[advances in heat transfer] advances in heat transfer volume 11 volume 11 || the overall convective...

The Overall Convective Heat Transfer from Smooth Circular Cylinders

VINCENT T. MORGAN

A'olzo~rl Measctreinent Lnbolrrlory,' C S f RO, Chippendale, Spine?), A ustralin

I. Int,roduct.ion . . . . . . . . . . . . . . . . . . . 199 11. Natural Convect,ion . . . . . . . . . . . . . . . . . 200

A. Horizont,al Cy1indc.r . . . . . . . . . . . . . . . . 200 B. Inclined Cylinder . . . . . . . . . . . . . . . . 210

111. Forccd Convect,ion . . . . . . . . . . . . . . . . 212 A. Cylinder ait,h Crossflow . . . . . . . . . . . . . . 212 B. Yawed Cylindrr . . . . . . . . . . . . . . . . . 239

IV. Conhined Nat,rir:d and Forced C'onvcction . . . . . . . . . . 244 V. Conclusions . . . . . . . . . . . . . . . . . . . 250

Nonlcnclat,ure . . . . . . . . . . . . . . . . . . 251 R.rferrnces . . . . . . . . . . . . . . . . . . . 252 Note Added in Profif . . . . 264

I. Iiitroduction

Accurate knowledge of the ovrrall convective heat transfer from circular cylinders is of importance in a number of fields, such as boiler design, hot- wire anrmometry, and the rating of clcctrical conductors. Although a great amount of cxperiincntal work has been done during the past sixty years, anyone who has recourse to thc literature soon becomes aware of large dispersions in the puhlishrd results for both natural convection and crossflow forced convrction, and notices that the various correlations which have brrn published only partially chi inate these variations. Possible reasons for these differences arc' rxnminrd, and improved correlations are proposed. The correlation for forccd convection is given for the special case of air, but it can be extendcd to ot,her fluids by multiplying the function of

* Fornirrly National Standards Lnhratory.

199

200 VINCENT T. MORGAN

the Reynolds number by the appropriate function of the ratio of the Prandtl numbers.

Inclination to the horizontal, in the case of natural convection, and yaw, with forced convection, both introduce an axial component of flow, which causes a thickening of the boundary layer and a consequent reduction in the convective heat transfer. The magnitude of this reduction is examined.

One field which has not received as much attention as it deserves is that of heat transfer by mixed natural and forced convection. Previous work is reviewed, and improved correlations are proposed. These take into account the angle of the forced flow, with reference to the natural convective flow, and also the variations in the heat transfer equations over various ranges of the Reynolds and Grashof numbers.

Only smooth cylinders are considered here: it is well known that rough- ening of the surface can considerably enhance heat transfer especially with forced convection [63]. The convective heat transfer is found by sub- tracting the radiativc heat transfer from the total heat transfer. The heat transfer by radiation is equal to ?rDl~a(T,4 - Ta4) where u is the Stefan- Boltzmann constant and E is the effective emissivity. The latter depends on the material and the nature of the surface of thc cylinder and also on the nature and proximity of other bodies. It is assumed that the correct value of the radiation tcrni has been used in calculating the convective heat transfer. Any error in using an incorrect value for the radiative loss will usually be negligible for forced flow with higher values of Reynolds number, but may be considerable with high surface temperatures with natural convection.

The convective heat transfer can be expressed in general form by the equation

Nu = jl(Gr, Pr, Re, TJT,, Ma, Kn) ( 1 )

The Mach number, Ma, and the Knudsen number, Kn, are only applic- able in high velocity, compressible, and rarefied flows, which will not be considered here, hence

Nu = ji(Gr, Pr, Re, T,/T.) (2)

11. Natural Convection

A. HORIZONTAL CYLINDER

It may be shown from dimensional analysis [l-31 that the heat transfer from horizontal cylinders varies with the Grashof and Prandtl numbers.

HEAT TR.\NSFER FROM CYLINDERS 201

It is seen from Table I that scvcral correlations have been proposed, the simplest having the form [4],

( N u ) D % ~ = A I + &(Gr. Pr)2;tf (3)

where A1, B,, and m1 are constants and the subscripts D and f denote that the Nusselt and Grashof numbcrs are based on the diamcter, and the thcrmophysical properticis are taken a t the film temprrature Tr = +(Ts + TS). Values found by various workers for AI, B1, and the exponent ml are givcn in Table I. The nondimensional grouping (Gr. Pr)D,f is also known as thr Itayleigh number (Ra)D,f, but this symbol will not be used hrrc, because it may hc confused uith the Reynolds number (Re)D,t. A comparative measure of tht: disprrsion of the Nusselt numbcr for tl given Rayleigh number is the prrccnt coefficient of variation Ti equal to 100 x Etd dev/mcan. For thr rxperimental results, V varirs from 37, to 357,, dcpcnding on thc Raylcigh numbcr; whereas for the corrrlations given in the litcrature, 17 vnrirs from 57, to 26%. Thus, the corrdatiuns do not reduce the uncertainty in thc rchtion bttween the Nusselt numbcr and the Rayleigh number.

The wide dispersion in the published rxperimental results can be attributed to one or more of the following factors: heat conduction to the supports and thr teinpcraturr mrasurcment locations; distortion of the teniperaturp and vclocity fields by hulk fluid movements, thc use of under- sized containing chambers or thch prcserice of the temperature mrasurement system and supports; and temprraturc loading effects.

For axial conduction losses to b t i ncgligible, the aspect ratio Z/D should be sufficiently large. Champagnr et al. [ G l ] have shown that, for a cylinder without heated end-guard sections, the temperature is uniform over at least the center third of the cylinder whrn Z/D > 200, but it may be seen from Table I that this ratio was sometinies < 10. Also, it may bc seen from Fig. 1 that, when both the heat dissipated and the cylinder temperature are calculated from the voltage drop across the cylinder, as in hot-wire meas- uremcnts, then Z/D should exrccd lo5 for negligible conduction error, but it may be seen from Table I that the aspect ratio was rarely >104. The apparent increase in heat transfer with hot-wire measurcments can be calculated from the cmpirical equations

or

TABLE I : NATURAL CONVECTION FROM HORIZONTAL CYLINDERB

Ayrton and Kilgour Pe t,avel Kennelly et al. Wamsler Langmuir Davis Rice Rice Nelson Koch Nusselt Schurig and Frick Ackermann King Jodl bauer Jakob and Linke Herman n Martinelli and Boelter Mason and Boelter Lander Elenbaas Elen baas Jakob Gordon Senftleben Bosworth Kyte et al. Merk and Prins Cheng McAdams Collis and Williams

Air Air Air Air Air Air, gases, liquids Air Air Water Air Air, liquids Air Water Air, liquids Air Liquids Gases Water Wat.er Gases, liquids Air, gases, liquids Gases, liquids Air, liquids Air Air, gases, liquids Air, liquids Rarified air, gases General Air Air, gases, liquids Air

0.031-0.356 1.1

26.2 20.5-89 0.04-0.51

0.083-0.155 -

42.8-113.5 0.33 14-100 -

11.7-72.7 50 -

20-90 35

9.5-31.8 19.1

-

- - - -

0.203, 0.254 - -

0.078

6 .5

-

- 0.003-0.041

143-1640

25-500 24-102

59&7430 647-1 2 10

18.5

- 16-42

313 38-286 -

63-475 -

- -40 - 180

-5

48

-

- - - - -

-350, 440 - - 5900

-5000 -

- -

790-9000 403

4540 34-147

1960-24,700 236-7 17 -

13-29 512 20-152 -

45-286 2.8 -

55-140 4.3 -

17 - - - - -

386, 483 - -

1910

466

-

- 1000-20,000

1.51 2.84 1.13 1.41 4.84 1.34

1.46 1.07 1.28

-

- 1.10 2

d 1.08 M

2 1.22 l+

- z

-

1.06 .

1.04 1.05 - - - - 1.09 1.07 - - - 1.15

1.57

-

Etemad Lemlich van der Hegge Zijnen hlikheyev Mikheyeva

Fischer and Dosch Beckers el al. Kays and Bjorklund Tsubouchi and Sato Fand and &ye Fand and Kaye liehrov Zhukauskas et al. Deaver el al. Knudsen and Pan Chiang et al. Winkler Lemlich and Rao Fand Dyer Tsubouchi and Masuda Saville and Churchill Mabuchi and Tanakn Li and Parker Tsubouchi and Masuda Penney and Jefferson IIatton el (11.

Weder Gebhart and Pera Kravtsov and Polnitskii Bansal and Chandna Pomfret Engineering Sciences

[a31

[35] C361 [36]

[37]

C2291 [39]

c341

c381

c40 1 c411 [42] l43 1 C441 [4.i] c 4 ~ 1

c48 I c49 1 c501

~ 2 3 0 1

C47l

[51] c521 [537

[54] [.5.5] p e l C.571 [5S] [:is] CG0l

[ZlS] [231]

Air Air Air, liquids Air, gases, liquids Hot and cold air, water

and oils Air Air, liquids Air Air Air Air Iiarified air Water, oil Water .4ir General Air Water, glycerine Water Air Air General Air, liquids Water Toluene, oils Water, ethylene glycol Air Sodium hydroxide Air Air Gases, liquids Air Air, liquids

60.4, 63.5 1.01

0 .1041.9 0.01 5-245 -

0.0022-0.114 0.023-0.66

0 0049-0.061 57.4

19 22 .2

1.31-9.9 S 0.178

3 .i - - 1 .2.5

11.1 77.2 21.5 -

0.03-0.20 0.20

0.015-0.063 0.20

0.10-1.26 26.2 -

1.98

0.81, 1.26

-

-

-

I

9.5-4000 -

-

5260-27,300 - -

~6000-80,000 159

-23 52-400

2 . 1 4 -2000

23 - -

490 7.8

- -10,000

- 1250-8000

750 -2500-10,000

1493 52r~6.500

5 . 8 -

400 - - -

7.1, 7.5 917 93-6.55 - -

807-12,800 - 8 . 1

2460-8200 21 5 . 7

61-458 -20 860

G . 9 - -

367 7.8 -

1600-6500 -

737-4790 -

N 1600-6600 752 96-1190

4 .0 -

500 - - -

1.15 1.34 - - -

2.02 1.02 1.07 3.27 I .23 1.17 1.23

1.13 1.07

-1.1

- - 1.08 1.09

1.20 -

- -1

1.13 1.26 1.28 1.34 - - 1.93 - - I

(Continued)

TABLE I-Continued E3

52 Range of (Gr * Pr)o. f From Eq. (1)

Author Ref. From To -4 I B1 ml Remarksa

Ayrton and Kilgour Petavel

Kennelly €2 nl. Wamsler Langmuir

Davis Rice Rice Nelson Koch

Nusselt Schurig and Frick Ackermann King

Jodlbauer Jakob and Linke

Hermann

Martinelli and Boelter Mason and Boelter Lander

10-4 0.1

3 x 102 10-2

4.5 x 1 0 - 6

10-2

10-2

1.4

3 x 1 0 4

10-4

4 x 103

4 x 103 4 x 105

104 2.7 x 103

107

3.9 x 104 104

104

5 x 104 103

103 106

108

L* 1 . 3 X lo6

108

3 x 10-2 3 x 102 2 x 105

0 . 3 3.5 x 106

10-2 0.6

106 104

6 X lo6 66

4 x 105

8 . 2 x 105

6 X lo6 108

4.5 x 108 106 10'2

3.6 X lo6 108 10'2

5 x 108 Ld

4 . 3 x 106 5 x 106

107 109

0 1.61 0 1 .or)

0 0.562 0 0.945 0 0.480 0 0.81 0 1.12 0 0.47

0 0.97 0 1.32 0 0.412 0 0.286 0 0.502 0 0.57 0 0.14 0 0.53 0 0.13 0 0.480 0 0.555 0 0.129 0 0.424 0 0.402 0 0.59 0 0.74 0 0.49 0 0.12

E c

0.141 0.14 0.25 0.118 0.25 0.065 0.125 0.25

0.203 0.102 0.25 0.28 0.25 0.24 0.32 0.25 0.33 0.25 0.25 0.333 0.25 0.25 0.233 0.225 0.25 0.333

c

4

Elen baas Elen bans Jakob

Gordon Senftleben

Bosworth Kyte et al.

Merk and Prins

Cheng Mc Adams

Collis and Williams Etemad Lemlich van der Hegge Zijnen Mikheyev

Mikheyeva Fischer and Dosch Beckers et al.

