THI-E UNIVEIRSITY OF MICH IGAN College of Engineering Depa-rtt-.ent of Mechanical Eng ineering Cavitation and Multiphase Flow Laboratory Report No. UMICH 03371-17-T ON THE EROSION RATE - DROPLET SIZE RELATION FOR AN EXPERIMENTAL STAND WITH A ROTATING SAMPLE by J. Krzyzanow'ski', Typing and reproduction supported by: National Science Foundation Grant No. GK-730 May, 1972 Presently on leave as Visiting Scientist fron the Institute of Fluid Flowv Machinery of the Polish Academy of Sciences, Gdansk, Poland.

ABSTRACT There is still little knoxwnr about the droplet size-erosion rate patterns for wet steam. turbine st:cgeso It has been already proved t]h-a: the structure of the droplet stream may influence very much some of the impact paramleters governing the erosion of the turbine blading (ref. 1). In order to shed mlore lighlt on tlhe problem, a series of experiments was initiated at IFFM. In the test stand of I1FFM the structure of the droplet stream may be readily controlled. The aim of this report is to present the algorithnm relating the structure of the droplet stream in the mentioned stand 7ith some imnpact parameterSo Also an attempt is made toward preliminary evaluation of the experimental results preselced in (ref. 3)c Institute of Fluid Flow Machinery of the Polish Academy of Sciences, Gdarns3, Pol.andc

TA3BLE]1 Oi CONTENT'S Page 1. NOMENCLATURE................................ iii 2. INTRODUCTON.................................. 3. FORMULATION................................... 4 4. SOME RESULTS OF NUMERICAL CALCULATION.... 11 AND SOME EXPERIMENTAI RESULTS 5, CONCLUSIONS................................... i 6. ACKNOWLEDGE-MENTS........................... 17 7. APPENDICES...................................O 18 APPENDIX A............................. 8 APPENDIX B............................. 21 8. REFERENCES................................. 31 9. LIST OF FIGURES............................. 32 ii

1. NO MEE NCLA, A TUR- E A' N t: ~\ c L.L 2 r p* C c1 air velocity, rn/s C droplet velocity, m/s D mean dianeter of the rotor, rn Ds mean dianeter of the sai-aple, m s Yo o E -D /2 coefficent of droplet separation s E rationalized erosion rate r F see Fig. 3, m G constant,m /s 1 n(r. c) f Ar number distribution function,l/nn n Ar nT k factor of proportionality, see Fig. 4, mg/m L see Fig. 3, n n (r*.C!) numnber of the droplets of a given size r per unit area N number of all droplets per unit area N normalized erosion resistance a N 2 3 constants r* droplet radius, mn u circumferential velocity, m/s U water flux over the sanmple surface, nm /m s a UA see (18) and (B7) U nmean value of U, n1 /2 s iii

UA see (19) and (B39) U manaxiniurn insta. tal-a,)Lus rva.luc of the volumetric material less pei unit ar(ea eM and unit time m / n s UEiM see (23) and (Bi3.) - 3 Ue,4M mean value of U m /n ms eM UEM see (24) and (B12) w* droplci s velocity in the relative frame of reference, m/s normal conmponent of wxv, nn/s *N' W *N"N' I' AR AR M M mean values of wN, m/s W aair velocity in the relative frame of reference, m/s o Q, 7 see Fig. 2, Am loss of material eroded, mg AM* amount of water supplied onto the trailing edge of the flat plate per its length AR1, kg/s AR the length element of the trailing edge of the flat plate, m A(Am)/Al the maximum slope of the curve A mn -' (T ), mg/min. K parameter, see (BI) viscosity of air, kg/nMs o 5C.p n see Fig. 3, om see Fig. 8, P density of air, kg/mn3 P* density of water, kg/mn3 T tinie, riinut cs iv

2. INTRODUCTION There are some chaaracteristic features of the kincm01atics of a droplet motion in the axial gap of a stean-i turbine blading. The neiglborhood of the leading edge of the rotor blade is exposed to the impact of droplets whose radius is a function of the location of the blade surface element. The further away frorm the leading edge of the blade the surface element is located, the smaller are the maximum and mean value of the droplet radius. Also the angle of attack, which is a function of the inclination of the blade surface element and the droplet: size, change s Another characteristic feature of the droplet impact intensity is its strong dependence upon the structure of the droplet stream.lT This had beeni proved in (ref. 1), wh'ere the correlation betwveen droplet streairi structu.re and the impact parameters was considered, In the experinmental investigation of the material removal-tirne patterns for different material, these characteristic features of the droplet stream are usually not taken into account: 1. The droplets, no matter whalat their size, collide with the sample of the material under the samle angle of attack. Usually this angle is 90 degrees. 2, The structure of the droplet stream is usually relatively homogoenerLous, In general, little is known about this strluctuure and often not even the droplet strean-i is used in tests, but ratller a liquid jet,

-2It appeIrs, however, that both the angle of attack and the droplet structure have sublsta;ntial influence upon erosion rate patterns. As far as the influence of the alngle of attack is concerned, some information is already available. One comes roughly to the conclusion that the normal component of the impact velocity governs the erosion. Much less is known about the influence of the droplet size. Only recently (ref. 2) an attemprt wes made to draw some preliminary conclusions from the meager experimental data; it is there assumed that the drop size effect can be represented by a factor of the form 2 W* 2 = const where the constanlt represents a critical or threshold combination of velocity and droplet size, such th!at, for w N r.. constant no s.ignificant erosion occur S In order to shed -more light on the droplet size effect, among others, a test rig with rotating sir,.nple was built in the Institute of Fluid Fl'ow Machinery of the Polish.\cademy of Sciences. The sam-ple intersects once per rotation the cdrol-J1et stream generated in the aerodynamic wake of a flat plate. The particular feature of the stand is that the droplet streanmz structure many readily be controlled by changing the air velocity in the test section, the amount of \vater per unit tinme and unit width of the plate, the length of the plate, the sh:ape of the trailing edge of the plate, etc. Mr. B. WMeigle design,::d the s':ta'nd; he and Mr. HI Severin are in charge of the See extensive reference atd.ta in (refo 2).

-3experimentSo The outline of the stand is shown in Fig. 1. More details are available elsewhere, (ref. 3). The aim of this report is to present the algorithmrn relating the structure of the droplet streamn writhrhnaan values of some selected impact parameters. The structure of:lihe droplet stream is defined by the droplet size distribution function. This function is assumed to be known, and to depend upon the droplet size and air velocity. Particular attention is paid to the mean value of the amount of wrater impinging upon the unit area of the sample per unit time, the mean value of the product of this anount of water and (WN/2550), and the mean value of the impact velocity. A preliminary evaluation of the experimental results presented in (refO 3) is made,

