an experimental study of combined forced and free
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AN ABSTRACT OF THE THESIS OF
YongJin Kim for the degree of Doctor of Philosophy in Mechanical
Engineering presented on April 27. 1990
Title: An Experimental Study of Combined Forced and Free
Convective Heat Transfer to Non-Newtonian Fluids in
The Thermal Entry Region of a Horizontal Pipe
Redacted for PrivacyAbstract approved:
James R. Welty \,
The case of combined free and forced convection heat transfer
in non-Newtonian fluids was experimentally investigated in
uniformly-heated horizontal tubes with laminar flow in the thermal
entry region. Velocity profiles were fully developed at the entrance
to the heated sections of the tubes. Aqueous solutions of sodium
carboxymethylcellulose(CMC) were used; their behavior showed a
reasonably good fit into the power-law model, 't = K Yn. A range of
polymer concentrations was chosen to attain several degrees of
pseudoplastic behavior. The power-law constants, K and n, were
determined using a rotational viscometer.
The test sections used were made of copper with inside
diameters of 3.823 cm and 5.042 cm and lengths of approximately
300 cm.
Twenty two of 25 total runs displayed noticeable secondary
flows caused by buoyancy; when present, secondary flows caused
significant increase in the rate of heat transfer over the purely
forced-convection case. The local rate of combined convective heat
transfer with temperature-dependent properties was 84.5% higher
than for the temperature-independent case for the least viscous
solution, 5% CMC; and 35% higher for the most viscous solution,
8.3% CMC. A value of the ratio, Ra*/Gz2 -a- 1, appeared to be a
reasonable estimate as a criterion for the onset of secondary flow.
The correlation
Nub (Kw/Kb )o.140/81/3,) = 2.116 [Gzb + 0.0083 (Rab)o.751o.27
was found to relate the rate of heat transfer for flows with
temperature-dependent properties, free convection effects, and
non-Newtonian effects, with a coefficient of determination of
0.973.
An Experimental Study of Combined Forced and Free
Convective Heat Transfer to Non-Newtonian Fluids
in The Thermal Entry Region of a Horizontal Pipe
by
YongJin Kim
A THESIS
submitted to
Oregon State University
in partial fulfillment of
requirements for the
degree of
Doctor of Philosophy
Completed April 27, 1990
Commencement June 10, 1990
APPROVED:
Redacted for Privacy
Professor of N.__ . V charge of major
Redacted for PrivacyHead of Department of Mechanical Engineering
Redacted for Privacy
Dean of Graduat chool
Date thesis is presented April 27. 1990
Typed by YongJin Kim
ACKNOWLEDGEMENTS
I would like to express my heartfelt gratitude to Dr. James R.
Welty, my major professor, who provided constant guidance and
invaluable advice for my research, and arranged for my financial
support during the course of my graduate studies at Oregon State
University. Working with him has been an enjoyable and precious
experience.
I extend my thanks to the department of Mechanical
Engineering of Oregon State University, especially Dr. Gordon M.
Reistad for financial aid during this study.
My thanks goes to Dr. Robert E. Wilson who has provided me
with practical knowledge through his excellent lectures and served
as a member of my graduate committee. I wish to thank Dr. Dwight
J. Bushnell for his concern while serving as a member of my
graduate committee. I am also grateful to Dr. Stuart M. Newberger
for his outstanding lectures and serving as a member of my
graduate committee. My thanks also go to Dr. Charles E. Wicks and
Dr. Logen Logendran who have served as members of my graduate
committee.
I would also like to express my gratitude to the faculty
members of the department of Mechanical Engineering of Seoul
National University, Dr. Taik Sik Lee, Dr. Sung Tack Ro, and Dr. Jung
Yul Yoo, for their encouragement.
My profound thanks to Bea Bjornstad for her helpful hand in so
many different ways. She has been so nice to me. Personal thanks
to Nick for his friendly assistance in many occasions during
preparation of this research. Many thanks are also due to Page Orrie
who has assisted me during preparation of the research.
Many thanks go to those fellow students, Yoon-Su, Joe, Ziaul,
Salvador, Steve, David, and Jack for their friendship.
I would like to dedicate this dissertation to my wife, Hwa-
Ryoung, and my parents for their patience and unending support.
Finally, special thanks goes to Jesus Christ for His care.
TABLE OF CONTENTS
PageChapter 1. INTRODUCTION 1
1.1 Scope and overview 1
1.2 Classification of non-Newtonian fluids 41.3 Constitutive models for viscous fluids 7
Chapter 2. LITERATURE REVIEW 102.1 Combined convective heat transfer in steady,
laminar, horizontal pipe flow of Newtonian fluids 102.2 Laminar convective heat transfer to
non-Newtonian fluids 152.2.1 External flows 162.2.2 Internal flows 27
Chapter 3. EXPERIMENTAL DESIGN AND SETUP 483.1 Experimental design 483.2 Experimental setup 52
3.2.1 Flow loop system 523.2.2 Apparatus 543.2.3 Test section 57
(1) Design of heated section 57(2) Construction 59(3) Other details 62
3.2.4 Viscometer 68
Chapter 4. MEASUREMENT SYSTEM 704.1 Temperature measurement 704.2 Thermocouple calibration 714.3 Power measurement 734.4 Flow measurement 754.5 Viscometric data measurement 75
Chapter 5. EXPERIMENTAL PROCEDURE 775.1 Test fluids 77
5.1.1 Mixing of polymers 775.1.2 Properties 77
5.2 Test procedure 78
Chapter 6. DATA REDUCTION 806.1 Incorporation of temperature variation into the
power-law model 806.2 Dimensionless parameters 87
Chapter 7. RESULTS7.1 Viscometry 927.2 Heat transfer 97
Chapter 8. CONCLUSIONS AND RECOMMENDATIONS 118
BIBLIOGRAPHYAPPENDICES
A. Error analysisB. Viscometric dataC. Reduced test data and parameters
121
139150154
LIST OF FIGURES
Figure E
1.1. Flow curves on arithmetic coordinates fortime-independent fluids 5
3.1 Cross section of test tubes 5 0
3.2. Schematic diagram of test loop 5 3
3.3. Assembly at test-section entrance 5 8
3.4. Schematic diagram of layout of heating stripsfor small test section 6 0
3.5. Schematic diagram of layout of heating stripsfor large test section 61
3.6. Overall view of test apparatus 6 4
3.7. Test sections with heating strips attached 6 5
3.8. Test sections with surrounding insulation removed 66
3.9. View of heat weigh tank, feed tank, tank mixer,and pump 67
3.10. View of entrance sections, heat exchanger, andstatic mixer 67
3.11 Schematic diagram of the rotational viscometer 6 9
4.1 Schematic diagram of temperature measurementsystem 72
4.2 Schematic diagram of power supply circuit 74
4.3. View of rotational viscometer 76
6.1. Interpolation between adjacent constitutiveequations at a particular shear rate 8 3
6.2. Interpolation between adjacent constitutiveequations at a particular shear stress 83
7.1. Viscometric data for 5% CMC solution. From the top,data are for 16.8, 25.3, 35.1, 44.7, 54.3, 64.2, 74.1,and 86.4°C
7.2. Viscometric data for 6% CMC solution. From the top,data are for 16.1, 25.5, 35.3, 45.2, 54.8, 64.8, 74..3,and 86.6°C
7.3. Viscometric data for 7% CMC solution. From the top,data are for 15.8, 25.3, 35.0, 44.9, 54.9, 64.5, 74.5,and 86.3°C
7.4. Viscometric data for 7.5% CMC solution. From the top,data are for 16.7, 25.0, 35.2, 45.1, 55.0, 65.1, 74.2,and 88.2°C
7.5. Viscometric data for 8% CMC solution. From the top,data are for 17.3, 25.4, 35.5, 45.0, 54.9, 64.7, 74.4,and 86.0°C
7.6. Viscometric data for 8.3% CMC solution. From the top,data are for 16.4, 25.6, 35.5, 45.2, 54.7, 64.7, 74.2,and 86.0°C
7.7. Variation of the power-law constant, K, withtemperature. From the top, data are for 8.3%, 8%,7.5%, 7%, 6%, and 5% CMC solution
93
93
94
94
95
95
96
7.8. Comparison of present data for 5% CMC solutionwith the temperature-independent solution 99
7.9. Comparison of present data for 6% CMC solutionwith the temperature-independent solution 99
7.10. Comparison of present data for 7% CMC solutionwith the temperature-independent solution 100
7.11. Comparison of present data for 7.5% CMC solutionwith the temperature-independent solution 100
7.12. Comparison of present data for 8% CMC solutionwith the temperature-independent solution 101
7.13. Comparison of present data for 8.3% CMC solutionwith the temperature-independent solution
7.14. Heat transfer behavior for 5% and 6% CMC
7.15. Heat transfer behavior for 5% and 8.3% CMC
101
103
103
7.16. Comparison of present data from typical runswith 7% CMC solution in both test sections 104
7.17. Experimental data comparisons 105
7.18. Comparison of present data with the temperature-independent Newtonian solution 106
7.19. Wall temperature variations for 7.5% CMCin the small test section 109
7.20. Comparison of experimental results with Cochrane'snumerical solution 111
7.21. Comparison of experimental results with Cochrane'snumerical solution 111
7.22. Comparison with Bassett and Welty results 112
7.23. Comparison with results of Mahalingam et al 114
7.24. Comparison with Joshi's results 115
LIST OF TABLES
Table Page
5.1. Thermal conductivity values 78
7.1. Comparison with Cochrane's solution 110
A.1. Uncertainties in basic quantities 141
A.2. Statistical analysis of experimental results 149
NOMENCLATURE
A Cross-sectional area
cp Heat capacity
D Inside diameter
Do Outside diameter
g Gravitational acceleration
Gr Grashof number, p2r3gD3 (Tw Tb)/112
Gr* Modified Grashof number, p2pga4qu/kii2
Gz Graetz number, rhcp/kx
h Convective heat transfer coefficient
H Viscometer rotor length
AH Activation energy
AP pressure drop
k thermal conductivity
K Power-law fluid consistency index
n Power-law flow behavior index
L Test section heated length
Le Hydrodynamic entry length
Mass flow rate
M Torque
Mf Mass of fluid weighed
Nu Nusselt number, hD/k
P Power supplied to the test section
Pr Prandtl number, 11 cp/k
q" Heat flux
Volume flow rate
R Radius of test section, D/2
Rh Heater resistance
Rr Viscometer rotor radius
Rs Shunt resistance
Ru Universal gas constant
Ra Rayleigh number, Gr Pr = p213gD3 cp (Tw Tb)/kii
Ra* Modified Rayleigh number, Gr* Pr = p2/30:3D4ci cp/k2i
Re Reynolds number, 4m/nDi
Reg Modified Reynolds number,
ReJ3 =Dn V2K -11 p 8 ( n )11
2 3n+1)
ReK Modified Reynolds number,n
ReK =Dr'
VK2ri 3n-1 n
P
4
( 1-7.) (3n+1 )
tf Time for Mf to flow
T Temperature
Tb Bulk temperature
Ti Inlet temperature
Tw Wall temperature
u Velocity
V Average velocity, 4 m/p7rD2
WY Uncertainty (error) in variable Y
x Axial position measured from inlet
X+ Dimensionless distance, 2 (x/D)/(Re Pr)
13 Coefficient of thermal expansion
ii Shear rate, I du/dr I
i'1, i'2 Shear rates defined in Figure 6.2
ip Shear rate defined in Figure 6.1
8 Shear rate ratio, (3n+1)/4n
I-1, Viscosity
T1 Apparent viscosity, tiit
P Fluid density
t Fluid shear stress
ti, T2 Shear stresses defined in Figure 6.1
tip Shear stress defined in Figure 6.2
tw Wall shear stress
Tr Shear stress at the Viscometer rotor
S. Dimensionless heat flux, q"D/kTi
CI Angular speed of viscometer rotor
Subscripts:
b Local bulk condition
r Viscometer rotor
s Shunt
w Local wall condition
x Position, x
An Experimental Study of Combined Forced and Free
Convective Heat Transfer to Non-Newtonian Fluids
in The Thermal Entry Region of a Horizontal Pipe
Chapter 1. INTRODUCTION
1.1 Scope and overview
Transfer processes involving non-Newtonian fluids continue to
be important in many technical areas involving, for example, rubber,
plastics, synthetic fibers, petroleum, soap, detergents,
pharmaceuticals, biological fluids, atomic energy, cement, foods,
paper pulp, paint, light and heavy chemicals, fermentation
processes, oil field operations, ore processing, and printing.
Therefore, it is evident that an improved understanding of heat
transfer to non-Newtonian fluids is important in solving practical
engineering problems.
Specifying fully-developed flow prior to entering an
isothermally heated or cooled section, Graetz [50] first obtained an
analytical solution for forced convective heat transfer with steady,
laminar, horizontal pipe flows. The asymptotic solutions to the
Graetz problem are given by Knudsen and Katz [73, p. 372-373] as
follows:
In thermal entry region
x/rwNu =C1
Pe
In fully-developed region
NU = C2
-1/3x/rw
Pe< 0.01
x/rw
Pe> 0.25
2
(1.2)
where, for an isothermal boundary, C1 and C2 are 1.357 and 3.656,
respectively; and for uniform wall heat flux the values are 1.639
and 4.364, respectively. The result for the isothermal boundary in
the thermal entry region is essentially the same as that obtained
much earlier by Leveque [82].
Experimental data for the laminar flow of oils heated and
cooled in isothermal horizontal tubes were correlated by Seider and
Tate [137]. Adding a large number of isothermal data from other
investigations, they suggested the equation
0.14
NUD = 2.0 Gz1/3 (141-LW (1.3)
where all properties other than p.w are evaluated at the bulk
temperature. Since their equation gives mean values of the heat
transfer coefficient and the Graetz results give local values of the
heat transfer coefficient, direct comparison of the Seider-Tate
results with the Graetz results cannot be made. These results were
presented graphically in Welty, Wicks, and Wilson [163, p. 364]. The
3
Seider-Tate correlation is shown to give values for Nu slightly
above values from the Graetz solution.
Many experimental investigations dealing with free convection
heat transfer or forced convection heat transfer to non-Newtonian
fluids have been carried out over the last decade. Most analytical
work for non-Newtonian fluids has been concerned with laminar
flow for several geometries, such as the vertical flat plate,
parallel plane surfaces, flow through a horizontal cylinder and a
rectangular duct. Turbulence has been less tractable in cases
involving non-Newtonian fluids.
Experimental investigations on both free and forced convection
have been reported for both isothermal and constant heat flux
surfaces with these same geometries. More recently, studies of the
heat transfer performance of non-Newtonian fluids has been carried
out both numerically and experimentally in a rectangular duct [57].
Buoyancy effects added to conventional forced convection
causes a secondary, transverse flow to develop in pipe flows. While
a number of combined free- and forced-convection studies with
Newtonian fluids have been reported, the case of combined free and
forced convection with non-Newtonian fluids has not yet been fully
examined. Experimental investigations for mixed convection with
non-Newtonian fluids are thus needed and comprise the focus of
this thesis.
The phenomenon of mixed convection occurs when a forced
flow at relatively low velocity experiences heating. A buoyant flow
results from the density reduction of fluid near the heated surface.
4
This becomes coupled with the forced flow. The object of this work
is to investigate the combined free and forced laminar convection
to non-Newtonian fluids in the thermal entry region, with a fully-
developed velocity profile, in a horizontal, uniformly heated,
circular pipe. While the Graetz solution considers only the basic
conduction mechanism at the boundary, the present work also
includes non-Newtonian effects; temperature dependence of
transport properties, particularly viscosity; and secondary
transverse flows due to buoyancy.
The working fluids chosen were dilute aqueous solutions of the
polymer, Sodium Carboxymethylcellulose (CMC). The characteristics
of these fluids, used frequently by other investigators, are
reasonably well known. Their characteristic behavior is
pseudoplastic, and the degree of pseudoplasticity increases with
increasing concentration. Various concentration of CMC were used
in this work.
Heat transfer results are compared both with numerical
results for temperature-independent, non-Newtonian fluids and
with previous published experimental results on the effects of free
convection with non-Newtonian flows in forced convection.
1.2 Classification of non-Newtonian fluids
Most real fluids exhibit non-Newtonian behavior, that is the
stress-strain rate curve is nonlinear at a given temperature.
Skelland [152] and Metzner [98] discuss the classification of fluids
5
into three broad groups:
(1) purely viscous fluids
(2) time-dependent fluids
(3) viscoelastic fluids
shear rate, du/dy
Bingham plastic
Pseudopiastic
Newtonian
Dilatant
Figure 1.1. Flow curves on arithmetic coordinates for time-independentfluids
The classic types of fluid behavior are illustrated in figure1.1.
Newtonian fluids are a subclass of purely viscous fluids. Purely
viscous non-Newtonian fluids can be classified into two groups
(1) shear-thinning (pseudoplastic) fluids
(2) shear-thickening (dilatant) fluids.
In figure 1.1 the stress-strain rate behavior of an ideal
plastic-type fluid, which exhibits a yield stress, is shown for a
6
Bingham plastic. Typical examples of fluids with a yield stress are
plastic melts, ores, sand in water, cement, greases and detergent
slurries.
Most non-Newtonian fluids encountered in practice are shear-
thinning. These "pseudoplastics" display a "thinning" or decreasing
proportionality between shear stress and shear rate as the shear
rate increases. Highly dispersed molecules or particles in a
solution generally exhibit pseudoplastic behavior. A diminishing
apparent viscosity due to a decrease in interaction between the
particles results from progressive shearing away of solvated layers
with increasing shear rate. Pseudoplastics also show a limiting
viscosity at high shear rates and the flow curve becomes linear. In
shear-thickening (dilatant) fluids the proportionality between shear
stress and shear rate increases with shear rate. Since shear-
thickening fluids are relatively uncommon among fluids of
significant industrial importance, the study of these fluids is very
rare.
Time-dependent fluids usually consist of two groups,
thixotropic fluids and rheopectic (antithixotropic) fluids.
Thixotropic fluids have the characteristic that the shear stress
decreases reversibly with time at a constant rate of shear under
isothermal conditions. In contrast, rheopectic fluids show a
reversible increase in shear stress with time at a fixed shear rate
and constant temperature. Skelland [153] describes these fluids in
more detail.
Viscoelastic fluids retain both viscous and elastic properties.
7
Gradual dissipation of the stress in viscoelastic substances will be
generated when they are subjected to stress. Upon removal of the
stress imposed on viscoelastic substances, a portion of the
resultant deformation is gradually recovered. Elastic properties of
a viscoelastic fluid do not have a significant effect on fully-
developed laminar pipe flow. Thus, for a viscoelastic fluid behaving
as a purely viscous non-Newtonian fluid, the friction factor and
heat transfer coefficient may be predicted by the well known power
law relations (Hartnett and Kostic [57]).
1.3 Constitutive models for viscous fluids
The following four constitutive models are often used. The
equations are all empirical or, at best only semitheoretical in
nature.
(1) The Power-law model expressing shear stress as a function of
the rate of shearing strain in the form
tyx.K(du)nay)(1.4)
gives a simple empirical equation which includes two constants.
This relation is frequently valid at moderately high shear rates and
over several orders of magnitude, is frequently an excellent
approximation. However, it does not represent the behavior at
extremes of both very low and very high shear rates.
(2) The Ellis model, expressed as
1 ( du )tyx dy )+ C2 tyx
8
(1.5)
includes a correction for the deficiency of the power law at low
shear rates by adding a Newtonian term. It is more useful at low to
moderate shear rates, as in the case of very viscous fluids.
(3) The ideal or Bingham plastic model is of the form
( duTYx L-d-T) (1.6)
This model is frequently used with slurries which have either a true
or apparent yield stress ty.
(4) The fourth commonly-used model is the Powell-Eyring
expression which is
dut 1 ( Chi= C3 (iy-)
sinh IJ(1.7)
This model provides a reasonably correct prediction for fluid
behavior at the extremes of both low and high shear rates. Although
its use is inconvenient for analysis because it is not explicit in
shear rate, it serves as a good approximation formula over wide
ranges of shear rates.
More information is available for other constitutive models in
references such as Skelland [152, p. 8,9]. The power-law model has
9
been employed in the present work. It represents relatively good
agreement with flow behavior for the working fluids used in this
work and is simple in form.
10
Chapter 2. LITERATURE REVIEW
2.1 Combined convective heat transfer in steady, laminar,
horizontal pipe flow of Newtonian fluids
Free convection always exists when there is an unstable
temperature gradient in horizontal pipe flows. Therefore, free
convection effects may be important based on the criteria
concerning the relative significance of free convection to forced
convection which is the dimensionless group, Gr/Re2 [86]. This
ratio relates buoyancy forces to inertial forces in the flow. If this
parameter is near to one, mixed convective heat transfer is
generally known to occur.
The study in this thesis on combined convective heat transfer
for non-Newtonian fluids involves measurements in the thermally-
developing flow region with a uniform heat flux-wall condition. An
extension of the classical Graetz solution for Newtonian fluids we..;
done by Sellars, Tribus, and Klein [138] for the case of a uniformly-
heated or cooled surface. Siegel, Sparrow, and Hallman [149] also
obtained an analytical solution for forced convective heat transfer
in the developing thermal region for the Graetz problem withuniform wall heat flux.
Morton [110] has reported his theoretical analysis of combined
free and forced convection in the fully-developed region with flow
in horizontal pipes at low Rayleigh numbers. Assuming a constant
axial pressure gradient, he carried out an expansion in terms of a
11
parameter, Nu Gr/Re. His solution has validity for small values of
this parameter. Del Casal and Gill [33] extended Morton's solution
to include cases with very small Re. They assumed that the density
was temperature-dependent and that other properties were
constant. Their purturbation solution is valid for small values of
(Nu Gr)/(Pr Re2). Both of these schemes were limited to very small
heating rates.
Experimental heat transfer results allowing for the secondary
flow effect due to buoyancy in a horizontal tube with an isothermal
thermal wall boundary have been correlated by Colburn [27], Eubank
and Proctor [40], Jackson et al [66], Brown and Thomas [14], Kern
and Othmer [68], Oliver [115], Depew and August [36], and Yousef and
Tarasuk [169]. The improved equation obtained by Depew and August
is of the form,0.14
Num = 1.75 (-1±-b) [Gzab + 0.12 (Gzab Gria0prOae 6)0 . 8 8 ]1/3
(2.1)
where ab indicates the average bulk condition. They claimed this
equation to have ±40% accuracy.
Yousef and Tarasuk [169] examined the influence of free
convection with air in the entrance region of a horizontal tube.
Three regions in the entry region were considered; those were 1)
near the tube inlet where free convection is dominant, 2) further
downstream where forced convection becomes dominant, and 3) far
from the tube inlet where the mean Nusselt number becomes
constant. They found that free convection had a significant effect
12
on heat transfer near the entrance to the tube. Comparisons were
also made with several correlations used in combined convection in
an isothermal horizontal tube.
In the thermal entrance region for flow in horizonal tubes with
uniform wall heat flux, experimental work on laminar combined
convective heat transfer was carried out by McComas and Eckert
[95], Mori et al [109], Hussain and McComas [63], and Lichtarowicz
[83] for air; Shannon and Depew [141], Petukhov and Polyakov
[127,128], Kupper, Hauptmann, and lqbal [77], and Berg les and
Simonds [10] for water; and Shannon and Depew [140] for ethylene
glycol.
