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REPORT No. 726

THE DESIGN OF FINS FOR AIR-COOLED CYLINDERS

By AR NOLD E . BIE RMANN and H ER MAN H . E LL ER BR OC K, Jr.

SUMMARY

h analys-icw a 8 m ad e to de~erm ine the proportion oj

fcn .s made of alum inum ., copper, magneeium , and s teel

neces s ay to d im ipate mazinaum guantitie8 of heat for

d ifferent $n w id ths , jin w eigh ts , and air-$ ow cond ition s.

m e ana ly sia d 80 concern 8 thz d eterm ina tion of th .+?opti-

m um jln proportions w hen epecijied lim its are placed on

the fin d imens ion. The calcu la tion of the heat$ow in the

jns is bas ed on an experim enta lly ~erijied , theoretical

equa tion . The s urfa~ hd-tran8jer coe# ieients us ed

w ith th is equa tion w ere tak en from precious ly reported

experiment.

In atiition to the pre8entd ion of $n-des ign inform a-

tion, this inves tiga tion show 8 that optim um jin d im en-

s ions are inapprem ”ably a$eeted by the d i~erences in air

jlow that are obta ined w ith d ij$erent a ir-jlow arrange-

ment or by mall change8 in the length of & air+?ow

path. For a p“w n fin w eigh t, the highes t hea t tran8fer

can be obtained m“th fins of a magnem”um alloy ; pure

copper and a lum inu m-allq$n8 a re only 81 ightly inferior

to m agnes ium -alloy jtns and w “ll d iw ipa te 8ew ral tim es

m ore h ~a t th an s teel.

INTRODUCTION

Previous investigations on the subject of cooling

surfaces by means of metal fins have shown that the

heat dissipated can be expressed fairly accurately by

au equation involving the tin climensions, the thermal

conductivity of the metal, and the surface heat-transfer

coe5cient q (references I to s). Experimental

values of q have been determined for a wide range of

air-flow conditions (reference 5). From the infor-mation previously obtained, it is possible to mlculat e

the over-all heat transfer of finned cylinders in the range

of generaI interest.

The problem of b design for any one air-flow cnndi-

tion involves the determination of the fin proportions

that wilI give the greatest heat transfer for (1) a given

weight of fin material or for (2) a given flu width.

Previous investigations of optimum h proportions

(references 1 and 6) have generally been made to deter-

mine the great est heat transfer for given weights of

& material. In these investigations it has been shown

that, for every vaIue of fln weight zmd air-flow oondi-

tion, only one particular fin design will give a mmsimum

heat flow.

In reference 6, flndesign information was presented

for one cyIinder diameter and one baffle arrangement.

The more complete data of the value of q presented in

reference 5 make it possible to widen the range of fin-

design data. The object of the present report is to

give iindesign information for severaI conditions of air

flow, different cylinder diamete~, and severaI metals.

The criterions for excellence of & design have been

based upon the maximum heat transfer for a given fin

-weight and the ma.ximurn heat transfer for a given fh

width.

q.,

specific weight of tin material, pounds per cubic inch

cylinder diameter at fin root, inches

thermal conductivity of metal, Btu per squareinch through 1 inch per hour per ‘F

thermaI conductivity of cooling air, Btu per

square inch through 1 inoh per second per ‘F

equivalent length for straight tubo (PRJ, feet

weight of fins, ponncls per square inch of outside-

wrdl mea

surface heat-transfer coefficient, Btu per square

inch total surface mea per hour per ‘F tem-

perature di.Rerence between surface and irdet

cooling air

surfaoe heatAransf er coefficient, Bt u per square

inoh total surface area per hour per ‘F tem-

perature difference between surface and aver-age cooling air

average radius from center of cylinder to

finned surfaceR,

( )— feet

E+2XW12 ‘

R ,

s

radus from center of cykder to h root (D/2),

inches

average space between adjacent fin surfaces,

inches

401

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402

t~.. -

1“

wWf

9

P19

PZ9

Pa89

P(I9

v

P

AT ,

Af12

Ap~Otil

R EP O13T m . 726—NATIONAL ADVISORY COMMITTEE m ~ AERONAUTICS

a.vcwrbgethickness of ihs, inches

over-all heat-transfer coefkient, Btu per

square inch outeidc-.wall mea per ‘F tem-

perature diEerence between cylinder wall

and inlet cooling air per hour

velocity of cooIing air, feet per second

fin width, inchesw+ t/ 2 , inches

acceleration of gravity, feet per second per

second

specific weight of cooling air in front of cyl~-

cler, pounds per cubic foot

specific weight of coo~ air in renr of cylin-

der, pounds per cubic foot

average specific weigh~ of cooling air (pIg+

P47)/ 2, pounds per cubic featspecific weight of cooling air at 29.92 inches

of mercury and 80° F (0.0734 Ib per cu ft),

pounds per cubic foot

absolute viscosity of cooling air, pounds persecond per foot

equivalent angle of curvature, 180° minus one-

half the a.ngIe subtended by front opening

of baffle

pressure. cliffwence across cylinder, inches of

water

preasurc difference ca~sed by loss of velocity

head from exit of skirt of bafHe or jacket,

inches of water

total pressure difference ncross set-up (Ap, - t

ApJ , inches of water

ANALYSIS

In the cooling of engine cyh.ndem, two outatancling

requiremmts must be considered. One requirement t is

to protect the surfaces; the other is to reduce the knock

or the tendencies to preignition caused by hot surfaces

that come in contact with the fuel mixture. The

lubricated working surfaces must be kept at a suffi-

ciently low temperature to insure the maintenance of

an adequate oil film. High piston and cylinder-wall

temperatures usually cause sticking of the piston rings

and rapid wear of the piston rings and the cylinder wall.

An additional problem of cooling is the prevention of

undue distortion of the cylinder barrels, such as might

be caused by uneven temperature chstribut,ions. Al-

though difllculties arising from thermal distortion of

cylinder bamls have been alleviated to some extent

with specially ground pistons, it appears desirable to

retain round cyhnders by means of a uniform or other-

wise satisfactory temperature distribution.

Of the several available methods of securing untiorm

temperatures around the cylinder, two methods are of

particular intered. One method is so to distribute the

M ect,ive fin area as to ficltieve the desired temperature

distribution. The other method is to COUM t.hc tiir

velocitiw around the cylinder by means of butlles sur-

rounding the cylinder. In generrd, either of the fore-

going methods will result in some 10ss in tic maximum

over-all heat transfer otherwise obtained for the same

fln wei@t. In experimental work, it has been found

that baffles desi~med for maximum ovor-nlI heat flowwill not give a @form ternpernture distribution i-ml,

conversely, baffles desigg~ed to give a uniform tempera-

ture distribution do so with a considerable sac.fllce in

over-fall heat transfer. W’lmn the exhaust valv~ and

the piston crowns are somewhat centrally located with

respect to the finned surfaces that cool them, these parts

generalIy are better cooled when a maximum over-alI

heat transfer of $he finned surfaces is obtained at the

expense of a uniform temperature distribution. In the

present report, emphasis has therefore been placed on

providing fins for obtaining high over-all heat. transfer

rather than on securing uniform cylinder temperatures.’

The over-all heat-transfer coefficient 27has I.weu cal-culated from the following equution, which was derived

in reference 1:

‘ = * i : ( l + %) t ’ ’ n1)where a.= ~2g/ k~ t rmcl k= is the thwnml ~onductivity

of the metal (2.17 for steel; 7.66 foi. alum iniin Y idToy;

18.04 for copper; and 7.54 for magnesium aIIoy). Ill

this report aluminum and magnesium alloys arc refwvd

to as “aluminum” ancI “magnesium,” rcspcc(ively.

This equation has been cxpcrimenhdly verified

(references” 1 to 6) for fins of steel, copper, and ahlmi-

num alIoy. Experiments huve dso shown that equu-tion (1) holds equalIy weII for rectlanguIar or tapered

fins, provided that the average values of the fin thick-

ness and. space are used in the calculations.

It has been founcl (reference 5) that thu surfmr

heat-trmsfw coefficiiwt q can be correlated for cacfi

air-flow arrangement in terms of functions cleflning n

single curve and involving the flu dimensions, the

cylinder diameter, and the air-stream characteristics.

Thus, for cylindme in a freo air stream with and with-

cmt baffles and for cyIinc?ers at a 45° fin-plane/air-

Stream angle,

(..)

where V is the velocity of the freo air stream and, for

sylinders enclosed in a jacket and cooled by a blower,—.

