Solution_Manual_to_engineering_fluid_mechanics_7e

Solution_Manual_to_engineering_fluid_mechanics_7e - L|it |L...

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U@+n 4: 4, @ nU@+n 4, 5146 Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D18 iru zdwhu= surylghg lq sureohp vwdwhphqw XQ @ 7=37 RQ @ 4=64 43ÃU Q v2pS 43ÃT Q v2pS XQ @ <:; nj2pT RQ @ 4> 333 nj2pT iurp Wdeoh D16 iru dlu XQ @ 5=37 RQ @ 4=:9 43ÃV Q v2pS 43ÃV Q v2pS XQ @ 4=36 nj2pT RQ @ 4=58 nj2pT Ilqg fkdqjh lq ylvfrvlw| dqg ghqvlw| iru dlu dqg zdwhu Vroxwlrq iru zdwhu  @ XQ RQ @  @ XQ RQ @ <=39 43ÃU Q v2pS 55 nj2pT iru dlu  @ .3=5; @ 43ÃV Q v2pS 3=55 nj2pT 4; 5147 Lqirupdwlrq dqg dvvxpswlrqv iurp wdeoh D16/  WQ @ 4=;< 43ÃV pS 2v/  RQ @ 4=74 surylghg lq sureohp vwdwhphqw Ilqg fkdqjh lq nlqhpdwlf ylvfrvlw| iurp 43r F wr 93r F ydluGRQ6WQ @ +4=;< 4=74, 4< 43ÃV pS 2v 43ÃV @ 7=; 43ÃW pS 2v 5148 Lqirupdwlrq dqg dvvxpswlrqv iurp wdeohv D17/ Ilj1 D15 dqg Wdeoh D181 surylghg lq sureohp vwdwhphqw Ilqg Vroxwlrq +Q v2pS , +nj2pS , +pS 2v, Rlo +VDH 43Z, 6=9 43ÃS ;:3 7=4 43ÃV 4=7 4=: 53 nhurvhqh zdwhu 43ÃT +Ilj1 D05, 9=; 43ÃU <<6 43ÃW +Ilj1 D05, 9=; 43ÃX 5149 Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D16/ dluGSQ F @ 4=;4 43ÃV Q v2pS >  @ 4=84 43ÃV pS 2v iurp Wdeoh D18/ zdwhuGSQ F @ 4=33 43ÃT Q v2pS >  @ 4=33 43ÃW pS 2v surylghg lq sureohp vwdwhphqw Ilqg udwlr ri g|qdplf dqg nlqhpdwlf ylvfrvlwlhv ri zdwhu dqg dlu Vroxwlrq dlu @zdwhu @ 4=;4 43ÃS >  dlu @ zdwhu @ 48=4 54 514: Frpsxwhu Sureohp 0 qr vroxwlrq1 55 514; Wkh udwlr ri nlqhpdwlf ylvfrvlwlhv lv  [email protected] W  r Wr . V sr W  @ @ r r  Wr W . V s Wr  [email protected] sr W  Wr . V @ r s Wr W .V 56 514< Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D15/ V @444 N/ dlu +48r F,@41:; 43ÃV Q v2pS = surylghg lq sureohp vwdwhphqw Ilqg xvlqj Vxwkhuodqg*v htxdwlrq/ qg  dw 433r F Vxwkhuodqg*v htxdwlrq   @ W [email protected] Wr FV @ TXT [email protected] r Wr W FV SYY  @ 4=548 4=:; 43ÃV @ 5=49 57 SYYFRRR TXTFRRR @ 4=548 43ÃV Q [email protected] = 5153 Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D15/ V @ 4<; N surylghg lq sureohp vwdwhphqw Ilqg e| ghqlwlrq   @ Lghdo0jdv odz  sr W  r  @ @ r r  r s Wr Vxwkhuodqg*v htxdwlrq  sr @ r s  W Wr [email protected] Wr . V W .V vr 4  @ r 5  7:6 5;; [email protected] 7:6 . 4<; @ 4=585 5;; . 4<; dqg  @ 4=585 4=3; 43ÃV pS 2v @ 4=68 58 43ÃV pS 2v 5154 Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D15/ V @4<5r U1 surylghg lq sureohp vwdwhphqw Ilqg  dw 533r I Vxwkhuodqg*v htxdwlrq  [email protected] [email protected] 993 W Wr . V 84< . 4<5 @ @ 4=4<: @ Wr W .V 84< 993 . 4<5  @ 4=4<: 6=8< 43ÃX @ 7=63 43ÃX oei0v2iwS  r  59 5155 Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D15/ V @476r U1 surylghg lq sureohp vwdwhphqw Ilqg  dw 63r I Vxwkhuodqg*v htxdwlrq  r  [email protected]  [email protected]  4=8 7<3 Wr . V 84< . 476 W @ @ 4=69 @ @ Wr W .V r 4 84< 7<3 . 476 @ 4=69 4=55 43ÃT @ 4=99 43ÃT iwS @v= sr s  5: 5156 Lqirupdwlrq dqg dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg Vxwkhuodqg*v frqvwdqw Vroxwlrq Vroylqj Vxwkhuodqg*v htxdwlrq iru [email protected] jlyhv  Wr [email protected] 4 V r W @ [email protected]  Wr 4  Wr rW Vxevwlwxwlqj @r @ 4=:5 dqg Wr @W @ 6:[email protected]:6 jlyhv V Wr V @ 3=<97 @ 693 N= 5; 5157 Lqirupdwlrq dqg dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg Vxwkhuodqg*v frqvwdqw Vroxwlrq Vroylqj iru Vxwkhuodqg*v frqvwdqw  Wr [email protected] 4 V r W @ [email protected]  Wr 4  Wr rW Vxevwlwxwlqj ydoxhv  6=79 @ r 5=3: 43ÃX @ 4=9:4 43ÃX dqg 85; Wr @ @ 3=94<: W ;85 lqwr htxdwlrq iru [email protected] V Wr V @ 4=:4 @ <36r U 5< 5158 Lqirupdwlrq dqg dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg +93r F, Vroxwlrq Ylvfrvlw| yduldwlrq ri d oltxlg fdq eh h{suhvvhg dv  4  @ h{s e+ r W  4 , Wr Wdnlqj wkh orjdulwkp dqg vroylqj iru e jlyhv e@ oq +@r , R R, +W Wr Vxevwlwxwlqj lq wkh ydoxhv ri @r @ [email protected]=39: @ 3=497 dqg W @ 6:5 dqg Wr @ 644 jlyhv e @ 6763 +N, Wkh ylvfrvlw| udwlr dw 93r F lv 4 4  , @ 3=7;66 @ h{s^6763+ r 666 644  @ 3=7;66 3=39: @ 3=365 Q [email protected] 63 5159 Lqirupdwlrq dqg dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg +483r I, Vroxwlrq Htxdwlrq iru ylvfrvlw| yduldwlrq ri d oltxlg fdq eh h{suhvvhg dv   4 4  @ h{s e+ , r W Wr Wdnlqj wkh orjdulwkp dqg vroylqj iru e jlyhv e@ Vxevwlwxwlqj lq wkh ydoxhv ri @r @ 3=6< oq +@r , R R, +W Wr 43ÃT @7=76 43ÃT @ 3=3=3;;3 dqg W @ 9:3 dqg Wr @ 893 jlyhv e @ ;5<6 +r U, Wkh ylvfrvlw| udwlr dw 483r I lv 4 4  , @ 3=5<< @ h{s^;5<6+ r 943 893  @ 3=5<< 7=76 43ÃT @ 4=65 43ÃT [email protected] 64 515: D frpsxwhu surjudp lv ghyhorshg iru Vxwkhuodqg*v frqvwdqw iurp wkh htxdwlrq  SXT [email protected] 4 V r W @  XT [email protected] 5:6 4  SW r Fduu|lqj rxw wkh fdofxodwlrq iru hdfk gdwd srlqw dqg wdnlqj wkh dyhudjh iru Vxwkhuodqg*v frqvwdqw jlyhv V @ 45: N Wkh huuru lv ghqhg dv huu @ 433    r      r mfdof     r Wkh uhvxowv duh W+N, m r fdof huuru+(, 593 1<939 1346 5:3 1<<4 136< 5;3 41354 13;7 5<3 41383 144; 633 413:< 143; 683 4154: 1699 833 418;5 17;9 4333 517;< 414: 4833 6149; 6189 Wkh huuru lv ohvv 318( iru whpshudwxuhv xs wr 833 N exw kdv dq huuru ri ryhu 618( iru d whpshudwxuh ri 