an investigation into the development of laminar natural … · 2017-06-02 · 1 tabte oi,'...
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TIìE UNIVEIIS]'f Y OI ,Ô,DE],AIDI
IlEPAR:liviilNT 0Iì iUlrCI{A.l'l.l C/iL,tr¡ìiil.NEERIliG
A1\ Ii{\¡ES'IIGAIII0N lN:f0 T'llll I,r[r\,rliL0P}firhr'l- 0F i,AÌiÍIi\,¡tlì
NA'I'URA]. CCN\"']]C1'10]i ]N fl],JA]']]1] V]'ìR'I',I-CAI, DLICTS
b1'
.1 . Ross I)¡,'ç,r, . ll . Ìr. , .ì1 . Ec.
Tlrc..;i s f or .l.hr,r l-ìe¡.ree of ìloc-'i,or ol- l'h,î l c:;o¡rìry'
Or:t,rbc:i-. 'l 971
1
TABTE OI,' CONTENTS
SUMMARY
DECI,ARATION OIT ORI GINALITY
ACKNOWI,EDGE}fENTS
INTRODUCTION
Scope of the investigationList of synibols
TI-IEORE'I T CAL CONS IDERAT I OhIS
Gene ralLaninar f lorv equations
PLAIN.ENTRY DUCTS l\lITFI UNIFORM SIJRIIACETE}{PERATURES
(i)
Page1V
vii
vi ii
11
1
14
?
1.1I.2
2.7??
3.I3.2
3.43.5
7LI2
3
3.3
19
19
2,I
23252SaI results
General N
Nr-rsselt rr
deve 1 op eNussel t TIayer f1
Method ofTheoret i c.
usselt relationshipelationship for fu11yd flolvelationsirip for boundaryO1\I
solving the flow equations
4 I{ESTIìICTED_ENTRY DUCTS }IITI] UNIFORIU SIJR-FACE TEMPEI{A'|UIìES
4 .1, Intr"oclu.ct ion4.2 Nusselt relationshin for fu11y
developecl flow4.3 Nusselt rela'bi.ons)iip for bounclary
layer flolr¡4 . 4 lt{ethocl of soh, ing the f loru equations4.5 Theoretíca1 results
42
42
43
474B49
5 UNIFORM SURFACE I]EAT FI,UX DUCTS
5.4
Introcluc t ionGeneral Nusselt lelationshipNusselt relationship for srna1l Ray-leigh numbers based on the uniformheat f1ux
l{usselt relationship for boundarylayer flow based on the uniformheat flux
Nusselt relationship for sna11 Ray-leigh nunbers based on the neansurface tenperature
Nusselt relationship for boundaryLayer florv based. on the mean sur-face temperature
Method of solving the f1ow equationsTheoretical results
5.5
5.6
6 EXPERIT{ENTAL STUDY
G eneralApparatusProcedure
EXPERIMENI'AL RESULTS
7.t Plain-entry ducts r^¡ith unj-form sut:-face ternperatures
7 .2 Iìestric ted- entr:y ducts rn¡j th unif ormsurface temperatures
B CONCLUS IONS
APPENDICES
A þ-inite difference equations for unj-fornsurf ace tenpera ture duct-s
ß Relaxation solution for unifornt surfac:eternpera Lure duct s
5.15 .?.5.3
(ii)
Page
6ó
6666
69
t-)
5.75.8
6.16.26.3
94
9494
109
LI7
7B8081
r77
rz4
T4L
74
746
150
7
C Cornputer programmetemperature ducts
for uniform sr¡rface153
(iii)
D
E
F
Irinite differencesurface heat flux
equations for uniformducts
Page
1s9
160
163
1ó8
L73
Relaxation solutionheat flux ducts
for uni-form surface
Cornputer programnìe for uniforn surfaceheat flux ducts
Entry restrictj-ons yielding values of!"i/ L in excess of 0. B
REFERENCES
G
(iv)
SUN,IT,IARY
An account- of a theoretical and exper:imental stucly
of the development of laminar natural-convective flow
in heatecL veltical ducts is presented in this thesis.
The clucts studied were open-ended and circular in cross-
section, and the heated i-nternal sr¡rf ace either hacl a
uniforn temperature or dissipated heat unifornly. In
addition, the effect of restricting the flow at entry
was studied in the case of ducts rsith uniform surface
temperatlrres. In this ryork the f loru restrj.ction r4/as
provided by keeping the lower part of the duc-t at
anbient tenperature.
ììelationships between Nusselt and Rayleigh numbers
r\¡ere derived by relaxing the govcrning ec¿uations. The
genera.l solutions obtained. on a computer r./ere corrobor-
ated at both smal1 and large Iìayleigh numbers by theor-
eti.ca 1 analys j- s .
For unj.forin surface ternper.'atule ducts two distinc.t
la.rninar f Lorv legimes were observed " At sna11 Rayl eígh
nuntbers the f low was fu.11y developed and at large Ray-
1e igh numbers boundar::y layer flow oc--currecl. Au entry
restriction had the effect of lowering the Nusselt
(v)
number. Ilowever, it was found that the flow ceased to
be laminar when the entry restrictíon reached a certain
size ancl. that the s).ze of the largest entry restriction
for laminar florv decreased rvith increasing Raylei-gh
numbers.
Besides confirning the theoretical work, the
experimental studies supportecl the hypothesi.s that entry
restric-.tions that were too large to a1low lalnj-nar flow
throughout the duct prod.uced a mixing flow above where
the laniinar flow degenerated" Furthermore, it rsas
found that, if the entry restriction were very large , âî
unsteady open-thermosyphon flow occurred in the top part
of the duct. In this case, the open-thermosyphon flow
dissi¡rated heat in the upper part of the duct as did
the restricted-entry f 1ow in the lorver part.
Prandtl numbers greater than about 0.7 were foutd
to have only a. smal1 effect oll the theoretical rel.ati on-
ship between [he Nusselt and Rayleigh numbers. IÌowever,
the influence that the Prandtl nunber had on the rela-
tionship was shotvn to increase significantly as the
Prandtl number decreased below 0.7 "
In uniform surface hea.t flux ducts, it is not
(vi)
possible to obtaj-n fully developed flow because the
fllrid. is receiving heat along the entire length of the
duct. However, snal1 lìayleigh numbers produced a f 1ow
that resembled fu1ly developed f1ow. At the other
extreme, large Rayleigh numbers produced a boundary
layer flow. For boundary layer f1ow, it was found
that the Nusselt number relationship for the uníforn
surface temperature ducts could be satisfactorily used.
if the entire surface urere assumed to be at the mean
surface tenperature.
(vii)
DECLARATION OF ORIGINALITY
,This thesis contains no material that has been
accepted by another University for the award of a degree
or a diploma. Also, to the best of the authorrs
knowle<1ge and be1íef, this thesis contains no material
previously published or rvritten by another pelson, except
where due reference has been made in the text of the
thesis.
J,R. DyerOctober, t97t
(viii)
ACKNOWLEDGEMENTS
The worlt described in this thesis I^Ias carried out
in the Departnent of Ivlechanical Errgineering at the
University of Adelaide under the aegis of Professor Il.FI.
Davis. The auth.or is grateful t-o Professor Davis for
allowing hinr to undertake the research.
The author is greatly indebted to Dr J. Mannant,
Reader in the Department of Mechani.cal Engineering at the
Universit-y of Adelaide, f or his encouragenent and f or
his help and a.dvice on nany occasions.
The author thanks Mes-srs R.C. Fitton, R.H. Schunanu,
FI. Bode and otlre:: members of the staff of the Mechani-cal
Engineering. l\torkshop f or their valuable contributions to
the experi.nental work.
For making the experiment.al ducts and for carrying
out some of t-he tests , Nlessrs I . G. Stanley, S. G. Lìn,
C.E"T. O;ig and R"I(. Inverarity, forrner undergraduate
studerrts in the Departrnent of lt{echani ca7 Engineering at
tlre Unir¡ers ity of Ade.l.aide , are gi:atefu1ly acknolvledged .
For proof reading and helping to prepare the
material for publicatì-on, the author is much indebted to
his wife, Elizabeth.
(ix )
A special word of thanks is due '[o Mrs G.lvf . Duncan
who, when typing the text, kept a careful watch for
inconsistencies in the manuscript.
Finally, the author thanks the Australian Institute
of Nuclear Science and Engineering for provì-ding finan-
cial grants, without rvhich this ivork woulcl not have been
possible.
.t
L INTRODUC'I I ON
1.1 Sc e of the investi ation - In many industrial
applications, the internal surfaces of vertical ducts
have to be c.ooled by natural convection. Since there
is 1ittle information in the literature on the subj ect
of natural convectio¡ in open-endecl vertical ducts, it
ì^/as considered profitable to carly out a theoretical and
expelintental study into the mechanism of lani.nar natural
convection in snch ducts. The restrlts of this study,
including those which have prevìously Þeerl published1(L,2)' , are presetltecl iu this thesis.
All the work to be described was undertaken on ducts
of circular cross-section, and the following specífJ-c
probleins ilt natural-conyective heat transfer tr'er'e exam-
i-ned:
1. nuðts with uniform surface teniperatures and
p1.a-i n entri-es
2 " Ducts with uniform surface ternperatures and
restrictecl entries
3. Ducts with uníform surf a.ce heat f luxes and
plain entries
1 lìef crences;, shotvn on pagc 17 5 ,
L
Slrorvn in Fig " I.1 are sectioual t¡iet'¡s of these dlrcts.
It rr¡i11 be soen in Fig. 1.1(b) that tTre etltrl'restric-
tion is.sinrply ¿rn utrheat-ed sccCion of the duct"
Twr
S ui-f ccei emperoture
ï*
Unifcrro
surfoceIr rro
T*
To
Gt.cvit¡r
field
Plain-entry duct with a unif,orm surface temperatureRestricteci-entry duct r,{ith a unifo::m surface temperaturePlain-entry duct with a unifoïm surface heat flux
Uniformsurfoceheot
f tux
f
T"Arnbientiernpei-oture
.rlO
Éntryresiriction
Fig. l-.1 Diagrannatíc sectioned views of the heated ducts
T"
(a)(b)(c)
(¡¡
+
a
A
c
t.Z List of symbols
General:
p
Cross - sectional area of the cluct
.Internal surface area of the heated section of
the duct
Specif ic heat a.t constant pressure
Constant
Acceleration of gravity
Heat dissipated by c-otrvection from the bottom of
the cluct to a particular elevation per unit time
Thernal conductivity of the fluid
Lengt-h of the heated, section of tire cluct
Length of tlre unheated section. of the duct (entry-
re s tr i ct ion)
Pre s s rlre
Pressure defect, (p-no)
Flydr:ostatic prcssure
Prandtl number, ucn/k
Vc¡l^ume flot^¡
Radial coordinate
Dimens ionles s radial coordiuate , t /'rw
Terupelature
Velocity of the f luid in the x- clirection
Velclcíty of thc fluicl ii-l the raclì.a1 clirection
I
C
hX
c'b
k
o
o
p
P¿
po
Pr
q
rR
T
u
\r
5
V
X
Dimensionless fluid velocity in the radial direc-
tion" r v/v'wVertical coordinate; x=0 at the bottom of the
heated section
Coefficient of thermal buoyancy of the fluid
Dynani-c viscos ity of the f luid
Kinematic viscosity of the fluid , u/ p
Density of the fluid
Applicable only to ducts wi th uniform surfacetemperatures:
Grashof number based on both the heated length and
the radius of the duct, Iß(Tw-To)r$/v'zø
Grashof number based on only the heated lengt-h of
the duct , B ß (Tr- To ) .0 3 /v 2'
Dinensionless heat dissipated fl'om the bottoln of
the duct to a pa'rticular eleva-tion per unit time,
h*/ ocnv!,Gr (Tr-To)
D j.mens ionl e s s length of the hea ted s ecti on of the
duct , 1-/Gr
Dimensionless length of the unheatecl section of
the duct , 9" i/ 9.,Gr
Nu Nusselt nulnber or1 the radíus of the duct,
reduces to lnt/Tnt (Tw-To)k
ß
u
v
p
Gr
Gr
Hx
L
L
9"
I
bas ed
whichhrrr/A(Tr-To)k,
6
Nu Nusselt number based on the heated 1 ength of
the duct, ht.a./A(Tur-To)k, which reduces to
hr/ Zltr*(Tw-To) k
Dinensionless pressuTe , prlo/ pL2t2Gr2
Dimensionless pressure clef ect, pdrrT/ pL2t2Gr2
Dimensionless volume f 1ow, g/.1,vGr
Rayleigh nurnber:, Gr Pr
Reynol-ds numl¡er based on the radius of the dtlct,
u r /v or GrO?"/Il.rm \^/' W
Dimensionless velocity of the fluid in the
X-direction, vrf,/.0vGr
Dimensionless vertical coordinate, x/ p"G'r; X=0
at the bottom of the heated section
Dimens ionles s ternperature , (T -T ì / (Tw-To)
Applicable only to uniform sttrface heat flux ducts
Uniform surface heat fluxh.eat f lux,
on both the
duct, ußfrfr/v2.ok
9"
d
P
P
a
Ra
Re r
U
X
0
rF
Gr*
1
Dinensionless
Grashof number
radius of the
uni f ornr
bas ed
L /Pr
length and the
1_ The symbols for the dimensionless variables tha.t arenot cornmon to those for ducts having uniform surf acetemperatures are identi.f ied by :t f or variabl es baseclon the uniform heat flux and by + for va'riabtesbasecl on the niean tenperature of the surfac.e.
7
G'ä
Gr+
H?kX
Lrs
Nu*
N"t
P*
Ra*+
Ra
Re*T
u?t
Grashof number based on
duct, gßf 1,'/t'k
6a * / li[¡¡*
Dimens ionles s
the duct to a
only the length of the
heat dissipated froln the
partícu1ar elevation pe::
H*x
botton of
unit time,
h /lìa* f .Q,rX' I\t
H* f Nu*l 2
Dimensionless length of the duct, If G-rx
Nusselt number based on the radius of the duct,
fr /(T -T ìktr\rm o'Nusselt number based on the length of the duct,
1!t"/(T -T lkI^nI O'
Dinensionless pressure, pt[/ plzvzçt*z
Dimensionless pressure defect, pdrl/p.[2v2'çr*2-
Dimerrsionless volume f1olv, q/ 9"vGr*
Q*Nu*
Rayleigh number, Gr*Pr
ft¿x /)rfs*:
ReynoJds nunber based on the radius of the duct,
u r /v or Ç¡*Q:t !"/IIrmw' w
Dinensionless fluid velocì-ty in X-direction,
r2u/ l,vGr*hT
Dimens j.onless vertical coorclinate, x/g?tx; X*=0
at the bottom of the heated section
D"t'd
Q.*
Q*
x?t
I
0.å Dirnens j-oirl ess tcntperat.ur:e, ('I'--T") lc/fr,,
0 å'Nu*t-
0
9
CX
ct
me
IVM
hrt
Subscr:ipts:
On the vertical axis and at the top of the duct
0n the vertical axis and at elevatiofl x, X, or X'*
Defect (pressure)
Refers to the smaller diameter entrance used in
the experiniental study
Refers to the unheated section of the duct
jth row of the relaxation grici
kth colunn of the relaxation grid
Based on the heated length of the duct
Mean value
At mid-elevation
Refers to ambient conditions
Optinun value
Corrs t.ant pres sure
Based on the radius of the cluct
At the top of the duct
Refers to the internal surface of the duct
0n the internal surface, and. the mean value of
the variable
0n the j.nternal surface and at the top of the duct
0n the internal surfac.e and at elevation x, X or XIT
d
e
1
J
k
o
m
o
opt
p
rt
w
WX
10.
Al;blcvi a-ti,ons :
USI' Uni f or:m
USiIF Uri foi:lrr
surf¿lcc:
sur:lla ce
t.crnperat-r.:ie
he¿Lt- f i u>r
11.
2 THEORETICAL CONSIDERATIONS
2.L General
Heating the surfaces of the vertical clucts shown jn
Fig. 1.1 r^¡i11 procluce an upl,rard natural-convective f I ow
through the ducts " The buoyancy forces giving rise to
the flov¡ are produced by density differences iu tlte
gravitational field.
In the case of the partly heated cluct shown in
irig. 1-.1(b) , the f l.ow, and consequently the heat transf er,
will be reduced owing to the resistance presented by the
unheated section, If, however, the restriction exceeds
a certain st.ze, the flow will cease to be laminar and
fluid nay even enter the top of the duct. This produces
a f lorv in the upper part of the heated section similar tr,r
that in an open-thermosyphon duct (3) (a vertical duct
that is open at the top and closed at the bottom).
Elenbaas (4) established the heat dissipat-ing
characteristics of pl.aj.n-entry ducts of circular and
other cross - sectional shapes rvith unif c,rm surf ace temper-
atures. Thi-s was clone by transforining the re-suJts of
his theoretical stucly of natural-convective flow through
heated vertical channels f onned by tivo paralle1 an'J in-
finitely rvide flat plates (5). It was not possible,
however, to crbta iri I cmper:a-[ur:e ancl tte] ocity prof íl e:; in a-
L2.
cluct by using this method of solution.
Thí s short-corning of Ìllenbaasrs nethod (4 ) can now
lre renredied by relaxing the governing equations on a
high-speed digital computer. Hence in t.he present work
the development of laminar flow in vertical ducts of cir-
cular cross-section r¡as deternined by relaxation. The
¡nethod adopted lvas sinilar to that usecl by Bodoia and
Osterle (6) in their study of natural-convective flow
through heated r¡ertical channels.
2.2 Lamj nar f lorv equations
Throughout the analysis to be presented the follorv-
ing simplifying assulnptions have been made:
(a) The fluid is Nervtonian
(b) Fluid properties, except dens ity, are indepen-
dent of tetnperaturre
(c) Densì ty rrariations a1'e signif icant only in
proclucing the buoyar'cy forces
(d) Irlotv in the duct is steacly, inconpressible and
axi symme tr ical
(e) l{eat generated by intelnal fr j.ction is negli -
gible
(f) Pressu,re varjatiots in the duct ar.e too small
'l-o ha.¿e au.y effect on the properties of the
f lrii c1
13.
The entry restriction sholvn in Fi g . 1 .1. (b) was
chosen because it allc¡rved laminar f low to be produced in
both sectjons of the duct, and thereby the mathematical-
anal.ysis hras simplif i ed. It should be nofed that a sirn-
íIar type of flow restrjction rvas used by Dyer and Fow-
ler (7) in their analysis of the problern of the restlic-
ted-entry yertical channel.
Fig. 2.\ clescribes the system of cylindrical c.oor-
dinates used in the analysis. Using the assumptions on
page I?, the eqtiations of c.ontinuity, molnentun and energy
in cylindrical coordirlates for lani¡rar f lorv (B) are:
Continuity equation
{-
l4omentum equa'lions
ãuäx 0
1 àrvr ðr (2.r)
p uâuEx
+V ðuãr
Dpðx
aul__tarJ
â2u gp-q..17d^
+ T +
(2 "2)¿
âv +
+
p uðx
Vav'lF=J
1 ârvl- --tr ôr )
+.
(z .:5)
Energy equation
äTu äx+v âT k 1AT
rôra'rlt*=-l
1_4 .
(2 .4)[,"[a-'-
+ +ôr pcp
The fluid entering and leaving the duct is under'
hydrostatic pressure. However, as the flow is confíned,
the pressure ivithin the duct (p) will be less than the
hydrostatic pressure (no) at the sarne elevation. The
difference between the trvo pressures (n-Uo) will be
cal1ed the pressure defect (f¿) (ó).
Tlr.e hydrostatic pressure decreases with elevation(9) accorcl.ing to the equation
do'oAx ooB
Hence adding both
Eq" (2.2) yields
ðpo
ðx
(z .5)
and 0og to the right hand side of
p u T +ðuôx
+v âuãT
a (p -po)o^
âuDr
The last term on
is the buoyancy force.
u(oo-o) (2 .6)
the right hand side of Eq. (2.6)
It calr be expressed in terms of
X1_5.
)t
orf = const. Grovity
field
Ambient ftuid
To
l
r
T"
I
t
I
I
I
l
I
I
I
I
I I
Entryrestrict on
Fig. 2..I Systen of cylindrical coordj-nates
16.
the temperat-ure difference (T-To) by introducing the
coefficient of thermal buoyancy (9) which is definecl by
the equation
qqlATJ
(2.7)Þ Ip
p
As an approximation for the sma1l
caused by a temperature variation
relationship can be written:
(oo- o) = pg (T-To)
Substituting Eq. (2.8) into Eq. (2
clensity difference
(9), the following
6) yields
p r -t-ðuâx
äuðr
AVAR
o
ðrðu.|-V
âP.d
ôx
(2.8)
Dr
+ + (2 .s)
Eqs. (2.L), (2.9), (2.3) and (2"4) can be expressed
in the following climensionless f orms by using the dinien-
sionless variables in the List of Symbols on pages 4' to 10.