I

- 10-7 104 108

10s Large

7 x 10-5

10-4 10-7 10’ 5

Ld L d

2 . 7 x 102 10-4 10-2 1

104 109 10-10

1 .2 x 106 6 X lo2 10-6

5 x 102

0.4

2 x 10-8

10-3

2 x 107

3 x 10-5

- -

10-2 108 1012

108 3 x 10-2

109 101.5 109

1 . 2 x 103

I, d

Ld

10-2 1

5 x 102 1 0 9

10-3

109

2 x 107 1013

8 x 10-3

1012

1 . 3 X lo6 6 X lo3

5 x 102

108

1

0 0.49

0 1.0 0 0.52 0 0.126

I /

I P

e d

0 0.41 h h

I

1 f

0 0.491 0 0.436 0 0.70 0 0 948 0 1.08 0 1.08 0 0.525 0 0.129 0 0.675 0 0.456 0 0.45

0 1.18 0 0.54 0 0 . 133 0 0 .i 1 0 I). 8(i2 0 0.9s 0 0.912

k k

0.25

0.10 0.2.; 0.333

*

J

0

0.25 A

1

0.2.; 0.25 0.23 0.0768 0.104 0.145 0 .25 0.333 0.0,58 0.25 0.22

0.125 0.25 0.333 0.25 0.0678 0.08 0.0767

k

to (Continued)

TABLE I-Continued N

Range of (Gr - Pr)D. f From Eq. (1)

Author Ref. From TO A I B1 ml Remarks

Kays and Bjorklund Tsubouchi and Sat0 Fand and Kaye Fand and Kaye Itebrov Zhukauskas et a2. Deaver et al. Knudsen and Pan Chiang et al. Winkler Lemlich and Rao Fand Dyer Tsubouchi and Masuda Saville and Churchill Mabuchi and Tanaka Li and Parker Tsubouchi and Masuda Penney and Jefferson Hatton et al. Weder Gebhart and Pera Kravtsov and Polnitskii Bansal and Chandna Pomfret Engineering Sciences

P 2 9 1 c391 ~ 4 0 1 ~ 4 1 1 c421

c45 1

C43 1 C44 1

C46 I C47 I C48l C49 1 C50l [5l] [ 5 2 ] [53]

C54 1

C56l [57] [58] [S9] C601

C2181 C2311

12301

C551

2.8 x 105

104 2 x 104 10-7

1.5 x 104 0.2

7 x 104 Ld

7 x 10-9

10 1.8 X lo2 1.6 X lob

105 1.6 x 104

Ld

5 10-6 0.25

6 X 10a

5 x 10-3

4 x 10-3

3 x 10-7 2

0.5 10-4

-

7.5 x 105

4 x 104 6 x 1 0 4

7 x 10-2

108 2.5 X lo6

20 1.3 x 105

1.9 x 103

5.3 x 104

Ld 108

- 4 x 106

Ld 3

61 10 30 10

6 X lo6 4 x 10-6 45

10 109

-

0 0 0 0

0 0 0 0 0.7 0 0 0 0 0 0 0 0.36 0 0.525 0 0 0

0

1

0.53 0.25 0.812 0.0667 0.485 0.2.; 0.495 0.26

0.50 0.25 1.15 0.15 0. ri6 0.25 0.532 0.25 0.42 0.25 0.58 0.25 0.48 0.25 0.55 0.233 0.48 0.2.5 0.548 0.25 1.02 0.10 0.35 0.32 0.56 0.23 1.08 0.213 0.422 0.315 0.858 0.22 0.676 0.058 0.94 0.171

0.94 0.162

t 1

m m

7 l

E E Eb Eb C, E E, dJ

E M

E

T C Eb E

E T E Eb E E b

E, t E Eb Eb C E C

0

s e d

z Eb, U z

0

Footnotes lo Table I

a C, correlation; E, experiment; T, theory; U, uniform heat flux; t, multiply by (7'f/Ta)oJs4; 6, multiply by ((Pr)J(Pr).)'J.*5. All

') I.:quation derived by present :tut hor to f i t published dat.a. c

d L, laminar boundary layer.

1 (Nu)D,f = 0.685(Gr

properties at ambient (fluid) temperature.

= 2 / l n l l + 2/0.47(Gr * Pr);;,;].

(Nu);,$ e x p ( - G / ( N u ) ~ , r ] = (Gr - Pr)D,8 /2361#~ (Gr - Pr)D,,. For (Gr * Pr)D,, < lo4, +(Gr - Pr)D,, = 1 . Pr)~,of39(03/7'f)o.061 for < (Gr ' Pr)E,of39(Bs/Tf)0.061 < 5 X and (NU)o,f = l.O5O(Gr - Pr)k,y'-

(e,/TfjO for .j x 10-4 < ( ( i r - Pr):).,":2(e,/Tl)o ~8 < I .

3 0 K

i N u ) ~ = (T./T.)O expC0.0545 + 0.0922 In(Pr). - 0.0147(ln(Pr).jz + 0.118 In(Gr - Prjr) + 0.00485(ln(Gr - P ~ ) B . . ) ~ ] .


1

. 0- - s

L

\ . 0.1 6 : -

L 2" 0

0.01

1 I I I I , 1 I l l , , - ;\\ ."\ I I I I I I ' I J

- '\, 0 - - - '\%

<\ -

0 -

- - - -

j j

0

0

A

A Gebhort and Pera (58), (Gr, Prl0,' = 3 ~ 1 0 . ~ -

Collis and Williams (.207), (Re),,, 24

Gosse (/I.?!, (Re),,, = I

Collis and Williams (32), (Gr, Prl,,, =lo7 Gsbkart and Pero (581. LGr, Pr),,, = 3 x 10.'

- - - EO (4al - - EQ. (4b)

- 1 1 I I l l I I 1 I I , , , I I I I I 1 1 1 ,

10 10 lo4 105

in the range 300 5 I,fD 5 105. Various methods have been given to calculate the error due to conduction [79, 112, 121, 205-2071. Lowell [205] and Gosse [112] derived the approximate expression

h R , ' I 2 (4c) I G(Nu)D,r = g {

(Nu)D,r 1 XaRT(NU)D,r

where X, and A. are the respective thermal conductivities of the wire a t its midlength temperature and the ambient fluid; and R, and RT are the resistances of the wire at the ambient temperature and the operating temperature respectively. This assumes that the convection loss is uniform along the wire and, hence, gives too low a value for the conduction loss. Betchov [206], Collis and Williams [207], and Baille [251] have dealt with the case of nonlinear convection loss.

For minimum error in measuring the surface temperature of the cylinder with thermocouples, the junction should be small, and the wires should have small diameter, be well insulated, and be run in contact with an isothermal for a reasonable distance before leaving the surface [31]. There is insufficient detail in many papers to enable the possible thermocouple error or any error due to movements of the bulk fluid to be estimated.

HE.\T TR.4NSFEIt FROM CYLINDERS 209

One would anticipate that too small a space ratio D J D , between the height of the test chamber and the cylinder diameter, would lead to distortion of the velocity and tempwature fields, but no determination of the effect of the space ratio on thr (.onvrctive heat transfer appears to have been undertaken. It is impossihlv to dcducc this from the cxprrimcntal results given in Tablr I, bccausti low values of D J D are invariably asso- ciated with low values of the aspcirt ratio l /D. Beckmarin [24i] found that the incrcase in heat trttnsfcr n.as proportional to 1n(De/B) * * for 1.19 5 De/D 5 8.1 for air and carbon dioxide. Crawford [247) deduced from theoretical considerations that t hcre is insignificant distortion of the iso- therms and streamlines in air M ith D,iD = 57 and (Gr)o = 8, but until the effrct of the space ratio is fully determined, it would probably be advisable to use as high a value a s is practicable, say D J D 2 100.

It will be noted from Table I that somr authors have given nonz(w) values for thc constant A , in Eq. ( 3 ) , :tnd it is found that the rewlts with low Raylcigh numbers givrn by o t h c ~ authors can also br corrclntcd quite satisfactorily with nonzero va luc~ for .4,. This has led saint' writers to concludc that the convective hvat transfer remains finite as the Iiaylcigh number approaches zvro. Howlvrr, Ohman [GZ] has shown that the Nussrlt number at zwo Rayhigh numhcr, (Nu)D,f ,o should be1 zcro for an infinitely long, infinitely rcmiotci cbvlindcr but, for finite valurs of the aspect ratio / / I > and the sp:irc' ratio I )< / D , it should have the limiting value 2/ln(21/'D) or 2/ln(DJD), rcywctively. So far as is known, i n none of the expcrimrnts did eithw ratio rxccwl loJ, so that ( NU)^,^,^ should h a v ~ the limiting value of 0.164. In fact, five \vorliers [28, 38, 30, 42, 581 have measured (SU)D,~ = 0.27 a t ( ( : I - . €'r)D,, = lo-' and three of them [38, 39, 581 have measured ( S u ) o , f = 0.23 at (Gr. Pr)D,f = lWq. C'ollis and Williams [32] nicasurrd (Ni1)D.f = 0.18 a t (Gr. Pr)D f = 3.8 X which is approaching the limiting vnluc.

Nussclt [l] postulated from thv diffcrcntial equation of natural convection that the Nussclt numbvr bascd on surface temperaturcl, ( Nu) I).*,

should vary with a trmperaturc loading factor. Hcrmann [l8] latw showed that the experimental results of cwlirr workers C5-91 cxhibitcd such a temperature loading effrct with small Grashof numbers. Thc Nusselt numbrr (NU)D,~ was shonn to dvcwase by 22% with increasc in the factor ( T , / T , - 1) from 0 to 1 in th(1 r:tiigr 10-4 < (Gr)D,, < 10. The thcorctical reduction in Nussrlt numhrr was givtw as 19%. On the othcr hand, the present author has aimlyzed thv cyx.rirncmta1 results of Davis [S], Rice [4], Ayrton and Iiilgour [5], Wamslvr [S], Langmuir [9], lioch [ll], Jodlbauw [lG], Lrmlich [34], anti Fischrr and Dosch [37] on thc. basis of the film tcniperaturc, and thew appcars to be no systematic c>ffrct of temperature loading on NU)^,, for 5 (Gr Pr)D f 5 lo8 and


Airton and Kdgour I51 Rssu(ts for 0 74 mrn wire omitted

Tf1Ta - I 0 8 o T f / l a = 1 2 5 D Tf /Ta = I 50

-

1.01 5 Tr/T, 5 3.39. A typical analysis is shown in Fig. 2. There appears to be no other published assessment of the effect of temperature on the Nusselt number. Hatton et al. [56] assumed that NU)^,^ was increased by the same loading factor as they found with forced convection, i.e.,

After a careful analysis of all the published experimental data, together with possible errors, the author proposes the correlation between Nusselt and Rayleigh numbers for horizontal smooth circular cylinders given in Table 11. It is believed that the proposed correlation has a maximum uncertainty of f5% ovcr the whole given range of Rayleigh numbers, except possibly in the transition regions a t (Gr. Pr)o,f values of lop2, lo2, lo4, and lo7, since it is improbable that there will be abrupt changes in the gradient of the curve in these regions.

(Tr/T.) 0.154.

B. INCLINED CYLINDER

When a heated circular cylinder is inclined a t an angle a to the horizontal in still conditions, there is an axial component of flow and a thickening of

0 2 I I 1 I I I l l 1 I I I I I I l l I 1 1

I G r . Pr) D,

0 2 I I 1 I I I l l 1 I I I I I I l l I 1 1

I G r . Pr) D,

FIG. 2. Effect of temperature loading on the heat transfer by natural convection from wires in air.

HEAT TR.\NSFER FROM CYLINDERS 211

TABLE I1

PROPOSED C O R R E L \TION FOR NATURAL CONVECTION

FROM HORIZONTIL CYLINDERS

Range of (Cr - €'l)n,r ( N r l i D , r = &(Gr - Pr)"uf

Frnm TO €3 1 ,?a1

10-10 10-2 0.67.5 0.0.58 10-2 102 1.02 0.148 102 104 0.8.50 0.188 104 1 0 7 0.480 0.2.50 107 10'2 0.123 0.333

I0- '0 0 178 10-9 0 20.% 10-8 0 2:<2 10-7 0 2fi5 1 0 - 6 0 30:3 10-5 0 34ti 1 0 - 4 0.39fi 10-3 0.4*;2 10-2 0. .i 1 t i 1 0 - 1 0 . 726 1 1.02

10 1 .43 1 02 2.02 103 3 .11 104 4.80 I 05 8.54 106 15.2 107 27 .0 10s 33.0 109 12.5 10'0 269 10" 5x0 10'2 12.50

the boundary layrr, leading to a drcrcnse in the convective heat transfer, as seen in Fig. 3. This effcct is small for small inclinations, and ( N u ) D , ~ decreasw by only 8% as CY incwaschs from 0" to 45", but as thc axis of the cylinder approaches thr vertical, thr heat transfer falls rapidly. Howcver, if we hasc the Nussrlt and Grashof numbers on the vertical dimtmsion, i.c., diametcr D for a horizontal cylinder, D/cos CY for an inclincid cylinder and height H for a vertical cylindcr, then the Nussclt numbrr can hr success- fully correlated with the Grashof numbrr. It can be shown that, except as a + 90", the ratio of ( N u ) D , ~ with inclination a to that with CY = 0' is equal to (cos a)1-3ml whew ml is the exponent of (Gr. 1'r)D,f in Eq. (3) . It is seen from Fig. 3 that this agrcrs quite well with the author's [63] results and thosc of I h c h [l l] for air. However, Bosworth [27] found that in glycerine the hcat transfer d(.creascd by 20% when CY incrrasrd from 0" to 45".


Inclination to ho ruon ta l , a:

FIG. 3. Effect of inclination to the horizontal on the natural convective heat transfer from cylinders in air.

In the case where 1 >> D, it is possible to correlate (Nu)D,r for a vertical cylinder wit,h (Gr. Pr. D/H)D,~ . If the diameter is large, and hence the curvature small, the vertical cylinder can be approximated to the vertical plate, and the correlation for the cylinder is then similar to that for the plate [31]. With very fine wires, the boundary layer is large in relation to the diameter, and it has been found [64] that ( N u ) D , ~ is independent of height. One can also use the height H as the reference dimension instead of the diameter D. It is seen from Table I11 that various correlations have been proposed, the simplest having the form

(Nu)lr,r = A + &(Gr. Pr)?:, (5)

Experimental values for the constants Az, B2, and m2 are given in Table 111. It is considered that, for lo4 5 (Gr. Pr)H,f 5 10l2, the correlation given in Table I1 may be used, provided that the diameter D is replaced by the height H .

