C.P. No. 165 (15.548)

A.R.C. TechnIcal Report

MINISTRY OF SUPPLY

AERONAUTICAL RESEARCH COUNCIL

CURRENT PAPERS

A Theoretical Note on’ Effusion Cooled Gas Turbine Blades

BY

R. Staniforth

LONDON HER MAJESTY’S STATIONERY OFFICE

1954

. Price 4s 6d net

C.P. NO. 165

A theoretical note on effusion cmoled gas turbme blade3

- by.-

R. Stamforth

Solutions of both dynamc and therm1 bmm3ary layer equatmns have been obtcmcd for tw di.mxs~onsl m&her&l inompresolble lazmnsr flow over sem-mflmte wedges for a range of w&e angles lnLi mJectlon qmmtl- ties. These solutlonn arc epplled to the cstmatwn of coolmg Cvr mJec- tlon velocity required by an cffusmn-xmled turbine blade, usmg an

2y;:;:3 3 thod end also a r&hod silnlar to that dascrlbea by Eokerd')

tu be appllea Eo Proposals are @vcn enabling c-alculated isotherml results non-isotherml flow.

In the turbulent rcgire, a workmg b,ypr&hesis 1s given enablmg the heat transfer coefficients ena required ovcling sir veluczty to be cdcu- late& though the method rust be regarded as tcntatlve.

Detnlls of the application of the theory are g~vcn in the rain text whilst the full mthemdloal theory e.i-d wthods of sdutlon of the result- mg equations are given 111 the Appendxes.

The above troatrrent 1s applies t,: the &xi@ of two effusion cooled nozzle gude vanes for a h@ temperature gas turbine. In these designs, the "msulatlng" effect of the mJccte:cl cooling tir is such as to reduce the coeffxxent of heat trnnsfer by abwt one-third as cumpareil with the internally 020iOa case. The designs ohvw the neer3 for a great varwtion of cooling 81~ inJcctlon vzloclty wth chwdwxse posltiDn, if -form cool- mg IS to be achieved. Tho theory given Ln this mr;xsndum cannot yet be checked by corr;paxlson with experxxnt, experxcntnl data not being avcul- able.

hn abrdged version of this Dx.cuss~on on Heat Transfer (Lon&n

*"Effusion cooling" has been tentdxvely &pted at N.G.T.E. for cooling by injection of gas through a permable wall: "sweat coolmg" is being restricteato the lnjcctlon of liquid through a permeable wall anil "lnjec- tion cooling" is being used es the generx term.

x

1 .o

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

Intrduction

Theory of two dimenslond, incompressible,isotherml, lanrinar flow over sem-mfmite weilges mth gas mJectlon

Appllcatlon of theory to blades mth gas iniection

Description of attached fkgures

Applxatlon to non-isothermsl flow

Turbulent boundary layers mtb gas inJectIon

Procedure for calculatmg the heat flmi and required inJectIon velocity consistent vnth mmtaming a constant blade temperature

Design of sn effusion cooled nozzle guide vane for a high temperature gas turbme

Comments on practxnbildy of effusion cooling

Conclusions

References

w

5

8

Ii

13

13

14

16

!lE?ENDIDIcES

N". Title

I List of symbols 18

II Solution of the dynamic boundary lc+yer equations 21

III Note on the method of oomuting smiler solutions of the boun&v.-y layer equations 23

IV Solutxon of the thermal boundary loycr equation 26

V Note on porosity, permability and prcnnure itrop in smtered materrals 30

Title --

Solutions of uotherrd boundary lwer vrith gas LLnJectlon p : 0 (flat plate).

Solutions of isothcrml bout&x-y layer mth gx.3 1nJeotlon f = 0.2.

Solutions of mothed boundary layer mth gas mjeot1on @ = 0.5.

Solutions of uotherml boundary layer mth gas lriJeotlon p : 0.8.

Solutions 0f motherml bounday lawyer mth gas lnJectl0n p = 1.0 (stagmt1on pod).

FIR. No. 1

2

3

4

5

(4 4

6(b)

9

10

II(a)

II(“)

12

13

14(a)

Dependence of t!le ratlo - 4%

upon InJ0otlon a-d

the pressure dutnbutlon.

Dependence of the function T' J(gT

upon mject1on

and the pressure dxtrlbutlon.

Dependence of the mmentum thwkness A2 upon the pressure dmtrlbution and amount of m.Jectlon.

Graph for the evaluation of the quantity Ztx2 from the form parameter It* and the mJeotmn parameter C.

Graph for the evaluation of the quantity Zt 32

from the form paramter Xt* and the teniperature pam.meter.

Graph-for the evaluation of the inJection para- mter C for a given wedge mgle (R) and tempera- ture paramter.

Velocity jlstnbutlon over the surface of the w/700 nozzle guae vane. I

Graph of &P+ - -4 7,2

Hc, ctganst LL for the W2/7OO 51,2

nozzle gude vane mth no mjectlon.

Graph of E& against e for the W2/700

nozzle gmd vane effusion cook to 600'~.

Cooling au veloczty dmtrlbutlon requred to cool a nozzle ,guuie vmo to 600'C. 1n o. gas strewn at l,ooo"c. (apprommte s01ut10n).

F1g. No.

14(b)

15(a)

15(b)

16

17(a)

17(b)

18(b)

I?

-4-

ILLmmATIONs (Cont'd.) ,

Compsrxon of heat transfer with and without injec- t1on (approximately s01ut10n).

Coolmg air velocity distributzon requred to cool a nozzle gude vane to 6OOOG. 3.n a gas stream at 1, ooo"c .

1 Comparison of heat transfer vnth and irnthout injeo- tlon. J Relatzve wall thickness of an effusion cooled nozzle guide vane.

Detsds of a nozzle cascade 1 Velocity dxdrlbutlon over the surface of a nozzle blade.

Coolmg air veloolty rcqured to cool a nozzle blade to 600°C. u-t a gas stresm at l,ZOO%. Re = 2 x 105.

Comparzon of heat transfer vnth and mthout inJec- tlon.

Rclatx.ve wall thlokncss of on effusion oooled nozzle blade.

-5-

i .o Introduction

Cecausa of the complex nature of the equatwns governing the boundary layer, it is impossible with our present knowledge to obtain a m&hem&i- tally exact solution to the aerodynarnc design problems of effusion cooled blades. Me are therefore obliged to seek an apprcnumate method or en empirical nothod. As no accurate experimental date. concerning this problem have been published, the latter course cannot be explored.