4833N1 65 515; Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh Iljxuh D15/  @ 8=5 surylghg lq sureohp vwdwhphqw Ilqg vkhdu vwuhvv Vroxwlrq Udwh ri vwudlq 43ÃU oe [email protected] [email protected]| @ 43@[email protected],@45,vÃR Uhodwlrq ehwzhhq vwuhvv dqg udwh ri vwudlq  @ [email protected]| @ 8=5 43ÃU 66 43 7; @ 3=583 oe2iwS 515< Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D16/ dlu @ 4=<4 43ÃV Q v2pS iurp Wdeoh D18/ zdwhu @ 9=86 43ÃU Q v2pS /zdwhu @ <<5 nj2pT 1 surylghg lq sureohp vwdwhphqw Ilqg nlqhpdwlf dqg g|qdplf ylvfrvlwlhv Lghdo jdv odz dlu @ [email protected] @ 4:3> 333+5;: 646=5, @ 4=;< nj2pT Iru dlu  @ 4=<4  43ÃV Q [email protected] @ @ @ +4=<4 43ÃV ,@4=;< @ 43=4 43ÃW pS 2v Iru zdwhu  @ 9=86 43ÃV Q [email protected]  @ +9=86 43ÃU @<<5, @ 9=8; 67 43ÃX pS 2v 5163 Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D18/  @ 43ÃT Q v2pS surylghg lq sureohp vwdwhphqw Ilqg vkhdu vwuhvv dw | @ 5 pp Vwuhvv0vwudlq uhodwlrqvkls  [email protected]|  @ @ @ @ @ [email protected]| [email protected],+43,+| ,[email protected] vÃR [email protected],+43,+=335,[email protected] vÃR [email protected],+4::=8, vÃR +43ÃT Q v2pS ,[email protected],+4::=8, vÃR @ 3=5<9 Q2pS 68 5164 Lqirupdwlrq dqg dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg vkhdu vwuhvv dw zdoov Vroxwlrq x [email protected]| [email protected]| ,|dQ Q @ @ @ @  Q=R 433| +3=4 | , @ 43| 433| S 43 533| 43 vÃS [email protected]| ,|dQ=R @ 43 vÃR [email protected]| @ +; 43ÃV , 43 @ ; 43ÃU oei2iwS @ ; 43ÃU oei2iwS 0.10 Distance 0.08 0.06 0.04 0.02 0.00 Velocity 69 5165 Lqirupdwlrq dqg dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg pd{lpxp dqg plqlpxp vkhdu vwuhvv pd{lpxp vkhdu vwuhvv dw zdoo Vroxwlrq   pd{  pd{ @ gY @g|  + Y@ | , qh{w wr zdoo @ +43ÃT Q v2pS ,++4 p2v,@3=334 p, @ 4=3 Q2pS Wkh plqlpxp vkhdu vwuhvv zloo eh }hur/ plgzd| ehwzhhq wkh wzr zdoov/ zkhuh wkh yhorflw| judglhqw lv }hur1 6: 5166 Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D17/  @ 9=5 43ÃR Q v2pS surylghg lq sureohp vwdwhphqw Ilqg yhorflw| dqg vkhdu vwuhvv dw 45 pp iurp zdoo/ vkhdu vwuhvv dqg yhorflw| dw zdoo Vroxwlrq Yhorflw| x [email protected]{ xRSpp xRSpp @ [email protected],[email protected]{,+E| |S ,> E @ 3=38 p @ 4> 933 Q2pT @ +4> 933@+5 3=95,,+3=38 3=345 +3=345,S , @ 3=8;; p2v Vkhdu vwuhvv  Q  RS @ [email protected]| @ [email protected],[email protected]{,+E @ +4> [email protected],+3=38, @ 73 Q2pS @ +4> [email protected],+3=38 6; 5 5|, 3=345, @ 53=; Q2pS 5167 Ghyhors dq htxdwlrq iru wkh vkhdu vwuhvv  [email protected]| Hydoxdwh  dw | K Hydoxdwh  dw | Q @ @ @ @ @ @ @ [email protected]| [email protected],[email protected],+K 5| , . xw @K K= [email protected],[email protected],+K 5K , . xw @K [email protected],[email protected],K . xw @K 3 [email protected],[email protected],K . xw @K Revhuydwlrq ri wkh yhorflw| judglhqw ohwv rqh frqfoxgh wkdw wkh suhvvxuh judglhqw [email protected] lv qhjdwlyh1 Dovr xw lv qhjdwlyh1 Wkhuhiruh m k m A m Q m = Wkh pd{lpxp vkhdu vwuhvv rffxuv dw | @ K 1 6< 5168 Iurp vroxwlrq wr 5167  Vhw  3 | @ [email protected],[email protected],+K 5| , . xw @K @ 3 dqg vroyh iru | @ [email protected],[email protected],+K 5| , . xw @K @ [email protected], +xw @[email protected],, 73 5169 Ghulyh dq h{suhvvlrq iru sodwh vshhg  [email protected]| Wkhq/ dw | Vroyh iru xw Qrwh @ @ @ = = [email protected]| @ 3 dw | @ 3 [email protected],[email protected],+K 5| , . xw @K 3 = [email protected]| @ 3 @ [email protected],[email protected],K . xw @K xw @ [email protected],[email protected],K S ehfdxvh [email protected] ? 3> xw ? 3= 74 516: Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D17/ +6;r F,@6=9 43ÃS Q v2pS surylghg lq sureohp vwdwhphqw Ilqg qg wrutxh rq vkdiw Vroxwlrq Rsin Rdθ θ R ∆R gW gW zkhuh  @ @ @ @ @ @ gD @ @ @ u@ Wkhq gW @ gW @ W Wrutxh ugI u gD [email protected]| , @ + Y @ U, +$U vlq @ U, 6=9 43ÃS Q v2pS ,+43 5@93, udg2v+3=38 p vlq @43ÃT p, 4=;;8 vlq  Q2pS 5U vlq Ug 5US vlq Ug 5US vlq g U vlq  U vlq  +4=;;8 vlq ,+5US vlq g , 44=;7UT vlqT g @ 44=;7UT Q vlqT g @ 44=;7+3=38,T ^ [email protected], frv +vlqS  . 5,` Q @ 44=;7+3=38,T ^ [email protected],+ 4,+5, + [email protected],+4,+5,` @ 4=<: 43ÃT Qp 75 516; 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I  @ pd Z . gcY @^+G Z . + 3=4 3=5 g,@5` @ [email protected] 6=8 43ÃR Y ,@+3=8 43ÃT @5, @ Z d@<=;4 Vxevwlwxwlqj Y @ 3=8 p2v dqg d @ 47 p2vS dqg vroylqj |lhogv Z @ 4;=4 Q 78 5174 Lqirupdwlrq dqg dvvxpswlrqv dvvxph olqhdu yhorflw| glvwulexwlrq= gY @g| @ Y@| @ $u@| surylghg lq sureohp vwdwhphqw Ilqg udwlr ri vkhdu vwuhvv dw u @ 5 fp wr vkhdu vwuhvv dw u @ 6 fp/ vshhg ri rlo dw frqwdfw zlwk f|olqghu/ vkhdu vwuhvv dw f|olqghu Vroxwlrq   S @ T Y  @ gY @g| @ $u@| @ + 4 5@|,@+ 4 6@| , @ [email protected] @ 3=99: @ $u @ 5 3=36 @ 3=39 p2v @ gY @g| @ 3=34 79 [email protected]=335 @ 3=63 Q2pS 5175 Lqirupdwlrq dqg dvvxpswlrqv dvvxph d olqhdu yhorflw| glvwulexwlrq surylghg lq sureohp vwdwhphqw Ilqg wrutxh wr urwdwh glvn Vroxwlrq   Wrutxh gY @g| $u@| 3=34 8 [email protected]=335 @ 58u Q2pS u gD u+43u,5ugu @ 83uT gu Q=QV  83uT gu @ 83uU @7 Q=V @ Q Wrutxh @ 5=78 g Wrutxh @ @ @ @ @ Q 43ÃU Q p 7: 5176 Lqirupdwlrq dqg dvvxpswlrqv dvvxph