solucionario de welty, transferencia de masa

1. Solutions Manual
To Accompany
Fundamentals of Momentum,
Heat, and Mass Transfer, 4e
By:
James R. Welty
Charles E. Wicks
Robert E. Wilson
Gregory Rorrer

2. 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 u1.2 'V = ji X d<j J
cv'PCQJb) =f'cv;:[k(cos IsiYlI +2)x
+ ~('51Yl ICos I)yJ
.". 7PCQ)b):::;2,",,-2 [t (~f-2)X
+i-(51~ 2 ) 9]
1.3 'VT(.K;j) =To(£.f [-k,Ca5 ~ ccsL, ~) ~
-rt(Sit1 ~ 51nh ~)g J
IT(a.).) ::::To £k,[-b.(c.o~1 CO'!>hl)x
+t ('5 in ~~h I ) Y]
V7(o.) 6)= -r;,e.""~ [l~I~~+ e.-') 5{
-+- (~i." / )G.-l-') 9]2h
=Toe-~rC&>-:.1 ( 1+ e.-"2.)';
2.. L~
-to ~r1 ( I - i:,2.);1~
::Tc, (O.CJ~:l~; + .1?23 ~)
1.>.1 /(Jt",1j) of 'PROB~~/Yf
/-10,,"04t!/l1c.ouS.
p~;;J O~ PEo13I..~/IIf
13£ )lO/f!OtS UV't! ocrs/
12 -:E. ::: [ lof '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 IJID 'PROP£.Rrlz. s :
/z./I1?:£RA7U..e£ / D:E,A..)SIT~
PR£S.5 U. REI =':>Pfl:.CII=IC
J.IEA0 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 <2.;r - ~''( <2.~
de
•
• •
....
:::: - ([.r
Q.E. D.

3. 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 eax - (x2 +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
+(5ine.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 =>- Vf =/-x r?s
f'sVs +,4 V; Vs -X P
2
= fJ/i.
1'><' +~ ( I-X)
/.1/ ¢ =3 )(21.J + 1/gJ.
a) 7<1 = (6X'g)x r (3x 2 t- 8!fJ;
'V r/; (3) S) =Clo X f {,? g
.... " . e ....~s = cos e (2..)( + 5111 CZ!j
••• 1 ¢ • ~s 15 IN THz.. - 60 0
DIRcECTIO/!/.
'1¢"~ =(!biY'q)e.A' -r(3X
2
+lj'JeJ
•[cos e cZx + sin.e a:~l
fiT THE PO/NT (3.1 5).-
V ~ " a.s =(90 4 f 6"7 ~ )
• (cas -(,O ~ + sin-60~)
= 1-15 - 5'2.02 = -/3.02
1.12 FOR A PERF2CT CSAS/
P:;O KT
M
FRo/Yl '?ROB. J./~
fY -= I'm ( I - X)
I - "om )(
A
.... --p -::: fJ", (/- X)
/- & >(
p~
RI"M

4. 1.13 1p:= A,. sinB (I - -Fi )
a) V'1p:: d 'I' ~ + ~ ~ ~9
()r r r ae
:. -vP =AOSil1e(1 -~) ar~ e
WHIC.H REQUIRES -n-tAT
*-1V7p/ = ?e Ivlf'/ = 0
~lvVI =0: - 511'12.e (I-t-~)
+C05~(I-~:) = 0 (I)
~ IlIP! =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[ Jide 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 IO~ ~~$ 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·~

5. CHAPTER 2
2./ V'P=PS j dP.... ,..
d~ e~ =-pc:, e.':j
~t...clP = -p<] C' cl':l
~~ = (JSn
@ STANDARD CONOlTIONS
f' = 1:>/RT
FROM ~.15 f1.IR=O.0"7651 I~~
FROM ~. 15 -Po..~ =211h.2 ~
fr2
h- (21/ 6.2 1J.F/f/.2.)(32.1'1i.f 10.... rt/'5:2/bf)
- (0=((,51 Ihn1/A-~';('32.114 fr/s':J.)
= 2 T/6~9 A:
2.2 FOR A PER FEeT q AS)
'"'P=pRT
f3=;;(-¥)T ='p'RT =="P
2.3 ~=-dV d'P
P V =7
V L"P '. - A ___ 3000?S1 _ l
V - f3 - 30"., oc)o-psi - 100
.: 90 VOLUME CHANGE = '0/0
2.l.f
MERCURI{
A •
If
R='Po. tP"" 5(:2"); -PI ='"P.z
"P2. = ~ t-PK 5 ( s") ; "?.s ="'P<{
P'i = 'PA +Pw9 (~~)i 'P,", :: '?s
Po. -rPM ~(l~")=~+/tI3(2",)+~9(S:)
Bl :'?a, -rpw g~~, '12'/- 2" -: tS'·sj
:.'"PA =--Po. T 5. ~ 17:5 == s:~ I?S~
2.5'" 'VP::: peS-a) I.e.."PR e:sSURE
G'RADIENT IS IN THE DI'KEcT/ON
OF (9 - a); I :5o'aARs ARE ..l-
~ - a.). TH E BALLOO"-l
CST'R IN6 WILL
,.. ASSUME THE (~-OJ
g t:>1'REGTlON.
.'. 'B,AUoO"-l WILL
-0. MOVE Fo~WA'K!)
2.6 EQUATING: -p 13EFO~E ANt)
D~N6t ACc.:t= Lt:RI"O~ i
-p =P5~ 0 :=. P( <j ;-0.) 'jQ,
l:ica. == ~ L1 < ~'3To. -.Jo 0
:. MANOM£TC~ LEVEL
60E5 1:::>c)(.4J N.
2.1 MAt£; A~~ ~3A
Of2TI-4!;,~JJ. A
Is IN 1n.2; 3J~ ~
04.7-3?K~ g~h/(#
h,. 144 .11.7 I tf4.7A
~~ ./'2.2. -= 2.2 ':: 2'.£c; 1ft.

6. 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
~.,gt d,j 5'H,.0'"Pa{d4~ +a~~9HJ
~-~::(1-!i1-~)~-(2):02A(2'Z 2-
~- Ps=Z44,7pgf :: 1.70psI
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
2JI F -= '? A - '?o..~A ==,oj11 -(fR2
e.G.
PH:JO = 1000 I<~/~l) h= 2W1) l<=.3~
F :: S~46 ~~/s~ =- 5"5"'-1 b N
- I1c.p. -= vt -t- ~
1r'R2
·2.tM
C"fr'Rq
FOR A Ct~LE J Tlob:= - "'I
• v1 - 2M +1rR.~
" "le.. p. - qrr'R2 (~~)
= 2.0tl ~

7. 2./2 ASSlJh1ING, ArlY1()SPHERIC AIR.
TO 13£HAV'£ I DEALL ~
&- -~Q --lldg - r.) - "RT
LET T = a +b~
tJlTH ~/VGN INFORMI1TION,
T.: S30 - 2l/ fJ
7;
dE. - -g dlj
-p - ~5......:30~--2-J.!tl.ly/~
fOP .dE - <3 h (' -2lJd (':J/h)
1; -p - 2i{R 10 5"30- 2~(Y/h)
1n. E =- :ili- /'1 5"'0,
? 2.4R $""30
-p = J O. " "PSI·a.
"Po ":: "30.11". H~
.'. h. = q I g 2 ft.
2.13
:r: -. --:'-- --_.-. -
1'lr ='Pa To I'H;1.(J Cj ('-/") +-f'~ 9 (10")
PJ: =1':zz:
~ -~ = -L) a (~,'QrL). 'J (10")
rH"0 ":J r H9
:: '1.63 psi
:. PoIN! A HAS THE HI9HER
-P1?~:S5 UR.!. .
6
AIR
- ---- -- -
D.} ~i=~ = 0 ON TANK
Ptrd2
._ "Pa.tltl. lTd~ _ 2!JO =0 (I)
-1l- T
@ W'ATER L£VEL INSIDE TANK,
"P = P ai", -to PH:1,O ,](h-,:{) (2.)
FRoM (t) AND (2)...
h- y = 1.27S" Ft. (3)
ASSUM£ ISOTHE.R.MAL COM-
PRESSION <:)1= AIR IN TH£
TAN-
rg~ VrANk =-p[ ~/,qJ
P = "3 -p~+no. eo:3 -<j
5 UBST. of ClI) I N (I) G /VcS
y= CJ.12f+.-
•.. h = I. '3 ~4 q..
b)
AIR
"':-". -- .-,", ..
t
i1 !='y = 0
p =P....t- + 2.S'O+-f:
7rd2./J{
@ WATER LEVEL INSIOE TAN}(~
P = 'Pa.fnt .,.. ~H.:20 9 (3-g)
F = I q6 (3 -'1) -25"0

8. "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 CONTINU£S TO
BE ?U5HED TO qREAT£R DE'P~
THE VOLUME OCcUPIED 'BY THE
AIR IJEcR£AS£S 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". -= L6P'S;9'/ilL! + ~ ] alA
~Mo =0
~ <j(dF"w-d7=A) = 0
(<<{ lj lf9 (h -4t-Cj) - gb~JdA ::0
4 (" CtJ9 hIJ -1)(3 'I'f +;;9!I :L-16¥~JiJ
:=0
h = IS: I Y ff:
b) FOR A TR.IANc; ULAR GATE.~
dA: (LJrr. -Cj )d'j
2.11
£if(4~_tjl) [G!(h-4+y)-~#Jld~
=0
h = 15: YQ"1 If.
STA8LE "posmoN
(M =0)
Tl??ED
'POsITIO~
eM)
M ~ C. '1f''3 Ll){ o.os-d A a
- 2 Sa
y
,- }(J. ~a dx L,.os
= fig L" C.e (0.045"" - /2.)
M = -0.0 31 (. :zo~ rad)~J L r
2.11
Q;RESSUR~ 1~
7
THE 130UYANT FORC£ CAN 13E
OBTA'N~D "BY INl"E~RATION
OVER ,HE CURVED SURI=AcE.I oR.
By THE FOUOWINq "R£ASONIN~:
~F~=O 0LJ =I'IOIJ <3'71R2

9. 2.1'1
F':J = BOUYANT FORcE ON suB-
MERGED LO<:i + Q, WHERE
Q= 'w'l:lqHT O~ 420 N SHADED
'REGioN. '1IR~)
Q = C"R:1. - T IfJ5
F~ = fJJ7r1?~ of" ('R2. - ~)1'9
=,.og~ (J +-4FJ . ~
,0 [7!'+ I-if] =,f~ 7r (SI/'JCE F~ ='vJ)
p~ =~ +J- = J. 06'1
~ JI 17'1/:2,0
a.) FORCE 'R£'Q'D TO l/F=T BLOC.K
FREE O~ BoTTOM;
Z F~ = 0 = F -Flo -F....
= F - (ow 3 22. ?S'+-Pa..+...'P'~s'J
- (3',. 3'x.S)A: J
F= (31c 3')( 1'",,9 22.;s:.' +R+.o. + .5";.g)gc, 10
= 32/1971bf
b) FORcE 'REQ'D TO MAINTAIN
FREE Pes/noN:
2 F~ = 0 = ~ -(3'x 3' )(,S'),4 9
-r
-(3Ix~')[o4 +'Po.nj , h::.S'
F:: (3'x3'X-,S"')(A ~••,,) K3'1C 3'JR.+....
:: (ll.';)(-Pyy:;. 6 I~/R:S) +f9R"){2.J'&:.. 2 /~
F =J9'{LfO 16;
2.21
Tb'
J-
h == <i./5ft.
AS'SllMmows :
CD~~~ ~rtr"~ &u.-
®HzO Us&!- A~ ~LLT(!)p
y 12:= #
j liP dF". llpdA
e 1-
01. ~ dA:%z:~ eSWledf}
llf-.:5'1(h-kbso<+~~)
dFlf ~dFa:>5e

10. 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<i IN S'tslZrfS5
1_ 2. (/[)
.Ii ~ S£ -+ fc ItI;.)rL
J 3 D~ L~ (t llf-nZI
~ d:34~
h ~ 0.28/0 rn.
2.22 .J"D 6%
~ =L)9 =tfge Tde. r 0
(A'P_d(A~) =(!,.q~cI~ z:
)0 e~'w,e l,:9-,8
-A~
e ~ ::: /-,49 n
~~
tj~ = -,13 in (1- ~ if"): 300}oook{i-.O'fq
:. J'P = ILl190 P$;
DENSITY RATIO ..e= e.~ff= UW11/
~
fJ=l.tJ~7~
9
2.23 "BouVANC V FORCE = I'v=;~ =- 'F
dF =:E. "PRoVID(:::.D VOLUME'
dT T 'REMAINS C.ONSTANT
BoUYANT FORCE VARIES
INVERS£L-V WITH TEMP£RATVR£
OF THE /t1R. AT CONSTA-NT
VOLUME.
2.2.4 5.~. =I. 0:25 @ If£" rn.
~= I.O:25"!2,gh..
= (I. 02~Y /000'3.. V9. 101m V /8£nf)
>>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)DlUe.t1R; VA~ tbwN~.