(3)

where T’ is the velocity of the cooIing air between h

Rns. Figure 1 shows the variation of g with fin find

cylinder dimensions and air-stream characteristics for

Lhe four air-flow arrangements.

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THE DESIGN OF F INS F OR AIR -COOLE D CYLINDER S 403

1--@om ——+ - Cylf+ s fin) w (in.) t (in.) ,Cyf&r S (inj w (in) .! (in.)

LCyji;&r s fin] w (inJ t [m.)

0.330 f.47 0.270 a i7 .2 fo 0 .97 0 .040 v 0 .210 0.37 aff40

+ 2 ..Z70 1.47 .230 0 [7 .[10 .97 .040 ~ .29 .110 .37 .040 WI

I Pn 5

I IC?7

I n 1? . 13i J.zz .035 q 26

Ilo;~10/4

.077 1..?2 :033.048 /.22 .035 II

M@oO

m

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1,000 . I Ill ! ~, 1l-” .“P : 1 1 1 +

I%7

I I 1 I I t I I I II I I 1 t I I I I I II I I2 34 8 Ku 2 34 6 .l,om 2

I (a) H II [111111 I I34 6 /o,Gw

(a) Cyllnders i nfreeahs t r e a mjm bafiles.

Elaum l.–ReM1onbetween fsctm Invoking o, h dbnmdons, c@hder clfmeter, md a~~ Chuacterktks

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REPORT NO. 726—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

p 3%$.w Cf%$

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r . .97 .035 .9 ?K j% :% “:%;07 “.. — — ...m-.--- ---

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34 6 Km 2 34 6 I.m. .-~P.98=/12@@w”45 ‘-

(l) Qhlar inha airstream, 140”bnfika.

FIGURE.-Oonthmad.

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.

THE DESIGN OF l?lNS F OR AU+COOLE D CYLINDERS 405

I

-.

(c) Cyh.der h free a ir a- oylhder aria M“ to air stream.

k’iiiCB E L-Continued.

430134”42+7

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406 REPORT NO, 726—NATIONAL ADVISORY COMMITT3E FOR AERONAUTICS

w‘ , - m .

2 0 0 . 0 0 0 $

—

I UI

.07’7;; .048a .022

Mitt

f,00Q0OO

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6 0 0 . 0 0 0

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(d) Cylinder enclosed I n Jw k e t ,b I owe ro d n g .

FIGUREl.–CmdndwL

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THE DESIGN OF F INS F OR AIR-COOLED CYLLWEIW 407

1 0 , , I I

8

f?

# 1 I I I, I

4 t I 1AXllvrr, Y ) A-i 4w l ‘H

I I I I I I I I I

~ .4g_

.3

.2

~20 50 4(?

Vp,g,lb jS eC~~ ft

FIGURE2.-Effee t of weight vefoeity of emhg sir on pressure difference per unit

length of path h eew?.rsffin sIXJoes.

For any one air-flow arrangement for which the

pressure differences across the cylindem are available,

g can be determined as a function of the pressure

difference instead of the weight velocity of the cooling

air. Previous tests of cylindem enclosed in jackets and

cooled by a blower (references 5 and 7) have shown that

the pressure difference across the cylinder Apl for a

given iln space ancl weight velocity is proportional to

the length of flow path 1 which, in turn, is determined

by the cylinder diameter, the k dimensions, and the

jacket design. The pressure dMerences across the

cylinders for various lengths of path, fin spaces, and

weight velocities are presented in figure 2. From this

figure, it. can be shown that

3P, p../ pO=fa(s,f,w,D,I“pfl] (J l

ml, as

AP:P.r/ PO=.f#( I “Pfgl (5)

l“p~=js(*?jtjwjDjAptotal pau/d (6)

The weight of fins per square inch of outside cylinder-

wall area is given by the equation

wh ere Tf’m is the specific weight of the fin material

(0.282 for steel; 0.101 for ahuninum Y alloy; 0.322 for

copper; and 0.0648 for magnesium alloy).

f o a lI I I I I I I lit I Ill 8. in.- .CY -.04. –

F/ 1 y, --y -

1

-

/ / Ii -.ob - –i I )

// / IllI

Yp12 -–

‘oH—H+MM+

u, I t , # II I I I I

41 I I A I I I I 1 I It N I

I 1 Y # 1 /1 1 d I ,1 1/’:/f . & 4 I

3

2

‘1 2 34668[0 2a m 40VP,g, lb @C& b- ‘- “-

FIGURE.3 .-EEeet of weight WxI ty of moMng sfr on prmure diff erence aems12Med oylfnders for semml fin spares . Awrege dr. rWidtk 0.825fneh; cyllnder

dfameter, 4.04fnrhes; !3ntbfekness, O.CCMnch: blmer-socdbg set-up.

From equations (1) and (3), it is evident that, for a

given metaI,

u=f7(8,t,w,D,T”p@) (8)

When fin weight is more important than fin width, w

can be ebinated from equation (8) by means of equa-tion (7). Then

U=j8(s,t,M,D, ~“p~) (9)

The over-all heat-transfer coefficient LTcan be expressed

as a function of ApW PaJ~ by combining equations

(6), (7), and (9), or

U=f9(s,t ,M,D,A}, .,. l pa,/ po) (lo)

T@ method of obtaining optimum fins follo-ived in

this report is generally to hold const tint the vahm of

the variables tlmt are specified by the desi=m conditions

and, from a plot of U against values of the remaining

variables, to obtain the rest of the dimensions that

give maximum heat transfer.

The design of fins for given values of M and

Aputil p.,/% is more diflicult than for constant values

of M and J’Plg bec~use the length of the flow path and

the losses from the bidlle exit enter into the calculations.

Both the length of the flow path and the exit los&es

depend upon the fin dimensions and the baffle dwign.

A method of designing fhs for a constant pressure

difference, using an average length of flow path and

assuming that all the pressure di.Rerence is avaiIabIe

for cooling, would considerably simplify the calcula-

tions. Computations have shown that the difference

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408 REpORTNo. 726—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

/ .5

\

1 .3 \ \ - ./ / ./ 6

% .20/

/ \ ./ 2

f.I\ /

\ \ \ .Q 9[1 I / ~

\ ~ . 0/ 6~ .01

$II / /

t, in,..

s .9 \

\ ‘ .06 s , ;n .c‘. /

R\

;, 7 \ II I\

a / / /s

\ .04.5 11/ ‘

\

.3 \ //

—(a) “. ‘ ‘-

1

‘ .027 (b)—

./ 0.@ .04 ‘ .06 .08 .io o “:.04 .08 ./ 2 .16 . 2 0

Fin fhidmess, t, in. -- .5puce between 7%79, S , in .

(a ) SpeolEed& (b) SPCOIRed.

FIGURE4.—Variatfon of Uwith 8and t forconstant M, APIPJmi md D

betwwm the we igh t ve lo cit y based on the exact flow

path and including the exit Iosscs and the weight

velocity bnsed on an average flow-path length and neg-

lecting the exit losses has very little effect on the con%ct

fin proportions. Consequently, figure 3 has been used

to determine pressure differences for this report.

It is evident from equation (10) that, for given vahwaof cylinder diameter, weight of fins, and pressure drop,

U is R function only of s and t. As an illustration,figure 4 (a) shows a plot of U againsts tmd t for constant

values of M, Ap@a*/po,and D:” Figure 4 (b) i9 a cross

plot of figure 4 (a). EMer part of figure 4 clearly

shows that, for one pnir of values of s and t, the heat

transfer is a maximum. The peak values of the curves

of coustants shown in figure 4 (a) and of similar curves

plotted for other values of M and Aplp.,/Po are shown

in subsequent figures and are Iabeled “specified s”

curves. Similarly, the peak values of the curves of

constant values of t shown in &ure 4 (b) and of similar

cwwes plotted for othm values of M and Aplpa,/po are

shown in subsequent figures and are labeled” specified t“

curves. The specified s curves are used. when a

lower limit is set on the value of s and t!~e spec.hied t

curves are used wheu a lower limit is set on the value

oft.

For given values of M, D, Ap,pa,/po and a specified

value of t, the valuo ofs for which U is a maximu m can

also be found by setting the derivative of U with respect

to s in equation (10) equal to zero and solving the

resulting equation. In order to obt~in an expression

for the function in equation (10), an equation wouId

,

have tc.be fitted to the.curve in figure 1.. Making the

substitution. previously indicated in obtaining equa-

tion (10) would result in a complicated relationship.

