CALIFORNIA S'l'ATE UNIVERSITY~ NORTHRIDGE
LAMINAR FIU1 COt."'DENSATION ON THE OUTSID:t:: OF
"
A HORIZONTAL CYLHWER v7ITI:I FORCED VAPOR
FLOl'l AND NONCONDENSABLES PRESENT
A thesis submitted in partial satisfaction of the
requirements for the degree of Haste.r of Seienc.e in
Engineering
by
Donald Leo Eddy
/
June, 1976
..
,--·-~,--·~~--~·-··"-··~---·-~-~-···~"~--·-,~---~--~--·-~·"·---~---- ~--·--~--·-··'"·'-'"'~~--~··-~·~~--·----~---~"'"'!
I
I
l!
I
I
I
;
i
The thesis for Donald Eddy is approved:
.,
l
I
California State University, Northridge
June, 1976
'
...
r~·-,___
•
...
. .,.........-~~----·-..--......._.~........~~·. .~...,.,...-.-·~-·"'-"'"'""·-~~ .. ,-"'~""·-----=-«~·_,,.,,_=....._. . ,_»<!····~~..,...,.~.:.·!! ....~
r,-~,._.._.,,~.,.;ct>:r""""'-~-"'""'""'..,....,., ..,~""~"'~-,,""" ~-,.
II
ACKNOWLEDGMENTS
I
I
In completing this thesis, I have found that I have not only
learned about a small segment of engineering, hut also have grow-n from
l the interaction of the people I have been associated with.· In partie-·
·~'
ular, I wish to thank my advisor Veeder Scuth for his encouragement
I
!
1
and patience and the staff at Californi& State University,
l Northridge.
Finally, I must thank my mom, Iva Eddy, for the guidance and
love she gave me through::mt rny years in school.
the California State University ,
In addition, I thank
Northridge Campus Computing
Network for providing computer time and money.
II
!
.
!
'
L--·-~--------><a>'~·-·------•-•v-. -------•-"'~----·•-•••-'•----·-~-""--"~·----•••><-·-~••-•J
iii
r.rt<.« <.~.....,.~,....
......,,...,..~;,<..,..._._...,~..- •_-,..,~---"' ... -....-,..-~-~.._ ,.,.-xr...,...,...w_>:..s'.-~"-""'""·. ~"" ,,..,. ~ .-.,...,""""'""."'""''"""..,...'"-'"'~.,...,-"="'.,.. r.-.r.•-~-;: .r.t..-..">.R.c...:r•·•-.....il\.,._•·.-..-.....,._,....."""-,...._ > ·.--,u-"*'-'"""''""=..:~.•:•.'<r·•""~-'-"-...._.,
<
I
!
TABLE OF CONTENTS
lI
ACKNOWLEDGHENTS •
Page
iii
j LIST OF FIGURES.
v
i
I
! LIST
I
OF SYlvffiOLS.
vi
I
I ABSTRA.CT
ix
l
CHAPTER
I.
II.
IlL
IV.
INTRODUCTION •
1
THEORY •
6
RESlJLTS AND DISCUSSION
15
CONCLUSIONS AND RECOH.~NDATIONS.
REFERENCES
28
APPENDICES
31
Appendix
--Tabulation of RS!su1ts.
iv
32
..··--"..
r~-·--·--"·---,--·~·-··-"·-~~--
~'""·-·_,;.,_.--~----··-'"~---~?M--=-,----·"-·-~~-"-~--~-~·= ··~··"=.. ··-----...-.,
! '
LIST OF FIGURES
l Figure
I
'
I
l
II
I
Page
7
2.1
Schematic of system. • • • •
2.2
Dimensionless referer,ce heat transfer 't'esults.
3.1
Comparison of predicted to experimental results at the
stagnation ·point . • . • • • . . • • • • • • • . .
16
Comparison of predicted to experimental results at
angles of 45° and 90° on cylinder. .
....
17
3.2
i
l
l
3.3
Graph of Tw(8) for velocities of .96 and 5.8 ft/sec.
18
!
3.4
Graph of q(e) for velocities of .96 and 5.8 ft/sec ••
19
'f
3.5
Craph of hh(8) for velocities of .96 and 5,8 ft/sec.
20
I
v
Diffusion coefficient
Ratio of local to reference mass
tranference conductance
Mass transfer conductance for dry wall
shear at the stagnation point
Mass transfer conductance for strong
suction at the stagnation point
Mass transfer conductance
Heat transfer coefficient for flowing
water to inside tube wall
Pseudo heat transfer coefficient fer
free stream to outside tube wall
Conductivity liquid
Conductivity steel
Length of tube
Mass fraction vapor in free stream
Mass fraction
interface
vapo~
at liquid to vapor
Mass fraction H2o inside liquid interface
Nusault mass transfer rate
Free stream pressure
Nussult heat transfer rate at 8=0
Predicted heat transfer rate
II
I
Numerical heat transfer rate
qexp
Experiment heat transfer rate
(qpred/qNu)
Dimensionless predicted heat transfer ratio
l
L--~---~---~...-...·--·---~-------------. ----.,~-·~· ·----~----..--- ·--..----~-~-----............_._____ ]
v:i
Dimensionless numerical heat transfer
-ratio
Dimensionless experimental heat transfer
ratio
Reynolds number free stream
ri
Inside tube radius
Outside tube radius
Total themal resistance
Schmidt number
Free stream temperature
Tr
Reference temperature
Tsi
Local liquid vapor interface temperature
Twi
Local outside tube \<Tall temperature
Tc
Inside tube water temperature
u
Velocity in x direction
uoo
Free stream velocity
v
Velocity in y direction
X
Direction coordinate
y
Direction coordinate
Boundary taye:r: thickness
e
Angle from top cylinder
p
Density
Dynamic viscoity
Used to find reference temperature
Average heat of condensation
vli
r~---·~·~-~-·-•-•«~.~·o>h'V>--•·•-•«~···-"'' -><~-·~""~····>'• ""'"~•.w=. -~·~·
I
LIST OF SYMBOLS (continued)
j
T
Shear stress at interface
ii
13
Driving force for mass transfer
n
Suction parameter
Percent error between predicted and
numerical results
Percent error between experimental and
numerical results
e:g
Percent error between predicted and
experimental results
_Subscripts
Liquid
Vapor
Dry wall results
0
-1
Strong suction
.v_i::i,i
,---··~·--··-~~~-------~~-····'-~~«-~-·-·-~-~-·--·~··-~~--~~'~'"~"~~~-~
~ '
l
l
!
ABSTRACT
!
!.
LAMINAR FILH CONDENSATION ON THE OUTSIDE OF
A HORIZONTAL CYLINDER WITH FORCED VAPOR
FLOW M'D NONCONDENSABLES PRESENT
by
Master of Science in Engineering
June, 1976
I
A simplified computer solution for laminar film condensation
1
1with
..
forced flow and nancondensables (air-steam) present was modified ·
;
Ito include variable wall temperature and extend the solution beyond
!