11. Po lsT~~~.
~{sT~~~.
)(ls~Dt~~
~E.~10T~~~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(u9~1ftl
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
FoRC.E / UN IT LEI{6rH r: f=
f=:: f(p - Pa..n..)a'A
~= C~ 9f 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

12. =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 =~<.H(t+~
ZFx- =0 I ,', F:z =~(3e !:L'J.
2
AREA =~(~ Ht~ 4r tJfr,=i( 0.1-1 +-5,'")
ZF~:.o
Fa =f'9c [2rH t~ - & - .E.c;:J
2 2. ;z.J
F. =f'Sc. [2 rH - ~J2 .
I ,
TO FIND L.OCATION at: F =Ptjc.H~
Z ::z I
"'EED ,0 KNOW' C.4 PosrnoN
AND THEN TAKE MOMENTs.
~7___ f'<3~"!fr~
A 2
~t-C{5'_3~ Tf'<3.. r.."
T, ~'~'I :2
r+9
2-
~
2! M) ::I'Jc. ~"t S"o..rol. + !] rl
A ~h.t T!j "
"-~ 1Ir 1+ 7(J. r lJ
Ij, -r J
NoW ZM" =0, SO
MAw -f'~.. ~(2 rt-tl~+f'Sc. (lrH - ~1~ f.2r)
-fXJ4!:1-'1. - A J
CANCEL to<Jc. :;z.
A H'" =z,-l.H + 71:a.r3 - bT(,-3 - 11('3
.::. Z ,:;I. 6
A =f +:. +-Jf:r3 - £7Trl - 1.1...r:!
rr "f tf1 "riO. 3 H"
'DISTANCE FROH BoTTOM == t +-A
if =~ t- ('[~(-R-) +(~y r~l- ~1T"- 1]J
q= *+r[~(R-) - 5,lH(JtY].

13. CHAPTER L/
1././ V =IO~ r7x l,
AT (2,2)~ {j- =10 ex rJil~j
A UNIT VEcTOR IN -304DIREcnON
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' [112o.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' (1J·n )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

14. =Al 'LJ + 'TrDv-.!:..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 =!flc.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 5r;;;- ,
= -~3.S' k 0.39 :;; 6() mt'".
/.S'I
:. ~:: /;/) rniH.-4.....----- c)
IF THE PLutD VOLUME IS CONSTANT;
dV/ - dVIde 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.

15. •~. d M +Jr d rn =0 ~ E 0di: 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 4MOVES 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" SR21T'" 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 vmJ.0
( 1- Yl J7Y? dYl
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
WlHORe = f 1.1,; (6d) -f>1I; (3d)
titHOR 12- =-;:rll; (.3d)
.
~ == 21'L b == -2"oL v, b =-v-
~t
) d... == 2 W,side = 2 ifnr., d~
THUs -2,aLv + 21'fobv d.!:J=0
Lv = fob -z.r(y) d~
a.) -u{'f) =!-AVERAGE' A CONSTANT
L1.r = 14:vE b
••• "VAVE
=L v
b
b) 1T(ej) = c. ':J + Cz <j2
TO DcTE"RHINE C; AND Col ) USE

16. 80UN DARY CONDITIONS:
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 LV =C-Vd~
v~~::: -!:L.::...::v:.--_ _
q Lb[t" -(f)jd~
LET 1 = ~b )
1).,.,"0.)( = --.:L::...::,1/-,--_
l/ b1'(t-yt'-JcJ~
V"..."'.... =.:2 Lv,,-"' 2 b
4.Ju,

17. 4.16~~r--- - -....:.--,
t J J
----t>- 2ern I J -
I ,Scm , 2-
I I
L_~ - - 4crfl....J
MAss~ ~:: MAaSnPw 0,"
tQ, 2JAzl12 ~?AiUJ
").
1~'16~ 1TQ6)2.1~~O-'~
~
Uj = B.l5" w.~
~ _____ J
~AsS~ lA ~ MmJtor,.,<lJr
2 ~ ~
~;(I'Yo~f~-l~3"!j25 ~
V,3=5.15 ~
CV.-; --1
t r ----; t
" .~ . f-.: t
::- - - - J O.8mrn.
Zern
USlN~ llwSEBVRlON (): ~
As W2t1T9Ilu~CM 4.10
~ U".5 ~ +rsclm. ,,0
dt: c~
IT d';" .. tDJ .j. 5'Qu:.tlc: ~Cs.
~ =-j'A.V=-ywV Vs
CDf +QL( -:. ('ltV
® QL
: 0 V=G>'tI : 1.91 CM&-
i; QL
~ 0." V: ~.'fw.:1.1 ~
A.ZZ
~MowQre Is ~
lr~~sltr ~.12
"''i.=jrd2..o{~(eo-<)dl-
o
V.e :: ~:,"" Q= '-'4Ii... I> (0
kr ~oh I l;1J,.=fV, =IZC~

18. 4.24 v
17
I
@BaTVsc:nt (1f(G,~
~~ j'21l'L bV~t-r
4: Vec,er ~ v l-~ b
~~Is~
lTtJar=4lT~(~-Ctr;
~f~~4.'2!2
~
n1.lA7r " f Ilfe2'1fLd"(
o
:: 5'!'iTL blf~
~
0°0 lf~: ~b.
4 b

19. CHJPrE'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(1.0IV;)"vz -flAv,: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~
=,Pf1(v;xfc:/x+2fv,·'dx'-f!r.'~
=f'~.~..t + 2V;2. -1/ V;=>J
:. ~ "'l1..".:l ( ~,~
2i I I SJ~CS FROM 5:t~ ~ =3 vI)
o -=-(1",)(.,! .,~,)(8·OO ~) =179. gVal
P,-'P-;a. =t"(J79.e #- ~f'@.o)~
= 189 p~f=J.31p£; =9.MJ<A:,.
J8
~~ ill
r" ~ I.. 25"j'p$
F~ ®~
Q =Al~ = Az. V2 = 3ff%
Va =~ = ZS fp.s
I=~ = Kc.s:.V~ p(v.n)dA
I=x =(pvA),V; cos (-.300) -(0vA),(-l/)
=I'Q~ C~)+pQV;.= I·g'''f~
F"'j =(DvAkv;. Sin(-30
e
) =:-SI'Q.15
F~ =271 Ib( ~ i='j =-72.71bf
IF BLADE MOVES TO 12~ AT rSf?:;
RELATIVE' TO THE BLADE:
V; = -40fps ) 1.& = 40 fps
AT THE LEAVIN6 S£:CTJON:
... ...lJ = v;. r~l. t-4 blo.cle +V bo..cl~
=(34.~x -~)+40ex
-, ... A
= -(~.~ eX - ZOe~
(0 I = 77.'6' fps
" .r - - - -'"1 ".5"~
I Cov. f -
~---0-J
5.5" Lx
~.,. fdm =0

20. ~= /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 <::OOROI NATES
Bx= (O)dM -m:l.STI=(157.. }~)
a:f :s
X-{:2.57/~) =-404N
-Bx .:: 1<>4N
Z~~=~ ((( ~M +a- Vy drn
~~v. -';) - JJc.s.
o
11"'1 OF r=WID l~ TAWf< = 0
B'j =-(rn )( -~OT)=(/57.1~r07~
= II(/N
~=-I/Il ~
""':'CO
r-----,--
i EM
3 :<DL ____ J@
Q= Atr= ,,£2 ,A,=O.2SOQ.:a.~
A~ :: 0. IS";po
~~JC =fb:,.yf'(i-·")J~~dV
o
&= Jo. A,v,:r. +-f':l~ ~~
=,oQ (Vi-V,J=I'Q (~ - ~)
=,oQ1. (-k. -:k,)
19
=1-66> Ibf
.: THE TENSION IN TIlE ROPE
= ~ = 215 lbj:
COs30-
~ zt=J(:: ffc.~. vx!'(v.n)dA
5.:
P,A, - fi A2 + I='x ==-1A.2 ~~-;: A,1I,.l
=;:; Q(Vi-1f.)
~(p. D12 -liD;)+K =1'~2(*~-k)
Fx =--1f(P'D,2-P"~~)+~f'Q:a.(k-~
SINCE ATM05PHE'RfC PRessURE
CANCELS.....
~ =p.~ -;: 50 posiC)
P2 = P;2.~ = 5" p-s1S
F - -171(so.lq~XI)-(5'·IL/I{)--L-l
x - 7lC .:z3.O#fJ
+;f(O.i)(J.94X 9 )(;23.0~-I)
:: -5"630 + 392
:. F"x ~ - G238 U,f
b FX ~ .[l::J
<D ®
Fa ::' 60 PSf''l :: ?~.'i psi~
D. =3" :: .:lS"'
Q =i()(J ~oJ/mit1 =. iCJ2 R-~s
P2, =1'1.7 psi~
O2 ::- t.sll
~ Fx =KG'S. VxpCvon)dA

21. 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")lb
f
-,;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'~SVltAjYi~
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;(<10)1
~.I"W
~ -p. =lilT psf == ~76~,
lOO ~ -"---::;~
~ --"',---'~}
';'i:2"'~~JsA :3~
q
F= k5.vccs6dm=(-z-oX-i.Jz
-::-FaRcE ou F'LUfO)ALso ow PUl~
20
5.12
(f) c.v. @
;;---~
F[:o: ::eJl
#////77///7/7177////////77//1/
FOR COORDINATES F"IXE"D TO THE CON·
TROL VOLUME;
ZFJ(&' fl'S~,«v.n)dA+~dV
o
ZF;c= Fx
(( VA',£'(v.n)M =PK1f-}.f) Ga~e]2A)}c.,s.
- P [ v--v;,]2. A
Fx =fJA(v:Vc )~(C05"e-I)
TI4'S IS THE FORCE ON THE C.v. AT
THE' WHEELS. FoRCE ON VA NE
DUE TO wATER J:LOV,-
RK=pA (1T-1JC):t(j-~os~)
POWER TRANSFeRRED TO VAllE j
P; RxV"e ::.pA1JC (v-lJ"J' ( 1-c.cs~)
LET m= 1rc/v-
p= "A11'ttt (lJ"-1J"nt)~ ( I-GOS~)
FCR "Pmo..,c.J 1:= 0
I-~Ht +3ttt· =0 :. m= I aR ~

22. ttl::: I as MIN1MUM
.~ FOR "P='P~J v-* =~ ~a.)
THE VANE I~ AlTACHeO To A
VH£El of RADI(IS ... ; NoTE TWA1
ALL MAss HITS CA'RT
M«j == I'Av-r[1Tc.. (I-'-DSe)vc::ose-~
::I'Av-r(/-caseX"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 +5d~ =0
M =IfX T/,'~
Jd~ ::: -~~ (UNIT CROS~ ~EGT10N)
flz i +//"i-/iV; =0
X=Vw, ~=-~
~(1T"",-~) =fl';-~ (I)
.
MCt'1ENTUM
~ FJ( =: ~ rif-cJM +rifd.-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;. -;01{lA,
REA'RRAWGINq
(P.-~)A, - ~ (A;l-A.) t- p (~-t)
-f'9A<1 y =- /Y A, v,-(,;;:-tr,)
P.-~ t- (rs - P:2. )(~ -I)r'S~yl.
-= jJ1r, (~-v,)
~ = H r4"P ) -v;. =V,i'A1f)
A2 = At +.AA _
-AP ....(p-~)AA -~4Yf -;>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
A1=' 0 TO? ",,:l
v; == 12 tvJ/s
R== 12i KPa.~
A2.,= .113<1m':&.
LIi= ?'I{V tHis
11=1l.{5" J<P~

23. 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.~.vdm
R= r vdm - ( vd~= Fao~f·E.
OUT )n~ FLUID
=2 fa3Jf'~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:" /.<t3'l "5/W3S./ f+3

