Fundamentals of Momentum Heat and Mass Transfer - Welty - 4th edition Solutions Manual

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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<j J 
cv'PCQJb) = f'c v;: [k(cos I siYlI +2)x 
+ ~( '51 Yl I Cos I ) y J 
.". \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 ~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~ 
PR£S.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 <2.;r - ~''(\ <2.~ 
de 
• 
• • 
.... 
:::: - ([.r 
Q.E. D . 
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'><' +~ ( 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(3X2 + 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 PER F2CT CSAS/ 
P:;O KT 
M 
FRo/Yl '?ROB. J./~ 
fY -= I'm ( I - X) 
I - "om )( 
A 
.... --p -::: fJ", (/- X) 
/- & >( 
p~ 
RI 
"M 
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<] 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 = P 5 ~ 0 :=. P ( <j ;-0.) 'j Q, 
l:ica. == ~ L1 < ~ 
'3 To. -.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 t f4.7A ~~ ./'2.2. -= 2.2\ ':: 2'.£c; 1ft. 
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<~/~l) h= 2W1) l<=.3~ 
F :: S~46 ~~/s~ =- 5"5"'-1 b N 
- I 1c.p. -= vt -t- ~ 
1r'R2 ·2.tM 
C"fr'R q FOR A Ct~LE J Tlob:= - "'I 
• v1 - 2M +1rR.~ 
" "le.. p. - qrr'R2 (~~) 
= 2.0tl ~ 
2./2 ASSlJh1ING, ArlY1()SPHERIC AIR. 
TO 13£HAV'£ 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 - <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 
Ptrd 2 ._ "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 [ ~/,q J 
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 
"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". -= L 6P'S;9'/ilL! + ~ ] alA 
~Mo =0 
~ <j(dF"w-d7=A) = 0 
(<<{ lj lf9 (h -4t-Cj) - gb~JdA ::0 
4 (" CtJ9 h IJ -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 Say,- }(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 0 
LJ = I'IOIJ <3'71R2 
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= (3 1c 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 = J 9'{LfO 16; 
2.21 
T 
b' 
J-
h == <i./5ft. 
AS'SllM mows : 
CD~~~ ~rtr"~ &u.-
® HzO Us\&!-A~ ~LL T(!)p 
y 12:= # j liP d F". llpdA 
e 1-
01. ~ dA:% z:~ e SWledf} 
llf-.: 5'1 (h-kbso<+~~) 
dFlf ~dFa:>5e 
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 T de. 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 = ILl 190 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)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 =~<.H(t+~ 
ZFx- = 0 I ,', F:z = ~(3e !:L'J. 
2 
AREA = ~(~ H t~ 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 -~J 
2 . 
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~ T f'<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]. 
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' (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 
= 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 <j2 
TO DcTE"RHINE C; AND Col
) USE 
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,., <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';" .. tD J .j. 5' Qu:.tlc: ~ 
Cs. 
~ = -j'A.V=-ywV Vs 
CD f + 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.=j\rd2 .. o{~(eo-<)dl-
o 
V.e :: ~:,"" Q= '-'4Ii 
... I> (0 
kr ~oh I \l;1J,.=fV, =IZC~ 
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~ 
~ 
0°0 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",)(.,! .,~,)(8·OO ~) = 179. g Val 
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) -(0 vA), (-l/) 
=I'Q~ C~)+pQV;.= I·g'''f~ 
F"'j =(DvAk v;. Sin(-30e) = :-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 b\o..cl~ 
=(34.~x -~)+40ex 
-, ... A 
= -(~.~ eX - ZOe~ 
(0 I = 77. '6' fps 
" . r - - - -'"1 ".5"~ 
I Cov. f-
~---0-J 
5.5" L 
x 
~.,. f dm =0 
~= /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<>4 N 
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 EM3 : <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 ==-1 A.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 =. iCJ 2 R-~s 
P2, = 1'1.7 psi ~ 
O2 ::- t.s ll 
~ Fx = KG'S. Vx pC von)dA 
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;(<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 PU\l~ 
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; Rx V"e ::. pA 1JC (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 ~ 
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 (/-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 <1 y =- /Y A, v,-(,;;:-tr,) 
P. -~ t- (rs - P:2. ) (~ -I )r'S ~ y l. 
-= 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 
A 1=' 0 TO? ",,:l 
v; == 12 tvJ/s 
R== 12i KPa.~ 
A2., = .113<1 m':&. 