The climens ionless equations f or uuif orm surf ace
temperature (UST) ducts reduce to:
a trrlãx'l ,pg (r_ro)
AUäX
VR
+ + 0 (2.10)
UAU-^-i;-cl À
+ V ðU
'I¡
âP.CI
N---d^+.
a2oãltz
ð2uã-Rz-
1.
R+ AU
ãT+0
17,
(2 . L7)
(2.L2)
(2 .76)
âP.d
^r f)OI\=0
0 ðU +V
ðXa0ãT
1i{+1
Þî0d
^T)OI\(2 "t3)
(2 .1 4)
âPtCl
ã-x=F
1 'd tJ*R tìt + oå- (2 ' ls)
+ nL-m-]
Using another set of clirnensionless variables, the
equations for uniforn surface heat flux (USHF) ducts
recluce to :
au'ÂtX:T
âVAR
VF
+ + -0
u* **i * \, +Hn =
âPå
ãR-
+
ð20'*-q
n-Zdt(
a 2u*tR-z
+
0
ðe* ð 0'* 7
PTðR5-f* (2.17)
Tlre reason that sone of the syrnbols in Eqs. Q.f4) to
(2.17 ) are without an a.sterisl< is that they are conlmon to
both UST and USIIF ducts. (Synbols rvith an asterisk are
applicable only to USIIII ducts).
Terms containing rw/ LGr and rlv/r"Gr* rvere omi tted
u* -r- \/
18.
from the foregoing sets of dimensionless equations.
This is pernissil¡1e because in most practical situations
these parameters , rvhi.ch are raised to the second power,
are very nuch less than unity as illustrated j.n the
follorvi-ng example. A 2 in. dianeter duct has an inter-
na1 surface temperature of 2000F and stancls in a fluid
whose temperature is 100oF. Calculations show that:
for ai-r, r -52x10/ t"Grw
f or rvater, Tw/ LGr-ô3 x 10
In each case the value of the paranleter rvill be seen to
be very much less than unity.
19.
5 PLAIN-ENTRY DI]CTS WITI-] UNTFOIìM SURFACETEI{PERATURES
3.L Genera,r- Nusselt reT.ationshi p
The plain-entry cluct v¡ith a uniforln surface temper-
ature i-Llustrated in Fig. 1.1- (a) has a dimens j-onless
lerg-th (L) , rvhich, by def inition (see List of Synboi.s) ,
is tJre reciprocal of the Grashof number (Gr). This
Graslrrrf nunber is the product of gß(Tw-To)rfl/v2, rvhich i s
a Grashr:f number of the conventional form, and the ratio
rro/ t" (6) .
Since it has been assumed t-hat the dr:nsity of the
fluid does not val'y (see page LZ), the cLimensionless
florv volume (a) is constant throughout the duct and is
g iven b,r'
1_
J
a 2n URdR (5.1)
0
It f ollolvs f rorn Eq. (3 .1) that the dimens ionless heat
dissipated by the surface from the bottont of the duct tcr
a particul,ar elevat. j.on (X) is gir,'en by
1
HX
0
2TI, UORdR (s .2)
20.
The Nusselt nuurber of the duct with the radius as
the c.haracterlstic dinension is given by
h. rNu= A(T -T k
(3.3)l{I o
htNu= 2ne"(T -T )ko'(3.4)
tr\r
)
rvhere ha is the rate at which the internal sulface of
the duct dissipates heat. When the surface area (A) is
replaced by 2Tl.rro9", Eq. (3.3) becomes
Expressing h, in terns of the dimensionless rate (llr),
Eq. (3.4) becones
Gr Pr llNu=
zTl(3.s)
Repla.cing Gr Pr by the Rayleigh number (Ra), Eq. (5.5)
reduces to
Ra
t
Nu=FIt
(3.6)
Since the relationship between Nu and Ra alone is required,
Ec1s. (2.10) to (2.15) rvi11. be solved for the fo11ot,i'ing
bounclary condition.s based on the geometry of the duct
shorn¡n in Fíg. 2.7.
]I
Boundary conclitions
0 V=0, 0=l-, Pd=0.
2I.
(a)
(b)
(c.)
0. (d)
(e)
ä0AR
0. (f)
(e)
O<X<L and- R--1 U=0, V=0, 0=1, <0.d
0<X<L and R=0 V t d<0,
X=0 and R=1
X=0 and 1<R.-<0
X=L and R=l-
X=L and R=0
0<x-<L
U
u=$,
U
V=0, 0--0,
0 V=0, 0=1, Pd=0.
V=0, P d
a is a constant.
P.=0'.o
ð0_ãR-0
P
PðUãR.-
0,
0 P9=0,dT\
The assunption that the fluid enters the duct with a
uniform velocity (U=Q/n) hras ba.secl on the practice usually
adopted in analysing the developntent of forced-convectir,'e
f low in pipes (9) "
Befole beginning the general solution of Eqs. (2.10)
to (2,L3) , sirnplif ied solutions for the extreme values of
the ratio p"/Tw r^¡i11 be obtainecl. The Nussel t relation-
ships yielded by this analysis are useful for the corrob-
oration of the results of tlie general solution.
3.2 Nusselt lelationshil> :[or ful1y developed fl.otv
Ful1y clevelopecl flolv is obtainecl rvhen the ratio
?)
*/rw is made suffj.ciently 1arge. This rvi11 produce a
<1uct that has a small Ra.
l,Vhen the flotv is fu1ly devel-oped, the temperature
of the fluid is the same as that of the sul'face, that is
0=1, and the veloci.ty profile is parabolic" Thus, for
fully developed flow Eq. (2.If) reduces to
+ +1
U
Subst:ituting Eq. (3.8) into Eq. (3.7) yields
and the velocity at a radius R for a volume flow (a)
given by
# (1-R')
ðP¿
AX
ðPd
ðX
Ð2uã-R-¡
1
1ãURAR (3.7)
1S
(3.8)
(3.e)g_qII
Clearly, the positive pressure graclient in the upper part
of the duct ivilI decrease as the length of the duct is
inc.reased and in the Limit rvj.11. approach z.ero. Thus,
rnaking aPd/aX equal- t-o zero in Eq. (3.9) yields a = IL/8,
which will be the naxinuur value .Jf Q. Contparing Eqs .
(3.1) and (3.2) will shorv that both Q and ll* have the
23.
same value when 0--1 . Thus H* rvi11 be equal to n/ I for
fu11y d.eveloped flow and, sinc.e the fluid cannot absorb
any noïe heat when the ternperature of the fluid has
reached 0=1, Ht also will be equal to II/8. Substituting
Ht=n/8 into Eq. (3.6) gives the Nusselt relationship
which shows that for ful1Y
tionship exists between Nu
(3. i0)
developed flow a linear rela-
and Ra.
NuRa16
3.3 Nusselt rel.ationship for boundary Tayer flol
tsoundary Iayer flow occurs in a duct that has a
snralL 9,/r-ru ratio and consecluently a large value of 'Ra.
The temperature and velocity distrj.lLutions n.ear' the sur-
face rvil1 tre sinilar to those in the natural -conlrective
borrndary la¡,s¡ olt a heated vertical f lat surf ace " Thus,
if the duct is opened out to forn a t¡erti.cal f lat strrface,
there should be very 1itt1e clifference betrveen the rate
of heat tt'ansfer frorn the fTai- sur:Eace thus forned and
the duct. Therefore, Nu will be independent of the
radius of the duc-t and consequently can be expressed by
the equation
Nu
where Cr is
plyi.ng both
24.
a constant. Expanding Eq. (3.11) and multi-
sides by 9, gives
7-/4
gg (T,_-To) r 4h ï wW CrAIT -l )k v 9"o
rvhich, on elirninat"i ng T reduces to
1-
gß (T'*,-To) [3
t 1-P'ra L (3.L2)9"
w
h t" 1,-4
Cr Pr (3.13)v
or
Nu Cr Ra7, (3.14)
I, 9.
The subscript 1, in Eq " (3.74) indicates that the charac-
teristic climension is now only the length of the duct.
It shoul.d be noted that Eq. G.f4) has the same form as
the simple relationship for a \rert j cal. flat surface, for
which an accepted value of the constant is 0.59 (10) .
Hence putting C r equal to 0 . .59 in Eq. (3 .11-) gi ves the
foll-owing relationship betlveen Nu anrl Ra for a duct- in
rvhich bouildary layer f low occilrs:
t
Nu 0.59Ra1-/4 (3.1s)
25.
The Nu-Ra relationships for fully developed flow
and for boundary layer florv, given by Eq. (5"I0) ancl Eq.
(3.15), are illustrated in Fig. 3.L. The actu-al rela-
tionship should be approximately asyrnptotic to the two
straight lines in this figure.
3 .4 lr{ethod of sol ving the f low equations
In order to obtain Nu for all. laminar f 10rv values of
Ra, and to study the flow development, Eqs. (2.10) to
(2.L3) rvere solved for the boundary conditj ons given ill
Section 3.7 by rel.axation on the CÐC 6400 computer at the
Univers it¡' of Ade laide .
Fini.te difference forms of Eqs. (2.10) to (2.ß)
are given in Appendix A and the r:elaxation p::oceclure is
describecl in Appendix B.
3.5 Theoretical results
Exarnination of Eq. (2'.13) will shorv that the
Prandt l nuiliber (Pr) of the f lu j-d is a paraneteÏ of the
problem. Since ai-r is the f triid in nany natu.ral - con-
vectirre pïocesSeS, rnost of the computations ì,VeTe basecl oll
Pr = 0,7 . Howe\¡€ïr to study the inf luence that Pr has
on the Ng-Ra relationship, Some calculations were made
u-sing othcr' \ralr-tcs of Pr.
10
Nu
0.1
0.01
0.1 10 103 i04
Fig. 3.1 Nusselt number against Rayleigh number for laminar fully dev-eloped flow and laminar boundary layer flow in plain-entry ducts v¡ithunifcrm surface temperatures
loz
Ra
.l05
D.J
Nu = 0.59 Rol/1 Boundqrv lqver f lowretotionship,' Eq.(3-15)
Nu = Ro/16
P¡' = 0.7l¡/t = 0
Fuity devetoped f [ow relqtionship,Eq. (3-10)
?'1
F-ig. 3.2 shows the laninar f 1ow relationship
betrveen Nu and Ra for Pr = 0.7. For srna1l Ra, the curve
will be seen to be asymptotic to the relationship
Nu = Ra/16, which was derived in Section 3.2 for large
values of the ratio p"/Tw. Ilolever, for large Ra, it is
asymptotic to
Nu = 0.63 na% (3.16)
and not Nu = 0.59 Ru\, which was derivecl in Section 3.3
for sma11 values of the ratio l,/rw. The discrepancy
between the equations, horvever:, ì-s insignif icant bearing
in mind that the constant 0.59 was based on the vertical
flat surface relationship.
It wil.l be seen in Fig. 3.3 that, for Pr >/ 0.'I , Pr
exerts only a sma1l influence on the Nu-Ra relationship.
In f act, changing Pr from 100 to 0,7 produces only a z,ero
to a 10% reduction in Nu as Ra inr:reases from 1 to l-Oq.
However, it will be obse::ved that the inf luetrce of Pr
becones j-ncreasingly signi.ficant as Pr decreases belorv
0.7; rvhen Ra = 104, a change in Pr from 0,7 to 0.01
(Pr = 0.01 is a typical value for a liquid metal) pro-
duces a 40eo decrease ín Nu. 'I'hus for Pr >¿ 0.7 a
relationship of the forrn Nu = f (Ra) is satisfactory.
10
1
Nu
01
0.0r
0.1 102
Ro
1oo
Fig. 3.2 Nusselt number against Rayleigh-number for larninar flowin plain-entry ducts with uniform surface temperatures
10310 105
È..)oo
Nu=
Nu = 0.63Ro itl
\----.-Boundory loyer f low
Pr = 0.7
l¡/t = 0
Fulty devetoped fIow
\:..l\u
r0
0.1
nnl0.1 'io2
Rc
Fig, 3.3 Nusselt number against Rayleighof 0.01, 0"1, 0.7, 10 and 100 for laminarwith uniform surface temperatilres
number for Prandtl numbersflow in plain-entry ducts
IU10 101 105 -
ÞJ(o
Pr - 10 ond 100
Pi- = C,l
Pr = 0,7
Pr = 0,01
\lt =0
30.
For Pr < 0.7, ho\.vever, tlie viscous forces become l.ess
important in relation to the inertia and buoyancy forces
(9) and Eire relationship has the form
Nu f (Ra, Pr) (3 .17 )
Elenbaas's relationship for Pr = ø (4) has been reproduced
in Fig. 3,3, and it is interesting to note that the curve
lies ver.y close to the relationship derivecl in the presetrt
study for Pr = 10 and 100.
Fig. 3.4 shorvs the dinensionless volume florv (a)
and total heat transfer (FIt) plotted against Ra. Both
curÌres at small val.ues of Ra will be Seen to converge and
approach II/8, rvhich was sh.otvn in Section 3.2 to be their
maxilnum value,
Heat t.ransf el frotn the bottom to various positions
along the duct is presented in Fig. 3.5. It will be seen
that h*/h, rapidly approaches unity as the r¡alue of Ra
decreases bel otv 10 and t.hat h*/h, is almost independent
of lta in the boundary layer regine"
Fig . 3 .6 compares the tenperature ancl velocity
profile.s at tJre top of the duct for a wide rallge of Ra.
In this figur:e both ful1y developed and boundary Taye'r
prof iles will be seen. The 'temperature growth j-n the
P¡'
t¡/t
Ht
Futty developedÍ[ow
= o.7
-0
e
U,1
1o-'
1or
lor
0,1 10 102 103 10 10'
Ro
Fig. 3.4 Dimensionless flow (a) and total heat transfer (Ht) againstRayleígh nunber for laninar flow in plain-entry ducts with uniformsurface temperatures
(¡¡F
Futly developedf Iow
Boundory layer flow
0.05
Pr = O.7
11li = 0
-uhl
0.8
0.6
0,t,
o.2
Fig. 3.5 Ratio of the(Hx) to the total heatplain-entry ducts r,\¡ith
10 102
Ro
heat transfert rans ferunif orrn
001 103 i,01 rou
up to a particular eLevationagainst Rayl.eigh numbei fcr
surface temperatures[ht)
(¡ìÈ.J
3.3.
Vetocityot topof duct
Pr = 0.7
i¡/l = 0
x/t=1
Ro
Ro=
1,2 x Temp ottop olduct
1.2x
08
0.6
0.4
0.2
Vetocity
utTemperoture
er1,0x10
t0x
llx
0
l.lxl05
0
Fig . 3 . ó Laminar-f 1ow vel.ocity and tenperature -pro -
fiÏes for various Rayleigh numbers at the top ofplain-entry ducts lvith uniforn surface tenperatures
0.5
R
0.5
R
34.
fluid along the axis of the duct for these Ra is shown
in Fig . 3.7. It is interesting to observe in this figure
that when Ra reaches l-03 the temperature of the fluid in
the centre part of the duct rj-ses barely above ambient.
Fig. 3.8 shows the pressure defects along ducts for
a sna1l ancl a large value of Ra. The pressu1e gradient
for Ra - 0.2 will be seen to be constant above x/9-=0.05.
This condition, it should be recalled, hias assumed in
section 3.2 to derive the Nu-Ra relationship ror fu1ly
developed f 1or,v.
Heat transfer per unit f lorv area
For given values of the length. and surface tempera-
ture of a duct, there will be a radius whicfu maximises
the rate of heat transfer (ht) per unit flow area (a).
The optirnun raclius of the duct is determi.necl by the Iìa
rvh.ic-h yields the naximum value of
h. h.firr
tr{
(3.18)
on the right hand side of Eq. (3.18) in
forni yields
a
Expressing lta
dimensionless
h Gr:pc vl,lT -T I'1¡/ O-ttTa
H
\AT
(3.1e)
35.
1
8.*0*
0.8
0.6
0.lr
0.2
00 0,2 0.lr 0.6 0.8
x/t
1000
120
Ro = 0.2 Ro = 17
l¡/l = 0
Pr = 0.7
Fig. 3.7 Tenperature clevelopment along the axesof plain=entrÌ ducts ivith uniform surface tem-perätures for various Rayleigh numbers
Rqtii t
o
Pr = Q.74.2 ,
00.39
2.8 x 104, Pr = 0Jlr/t =G=
00.01
0
1
Pox 102
-2
-3 0
L
Pox 107
-10
-150 0.2 0.4 0-6 0.8 1
xlt,
Fig. S.8 Distribution of the dinensionless pressure defect (P¿) in plain-entry ducts with uniform surface temperatures for laminar fully developed
flow (Ra = 0.2) and laminar bound.ary layer flow (Ra = 2.8 x 10u)(^c¡\
From Eq. (3.5) ,
ßgL (Tw-To)Pr
5t.
(s .20)
(s .2L)
(3 .22)
Ht Gr 2lI NuPr
and substituting Eq. (3.20) into Eq. (3.1-9) yields
h
a2
NuPr
pc v1, (Tro-T'o)tr
w
Multiplying the numerator and denor¿inator of the right
hand side of Eq. (3.21) by 'fv.a yields
)lta
a-
Nu
'Æ\JV
h.
Nu
1-/2
pc/{a p
Inspection of Eq. (3.?.2) will show that, for given values
of ," and (T\^r-To) and for constant fluid propert j-es, the
terms to the right of Nu/'Ãa are invariant. Hence it
follorvs that
(s " 2s)
It rvill be seen in Fig. 3"9 that, for a gì-ven Pr,
the pa1ameter Nu/'/Ra has a naxinu-n l¡alue. Hence a duct
can have a Rayleigh number lRa .l for which the hea-t' opf'1:.ta.nsfer per urit f]orv aTea j s a ïìaxjmttm for given values
a
0.20
0.18
0.16
0.14 ,ooot
= '10 ond 100
0.i2 0.7
0.10 c.1
^ /ìô0.01
0.06
0.04 l¡/l = 0
0.02,n310
Ro
Fig. 3.9 Parameter Nu//-R-a, which is proportional to therate of heat transfer per unit flow area, against Rayleighnumber for plain-entry dncts with uniform surface temper.a-tures
D
Nu
"i-R.
i0IU
(/.¡oo
39
of 1, and (Tw-To). Fig. 3.9 shor,r¡s that Ruopt is almost
inclependent of Pr for Pr >. 0.7 but increases as Pr de-
creases below 0.7 . When Pr has the value 0.7
Ra 32 (3 .24)opt
arLd therefore the optimun radius of the duct is given by
4
r 32 v2 t,w opt gß (Tw-To) Pr (3 .2s)
It is interesting to note that both the present.study for
Pr = 10 and l-00 and the work of Elenbaas for Pr = oo (4)
gave 30 as the value of Raona.
Reynolds nurnber
Based on the ra.dius of the duct, Reynolcls number
(Rer) is given by
urRe* = il w (3.26)rV
The nean flow velocity (rr) is
r, = # G.z7)IV
40.
Substi tuting Eq. (3.27 ) into Eq . (3.26) yj-e1ds
(3.28)
Expressing q in Eq . (3,28) in dinrensionless ternrs yields
ReT
w
Gra]T
tRe (3.2s)I T
w
orñ_ Ra a 9"R*, = ffi Ç (3.30)
Fig . 3 . 10 shorvs the paraneter (Rer Pr r*/ L) plotted
against Ra.
Critical values of R", can be estimated i.n t-he
fol-l.owjng nranner. Since both natural convectj.on in ver-
tical ducts and forced convection j n pipes produce para-
bolic velocity profiles for fu11y developed laminar f1ow,
it is reasonable to a ssunÌe that the trvo f 1ow concli tions
will have the same cri tical value of Reynolcis number.
Hence the crj-tical R", for fu11y develope<L f1orv in ducts
shoul<l be about 1150 (9) . However, for bounclarl' Iayer
flow the crit.ical R"' is i ikeiy to be less than 1150
because the f l.oi^¡ has a less stable velocit¡z prof i1e, as
shown in lrig . 3 .6 .
Re" Prfr
10
10
1
0.11050.1
+tts
10 10?
Ra
103 100
Fig, 5.1-0 Parameter RerPr f,i{/I- against Rayleigh number for laminarflow in plain-entry ducts with uniform surface tenperatures
Pr = 0.7
t¡ /l =0
42.
4. RESTRICTED-ENTRY DUCTS 'IVITI] IJN]FORM SURFACE'I.bMYhRAT UKT S
4.L Introduction
The entry restriction, as shown in Fig. 1.1(b), is
provided by leaving the lower part of the vertical duct
unheated. The sketches of the velocity profiles in
Fig. 1.1(b) shorv that the f luicl enters with a uniform
velocity and that the flow develops in the unheated sec-
tìon. If the length of the unheated section (gi) is
large compared with the radius of the duct (r*), the flow
will be fu11y developed when it reaches the heated sec-
tion. It should be noted that, â5 the unheated section
presents a resistance to the f1orv, the pressure defect
will increase along its length.
A restríc.ted-entry duct can be classif ied by ttlto
dimensionless parameters: the Ra1'1sigh number (Ra) and
the ratio of the lengths of the unheated and heated sec-
tions (9"i/9,). It should be emphasised here that ," in
the denoniltator of Ra is the length of the heated section
(1.), and not the overall length of the duct ([i+[) .
