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XAMPLE 9.I3GIVEN A 100 km/hr (i.e., 28 m/s) wind blows past the watertower shown in Fig. E9.l3a.
FIND Estimate the moment (torque), M, needed at the base tokeep the tower from tipping over.
U = 100 km/hr= 28 m/s----)
D,= L2 mD.= 5 mb=15m
b)IFIGURE E9.13
9.67 The structure ihown in Fig. W.67 consists of threecylindrical support posts to which an elliptical flat-plate sign isattached. Estimate the drag on the structure when a 80 km/hr windblows against it.
IFIGURE P9.67
Ql. (a)
-L' sME2323D€fin€ the bomdrylayer thickness, 6
Describe 6e formaim ofthe bondry layer.
(5nrk+)
O) Ddermine tbe valuc of Cr md Cz in fte Table Q l. Show clerly yoE oalcotdioa
Tabte Ql: Ft*plde momentum inqgral rcsults forvcious assmed ltminarflow vclocity Profrcs
Piofil€ Charactert{RE, cr@ cD@
(r)
(D
I
Blasius solution
Lins
(v=z\\u 6)Prabolic
t;2!=zL-[ Zlu 6 \d/Sinwerre
g=*iokz-)u \2dl
5,00
3-46
5.48
0.664
0.578
0.7
o.655
1.328
Cr
r.ffi
t.3loe4
(lSnats)
( c ) Determine thc total pm'€r required to p'ull ec timb6 (diet€r 0.4 m md leogth f5m) witt rdocity O8 ds. Only half of the timbcr on ftewdEr sufaca Ncgld ttedr€ force due to fimt srrfrce.
Tatre cD..- =#, co*=H, p*,=lwfulm!,u--lxloam2 ls
(5 malcs)
Question:a) Determinetheequationfordragcoefficient67fora“one-seventhpower”turbulentvelocityprofile.
TheBlasiusresistanceequationisgivenas:
DE = 0.0225JKLMJKN
OP
b) Atimberispulledbyaboatalongtheriverwithforce200NasshowninFigureQ2.Onlyhalfofthetimberabovethewater.Determinethedragforceduetofrontsurfaceofthetimber.Takethedragcoefficientforturbulentderivedinquestion(a).Take,67(XYZ[\Y]) =
O.^L_`a,JEYba] =
OcccdeZf MEYba] = 1×10i^jk/mL
Answer:a)
nK=
oN
Op DE = 0.0225JKL
MJnN
Op
Diketahui;
r =nK
1 −nK
t
cuo =
772N
D = JKLuruw
= JKLuuw
7N72
Buatpersamaanserentak;
0.0225JKLMJKN
Op=
772
JKLuNuw
Selesaikan
N =0.3721w
z{O|
Diketahui
r =772N =
772
0.3721w
z{O|
=0.036w
z{O|
67 =D~uw
12 JK
L~�=
JKL uruw ~uw12 JK
L~�=2r�=2�
0.036w
z{O|
=0.072
z{O|
b)Reynoldnumberattheendoftimber
z{ =JKwM
=(1000)(2 + 1)(10)
1×10i^= 3×10p(turbulent)
PanjangaliranBLlaminar
z{Ö = 5×10| =JKwM
w =5×10| 1×10i^
(1000)(2 + 1)= 0.17 m
Lessthan10%fromtotallengthoftimberLaminarBLcanbeignored
Dragforcecausebysidesurface
á =12×2àâ×� = 6.286mL
ã7 = 6712JKLá =
0.072
z{O|×12JKL 6.286 = 65.1 j
Totalpullingforce= Dragforceatside+Dragforceatfrontsurface200 = 65.1 + ww = 134.9j
Question a) What is drag? And why do we usually try to minimize it? b) Assuming the velocity profile in the turbulent boundary layer as:
!"=
$%
&'
Determine the equation of drag coefficient () in term of Reynold number using momentum integral equation
*+ = ,"-./.0
And the Blasius resistance equation
*+ = 0.0225,"-5"%
&6
c) Figure Q2 shows a flat plate 3 m x 1.5 m is held in water moving at 5 m/s parallel to its length. Determine the power required to overcome the drag. Take , = 10009:/<= and > = 0.001?@/<-
()ABCDEBF =1.328
IJAKLMIJ ≤ 5×10PQR.