-43. FORM'U L.ATION, Before we fornimulate t-he conditions of the droplet impact wvith the rotating sample of the IFFM e>perir.nental stand, let us consider the kinemrnatics of the individual droplet, In son-e prior authors' papers, it has been indicated that the droplet motion in the aerodynamnic wake may be described with sufficie-nt accuracy by the equation: C =i [1 + A' 4- A+ + 2A'z] ^ c, is t+he droplet velocity in the absolute reference frame of coordinat:es, The relatio:nship between this velocity, the gas velocity, and drop]et size is shorwn in.'Fig. 2 Thle group of paramneters is relevant to the IFFM stand. The droplet velocity w': in the relative reference frame of coordinates: W* - U (2) and the droplet path in thlis reference f-rame are also shown. The conditions of the droplet inpact nmay be easily calculated under the assulr-ption that the droplet path shown in Fig, 2 does not change its shape in the neighborhood of the mamnp]e. This assumption has been evaluated in Appendix 1 in:-nmore detail. Let: us novi consJider first the amoun:j.Ct of water wvhich hits the unit area of the sarmpl.e per unit timie.,'i'This parameter of the droplet im.npact is wrorth considering because it has besen:' al.re ady shown that the erosion rate is proportional: to it. Let us assu-me tlht thle structure of the droplet streaml is;

defined by the droplet size distribution) function 1 n(r*, c) f(rPC) n *~ 1 Ar N f(rc) - -^ _(3) and that this function in the point where the sample intersects the droplet stream is given by the equation N2 -N r f (r*, c ) = N1 r e, N11N2,N3 f(c ) (4) Then the number of droplets of a particular size r, impinging upon the Ds surface element A R1 ~-, d Y of the sample per one crossing of the droplet stream is equal to D n(r,c ) L F = n( r,) L, -2 d sin c (5) _X_ ^ 12 cos a(rc ) The volume of the water carried by these droplets is equal. to D -, L, ) dcp sin (6) 1 r n(r*,c) 2 d sin cos C(rc ) (6) s inc e cos —(r,c ) \ u sin y 1os.(r c ( r ) (7) and the frequency vwith which the samiple crosses the droplet stream per unit time is u /h Do Hence, the volumetric flux relevant to this volume of water is equal to 5 U(rc,) -- IT r LN f rc ) si.n cp 1 6 a'~ l'" ^ ~-x nf *'1 * u sin T7 TD AR1 (8) The product LN mary bot celimrinated bly r:means of the continuity equation

-6for the liquid phase forml fordla for tlhe cross- section 0-0 of the drol)3et O4 3: ^ C, CA Z L * (r c i) I P (9)- r r -0 The sumnation, whiclh mray be replaced by integration is extended over all the droplet radii in the streamrn Instead of TEqo (9), one may write: oo 3~ 30(10) L)fr f(r c * (0 Hence AMxLN co 4 G 4 11 pr r3 f (r C, ) c (r,c ) c ] * n * 1 * * 1 * (ii) 3 o (il) For the particular droplet size the angle oc (r,, c,) nlay be expressed by the equati on: arc tan. (12) wh e r e (r C )- ucos ) w*(r*,) u 2sin 2 tann a( - sim sin 1 - -- - \ w*^( r* c) /{ u- sin y \ 2 (13) W- r., c) (13) The sign-function takes care of the appropriate sign of term tan c( The second part of the R-1H-S of the Eq,. (13) may be easily deduced from-i the geometric relationship sholwn in Fig, 2, It nust be remembered that for each drop]et size r, we obtain a different c< or, in other words, o(0=x (fr Cr) Hence, in order to validate the q — (S) for all droplet sizes, the sinl~ functioiln in it must be replaced by function Fl(P), such that (see fig. 3):

-7F (cp) =0 for c < cp ( r c) I- n* 1 2 nK ~ 2n 1 2- *n' ( 2 F()=sn [ cr r ) + ] for Q (r r, c) 2 - P n( 1 2 F (p) = for ^ > cp(r* c) + where: n( *i, ) c Pn(r 1ilc) + C(rill) - c( r iC) (15) i= 1,2, 3...., also, (Pn(r oc) - and o(r,c ) (r c ) (16) n *0 - 2 *0 1 *1 Rearranging (8) by means of (11) and (14) results in A AM r37 f (r, c) w( r, C il) o U(r* c ~CD) = ~ a —'' p. T AR oo IT D sin 7 P* 1 3 s f r f ( r *. ) c*( r*,c) dr o sin [c? - n(r, cl ) ] One establishes the. volumet'ric water flux U (c c ) for a surface a 1' element located by Y extending the integration of the value U (r, c, ) over the wh-ol.e.r-nge of droplet size. I-ence one finally

-8obtains: 3 1AM 1 r* n'r,e1 *( rc ac 1,.. p AIR I. -1 In order to eliminate the dependence o-f Y one may average U (c tP ) a 1 over the circumference of the sample IC Hence: p AR 1 D sin ( * ax c Ua( ) II o o rn( *rnfax'1) 2J r*- fn(r*,cl) c*(r,*C1) dr* * n *7 (rc1) (r,c ) gsin [y - cp (r*,c ) -] dr dcp *x fn~rrt C1) i~X( TX 9 - 1 n I' 1 2 * o - 1 Ah - cI ) ( 19) p AR1 + D ssin in The erosion rate is odbveiopuslny notf fl tLh oe nflu of water erpinging upon tce surfacuce elof lent of tle rtti sample. It is alse a. function of the ifnpa.ct velcity.- Thi-s reilatiosahip. may- be expressed as an empilrical equation inP tARe Co (refsn 4): Tm) — r 2/ f p( -0 w25 m/ n ) a'(20)

-9-'br a flat sanpl.e ti.:e cocff.ic:ie.nt F p. Other symnbols of Eq. (20) are explained onc Fig. 4. The mr'axi.nmuim ilnsntnta.neous erosion rate Am) is according- to (ref. 4) and E'q. (20) equlal to: AT A - k Ara - A,- N Va2550 el (21) a Here U represents the rna-cxilrnur instantaneous value of erosion rate. em These empirical relationships have been established for the flat rnateria.l s ample attacked by a, relatively horn-ogeneous (in size and direction) droplet stream. Application of these formulae for the case of cylindrical, rotating samples deimands as su m.ing that the erosion damnnage caused by the droplet groups of a particular;size mary be superim.iposed. That assumption has not yet been proved anic'd may b)e accepted here merely as a first approximation. Under thiis as suniption, th e e theoretical evaluation of the. mnax — imum inst.an'tareo uzs value of Trosion rate may be lased on the value: /w_( (3.c )\, ~ UeM(r.,c].,q7) = 5 U (r,5C ) * I 2 (22) its integral. extended over all droplet size: U r = ooU AMd 1 _ r3 f (r, ) w (r C n) o- 1 f rX n(r*,cl) e*(rTe )df r 1 * UeH( C, C - 5 Ue d~r - j -'^-;-^^ ^ ^ ^^ eM I eM ^ p AR TI D s in 7 - oo: 2550 ^ ~ (r- nr c) +' C-] d I AM (25 ) P A\T I) SJ.3i P - * I

- 1.0 and thle -me ian valu.e: co ( r,e ) ~- n1( *max'm x 1 2 eM 1 (r IT eM 0 *mrax 1r 2f fr, (r.,. ) c (r, ) dr. 2 (2 * Rp AR D sin y 7 2 1 s The relevant prograrns of:Lu137c ncal Cc aeulat:i.ons of these inmpact pararneta e r s are presen.ted in Append';x A.A.;il In Appe'rdci; 13- i. addition, the equations:1(: local a nd me. n value of t;e i nA., ct veloc ity ar e give n.

