Fluid Mechanics

Fluid Mechanics

(Fluid Mechanics)

.. 2553

Fluid Mechanics

Fluid Mechanics

.

Fluid Mechanics

i Fluid Mechanics

Contents

1 (Basic concept and Fluid property) 1-1 1.1 1-1 1.2 (Dimension) 1-1 1.3 (Unit) 1-2 1.4 1-4 1.5 (Basic concept and Fluid property) 1-5 1.5.1 (Density) 1-5 1.5.2 (Specific weight) 1-5 1.5.3 (Specific gravity) 1-5 1.5.4 (Specific volume) 1-6 1.5.5 (Viscosity) 1-6 1.5.6 (Compressibility) 1-13 1.5.7 (Surface tension) 1-14 2 (Fluid static) 2-1 2.1 (Pressure) 2-1 2.1.1 (Pressure at a point in fluid) 2-2 2.1.2 (Variation of pressure in static fluid) 2-3 2.1.3 (Measurement of Pressure) 2-6 2.1.4 (Pressure Units) 2-6 2.1.5 (Pressure gauge) 2-8 2.2 (Pressure Force on a Plane Surface) 2-11 2.3 (Pressure Force on a Curved Surface) 2-18 2.4 (Buoyancy Force) 2-27 2.4.1 2-30 (Stability of Floating and Submerged Bodies) 2.5 2-36

(Variation of fluid pressure in moving container) 2.5.1 2-36 (Fluid pressure in Linear moving container) 2.5.2 2-42 (Fluid pressure in angular moving container)

ii Fluid Mechanics

Contents

3 (Basic of flow theorem) 3-1 3.1 (Flow classification) 3-2 3.2 (Flow analysis with Control Volume method) 3-5 3.3 (Reynolds Transport Theorem) 3-7 3.4 (Mass Conservation) 3-11 4 (Energy equation) 4-1 4.1 Euler (Eulers Energy equation) 4-1 4.2 Bernoulli (Bernoullis equation) 4-3 4.3 (Energy equation) 4-12 4.3.1 (Head loss) 4-12 4.3.2 (Pump) 4-21 4.3.3 (Turbine) 4-24 5 (Momentum equation) 5-1 5.1 (Linearly Monentum Equation) 5-2 5.3 5-14 (Momentum equation for moving control volume) 6 (Flow in Pressure Conduit) 6-1 6.1 (Behavior of flow in pipe) 6-2 6.2 (Entrance Flow Development) 6-3 6.3 (Friction head loss or Major loss) 6-5 6.3.1 6-7 (Friction factor for larminar flow) 6.3.2 6-9 (Friction factor for turbulent flow in smooth pipe) 6.3.3 6-12 (Friction factor for turbulent flow in rough pipe) 6.4 (Minor loss) 6-14

iii Fluid Mechanics

Contents

7 (Open channel fFlow) 7-1 7.1 (Type of channel) 7-1 7.2 (Open channel flow classification) 7-2 7.2.1 7-2 (Classified by flow pattern) 7.2.2 (Classified by stage of flow) 7-5 7.3 (Basic equation of open channel flow) 7-6 7.3.1 (Continuity Equation) 7-7 7.3.2 (Energy Equation) 7-8 7.3.3 (Momentum Equation) 7-9 7.4 (Steady Uniform Flow) 7-10 7.5 7-15 (Specific energy and steady rapidly varied flow) 7.6 7-2 (Momentum function and steady rapidly varied flow) 8 (Dimension analysis and Similarity) 8-1 8.1 (Dimension analysis) 8-1 8.1.1 (Dimension and Unit) 8-2 8.1.2 8-4 (Dimension analysis by Buckingham Pi Theorem) 8.2 (Similarity) 8-13 8.2.1 (Similarity analysis) 8-13 8.2.2 8-15 (Dimensionless term in Similarity analysis) 8.2.3 (Case study of similarity analysis) 8-18 (Exercise) E-1 1 E-1 2 E-3 3 E-11 4 E-13 5 E-15 6 E-17 7 E-20 8 E-21 R-1 A-1

iv Fluid Mechanics

Contents

1-1 Fluid Mechanics

Fluid Property

1

1.1 (Fluid mechanics) (Fluid Statics) (Fluid Dynamics) 1.2 (Dimensions) 2

1.2.1 (Primary Dimensions or Basic Dimensions) 4

- (Mass) M - (Length) L - (Time) T - (Temperature)

1.2.2 (Secondary Dimensions) (L) (L) (L) L3 (L) (T) L/T 1.2

1.1 (F) F = ma m () M a () LT-2 F () MLT-2 Ans

1 (F)

1-2 Fluid Mechanics

Fluid Property

1.2 - T (N) T-1 Ans -

T () T-1 Ans 1.2

1.3 (Unit) 2

- System International Unit SI SI - British Gravitational System BG

SI 1.1 1.2 1.1 SI BG

1.2

1-3 Fluid Mechanics

Fluid Property

1.3 Newton F = ma N (Newton) m () M kg a () LT-2 m/s2 F () MLT-2 kg m/s2 1 N = 1 kg m/s2 Ans

Prefixes

Prefixes

2.5 (km) 2103 2,000 (m) k 103 1.5 (mm) 1.510-3 0.0015 (m) m () 10-3

1.3 Prefixes

1-4 Fluid Mechanics

Fluid Property

1.4 (Fluid) (Liquid) - (Free surface) ( ) (Gas)

1.1

1-5 Fluid Mechanics

Fluid Property

1.5

1.5.1 (Density or Mass Density) (mass) (Volume) ( rho)

m

VolumeMass : ./.. (kg/m3)

4OC 1 1,000 ./.. (W) 1.5.2 (Specific Weight) (Weight) (Volume) ( gamma)

g mg

VolumeWeight : /.. (N/m

3)

4OC 1 9,810 /.. (W) 1.5.3 (Specific Gravity) 4 OC 1 S SG

wwww gg

WWSG

:

4OC 1 1

1-6 Fluid Mechanics

Fluid Property

1.5.4 (Specific Volume) (Volume) (mass)

1

MMassVolume : ../. (m3/kg)

1.5.5 (Viscosity) (Deformation) (shear stress) (Viscosity) (Friction) 1.2 (Cohesion Force)

1.2 1.2 (AD) V (BC) V+v v t = 0 ABCD t = t ABCD

yatan

yatan

= tva

1-7 Fluid Mechanics

Fluid Property

yv

tytv

--------- (1.1)

t

Rate of shear strain () Shear stress ()

t

yv --------- (1.2)

Shear stress

dydv --------- (1.3)

1.3 (Newtons equation of viscosity) ( mu) (Dynamic Viscosity) (Absolute Viscosity) FL-2T N s / m2 lb sec / ft2 dv/dy

1.3 (dv/dy)

1-8 Fluid Mechanics

Fluid Property

1.4 (Absolute Viscosity)

(Kinematic Viscosity) ( )

--------- (1.4) L2/T m2/s ft2/sec

1-9 Fluid Mechanics

Fluid Property

1.5 (Kinematic Viscosity)

1-10 Fluid Mechanics

Fluid Property

dv/dy (Newtonian fluid) dv/dy (Non-Newtonian fluid) 3

1) (Dilatant fluid) 2) (Pseudoplastic fulid) 3) (Plastic fluid)

(yield) Newtonian

1.6 (dv/dy)


Fluid Property

1.4 plate (F) plate v ()

a

v0a0v

yv

dydv

av

dydv

F = A A = ( 2 ) A = 2 (2b b) = 4b2

F = t A 24bavF

avb4F

2 Ans

1.5 parabola 6 m/s 3 m

Parabola (v-k) = C(y-h)2 ; (h,k) parabola

parabola y = 0 ; v = 6 h = 0 m k = 6 m/s

(v-6) = C(y-0)2 y = 3 ; v = 0 (0-6) = C(3)2 C = -(6/9) = -(2/3) 232 y6v y

dydv

34 y = 3 134 s43dy

dv

= = (910-5 m2/s)(1000 kg/m3) = 0.009 kg/s m ( N s/m2)

dydv = (0.009 Ns/m2)(4 s-1) = 0.036 N/m2 Ans


Fluid Property

1.6 1 ./ 2 ./ ()

bV

dvdv

24W m sNmskg72.0000,19.0108SG 2m

kN32.40005.0372.0

bV

20.005.032.4DLAF = 135.7 N Ans


Fluid Property

1.5.6 (Compressibility) () Compressibility (Modulus) () Bulk Modulus (k)

d

dpk --------- (1.5)

dp = d = =

2

- (Incompressible fluid)

- (Compressible fluid)

1.7 Incompressible fluid Compressible fluid


Fluid Property

1.5.7 (Surface tension) ws LF --------- (1.5)

Fs = (N) = (N/m) = () Lw = (m)

1.8

1.9

2 (A) (C) 2 1.10 () ()


Fluid Property

1.10

1.10 () ()

1.11

P Pa r 1.11 () Fs = 2a2 rPrP r2 = 2a rPP aPP

2r --------- (1.6)

1.11 () r Fs = 2a2 rPrP r22 = 2a rPP aPP

4r --------- (1.7)

1.6 1.7 4


Fluid Property

Capillarity

1.12 Capillarity

r

cos2h --------- (1.8)

r = =

1.4

2-1 Fluid Mechanics

Fluid Static

2

(Fluid Statics)

2.1 (Pressure) ( FL-2 ML-1T-2) dF dA A

dAdFP --------- (2.1)

A

AFP --------- (2.2)

SI / (N/m2)

2.1

2-2 Fluid Mechanics

Fluid Static

2.1.1 (Pressure at a point in fluid) 2.2

2.2

maF Y sindxdlPdxdzPy = yadxdydz21 dzsindl dxdzPdxdzPy = yadxdydz21 PPy = yady21 dy 0 0ady21 y PPy --------- (2.3) Z dxdydz21cosdxdlPdxdyPz = zadxdydz21 dycosdl dxdydz21dxdyPdxdyPz = zadxdydz21 PPz = dz21adz21 z dy 0 0adz21 z PPz --------- (2.4) 2.3 2.4 (Pascals law) Static Pressure

2-3 Fluid Mechanics

Fluid Static

2.1.2 (Variation of pressure in static fluid) 2.3 P

2.3

Y zx

2y

yppzx

2y

yppFy

zyxypFy --------- (2.5)

Y X zyx

xpFx --------- (2.6)

zyxzpFz --------- (2.7)

2.5 2.6 2.7 kFjFiFF zyxs

zyxk

zpj

ypi

xpFs

--------- (2.8)

k

zp

jyp

ixp

p --------- (2.9)

2.8 2.9 3 zyxpFs --------- (2.10)

2-4 Fluid Mechanics

Fluid Static

kzyxkw- --------- (2.11) amF azyxkwFs --------- (2.12) 2.10 2.11 2.12 azyxkzyxzyxp akp --------- (2.13) 2.13 () 0 2.13 kp = 0 kk

zpj

ypi

xp

= 0 --------- (2.14)

X 0i

xp

--------- (2.15)

Y 0j

yp

--------- (2.16)