It has been shown(&) that exact solutions of the isothermal bounw layer equations can be obtaned for flw over serm-infirmte wedges and flow through oertain k.~nds of channels without fluid injection. As fluid injection and abstraction alter s in no way the mathematical reasoning lead- ing to the above conclusion, we CNI obtain a range of corresponding solu- tions with inJection and abstraction, a special ease of which, with no injection, being the solution usually quoted.

It is proposed to apply these solutions to a body of more complex shape such as a turbine blade by approximating the velocity distribution over its surface to either a single wedge or a scrios of wedges.

2.0 Theory of two &mensional, incompressible, isotheri"dl le.m~nar - flow over serm-infinite wed,ges vnth gas inJection

For flow over a sema-x&x&e wedge, leyer equations resolve to a non-linen totel EtLfferential equation

f"' (v) + f (VI) f" (1) =P { f' (?I)* - 1) . . . . . . (1)

Fnth boundary oonditions

n = 0, f' (?;j = 0, f(n) = c

rl + 00, f' (11) = 1

and where ri is the non-dimensional distance normal to the surface and f' (v) is the dimensronless velocity parallel to the surface (see Appendix I for list of syllbols).

For oompletcness a development of this equation from the boundary layer equations IS given in equation have been published, were calculated by the method outlined by I. Fox

The temperature boundary layer solution can be conputed from the following equations knowing the values of f(n).

e=Jv 4 J(n) = s

" e-F('l) ilq, F(V) = Pr T -T,,

J(m ' JO

f(n) an, e = - (2) 0

Tg-Tb "

Appendix IV shows the derivation of the above solution. Further mathematical manipulation required to convert the solutions into a more practicable form is also inclu&d in this Appendix.

Some known solutions of equation 1 are shown graphmally ~1 Figures I to 5 for a Prardtl nuder = 0.71. Other aharacteristics of the hoe

-6-

and 4, the mmntum thickness were cslculated and

are plotted m Figures 6 and 7. From these graphs, the dependence of the boundary lsyer thickness and the heat transfer coefficient upon the pressure distribution and lnjeotion coefficient can be clearly seen.

3.0 Application of theory to blades with gas injection

Two n&hods of application of the above results to the solution of sny blade design problem may be used, the one chosen depending upon the accuracy desired. The sl&est is to replace the given velocity distribution over the blade surface by 8. singl.e curve correspontig to the flow over a semi- infznite weage. The chdioe of wedge angle rm be m%de by trial and error or by plotting the veloolty profile on 1ogarzthnLa axes and determining the

average slope of the graph. TLS gives n = -k 2 p which 1s the parameter govern-

mg the wedge sngle. If we require the boQ to be cooled to a uniformtet@za- turn, the injection psramder C is constant (because z.n Appendix IV equation 10, J(w) is inclepcndent of x) and therefore me knovr inndJ.ately the velocity of injection at any point, and the local heat transfer coefficients.

This approximation leads to the largest error at the lea&.ng edge although tk OM be reduced by calculating the exact heat transfer and inJectzon qusntzties at the nose &s in Section 7 and f&g in the curves to vlc1ud.e this point.

A second and more accurate m&hod, suggested by several writers, is to split up the profile rnto a large nuder of sections snd fzt a wedgs velocity distribution to each piece. The sections sre joined together by assuming the continuity of a function of the bounw layer. A.s it 1s inws- sible for one paramder to describe fully all the ohwscterist~cs of a bounilary lsyer, there remains the choice of a parander which, as well as agreeing with other methods and experimnts, is slso oonvenlent to use. The psrsmsters usually adopted are the boundary lsyer thictiesses of vtich the following are the four most aomnon:-

1. Displacerrent thickness S"

2. hlomentum thiclmess A2

3. Nominal thickness

4. Temperature displacement thickness 6t*

The first two fun tlons sre inconvenient to use for the sub'ect of this note as the relation Z* 2 tohW is saioiguous for high values of@ bQ are functions of the dynam~.c boundary leyer oorrespoding to Zt x2

anah*

the temperature.boud.sry lsyer). andAt* in

The temperature cksplaoement thickness has been used belanr although no doubt the nondnal boundzy layer.Wss-muld be just 09 suitable.

-7-

In this analysis dencrlbea in more cletcll m Sdctl3n 7.1, we prowe& to draw a graph uf the d~rxxs~onless temperature displacenent thickwss

y 6 against the cllstsnca x,ls frm the bl,de &u&nation pdnt, using

the z.socl2ne netlld. Equations 8 znd 9, Appendix LV, are pzrtioularly suet- able for solution by thu isocllne method, as at nny point on the graph, i.e.,

knaving g &yaab~) we cnn calculate the function of AtH frx~~ S s

Equzttlon 8, Appendix IV. Frona chart, descrlbedlater (Figure g), we CN~

5btIun the value of (1-p) Ztx2 a cI;t*/s GJ to be obt-ea enablmg - A 6. (;c/s)

(Equation 9, Appendix IV). Thus we ray plot a series of short lines through

selected values of "" -y& at each x/s station (the positions of these

should of course be carefully estlrrated), each llno berng at the slope pre- dicted.

As the initial value of 6t" -y-i- at the stagndxon point (9 -1) is

known fromEquatxm 8, Appendix IV, anal as tho slope there 1s knuvm to be zero, we can sketch the most probable path of the cuwe (see Flgurc 12, no mjcctlon, Figure 13 uath mjectlon).

4.0 Descrlptlon of attached fwures

The most convwucnt co-abates for obtanmg the funotlon (1-p) Zt"* are ?e sad %X2, the difference between the co-otinates being the deslrod function end the inverse slope being p. The calculated values of the above co-or&nates were plotted and grs.phed, the paranzeter bemg C (Figure 5). The rnJeotlon parsxetcr C is difficult to evaluate at any point except by trial end error because lt is a funotlon of x, the orlg~ of which 1s usually unknovm. The paramder is therefore changed usln[,Equation 10, Appenbx IV, and the graph re-plotted vnth the temperature ratlo as a p‘arameter (F'lsre 9). Other parameters could be employed but they suffer from a hsadvantage u that they have szngular points.

5.0 Appllcatlon to non-z3othermal flow

There are two methods m general use for co-relating non-isotherm1 and isothermal heat transfer cocfficrents. One method is to choose the tenpratures at whxh the physxal data arc taken such that the relattlon-

ship G -6

can be xxlependent of the blade-gas absolute temperature ratro.

