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!"#$%&
()*+, -./012/
34* 54+ 6478 94.2:7,; ?$@#AB#ACD:71> E&$E FF&%
G24+)> H2)4*I70*@1,*@/J
3:: -*7 -*78 31/)+1; ?$@#AB$A$#CD:71> EKL% EK&K
G24+)> //)::I70*@1,*@/J
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!:*M/1 ;*0)+71
64=M:/=:N+= ()*+, 61=D47+=/ O70M:,*=P:7 64//8 G71MJ. 47,
6:2170*2 B4)47=1/
()*+, ():H +7 C+N1/ Q+217/+:74) R74)./+/ C*2N/ 47,
!:2NM1//:M/
6+=M:/=:N+= ()*+, 61=D47+=/S!:T1M1, U. QM@ 31/)+1 3::V
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WM4,+7J
!:7P7*:*/ R//1//2170 X&%Y (+74) GZ42+74P:7 X K%Y
[1Z0U::\
!"#$%& ( ]421/ ;@ 5+)\1/)"*+,-$.& ( C14M/:7/012 (
LKF%$&$^F#$#F
()*+, 61=D47+=/ _:M !D12+=4) G7J+711M/
H+0D 6+=M:`*+,+=/ a !(Q #7,
1,
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CA policy
If student misses CA due to following reasons: valid MC (not
from Chinese doctor)
passing away of immediate family (parents,
siblings,grandparents)
participate in an activity representing NTUthen the CA component
will be counted towards thefinal exam. There will be no makeup
CA.
Otherwise student gets 0 for CA.
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Tentative Schedule - Tutorial Week- Quiz
- Holiday
Week of Mon Tue Wed Thu Fri
11/8/2012 NationalDay LT LT
18/8/2012 LT LT
25/8/2012 LT LT Union Day
1/9/2012 LT LT
8/9/2012 LT LT
15/9/2012 LT Quiz
22/9/2012 LT LT
29/9/2012 RECESS WEEK
6/10/2012 Hari RayaHaji LT LT
13/10/2012 LT LT20/10/2012 LT Deepavali27/10/2012 LT LT3/11/2012
LT LT
10/11/2012 LT LT
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()*+, 21=D47+=/
()*+, 21=D47+=/ +/ 4 /0*,. :_ `*+,/ 47, 0D1_:M=1/ 4=P7J :7
0D12@
()*+, /04P=/ b -0*,. :_ `*+, 40 M1/0
()*+, ,.742+=/ b -0*,. :_ `*+, +7 2:P:7
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5D. /0*,. `*+, 21=D47+=/c
!D12+=4) G7J+711M/C10M:=D12+=4) +7,*/0M.CD4M24=1*P=4)
+7,*/0M.
B+:21,+=4) +7,*/0M.C:H1M N)470/
?:7A!D12+=4) G7J+711M/R+MN)471d!4MdB:40 ,1/+J7
B+:21=D47+=/-N:M0/5140D1M _:M1=4/0
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GZN1=01, 314M7+7J ;*0=:21/
RU)1 0: T+/*4)+e1 `*+, `:H NM:U)12/ 47, /10*N4NNM:NM+401
J:T1M7+7J 1f*4P:7/
g7,1M/047, 0D1 ,+h1M17=1 U10H117 )42+74M47, 0*MU*)170 N+N1 `:H
NM:U)12/ 47, /:)T1,1/+J7 NM:U)12/ +7T:)T+7J N*2N/ 47, 0*MU+71/
g7,1M/047, 0D1 =:7=1N0 :_ ,M4J :7 ,+h1M170J1:210M+1/ 47, 4NN).
+7 ,+h1M170 ,1/+J7NM:U)12/
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5D40 +/ 4 `*+,c
W4/c 3+f*+,c -:)+,/c CM:N1MP1/ 0D40 =)4//+_. 4 /*U/047=1 4/
`*+,
Q1_:M24U)1c
():H4U)1c !:7P7*:*/ ND4/1c
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Q1i7+P:7 :_ 4 `*+,
R `*+, +/ 4 /*U/047=1 0D40 ,1_:M2/ =:7P7*:*/).*7,1M 0D1
4NN)+=4P:7 :_ 4 /D14M S047J17P4)V _:M=1
-:)+,/ =47 ,1_:M2 *7,1M /D14M _:M=1/ U*0 =47 M10*M7
0: +0/ :M+J+74) _:M2 HD17 0D1 /D14M _:M=1/ +/M12:T1, ()*+,
7:M24)). 04\1/ 0D1 _:M2 :_ )+f*+, :M J4/
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CM:N1MP1/ :_ `*+,
()*+, /04P=/ Q17/+0. j+/=:/+0. -*M_4=1 017/+:7 64// j:)*21
[12N1M40*M1 CM1//*M1
()*+, 21=D47+=/ j1):=+0. -0M1//
G4=D NM:N1M0. =47 =D47J1 H+0D ):=4P:7SZ8.8eV 47, P21 S0V
0
0
0
0
),,,(
=
=
=
=
=
dz d
dyd
dxd dt d
t z y x
!
!
!
!
! !
/014,. /0401
?:71 :_ 0D12 T4)+, k&AQ+217P:74) `:H
$ :_ 0D12 T4)+, k#AQ+217P:74) `:H
# :_ 0D12 T4)+, k$AQ+217P:74) `:H
R)) :_ 0D12 T4)+, k c
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[+21)+71/> )+71 _:M21, U. 4,l4=170 `*+, N4MP=)1/ +7 4 `:H
i1),40 4 J+T17 +7/0470
C40D)+71/> N40D :M 0M4J1=0:M. 0M4=1, :*0 U. 4 2:T+7J N4MP=)1
-0M14\)+71/> )+71 _:M21, U. N4MP=)1/ N4//+7J 0DM:*JD 4 iZ1,
):=4P:7 +7 4 /N4=1 :T1M 4 N1M+:, :_ P21 -0M142)+71/> )+71
047J170 0: ,+M1=P:7 :_ `:H 40 1T1M. N:+70 +7
`:H i1), S047J170 0: T1):=+0. T1=0:M +7 `:H i1),V C40D)+71/8
/0M14\)+71/ 47, /0M142)+71/ 4M1 +,17P=4) *7,1M
/014,. /0401
[+21)+71/ T/ C40D)+71/ T/ -0M14\)+71/ T/ -0M142)+71/
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Pathlines (Red) vs Streaklines (Blue) vs Streamlines (Grey)
[Wikipedia]
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(:M=1/ 47, /0M1// -0M1// +/ _:M=1 N1M *7+0
4M14 ?:M24) /0M1// (:M=1 4=0/ N1MN17,+=*)4M
0: /*M_4=1 [17/+)1 !:2NM1//+T1 S1@J@
NM1//*M1V -D14M -0M1//
(:M=1 4=0/ 047J17P4)). 0:
/*M_4=1 (:M 4 J+T17 /D14M M40180D1 /D14M /0M1// +/,+M1=0).
NM:N:MP:74) 0:0D1 T+/=:/+0. :_ `*+,
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CM1//*M1 O7 D.,M:/04P= /+0*4P:7/ S`*+, 40 M1/0V
`*+, 2:)1=*)1/ 4M1 +7 4 /0401 :_ =:2NM1//+:7 ,1JM11 :_
=:2NM1//+:7 +/ 214/*M1, U.
