AN ABSTRACT OF THE THESIS OF
Seung-Ho Hong for the degree of Doctor of Philosophy in
Mechanical Engineering presented on September 21. 1998.
Title: An Experimental Study of Friction Factors and Mixed
Convection in the Thermal Entrance Region of Vertically
Narrow, Horizontal Rectangular Channels for Different
Heating Conditions and Aspect Ratios.
Abstract approved:
Redacted for privacy
Heat transfer and fluid mechanical behavior of water
flowing horizontally in vertically-narrow rectangular
channels was studied in this work.
Friction factor and
Nusselt number variations were determined experimentally
for four different wall heating conditions, and for aspect
ratios(height to width) of 5,
4,
3,
2, and 1.
Wall
heating conditions examined were: all surfaces heated;
three surfaces heated, the top adiabatic; one side surface
heated the others adiabatic; and the bottom surface
heated, the others adiabatic.
Friction factors, in laminar flow, varied as
predicted from analysis.
The critical Reynolds number
varied linearly with ln(Dh).
Local Nusselt numbers were determined, at each aspect
ratio and heating condition, as functions of Reynolds and
Rayleigh numbers.
Mixed convection was the result of
buoyancy-induced secondary flows.
Local Nusselt numbers
decreased in a manner common to pure forced convection,
reaching minimum values some distance from the entrance,
then increased due to the presence of the secondary flows.
For given aspect ratio, local Nusselt numbers were found
to increase and the thermal entrance lengths decreased,
with increasing Rayleigh numbers.
In the thermal entry
region, for all heating conditions except the bottomheating case, local Nusselt number behavior showed minor
dependence on aspect ratio.
©Copyright by Seung-Ho Hong
September 21, 1998
A11 Right Reserved
An Experimental Study of Friction Factors and Mixed
Convection in the Thermal Entrance Region of Vertically
Narrow, Horizontal Rectangular Channels for Different
Heating Conditions and Aspect Ratios.
by
Seung-Ho Hong
A THESIS
submitted to
Oregon State University
in partial fulfilment of
the requirements for the
degree of
Doctor of Philosophy
Completed September 21, 1998
Commencement June, 1999
Doctor of Philosophy thesis of Seung-Ho Hong presented on
September 21, 1998
APPROVED:
Redacted for privacy
Major Profess
reps seSting Mecha ical Engineering
Redacted for privacy
Head of Department of Mechanical Engineering
Redacted for privacy
Dean of Gradu
e of School
I
understand that my thesis will become part of the
permanent collection of Oregon State University libraries.
My signature below authorizes release of my thesis to any
reader upon request.
Redacted for privacy
Seung-Ho Hong, Author
ACKNOWLEDGMENTS
I would like to thank my major professor, Dr. James
R. Welty, who guided me through this work.
Thank you Sir.
I wish to thank Dr. G.M. Reistad for his financial
support and other committee members.
TABLE OF CONTENTS
Page
Chapter 1. INTRODUCTION AND LITERATURE REVIEW
1
1.1 Introduction
1
1.2 literature review
3
1.2.1 Friction factor in rectangular
3
channel
1.2.2 Heating all surfaces of a rectangular
8
channel
1.2.3 Heating bottom surface of a rectangular
12
channel
1.2.4 Other related issues in rectangular
16
channels
Chapter 2. EXPERIMENTAL APPARATUS AND PROCEDURES
19
2.1 Introduction
19
2.2 Design range
20
2.3 Flow loop
20
2.4 Channel design
22
2.4.1 Hydrodynamic entrance length of
rectangular channel
2.4.2 Thermal entrance length for
hydrodynamically developed flow with
uniform wall heat flux
2.4.3 Channel construction
22
23
26
2.5 Heater
30
2.6 Micro Pressure gage
36
2.7 Thermocouple installation and temperature
measurement
39
2.8 Flow rate measurement
42
2.9 Data reduction
42
2.9.1 Data reduction for friction factor
2.9.2 Data reduction for local Nusselt
number
42
44
TABLE OF CONTENTS(Continued)
Page
Chapter 3. RESULTS AND DISCUSSION
48
3.1 Friction factors
48
3.2 Heat transfer
53
3.2.1
3.2.2
3.2.3
3.2.4
Whole surface heating
Three-surface heating
Heating of one vertical surface
Bottom surface heating
Chapter 4. MULTIPLE REGRESSION MODELING
53
68
81
94
106
4.1 Multiple regression
106
4.2 Modeling
107
Chapter 5. CONCLUSION
120
BIBLIOGRAPHY
122
APPENDICES
129
APPENDIX
APPENDIX
APPENDIX
APPENDIX
A.
B.
C.
D.
Uncertainty analysis
Friction factor data
Sample heat transfer data
Multiple regression model printouts
using the SAS program for backward
elimination
130
136
139
201
LIST OF FIGURES
Figure
Page
1.1 Coordinates of rectangular channel
3
2.1 Schematic diagram of flow loop
21
2.2 Local Nusselt number variation with different
aspect ratios [16]
25
2.3 Schematic diagram of experimental channel showing
thermocouple and pressure-tap location
27
2.4 Channel assembly with heaters
28
2.5 Test channel insulation
29
2.6 Cross section of test channel with detailed view
of heater
34
2.7 Power connection with heater
35
2.8 Micro-pressure-difference measuring equipment
37
2.9 Thermocouple implantation
41
2.10 Graphical scheme used in determining the mean
temperature for total surface heating with
aspect ratios, b/a=3, 4 and 5
45
2.11 Graphical scheme used in determining the mean
temperature for total surface heating with
aspect ratios, b/a=1 and 2
46
3.1 Comparison of experimental and predicted friction
factors with different aspect ratios
49
3.2 Variation of critical Reynolds number, Re, with
aspect ratios
50
3.3 Variation of critical Reynolds number, Re, with
hydraulic diameter, ph
51
3.4 Values of fRe over a range in aspect ratios for
hydrodynamically fully-developed flow in
rectangular channels
52
3.5 Temperature variations with dimensionless axial
distance for different aspect ratios, (a)b/a=1
(b) b/a=4
54
LIST OF FIGURES(Continued)
Figure
Page
3.6 Comparison of circumferential temperature
distributions for Ra=3x104, Re=602, Pr=6.72
with numerical results[16] for square channel
55
3.7 Local Nusselt number variation in the thermal
entrance region of horizontal square channels;
uniform heat flux on whole surface
58
3.8 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular channels
(aspect ratio b/a=2); uniform heat flux on whole
59
surface
3.9 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular channels
(aspect ratio b/a=3); uniform heat flux on whole
60
surface
3.10 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular channels
(aspect ratio b/a=4); uniform heat flux on whole
61
surface
3.11 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular channels
(aspect ratio b/a=5); uniform heat flux on whole
62
surface
3.12 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular
channels; uniform heat flux on whole surface with
different aspect ratios for Rayleigh number
Ra=lx105
64
3.13 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular
channels; uniform heat flux on whole surface with
different aspect ratios for Rayleigh number
Ra=2x105
65
3.14 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular
channels; uniform heat flux on whole surface with
different aspect ratios for Rayleigh number
Ra=4x105
66
LIST OF FIGURES(Continued)
Figure
Page
3.15 The effect of Rayleigh numbers and aspect ratios
on the location of minimum Nux for whole heating
condition
67
3.16 Temperature variation with axial distance Z for
aspect ratio b/a=1 in bottom- and side-surfaces
heating condition
68
3.17 Circumferential wall temperature distribution
for aspect ratio b/a=4
69
3.18 Local Nusselt number variation in the thermal
entrance region of horizontal square channels;
uniform heat flux on bottom and both side
surfaces
71
3.19 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular
channels(aspect ratio b/a=2); uniform heat flux
on bottom and both side surfaces
72
3.20 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular
channels(aspect ratio b/a=3); uniform heat flux
on bottom and both side surfaces
73
3.21 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular
channels(aspect ratio b/a=4); uniform heat flux
on bottom and both side surfaces
74
3.22 Local Nusselt number variation in the thermal
entrance region of horizontal rectangular
channels(aspect ratio b/a=5); uniform heat flux
on bottom and both side surfaces
75
3.23 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular
channel; uniform heat flux on bottom and both
side surfaces with different aspect ratios
for Rayleigh number Ra=1x105
77
3.24 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular
channel; uniform heat flux on bottom and both
side surfaces with different aspect ratios
for Rayleigh number Ra=2x105
78
LIST OF FIGURES(Continued)
Figure
Page
3.25 Local Nusselt number variation in the thermal
entrance region of s horizontal rectangular
channel; uniform heat flux on bottom and both
side surfaces with different aspect ratios
for Rayleigh number Ra=4x105
79
3.26 The effect of Rayleigh numbers and aspect
ratios on the location of minimum Nux;
uniform heat flux on bottom and both side
surfaces
80
3.27 Temperature variation with axial distance, Z, for
aspect ratio b/a=1 in one side wall heating
81
condition
3.28 Temperature variation with axial distance, Z, for
aspect ratio b/a=4 in one side wall heating
82
condition
3.29 Circumferential wall temperature distribution
for aspect ratio b/a=4 in one side wall heating
condition
83
3.30 Local Nusselt number variation in the thermal
entrance region of horizontal square channels;
uniform heat flux on one side wall
84
3.31 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular
channel(b/a=2); uniform heat flux on one side
wall
85
3.32 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular
channel(b/a=3); uniform heat flux on one side
wall
86
3.33 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular
channel(b/a=4); uniform heat flux on one side
surface
87
3.34 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular
channel(b/a=5); uniform heat flux on one side
wall
88
LIST OF FIGURES(Continued)
Figure
Page
3.35 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular
channel; uniform heat flux on one side wall,
with different aspect ratios, for Rayleigh
number Ra=1x105
90
3.36 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular
channel; uniform heat flux on one side wall,
with different aspect ratios, for Rayleigh
number Ra=2x105
91
3.37 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular
channel; uniform heat flux on one side wall,
with different aspect ratios, for Rayleigh
number Ra=4x105
92
3.38 The effect of Rayleigh numbers and aspect ratios
on the location of minimum Nux; uniform heat
flux on one side wall
93
3.39 Bottom center surface temperature and bulk
temperature of water variation along Z
94
3.40 (a)Typical temperature difference between bottom
wall and bulk water temperature and (b)Local
Nusselt number variation with axial distance;
constant heat flux on bottom surface
96
3.41 Local Nusselt number variation in the thermal
entrance region of a bottom heated horizontal
square channel
98
3.42 Local Nusselt number variation in the thermal
entrance region of a bottom heated horizontal
rectangular channel(b/a=2)
99
3.43 Local Nusselt number variation in the thermal
entrance region of a bottom heated horizontal
rectangular channel(b/a=3)
100
3.44 Local Nusselt number variation in the thermal
entrance region of a bottom heated horizontal
rectangular channel(b/a=4)
101
LIST OF FIGURES(Continued)
Page
Figure
3.45 Local Nusselt number variation for bottom heating
for different aspect ratios and Rayleigh number
(a) Ra=105, (b) Ra=2x105,
(d) Ra =4x105
(c) Ra=3x105, and
3.46 Ideal and multiple stage secondary flow
102
105
4.1
Comparison between experimental data and multiple
regression results for the whole-surface-heating
112
condition
4.2
Comparison between experimental data and multiple
regression results for the three-surfaces-heating
114
condition
4.3
Comparison between experimental data and multiple
regression results for the one-side-surface117
heating condition
LIST OF TABLES
Table
Page
2.1 Experimental range
20
2.2 Entrance length for a rectangular channel for
W=5.75, Re=2300
23
2.3 Electrical resistance of heaters with different
aspect ratios
33
2.4 Water head difference for Re=100, L=2.75 m, and
T=20°C
36
2.5 Location thermocouple
40
2.6 Order of regression to calculate mean surface
temperature
47
3.1 Experimental and predicted fRe for fullydeveloped laminar flow for different aspect
ratios
52
3.2 Dimensionless distance, Z, to minimum Nux with
different Rayleigh numbers, Ra, and aspect ratios. 67
3.3 Dimensionless distance, Z, to minimum Nux with
different Ra and aspect ratios; uniform heat
flux on bottom and both side surfaces
80
3.4 Dimensionless distance, Z, to minimum Nux with
different Ra and aspect ratios; uniform heat
flux on one side surfaces
93
4.1 Multiple regression for whole surface heating
case of square channel
109
4.2 Regression coefficient for whole-surfaceheating case
110
4.3 Regression coefficient for three-surfacesheating case
111
4.4 Regression coefficient for one side-surfaceheating case
111
LIST OF APPENDIX TABLES
Appendix
Page
A.1 Uncertainty of measurement error variables
131
A.2 Uncertainties of friction factor for different
aspect ratio and Re
133
A.3 Variable uncertainties
135
NOMENCLATURE
a, b
channel width and height, m
A,
channel cross section area, m2
Aq
heating surface area, m2
Cp
specific heat at constant pressure, J/kg.K
Dh
hydraulic diameter, m
g
gravitational acceleration, m/s2
h
pressure head, m
ID
inside diameter, m
f
friction factor, dimensionless
f'
modified friction factor, dimensionless
L
channel length, m
L *th
dimensionless thermal distance
m
mass flow rate, kg/s
Nu
Nusselt number
OD
outside diameter, m
p
pressure, N/m2 or Pa
Pr
Prandtl number of fluid
heat flux, W/m2
Ra, Ra*
Rayleigh number and modified Rayleigh number
Re
Reynolds number
Re*, Re+
modified Reynolds number
t
time, s
T
Temperature °C
u
velocity in x direction, m/s
u
average velocity, m/s
NOMENCLATURE(continued)
x, y,
z
rectangular coordinates, m
W
weight, kg
Z
dimensionless axial distance, L/(Dh.Re.Pr)
Greek symbols
13
volumetric coefficient of thermal expansion, K-1
O
angle(degree)
8
dimensionless temperature difference,
(T,-Tmwx)/(q"Dh/K)
K
thermal conductivity, W/m.K
A:
viscosity, kg/sm or Pas
kinematic viscosity, m2/s
P
mass density, kg/m3
(0*
geometric function(Equation 1-7)
(0
functional relation
O
electric resistance, ohms
e
(Ra-Ra,)/ Ra,
Subscripts
c
critical
f
fully developed
h
hydraulic
i, in
inlet
m
mean
msx
local mean surface
NOMENCLATURE(continued)
mwx
local mean water
n
non-circular
(n,
c)
non-circular critical variable
(c,
c)
circular critical variable
o
onset
out
outlet
s
surface
th
thermal
by
hydrodynamic
w
water
To my parents, brother and sister
Thanks
AN EXPERIMENTAL STUDY OF FRICTION FACTORS AND MIXED
CONVECTION IN THE THERMAL ENTRANCE REGION OF VERTICALLY
NARROW, HORIZONTAL RECTANGULAR CHANNELS FOR DIFFERENT
HEATING CONDITIONS AND ASPECT RATIOS
Chapter 1
INTRODUCTION AND LITERATURE REVIEW
1.1 Introduction
Mixed convection flows which are combination
of
forced and free convection flows arise in many heat
transfer processes in engineering devices and in nature.
Mixed convection occurs when the effect of the buoyancy
force in forced convection or the effect of forced flow in
free convection become significant. The tendency of mixed
convection process is pronounced when the temperature
difference in low velocity forced flow is large.
In
internal horizontal laminar flow as well as inclined
laminar flow, it is well known that the temperature
difference in the gravitational direction induces
secondary flow along the main stream and increases the
heat transfer rate.
It is, therefore, important to
understand the differences in thermal characteristic
values in mixed convection relative to values of pure
natural and pure forced convection.
2
With increased technological development, the size of
heat-related equipment, including heat exchangers, is
decreasing.
This trend often requires different channel
shapes from the currently-predominant circular case.
At
small scales, one of the easiest shapes to fabricate is
the rectangle. Unlike flow and heat transfer in circular
channels, studies of thermal characteristics associated
with fluid flow through rectangular channels are limited.
Additionally, most heat transfer textbooks list Nusselt'
numbers for thermally fully-developed flow in rectangular
channels which do not include buoyancy effects of the
fluid.
In heat exchangers, channel lengths are often not
sufficient to achieve fully-developed flow and heat
transfer.
Therefore, the thermal behaviors affected by
buoyancy forces in the thermally-developing and fullydeveloped regions are often more important than the
correlations which do not count buoyancy effects.
The purpose of this experimental study was to produce
empirical results for mixed laminar convection in narrow
vertical rectangular channels with different aspect ratios
for a range of Reynolds numbers, Rayleigh numbers, and
wall heating conditions using water as the working fluid.
3
1.2 Literature review
Although less numerous than for the circular tube
case, several papers have been published dealing with
rectangular channel flow during the past few decades for
different channel sizes, aspect ratios, heating conditions
and different working fluids.
In this chapter, published
studies for rectangular channel flow are reviewed.
Figure
1.1 shows the typical coordinates for a rectangular
channel.
b
Figure 1.1 Coordinates of rectangular channel
1.2.1 Friction factor in rectangular channel
A number of published studies [1-5] suggest that, at
constant Reynolds number based on hydraulic diameter, Dh,
the friction factor, f, increases monotonically with
increasing aspect ratio.
For such behavior one would
conclude that the hydraulic diameter is not the proper
4
length scale to use in the Reynolds number to insure
similarity between circular and other non-circular
channels.
The fully-developed velocity profile for rectangular
channels has been evaluated as[1]
-_
1
dp a2
4
7.1
dx
n =1,3,..
yn- 1)/2
cosh(nity/a)
n3
cos nIcz
a
co sh(inc b Ia)
(1--1)
and the corresponding mean velocity is[1]
n"
2a
(dp)a2{1_ 192 a =1 tank
7r5
12
µ
0 n=1,3,. n5
1
in
c b c
(1-2)
The Fanning friction factor, based on equation(1-2), is
expressed as[1,2]
24
fRe-
(1+a) 2 (1_192(a)
b
775
t 1 tanh(nnb ))
b n=1,3,.. n5
(1-3)
2a
The factor, fRe, in equation(1-3) may be closely
approximated, within +0.059,5, by the following empirical
equation [1]
.
fRe =
24 [1-1.3553 (b I a) + 1.9467 (b a)2 1.7012
(b a)3
(1-4)
+0.9564 (b /a)4
0.2537 (hia)5
I
5
Shah and London[1] include the results of several
investigators who have examined this case.
Hartnett et al.[3] compared published fluid flow
relationships for rectangular and circular channels in
turbulent flow with results from exact solutions of the
Navier-Stokes equation in the laminar flow regime.
These
studies considered several channel configurations, aspect
ratios, and hydraulic diameters.
They concluded that, for
Reynolds numbers between 6*103 and 5*105, circular channel
correlations accurately represent the friction coefficient
for flow through rectangular channels for all aspect
ratios.
Later, Jones [4]
introduced a modified(or laminar
equivalent)Reynolds number, Re*, and by using this number,
he demonstrated that conventional circular channel methods
may be readily applied to rectangular channels,
eliminating large errors in estimating the friction
factor.
The relation between friction factor and modified
Reynolds number in rectangular channel yields the same
friction factor relation as would be obtained for a
circular channel.
The relation is
f
64
Re*
(1-5)
where Re* is the modified Reynolds number, expressed as
(1-6)
Re * =
where q* is a geometric function given by[4]
( T.)
2
=--
1+
b
192 b
2
1
a
E
1
n=0
7C) a
(2n+1)5
tanhj
(2n+1)7ta
2b
(1-7)
Jones[4] cautioned that this approach can be especially
troublesome for high aspect ratio channels.
Obot[5] used another simple approach to explain the
friction characteristics of non-circular channels, a
Critical Friction Method.
The modified or reduced
Reynolds number, Ref, and friction factor, f% for
isothermal conditions were expressed as
Re'=*RRen- ()Recw
L
n
(1-8)
2
=
(
Apn,c)(
Re c,,
R
n,c
f
(1-9)
inn
fc,c
n,c
(1-10)
7
where the subscripts n,
(n,c), and (c,c) refer to the non-
circular, non-circular critical variable, and circularcritical variable respectively.
By this approach, the
reduced data show remarkable consistency with circular
channel relations.
However, as pointed out by Obot[5] in
his conclusions, there is a definite limitation on the use
of the proposed method for friction factor calculations,
i.e. a reliable and consistent trend with increasing
height-to-base aspect ratio or diameter ratio could not be
established for the critical Reynolds number, Re,.
Due to the increased capability of manufacturing
micro-scale structures, flow characteristics in microscale rectangular channels have received considerable
attention recently.
Tuckerman and Pease[6], and Wu and
Little[7] studied fluid flow through micro-scale channels
and their results for friction factors were higher than
those predicted by classical theory.
however, carried out by Pfahler et al.
Other experiments,
[8] and Yu[9]
showed different results in which the friction factors in
the laminar region were lower than those predicted by the
Navier-Stokes equations.
Peng et al.[10] obtained
friction factors with hydraulic diameters of 0.133 mm < ph
< 0.367 mm and aspect ratios of 0.333 < b/a < 1.
They
found the critical Reynolds number to be approximately
200-700, and friction factor behavior for both laminar and
8
turbulent flow to deviate from the general classical
theory.
Despite these painstaking and contradictory
efforts, no generally valid method is available for
predicting single-phase flow behavior through rectangular
channel.
1.2.2 Heating all surfaces of a rectangular channel
Clark and Kays [11] studied laminar flow heat
transfer in a rectangular channel. In their experiments
uniform wall temperatures and uniform wall heat fluxes
were applied for a square channel and a uniform wall
temperature was established for a rectangular channel with
an aspect ratio a/b=2.62.
Their results were presented as
a correlation for the Nusselt number in the following form
Nu
=
4){RePrl
L/Dh
Such a correlation does not express the relation between
Nusselt number and Rayleigh number in low Reynolds number
flows where secondary flow effects are significant.
Sparrow and Siegel [12] studied convection heat
transfer with hydrodynamically and thermally fullydeveloped laminar flow in non-circular channels using a
variational method. They evaluated the temperature profile
in a square channel using a velocity profile of the form
9
2
{X2 t) 112
1
0.2808(
a
2
2
2
2
105844(-'-') + 0.1185(X2
+y2)1
which agrees closely with the exact velocity solution for
pure forced flow.
Based on the velocity profile expressed
as equation (1-1) they computed Nusselt numbers for two
cases; a square channel with uniform wall heat flux in the
axial direction and uniform peripheral wall temperature,
and a square channel with uniform heat flux in the axial
and peripheral directions.
Schmidt and Newell [13] investigated the Nusselt
number for fully-developed flow through rectangular
channels with various aspect ratios by numerical methods.
In this study, they used both uniform wall temperature and
uniform heat flux boundary conditions.
A uniform wall
temperature was assumed for the conducting surfaces at
each cross section.
effects.
They did not account for buoyancy
Schmidt and Newell's [13] results for a square
channel and for some other aspect ratio rectangular
channels agree with those of Clark and Kays[11].
Chandrupatla and Sastri [14] examined laminar forced
convection heat transfer with a non-Newtonian fluid in a
square channel and evaluated the limiting Nusselt numbers.
10
Their results, obtained numerically, show good agreement
with those for Newtonian fluids.
They also obtained a
relationship between Nusselt number and Graetz number, Gz,
in the thermally developing region.
Cheng and Hwang[15] considered combined free and
forced convection for steady, fully-developed laminar flow
in horizontal rectangular channels for thermal boundary
conditions of axially uniform wall heat flux and
peripherally uniform wall temperature at each axial
position.
Using a successive overrelaxation method they
obtained results for temperature distributions, velocity
distributions, streamlines, and isotherms for several
aspect ratios.
Their graphical results for streamlines at
different aspect ratios indicated that the non-dimensional
location of the center of secondary flow changes with
different aspect ratios.
Cheng, Hong and Hwang [16] numerically investigated
the buoyancy effects on laminar flow in the thermallydeveloping region of horizontal rectangular channels with
uniform wall heat flux for
Prandtl numbers ranging from
10 to infinity, which allowed them to neglect the
convective terms in the momentum equations.
Their
graphical results show that the variation of local Nusselt
number along the channel agrees with the case of pure
11
convection until encountering the onset of secondary flow.
After reaching a minimum value, the local Nusselt numbers,
Nu, increased slightly and remained constant.
Their
investigation showed that the thermal entrance length
decreases with increasing Rayleigh number and that the
local Nusselt number in the thermally fully-developed
region increases with increasing Rayleigh number.
Later, Abou-Ellail and Morcos [17] investigated same
problem as Cheng et al.[16] and included the convective
terms.
They concluded that Nusselt numbers, for different
Prandtl numbers, approach the same asymptotic values as
for the fully-developed case [17].
Relative to the number of numerical and analytical
studies of mixed laminar flow through rectangular channels
experimental studies are rare.
Morcos et al.
[18]
investigated the heat transfer problem of mixed laminar
convection in the hydrodynamic and thermal entrance region
of inclined as well as horizontal rectangular channels
having aspect ratios, b/a=0.375 and 2.667.
For their test
section an aluminum rectangular channel having an outer
dimensions of 20 mm x 10 mm, a wall thickness of 2 mm, and
a total length of 2.25 m was used.
Their experimental
results yielded the following correlations in the fully
developed region.
12
For b/a=0.375
0..
Nu = 1.05Ram
135 [
27
1 +
3.27Rei0
+30)]
(1-13)
1 +
2.21 Re; '3.21 sine cos(0 +30)]
(1-14)
sine cos(13
and for b/a=2.667
Nu = 0.95Ran,135[
for 5x104 < Ram < 106, 100 < Re, < 500, and 0 deg < 9 < 45
deg.
Chou and Hwang [19] examined the Graetz problem
including the effect of natural convection in a horizontal
rectangular channel with uniform wall heat flux using
vorticity-velocity methods and presented the effects of
Prandtl numbers on local Nusselt numbers for the same
Rayleigh number.
Yan [20] also used the vorticity-
velocity method to simulate the mixed convection problem
in an inclined rectangular channel for hydrodynamically
fully-developed and thermally-developing flow.
1.2.3 Heating bottom surface of a rectangular channel
Over the past decade several investigators have
studied flow and thermal problems in mixed convection for
bottom-heated horizontal rectangular channels to provide
an even deposition layer in chemical vapor deposition
13
(CVD) reactors and to show the relationships between
Reynolds number, Rayleigh number and other variables which
affect flow patterns.
Early experiments conducted by Ostrach and Kamotani
[21, 22] indicated that vortex rolls are generated and
become irregular for Ra> 8000 and, for Ra> 18352, a second
type of vortex roll emerges whose wavelength is only half
that for the first type, i.e. A=b.
Within the
experimental range, 2.2x104 < Ra < 2.1x10 5 and 50 < Re <
300, using air as a working fluid Kamotani et al.[23]
found that the heat transfer rate and thermal entrance
length are influenced not only by the Rayleigh number but
also by the ratio of Re2/Gr. Additionally the resulting
thermal instability increased the heat transfer rate as
much as 4.4 times.
Using nitrogen, Chiu and Rogenberger[24] concluded
that, in the range 1368<Ra<8300, 15<Re<170, and a/b=10,
the thermal entrance length, Lth,o,
for the onset of
buoyancy-driven instability was
Lth,c, =
(0.65 ± 0.05
)He-°.44±0.01 Re 0.76±0.02
(e>0)
(1-15)
and the thermal entrance length, Lth,f, for fully-developed
longitudinal convection was
14
Lthf =(0.68±0.07)HC° .69±0.02 Re 0.96±0.03
(e>0)
(1-16)
where c=(Ra-Ra,)/Ra, represents the reduced Rayleigh
number, and Ra, represent the critical Ra for RayleighBenard convection.
They also observed that longitudinal
convection rolls are unsteady due to an admixture of
transverse convection rolls.
Incropera and coworkers [25-29] conducted a series of
experiments to examine the effect of buoyancy forces in
horizontal rectangular channels with uniform heat fluxes
applied at the bottom and/or top surfaces.
With a uniform
heat flux applied at both bottom and top surfaces [26,
27], visualization results showed the existence of a
buoyancy-driven flow which strongly influenced the heat
transfer rate at the bottom surface but had a weak
influence at the top surface. These studies also showed
the lower heat transfer rate on the top surface relative
to that at the bottom to be caused by boundary-layer
laminarization, while the higher heat transfer rate on the
bottom surface was due to free convection in transition
flow and in turbulent flow.
The heat transfer rate at the
top surface was correlated by an expression for pure
forced convection, while the bottom surface heat transfer
rate was correlated by an expression for mixed convection.
In the case of a uniform heat flux applied on the bottom
15
surface alone, the length-to-onset changed with the
Grashof number and Nusselt numbers increased to a maximum
value after onset and then assumed a fully- developed
value slightly lower than the maximum value [28, 29].
Concurrent with experimental studies, many numerical
simulations were also conducted to predict flow patterns
and axial variations of Nusselt numbers along the channel.
Evans and Greif [30], using semi-implicit and SIMPLER
methods for a two-dimensional case, showed that a thermal
instability in the fluid can result in transverse
traveling waves which enhance the heat transfer rate from
50 to more than 300 percent above conditions without
traveling waves.
Later, Evans and Greif [31, 39] extended
their simulation to a three-dimensional case for an aspect
ratio a/b=2, at supercritical Rayleigh numbers with
adiabatic side walls.
Hosokawa et al.[32] used the conditional Fourier
spectral method, for an aspect ratio a/b=2, and found that
unsteady flow states, occurring at rather high Grashof
numbers, provide favorable conditions for spanwise uniform
temperature distributions and Sherwood numbers on the
substrate at the channel bottom.
16
To study the buoyancy-force effect in a mixedconvective flow of air, Hunag and Lin [33, 34] solved an
unsteady three-dimensional model by a higher-order finite
difference numerical scheme for various conditions and
found that the buoyancy induced laminar-to-turbulent flow
transition follows the Ruelle-Taken route.
Lin and Lin[35, 36] compared experimental
observations with numerical simulation results and
observed the generation and merging of longitudinal vortex
rolls in the channel and the effect of buoyance-to-inertia
ratio; more rolls were generated and they subsequently
merged into fewer and larger rolls at higher buoyancy-toinertia ratios.
1.2.4 Other related issues in rectangular channels
Several other investigators have addressed heat
transfer problems related to rectangular channels for
various boundary conditions and configurations.
To
simulate the boundary condition associated with thermal
radiation from a plasma in nuclear fusion reactors,
Kurosaki et al.[40] applied a uniform heat flux on one
side wall and showed the entrance length to be longer than
for the case of all walls heated.
17
Javeri [41] studied the influence of the local wall
heat flux, which is a linear function of the local wall
temperature, on laminar forced convection in the thermal
entrance region using Galerkin-Kantorovich methods and
evaluated the influence of Biot number on local Nusselt
numbers.
Bhatti and Savery [42] used a uniform wall
temperature boundary condition for the problem of
hydrodynamically and thermally developing flow fields and
developed a correlation between dimensionless thermal
entrance length, Z, and all Prandtl number fluids as
Z=0.102*Pr.
Yan [43] employed a vorticity-velocity method with
the Du Fort Frankel scheme to examine the case of
hydrodynamically-developed and thermally-developing flow.
Nakamura et al.[44] studied Nusselt number variations
with high-viscosity Newtonian fluids for a boundary
condition of uniform wall temperature.
Lin et al.[45] considered a uniform temperature on
the bottom surface and conducted numerical simulations to
show the influence of buoyant forces in the thermal
entrance region.
18
After the introduction of microchannel heat sinks by
Tuckerman and Pease [6] many investigators [8-10, 47-61]
have examined this problem.
The predominant channel shape
in micro-scale heat exchangers is the rectangular channel.
Peng and his coworkers [10, 49-52] conducted a series of
experiments to determine thermal and hydrodynamic
characteristics of fluids flowing in rectangular
microchannels.
Their experimental results showed thermal
and hydrodynamic characteristics of fluid flow through
microchannels to be different from those with
conventional-size channels.
Mala et al.[53] suggested
this variation is due to the presence of an interfacial
electric double layer(EDL) at the solid-liquid interface.
19
Chapter 2
EXPERIMENTAL APPARATUS AND PROCEDURES
2.1 Introduction
As mentioned in the first chapter, the purpose of
this study is to examine the hydrodynamic and heattransfer behavior associated with fluid flow in
rectangular channels with different aspect ratios.
The
experimental apparatus was designed to satisfy the
following conditions:
1. Minimum fluctuation of inlet temperature and
velocity,
2. Hydrodynamically fully-developed laminar flow at the
thermal entrance,
3. Uniform wall heat flux downstream from the thermal
entrance,
4. Minimum axial and peripheral heat transfer,
5. The thermal entrance region must be of sufficient
extent to exhibit significant local Nusselt number
variation,
6. Single phase flow along the channel.
The above requirements will be discussed in more detail in
the following sections.
20
2.2 Design range
Experiments were performed to evaluate friction
factors in fully-developed laminar flow for different
aspect ratios and hydraulic diameters, and the thermal
behavior of single-phase flow in the hydrodynamically
fully-developed and thermally-developing flow region of
rectangular channels.
Table 2.1 shows the range of
experiments with different boundary heating conditions.
Table 2.1 Experimental ranges
Thermal experiment
Friction factor
b/a
1,
Re
2,
3,
4, and
100 - 3500
Ra
5
1,
2,
3,
4, and
5
200-800
104
2*106
2.3 Flow loop
Figure 2.1 shows the schematic diagram of the
experimental flow loop.
City tap water was used.
The
maximum temperature variation of tap water over a 24-hour
period was less than 5°C.
In the test loop water flowed
into a reservoir with a float-valve-assembly to maintain a
constant water level.
The water level difference between
reservoir water surface and outlet drain valve was
approximately 51 inches which was sufficient to produce
Reynolds numbers up to Re=5000 for the range of
experimental channel dimensions investigated.
Water level
control float valve
1111, Water reservoir
Micro-pressure gages
Outlet mixing
chamber
Flow rate
control valve
Developing channel
Test channel
15565524
Water from
tap
Drain valve
Thermocouples
Rotameter
Data logger
Variable transformer
Figure 2.1 Schematic diagram of flow loop
Drain valve
Elctronic balance
for flow rate
measurement
22
The water then passed through a flow-rate control valve, a
rotameter which was used to read approximate flow rates
(accuracy +590, a hydrodynamic development channel, the
test channel, a mixing chamber and an outlet section where
bulk fluid temperatures were measured.
Flow rates were
calculated using accumulated weight, time, and channel
inlet temperature.
The test channel was covered with a
5mm thickness of fiberglass, two layers of 25.4 mm(1 inch)
styrofoam boards to minimize conduction and convection
heat losses, and aluminum foil for radiation suppression,
Figure 2.5.
2.4 Channel design
2.4.1 Hydrodynamic entrance length of rectangular channel
The hydrodynamic entrance length for internal channel
flow is generally defined as the distance over which the
velocity profile reaches 99 percent of its fully-developed
value when the entering flow is uniform.
Unlike a
circular channel, the entrance length of rectangular
channels change with different aspect ratios.
Several
investigations have been undertaken to examine the
hydrodynamic entrance length of rectangular channels.
For
the present experiment the numerical results of Wiginton
and Dalton [ref.l] were adopted.
Their results are in
23
good agreement with those of other experimental studies.
Dimensionless hydrodynamic entrance length is defined as
Lfull
L
h''
(2-1)
Dh * Re
Table 2.2 shows the dimensionless entrance lengths for
different aspect ratios and corresponding channel lengths
for a width a=5.75 mm at Reynolds number Re=2300.
Table 2.2 Entrance lengths for a rectangular
channel for W= 5.75, Re = 2300
b/a
Lhy
Dh (mm)
Lifun (mm)
1
0.09
5.75
1190.25
2
0.085
7.667
1498.9
4
0.075
9.2
1587
5
0.08
9.583
1763.3
Based on these values, 2.032m (80 inches) was selected as
the hydrodynamic entrance length for all aspect ratios in
this work.
2.4.2 Thermal entrance length for hydrodynamically
developed flow with uniform wall heat flux
The thermal entrance length, Lth, is defined as the
axial distance required to achieve a value of the local
Nusselt number Nu, which is within 5 percent of the
fully-developed value.
24
The dimensionless thermal entrance length is expressed as
Li.;,,
LM
LM
D hPe
D hRe Pr
(2-2)
Unlike pure forced convection internal flow, the thermal
entrance length in mixed convection depends on Rayleigh
number as well as channel aspect ratio.
Previous
investigations [16-20] have demonstrated these trends.
For the present study, the results of Cheng et al.
[16], which is the only published work for aspect ratios
b/a=2 and b/a=5, were used to determine test channel
length.
Figure 2-2 from Cheng et al.[16] shows the local
Nusselt number variation with different aspect ratios and
Rayleigh numbers.
One may observe that an aspect ratio of
b/a=5 has a larger dimensionless entrance length than does
a square channel for the same Rayleigh number.
Based upon
the values in Figure 2.2, the present channel length,
2.737m, should be sufficient for Rayleigh numbers grater
than 105 at Re=500 and an aspect ratio of a/b=0.2.
25
Shannon and Depew[64 maximum Roylelgh number
33
3x103
Wet
2x 103
30 r
Wok73.091
0.5x105
2.5
Ro 3x105
6x10
4
3x104
5x103
1
1
4
2
12
1
I
I
1
1 1111
I
6
2
1
Iae
1
1
1
1
z Z/4
I-
I
I
4
101
1
1
4
I
1
1
1
6
1
MO
-Ft 1111
I
T
6
1-.1031
2
Iva v. a
Ra = 3)(105
103
6x104
4
3x104
0
3-
I
0-3
2
1111111
1
4
2
6
10-2
1
1111111=111111M
I
4
10
4
MO
z
Figure 2.2 Local Nusselt number variation with
different aspect ratios[16]
26
12
I
I
I
111111
I
T
T 1-1-1111
F
I
11111
I
bit), 7-- 6
Ro3x105
6x le
3104
3I
10
2
1
4
I
1
6
1
1
1
102
2
4
itilti
6
10
104
5x10
I
1
2
1
4
1111i
6
I0
Figure 2.2 (continued)
2.4.3 Channel construction
Figure 2.3 shows a schematic diagram of the test
channel.
Polycabonate, which can withstand temperatures
up to 120°C, was used as a channel material to prevent
deformation near the boiling temperature of water.
The
channel can be thought of as consisting of four parts: an
inlet mixing chamber, a velocity developing channel, the
test channel, and the outlet mixing chamber.
Both mixing
chamber were attached mechanically at the ends of the
channels.
The velocity-developing section and test
section were aligned using aligning-pins at the joints.
To prevent water leakage, a silicone rubber gasket was
used between channel sections.
Water out
Inlet temp. & pressure
measuring point
2.15rm
(Pressure measuring distance)
Somum
I
I
Go (PA rlY
2.03 frn
(Velocity developing channel)
2.939 n'n
(Thermal test channel)
Outlet pressure
measuring point
Coated
thermocouple
Water in
Brass tube
(1/16" OD)
Detail of inite temp. &
pressure measuring tap
Detail A: Outlet temp.
Measuring method
Figure 2.3 Schematic diagram of experimental channel
showing thermocouple and pressure-tap location
Kapton tape
Top stainless
steel heat
Side stainless
steel heater
Side stainless
teel heater
41,
Bottom stainless steel heater
Figure 2.4 Channel assembly with heaters
Aluminum
foil
Styroform
iberglass
Polycarbonate
channel
Wood stand
Figure 2.5 Test channel insulation
30
A moveable thermocouple, inserted in 1/16" OD brass tube,
was used to measure the inlet temperature and to minimize
flow disturbance.
The channel assembly with heater is
shown in Figure 2.4.
Figure 2.5 shows the thermal
insulation configuration.
2.5 Heater
In heat transfer experiments, it is difficult to
achieve the condition of a uniform wall heat flux boundary
due to heat conduction along the channel wall.
The most
common method for achieving a uniform wall heat flux
involves using an electrically-heated metal wall.
This
method involves either using the metal wall itself as an
electric-heater or wrapping the wall with an electric
heating coil.
Shannon et al.[62, 63], Sparrow et al.[64, 65], Krall
et al.[66], and Koram et al.[67] used a metal wall as the
electric heater in their experiments.
They used thin
stainless-steel circular tubes in their tests.
Peng et al.[49-52] fabricated a micro-size
rectangular channel in a stainless block and used the
stainless block as an electric-heater to provide a uniform
wall heat flux condition.
31
A study of mixed laminar convection in the entrance
region of inclined rectangular channels was undertaken by
Morcos et al.[18].
A study of combined and free
convective heat transfer to non-newtonian fluids in the
thermal entry region of a horizontal pipe, done by Kim
[68] employed a metal pipe wrapped with a heating coil to
supply a uniform wall heat flux to the working fluid.
As
mentioned by Baughn et al.[69] these two methods give
satisfactory results in some situations such as when wall
conduction is not too significant or the heat transfer
coefficient of the working fluid is relatively high
compared to the wall conductivity.
Thus, to reduce wall
conduction effects, it is important to make the wall and
the heating elements as thin as possible.
Rather than having a metal surface contact the
working fluid, Baughn et al.[691 suggested using a metal
heating element wrapped with a thin low-thermal
conductivity material.
concept.