The -ivo~kinvolved in salving the resulting equation for

the optimum value of s would be .considcrahly more

than the work of obtaining figures 4 (a) and 4 (b) and

picking the values of optimum g and t from thesecurves:

PIots of tk type shown in figure 4 were obtained for

other \ralues of M and Aplpa,/m by means of figures 1

md 3 “imd equations (1) and (7). For each value of s

and tin figure 4, the associated valuo of .Wcm IN crd-

culated from equation (7). The heat-transfer codi-

cient lZ can also be plotted against t for various values

of w and the optimum value of t can be obtained for

the -urn vaIue of U for each value d spccificd w.

In thie case, the value of s is unrestricted and may be

obtained from equation (7).

Figqqe 4 shows that, for given values of .Aplpa,/poand

M, definite values of s and t exist for which U is a

mmiimum. Although these vahw.s of s and t nmy be

outside. the practioabIe manufacturing range, a l~tidc

range Qf6ns becomes available for values of U 5 percent

below the maximum. Figure 5 is a cross plot of figure

4,4 having been ylotted against i!for several percentage

of maximum U. It is evident from figure 5 that a

single -pair of values of s and t represents tho optimum

fin design as indicated by 100 percent U. Ii case the

manufacture of these fins is impracticable because .sand

t are too small, some sacrifice in U must bo made if the

fin weight is to remain constant. I?or example, when

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THE DESIGN OF F INS F OR AIR-COOLED CYLINDERS 409

U is decreased to 95 percent of the mtium value, -an

infinite number of pairs of values of s and t wiII give

this heat tramsfer as shown by the points on the 95-

pert.ent Iine. When the WUm va]ue of 8 is limited

by mrmufmturing reasons, the points on line A give

the ma.timum values ofs and, if the value of t is limited,

the points 011 line B give the maximum values of t.

The. shaded area between A and B is the onIy region of

practical interest on these curves because the h in

this region give the incLicated heat transfer with the

highest values of s and t. The specified s Cmves in

this report correspond to the values of s and t along

the line A of graphs of the type shown in figure 5; the

specified t curves correspond to the vaIues of s and t

along the line B of graphs of the type shown in figure

5. As is evident from figure 5, reasonably close approx-

imations of values of g and t lying between the specfied

s line (A) and the speciiied t line (B) can be obtained

by assuming a straight Iine between correspondingheat-transfer points on lines A and B. Similarly, in

the charts presented later in this report, values of fin

dimensions lying between the specified s and the speci-

fied t charts may be appro.ximatecl by assuming a

linear relationship between similar dimensions on each

chart.

I’ery often the fin width is of more importmce thin

the iin weight. In certain types of engine, such as

in-line engines, the small distance between cylinders

places a restriction on maximum k width. From

equations (6) and (8), U can etident.ly be written as a

function involving width instead of weight.

t ?=j]o(s , f,w , ~,Ap to(a zPuu / AJ (11)

For a given cylinder diameter, flu width, and pres-

sure drop, U is again evidently a function only ofs and

t and curves similar to those in figure 4 can be plotted.

The curves of optimum I-Inproportions for specified s

rmd t are obtained in a manner simiIar t.o that preciously

indicated for the case in which weight was the criterion

and are shown Iater in the report. The specified g

curves represent the best fins that ful.fll.l the restrictions

placed on $ and w; the speciEied t curves represent the

best fibs that fulfill the restrictions placed on t ancl w.

When the flndesigg information is applied to engine

cylinders, the tin proportions may be determined froman average ~alue of the surface heat-transfer coefficient

q for an entire cylinder circumference or may possibly

be determined for wch portion of the cylinder circum-

fermce from the local heat-transfer coefficiertts. A

most airc.raf t-engine cylinde~ are composed of sevimd

cylindrical areas, it is believed to be most practicable

in applying the fidimension information to consider

each of these areas separately. The outside-wall sur-

face of a conventional cylinder can thus be considered

as five separate areas: The barrel, the lower head, the

intake-valve stack, the exhaust-valve stack, and the

curved surface betvreen the intake-vahe and the

exhaust-vaIve stacks. Further retiemeut that might

be obtained by the consideration of smaller areas is

believed unwarranted in view of the impracticability of

changing flu sections and spacing from one point to

another around a cylindrical surface.

In heat-transfer investigations, the heat-transfer

coefficient is customarily based on the difference

between the surface and the average fluid tempera-

tures. The problem of determining fin proportions is,

however, -rev much simplified ~“hen the coefficients are

based on the intake cooling-air temperature. V7hm the

t, in.

FIormE &-VarMon of a and t for several percentages of maxhnnm heat-transfer

oxt%ckt. Constant M and APIP.JP,.

mer-all heat-transfer coefficient U is calculated from

be surface heat-transfer coeficimt based on the aver- _

~ge air temperature, it is necessary to determine the

temperature rise of the air, which in turn depends upon

%e value of t~ being determined. In the present report,

;he o-cer-alI he~t-transfer coefficients have therefore

>een based on the intake-air temperature.

Equations (10) and [11) show that U is a function of

&e cylinder diameter. In most of the calculations of

& report, the length of the flow path was that for a

L66-inchdiameter cylinder; this value is a representa-

tive average of the various diameters of the cylindrical

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410 mMfiT No, ‘726–—NATIONAL ALJV”ISORYCOMMIMEE FOR AERONAUTTW

portions of a range of conventional aircraft-engine

cylinders. The diameter of the valve stacks is usually

much less thrm 4.66 inches whereas, except for small

cylhders, the barrel and the head diameters are larger.

In cases where the length of flow path is greatly diflerent

from that for a 4.66-iich-diameter cylinder, corrections

can be made for differences in the temperature rise of the

rmoling air. Most attempts to obtain g-mater accuracy

than ca n be obtained by using the data for the 4.66-

inch diameter are unwarranted because calculations

have shown that an appreciable chmge in flow-path

length is required to effect much change in fin propor-

tions although such a change in flow-path length will

change the absolute values of. U. In all calc~ations

for determining g, the viscosity of the cooling air p

was assumed to be 130X 10-7 pouncls per second per foot

and the thermal conductivity k= to be 3.4X 10-7 Btu

per inch per second per “F. The effect of variation of

both ~ and kc on the optimum fin proportions with the

4.0 ..

32-iz>2.4

G<~ L6G

S .8

0

FIOCREfL—Varlatlonof rnexlmum heat txe.neferwith ~wr required for coding onpressuredlfferenca and Powwrbes@. FInvolume, 0.46cubio Inoh per sqnere fnohwall arm; orikr lon, h wefght .

temperature range encountered is inappreciab~e, and

the assumed valuea give resulkthat are accurate enough

for alI practical purposes,

An exact determination ofoptimum fin proportions

would require a diilerant solution for evely condition of

air flow that might be mused hy differences in baffle and

cowling design. Furthmnore, in order to cover the

problem completely, it would also be necsssary to

determine fin proportions for variations of f3n weight,

fin dimensions, pressure drop, weight velocity, andpower to COOL It has been necessary, in order to limit

the scope of the present report, to choose for the final

calculations a limited range of conditions believed to be

of the greatest practical interest. The problem has

been simplfied, where possible, by eliminating several of

the variabh?s having little or no effect on the b

dimensions. .The determination of the most important of the fore-

going vrtriablei will depend upon their application.

For example, when the coding-air flow through an

engine is. induced by the movement of the airplane

thiough the air and the slipstream from the propeller,

the pr~ure di.flerence available for forcing the air

over the cylinders may be insufficient for cooling at the

power output clesired. In this case, it may be desirable

to aclcl fin weight to obtain sufficient cooling with

the limited value of the pressure drop. When the

cooling air is supplied by a blower, a wide range of

pressures may be available and the power required for

cooling may be equal]y as important as the pressure

drop or the weight of the fh.

The_determination, of optimum fin dimensions for

constant weigh~velocity and power conditions is of

in~erest only in special cases. If the cooling air is

furnished by a blower and the power required for

cooling k used as a criterion of fin dosign, tho total

blower power and not the power required to force tthc

air across the cylinclcr should be used. The cfficicmcy

of a blower is particularly dependent on tho prc%surc

difference used and, if fins are designed for a constant

cooling-air power, the pressure clifferenco required may

be such as to Lio b a my-y inefficient part of LLCpower

curve of the blower.

The resultg of calculations to dotwmine optimum fin

dimemions for constant pressure drop, constrmt weight

velocity, and constant power conditions show thnt, ill

general? the optimum fln spnco or thickness chaJigcs

with t@ediflercnt bases. The desired values ofs, whcm

t is specified, are somewhat smdlcr for constant weigh L

velocity and power than for constunt pressuro difference.