Using a boundary layer model, Siegwarth and co-workers [151]
studied the effect of secondary flow for the cases of Pr=1 and
P rw) in a heated horizontal tube. The temperature at the inside
surface did not vary circumferentially when a uniform heat flux
was specified at the outer wall of the tube. They obtained a
solution for the case Pr---)oo in the fully developed region as follows:
Nu = 0.471 Ra1/4 (2.2)
which agrees quite well with the experimental data of Siegwarth
and Hanratty [150] using a 2.5-inch I.D., 1-inch thick aluminum pipe.
Ra indicates p213ga3AT cp/kg where a means tube radius. It is noted
that the boundary condition used [151] is different from the uniform
circumferential heat flux condition applied to thin-wall tubes.
A theoretical analysis of combined forced and free convective
heat transfer in laminar boundary layer flows was done by Acrivos
13
[2]. The momentum equation, coupled with the energy equation
including the gravitational force term in two-dimensional flows
was solved. He used GrPr1/3/Re2 for large Prandtl numbers and
Gr/Re2 for Pr«1 as controlling parameters which indicate the
relative importance of forced and free convection.
The experimental investigation by Morcos and Berg les [108] on
combined convection in fully developed laminar flow throughout a
uniformly-heated horizontal pipe yielded the following correlations:
Nuf = {( GrP11."
(4.36)2f
+ 0.055(2.3)
for 3x104 < Ra < 106, 4 < Pr < 175, and 2 < Pw < 66 where Pw is tube
wall parameter, hdi 2/Kwt. d1 and t indicate inside tube diameter
and tube thickness, respectively. f means the evaluation of
properties at fluid film temperature. Combined laminar convection
has also been investigated experimentally, in uniformly-heated
horizontal tubes, by Depew, Franklin and Ito [35].
Numerous analytical solutions on combined convective heat
transfer have been proposed for fully developed Newtonian flow by
Faris and Viskanta [41], Hwang and Cheng [64], Newell and Bergles
[112] and Hong, Morcos, and Bergles [62]. Cheng and Ou [19]
investigated combined laminar convection in the thermal entrance
region of horizontal tubes with uniform wall heat flux for large
Prandtl number fluids. From their theoretical results, asymptotic
Nusselt numbers for the fully-developed case were evaluated as
Nu. = 4.36 + 0.286 (Ra)1/4
14
(2.4)
Nu. = 3.4 + 0.303 (Ra*)1/4 for Ra*>10 (2.5)
where Ra = gf3a4q"/v2k and Ra* = gf3(Twavg-Tb)(2a)3/vk. a and v are
the radius of tube and the kinematic viscosity, respectively. Nu,. is
the asymptotic Nusselt number.
Analytical solutions have been examined by a number of
investigators for combined convective heat transfer in a horizontal
pipe with an isothermal boundary. Of note are papers by Ou and
Cheng [120], Hieber and Sreenivasan [60], and Hieber [59]. Hieber
and Sreenivasan obtained a solution using the large-Prandtl-number
assumption in the entrance region where the velocity and the
temperature fields are developing simultaneously.
Yao [167,168] obtained asymptotic solutions for the developing
entry length problem by perturbing the solution for developing flow.
Hishida, Nagano and Montesclaros [61] generated numerical
predictions for combined convection in the entrance region of an
isothermally heated horizontal pipe, which are reported to be
applicable to a fluid of arbitrary Prandtl number. Hieber [58]
attempted to establish a correlation which includes all
experimental heat transfer results for laminar mixed convection in
an isothermal horizontal tube. His correlation was found to be good
for (Gr Pr)1/4 44. Data beyond this range fit the laminar
correlation of Eubank and Proctor [40] quite well.
In recent years Coutier and Greif [29] have investigated
15
laminar flow and heat transfer within a horizontal isothermal tube
both experimentally and theoretically. Since free convection
effects coupled to axial motion create a three-dimensional flow
inside the horizontal tube, they analyzed the flow behavior and the
heat transfer using a three-dimensional numerical model. Their
numerical results for the temperature profiles in a developing flow
throughout the entire length of the short exchange tube agreed well
with the data from their experiments and with numerical data from
Ou and Cheng [120). Secondary flow was observed over the entire
range of conditions they considered.
2.2 Laminar convective heat transfer to non-Newtonian fluids
The non-linear relationship between the stress tensor and the
deformation strain tensor makes convective heat transfer to non-
Newtonian fluids difficult to model by numerical and analytical
methods. Many empirical models [152, p. 6-9) have been proposed in
the literature for non-Newtonian fluids. Most commonly used for
research on fluid dynamics and heat transfer of non-Newtonian
fluids is the power-law or Ostwald-de Waele model
Txy = -K (du/dy)n (2.6)
where K is the consistency index and n is the flow behavior index.
For Newtonian fluids K is the coefficient of viscosity and n is equal
to 1. The present study also employs the power-law model.
16
The simple classical problems of fluid dynamics and heat
transfer up to about 1960 have been well reviewed by Metzner [98]
and Metzner and Gluck [100]. Bassett [8] and Shenoy and Mashelkar
[147] have also reviewed in detail the publications on convective
heat transfer to non-Newtonian fluids. The following sections will
be limited to a review of laminar convective heat transfer to non-
Newtonian fluids in both external and internal flows.
2.2.1 External flows
The theoretical analysis of flow past an external surface with
power-law fluids under laminar forced convection has been
presented by Acrivos, Shah and Petersen [4]. They applied
similarity variables to the momentum equations, and estimated
local heat transfer rates using Lighthill's approximate formula. It
was assumed that the thermal boundary layer is much thinner than
the shear layer, i.e., their modified Prandtl number was very large.
Schowalter [136] showed that similar solutions exist for
external flows of pseudoplastic power-law fluids. For two-
dimensional flow the solutions were analogous to those obtained
for Newtonian fluids. For three-dimensional flow, however,
similarity solutions were much more restrictive than for Newtonian
fluids.
A numerical solution for forced convection flow of power-law
fluids under a right-angle wedge with an isothermal surface was
obtained by Lee and Ames [80] by solving the boundary-layer
17
equations. The transformation group method, which has been
described by Ames [6], was used for their numerical approach.
Transformations of the boundary-layer equations for cases of
a steady-state boundary layer, an unsteady-state bounday layer, the
thermal boundary layer, simultaneous free and forced convection,
and free convection in general for pseudoplastic and dilatant fluids
has been presented by Berkowski [11] without completing to analyze
rigorous solutions and the implications of various dimensionless
groups encountered for different problems.
Denn [34] extended boundary-layer solutions to include the
entire class of wedge (Falkner-Skan) flows for elastic fluids with
shear dependent viscous and normal stress relations. He obtained a
set of ordinary differential equations for the boundary layer flow of
a general elastic fluid away from the leading edge of the submerged
object. It was concluded from his several solutions that the effect
of fluid elasticity on drag depends on both the system geometry and
fluid parameters.
Tien et at [158] first investigated the thermal instability of a
horizontal layer of an inelastic power-law fluid heated from below.
Critical Rayleigh numbers were obtained as functions of the flow
behavior index; the critical Rayleigh number was shown to decrease
with the flow behavior index. It was assumed that instability
would occur at the minimum temperature gradient at which a
balance can be steadily maintained between the kinetic energy
dissipated by viscosity and the internal energy released by the
buoyancy.
Ozoe and Churchill [121], Parmentier et at [123], and
18
Parmentier [122] have also investigated the problem of free
convection in a horizontal layer of an inelastic non-Newtonian fluid
heated from below. Shenoy and Mashelkar [146] have suggested that
the Ellis fluid model (see chapter I.), as used by Ozoe and Churchill
[121], is superior to the model used by Tien et al.
A theoretical study of free convective heat transfer to power-
law fluids with steady, laminar boundary layer flow from
arbitrarily shaped two dimensional or axisymmetric bodies was
recently carried out by Chang, Jeng, and Dewitt [15]. The effect of
the convective term in the governing momentum equation was
included even though non-Newtonian fluids generally have high
Prandtl numbers. Universal functions, which are independent of
geometry, were obtained.
Parmentier [122] and Parmentier et al [123] emphasized that,
for pseudoplastic fluids ( 0.3 < n < 1 ), the structure of thermal
convection cells at steady state is the same as for Newtonian
fluids. They also found that for values of n below 0.3 the fluid
deformation tends to become more localized and significant regions
of stagnant fluid develop.
Pierre and Tien [129] and Tsuei and Tien [159] have obtained
experimental results for free convection of a non-Newtonian fluid
in a horizontal layer
19
Vertical flat plate
Laminar free convective heat transfer to power-law fluids
was examined by Acrivos [1] for the case of a vertical plate with
constant temperature. He developed an expression for the local
Nusselt number from the exact asymptotic solution of the
appropriate laminar boundary layer equations. His solution can be
applied to a two-dimensional surface or a surface of revolution
about an axis of symmetry when the power-law Prandtl number,
pcp )( K 2 /(n+1)L(n-1)/2(n+1) [gi3 (TwT.:,)]3(n-1)12(n+1)
is greater than 10.
An experimental investigation of free convective heat transfer
in a non-Newtonian fluid was made by Reilly et al [131]. Their
results were consistent with Acrivos' [1] analytical results.
Na and Hansen [111] have shown that similarity solutions for
laminar free convection of non-Newtonian fluids can be obtained by
using group theoretic methods. Studies on laminar free convection
to non-Newtonian fluids have been also conducted by Tien [155],
Dale [30], Dale and Emery [31 ], and Kleppe and Marner [71]. Tien
[155] obtained approximate solutions using an integral method and
the power-law model with high Prandtl number situations for cases
of an isothermal plate, a non-isothermal plate, and a plate with
uniform heat flux.
Dale and Emery [31] measured and predicted numerically the
20
local heat transfer, temperature, and velocity distributions
between a vertical constant flux plate and several concentrations
of pseudoplastic fluids. Flow indices varied from 0.395 to 1.0, and
the fluid consistencies were 30 to 2300 times those of water. The
heat transfer in their study was expressed as the following
generalized non-Newtonian correlation:
where
Nux= C (Gr* Pr*x)(3n+2) (2.7)
* gi3(1Tjx(n+2)/(2n)Grx =
K )2/(2n)CpJ (2.8)
* C (K )1/(2-n) 2(1-n)/(2-n)X -_ Pk X
(2.9)
By transforming the governing partial differential equations
into ordinary differential equations, Chen and Wollersheim [18]
solved the problem of laminar free convective heat transfer for the
case of a vertical plate with uniform heat flux. Solutions for
constant heat flux and variable wall temperature were also
obtained for laminar free convective heat transfer to a power-law
fluid from a vertical flat plate by Shenoy [142].
Shenoy and Mashelkar [147] have discussed the appropriate
boundary and compatibility conditions in detail. These conditions
21
were not satisfied for the velocity and temperature profiles used by
Tien's [155]. These conditions were later used by Shenoy and
Ulbrecht [145] who obtained an integral solution for laminar free
convective heat transfer from an isothermal vertical flat plate to a
power-law fluid.
A theoretical analysis for laminar mixed convective heat
transfer to inelastic non-Newtonian fluids in external flows has
been studied by Kubair and Pei [75] for the case of a vertical flat
plate with constant wall temperature. They concluded that:
1. the combined effects of free and forced convection for non-
Newtonian laminar boundary-layer flow are satisfactorily
characterized by the dimensionless group, P = Gr'/Re'2/(2-n),
where a modified Reynolds number
pu2-nxnRe'
and a modified Grashof number
jx(n+2)/(2-n)Gr' =
2/(2-n)
(2.10)
(2.11)
2. based on the asymptotic limits in pure flows, the effect of
free convection is negligible for P < 0.1 and the effect of
forced convection is apparent even at P = 5.0 for Newtonian
fluids. At higher Prandtl numbers the free convection effect
22
may be predominant at somewhat lower values of P both for
Newtonian and non-Newtonain fluids.
3. for opposing flows the phenomenon of zero shear is strongly
influenced by non-Newtonian behavior.
However, a number of errors were later found in their analysis.
Shenoy [143] pointed out that the controlling parameter, P, would be
constant only under limited conditions, which Sparrow et al [153]
have specified and explained in the case of Newtonian fluids under
combined convection flow. He also indicated that the continuity
equation is not satisfied by their dimensionless groups. Their
statement that their boundary equations reduce correctly to the
Newtonian forms and does not have this limitation was found to be
incorrect.
Churchill [24] has presented a correlation for laminar
assisting combined convection flow of Newtonian fluids, which is
also valid for non-Newtonian power-law fluids. The form of the
correlation is
Nu3 =Nu3x,F +Nu3x,N
x,M (2.12)
where Nux,M' Nux,F' and Nux,N are local Nusselt numbers based on
the local distance x on the heat-transferring surface for mixed,
forced, and free convection, respectively. Later Ruckensten [132]
has established his correlation [24] using an approximate
interpolation procedure. Using the form above, Shenoy [143] has
proposed an equation for predicting the combined convective heat
23
transfer rate to the flow of power-law fluids past an isothermal
vertical flat plate. For pure forced-convective heat transfer the
correlating equation is from Acrivos et al [4]. The relation of
Shenoy and Ulbrecht [145] was used for the free convective heat
transfer term. Greatest accuracy for non-Newtonian fluids would
be cases of large Prandtl numbers.
Although the power-law model, which is a two-parameter
model, is popular for solving many engineering problems, the
Sutterby and Ellis model (three-parameter model) turns out to be
more useful, particularly when the stresses or strain rates are
small. Laminar free convection from a vertical flat plate has been
analyzed by Shenoy and Mashelkar [146] who used these two time-
independent models.
Lin and Shih [84] analyzed laminar mixed convection vertically
static or moving plates and power-law fluids for both the
prescribed surface temperature and prescribed wall heat flux cases.
The local similarity method [85,86] has been employed to
demonstrate nonsimilarity due to buoyancy effects, non-Newtonian
behavior, and moving boundary conditions. They showed that the
method of local similarity is useful for the study of power-law
fluids.
Vertical cylinder
Recently, Wang and Kleinstreuer [162] numerically investi-
gated laminar mixed convection with power-law fluids adjacent to
24
vertical slender cylinders. Using a coordinate transformation and
an implicit finite-difference method, they solved the non-similar
problem to examine the effects of transverse curvature, power-law
index, buoyancy parameter, and generalized Prandtl number on the
local heat transfer and skin friction coefficients. Thermal boundary
conditions used were both constant temperature and uniform heat
flux.
Horizontal cylinder
An experimental investigation for free convective heat
transfer from a horizontal cylinder to moderately elastic drag-
reducing polyethylene oxide solutions has been carried out by Lyons
et al [88]. They observed a decrease in Nusselt numbers, compared
to Newtonian fluids, with increased polymer concentration without
any quantitive comparison.
Local free convective heat transfer from a horizontal
isothermal cylinder to non-Newtonian power-law fluids has been
investigated numerically and experimentally by Gentry and
Wollersheim [47]. The local Nusselt numbers obtained experi-
mentally showed good agreement with their [47] similarity and
integral solutions.
Using concentrated corn starch suspensions in aqueous sucrose
solutions, Kim and Wollersheim [70] obtained experimental data for
free convection from a horizontal cylinder to dilatant fluids with
both isothermal and uniform heat flux surface boundaries. Their
data for an isothermal horizontal cylinder were in excellent
25
agreement with the theoretical solutions presented by Gentry and
Wollersheim [47].
In the case of external flows of viscoelastic fluids Shenoy
[144] obtained an expression for combined laminar forced and free
convective heat transfer in the stagnation region of an isothermal
horizontal cylinder using an approximate procedure. The correlation
which was obtained using a similar manner to Ruckenstein [132] and
Shenoy [143] is of the form:
NU3avR,M = u3avR,F + Nti3avR,N (2.13)
He noted the effects of viscoelasticity as well as free convection
to increase the heat transfer. Based on the radius of the cylinder
indicate average Nusselt number forNU3avR,M' Nu3avR,F and NU3avR,N
mixed convection, forced convection, and free convection,
respectively.
Ng and Hartnett [114] recently reported an experimental study
for free convection to horizontal wires, whose diameters were of
the same order or smaller than the boundary layer thickness of free
convection. Test fluids were Carbopol 960 and Carbopol 934, which
are both pseudoplastic. In order to reduce the results for the three
temperatures tested to a single curve, a reduced shear rate, 7r = 7
exp(B/T), which is a form proposed by Christiansen et al [23], was
introduced. The constant B was determined experimentally. By
transforming Acrivos' [1] analytical results into a new set of
26
dimensionless Ng and Hartnett [113] showed that the Nusselt
number may be expressed as
Nu = C RaN 1/(3n+1) (2.14)
where RaN indicates the Rayleigh number for power-law fluids. A
final equation based on their experimental data is
Nu = (0.761 + 0.431 n) RaN
Sphere
1/{2(3n+1)} (2.15)
A solution of the two-dimensional boundary layer equations
obtained by Acrivos et al [4] has also been presented for forced
convective heat transfer from a sphere to power-law fluids at large
Reynolds numbers. By using a Mangler-type transformation [135],
Acrivos [1] has provided and extended the theory for free convective
heat transfer from a two-dimensional surface to the three-
dimensional axisymmetric case.
Amato and Tien [5] carried out an experimental investigation
on free convective heat transfer from isothermal spheres to
aqueous polymer solutions of CMC-7H and Polyox WSR-FRA. The
local heat transfer variation on a sphere as a function of the
angular distance from the stagnation point showed good agreement
with the values of Acrivos [1].
Yamanaka and Mitsuishi [166] examined combined forced and
27
free convective heat transfer from spheres to aqueous solutions of
some polymers. They correlated their experimental data with an
empirical equation obtained by extending the Newtonian correlation
for combined convective heat transfer using the power-law model.
They applied the method proposed by Acrivos and Goddard [3] to
derive this equation. Solutions of 2.61% MC (methylcellulose), 5.52%
CMC, 0.74% SPA (sodium polyacrylate), and 1.48% PEO (polyethylene
oxide) were used as test liquids. Their equation correlated the
experimental data at Peclet numbers below 1000 and small
Reynolds numbers with a maximum and a mean deviation of 67.7%
and 29.3%, respectively. Since polymer solutions generally have
high Prandtl numbers, Peclet numbers are large in most cases.
2.2.2 Internal flows
For external flows the boundary development layer is
continuous. However, for internal flows, such as flows through
heated or cooled tubes, the boundary layer development is
constrained. While very few studies involving free convective heat
transfer to non-Newtonian fluids in internal flows have been
conducted, many studies on forced-convective heat transfer to non-
Newtonian fluids have been carried out by numerous investigators.
Parallel plates
Temperature profiles for flow between two parallel plates,
with one stationary and the other moving at constant velocity, have
28
been obtained by Tien [156]. He has provided temperature profiles
for two cases: both plates maintained at constant temperatures, and
one plate maintained at constant temperature while the other is
insulated.
Tien [157] also extended the asymptotic solutions of the
classical Graetz-Nusselt problem to the case with a non-Newtonian
fluid flowing between parallel plates. Instead of an exact velocity
profile, the approximation developed by Schechter [133] was used.
Both studies [133,157] were carried out for the Power-law model.
Matsuhisa and Bird [94] summarized many solutions to flow
problems using the Ellis model. Problems considered were
isothermal flow between flat plates, in circular tubes, in a film
flowing along an inclined plate, in annuli (axial, tangential and
radial), and nonisothermal flow in circular tubes with both uniform
heat flux and constant temperature wall conditions.
A numerical approach was attempted by Vlachopoulos and John
Keung [161] to examine heat transfer to a power-law fluid flowing
between parallel plates with constant wall temperature. As the
flow index, n, increased, the bulk temperature and the local Nusselt
number were observed to decrease. These results agree with Tien's
[157] work. It was also concluded that viscous dissipation had a
significant effect on heat transfer in a parallel plate channel.
Horizontal pipe
Analytical studies by Lyche and Bird [87] and Schenk and Van
29
Laar [134] can be considered as extensions of the classical Graetz-
Nusselt problem. They used velocity profiles appropriate to the
power-law model and the Prandtl-Eyring model, respectively. A
separation of variables similar to that used by Sellars, Tribus, and
Klein [138] for a Newtonian fluid was employed in Lyche and Bird's
work for a power-law pseudoplastic fluid.
By using ammonium alginate, applesauce and banana puree
which are predominantly pseudoplastic in character, Charm and
Merrill [17] investigated the heat transfer behavior of these fluids
in streamlined flow.
Analytical studies employing the power-law model have also
been conducted by Whiteman and Drake [164], Wissler and Schechter
[165], Pawlek and Tien [125], Foraboschi and de Federico [42] and
Kumar [76].
The analogy between heat and momentum transfer was
extended by Metzner and Friend [99], to include non-Newtonian
fluids in turbulent flow through circular tubes. The analogy was an
extention of the theoretical relationship for Newtonian systems
suggested by Friend and Metzner [45].
Data for a polymer solution of water-carbopol were acquired
by Metzner and Gluck [100]. They accounted for buoyancy effects in
non-Newtonian flows in an isothermal circular pipe. The equation
Eubank and Procter [40] proposed was modified using the power-law
model in the following form:
0.4 1/30.141 /33n+1
Nup = 1.75 (- (K) [G, + 12.6( Grw Pr, IT)
30
(2.16)
where Ko and Kw are the consistency indices evaluated at the bulk
and wall conditions, respectively. All physical properties were
evaluated at wall temperatures and wall shear rates. Gee and Lyon
[46], Griskey and Wiehe [51], Forsythe and Murphy [44], and Collins
and Filisko [28] also took polymer melt data in circular tubes with
isothermal walls.
A temperature-dependent power-law model was used by Hanks
and Christiansen [54], Christiansen and Craig [22], Korayem [72], and
Forsyth and Murphy [44], and by Cochrane [25]. Christiansen and
Jensen [21] used a temperature-dependent Powell-Eyring model in
obtaining their solutions. Christiansen and Craig proposed the
following simple temperature dependent equation to represent the
pseudoplastic rheology:
ti = K [S exp(AH/RuT)]n (2.17)
where S is shear rate, Ru is the universal gas constant, AH is the
activation energy per mole for flow, and k, n, and A H/R u are
empirical constants which are presumably independent of
temperature. They solved the governing equations numerically for
heating of both Newtonian and non-Newtonian fluids, for steady,
laminar flow and heat conduction to a flow without free convection
and thermal energy generation in circular tubes with constant
31
surface temperature. They also took experimental data for two
pseudoplastic fluids, a 3% water suspension of CMC and a 0.75%
water solution of CPM (carboxypolymethylene), under conditions
where free convection was negligible. Comparisons between the
estimated Nusselt numbers and measured Nusselt numbers gave a
mean deviation of ±7%.
By adding the possibility of viscous dissipation to the power-
law model, Gill [48] obtained a series solution. The eigenvalues are
functions of a rheology parameter, n. For each new n, a new set of
eigenvalues must be calculated.
Instead of viscous dissipation, slurry flow with uniform heat
generation was considered in the region of Gzx <25n by Michiyoshi
and Matsumoto [102]. Using the Bingham plastic model, Michiyoshi
[101] and Michiyoshi et al [103] also investigated the heat transfer
for slurry flow in both the fully-developed and the thermal entry
regions with internal heat generation.
