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Transkrypt
48 full
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U FFI U III '/3, '!J!'/3, '/3, - $ &)512 $)512 ?/,PD I I#UZ I I#VF I I##U I I#`[ I IV%W I IV[# I I#`V I I#`% I I`IU _ JI `Z JV [# J` `I J[ `I JU `` JU# %[ JU VV JV IZ JU# W 53+6 ### V#` UFZ ZV VIF WF ##% V[U UIF ?P1 ZWI# WWWZ F[%% %F%# WIIZ %U`F ZWUW Z#IF FI%I 53+6 ##% VVI U`I FV VUI FF #`I JJJ UI` ?P1 ZFII W`%I F%`I %UII F[%I %UII ZWII ZVWI FIII ^<?3/,)?! B"! /*.+;*/.+! */1>+:+! 8+6>/7L! 3(,+.14! >,)./8)D/! ">+3),+! #$U&$!N! ;),4! *(,41>6/8+.(+! 8:+7;(8/7;(! ?)>/0< ! 8! >+3),(! #$V! *+?()1*;*/.+! */1>+:+! 698.()-! (,/7L! ?/,(! D+*/8<;B! 26/045>98! 0)>/.+;C(! />6*<?+.+! *!0/5:+0.<;B!/3,(;*)d!"'/3,&$! K+5!8(0+L!*!>+3),(!#$V !2/?(?/!26*<CO;(+!3+60*/!426/1*;*/.)D/!698.+.(+!6/*5:+04!8<34;B/8)D/ !0,+!8(),4! =N ! 698.()-! /! *69-.(;/8+.<?! 3(,+.1()! >,)./8<? ! 4*<15+L! ?/-.+! 3+60*/! 0/36M! *D/0./7L! /1*+;/8+d! 26*<3,(-/.<;B! *! 0+.<?(! )512)6<?).>+,.<?($! N)! 81*<1>5(;B! 26*<2+05+;B ! 0/5:+0./7L! ?)>/0<! ?+,)C)! 86+*! *)! 12+05()?! 8(),5/7;(! DO1>/7;(! =N$! e8(M*+.)! C)1>! >/! .()8M>2,(8()! *! .()7;(1:/7;(+?(! 2/81>+CM;<?(! 26*<! 26*<3,(-/.<?! /56)7,+.(4! 15:+04! ;B)?(;*.)D/! 26/045>98$! f98.()-! 8+6>/7L! 2+6+?)>64! A! 2/8(..+! 3<L! ?/0<g5/8+.+! >+5 ! +3<! 48*D,O0.(L! 82:<8! *+?(+.<! *+8+6>/7;(! (! 8<1>O2/8+.()! 2/1*;*)D9,.<;B! 1431>+.;C(! "X] !bV !]b !']!(!(..)&!8!15:+0*()!26/045>98!26*)?(+.<$ E/0CO;()!1*)61*)D/!/?98().(+!*+D+0.().(+!1*+;/8+.(+!2+6+?)>698!26*)?(+.<!0)>/.+;<C.)C!*+!2/?/;M!?)>/0! 26*<3,(-/.<;B!".2$!hUUJU#i& !8<56+;*+!2/*+!6+?<!26*)01>+8(+.)C!26+;<$!'+!2/01>+8()!;B+6+5>)64!26*)01>+8(+.)C! 2/8<-)C! ?)>/0<! 26*<3,(-/.)C ! /6+*! 48*D,O0.(+CM;! 8:+7;(8/7;(! ?)>/0! 7;(1:<;B! 2/5+*+.<;B! 2/.(-)C ! ?/-.+! 815+*+L!.+1>O24CM;)!8129,.)!),)?).><!")>+2<& !C+5()!1M!;B+6+5>)6<1><;*.)!0,+!81*<1>5(;B!?)>/0!8<*.+;*+.(+! 1>+.4!26/045>98!26*)?(+.<!.+!2/01>+8()!15:+04!;B)?(;*.)D/!(!1>+.4!>)6?/0<.+?(;*.)D/!8<C7;(/8)D/!=N$! N!1*;*)D9,./7;( !?/-.+!8</06O3.(LG "j&! ./01231451( 6/7+89( ,:1;5,<41=>( &0>89/?@A! "?)>/0M! 26*<3,(-/.M! ,43! 2/26*)*! +.+,(*O! *+,)-./7;(! +,D)36+(;*.<;B !C+5()!*+;B/0*M!2/?(O0*<!8+6>/7;(+?(!1>O-)d!?/,/8<;B!2/1*;*)D9,.<;B!15:+0.(598&k "l&!-0<B&56+451(A7+2,5A>2,5(1410=1?B,<4B,:(&0>89/?>;(&0<1;5+4B$!N!?)>/0*()!26*<3,(-/.)C!2/26*)*!/56)7,).()! 8+6>/7;(!8129:;*<..(5+!A !8!?)>/0+;B!7;(1:<;B!2/26*)*!/56)7,).() !.+!2/01>+8()!>+3,(;!>)6?/0<.+?(;*.<;B ! 8+6>/7;(!).)6D((!8)8.O>6*.)C !).>+,2((!(!).>6/2((!/02/8(+0+CM;<;B!0+.)C!>)?2)6+>46*)!?()1*+.(.<k "X&! ./01231451( <+31C4>2,5( D<B,<4B,:E( F+/51( ;+FG( ,:+0+/?10B<>A+H( 6?+4( &0>89/?@A( &0<1;5+4B$! N! ?)>/0*()! 26*<3,(-/.)C !1M!>/!)?2(6<;*.)!*+,)-./7;(!"#$Z& !"#$[&$!N!?)>/0+;B!7;(1:<;B !/02/8()0.(5()?!><;B!8*/698! C)1>!2/0+.()!*+,)-./7;( !C+5()!2/8(..<!12):.(+L!26/045><!0)>/.+;C( !8<34;B4!,43!12+,+.(+$!]D9,.)!*+D+0.().(+! (.>)626)>+;C(! 5/d;/8)D/! 1>+.4! g*<;*.)D/! 26/045>98! 26*)?(+.<! 26*)01>+8(/.)! */1>+:<! 8! 24.5;()! V$! E/26+8.)! 1m/6?4:/8+.()! /02/8()0.(;B! *+,)-./7;( ! C+5()! 2/8(..<! *+;B/0*(L! 2/?(O0*<! 2+6+?)>6+?(! >)6?/0<.+?(;*.<?(! 26/045>98! 26*)?(+.<! C)1>! .()*3O0.<?! )>+2)? ! 5>96<! 2/8(.().! 2/26*)0*+L! 26/;)1! (,/7;(/8)D/! /56)7,+.(+! 1>O-)d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m6+D?).>+?(!?/,)54:!8<C7;(/8)D/!?+>)6(+:4 ! 6/0.(5+?(! /6+*! (..<?(! 1431>+.;C+?(! ;B)?(;*.<?( ! C+5()! ?/DM! 3<L! >8/6*/.)! 26*)*! +>/?<! 2()68(+1>598! 15:+0/8<;B$ n,+! 5+-0)C! *! 6)+5;C(! ;B)?(;*.<;B! *+;B/0*M;<;B! 8! ?()1*+.(.()! ?/-)?<! *+2(1+L! C)C! 698.+.()! 1>);B(/?)>6<;*.)! IUJU!Y!IVJV!Y!o!Y!IfJf! !IU-U!Y!IV-V!Y!o!Y!Ifp-fq! ! "`$U& D0*()G!J(!\!1431>6+>< !-(!\!26/045>< !R !Rq!\!/02/8()0.(/!(,/7L!1431>6+>98!(!26/045>98$! N*+C)?.)!6),+;C)!2/?(O0*<!8(),5/7;(M!43<>54!1431>6+>98!(!26*<6/1>4!26/045>98!/56)7,/.)!1M!26*)*!8+6>/7;(! 8129:;*<..(598!1>);B(/?)>6<;*.<;B 8Kj!G!8Kl!G!8KX!G!o!8Kr!G!8Ks!G!8Ke!G!o!S!Lj!G!Ll!G!LX!G!o!Lr!G!Ls!G!Le!G!o! ! "`$V& K+5!8<.(5+!*!*+,)-./7;(!"`$V& !1>/14.)5!26*<6/1>4!,(;*3<!?/,(!0/!8129:;*<..(5+!1>);B(/?)>6<;*.)D/!C)1>!0,+! 81*<1>5(;B!6)+D).>98!C)0.+5/8<$!N(),5/7L!><;B!*?(+.!/*.+;*+.+!C)1>!,(;*3M!2/1>O24!6)+5;C(!0M dxi "`$#& = dξ ! νi ! t8*D,O0.(+CM;! ,(;*3O! 2/1>O24! 6)+5;C( ! *?(+.<! 2/>).;C+:4! ;B)?(;*.)D/! 45:+04! *8(M*+.)! *! 26*<6/1>)?! 3M0u! 43<>5()?!2/1*;*)D9,.<;B!15:+0.(598 !?/-.+!*+2(1+L!*+!2/?/;M!*+,)-./7;(! r r ∑ µi dxi = ∑ν i µi ⋅ dξ ! ! D0*()G!µi!J!2/>).;C+:!;B)?(;*.<!iJ>)D/!15:+0.(5+$ N+64.)5!1+?/6*4>./7;(!26*)3()D4!6)+5;C(!?+!2/1>+L!hU`i! A!v!I! ! 2/0;*+1!D0<!1>+./8(!698./8+D/8)?4!/02/8(+0+! A!S!I! ! D0*()G!A J!2/8(./8+;>8/!;B)?