2.15 2.16 X Y 0

2-5 Fluid Mechanics

Fluid Static

Z kk

zp

dzdP --------- (2.17)

2.17 Z 0

2.4

2.17 dzdP 2

P

1PdP = 2

z

1zdz

P2-P1 = -(z2-z1) zP --------- (2.18) 2.18 z z

2-6 Fluid Mechanics

Fluid Static

2.1.3 (Measurement of Pressure) 2

- (Absolute pressure) (absolute zero pressure)

- (Gauge Pressure) (mean sea level : MSL) 0

2.5 2.1.4 (Pressure Units) SI (N/m2)

- SI (Pa) (N/m2) (m of water ; mm.Hg) (bar) 105 (105 N/m2) - BG / (psi = lb / in2) (ft of water ; in.Hg)

* atm (Standard atmospheric pressure) 1 atm (1.013105 N/m2) 1 atm = 1.013105 N / m2 (Pa) = 14.7 lb / in2 (psi) = 1.013 bar (1bar = 105Pa) = 760 mm.Hg = 29.9 in. Hg = 10.33 m of Water = 33.9 ft of Water

2-7 Fluid Mechanics

Fluid Static

2.1 2 - 0.5 . - 0.5 . Pa = 1 atm

P1 = P2+w(h1-h2) P2 = Pa+oil(h2-0) Pa = 1 atm = 1.013 105 N/m2 P2 = (1.013 105) +(0.85w)(0.3-0) = 103801.6 N/m2 P1 = 103801.6 +w(0.4) P1 = 107725.6 N/m2 Ans P3 = P2+w(h3-h2) = 103801.6+w(0.2) 0.5 m P3 = 105763.6 N/m2 Ans

= 107725.6-101300 = 6425.6 N/m2 Ans 0.5 . = 105763.6-101300 = 4463.6 N/m2 Ans

2-8 Fluid Mechanics

Fluid Static

2.1.5 (Pressure gauge)

2.1.5.1 (Barometer) 2.6 ( ) () (absolute pressure)

2.6

Pair = Pv+ h Pv = 0.000023 lb/in2 Pair = h --------- (2.19) ( : mm.Hg) 2.1.5.2 (Manometer) barometer (gauge pressure) 3

2.7 U

2-9 Fluid Mechanics

Fluid Static

- (Piezometer) 2.7 ()

A PA = h --------- (2.20) A PA = Pair+h --------- (2.21) - U (U-tube Manometer) Piezometer

(h) U Gage Fluid ( ) h 2.7 ()

A PA = 2h2-1h1 --------- (2.22) A PA = Pair+2h2-1h1 --------- (2.23) - (Incline Tube Manometer) manometer

2.7 () A PA = 2(L sin)-1h1 --------- (2.24) A PA = Pair+2(L sin)-1h1 --------- (2.25) 2.1.5.3 (Bourdon gauge)

2.8


Fluid Static

2.2 B A 25 mm-Hg ( m)

1 mm-Hg = (110-3 m.)SGHgw = 0.00113.69810 = 133.4 Pa A PA = 25 mm.Hg = 25133.4 = 3335.4 Pa = 3.34 kPa B PB = PA + (0.15) w + (0.30) HG - (0.45) Oil = 3335.4 + (0.15) (9810) + (0.30) (13.69810) - (0.45) (0.809810) = 41.31103 Pa = 41.31 kPa Ans 2.3 U 3 (Unknown fluid) 0.045W + 0.015 0.035 0.015W 0.005Oil = 0

(0.045-0.015)W 0.005(0.8)W + (0.015-0.035) = 0 (0.03-0.004) W = 0.02 = 1.3W

= (/g) = (1.3W)/g = 1.3W = 1.3(1,000) = 1,300 kg/m3 Ans


Fluid Static

2.2 (Pressure Force on a Plane Surface) (Static Pressure) 2.9

2.9

(Pressure Force : FR) (Center of Pressure : CP) 2.10 A X Y h

2.10


Fluid Static

C (Centroid of Surface) CP (Center of Pressure) yC Y yP Y hC () hP () dAhdF

AR dAhdFF

h = y sin A

R dAsinyF

AR ydAsinF

AyydA cA

( X) sinAyF cR hc = yc sin AhF cR --------- (2.26) Y (yR) O X PRyF =

AdFy

= A

dAhy =

A

2 dAsiny

Py = R

A

2

F

dAsiny

= sinyA

dAsiny

c

A

2

Py = c

A

2

Ay

dAy --------- (2.27)

xA

2 IdAy ()

Py = c

X

AyI --------- (2.28)


Fluid Static

X (IX) XI =

2cXC AyI --------- (2.29)

IXC X X Py =

c

2cxc

AyAyI

c

xccP Ay

Iyy --------- (2.30)

2.11 Y O Y PRxF =

AdFx

= A

dAhx =

AdAsinxy

Px = R

A

F

dAsinxy

=

sinyA

dAsinxy

c

A

Px = c

A

Ay

xydA --------- (2.31)

xyA

I dA xy ( XY) XY (IXY) ccxycxy yAxII

Px = c

ccxyc

AyyAxI


Fluid Static

Y cc

xycP xAx

Ix --------- (2.32)

( Y : Ixyc = 0)

2.12

2.11 2.2 PAFR --------- (2.33)

2.13 O X PRyF =

AdFy

= A

dAPy

P FR=PA

Py = PA

dAyPA


Fluid Static

A

dAyy AP

--------- (2.34)

2.34 A

dAyA

X cP yy --------- (2.35) O Y PRxF =

AdFx

Px = PA

dAxPA

A

dAxx AP

--------- (2.36)

2.36 A

dAxA

y cP xx --------- (2.37) 2.35 2.37


Fluid Static

2.4 1.2 m 1.8 m F P ()

FR = hCA hC = 3.9 sin 60O = 3.38 m A = 1.2 1.8 = 2.16 m2 FR = W(3.38)(2.16) = 98103.38 2.16 = 71,621 N = 71,621 kN

cc

xcP yAy

Iy ; 4

33

xc m583.0121.81.2

12bdI

m97.33.93.92.16

0.583yP MHinge = 0 0.97FR = (1.8F) F = (0.97FR)/1.8 = (0.9771621)/1.8 = 38,596 N Ans


Fluid Static

2.5 1.0 m 1,500 kg (h) 0.9

F = F1+F2

F1 P = Oilh F1 WW2Oil1 h7069.0h149.0hAF

---- (1)

F1 m5.0

20.1

2Dy 1P ---- (2)

F2 F2 wW

22CW2 2356.00.142

6.0AhF

---- (3) ( m3.0

53

20.1sinyh 2C2C

)

F2

AyIyy

2C

2XC2C2P

m5.020.1y 2C 4442XC m0491.0640.164DI

m625.00.1

45.0

0491.05.0y

22P

---- (4)

0MO 0aWyFyF 2P21P1 625.02356.05.0h7069.0 WW = 81.9500,14.0 0.353h + 0.147 = 0.6 ---- (5) (5) h = 1.283 Ans


Fluid Static

2.3 (Pressure Force on a Curved Surface) 2.14

2.14

(FH) (FV) 2.15

2.15 AB

2.15 abc abfd 2.15 () F1 aced F2 bcef 2.15 () 0 0FX Fx = F1 Fx F1 acde Fx (FH) yz (F1) aced 2.15 () 2.16


Fluid Static

2.16

0 0FY

Fy = F2 + W Fy W F1 bcef Fy (FV) xy (F2) bcef (W) 2.15 () 2.17

2.17


Fluid Static

2.6 4 . 2 .

ABC

0FX FX = F1

F1 AC ACCW1 AhF m5.3

225.2h:m842A C

2AC

F1 = 85.3W = W28 FX = W28 (FH) FX FH = W28 = 274.68 kN Ans c

cACP yyA

Iy

1224I:m842A:m5.3

225.2hy

3

XCACCC

m60.35.35.3812

24

y

3

P

Ans 0FY FY = F2 + W F2 BC BCBC2 APF 2BCWBC m842A:5.2P WW2 2085.2F


Fluid Static

F2 BC m1

22

x2 C W W = W2 4241 = W4 W ABC

m85.0

38

324x3 C

Fy = WW 420 = W420 (FV) Fy WV 420F = 319.48 kN Ans

FV FV F2 W xFV = 85.0W1F2 x =

W

WW

42085.04120

x = 0.94 m C Ans F = 2V

2H FF

= 2WW2W 42028 = 421.33 kN Ans tan =

H

V

FF

=

W

W

28420

= 49.31O Ans


Fluid Static

2.7 4 . 2 . ABC

0FX FX = F1

F1 AC ACCW1 AhF m5.3

225.2h:m842A C

2AC

F1 = 85.3W = W28 FX = W28

(FH) FX FH = W28 = 274.68 kN Ans c

cACP yyA

Iy

1224I:m842A:m5.3

225.2hy

3

XCACCC

m60.35.35.3812

24

y

3

P

Ans 0FY FY = F2 - W F2 BC BCBC2 APF 2BCWBC m842A:5.4P WW2 3685.4F F2 BC m1

22x2 C

W ABC


Fluid Static

W4 ADBC W4 = W422 = W16 W4 ADBC m1

22x4 B

W5 ADB W5 = W2 4241 = W4 W5 ADB

m85.0

38

324x5 B

Fy = WWW 41636 = W420 (FV) Fy WV 420F = 319.48 kN Ans

FV FV F2 W4 W5 xFV = 85.0W1W1F 542 x420 W = 85.04116136 WWW X =

420

85.041636

= 0.942 m B Ans F = 2V

2H FF

F = 2WW2W 42028 = 421.33 kN Ans tan =

H

V

FF

=

W

W

28420

= 49.31O Ans


Fluid Static

2.8 AB 4 . 2 . F B ABC

F1 F1 = AhCW 2C m842A;m5.315.2h F1 = 85.3W = W28 F1 c

cACP yyA

Iy

43

CC m667.21242I;m5.3hy

m60.35.35.382.667yP

F1 A 1.1 m F2 W F2 BC APF BC2 F2 = 85.4 W = W36 F2 BC : x2 = 1.0 m C W ABC W = 4241 2W = W4 W ABC : x3 = m85.03

24 C A 0 21 F0.1F1.1 = F0.2W85.0 WW 360.1281.1 = F0.2485.0 W F =

212.56 W

F = 275.26 kN Ans


Fluid Static

2.9 AOB 5 . 2 . 3 1 F ( 0.75) ABC

F1 + F2 F1 AC F1 APO 521 O WW 5.71075.01 F1 AC m5.215.01y1 F1 O 1.0 m

F2 AC AhF CW2 2C m1052A;m5.15.01h F2 = WW 15105.1 F2

CACC2 yA

Iyy

43

CC m333.31252I;m5.1hy

y2 = m722.15.110333.35.1

F2 O 2.5 - 1.72 = 0.78 m F4+W F4 BC APF BC4 F4 = 5215.0 OW = 1075.05.0 WW = W5.12


Fluid Static

F4 BC F4 O m0.1x4 W ABC W5 AOBC W5 = 522W = W20 W5 AOBC W5 O m0.1x5 W6 ABC W6 = 5241 2W = W5 W6 ABC W5 O

m85.0

324x6

O 0

621 W85.0F78.0F1 = F2F1W1 45 W585.01578.05.71 = F25.121201 W

WW 5.3255.32 = F2 F = WW 025.02

05.0

F 245.25 N Ans


Fluid Static

2.4 (Buoyancy Force) ABCD 2.18 DAB F1 DCB F2 F1 HE F2 FG

2.18 ABC CDA ABC CDA ABC CDA (FB) 2.19 Fb W ABCD ABCD

2.19

2.19 AB DC A AB F1 = Ah1 DC F2 = Ah2


Fluid Static

Fb = WFF 12 = WAhAh 12 Fb = WAhh 12 --------- (2.38) 2.19 Ahh 12 Box Box = Ahh 12 --------- (2.39) W W = Box --------- (2.40) 2.39 2.40 2.38 Fb = BoxBox Fb = --------- (2.41) Fb Fb