AlternatIvely, tho relation Is diii

can be calculated using free stream con&-

tlons and a correction fmtor introduced which 1s a function of the blade- gas e.bsoluta temperature ratio.

-8-

If the first method were used, the relevant values of the physical data used for the calculation of the injection parameter C would be the sub- pot of pure speculation. Rather than adopt this practice, it was decided to adopt the second process, first determining the heat transfer coefficient for isothermdl flow at the main stream temperature, correcting only for the change in Cp of the cooling fluid and then applying this heat transfer coefficient to the non-isothermal case with a suitable correction for the absolute temperature ratio.

This procedure leads us to a modified value for the temperature psra- meter

(‘pi, - T$ cp ‘3.c. Pr

=(‘& - Tb) cpg

. . . . . . . . . . (3)

Heat transfer coefficients and cooling au mass flows are modified bjj a

factor depending on the ratio 2 Tg'

the value of this function being taken as

f(i#z O.,+ 0.j (4 . . . . . . . . . . . . (4:

Tb for 0.5~~~1. This equation lmst be considered approximate as it is 6

deduced from only one set of experimental data (11).

6.0 Turbulent boundary layers with gas injection

At present, nothing is knox-n regarding the behaviour of an undeveloped turbulent boundary lsyer when a gas is injected at the wall wxth regard to modification of its velocity profile, thiokness, and heat transfer, although a little information is available on the he t ransfer to the wall of a porous tube with undeveloped turbulent fd. floe It appears that the Nusselt number is reduced only a small amount by air injection. We oould therefore expect that the heat transfer in the turbulent regime of the blade would be somewhat less than that calculated for no injection; the actual fraction depends on the amount of injected air. It is suggested that the coefficients should be calculated on a b s s t at there is no injection in the turbulent region using Young's 6-f d. n&hod 3 3 This is probably an over-estimate of the coolant required for vanour, reasons and shouldbe motifled. as experi- mental data becomes available,

It should be noted that Young's r&hod postulates sudden transition to turbulence whereas in praotice it is found that transition takes a finite tistanoe to oomplete.

7.0 Procedure for caloulating the heat flow and required injection velocity oonslstent wxth mnintainng a constant blade temperature

In order to calculate the boundary layer thickness and heat transfer coefficients, it is first essential to obtain accurately the potential velocity distribution over the blade surface by either experiment, oalculation or

electrical snalogy . Frond&a3 .IsI,s U/u0 and tzy" a-e tabulated at s

close regular intervals.

At the nose ',$' 1s csloulated from the nose curvature using the s

followmg equation derived from elementeJ potential flow theory:

a. u/u, s u li l r2-- . . . . . . . . . .

a ys (5)

R U out

the rat.tlo ' L= -bang calculates from the blade angles <asswkng ~noompres- u out

able flow.

7.4 Proccaure

(1) Given Tb, T,, and Tg c,dculatc the

ratio (Tb -'cl 'Pb, c Pr (Tg - Tb) Cp

g

(2) At the stagnation pokt where p = 1, we can obtaxn the function ht* usxng Flgurc 9 ana the terrperature ratlo calculated above.

This enables us to deterane y,& at this point by the use of

Equation 8, Appenda IV. We then proceed to complete the curve as described at the end of Section 3.0 (using the method of ~socllnes) over the whole of the ooncave surface and up to the point of w.xxxum velocity (or ~~lnlrmm pressure) on the convex surface.

We then tEbbulat.ta the wlues of y& at the chosen pomts, ana

proceed to calculake E and i& as in Table I:- -m

- 10 -

-11 -

The pomt of trsnsltmn to turbulence on the convex surface 1s or minimm pressure. We can assuxd to be the pomt of maxmum velcclty

then calculate the momentum thxdness of the ltinar boundary lsyer at thx point as follows:-

: Re, s

=~~~~ ,. . . . . . . . . (6)

A2 being read from graph 7. The heat transfer coefflclents in the turbu- lent region are now calculated as in References 13 and. 14.

As before

;Tb-Tc) %'b,.

Tg-T,) Cpg

. . . . . . (7)

Tb The value of the function f, ;I;- 1s unknown. It is therefore assund that

0 g thus function 1s the S~IX as ‘that used III the lsnnner flow case.

8.0 Desxp?l of an effusion cooled nozzle -de vane for a hip! temperature pas turbine

As the blade under consderatzon 1s subject to turbulent flow as well as larmner, it 1s tifflcult to rrnlntiun a constant blade temperature under all operating Reynolds numbers.

The blade nrust be desxgned at the worst estxmted con&tlons so that under other IIUXX favourable conditions, the blade ~~11 be over oooled. The potential velocity distribution over the surface of the W2/700 nozzle guide vane cascede, the first exanrple taken,together with relevant physlcd dilnenslons, are given in Figure 11.

As further design data 1s unavallsble concerning the example given, It has been assumed that the deszgn con&.t.tlon is Re = 2 x 105.

The pressure tistrlbutlon round the blade was obtaned assurmng the flow to be Incompressible. The outstsnting design figures are tabulated belcw-

Equxvalent gas temperature at exit (Tg + 0.86 EL,) 1,ooo~c.

Total I, II ,, II 1,012%.

Stat10 'I t, II ,I 927'C.

Coolant temperature 60%.

Mexxmmblade temperature 600%.

- 12 -

Static pressure at exit 25 IbJsq. in. gauge

Cooling sir pressure 40 lb./sq. m. gauge

Exit gas velocity 1,465 f.p.s.

Tenrperature parameter 0.86

The requlred solution of the botiary Layer equations has already been obtsined(F~gure 13).

If the blade material is of constant permeablllty snd if the pressure distribution, mternal pressure and the injeotlon velocltl profile are known, the relative wall thloknesses can readdy be calculated (Figure 16).

The scale of thus graph 1s dependent upon the perrreabillty of the blade materxd.

With a permeability of 5 x 10-'0in.2, the rcqulred thxkness at the stagnation point 1s approxllrately 0.004 m. gxvlng a mzx~m.ux blade thickness of 0.077 in. It mqq bc neccssargr to increase the wall thxkness at the nose to ease stressing and manufacturmg ddfloultics. This msy be aohievcd by using a larger blade nose m&us, a higher ooolsnt pressure, or by allowing the nose temperature to rise above that in the speolfxation.

Alternatively, the wall ccn be made of a m&erKL of variable perme- abdity using a constant blade vail thickness.

