LAMINAR-FLOW HEAT TRANSFER AND PRESSURE DROP
IN TUBES WITH LIQUID SOLIDIFICATION
A THESIS
Presented to
the Faculty of the Graduate Division
by
Ronald D. Zerkle
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy in the School
of Mechanical Engineering
Georgia Institute of Technology
September., 196^
LAMINAR-FLOW HEAT TRANSFER AND PRESSURE DROP
IN TUBES WITH LIQUID SOLIDIFICATION
Approved
S
Date approved ty Chairman:
A^djC^J^c^
2
/961/
Acmwimmms
The author wishes to express sincere gratitude to 3>r. J. E.
Sunder3jancL for his guidance a/ad encouragesient in the preparation of
this thesis? and for his inspiring example aa a teacher and friend«
The author wishes to acknowledge the assistance of Messrs. L. A.
GaralXi, C. R. Bannister, J. G» Doyal, and L, w. Gleason in constructing
various equipment for this investigation, and the financial support of
the Georgia Institute of Technology and the Ford Foundation during his
doctoral program
In addition, the author wishes to thank his family for their
patience and encouragement.
XAELE COT CCHTEMTS
AOK^WIEDQMIS
. , .
• „ . „ . . . «
8
LIST OF imOTRATOTS . . „ . .
L I S T OP TABLES. . . . . . .
SUMMARY o . .
. . . . . .
o . . . . . . . .
L I S T OF SYMBOLS . , , . ,
. , .
e
D
o .. . .
. . . . .
.
CHAPTER
I.
IHKRQIXJCEION
. . . o . o . , .
Review of t h e L i t e r a t u r e
. . .
Statement of t h e Problem . . .
II.
MfiffiHEMft.TIGAL ANALYSIS. . . . .
Fluid Velocity Distribution. »
T«aperatm*e D i s t r i b u t i o n . . .
.Heat T r a n s f e r Rate
. . , . , , .
L i q u i d Bulk T e m p e r a t u r e . . . .
SolM-Fhrnse S h e l l P r o f i l e .
. ,
Presetire D i s t r i b u t i o n • , . . .
Susmaxy of t h e R e s u l t s . . « .
III c
BCFERIMEa&EAL AHAESBIS
V i s u a l ExperSjaeat
. . . .
. • . . .
C i r c u l a r Tub* Experiment
. . .
ciiAJ ma
IV.
RES • : . • .
.- . . . . . . . . . . .
Theoretical A n a l y s i s .
Visual Experiu^nt
» . > .
Semi-Empirical Solution
VI.
COHCI.USXO^S
.
. . . .
. . . . . . • - *
CirouJar Tut*-:- t'xperl^.eat.
V,
c , , ,
.
. . . . «, - . .
. . . , . . . , . « .
. . . . . . . . . . . . . . . . .
RECOMMEHnAffilONS . . . . .
. . .
. . . . .
APPENDIX
A,
TABULATED' T H E Q R m C A L KESUJ-TS
........
B*
PROPERTIES QP WATER AND ICE
. . . * . „ . ,
CIRCULAR TUBS EXr'^IMl',Ni,A.L LaVTA . . . . . .
D.
DERIVATION OF THE EQUATION RELATING
E,
DERIVATION OF THE RELATIONSHIPS BBTWEEH I*, q*
AND (Nu)
'-.am
...,....,..,.,..,
Without S o l i d i f i c a t i o n .
With S o l i d i f i c a t i o n
BIBLIOGRAPHY
6* TO T*
. . . . . . . . . . .
. . . . . . . . . . . . . .
, . , . . . „ , . . . . . . . . . . . . , . . . <
VITA. . . . . . . . . . . . . . .
o ,
t
. .
. ,
c
. . .
.
LIST OP ILLUSTRATIONS
Schematic Diagram of the Solidification
of a Continuous Casting. . • . . . . . * . . «
Sectional View at the Thermal Eat trance ox
a Circular Tube with Solidification. , . . . .
Axial Velocity Distributions in Tubes.. . . . <.
Visual Experiment Apparatus, . „ < . . . . , . «
Visual Experiment Test Section and
Coolant Tanks, „ . . . . . . . . . .
t
„ . ..
Circular Tube Experimental Apparatus . „ , . , „
Test Section Inlet Connection,
. . . . . . . .
Tect Section Outlet Connection
. . . . . . . .
Theoretical Dimensionless Bulk Temperature,
Heat Transferf and Radius of the LiquidSolid Interface, . . . „ , . . . . . . . . . ,
Theoretical Fressure Drop Versus Axial Position
Photograph of Ice-Shell Profile vithin Test
Section of Visual Experiment . » » . . .
».
Sketch of Ice-Shell Profile wiuhin Tent Section
of Visual Experiment
. . „ . * . , « . . . .
Comparison of Experimental and Theoretical
Dimensionless Balk Temperature . . . . . . , ,
Comparison of Experimental and Theoretical
Dimensionless Pressure Drop,.. . ., ,
.
.
Comparison of Experimental Data and Calculated
Results Using Equations (64-66), . ,. , .
,
Bulk Temperature Distribution for Water Fio^ In
a Tube vith No Solidification. . • . , . , . ,
';
LIST OF ILLUSTRATIONS (Continued)
Figure
17«
Page
Comparison of T h e o r e t i c a l and Serai-Erapirl c a l
R e s u l t s for t h e Radius of t h e L i q u i d - S o l i d
Interface, . . . . . . . .
, - , . . . . . • . . < ,
C
M
LIST OF CABLES
Eigenvalues and C o n s t a n t s Occurring i n the
Graetz Problem.. ,. . . » .. <, , . . .. . . . , .
Dimensionless Heat T r a n s f e r and. Liquid Bulk
Temperature Versus Dimensionless Tube Length.
Dimensionless Temperature Gradient i n t h e
Liquid and Radius of t h e L i q u i d - S o l i d
I n t e r f a c e Versus Dimensionless Axial P o s i t i o n
I ( z •) for T w ~ 0 , 5
i f z } for T
-
1.0
V
l ( z ) for T* « 1,92
w
r ( z * ) for T* - 2.0
L2
Dimensionless P r e s s u r e Drop Versus Dimensionless
A x i a l P o s i t i o n for l " - 0, c ; , „ „ „ , , ,. , . .
v
Dimensionless Pressure Drop Versus Dimensionless
-- L.O « . » „ « .. . . . .
Axial Position for T
1*7
Dimensionless P r e s s u r e Drop Versus Dimensionless
A x i a l P o s i t i o n for T ' = 2 , 0 . , , , ,
P r o p e r t i e s of Water „
0
„ .
0
„
., « , , „ , « ., » , „ ,
T h e r m a l C0riai.1cti.vit.> o f I c e , . < „ „
0
u
c
„
0
Experimental Data for Watex* Flaw i n 3 H o r i z o n t a l
Tube with L/D ~ 19 • » , . ,
. . . . . . .
.
Experimental Data for Water Flow .1 n a H o r i z o n t a l
Tube with L/D - 53.75 . -
. . . . . , • . . „ . „
viii
SUMMARY
The effect of liquid solidification at the inner surface of a
circular tube upon laminar-flow heat transfer and pressure drop is
investigated analytically and experimentally for steady-state conditions.
The liquid flowing through the tube is assumed to be Newtonian and
incompressible, and physical properties are assumed tc be independent
of temperature for any given phase. The liquid is isothermal and has
a fully-developed velocity profile at the thermal entrance; i.e., the ;
tube section where cooling begins. Following this section the wall
temperature is constant and lower than the liquid freezing temperature.
A solid-phase shell is thus formed which constricts the liquid flow
and produces a two-dimensional velocity profile.
Theoretical expressions for the bulk temperature distribution,
heat transfer rate, radius of the liquid-solid interface, and axial
pressure distribution are determined from a mathematical analysis of
the continuity7 momentum, and energy equations«
The mathematical
analysis neglects axial heat conduction, viscous energy dissipation^
radiant heat transfer, and body forces, it also assumes that the
fluid flow remains laminar and the axial component of the fluid velocity
retains its parabolic profile throughout the cooling region. i'he
solution to the resulting form of the energy equation is obtained
through use of a variable transformation, which transforms it to the
classical Greetz equation for which the solution is known-
The axial
pressure distribution is determined by solving an approximate integral
form of the axial momentum equation-
The experimental analysis is divide:! inio
experiment and the circular tube experiment*
two parts, the visual.
The purpose of the visual
experiment is to observe the solidification of a liquid flowing through
a closed conduit. Visual observations of the solidification of water
flowing through a horizontal, rectangular conduit with plexiglass 1 jp
and bottom and copper side walls substantiate basic assumptions concerning the characteristics and appearance of the solid-phase shell.
The purpose of the circular tube experiment is to measure variables which enable a comparison to be made between the experimental and
theoretical results. A comparison between the theoretical results and
experimental data taken from two horizontal test sections of different
length shows that natural convection within the liquid can produce a
considerable variance in results .
A semi empirical method lor obtaining approximate solutions is
also presented along with an illustrative example,which demonstrates how
natuial convene Ion affects the sol id-phase shell piofil/-.--. This method
can be used for either laminar ui turbulent flow, and for fluids other
than Newtonian liquids .
LItiS
OF' S3O4B0&3
Cp
specific heat at constant pressure; B/lUa-^F
Cn
Constants occurring id the Grsetz solution
I)
Inside tube diameter, ft
g
Gravitational acceleration, ft/sec^
Qraahof Number, g D 5 6 R T
*laoi
O
+ T. )/2 - T,7I /'J % d t e n s i o n l e s s
Arithmetic mean heat transfer coefficient,, equations
("D) or ( 8 1 ) , B/hr«ft 2 -°F
I
Dimensionless integral, equatipa (60)
k
The rmal conductivity, B/hr - ft - c $'
k^
Thermal conductivity of the liquid
k,
fChexraal conductivity of the solid #haee
ij
L
Test section length
m
Mass flowrate, Ibn/aln
R
Indieial notation,, n = 0,, 1, Z, . . .
(Uu)SKI
Aritigaetie »eaa Busselt dumber, equation (c"7}
p
Pressure j in« ELO
p
-o
p*
J?r
Pressure at % r D
Diwansionlsss pressure drop* equation (-.