Using the power-law model and a numerical solution of the
continuity, momentum, and energy equations for two dimensional
flow, McKillop [96] obtained results on convective heat transfer for
pseudoplastic fluids. For the case of fully developed entry flow, a
7.5% increase in Nux for a change in n from 1 to 0.5 and a 119%
increase in Nux for a change in n from 1 to 0 were observed at Gzx =
100n.
An experimental study for heating and cooling of pseudo-
plastic fluids (water-CMC, water-carbopol, water-polyox and ethyl
alcohol-carbopol) under isothermal surface temperature heating and
32
cooling conditions was carried out by Oliver and Jensen [118]. They
showed that the free convection effect did not depend on the 'L/D
ratio, and the correlation equation suggested by Metzner and Gluck
[100] did not fit their data. They presented the following
relationship,
0.14
Nu!) = 1.75(K
-) [Gz + 0.0083 (GI., Prv,)0.75,1/3
,, (2.18)
They showed that the heat transfer rate was affected more by
buoyancy than by either non-Newtonian or temperature-dependent
viscosity effects. They claimed that less viscous non-Newtonian
fluids increase heat transfer by as much as 100%.
Inman [65] obtained power-law solutions for the fully-
developed region and circumferentially-varying heat flux, using a
rheological model that is not temperature-dependent.
Using the Ellis model, laminar heat transfer to non-Newtonian
fluids was studied for both isothermal uniform heat flux boundaries
by Mitsuishi and Miyatake [104]. They obtained eigenvalues for 5
values of the flow behavior index, n, for each solution.
Laminar heat transfer in a circular tube with uniform wall
heat flux was investigated by Mizushina et al [107] employing the
power-law with temperature-dependent consistency (K), and
applying a correction term (Kb/Kw)0.1/n" to Bird's [12] asymptotic
solutions. They also took data for glycerol, which is Newtonian, and
aqueous solutions of CMC in the range, 10< Gzx < 300, which
33
includes the end of the developing region and the beginning of fully-
developed conditions. Significant data scatter suggests that
possible buoyant effects were not considered.
Lyon's [89] solution was extended to include radial-dependent
heat generation by Sestak and Charles [139], using the power-law
model. For the fully developed region with viscous dissipation, they
obtained the relationship,
BNu
B(n)
1 n B(n)Br'
8(3n+1) (2.19)
where B(n) is the power-law solution for the thermal entry region
by Bird [12] {where Nux = 1.412 81/3 Gzx}, and 8 = (3n+1)/4n, and Br'
is the modified Brinkman number. Note that dropping non-
Newtonian effects on the shear rate (8) out of Bird's solution leads
to the solution for the Newtonian case using temperature-
independent properties. Lyon [89] also developed equations for
uniform heat flux at the wall which are applicable for any
temperature-independent rheological model.
The work of both Mitsuishi and Miyatake [105,106] and Pigford
[130] extended the Leveque solution to include the effects of
variable viscosity and buoyancy for the flow of viscous fluids in
vertical tubes with an isothermal boundary condition. Leveque [82]
obtained the same results as those of Graetz in the thermal entry
region by using a linear velocity profile in the vicinity of the wall.
His analysis presented the following correlation:
Num = 1.75 81/3 Gz1/3
34
(2.20)
where 8 the shear rate ratio is equal to iew/8V/D with 7 W a function
of Gz,11b/11w, and Gr. The quantities 7 W and ri represent wall shear
rate and apparent viscosity, respectively. By evaluating 8x at the
axial mean wall temperature up to that point, they attempted to
compensate for the inaccuracy that resulted from specifying a
uniform shear all along the pipe wall. The difference between
Bird's result and their solution is only in the method used to
evaluate 8.
Han et al [53] reported the results of a study on axial pressure
distributions for the flow of molten polyethylene and polystyrene
through a circular tube (L/D = 4) at shear rates from 100 to 500
sec-1.
Cochrane [25,26] presented a numerical solution for the
coupled energy and momentum equations describing steady state
laminar flow of temperature-dependent power-law fluids. Pipe
flow and channel flow between two flat, parallel plates were
considered. A rheological model, ti = K eAH/RuT)n, was employed
to obtain results for both heating [25] and cooling [26]. When values
for dimensionless heat flux (4) = q"D/KT;), n, Pri, and AFI/RuTi were
taken as 2.0, 1, 1000, and 5, respectively, Nux increased by 5% at
Gzx = n/4 x 105 and by 14% at Gzx = 71/4 x 102 compared to the
temperature-independent solution.
Khabakhpasheva et at [69] and Kutateladze et at [78] showed
what was apparently the same set of data for the flow of a 1%
35
what was apparently the same set of data for the flow of a 1%
aqueous solution of polyacrilamide which is viscoelastic. No
elastic effects were found in circular tubes where conditions were
those of fully-developed, steady, laminar flow without free
convection.
Using temperature-dependent power-law fluids, the boundary
layer equations for flow in the entrance region with a uniform
velocity specified at the entrance were solved by Bader et al [7].
Their results were compared with data taken for aqueous
hydroxyethylcellulose and sucrose aqueous HEC alone. Entrance
Prandtl numbers ranged from 144 to 270, and the two fluids with n
= 0.85 and n = 0.62 were used. Their results, however, were hard to
interpret.
For fully-developed turbulent flow of power-law fluids at
large Prandtl numbers, a new correlation on heat transfer rates was
made by Krantz and Wasan [74].
By using a non-linear plastic (t = To + no in), Shul'man et al
[148] analyzed convective heat transfer in a circular pipe, to
include dissipation. The boundary layer equations with a uniform
wall heat flux were solved by Etchart and Welty [39], using a
temperature-dependent Powell-Eyring model. Rheological data for
several non-Newtonian fluids taken from the literature were used
in their analysis. They found that temperature effects on viscosity
had a major influence on heat transfer for the fluids analyzed. Heat
transfer was also affected, but to a lesser degree by non-Newtonian
flow behavior. They stated that the effect of viscous heat
36
dissipation on heat transfer would be significant for highly viscous
fluids with viscosities exhibiting moderate to heavy temperature
sensitivity. They also observed that the pressure drop is
significantly reduced when a fluid exhibits a temperature-
dependent viscosity.
Payvar [126] obtained fully developed temperature distri-
butions and asymptotic Nusselt numbers for three popular models -
the power-law, the Bringham plastic, and the Ellis model for a
uniform wall heat flux, considering the effect of viscous
dissipation using a procedure similar to that which he used for
parallel flat plates.
By employing a temperature-dependent non-linear plastic
model, the momentum and energy equations in the absence of radial
velocity terms were solved numerically by Forrest and Wilkinson
[43]. They obtained results for both heating and cooling. Viscous
dissipation effects were shown in their additional results.
For the thermal entry region flow, Bassett and Welty [9]
initiated an experimental study to determine the heat transfer rate
to laminar, forced flow of pseudoplastic fluids in a uniformly
heated circular pipe with fully developed velocity profiles at the
entrance, where Graetz number values varied between 240 and
38,000. They found that the local heat transfer rate was influenced
by temperature-dependent viscous properties much more than non-
Newtonian effects. They also observed that the intensity of
secondary flow patterns decreased with more viscous fluids and
increasing flow rate, and increased as the flow moved downstream.
Effects of viscous heating were not detected in the flow of fluids
37
tested with Brinkman numbers greater than 4.22 x 103.
Experimental data, exclusive of two out of 26 runs, where free
convection was explicit in the results, were correlated as
Nux = 1.85 Gzx1/3-0.0318x (2.21)
which determines the heat transfer rate within 10% with a mean
error of 3.57% for flows without significant free convection
effects. Their experimental investigations led to conclusions that
the local wall shear rate controls the heat transfer rate and that
the shear rate is more profoundly influenced by temperature-
dependent viscous properties than by non-Newtonian behavior. They
also accounted for secondary flow due to buoyancy which can affect
the rate of heat transfer substantially far upstream of the usual
onset of full thermal development.
Laminar convective heat transfer of power-law pseudoplastic
fluids (methyl ether of cellulose, carboxypoly-methylene) in
circular conduits was investigated by Mahalingam et al [90), using
boundary conditions of uniform heat flux with a step change in heat
flux. Variations in the shear stress with shear rate for
temperatures between 80 F and 150 F were plotted for each
concentration to evaluate n and K. The exponent, n, was found to be
practically independent of temperature except for two
concentrations of methocel. Even these n values did not vary
greatly with temperature. So it was claimed that n may be assumed
a constant with temperature in any theoretical analysis. The
38
factor, K, was observed to decrease with increasing temperature
for each concentration. One of Bassett's conclusions [8] noted that
temperature-dependent fluid rheology is generally more important
in regard to the rate of heat transfer than the degree of
pseudoplasticity. The coefficient, K, was correlated as K = a ebt
where a and b are constants. Their [90] final correlation accounting
for pseudoplasticity, radial viscosity variation with temperature,
and free convection is of the form:
Kw
Nub 1)-1/3
= 1.46 [Gzb + 0.0083 (Grwprw)0.75]1/3(rb- 1-
(2.22)
where the absence of L/D in the free convection term is noted.
From a comparison of the numerical results with experimental data,
they concluded that analytical methods already available for the
Newtonian case may be extended to the non-Newtonian case.
Lakshminarayanan et al [79] studied heat transfer for heating
and cooling of pseudoplastic fluids in turbulent flow through
circular tubes. Experimental heat transfer data were correlated for
the heating and cooling of various aqueous polymer solutions
obeying the power-law model, allowing for the effect of
generalized Reynolds number and Prandtl number. They proposed the
following correlations for n' = 0.86-0.98, NRe,gen = 5000-22500, and
= 9-32:NPr,gen
N = 0.0710(N )-0.67 for heating (2.23)St Re, gen) Pr, gen
39
-0.33(N )4).67 for cooling (2.24)Nst = 0.0440(NRe, gen')
Pr, gen'
where
NRe, g en =DTIV2n p
8,1_1K( 3n+14n
Cp
KC 34n)11( 8V )11-1
Pr' k 4n J D J
(2.25)
(2.26)
n' indicates the slope of log versus s log (8V/D) plot.
Bird, Armstrong and Hassager [13] expressed laminar heat
transfer results in the thermally developing region by the following
asymptotic relationships:
for uniform heat flux
3n+1Nux 1.41 (4n
1
Gz3=(2.27)
where Gz > 25n, and for constant temperature
1
33n+1 )Nux = 1.16 (-4n
(2.28)
where Gz > 33n. The factor, [(3n+1)/4n]1/3, accounts for non-
Newtonian effect.
Gottifredi et al [49] developed a simple new analytical
approach to estimate local and mean heat fluxes for a constant wall
40
temperature to a fluid flowing in the laminar regime. They claimed
that their procedure could be applied to the analysis of both plane
and cylindrical geometries with the axial velocity distribution
given as an analytical function of position without any limitation on
the range of Graetz numbers. A marching technique was used and
comparisons were made with previous numerical estimates. The
agreement was very good.
Joshi and Bergles [67] investigated heat transfer with laminar
flow of two water-methocel pseudoplastic solutions in a circular
tube. Their following correlated equation was in generally good
agreement with the experimental data of Bassett and Welty [9]:
Nu vp,n
( Nuvp y Nucpm
Nu cp,n )\ Nucp,n=1
-0303 1
4.36 1 + (0.381 X+ )818
(2.29)
where, vp, and, cp, mean variable property and constant property,
respectively.
Cho and Hartnett [20] have described heat transfer and fluid
mechanics for non-Newtonian fluids in circular pipe flow. Laminar
heat transfer in the thermal entrance region of a circular pipe was
also reviewed by tabulating empirical correlations for uniform heat
flux and isothermal boundaries. Friction and heat transfer for
viscoelastic fluids were studied experimentally in turbulent pipe
flow of viscoelastic aqueous solutions of polyacrilamide by
Hartnett and Kwack [55], employing variables such as polymer
41
concentration, polymer and solvent chemistry, pipe diameter, and
flow rate. They also included a study of degradation effects. They
concluded that the friction factor and the dimensionless heat
transfer rate of the flowing fluid are determined only by the
Reynolds number, the Weissenberg number, and the dimensionless
distance. They suggested that their approach should be applicable
to other polymer solutions.
Vertical cylinders
Heat transfer studies on the vertical turbulent flow of water-
clay, water-powdered aluminum, ethylene glycol-graphite, and
ethylene glycol-aluminum slurries were carried out by Orr andDallavalle[119] for an isothermal wall. Their final correlation was
expressed as
)0.80( )0.33( )0.14
(11n3 0.027k Ps ) ks (2.30)
where s means suspension or slurry.
De Young and Scheele [33] and Marner and Rehfuss [93] studied
free convection in vertical pipes with constant heat flux conditions
for both upward and downward flow. Numerical solutions were
obtained for a power-law fluid. De Young and Scheele showed,
graphically, that the ratio Gr/Re at the maximum velocity varies
with the pseudoplasticity index n for both heated upflow and heated
downflow. It was seen from their figures that pseudoplastic fluids
42
set up flow instabilities earlier due to buoyancy effects. For
heated downward flow, opposite effects for the case of heated
upflow could not be predicted for a fluid with a power-law index
(less shear thinning) greater than n = 0.5 at a given value of Gr/Re.
It appeared that the critical Gr/Re increased with n and that, for
both upflow and downflow, the limiting Gr/Re was higher for
dilatant fluids than for pseudoplastic fluids. Unlike the case of
heated upflow, a decrease in the Nusselt number was found as
pseudoplasticity increased.
Marner and Rehfuss [93] showed three fully-developed velocity
profiles for n = 0.5. The velocity profile became more distorted
with an increasing Gr/Re. Nusselt numbers, flow stability, and
pressure drop were significantly affected by these velocity
distribution changes. As the pseudoplasticity index, n, decreased
the Nusselt number increased very significantly for a given value of
Gr/Re. While non-Newtonian effects tended to reduce the Nusselt
number in contrast to buoyancy effects for dilatant fluids, with
pseudoplastic fluids the buoyancy effects and non-Newtonian
behavior tend to increase the Nusselt number. They also observed
that the pressure drop increased with buoyancy, and that this effect
was much more significant for dilatant fluids than for
pseudoplastic fluids.
Combined convective heat transfer in a vertical tube with
upward flow for isothermal walls was investigated both
theoretically and experimentally by Marner and McMillan [92]. They
observed a point of maximum velocity profile distortion to exist
43
where an increase in dimensionless axial distance increased the
local Nusselt number. They also found that the pressure drop
increased with Gr/Re for all values of n. They obtained
experimental heat transfer data for carbopol solutions, which were
in agreement with theoretical predictions within ±15%.
Using a finite difference method to solve the coupled
continuity, momentum, and energy equations, Marner and Hovland
[91] investigated the simultaneous effects of viscous dissipation
and combined free and forced convection of power-law fluids for
fully-developed laminar flow in a vertical tube with uniform wall
heat flux. All properties were assumed to be constant except for
density which was considered a function of temperature. Velocity
profiles and Nusselt numbers were obtained as functions of the
flow behavior index(n), Gr/Re, and the product of E and Pr where E is
the Eckert number. Their results showed that the velocity profile
was distorted due to viscous dissipation. The Nusselt number
decreased while the friction factor increased with increased
viscous dissipation. Their numerical solutions are quite restrictive
because the power-law consistency index was assumed to be a
constant.
Rectangular duct
Fully-established friction factors for both purely viscous and
viscoelastic fluids in laminar and turbulent flow were measured in
a square duct by Hartnett et at [56]. Laminar friction factors for
44
both non-Newtonian fluids were in good agreement with the
equation for laminar circular pipe flow, 16/Re*. Turbulent friction
factors for purely viscous power-law fluids flowing through
rectangular ducts were in good agreement with the Dodge and
Metzner equation [37] for circular pipe flow of power-law fluids
with Re' replaced by Re*. Asymptotic friction factors for
viscoelastic fluids in turbulent flow through rectangular ducts
correlated well with circular tube expressions when compared at a
fixed Reynolds number, pVDh/rla, based on the apparent viscosity
evaluated at wall , and Dh is the hydraulic diameter. It was also
observed that the asymptotic fully-developed turbulent friction
factors in a square duct are much higher than the results of the
corresponding circular pipe at the some values of Re* (the Kozicki
Reynolds number).
Laminar heat transfer to a viscoelastic fluid flowing through a
rectangular channel was recently investigated by Hartnett and
Kostic [57]. In general, viscoelastic fluids behave as purely viscous
non-Newtonian fluids in fully-developed laminar pipe-flow because
the elastic properties of the fluid are not significant. High heat
transfer rates were measured compared with values for a purely
viscous fluid or a Newtonian fluid. Mena and co-workers [97] also
found higher heat transfer coefficients for viscoelastic fluids in
laminar flow through rectangular ducts than those for Newtonian
fluid. The elastic effects of a viscoelastic fluid, that make the
normal forces act differently on the boundaries, cause secondary
flows which produce a significant increase in heat transfer. The
friction factor, however, was unaffected by the presence of
45
elasticity in both studies [57,97]. They used an aqueous
polyacrylamide solution in a 0.5-aspect-ratio-rectangular-duct.
Rayleigh numbers ranged from 5,000 to 50,000 in their heat
transfer measurements. By comparing Nusselt numbers for the
viscoelastic fluid runs with the values for the water runs, they
concluded that a secondary flow, which was produced along both
upper and lower walls under the thermal boundary conditions given,
increased the heat transfer. They also found the effect of
elasticity on the pressure drop of a viscoelastic fluid to be
relatively small. Their measured dimensionless pressure drop
agreed well with the fully established friction factor, f = 16/Re*,
where
Re* = pVDh/[8 (a+/2-n)K](2.31)
and the constants a and b are 0.7276 and 0.2440, respectively. For a
circular duct a = 0.75 and b = 0.25 reduces the Re* to the
generalized Reynolds number introduced by Metzner [1965].
Tachibana and co-workers [154] reported a numerical analysis
of steady laminar flow of an inelastic power-law fluid (n < 1) in the
inlet region of rectangular ducts using finite-difference methods.
They also investigated experimentally the axial pressure
distributions from the entry region up to the fully-developed region
in rectangular ducts. Their results are as follows:
(a) In the inlet region with a power-law fluid the pressure
46
drop and the velocity in the duct center are smaller than with
a Newtonian fluid. Both pressure drop and velocity increase
with an increase in the the increasing power-law index. As in
the case of a Newtonain fluid, however, both the velocity in
the duct center and the pressure drop of a power-law fluid
decrease with an increase in the aspect ratio of the duct.
(b) The velocity profile in a power-law fluid is flatter than in
a Newtonian fluid.
(c) The entry region, with a power-law fluid, is longer than
that for a Newtonian fluid and decreases with an increasing
power-law index.
Others
An experimental study of the secondary and main flows was
carried out for viscoelastic polymer solutions, in an abrupt 2-to-1
circular expansion, by Halmos and Boger [52]. They measured the
developing centerline velocity and reattachment lengths for flow
through the abrupt circular expansion and observed that the
predevelopment of the flow field for viscoelastic fluids, which
decreases the size of the secondary cell, is directly related to
elasticity. It was also shown that the reattachment lengths of the
secondary cell are 13% and 29% less than those for inelastic fluids
at the same conditions. The size of the secondary cell is always
equal to or smaller than the inelastic prediction. They did not find
any significant deviation from inelastic behavior for We < 0.047.
47
We =VO
(2.32)
where 0, the Maxwell relaxation time, is expressed as
Pll -P22
2ti2Y (2.33)
V and D indicate the average velocity in the upstream tube and the
upstream tube diameter, respectively. P1
P22 means the normal
stress difference. For We > 0.12 the onset of flow instabilities
were detected in their measurements.
Non-Newtonian heat transfer in non-circular tubes has been
studied by Oliver [116] and Oliver and Karim [117]. Oliver and Karim
showed experimentally that flattened tubes produced higher
laminar-flow heat transfer coefficients than tubes of circular
cross-section. They stated that this increase is due to an increase
in the tube-wall shear rate and secondary flow patterns which
develop in viscoelastic liquids. The effect of secondary flows was
largest for a 1.5-aspect ratio. The aspect ratio was defined as the
ratio of the length of major axis of the flattened tube cross-section
to that of the minor axis.
48
Chapter 3. EXPERIMENTAL DESIGN AND SET UP
3.1 Experimental design
Appreciable free convection is known to occur with internal
laminar flows where Graetz numbers are greater than 10. To
achieve this value one needs either very low flow rates or very long
heated sections. The length of the test sections that could be
accommodated in the available facility was 10 ft. The entrance
sections were selected to be approximately 5 ft for flow
development. These entry lengths were found sufficient to ensure
fully-developed flow for all flow conditions used.
The flow remained straight from the entrance section inlet
through the exit of the test section. As discussed earlier, the test
fluids were dilute aqueous solutions of Carbose D-65. Their
behavior is pseudoplastic, and the degree of non-Newtonian flow
behavior increases with increasing concentration of solute.
Aqueous CMC displays flow behavior indices, n, as low as 0.6, and
maintains consistent behavior over a wide range of shear rates.
These fluids are non-toxic and their characteristics are well known.
They are also relatively inexpensive. They have been used
extensively by other non-Newtonian investigators.
Test section diameters of 1.5 and 2.0 inches were used in this
work. In general, smaller-diameter test sections have a large-
wall-thickness to diameter ratio, increasing the possibility of
significant heat conduction in the wall. Since the buoyancy
49
parameter Gr* varies directly with D4, smaller diameters also
reduce buoyancy effects.
The maximum possible Reynolds number was estimated for
each test section using the following Newtonian model, which is
conservative:
Le = 0.0575 D Re (3.1)
where Le is an entry length for fully-developed flow and D is inside
diameter of the test tube. The maximum Reynolds numbers for
fully-developed flow were 708 and 529 for the small and large test
sections, respectively.
The pressure drop along the entry and test sections was
estimated for this flow rate using the relationship [152, P 110],
AP=2K(324--II VnLri Rn+1
(3.2)
Values for AP were always found to be within the maximum
allowable pressure drop considering flange design, plastic wall
strength, pumping losses, and the possibility of viscous dissipation.
To avoid two-phase flow, the maximum design wall
temperature was chosen to be 185°F (85°C). Inlet temperatures
were adjusted to room temperature to minimize the possibility of
heat transfer to or from the fluid in the flow development section.
To reach this wall temperature for a range of flow rates, the
maximum power input required was estimated based on the Graetz
50
solution. This power input was scaled up by 50% to account for an
increase in heat transfer due to property variation with
temperature. The largest design value of input power was 6576
watts for the largest flow rate.
Therm couple Junction
Small Test Tube
Heating Strip
Large Test Tube
Figure 3.1. Cross section of test tubes
Axial locations selected for thermocouple placement were 3,
9, 18, 30, 48, 68, 91, and 117 inches from test section entrance.
These positions were chosen, based on the Graetz solution, to
capture important temperature information. The temperatures at
the bottom, side, and top of the tube wall at each axial position
were measured to examine the possible presence of buoyancy
effects. Figure 3.1 shows both test sections schematically.
Thermocouples could not be placed precisely at the positions 90°
and 180° in the small test section and 90° in the large test section
51
due to the equally-spaced heating strips. They were placed as close
as possible to these positions which were positioned at 0°,
98.2°,and 163.6° from the bottom in the small test section and at
0°, 102.9°, and 180° from the bottom in the large test section.