- 11-. 4. SOME RESULTS O -, )U'IR..,A C.LCULATIN AND SOM I E XPE —. INIL, F TJ, AI R ES U TI. I'.'S The theoretical analysJis'pr.e nted in the previous section a.d.in Appendix B had been trgiggered by in: re sting epe rir-en'tal re sults of B WTeigle and HPi Severin (ref. 3). T'ihey had fou.lnd tha.t for constant mass flux of water, G, supp)ied per unit: width per unit tine onto the plate, for constant circurnfere:ntial velocity, U, of the sample but cha-.ng — ing gas velocity c in the range of c - 60. thro og. h 75, 92, 153 " -I S s the erosion rate chJ.a.ri e s ub')s ta-n Ii'i11y A fra. grn'-e -ft of the exper iiriental results in th;e form,) of the rela.tion t.Ai. f(Z) is s:how\vi n Figo 5. The':"(: is also showvn thle dr ople t size distmib1ioiL functions.elevant to the gza.s velocity c 1 It is apparent thait v-,i:.' the incr.ease of gas velocity c1 tlhe dispersion of the droplets ic:r- reac.:-es, and the representative mn'ean values. of the droplet size r r- (r. 0) dr* (25) decreases, a.s does tile erosic.l: rJ.te Ire-pr:esented by the ratio A(Am)/AT defined in F-ig. 4. This significant clhange i:n er: i n rate may be a result from- several different effects. One of thes-: r"y. be the cha-nge in am.iount of water i:.npaceting the sample-n-.c per unit. a.i a-. u a..it timeo As a m.atct:er of fact, I-ey'mann's empirical relation'!' (] Eq. 20) say},'S- t'hat th' e mla.>x.i.mr-lum instant:ainGeous

value of erosion ratLe-./(Ar)/z^'' U- is propc)rtiona]l to the volulrlret-ic water flux IU over the eroded surfac. e of the sa.-mpl]e. The oth.er reasLon a for the change of experirmental value of A(iAm)/AT may be the change in the W1 5 term U (-2 --. ) It is also proportional to A (Atrn) / a 2550 Numerical calculation presented in this section have been perform-ed for the following set of pa-ramete-rs relevant to the stand of IFFM: c 1 60, 75, 92, 153 n-)/s u - 200 rm/s 7 - 80 D - 7~10 m s Ar - 10o 10 m 6( - 1.85 ~ 10 kg/ms P - 1.205 kg/ ns P* - 1000 kg/mns z - 0.3 m In Fig. 6 the results of numnerica.l calculat.ons of'UA and UliM ar'e presented. Both U.A and UJ-M a.re proportional to U and U respectively a e M (see Eq. 19, 24), UAT and UEt:i. decreaase, as does the experimental va.-ue A(A-: )/r wVhen c iincreases, The trend of change of both theoretical and expelrimentazl va-lues is thel same1. H'owv —ever, it mray be readily slh'iowin that c n As ao U 7P m/s changes nmchic faster t:la.n do IU.A^ / UAC r a-7rnl 5lE C /UE MC ^1 ^1 " ^1 ^1^]

-13For c 1 153 rn/s tlere is a rea.-.dy a difference in order of rnagnitudo betweenthe theoret:i.cal ancd expe r' nei nt.] valueSo. This diffe:ren:1ce clearly may not' be explained as the consequenc o' aver agi-)nl o or the- ass u.Irptioon conce.rling the superposition of eros-ion caused by the droplle t group s of particula.r s ize In searching for a rea.son for thlis discr:epancy, att:ention nimst: be paid to the problenm of the correl].a —.tion betwCeen the dr-oplet stream structure a.nd the erosion ra.te patte:rn. T!he prog.ram of experi-rrlenfts of Bo Wefigle andc M. Severin is parti cula3ly suitabl3e for these investigations because their stand makes it possi blel e to c:hlage thie droplet stream stru-cture easily for almost unchar ng ed m ean ian mrp a.c(-i vrloci.-ty (Fig, 6 ) Little is known, h, oweve-r, about the droplet size-erosion rate rel]ation.-~ ship, and only in exceptioonal epleriments was the droplet size investig.td c Hovwever, it. has belen indricated (ref, 2) thatll the droplet size is proba.blyJ rel ated to the.axi.im inst.ta. eo- s vatlue of rati onalized e r osion ra te, through the influlence on its; threshold valueo The rationalized erosion rate has been defi ned inr ( ref 2) as f o 1 lows: Volumne of nia-re2.ac. l l]ost per u.nit; area per time E r'Volurme o:'i-.'uid iinpinged per uJi.t a:re-a per unit tinre In the nomenclature of this:report: Vol-umne of ma.t eril lost.er unit area. per unit time S' A(~m l T L and Volunime of liquid:impi.n d per unit a.rca pe unit time'~' AM E ( U ) - a 1' D sin 7A

-14In Ref. 2 Heyrnlann suggcsts ii' ci:oplet sizecerosion rate dependence in the form of t: I. e f o 1. o i nl p o ss i) I e r e la t.i o s E= f -L 1 or = f w r 1 *N ~2 S o r 2 - f -N "1 (26) For both of these relations, -w:.hic'h s.trictly speaking, are not ye(t fully G confirmed by experiment, t.he ter..-is 2 and G/r relate w r *N - to the threshold ilmpi)act velocciy,', such that for ^ < v no erosion results. G m. ay b:e conu:si'dered as a mnaterial constant INowT, the experimental data.. of (ref. 3) for an alurninurrl sa.rple e, part:.ially quoted in Fig. 5, ila.y be used t:o dete:rmrnine this consttant GC, anld thlls t'le relationship between, N and rL It ma.y be done by means of thpe plot E = f(r,() because in the experimentls inenti:onled the mean v'alue of the r impact velocity w was wcCal'o;st co:nstant, In fact, for u - 200 rn/s the "P N rne an irnparct velocity cha.:ge: only 1e e tveen N 1 11]9; 117.1; 15. 5 and 113. 5 m/s for c -- 60; 75; 92 and 153 m/s respectively. Thus, frolm the appropriate:; extra..li-ation of the functicn = f(r,) (see Fig. 7), for a given impa.ct ve.ocity r;. = -16 m/s, results tlhe (.. () - 6 threshold droplet size r:..: 4!4.1 I.a0 J Hence, for the material. considered G- - d^'e - 1 - 1i8r 9 1 s8

"he relatio~n (iw T he reflaftion =: if (rc) fvor an. a.-l.. o-a i-Y- saI-np]e ulscc in (ref. 3) is shoIvt./:i in the Fig. 6, 11he rc sui.s of t.he Bu sclh and.lo.:ift eC:-xperin:lents (ref. 2) n-ry be here recollected. Th.ey obt'ainie. for th -e r- ir-eoe alurninull the thre shold velocity w - 33 n/s ]-ice,'L:fe correspl(-:l-diln. droplet size dn would -3 be the order 1. 10 mn: which apl' pears qu.al;ilatively rea sona.ble for the ra-in droplet size. It has to be pointe-d outl: here that- -inl t{his rep;Jort on-lly a small part of:-lhe results of (ref. 3) has been used. Inn addition, in: assessment of r, and G, It, C the results for only one velocit:y w h wave bee.n. used. More extensive experimental] data are needed t:o.shed mrore light on the problelro As far as the author i's i-lnfor ited, thle p]-ro lgram of releva-nt experiments is be-. jg continued in Ii\!MMI.L1 5. CONCLUSIONSE 1. The e expei r im.ental i:: e a tga.t lon of B. W.v ].e. and cI t Seve. in, publ.ished in (ref. 3), i.ndic ated sur;b Js: ta. n cl. d c pCe' nd. ic e of tlhe deros ion da-l:l.age - time patterns upon tihe dro plt sizc, 2. These e.xperi-nenta- l r esuti s o1F I F T'A deserve certainly soimne melore consideratiolno To nl-ake it:po-si:'ble:., a. tl-eoret'i.cal rmodel of droplet i-mp'-.,act for the e]xperi-mental staid of h l been pr-: aented in this report. Attention ha.s beetn particuaih:oy paid to Cthe cai.cul.ions of thel m-ean va.lue of the volumetric wa.ter flux U o7ver the sa -nie surfa cce Al.so th.e mean w.-i 5 valuer of the p oduct of water flux,; and: ( ---; —.) a w 1. as tJhe r ee:an imnpact veloc7ity - have beeni. a (cula_ td