24. ~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 VELOC.TV AT 12 RELATIVE
To TH E: IMPe:LLE"R = <51
~:' &CO~. ( ~)(.~6S"q fP
"- K11f ) bas - cy.1 I
= 10. ;J.1:l tp!>
THE ABSOLUTE" VcLoc.ITV (TAN6EuTJ
= Qr;- ~ =82.38-/0.~1;) =-1'J.17fps.
t~= 10.21'2.
'I-.-- • 8.2."38'
VR~IO.""''l.
TORGU E: = r~ 'tflS5. f' VR A
T~ .l . ?2./? •.!d- . Io.:u~ {2frsY-'J
12 32.1?II 14{C/
= 2()1/. 7' i+-Ib
f
1=bWS< :: WI = 45: J" np
23
to..l1o( =3E:§' =.?03" oc == 3So
;l<l.11
a) e. = 125"0
AXIAL. L04l) ~ F c }c:s.vdm
~:= J. 71 cfs v= <S2.M
V': (i·?fX I'lei) = 2 't?(. fps
~('&-I)
vb)
LOAD = (t.?lX6'IX2 I. 1') : TT J~
'3:1. J74{
,
I------··-=r
[ t ~--Lo=-
r 1.2in
I
---I .'---'"L..- _ _
I c.v.
- - - - y
~ M~ =M~ TCR'<UE ON ~PRJNI<LE~
I3Y SHAFT:
{{ I ; x~1 ~ p(v.y,JJA=2(pAv't-r;lJ)~."5.
M, = -2l'Av2
rfj
= -2.('2.~1)'Tf( ~f(C(OO)!s.
<jc.
=-1. ~ U:)f-ff
5:13 T=J(rxv)d.t:. =-"'R(lt'si~
1= Mf
-w~)'p2Al.fr
Mf =2AfJ lIr R (vr 'Sit1ol. - wR)

25. 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 IVsr _______ ~-v.
I I
L
2F =1k<iyp(vo
n)dA +;t}[2pdV
FaR caQR'DI NATE F=1X'G'£) TO cAR
I,." )(- DIRECTlotJ Z Fx = l;
r( V;r ,LXO.n)dA=pAj 11 (-'1>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 X0)
~ fcr v. IJdV=O
dl: )J)c.v. Cj r
:. F!j =,0Ac. 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"~ ::- fV'j dm
F:pv2. h sin ~
b) XMr :: )(0 X O)z dWi
F"~l =~ 1!f'Va -~ V";,vb
.:: pv2 hsiYloc.R= ~,Pv.Ji: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

26. 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'<j11
:z
L5
R= ? (h2 -L:a)-f'h~(1J[ -I)
529
Q,~j)()ny
V.h( =Vz h-z-
M.o~UJ-.1
~h.I~n.Z= ~Al)~
?"'Sohjz ,~ =5'0h,h
Wt,AlilC :z Jv, hi(V2 -V,)
~
~.-i~~" ~V.hJ (V2-V,)
~GNnw(rr V2. 2~h/h~
dn~ (ft>~:) =V;~h, (~,-h~
2
~1... ~I-Ya.
h.:th, h~- 2v,~=0
~~h2
~l = ~ (~-"("'-~f -02 gh,
~ ~U1i).)vrrY
Vz=1~'~~ {1~~V;~k,)

27. 5."3D r - - -,
USD..r:;T~ l~ ,; I
~L~j. ; f:~~1I t : I
M=s>Ah Vz - n:
m, '"gAV: -~AIi L_k' Y
~ + *dt<A, :.0 c.v.
. .
~ gA n--sAtt:rD
~(JJtff1 ls SAr{~USD
tv1~ LUrr14 tA$ +
Lf(f " ~ -t ~lJ~dM
~~~5 _Asas ~lQAL
-~~~+sA%ih~)fh2
~u:r Is 'kz -~
5.3l
USWG-rut;. c~v. Af!CM5 Wrrf4
II To TH-g ~(~Ha>w)
LM~ :: Ccrxu)t>e dM
j J ' l:
m01n~
V~ ~ := ZL19 +t.;~
,A: y~ +l:L
=O.442 W~"L
TAk:l1J<l ~ ~ 6
W~Tb>TW6~
-3'P2 A4T:3VSAV
- 3'4~l ~T = B.S
T :: 40.3 R-~bF
5,~ r - - - - - --,
2
v,----+- _+
L-
e.v-)J - - - - -- - -t
Hg
roo.TH& C.V. AfDIE
L~: ~~d~
~+~A,-~At%~(V2-VI)
fj~ V2 ~ ~Sf;{2~ Of
~J~~~JD
~.~~~
'9,-fz :Tf4Js~ k ~
o:=>&T~~~*
~ Is (Ll~ ~ As g~~)
Pc+Jwa(L4-t)=~tS'-t+~tz-
~-~ :z ~~(J~fJw)
=71.~~ ZPa.-

28. ~ 'J'
Q -:::.A,V, 'Z 'IT (.08)5 =0.02.51 mi4-
rh: gQ :z 2'5_15 tzrs
V2 :r V, ~~J'" (2.8 M,i
~ 2 m(V2-~)+~-~)A,
~ 2 19<O ~ 3«) "2 55coN
5.??J
SUSCE ~S~ 15 Cbv~
WLW = ~CUT=S<4Xt)LO st~
tno..q =eo] s/~
mOOT ::I :25'0)fJ-dY
o
=2J152fa-cps~'(}:IY
()
~2glJ2[Z-~]
~: lf2 = ~ 1!- :z 55.0 ~~
4'[-8
2..7

29. CHAPTER b
I--~
CD I I I.Z..,.= 22 -2
---, IT '
P,=:/S-'"~L -t-_J ~:: 175"KR..
0, =.25Itt D:z=alS'2 ~
WORK
,.- ~S(et!)d.;,+ ~ rrC..:;{dV
o J~S. ~~-
:. -1r= (et~J ~ -(erl), 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
<H: dot n
W d <- = (no - u) WI IN
~t
C", II =(Cp To -Cv ~Es.)0Av ),N
dt flV
~= (§; To -1R)CAv)'N
V V
To :::. ~N 1" V,./
2Cp
::: S30 ti'I()J~{J. 35ST:1') 'B.p..
(:)X. 21(){ 32.Jr'() f.l.lhf/'os"l{t(xl()~j
=- >31.01 ~R
~ = (1.1{ ·5'31.0{ -5"30)71(~tOlo)
~t ~ 10
::: 2"'.io/
= 61!J "E'5
6.1

30. 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
=IO·lq" = 0.029'7°F
'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~lf' 0 j
Ps =-aws/dt- t'ii tell' -,-LJa (!.jA-~ )
AV 2 Tf- J B
6.6
=(-2010 t 3 g5"0 +-3CJ 25" -I- q~7)
== G702 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?iS-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

31. -~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; (CvTz
+V.l
2
r ~ )
2 ,.0
AS T, =- 7;) p, ="P3
e=vT, l' ~rv,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.{~ -U
A
I' Z
PA -~ - 10','" + .liSP.
109 -:2 (3J. J =?'I)

32. ~-""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 - ':i1. ) ::. 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 = rvi! dtH ;:r ( p(O. Vi) V~dAc.s.
-F -Lv' +PA(A =-111 VzlA +~(o)
USE GAGE PRESSURE ~ R~5Um
FROM 6.10
r: := -t.J of- PA/A + mVz 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-)
t4 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 Gtves lis P=PCVEL.)
viE Do NOT KNOw THE VELO-
ern' VARIATION ALON6 THE' LID.
CD ®
6.12 Q=6~ AIR
~'S :lCLCOHOL
fJ=· ~J.I:tQ
"P.-~ =0.1 WI <l.lcohol :: n L/•'l~Pc.
A.="'U'(6)2 =.2'83..,.,~
L/
VI = ~/A, = 6",,'!./~/'j.'83 ",.,:1 =21.22 "]If;
f + ¥l.+tj rl =, -t lj2+ ~ ~2.

33. - -')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_ VJ. '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;20 ( I' fig0 '
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. =<j d - '118 -"Pa.+It1
Z f'"~.o
a) B£11J£EN STATION lA (AFTER
AIR IS INTRODUCED) E @I
'R ,,2 :l
~ +.!2 -t <j l,2 == ?A + V,A t SZ,.A
PM 2 PM2
~ = 'Pa.+m ~ V,2:: II,A I Z~ -2'A =htd
.1. '"RA - Po..i,"
CONSERVATION OF MASS ..
. . .mA,R +WI H~o == ~~I)(
~AIR +fH~o A11113 = PM1r A16.
tMH:z.o » ...nAIR (O£J.Js,T'{ 'RATOa:.' IO!)
.: ~= 2 V2 (Q.fIXI.K)('-P;r]J ~
- 5'. 9~ IHls
6J~
-h- 6' ·1
1(- ~"-!jf/=f{c5.(e t ~)f'(v.n)JA
31

34. 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<:l.('1o-:2B~=~CJ.i+2c~p~12.21£
[I<, fK,. ~y=[3D2.Z +.2~(S~=2l/.99 ~
.: r= -~o.;J.S(~.J? -:2¢,9'1)
:::: 109. S- S :: 1.125"',..,;,..
b) "'P. =B -FrS" = 136./3 ~
'F?z ~1'o -~~ =(1.21 ){24)1.=3'1'l.<fift,..
.2
~?=B -? ='3Ql,1I1- /36./3
= :J/~. 3S- Po..
AH = A"P = :2IJ."35P..
o.16~oj (O.1~jJOOO~/m3'Xq·f1~
I
6ft
APPLY CoNSERVATION Or: MASS
~ + d~ =0TO TANK: ~M f"t c.s.
M =Tlo2.hp
4
~M ='1l"O:1.f' dh (c1m= 'i/d:2..pvc
;it ~ at' )c.'i. LI
'7IDf dh +-r-r#dlp ~ =0
i.J at 7(
4h d
4
...I t-Uc-O~t- 0" -
APPLY' 'BERNOULLJ ('QVATION =
AJ BITtJ£EN SURFACE i 8I
~ - 'i1t'AC = - lie.
4_'3 H
,a :2
"'S.) "BETtJEEN '5VR~AcE' ~ C
Pc ='Ps =~T~} Vc.2 =2 <j L
.: 'Ps -i?s = -(L +11) :: -It! Ff
f'9
Vc :: ,I;2.qL = 25: 3~ f'ps

35. Q =Av =trd:2.Vc. = O.I3i! '(:t3/ 5
~
~ +..£ Vc: :0 0
dt: 0'"
Vc -: y2'jL - (ho -h) ho=I1@t=o
LET 11-h, =z 4h = d~
I dt dl:
dZ! d'l...,r
dl .,. D~ V2'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
AP =(- 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<J = VJ' 'S,' 11 30
.". ~. :5.'25'"15
Q = VJ' A. ::. 4.42.10""" m?s
HEAl> = 'j + ~."J.
.19
=.6+1.6'
A
Br;:T1JEEJ.J CD f ® :
=2.2.1 1ft
c
~~, + v;i r ]( = '3 <12 +- V;z'" t- 'PI
oP.2 .~ :z;p
Vi = Y;2'3 (tj,-~hJ = 35". q fps
Q= A;z V:z. =0.733 .f13/s a.)
VA :: Va=Vc. =Vt:> =A.l U - ~
A ~ -4'
="/.9rS -Fps b)
BETWeEN CD *A :
?A =-"B+TAt 1-,.0<3 (~I -<:1.1) -,0 Yd.2-
, ::z
= 2'1.12 pSI
'5LMILA~Lc,lJ
~ =11> = '2;:;.12 p'S~
'Pc: = 1£#, IS- P~'t
6.21

36. A =".193 fI'J. ~ v= ~ -= 6.5S fps
,+g: +':11 <j -= "P.1 t lfl"+~2 (),p- P ()
11.= -f:1.Q
- V?_ -b(3.l.1'~(/) - (US)).
I' J 2"- --r
=-t.1,.'L ~:
p.::-(62.'1 1~)(f.".2. AJ~) =: -2.i,7ps;~
~lf ,,~'" /
STATION : AT SURFACE O~ 1-1-,0
P. ::'P~", ) ), -= 0 ~ '1,:: 0
STATION : AT PUMP IN LET
':h = q' I ~ ='P....
~rAl +0 +0 = '('+ Et t 1I~:t -1-4'
~ ~ 2",
v:z.":Z.. _ BT.t -B- - "8:JLI,/U'/.7-.2'O.)
~ - p~ 62.'1
-~ = :2S:~
V:l.= ~().¥tpS ~ A='iT:Ja: :2'7.1 ~"2
l./
Q= All ="7.Fo ct5 c..I
Q:: (7.loX(0)( J;;r)= ~sol ;;W~
6.23 I=RoM DATA OP PRoBLEM S::ZOj
VElCXny LEAVJN6 IMPEUER
J 'lIr =I".~:z. fps
I
Vt; = 10.22 fp$ J w r: : 1,.:2. fps
.,.--
t10.2:2 .fps
L.._ _ _ _ _ _ _ '7:2 -rps
35
HEAD:. v~ -= S:21"O = Z2.S-f}-
.2~ 6'1.Q
~'P= 11'112 =5'279psf =3'.~i
~I{ THRUST - Q V I V - AnJs
hp -- Q Ah
• Til RUST -.. Q. {i;i; _ ~
.• hp Q AIt yAh
.: HIGH VOLUME', LoW PREcs.S()~E'
PUMP.
b.'2~ 1>= S'Ops,'3
D: Il I"
CD A=CZT'/., ~2.
V, =3.6'l,f,,!o
Q::. 1.9 cfs S.<i. - 0.'0
hL = 'B-~ + v, :2, _ Va.J.
P:J :lj
=(1'141)(45") ,'I -"1211/
+(61.'IJ.O. "S) 6 't. 4
= 130 - II :2 ':: J g ft.
6.2'
FROM 6.~
~
V 8 = ~cl- -P,~ - ~,.",
:z. f' tt~0
PIA - P,mM = tj(h+cl)
,oM
So "PIa -1?1A =P+I:a,o <3 d - f.~B:z.f'~0
-~ <j (h+d)

37. ACROSS SECTION ONEJ THE VEL-
OCITY CHANq£S BY A FACTOR
OF ABOUT 2. UNLESS THE:
MOM£NTUft1 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£~L£CT 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 q2 % REOlXTION.
/'.2'1
tM ='pAh
::f' 7:;r'l.(hfho)
~ =p'TIJt dh
d-t .~ dt
~t :: ~fi1 vC(At =/l![t.:J.";29h
~ :: -d 20 .bur = -d~ i ~Cjh
t)2- D~
Ch~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!EN A ~ E
} -r V.A
2
of- ~ 2!A = ~ f Ve;.2. + «j rE
2. 2 Pz. 2
-PE :"Pc: -I;' ~ L2 I ~ =V
~ = <:JL2 t "Pc _ Fa ~ L~ -rJ!." _ UA:l.
Ii Ii If 2"2

38. 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<1 FROM 6.2g'
~ -Pc; :: -(! Va:4 ,
':r
""p. - p =,.g U:l +9L2 (~-f.') -~ ~~A C:2 :. I :l 2
CONT/N VITC/: p. 43 ::l) ~ :: ~ VI 12 _
.'.e. AHEATER
AsrAo<
Vg':l,. : a:l)lJll. ~ LI. l. =JL:
- R:1. ;4 ~~
I
~-Pc. :: -A(~rz~l
:1( f?j 'R.1.
'R
"PA-"Pc ~ ~ }Ll. t~ L-'J.(~-f.) -IiVl
2 ~
ACROss f.fFATER (PRESSURE ORaP)
~ F« s rVx drn Vs-f--i-UA
-P1 1---0.
?s ~--' <j
(P-s-"PA)K=I'fi'Va (VA -Us)I
.. 'P~-?A =t!' JL(V -U ~)c: ~'lf_A)
2 'R R 1< It -;<:1. l Po
COMBINING WITH 'BER tJOULL, EQUA.
Ii "L~(I_ ~L) = -p,r8.)2~l. _ ..121~
1(.1. ~ 2"lp' 1<1 '2
'3 L:z. C~ -~ )f t1 V1
2"R2.
v:L(1. - ~~ of-.!. t; ~-t" ti -A )
"R 'R"'Pa 1'Ra. If 2 :zRl.
= <jL:a (p. ;l! )
v 2
~ 2tj L :z. (P./,p:z. -I)
~.3{)
1+ I-~~
"'R~
USINe; C. V. AtdOUJ-ll> ~ u,..s.
e+Py::: ~. At-~
"8tV !::: 9D(J J l)~::: LfoUr
So
~~~ d~IY'::; Uot.:rr t-~~
.: AU "" coAl"" dA~
CU--=C'f=4~DD S/~'IS
AT:79.<a(~ l~'" ~ ~
s: 4~r
AT~O.~5P~