NM1//*M1 S/=4)4M f*47P0.V B. ,1i7+P:78 ! k " d # SD4/ *7+0/ :_ 4
/0M1//V [D1 T4)*1 :_ NM1//*M1 40 4 N:+70 +/ +7,1N17,170
:_ 0D1 :M+1704P:7 :_ 47. 4M14 4//:=+401, H+0D+0 S5:M\ 0D+/ :*0
H+0D M1_1M17=1 0: (+J@ $@mV
O7 2:T+7J `*+,/8 0D1 7:M24) /0M1//1/+7=)*,1 U:0D 4 NM1//*M1 47,
1Z0M4 /0M1//1/=4*/1, U. 0D1 2:P:7 :_ 0D1 `*+,S,+/=*//1, )401M +7
!D4N01M mV
RU/:)*01 NM1//*M1 k J4*J1 NM1//*M1 n402:/ND1M+= NM1//*M1
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B4/+= )4H/ [: *7,1M/047, 0D1 U4/+= NM+7=+N)1/ 47,
=:7=1N0 :_ `*+, 21=D47+=/8 H1 711, _:M2/:_ 0D1 _:)):H+7J>
!:7/1MT4P:7 :_ 24// S!:7P7*+0.V
?1H0:7
/ /1=:7, )4H :_ 2:P:7
[D1 iM/0 )4H :_ 0D1M2:,.742+=/ C*0 /+2N).8 0D1 )4H/ :_ `*+,
21=D47+=/
42:*70 0: =:7/1MT4P:7 :_ 24//8 171MJ.847, 2:2170*2@
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Q+217/+:7/ 47, g7+0/-./012/ :_ Q+217/+:7/ Q+217/+:7/>
ND./+=4) f*47PP1/ S1@J@%& '& (& ) V o( p8 o%p8 o' p8
47, o) p o" p8 o%p8 o' p8 47, o) p
o" p8o( p8 o%p8 o' p8 47, o) p
317J0D 47, P218 47, 24// :M _:M=18 4M1 !*+,-*./0-1+23 4+,561+761
S_M:2 HD+=D :0D1M/ =47 U1_:M21, S,1M+T1,VV +7 4)) ,+217/+:74)
/./012/ +7=:22:7 */1@
-./012/ :_ g7+0/ 630[q-O S\J8 28 /8 rVst q=J/ S=2AJA/Vs
(30[qBM+P/D WM4T+04P:74) S)U_ 8 u8 /8: 9Vs (630[qG7J)+/D
G7J+711M+7J S)U
_ 8 )U
28 u8 /8: 9Vs
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CM1_1MM1, -./012/ :_ g7+0/ -O S\J8 28 /8 rV SM1_@ [4U)1 $@FV
$ ? v $ \J w 2d/ #
BM+P/D WM4T+04P:74) S)U_ 8 u8 /8: 9V$ /)*J v $ )U _ w/#du
$ /)*J v @# )U2 R)H4./ SM1N1408 4)H4./V +7=)*,1 0D1 *7+0 H+0D
47.
17J+711M+7J f*47P0. .:* D4T1 HM+x17 ,:H7@
O_ HM+x17 H+0D:*0 *7+0/8 :*M f*47P0. +/2147+7J)1//@
!:7T1M0 0D1 *7+0/ */+7J 4NNM:NM+401 =:7T1M/+:7_4=0:M/ S[4U)1
$@$$V@
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GZ42N)1 $@$> g7+0/ !:7T1M/+:7GZNM1// Em 2ND +7 S4V \2dDM8 47,
SUV 2d/S4V
SUV
hr
km6.041
m1000
km1
ft
m3048.0
mile
ft5280
hr
mile65
hr
mile65 =!!!=
sm06.92
3600shr
ftm3048.0
mileft5280
hr mile65
hr mile65 =!!!=
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Q17/+0. 47, /N1=+i= JM4T+0. :_ 4 `*+,
Q17/+0. ,1N17,/ :7 24// :_ 47 +7,+T+,*4)2:)1=*)1 47, 0D1 7*2U1M
:_ /*=D2:)1=*)1/ 0D40 :==*N. 4 *7+0 :_ T:)*21 (:M )+f*+,/8 ,17/+0.
,1N17,/ NM+24M+).
:7 0D1 N4MP=*)4M )+f*+,8 47, 0: 4 /24))1M1Z0170 :7
012N1M40*M1
-N1=+i= JM4T+0.81 :_ 4 `*+, +/ 0D1 M4P: :_0D1 ,17/+0. 0: 0D1
,17/+0. :_ 4 M1_1M17=1`*+, 40 /:21 /047,4M, =:7,+P:7 S1@J@/N1=+i=
012N1M40*M1 47, NM1//*M1VS 89: V>
(:M )+f*+,/8 89: +/ */*4)). 0D1 ,17/+0. :_H401M 40 ^y! (:M
J4/1/8 0D1M1 +/ 7: /+7J)1 /047,4M,
S=4M1 2*/0 U1 04\17 HD17 +701MNM1P7JN*U)+/D1, ,404V
P&Tsameat both
33
3
g/cmorkg/misunit
LM
][volume
mass
!!"
#$$%
&=
=
==
air
gas gas
SC
s
s
'
'
'
'
'
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Q17/+P1/ :_ J4/1/ (:M +,14) J4/1/ S !; k 6 ?-1k $V
! !:2NM1//+U+)+0.> 4 214/*M1 :_ 0D1 M1)4PT1 T:)*21 =D47J1
:_`*+, :M /:)+, 4/ 4 M1/N:7/1 0: 4 NM1//*M1 S:M 2147
/0M1//V=D47J1
! U*)\ 2:,*)*/ S:M +7=:2NM1//+U+)+0.V>
ww M RT p
M V n
== !
ww M ZRT
p M
V
n== !
V
pV
!
!"=#
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j+/=:/+0. j+/=:/+0. S:M ,.742+= :M 4U/:)*01 T+/=:/+0.V +/ 4
214/*M1 :_ +701M74)8 _M+=P:74) M1/+/047=1 0: `:H*7,1M 47 4NN)+1,
/0M1//@
?1H0:7+47 ()*+,/ 6:/0 :_ 0D1 =:22:7 `*+,/ S1@J@ H401M8 4+MV
3+714M `*+,/> /D14M /0M1// NM:N:MP:74) 0: /D14M M401t
47, 0D1 NM:N:MP:74)+0. =:7/04708 z +/ 0D1 T+/=:/+0. :_0D1
`*+,
z D4/ *7+0/> \Jd2w/ :M Jd=2w/ :M C4w/ S$ Jd=2w/ k $N:+/1t $
C4w/ k $ ?w/d2 #8 47, $ =17PN:+/1 S=CV k%@%$ C8 HD+=D +/ 0D1
T+/=:/+0. :_ H401M 40 M::2012N1M40*M1 V
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j+/=:/+P1/ :_ )+f*+,/! W171M4)). T4M. 4NNM:Z+2401). H+0D
4U/:)*01 012N1M40*M1)
S,1=M14/1/ H+0D 012N1M40*M1V [D1M1 +/ 4 =1M04+7 2:)1=*)4M
+701M=D47J1 U10H117 4,l4=170
)4.1M/ +7 )+f*+,/ b U*0 4/ 0D1 2:)1=*)1/ 4M1 /: 2*=D =):/1M0D47
+7 J4//1/8 0D1 =:D1/+T1 _:M=1/ D:), 0D1 2:)1=*)1/ +7N)4=1 2*=D 2:M1
M+J+,).
!:D1/+:7 N)4./ 47 +2N:M0470 M:)) +7 0D1 T+/=:/+0. :_ )+f*+,/@
O7=M14/+7J 0D1 012N1M40*M1 :_ 4 `*+, M1,*=1/ 0D1 =:D1/+T1
_:M=1/ 47, +7=M14/1/ 0D1 2:)1=*)4M +701M=D47J1@
91,*=+7J =:D1/+T1 _:M=1/ M1,*=1/ /D14M /0M1//8 HD+)1+7=M14/+7J
2:)1=*)4M +701M=D47J1 +7=M14/1/ /D14M /0M1//@
! 0: 4 J::, 4NNM:Z+24P:7 4M1 +7,1N17,170 :_ NM1//*M1T baeT ba
lnorlnln +!+!
