561 .798 1.069 1.363 1.590 1.772 1.880 1.909

 

Table l (cont’d)

picture 29

17
9
32.1

0.20667
0.128 .250
0.31

0.139 .267
0.41333
0.143 .295
0.51667
0.146 .326
0.62
0.143 .360
0.72333
0.158 .386
0.82667
0.155 .416
0.93
0.142 .399
1.03333
0.136 .379
1.13667
0.130 .344
1.24
0.120 .295
1.34333
0.117 .264
1.44667
0.113 .234
1.55
0.110 .218
1.65333
0.105 .201
1.75667
0.098 .171

.367
.392
.432
.488
.558
.632
.689

153

.574 .898 1.187 1.463 1.679 1.857 1.976 2.024
.616 .905 1.166 1.393 1.617 1.802 1.951 2.024
.658 .946 1.192 1.425 1.642 1.814 1.951 2.024
.756 1.039 1.270 1.479 1.680 1.820 1.939 2.024
.851 1.129 1.346 1.536 1.705 1.831 1.944 2.024
.934 1.187 1.407 1.586 1.724 1.843 1.943 2.025
.984 1.214 1.450 1.606 1.726 1.845 1.939 2.024
.730 1.019 1.272 1.461 1.615 1.735 1.839 1.937 2.016
.716 1.011 1.256 1.461 1.611 1.729 1.835 1.928 2.009
.665
.589
.494
.432
.382
.351
.332
.287

.971 1.226 1.437 1.594 1.714 1.826 1.923 2.009
.873 1.157 1.388 1.559 1.694 1.802 1.915 2.010
.740 1.069 1.286 1.502 1.660 1.792 1.910 2.012 -
.630 .903 1.180 1.415 1.609 1.763 1.897 2.017
.549 .770 1.046 1.332 1.556 1.737 1.880 2.010
.504 .702 .973 1.259 1.5 1.711 1.868 2.009

.472 .661 .892 1.201 1.463 1.708 1.863 2.008
.434 .631 .912 1.190 1.478 1.726 1.892 2.017

 

Table 1 (cont’d)

picture 31

17

8

32.1

19.4
1.0383
1.928
1.928
0.10711
0.094 .221
0.21422
0.096 .263
0.32133
0.107 .274
0.42844
0.108 .292
0.53556
0.097 .333
0.64267

0.74978
0.097 .372
0.85689
0.099 .363
0.964
0.098 .348
1.07111
0.090 .315
1.17822
0.090 .268
1.28533
0.088 .225
1.39244
0.086 .198
1.49956
0.076 .172
1.60667
0.075 .152
1.71378
0.076 .132
1.82089
0.075 .114

.381
.439
.472
.513
.568
0.096 .359 .
.733
.734
.71 1
.641
.526
.445
.376
.321
.274
.231
.196

.656

154

.988 1.384 1.652 1.816 1.917 1.949

.703 1.006 1.341 1.589 1.772 1.891 1.951
.735 1.019 1.316 1.565 1.736 1.867 1.958
.799 1.079 1.341 1.572 1.733 1.859 1.957
.877 1.153 1.410 1.596 1.741 1.860 1.952
.950 1.223 1.461 1.627 1.747 1.861 1.950
1.015 1.277 1.489 1.637 1.753 1.854 1.947
1.048 1.294 1.497 1.642 1.752 1.851 1.943
1.045 1.294 1.496 1.633 1.745 1.847 1.943
.977 1.238 1.460 1.611 1.734 1.837 1.944
.873 1.154 1.410 1.578 1.718 1.827 1.944
.738 1.050 1.336 1.528 1.692 1.818 1.943

.612
.514
.440
.375
.321

.919 1.226 1.471 1.662 1.809 1.940
.779 1.095 1.394 1.629 1.795 1.935
.675 1.000 1.330 1.590 1.778 1.933
.586 .907 1.262 1.552 1.752 1.933
.537 .878 1.260 1.588 1.796 1.945

 

Table 1 (cont’d)

picture 30

17

9

32.1

19.4
1.0383
1.883
1.883
0.10461
0.050 .067
0.20922
0.055 .091
0.31383
0.064 .114
0.41844
0.071 .146
0.52306
0.073 .180
0.62767
0.081 .211
0.73228
0.083 .235
0.83689
0.085 .246
0.9415
0.093 .256
1.04611
0.100 .243
1.15072
0.106 .238
1.25533
0.109 .227
1.35994
0.111 .207
1.46456
0.112 .192
1.56917
0.109 .181
1.67378
0.109 .175
1.77839
0.107 .169