I
.
/the boundary layer separation point to the bottom of a horizontal
II cylinder.
Local condensation heat transfer rates for isothermal and
I
l variable wall
temperature distribution based on experimental results
I
f
! up
l
to an angle of 90° on the cylinder were compared to an experimental
I
wo~k
and to a numerical boundary layer solution for the stagnation
l po1nt.
I
1.
It was determined that increasing the velocity increases the
t
I heat and mass transfer rates substantially.
i
2.
Decreasing the mass fraction of ail· increased the heat and mass
transfer rates.
3.
Decreasing the bulk to wall temperature increased the heat and
mass transfer.
I
heat
4.
The percent difference between experiment and theory for the
tr~nsfer
at the stagnation point were in agreement within 7% for
Lal!__~s:_~ t~sted.
t
)
I
---·-·---------··-----··-··----~-----·---·-··-~"-"_J
ix
r-~-5 .-~- T~~~~~r~~nt diff·::·~=c ~-=:~~~n :::::~::-:~~~··~:::::;~~;::,.. ,~~~'"'"'']
l
'
I heat transfer at angles of
I ively
for velocities greater than 5 ft/sec~ but were as high as 97%
I and 139% for
I'
6.
45° and 90° were w·ithin 3% and 20% respect_:
the low velocities (.96 ft/sec).
The variable wall temperature distribution did not increase
the accuracy of the results.
I
I
7.
No error esti:r1ations were
performed on the ove.rall heat and
mass flux rates due to insufficient experimental data for the range of
parameters tested.
l
I
I
I
I
l
I
I
I
X
CHAPTER I
INTRODUCTION
The prediction of the rate of condensation of vapor onto·a
surface is important and useful when dealing with steam power plants,
refrigeration processes, distillation, heat pipes and salt water conversion.
Over many years different models and theories
have been
proposed to aid in the design of heat and mass transfer equipment.
One of the biggest problems that has to be dealt with, j_s
when there is more than one gas present in a system.
Hhen this h.,.p-
pens condensation is usually inhibited.
Different forms of condensation
can take place.
Dropwise
condensation is not dealt with because of the unknown surface conditions and because it usually only occurs until the surface is
cleaned by the condensing vapor.
is usually modeled.
In the literature, film condensation
Film condensation can also be classified as
turbulent or laminar depending on the momentum of the liquid film.
Turbulent condensation theory has not progressed to the same
level of sophistication as laminar film condensation theory, due
to
the difficulty in making experimental measurements (4). Laminar
film condensation, however, can be applied to a wide variety of appli-
1
2
.
.
r"~-~~- ~----·~"·¥~~~-·"=·~"·-,-·--·--~--~"~-=-·"-·~-·"-~·~~=- ·~·~-~~·-·.,··~-····-··"·-
. -····. . .".
."_1
~--··~·""=··-·--"·" ......._.•• __
\
Laminar Film Condensation
II
A short history of the development of the the0ry for laminar
film condensation will be presented.
! the
I
This is done to familiarize
reader with past accomplishments and to show which direction this
I'
! work is preceding.
I
l
'·
Laminar film condensation theory was first presented by
Nussult (1) in 1916, for various configerations with pure steam vapor
with no velocity.
N11ssult postulated that both energy convection
arid fluid acceleration inside the condensate layer could be disregarded, with a balance betv7een gravity and shearing forces detenn_ining the heat transfer.
Rohsenow (5) improved Nussult's theory by using energy convection in his formulation on a control surface using an energy bal-·
ance, which 1·esults in a partial differential intergal equation that
(
I
I!
is solved by iteration.
Sparrow and Greg (6,7) went a step further and included fluid
acceleration and energy convection in their boundary layer formula-
/ tion and reduced the partial differential equations to ordinary dif1
li
ferential equations by a similarity transformation.
These equations
' were then solved numerically.
Acrivos (4) then treated the problem of mass transfer in
1
'.' forced and free convection laminar boundary flows with large inter1 facial velocities directed towards the surface.
i
'!
I
He used asymptotic
expressions to account for the difference in mass and heat tr&nsfer
I when a finite interfacial velocity due to phase conversion is directed
II
<....-_...,.._...,.,._., ... ........_,,.,....~...,_,_..~,-"""'..,.,_,~~•~•..,.., .. ~..-..,--..-•-,.._,.,.,_._._,.,-.c_ _ .....,__...___, ____ , _
I
__..._~...,..,.n,_......,._:>1....__.__,.
,
_ _ ,...,..~-----~--~~ ...-·,..---"""'-"'-,...~--..."'~*"'",.."-''""-,,_,....,.:
3
~--------·-·-~-~--~V~·••·•-••••<>~:-~''""'-~~~~·-•~w-~--~~---~·-~•~•--'•="~·~---·•"•·'~"-''''-''-~-~··•~"''·="·~~•• •-•·•>--.
!
I!
Il
l'
1
toward the surface.
The noncondensable gas problem then was treated by Sparrow
and Lin (9).
It was shown that a very small amount of noncondensable
gas in the bulk of the vapor can cause a large build up of nonconden-
1
! sable at: the liquid vapor interface.
,II
-·-
This lowers the partial pressure
and temperature which is the driving force for the heat transfer, 2.nd
consequently decreases the condensation rate.
The conservation of
species equation could::now be included with the development of boundary theory.
Other papers were written (Shekriladze and Gomelauri (10))
which introduced
laminar film condensation with forced flow in the
vapor, along with using the asymtotic shear to represent the effect
of the flowing vapor. It was found that along with vapor removal in
the boundary layer, the point of boundary layer separation can vary
depending on the incoming free stream velocity.
This can change the
heat transfer coefficient markedly behind the separation point.
Both
vapor suction and the moving vapor liquid boundary help to move the
separation point towards the backside of the cylinder.
Early sepa-
ration brings a sharp decrease in the heat transfer rate behind the
separation point caused by vapor reversal of flow opposite gravity
forces, increasing film thickness and static pressure
decreas~
result-.
ing in a_ small temperature difference across the film.
Denny, Mills (11,12) next analyzed the nonsimilar problems of
i
a vertical plate and horizontal cylinder with forced convection of a
I
pure
i
I
sa~urated
!!
!
vapor using the assumption of asymptotic shear.
!
L_____- - - - - - - -----·-------------------~~-----------·~--~----··--'·-~-~------------~-~
4
t~---·--~~_,3~"""'~-'-1«_.....,_,,,_..._,.~~-'\".r.\0....,,..,..<
..... __,_...
~-~~
..,.,.,_..,.,.,.'1<'-Vr....,_..... ••
l By introducing a modified version of the Pantankar Spalding
i
I
~
"<'>,.LC""'',...,','I"._!"•''·<'•-..,_~~--''"~''"'-r"",.~
(13)
finite difference method they solved the two-dimensional boundary
equations for the liquid film.