111. Forced Convection

A. CYLINDER WITH CROSSFLOW

For a cylinder in crossflow, dimensional analysis [l, 771 suggests that the convective heat transfer varies with the Reynolds and Prandtl numbers,

TABLE I11

N.ITUR.\L CONVECTION FROM VERTIC.\L CYLINDERS

Author Ref. Fluid D (mni) I I D Max T r / T ,

Griffiths and 1)avia Koch King Jakob and Linke Carne Eigenson 3Iueller Clenbaas Touloukian ct al Benftleben Kyte et al McAdams Sparrow and Gregg Le Fevre and Ede Millsaps and Pohlhausen Kreith Eshghy Nagendra et u2 Fujii e l nl Hanes:tn and Kalish Bot teriianne Engineering Sciences

[65] C l l l [15]

C66l

p .1 t68l [26] C28l

C69 1

c171

c67 1 c64 1

c311

c701 c711 1721

c76 1

173 1 c74 1 [75]

[216] [231]

Air 174 Air 14-100 Air, liquids -

Air 3.5 iiir 4.7-76.2 Air 2.4-.58 A i r 0.11-0.69 Air , gases Water, ethylene glycol 69.7

Air, gases 0.078

Gases General General - Air Air Water Water, oils 82 Air, fluorocarbon gases 2.5.4 Air

-

Liquids -

Air, liquids - .-

-

-

- 0.w3.0

-

Air, liquids -

0.87-1.7.2 1.17 20- 1.52 1.28

4 . 3 1.06 8-127 1.22

e50-140 - 1480-2720 1.14

- -

- - 2.2-13.2 1.16 - -

5430 1.28 - - - -

- - - -

- - - -

9.5-3050 1 .06 12.2 1.19 3.12 1.34 - 1.05 - -

(Continued)

TABLE 111-Continued

Range of (Gr - Pr)H. f From Eq. (2)

Author Ref. From To -4 2 B* 111 2 Remarks0

Griffiths and Davis

Koch King

Jakob and Linke

Carne

Eigenson

Mueller Elenbaas Touloukian et al.

Senftleben Kyte et al. Mc Adams

Sparrow and Gregg Le Fevre and Ede Millsaps and Pohlhausen Kreith

Eshghy Nagendra et al.

107 109

104 3.5 x 107

104 108

2 x 10' 2 x 108 < 10s

1010 -

104 2 x 108 4 x 10'0

10-11 -

104 109 L a

L L 105 109 107

1

1 0 9

3 . 3 x 107

10" C

10'2 108 10'2

2 x 10s 2 x 10"

109 10"

109

-

4 x 10'0 9 x 10" - 10-5 109 1012 L L L 109 10'2

5 x 107 1

0 0.67 0 0.0782

0 0.5.5 0 0.13 0 0.555 0 0.129 0 1.07 0 0.152 0 0.48 0 0.148

c

d d

B

0 0.726 f f

0 0

h h

0 0.59 0 0.129

1 i

k k

0 0.555 0 0.021 0 0.56 I 1

0.23 0.357

0.23 0.33 0.25 0.333 0.28 0.38 0.26 0.333

d

c

0.26 f

0

h

0.25 0.333

i

k

0.25 0.40 0.25

1

E b

Eb E*

E ls

E $

0 .a Eb C 0

E E E C C T T T C C E c, E

Fujii et al. C7-51 10'0 2 x 10'2 E Hanesian and Kalish C76 1 106 108 0 0.48 0.23 Eb

Bo ttemanne C2161 8 X lo4 4 x 107 0 0.56 0.25 E C Engineering Sciences P 3 1 1 - -

a C, correlation; E, experimental; T, theoretical; L, laminar boundary layer. 6 Eqmtion derived by present aut,hor t'o fit) experimental results. c ( N u ) D , ~ = 0.00562(Cr Pr * H/D)$ ' , l ' 2 ; lo9 < (Gr Pr - H / D ) D , f < 1011.

(Nu)D,f = I.O(Gr Prj;.:'; < (Gr P r ) D , f < 10-2. NU)^.^ e x p ( - 2 / N u j ~ , f = 0.6(Gr - Pr * D/H)))',;.

f ( N u ) H , f = 0.0674(Gr~(Pr)' "]::'3

where

(Gr * Pr):;.: . 3.34 H X = l n l + - i i D

( N u ) ~ , r = 2/ln(l + 4.47/(Gr - Pr - D/H)R,YI.

4 7 G r H ' (pr)z I" 4(273 f 315 Pr) H .- j NUH = 3 -{ 5(20+ 21 Pr) } + % ( 6 4 + 6 3 P r ) D

4.22 (Pr)'I2 H ' N U ~ = . - [Gre(4 + 7 Pr) D '

( N u ) D , ~ = 0.87(Gr - Pr - D/H)L,','; ( N u ) D , ~ = 1.3(Gr - Pr - D/H);,i6; ( N t 1 ) D . f = 0.67(Cr - Pr - D/H)Yi,:5;

'* NUH = 0.017(Gr Pr)~',~(va/v,)O-2l.

1W4 < (Gr - Pr * D1ff)o.f < 5 X lop1; 5 X lo-* < (Gr - Pr D/H)o,r < lo4;

lo4 < (GI * Pr - D / H ) D , ~ < lo6.

(Nu)H,* = (T,/T,)0.'75 exp[-2.95 + 1.02 In(Gr - PrjD,, - 0.0829(ln(Gr - Pr)D.,)* + 0.00267(ln(Gr - P ~ ) D , ~ ) ~ ] .

x m * e


a commonly used relation being

(Nu)o,f = {CI + D~(Re)$,fl (Pr)fP ( 6 )

where C1, D1, nl, and p are constants. It has been found experimentally that the exponent p lies between 0.3 and 0.4, the lower value being more commonly used, although Zhukauskas [225] prefers the value 0.37. For air, (Pr) is approximately constant for moderate temperature rises and has the value 0.7; hence

(n'u)o,f = CZ + Dz(Re)E,f (7)

Values obtained by various workers for the constants Cz, Dz, and n1 are tabulated in Table IV, and published correlations are given in Table V. The percent coefficient of variation of the Nusselt number for a given Reynolds number for the experimental data varies from 10% to 29%, depending on the Reynolds numbcr, whereas that for the various correlations varies from lOy0 to 46%. Hence, the correlations do not help to reduce the uncertainty in the relation between Nusselt and Reynolds numbers.

It is also seen from Table IV that some authors give nonzero values for the constant Cz, as in the case of natural convection. As the Reynolds number approaches zero, the Nusselt number for an infinitely long, infinitely remote cylinder should also approach zero, but Cole and Roshko 11411 have suggested that the heat transfer from fine wires should be given by the solution of thr heat conduction equation for an ellipsoidal surface a t constant temperature. For finite values of the aspect ratio and the space ratio, (Nu)D,f,,, would have the limiting value 2/ln(2Z/D) or 2/ln D,/D, respectively [62). Since the aspect ratio 1/D did not exceed lo4 in any of the published experimental results, ( Nu)D,f,o should have the theoretical limiting value 0.20, whereas the lowest value reported [58, 1151 is 0.30 a t (Re)D,f = after correction for axial conduction to the end supports. This discrepancy may be due to weak natural convection currents: Mahony [208] concluded from a theoretical analysis that two-dimensional natural convection prevails when (Gr) g,fZ/D >> 1, whereas three-dimensional conduction prevails when (Gr)g,fl/D << 1. The effcct of the aspect ratio on the heat transfer a t small Reynolds and Grashof numbers is shown in Fig. 1.

Apart from the effect of aspect ratio, other possible reasons for the large variations in Table IV arc wake and blockage effects due to the wind tunnel, and turbulence in the flow and temperature loading.

Several methods for correcting for solid and wake blockages have been given. Lock [154] used the method of images to calculate the increase in stream velocity due to solid blockage, and Thorn [219] that due to wake blockage. Combining these corrections, the free-stream velocity U , for a

HE.\T TIZ.\~NSFER FROM CYLINDERS 217

closed-throat wind t u n n d is givcw by

Urn = U T ( ~ + 0.25 C'D(D/DT) + 0 . 8 2 2 ( D / D ~ ) * ) (Sa)

whcrc UT is the mcwwcd vdocity of the flow in the tunncl, DT is the tunnel diameter, and CD is thr drag coc.f€icic.nt. Vincenti and Graham [153] used thc method of superposition to dvrivca a similar equation:

Urn = U T { 1 + 0.321 C"D(D/DT) + 1 . 3 5 6 ( D / D ~ ) ~ } (8b)

The drag coefficient varics with the Reynolds number, as shown in Fig. 4, and with blockagc ratio, D/L)T, turbulcnce intensity, Tu, and aspect ratio, Z/D [252] . For 10' 5 (Rc)n ,r 5 lo", Z/D > 10, D/D, << 1, and T u << 1, C D 'v 1.2, hence Eq. (811) bwoiric~s

Urn 'V l i r ( 1 + O.~%(D/DT) + 1.356(D/D~)'} (9)

Ihudscm and Katz [155] suggvsted that Urn = UT(1 + D/DT)1 /2 , and

0 1 1 10 10' m3 10' 10' Id

(Re',. f

FIG. 4. Variation of drag coeficimt with Reynolds number. (Mean dat,a of Relf [220], Von Wieselbergpr [221], Tril(o11 [222] and Humphreys [223] )

TABLE IV

FORCED CONVECTION FROM CYLINDERS IN CROSSFLOW: EXPERIMENTAL DATA

100 D/DT 100Tu Max Author Ref. Fluid D (mm) 1/D (%I (%I T d T e

Kennelly et al. Kennelly and Sanborn King Hughes

Gibson Davis Reiher Lohrisch Ulsamer

McIntyre Hilpert

Klein Griffiths and Awbery Goukhman et al. Small Krujilin and Schwab Benke Kruj ilin Schmidt and Wenner Piret et al.

[7] Air [78] Air [79] Air [SO] Air

[Sl] Air [82] Water, oils [83] Hot air [84] Air [85] Air

[86] Air [87] Air

[88] Hot air [89] Hot and cold air [90] Air [91] Air [92] Air [93] Air [94] Air [95] Air [96] Water

0.02 0.114

0.028-0.153 4.3-155

9.25, 95.2 0.10-0.20 4.6-28 - -

6.2, 10.3 0.02

0.02-0.10 0.05-25

2; -90 44-180 -

32-76 35

114 44 7, 9

23.6-300 50-230

0.025-0.07

197C-3970 2810

1290-1 520 1 .3-233

12.8, 132 247471

14.3-87 - -

9.8, 16.4 5120

1625-5120 2&3 170

0.95-11.4 5.6-20

- 8-19.2

17.2 5 .3 7.6

-100 2.7-42.4

1-12 350-1000

- -

- 0.47-1.1 1.1-17

0.51, 5 .2 -

2.9-17.5 6 . 2

18.2 46.0 7.6, 12.7

0.0079 0.0079-0.04

0.02-10 10-60

17.6-60

-

- 5.2-12.5 2.5

18.8 14.7 3.2, 4.1

1 6 . 7 4 1 . 7 0.067, 0.19

1 .7-21.5

- 1.18 1.67 2.73

- 1.15 - 1.15 - 1.14 5 - 0.76 M

- -

- 1.10 2 - 3

.e z

- - -

0 - 1.08 - 1.14

- 0.87 - 1.14

- 1.12 - 1.14 - 1.8 - 1.14 - 1.14 - 1.15

- I

Comings et al. Winding and Cheney Zapp Billman et 01. hlaisel and Sherwood Eckert and Soehngren Scadron and Warshawsky Laurence and Landes Berry el al . Snyder Cheng Franklin Kazakevit ch Churchill and Brier Brim et nl. van der Hegge Zijnen Glawe and Johnson Gosse Broer et al. Chung and Algren Kays and Bjorklund Collis and Williams

Shchitnikov Shchitnikov Srenivasan and Raninchandran Perkins and Leppert van Meel Ilavies and Fisher Parnas Delleur

Air Air Air Air Air, gases Air Air Air Air Air Air Air Air Hot nitrogen rZir ,4ir Hot gas Air Air Air Air Air

Hot air Air Air Water, el hylerle glycol Air Air Air Air

9.5, 31.8 38 60.2 4.82

13.5, 14.2 12.7-38.1 0.22-1.41

0.00.5 0.81

31.8 6 . .i

50.8 22 2.5.4 .; , s

0 . 1 , 9.04 0 . .70-1.14

0,026-0.49 0.004, 0.007

28.6 57.4

0.003-0.054

- I

8 . 7 3 3.2-9.5

23.2 0.0025-0 I 003

0.03 0.004

14 3, 40 8 5 .9

64 2.3-4 0

6-18 -

400-2300 330

12.8 186

8 11 4

3 37 .i, G O

37 G-.i.i.i 30

713, 1230 102-2430

21 4 8 . 1

2070-8660

-

-

17.5

6 9 21 3-64

400-1 200 1000 100

3.3 , 11.3 24 13.2 3.i. 3

13.3 , 13.9 10.8-32.3

2-6 0.0033 4 . 5 9 . 3 3 . 2

12,.i 8 . 8 --

.) I

1.7 , 2 7 0.02.5, 22 G

0.032-0.98

-

-

11.2 11.3

8 . 6 X -0.015 10 -

4 . 3

4 . 6

0 . 5 0.0033

10-39

-

1-3 -

0.9 -

-4 -

-

Low 3 . 8 -

-

-

- Y:!