Several studies employed this
Seban[46] used bakelite plastic attached to a
nichrome ribbon electric heater to study the heat transfer
with turbulent, separated flow of air downstream of a step
in the surface of a plate.
Baughn et al.[69] employed a
transparent thin polyester sheet on which
by vacuum deposition.
gold was coated
The thickness of the gold coating
was thin enough to be used as an electric heater.
32
Incropera et al.[25] use stainless steel foil as a heater
shielded from the working fluid by a mylar/adhesive
composite sheet.
In the present experiment, the heater design shown in
Figure 2-6 was used.
To produce uniform surface heat flux
and to minimize axial and peripheral direction heat
conduction, a 1.5 mil, 18-8 stainless steel foil was
coated with acrylic adhesive and attached to a
polycarbonate guide wall.
The total thickness of
stainless steel foil, with adhesive, was 5.75 mil.
Stainless steel heaters were mechanically joined to flat
copper wire as shown in Figure 2-7.
Heaters on each
surface were independent from each other and were
connected in different combinations to provide different
heating conditions.
parallel.
The heaters were connected in
Electrical insulation between the
bottom
heaters and side heaters was achieved by covering
both
side heaters with 0.05 mm Kapton® tape with silicone
adhesive.
Table 2.3 shows the resistance of the heaters
for different aspect ratios. The resistance difference
between heaters were within +0.1 0 for all cases.
33
Table 2.3 Electrical resistance of heaters
with different aspect ratios
Resistance (Q)
Bottom and Top heater
9.25
1
9.25
2
4.625
3
3.08
4
2.31
5
1.85
Side heaters
b/a
9,35afNA
035 no en
Kapton tape(0.05mm)
C
vi
S301 Stainless steel heater
(0.09mm)
Acijlic adhesive(0.05mm)
o3
Polycarbonate
Detail of A
S-40,
(unfttmorm)
Figure 2.6 Cross section of test channel with detailed view of heater
To power
Copper
conecter
Stainless steel
heater
Figure 2.7 Power connection with heater
Lk)
u-1
36
2.6 Micro pressure gage
The pressure head difference between two points in a
rectangular channel is expressed as
2
L U rn
Al I =
(2-3)
ph g
where f is the friction factor, which is a function of the
channel aspect ratio and L is the distance between the two
pressure measurement points.
Table 2.4 shows the
pressure-head-difference for water with a Reynolds number,
Re=100, channel length, L=2.75m, for several aspect ratios
and hydraulic diameters.
To achieve an uncertainty below
50, a pressure gage which can be read to 0.03 mm is
necessary.
Special pressure measurement equipment was
designed to meet this requirement.
Figure 2.8 shows the
micro-pressure measuring equipment used in this study.
Table 2.4 Water head difference for
Re=100, L=2.75m and T=20°C
Aspect ratio
Dh(mm)
oh(mm)
1
5.75
3.3
2
7.667
1.95
3
8.625
1.5
4
9.2
1.33
5
9.583
1.225
Micrometer
moving
head
Guide glass
tube(25mmLD.)
Stainless
steel needle
To outlet
pressure tap
To inlet
pressure tap
A-1
Figure 2.8
Micro-pressure-difference measuring equipment
38
The gage consisted of two independent measuring probes
consisting of stainless steel needles.
The length of the
probes allowed reading up to maximum of 50 mm.
The probes
were positioned by electronic micrometer heads(range 0-25
mm Mitutoyo® electronic micrometer head) which could be
read as small as 0.001 mm.
To minimize fluctuations in
water levels, glass tubes with inside diameters of 23 mm
were used as guide tubes and plastic tubing(ID=3mm) was
used to connect the pressure taps to the pressure gage.
A step-by-step method of measuring pressure difference is
given below:
a.
To set the zero point of both probes, the two glass
guide tubes were connected to each other, so that the
water levels in both tubes were equal.
b.
The probes were adjusted until their end points were
in contact with the water surface.
The zero-setting
button of the micro-meter was then set.
This step
was done by and was accurate within 0.002 mm.
c.
The two glass tubes were then connected to locations
to measure inlet and outlet pressures.
d.
Subsequently the probe contact points were adjusted
to touch the water surface.
The difference between
these two readings was the desired
difference.
pressure
39
2.7 Thermocouple installation and temperature measurement
To minimize reading errors across the experimental
temperature range, from room temperature to up to 120°C,
30 gage copper-constantan type-T thermocouple wires,
supplied by Omega®, were used.
Thermocouple beads were
pressed into disk shapes of 1 mm diameter and 0.5 mm
thickness to increase the contact area.
Thermocouples
were carefully inserted into drilled holes and fixed with
epoxy.
Figure 2-9 shows the thermocouple positions used.
The acrylic adhesive between the heater and guide wall
also served as an electrical insulator between the heater
and the thermocouples.
A total of ninety-six
thermocouples were positioned on the test channel.
Table
2.5 and Figure 2.9 show the thermocouple locations in the
test channel.
A Hydra Data Logger(model 2625A) made by
FLUKE® was used to read temperature.
40
Table 2.5 Locations of thermocouples
Surface
Peripheral
location(mm)
Axial
location
(inch)
Top
Center
1,2,4,8,16,2
4,54,94
0.8, 2.9, 5.2, 11,
Side
Same as top
16.5, 22.5, 28
(from bottom corner)
1,2,4,8,16,1
9,22,25,28,3
Bottom
Center
1,34,37,40,4
3,46,49,52,5
5,58,61,64,6
7,70,73,76,7
9,82,85,88,9
1,94
apton tape
Stainless steel heater
Acrylic adhesive
A
A
@c====="
T-type thermocouple
Wire:0.125mm Dia.
bead dia:0.5mm,
thickness:0.2mm
I
0
Cross section of A - A
o
0
Also top and bottom surface
thermocouple implantation
Figure 2-9. Thermocouple implantation
42
2.8 Flow rate measurement
Flow rates were measured using accumulated weight of
water over time.
An electronic balance, capable of
reading to 0.1g, and a 1/100-second stop watch were used
to measure accumulated weight and time respectively.
A
rotameter located at the inlet was used to read
approximate flow rates.
2.9 Data reduction
2.9.1 Data reduction for friction factor
For hydrodynamically fully-developed flow in a twodimensional channel, fRe is defined as
fRe =
2p,u dx
(2-4)
where x is the axial coordinate along the flow length of
the channel.
In a rectangular channel, hydraulic diameter
ph and Reynolds number Ren are defined as
4
4ab
D,
2(a+b)
(2-5)
h
ut,Dh
ReD
h
V
(2-6)
43
Equation(2-4), solved for a channel of length L, becomes
f
AAP
(2-7)
2pur,,L
The average velocity, um, and pressure difference, AP,
were calculated using measured flow rate, cross-sectional
area and pressure head difference over the length, L,
according to
(Wt)
pab
M
pA,
AP pg(h2
--=
h1) =
(2-8)
pgLh
(2-9)
After substituting these expressions into equation (2-7),
the friction factor is
j
A
1 Dh
g(pA,)2
t2
(2-10)
w2
Unlike circular channels for which fReE64,
rectangular channels have different values for fRe
depending on the aspect ratio, b/a [1].
In this study,
experimental results for the friction factor were compared
with values calculated using equation (1-4)[1] which is
repeated here for convenience,
44
f Re = 24 [1
1.3553 (b I a) + 1.9467 (b a)2
1.7012 (b /a)3
(1-4)
+0.9564 (b a)4 0.2537 (b a)5]
2.9.2 Data reduction for local Nusselt number
The local averaged Nusselt number is defined as
Nu x-
Dh
hxD
(2-11)
The input heat flux to the test channel was computed based
on the actual energy balance between the inlet and outlet
of the test channel
1
(2-12)
A
where Tm and Tout are bulk temperatures at the inlet and
outlet of the test channel, respectively, and Aq is the
heating surface area.
The specific heat, Cp, was
evaluated at the inlet bulk temperature, Tin.
Local bulk
temperatures, Tmwx were calculated as
Ae x
Cp L
Tin
Surface temperatures were calculated using the onedimensional conduction equation
(2-13)
45
1 AT
Aq R
(2-14)
where R is the thermal resistance between the fluid
contact surface and the thermocouple.
Local mean surface temperatures, Tms, were calculated
using statistical high-order polynomial regression and
numerical integration.
For aspect ratios of b/a=3, 4, and
5, second-order polynomial regression, T= bO +blx +b2x2, was
used to evaluate the vertical surface temperature.
The
mean temperatures of the bottom and top surfaces were
determined using averages between the bottom-center and
near-corner temperature, and top-center and near-corner
temperatures.
Figure 2.10 illustrates the graphical
scheme to determine mean surface temperatures for aspect
Figure 2.10 Graphical scheme used in determining
the mean temperature for total surface heating
with aspect ratios, b/a=3, 4 and 5
46
For aspect ratios of b/a=1
ratios of b/a=3, 4, and 5.
and 2, a fourth-order polynomial regression,
T= bO +blx +b2x2 +b3x3 +b4x4, was used between the bottom-center
and top-center as illustrated in Figure 2.11.
Forth-order polynomial
Regresshion line
6
1
Temperature
measuring points
Temperature measuring point
Figure 2.11 Graphical scheme used in determining
the mean temperature for total surface heating
with aspect ratios, b/a=1 and 2
The order of regression for heating conditions, and aspect
ratios are listed in Table 2.6.
The modified Grashof number, for the case of
specified heat flux is defined as
[3
Gr*
gDh q
//
(2-15)
V2 K
47
and the corresponding modified Rayleigh number is
Ra* = Gr* Pr
(2-16)
Fluid properties in these parameters were based on the
inlet bulk temperature.
Table 2.6 Order of regression to calculate
mean surface temperature
Heating
condition
Aspect
ratio
2
1,
Total
3,
4,
5
2
1,
Three Surface
3,
Side surface
Bottom surface
4,
all
5
Regression apply
range
Order of
regression
Bottom through
top surfaces
4
Side surface
2
Bottom and side
surface
3
Side surface
2
2
Use local temperature for Nux
48
Chapter 3
RESULTS AND DISCUSSION
3.1 Friction factors
Variations in values of friction factors with
Reynolds number over a range of aspect ratios in
rectangular channels were evaluated and experimental
results, based on equation(2-10), were compared with
values calculated using equation (1-4) for fully-developed
laminar flow.
Figure 3.1 shows the friction factor variation with
Reynolds number for five different aspect ratios. The data
points and continuous lines represent experimental results
and equation (1-4) respectively.
As seen in the figure,
the friction factors, f, for each Reynolds number decrease
linearly until the Reynolds number reaches a value near
2000.
At values of Reynolds number approximating 2000,
the linear variations no longer occur and the flow changes
from laminar to transition flow.
The experimental results suggested that the critical
Reynolds number for the channel geometries studied vary
little with aspect ratio.
Obot[5] investigated published
works to find relationships between channel aspect ratios
49
1
b/a=1
0.1
0.01
0.001
1E1
1E4
1E3
1E2
1
b/a=2
0.1
0.01
0.001
1E1
1E4
1E3
1E2
1
b/a=3 r
0.1
Oa
0.01
0.001
1E1
1
0.1
0.01
1E3
1E2
1E4
111111111111111111111111111
b/a=4
INA
111111
1111111111111111
iiii=111111
1111
0.001
1E1
1E2
1E4
1E3
1
b/a=5
0.1
Y.4 P
0.01
0.001
1E1
1E3
1E2
1E4
Re
Figure 3.1 Comparison of experimental and predicted
friction factors with different aspect ratios
50
and the critical Reynolds numbers for rectangular
channels.
He compared the data with the laminarThis comparison
equivalent-diameter concept of Jones[4].
showed good agreement.
The critical Reynolds numbers
obtained by Peng et al.[10] for micro-scale channels,
however, deviated from the trends of Jones[4].
The relation between critical Reynolds numbers for
various
aspect ratios are plotted in Figure 3.2 with
available data [4, 5, 10].
All data were taken for
rectangular channels with abrupt entrance configurations.
A
Peng e 1 ol
tt
0 met [5]
Present
12
10 --
Rec.c=2100
(circular tube critical
Reynold no.) [5]
A
A
ti_8
u
--...
d
6
A
U
w
W4
A
Jones[4]
A
2
00
0.2
0.6
0.4
0.8
1
alb
Figure 3.2 Variation of critical Reynolds number,
Re, with aspect ratios
51
As seen in Figure 3.2, there is no apparent trend between
critical Reynolds numbers and aspect ratios. The critical
Reynolds numbers obtained by Peng et al.[10] are all
higher than other values at similar aspect ratios.
Another parameter of interest is the hydraulic
diameter(Dh).
Rec and ln(DO.
Figure 3.3 shows a linear relation between
This relation can be expressed as
Re, = 1140
+4801n(Dh)
(3-1)
and all data obtained fall within the range, Rec+200.
3000
2500
2000
c.)
ar 1500
re
1000
500
A
Peng et al[10]
®
Present
III
0
0.1
Obot [5] - Regression
,
10
1
100
Dh(mm)
Figure 3.3 Variation of critical Reynolds number,
Rec, with hydraulic diameter, ph
52
Figure 3.4 and Table 3.1 shows the relationship
between fRe and aspect ratios in fully-developed
rectangular channel.
25
23
21
CL 19
17
15
13
0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
a/b
Figure 3.4 Values of fRe over a range in aspect
ratios for hydrodynamically fully-developed
flow in rectangular channels.
Table 3.1 Experimental and predicted fRe
for fully-developed laminar flow
for different aspect ratios
b/a
Prediction
Experimental
Data
Difference
Eq(1-3)
1
14.227
14.23
+0.3
2
15.548
15.69
+0.9
3
17.09
16.76
-1.9
4
18.23
18.68
+2.48
5
19.07
19.22
+0.8
(o)
53
The solid line in Figure 3.4 represents values calculated
using Equation (1-4). Uncertainty values of measured
friction factors are less than 3.3 percent in all cases
and the differences between experimental predicted values
are less than 2.5 percent, which is less than the
uncertainties in the measurements.
3.2 Heat transfer
Buoyancy forces in channel flow induce different
thermal and flow characteristics compared with cases of
pure forced flow.
Several investigators have studied
these cases of mixed convection both numerically and
experimentally.
In this chapter, the experimental results
for mixed convection flow through rectangular channels
with different aspect ratios will be discussed.
The
following sections are organized according to the kinds of
boundary conditions applied.
3.2.1 Whole surface heating
Typical axial temperature variations at the centers
of top, side and bottom surfaces are plotted along with
fluid bulk temperatures and average local surface
temperatures as functions of dimensionless axial distance,
Z, for aspect ratios of b/a=1 and 4, in Figure 3.5.
effect of buoyancy is shown clearly in this figure.
The
54
28
Re:600, Ra:3*10"4
13/a=1
27
U
26 -
AVERAGE SURFACE TEMP.
25
A
o. 24
E
Top
23
Side
Fluid bulk
Temp.
22
Bottom
21
0
0.04
0.02
0.06
0.08
0.12
0.1
(a)
28
Re:316, Ra:10^5
bia=4
27
26
AVERAGE SURFACE TEMP.
C.)
1725
=
24
A--
E 23
Top
22
Side
0-Fluid bulk
Temp.
21
Bottom
20
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
(b)
Figure 3.5 Temperature variations with dimensionless axial
distance for different aspect ratios,(a)b/a=1, (b)b/a=4
The cooler water outside of the thermal boundary
55
core descends and attenuates the rate of temperature
Conversely heated water
increase of the bottom surface.
rising upward enhances the rate of temperature increase of
the top surface. As seen in Figure 3.5(b), the temperature
of the bottom surface can actually be less than the bulk
temperature of the water.
The circumferential dimensionless temperature, Os, at
dimensionless distances, Z=0.00218,0.05668, and 0.01024,
for a square channel is compared with numerical results of
Cheng et al.[16] for Z=0.00192, 0.00554, and 0.097 in
Figure 3.6.
Experimental results show good agreement with
1.2
Present work
1-
CI
Z=0.00218 0 Z=0.05137
it,
Z=0.1024
Circumferential Location
Figure 3.6 Comparison of circumferential temperature
distributions for Ra=3x104, Re=602, Pr=6.72 with
numerical results [16] for square channel
56
numerical results of Cheng et al.[16].
Conduction effects
in the channel wall produce temperatures at the top and
the corners which are lower than numerical predictions.
The differences between experimental data and numerical
predictions for center temperature differences between the
top and bottom surface at the same distance, Z, become
large with increasing values of Z.
Therefore, the
influence of secondary flow becomes larger than predicted
from numerical analysis and the thermal entrance length is
also increased.
Local average Nusselt numbers, Nu, in the thermallydeveloping region, for aspect ratios of b/a=1, 2,
3,
4 and
5 are presented in Figures 3.7 through Figure 3.11 for
different Reynolds numbers and Rayleigh numbers as
functions of dimensionless axial location, Z.
In Figure
3.7, 3.8, and 3.11, the experimental data are compared
with available data including laminar forced convection
cases [16,17].
Local Nusselt numbers in the thermal entrance region
vary with the fluid Prandtl number [17].
Higher Prandtl
number values produce lower Nusselt numbers for the same
Rayleigh number and dimensionless axial distance, Z, in
the thermal entrance region.
However, all curves for
different Prandtl numbers approach the same values for
57
Nusselt numbers corresponding to the fully developed
condition[17].
The results of Chow and Hwang [19] differ
from those of Abou-Ellail and Morcos [17].
Chow and
Hwang[19] reported that local Nusselt numbers in the
entrance region have the same value for all Prandtl
numbers.
However, in the results of Abou-Ellail and
Morcos[17], along the channel axis, higher Prandtl numbers
produced higher local Nusselt numbers at the same Rayleigh
number and axial location in a square horizontal channel.
The experimental results depicted in Figures 3.7, 3.8, and
3.11 are in reasonable agreement with published numerical
results; they tend to be higher than the numerical
results.
The values of Z for
minimum Nux show good
agreement with numerical results at the same Rayleigh
numbers.
Any deviations are likely due to measurement
errors.
Figures 3.7 through 3.11 show the local average
Nusselt number variation along the channel axis for given
values of Reynolds and Rayleigh numbers to decrease with
the dimensionless axial distance Z, reaching a minimum
value, and then to increase.
In the thermal entrance
regions Nusselt number variations follow those for pure
forced convection.
Thus, in the entrance region, forced
convection dominates the thermal transport.
The monotonic
decay associated with laminar forced convection
[Re: 60 0 800
17
Ra:
310^4
15
"101'4
101'5
A
13
9
7
b/a=1
a5.75 mm
b 5.75 mm
5
3
0.0001
a
Apply uniform heat flux to
whole surfaces
I
I
III 11
0.001
C. en
I
cii
I
0.01
0.1
1
z
Figure 3.7 Local Nusselt number variation in the thermal entrance region of
horizontal square channels; uniform heat flux on whole surface
co
Re: 400 600 800 1000
17
Ra.
6*10^4
c\
15
A
10^5
0
2*10^5
13
A
4'101'5
...
o
9
Niko
El
".94-11pr-
b:11.5 mm
,,,,,,
b
,.....i....,
a
3
0.0001
Nis
Apply uniform heat flux to
whole surfaces
I
I
I
1111
0.001
,111
1411IVr
a:5.75 mm
5
ITIEMI!
11111**M
b/a=2
7
Al
I
go, g /Of [IQ
r . ...4/*
:16)
I
0.01
0.1
1
z
Figure 3.8 Local Nusselt number variation in the thermal entrance
region of horizontal rectangular channels(aspect ratio b/a=2);
uniform heat flux on whole surface
1
I
R e:
17
15
0
13
9
b/a=3
i
I
IIIII
400 500 600 800
Ra:
\
10^5
210A5
4'10 ^5
GO
0
A
0
75
7
5.75 mm
a:
b:17.25 mm
5
3
0.0001
I
I
I
till
Apply uniform heat flux to
whole surfaces
Mill
0.001
i
I
0.01
0.1
1
Z
Figure 3.9 Local Nusselt number variation in the thermal entrance
region of horizontal rectangular channels(aspect ratio b/a=3);
uniform heat flux on whole surface
C71
0
1111
Re:3 00 400 500
17
Ra:
101'5
15
E1-0
0^5
4'101'5
A
13
b/a=4
9
4i
7
5.75 mm
a:
b:17.25 mm
5
3
0.0001
Ii111
I
Apply uniform heat flux t 0
whole surfaces
I
I
I
IIII
0.001
0.01
0.1
1
z
Figure 3.10 Local Nusselt number variation in the thermal entrance
region of horizontal rectangular channels(aspect ratio b/a=4);
uniform heat flux on whole surface
1
Re: 200
17
Ra:
1
1
1
4 00
2*10^5
0
0
4*-10^5
A
101'5
15
1
I
I
I
13
I
I
9
7
b
a: 5.75 mm
b: 28.8 mm
5
a
3
0.0001
voi
Apply uniform heat flux to
whole surfaces
1
I
I
1111
0.001
1
1
0.01
0.1
1
z
Figure 3.11 Local Nusselt number variation in the thermal entrance
region of horizontal rectangular channels(aspect ratio b/a=5);
uniform heat flux on whole surface
63
is attenuated after the flow becomes destabilized by the
buoyancy force.
The subsequent increase in the strength
of secondary flows causes an increase in the local Nusselt
number after reaching a minimum.
Figures 3.7 through 3.11
also show the effects of Reynolds number and Rayleigh
number.
At the same aspect ratio, an increased Rayleigh
number is associated with higher values for local Nusselt
numbers and shorter thermal entrance lengths.
Reynolds
number influences, however, are in agreement with
published results [16-19] showing no local Nusselt number
variations with Reynolds number.
The same experimental data shown in Figure 3.7
through 3.11 are plotted in Figures 3.12, 3.13, and 3.14
to illustrate the effects of aspect ratio for common
Rayleigh numbers or Ra=105, 2x105, and 4x105.
For a
specific Rayleigh number, the strength of secondary flow
after a minimum Nux is continues to vary as a function of
aspect ratio.
Detailed representations of the regression
lines in the minimum Nux region are shown with each
figure.
As depicted in these figures, values of local
Nusselt numbers in the thermal entry region do not vary
strongly for different aspect ratios.
increasing aspect ratio from b/a=1 to 5
The effect of
is one of
increasing the axial location of the minimum Nux.
7.5
15
z
13
65
MEMO
Mani
NN:411111111
5.5
0.1
0.01
z
9
b/a
Re:200
300
400
500
600
800
0
7
0
E
1
2
3
+
EEI
A
A
2:3
X
0
Ra=1*10"5
5
0.0001
r
0.001
I
I
0.01
0.1
1
z
Figure 3.12 Local Nusselt number variation in the thermal entrance region
of horizontal rectangular channels; uniform heat flux on whole surface
with different aspect ratios for Rayleigh number Ra=1x105
15
x=
7.5
Z
13
7
A
6.5
Z
9
4
5
400
A
83
500
A
b/a
A
a
7
0.1
0.01
11
->
<--
U
0.001
I
111
1000
Ra=2*10"5
0.0001
0
600
II
3
Re:200
800
5
2
1
0.01
0.1
1
Z
Figure 3.13 Local Nusselt number variation in the thermal entrance region
of horizontal rectangular channels; uniform heat flux on whole
surface with different aspect ratios for Rayleigh number Ra=2x105
9.5
18
x
9
z
16
8.5
14
8
0.005
0.035
0.025
0.015
z
0 12
111\k
10
b/a
1111111111M11111
F11111111
2
3
4
5
Re:400
0
500
800
EB
1000
LAI
6
0.0001
Ra=4*10^5
HI
0.001
I
II
0.01
0.1
1
Figure 3.14 Local Nusselt number variation in the thermal entrance region
of horizontal rectangular channels; uniform heat flux on whole
surface with different aspect ratios for Rayleigh number Ra=4x105
67
Table 3.2 and Figure 3.15 show the effects of both
Rayleigh number and aspect ratio on the location of the
minimum Nux.
Table 3.2 Dimensionless distances, Z, to minimum Nux with
different Rayleigh numbers, Ra, and aspect ratios
b/a=5
Ra
b/a=4
b/a=3
b/a=2
b/a=1
0.02995
30000
60000
100000 0.166493
0.03297 0.026369
200000 0.044354
0.02467
0.02715
0.02078
0.02786
0.01552
0.01794 0.015577
400000 0.025936 0.017809 0.011415 0.009311
0.2
0.15
1,1 0.1
0.05
0
30000
60000
10000
0000
0000
b/a=1
Ra
Figure 3.15 The effect of Rayleigh numbers and aspect
ratios on the location of minimum Nux for whole
heating condition
68
3.2.2 Three-surface heating
In this section, results are presented for the case
of a uniform heat flux applied to the bottom and both side
surfaces while the top surface was insulated.
There are
no reported results for this situation for conventionalsized channels.
These type boundary conditions are common
in micro-sized rectangular channels.
Tuckerman and
Pease[E10] and Peng et al. [E4 -8] used these boundary
conditions for their study of micro-size rectangular
channels.
Figure 3.16 shows typical center-temperature
variations of the bottom and side walls, and bulk
34
Re:795, Ra:10^5
b/a=1
32
30
AVERAGE SURFACE TEMP.
328
co
a 26
E
I 24
Side
Fluid bulk
Temp.
22
Bottom
20
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
z
Figure 3.16 Temperature variation with axial distance Z
for aspect ratio b/a=1 in bottom- and side-surface
heating conditions
69
water temperature as functions of dimensionless distance,
Z, for a square channel.
Figure 3.16 shows that buoyancy
affects the bottom surface temperature for values of Z
grater than 0.005.
Figure 3.17 shows circumferential wall temperature
distributions at several axial locations.
-5
0
Bottom
0
2
1
3
4
5
6
7
8
9
Circumferential location
Figure 3.17 Circumferential wall temperature
distribution for aspect ratio b/a=4
As expected, the temperature distributions for this
condition differ from those with all surfaces heated as
seen in Figure 3.6.
With all surfaces heated, the upper
corner temperatures are always higher than at the lower
corners.
With an adiabatic upper surface, however, the
70
lower corner temperatures remain higher than those of
upper corners in the thermal entrance region.
Secondary
flow, induced by buoyancy, causes temperatures at the
upper corners to increase with axial position, eventually
becoming higher than those at the lower corners.
As seen
in Figure 3.17, upper-corner temperatures become higher
than at the lower corners for Z > 0.02 for the given
conditions in a square channel.
Figures 3.18 through 3.22 show local Nusselt number
variations for a range in Reynolds numbers, Rayleigh
numbers and aspect ratios in horizontal rectangular
channels having an adiabatic top surface.
The general
variations of local Nusselt numbers along the channel axes
are similar to those cases shown in Figures 3.7 through
3.11.
As Rayleigh number is increased, the minimum local
Nusselt number increases and the axial distance of the
location of minimum local Nusselt numbers decreases.
1
1
Re:
17
Ra:
3'101'4
15
600 800
t
0
6101'4
C
101'5
A
13
A
9
7
a
5
0
Three sides constant heat flux
other side is adiabatic
b/a=1
3
0.0001
1
1
1
0.001
1
I
I
I
I
I
0.01
0.1
1
z
Figure 3.18 Local Nusselt number variation in the thermal entrance region
of horizontal square channels; uniform heat flux on bottom
and both side surfaces
I
Re:
17
I
I
I
400 600 800
Ra:
5.7'101'4
15
10^5
2'101'5
13
A
4'101'5
x 11
z
Three sides constant heat flux
` -other side is adiabatic
/a=2
3
0.0001
0.001
0.01
0.1
1
z
Figure 3.19 Local Nusselt number variation in the thermal entrance region
of horizontal rectangular channels(aspect ratio b/a=2); uniform
heat flux on bottom and both side surfaces
I
Re: 400
18
Ra:
10A5
A
16
I
6 00 8 00
O
2*10A5
4'1 OAS
A
14
x 12
z
0
10
8
0
Three sides constant heat flux
other side is adiabatic
6
b/a=3
4
0.0001
0.001.
0.01
0.1
1
z
Figure 3.20 Local Nusselt number variation in the thermal entrance region
of horizontal rectangular channels(aspect ratio b/a=3); uniform
heat flux on bottom and both side surfaces
i
Re:
18
Ra:
10A5
16
0 0
2*10AS
4*10A5
14
x 12
300 400 600
A
4
a
10
8
4
0
Three sides constant heat flux
other side is adiabatic
6
b/a=4
4
0.0001
0.001
0.01
0.1
1
Figure 3.21 Local Nusselt number variation in the thermal entrance region
of horizontal rectangular channels(aspect ratio b/a=4); uniform
heat flux on bottom and both side surfaces
1
Re:
I
lilf
300 400 600
18
16
Ellin Mill
14
x 12
11111111
CIIIIIIM111111111111
z
10
1111111
111111111
itsj
Er II
Three sides constant heat flux
other sides are adiabatic
o
A
It
I
NONNI
e
110
b/a=5
4
0.0001
0.001
0.01
z
0.1
1
Figure 3.22 Local Nusselt number variation in the thermal entrance region
of horizontal rectangular channe]s(aspect ratio b/a=5); uniform
heat flux on bottom and both side surfaces
76
Figures 3.23, 3.24, and 3.25 compare local Nusselt
numbers at different aspect ratios for Rayleigh numbers of
Ra=105, 2x105, and 4x105 respectively.
One would expect
the strength of secondary flow in an asymmetrically heated
rectangular channel to be higher than in a uniformlyheated channel.
Accordingly, heat transfer rates are
higher than for the case of uniform heating.
impossible to compare
It was
Nusselt numbers in the thermally
fully-developed region because of an insufficient test
channel length.
Minimum local Nusselt numbers for both
boundary conditions show different values; minimum local
Nusselt numbers, for all surfaces heated, are lower than
for the three-side heating case.
Detailed figures in the
upper right corners of Figures 3.23, 3.24, and 3.25
provide additional information near the axial locations of
minimum Nux for the different aspect ratios.
Table 3.3
and Figure 3.26 show the effects of both Rayleigh number
and aspect ratios on the axial distance to
minimum Nux.
8.5
17
15
Z 7.5
7
13
6.5
0
0.06
0.04
0.02
z
9
7
7MT
b/a
Re:300
400
600
800
7.1
1
2
3
El3
11.
0
0
4
5
A
V
,L
7
Ra=10^5
3
0.0001
0.001
0.01
0.1
z
Figure 3.23 Local Nusselt number variation in the thermal entrance region
of a horizontal rectangular channel; uniform heat flux on bottom and
both side surfaces with different aspect ratios for Rayleigh number Ra.1x105
monist
NMI=
19
17
IMM1111
1111.421111111
15
0
0.03
0.02
0.01
13
x
z
11
MEM
IP
110
11499111
-411111M
111
2/11111=1111111
2
3
4
600
800
5
V
p
400
tIN-7:111Pfr
LI
b/a
Re:300
A
NI
EB
0
Ra=2 10^5
5
0.0001
0.001
0.01
0.1
1
z
Figure 3.24 Local Nusselt number variation in the thermal entrance region
of a horizontal rectangular channel; uniform heat flux on bottom and
both side surfaces with different aspect ratios for Rayleigh number Ra.2x105
10
19
9.5
x
17
z
9
15
8.5
x
13
0.005
N
0.015
0.01
0.02
z
b/a
9
AA
3
4
5
Re:300
o
400
0 A
600
A
800
7 ----ARa=4*10^5
5
0.0001
0.001
0.01
0.1
1
z
Figure 3.25 Local Nusselt number variation in the thermal entrance region
of a horizontal rectangular channel; uniform heat flux on bottom and
both side surfaces with different aspect ratios for Rayleigh number Ra=4x105
80
Table 3.3 Dimensionless distance, Z to minimum Nux with
different Ra and aspect ratios; uniform
heat flux on bottom and both side surfaces
Ra
30000
58000
60000
100000
200000
400000
630000
b/a=5
b/a=4
b/a=3
b/a=2
b/a=1
0.026
0.023878
0.018096
0.046303 0.029767 0.018808 0.014438 0.012096
0.024728 0.015974 0.012971 0.009264
0.013244 0.008641 0.008451 0.005222
0.009819
0.05
0.04
0.03
N
0.02
0.01
b/a=2
60000
20000
Ra
Figure 3.26 The effect of Rayleigh numbers and aspect
ratios on the location of minimum Nux; uniform
heat flux on bottom and both side surfaces
81
3.2.3 Heating of one vertical surface
In this section, results are presented and discussed
for the case of a uniform heat flux applied to one
vertical side wall of a horizontal rectangular channel,
for several different aspect ratios.
Figures 3.27 and
3.28 show typical temperature variations at top and bottom
corners and the center of the side wall along with the
overall average wall temperature and bulk temperature of
the fluid along the flow axis for aspect ratios b /a =l and
4.
Following the onset of secondary flow, the temperature
of a portion of the channel wall decreases.
and 3.28 show this behavior clearly.
Figures 3.27
For an aspect ratio,
b/a=2, the temperature of a bottom corner decreases
23
Re;590, Ra=3.190^4
b/a=1
Top Corner
22
Bottom Corner
20
B
Center
Fluid Bulk Temp.
everage Tw
19
0
0.02
0.04
0.06
0.08
0.1
z
Figure 3.27 Temperature variation with axial
distance, Z, for aspect ratio b /a =l in one
side wall heating condition
82
22
Re:400, Ra:10^5
b/a=4
Top Corner
21
Average Tw
e
4F
E
Bpttom Corner
Center
20
CI)
E
19
Fluid BA
Temp.
18
0
0.02
0.04
0.06
0.08
0.1
z
Figure 3.28 Temperature variation with axial
distance, Z, for aspect ratio b/a=4 in one
side wall heating condition
between Z=0.03 to Z=0.055 and then increases again.
Similar trends are observed for an aspect ratio of b/a=4
in Figure 3.28.
The temperature variations in this
experimental study show similar trends to those induced by
secondary flow patterns as studied by Kurosaki and
Satoh[40].
Figure 3.29 shows vertical wall temperature
distributions at different axial locations.
In the
thermal entry region, temperatures vary symmetrically with
respect to the horizontal mid-plane.
Following the onset
of secondary flow, this symmetric behavior is lost.
83
22
Re=400
Ra=19A5
z.Ton
21.5 ---Pr=7.-29
;-4
b/a=4
2 =0.049
21
U
13 20.5
2
0.
20
E
0
19.5
Z= 0.0038
0
Bottom
19
Comer
Z=0.0019
18.5
0
1
2
3
4
Circumferential Location
Figure 3.29 Circumferential wall temperature distribution
for aspect ratio b/a=4 in one side wall heating condition
Figures 3.30 through 3.34 show variations in the
local Nusselt number for different values of Re, Ra and
aspect ratios in horizontal rectangular channels with
uniform heat flux applied to one vertical side wall only.
In Figure 3.31, the Nusselt number variation for pure
forced and mixed convection, with an aspect ratio b/a=2,
studied by Kurosaki and Satoh [40] is compared with
present experimental results.
The experimental results in
the thermal inlet region show good agreement with the
forced convection case [40] until the onset of secondary
flow.
Direct comparison with the numerical results [40]
for mixed convection cases cannot be made due to the lack
of published information.
I
1
Re: 400 600 800
19
Ra:
3"10A4
17
0-0
6-10A4
10A5
A
15
x 13
z
11
b/a=1
5
0.0001
One side constant heat flux
other sides are adiabatic
0.001
0.01
0.1
z
Figure 3.30 Local Nusselt number variation in the thermal
entrance region of horizontal square channels;
uniform heat flux on one side wall
1
I
i
ill
Re: 400 600 800
19
Ra:
5*10^4
MEI1
ME Re.iiii
g rsi iii ii
17
"ill
1
15
C\
x 13
z
gi
.111 im
6-
210^5
0
A
L,
4101'5
1111..
, .
,c,,w/ffIL kb
Re =2.98
1111111.11,111/11
9
7
X/0
NN,
11
10^5
EIMI
MillillIrtil
1116Spillalli
1111
I
_One side constant heat fluxlt711.11111111
b /a=2
other sides are adiabatic
Ml'
o
el cony. only
No
5
0.0001
0.001
0.01
0.1
1
z
Figure 3.31 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular channel(b/a=2);
uniform heat flux on one side wall
I
1
I
I
iii
Re: 400 600 800
19
Ra:
10A5 g
17
2"10A5
410A5
A
R
15
x
13
m
z
11
9
7
5
0.0001
bla =3
One side constant heat flux
other sides are adiabatic
0.001
0.01
0.1
Z
Figure 3.32 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular channel
(b/a=3);uniform heat flux on one side wall
1
I
111111
Re: 400 500 600
19
Ra:
17
0
2*10^5
0
4'101'5
A
15
10^5
0
A
A
x 13
z
A
11
A77'
.71
9
7
One side constant heat flux
b/a=4 other sides are adiabatic
5
0.0001
0.001
0.01
0.1
z
Figure 3.33 Local Nusselt number variation in the thermal
entrance region of a horizontal rectangular channel
(b/a=4); uniform heat flux on one side wall
1
Re: 200 400
19
Ra:
101'5
17
2*10^5
4*10^5
15
8*10^5
O
A
V
x 13
z
11
9
7
/a=5
5
0.0001
One side constant heat flux
other sides are adiabatic
H
0.001
Il
0.01
0.1
z
Figure 3.34 Local Nusselt number variation in the thermal
entrance region of s horizontal rectangular channel
(b/a=5); uniform heat flux on one side wall
1
89
Figures 3.35 through 3.37 show axial variations of
local Nusselt numbers at different aspect ratios over a
range in Rayleigh number.
These trends are similar to
those shown previously for different heating conditions.
Table 3.4 and Figure 3.38 show the effects of both
Rayleigh number and aspect ratio on the axial location of
the minimum value of Nux.
10
9
19
x
8
z
17
a
7
15
6
0.01
A
0.02
0.05
0.04
0.03
z
x 13
z
11
b/a
1
2
3
4
9
A
400
0
600
800
7
V
A
0
Ra=1 0A5
5
0.0001
0.001
0.01
0.1
5
v
Re:200
1
z
Figure 3.35 Local Nusselt number variation in the thermal entrance region
of a horizontal rectangular channel; uniform heat flux on one side
wall, with different aspect ratios, for Rayleigh number Ra=1x105
10
9
z
8
7
0.005 0.01
0.015 0.02 0.025 0.03
z
0
b/a
2
400
V
4
5
A
500
600
3
v
o V
Re:200
ai
,n,
800
<1RAI=r10115
1111
1
z
Figure 3.36 Local Nusselt number variation in the thermal entrance region
of a horizontal rectangular channel; uniform heat flux on one side
wall, with different aspect ratios, for Rayleigh number Ra =2x105
11
19
10
z
17
9
8
15
0.005 0.01 0.015 0.02 0.025 0.03
z
x
3 13
z
7
ed
11
b/a
2
3
4
V
400
9
.L
500
600
5
7
R e: 200
0
A
800
Ra=4*10A5
HI
0.001
I
I
0.01
0.1
1
z
Figure 3.37 Local Nusselt number variation in the thermal entrance region
of a horizontal rectangular channel; uniform heat flux on one side
wall, with different aspect ratios, for Rayleigh number Ra=4x105
93
Table 3.4 Dimensionless distance Z to minimum Nux
with different Ra and aspect ratios;
uniform heat flux on one side surface
Ra
b/a=5
b/a=4
b/a=3
b/a=2
30000
b/a=1
0.04165
50000
0.0256
60000
0.0213
100000
0.0343
0.03015
0.0228
0.0165
200000
0.0212
0.0182
0.01462
0.01291
400000
0.0144
0.01232
0.0108
0.0085
800000
0.00932
0.01494
0.05
0.04
0.03
N
0.02
0.01
60000
Wa=2
20000
Ra
Figure 3.38 The effect of Rayleigh numbers and aspect
ratios on the location of minimum Nux; uniform
heat flux on one side wall
94
3.2.4 Bottom surface heating
In this section, results are presented and discussed
for the case of a uniform heat flux applied at the bottom
surface of a horizontal rectangular channel for several
aspect ratios.
The remaining surfaces were insulated.
Experiments were performed at two values of Reynolds
number Re=400, and 600, with various Rayleigh numbers,
3x104 < Ra < 6x105.
Figure 3.39 shows a typical axial temperature
variation at the center of the lower channel surface for
an aspect ratio of b/a=1, with Re=400, and Ra=105.
Figure
3.40 shows the temperature difference between the bottom
30
28 Bottom temperature
26 24 Bulk temp. of water
22 20 t
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Figure 3.39 Bottom center surface temperature and bulk
temperature of water variation along Z
95
surface and the bulk water and local Nusselt number
variation for the same case as Figure 3.39.
In the present experiment, without flow
visualization, it is difficult to verify the existance of
secondary flow in the narrow channel.
The local Nusselt
number variation along the channel suggests that there is
a different flow-pattern shape.
As seen in Figure
3.40(b), a monotonic decrease in Nusselt number,
characteristic of pure forced laminar convection, exists
in the thermal inlet region.
After flowing some distance
warm water, heated by the bottom surface, rises upward and
relatively cool water, in the upper part of the channel,
descends.
This mechanism attenuates the rate of
temperature rise and after a short distance, secondary
flow causes a decrease in the temperature of the bottom
surface over the range Z=0.02 to 0.04.