Fr&ndta Humbsr, w / a , dinenslonXees
q
Heat transfer rate, B/kr
q*
Dinansionless heat transfer,, equation (42) car (65)
r
Space coordinate accrual to tube asls* ft
Disettsloataas radial coordinate* ft£uat.io& (£#)
LIST OF Sn&QLS (Continued)
R
Inside tube radius^ f t
Ke_,
p
R
Reynolds Staber* VD/v , d:tansioal«ss
R'
Derivative of eij^nfunctios occurring in the Gfraetz
Ei^nfunefcions occurring in the Gr&ets solution
n
°
solution
T
T«ffiperature, °C or a P
T
Biraensionlesa temperature-.» equation (23)
2L
Liquid bulk teiaperature^ equation (¥5)
o
Diaensiouless feul& tetqperaturej equation (4y) or ( 6 l )
Tf
Freezing temper&ture of the l i q u i d
TT
SJeiBperature a? the l i q u i d
T
Ssajperature of the l i q u i d a t z z 0
Ta
Teagperature
T
Tube wall-temperature
T_
Diia&nsionless parameter, equation (f?3)
JM
STCC
the s o l i d phase
Component of velocity in the r direction,, f t / s e c
v
Ccraj>onent of velocity in the 2 direst-ion^. f t / a e o
V
Averags f l u i d velocity at: s ; 0, f t / s e e
Spaee ©©ordinate along tube axis , f t
/''
DiaeusionXegs a x i a l p o s i t i o n , equation (2^)
a
Thermal d i f f u s i v i t y , fc/p £ , f t / s e e
P
&
Coefficient of vc&uuetrie expansion, l/°C
6
*
Badiue of the liquid-solid interface? ft
DiiieasiQnless radius of the liquid-solid interface, equation (•
Xii
LIST CF SXmO'lB (Cotytisiued)
n
D?j&?nsion.le3s r a d i a l coordinate ? equation (30)
A
Ei^nvaXues oceurrtog in the Graetz solution
y
Dynamic v i s c o s i t y , Ihm/hr-ft
p
Dyriaaaie v i s c o s i t y evaluated a t ( l . + T v )/2
u ,
W
Dynamic v i s c o s i t y evaluated a t T
v
2/
Kxne»atic viscosity9 ft /see
p
Density, ITqaa/ft
V
Other symbols or subscripts that might be used are defined in the
text.
CHAPTER I
INTRODUCTION
The process of solidification or freezing of materials has considerable importance in many technical fields. For example, the freezing
phenomenon must be confronted in the making of ice, freezing of foods.,
and casting of metals. But the freezing of materials does not always
meet with approval. Frost formation on refrigeration cooling coils,
ice formation in water mains, and solidification of molten metals and
salts in nuclear-reactor heat exchangers, for example^ are all detrimental to the transfer of heat and mass. Also,, in the design of space
satellites and other space vehicles, consideration must be given to the
possibility of undesirable phase changes occurring in hydraulic systems
and heat exchangers which may be subjected to extreme environmental
temperatures.
Since the solidification of materials flowing in closed conduits
is of such practical importance, the purpose of this investigation is to
study the effects of a liquid-solid change of phase upon laminar-flow
heat transfer and pressure drop in circular tubes,
Review of the Literature
Many investigations of l&minar forced-convection heat transfer
in conduits without solidification are presently available in the literature, The solution of the original Graetz problem of forced convection
heat transfer with laminar3 fully-developed flow of & constant-property
.-..-.
fluid through an isothermal circular tube Is reviewed by Jacob (l)., The
Graetz problem has been extended by many authors and in a variety of ways c
The most significant of these are as fellows: K&ys (2) considers hydrodynamically developing fluid flew; Schneider (3) and Singh (4) study the
effect of axial heat conduction1 Sellers^ et al. (5) and Schenfc and
Dumore (6) consider various thermal boundary conditions;, Brinfeman (7)
investigates the effect of viscous dissipation,; Teissier (6) and Yang (9)
consider variable fluid properties; and Sparrow and Siegel (10) study
the transient heating problem. The corresponding problem of fluid flow
between flat parallel plates has also been thoroughly studied, 0
A presence of natural convection within the fluid is a main source
of disagreement between the analytical solution of Graetz and experimental
results. Due to a resultant complication of hydrodyn&mic flow patterns/most
studies of combined natural and forced, convection in horizontal circular
tubes have been empirical or semi-empirical in nature. Discussions of
r,he Graetz solution and empirical correlations are presented by McAdama
(ll) and Knudsen and KatE (12). Beeent contributions to the literature
have been made by Jackson, et al» (13) &&& Oliver
{lk)*
The existing literature concerning the solidification of materials
can be conveniently separated into the following two categoriest
1.
Solidification of a material 'which is stationary relative
to the heat sink,
2o
Solidification of a material which is in motion relative to
the heat sink,
Carslaw and Jaeger (15) review problems of *:he first category,
which involve a one-dimensional solidification front, and Poets (16) investigates the case of a ts?o -dimensional solidif lost ion front _
Problems of the second category are studied by Gtachev (17),
Horvay (id), Bueckner and Horvay (19), and Whitehurst (20), where the
heat sink is a flat plate. Tkaehev investigates the flow of a viscous
f luio. around a plate during freezing of the fluid onto the plate. The
rate of freezing Is assumed to be constant over the length of the plats.
Solutions are obtained for both laminar forced convection and laminar
free convection through use of approximate integral forms of the NavierStokes and energy equations. Horvay solves the one-dimensional problem
of freezing of a growing liquid column onto a flat plate through use of
an electrical circuit analogyd
Bueckner and Horvay investigate the
freezing of an inviscid fluid onto a moving flat plate under the assump tion of a constant rate of .freezing along the solidification front,
Whitehurst presents a theoretical and experimental investigation of
frost formation from humid air to a .metal plate during laminar frea
convection,
Studies of fluid flow in conduits with internal solidification
are made by Brush (21), Chen ana Eofasenow (22),
Veynik
Hirschberg (23),, and
{2k}.
Brush discusses the principles governing the freezing of water
in main*.*
Be divides ice into three kinds, based upon the manner in
which it formsy ±,e0}
surface 2.oe, frazil ice, and anchor ice, Surface
ice is formed by heat conduction frcm water to a heat sink, Frasil ice
is the form which appears in running water when the temperature of the
water falls 'below the freezing temperature, and where an ice sheet cannot
form due to the agitation of the water. Anchor iee is found attached
to the bottom of a river or stream,
it results, according to Brush,
k
from the cooling of the bottcea by radiation, and the resultant freezing
of the water which com.es into contact with the surfacest which have been
cooled below the f reezing temperature«
Brush states that the formation of ice in water mains is dependent upon the temperature of the water and the flow rate.
If the
wail temperature is reduced below the freezing temperature, a coating
of surface ice is formed on the inside of the pipe. If the flow rate
is reduced through an increased, pressure drop, the water in the main
is more readily cooled, and the ice coating increases in thickness.
It is probable that where the flow rate is high, ^he water is cooled
slightly below the freezing temperature and frazil ice is formed which
eventually clogs the main, stopping tne flow and the whole mass of
water in the m a m freezes*
Brush states further that freezing of water
.mains, which lie above the frost line^, can be prevented only by having
such a .mass flow rate that tne extent of heat, removal is not sufficient
to cause the water temperature to fall below the freezing temperature„
No analytical or experimental observations are presented by Brush,
Chen and Rohsenow present a combined experimental and theoretical
study of the heat, mass, and momentum transfer inside tubes in which
the condensable component of a gas mixture is removed from the stream
by causing it to solidify as a frost on the tube inner surface. 'The
transient formation of frost frees a turbulent gas stream is considered,
The authors show that the surface roughness of the frost is the single
most important factor in the determination of heat transfer and pressure
drop. Their theoretical predictions are in only fair agreement with
experimental results due to the assumptions utilised to solve this
,
extremely complicated problem, and to the fact that the physical properties of frost vary greatly -with the rate of formation.
Hirschberg analyses the complete freezing of pipes through which
a liquid flows. By assuming laminar flow, constant inlet pressure, constant heat transfer coefficient between the liquid and the ice layer,
resistance to flow inversely proportional to the decreasing mass flow
rate, and solid-phase shell thickness uniform throughout the cooling
length, a relation is determined between the parameters for complete
freezing of pipes. Hirschberg presents no experimental verification of
his theoretical results, but experimentally determined freezing times
for various stagnant liquid solutions in pipes and the pressure resistance of these plugs are presented.
Veynik analyzes the continuous casting of metals. The continuous
casting process consists of pouring molten metal into a water-cooled
mold, and then continually removing the solidified metal from the mold.
Figure I shows a schematic diagram of the solidification of a continuous
casting. Veynik assumes steady-state conditions, constant properties,
all heat transfer coefficients constant, constant temperature of crystallization, and negligible axial heat conduction and natural convection.
Also, by assuming a uniform cross-sectional temperature in zone II (based
on experimental evidence) and a linear radial temperature distribution
in zone ill, theoretical relations are determined for a', z' , d and the
temperature field in the casting for flat, cylindrical, and annular
castings.
in summary, it can be stated that previous investigators have
analyzed convection heat transfer in ducts without solidification, and
MOLD
DISTRIBUTION
FJHHEL
imav
I - PCURIP3- ZONE
II - ZOBE OF SOPERBEAT REMOVAL
III - SOLIDIFICATION ZONE
III3 - LIQUID mZAL SURROUMDED W FROZEH CRUST
17 - ZOME OF r^JLLY-SOLIDIFISB METAL
Figure 1=
Schismatic Diagram of the Solidification of a
Continuous Casting*
1
solidification of materials at rest relative to the heat sink-
However,
there is still much work to be done with regard to solidification of a
material which is in motion relative to the heat sink*
More specifically,
there has been, tc the author's knowledge, no satisfactory analysis of the
effect of the solidification of a liquid flowing in a duct upon the heat
transfer and pressure drop.
The object of this thesis is to present theoretical and experimental studies of the effect of a liquid-solid change of phase upon the
heat transfer and pressure drop in circular tubes. The investigation is
made for laminar flow and steady-state conditions,
otatement of the Problem
This investigation is concerned with finding the steady-state
temperature distribution, radial thickness of the solid phase, and
axial pressure distribution in a liquid flowing through a circular tube
with solidification at the inner wall surface. The liquid is assumed to
have a uniform temperature and a fully-developed velocity profile at the
thermal entrance.; i.e., the tube section where cooling begins. Following
this section the wall temperature is constant and lower than the liquid
freezing temperature, The liquid is cooled as it flows through the remaining tube length by convective heat transfer to the surface of the
fused liquid, which has formed on the tube walio
As the liquid proceeds
along the tube, its mean temperature approaches the liquid freezing
temperature, and the thickness of the fused-liquid shell increases. Due
to this constriction of the flow area, the fluid is accelerated,, thereby
producing a two-dimensional velocity profile, The physical model is
shown in Figure 2.
HEAT TRANSFER
BEGINS A T z = 0
Figure 2,
Sectional View at the Therioal Entrance of a Circular Tube with Solidification,
CD
>
The problem is described mathematically by the conservation of
mass, momentum, and energy equations along with appropriate boundary
conditions. The following assumptions are made in this investigation;
1.
Steady-state conditions prevail.
2.
The liquid flov is everywhere laminar, and is fully-developed
and isothermal at the thermal entrance,
3.
The liquid is Uewtonian and incompressible.
4.
Physical properties of each phase are independent of
temperature.
5.
Axial hea^ conduction, viscous energy dissipation, radiant
heat transfer, and. body forces are negligible.
6«
The liquid freezing 'temperature (Tf) Is constant
7.
The tube wall has negligible thermal resistance, and a
temperature (T ) in the cooling region, which is constant
and lower than the liquid freezing temperature.
8*
The solid-phase shell is smooth, homogeneous, and isotropic>
and has araonotonlcaliyincreasing thickness beginning at
the thermal entrance section0
After Imposing the above assumptions,, the continuity, momentum,
and energy equations are reduced to the following forms, The continuity
relation can be expressed as
i i.r 11.+ .,- = 0
r
3 r
3z
or
2
J„ v *
:
•
cu
The momentum e q u a t i o n f o r t h e r - d i r e c t i o n
9v
r
i
v r ~3r
j£ +
p
3r
is
i3r A
v
^vi a V|
UJ
3.ad f o r the % - d i r e c t i o n i s
fi
9v„
3V'
r —;r—"*• + v -r-~
3_r
2 3?
p
«^
-tJglJ!^
i ~2_
r 9
r V
Sz
3 r/
00
3z2
The "boundary c o n d i t i o n s a r e
(?)