(;*.) i =1 i =1 "`$`& "`$%& "`$F& r A = − ∑ν i µi! ! "`$W& i =1 N!26*<2+054!45:+098!12):.(+CM;<;B!(0)+,.)!698.+.()!1>+.4!2/>).;C+:!;B)?(;*.<!?/-.+!*+2(1+L!*+!2/?/;M! *+,)-./7;( µi!"" (& &!S!µi!!"" (&&!Y!N"!T!,.!&i! ! "`$Z& D0*()G!µi!!J!2/>).;C+:!1>+.0+6>/8< !&i!J!;(7.().()!2+6;C+,.)$ N+64.)5!"`$F&!26/8+0*(!898;*+1!0/!*+,)-./7;( r 1 r ∑ν i ln pi = − RT ∑ν i µi ⊕ i =1 i =1 ! ! "`$[& C !"#$%&'()*' 5>96M!?/-.+!*+2(1+L!C+5/! ∏ piν i − =e 1 r ⊕ ∑ν i µ i RT i =1 ! ! ;/!26/8+0*(!0/!26+8+!0*(+:+.(+!?+1!hU` !U%i!*+2(1<8+.)D/!8!2/1>+;(! "`$UI& i ∏ piν i i = K (T , p ) ! ! "`$UU& D0*()G − K (T , p ) = e 1 r ⊕ ∑ν i µ i RT i =1 !! "`$UV& 8(),5/7L!OP"E(&Q(./1(!.+*8O!1>+:)C!698./8+D(!6)+5;C($! e+,)-./7;(!"`$UU&!?/-.+!8!12/193!3)*2/76)0.(!8<5/6*<1>+L!0/!/56)7,).(+!*+,)-./7;(!2/?(O0*<!1>O-).(+?(! 2/1*;*)D9,.<;B!15:+0.(598!8<1>O24CM;<;B!8!26/045>+;B!26*)?(+.<!?+>)6(+:4!8<1/5/).)6D)><;*.)D/$ #$D$%E452(%=)<1,36%<'B,9-.'23%.%34(,F)-.)%()+,7;.%/:0.G9=4%<1GF)-.,0.%/(:96'123%/(=)0.,-4% =,%/:0:7?%<1,B478%(23-:3,>.% n,+!?+>)6(+:4!/!/D9,.<?!15:+0*()!;B)?(;*.<?!X+bb];'d !C+5/!2/01>+8/8)!15:+0.(5(!26/045>98!26*)?(+.<! 26*)8(0*()L!?/-)?<!.+1>O24CM;)!1431>+.;C)!;B)?(;*.)G!X] !X]V !bV] !bV !]b !b !]V !] !' !'V !']$!e3(96! 6)26)*).>+><8.<;B!6)+5;C(!;B)?(;*.<;B!?/-)!3<L!26*<CO><!8!2/1>+;( "`$U#& H + O2 ⇔ OH H + H ⇔ H2 CO + H2 O ⇔ CO2 + H2 O + O ⇔ O2 N + N ⇔ N2 1 1 N 2 + O2 ⇔ NO 2 2 yO y H p = K OH (T ) ! yOH y ! yH 2 p = K H (T ) ! y H 2 y 2 ! yCO ⋅ y H 2 O yCO 2 ⋅ y H 2 =K CH (T ) ! ! yO 2 p = K O (T ) ! yO 2 y 2 yN 2 p = K N (T ) ! y N 2 y 2 yO 2 1 / 2 y N 2 1 / 2 yNO = K N O (T ) ! "`$U`& "`$U%& "`$UF& "#$%&' "#$%(' "#$%)' *+,-./ yi 0 -1234 521- i67.*2 89:;+<-9; = >.+<287?. 5;8@ 5-.8,;<-<@$ AB,.+87;=-2<@ ,C-DB & BD=<;E =@?,.BFG>. -1234 <-.,;1.H<@?I 2+ 8-.C-. B.;9?>-! 97DB. 52*J ,;?I2+,-4 = 5-.8,;<-<-.$ A2,287;:. B.;9?>. 52*J C@4 =@B;H2<. F2FB,., B.;9?>. "#$%K 0 #$%)'$ L;?I2+,- F2<;+72 y = yO + yO 2 + y H + y H 2 + y H 2O + yOH + y N + y N 2 + y NO + yCO + yCO 2 ! "#$MN' O123?- 521- F28,?,.*D1<@?I 89:;+<-9D= 8F.:<-;4 5G8,J BD=<;<-; C-1;<8G 5;8@! =@<-9;>J?. ,. 89:;+G F-.B=-;8792=.*2 5;7.B-;:G =@>3?-2=.*2 ! !"#$%&'(#)*!+&,-)./#0(.1'#23)()4&"&0#5"67)5#"03,8'&0(.1'8./)4"3,-!567)999 yO + 2⋅ yO 2 + y H 2O + yOH + y NO + yCO + 2 ⋅ yCO 2 = bO y H + 2⋅ y H 2O + yOH = bH ! y N + 2⋅ x N 2 + y NO = bN yCO "#$M%' + yCO 2 = bC P@5C21;5- bC! b ! b! - b" 2,<;?,2<. ,287;:@ 2+F2=-.+<-2 -123?- 521- =Q*1;! =2+2BG! 71.<G - ;,27G = >.+<287?. 5;8@ 5-.8,;<-<@ ",=@91. % 9*'$ R=,*1Q+<-;>J? ,;1.H<23?- =@<-9;>J?. , BD=<;E "#$%K 0 #$%)'! BD=<;<-; C-1;<8G 5;8@ FB,.98,7;:?-4 52H<; +2 F287;?- p yO 2 K O2 (T ) + 2⋅ yO2 y yO2 1 / 2 y N 2 1 / 2 K N O (T ) + y H 2 yO2 1 / 2 y 1 / 2 ⋅ + K H 2 O (T ) p + K CH (T ) ⋅ yCO2 ⋅ y H 2 yO 2 K O2 (T ) ⋅ y H 2 K H 2 (T ) y 2 p + K OH (T ) K H 2 O (T ) p 1 / 2 ⋅ + 2 ⋅ yCO 2 y H 2 yO2 1 / 2 y = bO yO 2 K O2 (T ) ⋅ y H 2 K H 2 (T ) y 2 y H yO 1 / 2 y 1 / 2 p y H 2 K H 2 (T ) + 2⋅ 2 2 ⋅ + K H 2 O (T ) p K OH (T ) y p = bH ! yO 1 / 2 y N 2 1 / 2 p y N 2 K N 2 (T ) + 2⋅ x N 2 + 2 K N O (T ) y = bN K CH (T ) ⋅ yCO2 ⋅ y H 2 K H 2 O (T ) p 1 / 2 ⋅ y H 2 yO2 1 / 2 y + yCO 2 "#$MM' = bC F2+?,;8 *+@ BD=<;<-. "#$MN' =@B;,- 8-Q <;87QFG>J?2 y= + p p yO 2 K O2 (T ) + yO2 + y H 2 K H 2 (T ) + y H 2 y y y H 2 yO2 1 / 2 y 1 / 2 ⋅ + K H 2 O (T ) p yO 2 K O2 (T ) ⋅ y H 2 K H 2 (T ) y 2 + K OH (T ) p yO 1 / 2 y N 2 1 / 2 p y N 2 K N 2 (T ) + y N 2 + 2 K N O (T ) y + K CH (T ) ⋅ yCO2 ⋅ y H 2 ! "#$MK' K H 2 O (T ) p 1 / 2 ⋅ + yCO 2 y H 2 yO2 1 / 2 y L;1.H<23?- "#$MM! #$MK' 7=2B,J G9:;+ S6?-G BD=<;E , &6525; <-.=-;+25@5-/ TUM! VM! UM! WM! y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`! # _ N!%N%KMS' X.98F F2C1 [bA;] F.98F % X2C1 [5a8] P9:;+ 9*a5K [%(] [%&] 5a8 [%( ] [%&] bA; VM cMUM N!())# %)K%!& %)#% %)N) %!\%% %!\)M %!\) VM c UM N!\(SM MK%)!\ MKKK MKMN MKMN %!(N# %!(S# %!(NS %!(S MVMc UM N!#)N) M(K\!S M(SK M(K& M(MS %!)N# %!((S %!)N# %!(( KVMc UM N!K((( K%&\!# K%)& K%&\ K%S\ %!((\ %!(\# %!((\ %!(\ #VMc UM N!KM&S K#NM!\ K#MS K#NK KK## %!(%S %!(M# %!(%S %!(M SVMc UM N!M(\& KSS\!( 6 %!&M( Z 7;C.1- #$%! =@BDH<-2<; ,287;:; 5-.8,;<-<; 87.?I-25.7B@?,<; "MVM c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de.97 7.< +27@?,@ BD=<-.H -<<@?I 5-.8,;<-< =@CG?I2=@?I$ Z 7;C.1- #$K FB,.+87;=-2<. ,287;:@ =@<-9- ;<;1-,@ F;B;5.7BD= +.72<;?>- 5-.8,;<-< F;1-=2=26F2=-.7B,<@?I$ f2,F;7B,2<. ,287;:@ 7@F2=. *;,@$ A2+;<. =@<-9- +27@?,J 89:;+D= 87.?I-25.7B@?,<@?I$ U+F2=-.+<-2! = 9;H+@5 FB,@F;+9G! F2+2C<-. >;9 = 2+<-.8-.<-G +2 5-.8,;<-< =2+2B271.<2=@?I! 29B.31-4 52H<; 89:;+@ 2 5;98@5;1<@?I F;B;5.7B;?I +.72<;?