Figure 3.39 shows
this phenomenon clearly; local surface temperature drops
from 26.3°C at Z=0.02 to 25°C at Z=0.04.
In this region
the local Nusselt number rebounds after reaching a minimum
value.
After a minimum value is reached as seen, in
Figure 3.40(b), the local Nusselt number increases to
about 160 percent of the minimum value before beginning to
fluctuate within about 14 percent of the maximum value,
approximately 14, with Z>0.04.
96
4.5
Re=400
4
Ra=1 OAS
b/a=1
Z. 3.5
E
).< 3
F-
2.5
2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.1
0.12
0.14
0.16
(a)
18
Re=400
Ra=10^5
b/a=1
16
14
12
10
0
0.02
0.04
0.06
0.08
(b)
Figure 3.40 (a)Typical temperature difference between
bottom wall and bulk water temperature and (b)local
Nusselt number variation with axial distance;
constant heat flux on bottom surface
97
As the core temperature of the secondary vortex flow
exceeds the inlet temperature following the maximum value
of Nux, its ability to cool the bottom surface is
diminished.
This conceptual interpretation is confirmed
by the numerical result of Huang and Lin [34]
in
horizontally flat rectangular channels(aspect ratio of
a/b=4).
Fluctuations following the maximum value may be
expressed in two ways; the uncertainty of measurement and
an unstable secondary flow.
In the typical case, depicted
in Figure 3.40(b), measurement uncertainties following the
maximum local Nusselt number are less than ±5 percent.
Figures 3.41 through 3.44 show axial variations in
measured values of the local Nusselt number at the center
of the bottom channel surface for aspect ratios, b/a=1, 2,
3, and 4.
Each figure shows a Rayleigh number effect.
With an increasing Rayleigh number the thermal entrance
length decreases and the minimum local Nusselt number
increases.
Similar trends were observed in several other
experimental studies for bottom-heated horizontal
rectangular channels [23, 25, 28, 29, 35].
As with the
results for the different heating conditions discussed
earler, the effect of different Reynolds numbers is
negligibly small.
98
0
22
Re=400
bia=1
Ra:310"4 0 6104
0--
--s 10"5
rum
18
10
6
Constant heat flux at bottom
surface. Other sides are adiabatic
2
I 4 04+14
111-1-14
-1
I
0.0001
11111,
I
I
0.01
0.001
1
(a)
22
Re=600
Ra:610^4
0-- 105
290^5
b/a211
18
14
x
10
6
Constant heat flux at bottom
surface. Other sides are adiabatic
2
0.0001
i
i
0.001
iiii44
0.01
1
I
I
I
0.1
I
I
11111
1
(b)
Figure 3.41 Local Nusselt number variation in the
thermal entrance region of a bottom heated
horizontal square channel
99
20
Re=400
b/a=2
0
Ra1fY`5
2'10.5 a 310.5 --0--
410"5
16
x 12
3
8
4
Constant heat flux at bottom
surface. Other sides are adiabatic
q"
0 r
I
1-1 111.44.i
0.0001
1
1
1
111-1-1
0.001
i
I
144.44
I
I
1
11111
0.1
0.01
1
(a)
20
e
Re=600
b/a...2
0
Ra:210^5
16
390.5
0
490.5
8
4
Constant heat flux at bottom
surface. Other sides are adiabatic
q"
0
0.0001
i+++
0.001
I
I+
0.01
1
1
l
1MI
a 1
(b)
Figure 3.42 Local Nusselt number variation in the
thermal entrance region of a bottom heated
horizontal rectangular channel(b/a=2)
100
22
Re=400
b/a=3
20
0 Ra:390"5 0 410.5
610^5
18
16
14
12 410
8
Constant heat flux at bottom
surface. Other sides are adiabatic
6-
I
4, 14-1044
0.0001
i
i
0.001
0.01
ii lit-4i
0.1
z
22
.
.
.
---Re=600
Ra:310"5 0-- 41045
20
i0-- 690^5
18
16
x
14
12
ii
10
Constant heat flux at bottom
surface. Other sides are adiabatic
8
6
0.0001
4
I
i 11.44+4
0.001
i
4
i
0.01
0.1
1
Figure 3.43 Local Nusselt number variation in the
thermal entrance region of a bottom heated
horizontal rectangular channel(b/a=3)
101
18
16
Re=400
b/a=4
Ra:31045 0 Ra:41045 OD Ra:610"5
14
x
12
10
8
Constant heat flux at bottom
surface. Other sides are adiabatic
6
4
1-4 144
0.0001
18
I
I-1 HI
I
I
I
0.1
0.001
III
1
:
Re=600
16
bia=4
0
Ra:41045
14
RaV1045
12
10
8
6
4
0.0001
Constant heat flux at bottom
surface. Other sides are adiabatic
I
0.001
0.01
1
(b)
Figure 3.44 Local Nusselt number variation in the
thermal entrance region of a bottom heated
horizontal rectangular channel(b/a=4)
102
Figures 3.41 through 3.44 show that the fluctuation range
after decay of the strength of secondary flow increases
with increasing Rayleigh number for all aspect ratios.
Figure 3.45 shows the effect of aspect ratio on Nux
for several Rayleigh number at a Reynolds number of
The channel having a smaller aspect ratio
Re=400.
exhibits higher local Nusselt number values in the
thermally fully-developed region.
22
I
I
i
I
I
Ra=10^5
Re=400
18
14
x
=
Z
10
'N,L...eireas°
6
2
0.0001
0 001
0 01
0.1
I
Z
(a)
Figure 3.45 Local Nusselt number variation for bottom
heating for different aspect ratios and Rayleigh
number(a)Ra=105, (b) Ra=2x105, (c) Ra=3x105,
and (d) Ra=4x105
103
22
I
e
I
Ra=2*10"5
Re=400
AR= 1
18
AR= 2
14
x
10
6
2
0.0001
0 001
0.01
0.1
(b)
22
Ra=3*10"5
Re=400
18
x
14
10
6
2
0.0001
0 001
0.01
0.1
(c)
Figure 3.45 (continued)
1
104
22
I
1111111
Ra=4*1 0'15
Re=400
AR=2
18
AR=3
x
14
11111111111111111111WESKI
111111111A.TWAPIMPTi
z
z
__4--
AR.4
10
II
6
I
2
0.0001
0 001
0.01
0.1
1
(d)
Figure 3.45 (continued)
In the thermally fully developed region of Figure 3.45(a),
the local Nusselt number for a square channel is seen to
be approximately 100 percent higher than for a
with an aspect ratio of b/a=2.
channel
This phenomenon, i.e. a
rectangular channel displaying a higher local Nusselt
number than for a high-aspect-ratio channel, is counter to
an ideal secondary flow mechanism; only one pair of
secondary flow cells exists along the channel direction.
Therefore, a reasonable explanation for this phenomenon is
the presence of multi stage secondary flow in the channel.
105
Figure 3.46 shows a case of multi stage
secondary flow.
In a narrow vertical channel, secondary flow near the
bottom surface acts as a barrier
preventing contact
between the bottom surface and the downward flow from the
upper part of the channel.
Therefore, the bottom surface
temperature becomes higher than for a one-pair secondary
flow case.
Ideal secondary flow pattern
Multi stage secondary
flow pattern
Figure 3.46 Ideal and multiple stage secondary flow
106
Chapter 4
MULTIPLE REGRESSION MODELING
4.1 Multiple regression
Regression is a highly useful statistical tool for
developing a quantitative relationship between a dependent
variable and one or more independent variables.
It
utilizes experimental data on the pertinent variables to
develop a numerical relationship showing the influence of
the independent variables on one or more dependent
variables of the system.
In the previous chapter, experimental results were
regressed using a simple linear regression technique
between local Nusselt number Nux and dimensionless axial
distance Z.
However, as seen in these plots, local
Nusselt numbers also change with Rayleigh number, Ra,
Reynolds number, Re, and aspect ratio, b/a.
Therefore, it
is desirable to express the experimental results as a
simple relationship between dependent variable Nux in
terms of independent variables, Ra, Re, and b/a.
The
algebraic equation for multiple linear regression is of
the form
107
Y = ao + ai xi +a2x2 +
+cycn
or
(4-1)
aixi
Y = a()
i=i
where Y, x, and a, are the dependent variable,
independent variable, and correlation coefficients,
respectively.
A commercial statistic analysis program,
SAS, was used to perform a multiple regression.
Using the
SAS program, different sequential variable selection
procedures-forward selection, stepwise regression, and
backward elimination
regression.
can be performed to do multiple
In this work the backward elimination
procedure was used.
Backward elimination begins with all
regressors and eliminates one at a time.
The first
removed is the regressor, which results in the smallest
decrease in R2.
The procedure is continued until the
candidate regressor for removal experiences a partial F
value which exceeds the preselected Fout.
4.2 Modeling
As seen in the figures showing experimental data in
the previous chapter, the local Nusselt number is a highorder logarithmic(ln) function of Z and the changing
minimum Nux with Ra shows the relation between Ra and
ln(Z).
Regression was conducted at each aspect ratio.
108
The multiple regression model may be written in the
following form for the whole- and three-surface heating
cases
Nux=a0+aiZi+a2Z2+a3Z3+a4Ra+a5Re +a6I1+a7I2+a8I3
(4-2)
and for the side surface heating case
Nux=ao+alZi+a2Z2+a3 Z3 +a4Z4+a5Ra+a6Re+a711+a8I2+a9I3
(4-3)
where a1...a9 are correlation coefficients, Zi=ln(Z),
Z2=Z12,
Z3=Z13,
Z4=Z14,
I2=Ra*Z12, and I3=Re*Z1.
Table 4.1 presents annotated computer printout, using the
SAS program, for backward elimination for the wholesurface-heating case of a square channel.
In the final
model which shows R2=97%, and Cp=7.18, the regressors Re
and 13 were eliminated indicating that Reynolds number
does not affect to the local Nusselt number along the flow
path.
Other cases show similar trends.
All printouts of
multiple regression results are listed in Appendix D.
Coefficients of the multiple regression are listed in
tables 4.2, 4.3, and 4.4 for whole-surface-heating, threesurface-heating, and one side-surface-heating case,
respectively.
109
Table 4.1 Multiple regression for whole surface
heating case of square channel.
Backward Elimination Procedure for Dependent Variable NUX
Step 0
All Variables Entered
R-square = 0.97599787 C(p) = 9.00000000
Sum of Squares Mean Square
F
Regression 8
Error
30
Total
38
271.17444815
6.66883040
277.84327855
33.89680602
0.22229435
152.49 0.0001
Variable
Parameter
Estimate
Standard
Error
Type II
Sum of Squares F
Prob>F
17.87695829
8.38833480
1.59459269
0.07563550
0.00018050
-0.00527024
0.00005623
0.00000435
-0.00111985
3.61515692
1.99720961
0.41980579
0.02998997
0.00004398
0.00367342
0.00001817
0.00000178
0.00075869
5.43577367
3.92132316
3.20723741
1.41392901
3.74413838
0.45755827
2.12861952
1.32274888
0.48429738
0.0001
0.0002
0.0007
0.0172
0.0003
0.1617
0.0042
0.0208
0.1504
DF
INTERCEP
Z
Z2
Z3
RA
RE
12
13
Step 1 Variable RE Removed
DF
Regression 7
Error
31
Total
Variable
INTERCEP
Z
Z2
Z3
RA
11
12
13
38
Prob>F
24.45
17.64
14.43
6.36
16.84
2.06
9.58
5.95
2.18
R-square = 0.97435105 C(p) = 9.05834418
Sum of Squares
Mean Square
F
270.71688987
7.12638868
277.84327855
38.67384141
0.22988351
168.23 0.0001
Parameter
Estimate
Standard
Error
15.13359953
7.97578310
1.60795121
0.07371401
0.00015523
0.00004760
0.00000367
-0.00008085
3.11999611
2.00985466
0.42680674
0.03046718
0.00004098
0.00001744
0.00000175
0.00023001
Prob>F
Type II
Sum of Squares F
5.40859150
3.62013842
3.26280412
1.34568471
3.29812135
1.71297385
1.01136193
0.02840280
23.53
15.75
14.19
5.85
14.35
7.45
4.40
0.12
Prob>F
0.0001
0.0004
0.0007
0.0216
0.0007
0.0104
0.0442
0.7276
110
Table 4.1 (continued)
Step 2 Variable 13 Removed
DF
Regression 6
32
38
Error
Total
Variable
INTERCEP
Z
Z2
Z3
RA
12
R-square = 0.97424882 C(p) = 7.18611529
Sum of Squares Mean Square
F
270.68848708
7.15479147
277.84327855
45.11474785
0.22358723
201.78 0.0001
Parameter
Estimate
Standard
Error
Type II
Sum of Squares F
15.07075908
7.87803096
1.59755070
0.07287733
0.00015525
0.00004741
0.00000365
3.07191719
1.96307327
0.41990855
0.02995521
0.00004042
0.00001719
0.00000172
5.38143684
3.60088580
3.23628578
1.32338782
3.29902382
1.70088218
1.00420880
Bounds on condition number:
6211.849,
Prob>F
Prob>F
24,07
16.11
14.47
5.92
14.75
7.61
4.49
0.0001
0.0003
0.0006
0.0207
0.0005
0.0095
0.0419
69898.99
All variables left in the model are significant at the 0.1000 level,
Summary of Backward Elimination Procedure for Dependent Variable NUX
Variable Number Partial
Step
Removed
In
R**2
0.0016
0.0001
1
RE
7
2
13
6
Model
R**2
F
C(p)
0.9744
0.9742
9.0583
7.1861
Prob>F
2.0583 0.1617
0.1236 0.7276
Table 4.2 Regression coefficient for
whole-surface-heating case
Nux=a0+a1Z1 +a2Z2+a3Z3+a4Ra+a6I1+a,I2
b/a
al
ao
a2
a 3
a4
a6
a,
1
15.0707
7.878
1.59755
0.072877
0.00015525
0.00004741
0.00000365
2
15.5925
7.14308
1.38887
0.065429
0.00005462
0.00001633
0.00000133
3
16.33315
8.01813
1.70935
0.09436
0.00005664
0.00001903
0.00000168
4
14.3701
5.71728
1.07891
0.04407
0.00001378
0.000002
0
5
5.3831
0
-0.07896
-0.03468
0.00001282
0.00000214
0
111
Table 4.3 Regression coefficient for
three-surfaces-heating case
Nux=a0+a1ZO-a2Z2+a3Z3+a4Ra+a6I1 +a7I2
b/a
ao
al
1
20.25698
11.467
2.4792
0.14669
0.000219
0.00007229
0.00000628
2
21.7928
11.501
2.2994
0.12157
0.00011418
0.00003264
0.0000024
3
31.2645
16.9536
3.4501
0.19675
0.00009353
0.00003054
0.00000245
4
7.7336
1.04178
0
-0.04234
0.00005283
0.0000157
0.00000118
5
14.246
5.8215
1.3005
0.07142
0.00003355
0.00001036
0.00000081
a3
a2
a4
a6
Table 4.4 Regression coefficient for
one side-surface-heating case
Nux=ao+a1Z1 +a2Z2+a3Z3+a4Z4+a6Ra+a7I1 +a8I2
b/a
a1
1
5.0856
2
11.56
3
20.066
4
5
a2
a4
a3
a7
a5
a8
0
-0.03226
0
0.00018
0.0000679
0.00000665
-1.407
-0.361
-0.0216
0.000044
0.00001158
0.0000008
8.302
1.3156
0
-0.004865
0.0000508
0.00001567
0.00000125
16.7
0
-2.8
-0.7475
-0.04929
0.0000226
0.00000418
0
-7.732
-17.131
-6.833
-1.105
-0.05865
0.00000661
0
-0.00000018
0
Figures 4.1, 4.2, and 4.3 show multiple regression lines
with corresponding experimental data.
lines show good agreement.
All regression
112
Re: 800 800
17
_
Ra:
a
'310.4
15
E3
16'10"4
o
10"5
A
A
13
g 11
ail
z
1
.
9
7
...../.
_b/a'1
_
5
po,
a:5.75 mm
b:5.75 mm
a
P
414"AlOPPIr
Apply uniform heat flux to
whole surfaces
I
3
I
1111
1
0.0001
I
1
0.001
0.1
0.01
1
z
800 1000
lle: 400
17
Ra
I0
15
690.4
0
A
10"5
r
0
4 r'
13
la
t
1 1).5
4'1045
g 11
9
7
bra2
'
....Ins7
5
,
..::::;k:l*,
3
0.0001
'Ir.
14444.
a:5.75 mm
b:11.5 mm
Apply uinVOrm heat flux to
whole surfaces
1
1
1
1111
0.001
I
I
0.01
0.1
z
Figure 4.1 Comparison between experimental data
and multiple regression results for the
whole-surface-heating condition
113
Re: 400 500 600 800
17
9
15
0
13
xz 11
z
9
b/a=3
11111111011411119.01111!
7
Mend
1.1
Ma
Apply uniform heat flux to
5.75 mm
b:17.25 mm
a
5
a
3
71171'7
0.0001
1.101111111
111111111
I
I
0.001
0.01
0.1
1
z
17
15
13
x 11
II
111111111101111111
Re: 300 400 500
Ra:
.11111..1111111.111 1°A5
IMMIIIIIIIMIIIIIIIIMIIIIII 2.113'5
1111111111111111MIMIIIIIIIMMIII 4.1°5 NORIO
1111111111111111111111111
11111111
11111111111111111111111111111111111
111-1111111111MEIMMATME1111111111111111
7
1111111111MMIIRMIFIKEIRMIIIIIIIIII
lill I
1111111111KIEMEMMEMIAIII
ll
3
0.0001
5.75 mm
b:17.25 mm
a:
Apply uniform heat flux to
whole surfaces
1
1
1
1111
0.001
1
11111=11111111111
1
0.01
0.1
z
Figure 4.1(Continued)
1
114
Re: 200 4 00
17
Ra:
I
10^5
I
15
2110'15
I
4'10.5
13
x 11
b/a=5
9
7
---a: 5.75 mm
WI
b: 28.8 mm
5
.
.
Apply uniform heat flux to
whole surfaces
I
3
I
11111
0.0001
I
I
0.01
0.001
0.1
Figure 4.1(Continued)
I
Re: 600 800
I
17
IIRa:
15
13
1.11111111111111111111111.11111111 3"0" bill1111
IM111111111111111111111111111111 6-10'4 1:11111111
1111111111111111MMIIIIIIIIMMIIIIII
10"5
13111
x 11
9
11111111111111111
1111
11111111111110.1111ifiliiiiiiinelirlillM111111
1111.10111111111.110111hIMMICHUMMI111111
iiiil
wArii Three
b/a =1
3
0.0001
1111691111111MIIIIIIIII
constant
111111.1111111111.1111.111111
Zesideisadial'atic
des
I
I
11
0.001
I
I
1111
0.01
0.1
1
Figure 4.2 Comparison between experimental data
and multiple regression results for the
three-surface-heating condition
115
Re: 400 600 800
17
Ra:
15
11"11-111M
B
5.7'10.4
2.10.5
11 113111111
=111111 WA11111111
13
Ti
10^5
11 11111111111
a
4
4'10^5
x 11
IMIUMEN101111
1,6111111311/
9
a
7
-I>
b
Three sides constant heat flux
other side is adiabatic
5
iwler
I..- LAMM
111111111
bra=2
3
0.0001
0.001
0.01
0.1
1
Re. 400 600 800
18
Ra:
10'5
16
6
r10^5
6.)
a 0
4.10.5
14
x 12
10
a
8
Three sides constant heat flux
other side is adiabatic
6
bra=3
4
0.0001
0.001
0.01
0.1
Figure 4.2(Continued)
1
116
Re: 300 400 600
18
Ra:
10"5
16
2'10"5
A
4'10"5
14
x 12
a
10
8
b
I>
Three sides constant heat flux
other side is adiabatic
6
b/a=4
4
0.0001
0.001
0.01
0.1
1
Re: 300 400 800
18
Ra.
10"5
16
r10"5
4'10'5
e
a
14
6.3'10"5
x 12
2
10
Three sides constant heat flux
other sides are adiabatic
6
bia=
4
0.0001
0.001
0.01
0.1
Figure 4.2(Continued)
1
117
moms
mono
=mon
0.0001
0.001
0.01
0.1
1
Re: 400 600 800
19
Ra:
17
11111111111111111111111111=11111111111
15
10r5
11.1.1111111P.M.111111111111111 2" °^5
1111111
11111111
INN
11111111111111111111111111111111111111111111111 rioA5
x 13
z
11
1111111111111111111111
1111111
0111.11111111111116111111111111111111
b/a=2
5
0.0001
One side constant heal flux otter
sides are aciabatic
0.001
1411111111111110111111111111
II
0.01
0.1
1
Figure 4.3 Comparison between experimental data
and multiple regression results for the
one-side-surface-heating condition
118
Re: 400 600 800
19
a
Ra:
10.5
17
290.5
4'10.5
15
A
A
x 13
z
11
9
7
b/a= 3
One side constant heat &a other
sides are aciabatie
5
0.0001
0.001
0.01
0.1
1
z
Re: 400 500 600
19
Ra:
10.5
17
290.5
410"5
15
0
0
A
x 13
z
11
9
7
5
0.0001
One side constant heat sea other
b/a=4 sides are *dentine
0.001
0.01
0.1
z
Figure 4.3(Continued)
1
119
Re: 200 400
19
Ra:
10.5
17
2'10.'5
4.10^5
15
8'10^5
A
x 13
z
11
NYV
A
9
7
5
0.0001
4s
''- -One side constant heat flux
b/a=5
other sides are adiabatic
I
I
III
I
0.001
1
I
0.01
0.1
z
Figure 4.3(Continued)
a
D
A
120
Chapter 5
CONCLUSION
This work has involved an experimental study to
evaluate friction factors for laminar flow and mixed
convection heat transfer for laminar flow in the thermal
entrance region of vertically narrow horizontal
rectangular channels over a range of heating conditions
and channel aspect ratios.
The effects of hydraulic
diameter, aspect ratio, Rayleigh number
number have been examined in detail.
,
and Reynolds
A summary of the
major experimental results is the following:
(1)
In hydrodynamically fully-developed laminar flow the
friction factor, f, is a function of the aspect ratio
of the rectangular channel; the relation between fRe
and aspect ratio is in complete agreement with
theoretical prediction.
(2)
In rectangular channels, the critical Reynolds number
varies linearly with the logarithm of hydraulic
diameter.
(3)
Until a certain axial location is reached, the local
Nusselt number varies as if the flow were one of
purely forced convection.
121
(4)
Following the onset of buoyancy-induced secondary
flow the strength of the secondary flow increases
causing an increase in the local Nusselt number
compared with totally forced convection behavior.
(5)
Local Nusselt numbers increase with increasing
Rayleigh number.
The effect of aspect ratio is not
observed until a minimum local Nusselt number is
reached.
(6)
The effects of increasing Rayleigh number and
decreasing aspect ratio cause the thermal entrance
length to decrease.
(7)
For equal Rayleigh numbers and aspect ratios, the
local Nusselt number varies with heating condition in
the boundary order: side heating > three surface
heating > whole surface heating.
(8)
Local Nusselt number variation along the flow
direction is independent of Reynolds number.
(9)
For the bottom-surface-heating condition,
dimensionless thermal entrance length is a function
of Rayleigh number.
The local Nusselt number
increases with decreasing aspect ratios in the fullydeveloped region.
122
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128
61.
Landram, C. S.,"Microchannel flow boiling mechanisms
leading to burnout," HTD-Vol. 292, Heat Transfer in
Electronic Systems, ASME, pp.129-136, 1994.
62.
Shannon, R. L., and Depew, C. A.,"Forced Laminar Flow
Convection in a Horizontal Tube With Variable
Viscosity and Free-Convection Effects," Journal of
Heat Transfer, vol. 91, pp. 251-258, May 1969.
63.
Shannon, R. L., and Depew, C. A.,"Combined free and
forced laminar convection in a horizontal tube with
uniform heat flux," Journal of Heat Transfer, vol.
90, pp. 353-357, May 1968.
64.
Sparrow, E. M., Koram, K. K., and Charmchi, M., "Heat
Transfer and Pressure Drop Characteristics Induced by
a Slat Blockage in a Circular Tube," ASME Journal of
Heat Transfer, vol. 102, pp. 64-70, 1980.
65.
Sparrow, E. M., and Kemink, R. G., "Heat Transfer
Downstream of a Fluid Withdrawal Branch in a Tube,"
ASME Journal of Heat Transfer, vol. 101, pp. 23-28,
1979.
66.
Krall, K. M., and Sparrow, E. M., "Turbulent Heat
Transfer in the Separated, Reattached, and
Redevelopment Regions of a Circular Tube," ASME
Journal of Heat Transfer, vol. 88, No. 1, pp. 131136, Feb. 1966.
67.
Koram, K. K., and Sparrow, E. M., "Turbulent Heat
Transfer Downstream of an Unsymmetric Blockage in a
Tube," ASME Journal of Heat Transfer, vol. 100, pp.
588-594, 1978.
68.
Kim, Y. J.,"An experimental study of combined forced
and free convective heat transfer to non-Newtonian
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Baughn, J.W., Takahashi, R. K., Hoffman, M. A., and
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129
APPENDICES
130
APPENDIX A
Uncertainty analysis
This section discribed the uncertainty analysis of
the experimental results related to the friction factor
and local Nusselt number of the rectangular channela.
The uncertainty of experimental results is the sum of
individual uncertainties of measurements.
The concept is
to let the final experimental result, f, be a function of
the n independent variables
f f (x
x2,
x2,
x3,
x3,
,
x,
Thus,
x )
(Al)
The overall uncertainty, Wf of the experimental result, f,
is computed according to
2
wf=
of (0)
ax1
where col,
co2,
,
1
+(af (0) 2
ax2 2
+(af co) 2
axn
n
co, are the uncertainties in the
independent variables.
(A2)
131
1. Uncertainty of Friction Factor
To compute the uncertainty of the friction factor, f,
for a rectangular channel, equation (2-10) was considered
as follows;
f=
ph g(p A )2
1
2L
1
a3b3
L (a+b)
where Ah=(h1-h2)
.
t2
2
gp2Ah
(A3)
t2
pri2
The friction factor f is observed to be
function of L, a, b,
p, oh, time t, and weight W.
The
measurement errors of each variable in this experiment are
listed in table A.1.
Table A.1 Uncertainty of measurement
error variables
Variable
Measurement Error
wi,
+ 1 mm
wa
+ 0.05 mm
wb
± 0.05 mm
cop
± 0.002
96-
wAh
± 0.002 mm
wt
± 0.02 sec.
cow
+ 0.1 g
The uncertainties of each variable are computed using
equation (A2).
become
The partial derivatives for each variables
132
f
of
(A4)
L
af 2a +3b
as a(a +b)
af 3a +2b
ab b(a+b)
of
f
aL
p
af
f
aoh
Oh
af
f
at
t
f
aaw
2
(A5)
f
(A6)
(A7)
(A8)
(A9)
(A10)
Prif
Substituting equations (A4) through (A10) into equation
(A2) gives
W2
f coy ±((2a+3b)f coy +((3a+2b)f (a)2 +(2 f 6),12
a(a+b)
b(a+b)
b
(All)
(
wAh)
+
(21 6))1-(
t
2
fW
2
caw
133
Table A.2 lists the overall uncertainties in f for three
Reynolds numbers and aspect ratios.
Uncertainties in the
friction factor for all cases examined are listed in
Appendix B.
Table A.2 Uncertainties of friction factor
for different aspect ratio and Re
Re
b/a
1000
100
2000
1
+ 3.12
%
+ 3.31
%
+ 3.12
2
+ 2.59
%
+ 2.55
96
+ 2.8 %
3
+ 2.5 %
+ 2.49 %
+ 2.49 %
4
+ 2.5
.36
+ 2.5
+ 2.5 96
5
+ 2.5
96
+ 2.5 %
96
+ 2.51
%
%
2. Uncertainty of local Nusselt number
The local Nusselt number Nux, in equation(2-11), can
be written as
Nux = Cp
ab
+
W ATmo
t ATinswx
(Al2)
or
Nu x= f (Cp , K, a, b ,W,t, ATnio_ and ATmswx)
Where ATmoi = Tout
Tin and ATmswx =
Tmsx
Tmwx-
(M3)
134
Uncertainties in Cp and k, based on a local temperature
uncertainty of ±0.1 °C, are 0.002 percent and 0.03 percent,
respectively.
The partial derivatives for each variable
become
aNu
Nu
aCp
Cp
aNu x
(A14)
Nu x
(A15)
aK
aNu
ba
as
a(a+b)
aNu x
ab
ab
b(a+b)
amix
Nu
Nu
AW x
aw
aNu x
Nu x
at
aNu
aATmoi
(A16)
x
(A17)
(A18)
(A19)
Nu x
A Tmoi
(A20)
135
aNu x
Nu x
aAT,
ATmswx
(A21)
The overall uncertainty for the local Nusselt number, coNux
thus becomes
(6)Nz7 )2
Arttx
(W )2
(A)
)2 +(
b-a
a(a+b)
Cp
)2 +
a-b
b(a+b)
2
2
(A22)
(63,6 Ty61
A Tmoi
OT
Here, the properties Cp and k are counter values of the
inlet and local bulk temperature of water, respectively.
Each variable uncertainties are listed in Table A3.
Table A.3 Variable uncertainties
Variable
Uncertainty
6)Cp
0.002%
6)k
0.03%
(A)ATmoi
+0.1°C
6.3 A Tmswx
+0.1°C
Uncertainties in the local Nusselt number for all cases
examined, are listed in Appendix C.
136
APPENDIX B
Friction factor data
Aspect 1
Re
93.70
Aspect 2
Cf
Theory
Cr Re
Wcf(%)
Theory
0.1499
Cf Re
0.1519
14.05
3.12
83.98
0.1883
107.98
0.1852
0.1291
15.82
0.1318
2.58
13.94
110.48
0.1654
15.41
0.1288
2.59
14.24
94.07
116.89
0.1638
0.1289
3.12
3.13
123.45
0.1320
0.1331
15.42
0.1153
14.01
3.11
142.24
138.29
0.1135
0.1013
0.1081
0.1094
15.38
0.1029
14.00
3.11
153.82
196.78
0.0910
15.44
14.00
3.10
227.08
186.96
0.0685
0.0761
15.19
0.0761
14.24
3.10
205.45
0.0692
0.0554
15.37
0.0693
2.54
14.21
3.09
280.84
319.03
254.27
0.0567
0.0488
15.25
0.0560
2.54
14.41
3.09
365.11
282.41
0.0511
0.0504
0.0433
2.54
14.43
3.08
3.08
416.14
0.0442
0.0426
0.0374
15.45
328.54
347.40
15.63
0.0406
0.0320
15.87
0.0410
2.54
2.54
14.10
0.0379
0.0281
0.0377
15.88
14.29
3.09
3.08
553.70
377.64
415.11
0.0344
0.0236
15.69
0.0343
14.26
3.08
440.89
505.63
735.88
0.0323
0.0287
0.0211
15.31
0.0323
14.23
3.08
817.55
0.0190
15.66
0.0281
2.54
14.51
3.08
602.00
0.0233
0.0176
15.35
0.0236
14.00
3.08
647.93
884.20
988.43
0.0217
0.0220
2.54
2.54
14.09
3.08
712.19
1,120.83
0.0200
0.0187
0.0200
14.21
3.08
1,265.34
0.0183
14.54
3.08
862.17
1,031.79
1,337.02
0.0170
0.0165
0.0138
14.63
3.08
1,514.73
13.86
3.32
1,532.32
1,159.31
0.0123
14.18
3.27
1,377.53
0.0122
0.0108
1,650.78
0.0103
14.82
3.20
1,670.88
1,958.00
0.0088
0.0085
14.68
3.18
1,670.88
2,099.62
0.0088
0.0085
14.68
3.18
1,852.96
2,110.40
0.0096
0.0093
0.0077
0.0076
3.15
3.15
2,183.87
0.0784
0.0669
0.0547
0.0478
0.0423
0.0376
0.0327
0.0287
0.0238
0.0208
0.0192
0.0174
0.0156
0.0138
0.0125
0.0116
0.0106
0.0102
0.0096
0.0086
0.0085
0.0097
0.0097
0.0791
0.0925
2.56
2.55
2.55
2.54
2,201.76
0.0101
0.0128
0.0117
0.0074
2.68
3.11
0.0105
0.0118
2.65
0.0131
2.60
2.59
778.14
1,863.92
1,915.90
0.0134
14.53
Re
485.75
659.46
Cf
Wcf(%)
2.54
2.54
2.54
0.0157
15.46
0.0139
15.46
0.0123
15.78
2.54
2.55
0.0116
15.46
2.55
0.0103
0.0102
15.98
2.55
15.64
0.0094
0.0079
0.0074
15.78
16.89
2.55
2.55
2.77
17.76
2.77
0.0074
2.68
0.0071
2.67
2,174.24
2,403.14
2,910.82
0.0121
3.12
2,350.77
2,728.24
2,901.33
0.0113
0.0103
3.12
3,292.91
0.0141
2.61
3.13
3,426.87
3,777.07
3,517.78
0.0123
0.0109
0.0107
3.13
3.13
3,892.97
0.0124
2.59
2.58
4,190.62
4,935.71
0.0107
0.0097
3.12
1,961.48
3.13
Avg
14.2748
2.59
Avg
15.6929
137
Aspect 3
Aspect 4
Cf
Theory
crRe
Wcf(%)
Cf
Theory
Cf Re
62.70
0.2639
0.2726
16.55
2.54
53.58
0.3421
0.3403
18.33
Wcf(%)
2.59
78.36
0.2092
0.2182
16.39
2.52
61.71
0.2912
0.2955
17.97
2.56
92.59
0.1784
0.1846
16.52
2.51
81.42
0.2231
0.2240
18.17
2.53
101.40
0.1637
0.1686
16.60
2.51
105.06
0.1724
0.1736
18.11
2.51
104.56
0.1611
0.1635
16.84
2.51
121.58
0.1511
0.1500
18.38
2.51
127.79
0.1221
0.1338
15.60
2.50
161.81
0.1128
0.1127
18.25
2.50
181.43
0.0903
0.0942
16.39
2.49
173.14
0.1050
0.1053
18.18
2.51
230.16
0.0682
0.0743
15.70
2.49
202.62
0.0905
0.0900
18.34
286.31
0.0564
0.0597
16.14
2.49
216.92
0.0849
0.0841
18.42
2.50
2.49
313.17
0.0534
0.0546
16.74
2.49
217.41
0.0846
0.0839
18.39
2.49
365.08
0.0458
0.0468
16.71
2.49
247.31
0.0737
18.40
2.49
390.89
395.96
0.0437
0.0437
17.09
290.13
0.0628
18.07
2.49
0.0425
0.0432
16.82
2.49
2.49
335.54
0.0744
0.0623
0.0540
0.0543
18.13
2.49
455.17
0.0362
0.0376
16.48
2.49
381.34
0.0478
18.22
2.49
467.03
0.0355
0.0366
16.56
2.49
434.06
0.0424
0.0478
0.0420
18.39
2.49
529.30
0.0320
0.0323
16.92
2.49
449.28
0.0407
0.0406
18.29
561.40
0.0296
0.0305
16.64
2.49
500.73
0.0370
18.54
2.50
2.49
583.71
0.0281
0.0293
16.41
2.49
583.27
0.0313
0.0364
0.0313
18.28
2.50
648.88
0.0261
0.0263
16.96
2.49
630.16
0.0289
0.0289
18.21
2.49
725.34
0.0230
0.0236
16.71
2.49
679.50
0.0267
0.0268
18.17
2.50
801.19
0.0213
0.0213
17.04
2.49
767.63
0.0238
0.0238
18.30
2.50
885.94
0.0191
0.0193
16.93
2.49
848.03
0.0220
0.0215
18.65
2.50
986.90
0.0170
0.0173
16.76
2.49
878.95
0.0209
0.0207
18.35
2.50
1,136.40
0.0150
17.04
2.49
0.0190
18.01
0.0139
17.08
2.50
950.09
957.46
0.0192
1,225.63
0.0150
0.0139
0.0191
0.0190
1828
2.50
2.50
1,407.59
0.0123
0.0121
17.32
2.49
1,063.07
0.0171
0.0172
18.23
2.50
1,652.43
0.0108
0.0103
17.90
2.49
1,142.53
0.0161
0.0160
18.40
2.50
1,743.01
0.0098
0.0098
17.01
2.50
1,357.89
0.0135
0.0134
18.33
2.50
1,833.86
0.0095
0.0093
17.33
2.50
1,431.31
0.0141
0.0127
20.14
2.51
1,835.54
0.0093
0.0093
17.12
2.49
1,439.49
0.0135
0.0127
19.45
2.51
1,939.83
0.0089
0.0088
17.35
2.49
1,448.68
0.0136
0.0126
19.77
2.51
2,237.22
0.0084
0.0076
2.49
1,578.69
0.0128
0.0116
20.13
2.51
2,353.11
0.0091
0.0073
2.50
1,614.16
0.0120
0.0113
19.45
2.54
2,699.00
0.0108
2.62
1,745.50
0.0120
0.0104
20.91
2.51
3,007.99
0.0117
2.57
1,970.65
0.0112
0.0093
22.02
2.50
3,078.24
0.0135
2.55
2,190.52
0.0108
0.0083
3,703.59
2.53
2,462.24
0.0131
2.66
3,911.45
0.0134
0.0128
2.53
2,861.91
0.0142
2.58
5,120.70
0.0114
2.55
3,241.28
0.0151
2.55
3,801.08
0.0132
4,788.14
5,670.58
0.0111
2.54
2.54
2.58
Re
Avg
16.76
Re
2.51
0.0105
Avg
18.68
138
Aspect 5
Re
Cf
Theory
Cr Re
Wcf( %)
100.26
0.1667
0.1903
137.16
0.1369
0.1391
18.78
2.51
138.76
0.1346
0.1375
18.68
2.51
0.1008
0.1014
18.97
2.51
2.51
188.12
16.71
188.12
0.1050
0.1014
19.75
232.80
0.0839
0.0819
19.52
2.50
233.68
0.0820
0.0816
19.16
2.50
330.17
0.0579
0.0578
19.12
2.50
330.96
0.0579
0.0576
19.18
2.50
384.33
0.0506
19.44
2.50
385.76
0.0503
0.0496
0.0495
19.41
2.50
466.83
0.0417
0.0409
19.48
2.50
467.35
0.0415
0.0408
19.37
2.50
529.38
529.83
0.0361
0.0360
19.11
2.50
0.0361
0.0360
19.13
2.51
607.64
0.0316
0.0314
19.19
2.50
607.85
0.0315
0.0314
19.16
2.50
661.20
0.0292
0.0289
19.30
2.50
661.93
0.0292
0.0288
19.32
2.50
771.95
0.0253
0.0247
19.51
2.51
773.72
0.0253
0.0247
19.60
2.51
930.78
0.0205
0.0205
19.04
2.51
931.13
0.0204
0.0205
19.03
2.51
1,039.02
0.0185
0.0184
19.19
2.51
1,040.32
0.0184
19.19
2.51
1,207.42
0.0160
0.0183
0.0158
19.30
2.51
1,211.69
0.0159
0.0157
19.30
2.51
1,433.37
0.0136
0.0133
19.54
2.51
1,436.65
0.0136
0.0133
19.58
2.51
0.0115
0.0110
19.96
2.51
1,775.03
0.0112
0.0107
19.95
2.51
2,005.00
0.0101
0.0095
2.51
2,178.91
0.0095
0.0088
2.51
2,305.33
0.0085
0.0083
2.52
2,511.36
0.0108
2.67
2,643.59
0.0105
2.66
3,145.01
0.0108
2.59
3,601.84
0.0108
2.58
4,237.02
0.0104
1,729.96
2.57
Avg
19.22
139
APPENDIX C
Sample heat transfer data
140
Aspect Ratio: 1
Heating Whole Surfaces
1
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Z
Top
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
2
25.4
1.12E-03
23.69
23.70
23.30
23.60
23.69
23.76
21.99
1171.94
11.12
0.64
5.74
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Z
25.40
1.10E-03
Top
3
2
23.80
23.40
23.60
24.20
23.90
23.89
22.09
1,143.33
10.84
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
3
0.61
5.65
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Z
25.40
1.09E-03
Top
3
2
22.09
22.14
22.04
22.24
21.99
22.20
21.34
1,228.47
11.67
1.39
11.87
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
0.00575
0.00575
0.00575
2.73685
50.8
2.24E-03
24.19
24.30
23.90
24.20
23.69
24.28
22.08
942.29
8.94
0.42
4.66
101.6
4.47E-03
24.79
25.10
24.70
24.90
23.99
24.94
22.25
772.69
7.32
0.28
3.87
9.91
6.61
203.2
8.95E-03
25.79
26.00
25.50
25.90
24.59
25.90
22.61
23.31
631.28
5.97
0.19
3.22
3.21
Gr
Ra
Pr
101.60
4.41E-03
24.40
24.50
24.50
24.90
24.20
24.68
22.35
883.27
8.37
0.37
4.42
203.20
8.82E-03
25.20
25.60
25.40
25.90
24.90
25.70
22.69
685.23
6.49
0.23
3.49
Re
Gr
Ra
Pr
101.60
4.36E-03
22.49
22.84
22.64
22.74
22.39
22.75
21.47
824.79
7.83
0.64
8.12
406.4
1.79E-02
26.19
26.40
26.70
27.10
25.69
26.60
630.26
5.97
0.19
Re
0.00575
0.00575
0.00575
2.73685
50.80
2.18E-03
22.29
22.44
22.34
22.44
22.19
22.42
21.39
1,018.67
9.68
0.96
Pr
Gr
0.00575
0.00575
0.00575
2.73685
50.80
2.21E-03
23.90
23.80
23.90
24.60
24.00
24.20
22.17
1,017.19
9.64
0.49
5.05
Ra
597.43
9.09E+03
6.01E+04
Re
203.20
8.72E-03
22.89
23.14
23.04
23.24
22.79
23.16
21.65
696.23
6.61
0.46
6.95
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
711.2
3.13E-02
28.49
28.20
27.30
27.10
25.59
27.56
24.37
651.28
6.14
0.20
3.31
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
607.42
9.10E+03
6.01E+04
6.60
406.40
1.76E-02
27.10
27.40
26.60
26.40
24.70
26.70
23.38
621.33
5.87
0.19
3.20
711.20
3.09E-02
28.90
28.00
27.20
27.20
25.50
27.56
24.42
656.07
6.19
0.21
3.36
23.84
23.64
23.84
23.29
23.80
22.00
583.79
5.54
0.33
5.94
1,320.80
5.73E-02
31.00
30.00
28.80
28.50
27.30
29.26
26.49
743.99
6.98
0.26
3.77
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
602.73
4.35E+03
2.93E+04
6.72
406.40
1.74E-02
23.59
1320.8
5.81E-02
30.99
30.60
29.50
28.50
26.59
29.29
26.48
740.20
6.95
0.26
3.72
711.20
3.05E-02
24.79
24.54
24.34
24.54
23.59
24.51
22.52
529.02
5.01
0.27
5.46
1,320.80
5.67E-02
26.19
26.04
25.04
24.84
23.89
25.40
23.57
574.32
5.43
0.32
5.86
21.9
31.4
121.3
36.86
2387.6
1.05E-01
34.19
33.70
32.50
31.60
29.89
32.45
30.19
917.86
8.54
0.39
4.55
22
31.3
106.8
31.99
2,387.60
1.04E-01
33.50
33.40
31.90
31.60
30.30
32.44
30.11
886.30
8.25
0.37
4.43
21.3
26
117.2
34.78
2,387.60
1.02E-01
27.79
27.44
26.54
26.34
25.49
26.87
25.40
715.79
6.74
0.48
7.13
141
4
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Z
Top
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
5
25.40
1.11E-03
50.80
2.21E-03
22.00
22.16
22.16
22.26
22.00
22.16
21.34
1,240.33
11.78
1.46
12.43
22.20
22.36
22.36
22.46
22.10
22.34
21.39
1,061.92
10.09
1.08
10.70
a(m)
b(m)
Dh(m)
0.00575
0.00575
0.00575
2.73685
Lth(m)
L(mm)
Z
Top
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
L(mm)
Z
Top
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
25.40
7.87E-04
50.80
1.57E-03
20.36
20.29
20.29
20.49
20.36
20.42
19.64
1,493.79
14.26
1.86
20.36
20.49
20.49
20.69
20.46
20.59
19.67
1,280.47
12.22
13.01
6
0.00575
0.00575
0.00575
2.73685
a(m)
b(m)
Dh(m)
Lth(m)
25.40
7.90E-04
20.45
19.88
19.98
20.18
20.25
20.11
19.34
1,548.32
14.79
1.95
13.18
1.37
11.24
Re
593.41
Gr
4.20E+03
2.82E+04
6.72
Ra
Pr
101.60
4.43E-03
22.40
22.86
22.66
22.76
22.40
22.61
21.47
889.83
8.45
0.76
9.04
203.20
8.86E-03
406,40
1.77E-02
711.20
3.10E-02
22.90
23.16
23.06
23.26
22.80
23.05
21.64
719.74
6.83
23.60
23.76
23.76
23.86
23.50
23.75
21.98
23.90
24.66
24.46
24.56
23.70
24.46
22.50
515.99
4.89
0.27
5.54
0.51
7.43
Re
Gr
Ra
Pr
101.60
3.15E-03
20.56
20.99
20.99
21.19
20.76
21.00
19.74
931.12
8.88
0.74
8.37
0.00575
0.00575
0.00575
2.73685
20.96
21.49
21.49
21.79
21.26
21.58
21.76
22.29
22.19
22.49
21.86
22.33
20.16
542.26
5.17
0.28
5.32
19.88
691.22
6.59
0.43
6.46
Ra
Pr
20.35
20.08
20.28
20.68
20.35
20.42
19.37
1,149.13
10.98
1.09
9.93
20.35
20.28
20.48
20.98
20.45
20.64
19.44
1,003.25
9.58
0.84
8.76
203.20
6.32E-03
20.85
20.98
21.08
21.58
20.95
21.27
19.59
714.24
6.82
0.44
6.49
711.20
2.20E-02
22.76
22.89
22.79
23.09
22.16
22,93
20,59
21.55
21.98
21.88
22.18
21.55
22.02
19.88
559.63
5.34
0.28
5.33
1,320.80
4.09E-02
24.46
24.29
23.59
23.39
22.26
23.75
21,43
21.3
25.9
126.8
38.22
2,387.60
1.04E-01
27.60
27.26
26.36
26.26
25.30
26.74
25.31
710.39
6.69
0.49
7.34
19.6
23.4
127.4
27.44
2,387.60
7.40E-02
25.96
25.59
24.69
24.39
23.26
24.92
22.92
584.67
5.53
500.41
4.76
506.11
0.24
5,02
0,24
5.06
0.31
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
19.3
23.2
149.8
32.4
1,320.80
4.11E-02
24.15
24.08
23.28
23.08
22.05
23.50
21.18
2,387.60
7.42E-02
25.75
25.38
24.38
23.98
22.95
787.76
4.09E+03
2.90E+04
7.10
406.40
1.26E-02
1,320.80
5.76E-02
25.60
26.06
25.16
24.96
23.90
25.37
23.52
547.95
5.18
0.30
5.83
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
796.74
4.12E+03
2.90E+04
7.04
406.40
1.26E-02
Gr
101.60
3.16E-03
573,72
5,44
0.33
6.06
203.20
6.30E-03
Re
50.80
1.58E-03
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
711.20
2.21E-02
22.55
22.68
22.58
22.78
21.75
22.63
20.31
516.52
4.92
0.25
5.02
4.81
518.01
4.92
0.25
5.03
5.64
24.61
22.70
628.04
5.95
0.35
5.84
142
7
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Z
Top
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
8
25.40
7.96E-04
50.80
1.59E-03
101.60
3.18E-03
22.12
21.49
21.59
21.99
22.12
21.88
20.37
1,539.07
14.66
0.99
6.75
22.22
21.99
21.99
22.49
22.22
22.32
20.44
1,236.43
11.78
0.65
5.48
22.72
22.79
22.79
23.09
22.42
a(m)
b(m)
Dh(m)
0.00575
0.00575
0.00575
2.73685
Lth(m)
L(mm)
Z
Top
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
9
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
22.91
20.58
997.99
9.50
0.43
4.49
Re
803.01
Gr
Ra
8.74E+03
6.04E+04
Pr
6.91
203.20
6.37E-03
23.42
23.79
23.79
24.29
23.22
23.96
20.86
750.44
7.14
0.25
3.49
406.40
1.27E-02
24.92
25.19
24.89
25.19
24.02
25.10
21.43
632.77
711.20
2.23E-02
27.02
26.29
25.69
25.79
24.22
26.00
22.27
623.66
6.01
5.91
0.18
3.03
0.18
2.99
Re
790.01
Gr
8.63E+03
6.02E+04
6.97
Ra
Pr
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
1,320.80
4.14E-02
29.12
28.49
26.59
26.29
24.62
27.37
23.97
682.83
6.45
20.3
27.9
162.1
35.23
2,387.60
7.48E-02
32.12
31.19
29.29
28.59
26.92
29.83
26.93
801.14
7.51
3.22
0.28
3.69
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
20
27.8
142.9
31.34
0.21
25.40
8.02E-04
50.80
1.60E-03
101.60
3.21E-03
203.20
6.42E-03
406.40
1.28E-02
711.20
2.25E-02
1,320.80
4.17E-02
2,387.60
7.54E-02
21.61
21.91
22.51
23.21
24.51
26.41
28.81
31.91
21.47
21.37
21.67
21.97
21.77
22.07
22.47
22.37
22.57
23.47
23.37
23.97
24.57
24.27
24.87
25.07
24.77
26.07
28.27
27.27
26.87
30.47
28.67
28.27
21.81
22.01
22.31
23.11
24.11
24.91
25.21
27.51
22.54
20.29
1,049.97
10.00
0.46
4.63
23.71
24.76
21.16
656.04
6.24
0.19
3.06
25.83
22.03
27.44
23.76
642.84
6.07
0.18
3.01
29.52
26.80
870.20
8.16
0.32
3.90
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
20.8
33.6
139.8
31.19
1,320.80
4.26E-02
34.79
32.64
29.74
29.84
27.59
31.48
26.98
846.43
7.94
0.19
2.36
2,387.60
7.69E-02
38.99
36.14
34.34
34.34
31.59
35.37
31.97
1,119.84
10.38
0.32
3.04
21.66
20.07
1,488.33
14.19
22.10
20.14
1,208.51
11.52
0.91
0.61
6.43
5.27
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Z
Top
0.00575
0.00575
0.00575
2.73685
25.40
8.18E-04
23.79
22.44
22.94
23.74
23.59
23.50
20.92
1,476.54
14.04
0.56
3.95
0.00575
0.00575
0.00575
2.73685
50.80
1.64E-03
23.99
23.64
23.64
24.64
23.79
24.29
21.04
1,171.83
11.14
0.35
3.17
20.58
754.71
7.19
0.25
3.44
Re
Gr
Ra
Pr
101.60
3.27E-03
24.79
24.84
25.04
25.44
24.19
25.07
21.28
1,004.33
9.54
0.26
2.75
203.20
6.55E-03
26.19
26.54
26.44
27.04
25.29
26.69
21.75
771.56
7.32
0.16
2.17
621.31
5.89
0.17
2.93
791.86
1.50E+04
1.03E+05
6.82
406.40
1.31E-02
28.49
28.24
27.64
28.04
26.09
28.06
22.70
711.15
6.73
0.14
2.03
711.20
2.29E-02
31.49
29.34
28.54
28.94
26.49
29.21
24.13
749.69
7.08
0.15
2.12
143
Heating Three Surfaces
1
L(mm)
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.00575
0.00575
2.73685
25.40
1.09E-03
21.72
21.82
22.02
3
2
1
Bottom
Re
Gr
Ra
Pr
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
599.69
4.45E+03
3.02E+04
6.78
50.80
2.17E-03
21.92
22.12
22.32
101.60
4.35E-03
22.12
22.42
22.62
203.20
8.69E-03
22.52
22.72
23.02
406.40
1.74E-02
23.12
23.42
23.72
711.20
3.04E-02
23.82
23.92
24.22
1,320.80
5.65E-02
24.72
24.52
24.52
.4229.