Whens is specified, the vrducs of tmo generally lowest

for the constant weight-velocity condition md highosL

for thp constant power condition.

Although the optimums and t I-Weomewhat differm Lfor the conditions of constant-prrssure diffcrcncti and

co~tant. po~~’er to ~ol PI, th~ diff~rence b~h~e~l] thl!

heat transfer obtained for n given power to cool and the

heat transfer obtained for a given pressure drop is vmy

Slightl_m is shown in figure 6. These curves wero ob-

tained by determining the optimum fin designs for

several constant assumed powers and prcssuro diffw-

ences. In these calculations, tho fin weight \vas held

constant for each metal. The slight dillerence in U

showm by these ourvw mrdws the design of fins from Q

power-to-cool basis of little interest. An advrmtagc of

fins designed on a pressuredifference basis is hit the

optimum thickness and space me greater than for hdesigned on a power-to-cd bmis.

Op@mm fin designs were also Mermirml for three

air-flow arrangements: Cylindem in a frco air stream,

with and without baffles, and cylindom at a 45° fin-

plane/air-stream angle. The calculations were based

on a constant fin weight rmd a constant air-stream

velocity. These results show that diffmnccs in air

flow caused by these rLir-flow rn-mngomente do not

materially affect the best h dimcnakms.

Frcirn the foregoing results, it is believed that fin-

design information for cylinders enclosed in a jacket

will apply with reasonable accuracy to otlwr conditions

.

8/7/2019 19930091804_1993091804


TH E DESIGN OF F LYS F OR AIR -COOIiED CYLINDER S 411

of flow as ctiused by different baffle arrwgements. As

the power-to-cool and the weight-velocity bases are of

interest only in special cases, the fidesign data sub-

mitted in this report have been calculated for constant

vaIues of pressure difference across a cylinder-jacket

arrangement. Although the material of this report

was derived from data for cyl.inclricd barrels, the srndleffect of different air-flow chrtracteristics on the opti-

mum fin dimensions would appear to justify the appli-

cation of the material of the report to complicated

shapes such as cylinder heads.

The conditions covered in this report with il.u weight

as the basis of design m-e listed in table I and the con-

ditions with fin width as the b8sis of design are Iisted

in table II.

TABLE1.—CON7DITIONS FOR ‘WHICH C.4LCULATIONSJICEREMADE USING FIN WEIGHT AS CRITERION

+

steel........-.

Alurdnum..-.

t 1, “n. I ,,:1,3

466 1259z }

1,4,&12

40643“ .” * “ .U

.20 .0.1

i$ .s -

&o* 0.0040.2a .om

.0’55I1,48,12

i~ .lm?

{

0.02

+ x!!!!!’

Comr........l 4fM

M8gndum._. 4601{.Z1

I “ - - - -0 1 . 3 0

0778}

4

R .12Q0 ITABLE II.— CONDITIONS F OR WHICH CALCULATIONS

WERE MADE USING FIN WIDTH AS CRITERION

F !n rmterlal

Steal.........

Alurdmun. ----

pe.ssed#

of wa t e r ) .

4

4

The values of the over-all heahtransfer coefficients

U for the various conditions listed in these tnbles and

from which curves such as shown in figure 4 -mm drawn

have been tabulated in nine tabks, which are available

upon request from the National Advisory C!ommittec

for Aeronauti~.

The values of the peaks of the curves of the type

shown in figure 4 were used in plotting the fial charts

which are presented Iater in the report and in which

both fin -weight smd h width are used as criterions.

The peaks of some of th e cu rves of the type shown in

figure 4 are fairly ffat; the values of 8 and t may con-

sequently be varied somewhat without changing the

heat transfer.

OPTIMUhl FIN DESIGNS WITH LIMITED FIN WEIGHT

SPECIFIEDSPACE AND TEICKXESS

Figures 7, 8, 9, and 10 show the relation between the

optimum fin dimensions and U den s or t is specified

for both steel and aluminum with fin weight as the

oriterion. As each graph is for a constant weight of

materkd, it is apparent that the peaks of the pres-sure-

dMerence currcs represent the Iin designs that will

give the maximum heat transfer for the given weight

and pressure difference. The values of U, s, t, and w

at the peak point are the same for both the specified

s and the specified t charts.

Several c.haracteristios of these graphs are of par-

ticular interest. The wide range over which both sand t may be varied without much change in U is very

noticeable, especially for steel at 10-wfin -weights and

lo-iv pressure differences. In geDeraI, the peak point

of U occurs at sm a ller va luw ofs a n d t as the pressure

difference is increased. The h indicated by the peak

points, particularly for aluminum, are generally too

thin for practical use. Although the value of s can be

varied over quite a range without affecting maximum

U, itmay be desirable in some engine instalhtions to

limit &to small values in order to hnve a minimum vol-

ume of air passing through the engine cowling.

Information similar to that already presented for

steeI and aluminum fh is ahoycn for copper and mag-nesium fins for a pressure difference of 4 inches of

water in figures 11 to 14. Copper is of particular in-

terest owing to its high thermal comluctitity, and the

use of magnesium is significant because of its low weight

combined with fairly good thermal conductivity. A

comparison of the proportions of fins of steel, mag-

nesium, aluminum, and copper for mwtium heat

transfer shows that fins of metals hnving a high thermal

conductivity are a~t.remel-y thin. A comparison of the

maximum heat transfer obtainable with steeI, alum-

inum, copper, and magnesium is shown in @u.re 15 for

d&rent fin -weights. These data were taken from the

peak points of the curves of figures 7 to 14. Magnesium

alloy of the thermal conductivity chosen hns a slight

advantage over the other metals; whereas copper and

aluminum, although somewhat less effective than

magnesium, are equally good, both being several times

as ef%ctive as steel for a given fin weight. A plot similar

to that of figure 15 could be made showing maximum Z7

against width of fins as the criterion. Such a plot

would show a defhit e advantage for copper with

d-uninum, magnesium, and steel following in the order

Dftheir relative effectiveness.

8/7/2019 19930091804_1993091804


412 RMPORTNO. 726—NATIoNAL ADVISORy CQldMl~EE FoR ~RoN-K~lcS

Lo

~.9<

L>.8..b$7 .“:

c..J i

(a) I13’4 i l+,!------ --

.6$6 I I

.10 .14 .18 .2? ;2B .30 .34 ..38. 42.Space beh+eecu%sis, in.

.+&e & fween fins, S.;~. _

(a) F IDweight M, 0.011Spound per Wuare hmb wall area. _ lb) Fin wotght M,0.0&34ound pm square [richwall area.

.

\Ri\ ‘ .- ‘\{ I

Spcn? between fins, s, in.

“$: “.““~/6.m ”

s

.

..=

----

.

(c) Ftn woIght M, 0.1289pound per square Inch walI area. (d) Fln weight Jf, 033S4pound m Waare incII wa~ mea.

FIQCRE7.—Opt1mmn dimensions forsteel Llnswith spooMedM thicknees. (Yiterlon, fin weight.

/.f

Lo

.$.9

L

“>4.8

$3.7

Gb“6

5

:$22 .096 ,0/0 .014 .018 .0.22 .Q26 ,030..034 .038Fin fhickness. L, in.

..7 ~m

t4!-- . ~ =

- ~ -M. _ .-:.._., –

-f ~10‘7 ).1 - 04

’5 0 ....0 38 .0 !6 . ,.Q 24 .cW .0 40 .0 48 ‘.0 56 . .&7 4. ..0 72Fin fhickne.ss, i, in.

(a) Fin weight M, 0.0118pound per xnmra tnch wall oren. (b) Fln weightM’0.0M4 pound per square tncb wail are+.

FIGURE8.–Optimum dimenefone forsteel fins with apodlied flnspaca. Criterion, fin weight.

8/7/2019 19930091804_1993091804


TH E DESIGN OF F INS F OR AIR -COOLWI CYLINDERS 413

#[ I I nzlI .U61 1 I m-12 in. 1111111

I I I

~ v I I A I I I I

I /t 1 I I I I I I

.012 .&O .028 .036 -044 “.CkT2.060 .068 .076Fin thickness, t. in

(a)FintieigMV,0.1239pound per square fnoh WEUsres.

FrorRE S.-Optimum dimensions fmsteel fins with

L-Y- “E4.- L-Lk~

“4x

I If1. 1

— I 1 t t I 1(e.)