BF --------- (2.42)


Fluid Static

2.10

0Fy W = WBOilB FF = WWOilOil = WWOil 8.0

W

=

WOil 8.0

SG =

35.035.03.015.035.03.08.020.035.03.0

SG = 0.886 Ans 2.11 1.01.00.5 m3 20 kN

T = F1 + W F1 F1 = PA = 112 W = W2 T = 3W 10202 = 39.62 kN Ans

T = W - FB FB = W5.011 = W5.0 T = W3 5.01020 = 15.095 kN Ans


Fluid Static

2.4.1 (Stability of Floating and Submerged Bodies)

3 - Stable Equilibrium - Neutral Equilibrium - Unstable Equilibrium 2.4.1.1 (Stability of Submerged Bodies) 2.20() (C) (G) (Stable)

2.20 (G) (C) (Unstable) 2.20() 2.4.1.2 (Stability of Floating Bodies) (Metacenter) (Metacentric Height) G C


Fluid Static

(Metacenter) M 2.21 () 2.21() 2.21()

2.21

2.22

(Metacentric Height) 2.22() GM


Fluid Static

(GM)

2.23 2.23 () C C ABFE = ABFE + BBO - AAO xFE'B'A = 3AO'A2BO'B1ABFE xxx x Y FE'B'A 1x Y ABFE 2x Y BO'B 3x Y AO'A ABFE Y 0x1 xFE'B'A = 3AO'A2BO'B xx xFE'B'A = AO'ABO'B dxdx --------- (2.43) 2.23() d = dAtanx --------- (2.44) 2.44 2.43 xFE'B'A = AO'ABO'B dAtanxxdAtanxx xFE'B'A =

AO'A2

BO'B

2 dAxdAxtan --------- (2.45)

AO'A

2

BO'B

2 dAxdAx B'AOB'A

2dAx ( z) GHIJ 2.23 () 2.43 xFE'B'A = ZItan --------- (2.46)


Fluid Static

2.23() FE'B'A x = ZItan tan

x =

ZI --------- (2.47)

2.23() tanx

(C) (M) (Metacentric Radius) CM

ZICM --------- (2.48)

CG (C) (G) (M) CM CG CGCMGM --------- (2.49) GM (+) GM (-)


Fluid Static

2.12 30 . 60 . 30 . 318 N

0 W = FB = W 318 = Wd6.03.0 d = W6.03.0

318 = 0.18 m

(G) m15.023.0

(C) m09.0218.0

CG = 0.15 0.09 = 0.06 m = 18.06.03.0 = 0.0324 m3 Y X

ICM CM I I

43

X m0054.0126.03.0I 43Y m00135.012 6.03.0I

Y m4167.00324.000135.0

CM 048.009.0042.0CGCMGM GM (-) Ans


Fluid Static

2.13 Platform 0.250.30 . 10 . Platform 600 ./..

0 W = FB ----(1) W = 1025.03.02g = 75.02g600 = 8.829 kN

FB = Wd103.02 = Wd6 (1) 310829.8 = Wd6 d =

W

3

610829.8

= 0.15 m

(G) m125.0225.0

(C) m075.0215.0

CG = 0.125 0.075 = 0.05 m Y X

43X m50212

103.0I

433

Y m26.112106.0

12102.1I

Y m4.11015.03.02

26.1ICM Y 35.105.04.1CGCMGM GM (+) Ans


Fluid Static

2.5 (Variation of fluid pressure in moving container)

(dv) ( 2.13) 2.13 kp = a X Y Z

xP

= xa --------- (2.50)

yP

= ya --------- (2.51)

zP

= zag --------- (2.52) 2.50 2.51 2.52 X Y Z 2.24

2.24

2.5.1 (Fluid pressure in Linear moving container)

2.25


Fluid Static

Y Z 2.25 a 2 yz ( y , z ) ( y+dy , z+dz ) 2.24 dP = dz

zPdy

yP

--------- (2.53)

2.26 2 yz

2.51 2.52 2.53 dzgadyadP zy --------- (2.54) 2.54 0 dP = 0 2.54 0 = dzgadya zy z

y

aga

dydz

--------- (2.55) 2.55 2.27

2.27


Fluid Static

(a=0) 2.55 0 2.1.2 ay az 2.54 0 2.52 dzdzgdP 2.17 2.1.2


Fluid Static

2.14 1.20 . 0.45 . 1.45 .

-

- - 3 m/s2

2.53 z

y

aga

dydz

z : 0az

ga

dydz y ---- (1)

45.1

15.015.0dydz

(1) 45.115.015.0 =

gay

ay =

g45.115.015.0

ay = 2.030 m/s2 Ans

2.54 dzgadyadP zy CzgyaP y ---- (2) (2) 4 P y z C

E ( ) D 0 C D : y = -0.725 ; z = +0.15 ; P = 0 (2) 0 = C0.15g0.725-2.03 C = 0 A : y = -0.725 ; z = -1.05 (2) PA = 01.05-g0.725-2.03 = +11.77 kPa Ans B : y = +0.725 ; z = -1.05 (2) PB = 01.05-g0.7252.03 = +8.83 kPa Ans


Fluid Static

3 306.0

g3

dydz

DC D m222.0725.0306.0y

dydzz

DC ABFG d m072.015.0222.0bzd = 072.005.145.045.1 = 0.638 m3 Ans

2.15 a

- -

2.55 zy

aga

dydz

2

10.2

5.05.0dydz

a54cosaay

a53sinaaz

21 =

a53g

a54

a = g115 = 4.459 m/s2 Ans 2.54 dzgadyadP zy CzgayaP zy D E 0 C


Fluid Static

E : y = +1.0 ; z = 0 ; P = 0 (2) 0 = C0gg1.0g 1155311554 C = w114 A B A : y = 0 ; z = -1.5 PA = w1141155311554 1.5-gg0g = w1116 = +14.269 kPa B : y = +2.0 ; z = -3.0 PB = w1141155311554 3-gg2g = w1120 = +17.836 kPa B 17.836 kPa Ans


Fluid Static

2.5.2 (Fluid pressure in angular moving container)

z z 2.28()

2.28

(Polar Coordinate) r Z ( 2.53)

dP = dzzp

dp

drrp

--------- (2.56)

5.24()

rr maF radzddrrrddz2dr

rp

prddz2dr

rp

p

22r rr-arp --------- (2.57)

maF

adzddrrdrdz2dp

pdrdz2

dpp

rap

0a 0p

--------- (2.58)


Fluid Static

zz maF radzddrrdzddrrrddr2

dzzp

prddr2dz

zp

p

zagzp --------- (2.59)

2.57 2.58 2.59 2.56

dzagdrrdp z2 --------- (2.60) 2.60 0 dP = 0 2.60 dzagdrr0 z2

z

2

agr

drdz

--------- (2.61)

2.61 2.61 dr

agr

dzz

2

--------- (2.62)

2.60 Crag2z

2

z

2

--------- (2.63)

2.63 (Paraboloid) 2.27


Fluid Static

2.27

0 z = C C r


Fluid Static

2.16 z 0.4 m/s2 - -

( O) ( A) r = 0.6 m z = 1.0 m

2.63 Crag2

1z 2z

2

C = 0 m. s/rad53.706.04g2

10.1 22

Ans

2.58 dP = dzagdrr z2 P = 1z22 Czagr 2

1

C1 O O : r = 0 ; z = 0 ; P = 0 C1 = 0 D r = 0 ; z = -0.5 PD = 05.0-4g53.70 22 2

1

= 6.905 kPa B r = +0.6 ; z = -0.5 PB = 05.0-4g53.70.6 22 2

1

= 17.11 kPa 17.11 kPa Ans

( O) ( A) r = 0.6 m z = 1.5 m 2.61 Cr

agz 2

z

2

C = 0 m.


Fluid Static

s/rad73.1006.04g2

15.1 22

Ans

***

= = 25.12.145.12.14 22 = 0.848 m3 = = 848.00.12.1

42

= 0.283 m3 Ans

3-1 Fluid Mechanics

Flow Theorem

3

Kinematics of Fluid Flow

- Streamline streamline

- Streakline

- Pathline () streamline streamline streakline pathline ( 3.5) 3.1 (Pathline) x streakline 3.1() streamline 3.1 ()

3.1 streamline streakline pathline

3-2 Fluid Mechanics

Flow Theorem

3.1 (Flow classification)

( ) 5

3.1.1 (Real Fluid and Ideal Fluid) (Real Fluid) (Ideal Fluid) ()

3.2 () ()

3.1.2 (Compressibility) (Compressible Fluid) 3.3() (Incompressible Fluid) 3.3 ()

3.3

3-3 Fluid Mechanics

Flow Theorem

3.1.3 Steady Flow Unsteady Flow

3.4 Steady flow Unsteady flow 3.1.4 2

- (Laminar Flow) ( StreamLine) ( 3.5 )

- (Turbulent Flow)

( 3.5 )

3.5

3-4 Fluid Mechanics

Flow Theorem

3.1.5 2

- (Rotational Flow) - (Irrotational Flow) Irrotational Flow

3.6

3-5 Fluid Mechanics

Flow Theorem

3.2 (Flow analysis with Control Volume method) 3

- (Differential Approach)

- (Finite Region)

- (Dimensional Analysis and Modeling)

(Control Volume Approach)

- (System)

- (Surrounding) - (Control Volume)

- (Control Surface)

(Fix Control Volume) 3.7 () (Moving Control Volume)

3-6 Fluid Mechanics

Flow Theorem

3.7 () (Deforming Control Volume) 3.7 ()

3.7

3-7 Fluid Mechanics

Flow Theorem

3.3 (Reynolds Transport Theorem)

(General Conservation Equation)

3.3.1

3.8

3.8

B =

3.7 () t = t BSYS(t) BCV(t)

BSYS(t) = BCV(t) --------- (3.1) t

ttBsys = ttBttBttB IIICV --------- (3.2) ttBI ttBII

t B

tBsys

=

t

tBttB syssys

--------- (3.3)

