REPORT No. 726

THE DESIGN OF FINS FOR AIR-COOLED CYLINDERS

By ARNOLD E. BIERMANN and HERMAN H. ELLERBROCK, Jr.

SUMMARY

h analys-icwa8 made to de~ermine the proportion oj

fcn.s made of aluminum., copper, magneeium, and steel

necessay to dimipate mazinaum guantitie8 of heat for

different $n widths, jin weights, and air-$ow conditions.

me analysia d80 concern8 thz determination of th.+?opti-

mum jln proportions when epecijied limits are placed on

thefin dimension. The calculation of the heat$ow in the

jns is based on an experimentally ~erijied, theoretical

equation. The surfa~ hd-tran8jer coe#ieients used

with this equation were taken from preciously reported

experiment.

In atiition to the pre8entdion of $n-design informa-

tion, this investigation show8 that optimum jin dimen-

sions are inapprem”ably a$eeted by the di~erences in air

jlow that are obtained with dij$erent air-jlow arrange-

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

path. For a p“wn fin weight, the highest heat tran8fer

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

copper and aluminum-allq$n8 are only 81ightly inferior

to magnesium-alloy jtns and w“ll diwipate 8ewral times

more h~at than steel.

INTRODUCTION

Previous investigations on the subject of coolingsurfaces by means of metal fins have shown that theheat dissipated can be expressed fairly accurately byau equation involving the tin climensions, the thermalconductivity of the metal, and the surface heat-transfercoe5cient q (references I to s). Experimentalvalues of q have been determined for a wide range ofair-flow conditions (reference 5). From the infor-mation previously obtained, it is possible to mlculat ethe over-all heat transfer of finned cylinders in the rangeof generaI interest.

The problem of b design for any one air-flow cnndi-tion involves the determination of the fin proportionsthat wilI give the greatest heat transfer for (1) a givenweight 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 shownthat, for every vaIue of fln weight zmd air-flow oondi-tion, only one particular fin design will give a mmsimumheat flow.

In reference 6, flndesign information was presentedfor one cyIinder diameter and one baffle arrangement.The more complete data of the value of q presented inreference 5 make it possible to widen the range of fin-design data. The object of the present report is togive iindesign information for severaI conditions of airflow, different cylinder diamete~, and severaI metals.The criterions for excellence of & design have beenbased upon the maximum heat transfer for a given fin-weight and the ma.ximurn heat transfer for a given fhwidth.

q.,

specific weight of tin material, pounds per cubic inchcylinder diameter at fin root, inchesthermal conductivity of metal, Btu per square

inch through 1 inch per hour per ‘FthermaI conductivity of cooling air, Btu per

square inch through 1 inoh per second per ‘Fequivalent length for straight tubo (PRJ, feetweight of fins, ponncls per square inch of outside-

wrdl measurface heat-transfer coefficient, Btu per square

inch total surface mea per hour per ‘F tem-perature di.Rerence between surface and irdetcooling air

surfaoe heatAransf er coefficient, Bt u per squareinoh 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

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REPO13T m. 726—NATIONAL ADVISORY COMMITTEE m~ AERONAUTICS

a.vcwrbgethickness of ihs, inchesover-all heat-transfer coefkient, Btu per

square inch outeidc-.wall mea per ‘F tem-perature diEerence between cylinder walland inlet cooling air per hour

velocity of cooIing air, feet per secondfin width, inchesw+ t/2, inchesacceleration of gravity, feet per second per

secondspecific weight of cooling air in front of cyl~-

cler, pounds per cubic footspecific weight of coo~ air in renr of cylin-

der, pounds per cubic footaverage 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 openingof baffle

pressure. cliffwence across cylinder, inches ofwater

preasurc difference ca~sed by loss of velocityhead from exit of skirt of bafHe or jacket,inches of water

total pressure difference ncross set-up (Ap, -tApJ, inches of water

ANALYSIS

In the cooling of engine cyh.ndem, two outatanclingrequiremmts must be considered. One requirement t isto protect the surfaces; the other is to reduce the knockor the tendencies to preignition caused by hot surfacesthat come in contact with the fuel mixture. Thelubricated working surfaces must be kept at a suffi-ciently low temperature to insure the maintenance ofan adequate oil film. High piston and cylinder-walltemperatures usually cause sticking of the piston ringsand rapid wear of the piston rings and the cylinder wall.

An additional problem of cooling is the prevention ofundue distortion of the cylinder barrels, such as mightbe caused by uneven temperature chstribut,ions. Al-though difllculties arising from thermal distortion ofcylinder bamls have been alleviated to some extentwith specially ground pistons, it appears desirable toretain round cyhnders by means of a uniform or other-wise satisfactory temperature distribution.

Of the several available methods of securing untiormtemperatures around the cylinder, two methods are ofparticular intered. One method is so to distribute theM ect,ive fin area as to ficltieve the desired temperature

distribution. The other method is to COUM t.hc tiirvelocitiw 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 maximumover-all heat transfer otherwise obtained for the samefln wei@t. In experimental work, it has been foundthat 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 inover-fall heat transfer. W’lmn the exhaust valv~ andthe piston crowns are somewhat centrally located withrespect to the finned surfaces that cool them, these partsgeneralIy are better cooled when a maximum over-alIheat transfer of $he finned surfaces is obtained at theexpense of a uniform temperature distribution. In thepresent report, emphasis has therefore been placed onproviding fins for obtaining high over-all heat. transferrather than on securing uniform cylinder temperatures.’