6,::.; = 0
..' d , a )
^,(o,z)
(6)
: 0
(7)
- •:
8 v«
Tr £ • ( 0 , * )
v (r.,0;
= o
» [a - <rf]
P I * . - ; - iv
(9)
^
ti-
-•
The e q u a t i o n f o r t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e iiquiot
a own &t ream froaa t h e t h e r m a l e n t r a n c e i s
7
r
-^- +
3r
'.-g= *[\M* §)]
( - • : • ;
for which the boundary conditions are
l(r,0; , ,
(12)
11
( 6,*) = * f
(13)
The temperature in the solid-phase shell is described by the
equation
i-(*«E) : o
d.r *
(ih)
ar'
together with the boundary conditions
T(B,») ; T w
(15)
An additional relation necessary for the determination of 6(z)
is obtained from, an energy balance between the liquid and the solid-phase
shell. Therefore,
\ir(^
: k
s
F>5'2!
(w)
The analytical investigation presented in Chapter II is concerned
with solving the above equations subject to the prescribed boundary conditions .
CHAPTER II
MATHEMATICAL ANALYSIS
The analytical solution consists of finding the fluid velocity
distribution, the temperature distribution, the radius of the liquidsolid interface 6(z), and the axial pressure distribution.
;he primary problem is to determine the fluid velocity distribution through a duct with variable flow area. After the fluid velocity
is determined, then the temperature distribution and 6 (z) can be found,
But solving the system of equations (l-17) is complicated by the fact
that the momentum and energy equations are coupled by their boundary
conditions, i.e., the varying flow radius 6 (z).
Due to this coupling,
numerical or other approximate methods must be utilized to obtain a
solution.
Fluid Velocity Distribution
It is assumed in this investigation that the axial component of
the fluid velocity retains its parabolic profile throughout the'cooling
region. There are two effects which lead to this assumption0
The first
is that cooling of a liquid flowing in a tube tends to decrease the
velocity near the wall and increase the velocity in the central region.
This produces a velocity profile of the form represented by curve A in
Figure 3«
The effect is studied for flow in circular tubes of constant
diameter by Yang (9) • It depends upon the fact that the viscosity of
a liquid increases with decreasing temperature, Since liquid being
CURVE A: VELOCITY DISTRIBUTION FOR COOLING OF A LIQUID IN A
TUEE WITH CONSEAHT DIAfflSER
B: VELOCOT DISTRIBUTION FOR ACCELERATIWI-, ISOTHERMAL
FLOW IN A TUBE WITH DECREASING DIAMETER
C: VELOCBTSt DISTRIBUTION FOR raLIZ-DEVELOPED, ISOTHERMAL FLOW IK A TUBE WI.UH CdrSTAST DIAMETER
Figure 3 .
A x i a l V e l o c i t y D i s t r i b u t i o n s I n Tubes.
?.ll
cooled in a tube has a lower temperature near the wall, its viscosity
is correspondingly higher there, resulting in a profile which differs
from the parabolic profile occurring in isothermal flow.
The second effect is that the velocity profile of a fluid flowing
through a duct with decreasing diameter is flatter than a parabolic profile . This is due to the fact that when a viscous fluid accelerates
through a decreasing flow area, the increase of momentum must he transferred from the fluid near the wall to the central region.
Such a pro-
file is represented by curve B in Figure 3The above two effects tend to be offsetting in the case of liquid
flow with solidification. For fully-developed isothermal flow in a cir
cular tube, both effects are nonexistent and the velocity profile is
parabolic. But if a difference exists between the mean liquid temper
ature and the wall temperature, such that solidification takes place,
then both effects are introduced.
It is assumed here that the combi-
nation of these effects is sufficiently negligible or that they might
even cancel one another such that the velocity distribution remains
parabolic. Thus, it is possible to satisfy the continuity relation
and boundary conditions ($-9),
but the momentum equations (3^0 will
only be approximately satisfied.
It is assumed that the axial component of the liquid velocity
has a quadratic form
v.,(r,s) - a(z) +b(z)r • c(z)r2
z
where a(z), b(z). and c(z)
(l8)
are to be determined from
v ( 6 }zj
:0
(6)
15
-^f (0,«) -. 0
(8)
16
2 i
v rdr = R2V
vn
'o
(2)
z
The quadratic coefficients satisfying relations (6),
a(z) : 2V(R/ 6 ) 2 ;
b(z) r 0; and c(z) = -2VB /6
(8), and (2) are;
. Substitution of
these coefficients into equation (l8) results in the following expression for the axial velocity component
vz(r,z) = 2 V ( R / 6 ) 2 [ l - (r/*) 2 ]
Equation (19) satisfies boundary condition (9), since
(19)
6 :fiat z : 0 ;
and (19) reduces to
vz(r?0) z 2V [l - (r/R)2]
(9)
From the continuity equation (l)
vr(r,z) 1 - -
J
r -y-| dr
(20)
Substituting equation (19) into (20) and integrating yields for the
radial velocity component
vjr ? z) r 2V ^
^
Q
- (r/ 6} 2 ]
(21)
6
Equation (21) satisfies relations (5) and (7), sirtee for r z 5 and r 1 0,
(21) reduces to
16
vr( 6,2) = 0
(5)
vjO?z) z 0
(7)
Now that the fluid velocity distribution has 'teen determined in
the form of equations (19) and (21), the next step is to find the temperature distribution.
Temperature gistribution
After substituting equations (19) and (21) into equation (11),
which describes the temperature in the liquid downstream from the thermal entrance, equation (11) leeernes
W(R/«)2[l-(r/02] [I 2£-|! -HI
2
r a r a ^+ 1 3T"
l t ? rTF
< 22 >
(0£ r <, 6 ,z > 0)
for which the boundary conditions are
T(r,0) = T Q
(12)
T( &,z)
(13)
= Tf
Equations (22) s (12), and (1,3) car* be put into dimensionleBs form
by making use of the folloving definitions:
T* = (•? - T f )/(? o - T f )
z* : W p
ReT,,Pr z za /R2V
r* : r/R
(23)
(Sfc)
(25)
(26)
6 * -" 6
The quantity T^ is the dimensionless temperature, z
is the dimension-
lessi axial position variable, r-* is the dimensionless radial position
variable, and 6
*
is the dimensionless radius of the liquid-solid inter-
Substitution of the above dimensionless quantities into equations
(22), (12), and (13) changes them to
(2'
*2
T*(r*,0) = 1
(28)
!lP(6*, z*) z 0
(^9)
This system is complicated by the appearance of the unknown 6
. There •
fore, it is proposed that a variable transformation be introduced which
will at least make boundary condition (29) more tractable} i.e., define
nrr*/a*
Applying the chain rule of partial differentiation to the variable trassformation of T*(r*, z*) to T*( n , z*) gives the following relations:
L»r*J * ' 6* L 8 n J.*
2^fc
ay
3 r*
*2
3n
J
18
[
. [ag*-]
3T*]
3 a* J * "
_n_ d6_* [ 3 ^
bz*J
S*
dz*
Ll)n
(33)
J*
After substitution of equations (30-33) into equations (27-29)
the resulting system is
t(l - n 2 )
az *
L an
d
* — —
I , (0 < n < !,**>. 0) (3*0
n 3n
T*( n ,0) : .1
(35)
**(l,s*) = 0
(36)
Not only is the unknown 6* absent from equations (3^-36), but this
system, describing the dimensionless temperature distribution in the
liquid, also has the same form as the system describing the Graets problem, Thus, by means of the variable transformation (30), the system
(27-29) is transformed to the system (3^-36) for which the solution is
known.
It is
T*( n,**) : X
n=0
<UL( n)exp(- *?**/&)
n
(37)
s
The temperature gradient at the liquid-solid interface is
[•• i f ( 1 ' z i
:a
f 0 [-w^wh] ™(< */2)
and the integral of the temperature gradient along the liquid-solid
interface is
^36)
19
#
Jn
OP
3 T*
3n
U,z*)
ds*rlf
X
n=0
E-Cn<(l)^j(Xn}
lX
(39)
- exp(-X ^z*/2)]
The quantities A > C »fi. and (-C R'(l)/2) are given by Sellers,
et_ al.
(5) for all values of a. The quantities A
and (-C R*(l)/2) are
reproduced in Appendix A.
Heat Transfer Rate
The rate of heat transfer from the liquid for a tube of length 2
q ; 2
^X
L"
»*L
&
^7{6>z)]iz
(ko)
From equations ( 2 3 ) , (25), ( 2 6 ) , aid (31)
[• 8 ^(«,»)]=<* 0 -V [ - | f a,/)]
(ia)
If the dimensionless heat t r a n s f e r r a t e q# i s defined as
,* :
.2,.
9 /*K<Vp
then combining equations (ho),
(hi),
ep(To -
V
(42)
and (k2) gives the following rela-
tion for the dimensionless heat transfer rate:
,*
q
I 2
Jo
L" ~ <i»- >j «•
Therefore, the r a t e of heat t r a n s f e r fran the tube can be detewained
<«)
20
analytically through use of equations (**3) and (39)•
Liquid Bulk Temperataxe
An equation for the liquid bulk temperature is derived as follows
From an energy balance applied to a tube of length z
irR^Vp C T Q = q + p
/
Jn
The liquid bulk temperature T, is defined as
rp
-
——
bD ~ 2 /
BTV ^ 0
{k5
Tv rdr
z
Combining equations (hk) and (i*5) and solving for T, yields
T
b
= T
o " V^H 2 V'pc p
(1*6)
If the dimensionless bulk temperature T? is defined as
T* : < T b - T f )/(T Q - I )
(1,7)
then rearranging equation (46) into the form of equation 0+7) and substi
tuting equation (k2) gives the following relation for the dimensionless
bulk temperature j
T* : 1 - q*
(hS)
Solid-Phase Shell Profile
The radius of the liquid-solid interface
from equation (17).
5(z) can be determined
»\
\T'?
3 T
( 6
-
ik
}
s F T <4>z)
U7)
Fraa equation (Jrl); the temperature gradient in the liquid at the liquid
solid interface is
(T - T p ) f 3 'T /-, * n
9T
•v.-
n,M
""a
The temperature distribution rin the solid-phase shell is found
by solving equation (l^) with boundary conditions (15) and (l6). The
solution to equations ilk),
(15)> and (l6) is
r lT +
f
s
fef " V - ^ V
In 6
(50)
and the temperature gradient in the solid at the liquid-solid interface
3T
, (T„ - T...)
?^
,-i
-
6
; • • . * •
^
^
In 6
Substituting equations (^9) and (51) into (17) and rearranging
yields the following expression for the diaiensiotiless radius of the
liquid-solid interface:
[«•<*>] ^ : ^ f V [- i f (I-*)]] (58)
where the dimensicnlsss parameter (f* is defined as
w
5* = kj,(Tf '^,.S' : ,j, ;i - .,.;
(53)
-[Therefore, the radius of the liquid-solid interface can be determined
analytically through use of equations (52) and (38}.
Pressure ^iBtributlqn
A solution for the axial pressure distribution in the tube can
be determined by means of an approximate integral form of the axial
momentum equation.