Keeping these design criteria in mind, flow rates for each fluid
and each test section were calculated for the desired range in
performance parameters. Six concentrations of CMC were employed
to cover the range in flow behavior, 0.6 < n < 1.0. In each run, the
power applied was that necessary to achieve a maximum wall
temperature of 185°F (85°C). Local heat tranfer rates were
collected for 127 < Gzx < 27474 and 5832 < Rax < 238011.
52
3.2 Experimental setup
3.2.1 Flow loop system
Figure 3.2 shows a schematic diagram of the entire test
apparatus. The pump drew fluid from the feed tank through a short
span of 2-inch piping. The fluid then flowed through 1.5-inch piping
to the tube side of the heat exchanger. Tap water was used on the
shell side for cooling. The test fluid continued to flow through 1.5-
inch piping to a static mixer which was used to eliminate any
temperature gradients. The fluid then passed through a 10-inch-
long viewing section. The bulk temperature was measured just
after the viewing section. The flow then entered one of the two
test sections. These were uniformly heated by electrical
resistance heaters attached to the walls. Wall temperature
measurements were taken at various axial positions along the test
section. The temperature measuring system consisted of
thermocouples, a reference junction, and a Leeds and Northrup K-4
potentiometer. A data acquisition system was used to record the
other miscellaneous temperatures. The data acquisition system
was not employed for wall temperature readings due to its
unacceptable accuracy.
Another static mixer was located at the exit to the test
section just before measuring the bulk temperature. The flow was
then directed to a weigh tank where the flow rate was determined.
The fluid entering the pump was mixed in the tank to keep it in a
Enclosure Boundary
Entrance Sections
--XView
Window
StaticMixer
Thermocouples
110 0 0 0 0 0Test Sections
Heat Exchanger
(EI
To Drain City Water-ix
By -Pass
Pump
Weigh Tank
Tank Mixer
001:40.1*
11
F77774:771.777:---1)
Figure 3.2. Schematic diagram of test loop
IIII
mominminominiomort
4a
Feed
Tank
Scale
54
uniform state.
The enclosure surrounding the apparatus extended from a point
1 ft upstream of the inlet flanges to the end of the static mixer
beyond the outlet of the test section. It was 24 inches wide, and 18
inches deep, and was filled with loose, vermiculite insulation.
3.2.2 Apparatus
Details of the apparatus are presented in this section.
Tank mixer
A 240 V DC motor, rated at 1/3 HP at 1750 rpm drove the tank
mixer through a 5 to 1 gear reduction unit. A 110 volt variable
transformer coupled to a 2:1 step-up transformer and a full wave
bridge rectifier supplied power. A 4-inch diameter, 3 blade, paddle
stirrer, which was scaled up from a design recommended by Union
Carbide Corp, was used for the mixing of water-soluble polymers.
Pump
A Moyno, 2L4, "progressing cavity" type was used. It had a tool
steel rotor and a Buna N rubber stator. The rotor was 18 inch long
and 1.5 inch in diameter. The fluid was pushed along in a cavity as
the rotor turned within the stator like a screw conveyer. The
manufacturer rated this pump at 24 gpm at 1200 rpm for fluids
whose viscosity ranged from 1 cp to 1000 cp. The output was
reported to be reduced to 9 gpm at 450 rpm for a 2500-5000 cp
55
fluid. As the exit pressure was increased to 80 psig, it was also
noted that these outputs dropped to from 24 to 22 gpm and from 9
to 6.7 gpm, respectively. Although the capacity was reduced by
increasing pressure and/or increasing viscosity, it provided a
steady, almost positive displacement flow. It also provided a flow
with less shearing action than a centrifugal pump or a gear pump.
A 240 volt DC motor rated at 2 HP and 1750 rpm drove the
pump. A V-belt and sheaves were used to reduce speed. Bassett [8]
noted that a 3.061 speed reduction resulted from the sheaves whose
diameters were 2.65 and 8.0 inches. Power was supplied to the
motor armature by a 220-volt variable transformer. Smoothing was
provided by 2000 gf of capacitance. Power was supplied to the
field with a line voltage of 208 V and a full-wave bridge rectifier.
Rectifier output was smoothed with 20 gf of capacitance.
Heat exchanger
The shell-and-tube heat exchanger with 2 tube passes had
about 20 ft2 of heat exchange area. The tubes were 5 ft long and
3/4 inch in diameter. City water was used in the shell side.
Static mixers
Construction of the static mixers was based upon a design
patented by Kenics Company. It consisted of a series of "bow tie"
elements. These elements were fabricated from 0.10 inch, annealed
aluminum sheet. They were formed from rectangular pieces 1.939
inches wide and 3.25-inches long by holding one end while the other
56
end was twisted 180 degrees. The number of clockwise elements
was equal to that of counter-clockwise elements. The ends were
notched and joined at right angles with epoxy glue. Element twist
directions were alternated in the joining process. This mixer was
designed to split, develop, and turn the flow with each new element.
A total of N elements would divide the flow into 2 N strata, the size
of each being D/2 N. Ten elements coated with epoxy paint were
inserted into a 2-inch, "hi-temp" PVC pipe 30 inches long.
Entrance sections
A slip-on type of standard 150 psi PVC flange was cemented on
each entrance section for attachment to the test sections.
Test sections
Stainless steel sleeves were silver-soldered to the both ends
of each test section and threaded. In addition to providing positive
mounting, the use of stainless steel helped to minimize heat loss.
A transition bushing was constructed from CPVC to provide the
minimum disturbance of flow between two different inside
diameters of both sections. PVC flanges connected the test
sections to other portions of the system.
a. Small section
A 10-ft, 13/32-inch-long piece of 1.505-inch ID, hard drawn,
copper tubing was used as the basic structure for this test section.
Its outside diameter was 1.625 inch; the wall thickness was 0.060
inch.
57
b. Large section
A 1.985-inch ID, 0.070-inch wall copper tube, having a length
of 10 ft, 7/32 inch, was used; the outside diameter was 2.125 inch.
A diagram of the test section assembly is shown in figure 3.3.
Power supply for heater
A Sorensen model DCR 300-35A, regulated DC supply, was used
to supply power to the heating elements. The rated output was 0-
300 volts and 0-35 amps.
3.2.3 Test section
(1) Design of heated section
Methods for providing energy input considered were Jou lean
heating of the pipe wall and wrapping the pipe with resistance wire
or ribbon. Joulian heating of the pipe wall was attractive, however,
very high currents would be required to achieve the needed power
level. This was a problem that could not be solved. The method
chosen was to attach heating strips to the tube surface. These
closely-spaced longitudinal strips were made . of Kanthal A-1,
produced by Kanthal Corporation. The Kanthal strips were 3/8-inch
wide and 0.035-inch thick. The electrical resistance of this
material is 0.05219 ohm/ft. Although this method required a great
deal of construction time and careful work, there was a lower risk
of developing local hot spots than with helically-wound wire or
ribbon, and the uniform heat flux boundary condition could be
Do,E
PVC flange(socket)
PVC PIPE
Di,E
PVC flange(threaded)
Stainless steel sleeve
Copper tube
Di,T
:/"wl
Rubber gasket
Figure 3.3. Assembly at test-section entrance
Do,T
59
achieved.
Two design criteria were established. One was that the
heating element area be near to that of the inside tube surface. The
other was determined by the capacity of the available power supply.
Overall heater resistance was targeted in the range 3.7 < R < 12
ohms.
The heater for the small test section consisted of eleven
119.219-inch strips. A simple series electric circuit was used to
supply energy; connections were made by silver-soldering copper
strips at the ends. The total resistance was 5.7035 ohm. The
heater area and the design spacing were 87.2% of the inside wall
area and about 95 mils, respectively. For the large section 14 of
the same size strips were used; each strip was 118.906-inch long.
The electric circuit was made by connecting the ends as was done
with the small section. The total resistance was 7.2400 ohm. The
heater area is 84.2% of the inside area of the tube, and the design
spacing was about 107 mils.
Figures 3.4 and 3.5 show schematic diagrams of the layout of
the heating elements for both test sections.
(2) Construction
Heating elements had to be electrically insulated from the tube
wall. In order to provide good insulation, a 3.5-mil double-coated
polyester film with a rubber-resin adhesive on each side, with a
width of 1.5 inches, was applied in longitudinal strips along the
outside wall of the copper tubes. This tape was specified by the
61
: Flattened 6 gage of copper wire: 8 gage of copper wire
Figure 3.5. Schematic diagram of layout of heating strips for largetest section
62
manufacturer to have a high dielectric strength and capable of
withstanding 130°C. A 1.25-inch width of woven glass tape was
placed longitudinally over the polyester film. The glass tape was
able to withstand temperatures up to 180°C. Using the thermal
conductivities of each material (0.24056 W/m°K for the poly-
esterfilm and 0.16666 W/m°K for the woven glass tape) the
temperature drop across the insulation was estimated to be about
19°C for the maximum heat flux used in the present study.
A formidable effort was required to attach the heating
elements onto the tape while maintaining even spacing. Each strip
was 10-ft long The desired spacing was accomplished after many
trials.
Electrical connections between heating strips were made by
silver-soldering a copper strip across the ends. This process was
repeated until a complete series curcuit was made. Special care
was taken during the soldering to minimize damage to the
insulation.
Precise locations for the thermocouple attachments were
measured and noted on the tape. Then the same polyester film was
wound 2 times around the heating elements. Additional polyimide
film was wound once over the entire tape for a better thermalinsulation. Marks for the thermocouple positions were visible
through this transparent tape.
(3) Other details
Holes were cut in the insulating tape, at desired thermo-
63
couple locations. The insides of the holes were cleaned and the
thermocouples were carefully attached to the bottom wall of the
hole with a paste that was both highly conducting and temperature
resistant. Each thermocouple was also covered with an epoxy for
added strength.
In each test section 24 thermocouples were placed at axial
locations, 3, 9, 18, 30, 48, 68, 91, and 117 inches downstream from
the beginning of the heating section. An additional thermocouple
was mounted at the bottom of the each test section just prior to
the exit. Readings for this location were monitored by the data
acquisition system to establish that the maximum design wall
temperature was maintained. Each thermocouple was at a depth of
30 mils. Three thermocouples were placed on the top, side, and at
the bottom of the tube at each axial position. A relatively low
thermally-conductive epoxy, Omegabond 100, was also used for
permanent bonding of the thermocouple. Additional Omegabond 100
was packed around the exposed thermocouple wires for thorough
insulation from the heating elements. Finally two coats of
polyurethane insulation were sprayed over the entire test section.
Photographs are shown in Figure 3.6 through Figure 3.10, which
show respectively: 1) an overall view of test apparatus, 2) test
sections with heating strips attached, 3) test sections in place
with the loose insulation removed from the enclosure, 4) weigh
tank, feed tank, and pump, 5) entrance sections, heat exchanger, and
static mixer.
64
Figure 3.6. Overall view of test apparatus*
*Numbers in the figure indicate the following:1. weigh tank 2. tank mixer 3. scale 4. entrance sections5. switch boxes 6. power supply 7. ice bath container8. potentiometer 9. standard cell 10. constant voltage supply11. viscometer 12. data acquisition system
67
Figure 3.9. View of weigh tank, feed tank, tank mixer, andpump
r r 1
imp
:o t rrt ,Tx..
Figure 3.10 View of entrance sections, heat exchanger, and staticmixer
68
3.2.4 Viscometer
The Viscotester VT500 manufactured by the Haake Company in
West Germany, a kind of rotational viscometer, was used for
evaluating the non-Newtonian character of the test fluids. The
viscosity measurement system consisted of the viscometer, a
tempering vessel, rotors and beaker, a constant temperature
circulator, and a temperature sensor. Its speed range varied from 2
to 500 rpm.
The principle of rotational viscometers is to measure the
resistance of a fluid against the inner cylinder which rotates at a
defined rotational speed while the outer cylinder is held stationary
(Searle system). It develops a Couette flow in the annular gap
between the rotating inner cylinder and stationary concentric outer
cylinder. Figure 3.11 shows a schematic diagram of the rotational
viscometer.
70
Chapter 4. MEASUREMENT SYSTEM
4.1 Temperature measurement
Forty eight wall temperature readings and the inlet and outlet
fluid temperature readings were taken at each operating condition
by means of a potentiometer. Ten additional temperature readings
were recorded using a data acquisition system. This system
consisted of an HP 3421A data acquisition/ control unit, an HP-87
computer, and an HP-IB interface.
Copper-constantan thermocouples insulated with teflon were
used. Appropriate lengths were cut to fit each location of the
measuring site, the potentiometer, the data acquisition system, ice
bath, and switch boxes. Copper-constantan thermocouples were
also used for the other 10 measurement locations including those
for the inlet and outlet fluid temperatures.
Twenty five thermocouples were attached to each test section,
all but one were wired to the potentiometer through the
thermocouple switch boxes and ice bath reference junction. The
other was mounted at the bottom of the wall just prior to the end
of the test section. In order to check the wall temperature and
assure steady state, this one was connected to the 3421A data
acquisition/control unit. A null detector, constant voltage supply,
and standard cell were used in conjunction with the Leeds and
Northrup 7554 type K-4 potentiometer. Figure 4.1 shows a
schematic diagram of the temperature measurement system.
71
Thermocouple probes were used to measure flow temperatures
at 4 locations, these being in the main flow at the outlet of the
feed tank, just prior to the entrance sections, just after the outlet
mixer, and at the inlet to the shell side of the heat exchanger.
Another thermocouple probe was used in an auxilary capacity to
measure the room temperature near the entrance sections.
Five thermocouples were also attached at the following
locations: 6 inches upstream of the inlet on the top of the outside
surface of each entrance section, 1 inch downstream of the last
wall thermocouple on the outside wrap of each test section, and in
the vicinity of the end of the entrance sections.
To verify steady state operation of the flow system, the wall
temperatures at the bottom near the end of the test section, at the
inlet to the entrance section, at the inlet to the heat exchanger, and
on the outside wrap of the test section were continuously checked
using the data acquisition system. The system required less than 2
hours to reach steady state.
4.2 Thermocouple calibration
Calibration of each thermocouple was made using the same
circuit as shown in Figure 4.1 before they were mounted in the
apparatus. They were calibrated using a Fluke 2180A RTD digital
thermometer, the Haake constant temperature circulator, and the
K-4 potentiometer. The maximum error of the thermometer was
±0.1 °C over the temperature range used during the calibration.
StandardCell
Small Test Section, Inlet, Outlet Temp.Cu (26)
( 24 ) Large Test Section
/
Miscellaneous( ).
Constant NullVoltage DetectorSupply
11011111b)
TC TC TC TC
Switch Switch Switc Switch2p-12T 2p-12T 2D-1 2p-16T
K-4 potentiometer TC Sw
ReferenceJunction
Cu
Ct
AcqusitionSystem
S
tch
Figure 4.1. Schematic diagram of temperature measurement system N
73
Each of the test section thermocouples was calibrated by
taking at least 8 data points at temperatures near 15, 25, 35, 45,
55, 65, 75, and 85 °C. Calibration data for each thermocouple were
interpreted employing a third-order polynomial regression. The
largest standard deviation among the thermocouples used with the
small test section was 0.16. The largest for the large test section
was 0.12. A statistical package, Statgraphics, used for statistical
analysis of the calibration data showed that the coefficient of
determination was very close to 1.00, at the 99% level of
confidence, for all cases. A typical value was 0.998.
4.3 Power measurement
A sketch of the power supply circuit is shown in Figure 4.2. As
shown in the figure, the heater for each test section, power supply,
and a 50mv/30amp shunt were connected in series. A Sorensen
DCR-35 DC power supply provided energy to the electrical heating
strips. The power input to each heater was limited by two 30 amp,
250 AC fuses in the fuse box.
The power supply and the shunt voltages were measured
periodically throughout the experiment. For an accurate voltage
reading the shunt voltage was measured using a Fluke 8200A digital
voltmeter. This instrument has a specified accuracy of ±0.05mv.
The power dissipated in the test section, P, is given by a simple
application of electric circuit theory.
P = (Vs/Rs)2 Rh (4.1)
74
Shunt
;:ef,.:
Small Test Section
Large Test Section
Fuse BoxAMP
Power Supply
A.C. voltage input
Figure 4.2. Schematic diagram of power supply circuit
75
where Rh, the heater resistance, was found to be 5.70 ohms for the
small test section and 7.24 ohms for the large test section. The
terms, Vs and Rs, indicate shunt voltage and shunt resistance,
respectively.
4.4 Flow measurement
Flow measurement was done using a straightforward procedure
of timing the accumulation of a known mass in a weigh tank.
4.5 Viscometric data measurement
The Haake VT500 rotational viscometer, with a tempering
vessel connected to the constant temperature bath, was used to
measure rheological values throughout the experimental work.
Rheological values of several concentrations of the polymer
solutions were evaluated at fixed temperatures near 15, 25, 35, 45,
55, 65, 75, and 85 °C before each experiment. Measured torque
values were used to obtain absolute values of shear stress, using
the relationship
27c2 H (4.2)
where 'Cr is shear stress at the rotor, M is the torque, Rr is the
radius of the rotor and H is the rotor length. A photograph of the
viscometer is shown in Figure 4.3.
77
Chapter 5. EXPERIMENTAL PROCEDURE
5.1 Test fluids
Carbose D-65, purchased from Carbose Corporation, which is
used in the formulation of powdered laundry detergent, was used for
the present experiment. Carbose is an industrial grade of carboxy-
methyl-cellulose (CMC). Its composition is approximately 65% CMC,
25% sodium chloride, 8% sodium glycolate, and 2% water.
Concentrations of the fluids used for the current work were 5%, 6%,
7%, 7.5%, 8%, and 8.3% by weight. The higher concentrations were
prepared by adding more solute to the previous batch.
5.1.1 Mixing of polymers
Circulation of the fluid for several hours during the mixing
process caused considerable degradation by breaking down the fluid
polymer chains. Because of this, viscometric data for every batch
were taken after complete mixing was achieved.
5.1.2 Properties
The properties of the test fluids except the viscosity were
taken to be the same as those of water. It has been found that the
density and specific heat for aqueous CMC solutions are equal to
those of pure water, and the thermal conductivities have been
reported to be within 1 to 3 % of pure water. The thermal
conductivities of non-Newtonian aqueous solutions were also
78
measured by Lee, Cho, and Hartnett [81]. Their data showed good
agreement with the corresponding values for water up to
concentrations of 10,000 wppm (parts per million by weight). Table
5.1 provides values of thermal conductivities for water and aqueous
CMC solutions.
5.2 Test procedure
Six aqueous solutions of CMC were used in this work.
In order to check out the viscometric data for the working
fluid, a fluid sample picked up from the weigh tank was evaluated
during every test run.
Table 5.1. Thermal conductivity values
Liquid
K
TempW/m K
(°C)
c (wppm)20 30 40 50
Water 0.593 0.612 0.627 0.645
CMC
1,000 0.576 0.603 0.632 0.648
10,000 0.582 0.611 0.637 0.665
79
The power input to each test section at a fixed flow rate was
adjusted so that the test section might be brought to a maximum
wall temperature of about 85°C. The insulating tape did not
overheat at this condition. Values of power were evaluated by the
potential drop across the main shunt located between the heater
and the power supply.
Steady state was usually achieved in less than two hours
following start-up of the system. An inlet temperature near
ambient was maintained by adjusting the flow rate of tap water
entering the heat exchanger.
Data were obtained at the mass flow rates listed in Appendix
C. After reaching steady state, flow rates were measured using the
beam balance. The time for ten pounds of fluid to accumulate was
evaluated at the lowest flow rate. At the higher flow lates, 15 or
20 pounds of fluid were timed. The average time for 3 trials was
recorded.
Ambient temperature as well as those of the coolant inlet,
thermocouple conduit, heater wrap, each entrance section wall, tank
outlet, inlet bulk, and outlet bulk temperature were recorded during
each run along with test-section conditions.
80
Chapter 6. DATA REDUCTION
6.1 Incorporation of temperature variation into the power-law
model
The power law model, = K 1n, has been used in the present
study where values of T r, the shear stress at the rotor, were
obtained using the equation M = 27tRr2H 'Cr as discussed earlier.
Torque values were measured at several different rotor speeds for a
fixed sample temperature. Power law constants, K and n, were
determined by following the single-bob method as described by Van
Wazer [160].
n = d log "[rid log S2 (6.1)
A straight line on a log-log plot of f2 versus T indicates that the
flow curve fits the simple power law. Once the value of n is
determined , a value of K for a given angular speed of the rotor can
be obtained using the relationship
=2/n
2K1 /n 1( Rr (6.2)
Averaging these K values at the angular speeds, one can obtain the
power-law constant, K, at a given temperature. The terms, SI and Rc,
represent angular speed of the rotor and the radius of the cup of the
81
viscometer, respectively.
To include temperature effects it was necessary to use an
interpolation relationship based upon the Arhennius expression
c e-AH/RuT (6.3)
where C is a constant, AH is the activation energy, Ru is the
universal gas constant and T is the absolute temperature. At a
particular shear rate, Y, for the temperature, T, in the range T1 < T _<
T2, the following relationship is obtained:
ti e AH/RuT = ti1 e AH/RuTi = T2 e 6,1-1/Ru T2 (6.4)
Assuming AH is constant over this small temperature range, one
gets the following equations from equation (6.4),
and
6,1-1 Ti T2In(T2T1)
1 1
t=)
CF:
(6.5)
(6.6)
Substituting equation (6.5) into equation (6.6), one obtains a
discrete value of i for particular values of Y and T in the form
82
T2
T-T1
&.
tilT T -T
2 1
(6.7)
Interpolation between adjacent constitutive equations is
illustrated in Figure 6.1. The shear rates and interpolated shear
stresses fit the power-law model quite well. The constants K and n
were determined from a linear regression of the log of these values.
Etchart [38] showed that the axial pressure gradient remains
constant for the entire thermal entry region. The wall shear stress,
must thus remain constant also.
The vertical interpolation was used for a specified shear
stress. The complete form is obtained by replacing i with Y in
equation (6.7).
Y=
T2 ( T-T1
. T T2-TiY2
(6.8)
Figure 6.2 shows in graphical form, how the shear rate at a
particular shear stress and a temperature between T1 and T2 is
obtained.
Considering the wall shear stress to be constant along the test
section, local wall shear rates for this wall shear stress and local
wall temperature were determined using equation (6.8).
83
ifp
'C 'C2
Shear Stress
Figure 6.1. Interpolation between adjacent constitutiveequations at a particular shear rate
Ti T1 < T2
I
TP
Shear Stress
Figure 6.2. Interpolation between adjacent constitutiveequations at a particular shear stress
84
An apparent viscosity representative of the bulk flow
conditions was estimated using "mixing cup" analogy. The apparent
viscosity, r1B, was expressed as:
$ U d A2it
SA u dA Q 0u r dr
(6.9)
where A is the cross-sectional area, and u is the local velocity. For
a cross section
thus
= constantr
(6.10)
(6.11)
Using equation (6.11) and the definition of equation (6.9) becomes
Tr D2 fw 1-2orT1B = t
2 Q 0 1,*
(6.12)
Now one can develope the velocity, u, using the following
relationships, see Skelland [152, p 110],1
n
tR3 3n+1 K (6.13)
and
where p is a function of local bulk temperature.