I

f .08 J2 ./6 .20 .24 -28 -32 .36Spoce between t%s, s, in.

(a) Flu w e igh t . 11 ,0 .0 01 0 p ou nd p er s qu er e Inch wait rues

S#boce between II., s, in.

(c ) Fin w eight M, 0.M55ound per square inch wali ares.

Fin ihickness, i, in,

(d) Flu weight .?if,0.33S4pound per square inch wail area.

SPeCIFIedin space. CrfterIon. flu weight-Continued.

h) Fin weight J f, 0 .IE 02 pound pcr squsre inch waHarea,

Spoce between fibs, s, in.

(d) Fin weI@t M, 0.1212pound pa sqnere inch waifarm.

FIGnE 9.—Optimum dimensions for siwnfmun b with qxcifled dn thfdums. Crftwfon, dn wefght.

8/7/2019 19930091804_1993091804


414 REPORT NO. 726—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

2.0

1.8

f.6

&$1.4

“Gq L?

2

~ Los

.8

.6

0 .00.2 .004 .006 :LZ78 .0?0 .0/2 .014 .016 .0/8Fin fhickness, t, in.

(a) F in weight Af, 0.03LOpmnd pm sauaro Inoh wall area.

3.6

3.2

$2.8

‘yc

‘&2.4q

3~ 2.0

s

/ .6

f.z

.004 .008 .012 .016 .0 .20 .OM .028 .032 .036 .040Fin thickness, t, in.

(o) Fin weight M, 0.046Spoun dpersqua rehch wal la r ea .

\ “a

I 1,-I’LJUI+WWWH

Zb’1, 6

/4

.._Q4 .008 .012 .di ..-Fin fhickness, t, in. -

II I I —

I.LGULLJJL - I t “1:—.. .-

AP,Pa.fPo: _(~)— ,7.5 I

1.~~n. woter

1

‘W .020 .(.??4 .028 .032 .036

(b) FIn weight M, 0.0X12wund pm square inch WUUrea

--fin thickness, t, in.

‘td) Fin weight M, 0.1212pound per square Inch wall .smn

Fmmrt 10.—Optfmum dimensions for abr[mrm fins with spaolded fm space. Crit8rlon, dn wsfght.

34

3 . 0

$

22. e“\,$

0.2.2*

-1.

s

/.4

/ 0

.Or.uu Iz .lb .Zu .i

I I I 1 I I I i I 1 I I I I I 1 I J-- ,- ,,. --?4 .28 ,32 .36 :40

Space between fins,s, in.

3!

., 3.:$

g2.

3“; $.2

3z 1..5

L

/.

.0.—004 ~ .012 .0/6 ,020 .024 .02ff .032 .036Fin” thickness, t, in.

FIGUIIEIl. -Optimum dimensions for copper fins with specflled fln thfclrnms. FIOURE12.–Optimum dhnenaions for copper flue with spadled fln space, Crlta-0rft8r@ IMWtIfght;AW.JPW 4 hOhMOfWateI. rion, dn wefght; Afhp.Jp w 4hohesofWa&.

8/7/2019 19930091804_1993091804


THE DESIGN OF F1 .IWFOR AIR -COOLED CYLINDERS 415

3.6 I 11/ II;M, lb f sq in .walloreu

~zJ30

-.30111111~

\ k 1 1 I I

\ ./ ~.a90\l I I I

>

‘;2.4

q

220

1.2

,8.C i2 .f6 .20 24 28 X 36 .40,bEEe-4 .08

k ll . rR E 13.—0pthnurrr dhnen.s.sne for rnegn~run EmgwftJI spwf&j fsn thI& II(W.Crltcz[on, En weight; ARW.Jp e 4 Inches of watm.

ItllMM, lb /sq h.- —

A .20, WiYfl m-es.

p+m

>- [8

[4

YtTKH-liii,iilllllllo .010 .020 .030 .040 .050 .O@ .070 .080 .0s0

Fin fhickness. t. in.

FIGURE 1 4.-Op tlr mu n Ah n en sfon s fcwrnagneshmr fins wi th specIEed h s~eeCrfterion, fin weight: Afrw.Jp,. 4 Incbosof water.

.

0 .04 .08 J2 ./6 .20 .24 28

Fin weI&r ifM . Ibjsq in. WU17ureu

FIGURE]6.—VarfatIon of maxluurm o.m?r-e.11eat-transfer codcfantithmfght offins for several flu materhls. Crtterkm, fin wdght; A IM. JP@ 4 f rrch rs ofwater.

8/7/2019 19930091804_1993091804


416 REPORT NO. 726 -NATIONAL ADVISORY C .03 fMITrEE FO”RAER.ON.~UT~CS

-- .

m--:-,I

0 .04 .08 ./2 .16Fin weight, M, Ib/sq in watt area

(a ) Prm s uro dh%reneePIP.J PW 1 h ICh of W ater.

J6

.14

.12

d,.

~ ./0

.08

.06

.040.04 .08 ./2 .16 .20 .24 .28” “.32 “.36

fin” weigh f, M, Ibjsq in wall or~o.-

-.18J“U78

;- r...

1!] !

! - ‘“’”!3

G—- ...: “:02: :;:/ 6 r ---

/ .0 _ Heut transfer

11 *f I .06percen< of

,. , ,, no + :-.- ma.vimum;,

-—k”, ,

.061 ‘:0 ‘1~ “-? “ ‘ .’

I I I I I

1 [111111”+?)+

o -–.04 .08 .12 ./6 .20 24 .28: “.3? “,“36f i n weifl t, ~, Ibjsq in WOII urea

(h ) Pre ss u re rlt fk w n ee A PIP. JPC , 4 Inch= ofwaler

“-/6

14

,.,.

.. .

.:.

;.:’ j- ‘ J,08 J2 J6 .20 .24 .28 .32 ..36

. . ..- .“. Fin wej”gh L M, Ib / s@in . wal larea

(c ) Pre s s ur e d if fe renwPIP.W ’PW 8 h ck s of w afer. (d ) Prw uro d ifference AfhP. JPr 12 InchcwOfwatOr.

FIGURE10.-Optlmum dimensions ofsteel fins for vrxious pereentagm of maximum bent tramfer. SPWKM fln thlcknc<s: oriterion, En wesfht.

Fin weigh t M, Ib / s q m . w a ll ore o

“ l%

.05

e--.04%’

.m

-: - “ . 0 2

I I I I 1 I 1 I 1 t , , ,i Ri

. 0 8 . / 2 . / 6 ’ .20 .24 .28 .32 .38Fi% weight, M, Iblsq in ml are~

(a) Pressure dlfferem.eAfAP@j 1tneh ofWater, (b) Pressure ditlerenm A IJ IP. JPV 4 Id le s OfWaLW.

FIOUEE 1 7.—Op tfm u m d im en sion s o f s t eel d ns for m rlou s f3 er e3 nW.geJ of M8 1h t3 UIIi h ea t t ra n3 for . Sp eelfm d fin s pa ea er lt er ion , fin wd gh t.

8/7/2019 19930091804_1993091804


THE DESIGN OF FINS FORATR-cOOLEDC1-LINDERS 417

Fin weiqht, Afi lb/sq in wd oreo

.061 I 1 1 I , I

1 1 h 1 1 1 , Ii

.U[ I

- ki-

0 .CU .08 J2 J6 .20 .28 .32fin weig$}, M, Ib/sq ik24woJ ores

.36 “

(c) Pressure difference APIP.Jpt, 8 inches ofwater. (d) Pressure dlfferenw APIP.J PW 12 k h f!s Of W ater.

F taoR~17.--Optfmum dimenskwrs ofsteel IIDSfor varkme pereentagea ofmaxtmrrnr heat transfer. Speef6ed 5 SPX erIMon, fin wetght-Continued.

(a)

? ./2 ./4 .16 .18Fin wetgh~ M, lb/sq in. wdi oreo

(a) Pr mu r e d ffTe re nc e A PIP. .J P, , I fneh of water.

.26

.22

/8

e“;L4

./ 0

.08

.020.02 .04 .m .08 JO .12 .f4 .16 ./8

Fin weight M, lb/sq in. wolf or=

.28

.24

m

c.~. 16

./2

I ~ . f I !

# , I

-U4Q .C12 .04, , 1 1 1 I c 1 I I I I I.06 .08 ./0 ./2 ./4 -f6 .18

fin weigh( M, fb/sq in, wdi oreo

(b ) Pr es s u re dMerenee A PIP. JP6 , 4 fnches o f w at e r.