yhorflw| surohv duh olqhdu surylghg lq sureohp vwdwhphqw Ilqg ghulyh dq htxdwlrq iru gdpslqj wrutxh Vroxwlrq Uhodwlrq ehwzhhq vkhdu vwuhvv dqg yhorflw| judglhqw  @ gY @g| @ u$@v Rq dq hohphqwdo vwuls ri duhd ri udglxv u wkh glhuhqwldo vkhdu irufh zloo eh gD ru  +5ugu,1 Wkh glhuhqwldo wrutxh zloo eh wkh surgxfw ri wkh glhuhqwldo vkhdu irufh dqg wkh udglxv u ru gWrqh vlgh gWerwk vlghv @ @ @ @ W @ u^ +5ugu,` u^+u$@v,+5ugu,` +5$@v,uT gu +u$@v,uT gu G@S Q +7$@v,uT gu @ +4@49,$GU @v 7; 5177 Rqh srvvleoh ghvljq vroxwlrq lv jlyhq ehorz1 Dvvxpswlrqv= 41 Prwru rlo lv VDH 43Z063> wkhuh  zloo ydu| iurp derxw 5 43ÃU oei0v2iwS wr; 43ÃT oei0v2iwS +Ilj1 D05, 51 Dvvxph wkh rqo| vljqlfdqw vkhdu vwuhvv ghyhorsv ehwzhhq wkh urwdwlqj f|olqghu dqg wkh {hg f|olqghu1 61 Dvvxph zh zdqw wkh pd{lpxp/ udwh ri urwdwlrq $/ wr eh 6 udg2v1 Duelwudu| ghflvlrq= 41 Ohw k @ 7=3 lq1 @ 31666 iw 51 Ohw L1G1 ri {hg f|olqghu @ <133 lq1 @ 31:833 iw1 61 Ohw R1G1 ri urwdwlqj f|olqghu @ ;1<33 lq1 @ 1:74: iw1 Ohw wkh dssolhg wrutxh zklfk gulyhv wkh urwdwlqj f|olqghu eh surgxfhg e| d irufh iurp d wkuhdg ru vpdoo gldphwhu prqrodphqw olqh dfwlqj dw d udgldo glvwdqfh uv = Khuh uv lv wkh udglxv ri d vsrro rq zklfk wkh wkuhdg ri olqh lv zrxqg1 Wkh dssolhg irufh lv surgxfhg e| d zhljkw dqg sxooh| v|vwhp vkrzq lq wkh vnhwfk ehorz1 Pulley hr c ∆r W Wkh uhodwlrqvkls ehwzhhq > uv > $> k> dqg Z lv qrz ghyhorshg1 W @ uf Iv +4, zkhuh W @ dssolhg wrutxh uf @ rxwhu udglxv ri urwdwlqj f|olqghu Iv @ vkhdulqj irufh ghyhorshg dw wkh rxwhu udglxv ri wkh urwdwlqj f|olqghu exw Iv @ Dv zkhuh Dv @ duhd lq vkhdu @ 5uf k  @ gY @g|   Y @ u zkhuh Y @ uf $ dqg u @ vsdflqj Wkhq W @ uf + Y @ u,+5uf k, @ uf +uf $@ u,+5uf k, 7< +5, Exw wkh dssolhg wrutxh W @ Z uv vr Ht1 +5, ehfrph T Z uv @ uf $+5,k@ u T ru  @ +Z uv u,@+5$kuf , +6, Wkh zhljkw Z zloo eh duelwudulo| fkrvhq +vd| 5 ru 6 r}1, dqg $ zloo eh ghwhuplqhg e| phdvxulqj wkh wlph lw wdnhv wkh zhljkw wr wudyho d jlyhq glvwdqfh1 Vr uv $ @ Yidoo ru $ @ Yidoo @uv = Htxdwlrq +6, wkhq ehfrphv ST  @ +Z@Yi ,+uv @uf ,+ u@+5k,, Lq rxu ghvljq ohw uv @ 5 lq1 @ 31499: iw1 Wkhq  @ +Z@Ii ,+3=499:S @=6:3;T ,+3=33749:@+5  @ +Z@Yi ,+=35::<@=383<;,  @ +Z@Yi ,+4=3;8 43ÃT , oei v@iwS 3=897 =6666, H{dpsoh= Li Z @ 5r}1 @ 31458oe1 dqg Yi lv phdvxuhg wr eh 3157 iw2v wkdq  @ +3=458@3=57,+4=3;8 43ÃT ,> @ 43ÃU oei v2iwS Rwkhu wklqjv wkdw frxog eh qrwhg ru frqvlghuhg lq wkh ghvljq= 41 Vshfli| glphqvlrqv ri doo sduwv ri wkh lqvwuxphqw1 51 Qhjohfw iulfwlrq lq ehdulqjv ri sxooh| dqg rq vkdiw ri f|olqghu1 61 Qhjohfw zhljkw ri wkuhdg ru prqrodphqw olqh1 71 Frqvlghu ghjuhh ri dffxudf|1 81 Hvwlpdwh frvw ri wkh lqvwuxphqw1 83 5178 Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D18/ H @ 5=5 43` Sd surylghg lq sureohp vwdwhphqw Ilqg yroxph diwhu suhvvxuh dssolhg Vroxwlrq Uhodwlrq ehwzhhq prgxoxv ri hodvwlflw|/ dssolhg suhvvxuh dqg yroxph fkdqjh H@ s@+ @, YY Ilqdo yroxph 5 43W @+ @4> 333, Y 43` @ @ Y +5 43W 4> 333,@+5=5 Y  @ <<<=3<ff Y ilqdo @  Y 5=5 84 43` , @ 3=<3<ff 5179 Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D18/ H @ 5=5 43` Sd surylghg lq sureohp vwdwhphqw Ilqg suhvvxuh lqfuhdvh wr uhgxfh yroxph e| 4( Vroxwlrq Uhodwlrq ehwzhhq prgxoxv ri hodvwlflw|/ dssolhg suhvvxuh dqg yroxph fkdqjh H@ s@+ Y @Y , Iru 4( yroxph fkdqjh 5=5 s@+ 4@433, 43` @ s @ 5=5 43X Q2pS @ 55 PQ2pT +lqfuhdvh, 85 517: Uhihu wr Ilj1 509+d,1 Wkh vxuidfh whqvlrq irufh/ 5u / zloo eh uhvlvwhg e| wkh suhvvxuh irufh dfwlqj rq wkh fxw vhfwlrq ri wkh vskhulfdo gursohw ru s+uS , @ 5u s @ 5@u @ 7@g 86 517; Lqirupdwlrq dqg dvvxpswlrqv dvvxph hhfw ri wklfnqhvv lv qhjoljleoh surylghg lq sureohp vwdwhphqw Ilqg qg suhvvxuh glhuhqfh dfurvv exeeoh Vroxwlrq Irufh edodqfh 2 x 2πRσ p [ sUS I @3 5+5U, @ 3 s @ 7@U sUpp udgI @ +7 :=6 43ÃS Q2p,@3=337 p @ :6=3 Q2pS 87 517< Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D18/  @ 3=3:6 Q2p surylghg lq sureohp vwdwhphqw Ilqg pd{lpxp pdvv ri exj Irufh htxloleulxp [ IS @ 3 vxuidfh whqvlrq irufh vxssruwlqj exj 0 zhljkw ri exj @ 3 Frqvlghu furvv vhfwlrq ri exj ohj= Cross section of bug leg Surface tension force on one side of leg θ FF Assume θ is small Then cos θ =1; F cos θ= F IW @ @ @ @ , p@ +5@ohj,+9 ohjv,c 45c 45+3=3:6 Q2p,+3=338 p, 3=3376;Q IW pj @ 3 +3=3376;@<=;4, 4> 333 @ 3=779 jp 88 5183 Lqirupdwlrq dqg dvvxpswlrqv iurp Wdeoh D18 vxuidfh whqvlrq ri zdwhu lv 31338 oei2iw surylghg lq sureohp vwdwhphqw Ilqg dprxqw ri zdwhu froxpq gxh wr vxuidfh whqvlrq hhfwv Vroxwlrq k @ 7@+g, @ 7 3=338@+95=7 g, @ 6=54 43ÃU @g iw1 g @ 4@7 lq1 @ 4@7; iw1> k @ 6=54 43ÃU @+4@7;, @ 3=3487 iw1 @ 314;8 lq1 g @ 4@; 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Y Y D 3 @ g@gw+Y . YW DD , 3 @ YF DF . YW +@7,+;S 9S , 3@ 6 +@7,9S . YW +@7,+;S 9S , YW @ 43;@+97 69, @ 6=;9 iw2v +xszdug, 5;6 7199 Lqirupdwlrq dqg dvvxpswlrqv D urxqg wdqn +G @ 7 iw, lv ehlqj oohg zlwk zdwhu iurp d 4 iw gldphwhu slsh1 Lq wkh slsh/ Y @ 43 iw2v1 1 surylghg lq sureohp vwdwhphqw Ilqg Udwh dw zklfk wkh zdwhu vxuidfh lv ulvlqj= YU Vroxwlrq Dsso| wkh frqwlqxlw| htxdwlrq dqg ohw wkh f1v1 pryh xs zlwk wkh zdwhu vxuidfh lq wkh wdqn1 [ g . Y Y D 3 @ g@gw FY 3 @ g@gw+kDW , ++43 . 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TRV Exw TSQ @ 5TRV 3=63 @ 5TRV . 