.144
.220
.293
.353
.414
.491
.562
.596
.590
.527
.464

.354
.3 12
.277
.261
.254

.293
.413
.512
.600

.648
.535

.413
.383
.380

155

.590 1.003 1.395 1.681 1.841 1.940 1.991
.702 1.087 1.410 1.644 1.808 1.926 1.989
.819 1.159 1.442 1.642 1.792 1.917 1.992
.927 1.218 1.457 1.643 1.797 1.917 1.992
.698 1.009 1.293 1.497 1.670 1.808 1.922 1.994
.824 1.124 1.362 1.559 1.704 1.820 1.927 1.992
.928 1.212 1.417 1.606 1.730 1.834 1.935 2.001
.970 1.236 1.446 1.616 1.737 1.836 1.932 1.997
.944 1.233 1.457 1.621 1.740 1.837 1.935 2.000
.881 1.197 1.438 1.609 1.732 1.838 1.934 2.003
.774 1.126 1.390 1.585 1.728 1.839 1.939 2.009
.994 1.293 1.540 1.704 1.832 1.938 2.01

.843 1.170 1.452 1.656 1.815 1.933 2.013
.709 1.007 1.341 1.594 1.792 1.936 2.019
.618 .895 1.228 1.520 1.762 1.936 2.021
.572 .848 1.181 1.484 1.751 1.946 2.022
.592 .898 1.264 1.564 1.831 1.972 2.023

 

156

Table 1 (cont’d)

picture 345

17

8

32.1
19.4
1.0384
1.928

0.2142
0.036 .115
0.3213
0.036 .148
0.4284
0.036 .172
0.5356
0.036 .199
0.6427
0.036 .248
0.7498
0.036 .343
0.8569
0.036 .435
0.964
0.036 .428
1.0711
0.036 .326
1.1782
0.036 .261
1.2853
0.036 .190
1.3924
0.036 .159
1.4996
0.036 .128
1.6067
0.036 .112
1.7138
0.036 .093
1.8209
0.036 .085

.2 .387 .8 1.182 1.463 1.653 1.795 1.902

.227 .430 .817 1.165 1.437 1.628 1.778 1.902
.269 .514 .921 1.238 1.475 1.641 1.780 1.902
.335 .678 1.062 1.337 1.535 1.681 1.794 1.902
.431 .903 1.210 1.430 1.588 1.699 1.798 1.902
.662 1.074 1.328 1.502 1.624 1.718 1.807 1.902
.852 1.194 1.4 1.543 1.646 1.730 1.809 1.902
.934 1.259 1.435 1.559 1.657 1.735 1.813 1.902
.986 1.279 1.441 1.566 1.657 1.736 1.811 1.902
.851 1.236 1.425 1.559 1.654 1.735 1.809 1.902
.619 1.143 1.379 1.525 1.635 1.724 1.806 1.902
.403 .925 1.261 1.469 1.604 1.706 1.797 1.902
.301 .591 1.081 1.366 1.549 1.679 1.784 1.902
.249 .423 .811 1.206 1.454 1.623 1.766 1.902
.218 .349 .618 1.029 1.336 1.579 1.755 1.902
.195 .322 .527 .9 1.2581541 1.7591902

.194 .330 .540 .912 1.311 1.590 1.783 1.902

 

157

Table 1 (cont’d)

picture 340

17

9

32.1
19.4
1.0384
1.856
1.863
.1031
.0 .031
.2062
.0 .049
.3093
.0 .061
.4124
.0 .074
0.5156
.0 .102
0.6187
.0 .135
0.7218
.0 .171
0.8249
.0 .211
0.928
.0 .244
1.0311
.0 .238
1.1342
.0 .201
1.2373
.0 .177
1.3404
.0 .15
1.4436
.0 .134
1.5467
.0 .125
1.65