-

0 . 2 4 . 3 -

-

-

1-3 -

0.08

2.4 2.4

2 .9 0.2

LOW

0 . 2

-

-

1.13

1.12 1.23

-1

-

-

-

1.24 1 .23 1.19 1.09 1.15

0 .62

1.17

2.00 -1.15

0.94 1.07 1.50

-

-

-

- 1.20 1.14

1.07 1 38 2.27 1.32

-

(Continued)

TABLE IV-Continued

Fand Tsubouchi and Masuda Hodgson et al. Ahmed Fairclough and Schaetzle Oosthuizen and Madan Hatton et al. Johnson and Joubert Lewis KostiE and Oka Koch and Gartshore Martin Fand and Keswani Petrie and Simpson Andrews et al.

Wylie and Lalas Galloway and Sage Zhukauskas Bradbury and Castro Krall and Eckert Saito and Kishinami Nishioka Dahlen Isaji and Tajima Baille

Water Toluene, oils Air Hot gases Air Air Air Air Air Air Air Air Water Air Air, N2-CH4

Air Air Air, water, transf. oil Air Air Air Air Hot air Humid air Air

11.1 0.031, 0.065

76 - 0.079

19-38 0.10-1 .26

152 76

100 0.005

35-55 11.5 19

0.0038, 0.005

9 38.1 12

0.0053-0.011 4.73

0.0038 10

14.6 3-80 0 . OO,?

7.8 485, 910

1 .3 - - 8-16

96-1190 2 6 5

230 -

2.2 5.7

24-1300

-10 8 8.3

6.3 270-2000

14 650

13.9 3.8-100 120-800

0.128 0.031, 0.079

25 -

Low 4.7-9.4 0.2-2.5 9.1 5 0 . 2 0.0016

10 13.3 6.3

6.2 x 10-4, 8 .2 x 10-4

2.4 50 24

15.8 6.7 0.019 7.2 1-26.7 0.0007

-

- - 2.6 - -

<0.7

-1

-

- 2-8 - 0.3

Low 5.78

Low

- 1.3

Low

3.2 0.5

< I

-

-

-2 Low

1.01 1.20 1.05

2.87 1.22 1.34 1.14

-

- - 1.40

1.14

2.30

-1.00 1.10 1.18 2.05

-

-

- - 1.32 0.63

1.44 <1

TABLE IV-Continued

Range of (Iie)u, f From Eq. (.i)

Aii t hor Ref. From TO (12 D2 nl Remarks0

Kennelly et aE.

Kennelly and Sanborn King Hughes

Gibson Davis Reiher Lohrihch Ulsanier

McIn tyre Hilpert

Klein Griffiths and .4wbery Goukhmxn el al. Small Krujilin and Schwab Belike

c7 1 10 40

c781 .i 0 C79 1 6 x C80l 700

4 x 103 C81l 102

cfi3 1 4 x 102 C84 1 3 x 109 C833 0 1

C86l 7 x 102 c871 1

[82] 18

4 .i 0

4 40

4 x 103 4 x 104

1 8 x 103 [89] .-) x 103 [SO] 1 5 x 104 “311 7 x 102 [92] :3 x 103 [93] 2 5 x 102

8 x 102

40 180 1 .i0 30

4 x 103 4 x 104

2 . 3 x 103

6 x 103 4 x 104

170

4 3 0 1 W

4 40

3 .5 x 104

4 x 103 4 x 104 4 x 105

105

G x 104 8 x 104

3 x 103

2 . 3 x 104 6 . 3 X lo‘

8 x 102

0 0 0 0.40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.82 0.473 0.64 0.48 0,558 0,223 0.187 0.56 0 . 3 3 0,282 0.875 0.764 0.536 0.313 0.891 0.821 0.615 0.174 0.0239 0.328 0.18 0.161 0.299 0.35 1. 15 0.219

0.36 0.51 0.50 0.50 0.482 0.393 0.63 0.49 0.36 0.58.5 0.303 0.41 0.50 0.60 0.330 0.385 0.466 0.618 0.80<5 0.56 0 67 0.64 0.638 0.57 0.38 0.629

x m > c3

-3 a

a E

z 2 U m

N (Continued)

h3 E3 E3

TABLE IV-Continu.ed

Range of (Re)n. f From Eq. (5)

Author Ref. From To c2 Dz n1 Remarksa

Krujilin Schmidt and Wenner Piret et al. Comings et al. Winding and Cheney ZaPP Billman et at.

Maisel and Sherwood Eckert and Soehngren

Scadron and Warshawsky Laurence and Landes BerFy et al. Snyder Cheng Franklin Kazakevi tch Churchill and Brier

Brun et al. van der Hegge Zijnen

Glawe and Johnson

6 X lo3

0.1 4 x 102 6 X lo3

7.5 x 102

6 X lo2

5 x 103

3.9 x 104

2.8 x 103

20

2.5 X loz 1.5

1.5 x 102 8 X lo3

1.3 x 103 104

5 x 103 6.4 X lo2

6 X lo3 .5

20 450

1 . 3 x 105 4.2 x 105

2 x 104 3.5 x 104 1.1 x 105 2.8 x 103

5 x 103 3 x 104

3 x 104

1.5 x 103

104 4 x 104

3.5 x 104 5.1 x 103

1.5 x 104

3 x 103

10

6 X lo2

70

2 x 104

50 80

0 0 0 0 0 0 0 0 0 0.43 0 0 0.19 0 0 0 0 0 0 0 0 0 0.35 0

0.27 0.60 0.1.57 0.64 0.86F 0.28 0.232 0.63 0.16 0.67 0.50 0.54 1.13 0.38 0.040 0.81 0.322 0.57 0.48 0.50 0.57 0.473 0.385 0.515 0.51 0.50 1.13 0.38 0.278 0. g.5 0.161 0.65 0.174 0.65 0.246 0.60 0.535 0.50 0.66 0.48 0.136 0.65 0.68 0.41 0.43 0.50 0.428 0.50

b

Ob 0

M b

b

b

M

b

0 6

b

b

Of b

Broer et al. ~ 1 1 3 1

Chung and Algren ~ 1 1 4 1

Collis and Williams ~ 1 1 5 1 Kays and Bjorklund P291

Shchitnikov Cl161 Shrhitnikov ~ 1 1 7 1 Srenivawn and I~amachandrwi [118] Perkins and Leppert [119] van Meel c1201

Davies and Fisher c1211 Parnas [I221 Delleur ~ 1 2 3 1 Fand ~ 1 2 4 1

Tsubouchi and Masuda [125]

Hodgson et al. C1261

Fairclough and Schaetzle C1281 Ahmed ~ 1 2 7 1

Oosthuiaen and Madan ~ 1 2 9 1 Hatton el al. [56]

Johnson and Joubert, ~ 1 3 0 1

0 .2 3

40 0.01 1

2.5 X lo3 3 X lo3 2 x 10-2 44 104

2 x 1 0 4 2 .3 X lo3

40 3 X 103

1 10 1

1 . 1 x 104

3 x 10-2

3 x 104 4 2

0 . 6 0 .6 4

16

103

4 x 104 1 . 3 x 105

3 40

700 1 4

1.1 x 1 0 4 2 x 104

ri x 104 1.2 x 105

105 3 .7 x 104

44 140

1 ..j X lo4

30 60 3

6 . 3 X lo4

200

1.2 x 105

3 x 103

40 8

45 4 40 4 5

1.3 x 105 3 x 105

0 0 0 0 0 0 0 0.24 0 0 0 0 0 0.35 0 0 0 1.148 0.32 0 0.36 0.36 0 0.207 0 0 0.384 0 0 0 0 0

0.920 0.841 0.592 0.89 0.89 0.482 0.31 0.56 0.48 0.118 0.143 0.226 0.26 0.50 0,332 0.33 0.823 0.726 0.50.5 0.455 0.38 0.61 0.15 0.497 0.57 0.464 0.581 0.95 0.81 0.786 0.38 0.063

0.276 0.384 0.482 0.21 0.35 0 33 0.56 0.45 0.51 0.67 0.67 0 60 0.30 0. .50 0.36.5 0 3.57 0.50 0 50 0.52 0.53 0.50 0.45 0.67 0.46 0.50 0.50 0.439 0.30 0.38 0.392 0.56 0 .72

b

b . c

Ok

b

f 0 . 2 3 (Re);’,:, F

b

1

b

+0.0004 (Re)o,r @ b

b

b

b

b

tQ (Continued)

TABLE IV-Continued

p3 h3 From Eq. (5) t+

Author Ref. From To c2 D2 n1 Remarks"

Lewis KostiE and Okn Koch and Gartshore Martin Fand and Keswani

Petrie and Simpson Andrews et al. Wylie and Lalas Galloway and Sage Zhukauskas

Bradbury and Castro Krall and Eckert

Saito and Kishinami Nishioka Dahlen Isaji and Tajima

Baille

5 .7 x 104 1 .2 x 104

1.8 x 104

4.0 x 103

1 . 2

4 x 102

1.5 x 10-2 102

2 .7 X 109 8

0.5 5

50 2.5 x 102

2 x 102 7.6 X lo2

7 x 102 1 .5 x 101

103

10

1.2 x 105 4 x 104

4 .8 x 104 3.7 x 103

3 . 3 x 104

2.5 x 103 3.8 x 104

103 2 x 105

5 x 103 s x 103

8.4 x 103 7 x 102 1 x 105

4.1

20

12 50

0.2

0.6

0 0 0.72 0 0.231 0 0 0.34 0 0 0 0 0.24 0 0 0 0 0 0.14 0 0

0.36 0. .58 0.286 0.60 0.80 0.4.5 0.48 0.50 0.633 0.50 0.665 0.494 0.042 0.76 0.65 0.45 0.61 0.471 0.167 0.637 0.52 0.47 0.185 0.62 0.56 0.45 0.93 0.37 0.64 0.46 0.64.5 0.551 0.692 0.155 0.260 0.633 0.53 0.50 0.51 0.50 0 0.45

M *

+0.0096 (Re)%,?

b

Ob

a F, whirling fork; M, mass transfer; 0, open-jet tunnel; U,

Equation derived by present author to fit experimental results.

Multiply by (T./T,) A and Y a t surface temperature.

uniform heat flux; X, linear-motion wire.

c a t surface temperature.

f Multiply by (T,/T,)OJ2; g X and v a t fluid temperature.

Multiply by (T./T,)0J24.

and v at fluid temperature.

a Multiply by ( T r / T a ) .I7. i Multiply by ( p J ~ ~ ) l / ~ ,

h Multiply by (T./T,)0J85. Multiply by (vt/v,)OJ5.

m Multiply by (Tr/T.)0.154. n Multiply by ( T f / T . ) --0.6'.

0 Cz = 0.24[1 + $ exp ( (300 - 1/D)/80 + 0.341/D]] X [1 - 0.068 exp((250 - l /D)/120)R~/RJ;n, = 0.56[1 + 0.28exp((- 1/300D) 1 1.

TABLE V

FORCED CONVECTION FROM CYLINDERS IN CROSSFLOW : CORRELATIONS AND THEORY

Range of (Re)u . f From Eq. ( 5 )

('2 D2 n: Remarks4 Author lief. From T O

Boussinesq Rice

King, L. V.

King, W. J .

Ulsamer

Krarners Sailer and Drake Mc Adams

Franklin

Cole and Roshko Engineering Sciences

van der Hegge Zijnen l>oughs and Churrhill Beckers el nl. Kutateladze

[138] ClOl

[15]

[X;,]

[ S 5 ]

C38 1 [ 1421

[143]

0.3 70

>>o. 1 3

300 0 . 1 .i 0

5

-

0.1 103 103 104

LOW

2.6 10.5 10-2

10-2 .i00

> 80

5 x 103 > 5 x 104

70

<<o. 1

300

3 0

1000 > 300 1000

105

-

4 x 104

104

5 x 104 104

1 . 6 x 105 6 .2 x 103

5 x 105

106 Low

106 26 80

5 x 103 5 x 104

0 0 0

0.32 0 0 0 0 0.39 0 . 5 0.32 0 0 0

b

d

c

0.35 o

0 0 0 0

f

0.86 0.234 0.466

0.67 0.764 0.282 0.794 0.523 0.510 0.435 0.43 0.24 0.468 0.109

D

c

d

e

0.50 0.46

0.81 0.695 0.197 0.023

f

0.50 0.667 0.50

0.50 0.41 0.585 0.38.5 0.50 0. rio 0.50 0.52 0.60 0.504 0.662

b

d

0.50 0.30

r 0.40 0.46 0.60 0.80

T

T

T

T Gases Liquids +0.001 (1te)D.r i-0.00128 ( R e ) D , (

z m ;P e

2 E 2 U m

(Continued)

TABLE V-Continued

Range of (He)o. f From Eq. ( 5 )

Author Ref. From To c2 0 2 nl Remarks.