The over-all heat-transfer coefficient 27 has I.weu cal-culated from the following equution, which was derivedin reference 1:

‘=*i:(l+%)t’’nl’(‘1)where a.= ~2g/k~t rmcl k= is the thwnml ~onductivityof the metal (2.17 for steel; 7.66 foi. aluminiin Y idToy;18.04 for copper; and 7.54 for magnesium aIIoy). Illthis report aluminum and magnesium alloys arc refwvdto 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 taperedfins, 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 surfmrheat-trmsfw coefficiiwt q can be correlated for cacfiair-flow arrangement in terms of functions cleflning nsingle curve and involving the flu dimensions, thecylinder 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, forsylinders enclosed in a jacket and cooled by a blower,—.

(3)

where T’ is the velocity of the cooIing air between hRns. Figure 1 shows the variation of g with fin findcylinder dimensions and air-stream characteristics forLhe four air-flow arrangements.

THE DESIGN OF FINS FOR AIR-COOLED CYLINDERS 403

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REPORT NO. 726—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

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THE DESIGN OF l?lNS FOR AU+COOLED CYLINDERS 405

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406 REPORT NO, 726—NATIONAL ADVISORY COMMITT3E FOR AERONAUTICS

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THE DESIGN OF FINS FOR AIR-COOLED CYLLWEIW 407

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FIGURE2.-Effeet of weight vefoeity of emhg sir on pressure difference per unitlength of path h eew?.rsffin sIXJoes.

For any one air-flow arrangement for which thepressure differences across the cylindem are available,g can be determined as a function of the pressuredifference instead of the weight velocity of the coolingair. Previous tests of cylindem enclosed in jackets andcooled by a blower (references 5 and 7) have shown thatthe pressure difference across the cylinder Apl for agiven iln space ancl weight velocity is proportional tothe length of flow path 1 which, in turn, is determinedby the cylinder diameter, the k dimensions, and thejacket design. The pressure dMerences across thecylinders for various lengths of path, fin spaces, andweight velocities are presented in figure 2. From thisfigure, it. can be shown that

3P, p../pO=fa(s,f,w,D,I“pfl] (Jl

ml, asAP: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

where Tf’m is the specific weight of the fin material(0.282 for steel; 0.101 for ahuninum Y alloy; 0.322 forcopper; and 0.0648 for magnesium alloy).

foal I I I I I I I lit I Ill 8. in.- .CY -.04. –

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

FIGURE.3.-EEeet of weight WxIty of moMng sfr on prmure difference aems12Med oylfnders for semml fin spares. Awrege dr.r Widtk 0.825 fneh; cyllnderdfameter, 4.04fnrhes; !3n tbfekness, O.CCMInch: blmer-socdbg set-up.

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

u=f7(8,t,w,D,T”p@) (8)

When fin weight is more important than fin width, wcan 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 expressedas 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 inthis report is generally to hold const tint the vahm ofthe variables tlmt are specified by the desi=m conditionsand, from a plot of U against values of the remainingvariables, to obtain the rest of the dimensions thatgive maximum heat transfer.

The design of fins for given values of M andAputil p.,/% is more diflicult than for constant valuesof M and J’Plg bec~use the length of the flow path andthe losses from the bidlle exit enter into the calculations.Both the length of the flow path and the exit los&esdepend upon the fin dimensions and the baffle dwign.A method of designing fhs for a constant pressuredifference, using an average length of flow path andassuming that all the pressure di.Rerence is avaiIabIefor cooling, would considerably simplify the calcula-tions. Computations have shown that the difference

408 REpORT No. 726—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

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

.-

betwwm the weight velocity based on the exact flowpath and including the exit Iosscs and the weightvelocity bnsed on an average flow-path length and neg-lecting the exit losses has very little effect on the con%ctfin proportions. Consequently, figure 3 has been usedto 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 crossplot of figure 4 (a). EMer part of figure 4 clearlyshows that, for one pnir of values of s and t, the heattransfer is a maximum. The peak values of the curvesof coustants shown in figure 4 (a) and of similar curvesplotted for other values of M and Aplp.,/Po are shownin subsequent figures and are Iabeled “specified s”curves. Similarly, the peak values of the curves ofconstant values of t shown in &ure 4 (b) and of similarcwwes plotted for othm values of M and Aplpa,/po areshown in subsequent figures and are labeled” specified t“curves. The specified s curves are used. when alower limit is set on the value of s and t!~e spec.hied tcurves are used wheu a lower limit is set on the valueoft.

For given values of M, D, Ap,pa,/po and a specifiedvalue of t, the valuo ofs for which U is a maximum can

also be found by setting the derivative of U with respectto s in equation (10) equal to zero and solving theresulting equation. In order to obt~in an expressionfor the function in equation (10), an equation wouId

,

have tc.be fitted to the.curve in figure 1.. Making thesubstitution. previously indicated in obtaining equa-tion (10) would result in a complicated relationship.The -ivo~kinvolved in salving the resulting equation forthe optimum value of s would be .considcrahly morethan the work of obtaining figures 4 (a) and 4 (b) andpicking 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 1md 3 “imd equations (1) and (7). For each value of sand tin figure 4, the associated valuo of .W cm IN crd-culated from equation (7). The heat-transfer codi-cient lZ can also be plotted against t for various valuesof w and the optimum value of t can be obtained forthe -urn vaIue of U for each value d spccificd w.In thie case, the value of s is unrestricted and may beobtained from equation (7).