.0 .113
1.7529
.0 .093

.124 .245 .485 .871 1.24 1.5621.7281.8691.934
.149 .276 .539 .923 1.223 1.473 1.679 1.839 1.932
.176 .33 .653 1.023 1.287 1.516 1.7 1.844 1.928
.21 .42 .847 1.157 1.383 1.585 1.731 1.841 1.927
.288 .624 1.038 1.278 1.474 1.642 1.756 1.856 1.925
.391 .8621.16 13731.54 1.671 1.7721.86 1.924
.564 .993 1.244 1.433 1.575 1.688 1.78 1.862 1.922
.707 1.073 1.303 1.471 1.595 1.699 1.787 1.867 1.918
.735 1.091 1.312 1.481 1.599 1.699 1.785 1.867 1.918
.675 1.064 1.308 1.473 1.592 1.695 1.782 1.866 1.918
.518 .9721.2561.441 1.5721.68 1.77 1.86 1.919
.384 .777 1.148 1.377 1.538 1.658 1.76 1.855 1.92
.309 .547 .98 1.276 1.483 1.629 1.746 1.853 1.924
.261 .435 .78 1.1281.401 1.59 1.7281.8461.926
.24 .388 .647 1.013 1.323 1.549 1.719 1.847 1.931
.225 .365 .59 .94 1.263 1.523 1.72 1.85 1.935
.212 .355 .566 .907 1.257 1.5431.75 1.8741.941

 

Table l (cont’d)

picture 342

17

9

32.1

19.4
1.0384
1.935
1.935
0.1075
0.064 .138
0.215
0.064 .169
0.3225
0.064 .187
0.43

0.064 .205
0.5375
0.064 .237
0.645
0.064 .281
0.7525
0.064 .334
0.86
0.064 .390
0.9675
0.064 .383
1.075
0.064 .324
1.1825
0.064 .269
1.29
0.064 .229
1.3975
0.064 .199
1.505
0.064 .184
1.6125
0.064 .167
1.72
0.064 .151
1.8275
0.064 .135

.242
.282
.316
.36

.429
.567

.549
.433
.357
.312
.275
.245
.222

158

.385 .643 .963 1.291 1.556 1.750 1.855 1.927
.438 .704 1.008 1.302 1.522 1.71 1.832 1.927
.493 .798 1.087 1.341 1.547 1.7 1.824 1.927
.58 .914 1.177 1.401 1.58 1.72 1.826 1.927
.737 1.052 1.291 1.481 1.632 1.733 1.831 1.927
.905 1.174 1.378 1.542 1.658 1.749 1.841 1.927
.705 1.009 1.245 1.434 1.567 1.672 1.760 1.840 1.927
.795 1.080 1.300 1.467 1.588 1.681 1.763 1.842 1.927
.772 1.080 1.316 1.472 1.589 1.681 1.764 1.843 1.927
.694 1.040 1.280 1.446 1.572 1.668 1.756 1.837 1.927
.920 1.205 1.402 1.539 1.648 1.743 1.881 1.927
.765 1.110 1.330 1.497 1.622 1.729 1.827 1.927 '
.576 .953 1.224 1.428 1.586 1.707 1.818 1.927
.478 .774 1.097 1.347 1.543 1.690 1.813 1.927
.416 .660 .982 1.277 1.496 1.666 1.807 1.927
.377 .584 .895 1.209 1.461 1.642 1.800 1.927
.347 .563 .874 1.202 1.472 1.684 1.836 1.927

 

Table 1 (cont’d)

picture 343

0.9190
0.121 .423
1.0211
0.121 .348
1.1232
0.119 .286
1.2253
0.119 .247
1.3274
0.116 .220
1.4296
0.118 .200
1.5317
0.116 .182
1.6338
0.116 .171
1.7359
0.115 .163

.3 10
.370
.433
.507

.525

159

.950 1.363 1.623 1.819 1.928 2.018 2.045
.595 1.023 1.369 1.593 1.777 1.901 2.007 2.041
.730 1.115 1.390 1.601 1.762 1.885 1.996 2.041
.896 1.218 1.443 1.633 1.775 1.888 1.989 2.041
.644 1.038 1.312 1.513 1.671 1.786 1.894 1.985 2.037
.811 1.182 1.390 1.578 1.707 1.802 1.895 1.979 2.030
.960 1.250 1.442 1.6061.716 1.809 1.895 1.975 2.029