1
They simplified the computations for
the more general two-phase problem of variable property, forced flow
1
with matching interfacial conditions by proving by numerical experi1 ment that the reference temperature concept along with the constant
l
;I property
!
}
' i Denny,
Nussult analysis could be used for a variety of fluids.
Hills and Jusionis (14) then applied these concepts to solve
for forced flow down a vertical flat plate for the nonconder;sable
gas problem.
Also in this paper a simple formula was presented to
predict condensation rates.
South (3) then extended the work of.Denny, Mills and Jusionis
and applied their. concepts to a horizontal cylinder.
He used Patankar
Spaldings methods, a forvmrd marching solution procedure based on
finite difference analogs to solve the governing conservation equations to obtain solutions to an isothermal tube up to the point of
boundary layer separation (~Vl08o).
Rauschers (2) experimental works
later verified that South's numerical solution to the problem was
1
I
I within
7% of the actual heat flux for the cases tested up to the point
I
'
of boundary layer separation (108°) on the cylinder.
South also for-
malized the earlier mentioned simplified equation for condensation
I
rates and applied them at the stagnation point on the cylinder.
II
The next logical step in the progress of laminar film condensation is to extend the work of South using his simplified for-
mulation to include variable wall temperature, and solve for the
. total heat and mass transfer rates by including the region between
i
.
i
--·~-----><·----·--------~---------·---"·--·-·-··--·-----"·--·--·~-"'
5
separation (61080) and the rear stagnation point (6=180°) of a horizontal cylinder.
~;
i
.
..
~..,,,..........,.,.>=»..""-"-"""'-""'"-'>·""""'""'~•....,.,..-"""""""" "'"~>"""'""""-~•"'-'""'-"<'""'""""'~~_.,.._,.._,.,-'>"•4"""'.;:~,....,_~a
I
I
CHAPTER II
FORMULATION
The situation to be dealt with is a cylinder in transverse
l dm·mward
I
lA
I
cross flow of steam 'vith a small percentage of air present.
schemetic representing the physical model and coordinates is shown
I in figure 2.1.
The x andy coordinates run parallel and perpenditular:
Ito the surface with corresponding u and v velocities.
!
Cold water flmvs through the tube at a constant velocity so
l
·f that the heat and mass transfer between it and the saturated vapor
I on
I
\
the outside of the tube are at a steady state condition.
A liquid
and vapor boundary layer are then formed around the cylinder with an
j interface
la
i
forming between the condensed liquid and vapor, prcducing
shearing stress between them.
The vapor outside the boundary layer
I
is assumed to obey potential flow theory (Eqn 1), with Tee, Pco, :Mlco
and Uco representing the free stream saturation temperature, pressure,
I
mass concentration noncondensable and velocity.
l
U == 2Ucosin8
I
i
I
j
I
(1)
The tube wall temperature Twi varies with the angle 8 as does
the film temperature Tsi and liquid and vapor boundary layer thicknesses
oi
and ov.
These four parameters then have to be found from
conservation equations and boundary conditions.
Governing Equations:
The Nussult assumptions are assumed
valid in the liquid film, where convection of energy and acceleration
are neglected.
1. . . . . ._'"""'. .......,..,,.=----.-·--.. .- ......
The conservation of mass, momentum, energy and species
_-~.,......-~-........,.
. . . .,..,...._..__. . . . . . ._..
c-o.......---~- ..
6
~---=-,·---O>.--------.,._,...-.,_..,~.,.,.-.,._.....--.~-~"""~ ........ ~~-~·"··.. ~------- . . ------, ..... -.~..... ,~ . .·'
..
I.
Figure 2.1
~
l-.,_~_._.-.;.,........,._.,,~,__..,....,..,......='>...
,"o--<t-..<a,...........
,_~
Schematic of Cyllnder.
..,.,._, _ _ _ _ _ _ ._.. ..___,_,..•...__._, .... _ .__ _. _
_,.,.._..,,=,.~-.-·-._.,,.~...,_.,.....~~--0~"<".......... ·~·~..... ~..-~o-Y...-~•··~-e... -<F'~~'·'>"'""-'-"''""· L~ • • .. ~,..,.,,,.., "~"' ' '«»~~-~ ·--~·
'7
I
"'-'•<
""'''""'•'••oJ
8
#"=-~.,.~.ff.o;,..~'r¥-><'""<_..,..,.,__.._,-..,..,.__..,......, _ _ _,~."""-"'":.ff..,.o:<:,....,...,..,.,~-?~~----~,..-'<W'~--_,_....-.-..-.,-•...,,.._~...._.....,
r
I
I then
__ ,.-,.~~·-'•._""'u•><·h.....,....<"".....,..."""""'W""'"""'"""'"·-l"=~,..•...-.:~=~~·:>-.1<,.~<,·j
j
.
l
are:
·
au1 Jax.+ av1 /ay
I
l
2
a Ut/ay
Il
2
=
l
o
(2)
-(gp sin8/V!-l/u 2 dp/dx)
.a 2 T
/'dy
2
0
(3)
( 4)
l
Second order effects such as thermal diffusion, diffusion thermal
and viscous dissipation and compressibility have been assumed negligible.
The liquid properties are evaluated from the reference temperature concept presented by Denny and Mills (12)
i
Tr
1 where a=
= Tw + Ci: (Tco-Twi)
(6)
• 33 for steam air mixtures.
The equation for heat transfer in the liquid fiL'Il is then
1
q(e)
= K:t(Tsi-Twi)/8!(8)
(7)
The entire temperature distribution in the tube wall will be found by
I
l
numerical finite differencing, using a Jacoby elimination method in
order to find Twi once the heat and mass transfer coefficients have
been determined
q(e) /L
=
hc2
7f
ri(Tw(inside)-Tc)
2TIK
s
(Twi-Two)/ln(ro)
(ri)
(7)
The film temperature Ts(8) then is the unknown to be solved by equat;
I
'
ing the liquid-side and vapor-side expressions for q(e) the heat flux.;
I
'
l
.
l
l
L--~~·~--~----~-----------·-----·~·-~·----·--·-------. ·--·· --.. ···-·------..·---····-······-.J
9
-
q/A.
=
e
J K; (Tsi-Twi) rde
0
]_
=
~
0
'
I\
fo
p 21
0
u2 i(o,Tsi)dy
(8)
is a smoothly varing function of 0
~
at 8=0
I'1 at
8==0
i at
e=1T
d(q/:\ /de ·- o
l
q~
is then approximated by a parabola
q/X
=
a
+
be
+
ce 2
I
From the boundary condition
I
Then integrating th2 L.H.S. of eqn (8)
I
I
fe
I!
o
.
q/A
I Substituting
de
=
(9)
(q/I) e(l-l/3(e/7i)2)
0
eqn (9) into (8) the resulting eqn for
o2
frcm integrat-
j ing
(3) is
,
1
o~ 3 + '(3Tsi/2l1Q.B1 sine)o/-(3(q/I ) 08(1-l/3(e/1T) 2 )/pg_Bl sine)
=
0
'
(10)
where
!
