409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A Q i k L kA Quick Look
Reynolds’ Transport TheoremLinear MomentumLinear Momentum
Newton’s law of motion The linear momentum theorem
Applications of the Linear Momentum Theorem Propulsion, wind turbines, wakes/jets, turbine blades, free surface flow
Angular MomentumAngular Momentum Newton’s law of angular momentum The angular momentum theorem
Applications of the Angular Momentum Theorem Centrifugal pumps/compressors, axial flow turbines/compressors
Momentum Conservation
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
R ld ’ T ThConsider a material volume m containing the same fluid as it
Reynolds’ Transport Theorem
moves and deforms following the motion of a fluid at three successive times t1, t2 and t3.
m mm
t1 t2 t3
Consider a fluid scalar property b/unit mass which might be a thermodynamic t h i t l th l it t fproperty such as internal energy or enthalpy per unit mass or components u,v,w of
velocity . The material volume m, unlike the control volume , moves with the fluid
V
Momentum Conservation
enclosing the same fluid particles.
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
Li M
Reynolds’ transport theorem
Linear Momentum
Reynolds transport theorem
dSnVbdbddB
S
dSnVbdbdtdt
T k hTake , thenVb
dMd
S
dS)nV(VbdVdtd
dtMd
Momentum Conservation
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
E lExample - Liquid Flow from the Tube
dSnVddtdmass
S
Atube
dtdLV
dt
L(t)
momentum
2
S
Ld
dS)nV(bVdVdtd
dtdM
V
2
2
dtLdAL
Momentum Conservation
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
N ’ L f M i (1)Apply Newton’s law of motion to the material volume that
Newton’s Law of Motion (1)Apply Newton s law of motion to the material volume that
coincides with our CV at a particular time Determine the forces that act on the fluid within the CV as
p
SFdS)np(
g
S
Fdg
FdS
gg
Then obtain the linear momentum theorem as
dS)nV(VdVdtd
S
Momentum Conservation
dgdSdS)np(
SS
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
N ’ L f M i (2) When the inflow and outflow streams are identified, the momentum equation takes
Newton’s Law of Motion (2)
the form
dgdSdS)np()Vm()Vm(dVdit
dgdSdS)np()Vm()Vm(dVdt SSinout
So far we assumed that the control volume contained only the fluid of interestSo far we assumed that the control volume contained only the fluid of interest. However, it may simplify the analysis to select a fixed control volume that encloses solid objects, then add the external force to the momentum equation as
inout )Vm()Vm(dVdtd
exSS
FdgdSdS)np(dt
Momentum Conservation
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
E lPatm
Example - Fire Hose Nozzle (1)
PatmFex
steady
Q=150 gpm
ρD=1 inch
inout
FddSdS)(
)Vm()Vm(dVdtd
exSS
FdgdSdS)np(
2
2Q4F
Momentum Conservation
2ex DF
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
E lmomentumHorizontal
Example - Fire Hose Nozzle (2)
Patm
.B.NFA)pp()Vm()Vm( cinoutininout
DdInviscid Flow
Fc Pin
equations'Bernoulli
)AA(pApAp inatmatmatmatminin
Pout = Patm
2Vp
2Vp 2
outout2inin
mAVAV
mass
outoutinin
1
AA
AA
21QVF
in
out
out
inoutc
Momentum Conservation
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A li i f h Li M Th (1)Rocket Engine Compressible Flow of combustion gases
Application of the Linear Momentum Theorem (1)g
FFuel burns at m = 2kg/s
.Ve =200m/s
Far enough down stream so that the plume pressure =atmosphere pressureatmosphere pressure
)Vm()Vm(dVdtd
inout
mkg
FdgdSdS)np( exSS
Momentum Conservation
N400sm200
skg2VmF e
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A li i f h Li M Th (2) Jet Engine
Application of the Linear Momentum Theorem (2)g
Vf= 250 m/s
Ain FVe
= 500m/s= 250 m/s F
mf
.