LIi = ?'I{V tHis 
11 = 1l.{5" J<P~ 
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~f·E. 
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:" /. <t3'l "5/W3S./ f+3 
~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 = <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~ {2fr sY- 'J 
12 32. 1 ?II 14{C/ 
= 2()1/. 7' i+-Ibf 
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 'IX 2 I. 1') : TT J~ 
'3:1. J74{ 
, 
I------··-=r [ t ~--Lo=-
r 1.2 in 
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) 
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<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 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' <j 11 
: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)~ 
?"' So hjz ,~ = 5'0 h,h 
Wt,AlilC :z Jv, hi (V2 -V,) 
~ ~.-i~~" ~V.hJ (V2-V,) 
~GNnw(rr V2. 2~h/h~ 
d n~ (ft>~:) = V;~h, (~,-h~ 
2 ~1... ~I-Ya. 
h.: th, h~- 2 v,\~ =0 
~~h2 
~l = ~ (~-"("'-~f -0 
2 gh,\ 
~ ~U1i).)vrrY 
Vz = 1~'~~ {1~~V;~k,) 
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<A, :.0 c.v. 
. . 
~ gA n-- sAtt:rD 
~(J\Jtff1 ls SAr{~USD 
tv1~ LUrr14 t A$ + 
L f(f " ~ -t ~lJ~dM 
~~~5 _Asas ~lQAL 
-~~~+sA %ih~)fh2 
~u:r Is 'k z -~ 
5.3l 
USWG -rut;. c~v. Af!CM5 Wrrf4 
II To TH-g ~(~Ha>w) 
L M~ :: Ccrxu)t>e dM j J ' l: 
m01n~ 
V ~ ~ := Z L 19 +t.;~ 
, A: y~ +l:L =O.442 W~"L 
TAk:l1J<l ~ ~ 6 
W~Tb>TW6~ 
-3'P2 A4T:3V SAV 
- 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 +Jw a(L4-t)= ~tS'-t+~tz-
~-~ :z ~~(J~fJw) 
= 71.~~ \ZPa.-
~ '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<O ~ 3«) "2 55 coN 
5.??J 
SUSCE ~S~ 15 Cbv~ 
WL W = ~CUT=S<4Xt)LO st~ 
tno..q = eo] s/~ 
mOOT ::I :2 5'0) fJ-dY 
o 
= 2 J152 fa-cps ~'( }:IY 
() 
~2glJ2[Z-~] 
~: lf2 = ~ 1!- :z 55.0 ~~ 
4'[-8 
2..7 
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 
<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. '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~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 <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. 
- -' ) 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. = <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 S Z,.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 A 11113 = PM1r A 16. 
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 
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~ =0 TO 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 8 
I 
~ - '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 
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<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' 
= "/.9 rS -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 
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,7 ps;~ ~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 
.,.--
t 10.2:2 .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) 
ACROSS SECTION ONE J THE VE L-
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 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 
~ = <:JL2 t "Pc _ Fa ~ L~ -rJ!." _ UA:l. 
Ii Ii If 2"2 
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 + 9 L2 (~-f.') -~ ~ ~ 
A C:2 :. I :l 2 
CONT/N VITC/: p. 43 ::l) ~ :: ~ V I 12 _ 
.'.e. A HEATER 
AsrAo< 
Vg ':l,. : a:l)l Jll. ~ LI. l. = JL: 
- R:1. ;4 ~~ 
I 
~-Pc. :: -A (~rz~l 
:1( f?j 'R.1. 
'R 
"PA -"Pc ~ ~ }Ll. t~ L-'J.(~ -f.) -Ii Vl 2 ~ 
ACROss f.fFATER (PRESSURE ORaP) 
~ F« s r Vx 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,r 8.)2 ~l. _ ..121~ 
1(.1. ~ 2"lp' 1<1 '2 
'3 L:z. C~ -~
) f t1 V 1 
2"R2. 
v:L (1. - ~ ~ of-.!. t; ~ -t" ti - A ) 
"R 'R"'Pa 1 'Ra. If 2 :z Rl. 