If equation (k)
is multiplied by r and integrated
from r : 0 to r : 6 , arid if it is assumed that p = J?(z)j then the
following integral form results:
h j f *!?* *lr& = *« T ^ 5 •»)+ - f0& V? r d r
(5*0
Substituting equation (19) into (54) and rearranging yields
1 dp _ j*gS?f 46 ^-B^v f2p dz ~ ^ 5 " dz " ™^T" L
1/d 6\ 2 1
2
*d2' J
(55)
If the dimensionless pressure drop is defined as
*
F :
P -P
'19 ._,
r?/a
(56)
then the dimensionless form of equation (55) is
a£: . J g _ 4«_ + i*r Pi • i(4i_) 1
(5?)
If it is assumed that
Hl&) << *
(»
then integration of equation (57) yields
p*(**) = -±-j(1 - 6 **) * l6Pr I U * )
36*^
(59)
where the quantity l(x*) is defined as
Kz*) -- r
do
z
(6o)
—j
6
Therefore, the axial pressure distribution in the tube can be determined
analytically through use of equations (59) and (60).
Summary of the Results
The results of the theoretical analysis presented in this chapter
are summarized as follows»
The rate of heat transfer frem the tube is
described by equations (43) and (39)> the liquid bulk temperature by
equation (48), the radius of the liquid-solid interface by equations
(52) and (38)^ and the axial pressure distribution by equations (59) &&&
(60).
Solutions to equations (3&), (39)> and (66)
aid of a digital computer*
were obtained with
Equations (38) and (39)t which involve infinite
series,, were evaluated by summing the iTirst forty terms of the series <
=
Equation (60), which involves an integral., was evaluated numerically by
means of the Simpson method for approximating an integral. After equations (38) ? (393* &Bd (60) were evaluated,, solutions to equations (43)?
(^8); (52)j and (59) were obtained merely by substitution«.
Solutions to the above equations are presented in a graphical
form in Chapter IV, and in a tabular fovm in Appendix Ac
CHAPTER III
.EXPERIMENTAL ANALYSIS
In this chapter descriptions of the experiments devised for
studying the effects of solidification of a liquid flowing through a
closed conduit are presented. Two separate experiments were carried
out for this purpose. The first involves the visual observation of a
liquid-solid change of phase with flow through a rectangular duct.
The second experiment involves the measurement of variables of interest;
with liquid flow through a circular tube; so that a comparison can be
made between the experimental and theoretical results.
Visual Experiment
The object of the visual experiment is to observe the solidification of a liquid flowing through a closed conduit.
Such observa-
tions could not only substantiate basic assumptions, but could also
contribute to a better design of the second experiment involving the
measurement of variables. Several questions to be answered by this
experiment are;
1.
What Is the appearance of the solid phase shell?
Does
the solid phase have a monotonieally increasing thickness
beginning at the thermal entrance?
2.
What is the physical nature of the solid phase?
Is the
solid phase homogeneous, smooth, and hard without the
oecurranee of imperfections and gaseous voids?
1/k I N . THICK RUHBER
GASKET (INLET AND OUTLET)
CONSTANTHEAD
TANK
OUTLET
DUCT
>w-
PUMP
Figure k,.
V i s u a l Experiment A p p a r a t u s ,
ro
U1
26
3-
Approximately how much time is required for the solidification to reach steady state; i.e., when does the solid
phase shell-thickness increase no more at any position along
the duct?
In order to carry out the visual observation, a horizontal tast
section of conduit with a rectangular cross-section was constructed. The
two vertical sides of the test section were made of thin copper sheets.,
and the two remaining sides were l/k
inch-thick clear plexiglass strips.
Water entered the test section flowing in a steady, laminar, and fully developed manner. The entrance temperature of the water was uniform, and
the copper walls of the test section were cooled and held below the freezing temperature of water.
The experimental apparatus consisted of a constant-head, tank,
a flowmeter, a long rectangular entrance duet, the test section, a short
outlet duct with a throttling valve attached, and a heat exchanger with
its associated pump and piping*
A schematic diagram of the assembly is
„iown in Figure k.
The function of the entrance duct was to establish a fullydeveloped water flow.
1% had a length of ten feet and a cross-sections
identical to the test section. The entrance duct was constructed from
galvanized sheet metal. The outlet duet was similar iu construction but
only 18 inches long.
The test section is shown in Figure 5• Tanks constructed from
galvanized sheet metal were attached to each side of the test section.
Acetone was circulated through the tanks and cooling system to achieve
a test section wall-temperature below the freezing temperature of wat®r=
VMIT
THERMOMETER
ACETONE
(EACH SIDE)
— ACETONE (EACH SIDE)
WATER
Figure 5e
Visual Experiment Test Section and Coolant Tanks
28
A 3 A HP centrifugal pump circulated the acetone through the
test section tanks and the heat exchanger. The heat exchanger, consisting of looped aluminum tubing, vas immersed in a dry ice and acetone
bath, A by-pass valve was utilized for regulating the acetone flowthrough the exchanger, and in this way, controlling the wail temperature
of the test section The procedure followed in the visual experiment was to first allow
the water to flow through the system at the entrance temperature.
After
the flaw rate stabilized and all air pockets were removed, then the
acetone was started to circulate through the cooling system-
1'he temper-
ature of the test section walls quickly dropped to a value dependent
upon the regulation of the by-pass valve in the cooling system.
Ice
then began to form on the test-section copper walls. Eventually a
steady-state condition was reached when the rate of heat transfer from
the water was equal to the rate of heat transfer through the ice, and
the formation of ice was thus halted.
A discussion of the observations made during the visual experiment is presented in Chapter IV.
Circular Tube Experiment
The purpose of the circular tube experiment is to measure variables which enable a comparison to be made between the experimental and
theoretical results of this investigation-
In order to make this com-
parison, it was endeavored to have the following conditions exist as
nearly as possible within the experimental system;
1. The flew of liquid into the test section (cooling region)
should be steady* laminar, fully-developed, and isothermal,
2,
The tube wall -temperature in the test section should be
constant and lower than the freezing temperature of the
liquid flowing through the tube.
3»
Steady-state conditions should be achieved.
The experimental system is shown schematically in Figure 6. The
system consisted primarily of a constant-head tank, an inlet tube, two
horizontal test sections of different length, outlet piping, a cooling
system, two pressure taps,, and two thermocouples.
The constant-head tank was used to obtain a steady flow of water
into the system.
It was a 55-gallon drum fitted with an overflow pipe
approximately four feet above the test sect.ion and with an outlet at
its bottom.. Tap water was pumped into the constant-head tank from a
l6^> -gallon, polyethylene-lined reservoir by means of a small centrifugal pump. A gate valve was situated between the constant-head tank
outlet and the inlet tube.
The purpose of the inlet tube was to obtain a fully-developed
flow of water into the test sections. The inlet tube was 1,3 feet long
and was constructed from the same tubing as were the test sections,
The length to diameter .ratio for the Inlet tube was approximately 102.
A i/8 inch I.D. pressure tap vs^ located on the side of the inlet tube
and two inches from the end nearer to the test sect ons*
rhe inlet tube
was insulated from its environment by a one--inch thick layer' of fiberglass covered by wrappings of asbestos tape and aluminum foil.
Two test sections of different length were constructed from 1 1/2
inch, thin-walled, hard-draira copper tubing (1-526 in- I.B., 1»627 in,
O.D.) jacketed by 2 l/2 inch galvanised steel pipe.. The test sections
were 29 and 82 inches long -with length zo diameter ratios of 19 and
53 • 75 respectively.
Copper flanges were soldered, to each end of the
test sections. One-inch thick plexiglass insulators were situated between the test section copper flanges and the inlet tube and outlet pipe
steel flanges-
These connections are shown in Figures "J and 8 =
Interchangeability of 'the test sections and. alignment of their
connections were achieved in the following way-
A steel flange, plexi-
glass insulator, and two copper flanges were first bolted together. Two
dowel pins were then put into this assembly, and each piece was marked
to indicate its position. The assembly was then placed in a lathe and
a hole with the same diameter as the inside dimeter of the copper tubing
was bored. Then the plexiglass was removed and a hole with the same diameter as the outside diameter of the copper tubing was bored through the
steel and copper flanges. This procedure was carried out for both the
inlet and outlet connections*
Water lea&age between the flange and plexiglass interfaces was
prevented by placing small, continuous beads of Permagum sealant between
them.
of 2 l/k
Grooves l/k
inch wide, l/8 inch deep, and with an inside diameter
inch, cut into the plexiglass faces, accumulated any excess
sealant and thus prevented the sealant from squeezing into the water
passagewayo
The test sections, end connections, and outlet piping were insulated in the same manner as the inlet tube*
The outlet piping consisted of a 1 l/2 in* steel nipple, a 1 l/2
in* steel elbow reduced to a 3/8 in. steel nipple, a 3/8 in, throttling
valve, and finally another 3/8 in* steel nipple. A pressure probe constructed froa l/8 in. 0*D„ and l/k
in* O.D« copper tubing was inserted
into the liquid passageway through a hole drilled in the elbow*
The
*—n PLEXIGLASS
ifHZ BOLT (4)
2 ^ STEEL PIPE
INLET TUBE
STEEL FLANGE
COPPER FLANGE
-»TJ h—l-H^KFigure 7.
Test Section Inlet Connection,
PLEXIGLASS
^
^
^
^
^
^
^
^12
BOLT (4)
^
2-i STEEL PIPE
£•
jg
Mi*
-Jx2 DOWEL PIN ( 2 ) - ^
1-
OUTLET PIPE
STEEL FLANGE
COPPER FLANGE
-Hi
Figure 8.
Test Section Outlet Connection.
i^STltL
ELBOW
#
probe is also shown in Figure 6.
It had a solid blunt tip, and four
0.(Ao in* holes were drilled around its circumference one inch from the
tip. When the probe wa^ inserted into the water passageway, these holes
were located at the test section outlet.
The test section -cube wall-temperature was measured 'by two
Minneapolis-Honeywell^ six--inch, Mego-Pak, Type T thermocouples . Thermocouple T, entered the test section through the top and rested against
the tube wall 1 1/2 inch from the test section inlet. Thermocouple Tp
was positioned in the ^ame manner as VC , but was located 3 inches from
the test section outlet. The thermocouple voltages were determined by
means of a Leeds and Northrup millivolt potentiometer (Cat. No, 8686,
G.I.T. 8159^)•
The thermocouple readings were compared with a standard
thermometer (Scientific Glass Apparatus Co,, No. IC&k, Range -10 to+60°C
in 0.2°) at the ice point and at room temperature, and agreed within
0.1°C.
It is believed that these thermocouples .have an error less than
0.2°C throughout the temperature range experienced in this investigation,
Acetone from the cooling system entered the annular space of the
test section horizontally at each side and 1 1/2 inch from the test
section inlet. After flowing through the annular space at a high flow
rate, the acetone exited through- the top of the annular space 1 1/2
inches from the test section outlet..
The cooling system consisted of a one HP centrifugal pump, heatexchanger, by-pass gate valve, and l/2 in. motorized throttling valve.
The pump circulated acetone through the heat exchanger and by-pass line
into the test section annular space. The by-pass and motorized valves,
acting together, regulated the acetone flow through the exchanger, and
35
in this way,, controlled the tubs wall-temperature in the' test section.
The heat exchanger was simply looped aluminum tubing immersed in a
dry ice and acetone bath. The heat exchanger consummed approximately
30 pounds of dry Ice per hour.