Considering the extreme values of the ratio 9"i/ f-,
if !,i/ L is macle equal to zero the duct will become a'
plain-entry <1uct and if the ratio is made infinitely
large the duct r,vi1l closely resemble an open thermosyphon
duct (3).
43.
As in the case of the analysis of plain-entry ducts,
the Nu-Ra relationships for fu1Iy developed florv and for
boundary layer florv wi11 be derived to corroborate the
general solutj.on for the extreme values of Ra.
4.2 Nusselt relationship fo:: fu1.1y' developed flow
and
made
have
the
for
To obtain fu11y develope<l florv in both the
the urrheated sections, the latios 9"/r* and L
J-arge. The large L/rw means that the duct
a snall Ra.
The nomenturn equation for ful1y developed
heated section is the same as that given by
the p1.ai,n-entry duct narnely,
heated
./9" area'
wi 11
flotv in
Eq " (3.e)
(4.1)
1S
the
the
aP.cl
AX1 !_q
ÏT
heatecl
The correspouding equation for the unheated section
derived in the sane rranner as Eq. (3"9), except that
dimens ionl es s temperature ( 0 ) i.s lnade equal to zero
instead of to one. Thus t-.he uromentum equation for
unheatecl secti-on becones
aP.d
ðX8Q1I
unheated
(4.2)
According to Eqs.
in Fig. 4.L, the
cons tant. Ilence
unheatetl s ection
(4.1) and (4.2), and as illustrated
pressure gradient in each section is
the pressure defect at the top of the
wi 11
But
in
and
H isx
which
44.
(4 .3)
(4.s)
that
equal
the
have
L
1S
!q]I 1
PdrX=o
and at the bottom of the heatecl section the pressure
defect is
d rX=o n9['
P L (4 .4)
Since the pressure defects given by Ec1s. (4.3) and (4.4)
are equal, it fo1lor,vs that
Ia L
I 1l. +-
Eq. (4.5) shotvs that for fu1lY
of a is a function of only the
the two sectiotrs of the duct.
Inspection of hlqs. (3.1)
for 0=1, a and H* have the
to llt for that part of the
f lorv is ful1y developed ancl
developed flow, the value
ratio of the lengths of
and (s.2) show
s ame value .
heated section
therefore Ht Q will
X45.
l-!ËA'tIDS Ë CTION
lJNil[i\Ti:.DSECl'ION
Fig. 4.L Diagrarnnatic clístribution of thediltensionles.s plessur:e defcc't (Pd) ilt a rc-stricl-eðr-etttry duct in rvhich the flow in each
section is laminar and ful 1y developed
T*
T"
ft
Fà'= P*B
the same value. Substituting Ha for a in Eq. (4.5)
yie 1ds
46.
(4.ó)iTrIt L1
8 1+9"
and substituting Eq. (4.6) into Eq. (3.6) gives the fo11-
owing relationship betrqeen Nu and Ra fo:: fu11y developed
larninar f low:
NuRa (4.7)
L6 1
It is interesting to note that as 9,i/L approaches zero
Eq. (4.7) reduce-s to Eq. (3.10), ruhich is the Nu-Ra re-
lationship for a plain-entry duct. Obviously, Eq. (4.7)
cannot be valid for all values of f.i/L. For example, if
!. i/ L hrere to approach inf ini ty , Nu , a ccorcling to Eq . (4 .7 ) ,
woul<l approach zero. In practice this worlld not be the
case, because an open- thernosyphon f l.olv (3) would develop
in the upper part of the duct and heat woul<l be dissipated.
accordingl)'. A discussion olr the Targest values of Li/ L
giving laninar flor^¡ will be deferred until a later section.
1,"
L+
47.
4.3 Nusselt relationship for boundary layer: flow
To obtain boundary layer flow in the heated section,
the ratio 9,/ r-. is made snall . Thus Ra will be 1arge.,wIt is reasonable to assume that the form of the
Nu-Ra relationship will be the same as that for plain-
entry ducts. Ilowever, the values of Nu lvil1 be snallerolving to the entry restriction reducing the flow rate.
Thus it folloivs fron Eq. (3"11) that the Nusselt rela-
tionship is givcn by
7-
Nu Ct- Ra (4.8)
wlrere Cz is a constant for a given value of f.i/ L. The
rnaximun value of C2 will be 0.63, i'rhich is the value
given by Eq. (3.16) for the case of 9"i/9, = 0. Florvever,
the lower linit of Cz can be obtained only by solving
the flow equations. This will be done in a Tater
section.
48.
4.4 }4ethod of solving the f1ow equations
The procedure used to solve Eqs. (2.I0) t.o (2.f5)
to give the general solution for the follorving boundary
conditions is described in Appendix B.
Boundary conditions
X=-L. and R=1 U=0, V=0, 0=0, I'd=0.1
X=-L and 1<R<01
u=$, v=0, o=0, P.1=o .
0 V=0, 0=1, Pd.0 .
-L <X<0 and R=1 U=0, V=0, 0=0, Pd.O.1
-L.<X<0 and R=01
0<X<L and R=1
V 0 0=0,
U
U
--ðRP.1'o
'
U
o
Ò
AR0,
(a)
(b)
(c)
(d)
0. (f)
(e)
0. (h)
(i)
0
(e)
O-<X<L and R=0 V=0,
X=L and R=1 U=0, V--0, 0=1, Pd.=0.
X=L and R=0 V=0,
-L <X<L a is ccnstant.1
Pd.0 ,a0_ãR-
dU
-=ARffi=oP.1= o '
49.
4. 5 Theoretical results
Fig. 4.2 shorvs Nu as a function of botlr Ra and the
volume flow (a) for Pr = 0.7. T'he top curve describes
the r:elationship f or plain-entry ducts (9"r/ 9" = 0) , and
branching off it are constant volume flow (a) curves for
restricted-entry ducts. In the fu111. developed regime,
Nu will be seen to be proportional to Ra for a constant
value of Q. The explanation for this is obtained from
Eqs. (4.5) , (4.6) and (3.6) . Accorcling to Eq.s. (4.5)
and (4-.0) , Q and Ha have the same value when the flow is
ful1y developed. Hence it follows from Eq. (3.6) that
Nu for snall Ra is propor:tional to Ra for a given value
of a.
The relatj.onship between 9"i./ L and Ra for various
values of a is shown in Fig. 4.3. As Ra decreases, it
wj,ll be observecL that the constant a curves become inde-
penclent of Ra and that the value of Q is deternined onl1'
b)' the ratio I'i/ L. This f inding is consistent with
Eq. (4.5). Further, ãt large values of Ra it will be
seen that the curves fot a < 0 . 021 terminate v¡hen 9" j / L
is between 2 and 4.
The Nu-Ra relatiorr.ship in a lnore practical form,
rvith f"i/ L instead of a as the third parameter, is shotçn
i0
Nu
0,01
0.1 10 1C2
Rq
103 101 r0'
Fig. 4.2 Nusselt number as a function of the Rayleigh numberand the dinensionless flow (a) for laminar flow in restricted-entry ducts Cn
0
Restricted-entry ducts, Q - const
Ploin-entry ducts, [/t = Q
0.021
0!
oF - n"
Uf,
0.2 0.1 0-05 0.
.-Qcurve terminetes
Futly developedf low
= 0.02
Boundory Iayei- f Iow
Iri t
'10
0.1
c.01
0.1 1 10 102 103
Rq
Fig. 4.3 Relationshi-p between +'he diniensionless f low (Q) ,
of the lengths of the unheated and heated sections (Li/ ,,)
Rayle:-gh number for lanj-nar flow in restricted-entry ducts
101
the ratioand the
ios
U1F
iir llt
r
+ - curve terminotes
+
+
Q = 0-35 t'l 0.2 0.1 0. 5 0.03 0.01 0.005
0.02
52.
in Fig. 4.4. (In cross-plotting this figure from
Figs . 4 "2 and 4.3, the naximun value of ,. j./ L j-n the
boundary layet regime was taken to be Z). T'he relatjon-
ship agïees with Eqs. (4 "7) and (4. B) , rr'hich ll'ere clerived
for fulJ"y developed flow and boundary layer flow respec-
tively. As the entry restrj.ction increases, the fully
develope<l flot¡ regine will be seen to extencl to larger Ra.
This is readily explained by the lact' that the flow
becones more developed as the florv rate decreases.
Iìurther, the maxirnum value of 9"i/ L for a given Ra
wil1 be seen in ]ìig. 4.4 to decrease progr'essively from
18 j-n the fu1ly developed flow regirne to 2 ín the bounc'lar:y
1a1's1' regime . Helrce i-t f ollolvs that if a duct has a
ratio of f,î/ 9., greater thalt the values shotvn in Fig. 4,4,
the f1or,¡ tçill nc¡t be lanlinar. The reason for tliis tr¡i1l
be discussed later.
A¡.r entrlz-Testriction can reduce the value of Nu
for fu11y developecl laninar florv by a factor of 19 as
shown by the follorving eqllations:
NuRa16
when t /e" (4.e)1
0
Ra764-
e"./9"t'Nu when 18 (4.10)
10
Nu
0.1
0.01
0.1 1 10 102 to3 101
Ro
Fig. 4.4 Nussel-u number as a function of the Rayleigh number
and the ratio of the lengths of the unheated and heated sections(f"i/ !,) for laninar f low in restricted-entry ducts
1os
Lrt(¡¡
Pf=co
Pr = 0,7
Ptain-entry ducts
Restricted- entry ducts
Pr = 0.7
I Boundory loyer flow
P I
oo
Open - therrnosyphon ducts,Lishthitt (3 )Lestie ond Mqrtin (11)
t/t 0.1 12 4 6 8rC1s ,yFutty cievetoped flovr
However, for laminar boundary
decreased by a factor of only
following equations:
Nu = 0.63 P,a% r^¡hen 9"
Layer flow, Nu can be
1. 5 as shown b1' the
/e" 0
54.
(4.11)
(4.1.2)
a
1-
Nu 0.48 Iìa rvhen L /t. J1
The laminar f1ow Nu-Ra relationshlp in Fig. 4.4 j-s
valid only if it lie.s above the relationship for open-
thermosyphon ducts. In order to test this criterion,
the theoretical relationships bet-.ween Nu and Ra f or open -
therntosyphon ducts which L\¡ere derived by Lighthill (3)
for Pr = o ancl Leslie and Martitr (l1) for Pr = 0.7 and 1
are also shown í.n Fig. 4.4 (the latter dicl not provicle
data on non-sinilarity f lorv) . I t rr¡i11 be observed that
f or Ra < 4 x 10 3 these curves li-e belotv those f or the
r:estricted -en'lr:y ducts . Consequently, f or any Ra belotv
4 x 103, a restricted-entry duct fol r,vhich the ratio t"i/ 9"
is within the range of values allolved by Ììig. 4.4 wj-ll
procluce laminar florv. 0n the other hand, the situation
for Ra > 4 x 10 3 is ìnore complicated. Cornparing Fig. 4,4'
with Fig. 3.3, it- wiil be found that for Pr > 0.7 tlle
curves for open-thernosyphon ducts 1ie just below those
55.
for: restticted-entry ducts. Therefore, for: Pr > 0,7
laminar boundarl' 7-ayer flow for ducts with small entry
restrictions appears possible. ln the case of Pr - 0.7 ,
it wil.1 be observecl in Fig. 4.4 that the opçn-therrnosyphon
curve almost coincid,es with the curve for !"i/L = 0.1.
However, Lighthil l (3) shorved th.eoretically and Martin and
Cohen (LZ) experimentally that for Pr less than about L,
tlre 1anínar bou.nclary Taye'r regime of l,he open-thernosyphon
duct cannot occul and its p1ac.e is taken by the ùnpeding
f lorv regime . In this regirtte inter'- action betrveen the
uprr'arc1 and downward movittg streatns reduces the rate of
flolr¡ in and out of the duct, atld this causes a markecl re-
duction in the rate of heat. transfer. The resulting
lower Nu-Ra relat-ionship persists until Ra is sufficiently
large for a turbulent boundary layel regime to esiablish
itself . Flence , it would a.ppear that the relationships
shorn/n in Fig . 4 .4 f or restrictecl-entry ducts f or Pr = 0. 7
are valid.
It rr'il1 be recalled in Fig. 4.4 that the 9"i/ L
curves ranging froln 4 to 18 are not continuous, Each
curve reaches a maximum Ra, which inc.reases with clecLeas i.ng
,,i/ L. An explanation fol this rcsult can be obtained b)'
consiclening Figs. 4"2, 4.5 and 4.6,
00
56 "
Ve,locity
Ux102
Tempc'roture
0
00
0 0
1
000
RodiusR
Fig. 4.5 DiLnensionless tenperature [0) and
veiocity (U) profiles for'laminar florv in a
restricted-"r.t',,1' cluct in which t¡e velocityó" the axi s fa.1l s to zero at the top of theduct
Bottom of heoted :;ection
0
I
Rq = 1180Pr = 0.'/0 = 0-021t¡/t = 3.37
i
0.05
VelocityTemperoture
x/[
00
57.
VetocityUx 102
Tenrperature0
00
1
00
1
000
RctdiusR
Fig. 4.6 Dinensionless tenrperature (0) and veloci-ty(u) profiles for laminar flov¡ in a restricted-entrycluct^ in which the vel ocit-y on the axis f alls to zerone ar the bottoni of the heateci section and fu1l y clev-el oped f 1ow occuïs near tlle t-op of the duct
1
Bottom cf heotecl section
l
0
t
RoPrati/ t
6l'
0.0216,'r
0:t
vVelocit
58.
In Fig. 4.2 it will be observecl that the constan't
a curves ranging fron 0.021 to 0.3 enanate from the curve
9"i/ L = 0 and continue into the fully developed f lorv
region. On the other hand, the cul'ves which have a
values less than 0,021 terminate in the boundary Tayet
region. In an atternpt to obtain Nu-Ra values beyond the
termination points, it r{as found that the relaxation
(descrj_bed in Appendix B) ran out of control. It rvas
conjectured that the flolr' sit.uati.on had changed when the
point of terminatj-on was reached' To explain t.his
phenonenon further, the temper¿rture and velocity prof ilers
in Figs. 4.5 and 4.6 f or the continuous curve a =' 0.021
wil-1 be examinecl. It will be seen in Fig . 4. 5 that the
velocity clistribution for Ra = 1180 changes from a fully
developed parabolic profile '¿-c the bottom of the heated
sectio:n to a boundary layer profile. At the top of the
duct the veloc ity on the axis ha"s f al1en to zeTo. The
ef f ec E on 1-he r¡elocity and tenperature prof il es of
reducir-rg Ra to 64 and- maintaiuing the sanìe value of
a (Q=0. 027) is shot,¡n in Fig. 4.6. Again the velocity
profile is parabolic at the enLty to the heated section,
but in this case the bounclarl' 'ra'yet prof i1e with ze{o
velocity on the axis occurs near the bottoln of the heatecl
59.
secti.on at x/t = 0.05. Further up the duct, v€locities
in the centre region begin to increase aud the f1uic1
reaclres the top of the duct with a fu1-1-y developed profile.
It should be noted in studying ¡-ig . 4 .6 that there i.s a
radiaT outward component of the velocity of the fluid
between x/.Q, = 0 and 0.05, and a radiaL inward component
above x/ 9" = 0.05 to the fornation of the fully developed
f1ow. On the other hand, by inc.reasing Ra above 1180
and maintaining a = 0.027, boundary layer flow rvas pro-
duced similar to that shown in Fig. 4.5 except that the
r¡elocity on the axis did not drop to zero at the top of
the duct.
Cons j-deration lvill now be given to the curve
a = 0,020, which according to Fig. 4,2 terminates at about
Ra - Z x 103 . At tltis value of Ra, the velocity pi:of iles
were similar to tho-se already shorvn in Fig. 4.5 with the
velocì ty dropping to zero at the top of the duct. For
the same a of 0.020 and a larger flow restriction to give
a sma11e:: Ra, boundary layer flow with zero velocity on the
axis was found to occur below the top of. the duct. Horv-
ever, instead of the velocity increasing above this
point as it did for the curve a = 0.02tras shown in
Fig. 4.6, velocities -Ln the centre region of the duct
ó0.
becane negative and the conputer solution ran out of
control- because reverse flow is incompatible with in-
creasing tenperatures. Under these conditions, laminar
f l.orv could no longer be sustailted and the f low degenerated,
as i11ust::ated in Fig . 4.7, i-nto rvhat r'1i11 be ref erred to
as mixing flow. Describing it in another waY, degenera-
tion occì,ITs rvhen the laminar boundary layer is sti1l
g,rolving but there is no flow in the centre region fron
which the boundary Tayer can draw fluid. This condition
can occur only when Q is less than 0.02I.
In Fig. 4 "8, showing the Lrliessure def ect along a
restricted-entry duct for a. snal1 Ra, it will be observed
that the pressure gradient in each sectiotr is almost
constant. This dj stribution of the pressure defect is
in acc-ord rvith the assunption nade in Section 4.2 to
derive the Nu-Ra relationship for fuJly cleveloped f1orv.
Heat transfer per unit flow area
The heat transfer per unit f1ow ãTe-'ã for a restrrc-
ted-entry duct is given aLso by Eq . (3.22) . It follolvs
fron Eq. (3.22) that, if the tenperature excess of the
heated surface and the lengths of the tlvo sections of
the duct ale constant and the fluid properties do not
I
\1\\r, \
(
f r fr\ÈÍ \\
.N /i-\
61.
Mixinçi f lc¡w
Heatedsect ion
Larninar fIorv
Unhentedsection
Fig. 4.7 Sketch sholing the clegenerati.onof laninar fl.olv into nixing f1or,u in a ductwith a 1.arge entry rest-riction
T,
I I I
I
I
t.
linheci.ecisection
Ro = 1.8, Pr = 0.7
i¡/t = 2'60 = 0-'i
L
iìRatedsection
c
-0.1
-c2
-0,3^o -u'b -0,/, -'J!1
Fig. 4.8 Distribution cf the pressure defect (P¿) in a
restrr-cted.-entiy duct in which the flow is laminar and
fu1-ly d.eveloped j-n each section
0 4.2 0.4
X
o'\N)
63.
vary, the heat transfer per unit flow area will be pro-
portional to NLy'/M. In Fig. 4.9 it will be observed
that the optirnurn value of Ra (O"opa) increases with the
ratio Li/t, ancl coincides with the maximum values of Ra
for 9"i/L > 6.
Reynolds number
The relationship between Reynolds number lRer) and
Ra is the same as that for plain-entry ducts and is given
by Eq. (3.30). As shown in Fig. 4.10, Rer decreases rvith
increas ing values of the ratio t"I/ !,. This is explained
by the fact that an entry restriction reduces the floiv
rate
/P^Nu
o.i8
0.16
0.i4
0.12
0.1c
0.08
0.c6
0.04
4.02
10 IUI
10CI
С
Fig. 4.9 Paraneter' Ì\iu/rÑ-, whj-ch is proportional to the rateof heat 1*ransfer per unit fl-or,¡ atea, agalns+, Raytreigh numberfor rest::icteci-entry <iucts with uniform surface temperatures
o.Þ
,/l
?r = 0-7
tt
I
1.
0
6
4
i0I
\\
åooot
x - curve tei'mino
-)
1G2
10
ne erf
00î
Fig. 4.70 Paramqterconstant values of L
ducts
i02
Ra
103L
l^
RerPr rn /L against Rayleigh number for/ 9" for laminar flow in restricted-entry
10t0.1
o,\úl
1
)v
t;/[ =
¡
L
F
15
i8
Pr = O,7
ó6.
5 UNIFORM SURFACE HEAT I-LUX DUCTS
5.1 Introduction
fn this section vertical ducts of circular cross-
section rvith uniforrn heat f luxes will be considered.
As shorsn in Fig. 1-.1(c) it wj 11 be assumed that the
fluid enters the duct rvith a uniform velocity and thatthe unknown temperature of the surface increases from
anbient at the botton of the duct.
5.2 General Nusselt relationship
In orCer to obtain the relationship between the
Nusselt and Rayleigh numbers the follor,ring theoretical
analysis is a<lopted.
The climensionless volume florv (Q*) and the rate of
heat transfer (FIi) from the bottom of the <1uct to eleva-
tion X* are given by
1
Q* = 2II' U*R dR
o
(s.1)
and
1
H*X
o
2r, U*O*R dR (s.2)
67.
HI can also be expressed in tetnts of the dimensionlessx
uniform heat f lux. (F) as fol1or^¡s:
(s,3)
Since the surface temperature varies along the duct,
a refer:ence tenperature has to be adopted. The mealì
surface tenperature given by
9"
fT -T ìdxt l^tx O'T o (s.4)
provides a suitable reference tenperature (14).