3o Conm.p'. r i so o) f ex'-''ip. n-:e tal.1 and tlheor' ct:cal dilal{ta i:ndicacat cl tchat the experirnentall. establ. ished (rat tio)lized:, e e rosion ra.., E, charJCes r faster \vith the gas velocit y c thlan theoretically calculated water flux 1....::tN 5 U and the mean val'e IU 1' h e prod uct U (') a eM a 2550) 4, It is likely thaft v-a.riatij-ton in dnroplet size caluses this discrepancyo 6 The threshold dropJle size of ordalr. r - 4-44 10 m for mean imnpact velocity w - v 116 rn/ for soft al.u-minuni has been established -b N under the assumption (ref. 2) that the droplet size effect can be repres entiedC by a factor of the form: 2 v f,? r 1 const /G 5, The rmodcel presented ofC: t -he droplet impa.ct indicate tht the e pe - mental stand of lFF:/I is part'icularly.3: suitable for expe ri-.ients- designed to shed ore light. on:he pr oblem.1 of cdr r op et iz. ecffe ct:.5 1 It provi.des th"e p.osibili.ty of t'he exception.al change of the -6 mean cdiro:J. s rize b-et tvween r, -'v 50.10 m up to about 6 r - -, 00, 10 "11 5 2 The mean lu of tvhe. im. l pact velocity J.mays' be controllecd indepen d e ntly. t, 5 3 The dropit: - i; t ea s trjucture seens to approac h {;ell strea -om structu -re i-.;.: c7:e ste a3.m turbXine, 6, Furtherh inve it:a of the rela.t:ionshi p between the erosion resistanc an tC1 dlle t oc ae of iI po rbance eca.use in (ref. 1) it has been a.l redy shown that thar mp tA:.'.., J l - rc f a -r lst e i te r.. e s i s er ious ly i:rl e ncr by t}he droplett.stream:- 1; iC h-}~-l1-z. t:il Thus:r the control of the droplet stream

strulcture m nay offer a. powerf. l etho for I the prof.:e ctior of ste alm) turb:ie!) bladi. ng 6 ACKNO T LE DGME NT s This report has )been preparedl during the author's sabbatica] year in the United States, twhliclh was a.rranged in the exchange progra.m betwveen the Polislh Acc demy of Science s Wa, -i. rsaw, and th.e Na.tiona.l Academy of Sciences, Wa.shinigton, D. C.'IThe -.uthor apppreciates the assistance of his hosts in the United Stat. r: Pr ofecssor Jo R. Moszy nski of th e Ulni ve r:. tyr of Delaware, a.nd Pro'fesso.r ]F' G. Itla:,rrnJitt of th-e University of \Michiga;ca.. The author is also inde"bt-:ed to Ihns collea-gues Mr. B. Weigle and hMr. fI Severin for sui-)pplxying son-ic of t:heo ec. xperin.ntal resrlt ts before pub, 1cation(ref. 3).o F'i.nally, the a.utor is gcateful to t1r, FPrank. Heyr-an for numerous dis.cuss; ions. wlhic s tiul.ated prepa ra1i: ion- of t1his reporft.

7. A PPEJICS -,S Th e p ro g rati'-n. of -:1'r:a c3. ucn r t ons. e, en f! r I ate c in A1 ol for the Burroughl]s B 5500 c un -)c.l r Te Iext (of the program. is quolted below. It makes it possible Lt c:.., (r.,c ), o(:rc ) T (l), lr,) *'1 *- 1.~1! —ma1x1. UA(ec.), UEM(c 5Cy), IA(Ce) UiLJ(r) according to the equation, ( Z; ), (15) (18), (23), (19), and (24), re spe ctivel y. The input data hav-e t-o be isec - en.l-ced according to the sequence nunmbers: 900 an:d 1000 wh ere: c - C u U, - i G \, It - T P - OC 11 1 p - RO1, z= Z, 1 ^ - The output da-ta are b'inog'pr,.:)i. nt' *ed' -l: t" ce re sulting from sequence nu -.iers: 2350, 250. ), 50, a':d 5800 where: rTI Cr ()(r T'c) +ll/2 r* - R[']m.x'.r el) -2 FIL-r,'- [ ("' ALFDGi[K] r - J CWfl (r *flhOX' e ) + - a FIMAIKIDG -- -:D(D D[ LI J[ I max N, cq(N) - FIDGR[],, (c ) [ U'M(cl ) iM[N, tc ) -r U~[rK ], orEM( e ) L8TL h[... 1 - V'->Ir [I Lx j ) Ir

l () BG0 P ~00 COMH 0 T B-A D AV 1 A. I 1:KiJAP.. PU0BKI N r K(4I ( i (C 1I)I ('f)0 C AiL,],.:!['T:',D A(',,', I (,1 t.,.;',! ].....A t.....} t,; i I A i ~300 H EAL. 1 G h CAfA, V I,N 33... II.O.'rO 1), CI ALK-, i GAL I' 350 F IIAX D.l:,'i F I!, F.',.:'' I l.) hL-T i ].J.)G GLA } (^ (G Al.. i A/ 3, C At...A. 360 CALKA 5, CA:LMA 6.; /40C I0 [. T' }'G F K.,'.7; hl N; -500 LABE..L h.Tt. 600 ARRAY A,A FN" - J C I } I,,JL.{, It0 s, U;...t. 650'TGI'C L F-A A, f A. F INJ AL AD FI i" f7 },,' i i?N }: 5 1, 660 A3. A3iA Al. 5:., U h..Ii' }SR I, F I. DL'H [. I21 I; 70)0 F' l.. l A (33 Hi,3''. ( OTL 9 ) 80 C) FO:-i AT YU.iR A'i': - ( hi 4 3 ); 85( F 0R MAT IFO-iMfAT2 (X ) I 900 (:TI:E HREALI t T A33., CA GIIJ, G Ai4,AA, DR.,'I HO., Oi. IZ); 100 ) FRiAD (tJAK33, /^N. 1I.N22. N33); 1 019 CALKAl:=0.; 1020 CALKA2: =0-O 1100 K}:=2 1200 RHK-1 3:0; 1205 G [K —1 3:=0 12 1 0 H K- 3 =O 1220 I K-1 3:: 0J 1230 tJK(-I1 3:=0; 1300 F 0R.: -2 S ETIP 1 DNTIL 51 )D 1400 BFEG I 1500 R] C K = CI: K-1 3 D + 1600 FN } K 3 I \: 3 *N22\ E XP (-N 33\R K ); 1700 A1I.[K3 S O). i (( \tN I \O)/ (HRiLK <:3\OI-1 2\,1 )); 1800 CK R I: 8o \C G I (1 1 /(I + A1 3 \. J + 1900 SQPT (C C 1 ( 2\ 2 \I \ 1[}(] \ ) )I 2 >) 1910 W(RKH JRK I( S T (.: I ( 2 ( S IN () A -,2+ CKR}( 3- 1.I \ COS ( GANA) ) 2- ); 2000 G( 1 K 3 * C 1 [ 3 U 3 \ f'iJ I K 3; 20 50 H K 1 ], I[ K ] -; 3 \ C KR I[ K 3\ F'N K 3, 2055 OCI I;RI K I vKKR.RC IAK3\FNIP CK3 I; 21 00 CALKA A ( } 1C K- I3+ Ki 3 )/2\DH}: 2200 IX K3 I t l -13 +C- LKAI; 2.300 CALK A 2 U- t = ( GC i 3 C+ K 3 ) /2 \ DR; 2310 J I K i:= J C- 31 ] + rA L KA 2; 2320 TGAL FA t K =K S I (t ( SIN ( (I CK}(C K - i COS ( HAMi ) ) /JK}X K 3 ) ) 232 3 Wt,}K3 / ( tAMA) \ SO T'' ( IA -N t ) \ ( 2 S IN ( GA ) )2 / 232 6'K:I, I 3 ]2 ); 2330 A.LF' K3: = AF ('ir: ( T (LF,, F A K 3 2333 IF 1 }I L 2 2336 THi N F 1:=2 1 I * 57079 P339 9LLS i F I N' C }:'FI N, C I- 1 3 + AL F A K- 1 - ALF'AC 3[;.23 40 AL F'AD(.- K C -: -= AL FA 1( \360/. f 28 3185 23 45 F' I N i)RP 1 K]3 = - F I r j 3\360 /6 283 85; 2350 WR, I t'3. J( F, AL 3 3 F ri'T I t K.; i N1 ) i (, I AL F-A D:R }- ); 2360 END1) 2370- F' i M hX = F" I:; [ C 51 +! * 07 96; 238() D.) EL, TA F'I = - F TAX/ x _,2i; 2385 DELT'IA F'I DGH: = T}L1i.AF I \360 / 62 8318 5; 2390 J) j 51: ]=3 t 1 C 3. < 3 2400 FI'MiAXDGHs..-..MHAX\;-)0/O.23.3185; 2450 tt I lF, ( J I AT 3 3F l FI.'i<:.T2 ); 2500 WIT F(,JAK33.,FO A I.,J[! 51, I 7 AX P, i)).I^ LT AF"IDG ).; 2600 FOR. N,2 SF=".1:P I ^TIL 21 DO 270 () B. I