3-8 Fluid Mechanics

Flow Theorem

3.1 3.2 3.3

tBsys

=

t

tBttBttBttB CVIIICV

t

Bsys

=

t

ttBt

ttBt

tBttB IIICVCV

--------- (3.4)

t 0 t

Bsys

DtDBsys

t

tBttB CVCV

t

BCV

3.4

DtDBsys =

t

ttBt

ttBt

B IIICV

--------- (3.5)

t

ttBI

t

ttBI

InB

t

ttBII

t

ttBII

OutB 3.5

tDBD sys = OutInCV BBt

B

--------- (3.6)

3.6

OutInCVsys BBt

BtD

BD --------- (3.7) 3.7 (Raynolds Transport Theorem)

QIn/Out = (-)

tOut/In

b = ( ) b =

mB

3-9 Fluid Mechanics

Flow Theorem

( InB ) In

InIn

InInInInInIn btt

btbmB

InB = InInIn bQ --------- (3.8) ( OutB ) Out

OutOut

OutOutOutOutOutOut btt

btbmB

OutB = OutOutOut bQ --------- (3.9) 3.7

OutOutOutInInInCVsys bQbQtB

tDBD

--------- (3.10)

3.3.2

3.9

3.9

syssyssys dbdmbB

sys

sys dbdtd

DtDB --------- (3.11)

CVCV

CV dbdmbB

CV

CV dbdtd

dtdB --------- (3.12)


Flow Theorem

3.9 () t+dt (II) nvvn

--------- (3.13) v = n = 3.9 () tdAbvB nOut --------- (3.14) dAbv

tB

B nOutOut --------- (3.15)

OutCSnOut dAbvB --------- (3.16)

tdAbvB nIn --------- (3.17)

nvnv dAbv

tBB nInIn --------- (3.18)

InCSnIn dAbvB --------- (3.19)

( 3.7)

tD

BD CV = InOutCV BBtB

]

=

InCS

nOutCS

nCV dAbvdAbvt

B

CS

nCVsys dAbvt

BtD

BD --------- (3.20)

CS

nCVsys

dAbvdbdtddb

dtd --------- (3.21)


Flow Theorem

3.4 (Mass Conservation)

3.7

OutInCVsys MMtM

tDMD --------- (3.22)

0tD

MD sys 0 =

OutIn

CV MMt

M

OutInCV MMt

M --------- (3.23) 3.23 (Mass Conservation)

(Continuity Equation) 3.21

1mmb 0db

dtd

sys

0dAbvdbdtd

CSn

CV --------- (3.24)

3.24 (Mass Conservation) 3.20

mb

()

CS

nCVsys dAvttD

D --------- (3.22) 1

CS

nCVsys dAvttD

D --------- (3.23)

(Incompressible Fluid) 0

tDD sys

CSn

CV dAvt

--------- (3.24)

InCS

nOutCS

nCS

n dAvdAvdAv 3.24

tCV

=

InCSn

OutCSn dAvdAv


Flow Theorem

tCV

= OutCS

nInCS

n dAvdAv

QdAvCS

n

OutInCV QQt

--------- (3.25)

(Fix Control Volume)

(Steady Flow) 0

tCV

OutIn QQ --------- (3.26)


Flow Theorem

3.1 Y 0.1 cms 0.3 cms (SGAlcohol = 0.8)

OutInCVsys BBtB

tDBD

Fix Control Volume

OutIn

CV MMt

M 0

tMCV

InM = AlcoholAlcoholww QQ = ww 8.03.01.0 OutM = MixMixQ MixMixwlw Q8.03.01.00 ----- (1)

OutIn QQ Alcoholw QQ = mixQ (0.1) + (0.3) = mixQ mixQ = 0.4 cms (1) 0 = 4.08.03.01.0 Mixwlw Mix = w4.0

8.03.01.0

= w85.0 = 850 kg/m3 Ans


Flow Theorem

3.2 1 l/s 6 mm 100 mm (V) () 600 rpm

OutInCVsys BBtB

tDBD

Moving Control Volume

OutInCV MMt

M ----- (1) 0

tMCV

InM = inwQ = w001.0 OutM = OutwQ2 QOut = WAOut = W006.04

2

W OutM =

W006.0

42 2w

(1) 0 = w001.0 -

W006.0

42 2w

W = 2006.0

42

001.0

= 17.68 m/s

UWV U = R = 1.0

606002

= 2 m/s

V = (-17.68) + (+2) = -11.40 m/s Ans


Flow Theorem

3.3 (Plunger) 500 .. 300 cc/min 0.1

OutInCVsys BBt

BtD

BD Deforming Control Volume

OutIn

CV MMt

M ----- (1) 0MIn OutM = Q1.0Q

(1)

tMCV

= Q1.0Q0

tMCV

= - Q1.1 ----- (2)

tMCV

= td

d CV = tdALd

t

MCV

= td

dLA ----- (3) V =

tddS =

tddL ----- (4)

(3) (4) (2) VA = Q1.1 V =

A

Q1.1

= 6

6

1050060

103001.1

V = 0.011 m/s Ans


Flow Theorem

3.4 500 / 1.0015 1.0012 1.6552 ()

OutInCVsys BBtB

tDBD

Fix Control Volume

OutIn

CV MMt

M ----- (1) InM = ininQ = 0.10015.1 W ----- (2) OutM = outoutQ = 0.10012.1 W ----- (3)

t = 0 Control Volume 1 2 t = t Control volume 1 2 CVM = 211 mmmm = 2 mm = 22 = 2 = 10000012.16552.133600 CVM = 7,063,200 kg = 7,063.2 ton Control volume

tMCV

= t

MCV

= t

102.063,7 3

----- (4)

(2) (3) (4) (1) t

102.063,7 3

= W0012.10015.1

t = 0012.10015.12.063,7

= 47,088,000 = 545 Ans

4-1 Fluid Mechanics

Energy Equation

4

3 (Incompressible Fluid) Streamline Steady flow

Leonhard Euler (streamline) Bernouli Euler

4.1 Euler (Eulers Energy equation)

streamline 4.1

4.1

streamline as

- dAP dAds

sPP

- dsdAddW 2 amF

4-2 Fluid Mechanics

Energy Equation

samsindWdAdss

PPdAP

--------- (4.1)

dtdVas t,sfV

tV

sVV

dtdt

tV

dtds

sVa

s

--------- (4.2)

(Steady flow) 0tV

dsdV

sV

4.2

dsdVVas --------- (4.3)

dsdAdm --------- (4.4) 4.3 4.4 4.1

dsdVVds

dsdzdsds

sPPP

dsdVVdsdAsindsdAdAds

sPPdAP

Steady flow dsdP

sP

dsdVVds

dsdzdsds

dsdPPP

0

gdVVdPdz --------- (4.5)

4.5 Leonhard Euler

4-3 Fluid Mechanics

Energy Equation

4.2 Bernoulli (Bernoullis equation)

Bernoulli Leonhard Euler

0g

dVVdPdz

C

g2VPz

2 --------- (4.6)

g2

VPzg2

VPz222

2

211

1 --------- (4.7) 4.7 Bernoulli Bernoulli Bernoulli

streamline 1 2 4.7

4.2 Bernoulli

4.6 4.7 (L)

L:z LTML

LMLTTML

FL:P 2222

22

2

L

LTLT:

g2V

2

212

= LFFL:

WeightEnergy

Bernoulli

4-4 Fluid Mechanics

Energy Equation

(Energy Head) (Head)

Z (Potential head or Elevation head)

P (Pressure head)

g2V2 (Velocity head)

(Steady flow) (Ideal Fluid) (Incompressible fluid) streamline streamline 4.2 (Energy Grade Line ; E.G.L.) (Hydraulic Grade Lime ; H.G.L.)

( ; Static head Piezomatic head)

4-5 Fluid Mechanics

Energy Equation

4.1 1.0 . 0.7 . 0.2 . 10 . () Bernoulli A C

g2

VPz

2AA

A = g2VP

z2CC

C (Datum) 0 A : zA = +0.7 m ; PA = 0 ; VA = 0 C : zC = +0.2 m ; PC = 0 (1) 007.0 =

g2V02.0

2C

VC = 3.13 m/s

s/l29.12s/m012.013.310.0VAQ 324CC Ans 4.2 A B A 10 . 68.67 kPa B 7.5 . ( B) Bernoulli A B

g2

VPz

2AA

A = g2VP

z2BB

B ----- (1) (Datum) 0 A : zA = 0 ; PA = 68.67 x 103 Pa B : zB = 0 ; PB = 0 (1)

g2V1067.680

2A

3

=

g2V00

2B

0957.6g2 = 2A2B VV ----- (2) BB

2

B

2

A

BB

A

BABBAA V5625.0V10.0

075.0VDD

VAA

VVAVAQ

(2) 957.6g2 = 2B2B V5625.0V VB = 14.13 m/s

s/l42.62s/m06242.013.14075.0VAQ 324BB Ans

4-6 Fluid Mechanics

Energy Equation

4.3 AG A-B 15 . C-G 10 . A 39.24 kPa G A 78.3 l/s

GCGCBABA VAVAQ sm43.4

15.00783.0V;sm97.9

1.00783.0V 2

4BA2

4GC

m50.8g2

43.445.3g2

VPz.L.G.E

22AA

AA

m5.70.45.3Pz.L.G.H AAA

G A

g2VP

z2GG

G = g2VP

z2AA

A

g2

97.9P5.6

2G = g2

43.41024.395.323

GP = -3.06 m

m5.8g2

97.906.35.6g2

VPz.L.G.E

22GG

GG

m44.306.35.6Pz.L.G.H GGG

F A

g2VP

z2FF

F = g2VP

z2AA

A

g2

97.9P5.62

F = g243.41024.395.3

23

FP = -3.06 m

4-7 Fluid Mechanics

Energy Equation

m5.8g2

97.906.35.6g2

VPz.L.G.E

22FF

FF

m44.306.35.6Pz.L.G.H FFF

E A

g2VP

z2EE

E = g2VP

z2AA

A

g2

97.9P0.52

E = g243.41024.395.3

23

EP = -1.56 m

m5.8g2

97.956.10.5g2

VPz.L.G.E

22EE

EE

m44.356.10.5Pz.L.G.H EEE

D A

g2VP

z2DD

D = g2VP

z2AA

A

g2

97.9P5.32

D = g243.41024.395.3

23

DP = -0.06 m

m5.8g2

97.906.05.3g2

VPz.L.G.E

22DD

DD

m44.306.05.3Pz.L.G.H DDD

C A

g2VP

z2CC

C = g2VP

z2AA

A

g2

97.9P5.32

C = g243.41024.395.3

23

GP = -0.06 m

4-8 Fluid Mechanics

Energy Equation

m5.8g2

97.906.05.3g2

VPz.L.G.E22

CCCC

m44.306.05.3Pz.L.G.H CCC

B A

g2VP

z2BB

B = g2VP

z2AA

A

g2

43.4P5.32

B = g243.41024.395.3

23

BP = 4.0 m

m50.8g2

43.445.3g2

VPz.L.G.E

22BB

BB

m5.70.45.3Pz.L.G.H BBB E.G.L. H.G.L.

