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Solutions Manual To Accompany
Fundamentals of Momentum, Heat, and Mass Transfer, 4e
By:
James R. Welty Charles E. Wicks Robert E. Wilson Gregory Rorrer
C.l1APTf:.R ,
1.1 n = L/ >( 1020 /i,,;, V::''''; J< ~ R-r = /.32 x 10'" il1/5
A = r ( 163 ;.,,/-,:. NA = -'- n OA = 1.04 x IO lg/s
4
P ~p II + dP u 1.2 'V = ji X d-:.1 ( 1+ e.-"2.)'; 2.. L~
-to ~r\1 ( I - i:,2.);1 ~
::Tc, (O.CJ~:l~; + .1?23 ~)
1.>.1 /(Jt",1j) of 'PROB~~/Yf /-10,,"04 t!/l1c.ouS.
p~;;J O~ PEo13I..~/IIf 13 )lO/f!OtS UV't! ocrs/ 12 -:E. ::: [ l\of 'S a 'l
-=' v z f~ J
1.:5 IS
1.2 WILL IF
O~
IF ,)IE: Co/VVcte.S/ON rr'1C70R/
je I IS USe.D, ~ ~ [ csk35 j.
I.S ~LOl.J PROP~RT/g s: SrR~S5 / PRE.S5UR.~ ($~ADIE./v~ II~LocrrY.
Pi IJ ID 'PROP.Rrlz. s : /z./I1?:RA7U..e / D:E,A..)SIT~
PRS.5 U. REI =':>Pfl:.CII=IC J.I EA0 SPEED of Sou/1/D.
I. {, 9
'" ('" I A l" \ "-e. r:2 ~ r x ex + er ~ e. ~ = COS e~.x + sin g e.~
~ ;:( ~!,\" e); + I~ IS! e.~ = -sin e ex + GOes. e ~j
.. Q. E. D.
A
- ~e
d e.s e'" e'" __ = -cos
1.8 {L = ~ ~ + ~ ~ aX" ax or ax ae ~=~a+~.a.-oy ay dt' (39 de
r2= )(:1+Lj:1 ) 9= 1:a.n-1 - dr - )( II e e ax - (x 2 + yt)'2 = r' C:S = cos ~ = - y = - ("'51118:- sine a~ )(2 + y~ ("2 '('
dr=sinG .afr= case ~ I ay t'
-sineL --r- ae + case L
r" ;;8
1.9 'V = ~ a.x + L i. + a- Q.;: ;)x d~!i 02
= (cos8?r - Si~e~) ax +(5 ine.L of- CO~e ~ '\ e
;;, r- c}9) ~
+ ~ Qc ;)2 = (ex case +- ~~ Sirl e)~
+ y!:- (- e.x Si118 + cZ~ case) 2e + er..a.. ae
"". V= a.,.~ J .... .a... A ;; dr +- f c2.e (}E:1 + er a2 .
1.10 MASS 01=' SOLID = If Vs II /I FLUID == It If
x = p~ Vs =>- V f = /-x r?s f'sVs +,4 V; Vs -X P
2
= fJ/i. 1'>
1.13 1p:= A,. sinB (I - -Fi ) a) V'1p:: d 'I' ~ + ~ ~ ~9
()r r r ae
:. -vP = AOSil1e(1 -~) a r~ e
WHIC.H REQUIRES -n-tAT
*-1 V7p/ = ?e I vlf'/ = 0 ~lvVI = 0: - 511'12.e (I-t-~)
+C05~(I-~:) = 0 (I)
~ I \lIP! =0 : "!>lrleCOSe[C1 +~ f -(1- ~:YJ::o (2)
FRo;n (2) 5i"e case . L/a"4 = 0 r':4
FoR. a;l 0" r'" 0;' sin e cos e = 0 . e - 7r - 0" _
2
(3") IAlTO (I)
G)=O: 1- a~ :. 0 r2.
.. a=f"
IhI PasS/SL ~ ,
:. C/.)AJDIT /ot1JS .4Rs. . e =0; i=a...
(3)
IS
3
~ ~ = -jfJv;.'-[~~r] ~
ffE? A- _.J. LJ v: 2.. r2;(l[ J i de fL.2 - 2 r oC L - L 3 if
- -~. - ... -------------,-,15
lAKE R =- i AnA. Wrm s:'&, : 1.01 ?= '3001 (1.01):' 3CCD:; 217 AiM
L\(o
AT Co~~ntNTlEMP,) .?""' \/ fOe I O~ \~~$ l,v 9 / A lO'r\~EA$E IN PI~ ~o
t.l7 DEJJSrry:: \1 m W"'~ m
1'5 MOL&:::u...A12. Wi;(ah'T. ;.tr 2.~Q:;()'-, wt ~ W1~ \~ . ~ (lAwf>
n ,:: n ?2SO,,~1 2~,,~ CS).L
j>s .. l-
= 2.5'10" ~(e~ ln 3
~lc;H AL.TrT'UDc Utrt~ IVf\~ G Is CVr~~. n~ L~'(O~
CHAPTER 2
2./ V'P=PS j dP.... ,.. d~ e~ = -pc:, e.':j
~t... clP = -p
2.6 A
8
Hg
I q -= ~ -JOlt. ~ 10 I
Pc ~~+Yu20a~ 'PD '" \/" -Sllff ~ I
B -~ " J~ d (1)- 5511;{j-IOgOt" 0 Pp = 'iIM I JOII. :O,6JH2-D ~ qf({!5 ~~~/ ~-~ =: ~(I)- !':>'G24 =%.8fsP
2.9 Air
~., g t d, j 5'H ,.0'" Pa {d4 ~ +a~~9HJ ~ -~ :: (1-!i 1-~) ~ -(2 ):02 A
(2'Z \2-
~- Ps= Z44,7pgf :: 1.70 ps I 5
ST~JJG troM 16wr Aj
~ = 'B.. -dl ~ j'H2.0 5ru2rtA-q ~ ~ B./
~ '" 'Pe-~,+dlfd3~~.Jd2Jf~ fuwtr~,
~-?g "'d2iJf~-(d2"d3)Off1p
ExPOt;$S"!M4 d IN .r~
.~ -~ = SZ.7p'; = 0.227 ~J
2 JI F -= '? A - '?o..~A == ,oj 11 -(fR2 e.G.
PH:J O= 1000 I
2./2 ASSlJh1ING, ArlY1()SPHERIC AIR. TO 13HAV' I DEALL ~
&- -~Q --ll dg - r.) - "RT
LET T = a + b~ tJlTH ~/VGN IN FORMI1TION,
T.: S30 - 2l/ fJ 7;
dE. - -g dlj -p - ~5 ......:30~--2-J.!tl.ly/~
fOP .dE -
"I>y ANALYSIS SIMILAR TO 0..) ~
OBTAI N (3-,)) = 2.? FI. .e o F: Iq6 (:2.1) -250 == 293.6 /bf
2.JS AS THE TANK CONTINUS TO BE ?U5HED TO qREATR DE'P~ THE VOLUME OCcUPIED 'BY THE AIR IJEcRASS AND 77IE 750UYANT FORCE THUS DE-
CREASES.
'BOUYANT FORcE = 2Solbl DISPLAcED VOLI.JM O~ H2 0 = 250 = 1. 01 /f;3 ~o9
!iSSUMINej AIR To -SEHAtI IDEALL Sol AND ISOTHERMAL COM?RESS/oN/
~+m A (3 P. ) In -P(4.01 (j.. f) .::z 'C..-fnt f-(-J9 c)( 1. 0/)
Z:: '15: 'i'l' Ii-. TOP WILL. 8E ('IS": i?~ 4. 01 ff3)
7Tlg;:.)
eo T()P J S ~~ b fI. EELctU J6 ~
o
THE HEI6HT O~ nlE WitTER co-LUMN FROM rHE DIF;::: ElEMENT
IS h - Jll-g Q.) FOR A REcTAN6ULAR 6Ar&) d,4== ~'d~
clFw == [ ~ 9 (h-4ry) -rPetIfCJ dA d F". -= L 6P'S;9'/ilL! + ~ ] alA ~Mo =0
~
2.1'1
F':J = BOUYANT FORcE ON suB-MERGED LO
h fim" ~ ~ (h-~+~)sm~ Z7reSd C(
= - (~-~\t~ t 2 (\+aJDC) '2 ~
'rJH~ Fa =0 ~~ e~0(+2Q Q-fQJz1)
'3 SWI,"l.()(
~~~
'5(t"\ D( '" 'PIcL Q:)~ "/'- rJfct 2 n -: 11- rJid" /-+ (1- V%t.)3~ - + ----=--:.~ d 2 3DYct ~
E.>c.?At-JDHS>is'' sa.J.... '/ ::: t. ~to X/O /, ~ = /760 kPa...
'" ~ a.
2.25
"'- ' Fan ~ -,.----------..-----
0.30 m
/J.P" J"20 6l1.k = 999 k19.B\ Wl (?2')""-.
m~ $~ 10'1
::. 2.4G Pa..
~ 'D'J::~ Is~ ~TfUSRA;~~ AN)D lUe.t1R; \VA~ tbwN~.
Po lsT~~~. ~{sT~~~. )(ls~Dt~~
~E.~ 10 T~ ~~TU5 \VA:Jr;e ..
B,i?H~~::: 1?""~l"q~)f~ 'UT2. (1 IZ ~~
~-~ 2 Ylla3a - ~~6~ == 7O.fo7- 15.~ .. 55.1 pJ
10
T. AI THe. CENTER of THE EARTH} z:.:R
Pc :: r;.+n\ T ffjo ~ NEGLECTINq ~+,."
~ =- P90 R :: ~ 6 '1 /()10 ~ .9.107'" 2 ml ~
6330./0 3 ",
= 352 x/o9 k::, m5~
2.2'1 :-IT H,.,O 12' p= 2 ~k.!.'3~/1+.3
+ MUO 10 ' p= 4 '5(u9~ 1ft l t "'B
t=A -~~ =~'J 12 := 24 j 'Pa -Pa..,"", = 2l/'j -1-40, = 6L(j 'Po" -H .. tM. = ,ag ~ PAS -Po..~ = I.l ~ 12 +tE J (f-12)
H:z.O lit
F oRC.E / UN IT LEI{6rH r: f= f=:: f (p - Pa..n..)a'A ~= C~ 9 f df + f~j'2 t(! j(l-l2})dc
12-
F'=Pw~( Iq2 kl)-t-~ ~ (s-OJi.'1
F::.2~ ,/9; +l/'9'S'o = 18,790 Ibf
= fJw9 (~6 f20l{O) +-~ j (2.q~3 -20L/O)
= 2 'i~ '106 H-'/bf 2 == IS". 3S- ff:
2.30 FREE !BoDY ~~~
(FORM INCLUDED)
FORCE=f'~H Am e ~ 2
:3
~ESSURE FORCE =~
CHAPTER L/
1././ V = IO~ r7x l, AT (2,2)~ {j- = 10 ex rJil ~j A UNIT VEcTOR IN -304 DIREcnON
IS ~I e"'_Y3A I - 2' ex -2 ~~ COMPoNeNT IN e. DIRECTION = e v - (Y3" 1 A ) !J - 2" eN -2 lZ:s .~/ofZx +I'I~) = 5"13 - 7 = 1. 66 fps.
l/. 2 {} = 10 ~x + 2 x .~'J 0.) ~ = ~ = :Y..
>( \1")( /0
/0 d'J = 2>f 10 'J = }{2 .,. C j (2, I) .'. c = ~ >(2 _ /0'] + 6 = 0
Yl )~2J (1,0) )(
Q= IT ,.,,3 ::.
~ CONTROL r - - - I VOWME:
i~- --+ -~-r!' L.. ____ J
12
FOR C.v. SHOWN; f)c.s.f'(v.nJdA ... k fVdV =0
o
V= 'i (I-fi) fp:. ..
)5('.s/,Cv.MdA = )fA/,(v.r1)dA
+ ~~A1P(V-~)c:lA
= f' [112 o.ve.. A;l - )oRq(l- n) :2'rrrd~= 0 ~,.. q1r R:2. (f-.,"- -I) 1J2 o.lI'c.., = ~ .. = I 2 $ fps
11 (1.5"~
l/.'1
V;= .of,., [J'--'-____ ;::w He.s. f'(v.n)dA::" ))Ai f' (1Jn )dA + ),40 f' (V-. n)cl A
= - f)A' fJvdA + (( pvcos30acJA , JJA.
= -~/J"Ah +~1.T~s30"A)o = 0 ~. =,q, , it - ,/
lie - ..,
.~ V; = A: ,,;;. = 1/6. /9 fps Ao ("D~"30 .