The Nusselt nurnber of the duct (Nu*) with the
radius as the characteristic dimension is given by
rNu* =
w (s.s)
which reduces to
frl\u^ =
w (s.6)(T -T ìK' hr-m o'
lr{ultiplying Eq. (5 . 6) by the dimensi onless mean surf ace
temperature (0i*), r,thich is given by
FH¡t = 2TÍ. X*x
1üm
k-
0* =t^/m
(T -T lkt r,vm o'tr
ltr
68.
yi e lds
Nu* = +_ (s.B)U
wÏn
By definition, the overall dinensionless rate of
heat transfer (FIT) is
htHt
FIït
(5.9)T
2TlRã-f
9"rhr
Substituting 2[t.u t, f for ha in Eq. (5.9) yields
(s.10)
where Ra* is equal to Gr* Pr.
It should be noted that dividing Eq. (5.10) by the dimen-
sionless surface area (ZIIL*) and replacing L'¿r by 7./ Grx
yields the dirnensionless heat flux (F) . Thus
Lr = Pr (5.1-1)
Eq. (5.1L) states that the dimensionless heat flux is
sinply the reciprocal of the Prandtl number.
From Eqs, (5.8) and (5.10) the general Nusselt relation-
ship is obtained namely,
Ra* nä
zR-o-r-*wn
Nu* = (5.1,.2)
69.
The strong sinilarity between Eq. (5 " 1-2) and Eq. (3.6)
for UST ducts should be noted.
Although hitherto it has been irnplied that the
uniform heat fl-ux (f) is known, in practice only the
temperatul'es along the surface may be available. There-
fore, Nusselt number relationships will be clerived for
both a known uniform heat flux ancl a knorvn surface
tempera,ture.
To corroborate the general solution, lantinar flow
Nusselt nunber relationships for extreme values of the
RayleigXr nurnber will be established in the sections that
folloiv-
5.3 l'åusselt relationship for sna1l lìayleigh numbersb¿rsed on the uni.forn heat f lux
$ma1l Rayleigh nunbers are obtained by naking the
ratio n/r-- sufficiently 1arge. In contrast with UST'ürduc.ts, fully cleveloped florv cannot be achieved in USHF
ducts. The rnean ternperature of the fluid rvil1 always t.ag
behind that of the surface because the fluid is receiving
heat along the entíre lengtìr of the cluct. Llowever, to
simplify the analysis, the velocity distribution lvill be
assumed to be f,u1ly devel-oped and the raclial temperatu::e
gra-dients 1"ó be negligible. Thus tl'le dilnensionless
vertical component of the vel ocity at a radius R willbe given by
u* î*,t Rr)
70
(s.13)
(s.14)
and the dimensionless tenperature of the fluid at eleva-
tion X* and radius R by
0*= 0rtx
For flow thatequation, Eq.
= 0*WX
is alnost fu1ly developed, the nomentun
(2.75) recluces to
âP4d
ã-xFa 2u*
Uã-R-Z äR
1R
d.&
+ 0*
Substituting Eqs. (5.13) and (5.l-4) into Eq. (5.15)
yie lds
aP{d
ãFf 0*
(s.1s)
(s.16)IVX
9q]I
For a uniform velocity pi:ofile, 0$* will j_ncrease
Tinea'rly along the length of the duct , and hence itfollows from Eq. (5.16) that APå/AX* rvi11 also increase
linearly. FurtherltÌore, since På is zero at both the top
and tlre bottom of the duct, ðPå/äX*, as illustrated inFig. 5.1, rvill change sign at nid-elevation (X* = L* /Z).
onci
*dPo
dx-
oi
7L.
0.5xll
00.5 1
x/t
5.1 Diagramrnatic distribution of the sur-'temperature (0rI*), tlte fluid temperature (01),
00 I
0
Fig .
faceand
heatthe prcssule gradientf1.ux dur:t for ¿l srnall.
(d På/.1xo ) in a uniformnurnber:llayl eigh
'7?
Also at nid- elevation, since 0rT* increases 1i.near1y f rom
the bottorn to the top of the duct, the temperatur:e of
the surface will be equal to the mean surface tetnperature
(e-**) . Henc-e Eq. (5.16) at micl-elevation reduces to' w]n'
0*wln
(s.17)
The rate at which the lower half of the surface dissipates
heat (Hj^) is given by- me'
H]^ = Q*o-*_ (5.1"8)me ' wrn
Using Eq. (5.18), Q* in Eq. (5.17) can be replaced by
II* / 0* to vieldme' hrm
gqJI
Since the heat flux is unifo::n along the duct
H*me
H--Tt
Using Eq. (5.10) , Eq. (5.20) becomes
Lr* = Il^'me lìa*
and substituting Eq. (5.21) into Eq. (5"19) gives
0*I4ln
(s.1_e)
(s.20)
(s.21)
0*wrn
(s.22)
Fina1ly, substituting Eq. (5.22) into Eq. (5.8) yields
the Nusselt relationship
73.
(s.23)
(s .24)
Nu*
Eq. (5,23) shows that, for smal1 Ra*, Nuo is proportional
to 'Æx. In the case of the UST duct, it will be re-
called that Nu is proportional to Ra fclr sma11 Ra.
It is interesting to note that according to Eqs.
(5.17) and (5.22) the relationship between Qx and small
Ra* is given Úy
QN
5.4 Nusselt relationshi for bounda Ia er fl.ow basedon e un1 oTm eat
Boundary Iayer flow will occur if the ratio l"/rw is
ma.de sufficieitly sma-11. Cousequently, Ra*' will be
1arge. As in the case of the usT d¡ct, it is reasonable
to assuflre that the rate of heat transfer will be the
same as that from an equivalent vertical f.7.at surface.
Therefore Nu* will be independent of the radius of tlie
duct, and this requirenent is achieved if
1
5Nu* = C: (Ran) (s.2s)
where Cs is a constant" Expanding
(5.25) and multiplying through by t"
both sides of Eq
yields
74.
(s.26)
the
and
ver-
for
N"'f ca (Ra[)I
the characteristic dinension. Sparrotv
derived the follorving relationshi.p for a
surf ac'.e dí s s ipa t ing a unif orm heat f 1ux
where
duct
Gregg
ti c-a1
Pr=
15 nOW
(14)
f1 at
0.7:
the subscript f, indicates that the length of
IN"l = 0,62 ¡na[)T (s.27)
Eq. (5,26) r,rri11 be seen to have the sane form as Eq.
(5,27), Ilence, if Cg in Eq. (5.25) is assurnccl to be
ec¡ual to the constant in Eq . (5.27) , the relationship
between Nurtand Ra* for laltinar bor-rndary layer f1ow in a
vertical duct is given approximately by
Nu* 0 .62 (Ra* ) (s.28)I5
5.5 Nusselt relationshi f or :;mal 1 Ra l.ei h numbersase orì t e nean suT ace temperature
In order to obtain the Nusselt nutnber relati.onship
based on the mean surfa-ce temperature, it is necessary ta
establish a Grashof number basecl on the mean surface
temper¿-Lture ¡C;r+) . As shown by the f ol.lorvi ng set-. of
/5.
equations, Gr* is obtainecl by dividing Gr* by Nu* (14) .
ï
gßfr 5 lT -T )k'wn o'Gr w (s.2e)v'Rk frw
+ Gr*Nu*G
+
I s ß (rwm-To)r$Gr
V o\)
Hence the Rayleigh number based on the mean surface
temperature is
Ra*ñü*
(s.30)
Introducing Eq. (5.30) int.o Ec1. (5.23) yields for sna11
Nu* =Ra-T- (s.31)
comparing Eq. (5.31) with Eq. (3.10) will reveal that the
Nusselt number of the USIIF duct is twice that of the UST
ducl-. The following analysis ivi1l explain the reasoll
for thi.s.
Multiplying 0* by Nu* yields a dinensionless tent-+perature (0-) rvhich j-s based on the nean surf ace tenp eTa'
ture of th.e cluct as shown by the following set of
equations.
+Ra
+Ra
+
r
+e
+0
0 ?t Nu*
76.
(s.s2)f
w(T 'T lk'wn
T-To
T -Thrln o
It was shown in Section 5.3 that Ofu for small Ra* is
approximately equal to the tentperature of the surface at
mid-elevation, which in turn is approxinrateLy equal to
the temperature of the fluid at the sane elevation.
Hence it f o1lows frorn Eq. (5 .32,) that the dinensionless
fluid temperature at mid-elevation is given approximate-
Ly bv
0+ +
wn1 (s.33)
me
Since, â.s shown in Fig. 5.1, the temperatures of the
surface and the fluid will incr:ease llnearly lvhen the
f 1ow j-s approximately fu1ly developecl, the temperature
of the fluid at the top of the duct is given by
+
wln(s " 54)20
Fr:orn Ec1. (5.17)
+0
e
+ot
Q,r = å ui*
?
(s.3s)
77.
a (s.3ó)
Horvever, the product of Q* and Nu* gives the dimension-
less flow (Q+) based on the mean surface temperature, as
shown by the following set of equations.
Q*
QN ¡-f--e¡- Nu* (s.37)
= Q* Nu*
and using Eq. (5.8) to replace Ofo in Eq. (5.35) yields
I8-
*Nu?t
Substitutì.ng Eq. (5.37 ) into Eq. (5.56) yields
Ig
The heat dissipated by the sur:face of the cluct
H;=Q*o;
Ilence using Eqs. (5.34) and (5.38) , Eq. (5.39) becones
Q*
(s.38)
r_s
(s.5e)
(s.40)
In Section 3.2 it was shown tha| the heat dissipaled by
the sur:face of a UST duct in which the flow is ful1y
Q*
Hi =l
78.
developed is
H
Flence it follows
for sma11 values
An inspection of Eqs. (5.40) and (5.41) will show that
(s .42)
Eq. (5.L2) can be written
Ra HNu* = 2II
(s.43)
]I8-t
2H,Hi
(s.41)
from Eqs. (5.42) , (5.43) and (3.6) that,+of Ra ancl Ra, Nu* is trvice Nu.
Since Hl = H] (N'*) 2,XX
+ +
t
5.6 Nusselt relationshi. for boundar 7a er flowase on e mean sur ace temperatttre
To obta.in the relationship between the Nussel t
number and the Rayleigh nuntbe:: based on the rnean surface
temperature for boundary 1-ayer flow, Eq. (5.30) is
introcluced in Eq. (5.2"8) . This yields
Ntrt = O.ss(Ra+)k (5.44)
An inspectj-on of Eqs. (5.44) and (5.15) will show that
for boundary layer flolv the Nusselt nunbers of a USHF
79.
duct are aLrout 7eo less than those of a UST duct. This
neans that using the mearr surface tenperature in the
UST duct relationship (Eq. (3.16) ) rvi1l give a good
estinate of the Nusselt number for a USFIF duct for
boundary layer f1ow. Sparrorv and Gregg (14) reported
the same finding for a vertical flat plate with a uniform
heat f1ux.
80.
5.7 Method of solving the f1or,¡ equations
In orcler to obtain Nusselt numl¡ers for all laminar'
florv values of Rayleigh nunbers Ìrased on both the uniform
heat flux and the mean surface tenperature, Fìqs. (2.14)
to (2,77) were solved for the follovring boundary condi-
tions.
Boun<lary conditions
X*=0 and R=1 U*=0, V=0, o*=0, På=0. (a)
u* -$i V=0, 0*=0, På=0X*=0 and O-<R<l-
0<X*<L* and R=l U*=0, V=0,
0<X*<L* and R=0 V=0,
x*=L* and R=1 Ux=O, V=0,
X'Ì=L?k and R=0 V=0,
0-<X**<L* Q* is constant.
The f in i te clif f e rence f orns of
and (7,.L7) are gi.ven in Appendix D,
procedure is described in Appenclix E
=0,ðR
Eqs. (2,L4) , 12.15)
and the relaxation
oä'o '
au'*ðR
d u'"AR
(b)
(c)
=0 . (d)På'o '
=0,
P{=0.o
Pä=0, au*ðR
(e)
=0. (f )ð 0'*
(e)
8l- .
5.8 Theoretical results
Relationships between the Nusselt number and t.he
Rayleigh number for Rayleigh nuntbers based on both the
uniform heat flux (na*'; and the mean surface temperature
(Ru*) are shown in Figs. 5.2 and 5.3 respectively' It
will be seen that the curve basecl on the uniform heat
flux agïees satisf actorily with Eqs. (5.23) and (5 ' 2:8)
for the extrerne r¡alues of Ra* and the curve based o11 the
mean surf ace tentperature wi-th Eqs. (5.31) and (5 .44) f or
extrerne l¡alues of Ra*. For bounclary layer: flow the gen-
eral. 3olution yiel.ded the following relationships:
Nu* = 0.67 Ra*s (s.4s)
and
Nu* = 0.61_ Ra+L4 (s.4ö)
It is interesting to note tha.t t.he relationshì.ps based
on a vertical flat surface dissjpating a uniform heat
f lux, Eq.s . (5 .27) and ( 5 . 44) , have constants that are
about 1.0% smaller than those in Eqs. (5.4 5) and (.5.4é) .
The usT duct relationship is also p1 otted in Fig.
5.3 and it will be seen that for boundar¡' Laye'r f low the
two curve-s are almost identical. This means, of courSe,
that for bounclary Layer florv the Nusselt number relation-
10
Iti\ U
0.1
0.0i0.1 10 ,"02
Rd
1^3 1o¿ 1os
Figfor
5.2 Nusselt nunber (Nu*) against Rayleigh nu¡nber (Ra*)uniform surface heat flux ducts
ooñ
Nut= 0'67 Ros
Nu-= ,E
1r5
Pr = 0.7Flow resembtingfulty Ceveloped flow
Boundory loyerf low
10
irU
iN u)
0.1
0.Ð1
u-l 10
Fig. 5.3 Nussel-t number [Nuo)on th.e nean surface tenperatureflux <iucts
( Ro)
against Rayleigh number based
[R"*) for uniform surface heat
io2
RG'
103 1o' '¡0u
oour
Ro*
Nu*= O'61 Ro't/a
Uniforrn surfoce heot ftuxducis
Nuo=
ÞBoundory loyer flcw
Unifoi'm surfoce temperoture ducts (Fí9. 3.2),Nu = f (Ro)
Pr = 0.7Flow res.-rnblingfully devetoped f low
84.
ship for UST ducts could be used if the entire surface
\^rere assumed to be at the nean surface temperatu1e.
Fig. 5.4 compares the dinensionless temper:atures
along the surface (OiT*) of the duct with those along the
axis (0å"). It rvill be observed that for small Ra*
these tenperatures increase linearly as assumed in
Sectiol 5.3. Llowever, for large Ra*, it is interesting
to note that the temperature di.stribution on the surface
approaches that of a usHF vertical flat surface (14).
This findi.ng is in accord with the assumption nade in
deriving the relationship between Nu* and large Ra* in
Section 5.4.
In Fig. 5.5, Qn, Ht, e$t and 0'ï, are plotted
against Ra*. unlike Q and tl, for usT ducts, which are
illustratecl in Fig. 3.4, Q* and rI[ do not asymptoLtcaLLy
appr:oach a comnon value at sma1l Ra* . In fact, Qo' f or
small Ra* lvill be seen to approach the relationship
give' by Eq. (5.24) nanely, Q* = I/ ÆIiä*, and Hf for a1'I
Ra* to be inversely proportiona.l to Ra* in accordance
with Eq. (s.10) .
Figs. 5.6 and 5.7 shorv the stages of developtnent
of the tenpe::ature ancl velocity in ducts having a snal1
and a large \¡alue of Ra* respectively. The shapes of
Surfqce
90
c.08
1.2
( 14)
5.5x 103
Ro!= 2.7x105
Vei'ticot f lot surfoceSporrow cnd Gregg
5.5x
90
't ')t,L
Pr = 0.7
103
Ftuid onoxrs
0.8
0.6
0.2
0.t
oå¡*Uwt
Q;"
=-U*t
0.1*
0.6
0.2
o.4
000 0-2 a'4 0.6 0.8 1
x/t
Fig. 5.4 Distributions of the dimensíonless temperatures of thesurface (0**) and of the flr:irL on the axis (0Ë*) of uniforn heat
flux ducts; the distribution of temÐerature of the surface of a
rrert ical' f 1at surface (14) is shc',,'ll f or coínllarison
i.¡l
ü9*t
102
ic-2
g*
^*
*iì.
m
0.j IU l02 103 io1 105
IR c
Fig. 5.5 Dimensioniess flow [Qo), total heat transfer (HT), tem-perature of the surface at the top te*t) and mean surface tenpera-ture t0¡*) for uniform surface heat fiux ducts against Ra*- 'wTl'
ooo\
Veloc ityTemperoturex/[ =1
0.s
0.1
Ro' = 0'08
Pr = 4.7
2
22
21
J
12 ?0
'11
210
03
2/l
Vetocîty
^U¿
gl.
3
Temperoture
e2
0
2
0'l 0.5 0.s0
Rodius
R
Fig. 5.6 Dimensionless temperature (e*) and velocityprofiles in a uniform surface ireat flux duct for a srrraRayleigh nunber
(u* )11
oo-¡
c.1
Temperoture
x/l =1
0.5
Vetocity
Pr = O:7
l*2
Temperoture
e'x 10
21
01
t4
Vetocity
Unx'iOa
20
20r,
02
000.5 0.s
Rodius
Fig. 5.7 Dimensi.onless tenperature (e*) and velocity '(U*)prófiles in a uniform surface heat flux duct for a largeRayleigh nurnber
0
o
aocþ
89.
the temperature and velocity profiles will be seen to
justify the assumptions nacie in Sections 5.3 and 5,4
to derive the Nu*-Ra* relationship for the extrene
values of Ra*.
Fig. 5.8 shows the pressure defect along clucts
having a sma1l and a large Ra*. The zero pressure gtad-
ient which will be observed to occur half rvay along the
duct rvith the sma11 Ra* is in accord with the assumption
nac1.e in Sect ion 5 . 3 .
Fleat transfer per unit flow area
The USIIF duct, unlike the UST duct, does not have
a radius for which tlie heat dissipation per unit flow
area is a rnaxiniurn. This can be shown by the following
analysis.
The heat dissipated per unit area is
h lt,Fi-Tz
w
and substituting 2Tl,r*
of Eq. (5.47) yields
9"f for on the right hand side
t (s.47)a
h.
hZ e"f
at
rw
(s.48)
Ro* = 0.C8, Pr = 0.7
o" -- L-o
Ror = 2.7 ¡ 1gs, P¡ = 0'7
Q* = 0.001
I
dP.
¿x'
10
5
0
-i0
-20
R
Pjx tOe
-10
-15o 0.2 0.8
x/t
Fig. 5.8 Ðistribution of the dimensionless pressure defect(På) in uniform sulface heat fLux ducts for a srna11 a:ed a
large Rayl.eigh nunber
0
tr
-10
dP;I
dX
IPd
-30
c
0.60.lr
(oO
91-.
From Eq. (5.48) it will be seen that, for given
values of the length of the duct (!,) and the uniform
heat flux (f), the heat transfer per unit flow area is
sinply inversely proportional to the radius of the duct.
Reynolds number
Based on the radi.us of the duct, Reynolds number
of the uniform heat flux duct is given by
ïw (s.4e)
unRe*r V
Expressing'u* in terms of Q* yields
tlr^ n
(s. s0)T
w
oï
Ra* (s. s1)n T r
't^i
Fig. 5.9 presents the relationship betrveen (IfeN Pr rr/ 9')
and Ra*.
Since for small Ra* Fig" 5.6 shows that the vel-
ocity profile is almost fully developed, it is reasouable
to a.ssume, as in the case of the UST duct, that the
critical value of Reynolcls number v¡i11 be approximately
the same as that for forced-convective flow in pipes.
¿
ok
r
-,k =ï
Re
Re
102
10
Ref Pr r*/l
0.1
,l
Pr = Q.7
ñ1 lc2
Rqr
Fig. 5.9 Parameter RelPr fil/!- again-st Rayleigh number forlaminar flow in uniform surface heat flux ducts
C
l0'd,l03101
(.oNJ
93.
Thus for small Ra*, the critj-cal Re| will be approx-
imately 1150 (9). However, for large Ra*, the critical
value is likely to be sma11er, because the velocity
profile has a less stable shape as shown in Fig. 5.7 .
94.
6 EXPERI}4EN'TAL STUDY
6.1 General
Experiments were conducied to confirn some of the
theoretical f inclings. The tests were carr j.ed out on
plain- and restricted-entry ducts lvith unifcrm surface
temperatures.
With air as the working fluid, a range of Rayleigh
numbers vías studied which extended from the fully devel-
oped regirne to the boundary layer regime. The stucly
inclucled tests with entry restricti ons tr'hich were larger
than the theoreti.cal maxima for laminar f low shorvn i.n
Fig. 4.4 and, for compari.son, tests on open-thermosyphon
ducts.
Gross hea.t transfer data r./ere souglit and also tem-
peratures of the fluid along the axis" Owing to tireir
snalJ-ness, no attempt was nade t'o measure pr:essure defects
and veloci.ties.
6.2 Apparatus
Three rluct.s of different dianeters, details of
Table 6 " 1, lvere recluired to study a
Iìa from 10 to 1-0't.
wlrich are gi-ven
range oF values
1n
of
95.
Table 6.1
Experinental duçt¡
Characteristicof the duct
Noninalinternal dianeter
of the duct(in. )
',LL'41 ZA'r.li
Internaldiameter
(in.)