3
ff
s/m500Vm/kg40
s/kg2ms/m250V
2in
ea
m0.1A
s/m500Vm/kg4.0
FddSdS)(
)Vm()Vm(dVdtd
inout
s/m250s/kg100s/m500s/kg102VmVmF
FdgdSdS)np(
finout
exSS
Momentum ConservationN106.2
s/m250s/kg100s/m500s/kg102VmVmF4
fineout
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A li i f h Li M Th (3) Propellers Dp
Application of the Linear Momentum Theorem (3)p
Patm PoutPin
Vp (propeller)
Patm
VwVf
p (p p )F
CV1(Wake)(Flight)
1C
CV2
A(1) (2)ppoutin mAV)VA()VA(
mass
pppoutin
)Vm()Vm(dVdmomentum
)()(
exSS
inout
FdgdSdSnp
)Vm()Vm(dVdt
Momentum Conservationpinout A)pp(F
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A li i f h Li M Th (4)
i'B lli2C
Application of the Linear Momentum Theorem (4)
2Vp
2Vp,
2Vp
2Vp
equations'Bernoulli2
wa2
pout2
pin2
fa
F000VmVm0
equationMomentum2222
F000VmVm0 fw
p )())((A
Force
fwppfwpwp
FVPFV)(AVPPower
)VV(VA)VV)(VV(2
F
ff
fvpinoutppp
V2FVPefficiencyopulsivePr
FVP,FV)pp(AVP
Momentum Conservationfw
f
p
f
p
vprop VV
V2FVFV
PP
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A li i f h Li M Th (5)Wake Vf
Application of the Linear Momentum Theorem (5)
F
fVf
VFAw
Vw
Vf
x
inout )Vm()Vm(dVdtd
exSS
FdgdSdS)np(
)VA(/F41VV 2
Momentum Conservation2
)VA(/F41VVV fwwff
w
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A li i f h Li M Th (6) Jet
Application of the Linear Momentum Theorem (6) Jet
Jet Source
Entrainment
As
Aj VjVs
s
inout )Vm()Vm(dVdtd
exSS
FdgdSdS)np(
A
Momentum Conservationjj
sssj A
AVV
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A li i f h Li M Th (7) Flow through a sluice gate i
Application of the Linear Momentum Theorem (7) Flow through a sluice gate
Paair
F gUpstream
hout PaVin
hinwater2
in
h
0 a hg21dz)pp(in
Vout
a
inout )Vm()Vm(dVdtd
)W/F(widthperForce
exSS
FdgdSdS)np(dt
M
222
outoutinin
h1F
)Wh(V)Wh(VMass
Momentum Conservation
in
out2outout
2out
2in h
h1Vh)hh(g21
WF
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A li i f h Li M Th (8)Bernoulli’s equation
Application of the Linear Momentum Theorem (8)
gh2
Vpgh2
Vpout
2outa
in
2ina
dhhhg2V
outin
2in2
out
hhgh2
hh1h)hh(g
21
WF
and2inout
out2out
2in
)hh(hh4)hh)(hh()hh(2
ghhh2W
outinoutinoutin2out
2in
outin
outinin
)hh(2)hh(g
outin
3outin
outin
Momentum Conservation
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A li i f h Li M Th (9)Hydraulic Jump
Application of the Linear Momentum Theorem (9)y p
hh1Vh)hh(
2g0
MomentumMass
out2outout
2out
2in
subcriticalsupercritical
gair
F = 0 11hh
hh
21
hV
h)(
2
inin2out
inoutoutoutin
p
V
hin water Vout hout
11hh
hh
21
ghV
hh2gh
outout2in
outoutout
Vin
gatesluicethebeneathflowtheForjumpichydrodynamacrosshh
hh2gh
inout
ininin
11)h/h(
)h/h(2ghV
gatesluicethebeneathflowtheFor
outin
2outin
out
2out
Momentum Conservation
)(g outinout
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A l M ThNewton’s Law of Angular Momentum
Angular Momentum Theoremg
Multiply the linear momentum by R
FR)V(dR
FR)VmR(dor
FR)Vm(dt
R
FR)VmR(dt
or
Define the angular momentum as
d)VR(H
A l R ld ’ hApply Reynolds’ transport theorem as
dS)nV)(VR(d)VR(dHd
Momentum Conservation
S
dS)nV)(VR(d)VR(dtdt
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
E lKnown for the inflow at R1 andVDetermine)a(
Example - Steady Swirling Inward Flow (1)
R1
11 cosV 11 sinV
11
1
RtoRfromstreamlineaalongppDetermine)b(
andVDetermine)a(
.
1
R
V1
sinV
cosV r
1
i)sinV(i)cosV(V
havingflowInward
R
V
equationmomentumAngular
)1(d)gR(dS)R(dS))np(R(
dS)nV)(VR(d)VR(dtd
S
Momentum Conservation
)1(d)gR(dS)R(dS))np(R(SS
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
E lequationonconservatiMass
Example - Steady Swirling Inward Flow (2)
)2(0dS)nV(ddtd
S
sinVRsinRV)2(sincosVRsincosVR)1(
111
112
121
22
VRV 11
1
VpVpequations'Bernoulli
VR
V
22
1
RV
2Vp
2Vp
22
11
Momentum Conservation
1RR
2Vpp 11
1
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
E lMoment of external forces
Example - Sprinkler (1)
inout
Td)gR(dS)R(dS))np(R(
)VmR()VmR(d)VR(dtd
exSS
Td)gR(dS)R(dS))np(R(
Angular speed of sprinkler and velocity V of fluid stream relative to ground
V
α
Vθ =Vcosα
T i)RcosV(i)sinV(V r
)VR()VR(d)VR(d
RAΩ R
exSS
inout
Td)gR(dS)R(dS))np(R(
)VmR()VmR(d)VR(dtd
Ω
VAR2Tcos
RV
2
Momentum Conservation
V VAR2R
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
E l Convert the sprinkler to a reaction turbine by attaching an
Example - Sprinkler (2)
electric generator that applies a restraining torque to the rotor and absorbs an amount of power.
Pfor)b(
turbineofP)a(maxout
out
out )RcosV(RmTP
)RcosV(RmT)a(
vesselpressuretheinfluidtheofp)c( s
2
outmaxout
out
0d
dPwhenoccursP)b(
)cosV(
2maxout
equations'Bernoulli)c(4
)cosV(mP
2a
2in
2in
2)R(p
2Vp
2V
q)(
Momentum Conservation
22as )R(V
2pp
409.319A
Systems Engineering HydrodynamicsDepartment of Nuclear Engineering, Seoul National University
A li i f h A l M ThAxial Flow Turbines / Compressors
Application of the Angular Momentum Theoremp
(1)
Stator V2zV2
2zz
equations'Bernoulli)R(miRmiPpowerThe
(2)
Stator
izΩ R
2z
~ 222
21
2221
V)R(Vpp2
V2
Vpp:stator
(3)
rotor V3
Ω R
V3 z2332
equationonconservatiMass2
V2
)R(2
Vpp:rotor
231
3z21
)R(pp
VVV
Angular momentum equation
inout )VmR()VmR(d)VR(dtd
Momentum Conservation
exSS
Td)gR(dS)R(dS))np(R(