-20-.2-7 U i) ) 3j;' { 270a t.. I]; 270/4 I i 1: 1 j: 0.; 2706 9AD O. t.: I; 27 ObU AlJFMSHa I l3 1 (0 WiKT tC 3 ^ - WR C Kt 3 \ S I N (I I [ NI 3 - AL FA I 2 3 +AL PA l ); 3200 0 13:-0 3210 liAC 3: =0; 3250 PF I3. O(, 32.60 iE t.,F 1 3 =0.; 3300 IF F I'j L S ( F I f\J C 1 o 570 79b) 3400 TI } {FN IMA K3 =S 3500 ELSE IF I [ g I'TH ( FIFI.N [ I3 + I.570796) 3600 TH[ N:EK, I K = 3800) 1 570 7 96); 39()00 (AL1(A3: ( i C c t K- 1 3 / 2\D,i; (()0 ()) A (CK 3:= IA CI- I.3GALK C A3 4100 I F l r N I[iJ3 LSS ( EIt'JL;K] - I570796) 42 )0 THN'I [1 =P t0 4300.EI_ S F IF El CI 3 G T ( FI N K] + I.5707' 96) 4400 I-tH Ei FPL 1( 3 4500 EL, S [K]I/ I1 \ C 1 3 \0 K3\ (WK NC 3 /2 () ) * 5 \ N 4600 Si10 N (.'F.I C N I - I C K 1 5 t0 7 9; 470()0 G(ALlA(: =A (1'CK 3+I 3PK- 1 1 ) / 2,\ I.; i480C0 0 H AEM C K I l ~ h - IC I + GCAL KA 4.;'5000 F.' 50 50() IL IT I J A ( I lc'i 3F3 FO R M A'-T I. I N. FI DG RN 3 A N AC 5S 1 3] UF 1ENFC 51 3 I ); 51 00 A3C N I I/ PIX A tC 5 3; 5200( A CIJ3: 1 / I-IAX\U 51 t 3; 5300 ) C AL KA5 A (i N +' 3 N+AF 1 t ) /,2\D E) FLT IA FI; )4 00 UASHf C N 3 =ASH N- 3-i GALKA 5; 5500 ( CALA6 3 - - =( [ ( + - ) \ }1D F TA F I; 56:00 U EI SH IC 3: ^ iJ 1. SR F ij t 1 3 Cf ALKA b; 5700 N).N D; O5800 WRI T. J (4AK` 3 3, FO'.AT I,, ASU 2 1 ]3 I I iUEI SHF 2IP 3 ); 59.00 GO TO E.TI; 6)00 EtN I).

- ZL] — C-, i., c CA..) - -i, In s-,ectio,'nM Z one.'.~, ha;:,s (acceopted c.'te sin'pici'i- a.ssurmpt',i:J.on th.aLt t0I-e presence of the round- s e ssh w le1 Aoes Tnot caor.e tlhe dr-ople t path.-S in the sq~ a m'pl1 e' 11. jb Iei C,". C bo 1. oo-:'o Io I e t'...... in. the sa.3nlIe J ( J..', C) C.i -o "h'Je vaJilit:) of t isC assn mptio ha. s. to!be checked. Ulnfortua.t el, tII solut on of thie eql..ion oIcf the droplet pat>l for the case lund-er c c.nside rat:i.on cdoes notC exist The re do exist, however, such sollutiouls f:or th.e case ~whiere(-ie' the dclopleCt ve;ltcity aind its dlirectioan far from: the circulla.r oIbs:tc cl is equal. to the: gas vrcl)city (ref. 5, 6)o Let us observe, thonug 1., th. at in. t-he ca. se of a. r. a t:iLng s an:p]e 1o Pr smnal] drolplets,':e e l th re se:nce of the sample may io::fluence substauti a ly t drople.t p a., the differ: ces between the gas a - C3d. the- droplt:' vel s fcar fro-) [e sa:,pl?-.e arc' sn-al.]L 2, For large d-rop'.)J th] dJf f1 nces maybe a:- rge, but the pr-se er' o.f tI e *S CA -I P:I dC o S n' I 1 -U[ CT, C l- OL C -'LCC of thle samr~ple does not inf]lue.nice the dropleti: path. One mingoit eC-Tp)ect tthen, tl a- fo'r ccrtai- o. convenienti a. rranc;genl-ents of pa'rameters, one cculd a-pply a.n already eis. ti,, s,-.Lil:.olu for the- - c-ase underconsidoed rati.on(. Let u"s ch.eck whio?.:Le _hr this is notI poss. ible int our caseo )/ a'n d m ( ig. g 8) Both ar.c funt'Jctions of the para met.e-rs 2 2 P x- p w 9 * S/2 16' p ~ 1) /2' 18s ) -:'-:'(1'32). p^ (i)