- 47, 0 4M1 T+/=:/+0. N4M42101M/ S[4U)1 $@^V
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j+/=:/+0. :_ J4/1/! O7=M14/1/ H+0D +7=M14/+7J 012N1M40*M1
[D1 2:)1=*)1/ :_ J4//1/ 4M1 :7). H14\). \1N0 +7 N:/+P:7
U.2:)1=*)4M =:D1/+:7 S4/ 0D1. 4M1 /: _4M 4N4M0V@
R/ 4,l4=170 )4.1M/ 2:T1 U. 14=D :0D1M 0D1M1 +/ 4
=:7P7*:*/1Z=D47J1 :_ 2:)1=*)1/@ 6:)1=*)1/ :_ 4 /):H1M )4.1M 2:T1
0:_4/01M )4.1M/ =4*/+7J 4 ,M4J8 HD+)1 2:)1=*)1/ 2:T+7J 0D1:0D1M H4.
1Z1M0 47 4==1)1M4P:7 _:M=1@ 640D124P=4)
=:7/+,1M4P:7/ :_ 0D+/ 2:2170*2 1Z=D47J1 =47 )14, 0:?1H0:7 / )4H
:_ T+/=:/+0.@
O_ 0D1 012N1M40*M1 :_ 4 J4/ +7=M14/1/8 0D1 2:2170*21Z=D47J1
U10H117 )4.1M/ H+)) +7=M14/1 0D*/ +7=M14/+7JT+/=:/+0.@
! O7,1N17,170 :_ NM1//*M1 47, ,17/+0. j+/=:/+0. H+)) 4)/: =D47J1
H+0D NM1//*M1 b U*0 *7,1M 7:M24)
=:7,+P:7/ 0D+/ =D47J1 +/ 71J)+J+U)1 +7 J4//1/@n
T
T
!!"
#
$$%
&'
0
0 @% k T+/=:/+0. 40 4U/:)*01M1_1M17=1 012N1M40*M1) %
S[4U)1 $@mV
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614/*M12170 :_ T+/=:/+0. 9D1:2101M
j+/=:2101M
!:4Z+4)A=.)+7,1M9D1:2101M
C4M4))1)AN)4019D1:2101M
!:71A47,AN)4019D1:2101M
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-*M_4=1 [17/+:7 017,17=. :_ 0D1 /*M_4=1 :_ )+f*+, 0: U1D4T1 )+\1
4
/0M10=D1, 1)4/P= 212UM471 =4*/1, U. 4xM4=PT1 _:M=1/ +7
)+f*+,
0H: 7:7A2+Z+7J `*+,/ S1@J@8 4 )+f*+, 47, 4 J4/V H+)) _:M247
+701M_4=1
2:)1=*)1/ U1):H 0D1 +701M_4=1 4=0 :7 14=D :0D1M H+0D_:M=1/ 1f*4)
+7 4)) ,+M1=P:7/8 HD1M14/ 0D1 2:)1=*)1/714M 0D1 /*M_4=1 4=0 :7 14=D
:0D1M H+0D +7=M14/1,_:M=1/ ,*1 0: 0D1 4U/17=1 :_ 71+JDU:M/ +@1@
0D1+701M_4=1 4=0/ )+\1 4 /0M10=D1, 212UM471
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017,17=. :M 1h:M0 :_ 14=D2:)1=*)1 40 0D1 /*M_4=1 0:M14=D 0D1
=170M1 =4*/1/ 0D1`*+, 0M. 0: 04\1 0D1 /D4N10D40 H+)) D4T1 0D1
JM1401/07*2U1M :_ 2:)1=*)1/714M1/0 0D1 /*M_4=18 4/ND1M1
:0D1M _:M=1/ /*=D 4/ JM4T+0.:u17 :NN:/1 /*M_4=1017/+:7 _:M=1/ /:
0D40 0D1`*+, 0M+1/ 0: ,1=M14/1 +0/
/*M_4=1 4M14 0: 4 2+7+2*2 ,1N17,/ :7 )+f*+, 0.N1 47,0.N1 :_
/*M_4=1 H+0D HD+=D +0/D4M1/ 47 +701M_4=1
{ D4/ *7+0 ,.71d=2 :M ?d2
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F !
= ! L = ! !2 " r
P1 " P 2 = !
1
r1
+1
r 2
#
$%
&
'(
r
P 2
P 1
F ! F !
"
#h
-
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r1
r2
Note: r1 = r2 =/= radius of capillary tube
r1
r2
The meniscus in a spherical shapedcapillary tube has the radius
ofcurvature that extends to form asphere.
Top view of the centerplane of the sphere formedfrom the radius
of curvatureof the meniscus.
-
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r1
r2
r1r2
Elliptical shaped capillary tube
Meniscus forms an ellipsoid basedon the two radius of curvature
at theelliptical shaped capillary tube. Top view of the center
plane of
the ellipsoid formed from theradius of curvature of
themeniscus.
-
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r
P 2
P 1
F ! F !
"
#h
Force balance on the liquid column:
F ! =
! ! 2
" rF g = # g" r
2 " hUpward force:Downward force:
F ! = F g
! h = 2 ! " gr
F !
cos " = ! !2 # r !cos "
F ! cos " = F g = V # g
V = $ r 2 ! h
" ! h =2 ! cos "
# gr
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".,M:/04P=/ S()*+, /04P=/V
".,M:/04P=/ +/ 4U:*0 NM1//*M1d_:M=1 1Z1M01,U. 4 `*+, 40 M1/0
!:2N*01 _:M=1/ :7 /*U21MJ1, :Ul1=0/ Q1T1):N +7/0M*2170/ _:M
214/*M+7J NM1//*M1 Q101M2+71 _:M=1/ ,1T1):N1, +7 D.,M4*)+=
/./012/
+7 4NN)+=4P:7/
-
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CM1//*M1 4M+/1/ ,*1 0: 0D1 =:7P7*4) 0M47/_1M :_
2:2170*2 _M:2 +7,+T+,*4) 2:)1=*)1/,*M+7J =:))+/+:7/ H+0D 0D1
H4))/
RU/:)*01 NM1//*M1 _:M=1 N1M *7+0 4M14 0D40 0D1 2:)1=*)1/
+7/+,1 4 =D42U1M 1Z1M0 :7 0D1 =D42U1MH4))/@
4)H4./ N:/+PT1 W4*J1 NM1//*M1
0D1 ,+h1M17=1 U10H117 0D1 4U/:)*01NM1//*M1 47, 0D1 402:/ND1M+=
NM1//*M1S=D42U1M +/ +221M/1, +7 0D1
402:/ND1M1V =47 U1 71J4PT1 +_ NM1//*M1 +7 =D42U1M
+/ )1// 0D47 402:/ND1M1 402:/ND1M+= NM1//*M1
+/ $%$8%%% ?d2# 40 /14A)1T1) S$ C4 k $?d2
#V
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CM1//*M1 T4M+4P:7 +7 4 /04P= `*+, W4*J1 NM1//*M1 k RU/:)*01
NM1//*M1 b R02:/ND1M+= NM1//*M1 O7=:2NM1//+U)1 `*+, *7,1M
JM4T+0.> gH p ! ="
N:+70$
N:+70#
H
CM1//*M1 40 N:+70$ k $ 402
CM1//*M1 40 N:+70# k $ 402 n! ?A
CM1//*M1 ,+h1M17=1 U10H117 N:+70$ 47,N:+70# k N$AN# k A! ?A
( )2121
1
2
1
2
z z g p p
dz g dp g dz dp
z
z
p
p
!!=!"
!="!= # # $
$ $ z
c
c
c
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B%A9
C! D-'5*
?EFG
! -+* ?E
HG
! 7+> ?E
IGB
-',B C/ ! D-'5* EJ K ! D-'5* EF G! -+* EH 3LEI
(:M /+2N)+=+0.8! -+* =47 U1 71J)1=01, U1=4*/1! -+* || !