22.47.
21.14
23,51
21.55
564.60
23:97
21.96
551.13
24 4(3
832.51
22:93
21.27
711.73
22.79
686.95
24.23
856.67
0.63
7.99
0.47
6.98
0.31
0.30
5.67
0.44
6.77
0.66
8.20
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
21.1
....
.. ..
Tms
Tmw
21.94
21.03
1,222.25
hx
21.07
978.46
Nu(x)
1162`
930
W(Nux)
Error(%)
1.32
11.37
0.86
9.24
2
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
3
2
1
21
24.7
116.5
34.49
0.00575
0.00575
0.00575
2.73685
Re
Gr
Ra
Pr
5.77
595.20
8.78E+03
5.94E+04
6.76
2,387.60
1.02E-01
25.82
25.52
25.62
... ,
28.4
123.1
36.81
25.40
1.10E-03
22.46
22.86
23.16
50.80
2.20E-03
22.76
23.16
23.66
101.60
4.39E-03
23.36
23.76
24.16
203.20
8.78E-03
24.16
24.46
25.06
406.40
1.76E-02
25.46
25.56
25.76
711.20
3.07E-02
26.16
26.06
26.36
1,320.80
5.71E-02
27.66
27.26
27.16
2Z95.
2:39
21.17
1,213.20
21.24
1,017.76
13,85
21.37
872.22
21.64
723.64
22.18
654.43
23.00
712.90
24.62
917.35
27.47
1,091.18
11,63
0.67
9t`r7
0.35
4.26
0.25
3.62
0.21
0.24
3.57
0.39
4.46
0.53
5.23
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
20.65
23.4
182
40.38
1,320.80
4.23E-02
23.92
24.02
24.12
2,387.60
7.65E-02
24.92
24.82
24.72
23.91
2450;
21.98
23.05
710.17
2,387.60
1.03E-01
30.06
29.46
29.66
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
5.78
3
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
3
2
1
0.47
4.90
0.00575
0.00575
0.00575
2.73685
Re
Gr
Ra
Pr
25.40
8.13E-04
21.22
21.42
21.62
50.80
1.63E-03
21.42
21.52
21.82
101.60
3.25E-03
21.62
21.82
22.02
20.68
1,262.68
20.70
1,099.02
1.45
12.06
1.11
203.20
3.32
793.35
4.27E+03
2.92E+04
6.85
21.92
22.12
22.42
406.40
1.30E-02
22.37
22.72
23.02
711.20
2.28E-02
23.12
23.22
23.62
20.75
932.05
20.85
792.22
21.06
626.77
21.36
542.06
568.01
0.82
9.24
0.61
0.32
6.13
0.34
6.32
6.51 E -03
Bottom
Tms
Tmw
hx
Nu x)
W(Nux)
Error(%)
i$;*
10.65
8.08
0.40
6.77
.611Z
0.50
7.42
144
4
a(m)
b(m)
Dh(m)
0.00575
0.00575
0.00575
2.73685
Lth(m)
Re
Gr
Ra
Pr
795.18
8.84E+03
6.06E+04
6.86
25.40
8.10E-04
50.80
1.62E-03
101.60
3.24E-03
203.20
6.48E-03
406.40
1.30E-02
711.20
2.27E-02
2
22.01
22.01
22.41
23.01
1
22.41
22.21
22.31
22.91
23.01
23.51
24.21
24.31
24.91
25.11
25.41
25.41
25.91
L(mm)
Bottom
..
Tms
Tmw
hx
23: 31
...
.
..
..
.
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
20.6
26.3
217.2
48.02
1,320.80
2,387.60
7.62E-02
4.21 E -02
26.51
26.21
26.31
28.61
27.91
28,01
...
20.65
1,337.74
20.71
20.81
1,133.78
1,006.73
21.02
828.37
21.45
693.36
22.08
671.90
23.35
836.75
25.57
981.21
0.78
6.12
0.57
5.27
0.45
4.75
0.32
4.03
0.23
0.22
3.43
0.32
4.06
0.43
4.64
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
20.3
30
218.6
48
Nu(x)
W(Nux)
Error(%)
5
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
2
0.00575
0.00575
0.00575
2.73685
3.51
Ra
794.80
1.47E+04
1.02E+05
Pr
6.91
Re
Gr
25.40
8.04E-04
22.20
22.60
50.80
1.61E-03
22.90
23.20
24.00
101.60
3.22E-03
23.50
24.30
24.90
203.20
6.43E-03
24.90
25.50
26.50
406.40
1.29E-02
26.80
27.10
27.40
711.20
2.25E-02
27.80
27.50
28.00
1,320.80
4.18E-02
29.90
28.80
28.80
2,387.60
7.56E-02
33.10
32.20
32.20
20.39
1,443.95
20.48
1,211.49
20.66
1,021.71
21.02
815.25
21.74
743.98
22.82
836.24
24.98
1,112.05
28.76
1,216.06
0.53
3.83
0.38
3.26
0.18
2.33
0.15
2.16
0.19
2.37
0.32
3.02
0.37
3.27
Tin(c)
Tout(c)
Weight(g)
Time(sec)
19.6
34.21
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
0.27
2.81
Heating One Side Surface
1
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
2
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
W(m)
0.00575
0.00575
0.00575
2.73685
H(m)
Dh(m)
L(m)
Re
Gr
Ra
Pr
589.91
4.50E+03
3.17E+04
7.04
25.40
1.06E-03
20.34
20.34
20.34
50.80
2.13E-03
20.54
20.54
20.64
101.60
4.25E-03
20.64
20.84
20.74
19.61
19.63
1,384.22
19.65
1,144.01
1.25
11.45
0.89
10.15
0.65
1,783.86
E703
2.67
15.67
203.20
8.51E-03
21.04
21.14
21.04
21
117.6
406.40
1.70E-02
21.54
21.59
21.34
711.20
2.98E-02
21.94
21.84
21.69
1,320.80
5.53E-02
22.44
22.04
21.64
2,387.60
9.99E-02
22.84
22.34
21.94
19.70
19.81
922.879
742.72
19.96
686.48
20.28
720.84
20.82
825.90
0.58
8.94
0.62
9.10
0.76
9.63
13,2E
1.71
12.97
9.21
145
2
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
0.00575
0.00575
0.00575
2.73685
590.52
8.68E+03
6.11E+04
7.04
Re
Gr
Ra
Pr
Tin(c)
Tout(c)
Weight(g)
Time(sec)
19.6
22.3
120.2
34.93
1,320.80
5.52E-02
24.33
23.63
22.83
2,387.60
9.98E-02
25.33
24.43
23.63
1
25.40
1.06E-03
20.93
21.13
20.93
50.80
2.12E-03
21.43
21.43
21.53
101.60
4.25E-03
21.73
22.03
21.83
203.20
8.50E-03
22.43
22.43
22.33
406.40
1.70E-02
23.33
23.13
22.63
711.20
2.97E-02
23.63
23.23
23.03
Tms
Tmw
hx
19.63
1,720.83
19.65
1,374.97
19.70
19.80
946.25
20.00
799.36
20.30
829.07
20.90
900.55
21.96
975.79
1.30
7.89
0.88
6.69
0.61
5.81
0.48
5.33
0.38
4.92
0.39
5.00
0.45
5.20
0.50
5.42
Tin(c)
Tout(c)
Weight(g)
Time(sec)
19.5
24
153.2
3
2
. ........
Nu(x)
W(Nux)
Error(%)
3
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
1
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
4
2
1
Tms
Tmw
hx
Nu x)
W(Nux)
Error(%)
593.11
Gr
1.44E+04
1.02E+05
7.06
Ra
Pr
22.00
22.40
22.70
203.20
8.44E-03
24.30
24.30
23.90
406.40
1.69E-02
25.40
24.80
23.90
711.20
2.95E-02
25.60
25.00
24.65
1,320.80
5.48E-02
26.80
25.50
24.60
19.54
1,834.81
19.58
1,486.98
19.67
1,101.11
19.83
940.43
20.17
898.44
20.67
943.70
21.67
1,049.42
0.87
4.96
0.60
4.22
0.36
3.47
0.29
3.18
0.27
0.29
3.18
50.80
2.11 E -03
44.22
2,387.60
9.91 E -02
28.50
27.00
25.90
3
23.43
1,118.56
.....................................................
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Re
101.60
4.22E-03
23.00
23.60
23.30
25.40
1.05E-03
21.50
21.90
21.80
2
0.00575
0.00575
0.00575
2.73685
0.00575
0.00575
0.00575
2.73685
3.11
591.70
2.82E+04
2.00E+05
7.08
Re
Gr
Ra
Pr
24.44
24.94
25.14
101.60
4.22E-03
26.64
26.74
26.04
203.20
8.44E-03
28.64
27.64
26.84
406.40
1.69E-02
28.84
27.64
26.94
711.20
2.95E-02
29.54
28.44
27.94
19.48
1,990.31
19.57
1,533.25
19.73
1,191.19
20.06
1,069.94
20.72
1,162.66
1,19247
0.51
0.32
2.18
0.21
0.18
1.72
0.20
0.21
1.81
1.84
25.40
1.05E-03
23.54
23.74
23.64
2.68
50.80
2.11 E -03
1.84
21.71
0.34
3.37
0.37
3.50
Tin(c)
Tout(c)
Weight(g)
Time(sec)
19.4
28.3
153.2
44.22
1,320.80
5.48E-02
31.64
29.94
27.84
2,387.60
9.91E-02
34.84
32.34
30.74
23.70
1,310.16
27.16
1,510.38
0.24
1.96
0.31
2.16
146
5
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.40
0.00
21.31
21.31
21.01
2
1
Tms
Tmw
hx
Nu x
W(Nux)
Error(%)
6
3
2
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
1
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
1.41E+04
1.00E+05
7.10
Ra
Pr
406.40
0.01
0.01
711.20
0.02
1,320.80
0.04
2,387.60
0.07
21.91
22.51
22.41
23.11
23.51
23.31
24.61
24.51
23.71
25.01
24.51
24.21
26.01
24.71
23.81
27.11
25.61
24.71
19.79
894.99
24.56
20.16
937.17
2483
19.55
1,070.27
20.89
1,047.86
25,78
22.18
1,14572
0.32
3.73
0.34
3.79
0.39
3.95
0.66
6.05
Tin(c)
Tout(c)
Weight(g)
Time(sec)
21.6
23.2
83.5
35.87
.
1.25
6.05
0.87
0.60
4.52
a(m)
b(m)
Dh(m)
25.40
1.58E-03
22.37
22.27
22.27
22 29
21.61
1,465.72
13.922.24
16.08
0.00575
0.00575
0.00575
2.73685
50.80
3.16E-03
22.47
22.47
22.47
22.44.
21.63
1,221.25
11.59
1.60
13.83
19.3
22.6
159
33.8
203.20
19.42
1,385.68
5.21
Tin(c)
Tout(c)
Weight(g)
Time(sec)
101.60
0.00
1.72:
Lth(m)
2
21.91
Gr
19.36
1,749.17
a(m)
b(m)
Dh(m)
L(mm)
50.80
0.00
21.61
(1.79)
801.51
Re
19.33
2,160.85
Lth(m)
L(mm)
0.00575
0.00575
0.00575
2.73685
0.41
3.99
Re
419.51
Gr
4.21E+03
2.81E+04
6.67
Ra
Pr
101.60
6.32E-03
22.57
22.67
22.57
22
21.66
1,052.07
203.20
1.26E-02
22.87
22.87
22.87
406.40
2.53E-02
23.27
23.27
22.97
711.20
4.42E-02
23.47
23.37
23.27
21.72
859.60
21.84
715.84
1,320.80
2,387.60
8.21 E -02
1.48E-01
23.97
23.57
23.27
24.47
24.17
23.87
22.02
728.59
22.37
805.95
23.00
835.67
579
6<91
TO4
0.65
9.56
0.67
9.66
0.78
10.27
416.40
8.83E+03
5.90E+04
6.69
7
Tin(c)
Tout(c)
Weight(g)
Time(sec)
406.40
2.54E-02
24.89
24.59
24.09
711.20
4.44E-02
25.09
24.79
24.49
1,320.80
8.25E-02
25.99
25.29
24.69
21.75
880.12
22.00
825.42
22.38
876.84
23.14
969.15
0.43
5.13
0.39
4.92
0.43
5.12
0.50
5.49
,
999
1.23
12.33
0.00575
0.00575
0.00575
2.73685
0.87
10.70
Re
Gr
Ra
Pr
25.40
1.59E-03
22.79
22.89
22.89
50.80
3.17E-03
23.29
23.29
23.39
101.60
6.35E-03
23.49
23.79
23.59
21.53
1,563.42
21.56
1,221.65
21.63
1,025.37
1.19
8.03
0.76
6.54
203.20
1.27E-02
24.19
24.19
23.99
0.83
10.51
21.5
24.9
79.8
34.45
2,387.60
1.49E-01
27.39
26.59
25.89
:Ulan
,.
..
.
24.47
980.63
...
0.56
5.72
0.51
5.53
a(m)
Re
arC 00575
l
0.00575
D
0.00575
Lth(m) 2.73685
hx
L(mm)
25.40 040:.
50.80
101.60 fr;66640
1,271.65
1,063.44
801.10
729.19
669.13
690.54
.
939.80 00606:
1,016.00
1,092.20
23.55 &Alai:
1,168.40
:23.6
1,244.60 00
23
23.68
1,320.80
1,397.00
MTilt:
22 D
23.68 f
009
2,006.60
2,082.80
2,159 00 0
2,235.20
2,311.40
2,387.60
;
23.78
3.88 22i
8
23.98
24.08
6 .94
669.61
738.20
713.90
738.31
792.52
764.58
792.65
822.87
23.78
23.78
23.78
23.88
alui
23.98
24.08
669.23
690.64
669.32
690.74
713.57
690.84
669.52
,,
-A
855.47
8 .01
9 .10
8 .94
855.79
8 .30
929.47
89128
Gr
9°°E+83
Ra 6.05E
+04
6.72
Pr
°0575
0.00575
D:,
Lth(m) 2.73685
)
78.5
34.7
smi
Error
406.57
Re
0.00575
b(m)
23.1
L(mm)
hx
1,419.20
50.80
23 ,3
1,280.01
101.60
203.20
304.50
406.40
23.5
1,016.02
.85
759.03
768.05
7 24
8 59
9
975.03
''F1?#24.6
482.60
558.80 °
gRcIf4.67
635.00
711.20 119416.452124.47
787.40 0 06010 24.47
8 63.60 9:T.104.37
939
Pg!*4.37 22
00646.4
1,016.00 00;424.57
1,092.20
"1424.
64r
1,168.40
1,244.60
1,320.80
64124.87
1,0
1,126.86
5
1,188.61
1,127.13
1,127.27
1.071.83
6bdiii?4.87
6.6200..:1.2255:0177
11
111 ,:g1277722.:950197
bail
1,549.40 [3 x9825.37
1,072.44
1,625.60
1,701.80
1,072.56
1
Z006.60
07Z89
567
1 '072.81
.67
84 1,190.57
1,072.93
1,854.20
yr
1'48.72
1.12963
uU
1142
Tin(c)
Tout(c)
W(g)
t(sec)
x:::w (W(Nu) tif.
25.40 Pottl,2,2 a
764.46
23.
1'473.20
1,549.40 0;0066 23.
1'625.60
1,701.80
1,778.00
1,854.20
Nu)
21.3
1,409.66
2°3.20
304.80 0
406.40 s."
482.60
558.80
635.00
711.20
787.40
863.60
Tin(c)
Tout(c)
W(g)
t(sec)
404.63
48 +03
Or 4.48E
-+83
R a 3.01E+04
6.72
Pr
1,129.17 1
1,129.31
1,191.33
1,191 .45
21.3
24.9
1Z
r
a(m)
0.00575
396.92
Re
b(m) 0.00575
Gr 1.49E+04
Dh(m) 0.00575
Lth(m) 2.73685
Ra 1.01E+05
6.76
Pr
L(mm
Tin(c)
Tout(c)
W(g)
t(sec)
hx
x
203.20 ....:f 10125.9
841.21
304.80 0t 1:8:26.23
::6.03
406.40 6.:...
827.97
916.85
1,625.60 0;;105357.33 2 78: 1,443.49
1,701.80 011iY2027.43
06:: 1,485.89
27.83
1,360.75
1,778.00
1,398.36 :
1,854.20 04-- 27.
1017.93
1,930.40
2,006.60
2,082.80 Q 1 48E :28.33
2,159.001'
2,235.20 DJt 8628.73
2,311.40 0114*: 8.
2,387.60
1
1 496.70 1
1,422.87
1 1,464.04
1,393.32
1,432.78
:. 1,474.54
1,584.31
Gr 2.95E+04
Ra 2.00E+05
6.78
Pr
TO*
hx
1,
.57 1
1,302.841
.
1,078.43:1
::::
1,041.17 ':::.
::
,....1: 1,289.45
1,550.09
482.60 0031:0727.76 2i>17 ;: 1,598.20
558.80 400128.16
1,578.41 .
635.00 6.44.61028.16
1,703.97 ::
1,573.31
,:::::.:,:,,,,,f
711.20 07418.
787.40 0400i429.26
863.60 0)0%429.1
:: 1,554.13 .
1,71 4.87
939.80 0:29.86512 1,582.60
:
1,464.72
1,342.98
1,379.60
1,316.62
1,459.19
x
L(mm)
25.40 ,.:.......126.76
:::: w::::.:,,,,:,,
::: 1,470.30
1 255 55
0: 1,334.26
1,321.13
1,320.80
1,397.00
1,473.20
1,549.40
Dh(m) 0.00575
Lth(m) 2.73685
Tin(c)
Tout(c)
W(g)
t(sec)
398.50
Re
406.40 0:9mA27.56
*25.83
.. 1,010.85 :.:.
.32 :
7 ; 1,1
558.80 0;0;85225.63
25. 3 22.54 1,475.92 .,!..:::'
635.00
': 1,404.07
15 33
711.20
15 33 2 85: 1 503.24 i:
787.40
': 1,486.58
863.60 ...1..........7:25.53
1,092.20 .:-......-516.33
26.
1,168.40
16.
1,244.60
0.00575
0.00575
101.60 00065428.26
203.20
18.06
304.80
482.60
939.80 0004005.73
1,016.00 03*5846.33
35.11
a(m)
b(m)
50.80 0.3*1.
50.80 gogog4.13::::.......:: 1,261.35
967.43
27.3
78.3
(Nu) poot.m)
: 1,326.22
101.60 0;5825.13
21.1
0.51
0.
0.
0.59:
0.58
0.61
0
0.
1,
1,016.00
...30.
, 1,
1,092.20 002031::'30.36
1,558.19
1,168.40 ..,
....,41
1,719.82
1,244.60
.86
.,
::::
1,586.81
31 56
1,320.80 4.I
31.662,7.::2$:: 1,674.63
1,397.00
1,691.03
1,473.20 :.::::...:.:.
1,596.26
1,549.40 00974
1,647.33
32.76
1,625.60
, 1,626.31
,
1,701.80 1...:.:.... 33.16 ..,,,,,::::.:::
.33.86 2t99 1,506.87
1,778.00 0111
1,657.52
1,854.20
;33.76
1,673.57
34.06
1,930.40
.::
1,861.47
2,006.60
1,747.28
2,082.80 0 0 Q 34.56
1,684.09
2,159.00 ., :30..35.
,:::.
1,590.08
2,235.20
1,758.75
2,311.40 ...41435.
1,867.29
2,387.60
.
0.59
0.54
0.57
i
.
%.
:.
21
33.3
71.4
31.81
Nu) Error(%
0.24
0.24
0.17
0.16
0.24
0.33
0.35
0.34
0.
0.34
0.33
0.
0.34
0.31
0.37
0.37
0.36
0.31
0.37
0.38
0.
0.41
0.38
a(m)
Re
0.00575
b(m) 0.00575
Dh(m) 0.00575
Lth(m) 2.73685
L(mm
Pr
50.80 OPPC..
101.60 o
203.20 4 ::::' 423.26
.66
1.092.20 0'..009)..
1,168.40 404022 .
1,244.60 4'0'324.1
24.26
2324.16
24.06
1,549.40
:24.16
1,625.60
1,701.80 0 54 24.26
1,778.00
4.36
1,854.20
4.26
1,930.40
24.36
2,006.60
2,082.80 g:00.1551 4.
:%:.124.66
2,159.00
2,235.20
:24.76
2,311.40
2,387.60 011
20.7
23.1
131.9
38.41
u* W(Nu) Ertir
hx
1,522.34
1,444.30
1,170.03
919.14
813.05
a(m)
b(m)
0.00575
0.00575
Re
Dh(m) 0.00575
Lth(m) 2.73685
L(mm)
25.40
50.80
Ra 1.01E+05
Pr
6.84
x
::
4.
hx
23.1
7
795.03
713.16
754.13
721.06
635.00 0.010:25.89
711.20 0.;04gz.
787.40 (00425.69
713.27 F
863.60 q.p.ge:t:25.39 2 :
939.80
x:25.3922
7
1,016.00 040025.59 2
1,092.20 0;00425.39
1,168.40 0;04845:25.49
.72
924.27
911.51
979.52
1,058.50
1,041.79
1,075.
1 058 74
1 042 02
.. 1 076 24
1,172.35
613.47
Gr 1.48E+04
203.20 0 p843::25.
304.80
772.02
763.09
810.19
939.80
1,016.00
1,397.00 43f3
Tin(c)
Tout(c)
W(g)
t(sec)
754.01
558.80 0.44424.
635.00 04004241
711.20 4Q4924.
787.40 ompo:24.
863.60 4040:24.36
1,320.80
6.84
x
................
304.80
406.40
482.60
605.19
Gr 8.56E+03
Ra 5.85E+04
1,244.60 00015E2 .19
894.05
897.04
1,115.62
1,154.3810 55
1,124.99
1 240 60
1 246 37
1,442.73
1,347.47
1,404.43
0.51
1,320.80 otvo .49
0.57
0.64
0.62
5.49
1,397.00 0 5T
1,473.20 0 0910 25.59 22:0 1,411.83
1,549.40 0 ;04:425:25.99 2313
1,625.60 0 06
1,701.80
:2
1,276.04
1,327.01
1,245.99
1,778.00 ......101313.26.39 2
1,854.20 007$89::26.39
1,930.40 930gg$5.99..
1,251.81
1,300.82
1,579.35
1,059.21
2,159.0011;08853:1':27.19
1,042.48
1,076.73
1,059.45
2,235.20 4444.927.
1,360.70
1,195.24
1,163.75
1,168.83
2,311.40 43;0.9gtg.27
1,29413
27.
1,346.58
1 151 89 :1
2,006.60 0:080Z.V26.49
1,132.141
2,082.80 4440726.99
2,387.60 4-
Tin(c)
Tout(c)
W(g)
t(sec)
20.7
24.8
121
34.76
eri*titi
a(m)
b(m)
Dh(m)
Lth(m)
101.60
203.20
304.80
406.40
482.60
558.80
635.00
711.20
787.40
863.60
939.80
1,092.20
1,168.40
1,244.60
1,320.80
1,397.00
1,473.20
1,549.40
1,625.60
1,701.80
1,778.00
1,854.20
1,930.40
2,006.60
2,082.80
2,159.00
2,235.20
2,311.40
2,387.60
Gr
Ra
Pr
Tsx
L(mm)
25.40
50.80
1 ,01 6.00
Re
0.00575
0.00575
0.00575
2.73685
:WO&
25.71
25.21
26.81
28.31
28.41
28.51
28.31
27.81
27.21
27.61
28.01
27.61
27.91
28.41
28.51
.01
28.81
.51
.71
.11
30.11
.21
.41
.81
81
31.31
91
31.21
31.41
31.61
31.81
31.81
598.66
2.91E+04
1.99E+05
6.86
hx
1,492.86
1,686.27
1,273.00
1,059.05
1,091.07
1,125.09
1,202.76
1,362.25
1,604.05
1,548.20
1,496.10
1,711.23
1,684.78
1,588.93
1,634.25
1,543.91
1,694.03
1,532.00
1,541.78
1,490.11
1,561.71
1,605.48
1,616.22
1,559.53
1,638.14
1,547.38
1,778.64
1,750.08
1,762.85
Tin(c)
Tout(c)
W(g)
t(sec)
20.6
28.9
164
48.16
(Nu)
0.33
0.41
0.25
0.19
0.
0.21
0.23
0.28
0.37
0.35
0.33
0.42
0.41
0.37
0.38
0.35
0.41
0.34
0.35
0.33
0.35
0.37
0.38
0.35
0.38
0.35 :
0.44
0.43
0.
1,775.81
0
1,788.96
1,893.17
0.
O.
119
151
Aspect Ratio 2
Heating Whole Surfaces
1
L(mm)
Z
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.0115
0.00766
2.73685
Gr
Ra
Pr
25.40
1.21E-03
50.80
2.42E-03
101.60
4.85E-03
Top
22j 6
2.2 tj
2440
4
22.20
21.90
22.00
22.30
22 30
22 10
22 10
22 60
22 30
22.26
21.29
712.33
:::9::)
0.95
10.49
22.70
22 40
22.50
22 70
3
2
1
Bottom
2110
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
22.10
21.24
809.80
2
L(mm)
Z
Top
4
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
3
:::4020M
1.22
11.87
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
2140
22.57
21.37
579.88
:::::::706:::::::::
203.20
9.70E-03
23 00
23 20
22 80
22 80
23 10
22,60
23.00
21.54
476.01
;
::::
0.63
8.64
:::
0.43
7.20
0.00575
0.0115
0.007666
2.73685
25.40
8.03E-04
22,14
22 08
21,58
21 78
22,18
21,95
21.95
20.65
926.83
50.80
1.61E-03
101.60
3.21E-03
203.20
6.42E-03
22.28
21.78
21.98
22.48
22:05i
22.15
20.70
830.54
22.78
22.18
22.38
22.78
23.58
22.68
22.98
23.38
22:06
23.10
20.99
572.34
::::ttlk::
0.93
7.92
a(m)
b(m)
Dh(m)
25.40
5.99E-04
21,47
21 77
21 07
21 17
21 67
21 54.
21.49
20.34
1,056.61
-::::i:m42.:.:::
1.21
9.03
::::40$*
0.75
7.15
Ra
Pr
22:2$
22.53
20.80
695.59
1:::::$.4
0.00575
0.0115
0.007666
2.73685
50.80
1.20E-03
21,,,e7,::
21 87
21 27
21 37
22 07
21 74
21.71
20.37
911.95
7.90
0.37
5.12
0.59
6.46
0.41
7.00
711.20
2.25E-02
2135
ao$
23.75
21.39
24.35
21.98
508.12
510.21
::::::::00::::::::
23.07
22.37
22.57
22.97
406.40
9.59E-03
23,64
23 57
23 07
23 27
23 47
2247
2154
22.77
20.60
560.58
23.32
20.89
220:
::
0.37
5.25
502.11
:.:.:
1,320.80
6.30E-02
2,387.60
1.14E-01
26 10
25 90
24 30
24 10
24 20
23,60
24.80
23.42
503.06
28j0.:
4.83
25;10::
26.41
25.21
578.54
:::::::Z26:
0.48
7.57
0.63
8.62
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
20.6
25.9
165.8
32.28
1,320.80
4.17E-02
2,387.60
7.55E-02
27 28
24 58
24 28
24 38
23 35
25.36
23.16
547.46
3)28 :
29.28
26,38
26.18
26.18
20$]
0.34
4.93
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
20.3
24.3
280.4
40.78
711.20
1.68E-02
24,74
24 57
23 47
23 47
23 77
22,74
23.87
21.34
481.37
1,320.80
3.12E-02
26 64
26 37
24 07
23 77
23 97
22 64
24.72
22.23
489.29
2,387.60
5.63E-02
0.29
4.69
0.29
4.74
:::i:::::00::::::::::::::::::::::::::0110:
0.30
4.63
0$w .43C..............
0.31
27.60
25.90
25.70
25.80
27.26
25.22
592.06
7:CZ
0.39
5.27
6.91
203.20
4.79E-03
21.2
25.8
114
33.45
27 66:'
25.38
23.88
23.78
23.98
800.00
1.45E+04
1.00E+05
Pr
,1*..A7
461.41
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
:::::::i5AW::::::::::::::::::1:,K.
24.18
23.38
23.58
23.88
0.30
4.65
Ra
00W1040.:
0.91
q:::::14$::::::
Gr
22 37
21 67
21 97
22 37
21,94
22.13
20.45
724.43
21.88
454.67
::::::: 16M
0.40
24::45
Re
101.60
2.40E-03
23.41
406.40
1.28E-02
:..
0.54
6.08
23,80
23.70
21 20
23,30
23 40
22,90
711.20
3.39E-02
24 70
24 50
23 50
23 50
23 70
23,10
23.90
22.40
601.99
1.48E+04
1.01E+05
6.86
Re
Gr
24*
406.40
1.94E-02
6.91
a(m)
b(m)
Dh(m)
Lth(m)
Lth(m)
L(mm)
Z
Top
4
405.38
8.99E+03
6.06E+04
6.74
Re
wor:
27.87
25.17
24.97
24.97
2:314:
25.99
23.79
553.59
........
0.36
5.20
152
4
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Z
Top
4
3
2
1
Bottom
Tms
Tmw
hx
Nu x
W(Nux)
Error(%)
5
25.40
6.15E-04
23-39
23 89
22 99
23 29
23 99
50.80
1.23E-03
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
6
L(mm)
Z
Top
4
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
3.01E+04
2.03E+05
6.74
Ra
Pr
23,19
24 29
23 39
23 69
24 39
2; 09.
25,f6
25 09
24 09
24 39
24 89
23.12
2342
2382
26,39
24 89
25.19
25,49
23,19
23.55
21.27
1,018.99
23.94
21.34
894.28
24.59
21.49
>:> 12:9
0.59
4.59
a(m)
b(m)
Dh(m)
22 30
22 80
3
799.21
Gr
203.20
4.92E-03
25.40
4.68E-04
Top
4
Re
101.60
2.46E-03
Lth(m)
L(mm)
0.00575
0.0115
0.007666
2.73685
.
.
.
411$:::::::
0.46
4.07
25.41
748.31
::0::r.