Since the velocity u 1 is a function of o Q. + and
T
(shear stress)
I
j an expression relating the dry wall shear (i.e. Iit+o) Ti,o to the
asymptotic limit of very strong suction (i.e. ,.__.,) was derived by
I
!
ll.
South and Denny (15) for a stagnation point.
This was accomplished
by applying the results from Acrivos (8) expression for mas~ transfer
I
L.~.QI_lgJ,t~t:?l1.q'!e_.,_,••----··--·-·--·-~~•--•~"·----•c.~ --~-~-··--~----·---··•--•·-~----•-•••e•-~·"·'~"•'·"•"""'-·'·
10
r--~,--~--·------~-·~·--#-~
I
'
g = (go3f2 +
..
.......")
·--·~·-···tt~-···~,--~,----~--··---~~--~--··=··.,~¥"~·
;
(~Bg-1)3/2)2/3
(11)
l
i in
l
the same manner to the shearing force
n
n 1/n
0 + '( . -1)
!
'
1.'
"There for the stagnation point
I
(12)
= (T i
T
I
j
I
g
I
I
!
1"
0
=. 1 m g =
~0
li 2 s-6
.81 poouoo/R
~
2pooUoo/R~/ 2 Sc~/ 2 (1/(l +
g-1 =
(13)
Sc00 )(1 +S))l/2
(14)
I
! and
from Schlicting (20)
!
l
l
Iwhere n
=
2
• ')
Ti,o
= 3.486 poo Uoo /Rer 1 / - (x/r)
T.1,......
==
1
-n1 (Uoo-UV1. )
(15)
(16)
1. 3 75 predicts closely the real shear ( 15) •
l
l greater than zero degrees, dry wall shear
Ti
For angles
is found from Schlicting.
'i,o = poo Uoo 2 / Rer 1 / 2 (3.4860 -1.3668 3 + .14685-.009158 7
+
.oooozlse 9 -.oooo575s 11 )
(17)
Vapor Side q
The liquid side problem then is coupled to the vapor side
problem by equating
I
q = -X
l
i for the vapor side to equation (7).
S
(18)
Sg(S)
The mass transfer driving force
= Cm 1 ' -m 1 ,s )/(n1 ,s -m 1 , u>
00
(19)
is known for a giyen Tsi and thermodynamic constraints.
fer conductance is found from eqns (11, 12, 13).
The mass trang-
The interfacial
Tsi is then found by iteration between equations (7) and (18).
l
I
1
To obtain local results for q(8), numerical results (3} for
q/qN
are taken from figure 2.2 by fitting a parabola to the general
ru
l shape.
Il
A small correction factor (the suction parameter) is included
for correlation of g, to account for local variations in the shape
I
_\,,_,__",.....,...._..,. __.......... _,...,.,.,...,.,...............
,
...,.,._~~---------.,--...,_........,.,.,.~9.....,_~,.,..-·,....._~..,..__..._,._,..._~~_,...~•...., ..--.-c._.,._,..._..,,,..,._,..,-_.._,,..__.._._ _ _ _ O',....__ .-..-..,.,-._..<--.o(~
:------·--·'---·----------... ·- ... ....- ....____
.
,---~-.-~--·-···-·---. .
~
-.
1
I
!
l
"
I
1.0
I
I
I
-
i
Il·
!
!l
~
I
o-1
I
a .6 t-
l!
Uoo=
·50 ft/sec
0
T~-T,.,11 = 20 F
m·l· co= ~ 99
'l
'
.
P~= 2116.224
I
I
II
--:....__~~--·
.8
!
...,......
rl'
!·
.4
'
I
i
i
I
I
.2
I
f'
I
I
0
10
20
ij
~
0
I
90
'
0 (0)
t
Figure 2.2
Correlation pata.
•
I
106
I
12
Rer 1 t12
=
(2o)
(Ki3 g pi2 (Tro-Tw)3/4~i RA3(4j3
8
!o- sin
1 / 3 ede) 1 / 4
(21)
(22)
!I The
local mass transfer conductance g is related to the reference g
;
I
; by
t
F(n) -- 2 -(n/nref (Bref/B) 1 I 2 ) 1 I 5
l
l
l where
l
the reference conditions are:
Uco
I
= 50 fps
0
(Too-Tw 0 ) = 20 F
I
I
(23)
m
l,ro
r
0
.99
=
= .03125 ft
!
I
l
I
I values
l
P""
B = .8
Overall heat transfer rates are then calculated from local
by integration to obtain
-
q
I
14.7 psf
= 2R
f
0
7T
q(S)d9
Local values for the heat transfer rate q(S) and mass flow rate
m(S) beyond separation (i.e. 108°) are then determined by combining
j a liquid film resistance o2 /K 2 with a cubic equation fitted between
I
the end points of a pseudo heat transfer coefficient found from
L-~--~-·-
.
r=·~--·<·----~---~--~-~-··--·--...;..-·~·~"·~-~- ~-·~---~~-·--~---~"-"'--'·=•o•-~•·<~~"········~-1
h(8)
1
ljwith
=
rnA./ 27rR 0 (Too-Twi)
(25)
;
end conditions
I
= h(l080)
at
h(8)
at
d(h(8)/d8 = d(h(l080)/d8
=0
at
8=
7f
h(7f)
at
8=
7f
d(h(7r))/d8
=0
l
i then for e > 108° the total resistance is
I
(26)
Rtotal = (1/h(B)
i~ This
was done because the heat and mass transfer rates drop off
I
j rapidly
behind the separation point, and its been determined that
j
! a maximum of 35% of the total heat transfer occurs beyond this point
l depending
on free stream conditions (10).
I
All values for the calculated heat flux q(8) were normalized
to the Nussult heat flux qNu,e=O where
qN-L1,e=o(8)
=
4
(2Kn 3 (Too-Tw) 3Xgpn(Pn-P )/3D]J,q,(--) sin - 4 / 3 8
X.
X.
!<.
V
! 8 sin lf 3 8d8)
0
.J
3
(27)
so that it would be possible to compare the results to existing
experiment (2) and theoretical (3) works.
i
In this paper South's (3) simplified equations are used to
Ii predict
heat and mass transfer
!
I
~ates,
using nonlinear wall temperature
distribution based on conservation equations.
I
The effect of drast-
ically reduced heat and mass transfer rates behind the separation
point is also taken into account as an effective additional liquid
Il
l resistance with known boundary conditions on both sides. The local
I
heat and mass transfer rates will then be compared to experimental
j (2) results at oo,
!
450 and 900 angles on the cylinder and to
'-·-------··---·.·-'-'"""W•>" •'-•·-·---~--~·••·<•> ~----•- ·-•»·---~·--•-••·------··----------~·"
----..·------··----__!
..
14
r::,~~eti·::l -~:=~~~~~t-:,~-~~~~·=~vaJ:es ::-::en-:::ed~·~:·=···j
l
I be
reasonable when the local values correspond to experimental
;
results.