D-'5*
! 7+> C! D-'5* /EJ KEF 3LEI
17+> C ! 7+> / ! D-'5* = /EJ KEF
3LEIC/FMNOIMN3LHMPCNMQN
GZ42N)1 b CM1//*M1 U4)47=1 :_ 4 247:2101M (:M 4 gA0*U1 247:2101M
/D:H7 +7 0D1 iJ*M18
HD40 +/ 0D1 /N1=+i= JM4T+0. :_ :+)c
H401M
4+M
:+)
CM:N1M0. :_ 247:2101M> [D1 NM1//*M1 40 0D1 /421 D1+JD0 :_
0D1247:2101M =:771=01, H+0D 0D1 /421 `*+, /D:*), U1 U4)47=1,
C402 C402
$
#
D$ k #@m u
D# k %@m u
D& k $@% u
D k &@% u
) 340 ) 540
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!:2NM1//+U)1 `*+,> 8}=:7/0470 S1@J@ 4 J4/V Q17/+0. :_ J4/1/
J171M4)). ,1N17,/ :7 C8 [ O,14) W4/ 1f*4P:7 :_ /0401 4==*M401).
2:,1)/ 0D1
U1D4T+:M :_ 2:/0 J4/1/ *7,1M 17J+711M+7J =:7,+P:7/
w M RT
pnRT pV == ! or
dp = ! ! gdz = ! pg RT
M wdz " dp
p p0
p
# = ln p p0= ! g
RT M w dz
0
z
#
" p=
p0 exp ! g RT M w z
$%&
'()
ln p p0
= ! g R
M wdz
T z( )0
z
#
(:M =:7/0470 012N1M40*M1>
(:M 012N1M40*M1 0D40 T4M+1/H+0D e S [k[SeV V>
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".,M:/04P= _:M=1 :7 /*U21MJ1, /*M_4=1 ?10 _:M=1 :7 4 ,42 :M ):=\
J401 :_ 4 =474)
-:21 _4=0/> ?: /D14M /0M1//1/ +7 4 /04P= `*+,8 0D1 D.,M:/04P=
_:M=1 4=0/
7:M24) 0: 0D1 /*M_4=1@
[D1 M1/*)0470 _:M=1 4=P7J :7 /*M_4=1 +/ _:*7, U. /*22+7J
0D1=:70M+U*P:7/ :_ 0D1 +7i7+01/+24) _:M=1/ :T1M 0D1 17PM1 4M14@
R0 ,1N0DE8 0D1 NM1//*M18 ! k !%n 8?E
h1
h2
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0for2
1
21
2
2
2 2
1
2
1
2
1
==!"
#=
===
=
=
$ $ $
$
h gwhh
gw
hdh gw ghwdh ghdA F
gh p
pdA F
h
h
h
h
h
h A
A
% %
% % %
%
-+7=1 ! +/ 7:0 =:7/0470 0DM:*JD:*0 0D1 4M148 +701JM4) 210D:, +/
711,1,@
gh p p ! += 0
w
h
h1
h2
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".,M:/04P= _:M=1 :7 /*U21MJ1, :Ul1=0
-+2+)4M ,1M+T4P:7 4/ /*U21MJ1, /*M_4=1
V g dV g hdA g ghdA F
gh p
pdA F
V A A
A
! ! ! !
!
====
=
=
" " "
"
V g F buoyancy ! =
Weight ofdisplaced fluid
Pressure of fluid
RM=D+21,1/ NM+7=+N)1
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B*:.47=. B*:.47=. +/ 0D1 710 T1MP=4) _:M=18 ,*1 0: )+f*+,
NM1//*M1 SD.,M:/04P= NM1//*M1V8 4=P7J :7 47 :Ul1=0+221M/1, +7 4
)+f*+,8 :M `:4P7J :7 0D1 )+f*+, //*M_4=1@
[D1 /*U21MJ1, :Ul1=0 711, 7:0 U1 4 /:)+,
".,M:J17 U*UU)1/> M+/1 /):H). +7 4 /*/N17,+7J `*+,S1@J@
H401MV 4/ 0D1. 4M1 /H1N0 4):7J U. 0D1 `:H C:/+PT1). U*:.470
5401M ,M:N)10/> /+7\ +7 :+) ?1J4PT1). U*:.470
?1*0M4)).AU*:.470 N4MP=)1/ 4M1 */1, 4/ `:H 0M4=1M 0:
0M4=1 `:H N4x1M7/ */+7J T+/*4)+e4P:7 01=D7+f*1/
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B*:.47=. (:M=1> R+M [: HD40 012N1M40*M1 2*/0 0D1 4+M +7 4 D:0
4+M U4))::7
S4NNM:Z+2401, 4/ 4 /ND1M1V U1 D1401, 0: )+u 4 U4/\10 ):4,c
Weight of plastic skin of
balloon is negligible Volume of basket is negligible
F buoyancy = ! air gV balloon
F y = 0!
F y = F buoyancy ! W hot air ! W load"= ! atm gV ballon ! ! hot
air gV ballon ! W load = 0
! hot air = ! atm ! W
load
gV ballon
patm MW air
RT hot air =
patm MW air
RT atm! W load
gV ballon
Apply ideal gas law:
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B*:.47=. _:M=1> O22+/=+U)1 `*+,/
What fraction ofwood is belowthe gasoline-water interface?
gasolineV
water V
! gasoline gV gasoline + ! water gV water ! ! wood gV gasoline !
! wood gV water = 0
V gasoline ! wood ! ! gasoline( )= V water ! water ! ! wood ( )V
gasolineV water
=! water
! ! wood ! wood
! ! gasoline=
1 ! 0.960.96 ! 0.72
= 0.167
Fraction of wood in water =1
1 + 0.167 = 0.857
Woods =0.96
Gasoline s =0.72
Water
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B4/+= Gf*4P:7/
[: *7,1M/047, 0D1 U4/+= NM+7=+N)1/ 47, =:7=1N0 :_`*+, 21=D47+=/8
H1 711, _:M2/ :_ 0D1 _:)):H+7J>
6%7-.&89:%7 %; %7? +9= %;
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-
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[.N+=4) 21=D47+/2/ :_ +701M4=P:7 U10H117 /./012 47,/*MM:*7,+7J/
R `:H+7J /0M142 1701M+7J 47,d:M )14T+7J 0D1 /./012 R =:704=0 _:M=1
:7 0D1 U:*7,4M.8 7:M24) :M
047J17P4) 0: +0 S:M 0D1 /0M1//V R U:,. _:M=1 ,*1 0: 1Z01M74)
i1), 0D40 4=0/0DM:*JD:*0 0D1 /./012 S1@J@ JM4T+0.V g/1_*) H:M\ /*=D
4/ 1)1=0M+=4) 171MJ. 1701M+7J 4
2:0:M :M /D4u H:M\ )14T+7J 4 0*MU+71
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W171M4) !:7/1MT4P:7 34H ?:7AM14=P7J -./012 S~ k 24//8
171MJ.8
2:2170*2V
914=P7J /./012 S~ k 24//8 + k /N1=+1/V
GZNM1//1, :7 0D1 U4/+/ :_ *7+0 P218 :M M401/
systemout in X X X !="
i system
idestroyed
icreated
iout
iin X X X X X !="+"
dt
dX x x systemout in =! !!
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B4/+= 34H/ _:M 4 -./012 !:7/1MT4P:7 :_ 64// S1@J@V
64//8 ( 8 :_ 0D1 /./012 +/ =:7/0470
! ! ==="
=
#$%
system system V M system
systemout in
system
V d dm M dt
dM mm
dt
dM
&
0
!!
