College of Fnginezering Mecllanical 'Engilnee ring Departnment Ga~itation and Muliphase Flow Laboratory Renort No. UMICH- 012449-19-I CA VITA TION DA MA GE (In partial fulfillment of ME 490) by Kin- Yum Yue Approved by F. G. Hammitt Financial Support Provided byNational Science Foundation Grant No. 40130 April 1975

2t NJ @ Ft f3 tVX9 O RI h/ A c\. A "I e ly t a QRScaZ PT IOlA CWMK4 avt & R s an<Cct4t &oJ.ke AA. 'C \ Pr Jt _t.& QA iA*A.. di Vowae,~, at ~iJn ava Jdi~ CI2GE~cw A~~~~~3PL~Pr p~t LR c~c) Aat~ UMC MG& Y~ck it nh *-%s XR *nuc~~u do~c JUL MblD tmAK( r4C~ )+p c LL) cAr jJcC~vf'Jlc,) f, a t c Aa.tRv f&~L va v A jWp~ vv-ur1 vry AJvLO C 1 -J AYT" r o~ " c,/v,,cAc h~z~t~r~ ve iy~~o~ldaC/ cY Wn YV(I- -On

NJv - u\ J'l 47e e6 wL'- y U 0................... i — -_- I _____ I............ " r >QYo,-n I --- l-a~z__ __j

SR U LT G AL- lloo Jce -LL{AC o C&wnXc-tt{;r 4 ct rtL nm corxcuI? wE Q - av~ o& A os CC(4J~,XoLZ C AJX L7 )UCS curt~~~~~~~~~~~~~~ uC- L S S VaA ~~-d~ -~ ~;bo~~t~rC uwh SDI-1 cta-,nwu C Ck-~t W-Ur/vi ~ CCAV AMur~ StRA CO c l 3e d b S(L n, 4~ IL, -vrCt~ l t we Jcv at-c A~.. ct d~zn. Jl f L ~ ~AZ~C~ 4f %N t1m P. Ao Vc. qitt Ji?"a" ~tK. ~ J~ O~ltn ACtbW~QL~ CQCLtc OA & 8A ~~H D P ~ 1 L c (JiL, A AX &c ~ &'Al -ZOQLU~ CE CN 4 CTJ~ 'lb j C tRL M~~J1 c% p lR JmcvscuY cXE Ssh 4D vm~-ac&.LO'o 4e - 9 G ~-(Do nru we 430.er\L, L A~ cAg~grLt % c IYDR~oC seyLa ~cct A we ~~IJ. JLe~ C~x <R iPRv ~3 nlunk t@ 4ut do <t &n A w 42. d ~tL4 ~- W A UtRLZ i 4 r.

~~~~~~~~~~~~o cL~r t1c L C:- eJtat CU tn Jw O s A 014- ccndlfvh CLA Aet AA-bl O OM I -e;,l~aL a L RJA ci g t~ ~ z cfw -e-rs s~.,, orrLC v D, - 0, aej~tA Cl-r-, wU tl%. A_;w.~~P,3.e rel riO.OtCL~ *i y*9 3, L4, S &UD 6. CON C US io rS 5tA ~ 7, of tcailclS4n clof~ we Acd' Aetc O-Art ~d ~%~- 4L- %W ~t~ Vy C kamooua, OXL;t "< 48A., 4tj rr m jZ; ~ ci ~~~ t at w S~~w-OrQ, ~t~z~e CUCC C4n ~ K4,4" jv fcar AA\C yezLa. VJtAe It C -L, t - 3, wariAX w~4, oL 4 4 Jo et 2tQ v _aX ijt: tCwti ~ L4&,S 2 WX DU. J, 0 & 4t 1 ffi 41xQ.C~x4 %_ _z _eddjc -rCtZC~- _fFZ nlPI~TCO t/tUX t<7.; st^ _ Q

.TIAL N_____ S -S -e RUN;1 NGS CO NtI.Tl('S ~"~a~. L c-b I fO r - i., % Pressure / ATM' a _____...........lb -o.. - - - 7..TornuDe-.. - ___;J -. sce t,'reparat ion DATA It''%.- I ' ''1 A L c_____ ~___, 4 - _-'- _,. "' F ___ _ ' _ J__ _ _; I rva Ti 1I os! s ID I 5 _______ _______________ ____ _ __,_ _ _,_ ___- _ - ________ ______ _______ —?C __ _ ___ _ _ _ —__ — __ 3~o _ ________o__?_ _i___ __ __ __ ____ _ i ___ __ _ _______ ___ _ __ _

_':- _J~ - -- — ~- I — RUn:-.h.,OG NE 1________ __ Jn -T 4$.5 _ cr.ture 75 F I.c.e _ _epation I,ctor _ _s7y ' - 7;:, c,..,/3 A _......... 1. Factor __ _ _ -............. t-e'l2i ( 4- _ _ _ _ _ ____ C\ r, *>3 I t /7r.,-.1 _ _ I.I I L- ' ' 7-S - e — - _l _. I I _ _ - II-_ _ _ _ _ _ __ _ _ _ _ _ ___ _ __ _ _ _ _ _ _ _ _ __ _ a, 5 _r t _.1,_ _ i I i I t__ ' ---_.......