The percentage ooollng air required IS dependent upon the Reynolds number, the oaloulated flgurcs being 1.22 per cent at Re = 5 x 105 and 1.71 per oent nt Re = 2 x 105.

These figures are higher than would be usual in gas turbine practice because of the close spacing of the blades.

thiokn%!lFy ooess was repeated for n similar type of blade of nnxh grcatcr ns it was conadered that the first blade, which was designed

for M uncooled turbxw, was unslutable for this n&hod of ooolmg.

The relevant design data is as follows:-

Design Reynolds number 2 x 105

Equivalent gas temperature at exit (Tg + 0.86 ev) 1,200°c.

Total " II II I, 1,207OC.

stat10 " 0 0 II 1,15Y%.

Coolsnt temperature 60%.

Mxci~rn blade temperature 600%.

Static pressure at exit 28.4 lb./sq. In. gauge

Coolmg sir pressure 54 lb./sq. in. gage

Temperature parameter 0.555

The corresponding figures sre Figures 17, 18 and 19.

- 13 -

The cooling am qusntlty required to effusmn cool the bide 1s 1.12 per cent and the quantity of am to mdx-eotly cool the blade at 100 per cent ooolmg effxlency 1s 1.56 per oent. In Figure 19, the units of wall thickness are 0.004 in., for a mterml of the sam permabll~ty as mentioned previously.

9.0 Cements on practmabdlty of effusion coolmg

It cm be seen by reference to Figure 16 that it would be ddficult to mmufacture such a blade mth thm sections as are required at the lead- ing and trailing edges.

A great deal could be done in ,dlevlatmg the design problem in a high tmperature turbine blade, tireotly coole& or smat cooled, by investl- gatlon into unorthodox blades of relatlveljr large thickness with a vie= to their use m smdur cleslgns to the above.

In ooolmg a blade dzrectly by mjcctmg ooolmg am through the blade walls, the heat transferred from the gas to the b1ad.e walls 1s only about 2/3 of that transferred to a blsdcir: at the ssm tenperatura but mternally cooled. The saving of ooolmg eu: woulil probably be greatcr than l/j because of the ddficulty m obtszmmg nuf'fxlant heat trvlsfer m the blade usmg mctirect mthods to enable the coolmg air to be efficiently used.

Provding cam is taken m flltermg the cooling air, trouble due to overheating of the blade arlsmg fronblcckmg of the pores of the mtal vnth foreign mtter shoul(l not cause ana diffxulty.

10.0 Conclusions

A process has been outlined for the thermal design of an effusion cooled gas turbme blade such as vioulii bc employed m a high temperature gas turbmc.

To Xxs Id. G. Kennard for the caloulatlons of the solutions of Equation 6, Appendix II, on which tlvs work IS based.

- IL -

Ns.

1

7

8

9

to

II

12

Author(s)

E. Eckert

R. Staniforth

w. Mangler

S. Goldstem (Eddor)

H. Sohlichtmg end

K. Bussmsnn

M. Schaefer

w. liangler

H. Holstein

L. Fox

S. J. Andrcms cd

I?. '2. Bradley

P. Duwez and H. L. Wheeler

13 H. B. Squire

RTX%REXKFS --

Title -~

Dm Berechnung des W&diber~s.nges m der la&wren Greneschicht umstrornter I&per. V.Y.I. Forschungscheft No.416. Berlm 1942.

Boundary Layers: 1002, Gmh235.T.

approximate n&hods. R. & T.

Gontrlbution to the theory of effusion coolmg of gas turbme blades. General Dlscusslon of Heat Transfer, Section V - Specm.1 Problems. London Conferenoe at the Institute of Mechanical Engineers II-13th September 1951.

Boundary layers: Steady lsmnar boundary layers: Special exact solutions. KA.?. Volkenrode AVA Monograph. R. & T. 1001, GDC/3234.T.

LIodern developments In fluid dynamos. Camb. univ.~ress (1938), p.141.

Exskte Lblsungen f%r die lanare Grenzachlcht mLt Absaugung und Ausblasen. Schrlften der Deutsohen Akadetie der Luftfshrtforschung. 7B (1943), Vol.2.

Laminar boundary layer for the potentld floPi U = Ulxmunth suction and blmng. U.IS.2043

(19443.

Laminar bounQry layer vnth suction and bla'nng. U.ri.3087 (19443.

Slrmlar 1auuns.r boundary layers at permeable walls. u.Xl.3050 (1943).

The solution by relsxation methods of Or&.naQ differentId. equations. Proc. Camb. Phil. SOC. Vol.45, Part I. January 1949, p.50.

Heat transfer to turblnc blades. N.G.T.E. Memorandum No. X.37. A.R.C. 12,070. October, 1948.

Fxperlmentnl study of cooling by lnflltration of a fluid through a porous rratenal. Journal of the Aeronautrcal Sciences, Vo1.15, No.'. Septsder 1948, p.508.

Heat transfer calculation for aerofolls. A.R.C. R. & M. 1986. November (1942).

- 15 -

s.

Ir,

Author(s)

H. B. Squre a-d

A. D. Young

Tlt1.e

The calculatmn of the profile drag of aero- fOll.53. A.R.C. R. B 16. 1838. Novenlber 1937.

15 S. J. Andrews An expermentd investlgataon of a thick aero- and foil nozzle cascade. N.G.T.E. Menmrmdum

F. 'ii. Schofield No . ia. 814.. A.R.C. 13632. Kay 1350.

16 K. v. southwell Relaxation methods. Clarend.onI?ress (194.6).

- 16 -

Experimental investx&lons

"Coolmg by forcing a fluid through a porous plate m contact wth a hot gas stream". Presented at the IIeat Transfer and Fluid Mechanics Institute. Published by A.S.M.E., May, 4949.

"Heat transfer measurerrents in a nitrogen xveat cooled porous tube". Progress Report 4-48: Pasadena, Jet Pro- pulslon Laboratory, 6th November, 1947.

"Experimental study of coolxng by xd?dtratlon of a. flud through a porous metal". VIth International Con- gress for Applied Mechanics. Parrs, September, 1946.

"Heat transfer to a porous wall through which a gas 1s inJected". VIIth Internntlonal Congress for Applwd Meohanics. London, Septeher, 1948.

"The powder mdallurgy of porous nrztals and alloys having a controlled porosity". 4.i.Ni.s. Tech. Pub. 2343, 1948.