Figqqe 4 shows that, for given values of .Aplpa,/po andM, definite values of s and t exist for which U is ammiimum. Although these vahw.s of s and t nmy beoutside. the practioabIe manufacturing range, a l~tidcrange Qf6ns becomes available for values of U 5 percentbelow the maximum. Figure 5 is a cross plot of figure4,4 having been ylotted against i!for several percentageof maximum U. It is evident from figure 5 that asingle -pair of values of s and t represents tho optimumfin design as indicated by 100 percent U. Ii case the

manufacture of these fins is impracticable because .sandt are too small, some sacrifice in U must bo made if thefin weight is to remain constant. I?or example, when

THE DESIGN OF FINS FOR AIR-COOLED CYLINDERS 409

U is decreased to 95 percent of the mtium value, -aninfinite number of pairs of values of s and t wiII givethis heat tramsfer as shown by the points on the 95-pert.ent Iine. When the WUm va]ue of 8 is limitedby mrmufmturing reasons, the points on line A givethe 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 ofpractical interest on these curves because the h inthis region give the incLicated heat transfer with thehighest values of s and t. The specified s Cmves in

this report correspond to the values of s and t alongthe line A of graphs of the type shown in figure 5; thespecified t curves correspond to the vaIues of s and talong the line B of graphs of the type shown in figure5. As is evident from figure 5, reasonably close approx-imations of values of g and t lying between the specfieds line (A) and the speciiied t line (B) can be obtainedby assuming a straight Iine between correspondingheat-transfer points on lines A and B. Similarly, inthe charts presented later in this report, values of findimensions lying between the specified s and the speci-fied t charts may be appro.ximatecl by assuming alinear relationship between similar dimensions on eachchart.

I’ery often the fin width is of more importmce thinthe iin weight. In certain types of engine, such as

in-line engines, the small distance between cylindersplaces a restriction on maximum k width. Fromequations (6) and (8), U can etident.ly be written as afunction involving width instead of weight.

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

For a given cylinder diameter, flu width, and pres-sure drop, U is again evidently a function only ofs andt and curves similar to those in figure 4 can be plotted.The curves of optimum I-In proportions for specified srmd t are obtained in a manner simiIar t.o that preciouslyindicated for the case in which weight was the criterionand are shown Iater in the report. The specified gcurves represent the best fins that ful.fll.lthe restrictionsplaced on $ and w; the speciEied t curves represent thebest fibs that fulfill the restrictions placed on t ancl w.

When the flndesigg information is applied to enginecylinders, the tin proportions may be determined froman average ~alue of the surface heat-transfer coefficientq for an entire cylinder circumference or may possiblybe determined for wch portion of the cylinder circum-fermce from the local heat-transfer coefficiertts. Amost airc.raf t-engine cylinde~ are composed of sevimdcylindrical areas, it is believed to be most practicablein applying the fidimension information to considereach of these areas separately. The outside-wall sur-face of a conventional cylinder can thus be consideredas five separate areas: The barrel, the lower head, theintake-valve stack, the exhaust-valve stack, and thecurved surface betvreen the intake-vahe and the

exhaust-vaIve stacks. Further retiemeut that mightbe obtained by the consideration of smaller areas isbelieved unwarranted in view of the impracticability ofchanging flu sections and spacing from one point toanother around a cylindrical surface.

In heat-transfer investigations, the heat-transfercoefficient is customarily based on the differencebetween the surface and the average fluid tempera-tures. The problem of determining fin proportions is,however, -rev much simplified ~“hen the coefficients arebased on the intake cooling-air temperature. V7hm the

t, in.

FIormE &-VarMon of a and t for several percentages of maxhnnm heat-transferoxt%ckt. Constant M and APIP.JP,.

mer-all heat-transfer coefficient U is calculated frombe surface heat-transfer coeficimt based on the aver- _ .._.~ge air temperature, it is necessary to determine thetemperature 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

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

portions of a range of conventional aircraft-enginecylinders. The diameter of the valve stacks is usuallymuch less thrm 4.66 inches whereas, except for smallcylhders, the barrel and the head diameters are larger.In cases where the length of flow path is greatly diflerentfrom that for a 4.66-iich-diameter cylinder, correctionscan be made for differences in the temperature rise of thermoling air. Most attempts to obtain g-mater accuracythan can be obtained by using the data for the 4.66-inch diameter are unwarranted because calculationshave shown that an appreciable chmge in flow-pathlength is required to effect much change in fin propor-tions although such a change in flow-path length willchange the absolute values of. U. In all calc~ationsfor determining g, the viscosity of the cooling air pwas assumed to be 130X 10-7 pouncls per second per footand the thermal conductivity k= to be 3.4X 10-7 Btuper inch per second per “F. The effect of variation ofboth ~ and kc on the optimum fin proportions with the

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

temperature range encountered is inappreciab~e, andthe assumed valuea give resulkthat are accurate enoughfor alI practical purposes,

An exact determination of optimum fin proportionswould require a diilerant solution for evely condition ofair flow that might be mused hy differences in baffle andcowling design. Furthmnore, in order to cover theproblem completely, it would also be necsssary todetermine fin proportions for variations of f3n weight,fin dimensions, pressure drop, weight velocity, andpower to COOL It has been necessary, in order to limitthe scope of the present report, to choose for the finalcalculations a limited range of conditions believed to beof the greatest practical interest. The problem hasbeen simplfied, where possible, by eliminating several ofthe variabh?s having little or no effect on the bdimensions. .