.914 1.248 1.454 1.603 1.718 1.807 1.890 1.960 2.020
.764 1.191 1.420 1.587 1.705 1.802 1.885 1.960 2.022

542 1.041 1.336 1.536 1.670 1.782 1.876 1.953 2.022

.350
.305
.274
.252
.242

.540
.454
.392
.354

.416 .772 1.201 1.446 1.617 1.745 1.854 1.942 2.019
.984 1.307 1.525 1.693 1.831 1.931 2.019
.722 1.116 1.394 1.617 1.786 1.910 2.011
.574 .918 1.236 1.506 1.727 1.887 2.011
.512 .800 1.108 1.423 1.656 1.861 2.013
.510 .795 1.090 1.420 1.646 1.878 2.017

 

Table 1 (cont’d)

picture 445

17

9

33.2

19.6
1.0408
1.924
1.955
.1069
.045 .083
0.2138
.044 .095
0.3207
0.047 .115
0.4276
0.054 .138
0.5344
0.060 .172
0.6413
0.067 .229
0.7482
0.069 .324
0.8551

.153
.170
.207
.248
.345
.684 1.163 1.360 1.500 1.612 1.693 1.777 1.843 1.918
.966 1.272 1.436 1.548 1.643 1.717 1.783 1.844 1.917

.532 1
.963 1

160

.248 .485 .905 1.231 1.496 1.671 1.807 1.912
.282 .580 .983 1.279 1.483 1.653 1.787 1.911
.362 .788 1.140 1.367 1.540 1.683 1.800 1.912

.048 1.287 1.468 1.606 1.722 1.816 1.912
.230 1.419 1.566 1.665 1.753 1.830 1.918

0.072 .518 1.109 1.329 1.473 1.576 1.659 1.727 1.793 1.847 1.916

0.962

0.074 .485 1.164 1.360 1.483 1.586 1.661 1.730 1.789 1.849 1.917

1 .0689

0.076 .326 1.079 1.340 1.478 1.577 1.657 1.727 1.786 1.847 1.915

1.1758

0.077 .228 .644 1.285 1.443 1.555 1.644 1.712 1.783 1.843 1.914
327 1.123 1.365 1.509 1.618 1.698 1.770 1.842 1.911

1.2827
0.069 .182
1.3896
0.059 .148
1.4964
0.051 .125
1.6033
0.052 .114
1.7102
0.045 .107
1.8171
0.035 .101

.252 .479 1
.213 .338 .
.192 .292 .
.186 .292 .
.184 .301

.230 1.432 1.569 1.669 1.756 1.837 1.911
908 1.284 1.490 1.628 1.735 1.830 1.909
544 1.074 1.359 1.569 1.710 1.823 1.909
454 .881 1.249 1.495 1.687 1.822 1.910

.484 .853 1.215 1.511 1.711 1.837 1.908

 

Table 1 (cont’d)

picture 440

17

10

33.2

19.6
1.0408
1.851
1.861
0.10283
0.073 .111
0.20567
0.075 .122
0.3085
0.076 .133
0.41133
0.081 .152
0.51417
0.083 .168
0.617
0.086 .195
0.71983
0.088 .231
0.82267
0.091 .268
0.9255
0.090 .286
1.02833
0.091 .261
1.13117
0.088 .230
1.234
0.088 .202
1.33683
0.088 .185
1.43966
0.087 .173
1.5425
0.087 .166
1.64533
0.086 .164
1.74817
0.084 .152

.179
.198
.212
.250
.297
.377

.267
.290
.318
.392

.290 .430
.264
.249
.241
.233

.375
.343
.341
.335

161

.418 .723 1.085 1.382 1.638 1.807 1.935 2.001

.445 .798 1.139 1.390 1.617 1.786 1.915 1.998

.545 .984 1.260 1.474 1.662 1.806 1.918 1.992
.829 1.176 1.400 1.583 1.730 1.840 1.931 1.994
.559 1.092 1.335 1.515 1.658 1.771 1.861 1.937 1.993
.888 1.236 1.445 1.589 1.706 1.797 1.875 1.942 1.991
.546 1.091 1.341 1.518 1.634 1.731 1.810 1.880 1.944 1.991
.720 1.169 1.414 1.551 1.654 1.743 1.817 1.883 1.943 1.989
.743 1.205 1.432 1.561 1.662 1.745 1.818 1.884 1.945 1.991
.566 1.156 1.412 1.549 1.656 1.739 1.812 1.883 1.945 1.992
.420 1.048 1.349 1.512 1.630 1.725 1.802 1.877 1.942 1.995
.334 .611 1.225 1.435 1.585 1.695 1.786 1.869 1.939 1.996
.966 1.305 1.500 1.651 1.761 1.855 1.939 1.999
.610 1.105 1.379 1.577 1.719 1.834 1.927 2.003
.5 .854 1.219 1.479 1.673 1.815 1.924 2.005