n?.AL..._ II_.' S'lIZ_ rar-:ber - -- -~ - -- - ---- RUNrt.1 NG CO NITI (:' S 3rn 7 4 Poner x 2 J Pressure / 7A N po; Da~, ~ - S - 7 5. Torqua SO 9o e/ F _es _, _ ~ _ _ _75 I _n. _ _o- 90 _tcr,-in irf._ce Pr"paration 'nsi ~ty ' A | Ar.2R~I~ 2.7/ ~1,/crJ Are DP Factor 'D.n'rent s.. L - /rn - 2l _ __~______ DATA F TriU:.uea,.2 |;;H In t:; CIr.u1. -fivc nlt.crval Tir | Loss Loss L e -P { 2._ __ ____ _ _ __ _ _ _ _ _ _ _ _ _ I _C { r7 tC_ _ _ i _ _ _ __ _ _ _ __ _ ___ __ _________ 7. i.6...1 ____ _~. _ I. - 1 I! I. I i

T'j AL,.'c d.st~~~iat 4 s s W yr _~~~D~,. S+ZE riar,:ber 7 IRU N.I N CONDITIONS 5 ~~~~~~~~~~~~~~~~~_ I,... = --- osrn 7* 1P ure PoT-,er r 3. 7 I Pressure / A T M?p-o o:.Date _ _ ____1........Torque rcru-mrn urxfoce,'rep3rat Ion BASIS FOR C.ALCUi,,LO;S_. snsity _ ^ Are& -'-.0. 2~~~~~~~tb t, 7 3. 1c _.......... rt...... W..... *........... __~~~~~~~~~~______ _ _ _Q~: D)ATA i Facr;......................... _ C lntcrvral I TI i re Loss s 1 Loss. _ _ C _~~~~~~ — - Z>v_ - 1 R< 5 I 1 __________________________ 3 4 _, ~ __t _- 7 S i_ __ _ — t. era:,'.. 7~o e ~ I. {~~- v ~!,,!,,7-,o _ [... ________- / ________ _______. _ ______'. _ _~~~~~~~ _ _ I L_ l._l I, j t -1!,. e -. ____!! L__ II__<___ I_.. ___ i_ L _ 1__ _,______! _-_I I

MATERIAL. D4AMIAGE SHGET aate r~al AL ulo RV)NNINO CONDITIONS t Hot rPi nr. 5 @ 5. Te epratu re 73 J Power 5: P.ressure.T AppraX. DaJte 3- 7' Torque 1 PFre run BASIS FOR CALCLA TIONS b;*enuS.t/ r > *7 I r.. / ~. i~ A rea _ MDP Fwctor \Afr /1. F/$4A tntrn... - - - 18 -T WegahL..2. D Tsme jCumulative Weight Cumuiativ Culati Inter va I Time Los I...._._ Sr 2r ' "7............ -m..