Baldwin el al. Hsu Richardson Richardson Illingworth Tsubouchi and Masuda Kassoy Hieber and Gebhart Dennis et al. Fand and Keswani Ototake

Zhukauskas

Piercy and Winny

Kibaud and b u n

[a251

10-2 -

100 1

Low 0.5

Low Low 10-2 10-2 1 0 2

1 40

5 x 103

103 2 x 106 4 x 10-2

Low

3 x 105

105 105

103

<500

Low

Low Low

40 2 x 105 5 x 103 5 x 105

103 2 x 105

40

106 1 .o

Low

0 0.43 0 0.37

0.36 g

h

i k

1

m

0 0 0 0 0.44

0 n

0.431 0.48 0.37 0.37

0.48 0

h

i k

1

I

0.66 0.45 0.23 0.067 0.59

0.474

0.50 0.50

0.50 +0.057 (Re);',; 0.50 +0.057 (Re);'.: =: 0.50 3

I 3 T

h T .e ' T i T

I T T

k

0.40 0.50 0.60 0.70 0.50

T 0.50 T

1:

Footnotes to Table V

a T, theoretical; y, Euler’s ronstant = 0.577; L, laminar boundary layer. (Nu)D,f = 2/11 - y - 111 2(Re - Pr)o , f ) . 2/ (Nu)n . f = ln(S/(I le - Pr)n, f} - y. (Nu)D,, = (Ts/T,)0.L75 exp[-0.179 + 0.252 ln(lte)o.n + 0 . 0 5 2 8 { h ( I < e ) ~ , , ) ~ - 0.00666( ln( I te )~ , , )~ + 0.000335(In(Ite)~, ,)~]. (Nu)D,, = ( p 8 / p 8 ) 0 . * 3 7 exp[-0.186 + 0.338 ln(Re)D,, + 0.362 ln(Pr), + 0.0131( ln( I te )~ , . )~ - 0.00926(ln(Pr),Jz]. ( N u ) D , ~ = 0.1“ (1.25 + 0.19log D ) log Prf - log D } + Iog(Ile)D,f]*.~ + O.%(Gr)L’,?’.

c

f

x P

Y 21 s 2

and t = D (Ite)o,,/4

( N u ) D ~ = 1.38 + O ( ( l n Re,,t.l]

E:

$

z tl

[l - 2

In(S/y(Re * Pr)DlSJ Inl8/r(Re - Pr)L.sl

for Pr = 0.72.

temperature.

where x = 0.247 + 0.0407 (l le)

( N i i ) ~ . , = -2g0’(0), where go’(0) is the first derivative of the Fourier cosine transformation of the function defining the surface

(Nu)D f = (0.184 + 0.324(Ile)L,:+ 0.291(Re)Z,,}(Tr/T,)017 ,ya.

’“ (Nu)u,r = 0.5 + O.253(Re - Pr)h’,i+ 0.0214(Re)~ ,~(Pr) : /3 .

for ( N i i ) ~ ( N i i ) ~ , , = 0.7183(Re - Pi-);‘,:

= -2[1 + 0.123(lte - Pr)i , ,{ ln( l te - Pr)D,,/8 + + I ] i (In(He - Pr)D,./8 + y) for (Ice - Pr)D,, << I , and (Re - Pr)o.B >> 1.


10''

Robinson et al. [156] that Urn = UT( 1 + ( D / D T ) ~ ' ~ } . Perkins and Leppert [119] proposed the use of a mean flow area, giving U , = uT/[1 - (s/~)D/DT]. Mikheyev [36] used the equation U , = UT/ (1 - D/DT) , whereas Akilbayev et al. [226] derived the expression Urn = U T { ~ + 1 . 1 8 ( D / D ~ ) ~ } ~ . All these proposed corrections are compared with each other and with the experimental results of Oosthuizen and Madan [129] and Akilbayev et al. [226] in Fig. 5, where it is seen that there are appreciable differences in the corrections given by the various methods. The method due to Vincenti and Graham is used here.

If CZ is zero in Eq. ( 7 ) , the increase in the Nusselt number due to solid

Exprrimenlal resulls 01 Oorlhuizen and Madan (/29)

X Exprr imrnta l resul ts 01 Akilbavev et 01 I2261

I . ( I t 1 I 1 I I I I I I , ,

lo-' loo

FIG. 5. Comparison between various methods for correcting for solid and wake blockages.

H E ~ T TRANSFER FROM CYLINDERS 229

and wake blockages is found from

~ ( N u ) D , ~ / ( N u ) D , ~ = ( 1 + 0 . 3 8 5 ( D / D ~ ) + 1.356(D/D~)')"' - 1 (10)

In the case of an open-jet tunnel, if the flow is essentially two-dimensional, wake blockage is usually taken to be zero. When the solid blockage correction [154] is applied, the frce-stream velocity is given by

U , = V,( 1 - 0.411(D/D~)'}

6(Nu)D,f/(NU)D,f = ( 1 - 0 .411(D/D~) ' }~ l - 1

(11)

(1%

The decrease in the Nusselt number is then found from

The effect of free-stream turbulcnce has been studied by a number of workers, as seen in Table VI. Both the intensity Tu and the scale A can affcct the laminar boundary layw, and hence the heat transfer. In the usual range, say 0.2 mm < A < 60 min [171], the effect of the scale appears to be negligible [164, 1661. van dcr Hcggc. Zijnen [159] reported an optimum increase in heat transfcr a t about A / D = 1.6 and attributrd this t o a resonance between the frtqumcy of the frrc-stream eddies and the frc- quency of the eddies shed from the cylinder. However, Mujumdar and Douglas [209] and Hinze [253] found 110 such systematic effect. On the other hand, the intcmity has a profound dfert. The increase in heat transfer due to th(1 intensity of turbultwc- in the direction of flow can be expressed by

6 ( Nu), , I / ( Nil) D , f = C3 (Re) 5,dTu) * (13)

where C3, a, and 6 arv constants. With turbulent flow, the dependence of the in(-reasc in local hcst trailsfor on the Reynolds number appears to be proven for the stagnation point and, possibly, the front surface of the cylinder [167, 2091. However, the scattchr in thc experimental results is so large that the exact form of t h c x d(ymidcnce is still in doubt, a variation with (Re)& being most favorcd [167, 172, 1731. Some authors correlate the increase in heat, t ransfer :qainst a so-called turbulence Reynolds number [(Re),,,. Tu], but this swms rather restrictive. For the overall heat transfcr from a cylind(.r, t h(* cixistcwce and form of any depc'ndence on ( I < c ) ~ , f cannot be vstablished, lwrsusr of insufficient rcwlts. For a value of ( R c \ ~ , f in the rvgion o f 1 0 4 , it has been found [97, 1591 that G(NU)D.,/(NU)D,I is equal t o 1 . 2 1 ) ( T ~ ) ' / ~ for 0.01 5 Tu < 0.03, and 2 . 4 2 ( T ~ ) ' ' ~ for 0.03 5 Tu 5 0.12. Thc discontinuity a t Tu = 0.03 may be due to increasing span\\isc. vffvcts i n the flow [248]. (Scc Note Added in Proof on p. 264.)

Thc c~dculated combined cffvcts of closed-jet tunnel blockages and free-strcam turbulenw arc. shown in Fig. 6. To compare measured increases in heat transfer duc. to these affects with the calculatcd curvrs, it is neccs-

230

- 100 Tu -

L 9 - 12 ( 99 )./ 12% I

VINCENT T. MORGAN

FIG. 6. Combined effects of tunnel blockagcs and stream turbulence on the convective heat transfer from cylinders in crossflowing air. For the cxperiment,al points, the first number is the percent turbulence intensit.y, and the number in parentheses is the refcr- ence.

sary to know the absolute value of the Nusselt number for a specific Reynolds number with zero intensity of turbulence and negligible blockage error. Possibly some of the results obtained by Hilpert [87 J with an open jet were obtained with conditions coming closest to these requirements. For his tests on wires, in the range 1.14 I (Re)D,f 5 1550, the ratio D/DT varied from 7.9 X so that blockage errors were negligible. However, for his tests on tubes, in the range 507 I (Re)D7f I 207 000, D/DT varied from 0.012 to 0.36, so that the error in (Re)D,f due to blockage varied from +0.025% to +2.19%.

There is another error in Hilpert’s data which partly compensates the blockage error. He, and probably most of the workers up to about 1955, used the Landolt-Bornstein [174] values for the thermal conductivity of air. These values are 2-3y0 lower than those now generally accepted [175-1771; hence the values Hilpert gave for (Nu)D,f are %3% high. It should be noted that Hilpert used the integrated mean values for the thermophysical properties, e.g., A,,,, = {I/ (T. - T,) ] j’E X dT, but the

to 4 X

TABLE VI

EFFECT OF INCRE.LSE I N FREE-STREIM TURBULENCE ON H E . i T TR.\NSFER FROM CYLINDERS IN CROSSFLOW

Iieiher Comings el a2. Maisel and Sherwood ZaPP Giedt van der Hegge Zijnen

Gosae Kestin and hlaeder Schnautz Seban Sogin and Subramanian Kestin et al.

Morishita arid Nornura Smith Kestin and Wood

hZizrishina et nl. nlujrnndar and Douglas Errdoh et d.

- 0.9.52, 3 . 17

1.42 6 .03

1.63 1.63 1.63 0.00.5

10.2

10.6 6.2-8.9

3.16 10.7 10.7

5 . 0 20.4 7.62

1 .0 1.27-3.18

1 .0

14.3-87 14.3, 40 2. .5-4.0

-5 .9 5 . 2

93-3.5.5 93-5% 93-555

102-2430 7 .6

1.55-15.6 4 . 8 1 . 4 4 .7 4 . 7

8, 10.5 7 .3 7 . 3 8

8.8-20 2.0

-

2 . .i 2 .41 , 11.3

1 3 . 9 13.2 9 4 . 1 4.1 4 .1 0.1

19.1 44.3-63.8

13.9 13.1 13.1

s.3 13.3 9.4 9.4 6 .3

4 . <5-11. 4 7 .7

- -30a 40-70 -11"

2.4-4.7a 176 11.3 80

440- 1060 1 ,6-d , 4 16-238

5 Q 4.56 3.0.

0.6-2.2 3 .10 - - -

0.6-8 -

(Continued)

TABLE VI-Continued

N Co

Increase in heat transfer h3

Reiher Comings et al.

hfaisel and Sherwood ZaPP Giedt van der Hegge Zijnen

Gosse Kestin and Maeder

Schnautz Seban Sogin and Subramanian Kestin et al.

Morishita and Nomura Smith Kestin and Wood

Mizushina et al. Mujumdar and Douglas Endoh et al.

180(t3400 1 . 2 5800 1 3800 2 .1

104 <3 104 3.5 105 0.9 105 1.2- 104 0 . 3 104 0 . 3 104 0 . 3 102 1 . 1

1 . 5 x 105 0.8 0.8 0.5

1 . 5 x 105 -1 2.18 x 105 0.8

1 .4 x 105 1 1 0.24

6 X lo4 0.1 105 0.15 105 2.8 104 0.8

3 . 2 X 10'2.17 X lo5

5 x 104-2 x 105

6 .7 X lo3, 8.8 X lo3 2 5 x 103 -2.2 x 104 1

-5 a

7 10.4

> 7 11 .o 12.0 3.55 2 . 1 5 . 1

10.1 4 2.6 4.3a

3" 2.4 2 4.3a 5.8 5.1 7 . 3 7 . 3 2.3

11

10 20

15 -25

23 16 l i b

41 14 15 28 37 12

-15 -15

50 17 36 20 20 30 * 32 52b 1 1 6 22b 23 54

15 24 26 16 24 43 10

5

12 3

15 28 46 M

z 0

10 j

z 10 ?

15

0 50 10 !a

1: 6 13 31 32 42 12 9

30 55

a Estimated from [169,170]. b Mass transfer. c Open-jet tunnel.

HEx'r TH.ZNSFEI~ FROM CYLINDERS 233

diff crciicvs betwecw t h ( w v:iluc\s : u i d tho mcan film temperature values are nc.gligit)lv for Hilpwt's t,wt coritlitioris. The correctvd corrt4ation hc twcn Nussdt and Iteynolds nunibcw for Hilpcrt's results is givcw in Table VII. It is intcwhsting that Fnnd mid Jiwwaiii [239] have also rcw.iitly noted indcpcmdcntly that Hilpert u s d v:iluc~s f o r the thcrmophysical propertics of air Q hic-h have sinw hwi supvrscdcd. They give new va luc~ for thv constants II2 and n1 using n i o r ~ niodorii, not the most r (wnt , data for p and A; t hyv do not considw hlorkagc. cwors. The correlation proposcd hv tho prcwnt author for lic~yriolds iiuriil)cw ranging from lop4 t o 2 x lo" is set out in Tahlr VIII. I t is thought that this correlation, which talws account of all known probablr ('rrors, ha:, :I in:r.ximum uncertainty of *so;,, except possibly in the vicinity of thv tr;insition points.