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[M47/+170 64// B4)47=1 47, -014,. -0401 _:M 7:7AM14=P7J
/./012
+_ /014,.A/0401 /+0*4P:7/
1Z017/+T1 NM:N1MP1/ +7 4 /./012 4M1 =:7/0470 :M,: 7:0 =D47J1
H+0D P21
R M+T1M H+0D 4 `:H M401 0D40 +/ =:7/0470 H+0D P21 R 047\ 0D40 +/
,M4+7+7J 0DM:*JD +0/ U4/18 U*0 +/ 4)/:
/*NN)+1, H+0D 47 +,17P=4) `:H M401 :_ )+f*+, 0DM:*JD 47+7)10
N+N18 /: 0D40 0D1 )+f*+, )1T1) +7 0D1 047\ M124+7/
=:7/0470 H+0D P21
dt dM mm systemout in =! !!
0=dt
dM system
Steady-state mass balance equationis also known as Continuity
Equation
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!:70M:) j:)*21 S:M ;N17-./012 +7 [D1M2:,.742+=/V RMU+0M4M.
T:)*21 +7 /N4=1
0DM:*JD HD+=D `*+, `:H/ W1:210M+= U:*7,4M. +/ =4))1,
=:70M:) /*M_4=1 SM14) :M+24J+74M.8 40 M1/0 :M +72:P:7V
O7 `*+, 21=D47+=/8 +0 +/ 2:M1=:7T17+170 0: _:=*/ 4x17P:7:7 4
=:70M:) T:)*21 0D47 :7 4iZ1, +,17Pi4U)1 f*47P0. :_
24//
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GZ42N)1 b 64// U4)47=1 _:M 047\ 1T4=*4P:7
( )
V
Qt
p
pdt
V
Q
p
dp
pV Q
dt dp
RT p M
Qdt dp
RT M
V
dt d
V dt dV
Qm
dt V d
dt dM mm
p
p
t
ww
out
systemout in
!="!="
!="!="
+=!=!"
==!
# # 0
0ln
0
$ $ $
$
!
!! RT
pM V
nM ww == !
hrs56.2s9210 ==! t
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GZ42N)1 b 314\4J1 40 4 N+N1 l:+70
S314\VR//*2NP:7/>S$V /014,. `:HS#V +7=:2NM1//+U)1 `:HS&V
*7+_:M2 NM:N1MP1/ 40 4)) /1=P:7/
m/s5.4
0
2
433112
4332211
!=
!!
=
=!"!"!"
=!
AQ Au Au
u
Q Au Au Audt
dM mm systemout in !!
! C D EFC
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[47\ )14\4J1
":H ):7J H:*), +0 04\1_:M 0D1 047\ :_210D471 0: ,M:N _M:2m%% \C4
0: ^%% \C4c
610D471
j k $% 2 &
[ k #& !
Leakage rate:
RT
pAm 66.0=!
Hole size = 10 -7 m 2
!min ! !mout =dM system
dt ! !mout = V
d ! dt
! 0.66 pA RT = V RT dpdt
dp
p p0
p
" = ! 0.66 A RT V
dt 0
t
"
t =1.52 V
A RT ln
p0 p
=
8.6 # 104
s $ 1 day
R = 516 J/kg-K
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!:7/1MT4P:7 :_ G71MJ.
[M47/_1M +7T:)T1/ U:0D 24// 47, 171MJ. G71MJ. 0: 0D1 /./012 k
"140
G71MJ. U. 0D1 /./012 k 5:M\
-./012
4( +6
4( 7T'"140 4,,1,8 4W
5:M\ ,:718 4X
dE dW dQ =!
upup )) 22
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5:M\ ,:71 U. -./012 :7 +0/ -*MM:*7,+7J/
-D4u H:M\
():H 5:M\ 5:M\ ,:71 U. ?:M24) -0M1//1/ SNM1//*M1V 40 !- 5:M\
,:71 U. -D14M -0M1//1/ 40 !-
;0D1M H:M\ S1@J@ 1)1=0M+=4)8 1)1=0M:24J71P=V
other f s W W W W ++=
system
out
out
in
in
u gz e M d dW dQ
u gz
pedM
u gz
pedM
!"
#$%
&''(
)**+
,++=-+
''(
)**+
,+++-''(
)**+
,+++
2
22
2
. .
O701M74)
G71MJ. ():H5:M\
C:017P4)G71MJ.
r+71P=
G71MJ.
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5:M\ D4/ /421 *7+0 4/ 171MJ. C:H1M +/ ,1i71, 4/ M401 :_ H:M\ 47,
=47 U1
1ZNM1//1, ,+h1M170). _:M ,+h1M170 /./012/> ():H+7J /0M142>
24// `:H M401 ! H:M\ N1M *7+0
24// (:M=1 ,+/N)4=12170> _:M=1! ,+/N)4=12170 T1):=+0. 9:04P7J
/D4u> [:Mf*1 ! 47J*)4M T1):=+0.
C*2N> T:)*21 `:H M401 ! NM1//*M1 +7=M14/1 -O *7+0 _:M N:H1M
+/ 5 :M ]d/
D4 5 M\ W
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-D4u 5:M\ Q1i71, 4/ H:M\ :0D1M 0D47 `:H H:M\
g/*4)). ,:71 0DM:*JD 4 /D4u !:22:7). 4//:=+401, H+0D 4 N*2N :M
0*MU+71>
C*2N 64=D+71 0D40 ,:1/ H:M\ :7 0D1 `:H8 0D1M1U. +7=M14/+7J
0D1 171MJ. :_ 0D1 `:H (M:2 0D1M2:,.742+= ,1i7+P:7> 71J4PT1
H:M\
5D17 1 [*MU+71
64=D+71 0D40 1Z0M4=0/ 171MJ. _M:2 0D1 `:H8 ,:+7J H:M\:7 0D1
/*MM:*7,+7J/ (M:2 0D1M2:,.742+= ,1i7+P:7> N:/+PT1 H:M\
5D17 1
sW
pt s W W W !=
suppliedW W p p ! =
t t W W ! =obtained
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( )upup !!!! $' *-*-22
-
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-1N4M401X 1 +70: X
! 47, X
' 47, =:7/+,1M Y k $8
0D17 ,+T+,1 0D1 171MJ. 1f*4P:7 U. g m!
( )
g m
Q
g
ee
g m
W
g m
W
g
u z
p
g
u z
p pt
!
!
!
!
!
!!!=
""$
%%' !!(
)
*+,
-""#
$%%&
'++!""
#
$%%&
'++
12
2
22
2
2
11
1
22 . .
Head lossPump headTurbine head
L pt hhh g u z p
g u z p =+!"
#$%
&' (()
*++,- ++!(()
*++,- ++
22
2
22
2
2
11
1
. .
"14, ):// =47 U1 =:7/+,1M1, 4/ 171MJ. ):// ,*1 0: :T1M=:21
:_ _M+=P:7
( ) QeemW u gz pu gz pm s !!!! !!=!"#
$%&
(()++,
++!(()++,
++12
222
2111
1
22.
/ .
/
Specific Weight = $" g
GZ42N)1 b C*2N+7J 7AC170471
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GZ42N)1 b C 2N+7J 7AC170471
3 in ID pipeFriction loss =
Pump efficiency = 75%um = 25 ft/sn-pentane density = 39.3 lb
m/ft3
!"
#$%
&2
22 ft5.2
s
um
Find the power required to drive the pump The pressure at the
inlet of the pump The pressure at the outlet of the pump
4.5 ft
4 ft
40 ft
Supply tank
Storage tank
R//*2NP:7/
Storage tank
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-014,.A/0401 O7=:2NM1//+U)1 `:H ?1J)1=0 _M+=P:7 :_ 0D1 /D:M0
N+N1 40 0D1 +7)10 :_ 0D1 N*2N [4\1 0D1 /*NN). 047\ 4/ S$V 47,
:*0)10 0: /0:M4J1 047\ 4/ S#V B:0D S$V 47, S#V 4M1 :N17 0:
402:/ND1M1 -*NN). 047\ +/ )4MJ1 =:2N4M1 0: 0D1 N+N1
( )
ft2.982.322
2540
2.32
255.2
2
5.2
22
2
2
21
2
2
=!!"