@><@?I$ P9:;+ F2=-.7B,; FB,@>52=;<@ = ?,;8-. 2C1-?,.E FB,.+87;=-2<@ ,287;: = 7;C.1- #$M !"#$!%&')'% P9:;+ F2=-.7B,; FB,@>52=;<@ = 2C1-?,.<-;?I "$N _ M)(!%S `' R+,-;: 2C>Q723?-2=@ `2<?.<7B;?>; 5212=; P9:;+<-9 g 521a5K 521a9* h,27 &(!%N K%!)MK M\!)\& i1.< MN!)S (!S\K &!MK# hB*2< N!)S N!K(( N!KM( !"#$!%&'*'% h<;1-,; F;B;5.7BD= +.72<;?@><@?I 5-.8,;<-< F;1-=2=26F2=-.7B,<@?I R+,-;: % & #T6^ A;1-=2 2C>Q723?-2=@ 9*a5K 5a8 bA; g Z2+DB VM M)!SK N!(S(\ %)\\!# %!S(M b.7;< TV# )!#( %!%KK& %(N%!M %!&#K h?.7@1.< TMVM &!&K %!%&#\ %(\#!S %!)K) d7@1.< TMV# \!SK %!%(%# %(MM!N %!(\% d7;< TMV\ S!\S %!%(\K %(NN!& %!(MK AB2F;< TKV( #!NM %!MN(& %&)(!N %!(SN kG7;< T#V%N K!%M %!MM%N %&)\!K %!(\\ i1.<.9 .7@1.<G TMV#U &!&K %!%(N) %(KN!\ %!)\N V.98;< T\V%# M!%\ %!MK#K %&)#!S %!((M V.F7;< T&V%\ %!(& %!MK(M %&)#!N %!((& $T6^ ` jeV"M)(!%S' b^a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je '! +2+;7<-; =;B7234 .<7;1F-- 7=2B,.<-; 2,<;?,;! H. F2+?,;8 =@7=;B,;<-; +;<.> 8GC87;<?>- , F-.B=-;879D= = 87;<-. 87;<+;B+2=@5 T"8'! VM! UM! -7+$! <;1.H@ +2 G9:;+G +287;B?,@ .<.B*-Q$ d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e2B5;?>- 2 89:;+,-. ?I.5-?,<@5 FB2+G97D=! =;B7234 =@9:;+<-9; F21-7B2F@ -<<. +;<. 2 87;<-. 7.B52+@<;5-?,<@5 FB2+G97D= FB,.5-;<@$ b.72+; =@92B,@87;<-; 87;:@?I BD=<2=;*- B.;9?>- ,;?I2+,J?@?I = FB2+G97;?I FB,.5-;<@ ,;=-.B; 8,.B.* <-.+2*2+<23?-$ ^.+<J , <-?I 8J 29B.312<. =. =:J?,.<-G +2 89:;+G FB2+G97D= +2+;792=@?I 8GC87;<?>-$ Z@5;*; 72 FB,.CG+2=@ B2,=-J,G>J?.*2 G9:;+G BD=<;E "#$MM'$ L 921.- =FB2=;+,.<-. -<<.> F287;?- BD=<;<-; 87;<G! <-H BD=<;<-. 87;<G *;,G -+.;1<.*2! =@5;*; =@B;H.<-. FB;=; +,-;:;<-; 5;8 "#$%%' = 2+<-.8-.<-G +2 127<23?- [%#]! <-. ,;3 ?-3<-.<-;$ W;>F2=;H<-.>8,J C2+;> <-.+2*2+<23?-J 5.72+@ 2F;B7.> 2 =@92B,@87;<-. 87;:@?I BD=<2=;*- >.87 CB;9 52H1-=23?G=,*1Q+<-.<-; =@87QF2=;<-; = 5-.8,;<-<-. 8GC87;<?>- = 87;<-. 892<+.<82=;<@5! ?-.9:@5 1GC 87;:@5$ b2H<; 72 G?,@<-4 >.+@<-. = 8F28DC F23B.+<-! F2FB,., C;+;<-. 87QH.<-; F;B <;+ F2=-.B,?I<-J e;,@ 87;:.>$ Z 9;H+@5 FB,@F;+9G! =@5;*; 72 2FB;?2=;<-; 2+BQC<.*2 729G =@,<;?,;<-; B2,=-J,;<-;! <F$ +1; G=,*1Q+<-.<-; T"8'! h1MUK"8'$ ^;9 =@<-9; , FB,.+87;=-2<@?I G=;*! 29B.31;<-. 87;<G 5-.8,;<-<@ <; F2+87;=-. C;+;<-; F28,?,.*D1<@?I B.;9?>?I.5-?,<@?I ,;?I2+,J?@?I = G9:;+,-. >.87 5.72+J 5;:2 .1;87@?,<J! ,=:;8,?,; = FB,@F;+9G 5-.8,;<-<@ =-.12e;,2=.>! ,;=-.B;>J?.> 8GC87;<?>. = e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dS ≥ Qel ! T "S$%' *+,-./ '( 6 .1.5.<7;B<@ FB,@B287 .<7B2F-- <;725-;87 Q.1 2,<;?,; ?-.F:2 FB,.5-;<@ .1.5.<7;B<.> ",;F-8 BDH<-?,92=@ 2+<28- 8-Q +2 5;:@?I ,5-;< 87;<G G9:;+G'$ R=,*1Q+<-;>J?! H. ') _ Q.1 c *.1 ! "S$M' *+,-./ ') 0 ,5-;<; .<.B*-- =.=<Q7B,<.>! *+, 0 FB;?; .1.5.<7;B<;! =;BG<.9 8;52B,G7<23?- 9;H+.*2 FB2?.8G ,;?I2+,J?.*2 = 5-.8,;<-<-.! = 7@5 =;BG<.9 8;52B,G7<.*2 FB,.C-.*G B.;9?>- ?I.5-?,<@?I! 52H.5@ ,;F-8;4 = F287;?- dS − dE + pdV − W el ≥0! T "S$K' Z ,;1.H<23?- "S$K' =@BDH<-2<; ,287;:; FB;?; <-.2C>Q723?-2=; *+, ! FB,@ ?,@5 *+, _ 0 #'- c *+, ! "S$#' Z@BDH<-.<-. G+,-;:G FB;? <-.2C>Q723?-2=@?I 52H. C@4 <-.,CQ+<. B2,F;7B@=;<-G G9:;+D=! = 97DB@?I FB2?.8 FB,.B.;*2=;<-; 5;7.B-;:G =@8292.<.B*.7@?,<.*2 ,;?I2+,- = =;BG<9;?I <-.;+-;C;7@?,<@?I! FB,@ =@5-;<-. .<.B*-- , 272?,.<-.5$ L;1.H<234 "S$K' 52H. C@4 ,;F-8;<; = F287;?') c #'- 0 $'( l *+, ! "S$S' 1GC '. l *+, ! "S$\' *+,-./ ."$!#' _ ) c #- 0 $(/0/ /0 $(/1/.<7;1F-; 8=2C2+<; "F27.<?>;: m-CC8;'! FB,@ ?,@5 /0/) c #- 0 .<7;1F-;$ Z;BG<.9 "S$\' =@B;H; 2*D1<J ,;8;+Q 29B.31;<-; 9-.BG<9G FB,.C-.*G FB2?.8G Y,@926?I.5-?,<.*2 2B;, =;BG<9D= G87;1;<-; 8-Q BD=<2=;*2=.*2 87;<G G9:;+G$ L;F-8 "S$\' >.87 2*D1<-.>8,J e2B5J ,;1.H<23?- "#$S'! "#$\' 29B.31;>J?@?I 9-.BG<.9 FB,.C-.*G +;<.> B.;9?>- ?I.5-?,<.>$ Z;BG<.9 "S$\' 52H<; =@92B,@87;4 +2 =@,<;?,.<-; BD=<2=;*2=.*2 87;<G 7.B52+@<;5-?,<.*2 - 2+F2=-;+;>J?.*2 5G 89:;+G ?I.5-?,<.*2 FB2+G97D= FB,.5-;<@ =@CG?I2=.>! =3BD+ 97DB@?I =@87QFG>J 89:;+<-9- = e;,;?I 892<+.<82=;<@?I! ?-.9:.> - 87;:.>$ #F#%GH5(0BA53,-.)% 9,4,-.,% 5'()*+)-.,% (23-53,:53):5% 8'A,4B% /(54B'123% /(9)0.,-;% -,% /5481,3.)%0.-.0,+.9,67.%/51)-67,AB%1)(054;-,0.69-):5%B'A,4B A27.<?>;: 7.B52+@<;5-?,<@ G9:;+G ,=-.B;>J?.*2 ng *;,2=@?I 2B;, n8 89:;+<-9D= = e;,-. 892<+.<82=;<.>! 52H.5@ ,;F-8;4 ,; F252?J ,;1.H<23?- [%)] !"#$%&'(#)*!+&,-)./#0(.1'#23)()4&"&0#5"67)5#"03,8'&0(.1'8./)4"3,-!567)999 ng g (T, p, x) = ∑ i=1 n g +ns =$ x xi H i (T,p ) + Δ f H i (0 K ) − T Si (T,p ) + RT ln i x ∑ [ xi H i (T,V ) + Δ f H i (0 K ) − T Si (T,p ) ] ! "S$&' i=n g + 1 P@5C21;5- 2- 2,<;?,2<. ,287;:@ =-.1923?- 87QH.E 5212=@?I F28,?,.*D1<@?I B.;*.