! ~,.~~~~~~~~~~~~~~............ -~ —~-~~~~~ ~~ ~ - ~ - -I~~-rrr- ~- — t --- l~ — r-+-~,...........,....................... I~~ ~... ' ' ' -..L~.~_~ID/. 7~ i~ '- ' — - -~ - ~r - - - ~ i ~ ':- o'> —",-~-,'r'e ~:~-._I N lC F,~ ~XC —.~~-.~-:~. —~.- --- -- -': ~ —I. ---....~ ~.i- -....~.... ~:-:-::~~-.... - ~~~ ~~~~~Oc --- —~~~~~~~~~~~~~~~~~~~~~~~~~~-.::~-:: ~~ ~~. ~ - ~ -~~~' i:-........... ~........ -... -.. "..- i.!:- ~ -;;...... -.... ~~~.........,.., ~t -f:~ _: I:...~ NAG~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~-........ ~~~Z D-j/~~~~~~~~~ ---.,' -—.-....~C y ----- ~q~.~ J~~~~~~~~~~~/ I "...,............~3............ Q~~~: ~~ ' -.,.._... ' ~~~~~~~~~~~~~~~~~~,.~; " I /.,- _ C-.~'o-.~..-, > ~0 - ' I: —...~.-.I....c....._. _ -............~_...~.........,.. _~..I_._._.............. /. _... _ ^. —.. — I.. _- - I.-..-. —...-..-.~. - - - -..-. I.... _...... 1......-.-. —.~l -........ i~~~~~r —',' ~ - ~~~^ — ~iu..................i~ ~ I-::"...... ' M - - U - ~ ~ ~ I - - ~ - - - - -- - - ~ L - I- - ~ ~ -- ~.............. O ~ ~ ~~ J......:~.. ~.- ' rlM~~~~~~~~~ C~~~~~~~lh)) TrG, 1~~~~~~~~~~~~~~~~~~~.

.L 1-i-i-I-Iii- t~~~~~~]i III' 1~~~~...j...!::"' -~-?-"... F~I-[ L. [ 7.__._ I~~~~~~....I r~~~~~~~~~~~~~I j..... _~ M:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:....~~~~~~~~_ r —T- 1, -~-7 ---!!:_!-I ~J..~~ ~~~'....,,. 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! ' i - rII i'; 1 I,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 11. ~ ~~~~~~~ I i:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I'I I~~~~~~~~~~~~~~~~~! 1~~~~~~~~~~~~~~~~' i:' 1~~~~~~~~~~~~~~~~~~~~~~~~~~I - I i jL I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ j~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ r~ i~ 3i9I —~~~~~~~~~~~~~~~~~~ ~~~~ ~I I I~~~i~,!l i-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'' I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~] Ii!!! ~~i 1.1..!1...~: i,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I ~II ' I~~~~~~~~~~~~~~~~~~~~~I'!- il"'r,, 1 ~Ili-~~~~~~~~~~~~~~~~~~~~~~~~~..... —L ~~~~~~~~~~~~~~~~~!..... I2" [ '~~~~~~~~~~~~~~~~~~~~~~~~ -1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' ' I1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~: [,, "TT 1 1~~~~~~~~~~~~~~~~~~~~~~~~~~.~..., ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ ~~~~,-r~

CAVfATIONJ AMAG6 ON CADI I tNUM PLATL SVFC MNt I: 1-0...........XK4 *,,'NC~!' iS {-z )o )< V MA;w ~XtAt& ~~ I-s,,:o. ~ pO WA ' A 1 * S 4. '333 1! 5 T1VE 1 V Yjantr 'K ' 1t V% 4 '77~~~~~~~~~~~~~~~~~~~~~''~~~~~~ NAChNJIr-ICATIMA %4to -~~~~~~ vs wt. Th~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n......7..$.. 10~~~~~~~~~~~~~~~~~~~~~~~~~~...... M............