= <jL:a (p. ;l! ) 
v 2 ~ 2tj L :z. (P./,p:z. -I) 
~.3{) 
1+ I-~~ 
"'R~ 
U SINe; C. V. AtdOUJ-ll> ~ u,..s. 
e+Py::: ~. A t-~ 
"8 tV !::: 9D(J\ J l)~::: Lf oUr 
So ~~~ d ~ IY'::; Uot.:rr t-~~ 
.: AU "" coAl"" d A~ 
CU--=C'f=4~DD S/~'IS 
AT:79.<a(~ l~'" ~ ~ 
s: 4~r 
AT ~O.~5P~ 
6>.'31 
tJ~ur;a-£U~ 
TW*~C»A f 
-OFT~ 'JJC1DM~ AI CZ-,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. .z 7(,]w> Z -:it 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 h t.. :zoD JOe ~ 
A~ h1.>D FOe 5>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;.CE6·roLcfs 
~.'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 
<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~ 
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<s mole CONTAINS 6.0;25" • /02J, 
MOLECULES .? HAs A VOLUME 
O~ 22.4 m l 
6.025". IOU. MOLECULES 
2;;.~ W\3 
=2.6Z·/02S mol = 7.6"3'10 23 mol 
m 3 fP 
- -/Z c - 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 
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 v1 OU tct'" bouJ'\d o¥"~ :: 0 
T = ","!bl = p. rw (LINEAR PROAL.£) 
d':i t:. GAP 
. ~O'~TANce 
:. i~ = 'rvA =0'~4'XrwX27rrh) 
~ = s: S"? 52 l!.:!- = 5.S''l W 
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 
N\TRO<:iEN £Ah<. = cU.S' 
~ = I.ql 
.~ -O,a = 1.I<1J{2 (UIYEAR INTER?) 
M = II. S 1./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 
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 A AX ..di: 
At-+O 
= lit1li-r '1r (>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 £ = b8tJ·9.1/-;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 ;<" 41"' = 21.7 rsf4\ 
L D 
7.2. \ , 
~% 0·76 ·ID- 144'5 
-, 
jLl2():Z: D.~~ .. (0 ., 
Xe~:a-5l% 
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 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 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<~) 
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 
'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.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~~ 
A LINEAR PROFI LE .' 
OR 
8.6 1 P.,; 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'"f 2jJ.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 
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 
BERNOU lLf 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 (9 AZ _ Va::1.) 
0'1 L;u 
Q= 1rD~Li," ~ = </Q 
q) 7rD~ 
1) = 1iDl/ [~t.r -~] 
121 G. L 11" 2 D A# 
?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)( = ~ + C J. Y -,.0'3 y~ 
:2p 
'B.C. (J) C, = -V 
B.c. (2) C2 =: ~ 
A 
b) v.. = -V-t ~~~'T J(~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 C r) ~ 
~= ~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 -;:-
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<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"Ph3 
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 INI\JER 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.] 
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 
<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::l5 T I-tG I\J s:r 
5\ 
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<A r --t d (rT):O 
U dr 
to ry ~ dU;lcir 
(. "" /'\..-.. :.\ ~y'G i ! 1'""'~~o.lJ L.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 ~ 
'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& 
3'21 = 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. 
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'" 9€e, = 0, dee = -e
r , 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, <j t C:2 
2)-' ~X 
B.C. @ C:j = :t L ~ Yx ,0 
~=O C2 = -..L ~ L'J. 
~"" ~J( 
:. Vx =..L BP (u';l_L;2) 
;:?~ ~ .J 
'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<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", + Vz dVi?l ~x ay ~i'J 
=P9
x
- ~p - d I2 IJ/~ +;~+-~~ 
ax cfll3'\=;lX' dCj 9r')J 
+.£...(,ud~) of- C1 (.u a~)+~ 1M d~) 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) 
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<S 
'PRE'SSVRE • c) 
9.1l/ FoR I-DIM STEADY FLOW; 
Vx = VX' (X") V'1 == Vz = 0 
NEGLECTINg 9) 
pVx ~ = -p of" 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; 
~ [1 a (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 i 
ap il 1 il 1 a , 2 • i, 
=--+P8r+/J-[-(--( ,))+,a7{_~~+ij;,] 
ar ar r ar r Iii? ryao Pz 
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~ \.-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?~ ~, 
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).AiJ 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<)~ + 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 V 0/ J {bsnJ.J<XTY 
&l (s v..Gzo DJZ Vf-:'O 
Ust~ ~ 
2'1 + P.l ~O 
~' ayJ/ 
\0)6 - '0 X ::30{) 
lL 2 ~ := Cl!P. :z 5~Z 
x ~ ar 
/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 .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. = 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) <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 
WHEN '-P=O j '1 =0 OR <j = !".J3 x 
----l-----":~~x 
4/::0 
b) ¢= u,,¥! 