0.33
3.49
0.00575
0.0115
0.00766
2.73685
50.80
9.37E-04
21 30
21 50
22 60
22 16
22.20
19.96
1,129.58
22.50
20.02
1,020.38
0.68
4.73
0.56
4.33
21.78
639.20
::::''
0.25
3.06
:
Re
Ra
Pr
1,320.80
3.20E-02
32,42
32.99
27 49
26,89
26.79
24 82
28.70
24.96
621.26
.1A:
0.23
2.99
2,387.60
5.78E-02
24 52
27.15
23.23
591.59
.iiii1A C
..."
0.22
2.88
0.21
2.87
406.40
7.49E-03
711.20
1.31E-02
2wit
280$
23 80
22 10
22 40
23 30
26 10
25 00
24 80
24 90
28 00
25 10
25 00
25 50
23.10
20.14
855.07
24.37
20.38
634.47
25.18
20.87
586.33
26.07
21.59
564.59
1,320.80
2.44E-02
I1 ;15
31 10
26.20
25 60
25.70
23,56
27.50
23.04
566.84
0.20
2.73
0.20
2.74
2166
;
C
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
203.20
3.75E-03
24,36
25 10
23,70
24 30
24 70
101.60
1.87E-03
21.2
29
286.9
42.7
711.20
1.72E-02
29 32
29 49
26 19
25 89
26 29
1,012.55
2.89E+04
2.02E+05
6.99
Gr
agO
23.00
21.60
21.80
22.90
406.40
9.84E-03
21' $2
27 29
25 99
25 79
25 99
24 42
26.27
22.36
593.30
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
3.73
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.0115
0.007666
2.73685
25.40
4.54E-04
50.80
9.08E-04
101.60
1.82E-03
2453
2478
-208
24, 1 '''''':''
23,88
22,18
22 58
24 18
22;35
23.32
18.93
1,267.75
:::::::::4C18::::::
0.39
2.41
23 08
2348
25 08
22 08
24.20
19.06
1,082.60
:i:::]:13:8.(M
0.29
2.10
Re
Ra
Pr
27 08
24,48
25 18
25 98
23 16
25.72
19.32
869.26
.,
0.21
2.80
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
1,014.85
5.72E+04
4.11E+05
7.19
Gr
35.69
29.99
29.19
28.89
2582::
31.30
28.00
704.28
:
: :1*0:
0.29
3.31
19.9
26.4
238.4
27.13
2,387.60
4.40E-02
3330.
27 80
27 30
27 po
24;68
29.30
25.57
678.35
ItAti
'' ''
0.24
2.96
0.41
3;3:9:
203.20
3.63E-03
-2666
29 18
26 38
26 58
27 18
23 38
27.23
19.83
752.46
406.40
7.26E-03
32,38
30,98
27 78
27 58
27 76
24,54
28.67
20.86
713.14
711.20
1.27E-02
36,08
35 08
27 68
27 28
27 88
24,88
30.12
1,320.80
2.36E-02
38 78:
40 08
29 68
29 28
28 98
2$ 2IY
32.70
22.41
25.51
722.20
774.03
:::::::00$0
:::::::11k...
..alf
0.19
0.15
0.14
1.75
1.57
1.51
0.14
1.52
0.16
1.60
::: ..?.EKW:::::ii:i'ii:i:::::::S.:$V::::::::
0.26
3.11
18.8
32.7
253.4
28.03
153
Heating Three Surfaces
1
L(mm)
Z
4
3
2
1
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.0115
0.007666
2.73685
25.40
1.21E-03
21 19
20 99
21 09
21.59
50.80
2.42E-03
21 29
21 09
21 19
21 69
Bottom
21.4
21,4
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
21,1$
20.34
-.21. 26
866.82
20.37
777.23
1.38
12.56
1.12
11.32
fOt
2
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.40
7.98E-04
20.79
20.59
20.69
4
3
2
21...09
1
tow
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
Z
4
3:
2
1
Bottom
Tms
Tmw
hx
Nu x)
W(Nux)
Error(%)
101.60
4.84E-03
21.49
21.39
21.59
21.89
203.20
9.68E-03
21 89
21 89
22 09
22,29
2tz 0
...21 Ake
21AS
20.59
518.52
:04
;z:-.54
20.44
644.09
::::AfC:
0.78
9.49
.
406.40
1.94E-02
22 39
22 39
22 49
22 59
711.20
3.39E-02
23.09
22.69
22.69
22.79
2239
IVO
2329
20.88
466.89
21.31
508.76
22.18
584.02
ma
0.42
7.79
7.11
Re
Gr
Ra
Pr
101.60
3.19E-03
21 09
20 89
21,09
21.39
203.20
6.38E-03
21 49
21 29
21 49
21 79
21,0
21,29
ff.
:::$420::::::
0.51
22.fig
, ... ::::
0.49
7.66
595.94
8.23E+03
5.74E+04
6.97
406.40
1.28E-02
21.89
21 79
21 99
22 19
1,320.80
6.30E-02
24 19
23 39
23 29
23.39
..:..
20.3
24.2
139.5
40.99
2,387.60
1.14E-01
25,49
24.79
24.69
24.69
9
23.70
679.81
,',:: 8,8>.
,..:
0.64
8.67
0.85
9.98
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
20
22.6
221.8
42.99
1,320.80
4.15E-02
23.19
22 59
22 49
22 69
2,387.60
7.50E-02
24.29
23.49
23.39
23.49
711.20
2.23E-02
22.39
22.09
22.29
22.49
.20
20.05
889.28
20.10
724.95
20.19
576.39
20.39
470.95
20.68
467.72
21.25
538.00
22.27
593.26
1.79
1.48
13.06
1.00
10.88
0.66
8.96
0.46
7.65
0.45
0.58
8.48
0.69
9.17
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
20.9
25.2
209.3
40.75
1,320.80
4.18E-02
25.99
24.79
24.49
24.59
2,387.60
7.55E-02
27 29
26 19
25 79
25 89
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
6.91
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
20.02
982.23
14.31
3
Pr
Gr
0.00575
0.0115
0.00766
2.73685
50.80
1.60E-03
20.89
20 69
20.79
21.19
Ra
395.96
8.39E+03
5.80E+04
Re
25.40
8.04E-04
22.29
21.99
22.14
22.49
"":0
20.94
969.93
0.00575
0.0115
0.007666
2.73685
50.80
1.61E-03
22.29
22.19
22.29
22.89
869.29.......
Re
Gr
Ra
Pr
101.60
3.21E-03
22.59
22.59
22.79
23.09
203.20
6.43E-03
23.09
23.19
23.29
23.69
21.06
706.90
21.22
580.83
7.61
606.42
1.48E+04
1.00E+05
6.80
406.40
1.29E-02
23.79
23.79
23.89
23.99
711.20
2.25E-02
24.69
24.09
23.99
24.29
21.54
22.02
574.62
543.01
7:27:a
1.06
8.59
0.86
7.77
0.58
6.46
0.40
5.48
0.36
5.19
0.39
5.43
22.98
684.45
6:64g:
0.54
6.29
857.52
.:10 110
0.83
7.67
154
4
L(mm)
Z
4
3
2
1
Bottom
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.0115
0.007666
2.73685
25.40
8.08E-04
23.17
22 97
23 27
24 17
50.80
1.62E-03
23.47
23:37
23.57
24.57
013,21
2325-
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
5
3
2
1
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
L(mm)
2
1
101.60
3.23E-03
24 27
24 17
24 57
25 07
2.01
20.96
887.67
21.12
729.03
0.55
4.39
0.44
3.95
a(m)
b(m)
Dh(m)
""i#0
50.80
1.19E-03
21.44
21.24
21.44
21.94
321
101.60
2.38E-03
21.84
21.64
21 84
22,24
22.90
20.76
1,122.09
20.82
1,013.67
0.70
4.93
0.58
4.50
1,320.80
4.20E-02
30.37
27 67
27.07
27 17
2,387.60
7.60E-02
33.07
30.87
30.17
29.97
201
52f
2 ki21
23.06
759.97
27.53
25.00
932.25
406.40
1.29E-02
26.67
25.97
25.67
25.87
711.20
2.26E-02
28 07
26 17
25 87
26 27
25;72
22.09
650.44
N,It
Mil
0.25
3.00
3054:
28.39
1,097.51
:9.5.9,,.....
:1::::.1:
:::::::'::41.0:
0.33
3.44
0.48
4.13
0.66
4.80
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
20.1
23.6
247.3
35.78
1,320.80
3.10E-02
25 04
23 94
23 64
23 74
2,387.60
5.60E-02
26.04
24.84
24.54
24.44
800.27
1.50E+04
1.04E+05
6.95
711.20
1.67E-02
23 64
23 34
23 44
23 64
406.40
9.53E-03
22 94
22 84
23.14
23 24
*;72
444
122.0
7...21,42.
20.36
624.10
20.62
554.13
554.08
21.79
639.38
0.45
5.64
0.36
5.18
0.36
5.18
0.46
5.74
0.74
7.08
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
20.7
27.3
0.73
6.99
50.80
1.21E-03
23.36
22.86
23.06
23.96
20.8
29.5
208.5
40.85
203.20
4.77E-03
22 34
22 24
22 44
22 94
1.05
8.30
25.40
6.03E-04
23.16
22.46
22.86
23.66
0.24
2.99
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
601.14
2.94E+04
2.01E+05
6.82
::::::0..tM.:::::::::::::::02b
Pr
1.26
Lth(m)
2x;10
21.45
645.82
Ra
20.16
1,000.55
0.00575
0.0115
0.007666
2.73685
*77
Gr
20.13
1,101.26
a(m)
b(m)
Dh(m)
.,
Re
21,
20.23
818.90
9.04
203.20
6.47E-03
25 37
25.27
25 37
25.77
0.31
3.31
0.00575
0.0115
0.007666
2.73685
Bottom
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
Pr
20.88
996.04
01,1
6
Ra
2 F 36
25.40
5.96E-04
21 24
21 14
21 34
21 74
21.32
4
Gr
23Z2
Lth(m)
L(mm)
jni
Re
Re
Gr
Ra
Pr
203.20
4.82E-03
24 86
24 56
24,86
25 06
aft'
2.74
21.01
803.86
2.96E+04
2.03E+05
6.84
2 &1
MO
711.20
1.69E-02
27 26
25 86
25 76
26 06
24.8
20.95
828.97
21.19
689.62
21.68
639.97
22.42
690.63
0.40
3.79
0.29
3.27
0.25
3.09
0.29
3.27
101.60
2.41E-03
23 96
23 56
23 96
24 46
asp
2252
406.40
9.65E-03
25.86
25.46
25.46
25.56
23.15
831.31
251.1
36.7
1,320.80
3.13E-02
29.36
26.76
26.46
26.36
2,387.60
5.67E-02
30.86
29.46
28.36
27.66
23.89
846.56
28.70:
26.46
1,070.31
::16.6
.
0.41
3.85
0.63
4.72
155
7
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Z
4
3
2
1
Bottom
0.00575
0.0115
0.007666
2.73685
801.11
Gr
5.87E+04
4.03E+05
6.87
Ra
Pr
25.40
50.80
101.60
6.02E-04
1.20E-03
2.41E-03
23 49
24 29
24.49
24 29
25 09
26.29
24.49
25 39
26.89
26 39
27 19
27.79
25,28
26,a9
24,51
25 43
25:34::
20.62
20.75
20.99
1,156.88
1,034.22
905.94
':.*06:.1..
Miti:::::: ..1...
0.37
0.30
0.23
2.53
2.29
2.04
.
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
Re
203.20
4.81E-03
25.29
27.39
26.99
28.09
.. ........
.
zen
21.49
893.19
:411:::
0.23
2.02
406.40
9.63E-03
29.39
27.59
28.39
28.99
.
.
.
_.
27::aa.
22.47
894.43
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
250.1
36.5
711.20
1.68E-02
31.49
28.69
27.89
28.19
1,320.80
3.13E-02
32.09
30.69
29.19
28.29
Aii
0.19
2351:i
23.96
1,063.58
216
6
26.92
1,704.55
32.10
4,625.00
:::::::::.....
0.23
2.02
20.5
33.8
::::
::::2: ]....
0.31
0.77
2.34
3.61
2,387.60
5.66E-02
32:99
33.49
32.89
33.89
::::::
:::::
.:5::
:
..
5.48
9.58
Heating One Side Surface
1
L(mm)
4
3
2
1
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
2
L(mm)
4
3
2
1
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.0115
0.007666
2.73685
25.4
8.03e-04
21 44
21 24
21 34
21 44
21,32
20.22
1174.82
50.8
1.61e-03
14
1.57
10.49
IA::
595.409
Gr
1.52E4-04
Ra
1.057E+05
6.93
Pr
406.4
1.28e-02
22.94
22.54
21 64
21 44
21 44
21 54
101.6
3.21e-03
21.94
21.84
21.84
21.74
203.2
6.42e-03
22.34
22,24
22,14
21 94
2149
ZNOO
2.2.23
20.24
1032.16
20.27
814.79
20.34
685.62
22.34
22.14
22:$7
20.48
621.48
1.25
0.85
9.56
8.21
0.65
7.47
0.56
7.13
I
a(m)
b(m)
Dh(m)
Lth(m)
Re
W(m)
711.2
2.25e-02
23.44
22.64
22.34
22.34
474
20.69
634.45
1320.8
4.10e-02
24 14
22 84
22 54
22 14
23,04
21.12
673.45
20.2
22.1
223.1
43.49
2387.6
7.55e-02
24.54
23.44
22.94
22.64
23 53;
21.86
774.36
g.Ao
0.00575
0.0115
0.007666
2.73685
H(m)
Dh(m)
L(m)
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
Re
Ra
600.3877
2.83E+04
1.982E+05
Pr
7.0051
Gr
0.58
7.20
0.63
7.41
0.78
7.97
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
23.4
237.8
45.53
1320.8
4.10e-02
2387.6
7.40e-02
19.8
25.4
7.88e-04
50.8
1.58e-03
101.6
3.15e-03
203.2
6.30e-03
1.26e-02
22 41
21 81
22.01
22 11
22.61
22.21
22.21
22.41
23.41
22.91
22.81
22.61
24,21
23 61
23 31
23.01
24.91
24.01
23.51
23.01
25 61
23 91
23 51
23 21
26.91
24.51
23.91
23.31
27.91
25.51
24.71
24.11
23M
23,06
20.07
691.58
24:98
20.33
667.07
ziti 7
2.+00
21.54
26:84:
22.94
861.72
22,00
201
19.83
1153.19
19.87
1022.63
1.:1407
0.79
5.40
1301
0.64
4.96
19.93
809.52
406.4
711.2
2.21e-02
20.74
727.47
847.
0.44
4.28
0.35
3.94
0.33
3.87
0.37
4.04
743.01
X41
0.38
4.08
40:8:
0.48
4.44
156
3
L(mm)
a(m)
b(m)
Dh(m)
Lth(m)
25.4
1.20e-03
21 86
21 66
21 66
21 76
4
3
2
1
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
4
3
2
1
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
5
4
3
2
1
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
Gr
Ra
Pr
101.6
4.80e-03
22.36
22.16
22.16
22.06
2'240
20.50
905.71
724.61
1.05
8.63
0.95
8.24
0.64
6.96
a(m)
b(m)
Dh(m)
20.45
0.00575
0.0115
0.007666
2.73685
406.4
1.92e-02
23.36
22.86
22.56
22.26
2250
2217
20.60
628.55
20.80
595.27
711.2
3.36e-02
23 76
22 86
22 66
22 56
23,ot)
21.10
648.80
0.50
0.46
6.10
0.53
6.45
6.31
Re
Gr
Ra
Pr
101.6
4.86e-03
21.54
21 49
21 44
21 44
21,40
20.65
705.32
203.2
9.71e-03
21.84
21 74
21,64
21 64
21,71
20.70
567.64
2.29
2.05
18.41
17.51
1.28
14.28
0.89
12.37
2t2T:
20.61
981.11
a(m)
b(m)
Dh(m)
25.4
5.95e-04
22.02
21.72
21.82
21 92
21 86
19.93
1323.90
0.00575
0.0115
0.00766
2.73685
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
400.426
1.48E+04
1.019E+05
6.89
203.2
9.60e-03
22.86
22.56
22.36
22.26
50.8
2.43e-03
21 34
21 24
21 24
21 34
21.26
20.62
922.58
Lth(m)
L(mm)
Re
20.43
958.54
25.4
1.21e-03
21.34
21.14
21.24
21.24
4
50.8
2 40e-03
21,96
21 76
21.76
21 86
21,81
Lth(m)
L(mm)
0.00575
0.0115
0.0076667
2.73685
Re
Gr
Ra
Pr
398.066
7.18E+03
4.920E+04
6.855
406.4
1.94e-02
22 14
22 04
21 84
21 74
22 iv.
20.79
486.05
0.70
11.31
711.2
3.40e-02
22 54
22 04
21 94
21 84
22.12..
20.94
494.56
0.72
11.42
797.23
2.84E+04
1.983E+05
6.98
2136
0.61
6.81
0.77
7.54
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
20.6
21.9
170.5
50.2
2387.6
1320.8
6.31e-02
22 94
22 24
22 04
21,84
1.14e-01
21.23
528.69
23 34
22 74
22 44
22 24
22.78:
21.73
560.90
0.79
11.85
0.87
12.28
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
19.9
22.6
135.4
19.57
1320.8
3.09e-02
26 52
24 32
2387.6
5.59e-02
27.12
25.02
24.32
23.72
.
22,05
22 67
20.00
929.76
23 31
nag
20.10
773.40
20.30
678.38
705.51
23 12
24 60
21.20
730.72
0.62
5.28
0.48
4.85
0.40
4.62
0.42
4.68
0.44
4.74
19.95
1182.09
25.36
24 16
23 66
22.76
807.89
203.2
4.76e-03
23 72
23 32
23 02
22 82
24 12
20.60
2387.6
1.13e-01
702.21
101.6
2.38e-03
23 02
22 52
22 62
22 52
711.2
1.67e-02
25 32
23 92
23 52
23 42
23.1
156.9
45.7
1320.8
6.24e-02
24 66
23 26
22 86
22 56
23 46
21.70
50.8
1.19e-03
22 32
21 92
22 02
22 12
406.4
9.52e-03
24 52
24 02
23 52
23 02
20.4
2648:
22.26
820.70
[6:03
1.09
6.50
0.91
6.04
0.52
4.98
157
6
L(mm)
4
3
2
1
Tms
Tmw
hx
Nu(x)
W(Nux)
Error(%)
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.0115
0.007666
2.73685
25.4
5.91e-04
24.33
23.53
23.73
23.83
50.8
1.18e-03
24.73
24.13
24.13
24.33
446
206
101.6
2.36e-03
26 13
25 43
25 23
24 73
203.2
4.73e-03
27 33
26 53
25 63
25 03
4 46
19.95
1305.25
20.00
1168.04
26 62
20.10
922.39
0.53
3.22
0.45
3.00
Re
Gr
Ra
Pr
0.31
2.64
802.77
5.72E+04
3.994E+05
6.986
406.4
9.45e-03
28A3
26/3
20.30
819.62
25 93
25.13
26-66
20.70
811.76
0.26
2.50
0.26
2.49
711.2
1.65e-02
29.43
26.63
25.93
25.83
Tin(C)
Tout(C)
Weight(g)
Time(sec.)
19.9
25.3
260
37.32
1320.8
3.07e-02
31 53
27 53
2387.6
5.55e-02
32.43
28.83
27.53
26.43
2E12*
2533
2800
27:07
21.30
866.86
22.51
24.61
909.90
1079.90
0.28
2.56
0.30
2.62
0.39
2.86
Heating Bottom Surface
Gr 4.27E+04
Ra 2.95E+05
6.89
Pr
x WOO
L4rilm gggiggie
25.40p
Tin(c)
Tout(c)
W(g)
t(sec)
400.62
Re
a(m) 0.00575
0.0115
b(m)
Dh(m) 0.007666
Lth(m) 2.73685
Nu) fro$041
hx
20320
iiiiiii 24.76 ......i
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25.17
304.80
24.66
40640 ' s00t
482.60
558.80
24.56 :
993.87 *$.
0.54
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0.38
0.34
0.43
0.48
24.6
996.26
0.
50.80
101.60 .
23.76
1 1,070.71
23.964 1044
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i:
711.20 .:1:::1.
24
787A01;::::......
25.1
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794.37 ::
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930.291
863.60 :1 NW: 25.16
939.80 Q 0025.36
1,016.00 0 EJ479925.
25.2
1,09220
0.51
1,077.26 :....
2
1,244.60 ...M. 25.36 :..........
1,048.26 ;...
1,116.85 :::::.
1320.80 Agc 25.56 :
25. 1 085 72 1.
1397.00 iii060650925.66
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1,473.20 .12925.86 #73t 1,058.97
1,030.93G2
1,549.40 191 2
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1,1
1,701.80
1,778.00
1,85420
326.
1,930.40
2,006.60
26.86
2,082.8V.
2,159.00 13 313 0. 26.
26.76
2,235.20
2,311.40;(9(926.86
2,387.60
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0.49
0.47
0.49
0.47
0.47
0.54
0.52
0.57
0.55
0.55
0.53
0.51
0.53
0.56
1,069.89 13
1,041.29
1,04351
054
1,07824
0.54
0.50
0.64
0.58
0.58
0.58
1,019.15
1,192.83
1 120 93 .
1 123 97
1,127.03 :
Re
a(m)
0.00575
0.0115
b(m)
Dh(m) 0.007666
Lth(m) 2.73685
L(mm)
25.40
50.80
809.46
Gr 148E+05
Ra 1.02E+06
Pr
6.86
Tsx
.
Tin(c)
Tout(c)
W(g)
t(sec)
hx
....
2,327.79
2 310 59
1 654.81 20 9
101.60
203.20
304.80
406.40
482.60
'
1 694.92
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Lth(m) 2.73685
L(mm )
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L(mm)
25.40
50.80
Tin(c)
Tout(c)
W(g)
t(sec)
609.94
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0.82
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304.80
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870.26 it'
1,397.00
1,473.20
1,549.40
1,625.60
0.69
0.68
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Ra 2.93E+05
8.29
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Tout(c)
W(g)
t(sec)
793.88
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726.82
674.77
2.74
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Tout(c)
W(g)
t(sec)
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25.27
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2,387.60
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Tin(c)
Tout(c)
W(g)
t(sec)
20.05
22.7
235.9
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0.95
0.89
0.66
0.50
0.45
0.47
0.54
0.59
0.73
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44,5496
0.72
0.69
0.76
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4.91
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480
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5.18
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45:9200
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b(m)
0.00575
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Ra 6.10E+05
Dh(m) 0.007666
Lth(m) 2.73685
6.89
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L(mm)
25.40
1,795.16
1,578.91
1,294.69
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37.95
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2.99
0.71
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0.81
2.27
3,73
342
310
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3,52
1.71
6.27
4,66
2.14
513
1.73
4.130
1.11
1.76
1.23
0.83
0.87
1.08
0.95
1.84
4.87
2.57
3.14
0.00
2.23
4.59
8.60
340
472
4,10
3.58.
3.61
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372
4.112
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$
6.09
'ERR
6.23
726
979
166
Aspect Ratio: 3
Heating Whole Surfaces
L(mm)
a(m)
0.00575
Re
395.71
b(m)
0.01725
Gr
1.54E+04
Dh(m)
Lth(m)
0.008625
Ra
1.089E+05
2.73685
Pr
7.06
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
19.5
25.2
192.5
41.64
25.4
50.8
101.6
203.2
406.4
711.2
1320.8
2387.6
1.05e-03
2.11e-03
4.22e-03
8.43e-03
1.69e-02
2.95e-02
5.48e-02
9.91e-02
21 54
21 62
21 02
21 02
21 32
21,52
21,14
22,32
23 34
23 22
22 02
22.12
22.22
22 42
24 64
24 32
22 72
22 52
22 52
2922::
22 72
26,44
26 42
23 92
23 22
23 22
23 52
21.64
21,84
22,54
Tap
2084
21..04
Side 5
21 12
21 32
4
3
20 42
20 42
20 62
20 72
2
20 62
20 82
1
21 07
21 22
60110111
20,74
20 94
Tm(side)
20.55
Tms
Tmw
hx
20.61
20.86
21.23
21.74
22.35
22.99
24.17
19.55
19.61
19.71
19.92
20.35
20.98
22.25
24.47
827.24
695.97
577.06
481.05
437.22
436.63
457.06
549.58
22 32
21 32
21 62
21 82
22 12
21,54
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28 42
25 62
25 12
25 02
25 02
24,14
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26.07
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W(Nux)
Error(%)
2
L(mm)
1.14
0.81
9.62
8.15
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0.56
6.84
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8
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0.40
0.33
0.33
0.36
0.51
5.79
5.31
5.31
5.53
6.54
a(m)
0.00575
Re
492.88
b(m)
Dh(m)
Lth(m)
0.01725
Gr
1.51E+04
0.008625
Ra
1.074E+05
2.73685
Pr
7.137
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
19.1
23.7
242.4
41.7
25.4
50.8
101.6
203.2
406.4
711.2
1320.8
2387.6
8.37e-04
1.67e-03
3.35e-03
6.70e-03
1.34e-02
2.34e-02
4.35e-02
7.87e-02
Top
20,64
20 54
21,04
21,71
22,54
23,74
25.44
28,01:
Side 5
20 51
20 71
21 11
21 71
22 41
23 41
25 11
2711.
4
19 91
20 11
20 41
20.71
21 31
22 01
22 91
24 51
3
20 01
20 11
20 51
21,01
21 61
21 91
22 51
23 81
2
20 21
20 31
2071
21 31
21 81
22 01
22 61
23,81
1
20 61
20 71
21.01
21 61
21.91
22 11
22 71
23 91
2044
2054
2074
21 14
21,24
21 44
21 94
23 04
Bottom
Trr(side)
Tms
Tmw
hx
W(Nux)
Error(%)
41
20.19
.
... .
r
.:
19.14
19.19
849.75
764.02
...
Ititi
::
20.35
WV:::::
20.68
.
:
::
,90::::
21.16
:::::
1 , 00
.
:::
,14::::::
:
4,0
22.30
23.25
24.83
20.30
442.69
21.32
460.25
518.77
19.27
19.44
19.78
630.87
517.73
455.79
::::1 1Q.....
:4.:::::
21.73
7..42 .::::::::
652
:t
23.11
1.20
0.97
0.67
0.46
0.36
0.35
0.37
0.46
9.82
8.88
7.44
6.24
5.59
5.46
5.64
6.25
167
3
a(m)
b(m)
0.00575
Re
502.40
0.01725
Gr
2.93E+04
Dh(m)
0.008625
2.73685
Ra
2.057E+05
Pr
7.0242
Lth(m)
L(mm)
25.4
50.8
101.6
8.34e-04
1.67e-03
3.34e-03
203.2
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
406.4
711.2
1320.8
19.7
28.1
222.5
38.09
2387.6
6.68e703...,,..134e:02.......2.34e-02........4.34e702.......7.s4e7p.
Top
21.82
22 22
212
2439
28,02
28,22
3092
3509i
Side 5
22.09
22 59
33.79
21 69
24,79
26 49
29.19
3
21 79
25 69
23 89
23 89
30 39
21 39
21 39
24 29
22 89
27 49
4
24 09
25 09
27 69:
2
21 69
22 04
23 19
22 29
22 39
22 79
23 79
23 99
24 99
27 49
1
22 29
22.64
23.69
23 89
24 29
25 19
27 49
21 72
21,92
23 19
22.22
22.82
22,52
22,92
23,62
26,02
BOtt001
23 29
23 39
.:.
Tm(side)
;::
Tms
Tmw
hx
.. ..
21,507 . ...
21.62
.. .
21 90
21.98
.
19.78
19.86
882.79
766.39
.
1263 ::::::::::::::::::::::::w4c
W(Nux)
Error(%)
0.70
5.56
4
L(mm)
Top
Side 5
4
3
2
1
Boom
Tm(side)
Tms
Tmw
hx
. ..
22,50?:::........ ... 21.15.. .......... . .24,19.
24.21
22.60
23.38
..
0.53
4.87
:::-.:.(:41.4-M:
..i.Z ,,.
:::::::::::::920:
0.39
4.07
3.51
3.32
3.40
3.67
4.21
596.21
Gr
0.008625
Ra
Lth(m)
2.73685
Pr
2.77E+04
1.968E+05
7.0994
25.4
50.8
6.96e-04
1.39e-03
...... ...
21 63
21 60
20 80
20 80
21 10
21 80
21 43
21.73
22 00
21 00
2263
21 10
21 40
21 70
101.6
2.78e-03
.....
22.70
21 60
2200
22 00
22 40
21.63
21 83
...... ..21........... .21,15..................... .21.92.
21.11
21.41
21.98
.
19.56
920.78......................810:96..........661,24
5.28
'...]:::;;;::;,,,.:,.-:.-:,::::::::.:
0.29
Re
5.94
27.03
654.25
0.25
0.01725
0.61
23.75
560.35
0.24
0.00575
0.78
21.88
0.27
0.42
4.39
19.3
26.2
236.9
33.85
203.2
406.4
711.2
1320.8
1.11e-02
.......
1.95e-02
3.62e-02
23,70
23 60
25,33
24.90
23 10
23 30
27,13
26 40
23 90
29,73
29 30
25 20
23 50
24 20
27 20
26 10
23 40
23 50
23 50
24 20
26 1 0-,
2370
24 50
26 00
22 23
22 53
23,03
.4.1.7................. . .
....479..........
22 20
22 60
22 80
23 20
22.33
.
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
5.57e-03
.........
.4,.
.
2.
4.....................
.
.33,60 :
31 50:
24.93
....
.
27.5:
27.85
23.56
24.30
19.81
20.32
21.09
22.63
25.32
545.10...........496.42........... 499.89
529.54
633.12
709
1:::!iiiiiiiiiiii:;::::::::::i::::::::::::::::E1I
25.66
2387.6
6.54e-02
22.76
13.3Min1.1..04;::::::::::::$;:i:::::::947 :?.:T:iii:::::::::::::....: ..::::
W(Nux)
Error(%)
26.66
0.37
b(m)
Dh(m)
19.43
.2925:
29.52
.
513.24
...6.e4.::::::::::::::::::::::::.
::::]:::::41::::::::::::::::::
. zoo... .............:.. .20,51. ................... .
25.06
20.01
20.95
20.32
629.70
533.07
500.09
,..._,,,,,_:::::::::::::....,:::::::::::::::::
a(m)
19.36
.................
.:::fig-::::;::.::--:"::::.".:'
0.29
0.24
0.25
0.27
0.38
3.72
3.45
3.47
3.63
4.23
168
5
L(mm)
a(m)
b(m)
0.00575
Re
797.35
0.01725
Gr
5.74E+04
Dh(m)
0.008625
Ra
Lth(m)
2.73685
Pr
4.063E+05
7.0806
25.4
5.22e-04
50.8
1.04e-03
....................................
Tin(c)
Tout(c)
Weight(g)
Time(sec.)
19.4
30
279
29.88
101.6
203.2
406.4
711.2
1320.8
2387.6
2.09e -03
4.17e-03
8.35e -03
1.46e702
2.71 e -02
4.90e -02
Top
23,04
2344
25,14
27 17
29,74
33,14
37 34
43 47
Side 5
23 57
24 17
25 27
26 87
31 97
36 27
39,37
4
22 57
24 67
25,17
29.27
31 77
25 97
26 77
29 77
2
22 67
22 77
23 17
23 77
23 77
27 27
3
22 07
22 17
24 27
25.27
28 87
26 17
25 97
25 67
25 77
26 67
29 37
1
23 87
24 17
24 87
25 87
25 87
26 17
27 07
29 07.
22 74
23,04
2334
23,74
23,64
23.54
24,54
22.58
23.09
24.17
25.36
26.54
27.65
29.66
32.79
19.50
19.60
19.79
20.19
20.97
22.15
1067.69
941.81
751.65
636.24
590.97
597.90
24.52
639.16
28.65
793.52
BOttOrTi
Trn(stde)
.....241.5...,
Tms
Tmw
hx
A
W(Nux)
Error(%)
:,,z,a
...
265. . ....
0.52
0.41
0.27
:ii0 30.
0.20
0.18
0.18
0.20
0.29
3.41
3.05
2.52
2.20
2.08
2.10
2.21
2.63
.
Heating Three Surfaces
1
L(mm)
a(m)
0.00575
Re
397.0
b(m)
Dh(m)
0.01725
Gr
1.54E+04
0.008625
Ra
1.091E+05
Lth(m)
2.73685
Pr
7.0806
Tin(c)
Tout(c)
Weight(g)
Time(sec)
19.4
24.4
195.4
42.03
25.4
50.8
101.6
203.2
406.4
711.2
1320.8
2387.6
1.05e-03
2.10e-03
4.19e-03
8.38e-03
1.68e-02
2.93e-02
5.45e-02
9.85e-02
Side 5
2052
24 32
26 12
23,42
25 02
22 22
22 92
24 52
2
20 52
2062
22 22
22 92
24 42
1
20,92
21 02
21 02
21,42
21 42
21 62
22 02
21 62
21.92
22.02
22 32
3
20 92
20 62
20 82
22 92
20 12
20 32
20,62
20 32
20 42
21 42
4
21 92.
22.12
22 42
23 12
24 52
2014
20 34
21374
21,44
21,44
21.74
122.34
23.74
20.34
20.53
20.86
19.45
19.49
19.59
21.35
19.77
987.42
850.00
693.42
559.27
Bat=
21 02
Tm(side)
Tms
Tmw
.:.:.:.21.$72::.:
,:::::::?w.
...
W(Nux)
Error(%)
A.-
:::::%9::::::::::
.:
-::::::U. O.::::
.:.:.:.:22.::
:.:.:.:.:.:.:.:.:.:.:.:.,Z.6.
.:.:::.:.:.:.::::: : :24,99.
21.83
22.34
23.28
20.14
522.08
20.70
21.81
23.76
536.37
603.02
791.81
...:::46::
:7.:88.:i:
:::
,
24.88
.4::
1.61
1.20
0.81
0.53
0.47
0.49
0.61
1.03
11.37
9.85
8.12
6.66
6.26
6.41
7.13
9.20
169
2
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Side 5
4
3
2
1
Bottom
Tm(side)
Tms
Tmw
hx
W(Nux)
Error(%)
3
25.4
7.23e-04
21 27
21 17
21 07
21 37
21 77
21 WI
21 22
21.27
20.53
1036.48
AW
2.05
13.88
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
6.92e-04
Side 5
21 31
21.01
21.01
21.41
22.01
21014
4
3
2
1
Bottom
Tm(side)
.2t ..1 45
Tms
Tmw
hx
21.18
19.65
1033.07
::14IC
W(Nux)
Error(%)
1.00
6.78
4
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
'Side 5
4
3
2
1
Bottom
Tm(side)
Tms
Tmw
hx
W(Nux)
Error(%)
0.00575
0.01725
0.008625
2.73685
50.8
1.45e-03
21.37
21.37
21.27
21.47
21.87
21
2t 4E,
21.42
20.56
894.78
:: .
II . li.....ti::::::::-
1.55
12.10
Ra
Pr
2i-q2.0.
21.65
20.61
742.06
::400k.
203.2
5.78e-03
21.97
21.77
21,97
22 17
22,47
406.4
1.16e-02
22 37
22 17
22 47
22 57
22 87
711.2
2.02e-02
22 97
22 57
22 77
22 77
23 07
22i8
ZIA:
# 2,1t
2Z.W.
4,$$.
-.47.
21.98
20.72
612.44
22.37
20.95
539.48
22.72
21.28
535.90
::0:1::
10.21
Re
Gr
Ra
Pr
0.60
7.77
101.6
2.77e-03
203.2
5.54e-03
406.4
1.11e-02
22 11
21 81
22 01
21 66
22 36
22 76
22,14
22 71
22 31
22 81
23 01
23 51
23 61
23.21
23.51
23 51
23.71
22 44
23.425
23.34
20.48
21*
Z..4,16
21.46
19.71
898.81
;:1287
0.77
5.97
22
22.01
2142
2056
214
20 69
20 79
21 09
21 69
2Z0.0
22.64
20.04
605.94
....
19.82
718.70
it.i.,t280:::::: :::8;67-m::::
0.50
0.37
4.89
4.23
0.00575
0.01725
0.008625
2.73685
50.8
1.04e-03
20 89
22. 4
Gr
Ra
Pr
210
0.31
3.91
406.4
8.32e-03
22.89
22.39
22.89
23.09
23.29
22.69
19.78
561.82
21378
20.88
19.19
21.39
19.27
965.11
771.01
22:01
22.02
19.44
633.98
711.2
1.94e-02
0.87
0.58
0.41
6.31
5.21
4.47
23.87.
23.97
24.15
23.12
744.92
1.08
10.25
19.6
25.5
226
32.11
1320.8
3.60e-02
2387.6
6.51e-02
24 91
24 11
23 61
23 61
23 91
26 91
25 31
24 21
24 31
24 41
28.81
27.21
26.01
25.71
25.81
22.*
23,04
23.87
21.13
576.65
24.91
22.45
639.50
26.65
24.75
828.93
0.33
4.06
0.40
4.42
0.65
5.54
al
Tin(c)
Tout(c)
Weight(g)
Time(sec)
711.2
1.46e-02
23.89
23.29
23.29
23.09
23.39
.2004
0.33
4.09
1320.8
2.70e-02
25 89
24 39
23 59
23.49
23 79
19.1
23.7
206.3
22.04
2387.6
4.89e-02
27 49
25.59
24 59
24 49
24 59.
22*
23,4;
24.27
25445:
24.11
25.24
21.32
585.72
767.54
Zt44:::::::::
S.:1
i1:;10.:89::
0.32
4.04
0.35
4.22
75
:00C
1.13
7.12
2387.6
6.80e-02
24.97
24.27
23.97
::::::..:::7;ff7.:::::::::
203.2
4.16e-03
22.09
21.69
22.09
22.39
22.89
..
20845
0.63
7.96
Tin(c)
Tout(c)
Weight(g)
Time(sec)
793.65
2.77E+04
1.977E+05
7.137
Re
101.6
2.08e-03
21 39
21 29
21 29
21 59
22 09
551.21
20.5
23.5
197.9
29.28
:.::::Z.., . :..
0.59
7.73
603.90
2.81E+04
1.976E+05
7.043
50.8
1.38e-03
21 51
21 26
21 41
22 21
1320.8
3.76e-02
23 97
23 47
23 17
23 07
23 27
22$3.
2.44
23.33
21.95
555.24
..:7.10.E::::::
0.76
8.64
1.08
0.00575
0.01725
0.008625
2.73685
25.4
5.20e-04
20 79
20 39
20 49
20 79
21 39
20.62
19.14
1106.45
Gr
101.6
2.89e-03
21 57
21 57
21 57
21 77
22 07
1 ea
Tin(c)
Tout(c)
Weight(g)
Time(sec)
592.66
1.49E+04
1.027E+05
6.87
Re
23.26
20.30
552.08
23.11
0.57
5.20
170
5
a(m)
b(m)
Dh(m)
0.00575
0.01725
0.008625
2.73685
Lth(m)
L(mm)
25.4
5.13e-04
22 03
21 63
21 73
22 23
23.48
2244.
Side 5
4
3
2
1
Bottom
Tm(side)
Tms
Tmw
hx
50.8
Ra
Pr
203.2
4.10e-03
24 63
24 13
24 93
24 93
25 63
262f
21.,876
228$
22,4
2151
24 6
21.92
18.89
1117.88
22.44
18.97
977.60
23.43
19.15
22 43
22 03
22 33
22 73
23 93
404.k::
W(Nux)
Error(%)
Gr
101.6
2.05e-03
23 43
23 13
23,43
24,03
24 63
1.03e -03
Z::::
0.56
3.50
2345
406.4
8.20e-03
26.33
25 83
25.53
25.43
25.73
MA
25 768
25.55
20.20
631.84.
24.50
19.50
676.65
7.Pl.P.P..
000..a:
:::::::::ft 04::%
Tin(c)
Tout(c)
Weight(g)
Time(sec)
798.56
5.57E+04
4.007E+05
7.1934
Re
0.44
0.30
0.22
3.11
2.61
2.31
711.2
1.44e-02
28.63
26.63
25.53
25.43
2:93
255
:::::::1::00::::::::::::::i:::::::::::RS:::.
0.20
2.20
0.22
2.31
1320.8
2.67e-02
31 73
28 53
26 43
26 13
26 63
2387.6
4.82e-02
33.93
30.43
29.03
28.43
2441
2Ch
28
27.65
23.34
785.45
26.25
21.24
676.56
.
18.8
28.2
324
34.16
::1illiC
28.4.3.