j C) 0 In 71 i....71. -.:0000 00000.0.0i iii:..A... i 7* 00000...00000.....-iiii??ii iisiii0iiiiiiiiiiiiiiii iiii-00iiiii '."'':',&.iiiiii.g~iiiiii..iiiiiiiiii ', iiii,' ',' '??!.iiB...,-, -iiiiiiiiiiBB' ii"'... ii ~iiiiiiiiiiiii ii&iiii&ii'iii'ii

.................................-.......................................................................................................................................................................................................................................................................................................................................................................................... Lo.................................................................................................... j...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................-........................................................................................ --..................................................................................... -- --.......................................................................................................................................................-................................................-...............................................................................................-......................................................................................................................................-................-.....................................-.................................................................................................................................................................................................................................-......................................................................................................................................................................I..'', I I ---:............-................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... X.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................-................................................................................................................................................................................... 1. I:I'll,............................ - —::::.:::::............................................................................................... -x:....................................................................................... lxx....... I................................................................................................................................ II....................................... I I.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... X X.. -X.,.......................................................................................................................-.........................................I............................................................................................................................................................... I —............................................................................................................................................................................................................................................................................................I......................................................................................................................................................................... - xx........................................-.............................................................................................................X.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................I.............................................................................................................................................................. I..........-...................................................................................................-........................................................................................................................................................................................................................................................-...........I............................................................................................................................................................................................................................................................................................................................. X, X...'I..:.:. X........................................................................................................................................ :-........................................................................................................................................................................................................................ --..................................................................I.....................................................I.............II I I.. - - ........................... XXX............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. I...............'....................."......'....,.",.....,..........'.... - ...: X..............................................................I....... -..l,:,:::,:: a I.. I...........................................................................................................................I..................................... I.................................................................................................................................................................................................................................................................................................................................................... I.-I.I.. ----................................................. --.............................. X..........................................................................................................................................................................................................................-...............................................................................................................................................................................................................-................................................................................................................................. I.1:-.............................................................................................................................................................................................................................................. ---.......................................................................... 11.1...................................................-.............. -- ----.............-.........................................................................................-.................................................................. 11......................................... -- -—....................................................................... --..................-...................-........................................................... 11.................-.......................................................................................................................................................................................................................................................... ----.........................................................................................................................................................................-................................................................................................................. 1.111.11,1111.1'..'.11.11.1'..I....................................................................................................................................................-..................................................................................................................................................................................................................................................................................................-............................................................................................................................................................. - - X.,..................................................... xX-:...:::,XX......................................................................................-..................................................................................................................................................................... --............................-.................................... 1-11.1-111 --—..................... 1.11.11....................................................................................................... --- --.......... --............... -:.............................................................................................................................................................................................................................................................................................. — Ill 11-11.1.1-1-............................................................................................... —.......................-........................................................................................................................................................................................................-........................-..........................-........................................................................-.................................................................I................................................................................................................................................................................-.............-......................... 11..................................................................................................................-........... --....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... 11.1................................................................................................................................................................................................................................................................................... 11...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................-.....................................................................................................................................................................................................................................................................................................................................-...................................................................................................................-............................................................................-......................................................................................................................................................................................................................................................................................................................................................................................................................................................-..................................................................................................................................................................................................................................................................................................