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Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Xvlqj wkh vwhs0e|0vwhs phwkrg I G Y   nv $ PO W O O W P O P OW O R W 5 6 I G P W I G Y G GT G nv G $ W P P W 3 R W R Y G 5 7 R W 5 I Y G 5 5 3 R W  G nv G $ 5 R W 3 R W  Y G 3 $G Y 3 Ohqjwk lv uhpryhg lq wkh uvw vwhs zlwk G> pdvv lq wkh vhfrqg vwhs zlwk GT dqg wlph lq wkh wklug vwhs zlwk Y @G= Wkh ixqfwlrqdo irup lv Y G nv $G I >> , @ i+ Y S GS  GY Wkhuh duh rwkhu srvvleoh irupv1 98: ;1< Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Xvlqj wkh vwhs0e|0vwhs phwkrg IG Y  E  xQ O{ PO W O W P O O P OW O W O 5 6 R IG Y E R  Y 3E x Y O{ E IG E Y E E T P W R W P IG E Y E W R W E x3 E O{ E P W R W 3  E3 x E O{ E W R W 3 5 7 5 5 5 5 3 3 3 3 Ohqjwk lv uhpryhg lq uvw vwhs zlwk E> pdvv lv uhpryhg lq vhfrqg zlwk E T dqg wlph lv uhpryhg lq wkh wklug zlwk Y @E= Wkh ixqfwlrq irup lv  xQ O{ IG >> , @ i+ Y S E S Y E Y E Rwkhu irupv duh srvvleoh1 98; ;143 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Xvlqj wkh vwhs0e|0vwhs phwkrg Äs Äc  Y G P OW P OW O W O 5 5 Äs G S Äc G Y G P W P W R W 5 Äs G Äc  W Y G Äs G Äc Y W R 5 3 R Ohqjwk lv uhpryhg lq wkh uvw vwhs zlwk G> pdvv lv uhpryhg lq wkh vhfrqg zlwk G dqg wlph lv uhpryhg lq wkh wklug zlwk Y @G= Ilqdoo| zh kdyh s GS @F c Y ru Y s @F S c G 98< ;144 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Xvlqj wkh vwhs0e|0vwhs phwkrg s G q T  P OW O R W O W P O 5 6 6 sG q T G GT W R W P 6 5 R Äs G W q T G P W W R W 5 6 R 5 R Äs qG T qG 5 6 3 3 Lq wkh uvw vwhs/ ohqjwk lv uhpryhg zlwk G= Lq wkh vhfrqg vwhs/ pdvv lv uhpryhg zlwk GT dqg wlph lv uhpryhg lq wkh wklug vwhs zlwk q= Wkh ixqfwlrqdo irup lv T s @ i+ , qGS qGT 993 ;145 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Xvlqj wkh vwhs0e|0vwhs phwkrg i s U  n R W P OW O P O 3 5 6 R R R i s  U W O W O i s U5 W n 3 n 3 5 5 W i t U s 5 3 5 n 3 Lq wkh uvw vwhs/ pdvv lv uhpryhg zlwk = Lq wkh vhfrqg vwhs/ ohqjwk lv uhpryhg zlwk U dqg/ qdoo|/ lq wklug vwhs wlph lv uhpryhg zlwk s@US = Wkh qdo ixqfwlrqdo irup lv u  @ i +n, iU s 994 ;146 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh qrqglphqvlrqdo irup ri htxdwlrq Vroxwlrq @ d e Gf fg @ Od +P@OT ,e Of +O@W ,g @ OdÃTeFfFg P e W Ãg I P O@W S Htxdwlqj srzhuv ri P> O dqg W / zh kdyh W =g P =e O=4 4 d.f d @ @ @ @ @ @ 5 4 d d 5 5 6.f.f 6.f.5 f Wkhuhiruh/ I @ CSÃfD Gf fS I@+fS S , @ i +G@, Dqrwkhu ydolg dqvzhu zrxog eh I@+fS GS , @ i +G@, 995 ;147 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg dq h{suhvvlrq iru Y Vroxwlrq Xvlqj wkh vwhs0e|0vwhs phwkrg Y c   O W O P O P W 6 Y c   Y c O W O W O R W Yc @ 45   @ @ 45 45 3 T S c W 5 6 5 Lq wkh uvw vwhs/ pdvv lv uhpryhg zlwk  1 Lq wkh vhfrqg vwhs/ ohqjwk lv uhpryhg zlwk c dqg lq wkh wklug vwhs/ wlph lv uhpryhg zlwk cT @= Wkh ixqfwlrqdo irup lv u Y c @F  ru u Y @F 996  c ;148 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Xvlqj wkh vwhs0e|0vwhs phwkrg W  $ V G PO W P OW R W O O 5 5 W G G $ V G 5 P W P W R W 3 W G 5 6 $ V G R W R W 3 W G $ 3 V G 3 6 Lq wkh uvw vwhs/ ohqjwk lv uhpryhg zlwk G1 Lq wkh vhfrqg vwhs/ pdvv lv uhpryhg zlwk G dqg lq wkh odvw vwhs/ wlph lv uhpryhg zlwk $= Wkh qdo ixqfwlrqdo irup lv V W @ i+ , GT $ G 997 ;149 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Xvlqj wkh vwhs0e|0vwhs phwkrg k w     g O W P W5 P O6 P O5 W 5 P OW O k g w  gT gS g 3 W P W5 P P W5 P W k g w  g W  g  g W R W 6 5 k g 3 R w g 3 R w g w g 3 3 3 W 5 5 5 6 5 5 Lq wkh uvw vwhs/ ohqjwk lv uhpryhg zlwk g= Lq wkh vhfrqg vwhs/ pdvv lv uhpryhg zlwk gT dqg lq wkh qdo vwhs/ wlph lv uhpryhg zlwk w= Wkh qdo ixqfwlrqdo irup lv wS wS w k @ i+ T > > , g g g gS 998 ;14: Iluvw/ hvwdeolvk wkh glphqvlrqv ri wkh yduldeohv= ^ s` ^` ^Y ` ^` ^ ` @ @ @ @ @ I@OS @ +P O@W S ,@OS @ P@+W S O, P@OT > ^` @ I W @OS @ +PO@W S ,+W ,@OS @ P@OW O@W > ^HY ` @ I@OS +P O@W S ,@OS @ P@OW S I@O @ +P O@W S ,@O @ P@W S I@OT @ +P O@W S ,@OT @ P@OS W S Dvvxph h i +> O> Y> > Hy > >  , @ d Oe Y f g HY i  j , ^ s` @ P@W S O @ +P@OT ,d Oe +O@W ,f +P@OW ,g +P@OW S ,h +P@W S ,i +P@OS W S ,j Htxdwlqj srzhuv ri P> O dqg W / zh kdyh O = 6d . e . f g h 5j @ 4 P = d.g.h.i .j @4 W= f g 5h 5i 5j @ 5 6d . e . f @ g . h . 5j 4 d@4ghi j f @ 5 g 5h 5i 5j Vroylqj wkh deryh wkuhh htxdwlrqv iru e |lhogv e@ ru g i .j h s @ RÃgÃhÃi Ãj OÃgÃi Fj Y SÃgÃShÃSi ÃSj g HY i  j s @ Y S +@OY ,g +Hy @Y S ,h +@OY S ,i + O@Y S ,j s@Y S @ +@OY ,g +Hy @Y S ,h +@OY S ,i + O@Y S ,j Duelwudu| ydoxhv fdq eh jlyhq wr g> h> i dqg j dqg wkh frpelqdwlrqv ri yduldeohv zloo vwloo eh glphqvlrqohvv1 Wkhuhiruh/ ohw g @ 4> h @ 4@5> i @ 4 dqg j @ 4 wr |lhog s s@Y S @ i +OY @,> Y@ Hy @> OY S @> Y S @  , 999 ;14; Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg glphqvlrqdo dqdo|vlv iru hurvlrq udwh/ h Vroxwlrq Xvh h{srqhqw phwkrg b h @ i +Eu> > H> Y> g> Ps > G, zkhuh ^h` @ P@+OS W , > ^Eu` @ glphqvlrqohvv ^H ` @ P@+OW S , > ^ ` @ P@+OW S , b ^Y ` @ O@W > ^g` @ O > ^Ps ` @ P@W > ^G` @ O   Y  g P % G ` b , ^h` @ ^H s SW , @ +P@+OW S,,+P@+OW S,, +O@W , O +P@W ,%O P +O P = 4@..% O = 5@.    W = 4 @ 5 . 5 .  . % Xvh >  dqg % dv xqnqrzqv .3.% @ 4  .3 @ 5 5 .  . % @ 4  .. 5 +4, =  . % @ 4  +5, . +6, = 6 . % @ 6 6 .  .  +5, . +6, +4, = 5 @ 5 5 .  .  @ %@ @ @ h@ 4  . + . ,@5  . 4  @ 4 .  ++ . ,@5, . 4  @ + . ,@5  5.   4  . ++ . ,@5, 5 .  + . , @ 4 ++ . ,@5, bà i +H CRà FCCFD@SD  Y ÃRÃCCFD@S g Ps CCFD@SD G > Eu 99: +8, +9, +:, ru b b hY @H @ i +@H> HgS @+Y Ps ,> HGS @+Ps Y ,> Eu, Dowhuqdwh irup= b hY@H @ i +@H> HgS @Y Ps > g@G> Eu, 99; ;14< Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg glphqvlrqohvv uhodwlrqvkls iru suhvvxuh fkdqjh Vroxwlrq Xvlqj wkh vwhs0e|0vwhs phwkrg s T   G g P OW O W P O P OW O O 5 6 6 sg T g6 gT g G g P W R W P P W 3 5 Äs g T g W R W  g G g W 3 5 6 5 R Äsg R g T G g 5 T 7 5 3 3 3 Ohqjwk lv uhpryhg zlwk g lq wkh uvw vwhs/ pdvv zlwk gT lq wkh vhfrqg vwhs dqg wlph zlwk T@gT lq wkh wklug vwhs1 Wkh qdo irup lv g G sgU >, @ i+ TS T g 99< ;153 Wkh ylvfrxv irufhv duh uhodwlyho| vpdoo1 9:3 ;154 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Xvlqj wkh h{srqhqw phwkrg Y d @ e f g Gh j i is Zulwlqj rxw wkh glphqvlrqv  d  e  f   i g P O P P hO @ +O, W OT OT OW WS Vhwwlqj xs wkh htxdwlrqv iru glphqvlrqdo krprjhqhlw| O = d @ 6e 6f g . h . i P = 3@e.f.g W = d @ g . 5i Vxevwlwxwlqj wkh htxdwlrq iru W lqwr wkh rqh iru O jlyhv 3@ 6e 6f 5g . h 3@e.f.g i Vroylqj iru h iurp wkh uvw htxdwlrq dqg f iurp wkh vhfrqg htxdwlrq h @ 6e . 6f . 5g . i f@ g e dqg wkh htxdwlrq iru h ehfrphv h@ g.i Vxevwlwxwlqj lqwr wkh ruljlqdo htxdwlrq Y gFSi @ e ÃgÃe g GÃgFi j i is 9:4 Froohfwlqj whupv   i  e  Y s G g i Gj @ S  Y s Wkh ixqfwlrqdo htxdwlrq fdq eh zulwwhq dv Y s G i Y s @ i+ >,  s jG 9:5 ;155 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Wkh ixqfwlrqdo uhodwlrqvkls lv Y @ i +o > o > G> > j, Xvlqj wkh vwhs0e|0vwhs phwkrg Y o o G  j O W P O P OW O P W O W 6 5 5 R Y G o GT o G W P P W o o G  j G P W R W  o G j G 5 5 R Y G W 5 6 R W R W R W 5 sY jG o G o @ 65 j 3  o G j 5 @ 45 3 3 5 Lq wkh uvw vwhs/ G zdv xvhg wr uhpryh wkh ohqjwk glphqvlrq1 Lq wkh vhfrqg vwhs/ o GT zdv xvhg wr uhpryh s wkh pdvv glphqvlrq dqg qdoo|/ lq wkh wklug vwhs/ j@G zdv xvhg wr uhpryh wkh wlph glphqvlrq1 Wkh qdo ixqfwlrqdo irup fdq eh h{suhvvhg dv  S Y s , @ i+ S oT > o G j o GS j jG 9:6 ;156 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxs Vroxwlrq Wkh ixqfwlrqdo uhodwlrqvkls lv p @ i +G> > s> , b Xvlqj wkh h{srqhqw phwkrg/ zh kdyh pd @ Ge f sg h b Zulwlqj rxw wkh glphqvlrqdo htxdwlrq  P W d @ Oe  P OW f  P OW S g  P OT h dqg wkh htxdwlrqv iru wkh glphqvlrqv duh O = 3 @ e f g 6h P = d@f.g.h W = d @ f . 5g Vxevwlwxwlqj wkh htxdwlrq iru wlph lqwr wkh htxdwlrq iru pdvv |lhogv wzr htxdwlrqv 3@e f 3 @ g.h g 6h ru g @ h dqg wkh uvw htxdwlrq ehfrphv 3@e f 7g ru e @ f . 7g Vxevwlwxwlqj edfn lqwr wkh ruljlqdo htxdwlrq pfFSg @ GfFUg f sg g b 9:7 Froohfwlqj olnh srzhuv jlyhv  pS b U s G g  @ G p b f D ixqfwlrqdo uhodwlrqvkls lv s G p b , @ i+ p b  sGS Wkh ixqfwlrqv fdq eh frpelqhg wr irup  p b s S @ i + s sG ,  sG 9:8 ;157 Wkh vljqlfdqw glphqvlrqohvv qxpehuv zrxog eh Uh|qrogv qxpehu dqg Iurxgh qxpehu ehfdxvh ri wkh ylvfrxv gudj irufhv rq wkh erg| dqg wkh zdyhv wkdw zrxog eh surgxfhg e| wkh sur{lplw| ri wkh erg| wr wkh zdwhu vxuidfh 9:9 ;158 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Wkh ixqfwlrqdo uhodwlrqvkls lv s @ i +G> O> > > , Xvh wkh vwhs0e|0phwkrg s G v    P OW O O 3 P OW P O 5 6 sv G Äv P W 3 Äs Äv G Äv  v  vT 3 P W P   Äv5 5 5 R ÄsÄv 3  W 3 5 R  G Äv 5 5 3 3 3 W Lq wkh uvw vwhs/ wkh ohqjwk zdv uhpryhg zlwk G= Lq wkh vhfrqg vwhs/ wkh pdvv zdv uhpryhg zlwk  vS = Lq wkh wklug vwhs/ wlph zdv uhpryhg zlwk @ vS = Ilqdoo| wkh ixqfwlrq irup lv s G sv @ i+ > ,  v 9:: ;159 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh  jurxsv Vroxwlrq Wkh ixqfwlrqdo uhodwlrqvkls lv IG @ i +> Y> V> $ , Zulwlqj rxw wkh glphqvlrqdo sdudphwhuv xvlqj wkh h{srqhqw phwkrg d IG @ e Y f V g $h Lqfoxglqj wkh glphqvlrqv  PO WS d  @ P OT e  f  h O Sg 4 O W W Zulwlqj wkh htxdwlrqv iru glphqvlrqdo krprjhqhlw|/ P = d@e O = d @ 6e . f . 5g W = 5d @ f . h Vroylqj iru d> e dqg f lq whupv ri g> dqg h jlyhv d @ g h@5 e @ g h@5 f @ 5g 5h Vxevwlwxwlqj lqwr wkh ruljlqdo htxdwlrq gÃh@S @ gÃh@S Y SgÃSh V g $ h IG  IG Y S V g # @ 9:; 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4714 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkuxvw rewdlqhg iurp surshoohu Vroxwlrq FW @ IW @GU qS @ 3=37; Wkxv/ IW @ 3=37;GU qS @ 3=37; 4=38 44<< 6U +4> 733@93,S @ 5> 556 Q 4715 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkuxvw rewdlqhg iurp surshoohu Vroxwlrq YQ @qG @ +;3> 333@6> 933,@++4> 733@93, 6, @ 3=64: Iurp Ilj1 4706/ FW @ 3=3531 Wkhq IW @ 3=353 GU qS @ 3=353 4=38 6U +4> 733@93,S @ <59 Q Dovr iurp Ilj1 4706/ Fs @ 3=344 dqg S @ 3=344GV qT @ 3=344 4=38 4533 6V +4> 733@93,T @ 68=: nZ 4716 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkuxvw surgxfhg e| surshoohu Vroxwlrq q @ 4> 333@93 @ 49=9: uhy2vhf YQ @ 63 psk @ 77 isv YQ @qG @ 77@+49=9: ;, @ 3=66 Iurp Ilj1 4706 FW @ 3=34;5> Fs @ 3=344 vr IW S @ 3=34;5 @ 3=344 3=3357 3=3357 ;U 49=9:S @ 7<=: oei ;V 49=9:T @ 7> 338 iw0oe2vhf @ :=6 ks Li YQ @ 3> FW @ 3=37:8 vr IW @ 3=37:8 3=3357 ;U 4534 49=9:S @ 463 oei 4717 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg dqjxodu vshhg ri surshoohu Vroxwlrq YQ @ 63 psk @ 77 isv Iurp Ilj1 4706/ dw pd{lpxp hflhqf|/ YQ @+qG, @ 3=5;8 q @ 