#$$%
&'
((('=
)*
+,-
.(((
=
p
p
h
g u
z z g u
h
L pt hhh
g
u z
p
g
u z
p=+!"
#
$%&
'((
)
*++
,
-++!((
)
*++
,
-++
22
2
22
2
2
11
1
. .
[ ]ft5.22
g u
h m L =
Friction loss check unit
4.5 ft
4 ft
40 ft
Supply tank
( ) slb2.48
123
4253.39 m
2=!!== " # Aum m!
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4.5 ft
4 ft
40 ft
Supply tank
Storage tank
kW56.8slbft
317,6
ftlbslb
2.321ft
75.03.982.322.48
f
m
2f s
fts
lb
supplied
2m
=
!=
!!"
##$%
&&'(
)!!""=
=*= p
p p p
W W ghmW
+
!!!
( ) s
[: i7, 0D1 NM1//*M1 40 0D1 +7)10 :_ 0D1 N*2N8 4NN). 171MJ.
U4)47=1U10H117 0D1 /*NN). 047\ S$V 47, 0D1 N*2N +7)10 S#V>
( )
( )
psig42.1
02.322
255.4
2.322.32
3.39
1440
02
2
22
2
221
21
!=
=
"
!++
"
"!
=!
+!+!
p
p
g u z z
g p p
#
L pt hhh g u
z p
g u
z p
=+!"#
$%&
'(()
*++,
-++!(()
*++,
-++
22
2
22
2
2
11
1
. .
[: i7 0D1 NM1//*M1 40 0D1 :*0)10 : 0D1 N*2N8 4NN) 171MJ
U4)47=1
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[: i7, 0D1 NM1//*M1 40 0D1 :*0)10 :_ 0D1 N*2N8 4NN). 171MJ.
U4)47=1U10H117 0D1 N*2N +7)10 S$V 47, :*0)10 S#V>
( )
( )
psig2.25
02.985.0
2.322.323.39
14442.1
0
2
2
2121
=
=+!
"
"!!
=+!+!
p
p
h z z g
p p p #
L pt hhh g u z p
g u z p =+!"
#$%
&' (()
*++,- ++!(()
*++,- ++
22
2
22
2
2
11
1
. .
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(+7, 0D1 M401 :_ N:H1MJ171M4P:7@
hL=1.5 m" t = 0.87# = 9810 N/m 3
Q = 14.1 m 3/s
L pt hhh g u
z p
g u
z p
=+!"#
$%&
'(()
*++,
-++!(()
*++,
-++
22
2
22
2
2
11
1
. .
Assumptions:
Pressures at point 1 and point 2 are both atmospheric
Velocities at point 1 and point 2 are nearly zero
No pump
-
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ht = z1 ! z2( )! h L = 61 ! 1.5
= 59.5 m
!W t = !mgh t !W obtained = ! t !W t = ! t Q" ht
= 0.87 " 14.1 " 9810 " 59.5
=
7.16 " 106
W= 7.16 MW
B1M7 *))+ / Gf*4P 7
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B1M7:*))+/ Gf*4P:7 R M1,*=1, _:M2 :_ G71MJ. Gf*4P:7 U. +2N:/+7J
4 =1M04+7
4//*2NP:7/> ():H +/ /014,. S4NN)+1/ 0: U:0DV ()*+, +/
+7=:2NM1//+U)1t +@1@ ,17/+0. +/ =:7/0470 S4NN)+1/ 0: U:0DV ?: H:M\
1h1=0/ S:M /D4u H:M\/Vt +@1@ 0D1 `*+, 71+0D1M N1M_:M2/ H:M\
S4/ +7 0*MU+71V8 7: H:M\ D4/ U117 N1M_:M21, :7 +0 S4/ +7 4
N*2NV
():H +/ _M+=P:7)1//t +@1@ +7T+/=+, S7:7T+/=:*/V `:H S7: /D14M
1h1=0V :M7: D14, ):// RNN)+1, U10H117 0H: N:+70/ 4):7J 4 /0M142)+71
S51 H+)) M1T+/+0 0D+/
N:+70 HD+=D +/ +2N:M0470 +7 ,1M+T+7J 0D1 B1M7:*))+ / Gf*4P:7V
-0M+=0).8 0D1 `:H N+N1 +/ 4 /24)) /0M142 0*U1 17=):/+7J 4
/0M142)+71t +@1@Y k $ S:M 0D1 `:H +/ *7+_:M2 40 14=D =M://
/1=P:7 +@1@0D1 T1):=+0. +/ =:7/0470V
2
2
221
2
11
22 gz
u p gz
u p++=++
! !
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G4=D 01M2 =47 4)/: U1 1ZNM1//1, +7 01M2 :_ )17J0DS +T+ +7J 14 D
U ? V8 47 M1NM1/170/ 4 1M04+7 0 N1
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S,+T+,+7J 14=D U.? V8 47, M1NM1/170/ 4 =1M04+7 0.N1 :_D14,>
NM1//*M18 1)1T4P:7 S:M /04P=V8 T1):=+0.
2
2
221
2
11
22 z
g
u p H z
g
u p++==++
! !
Pressure increase across apump, ! p = head increase! h = ! p /"
g
(energy equation applies)
2
streamlineaalongconstant,2
2
=++ z g
u p
!
RNN)+=4P:7 : B1M7:*))+ / CM+7=+N)1> 214/*M12170 :
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RNN)+ 4P:7 :_ B1M7: ))+/ CM+7 +N)1> 214/ M12170 :_/04P= 47,
/04J74P:7 NM1//*M1/ SC+0:0 [*U1V
BG M1)401/ =D47J1/ +7 /N11, 47, NM1//*M1 4):7J 4/0M142)+71 +7 47
+7=:2NM1//+U)18 _M+=P:7)1// `:H@
-04P= CM1//*M1 S0D1M2:,.742+= NM1//*M1V CM1//*M1 /117 U. `*+,
N4MP=)1 4/ +0 2:T1/
Q.742+= CM1//*M1>2
2
1u !
Stagnation Pressure b pressure obtained when a flowing
fluid is decelerated to zero velocityby a frictionless
process
b measured using a stagnationpressure probe (pitot tube)
C+0:0 [*U1> */1 *) :M 214/*M12170 : T1):=+0
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C+0:0 [ U1> /1_ ) _:M 214/ M12170 :_ T1): +0.:_ 2:T+7J =M4u
/*=D 4/ 4 U:40 :M 47 4+MN)471
2
0
22
00
2
1
22
u p p
u pu p
!
! !
+="
+=+
stagnationpressure static
pressure
dynamicpressure
( ) !
p pu
"=#
02
-04J74P:7 NM1//*M1 +/ 0D1 0:04) NM1//*M1
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-04J74P:7 NM1// M1 +/ 0D1 0:04) NM1// M1+7=)*,+7J 0D1 /04P=
NM1//*M1 47, ,.742+=NM1//*M1 SHD17 \+71P= 171MJ. +/ =:7T1M01,+70:
NM1//*M1 171MJ.V
GZ42N)1 b -014,. 4+M `:H 0D:*JD 4 7:ee)1
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GZ42N)1 b 014,. 4 M :H 0D: JD 4 7:ee)1
Nozzle: bydefinition is adevice foracceleratingflow.
-+7=1 0D+/ +/ M1J4M,+7J J4/ `:H8 HD+=D +/ 4 =:2NM1//+U)1 `*+,
U.740*M18 H1 711, 0: =D1=\ +_ H1 =47 4//*21 +7=:2NM1//+U)1 `:H@
Introduce a dimensionless number: Mach Number (M) = V/c;V = flow
speed of gas, c = local speed of sound (~343 m/s in air)M < 0.3:
gas flow can be treated as incompressible as the density change is
S$V /014,. `:H@S#V +7=:2NM1//+U)1 `:H@S&V _M+=P:7)1// `:H@S^V
`:H 4):7J 4 /0M142)+71@SmV M1/1MT:+M +/ )4MJ1 =:2N4M1,
0: 0D1 /+ND:7@
SEV C:+70 $ 47, # 4M1 U:0D 40402:/ND1M+= NM1//*M1
5D40 +/ 0D1 T1):=+0. :_ H401M )14T+7J 0D1 /+ND:7c5D40 +/ 0D1
NM1//*M1 :_ H401M 40 N:+70 Rc
Free jet
2
2
221
2
11
22 gz
u p gz
u p++=++
! !