.479 .740 1.104 1.396 1.623 1.805 1.923 2.011

.476 .715 1.057 1.364 1.629 1.818 1.938 2.015

162

Table 1 (cont’d)
picture 443

17

9

33.2

19.6

1.0408

1.863

1.880

0.1035

0.020 .069 .149 .277 .5541.1051.4741.698 l.8321.9221.963
0.207

0.033 .1 .194 .335 .6591.1831.4741.6751.8011.9001961

0.3105

0.046 .128 .239 .402 916129515251684180719011.961

0.414

0.047 .154 .290 5821.1591.4101.5861.7141.8191.9061963
0.5175

0.060 .198 .387 1.013 1.317 1.513 1.639 1.743 1.831 1.908 1.962
0.621

0.067 .263 .631 1.228 1.438 1.5741.682 1.7661838 1.908 1.958
0.7245

0.068 .349 1.0161.3191.494 1.6101.697 1.775 1.8441908 1.957
0.828

0.068 .459 1.1161365 1.518 1.625 1.708 1.7801.8441.9041.954
0.9315

0.068 .420 1.0921367 1.5201.626 1.709 1.781 1.845 1.9041954
1.035

0.068 .298 9691.328 1.491 1.607 1.7021777 1.8441.9001.955
1.1385

0.068 .220 .493 l.2241.4401.5741.677 1.762 1.835 1.895 1.955
1.242

0.1316; .172 .328 .9911.3391.513l.6391.7371.8201.8881.953
1. 4 5

0‘22; .145 .257 .4381.1151.3981.5661.6951.7961.8751.946
1.

0.058 .122 .212 .338 .7071.1881.4471.631 1.7601.8611.940
1.5525

0.1352 .112 .186 .283 .462 9241.2851.523170618381930

1. 5

0.9535100 .169 .252 .394 .6961.1031.4021.6451.8101.946

02053 .099 .164 .251 .397 .6811.019l.3851.6291.8171.943

-w... -_ .lnfiy

 

‘1

 

Table 1 (cont’d)

picture 41

17

10

33.1

19.8
1.0396
1.853

1.87
0.102944
0.049 .074
0.205889
0.047 .079
0.30883
0.044 .087
0.411778
0.042 .096
0.52372
0.040 .105
0.617667
0.044 .124
0.72061
0.044 .169
0.823555
0.041 .251
0.9265
0.045 .270
1.02944
0.046 .193
1.13239
0.050 .152
1.23533
0.048 .125
1.3383
0.044 .115
1.44122
0.042 .104
1.54417
0.040 .098
1.6471
0.049 .1
1.75005

.127
.139
.153
.170
.208

163

.202 .330 .632 1.016 1.291 1.496 1.647 1.764 1.858
.218 .346 .734 1.110 1.337 1.509 1.646 1.760 1.854
.234 .411 .976 1.250 1.429 1.575 1.688 1.777 1.861
.281 .869 1.214 1.396 1.530 1.635 1.719 1.789 1.861
.445 1.189 1.373 1.500 1.595 1.670 1.739 1.797 1.858

.286 1.057 1.319 1.455 1.555 1.631 1.691 1.750 1.801 1.854
.531 1.206 1.397 1.503 1.583 1.650 1.703 1.754 1.804 1.862
.820 1.280 1.437 1.528 1.601 1.659 1.712 1.761 1.807 1.860
.961 1.320 1.454 1.539 1.608 1.663 1.712 1.762 1.810 1.853
.533 1.308 1.446 1.537 1.604 1.662 1.711 1.763 1.810 1.853
.303 1.249 1.424 1.519 1.594 1.657 1.708 1.762 1.809 1.860
.224 1.122 1.366 1.483 1.576 1.643 1.701 1.758 1.814 1.858