3615:
29.88
27.00
1174.49
:::40.52::
0.29
2.59
0.60
3.66
Tin(c)
Tout(c)
Weight(g)
Time(sec)
21.2
156
33.53
Heating Side Surface
1
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Side 5
4
3
2
1
Tms
Tmw
hx
NOW
W(Nux)
Error(%)
Side 5
4
3
2
1
Tms
Tmw
hx
W(Nux)
Error(%)
Gr
Ra
Pr
25.4
1.05e-03
20 37
19 82
19,92
20,12
20 32
20.03
19.12
950.69
50.8
2.09e-03
20 02
20 02
20 22
20 32
20.15
19.14
856.17
101.6
4.18e-03
20.52
20.32
20.27
20.42
20.47
20.38
19.18
720.12
1.63
11.98
1.35
10.99
0.99
9.60
2042
394.49
1.47E+04
1.047E+05
7.137
Re
203.2
8.37e-03
21 02
20 52
20 62
20 72
20 72
20.69
19.26
603.60
:AW
2
L(mm)
0.00575
0.01725
0.008625
2.73685
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.01725
0.008625
2.73685
25.4
7.00e-04
20.72
20.32
20.42
20.62
20.62
20.48
19.71
1133.24
::::::402::::::::
2.42
14.95
50.8
1.40e-03
20.72
20.52
20.52
20.62
20.72
20.60
19.73
988.02
44,
0.73
8.46
Re
Gr
Ra
Pr
101.6
2.80e-03
21.02
20.92
20.82
20.92
20.92
20.90
19.75
752.19
::4017::
1.91
1.21
13.50
11.27
406.4
1.67e-02
21.42
20.92
20.92
20.92
20.82
21.02
19.41
538.26
'4.:t
0.61
7.85
598.85
1.55E+04
1.090E+05
7.0242
711.2
2.93e-02
21 92
21 32
21 02
20 92
20 82
21.29
19.65
528.04
19.1
570.78
2387.6
9.83e-02
23.42
22.32
21.62
21.62
21.42
22.20
20.93
682.67
"':......t$if
::::m
::::::.A:74:
0.59
7.75
0.66
8.15
1320.8
5.44e-02
22 72
21 62
21 22
21 12
21 12
21.63
20.11
Tin(c)
Tout(c)
Weight(g)
Time(sec)
0.90
9.22
19.7
21.1
233.6
33.55
406.4
711.2
1320.8
2387.6
203.2
3.64e-02
1.96e-02
6.58e-02
5.60e-03
1.12e-02
23.32
22 92
21 32
21 82
22,22
21 42
21 82
22.12
22.42
21 02
21,52
21,72
21.92
21 12
21 52
21.82
21 52
21 52
21.62
21 22
21,52
21 52
21.62
21 22
21 32
21.75
22.00
22.32
21.15
21.50
20.92
19.91
20.06
20.38
19.80
641.53
542.39
512.12
531.62
617.41
::::::04:3ft.
:IgAM: ::-.7:::::. :::::::769::::::::::::::::::::::::.8P:::
0.74
0.68
0.72
0.89
0.95
10.12
10.32
9.52
9.30
9.44
171
3
L(mm)
Side 5
4
3
2
1
Tm(side)
Tms
Tmw
hx
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.01725
0.008625
2.73685
25.4
6.99e-04
21 30
21 00
21 00
21 10
21 20
50.8
1.40e-03
21 40
21 20
21 20
21 30
21 30
101.6
2.80e-03
22,00
21.80
21 70
21.70
21 60
203.2
5.59e-03
22 50
22 30
22 30
22 00
21 90
406.4
1.12e-02
23 30
23.00
22 60
22.30
22.10
711.2
1.96e-02
24.10
23.20
22.60
22.40
22.20
1320.8
3.64e-02
25 10
23 70
22 70
22 40
22 20
2387.6
6.57e-02
25.80
24.40
23.20
22.90
22.50
21.07
19.62
21.30
19.65
972.09
21.79
19.70
768.44
22.28
19.79
646.67
22.80
19.99
23.04
20.28
580.84
23.42
20.85
625.92
24.02
21.87
746.19
6.16
0.52
5.59
Gr
Ra
Pr
1110.51
S9D ......................................................
1.26
1.00
0.68
W(Nux)
Error(%)
7.93
4
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
6.85e-04
22 50
22.20
22.20
22.40
22.40
Side 5
4
3
2
1
TM(side)
Tms
Tmw
hx
2228.
22.29
19.45
1129.54
118
W(Nux)
Error(%)
0.66
4.05
5
a(m)
b(m)
Dh(m)
Lth(m)
7.19
0.00575
0.01725
0.008625
2.73685
50.8
1.37e-03
22 80
22.70
22.70
22 70
22.70
22,72
22.72
19.49
995.52
142
0.53
3.70
Tin(c)
Tout(c)
Weight(g)
Time(sec)
598.13
2.86E+04
2.012E+05
7.043
Re
570.61
19.6
22.2
223.7
32.09
.... .......................................................................................................
0.43
5.26
0.44
5.30
606.86
5.60E+04
3.967E+05
7.0806
Re
Cr
Ra
Pr
0.49
5.50
0.64
6.05
Tin(c)
Tout(c)
Weight(g)
Time(sec)
19.4
24.5
246.6
34.7
1320.8
3.56e-02
29 10
26 60
24 70
24 30
23 80
26
26.00
21.86
2387.6
6.44e-02
30 40
27 90
2580.
25 10
24.40
101.6
2.74e-03
23.90
23.80
23.60
23.40
23.00
203.2
5.48e-03
25,00
24,70
24 10
23,50
23 20
406.4
1.10e-02
26.30
25.00
24.30
23.80
23.30
711.2
1.92e-02
27 40
25 50
24 40
23 90
23 70
23676
24352
244:78$
25 22
23.68
19.59
785.70
1126
0.36
3.17
24.35
19.78
702.09
24.80
20.16
692.20
25.22
20.73
714.37
1005
990
1020
1104
0.30
2.97
0.29
2.95
0.31
0.35
3.15
0.49
3.60
Tin(c)
Tout(c)
Weight(g)
Time(sec)
19.4
23.3
288.8
31.12
0.00575
0.01725
0.008625
2.73685
3.00
792.47
5.60E+04
3.962E+05
7.0806
Re
Cr
Ra
Pr
775.81
272
27.20
23.85
958.24
:
L(mm)
25.4
5.25e-04
50.8
1.05e-03
Side 5
101.6
2.10e-03
203.2
4.20e-03
406.4
8.40e-03
711.2
1.47e-02
1320.8
2.73e-02
2387.6
4.93e-02
22 11
21 91
21 81
22 01
22 11
22 41
23 51
23 31
23 11
23 11
22 81
24 31
24 11
23.81
23 31
23 11
25 61
25 01
24 11
23 71
23 31
26.71
25.11
24.21
23.81
23 51
28 31
25 91
24 41
23 91
23 61
29 51
26 91
25 01
24 51
23 81
21 9
22,27
22.27
2485
:255
24.85
26,83:
26.33
22.80
908.92
4
3
2
1
Tm(side)
Tms
Tmw
hx
Nux
W(Nux)
Error(%)
22 26
22 31
22 31
22 41
232
2187
19.47
1146.11
23.20
19.54
877.21
23.87
19.69
767.00
24 57
24.57
19.98
698.42
722.72
25.50
21.28
760.19
1864
642
1256
1096
999
1033
1084
1291
0.90
4.82
0.73
4.42
0.47
3.78
0.39
3.54
0.34
3.40
0.36
3.45
0.38
3.52
0.50
3.85
21.90
19.44
1301.39
20.41
Aspect Ratio: 3
Gr 5.74E+04
Ra 4.06E+05
Pr
ii iiiii.l111111111
11111111111
25.40
50.80
101.60
203.20
304.80
23.
406.40 01
482.60 10001263.
558.80
23.
23.74
635.00 0.O
879.01
24 04
939.80 ;;59
*:24 04
Ot24. 4
24.
1,016601..
1,092.20 044.61:304.24
1,168.4010 44004.
1,244.60 iii:004:*
1,320.80 giNgggpa.
1,397.00 ;04
1473.20
1,549.40
*:
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7 24.64 2G 55
24.74 211]3
4. 64 2100<
.:1,
445
4+42
904.20 I2 8
949.981
0.49
0.54
0.58
0.59
0.55
0.58
0.58
0.55
0.57
0.54
0.00
0.57
0.55
0.58
0.56
0.55
0.57
0.56
0.60
0.59
0.55
0.63
0.60
0.59
0.57
0.60
0.60
0.64
000.63 004:
ass-
867.66y
775.601i
24
....
0.51
3z 839.98
....::
19.4
22.1
174.9
38.18
W(Nu)
1111111
IIIIiq1iN
0.68
411
0.63
897.05
845.19
885.05
879.18
850.77
.
.
W(g)
t(sec)
7.08
1.46
1.77
793.94
775.26
8 .34
711.20
787.40
863.60
Tin(c)
Tout(c)
391.18
Re
0.00575
a(m)
b(m) 0.01725
Dh(m) 0.008625
Lth(m) 2.73685
.54
845.49
885.39 :on
856.59
851.08
867.97
862.32
936.171
904.032;
897.90 f..
868.29
910.42 t
481
459
a(m)
b(m)
Gr 8.56E+04
Ra 6.06E+05
Dh(m) 0.008625
Lth(m) 2.73685
Pr
25.40 0,00103 24.47
50.80 0.00206! 24.77
101.60 0.0041Z 25.47 1
203.20 040824 24.57 1
304.80 061236 24.87
:24.87 1
24.87
4.62
406.40
482.60
453
558.80 i;
4,611
635.00
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fity.425.07
787.40
25.27
25.57
Asp
484
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452
863939.6080
ERR
016.00
4.60
453
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4+63
4.55
4,54
456
4.e4
4.63
453
4+70
4 64
4,63
457
465
4.52
25.77
.;
Tin(c)
Tout(c)
W(g)
t(sec)
:NO.k:1W(Nu) qftpre4'
0.46
328
0.43
322
0.37
3,10
0.48
&32
0.46
973.86
323
0.48
.69 1
3.31
1,
0.49
1,025.461
3.34
5.9914: : 0.46
329
1629.16
049
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971 48 1
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1,013.36 :
974.80
872.24
0.48
0.45
0.48
0.46
0.00
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1,
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297.00 6,
23.3
175.4
37.08
hx
0.51
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19.4
974.21
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827.63
1,004.83
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1,066.93
1,003.31
1,026.11
1,027.951..
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1,076.99
1,079.02',
7
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1,014.
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1,133.18 .
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6.37
06. 7
1,085.18 :
630.40
87 2 26 1,063.65
1,021.25
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1,091.40
7
Z159.00
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4.64
4'72
7.08
Tsx gow
Z
L(mm)
403.94
Re
0.00575
0.01725
0.47
0.52
0.47
0.49
0.49
0.48
0.52
0.52
0.48
0.56
0.53
445
322
II/
3,33
3 zs
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IV
33,
3.40
3.31
3.34
3.34
332
3A1
141
333
3.48
3.42
0.51
1177 0.000 11:09065913
2,311.40
2,387.60
07 :
1,149.14
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a(m)
0.00575
b(m)
0.01725
Dh(m) 0.008625
Lth(m) 2.73685
Re
406.54
Gr 4.41E+04
Ra 3.15E+05
Pr
7.15
25.4e ;t)f1Q121.74
21.94
101.60 r22.
203.20
304.80
406.40
482.60
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980.481
918.19
766.90 I
700.38
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558.80 31=0
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787.40
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1,244.60
434
1,320.80
1,397.00
1,473.20 ::::04$479
1,549.40
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1,625.60
23.74
1,701.80
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1,778.00
1,854.20
23.94
1,930.40
2,006.60
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24.14
2,082.80
2,159.00
24.14
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2,311.40
2,387.60
19.05
21.1
164.7
34.31
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L(rtwn)
223520:c
Tin(c)
Tout(c)
W(g)
t(sec)
24.14
24.04
23.94
786.59
776.57
789.95
779.84
769.99
760.38
730.05
702.04
6.46
744.86
779.73
769.88
760.2810
750.91
741.77
776.35 1,
766.59
735.761
769.78
7 .18 it
3.00
741.67
753.87
766.48 1
803.471E
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0.87
0.79
0.63
0.56
0.57
0.63
0.65
0.64
0.65
0.64
0.63
0.62
0.59
0.00
0.64
0.60
0.64
0.63
0.62
0.61
0.60
0.64
0.63
0.59
0.63
0.62
0.63
0.60
0.61
0.62
0.66
0.71
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3.70
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469
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6.16
$ 07
a(m)
b(m)
Re
0.00575
0.01725
Tin(c)
Tout(c)
407.02
Gr 1.45E+05
Dh(m) 0.008625
Lth(m) 2.73685
Ra 1.03E+06
Pr
W(g)
t(sec)
7.14
'(x4
L(mm)
Z ,': Tsx :MAW hx
1,118.45 1
25.40 D.00101 26.81
1,084.70 I
50.80 00020327.11
1,178.03 :.
10150 000406 26.61
1 237 50
203.20 0.00011 26.51
1 210 98
30450
26.91
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406.40 .
482.60 4
26.41 2043.: 1,395.93 i
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1 A59.65 2
635.00 420425.81
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711.20
26.11
787.40
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2621
1,712.73
863.60
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939.80 0.03751 26.91
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1,092.20
1,441.20
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1,168.40
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1,244.60
1,320.80
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1,397.00
2,319.61
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1,
1,47320 ....*.f05664:27.31
2,366.55
1549.40 :;0;06184:26.51 22
1,931
1,625.60
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1,778.00
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1,854.20
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1,930.40
27.61
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2,159.00
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2,23520
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19.1
25.8
164.7
34.31
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0.33
2,04
0.31
201
0.35
0.38
0.37
0.49
0.45
0.45
0.59
0.57
0.40
0.62
0.53
208
0.47
0.54
0.54
0.74
1.03
0.71
1.07
0.75
0.65
0.69
0.69
0.98
0.98
0.85
0.85
0.46
0.43
1.04
2.13
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233
2.26
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2040
2A6
218
254
248
210
241
2.41
220
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2.88
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2.75
2.89
208
2.80
3.07
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2.80
200
2.28
2.24
414
0.00575
a(m)
b(m) 0.01725
ph(m) 0.008625
Lth(m) 2.73685
L(mm)
25.40
50.80
101.60
203.20
304.80
406.40
482.60
558.80
635.00
711.20
787.40
863.60
939.80
1,016.00
1,092.20
1,168.40
1,244.60
1,320.80
1,397.00
1,473.20
1,549.40
1,625.60
1,701.80
1,778.00
1,854.20
1,930.40
2,006.60
2,082.80
2,159.00
2,235.20
2,311.40
2,387.60
Tin(c)
Tout(c)
W(g)
t(sec)
397.37
Re
Gr 2.83E+05
Ra 2.02E+06
Pr
7.14
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28.13
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1..19
Tin(c)
Tout(c)
W(g)
t(sec)
W(Nu) :rnbt(%)
hx
X ':<)'#n*
25.40
a;.cfga:
50.80
1455
0.44
0.68
0.51
599.84
Re
Dh(m) 0.0086
us
115
1St
0.37
0.57
0.00575
0.01725
126
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0.31
35.13
35.93 2
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180.9
38.6
1,15
1.54
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1,958.49
1,868.33
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2,552.61
1,230.39
2,570.76
2,134.53
1,484.12
2,454.46
1,751.94
a(m)
b(m)
19.1
Nu) Em
hx
1,245.21
Z 118.19
32.5
905.56 a
890.84
851.40
916.15
874.49
942.94
926.99
WI
0.85
0.83
0.79
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0.56
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26.77 978 1,189.27
101.60 fr:iPtIr
19.7
:.....; 2,662.37 :
....:;28.70 -....
249
382
1,397.00 iiP.;:M.T.,C2550 Opyg 5581.12 i...............
.10 :,
221
227
2,62
2,20
211
2,865.59 406
...
2,08280 Ot 1 AO
4,426.641
;
2,382.84 33;70;,
: 130233 L
3A29.67 ....
1.29
2,67727 :
0.87
113650 44;
1.12
179
Aspect Ratio: 4
Heating Whole Surfaces
1
L(mm)
a(m)
0.00575
Re
318.67
b(m)
0.023
Gr
1.44E+04
Dh(m)
0.0092
Ra
1.004E+05
Lth(m)
2.73685
Pr
6.9490
25.4
1.25e-03
Top
21,23
21 34
Side 5
4
2
20 84
20 84
20 94
1
21 34
3
21.0
Bottom
50.8
2.49e-03
101.6
203.2
406.4
711.2
1320.8
2387.6
4.99e-03
9.97e-03
1.99e-02
3.49e-02
6.48e-02
1.17e-01
21.43
21 64
21 04
20 94
21 04
21 54
21,73
21 84
21 14
21 24
21 44
21 84
22,23
22 24
21 54
21 54
21.74
22.14
2303
2543
27,73,
25 44
24 04
27.44
22,04
22 04
22 14
22 44
23,83
23 54
22 64
22.54
22 54
22 64
23 44
23 34
23 44
25 14
24 84
21 53
21:379...
21 63
22,13
21.43
21.63
21,72
21.76
22.25
22.75
20.28
20.46
20.83
21.37
22,93
23,8
23.88
22.46
521.02
460.42
421.38
434.03
423.51
22.94
25.84:
24 84
2423
0.98
0.71
0.56
0.47
0.50
0.48
0.61
11.98
10.50
8.97
7.99
7.36
7.57
7.40
8.34
22593
25,824
25.61
24.37
481.85
iZT:
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
34.51
1.29
20.15
2
25
158.3
704.66
20.99
NOM
W(Nux)
Error(%)
20.1
21 23
21,123
21.16
20.19
614.15
TM(Side)
Tms
Tmw
hx
Tin(c)
Tout(c)
Weight(g)
Time(sec)
0.00575
Re
394.51
0.023
Gr
1.47E+04
0.0092
2.73685
Ra
1.046E+05
Pr
7.1182
25.4
50.8
101.6
9.83e-04
....... ................
1.97e-03
.............
3.93e-03
.....
406.4
203.2
7.87e-03
......
.
.......
.
..
Tin(c)
Tout(c)
Weight(g)
Time(sec)
711.2
1.57e-02
2.75e-02
.. .. .. ........ ....
..
....
.
.
.
19.2
23.5
243
41.88
1320.8
2387.6
5.11e-02
...............
9.24e-02
Top
20.21
20 41
20,71
21,31
24,51
26 et
20 41
20 61
20 81
21 41
22 21
22 01
23 11
Side 5
23 01
24 41
26 31
4
19 96
20 11
20 31
20 61
21.11
21 71
22 81
24.51.
3
19 91
20 01
20 31
20 61
21.11
21 61
22 41
23.71
2
20 11
20.21
20 51
20 91
21 31
21 61
22 31
23.41
1
20 51
20 71
20 91
21 31
21 51
21.81
22 41
23 51
Bottom
2021
20 41
20,61
20,91
21 01
2121
21.91
22.80;
Tm(side)
Tms
20,1
20.12
20.24
20.53
21.36
Tmw
hx
19.24
19.28
19.84
20.32
21.28
22.95
752.36
19.36
565.97
20.94
19.52
466.32
435.47
416.04
421.27
457.32
U(xy::
W(Nux)
Error(%)
21.,6....
21.83.....
21.91
..
2270
22.85
....
24 3:
24.40
.:::::::,,,,
044:,::
#:W.
1.33
1.13
0.77
0.53
0.46
0.43
0.43
0.51
11.59
10.67
8.86
7.43
6.99
6.71
6.79
7.30
44...:
:::::64
043::
180
3
a(m)
0.00575
Re
398.111
b(m)
0.023
Gr
2.85E+04
0.0092
Ra
2.042E+05
2.73685
Pr
7.1558
Dh(m)
Lth(m)
L(mm)
Tin(c)
Tout(c)
Weight(g)
Time(sec)
19
27.4
246.2
41.85
25.4
50.8
101.6
203.2
406.4
711.2
1320.8
2387.6
9.69e -04
1.94e-03
3.88e-03
7.75e-03
1.55e-02
2.71e-02
5.04e-02
9.11e-02
3422
23 93
21 13
2
20 63
21 53
2223
26 13
24 73
24 13
1
21 53
22 23
24 13
26 23
20,82
21 22
21,42
22.63
21.52
25.12
24 63
22.83
22 73
22 63
22 63
30 02
29 53
20 43
23.52
23 23
21 83
21 73
27,22
26 63
3
21.37
21 83
20 68
20 63
20 83
21 73
22,22
22 43
4
21.12
21 53
20 43
21 62
22.12
22,92
25.74
Top
Side 5
Bottom
Tm(side)
Tms
Tmw
hx
21 13
21.03
21.56
22.23
22.99
23.97
19.08
19.16
19.31
19.62
20.25
21.18
23.05
772.90
702.22
583.36
503.68
479.01
470.68
488.72
0.71
0.59
0.41
0.31
0.28
0.27
0.29
0.38
6.03
5.50
4.63
4.05
3.87
3.81
3.94
4.50
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
.
0.00575
Re
400.64
0.023
Gr
5.56E+04
0.0092
Ra
3.992E+05
2.73685
Pr
7.17
.
Tin(c)
Tout(c)
Weight(g)
Time(sec)
25.4
50.8
101.6
203.2
406.4
711.2
9.60e-04
1.92e-03
3.84e-03
7.68e-03
1.54e-02
2.69e-02
485
25,65
4
23,05
23 38
21 78
23 98
22.18
25 08
23 08
31 35
29 98
26.28
3
21 68
22 08
2
22 18
24 38
1
23 98
22,55
22,78
24 18
22 98
23 88
27.05
27 18
24 38
24 58
$5.15
33,68
28 08
26 18
25 18
25.28
23.55
Top
Side 5
Bottom
Tm(side)
Tms
Tmw
27 53
26 53
20.78
2216
.
.
4
22.93
23 13
25 12
28-4
28.65
26.33
564.86
2096
It
W(Nux)
Error(%)
23 23
32 93
29.33
1320.8
4.99e-02
18.9
35.3
243.9
41.1
2387.6
9.03e-02
:
25 28
::..,...
40 35
4 55i
38 88
44 98
32 18
29 18
38 68
27 78
32 78
27 68
32 58.
30.75
34 88
24,38
24 58
24.48
24 68
2285
22.75
22,75
22 85
22 8
23.63
23.74
24a
25.$5
26.18
27,00.
27.81
25,65
30 84
31.27
20.12
539.88
21.34
534.04
23.16
26.81
33.21
556.14
580.46
667.27
36 $ti
22 275
22.38
22.91
19.05
19.20
19.51
777.56
698.26
610.93
0.37
0.30
0.23
0.18
0.18
0.19
0.21
0.27
3.11
2.82
2.50
2.24
2.22
2.30
2.39
2.70
24.91
37.09
43:
W(Nux)
Error(%)
181
5
a(m)
b(m)
0.00575
Re
497.20
0.023
Gr
2.78E+04
Dh(m)
0.0092
Ra
2.014E+05
Lth(m)
2.73685
Pr
7.24
L(mm)
Tin(c)
Tout(c)
Weight(g)
Time(sec)
50.8
101.6
203.2
406.4
711.2
1320.8
2387.6
1.53e703
3.06e-03
6.13e-03
1.23e-02
2.14e-02
3.98e-02
7.4e702
21 21
21.,01
21 81
23,01
2451
2841
21 12
21 12
22 02
23,82
25 62
4
19 82
20 52
3
19 72
21 92
21 92
22 92
22 62
2911
28 12
24 72
23.52
2
20 02
20 02
20 02
20 12
20 62
22 62
21 02
21,02
21.52
2322
21.02
21 32
2091
21 77
22 22
22 22
22 22
22 22
1
22 42
2121
21.31
21 -31
21.61
23.32
2Z21
Bottom
20.61
... .
20 I . .....
6
L(mm)
Top
.
20 42
....... .. ....... ... 444. ....... .. 21 44 .. ......... .. 2.2.2. ...... ............. ... 4... ....... ...2420
.
20.26
20.43
20.97
21.60
22.40
23.20
24.64
18.56
18.63
18.75
19.00
19.51
20.27
21.78
791.10
74426
4242 :
W(Nux)
Error(%)
6.09
5.76
0.44
4.77
0.27
0.26
0.27
4.15
3.80
3.75
3.84
0.00575
Re
502.76
b(m)
Dh(m)
Lth(m)
0.023
0.0092
Gr
5.61E+04
Ra
3.973E+05
2.73685
Pr
7.08
23..71
.. .
Tin(c)
Tout(c)
Weight(g)
Time(sec)
26.,63E
26.98
24.43
528.31
::::::::797:::
0.34
4.23
19.4
32.1
277.6
37.72
25.4
50.8
101.6
203.2
406.4
711.2
1320.8
2387.6
7.76e-04
1.55e-03
3.10e-03
6.20e-03
1.24e-02
2.17e-02
... ...
4.03e-02
7.29e-02
2388
25,38
27,28
3338
23 82
24 92
2672
32 12
37.98
36 52
41.92
22 22
23 12
24 12
27 42
3062
35 82
22.22
23 12
26 02
2402
2308
22.98
28 02
26 72
26 72
24.78
32.22
22 72
24 32
24 32
24 52
24.72
22,88
30 08
29 02
25 92
25.52
24 82
24 82
22 O.
Tms
Tmw
hx.
22.59
22.94
23.1
23.80
19.52
19.64
809.90
752.92
1
0.33
a(m)
224
2
24 82.-
..
0.66
Bottom
3
2742.
25 72
24 82
.1:31407 ::::::::8:!:0 .................................................................................... 698
...... ::ii:::::::::::::::::::f..::itir
0.74
Trn(sgle)
4
31 22
516.89.............464,64..........457,86..................170.81....
23.88
23 42
21 92
21 92
22 22
24 12
22.78
Side 5
36.24
25.4
Top
Tms
Tmw
hx
25.3
269.4
7:66e-04
Side 5
Trn(Side)
18.5
..
.
24 62
.
.
25 12
25 32
4436:
30 42
30 12
2308
23.58
24.75
26.01
20-147.
27.25
29.83
34.37
19.87
20.34
21.29
22.70
25.53
633.12
563.53
545.67
577.96
30.48
638.64
4,6.7
.
.
525.71
2628
... ,44
.............:
Pti4.
W(Nux)
Error(%)
:147=:::
:.n..:40::::::::::::::::::::::::::*
.
'''....140.
:::7:99::::::::::::::::::::::::::::t..
......
;:. ..
:2a.
]]:::g§1:::
0.42
0.37
0.26
0.21
0.19
0.20
0.22
0.26
3.40
3.18
2.72
2.46
2.32
2.40
2.52
2.74
182
Heating Three Surfaces
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
Sidi 5
4
3
2
1
Bottom
Re
Ra
Pr
Tout(c)
Weight(g)
Time(sec)
50.8
2.45e-03
101.6
4 91e-03
203.2
9.82e-03
19 51
18 91
18 81
19 21
20 01
19.51
19.01
18:91
19.21
20.01
19.71
19.31
19.31
19.61
20.31
20 11
19 61
19 51
20 11
20 51
21161
21 61
20.11
20 21
20.41
20 81
20 81
20 71
20 81
21 01
21 71
21 71
21 91
la $0
100
moo
243:29
2050
21.$0
..1925
19,53
19.56
18.38
567.50
2010
1985
19.19
18.25
709.29
19.30
18.29
664.59
1.17
10.78
1.03
10.13
0..,
W(Nux)
Error(%)
18.2
307.88
1.35E+04
9.835E+04
7.3062
Gr
1.23e-03
.1106/d0Vii:
Tms
Tmw
0.00575
0.023
0.0092
2.73685
a(m)
b(m)
Dh(m)
19.87
18.56
512.38
406.4
1.96e-02
0.62
7.92
2387.6
1320.8
6.38e -02
1.15e-01
24.91
24.11
23.51
23.21
23.31
23 11
22 21
20,34
20.33
18.93
479.17
20.87
19.47
479.25
21.94
20.56
488.96
0.55
7.45
0.55
7.45
0.57
7.59
:z0.5...g
::::
0.76
8.72
711.2
3.44e-02
23.1
199.6
43.06
:::
20.91.:.:: :::::
:::::24:0:1
,A17,45:::::
Tin(c)
Tout(c)
Weight(g)
Time(sec)
23.70
22.47
547.34
::::::::tt.:
0.70
8.43
Lth(m)
0.00575
0.023
0.0092
2.73685
L(mm)
25.4
9.68e-04
50.8
1 94e-03
101.6
3.87e-03
203.2
7.74e-03
406.4
1.55e-02
711.2
2.71e-02
1320.8
5.03e-02
2387.6
9.10e-02
Side 5
19.61
19.51
19 41
19 61
20 31
19 61
19 61
19 51
19 91
19 81
19 71
20 01
20.61
20 41
20 11
20 11
20 41
20 91
20 91
20 51
20 61
20 91
21 21
21 71
21 11
21 21
21 21
21.51
23.11
22.31
22 01
22.01
22 11
24.51
23.81
Bottom
X,10
2010
aSO
19 7
19.74
18.82
731.27
20 295
19.95
18.89
630.24
,
20.31
21,2$
21.25
22.1$
22.09
19.04
523.66
20,70
20,749
20.74
19.32
470.29
Z,00
19,59
19.64
19V
'W,M.
Tm(side)
Tms
Tmw
hx
19.75
444.89
20.61
22.11
451.25
507.47
1.26
11.24
0.94
9.78
0.66
8.26
2
4
3
2
1
W(Nux)
Error(%)
3
18.79
786.49
Side 5
4
3
2
1
Bottom
Trn(side)
Tms
Tmw
hx
20 31
..-1
1.45
12.05
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
1971
Gr
Ra
Pr
*V
.. .
:.:01.
0.00575
0.023
0.0092
2.73685
25.4
9.58e-04
20 56
20 66
20 36
20 66
21 96
50.8
1.92e-03
20 66
20 66
20 46
21 34
20 89
20.75
21.134
19.16
776.98
0.77
6.49
Re
Gr
Ra
Pr
WAC:::::
0.54
7.51
::::*
0.49
7.16
23.21.
23.11
23.21
2
23.43
::::0Atin
0.50
7.25
Tin(c)
Tout(c)
Weight(g)
Time(sec)
403.89
2.70E+04
1.927E+05
7.137
18.75
22.6
257.9
43.81
0.62
8.03
19.1
26.1
234.6
39.4
203.2
7.66e-03
22.06
21.56
21.46
22.06
22.56
406.4
1.53e-02
23 06
22 26
22 46
22 56
22 76
711.2
2.68e-02
24 46
23 46
23 06
22 76
23 06
1320.8
4.98e-02
25 16
24 26
23 86
23 86
2387.6
9.00e-02
29 16
27 86
26 66
25 86
25 80
21j
261
22 14
4.94
2494
21..1945
22 54
247
27.,08
21.80
19.62
563.60
22.47
20.14
528.76
24.54
22.48
596.98
26.87
19.23
786.67
21.23
19.36
659.60
23,291
23.19
20.92
542.86
0.79
6.57
0.56
5.57
20 81
21 76
20.74
20.79
101.6
3.83e-03
21 26
21 06
20 86
21 36
22 06
21 VA
8.W
Nu(x)
W(Nux)
Error(%)
396.04
1.42E+04
1.022E+05
7.20
Re
0.42
4.83
*14
::::006:
0.37
4.56
0.39
4.67
26 66.
::OAV::::
0.46
5.09
25.21
741.23
::::4011:
0.69
6.21
183
4
L(mm)
Side 5
4
3
2
1
Bottom
1.M(sIde)
Tms
Tmw
hx
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.023
0.0092
2.73685
25.4
9.79e-04
22 85
22 15
21.85
22.65
24 85
50.8
1.96e-03
23 15
22 45
22 35
23 45
24 75
28,40
010
19.73
838.34
22.97
19.86
776.23
W(Nux)
Error(%)
5
0.46
3.59
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
2
25.4
6.35e-04
19 59
19 09
18 99
19.29
1
21
Side 5
4
3
09
aft
Bottom
107::
Tms
Tmw
hx
19.49
17.95
909.28
0.40
3.34
101.6
3.92e-03
23.95
23.25
23.25
24.55
24.85
203.2
7.84e-03
25 25
24 45
24 85
24 45
24 85
23,*
23,10
23.72
20.12
670.41
6
L(mm)
Side 5
4
3
2
1
0.95
6.78
a(m)
b(m)
Dh(m)
24.56
20.64
614.29
::.9,
0.30
2.92
0.25
2.70
Re
Gr
Ra
Pr
25.45
21.68
638.46
:R70::.::
0.27
2.80
Tin(c)
Tout(c)
Weight(g)
Time(sec)
711.2
2.74e-02
30 15
27 75
26 15
25 25
25.35
1320.8
5.09e-02
34.15
31 15
28.95
27.65
27.45
2400
WM
0.34
Tin(c)
Tout(c)
Weight(g)
Time(sec)
711.2
20 19
2O1
1:94:::::::::::::::::
19.56
20.03
241*
4,0
MO
20 6'5
20.67
18.29
589.10
21.41
22 0$
21.95
18.00
895.60
0.92
6.69
.
18.10
726.08
><11'3 ><<
pg.4....:
0.62
5.54
0.42
4.64
Lth(m)
0.00575
0.023
0.0092
2.73685
25.4
6.24e-04
20 78
20 68
20 08
20 58
22 98
50.8
1.25e-03
21 28
20 78
20 38
20 78
22 98
101.6
2.49e-03
22 18
21 48
21 18
21 98
23 98
Re
Gr
Ra
Pr
203.2
4.99e-03
23.48
22.48
22.58
23.88
24.38
224
Bottom
2238
Tm(side)
Tms
20 72
20.87
22.28
20 942
21.06
21 836
Tmw
hx
17.80
910.25
17.89
882.64
0.48
3.43
0.45
3.34
21.36
18.69
523.62
:]:*.01:i:i::::;i:
0.34
4.22
...'s:
18.09
731.33
23.08
18.47
607.59
24.10
19.24
576.57
0.32
2.84
0.23
2.44
0.21
21.91
1320.8
3.30e-02
25 09
23.39
22.69
22 49
22 59
1.78e -02
22.99
21,99
21.99
21.69
21 89
19.28
523.49
0.34
4.22
0.50
3.79
17.9
23.2
280.5
31.34
2387.6
5.97e-02
27 29
25 49
24 39
23 69
23 59
2$40
231$
VW
23.00
20.46
24.69
22.52
646.74
551.31
A.#1i:ii::::::i
24 0
----'10.41
g.10.0
0.37
4.40
0.49
5.02
Tin(c)
Tout(c)
Weight(g)
Time(sec)
598.23
5.34E+04
3.952E+05
7.40
406.4
9.98e-03
25.18
24.28
24.38
23.88
24.08
0.40
3.38
3.11
203.2
5.08e-03
20 89
20 39
20 19
20 89
21 79
SS
2387.6
9.21e-02
39.15
36.75
34.25
32.45
32.05
401351439:a01148:
590.39
2.73E+04
2.009E+05
7.3626
406.4
1.02e-02
21 69
21 09
21 19
21 69
21 89
19.6
33.6
253.8
43.53
M et5i .................2.94........... MO
26.59
29.43
34.53
23.24
26.36
31.81
718.98
785.00
888.30
101.6
2.54e-03
20 29
19 69
19,59
19.99
20.99
:14:
W(Nux)
Error(%)
406.4
1.57e-02
27 45
26 05
25 45
24 75
25 05
2320
24,7- :::::::::4,6a:
.
0.00575
0.023
0.0092
2.73685
50.8
1.27e-03
19 79
19 29
19 19
19 39
20 49
5.55E+04
3.911E+05
7.043
Pr
5
Nu
400.21
Gr
Ra
22.52*::: 22:9$2.:::23.7:77
22.61
Re
17.7
28.1
301.2
33.06
711.2
1.75e-02
27 88
25 78
24 78
23 78
24 18
22
1320.8
3.24e-02
31 18
28 28
26 28
25 08
25 18
2308
258
25 2
24.94
27,1
3030
20.40
616.28
26.73
22.72
696.93
29.93
26.77
887.67
0.23
2.46
0.29
2.72
2387.6
5.86e-02
34 68
32 18
2948
27 88
27 68
1332
W(Nux)
Error(%)
2.34
0.45
3.36
184
Heating Side Surface
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
9.47e-04
19.40
19.20
19 10
19 10
19.40
Side 5
4
3
2
1
Tms
Tmw
hx
19.21
18.52
981.09
.:::4$.0.,:,
2.35
W(Nux)
Error(%)
15.61
2
L(mm)
Side 5
4
3
2
1
Tms
Tmw
hx
W(Nux)
Error(%)
0.00575
0.023
0.0092
2.73685
50.8
1.89e-03
19.40
19:20
19.20
19.20
19.40
19.26
18.53
934.87
Gr
Ra
Pr
101.6
203.2
3.79e-03
7.58e-03
19.60
19.9.0
19.40
19.70
19.40
19.70
19.70
19.50
19.50
19.60
19.47
19.73
18.56
18.63
750.52
.616.40.
...........................
::444Z:',.:,P:
2.14
14.98
:::::1149.,,,::::::::::::-----:::::,,
1.44
12.53
406.4
1.52e-02
20.30
20.10
20.10
19.90
19.70
20.05
18.75
522.59
711.2
2.65e-02
20.90
20.50
20.30
19.90
19.80
0.78
9.70
0.72
20.31
18.94
497.08
1320.8
4.92e -02
21.60
21.00
20.50
20.20
19.90
20.67
19.32
503.97
,.:-::
1.02
10.83
0.00575
0.023
0.0092
2.73685
25.4
9.34e-04
19.58
19.18
18.98
19.18
19.78
19.25
17.83
101.6
3.74e-03
19 98
19 78
19 63
19 88
19 83
19.80
17.93
755.59
203.2
7.47e-03
20.58
20.28
20.28
20.28
19.88
20.29
18.06
633.76
406.4
1.49e-02
21.48
21.08
20.88
20.38
19.78
997.01
50.8
1.87e-03
19.58
19.28
19.18
19.38
19.68
19.37
17.86
942.87
1.17
7.62
1.05
7.27
0.70
6.08
0.52
5.34
Re
Gr
Ra
Pr
18.5
20.2
236.5
39.34
2387.6
8.90e-02
22.10
21.60
20.90
20.40
20.10
21.07
19.98
625.97
:::::::.15.5:
a(m)
b(m)
Dh(m)
Lth(m)
9.41
400.43
2.73E+04
2.014E+05
7.3814
20.81
711.2
2.62e-02
22.48
21.48
20.88
20.28
19.88
21.06
18.32
568.55
18.71
601.61
0.43
4.96
0.47
5.15
0.73
9.48
1.05
10.95
Tin(c)
Tout(c)
Weight(g)
Time(sec)
17.8
21.3
252.2
41.45
1320.8
4.86e-02
23.68
22.48
21.38
20.58
20.08
21.66
19.49
651.33
...
0.54
5.44
2387.6
8.78e-02
24.58
23.58
22.38
21.28
20.58
Tin(c)
Tout(c)
Weight(g)
Time(sec)
18.2
20.9
253.9
33.52
22.41
20.85
908.61
0.97
7.05
Lth(m)
0.00575
0.023
0.0092
2.73685
25.4
7.51e-04
50.8
1.50e-03
101.6
3.00e-03
203.2
6.01e-03
406.4
1.20e-02
711.2
2.10e-02
1320.8
3.91e-02
2387.6
7.06e-02
19 91
19 61
19 51
19 71
20 01
20.31
20.01
19.91
20.11
20.21
20.91
20.51
20.41
20.61
20.41
21.61
21.21
21.21
20.81
20.31
22 71
21 71
21 21
20.61
20.41
23 71
22 41
21 61
20 91
20 41
24.41
23.31
22.21
21 31
20 71
Tms
Tmw
20 11
19 61
19 41
19 71
20 31
19.71
18.23
20.06
18.30
772.32
20.55
18.40
632.28
21.09
18.60
546.03
21.40
18.90
543.99
:::::::02::
0.46
5.48
21.84
19.50
581.56
22.46
20.56
3
L(mm)
Side 5
4
3
2
1
a(m)
b(m)
Dh(m)
Tin(c)
Tout(c)
Weight(g)
Time(sec)
402.09
1.40E+04
1.018E+05
7.2498
Re
hx__
915.25
19.68
18.25
950.50
Nux
1403
1457
W(Nux)
Error(%)
1.08
7.70
1.16
7.93
Re
Gr
Ra
Pr
503.11
2.73E+04
1.993E+05
7.3062
040:.=
0.81
6.80
0.58
5.97
0.46
5.49
:044t
0.51
5.69
713.61
....tat
0.70
6.45
185
4
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
7.52e-04
22 10
21 40
20 90
21 80
22.60
Side 5
4
3
2
1
Tms
Tmw
hx
5
L(mm)
Side 5
4
3
2
1
Tms
Tmw
hx
1
Tms
Tmw
hx
W(Nux)
Error(%)
0.55
3.87
0.63
4.09
Re
Gr
Ra
Pr
101.6
3.01e-03
22 70
22 20
22 00
22 80
22.10
22.29
18.70
771.10
11.80
0.40
3.37
Lth(m)
0.00575
0.023
0.0092
2.73685
25.4
6.19e-04
19 29
19,59
19 29
19 49
20.09
19.52
18.22
1080.06
50.8
1.24e-03
19 29
19 79
19 59
19 59
20.09
19.68
18.24
975.94
101.6
2.48e-03
19 59
20.09
19 99
19,99
20.19
1498
1.23
8.22
a(m)
b(m)
Dh(m)
1.47
8.85
6
4
3
2
1535
15
W(Nux)
Error(%)
Side 5
1431
18.55
°t!