Heatecllength(in.)
l{a11t.hickne s s
(in. )
It{aterial
Externalsurfac ef inis;h
0.999 2.240 3.737
48 48 48
0"r2,8 0.128 0.128
Aluminium
Hard anodised
As illustratecl in Fig " 6.1, the ducts were heated
in three sections by independent electrical. resj-stance
elenents and the external heat losses l.\rere reduce d by
sulrounding the ducts with fibre-g1ass insulation. An
insulation thickness of 6.5 in. was cho.sen as a suitable
compronrise between satisf actory insulati on and the oveT-
al1 size o1-- the dttct. Fig. 6.2 shr:r'¡s the lotver' heating
96
Leads toheoting etements
Therrnocouplewires
2
3
I,
5
1 - Ductlnsutation
f ndependenthectingelements
6 - Betl-mouth7 - Epoxy resinI - Thermocouple
junction
The rnrocouplejunct iorr
rnountirrg
Fig. 6.1 Sketchsurface te:nPeratuthree sections bYe lenents
of an experinental uniformre duct t-hat i.s heatecl in
independent resistance
97
Fig. 6.2 Lower Part of the 3
without the insulabion
3T
in. dianeter duct
98.
element of the 3'lt* in. dianeter duct, and Fig. 6.3 the
same duct after the addition of the insulation.
The design of the heating elenents was l¡ased on
calculated values of the 1oca1 rates of heat dissipation
and hea'[ losses through the insulation. T]re details of
the elements are given in Tables 6 .2 , 6 .3 ancL 6 .4 . To
forn the elenents, the bare resistance wire v{as wound
directly on to the hardlanodised external surface of each
of the two srnaller dianeter duc'ts . Thj-s rùas poss ible
because the harcl anodising provi-ded a surf ace tliat was
electrically non-conducting. After the turns of a
coil rvere correctly positi oned at 'the specif ied pitch,
they were held in place by an epoxy resin adhesive. To
position the coils tnore readily on the 37r- ín. dianeter
duct, shallorv helical grooves l^/ere machined arouncl the
wall.. Unf ortunately, the edges of the gl'ooves ï/ere
ina c1e quat e 1y raclius ed and cons equent 1y the anodi s ed sur -
face was dama-g;ec1 in several places v¡hen the el-enents 1Á/ere
being rvonnd. The problen wa.s ovetcome, as shown in
Fig. 6.2, by insulating the v¡ho1e surface wit.h fibre-glass
tape bef ole reruin<ling the elements.
The 240 vo1ts , 50 cycles per second , A.C . mai.tls
provi-ded the po.hrer: supply, and, as shown in Fig" 6.4,
99.
Fig. 6
glass.3 Sf; ,n. cliameter duct surroutlded by fibreinéulation
Vqríoc S tep - downtrq nsf orm e r
3 Amp.fuse
Heotingetement
Top
fd ídd te
Botiorn
?40 V. A.C.
?/.0 v. A.c.
2/r0 V. A.C. 50 V. rnox
fig. 5.4 Circuit for controLling the three independentheating elements H
O<)
tlil
! R
1
the voltage across each element rvas controlled by a
variac. Since the breakd,own voltage of the anodised
surface cf the clucts was betlveen 20C and 300 vol ts, it
was decided for safety reasons to linit the maximum
voltage acrosS each element to 50 vo1ts. Calcul.ations
slrowed t.hat the induc.tive react¿lnce of the elements was
negligible compared with the resj-stance, and therefore
voltages and currents were in phase.
Three gradecl heating elements on each duct provided
adequate flexibility for controlling the temperature of
the surface; examples of the unifornity of the surface
temperatures achieved are shown jn Fig. 6.5.
Temperatures along the internal surface of each
duct r\rere measured by a number of 30-gauge copper-
constantan t-hermocouples (Leeds Ê Northrup catalogue no.
30-5-1). The thermocouple junctions were cenented in
grooves nachined jn the rval1, as jllustrated in Fig. 6.7;
this method of mounting placed the junctions close to the
internal surface and allorved the wires leaving each
junction 'to l¡e run along an isotherrn fo'r a. shor:t distance.
The locatioirs of the j urrctions aTe given in llable 6 . 5 .
It rvi11 be Seell that five thermocouples lveIe placcd under
each heatì-ng elelttent on the 1 in. diameter: düct, which
\{ia-s the f irsl- of thc three clttct.-s to 'be br-l j-.1 t. Flolver¡er,
160
150Surf oce
Temperoture"F Bc
70
10 c.2 4.4 U,O 0.8
xllFig. 6.5 Examples of the uniforrnity of the temperatures achievedalong the surface of the 214, in. dianeter duct;' the temperatureswere measured by thermocouples located in the wal1 at the endsand the niddle of each heating element H
b.J
Ii/ i
Pr c.7
Ro = 530
Ro = 182C++--4
44- +
+
+
+
U
Duct - 2.24in. dio. x 48 in. long
+ã +-++
+
l_05.
'Tab1e 6.2
Heat elernents for the 1n dianeter duct
Heatingelenent
Locationfrom bottom
of duct(in.)
Wirematerial,
Numberof
turns
Windingpitch
Iturns/in. )
I{Iiredi a:n .
(in.)
Top
Midd le
Botton
48 .0to
24.t
4.8to0
Eureka 0.056 ?,Ls
Eureka 0.040 734
Eureka ' 0;A?.2 40
L3to3
3toI6to11"
24.0to4.8
104.
Table 6.3
Ileating elem ents for the 2h in. diameter duct
Heatíngelement
Locationfrorn botton
of duct(in.)
I\¡iremateri a1
Nunberof
turns
Windingpitch
(turns / in.)Wirediam,(in')
Top
Middle
Bottom
48.0to
36.0Nichrone 0.028 1-20
Nichrome 0 . 064 2,7 0
Nichrorne 0.056 90
L4to10
9to1.2
6to2g
36.0to9.0
9.0to0
1_05.
Tabl.e 6.4
Heatin elenents for the in, diameter duct
Heatingelement
Locationfrom botton
of duct(in.)
Wirematerial
Nunberof
turns
Itrindingpitch
(turns/in. )
trViredíam.(in"l
Top
Middle
Botton
48.0to
36 .0
36 .0to9.0
Nichrome 0.036 IAz
Nichrone 0. 056 135
Nichrome 0.048 5B
10toI
5
6toI
9.0to0
106.
tests with the duct indicated that only three therlno-
couples under each element would have been qui-te satis-factory and therefore a total of nine thermocouples
instead of fifteen was used on each of t.he two larger
ducts.
Ambient aLT tenperatures r^/ere obtai.ned from tr'¡o
thernrocouples located 20 in. from the axis of the duct
and 10 in. above and below the heated section.
Temperatures along the axis of the duct were meas-
ured by a travelliirg thernocoupl.e, ivhich is illustrated
in Fig. 6.6. It shonld be noted that the lower actuating
cord of the travelling therrnocouple was posi tioned to
move along the surface of the cluct so as not to disturb
the f 1or,u approaching the j unction.
The thermocouple E.M.F.'s hrere measured by a
potentiometer, which was coupled to a nu1l detector to
give greater sensiti-vity. A recorcling potentiometer was
used to observe the directjon of surface tenperature move-
ments when equilibrium conditj ons rrere being established
and to reco::d the unsteady temperatures that were encoun-
tered within the duct.
In planning thc tcsts on rcstrictcd-cntry ducts, a
serious problem concerning the overal.l lengths of the
duc:t-s was cnc.oun1".erecl. Duct-s that rrrere lolrp.er than
Hypodermic tube
Thermocoupleju nction
Fig. 6. óthe axis
Tension spring
Cot to n cord
Trarrelling thermocouple for measuring ternperatures alongof a duct
Ha--¡
Recordingpotentiorneter
108.
Table ó.5
Locations of the thernocouple junctions along the ducts
Distances of junct-'rons frombotton of duct
(in. )
I
Thermocoupledes ignation
1 in.diameter
duct
ZLc in.diame ter
duct
-4. -5rry IfI .
diameterduct
1514T31.2
1110IB
7
65432
1.
47 .842 .0JO. /,
47 .847 .456 " 029 .7?4.tr24 .010 )
1_4.59.85.0
47 .B42 .036 .2,33;l22 .5a)
35.922 .5o')
8.94.50.4
8.94.50.4
4.63.72.4L.20"5
1 The junctions are grouped accorcling to the ireatingelements under which they are located.
109.
about 100 in. could not be ac.con'rnoclated in the labor atoty
and theref ore, because 48 in. long unheated sect.ions \vere
to be used, the largest value of the ratj.o 9,í/L that it
was possible to obtain directly was about unity. To
obtain the equivalent of a larger 9,r/ l-, a srnaller diam-
eter restrjction, as illustrated in Fig, 6.7, was fitted
to the bottorn of a 37 in. long unheatecl section. The
equivalent length (¿i) of the combined restriction lvas
estinated by a procedure described in Appendix G. The
heated and unheated sections of each cluct were joined by
a coupling made of nylon so that heat transfer to the
unheated secti-on wa s ninimal .
To ¿rllow the air to enter the duct smoothly, a
sma1tr bell -mouth, examples of which are shorvn in Figs.
6,2 and 6.8, was fitted. It was nade of nylon to mi-n-
inise heal-ing the air before i r reac.]red the actual duct.
The ducts r\rere su-spendecl in a laboratory in r,vhi.ch
the diurnal temperature variations rvere relatively smaIl.
A large draught-shield protected the ducts fron any
abnormal ail movenients.
6.3 Procedure
To carry out a test, the heated surface of the
Unheotedsection l,= 37 in-
Heoted sect¡on
110 .
Smo[[erdiometerduct
or
0.5 ín.
0.2 in. R.
Smotl dionreterentrqnce
l2
Fig. 6 .7 Addismaller dianetof r,i/ L greate
Ty restrictions withhe duct to give values
entionalhaners t
r than 0
tt8
111 .
Fig. ó.8 Smaller diameter duct with be11 -mouth entrancefi.tted to the 37 in. long unheated section; this combinedentry restriction gave an equivalent I'i/ L of about 48
LIZ.
duct had to be at a uniform tenperature that was j.n
equil ibriurn with its surroundíngs. Obtaining this corl-
dition was a tedious task, taking from 6 to I hour:s to
achieve. It was expedient to raise the temperature ofthe surface quickly al¡ove that r:equirecL, and then toreduce the voltages across the elements to the estimated
val.ues. 'Iemperatures along the surf ace were moni tored
by a recorcling potentiometer to deternrine the direction of
their movelnents and the voltages r¡ere ad justecl. accordingl)'.
To ensure that the tenperatures were jn equilibrium, the
readings for the test were not taken ttntil 2 hours afterthe f inal voltage-s had been established. As voltage
stabilisers were not available, very srnal1 adjustments to
the variacs rvere occasional,ly necessar)r to keep the
voltages constant.
The rate at which the Ëinternal surface of the duct
dissipated heat lvas obtained indirectly hy subti:acting
the heat losses thr:ough the insulation from the power
in.put to the Ìreating elements. The relationship between
the heat 1osses and the surface temperature v/as obtaj_ned
by operating the duct at va-rious temperatures with the
ends piugged and the bore fillecl r'rith pieces of fibre-g1ass
insul.ation to preveul. internal air movemeltts. Since
1t_3.
calculations sholved that. for the temperatures under con-
sideration the heat losses frorn the exposed surfaces of
the plugs \,rere very approximately equal to the heat
radiated by the internal sur:face through the ends rvhen
the cluct was open, the heat dissipation of the ducb was
treated as pu:rely convective"
The nìear temperature of the surface of the duct
was obtainecl b¡' averagi.ng the temperatuTes recorded under
each heating element an<1 th.en weighting these a\rerages
according to the lengths of the elements.
Axial tetnperature me¿ÌsuTements lvere not ma.de trntil
after the surf ace '[emperatures had bee¡r neasurecl because
the travell-ing thermocoupl.e slightly impeded the fiow an<1
thereby brought about a small irrcrease in the surface
temperature.
For most tests, the temperature difference between
the surface of the duct and the ambient air was keptA
within t.he l.ange from 10 to 12(]For" T'emperature dif f er-
ences less than 1OFo could not be obtained with suffic-
ient accvTacy because the subtraction involved magnifiecl
smal1 elîrors in the measur:ed temperatures. On the other
hancL, lteither Ìvere large ternperature tlif f erences suitablc
1t 'I'c,,tirol:a tut'¡l clj f icrcnccs :lïc .shot'¡ll thtts , Ììo
250
200
150T*-t"T
hrz F? f t4'¡00
50
00 100
(T*-To) Fo
Fig. 6.9 Parameter (Trnr-To) /v', which is contained in th.e
Rayleigh nunber, against the difference in temperaturebetween the surface and the anbient air (Tw-To)
50 150 200
PHè
Dry o,ír qt otmospheric pressure.
To=ô0oF
1_15.
because, as shown in Fig. 6.9, (Tr-To)/v' and consequently
Ra increas e at a dininishing rate r+ith temperature
difference. Furthermore, large temperattire differences
would violate the condition on Page 1,2 that the density
variat ions should be srna11 .
For evaluating Nu and Ra, the aír prope::ties,
except the coefficient of thernal buoyancy, were based on
the tenperature of the surface of the duct (9). The
coefficient of thermal buoyancy (ß) of the ait was ob-
tained f rom the f o11oh¡ing relationship, which rvas clerived
for a perfect gas (9).
ß1 (6.1)
SO u e empeïa othe ail entering duct
uTethe
Inspection of Fig" 6.10 will shorv that the pro-
perties of dry a"ir at atmospheric pressure, unlilce those
of the ideal fluid considered in the theory, vary sig-
nificantly with temperature.
To study the be-haviour of the florv ilito and out of
the duct, smoke was carefully injected into the ajr. The
snoke was produced by passing air through a smoke gener-1
ating tube'.
The smoke genei:a'tì.ng l<it was manufacturecl by Drägerwerkof Lübeck, 1Vest Germany"
1
1.0
1.2
008
0.8 0.07
116 .
0.020
0.0f B
k
Btu/hr ft F"
0.0i6 0.8u
f t2/hr0.05
e
lb/f t30.05
0 0/,
0.6
O./+
0.7 Pr
0-0U, 0.6
100 200
-o_lemp3,roture l-
300
Fig. 6.10 Properties of dry air at atmosphericpressure. (Thermal conductivity (k) , kinenaticviscosity (v) and density (p) \{ere taken fromM. Jakob and G.A. Ëlarvkíns, Eleniel.ts of Heat 'I'rans-fer, 3rd Ed. , John l{iley and Sons Inc. , New York,l-959 and Prandtl number (Pr) from A.J. Chaprnan,Heat Transfer, Znd Ed., the Macmilla.n Company,New York, 1967)
7r7 .
7 EXPERIMENTAL RESULTS
7.L Plain-entry ducts with uniform surface tenperatures
Experi-nental values of Nu for plain-entry duct-s
rvith uníforn surface ternperatures are presented in Fig.
7 .1. Air was the convective f luid. It rvi11 be seetl
that these results compare satisfactorily witìt the theor-
etical larninar f lorv relationship reproduced f ron F j g. 3.2
and also with the experimental results of Elenbaas (4).
Smoke studies revealed that for Ra in excess of
a.bout 103 the out-flowing plurne became less larninar in
appearance. Hence, an inspection of Fig. 7.t will shorv
that for sone of the tests on the 2'4 in. diameter duct
and for all of the tests on the 34 ín. diameter duct the
f lorv was transitional . These f indings a.Te in accord
with the following critical values of Ra obtained from
the theoretical analysis: using 1150 as the critical
val.uc of Re* (see Section 3.5) , Fig. 3.10 yields crlticalrRa of 3 x !0', Z x 103 and 4 x 103 for the 1 in., Tta in.
and 3f in. diameter ducts respectir¡e1y.
In Fig . 7 .7, it is interesting to observe that for
large Ra the experimental values of Nu are smaller than
those given by the theoretical curve. T'his is attributed
to the mi-xing ivhich took place in the transitional flow
bringing about a r:edltction in the f I ot',' rate and, colìse-
ooo
a0+
9'.
Elenbcos (a) - v
9Ducts - /.8 in. long, ti/t =0
" - 1'00 in. dio.
" - 2.24 in. dio.o
- 3:Ì1in. dio.Theory (Fi9.3.2)
Pr=07
10
Nu
0.1
0.01
0.1 10 i02
Rq
Fig. 7.7 Nusselt number against Rayleigh number for the 1- in.,^L -32| in. a.nd 3| rn. dia¡reter plain-entry ducts with uniform sur-
IT
face temperatures
l01rn3I l05
tsH
119 .
quently, the rate of heat transfer. Tlie validity of
this deduction is strengthened by the fact that for the
same range of Ra there rvas less mixing in the air emerging
from restricted-entry ducts with t,i/L = 0.8 and, as will
be seen later, there lvas better agreement between the
experimental and theoretical values of Nu.
Fig. 7.2 sholvs temperatures measured along the
axis of the 1 in. diameter duct for Ra = 7I and 69. As
the air approaching the entrance of the duct was heatecl
by the underside of the insulation and by the bell-mouth,
temperatures obtainecl from the theoretical analysis fcr
comparison were basecl on the slightly higher entry ternper-
atures. Although their trends weïe siniTar, the experi-
mental tenperatures lvi1l be seen to be stil1 higher than
the theoretical for a number of reasons. First, the
travelllng thermocouple slightly impeded the flolv anrl
therefore, as the heat input renrained const.an[, the sur-
face temperature and temperatures in the air stream lvere
raised. Secondly, althougb the val.ues of Ra rr'ere well
below the critical Ra of 3 x 10', some nixing dicl occur in
the air stream. This nixing would have reduced the rad.-
ial temperature gradients in the cerrtre part of the duct
and thereby would have raised the tenperatures along the
axi s. Thirdly, the increasing t-emtreratuLe in the
1
j;- -T"
T* .L
0.8
0.6
A-¿r
a.2
0I0 0.2 0.t- c,6 0.8
xlI
Fig. 7.2 Temperatures aLong the axis of the 1 in. dianeterplain-entry duct with uniform surface tenperatures; the Ëtheoretical- curves were based on the actual entry temperatures È
Theoi-y -Rq =69T,"-To = 114t'
RqT*-1= 9'6 F"'
48 in. Iong
o
+
o
+
+
+
o
I
ô
+
a
o
+
o
++
o
+
o
+
Duct - 1in.diq. x
Pr = C.7
ti/[ -0 o
LzT.
therntocouple wires directly al¡ove the junction would have
produced a temperature reading that lr¡as too high.
Although all the tests conducted on the 1 in.dianeter duct yielded Ra belolv the critical value of
J x I0', it rvas surprising to find that temperatures along
the axis f luctuated at randon as shorvn in Fig. 7 .3.
Turning to Fig. 7 .4, it lui11 be seen that the maximum
variation in tenperature recorded over 1.0 minutes was as
large as LL% of (Tr-To) ancl that this occurred at
x/ L = 0.3. fn actual fact, the tenperature variationswould have been in excess of 1,Leo of (Tr-To) if the thermo-
couple junctj-on had had a sufficiently large frequency
response to enable it to fo11ow the temperature changes
accurately. Since smoke stuclies revealed that the flow
pattern at ent::y was axisymmetric and also va.ried lvith
tine, it was deci.ded- to shi e1d the entrance from the in-
fluence of stray air movernents in the room. Hence a
cylindrical shield, 7 in. in diametel and B in. long, hras
concentrically attached to the botton of the duct. As a
result , it rvas f ound tha t the arnplit-udes of the f luctua-
ticrns Ì{ere reduced by about 50eo " This nodif ication to
the duct cl.early shorved that the tenperature fluctuations
were brought about by f lorv irregulari.ties in the air as
722.
T., - To
Tr, - ïo
t0
OB
0.6
0.r,
0.2
00 200 400
T imeSeconds
600
trr igofunia1-
1v)
. 7 .3 Tenperatltre f luctuations on the axisthe 1 in. díameter plain-entry duct with aforn surface temperature. (The temperatureseach elevation 1{ele not recorcled sinultaneous -
. The Raylei-gh nunbel- was smaller than the
Duct - 1in, cfio. x 4B in. tong
Ro =72, Pr =0.7li/l = 0
0 85
transitional value of 300
+
+
+
+
\+
+
+
+
+
Duct - 1 in. dia. x trÙ in, tong
Ro = 72, Pr = 0.7
lí/t = 0
12
I
al-" ô,x 100/
Tf16lw- lo
I
00 0.6 0.8
xll
Fig. 7.4 Distribution of the largest tenperaturevariations in Fig. 7.3
0.2 0.t,
FN?(.¡
L24.
it entered the duct. More eviclence rvi1l be presented in
a later section, where it rr'i11 be shor'¡n tJrat the tempera-
ture fluctuations in. the 1 in. diameter restricted-entry
duct with 9, i/ l, - 0 . B were quite insignif icant .
It is interesting to note that ternperature fluctu-
ations along the axis of the cluct \^/ere .simulated by
solving the flow equations for two different velocity
prof iles a'L entry which yielded the sane value of Ra.