t -. 1o0 the pa ran-i c.';.,'1 %a;,t L()1t io Sthe S{:I of Irivi and'X 200 rn/-' w' -> 106 in 5 10 O0 40 60 80 1(0 1.000 K. ~ 17, 2 69 0 286 915 2540 4520 6900 69000 E ~ 0. 830 0. 920 9977 0. 990 1 0 1 0 1. 0 1o 0 c(p Gra d L 450 5.5 1.50 1. 560 1.564 1. 66 1o 568 o 5 m It appIe ars thai: in). the case nA-.I de-r cc) J-.(1icraic):i o nlyi y for drople ts s asal.aie *-6 than about' r - 50o 10 6':( is ithe rddroplet pat slig,']tly rod lif(cd ine L. t(e neighborlhLood of:the1 sa l-rip For.tee sa: ll dropclets, the aplicai of tihe assl.lii.:on 1 l.. t. ca ople1t an.,s ve lcit ar froml th:e obstacl. cdoe: no. differ s'g fiea' see to befly jstified:,I..; A: f ut I I Y- u's 1f i o QOne may aii n as; in Sc l:0;o 3o. hen. 1 i,( a' r * f. (r, e ) Ty(:.' c ) The new el.e l'n a iv;l-s of thiele ox1O en - 1: ac u ri' a-L ex: c apineda i an Figo 80 Nowy aoo, (-. -, () IT f 0x AR I' i) s. TjI D 7/2 D D/2 I)

fie) (-..) - C i 0, ( -'........-'1 sir ~(-7 (-f c 0) =~ for' > C T i enc:. - Ay-(r:U....) ]. q;o(n <:.....s- C r-Ic par amete r, of t o l: pt: 2 e O' ad t r a,.v J ~ are o a 1'i U" AM> 0O r m (r ec )w (rc),, __ I ^ I ^ __J^ ^^^'^J> 3. 31 I Ta- T. r-= --.. o _ —......r,.. ~ ~".- (li*f J ( ~'ol| (, e7i ) ^ 9) r' C l _ l c(c r. II. UA(c q.) [~ Ii n K> p AR1. B s8n 1' (BJ7)::,,: o ~:.::(r~, c. ),, (~o),._- x- ],, -n —.''-x~ 1 m 1 U~~~~~~C ~ (l -',............................ Cf;.......'f~~ CO':'~" A~ ]i""D 2 q: _l_,~i,.3u

- Z4,I e rC a. gain the cos. func). c t tl3.'.. — i; sc I. th 1,at. 714 rj- ('-' cI --- (~c fo r < (l ( K' co aS ( T. 0 for p > cP~ +P (Ol The omean aluv e Ua(c ) of the flictiaon U (cp) calculated f or 0 <q< p< (g+"(- V ) r rmray bhe olbtai ned fromar:..-ax -:i 1 A<i. 1 (<- 9) ).c a 1' 1n 0 <( ) - --- (q. Uq'k L ciax ~~~a~ ~~~~~~~(C[n M r c ~>naxC 1 J r-x fl((r,c1)c (rcl)dr -1 p*, R1 I[ D s -in 7ra - ir9 ~ ( o~ ) c a f(c ) r 3 / f( r f) (r.e ).- ( c) ~, cos )~ -cp ir 0 D s2 2c) 2qi n n P AR Il D sin _ (n + ) ) n m r,c m3 / (I' \ )c " a__., 1 f r, f ",c c rc: o )dr a

In o d (i:r o) c..c. A'i _ a U J:, a.,s. o t.,.. in t ac count lt a1. Ic.l.' e E' w.11 -X-TQ (in a 2 2 l AIM I r* f (Cc )w (r c ) y U (c c) -".. I......fI' 10 aC~J~r;('a dyo,, -—: — ~ i I~ j eM' 1'' p A.RMg Tl 1D sinD 7. J 3 > tx,(goO 1: 2i.. Fo(:-1 A, )R - i ).s-i;' and: 1L _M _ 1 ) 7-.' * -1 - - ).1 —''- 1 (eo,'.' s' p R''; i1. |'';'". 1,- 1 1'- 71 Si. in. i C - (B) 0 L_ —--— j: ) I\ ). ~ )__ o. uc,_,,(ol) -

S~~~ ~~~~~~~ S ~_ C...4 0.. ~:_ -2~~~~~~~ C) ^^' ^^.' ^ ^^ 11 i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~11 - iJ~~~~~~~~~~~~~~~~~~~~~~~~~- t; 0^ - l' —-P'. --: - 0 h -...,' o*' -'. i ~ o i~;,_1,_, ^.:..'O - _ 1~- -,- - -.~ _ + f~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~ * J P- -.o? ^...... _. " -... H. L., 2 -: z\, s O'~.V, -. _ ~',- >,i C) >),.;_'~ ~~i-'~i -~.,.. CD,..._~.*.._..

para.::- "',':;-:.S i;'n.c.o''. 4'.- The.''".''.s ar.-c:;ov,l s.O is n thatl s7 CCciCo:io The i""ti: d h t) e Mq c CCoC( (I to the se, c._ U..C..e C nu 1. C be.: 900, 1000,1004. wher C ~ ~.1 /) Cl ="I'... =. U,'>'' 5' v';A ]' /2 t- 1"', the'amioul'nt of t-.ermis in ukhe (r-,) ar:.:.y'', (t q)arr';2,r -, (Z,')~.IrX7 - g Z3, respec-'ivel-y, Ar -A DR, 1- 1MI p =:0O, (CCP -l F IVi[S]) for.C I K S E <Z numbrs1)V: j 2.( 2350 500 ), 5050(), and-i 580 1,e Ee: 5 ) CE( -^) -Fi l': ) - [Z] ( - -. F ZX;'. The array (, F, ) ealled,/>.1 ay be s'ored..i:? oI.Oi- tape 0o, 35: ci,.:: m130 rn(0O r C o -, C e G — ej If. (2 oti pe po 0' m a t 1 11 CteC,3. (( _. ). Ap~o ='[.l::ia - I-.-LTdAF:: ]. ip[N]::,c - iFI.!Gt[Nu], UTt,(e.'o_,) - IJ.A,[Z: L, TEMI(c^,q;) - E L..) -r t;'K., ( C C ) - CI- AS[-L]' n.u-Iii:S 1 r''[- AR I'1 W I 3 s A S.'A -.,' r'. ['. —'' J 33R [l.', C L _ 3-N' 1'i)~~~~~~1 -1- 1 -- C, -(.. [L ] JR L(, A R WIFRITn *.) iRV....