77@+3=5;8 ;, @ 4<=63 usv Q @ 4> 48; usp 4535 4718 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkuxvw dqg srzhu rxwsxw Vroxwlrq Dw pd{lpxp hflhqf| FW @ 3=356 dqg Fs @ 3=345 IW S @ 3=356 3=3357 @ 3=345 3=3357 9U 9V 58=:6S @ 7:=7 oei 58=:6T @ 6> ;48 iw0oei2v @ 9=<7 ks 4536 4719 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg gldphwhu ri surshoohu dqg vshhg ri dlufudiw Vroxwlrq S  W W 6<5 G @ @ @ @ @ @ O O@+FO V , YQS YQ @ @ @ @ Fs qT GV s@UW @ 93 43T @++5;:,+5:6,, @ 3=:99 nj2pT FW qS GU Gudj @ Oliw@63 @ +4> 533,+<=;4,@+63, @ 6<5 Q +3=358,+3=:99,+6> 333@93,S GU 4=9< p Z @ FO +4@5,YQS V +YQS @5, @ +4> 533,+<=;4,@++3=73,+43,, @ 5> <75 +5> <75,+5,@+3=:99, @ :> 9;4 ;:=9 p2v 4537 471: Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg pd{lpxp doorzdeoh dqjxodu vshhg Vroxwlrq Ywls Ywls q Q @ @ @ @ 3=<f @ 3=< 668 @ 634=8 p2v $u @ q+5 ,u 634=8@+5u, @ 634=8@+G, uhy2v 93 q usp G +p, Q +usp, 5 5> ;:< 6 4> <4< 7 4> 773 4538 471; Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg dqjxodu vshhg ri surshoohu Dw pd{lpxp hflhqf| YQ @+qG, @ 3=5;8= Wkhq q @ YQ @+3=5;8G, @ +73> 333@6> 933,@+3=5;8 Q @ 4<=8 93 @ 4> 4:3 usp 4539 5, @ 4<=8 uhy2v 471< Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkuxvw dqg srzhu lqsxw Vroxwlrq Dw pd{lpxp hflhqf| FW @ 3=356 dqg Fs @ 3=345 IW S @ 3=356 4=4 @ 3=345 4=4 5U 5V 453: +4<=8,S @ 487 Q +4<=8,T @ 6=46 nZ 47143 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg lqlwldo dffhohudwlrq Vroxwlrq FW @ 3=37; Wkxv IW @ 3=37;GU qS @ 3=37; 4=4 5U d @ I@p @ 568@633 @ 3=:;5 p2vS 453; +4> 333@93,S @ 568 Q 47144 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh glvfkdujh Vroxwlrq q @ 4> 333@93 @ 49=9: uhy2v kj@GS qS @ 6 <=;4@++3=7,S FK @ +49=9:,S , @ 3=995 Iurp Ilj1 4709/ FT @ T@+qGT , @ 3=958= Wkhq T @ 3=958 49=9: +3=7,T @ 3=99: pT 2v 453< 47145 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg glvfkdujh dqg srzhu ghpdqg Vroxwlrq q FK iurp Ilj1 4709/ FT T FT qGT Srzhu @ 9<3@93 @ 44=8 uhy2v @ kj@+qS GS , @ 43 <=;4@++3=:45,S +44=8,S , @ 4=9 @ 3=73 dqg Fs @ 3=:9 @ 3=73 44=8 3=:45T @ 4=99 pT 2v @ Fs GV qT @ 3=:9 4> 333 3=:45V 44=8T @ 544 nZ Iurp Ilj1 4709/ FT @ 3=73 dqg Fs @ 3=:9= Wkhq T @ FT qGT @ 3=73 S @ Fs GV qT @ 3=:9 44=8 3=:45T @ 4=99 pT 2v 4> 333 4543 3=:45V 44=8T @ 544 nZ 47146 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg glvfkdujh dqg srzhu uhtxluhg Vroxwlrq G @ 68=9 fp q @ 44=8 uhy2v Hqhuj| htxdwlrq iurp wkh uhvhuyrlu vxuidfh wr wkh fhqwhu ri wkh slsh dw wkh rxwohw/ [ kO sR @ . YRS @+5j , . }R . ks @ sS @ . YSS @+5j, . }S . 53 . ^TS @+DS 5j ,`+4 . iO@G . nh . ne , O @ 97 p ks @ 54=8 Dvvxph i @ 3=347> ue @G @ 4= Iurp Wdeoh 4306/ ne @ 3=68> nh @ 3=4 ks FT ks @ 4=8 . ^TS ++3=347+97,@3=689, . 3=68 . 3=4 . 4,`@^5+<=;4,+@7,S +3=689,U ` @ 4=8 . 53=75 @ T@+qGT , @ T@^+44=8,+3=689,T ` @ 4=<6T @ FK qS GS @j @ FK +44=8,S +3=689,S @<=;4 @ 4=:4FK T+pT @v, 3143 3148 3153 3158 3163 3168 FT 314<6 315;< 316;8 317;5 318:; 319:8 ksR +p, 41:3 41<9 5165 51:; 6167 7133 FK 5138 41:3 4188 4158 31<8 3188 ksS +p, 6183 51<4 5198 5146 4195 31<7 Wkhq sorwwlqj wkh v|vwhp fxuyh dqg wkh sxps fxuyh/ zh rewdlq wkh rshudwlqj frqglwlrq= T @ 3=55 pT 2v Iurp Ilj1 471: S @ 9=8 nZ 5 hp(m) 4 pump curve 3 2 1 system curve operating point 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Q (m 3 /s) 4544 47147 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg glvfkdujh dqg srzhu uhtxluhg Vroxwlrq Wkh v|vwhp fxuyh zloo eh wkh vdph dv lq Sure1 47146 FT ks @ T@^qGT ` @ T@^48+3=689,T ` @ 4=7;T @ FK qS GS @j @ FK +48,S +3=689,S @<=;4 @ 5=<4FK T 3153 3158 3163 3168 FT 315<9 316:3 31777 3184; FK 4198 4188 4168 4148 ks 71:< 7184 61<5 6167 Sorwwlqj wkh sxps fxuyh zlwk wkh v|vwhp fxuyh jlyhv wkh rshudwlqj frqglwlrq> T @ 3=65 pT 2v FT @ 4=7;+3=65, @ 3=7:7 Wkhq iurp Ilj1 4709/ Fs @ 3=:3 Fs qT GT  @ 3=:3+48,T +3=689,V 4> 333 @ 46=8 nZ 5 Pump curve hp(m) 4 3 System curve 2 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 2 Q (m /s) 4545 47148 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg srzhu uhtxluhg Vroxwlrq Dw pd{lpxp hflhqf|/ iurp Ilj1 4709/ FT @ 3=97> Fs @ 3=93> dqg FK @ 3=:8 G @ 4=9: iw q @ 4> 433@93 @ 4;=66 uhy2v T @ FT qGT @ 3=97 4;=66 4=9:T @ 87=9 fiv k @ FK qS GS @j @ 3=:8 4;=66S 4=9:S @65=5 @ 54=; iw S @ Fs GV qT @ 3=93 4=<7 4=9:V 4546 4;=66T @ <6> 449 iw0oei2vhf @ 49<=6 ks 47149 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg srzhu uhtxluhg Vroxwlrq Dw pd{lpxp hflhqf|/ iurp Ilj1 4709/ FT @ 3=97> Fs @ 3=93> FK @ 3=:8 T @ FT qGT @ 3=97 78 3=8T @ 6193 pT 2v k @ FK qS GS @j @ 3=:8 78S 3=8S @<=;4 @ 6;1: p S @ Fs GV qT @ 3=93 4> 333 3=8V 78T @ 4> :3< PZ 4547 4714: G @ 47@45 @ 4=49: iw q @ 4> 333@93 @ 49=: uhy2v SGS@j @ FK +49=:,S +4=49:,S @65=5 @ 44=;FK iw k @ FK q T @ FT qGT @ FT 49=:+4=49:,T @ 59=8FT fiv FT 313 314 315 316 317 318 319 FK 51< 5188 513 41: 418 415 31;8 T+fiv, 3 5198 816 :1<8 4319 4615 481< k+iw, 6715 6314 5619 5314 4:1: 4715 4313 40 35 Head, ft 30 25 20 15 10 5 0 2 4 6 8 10 12 D ischarge, cfs 4548 14 16 18 4714; G @ 93 fp @ 3=93 p Q @ 9<3 usp @ 44=8 uhy2v Wkhq k @ FK GS qS @j @ 7=;86FK T @ FT qGT @ 5=7;7FT FT 313 314 315 316 317 318 319 FK 51<3 5188 5133 41:3 4183 4153 31;8 T+pT 2v, 313 3157; 317<: 