Vx =- t} ¢ = u., 'j -= ~ 
ax T ~g 
~= ~ = ~X' = -~ 
afj 1: ax 
¥= ~ u 2 + f(:x l ' '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~ fc3Vf(: )TUQ) LQ 'f~~ (6r9A~04~ +~~) ~ 
~ ,2: r ~ 1 ~ ASQ-ULtS 
2s~~ 
8/~ r ;'PLar 
'(' 
lO.t2- t- 0 -= lf
tD
r$ln9 + Q S 
2i 
6'( OeFlAJlTr61J r~OJ ~ If'! ~>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<l 
A~ If(a,O): -1 e 
47f(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 
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/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; = 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 
~=[~ ·?f·jff.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<, 112 =/3 
'IT3 = M 0. LJ., ~ c C¥MX 
I = M~ Lb (LlP')c M L)lt ~ 
a. = -1 J 10= -/ I C = -/ 
1TS = C I'Mtl.X 
ML~ 
1lq = ~Q.Lb~cf ; 
71S = twf' L b~ 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 
(~) = ~; =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 'H
2
o = 10°C 
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 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. 
cl 2 a 
<"i73 = olAa. Vp~. rrr;=~ =_, 
J f'd~o. my 
I'" (K.. I J;L ) lReJ) = 0 .. a.) 
,. da. d 4o. 
VAR II 0. AND/oR d 
Tl-IEij ~oR FIXED VALUES oT: ~ 
'RoT I:>Af;1Q. vs. ~a... b) 
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 
COR E" G Ro()~ -= P N 1) 
G 11, = p a ~ 10 D C 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- fV\lt: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 
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 <;(3 
tLl6 FOR DYNAMIC SI MlLA'RIT'I,I ) 
~ ~ :. lRe. Fa L...L ScALE 
:. tFlM == UF.S. LF.s. JM 
\.- r\\ J t=: '5 • 
VM = 60~(~)F.10~ = 2LfO r)'\ph 
2. 10-5 
/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 
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. <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:L\to\ PrM 
FF.S. :: 2'lS (,. ql./)(-..-L )2(:2.ct 5'12N 
\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 
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 @ 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 
V fp~ 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 
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.\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 P Vooa'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 
.: 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~(( 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'1\xtAX 
-Lip V/d~\x - VOJ(~:f Vx ~~ IXtAX 
- tdpvxd'j}x -V'joAX) 
, 
REARRANGING f D1VIDINq "BY AX: 
-'P\X+AX -'P'~ JIXtAX 1" (Pi XtAX 
~X 2 
-1'\ pI -XI1)(tAX -~IX)-T ~+ I)(. AX 0 
2 
== j:pv/dY\XtAX - S: pV/dlj\x 
~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 (J VCXI~ 
12.13 FOR THIS "REGIME) 1<& <:103 
1I?e.= Dv @ 60GF J:zlL{t·IO-S"~2. J - J S 
"Re. = oba Y ; ~ =I J '1'::0.001/6 ~5 
~=\as '1'== \.1"3 Wo/s I 
.: 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[ 
~ = 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. 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. '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<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'R A6 = ~ 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' 220 7 
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, au I ~ %0.4 
b_1kt~ ~ = ~ SN'LC't> A 12-
<' Mi='0 - ??~u .. ~~(··_/~ 
.) - . __ C)\...... ~if 
JD:: o-lZ8I.16(1~.33}i"-ID7 
2. i44 
ID :? 0 .42 l~f 
C. LAMiJJAe ~LA$J2. 
(As D'5d..~ kr Ytr; END 
'at: ~ 12.2). 
It.2.4 :} 
J}~ 0.16.1 ·/0 ~~ . S'T4rlTtN~ 
AT ~ -r7.5·'cf I Va 92J~ frs 
2-1Dz~~V ~~~== V20/$,@ 
~.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 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. (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 
ENER§V ~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