. 2 0 *

J 4-” I t~..

A7 / ’ i . 1 5 ’ + , ’ ‘ “. =90

~- 85 I

–/ 00 -,i - I

I I 1 , I ,

II

(d) -1,

9 ./2 ./4 .f6 /eFin weigb~ M, lb /sq in. wu# oreo

(c) Pressure difference APIP. JP*, 8 f noh ee o f w a te r . (d ) Pr es m re A1 .1 9e re nr t A pW .J P@ 1 2 h mh ee of water.

F1ouaE IS.-Optimum dimensions ofrdumhrum fins for mrfons percentages ofmexfmnm heat transfer. Spe&%d dn thickness; erfterlon, fin wefght.

8/7/2019 19930091804_1993091804


418 IiEPORT NO. 726-NA’ITONJkL ‘ADVISORY CO~IT1’EE FOR AERONAUTICS

.10

,08

,$G5

.

,04

,02

0Fin weigh ~ M, Ib/sq in, wolf area

.fo

.08

$:a5

c

.04.

.02. .

..”0

( a ) p r e s su r e lf le r cn e ePAJPO, 1 k h Of~’ater. (b) Pr o. wr e d h Te re ne o APIP. ./ Pu , 4 In ch e s OfWate r

. f2

.10

.08

.$06

%

.04

.02

0 .02 .04 .06 .06 ./0 ./2 ./4 .16 .[8Fin wagh ( M, lb /sq in. wofl urea

.06

.05

:04

f.. .*J23

.02

;01

..o _

FFn weigh< M, lb / sq in. WOII area,

(c) Pre?sure dlflerenm APIP.~P@8 inebw OfWater. (d) Pressure difference APIP.JP,, 12 inch or W@

F IGURE 19 .—Op t lm um dirnenslmrs of ahnnfnum fins for verimrs peramtsges ofmaximum heat trar.rs4er. Spc.dfledSu space; M.erion, drrwclght.

8/7/2019 19930091804_1993091804


TH lil DE SIGN OF F INS F OR AIR -COOLE D CYI .J NDE ~ 419

,

:4

:0

,60.Of ,02 .03 .04 .06 .08 .07 .08 .09

Fin thickness, t, in.

FIOGRB21. -Opthnum dhnensfom for alnmimnn fins wi th spechled dn width.

Crit erion , t in weigh~ A PIP. dPw 4 f nch es o fwa t e r ;op t hn muf i SPW@Wns t it a t

0.072In ch .

S@ze between fins, s..A

(a) Speclflad tchart .

FIGUaE 21.--Optimum dhnensfone forsteel dns

.34

.30

k 26L“\

s pz

~

; 18

~.

L4

/.0

.02 .04 .06 .08 .10 J2 .[4 ./8 .18 .20Spuce befween fins. s, in.

( a ) Sp s o i a e d0h8 r t .

The data in figures 7 to 10 have been cross plotted in

iigures 16 to 19 with the fin weight as abscissa. These

plots show the lin dimensions for maximum 27and also

the fin dimensions when certain percentage reductions

in maximum Z7 are allowed in order to obtain easily

constructed fins. The usefulness for design purposesof the data plotted in this manner will be shown later.

Figures 16 and 18 show that, for a given pressure

difference, the optimum spacing remains practically

constant for mmirnum heat tLransfer over a large range

of fin weights. The same is true for the optimum

thickness as shown in figures 17 and 19 at the higher

pressure differences. The optimum spacing for the

ma.simum heat transfer at a given pressure difference

is approximately the same for steel ancl aluminum over

a large range of fin weights.

f.8

~f.4~

“ :L2.$

$/ . o>Gs .8

.6

.4

0 .02 .04 .06 .CA9 ./0 .12 ./4 :16 .18Fin fhickness, t, in.

(b ) 6pdfkd 8c h a r t .

Crit erion , f in wfdtW,ARjP. ~P@ 4 ImX re s O fWa le r.

.3.4

3“

~aG.“\.~ ~

$2 J!!m

sL4

Lo

o .02 .04 .& .C@ Jo .12 ./4 ./6 .J8Fin fhickness, f, in.

(h) Spaowd$Chti

FIGURE Z2.-Opthnum dimen.dons for alrudnrtm flna. Criterion, ti width; AlhP.s@, 4 Inches ofwater.

8/7/2019 19930091804_1993091804


420 R EP OR T NO. 726—NATIONAL ADVISOR Y COMMIT TE E F OR AE RONAUTI CS

SPECIFIEDWIDTH

In certain cases of fm des ign , lin it in g vahws of th e

fin wid th w will be m ore im portan t th an lim it in g vahm s

of ~ or t . One ch a rt for a lum in um fin s in wh ich w is

speciJ led and in wh ich fin weigh t is th e cr iter ion is

shown h figu re 20. In the calculations for figure 20,

U was found to be a maxihmrn when s was 0.072 inchregardless of fin width or tin weight. This figure also

shows that, for every fin width, oIIIy one fin thickness

and b weight give a maximum beat transfer.

OPTIMUM FIN DZSIGNS WITH LIMITED FIN WIDTH

For certain cases in which fin width is limited, it will

be desirabIe to obtain a mtium heat transfer irre-

spective. of fin weight. The fins for the adjacent sur-

faces of the cylinder heads of in-line engines are an

example of this type. In this case, an addition of fin

width may cause a corresponding increase in the length

of the engine,

Curves of maximum ~ when fin width is limited are

shown in figures 21 and 22 for steel and aluminum,

respectively. In these figures, the h spacing corre-

sponding to the peak point of each width of curve

slightly decreiwa as the fin width increases. The

optimum spacing for both steeI and aluminum is approx-

imatesly 0.07 inch for the rtmge of ii.n widths shown.

The fin dimensions at the peak points of each constant-

width curve are the same whether fin space or fin

thicknass is specified.

When fine of different metals are compared on the

basis of width as the criterion, metals of high t@mnaJ

conductivity me obviously superior, For this reason,

copper should prove of definite advantage in applica-

tions where w is limited.

APPLICA~ON OF RESULTS

The following examples are intended to illustrate not

only the use of the material of the report but also

possible improvements in fin design. For simplicity in

the solution of these exampIes, it will be assumed that

the total heat from the cylinder chmgea inappreciably

with change in cylinder tsmpernture.

Two methods of design@ h for anew engine cylin-

der are possible. One method consists in obtaining the

ratio of the hat-transfer coefficient required to cool thenew cylinder to the heat-transfer coefficient of an exist-

ing cylinder from a consideration of reIative power and

size of the cyIinders and then in obtaining a & dasign

that ghws this ratio of the heat-transfer coefficients for

fins Iocated at simiIar positions on the two cylinders.

The heat-transfer coefficients of the fins on both cylin-

ders can thus be detmnined from the data given in

this report.

The second method consists in estimating the

quantity of heat to be dissipated and in using the heat-

transfer coefficients given in this report for obtaining

t h e & dimon&ions. ‘k the fins were-tested un&rso”mc-

what different conditions of air flow than may exisL in

flight- and, furthwnore, as the estimation of the heat

to be..&ssipated is rather indeterminate, the accuracy

of the second method is questionable, In the firs~

method, however, cliffwenccs in flow conditions should

not appreciably chauge the relative heat-t,ransfrr COC41Lcient “of different fins when both cocfficienk arc used .

under the same conclitiona. The first method is tllt,rc-fore believed to be more reliable.

Example 1,—h’t it be required LO lower tho WR1l

temperature of m aluminum-alloy, cylindrical surftiw

havings = 0.142 inch, t = 0.08 inch, u) = 0,6 inch, and

Zl = 7.0 inches from 480° F to 380° F, assuming a

specific weight of the air pad of 0.0734 pound pm cubic

foot, ai air temperature of 80” Y, and u pressure diffm-

ence, Apl, of 4 inches of water. Let it also be aasumcd

that both minimum ~ weight and narrow fin width

are deiirable and that, for manufacturing reasons, s and

t shall not be less than 0.08 inch and 0.03 inch, respec-tively. The final choice. of fin dimensions will be made

upon inspection of the sevcrrd resulting h designs.

& previously stwted, the gmphs of this report arc

for a D of 4.66 inches. Other diamct.ers will affect U

but will not materially affect the fin dimensions. Any

change in U ef fected by changing fin dimensions for

either of two different diametem will cause a propor-

tional change in U for either diameter; this fact will lx

demonstrated in the present example.