31:78 31<<7 41575 417<3 k+p, 4714 4517 <1: ;16 :16 81; 715 16 14 Head, m 12 10 8 6 4 2 0 1 3 D ischarge, m /s 4549 2 4714< Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg khdg dqg glvfkdujh dw pd{lpxp hflhqf| Vroxwlrq G @ 3=6:4 5 @ 3=:75 p q @ 5> 466=8@+5 93, @ 4:=:: usv Iurp Ilj1 47043/ dw shdn hflhqf| FT @ 3=454> FK @ 8=48= Wkhq k @ FK qS GS @j @ 8=48+4:=::,S +3=:75,S @<=;4 @ <4=6 p T @ FT qGT @ 3=454+4:=::,+3=:75,T @ 3=;:; pT 2v 454: 47153 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg srzhu qhhghg wr rshudwh idq Vroxwlrq T @ Y D @ +93,+@7,+4=5,S @ 9:=; pT 2v FT @ T@+qGT , @ +9:=;,@++4> ;33@93,+5,T @ 3=5;5 Iurp Ilj1 47049 Fs @ 5=9= Wkhq S @ Fs GV qT @ +5=9,+4=5,+5,V +63,T , @ 5=:3 PZ 454; 47154 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg glvfkdujh wkurxjk slsh Vroxwlrq } @ 783 Dvvxph 699 @ ;7 p k @ <3 p^A } `> wkhq iurp Ilj1 470</ T @ 3=57 pT 2v Y @ t@D @ 3=57@^+@7,+3=69,S ` @ 5=69 p2v> nv @G @ 3=33345 Dvvxplqj W @ 53 F> Uh @ Y G@ @ 5=69+3=69,@43ÃW @ ;=8 43V iurp Ilj1 430;/ i @ 3=347 ki @ +3=347+943,@3=69,++5=69,S @+5 <=;4,, @ 9=:6 p k r ;7 . 9=: @ <3=: p iurp Ilj1 470< T @ 3=56 pT 2v> Y @ 3=56@++@7,+3=69,S , @ 5=59 p2v ki @ ^3=347+943,@3=69`+5=59,S @+5 <=;4, @ 9=4; p vr k @ ;7 . 9=5 @ <3=5 p Y @ 3=56@++@7,+3=69,S , @ 5=59 p2v dqg iurp Ilj1 470< T @ 3=558 pT 2v 454< 47155 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh glvfkdujh Vroxwlrq G @ 3=6:4 p @ 4=54: iw q @ 4> 933@93 @ 59=: usv k @ FK qS GS @j vr FK @ 483+65=5,@^+59=:,S +4=54:,S ` @ 7=8: iurp Ilj1 47043 FT @ 3=478 wkhq T @ FT qGT @ 3=478+59=:,+4=54:,T @ 9=<; fiv 4553 47156 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg pd{lpxp srvvleoh khdg ghyhorshg Vroxwlrq FK @ K j@GS qS Vlqfh FK zloo eh wkh vdph iru wkh pd{lpxp khdg frqglwlrq/ wkhq  qS K ru KR>VQQ @ KR>QQQ KR>VQQ @ 435 +4> 833@4> 333,S 5=58 @ 55<=8 iw 4554 47157 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg vkxwr khdg Vroxwlrq K  qS vr KTQ @KTV=W @ +63@68=9,S ru KTQ @ 437 +63@68=9,S @ :6=; p 4555 47158 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg glvfkdujh zkhq khdg lv 83 p Vroxwlrq FK @ kj@+qS GS , @ 83+<=;4,@^+58,S +3=73,S ` @ 7=<4 iurp Ilj1 47043 FT @ 3=469 wkhq T @ FT qGT @ 3=469+58+3=73,T @ 3=54; pT @v 4556 47159 Lqirupdwlrq dqg Dvvxpswlrqv iurp wdeoh D17  @ ;47 nj2pT surylghg lq sureohp vwdwhphqw Ilqg rz udwh/ suhvvxuh ulvh dfurvv sxps dqg srzhu uhtxluhg Vroxwlrq Q @ 8> 333 usp @ ;6=66 usv iurp Ilj1 47043 dw pd{lpxp hflhqf| FT @ 3=458> FK @ 8=48> Fs @ 3=9< T @ FT qGT @ +3=458,+;6=66,+3=53,T @ 3=3;66 pT @v k @ FK GS qS @j @ +8=48,+3=53,S +;6=66,S @<=;4 @ 478=; p S @ Fs GV qT @ +3=9<,+;47,+3=53,V +;6=66,T @ 437=3 nZ 4557 4715: Wkh glvfkdujh frhflhqw lv ghqhg dv FT @ T@qGT Wkh urwdwlrqdo vshhg lv 4:83293@5<15 usv1 Wkh gldphwhu iru hdfk lpshoohu lv 31749: iw/ 3178; iw/ 318 iw/ 31875 iw dqg 318;6 iw1 Rqh jdoorq shu plqxwh lv 3133555; iwT @v1 Vr iru hdfk lpshoohu/ wkh frqyhuvlrq idfwru wr jhw wkh glvfkdujh frhflhqw lv 8% 8=8% 9% 9=8% :% jsp jsp jsp jsp jsp 3=33438 3=333:<7 3=333943 3=3337:< 3=3336;8 Wkh khdg frhflhqw lv FK @ Kj qS G S Wkh frqyhuvlrq idfwruv wr jhw wkh khdg frhflhqw duh 8% 8=8% 9% 9=8% :% iw iw iw iw iw 3=54:8 3=4;33 3=4843 3=45;8 3=4444 Wkh shuirupdqfh lq whupv ri wkh qrqglphqvlrqdo sdudphwhuv lv vkrzq rq wkh judsk1 6 Head coefficient 5 4 5 inch 5.5 inch 6 inch 6.5 inch 7 inch 3 2 1 0.00 0.02 0.04 0.06 0.08 0.10 D ischa rg e coefficien t 4558 0.12 0.14 4715; Wkh urwdwlrqdo vshhg lq usv lv q @ 833@93 @ ;=66 usv Wkh glvfkdujh lv jlyhq e| T @ FT qGT @ FT +;=66,+4=85T , @ 5<=5:FT +pT @v, Wkh khdg lv jlyhq e| k @ FK qS GS @j @ FK +;=66S ,+4=85S ,@<=;4 @ 49=69FK +p, FT 3 3=37 3=3; 3=43 3=45 3=47 3=49 T FK 3 8=; 4=4: 8=; 5=67 8=:8 5=<6 8=9 6=84 8=58 7=43 7=; 7=9; 7=3 k <7=< <7=< <7=4 <4=9 ;8=< :;=8 98=7 100 95 Head, m 90 85 80 75 70 65 60 0 1 2 3 3 D ischarg e, m /s 4559 4 5 4715< Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg vxfwlrq vshflf vshhg Vroxwlrq Qvv Q Q S VK T Qvv @ @ r @ @ Q TR@S +Q S VK ,T@U 9<3 usp 47=: svl 5=64 iw2svl kydsIsuhvvI r 66 iw 3=55 pT @v 597=5 jdoorqv2v 93 v2plq @ 6> 7;: jsp 9<3 +6> 7;:,R@S @+66,T@U @ 5> <93 Qvv lv pxfk ehorz ;/833> wkhuhiruh/ lw lv lq d vdih rshudwlqj udqjh1 455: 47163 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg w|sh ri zdwhu sxps Vroxwlrq Q qv @ 4> 833 usp vr q @ 58 usv> T @ 43 fiv> k @ 63 iw s @ q T@^j T@U kT@U ` @ +58,+43,R@S @^+65=5,T@U +63,T@U ` @ 3=79 Wkhq iurp Ilj1 47047/ qv A 3=93> vr xvh d pl{hg rz sxps1 455; 47164 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg w|sh ri sxps Vroxwlrq q T k qv @ @ @ @ 58 usv 3=63 pT 2vhf ; phwhuv s q T@^j T@U kT@U ` @ +58+3=6,R@S @^+<=;4,T@U +;,T@U ` @ 3=85 Wkhq iurp Ilj1 47047/ qv ? 3=93 vr xvh d pl{hg rz sxps1 455< 47165 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg w|sh ri sxps Vroxwlrq Q T k qv @ @ @ @ @ 4> 433 usp @ 4;=66 usv 3=7 pT 2vhf :3 phwhuv s q T@^jT@U kT@U ` @ +4;=66,+3=7,R@S @^+<=;4,T@U +:3,T@U ` +4;=66,+3=96,@^+8=87,+57=5,` @ 3=3;9 Wkhq iurp Ilj1 47047/ qv ? 