Q = Avo = o. S'S-S' fI?,s
= Al 'LJ + 'TrD v-.!:.. 2. = V-(-n:~1 + 1J' 0 L) 2L 2 ' V
V- = ITrO,,/q 'XV;) == V; "irDa r ~ I + "'D ~ 1
= 1.1-1 ~/s
L ( ~~S-)4-G~tl + . J 6) Q = 12.1" cf~ a.) 1)"= Q = 12.1(, == 5.~~ fps
A '7r(fft
l{. -; ffe.s. ,o(o-.n)dA + ~ 5f{~dV ~ 0 fL:$.,a(v.n~ = mout - w'irt = IQ.2 '2 (1Y-)= 0
?lfl /
.2.M=o ~t
M:" 70TAL MASS IN TANK IF-" 5:: SAL7 IN TANK AT ANY TIMe"
If ,o(fJ'.n)dA =: /9.2(F) - 2(f. 92) ~s. M
~ ffL pdll = !fl c.v. 13
:. ~ T let. 2 5 - 3. iLl :: 0 dt M
",(
S -,q.:z.t) =~(l-e.J;iI
M= Z33 I~ I='O"R t = '00 ""i"" S = 15'0 Ibrt\ .. 0..) F' OR -t: =:> oJO S = I 6 b. 6 Ib"" .. . b) (t.l (S::z __ d;.,..s __ _ Jl; dt = Js 3. ~q - 19!.3 s
, I tI\
t -t = -1:1 .,.. 3.Zl/ - ~ 5:, 1 I ,Q.4 M
'3.!q - IQ.2 5 r;;;- ,
= -~3.S' k 0.39 :;; 6() mt'". /.S'I
:. ~:: /;/) rniH.-4.....----- c)
IF THE PLutD VOLUME IS CONSTANT;
dV/ - dVI de I - cit 2
AI if. = A~ 'Vi 1.1,. : V; ~ = V; (~ r o.:l = A., ( -. r
~
11;.:: 2 ( ..;fr ) 2. = 127 fps 0.2, = 5'( 6l/) = 320 fpsl.
1/.9 STEADY I=LOW .r. Jfc.tCv.n)dA:.O oR (r d (p-.rft) = 0 . pA = c.onsi.
JJc.s )
d (evA) = cJA f EY -r.:!..E:::. '0 ~ V-A A "IF ,0
:. Q. D.
~. d M + Jr d rn = 0 ~ E 0 di: nees :. '"to
'1.1/
Vi ~ l~", -; v. =0 ~LIIP.
I"' 1 "I -;( 'j
fls.pCi/".n)dA -1" ;~av ~O o
CONTROL VOLUME IS FIXED TO
WAVE FRONT 4 MOVES WITH VELOCITY v,.., TO THE R If$HT. -,.0. A~ T ~ A (v-m -1I;.J = a
:. 112 = Ym ( I -~)
'/./2 v = f r -vdA = ifmo.x" SR 21T'" rt- ,..lY7 dr
'11 R~ 0 L' R'J
LET ;z = % de = d YR (' I
v-= 2~Jo ~(I-zf' de
LET q = , - & J d 7. = -d ~ ". = - 2 vm J. 0 ( 1- Yl J 7 Y? d Yl
l/. 13
=!1.J.v: (,0 ma;x
.0 V = o. ilt'1I"mc.x ~I
'4
[ p(v.\)dA + ~ rrvpdv= 0 Cos, ~c.~
o 'STEAOy F~w
K ~(-o-.n)dA = -1'11; (6cO + WtHoRlc. lb.,. 13J
+ 2. jl ]!;. .ydy = 0 o 3d
WlHOR\e = f 1.1,; (6d) -f>1I; (3d)
tit HOR 12-= -;:rll; (.3d)
. ~ == 21'L b == -2"oL v, b =-v-~t
) d ... == 2 W,side = 2 ifnr., d~ THUs -2,aLv + 21' fob v d.!:J=0
Lv = fob -z.r(y) d~
a.) -u{ 'f) = !-AVERAGE' A CONSTANT L 1.r = 14:vE b
"VA VE = L v b
b) 1T (ej) = c. ':J + Cz
80UN DAR Y CONDITION S:
LJ"(h) = 0) V"~) = V,."o.x
0= C. b + C 2 b~
11" mo.x = C, b .,. Cz b2. "2 1/
C :. l/ V",&1L C ... = -~ 11' wrCJ.X .. , ) ~ b b2.
11" ~ q V-*~" [t ~ (~t] . b slnce L V = C -V d~ v~~::: -!:L.::...::v:.--__
q Lb[t" -(f)jd~ LET 1 = ~b )
1).,.,"0.)( = --.:L::...::,1/-,--_
l/ b 1'( t -yt '-J cJ ~ V" ... "'.... = .:2 Lv ,,-"' 2 b
4.Ju,
4.16~~r--- - -....:.--, t J J
----t>- 2 ern I J -I ,Scm , 2-
I I
L_~ - - 4crfl....J
MAss~ ~:: MAaS nPw 0,"
tQ, 2JAz l12 ~?AiUJ ").
1~'16~ 1TQ6)2.1~~O-'~ ~
Uj = B.l5" w.~
~ _____ J
~AsS ~ lA\ ~ Mm Jtor,.,
4.24 v
17
I
@BaT Vsc:nt (1f(G,~
~~ j' 21l'L b V~t-r
4: Vec,er ~ v l-~ b
~~Is~
lTtJar = 4lT~(~ -Ctr; ~f~~4.'2!2
~
n1.lA7r " f I lfe2'1fLd"( o
:: 5'! 'iTL blf~ ~
00 lf~: ~b. 4 b
CHJ\PrE'R S'
5'.1 ffc.s.p(O.n)dA = 0
5: pv,dx :: 2 [p~lfdHJ.:C~ dX] #v, =3~
.: ~ ::' ; V, = 26. "1 fps
5:2 1?l( = fls.VXp(v.n)dA = f>A~2. -~A V;2
= I'A (1.0 I V; )" vz -flA v,:1 Tx == -~ = ",A V; (I.OZv;.-V,)
= - ~030S" ~XIO.UJ-A300~X6i8 ft) '32. H l./ IbM ~
lb~ Sa
=50IOlbf
5'3 Xl=x = Sfc.s. ~p(v.n)dA ASSlJME VN IT L NG-rn
f: (p, -~ )dy - DRA~ =,Pf 1 (v;xf c:/ x+ 2 fv,'dx' -f!r.'~ =f'~.~ . .t + 2V;2. -1/ V;=>J :. ~ "'l1..".:l ( ~,~ 2i I I SJ~CS FROM 5:t~ ~ = 3 vI)
o -=-(1",)(.,! .,~,)(8OO ~) = 179. g Val
P,-'P-;a. =t" (J79.e #- ~f'@.o)~ = 189 p~f=J.31p; =9.MJ
~= /S7.t ~/s caltlTROL VOUJME MOVES AT
V= 4.S-~ e.x. MEASURE nUID VELOCmES RELATIVE: TO TANK
~9 = FoRCE' O~ FLUID BVTANK
~I='x = 2.. ((( v~M + (( VKdm dI: J))c.v. J)c.~.
Bx ==-if(vx M) + (-m)(2. S"7J %) FLUID IN TANk #-lAS 0 VEL/JC.ITlI
R~LATIVE TO
Jr ("'P. D. :z _ n 0::1.) _ r:" _ LJ LJ(i1.{ L _ J..~ LI \ 'r S ~ rx -iff' \ ~ OJ. , Fx = f (?I.;r. 9-/11.7 2.25") lbf
-,;p (1. 9L{'Q ~l{ .qr) 'bt FA' = 5'02 - 9f.1 = L,J() Ibf
5:9 ~,--; -~~V;
ZFx = JL.s.~f>(1/nkIA +~~ o
== P. AI - 'FiA~
['.5 ~f'(-v.nJd4 =f'~ ~lr'~S VltAjYi~ p.-~ = ;O[V;l- :l - ~i1f;?] BY CONSERVATION OF MASS,
(( fJ (v.,j)dA :::0 Jlc.s.
1'~'Vi -~(As~ flliVj) =0
~ :: ~s lIS + ~ 15- :: :!'1 (iO/ps)t:~'{~ a) ~ = li+ps h) B. -P, =If(Jtl - :7{1oa) .. '.0;(
ttl::: I as MIN1MUM .~ FOR "P='P~J v-* =~ ~a.) THE VANE I~ AlTACHeO To A \VHEl of RADI(IS ... ; NoTE TWA1 ALL MAss HITS CA'RT Mj == I'Av-r[1Tc.. (I-'-DSe)vc::ose-~
:: I'Av-r (/-case X"t1C -u-) m= VC/v- .
)
?= ~A v-3 ( ,- ccse)(m:l.-m) @-o' IAA_ ~ dMot - ) nl - '"'-
.: FOR -P-P,..,ClX I clc:N -11 Q.E. D. =< b)
5:13
CONSERVATION ~ MASS
~M + 5 d~ =0 M = If X T/,'~ Jd~ ::: -~~ (UNIT CROS~ ~EGT10N) flz i +//"i-/iV; =0 X=Vw, ~=-~ ~(1T"",-~) =fl';-~ (I) . MCt'\1ENTUM
~ FJ( =: ~ r if-cJM + r ifd.-H "Pa-Pa :::?t (~x&)- V;tf~ = ~ ~ l.C ~1f4 = ~A (vw -zr) FROM (I) ';;,.02 (~-~s) = A' It, 1..[ .:1i-P, -A~1&
2.1
COAISERVA-not( OF MAss!
V,A. =~A2 MaM~NTUM :
2F=f-oolm P.A, + ?(A:l-A.)-~~-~AASf> ~~ = /rv;.2A;. -;0 1{lA, REA'RRAWGINq
(P.-~)A, - ~ (A;l-A.) t- p (~-t\) -f'9A v,IJ .A, I
As AV-o-+dYJ AP-dP ~
A ~ A .1 (p-aJAA ~o So -dP _pgclY -=..ov-cIv
:. dp of-;nrdv- t-~ dY :: 0
CD A 1=' 0 TO? ",,:l
v; == 12 tvJ/s R== 12i KPa.~
A2., = .113
Q= Av; -= o.3l/"I'4 Ht"l/s m /,Q= 841.1.1 k'~/.s g FJ( = If"V;c a.-H Fr +P.A, -P~A.;\ CD'5I)-~(v;.cose-y;) Fx = R" -=/,Q("5~e-v;).,.p~cose
-BA, R" = (ZCff.1X-S".S'22) + 1l/2~o - 9~9. 6
::: S"OS:S-JJ
F'3 - ~A2 s;,,8 =;:JQ. (1{ sine -0)
F~ ~ "R~ -':jJQv;..s,'r18 .... B.A..1 sine
= 31?$ +r:a2 'R~ = H, 3GJS' N
tiAj
SEC.TION ~ + 'P.1 A2 - P3 A3 ::. ~ (\Jj -1,.1;.) Fx = ttl (lJi -1Ii)-rf3A:J - P~A.:z. o-A +"R1"m (A3 -A~) - F)(=O o-A = Fx.-RiMtAs + Pc-mtA:z. o-A:: m (Vj-v;,).,.(p~-Rn.,)A3
~-'P~)A.2 o-fJrof.)=E!' (33ao) t U:~~(:l~t
3!U ~
- SHi. ~ ~(12)
0--= I Z2/ psi (COMPRESS/Otl) A.UID o;-A, N02lLE
~
-SECTlON CD 2Z
F)( + P.A,-~~ = 1M (u;. -11,) F'x= mClI:z.-t()-p'A, + P~A:a. o;A. +~A:a""~ +Po..t-.(A.-Aa)-=o 0; A, = -Ii- -O',iA2 -l'o.~ (t\,-A.1.) 0; A, = P. A, - BzA.l -m(V2 -11,)
- o.iA;l - ?G..~ A. + ~ilt'A:z. 0-= L/922 ps; (IE~SION)
-~---=-.::..-----,
Z F' = rc.~. vd m R= r vdm - ( vd~= Fao~fE.
OUT )n~ FLUID
= 2 fa 3Jf'~2(~)\~j t-f'ifo.l3ol ....
--;l, V;;2 bd (MOMEN1UM OUTSIDE'
R::'2,o~2d..,. 3p~~- 6/,v;,2d
:: -r>Vold FoRcE o~ CVLlNDER.: -R=l't{/J
S'Jt ~ -~--tJ;=10fps ~
0.) AIR: lfw = 1130 fps 1'= O.0023? slc..c.gs./Pf3
A'P= ~ -P. = Pal.{." ~ :(.OO:131XIl30)(,q -
=:26.KQpsf = O. 116 'i!ps;
b) \VATER: VW = 11100 fps p:" /.
~rp= (l,q37X 41100)( 10) = q,} oi'o psf : ~'33 ps;
S.!1 3 ~VALVE OPEN
FoR AN OBSERVER. M() V/~ AT
3M/s} THE SITUATION LOOkS UK'
~O ~lfw-3 4-V"~-3 ny~ W/HcH IS :nJs.r LIkE! PRaa. 5".13. SINCE Vw ~ 11133 mIs, Vw -3 = IQ30 w.J.s ~ P == r'V",./A'V= (looafl 'l30}(3)
= 4287 KPc.