()STFUTAVES =0.074
IJA&P
KLMIJ ≥ 5×10P
Answer:
/ =9110
%
* = * = ,"-./.0
= ,"-..0
.110
% = ,"-9110
.%
.0
Diberi dari penyelesaian Blasius ;
* = 0.0225,"-5"%
&6
Oleh itu ;
% = 0.42565"
&P06P =
0.42560
(IJ)&P
Shear stress ;
* = 0.0225,"-5"%
&P
Masukkan nilai % ;
* =0.02795
&P,"-
"&P0
&P
Drag force ;
)̂ = *_.0 =0.02795
&P,"-
"&P0
&P
_.0`
a= 0.0349,"-_
5&P
"&P
b6P
Drag coefficient ;
() =
)̂12 ,"
-c=
0.0349,"-_ 5&P
"&Pb6P
12 ,"
-_b= 0.0698
5&P
"&Pb
&P
=0.0698
IJ&P
, = 10009:/<=, > = 0.001?@/<-
Nilai nombor Re,
IJ =,"0>
=(1000)(5)(3)
0.001= 1.5×10d = Turbulent
Re number at transition ;
5×10P =,"0`>
0` =5×10P>,"
=5×10P (0.001)(1000)(5)
= 0.1<
Hanya 3.3% sahaja kawasan akan mengalami laminar boundary layer. Oleh itu laminar boundary layer boleh diabaikan
Diberi dari soalan, nilai () untuk turbulent,
() =0.074
(IJ)&P
=0.074
(1.5×10d)&P
= 2.71×10m=
Drag force ;
)̂ = ()12,"-c = 2.71×10m=
12
1000 5 - 3 1.5 = 152.4 ?
Tetapi BL akan berlaku dikedua-dua belah permukaan plat, oleh itu jumlah drag force ;
)̂ = 152.4×2 = 304.8 ? Power = Drag×Velocity = 304.8×5 = 1524 Watt
Question1a) Transitionfromlaminartoturbulentboundarylayerflowactuallyoccursoverafinitelengthofthe
surface.Duringwhich the velocityprofile andwall shear stress adjust from laminar to turbulentforms.Ausefulapproximationduringtransition is that themomentumthicknessof theboundary
layer remains constant. Assuming constantmomentum thickness, find the ratio (CDEFGEHIJDCHKLMJKF
) fortransitionfromaparaboliclaminarvelocityprofiletoa“1/7=power”turbulentvelocityprofile.
b) Thevelocityprofile ina turbulentboundary layeroften isapproximatedby the ‘”1/7power law”equation.EvaluateV = W∗/Yforthepowerlowprofile
Answer:(a)Paraboliclaminarvelocityprofile,
\] = 2_ − _a
Momentumthickness,
Ycdefgdh = W i(_) 1 − i(_)j
kl_ =
215 Wcdefgdh
1/7-powerturbulentvelocityprofile,
\] = _
jo
Momentumthickness,
Ypqhrqcsgp = W i(_) 1 − i(_)j
kl_ =
772 Wpqhrqcsgp
WpqhWcde
=215×
727 = 1.37
(b)1/7-powerlaw,
\] = _
jo
Where,
Y = W772
W∗ = W18
V =W∗
Y =W 18
W 772
= 1.286
Question1
a) Explainclearlywithsketches,theboundarylayerthicknessforaflowpastaflatplate,andwriteamathematicexpressionoftheboundarylayerthickness
b) Fluidhavingadensityof1.24BC/EFandkinematicviscosityof1.5×10JKEL/Mflowingwithvelocity4.0E/Moveraplateof0.5mlongand0.6mwide.ThevelocityprofileoftheflowisintheformofP = R + TU + VULDetermine:
i. Thevelocityprofileii. Theboundarylayerthicknessatthetrailingedgeoftheplateiii. Theshearstressatthemiddleoftheplateiv. Thedragforceononesideoftheplate
(20marks)
Answer:(b) P = R + TU + VUL
i. Denganmenggunakanboundarycondition:U = 0, P = 0
U = [, P = \
U = [,]P]U
= 0
P\= 2
U[−
U[
L
ii.