Theoretical investigations

"The flow of gases through porous m&al compacts". "Engineering", ~01.167, X0.434, April, 1949, pp.291-292.

"An er~~r~oal equation for heat transfer in a tube sweat-cooled with a gas". Progress Report I‘To.4-49, Pasadena, Jet Propulsion Laboratory, 19th November, 1947.

"A swplifled theory of porous wall ooollng". Progress Report No.4-50: Pasadena, Jet Propulsion Laboratory, 24thNovember, 1947.

"Heat transfer m sweat-cooled porous metals". Journal of Applxd Physxs, Vo1.20, Januay, 1949.

"Some observations on the mechanism of sweat-cooling". i+oc. VIIth International Congress for Applied Meohsnics. London, Scptembcr, 1948.

"On sweat-cooling". Journal of Aoronautloal Sciences, ~01.16, No.1, January, 1949.

"Porous cooling". Journd of Aeronautlea Scxnces, Vol.16, No.4, April, 1949.

"Stablllty of the boundary layers and of flow m entrance seotlon of a channel". Journal of Acronauti- cal Sciences, Vol.15, ~0.8, Aupst, 1948.

Authod sj

M. Jakob and I. B. Fieldhouse

P. Duwee and H. L. Wheeler

T . Duwez

1;'. Duwez and H. L. Wheeler

P. Duwez and H. E. hIartens

P. Grootenhuls

Ii. L. Wheder

W. D. Rsnnie

S. Weinbaum aid H. L. Wheeler

P. Grootenhuls and

N. P. W. hIoxe

E. A. Richardson ,

L. Lees

E.Hahnemann and j, C. Freemn

- 17 -

DIEGIOG~HY (Cont'd.)

AU-Lhor() -

"The stability of the lsminar boundary layer vrith L. Lees injection of a cool gas at the wall". Princeton Um- verslty, Aeronautical Engineering Laboratory. Report No.139, 20th May 1948, P.16252.

"Heat transfer in lnrlnnsr compressible boundary ls,yer s. w. Yuan on a porous flat plate with flurd inJection". Journal. of the Leronautio,d Sciences, ~01.16, No.12, Decerrber 1949.

"Sweat-cooling". The Engxwsr, 2!+th%ebruaq, 1950. N. P. !;. Moore P.230. and

I?. Grootenhuls

"Sam factors m the use of high temperatures m gas turbznes" presented at the InstOxte of 31echnnloal Engineers, 27th Januw~ 1950.

T. \:. F. Rrovm

"Heat transfer oharadenstlcs of coded gas turbine H. L. Ellerbrcok bides" . General Discussion of Heat Transfer Seotlon. V.- Speclel problems. London Conference at the Insti- tute of Mechanical Engineers, II-13th Septeirber 1951.

"Heat transfer zna temperdxre profiles m l,annnar boundary layers on a sweat-czolu4 wall". Tech. Rept. X0.5646 irlr &Weriel Conw.md, 3rd Novedxr 1947.

E. Eckert

"A theoretwal and experimental investigation of J. Friednxm Rocket Motor swent cooling". Journal Amer. Rocket SOO.

No.79, Doomher 1949, pp.147~454.

"Exact solutions of the laninar boundary layer cqua- B. Brown tlons for .a porous plate vnth vnriable fluid properties and a pressure grsdxtint in the ma=n stream". &per presented before the fxrst U.S. National Congress of 1qplledMechanics, Chicago, Illmols, II-l&h June 1951.

- 40 -

APPENDIX1

SJmbols

c

c

f(n)

Fbd

J( *rl)

x

m

NU

P

Pr

9

Q

R

Re

s

Dimensions

Blade chord ft.

Injection parawter defined in Equation 9, Appendix II. Dimensionless

Speclflo heat at constant pressure cm lb.-' oc.-'

Heat trsnsfer correction fz temperature ratlo in a l-sr boundary layer defzned. in Equa- tlon 13, Appen&x IV. Dimensionless

Heat transfer correction for temperature ratlo 111 n turbulent boundary layer defined mEqua- tion 7, main text. Dimensionless

Funct+on of q defined by Equation 7, Appendix Il. Dimensionless

Function of Q defined by Equation 3, Appendix IV. Dimensionless

Function of q defined by Equation 3, Appen&ix IV. D1menslonless

Constant in Equations 4 and 5, Appenillx II.

Wedge parameter = L 2-p

Nusselt number y

Dependent vanable inEquation 2, Appendix III = f' (a).

Prsndtl number 9

Coolant mass flow per unit area per sec. q VP

Mainstream mass flow per unit area per sec. at

blade exit q Uop i.e. Up at exit [ 1

Blade nose radius

Reynolds number 3 P

Dlstanoe from stagnation point to trailing edge m%sured along curved surface, or sn arbitrary standard length in the case of awedge.

Temperature (absolute).

Dimensionless

Dzmensionless

Dimensionless

ft. sec.-'

Dimensionless

lb. f't.-2 sec.-'

lb. ft,-2 sec. -1

ft.

D~mens~K~ess

ft.

OK. -1

Velocities parallel to surface in boundary layer. ft. sec.

Velocity parallel to surface at edge of bounw layer. ft. sec.-'

- 15 -

APFmDIX I (Co&'&)

Y

v

x

Y

2tW

a

P

Y

6t*

*2

E

A

?bt"

P

8

CI

11

*

v

Syubols

DlmeIlSlOnS

Velocity norml. to surface m boundary layer. ft. sec.-'

Vclomty n0rm.l to surface at wall. ft. sec. -1

Distance masured along the surfem ft.

Distance measured nom&L to the surface ft.

Dimensionless temperature boundary lwer Muck- ness defined. by Equation 6, Appenhx IV.

Heat transfer coefflcmnt

r ncluded wedge angle

h 1 Constant in Equation 4, Appenbx II

Temperature bowdary layer thlotiess

Dmzmsionless momntumthiclmess

Dmenslonless

cm ft.-2 oc.-' sec. -1

Dimensionless

Constant inEquatlon 5, A~pend5.x II

Thermal concluctivlty

Form parsmeter defined by Xquatlon 8, Appendix IV.

Density

Xmensionless temperature m boundary leyer dofined in Appendix III.

Viscosity

Dimnsionless distame perpendicular to sur- i'sae defined by Equation 7, Appetix II.

Stream function, see Equation 7, Appendix II.

Dynado vlsoosity

ihmAnsmnless

ft.