5:20 FOR STEAJ)Y F~W EMz=ffcfxOz)c1ti "THE RADIAL VE LOC.TV AT 12 RELATIVE To TH E: IMPe:LLE"R =
s~ ~ ) L ~ ~ t II I x
i--3' + ' /" ----1 XM2 = ff~s('-xV~ -~o)l'(v.A)dA
:: sq r( -\I La (v)-t dr 3
= -~v:l.t [rP5]~=7'V~t(36) V ~ /') g L. t = 4-;~ = .::l.. I.:t = c~/ -.
bt'
.: M~ = 595"8 ft -lbf
, I + t f I I IVs r _______ ~-v. I I
L
2F = 1k C'5 .. . . -f'Ac V~ (-ve.)
& ~)fc.v. V)C pdV=:. ('(Ae Vs :-Aj~')(O) :.FK::f'[Ac.V;Vc -Aj1f2.] I)J y- DIREC.T/ON l:F~ -= F'j f~c.s.V 'j p(-(J;')dA :: 6oA~ vsX-~)
;- CfJAi Vi X 0)
~ fcr v. IJdV=O dl: )J) c. v. Cj r
:. F!j = ,0 Ac. vs:z. . FORCE OF FLUID ON CAR" R = -~
24-
. -mitt = ~out plfh = pv-(o. +b) :. b-o-. = hcosO(.J b+o.. = h b = h( 1+ cascX)
2. .
0. -= h (I - cos ex. ) :2
ZF"~ ::- f V'j dm F:pv2. h sin ~ b) XMr :: )(0 X O)z dWi
F"~l = ~ 1!f'Va -~ V";,vb .:: pv2 hsiYloc.R= ~,Pv.J i:lf'V.:l.
,,_ I , (o.':l-b~) .{ - 2 h ~l"""'-
= J:lI(.Y -2coso(-f ~ ?f-2cOSc(~ 1I 2 h sino(.
f= 11 cotcX 2
H2 = h~ + 2 v-:1 n/9
H -= Yh~+(2V~hY9
b) USING CONTRoL VOLUME-lI) ~Fx= f(t:.J(xl'(O-.J1)dA f~V=i'
o
ZFx= P.A, -~Az. -""R=YnAV}( 1=1h - P:lL -1< ~ phYa(~-V.)
R= Pah -~L-,ohv.2('X-I) F'ROM HYDRO"STAT/C:~
p=: ~ ~i"c:X 1. )
='f) 1.=f' 1,. =~ -P. -=1'
5."3D r - - -, USD .. r:; T~ l~ ,; I ~L~j. ; f:~~1
It: I M=s>A h Vz - n: m, '" gA V : -~ A Ii L _k' Y ~ + *dt
~ 'J' Q -:::.A,V, 'Z 'IT (.08)5 =0.02.51 mi 4-
rh: gQ :z 2'5_15 tzrs V2 :r V, ~~J '" (2.8 M,i ~ 2 m(V2-~)+~-~)A,
~ 2 19
CHAPTER b
I--~ CD I I I.Z..,.= 22 -2 ---, IT '
P,=:/S-'"~L -t-_J ~:: 175"KR.. 0, =.25 Itt D:z=alS'2 ~
WORK
,. -~ S( et !)d.;,+ ~ rrC..:;{dV o J~S. ~~-
:. -1r= (et ~J ~ -(e rl), m = ~[, ~~-u.2. R ';"I :l lU,,-U, + 2"- + ~'i + ~(~-i')J S/ijCE 12 =- T. U2 .: U, ,/ m =f'Q =1025.'2(= 21S.2Sk1ls
V, = Q = 4.278 M/!) AI
V'- = Q = 1/. S7~ m;~ A'l
P, ~ IOl/3Jdt- IS",HjAf- '3~'!?I;) oJ ~ :M.92 il1 ~ J
= 'i/~ 32 b Po. SU&STITUTION YIELDs
~$W '2C cH = ~J1916 AI.., = 35.9 KLJ
5
MIUUS SI~}J I NDICATES WORK INTO FLUID.
FlUID APPEA'Rs TO BE H 0 A
;z. ,I
SSllME' NO 'PHAsE CHAf-IGE; THEN 11 - LJ V-l) '2.-r,) .1.- I
n-\ de.( T U ~ :z homl~1 u = CvT
6. 'I sa _ J'fAl .:: ((ore. + ~ )(0 il )dA de cit )lC.~_
r~ ((f 4dV ~t >f. v.-' o
JL.ie~ +-'%)I'(v.n)dA =0 .
tri. +t(I+~ =f+~ t~~ r} U:z. - (..(. = P. -~ == Cv .1T
to AT :: ?,-J=i
fJCv c --I ~
v Ib,.,eF
= IOlq" = 0.029'7F '2.'1 (IX??l)
6s - ~~s = ~c~+-~)f'(O'n)dA
[Js = gQ. 6"5"0 = qo~ 000 ff. ~tf tit . K2
o
~ (e.T~)o(v.n)dA =/Avf>1- _~l. ~~. 2
+.! ('Po-~) +- ~ (l:\~ -'jA) + C~l f' 0 j
Ps = -aws/dt- t'ii tell' -,-LJa (!.jA-~ ) AV 2 Tf- J B
6.6
= (-2010 t 3 g 5"0 +- 3CJ 25" -I- q~7 ) == G 702 psf"" = ~6. S" psi""
CD
D-IO" ,7 li=LiO~
1?= -6ps'I~
29
- [lJs = M~f'Pl. -~q~ + 111 :I._l{t+-~~ l de [- (->9 .2 'j IJ
= (lXr.:J.l/) .trnO+b)/I{I(.r, t 53.7-:2'.0 +51 ~L - 62.1./ 9'" t;(32.J~'I) j
- dW.s = 2?i S-I Ff -If". dol: 5
=: 5"D.6 hp
6.1 ~ ____ _ CD (fW-u----==----
Ah:: 2.5' em.
P. -B = 25"",. 10/, 3:2S i=h./a;fns 102 10. 33 mH;.ola.+m
== 2l( 5'.;; I=b. v/' = .3.& = :2 2'15':2 = ~02 ~
P h22 s~
V.:. 20. 0 I+t/s
Q = Av= 7TG3)2fl-O) = I.t/I? mJs -q-
== 5'0.0 ft3/S
-~w= I.:U~ I.'1l1m3 202 ",~ de M~' 5* 2 SS
= 346 "" oR .3Yb KW = 0.465 hp
6.i ENERGY EQUATION - STE.ADV ,..:" hOI + 1IM31103 = Wt:z ho:{ A'S p=c/ v,AI(CvT. + ~~ t~)
t V3 113 (C", 73 + ~3~ t ~3 ) = ~ A; ( Cv Tz
+ V.l2 r ~ ) 2 ,.0
AS T, =- 7;) p, ="P3 e= v T, l' ~ r v,A, f V:.A.3) t A, v,3 t"A!.vf
,u 2:2
= V:zA2 (Cv7i t ;'2 +- ;) FRoM CONTINUITY;
V, A, + tI, A3 = v2,.A:z J 112 = V. -r 1r, ~ ~A2 ~V (T;1 -T.) + P=; P,] = ~,~ lft~ + A3tJjV;~ _ ~ Vl.1&2.
2 :2
CAN ELIMINATE Z6. Cv (72 -T,) = P,-'f;i + V. .!:1,.2 + AJV] V3~
,.0 1f:2 A;:~.:l _ V.14
T So Cv(-r; -T,) = P. -P:a. + I 1I,;\
r' I t A3~ 2 A,V,
+ A-!lI3 V:::)" l M( It 11:):1. _~-_~ --v. 1+ 3 3 A3 ~ tA, v..4 ;;. I A,'if
6.9
MOMENTUM:
f>. -~ )A. = p~2A, -1'1f,'lA,
P,-~ = p
-,0 t!i '4 A3 ccs.G
+ ~2[43~ ~ -2 VI - 2 b, c.osB I ... Ihv3 A.
A,V,
p(et~)(v.n)dA=O Yc.'S. ,-lL ~ V ':1 ve, -:4 + U13 - .lit t "Ps -~ :::a 0
2 ;z>
VA ::= Q = 3f1o/s =3. B2 fI'/s AA 701 (19-):1
Us = Q = ~ tI~ = '-IlI.4 = 15.28 fl;ts Ag /fg
~ -~ = Vs:l. -'{12. + c.{~ -UA I' Z
PA -~ - 10','" + . liSP. 109 -:2 (3J. J =?'I)
~-""B3 = 2./S' Q. of flt.\id fCj
~ = Z f1- + 2. IS A of flt.\ic:I
= 4.15 w- at flu.lcl. ...-I:t!.{
6.JO ----=-t
VA: 2.1{'5'"Y
~:l:: lI.JI V4 2
"Us = 3.gj V 1:!..'-= 1.:3 V2 2
FaR TI-I~ COlJTRoL VOLUME SHCWN;
~ -~ -{if:: f{(er ~>,(V. n)cM o 0 t. 0 t +~~C.S. pdV
~t o 'C.v.
~ -'PH + VA :l. - Vl. '2.. t ~ (l1A - ':i 1. ) ::. 0 fJ .L
(IO./l/L/ Ibf/f+ 2 ) + y2( 1.11-"7.3) 6:2.'1 lh""/f+3
~G
+ "32. r;L/ (-;" 1 -:! 0 y2 = 1'/2.3 y = 13.5" ~
6./1
~LUIO
WEltiHT
31
Z~r = r vi! dtH ;:r \( p(O. Vi) V~dA c.s. -F -Lv' + PA(A = -111 VzlA + ~(o) USE GAGE PRESSURE ~ R~5Um FROM 6.10
r: := -t.J of- PA/A + m Vz fA w:: p Q VOLUME
-= 6:l.l./nr~~.s + 'lI) =1/1. / /hi ( ~S-7" j.'1
!/rIA = IIlAz :S'1.51(2.7r'12 '-b-) t\4 t-1T. 1~ -;
lIi IA :: 3Z.6Q fps F:: -111./ r 10 '!!.1.+ 62.'{[,3S'X3r..,
I.{ 52.lrl(
F = 1399 16[ ON FLUID FoRCE oAt LID IS /39'K Ihr 1
b) THF FORCE ON THE LID IS
THE INTF6RAL 01= THE' PRFS-SURE OVe;K THE' AREA O~THE LID. WHILE "'BE-RNOULLIS EQ UAT7 0 N G t ves lis P= PC VEL.) \viE Do NOT KNOw THE VELO-ern' VARIATION ALON6 THE' LID.
CD 6.12 Q=6~ AIR
~ 'S :lC\LCOHOL fJ= ~J.I:tQ
"P. -~ = 0.1 WI
- -' ) A;:l
p'-g = 7gJ.f. 86 N/rn2 = 6LfO.2~ P 1.226 ~/m3
6'10.2 = ~~r-J (2;2)~r-~
~t=3."ZLfI Az= .510A, =a./II'1ml. ~ =. tl29 n1
6.13
--V. ::L _ V J. 'j"') l. 1+ i2.-p,= 0
2 P V, = 5:1 fps I U;1 = U.S" ~
11-p. = -2.0'15' Fl- of H2 o ,.oj = -2 .a45 " H;2 0 ( I' fig 0 '\
13.b "H;1a)
= -0. /5"05' ' H-S -: -/.1r' Hj MANOMET~R 'READING' 'S GREATER
AT (J)
6.JlI
h ~4---~ .. '~----------
d .. ." CD :.': ..... ~~- AIR
USE: SfRNOVLLI QUATION BfJ1JE'EN A) SUR~AC (5)*0 BEFORE AIR
IS INTRODtJc.e:D (ST'A. fa)
Ps .,. YJ.~ + S ~5 = 'P.s ~ V'B~ + '3 i!,.B P 2 P 2 ~ 2. =
IAJllfRE K, = 2 A P J K,. = :2~ P
t: -~ ~ ~K' +K~(Yo-2)'fs -(KI+K1YO~
~AT = Gbf = 40.2~ '5~ K2A-r ~ ft
.,fP = (5"-3)09:- 46. '6 ~ t}l.
't(,=(2Xlf~.~)(r~.'Xq)(3:;.JflI) 30 zfP' . (. KS") (.1S-)(12)(~2.") = J. S4
~"'J
Q = Av =trd:2.Vc. = O.I3i! '(:t3/ 5 ~
~ +.. Vc: :0 0 dt: 0'"
Vc -: y 2'jL - (ho -h) ho=I1@t=o LET 11-h, = z 4h = d~
I dt dl:
dZ! d'l...,r dl .,. D~ V 2'j L (I + z) = 0
-3 -( dr :: -d':1.Vl~L (' dt )oYlf2A. D:l Jo ~~. @ t=o h". 1,. ,/B t =T ho-S, = 3'
2L '/,- 24. . = -::1.. ~ '{JaL T ;-3
o 0:1. J
T- JL D~ ( ) - d.1 Y;2c.3l I - ..; I - 3//..