L(mm)
. 40
50.8
1.50e-03
20.58
21.50
21.10
21.60
22.20
21.36
18.60
. 002.50
21.51
W(Nux)
N( Ni
Error(%)
0.00575
0.023
0.0092
2.73685
a(m)
b(m)
Dh(m)
Lth(m)
0.00575
0.023
0.0092
2.73685
25.4
6.42e-04
22.25
21 55
21 45
21 95
22 95
21.74
19.04
896.90
50.8
1.28e-03
2387.6
7.07e-02
30 40
28 60
26 20
24 20
23.00
26.64
23.30
651.90
1320.8
3.91e-02
28 90
26 40
24 60
23 20
22.30
25.12
21.15
697.25
997
966
1024
1060
0.30
3.03
0.29
2.97
0.32
3.08
0.33
3.16
12.52
0.44
3.54
18.91
Cr
Ra
Pr
406.4
9.91e-03
21 29
21 09
21 09
20 79
20.09
20.98
18.54
813.45
203.2
4.96e-03
20 39
20 59
20 39
20 49
20.19
20.47
18.37
668.36
1247
1024
880
0.91
0.66
6.47
0.53
6.00
7.27
Re
Cr
Ra
Pr
Tin(c)
Tout(c)
Weight(g)
Time(sec)
609.99
2.82E+04
2.058E+05
7.3062
574.71
711.2
1.73e-02
22 29
21 79
21 39
20 69
20.29
21.39
18.80
541.90
1320.8
3.22e-02
23 29
22,29
21,59
20 89
20.29
21.76
0.48
5.84
601.29
5.26E+04
3.763E+05
7.1558
406.4
1.03e-02
19.08
934.30
203.2
5.13e-03
23 75
23 15
23 05
23 55
22 45
23.05
19.30
646.79
593.57
24 75
23 85
22 85
22 15
23.93
20.07
626.86
1372
1429
1188
969
906
956
0.61
0.66
4.60
0.47
4.02
0.36
3.66
0.32
3.50
0.34
3.60
4.47
21 65
21 55
21 45
22 55
21.67
24
263.8
34.86
711.2
2.11e-02
26 90
24.80
23.80
22.70
21.90
24.05
19.93
671.02
101.6
2.57e-03
22.65
22.35
22.25
22.35
22.65
22.33
19.15
762.47
2195
18.5
406.4
1.20e-02
25.60
24.40
23.60
22.70
21.70
23.69
19.32
632.28
203.2
6.02e-03
23 90
23 20
23 60
22 80
21.90
23.15
Re
20.01
18.29
Tin(c)
Tout(c)
Weight(g)
Time(sec)
506.14
5.72E+04
4.146E+05
7.2498
711.2
1.80e-02
2625
24 45
23 85
22 95
21 95
23.69
19.61
.
18.2
20.5
282.4
30.75
2387.6
5.82e-02
23 69
2299
22 09
21 19
20.39
19.31
22.21
20.21
573.64
700.28
0.53
5.99
0.71
Tin(c)
Tout(c)
Weight(g)
Time(sec)
19
23.1
1320.8
3.34e-02
28.05
25 85
24 35
23 25
22 35
24.71
20.98
649.26
88
0.36
3.67
6.64
334.8
37.68
2387.6
6.03e-02
29 05
27 25
25 35
23 85
22 65
25.66
22.58
785.88
1191
0.49
4.10
Heating Bottom Surface
L(mm) if:::::::::V:::::::::: Tsx
........ ........ .
Gr 4.02E+04
Ra 2.95E+05
Pr
7.34
1.010
25.40 :',....:.:,..)2
5020 .
10120 ,
20320 gg.000t0
304.80 O0230, 22.20 829
406.42 0,01.04".
482.60 goggo2
., :
787.40 ;
863 20 .
939.80
*A
.
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23.301f1>
01
4
522.34
521.88
0.37
4,62
0.37.
505.41
501.99 :
036
422
422
48q
422
491.98
513 27
49734
4;52
101.60 '...:'. i0 .i 23.201
203.20 00..1 0 22.80'1
304.80 04212 23
406.40 0-030td23.501 1
23.70 1
23.80 4
711.20 00.
23.901
421
4.44
40
23. 0 :.
558.80
635.00 0,..
24.10 ;..
863.60 ...,,
...........
939.80 #3;13
24.1
4.42:
1,168.421(7.-
589.85
0.33
4,5,/
602.11
032 .
033
4.51 1
1,244.60
1,320.80
421
1397.00
0.32
4,150
471as...
033
1,473.20
1,549.40
as:
44'4
0.32
0.32
421
1225.60
4,52
473.86 74 033
424
421
442
450
1,701.80
1,778.00
1,854.20
1,930.40
2,006.60
2,082.80
2,159.00
2,235.20 0.1;6515:
2,311.40
2,387.60
1,701.8011:208024.00
0.32
:24.10 ;:1976
2,159.00124.50
0.35
0.33
0.33
4, 7T
25.40 0 00155 22.70
50.80 ts 75 23.001
1,092.20 :..;,....
478.07 731
4.22
2,08Z80
7.25
450
1,397.00010 0 23.601 33
21,00693%040 1,111?21
Pr
4,84
479.24 7
470.11
1,854.20 0;;1
202.91
Gr 5.67E+04
Ra 4.11E+05
1,016.00 0.07507
1,320.80 0 09995 23.6019
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Re
452
7 >31
465.32 712
473.12 I
1,778.00004.51:
0.35
0.35
0.35
0.34
06
3
1,168.40 0010 23.5019 1
1,244.60 099.40:23.50 t61':
1,473.20
1 249AO
464
450
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121200 00789023.301697
1,092.20 0 ?08
b(m)
0.023
Dh(m) 0.0092
Lth(m) 2.73685
45.1
0.37
0.32
495.27 ].,....
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a(m) 0.00575
.
61
71120::*:.:::,.,.....,:.:.:::
coratit
18
20.6
133.4
hx
:No W(Nu) ittott141
466
0.38
52720
516.89
463.90
'',
558.80
Tin(c)
Tout(c)
W(g)
t(sec)
195.56
Re
a(m) 0.00575
b(m)
0.023
Dh(m) 0.0092
Lth(m) 2.73685
7A0 '.
464.93'3
031
451.82
9.16 7
0.31
0.31
0.31
0.33
0.32
2,235.20 0.1
70:1
2,311.40 ::,.....*.
....
474.60 ....,.....
2,387.60
ki.241
471.5710
032
4447:
4AR
442
422
452
Tin(c)
Tout(c)
W(g)
t(sec)
18.5
21.9
139.1
45.85
a(m)'07023'
Re
Gr 8.41E+04
Ra 6.07E+05
b(m)
Dh(m) 0.0092
Lth(m) 2.73685
Pr
Tsx
8004.24 it
L(mm
25.40
50.80 0223.94'
101.60
203.20
."---43.94
823.74111 18
4.
43:80
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482.60
752.91
743.98
696.74
787.40 oat
863.60
689.09
705.72
939.80 te
1,016,00
x"-6.
27.2
27.84
27.
27.74
27.94
1,701 .80
?.4392 26.14
2,159.00 00656
Tin(c)
Tout(c)
t(sec)
W(g)
28.
8.
04:4
29.0422 7f1
2,235.20
2 , 3 11.40 416414
2,387.60
a(m)
b(m)
Dh(m)
18.7
23.6
138.3
45.15
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Gr 1.42E+08
0.0092
43:845.4884°0 PliatiL446',1277.:6812
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863.60 06.468666.;:i4: 7.9
1,016.00 666i:4:28.12
1,092.20
0.31
709.21
0.30
0.27
1,701.80
0.31
1,778.0013;:
0.29
0.29
0.28
0.26
Ell
704.74
0.25
0.33
0.29
.
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76387514000
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k
4:49°.52
1,014.67
852.36
62
745.02
958.19
1,038.67
1,154.26
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1,031.97
42 4
1,073.30
1,041 20
8 .18
891.66
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31.
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M85482..82°0
2,159.00 01
2,235.20 or
22,338171.6400
Tin(c)
Tout(c)
W(g)
t(sec)
W(Nu)
19.2
27.1
156.3
51.04
4
8 .77
935.01
1,015.75
..,...
,397.00
1,473.20
1,549.40
0.
7.12
50.80
0.31
655.22
731.70
697.96
690.28
682.77
653.30
636.26
629.87
767.60
Pr
hx
727.98
719.63
766.13
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1,397.00
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1,701.80
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Gr 2.84E+05
Ra 1.99E+06
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1,492.37
1,520;14
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1,36533
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00.6688431
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0.76
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NuWNu Error :A
hx
Tsx
L mm
"
Tin(c)
Tout(c)
W(g)
t(sec)
604.54
Re
b(m)
0.023
Dh(m) 0.0092
Lth(m) 2.73685
412
1,027.33 ":'
1,057.83
410
411
4 Is
4.08
4 15
1,280.01
971.32
1,124.47
4,03
4.15
4.14
00.663707
1,114.36 :;.'
416
397
0.70$
11:010741..4323
2,311 .40
006i:527.38
0.72
0.50
0.57
1,119.00
821 85
28.1 8
328.1
a(m)
1,144.52
82035 1 129241721
558.80
635.00
711.20
19.8
22.5
wow) Eh&
...... ...s
nx
1-(mm
Tin(c)
Tout(c)
W(g)
t(sec)
.7
1176991.3105 ::
55
1,209.12
2,485.63
2,231.40 45,4
0.37
0.67
0.62
0.41
1.07
.92
244
2.47
2.25
Z39
3,00
237
21k
2,00
2,81
223
2.34
2-28,
285
2.72
193
Aspect Ratio: 5
Heating Whole Surfaces
1
a(m)
0.00575
Re
199.44
b(m)
0.0288
Gr
1.46E+04
0.0095861
Ra
1.001E+05
2.73685
Pr
6.83
Dh(m)
Lth(m)
L(mm)
25.4
0.0019433
...
21,75
21 73
Top
Side 5
101.6
203.2
406.4
711.2
1320.8
2387.6
0.015547
0.031094
0.05441
0.101055
0.18267
21 96
21.93
21 53
22.36
22 23
21 73
23,95
23 73
22 63
25,60
24 93
27.06
27 03
23 53
25 33
21 73
21 73
22,96
22 73
22.03
21.93
22 43
23 13
22 03
22 48
22 06
22 48
23 03
24 63
24 43
22 83
23 33
24 63
22.36
22.86
24,06
30.26
29 83
28 23
27 23
26 73
26 63
26 06
2225
22,S
22.86
2S$
25
25.09
27.61
21 33
2
1
21 33
21 93
21 43
21 43
22 03
Bottom
2t56
21 76
22 33
21.96
Trn(side)
21,46
Tms
Tmw
hx
21.51
21 64
21.68
20.82
566.03
21.9
21.94
20.94
483.15
435.21
1.38
1.06
0.77
A,
0.63
13.44
11.79
10.10
9.13
22.29
21.18
1O26:
2
L(mm)
a(M)
b(m)
Dh(m)
Lth(m)
22.38
23.81
26.33
389.35
378.74
378.00
0.50
0.48
0.47
8.20
7.99
7.97
.15,.
0.54
Re
199.74
Gr
0.009586
Ra
2.83E+04
1.991E+06
2.73685
Pr
7.0242
'OAS
Tin(c)
Tout(c)
Weight(g)
Time(sec)
121.3
34.77
25.4
50.8
101.6
203.2
406.4
711.2
1320.8
2387.6
3.78e-03
7.55e-03
1.51e-02
3.02e-02
5.29e-02
9.82e-02
1.78e-01
24 70
24 13
22 53
22 43
22 53
22,83
22,10
26,80
29.40
28 73
33,00
39 40
32 93
38 63
25,83
29.43
35.63
23 43
23 23
24 73
27 73
33 13
24 23
26 83
31 73
23 33
22,20
24 13
26 43
30 93
23,10
25.30
29,70
22.72
22.83
20.69
23.6
252
23.92
25.37
28,25
28.40
26.17
463.00
2280
23.70
21 83
22 43
4
21 13
3
2
21 13
21 13
21 43
21.33
21 33
23 18
22 03
1
21 93
22.53
BOttOM
21 50
21,35
21.42
21,70
21,7
21.77
21 93
22 03
22 83
21 90
22.26
22.35
26 23
23 93
33.5
33.67
31.39
19.82
19.95
649.38
566.17
20.20
480.34
483.51
464.31
0.66
0.50
0.36
0.37
0.34
0.35
0.33
0.31
6.35
5.56
4.74
4.77
4.59
4.66
4.58
4.48
21.69
23.18
471.51
7
W(Nux)
Error(%)
19.7
33.1
1.89e-03
22.00
hx...
21,5
21.66
0.0288
Top
Tms
Tmw
23.62
403.43
0.00575
Side 5
Trn(side)
29
0.007773
21 33
W(Nux)
Error(%)
98.6
50.8
3
20.76
20.7
27.15
0.003886
4
646.65
Tin(c)
Tout(c)
Weight(g)
Time(sec)
452.50
7.00 :.
194
3
a(m)
0.00575
Re
395.58
b(m)
0.0288
Gr
1.47E+04
0.009586
Ra
1.019E+05
2.73685
Pr
6.95
Dh(m)
Lth(m)
L(mm)
20.1
23.5
200.7
29.33
25.4
50.8
101.6
203.2
406.4
711.2
1320.8
2387.6
9.64e-04
1.93e-03
3.86e-03
7.71e-03
1.54e-02
2.70e-02
5.01e-02
9.06e-02
21,34
21 74
2680
24 34
23 17
24 37
28 24
26 17
22,07
21.87
22 97
22 67
24 67
23 97
21 97
22 67
23 77
22 37
22 97
23 87
21.74
22,34
23,24
23.06
21.74
24.37
23.07
Top
2084
2094
Side 5
20 97
21 17
4
3
20 57
20 57
2
20 62
20 77
20.67
20 77
1
21 17
21,27
2074
2074 . ....
20,94
80110471
TM(side)
Tms
Tmw
hx
Tin(c)
Tout(c)
Weight(g)
Time(sec)
.........
.. ...
21,47
21 07
21 77
20 97
21.17
2097
21 17
21 57
21 82
22.44
22 47
21 57
21 57
21 57
22 07
21 24
21,34
21 44
21 27
20,9.. ......... ... 21, ............. ...21.,.O. .. ......... .. 21,75.
. ..............
20.75
20.91
21.13
21.39
21.78
22.28
20.13
833.10
20.16
20.23
20.35
20.60
20.98
13:24::::::::::::
390.77
393.34
397.23
567.72
495.49
436.90
..::::ii:ii:::::::::::::::::::::::::::::::::::::::::::..;.:::::::::::::
691.20
195
5
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
0.00575
Re
398.11
0.0288
Gr
5.65E+04
0.0095861
Ra
3.990E+05
2.73685
Pr
Tin(c)
Tout(c)
Weight(g)
Time(sec)
7.06
19.5
33.1
224.2
32.09
25.4
50.8
101.6
203.2
406.4
711.2
1320.8
2387.6
9.42e-04
1 88e-03
3 77e-03
7 54e-03
1 51e-02
2.64e-02
4.90e-02
8.86e-02
29.36
29 15
25 85
25 55
25 25
32.86
37 76
32 15
36 90
27 85
31 55
26 45
29 25
25 65
27 85
24.65
22.26
25 05
25 35
27 25
44 96
43 55
38 15
34 35
32 25
31 25
22,46
24,56
26,46
245
257
24.48
25.73
22,96
26.67
27.03
Top
22 38
23.36
24.96
26 46
Side 5
22 75
4
22 05
21 95
24.95
23 45
3
23 85
22 55
22 45
26 05
24 15
24.15
2
22 05
1
23 95
22 55
24 55
Bottom
2186
2176
2266
Trn(side)
22.4
22.35
22,9
20.51
21.52
23.03
26.06
31.36
529.79
498.54
526.38
530.27
536.06
Tms
Tmw
hx
23 35
23 45
24 45
24 65
29.8
35
30.03
35.28
19.63
19.75
771.15
662.11
23.8
23.80
20.00
553.53
0.47
0.35
0.25
0.23
0.20
0.22
0.22
0.23
3.79
3.29
2.80
2.69
2.55
2.68
2.70
2.72
22.93
:84.Q:
W(Nux)
Error(%)
Heating Three Surfaces
1
L(mm)
a(m)
0.00575
Re
301.39
b(m)
Dh(m)
Lth(m)
0.0288
Gr
1.27E+04
0.009586
Ra
9.911E+04
2.73685
Pr
Tin(c)
Tout(c)
Weight(g)
Time(sec)
7.82
15.8
20.8
190.1
32.88
25.4
50.8
101.6
203.2
406.4
711.2
1320.8
2387.6
1.12e-03
2.25e-03
4.50e-03
8.99e-03
1.80e-02
3.15e-02
5.84e-02
1.06e-01
Side 5
17,22
17 22
17 32
17,82
18.72
19 52
21 22
23.62
4
3
2
16 62
16 72
16 87
17 22
17 77
18 47
22 22
16 52
16 52
16 72
17 12
17 67
18 32
16 42
16 52
16.82
17 12
17 82
18 42
1992
1942
1942
21 22
1
17 62
17 67
18 02
18 32
18 72
19 02
20 02
21 4;
17,t0
17`,60
0,89
1840
1810
1 MR
18,50
18,62
19.40
Bottom
Tm(Side)
Tms
Tmw
hx
W(Nux)
I 7..,Oft. . .....
21 52
200
16.77
16.90
17.13
17.47
18.03
18.61
19.61
19.77
15.85
15.89
15.99
16.17
16.54
17.10
18.21
20.16
753.56
691.78
611.38
536.28
447.47
452.18
1
1.33
1.6 0832.
.. ...
.... .
.
1.13
....17,42.. . ......
.... .
462.28
.
. . _._
.. .
.
_Z.1 . ...7.0.::
21.71
."...,:.....,:fx::
0.69
0.53
0.51
0.48
0.49
7.96
7.05
6.95
6.74
6.81
0.88
Error(%)
11.00
10.13
9.01
196
2
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
1.14e-03
18 52
17 92
17.67
17 42
19.02
Side 5
4
2
1
Bottom
Tms
Tmw
hx
/606
18.01
18.29
16.29
710.14
a(m)
b(m)
Dh(m)
25.4
1.13e-03
20.66
19.96
19.56
19.36
22.36
3
2
1
Bottom
ide)
Ti
Tms
Tmw
hx
20.25
16.39
726.71
W(Nux)
Error(%)
4
3
2
1
0.58
5.12
25.4
9.51e-04
20.40
20.50
20.40
20.30
20.60
21.01
21.36
22.56
1320.8
5.91e-02
26 52
24 72
23 62
22 92
22 67
190
19,76
21 65
21 16
23 6
23.56
21.07
571,14
2452
4:
20.40
17.63
513.87
8 22:
4.11
3.79
21.48
18.78
526.28
0.32
3.87
0.38
4.17
Tin(c)
Tout(c)
Weight(g)
Time(sec)
303.29
5.35E+04
4.134E+05
7.722
406.4
1.81e-02
27.56
25.16
24.26
23.66
23.56
711.2
3.17e-02
30 76
27 86
26 16
24 76
24 16
1320.8
5.88e-02
36.36
32.76
30.76
28.86
27.16
22,13
249
21,98
26,27
300
22.79
17.70
551.73
24.20
19.20
561.64
25.88
21.45
633.50
677.91
10,03
1ki2
0.24
2.39
0.27
2.54
0.19
2.11
Re
Gr
Ra
Pr
203.2
7.61e-03
20.90
20.90
20.90
20.90
21.40
0.19
2.15
406.4
1.52e-02
21.50
21.20
21 20
21.30
21.80
30.09
25.95
Tin(c)
Tout(c)
Weight(g)
Time(sec)
396.80
1.51E+04
1.062E+05
7.0242
711.2
2.66e-02
22.00
21.70
21.70
21.70
22.10
16.1
26.4
187.8
32.81
2387.6
1.07e-01
31.02
29.12
27.62
26.52
25.92
2.
'27;7::
27.41
25.09
612.80
:;:i:04,2::
> > ::::::MC
203.2
9.05e-03
23.76
22.66
22.86
22.96
23.06
2.39
101.6
3.80e-03
20.50
20.60
20.60
20.70
21.10
711.2
3.18e-02
23 32
22 02
21 42
21 12
21 17
0.31
100.2148
0.00575
0.0288
0.009586
2.73685
50.8
1.90e-03
20.40
20.50
20.50
20.50
20.80
E 4;
406.4
1.82e-02
21 62
20,42
20 32
20,32
20.57
0.37
Ra
Pr
7
21.37
16.95
635.30
0.28
2.56
>9
Gr
2diie
0.32
2.70
><>
Re
101.6
4.53e-03
21 76
21 06
20.67
16.57
685.90
203.2
9.10e-03
19.82
19.22
19.12
19.32
20.12
1 640
19;438
19.40
16.86
561.99
0.46
4.55
50.8
2.26e-03
21.16
20.26
20.06
20.06
22.56
'Atom
a(m)
b(m)
Dh(m)
:100 >:
0.00575
0.0288
0.009586
2.73685
1166
Lth(m)
4
101.6
4.55e-03
18 92
18 42
18 32
18 42
20 02
16,722
18.75
16.48
626.14
Lth(m)
4
Pr
16..*
5.62
Side 5
Ra
1 .6246
0.71
L(mm)
Gr
1:08
1Z5 M : illa
3
Side 5
50.8
2.27e-03
18 62
18 02
17 92
17 82
19 42
Tin(c)
Tout(c)
Weight(g)
Time(sec)
300.73
2.68E+04
2.077E+05
7.74
Re
17;977
16.20
782.51
W(Nux)
Error(%)
L(mm)
0.00575
0.0288
0.0095861
2.73685
1320.8
4.94e-02
23.10
22.60
22.40
22.40
22.70
0.43
4.45
16.2
36.4
204.1
35.45
2387.6
1.06e-01
45.26
41.56
38.16
35.56
33:76
37.53
33.82
756.12
0.33
2.80
19.7
23
240
34.63
2387.6
8.94e-02
24.40
23.90
23.60
23.50
23.50
..
Bottom
t.li.#
ZOO
20:06
Tms
Tmw
20.40
19.73
819.84
20.53
19.76
716.88
20.69
19.82
632.70
20.97
19.95
536.85
21.34
20.19
479.77
W(Nux)
Error(%)
1.98
15.18
1.52
13.36
1.20
11.88
0.87
0.70
9.23
2:1
21.77
20.56
456.07
V%
....
21.29
452.18
2169::
23.62
22.58
529.02
0.63
8.76
0.84
10.07
22.51
::11C
10.21
0.64
8.83
197
5
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
9.48e-04
21 53
21 13
21 03
21 03
21 93
0.00575
0.0288
0.009586
2.73685
Gr
Ra
Pr
23.63
22.73
22 73
22 88
23.28
2387.6
8.91e-02
29 63
28 23
27 13
26 43
25 93
21,10
2220
40.60.
23,40
moo
22.24
20.36
22.89
20.82
502.96
23.59
27.02
500.00
24.85
22.89
529.90
608.37
0.41
0.40
5.13
5.11
0.45
5.38
0.58
6.10
22.43
22.03
22.03
22.13
22.93
=OD
214
Tms
Tmw
21.25
19.96
801.84
21.44
20.02
727.52
21.76
20.13
636.64
553.51
1.01
0.83
7.90
7.21
0.64
6.36
0.49
5.59
4
3
2
1.
26.1
227.2
32.68
1320.8
4.93e-02
26 78
25 53
24 83
24 43
24 43
406.4
1.52e -02
Bottom
Side 5
19.9
711.2
2.65e-02
24 73
23 83
23 53
23 43
23 53
203.2
7.59e -03
50.8
1.90e-03
21.63
21.23
21,23
21 13
22.23
21..76
101.6
3.79e-03
21 73
21 53
21 53
21 53
22 63
Tin(c)
Tout(c)
Weight(g)
Time(sec)
399.97
2.91E+04
2.031E+05
6.98
Re
21.51
25.31
f.0
W(Nux)
Error(%)
6
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
9.62e-04
23 70
23 30
22 90
22 80
24 40
Side 5
4
3
2
1
Bottom....
0.00575
0.0288
0.009586
2.73685
50.8
1.92e -03
23 60
23 40
23.25
23.10
24 80
4,3Z
Gr
Ra
Pr
101.6
3.85e-03
24 30
23 95
23 90
23 90
25 20
28,92
203.2
7.70e-03
25,60
24 80
24.90
25.20
30
2825
Tin(c)
Tout(c)
Weight(g)
Time(sec)
400.73
5.93E+04
4.079E+05
6.873
Re
V
406.4
1.54e-02
28 00
26 40
26.10
25 90
25,90
711.2
2.69e-02
30 10
28 20
27 00
26 40
26 20
1320.8
5.00e-02
33 90
31 20
29 60
28 30
27 60
2387.6
9.04e-02
39 15
36 65
34 40
32 60
31 60
2:44.2
24,71
02
4,82
.... .
Tms
Tmw
hx
23.21
20.61
23.57
20.72
771.12
703.11
0.49
3.98
0.41
24.14
20.95
628.35
24.99
21.40
557.47
26.07
22.30
531.68
27.04
23.64
590.41
29.35
26.34
666.20
0.33
3.30
0.26
2.96
0.24
2.84
0.29
3.12
0.36
3.48
.:(40.44
W(Nux)
Error(%)
7
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
2
25.4
5.63e-04
18 00
17 50
17 20
17 20
1
18400
Side 5
4
3
3.65
0.00575
0.0288
0.009586
2.73685
50.8
1.13e-03
18 00
17 60
17 40
17 40
18 80
Re
Gr
Ra
Pr
101.6
2.25e-03
18 10
17 90
17 80
17 80
1940
is*
Bottom
ltos
'T.6000Y
17 55
17.60
15.95
879.34
803.24
18 13
18.18
16.09
695 17
0.90
6.39
0.76
5.89
0.58
5.20
Tms
Tmw
hx
W(Nux)
Error(%)
17,745
17.80
16.00
20.5
32.6
219.2
31.93
Tin(c)
Tout(c)
Weight(g)
Time(sec)
604.03
2.67E+04
2.079E+05
7.797
203.2
4.50e-03
18 80
18 50
18 40
18 40
19 90
406.4
9.00e-03
20 30
19 20
19 20
19 40
20 30
711.2
1 58e-02
21 20
20 20
20 10
20 20
110:98
1888
18 72
18.74
16.29
590.32
19 55
19.49
16.67
515.00
0.43
4.54
0.34
4.08
34.4
33.96
31.06
690.
40086
0.38
3.60
15.9
21.1
268.7
23.25
1320.8
2.93e-02
23.40
21 90
21 30
21.10
21 00
2387.6
5.29e-02
26 50
24 30
23 20
22 60
22 20
1943
IVO
20.33
20.23
17.25
487.77
21.53
21.37
28,84
23 44
489.88
0.31
0.31
3.92
3.93
20 SO
18.41
23.21
20.44
523.57
0.34
4.13
:
198
8
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
5.66e-04
19.60
18.90
18:70
18:30
Side 5
4
3
2
1
Bottom
ej
Tms
0.00575
0.0288
0.009586
2.73685
50.8
1.13e-03
19 80
2050
19 10
19 00
18 80
21 30
10,35
%OS
...1.9:,07...
19.10
15.80
882.09
Tm w
hx
19.52
15.89
....801.89
:.:.. W.,.:::i:.E.:.::.:::::42:".-.
0.46
3.23
W(Nux)
Error(%)
9
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Side 5
4
3
2
1
25.4
5.59e-04
21 10
20 35
19 90
19 60
23 50
Bottom
21,47
..
W:,.7._.
Tms
Tmw
hx
20.77
15.65
891.40
iF
4
0.31
2.13
W(Nux)
Error(%)
0.38
2.98
Re
Gr
Ra
Pr
101.6
2.26e-03
20 20
19 70
19 70
19 60
22 00
203.2
4.53e-03
21 50
20 85
20 70
20 90
22 60
21126
20,46
-15.. .
...
:::::...OW,......
2.70
0.00575
0.0288
0.009586
2.73685
Re
Gr
Ra
Pr
2.24e -03
21 50
20 80
20 50
20 50
24 40
22 20
21 60
21 60
21 70
24 60
2'1,92
....ZI,
..
21.40
15.80
815.16
::::W
0.26
1.98
101.6
...22.75....
22.55
17.26
549.85
t..." ..
0.19
2.20
0.24
2.42
0.31
50.8
1.12e-03
00,66
21.15
16.48
622.89
16.09
714.90
....1::
406.4
9.05e-03
24 30
22 30
22 45
22 70
22 90
21.22........
20.16
406.4
8.95e-03
28 70
26 00
25 80
25 40
25 00
2$.1
22 12
210
:401::...:.
22.17
23.64
16.11
16.71
752.33
658.17
10 55
...,,,,0.18
1.68
, 0%:::
0.22
1.86
1320.8
2.94e-02
30 18
27.10
25.50
24,40
24 00
711.2
1.58e-02
26 30
24 40
23 90
23 20
23 40
21,
2249
25.46
20.77
620.05
23.75
18.43
sasss..
'",i:t1.2::,
0.19
2.19
1.56
23,85
28.90
24.86
719.29
0.31
2.41
2.72
Tin(c)
Tout(c)
Weight(g)
Time(sec)
15.5
31.8
319
711.2
1 57e-02
27.51
1320.8
2 91e-02
36 80
31 60
30.50
27.80
26,70
32 GO
28 80
26 80
25 90
25 30
an
4,07
.
.......
26.88
19.74
638.75
29.45
23.37
749.79
]!!!':10Ak
::::1:1Ait:
0.17
1.64
2387.6
5.32e-02
35 00
32 30
29 30
27 35
25 90
0.24
:27..,g0......,....
25.57
17.92
596.83
:10
0.15
15.7
26.2
317.6
27.67
....29,41..
....23.95.........
599.78
7.98E+04
6.304E+05
7.89
203.2
4.48e-03
24 20
23 20
23 30
24 00
24 70
:2Z2::
Tin(c)
Tout(c)
Weight(g)
Time(sec)
596.79
5.22E+04
4.094E+05
7.847
0.22
1.85
Heating One Side Surface
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Side 5
4
3
2
1
Tms
Tmw
25.4
1.87e-03
21 71
21 61
21 51
21 51
21 71
21.59
20.93
795.50
g*:.1::.::.::
W(Nux)
Error(%)
1.97
15.58
0.00575
0.0288
0.009586
2.73685
Re
Gr
Ra
Pr
50.8
3.74e-03
101.6
7.48e-03
21 81
21 71
21,71
21 71
21 71
21 91
21.91
21 91
21 91
21.91
21.91
21.72
20.95
683.62
400,
1.46
13.51
21.00
579.48
203.2
1.50e-02
22 41
22 31
22 31
22.31
21 91
22.25
21.11
459.71
20.9
23.7
128.8
36.44
2387.6
406.4
2.99e-02
711.2
5.24e-02
1320.8
9.73e-02
23 01
22 71
22 61
22 41
21 91
23 61
23 21
22 81
22 01
24 51
23 91
23.51
23 01
22 41
1.76e-01
25.71
25 11
24 51
24 01
23 21
22.50
21.32
443.36
22.76
21.63
463.66
23.38
22.25
467.23
24.40
23.34
496.58
0.70
9.54
0.71
9.61
0.79
10.13
2251
0..
4.10.::.::::::::::
1.07
11.62
Tin(c)
Tout(c)
Weight(g)
Time(sec)
208.35
1.62E+04
1.099E+05
6.798
0.69
9.48
0.65
9.19
199
2
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
1.88e-03
23 43
23,33
23 23
23 23
23 13
23.24
20.60
738.57
Side 5
4
3
2
1
Tms
Tmw
hx
0.00575
0.0288
0.009586
2.73685
50.8
3.77e-03
24,03
23 83
23 73
23 83
23.23
23.71
20.69
647.31
*t::;Y.Z.V
ROY
W(Nux)
Error(%)
0.46
3.95
3
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
25.4
1.86e-03
27 02
26 62
26 62
26 52
25 82
26.49
21.39
743.49
Side 5
4
3
2
1
Tms
Tmw
hx
0.36
3.50
Gr
Pr
25 03
24.63
23.33
24.85
21.28
546.63
,.
0.26
3.02
Ra
Pr
28 92
26 42
27 72
25 82
27.98
21.95
628.85
203.2
1.49e-02
31 52
29 72
28,62
27 82
25 72
28.43
22.69
661.31
0P
W(Nux)
Error(%)
0.25
2.11
4
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Side 5
4
3
2
1
Tms
Tmw
hx
0.21
1.94
0.20
1.83
1.91
Re
Gr
Ra
Pr
25.4
9.38e-04
50.8
1.88e-03
101.6
3.75e-03
20.71
20 61
20 51
20.41
20 81
20 91
20 81
20 61
20 61
21 01
20.58
19.43
936.33
20.75
19.45
832.97
21 31
21 11
21.01
21 11
21 11
21.11
19.51
675.91
Nigit
W(Nux)
Error(%)
0.18
0.00575
0.0288
0.009586
2.73685
203.2
7.51e-03
21.91
21 71
21 61
21 71
21 31
21.64
19.62
534.31
401.0%
1.39
9.35
1.12
8.47
0.77
7.17
711.2
5.28e-02
29 73
28 03
26 53
25.33
23 73
26.36
23.23
623.26
.
0.28
3.14
:::::::
0.52
6.06
406.4
2.98e-02
34 62
31 52
29 82
28 42
26 12
29.48
24.18
716.96
..48:,
0.23
2.04
$:::::::::::::::::::':::::::::::::A
711.2
5.21e-02
37 52
34 32
31 82
29 62
26 62
31.63
26.42
728.73
406.4
t
0.41
5.57
1320.8
9.67e-02
43 42
39 62
37 22
34 02
29 52
36.15
30.90
722.77
1:41m::::
0.24
2.07
20.5
31
129.7
37.02
2387.6
1.77e-01
37 13
35.43
33 23
31.33
28 73
32.80
29.66
621.58
.
0.32
3.36
Tin(c)
Tout(c)
Weight(g)
Time(sec)
0.33
3.38
21.2
41.3
143.3
40.27
2387.6
1.75e-01
52 72
49 42
45 52
41 82
36.12
44.06
38.74
712.56
:20:
:00:
0.23
2.06
0.22
2.03
Tin(c)
Tout(c)
Weight(g)
Time(sec)
398.8527
2.87E+04
2.036E+05
7.0806
1.50e-02
22 91
22 31
22,31
22 01
21 41
22.14
19.83
467.92
1320.8
9.80e-02
32.63
30.83
29.03
27.53
25.33
28.73
25.57
617.10
0.33
3.39
211.33
1.20E+05
8.101E+05
6.742
Gr
2942
572.21
5......
Re
101.6
7.44e-03
406.4
3.02e-02
28 08
26 53
25 63
24 83
23 43
25.47
22.06
203.2
1.51e-02
26 28
25.53
:::57Z.,..
0.26
3.03
0.00575
0.0288
0.00958
2.73685
50.8
3.72e-03
28 02
27 42
27 32
27 52
25 82
27.19
21.57
675.52
5.78E+04
3.974E+05
6.87
Ra
101.6
7.54e-03
24 83
24 63
24 73
24 63
23 33
24.44
20.89
549.74
......
Tin(c)
Tout(c)
Weight(g)
Time(sec)
204.511
Re
19.4
22.3
229.5
32.71
711.2
2.63e-02
1320.8
4.88e-02
2387.6
8.82e-02
23 71
23 01
24 81
23 71
23 01
22 51
21 51
26.31
25 11
24 11
23.31
22 11
22.95
20.80
502.06
23.99
21.93
524.09
22 51
22 01
21 41
22.42
20.15
477.22
7:50::::
0.43
5.64
7:,
0.46
5.82
0:
0.50
5.98
200
5
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Side 5
4
3
2
1
Tms
Tmw
hx
W(Nux)
Error(%)
6
25.4
9.34e-04
22 15
21 85
21 65
21 65
22 05
21.82
19.55
923.24
1459_
0.70
4.79
a(m)
b(m)
Dh(m)
Lth(m)
L(mm)
Side 5
4
3
2
1
Tms
Tmw
hx
W(Nux)
Error(%)
25.4
9.40e-04
25 29
24 89
24 49
24 19
24.59
24.59
19.80
878.33
is0:
0.33
2.35
0.00575
0.0288
0.0095861
2.73685
50.8
1.87e-03
22 40
22 15
22 05
22 05
22 25
22.14
19.60
825,67
:-:1 .M4:.
0.57
4.37
Re
Gr
Ra
Pr
19.71
203.2
7.48e-03
24 05
23.75
23 75
23.85
22 40
23.57
19.92
696.33
5.72.39
101.6
3.74e-03
22 95
22 75
22 70
22 85
22 35
22.72
::'11 4)
0.00575
0.0288
0.0095861
2.73685
50.8
1.88e-03
25 89
25 29
25 09
25 09
24 59
25.13
19.91
805.43
Gr
Ra
Pr
'I04:
0.28
2.19
0.30
3.32
Re
101.6
3.76e-03
26.79
26 49
26 59
26 89
24 69
26.34
20.12
675.66
0.21
1.93
406.4
1.50e-02
26 15
24 75
24 25
23 85
22 55
24.15
20.33
548.37
203.2
7.52e-03
29 19
28 69
27.89
27 09
24.69
27.43
20.54
609.91
001:.
0.17
1.80
711.2
2.62e-02
27 05
25 55
24 65
23 85
22 55
24.51
20.96
589.05
9g4,
? <:9;1.Q.,. ::::.
0.42
3.82
Tin(c)
Tout(c)
Weight(g)
Time(sec)
401.5463
5.63E+04
3.976E+05
7.0618
0.28
3.23
0.32
3.38
406.4
29 29
28 29
27 19
24 99
28.00
21.38
634.69
'' Ca
0.19
1.85
::::::::APT::
0.36
3.58
Tin(c)
Tout(c)
Weight(g)
Time(sec)
401.34
1.15E+05
8.094E+05
7.0242
1.50e-02
32,09
1320.8
4.86e-02
28 95
26 95
25 65
24 55
22.85
25.49
22.20
637.34
19.5
25.1
262
37.18
2387.6
8.78e-02
32 05
29 85
27 85
26 45
24 15
27.69
24.39
633.64
00
0.36
3.56
19.7
31
265.6
37.89
711.2
2.63e-02
33 89
31 39
28.79
27 29
24.99
28.80
22.64
681.90
1320.8
4.89e-02
38 09
33 69
31 29
29 19
25 79
31.00
25.15
718.86
2387.6
8.83e-02
43 69
39 59
36.69
33 29
0.21
0.23
2.02
0.21
It
1.94
28.19.