The difference betrr¡een the axial- temperatures obtained at
each elevation s irnul ated the aurplitude of the ternperature
fluctuations. An example of simulated temperature fluc-
tuations is presented irr Fig . 7 .5. Cornparing Fig . 7 ,4
with Fig. 7 .5, it rvi1l be seen that the sinuiated temper-
ature r¡ariations along the axis are rernarkably s imi1ar to
those which were measured. In each case, the largest
variation occurred at appr:oximate1-y x/I' = 0.3.
7 .2 Restricted- entry duct.s witlt urtif o::n surf acetemperatures
The heat dj-ssipating char¿rcteristics of the res-
tricted-entry ducts are presentecl in Figs. 7.6 and 7,7
for the 1 in. cl j.ameter c1uc.t- ancl the 2h in. and 3þ+ :-t't,,
d j-arneter cluc.ts respectively. A1so shor,vn in these f igures
Trxz
0.1?
0.10
0.8
0.6
4.4T*-1
0.2
00 0.2 0.4 0.6 0.6
x/t
Fig. 7 .5 sirnulated distribution of the largest ternperaturevariations on the axis i Tcxl and T cx| are the tenperatureson the axis for entry veiocity p:rofiles l- and 2 respectively
î.0
HN)
3
Ro =
Pr =
3lr
0.7
Entryvelocitypnof ile.
10
t\ì¡ ,
0.1
0.01
0.1 10 10 loo
Fig. 7.6 Nusselt nunber against Rayleigh nunber for the 1 in.dianeter restricted-entry duct witli uniform surface ternperatures
io2
Ra
105
PN)o\
l-ieated section - 1in. dio. x 48in. tong
Pr = 0.7
li /t = 0.9
L Ia,^
\Open thermosyphon ciuct,I-eslie ond Martin (11)
Pr = O.7,
Theory (7iç. 4.4) 'i.e* il+0pen thermosyphon duct
12+
+
7+ tì/r> 20
iill=O i 2 4 6 810i5 to v - Nu ond Ro for l¡/t>20bosed on (T*-T.r)
10
Nu
c.1
0.01
01
Fig. 7 .7 Experirnental Nusselt_-f7for the 2o! anð. Si in. diameteruniform surface tenperatures
?
IU 101
numbers against Rayleigh numbe¡:srestricted-entry ducts with
0
D^i05
Pb..){
.-t1/t =0.8v-ti/[ =8"-t¡/[ =48+ - op€n therrnosyphon duct
ro
Þ +
-{r a +
T +
iA oo
lmpedíng flowrecime [/r-= 59
 - open thermosyphono'/
âooi+I
duci, itlortin qndCohen (12)
^Theory (Fig. a.a)
Duct diqmeter
* 2.24 in. J./4 rn. *Lominor ftowreornne Heqted tength L = 4S in.
l.¡l /r=0 i 2 4 691015i8
Opt-n ihei-r-ncsyphon ductsÐ._1 Lighthill (3) P¡- = 0:7
L28.
f or comparison is the theoreti cal Nu-Ra. relat j-onship in
Fie. 4.4.
I¡r the following discussions 01ì the results, it
should be noted that the values of Li/L exceeding 0.8 are
only uppto*itut" because unfortunately, for the reason
explained in Section 6.2, it was necessary to use a
smaller diameter restriction at the bottorn of the 37 in.
long unhea.ted section.
Laminar flow
Smoke -studies -showed that tÌre f lorv i-n restricted-
entry du-cts l^¡as laminar f or those tests that yielded
values of Nu r,vithin the theoretical laminar f 1ow regime.
Referring to Figs. 7.6 and 7.7, it rçi1l be seen that the
results of these tests compare satisfactorily with the
theoretical relationship.
Fig. 7.8 conpares for trvo values of Ra the tempera-
tuLes measured along the axis of the 1 in. diametei' duct
having 9"i/L = 0.8 wjth those obtained from the theoret-
ical analysis. In each case the experirnental and theor-
etical temperatulîes will be observecl to displ.ay sinilar
growth patterns. It should be notecl t,hat nea1^ the top
of t|e duct the tetnperature of the surface was slightly
1
T _T'cx
T* -To
0.8
0.6
c4
0.2
U a2 0./, 0.6 0.8
vll
Fig. 7.8 Temperatures along the axis of the 1 in. dia-meter restTicted-entry duct; the theoretical curvesϡere based cn the actual entry temperatures
PN)(o
r;-L = 9.5 Fo
+
+
+
+
+
+
+ + +
Ro =1i
RqT* -1" = 128 F'
Theory -
IÐ
o
oo
o
o
o
o
o
+ + + +
+
ê
Prtï/i
-- 0.7
= 0.8
Duct - 1 in. dio.x 48 in. heated sectio
1_30.
lower owing to the heating element not fu1ly coìnpensating
for the external heat 1oss. A cornpa.rison of Fig. 7 .2
with Fig. 7.8 will show that the r:estricted-entry duct
for the same value of Ra. produced higher temperatures
along the axis than the plain-entry duct. The explana-
tion for this is that the entry restrrction reduced the
f loiv rate H'hich, in turn, led to greater: f lorv development.
The arnpli.tudes of the axial ternper:atttre fluctu-
ations shown in Fig. 7.9 will be seen to be cotlsiderably
less than th<¡se in Fig. 7.3 for a plain-entry duct. The
smaller ternperature variations were brought about by the
37 in. long unheated sectjon smoothing the flolv before i-t
reached the heated section.
Open- thermosyphon ducts
Sone tests 'hiere carried out lr'ith the ducts closed
at the bottom to yi,eld 9,i/L - æ. The results of these
tests on open-thermosyphon flor.^¡ are presented in Figs.
7 .6 and 7 .7 and will be seen to agree satisfactorily vrith
the experimental work of lr{artj-n and Cohen GZ) and rvj-th
the theory of Lighthill (3) ancl of Leslie and Martjn (11).
Snoke studies revealed tliat r'¡hen Ra tvas below
4 x 702 the open-thermosyphon flotq was laminar. Ilorvever'
1.0131 .
OB
0.6
T.r -T"
T',u -To
0.4
0.2
200 400Tlrne
Seconds
600
FiS. 7 .9 Tenperature fluctuations on the axis of the1 in. d.iameter restricted-entry duct " (The tempera-tures at each elevation were not recorded sinultan-eously) . Conparison of this f igure rvith -¡ig. 7 .3
^rvill sirorv tha{- the entry restrict-ion snootlied the fl-olvin the ireaUed sectiort
00
Duct - I irr. dio. x 48 in. lreoted sectíon
Ro = 69, Pr = 0-7
ti/t = 0'B
x/t = 0.Bb
0.50
0,19
0.07
132
for Ra > 4 x 102 t]ne dor,vn- and up-flowing streams were
observed to mix with each other. This interaction
between the streams produced the ímpeding flow regime
which gave lorver values of Nu (3, 12, 13) .
It should be noted 1.hat, âs in other work oll open-
thernosyphon ducts (3, 7L, 12, LS) , lrlu and Ra have been
based not on (Tr-To), which was usecl in the case of plain-
and restricted-entry ducts, but on (Tro-T.t) . Since the
temperature on the axis at the top of the duct (T.t) t\¡as
higher than that of the ambient air (To), (Trr-Tct) rvas
less than (T -Tl\I o
Conbined upward ancl open-therrnosyphon flow
When the entry restTiction l/as very 1arge, it rvas
found that aî open-thermosyphon fI.or.v occurred in the
upper part of the heated section of the duct. Smol<e
studíes revealed that air still entered the botton of the
duct. Thus heat was d j.ssipated by a restl'icted-entry
flow in the -tot,ver part of the heateci section and b)' an
open-therno-syphon flow in the upper part.
In Fig. 7 .6 it will be seetì that a 1 in. diameter
duct with 1,, / r. >>1',
the sarne duct operating as an opell- th.ernosyphorl cluct .
)
1.33 .
It should be noted that since a smaller rjiametel entrance
was used (see Fig. 6.7) a more accuïate value could not
be given to 9"i/t, for reasons .stated in Appendix G.
In cornparing the values of Nu for t"i/9" >>
those for the open*thernosyphon duct in Fig. 7 .6, it is
inportant to realise that the temperature of the air
entering the top of the restric-ted-entry duct r+as higher
than ambient. This was brought about by the hot out-
flowing air heating the air space above the duct. T'hus
the problem involved not one, but two temperature differ-
ences. To be consistent with the other tests on res-
tricted- entry ducts , the temperature <1if f erence f or calctr-
lating Nu and Ra was based on the tenperature of the
air (To). On the other hand, basing the temp-
clifference on the temperature of the air entering
top (T.t) , the results for !"i/ L >>
7 .6 to be sÍmil-ar to tho.se f or the open-thermo-
dlrct . Thi s was because the open- thernosyphon f lorv
predominant heat transfer nechanism.
In Fig. 7 .70, shoviing temperatures along the axis
heated section of the l- in. diameter duct with
20, it will be observed that f-rom x/9' = 0.15 to
ai.r stream was at the same temperature as the
anb ient
eratur e
at the
in Fig.
syphon
was the
of the
9". / r, >>1'
0.8 the
+ + l"+t
+f+
+
Duct - 1 in. dio. x 48 in. heqted sectionRc = S3, Pr = 0.7
lilt >> 20
f+
t
f - Ftuctucting temperotui'e
+
T_Ttcx ¡ô
T* -'L
0.8
0.6
0.4
0.2
0
0.60 0.2 0.1* 08
x/l
Fig. 7.L0 Tenperatures a1on.e the axis of the 1 in. diameterrestricted-entry duct; the entry restriction was sufficiently -large for air to enter the duct at the top H
1
135.
surface. Hence it follolvs that heat dissipati.on took
place only near the two ends of the heated section"
Turning to Fig. 7.7, tests on the 2t'4 and 3'k Lî'
dianeter ducts rvith t i/ !, = 48 also produced a colnbinecl
upward and open-thermosyphon f1ow. Although most of the
values of Nu for f.i/ 9" = 48 will be seen to be only
slightly 1ar:ger than those obtained for the open-therno-
syphon ducts, the rates of heat transfer were not similar'
The reasons for this are as follows. First, higher
temperatureS urere experienced in the open-thermosyphon
ducts and therefore the properti.e-s of the air in these
tests were quite different. Seconclly, Nu and Ra for the
open-thermosyphon ducts were basecl on a different temper-
ature clifference namely, (Tw-Tct). An inspection of
Table 7.1 will show that for approximateLy the sane value
of Râ, the open-thermosyphon duct dissipatecl 1,1,0"ó more
heat than the restricted-entry cluct with 9"i/ L = 48.
Furthermole, it is interesting to observe in Table 7 .2
that, despite the opeD-thermosyphop flolv, the lower part
of the heatecl section of the restricted-entry duct with
,.í/g, = 48 was responsible for the greater part of the
heat cliss j.pati on. Of course, if the entry restriction
hacl been increased, the open-therrnosyphon flow would have
as sumecl grea ter s ignif icance .
1,s6.
Table 7 .1-
Comparison betv¡een the heat direstricted-entry duct with Li/an open-thermosyphon duct ;214 in. in diameter and the heawere 48 in. long
ssr.pI-
theted
ated by a48 and byducts rveres ect i. ons
Ileat trans f erdata
Restri.cted-entry ductf.i/k = 48
Open-thermosyphon
duct
Ra
Nu
1_
1
Tw
(" F)
To
("i..)
Tct('F)
Tenperaturedifference
(F')
Heatdiss ipated(Btu/hr )
68
98 (av.)
1060
0.66
216
75
1,26
(Tw-Tcr)
90
26 .5
1_040
0.7r
11,2
=44
tz.7
rT-TIo'
1 roperties for calcut tlre tetnpclature o
ing Nu and Ra werehe surface.
The fl.uievalu ate
'Latfrdp
da
737 .
Table 7.2
the total heat that rvas dissipatedPercentage ofby each heatinin diameter anlong.
gd
element; the ducts were 2h in.the heated sections u¡ere 48 in.
Dis tanceaborre the bottonof the heatecls ecti on
(in-)
Percent-age ofrate of heat
Restricted- entryduct with9../9" = 481'
Ra=1040
the totaltr ans fe r0pen- thermo -syphon duct
Ra=1-060
36 to 48
9to360to9
7
27
66
95
5
0
Total 100 100
Temperatures along the axis.of the heated sectj-on
of the 21-4, in. d"ianeter duct with g"i/ t, = 48 f or Ra -= 104 0
are presented in Fig. 7 .II. It will be observed that
the temperatures were unsteacly rvith the largest fluctua-
tions oc.curring near the bottom of the heated section and
at the top of the duct . A't x/ 9" = 0 .04 and 0 .I2 the
fluctuating temperatures rtleTe caused by the laninar: flow
degenerating j-nto a mixing f lorv as illustrated in Fig. 4.7
Duct - 2.21+in diq. x 48 in. heoted sectionRe=1040,Pr=0.7ti/t = 48
L Xa.s
0.19@
Top of cjuct, x/[ =i 4.12
c ¿UUSecond s
4C0 0.040
T"
1
T.r -LTro -To
0,s
0
Tirne
Fig. 7 .7L Tenperature fluctuations on the axis of the2L+ in, <iiaineter restricted=entry du.ct. (The temperaturesat each elevation were not recorded simultaneously). Thetemperature at the top of the duct was reduced by theunsteady open-therrnosyphcn flow
Hu.tco
139 .
At x/ g = 1 the entering air in the open-thermosyphon
flolv rrras responsible for the lower mearì temperature, and
flow visualisation studies revealecl that the large tem-
perature fluctuations were caused by the aír being
alternately drawn in and pusherl out.
For the test on the restricted-entry duct des-
cribed in Table 7.L, Reynolds number ¡ner) of the uptvard
florv in the lolver part of the heated section can be
obtaj,ned as f ollows. In Fig. 7 .1,1, it will be seen that
between x/.0 = 0.37 and 0.87 the air had almost reached
the ternperature of the surface. In addition, the heat
dissipated up to x/ 9, .= 0.75 was 3.47 watts. A heat
balance yielded t.he follolving equation.
h*¡ 9"=0.75 IIr 2 (7 .L)unt
p c )( T -Tw w o
Solving Eq. . (i.1) gave a mean f low veloci ty (tn,) of
570 ft/hr. and from Eq. (3.26) lìe, was found to be 80.
Re, being very rnuch smaller than the critical value of
l-150 supports the explanation given in the previous para-
graph for the tenperature fluctuations in the lower patt
of the heated section.
p
140.
N{ixing flow
If the entry restriction rvere too large for
lamj-nar flow to be sustai.ned yet not large enough for an
open-thermosyphon flow to develop at the top, a rnixing flow
occurrecl above the d.egeneration of ifr" laminar f 1ow, as
illustratecl in Fig. 4.7. The 2!a ln. diameter duct with
9"i/L = I produced such a mixing flotu. Flow visualisa-
tion studies for these tests showed that considerabl e
nixi.ng took place in the air strean as it ernerged frorn
the top of the duct with the intensity of mixing incr:eas-
ing rvith Ra. In Fig. 7.7 the experimental values of Nu
for the 214 in. diameter duct with Li/L - 8 will be seen
to approach the theo::etical laminar flow regime just
below the clrrve derived for the same vaLue of 9"i/L.
This discrepancy could have resultecJ from errol's inherent
in the rnethod of estimating Li/ L (see Appendix G) and
from errors .arising out of the use of sma1l temperature
differences in orcler to obtain values of Ra less than
6 x 702.
It is considered that ducts rsith values of 9"i/ 9"
just larger than those allorved by the laminar flow regime
nierit further investigation to obtain a better under-
standing of the heat transfer mechanism in the mixing flon
regine.
1.47.
I CONCLUS IONS
Fron the theoretical investigations into laninar
natural -convective f lorv in vertica1 ducts described in
this thesis, the following conclusions hlere drawn.
Plain-entry ducts with uniform surface temperatures
For Ra <
oped and
RaNu=Í6
1 and Pr 0.7, the florv rvas ful1y derrel-
(8.1)
At the other extreme, for Ra
boundary layer flolv occurred
Nu = o. 63 RaLo
For internediate values of Ra
Fig. 3.2.
The Prandtl nunber of
only a smal-l influence on the
Pr >/ 0.7 but an increasingly
decreased below 0.7 as shown
The heat transfer per
be a ntaxirnuin when Ra = 32 for
that as Pr decreased belol 0.
increaserl.
(8.2)
Nu can be obtainecl flom
the fluid was found to have
Nu-Ra relationship for
signi.f icant j nf luence as Pr
in Fig. 3.3 "
unit flow area h/a.s founcl to
Pr = 0.7. Fig. 5.9 sÌrows
7 the optirnum value of Ra
>3x
and
L03 and Pr = 0.7 ,
r42
Restricted-ent duc-ts with uniform surfacetempera ures
An entry restriction in the form of an unheated
section of the duct reduced the flow rate and consequent-
ly Nu lvas dependent upon both Ra and the ratio of the
lengths of the unheated and heate<l sections (9,í/l-). The
largest value of t"i/t yielding larninar flow was shown to
decrease from 1-8 in the fully cleveloped flow regime to
about 2 in the boundary layer regime.
For Ra < 20, Pr = 0.7 , and t"i/ 9. = 18 (the largest
value of the ratio yielding ful1y developed flow)
Nu= Ra304
(8.3)
At the other extrerne, for Ra > 3 x 103, Pr - 0.7 and
9,i/ 9, = Z (the largest value of the ratio yielding boundary
layer flow)
t4
Nu = 0.48 Ra
For intermediate values of P.a and other laminar flow
(8.4)
values of 9,.]- /r,, Nu can be obtained fron Fig. 4.4
Uniform heat flux ducts
The relationships between Nusselt and Rayl eigh
numbers for uniform surface heat flux ducts were derived
L43.
in terrns of both a known heat flux and a knolvn mean sur-
face temperature.
In terms of the uniform heat f1ux, the relation-
ships obtained for laminar flolv are:
For Ra* < 0¡1 ancl Pr = 0.7, a flov¡ resenbling fully dev-
eloped flow occurr:ed and
Nu* =Ra*8
(B.s)
(8 .6)
(8.7)
(8.8)
At the other extreme, for Ra* > B
boundary layer flow occurred and
l-03 and Pr = 0.7 ,X
Nu* = 0.67 (Ra*¡å
In terms of the mean surface ternperature, the
corresponding relationships are :
For Ra* < 0.9 and Pr = 0.7 ,
Nu* =Ra6-
and for Ra* > 2 x l-03 and Pr = 0.7,
Nu* - o. ot ¡na+;%
+
It is interesting to note the similarity between Eqs.
(8.2) and (8.8) . Nu* for intermediate values of Ra*
and, Ra+ can be obtained from Figs . 5.2 and 5.3 ïespec-
tively.
L44.
The experi.mental investigation yielded results
that agreed sati-sfactorily with those obtained theoret-
ically for laminar f1ow. Although air was the only
fluid used in the expeïiments, it is reasonable to infer
frorn these results and from those of other -studies in
natural-convectj.ve heat transfer that the theory is also
valid for Prandtl numbers in excess of 0.7 (9) .
An i-nteresting experimental finding was that the
tenperature of the air it plain-entry ducts fluctuated
even though the flow was nominally laminar. Since the
f luctuations I4IeTe caused by f l.ow irregularities as the
air entered the duct., they 1,üere much less apparent in
restricted-entry ducts owíng to the snìoothing effect pro-
duced by the unheated section.
The experimental lvork rvas c.arried beyond the
laminar flow regime. Tests Ievealed tha.t if the ratio
of the unheated to the heated length of a restricted-
entry duct (9"i/ 9,) l,vere greater than that allowed by the
l.aminar florv theory, laminar f low entering the heated
section degenerated into a nixing f1olv, as illustrated
in Fig. 4.7 . Furthermore, if the restr:iction were very
1arge, âr ulÌsteady open..therrnosyphon flow was found to
occur in the top part of the duct. As a result, heat
1.45.
was dissipatecl in the lower part of the heated sectì-on by
the uprvard florv through the restliction, and by the open-
thernosyphon flow near the top of the duct. Unfor-
tunately, it was not possible to ca.rry out a suff icient
number of tests to establish the values of Li/ L at rvhich
open-thernosyphon flow began.
As large entry restrictions are of some practical
interest, it would be profitable in future work to study
the behaviour of nixing flow and of combined uprvard and
open-thermosyplr.on f 1ow in transparent ducts (15).
Although the only entry restrictions corlsidereci in
this work were unheated sections of the duct, it would
be perrnissible, in the absence of any other information,
to transform other entry restrictions into an unheatecl
section of the duct and use the Nu-Ra relationships
presented in this thesis.
L46.
APPENDIX A
Finite difference e uations for unifor-m surfaceempera uTe uc S
Since the flow is assurned to be axisymrnetrical,
the relaxation can be performed on a two-dimensional grid
passing through the vertical axis of the duct as shown in
Fig. 4.1.