= 105"1/ s == '30.9 /H;".
6J185~=/21;..7f.ps~ T:l/o-p
PAr ... = p~j"~J(1O.?.3 I~ "=2051.11!1l. ~} fl.2
fJ=:E ::. :205"1./";') =. OO~ 39;< slu, RT (,115" ~()O it!
PAlM + ~2t7}2._:E t 020P f> ;( -,0 .2
A P = (- 00~3'P 'l!5912 - #l1/()o) = 1.3(,6psf =O.tXBSpsI3
P= :J.05"I.lr.r/.3" =2052:5 ?sf :: 14.2Cpsi
6.19
Vox- = ~. Cbs 30- ; VD
A = ".193 fI'J. ~ v= ~ -= 6.5S fps
,+ g: + ':11 = S'Ops,'3 D: Il I"
CD A = CZT'/., ~2. V, = 3.6'l,f,,!o
Q::. 1.9 cfs S.
ACROSS SECTION ONE J THE VE L-OCITY CHANqS BY A FACTOR OF ABOUT 2. UNLESS THE: MOMNTUft1 OF THE AIR IS SUF-FICIeNTLV LARGE -ro ACC[LERATE THE FLO~ THERE WILL 'BE A PRESSURE: DROP, A CHECK OF AIR VELOCITV "REQUIRED YIELDS SUPERSONIC AIR SPEEQ, THUS WE MAv N~LCT THE AIR MOMENTUM.
t AmP. AIR : v, Jj~~ l. BEFORE AIR ~ 4a -t -~ -l
rl~
ZFl: = ) Vz drM
fiB -11 ... ) A = m (v2 -V,B) =~A~ (V2 - ~ '4)
,a~o
"Prs-"PlA =-~2.(I_g ) . fJu,p
TOGE.THER WITH BERNOULLI EQVA.
~ vt(J -~o)= CJd~oU-tJI i-~)] I Ll 2. v.l.
-- 1M J.. 2~
~,. = c;;!d PH~O r, -~(\t ~dn p,... II - A4 )
\ 2~o .,..---:--~-----:-
'6 ~ 9. iIXI.8~X' -1 !) :s 3.113 IM/s I-~
A q 2 % REOlXTION.
/'.2'1
tM ='pAh :: f' 7:;r'l. (hfho) ~ = p'TIJt dh d-t .~ dt
~t :: ~fi1 vC(At = /l![t.:J."; 29 h
~ :: -d 20 \.bur = -d~ i ~Cjh t)2- D~
C h~dh = (~-d~ ~ dt 2~ Jo D~ ~ :n~ I;,:: -~ vSS t -= :2(J-m~-b.S&3 t: -(-G.st>3X/5)2. - /''-.L7
- ~T sec i~X-s'J.r~i{) ( ;Y;:2.t .
= /l0.~ Ift;K
I~ ~~ A p'~-_....J ,
1FIlJf!E N A ~ E
} -r V.A2 of- ~ 2!A = ~ f Ve;.2. + j rE 2. 2 Pz. 2
-PE :"Pc: -I;' ~ L2 I ~ = V ~ =
50 ~--Pc = -Va~, Ii :2
"PA --Pc. = ~ L2, (1- P.) t Jl..2. ~ 7J 2
ASSUM INq Va ~ 0 = ~ ~ THEN Pa = -Pc =P"
~ = ~ L:z. (~ -0
6.2
6>.'31 tJ~ur;a-U~ TW*~CA f -OFT~ 'JJC1DM~ AI CZ-,1 ~AP T~~ ~TC46 V~~L.
~~ '8-
0.33 GJerr(].,f~ NbPrF 16.D
&fo.O.JLL{ ~ ~$CC
c.
! -t Y2,."1. ~ls;- = R: + ~~4hL ?J 2(1 J78 2(j
k)Zl.c) ~ ~ \?: : ~~ J ~zO 25-2:c :::Z H
LJ:'1.. '4 ~ + hL = H= 367 ~ 2(5 t (] W~ ~ 2 '0/ (Je Z 9.32f;-f~ q 2 A l& ::: f;.CE6roLcfs
~.'34-LSvg:; fu (;. \() ~ FOe
A C. v. &J:I03Wc:fr~ ~p
-?JA = ~ ?-?~% Ail? at g ~
= ~ fp'lM:~tR Q -z fr:) Z3Yrm : l.~ ~
Q4-%a) $re Qf/?;;: (1).\44 1.25~ ~
~'FV\
}/0cur;I2 % ~ :z' fBI U ~fJfl .. ~ s fO'~
E ". Y~:r ~ ::.4.1{, 10 !!!c. .F.".... 59
USl~~t~ ~~Aw9~A
~ J..(~I/l-=. a. +w-tS ~ 20 ~ 20
~:~=: ~"~%2WV
~~=-2:5VW=L~VW e ~ (%~vw \1')P~
b.% Lcr V-=A~Br:V(r)
V(Io) = 0 2 A +f>G :0 A:;z -~ro I V=B(r-ra) V(~-)2 wdh ::z -etr{ -~)
., ~ V(r)zu.rl r- G 2- (.-_~
( 10
2 &T~JJ ~ AJJ.t> Z I
~ ,. t.5 .... t$ '% nL-~ ~ + o;'l.-, of ~z. S(3 20 ~ ~ ~ 2 1< u: zl) r. --"t -z U
a --I;> I S 2-
0:2
~ 2+hL%~
J kl-~D}lo ~l, t52 % @.::: (4,0 Mt M:: yAUC :. Ill) ~
~@ h~'%()J ~
L1:2~ A~A~ } 1.6
~ z L(O~
~ I'!L "3~ }.b ~ ~V~tfz
0"2. '" ~ '" .g,291'V1i;
10/),:6,( ~
'~ hL~~J~t:'6
I-I~ V: LY2 }%i~ " lJ ~ h 1. 1-t...2.~% 3th
C1-lAPTEQ 7 7. I ~'lS{~ ~? Aa; ~lN6D ~ ~x. I .
T OF? ~e I A~ Z6, ) ~ & .0177 G::O
80
ICO
. {){)70L
.cor28
.r.m
I
.Dloi \
.W'S
i eo
z~ Foe WAn3f2 C(IV '}
Q.~::: 11.32. \ 2D-1DJ ~ /~ ::.' 1 ;::. :> 7 '-\'32 PI+O O. ~ 10'" . ~~~:Z70%
4l
7. W t6e. An2- Q'" 0 QI40 :::}d32. _ LIS:> .10-5
Q32 /4M - (,34::).IC-5
QIM) ~O.~2 41'32
~b~z-(3~~!
7.4 z = N c = No. OJ=' MOLECULES
4 CROSSING A PLANE W = MOLECULE'S / UN IT VOL)M~ I t
1.'7
"IV:::: 2,," 1/".2 ~ E' + ,.37 J
III ~
iJ)
~r ~7
L_ / a.) dUX aU; a'j ax
" [J---: I I I J .... -
2-0IMENSIONAL (;(~y) FLO\..!
Vi! =0 OZz =0 av 0 ;)J{z = -.. 12-)( =0 =?;,z av~ __ o __ ~ ;?ff6X)AT: - \I"r ()()At= dl4 AJ(~o A>f .1t ax 41: .. 0
FOR 3-D\M: BOrn AXIAL STRAIN RATE AND VOLUME CHAN4E RATE ARE EXPRE SSED AS
a~ t d 1.1~ + JV2 (5EE p~oa ct.3) ax ;;9 dZ . .
7.14
r- z plo.ne. Z
B-2 PLAN~
V"-8 -PLAN~ r
" .61:
= ~~O rVrlGt6B -Vr Ie Ar~O t )"'11$
+ r( ~ 1,,+6,. - ~J,.)l ~r J
= ..!.. C?Vr -r r l.. (~) r;Ie dr-~r
-,.J rr IT d~ f- r d (Ve)\l .. 're = 1/9,. =p tr ;;e ar r IJ
?J5 H ~1T 1 ~ 1 t i
'j/t ~h f LL 1 --1 E l--
E=.oIGMt -1 0 t-RESISTINCi FORCE' = f:z F2 = STdA = fcr1l'Ddh=rJrDh 'I = ,P dv- = ::y:
elY ~ .. F = n v-7tDh =.aV'.jl'ltDh.
Z --E ,-- E.
F;i' = 1000 C.osX 3:t ./O-3)/olS-J7f l"lo""
.S~2 (3./t)
= {C07a, f\J
"lib Fr. = W FROM PREVIOUS PROBLEM
Fz ::;; v"'}:: 7TDh .I ALSo" Lv' = ;7\9 E
:. jJ ,/ :u7r D h = I')'t~i e
1.T:: m9 = b8tJ9.1/-;o-r pif7rDh a5"OU1XIO~~.~
".. = o. 7 f:>6 nils
dA: .... d'dL
dL =~ sin 01..
M = .M~ ,..3d~dr l A1ij,1f h SIf1C1C 0 0 .: M = 1T'~ w D.y
32 h '51"''''
lvzO ~
(?t-~)~: \'Trl)L%(J)
4P ;
CHAPTER g
t.1 - = 32"uv. Q cJx 0 2 v'1f!i 4:: ~Q dx 7T IfI
Q. =KD.' Q2::: K ~i
K = (-:&\Jr . dxjmp.
02 = 20. Q4 ::: 16Q. ~ CHANqE = Q.2. -Q . 100
Q,
= 1500 ~ INCREASE.
K.20RlG/NAI STARr 4O~Km=-_=EN=O:.-
'~Km Nat 22Km I ~-------@-
-dP - -.1P dX - T
ORIGINAL: ,;, ::: KD" (-:~)
-t1lf.3 = LI.) ~ K 041
Nat: () = rHH -A'f.2= LI'l~ L 1'2 K 1)'1 J K[)'I
f..I1.P) =. t\1N/a..,1 -.dl,?3 = L.2.3 mN t L J3 KO"* 2K04f
SINCE - 11"P..310l0 = -41P..1/Na.J-Alii Lu ma _ L .. :a. ';'H + L.:1-3 mN
K D'f - K [)'12K ()'I
46
13 CONSlOR THE CYLINDRICAL
SHll ELEME NT
~I __ -L ar .,
THE SAME ANAL VSI5 AS IN SEC. S.I OF THE TEXT LEADS TO;
4..(rTJ = rAP a) dr L LIT R ::: OUTSIDE D
KR= INSIDED (K.(,I)
) def'Y) = ) t? rdr r"T;: t&""P r~ + C,
2L
,., = -; dv = .1'Pr + C, dr 2 L. r
J dv= -.1P (rdr - Sf d,.. :2L.,.u JAr
V = -.A!:.,...1. _ e,k r + co2 I./)-' L ;U
"B.C. v=o @ r ="R J r. KR C. & ~ R~(J-I
'Tf.)( = ~ dVt :: ~ l: + c. dr dx 2 ...
V.{:: -L. 4E r2 t S. .t.-c r + C;z lfA dx oM
B.C. Vx O@ r co.Q. , ~ = V@ r = d :L ;z
C, = -A r; -L.. dJ>( 11 k~ LV+ J6,.u dK D~_da~
d .
C2.= -2-dP ~_ C, kl2. ~A dX 'I .M :z
F :: fA = rC1id .1)
.: F::. 17" d,uf} :L (V + ..L. 2,!: d..~ l! 16,M x
d
(D~-cF) + d &J ~dx
FOR CONCENTRIC FLOW IN THE. e DIRECTION J Vr =0) Vg = F(r)
50 Ire = A r~(Ve) d... ,..-;
Pte Ar41i! X Fe :II 0 I P~ ::. ple.+c16 7r ~eAZ'lr+t.r -,(,48621,. = 0 SO THAT Tr = COt.lSTANT OR )l ri .4..(~) = C J d(lfG) = ~ 4r dr r" r A (2-
INTEGRATION YIELDS:
~=c, ... -
47
lic. Va '1:1 0 @ yo = ""'ROc.TTR 0= e,Ro-S&
M
Va =(JRu'Nf~ @ r= 'RuIHER
:. w'Ri= C,R; -.. A
C =# CiRca c, = wI
m= pAri = ~7rO~ 41P 02. = 1TD;..oilP 'I L 32r' lli Lp
= 1J.2
GOVERNING EQUATION JS
!L (~)( ) - JfP -= 0 dy dx
FOR NWTONIAN FLUIDS IN LAMINAR FLO~ -;;x -=fol d ~
dy V - -'- d? tj~ 1- C, u t C;z . x -:J"u d}( Z;"J
ae. @ INTR~ACf (@ y =0) I) Vr = V:zz: 2) '~r = ~XIr ~I dVg -=,I-{Jl d Uxrr
dy d~
1.1
g:p = ;11 d:l.~ dX dlf \Ix =...L ~ 92.1" c1 Y + ~
2"a elx M
B.c. Vx =0 @ y=O UK = V @ ~=h
j~t ) 7 J 7 J; II c, = A ( v- 2:!: dP)
., .2"u. dx
C~=O
FOR ~>') =D = 0 " d Vx ) = 0 ) Y d Y 4=0
c, =0
:. dP _ 2).( V dX' - h~
CONTUJUITY: dP + ~ (Plh) = 0 dt ax
MOMENTUM: dVK + Vt JI! =:1 aP at ax fJ ax
5
til T 'ieM
'f' + S5'e", @ 1
BERNOU lLf f='ROM S TO I
P5 + ~ +
8.~
~b"/ . / ---
(NO PRESSURS CHANGE IN e DIRECTION ).
g.J!; CONSIDER "PASSAGE AS A STRIP
CONSIDER FREE BODY OF 8.MENT Y n-I -AX/ , Iy-t~ z.~ =0 pi&'-I-I:';LF1A'(.1 (NO MOMENTUM IX -~- X+~
FLUX) Tty'.AX"
)(
(P\X -1'lX-rAX)AY +- Ci'f:S+.6'1-1ifj)6X =0 .