35.60
29.56
695.62
1.97
201
APPENDIX D
Multiple regression model printouts using
the SAS program for backward elimination
202
Whole heating. b/a=1
Backward Elimination Procedure for Dependent Variable NUX
Step 0
All Variables Entered
DF
Regression 8
Error
30
Total
38
Variable
INTERCEP
Z
Z2
Z3
RA
RE
I1
12
13
Sum of Squares Mean Square
Regression 7
Error
31
Total
Variable
INTERCEP
Z
Z2
Z3
RA
Il
12
13
38
F
Prob>F
271.17444815
6.66883040
277.84327855
33.89680602
0.22229435
152.49 0.0001
Parameter
Estimate
Standard
Error
Type II
Sum of Squares F
17.87695829
8.38833480
1.59459269
0.07563550
0.00018050
-0.00527024
0.00005623
0.00000435
-0.00111985
3.61515692
1.99720961
0.41980579
0.02998997
0.00004398
0.00367342
0.00001817
0.00000178
0.00075869
5.43577367
3.92132316
3.20723741
1.41392901
3.74413838
0.45755827
2.12861952
1.32274888
0.48429738
Step 1 Variable RE Removed
DF
R-square = 0.97599787 C(p) = 9.00000000
24.45
17.64
14.43
6.36
16.84
2.06
9.58
5.95
2.18
Prob>F
0.0001
0.0002
0.0007
0.0172
0.0003
0.1617
0.0042
0.0208
0.1504
R-square = 0.97435105 C(p) = 9.05834418
Sum of Squares
Mean Square
F
270.71688987
7.12638868
277.84327855
38.67384141
0.22988351
168.23 0.0001
Parameter
Estimate
Standard
Error
15.13359953
7.97578310
1.60795121
0.07371401
0.00015523
0.00004760
0.00000367
-0.00008085
3.11999611
2.00985466
0.42680674
0.03046718
0.00004098
0.00001744
0.00000175
0.00023001
Prob>F
Type II
Sum of Squares F
5.40859150
3.62013842
3.26280412
1.34568471
3.29812135
1.71297385
1.01136193
0.02840280
23.53
15.75
14.19
5.85
14.35
7.45
4.40
0.12
Prob>F
0.0001
0.0004
0.0007
0.0216
0.0007
0.0104
0.0442
0.7276
203
Step 2 Variable 13 Removed
DF
Regression 6
Error
32
Total
38
Variable
INTERCEP
Z
Z2
Z3
RA
12
R-square = 0.97424882 C(p) = 7.18611529
Sum of Squares Mean Square
F
Prob >F
270.68848708
7.15479147
277.84327855
45.11474785
0.22358723
201.78 0.0001
Parameter
Estimate
Standard
Error
Type II
Sum of Squares F
15.07075908
7.87803096
1.59755070
0.07287733
0.00015525
0.00004741
0.00000365
3.07191719
1.96307327
0.41990855
0.02995521
0.00004042
0.00001719
0.00000172
5.38143684
3.60088580
3.23628578
1.32338782
3.29902382
1.70088218
1.00420880
Bounds on condition number:
6211.849,
24.07
16.11
14.47
5.92
14.75
7.61
4.49
69898.99
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Variable Number Partial
Step
Removed
In
R**2
Model
R**2
0.0016
0.0001
0.9744
0.9742
1
RE
7
2
13
6
C(p)
9.0583
7.1861
F
Prob>F
2.0583 0.1617
0.1236 0.7276
Prob>F
0.0001
0.0003
0.0006
0.0207
0.0005
0.0095
0.0419
204
Whole heating, b/a=2
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.98914104 C(p) = 9.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
DF
Regression 8
Error
Total
38
46
Variable
INTERCEP
Z
Z2
Z3
RA
RE
12
13
Sum of Squares Mean Square
301.57566490
3.31074883
304.88641373
37.69695811
0.08712497
Parameter
Estimate
Standard
Error
16.05385904
7.31673138
1.42620810
0.07004299
0.00006075
-0.00146167
0.00001839
0.00000149
-0.00032574
1.52029219
1.04421715
0.23224825
0.01666265
0.00000859
0.00125676
0.00000310
0.00000027
0.00024271
Bounds on condition number:
Step 1 Variable RE Removed
DF
Regression 7
Error
39
Total
Variable
INTERCEP
Z
Z2
Z3
RA
12
13
46
7053.968,
F
Prob>F
432.68 0.0001
Type II
Sum of Squares F
9.71510279
4.27755118
3.28551555
1.53951533
4.35959363
0.11785121
3.06234488
2.63716053
0.15692186
Prob>F
111.51 0.0001
49.10
37.71
17.67
50.04
1.35
35.15
30.27
1.80
0.0001
0.0001
0.0002
0.0001
0.2521
0.0001
0.0001
0.1875
120252.4
R-square = 0.98875450 C(p) = 8.35266854
Prob>F
Sum of Squares Mean Square
F
301.45781369
3.42860004
304.88641373
43.06540196
0.08791282
489.86 0.0001
Parameter
Estimate
Standard
Error
Type II
Sum of Squares F
Prob>F
15.76883914
7.30149352
1.42086580
0.06793874
0.00005499
0.00001661
0.00000136
-0.00005339
1.50717928
1.04884527
0.23325034
0.01663885
0.00000705
0.00000271
0.00000025
0.00006415
9.62325144
4.26042354
3.26222393
1.46568418
5.35028339
3.30552689
2.64007028
0.06091058
0.0001
0.0001
0.0001
0.0002
0.0001
0.0001
0.0001
0.4103
Bounds on condition number:
7051.209,
97461.65
109.46
48.46
37.11
16.67
60.86
37.60
30.03
0.69
205
R-square = 0.98855472 C(p) = 7.05178587
Step 2 Variable 13 Removed
DF
Regression 6
40
Error
46
Total
Variable
INTERCEP
Z
Z2
Z3
RA
12
Prob>F
Sum of Squares Mean Square
F
301.39690312
3.48951061
304.88641373
50.23281719
0.08723777
575.82 0.0001
Parameter
Estimate
Standard
Error
Type II
Sum of Squares F
15.59256842
7.14308211
1.38887570
0.06542926
0.00005462
0.00001633
0.00000133
1.48648757
1.02746713
0.22917736
0.01630050
0.00000701
0.00000268
0.00000025
9.59880760
4.21638193
3.20396573
1.40555207
5.29913837
3.24526302
2.58328106
Bounds on condition number:
6859.779,
110.03 0.0001
48.33 0.0001
36.73 0.0001
16.11 0.0003
60.74 0.0001
37.20 0.0001
29.61 0.0001
81358.29
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Step
Variable Number
Removed
In
1
RE
2
13
7
6
Partial
R**2
Model
R**2
C(p)
0.0004
0.0002
0.9888
0.9886
8.3527
7.0518
F
Prob>F
Prob>F
1.3527 0.2521
0.6929 0.4103
206
Whole heating. b/a=3
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.99400933 C(p) = 9.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
DF
Regression
Error
Total
Variable
INTERCEP
Z2
Z3
RA
RE
II
12
13
Sum of Squares
Mean Square
215.06136443
1.29612581
216.35749024
26.88267055
0.04320419
8
30
38
Parameter
Estimate
16.86140101
8.15911618
1.71427700
0.09489930
0.00005861
-0.00162825
0.00001953
0.00000168
-0.00040144
Bounds on condition number:
Step 1
Regression
Error
Total
Variable
INTERCEP
Z2
Z3
RA
12
13
Type II
Sum of Squares
1.46116370
0.87582880
0.19185094
0.01386279
0.00000503
0.00243010
0.00000183
0.00000017
0.00048950
5.75327947
3.74950290
3.44953101
2.02465213
5.87293510
0.01939623
4.91376006
4.27029809
0.02905767
Prob>F
0.0001
0.0001
0.0001
0.0001
135.93 0.0001
0.5080
0.45
113.73 0.0001
98.84 0.0001
0.4186
0.67
133.16
86.79
79.84
46.86
R-square = 0.99391968 C(p) = 7.44894328
Mean Square
215.04196820
1.31552204
216.35749024
30.72028117
0.04243619
Parameter
Estimate
16.39480262
8.09541247
1.72109596
0.09526029
0.00005675
0.00001918
0.00000168
-0.00008858
Bounds on condition number:
F
112277.4
Sum of Squares
7
31
38
Prob>F
622.22 0.0001
Standard
Error
7473.055,
Variable RE Removed
DF
F
723.92 0.0001
Standard
Error
Type II
Sum of Squares
1.27307510
0.86287997
0.18987041
0.01372865
0.00000415
0.00000174
0,00000017
0.00014560
7.03786243
3.73519794
3.48684013
2.04317032
7.91807660
5.16731490
4.27297357
0.01570699
7452.026,
94204.85
Prob>F
F
F
Prob>F
165.85 0.0001
88.02
82.17
48.15
0.0001
0.0001
0.0001
186.59 0.0001
121.77 0.0001
100.69 0.0001
0.37
0.5474
207
R-square = 0.99384709 C(p) = 5.81249579
Step 2 Variable 13 Removed
Regression
Error
Total
DF
Sum of Squares
Mean Square
6
32
38
215.02626121
1.33122904
216.35749024
35.83771020
0.04160091
Z2
Z3
RA
II
12
861.46 0.0001
Parameter
Estimate
Standard
Error
Type II
Sum of Squares
F
Prob>F
16.33315032
8.01813342
1.70935567
0.09436341
0.00005664
0.00001903
0.00000168
1.25648389
0.84503788
0.18701901
0.01351427
0.00000411
0.00000170
0.00000017
7.02957179
3.74538964
3.47533112
2.02826571
7.90241106
5.19186962
4.25772398
168.98
90.03
83.54
48.76
189.96
124.80
102.35
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
Variable
INTERCEP
Z
Prob>F
F
7375.049,
Bounds on condition number:
79603.85
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Step
Variable
Removed
Number Partial
In R**2
1
RE
7
2
13
6
Model
R**2
0.0001 0.9939
0.0001 0.9938
C(p)
7.4489
5.8125
F
Prob>F
0.4489 0.5080
0.3701 0.5474
208
Whole heating. b /a =4
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.98554057 C(p) = 9.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
DF
Regression 8
Error
39
47
Total
Variable
INTERCEP
Z2
Z3
RA
RE
11
12
13
Standard
Error
13.10411310
5.61679979
1.13851275
0.05122819
0.00001680
0.00314826
0.00000372
0.00000020
0.00047820
1.31561606
0.83626099
0.18406763
0.01330774
0.00000296
0.00188463
0.00000133
0.00000014
0.00038755
6202.164,
Sum of Squares
Regression 7
Error
40
Total
47
141.10569894
2.15229383
143.25799276
Z2
Z3
RA
RE
12
F
Prob>F
99.21 0.0001
45.11 0.0001
38.26 0.0001
14.82 0.0004
32.21 0.0001
2.79 0.1028
7.87 0.0078
2.04 0.1611
1.52 0.2246
5.26941344
2.39607026
2.03201211
0.78707294
1.71080878
0.14821522
0.41807749
0.10839748
0.08086533
87814.72
R-square = 0.98497610 C(p) = 8.52249901
DF
INTERCEP
Prob>F
332.28 0.0001
Type II
Sum of Squares
Parameter
Estimate
Step 1 Variable 13 Removed
Variable
17.64832053
0.05311355
141.18656426
2.07142850
143.25799276
Bounds on condition number:
F
Mean Square
Sum of Squares
Mean Square
20.15795699
0.05380735
F
Prob>F
374.63 0.0001
Parameter
Estimate
Standard
Error
Type II
Sum of Squares
F
Prob>F
13.74774882
5.72639110
1.13336674
0.05138629
0.00001765
0.00093395
0.00000397
0.00000021
1.21564953
0.83694438
0.18521836
0.01339375
0.00000290
0.00057938
0.00000132
0.00000014
6.88157695
2.51889685
2.01471877
0.79201172
2.00103724
0.13981544
0.48623497
0.12287573
127.89
46.81
37.44
14.72
37.19
2.60
9.04
2.28
0.0001
0.0001
0.0001
0.0004
0.0001
0.1148
0.0046
0.1386
Bounds on condition number:
6198.98,
75896.21
209
R-square = 0.98411838 C(p) = 8.83595254
F Prob>F
Mean Square
Sum of Squares
423.43
0.0001
23.49713720
140.98282320
0.05549194
2.27516956
Step 2 Variable 12 Removed
DF
6
Regression
Error
41
Total
47
143.25799276
Parameter
Estimate
Variable
INTERCEP
Z2
Z3
RA
RE
14.33701332
5.91383627
1.12210757
0.04712849
0.00001350
0.00090687
0.00000199
Bounds on condition number:
Step 3
Regression
Error
5
140.85087041
2.40712235
143.25799276
28.17017408
0.05731244
Variable
Z2
Z3
RA
49.50
35.65
12.56
0.0001
0.0001
0.0010
219.01 0.0001
2.38
0.1307
117.23 0.0001
6051.458,
F
Prob>F
491.52 0.0001
Type II
Sum of Squares
Standard
Error
14.37010514 1.18812296
0.84435594
5.71727812
0.18886766
1.07891100
0.01336299
0.04407367
0.00000091
0.00001378
0.00000019
0.00000200
Bounds on condition number:
150.34 0.0001
R-square = 0.98319729 C(p) = 9.32030523
Mean Square
Parameter
Estimate
Prob>F
56453.59
Sum of Squares
42
47
F
8.34251828
2.74683490
1.97808893
0.69704555
12.15338492
0.13195279
6.50547202
DF
Total
INTERCEP
1.16929761
0.84055856
0.18794317
0.01329744
0.00000091
0.00058810
0.00000018
6188.949,
Variable RE Removed
Type II
Sum of Squares
Standard
Error
8.38389801
2.62770364
1.87027272
0.62344785
13.22756252
6.56270440
F
Prob>F
146.28 0.0001
45.85
32.63
10.88
0.0001
0.0001
0.0020
230.80 0.0001
114.51 0.0001
45987.97
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Variable Number
In
Step
Removed
1
13
7
2
3
12
6
RE
5
Partial
R**2
Model
R**2
0.0006 0.9850
0.0009 0.9841
0.0009 0.9832
C(p)
8.5225
8.8360
9.3203
F
Prob>F
1.5225 0.2246
2.2836 0.1386
2.3779 0.1307
210
Whole heating. b/a=5
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.98941280 C(p) = 7.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
DF
Regression
Error
Total
Z3
RA
12
Standard
Error
4.25390493
-0.78909930
-0.24387619
-0.04536984
0.00001475
0.00000311
0.00000011
0.82620983
0.58811284
0.13888535
0.01081140
0.00000305
0.00000143
0.00000015
3186.64,
Bounds on condition number:
Regression
Error
Total
Variable
INTERCEP
Z
Z2
Z3
RA
138.41992165
1.50244557
139.92236721
27.68398433
0.04418958
Parameter
Estimate
Standard
Error
4.50683058
-0.69955690
-0.24664746
-0.04712420
0.00001277
0.00000214
0.73329993
0.56890403
0.13773831
0.01042128
0.00000097
0.00000020
Bounds on condition number:
0.0001
0.1888
0.0884
0.0002
23.36 0.0001
4.75
0.47
0.0366
0.4982
36655.77
Mean Square
5
Prob>F
26.51
1.80
3.08
17.61
1.19000639
0.08081583
0.13841412
0.79054247
1.04868193
0.21313872
0.02105905
Sum of Squares
34
39
F
R-square = 0.98926229 C(p) = 5.46912030
Step 1 Variable 12 Removed
DF
514.00 0.0001
Type II
Sum of Squares
Parameter
Estimate
Variable
INTERCEP
Z
Z2
23.07349678
0.04489050
138.44098069
1.48138652
139.92236721
6
33
39
Prob>F
F
Mean Square
Sum of Squares
3183.935,
F
626.48 0.0001
Type II
Sum of Squares
1.66916399
0.06681714
0.14169797
0.90357874
7.61735720
4.83665752
24340.96
Prob>F
F Prob>F
37.77 0.0001
1.51
3.21
0.2273
0.0822
20.45 0.0001
172.38 0.0001
109.45 0.0001
211
R-square = 0.98878476 C(p) = 4.95756756
Step 2 Variable Z Removed
Regression
Error
Total
DF
Sum of Squares
Mean Square
4
138.35310451
1.56926271
139.92236721
34.58827613
0.04483608
35
39
Parameter
Estimate
Variable
1NTERCEP
Z2
Z3
RA
Standard
Error
5.38310460
-0.07895599
-0.03468390
0.00001282
0.00000214
Bounds on condition number:
0.17419524
0.01948971
0.00251856
0.00000098
0.00000021
62.82862,
F
Prob>F
771.44 0.0001
Type II
Sum of Squares
42.81743771
0.73584633
8.50310399
7.68686984
4.84454231
F
Prob>F
954.98
16.41
189.65
171.44
108.05
0.0001
0.0003
0.0001
0.0001
0.0001
575.3773
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Step
Variable Number
In
Removed
1
12
5
2
Z
4
Partial
R**2
0.0002
0.0005
Model
R**2
0.9893
0.9888
C(p)
5.4691
4.9576
F
Prob>F
0.4691 0.4982
1.5121 0.2273
212
Three surfaces heating. b/a=1
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.97813283 C(p) = 9.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
Regression
Error
Total
DF
Sum of Squares
8
193.60930324
4.32833676
197.93764000
31
39
INTERCEP
Z
Z2
Z3
RA
RE
II
12
13
Standard
Error
21.63453611
11.93024861
2.51152880
0.15002443
0.00022404
-0.00211676
0.00007413
0.00000640
-0,00059939
2.63984512
1.58216883
0.33557607
0.02370406
0.00002596
0.00213994
0.00001150
0.00000119
0.00044171
Bounds on condition number:
6589.445,
Error
Total
Variable
INTERCEP
Z2
Z3
RA
II
12
13
173.33 0.0001
Type II
Sum of Squares
F
Prob>F
67.16 0.0001
56.86 0.0001
9.37772793
7.93876028
7.82084550
5.59291166
10.40352334
0.13661548
5.80501616
4.06882231
0.25709227
56.01 0.0001
40.06 0.0001
74.51 0.0001
0.98
41.58
29.14
1.84
0.3302
0.0001
0.0001
0.1846
94147.7
DF
Sum of Squares
Mean Square
7
32
39
193.47268776
4.46495224
197.93764000
27.63895539
0.13952976
F
Standard
Error
Type II
Sum of Squares
20.57035742
11.82615003
2.53784374
0.15121634
0.00022003
0.00007302
0.00000634
-0.00018334
2.40986637
1.57813342
0.33440728
0.02366544
0.00002563
0.00001144
0.00000118
0.00013488
10.16633910
7.83549027
8.03609656
5.69685471
10.28552871
5.68680845
4.00364355
0.25780618
6548.032,
81298.88
Prob>F
198.09 0.0001
Parameter
Estimate
Bounds on condition number:
Prob>F
R-square = 0.97744263 C(p) = 7.97845434
Step 1 Variable RE Removed
Regression
24.20116291
0.13962377
Parameter
Estimate
Variable
F
Mean Square
F
Prob>F
72.86 0.0001
56.16 0.0001
57.59 0.0001
40.83 0.0001
73.72 0.0001
40.76 0.0001
28.69 0.0001
1.85
0.1836
213
R-square = 0.97614017 C(p) = 7.82488914
Step 2 Variable 13 Removed
Regression
Error
Total
DF
Sum of Squares
6
193.21488158
4.72275842
197.93764000
33
39
Parameter
Estimate
Variable
INTERCEP
Z
Z2
Z3
RA
11
12
20.25698244
11.46698410
2.47924895
0.14669292
0.00021906
0.00007229
0.00000628
Bounds on condition number:
32.20248026
0.14311389
225.01 0.0001
Standard
Error
Type II
Sum of Squares
2.42942757
1.57571204
0.33584946
0.02372932
0.00002594
0.00001157
0.00000120
9.95000695
7.57925473
7.79888933
5.46927115
10.20328458
5.58520820
3.93569490
6439.227,
Prob>F
F
Mean Square
F
Prob>F
69.53 0.0001
52.96 0.0001
54.49
38.22
0.0001
0.0001
71.29 0.0001
39.03
27.50
0.0001
0.0001
68469.3
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Step
Variable
Removed
1
RE
2
13
Number Partial
In
R**2
7
6
0.0007
0.0013
Model
R**2
C(p)
0.9774
0.9761
7.9785
7.8249
F
Prob>F
0.9785 0.3302
1.8477 0.1836
214
Three surfaces heating, b/a=2
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.96536842 C(p) = 7.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
Regression
Error
Total
Variable
INTERCEP
Z2
Z3
RA
12
DF
Sum of Squares
Mean Square
6
46
52
303.50989816
10.88809884
314.39799700
50.58498303
0.23669780
Standard
Error
21.79279582
11.50123886
2.29940794
0.12157419
0.00011418
0.00003264
0.00000240
2.42786185
1.63179399
0.35733326
0.02520195
0.00001005
0.00000390
0.00000037
6452.103,
19.07094361
11.75852404
9.80120310
5.50818723
30.55650132
16.57405020
10.17214136
72742.13
All variables left in the model are significant at the 0.1000 level.
Prob>F
213.71 0.0001
Type II
Sum of Squares
Parameter
Estimate
Bounds on condition number:
F
F
Prob>F
80.57 0.0001
49.68 0.0001
41.41
23.27
0.0001
0,0001
129.09 0.0001
70.02 0.0001
42.98 0.0001
215
Three surfaces heating. b/a=3
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.98423579 C(p) = 9.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
Regression
Error
Total
DF
Sum of Squares
Mean Square
8
31
273.31979712
4.37768263
277.69747974
34.16497464
0.14121557
39
Parameter
Estimate
Variable
INTERCEP
Z2
Z3
RA
RE
12
13
30.52686060 2.37398170
16.74632664 1.53552817
0.32888597
3.42711405
0.19336736
0.02297721
0.00000818
0.00008933
0.00223189
0.00210126
0.00002915
0.00000322
0.00000030
0.00000236
0.00040885
0.00052887
7231.763,
Bounds on condition number:
DF
Variable
1NTERCEP
Z2
Z3
RA
12
13
Type II
Sum of Squares
23.35028215
16.79600973
15.33377197
10.00125322
16.84084883
0.15931883
11.60655271
8.68060772
0.23629597
Mean Square
39.02292547
0.14178130
273.16047829
4.53700145
277.69747974
Parameter
Estimate
Standard
Error
Type II
Sum of Squares
30.99405739
16.69428187
3.39862029
0.19274694
0.00009286
0.00003011
0.00000242
0.00011264
2.33754594
1.53781731
0.32844601
0.02301575
0.00000749
0.00000309
0.00000030
0.00011682
24.92616684
16.70878732
15.18085523
9.94360404
21.77745145
13.46357773
9.36860564
0.13181115
Bounds on condition number:
F
Prob>F
165.35 0.0001
118.94 0.0001
108.58 0.0001
0.0001
70.82
119.26 0.0001
0.2964
0.0001
0.0001
0.2054
1.13
82.19
61.47
1.67
107509.7
Sum of Squares
7
32
39
241.93 0.0001
R-square = 0.98366207 C(p) = 8.12819590
Step 1 Variable RE Removed
Regression
Error
Total
Standard
Error
Prob>F
F
7183.649,
91824.85
Prob>F
F
275.23 0.0001
F
Prob>F
175.81 0.0001
117.85 0.0001
107.07 0.0001
70.13
0.0001
153.60 0.0001
94.96
66.08
0.93
0.0001
0.0001
0.3422
216
Rsquare = 0.98318741 C(p) = 7.06159969
Step 2 Variable 13 Removed
DF
Regression
Error
Total
Sum of Squares
Mean Square
273.02866715
4.66881260
277.69747974
45.50477786
0.14147917
6
33
39
Parameter
Estimate
Standard
Error
31.26450293
16.95365486
3.45015833
0.19675057
0.00009353
0.00003054
0.00000245
2.31818288
1.51249263
0.32372197
0.02261396
0.00000745
0.00000306
0.00000029
Variable
INTERCEP
Z2
Z3
RA
12
Bounds on condition number:
6993.392,
Prob>F
F
321.64 0.0001
Type II
Sum of Squares
25.73357771
17.77594288
16.07038207
10.70955386
22.28670500
14.13185713
9.77088301
F
Prob>F
181.89
125.64
113.59
75.70
157.53
99.89
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
69.06
76573.03
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NW(
Variable Number Partial
Step
Removed
1
RE
2
13
In
R**2
Model
R**2
C(p)
7
6
0.0006
0.0005
0.9837
0.9832
8.1282
7.0616
F
Prob>F
1.1282 0.2964
0.9297 0.3422
217
Three surfaces heating, b/a=4
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.97518713 C(p) = 9.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
Regression
Error
Total
DF
Sum of Squares
Mean Square
8
33
41
153.75385182
3.91214511
157.66599693
19.21923148
0.11854985
INTERCEP
Z
Z2
Z3
RA
RE
12
13
Bounds on condition number:
Step 1
Variable
INTERCEP
Z
Z3
RA
RE
11
13
7365.072,
1.52959707
0.16785043
0.04893632
0.07414222
11.14267921
0.16162198
4.45078600
2.07863476
0.12215969
Sum of Squares
Mean Square
7
34
153.70491550
3.96108143
157.66599693
21.95784507
0.11650239
Parameter
Estimate
12.90
1.42
0.41
0.63
93.99
1.36
37.54
17.53
0.0011
0.2426
0.5250
0.4347
0.0001
0.2513
0.0001
0.0002
0.3174
1.03
1.11067298
0.35847392
0.00575563
0.00000533
0.00206865
0.00000254
0.00000028
0.00044703
1031.367,
F
4.68948495
0.78822572
6.57101105
11.13174091
0.15394442
4.40371960
2.03163128
0.12496785
13611.56
Prob>F
188.48 0.0001
Type II
Sum of Squares
Standard
Error
7.04662816
0.93242828
-0.04322566
0.00005209
0.00237794
0.00001560
0.00000119
0.00046298
Bounds on condition number:
Prob>F
104936.8
DF
41
F
R-square = 0.97487676 C(p) = 7.41279104
Variable Z2 Removed
Regression
Error
Total
12
2.32643379
1.65644600
0.39082962
0.03040855
0.00000542
0.00208891
0.00000258
0.00000029
0.00045101
8.35658260
1.97100626
0.25110341
-0.02404795
0.00005252
0.00243904
0.00001581
0.00000121
0.00045783
Prob>F
162.12 0.0001
Type II
Sum of Squares
Standard
Error
Parameter
Estimate
Variable
F
F
40.25
6.77
56.40
95.55
1.32
37.80
17.44
1.07
Prob>F
0.0001
0.0137
0.0001
0.0001
0.2584
0.0001
0.0002
0.3077
218
R-square = 0.97408414 C(p) = 6.46692858
Step 2 Variable 13 Removed
Regression
Error
Total
DF
6
35
41
Parameter
Estimate
Variable
INTERCEP
Z
Z3
RA
RE
II
12
Bounds on condition number:
Error
Total
5
36
41
Variable
INTERCEP
Z3
RA
12
219.25 0.0001
Prob>F
F
61.71 0.0001
7.20413526
1.04869683
6.45819076
11.70377544
0.03796348
4.52656838
2.03451986
8.98 0.0050
55.32 0.0001
100.25 0.0001
0.33 0.5722
38.77 0.0001
17.43 0.0002
11145.43
Mean Square
30.70839683
0.11455591
Sum of Squares
153.54198417
4.12401276
157.66599693
Parameter
Estimate
Standard
Error
7.73367759
1.04178290
-0.04234042
0.00005283
0.00001570
0.00000118
0.93574600
0.34158397
0.00560432
0.00000523
0.00000251
0.00000028
Bounds on condition number:
Prob>F
R-square = 0.97384336 C(p) = 4.78716081
Step 3 Variable RE Removed
Regression
0.96886487
0.34508322
0.00566729
0.00000528
0.00054569
0.00000253
0.00000028
1026.568,
F
Type II
Sum of Squares
Standard
Error
7.61090605
1.03426312
-0.04215149
0.00005287
0.00031118
0.00001578
0.00000119
DF
Mean Square
25.59665794
0.11674427
Sum of Squares
153.57994765
4.08604928
157.66599693
1023.271,
F
Prob>F
268.06 0.0001
Type II
Sum of Squares
7.82480753
1.06555772
6.53856047
11.68913229
4.49413965
2.00956901
Prob>F
F
68.31
9.30
57.08
0.0001
0.0043
0.0001
102.04 0.0001
39.23
17.54
0.0001
0.0002
9254.278
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Step
Variable Number Partial
Removed
In
R**2
1
Z2
7
2
13
6
3
RE
5
0.0003
0.0008
0.0002
Model
R**2
C(p)
0.9749
0.9741
0.9738
7.4128
6.4669
4.7872
F
Prob>F
0.4128 0.5250
1.0727 0.3077
0.3252 0.5722
219
Three surfaces heating. b/a=5
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.91812526 C(p) = 7.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
Regression
Error
Total
Variable
INTERCEP
Z2
Z3
RA
II
12
DF
Sum of Squares
Mean Square
6
63
69
226.92316478
20.23610123
247.15926601
37.82052746
0.32120796
14.24619290
5.82156671
1.30049235
0.07142426
0.00003355
0.00001036
0.00000081
Bounds on condition number:
2.33508414
1.52486729
0.32986687
0.02345036
0.00000536
0.00000220
0.00000021
5190.967,
11.95580624
4.68168244
4.99257387
2.97974144
12.56496730
7.14556421
4.63332537
57662.41
All variables left in the model are significant at the 0.1000 level.
Prob>F
117.74 0.0001
Type II
Sum of Squares
Standard
Error
Parameter
Estimate
F
F
Prob>F
37.22 0.0001
14.58 0.0003
15.54 0.0002
9.28
0.0034
39.12 0.0001
22.25 0.0001
14.42 0.0003
220
Side surface heating. b/a=1
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.98380533 C(p) = 8.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
DF
Regression
Error
Total
Z2
Z3
Z4
RA
12
0.44168666
-6.42088702
-2.68974002
-0.46688343
-0.02388249
0.00014213
0.00005043
0.00000485
225357.2,
Bounds on condition number:
Error
Total
Variable
INTERCEP
Z2
Z3
Z4
RA
12
1.83 0.1843
2.30 0.1374
1.95 0.1708
33.58 0.0001
19.82 0.0001
15.83 0.0003
3133510
Sum of Squares
Mean Square
6
484.17968913
8.25997470
492.43966383
80.69661486
0.20649937
Parameter
Estimate
0.86458397
0.22183827
0.05801117
0.00427385
0.00002462
0.00001135
0.00000122
7925.023,
F
Prob>F
390.78 0.0001
Type II
Sum of Squares
Standard
Error
6.61756121
-0.35453366
-0.10986119
-0.00430800
0.00014073
0.00004935
0.00000470
Bounds on condition number:
0.01 0.9340
1.39 0.2449
0.00141971
0.28507733
0.37355583
0.47037197
0.39816274
6.86671291
4.05315750
3.23666355
DF
40
46
F Prob>F
R-square = 0.98322642 C(p) = 7.39412650
Step 1 Variable Z Removed
Regression
5.30083647
5.43805927
1.99004452
0.30783500
0.01711510
0.00002453
0.00001133
0.00000122
Prob>F
338.46 0.0001
Type II
Sum of Squares
Standard
Error
Parameter
Estimate
Variable
INTERCEP
Z
69.20925235
0.20448455
484.46476646
7.97489737
492.43966383
7
39
46
F
Mean Square
Sum of Squares
F Prob>F
12.09763021
0.52742505
0.74060058
0.20981295
58.58 0.0001
6.74768216
3.90746444
3.06780353
32.68 0.0001
18.92 0.0001
14.86 0.0004
82058.95
2.55 0.1179
3.59 0.0655
1.02 0.3195
221
R-square = 0.98280035 C(p) = 6.42018424
Step 2 Variable Z4 Removed
Regression
Error
Total
DF
Sum of Squares
Mean Square
5
41
483.96987618
8.46978765
492.43966383
96.79397524
0.20658019
46
INTERCEP
Z2
Z3
RA
12
6.02786997
-0.15227548
-0.05285004
0.00014504
0.00005140
0.00000490
873.6657,
Bounds on condition number:
INTERCEP
Z3
RA
12
0.1152
Standard
Error
Type II
Sum of Squares
5.08565811
-0.03226219
0.00018024
0.00006790
0.00000665
0.25495629
0.00169198
0.00001066
0.00000451
0.00000052
85.30754147
77.95102174
61.24889696
48.69739839
35.49103225
Bounds on condition number:
137.0981,
Prob>F
563.71 0.0001
120.85871715
0.21439989
Parameter
Estimate
Variable
2.59
16.78 0.0002
35.77 0.0001
21.20 0.0001
16.64 0.0002
F
Mean Square
483.43486859
9.00479524
492.43966383
42
46
89.62 0.0001
18.51446388
0.53500758
3.46712879
7.38988427
4.37966320
3.43753646
11263.39
Sum of Squares
4
Regression
Error
Total
Prob>F
F
R-square = 0.98171391 C(p) = 7.03655594
Step 3 Variable Z2 Removed
DF
0.63672608
0.09462248
0.01290044
0.00002425
0.00001116
0.00000120
Prob>F
468.55 0.0001
Type II
Sum of Squares
Standard
Error
Parameter
Estimate
Variable
F
F
Prob>F
397.89 0.0001
363.58 0.0001
285.68 0.0001
227.13 0.0001
165.54 0.0001
1009.468
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Variable Number Partial
Step
1
2
3
Removed
In
Z
Z4
Z2
6
5
4
R**2
0.0006
0.0004
0.0011
Model
R**2
0.9832
0.9828
0.9817
C(p)
7.3941
6.4202
7.0366
F Prob>F
1.3941 0.2449
1.0160 0.3195
2,5898 0.1152
222
Side surface eating. b/a=2
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.98053538 C(p) = 8.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
DF
Regression
Error
Total
7
30
37
Sum of Squares
244.89102101
4.86133563
249.75235664
Mean Square
34.98443157
0.16204452
Prob>F
F
215.89 0.0001
Type II
Standard
Parameter
Sum of Squares
Error
Estimate
0.00225740
7.23740005
0.85422005
0.36027778
-10.19660623 6.83839343
0.71342322
2.29470818
Z2
-4.81486270
1.07933877
0.32575453
-0.84072154
Z3
1.22299807
0.01665711
-0.04576098
Z4
2.66565999
0.00001027
0.00004167
RA
1.13411494
0.00000408
0.00001080
0.58289893
0.00000039
12
0.00000073
4836789
Bounds on condition number: 344552.8,
F Prob5F
Variable
INTERCEP
Step 1 Variable Z Removed
DF
6
Regression
Error
Total
31
37
0.01
2.22
4.40
6.66
7.55
0.9068
0.1464
0.0444
0.0150
0.0101
16.45 0.0003
7.00
3.60
0.0129
0.0675
R-square = 0.97909284 C(p) = 8.22332591
Sum of Squares
244.53074323
5.22161341
249.75235664
Mean Square
40.75512387
0.16843914
F Prob>F
241.96 0.0001
Type II
Standard
Parameter
Sum of Squares
Error
Estimate
25.38540729
11.55758140 0.94144814
7.59434703
0.20954407
-1.40701615
7.68044377
0.05351130
Z3
-0.36134093
5.26963010
0.00385691
Z4
-0.02157288
0.00001035
3.04393966
0.00004401
RA
0.00001158
0.00000413
1.32668658
0.00000039
0.69067531
12
0.00000080
Bounds on condition number: 8944.524, 99711.67
F
Prob>F
150.71 0.0001
Variable
INTERCEP
Z2
45.09 0.0001
45.60 0.0001
31.29 0.0001
18.07 0.0002
7.88
4.10
0.0086
0.0516
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Variable Number Partial
Step
1
Removed
Z
In
6
R**2
Model
R**2
C(p)
0.0014
0.9791
8.2233
F
Prob>F
2.2233 0.1464
223
Side surface heating. b/a=3
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.98309806 C(p) = 8.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
DF
Regression
Error
Total
7
31
38
Sum of Squares
311.41908761
5.35408031
316.77316791
Parameter
Estimate
18.77593074
7.10055004
0.91952458
-0.05524197
-0.00763799
0.00005065
0.00001564
0.00000125
Variable
INTERCEP
Z2
Z3
Z4
RA
12
Step 1
Variable Z3 Removed
DF
6
Regression
Error
Total
32
38
F Prob>F
257.59 0.0001
Type II
Standard
Sum of Squares
Error
0.78953574
8.78167119
0.13521612
8.02489491
0.02124302
2.62190431
0.00396217
0.36472375
0.02991833
0.01835151
10.32022961
0.00000655
5.69814818
0.00000272
3.78075181
0.00000027
444265.1,
Bounds on condition number:
Mean Square
44.48844109
0.17271227
F
Prob>F
4.57 0.0405
0.78 0.3831
0.12 0.7282
0.02 0.8806
0.17 0.6801
59.75 0.0001
32.99 0.0001
21.89 0.0001
6264546
R-square = 0.98308555 C(p) = 6.02294089
Sum of Squares
311.41512543
5.35804248
316.77316791
Mean Square
51.90252091
0.16743883
Prob>F
F
309.98 0.0001
Type II
Parameter
Standard
Error
Sum of Squares
Estimate
15.16647553
20.06587730
2.10835581
8.03214808
1.19865039
8.30195352
8.29352085
0.18693182
Z2
1.31560231
2.42792077
Z4
-0.00486538
0.00127770
10.48855253
0.00005076
0.00000641
RA
5.75667074
0.00001567
0.00000267
3.80064864
0.00000026
12
0.00000125
1844.37, 28671.87
Bounds on condition number:
Variable
INTERCEP
F
Prob>F
90.58 0.0001
47.97 0.0001
49.53 0.0001
14.50 0.0006
62.64 0.0001
34.38 0.0001
22.70 0.0001
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Step
1
Variable
Removed
Z3
Number
In
Partial
R**2
Model
R**2
C(p)
6
0.0000
0.9831
6.0229
F
Prob>F
0.0229 0.8806
224
Side surface heating. bla =4
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.96159672 C(p) = 8.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
DF
Regression
Error
Total
Variable
INTERCEP
Z2
Z3
Z4
RA
12
Sum of Squares
7
35
42
236.83090217
9.45831350
246.28921567
Standard
Error
14.42064761
-0.34052473
-2,49597781
-0.67341043
-0.04529436
0.00003928
0.00001194
0.00000084
11.06334825
10.41558024
3.52154501
0.50542463
0.02603220
0.00001073
0.00000484
0.00000052
409230.8,
INTERCEP
Z2
Z3
Z4
RA
II
12
1.70 0.2009
0.00 0.9741
0.50 0.4832
1.78 0.1914
0.45913671
0.00028885
0.13575649
0.47972520
0.81811093
3.62255765
1.64786379
0.70734672
3.03 0.0907
13.41 0.0008
6.10 0.0186
2.62 0.1147
5868879
DF
Sum of Squares
Mean Square
6
36
42
236.83061331
9.45860235
246.28921567
39.47176889
0.26273895
Parameter
Estimate
1.47871879
0.35806889
0.08884814
0.00600369
0.00001032
0.00000466
0.00000050
13006.89,
F
26.24484894
11.62190973
14.37334920
14.41323797
3.81827956
1.73602006
0.74670805
132127.5
Prob>F
150.23 0.0001
Type II
Sum of Squares
Standard
Error
14.77901183
-2.38145896
-0.65715091
-0.04446688
0.00003935
0.00001197
0.00000084
Bounds on condition number:
F Prob>F
R-square = 0.96159555 C(p) = 6.00106888
Step 1 Variable Z Removed
Variable
125.20 0.0001
Type II
Sum of Squares
Parameter
Estimate
Bounds on condition number:
Regression
Error
Total
33.83298602
0.27023753
Pmb>F
F
Mean Square
F
Prob>F
99.89 0.0001
44.23 0.0001
54.71 0.0001
54.86 0.0001
14.53 0.0005
6.61 0.0144
2.84 0.1005
225
R-square = 0.95856371 C(p) = 6.76422341
Step 2 Variable I2 Removed
Regression
Error
Total
DF
Sum of Squares
Mean Square
5
236.08390526
10.20531040
246.28921567
47.21678105
0.27581920
37
42
Variable
INTERCEP
Z2
Z3
Z4
RA
Standard
Error
16.70239989
-2.80135332
-0.74752544
-0.04929578
0.00002260
0.00000418
0.96383327
0.26357242
0.07259514
0.00540605
0.00000285
0.00000060
8271.643,
171.19 0.0001
Type II
Sum of Squares
Parameter
Estimate
Bounds on condition number:
Prob>F
F
82.82823797
31.15733652
29.24565395
22.93418564
17.28127590
13.58370557
F
Prob>F
300.30 0.0001
112.96 0.0001
106.03 0.0001
83.15
62.65
49.25
0.0001
0.0001
0.0001
62702.21
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Variable Number
Step
Removed
In
1
Z
6
2
12
5
Partial
R**2
Model
R**2
C(p)
0.0000
0.0030
0.9616
0.9586
6.0011
6.7642
F
Prob>F
0.0011 0.9741
2.8420 0.1005
226
Side surface heating. b/a=5
Backward Elimination Procedure for Dependent Variable NUX
Step 0 All Variables Entered R-square = 0.95027200 C(p) = 8.00000000
NOTE: The model is not of full rank. A subset of the model which is of full rank is chosen.
DF
Regression
Error
Total
7
47
54
Sum of Squares
192.14282364
10.05488796
202.19771160
Mean Square
27.44897481
0.21393379
Prob>F
F
128.31 0.0001
Type II
Parameter
Standard
F Prob>F
Error Sum of Squares
Estimate
Variable
4.20 0.0461
0.89791700
-7.20184019
3.51532167
INTERCEP
19.35 0.0001
4.13964996
-16.83099740 3.82620057
Z
21.18 0.0001
4.53149724
1.47386134
-6.78324962
Z2
21.36 0.0001
4.56999963
0.23814194
-1.10066277
Z3
18.12 0.0001
3.87557112
0.01370892
-0.05834871
Z4
7.95 0.0070
1.70053535
0.00000188
0.00000530
RA
0.51 0.4790
0.10893980
0.00000094
-0.00000067
5.72 0.0208
1.22412818
0.00000011
-0.00000026
12
Bounds on condition number:
133987.6,
R-square = 0.94973322 C(p) = 6.50922205
Variable Il Removed
Step 1
DF
Regression
Error
Total
Variable
INTERCEP
Z
Z2
Z3
Z4
RA
12
6
48
54
Sum of Squares
192.03388384
10.16382776
202.19771160
Parameter
Estimate
-7.73235684
-17.13143228
-6.83296339
-1.10484913
-0.05865201
0.00000661
-0.00000018
Bounds on condition number:
1843826
Mean Square
32.00564731
0.21174641
Standard
Error
3.41819926
3.78347523
1.46466824
0.23684946
0.01363209
0.00000044
0.00000002
133906.3,
F
151.15
Type II
Sum of Squares
1.08354034
4.34131819
4.60845891
4.60762586
3.91973398
47.99766795
21.80905191
Prob>F
0.0001
F
Prob>F
0.0283
5.12
20.50 0.0001
21.76 0.0001
21.76 0.0001
18.51 0.0001
226.68 0.0001
103.00 0.0001
1573560
All variables left in the model are significant at the 0.1000 level.
Summary of Backward Elimination Procedure for Dependent Variable NUX
Step
1
Variable Number
In
Removed
6
II
Partial
R**2
0.0005
Model
R**2
0.9497
C(p)
6.5092
F
0.5092
Prob>F
0.4790