The determination of the most important of the fore-going vrtriablei will depend upon their application.For example, when the coding-air flow through anengine is. induced by the movement of the airplanethiough the air and the slipstream from the propeller,

the pr~ure di.flerence available for forcing the airover the cylinders may be insufficient for cooling at thepower output clesired. In this case, it may be desirableto aclcl fin weight to obtain sufficient cooling withthe limited value of the pressure drop. When thecooling air is supplied by a blower, a wide range ofpressures may be available and the power required forcooling may be equal]y as important as the pressuredrop or the weight of the fh.

The_determination, of optimum fin dimensions forconstant weigh~velocity and power conditions is ofin~erest only in special cases. If the cooling air is

furnished by a blower and the power required forcooling k used as a criterion of fin dosign, tho totalblower power and not the power required to force tthcair across the cylinclcr should be used. The cfficicmcyof a blower is particularly dependent on tho prc%surcdifference used and, if fins are designed for a constantcooling-air power, the pressure clifferenco required maybe such as to Liob a my-y inefficient part of LLCpowercurve of the blower.

The resultg of calculations to dotwmine optimum findimemions for constant pressure drop, constrmt weightvelocity, and constant power conditions show thnt, illgeneral? the optimum fln spnco or thickness chaJigcs

with t@e diflercnt bases. The desired values ofs, whcmt is specified, are somewhat smdlcr for constant weigh Lvelocity and power than for constunt pressuro difference.Whens is specified, the vrducs of tmo generally lowestfor the constant weight-velocity condition md highosLfor thp constant power condition.

Although the optimums and t I-Wesomewhat differm Lfor the conditions of constant-prrssure diffcrcncti andco~tant. po~~’er to ~ol PI, th~ diff~rence b~h~e~l] thl!heat transfer obtained for n given power to cool and theheat transfer obtained for a given pressure drop is vmySlightl_m is shown in figure 6. These curves wero ob-tained by determining the optimum fin designs forseveral constant assumed powers and prcssuro diffw-ences. In these calculations, tho fin weight \vas heldconstant for each metal. The slight dillerence in Ushowm by these ourvw mrdws the design of fins from Qpower-to-cool basis of little interest. An advrmtagc offins designed on a pressuredifference basis is hit theoptimum thickness and space me greater than for h

designed 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 basedon a constant fin weight rmd a constant air-streamvelocity. These results show that diffmnccs in airflow caused by these rLir-flow rn-mngomente do notmaterially affect the best h dimcnakms.

Frcirn the foregoing results, it is believed that fin-design information for cylinders enclosed in a jacketwill apply with reasonable accuracy to otlwr conditions

...

THE DESIGN OF FLYS FOR AIR-COOIiED CYLINDERS 411

of flow as ctiused by different baffle arrwgements. Asthe power-to-cool and the weight-velocity bases are ofinterest only in special cases, the fidesign data sub-mitted in this report have been calculated for constantvaIues of pressure difference across a cylinder-jacketarrangement. Although the material of this reportwas 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 complicatedshapes such as cylinder heads.

The conditions covered in this report with il.u weightas the basis of design m-e listed in table I and the con-ditions with fin width as the b8sis of design are Iistedin table II.

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

+

steel.........-.

Alurdnum..-.

t 1, “n. I ,,:1,3466 1259

z }1,4,&12

40643“.”* “.U

.20 .0.1

i$ .s -&o* 0.0040.2a .om

.0’55I

1,48,12i~ . lm?

{

0.02

+ x!!!!!’

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

M8gndum._. 460 1{.Z1I“----I.01.30.0778}

4R .12Q0 I

TABLE II.— CONDITIONS FOR WHICH CALCULATIONSWERE MADE USING FIN WIDTH AS CRITERION

F!n rmterlal

Steal..........

Alurdmun. ----

pe.ssed#

of water).

4

4

The values of the over-all heahtransfer coefficientsU for the various conditions listed in these tnbles andfrom which curves such as shown in figure 4 -mm drawnhave been tabulated in nine tabks, which are availableupon request from the National Advisory C!ommittecfor Aeronauti~.

The values of the peaks of the curves of the typeshown in figure 4 were used in plotting the fial chartswhich are presented Iater in the report and in whichboth fin -weight smd h width are used as criterions.

The peaks of some of the curves of the type shown infigure 4 are fairly ffat; the values of 8 and t may con-sequently be varied somewhat without changing theheat transfer.

OPTIMUhl FIN DESIGNS WITH LIMITED FIN WEIGHT

SPECIFIEDSPACE AND TEICKXESS

Figures 7, 8, 9, and 10 show the relation between theoptimum fin dimensions and U den s or t is specifiedfor both steel and aluminum with fin weight as theoriterion. As each graph is for a constant weight ofmaterkd, it is apparent that the peaks of the pres-sure-dMerence currcs represent the Iin designs that willgive the maximum heat transfer for the given weightand pressure difference. The values of U, s, t, and wat the peak point are the same for both the specifieds 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 verynoticeable, especially for steel at 10-wfin -weights andlo-iv pressure differences. In geDeraI, the peak pointof U occurs at smaller valuw ofs and t as the pressuredifference is increased. The h indicated by the peakpoints, particularly for aluminum, are generally toothin for practical use. Although the value of s can bevaried over quite a range without affecting maximumU, itmay be desirable in some engine instalhtions tolimit &to small values in order to hnve a minimum vol-ume of air passing through the engine cowling.