L \it i ) h...........il...l. \ C * [,-f ] ~ j [ i)...i ] H O'!, ~.[ )T ] N..i \i] H \C * [ M', ] H i ] 0: 7.. 9 03 f.C.{3 )\l94^ C f: 1 [>(jJ$) D 0 0 8t'v/l^sI 0I 00 r J.) 0(9J661 f J?:. — f 0j, 0 f:) 6.'-[ (, IT. E 3 I I I + ( 1:, ] I a: )4- C [[ 3 - A: ) 0 6I I \( [1,3] I l — [: 1 1:tI Nl.~ -)/( C,. II I > I _ - 1 1 1, I itI.. )=: [ f] >i'J 5; )61 NI[' i [ t 3 /.1{:{ S:;' t [. i I. 1: I.' 00 61[:f~GLT[I^~~ O J, 09 JN'I-:!L[, ]t: ~1 >i > J-O 1 3,61 I =:,L:;, L: OL6l f 0 -=: 7J, ^ 9 61 fL4 L 0 9."t7 0961 E1 rH, ( I ~.13 + J I M- O )..1 -,. 76 1 \ ( [ 1, I a: - C I +, J, 1 1 + ) / ( [ 1 3 I'-I - ~ I + I I -? ) = I- I I I 01 t61. 1'; 09.'J, 00 H IJj 9 1.',4 H, J ] F Ci-: -. 9 [. I i 1VO6)tI V I - 0I 0'61 f(,-dH\ l 1.\6 ) / ( [ )Y ] 4'. \. )'. I () I - ) -: L I.i 16 [ 1 I( ( ] \2V 10 \ * C I3 )/( i \ ) >l-',. ^ 1^ I 06 i 6 ( f('I [{ ] \+,^~ - \+. [ 5, [ j J,;',.:4 0091 I I'.AfH ^ I -. 0091. O I On{- 1'2;TcI[l i c[5[]HS; i O^,t OO a- * I I-"] I >>,I'Io V1l.ff) 0081 q i ^-'Wi>i: l (:; 0. 1 i1 (( I7 r "f'i [ c - (':' f(o::' 4-1435) ~i0O'^1 f0 2 -: " ~]'4"it VO 0(),01, S X 1' I F.., If I E.D I P.; O- 0101 "'- -:t:001 $? ( C O3 H i 9 7 i b:) I' } C S 1 - M ~ /. tV f ) 0 VH H t? 0 0 1(9( j ( e' ii " d o S'.( i.f (, (a )O o f ( [ 1 13') ol r[- 0 J. 003 i-') 0' J. - 0.C'{ 0 I. I I' 0 1".Y-I4'I1.. I'I'7. -I I I f'i -.^,:U' ( O r 0 I r:., ) "I,~V I:..:.~:':{'.q f 5'^ i Cl I- ( P' 4 "iI T, ff 01[,'V1100'ci'kf'l 0.'W'7rF19S~'I'9'^ Jf'';> ti* ^ai'' j >'6 I " 09 (9 C C('OO I 3 4-;..'i.,: r ["E ( [: 1 0l 099 9 VH (.099' 9119>177)'1^ 791" t/l' i Jr U II- "1 - C " 0 (9 f~( 111(Ui 97H^~) 109 ~t I0(.;:^ I i>:.0> [.i fO r(1'i-1f^r'/. -11>'1.,'1 >.> ( 4i;,i 1'.-O 00 9'ri>e -~,'i::[ilr ~':'r-." —f "'",. ~?.ti (.)

- 2921 0 0 GALA I:) ( - 3 {'l I ) /P\ D,"23 1 C. [ }{ J i [ < I. -- r ] C A..,' I 2320 TG""iFA- J:.; T...... ( (;('-\CS(CGAHL, A) ) /fK 3 ) )\ t232 3 fK J [ - / (,' - I. ( (: iAA) ) \,', S:'K (1 2 \ ( SIN ( GAMIf ) ) ~-2 / 2326 i,.K' 2; 2330 A FA [ -AC I. (T(-A,l ( A ) 3 ); 3:333!r I-: L 21 2336 T3-: T3'J 1 i [ l 7 796 2'? 339,L., S T.J.1 E f i -F itIFEK].F1ii(-I 3-+AL FA t K-1 T E 3-AL ]FAK;.; 230 AL FA. )'-' ^' A L FA e K iX 3 \ 6 6. 2831 5; 23/45 F' I N': G1 I 1: F I. K \ 3I 6 I 0/6 +2831 85. 23 50 W I TS ( fK'3 F T i }I]. i'I' l 1'A D.F i. 236 I 5 ID 2370 F5 5 1 I A F Ii Jt Z 3 I 23 60 -2380 EDL T I 3I t: F"' I. / ); 238 3 s l 7 P I 238 5 D -LTAI T DGR: DF.. TAFI T \3 0/62831 8 5 23 90 Z C Z I ) J 3'] 333?3; 2Z I( ) F I >X }) i -' F I T A X \ 3 6() / 6. C' 31 8 5; 24 50 I T T ( t A -, 3 I3 FI' it-'s, A" i ); 2500 0- R.! T7 F ( JA AK3 i' T I. Z. J I L I ]. l;I IX, AG].L, T At ) 2 2600 FO- fJ' STE 1' I i L Z DO') t2701 FTIC 3 -— 0; 2702 If3l t I, 2704f?: AS::2 1.0: 2 7 (0 4-' SI,t", I l 3, - s 3 2706 A/' 3 3 I 0; 2708, t 5' [ I ] (I I S 2710 Af5[I t ]: 271/4 A6 F I 1 -^; 2 7 1 6 SW!;.:.' 1 f 3 - rI 2800 F I F' I N- I I +D.TA F' T -2850 F' 1 D: I N \ 3 60/ t 2318 5 2900 Fi' O) I t T' i'TIL ZI L I 1)DO 3000 B.I't 31 50 \C OS 1 570 79 6 /FT F,'IiK \( F' LI ) 33155 FTN.CK3))) 32 00 (. l] 0) 32.1 r0A f: 1 3 e t C:. 3250 T P13{ l3: i 0; 320 i] I IF:..Cl 1: ('i3vO 3265 f1Ff 3O1 E 328 K N S i L.'. 1K 3 ) 0 3200 THF I 3 " 10' 3500. 7 5,. I l G ( F".3' FLX+ T, K 3 ).

-3 036.) 0 T. it' x ^. ). =3 I 3700, j}i,'!1:K - 1 / 4 1./I 3 i ]\O I \ [ ] \ I 570796/FI [[ ] 3800 \C O (1 570 79 6/FIi C \ ( I C iJ 3- l J ) <<t;.(){. r i3900g C A L K A 3. ( v.. K /2\ iDF 4000 I iA C..:l I-': W I -A U+,i K 3/ K 4100 IF[ F l ]' =: (I/; Ti'... (. ] ": -Fl ( i 8I ) 4200 THN i i ^ 4300 I;l.)SoL I} F'1CtN; G~ it", (FI?.CPK< +F"1 IPI[E.) I 3 /400 T-I} PT- [,' 4 50(0j PS F P -C1,, -1: \O.3\ \I I \} 570 796/'IP I',[(] /-550 \()S (I. 5 970796/F!Ir C K 3I\ ( F' ICJ ]- F IC} 3 ) ) 4600 \, (, i [ /p 5s ) / 5 4700 (,LA. C 4: -( 1: }: I 3 )/ 2D\ DV 48 0.i T.l ( ]-l t.- I 3 C A L JA 4 Z480 1 [ I ]: = T.-PN [ ): \ Fj C - ]i 4820 ALiA 7 (i= (PI I [U 1 ] 3 ] 2\0DH..3... 3 3iiR11.... ( 1 L'- A 7; /480 02C?(:1:^ W -X&z} R C i JE1XF \. C K 3 ~ 3 \ It ~ K 3 _. iB850 L I> A 8 ( ) - rF I ^ ) i 3 )/2 \ DF /J C JI 1 3 3 4960 1{2VA:F 1 ~ I^'~,A->'. - I 3+CAL-A8 5Y: 5.^) 5W.'I1... I tr 3? T -A 3 3. fC' I ( I A 3 R..E FO. IRMA ( i Ii, D i3 A C Z: 7 1 3. Z I 5055'1'-'11A F7'L 3, VWKRiN=AS C L \ 3); 510 A) 3< Il A:: -/ I,' [\ I J. L +.'L 3" 5200:.......... ) /\ 0/) 1: I / I 1 *, 5300 ) I 0 3 )/2\ D;TA I 5400 A S C N 3.!, HA S iRV C I — I C (. [ A I']; 5b50 (0 A 6 ( 1' 3 1 A * N i 3 )/ \ 1 3) I A I J 5600 T j 5 S. { A 6!{ 1 561 0 A 5 [ |: ^ I /'F! t. \ |f,7.. [ CZ i J 3 5620 A C N I / I t' \ A fS 1 1...56 30 C: (A' i', ) / ^ \ DEL- TA IF; 5640 WI A S' F. i:"' [ w' ]::iAS: [ N.-. 3I A + (.. 565 0 I('. AIO (P (A6 3-ACi) 3 )/2\D 3LTAFI; 566l0 \. iI')S.. P' J P^ASSdL1 I 3+CIAL}AIT 1,,5700 E.N0 f:580 I T ( j0''.A'i' I - Z23 l, (JI1S [ Z 3I, 1 A? 2:1 58 0 V [HASS1: - r[ 3 /74, 200 17..7i... 0...30. I 5 3 900 C 0 1 O6 7) 0 115". ), 5610 (460?.q/ l"].-' 7'00 1^ 0 1 8{* f ) (,) d-~,'' -fi-,jbv l,;r 00 * t,......bb?-t