DIVIDE BY AX Ay f TAKE LIMIT E' = d'P = f'B-"B.. _ to?
C1
SHOWN BELOW' ARE THE' VELO-CITY PROFILES FOR 3 CASES;
~ =-1 IS MAX. Ei=1=IClENCY 'RSlh 3
~ = 1 IS MAX. FLoW' @ ZERo "'R.Sl h 2 AI=>
I
.6
. tt
.2
B,17 1 \V ~ ~~ -- --~~I- l' A'S S'~ Tul: A1 1- r l' r 'rlrt .. " \ L_ _. 1_ - _ L _I !VET ~lC4L . I I t { j ~'T~ f1..vx.- t ; Ar l I"S~ SO L f::tD
i:
W' +- 27f f( f A.t- -1'cr"r'l{ Lxi r.O rt.(r r
~~
\II 2. S(j 111 ( Are 4Fra --rA~w::t lKG LMtT As Ar ~o
J
'1.1
CHAPTER q
e~ c1r r
(I)
+ &~~!.:dV = 0
[c~o.n)dA ~ PVr(~iI'"A(:;)lr"~H'
-l'vr(~r~e)lr + f'\l9(~r~Z)/GtA6
-pVe(JlrAi)le +pV~(r~~r)Jz+4&
-p* (r"6BJlf)J z L~ fJdV = E.. p(rAGtirJlZ) de JJ c.v. at 5lJ8STITUTE INTO (I) I D'V'D~ -sy
(r A,edf'A Z~ TItEIJ TAKE' LIMIT As Afj 6e~ A~ - 0
!.S-(rV,.) +..!.. ;>Vs + dV2 = 0 r (' Y' d9 ac
q ,. I" ,.. ~ .2 V = Vx ~;c T ~ ~ t Vz fC.~ t'7_~ fj dn ;}".. y - ~ 1t.)C + ~ "''.1 t aZ eJ
(0 V) = Vx !x(il( .e;r) + tI~~(~.~)
+ Vi!~( ez -i!} NOTE: ei' e~ :: 0 IF i.;I~
=, IF i. = i :. (fiV):: Vll'~ t V~~ + Vi! ~
53
(0 \7) TELLS TH E ~ATE Ol=' CHA~bE "DVC TO MOTION.
~D3 , 2
t
CONSIDER 2- DIM. PLOW
CHANGE'S IN VOLUME = (1'2'XW) -I - ( i2 )( '32) ./ 12 = AX~ 32 = Ay 1'21 = A X t [V)(J
42 = ~o + d r dO + de ;;0 dt ~ di dr" dt de
~ = ~V~ t 1" aVe p- + v. ae dr d r ,- ~ r -e r W
-T Ve~ie dr
~ -= ~ ~ +a\j~ +Vt"det"+ Ye~ dB ae I'" ae ~ ae
~'-' ,.. ~-:= :;}er ~ = ee~ = 0 dr ~e ar dr
(}e,. = - ex slY'e t ~ case = es ()6
SIM'LA~L'" 9e, = 0, dee = -er , d(' ';)8
HENCE I i'
~ = ;;Vr i(" -t dVe e dr dr ar Q
A
: = (~ - Ve)~r i-~~ t-V0 ~ FoR Q.? To BE 'D V ; ~ = Vr
dt bt dt
de = w = Ve d+: r .. -:g2 ::. ~ -rIV('dVr- -t!!dV,. _~~
I.. ~~. dr r ~ rtr ~v. ~VB r V9JVg t vr Ve)~_ 'l r Jr r;;Je r ""'S
ct.S USUJ6 THE n",COMPRESS\'BLE Fo'RM OJ:: "THE NAVIE'R-CST~ES
EQlJS; D v = - V1=>+ vV2 y Dt" P
a.) FOR SlW\ALL V; ALL TE'RMS OF
~( %f + 0 +vv) ARE SMALL 'RELAnVE' TO THE' orHE~S
?RESENT.
l,) F"OR V SMALL. I3UT V LJm6EJ. TH "P'Rot>CJG"T at=' SMAll V AUt:> 2~ ORDER OF l-AR~E :; MAI{"BE
SI6~IF1C.AklT COM1=>AR~D TO THE
REMA/NltJ~ 7eRM5.
~ = ~ - 1. V'P + J V:2 0 Dt P
if T Vx~~ T ~~ = '3x -;~ () 0 ()
:2 :;}~v. 'V Vx =~ = 1 2P
~'1~ ,.u 'dX
~=J..~Cj+-C dlj j.J. ~ I
+ v'V':Z~
~ =-L Q)P ':12 +e,
'V. V= 1..(w"R2)= W'R:ld (,) - 0 r Ole r r- --y:- de r -
:. CONTINUITY IS SATISF/E!:>.
'i.j Vp=:! + ~t' = - V~ Dt ~ d
9./3 FoR THE INCOMPREsSIBLE
LAMINAR CASE)
OV ".. rli:) J 2-Dt"=g-7+ V 11 FOR 9 NEtSU6~LE') "D V == -yp ... Jv:2v Dt I'
VECTOR "'PRoPERTIES DETE'R-
f.lUAIm . 'BY V E VP wHIC.H I ~
ARE IN"ITRDFPENDENT; i.e. CAUSE i EFFECT. , .: MLJST LIE IN SAME -PLAN~
La')
IN ABSENC.E O~ V/SCO()S
FORCE'S Dv _ -V? -- -I:>t .fJ Dv ~E'7E?M'NE'D 0"1 -\JP~ Dt J HAS Pes/TIVE SENSE 6lVEN ,
"BV -VP OR DIRECTIoN Ot='
DE'CREASIN6 -PRESSURE -b)
SIMILARLY" ANY FLUiD .... '5TA11C
OR MOVINq.... HAS nils SAME' 'N~WENCS e WILL MOtIF oR
I
TEN'D To MoVE IN THE
DtR~CTION 01= D~CRf'AS/N
Tw~&~
~ (P~3lYi)~ Jqr or 2- 0
~IB 2 Sf.t1(.o Tw @ % -lfe e,., ~
dt r tre :zfCr) ~ trrz~ -z0.
U-sUKt ~-\6"~/~ T~ TJ.tt5~ \.-sTt5 Lgr W~~(Dg ~Tf4tS E"nlir-JS ~.
, direction
(av, av, v. av, v/ + v av,) -+v-+---- z p al ' a, , aIJ, az
ap [a(1 a ) 1 iv,_2 av.+iv,] = --+pg.+/L - - -(rv,) +,. -=aIJ ? iJIJ a? a, a, , iJ, ,
9.{9 ~ ;- 5" '( fS.DS
IJ direction
(iJv.+.LV'+~~+~+Vz av,;( Pat /fiJ, "iJIJ /, Tz) laP [iJ (1 iJ ) ~ 2~+1z!J = -- -+pg.+/L - - -iJ (ro.) +,. () +? IJ z . 1 iJ(} a, , r ,
~ ~ &~ ~ %(je~O W~
~ % peJi e..(n~\ at or\: 1 ~ ~))
9_a? A~
CHAPTER 10
10.2 b...;;. ~ ~r t3 ~ tt~t
Wi! = ~ (0( +,s) cAt. 2
: lirK ifQ.r1-'1 (rv81rtAt"-r~I ... ) At ... O r Ar
At:
.At .,. to..n-I (V ... le+A8 -Vr!e).Ai
J r'Ai9 IN THE' LIMIT,' TAN ~=Z
G.;Z = li~ .J{r Vgl r+Ar - r ~ r r) ~} .... ~ Ar t.z
- Vr 18+6.9 - V ... 19 rAe
5e
= 1 ~ (rVe) - 1. ~ r ;;r r C1e Wz = ~V9+!(\h _dVc)." Q.D. ar r' I) ;]9
JO.3 d '1'= -~dx +vxd~
== -(VQOsirtoc)dx +(~cosoc~
0/= -Voo(sirlol.)X + ~(coso
/O.b IN CO~E d"P _ "v~ i.e. w- -p'IW " - - -tr-- - ()t
Dv:-V:l.C,. Of: r
dt' r
V= Vmruc i "00 ?l~) -"P(o}: P UM~{"Rrd(' = e.lJttt 2
1(2. Jo 2 I1?ROTATIONAL: (r ~ R) 'P + U2 : 'Poe, p 2 P J '\1= Vt'H '"R r PoO - PC R) -= IZ u..,. 2-
2
THU 5.1 "'Pot::) -1(0 J -=,tJ \J w?
So
:. U2= ~fP =-~ VtM=126f~
,.0 .00200 -1=> = !!~2: I'~~ (~ t-= IOpsf pUm2=3~ .... ~ : "$l = J.q
'R2 20 f.:: 13=1.5'
SO T/rvtE % (31'.5: 131.5 = /.5'6 5 V l?
c.) IN CoRE B -= P-r ~J.=r:jP';
fJ V2. = P \1'".r: -= :3~ r-t Ra "R~
"Po = (2116-3i ) +-g~~ Po = 211' -3'l (I - t'.t/R~) VARIATION = '3"'8 'Psf
59
10.7 VI' = \'/oe CoS e (\ - ~~) ALONG STA6NATlON STREAMllfltS
e=-~
~ Vr- = -\Jo,o ( l- ~)
b) ~\lf' = - 2 tLo a.a 9Vrl _ -2lbo ~r --;:-3) ar: c:a. - T
10.9 "Pi" pU2 = CONSTANT 2
I~ ~ -=+>00" V 2 = Vo02. HENCE J \ Voa I -= \ vel:: 2 \b, si)19 sl~e =.5 .". e%! 30; ! ISOO
10./0 a)
WHEN '-P=O j '1 =0 OR o f:ao Is-W& lwe. 8=6 \.e:Tltl; ~v,;; \( Ax,$'. ~ li_~O (f"~ ~ G:;irr To L~) I OIJl; GB"~ Yzee ~ \2~ Q.
St, 0 '( Zilttl r ~2IilMit(.a \
e-+o ~w.e)
10./3 ~VRCE AT ORI61N !.p= Wte p2Tr
m == SOURC. STRE N6TH
FREE sTR EAM tp = Vc:o y TOGETHfR tV= Va) g t- ~e
211"1'
Vr =! ~tp = VcO C05e +..ttL ,.. a e :.27Y' r
~~ r'sine
Or' =0 @ $=11 AT a =71 r= ~ _ Q.
zrr P l.4x,- 2'ii~
lall.{ As ~iP = pDv Dt
=pBft V(~2)-o. (V'xv~ /WD FLOW 15 STEADY AND lR'RoTA-
TJONAL I Vp = - P \l (~)
OR v?= -p vVv BUT AT STAGNATION PolNT v=o HENCE vP=o
10./5'
LIFT FoRCE : F~
d,cj = d 1= ~iV1e = CR '..1 -~ t ." mosiY\ede 1~lae . t'ov ~10'.e) '"' (ir
F~= Jo (1=>~-BYR'5IY\ed8
FROM BERNOULLI EQUATION
61
-p + tpv::t = CONS.TANT
'Poe t' f p uc%)~ = -p i- f pV2
O~ TUE HUT v= 2 (/a:I 'SI'rtlt
:. P="Poo t-! PVoD~[t-l{'5;",~e]
F ~= ro t'lleo2E-l/~;n~~ +45;~ 'R~'rle de
F~ = 2 f' 14' Rr'LSin~e-~iYl~Iv\~ (;) de
F~= 2.Rp~ [j- - 2'5," 2eoJ ~~ = 0 WHE'N
10.16
~~TIDkJ 1blJ:r'S tLc~ S~ Bv QacLe.5.
tf= - Ie( ~ r ) ue = ~ l.J~6,lJ Z'ti 21Tr
Ol2(qw 15 Itr \krtk: . .~ ~ f(1,o) ~ K ~ K.