The rcwlts of 13 worlwrs bvho have given the intensity of turhulcnce in the wind tunriel t hcy used haw Imm analyzed. At the Itcynolds numher 2 X lo4, thr mcasurvd Nussdt iiuinlwr varics from 65 to 119 mith the mean valuca 100.8 :iiid the st;iiid:wd dcviation 12.3. After correcting for blorkagc. and turhdmcc~ vrrors, t ti(. range becomes 68-93, the rncm 79.9 and thv standard dwiation 7.2. Ttic value calculated from Tablr VIII is 78.1. ICo tc.mpcraturtL loading cwrrwt ion has been app1ic.d' h ~ ~ a u s e of uncertainty in its v a l u ~ , particularly : L t high Reynolds numbers; this is discussd later. At most, such :t (-orrrc.tiori would amount to I(

Thr rv)rrcbctcd values for ( S u ) u , f at (12e)D,r = lo4 and 105 arc' 50.4 and 244, rc y)clctively. IJsing th (w vnluw as reforcnce, the incrtmc.nta1 hcat transfvr 6(Xu)D,f/( NU)^ I for t l i r b results of various worlwrs using mind

T A B L I ~ ; VII

CORREC'I'IONS TO IIILPEH I'S [X7] (:OKHEL \TION FOR CROSSFLOW FORCED CONVECTION

FROM ( ' Y I ~ I N D E R S IN AIR"

1 4 0.891 o.:Ml I 4 d I

4 40 0.821 0.:1X.S 4 35 0.7'3.5 0.384 40 4 X lo3 0.61.; 0,466 3 5 5 X lo3 0.683 0.471

4 X lo3 4 X lo4 0. 174 0 . l i l X 5 X lo3 5 X lo4 0.148 0.633 4 x 1 0 4 4 x 1 0 6 0 . o ' ~ : ~ o.~o: , 5 x 104 2 . 3 x 1 0 5 0.0208 0.814

a (Nu)a.f = D,(Re)$,. bol m. Too few data.

* Hilpert used the symbol c. Hilpert used the sym-


TABLE VIII

PROPOSED CORRELATION FOR CROSSFLOW FORCED CONVECTION FROM CYLINDERS IN AIR

(1le)n.r (Nu1D.r = Dz(Re)&

From To DZ nl

10-4 4 x 10-3 0.437 0.089.5 4 x 10-3 9 x 10-2 0.56ii 0.136 9 x 10-2 1 0.800 0.280

1 35 0.795 0.384 33 5 x 103 0.583 0.471

5 x 103 .5 x 104 0.148 0.633 5 x 104 2 x 105 0.0208 0.814

10-4 0.192 10-3 0.23.5 10-2 0.302 to-' 0.420 I 0.795

( 1 k ) D . f (Nu)D. f

10 1.92 102 5.10 103 1.5.1 104 50.4 105 244

tunnels having ranges of blockage values and flows with differcnt intensities of turbulcnce are shown in Tablc IX and Fig. 6. It is seen that, in general, the agreement bttwctn measured and caIculated values is reasonably good, except for values of D / D , in excess of 0.2, when calculated values are too low, becausc the drag coefficient increases with increasing DIDT [252]; hence Eq. (8) underestimates the free-stream velocity. The effect of increasing the intensity of turbulence in a wind tunnel is shown in Table VI, and herc again measured and computed incremental heat transfers agree quite well in most cases.

Hilpcrt [87] recognized that the tcmperature of the surface of the cylinder can affcct the htat transfcr coefficient by altering the fluid propcr- ties. Most of his mcasuremcnts were madc with the surface a t 100"C, but he also studied the effcct on heat transfer of increasing the surface tempcra- ture to 1014°C in the range 33.8 5 ( R C ) ~ , ~ 186. He concludcd that the heat transfer coefficient was increased by thc factor (T,/T,) n1/4 where T, and T , arc the absolute surface and ambient temperatures respectively, and n1 is his value of the exponent in Table VII. For his tests, n1/4 equalled 0.116.

HEAT TR.~NSFER FROM CYLINDERS 235

Although his own results spanned only a limited range of Reynolds number, Hilpert also used thc rrsults of King [79] for surface temperatures up to 1200°C and for 0.06 5 ( I t f ’ )D, ( 5 49 to support this relationship. On the other hand, Gosse [112] gave thc temperature loading factor as (T,/T,)0.’24 for surface tc>mpcraturw up to 725°C and for 40 5 ( R e ) D , r 5 300, while Parnas [I221 gavc flit, factor ( T,/Ta)0.085 for surface temperatures up t(J 800°C and for 5.4 5 ( I t r ) ~ f 5 24.3. Churchill and Brier [l09], who uwd a stream of hot nitrogen, gave the factor as (T,/T,)O l2 for 570 5 5 5070, but their derivation of the value 0.12 fur the exponent is not convincing and thc aspect ratio they used was only 3. The results given in th tw four papvrs have been reanalyzed in tcrms of the film tempcraturc, Tf = f ( T , + T , ) , hy the present author and are given, togrthcr with thosch of othcr w o r h s , in Table X. The temperature loading factor is here equal to ( Tr/Ta) (1 , and it is seen from Table X and Fig. 7 that the exponent q is not ronstant, but appears to decrease with increasing Reynolds number, as proposed by Parnas [l22], and with decreasing aspect ratio, as noted by Andrws et (11. [137]. Equally disconcerting is the fact

03

0 2

0

0 1

0

I , I I 1 I , I 1 ,

(Re)D, f

FIG. 7. Effrrt of I?cynold- niunhrr on cxponent of tempernturr IoiLdinR f:irtor with forced coiivcctiori.

til W Q,

TABLE IX

INCREMENTAL HEAT TRANSFER IN WIND-TUNNEL MEASUREMENTS

G(Nu)D. r 100 Tu 100 -

100 D/DT ( N u ) ~ , f Calc. Mew. Author Ref. (Re)D. f (9%) LID (%) (%) (%) Remarkso

Hughes Gibson Reiher McIntyre Klein Griffiths and Awbery Goukhman et al. Small Krujilin and Schwab Benke Krujilin Schmidt and Wenner Comings et al.

Winding and Cheney Giedt ZaPP

Billman et al. hlaisel and Sherwood Berry et al. Cheng

C95l C97 1

104 103 103 104 104 104 104 104 104 103 104 104 103 104 104 104 105 105 105 103 104 103 104

2.1, 5..5 13.3 2.9

12.7 -10 5.2-12.5

2.5 18.8 14.7

3.2, 4 .1 1.7-21.5

16.7-41 .7 3 .3

11.3 11.3 24 9

13.2 13.2 25.3 13.3 4.5 3.2

19.6, 50.8 12.8 87 9.8

8-19.2 17.2

-5 . 3 7.6

-

-100 2.7-42.4

1-12 40 14.3 14.3 8.0 5.2 5.9 5 . 9

2.5 64

3.50 186

0 14 16 51 9

62 14

213 28 14 15

21 26 48 47 31 18 66 7

17 .5

22

9.6

< 0 . 2 1.5 1.2 8.5

-0.4 12.5 0.9

2.9 0.9 1.2

<0.2 2.0 2.7 8.0 6.2 3.9c 1.1

12.5 <0.2

1 .o 0.3 2.1

> 1.5

- < 3 <3 >7 - 3.8 0.9

12 -

3 . 5 3 .8 -

0

M

Franklin [lo71 104 12.5 8 . 0 30 3 . 5 - Kazakevitrh Cl0f31 104 8 . 8 11.4 18 1 . 3 Zhuknuskas [224] 104 24 8 . 5 1 1 Churcliill arid Brier [ 109 1 103 .> 7 3.0 21 k . 1

Brun et 01 . C l l O l van der Hegge Zijnen [I591 104 4 . 1 24.5 6 0 .3 0 .2 Gosse [I121 lo2 0.31, 0.61 164, 318 5 Chung and Algren ~ 1 1 4 1 10' 11.2 21.4 24 2. .> 1-3

x Collis and LViIlianis [ I133 102 0. 1 2070 - 2 . 3 < 0 . 2 0.0x

Schnaut z [lrjl] 104 I 3 13 . ti 8 0 . 3 0 . 5 31 n 10' 1 :i 2.1 63 l l . . i 11 .0 3f $

2 Srenivasan and Iiamac.handran [l 181 104 4 . 3 17.3 8 . .> 0 .4 Perkins and Leppert [ l lO] 10' 30 21.1 40 6 . 0 2 . 9 $ van 3Ieel [I201 10' 4 .6 (i . 9 1 3 0 . 8 0 . 2 9

Hodgson ef al . [12fj] 10' 2.j 1.3 3 7 3 . .5 2 .6 z Jtshnson and Jouhert [13OJ 103 9 . 1 2 j: < 0 . 2 -1

z Oosthuizen and 11adari [ l % Y ] 10' 9 . 4 8 4 . 6 < 0 . 2 <0 .7 Morishita and Noiii~ira [ 16.i] 105 8 . 3 - 6 . .i 0 . 2 0.2

105 8 . 3 38 5 . 7 5.8 a Kestin and Wood ~ 1 6 7 1 1 0 5 9 .4 7 .4 1 <0.2 0 .15 M E

105 9 . 4 7 .4 53 10 7 . 3 M 3 Kosti6 and Oka [132] 104 20 5 36 4 . 0 2 .8 M

E Mujunidar and Uouglas c209 1 -10' 4.5-11.4 8.8-20 3 0 4 . 0 0.5

Galloway and Sage [21.5] 104 50 8 11 3 . 5 1 . 3 0 I y a j i arid Tajiina [2443] 10' 6.:< 1.;. R 1 % 1 ..i -2 M

-

< 0 . 2 < I -2

104 2 . 7 37.3 3 . .i <0 .2 -

0 .3 -

Shchit nikov CllSl 104 -10 - 23 2 .4 2 .4 -

32

-3 z

-

Martin [134] 104 10 16 0 . 9 0.3-0.6 -

- 10' 4 . -5-1 1 . 4 8.8-20 44-70 6-14 10

'1 11, mass ti,:msfer; 0, opeii-jet tunnel . * Scliniidt. aiid Wenner used Hilpert'h [87] opeii jet. Using Dryderi ef a!. [IGB] and Bnines and Pet.erson [170]. t3

2

TABLE X

TEMPERATURE LOADING FACTOR (T,/T,) 1 WITH FORCED CONVECTION

Range of (Re)o, f Range of Tr/T ,

Author Ref. I /D From To From To Exponent q

King Hilpert Churchill and Brier Gosse Collis and Williams Kays and Nicoll Parnas Hassan and Dent Hatton et al. Davis and Davies Koch and Gartshore Andrews et al.

Ahmed

[79] 1290-1520 c87 1 714 ~1091 3 [112] 102-2430 [115] 2070-8660 P331 4.1

1000 - c1221

~1781 C56 1 96-1190 c204 1 380 C1331 230

-300 >400

El371 -200

~1271 6.7

0.06 33.8

0.2 0.02

5 . 3 5 0.6 0.1 1.2 0.2 0.2 0.2 4

570

4.2 x 103

49 186 5070 655 140

2 x 104 24.3 50 45 3.4 4.1 4.6 4.6 14.7 40

1.36 3.03 1.07 2.74 0.63 0.75 1.02 2.22 1.05 1.50 1.04 1.96 1.06 2.27 1.15 2.29 1.05 1.35 1.25 1.63

1.06 1.54 1.06 1.54 1.09 2.28 0.61 0.83

- -

0.14-0.21 Q - 6

b 5 -0 5 2

M 0.126 2

0.14-0.21 Q - 6

b 5 -0 5 2

M 0.126 2

H 0.17 0.2ga -4

0.12a z 0.25. 5

0 z 0.17 0.154 0

-0.67 --0.5a w-0.1a

0 0.15

a Value deduced by present author from published data. See Fig. 8.

HEAT T H . 4 N S F E R FROM C Y L I N D E R S 239

that Hilpert's results [S7] and some of those of King [79] indicate that q increascs with increasing values of Tf/T,, as seen in Fig. 8. It is clear from the preceding discussion that more experimental work is essential before these interesting effects can be separated and quantified. It should be noted, however, that the majority of the tests reported in the literature (see Table IV) were made with values of T f / T , < 1.2, so that corrections for temperature loading are less than 37,,.

€3. YAWED CYLINDER

It is found that the convc4vc heat transfer is reduced as the yaw increascs, i.e., as the angle + betwren the fluid stream and the longitudinal axis of the cylinder is reduced from 90". Most of the published experimental results have been obtained with fine wires in anemometric studies, see Table XI. Scars [197] and Jones [I1981 have shown that the components of the flow vdocity normal to and pnmllcl with the cylinder axis are independent

1 2 5

1 20

I15

cr \

I-!! \ I-!' - 110

1 0 5

1 I I I

1 0 1 2 1 5 2.0 2 5 3 0

Tf 1 'a

FIG. 8 Variation of trmpmxturc. londtng Fartor with Tr/T. for forced convection.