The motorized valve was driven by a Minneapolis-Honeywell
M930B Actionatcr motor. Thermocouple T. was connected to a Honeywell Brown Electronic strip chart proportional controller with an integrally
mounted Electr-O-Lina control unit (G.I~T« 6213b) • The control unit
operated the motorized valve, keeping the temperature T., within *Oal°C
of a value preset on the controller 0
The pressure drop of the water flowing through the test section
was determined by measuring the difference in the vertical height of
water columns connected to the inlet pressure tap and the outlet pressure probe. A Gaertner Scientific Corporation cathetometer (G»LT.
28191), which can be read to the nearest 0*005 om., was used to make
this measurementc
It is estimated that the pressure drop across the
test section was determined with an error less than 0.010 inch of wfcer.
The water temperature in the constant-head tank, f , the bulk
temperature of the water leaving the system, T, , and the mass flowrate
of the water were measured in addition to T„ j 3L* ani tine pressure dropacross the test section 0
Temperatures T
T
and 1' were measured with the standard thermometers
was determined by simply immersing the thermometer into the tank, and
T. was determined by immersing the thermometer into a Dewar flask into
which the water from the system was flowingo
in order to check the
methods of measuring T , IV, T, , and T- 9 water approximately 5 C lower
than the ambient temperature was allowed te pass through the apparatus
while the coding system was not in operation. This test confirmed ^hereliability of the temperature measurements, sicca all readings ©greed.
The mass flowrate of the water was determined by measuring with
a stopwatch the time required to fill a one-gallon bottle. The filled
bottle was then weighed on a balance which can be read to the nearest
0.01 pound.
It is estimated that mass flowrate measurements .made in
this manner had an error of less than one per cent.
The procedure employed for obtaining data in the circular tube
experiment was as follows. Before each series of runs, water to be
used was allowed to stand in the constant-head tank and reservoir for
at least 12 hours. This permitted air to leave the water and allowed
the water temperature to become very nearly equal to the ambient temperature . Then the motorized valve controller was adjusted to a desired test
section tube wall-temperature, and water was started flowing through the
systemo
After the water flow stabilized at a desired flow rate* the
cooling system pump was started and time allowed for each of the temperature readings (T , T, > T,, and T~) to "become constant. Then the temperatures^ mass flowrate} and pressure drop across the test section (when
sufficiently large to be measured accurately) were recorded. The water
fiowrate was then decreased, and tine was ag&in allowed for steady-state
conditions to be reached before new readings were recorded. This procedure was repeated for a series of flowrates and test section tube walltemperatures 0
Experimental runs were made with tube wail -temperatures both
lower and slightly higher than the freezing temperature of water.. This
permitted a v&zy important evaluation of T£SB effect of natural convection and variable fluid properties* arid enabled their Interact ion with
solidification effects to be properly assessed,
A discussion of the results of the eircul&r 'Lube experiweat is
presented in Chapter XV, properties of vater and ice are presented in
Appendix £_, and experimental data are presented in a tabular font in
Appendix C.
RESULTS
Theoretical Analysis
Numerical results of the analysis presented in Chapter II are
shewn graphically in Figures 9 $,ud X0«
The dimensionlsss bulk temper-
ature T, , the dimensionless radius of the liquid-solid interface
6*
with the reciprocal of the parameter 9** as an exponent, and the dlmensionless heat transfer rate q* are plotted versus the dimensionless
axial position variable z
in Figure 9.
These curves show that as z
increases, the liquid bulk temperature and radius of the liquid-solid
interface decrease, and the heat transfer rate increases. Also,, if
the parameter T* is increased., then the radius of the liquid-solid
interface is decreased. The numerical results for T? are computed frcm
equation (^8), results for ( 5 * )
& ^ ccanputed from equations (52)
and (38), and results for q* are computed frcm equations (^3) and (39)°
In Figure 10 the dimensionless pressure drop p
sus z
is plotted ver-
for values of the Prandtl Bomber equal to 5*0 and 10,0; and for
T* equal to G*5„ 1.0, and 2.0. These curves illustrate how p* is inw
creased when 2.* FT, or 5* is Increased. The numerical results for p
'
•
w
are computed from equations (59) & ^ (60) °
The most notable result of the theoretical analysis is that the
dimensionless bulk temperature and heat transfer rate (with solidification) are independent of the tube wall-temperature and rsdius of the
liquid-solid interface, Therefore, if T* and q* for the case of T w
I-Oi
1 JH
Lkfll
vC
A
7 \r
T^
,T*
^ - s ^
/
T
\
NN
0*8
_J
PTK.
1
i
ft-fi
U w
^
Tries*)
'
i
Mil
j 11
—j-—t
0-4
!•
„ k
L2k
'w 1
- •
t"' "'! • r
i
j
1
1 !1
if"
I
- —-••-
r-q
Figure 9,
1 ;1 • »
i j*
11 <
i i I
T4 i
MLf
J/U
S]
^ J
0-01
1
J
!
:TT-
/
1
"
0-2
0
0-( ) 0 I
f N
' r ^i_
1'
/
pf I M x
1 i iKf^S
Y\\ \
1
1
1i
/
| \
KV
^ 1/
1
1
I
\
M-
11 i'
i t i t
L j | i
\
Li Li
\
K
•
1
ri
L i t
XX ZAJ M M
\\tf
V
_>LTH
M FXtl
1
— \
1 t 1
L i 1-1
0-1
rTn
i-
T h e o r e t i c s ! Dimensiouless Bulk Temperature, Heat
Transfer, aud Kadius of the Liquid-Solid Interface,
ui
hO
0001
/Igure i.0,
iiitiorecxct* /recsai r e Drop Versus; /-.^is.
greater than T f (no solidification) are defined as
< . - ( V *w)ArV-V
q* : q / , A p e p ( T 0 - Ty)
^
(6S)
then the dimensionless bulk temperature and. heat transfer carves of
Figure 9 also represent solutions to the Graetz problem*
Visual Experiment
The object of the visual experiment is to observe the solidification of water flowing through the test section described in Chapter
IIIo
A photograph of an ice-shell profile formed within the test sec-
tion under steady-state conditions is shown In Figure 11, and a labeled
sketch of the photograph is shown in Figure 12. The photograph is
taken from above the test section, while looking toward the test section
inlet. Floodlights placed above and. below the test section illuminate
the water passageway„
Conditions within the experimental, apparatus at the time of the
photograph were as follows; Water temperature at the test section inletwas about 55 * l acetone temperature in the- test section tanks was about
10 JP 1 water flowrate from the constant-head tank was about 0»5 gallon
per minutej and the Reynolds Number in the inlet duct, based on the hydraulic diameter j was about ^70.
The photograph shows a gradual increase of the ice thickness
beginning at the test section inlet 3 and a very smooth appearance of
the ice-water interfaces. The ice was observed to be hard, clear, and
homogeneous except for the presence of a small amount of air bubbles
Figure 11.
Photograph of Ice-Shell Profile within Test Section of Visual Experiment.
rv>
EDGE VIEW OF
ICE-WATER INTERFACE
AIR KIBBLES AT TOP
SURB&CE OF WATER
TCP EDGE OF
TEST aECTICN INLET
F i g u r e 12o
Sketch of I c e - S h e l l P r o f i l e w i t h i n Test S e c t i o n of Visual Experiment,
tr
UJ
within the lee. The volume of the air bubbles relative to the ice volume
was negligible, The hardness of the ice was verified by an attempt to
scratch its surface with a piece of wire, which was inserted through the
air vent=
A labeled sketch of the photograph is given in Figure 12, The
sketch shows an edge view of one copper wall and,, since the photograph
is taken from an angle, the top and bottom edges of the test section inlet o Since the temperature of the plexiglass top and bottom walls was
not held below the water freezing temperature, the ice-water Interface
had rounded top and bottom corners., The edge of the rounded top corner
is shown in the sketch along with an edge view of the ice-water interface . Air bubbles which floated to the top surface of the water are
also shown«
The ice formation along the entire length of the test section required approximately 20 to 30 minutes to reach the steady-state condition,
This time interval was estimated by simply observing when the growth of
the ice-shell thickness had apparently stopped,,
In generalt the observations made during the visual experiment
substantiated basic assumptions concerning the character and appearance
of the solid-phase shell. The visual experiment was also helpful by
contributing to the design of the apparatus and to the formulation of
the procedure used in the circular tube experiment„
Circular Tube Experiment
The purpose of the circular tube experiment described in Chapter
III is to measure variables which enable a comparison to be made between
the experimental and theoretical results of this investigation. The
h5
variables used for this comparison are dlmansjLonl&ss bulk temperature
and dimensionless pressure drop. Plots of the experimental data and
theoretical results are shown in Figure 13 and Ik.
The experimental
data are given in a tabular form in Appendix Co
The experimental and theoretical dimensionless bulk temperatures
T, are plotted versus the dioensionless axial position variable z in
Figure 13. Curves A, B^ and C represent data taken frcm the test section with an L/D of 19j and curves D^ E., and. F represent data taken
from the test section with an L/D of 53°75=
Curves A and D represent data taken with tube wall-temperatures
slightly warmer than the freezing temperature of water (no solidifieation).
Therefore, the difference existing between the theoretical curve and the
curves A and D is due to the effect of natural convection and variable
properties, which increases heat transfer and^ thereby, lowers bulk
temperature.
Curves B, C, E 5 and F represent data taken with tube walltemperatures colder than the freezing temperature of water. Therefore
the difference existing between these curves and curves A or D is due
solely to the effect of solidification.
The tube waJLl-temperatures of curves C a?sd F are colder than
those of curves B and E 0
Since curves C and F lie above B and, E, this
illustrates that increased solidification reduces the effect of natural
.onvection, and results in a corresponding reduction of heat transfer
and increase of bulk temperature.
It is apparent that this increase
of bulk temperature with increase of solidification is greater for larger z
(larger z/D or lower mass flowrates), and in the case of curve F9
1-0
I
o-Ll ""
J j ys^
0*
-^,
T^\
"~TH
/(GRAETZ SOLUTION)
^^^
p*^v
Bw
^C' I
s^
/
^
0-6
rv
^^
/D
TjsjiPH.
.
SS
/N
F
i
N^
"F
/
0-4
|
^
I
IN
1
LTEST S>ECTK3N: U D*
j
Ny
^v
("CURVE , A: FOR RUNS (1-3); Tw > T|
B: FOR RUNS (4-6); Tw < Tjp
C: FOR RUNS (7-'tt*. T„ <T#
02
*
i
i
i l 7 i i ir
TEST SE
pCURVE D: FOR RUNS (10-12); > > Tf
E FOR RUNS 0 3 - O K \ , < Tf
IF: FOR RUNS (16. 17); TL < T«
1
1
001
Figure 13.
L_i
i 0005
i i
ITI
i
0-01
i—
i
—
-
t m>
005
£• • >.. „ J L _ --*•
0-1
Comparison of Experimental aod Theoretical Dimensionless Balk Temperature.
N
96
fo
— ~--"-
*
'iV
s,
L
i .
,, 1, mOv K
NN
.
1
, _ . . _ . .
——
, ., .
•i i
s.
it,
.
r ^
N
i
.
.
...
0
-^
......
*
i
o —H
-
s
~
_.
— t -^
* a5-*L 1 * _""
"\
\
~t~
*ct—
_ _,
"
—
•
—
111
"^ .