The wall shear stress can be expressed as
tiW =m 3n+1
pR3 n
85
(6.14)
(6.15)
Also from Skelland [152, p. 110] the pressure gradient may be
expressed as
with
AP2K
( 3n+1 Nin
n Rn+i
V= m
7tpR2
(6.16)
(6.17)
In equation (6.16) AP indicates the pressure drop along the test
section of length, L, and V and R indicate the mean velocity of flow
and inside diameter of the test section, respectively. Combining
these two equations we obtainn
AP m 3n+1 1= 2K
L(
it pp n R3 n +1(6.18)
Skelland [152] also gives the expression for the velocity profile as
i.
(_AP )7 n [R(n +1 Vn r(n +1 )/ni
2KL n+1
86
(6.19)
The final velocity is obtained by substituting equation (6.18) into
Equation (6.19) to yield:3n +1
m (u np n +1
3n+1A R)( 1 n [R(n+1)/n r(n+1)/n
]= --
Using the relationshipdu
7. = dr
The rate of shear can be expressed as
3 n+1
Ili ( 3n+1 11( 1 ) n lin7 = ---1) LT) ricp L n+
Recalling equation (6.11), one can gettwdt =R
dr
2
T2 = ( TviR
) r2
(6.20)
(6.21)
(6.22)
(6.23)
(6.24)
Employing equations (6.20), (6.22), (6.23) and (6.24) and simplifying
equation (6.12), we get
law ( n ) 3
11B 2Q 3n-1 ) (6.25)
87
where tic may be determined at the entrance section. Using equation
(6.15) for tw the final expression for 71B becomes
K Iile=
nTcp n )(3n+1) R3(1n)
3n-1 n
(6.26)
Equation (6.26) is valid for n > 1/3; and K and n were determined at
local bulk conditions.
6.2 Dimensionless parameters
A number of dimensionless parameters were used in analyzing
the data. The local Nusselt and Graetz numbers are defined as
and
q" DNux
k (T, Tb)
Gz =m cp
k x
(6.27)
(6.28)
Values of Nu were computed for the cases with k evaluated at the
local bulk temperature and at the wall temperature. Properties in
Gz were evaluated at the local bulk temperature. The local bulk
temperatures were calculated using an energy balance
Tb T = P x
L m c
88
(6.29)
where P = q "/(7t D L). D indicates the inside diameter of the test
section.
Grashof numbers, defined as
P2 13D3 (rw_Tb)
Grx2
(6.30)
were obtained based on the properties, p and 13, at the local bulk
conditions and at the wall conditions.
Prandtl numbers,
cPPr =
k (6.31)
were evaluated based on 4 separate conditions; the entrance bulk
flow conditions, the entrance wall conditions, the local flow
conditions, and the local wall conditions.
Using the apparent viscosity, one can represent Reynolds
number as
Re4mD (6.32),
which can be transformed to a modified Reynolds number, ReK as
follows
4 ( 3n-1 n )nReK = Dn V2-11 PK 2nl n ) 3n +1)
89
(6.33)
For n=1, ReK is identical to the standard Reynolds number.
For tube flow of power-law fluids, the Reynolds number [152, p.
110] is
Dn V2-13 p 8 ( n )nRe,B K 2n 3n+1 J
Rayleigh numbers given by
(6.34)
Rax = Grx Prx (6.35)
were evaluated at both the local bulk and the local wall
temperature conditions.
Modified Grashof numbers, expressed as
rn2R D4q,,
Gr =1021
and modified Rayleigh numbers
Ra* = Gr* Prx
(6.36)
(6.37)
90
were computed at local bulk flow conditions.
Emperical power-law constant ratios, Kw/Kb, and shear rate
ratios given by
S= Y,,,, 3n+1
( 8V ) 4nD ) (6.38)
were obtained based on 3 separate conditions, the entrance bulk
flow conditions, local bulk flow conditions, and local wall
conditions.
Brinkmann numbers, expressed as
Br =1C2 D4 k Twi
1611 Q2
were determined at the entrance wall temperature
The dimensionless flux, given by
q" D9 k Twi
(6.39)
(6.40)
was computed with k evaluated at the entrance wall temperature.
The following two dimensionless groups, 7t1 and 7c2, were
employed for a final correlation of the present heat transfer
results;
and
0.14 _1/37C1 = Nub (Kw/Kb) Sw
7E2 = Gzb + 0.0083 (Grb Prb)(175
91
The dependent parameter, nl, was well expressed as a function of
7c2 in this study. Radial viscosity variation with temperature was
considered by including the bulk wall empirical power-law constant
ratio, Kw/Kb, in ni in place of the bulk wall viscosity ratio used for
the case of Newtonian fluids. Free convection effects have been
reported as independent of D/L. We noted that the factor, D/L, was
not considered for the free convection effects in n2, which consists
of the forced convection term and a free convection correction
term. Mahalingam et al [90] correlated their 23 data points using
dimensionless groups quite similar to these two; the only
difference is that their values in Gr and Pr in 1t2 were evaluated at
local wall conditions rather than bulk flow conditions. A better
correlation, however, was obtained in the present work using Gr and
Pr evaluated at bulk flow conditions.
The following dimensionless group, a ratio of buoyancy forces
to inertial forces,Gr
Reg
was used to relate the significance of free convection to forced-
convection effects.
92
Chapter 7. RESULTS
7.1 Viscometry
Viscometric data for the fluids used in this work are plotted
in Figures 7.1 through 7.6. The plots of log ti vs log SI are similar to
at - 7 plot, where CI indicates the angular speed of the viscometer
rotor. Each set shows data obtained at a single temperature. For
fluids with low viscosity at higher temperatures and lower rotor
speeds the digital readings from the viscometer were often too low
to be meaningful. For more viscous fluids at lower temperatures
and lower rotor speeds, data were also not always obtainable
because readings were often too high to be measured. Thus, fewer
than 10 data points are shown at every temperature for every test
fluid. All viscometric data fit the power-law model reasonably
well as evidenced by straight lines on log i log S2 plots at each
temperature.
For a given fluid the fluid behavior index, n, tends to increase
with temperature and the empirical constant in the power-law
model, K, is greatly temperature dependent. Figure 7.7 shows that
K decreases with increasing temperature, and increases with
increasing solute concentrations. At constant temperature the fluid
behavior index, n, appears to decrease with concentration and K
increases with concentration. While Bassett [8] observed a
degradation of his test fluids with time of operation, there was no
apparent degradation in this work. Sodium chloride, which is one of
93
Figure 7.1 Viscometric data for 5% CMC solution. From the top,data are for 16.8, 25.3, 35.1, 44.7, 54.3, 64.2, 74.1, and86.4°C
0Log Si
1
Figure 7.2. Viscometric data for 6% CMC solution. From the top,data are for 16.1, 25.5, 35.3, 45.2, 54.8, 64.8, 74.3, and86.6°C
94
Log SI
Figure 7.3. Viscometric data for 7% CMC solution. From the top,data are for 15.8, 25.3, 35.0, 44.9, 54.9, 64.5, 74.5, and86.3°C
Log LI
Figure 7.4. Viscometric data for 7.5% CMC solution. From the top,data are for 16.7, 25.0, 35.2, 45.1, 55.0, 65.1, 74.2, and88.2°C
Cr)0
95
Log L2
Figure 7.5. Viscometric data for 8% CMC solution. From the top,data are for 17.3, 25.4, 35.5, 45.0, 54.9, 64.7, 74.4, and86.0°C
4
3
rn0-J
2-
0
Log CI2
Figure 7.6. Viscometric data for 8.3% CMC solution. From the top,data are for 16.4, 25.6, 35.5, 45.2, 54.7, 64.7, 74.2, and86.0°C
50
96
40
C)30
1:1
20
10
0 20 40
Ternp(°C)
60
.-44.0. IN_ re.
80 100
Figure 7.7. Variation of the power-law constant, K, withtemperature. From the top, data are for 8.3%, 8%,7.5%, 7%, 6%, and 5% CMC solutions.
the components of the current polymer solution, may tend to
stabilize the fluid solutions that were used. Evaporation of the test
fluid did occur during the experiment. About 800 ml of water were
added to the fluid approximately 12 hours after the initial run for
each group. A complete listing of the viscometric data is included
in Appendix B.
97
7.2 Heat Transfer
Obvious effects of buoyancy on the rate of heat transfer were
observed for most runs except for the case of high flow rates in the
smaller test section. Nusselt numbers were determined using K
evaluated both at the local bulk temperature and at the local wall
temperature. Campo and Schuler [16] observed that deviations
between the internal and external temperatures of a tube wall for a
ratio of Do/D = 1.2 and Pe = 500 are unnoticed whenever the ratio of
the thermal conductivity of the wall to that of the fluid is equal to
or greater than 10. They also indicated that this negligible
difference prevails for high values of the ratio and small values of
Do/D . The ratio in the present study was found to be 568 at 20°C
for Do/D = 1.08 and 1.05. Thus, according to their criteria the
conditions used in the present study are such that we may neglect
the temperature difference across the wall.
Buoyancy effects, non-Newtonian behavior, and radial variation
of viscosity were considered in generating a final correlation for
heat transfer behavior. A better fit was obtained using bulk
conditions. The heat transfer results were obtained under the
following range of conditions:
K (dyne secn/cm2) = 0.21 48.5
nb = 0.662 0.838
nw = 0.689 0.959
6o = 1.048 - 1.128
ow = 1.011 1.113
98
= 26.05 42.17
Gz = 127 27474
Prb = 1532 25191
Prw = 132 16642
Re = 0.44 - 29.77
Grb = 0.3 155
Gr = 10 36383w
Rab = 5832 - 238011
Raw = 45638 5268414
Bri = 6.4x10-6 1.7x10-4
where 8 = (3n+1)/4n and the dimensionless heat flux, 4) = P /(lrLkTbj).
The subscripts, b, w, and i, mean bulk, wall, and entrance
conditions, respectively.
Figures 7.8 through 7.13 show results for the present
experimental work. The solid lines below the data represent the
equation,
Nux = 1.412 81/3 Gz x1/3 (7.1)
for a non-Newtonian solution, which was obtained by Bird [12],
where all properties are temperature-independent. Every figure
shows higher heat transfer rates for the current experiments
except at high Graetz numbers where buoyancy effects were
relatively small. The least-viscous solution, 5% CMC, shows values
which are 84.5% above the temperature-independent property
solution, and approximately 87% greater than the temperature-
independent property Newtonian solution (n =1) at Gz = 231. Data
100
z
100 1 000
Temp.-Independent Solution
10000Gz
99
100000
Figure 7.8. Comparison of present data for 5% CMC solution withthe temperature-independent solution
100
2
1
100 1 000
Temp.-Independent Solution
10000Gz
Figure 7.9. Comparison of present data for 6% CMC solution withthe temperature-independent solution
100000
100
Q
- 10z
100 1000
o Run 7
Run 8o Run 9
Run 10
Run 11
o Run 12
Temp.-Independent Solution
10000
100
100000Gz
Figure 7.10. Comparison of present data for 7% CMC solution withthe temperature-independent solution
100
--- 10
z
100 1 000
Run 13Run 14Run 15
Run 16Run 17Temp.-Independent Solution
10000 1 0 0 0 0 0
Gz
Figure 7.11. Comparison of present data for 7.5% CMC solution withthe temperature-independent solution
74°
z
101
a Run 18
Run 19o Run 20o Run 21
Temp.-Independent Solution
Gz
Figure 7.12. Comparison of present data for 8% CMC solution withthe temperature-independent solution
100
1
100 1000Gz
Run 22
Run 23Run 24Run 25Temp.-Independent Solution
10000 1 0 0 0 0 0
Figure 7.13. Comparison of present data for 8.3% CMC solution withthe temperature-independent solution
102
for the most viscous solution, 8.3% CMC, are 35% higher than values
for the temperature-independent property solution and about 51%
higher than the temperature-independent property Newtonian
solution (n = 1) at Gz = 140. For values near Gz = 9000 the
experimental heat transfer values were close to those of the
temperature-independent property solution, and above G = 9000 the
heat transfer values were lower than those of the temperature-
independent property solution. For a 5% CMC solution the measured
heat transfer rate was approximately 7% lower at Gz = 9236. For
the case of 8.3% CMC solution experimental values are roughly 12%
lower at Gz = 12433 in the small test section and about 1% lower at
Gz = 12349 in the large test section than those of the temperature-
independent property solution. The large test section is associated
with larger buoyancy effects than the small test section. All
figures show similar tendencies of Nusselt number variation with
Graetz number. The current results correlate with different slopes
than the temperature-independent property solution. Experimental
results of this study are above the temperature-independent
property solution over wide ranges of Gz.
Comparisons among different fluids with similar flow
conditions and buoyancy effects are observed in Figures 7.14 and
7.15. Values for 5% CMC show higher heat transfer rates, due to
greater buoyancy effects, than 8.3% CMC. Less viscous fluids show
greater buoyancy effects.
A comparison between runs in both test sections with the
same fluid, 7% CMC, are shown in Figure 7.16. It is clear that
z
z
100
10
o rib
- 1171111
Et
d
1111111
CMC 5%
CMC 6%1
100
100
10
1000Gz
10000
103
100000
Figure 7.14. Heat transfer behavior for 5% and 6% CMC
o
1111.1.
1 11.1.1111 II 111,
CI
O CMC 5%
CMC 8.3%
100 1000Gz
10000 100000
Figure 7.15. Heat transfer behavior for 5% and 8.3% CMC
z
100
10
0 m
1;1
ci Large Section
Small Section
1 1 I
100 1000Gz
10000 100000
104
Figure 7.16. Comparison of present data from typical runs with 7%CMC solution in both test sections.
somewhat higher Nusselt numbers were obtained in the large test
section due to larger buoyancy effects. Figure 7.17 shows all data
obtained in both test sections. The solid line represents the
temperature-independent property solution in all cases. Data from
both test sections show similar tendencies for Nu/81/3 vs Gz. The
heat transfer rate was observed to increase regularly with Gz, with
values higher than for the temperature-independent solution up to
Gz = 10000. Buoyancy effects on heat transfer were greater for low
values of Gz, and the deviation of the actual heat transfer rate from
the predicted values was larger for the lower value of Gz.
In Figure 7.18 the current data are also compared with the
100
z 10
1
Temp.-Independent Newtonian Solution
100 1000
Gz
10000
1 06
100000
Figure 7.18. Comparison of present data with the temperature-independent Newtonian solution
107
temperature-independent Newtonian solution (n = 1). A behavior
similar to the temperature-independent case is observed. As seen
in these figures, non-Newtonian effects accounted for in the
parameter, 81/3, had a relatively small effect on heat transfer
compared with buoyancy effects.
No obvious effect of viscous heating on heat transfer rates
was observed in the present study. All Brinkman numbers were
very low, the largest value for Bri being 1.7 x 10-4. Lower heat
transfer rates near the entrance of the heated section compared
with temperature-independent property Newtonian cases are
possibly caused by non-uniform heat flux distributions.
For uniform heating, the wall shear rate should be increased
and become more intense as flow progresses downstream,
presumably until a maximum wall shear rate is reached. The
present data are consistent with this expectation.
A systematic error occurred in readings of the wall
temperature at the bottom and side of the second axial position
upstream (22.86 cm from the entrance) in the small test section
and at the bottom of the fourth axial position upstream (76.20 cm
from the entrance) in the large test section. This was evaluated to
be due to inadequate thermal insulation of the thermocouple
junctions from the heating elements at these locations.
Corrections for wall temperature readings at these locations were
made using an interpolation based on the log-log plot of wall
temperatures measured at the other axial positions.
Evidence of secondary flow was found with both test sections
in the form of oscillations in wall-temperature readings,
108
particularly those at the top; and as differences among the top,
side, and bottom readings of wall temperature at the same axial
locations. The intensity of oscillations and temperature
differences increased at positions further downstream. The
temperature difference between the top and bottom thermocouple
readings near the test section outlet varied from 2.15 °C to 9.41 °C
in the large test section, and from 1.47 °C to 7.66 °C in the small
test section. Figure 7.19 shows a typical wall temperature profile.
A general criterion for the onset of significant secondary flow
due to buoyancy has not yet been established due to inadequate
amounts of data. The criterion cannot be established from the
results of this work, however, the onset of secondary flow appeared
to occur with the 3rd downstream data point for the large test
section and the 4th downstream data point for the small test
section. Using estimated values of Ra* and Gz at transition the
parameter Ra*/Gz2 varied from 0.1, for Ra* = 3.20 x 105 and Gz =
2069, to 1.9 for Ra* = 1.84 x 106 and Gz = 981. A value of the ratio,
Ra*/Gz2, near unity appears to be a reasonable estimate as a
transition criterion. To provide greater precision on transition
more detailed data would be necessary.
Comparisons with the numerical results obtained by Cochrane
[25, p. 136] are shown in Figures 7.20 and 7.21. The Cochrane
results apply for a non-Newtonian fluid at the following conditions:
8i = 1.083, Prwi = 1000, AH/RuT = 5.0, 4 = 1.0, and Br; = 0.0. In these
parameters Ru is the universal gas constant and OH is the activation
energy per mole for flow. A close match of these parameters for
1 00
109
80 -
H
40 -
20
I
V
i
a
V
ci
e0
O Bottom
Side
Top
Bulk Temp.
a
rt
0 100 200
Axial Position, x (cm)
300
Figure 7.19. Wall temperature variations for 7.5% CMC in the smalltest section
110
pseudoplastic behavior was not possible; the present runs showed
higher heat transfer rates due to obvious buoyancy effects. It is
thus clear that numerical solutions should include buoyancy effects.
Table 7.1 shows the comparison of 2 typical runs with Cochrane's
predicted values. The conditions for Run 7 and Run 10 are Si = 1.081,
Prwi = 4932, AH/RuT = 5.6, 0= 38.6, and Br; = 0.0, and 8; = 1.080, Pry,,;
= 3968, OH /RAT = 5.6, (I) = 35.2, and Br; = 0.0, respectively.
Table 7.1. Comparison with Cochrane's solution
Run 7 Run10 Cochrane
Gz Nu G7 Nu G7 Nu
325 14.98 327 13.69 275 9.77420 15.54 422 13.90 396 10.97564 16.51 567 15.20 575 12.33801 17.50 805 16.67 841 13.89
1285 19.59 1291 19.24 1234 15.782145 22.64 2154 21.39 1817 18.01
4295 26.60 4312 25.43 3961 23.0812892 32.34 12942 31.62 12854 34.25
Figure 7.22 shows a comparison of the present experimental
results with those of the Bassett and Welty [9] correlation which
does not account for free convection effects. The present data
follow a different slope from their correlation. At the highest
value of Gz (Gz = 27,000) the present results are 9% lower than the
Bassett and Welty prediction, however the present data generally
show higher heat transfer rates up to 42% higher at Gz = 127.
In Figure 7.23 the present heat transfer results are compared
1 1 1
0 Conditions:8j=1 .081
Pr Wi=4932A H/RuT = 5.6X38.6Br-0 0
i
Gz
Figure 7.20. Comparison of experimental results with Cochrane'snumerical solution
Gz
Figure 7.21. Comparison of experimental results with Cochrane'snumerical solution
1 1 3
with a correlation of Mahalingam et al [90] which was obtained
from 23 data points displaying buoyancy effects. The present data
display considerable scatter for Ra evaluated at the wall
temperature. The difference between the present data and their
equation ranged from 3.3% to 35.9%, with the mean deviation being
15.1% below their predicted values.
Figure 7.24 shows a comparison between the present heat
transfer results and those of Joshi [6l X+ < 0.013; our results
show higher heat transfer rates than his. The present heat transfer
rates are 22.1%, 16.3%, 12.3%, 38.7%, and 27.9% higher at X+ =
0.00078, 0.00096, 0.00179, 0.00320, and 0.0109, respectively. X+
indicates a dimensionless distance, 2 (x/D)/(Re Pr) or 7t/(2 Gz).
Excluding 3 runs, which showed noticeable buoyancy effects,
the best correlation for this work was obtained using the
dimensionless parameters,
=Gzb + 0.0083 (Rab)°.75
and
7c2 = Nub (KW /Kb )0.14 . 1 / 3 3
These parameters have been employed to correlate tube flow
data for a Newtonian fluid with free convection effects using the
viscosity ratio, µw /µb, instead of KW /Kb [115]. With this change
they were found to be valid for a non-Newtonian fluid also. A
better correlation was obtained in the present work using Ra
evaluated at bulk flow conditions while Mahalingam et al correlated
116
their 23 data points with Ra evaluated at wall conditions.
Considering radial viscosity variation with temperature, non-
Newtonian effects, both forced and free convection effects, the
correlation obtained in the present study is
, ,..1/3Nub (Kw/Kb)
o .14li/Ow = 2.116 [GZb ± 0.0083 (Rab)0.7510.27 (7.2)
which gave the value of the coefficient of determination, R2, of
0.973 and detailed information on statistical analysis of this
regression fit is included in Appendix I. The present data and their
correlation equation are shown in Figure 7.25.
1 00
loo 1000
Gz + 0.0083 R'75
1 0 0 00
Figure 7.25. Correlation of data from present runs
1 0 0 00 0
1 1 8
Chapter 8. CONCLUSIONS AND RECOMMENDATIONS
This experimental study has provided data for the combined
free and forced convective heat transfer to pseudoplastic fluids in
the thermal entry region of uniformly heated, horizontal pipes.
Viscometric data for the test fluids used were seen to fit the
power-law model. The following conclusions can be drawn:
1. The local rate of combined convective heat transfer for a
fluid with temperature-dependent properties, especially viscosity,
except at high Graetz numbers where buoyancy effects were
relatively small, is greater than equation (7.1) which is a
temperature-independent property solution.
2. The local rate of combined convective heat transfer is
84.5% higher than predicted by the temperature-independent
solution at Gz = 231 for a solution with 5% CMC, and 35% higher
than predicted by the temperature-independent solution at Gz = 140
for a CMC concentration of 8.3 percent.
3. The local rate of combined convective heat transfer of non-
Newtonian fluids where significant free convection effects occur is
greater than for purely forced convection due to their additional
free convection effects caused by secondary flows.
4. Local heat transfer rates are higher than those expressed by
the correlation reported by Bassett and Welty [9], which does not
account for free convection effects. For example results of this
study are 42% higher at a value of Gz = 127.
5. Both temperature-dependent fluid properties and free
1 1 9
convection effects are significant factors on the total heat transfer
rate. Heat transfer rates are increased by as much as 384% for Gzx
= 681 and Rax = 154100 over the classic constant value of 4.364 for
full thermal development. Non-Newtonian effects accounted for by
the term, 81/3,are relatively small.
6. Free convection effects become greater for less viscous
fluids due to their greater buoyancy effects; and this increase
becomes greater as the flow progress downstream.
7. Using estimated values of Ra* and Gz at transition the
parameter Ra*/Gz2 varied from 0.1, for Ra* = 3.20 x 105 and Gz =
2069, to 1.9 for Ra* = 1.84 x 106 and Gz = 981. A value of the ratio,
Ra*/Gz2, near unity appears to be a reasonable estimate as a
transition criterion.