240 VINCENT T. RqORGAN

of each other, and it has been suggested that the normal component of velocity should be used to calculate the heat transfer. If this is correct, the ratio (Nu)+/(Nu)+=w should vary only with some function of sin$. However, this is only true for laminar flow past yawed infinitely long cylinders; for wires of finit,e length, the effective velocity U is given by:

U2 = Ux2(sinz I) + F2 cos2 I)) (14)

where U , is the free-stream velocity and F depends on the aspect ratio Z/D. Bullock and Bremhorst [196], using the results of Friehe and Schwarz [193], have proposed the relationship:

(15) F = 0.29 - 0.0045 Z/D

for 200 5 Z/D 5 600. Substituting for U from Eq. (14) into Eq. (7),

(Nu)+ = Cz + Dz{ UxD(sin2 I) + F2 cos2 I))1/2yJpClr)nl (16) hence,

When Cz = 0,

It is seen from Table X I that, when the aspect ratio exceeds 400, the heat transfer varies with the sine of the angle of attack I), as theory predicts, except as I) -+ 0, when only the experimental results of Friehe and Schwarz [193], \vith a free-stream turbulence intensity of O.l%, indicate that this relationship is still maintained. A11 other experiments have yielded values for the heat transfer with axial flow equal to approximately 40% of those with crossflow. Assuming that there were no significant errors in measuring I), this could be explained by heat conduction to supports; or three-dimensionality in the flow, due to turbulence caused by the supports, vortex shedding, or turbulence in the free stream. The asymmetry in the axial temperature distribution [Sl, 121, 2041 and the longitudinal turbulence in the flow around the yawed cylinder [210, 2381 are of particular importance in this respect. Bruun [227] has shown that for Z/D > 400, the value of the yaw factor F is approximately 0.25 with perpendicular supports, but F -+0 with parallel supports. For Z/D < 400, F decreases with both increasing aspect ratio and increasing Reynolds number. Ciray [192], Nishioka C2-191, and Baille [251] also found that F decreases as the Rey- nolds number increases.

HEAT TR.~XSFER FROM CYLINDERS 24 1

TABLE X I : F O R C E D C O U V E C l I O N FROM Y.\WED C Y L l K r ) E R S

;2 I I t hor Ilef. n (In x 1 0 - 4 ) (11e)D. f 1/11

Sine law -

Grimison [I791 King c79 1

Yaglou [IS01 Lokshin L-1811 Ornatski [182] Mueller [ti41 Weske [ 1x31

Siinmoris ~ 1 x 4 1 Newman and Leary [IS51 Kronauer [ I X O ] Kazakevitch [ I OX] Cherig [:lo] Sandborn and ~ 1 x 7 1

Laurence

Zhukauskas Anantari:trayanan and

Shchitni kov Webster Delleur Hama Chu Ciray Davies and Fisher Chainpagiie el n l .

Ramacharidran

Friehe and Srhwarz [I!,:S]

Bruun [194] Kjellstrcm arid Hedberg [ l!$.j] Dickinan [21 I ] Sawatzki p 1 2 1 J6rgensen p1:31

Bruun p”]

Nishioka [249] Baille [a511

- 1-50

80-7400

-

- -

2-1 16 15-41 16-49

1-6 2.9

500-3 X 10‘

-

1300-9400 8-190

- 198-580

lo4-6 X lo4 O ( 1 )

1-5 -

-

2-3 10-50 5-30

3-150

0.03-25 - -

-

-

-1-12

1.5-15 0.01-0.3

-

1780

-3 -

- -

j000-2‘2,OOO 2.50 4 20

-100 390 -

11.4 187 100 270 390

3200

-

- 86-14.56

100 -400

200 400 200 430 GOO 100 200 500 400

-

- -

-

ia 230 600 200 4 00 600 600

160-343

(Continued)

TABLE XI-Continued

FORCED CONVECTION FROM YAWED CYLINDERS

From Eq. (18)

Author Ref. 0 10 20 30 40 45 50 cjo 70 75 80 F n1

100 Nu+/Nu+w for given $ (70)

Sine law King Grimison Taglou Lokshin Ornatski Mueller Weske

Simmons Newman and Leary Kronauer Kazakevitch Cheng Sandborn and Laurence

Zhukauskas Anantanarayanan and

Shchitnikov Webster Delleur Hama Chu Ciray

Davies and Fisher Champagne et al.

Ramachandran

Friehe and Schwars

0 53 50

44-53 - -

40 3 3 4 5 35-45 54-60 - - - 48 44 42 59

35-45

115

36-45

-

- - - -

63 -

26 7 1

70.7 64

52-77 60 60

75 72

72

72 76 73 77 84 66

-

-

-

-

-

- - -

73 71 -

79 73.2 70.8 70.7 78.4 72.7 71 .O

80.2 - - - - - -

82 82

81

-

- - -

81 85 a9 78 -

98 - -

82 80 -

- 81.5 80.4 80.2 85.3 81.5 80.3

84.1 81 75

81 81

80-100

- - - -

86

86 88 86

-

- -

83 -

- -

85 85 84 -

-

85.1 84.3 84.1 88.2 85.1 84.2

87.5 - - - - - -

89 89

88

- - - -

87 90 92 88 -

- - -

89 87 - -

88.3 87.7 87.5 90.7 88.3 87.5

93.1 91 - -

91 91

93 94

03

91 -98

93 92 96 93

-

-

-

98 - - 93 93 -

94 93.4 93.2 93.1 94.5 93.5 93.1

!Mi. 9 - - -

96 96

97 97

-

- - - - -

96 97 97 98 -

- -

97 97 97 -

-

97.1 97.0 97.0 97.7 97.1 96.9

99.2 - - - 99 99

99 99

-

- - -

98

98 98 99 99

-

-

103 - - 99 - - -

99.3 99.3 99.3 99.4 99.3 99.2

0 - - - - - -

0.1 0.3

0

-

a b

- e

c

c

- -

- 0.2

d

0 0.14- 0.36 0 0.22 0.1 0

e

c

e

-

0.5 - - ~

-

0.5 0.5 0 5 0 . 5 0.457 0.5 0.5 -

c

c

-

0.729

-

0.5 0 . 5

0.5

-

-

0.5 0.45 0.45 0.45 0.45 0.45 0.45

Bruun Kjellstronr and Hedberg Dicknian Sawatzkl Jergensen Bruun Nlshioka Raille

- 37 64 -

-

61 79 -

I a

- A - j

k - m

0.57.5-0.51 0.371-0.57G

0.4g5 - -

0.5 0.155 0.45

a 9 .a 0

where .4 = 0.42, > - 0 . 2 , c 2 0.21, and D .Y 0.14. d + <20", F = 0.14; "0" _< + 5 900, F = 0.

fsin?+ + F 2 ros2 = 1 - ( 1 - sirl1:?+i@(/:D); hence F = 0.29 - O.O043(Z/D) for 200 < 1,/D < 600. [See Billlock and €hem- g

E a horst [1961].

F = 0.44 - 0.035 +. F = (0,0505 - 0.000 415 ylj) i '2 for 36" 5 $ 5 90". 2:

ti M h P = 0.17 + 0.0043 $ for 20 _< + _< 90" and 1/D = 18 (deduced by present author).

i F = 0.14 + 0.0042 $ for 15 5 $ 5 90" and 1/D = 250 (deduced by present author). i F = 0.08 + 0.0021 $ for 1 5 _< $ _< 90" and Z/D = 600 (deduced by present author). k F = 0.22, 0.20, 0.17 for ( R e ) ~ , f rr 1, 2, 6, respectively (parallel supports); F = 0.43, 0.40, 0.35, 0.31 for (Re)D,( 'v 1, 2, 6, 12,

F = 0.18, 0.13, 0.05, 0 for (1le)o.r 5 1, 2 , f i , 12, respectively iparallel suppork); F = 0.26, 0.28, 0.27, 0.25 for ( I l e ) D . f 'u 1, 2, (i,

A 1, 2, ti, 12, reipertively (parallel supports); F = 0.25, 0.26, 0.24, 0.28 for (1te)u.r 5 I , 2, 6, 12,

$

respectively (perpendicular supports) ; 1/D = 200.

12, respectively (perperidic'ular supports) ; 1/1) = 400.

respectively (perpendirulnr siippnrts) ; 1/13 = (i00. "'F = 0.03, 0.03, 0 f ( ~ r ( I l e ~ , ,

n F = 0.94 - 0.8 log (Ite),j,f (deduced by present author). 0 F = 0.3 exp( -//400L>) + { 1 + , 5 Z j 7 0 ] - 1

-0.3exp(-Z/4000) + 11 + 22(Iie)D,rJ-'. t.3 * w


IV. Combined Natural and Forced Convection

At high Reynolds numbers, the heat transfer occurs mainly by the process of forced convection but, as the Reynolds number decreases, the contribution due to natural convection becomes significant.

In the past, there has been confusion over how to calculate the Nusselt number when both natural and forced modes of convection are prcsent. For examplc, McAdams [31] suggested that the heat transfers be calculatcd separately and the higher value used. On the other hand, van der Heggc Zijncn [35] proposed that the vectorial sum of thc Nusselt numbers with thc two modcs bc used. However, both these methods can result in considerable error, and approaches based on thc resultant flow vclocity, e.g., those by Hatton et al. [SS] and Borner [199], have led to greater success. Following these workws, wc consider that an equivalent Reynolds number (Rc*) for natural convection is found by equating the Nussclt numbers for natural and forced flows; from Eqs. (3) and (6),

wherek = { (B1/D1) (l’r)T-p]l’nl = { (B,/D,) (Pr)r’)l!nl (for air, Pr = 0.7). Values for B1, D2, ml, and nl arc given in Tables I1 and VIII. The exponcnt p for the I’randtl number is usually taken as 0.3. For simplicity, we now omit the subscripts D and f. Let the direction of forced flow be a t an angle 4 with the vertical dirc\ction of thc natural flow. The lattcr will be upward when thc cylinder is warmer than its surroundings, and downward when it is cooler. The effectivcl Reynolds number for the combincd flow is given by

(Re)& = {Re* + R c c 0 s 4 ) ~ + (20) 1.c.,

= (Re*)2 + 2(Ite*) (Rc) cos9 + ( R C ) ~ (21)

Thc total heat transfer is now found by using (Re),ff in place of in Table VIII.

Wc first considcr the boundary between pure natural convection and mixed convection. WP define thc limiting Grashof number (Gr) l im as that valuc of the Grashof number at which the total effcctivc Nusselt number NU)^^^ is 100 8% grvatcr than the Nussclt number corresponding to pure

HEAT TR.~NSFER FROM CYLINDERS 245

natural convection alonc~, ( N u ) N ; 100 6 is usually taken to be 5 5 . Then

hcnce,

Whm the forced flow aids t h r b natural flow., d, = 0", hence,

I<(%* = I<(./[ ( 1 + 6 ) l ' n - 11 (24)

( 2 5 )

I.C.,

( G r ) l l r n = [li(~/li( (1 + S ) l / n - 1 I l n l m

When the forced Ao\v opposcs thc natural flow, d, = 180", hence,

It(.* = I t r x / [ (1 + 6 ) l i n + 11 ( 2 6 )

(Gr),,,,, = [ I r t b / l k { ( I + + l)]n/m (27) I.C.,

With rrossflow, d, = 90", so that

RP* = Ii(h/j ( I + S)?'" - 1 I (28) 1.e.,

(Gr) I,,,, = [licb/k( ( I + 6)?ln - I 1 1 l /* n / m (29)

It can be seen that the liiniting Grashof number for d, = 90" is the harmonic mean of the limiting Cirashof numbers for d, = 0" and d, = 180". Equations for th r limiting Gr:ishof numhclrs for various ranges of Rayleigh and Reynolds numbers are givcri in Table XII.

We now considm t hc boundary Iwtween pure forced convection and mixed convwtion. Wc d&ne t l i ~ lirniting Reynolds number (Re) I , ,~ , as that value of the Reynolds numbcr for which the total effective Nussr:lt number is 100 6% greater than the Nusselt number corresponding to pure forced convection alone, (Nu)F; 100 6 is usiially given the value 576. Then

hence,

TABLE XI1

LIMITING GR.\SHOF . \ND IIEYKOLDS NUMBERS FOR M I X E D CONVECTION FROM HORIZONT.\L CYLIKDERS I N AIR=

9

(Gr - Pr)o, t (Relo r 0" 90 O 180"

From To From T O k m / r i n/m A4 Rc A 4 B4 '4 4 B*

10-10 10-2 10- 4 x 10-3 102 0.648 1.54 1.30 X low3 141 4.69 X 72.7 1.69 X 37.5 4 X 9 X 10-2 3.18 0.426 2.34 0.477 7.36 0.0629 3.10 8.28 X 1.31 9 x 10-2 1 0.506 0.207 4.83 8.04 X lo4 2.66 221 0.784 0.607 0.231 2

e 9 x 10-2 1 1.97 0.529 1.89 6.38 10.4 0.633 3 .05 0.0628 0.900 . 10-2 1 0 2 4 x 10-3 9 x 10-2 52.2 1.09 0.919 0.0571 121 0.0258 51.0 0.0117 21.5

1 35 1.67 0.385 2.59 47.4 12.3 1.33 3.10 0.0370 0.781 102 104 9 x 10-2 1 0.977 0.671 1.49 12.2 5.13 1.99 1..51 0.322 0.446

1 35 1.00 0.490 2.04 59.4 7.38 3.55 1.86 0.212 0.468 a 35 5 X lo3 1.93 0.399 2.51 49.4 17.7 1.21 4.03 0.0296 0.916

35 5 X lo3 0.548 0.531 1.88 202 5.02 12.4 1.14 0.762 0.260 104 107 1 35 0.213 0.651 1.54 232 1.57 27.9 0.396 3.35 0,0998

5 x 103 5 x 104 .5.57 0.395 2.53 7.70 69.5 0.125 13.6 2.02 X 2.68

5 X lo3 5 X lo4 0.635 0.526 1.90 288 7.92 13.0 1.55 0.590 0.305 5 X lo4 2 X 105 7.82 0.409 2.44 5.91 127 0.0813 21.9 1 . 1 2 X 3.79

107 1012 3,s 5 X lo3 0.0296 0.707 1.41 3330 0.271 411 0.0616 50.6 0.0140

a (Gr)o,f.lim = Ad(Re)g'T [see Eqs. (25), (27), and (29), 6 = 0.051; (Re)D,f,lim = &(Gr);!; [see Eqs. (33), (35), and (37), 6 = 0.05).