<7D=$ AB,.+ B2,=-J,;<-.5 ,;+;<-; 2 =@,<;?,.<-G 87;<G G9:;+G! =-.1923?- 7. 8J <-.=-;+25@5- ",5-.<<. <-.,;1.H<.'$ x _ [2%! 2M! n2ng! 2<*c%! n! 2n]! n _ ng cn8$ R 0 G<-=.B8;1<; 87;:; *;,2=;$ A27.<?>;: g"$! #! x' 2B;, =;B723?- 87QH.E 2- 2+<28,2<. 8J +2 >.+<2879- 5;8@ ",=@91. % 9*'$ je "N`' 2,<;?,; .<7;1F-Q 7=2B,.<-; = N 872F<- `.1=-<;$ AB,@ G=,*1Q+<-.<-G =-.1923?- .<7;1F-- 7=2B,.<-;! F27.<?>;: g"$! #! x'! ,;F-8;<@ = F287;?- "S$&' 29B.31; 7,=$ F.:<J "?;:92=-7J! -<7.*B;1<J' .<7;1F-Q 8=2C2+<J G9:;+G$ d<7;1F-; ?;:92=-7; G=,*1Q+<-; .e.97@ .<.B*.7@?,<. ,5-;<@ 89:;+G ?I.5-?,<.*2 G9:;+G$ Z;BG<9-.5 28-J*<-Q?-; BD=<2=;*- 7.B52+@<;5-?,<.> G9:;+G >.87! ,*2+<-. , "S$\'! 5-<-5;1-,;?>; F27.<?>;:G 7.B52+@<;5-?,<.*2 g"$! #3/x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a11 x1 + a12 x2 + ... + a1n xn =b1 a21 x1 + a22 x2 + ... + a2n xn =b2 ! M "S$)' am1 x1 +am 2 x2 + ... + amn xn =bm *+,-./ 456 6 =8FD:?,@<<-9- 29B.31;>J?. -1234 ;725D= 567.*2 F-.B=-;879; = ,=-J,9G ?I.5-?,<@5 2 <G5.B,. io 75 6 -1234 521- 567.*2 F-.B=-;879; = B2,F;7B@=;<.> -123?- 5-.8,;<-<@ ">.+<287?. 5;8@'o 8 6 -1234 F-.B=-;879D= =@87QFG>J?@?I = ,=-J,9;?I 7=2B,J?@?I 5-.8,;<-<Q$ L;+;<-. 5-<-5;1-,;?>- B.;1-,2=;<. >.87 = 29B.312<@?I =;BG<9;?I 7.B52+@<;5-?,<@?I$ A25-Q+,@ =-.1923?-;57.B52+@<;5-?,<@5- #! $! 9 "*+,-. 9 _ %a% 0 2C>Q7234 =:;3?-=;'! F2=-<<@ ,;?I2+,-4 ,;1.H<23?- =@<-9;>J?. , BD=<;<-; 87;<G! <F$ p ⋅υ = ∑ xi ⋅RT ! "S$%N' i >.H.1- >.87 B2,F;7B@=;<; 5-.8,;<-<; *;,2=; 8F.:<-;>J?; BD=<;<-. 87;<G *;,G -+.;1<.*2$ b.72+@ G=,*1Q+<-.<-; =F:@=G <-.-+.;1<@?I =:;3?-=23?- 8GC87;<?>-! >;9 =F:@= 2C>Q723?- =:;8<.> 521.9G:! =F:@= ?I;B;97.BG 2++,-;:@=;E 5-Q+,@521.9G1;B<@?I <; F287;4 BD=<;<-; 87;<G! G=,*1Q+<-2<. ,287;<J = FG<9?-. \$ fD=<;<-. 87;<G 29B.31; <-.892E?,2<@ ,C-DB F;B;5.7BD= "#! 9'! +1; 97DB@?I 52H. C@4 ,B.;1-,2=;<@ =;BG<.9 5-<-5;1<.> =;B723?- F27.<?>;:G g"$! #! x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ε − ε0 = p + p0 (υ0 − υ ) + Qυ ! 2 "S$%K' T-.F:2 =@CG?IG Q9! =@,<;?,;<. >.87 = =@<-9G B2,=-J,;<-; ,;+;<-;$ n Qυ = − Δ f H MW (298,15) − ∑ Δ f H i (298,15 K ) ! i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y I Ny%! yO! Pyng! y;8Q%! P! ynR -?1 6<*+38, S7;69>1 g"!!"! ' ,20B81 :0;0:7:! )8,-;03 ) /1+7;603: "#$T'! .+)5 ,8+1;09)3;019M ,6+3E?,;59M +*/;1;01:0 "#$U'$ K,- /)8?=-3: :1<3:1<59);5:! >32< <, )181-;03;03 ,.<5:1?0)19>0 /1+7;6,/3> NOJVOOR$ W38, +31?0)19>1 ,-.,/01-1 ,6+3E?3;07 :0;0:7: S7;69>0 m n "#$%#' Q( p, T , x) = g ( p.T , x) + ∑ λ i bi − ∑ aij x j ! i =1 j =1 8-)03X Y#%! #O! P! #:Z 2B :;,4;061:0 [18+1;83\1$ (.+,/1-)3;03 :;,4;06*/ [18+1;83\1 .,)/1?1 ;1 7/)8?=-;03;03! / .+,93203 :0;0:1?0)19>0 S7;69>0 g"!!"!x'! ,8+1;09)3L /5;061>B959M ) +*/;1L "#$U'$ ]1-1;03 :0;0:1?0)19>0 S7;69>0 Q"!!"!x' :,43 C5D +,)/0B)1;3 :3<,-B ;1>2)5C2)38, 2.1-67 "$%&&"&$%'(&$)&*%' NOG! O&R$ ^2<,<1 :3<,-5 ;1>2)5C2)38, 2.1-67 .,?381 ;1 /5);19)3;07 +,)/0B)1;01 .,.+)3) 0<3+195>;5 -,C*+ 6,?3>;59M .+)5C?043L y"6' = Ny%"6'! yO"6'! Pyng"6'! y;8Q%"6'! P! yn"6'R .,2)760/1;38, E902@38, +,)/0B)1;01 .+,C?3:7$ ( /5;067 +31?0)19>0 -1;38, 6+,67 0<3+195>;38, /5);19)1;3 >32< 6,?3>;3 +,)/0B)1;03 "6' = N+%"6'! +O"6'! P+ng"6'! +;8Q%"6'! P! +n"6'R! 6<*+3 >32< ?3.2)5: .+)5C?043;03: .,2)760/1;38, +,)/0B)1;01 E902@38,! / 2<,27;67 -, )C,+7 /5>E90,/38, y"6'$ _<+)5:1;3 +,)/0B)1;03 >32< .+)5>:,/1;3 >16, .,-2<1/1 -, 6,?3>;38, 6+,67 0<3+195>;38, y"6Q%' = "6'$! ( 93?7 +31?0)19>0 6,?3>;38, 6+,67 0<3+195>;38,! ;03)C=-;3 >32< ,6+3E?3;03 :3<,-5 -,C,+7 6,?3>;59M .+)5C?043L! <>$ :3<,-5 /5);19)1;01 )C0,+*/ "6'! 6<*+3 ?3.03> C=-B .+)5C?041D .,2)760/1;3 +,)/0B)1;03 E902@3! ;0`?0 -1;5 )C0*+ /5>E90,/5 y"6'$ aC5 <, 7)5261D +,)6@1-1 20= S7;69>= g"!! "! x' / 2)3+38 b15?,+1 /,6*@ )C0,+7 y"6'$ K,:0>1>B9 9)@,;5 +)=-*/ /542)59M ;04 -+780! ,<+)5:7>3 20= !"#$%&'(#)*!+&,-)./#0(.1'#23)()4&"&0#5"67)5#"03,8'&0(.1'8./)4"3,-!567)999 ∂g 1 n n ∂ 2g ( xi − y ( k ) i ) + ∑ ∑ ( xi − y ( k ) i ) ⋅ ( xk − y ( k ) k ) ! ∂ ∂ ∂ x 2 x x i k i i =1k =1 i =1 x=y x=y > n Y ( p, T , x) = g ( p, T , y ( k ) ) + ∑ "#$%c' A,4;1 )1.021D m n "#$%d' Γ ( p, T , x) = Y ( p, T , x) + ∑ λ i bi − ∑ aij x j ! i =1 j =1 e7;69>1 f ""!!! x' >32< .+)5C?043;03: S7;69>0 Q"!! "! x'! / ,<,9)3;07 .7;6<7 y"6'$(1+7;603: :0;0:1?0)19>0 S7;69>0 f ""!!! x' >32< )3+,/1;03 20= .,9M,-;59M ∂ Γ ( p, T , x) =0! ∂ xi "#$%T' 8-)03 i I %! $$$ ! n ]1?34;,E90 "#$%T' ,6+3E?1>B n +*/;1L! +*/;5 0?,E90 ;03/01-,:59M! >160:0 2B 2<=43;01 :,?,/3 .,2)9)38*?;59M +3183;<*/ +0$ (03?6,E90 "! ! .3@;01 +,?= .1+1:3<+*/$ ]181-;03;03 :0;0:1?0)19>0 S7;69>0 Q"!! "! x' +,)/0B)5/1;3 >32< -?1 -1;3> 6,:C0;19>0 .1+1:3<+*/ <3+:,-5;1:09);59M "! !