8. The dimensionless parameters, ni =Gzb + 0.0083 (Rab)°75
and 7t2 = Nub (Kw/Kb)014 (1/8w 1 /3) represent the data of this work
well, including combined free- and forced-convection heat transfer
behavior. Equation (7.2) expresses the rate of heat transfer for
flows with temperature-dependent properties, free convection
effects, and non-Newtonian effects .
9. Viscous heating for fluids with Brinkman numbers as high
as 1.7 x 10-4 did not influence heat transfer rates.
It is recommended that future work address:
1. Experimental investigations with highly-viscous pseudo-
plastic fluids to determine viscous dissipation effects
2. Flow patterns and associated heat transfer behavior for
turbulent flows of pseudoplastic fluids in uniformly-heated
120
horizontal pipes
3. The heat transfer behavior of dilatant fluids and Bingham
plastic fluids in uniformly-heated horizontal pipes
4. The development of alternative techniques to determine the
flow patterns characteristic of secondary flows in pseudoplastic
fluids
121
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140
ANALYSIS OF ERROR
This section includes the estimates for errors in measured
data and dimensionless parameters used in the present work.
For the case when Y is a given function of independent
variables X1, X2, X3,...,Xn; Wy is the uncertainty interval for the
quantity, Y; and W1, W2, W3,...,Wn are the uncertainty intervals for
the independent variables; if the uncertainties in the independent
variables are all given with the same odds, then WY is given as
2 2 -Ny 2 (W7 = ( W (kW3)ax, 1 ax2 2 3 wn) (A.1)
where Y = f(X1 ,X2,X3,...,Xn) and the independent variables X1, X2,
X3,..Xn are measured quantities. If Y is comprised of products and
quotients of the measured variables, then the following equation is
obtained from equation (A.1):
2 2WY N1S/3( Wy2 Wv'3 WXn
Y = 1 +
X2x2 2
2 A .n.2 31 X, (A.2)
The uncertainties from various sources for the basic
quantities used in generating the results in this thesis are shown in
Table A.1. The smallest value measured for the quantity is shown in
the last column. Entries under "Uncertainties other sources" are
as expressed by Bassett [8], and were obtained under the assumption
that fluid properties other than viscosity were equal to those of
141
Table A.1. Uncertainties in basic quantities
*Quantity
Vs
Measurement Correlation Uncertainty Smallest
Uncertainty Uncertainty Other Value
±0.1 my 34.86 myRh 0.1 %
Ei ±1.95 1.tv
Etc ±13.5
Mf ±0.1 btf ±0.5 sec
Tq 4.3%
Tv ±0.05°C
D ±0.005 in
L ±1/16 inx ±1/32 in
pk
cP
* Note: V
±0.0001 %±0.1 %±0.0004 ±0.5
800.5 gy
1989.5
10 lb16 sec
16.8 °C
1.505 in118.91 in
3.0 in±1 %±2%
±0.02 % ±2%heater shunt voltage, Rh - heater resistance, Ei inlet
thermocouple emf, Etc thermocouple emf, Mf - mass offluid weighed, tf - time for Mf to flow, Tv -viscometertemperature, Tq torque value of the viscometer.
pure water.
Power input to the heaters was directly related to shunt
voltage, shunt resistance, and heater resistance. It was determined
by2
VsP = () Rh
Rs (A.3)
142
An additional uncertainty (4.72%) in the power supplied to the fluid
resulted from heat losses to the enclosure. Thus, the square of the
relative error for the power, Wp, is
C
2wp )2
Nsiv )2 2WR )2 WR
P ) br:s /1h + (0.04720)2
= (0.00574)2 + (0.00000)2 + (0.00100)2 + (0.04720)2
= (0.04756)2
Calibration for the thermocouples also introduced uncertain-
ties due to the variation of the bath temperature, the accuracy of
the potentiometer, and measurement error. They were ±0.1 °C for
the smallest temperature (15.51°C), ±0.05 1.1.v for the smallest
voltage reading (640.4}.1v), and ±0.21.tv for this smallest voltage
reading respectively. Thus, the square of the relative error in the
calibration is
(0.00645)2 + (0.00008)2 + (0.00031)2 = (0.00646)2
Mfm=tf (A.4)
Using equation (A.4), the square of the relative error in the mass
rate is given by
m I
2
( WMfl2
Wtf)2
Mf ) tf )
= (0.01)2 + (0.03125)2
= (0.03281)2
The inlet thermocouple reading had an additional uncertainty
of 0.646% from calibration and 0.401% from conversion of emf to
143
temperature. Thus, the square of the relative error is
Wri ( w2
E, )+ (0.00646)2 + (0.00401)2
= (0.00244)2 + (0.00646)2 + (0.00401)2
= (0.00799)2
The wall temperature reading includes the same additional
uncertainties from the calibration and 0.3% from conversion of emf
to temperature. Thus,2
( WT )2Wpb (0.00646)2 + (0.003)2
Tw "Cc
( 0.00679)2 + (0.00646)2 + (0.003)2
= (0.00984)2
The uncertainties in the fluid properties k, p, cp, and 13 were
estimated in Bassett [8, p.124]. They are functions of temperature,
the variation being expressed in terms of polynomials. Since the
zeroth order term dominates the expression, the contribution to the
error in
significantly
from temperature,2
Wn
the properties from the error in the temperature is
reduced. Thus, assuming a negligible contribution
the result is:
= (0.0001)2 + (0.010)2 = (0.010)2
= (0.001)2 + (0.020)2 = (0.020)2
= (0.0004)2 + (0.005)2 = (0.0054)2
P2
2Wk
k2
2Wcp
C2
144
2
= (0.0002)2 + (0.020)2 = (0.020)2
The value of the power-law constant, n, has uncertainties
caused by measurement errors for torque and temperature of the
sample fluid of the viscometer. In the case of 5% CMC, which was
the least-viscous fluid, the smallest scale obtained for torque
values was 0.115 and its variation was ±0.005. The constant, n,
also had additional uncertainties from the errors produced in the
plot of log vs log T. Interpolation was made between 2 different
flow curves, and the errors in the plot at the temperatures for the
worst case were 1.32% and 1.19%, respectively. The error for a
flow curve obtained using interpolated shear stresses and shear
rates appeared to be quite small. Thus,2
(w ) = (0.04348)2 + (0.00298)2 + (0.0132)2 + (0.0119)2
= (0.04706)2
The power-law parameter, K, had additional uncertainties
brought by the use of equation (6.2) at two different temperatures
in the interpolation. The largest deviations of K from its mean
values for each temperature were 6.15% and 5.07%, respectively.
Thus,2
= (0.04706)2 + (0.0615)2 + (0.0507)2
= (0.09256)2
The wall shear stress, Tw, was evaluated using equation (6.15)
by replacing R with D/2. The square of the relative error of Tw was
1 45
estimated for the flow of the 5% CMC test fluid, with n = 0.774, at
its inlet temperature as follows:,2
( WK 112 ( 3nWD 12(
nWni n Wp
T =K)+L D )+ )m2
W{ In8m(3n+1) 1
ti
itpD3n , 3n+1
= (0.09256)2 + (0.00771)2 + (0.00774)2 + (0.00774)2 + (0.04698)2
= (0.10466)2
Using the power-law equation for the shear rate at the inlet
(tw)n=
The square of the relative error for the inlet shear rate is
2W. 2
(WKT1Z /I
+
\2TwWn
n2
(A.5)
= (0.13522)2 + (0.11959)2+ (0.06108)2
= (0.19057)2
According to Bassett [8, p.125], an uncertainty of ±5% is
introduced with the assumption of constant pressure gradient when
the wall shear rate in the test section is specified. Thus,
Yw
= (0.19507)2 + (0.05)2
= (0.19702)2
Recalling equation (6.29) for the calculation of the local bulk
temperature, the square of the relative error of Tb-Ti is
146
wTb_Ti )2 Wp )2 ( w, y ovo2Tb P ) x) L )
2 11,c12win ",[H 71
= (0.04756)2 + (0.01042)2 + (0.00053)2 + (0.01)2 + (0.0054)2
= (0.0500)2
Then the square of the relative error for the local bulk temperature
is
"Tb2 2
WTbTi ) (Tb -T;) "Ti2 2
Tb )Tb TbTi
= (0.01653)2 + (0.00535)2
= (0.01737)2
for a point near the exit. Near the entrance, Tb = Ti, so that2
Tb = (0.00799)2T b
and the square of the relative error of the smallest AT (Tb is
CWAT2 wr 2
WTb
OT) AT) ( OT)2 2 )2( 2
T
( WT,,, )
COT))Tv, ) "I*
Tb ) AT )=
41.712
2 21.3412= (0.00984)2(757)
+(0.00799) 2T.g.7 )
= (0.02015)2 + (0.00837)2
= (0.02182)2
Using equation (6.25) by replacing R and Q with D/2 and m/p,
respectively, the square of the relative error of an apparentviscosity, is evaluated as follows:
147
2 2 2 \2 )2(W=
2
(WI) (WP) 4- (3wrlWm
+(nOWnn-1)tww)
M
= (0.01)2 + (0.00997)2 + (0.01)2 + (0.03560)2 + (0.10466)2
= (0.11189)2
The uncertainties for the pertinent dimensionless parameters
can now be estimated:
Graetz number22 ( 2 2 2
miwcyz) W.
+(\11 TC1;) +(\14)+((W )Gz )m )
= (0.01)2 + (0.0054)2 + (0.02)2 + (0.01042)2
= (0.02525)2
Shear rate ratio is
8= (3n+1)/4n. (A.6)
Entrance shear rate ratio2
wnUt(3n+1)
( 0.04706 123x0.774 +1)
= (0.01417)2
Local shear rate ratio
( 2
8
W8= (0.01533)2
Nusselt number -
1 48
(WNu)2(WO
2(W112 (WO
2rWAT)2
Nu°) P ) L ) k ) )
= (0.04756)2 + (0.00053)2 + (0.02)2 + (0.02182)2
= (0.0560)2
Rayleigh number -
2 2 2 2 2 2 2
(?) (1pW) (14) ± C74) 443 ) ±(.1747.) ±(W1171)= (0.02)2 + (0.02)2 + (0.02)2 + ( 0.00997)2 + (0.02182)2 + (0.11189)2
= (0.11956)2
Modified Rayleigh number -
2 2 2 2( WRa* \
22Wp ) ( W13 ) ( 3WD ) ( WP ) (WO
2( WCP )2
p ) 13 ) rr) 7) +-"-e7))\ Ra*
±(2Wkk)2 +(Wn)2
= (0.020)2 + (0.020)2 + (0.00997)2 + (0.04756)2 + (0.00053)2
+ (0.0054)2 + (0.040)2 + (0.11189)2
= (0.13157)2
The relative errors in the Graetz and Nusselt numbers have far
less uncertainty downstream of the first local position, 3 inches
from entrance.
A statistical analysis of the experimental results is shown in
Table A.2. The coefficient of determination is defined as the ratio
of the sum of squares due to regression (E (YRi-Yrn)2) to the sum of
squres about Ym (Yj- Ym)2). YRi means Y value obtained by the
regression fit and Ym indicates the mean value of the sum of Yj
which are the data values from the measurements. If the fit of the
149
linear relationship YRi is perfect, then Y1 is equal to YRj and R2 is
equal to 1.0. If, on the other hand, the variation from the linear
relationship is nearly as large as the variation about the mean of Y,
R2 approaches zero. The statistic R2 is closely related to the F
value of the F test. When R2 is zero, F is also zero. As R2
approaches one, F approaches infinity. The table A.2 shows that the
coefficient of determination is nearly close to one and the F value
is considerably large. In this sense the equation (7.2) produced a
good fit.
Table A.2. Statistical analysis of experimental results
Regression Analysis Multiplicative model: Y = a Xb
Parameter Estimate Standard Error T Value Prob. Level
.Intercept 0.749469 0.0247576 30.2723 0.00000
Slope 0.272308 0.0034183 79.6618 0.00000
* Note The Intercept is equal to Ln a.
Analysis of Variance
Source Sum of Squares Df Mean square F-Ratio Prob. LevelModel 20.4057 1 20.4057 6346.01 0.00000Error 0.55950 174 0.00322
Total 20.96519 175
Correlation Coefficient = 0.986566 R2 = 97.33 percentStandard Error of Est. = 0.0567055
Note: the independent variable, X, and the dependent variable, Y, correspond to
Gzb+0.0083(Rab)°15 and Nu b(Kw/Kb)014(1 /8wi /3), respectively. R2 indicates
the coefficient of determination.
151
Unit : Sh. St. (Temp.) ; dyne/cm2 (C)
5% CMC Solution
s.2 (11$)
1.46
Sh. St. (16.8) Sh. St. (25.3) Sh. St. (35.1)
33.03
Sh. St. (44.7)
2.43 115.59 75.96 52.84 39.634.74 204.76 138.71 99.08 66.056.75 270.81 184.94 132.10 92.4711.29 396.30 284.02 204.76 151.9218.81 561.43 422.72 317.04 237.7831.38 799.21 614.27 468.96 356.6752.31 1129.46 898.28 686.92 541.61
(1/s) Sh. St. (54.3) Sh. St. (64.2) Sh. St. (74.1) Sh. St. (86.4)
2.43 26.42 19.824.74 46.23 39.63 29.72 19.81
6.75 66.05 52.84 39.63 26.42112.29 82.56 66.05 46.24
18.81 175.03 122.10 105.68 79.2631.38 267.50 204.76 171.73 125.5052.31 412.81 317.04 270.80 198.15
6% CMC Solution
a (1/s) Sh. St. (16.1) Sh. St. (25.5) Sh. St. (35.3) Sh. St. (45.2)
0.52. 52.84 39.63 26.42 19.810.87 79.26 56.14 39.531.46 118.89 85.87 59.45 46.24
2.43 184.94 128.80 92.47 66.054.74 323.55 227.87 165.13 118.896.75 429.33 303.83 217.97 158.5211.29 634.08 468.96 343.46 244.3918.81 885.07 686.92 508.59 376.4931.38 1221.93 964.33 726.55 554.8252.31 1334.21 1050.20 812.42
a (t rs) Sh. St. (54.8) Sh. St. (64.8) Sh. St. (74.3) Sh. 51(86.6)
0.52 13.21 9.910.87 19.82 13.211.46 33.03 26.42 19.822.43 46.24 39.63 33.034.74 85.87 72.66 52.84 33.036.75 115.59 99.08 72.66 46.2411.29 178.34 151.92 118.89 72.6618.81 284.02 237.78 178.34 138.7131.38 429.33 369.88 284.02 217.9752.31 640.69 554.82 435.93 336.86
152
7% CMC Solution
(1 (1/s) Sh. St. (15.8) Sh. St. (25.3) Sh. St. (35.0) Sh. St. (44.9)
0.52 105.68 72.66 52.84 39.630.87 151.92 105.68 79.26 52.841.46 224.57 158.52 112.29 82.562.43 336.86 234.48 171.73 122.194.74 551.52 402.91 290.62 204.766.75 713.34 521.80 376.49 277.4111.29 1017.17 766.18 591.24 426.0218.81 1089.83 832.23 627.4831.38 1169.09 911.4952.31 1268.16
Q (1/s) Sh. St. (54.9) Sh. St. (64.5) Sh. St. (74.5) Sh. St. (86.3)
0.52 26.420.87 42.93 33.03 23.12 19.821.46 59.45 46.24 36.33 29.722.43 92.47 69.35 52.84 42.934.74 151.92 118.89 85.87 72.666.75 204.76 158.52 118.89 92.4711.29 310.44 237.78 184.94 145.3118.81 475.56 376.49 284.02 224.5731.38 706.74 561.43 435.93 343.4652.31 1003.96 825.63 660.50 535.01
7.5% CMC Solution
f2 (1/s) Sh. St. (16.7) Sh. St. (25.0) Sh. St. (35.2) Sh. St. (45.1)
0.52 145.31 108.98 79.26 59.450.87 204.76 151.92 112.29 85.871.46 300.53 224.57 165.13 125.502.43 435.93 323.65 237.78 178.344.74 700.13 528.40 396.30 297.236.75 898.28 686.92 518.49 389.7011.29 1268.16 990.75 759.58 587.8518.81 1393.66 1089.83 852.0531.38 1202.11
a (1/s) Sh. SL (55.0) Sh. St. (65.1) Sh. St. (74.2) Sh. St. (88.2)
0.52 66.050.87 92.47 46.24 36.33
.46 138.71 69.35 52.84 39.632.43 224.57 99.08 79.26 59.454.74 297.23 171.73 132.10 92.476.75 449.14 217.97 171.73 125.5011.29 667.11 336.86 264.20 188.2418.81 964.33 501.98 396.30 284.0231.38 1354.03 739.76 634.08 435.9352.31 1070.01 878.47 667.11
153
8% CMC Solution
(1/s) Sh. St. (17.3) Sh. St. (25.4) Sh. St. (35.5) Sh. St. (45.0)
0.52 204.76 155.22 112.29 82.560.87 284.02 217.97 158.52 118.891.46 409.51 310.44 227.87 171.732.43 584.54 445.84 330.25 247.694.74 911.49 713.34 528.40 402.916.75 1142.67 904.89 683.32 515.1911.29 1268.16 977.54 746.3718.81 1367.24 1063.41
(i/S) Sh. St. (54.9) Sh. St. (64.7) Sh. St. (74.4) Sh. St. (86.0)
0.87 89.171.46 132.10 99.08 82.56 66.052.43 184.94 145.31 118.89 92.474.74 310.44 237.78 191.55 145.316.75 396.30 310.44 244.39 191.5511.29 581.24 462.35 363.28 284.0218.81 845.44 673.71 528.40 416.1231.38 1195.51 970.94 772.79 614.2752.31 1360.63 1122.85 911.49
CMC Solution
K2 (1/s) Sh. St. (16.4) Sh. St. (25.6) Sh. St. (35.5) Sh. St. (45.2)
0.52 250.99 191.55 138.71 105.680.87 343.46 260.90 191.55 145.311.46 498.68 373.18 274.11 211.362.43 696.83 525.10 389.70 300.534.74 1076.62 825.63 620.87 482.176.75 1347.42 1043.59 792.60 614.2711.29- 1443.19 1122.85 878.4718.81 1241.74
(1/s) Sh. St. (54.7) Sh. St. (64.7) Sh. St. (74.2) Sh. St. (86.0)
0.87 112.29 92.47 72.77 52.841.46 165.13 132.10 105.68 75.962.43 237.78 184.94 148.61 112.294.74 376.49 297.23 237.78 178.346.75 482.17 383.09 303.83 231.18
11.29 693.53 554.82 442.54 336.8618.81 990.75 799.21 647.29 495.3831.38 1400.26 1142.67 931.31 726.5552.31 1334.21 1056.80
154
APPENDIX C. Reduced test data and parameters
Computer programs for the data reduction includes the
evaluation of apparent viscosities of test fluids, their properties,
and related heat transfer parameters using the interpolation
scheme described in Chapter VI. An overall flow chart of the
computer programs is shown in Figure A.1.
/Input data includingviscometric data
[I
determines the power-law constants at 1
Regress.pas ;determineseach test temperature of the viscometer'L .