Also to find out if variable wall temperature is pertinent
to the problem, the wall temperature will be held constant so that
the heat flux for both the isothermal and variable wall temperature
distributions can be compared with the experimental
rest.~lts.
A
comparison of both methods based on the experimental results will
determine if variable wall temperature should be used in the analysis ..·
CW\PTER III
RESULTS AND DISCUSSION
The results of the analysis are displayed in tables
A.l - A.3 and figures 3.1 - 3.5.
The parameters studied in-
eluded velocities ranging from (.96- 5.80 ft/sec), saturation
temperatures (97.4- 122.5°F), air mass fractions (.0006 -·
.0598)) and bulk to wall temperature differences of (5.1
30.9°F). These specific input conditions were selected so that
the results could be compared directly with Rauscher's (2)
I
experimental results and South's (3) numerical results.
Initially input parameters were iterated upon until
1 3.1
I
!
I
I
conditions corresponding to Rauschers experimental analysis
~11ere
found.
The conductivity of the stainless steel tube was
taken as 11.0 Btu/hr ft F with a 3/8 inch outside tube radius
and a 1/4 inside tube radius.
The number of nodes used in the
I
l
I
program was limited to 2·0,10. on the circumference and 2 in the
I
radial direction due to increased computer time for larger ma-
l
trixes.
Il
version of the model on the CDC 3170 computer.
I
effect on the heat transfer results, while it was substantial for
I
the circumferential direction.
l
It took approximately 7 minutes to run the 20 node
It was found
that the number of nodes in the radial direction had little
Therefore the largest number of
nodes was chosen for the circumferential direction and the
1'
i
L ..- - · - - ·- -....·------~..---~-·-·-·-_. ...,_ ·-"·-~,·-----~~.........~~,--..- ..,·'"·-·--·<"···· .............- ..~.1
15
16
17
l
l
1
i
I
I
I
.
L-·-·--·---~--·
18
~
!
l
r-a
!
'-0
I
I
I
I
j
I
Ii
Ii
•
~!
,,r.
~
I
Ii
!
~I
t---··-·-··~~,.
"''
-.:;~---~-~~~~-·=·•_..-"
...
-
-
-
.
"'"""'--~~~_..,_...,,....,.._._..~----~-~-cu"'-'-_..,..'*"'..,._.,.........,...
__~><JW·-..r.:~,.,,=r.,,...,.._,~·"•"""'~"'l<~:.'>-1
~
20
21
3. 2
Condensation of steam at
the,~stagnation
point.
The predicted results for the stagnation point are presented in table A.l and figure 3.1 in terms of
~/qNu
8=0).
'
It is seen that q/qNu is reduced drastically when the mass
fraction of air is increased due to the air building up next
to the liquid layer (example runs 1-2).
the resistance to heat transfer.
This in effect j_ncreases
Also higher velocities tend
to increase q/qNu along with decreasing the bulk to wall tempi
l!
I
erature difference (example runs 21 and 22).
always be true (3), but occurred for the cases tested.
An e·.cror
analysis is performed with the percent error
I
l
(q/qNu,8=0)i- (q/qNu,e=O)j
xlOO
(28)
/(q/qNu 8=0).
'
J
where n=l, 2, 3 for En=l, i = predicted result, j
= numerical
experimental result, j
numerical
result and fon En = 2, i
II
This might not
result and for En = 3, i = predicted result, j
·result.
I
and E3 =
The maximum discrepancies were r:: 1
= experimental
= 4.11%,
c2
= -6.23%
+7.14% with absolute average deviations of 1.48%,
4.81% and 4.39% respectively.
The smallest error occurs for <1
as expected because the predicted results are based on South's
(3) numerical solution.
Fmr E2 and
E~,
the predicted, experi-
mental, and numerical solutions are in excellent agreement with
each other.
j
!
j
~
L. ~---~-·-----------=-·----------------------"~---,___________.1
22
..
r·-~~----··-··- ··--·~·-·-~=-~~~·-~h-··-·-""-"'··--~.,-
I.
3.3
...
.
~·-·------·""'~ -·~···"-·--·~···~=·"--~-~~"··
... . .. '"'···
~ .,~
--·······~·-
···-"''''"''·
Condensation from steam-air mixtures at l150 and 900.-
I
l
I
I
I
l
l
I
II
- Condensation heat transfer results are presented in this
section for angular positions of 45° and 900 on the cylinder
(figures 3.3- 3.5, table A.2), and compared to Rauscher's (2)
experimental ~esults.
These values are also presented in t.he
form of (q/qNu,e=O) to simplify tabulation and analysis.
The predicted results ·are compared to the experimental
(2) resdlts using equation (28) for the error analysis.
ing tables 3.1 to 3.2 it is found
Compar-
that q/qNu decreases as 6+900
for the experimental results, but increases significantly for
runs 1-18 and 24-25 for the predicted results.
19-23 still increase but at a smaller rate;
q/qNu for runs
Maximum errors
between predicted and experimental result for angles of 45° and
90° for runs 1-18 and 24-25 are-96.7% and -138.9%. ·For runs
19-23 they are -2.75% and -19.5%.
The reason for the large
errors for runs 1-18 and 24-25 is that the velocities are much
smaller than the reference velocity (figure 2.2), egns (20)-(23).
This effect tends to increase q/qNu as 8-+7T.
Although runs
19-23·are a factor of 10 smaller than the reference velocity,
the error tends to decrease sharply.
In figures 3. 3--3.5 a low
velocity run /118 (1 ft/sec) with a high mass fraction of air
( .0666) is compared to a high velocity run (/119, 5 ft/sec)
with a low mass fraction of air.
The temperature, pseudo heat
transfer coefficient, and heat flux profiles are quite different
·~
2 ~)
I
for the two velocities with larger initial slopes occuring for
!
the low velocity as 8
+
90°, and almost zero slop~s occuring for
the high velocity profiles. iVhen higher velocities (50 ft/sec)
I
were run in the program the beginning slopes were slightly
I
l
!
negative (as they should be) indicating that too high a velocity
!
was used for the reference velocity.
It should also be pointed out (figures 3-2, 3-4) that
q (8) and h (8) increase until
e
=
1080 for the low velocity rn1s
and then are forced to zero by adding the liquid res:i.stanced
o~/k~
to a cubic decaying equation for h(8) and by bringing the
wall temperature used in the equation to predict g(8) to the
stagnation wall temperature. 6R- is approximated by an increasing cubic equation.
i
This forcing of Tw (108°) to Tw (0°) tends
to make the graph of q (8) peak out abruptly at 108°.
Additional
.nodes did not improve the contour of the curves.
In table 3.4 the tota.l heat and mass fluxes are listed
for 1/2 of the cylinder.
I
I
It is seen from the above arguments
that the low velocity runs (1-18 and 24-25) are invalid.
By
comparing runs 18 to 19 it is found that the overall mass and
heat fluxes are larger for the low velocity runs.
This is in-
correct and is a consequence of the reference velocity being
I
too high to correctly model the profiles.