$ K03+/2)3 .,9M,-;3 S7;69>0 ,""! !! x' /5+141>B 20= ;12<=.7>B9, ∂ g (T , p, x) ∂ = H i (T , p ) + ∆ f H i (0K) - T Si (T , p ) + RT ⋅ ∂ xi ∂ xi ng ∑ x ln xk k x k =1 ! "#$%U' -?1 i I %! O! P! ng ,+1) ∂ g (T , p, x) = H i (T , p) + ∆ f H i (0K) - T Si (T , p ) ! ∂ xi "#$OJ' >343?0 i I ngQ%! P ! n (56,;7>B9 ,-.,/03-;0, .+)362)<1@93;01! ,<+)5:7>3 20= ".,+$ N%UR' RT ∂ ∂ xi ng x x ln k ∑ k x k =1 = RT ln xi ! x "#$O%' g<B- x ∂ g (T , p, x) = H i (T , p ) + ∆ f H i (0K) - T S i (T , p ) + RT ⋅ ln i ! ∂ xi x "#$OO' -?1 i I %! O! P! ng ! n x = ∑ xi ! "#$OG' i =1 K,-,C;03! ,C?09),;3 C5D :,8B -+7803 .,9M,-;3! /52<=.7>B93 /3 /),+)3 "#$%c'$ A,4;1 /*/9)12 ,<+)5:1D ;12<=.7>B95 76@1- +,)/0B)7>B95 >> 1 y 1 a 11 a 21 M a m1 1 !"#$%&'()*' 1 y2 O a12 K K M a1n g a 2n g M a m2 K a mn g 1 K K 0 − 0 K 0 − M 1 y ng a 22 0 1 0 M K − 0 1 y 1 y M 1 y a1n + 1 g a 2n + 1 g M K a1n + n g s K a 2n + n g s a11 a 21 K a12 a 22 K M M a1n g a 2n a 2n +1 g M K a1n + n g s 0 a 2n + n g s 0 K M −1 K a1n +1 g M M a mn + 1 K a mn + n g g s 0 K 0 g 0 0 y B1 − ln 1 x1 y x B − ln y 2 2 2 a m2 y M M M x yn ng g a mng x Bng − ln y ng +1 a mn +1 Bn +1 M g g × = M M x + n n a mng + ns g s B ng +ns x 0 b α 1 1 b2 α2 M M b m 0 α m 0 a m1 ! "#$O&' g)9)38*@5 /5.+,/1-)3;01 )1?34;,E90 "#$O&' );1?3`D :,4;1 / .+195 N%UR$ h6@1- "#$O&' )1/03+1 n Q - Q % ;03/01-,:59M! .+)5 9)5: ):03;;3 .6 2B ):,-5i6,/1;5:0 :;,4;061:0 [18+1;83\1 λ αk =− k ! RT "#$O#' 8-)03 / I %! P! -! (03?6,E90 B0 /52<=.7>B93 / .+1/59M 2<+,;19M 76@1-7 +*/;1L "#$O&' ,6+3E?,;3 2B )1?34;,E901:0 Bi = − H i (T , p) + ∆ f H i (0 K ) − TSi (T , p) RT ! "#$%&' ()*+, i - ! .! ng! ng/ ! .! ng+ n 0123) 456738 "#$%9' :;*63<3 73 6=*73>*,7+, *?+;4@ "1' = A!B"1'! !%"1'! .!ng"1'! !7(/B"1'! .! !n"1'C! 1D54= *(;)7+, * *3E3)3 :4*= 1D54, @123) "#$%&' *;ED32 *?@);637=! ED37;6+ <,:E*, :4*=?<+F,7+, :;E*@1+637,(; 4;*6+G*37+3 H>+E2,(; !! 37+F,<+ *?+54 !"1'! 6 ;)7+,E+,7+@ ); 1D54,(; 6=*73>*37, I,ED 4;*6+G*37+, "1'$ J56737+3 4;*6+G*@IG>, ; :;ED3>+ 373<;(+>*7,I ); "#$%9' ":4*= @6*(<K)7+,7+@ ;14,H<;7,I ;)L+,77;H>+ ;*73>*,8 + E:;E;?56 *3:+E@' 6=L+,7+37, I,ED 6 );7+,E+,7+3>M <+D,43D@4;6=>M! 7:$ A9! %9C$ N;ED3O :4*,)ED36+;73 6 :43>= A9C @6*(<K)7+3 6=EDK:;637+, E123)7+156 P3*+, ED32,I$ !"!#$%&'()&*'+*#,-&.+/&('+(#&(0('+(#-1,*23*'+(#415(06#)7*8+)&'*9-#:,-061;<.#+#4;('6#;*,= 8-0%'(8+)&'*9-#615(06#&(#:-8-)/#&,*061-.('*9-#615(06#,<.'(>! Q=:4;63)*,7+, @123)@ 456738 "#$%9' :;*63<3 73 ?,*:;H4,)7+, :4*,IH>+, ); ,D3:@ 6=*73>*37+3 4;*6+G*37+3$ R;?54 1;<,I7=>M *?+;456 EDKF,8 "1' = A!B"1'! !%"1'! .!ng"1'! !7(/B"1'! .! !n"1'C :4;63)*+ E+K 3F ); L;L,7D@! 6 1D54=L 45F7+>3 :;L+K)*= 6=IH>+;6=L :4*=?<+F,7+,L y"1' = AyB"1'! y%"1'! .yng"1'! y7(/B"1'! .! yn"1'C 3 ;D4*=L37=L 4;*6+G*37+,L "1' ED37+, E+K L7+,IE*3 ;) *3)37,I );123)7;H>+ 6=*73>*,7+3 EDKF,8 L;<;6=>M " !# n ∑ xi (k ) − yi (k −1) ≤ δ xy ! "#$%S' i =1 ()*+,T $ U 7@L,4 :;4*G)1;6= +D,43>I+$ 0123) 456738 "#$%9' I,ED @123),L <+7+;6=L! ); I,(; 4;*6+G*37+3 L;(G ?= 6=1;4*=ED37, D=:;6, L,D;)= 4;*6+G*=637+3 <+7+;6=>M @123)56 456738! I31 7:$ L,D;)3 ,<+L+73>I+ V3@EE3 A%B! %%C$ W3)37+, 6=*73>*,7+3 ?@ !"#$%&'(#)*!+&,-)./#0(.1'#23)()4&"&0#5"67)5#"03,8'&0(.1'8./)4"3,-!567)999 E123)@ >M,L+>*7,(; :4;)@1D56 6 )37=>M 634@713>M D,4L;)=73L+>*7=>M "*3)37, 634D;H>+ %! &' L3 I,)7;*73>*7, 4;*6+G*37+, A%#C$ N4*= :431D=>*7=L 6=1;4*=ED37+@ @123)@ 456738 "#$%9'! *34567; <+>*?3 456738! I31 + 7+,1D54, 623H>+6;H>+ @123)@! 6=7+13IG>, * :;ED3>+ 6=EDK:@IG>=>M 6 7+L 456738! L;(G ED634*3O 7+,);(;)7;H>+! @D4@)7+3IG>, E*=?1+, + );123)7, 6=*73>*,7+, 4;*6+G*37+3$ X31 6+3);L;! +<;H>+ ;:,43>I+ 3<(,?43+>*7=>M 6=L3(37=>M ); 6=*73>*,7+3 4;*6+G*37+3 @123)@ 456738 *36+,43IG>,(; n 456738 "+ n 7+,6+3);L=>M' I,ED :4;:;4>I;73<73 ); n%$ N4*= ;14,H<37+@ E123)@ :4;)@1D56 :4*,L+37= L3D,4+3256 6=E;1;,7,4(,D=>*7=>M * (4@:= YZ[Z\Z]! 73<,F= *6=1<, ?43O :;) @63(K 6=EDK:;637+, B# U %# E@?ED37>I+$ 0123) "#$%9' *36+,43 656>*3E ;) %^ZD@ ); _^ZD@ 456738$ `<;HO E@?ED37>I+ 6=EDK:@IG>=>M 6 *73>*G>=>M +<;H>+3>M *3<,F= 6 ;14,H<;7=L ED;:7+@ ;) 634@7156 :4*,L+37= + ;) E123)@ >M,L+>*7,(; 6=IH>+;6,(; L3D,4+32@$ a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b,(; D=:@ 45F7+>, 6 634D;H>+3>M 6E:52>*=77+156 <+7;6,(; @123)@ 456738! L;(G 6:2=63O 73 );123)7;HO ;?<+>*,8$ Q >,<@ @:4;E*>*,7+3 :4;>,E@ 6=*73>*37+3 4;*6+G*37+3! 6=:4;63)*;7= *;ED32 *4,)@1;637= @123) 456738 ABcC! 4567;63F7= @123);6+ "#$%9'$ W4,)@1;637= @123) 456738 L;F73 ;D4*=L3O :;:4*,* ,<+L+73>IK :;E*>*,(5<7=>M *L+,77=>M! L,D;)G D6;4*,7+3 1;L?