.189
.171
.163

.315 1.250 1.413 1.532 1.621 1.690 1.753 1.814 1.858
.267 1.014 1.280 1.457 1.579 1.668 1.747 1.816 1.855
.251 .448 1.090 1.347 1.513 1.642 1.737 1.820 1.852

.164 .254 .411 .879 1.229 1.433 1.602 1.729 1.822 1.852
0.039 .097 .161 .248 .405 8921.234 1.441 1.602 1.731 1.818 1.852

 

‘1

 

Table 1 (cont’d)

picture 43

17

11
33.1
19.8
1.0396
1.808

1.60711
0.075 .121
1.70755
0.076 .125

.157
.165
.178
.192
.226
.272
.384

.380
.295
.250
.219
.202
.191
.186
.188

164

.231 .355 .619 .973 1.230 1.454 1.634 1.760 1.859 1.918
.237 .362 .668 1.047 1.279 1.457 1.615 1.741 1.838 1.914
.255 .402 .900 1.187 1.377 1.530 1.659 1.762 1.842 1.914
.293 .562 1.127 1.340 1.489 1.610 1.708 1.786 1.850 1.913
.366 1.035 1.288 1.453 1.571 1.663 1.738 1.8 1.856 1.912
.666 1.199 1.406 1.528 1.619 1.694 1.757 1.812 1.861 1.911
.980 1.305 1.475 1.573 1.651 1.717 1.771 1.822 1.869 1.914
.522 1.073 1.361 1.501 1.592 1.665 1.727 1.778 1.828 1.872 1.915
.500 1.062 1.379 1.514 1.601 1.671 1.731 1.781 1.831 1.875 1.913
.951 1.355 1.502 1.597 1.669 1.731 1.783 1.833 1.879 1.926
.521 1.282 1.462 1.572 1.654 1.721 1.777 1.829 1.879 1.924
.385 1.133 1.386 1.522 1.621 1.699 1.762 1.822 1.877 1.929
.317 .533 1.242 1.426 1.562 1.660 1.739 1.809 1.871 1.926
.282 .418 .993 1.269 1.456 1.596 1.703 1.790 1.860 1.930
.269 .379 .658 1.084 1.329 1.502 1.653 1.765 1.849 1.931
.264 .379 .573 .936 1.210 1.425 1.603 1.743 1.846 1.936
.269 .403 .595 .886 1.162 1.395 1.598 1.751 1.862 1.947

 

165

Table l (cont’d)
picture 40

17

8

33.1

19.8

1.0396

1.808

1.815

0.10044

0.112 .145 .212 .322 .672 1.347 1.656 1.828 1.922 2.019
0.20088

0.116 .162 .240 .347 .8441.421 1.648 1.809 1.911 2.006
0.30133

0.120 .179 .268 .390 1.222 1.523 1.696 1.823 1.917 2.007
0.4018

0.123 .196 .290 .465 1.407 1.601 1.7401.845 1.921 2.006
0.5022

0.132 .232 .376 1.400 1.577 1.708 1.797 1.872 1.938 2.005
0.6027

0.149 .281 1.272 1.527 1.657 1.755 1.826 1.890 1.946 2.005
0.7031

0.149 .384 1.424 1.605 1.705 1.783 1.846 1.899 1.950 2.002
0.8036

0.150 .574 1.487 1.635 1.719 1.793 1.847 1.901 1.950 2.002
0.904

0.162 .476 1.501 1.642 1.723 1.793 1.850 1.901 1.951 2.003
1.0044

0.163 .349 1.465 1.621 1.714 1.788 1.844 1.900 1.949 2.007
1.1049

0.1393286 .518 1.572 1.686 1.766 1.829 1.890 1.942 2.006
1. 5

0.;318256 .377 1.477 1.635 1.735 1.810 1.877 1.935 2.002
1. 5

0.162 .237 .330 .516 1.537 1.676 1.778 1.865 1.929 2.001

0.159 .226 .302 .416 1.362 1.568 1.718 1.835 1.915 2.004
02154 .218 .283 .373 9861.3971.612178318892016
0.156 .215 .275 .357 .5541.2041.4881.7191.8652.021

1.7076
0.154 .212 .277 .361 .565 1.075 1.440 1.705 1.885 2.021

 

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