Information similar to that already presented forsteeI and aluminum fh is ahoycn for copper and mag-nesium fins for a pressure difference of 4 inches ofwater in figures 11 to 14. Copper is of particular in-terest owing to its high thermal comluctitity, and theuse of magnesium is significant because of its low weightcombined with fairly good thermal conductivity. Acomparison of the proportions of fins of steel, mag-nesium, aluminum, and copper for mwtium heattransfer shows that fins of metals hnving a high thermalconductivity are a~t.remel-y thin. A comparison of themaximum heat transfer obtainable with steeI, alum-inum, copper, and magnesium is shown in @u.re 15 ford&rent fin -weights. These data were taken from thepeak points of the curves of figures 7 to 14. Magnesiumalloy of the thermal conductivity chosen hns a slightadvantage over the other metals; whereas copper andaluminum, although somewhat less effective thanmagnesium, are equally good, both being several timesas ef%ctive as steel for a given fin weight. A plot similarto that of figure 15 could be made showing maximum Z7against width of fins as the criterion. Such a plotwould show a defhit e advantage for copper withd-uninum, magnesium, and steel following in the orderDf their relative effectiveness.

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

Lo

~.9<L>.8..b$7 .“:

c..J i

(a) I 13’4 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) FIDweight M, 0.011Spound per Wuare hmb wall area. _ lb) Fin wotght M,0.0&34pound pm square [rich wall area.

.\Ri\ ‘ .- ‘\{ I

Spcn? between fins, s, in.

“$: “.““~/6.m”

s

.—

~.. ..{.—-... . ,...=. -.:..

.+. ..-, .. ..-,:. :

. :,

.

... -A

. . .

----- --:.1 *.

.-. .. .

(c) Ftn woIght M, 0.1289pound per square Inch walI area. (d) Fln weight Jf, 033S4 pound m Waare incII wa~ mea.

FIQCRE7.—Opt1mmn dimensions for steel Llnswith spooMed M thicknees. (Yiterlon, fin weight.

/.f

Lo

.$.9L“>4.8

$3.7Gb“6

5

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

..7 ~m

t4!-- . ~ =

- ~ -M1L . _ .-:.._., –-fI ~10

‘7 ).1 - 04

’50 ....038 .0!6 .,.Q24 .cW .040 .048 ‘.056. .&74...072Fin 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 for steel fins with apodlied fln spaca. Criterion, fin weight.

THE DESIGN OF FINS FOR 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)FintieigM-V,0.1239pound per square fnoh WEUsres.

FrorRE S.-Optimum dimensions fm steel 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 weight .11,0.0010 pound per squere Inch wait rues

S#boce between II., s, in.

(c) Fin weight M, 0.M55pound per square inch wali ares.

.—---

,,. ..

Fin ihickness, i, in,

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

SPeCIFIedfin space. CrfterIon. flu weight-Continued.

i(-

h) Fin weight Jf, 0.IE02 pound pcr squsre inch waH area,

.._ -.=

Spoce between fibs, s, in.

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

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

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) Finweight 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, 0.046Spoundpersquarehch wallarea.

\“a

I 1,T-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 WUUarea

--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 .iI I I 1 I I I i I 1 I I I I I 1 I J-- ,- ,,. --

?4 .28 ,32 .36 :40Space 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@ IM WtIfght;AW.JPW 4 hOhM OfWateI. rion, dn wefght; Afhp. Jpw 4hohesofWa&.

THE DESIGN OF F1.IW FOR AIR-COOLED CYLINDERS 415

3.6 I 11/ II;M, lb fsq in.wall oreu

~zJ30-.30

111111r~\ 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

kll. rRE 13.—0pthnurrr dhnen.s.sne for rnegn~run EmgwftJI spwf&j fsn thI& II(W.Crltcz[on, En weight; ARW.Jpe 4 Inches of watm.

ItllM M, lb /sq h.- —

A .20, WiYfl m-es.

p+m

>- [8

[4

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

Fin fhickness. t. in.

FIGURE 14.-Optlrmun Ahnensfons fcw rnagneshmr fins with specIEed h s~eeCrfterion, fin weight: Afrw. Jp,. 4 Incbos of water.

.

0 .04 .08 J2 ./6 .20 .24 28Fin weI&r if M. Ibjsq in. WU17ureu

—

..

FIGURE]6.—VarfatIon of maxluurm o.m?r-e.11heat-transfer codcfantwithmfght offins for several flu materhls. Crtterkm, fin wdght; AIM.JP@ 4 frrchrs of water.

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

-- .m----,:--,I

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

(a) Prmsuro dh%reneeAPIP.JPW 1 hICh of Water.

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 .06 percen< of

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

-—k”,,, ,

.06 1 ‘:0 ‘1~ “-? “ ‘ .’ I I I I I

1 [111111”+?)+

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

(h) Pressure rltfkwnee APIP.JPC, 4 Inch= of waler

“-./6

14

,.,.

.. ..:.

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

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

(c) Pressure differenwAPIP.W’PW 8 hcks of wafer. (d) Prwuro difference AfhP. JPr 12 InchcwOfwatOr.

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

Fin weigh t M, Ib/sq m. wall oreo

“l%

.05

e--.04%’

.m

-:-“.02

—.