A
914MM47J1
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!"## %#&
'
!"## %#&
' [: ,101M2+71 0D1 NM1//*M1 40 R8 4NN). BG U10H117
N:+70 $ 47, R S:M U10H117 R 47, #V>
RNN). =:7P7*+0. 1f*4P:7>
914MM47J1
( )122 2 z z g u !!=
A A A
gz u p
gz u p
++=++ 22
2
1
2
11
! !
2uu
A =
( )2
2
211
u z z g p p A A ! ! ""+=
!4*P:7/ :7 0D1 */1 : B1M7:*))+ / Gf*4P:7
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!4 P:7/ :7 0D1 /1 :_ B1M7: ))+ / Gf 4P:7
-*U/:7+= ,+h*/1M S,+T1MJ+7J /1=P:7 :M /*,,17 1ZN47/+:7V 4,T1M/1
NM1//*M1 JM4,+170 )1, 0: JM:H0D :_ U:*7,4M. )4.1M 47, `:H
/1N4M4P:7 S40 U1/0 47 4NNM:Z+24P:7V
():H 0DM:*JD 4 24=D+71 1@J@ NM:N1))1M8 N*2N8
0*MU+718H+7,2+))
6:T+7J /*M_4=1/t +2N://+U)1 0: D4T1 ):=4)). /014,. `:H
47,/0M142)+71/
():H 0DM:*JD 4 D.,M4*)+= l*2N S4UM*N0 =D47J1 +7 ,1N0D 47,171MJ.
)://V j+:)170 2+Z+7J :==*M/8 +2N://+U)1 0: +,17P_. /0M142)+71/
R+M `:H 0DM:*JD D14P7J 1)12170 [12N1M40*M1 =D47J1/ =4*/1
/+J7+i=470 =D47J1 +7 ,17/+0. :_ J4/1/
!:2NM1//+U)1 J4/ `:H Q17/+0. =D47J1/ S( %@&V
6:2170*2
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6:2170 2 ?1H0:7 / -1=:7, 34H :_ 6:P:7>
4NN)+1, 0: 247. NM:U)12/ +7T:)T+7J _:M=147, 4==1)1M4P:7 +7 /:)+,
21=D47+=/
[D1 )4H +/ 1f*4)). */1_*) +7 `*+, 21=D47+=/_:M 474)./+/ :_>
_:M=1/ :7 N+N1 U17,/ 0DM*/0 NM:,*=1, U. 4 0*MU:l10
4==1)1M4P:7 :_ 4 M:=\10 10=
6:2170*2 0M47/ 1M
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6:2170 2 0M47/_1M
6:2170*2 +/ U1+7J 0M47/_1MM1, ,*M+7J 0D+/ NM:=1//@ [D1 `*+,
0:*=D+7J 0D1 0:N N)401 H+)) 2:T1 40 0D1 /421
T1):=+0. 4/ 0D1 N)401 HD+)1 0D1 `*+, 0:*=D+7J 0D1U:x:2 N)401
H+)) 7:0 U1 2:T+7J@ S7:A/)+N =:7,+P:7V
6:2170*2 +/ U1+7J 0M47/_1MM1, U. _:M=1/@
Moving plate
Stationary plate
Forceu x
t2t0 t1
5D17 0:N N)401 +/ 2:T1, U. 4_:M=18 0D1 )+f*+, U10H117 0D10H:
N)401/ H+)) 4)/: 2:T1U1=4*/1 :_ 0D1 +701M74)_M+=P:74) M1/+/047=1 +7
0D1 `*+47, 0D1 _M+=P:7 U10H117 0D1`*+, 47, 0D1 N)401@
x
y
6:2170*2 B4)47=1
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6:2170 2 B4)47 1 6:2170*2 k 24// Z T1):=+0.
(:M=1 =47 U1 T+1H1, 4/ 0D1 2:2170*2 `:H M401>
R 2:2170*2 U4)47=1 +/ 4 U4)47=1 :_ _:M=1/>
6:2170*2 U4)47=1 40 /014,. /0401 +7T:)T1/ 4U4)47=1 :_ 4))
_:M=1/>
2
22
2
11221121 u Au Aumum
dt
d F ! ! "="="==# !!
"M M
M
( ) ( ) 2 Auu Auumdt
umd F ! ! ===== !
"!
"M
TML ][== um
!!M
0==! dt d
F M
!!
(:M N+N1 `:H8 0D1 :M=1/ +7T:)T1 +7=)* 1/
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( ) Lg D DL p p Ddt
d F w 44
2
21
2!
" ! # ! $$$==%
M !
(:M N+N1 :H8 0D1 _:M 1/ +7T:)T1, +7 ) ,1/NM1//*M1 _:M=18 /D14M
/0M1//8 47, JM4T+0.
sw A!
[D+/ 1f*4P:7 +/ :71 H4. 0: ,101M2+71 0D1 /D14M /0M1//@ R0 /014,.
/040180D1 _:M=1/ 4M1 U4)47=1>
( )
( )44
044
21
2
21
2
Dg
L
p p D
Lg D
DL p p D
dt d
F
w
w
! "
# ! # "
#
$$
=
=$$$==% M !
O_ 4 =D47J1 +7 24// +/ +7T:)T1,8 0D1 2:2170*2
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U4)47=1 H+)) 711, 0: U1 M1HM+x17 4/>( )
dt
dmu
dt
dum
dt
mud
dt
d F +===
! M !
GZ42N)1 b 4==1)1M4P:7 _M:2 4 H401M l10Jet at velocity, u
Nozzle of area, A
Cart moving at velocity, v
Apply momentum balance:
( )vumdt
dM v
dt
dv M F !=+= !
M
Aum ! =!
( )vu A ! "
( ) ( )
( ) M
vu A
dt
dv
vu Auvu Avdt dv M
2
onaccelerati !==
!"=!"+
#
# #
GZ42N)1 b [DM*/0 : 4 [:. 9:=\10
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GZ42N)1 b [DM /0 :_ 4 [:. 9: \10
(+7, 0D1 _:M=1 711, 0: /*NN:M0 0D1 M:=\10@
u = 450 m/s" = 0.5 kg/m 3
m = 40 g
1 cm
F s?Draw force diagram on the rocket
mg
F s
m j u
N56.7
4504
01.05.081.904.0
0
22
!=
""
"!"=
"!=!=
=!+=#
$
% u Aumg ummg F
mg um F F
j s
j s y
!
!
-ve sign means F s should be downwards(prevent the rocket from
flying up)
GZ42N)1 b !:7=M101 `:H+7J +70: 4 =4M02400 k / 3
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(:M=1 U4)47=1>mu
m cart+concrete g
T
N5471
360060sin31.02400
60sin
0sin
N1080
60cos31.02400 60cos
0cos
2
2
2
2
=
+!!!=
+=
=
"+=
=
!!!==
="=
#
#
g m Au N N g mmu F
AuT
T mu F
cart
cart y
x
$ %
$
%
N
3 m/sFind(a)the weight on the scale, and(b) tension on the
cableat the instancethe weight of cart + concrete = 3600 Ncable
" = 2400 kg/m 3
Area of concrete stream = 0.1 m 2
GZ42N)1 b (:M=1 :7 4 N+N1 U17
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GZ42N)1 b (:M 1 :7 4 N+N1 U17,
(:M=1 ,+4JM421 m
Crude oilS = 0.94
2 m 3/sp = 75 kPa (gauge)
Find the force needed at the boltedflange to hold the bend in
place
m bend+oil g
mu
mu
Rx
R y
+
( )
( )( ) ( )( )
kN53.8
30cos15.0000,755.0
2940
cos1cos1
0coscos
22
2
=
!""#
$%%&
'((+
((=
!"#$
%&'
+=
!+=
=!!+!+=)
* *
+ , +
+ +
pA A
Q
Q pAum R
Rum pA pAum F
x
x x
!