( Interpon.pas
V
[Regress.pas
----..ri ' d e t e rm i n e the power-law constants at1 i
1 each bulk temperature of the working 1
1
in1
1 fluid n the test section. J./-
V
/WallInputTemperatures
V
IlnterTw.pas\I-I determine the power-law constants at
each wall temperature of the test section
[ Regress.pas
V
155
Apparent Viscosity
Heat Transfer Results
rfththitd
1
1
1 determines the heat transfer 1
1
parameters including wall 1
1
1 shear stress and shear rates 1
Figure A.1. Overall flow chart of Computer Programs
156
CMC(%) =Dia.(cm)
X(cm)
5 Run No. = 1= 3.823 Mass Rate(gr/sec)
Tb(C) V(poise) Kpb npb ATw(C)
= 101.15 Input Power(W)
Kpw npw Prb Prw
= 4645.4
Re0.00 21.84 2.729 4.264 0.774 21.84 4.264 0.774 1900.1 1533.0 12.357.62 22.12 2.695 4.170 0.776 50.95 0.842 0.894 1875.4 345.1 12.50
22.86 22.67 2.673 3.989 0.781 59.51 0.634 0.900 1858.7 255.1 12.6045.72 23.50 2.645 3.732 0.787 67.79 0.498 0.907 1836.9 201.0 12.7476.20 24.60 2.620 3.417 0.795 70.99 0.435 0.919 1814.8 183.4 12.86121.92 26.26 2.588 3.046 0.806 75.91 0.347 0.937 1786.4 158.2 13.02172.72 28.11 2.563 2.719 0.815 78.64 0.302 0.948 1761.2 144.5 13.15231.14 30.23 2.535 2.390 0.826 82.09 0.254 0.961 1733.3 129.2 13.29297.18 32.63 2.511 2.070 0.838 81.35 0.263 0.958 1707.3 132.3 13.42
X(cm) Gr Grw0.00 0.0 0.07.62 48.5 2549.2
22.86 64.0 6437.345.72 80.6 13233.976.20 89.6 16987.0121.92 103.7 25167.5172.72 114.1 31168.6231.14 127.2 40777-2297.18 130.1 36382.9Br. No. = 0.0000097
WSR WSS BuoyF Gz Nub Nuw19.6 43.181.3 43.1 0.31 9236.4 28.18 26.38
108.5 43.1 0.40 3076.8 22.03 20.34136.0 43.1 0.50 1536.4 18.28 16.70148.4 43.1 0.54 919.5 17.40 15.87171.1 43.1 0.61 572.6 16.19 14.73186.7 43.1 0.66 402.4 15.83 14.43208.1 43.1 0.72 299.3 15.34 14.00203.4 43.1 0.72 231.4 16.23 14.92unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC(%) = 5 Run No. = 2Dia.(cm) = 3.823 Mass Rate(gr/sec) = 164.66 Input Power(W) = 4931.9
X(cm) Tb(C) V(poise) Kpb npb ATW(C) Kpw npw Prb Prw Re0.00 22.22 2.386 4.136 0.777 22.22 4.136 0.777 1659.5 1342.4 22.997.62 22.40 2.359 4.076 0.779 50.93 0.843 0.894 1639.9 331.0 23.2522.86 22.76 2.347 3.959 0.781 55.77 0.679 0.906 1630.9 274.2 23.3745.72 23.30 2.330 3.791 0.785 62.32 0.602 0.897 1617.8 226.0 23.5476.20 24.02 2.312 3.579 0.791 67.31 0.509 0.906 1603.0 196.4 23.72121.92 25.11 2.287 3.284 0.799 73.06 0.398 0.926 1582.4 168.2 23.98172.72 26.31 2.268 3.038 0.806 76.68 0.334 0.940 1565.5 150.7 24.18231.14 27.69 2.249 2.789 0.813 80.60 0.274 0.955 1547.4 133.3 24.39297.18 29.26 2.235 2.535 0.821 79.83 0.284 0.952 1531.6 136.5 24.54
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 31.9 61.57.62 63.4 2741.0 121.0 61.5 0.12 15023.4 30.22 28.31
22.86 74.9 4838.2 144.9 61.5 0.14 5005.2 26.09 24.2645.72 91.3 8880.9 174.0 61.5 0.16 2500.7 22.04 20.3076.20 105.4 13507.6 198.8 61.5 0.19 1498.4 19.83 18.15121.92 123.5 21146.2 230.4 61.5 0.21 934.5 17.85 16.26172.72 137.1 28259.9 256.1 61.5 0.23 657.9 16.94 15.40231.14 152.8 38774.4 288.4 61.5 0.26 490.2 16.07 14.60297.18 155.4 35188.7 281.8 61.5 0.26 379.7 16.74 15.28Br. No. = 0.000023 unit ; WSR(1/sec), WSS(dyne/sq cm)
157
CMC(%) =Dia.(cm)
X(cm)
6 Run No. = 3= 3.823 Mass Rate(gr/sec) = 107.96 Input Power(W) =
Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw
4435.8
Re0.00 22.12 4.088 6.380 0.783 22.12 6.380 0.783 2843.8 2292.4 8.807.62 22.37 4.050 6.324 0.783 48.80 2.200 0.827 2815.3 700.3 8.88
22.86 22.86 4.036 6.215 0.783 55.84 1.572 0.851 2803.5 523.8 8.9145.72 23.60 4.019 6.055 0.784 63.51 1.262 0.868 2788.5 438.5 8.9576.20 24.59 4.009 5.850 0.784 67.50 1.155 0.867 2775.4 389.9 8.97121.92 26.08 3.985 5.514 0.786 74.27 1.017 0.856 2751.0 314.5 9.02172.72 27.73 3.955 5.097 0.791 79.48 0.625 0.913 2720.5 253.4 9.09231.14 29.63 3.924 4.662 0.796 83.75 0.423 0.958 2688.6 213.3 9.16297.18 31.77 3.903 4.221 0.802 83.87 0.418 0.959 2659.8 212.3 9.21
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 21.1 69.57.62 19.9 554.9 64.9 69.5 0.25 9849.1 29.34 27.59
22.86 25.4 1325.4 85.7 69.5 0.32 3280.7 23.48 21.8345.72 31.7 2432.6 101.1 69.5 0.40 1638.5 19.37 17.8176.20 35.5 3400.5 113.1 69.5 0.44 980.9 17.96 16.47121.92 42.3 6125.2 139.0 69.5 0.52 611.2 15.93 14.52172.72 48.4 10437.6 171.6 69.5 0.59 430.0 14.77 13.44231.14 54.3 15761.5 203.0 69.5 0.65 320.0 14.06 12.79297.18 56.1 15326.8 204.0 69.5 0.66 247.6 14.53 13.29Br. No. = 0.000017 unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC(%) = 6 Run No. = 4Dia.(cm) = 3.823 Mass Rate(gr/sec) = 172.82 Input Power(W) = 4575.1
X(cm) Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw Re0.00 22.52 3.640 6.290 0.783 22.52 6.290 0.783 2529.7 2039.5 15.817.62 22.68 3.608 6.255 0.783 48.30 2.257 0.825 2505.8 664.1 15.9622.86 23.00 3.601 6.185 0.783 52.27 1.841 0.840 2499.7 560.4 15.9945.72 23.48 3.589 6.082 0.784 58.53 1.454 0.857 2490.0 463.4 16.0476.20 24.11 3.579 5.948 0.784 63.70 1.255 0.869 2479.9 413.8 16.08121.92 25.07 3.565 5.754 0.784 70.18 1.098 0.862 2465.8 338.0 16.15172.72 26.13 3.547 5.499 0.786 75.42 0.914 0.868 2448.5 283.8 16.23231.14 27.36 3.524 5.188 0.790 80.66 0.560 0.925 2426.7 234.2 16.34297.18 28.74 3.509 4.860 0.794 81.55 0.516 0.935 2408.3 226.8 16.41
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 33.8 99.17.62 24.7 594.7 97.6 99.1 0.10 15752.8 31.19 29.3922.86 28.6 994.3 114.9 99.1 0.11 5248.2 27.28 25.5345.72 35.0 1841.6 137.5 99.1 0.14 2622.3 22.75 21.0976.20 40.5 2713.6 152.8 99.1 0.16 1571.6 20.11 18.52121.92 47.9 4841.4 185.4 99.1 0.18 980.6 17.60 16.09172.72 54.6 7742.3 219.4 99.1 0.21 690.7 16.06 14.63231.14 62.0 12663.0 264.4 99.1 0.23 514.9 14.81 13.44297.18 64.7 13439.9 272.8 99.1 0.24 399.2 14.89 13.55Br. No. = 0.000038 unit ; WSR(1/sec), WSS(dyne/sq am)
158
CMC(%) =Dia.(cm)
X(cm)
6 Run No. = 5= 3.823 Mass Rate(gr/sec)
Tb(C) V(poise) Kpb npb ATw(C)
= 224.99 Input Power(W) =
Kpw npw Prb Prw
4978.3
Re0.00 21.86 3.517 6.439 0.783 21.86 6.439 0.783 2449.0 1974.0 21.307.62 21.99 3.485 6.408 0.783 48.61 2.221 0.826 2425.2 624.3 21.5022.86 22.26 3.479 6.348 0.783 52.07 1.860 0.839 2420.1 540.8 21.5445.72 22.66 3.469 6.259 0.783 57.83 1.483 0.856 2411.8 452.8 21.6076.20 23.19 3.460 6.143 0.784 62.55 1.296 0.866 2403.2 409.5 21.66121.92 23.99 3.446 5.974 0.784 69.21 1.118 0.864 2390.5 336.5 21.74172.72 24.88 3.435 5.792 0.784 74.14 1.019 0.856 2378.8 286.3 21.81231.14 25.90 3.420 5.561 0.785 79.27 0.637 0.910 2363.7 240.5 21.91297.18 27.06 3.405 5.262 0.789 80.98 0.544 0.929 2347.3 227.2 22.00
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 43.9 124.67.62 26.7 701.5 130.5 124.6 0.06 20546.7 32.73 30.7722.86 30.2 1085.4 149.8 124.6 0.07 6845.9 29.20 27.2945.72 36.3 1923.8 177.1 124.6 0.08 3421.2 24.73 22.9076.20 41.6 2732.8 194.5 124.6 0.09 2050.8 22.06 20.30121.92 49.4 4865.2 234.4 124.6 0.10 1280.0 19.16 17.49172.72 55.7 7550.8 274.0 124.6 0.12 902.0 17.55 15.95231.14 62.8 11931.6 324.2 124.6 0.13 672.8 16.15 14.63297.18 66.5 13635.3 342.7 124.6 0.14 521.8 15.94 14.45Br. No. = 0.000062 unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC(%) = 6 Run No. = 6Dia.(cm) = 3.823 Mass Rate(gr/sec) = 292.57 Input Power(W) = 5174.3
X(cm) Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw Re0.00 21.50 3.364 6.521 0.783 21.50 6.521 0.783 2344.9 1889.8 28.977.62 21.61 3.333 6.497 0.783 48.91 2.187 0.827 2321.7 589.0 29.2422.86 21.82 3.329 6.448 0.783 51.47 1.917 0.837 2318.0 531.1 29.2845.72 22.14 3.319 6.375 0.783 57.08 1.516 0.854 2310.7 443.4 29.3676.20 22.56 3.312 6.280 0.783 61.03 1.353 0.863 2304.0 408.4 29.42121.92 23.20 3.300 6.141 0.784 67.27 1.160 0.867 2293.4 347.4 29.53172.72 23.91 3.290 5.990 0.784 71.92 1.062 0.860 2283.7 296.9 29.62231.14 24.73 3.279 5.822 0.784 77.28 0.767 0.889 2272.5 250.0 29.72297.18 25.65 3.273 5.627 0.785 79.11 0.646 0.909 2263.6 236.3 29.77
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 57.1 154.97.62 29.4 811.2 172.0 154.9 0.03 26748.2 33.20 31.1622.86 32.2 1112.5 189.8 154.9 0.04 8912.8 30.55 28.5545.72 38.6 1980.6 225.2 154.9 0.04 4454.9 25.90 23.9976.20 43.2 2654.2 243.0 154.9 0.05 2671.0 23.50 21.64121.92 50.9 4394.4 283.2 154.9 0.06 1667.7 20.48 18.71172.72 57.1 6748.4 329.3 154.9 0.07 1175.7 18.76 17.06231.14 64.6 10751.8 388.7 154.9 0.07 877.3 17.10 15.48297.18 68.1 12371.9 410.5 154.9 0.08 680.9 16.77 15.18Br. No. = 0.0001 unit ; WSR(1/sec), WSS(dyne/sq cm)
159
CMC(%) =Dia.(cm)
X(cm)
7 Run No. = 7= 5.042 Mass Rate(gr/sec) =
Tb(C) V(poise) Kpb npb ATw(C)
141.07
Kpw
Input
npw
Power(W) = 4726.7
Prb Prw Re0.00 21.50 9.091 12.976 0.756 21.50 12.976 0.756 6336.4 4931.2 3.927.62 21.70 9.008 12.861 0.756 47.37 5.073 0.778 6273.1 1490.5 3.0522.86 22.11 8.976 12.635 0.757 53.28 4.087 0.788 6246.8 1170.5 3.9745.72 22.71 8.937 12.304 0.758 59.27 3.370 0.795 6213.4 940.2 3.9976.20 23.52 8.891 11.878 0.760 65.68 2.755 0.802 6172.2 751.6 4.01121.92 24.74 8.837 11.271 0.763 71.78 2.202 0.813 6119.7 599.8 4.03172.72 26.08 8.803 10.709 0.764 75.78 1.967 0.816 6078.1 527.8 4.05231.14 27.64 8.775 10.160 0.764 80.22 1.867 0.810 6038.5 476.5 4.06297.18 29.39 8.751 9.579 0.765 83.69 1.794 0.806 5998.9 440.8 4.07
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 12.1 85.67.62 8.7 268.6 37.7 85.6 0.56 12892.2 32.34 30.44
22.86 10.8 562.4 47.5 85.6 0.69 4294.8 26.60 24.7945.72 13.1 1077.0 58.5 85.6 0.82 2145.2 22.64 20.9276.20 15.6 2038.3 72.5 85.6 0.97 1285.2 19.59 17.96121.92 18.4 3715.5 90.1 85.6 1.13 801.3 17.50 15.95172.72 20.5 5190.1 101.9 85.6 1.25 563.9 16.51 15.02231.14 22.9 6908.0 112.4 85.6 1.39 420.0 15.54 14.12297.18 25.0 8491.8 121.1 85.6 1.51 325.4 14.98 13.62Br. No. = 0.00002 unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC(%) = 7 Run No. = 8Dia.(cm) = 5.042 Mass Rate(gr/sec) = 300.74 Input Power(W) = 4794.6
X(cm) Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw Re0.00 21.63 7.518 12.902 0.756 21.63 12.902 0.756 5237.7 4077.1 10.107.62 21.73 7.465 12.847 0.756 42.80 5.994 0.773 5196.8 1536.7 10.17
22.86 21.92 7.446 12.739 0.757 47.85 4.983 0.779 5182.5 1244.1 10.2045.72 22.21 7.418 12.579 0.757 54.68 3.888 0.790 5161.8 952.0 10.2476.20 22.59 7.400 12.369 0.758 57.94 3.510 0.794 5145.4 849.8 10.26121.92 23.17 7.371 12.062 0.759 62.83 3.026 0.798 5120.6 720.6 10.30172.72 23.81 7.347 11.730 0.761 66.28 2.694 0.803 5097.1 639.4 10.34231.14 24.55 7.314 11.362 0.762 71.35 2.236 0.812 5067.6 534.3 10.38297.18 25.38 7.284 10.969 0.764 75.57 1.972 0.816 5038.5 466.9 10.43
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 25.8 151.07.62 10.5 196.1 65.0 151.0 0.10 27473.8 39.95 37.9622.86 13.0 391.6 79.5 151.0 0.13 9156.2 32.45 30.5445.72 16.5 897.1 102.7 151.0 0.16 4577.3 25.89 24.0876.20 18.3 1260.9 114.4 151.0 0.17 2744.4 23.76 22.00121.92 21.2 2043.9 133.9 151.0 0.20 1713.5 21.14 19.45172.72 23.3 2848.5 150.1 151.0 0.22 1207.9 19.71 18.06231.14 26.5 4645.0 178.5 151.0 0.25 901.4 17.85 16.27297.18 29.4 6691.1 203.3 151.0 0.27 700.0 16.61 15.09Br. No. = 0.000075 unit ; WSR(1/sec), WSS(dyne/sq cm)
160
CMC(%) =Dia.(cm)
X(cm)
7 Run No. = 9= 5.042 Mass Rate(gr/sec) = 221.27
Tb(C) V(poise) Kpb npb ATw(C) Kpw
Input Power(W) = 4774.0
npw Prb Prw Re0.00 22.85 7.711 12.231 0.759 22.85 12.231 0.759 5353.8 4180.5 7.257.62 22.98 7.651 12.161 0.759 45.40 5.461 0.775 5308.7 1489.0 7.30
22.86 23.24 7.626 12.024 0.759 51.44 4.368 0.785 5290.2 1164.8 7.3345.72 23.63 7.596 11.822 0.760 57.77 3.528 0.793 5267.2 919.1 7.3676.20 24.15 7.571 11.558 0.761 61.93 3.109 0.797 5245.0 797.0 7.38121.92 24.93 7.537 11.175 0.763 67.23 2.600 0.805 5213.5 661.2 7.41172.72 25.80 7.512 10.813 0.764 71.28 2.242 0.812 5187.4 571.3 7.44231.14 26.80 7.489 10.451 0.764 76.04 1.961 0.816 5161.1 491.5 7.46297.18 27.93 7.467 10.059 0.764 80.16 1.868 0.810 5134.5 446.0 7.46
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 19.0 114.37.62 11.2 229.7 50.5 114.3 0.21 20147.7 37.26 35.34
22.86 14.3 504.6 63.9 114.3 0.27 6714.4 29.60 27.7745.72 17.6 1039.6 80.1 114.3 0.32 3355.7 24.43 22.6876.20 19.9 1581.2 91.8 114.3 0.37 2011.5 22.04 20.36121.92 23.0 2670.4 109.8 114.3 0.42 1255.4 19.65 18.03172.72 25.6 3946.8 126.4 114.3 0.46 884.6 18.23 16.68231.14 28.7 5939.5 146.1 114.3 0.52 659.7 16.79 15.30297.18 31.6 7830.0 160.3 114.3 0.57 512.0 15.79 14.36Br. No. = 0.000042 unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC(%) = 7 Run No. = 10Dia.(cm) = 3.823 Mass Rate(gr/sec) = 141.75 Input Power(W) = 4388.3
X(cm) Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw Re0.00 21.83 7.317 12.789 0.757 21.83 12.789 0.757 5094.9 3968.0 6.457.62 22.02 7.254 12.685 0.757 46.30 5.279 0.777 5046.9 1304.7 6.51
22.86 22.39 7.228 12.479 0.758 52.55 4.196 0.787 5026.0 1019.1 6.5345.72 22.95 7.197 12.178 0.759 58.76 3.423 0.794 5000.1 815.3 6.5676.20 23.69 7.168 11.790 0.760 63.42 2.973 0.799 4972.3 697.6 6.59121.92 24.81 7.122 11.234 0.763 70.54 2.303 0.811 4930.3 543.1 6.63172.72 26.06 7.090 10.720 0.764 76.03 1.961 0.816 4895.7 456.3 6.66231.14 27.49 7.060 10.212 0.764 81.93 1.830 0.808 4862.0 395.8 6.69297.18 29.10 7.047 9.672 0.765 84.15 1.785 0.805 4834.8 375.8 6.70
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 27.9 159..07.62 5.6 142.8 80.1 159.0 0.13 12942.1 31.62 29.86
22.86 7.1 310.6 101.4 159.0 0.17 4311.9 25.43 23.7645.72 8.7 608.9 125.4 159.0 0.20 2154.0 21.39 19.7976.20 10.0 956.5 145.5 159.0 0.23 1290.5 19.24 17.71121.92 12.0 1905.3 185.1 159.0 0.27 804.8 16.67 15.22172.72 13.8 3048.4 218.9 159.0 0.31 566.7 15.20 13.83231.14 15.8 4559.5 250.8 159.0 0.35 422.3 13.90 12.60297.18 16.8 5176.5 263.7 159.0 0.38 327.3 13.69 12.44Br. No. = 0.000049 unit ; WSR(1/sec), WSS(dyne/sq cm)
161
CMC(t) =Dia.(cm)
X(cm)
7 Run No. = 11= 3.823 Mass Rate(gr/sec) =
Tb(C) V(poise) Kpb npb ATw(C)
295.75
Kpw
Input
npw
Power(W) = 5086.5
Prb Prw Re0.00 22.93 5.866 12.188 0.759 22.93 12.188 0.759 4071.7 3179.6 16.797.62 23.03 5.813 12.133 0.759 48.58 4.850 0.780 4033.4 1029.6 16.9522.86 23.24 5.801 12.024 0.759 52.05 4.272 0.786 4024.1 902.6 16.9845.72 23.55 5.783 11.863 0.760 56.99 3.614 0.793 4010.3 756.5 17.0376.20 23.97 5.765 11.652 0.761 61.24 3.174 0.797 3995.4 657.3 17.09
121.92 24.59 5.740 11.343 0.762 66.92 2.631 0.804 3973.8 544.1 17.16172.72 25.28 5.715 11.011 0.764 71.73 2.206 0.813 3952.3 462.6 17.24231.14 26.07 5.694 10.714 0.764 77.33 1.931 0.814 3932.7 392.7 17.30297.18 26.97 5.686 10.392 0.764 79.18 1.890 0.812 3919.1 374.8 17.32
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 58.2 266.77.62 9.6 247.5 169.5 266.7 0.03 26932.1 34.74 32.7522.86 11.0 376.5 192.1 266.7 0.04 8975.0 30.79 28.8645.72 12.9 650.7 227.3 266.7 0.04 4485.9 26.51 24.6476.20 14.7 994.5 259.9 266.7 0.05 2689.8 23.75 21.95
121.92 17.1 1716.3 311.4 266.7 0.06 1679.5 20.88 19.16172.72 19.3 2687.3 363.9 266.7 0.07 1184.1 18.99 17.34231.14 21.9 4251.3 425.9 266.7 0.07 883.7 17.18 15.60297.18 23.1 4802.9 445.4 266.7 0.08 685.9 16.82 15.28Br. No. = 0.00017 unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC( %) = 7 Run No. = 12Dia.(cm) = 3.823 Mass Rate(gr/sec)
X(cm) Tb(C) V(poise) Kpb npb ATw(C)
= 223.99
Kpw
Input Power(W) = 4869.4
npw Prb Prw Re0.00 22.45 6.390 12.446 0.758 22.45 12.446 0.758 4441.6 3464.5 11.677.62 22.58 6.331 12.375 0.758 48.60 4.846 0.780 4398.3 1086.3 11.78
22.86 22.84 6.315 12.235 0.759 52.61 4.187 0.787 4386.1 931.4 11.8145.72 23.24 6.293 12.027 0.759 58.23 3.479 0.794 4368.1 763.2 11.8676.20 23.76 6.270 11.757 0.761 62.84 3.025 0.798 4348.7 655.2 11.90121.92 24.54 6.237 11.364 0.762 69.37 2.403 0.809 4320.0 524.1 11.96172.72 25.42 6.207 10.957 0.764 74.65 1.994 0.818 4292.7 439.2 12.02231.14 26.42 6.183 10.587 0.764 80.48 1.861 0.810 4268.4 379.7 12.07297.18 27.56 6.172 10.187 0.764 82.60 1.816 0.807 4250.3 360.7 12.09
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 44.1 219.87.62 8.1 226.5 132.3 219.8 0.06 20422.6 32.70 30.79
22.86 9.4 367.3 153.3 219.8 0.07 6805.1 28.56 26.7145.72 11.2 676.1 185.3 219.8 0.08 3400.8 24.27 22.4976.20 12.9 1061.8 214.3 219.8 0.09 2038.7 21.70 19.99121.92 15.3 1990.8 265.5 219.8 0.11 1272.5 18.88 17.26172.72 17.4 3215.5 314.8 219.8 0.12 896.8 17.15 15.60231.14 19.8 4880.4 361.9 219.8 0.14 669.0 15.58 14.11297.18 20.9 5571.8 380.2 219.8 0.14 519.0 15.25 13.82Br. No. = 0.00011 unit ; WSR(1/sec), WSS(dyne/sq cm)
162
CMC(%) =Dia.(cm)
X(cm)
7.5 Run No. = 13= 3.823 Mass Rate(gr/sec) =
Tb(C) V(poise) Kpb npb ATw(C)
63.05
Kpw
Input
npw
Power(W)
Prb
= 3466.7
Prw Re0.00 21.41 13.833 20.761 0.715 21.41 20.761 0.715 9643.8 7034.8 1.527.62 21.74 13.718 20.500 0.716 45.42 9.288 0.741 9550.5 2449.7 1.53
22.86 22.40 13.652 19.990 0.716 54.08 7.045 0.750 9494.4 1738.0 1.5445.72 23.40 13.589 19.251 0.717 60.79 5.431 0.763 9431.4 1310.7 1.5576.20 24.72 13.527 18.316 0.719 66.08 4.437 0.774 9361.1 1059.2 1.55121.92 26.71 13.412 17.018 0.722 73.73 3.408 0.788 9242.5 811.4 1.57172.72 28.92 13.301 15.702 0.726 79.54 2.965 0.789 9122.5 676.3 1.58231.14 31.46 13.187 14.335 0.731 84.68 2.642 0.788 8996.0 579.4 1.59297.18 34.33 13.103 12.955 0.736 85.71 2.582 0.788 8879.3 562.1 1.60
X(cm)0.007.62
22.8645.7276.20121.92172.72231.14297.18Br. No.