In the finite difference forms of Eqs. (2.L0) ,
(2.tI) and (Z.LS) r,vhich follotnr, it will be seen that
separate equations have been written for points on the
axis of the duct. To write these equations, terms con-. 1âtaining i- fu , wh.-ich i-n finite difference forn cannot be
directly evaluated at R = 0, were reduced to $rbtLrHospital's ru1e.
Continuity equation
I'Vhen0<R<l-
)Rk*Rk*1
fuj,o*, *u,]jr],,lIoo)
+
+(u¡ ,k+1*uj,¡) - (ur r,\rr'5-l-k)
2AX 0 (A.1)
t47 .
The continuity equation in fj.nite Cifference form forR - 0 was not required to solve the flow equations.
Ilowever, it should be noted that when R = 0
V ,k 0 (^.2)
Momentum equation
l'Vhen0<R<1
J
UU -U
,k j-1,kAX
l +u¡,k*1-uj,u-t
2ARj -1,k
u ¡ ,k*7-2
*ui,k-1 nuj ouj -t (A. 3)
(A. 4)
U.J,K +
^RzAX
0.J k
'lllhen R
U j-1,k
0
t
U -U-.¡t , k+ 1- i 'k-t2AR
-FV j-1'k
UU2 J
-2U +llk i,k i 'k-1 +
kJAR'-
l-48.
uj,k-uj-1,kU
kk+11Þr
Energy equation
When0<R<l-
AX
U -2 0 +0
When R 0
+ \T"j-1rk
k-1
j-1,k
uj,kn1-oj,k-12AR
1Pr Rk ItJ 1rk
(A.s)
(A. 6)
H 0
I
j k j-1,k +VAXU j -1,k
2Pr
oi,k*1-2 uj,o +0i,k-1
ARz
!I
I
X
¡i
149.
HEATEDSECTiON
R
UNHEATEDS ECTIO¡.I
L
3
2
1
ç_ Wl\LL
Li
j-1
j
3
2
1
12 3 kknl 21
(¡I
Fig. 4.1 Relaxatio:r grid
150.
APPENDIX B
Relaxation -solution for uniform surface tem eratureuct s
Since laninar florv in a duct is unidirectional
the governing equations rrere solved by a step-by-step
relaxation (6, 16). With this method, each row of the
grid shorvn in Fig. AJ 'was relaxed in turn, beginning at
the botton of the duct.
For the bounclary conditions in Section 3.1 for
plain-entry ducts and in Section 4.4 for restri.cted-
entry ducts, the procedure is as follows:
1. Values are chosen for Pt, the volume flow (a)
(0<Q<II/8) and the length of the unheated secti.on
(Li) (Li=0 for plain-entry ducts).
2. 0n the botton row of the gri.d the pressure defect
(P¿r )
fluid
Q/II, and the radial velocity
Plain-entry ducts:
is set equal to zero, the tempera'bure of th.e
(0f,t) to zero, the fIow velocity (Ul,t) to
(vr , t) to zero.
3a.
For initial values of the tenperatures on Row 2,
U r,k is set- equal. to U1,k. The energy equation
in Appendix A is relaxed until satisfactory
values of OZ,L are obtained.. An over-relaxation
3b.
151.
factor of about 7.2 was found to be advantageous
i.n reclucing tlie nunber of i terations.
Restrictecl- entry ducts :
Throughout the unheated sectic;n U j ,U is zero.
The nomentum equation in Appendix A is relaxed
for the grid points on Ro'w 2. Both Ur,k and PdZ
are unknoln and therefore the equation is solved
for
4
UP -P
d2 d1,k
+ + C) A ÃR-z
5
6
rvlre::e C - 2 for 0 < R < l- and C = 4, for R = 0.
Since PaZ has the same value for each grid point
on Rcxv 2 and a js constant ttrroughout the duct,
Eq.(¡.r) in finite difference form is userl to sep-
arate the two unknown vari-ables.
The continuÍty equation in Appendi-x A is soh¡ed
directly for each point orr Rol.v 2 to give ,r,k.
The procedure is repelated for Row 3 and subsequent
rows until P¿ ceases to be negative. Linear
interpolation yields the el.evation at which I'.1=0;
this el-evation gives the top of tlie duc.-u.
The rate at which the fluid transports hcat across
€:ach l'orr' is obt¡rinecl. from Eq" (3.2) in f j.nj te
dii:f erence form.
7
r52.
B. The reciprocal of the dimensionless length of the
heated section of the duct (L) yields the Gra.shof
number, and the Nusselt number is obtained frorn
Eq. (3.6).
9. Restricted-entry dtrcts:
The Nu-Ra rel.ationship for restrictecT-entry ducts
is obtained by successj"ve sol.utions of the equations
for constant values of a and jncreasing values of
L..l-
The computer progranne, rvhich wi1.1 be presented
in Appendix C, r{as designed to place the top of the duct
between Rows 40 and 100. Since the changes in tenpera-
ture and velocity near the bottom of each secticrn t^Jere
relativell' 1arge, a smaller 1'9I^I spacing was used betrveen
Rorr's 1 and- 2I.
Twenty-one grid points per roì4r were founcl to be
satisfactory for all calculations. However, i.t is
interesting to note that trial computations using 47 grid
points per r:oiv yielded almost identical clata.
153.
APPENDIX C
Computer programme for uniform surface temperature ducts
The prograrune for uniform surface tenperature ducts
given in Fig.C.1 was originally written for a CDC 3200
digital comput-er and later modified to run on a faster
CDC 6400 conputer. The actual cornputing tine for a
solution on the CDC 6400 computer was usually between
5 and 60 seconds.
Infornation that is r:equired on the data cards
will be found in Statement No. 100 of the programme.
The symbols used in this statement are defj-ned as
follows:
Data Card Synbols
a
PR
DX
DCX
Dirnensionless f 1ow volume; 0<Q<lI/B
WT
Prandtl number
Dinensionless spacingthe first to the LLLth
Dimensionless spaciugthe llltlt ro$/ in each
of the rows fromrow in each section
of the rows above
s ect ion
for temperatures;Over - relaxation factorWT is usually I.2
Row at which the grid spacing changes front
DX to DCX; LLL is usuallY 2tLLL
III
PHI
XP
LX
TA
TB
LCO
1_s4.
Number of rons that are relaxed beyond thehighest rol having a negatí-ve Pdi III isusually 1
Angle of the duct to the vertical; PHI
should not exceed 0.2 and is zero for a
vertical duct
Dimensionless length of the unheatedsection
Nunber of rows in the unheated sectiort
Constant in the expression for the tem-perature of the heated surface; TA=1 for a
uniform surface tentperature duct
Constant in the expression for the temper-ature of the heated surface; TB=0 for a
uniform surface temperature duct
Controls the amount of data printed;LC0=1 gives Nu and Ra, and temperatures,velocities ancl the pressure defect for each
of the top three oï four rows, and LCO=O
gives, in acldition, temperatures, vê1oc-iti.es and the pressure defect fo:r each
tenth Tot^I
PlìOcRr'.r-l JRDu3lU (INFUT'TApË60ËillpU1'!0UTpU1 çtAPE6l=0Ui'F,UTl 1SS.C DYEk riEcr-t Filo Df FT U 0F /r I EL X2r+3çC NAIURAL COI.IvI..C rIOi! TIiROUGH Â CYLINÛERC hITH T[.AT VTLOCTTY PROFII.E
^T II..ILEI'
C V/iRI¡,ilt.[. uaLL ]-glltr OR Ul,lIFOfit'i Hf\LL f[¡jPC JRD,/ü3 t U
DIr.îË_r{slorlu(?r22) r'l (3¡2p) yV(2?l rSl.l {Zz) ¡l'u(?2)yUU(z?}rtiU(22) rXLr (Z?}DI t,rt,r{S I otlzrJ ( ¿¿ ¡, Áu ( 22) r tjtJ ( 22 } ! cu ( ?2 ) I yu ( 22 ) ç ou ( 22 ¡
I F 0lt¡¡¡r T ( 4 t- I ll "
3 ç F ç¿ . ? ¡ ? ï 4 r t 9 "
6 ç Ei . 2 q Í 4,t'¿.EL,J o 3 I I 2 I5 FOR;'rtT (,zltilr5,[rI3HJtìD,/{J3lu I3)
l0 F0Rr'ìAT(l¡r r5xr37FlÍ.¡A.TURAL C0t.¡VEClI0r.t THROUGH A CyL. IN0ER)l5 FC)t'll.,lìI ( lit r5xrlì5rrilITH Ft-AT vti_ocITy pR0Ftt-E A1 INLET)?O FOIII'iAT(IH I3X¡26I.IZI IAUAtLY SPACED STAT]ON5)25 FOÉtr+Â1ilFt ,5Xrl7HSTAlI0N 22 * rtALL)30 FORitAt(lt't rlixr24hsTtrTICrtt ¿ - cEt,,lìRt:-1._Ir,tE)32 FORì'lAT(ltt ¡5Arl7tlr¡ÀLL_ IEHP :i ¡r+81+X)35 FORí'r¡,T(Irl r5Xrl0litJATA CAiìD zr 8I0,3¡F.;"?t2Il+¡Fjc).6çE9"2rI4r2810.3r12)40 FOrll.rÅl (Ìtj r5Xr4HQ =FT.4rTXclBliUtJHtATEt) LEriCì111 =f9oA)45 F0kf ¡Á1'(l.t r5Xr4.tjpR =FZ.3r7Xr5t.lpHI =F7.3rT)(. 1
47 FOtrilÉT(ll-l r5Xrl3llvALL TEtlP ; Et0"3 ¡'¿lt +EIA.3c4fl c, X/l50 FOÊi¡1,ìT(I¡t r7Xr47HX DP,/DX P DIFi' HTlTRAl.lS FLOll UPr
l36x I ?Hí)2(r/DR¿ )
55 FOR:'4ÁÌ (/Iil r5Xr5Et 0,3 r I6 ¡9X ç I5ç 5Xr I5rË10.3)6 0 F O R r r,\T ( I I rt r 5 X ¡ I [: I 0 . 3 ; I 0 X ç 3 E I o . 3 r I 6 r g X ç I 5 o 5 X r I 5 r E I 0 . 3 )
65 FOH;'rÀT (IH r5Xr5fIù.3¡ l0X¡5[10.3]7U FOR¡IAT lllï ¡1 /,t44t11 GRr GRPR* NU PI r
ll7xt47tlLl/L frR 0 H Tl'J 0UT)FO[.]¡.rJ+Ì (/111 ¡5Xr5Ei0n3¡l0Xr5El0 "3]FO'{f 'AT
(/lH t 7X r26¡lTirl tIEAN GRI'tì Tlil.l NU TY¡M}FOrì:'ìÁT (r'lH tBt, r lJEl0.3/)RE¡tD I ôC r l ) 0 r PH I l,X ç DCX I þrT I [.[.L r I I I I Pl{I ¡ )iP r t.X ¡ TA c TB ¡ LCOLLLt-= IIt (rJ.80.0.)GoT036i)
¡05 HRITf(6Iç5)LLLI-IrJRI'l'E (61 ¡ l0 )' h'RITEtólrI5)TIRITt(61c20)tIR I TE (6l r 25ll/RIT[ (6t r3fl)HR iTI t61 r 32 )
l'JR I'l E ( 6 I r 35, tlr PR r DX r DCX r l{T r t.LL. r I I I ¡ PtlI ç XP r LX r TA ç l'B ç LC0lirl¡TEl61r40lrl¡xPþlfì I T f- ( 6 I r 4 5 I Ptì ç [)]'l IÞtR¡ T[ (6I r 47 ) 1A I l9},R¡TÊ:(6I¡50)DOI?0x;?;2?yet(-zYU I li );Y+. ¡¡ç\'(ií)å0.U ( I r l() aU (?r K) sQ/3'
" I /+I592ú54
¡20 T ( I r)il;T (2rK) :¡o 0r) 1-0.IF (LX"rtL"0ICì0ï0Ie2T(1r22)çTA
I eZ X=PslfÊ029-¡.' llBX-D/,AAË.0125/DX
Fig. C.1=1 Sheet I of 4 of the cotnputer ptro-granrnìe for p,lain- and restricted-entry ductswi th unif orm surf ace l"cmper:atu1'es
75BO
85100
PQÂ;$Q6.7PPPR8;PRÂü2.I I*t-s0LCÈe0LXLeLXLL=1HRITE (61 r60 I A rP¡ H¡ Q¡LLrLçLr D?UHRITE lóI¡65¡ (T ( I rl() eKsle¿¿¡hilll'E(6¡r65)tU(leKlrK=?¡22) !
Uflr?2)-U(2r22¡=0!CU(¿21Ë"001¿5uu(e¡=''Tu(2Ì=PRBAU(?)=UU(21=800o00126K=2r21DU (K ) s (YU (K+l ) råYU (K+l )-YU (K) $YU (K) , rll .570796327LLgLL + IX=X + DXIF (LL"LT.LX.urì.L.L,EO.LX) cOT0l30Í l2¡2?l:TA+TBnxIF (T (2r221 .LT.0r ) GOTOI0000Ì 35K=3rZlZu(K, E(V(K) -1./ (PRÈYU (K, ) ) /r¡uu(¡iÌ=PRA+7U((¡l{U(ñl=PR,\-ZU(ÁlTU(KlsrrnçKl/clxXtr(r.)åTU(KÌ+U(IrK)BU (K ) = (V (r() -I . /'( û (,t;l I /. IAU(l"il=4tì0.-BU(KlBU ( r, I =40 0. + tsU ( i( )
CU(K) =1./ (lU(X).tÌ00" )
SU(KlElU(K)cT(IrK)TtJ(K)=TU¡6¡+pRËlSU(2) =U (l rZ)çT lL ¡21 /OXTU ( 2 ¡ =U l1¡71 /aXoP iìBs2 rcu (21 -l "/ (u ( I ¡?l /DX+1600. )
XU ( 2l =t, ( I r ?) sl.i lL r?l /DXlF ( LL-LX) 170 r l36r 138LL= ILCå2 0L X:uGoT0t70LåÐN=3FT-1.001501'1=l rNL=l- + ISUf,ta0 o
T(2¡ll=T(2r3ÌD0l50i(K=1 r20 '
Ks2?-Kl(TEI'lP;r ( T I Zr Kð I ¡ nf,U (K l +T I 2 r K- I I r.UU 1 5¡ +SU ( K I I /TU ( K ITErlP:¡ÊTo ( l€HF-T ( ? o K I I
S U¡ì=5 Ul.1+ T El.lPnI El-lPT(?rl(l-T(?rKloTIMFcoilT I t'¡LtE
IF {t-L "
0T. I 00 0' lt¡10 o LX. EQ n 0 ) l',RI TE I 6l ¡ (¡5 ) (T (2 ¡ K I r K=2 ¡ 2? )
IF tSUl'l.LT. ¡ o g-!41 G0TCI 70F T*I.JT
1.só.
126le8
130
l3s
t36
l3B
140
I50
Fig, C.1-2 Sheet 2 of. 4 o1. the cornputer pro -gramne for plain* and Testricted-entry ductsì^¡itlì uniforîììt sul'face terlpel'atures
170
¡90
200
N=lG0T0l 4 0
R=0.D0teÍrK=Zr2t 1 5'7 .I (3çK) =T (?rK) GCOSF (PllI )
l(3¡K)=T(2rK)R:lìr0U (K! è (CU (K+l ) +çg 16 ¡ ¡SU (22) =0 "I=0I=l+lSUÞt=A;0.U(?rl)=U(Zr3)DO2I0KK=l r20K=2?- l'rKSU (l(,= (Xrl ('() ôU (2rK+l I råAtJ (K) +U (2çK-l)nBUtXl+T (3rKl ) ëCU (K)D02è0K-? g?lA=A+DU (Kl Þ ISU (K+l ) +SU (K) )
DPD t,': I ,r -ù ) /RSIJ (22 ) =l)P0x,/g()0.00230K:;/.21lEr'rP=sr j ( K ) -LTPDXöCr.r (K Ì
TE'"tf '=1 ¡¡.1P-t-r ( 2 r K )U(2rK)=ú(2rK) +TE|4P
S tJ i'l=S U r'1+ I Er¡ P tf T I l"lPlr tul. cT, I 0 0 ir, AIJD. LX. F-o. 0 ) t.lR I TE ( 61 r 65 ) ( u ( z r K ) ¡K=z ç ??IIF (:jUi4. GT. l .E-14 ) c0Tù200DQ?Ê.Ítll=2 t2!\V (K+1 I = (U (2rK+l ) +U (2r K) - (U ( I rK+L ) +U(IrK) I lÕ IYU (K) +YU (K+1 ) I {tAAV (K.I I = (V (K) +YU(¡() -V (Krl ) ) /YU (K+l )
P=i)PDXr¡l)X+PIF(LL..JE.I)GOTO2ò¿O,(-=l:iiXAA=.0125,/l,XÀ=0.flt¡?2|=Il?ç221=1.PI=PGÛ TÛ29I:IF (1,-L'GT.9g9
" rìl'lD. LX. Ê-Q. 0.AND "
(P+4 " CtlDP0XitDX ) . LT " 0. ) G0Ï0290
I F ( LC0, IA. I . Ar.¡D" Lr_. GT.4 ) G0T028óIF ( t_L. LT. 2î ) rtOl'O2e0IF (LL" l_0.LC. Àr.rDeLX. EQ. 0 ) 284 I 286LC=LC+ I tr
GOlt)29'rI F ( |'+¿r. ir+l_JP0x*D/. o 6T. n . . 0R. LL. EG. 999.0R. LL. EQ. ( LX- I ) ) 290 I 30 0
R:llr0 ¡DO292K=¡2 t2lr{=H+0U (K ) tt (U ( 2 ç K+ I } r¡l I ? I K+ I ) +U ( 2 I K ) çT (ZrK} IR=R+DU (t(¡ + J¡ (2rK+l ) +U (2rK) )
DZU= {1, (2 ¡3} -U (Zç2) ) +t800.þ,fì i 1E ( 6 I ç 55 )
^ I DPL);{ r P r H r fl I LL I L I I r DllU
iJf{ITE (61 r ú5) ( T (2r K) çt\=2t 221þlRiT[ (61¡65) (U (2çKl ti\=Z¡221T/Rl fÉ- (61 r65) (V (K) rK=?r??lIF(P.GT.0.l3u5r3l0lI=llâtIF{lI-III}310r350D03l(lK::2 ¡ 22I(IrK)=T(2çKl
Fig. C.1-3 Sheet 3 of 4 of the computer pro-granme for p1-ain- and restrictecl-entry ductsÍrith urliform surf ace tempelatures
210
?UJ
¿30
260
?az
284
28ó290
?tì?
3û03ù5
3t0
158.3a0
35ù
354
360
U(lrKl=U(?rK)PPa-PQP=QHP=Ì.1
lF (LL"llE.LL[,] G0T0I Z8Dx=DCx
^A:.01.25.jDXGOf rJ 128)i=X-P*l)X,/ (PP+PlGR=l o/xGRPIi =$fl* P¡1Q=Ql)+ (A-OP ) ¿PP / IPP+PlH=r-lP+ (H-HPl +PP/ IPP+PlTO=l (I t??l + (1 l?-t2?) -T ( I r2?'l Il,PP/ (PP+p¡f tJ;HTGRPR/{r.283 ¡ 853R=)\i; / X'tRITE (l:l rTn IWRI TE (6 I r 75 I X r GR r GRPR I FU r PI I R r PR r Qr llr 1'OT14= (TA+Tot /2"GRPÍ(=GtìPRt*TllFU=FUlTI'!IIRIIE l6l rBn)vTRITE (51 r 85 ) lf4 r GRPRr FUi.lRlTE(óIr47)TArTbl F ( ( l .-TO ) n+?2. LT.2. E-07 o AND.LL¡ GTe 39) GOT0l 00TI= ( ¡.-TA) //.l8=(Ttltt30.+lI!/31.DX=t,H X
LX=LXLLLI.L=LLLL+IIF(LL.LT.94)GOT0354DXEljBXn2.DCX=DCX ¡r2.GO t0 105IF(LL.Gf.4n)GoTOloS0X=[)rlX/Z rOCx=9'x I 2.GQ T0 irjSSl 0rEltt)
Fig. C.1-4 Sheet 4 of 4 of the computer pro-grãnnìe for plain- and restricted-entry ductswith unifor:n surface temperatures
159 .
APPENDIX D
Finite difference e uations for uniforn surfacetempe TA ture ucts
The f inite diffefence forns of Eq. (2.L4), (2.1-5)
and (2.17) become identical to those in Appendix A for
ducts with uniform surface temperatures after replacing
P¿ bY PTcl
UbyU*
XbyXt
0 by 0:k
and substituting V
(A. 6) .
j,k for vr-1,k in Eqs. (4.5) and
160.
APPENDIX E
Relaxation solution for un form surface heat flux ducts
Again a step-by-step relaxation was used but the
procedure differed slightly from that described in
Appendix B because the local surface ternperature of the
uniform heat flux duct had to be deterrnined concurrently
with the other tenperatures on each row.
An expression will now be derived which was used
to obtain the surface tenperature of a roüi. Fron Eq.