I
l
I
!
Larger heat and mass
flux rates should occur for larger velocities and smaller mass
fractions of air.
No other literature data could be found for
th~ range of parameters studied, but results ~re included to
l
L~.,._..,__ '"_.,_.•..---·~--·-·----------·- ---·"-·-----·-----~-·--·'-'"·--····---·-·------~--~-----~·-..J
24
r--·--~--·~·~"~""··-----~~~~--=~·~<•.·•~·-~••r.•j
l
complete this work.
J
Isothermal wall temperature results for runs 19-23
showed that the maximum discrepancies for the heat flux between
·it and the experimental results occured for run 19 with errors
of 7.8% and 15.6% at angles of 45° and 90° respectively.
The
maximum deviation between the variable wall temperature results
and experimental results occurred for run 22 and had errors of
2. 7 5% and 19. 5~~ respectively.
--~---
_,...,_ _____
·-·-·
__,.,.
~--
CHAPTER IV
SUM}~RY
1.
AND CONCLUSIONS
Heat and condensation rates for small air to steam·
mixture ratios with forced flow have been solved using simplified computed programs developed for the outside of a horizontal
cylinder.
The simplified correlation eliminates the need to
solve the complicated (Numerical) conservation equations in the
vapor phase.
2.
All results are presented in the form of a dimension-
less heat flux referenced to the classical Nussult solution for
pure vapor except for the overall heat and mass fluxes.
3.
It was demonstrated that increasing amounts of air
(non condensable) plays a decisive role in decreasing t.he heat
transfer rate, while increasing the velocity increases the heat
transfer rate.
4.
For steam-air mixtures the stagnation point values of
q/qNu were found to be in excellent agreement with experiment
(2) and also to a numerical laminar boundary analysis (3) that.
assumed the tube wall to be isothermal.
The maximum error
at stagnation between this work and the experiment was 10.55%
with an average deviation
of 4.39%, while the maximum error
between this work and the isothermal case was 4. 4%
age deviation
of 1.48%.
IL-~·--···-=·-·------~----~---··-··-~M-·----25
~vi th
an aver-·
26
5.
The agreement between this and- the experimental
results at angles of 45° and 90° were not satisfactory for
lmv velocities (less than 5 fps) and high mass fractions of
air (greater than 1/2%) due to the reference velocity used
for the dimensionless heat flux profiles (figure 2.2) being
too large (V=50 fps).
The maximum discrepancies were for
run 18 with velocity equal to .96ft/sec and were 96.7% and
i
183.9%.
The largest errors for runs (19-23) with velocities
greater than 5 ft/sec were 2.75% and 19.5%.
This indicates
that the computer program will be even better at higher velocities, and should be acceptable for engineering estimations
at these angles.
Different reference data should be used for velocities
less than 5 ft/sec if more accurate results are to be attained.
6.
The drop in wall temperature is a strong function
of free steam gas concentration for velocities used tn this
work.
7.
No comparisons can be made for the overall mass ·
and heat transfer rate results due to insufficient works
being published for this velocity range.
Low velocity runs
were invalid because too high a reference velocity was used
for the dimensionless heat transfer profiles. (figure 2.2)
..
27
,--~------~-~~~--~---~--~#~-~~-,--, ~~·---~·-~~-~· ·"-"~"= ., '
!
8.
It was determined that the variable wall temperature
distribution was no better than the isothermal wall temperature
profiles in predicting heat transfer for the cases tested (runs
19-23).
Maximum errors based on experimental results for angles of
45° and 900 for the isothermal wall temperature results (7.8%
and:15.6%) were about the same magnitude as the variable wall
temperature results (2.75% and 19.5%).
Also the finite differenc-
ing method used to find the variable wall temperature distribution :
proved to be time consuming in both programming and running the
computer model.
The computer running time would increase by a
factor of 4 by doubling the number of nodes,so that there was a
penalty for greater accuracy.
Therefore, based on these con-
elusions, it is recommended that the isothermal wall temperature
distribution be used for engineering calculations. At the stagnation point on the cylinder
the outer wall temperature is used
for the isothermal temperature profiles and is determined from a
heat balance between the free stream and cooling water.
~--~-~--~~--~----~~>---=·<=-0<.-~.,··---~=··~-·-·1V-·~-·-~-·~-~'~"·-~~=--~--~~-~--#-~""''~'-•
I
REFERENCES
I
I
IL
Nussult, W "Die Oberflachen Kundensation des lvasserdamfes."
Zeitschrift Vereins Deutscher Engenieure 60 (1916): 541-80
I
I
from Shav Ti Hsu, Engineering Heat Transfer. Princeton: D.
Van Nostrand, 1963.
i.
2.
Rauscher, J .lv. "An. Experimental Investigation of laminar film
condensation from steam-air mixtures _flowing doW11ward over
a horizontal tube," Unpublished doctoral dissertation,
University of California, Los Angeles, 1974.
3.
South, V., III
11
Laminar film condensation from binary vapor
mixtures on the outside of a horizontal cylinder." Unpublished doctoral dissertation, University of CalifQrnia, Los
Angeles, 1972.
4.
Colburn, A.P., "Calculation of condensation with a portion of
condensate layer in turbulent motion," Ind. Eng. Chem.,26,
No. 4:432 - 434, 1934.
5 . . Roshsenow, W.M., "-Heat transfer and temperature distribution in
laminar film condensation," Trans. A.S.M.E., vol 78, 1956,
pp 1645 - 1648.
6.
Sparrow, E.M., and Greg, J.L., "A boundary-layer treatment of
laminar film condensation," A.S.M.E., Journal of Heat
Transfer, February 1959.
I
I
L~------------------··-~---·---. ·----~-------·-J
28
29
r----~~~- -·~--~~·-R~~~-,~~.-.-~----~-~·-··~·--~~~--~~---·n~~·~-~=~-~-~M<>''"'''"'~'~'•••·•-·>'<
I
7.
Sparrow, E.M., and Greg, J.L., "Laminar Condensation heat transfer on a horizontal cylinder, "A.S.H.E., Journal of Heat
Transfer,
!
I
8.
I
November 1959.
Acrivos, A., "Mass transfer in laminar-boundary layer flows with
finite interfacial velocities, "Trans. A.I.Ch.E.,
!
6:4J.o~
414,
j
~
1960.
9.
Sparro·w, E.M., and Lin. S.H., "Condensation heat transfer in the
presence of noncondensable gas," A.S.M.E.
fer, 86:430 -
436~
~·
of Heat Trans-
1964.
10. Shekriladze, I.G., and Gomelauri, V.I., "Theoretical study of
laminar film condensation of flowing vapor, "Int. J. of
Heat and Mass Tran_?fer, 9:581 - 591, 1966.
il. Denny, V.E., and Mills, A.F., "Laminar film condensation of a
flowing vapor on the outside of a horizontal cylinder at
normal gravity", A.S.M.E., J. of Heat Transfer, 91:495-
501, 1969.