CMC(%) =Dia.(cm)
Gr Grw0.0 0.01.5 39.12.1 113.82.6 250.03.0 440.03.7 894.84.3 1433.55.0 2111.05.2 2177.4= 0.000017
7.5 Run No.= 3.823
WSR WSS BuoyF12.7 127.734.3 127.7 0.6547.6 127.7 0.8962.4 127.7 1.0876.6 127.7 1.2699.1 127.7 1.52
118.1 127.7 1.74137.2 127.7 1.96141.3 127.7 2.04
unit ; WSR(1/sec),
= 14Mass Rate(gr/sec)
Gz Nub Nuw
5760.7 25.64 24.241918.1 19.13 17.82957.1 16.16 14.92572.6 14.56 13.38356.4 12.74 11.64250.4 11.77 10.74186.1 11.12 10.16143.8 11.44 10.51WSS(dyne /sq cm)
= 141.75 Input Power(W) = 4038.6
X(cm) Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw Re0.00 23.81 10.072 18.953 0.718 23.81 18.953 0.718 6973.6 5104.9 4.697.62 23.98 9.986 18.831 0.718 47.65 8.638 0.744 6911.3 1886.0 4.7322.86 24.32 9.955 18.590 0.718 53.48 7.178 0.749 6886.9 1506.6 4.7445.72 24.84 9.917 18.235 0.719 59.78 5.651 0.761 6855.8 1170.0 4.7676.20 25.53 9.888 17.776 0.720 62.84 5.012 0.768 6825.3 1035.5 4.77121.92 26.56 9.830 17.114 0.722 69.30 3.965 0.780 6772.5 821.9 4.80172.72 27.70 9.776 16.412 0.724 74.44 3.336 0.789 6720.2 694.1 4.83231.14 29.02 9.710 15.646 0.726 80.77 2.883 0.788 6659.3 570.1 4.86297.18 30.50 9.668 14.831 0.729 82.72 2.759 0.788 6608.2 537.5 4.88
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 28.4 209.77.62 3.1 67.6 72.9 209.7 0.14 12874.4 29.70 28.1222.86 3.9 138.6 90.3 209.7 0.17 4289.7 24.09 22.5845.72 4.8 290.9 115.0 209.7 0.21 2143.3 20.07 18.6476.20 5.3 406.0 129.4 209.7 0.23 1284.1 18.76 17.36121.92 6.3 771.6 161.6 209.7 0.27 801.0 16.33 15.01172.72 7.2 1220.7 190.1 209.7 0.31 564.2 14.89 13.63231.14 8.4 2075.9 229.9 209.7 0.35 420.6 13.41 12.22297.18 8.9 2381.6 243.4 209.7 0.37 326.0 13.24 12.08Br. No. = 0.000063 unit ; WSR(1/sec), WSS(dyne /sq cm)
163
CMC(%) =Dia.(cm)
X(cm)
7.5 Run No.= 5.042
Tb(C) V(poise)
= 15Mass Rate(gr/sec) =
Kpb npb ATw(C)
55.79
Kpw
Input
npw
Power(W)
Prb
= 3537.4
Prw Re0.00 19.45 19.566 22.390 0.713 19.45 22.390 0.713 13720.8 9973.2 0.727.62 19.83 19.417 22.061 0.714 42.54 10.104 0.740 13588.5 3399.3 0.7322.86 20.60 19.321 21.419 0.714 51.60 7.614 0.748 13501.2 2350.8 0.7345.72 21.75 19.222 20.497 0.716 58.99 5.831 0.759 13399.2 1714.9 0.7376.20 23.28 19.121 19.339 0.717 65.19 4.579 0.772 13282.4 1308.5 0.74121.92 25.57 18.980 17.745 0.720 72.10 3.602 0.785 13115.4 1013.4 0.74172.72 28.13 18.831 16.158 0.725 76.02 3.215 0.789 12933.9 887.6 0.75231.14 31.06 18.668 14.536 0.730 80.19 2.921 0.788 12736.3 780.7 0.75297.18 34.38 18.516 12.929 0.736 82.55 2.770 0.788 12538.1 727.1 0.76
X(cm) Gr Grw0.00 0.0 0.07.62 1.5 43.0
22.86 2.2 136.445.72 2.8 329.076.20 3.3 666.3121.92 4.1 1289.8172.72 4.6 1771.3231.14 5.2 2403.8297.18 5.7 2751.SBr. No. = 0.0000064
WSR WSS BuoyF Gz Nub Nuw4.9 69.413.5 69.4 2.89 5123.9 27.50 26.0119.2 69.4 4.07 1705.5 20.10 18.7126.0 69.4 5.13 850.6 16.68 15.3733.8 69.4 6.13 508.6 14.76 13.5343.2 69.4 7.38 316.2 13.21 12.0649.1 69.4 8.25 221.9 12.75 11.6655.6 69.4 9.18 164.7 12.34 11.3159.6 69.4 9.78 127.1 12.49 11.51
unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC(%) = 7.5 Run No. = 16Dia.(cm) = 5.042 Mass Rate(gr/sec) = 141.75 Input Power(W) = 3723.1
X(cm) Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw Re0.00 21.18 14.033 20.945 0.715 21.18 20.945 0.715 9790.0 7138.7 2.557.62 21.34 13.949 20.818 0.715 40.13 10.842 0.739 9719.3 3028.9 2.5722.86 21.66 13.896 20.567 0.716 48.20 8.486 0.744 9679.1 2194.1 2.5845.72 22.13 13.851 20.197 0.716 53.86 7.094 0.750 9639.9 1756.5 2.5876.20 22.77 13.811 19.717 0.717 58.03 6.059 0.757 9599.5 1475.7 2.59121.92 23.72 13.755 19.020 0.718 63.46 4.893 0.769 9541.7 1175.5 2.60172.72 24.77 13.708 18.280 0.719 67.30 4.251 0.776 9486.3' 1016.1 2.61231.14 25.99 13.636 17.474 0.721 72.16 3.595 0.785 9412.0 857.6 2.63297.18 27.36 13.568 16.613 0.723 75.40 3.262 0.789 9336.7 770.2 2.64
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 12.4 126.87.62 2.6 43.4 27.9 126.8 0.40 12961.8 34.83 33.2522.86 3.8 129.4 37.8 126.8 0.57 4319.0 24.63 23.1545.72 4.6 255.6 46.8 126.8 0.69 2157.7 20.58 19.1676.20 5.3 417.5 55.3 126.8 0.79 1292.9 18.48 17.12121.92 6.2 773.4 68.9 126.8 0.92 806.5 16.36 15.05172.72 7.0 1137.3 79.3 126.8 1.02 567.9 15.25 13.99231.14 7.9 1787.6 93.3 126.8 1.15 423.3 14.00 12.79297.18 8.7 2350.5 103.5 126.8 1.25 328.2 13.41 12.24Br. No. = 0.000029 unit ; WSR(1/sec), WSS(dyne/sq cm)
164
CMC(%) =Dia.(cm)
X(cm)
7.5 Run No.= 5.042
Tb(C) V(poise)
= 17Mass Rate(gr/sec) =
Kpb npb ATw(C)
221.81
Kpw
Input
npw
Power(W) = 4339.8
Prb Prw Re0.00 23.35 11.409 19.285 0.717 23.35 19.285 0.717 7909.7 5786.8 4.917.62 23.47 11.324 19.199 0.718 45.09 9.389 0.741 7846.6 2284.9 4.9522.86 23.70 11.294 19.029 0.718 50.34 7.924 0.746 7824.3 1854.7 4.9645.72 24.06 11.258 18.777 0.718 56.20 6.522 0.753 7795.7 1473.6 4.9876.20 24.53 11.229 18.447 0.719 59.96 5.611 0.761 7769.5 1259.7 4.99121.92 25.24 11.187 17.965 0.720 64.85 4.638 0.772 7730.2 1033.0 5.01172.72 26.03 11.145 17.449 0.721 68.52 4.074 0.779 7688.5 908.0 5.03231.14 26.93 11.088 16.878 0.723 73.87 3.392 0.789 7636.7 757.7 5.05297.18 27.96 11.037 16.258 0.724 77.84 3.083 0.789 7585.9 668.6 5.08
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 19.4 161.97.62 5.0 93.7 46.7 161.9 0.21 20170.1 35.07 33.3322.86 6.3 185.9 56.9 161.9 0.25 6721.8 28.45 26.7845.72 7.7 375.7 70.9 161.9 0.31 3359.6 23.56 21.9676.20 8.6 584.6 82.5 161.9 0.35 2014.0 21.34 19.80121.92 9.9 1007.9 99.8 161.9 0.40 1257.1 19.06 17.57172.72 11.0 1434.5 113.0 161.9 0.44 885.9 17.73 16.29231.14 12.6 2352.1 134.5 161.9 0.49 660.9 16.01 14.64297.18 13.9 3284.3 151.8 161.9 0.54 513.0 15.02 13.70Br. No. = 0.000058 unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC(%) = 8 Run No. = 18Dia.(cm) = 3.823 Mass Rate(gr/sec) = 58.15 Input Power(W) = 3176.1
X(cm) Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw Re0.00 19.50 23.185 35.023 0.680 19.50 35.023 0.680 16256.0 11129.5 0.847.62 19.83 23.008 34.585 0.680 42.16 15.519 0.711 16101.1 3733.1 0.8422.86 20.49 22.891 33.727 0.681 50.75 11.666 0.722 15999.8 2605.3 0.8545.72 21.47 22.776 32.487 0.682 56.97 9.572 0.729 15885.3 2043.7 0.8576.20 22.79 22.657 30.915 0.684 61.94 8.209 0.735 15753.7 1697.7 0.85121.92 24.76 22.474 28.722 0.687 69.50 6.893 0.734 15557.6 1319.0 0.86172.72 26.95 22.299 26.501 0.690 75.22 6.171 0.732 15360.1 1107.0 0.87231.14 29.48 22.105 24.195 0.694 81.19 5.345 0.735 15143.3 926.5 0.88297.18 32.33 21.946 21.868 0.698 83.85 5.021 0.737 14937.1 857.8 0.88
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 11.9 188.27.62 0.5 15.2 33.4 188.2 0.66 5340.5 25.04 23.7022.86 0.7 46.9 47.1 188.2 0.91 1778.0 18.44 17.1945.72 0.8 94.6 59.4 188.2 1.12 887.1 15.68 14.5076.20 0.9 157.4 70.9 188.2 1.30 530.6 14.17 13.03121.92 1.2 313.9 90.3 188.2 1.59 330.2 12.33 11.28172.72 1.4 497.8 106.9 188.2 1.85 231.9 11.36 10.37231.14 1.6 787.3 126.9 188.2 2.13 172.4 10.54 9.61297.18 1.8 928.3 136.8 188.2 2.29 133.2 10.50 9.62Br. No. = 0.000023 unit ; WSR(1/sec), WSS(dyne/sq cm)
165
CMC(%) =Dia.(cm)
X(cm)
8 Run No.= 3.823
Tb(C) V(poise)
= 19Mass Rate(gr/sec) =
Kpb npb ATw(C)
185.07
Kpw
Input
npw
Power(W) = 4031.3
Prb Prw Re0.00 25.21 13.105 28.248 0.688 25.21 28.248 0.688 9038.7 6277.0 4.707.62 25.34 12.997 28.112 0.688 47.81 12.839 0.718 8962.6 2317.8 4.7422.86 25.60 12.966 27.842 0.688 51.92 11.234 0.723 8938.0 1976.2 4.7545.72 26.00 12.924 27.443 0.689 56.82 9.618 0.729 8904.6 1644.4 4.7776.20 26.52 12.879 26.922 0.690 61.51 8.317 0.734 8866.0 1388.2 4.79121.92 27.31 12.820 26.162 0.691 67.02 7.224 0.736 8813.0 1150.0 4.81172.72 28.18 12.766 25.346 0.692 71.34 6.661 0.733 8761.1 1001.5 4.83231.14 29.19 12.687 24.445 0.694 78.21 5.739 0.733 8693.8 813.4 4.86297.18 30.33 12.642 23.472 0.695 80.40 5.446 0.735 8641.1 763.5 4.88
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 37.7 342.97.62 1.9 42.6 96.9 342.9 0.08 16748.3 31.12 29.5622.86 2.2 71.6 112.8 342.9 0.10 5580.9 26.55 25.0445.72 2.6 126.8 134.5 342.9 0.11 2788.9 22.64 21.1976.20 3.0 209.7 158.1 342.9 0.13 1671.9 19.92 18.52121.92 3.5 -360.7 189.4 342.9 0.15 1043.5 17.52 16.18172.72 4.0 531.4 216.3 342.9 0.17 735.3 16.08 14.80231.14 4.7 952.6 264.2 342.9 0.20 548.6 14.12 12.91297.18 5.0 1117.9 280.8 342.9 0.21 425.7 13.79 12.61Br. No. = 0.00013 unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC(%) = 8 Run No. = 20Dia.(cm) = 5.042 Mass Rate(gr/sec) = 58.15 Input Power(W) = 3166.4
X(cm) Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw Re0.00 21.31 28.224 32.687 0.682 21.31 32.687 0.682 19682.9 13549.3 0.527.62 21.64 28.022 32.283 0.683 41.98 15.615 0.711 19510.8 4862.3 0.52
22.86 22.30 27.883 31.492 0.684 50.47 11.772 0.721 19390.2 3360.4 0.5345.72 23.28 27.738 30.347 0.685 57.14 9.522 0.729 19250.1 2564.9 0.5376.20 24.60 27.578 28.897 0.687 63.19 7.903 0.736 19084.6 2032.3 0.53121.92 26.57 27.373 26.875 0.690 69.44 6.901 0.734 18859.3 1659.3 0.54172.72 28.76 27.179 24.823 0.693 74.09 6.332 0.731 18632.3 1444.0 0.54231.14 31.28 26.977 22.689 0.697 78.27 5.731 0.733 18390.0 1269.6 0.54297.18 34.14 26.779 20.532 0.701 81.51 5.305 0.735 18140.0 1151.3 0.55
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 5.2 100.37.62 0.7 18.7 13.7 100.3 2.61 5313.5 27.34 26.02
22.86 1.0 60.0 19.5 100.3 3.69 1769.0 19.70 18.4745.72 1.3 131.7 25.2 100.3 4.60 882.7 16.35 15.1976.20 1.6 250.7 31.5 100.3 5.49 528.1 14.29 13.19121.92 1.9 436.0 38.3 100.3 6.50 328.6 12.80 11.76172.72 2.1 626.2 43.7 100.3 7.35 230.8 12.04 11.05231.14 2.4 860.1 49.5 100.3 8.15 171.5 11.54 10.61297.18 2.7 1073.5 54.4 100.3 8.82 132.6 11.37 10.48Br. No. = 0.0000094 unit ; WSR(1/sec), WSS(dyne/sq cm)
166
CMC(%) =Dia.(cm)
X(cm)
8 Run No. = 21= 5.042 Mass Rate(gr/sec) =
Tb(C) V(poise) Kpb npb ATw(C)
184.62
Kpw
Input Power(W) = 3948.3
npw Prb Prw Re0.00 24.88 17.196 28.595 0.687 24.88 28.595 0.687 11871.2 8239.0 2.717.62 25.01 17.087 28.459 0.688 43.13 15.014 0.712 11789.7 3521.3 2.7322.86 25.27 17.019 28.189 0.688 51.09 11.538 0.722 11741.7 2532.8 2.7445.72 25.65 16.964 27.790 0.689 56.22 9.801 0.728 11697.4 2073.1 2.7576.20 26.17 16.914 27.269 0.689 59.90 8.738 0.733 11652.0 1805.4 2.76121.92 26.95 16.842 26.509 0.690 64.90 7.523 0.738 11585.8 1505.1 2.77172.72 27.81 16.776 25.693 0.692 68.88 6.974 0.735 11520.1 1326.5 2.78231.14 28.80 16.694 24.792 0.693 73.87 6.358 0.731 11442.7 1137.3 2.79297.18 29.92 16.615 23.818 0.695 77.96 5.774 0.733 11364.0 1005.2 2.81
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 16.4 195.57.62 2.0 32.3 36.7 195.5 0.26 16716.6 37.92 36.33
22.86 2.8 97.4 50.3 195.5 0.38 5571.7 26.59 25.1045.72 3.4 180.5 60.9 195.5 0.45 2784.4 22.44 21.0176.20 3.8 270.8 69.5 195.5 0.51 1669.0 20.31 18.92121.92 4.5 455.1 82.7 195.5 0.58 1041.6 18.01 16.68172.72 5.0 651.3 93.4 195.5 0.64 734.0 16.61 15.32231.14 5.7 1002.6 108.2 195.5 0.73 547.5 15.10 13.87297.18 6.3 1400.4 121.9 195.5 0.80 424.9 14.12 12.94Br. No. = 0.000057 unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC(%) = 8.3 Run No. = 22Dia.(cm) = 3.823 Mass Rate(gr/sec) = 60.96 Input Power(W) = 3123.0
X(cm) Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw Re0.00 19.89 27.976 43.460 0.662 19.89 43.460 0.662 19592.3 12939.2 0.737.62 20.20 27.781 43.044 0.662 42.12 20.470 0.690 19420.6 4563.8 0.73
22.86 20.81 27.661 42.228 0.662 51.39 15.454 0.699 19316.6 3135.8 0.7345.72 21.74 27.582 41.037 0.662 56.41 13.452 0.702 19221.4 2599.2 0.7476.20 22.97 27.499 39.513 0.663 61.18 12.056 0.703 19108.4 2215.9 0.74121.92 24.82 27.363 37.353 0.663 68.56 10.021 0.707 18937.0 1725.1 0.74172.72 26.88 27.151 34.707 0.666 73.99 8.666 0.711 18702.4 1433.0 0.75231.14 29.25 26.847 31.673 0.671 79.83 7.155 0.721 18398.6 1169.4 0.76297.18 31.92 26.559 28.611 0.677 83.75 6.307 0.728 18093.8 1024.3 0.76
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 12.6 231.97.62 0.3 10.0 33.7 231.9 0.60 5592.9 25.06 23.74
22.86 0.5 32.9 48.1 231.9 0.85 1862.4 17.93 16.7145.72 0.5 56.9 57.6 231.9 1.00 929.2 15.77 14.6176.20 0.6 89.6 67.0 231.9 1.16 555.9 14.27 13.15121.92 0.8 178.3 85.2 231.9 1.42 346.0 12.40 11.36172.72 0.9 287.9 101.9 231.9 1.64 243.1 11.45 10.47231.14 1.1 479.8 124.1 231.9 1.88 180.8 10.60 9.68297.18 1.2 654.6 141.2 231.9 2.07 139.8 10.28 9.40Br. No. = 0.00003 unit ; WSR(1/sec), WSS(dyne/sq cm)
167
CMC(%) =Dia.(cm)
X(cm)
8.3 Run No.= 3.823
Tb(C) V(poise)
= 23Mass Rate(gr/sec) =
Kpb npb ATw(C)
136.53
Kpw
Input Power(W) = 3802.7
npw Prb Prw Re0.00 22.80 19.489 39.723 0.663 22.80 39.723 0.663 13532.5 8952.7 2.337.62 22.97 19.342 39.520 0.663 45.19 18.695 0.693 13420.5 3302.0 2.3522.86 23.30 19.293 39.117 0.663 50.56 15.847 0.698 13379.9 2671.2 2.3645.72 23.81 19.240 38.523 0.663 55.84 13.632 0.702 13331.6 2193.0 2.3676.20 24.48 19.185 37.747 0.663 60.75 12.174 0.703 13277.1 1861.1 2.37121.92 25.48 19.116 36.619 0.663 66.55 10.587 0.705 13202.9 1535.2 2.38172.72 26.60 18.999 35.088 0.666 71.30 9.307 0.709 13092.1 1306.4 2.39231.14 27.89 18.833 33.377 0.669 78.80 7.399 0.719 12949.0 1017.5 2.41297.18 29.34 18.714 31.559 0.672 81.50 6.778 0.724 12825.4 931.7 2.43
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 28.1 362.67.62 0.8 20.1 72.3 362.6 0.14 12432.5 29.87 28.3422:86 0.9 40.1 88.5 362.6 0.17 4142.1 24.33 22.8645.72 1.1 73.4 106.8 362.6 0.20 2069.3 20.67 19.2776.20 1.3 120.3 124.9 362.6 0.23 1240.0 18.22 16.88121.92 1.5 208.6 150.1 362.6 0.27 773.5 16.05 14.77172.72 1.8 323.4 175.3 362.6 0.31 544.7 14.71 13.48231.14 2.1 634.2 223.1 362.6 0.36 406.1 12.87 11.72297.18 2.3 786.5 243.1 362.6 0.39 314.9 12.51 11.40Br. No. = 0.00010 unit ; WSR(1/sec), WSS(dyne/sq cm)
CMC(%) = 8.3 Run No. = 24Dia.(cm) = 5.042 Mass Rate(gr/sec) = 63.05 Input Power(W) = 3195.4
X(cm) Tb(C) V(poise) Kpb npb ATw(C) Kpw npw Prb Prw Re0.00 20.42 36.027 42.748 0.662 20.42 42.748 0.662 25190.7 16642.3 0.447.62 20.73 35.796 42.344 0.662 40.92 21.218 0.689 24985.8 6129.3 0.4422.86 21.34 35.652 41.550 0.662 49.77 16.231 0.697 24857.3 4234.1 0.4545.72 22.25 35.525 40.392 0.663 56.31 13.484 0.702 24721.8 3283.2 0.4576.20 23.48 35.397 38.908 0.663 62.22 11.777 0.703 24566.1 2693.7 0.45
121.92 25.31 35.248 36.805 0.663 68.29 10.095 0.706 24361.3 2182.7 0.45172.72 27.36 34.942 34.073 0.667 73.06 8.882 0.710 24037.0 1845.9 0.46231.14 29.70 34.583 31.127 0.672 77.72 7.666 0.717 23664.8 1558.1 0.46297.18 32.36 34.227 28.149 0.678 81.20 6.844 0.724 23282.7 1373.8 0.47
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 5.7 134.77.62 0.4 11.5 14.6 134.7 2.11 5775.5 27.86 26.5122.86 0.6 37.9 20.8 134.7 3.04 1923.0 19.75 18.4945.72 0.8 80.3 26.5 134.7 3.77 959.7 16.45 15.2676.20 0.9 142.3 32.0 134.7 4.49 574.3 14.41 13.28
121.92 1.1 250.7 39.1 134.7 5.31 357.4 12.93 11.86172.72 1.3 384.1 46.0 134.7 6.03 251.1 12.09 11.08231.14 1.4 581.9 54.2 134.7 6.78 186.7 11.44 10.48297.18 1.6 776.1 61.2 134.7 7.39 144.3 11.18 10.26Br. No. = 0.000014 unit ; WSR(1/sec), WSS(dyne/sq cm)
168
CMC(%) =Dia.(cm)
X(cm)
8.3 Run No.= 5.042
Tb(C) V(poise)
= 25Mass Rate(gr/sec) =
Kpb npb ATw(C)
136.08
Kpw
Input Power(W) = 4177.5
npw Prb Prw Re0.00 24.02 24.880 38.273 0.663 24.02 38.273 0.663 17216.4 11397.0 1.387.62 24.21 24.709 38.059 0.663 44.22 19.235 0.692 17086.6 4461.0 1.3922.86 24.58 24.614 37.634 0.663 52.54 14.930 0.700 17014.7 3185.1 1.4045.72 25.13 24.526 37.008 0.663 59.17 12.621 0.703 16940.9 2515.1 1.4076.20 25.87 24.438 36.102 0.664 63.83 11.360 0.703 16855.9 2157.6 1.41121.92 26.99 24.263 34.566 0.667 70.08 9.617 0.708 16699.6 1735.5 1.42172.72 28.22 24.096 32.949 0.669 74.89 8.421 0.712 16543.7 1470.6 1.43231.14 29.64 23.906 31.198 0.672 80.38 7.028 0.722 16368.1 1212.1 1.44297.18 31.25 23.716 29.347 0.676 84.98 6.065 0.730 16187.1 1036.2 1.45
X(cm) Gr Grw WSR WSS BuoyF Gz Nub Nuw0.00 0.0 0.0 12.2 201.27.62 1.0 22.5 29.8 201.2 0.52 12349.0 36.40 34.7322.86 1.4 67.7 41.0 201.2 0.73 4115.0 26.03 24.4645.72 1.8 140.0 51.4 201.2 0.91 2055.8 21.35 19.8676.20 2.0 219.8 59.5 201.2 1.03 1231.7 19.11 17.68121.92 2.4 402.3 73.3 201.2 1.21 768.2 16.78 15.42172.72 2.8 624.6 86.1 201.2 1.36 540.9 15.45 14.15231.14 3.2 1031.0 103.8 201.2 1.53 403.1 14.16 12.93297.18 3.5 1531.3 120.9 201.2 1.69 312.5 13.32 12.15Br. No. = 0.000043 unit ; WSR(1/sec), WSS(dyne/sq cm)
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