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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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[.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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Documents

CH2103 Part1-1