12. Denny, V.E., and Hills, A.F., "Nonsimilar solutions for laminar
film condensation on a vertical surface, " Int. J. of Heat
and Hass Transfer, 12:965 - 979, J969.
13. Pat:ankar, S.V., and Spalding, D.B., "Heat and Mass transfer in
boundary layers", London, Morgan-Grampian, 1967.
JO
93:297 - 303, 1971.
!
I
-
..
15. South, V. III, and Denny, V.E., "The vapor shear boundary condition for laminar film condensation," A.S.M.E.
J. of Heat
Transfer, May 1972.
16. Sparrow, E.M. and Eckert, E.R.G., "Effects of noncondensable
gases on laminar film condensation, "Trans. A.I.Ch.E.,
7:473
477' 1961.
17. Bird, R.B., Stewart, W.E., and _Lightfoot,E.N.,"Transport
phenomena", New York, Wiley, 1963.
18. Keenan, J.H. and Keyes, F.G., "Thermodynamic properties of steam",
New York,
l~iley,
1936.
19. Abramowitz, M. "Tables
f 8
of the functions
o sinl/3 x dx and 4/3 sin4/3 ~ sinl/3 x dx
"
J. of Research, National Bureau of Standards, 47:288- 290,
1951.
20.
Schlichting~ H. t
McGraw~Hi11,
Boundary Layer Theory, 4th Ed., New York,
1962.
..
.
r~·--·~4~=~·"'·-~·~~ -~"'-"··-~---~·--·--·-·~~--~·-- ~~--"~-·~~~~-·"··
·w•.., .....
><W>···.~···l
!
I
j
i
l
I
I
l
)
'
I!
l
APPENDIX
Appendix--Tabulation of Results
31
r----·-·------------------------·-·-----.. --.. .------------.. - - - - - - - - -. ------------------ TABlE . 1 - - . . ---------------------.--.. ------- ____ ,______ ·--------·---·---------··------------..
--~-------~--~!
l
!
t
I"
!
'
I
i
STAGNATION POINT DATA FOR .STEAM-AIR MIXTURES
t
Run
Tc>6
Tw
ml
u
I
~
l
!
!
I
!
I!
OF
'i
•F
ft/sec
J!,
qpred
Btu
hr ft 2
qNu
Btu
hr ft 2
a
_,_
.9..
qNu
· qNu
- qNu
pred
exp
num
.9..
£,
£,,_
~'3
!
I
I
.;
%
%
%
I
!
!
'
~.
I
1 115.8
w
N
l
103.9
106.6
2 117.7
3 119.3 106.5'
4 121.4 107.2
' 5 122.3 107.2
6 122.5 - 106.6
7 122.9 105.2 .
8 111.1 97.1
9 111.7
96.4
10 111.7
94.4
94.5
11 112.5
12 113.5 94.8
13 113.9 94.9
14 114.9 94.8
15 116.9 94.5
16 117.6 94.1
.0080
.0050
.0093
.0124
.0143
.0162
.0181
.0273
.0304
.0363
.0377
.0388
.0412
.0436
.0490
.0517
1. 23
26564
27051
27421
27829
27753
27634
28434
19531
19466
18862
19183
19211
18610
18662
18501
1. 21
13385
1.74
1. 75
1. 65
1. 61
1. 53
1. 53
1. 51
1.52.
1.46
1. 39
1. 39
1. 36
f.31
1. 29
31065
29780
33056
35894
37710
39188
40525
.798
.859
.777
.724
36598
39922
41171
42411
43040
44852
.855
.908
.830
.775
.736
.705
. 702 '
.567
.532
.472
.466
.453
.432
.416
48700
50659
.380
.363
.365
.473
.463
.452
.435
.420
.386
.349
.371
34-425
.660
.851
.907
.802
.766
.727
.701
. 635
.672
. 0.47
0.11
3.49
1.17
1.24
0.57
4.46
.558
. 515'
.455
.440
.427
.418
. 4-02
• !164
0.53
.528
0.76
-0.21
0.65
0.22
-0.69
-0.95
4.11
-2.16
.690
-6.23 · 1.14
-5. 29 _ s. 1o
-3.12 6.82
-5.48 7.o4
-5.09 6.67
-5.84 6.82
-5.51 10.6
-1.06 1.61
-2.46 3.30
-3.81 3.74
-4.97 5.91
-5.53 6.09
-3.91 3.35
-4.29 3.4-8
-5.44 4.11
-5.92 4.01
I
I
1
I
I
!
I
1
a
l
'l
!!
l
I
l
'
l
.
-'
1
.. -
'
I
l_, __. ·-.. ._..__,.._______ "'-__. . ,. __,_,__,,. __.,.-:. . . .-.,. . .
''~
~-
II
~W•
_ ......
...
.
~ ,~ ~-<'~~-·:"'-·- --.-.~--·~-·- ~-~-·-•'''-• ~M••
••
0
•''
•o,~,.....,..,., •. ,,,.,,•,oo~ '' ,, ___ , _
TABLE
''
-~-~ .• '
,.., _ _
"''''~·--·•·>~ ,......._~..,,, _ _.,~e.o O~h-<o
"''"'
1
STAGNATION POINT Dl\TA FOR STEAM-AIR MIXTURES
1
~
I
l
i .
Run
I!
T"'
ml
u
I
I
!
!
'~
!
w
!'
w
!
17
18
19
20
21
22
23
24
25
· qNu
g_
g_
g_
qNu
qNu.
qNu
ft/sec
Btu
hr ft 2
hr
. pred
exp
-
88.8
.0589
85.3
.0666
.0006
92.3
100.2 . .. 0015
106~2
101.9 96.4
.0029
103.5 97.6
.0029
105.8 . 99.8
."0045
104.8 94.1
.0147
106.0 94.2
.0165
1.06
0.96
5.80
4.53
5.11
5.01
4.92
2.06
. L 83
17477
15822
17358
19543
17888
18943
189B9 .
21021
21238
57831
60616
15502
18197
16735
17772
18235
27560
29673
.302
.261
1.120
1.074
1.069
1.066
1.041
.. 763
.716
.293
.. 261
II
!
qpred
OF
OF
!
l'
T.;,
117.4
116.2
97.4
Btu
ft 2·
1.116
1.·059
1.056
1.023
1.014
• 7.27
.. 679
num
€~
es
%
%
%
-3.82 -6.68 3.07
-4.40 -4.40 0.00
0.36
---1.42
---1.065 0.38 -0.85 1. 23
1.063 0.28 -3.73 4.20
1.045 -0.38 -2.92 2.66
1.46 -3.32 4.95
.752
.706
1.42. ··3.82 5.45
.314
.273
-----
-------
average
absolute
error
·.
€,
1.48
4.81
4.39
~
i
~.
j
. ~
i
i
ry
r-----~-----~-
TABLE .. 2
1
i.
l
l
45 AND 90
l
I
e
l Run
l
1
j
l
.
l
.I
1
1
'
2'
2
~
3
4
4
5
5
6
6
7
-7
I .8
·I
i
8
i
I 9
i.
i
9
i'
~
10
l
l.
l 10
!