-318. REFEPRENCES 1o Jo Krzyzanowski, "The Correlation Between Droplet Stream Structure and Steam Turbine Blading Erosion, " Technical Report No, 147, Department of Mechanical and Aerospace Engineering, University of Delaware, December, 1971. 2. W, D. Pouchot, F. Jo Heymann, et al, "Basic Investigation of Turbine Erosion Phenomena," NASA Contractor Report, NASA C 1830, November, 1971. 3. Bo Weigle, Ho Severin, "The Investigation of the Relationship Between the Gas Velocity, the Droplet Stream Structure and the Erosion RateTime Pattern," (in Polish), Report of IFFM, No. 273/71, Gdansk, Poland, December, 1971. 4. F. J. Heymann, "Toward Quantitative Prediction of Liquid Imnpact Erosion, " ASTM STP 474, American Society of Testing and Materials, pp. 212-248, 1970. 5. Go Gyarmathy, "Foundation of the Theory of Wet —Stearn Turbine, " (in German), Juris-Verlag, Zurich,.1960. 6. Ro J. Brtun, W. Lewis, P. J. Perkins, J. S. Serafini, "Imnpingement of Cloud Droplets on a Cylinder and Procedure for Measuring LiquidWater Contents and Droplet Size Supercooled Clouds by Rotating Multicylinder Method, " NAS Report No. 125, 1955.

-32 9. LIST OF FIGURES Figure ge 1 Scheme of the experimental stand with the rotating sample 33 2 Sonicme information about the kinematics of the droplet in the IFFM experimental stand 34 3 Kinematics of the individual droplet impact 35 4 Characteristic material removaL -time pattern, nomenclature 36 5 a.) material removal-time patterns (soft aluminum) b) droplet stream structure 37 6 Some of the theoretically calculated droplet impact parameters 38 7 Threshold combir'.t ion of velocity and droplet size (soft alumrinum) 39 8 Droplet trajectories in the neighborhood of the sample, nomenclature 40

-33A r- -I z z A \ ra/ A-A: 1 c Air Stream -. c(rc ~~ - ~ -_ _ _ z c(z) / As _____ Fig. I

C, [misl C 1 3c 0. 8 1 \ c'^'mis]0C1 + A'z +Az+ 2A'zl2 100 c = 153 m/s \ _501,___ ____ c =c92m/s \A - 1 =275 m/s o=-c 60 m/s IGO 200 300 4,00I L L&J Al C1 n/s 60 15 92 153 302 ___ 1r, ji cljl~i r. 10 400 10 400 10 400 10 400 u = 200 m/s w1 m/s 198.5 199.0 200, 5 220.0 y 800 w, m. /s 197.51 197.5 198.51 197,5 200.5 197.0 214.5 197, L I~'a3 + I" 5 3 D -7o10 m a 0 + 30 - 4 7 - 4039- +10^ - __2_ - -1 83 -5 10A k a 0.337^1^3 2Q^00u 13^ 6 43 2 0^18 748 I= 1.85' 10 kg/ms 13o I g 21:8'+23o3' J 00 p =1.205 kg/m3 p,. = o1000 kgm Fig. 2 z = 0.,3 m

0 - -35o }!' " ^ o Jl^... >.,.....;.\.; W X,. ~/w.N(r. (:, w ( ) Ds 1 ^ \ 2 I / /\ A -~s^ ^'in [y' y - r., < )' n' is l fFig. 3

-3 6Am Fp Y U w r Am a N' Am ~k-( —) exp (-0O.25 T N 2550 Am Am A(Am) Ua N' 5 Ar(Qm'" ~ Refo [4] Tgo N T4 FeatT4 H g. Am f(r) Am

^200 Am UFig. 5 200 L- -Am s Cni ~ I.1. S [mg] i ] M 91 - u = 2o00m/s I. A(m) G = Const. lAA 100 - - _ ~ J i^ c 153nir/s I /::' 100 200 300 400 500 r[Emin ] T Ar ____. __ __ __ b) cm/s 75 92 116 153 m/ l f _ n M) l \n3 A(A0m) o 482 0 199 0. 076 0o 020 gw/l, 03l/m] { AT 31 __34 = 48. 8 - 1 rnM m T 143 222 318 374 i m In r = 78. o 10 m S~c1 m I sm 60 75 92 153,)~~~~ 1 023,____ 4_ 3_, 1 I ___" N1 2. 096 10 4 828 1014 50 07 10 5519- 10" N2 4, 602 2. 293 60 547 40 561 10 4\ \ - - A 45 i.O ~ ~It. N. 3. 039- 10 3 877 10 1 090 10 33 10 P 4 I 9 r - 08. 8 - 1On^ 5 TOO 200 0 r- 16 r,: [10 m]

-38DT (C) UEM(c,) UEM (ct) 1o -[1 1 ] [x 10 ] i -0 0 T I " 0 50 IU00 c lnls 50 100 150 C nm/s ~A ^uR~u = 300 rn s "N 120 A 0 a 50 mls ]s50 100 130 c s [mrs] / -160 [r/s1 / u %tl r 200 m/s 5 90 - ___j........_....!...._ 50 100 150 c m/s u - Const.=2GO r/ls 1.5 \/S rn(Ue)c, (UeM) ni __ _ _____i, c1. 0.c 75 mIs rn eM/ -, 75 mis 50 Lo).50c c [r n/si Fig. (6) c^. "75me c4:

A A(AmL n/- 0..... -- E (c r I UA 0.2 F [mg/mIn] ci rI rl' m = 88 Iu Om u C st 200 mr/s G. 1 w ~'" iN =-ConIst = 116 m/s Ar iu r) i n urn 0 100 200 300 d -2r [ri06r,,m m, rri M A [imo —'' G 1n, Aliuumin unm 10\.NC rd d 0 ^ rr.m: m "si u-2 = 00.t s1000 d 2r- [30/s U A \(A i/,:.y 0,, 2 [mg/min] P 2 [mg/min] 0. I I- O. I i Ou 0 w s u 2u0,-n PII t~f, uA 350 100G 0 50 100 wi,, NCI v, V.: (1- L ^: —, )!',':",',,N' W.,.."- r~

8 ^Ij -/ I / <We11^ /''."'" / " "i ~' X " ^-^ ^ \ /l\ ^ L -_____ - M \ / I\ / X X ES( A -) 0 i