39. 6>.'31
tJ~ur;a-£U~
TW*~C»A f
-OFT~ 'JJC1DM~ AICZ-,1 ~AP
T~~ ~TC46 V~~L.
~~ '<'lA5t-DS A~= M~
~ A'P:z 1?~ 9An. .
T~ ~CJJLL' ~~
We~ ~t5N J~ Iw>
TS(!; ~AP '(~
L:?, n 'L,
~~=. ~+l)~
s> Y 2-
TUO$
o:L
~:z AP ~ (y. .z7(,]w>Z - : i t Ie $
Y J>
~ /~ ~27~2.
T~ J7lL)W ~ l~ M=~Q)
Il-=- u: ·~2~ .~«r' ~rr
Q:: 76.7·Z4·"2. 1 49.4 ~'%'
[00
M., ~ <st.76 b~1s
9ol»~
-~. '" 1i+~~)~(e~~)J
38
".. "2-
h :: lJi-u;. + h-h£. - l 2.
23
lr:~JJ(XJU...( ~. is VAL-fP
hL.=D. Urgscg w~ Mu~
S~<n nL/6,Twcs <:;'(/E!;~
~ ~y Be; l.del1FG~
h2 ;, ~(Jt~B-l) 13:: 8~2.. ; -
;Jh,
~B~S IF'h~211,.~~
~{s
~ ~ ZUi (1.f./J-fB)
F
SOB<M(nm).J~ 1#T<2 ©
~'" 2 +~_L -(~
t'll -4 (0 26 2B
o~ ~LA>S ht.. :zoD JOe ~
A~ h1.>D FOe 5>8-

40. 0.33
GJerr(].,f~ NbPrF16.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/ (JeZ 9.32f;-f~
q 2 Al& ::: f;.CE6·ro
L
cfs
~.'34-
LSvg:; fu (;.() ~ FOe
AC. v. &J:I03Wc:fr~ ~p
-?JA= ~ ?-?~% Ail?
at g ~
=~ fp'lM:~tR
Q-z fr:) Z
3
Yrm: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

41. 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 Ua --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
<J Z%,
~% cfI/~f~)
U2..:Z l3~ ~
M, z Lce6lz~
~.3B II
~~L'-r ~-----.-<-- It-
(sT/.(lS~ z;>, ~
As I~l'iiG I ~
~ ~ '-~
~~d))
~,~ l7·S9 ~
&.. ~@ / -!2.%2~J
:0 ~zJ2~(!)z~t86~

42. C1-lAPTEQ 7
7. I ~'lS{~ ~? Aa;
~lN6D ~ ~£x. I .
T OF? ~e I A~ Z6, ) ~ &
.0177G::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<s mole CONTAINS 6.0;25" • /02J,
MOLECULES .? HAs A VOLUME
O~ 22.4 ml
6.025". IOU. MOLECULES
2;;.~ W3
=2.6Z·/0
2S
mol = 7.6"3'1023
mol
m 3
fP
- -/Zc - 7r RT =15'0'& fps
~ =? ''8' . /0;).3 • IS-O'S
21
= :2. 1'7 'IO~' mol
tP,~
7.5 7' =.,i/(d~.
dr )R
V; =1Tm~)(D -(~)J =2VAVEE-(~)J
d1,fx =-1./ "lJAvE Y'
dr ~
r; ::. -~."lA 1.TAVE = - 1920~
tf = -.~~3 i/:"/R'a @ tCo·p

43. 1.'7
"IV:::: 2,,"1/".2 ~ E¥' + ,.37
J<Z Raj
1,.('iv)= 2-;2v-2-..... f'+3(fj]
4. ('T'v1_- 0 J .r - I
d... J ~ - Y"3
APPLY ~/RST LA,! of THERMO
tfQ _ 6"LJ= (( (e.t~+~«r ~V
dt dt - Jk.?I'"'-U ;eJ~
~AU VISCOUS lJORK
~ = £lJe-
d-c dt
[!de :: (Tvl dA
dt J ~~~
IlTJ. b - ~ r'-'Irll1V cu"olo.~ - , .....
r v1OUtct'" bouJ'do¥"~ :: 0
T =","!bl = p. rw (LINEAR PROAL.£)
d':i t:. GAP
. ~O'~TANce
:. i~ ='rvA =0'~4'XrwX27rrh)
~ = s:S"?52 l!.:!- =5.S''lW
1.1 j = <fQ= kw2
dt
~=J<tJ,4
I
42.
<6 = k w} "7 =2 CcJ,
2-
~2 = 1/ <1,.
~ INCREk5E :. <t2. - ttl _100
<b,
= '300 ~a
19 ~ = 2. ~6 q3 -/06 -IMT
,252""
T:. J"15K tr = 3. 611 A
M =21 ..0."" = I.lq 42
NTRO<:iEN £Ah<. =cU.S'
~ =I.ql
.~ -O,a = 1.I<1J{2 (UIYEAR INTER?)
M =II. S1./75' ./()-6 Pc.·s
= 1I.5'5",P f>a·s
?JO ® L-I_ _ _>- 3.1 'o/S
CD I >- Lf ~s
CHOoSE: Cov. MQVlNq WITH SHIP I
~ F= ~ V d", STEADY r=101J
c:.s. -§
Ftk;d =~x -11" 0!.A.t5;2~'"
C" _ )oX
wrrn RESPECT TO Co'! MOVIN6 AT II!!!
~
Po" ;: 0
F~ = -Pix = -V;; m:z -(-.q H1/s)
Ir( lal ~/s)
F'fluid =qO ~'" =qO N
~
J=rl • I - -F_L'n~CI - ~IIP1
l='5hlp 1 = qON IN THE'
Wt;'GATlVE X DIRECTION

44. 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 __ ~
;?<j ---- "TZy = 0 = '<1r
AXIAL STRAIN RATE
=li"",;i lfJ((}(rAK)4t-u;,lX')..dT: =~
~K+O AX4t ~t
45.r~ a
43
VOLUN£ CI-IANlii E RATE
=Ii..,i!- AAKle+"t - A.1)t'~
4¥+ 0 AAX ..di:
At-+O
=lit1li-r '1r (>ff6X)AT: - I"r ()()At= dl4
AJ(~o A>f .1t ax
41: ..0
FOR 3-DM:
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

45. 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,. =ptr ;;e ar r IJ
?J5 H~1T 1 ~
1 t
i •'j/t ~h
f
LL1
--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.osX3:t ./O-3)/olS-J7f
l"lo""
·.S~2 (3./t)
= {C07a, fJ
"lib Fr. = W
FROM PREVIOUS PROBLEM
Fz ::;; v"'}:: 7TDh .I ALSo" Lv' =;79
E
:. jJ ,/:u7rD h = I')'t~i
e
1.T:: m9 £ =b8tJ·9.1/-;o-r
pif7rDh a5"OU1XIO~~.~
".. = o. 7f:>6 nils
dA: ....d¢'dL
dL =~
sin 01..

46. 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;<" 41"'= 21.7 rsf4L D
7.2. ,
~% 0·76 ·ID- 144'5
-,
jLl2():Z: D.~~..(0 .,
Xe~:a-5l%

47. CHAPTER g
t.1 -£ = 32"uv. Q
cJx 02
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.) ~
K041
Nat: (£) =rHH -A'f.2= LI'l~
L 1'2 K1)'1 J K[)'I
f..I1.P) =. t1N/a..,1 -.dl,?3 = L.2.3 mN
t L J·3 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 CONSlO£R THE CYLINDRICAL
SH£ll ELEMENT
~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
Jdv= -.1P (rdr - Sfd,..
:2L.,.u J A r
V = -.A!:.,...1. _e,k r +co2
I./)-' L ;U
"B.C. v=o @ r ="R J r. KR
C. & ~ R~(J-I<~)
4 L .t.... VK
~ = 4"PR:l _ 4 PR2.(I_~) k'R
~L. ~L AYK
:. V:: ~-Pl?:l.[i- rl._ (I-KaJ.tr1.]
'IAL R~ t.. '1K r
g.Lf i(rlrx) _rdP = 0
dr dx

48. 'Tf.)( =~ dVt :: ~ l: + c.
dr dx 2 ...
V.{:: -L. 4E r2 t S. .t.-cr + 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.TT£R
0= e,Ro-S&
M
Va =(JRu'Nf~ @ r= 'RuIHER
:. w'Ri= C,R; -..£
A
C =# CiRca c, = wI<:
R,-'Rc
~ =G)~ r _ w-"RiRo
Ri -"Ro I<' -'"Ro
=w"Ri"Ro (r _1::)1(0 -Rl 'Ro
= c...J~i ( I - r"fRo )
I-K~~
ALINEAR PROFILE .'
OR
8.6 1P.,; 207KR.·I:==::;j========@
- . 0.63501
<D
#:: 1/10 X/O-& ~; =. 0165"1<1 ~ms
JJ= 5"3.0 Ib"JR' ='l'l'i. <t5lJ ~/,"3
a.) INVISCID.; USE BERNOULLI EQN.
P. + V.Y+ ~ = B + ~~+. ~
P elf: '0/ I 7 T / <
P,-~ :: £11'
~ =y2pP J m=,aAv==w;o'"f2jJ.AP
Q= Av =1J]).'-Jj~P
1./ --r
b) VISCOUS" LAMlNARJ
-!!P ="32,.u it· -dP = ~p
dx 02. J dx L-
v: AP D2.
T 32...u