4-9 Fluid Mechanics

Energy Equation

Bernoulli - Venturi meter

Venturi meter

4.3 Venturi

4.3 Piezometer (H1 , H2) Static head Bernoulli 1-2

g2VPz

g2VPz

222

2

211

1 H1 , H2 static head 1 2

212212

22

2

21

1

VVHHg2

g2VH

g2VH

122

21

22211 VDDVVAVA

1DD

Hg2VV 4

2

11

--------- (4.8)

4.8 (CV) Hg2C

1DD

Hg2CV W4

2

1V

--------- (4.9)

CW 1

DD

C4

2

1

V


Energy Equation

- Pitot tube Pitot tube 4.4

4.4 pitot tube

4.4 Pitot tube Pitot tube Static head Velocity head (Total Energy Head) Bernoulli 1-2

g2VPz

g2VPz

222

2

211

1 H1 , H2 static head 1 2 0H

g2VH 2

21

1 Hg2VV 1 --------- (4.10) 4.10 (CP) Hg2CV P --------- (4.11)

4.5


Energy Equation

4.4 Venturi 1 2 D 0.5D h Ideal Fluid Bernoulli 1 2

g2VPz

g2VPz

222

2

211

1 z1 = z2

g2VVPP 21

2221

------- (1) 1W1 hP 2W2 hP hhhPP W21W21 ------- (2) Q1 = Q2 11

2

1

2

2

11

2

122211 V4VD5.0

DVDDV

AAVVAVA

------- (3)

(2) (3) (1)

hg2V15g2

VV4h 21

21

21

W

W

15

hg2V1 ------- (4)

15

hg2D4

VAQ 211

Ans


Energy Equation

4.3 (Energy Equation)

Bernoulli () () (Pump and Turbine) Bernoulli

4.3.1 (Head loss)

2

- (Major loss) (Friction head loss) hf

- (Minor loss)

(Fitting Devices) hm

4.6

Bernoulli

mf222

2

211

1 hhg2VPz

g2VPz ------- (4.12)


Energy Equation

4.7


Energy Equation

4.5

A-H

g2VPz

2A

w

AA = mf

2H

w

HH hhg2

VPz

PA = 0 , PH = 0 , VA = 0 Az = mf

2H

H hhg2V

z ------- (1)

m722111h HAf m75.12115.1h HAm (1) 0.15 = 0.70.7

g2V

0.02H

HV = 4.429 m/s

sm035.0429.410.04

VAQ 32HH A-B

g2VPz

2A

w

AA = BAmf

2B

w

BB hhg2

VPz

PA = 0 , VA = 0 s/m968.1429.415.010.0V

DDV 2

2

H2B

2H

B

m5.1h BAm m0h BAf 000.15 = 05.1

g2968.1P

0.102

w

B

w

BP = = 3.303 m


Energy Equation

B E.G.L. = m5.13g2

968.1303.310g2

VPz22

B

w

BB

H.G.L. = m303.13303.310Pzw

BB

B-C ()

g2VPz

2B

w

BB = mf

2inC

w

inCinC hhg2

VPz

m5.13g2

VPz2B

w

BB s/m968.1VV BC

m1h CBf m0h CBm 13.5 = 00.1

g2968.1P

0.102

w

B

w

inCP

= 2.303 m

Cin E.G.L. = m5.12g2

968.1303.210g2

VPz22

C

w

CC

H.G.L. = m303.12303.210Pzw

CC

- C

g2

VPz

2inC

w

inCinC = mf

2outC

w

outCoutC hhg2

VPz

m5.12g2

VPz

2inC

w

inCinC s/m968.1VV outCinC outCinC zz

m0h outCinCf m1h outCinCm 12.5 = 0.10

g2968.1P

0.102

w

outC

w

outCP

= 1.303 m

Cout E.G.L. = m5.11g2

968.1303.110g2

VPz

22outC

w

outCoutC

H.G.L. = m303.11303.110Pzw

outCoutC


Energy Equation

C D

g2

VPz

2outC

w

outCoutC = mf

2inD

w

inDinD hhg2

VPz

m5.11g2

VPz

2outC

w

outCoutC s/m968.1VV inDoutC

m0.1h inDoutCf m0h inDoutCm 11.5 = 00.1

g2968.1P

0.52

w

inD

w

inDP

= 5.303 m

Din E.G.L. = m5.10g2

968.1303.50.5g2

VPz

22inD

w

inDinD

H.G.L. = m303.10303.50.5Pzw

inDinD

- D

g2

VPz

2inD

w

inDinD = mf

2outD

w

outDoutD hhg2

VPz

m5.10g2

VPz

2inD

w

inDinD s/m968.1VV outDinD outDinD zz

0h outDinDf m0.1h outDinDm 10.5 = 0.10

g2968.1P

0.52

w

outD

w

outDP

= 4.303 m

Dout E.G.L. = m5.9g2

968.1303.40.5g2

VPz

22outD

w

outDoutD

H.G.L. = m303.9303.40.5Pzw

outDoutD


Energy Equation

D - E

g2

VPz

2outD

w

outDoutD = mf

2E

w

EE hhg2

VPz

m5.9g2

VPz

2outD

w

outDoutD s/m968.1VV EoutD

m0.1h EoutDf 0h EoutDm 9.5 = 00.1

g2968.1P

0.52

w

E

w

EP = 3.303 m

E E.G.L. = m5.8g2

968.1303.30.5g2

VPz22

E

w

EE

H.G.L. = m303.8303.30.5Pzw

EE

E - F

g2VP

z2E

w

EE = mf

2F

w

FF hhg2

VPz

m5.8g2

VPz

2E

w

EE

s/m428.4968.110.015.0V

DDVVAVA 2

2

E2F

2E

FFFEE

0h FEf m0.2h FEm 8.5 = 0.20

g2428.4P

0.52

w

F

w

FP = 0.500 m

F E.G.L. = m5.6g2

428.45.00.5g2

VPz22

F

w

FF

H.G.L. = m5.55.00.5Pzw

FF


Energy Equation

F G

g2VPz

2F

w

FF = mf

2inG

w

inGinG hhg2

VPz

m5.6g2

VPz2F

w

FF s/m428.4VV FinG

m0.2h inGFf 0h inGFm 6.5 = 00.2

g2428.4P

0.52

w

inG

w

GP

= -1.500 m

Gin E.G.L. = m5.4g2

428.45.10.5g2

VPz

22inG

w

inGinG

H.G.L. = m5.35.10.5Pzw

inGinG \

- G

g2

VPz

2inG

w

inGinG = mf

2outG

w

outGoutG hhg2

VPz

m5.4g2

VPz

2inG

w

inGinG outGinG VV outGinG zz

0h outGinGf m5.1h outGinGm 4.5 = 5.10

g2428.4P

0.52

w

outG

w

outGP

= -3.000 m

Gout E.G.L. = m0.3g2

428.40.30.5g2

VPz

22outG

w

outGoutG

H.G.L. = m0.20.30.5Pzw

outGoutG


Energy Equation

G - H

g2

VPz

2outG

w

outGoutG = mf

2H

w

HH hhg2

VPz

m0.3g2

VPz

2outG

w

outGoutG s/m428.4VV HoutG

m0.2h HoutGf 0h HoutGm 3.0 = 00.2

g2428.4P

0.02

w

H

w

HP = 0.000 m

H E.G.L. = m0.1g2

428.400g2

VPz22

H

w

HH

H.G.L. = m000Pzw

HH

Ans


Energy Equation

4.6 1 2 2.5 cm D Bernoulli A G mf

2GG

G

2AA

A hhg2VP

zg2

VPz ------- (1)

H (Datum) 0 A : zA = +1.3 m ; PA = 0 ; VA = 0 G : zG = +0.3 m ; PA = 0 m55.015.0215.01.0h;m3.015.015.0h GAmGAf (1) 003.1 = 55.03.0

g2V

03.02G

VG = 2.971 m/s

s/l6.0s/m0006.0971.2025.0VAQ 324GG Ans Bernoulli A D mf

2DD

D

2AA

A hhg2VP

zg2

VPz ------- (2)

D : zD = +0.8 m ; VD = VG = 2.971 m/s m25.015.01.0h;m15.0h DAmDAf (2) 003.1 = 25.015.0

g2971.2P8.0

2D

DP = -0.05 m

PD = -490.5 N/m2 Ans


Energy Equation

4.3.2 (Pump)

4.8

(hP) (HP) 4.8

Bernoulli

mf222

2P

211

1 hhg2VPzH

g2VPz ------- (4.13)

(Power ; PW) PW Ht

VolumeVolumeWight

tWorkP

PW QHP ------- (4.14) (PP)

WPP PP ------- (4.15)

P


Energy Equation

4.7 1 2 0.594 ../ 60 % A-F P

2A

w

AA Hg2

VPZ = mf2F

w

FF hhg2

VPZ

PH000.5 = mf hh000.25 ----- (1) BC sm50.1

71.0594.0

AQV 2

4BCB

DE sm50.255.0

594.0AQV 2

4DED

BC g2

V0.7g2

V0.1g2

V0.6hh2

CB2

CB2

CBCBmf

DE g2

V5.13g2

V0.4g2

V5.9hh2

ED2

ED2

EDEDmf

(1) PH = g2

V5.13g2

V0.75252

ED2

CB

= g25.25.13

g25.10.7525

22

= 25.1 m Pw = PQH = 1.25594.0 = 146,261.214 W = 146.26 kW Ans PP =

60.0kW26.146P

P

W = 60.0kW26.146

= 246.77 kW Ans


Energy Equation

Cavitations () (suction zone)

4.9 Cavitations

4.10 Cavitations


Energy Equation

4.3.3 (Turbine)

4.11

(HP) (hT) 4.9

Bernoulli

mfT222

2

211

1 hhHg2VPz

g2VPz ------- (4.16)

(Power ; PW) TW Ht

VolumeVolumeWight

tWorkP

QHTPW ------- (4.17) (PT)

TTP PP ------- (4.18) T


Energy Equation

4.8 +150 . +95 . 0.594 cms 75 % A-F T

2A

w

AA Hg2

VPZ = Tmf2F

w

FF Hhhg2

VPZ 00150 = mfT hhH0095 ----- (1) BC sm50.255.0

954.0AQV 2

4BCB

DE sm50.171.0954.0

AQV 2

4DED

BC g2

V0.16g2

V5.3g2

V5.12hh2

CB2

CB2

CBCBmf

DE g2

V2.15g2

V0.5g2

V2.10hh2D

2ED

2ED

EDmf (1) TH = g2

V2.15g2

V0.16951502

ED2

CB

= g25.12.15

g25.20.1695150

22

= 48.16 m Pw = TQH = 16.48594.0 = 280,635.06 = 280.64 kW Ans PT = WTP = 75.0kW64.280 = 210.48 kW Ans


Energy Equation

5-1 Fluid Mechanics

Momentum Equation

5

(Impulse momentum) 5.1

5.1

(Impulse momentum) (Reynolds Transport Theorem)

5-2 Fluid Mechanics

Momentum Equation

5.1 (Linearly Momentum Equation)

5.2

5.2

5.2 5.1 () 5.1 ()