~
j
l
I
l
I
I·
i
l
.!
11
11
12
i
I . 12
13
13
I'.
.Il
I
I
'
..,. ._;_., .,~~__,;-~--~-~--.......;.,.-~-~=·-~-~,.-~_:___ ·--··--~<>-~-- -~-·-"··-~- ·-·..·--·~~--~..·--·~-.,_...., -~........:.......... ···"~·=--=--·-- . ·>···"·-··---
,.
)
T
wexp
"F
103.6
90
102.2
45
106.1
90
105.1
45
106.1
90 104.9
45
106.9
105.1
90
45 106.9
90 104.9
45' 106.2
90 104.1
"45 105.8
90 103.5
45
96.3
93.5
90
45
95.9
90. 92.9.
45 . 93.8
"90.4
90
94.0
-45
90
90.5
45
94.3
90
90.7
45
94.4
90
90.8
45
0.1\TA FOR
T
wnum
•F
105.4
i03.0
107.5
104.9
108.4
105.9
110.0
107.4
110.7
108.1
110.7
108.2
110.2
107.7
102.1
101.0
102.3
101.2
101.8
101.1
102.3
101.7
103.4
102.5
10~.6
103.2
STEAM-AIR MIXTURES
q
-"-
a
g_
qNu ,9=0
qNu,S=O
Btu
?
hr ft-
pred
29044
32734
28828
32577
30418
34228
31833
35679
32508
36323
32980
36746
34903
38771
25161
27535
25982
28279
26780
28868
.935
1.054
.968
1.094
.920
1.035
.887
.994
.862
.963
.842
.938
. 861
.957
.731
.800
.710
.773
.671
-. 723
.671
.721
.660
.709
.642
.688
2763~
29674
27997
30054
27645
29591
I
exp
-
.837
. 792
.897
.851
.814
.771
.760
.714
. 723
.678
·.690
. 646
.664
.620
.. 575
.529
.533
.490
.472
.424
.456
• 411
.447
.400
.434
.391
€.
%
11.7
33.1
7.9
28.6
13.0
34.2
16.7
39.2
19.2
42.0
22.0
45.2
45.2 .
54.4
27.1.
51.2.
33.2
57.8
42.2
70.5
47.1
75.4
47.7
77.3
47.9
76.0
.
\,,".._,.-""_._""'...--""'~"'-'-..,..<~~-~'<....--"~""--·--~·--·--a.<.•..-"'w"'-.-<r•-...-,~..,..,~'"fC...-.._._~~...... ~,=- ...~---------"""""""'-· -.,..~.....,....,.,.""""'''"..,."'~-~;.v..,,..,,.~...,_..,,,..._..,~-~-....._- ..,,...,.;
34
.
f._,.__.,._,.,..,.._...,.....,..._,_W__........__-.-.,""-"""'""-"'·~,.,.-~.,;~-I<i"'"="~'>""'•~~~<'~.<,_ov-~riY-~""-..ue~-.;>......._:r..':'•"'-',_""""""~•-·... ,~~·~"·...-~_.,rl><4,>'"•~·~-·,~<6"<.,._"t<?."""'.Co-'<<c~<·=<"-'"".""""~•.-·,~-:>-'"'-'"..,._..,,.,,-..;;..,~-.'"'""•~"'•·~oc'.ct:•-• ,.,,.,c;,
i
.!
!
TABLE
!
I.
l
45 AND
j
I
e
(CI)
17
45
90.
45
90
45
90.
45
17
90
18
18
19
45
: 14
'
14
'
' 15
15
16
1
16
\
;
\
~
;
'
'
;
l
. i
19
20
45
20
. 21
. 90
23
·l
24
I
i
24
25
~
l
l .
l . 25
I
90
45
.90
21
22
22
23
.)
DAT.l'l, FOR STEAM-AIR HIXTURE$
.,_
l! Run
\.
go·
2
45
90
45
90
45"
90
45
90
45
90
T
wexp
. •F
94~1
.
Tw
4F
93.5
89.0
88.2
83.0
84.7
79.6
104.2
104.0
105.6
105.7
105.9
106.3
104.3
105.5
102.6
104.7
91.7
·92~1
91.3
99.5
99.0
95.7
95.3
97.0
'96.3
.99.2
98.6
93.4
92.2
93.5
. 90.6
90.2
94.0
89.5
91.9
q
pre·d
100.0
98.4
. 96.3
94.9
97.5
96.0
99.8
98."3
"96.3
94.5
97.1
95.3
Btu
g_
g_
qNu
9=0
. '
qNu ,8=0
%
hr ft 2
2red
28357
30255
.632
·. 675
.608
.644
.595
.628
. 561
.584
. 529
.547
1.125
1.273
1. 08"9
1.226
1. 086 .
1.221
29593
31354
30130
31828
32469
33779
32084
33149
17433
19727
19808
22302
18173
20439
19258
21669
·19371
21781
23951
26757
24947
27771
~
. 1.084"
1.219
1.062
1.194
.869
• 971
.841
.936
ex2
.415
.371
.377
.333
.363
.315
.304
.259
.269
·.229
1.148
1.120
1.089
1.058
1.082
1.070
1.055
1.020
1.049
1.014
.749
.718
.699
.665
52.3
81.9
61.3
93.4
63.9
99.4
84.5
125.5
96.7
138.9
-2.0
13.7
0.0
15.9
0.4
14.1
2.8
19.5
1.2
17.8
16.0
35.2
20.3
40.8
.
L..~---=........_>·--~--~---···---·-----·-·--~-'--~·-···-~·---···~~-~-·-,.·--'··. ·"---,..........~---- ..-··--········-·· . ··---····· ·-· ·
35
'
..
.
r··--~-~ -~-~~·~--·---~·~·~-~"-'"=·~·~,--~·-···~-~.,.-~.,~-'"-"'-""·-··~·"""~---~······;,.= ·~"'-"'''••=w····-·······~~-·-···
I
TABLE
l·
l
l
3
TOTAL MASS AND HEAT TRANSFER FOR 1/2 CYLINDER
.l
. l
l
Run
j
I
1
2
3.
4
5
6
7
8
9
I .
I
10
11
12
'13
14
15
16
17
18
19
20
21
22
23
24
25
ni
q
lbm
hr ft 2
Btu
-h
r ~2
.t
185.09
183.17
194.59
204.73
209.82
213.38
226.46
164.83
171.15
178.25
184.28
187.15
185.95
191.63
202.37
207..24
227.78
230.85
108.77
190590
188395
199998
210225•
215368
219047
232438
170274
. 176771
184163
190329
193204
191939
197727
208641
213616
234948
238345
123~76
127986
117535
124572
125515
158552
165778
113.35
. 120.23
121.29.
. 153.05
160.12
36
. I
113055
..
'.