(5.10) it follows that
H*=x2nPr xrt (8.1)
(E.2)
(8"3)
and according to Eq. (5.2)
2TI u?t 0* R dR
o
Conbining Eqs. (E.1) and (E.2) yields
L
x*pI u?t 0ât R dR
II* =X
o
t
1ól_ .
For the boundary conditions in Section 5.-/ , the
relaxation procedure is as follows:
1. Values are chosen for Pr and the l.olume flow (a) .
2. On the bottom ror,¡ f the grid shown in Fig . A .1
the pressure defect tpåf) is set equal to zero,
the temperatures of the surface (0f,Zf) and the
f luid (0T,t) to ze o, the f lorv velocity (Uf,t) to
Q/n and the radial velocity (Vf,k) to zero.
3. On the second Totv:
Uä,t is initj-ally made equal to UT,t
and
oi k is initially made equ.al to 0T,tt
4 The momentum equation (see Appendix D) is relared
for all fluid points on Row 2. Since both Uä,t
and PâZ are unknorvn at each point, the equat-ion
is relaxed for the variable
+ui J.k
where C=2 for 0<R<1 and C=4 for R=0.
As each point on Row 2 lnas the same value of Pä2,
5
6
L62.
the variables ard separated by using Eq. (5.1)
in finite difference form.
The continuity equation (see Appendix D) is solved-
for each point on Row Z to give ,, ,k.The energy equat-ion (see Appendix D) is solved
for all the points in the fluid on Row 2. Eq.
(8.3) is then solvecl to obtain a new estimate of
the surface tenperature 02,2t. This pl'oceclure
is repeated until all temperatures on the row are
satisfactori-1y relaxed.
Tire plocedure is repeated for Row 3 and subsequent
Torvs until Pd ceases to be negative. Linear
interpolation yields the elevation at which nå=O;
this elevation gives the top of the duct.
The reciprocal of the dimensionless length of the
duct yields the Grashof number. The Nusselt
number is obtained from Eq. (5.8).
7
B
The
Appendix
duct s 2I
computer programme
F. As in the case
grid points per rorr \^rere usecl.
will be presented in
of the prograrnnìe for UST
L63.
APPENDIX F
Computer programme for uniform surface heat flux ducts
The programme for uniform heat flux ducts is given
in Fig: F.1. Information that is requirecl on the data
cards will be found in Statement No. 100 of the pro-
grarrune; reference should be nade to Appendix C for the
definitions of the symbols. It should be noted that
the dimensionless volume flow (a and not Q* in the pro-
gramme) is no l.onger restricted to values less than
II/8 as in the case of uniform surface temperature ducts.
cccc
PROGRAFI JRO03I M ( IÌ.¡FìUT' TAPE6()= INPUT T OUTPUT I TAPE6 I ãOUIPIJT )
OYER MECH ENG DEPT U oF A TËL X439 COI'IPILE ON CDC64OO}.JATURAL COiTVECTION T¡.íROUGH A CYLINCJËR }IITII|',INIFORI'l I{ÀLL
'IEAT FLUX
!64.
JflD/03tM0Ir.lENsIONu l?g?21 rT(3r22) rV(?2) rSUl22l rTUf 22) rulJ(22) ¡vtll22) ¡xu(22)Dlr'rEÌISIONZU(?2) ç ¡tU(?-Z) rBU(22) rCU(22) rYU(22) rLìU(?2) 'lìU(2?lF0RMAT (4810.3 rF5.2r ?Itt çF9.6 r 89.2r I4¡ l2)FORMAT (/1 Hl I 5Xr 8HJR0/031¡.1)FOR¡4AT
' 1U ,5rXI3?}.iNATUI"ìAL CONVECTION THROUGH A CYLINDER)
F0Rr.tAT nH r5Xr35H'J¡TH FLAT VEuOCITY PR0FILE Al ¡NLET)FORMAT(]H ¡5XI22HUNIFOR¡1 TALL HEAT FLUX)TON¡4AT(IH I5ÃI26H2I EGUALLY SPACED STATIONS)F0nr'lAT(IH r5XrlTFtSTAlI0N Z2 - d.ALLtFORMAT(lH r5/.rZrrFlSlATI0N 2 - CENTRÊ-LIltEÌFOtìr'lAT ( I H r 5X r l0ilDATA CAnD 4E I 0
" 3 r F5 "2ç214 ç F9 ' 6 r E9 "2 s l4 ç !?l
F0RÌ4AT(lH r5Xç4HQ =f;7.3r7Xrl8HUNHEATED LENGTfI =E9.2)F0Rì1Al ( lX r5xr4HPR =F7.3r7Xr5rlPHI =F7.3)FORMAT(ll-l rTXr¿r4HX 0PIDX P DIFF HT/ÏRÂNS FLOllt
l36XrlTHDZLl/DXZ ll CIIECK)FORNAT (/lH r5Xç5El 0, 3ç l$¡9Xr I5r5Xr I5r2El0. 3)F0Rl4AI ( /lH I 5X, r I Ê- I 0 n3 ¡ I 0X r 3EI 0 . 3 r I6 r gXr I5 r5Xr I5r EI 0'3)F0Rr'1AT ( l¡t rSXr 5EI0 "3r I0Xr5El 0.3)FORI'IAT (/1ll rTXr 4¿rHL GR+ GRPR+ NU TIJM PI r
ll?XraTHLI/L PR O H TH OUT¡F 0 R f4 A T ( ,/ I H c 5 X r 5 E I 0 n 3 r I 0 X I 5 E I 0 . 3 )
F0ilr,lAT (/lH rTXr ¡ BHTw Þ1EJ\r,l GRPR Tl¿HlFoRMt T (,/lH rSXr ZEl0 "3)READ ( 60 y I ) Q r PR r0X r DCX r rrlT rLLL ¡ I I I rPHI I XP ç LXTLC0IF(0"8Q.0.)G01',O360t4RiTE (ól r5)li RITE (61ç I0lTSRITE(6IçI5)HRITE (61 r I8)URITE (6L rZ0lUTRITE (ól r?5)iiRITE (61. ¡30)uRI TE (6t r35lQ¡PR¡DX ¡DCXrl'lf rLLLr I I I rPHI I XP I LXTLC0vRITE(6i ç¿ro)QrxPI{RIIE(61I4f>}PRIPHIli RITE (ól ç50ìD0l20K=2ç22Y*K-?YU(K)=ltt"95U ( I rK) ::U (2 r Kl sQ/3,L4159?65t,T ( Ì rKl =T (2r(l rv (K) =0¿IF(LX.E0"o)lzt'122PI=00[::p;l-l=þlg=Q.DBXgDXÂÀs o 0l?5/tl,FRAs,¿00",/FRPlìBaPRAü? ¡II=L3OLc'10Lt-a INRI TE ( 6l ¡ 60 ) Åç P r l'loQr LLrL rL r DZU
Sheet 1 of 4 of theunifcr-ilì su1'face heat
I5
l0lsIB20?s3035404550
55ó06570
15BOB5
100
120
IzIl2?
Fig. F.1-1g1-amtne f or
computeT pro-flux ducts
l?6leB
HRITË ( ól r65) (T t I r K) sR=?ç2?lURIIE(ó1I65) (U(trK) rK¡?rZ?l 165.U(lr?2)eU{2rZZ)r0¡ï14r 0.CU(¿?)sr00I25UU (2) =lriU(2¡ *PRBÂU(Z)=,Bil(?l=800.DOl26K=?r?1.DU ( K t z (Yllt (K+ t l *YU (K+ I ) -YU ( K I èYU (¡í) l ã1 r57079ó327LLsLLó IIF (LL.GT,2OI
' GOTOI OO
XÉX+OXlF (LL.LT.LXoOR"LL.EG,LX) G0T0l30D0l35Ks3r?lzu (KÌ å (Y (K)-1./(PRSYU(Kl ) I /. IUU (K!:¡PRA+ZU (KluU (Kl -PRÂ-ZU (KlTU(K)=U(lrKl/0XXU(K)=TU(K)ðU(lrKìBU (K) = (V lKl -l '/YU
(K) I /. ¡AU (Kl =400"-BU(K¡BU(K1e400.+BU(KlCU(K) ;1./ (TU (K) t800. !
RU(KlslU(Klsl(l;K)TU(K)=Tt,(K)+pRBRU (21 *U(l c2l 6T (1 ¡21 /DXTU (21 =U (l re),/0xçPRBö2.CU (e) Ët '/ (U ( I c?l /OX+ 1600. ¡
XU (2¡ EU{l rZ) +U ll t?l /DXR=000190K-2r ?lT ( 3rK I *l (?r K) +C0SF (Pl'lI )
R=R+OU f l,r!ð {CU (Kûl ¡ +CU lKl )
SU(22)=0'Is0I=Iolsut4=AÐ 0 rU(?rlt::U(?r3)0021 0K*?rZISU (KI = ( XU (K) +U (? rK+ I ) rlAU (K) +U (2rK-l) +BU (K) +T (3¡K) I çCU ( K,D0??0t-2t2LA3A+0u lKl s (SU (K+I¡ c59 ¡6¡ ¡
OPDX= (A-0) /RSU (eA l =DPDX/800 e
D0?30K=Zr2lTEHP=SU (Kl -DPDXçCU (K)TË¡IP=TEllP-U(2rKlSUHSSUM+lIl'iP.tT ÉHPU(?rKlsU(2rKl +Tel'lPIF ( SUt! "LT.l "
E'"lZ) 250 ¡ 200DOZ6 0K;2 * ?0V (K+11a (U (2rK*11 oU (2rK) - (U l l rK+ ll +U{I ¡K} I l o (YU (K) +YU (K+l ) ) {iAA
V (t\+t ) s {V (K) nyU (Kl -V (l(+l) ) /YU(K+l IPÊ0PüXr'0/c.+ PIf (LL-LXl ?80ç?6?t?&4LLãILCal 0LTsO
Sheet Z of 4 of. theuniforn surf ace hea.t
¡30
135
190
200
el0
22a
230
?50
?60
2b?
Fig.F.I-2gramme for
computer pro-flux duct-s
GOt0eB0?ó+ Le0
N=3FT= l.t'ls6.2B3l853ÈXIPR
2ó5 FIR=0.DOZ66X=? e I 9
266 t1R=Htl{DtJ (K) * (U (2rK+ I ) çT l2 cK+ I } +U (2 r K) +T (2 oKl I
1ó6.
TEnr=¡ l? ç7LlI (?ç?t) s (t1-(HRòDU(20¡ 'rU(2r?0)
sT {2¡20} ) }/ í tÙU (20} +DU (21) ) (U (?s?l ) )
R= (TEHP-T (2rel l ¡ ë+r2
0OZblH=l çNL=L+ ISU14=RTlZrlÌ=T[2r31DO26?K=? c?0TEHp=(l (Z çK¡Ì ) {¡}tU (K} rl (Z íK-I I ¡,UU (K) rRU (K) I /TU (K¡
TEÞtP=FTë (TEl,!P-T (2rKl )
SUt{=SUfi+ T EHPåT EttPT (2rK);-T (?tK¡ ùTEltP
u67 COr¡T¡l'luEIF (5Ul4"LT'l.E-lê Ì 27 0 t26a
2óB FT=t¡TN=IGOT0265
?'10 f r?¡?Zt = (T (2 r Zt | {r TU (21 ) - ( T (2 r 20 ) *UU (el I .RU (21 ) ) ) /NU ( 2I )
Tl4=TH+ (T (I r22) +T (2r22) ) ÈDX
ZBs IF(LL.Eo'r)2Btç28? Ér
2gl OX=DBXA Âã. 0125.1DXX=0 rPl=PG0T0290
?B?. IF (LCO r EO. I "Al'l0.LL.GT. /i ) G0T0286
IF (LL'Ll.l0)?90¡283?S3 [F (LL"EO"LC) ?84 t786e84 LC;LC.l0
GOT029 0286 IF (P+è" 0$DPDXTDX¡Gl. 0 o .0R "LL n E0.20 I ) ?90 r300?9t RÊHCg0+
OO?9?r'z?¡ZlHC-llc+DU tKl n (u (?çK+l ) çT (2 oK+ I ) +U (? r Kl tT (2rKl I
29¿ RsR+DU(lit è (U(?rK+l I +U (2¡ K) lD?U;(U(?ç31 -U (ere) ) ç800 c
l{RITE(6I r55} Xr DPDXTP rHs R¡LL rL ¡ I r02UrHCl{RIIE(6I¡65} (T (ZrK) çKrTt2?li{RI lË (6! ¡ 65) (u (2 rKl tY'=l'¿2¡URITË(6Ì r65) (V (K) rKn2r ?2)
300 IF {P.GÎ.0.)305r110305 I fsIIrl
IF (lI-¡II)3I0¡3503lû 003¡0Küer?2
T(lr!{}sT(?rK}3e0 U(1rK)*U(?rKl
PFs-Pl{PsHI F (LL. ËG
" LLLI 340 I 33 0
330 GofolêB
Sheet 3 of 4 of. theunifolm surface heat
Fig.F.1-3g1'anìrne f or
computer pro-flux ducts
Fig. F.1-4for uniform
340 0X=DCXAA=n0l?5/ùXG0T0 I 2e
ttO 1¡a=It4l (2.èX)xËx-P.sDx/ (PP+P )
GH=1./XGRPR =GR +P RH;HP. (H-IJP) ÕPPl (PP+P)TO;f { I q?21 + lI l?t?21 -1 (Lt2?¡ ) çPPl (PP+PlFU--I "/TI4R=XPlXr{RIl[(6Ir70ll,{RI TE ( 6l I 75) Xr GR I GRPR ç f:U c P t r F r PR I Or H s f oGRPRSGRIJRö TI,I
HRITE(6IrB0l}JRITF: (6I ç85) IþíçGRFRc0ïo100
360 ST0PEND
767
of the computer PTogrammeflux ducts
Sheet 4sur face
of4heat
o
168.
APPENDIX G
Entry restrictions yielding values of ''i/L
in excess
Anentryrest.rictionlargerthan.thatgivenbythe
37 in. long unheated section of the duct was obtainecl,
. as shown in Fig. 6.1 , by using either a s¡naller dianeter
entÏance or an adclitional duct of snaller dianetel.
Smaller diameter enttance
A smaller diameter entrance is -suitable only for
the lamj.nar flow regime. The reason for this is that
for the other flow regirnes it is not possible to deter-
mi.nethevolumeflow(a)whosevalueisrequiredto
obtain the equivalent length of this additional entry
restriction..Tlreequivalentlengthofthecombincdentry
restricti.on èan be estirnatecl in the fÓ1lor'ving mander:
l.Thedimensionlessvolumeflov¡(a)isobtaj.ned
by plotting the experimental values of Nu and
,)
Ra on Fig. 4.2
Reynolds nuntber (Rer) is obtai-ned fr:on Eq '
(3.30) .
The mean flow velocity in the duct (tr) is
obtainecl f rom Eq. (3 -26)
3
4 For fully developed flow
section of the duct, the
unit length (Apf ) G7) is
ufr rsz^-Y22rReI^IÏ
The pressure drop
abrupt enlargement
(L7) is
D2 g"
ME C J¿-Z- Zr e Rere
e refers to the entrY
in the unheated
pressure drop per
169.
(G.1)
(G.3)
passage.
of the unheated-
Ap1
5 (¡pZ) produced by the
of the flow cross-section
1_ p (G .2)
where r is the radius of the entrY Passage.
A very much sna1leï pressure drop (¡pS) occurs in
the êntry passage. A bold assumption is made that this
pïessure drop is the same as that for fu11y developed
laninar f lot'¡. Hence
tJ2m
z-T2
^pz = +
e
Ap3
rvhere the subscript
The combined
section of length I'
given by
equirralent length
and the snaller
p
( ßi)
1dianeter entrance 1s
li=[t+Lp2 + aPs
aPt
770.
(G.4)
It is clearr f.rom the' foregoing equations that the
equivalent length of the combined restriction (li) is a
function of Rer. Hence it follows that Li will vary
with Ra" This dependence of [i on Ra can be seen by
inspecting the experimental results in Fig . 7.6i a
0 .400 in. diarneter entrance gave values of 9"i/ L ranging
fron 1.8 to 4.2, a 0.219 in. dianeter from 8.7 to 20 and-
a 0.098 Ín. diameter in excess of 20.
Additional duct of sntaller diameter
since the equivalent length of an additional duct
of snaltrer dianeter can be obtained ivithout requiring a
knowledge of the volume f1ow Q, this entry restri-ction is
suitable for tests in all flolv regimes. However, the
length and dianeter of the additional restriction should
be chosen so that the pressure drop produced by the
abrupt enlargement of the flow area is sufficiently sma11
to be ignoled.
Referring to Fig. 6.7 , the equivalent length of
the conbined restriction is estimated in the following
manner. The flow in each of the unheated sections is
assumed to be laminar and fully developed. Therefore
the pïessure drop (Ap+) in the larger diameter section,
designated by the subscript 1,'is (L7)
32ReT̂¿
L7L.
(G.s)
(c.6)
(G.7)
(c.8)
Ap4 = ",ir LLP-Z-Zr* 32
Re rI
and the pressure drop (^ps) in the smaller diameter
section, designated by the subscript 2, is
,J2 ^.m¿¡P5 = P -Z-
since tlie vol-ume flow in each section i.s the same, divid-
ing Eq. (G.s) bY Eq. (G.6) Yields
4
^p4
^\-
Lt
-Lz t*lThe equivalent length of the combined restriction is
given by
o
1 1
and using Eq. (G.7) gives
case, Eq. (G.9) shorvs that the
combined entry restricti.on (gi)
and consequently of Ra.
L72.
(c.e)
equivalent lengtli
is indepenclent
4
daxr + 0
i+l1 21
In
of
of
thi s
the
Re r
17 s.
REIìERENCES
1 J.R.
z
3 M
6 J.R
7 J.R
4
5
W. Elenbaas: The dissipation of heat by free con-vection (from) the inner surface of vertical'tubes of
-different shapes of cross-sectic¡n'
Physíca, vo1. 9, no. 8, pp. 865-S74 (1942)
Í1. Elenbaas: Heat dj-ssipation of parallel platesby free convection. Physica, vo1' I, no' 1,pp. L-2'8 (Le42)
R. Dyer: The development of laminar natural-cônvective flow in a vertical duct ofcircular cross-section that has a flow res-triction at the botton. Heat transfer 1970,vol. IV, Paper NC 2.8, Elsevier PublishingCompany, Amsterdan (1970)
J. Lighthill: Theoretical considerations onfrõe convection in tubes- Quart. Journ'Mech. and Appliecl Math., vol. 6, part 4,pp. 398-45e (1es3)
Bodoia and F.J. Osterle: Developnent of lreeconvection between heated verti-cal plates 'Trans. A.S.M.E. , Series C, Journ. of FleatTransfer, vo1.84, PP.40-44 (L962)
Dyer and J.H. Fowler: The developrnent ofnâtural convection in a partially-heatedvertical channel forrned by two parallelsurfaces. Mech. and Chem. Engg. Tlans'Aust., vo1. MCz, no. I, pp. LZ-76 (196ó)
Dyer: The developnent of natural convectionii a vertical cirèu1ar duct- Mech. andChen. Engg. Trans. I.E. Aust., vol. MC4,No.1, pp.78-86 [1968)
Schli.c.hting: Boundary layer thegry-' 4t\ ed',McGraw-Hí1l Bool< Coinpany, New York, 647 p'(1e60)
8 FI
9 E.C.R. Eckert.and Rtransfer. ZndNew York, 550
Drake: Ileat, [{cGrarv-Hi11(1sse)
10 . W. H. McAdarns : Heat transmission 'I¡lcGraw-Hi11 Book ConPanY, New(1es4)
11, . F.M Leslie and B.W. Martin: Laminaropen therrnosyphon, with specialsmall Prandtl numbers. Journalical Engineering Science, vo1 - 1
pp. 184-1e3 (19se)
lV. Martin and H. Cohen: Heat transferconvection in an open thermosyphonBritish Journ. of APPlLed PhYsics,pp. 91-9s (19s4)
Martin: Free convection in an open thermo-syphon, with special reference t'o turbulentflow. Proceeãings of the Royal Soci'ety ofLondon, series A, vo1. 230, PP. 502-530(1ess)
Sparrow and J . L . Gregg: La.minar free convec-tion front a vertical- þtate with a unifor¡nheat f1ux. Trans. A.S.M.E., vo1. 78,pp. 4ss-44a (1es6)
.M.ed.p.
Entry effectsFluid Mech.,
L74
and rnas sBook ConpãîY,
3td ed.,York, 532 p
flow in thereference toof Mechan-
, fto. 2,
by freetube.vo1. 5,
Lnvc¡1.
T2. B
13 . B .l\i
L4 . E .lvf
15. B.tV. h4artin ancl F.C. Lockwood:the open thermosYPhon. J19, PP. 246-256 (1964)
16. J.R
L7. V.L
Bodoia: Finite d if f erence anal ysis of c'on-fined viscous f1ow. Ph-D' thesis, CarnegieInstitute of Technology, LL7 p. (1959)
Streeter (ed.): Handbook of fluid' dynaniics,McGrarr¡-Hii1 Bôok Company, New York (1961)