I I I I 1 I 1 I 1 t , , , iRi

.08 ./2 ./6’ .20 .24 .28 .32 .38Fi% weight, M, Iblsq in ml are~

.—

.,

(a) Pressure dlfferem.e AfAP@j 1 tneh of Water, (b) Pressure ditlerenm AIJIP.JPV 4 Idles OfWaLW.

FIOUEE 17.—Optfmum dimensions of steel dns for mrlous f3ere3nW.geJ of M81ht3UIIi heat tran3for. Speelfmd fin spaea erlterion, fin wdght.

THE DESIGN OF FINS FOR ATR-cOOLEDC1-LINDERS 417

Fin weiqht, Afi lb/sq in wd oreo

.061 I 1 1 I , I

1 1 h 1 1 1 , I i

.U[ I

- ki-0 .CU .08 J2 J6 .20 .28 .32

fin weig$}, M, Ib/sq ik24woJ ores.36 ““”

(c) Pressure difference APIP.Jpt, 8 inches of water. (d) Pressure dlfferenw APIP.JPW 12 khf!s Of Water.

FtaoR~17.--Optfmum dimenskwrs of steel IIDSfor varkme pereentagea of maxtmrrnr heat transfer. Speef6ed 5 SPX erIMon, fin wetght-Continued.

(a)

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

(a) Prmure dffTerence APIP..JP,, 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 ~.fI !

# , I

-U4Q .C12 .04, , 1 1 1 I c 1 I I I I I

.06 .08 ./0 ./2 ./4 -f6 .18fin weigh( M, fb/sq in, wdi oreo

(b) Pressure dMerenee APIP.JP6, 4 fnches of water.

.20*

J 4-” I t~..

A7/’i .15’+ ,’ ‘“-. =90

~- 85 I–/00 -

,i - II I 1 , I ,

II

(d) -1,

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

—

.

(c) Pressure difference APIP.JP*, 8 fnohee of water. (d) Presmre A1.19erenrt ApW.JP@ 12 hmhee of water.

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

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) pressuredlflercneeAPAJPO, 1 kh Of~’ater. (b) Pro.wre dhTereneo APIP../Pu, 4 Inches OfWater

. 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@.

FIGURE 19.—Optlmum dirnenslmrs of ahnnfnum fins for verimrs peramtsges of maximum heat trar.rs4er. Spc.dfled Su space; M.erion, drr wclght.

THlil DESIGN OF FINS FOR AIR-COOLED CYI.JNDE~ 419

,

:4

:0

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

Fin thickness, t, in.

FIOGRB21.-Opthnum dhnensfom for alnmimnn fins with spechled dn width.Criterion, tin weigh~ APIP.dPw 4 fnches ofwater;opthnmufi SPW@Wnstit at0.072Inch.

S@ze between fins, s..A

(a) Speclflad tchart.

FIGUaE 21.--Optimum dhnensfone for steel 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) Spsoiaedt 0h8rt.

The data in figures 7 to 10 have been cross plotted iniigures 16 to 19 with the fin weight as abscissa. Theseplots show the lin dimensions for maximum 27 and alsothe fin dimensions when certain percentage reductionsin maximum Z7 are allowed in order to obtain easilyconstructed 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 pressuredifference, the optimum spacing remains practically -constant for mmirnum heat tLransfer over a large range ___of fin weights. The same is true for the optimumthickness 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 differenceis approximately the same for steel ancl aluminum over

a large range of fin weights.

f.8

~f.4~

“:L2.$

$/. o>Gs .8

.6

.40 .02 .04 .06 .CA9 ./0 .12 ./4 :16 .18

Fin fhickness, t, in.

(b) 6pdfkd 8chart.

Criterion, fin wfdtW,ARjP.~P@ 4 ImXres Of Waler.

.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 of water.

420 REPORT NO. 726—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

SPECIFIEDWIDTH

In certain cases of fm design, liniting vahws of the

fin width w will be more important than limiting vahms

of ~ or t. One chart for aluminum fins in which w is

speciJled and in which fin weight is the criterion is

shown h figure 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 alsoshows that, for every fin width, oIIIy one fin thicknessand b weight give a maximum beat transfer.

OPTIMUM FIN DZSIGNS WITH LIMITED FIN WIDTH

For certain cases in which fin width is limited, it willbe 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 anexample of this type. In this case, an addition of finwidth may cause a corresponding increase in the lengthof the engine,

Curves of maximum ~ when fin width is limited areshown 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 curveslightly decreiwa as the fin width increases. Theoptimum 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 finthicknass is specified.

When fine of different metals are compared on thebasis of width as the criterion, metals of high t@mnaJconductivity 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 notonly the use of the material of the report but alsopossible improvements in fin design. For simplicity inthe solution of these exampIes, it will be assumed thatthe total heat from the cylinder chmgea inappreciablywith change in cylinder tsmpernture.

Two methods of design@ h for anew engine cylin-der are possible. One method consists in obtaining theratio 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 andsize of the cyIinders and then in obtaining a & dasignthat ghws this ratio of the heat-transfer coefficients forfins 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 inthis report.