Z'ir(2a) 4~a. ~
Lfe (-.:tJ 0) '" -I(, e~ 4tT or c:IO
~10 e~1T) ra:sn = k=- Q. n~
S, Ar ~~JJA17o.~ VDlAF )("Z -~ -= - l.~ :::- D.02~j.t-\... '(: () Z'i1 tTll) Z'il9
b. &py lJ~(Gl~
Sr~N.crm ~1...Ll& l.5
~-:! I5QDrStnTI + Q1I -z ~ 2it Z
. ~vs
~ ~ ~ rSUA
C. A.7 Lt~s?~6. ~ ALL Tf; h.olV b~AT (,)0" W~ Q=~(2h) h:z ~ = I.q z D.Cf>33W\
2o-lt) 2-S d ~ tv1Axl),AJ.,W) ~ ~
'0,22. \.EIlMM~ '" V Ple)
ZE,=O d ,.GlT, p. ' GT
11 O-I:tr - ~ 1'P 5 Ilt8d9 =0 o
1/: ~TM +~ J(~:-U1-~ LJ:-2'tsm,.g
Iff "'~D ~M ~~')p.jlf2Ijf s2eJ9 I~T ~ (~. - ~A\1t'\ 'D +.2. O{y"z.. V .,J' ,j (, J ,LV
~O\za 1.257~N
T ~ 10.\ IzN
CHAPTER"
fI.r 1) (1..) cv ( 'Ii;)
11.2
p (MIL') Q (L~/tJ H ( L) n 9 (L/t
2) "P (MLYt3)
i = rt- r =- Z - 3 = S" CORE ($ROUP (p I "OJ w)
1l;=t1 (8Y INSPECT/ON) 11; = fQ, DbCJc H
.
V
D
P
(LIt) (L)
(MIL')
t= 5"-'3: 2
~=~
CORE GROtJP (D, V, p)
n; = D c4. Vb fC,)J.; 71i = Ai _ 1-r;rvjJ - 1Rc
7T;. = DQ, Vb pC: e ; rrr; = L I>
11.'3 ~'P (M/Li-a) D (L)
P (M IL3) Q (L3/-t) w (I It.) )A (MILt) t: 6-3=3 CO"RE 6 Ra./P (P.I 'OJ w)
r;r; = pa.. DbWC.~""p ; 11:= A'P I fJDV
112 -= p q Db w G Q j ~=~ 'O'3w
17; = f' a.. Db W c,,.a ; 77;= ~ PDaW
I/. T
tC4
tjI L ~ BV GEOMETRIC SIMILAR/TV:
d = .J2. v= J.3 V .I L 7r cl2J =..L 'Tl'D'2.L 1)~ _ 31 _ 3d
i/ 3 '4 d 2 - L - 13" :. ~:: (3)~ :: J.I./l/2'" a.) BY 'DIME:.N SIONAL ANALYSIs:
~ = 'DQ.w b pC 'P I = La. (Vc)b (MIL~t MJ: ~
-c 3 '-t=-5'" b:. -3 c=-I
J
:. 7(, = _:Po.--_ ,ow3 D5"
FaR DYNAMIC SIMILA'RITY:
'"P ( --P I p4.)3"C~ model - ,aW3 D6 proh1:'jpt
~=[~ ?fjff.r 3 I 3-$'~
= (3.3-~/"3j3 = 3-2/q :. l.)p:. O. 'T13 ........ t----- 1:,)
11.5" MODEl "PRoTOTVPE
D D ,,1> V V 20 knots p p p ).t A M F /Olbf F A 1)2 (bD)2
FOR DYNAMIC 51 MILAR ITY ;
~'" = 1~;p ) Dvpl == DVfJ( ,u rtf A- ?
~:: Vp(~ . fI;. .t)= 6vp , I I
.: v~ .= I 20 Krto-t5 a.) ALSO FOR DYNAMIC 5IMlLARm'
u.~ = EtA? ELA-I - fAI p U2 m - ; V" ? r?:: F,.. (}t. %. ti)= F~
I 3b J.ri. :. Fp = IOlhF
/I.' VAR tABLE C~o.x
0(
S M L p 9
1<
Cmo.x 0( f3 M L fJ 9 1< MOo I 0 I
L 2 0 0 0 I -3 , tOO 0 0 0 -2 0
.: ~ = '3 -..... ------b)
l= n-r' = Z-3=S .: No. OF" DIMENSIONLESS G'Roups ::- 5' -~.-------
'iT; -= 0
(~) = ~; =0.1 --. \1"", = _ 31 b V-p MODel SPEeD = 31. 6 dlo OF S?E'ro O~ FULL SCALE SHIP.
11.9 RJR SIMILARITV ReM =~FULL. SCALE
TMERATUR NOT GIVEN,!
ASSUME 'H2o = 10C
JH,.o = I. 3x/o-6 W1~
'MR = 2t;OC JAlR ~ ,.": >(105' ~ = 2.'1'l>C10-6 wsl
~
L V I - L vI . u. -, I J L J - IT .. ,..- vF.s. ~ F.s. ~ h~ .J L
VF,s. m
Um= 1l,2.4Cf/o-6 .q :: 122.3~ /.3 10-" 5
F"l. = .0262 h F.S.
11.10 NAV/ER- STOKES EQUATION;
Dv = Q _ vP + )) "12 V 'Dt ..J P
NONDIME'NSJONALl2JNG;
VC)C)2 DO ~ ,.. P' I ;2 r7"~ - - #' = S - VoO v r" L Dt ~L;:"'P~--
2 + J Voo 2 V'.,. 0:/1 La
DO~_ 9 L vp* J v~ o~ -- -- +-Dt* U~2 L~
~ = ..!.. - V~1l'~ ~~ fj>Jf: Dt Fr 1t?e.
11.11 SYM80l.. PIMEWSION
MASS TX COEF. K Lie 'DIffUSION CQEF. D Lo/t DISK DIAM. d L ANGULAR VEL a.. \/t DENSITV p M/L3 Visc.osrrv MILt
K D d 0.. P M M
U 0 0 0 I -: ) L 2 I 0 -3 t -I 0 -I 0 -) r:'3.1 V1=b, i.= 6-3=3
77;= d/o..~pnk; ~ = K cia:
~=..D. cl 2 a
IU2 SYMBoL PIMEW510N
FLOW RATE" Q. DIAMETER 'P 5HA~SPEE.D N VISCOSITV A 5U L RPM w \ It: VOLUME Q L~/t DENS\TY P M/L3 \I'ISc.oslTY M MILt r=3 Y\::.6 L=6-'3=3
" J
CORE (f\RDt)? : "'P"DP
~ = -P"''D 1o pc. W
~ = -pa.DIIa pc Q
~.1:... II. =pD~4i
'IT = ~ 2 P
/I. 11 AS5UMIN:S /NVISCID EQUATiONS)
~ DO = -tiP +PS Dt
MAKIN6 EQUATION DIMENSION-
LEss: V := +(~ ) t ~ I ~) to L L ~2.
OR JL = f(>< t~) 'f;L L"J L ~)5IZE" = 2 "" = .005~6 ~
"360
VE LOC lTV 1r _ \J ~ ~-~
Vm: ~~IA60 =. 422 ~/5
b) TIME- t: Uoc = canst. a'R 1: - 1:. L '1.0
t*- _ Lwt ~ - Lw.j.Le. -.l-f; - L P Vry. - r; LW' - J 1.
1\.20
IN he$r~NT(,~/~ Is W~
g W=-\7f+)AV?ll~ -I) f.!1t$(~
l~ o-~ ~ u-/~;\t~ v;,-l/L Ou;~AtJ\C
r "-
- ilt: \ QY.E.. ~~z~~+31~~-l) J L/I/ n.+-" .~(5c::Z)W' C 7 D
o!.-Il .. l~
T ~ GrO D="THE ~ vrry ~}./\To mt. hpzn~ -rt~t ls
I \.Z I E 0't cfr2-J I (L) S (M/k1 ) t (T)
TIJ~ Is ~ 9IMENSt:)N~"5S q~u? L..
t=.t -~rS
We.~ 1,. f'S.,. lz.EtIs @
~ug 4r ~ Z. tzEt cit S "8 r4
12_ ~ /:4\.i"2 5 It ~ ~H.&) L. z ('?/~
m~ d:4:~t~ A- Yr~/z, ~~---~-",-~~-~
69
1\.2'2. a (/v\A: ) )1- (J.vLT) J lM/L~) V (Lfr) d(L) V(L)
__ .\_ /_ \ f. r7 . ,r;:?.y< ~ I \~ \~~ ~ ~L.-~Lb..JI'" v.uy
~~~ ,let::5~, ~tVt~ l~(O~) I tv5
~~AJ d J Oy'1) YD ) .J-i~~ /
X> 3~ ~~ i\\l-g W~$
D) ~) V f() ~7 {JD~V 7, 1ilu:; J
JIrD
= ,~/ :lEi. ) Gli'l 'I \ /- 'j'V" )
1\_'25 5StEW VA2L~ .. A? (r!Lz.) ?(~-t!LZ.) Q( L'L/t ; L (L) Q(L) r2(L)
, "'\ net-I)
T~lJS T~ ~LD Sf; 4~~ VIA {~(D(;; WE UA~o 0/. 2i ~
'\) /0 ._A. .-7 L e 6JLt ~r j J;.. ~N h:X7C~ ~ Q 6.111;, , i.Mr Grow ~TAl.V~Q - ~Qh
CHAPTER 12
122 "DRAG = ~ pv2AR C]:) SO
Df =.!.pV2~C-r 2 ~L>-t
= ~ (. 002 3"Y1! rfa )(293.33/"( ito f 21100 (.011)(.75') Ibf
=202b~)(~O~p~} Ib f a) WHEN p=. 000 'T3? ~IUlj5/r;p. V= 500 ""P'1 Of ::' 3Q21./ I~t ( 5232 hp)
b) WHE~ P =A, (SEA L.EVEL) V= 200 mpk
Df = 202b llof (IO~ hp)
/2.3 SPHERE IS SIz.E OF A GOLF BALL 1ReC.R\T1CAl. == 2 lOS" AlR @ 20C J= }.1/9/0-5' mys ~D= 2./o'ii" V= :;'105..;2, l/ / D
If=:J./o'!i . f.'19/0-~ = 70. 9S Wf/S l/2/0-3
12.l/ GOLFBALL SIZE SpHERE
1)= ~ pV2 PI ~=.l(.OO23?~JJr([;g"'AJ 2. 4 12
= J.?bb . 10-S'"Cn V2 Ibof v= 1ReJ __ 1R.e.(.ISq)cIO-?)
15 l.bS /1'-= 1.'5"61R~
103
V fp~ 1s
12 S" ~ TRANSITION c::: 2:/05'
~x = ~ J X = JJRe.TRAf.ls V V
X = 1.l.Iq -/0-;; .2. /os = 0.099 vYt '30
12.6 F/UD V~ @ [[)5E OF B.L.
v~ = ~ ( Jxl.OO )~ ('It' -of)'1. = 5 OR ~ = 2~ (IO-8.2?92)= 0.1"
~ v;Re;c
@ T= loo"F JAiR =O.\ll163 HIs lRe.)C = x,\VtlO:: X"'. gg = ~O/S76 x"
v \2 .Il) '10.3
x" .s J 2 :3
'IRE-x
2/)J2~ 'IO,S":; '61,031
121,5'11
X" 2
v~ -Ff/s 0.5"32 0.376 O.U6 0.211
12.7 NO, "BERNOULLI's EQUATION IS NOT VALJD IN A '"REGlON OF
SEPARATED FL"OlJ.
12.9 Vx = C, +~ f:j +
.: 8" -= ttxo.lS.... ----------0.) ~
12.1/ VJ.= 2 V~ ~
Vx = VE Sir11T!i = 2VQlX SlYlltl;& :2E it u
p=~o - 2pt&,2 SIr12B =~-2"oVC: ~
~ = -L/PVaJ2 x dX 0.. '2.