1
al^Ss
^
1
n
•
M
r''
... \
^aVJ Ji ;h^ —
V
\
5
^ ->
' ^
i
""*"
P
— .
,
.|
— —
H
L ,,,
to
"k "^^fc^
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tm
• ^
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.
i
i
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,,
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.
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-
RUN
o
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"V \
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""
—«
RUN
m
.
^ k
9
D
^-"-^——- - - — —
•v
\
n
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1
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-
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"H
-
1ro
— W -
"*""
in,_.
•
—
.— m.
,
^
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.
.
1
,
.
i
.
.
.
1
.
_
p — k — ^
—L__
~~-
,
* "~
,_
,.,
, - I . . i C D "•
•
•
.
1 ...
i
..
I
_.
1
J
f[
1
—
—
1 1,
• -
kB
the bulk temperature is increased to su<oh an extent that it lies above
curve D.
An equation relating the radius of the liquid-solid interface
to bulk temperature is derived in Appendix .Do It is
I/T*
(6 )
r
«r?
L.
dz
(63)
= exp
Equation (63) shows that the radius of the liquid-solid interface is a
function of the gradient of the bulk temperature distribution.. The
gradient is negative with a value of minus infinity at the tube thermal entrance.
It increases as z increases, and approaches a value of
zero corresponding to complete closure of the tube by the solid phase.
The effect of natural convection upon the radius of the liquidsolid interface can be discussed through the implications of equation
(63)•
Near the entrance to the cooling region, natural convection com-
bined with solidification cools the liquid bulls: temperature lower than
what ifl theoretically predicted»
Therefore, the bulk temperature
gradient is smaller, which means that the radius of the liquid-solid
interface is larger than what is theoretically predicted. But further
along the tube, solidification and fluid acceleration tend to offset
natural convection, and a position is reached where the bulk temperature
gradient is larger than what is theoretically predicted.
Beyond this
position, the radius of the liquid-solid interface is smaller than the
theoretical result=
Thus, the overall effect of natural, convection
upon the solid-phase shell, is to make it thinner at small 2
at large 2, , than what is theoretically predicted.
and thicker
A comparison of theoretical diaeasionless pressure drop p' an$
experiaental results for rune 16 and IT is shown in Figure 1**. Theoretical results are plotted for a T* of 1.92 and for Prandtl numbers of 5°0>
7ol2^ 7*28, and 10.0* Experimental measurements for run l6 are p* = 1,6c.
s* = 0» 0333 s Pr ~ 7 c 12, and tf* : 1.92 J and for m a 17 they are p* i 22^0,
z* z 0.0526, Pr = 7.28, and T* r 1,92.
The theoretical p* values corresponding to rums 16 and 17 are
130 and too, with respective errors of -19 and -82 per cento
that the experimental values of p
The fact
are greater than the theoretical pre-
dictions can be partially attributed to the effect of natural convection
currents upon the water flow*
Combined forced and natural convection
in horizontal tubes produces a spiraling flow which increases viscous
shear and pressure dropo
Another factor is that sine© the effect of
natural convection on solidification at large 2
is to constrict the
flow area even more than what is theoretically predicted, the pressure
drop should, he correspondingly larger than the theoretical results. This
reasoning would explain the large pressure; measurements of experimental
runs 16 and 17, especially since the -per cent error is greater for run
17 at the larger z*.
In general, the comparison between the theoretical results and
data tatesn from the circular tube experiment shows that natural convection can produce & wide variance in results, It also demonstrates
that the theoretical result for T^ is an upper bound for liquid solidification in circular tubes, the theoretical result for 6 * is an upper
bound at large z", and the theoretical result for p
is a lower bound
at large z „ E^oation (63) further implies th&t if tlss
WJ&rt&Qa£"
atur« distribution in a ^ircul&r tube -with liquid eolldif iea-1 ic ..,
50
can be predicted or closely approximated., then the solid-phase shell
profile can be determined,
Such an approximation, considering the
effect of' natural convection vithin the foloving liquid, is presented
in the following section.
Semi-Sapir leal Solution
If the bulk teagperature distribution in a circular tube with
liquid solidification can be predicted or closely approximated, then
the solid-phase shell profile can he determined through use of equation
(63).
One »ethod of solution is to approximate the bulk temperature
distribution with empirical results for fluid flow without solidification. Thus, natural convection within the flowing liquid is taloen into
consideration„
An empirical equation due to Oliver {lh)
for laminar, fully-
developed flow entering a circular tube is suitable for such a solution
It is
(Hn)^ : l.TSfrjt vj vj0'*
[*A* * 5.tacf*fer Pr zfvf1]
'
(ft
where
.*
2
-*:< f c >«
(66)
2 *a"(»i) m
The material properties occurring in equation (6^) are to be evaluated at
the average of the inlet and bulk temperatures, except for ^ , which, is
to be evaluated at the tub© wall -temperature.. 'Eh® temperature different®
%l
in the Grashof Kumber is defined as the ayeraggs of the inlet and bulk
temperatures minus the tube wall-temperature. Derivations of the relationships between 3V, q , and ($u}, ^ are gives in Appendix E«
A comparison between circular tube data with ao solidification and
results calculated from equations {Ot-^S}
is shown in Pigure 15* rIb.e
difference between the calculated end experimental dSmensionless bulk
temperatures ranges from I*1*- to 7-3 per cent.,
In order to illustrate how natural convection affects the solidphase shell profile? a seal-empirical solution was carried out for the
following hypothetical problems
Water enters the cooling region of a
tube (1.526 in I = P O with a tempe.ra.ture of 2&°0S and, the tube walltemperature is colder than the freezing temperature of water so that an
ice shell is formed at the tube i-^aer surface,
Equations [&h-(£)
were used to calculate the bulk temperature dis-
tributions shown in Fi.gij.re 16 for inlet Reynolds lumbers of ;-00; 1000,
and 1500. Since the bulk temperature curves are intended to approximate
those for the above: hypothetical problem^ a tube wall-xamperature equal
to the freezing temperature of water was assumed for
the calculations.
The difference between the Graet? solution and the curves A, B* .and C
of Figure 1,6 is due to the effect of natural convection and variable
properties.
Warnerleal results for the radius of the ice-water interface9
corresponding to the bulk temperature curves A., B, G of Figure .1.6, are
plotted as curves fr, E, F in Figure 17
The numerical results were ob-
tained by approximating the bulk temperature curves of Figure 16 with
a series of second degree polynomials, and then substituting the gradients of the polynomials into equation (63)«
The gradients of the balk
1
ru
'
i
1
^ D » C T 7
GRAETZ
_5bL
^ - ^"
;T^
0-81
^
pvN
/
H—A
[
rb
'
~5jb— L-
p
V
VvJ
1
A
5 fjT-i
HJ
fin
1 fv
&
^
02
O
\TT
fD
i
I
TEST SECTION: L/D « I9LJ
CURVE A: FOR R U N S < I - 3 >; T«> T # - H
[• 'w 'f
li(64-66)_J
^
:J*£^
0-4
1
1
C*f\t
/
"*-
z£ y
0-61
Tl
S O L!._.—
ui ivn
^
XI
k
ns.
r
i
.. i
i
j
1 ^s_
1 ^s_
i
^J
•
^W
it
TEST SECTION*. L/D * 53-7 gl
I—CURVE ft P O » miwfi firs— 12):
Tlyg >
T*
H
t€»ff
^
*§
0-C
xa
Figure 15.
0~<3d"
r ™ .*
(
005
0-i
Comperison of Experiments! Data and Calculated. R e s u l t s Using Equations (64-66)
Njl
ru
^
0-8
- TXl
\ ,
^
0-6
%»
H
-IT
H
Pi JMLGRAETZ
Tr HN
L U
N
Ml
1 1 K.
x
}
0-4
LD»
SI
\W
M
1-526 in.
L T 0 - 28* C
U T - = 0*C
0-2
0
I
1
1
'
ICURVE
0001
i
i« » 500
rr\Jn
no
[C:
FOR R»Pn = 1500
I ' l l
Figure 16.
"•[-»
o~
M. .I
W W
m mnn
0-01
\
pi
H J{
txw\M
x}AI i
cW
L
1
i ~i" i • i
rMl
IR X
u
IL . a*
O'
\
rp\ ^
Ln
Ml
01
^
^
. i. i i n
1-0
Balk Tempers tare Distribution for Water Flow in a Tube with Ho Solidification.
I-O,
-••••
1
•
1
c
T
55^
C
A.Q
U O
^ • ^ ^
^
r»-fi
\J D
/
j THEORETICAL!
> ^
r>
V»L
>
^
vlV ^
v|
0-4
CURVE
•
u
~
•
- £F"
rF O
\jr\R
"P»an
r
'
I
v*l
^
%
0-2
n
v\
:
11
F1
S
WrUi\T
J
in
pF* FOR Rep = 1500
0
0001
I........I
Figure 17.
1 1..1 i n
001
0*1
Comparison of Theoretical and Semi-Empirieel Results
for the Radius of the Liquid-Solid I n t e r f a c e .
-•_•
55
temperature curves could also have been determined graphically or by
approximating the bulk temperature curves with a series of first or
higher order polynomials.
The semi-empirical curves Dv E, and F of Figure IT illustrate
very well the effect cf natural convection upon the radius of the
liquid-solid interface. The curves lie above the theoretical curve
for small z , and fall below the theoretical curve for large z . Since
solidification reduces natural convection effects at large z
}
curves
corresponding to the actual solution of the hypothetical problem would
lie below curves D, E, and F,
Thus,, the semi-empirical solution gives
a valuable upper bouad to the radius of the liquid-solid interface at
large z .
The semi-empirical method presented in this section not only
illustrates how natural convection affects the solid-phase shell profilef
but also provides, for large; z
}
an upper bound, which is even closer to
the actual solution than the theoretical results.
CHAPTER V
CONCLUSIONS
The effect of liquid solidification at the inner surface of a
circular tube upon laminar-flow heat transfer and pressure drop has
been investigated analytically and experimentally for steady-state
conditions.
Theoretical expressions for the bulk temperature distribution,
heat transfer rate, radius of the liquid-solid interface, and pressure
drop have been determined from a mathematical analysis. In the mathematical analysis it is assumed that natural convection vithin the
flowing liquid is negligible. The experimental results show that if
natural convection is not negligible, then a considerable variance between the theoretical and experimental results can exist.
Since the theoretical solution for combined forced and natural
convection is extremely diff icult to obtain, a semi-empirical method
for obtaining approximate solutions is developed, which accounts for
the added effect of natural convection. This method can be used for
either laminar or turbulent flow and for fluids other than Newtonian
liquids.
CHAPTER VI
EEcaummsnom
The literature survey prior to this investigation revealed that
very little research has been accomplished with regard to the solidification of a material flowing in a duet* This means that a host of problems can be recommended for future study. Among these are analytical
and experimental studies involving isothermal, convection, or radiation
boundary conditions with laminar, turbulent, or pulsating flow of a
liquid, gas, solution, or non-Newtonian fluid in a horizontal or vertical. duet under transient or steady-state conditions.
It is hoped that the results and observations reported in this
investigation will provide & suitable foundation for future related
investigations.