+73>I+ <+7+;6,I :;E*>*,(5<7=>M 456738 @123)@ "#$%9'$ Q >,<@ 6=,<+L+7;637+3 *L+,77,I !B 73<,F= :+,46E*, ' 456738 "#$%9' :;L7;F=O 1;<,I7; :4*,* "U(+B' ()*+, i - B! .! '! *3H 456737+,)')/ B :4*,* "UB'$ ]D4*=L@I, E+K 656>*3E − x1 ⋅ a11 + y x ⋅ y1 ⋅ a11 − (α 1a11 ⋅ y1 + α 2 a21 ⋅ y1 + K + α m am1 ⋅ y1 )⋅ a11 = − B1 ⋅ y1 ⋅ a11 + ln 1 ⋅ y1 ⋅ a11 y y y x − x1 ⋅ a2 1 + ⋅ y1 ⋅ a2 1 − (α 1a1 1 ⋅ y1 + α 2 a2 1 ⋅ y1 + K + α m am1 ⋅ y1 )⋅ a2 1 = − B1 ⋅ y1 ⋅ a2 1 + ln 1 ⋅ y1 ⋅ a2 1 y y ! y x − x1 ⋅ a m1 + ⋅ y1 ⋅ a m1 − (α 1a11 ⋅ y1 + α 2 a 21 ⋅ y1 + K + α m am1 ⋅ y1 )⋅ am1 = − B1 ⋅ y1 ⋅ am1 + ln 1 ⋅ y1 ⋅ am1 y y − x1 x + ⋅ y1 y − α 1a11 ⋅ y1 − α 2 a21 ⋅ y1 + K − α m am1 ⋅ y1 = − B1 ⋅ y1 "#$%d' y + ln 1 ⋅ y1 y 0123) "#$%d' *36+,43 L /B 456738$ R;)3IG> :;E*>*,(5<7, 456737+3 "#$%d' ); 1;<,I7=>M ng + n / i 456738 @123)@ "#$%9'! ()*+,T i - B! '/B ;D4*=L@I, E+K Bn g +1 Kα m amn α 1a1n α 2 a 2n B n g + ns x y x2 a12 + x3 a13 K+ xng a1ng + xng +1a1n g +1 K+ xn a1n − y1a11 − α 1a11 y1 ⋅ a11 − α 2 a 21 y1 ⋅ a11 K− α m am1 y1 ⋅ a11 1 b1 − B1 y1 ⋅ a11 + ln y1 ⋅ a11 y y x x2 a 22 + x3 a 23 K+ xng a1ng + xng +1a 2n g +1K+ xn a 2n − y1a 21 − α 1a11 y1 ⋅ a 21 − α 2 a 21 y1 ⋅ a 21 K− α m am1 y1 ⋅ a 21 = b2 − B1 y1 ⋅ a21 + ln y1 y1 ⋅ a21 y y M M M M M M y x x2 am 2 + x3 am3K+ xng a1ng + xng +1amng +1K+ xn amn − y1am1 − α 1a11 y1 ⋅ am1 − α 2 a 21 y1 ⋅ am1K− α m am1 y1 ⋅ am1 bm − B1 y1 ⋅ am1 + ln 1 y1 ⋅ am1 y y y x − B1 y1 − ln 1 y1 + x3 − x + y1 − α 1a11 y1 − α 2 a21 y1 K+ xn g K − α m am1 y1 x2 y y α 1a1n g +1 α 2 a 2 n g +1 M M Kα m amn g +1 M ! "#$%c' X31 6+)3O! ;D4*=L37= @123) 456738 "#$%c' 7+, *36+,43 *L+,77,I !B$ N;EDK:@IG> 6 :;);?7= E:;E5?! L;F73 );1;73O ,<+L+73>I+ 1;<,I7=>M *L+,77=>M !+! i - B! %! .! ng$ ?A !"#$%&'()*' e;F,L= 656>*3E ;D4*=L3O @123) 456738! 1D54= ?K)*+, *36+,432 nE / ' / B 7+,6+3);L=>MT !7(/B! . !7(/7E! !! *B! *%! . *LT α 1 ⋅ a1ng +1 M α 1 ⋅ a1ng + ns x ng +1a1ng +1 K + x n a1ng + ns x n +1a 2 n +1 K + x n a 2 n + n g g s g M x ng +1a mng +1 K + x n a mng + ns + x⋅ x + x⋅ 1 + x⋅ y y M x y ng ∑ y j a1 j j =1 ng ∑ y j a2 j j =1 ng ∑ y j a mj j =1 0 + α 2 a 2ng +1 + M + α 2 a 2 n g + ns + K + α m a mng +1 M K + α m a mng + ns Bng +1 B n g + ns ng ng ng ng yj + α 1 ∑ a1 j y j ( −a1 j ) + α 2 ∑ a 2 j y j ( −a1 j ) + K + α m ∑ a mj y j ( −a1 j ) b1 − ∑ B j y j + ln y j ⋅ a1 j y j =1 j =1 j =1 j =1 ng ng ng ng y + α 1 ∑ a1 j y j ( −a 2 j ) + α 2 ∑ a 2 j y j ( −a 2 j ) + K + α m ∑ a mj y j (−a 2 j ) = b2 − ∑ B j y j + ln j y j ⋅ a 2 j y j =1 j =1 j =1 j =1 M M ng ng ng ng y j y j ⋅ a mj + α 1 ∑ a1 j y j (−a mj ) + α 2 ∑ a 2 j y j (−a mj ) + K + α m ∑ a mj y j (−a mj ) bm − ∑ B j y j + ln y j = 1 j =1 j =1 j =1 ng ng ng ng yj y j − ∑ B j y j + ln K + α m ∑ a mj y j + α 1 ∑ a1 j y j + α 2 ∑ a2 j y j + y j =1 j =1 j =1 j =1 ! "#$_^' 0123) 456738 "#$_^' L;F73 *3:+E3O 6 :;ED3>+ Yf0 - " ! "#$_B' ()*+,T Y U L3>+,4* 6E:52>*=77+156g U U 6,1D;4 "1;<@L7;6=' *L+,77=>M 7+,*3<,F7=>Mg " U L3>+,4* "1;<@L7;63' 6=43*56 6;<7=>M$ ]):;6+,)7+; 0 M 0 a 1ng +1 C = a2n +1 g M a m ng +1 0 K K 0 0 M M 0 0 a1ng +1 M K a1ng + ns 1 ∑ y j a1 j y j =1 ng K a2ng + ns M 1 ∑ y j a2 j y j =1 M ng K am ng + ns K xn g +1 M x n g + ns x U= α1 α2 M α m 0 M a1ng + ns ng 1 ∑ y j am j y j =1 0 a2ng +1 a2 n g + n s ng ng j =1 ng j =1 ng j =1 j =1 ∑ a1 j y j (−a1 j ) ∑ a2 j y j (−a1 j ) ∑ a1 j y j (−a2 j ) ∑ a2 j y j (−a2 j ) M M ng ng j =1 ng j =1 ng ∑ a1 j y j (−am j) ∑ a2 j y j (−am j) ∑ a1 j y j j =1 ∑ a2 j y j j =1 B n g +1 M Bn g + n s ng yj b − ∑ B y + ln y j ⋅ a1 j 1 j j y j =1 ng W = b − B y + ln y j y ⋅ a 2 ∑ j j j 2j ! y j =1 M ng y b − B y + ln j y ⋅ a ∑ mj m j j j y j =1 n g yj yj − ∑ B j y j + ln y j =1 K am ng + ns ng K ∑ am j y j (−a1 j ) j =1 ng K ∑ am j y j (−a2 j ) , j =1 M ng K ∑ am j y j (−am j) j =1 ng K ∑ am j y j j =1 K am ng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hD'! @123) 456738 "#$_B' ?K)*+, E+K *36+,432 & 456738 "nE - B'$ X,F,<+ 4;*63F3L= 6=EDK:;637+, 6 :4;)@1D3>M :4*,L+37= Bc :4;)@1D56 6 ED37+, (3*;6=L! ;*73>*3 D; *L7+,IE*,7+ +<;H>+ 456738 * %#Z>+@ 6 @123)*+, "#$%9'! ); &Z>+@ 6 @123)*+, "#$_B'$ N4;>,E 6=*73>*,7+3 4;*6+G*37+3! :4*= 6=1;4*=ED37+@ *4,)@1;637,(; @123)@ 456738! :4*,?+,(3 6 73EDK:@IG>= E:;E5?$ N;)ED36G ); 4;*:;>*K>+3 +D,43>I+ E123)@ >M,L+>*7,(; L+,E*37+7= I,ED *32;F;7= *?+54 :4*=?<+F;7=>M 634D;H>+ EDKF,8 L;<;6=>M y"B' = AyB"B'! y%"B'! .yng"B'! y7(/B"B'C$ W3)37+, I,ED 4;*6+G*=637, :4*= *3)37=>M 634D;H>+3>M >+H7+,7+3 "%' + D,L:,43D@4= "&' 1D54, 6=7+13IG * :;ED3>+ 634@71@ h*=>*7,(; "6+K*56 h*=>*7=>M I31+>M :4*,?+,(3 :4;>,E'$ R<3 )37=>M %! &! y+"B'! i - B! .ng! Z EG ;14,H<37, 634D;H>+ :;E*>*,(5<7=>M ,<,L,7D56 D3?