The over-all heat-transfer coe.flit.ient for the original

cylinder is obtained from equation (1) m foIIows:

From figure 1, q can be determined from

~s’ka=-f(aaFrom figure 3 at s= O.142 inch, T~p,g=7,6 pounds per

second per square foot.

~7P1@~ = 7.6X0.1422

_ 12X 130X 10-7X4.t16’l’~_670_- --

From figure 1, qs/ k==63,200

=63200 XO.CYOOOO034!l 0.142

=0.1512 Btupcrsqurme

inch per ‘F per hour

#w’=w+g=0.6 +0.04 =0.64 illCh

a.=~==~(2 X0.1512)/(7.66X0.08) =0~703

tanh aw!=O.422 (Sco reference 1, fig. 15.)

‘ = , a : ( ’ + %) t ’ ’ ’1)

=1.03 Btu per square inch per ‘F per hour

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TH E D ESIGN OF J ?lNS F OR .41B-COO IiE D CYLLWDE RS . .

A reduction of tcmperahm from 480° F to 380° F The fin dimensio~s as .determiuecl for a Cl of_1,(37

requires B~iI per square “hch per ‘F per hour, ‘minimums of “

0.08 inch, and minimum t of 0.03 inch are as fol-

lows:

&Iutlon Chart Critcrirm

i~

(K.) & (:.)

Speetdedt (klg. 18 ) . . . . . . . . . We ht . .. . . . . . . . . . o.w; $1 3 “— az 0.03Speci fkda &g.19 ) . .. . . . . . . . . . .. 0.. . .. . . . . . .. . LIO3

1

.24 .03SpeeIt led w dg. 20). . . . . . . . . . . . . . .-. --do .. .. . . . . . .. . . .6s

4.0i2 .03

Specidedt dg.2?(ll . . . . .. . . . . Width . . . . . . . . . . . . .6 {{

.70 .w 6 .03 .SW31M 8 ‘I& 22 (b --------- --ado. _... . . .__ .io .137 .03

O~bdcy~dcr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..- Y . l.tz .m

J’rom the foregoing table, the designer can pick tho

combination of fin dimensions that will best suit his

particular requirements When the values for this

tabIe were calculated, solutions with minimum s=O.08

inch and minimum t= 0.03 inch were. not obtainable

from all charts. It was felt, however, that the inclusion

of the next best solution possible in these roses wouldbe of interest.

The percentage increase in U obtained can now becomparecI with the corresponding increase in U had a

diameter of 7 inches been used:

U for the original cylinder with a 11 of 7 inchw=

(1.f128Btu per square inch per ‘F per hour.

ZTfor solution 4 for a D of 7 inches= 1.28 Btu per

square inch per ‘F per hour.

If all fins are now assumed to be on a 7-inchdiameter

cylinder, the increase in U for the fin desibm of solution

4 over the original design is 38 percent. The corre-

sponding increase in U for solution 5 is 32 percent.

These vrdues check with fair accuracy the increase of

33 percent obtained for a diameter of 4.66 inches.

Example 2.—Let it be required to determine whetherthe fin design of solution 4 of example 1 will be ade-

quate for coding at an altitude of 23,000 feet, if the

same total heat is assumed to be dissipated with o

cylinder-wall temperature of 380° F and a pressure

di17erence Apl of 4 inches of water.

At 23,000 feet, the cooling-air temperature is —23° F,

uncl plg is 0.036s pound per cubic foot. From the fore-

going,

.,

1.28(380–80)

‘Ta’’i’ti’= (380+23)=0.953 Btu per square inch

per ‘F per hour

The weight velocity between the fins is proportiomd

to AP,p@ When p , g = O. 0 7 3 4 pound per cubic foot,

From figure 3, ~’PN=4.1 pounds per second per square

foot when Ap,=2 inches of water, PM= O.0734 pound

per cubic foot, and s= 0.095 inch. The value of U m

determined by equation (1), as in example 1, is 0.973

Btu per square inch per “F per hour, which is greater

than the lr required rmd therefore the fln design is

satisfactory. If the calculated ZThttcl been less than

that required (0.953), a ncw fin design would have

been necessary.

Example 3.—Let it be required to determine how

much the power of a cylinder having the fin dimensions

of the original cylinder of example 1 with a wall tem-

perature of 480° F can be increased without exceedinga walI tem pera tu re of 380° F by su bs t itu tin g a ncw fin

design having a value of s n ot les s th a n 0.14 in ch , of t

n ot les s th a n 0 .08 in ch , an d of w not grea ter th a n 1 .5

inches . k t th e coolin g-a ir tempera tu re rm d th e

pre s su re difference available for cooling be the same as

in example 1 . The possible solutions from the data

of this report, which are for a cylinder diameter of 4.66

inches, are:

‘“”on,))......H.~E “tm~:. . . . . . ----- Sp e”kied t(ag18 ))----------- w ha: . .. . .._.--_2. . . . . . . . ..- Spezlded 6 (El .19 ))-- _..... - . . . . . o------------3... -.. . . . . . Speoh%dw g. 3 0)------------- .-.-. do .. ----------- L64-. . . . . . . . . . Spwitledt(

6 ----------- s ~ified 8 (fig. 22 (b )) ______ ..-_do .__ . . . . ---- L6 :;

For solution 5, U is equal to 2.104 Btu per square inch

per ‘F per hour for a D of 7 inches.

The following equation, which expresses the power in

terms of the cylinder temperature and U, can be deri-red

from reference 8.

where

1 indicated horsepower

Th average temperature o~”er cylinder-wall sur-

face, ‘F

T-l inlet temperature of cooling air, ‘F

T , effective gas temperature inside cylinder, ‘F ‘

n’ an exponent

These calmdations indicate that the new fin design

shouId permit an increase in indicrtted power output of

4X1134”—g2——s

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422 R EP OR T NO. 72&NA’r IOI SAL ADYL50R T C!OMMI’I ’IXE F OR AE RONAUT ICS

almost twice that of the original value before the inch and 0.035 inch, respectively, rind of a maximum

specified temperature limit of 380° F is att ained. -wdue of w=2 inches will be illustrated. Tho six pos

Example 4.—The effect on Z7of decreasing the ~alue sible solutions available from the data of this repor

of s and t used in ~sample 3 to minimum values of 0.07 are:

solution

~1 c“’”on F itiow:iiw

l.-. --_-:-. S~oided t 5g. u (h ) . . ..-.-__ Wel hL_______2. . . . . . . . . - . S~eMedu fl 19(b)..i..-.-..._.80__ . . . .._.-

3----------- S~cl&dec g.20) . . . . . . . . . . . . . .---. do-----------4----------- Spediiedw @. Xl . . ..__.. _.__aO__. .__... ;5 . . . . . . . . . . . Spxjifled i ( g.22 a)) _.- . Z.. .Z TYidth----- -- -- -6 . . . . . . . . . . . S~oiSed a (fig. 22 ))-... ----_ --.. -ti------------- 2

If solutiou 4 is taken as the accepted design, the in-

crease in CTover the original design (U= 1.98) is 45

percent.

Example 6,—Let it be required to determine how

much the heat transfer of the steel barrel of a cylinder

h avin g an s of 0.115 in ch , a t of 0.026 in ch , a w of 0.5

in ch , and a , diameter of 4.66 in ch es can be in crea sed

wh en th r lim it in g dimen s ion s a re: Afin imu m 8, 0.07

inch; minimum t, 0.03 inch; and maximum w, 1.

inches.

With a Ap, pC,/ poof 4 i n c h e s of water, U is 1.08 Btu

per square inch per ‘F per hour for the original cylinder.

Only four solutions are possible from the graphs of this

report because the curve for steel with a specified w

and with weight M the criterion has been omitted fo

the sake of brevity.

Srdution melt criterion (i:)

i

(:) (& (lFd$:in.) (Btn/sq ~.pF/lw)

1----------- spceified i fl~ 16 (b )..__-_-. Tel t---------

[

~----------- %=mffleda fig. 17$------- __%______ :; %j :% :g - i%a--.--.-_-- SpeofUedt fig. 21 a))_. ------- Width .___ . .4. . . . . - -- -- Speci!3eds f ig. .21(b))--- -- -- - ._. .de ----- -- - i2 .Oio .041 .100 ;:%

Solution 4 gives a 32-percent increase of U over the

original cylinder. This increase of G- is, however, ob-

tained at the expense of an increase of fin reight of

460 percent.