The second method consists in estimating thequantity of heat to be dissipated and in using the heat-transfer coefficients given in this report for obtaining

the & dimon&ions. ‘k the fins were-tested un&rso”mc-what different conditions of air flow than may exisL inflight- and, furthwnore, as the estimation of the heatto be..&ssipated is rather indeterminate, the accuracyof the second method is questionable, In the firs~method, however, cliffwenccs in flow conditions shouldnot 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 WR1ltemperature of m aluminum-alloy, cylindrical surftiwhavings = 0.142 inch, t = 0.08 inch, u) = 0,6 inch, andZl = 7.0 inches from 480° F to 380° F, assuming aspecific weight of the air pad of 0.0734 pound pm cubicfoot, ai air temperature of 80” Y, and u pressure diffm-ence, Apl, of 4 inches of water. Let it also be aasumcdthat both minimum ~ weight and narrow fin widthare deiirable and that, for manufacturing reasons, s andt shall not be less than 0.08 inch and 0.03 inch, respec-tively. The final choice. of fin dimensions will be madeupon inspection of the sevcrrd resulting h designs.

& previously stwted, the gmphs of this report arcfor a D of 4.66 inches. Other diamct.ers will affect Ubut will not materially affect the fin dimensions. Anychange in U effected by changing fin dimensions foreither of two different diametem will cause a propor-tional change in U for either diameter; this fact will lxdemonstrated in the present example.

The over-all heat-transfer coe.flit.ient for the originalcylinder 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 persecond per square foot.

~7P1@~ = 7.6X0.1422_ 12X 130X 10-7 X4.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

THE DESIGN OF J?lNS FOR .41B-COOIiED CYLLWDERS . .

A reduction of tcmperahm from 480° F to 380° F The fin dimensio~s as .determiuecl for a Cl of_ 1,(37requires 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 0.03Specifkda &g.19 ) . . . . . . . . . . . . . . 0. . . . . . . . . . . . . LIO

3

1

.24 .03SpeeItled w dg. 20). . . . . . . . . . . . . . .-. --do .. .. . . . . . . . . . .6s

4.0i2 .03

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

.70 .w6 .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 thocombination of fin dimensions that will best suit hisparticular requirements When the values for thistabIe were calculated, solutions with minimum s=O.08inch and minimum t= 0.03 inch were. not obtainablefrom all charts. It was felt, however, that the inclusionof 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 adiameter 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 persquare inch per ‘F per hour.

If all fins are now assumed to be on a 7-inchdiametercylinder, the increase in U for the fin desibm of solution4 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 of33 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 thesame total heat is assumed to be dissipated with ocylinder-wall temperature of 380° F and a pressuredi17erence 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 proportiomdto AP,p@. When p,g=O.0734 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 poundper cubic foot, and s= 0.095 inch. The value of U mdetermined by equation (1), as in example 1, is 0.973Btu per square inch per “F per hour, which is greaterthan the lr required rmd therefore the fln design issatisfactory. If the calculated ZThttcl been less thanthat required (0.953), a ncw fin design would havebeen necessary.

Example 3.—Let it be required to determine howmuch the power of a cylinder having the fin dimensionsof the original cylinder of example 1 with a wall tem-perature of 480° F can be increased without exceedinga walI temperature of 380° F by substituting a ncw findesign having a value of s not less than 0.14 inch, of t

not less than 0.08 inch, and of w not greater than 1.5

inches. kt the cooling-air temperature rmd the _____

pressure difference available for cooling be the same asin example 1. The possible solutions from the data

.-

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

‘“”on;,)).......H.~E “tm~’“ :i1. . . . . . ----- Spe”kied t(ag18 ))----------- w ha: . . . . .._.--_2. . . . . . . . ..- Spezlded 6 (El .19 ))-- _..... - . . . . . o------------3... -.. . . . . . Speoh%d w g. 30)------------- .-.-. do . . ----------- L64-. . . . . . . . . . Spwitledt(6----------- s~ified 8 (fig. 22 (b)) ______ ..-_do .__ . . . . ---- L6 :;

For solution 5, U is equal to 2.104 Btu per square inchper ‘F per hour for a D of 7 inches.

The following equation, which expresses the power interms of the cylinder temperature and U, can be deri-redfrom reference 8.

where

1 indicated horsepowerTh average temperature o~”er cylinder-wall sur-

face, ‘FT-l inlet temperature of cooling air, ‘F

T, effective gas temperature inside cylinder, ‘F ‘n’ an exponent

These calmdations indicate that the new fin designshouId permit an increase in indicrtted power output of

4X1134 ”—g2——s

422 REPORT NO. 72&NA’rIOISAL ADYL50RT C!OMMI’I’IXE FOR AERONAUTICS

almost twice that of the original value before the inch and 0.035 inch, respectively, rind of a maximumspecified 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 reportof 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 45percent.

Example 6,—Let it be required to determine howmuch the heat transfer of the steel barrel of a cylinderhaving an s of 0.115 inch, a t of 0.026 inch, a w of 0.5

inch, and a, diameter of 4.66 inches can be increased

when thr limiting dimensions are: Afinimum 8, 0.07

inch; minimum t, 0.03 inch; and maximum w, 1.2inches.

With a Ap, pC,/poof 4 inches of water, U is 1.08 Btuper square inch per ‘F per hour for the original cylinder.Only four solutions are possible from the graphs of thisreport because the curve for steel with a specified wand with weight M the criterion has been omitted forthe 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 fig. .21(b)) -------- ._..de -------- i2 . Oio .041 .100 ;:%

Solution 4 gives a 32-percent increase of U over theoriginal cylinder. This increase of G- is, however, ob-tained at the expense of an increase of fin reight of460 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 dimensionsfor 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

%’r2)c21%kln’E=:;:K:2;:n-here subseripi a dCUOt.CS one cylinder; subscrip~ bdenotes 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 assumedas in the foregoing axamples that the total hen t fromth~cylinder changes inappreciably viith change in cylind-er temperature and, furthermore, that the ratio of aJaOis 1, which is justiiied except for thick-wall cylinders.Then

u= [–1(l/u)=“(T,- T=,),Vb= (l/v)b (T,– Tal).