49. m= pAri =~7rO~ 41P 02. =1TD;..oilP
'I L 32r' lli Lp
=1J.2
GOVERNING EQUATION JS
!L (~)( )- JfP -= 0
dy dx
FOR N£WTONIAN FLUIDS IN
LAMINAR FLO~ -;;x-=fol d ~
dy
• V - -'- d? tj~ 1- C, u t C;z
•. x -:J"u d}( Z;"J
ae. @ INT£R~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

50. BERNOUlLf f='ROM S TO I
P5 + ~ + <3 i!s = P. + ~ + Q r
p 2 P 2 -J'
~ =9.di!' - !t
P 2
l-IAGEN-POlSEUILLE EQN. FROM 1-2
(NEGLE.CTlNG, MOM£NTUM)
Fa -~ = 32-" 11." =11
L D2 L
32 .Ltv, :L :: f (9AZ _ Va::1.)
0'1 L ; u
Q= 1rD~Li," ~ = </Q
q ) 7rD~
1) = 1iDl/ [~t.r - ~]
121 G. L 11" 2 DA#
?12
tv
P941X4!:1A~
APPLY MOMENTUM THEOREM TO
THE ILLUSTRATED (LfMENT
~J=x =0
It-J THE UMIT dT ......pg -= 0
d~
2
~= -pg
dy~ ,Lf.
a.) "BOUN DA1W CON DITIONS:
@ <1 = 0 , Vx = -V (I)
@ I:} = h , 1=0 .: ~:~Ih =0 (2)
49
v)( =~ +CJ. Y -,.0'3 y~
:2p
'B.C. (J) C, = -V
B.c. (2) C2 =: ~
A
b) v.. =-V-t ~~~'TJ(~J{~J]
c) Q= "'evA= -ihfv+ alr~
- ~ h'"<:I"l cl~
2~ h J
Q=Uh-P9h"! +~3
:2.u 6.,u
= vh _,.o:rh 3
:3).(
3./3 0 = Ve Cr) ~
~= ~Ve 4!' ~ +Ve(r)~
dt dr dt
FOR FLUID dr=o:. dO': VeCr)d~
at di dt
d~ =~ X' ~. w= Ve ~2
dt } r
~ = ~(-~)
de r
:. d I I --U :2. ,..
- - e fl..,.
dt FLUID -;:-

51. 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"
)(
(PX -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<j ~x L - T
NoW '1=..udv 50 Ad2v= Ll"P
d~ d~:1 L
SOLUTION IS V=C, +S'i +A?y:2
2pL
50
BOUNDARY CONDITIONS;
@ '1 =0 V=R52 .0. c, ="R~
@ y =h V= 0 :. C2=R2-APh
h 2)lL
THUS
V=1?.l2(I-t)+ ~~"rr~r-*l
FLOW RATE
Q=i"'vd~= h (I ld~)= l<n.h_ A"Ph
3
o ..b 2 12p.L
HENCE AP= J~f L [~h - ~
EFFICJENCY: (il~)
'1.= "POWJ:R OUT = I' 62 P)
-POWER'N llR(L (-10))
To IS SHEAR STRESS ON FLUID
AT THE INNER WALL,
-r;; 'S SHEAR STRESS ON
THE INIJER WALL.
70 =.",a &1 =~~1l_ API,
- dy Ilj=o 11 2L
THUSJ 12~(R.o.h _ Q
~= G. 113 2 1
.a~r;u1?.Q. +!1.!!'~ Ql
L h Zt~z-1
~ = 12~ (1?~h _Q)
Sl.RjJ [¥+ tr~h -Q)]
"(= EE.. ~~h -Q)
"R.Q.n [L.JRSlh - 6G.]

52. SHOWN BELOW' ARE THE' VELO-
CITY PROFILES FOR 3 CASES;
~ =-1 IS MAX. Ei=1=IClENCY
'RSlh 3
~ = 1 IS MAX. FLoW' @ ZERo
"'R.Slh 2 AI=>
I·
.6
.tt
.2
<1
h
Q=O
ruu, .
-:(1 -.'f ~2 0 .2 .1 .6 .Y 1.0
V
-RQ
8.10
~QJTk"6~ ls-
(1~
~== 2J (V--P~PJ6
t::)
A}.JA~~ Of!" ~ta.c U 1m=;
~ A~~Y~mt;;
?~~ 51Ax::l5TI-tG IJs:r
5

53. B,17 1 V
~ ~~ -- --~~I- l'A'S S'~ Tul: A1 1-rl' r'rlrt..
" L_ _. 1_ - _ L _I
!VET ~lC4L . I I
t { j
~'T~ f1..vx.- t ; Ar l
I"S~ SO Lf::tD
i:
W'+- 27f f( fA.t- -1'cr"r'l{ Lxi r.O
rt.(r r
~~
II 2. S(j 111( Are
4Fra--rA~w::tlKG LMtTAs Ar~o
J<A r --t d (rT):O
U dr
to ry ~ dU;lcir
(. "" /'..-.. :. ~y'G i !1'""'~~o.lJL.Lb ~! AL,J ,j~-,~_ VV'~~~.
'jCJ, t 1.~ £L r%. "2 a~~
U'2, / dr-
AT ,-"z Q{ h 40;% ~
/ dr v
t.4US
pra::z ~ ~e~n):'r)
1~004~ ~w Wrr'i q(Q)~
~v:zr7
Ufo % 2t(c~1h).~+~O-4)
52
8.B
k- % QA-VL.-/ ~ z LTIM)Z
1.
tr z ,,(e{~ik6+~}1-e(~,~)
~ ~ 21v ~z ~

54. '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
:. (fi·V):: 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<tAXJ
y) -Vx(X,'iU&
3'2
1
=4<:!+[V'1(X+U)y-tAY)
- ~( X+AX J y)1 At
Q2x 3:2.) = AX ~y
0':2') (3~ ;).') =JlXJl~ +[V~ (X +t.~
~i-A'1)- V~(x+lI.)(J~l~x~t
i-[V1(XtAX,y) -Vx (X,!j)J~~~t
-1-[ ]At"
TIME RATE or: CHANGE OF VOLUME
AT A -POINT
=lim ~ V
t~}+o AX'A~ .1l1t
~t
:. t=1.UIO VOL. CHANGE =dV~ +dV",
d 'j G;)X
='I-v
"BUT v·0=0 FROM CO)ffIt-iUITY.

55. 42= ~o + d r dO + de ;;0
dt ~ di dr" dt de
~ = ~V~ t 1" aVe p- + v. aedr d r ,- ~ r -e r W
-T Ve~ie
dr
~ -= ~ ~ +aj~ +Vt"det"+ Ye~
dB ae I'" ae ~ ae
~'-' ,..
~-:= :;}er ~ = ee~ = 0
dr ~e ar dr
(}e,. = - exslY'e t ~ case =es()6
SIM'LA~L'" 9€e, =0, dee =-er, d(' ';)8
HENCEI
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 SlWALL 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
ifT Vx~~T ~~ ='3x -;~
() 0 ()
:2 :;}~v.
'V Vx =~ = 1 2P
~'1~ ,.u 'dX
~=J..~Cj+-Cdlj j.J. ~ I
+v'V':Z~
~ =-L Q)P ':12
+e, <j t C:2
2)-' ~X
B.C. @ C:j =:tL ~ Yx ,0
~=O C2 =-..L ~ L'J.
~"" ~J(
:. Vx =..L BP (u';l_L;2)
;:?~ ~ .J

56. 'V. V= 1.£.(w"R
2
)=W'R:ld (,) - 0
r Oler r- --y:- de r -
:. CONTINUITY IS SATISF/E!:>.
'i.j Vp=:¥! +~£t' = - V~
Dt ~ d<j ~~
o
= -V;(R, e9j~)= Pc ve-~~
~ p
AT ~= 100)000 tt-) V= 20) 000 fps:
I2£ =~~ 000 s+-/s Po e.-1I.91S"
DC- 22., 000 ff: a
= ~ (O.OI06)~ = a ()o96 Po
s
55
+'i]-()J~:)t 'J.~ VVx )
,.o[;~)( +~ dVt! + V'j dV", + VzdVi?l
~x ay ~i'J
=P9x
- ~p - d I2 IJ/~ +;~+-~~
ax cfll3'=;lX' dCj 9r')J
+.£...(,ud~) of- C1 (.u a~)+~ 1Md~)
ax ~)( ay (}X ~l ~ W
i-!x(M~)t~(Md~)+;0~)
tJOTF: JHElJ 'V-v=0 } A 15
CONSTAt-lT "TERMS '
~
~ (; ).{V- v) f '7. (M¥X)
AR£ 0 AND TERM ". (~'il Vx )
'BEcoMES ).).'0'2 V.x.
'1./2 GIVEN: f ~(rVr) +~ ~: =0
0.) f~ Ve=OJ ~(('v,..) = 0
.: r"'Vr (e) =Fee) J Vr =F{e)
r
b) IF Vr =OJ ~Ve.:: 0
-;;8
Ve =fCr)

57. 9./3 FoR THE INCOMPREsSIBLE
LAMINAR CASE)
OV ".. rli:) J 2-
Dt"=g-7+ V 11
FOR 9 NEtSU6~LE')
"D V== -yp ... Jv:2vDt I'
VECTOR "'PRoPERTIES DETE'R-
f.lUAIm .'BY V EVP wHIC.HI ~
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 oRI
TEN'D To MoVE IN THE
DtR~CTION 01= D~CRf'AS/N<S
'PRE'SSVRE • c)
9.1l/ FoR I-DIM STEADY FLOW;
Vx = VX' (X") V'1 == Vz = 0
NEGLECTINg 9)
pVx ~ =-pof" 4. [1.lp~tf"J.{itx
dx (jX <.it [~ 0.)( J oJ(
9J5" COI>JT/NUITV: ~ t :x(p'lx) =0
MOMENTUM: p(~ +V)C ~Va~: :tf
l'cu ax) ax
9.1"
Usu:4TwG; .c ~(()kJ A~
~rTtVG 'O:wAl ~ L1?~20
AND ~:: t<r) EC( E -~
y.!WS
z direction
(a"- .~ v. ~ ai)PWc+ v7ar +~+v,-r;
~ [1a (av,) 1¥or2
, ~2'J=- +pg,+/J- -- r- +, +
z r ar ar r 0
9..17 A5$OM,W~ IN ~t&ea
~WJ(bm(}.J()rrY ~ Yt6t.PS
rU"C :::.~~ l.J~GLS ~2~=O
~ E-4 TI!5U'S
r direction
rt' av, v.%t' i ¥Z)' 0 ~~p +v,-+- - +v,
t ar r 0 r z iap il 1 il 1 a , 2 • i,=--+P8r+/J-[-(--( ,))+,a7{_~~+ij;,]ar ar r ar r Iii? ryao Pz