3=56 vr xvh d udgldo rz sxps1 4563 47166 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg pd{lpxp vshhg Vroxwlrq Wkh vdih rshudwlqj Qvv lv ;/833/ ru ;> 833 @ QTR@S @+Q S VK ,T@U Wkh vxfwlrq khdg lv jlyhq dv 8 iw1 Wkhq dvvxplqj wkdw wkh dwprvskhulf suhvvxuh lv 471: svld/ dqg wkh ydsru suhvvxuh lv 31589 svl/ zh frppxwh Q S VK dv Q S VK @ 47=: svl 5=64 iw2svl .8 iw kydsIsuhvvI @ 6;=7 iw T @ 3=73 pT @v 597=5 jdov2pT @ 9> 333 jsp 93 v2plq Wkhq Q Q @ ;> 833 +Q S VK ,T@U @TR@S @ ;> 833 48=7@::=8 @ 4> 9;< usp 4564 47167 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg w|sh ri sxps Vroxwlrq qv q T k Y @ @ @ @ @ s q T@+jT@U kT@U , 43 usv 4=3 pT @v 6 . +4=8 . i O@G,Y S @+5j,> 4=5: p2v Dvvxph i @ 3=34/ vr k @ 6 . +4=8 . 3=34 53@4,+4=5:,S @+5 <=;4, @ 6=47 p Wkhq qv @ 43 s 4@+<=;4 Iurp Ilj1 47047/ xvh d{ldo rz sxps1 4565 6=47,T@U @ 3=:9 47168 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg gldphwhu dqg srzhu uhtxluhphqwv iru wzr eorzhuv Vroxwlrq Pd{1 dlu vshhg @ 73 p2v> Duhd @ 3=69 pS > q @ 5> 333@93 @ 66=6 usv> T @ 73=3 3=69 @ 47=7 pT 2v  @ 4=5 nj2pT dw 53 F Iurp Ilj1 4709/ dw pd{lpxp hflhqf|/ FT @ 3=96 dqg Fs @ 3=93 FT @ T@qGT> vr GT @ T@+qFT , @ 47=7@+66=6 G @ 3=;;5 p Fs 3=96, @ 3=9;9 pT @ S@+qT GV , vr S @ Fs qT GV @ 3=9+4=5,+66=6,T +3=;;5,V @ 47=5 nZ 4566 47169 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg gldphwhu dqg srzhu uhtxluhphqwv Vroxwlrq Yroxph @ 43V pT > wlph iru glvfkdujh @ 48 plq @ <33 vhf Q @ 933 usp @ 43 usv  @ 4=55 nj2pT dw 93 I T @ +43V pT ,@+<33 vhf, @ 44414 pT @vhf Iurp Ilj1 4709/ dw pd{lpxp hflhqf|/ FT @ 3=96> Fs @ 3=93 Iru wzr eorzhuv rshudwlqj lq sdudooho/ wkh glvfkdujh shu eorzhu zloo eh rqh kdoi vr T @ 88=88 pT @vhf wkhq GT G S @ T@qFT @ +88=88,@^43 @ 5=399 p 3=96` @ ;=;48 @ Fs GV qT @ +3=9,+4=55,+5=399,V +43,T @ 5:=9 nZ 4567 4716: Lqirupdwlrq dqg Dvvxpswlrqv iurp Wdeoh D08 iru phwkdqh U @ 84; M2nj2N dqg n @ 4=59 surylghg lq sureohp vwdwhphqw Ilqg vkdiw srzhu wr uxq frpsuhvvru Vroxwlrq Swk Suhi @ +n@+n 4,,TsR ^+sS @sR ,CnÃRD@n 4` @ +np@+n 4,,UWR ^+sS @sR ,CnÃRD@n b @ +4=59@3=59,+4,84;+633,^+4=8,Q=SW@R=SW 4` @ 98=9 nZ @ Swk @h @ 98=9@3=98 @ 434 nZ 4568 4` 4716; Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg yroxph rz udwh lqwr wkh frpsuhvvru Vroxwlrq @ @ @ T@ Swk Swk 45 nZ 3=9 @ :=5 nZ +n@+n 4,,TsR ^+sS @sR ,CnÃRD@n 7=4; 43U T :=5@74=; @ 3=4:5 pT 2v 4` @ +4=6@3=6,T 4569 < 43U ^+473@<3,Q=T@R=T 4` 4716< Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg wkh vkdiw srzhu Vroxwlrq Swk Suhi @ sR TR cq+sS @sR , @ pUWR cq+sS @sR , @ 4 b @ 447=9@3=8 @ 55< nZ 456: 5;: 5;; cq7 @ 447=9 nZ 47173 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg gldphwhu ri wxuelqh zkhho Vroxwlrq Dvvxph W @ 43 F Zulwlqj wkh hqhuj| htxdwlrq iurp uhvhuyrlu wr wxuelqh mhw/ sR @ . YRS @5j . }R @ sS @ . YSS @5j . }S . 3 . 3 . 983 @ [ kO S S 3 . Ymhw @5j . 3 . +i O@G,+Yslsh @5j , exw iurp frqwlqxlw|/ Yslsh Dslsh Yslsh @ Ymhw Dmhw @ Ymhw +Dmhw @Dslsh , @ Ymhw +3=49, @ 3=359Ymhw vr S +Ymhw @5j ,+4 . +iO@G,3=359S , @ 983 Ymhw S Yexfnhw G @ @ @ @ ^+5 <=;4 983,@+4 . +3=349 43> 333,@4,3=359S ,`R@S @ 43:=6 p2v S TYmhw h @ 43:=6+@7,+3=49,S <> ;43+43:=6,S 3=;8@+5 <=;4, @ 43=88 PZ +4@5,Ymhw @ 86=: p2v @ +G@5,$ 86=: 5@+693 +@63,, @51;8 p 456; 47174 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg Uhihuhqflqj yhorflwlhv wr wkh exfnhw Vroxwlrq V-VB j V-VB j V-VB j [ Wkhq Iexfnhw rq mhw @ T^ +Ym [ YE , +Ym Irq exfnhw @ Ym Dm 5+Ym YE ,` YE , dvvxplqj wkh frpelqdwlrq ri exfnhwv wr eh lqwhufhswlqj rz dw wkh udwh ri Ym Dm = Wkhq S S @ I YE @ 5Dm ^YmS YE S Ym YE ` Iru pd{lpxp srzhu surgxfwlrq/ gS@gYE @ 3> vr 3 @ 5D+YmS Ym 5YE , 3 @ Ym 5YE ru YE @ 4@5Ym 456< 47175 Frqvlghu wkh srzhu ghyhorshg iurp wkh irufh rq d vlqjoh exfnhw1 Uhihuhqflqj yhorflwlhv wr wkh exfnhw jlyhv [ Irq exfnhw @ TuhoI wr exfnhw + +4@5,Ym +4@5,Ym , Wkhq Irq exfnhw @ +Ym YE ,Dm +Ym , exw Ym YE @ 4@5Ym vr Irq exfnhw @ 4@5DYmS Wkhq S @ I YE @ +4@5,TYmT @5 Wkh srzhu lv 425 wkdw jlyhq e| Ht1 +47053,1 Wkh h{wud srzhu frphv iurp wkh rshudwlrq ri pruh wkdq d vlqjoh exfnhw dw d wlph vr wkdw wkh zkhho dv d zkroh wxuqv wkh ixoo glvfkdujh> zkhuhdv/ d vlqjoh exfnhw lqwhufhswv rz dw d udwh ri 4@5 Ym Dm = 4573 47176 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg R iru qrqvhsdudwlqj rz frqglwlrqv dqg pd{lpxp dwwdlqdeoh srzhu Vroxwlrq YuR $ @ t@+5uR E , @ 459@+5 8v{4, @ 7=34 p2v @ 93 5@93 @ 5 udg2v 41 d, R @ duf frw ++uR $@YuR , . frw  R , @ duf frw ++8 5@7=34, . 3=8::, @ 9=:; S @ duf wdq +YuS @$uS , @ duf wdq ++7=34 8@6,@+6 5,, @ duf wdq 3=688 @ 48=8 e, YR @ YuR @ vlq R @ 7=34@3=44; @ 6<=<: p2v> YS @ YuS @ vlq S @ 53=3 p2v S @ T$ +uR YR frv R uS YS frv S ,  6 53=3 frv 48=8 , @ 444=4 PZ S @ <<; 459 5 +8 6<=<: frv 9=:; f, Lqfuhdvh  S 4574 47177 Lqirupdwlrq dqg Dvvxpswlrqv surylghg lq sureohp vwdwhphqw Ilqg R iru qrqvhsdudwlqj rz frqglwlrqv dqg uhvxowlqj srzhu dqg wrutxh Vroxwlrq YuR YuS $ R R YwdqR YwdqS W Srzhu @ @ @ @ @ @ @ @ 6+5 4=8 3=6, @ 4=394 p2v 6@+5 4=5 3=6, @ 4=659 p2v> +93@93,5 @ 5vÃR duf frw ++uR $@YuR , . frw  R , @ duf frw ++4=8+5 ,@4=748, . frw ;8 , duf frw +9=99 . 3=3;:8, @ ; 58Q uR $ . 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This note was uploaded on 06/09/2010 for the course CIVIL 3502 taught by Professor Kimberlyhill during the Fall '08 term at University of Minnesota Crookston.

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