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

knowm values of (Z’fi—T%)= and (l/v)= at the pressure

difference available and from required values of( T,– T%)~ and (1/w)K The determination of b pro-

portions for obtaining the deairecl heat transfer for thenew cylinder UBis similar to that for the ot her examplespresented.

INCREASING THE COOLING BY USING HIGH AIRVELOCITIES ‘

ln the foregoing examples, impro~-ements in heattransfer have been mado by @reasing the. effective fin

~9URE28.-VarfeAfon of maximmn bent tmnefer with power requfred for moling.Criterion, h weight.

mea. Corresponding increases can be made by usingKlgher air velocities. In references 6 and 9, however, ithas been shown that, from considerations of power

THE DESIGN OF FfNS FOR AIR-COOLED CYLLWDERS 423

requirccl for coding ~1,” “the method of increasing the

fin arm is greatly superior to the method of using

higher ftir Velocities. Figure 23 has been prepared to

illustmte this same point for optimum h designs. The

power required for cooling vras calculated for the fin

designs giving maximum heut transfer for severed fin

weights from figures 16 (b), 16 (d), 18 (b), and 18 (d)

and is plotted in @e 23. I?or a given Z7, the power

requirccl is in some eases three times 8s great for n

pressure difference of 12 inches of water as it is for a

pressure difference of 4 inches of water for both steel

and aluminum. The fin weight corresponding to any

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 heattransfer, a greater effective h area should be used inpreference to increasing the air velocity. The problemof determining how much the fin weight should beincreased in order to decrease the power requirwl forcooling depends upon the particular enginwtirplanccombination involved.

INCREASING THE COOLING BY USING SHORT FLOW’PATHS

In the application of closely spaced fins, a definiteadvantage has been noted (ref erenem 4) in making flowpaths M short as prrmticable. Short flow paths increasethe heat transfer because of the lower air-temperaturerise and the higher weight veloc.it ies of the cooling airfor the same pressure difference as Iong flow paths. Ithas 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 notchange enough to affect appreciably optimum fin pro-portions. For very short flow paths, however, theoptimum fin spacing decreases as the flow pathdmreases~ as has been noted in reference 10.

Calculations have been made to compnre the optim-um fin spacing obtained with aluminum cylihdemhaving a flow-path length of appro.xinmtely 8 incheswit h the optimum tin spacing. obtained for cylindel=having u flow-path lmgth of 1 inch. In both cases, thepressure difference assumed was 1 inch of water- Thecorresponding weight velocities were obtained fromfigure 3 for the Iong-path cylinder (APIP=JPO= 1) andfrom 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 thesurface heohtrtmsfer coefficients at the front of thecylinders 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 inchof wall area was used for both cases.

The foIlowing table gives the optimum spuciugs andovcw-alI heat-trwmfer coefficients for the IO%Crmcl theshort paths for the several thicknesses Msurned.

Spegyd t

0:p

.02

:%

1. t“”

Opthnnl ;wIng I (Bt@q :~F/k) I

The foregoing tabIe shows that the decrease in tholength of path from 8 inches to 1 inch reduces thooptimum spacing to approximately one-half its originalvalue and increases the heat transfer a little more thantwice its original vwluc. It is thus apparent that shortpaths are advantageous anti thut the optimum findimensions are appreciably different for extreme differ-ences in the length of the flow- path. The difimdticsin the breakiig up of a lo~~ flow path into more thantwo 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 thespacing increasing and tlm thickness decreasing at verylow fin weights.

2. The optimum fin spac~~ n.nd thickness decreaseslightly with increase of the pressure difference.

3. For a given h weight, the highest heat transfercan be obtained with fins of a magnesium alloy. In thisrespect, pura coppm mid aluminum-alloy fins are onlyslightly inferior to maagnesium-alloy fins rtncl willtransfer several times more heat than steal.

4. For n given fin width, the highest heat transfercan be obtained with metals having a high ~herma[conduct ivit y. Of the metals considered, the highestheat transfer will be obtained when copper is used;aluminum, magnesium, and steel foIIow in the order oftheir 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.indem in an Air Stream. Rep.No. 488, NACA, 1934.

.

424 REPORT NO. 72 fiNAT10NAL ADVISORY COMMITTEE FQR AERONAUTICS -.

2. Schey, Osoar W., and Rollin, Vern G.: The Effeot of Baffles I 7. Rollin, Vern G., and Ellerbrock, Herman H., Jr.: Prmcurcon the Temperature Ilistrihution and Heat-~rwefer ]Coefficien& of Fined 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 HavingCloeeljJSpaced 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: CWlCtlEngine Cylindere. Rep. No. 612, NACA, 193S.

9. Campbell, Kenneth: Cylinder Cooling and D~ag of ltadialEngine Installation. SAX Jour., vol. 43, no, 6, Dec.1938, pp. 515-527.

10. Brevoort, Maurice J.: The Effect of Air-Passage Lcygth ONthe Optimum Fin Spacing for Maximum Cooliug. T. N.No. 649, NACA, 193S.

,