Dimensionless

Dimnslonless

CHU ft.-' oc.-' sec. -1

Dimensionless

lb. ft.-3

Dimensionless

lb. ft.-) sec. -1

Dimensionless

ft.2 sec. -1

ft.2 sec."

- 20 -

s

1

2

Refers to gas or flud

Refers to blade or body

Refers t> coolant

Refers to blade exit

Refers to value at point x, e.g. Nux = 7 Rex = 7

Refers to value at pout s, e.g. Nu, I: 0-s Res = o- u SP A u /

Refers to the convex blade surface

Refers to the concave blade surface

- 21 -

zWl?EXiDIX II

Solution of the &xuxio b cnmdary lcqer ectuations

The boundary lsyer equations for two dxnens~onal lncompresnible, isothermal lwinar flu# are:

au + av=o a~ ay . . . . . . . . . . -* (2)

rnth the baunw condxtxons

y = 0, u = 0, v = v

y,ca u=u . . . . . . . . . . . , (3)

It has been shown that m order that "sidlsr" solutions of the above equations w be obtained, the velocity &stribution over the surfaoe consuiemd rust be of the form

u = Kj C

(2y -p) $ 1 T-5 . . . . . . (4)

or, in the special case whore 2y -B = 0

U= K2e g$ (Reference 4) . . ., . . (5)

If Y = I in Equation 4, we obtain the velocity distribution as over a send- xdk-iite wedge of included anglepn. Although solutions have been obtaxxd for other values 02 Y, the only vslue of xnterest in this report is Y = 1, and prengmg from 0.1986 to 1 .oooo.

Equation 4, re-wmtten with Y = 1 gives us

UZK g 0 &-J .- . . . ^. . . (6)

Equations I, 2, may be readily oonverted into a non-linear total differential equation by ohsnging the variables from u, v, x, y, to 11 and f(n), the latter f'unotions being defined by:

Q "J2-p J=f(rl) . . . . . . . . (7)

- 22 -

l%Yl?PENDIx II (Cont'd.)

Hera + = f' (q)

and uv

v =J&y 7 f (17) + (p - J-i I) w (d)

and equatmns 1, 2, resolve to

*' t I (rl) + f (4 f" (4 = P {f' (I$ - I}

m.th boundary oondztmns

q = 0, f' (q) = 0, f (7-J = -,Jcyi ; J- + = c

q-+co f' (?l) = 1

...... (8)

...... (9)

This Equation my be solved by a relaxation process (see Appendix III).

- 23 -

Note on the method of coxnputm~ smnlar solutions

of the boundary layer equations

It has been previously shown that the xotherml dynarmc b0un-J layer equat.tlons may be transformed to the non-lmcar total dlfferentisl equuntlon(see Appendix II).

f”’ (Tl) + f Cd f" (d = P{f' ($2 -1j . . ., .. (1)

As equations of odd orders arc dif'flcult to solve by relaxatmn pro- cesses, equatum (1) is Integrated to g&ve the second or&r equatmn

ptl+pl~o,+~pd?+ {p?l)=o . . . . . . (2)

where f' (v), the dependent varmble, 1s replaced by p.

The quantxty is squared brackets (Squatlon 2) at <any pomnt ri is replacedby the sy&ol gll to smpllQ later fomulae.

We can replace Equatmn 2, by a fmite difference equatmn ('@ the equattlon becomng

pi + h(1 + 5 hgrl) t 19 - h(1 - & hg,,) - 2prl -@h* (p>,* -1) + A = 0 . . (3)

where A, the difference correction = - i?t2 63 6

-164 o 12

and the uterval 1s h.

Xquatlon (3) is solved by successive approxmatlons; the left hand side bemg called the resldusl R, and the relaxation process bemg to reduce this resuB&.. to zero. The relsxatmn equatmn or the equation conncctmg the change in resGLua.1 with the change m p is obtamad from this equation neglectmg the effect of a chmge in p on the &f'ference correctmns.

Thus the two relevant equations are:-

p,rl+ hjl + $-hgV)+ pi - h(1 - & hgll) - 2 pq - ph2(pq2 -1) t A = R . . (5)

Ap,+h(l+$hg7))+Apn-h(l-+hgr,)-2ApV-22:h2p$p+B ..(6)

n The mtegral

J pdn 1s evaluated in the later stages of the relaxa-

0 tion process using central bfference mtegratlon fcrmla.

-UC-

APFENDIX III (G:nttd.)

c

JO p&,=h 'fo+o+~E3,- '91,5o+

XT,480 (7)

‘fo, the first sum 1s adJusted so that the Integral at 77 = 0 is zero, 1.e.

‘f, = 16’, 12

-11 63 720 0 .-.** . . . . . . ., . . . . . . . . (8)

g,, can easily be evaluated kn&ng the above integral, as the value of f(0) = C is known.

Details of method of solution

1. Vdues of p were guessed at mtemals h. The interval h was chosen as either 0.5 or 1.0, the larger intervals being chosen when the value of 11 at vdmzh p approached unity was expected to be large.

If other solutions are available mth values of P and f(0) n&r to that required, a close approxlmatlon to the values of p may be obtaned by mnterpolatlon or extrapolation.

rl 2. The integral

J pd, in the first instance was obtolned using Simpson's

0

rule. The difference r&hod of xntegrtlon could not be used at ths stage because the difference table was unreliable.

3. The residuals (1.e. the left hand side of Equation 4) were calculated at each pomt, the ddferenoe oorreot.xon being neglected at this stage. One more figure was kept in the res.xLmls than =n the values of p.

4. The resdmls calculated in 3 were relaxed using Equation 6 to sln?ost zer3 negleotmg the dependence of the function g,- on p.

5. Steps 2, 3 and 4 were repeated until the successive values of p, correct to two deoliral places, were very nearly equal.

6. A differenoe tnblc was n&e of the function p up to the order at which the differences ceased to vary smoothly. It is necessary to estznde the dlffercnces at the begiang of the table by extrapolation. The process adopted was to plot a graph of the first order differences agslnst q and assuIT& the dlffarence zero at n = 0.25.

Any central differences roqulred were obtained from the arlthr&ical rrerm of the s&acent forward and. baokw.rd differences.

7. The integral I" Jo

p+ was integrated using the oentral bfference formula

(Equations 7, 8). One more figure was kept in the integral than in p.

8. The residuals were agiun calculated (Equations 3, !+) inoorporatlng as many terms of the difference correctzon as necessary.