- d"d'P = To + ~.(dpV/' 0.'1 dx ux Jo
-Veri)~ pvxd~ ~ = II d vx\ == 7r)J. Voa>:
o r ~ ~::o a. d
~ (" E P v)(~&~ == ~p v"'~\t (dX '2. ax )0 0.2 dx)o ,_ eos'Ti::5Lr dg - 2-
= 21' Voo:2 d (JX'J.) 0.2 c&
Vs ct)~ p V" d~
= 41' Voo2 X cI ra x"SiYl7i~ d~ 0..2 dx Jo :::zE
::: ~p Vco-:J. >< i.. (d"x) '11 0..2 dx
COLLecrrt-lq 'iE'RMSj
4p"&1{~x) = qreU ~ ')( + 2I'Voa2d(&~ Q. :1 a. t 0. :z olJC
- zpv!..?)( d- (bx) 7rc..'" &x
"fret $)( = 11 A VQ:l'X T ~,I)Val"Xo a.E ?~
- YJllof-$X + d clr~p vco~)C,. .. !PV~ ~ '1ic..::1 -ax[ a.~ ~o..:i.J
IN LIMIT AS )(-+0; dS ~ 0 a)(
12.12
11771/11111/11//777/7/7 ;( X'+dX
XF~(( VXpCOn1tAt~i(: /'3 '0 . .2 )(1.2)(O.O?6lfXIT):J.. \ l~ 2 32.11Lf
=.SOII"'f- .: Ft> = . S'Ollhr
~ ~ J; .... ~..-------- ~ b) VX:' aSlvt 6j "B.c. V.,.(lJ =-Vx[
~ = 51r1 rtr;:i ~:x (S} = 0 VXd 2~ .:J
b) FOR A CIRCULAR CYLINDER " 'L WHERE Vx$ = 2 VQ') 51 (\ 0. }
VxJe2._ 0.'11j)( 5" ~ T - V)(~ 0 V,,$
s: Sir15(~)d~) = = f S\Y\~~ d~ _ - Si\1 'I ~ cas ~ _! co.; ~ (Z +Slrt 2~)
5 ,s
= .! - J. cos. ~ {2. + S;r'l~~)-SiY1\cos~ 15 ,~ 5
= ~ ~ - G?s ~(gtq5in2~ t3sin~~] ~Gl2= o.'1f[1-cas~('irl5in:li T3~in'l~li
,) (2XI5) sin' x s;Y\~ a: 0..
et.=: o.47)~-casl (~tL('Sil! +3~j,,*~ ~ VOl> Sil"l6~
a.
c) VxJ;2 2 Va, sin ~
As x-o dR. xa. j
74
= 2'19 /s
6) IRe = IQ.l ::: VD J
U = (IQ:1X ISq'IO-~) = .IS fps I/l/~
THIS INDICATES THAT THE EXPRESSiON t5 VALID OVER
A WIDE 'RANGE O~ VElOCITIES (AT v= .ISfpsJ IT IS Nor VALID)
12.1'E D -= CoA ,oVco2. 2. 2 = . S-(2. '29 X I. 22G)(30)
:2
= 631.1'0 N
POWER = 30m (631.10)= 1095"o.Q\V s
= 25". '1l1p FoR b 'hI/s HEADWIND 1D = .S-(2.2~)(1.22b)C36)2.
2-
= '109. 6'1 N PoWER :. 30 !t1 (qoq. 6'i ")
s = 2721''1 W = 36.b hp
FOR 6 Y)1/s TAILLJ1ND lD= .5'(2.2'1)(,.2.2')(24)2-
2.
;;: "I (j/. 2. 'HJ
'PoWER -= 30 ~ (L104.2Q ~)
:::: 1212 \ 0'4]
'1
.: I> -= l.20J ~
12.2.1 JI)::: Co A t pV4 = =1.I1(irlb)i (.oo23nXI7(,)2
12.22 3 W=VV= t~VCoAJ2
fo" :: O.CXJ2(9i9 5(~/fP' ?ttO~ =6. ll' 220 7
a -, ~ ~ W=~ .2b'RttJ (lOZ."1).?i(25.i3J
!2:)
: 19.~ Yp
b W= gEE ~:: 15.70 tr 5>0
IZ.Z3 T ~\ F 6 -? "l
It.2.4 :} J}~ 0.16.1 /0 ~~ . S'T4rlTtN~
AT ~ -r7.5'cf I Va 92J~ frs 2-
1Dz~~V ~~~== V20/$,@
~.J64 V CD 7.5
10
15
2.{)
Z5
.2
gz.lB ~
12'Z.9( A6
'B4.% .47
2'45.82 .44
307.l7 .10
ilk
/ ;' Slt1(X)Thl
/ (~12.4) 1/
I f
/ I
11>
.072-
.1014
.(~
.128
.16k,7
lb
1b
1~.2S '2..
lFT:Z~fV ct~ .. C~ ~! I V~44.7~ L =: 5(\'2~(44,7)1.Q) (2.29)
-z f. 8CIE tJ (W~ (~a::of\-!) J
If.2~ W: S~ Ot; = o. 52~'b L= W~ ~gV'2,A~CL
Ae 2 ~.B{ 1~1. / C,-:% O. 224 ~l)5 gQ ~ L l42
V
.Qz{5LO~:: 240 @at/s ~Too kApID.lIJ.t ~ TQ\E.. Is O.372s.}J~ lY Qw~~ Is 88.5.
122'7
S~ U-Y(k',D)iG .. Tm; NAVl6Q- -~~ ~ At-T~ lJtu 1-
JL ~ :d?-tJ(Uf~ / 8y1 .r~d)( iJ( Y20
~A~\3
I~ .. ' ls- lYm % M5W WIND ~
~~~W~5PmD
12e2kjJ.a:me~y
12e=~ 2 128 % (~+Vt~~+tft.lf:' z ~ ~(lfa>-t-~~tr:(trm~V)
-1."-- -
4- ~ ~ l5~ L +- lSi,t.
T~(p
~~t~tJ ... V)+ l( ~~J" .. ~') Tug Clt~ ~ TD V (S
t.~,.~( V~2'(JtDV)
lW~dN fu ~mLt; DJZ ~vt;_~~T~
KWl5Tl~ ~tt &-r~ T~~R.u~ ~Nor-~.
'~:2. NOTb. T~-rUi; Buz. ~ Nor 'Stow ~1Z.L~ \)J ~ 155
77
I= ~f\J~~~jh/(Ja> TI-lOS
ZI?e :%.lJ~ .lY~~~'\~'L
""" O:(1~3r) I.,O.I (Iae~ 2:91t0e i~~
15.3 2CJl3A:z 044~ {o'2. cPs -1.
V~d.>* ~ ~-k2 =- 14524 ftk o.44It3!t~ ~
v.~ os O~C:1~ _(O.-t; !t~
V45 ~ '~57 _\O-s .ft~
a.~" ~7ft(z)(r4S2) ~14('87 t!>.~i . \0 -t; I
ak.1?O ~ \4}ffi)
b~ z 1\7_ ( \ ;: S 7m ~ rlD I.S7/ )
x 1Re.x dLJ~ ~~ G4't1 0 0 0 0
. I 21()5 .111 0.321
.5" .219 I. Jt' I .352 2.0'3 2 .'1Ql 3.SQ'
'I TRAHsmON Pol NT ~=2'ltfi
f.JAR
I 2.
X, me~
13. S' ~L = Lv = (Y2)( LlO} = \-:l~ ZOO J} .Isq .,0-3
0.) TlJRBULENT FLOW
Cfx -= O.OS?6 ( ';f2( x) 0.7.
efL = J. (L CL ax = O.o!;".f' (L -0.2 L..Jo TX L(Tr2)0 )( c;{x
= 0.07-2 == 0.006"07 JO.l/51
D'RA6" = 2(bJ P v2A Cfl: FOR 2 SIDES =(O.~2373X I600XI.5)CfL
= s.;r4L
=0.0392 lb.
b) LAMINAR FLOW eft. = (~~Y2 = 0.00375 DRAG = 5. ?CfL. :::=.0.0214 lb.
I = J+n FOR TURBULENT FLO\J~ d' _ O.3'g1 X - (""Rc).2
~---x
IF Y);: 1. /
13"'7 v= Q == .~b = 0. 3'1 Hot/s .7 A l((.u;)4.
T CALCULATE ~t f Vt ~
-2 r: J J~ 1';;' = a 022 5' f V;< ma.x --~--VXmAX ~~
FOR THE Y7~ POWER LALJ v = o:Z{? Vmo.x ("PRoB. '-1.12) :. VW\~ :- O. f 16 t\1/s
~h1o..x == t:J. 07!J WI J ':::110- b Hil)S ~ H~o
L J )-V _ I \'vxmc..x ~tl1Q.X' - I3.2ct
: . .jJ;. = O. '{It, ~.022!; = o.ori'll!! p 13.2' s
a) LAMlNAR SUBLA YE"R
~+= ~ijJ ~ ::: 5 J
~ = tj+)1 = 0.292 n1m ~,",(p
b) 'BUF"J:"ffi LAVI:R 30 > y+ >S' .: ~~)C = I.?SI./ Mtti\ AY = 1.~'2. ~tH c) CORE 1'5"-I.?" = 73.25'" .........
13.&' MOMENTUM ,-pV2 EN ERG Y ,,-.J P v3 @1i?e -= IO~ .&- =~S-Jk)CO.2,\
6L l-s i~ ~~S-J = 2. '3'1
v= V)f(~) MoMENTUM = pVoo2 f2.(-rJ MOMENTUM nuX' = f 2(iJ
P ~2 ENERV ~LUX = f3~) Y2f1~3
LAMINAR;
M = sin2.(i 1J) p Veo 2 L E = SiY13j~ 1r)
~pV~3 \-CL 2
~ ~It{~ rrt) ..M-~L &1- Z pVoo2.
0 0 0 .1 .J5'b .021./1/ .3 'I5S' .~ol .5 ?O1' SOO .1 Zq .195 .q .qq .t/f
E ~pV~
0
.OO3E
.Oqq
.355"
.?aK
.'i7 1.0 l. 00 J .00 /.00
79
~ M ~ q PV~'2. ~f'V~3 0 0 0 .~:z CJ()l/ .2S1
.ofq .1./92 .3f1S-.126 .5S3 .4112 ./68 .601 ~6b
.2.10 .6QO 5/2
.:25:2 .61~ .5SQ
ON 6RAPH: \) MoM-LAM 3) MoM-TURB 2) ENER6.Y-LAM q) ENER6'1-TlJRB
J.o
.X
.6 .q
.2
OL-~~~--~~---
() .2 .'1 .6 .8" /.0
M -L pV .2 ,~pVco3.
r3.Q "'ORA6 =CfL ~pv2A -2SIDE5 A = Tl/O2= SbOfP-D= ~(.OO20qqX205)2(S60)CfL
= 2~" J Eq .eft. Ibf-"IRe. = vL - 205? - 'Z IL/O 000
1) - 3~6'1'IO-7 - J .I D02.0~
0.) LAMINJR CfL = 1.32X' = 0. OOL/65' ~
D= 11.26 Ibl b) TURBULE'NT
O.OS'1& cfj( = (vx/lJf2
4L:: 0.072 :: O.Oo2QQ ~.2
L
13.10 ~)< = \0" COM"PA'RE' C" ~ CfX a) LAMlNAR ~:: S"x
fi& TURBULENT s:" _ 3?6 ~
'1" - 1Re.X
2
C;- _ .3=1,'ii"".).3 LJ ':UJ ~ - ~ If"e :: -"T7
b) LAMINAR Cfx= ~tt
TURBULENT CfJ( = . OS''7~ 'iRe)( .2
Cfr _ . 05'76 ~.3= S:'17 ef. - .66'1
1"3.11 Tl1R13ULENT: o _ o. 3~ C _ o. on. X - "Re".2 tt. - ~L..4.
1Re - V.L L-7 T= 2oDC, J=IO-b ~} 'IRe 2D. = Lj./OT D= fPV2 Cf A
=~ IOOOJIC)O''1200 =1I./0'7Cf N
efL = .00137 D:::r 5"l./, 'l2S" N
6' = .31(;, 20 = O-W3)')f -IL/.3cm ~.2
Bo
eft = 0.0000664
D= 265"6 N
$:. 5".2,0 =. S'/O-2~ ::" . 5" em fl//oJ
13.12 0.) J"~= 1,(1- ~.&)cl'j
= ~ s: (1- ~) d(~) LET ~ = t{
J~ (. ~1 I ~ = )0 ~ -(1J d~ = 1- lti;-
I +YI
= $ V1 (rt+lX (lt2) .
c) 2 + .E! = 2 + (21X"H'2) e (~)CV\)
== 2 + ~ Vl
13.11 J; = V, I cl Vxd"i 2) 2 de P 'X d;"t2t e e t-Vxcr G&
O.022S Vd f. I n )~ \ V)C~ (mIXr1t2)B
= vx~ d lJxJ"(2+3"\e +- v~ de ,dx 11 j dx
MULTfPL Y eY ekI AND NOTE eX! de = % Je 5N
;L deSfil + e5~(2 +3,,') d v'x~ 5' dx \14 n} clx
=0.0225 fr~+I~(Y1tiJ~ (~J~
rS./~ LET eS-lt = U EQUATIoN 15 OF J="oRM
~ + 'PC)() U = QU() ( l) WHERE ~} = ~(2t-3"\dIAtVxi
Li Yl"} r).y.