APPEHDIX A
TABULATED THEORETICAL RESULTS
and constants (-C R!(l)/2),? which were used
The eigenvalues X
tc evaluate equations {38) and (39)
> appear in Table 1. They are repro-
duced from Sellers, et al. (.5)*
Table 1. Eigenvalues and Constants Occurring
in the Graetz Problem
Q
X
(-CnB;(l)/2)
n
0
2,70k36kk
6.679032
1
10
2
o 67338
Ik
067108
3
I8066987
k
For n g r e a t e r than k,
X
n
(^E;(I)/£)
= kn
0.7lt879
0,<pkh2k
0.1*6288
0.^1518
0038237
8/3
z L o i a ^ A ^
3
The dimension.less heat transfer f ram the tube q ', and the diffieasionless liquid bulk temperature Tf are tabulated in Table 2 for various
at
J&
M.
values of the dimansionless tube length a . The values for q and T h
were computed from
f
Z
[- -if (1'Z#)] dZ* "" k ^ D-W 1 )/*] (» „)•*[!•«»(-» ^*/2)]
q* Z 2 f
" [-±f
(1,8*)] dz*
(4 3 )
m *
T
b
: 1
"
Table 2« Dimensionless Heat Transfer and
Liquid Bulk Temperature Versus
Dimensionless Tube Length
_1
t
1_
0.001
0.02267
0.97733
O0O025
0.005
0.01
0.025
0.05
o.o424o
0.06703
0.10480
0.18591
0.2821.4
0.95760
0.93297
0.89520
0.8lJf09
0.71786
0.1
0.2
0.3
0.5
0.7
1.0
0,41949
0o60299
0.72473
0=86670
0.93496
o o977i5
0.58051
0.39701
0.27527
0.13330
Oo06504
0.02285
The dimensionless temperature gradient in the liquid at the
liquid-solid interface, and the dimensionless radius of the liquid-solid
interface are tabulated in Table 3 for various values of the dimensionless tube length z . The values for the dimensionless temperature
gradient and radius of the liquid-solid interface were computed from
[- if(i^]
:2
% [-C,R;W/£] « ^ ' * t o
[6 *(«•)] l/v : axp £-1/ [- ±f (i,«*)] J
:3s;
(52)
ooooo
ro ro ro ro ro
own $?U3 ro
ooooo
ooooo
ooooo
ooooo
ooooo
o
ro ro H H H
H H H H
ONVn ^ U J
H
H
O O O O
ONVn p - ( j U
O O O O O
H O Q O O
o v o 00-3 ON
O
O
P O V O
CO-4
M O O O
O\0
CO-O
vn
O O O O O
O O O O O
O O O O H
h* \~* h-1 b-1 I-*
l-» I- 1 H fO rO
UJ u o U ) U i - P
4=*
VH ON ON ON ON
CD O PO vn=-4
=0 ~ J ~>j -v] CO
O U> ONVO fO
• P U> U> U l CO
COVO VO VO Q
T \ 0 4^'VO 4~'
O H
v n ON cp H v n
H
0 \ 4 ? O O
NO h~> ON-P'VO
jO £~' ON CO O
US 0 0 - P ' - P ' - > 1
=<] O v n 5-J CO
U.)
~3
^
CO VO O H ON
U) ro vo vn J?
Ul -f" O MD O
o
O O O O O
ro p-.~j ro-4
O 4^ co ro vo
COr
vo UJ ro Q co
O O O O O
ro
O O O O O
ro ro ro ro Fu
" 4 VO O H fO
v p w * ,o v n c o
P H L O CO-4
O VO 4-" --3 - 3
=P* Vn ON , _• • C
U i P O
H H* PO ro
O O < 0 COVO
rovn vo 4^vo
O =-3 -=4 CO iO
j
=r I vo o ON
fO U> 4=*
vo v i ro H o
o\\o \ o o o i
vn w v i H H
o co-^ r u i
u> 4^vo -p ro
\ O O W W f
O O O O O
O O O O O
O O O O O
CO
H
4^
-VI
U3
4r-^-4^4^4^
O tO 4^ Q\VO
H H CO ON Q
vn vn vn
U> UJ
CO 4 ^
Q ~4
- 9 CO
v n VO
W <LO
ON CO
-P* PO
- 4 CO
TO P"
*q ro <-4
ON H
Oo vo co ro co
CO CO 4T O VO
--5-J g y P * - ^
ON ON
H -p* oo ro
„0 ^ M ^_i
-j
•P*vn ON—3 co
H
^oooro
CO CO CO»<l LO
Covn vn —3 ON
O T O H COVn
v o ON ro H v n
4^
O
-4
vo
vn
ow
•f
Table 3 . (Continued)
2
- TrH
1
^
)
[S
(z )J
• w
0.27
0,28
0.29
0.30
0.31
0.56059
Oo54003
0052029
0030133
0 0^8311
Oo16799
Oo15696
00X4631
0.13606
Oo 1 2 6 1 9
0,32
0.33
0.34
Oo35
0.36
0*46558
0.W72
0ol*3250
0o4l68<
Oo40184
0.11674
0010769
Oo09905
0.09083
Oo08303
0.37
Oo38
0*39
o,40
0o4l
0038735
0*37339
Oo35995
Oo3h€99
0.3*51
Oo07565
Oo06869
Oo 06215
Oo05603
0005032
0.42
0.43
0 03221*8
0031089
0029971
0 028892*
0027856
Oo04501
Oo C&009
0o03556
Oo03140
Oo02760
0026855
0.25890
Oo2^960
0 .24063
Oo02414
Oo02102
Oo01820
OoOI567
o.kk
0<.k5
0.46
0.47
0M
0o49
Oo^O
The integral i(z') 1B tabulated in Tables k$ 5, 6, and 7 for 5*
equal to 0*5> loO, 1»92^ aad 2.0.
The values for l(z*) were obtained hy
application of the Simpson method of approximating an integral to the
relation
62
Table *.
l ( z * ) for T
= 0,^
#
itf)
z
I^Z )
0.002
OoOO*
O0OO6
O0OO8
0.01
OoOQ2510
Oo005^22
0o0Q8580
Oo0X19^5
000X5^98
0,23
1.825*1
0 .25
0o27
0o29
Oojl
2=31061
2 0 931*19
0,03
o»05
0o07
0o09
0.11
0.0**272
Oo10338
O.1780*
0,27018
0.33
0.35
0,37
0.39
6.30963
8.36013
0.38265
oJa
0,13
0.15
0.17
0.19
0.21
0051932
0068533
0o*3
0o*5
0**7
0.88763
1.13561
0.49
3.7*923
* .83*51
II028067
15-55196
21,97851
31-95:329
*7-95232
Jho5?656
120.51667
l.**222
Table 5-
*
z
l(z*) for T
= 1.0
I(z*)
zJ*
0.002
0.00*
0.006
O0OO8
OoOl
0.003179
0.007*23
0.012*11
Oo018075
0002*387
Qo23
0o25
0.27
0.29
0.31
0.03
0.05
0o07
0.09
0.11
OoO9898I
0.27*61
Oo 5 5 ^ 8
0098033
1,61*91
0.33
0o35
0.37
Oo39
0,*1
25500195
* 6 7 03000
89805915
182206967
0.13
1-15
0oL7
0.19
0o21
2.55176
3.93*20
5<9872*
o.*3
89800739*
220380677
z
9007297
13-79192
0o*5
00*7
0oi?9
I(z
)
21.16608
32099*Q*
5205*721
85.97^5
1*503002
3918.878*
58260.983
166755 *23
63
Table 6,
l(z*) for T v = I.92
z*
z
0.005004
Oo013506
0,025085
Oo039859
0,058042
0o002
0o004
O0OO6
O0OO8
0.01
0.03
0.13
O0I5
0.17
0.19
0o21
Oo44582
1076617
0.05
0,07
0o09
0o23
0.25
0.27
0.29
4.97990
12016942
27o6l282
Ooll
Table 7.
Z
*
O031
l(z") for T
11Z J
JU*)
60.2322
129*0958
27^-3233
598,5343
1327.8074
3048.8804
7321.4847
18561.735
50124055
145400«54
= 2.0
z*
I(s*)
00002
00004
00006
O0OO8
OcOi
Oo005210
Oo014243
Oo026703
Oo042762
Oo 062698
0.13
0ol5
O0I.7
Go19
0o21
80,0906
17608752
390-4637
87304525
200405950
0003
0005
0007
0.50889
2„08370
6005997
0o09
Ooll
15*25734
35065168
Q.23
0o25
0o27
0o29
0,31
4771,7037
11907 o64:i
31^56«967
88779062
270021.25
The dimensioobsss axial pressure drop p*(z*) for values of the
Prandtl lumber esgual to 1.0, 5°0, and 10,0 is tabulated in Tables 8, 9^
and 10 for T" equal to 0«5; 1°0^ and 2o0o
The values p° (35 ) were oom-
puted from
P V ) : M i - « *^)/3o * - i6Pr i(z*)
(59)
Table 80
Dimensionless Pressure Drop Versus Dimensionless Axial Position for T* - 0,5
t.r
1-
••"••
—
•
—
p*(«*)
z*
0.002
0.004
0.006
0,008
0.01
0o03
0o05.
0.07
0.09
0.11
0.13
0.15
0.21
0.31
0.4l
0.49
Pr = 1.0
Pr = 5 . 0
44.501
159-7k
877-01
5953*2
0.91269
1.5631
2.2156
2,8848.
3 u 5744
9 -1.982
O.7.H89
1.1295
1.5292
1.9296
2.3344
0.55125
0.78265
0.98008
1.1654
1.3424
2.8233
4.7588
7»0388
9^7727
13 -045
160999
21.819
P r 1 10
5&566
11=375
180433
27.064
37«53**
50.235
65 0680
136.81
469015
2283.7
13667
19.646
32.676
48.673
68.146
91»78l
.120,50
252,18
855-91
4042.0
23308
Table 9° Dimrusionless Pressure Drop Versus Dimensionless Axial Position for T* • 1.0
P*(z*)
*
Z-
0.002
0o0O4
0,006
0.008
0.01
0.03
0.05
0.07
0o09
0.11
0*13
0.15
0.21
0.31
Pr = 1.0
1 o2690
1o8739
2.4208
2.9500
3^772
9 01714
17 ° 862
30*499
48.864
75^618
Ilk 0 85
173*03
607^3
7581.4
Pr : 5 . 0
1.4724
2.3489
wy
OlC.JL^xJ^
4.1068
5*0380
15.506
35 ^ 3 7
65.998
111.60
178097
278.16
424.82
1490.5
16881
P r r 10
I.7267
2.9428
4,2080
5*5528
6.9889
23*425
57<-4o6
110.33
190,03
308 d 6
482.30
739*55
2593-9
28505
Table 10.
Biue&siofcless Pressure Drop Versus Dime as'legl e s s Axial Position for S* = 2.0
_ _
z
0.002
0.004
0.006
0.008
OoOl
0.03
0.05
0.07
0o09
0.11
0.13
0-1$
P- (z ). .