- 25 -

AFPENDIX III (Cont'd.)

9. The resdmls ngan relaxed to nearly zero as =n 5

IO. Steps 6, 7, 8 and 9 were repeated, mcreasmg the number of deciml places as the mouraoy of the solution moreases.

Conmnts on solutions obtained

For smll ad negat*tlvc values 0 f 0 the process took longer cs lLa-ge changes 1~ p were neocssaq to relax a smll residual. Also the resduals had. to be calculntcd to more deound places th,an would. be nee?teL If 0 were large. With negative values of fi the usual solutions could be obtau-d up to il crltlojl value of f(0) but the reversed flow solution ooul?i not be obtolncil unless the uutral estunate of p was ma& \nth very high accurnoy.

Khen 6 = 0, the boundary lc.yer thickness lnoreased very rapdly as f(0) approached -0.8. No solutions were obtanod for f(O)< -0.8 because of the extremely large values of q ct whxh p+l.

- 26 -

Solution of the themd bounbv~ layer equstion

When solving the differential equation controllmg the temperature field. for high speerl flow it 1s ususl to neglect those term conoernmg dx- sip&ion, dxslpatlon bang allowed for by consdering the effective tempera- ture of the movmg gas to be the static temperature plus 0.85 of its kinetic temperature at the edge of the boundmy lwer.

Thus the equation we have to solve is:

aT aT h a2T u-+v-=- -7j ax ay pep af

. . . . .* . . . . (1)

mth the boundary conrktlons

y = 0, t = Tb

y-rm t=Tg

Substituting the vsrxables n and f(n) from Appendix II and replacmg T -TjJ

T by 8, 0 bcmg Cefmed as T _ T , we transforn Equatlon$)into e simple 8 b

second degree, f-rrst order equatxon which can be resddy solved by sepsratmg

the variables snd integraturg.

Equation (1) becomes

i?fi + Pr f(v) -B aTI2

=o . . . . . . . . . . . . . . (2) a

mth the boundm,y conbtlons

And the solution for 8 is

e = 3; J(V) = rl

J e -F(l7) dq ; F(7) = Pr 0

- 27 -

AET?ixDIx Iv (Cont'd.)

We cm nlso fmd that . . . . . . . . (4)

It 1s impoptsnt to note that the abwe solution implies that Tb and Tg are constant. If one or both of these temperatures vary, mpxtant changes in the temperature profile an? heat transfer my occur.

We defme the displacerent thic!mess of the temperature boundary lwers as:

6P = r (I -0) (7y Jo

= m /$(I - e) a11 . . . . (5)

G

or, on writing ZtW for J

(1 - e) a17 0

it* = &F/F Zt,W

where Zt* 33 a functmn of C, fi and Pr . . . . . . . . . . (6)

As the origin of the boundsry layer 1s indetermnate, It 1s desir- able to elmnnate x from the above expresmon. Because the flow 1s that over a sem-lnfimte wedge we have

Dlfferentiatmg we obtam z = - 13 vu, S 2-p d(U/Uo) *- *- **

a (V, 1

Squsrmg Equation (6) end elminatmg x usmg (7)

p zt 2 a. (VJO) = .# a (X/s, .’ I

*. (7)

I. (8)

where ht* 1s a tictlon of C, 3 andPr only.

- 28 -

ApFmllIX Iv (Cont'd.)

Dlfferentiatmg (6) mth respect to x, ZtW and P being constant we obtain

Substituting for St* from (6) the above equation becomes

d (at*/,) Ji&-= (1 - p) ztx2 a (“/El) u/u, St*/, .JRy

. . . . . . (9)

In order that we my solve the boundary lsyer equations we still require a cotmectlon between G and Zt* or I-'. The expression for a constant surface temperature 1s obtained as follow:-

The heat balance equation for the surface coolmg 1s

V'3 CP (Tb - To) = a ( Tg - 'pb)

Substituting for a from(4) we obtain the relatzonship

. . . . . . (10)

Note that the left hand side is a function of C, 13 andPr.

The oonnectlon between C, Zt', St* and the functlon~.& for isothermal

flew is obtamed by elmmating x between the two equations.

2nd

v u,s I-

G zt’ - u v

0 -= - bt*/sfi . . . . . . (11)

- 29 -

mNDIx Iv (Cd’d.)

By a sindlar mans wa can shovthat:-

Tb - To Pr v = Tg -Tb ;?F . . . . . . (12)

If the fluw 1s non-isothermal, mtiications to the above equatums are necessary.

The heat balance equatmn becorres:

'pb 'Pb,c tTb - To) = a (Tg - Tb)

Assuming that both the heat transfer coefficient and oooling sir mass flow are reduced in the same proporl5.on by the temperature ratlo, this equatmn my be resolved into a slmlar form to equation 10, 1.e.

The values of j&m+%ayb e obteaned by mltrplymg the iso- Tb

therm1 value @ven by Equations (11) and (12) by a function of F, tcnta-

tlvely given by the equation: L~L3

. .

. .

. .

(13)

(124

- 30 -

AEmNDM v

Note on porosity. permabLLit.y and pressure drop in sintered mtenals

A porous material has two characteristics, the porosity and the perm- ability, the former bemg the ratm

Specific mavity of m&it - apparent specific qavltx Specdxc 8;ravlty of metal . . (1)

and the latter being defined by

hlnss flmr/unit area = permabillty e2 CI c3.l *- l ’ **

(2)

2 end. permeability has the uuts ft. . The permeability is constant only lf the flow is lamrner in the pores. As the temperature variatmn through the blade thickness is sm.11 the flow through the mtal m?ji be assumd to be isothermal. hence pressure drop

= MESS flmr/tit area x 1 x !.I man permenbillty x p nesn . . . . . . (3)

FIG. I. p- lNCLUD2D WEDGE ANGLE PARAMETER t -

-IT

0 I 2 3 4 9 6

. -I - -

0 I 2 3 4 5

SOLUTIONS OF ISOTHERMAL BOUNDARY LAYER WITH GAS INJECTION

p-0 (FLAT PLATE)

FIG.2 p ~ INCLUDED WEDtE AWL2 PARAMETERt=-&i $ “f f-

ll-

CPRANDTL ADVISER = 1x713

I.0 /

09

0.8 I /

07--

06 I I

I I Q*5- 04-

I I I I

I I I I 3 4 5 6

0 0 I 2 3 4 5 6

SOLUTIONS OF ISOTHERMAL BOUNDARY LAYER GAS INJECTION B =0.2.