!!
pA
pA
Top view
( )
( )
kN8.31
30sin5.0000,755.0
2940
sinsin
0sinsin
22
2
2
!=
""#
$%%&
'((+
((!=
""#
$
%%&
'+!=+!=
=!!+=)
* *
+ , +
+ +
pA A
Q pAum R
um R pA F
y
y y
!
!
Weight of the bend = 4 kNVolume of the bend = 1.2 m 3
GZ42N)1> [47\ QM4+7+7J S!4/1 OV
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R//*2NP:7/> -014,.8 _M+=P:7)1// `:H O7=:2NM1//+U)1 *+,
!M://A/1=P:74) 4M14 :_ 047\ +/ )4MJ1
3+f*+, ,1/=17,+7J /):H).> 7: _M+=P:74),+//+N4P:7 *7)1// `*+,
+/ T1M. T+/=:*/47, T+M0*4)). /014,. `:H
9:*7,1, 7:ee)1 :M :M+i=1> /0M142)+71/ N4M4))1) 40 1Z+0
47,
NM1//*M1 +/ *7+_:M2). 402:/ND1M+=
?: 1Z01M74) H:M\
Apply BE between points 1 and 2 along a streamline:
2
2
221
2
11
22 gz
u p gz
u p++=++
! ! gh A AuQ ghu
2
2
2
2
==!
=!
The fluid velocity is exactly the same as the velocity the fluid
would attain by fallingfreely from rest a distance h (Torriceli
s Equation) .
GZ42N)1> [47\ QM4+7+7J S!4/1 OOV
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R//*2NP:7/> -014,.8 _M+=P:7)1// `:H O7=:2NM1//+U)1 *+,
!M://A/1=P:74) 4M14 :_ :M+i=1
=:7/+,1M4U)1 =:2N4M1, 0: 0D1 047\/*M_4=1 [D1 3+f*+, H+)) ,1/=17,
H+0D
214/*M4U)1 rG 9:*7,1, 7:ee)1 :M :M+i=1>
/0M142)+71/ N4M4))1) 40 1Z+0 47,NM1//*M1 +/ *7+_:M2).
402:/ND1M+=
?: 1Z01M74) H:M\
Apply BE between points 1 and 2 along a streamline:
2
2
221
2
11
22 gz u p gz u p ++=++
! ! ?2 =u
With 2 unknowns; we need another equation to solve the problem.
Using the mass balance for the fluid, we can solve for u1 in terms
of
u2, A1 and A2 ; substitution in BE, we can obtain u2 and hence
u1.
[47\ ,M4+7+7J 0DM:*JD 4 /D4MN :M+i=1
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[47\ ,M4 7 7J 0DM: JD 4 /D4MN :M i 1
O_ 0D1 :M+i=1 +/ /D4MNA1,J1, H+0D 4M14 # > 0D1 =M://A/1=P:74)
4M14 :_ 0D1 l10 H+)) =:7P7*1 0:
=:70M4=0 4u1M +0 )14T1/ 0D1 :M+i=1 U1=4*/1 :_ +71MP4 0: 4T4)*1 -
40 4 ):=4P:7 \7:H7 4/ T174 =:70M4=048 HD1M1 0D1/0M142)+71/ 4M1
N4M4))1) 0: :71 47:0D1M
4 =:1
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!:7P7*+0. Gf*4P:7
B1M7:*))+ / Gf*4P:7 S4NN)+=4U)1 *N/0M142V
p1!
+ u12
2= p2
! + u2
2
2
p1 ! p2!
=u2
2 ! u12
2
=1
2
A12
A22 u1
2 ! u12"
#$
%
&' =
u12
2
A12
A22 ! 1
"
#$
%
&'
C$ C# C&
Unknown location
2
11213332211
and A
Auuuu Au Au Aum ==!=== " " " !
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C#R C&R
2* # 2* &
6:2170*2 B4)47=1>
C#R b C&R n 2* # n SA2*&V k %
$ ( )231
32 uu Am
p p !=
! !
CM1//*M1 ,M:N +7 ,:H7/0M142 /1=P:7>
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;T1M4)) NM1//*M1 ,M:N>
O0 =47 U1 /117 0D40 ! & +/ 4)H4./ )1// 0D47 ! $8+7,+=4P7J 47
:T1M4)) ):// :_ 171MJ. :M */1_*) H:M\@
( ) ( ) !!"
#$$
%
&'='='='
2
12
121123
1
32 1
A
Auuuuuu
A
m p p
( (
!
p1 ! p3!
=
p1 ! p2!
"
#$%
&'+
p2 ! p3!
"
#$%
&'
=u1
2
2
A12
A22 ! 2
A1
A2+ 1
"
#$
%
&' =
u12
2
A1
A2! 1
"
#$
%
&'
2
9:042101M/
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T4M+4U)1 4M14 2101M/
D4/ 714M). =:7/0470 NM1//*M1 47,,1N17, :7 =D47J+7J =M://
/1=P:74)4M14 0: +7,+=401 `:H M401
4M1 1Z0M121). /+2N)18 M:U*/0 ,1T+=1/0D40 =47 214/*M1 `:H M401/
:_ U:0D)+f*+,/ 47, J4//1/ `*+, `:H/ *N 0DM:*JD 0D1 04N1M1, 0*U1
47, /*/N17,/ 4 `:40 +7 0D1 =:)*27 :_`*+,
0D1 1f*+)+UM+*2 N:/+P:7 :_ 0D1 `:40+7,+=401/ 0D1 `:H M401 :7 4
24M\1, /=4)1
CM+7=+N)1 :_ ():H 614/*M12170 U. 4 9:042101M
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[D1 M:042101M ,1N17,/ :7 0D1 =D47J1 :_ 47 477*)4M 4M144 U10H117
0D1 `:40 47, 0D1 0*U18 HD+=D +/ 4 _*7=P:7 :_0D1 T1MP=4) ):=4P:7 :_
0D1 `:408 0: .+1), 47 1//17P4)).iZ1, NM1//*M1 ,M:N 40 4)) `:H
M401/
[D1 477*)4M 4M14 _*7=P:7/ 4/ 47 :M+i=1 :_ T4M+4U)1 4M14 [D1 `:H
M401 =47 U1 ,101M2+71, _M:2 0D1 1f*+)+UM+*2
N:/+P:7 :_ 0D1 `:40 j+/=:*/ 1h1=0/ 4M1 *7+2N:M0470 S0M*1 _:M
24l:M+0. :_
+7,*/0M+4)). +2N:M0470 `*+,/V
R7 474)./+/ :_ 0D1 `:H 47, 1f*+)+UM+*2 N:/+P:7 :_ 0D1 `:40
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2211 Au Aum ! ! ==!
2
2
221
2
11
22 gz
u p gz
u p++=++
! !
( ) ( ) 0122121 =!""$
%%'
!!!!+! Mg g M A z z mumu A p p f
( (
6:2170*2 1f*4P:7
?1J)1=0 /)+JD0 04N1M+7J :_ 0*U1
W:T1M7+7J 1f*4P:7/> !:7P7*+0. :M 24// =:7/1MT4P:7
B1M7:*))+/ 1f*4P:7
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![Marine Engineering Part1[1]](https://img.pdfslide.net/doc/110x75/577cd4d81a28ab9e78994467/marine-engineering-part11.jpg)
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