_ = `\L215
][]a
=2b\[
[ =5.48a
`\ab
=5.48(0.5)
(4)(0.5)1.5×10JK
= 7.5×10JF m
iii. Shearstressatthemiddleoftheplate
_ =2b\[
[ =5.48(0.25)
(4)(0.25)1.5×10JK
= 5.306×10JF m
_ =2b\[
=2 1.24 (1.5×10JK)(4)
5.306×10JF= 0.0280 gR
iv.
hi = _j]a = _j]a = 0.73 ` k \j
\lk=
m.K
m
n
m0.011899 p
Question:
a) Explainwhatyouunderstandbyboundarylayerthicknessanddisplacementlayerthickness.
b) Assumethatinthelaminarboundarylayer,theflowobeysthelawofshearstress> = @AB
AC,where@
istheviscosity,whichleadtotheprofileof(F − H) = I(J − K)L,whereUisthefreestreamvelocity,
u is the velocity at a distance y above theplate andk is a constant.Determine thedisplacement
thickness.
c) Usingtheanswerfrom(b),findthedisplacementthicknessattheboundaryconditionsy=0andu=0
Answer:
Displacementthickness,,J∗
J∗ = 1 −H
FSK
T
U
Velocityprofile,
F − H = I(J − K)L
BahagikansemuadenganU
1 −H
F=
I
F(J − K)L
J∗ = 1 −H
FSK
T
U
=I
F(J − K)LSK
T
U
=I
F(J − K)LSKT
U
=I
FJL − 2JK + KL SK
T
U
=I
FJLC −
2JKL
2+KZ
3U
T
=I
FJZ − JZ +
JZ
3
J∗ =I
F
JZ
3
Atboundarycondition
J∗ =I
F
JZ
3 (i)
AtK = 0, H = 0
(F − H) = I(J − K)L
F = IJL (ii)
J∗ =IJZ
3
1
IJL=J
3
Question
a) Whatdoesitmeantbydisplacementthickness?
b) ProvethattheVon-KarmanMomentumIntegralequationcanbewrittenas
BC = EFGHI
HJ
Where
BC :Wallshearstress
E :Fluiddensity
F :Freestreamvelocity
I :Momentumthickness
c) FigureQ1showsauniformflowpasta flatplate.DeterminethewallshearstressBCbetween
AD=0.5m.thevelocityprofileislaminarT
U=
V
W
XG, Z[\ = 0.15]andZ[^_ = 0.16].TakeE =
1.208de/]gandh = 1.81×10jklm/]Gforair.
d) DetermineCDforT
U=
V
W
Xo= ɳ
Xo
Answer:
(c)
E = 1.208de/]g Z[\ = 0.15]
h = 1.81×10jklm/]G Z^_ = 0.16]
HZ = Z^_ − Z[\
s
F=
t
Z
XG
= ɳXG
= 0.16 − 0.15
= 0.01]
HJ = 0.05]
BC = EFGHI
HJ
=
EFG
H
HJ
s
F1 −
s
F
W
u
Ht
=
EFG
HZ
HJɳXG 1 − ɳ
XG
W
u
Hɳ
=
EFG
HZ
HJɳXG − ɳ
W
u
Hɳ
=
EFG
HZ
HJ
2
3ɳgG −
1
2ɳG
u
X
BC = EFGHZ
HJ
2
3−1
2
= EFGHZ
HJ
2
3−1
2
= (1.208)(0.5)G0.01
0.5
1
6
= 1.007×10jg l/]G
Thevelocityprofileofturbulentboundarylayer
s
F=
t
Z
Xo
= xXo
Momentumintegralequation
BC = EFGHI
HJ
Blasiusresistanceequation
BC = 0.0225EFG
z
FZ
X{
BC = EFGHI
HJ= EFG
HZ
HJ
9
10−9
11= 0.0818EFG
HZ
HJ
0.0818EFGHZ
HJ= 0.0225EFG
z
FZ
X{
ZX{HZ =
0.0225EFGzX{F
jX{
0.0818EFGHJ
ZX{HZ = 0.275z
X{F
jX{HJ
4
5Zk{ = 0.275z
X{F
jX{J + �
Z = 0.3438zX{F
jX{J
{k
�_ =
0.0698
ÄÅÇ
X
k