<+>= 6E:52>*=77+156 Y + L3>+,4*= 6=43*56 6;<7=>M Q! *(;)7+, *, 6*;43L+ "#$_%! #$__'$ Q=*73>*3L= 4;*6+G*37+3 @123)@ "#$_B' 7:$ L,D;)G ,<+L+73>I+ V3@EE3 "A%B! %%C'$ Q 6=7+1@ 4;*6+G*37+3 @123)@ "#$_B'! @*=E1@I, E+K 6=*73>*,7+, 634D;H>+ *L+,77=>MT !7(/B! !!! !7(/7E! !! *B! *%! . *L$ \3 D,I :;)ED36+, ;14,H<3 E+K 634D;H>+ EDKF,8 :;*;ED32=>M E@?ED37>I+! *(;)7+, *, 6*;4,L x j (k ) = x y (k ) ⋅ y j ( k ) − α 1a1 j ⋅ y j ( k ) − α 2 a2 j ⋅ y j ( k ) + K − α m amj ⋅ y j ( k ) + Bj − ln (k ) y j (k ) y (k ) y j (k ) ! "#$_9' ()*+,T)+ - B! %! .! ng Q=*73>*;7, 634D;H>+ !+"1' :4*=IL;637, EG I31; :;)ED363 ); 1;<,I7,I +D,43>I+ *?+;4@ EDKF,8 L;<;6=>M yi ( k ) = xi ( k ) i = 1, K ng ; ! y (k ) =x "#$_#' (k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h*=>*7=L! 6 1D54=>M :4*,?+,(3 :4;>,E$ Q *3<,F7;H>+ ;) :;ED3>+ 634@7156 h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aDKF,7+3 :;*;ED32=>M E@?ED37>I+ 6=*73>*37, EG *, 6*;4@ "#$_9'$ Q=1;4*=ED37+, *4,)@1;637,(; @123)@ 456738 :;*63<3 73 H>+E2, ;14,H<37+, EDKF,8 E@?ED37>I+! 1D54, 45F7+G E+K L+K)*= E;?G ; 1+<13 4*K)56 6+,<1;H>+$ @C !"#$%&'()*' ?!# D+*+0*(3'*#.5(2)+.-2)+#46E4;(')F+#.#.(,6'1()7#.%4-1+)7#)+2'+*># +#;*8:*,(;6,# Q ED37+, 6=IH>+;6=L eQ H4,)7+, ;)<,(2;H>+ :;L+K)*= 3D;L3L+ EG 4*K)@ )2@(;H>+ 6+G*38 L;<,1@<347=>M$ b=L E3L=L! ;))*+32=637+3 L+K)*=L;<,1@<347, EG D31+,(; 4*K)@ 6+,<1;H>+! 1D54= ;):;6+3)3 6=D6;4*,7+@ D4632=>M 4,<3>I+ "6+G*37+3 L+K)*=3D;L;6, + L+K)*=L;<,1@<347,' :;L+K)*= 3D;L3L+ + L;<,1@23L+$ Q 6=7+1@ :4*,L+37= ),D;73>=I7,I (KED;HO L3E;63 ;H4;)13 @<,(3 6*4;ED;6+$ VKED;HO 6 :@71>+, YM3:L373ZX;@(@,D3 ;14,H<373 I,ED *3<,F7;H>+G ABZ_C ρC−J = k +1 ρ 0 MW ! k "&$B' ()*+,T $ U 6=123)7+1 :;<+D4;:= :4;)@1D56 ),D;73>I+ 6 :@71>+, YM3:L373ZX;@(@,D3$ b=L E3L=L! 6+,<1;HO E+2 * I31+L+ L;<,1@2= ;))*+32@IG 73 E+,?+, 6 ED4,h, ),D;73>I+ :4*,143>*3 *314,E 634D;H>+! :4*= 1D54=L 3D;L= + L;<,1@2= D6;4*G ED32, @:;4*G)1;637+3 "6+G*37+3! E+,>+ 14=ED3<+>*7,! ED4@1D@4= 3L;4h>*7,! + +77,'$ N4;)@1D= ),D;73>I+ EG 7+,6GD:<+6+, 73I:;6E*,>M7+,IE*=L ED37,L 6=E;1+,(; @:31;637+3 ;H4;)156 L3D,4+3<7=>M! I31+, E:;D=13L= 6 D,>M7+>,$ X,ED D; ED37 ;E+G(37= :4*= 13F),I ),D;73>I+ 23)@71@ L3D,4+32@ 6=?@>M;6,(;$ 6.1. G:+4#.5(2)+.-2)+#46E4;(')F+#)7*8+)&'%)7#.#.(,6'1()7#.%4-1+)7#)+2'+*>#+#;*8:*,(;6,#&(# :-8-)/#,<.'('+(#4;('6#HI$ Q <+D,43D@4*, :4*,)ED36+;7=>M I,ED 6+,<, :;),IHO ); @6*(<K)7+,7+3 6:2=6@ 6=E;1+>M >+H7+,8 + D,L:,43D@4 73 :343L,D4= :4;)@1D56 6=?@>M@ + ),D;73>I+$ N4*= ;:+E+, ED37@ :4;)@1D56 ),D;73>I+ + 6=?@>M@! I,)7=L * E*,4;1; ED;E;637=>M I,ED 456737+, ED37@ i,>1,43Zj+ED+31;6E1+,(;ZQ+<E;73 "ijQ'$ J56737+, ijQ 6 ;4=(+73<7,I :;ED3>+ *3:4;:;7;637,I i,>1,43 A%SC ( )− Va + V nb+1 RT1 , pV = 1 + xe x RT !"#$ %&'()*+ +,+!-."+ /)+ 0'%12&3+ 45+ 67895:;+ <5=+ >7?45+ '5305?@:;+ AB0454()+ !"#$+ 0@=5'3<)+ 67&7C()D8905+ &7+ AB0454(5+ E54+ &)A+ F55185"+ F+ 8'G')%B147HG(;+ G'I74+ "-V+ '7895I+ 06A705&'74@+ <5=7+ 30'%12&4(5<JG@+ 06I@0+ 8(I+ 6A'@G(J%5<JG@GK"+ L)&45=?);+<5=+8(2+7=5'5I7+0+4589264)<+6A5G@;+0@895AG'5<JG@>+&7+7C1(G')4(5+6A2&=7HG(+&)9745G<(+4(9A7%1(G)A@4@+ 7=5'5I7+8(2+C@:+AB0454()+7+67895G(+ ( ) pV = 1 + xe x = F ( x) , RT !"M$ N7A>5+ !"M$+ &5I5+ 67G'J9)=+ &518')>3+ A7'07<70(+ AB0454(5"+ O(89(5=708=@+ (+ F(1874+ P#QR+ 06A705&'(1(+ &15+ 7=A)H1)4(5+'>()44)<+ +S7A>3I2 x= K , 1 !"T$ 3 V ⋅T 6A'@+ G'@>+ 7=5'5I7+ 8(2;+ ?)+ 0()1=7H:+ K+ >7?)+ C@:+ 79A'@>545+ <5=7+ 83>7054()+ =70713>)4B0+ 0I5HG(0@GK+ 678'G')%B14@GK+ 8=I5&4(=B0+ >()8'54(4@+ <5=J+ 907A'J+ 6A7&3=9@+ &)9745G<("+ U589264()+ &7=74547+ >7&@V=5G<(+ 86787C3+ 30'%12&4()4(5+ 9)>6)A593A@+ (+ 3HG(H1747+ >)97&2+ 7=A)H154(5+ =70713>)4B0+ >7170@GK"+ W=93514()+ AB0454()+89543+XOF+89787054)+<)89+0+67895G(+PTR pV = 1 + x exp ( β x) , nRT %&'()+ !"Y$ !"#$%&'(#)*!+&,-)./#0(.1'#23)()4&"&0#5"67)5#"03,8'&0(.1'8./)4"3,-!567)999 x= k V (T + θ )α ! !"!$ , 6A'@+G'@>+!+<)89+=70713>)4)>+0@65&=70@>+79A'@>@054@>+<5=7+83>5+=70713>)4B0+>7170@GK+!( ng k = κ ∑ ni k i , !"Z$ i =1 [B0454()+ !"Y$+'50()A5+G'9)A@+65A5>)9A@+&7C()A54)+&7H0(5&G'514()*+#;+$;+%+(+&+"+F5A97HG(+9@GK+65A5>)9AB0+ 67'0515<J+ &765870@05:+ 67895:+ AB0454(5+ &7+ 0I5HG(07HG(+ 9)A>7&@45>(G'4@GK+ >()8'54(4@+ A)5=9@04)<;+ &7+ 76(83+ =9BA)<+ AB0454()+ <)89+ 6A')'45G'74)"+ F5A97HG(+ !(+ 8J+ =70713>)45>(+ GK5A5=9)A@'3<JG@>(+ (4&@0(&3514)+ 0I5HG(07HG(+ 678'G')%B14@GK+ 8=I5&4(=B0+ >()8'54(4@"+ F()1=7HG(+ n(+ 7'45G'5<J+ 7&670()&4(7+ (17HG(+ >71(+ ng 678'G')%B14@GK+8=I5&4(=B0;+n = ∑ ni\+ng+]+(17H:+8=I5&4(=B0+%5'70@GK+0+>()8'54(4()" i =1 ^89794@>+ &15+ 0@<5H4()4(5+ 0I5847HG(+ (+ '585&47HG(+ 89787054(5+ AB04(5+ 89543+ XOF+ <)89+ A7'659A')4()+ <)%7+ (49)A6A)95G<(+V'@G'4)<+P#_R" L5=+0(&5:+'+67895G(+AB0454(5+ !"Y$;+6A'@+&3?