-........................................................................................................................................................................................................................................................................................................................................................ --...................................................................................-................................................................................................................................................................................................................................................................................................................ ''I'll,........................-.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... —........................................................................................................................................ --..................................................................................... --..............................................................................................................................................................................-.........................................-.........................................................................................................................................................-..............................................................................................................................................................................................I.....................................................................................................................................................................................................................................................................I ---........... -1.1 ----................................................................................................................................................................................................................................-........................................................................................-.............................................................................................................................................................................................................................................. -- --...............................................................................................................................................................-..................................................................................................................................................................................................................................-..........II................................................................................................................................................................................................................................................................................................................................-..............................................................................-............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. --..........................-.............................................................................................................................................................................................-.......................................................................................................................................................-........... --............. 1.11-1...........................................................................................................................................................................................................................................................................................................-.......................................................................................................................................................................................... --................. ---........................................-........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................-.........................................................-.........................................................................................................................................................................-.............................................................. --...............................................................................................................................................................................................................................................................................-......................... --..................-...............................................................................................................-................................................................................................................................................. --.......................................................................................................................................I ---l-.........................................................................................................................................................................................................................................................................................................................................................-.......................................................................................................................................................................................................................-.................................................................................................................................................................................................................-..................................................................................................................................................................................................................................................................................... x1l................-.........................................................................................................................................................................................................................................................................................................................................................................................-.......................... X.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................-..............................................................................................................................................................................................................................................-...................................................................................................................................................................................................................... -x................................................................................................................................................................................................................................................................. 11.11................................-...................................................................................................................................................................-.................................-......................-........................................................................................................................................................................................................................................................................................-X............................................................................. --..............-...........................................::::::,.........................................-:,X-.........................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................................................................~~~~~~~~~~~~~~~~~~~~~~6 ~~~~~~i~~~~~~~~~~i~~~~~~~~~~il~~~~~~~~~~~~~~~i~~........ ~~)i~i~~~~i~i~~i~~~~~~~i~:~i~i~i~~i~i~ili~i IS~l~~li liicilliiiliilfi........... ~~i~~i~~i~~jii~~~~~~~~~i ~ ~ i~~j~~~i~~~~~~: ~ ~ ~ ~ ~ ~~l~~~~~iX~ ~ ~ ~ ~ ~ l~~~~~~i L~~ii~~~i~~i~~ii c..ll....... j:::::~~~~~~-:~~~:~~:::~~~:j::::~~~~~~:::~~~~:~~ ~ ~~;; ~ ~.......... ~~.~~j~~~~:~~:i:~~~~:~~:~~:: jjjj~~~~~lX~~~:i ~ ~Piiiil(Illi~~~~~~lii~~~iiililli:~......... ~~:~~~i~~~~~~i~~~~l~~li~~~~i~~~~i ~~~~lil i~~~~~li~~~~~~~i~~~~....... ~~~i~~~i~~~~i~~~~~~~~~~~~~l~~~~~~~~~~ii~~~~~i~~~~i~~~~~~~~' i~~~~~~~i~~~~iii~~~~~~~~X~....... Xii~~~ii~~iiii....................... ilciiciiilill~~~~~~~~~~~~~~li l~~~~iP~~~illlliilliill l ~ ~ ~....... Jiiiii~iii....................... ~~ii~~~iiii~~~~ii~~ii~~~ii~~~~i~~~~i~~j~~j~~~:~.............. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..................~~~~~~~~~~~~~~~~1E I~~~~~~ii~~i~~~i~~i l~~i~~~i~ ~ ii~~i~~~i~~i~~i~~i~~ii~~~ii~~.............................j i~j:~:~~ii~:~~iiiiiiiiiji..-......................................................~i ii ii ~ jii l liiiiiilil li~i~:illiii~ii ~~ij~i~,ij ~ i ~~i~ii~i.................................................j~i~ixi~i~ix~iiiriiiiiiii ''"'~:~:~:~:":~:~:~:~~~~~~~~~~~~~'~~~:~~~~:............... ~ ~~~:~~~~~~~':~~~~~~~~~5i~~~~~~~~~:~~~~~~~:.................................. ~~~~~~~~~~~~~i~~~~~.~~~~~~i~~~~i~~~ii~~~~~~i~~~~i~~~~~i~~~~i......................... ~:~::~::~jj:~j::~:~ j~i~:~~::~:~::j:: i~:i~~~l~~~i:~:~~~~~~I~~XI...........................iiii ~~~~~i~~~~i.~~~~~ ~~~~ij~~~~~~~jij~~~~~~jii~~~~~~~jij~~~~~~~~~~~:~~~~~~~~:~~~~ixi~~~~~~:~~~~:::~~~~~~:~~~~:: i~~~~~~~~:~~~i~~~~~~~~~ii~~~~~i~~~~~~i~~~~l~~~~~~~~I~~~~~~~~:~~~~lg~~~........................................................................................................... ~~~j~~~:~~~~~~~:' j~~~~~~:~~~:~~~~~~:~~~:~~~:~~~~:~~~:~~~: ~ ~ ~ ~ ~ ii~~~~iiji~~~~~~~~~~~i~~~i~~~~~~j~~~ii~~~~~~~i~ ~ ~ ~~ ~~i~~~~~~~~~~~~~~i~~~~~~:i ~....................... ~ ~~i~ji~~i~i~i~ili~~:~i~~l~i~i~~~iiiii ~ ~..................................... i:I:~:~:~:~::~:~~~::~~:i~:.......................i~ i~~~~~~i~~~~i~~~i~~~~:~~~i~~~~j~~~ii~~~~~i~~~~~~l~~~i~~~~................... ~ ~~~~~~~~~~~i ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~....................................... ~~~~~~~~~~~~i~~~~~~~~~~~ll~~~~~~~~~~~~~~~~~~~~~~i ~ ~ ~ ~ ~ ~..........................................~ ~ ~ ~ ~ ~ ~~~~~~~;b ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.....................................~.......................... ~~~~i~~~~~~i ~ ~ ~ ~ ~........................................ ii i iii............... iiiii c~l................................ -.......................................i ii l.....................::1i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ll~~~~~~~~~~~~~~~~~~~~~~~~~~~~ l~~~~~~~~~~~.................................................................~: ~ 8:~:::::::i~:::j:::i:::a j:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~......... i~~~~~ljl ~~~~~ ~~~~l~~~~:'jl~~~~~~~~~:.iiii'~~~~~~~~~~~~~lllllilllllliiil':~~~~~~~~~~~~~~~~~~~~~~~............................ -jj~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i ~ ~ ~ ~ ~ ~ ~ ~ ~.................. ~~~~,.............................:::::i::::i:-::::-:-:I: i-:::-:i:::.......................::i~-:-l: -::::::.................... X.X...............