58. Tw~&~
~ (P~3lYi)~Jqr
or 2- 0
~IB 2
Sf.t1(.oTw @ % -lfe e,., ~
dt r
tre :zfCr) ~ trrz~ -z0.
U-sUKt ~-6"~/~ T~
TJ.tt5~ .-sTt«5 Lgr
W~~(Dg ~Tf4tS E"nli£r-«»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~<.we: ~y t1..ow"
~~~~J
~[.L d. (r LJ~)l -:.D
dr r dr 'J
~ .LQ...(rtJe}=:~
ra-r
~~
'tJe ~ ~I fur ..... Cz-
Ar Q. LYe ~ 2.12.-.1
/tr l4 L1~ I4..a~
lJe".!fe~n,t(Q:Q2Q~OJ~~.
r(' )..J?~~,

59. 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
Jr'Ai9
IN THE' LIMIT,' TAN ~=Z
G.;Z =li~ .J{rVgl r+Ar - r~rr)
~}....~ 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<)~ +V'o
Ja'i V·O=1~(rvr)r..L~V9 =0
r ar (' ;;6
LET r Vr = ~lP ( r'~ 8)
()8
V'.O = J..r~ (tl) + 9Vsl.= 0
r Lar 'as O'6J
~ (9'1' tv.J-0 . I = -.g]J!- - 9 - •• viii
~ ~y- -ar
:. Vr =~ ~: } Ve = -!f
:. Q. E. D.
la~ rb 5 3 5 2-
1 -:: -- X. - xY
B....
S~ lJ":z V0/J {bsnJ.J<XTY
&l (s v..Gzo DJZ Vf-:'O
Ust~ ~
2'1 + P.l ~O
~' ayJ/
0)6 - '0X ::30{)
lL 2 ~ := Cl!P. :z 5~Z
x ~ ar

60. /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 - PCR) -= IZ u..,.2-
2
THU5.1 "'Pot::) -1(0J -=,tJ Jw?·
So
:. U
2
=~fP =- ~ VtM=126f~
,.0 .002<f
a.) MAX. WIN]) YfLOOTV =126 t?S
b) OSIN6 ""BERNOULLI
1=>00 -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. = P1'".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" V2
=Vo02.
HENCEJ Voa I -= vel:: 2 b,si)19
sl~e =.5 .". e%!30; ! ISO
O
10./0 a) <p =V... L [(::f-s:tJ
o=V rp =VX' €)( -r V~ e.~
VlI == ~ -= :3 U ( 2_u2)=a;;)( ---e >f -' ;)u
L-~ J
V'j -= BaS_ -6 Lbo )(9 -:: - ~cp
~~ - L-:t ~x
'IJ-= g ~ ( ,.,2~ _~) 1- f(X)
L2 :3
'"= 3 VoC) X.l. 5 t- '1(~)
La

61. WHEN '-P=O j '1=0 OR <j =!".J3x
----l-----":~~x
4/::0
b) ¢= u,,¥!
Vx =- t} ¢ = u., 'j -= ~
ax T ~g
~= ~ =~X' = -~
afj 1: ax
¥= ~ u 2 +f(:xl ' 'P= -.lx, )(.1 +~(u)
2L :l I) - 21. .J
WHEN 'V=O j '1 = :t X
~ '1'=0
------~+H~----X
c) ¢; = Vca L 1M ( )(2 +~~)
2
Vx = VaiL 2x _ d lJl
2"" )(2.+'j2. - Olg
VCj = VooL ~ --21
2 X~2. - - ~x
'I'= ~ ~o.."-f(.!i)+f(x)
2. x x
IV =- ~L ~~ to..,-'(~) T <3(<j)
:. 'IJ= Vd)L[tcl~tl(t)-io.n-I(~~
W~£N ~=O .)
x
10.11 Cf= 2r35ln~ J ~ e:J~ '# 0
~:D(Se~fc
3
Vf(:)TUQ) LQ 'f~~
(6r9A~04~ +~~) ~
~ ,2:r~ 1~ ASQ-ULtS
2s~~
8/~ r ;'PLar
'('
lO.t2- t-0 -= lftD
r$ln9 + QS
2i
6'(OeFlAJlTr61J r~OJ ~ If'! ~>o
f:ao Is-W& lwe. 8=6 .e:Tltl;
~v,;; ( Ax,$'. ~ li_~O
(f"~ ~ G:;irrTo L~) I OIJl;
GB"~ Yzee ~ 2~ Q.
St,0 '( Zilttl
r~2IilMit(.a
e-+o ~w.e)

62. 10./3 ~VRCE AT ORI61N !.p= Wte
p2Tr
m== SOURC.£ STREN6TH
FREE sTR EAM tp =Vc:o y
TOGETHfR tV= Va) g t- ~e
211"1'
Vr =! ~tp =VcO C05e +..ttL
,.. ae :.27Y'r
~~ r'sine
Or' =0 @ $=11
AT a=71 r= ~ _ Q.
zrrPl.4x,- 2'ii~
lall.{ As ~iP = pDv
Dt
=pBftV(~2)-o. (V'xv~
/WD FLOW 15 STEADY AND lR'RoTA-
TJONAL I Vp = -Pl (~)
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 ." mosiYede1~lae . t'ov ~10'.e) '"'
(ir
F~= Jo (1=>~-BYR'5IYed8
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.

63. 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(-.:tJ0)'"-I(, e~
4tT<l
A~ If(a,O): -1 e47f(l. tf
s;'~ 'f -= +~1r
VO#!J'Q(. ~ -t:t. 2.'iT
G2.
f
h.
-.1~___~--L
I ~5~ )6
Sr.tq.u..trl~'G/Xj
<1- lfa>rsut6 + ~
2'rr
0. _g..~AnClJ -g,AJT
O=-~D ¥1~s (Jr: ~2D
lJ: =..Lo~ _l.(~ + I.[r~
r r W r 2IT ~ Y
0: ~ - LJtp := - If. ~v.tb
e:> 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<9 +-~
211
WH~ e:1t~ rSUA8=y:z..
i?.J~ -~)~ $.0.007"1

64. C. A.7 Lt~s?~6. ~ ALLTf«; 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'P5Ilt8d9 =0
o
1/:~TM +~J(~:-U1-~ LJ:-2'tsm,.g
«
Iff"'~D ~M~~')p.jlf2Ijfs2eJ9
I~T ~ (~. -~A1t''D+.2. O{y"z.. V
.,J' ,j (, J ,LV
~Oza 1.257~N
T ~ 10. IzN

65. CHAPTER"
fI.r 1) (1..) cv ( 'Ii;)
11.2
p (MIL') Q (L~/tJ
H ( L) n
9 (L/t2) "P (MLYt3)
i =rt- r =- Z- 3 =S"
CORE ($ROUP (p I "OJ w)
1l;=t1 (8Y INSPECT/ON)
11; =fQ, DbCJc H
.
V
D
P
(LIt)
(L)
(MIL')
t= 5"-'3: 2
~=~
CORE GROtJP (D, V, p)
n; =Dc4. Vb fC,)J.; 71i=Ai _ 1-
r;rvjJ - 1Rc
7T;.=DQ, VbpC: e ; rrr; =L
I>
11.'3 ~'P (M/Li-a) D (L)
P (M IL3) Q (L3/-t)
w (I It.) )A (MILt)
t: 6-3=3
CO"RE 6 Ra./P (P.I 'OJ w)
r;r; = pa.. DbWC.~""p ; 11:= A'P
I fJDV
112-= p q Db w G
Q j ~=~
'O'3w
17; =f' a.. Db W c,,.a ; 77;= ~
PDaW
I/.¥
T
tC4
tjI L
~
BV GEOMETRIC SIMILAR/TV:
d =.J2. v= J.3 V
.I L
7rcl2J =..L 'Tl'D'2.L 1)~_ 31 _ 3d
i/ 3 '4 d2 - L - 13"
:. ~:: (3)~ :: J.I./l/2'" a.)
BY 'DIME:.N SIONAL ANALYSIs:
~ ='DQ.wb pC 'P
I= La. (Vc)b (MIL~t MJ:~
-c3
'-t=-5'" b:. -3 c=-IJ
:. 7(, = _:Po.--_
,ow3 D5"
FaR DYNAMIC SIMILA'RITY:
'"P ( --P Ip4.)3"C~ model - ,aW3
D6 proh1:'jpt
~=[~ ·?f·jff.r3 I 3-$'~
=(3.3-~/"3j3 = 3-2/q
:. l.)p:. O. 'T13........t - - - - - 1:,)

66. 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 MILARITY ;
~'" =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 - fAIpU2 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<
M O o I 0 I
L 2 0 0 0 I -3 ,
t O O 0 0 0 -2 0
.: ~ = '3 -.....- - - - - - b )
l= n-r' = Z-3=S
.: No. OF" DIMENSIONLESS
G'Roups ::- 5' -~.-------
'iT; -= 0<, 112 =/3
'IT3 =M0. LJ., ~ c C¥MX
I =M~ Lb (LlP')c ML)lt ~
a. =-1 J 10= -/ I C = -/
1TS = C I'Mtl.X
ML~
1lq = ~Q.Lb~cf ;
71S = twf' Lb~c 'R ~
11~ =1::1-
175 ="R-c.)
L
11.'1 IRe = L V J, = I '2..&, ~L .10-5' ,..,4-1) "llIa.,.... . oJ 70 5
@ 2iOK (~'1.6°F)
a.)~ASE'D O~ LEN~TH
~ = (r:s.'1X22.2)(/OS)=9.21_'cP
J.'3~76
b) "BASED OW ANTE NNA DIAM.
Re. =6.1/ ./0-3(2.2.2 )(102-
1.3¥?6
= I~ ;16" (1.97./0'1)
/1.1 JI. =COIJSTANT
~L
'/::a. U 4
.."., - p
---'-m Lp

67. (~) =~; =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
T£MERATUR£ NOT GIVEN,!
ASSUME 'H2
o = 10°C
JH,.o = I. 3x/o-6 W1~
'MR =2t;OC JAlR ~ ,.":>(105' ~
=2.'1'l>C10-6 wsl
~
LV I -LvI .u. -,I J LJ - IT .. ,..- vF.s. ~ F.s.
~ h~ .J LVF,s. m
Um= 1l,·2.4Cf·/o-6 .q :: 122.3~
/.3 ·10-" 5
F"l. = .0262
hF.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 Voo2
V'.,. 0:/1
La
DO~_ 9L v¥p* 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.
cl2
a
<"i73 = olAa.Vp~. rrr;=~ =_,J f'd~o. my
I'" (K.. I J;L )lReJ) =0 .. a.),. da. d 4
o.
VAR II 0. AND/oR d
Tl-IEij ~oR FIXED VALUES oT: ~
'RoT I:>Af;1Q. vs. ~a... b)

68. IU2 SYMBoL PIMEW510N
FLOW RATE" Q.
DIAMETER 'P
5HA~SPEE.D N
VISCOSITV A
5U<F. TENSION 0-
DENSITV P
o 0
I 0
o -/
L?/t
L
I-t
MILt:
M/t2
MIL:!.
r =3, r'l =b J L= b - 3 =3
CORE" GRo()~ -= P N1)
G
11, = pa ~10DC Q ----=-ir, ="ft:>:'
'112.= p~NbbC,L{ _11'2= pND~
"M
113 =p~NbDcO"" .-113 =pffiD3
11./3 tt L -t
M-rn M 0 0
'D- L D 0 I 0
P - m/L3 p , -3 0
9 - l/t2 9 0 -2
(7- fVlt:1 r:r 0 -2
BV INSPECTION
67
Jl.1L( M L t
V 0 0 -l
L 0 a
1) 0 0
P -3 0
T I -2
11"; =L/D ~ =~"J.pD2,
T
:. nD'W =t ( LID)
O'R V L Yf -= f (LID)
II.IS" ~YM. PIM.
POWER P M L2/t3
DIAMETEJ< 1> L
RPM w It:
VOLUME Q L~/t
DENSTY P M/L3
I'ISc.oslTY M MILt
r=3 Y::.6 L=6-'3=3
" J
CORE (fRDt)? : "'P"DP
~ =-P"''D1o pc. W
~=-pa.DIIapc Q
~.1:...
II. =pD~4i
'IT = ~
2 P <;(3
tLl6 FOR DYNAMIC SI MlLA'RIT'I,I)
~ ~ :.lRe. Fa L...L ScALE
:. tFlM == UF.S. LF.s. JM
.- r Jt=:'5 •
VM =60~(~)F.10~ =2LfO r)'ph
2. 10-5

69. /I. 11 AS5UMIN:S /NVISCID EQUATiONS)
~ DO = -tiP +PS
Dt
MAKIN6 EQUATION DIMENSION-
LEss: V := +(~ )t ~ I ~)
t£o L L ~2.
OR JL = f(>< t~)
'f;L L"J L
~)5IZE" = 2 "" =.005~6 ~
"360
VELOClTV 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' - J1. <i
tltt:: ..!1:. 1,,. = 3~ Hti..,.
If.t:t
II.JZ IRe. mode.l -==~ 'Pl'oto~f:e.
I'M =ArM. vp L? A
PP VIM L.W ?
OR '"Pw -M =1)401 )1"", vp Lp ~
Tp ,up v-,.., Lm
11./9 I=R =~~) r = &.
mode.! ~(( SCAle
V :2.sr""/s
L O.lfl n !2. '15" ~
N 45'0 rpwt
v"" =~ lM =O.qc;q V 671~'V L I J::S. =.~ ''5
F.S. F.e;.
b) TH-~05T: EM =E~.s.
~~a ) Fr:s. =F)I p v~ F'.~ Af'S
pv:Lto PrM