(H)

tDHD sys

=

OutInCV HHt

H

--------- (5.1)

H F = am =

dtVdm

dtF = Vdm --------- (5.2)

5.2 (Impulse momentum) dtF (Impulse) Vdm 5.2 5.1 dtFsys = syssys Vdm sysF

= DtHD sys

--------- (5.3)

5.3 ( sysF )

5-3 Fluid Mechanics

Momentum Equation

5.3 5.1 sysF

=

OutInCV HHt

H

--------- (5.4)

() (Incompressible Fluid) (Steady Flow)

tHCV

= 0 --------- (5.5)

H =

dt

Vm =

dt

V

H = VQ --------- (5.6) 5.4 InInInOutOutOutsys VQVQF --------- (5.7) 5.7 (Momentum Equation) (Incompressible Fluid) (Steady stage)

5-4 Fluid Mechanics

Momentum Equation

5.1 2 0.30 ../ 1 10 kPa 2

- - 1.5 Velocity head 1

-

Mmentum

InInInOutOutOut VQVQF ---- (1) OutOutOut VQ = 2w VQ ---- (2) InInIn VQ = 1w VQ ---- (3)

F F2 2 2 1

g2VPz

211

1 = g2VPz

222

2 ---- (4)

sm53.15.03.0

DQV 2

14214

1 ; sm12.625.03.0

DQV 2

14224

2 P1 , V1 V2 (4)

g2

53.11010 23 =

g212.6P 22

P2 = -7.56 kPa 1 2 kN96.15.01010DPAPF 2432141111 N44.37025.01056.7DPAPF 2432242222

+

FX F = 2X1 FFF = 44.370F1096.1 X3 F = XF44.2330 ---- (5)

5-5 Fluid Mechanics

Momentum Equation

(2) , (3) (5) (1) XF44.2330 = 12.63.053.13.0 ww XF = 0.953 kN Ans 2 1.5 Velocity head 1

g2VPz

211

1 = L222

2 hg2VPz

g2

53.11010 23 =

g253.15.1

g212.6P 222

P2 = -9.312 kPa 1 2 kN96.15.01010DPAPF 2432141111 N00.45725.01031.9DPAPF 2432242222 FX F = 2X1 FFF = 457F1096.1 X3 F = XF2417 ---- (6) (2) , (3) (6) (1) XF2417 = 12.63.053.13.0 ww XF = 1.04 kN Ans

5-6 Fluid Mechanics

Momentum Equation

5.2 A 5.0 2.5 ( A) 88.29 kPa 2

- - 1.5 Velocity head A

-

Mmentum InInInOutOutOut VQVQF ---- (1) OutOutOut VQ = Bw VQ ---- (2) InInIn VQ = Aw VQ ---- (3) F = XA FF ---- (4) (2) , (3) (4) (1)

XA FF = AwBw VQVQ ---- (5) (5) VA VB 1

g2VPz

2AA

A = g2VPz

2BB

B

g2V1029.880

2A

3

=

g2V00

2B

AA2

A

2

B

AA

B

ABBBAA V4V5.2

5VDDV

AAVAVAVQ

g2V1029.88 2A

3

=

g2V16 2A

VA = 3.43 m/s VB = 13.72 m/s Q = 24 05.043.3 = 0.0067 m3/s (5) XAA FAP = 43.3Q72.13Q ww FX = 43.372.130067.005.01029.88 w243 = 104.41 N

FX FX 104.41 N Ans

5-7 Fluid Mechanics

Momentum Equation

2 1.5 Velocity head A

g2VPz

2AA

A = L2BB

B hg2VP

z

g2V1029.880

2A

3

=

g2V5.1

g2V00

2A

2B

AA2

A

2

B

AA

B

ABBBAA V4V5.2

5VDDV

AAVAVAVQ

g2

V1029.88 2A3

=

g2V5.1

g2V16 2A

2A

VA = 3.27 m/s VB = 13.09 m/s Q = 24 05.027.3 = 0.0064 m3/s (5) XAA FAP = 27.3Q09.13Q ww FX = 27.309.130064.005.01029.88 w243 = 110.51 N

FX FX 110.51 N Ans

5-8 Fluid Mechanics

Momentum Equation

5.3 5.3 - 5.7 XInInInXOutOutOutX VQVQF --------- (5.8) YInInInYOutOutOutY VQVQF --------- (5.9) ZInInInZOutOutOutZ VQVQF --------- (5.10)

5.4 X-Y

5-9 Fluid Mechanics

Momentum Equation

5.3 1 150 kPa

- X-Y V3 outin QQ Q1 = Q2 + Q3 33AV = 21 QQ 3V =

3

21

AQQ ----- (1)

sm98.0196.055.05AVQ 324111

sm31.0031.0102.010AVQ 324222 (1) 3V = 24 15.0

31.098.0 = 018.0

31.098.0 = 37.2 m/s Q3 = Q1 - Q3 = 0.67 m3/s X-Y

2 3 2 3 P2 P3 1 2

g2VPz

211

1 = g2VPz

222

2

g25101500

23

=

g210P0

22

P2 = 112.5 kPa 1 3

g2VPz

211

1 = g2VPz

233

3

g2

510150023

=

g22.37P0

23

P3 = -529.42 kPa


Momentum Equation

X

xF = XininXoutout VQVQ ----- (2) XF =

O32X 45cosFFF

= XO

3322 F45cosAPAP = X

O F45cos018.042.529031.05.112 XF = kNF23.10 X ----- (3)

XininVQ = 0 ----- (4) XoutoutVQ = O33W22W 45cosVQVQ = OWW 45cos2.3767.01031.0 = 14.52 (kN) ----- (5) (3) , (4) (5) (2) kNF23.10 X = 14.52 0 FX = 4.29 kN X YF = YininYoutout VQVQ ----- (6) YF =

O31Y 45sinFFF

= O3311Y 45sinAPAPF = OY 45sin018.042.529196.0150F YF = kN14.36FY ----- (7) YininVQ = 11W VQ = 598.0W = -4.9 (kN) ----- (8) YoutoutVQ = O33W 45sinVQ = O45sin2.3767.0 = -17.62 (kN) ----- (5)


Momentum Equation

(7) , (8) (9) (6) 14.36FY = (-17.62) (-4.9) FY = 23.42 kN F = 2Y

2X FF

= 22 42.2329.4 = 23.81 kN

=

X

Y1

FFtan =

29.481.23tan 1 = 79.62O Ans

5.4 1 2 1.12 . -

- Mmentum X XInInInXOutOutOutX VQVQF ---- (1) F2 2 P2

g2VPz

211

1 = L222

2 hg2VPz

g2

381.986.580

2

= 12.1g2

VP0

222 ---- (2)

sm1234V4V15.03.0V

DDV

AAV 11

2

1

2

2

11

2

12

(2)

g2

381.986.580

2

= 12.1g2

12P02

2 P2 = -19.62 kPa


Momentum Equation

XF = XX21 FFF

= X542211 FAPAP = X245424 F15.062.193.086.58 = 4.438 - FX (kN) XoutoutVQ = 25422w VAV = 224w54 1215.0 = +2.036 (kN)

XininVQ = 111w VAV = 224w 330.0 = +0.636 (kN) (1) 4.438 - FX = (+2.036) (0.636) FX = 3.038 kN - Mmentum Y YInInInYOutOutOutY VQVQF ---- (2) YF

= YY2 FF = Y5322 FAP

= Y2453 F15.062.19 = 0.208 FY (kN) Youtout VQ = 25322w VAV = 224w53 1215.0

= +1.527 (kN) YininVQ = 0

(2) 0.208 + FY = (+1.527) FY = 1.319 kN

F = 2Y2X FF

= 22 319.1038.3 = 3.312 kN

=

X

Y1

FFtan =

038.3319.1tan 1

= 23.47O Ans


Momentum Equation

5.5 20 mm 7 m/s 5 cm 10 cm (V)

g2

VPz2

111 = g2

VPz222

2

g2

7002

= g2

V01.02

V = 7.14 m/s - - Mmentum X

XInInInXOutOutOutX VQVQF ---- (1) xF = -FX XoutoutVQ = 0 XininVQ = 0 (1) FX = 0 - Mmentum Y YInInInYOutOutOutY VQVQF ---- (2) YF

= YF YoutoutVQ = 0

YininVQ = QVW = 2W AV = 224W 702.0 (2) YF = 224W 702.00 FY = 15.3 N F = FY = 15.3 N Ans


Momentum Equation

5.2 (Momentum equation for moving control volume)

5.5 5.5() V = 12 VV --------- (5.11)

5.6 (V2) () 5.4() 1 = uV1 --------- (5.12) 2V = u2 --------- (5.13) 5.12 5.13 12 VV

= 12 V = --------- (5.14) 5.14


Momentum Equation

5.7

5.6 5.6 XF = XF XOutOutOut VQ = cosuvQ XInInIn VQ = uvQ XF = uvQcosuvQ XF = cos1uvQ --------- (5.15) YF = YF YOutOutOut VQ = sinuvQ YInInIn VQ = 0 YF = 0sinuvQ YF = sinuvQ --------- (5.16) 5.15 5.16 5.15 5.16


Momentum Equation

5.7 25 mm 10 m/s 45O M 1 kg (u) -

XF = inXoutX QVQV ---- (1) XF = -FX = -T

2025.0

4UVAUVQ

outX QV = OW 45cosuvQ = O22W 45cosuv025.04

inX QV = uvQW = 22W uv025.04

(1) -T = O2WO22W 45cosuv45cosuv025.04

T = O22W 45cos1uv025.04 (1g) = O22W 45cos1u10025.04 u u = 1.91 m/s Ans


Momentum Equation

(pump) (Hydraulic turbine) (Absolute path : V) (Relative path : ) 5.8 5.12

5.8 Radial-flow centrifugal pump

5.9 Axial-flow centrifugal pump


Momentum Equation

5.10 Impulse Turbine (Pelton Turbine)

5.11 Reaction Turbine Fransis (radial-flow)

5.12 Reaction Turbine Kaplan (axial-flow)

6-1 Fluid Mechanics

Flow in Pressure Conduit

6

() (Flow in Pressure Conduit) (Steady Incompressible Flow in Pipe) ()

6.1 - Closed conduit - Pipes - Duct - Pipes () - Fitting Devices () - Flowrate control devices () - Pump or Turbine ( )

6-2 Fluid Mechanics


6.1 (Behavior of flow in pipe)

.. 1883 (Osborne Reynolds) 6.2

6.2 3

1) (Laminar Flow)

2) (Turbulent Flow)

3) (Transition Flow)

3 (Reynolds Number ; Re) Re < 2000 2000 < Re < 4000 Re > 4000

6-3 Fluid Mechanics


Reynolds Number

VDVDR e --------- (6.1)

V = D = = = (Absolute Viscosity ; 3

Co22atWater100.1 )

= (Kinematic Viscosity ; 6Co22atWater

100.1 ) 6.2 (Entrance Flow Development) (Entrance length : LE)

6.2.1 (Entrance condition in laminar flow)