Pr = 1-.0
Pr z 5-0
^06322
6.0483
8.5750
11.315
14.325
66.493
3,9656
6.9597
10.284
14.052
l8o338
99*062
196.31
49Xo08
1136.1
252806
5f;38o8
12139
329067
878.92
2122*6
4810.4
IQ665
23449
Pr - 10
4.3824
8.0989
12.420
17**73
23-35^
139-77
496.36
1363.7
3333-1
7662.5
17072
37609
APP.EHDXX B
PROPERTIES CF WATER MB ICE
Property values for water and ice, which were used in the experimental analysis, are tabulated. 1 B Tables 11 and 12 for various temperatures o The property values are taken from Dorsey (25)» and are valid
at a pressure of 1 atm. The density values are taken frcm Table 93
(section III) of reference (25)* the viscosity frcm Table 82, the specific heat at constant pressure from Table 113> fc&e coefficient of volumetric expansion frctt Table 97, the thermal conductivity of water froo
Table 130 (data of Barratt and Kettleton), and the thermd conductivity
of ice from Table 208 (data of Jacob and Erk) 0
Table 11. Properties of Water
T
P
jLTan
0
C
0
5
10
1"
20
25
30
ft
3
62,k2
62.11.3
62 Al
62.38
62*32
62.2k
62,16
\i
c
3 x 1CP
p
Ibm
hr-ft
k.'3h0
3.675
3.169
2*770
2o?44l
2ol63
1=935
B
1
X
*z
B
lbto-*^
hr-ft-%
2.008
1.0C&
1.001
1,000
0.999
o 0 99B
0.999
0.320
CU ?25
Oo330
0-33*5
0.339
00$A
0*939
2.00
__A^
.........9*^2-
67
T&ble 11
E
J
VX i O
o
J»A. C
xu
S6C
0
5
10
15
20
25
30
1*931
1.635
iota
L2'%
1.088
0.965
O.865
ont issued
a*; 10 5
Pr
rt^
sec
Q.lkXk
0AW>
0,1^65
0.1^90
0.15:A
0 8 1538
0,1561
13 066
11,35
.63
,28
• 19
>28
eft.
Tta&xn&l Go^dustivlty of l e e
m
Orj
& ^ B / i i l * - ! T>.°P
•10
lo^94
X O ' .T'bk,
-20
-30
lJ*<&
loVft
68
CIRCULAR TGEE EXPERIME!»L
M m
Experimental daica for water flow in tubes with length to diameter
ratios of 19a^i53°75
T
ar
® tabulated in Tables 13 and Ik.
In the tables
appears as the average of Tn and T9<? Re n is the inlet Reynolds Bunber ,
and the material properties occurring in z* and Pr are evaluated at the
average of T
and Th<» The thermal conductivity of ice occurring in T
is evaluated at the average of T
tivity of water occurring in T
and
and Tf(0°0)5 and the thermal conducis evaluated at the average of (T, + T. )/&
Tfo
1?. o Experimental Dat-a for Water Flow in a
with L/D = 19
2
T
w
p
°G
°C
°C
On
°G
2.20
2o20
2°33
2*K8
23 0 3
2 0 oO
28,7
28o9
29o0
22.5
21=0
18.6
27,8
27-7
27° 7
23°2
28*4
28.0
T
Run
l
1
0»40
2
0ol+0
3
0*79
T
1.3
1 =3
1,6
h
-7-21
5
6
-7°o8
-7°08
-5*50
-6,6
-6.5
-6,3
7
8
-19*70
-19-70
- 1 5 062
-3.7-7
-5.90
-5°90
-15°89
- 1 7 08
210 5
T
0
£9
Table 13. (Continued)
RUB
1
ih
llm/miia
3
5 .19
3^3
1*92
k
5
6
ifo96
3.6l
2.1*6
7
8
1**95
3o62
9
#
Re
D
*
m
z
*b
1561
1C&0
583
0.008^6
Oo 01275
0.0226
O0857
0.797
O08IO
10$*
725
0.00879
Oo01207
Oo 01762
0,758
0-.671
1^93
1078
51h
0.00883
Oo01201
0o0226
0„8l7
O0768
0,62*9
m
"V
--
O0672
0-932*0,923
0,896
2o50
2 056
2 «59
Experimental Data for Water Flos? in
HoriEaat&i Tube with L/D r 53 075
2
Run
OQ
10
11
12
0*58
0,58
O066
T
V
T
0
Mf
%
On
°0
°c
2o59
2.59
I069
1.6
1.6
1.2
17*9
15 •*
9 06
28-0
28.2
28 o 0
13
lh
15
-80 00
-8.00
-7 "71
-5-37
-5-37
-6,16
-6.7
-6.7
-6,9
15 o4
12^5
8,8
26 o2
26 „k
2.6.9
16
17
-13.7 1 *
-13*87
-11061
-12*13
-12.7
-13.0
14 0 3
12 oO
26,5
27 o2
Table I H . (Coofcioued)
(P0-P)
Pr
0.177
0.990
7-12
7.28
Run
16
17
p
.
Table lk»
Run
m
ITm/min
10
11
12
5-18
Re
[Coatimi«dj
#
D
T*
z
0.0237
0.03^5
O0O679
Go 6l8
0.519
0.31^
—
1»783
15^0
1060
530
13
1^
15
5.18
3 = 37
1.892
ikTT
966
5h6
0.0235
O.0360
0=0639
0.588
o.kik
0.327
1.01
1.01
1.02
17
3.65
2.31
10^7
_6J2
0.0333
0*0526
0.^1
0.¥H
1.92
1*9?-
3*5h
iUPPEBPBC D
DERIVATION CF THE EQUATION RKIATIMG
6
TO T.
b
An equation relating the radius of the liquid-solid interface in
3 to the gradient of the "bulk temperature is
so1
presented in Chapter IV. It is
dri;.* -.
z exp
2
(63)
[•
d2* J
i s derived as follows.
From an energy balance
len{£th z
TTR^
pc T
P
=r
z
r
9T
r^-T-^R
2 TT RdZ +
P
X'
W0
IV 2 * r d r
(6?)
sea as
T>
WV
^0
and solving for X. yields
2%,
1^ = T +
o
m pe
P
z ,-8T
^o
8 r
[R,z) dz
An equation for the temperature distribution in the solid phase is
«> + (T - T ) te.fr/.S!
x
f
^f
V
in 6*
(68)
72
and the temperature gradient in the solid phase at the tube wall is
T -T
f
3T
'
•
:
8 (R,z) i —
3 r
(69)
R In <5
Substituting equation (6$) i n t o (68) and nois&ijKtJsion&lis&iBg y i e l d s
1 + 2T
^Jn0
In 5
*
(70)
D i f f e r e n t i a t i o n of equation (70) gives
d-P*
2T*
p _ _jw_
&z
(7i)
Ira 6
and rearranging equation (71) yields the desired relation
co
Equati
for fluids
£T*-|
w r exp 2/
(63)
(63). can be used for either laminar or turbulswfc flow^ m
than Ifewtonian liquids»
The primary assumptions are
heat conduction, constant density and spec&fic heat^
e distribution in the solid phase vhldtl can be suitably
described by equation (50)°
73
APPEHDIX: E
DERIVATION OF THE REIAT103SSHIPS EE^WKSH T*< c * . ML.
x
p* * '
'am
Without Solid.if i^ation
The relation between 3^ and <f without solidification is derived as
follows. Froa an energy balance applied to a tube of length z
p
/*% K
7T R T p c T
: q + p c
/
Tv 2 ir r d r
M
p O " M Pfc/Q
2
(72)
'
Vl
The liquid bulk temperature is defined as
R'T
a
^0
Combining equations (72) and (&5) and solving for T, yields
Tfe : To - q/ , R2V p c
(73)
The dimensionless bulk temperature T, and heat transfer rate q without
solidifieation are defined as
%t : (Tv- T )/(T - I )
b x b w/,r * o
w
q* I q/ ITR^V pe (T
*
^
P
- T )
O
(61)
(62.)
¥
,'iearranging equation (73) into the form of equation (6l) and substituting
equation (62) gives the following relation between T, and q*,°
T* - 1 - q*
relation between q and (Nu)
(7*0
without solidification is de«
a?31
rived as follows. The arithmetic mean heat 'transfer coefficient for a
circular tube of length z is defined as
Combining equations (62) and (75) yields
#
q
h Dz
SU&
: - j 2R Vp e
[(Tb - V/(T 0 - "i¥) - l ]
(76)
The a r i t h m e t i c mean Kusselt Number i s defined &t
fu)
z h D/k
'an
am '
Combining equations (2k),
(61), (77) > and (76) yields
q* r ^'(Hu^CT* • l)/2
Substituting equation (j4) ifflfeo (78) and solving for q
(78)
gives the
etwee'a 3T and
D
q* : 2z*(«a)J
Substituting equation (jh)
[2 * a*<»0 ]
into (79) and, solving for T? gives the follow
„*
ing r e l a t i o n between T, and ( l u ) . without
D
_* _. r~
(79)
solidification
ivl T.
*
[a-«*(*»)«]/[«••*(*)«]
Cto)
75
With Solidification
The relation
between ST* and q with solidification is derived
in Chapter II. It is
j., ~ x. — q
where
% - t \ - T f )/<«0 " T f )
(W)
q* I q/ir ifV pc»p(To - T f )
(k2)
The relation between q* and (Ku)
with solidification is derived
follows. If the arithmetic mean heat transfer coeffieient for a tube
length z. with solidification is defined as
h
*i :q/"Dz [K+ V / 2 " Tf]
then combining equations (k2) and (8l) yields
q* :
J « L _ r(T _ j. )/(T .T )+ fi
2ITV pe
L
D
r
°
(82)
*
If the arithmetic mean IBusselt Number is defined by equation (77) with
or without solidification, then combining eguatlooe (2h)f (Vjf), (77).?
and (82) yields
9* : # { u u ) m ( i * • D / 2
(83)
Substituting equation (48) into (83) and solving for q* gives the following relation between q* and (Mi) ;
?6
q* : 2Z*(m;)a]n/ [2* A f c ) ^ ]
W
Substituting equation (48) Into (84) and solving for T. gives the following
relation between T, and, (Hn)
D
with solidification J
cOi
It; can 'be seen that if a*, T*, h > and (Mi) are defined by the
^ ' fcr am
asi
equations in this Appendix,, then the same relationships exist
^JJ ft*> GUB& (Mu)
between
with solidification as without solidification,
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•
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81
V XJMHL
Ronald Dc ZerKLs was 'bora on Sovember 3> 1937,? A& Springfield,
Ohio. There he attended elementary and secondary schools, graduating;
from Northeastern High School in 1955 «
He entered the University of Cincinnati in 1955 and received a
degree in Mechanical Engineering in i960. While at the University of
Cincinnati he participated in the co-operative program, and thereby
gained two years of industrial experience at the International Harvester
Company in Springfield, Ohio
In i960 he entered the Graduate School of Northwesters University
and received the degree Master of Science in Mechanical Engineering in
1962, While at northwestern University, he received support from a
Walter P. Murphy fellowship.
During the summer of 1961 he was employed as a staff engineer in
the Chemistry Division of the Argonne National laboratory, and during
the summer of 19&2 he was employed as a research engineer at 'the Borg
Warner Research Center in Des Plaines, Illinois.
In the fall of 1.96?. he entered the Georgia Institute of Technology
to work toward the Doctor of Philosophy in Mechanical Engineering„
Sine®
then he has been a graduate teaching assistant and has received support
from a Ford Foundation fellowship.
He is a student member of the American Society of Mechanical
Engineers, and a member of the Societies of Pi Taw Sigma, Tau Beta Fi^
and Sigma Xi«
ID September, l$6l,
he married the former Sandra S. Sender, asd
they presently Imve two children, Mary Lisa a;ad David Karl*