FF.S. :: 2'lS (,. ql./)(-..-L )2(:2.ct5'12N
j.q'l .'10<} ~- .~I j
FF.s. =SZJ '300 N
ToRQUE:
Q --= FL :. QF:s. = QM(F"•.v LF.~
FMA LM
QF.S. % 20 (5'~30D) (-:2.({~)
:ailS . £/1 )
~ 25',5"/1 Nm

70. 1.20
IN he$r~NT(,~/~ Is W~
g W=-7f+)AV?ll~ -I)
f.!1t$(~
l£~ o-~ ~ u-/~;t~ v;,-l/L
Ou;~AtJC r "-
- ilt:
QY.E.. ~~z~~+31~~-l)
J L/I/ n.+-" «.~(5c::Z)W' C 7D
o!.-Il·.. l~
T~ GrO D="THE ~ vrry
~}./To mt. hpzn~ -rt~t ls
I.Z I
E 0'tcfr2-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 "8r4
12_ ~ /:4.i"2 5 It
~ ~H.&) L. z ('?/~
m~ d:4:~t~ A- Yr~/z,~~---~-",-~~-~
69
1.2'2. a (/vA:) )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 dJ Oy'1)
YD ) .J-i~~
/
X> 3~ ~~ il-g W~$
D) ~) Vf() ~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~lJST~ ~LD £Sf; 4~~
VIA {~(D(;; WE UA~o 0/. 2i ~
') /0 . _ A . .-7 L e6JLt ~r j J;.. ~N h:X7C~
~ Q6.111;, , i.MrGrow
~TAl.V~Q - ~Qh

71. CHAPTER 12
122 "DRAG = ~ pv2AR C]:) SO
Df =.!.pV2~C-r
2 ~L>-t
=~(.0023"Y1!rfa)(293.33/"(itof
21100 (.011)(.75') Ibf
=202b~)(~O~p~} Ibf
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 OFAGOLFBALL
1ReC.RT1CAl. == 2·lOS"
AlR @ 20°C 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
Vfp~ 1<e CD Dlbf
So
?5'
100
12S
1,0
115'
200
225"
2S()
2"15
3CO
325
350
lfOO
J.{3~ ?l5"0 .1.11 .021
611, ?'19 .If? .
Xb,505 .lft .0'03
108, J3J .11"1 .130
J 29,759 .lIb . ,'Z3
1'T30/0()I) •liS' . '311
21"/262- .'10 .li~2
25'9, ")15 ;"3 . lin
,-ZI/I'I2 .2 ·373
302" ':;{/4 . I .21&
31f6, 02.1 .08 .220
100 200 300 '100
.J -h:>s
12S" ~ 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.

72. v~ =~ (Jxl.OO)~('It'-of)'1.=5
OR ~ =2~ (IO-8.2?92)= 0.1"
~ v;Re;c
@T= loo"F JAiR =O.ll·163
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 + <:3 y~ +C"f ~3
Va-
"BOUNDARY CONDlnoNS:
(I) V)( f)} =0 C, =0
(2) V't( (F) =VxF
(3) av~ (d) =0
;/':1
(q) VA' ~~ T ~ dVx == -JP +..M~
dX" ~ c::tx 9lj:l
@ '1=0 .. v)( =V~=O
:. ;)2Vx/ =..L 4P = - I PVooa'4
~ ':S.2 'j=O ~ d x ;a cJ.x
... - ,FROM 13ERNOULLI EQVAnONV
:. ~ 2 VXI = -/00 cJ. Va1
d~.2 ~ =0 ~ d.X.
~$ = C2 (i) t C3 (iY-t C~{Jt
FROM (2)
FROM (3 )
}=RoM (~)
1= C2.+ C !>+C,{
0= Cz +2C3 +3C'i
-F2
dvao = 2C3
J ~X
_-. Vx =- ~..1 _1 (~)3
V1(S 2 $ ;a E'
+£2~(~ _2(Y)!f!!~)
4J dx lJ 0 It}/
/2.10 Ix = a. sin bg
Vx=o @'jzO
1J
Vx = Ve:%) @ ~= 6" v~= a.S;r1bE
~VX =0 @ f1=d
~~
o =Co-sbd:. bF='% :. a = '.100

73. .: 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/PVaJ
2
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
~ (" EPv)(~&~ == ~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)~pV" d~
= 41' Voo
2
X cI rax"SiYl7i~ d~
0..2 dx Jo :::zE
::: ~p Vco-:J. >< i.. (d"x)
'110..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$)( =11A VQ:l'X T ~,I)Val"Xo
a.E ?~
- YJllof-$X + d clr~p vco~)C,...!PV~ ~
'1ic..::1 -ax[ a.~ ~o..:i.J
IN LIMIT AS )(-+0; dS~ 0
a)(

74. 12.12
11771/11111/11//777/7/7
;( X'+dX
XF~(( VXpCO·n1tAt~i(<<~~dV
Jlc.s. oti).4C.v."""
g~=PJ"I)( -RtI)(t"AX
-tP}X+.4X +r1x (J1X -J/x)-1;AX
-----:2-- tAX
f1.s.vxpev.~)dA -= L[pVx:ld'1xtAX
-LipV/d~x - VOJ(~:fVx ~~IXtAX
- tdpvxd'j}x -V'joAX)
,
REARRANGING f D1VIDINq "BY AX:
-'PX+AX -'P'~ JIXtAX 1" (Pi XtAX
~X 2
-1' pI -XI1)(tAX -~IX)-T~+ I)(. AX 0
2
== j:pv/dYXtAX - S:pV/dljx
~x
-veo i!pVx d~I)(TAX - fa!'~d,:!~
+Voo~o
IN THE LIMIT AS dX-O
-a ~= To +I(,,~. +ale ~~pV:c!~
_ /, 4.(c3 P~~
CD ~)o
REPlAC1NQ J'f =1x(JVCXI~
12.13 FOR THIS "REGIME) 1<&<:103
1I?e.= Dv @ 60G
F J:zlL{t·IO-S"~2.
J - J S
"Re. =oba Y; ~ =I J '1'::0.001/6 ~5
~=as '1'== .1"3 Wo/sI
.: 0.0016 ~5 ~ V < . "3 ~/s
AIR. @ ~o·1=
V=1.5"9 ·10-41 er%;
R~ = (O.'2/12)(OZ) = Q220
1.S''l·/o-4/
FROM 'F16. 12.2 I CD ~ 1.:2
-Ft>: /'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[
~ =51r1rtr;:i ~:x (S} =0
VXd 2~ .:J

75. b) FOR ACIRCULAR CYLINDER
" 'LWHERE Vx$ = 2 VQ') 51 ( 0. }
VxJe2._ 0.'11j)( 5" ~
T - V)(~ 0 V,,$
s:Sir15(~)d~) =¢=fSY~~ d~
_ - Si1'I ~ cas~ _! co.; ~(Z+Slrt2~)
5 ,s
=.! - J. cos. ~ {2. +S;r'l~~)-SiY1cos~
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. x«a. 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. '29XI. 22G)(30)
:2
= 631.1'0 N

76. POWER = 30m (631.10)= 1095"o.QV
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<1 W =16.3hp
12./9 L= CL ~ pv2
A
;;: a4(1.22'}(~4.7)2('2.2'i)
~
::: 1122 N = 2~.2 bf
12.:20 D'RA6 =~ P V2 Co A~
FOR EQLJAL DRA~ AT TIlE
SAME S"PE ED <;A6R= Ct,AI~ATE
Ct)A~ = .5"(2.2<1 ,..2) -= 1.''lS"M~
CD APlATE = .01 .'iTt> 2. ["iRe. >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'2207
a -, ~ ~
W=~ .2b'R·ttJ (lOZ."1).?i(25.i3J
£!2:)
: 19.~ Yp
b W= gEE ~:: 15.70 tr
5>0
IZ.Z3
T ~ F 6 -? "l
<.:Ay =UJ 'v -p"? O.{{i8·fO ~
CL. ~:: '{Q:: 139.3 '2.~IiL
7 D,I~-{6-3
~ =2f)2, auI ~%0.4
b_1kt~~=~ SN'LC't> A12-
<'Mi='0 - ??~u..~~(··_/~.) - . __C)...... ~if
JD:: o-lZ8I.16(1~.33}i"-ID7
2. i44
ID :? 0 .42 l~f
C. LAMiJJAe ~LA$J2.
(As D'5d..~ krYtr; END
'at: ~ 12.2).

77. It.2.4 :}
J}~ 0.16.1 ·/0 ~~ .S'T4rlTtN~
AT ~ -r7.5·'cfI Va 92J~ frs
2-
1Dz~~V ~~~== V
2
0/$,@
~.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
Ae2 ~.B{ 1~1. / C,-:% O. 224
~l)5 gQ ~ Ll42
V
.Qz{5LO~:: 240 @at/s
~Too kApID.lIJ.t ~TQE..
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

78. ~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= ~fJ~~~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 ftko.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 17_ (§;:S 7m~ rlD I.S7/ )

79. x 1Re.x dLJ~ ~~ G4't1
0 0 0 0
. I 2·1()5 .111 0.321
.5" .2«19 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. (LCL 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(bJPv2A Cfl: FOR 2 SIDES
=(O.~2373XI600XI.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 FLOJ~
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';;'= a022 5'f V;< ma.x --~--
VXmAX ~~
FOR THE Y7~ POWER LALJ
v= o:Z{? Vmo.x ("PRoB. '-1.12)
:. VW~ :- O. £f 16 t1/s
~h1o..x == t:J. 07!JWI 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

80. 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
ENERGY ,,-.JPv3
@1i?e -= IO~ .&-=~·S-Jk)CO.2,
6L l-s i~~~S-J
=2. '3'1
v= V«)f(~)
MoMENTUM =pVoo
2
f2.(-rJ
MOMENTUM nuX' = f 2(iJP ~2
ENER§V ~LUX = f3~)
Y2f1~3
LAMINAR;
M = sin2.(i 1J)p Veo2 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 £-LpV•.2 ,~pVco3.
r3.Q "'ORA6 =CfL ~pv2A -2SIDE5
A = T·l/O·2= SbOfP-
D= ~(.OO20qqX205)2(S60)CfL
= 2~" JEq .eft. Ibf-
"IRe. = vL - 205·? - 'Z IL/O 000
1) - 3~6'1'IO-7 - J .I
• D02.0~
0.) LAMINJ¥R
CfL =1.32X' = 0. OOL/65'
~
D= 11.26 Ibl
b) TURBULE'NT
O.OS'1&
cfj( = (vx/lJf2

81. 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
=~ ·IOOO·JIC)O''1·200
=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

82. 13.1£1 J; =V, I cl Vxd"i 2) 2 de
P 'X d;"t2te et-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- fPr.x)clx5~(x>e ~P(lt}dxc(x
<
WHeAl vxr.s CONSTANT
?CX) = 0
U =Q x -t C..I c= ul){=o
I~J"
TM£:A<XS»tE1;> A~ tSb
~Tm:.. VIcM~IM~
S1tn>~ ~~ &t~
~b-r~~
~v,lk(~~~
~DtS~~<1~
hJ 8~£·

83. '3.I8VlC'''(~yJ=~O) t ~~./Lx
~Vx'L 7'l1!...2 d2.Vx'/.f+ ~ y + X~ 2 + ;;y2. 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~<j+ •.. +b, +2b2'1+b,3X",=O
COMPARING COEFFICIENTS OF
X-/X/ ¢ ~' TERMS.. WE OBTAIN
bJ=o} ~=0.l C 3 +2b2 =0
HENCE ~:'(XICj J= ~ y +Sy2
+ C3XY t-
V~I ()(,<1) = -C3 '{2 ...
U ... I / - -c C '{3 - c., c.. y~
)( v'j - 3 I ... 4
TAKINc$ TIME AV£RAQ£
VX ..V~ .. =-Cg C, '1.3 t- HIQ.H E'R OR DE/{
TERMS ~OR MIXlN6 L{;N6TH
"
THEORy fx'V~/ ~ ~2.
~VX =VM (~)Vn -'-oCj ;t"R ~
AS ~~o ~Vx~(;x,
c;1~
As Y=>"R ~ vx ~ v""
~'1 ttl<
BZ.