6.3

6.3 3

6-4 Fluid Mechanics


1) (Invicid core length : LI) (x) (y) Li

2) (Development length : Ld) Li

(x) (y) Li

3) (Developed flow) (Ld)

(y) (x)

(Entrance length : LE) LE = Li + Ld --------- (6.2) LE 0.065 D Re --------- (6.3) Li E41 L --------- (6.4) Re = D =

6.2.2 (Entrance condition in turbulent flow)

6.4

6-5 Fluid Mechanics


(Li Ld ) (Ld) 6.4 Re > 105

Li 10 D --------- (6.5) Ld 40 D --------- (6.6) LE 120 D --------- (6.7)

6.3 (Friction head loss or Major loss : hf)

6.5

R 6.5

g2VP

Z2

111 = f

222

2 hg2VP

Z V1 = V2 hf = 2121 ZZPP

--------- (6.6)

F = InInInOutOutOut VQVQ V1 = V2 F = 0 RL2sinALAPAP O21 = 0

6-6 Fluid Mechanics


O A

sinLPP 21 =

A

RL2O --------- (6.7)

6.5 L sin Z2 Z1 6.7 2121 ZZPP

=

A

RL2O --------- (6.8)

6.8 6.6 hf =

A

RL2O

hf = 2

O

RRL2

hf =

RL2O --------- (6.9)

(O) (V) () () (D) () (Dimensionless analysis) D V (repeating variables)

2O

V

=

D,

VD --------- (6.10)

( ) 6.10

2O

V

=

D,VD

VD (Re)

O = De2 ,RV --------- (6.11) 6.11 6.9 hf = RL2,RV De2 f = De ,R8 hf = RL28fV2

g2

VDLfh

2

f --------- (6.12)

6-7 Fluid Mechanics


6.12 (Henry Darcy) .. 1857 (Julius Weisbach) .. 1850 - (Darcy-Weisbach Equation) f (Darcy friction factor) (friction factor) f = 2

O

V8 --------- (6.13)

f = De ,R --------- (6.14)

6.3.1 (Friction factor for larminar flow)

6.6 ()

dydVr Vr r y y = R r

r =

drdV r --------- (6.15)

6.9 r hf =

r

L2

= rL2

drdV r

dVr = drrL2h f

Vr = C2r

L2h 2f --------- (6.16)

6-8 Fluid Mechanics


( 6.6) r = 0 Vr = Vmax C = Vmax Vr = 2fmax rL4

hV

--------- (6.17) r = R (Vr = 0) 6.17 Vmax = 2f R

L4h = 2f D

L16h --------- (6.18)

6.16 6.17 Paraboloid R Vmax Paraboloid V = 0.5 Vmax

6.7 Paraboloid 6.18 2V = 2f D

L16h

hf = 2D

LV32

= 22

VV

gD

LV32

2

hf = g2V

DL

VD64

2 --------- (6.19)

6.19 6.12

eR64

VD64f

--------- (6.20) 6.20 Hangen-Poiseuille law Hangen Poiseuille

6-9 Fluid Mechanics


6.3.2 (Friction factor for turbulent flow in smooth pipe) r (u(t)) (t) Re 6.8 (Re > 4000) r 6.9

6.8

6.9 r

*utu =

*yu --------- (6.21)

6.21 law of wall tu = y t u* = friction velocity

O y = R r () = kinematic viscosity

u* = O

2*u1 =

O



V2 22

*uV =

O

2V

--------- (6.22)

6.13 6.22

*uV =

f8 --------- (6.23)

6.10 r r 3

- (Viscous sublayer)

- (Outer layer)

- (Overlab layer)

*utu = 5.0*yuln44.2

*utu = 5.0*urRln44.2

--------- (6.24)

6.11 r

Q = dAtuR0



Q = VA ( V ) dA = drr2 ( r) V = rdr2tu

R1 R

02 --------- (6.25)

6.24 6.25 V = rdr2]5.0*urRln44.2[*u

R1 R

02

*u

V = 34.1*Ruln44.2

--------- (6.26)

6.23 6.26

f8 =

34.1

8fV2Dln44.2

f

1 = 02.1fRlog99.1 e --------- (6.27) .. 1935 Prandtl 6.27 Nikuradse ( Prandtl) 80.0fRlog00.2

f1

e --------- (6.27) (smooth pipe) ( f = [ Re ] )

6.12



6.3.3 (Friction factor for turbulent flow in rough pipe) Nikuradse (roughness : ) 3

5*u

(f = [ Re ]) 7*u5

De ,Rf

7*u (Fully rough flow or Complete

turbulence flow) Re

Df .. 1939 Colebrook

fRe51.2

7.3D

log2f

1 --------- (6.28)

.. 1983 Haaland Cloebrook Haaland 10-15%

Re91.6

7.3D

log88.1f

1 11.1 --------- (6.29)

(Fully rough flow) Karman

D

7.3log2

f1

--------- (6.30)

.. 1944 Lewis F. Moody Hangen-Poiseuille ( 6.20) Prandtl ( 6.27) Colebrook ( 6.28) Karman ( 6.30) Re D friction factor (f) 6.13



6.13 Moody Diagram

6.1



6.4 (Minor loss : hm) Minor Loss (Minor loss coefficient : k) (Velocity Head) 6.31

g2

Vkh2

m --------- (6.30) k 6.2 6.2 (Minor loss coefficient : K)

6.14



6.15 -

6.16

6.17



6.1 1 2 z

- 40 l/s

- 35 m

A B

g2VPz

2AA

A = mf2HH

H hhg2VPz

00zA = mfH hh00z Z = mf hh --------- (1)

B G fh = GBfh = g2

VDLf

2

=

g2V

10.020101010f

2

fh = g2V500f

2

--------- (2)

- (B : kB = 0.5) (Globe valve : kvalve = 10) 90O (E F : kE = kF = 1.5) (G : kG = 1) ( k 6.2) mh = g2

Vkg2

Vkg2

Vkg2

Vkg2

Vk2

G

2

F

2

E

2

valve

2

B

= g2

Vkkkkk2

GFEvalveB

= g2

V0.15.15.1105.02

mh = g2V5.14

2

--------- (3)

(2) (3) (1) Z =

g2V5.14

g2V500f

22

Z = g2

V5.14f5002

--------- (4)



- ( Z ) Q = VA V =

AQ = 24 10.0

04.0

V = 5.09 m/s

Re = VD =

6101 1.009.5 Re = 5.09X105 Wrought iron = 0.045 mm ( 6.1)

D =

10.010045.0 3 = 0.00045

Moody diagram f = 0.0175

V f (4) Z =

g209.55.145000175.0

2

Z = 30.70 m Ans

- Q 35 m ( Z = 35 m) Trial & Error f = 0.020 (4) 35 =

g2V5.14500020.0

2

V = 5.294

Re = VD =

6101 1.0294.5 = 5.29 X 105

D =

10.010045.0 3 = 0.00045

Moody diagram : f 0.0175 f

f = 0.017 (4) 35 =

g2V5.14500017.0

2

V = 5.464

Re = VD =

6101 1.0464.5 = 5.46 X 105



D =

10.010045.0 3 = 0.00045

Moody diagram : f 0.0172 f

f = 0.0173 (4) 35 =

g2V5.145000173.0

2

V = 5.446

Re = VD =

6101 1.0446.5 = 5.45 X 105

D =

10.010045.0 3 = 0.00045

Moody diagram : f 0.0173 f V = 5.446 m/s Q = 24 10.0446.5 = 0.0427 m3/s = 42.7 l/s Ans



6.2 12 . wrought iron 300 0.7 5X10-7 m2/s 10.3 kPa 1 . (1) (2)

g2VPz

21

O

11 = f

22

O

22 hg2

VPz 21 zz VVV 21

O

1

O

2 PP = hf

O

P

= g2

VDL

f2

O

P

= g2

VD000,12

f2

--------- (1)

10.3 kPa/km P = (10.3 kPa/km) (12 km) = 123.6 kPa (1) 98107.0

kPa6.123

= g2V

D000,12

f2

0.0294 = DV

f2

--------- (2)

wrought iron = 0.045 mm Q = 300 l/min = 5 l/s D = 10 cm

D

= mm100

mm045.0 0.00045

V = AQ

= 24 1.0

005.0 = 0.637 m/s

Re = O

VD =

7105 1.0637.0 5103.1 Moody diagram f 0.0195 (2) 0.0294 = 1.0

637.00195.02

. 0.0294 0.0791



D = 15 cm

D

= mm150

mm045.0 0.0003

V = AQ

= 24 15.0

005.0 = 0.283 m/s

Re = O

VD =

7105 15.0283.0 4105.8 Moody diagram f 0.020 (2) 0.0294 = 15.0

283.0020.02

. 0.0294 0.0107 D = 12.25 cm

D

= mm5.122

mm045.0 0.0004

V = AQ

= 24 1225.0

005.0 = 0.424 m/s

Re = O

VD =

7105 1225.0424.0 5100.1 Moody diagram f 0.020 (2) 0.0294 = 1225.0

424.0020.02

. 0.0294 0.02935 12.25 cm Ans



6.3 20 l/s 1.5 m + 25.0 m

- 65 %

- I ( BI 9 m)

H A P

2H

W

HH Hg2

VPz = mf

2A

W

AA hhg2

VPz

HP = mfAH hhzz --------- (1)

Q = VA VGE = GEAQ

= 24 10.002.0

= 2.546 m/s

VDB = DBAQ

= 24 075.002.0

= 4.527 m/s

G B GE DB GE GEeR =

6101 1.0546.2 GEeR = 2.546X10

5

= 0.15 mm ()

D =

10015.0

= 0.0015

Moody diagram fGE = 0.0225 GE DBeR =

6101 075.0527.4 DBeR = 3.395X10

5

= 0.15 mm ()

D =

7515.0

= 0.002

Moody diagram fDB = 0.024



fh = BDfEGf hh =

g2V

DL

fg2

VDL

f2DB

DB

DBDB

2GE

GE

GEGE

fh =

g2527.4

075.00.330240.0

g2546.2

1.05.70225.0

22

fh = 11.588 m (foot valve : kG = 2.0) (Gate valve : kvalve = 2.5) 90O (F C : kF = kC = 1.5) (B : kB = 1) ( k ) mh = g2

Vkkg2

Vkkk2DB

BC

2GE

valveFG

= g2

527.40.15.1g2

546.25.25.10.222

mh = 4.594 m fh mh (1) HP = 594.4588.115.10.25 = 42.682 m PW = PQH = 682.4202.09810 = 8374.208 Watt PP =

P

WP =

65.0

208.8374

= 12883.398 Watt PP = 12.883 k Watt Ans I A

g2

VPz

2I

W

II = mf

2A

W

AA hhg2

VPz

g2

VP5.15

2DB

W

I = mf hh000.25

W

IP = mf

2DB hhg2

V5.9 --------- (2)



I B fh = BIfh

= g2

VDL

f2DB

DB

IBDB

=

g2527.4

075.00.90240.0

2

= 3.008 m (kB = 1.0) mh = g2

Vk2DB

B

= g2

527.40.12

mh = 1.045 m (2)