Q(x): ~O.022S ~. IhQ\~ "tl)(m-2)j tvxiJ
Ux~ MAy VARY WITH X
EQUATIONO) J5 KNOWN AS A L1N~AR FIRST ORDER
DIFFERENTIA L EQUATION
T\-IE SOLUTION \5
u= eSA{
= e - f Pr.x)clx 5~( x>e ~ P(lt} dx c( x
<
WHeAl vxr.s CONSTANT ?CX) = 0
U = Q x -t C..I c= ul ){=o I~J"
TM:A A~ tSb ~ Tm:.. VIcM~IM~ S1tn>~ ~~ &t~ ~b-r~~ ~v,lk(~~~ ~DtS~~
'3.I8VlC'''(~yJ=~O) t ~~./L x
~Vx'L 7'l1!...2 d2.Vx'/.f + ~ y + X~ 2 + ;;y 2. 0 Z "
T ~VX'I X~ ~J(d~ 0
SiNCE AS V-Q ~/~a .. . ~ v/ (X, '!J) "../c, ~ T C,.~4 t-C3~ ~
;;;V' I ~ 'j=o etc. ~/ (x,~) :> 'j tb,. :j:l. + ~)(~
;} V'; 1 e. te. ~4j ~eO
FRoM CONTINUITY ;)V'/ + ;;Vtj' =0 ;;x d~
So~ C~"R ~ vx ~ v"" ~'1 ttl<
BZ.
CHAPTER ~
ru V = ~ = '0 ~'~~o.J ~s ~ (o~~'f ff)2
= l.fK tps \RQ. = \) V = V.l2'1/i l)( I. Ii) = 2 q 5"
JJ 8 -Io-S' LAMINAR
A P _ ,- - 2 rf L V 2 . . = \6 -- -"L - )~ - ) 't P 1) ~ :: 0.0542
A?:: 2 (.OSq:;J~ (1.II)2(SV 0.0'2 32.2
= 66-=1 tbf a q .b3 ps~ R-'2.
l'i.2 ~D~.111
2
A P -= \5" ps', Gl)ESS LAMINAR FLOlJ
g = '32.f
GIVEN "PIPE J
A'P -- .f P V 2 - f iA ~ P
FoR l=ULL'l' iU~ULE"NT J:'LOW,
f -= fUNCTlON O~ e/O ON L Y THUS ~?_ ~2-
P THE \t)EA 'BEHIND WATE~ CAL IB'RATION IS A'P. :: ~,~ B
~?2 w.~ P.
So t..~2 = A'PIIo.O (~02.J p ..... wHa,o ;:;0%
.:1 'D _ \'2 (~\~ 6:2.'1 - ,'" ~"nc;' 12 .- ~ 2t.3J ?o - 7. T..c r -'
1~.5 I 3.20 KM , ~= 6."1 '/0-6 jlM2/s p.: ZOo I K~ I ""~ v='.1~/5 D=.l"~ e = . 000 ,5 ff- L:: 32q DOO~ 'D e = \~~3 / ~e.:= llbJ Doo
~ = 13,Q"? ~ = .005'12
~ = l/t~1~ p~ t> ~~
::: ~(.OO51:2)(32q ~) (l.I):2. \: ,-:rl J ~Y()7
:: 5".1 M
~ = M~L:: Z'D11L (.'1/)2(1.1) S"~1.1{ ~ ~
= lq41 KW
W.G ~ 'P.: A"P, +- A ~ + 6 'ij I) A? MAltv' n> 'P'~E V 2. ~ "i:) \J 2-~+~=rpt~ :z. ,a T t#o A'P. -= \lp :z.
p ~
2) 6."PFRtc.notJ
A~ =ZK~~+t(f.f~~~ P 2
t; = ~PlZK or 'It! -'c J '3) A ~NOZ2~
:2 u.. ~ PI" _ VEX,T .:.L-+ r --2 T2. A P == V i-xfT _ vp:2 == ~ ~(V~ IT - i" p ~ ~ 2 Vp:1 I)
C.ONTI N U lTV / Al> Vp -= A\lOiZ ~ VE)OT :. A~::: \.?2.(Ap'2_r p -"2 AN:2 I} ADDIN6 T06ETHER
~:: Vp:l fi'"t" 2\( : Llf b + Ap -0 jJ 2 t: :f 0 Aw?' . 'J
K := !/ f! L~ ID 2K =;; -a:; t 3.1 +1.5 == 155
t)/e. == '=1:J,5'fOb) = I~
~y '" 1'1.5;
:. Q =Av =~~i~l~.Ul= .~cfs - 13.~ 6?M
~.1 AP = hI. : 2 tf 1: \)2 P D ~ := (4.'15"'1'4'1) = 3304+2/S~ fJ 62.'4/32.114
hL = 2 i" 2S0fjli/6OJ2= 31ls" . D &D~ DS
.. := ~ -::: O. lOS"," DS 31:2S
lR~ = Dv =- D(ui/bo) V \4 '10-" (1TD)t~)
:::: Li 75' ., oS D
ASSUME ft = .003 ... D5" = . 02 "gt.{ ~ = 3.i".2 x lOS ASS(),tt fr = DS = .0'3:1.'1
~= '3."72 /O~
~p = '1K V~ P9 .2~
D =0. t.f91 Fe
ff = . OOl'{ 5" .003'1;-
D = o. S"OI-l fI-
tf= .OO'S
- P:a. = "S. S I' ~
'"Pz =-~/q ?sf~ :: -'S~ 'P'$.~~
IL(./O
?>fTWEEN I E 2
4f"OIA UNE
'2" 1)IA LINE
I
V= Q = ~ '~'1 :::: 225fps A ~,~
De = i A~A = 2 '21 -.:: Z'''' = . "6t1 l' '32
~ = q~J 3~O ~ -= 0'5 ff
~ = L '3 33 ~ :: . 005;Z q I1P = p~ "T k v:z.
Dc.2. :l..
= .oO:2.3~ tJ . . 005'21. 22 (22.S) W3 2
:' ,~?7? }:)sf
= ,ocHq II U~O '2 II H~O -= 62.Q ;:>s.f
Iq.l'S CAST IRON e. = . ooo~5 9: Q= 3/0' ~Qj/do.~ = 4.6417 cfs %T = 'PoWER COST + Ulrr/AL COST
YR ~V'K5 + _DfI (INITIAL caSi)
$ - 'PoW'R COST + .II (INITIAL YR. - CO'!>T)
F'OWfR COST: ~ , it'60hrP KW"'r lj("'
?eWER = P. ':: iL[ht. +112]25':l5.KW .g(5S{)} 311/3 ~
= . oOl6QS ~GL t/l'~ KW PoWR t = 1.03q~ (~ Ib;Xhl fl75)
== 301. OS OlL t 175") ~,. 10" PIPE:
V= Q = ~,5JO R-/~ A
IRe. = (5'1' X~S)) = 4, 3 ~/OS l.b5 ~ 10-5
87
~ = '1'gO, .!- = JLI. O~61 ft =.oOSog , ff
hL.=4G b V~ = q25"270'bG22~ o ~ D:;2 S rrr2 1)'1
hL = 22" Q62.!i = 2QO.3 OS
jl = '30L05" (2QO. '3+ 175') ~r +.JI.(~gO'2'll.lf)
= S 153 '321 /
0..) SOO 3..23 ""'/5 'IRe. :;> aU~JO()O ~ = .Oo4~5 V= '3. 2;2q ~5
1Ke.= ~S/OOO OK, v~ '3.Cb WlJs Q -= O.O/erg 13/5
J~lb ~=~~ L fj !)~
- /1 r L Q,'J. - "1"t"~ D 2g (~"D2)'J.
= 32 ~.h.. Q~ ~.2. '3 pS
1Re. : VP = qG.. II rrr'D 0
ASSUM PTIONS
I)RUB6ER HOSE IS SMODTH "D"RAWN TUBING IS o'*'
2) NO OTI-I R LOSS ES IN HOSE
JbooF = I. 22 'lO-s Ftys ~:e. '11'2 p"5 = ff Q2. P 32. L
= f;Ql. = 1. 36:2>< IO-~ DS" i .OO?I
" => t!.~S'1 /MIs 'Re. = :~s ?bSI = 3.03 '/0'"
10" .
Q::: O. Z 3:2 IN\3/S
I
I'/./f T "~~
1
AP ="?.l-~ = ffj(- ~l. +Ac)
= fJ ~ ft' v l + Pc) Ai! v= Q = qO *3/5 = 4.S'i "'/5
A 7r( S..,p. 2j
,6i=> = 10 "'f8~. :>. ('4
T (q :fif? X6h"Z M4j
= lO'!.f'"?I:2.'1~ t b5f:7.:?]
~ = V"D = (4. ~Xs) = 1.''If"tI T 1.3'( 1< 10-5'
(ASSUM~ Tft2.0 -:::IOOC)
tve.: Iq 6 is
-L - -3b I f)9 tl. I - ~~l f.fi - . OS,aU'lfl-IO" {J.1J9~I~J J ::: IX .32./ tf = .OO29Z
A -p = 10 3 t6'112l{' . OOl9Z +{,"r1.~ = ,. 35'8 '/03 K1=b..
14.\9 fi!oM ~UbWlI1'3/tr,:I.~~
THe; 'Cb 4'rk:W ~ 1'a.oa;e Ie;
1P=Sl,JV~~~ ?e-P. ... ?e-Z}hJ U L 2~ s>~
'-!t;RE ~ to;; T,,~ fAILS" v~ AJJL)~Pta; kr~ Aa \.2.
-rt;I2M-6y-~
'1, V. t '\. t "" V~ " (~J iQ, Y /- A~!) 20 2. ~, L T
= -2.03~
lie. -90 -::(S-\)IOS ~ QS.S ~ )lej glOd
-c~ -2a 111 - 2fD. 0 W"\ nj.. = hL, ~ hL'2-W~~ 1) = (),'ZM/ (): (.
14.~ J
Eo:; O.QX)(5
e.,.o.OCO~
gf5f~ s ~ I Qz~t;~ to f
, ,-/ 0 1. I
~+~+~t i"~T~~1 ~- P,tJ:Vl ~ 4 - V ~ trl SCi '2(3 \tY
VIz V1 =~ = G_;4Cot~
Ck1. ~ ZS~8~) L~ = ZO.ll TuQ6ULbVr ~ HotJ / \J3~ H~ \4.7. ~6)-t
14.22-~14.~
4P " y?'f(Ar r ~h +It] gG' 20 ~ A~ V ~ Z I( '" (1.3 ~ (0(.7) ~ ~
(' I- A 1... tfO 25711'5; ij. ,19.';2
lUU5 -z. ~~
~:: 2~ [ et>.Z7~4.~
1Q~ :z ~,2 z O.()dr.t;; 92.47
z~~5%
92.
}4.f~ WJ2lT~~LLl(~) ~~~~~J)Ulr
(~~Pe ~ ?~~)
~'--= ~~-~e)- ut~ t30r AI" l~
hI- "''rtf ~ LYV~ ~ t
i. ~ t,.:c-l),~~ of 4L' 1,.: {plf, 23
kJow Q=ALJi, ~ :: \O.73~/~ T~
14.25
T~ H.a.u '" ~ 1t:4~~) ~ ~ = 240 1 No ~AeE;CI FIDM fiG; ~4~ 7.
93
~t _~ '7 G - 'bD F 7\ ~~ \ h \f2~ 1-~L v4S6 ~ -r +~+-0, 'tF2 O.(Ylg G,QS OlO\2..
'::::
\SA 19"sc.
'1-- 7;4.() .f)J,S f) . " - ...-('\ "/ \. I ~-~ ____ ~ __ ~T ~tf- V'I \QI01'L
'}oY7";'. ~o- ';J,S1-'-..S_' _ OIIIl11 + 14,4" L
L -:;- 0 f O~'I(O v0. -, .. _--- --------.----
f ... ~-k~t~ c. U,)S W/lMo,?~S v../~ \/Jeo~JM ') l~9 ::: 0-0?17 INIW\~ = b.T/R
tq'-;;: Jo'LS (t!S)~ 40s ~ TM\~ -:= ~~ -' 40'0 ~ 445 ~
\SJ\
fj I ?CO '
IS,IS
~ L 01\t; ~L\ -=: \2.f\ -= l4G Y':'Y4)0l0.1'O~1-
~~ \3.\{o ~ /w ~-6I.-tCil~ ~\b{ \~ ~SS~C:\l~M, ~~ ~ k~~'5\Os:
~ ~l-x ';bot. ~fl6~
Rf6b~ b.Tkt ~ 1,Qo -.= ~l3A \'-/w \) \~3
\ = _\ +-_\_ - I ~\\l I ~,\~ \1~A' \?-/~i~
~~. t\\I = IJA,s W
~\;f.. IJb.l \, lftJ /;;1* '. R,~ "" \0r\'\\ ~~-= O,~YZ3.)% t, :=: ~\f"Ir\ = OroolOS
?Ti~1 ~ 1. -:;; b \(:/\"1-. ~ 0 d \ S irG1-
~l\ ~,'j.
g,,( letJJGG'~~ \ y~L~~-\ -:;: CoI,1. b ~\~\~ __ ~~,~ U~
If~~P'~S ~tt\lt~ r;40~ 1..
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I Problem 24.9 Excel Spreadsheet
Solute
Lysozyme (A) Catalase (8)
dpore = DOA-H20 I DOS-H20 =
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