84. CHAPTER ~
ru V =~ = '0 ~'~~o.J ·~s
~ (o~~'f ff)2
=l.fK tps
RQ. = )V = V.l2'1/il)(I.Ii) =2 q 5"
JJ 8 -Io-S' LAMINAR
AP _ ,- - 2 r
f
L V2 . £. = 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
AP -= 5"ps',
Gl)ESS LAMINAR FLOlJ
g = '32.f<2 OR ~;p _ L.. 32Jv
AL '[)2 ~f - D-r c
V=A'P S .Q.D
fj'j2 L ~
=lS".I&ttf ~..J. ~IO~ 13.2~
5'? 32 ~o 2'Y f""ps
~ =~ = 13.2'1'. lOS:: l'3.fC1
Y 12·1
.: LAMI NA'R J:"LOW~
Q. =AV=1l'D2
y;== aOOO?22 CfS
'i c:J
oR Q =2. 60 ff~n1in,= "lettS ~~
(i)~N.'3 _________
Ar 2c?O~
;::-250-" 'D= .b2~ ~
2
S3
~ == ~o...~ J "P2 :. 300 <'?~
Q == O. 56 w.~
S
J= 4.~)(IO-b Ml2.J!:>
P=2'10 ~ /~3.
~ - ~ =(Pow£R)
1t cH
o =~V +ffCe.-rfrM.ito
P:: ~ [/2.2
+3.. 't 42 +uz p J 2 2
-Or+~dt,+U.,~
p= ~.;.[~¥,+?j;;+~-i:'lf"J
1L
= 2 tf .b.}L2
o ~
ff =ff (1Re) %)
1Rt!. == VI) - Q. 4 ~ 25'S 000
-;) - 'iro"'J "
e =.OOOlS fl., E:. ,oooo~'{
%=-3.bl~lo[~ +(3~D)~]
=: '(,./3 $i :: .003'Z1.{
V = I.f5 G ~/s r2=O,~!; J
~
h1.:: 2(.003~)(2i'O"O'l)(.~s"I)=1:l17
.62 ~
r>::2, -'P. .= 3'00, 000 -/~ ()OO ~ as.;z~
,o~ -aIO ,9. 'lo~
~~ =~p~ ::. if.XV1·¥;lO'.Sb ~ 4441
?= 4'141 (25',2 -250 r/2/":1)
::: '/,413 I 30b . IJ, "'1/5
= ~~ 13 KW DR.. 6"'Q,! hp

85. GIVEN "PIPE J
A'P -- .fPV
2
- f iA~
P
FoR l=ULL'l' iU~ULE"NT J:'LOW,
f -= fUNCTlON O~ e/O ON LY
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;'
1°2 .- ~ 2t.3J ?o - 7. T..cr -'
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! -'cJ
'3) A~NOZ2~
:2
u.. ~ PI" _ VEX,T
.:.L-+ r - -
2 T2.
AP == V i-xfT _vp:2 == ~~(V£~IT- i"p ~ ~ 2 Vp:1 I)
C.ONTINUlTV / Al>Vp -= AlOiZ ~VE)OT
:. A~::: .?2.(Ap'2_rp -"2 AN:2 I}
ADDIN6 T06ETHER
~:: Vp:l fi'"t"2( : Llf b +Ap -0jJ 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;<) USIi v= 1.570
/0-5.
Vp2 == 1Cfl1 lSp ~ 9.iripc'
35.CY2 +I0:2'-10fr} r
FC)R 'DIe. = 1.2.SZt)lR.c ~ 36 12") ~;J
~- ::: }8.35 I fF= .005~1
U:: 9.81 fps lRe. ::::3~09(/~Vp
e4

86. :. 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/6OJ
2
= 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<l1
ASSUME "PZ = PATM = lor kPo.
v2 ::. ~2 = 16 Q2.
A2 11:2. 1)"
CK
85
_pA?=2K 16 (i?2
71"'''' Dc{
Q~ 2 'Ir~ D~~ P 712 135'
32 K; 32 (s)SK
OPEN
~ CLOSED
'12 CLOSED
"3,A a.os ED
."3
Q
,"5
.2
"!I
.1
K
.JS
.KS
~.q
20
Q
.2lff
.125
.055
.02.s'i
Yz ~'1 OPEN
MODIFIED 'BERNOULLI ffiUATION
BETWEEN I fZ
'P. ~ v.~ ={'= ?2 1" y{+ 22 t- hL
p§ -. 29 T i'; fXj 2~
o V. 2. L ~
~::. ~.,..Z2 +Llff -;;1.
f'3 2g 0 ;;z.~
= Zz. +- :1 (1+ 4~ L)
2~ C
Q::. Soo tS?~ ::. 1.1l c..fs
v= Q : 5.b?~5 'D = 3333
A e.
'i<.e = ~ = So 6:t ',S ·,a~= 1.D3·1()~
II I.'t ' - r.:;oo F
l
% :.1'5'.136 ff =.00 '(q
-1'2, = '3' .f ~:l.(it 'I". O~'1' ~)
~:2~ .S

87. -P:a. = "S.S I'
~
'"Pz =-~/q ?sf~ :: -'S~ 'P'$.~~
IL(./O
?>fTWEEN I E2
4f"OIA
UNE
'2" 1)IA
LINE
I
<j('12 -~,)of-V:22_v.2+ 'P:a.-P,::: 0
2 to
-20~ T '1/2. + 1'z -?..rrM -="2 P
BETWEEN 2.f3
~(~~ - t;i2.)-t v·i -V2.. 1'3 -P:a. .J.hL=o
2 P
2. a"
V3 - V2 +"PATM -'Pa t- hL =0
2 P
ADDING : -20~ + V32. +hL =0
2
V"3 =A:a V2. = t4 ~
Aa
nL = ( IS • O.? +-J) V
2
... 2 t; DL V
2
~ 2
ELBOWS (ENTRANCE
::r~+ 2f JJi"lv2
Lz f Q/,2J
== (2.25' of- 6"QOf;}v1
-6'4~ r(2.15"t 6QO)V2. =0
ASSUME f ::: 0.006 •
V2. =6£1l4 = 1,2q.5'7
2.i5 r'i.1'1
V::::. IL'31 fps 1Re..::: 0/.2 (U. 3t)
'.22 'IO-S'
1Re. = '3.1'10
5
fj:. = .OO~£ -~y
V2.= ~~~+3.J :::/l0.S" /: 10.Lf~1Ps
~= 2."l·/o5 tf -: .o~t; Ok
FlOW 'RATE = 1F(t fl.1,0.<f9
== o. q,S c f's
"'.11 UESTIMATE: VELOCITY USING
&RNOULL'''s EQUATION
v 2 =2 q An v~ 25' fps (A11 =10')
1UI5 IS A MAX. WITH NO LOSSES
USINq hL = K3L.l == V2 DUE TO A
2 2-
SUDOEN OPENIN6 AT ENTRANCE
(SEE (;c. Ii -p.79) v2 =SAh
V= Fl.i-t'P5
2) GtJE"SS v= 10 fps
WITH WATER @ ~OoF ~-=b10c0
I ~
tf =.004'1
:1 f' 2. v1 v:1
a~ =Zh + Y.. ='h L ~ +_ +-J I L Z t5'2. 2 .2
::11(2 + t4 fr~)
.!: :: 2~6
'[)
(FULLY DE:VELOPEO)
'i~~ =5".<-11
D
v-= 't 32 fps (CLOSE EfJOUtiH)
Q =AV =1fL~q)C'1.32) =.0 5J cfs
=,q.'H, G'PM
PRESSURE AT "PoJNT'B
ASSUME Z-R O~PIPE BEFORE B
Z. + ~ +1r. = HL -+ Yi+;'+ ~
l 2.; Pj ~ 14 I PS
ZPRO ON SlJ1<F'ACE'
~ = -'4 - vi -hL =-1./-Y'"~+4t;.!:.)
P3 ~~ .2sl" fc
: -'1- J.3S'O(3.n)::- -q_2~ ff
P"B = -4 'f>'51'~

88. V= Q = ~ '~'1 :::: 225fps
A ~,~
De = i A~£A = 2 '21 -.:: Z''''=."6t1
l' '32
~ = q~J 3~O ~ -= •0«'5 ff
~=L'333 ~ :: . 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
'2II 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
PoW£R t =1.03q~ (~ Ib;Xhl fl75)
== 301. OS OlLt 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~ =q·2·5"270·'bG22~
o ~ D:;2Srrr2 1)'1
hL =22" Q62.!i = 2QO.3
OS
jl = '30L05" (2QO.'3+ 175')
~r +.JI.(~gO'2'll.lf)
=S 153 '321 /<:!t'""/
12" "PIPE
v= ~ = 5.9/0 ~/s
A
~= ~.~7f·lo5'
.~ =1/16 / 'iff::1'I.2!)3
+~ = .OOQ92.
hi. = 22962 off :=. It~. 03'
!)S"
~r =~oI.05"(.J.n.o3)
-+- .11 4(5"2'80.2'1'1.7)
=If, IO~ 787
1<1" "'P I'P~
v= <:) = Ll.31..[2 fps
A
1Re. -= 3.07' -/05
'D - l"~" .l- - ItJ 4~ fr=.OO~X'Je: - ;)T oJ " 'f.t;: - 1. I r
hL =5"/.10
$/ :: 30J.OS(:226.IO)
/~r
+ .{J-CS2"'!o·2 ·'b.n)
= ~ B7,&82 /~r
.", Iq" -PJ'P~ HOST ECONOMicAL
6" I (C Lf•24"

89. 0..) SOO<3~W= ~ :: I.ll~ cfs
~(7.1/6)
=G2.
A:.1rD~ A, = O.JQb3 ff2
l{ A2.,=O.lq 61 Ff2
v= 5'.b~.fpS (IN ALL 'PIPES)
b) I~ ~ Io..~t loofl~!::: .00171
n
~ = 5~ J= 1.22 'IO-~e.
'Re =~= O.S(:;.6Z) =2.33 'IO~
J (.22 '/O-~
J1'ff =13.1'3 fr.:: .005'8
A'P = Llff ~ ~~ ~=1-f'V:1=3/.'3~
t = 200 A?= l./{.oot;1)(~ooK31.3)
::: I'iS.2 ,?sf
41P :. 1~5".J. p-;.f ;::: J.0041' psi
t-HOOlE 200' e =O.OO;2.~1
1:>
J:? - ~I fo 'U'l_ =1.6l! ·/oSe- ~
..L :: 1:2. SCI t,.= .00 6 :3 "
~f
L. :: 21"3 t,.'P ='-I~OOb'3(:'X.2! ~X'3I."3)
t>
=22Spsf =1.S'Tp~i
C) TOTAL ~p= l.ooY +I.S? +-I.ool'
= 3. Sg-ps;.
1..-.2 ... CAST I1<ON
;SO~ • lAP: ~
.ob7 COMMERCIAL
ST~EL
A"P =p Lf t+ J:. ~~=10
3
IS-O· 2 v"-If
1:>2 D
.:. ~.'OS' V2.+
f
=2-' -lOS
D
'IRe = Vb =I06 VD
'T
a.)D=.2M "~+f=·l<{
l<e. =2 '10 5 V e = .000 !"S"6'
'D/e. =??:z I ~ = 1~.:213./ ff=.005"73
'iRe =>c:o .: V => '.O~ ·/Ol» Of<
Q.= 0.163 m3/s
b) V2 t.f-':: .0'16 q
'iRe. = b~OOOV e = .000 IS"H
"l1'e =1£165 I~ =13. 53<{
++=>.00 'S~b ~e- ~ 00
FoR ~= .DOSO V=> 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

90. =f;Ql. =1. 36:2><IO-~ DS"
i<e = ~ = 12.5":2.'°5 Gl
7b~ p
a.) "'D :=. Y:2 'I ~Q':I.= 2.301 -/O-b
'iRE! =2."S0'/" 0' Q.
fASSUME -= .OOSJ
'iRe. = 5"32.10 fCALL. = .005/1
.: Q. == .02. J:2:2 c::. fs =- I. 2 =t~ c f~
.;. -= I. '32S U:JrK/s
b) u= ~c./' ffQ:J = Fl.4"6·10-1:.
iRe.. =I."6Q _/06Q
fASSUME =. OOt.(4t;;
Q =.Ob.26 ~ 1R~ ::: IOq" b I q
+CALL = .00 1I~).
:. Q== .06:2.<f0 cfrs = '3. 77'iI cfW1
~ = 3.'1215 tlo""'5
11I.1? A r - _
(I)
89
~ + V¢+eA =~ +~+Z'g -rhl..
~ ~ ~ 719'
EQUAL
Pi! -~ =ZA -L'a -hL (2)
,09
~ = h3 (3)
,.os
..../ V2.
h3 - ~J +f'A -t- ~ = ~ - 2-B - hL
1./:1. L
- +- YlL = E, -zB -11.3
~
2. =- 60 1M %::. 2.2.,.,
~( I t 'I tf ~) =30... 1..3=l!'...
~ .ff~ = q(. ooct)XQ = 3.65"7
.3S
v'l. _ 30
- --- v=IJ. .b'8 Kifs
~ <f.bS;
Q= 1.2l W1 3/s -......I t - - - - -
b) ~ =.2S"0
off ::;:> .OO?I
" => t!.~S'1 /MIs
'Re. =:~s ·?bSI =3.03 '/0'"
10" .
Q::: O. Z3:2 IN3/S
I
I'/./f T
"~~
1

91. AP ="?.l-~ = ffj(- ~l. +Ac)
=fJ~ ft' €vl +Pc)Ai!
v= Q = qO *3/5 =4.S'i "'/5
A 7r( S..,p.
2j
,6i=> =10 "'f8~. :>. (</.Si)>'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 6is
-L - -3b I f)·9 tl. I - ~~l
f.fi - . OS,aU·'lfl-IO" {J.1·J9~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 L2~ s>~
'-!t;RE ~ to;;T,,~ fAILS" v~
AJJL)~Pta; kr~ Aa .2.
-rt;I2M-6y-~
'1, V.t '. t ""
V~" (~JiQ,Y/-A~!)
20 2. ~, L T
= -2.03~
lie.-90-::(S-)IOS
~ QS.S ~
)lej glOd
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