The foregoing exa.mplcs illustrate methods of improv-

ing t.]]c fin design of a given cylinder. Another problem,

as has been noted, is the determination of fin dimensions

for a new cylinder design. For practical purposes, thesolution of such a problem maybe determined as follov-s.

From rcfcrencc S it can bc shown that

%’ r 2 ) c 2 1 %k l n ’ E= : ; :: 2 ; :n-here subseripi a dCUOt.CS one cylinder; subscrip~ b

denotes another cylinder; al, inside cylinder-w-all area;

au outside cylinder-wall area; Ta, Z’=l,Z’fl,1, and n’ have

been previously defined in example 3; and v is dis-

placement volume. For simplicity, it will bc assumed

as in the foregoing axamples that the total hen t from

th~cylinder changes inappreciably viith change in cylind-

er temperature and, furthermore, that the ratio of aJaO

is 1, which is justiiied except for thick-wall cylinders.

Then

u=

[ – 1l / u ) =(T ,- T=,),

Vb= (l/ v)b (T,– Ta l).

k’rorn the pressure Werence available for cooling,

U= can be determined from the fin dimensions for an

&isting cylinder from the material of the present report.

me foregoing equation can then be sol-red for Ub from

k n owm va lu es of (Z’fi—T%)= and (l/v)= at the pressure

difference available and from required values o

(T,– T%)~ and (1/w)K The determination of b pro-

portions for obtaining the deairecl heat transfer for the

new cylinder UBis similar to that for the ot her examples

presented.

INCREASING THE COOLING BY USING HIGH AIRVELOCITIES ‘

ln the foregoing examples, impro~-ements in heat

transfer have been mado by @reasing the. effective fin

~9URE8.-VarfeAfon of maximmn bent tmnefer with power requfred for molingCriterion, h weight.

mea. Corresponding increases can be made by using

Klgher air velocities. In references 6 and 9, however, it

has been shown that, from considerations of power

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THE DESIGN OF F fNS F OR AIR -COOLED CYLLWDERS 423

requ irccl for coding ~1 ,” “th e m ethod of in crea s in g th e

fin a rm is grea t ly su perior to th e meth od of u s in g

h igh e r ft ir Ve loc it ie s. Figu re 23 h as been prepa red to

illu s tm te th is s ame poin t for op timum h des ign s . Th e

power requ ired for coolin g vra s ca lcu la ted for th e fin

des ign s givin g maximum h eu t t ran sfer for s evered fin

weigh ts from figu res 16 (b), 16 (d ), 18 (b ), an d 18 (d )

an d is plotted in @e 23. I?or a given Z7, th e power

requ irccl is in some eases th ree t im es 8s grea t for n

pres su re differen ce of 12 in ches of wa ter a s it is for a

pres su re d ifferen ce of 4 in ch es of wa ter for both s teel

an d a lum in um . The fin weigh t correspond in g to an y

w-due of U can be obtained from the original figures.

From considerations of the power required for cooling,

it is thus appmen t that, in order to increase the heat

transfer, a greater effective h area should be used in

preference to increasing the air velocity. The problem

of determining how much the fin weight should beincreased in order to decrease the power requirwl for

cooling depends upon the particular enginwtirplanc

combination involved.

INCREASING THE COOLING BY USING SHORT FLOW’PATHS

In the application of closely spaced fins, a definite

advantage has been noted (ref e renem 4) in making flow

paths M short as prrmticable. Short flow paths increase

the heat transfer because of the lower air-temperature

rise and the higher weight veloc.it ies of the cooling air

for the same pressure difference as Iong flow paths. It

has been noted in the present report that, for the rangeof cylinder diametera and fin widths used on conven-

t iomd aircraft-engine cylinders, the flow path does not

change enough to affect appreciably optimum fin pro-

portions. For very short flow paths, however, the

optimum fin spacing decreases as the flow path

dmreases~ as has been noted in reference 10.

Calculations have been made to compnre the optim-

um fin spacing obtained with aluminum cylihdem

having a flow-path length of appro.xinmtely 8 inches

wit h th e optimum tin spacing. obtained for cylindel=

having u flow-path lmgth of 1 inch. In both cases, the

pressure difference assumed was 1 inch of water- The

corresponding weight velocities were obtained fromfigure 3 for the Iong-path cylinder (APIP=JPO=1) and

from figure2 for the short-path cylinder [(Ap,pU,/pJ/l= 1].

The over-all heat-transfer coefficients for the shor~

path cylinder were calculated from the values of the

surface heohtrtmsfer coefficients at the front of the

cylinders tested in the work reportd in reference 5.

The fin weight was taken as the criterion in these cal-

cultitions rmd a weight of 0.0455 pound per square inch

of wall area was used for both cases.

The foIlowing table gives the optimum spuciugs and

ovcw-alI heat-trwmfer coefficients for the IO%Crmcl the

short paths for the several thicknesses Msurned.

Spegyd t

0:p

.02

:%

1. t “ ”

Op t h n n l ;w I n g

I(B t@q :~F/k)

I

The foregoing tabIe shows that the decrease in tho

length of path from 8 inches to 1 inch reduces tho

optimum spacing to approximately one-half its original

value and increases the heat transfer a little more than

twice its original vwluc. It is thus apparent that short

paths are advantageous anti thut the optimum fin

dimensions are appreciably different for extreme differ-ences in the length of the flow- path. The difimdtics

in the breakiig up of a lo~~ flow path into more than

two paths in parallel presents some practical objections.

CONCLUSIONS

‘17hecharts presented in this report indicate thttt:

1. The fin sparing and the fin thickness for maximum

heat transfer at a given pressure dfierence are pr~~cti-

ca.lly constant for o large range of fin weights, with the

spacing increasing and tlm thickness decreasing at very

low fin weights.

2 . The optimum fin spac~~ n.nd thickness decrease

slightly with increase of the pressure difference.3. For a given h weight, the highest heat transfer

can be obtained with fins of a magnesium alloy. In this

respect, pura coppm mid aluminum-alloy fins are only

slightly inferior to maagnesium-alloy fins rtncl will

transfer several times more heat than steal.

4. For n given fin width, the highest heat transfer

can be obtained with metals having a high ~herma[

conduct ivit y. Of the metals considered, the highest

heat transfer will be obtained when copper is used;

aluminum, magnesium, and steel foIIow in the order of

their respective effectiveness.

1. ANGLEY lIEMORIAL ~ERONALJTICAL LABORATORY,

NATIONAL ADVISORY COMMI~EE FOR AERONAUTICS,

L.4NGLEY FIELD, ~TA., June 28, / 939.

REFERENCES

I. Elisrmann,Arnold E., and Pinliel, Benjamin: Heat Transferfrom Finned Metal Cyl.indemin an Air Stream. Rep.No. 488, NACA, 1934.

.

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424 R EP OR T NO. 72 f iNAT10NAL ADVISOR Y COMMITTE E F QR AE RONAUTICS

2. Schey, Osoar W., and R ollin , Ve rn G.: Th e Effeot of Ba ffles I 7 . Rollin, Vern G., and Ellerbrock, Herman H., Jr.: Prmcurcon the T em p er a t u re I li st r ih u t io n and Heat-~rwefer ]Coefficien& ofFined Cylinder>. Rep. No. 511,NACA, 1934.

3. Schey, Oscar W., and Ellerbrock, Herman H., Jr.: BlowerCooling of Finned Cylinders. Rep. No. 587,NACA, 1937.

4, Biormann, Arnold E.: Heat Tramfer from Cylinders Having

CloeeljJSpaced Fins T. N. .No. 602, NACA, 1937.5. Ellerbrock, Herman H., Jr., and Biermann, Arnold E.:

Surface Heat-Transfer Coefiicienti. of Finned CyIindera.Rep. No. 676, NACA, 1939.

6. Biermann, Arnold E.: The Design of Metal Fins for Air-Cooled Engines. SAE Jour., vol. 41, no. 3, Sept. 1937,pP. 388-392

Drop aaross Finned Cylinders Enclosed iu a Jacket.T. N,

No. 621, NACA, 1937.

8. Pinf@ Benjamin: Heat-Tranefer Procescs in Air:CWlCtl

Engine Cylindere. Rep. No. 612, NACA, 193S.

9. Campbell, Kenneth: Cylinder Cooling and D~ag of ltadialEngine Installation. SAX Jour., vol. 43, n o, 6, Dec.

1938, pp. 515-527.

10. Brevoort, Maurice J.: The Effect of Air-Passage Lcygth ON

the Optimum Fin Spacing for Maximum Cooliug. T. N.

No. 649, NACA, 193S.

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