HEIT TR~NSFER FROM CYLINDERS

When thc forced flow aids the natural flow, 4 = O", hence,

(Re) = Re*/[ ( 1 + 6)1'n - 11

i.e.,

( R v ) ~ , , , , = [k((:r)m/n]/[(l + 6)1 'n - 11

When the forced flow opposcs the. natural flow, 4 = 180"; hence,

Re = lCP"/[( 1 + 6 ) l / n + 11

IS. ,

(1Ic)llln = [ l , ( t~ r )m/~~] / [ ( l + 6)1'n + 11

With crossflow, 4 = 90") so that

( I < ( . ) = RP*/{(l + 6 ) 2 / " - I)"'

hencr,

(11~) I , , , , = [ A ( ( :r) m/7L]/ { ( 1 + 6)z/n - 1 ] I / ?

Theoretical and expcrimc~ntal va1uc.s of thc limiting I t c y ~ d d s number with 6 = 0.05 arr comp:irrd in 14g, !I for 4 = O", 90") and 180". Thcrc is secn to bc good agrctc\mciit for 4 = 90", but poorer agreemait for 4 = 0" and 4 = 180". It is clrar that t lw quadratic relationship brtwwn (Re)llm and G r proposcd by Sparrow n i i d Grrgg [203] is only an approximation. This fact has also l iwn notcd by Sharni:t and Sukhatnic [201].

An iiitcwsting CBSP is that of n horizontal circular cylindcr vibrating in a vertical plane. Thcx cquivalvnt Itc.ynolds number due to thv vibration, (R(3)v31,) then acts altcmmtcly in the dirclctions 4 = 0" and 4 = 180" with resprct to thr cqiiivalcnt Rcbyriolcls niiinbcr for pure natural convection, He*. Thr rffcctivci I<c.ynolds iiunilwr is givm by

Thc limiting valuc. of thc~ vihrat ion Rcyiolds number is given by Eq. ( 3 7 ) . This rrsult could RISO h a v ~ l m k i t o1)tainrd by taking thc 1i:irmonic niean of the liniiting valuvs with 4 = 0" mid 4 = 180" from Eqs. (33) and ( 3 5 ) , rcspcictivcly. It is swn from Pig. !Id that there is rrasonablt. agreement bctw(w thcwetir:il :md cxpc.riiricnta1 rvsults for this caw. Equations for the limiting Reynolds numbrr for various rangcs of Raylrigh and Reynolds nunibrr arc given i n Tahlc XII .


Iheorelical / R e ) ,,T

i t 1

theoret ica l

(dl

FIG. 9. Comparison Mween experimental and calculated values of the limiting Rey- nolds number with mixed convection:

x Martinclli and Boeltcr [19], water 0 Collis and JVilliams [115], air V v Deavrr et al. [44], n a k r 0 Shine [ZOO], air A Penney and Jeffrrsori [55] , water A Sharma and Sukhatme [201], air

0 + Gebhart and Pera [SS], air and (3 Fand and Keswani [236], water

Mason and Boelter [ZO], air Oosthuieen and Madan [129], air

and ethylene glycol Hatton el al. [56], sir

silicones

# Oosthuizen and Madan [202], air

The ratio between the effective Nusselt number with mixed convection to that with forced convection alone is as follows:

HEAT TRANSFER FROM CYLINDERS 249

FIG. 10. Coinparison bet,wcen eup~~riinrrital and calculated effectivr Nrisselt numbers for mixed convection, 9 = 90":

1 Eq. (41), stir 2 3 4 Eq. (41), watw 5 Fand and Keswani'b [236] 15q (17) (ZOIW 2), water. 6 F:md and Keswaiii's [236] rkl (25) (zone 4), water.

1)ata of Sharmn and Siikhatiric [20l], air. 1)nt:z of Oosthuiatm and M ~ d , i i i [202], air.

0 0

Laminar bouiidary l q w , data of Fnnd and Keswani [236], water. Srparated boundary layer, datn of Fnnd and Keswani [236], water.

VINCENT T. MORGAN

Experimental results for crossflow of air and water, i.e., with 4 = go”, are compared with curves derived from Eq. (41) in Fig. 10. It is seen that agrecment is fairly good. It is interesting to note that the two flow regimes noted by Fand and Iicswani [236], which they called the “laminar boundary layer” and the “separated boundary layer,” do not produce very different effective heat transfers for equal values of G r / ( n ~ ) ~ .

There is poorer agrecment between measured and calculated values of heat transfer with parallel flow ( 4 = 0”) and even worse agreement with counterflow ( 4 = 180”). Fand and Keswani [236] suggest that this may be due to the complcx secondary flows caused by the walls of the wind tunnel.

V. Conclusions

It has been shown that the wide dispersion in the published experimental data for tht. heat transfrr from smooth circular cylinders by natural and forced convection can be attributed to various factors associatrd with the experiments. The error due to heat conduction to the supports is particularly important with natural convection, especially where the heat loss and the temperature rise of the cylinder are calculated from the voltage drop across it. This error can be corrected with the use of Eq. (4) .

There is insufficient information in many papers to estimate the possible error in measuring the trmperaturc of the surface of the cylinder. Hou-evcr, the precautions requircid when thermocouples are used ought to be \wll known.

A common cause of (wor is the use of too small a space ratio, so that the temperature and velocity fields are distorted. To reduce this error to less than lx, the space ratio Dc/D for natural convection or D T / D for forced convection, should c.xceed 100. The error caused by blockage with wind- tunnrl measurements can be calculated from Eq. (10) or Eq. (12) depending on the type of tunnel.

One of the greatest sources of error with forced convection is the failure to allow for the effect of stream turbulence, as seen from Table IX. This error can be computed from Eq. (13) or Fig. 6.

The effect of temperature loading depends on the type of Convection. With natural convection, and when the Nusselt number is based on the mean film temperature, there is no evidence for its variation with temperature for constant Rayleigh number. On the other hand, with forced convection, the Nusselt number increases with (Tf/T,) Q for constant Reynolds number. The exponent q is not known precisely, but is about 0.17. However q appears to decrease with increasing Reynolds number and may increase

H E A T T R A N S F E R FROM C Y L I N D E R S 25 1

with increasing T J T , . It also pussib1 y dccrcascs with dccreasing aspect ratio, 1/D. In most of the cxprrimrntal work described in thr literature, T r / T , did not exccrd 1.2, so that the correction for temppraturc loading is less than 37,. Thr rffrcts of drafts with natural convcction tests, and natural convcction during forccd ronvwtion tests a t low Reynolds numbers, arc often overlookrd. Equations arc' givrn in Section IV for calculating the total heat transfer in both caws. The limiting Grashof and Reynolds numbers, which depend on thc rcllativc ranges of the two numbers, may be determined from Table XI1 for horizontal smooth circular cylinders in air.

Inclination to thc horizontal, in thr case of natural convection, or yaw, in tho case of forcrd eonvwtion, introduces an axial component of flow, leading to thickening of tho boundary layer and a decrease in heat transfer. With natural convcction, thc Nussrlt and Rayleigh numbers can be correlated on thr basis of the vertical dirnrnsion, i.e., diameter D for a horizontal cylinder, D/cos a for a cylindvr inclinrd a t angle a and height H for a vertical cylindcr. With forccd convcction, the Nussclt numbrr is only proportional to the sinc of thr anglc of attack 9 for sufficiently large values of the aspect ratio Z/D. For loww values of Z/D, turbulcnce and vorticity caused by the supports upset this proportionality by introducing three- dimensionality into the flow.

Improved correlations arc givcln for the variation of the Nusselt number with th r Rayleigh and Reynolds numbers for natural and crossflow forced convection respectively in air.

NOMENCLATURE

C D

D e F 9

Gr h

H k

Constmtq (with nurncwcd

Constants Specifir hrat capacity nt con-

stant pressure (joulcs kg-'

subsrript )

K-1) I h g corfiricnt 1)iarnetc.r (m) Base of natural logarit hnis See Eq. (14) Acceleration due to gravity

(msec-a) Grashof number, DJfIBgy2c(-2 Heat transfrr corffiricnt

Height (m) See Eq. (19c)

(watts m-* K-1)

1 m, n

Nu P Pr Q Iln

T

Tu U

Length (m) Constants (nitlh numerical

subscript) Nusselt number, hDX? Constant Prandtl number, cpX-'

Exponent (see Figs. 7 and 8) Rayleigh number, D388gcy2.

Reynolds number, UDyc(-' Effective Reynolds number

Thermodynamic temperature

Turbulence intensity, u / U Root-mean-square velocity

fluctuation in direction of flow (msec-l)

M-lX-l

[see Eq. (19b) 1

("a

VINCENT T. MORGAN 252

U

(1

P

Y 6

x e

A 1 V

Mean velocity in direction of flow (msec-1)

GREEK SYMBOLS

Angle of inclinat,ion relative to horizont,al (degrees)

Temperatmure coefficient of volumetric expansion (K-1)

I)ensity (kgm-3) Defined in Eqs. (22)and (30) Temperat#ure rise (OK) Thermal contluctivity (watts

1 Scale of turbulrnce (m) Ilynamic viscosity (Nsrnp2) Kinemat,ic viscosity (m2

m-l K-I

scc-1)

+ 1c.

a

crit D eff f H lim m

T vib

C

S

m

Anglc relative to direction of natural flow (degrees)

Angle of attack (degrees)

SUBSCRIPTS

Ambient Chamber Critical 1)iamctcr EfIective Film Height Limiting Mean Surface Tunnel Vibration Free stream

REFERENCES

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2. A. H . I)avis, Nat,ural convectivc cooling in fluids. Phil. Mag. 44, 920-940 (1922). 3. A. H . Davis, Natural convective cooling of wires. Phil. Mag. 43, 329-339 (1922). 4. C. W-. Rice, Free convection of heat in gascs and liquids. 11. Trans. Amer. Znst.

Elec. Eng. 43, 131-144 (1924). 5. W. E. Ayrton and H. Kilgour, The thermal emissivity of thin wires in air. Phil.

Trans. Roy. SOC. Lonilon, Ser. A 183, 371405 (1892). 6. J. E. Pet,avel, The h(xt dissipated by a platinum surface a t high temperatures.

Phil. Trans. Roy. Soc. London, Ser. A 191, 501-524 (1898); 197, 229-254 (1901). 7. A. E. Konnelly, C. A. Wright, and J . 6 . van Bylevelt, The convection of heat from

small copper wires. Trans. Amer. Znst. Elec. Eng. 28, 363-393 (1909). 8. F. W-amsler, Die Warmcabgabc geheiztcr Kiirper an Luft,. V D I (Ver. Deut. Ing.)

Forschungsh. No. 98/99 (1911). 9. I. Langmuir, Thc cotivcction and conduction of heat in gascs. Trans. Amer. Inst.

Elec. Eng. 31, 1229-1240 (1912). Also Phys. Rev. 34, 401422 (1912). 10. C. W. Rice, Free and forccd convection of heat in gascs and liquids. Trans. Amer.

Inst. Elec. Eng. 42, 653-701 (1923). 11 . W. Koch, Uber die W’iirmeabgabe geheizter ltohre bei verschicdcner Neigung dcr

I~ohrachsc. Gesundh.-Zng., Beih , Ser. 1 No. 22, 1-29 (1927). 12. W. Nusscalt, Die Wiirmc&gabe cincs waagrecht liegenden Drahtcs oder Itohrcs in

Flussigkciten und (hscii. VDZ (Ver. Deut. fag.) Z. 73, 1475-1478 (1929). 13. 0. It. Schiirig and C. W. Frick, Heating and current carrying capacity of barc con-

duct,ors for outdoor sorvice. Gen. Elec. Rev. 33, 141-157 (1930). 14. G. Ackcrmann, Die Wiirmcabgabe cines horizontalen geheiztcn Rohres an kaltes

Wassrr bei natfirlichrr Konvcktion. Forsch.. Geb. Zngenieurw. 3, 42-50 (1932). 15. W. -1. King, The bmic laws and data of heat transmission. Mech. Eng. 54, 347-353,

410414, 426 (1932).

(1915).

H E . ~ T TR~NSFER FIWM CYLINDERS 253

16. K . .Jotllhauer, Ijas Tcmperat.ur- 11n(1 (;c.scli~~,indigkeitsfeld um eiii gcheiztcs Iiohr bei freier Konvektion. Forsch. Gch. Ingeniertrw. 4, 157-172 (1933).

17. M. .Jakob and W. Linkr, I>cr 11 meiibrrgang beim Verdampfen voii Fliissigkeiten an senkrechten und waagrrerhtcn Flaschcn. Phys . Z. 36, 267-280 (1935).

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NOTE ADDED IN PROOF

It should be noted lhat Boulos and Pei [234] found that the overall heat transfer in turbulent flow was 10-20:; higher with the constant heat flux condition than with the constant wall temperature condition for 3 x lo3 5 (Re)o,r 5 lo4 and 0.01 5 Tu 5 0.06. Most of the experimental data in the literature have been obtained using the constant wall temperature condition

[advances in heat transfer] advances in heat transfer volume 11 volume 11 || the overall convective...

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