FIG.3

B = INCLUDED WEDGE ANGLE 17

PARAMETERC=-fii:@

(PRANDTL NUMBER =o.~l]

--

07

0.6

cl.5

0.4

03

02

0. I

0 I 2 a 4 6

:; I

I- c” ‘I al

06

07

06

05

04

03

02

01

0 0

SOLUTIONS OF ISOTHERMAL BOUNDARY LAYER WITH GAS INJECTION B=O-5.

I-0

o-9

0.0

0.7

0.6

IO

0.9

06

07

0.6

0.5

0.4

o-3

0.2

0. I

0

FlG.4. = INCLUDED VVEDQS ANGLE PARAMETER C=

l-r

I 2 3 4 9 6

0 I 2 3 4 5 6

SOLUTIONS OF ISOTHERMAL BOUNDARY LAYER WITH GAS INJECTION J3=0-8

FIG.5. p:INCLUDE; WEDGE ANGLE PARAMETER C= fl s E

(+RANDTL NUMBER=D~I)

0.9

n.0

-7

06

0.5

04

03

02

01

09

00

07

06

05

04

03

02

0 I

4 6

SOLUTIONS OF ISOTHERMAL BOUNDARY LAYER WITH GAS INJECTION

p=i (STAGNATION POINT)

FIG .6.

1

Nu% CaIDEPENDENCE OF THE RATIO= UPON

:TION & PRESSURE DISTRIBUTION

0-I L 0

a-2 ? 4 0.6 0.6

DEPENDENCE OF THE FUNCTION +i.=,UPON INJECTION & PRESSURE DISTRIBUTION.

-0’

6\ h 1

t

\

t

\

C I I

04 0% 0 &WED? ANGLE)

I.0 .

DE PENDENCE OF THE MOMENTUM THICKNESS A, UPON -&lE - PRESSU,RE DISTRIBUTION 1 ..A.

& MAOUNT OF INJECTION

m 6

m 0 0 -; n n

FIG. 8.

nJ

GRAPH FOR THE EVALUATION OF THE “e QUANTITY Zi’FROM THE FOR PARAMETER

ht*AND THE INJECTION PARAMETER C

\

\’ \’ \’ \ \, \,

FIG. 9. - $

9 R

GRAPH FOR EVALUATION OF THE QUANTITY

2”” FROM THE FORM PARAMETER ht* AND THE TEMPERATURE PARAMETER.

FIG .lO.

GRAPH FOR THE EVALUATION OF THE INJECTION PARAMETER C FOR A

GIVEN WEDGE ANGLE@)AND TEMPERATURE PARAMETER

F lG.h.

I I

IO

09

00

67

00

OE

0.4

3IP 0.3

02

0, I

0

L 1 4”50’

@)~ETAILS 0~ w2/700 NOZZLE GUIDE VANE CASCADE.

0 01 D2 O-3 0.4 0;5 0.6 0 7 0.6 0 9 I 0 g,,2 SI = I 173c. 52 = I ossc

(b) VELOCITY DISTRIBUTION OVER THE SURFACE OF THE W2/700

NOZZLE GUIDE VANE.

GRAPH OF SI,a “fi AGAINST i!i,z FOR THE

W2/700 NOZZLE GUIDE VANE WITH NO INJECTION

FIG. 13

T 0

0

N 6

n 0

d ti

VI 0

ca 0

I- O

ul 0

al 0

0

HI2

GRAPH OF be czF AGAINST ~7, FOR THE

W2/700 NOZZLE GUIDE VANE EFFUSION

COOLED TO 600°C

NOTE ALL PHYSICAL CONSTANT AT FREE STREAM CONDITIONS

FlG.i4.-

-39

CONCAVE SURFACE CONVEX SURFACE 3 0 --__-~ ----- ---

25

I I I I I I

IO 08 0.6 04 02 0 0 2 04 0 6 00 I I. r

S2 5,

I

(ct) COOLING AIR VELOCITY DISTRIBUTION REQUIRED TO COOL A NOZZLE GUIDE

VANE TO 6OO’C IN A GAS STREAM A T i.ooo”c (APPROX .S~LUTI ON)

NO INJECTION

INJECTION

IO 0 0 0.6 OS4 0 2 0 0~2 04 06 08 IO 3c II

&)C~MPARISO~? OF HEAT TRA~FER WITH & WITHOUT INJECTION (‘APPROX.SOLUTION)

NOTE :- ALL PHYSICAL CONSTANTS AT FREE STREAM

CONDITIONS

CONCAVE SURFACE

I-lb.13.

CONVEX SURFACE

01 IO 00 0 6 0 4 02 0 0 2 0.4

r 0:” 0.0 IO

(a) COO&i AIR VELOCITY Dl~-iRIB”TION REQUIRED TO COOL A NOZZl% GUIDE

VANE TO 600°C IN A GAS STREAM OF 1000°C

IO 0 6 0.6 04 0 2 0 02 0.4 06 0.0 I.0

C~XOMPARISON OF HEAT TRANSFER WITH AND WITHOUT INJECTION

c

F

-

RELATIVE WALL THICKNESSES OF AN EFFUSION COOLED NOZZLE GUIDE VANE

F IG.16. J

D 3

” 3

3

D

3

(aJDETAILS OF A NOZZLE CASCADE

FIG.17

04. 05 0.6 0.7 0.8 0 9 I.0 I I I.2

SURFACE OF A NOZZLE BLADE

NOTE AU PHYSICAL CONSTANTSAT FREE STREAM CONDllYONS. FIGIS.

I I I I I I I I I I I I I

COWA% SURFACE CONVEX ZRFACE

Ca3COOLlNG AIR VELOCITY REQUIRED TO COOL A NOZZLE BLADE TO 600”IN A

GAS STREAM AT l200’C R&x105

3.0

-2.5

CONCAVE SURFACE CONVEX SURFACE

(&I COMPARISON OF HEAT TRANSFER WITH &WITHOUT INJECTION

RESTRICTED

- .-

!i t!r E

ri t w - z!

\

i -2 PI

2

--

1

-

W E -- 2 111

-.

_;

-

RELATIVE WALL THICKNESSES OF AN EFFUSION COOLED NOZZLE BLADE

REST4ICTED

C.P No. 165 (15,548)

A.R C Technical Report

CROWN COPYRIGHT RESERVED

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1954

S.O. Code No. 23-9001-65