@GK+05A97HG(5GK+7C<297HG(+3=I5&3;+9<"+&15+%5'B0+A7'A')&'74@GK;+ AB0454()+89543+XOF+6A')GK7&'(+0+AB0454()+89543+%5'3+(&)514)%7"+`15+&3?@GK+V+65A5>)9A+ +895<)+8(2+C70()>+ C1(8=(+')A3"+/+=71)(+6A'@+&3?@GK+%2897HG(5GK+>5870@GK;+05A97H:+676A50=(+45+4()(&)5147H:;+7=A)H1545+6A')'+ 6A505+89A742+AB0454(5+ !"Y$;+'45G'JG7+0'A5895"+a'I74+0@89263<JG@+67+6A50)<+89A74()+AB0454(5;+=9BA@+7=A)H15+ 06I@0+4()(&)5147HG(+7HA7&=5;+>5+C70()>+67895:+0@=I5&4(G'J"+ /%7&4()+'+0I5847HG(5>(+S34=G<(+0@=I5&4(G')<;+6A50J+89A742+AB0454(5+ !"Y$+>7?45+6A')&8950(:+0+67895G( pV ( β x) 2 ( β x) 3 = 1 + x 1 + β x + + + .... , nRT 3 ! 2 ! !"Q$ 13C pV x3 x4 = 1+ x + β ⋅ x2 + β 2 +β3 + .. , nRT 2! 3! !"_$ 30'%12&4(5<JG+8678BC+7=A)H1)4(5+ + 0'BA+ !"!$$;+79A'@>3<)+8(2 pV K (T ) K (T ) 2 β 2 K (T ) 3 β 3 K (T ) 4 = 1+ +β + + + ....... , 2! V 3 3! V 4 nRT V V2 !"bc$ %&'()* ng κ ∑ ni k i K (T ) = i =1 (T + θ )α , !"bb$ L5=+0(&5:+'+67895G(+AB0454(5+ !"bc$+>5+747+S7A>2+54517%(G'4J+&7+0(A(514)%7+AB0454(5+89543"+d7&+0'%12&)>+ 67895G(;+ 0(A(514)+ AB0454()+ 89543+ 7&670(5&5+ '585&4(G'@>+ 0@>7%7>+ 676A5047HG(+ AB0454(5+ 89543+ %5'B0+ A')G'@0(89@GK;+0@4(=5<JG@>+'+6A50(&)I+9)A>7&@45>(=(+8959@89@G'4)<+PMc;+MbR+ pV B (T ) C (T ) D(T ) = 1+ + + + ....... , RT V V3 V2 !"b#$ d78'G')%B14)+G'I74@+AB0454(5+0(A(514)%7+67'0515<J+45+30'%12&4()4()+06I@03+0()17=A794@GK+7&&'(5I@05D+ >(2&'@>71)=315A4@GK"+ `A3%(+ 086BIG'@44(=+ B '$+ 7=A)H15+ 3&'(5I@+ 0478'74)+ 6A')'+ '&)A')4(5-7&&'(5I@054(5+ &0BGK+>71)=3I;+C '$+]+9A')GK;+( '$+]+G'9)A)GK;+(9&"+ d7AB043<JG+A7'0(4(2G()+AB0454(5+XOF+ !"bc$+'+AB0454()>+0(A(514@>+ !"b#$+>7?)>@+'5305?@:;+?)+AB0454()+ XOF+ >7?)+ C@:+ '(49)A6A)97054)+ <5=7+ 86)G<5145+ S7A>5+ AB0454(5+ 0(A(514)%7"+ d78'G')%B14)+ 086BIG'@44(=(+ 0(A(514)+>7%J+C@:+0@A5?74)+<5=7+7&670()&4()+=7>C(45G<)+895I@GK+AB0454(5+XOF+P#_R " !"#$%&'()*' ng κ ∑ ni ki , B(T ) = K (T ) = i =1 (T + θ )α C (T ) = β ⋅ K (T ) 2 ; D(T ) = !"bM$ β2 K (T ) 3 ; itd. , 2! !"bT$ eC)G47H:+0@?8'@GK+G'I74B0+0+A7'0(4(2G(3+AB0454(5+XOF+0+8')A)%+f5@17A5+ !"bc$+08=5'3<);+?)+AB0454()+ 67'0515+ 45+ 30'%12&4()4()+ 06I@03+ <)&47G')84)%7+ 7&&'(5I@054(5+ &0BGK;+ 9A')GK;+ G'9)A)GK;+ 6(2G(3+ (+ 0(2G)<+ >71)=3I"+g9J&+AB0454()+89543+XOF+<)89+6A)&)89@47054)+&7+76(83+8954B0+>59)A((+7+%289@>+365=7054(3;+%&'()+ ')+0'%12&3+45+C1(8=7H:+>71)=3I;+0@89263<)+45=I5&54()+8(2+6B1+7&&'(5I@05D+67GK7&'JG@GK+7&+8J8(5&3<JG@GK+ ')+87CJ+G'J89)G')="+`15+3'@8=54(5+676A504)%7+76(83+0I5HG(07HG(+>()8'54(4@+7+&54@>+8=I5&'()+GK)>(G'4@>;+ 086BIG'@44(=(+0(A(514)+670(44@+C@:+&7CA54)+95=;+5C@+7&670(5&5I@+67895G(+679)4G<5IB0+>(2&'@+>71)=315A4@GK+ 678'G')%B14@GK+ 83C8954G<(+ GK)>(G'4@GK;+ 0GK7&'JG@GK+ 0+ 8=I5&+ >()8'54(4@"+ L5=+ 0(&5:+ '+ 67895G(+ '51)?47HG(+ !"bT$;+086BIG'@44(=(+0(A(514)+<5=()+0@4(=5<J+'+0I5HG(07HG(+AB0454(5+89543+XOF;+670895<J+<5=7+7=A)H174)+ =7>C(45G<)+895I@GK;+=9BA)+65A5>)9A@'3<J+AB0454()"+ g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i"+ L5=+ 0@4(=5+ '+ 0I5847HG(+ AB0454(5+ XOF;+ =9BA@GK+ (1389A5G<J+ <)89+ A7'0(4()G()+ 0+8')A)%+f5@17A5;+678'G')%B14)+65A5>)9A@'5G<)+AB0454(5+XOF+7&670(5&5:+C2&J+>()8'54(47>+7+&54@>+9@6()+ 8=I5&3+GK)>(G'4)%7+7A5';+'5GK703<J+676A5047H:+0+7=A)H174@>+6A')&'(51)+'>(54+%2897HG(+>5870@GK;+G(H4()D+ (+9)>6)A593A+A7'659A@054@GK+>()8'54(4+ 6A7&3=9B0+&)9745G<(+(+0@C3GK3$"+ dA50(&I70)+ 7=A)H1)4()+ '5=A)83+ 05?47HG(+ &54)<+ 65A5>)9A@'5G<(+ XOF+ <)89+ 67&8950J+ &7+ 3'@8=54(5+ 0@87=()<+ &7=I5&47HG(+0@4(=B0+7C1(G')D+0@=7454@GK+6A'@+<)<+0@=7A'@8954(3" 6.2. #$%$&'(%)*$+,'-%./0$01$-BKW d7895:+ AB0454(5+ XOF+ !"Y+ h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i7?45+0@AB?4(:+9A'@+ 86787C@+&7C7A3+05A97HG(+65A5>)9AB0+AB0454(5+XOF* j$+]+%)7>)9A@G'4@\+6A'@+7=A)H154(3+05A97HG(+=70713>)4B0+>7170@GK+0@=7A'@89@054)+8J+&54)+&79@G'JG)+ %)7>)9A((+>71)=3I+678'G')%B14@GK+8=I5&4(=B0+>()8'54(4@\ W$+]+5&(5C59@G'4@\+05A97H:+=70713>)43+7&670(5&5<JG)%7+&54)<+83C8954G<(+GK)>(G'4)<+0@'45G'545+<)89+45+ 67&8950()+6A')C()%3+5&(5C59@+3&)A')4(70)<+9)<+83C8954G<(\+ `$+ ]+ &)9745G@<4@\+ 65A5>)9A@+ AB0454(5+ XOF+ &7C()A54)+ 8J+ 95=;+ 5C@+ 3'@8=5:+ '%7&47H:+ 0@4(=B0+ 7C1(G')D+ '+)=86)A@>)49514()+7=A)H154@>(+05A97HG(5>(+65A5>)9AB0+&)9745G<(;+'0I58'G'5+'+6A2&=7HG(J+&)9745G<(" F+ 1(9)A593A')+ 6A')&8950(74@GK+ '7895I7+ 8')A)%+ >)97&+ 7=A)H154(5+ ')8950B0+ 65A5>)9AB0+ AB0454(5+ XOF"+ L5=7+ 45<C5A&'()<+ GK5A5=9)A@89@G'4)+ (+ >5<JG)+ 45<0(2=8')+ '45G')4()+ 6A5=9@G'4)+ 0@AB?4(>@+ 4589263<JG)+ 65A5>)9A@'5G<)* XOFki*+65A5>)9A@'5G<5+76A5G70545+6A')'+i5&)A5+PTR; 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