W

IP =

045.1008.3g2

527.45.92

= 12.508 m PI = W508.12 = 122.703 kPa Ans



6.4 +210.0 .. +125.5 .. 0.5 cms 55 %

A I

g2VPz

2A

W

AA = mfT

2I

W

II hhHg2

VPz

HT = mfIA hhzz --------- (1) Q = VA VBE =

BEAQ = 24 50.0

50.0 = 2.546 m/s

VFH = FHAQ = 24 75.0

50.0 = 1.132 m/s

B H BE FH BE BEeR =

6101 5.0456.2 BEeR = 1.228X10

6

= 0.20 mm ()

D =

50020.0 = 0.0004

Moody diagram fBE = 0.016 FH FHeR =

6101 75.0132.1 FHeR = 8.49X10

5

= 0.15 mm ()

D =

75015.0 = 0.0002

Moody diagram fFH = 0.015



fh = HFfEBf hh =

g2V

DL

fg2

VDL

f2FH

FH

FHFH

2BE

BE

BEBE

=

g2

132.175.0

35015.0g2

546.25.0

150016.022

fh = 1.632 m ( : kB = 3.50) (Gate valve : kvalve = 0.39) 45O (C D : kC = kD = 0.20) 90O (kG = 0.30) (H : kH = 1.00) mh = g2

Vkkg2

Vkkkk2FH

HG

2BE

valveDCB

= g2

132.10.13.0

g2546.2

39.02.02.050.322

mh = 1.502 m fh mh (1) HT = 502.1632.15.1250.210 = 81.366 m PW = TQH = 366.815.09810 = 399.100 k Watt PP = WTP = 125.40055.0 = 219.050 k Watt Ans



(Pipe in parallel) 6.18

6.18

6.18 1 2 ABCDEFG ABCDHIEFG ABCJKLG ( A) ( G) 6.18 BCJKLmfBCDHIEFmfBCDEFmfBA hhhhhhzzz



6.5 Fully rough flow (complete turbulent)

A G ABCDEFG ABCDHIJG

g2

VPz2A

W

AA = DFmfBDmf

2G

W

GG hhhhg2

VPz GA zz = DFmfBDmf hhhh --------- (1)

g2VP

z2A

W

AA = DJmfBDmf

2G

W

GG hhhhg2

VPz GA zz = DJmfBDmf hhhh --------- (2) (1) (2) DFmf hh = DJmf hh --------- (3) (1) (2) GA zz2 = DJmfDFmfBDmf hhhhhh2 --------- (4)

Fully rough flow f D

0235.0f002.0100

20.0D BDBD

0285.0ff004.0

5020.0

DD DJDFDJDF

BDmf hh = g2Vk

g2V

DLf

2BD

DF

2BD

BD

BDBD

= g2V5.15.0

10.0100235.0

2BD

= g2

V35.42BD



DFmf hh = g2Vk

g2V

DLf

2DF

DF

2DF

DF

DFDF

= g2V0.139.05.19.0

05.0150285.0

2DF

= g2

V34.122DF

DJmf hh = g2Vk

g2V

DLf

2DJ

DJ

2DJ

DJ

DJDJ

= g2V15.139.05.19.0

05.0200285.0

2DJ

= g2

V69.162DJ

(3)

g2V

84.92DF =

g2V

69.162DJ

VDF = (1.302) VDJ --------- (5) (4) 12302 =

g2V69.16

g2V34.12

g2V35.42

2DJ

2DF

2BD

36 =

g2

V69.16g2

V919.20g2

V70.82DJ

2DJ

2BD

36 =

g2

V609.37g2

V70.82DJ

2BD --------- (6)

QBD = QDF + QDJ BD24 V1.0 = DJ24DF24 V05.0V05.0 (4) VBD = VDF + VDJ (5) (4) VBD = (1.302) VDJ + VDJ (1.738) VBD = VDJ --------- (7)



(7) (6) 36 =

g2V738.1609.37

g2V70.8

2BD

2BD

36 = g2

V738.1609.3770.82BD2

VBD = 2.403 m/s VDJ = (1.738) 2.403 = 4.176 m/s VDF = (1.302) 4.176 = 5.438 m/s QBD = 403.210.0 24 = 0.01887 m3/s = 18.87 l/s QDF = 176.405.0 24 = 0.00820 m3/s = 8.20 l/s QDJ = 438.505.0 24 = 0.01067 m3/s = 10.67 l/s Ans

7-1 Fluid Mechanics

Open Channel Flow

7

(Open Channel Flow) (Atmospheric pressure) (Gravity)

7.1

7.2

7.1 (Type of channel)

2 - (Natural channel)

(Non-Prismatic channel) ( 7.1)

- (Artificial channel) (Prismatic channel) ( 7.2)

7-2 Fluid Mechanics

Open Channel Flow

7.2 (Open channel flow classification) 2 (Type of flow) (State of flow) 7.2.1 (Type of flow)

2 1)

a) (Steady flow) (y) (V) (Q) (A)

0Q,v,A,ydtd

b) (Unsteady flow)

0Q,v,A,ydtd

2)

a) (Uniform Flow : UF) (y) (A)

b) (Varied flow Non-Uniform flow) 2 - (Gradually Varied Flow : GVF)

- (Rapidly Varied Flow : RVF)

7.3

7-3 Fluid Mechanics

Open Channel Flow

- (Steady Uniform flow)

Steady uniform flow

7.4 Steady uniform flow

- (Steady Gradually Varied flow) Steady gradually varied flow

7.5 Steady gradually varied flow

- (Steady Rapidly Varied flow) Steady rapidly varied flow

7.6 Steady rapidly varied flow

7-4 Fluid Mechanics

Open Channel Flow

- (Unsteady Uniform flow) Unsteady uniform flow

7.7 Unsteady uniform flow

- (Unsteady Gradually Varied flow) Unsteady gradually varied flow

7.8 Unsteady gradually varied flow

- (Unsteady Rapidly Varied flow) Unsteady rapidly varied flow

7.9 Unsteady rapidly varied flow

7-5 Fluid Mechanics

Open Channel Flow

7.2.2 (State of flow)

7.2.2.1 (Reynold number : Re) Reynold number (Vicous

force) (Inertia force) ( 8) Re

VRRe --------- (7.1)

R = (Hydraulic radius) V = =

3 - (Laminar flow)

500 - (Turbulent flow)

2,000 - (Transitional flow)

500 2,000

7.2.2.2 (Froude number : Fr) Froude number

(Gravity force) (Inertia force) ( 8)

gDVFr --------- (7.2)

D = (hydraulic depth) g =

3 - (Critical flow) 1

- (Subcritical flow) 1

- (Supercritical flow) 1

7-6 Fluid Mechanics

Open Channel Flow

7.10 7.3 (Basic equation of open channel flow)

(Ideal fluid , Incompressible fluid) (Steady flow)

- (Depth :y) - (Top width :B) - (Hydraulic depth :D)

BAD ( yD )

- (Wetted perimeter :P)

- (Hydraulic radius :R)

PAR

7.11

7-7 Fluid Mechanics

Open Channel Flow

7.3.1 (Continuity Equation)

OutIn

CV MMt

M 7.12 (incompressible fluid) (steady state)

tMCV

= 0

InM = 1Q = 11VA OutM = 2Q = 22VA 0 = 2211 VAVA 11VA = 22VA --------- (7.3) 11 VyB = 22 VyB 11Vy = 22Vy --------- (7.4)

7.12

7-8 Fluid Mechanics

Open Channel Flow

7.3.2 (Energy Equation) 4 (Energy head) streamline (Elevation head) (Pressure head) (Velocity head) (Static head) 7.13

7.13 static head 1 2

yzhzhzPz 111111

yzhzhzPz 222222

static head

L22

22

21

11 hg2Vyz

g2Vyz --------- (7.5)

hL = yi = i zi = i Vi = i sf = sw = so =

7-9 Fluid Mechanics

Open Channel Flow

7.14

z 7.5 (y) 7.3.3 (Momentum Equation) 5 7.14

7.15 Control volume

7.14 5.7 sysF

= InInInOutOutOut VQVQ fFFF X21 = 12 QVQV F FX + f F = 2211 QVFQVF = 222111 QVAyQVAy

F =

2

2

221

2

11 gAQAy

gAQAy --------- (7.6)

7.6


Open Channel Flow

7.4 (Steady Uniform Flow) 7.16 (Sw) (So) (Sf) 7.17

7.16 Uniform flow

7.17 Uniform flow

7.16 F1 F2 f = W sin (L) ostansin PL = tanAL =

PLsAL o

= oRs --------- (7.7) (Wall shear stress)


Open Channel Flow

.. 1773 Antoni Chezy (V) V2 = 2KV (K ) --------- (7.8) 7.8 7.7 2KV = oRs V =

KRso

oRsCV --------- (7.9) 7.9 Chezy Chezy (Chezys Formula)

KC

Chezy (Chezy coefficient) .. 1890 Robert Manning Chezy Chezy Manning (Mannings roughness coefficient) 7.10 7.11 SI C = 61R

n1 --------- (7.10)

BG C = 61Rn49.1 --------- (7.11)

n Manning (Mannings roughness coefficient) Chezy ( 7.1) 7.9 Chezy Manning SI 2132 SR

n1V --------- (7.12)

2132 SRAn1Q --------- (7.13)

BG 2132 SRn49.1V --------- (7.14)

2132 SRAn49.1Q --------- (7.15)

7.12 7.14 Manning Manning (Mannings Formula)


Open Channel Flow

7.1 Manning (Mannings roughness coefficient)


Open Channel Flow

7.1 2 m. 0.015 0.001

2m162224ymybA m94.1221224m1y2bP 22 m24.1

94.1216

PAR

Manning V = 2132 SRn1

= 2132 001.094.12n1 --------- (1)

... Manning 0.015 (1) V = 2

132

001.094.12015.01 --------- (2)

Q = VA Q = 16001.094.12015.01

21

32

= 38.88 cms Ans 7.2 7.1 29 cms (yn) 7.1 nn ymybA 2n m1y2bP 2

n

nn

m1y2b

ymybPAR

Manning

Q = 21032

SARn1 = 210

32

2n

nnnn S

m1y2b

ymybymybn1

29 = 2132

2n

nnnn 001.0

21y24

yy24yy24015.01

yn ( trial & error) yn = 1.73 m Ans


Open Channel Flow

7.3 4 m. (Main channel) (Floodplain) 0.015 0.035 0.001

3

Main Channel : 1 21 m5.435.1145.22

144A

m18.15215.224P 21

m87.218.155.43R1

Q = 211032111

SRAn1 = 2132 001.087.25.43

015.01

= 185.20 cms Floodplain : 2 3 232 m38.185.12

5.1410AA

m74.14315.110PP 232

m25.174.1438.18

RR 32 Q2 = Q3 = 21 203222

2SRA

n1 = 2132 001.025.138.18

035.01

= 19.27 cms Q = Q1+Q2+Q3 = 185.20+19.27+19.27 = 233.74 cms Ans


Open Channel Flow

7.5 (S

Documents

Fluid Mechanics