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Nozzle Study
Yan Zhan, Foluso Ladeinde
April 21, 2011
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Outlines
• Region of interests• Bend Combinations without nozzle
– Bend Combinations Without Nozzle– Turbulence Model Comparisons– Mercury Flow in Curved Pipes – Discussions
• Bend Combinations with Nozzle– Bend Combinations With Nozzle– Mercury Flow in Curved Nozzle Pipes – Discussions
• Appendix
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Region of interests
Hg delivery system at CERN
Simulation Regime
Interests: Influence of nozzle
& nozzle upstream on the jet exit
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Outlines
• Region of interests• Bend Combinations without nozzle
– Bend Combinations Without Nozzle– Turbulence Model Comparisons– Mercury Flow in Curved Pipes – Discussions
• Bend Combinations with Nozzle– Bend Combinations With Nozzle– Mercury Flow in Curved Nozzle Pipes – Discussions
• Appendix
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Bend Combinations Without Nozzle (1)
Geometry of bends with varying anglesNondimensionalized by pipe diameter (p.d.);φ range: 0⁰, 30⁰, 60⁰, 90⁰.
Mesh at the cross-sectionnr × nθ =152×64 (the 1st grid center at y+≈1)
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Geometries without nozzle: (a) 0⁰/0⁰ ; (b) 30⁰/30⁰; (c) 60⁰/60⁰; (d) 90⁰/90⁰ nx =40 +nφ+20+ nφ +100where nφ depends on the bend angle φ
Bend Combinations Without Nozzle (2)(a)
(b) (c)
(d)
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Turbulence Model Comparisons (2)
Sudo’s Experiment (1998) for 90⁰ bend*
•K. Sudo, M. Sumida, H. Hibara, 1998. Experimental investigation on turbulent flow in a circular-sectioned 90-degrees bend, Experiments in Fluids. 25, 42-49.
uave = 8.7 m/s; Re=6×104; ρair = 1.2647; μ air = 1.983×10-5; Pr= 0.712
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Inlet
Outlet
Turbulence Model Comparisons (3)
Mesh at the cross-sectionnr × nθ =80×50 (the 1st grid center at y+≈1)
Mesh for the test pipe
nφ =75 (dφ=1.2⁰)
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Turbulence Model Comparisons (4)
Longitudinal distribution of wall static pressureCp pressure coefficientpref reference value of p at z’/d=-17.6
RKε is the best of the three in simulating curved pipe flow
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Mercury Flow in Curved Pipes (1)
• Steady incompressible turbulent flow
• Boundary Conditions– Inlet: Fully developed velocity;
– Outlet: Outflow;
– Wall: non-slip
• Schemes– 3rd order MUSCL for momentum and turbulence equations
– SIMPLE schemes for pressure linked equations
• Convergence Criterion– 10-5
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Mercury Flow in Curved Pipes (2)
• Turbulence Characteristics– Turbulence Level (Flutuating Velocity)
– Momentum Thickness (Mean Velocity)
Measure of the momentum loss within the boundary layer due to viscosity.
mm uk
uuI 3/2
=′
≡
∫∫ −=⇒−= R
t
R
t drUu
UudyuUuUU
00)1( )( θρρθ
Small I
Big θt
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Mercury Flow in Curved Pipes (3)
• Turbulence Level (1)
Fig. Two shown planes at the pipe outlet
Pipes Imean at outlet
90⁰/90⁰ 5.204039%
60⁰/60⁰ 5.323943%
30⁰/30⁰ 5.368109%
0⁰/0⁰ 4.649036%
Table. Mean turbulence level at pipe outlet Mean Iin=4.600041%
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Mercury Flow in Curved Pipes (4)• Turbulence Level (2)
Turbulence intensity distribution at the pipe exit(a) Horizontal plane (from inner side to outer side )
(b) Vertical (from bottom to top )
(a) (b)
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Mercury Flow in Curved Pipes (4)
• Momentum Thickness
Momentum thickness distributionalong the wall at the pipe exit(a) 0⁰/0 ⁰ bend (b)30⁰/30 ⁰ bend (c) 60⁰/60 ⁰ bend (d) 90⁰/90 ⁰ bend
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Discussions
• Bend Effects– Bend and turbulence level
• Bend enhances the turbulence level but not too much;– The 0⁰/0⁰ bend has the lowest turbulence level;
• Symmetry I for the 90⁰/90⁰ bend;
– Bend and θt
• Bend effects θt not linearly with the increasing bends– The 60⁰/60⁰ bend seems like a turning point
• Less uniform θt distribution in larger bend– θt is bigger near the inner side and smaller near the outer side
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Outlines
• Region of interests• Bend Combinations without nozzle
– Bend Combinations Without Nozzle– Turbulence Model Comparisons– Mercury Flow in Curved Pipes – Discussions
• Bend Combinations with Nozzle– Bend Combinations With Nozzle– Mercury Flow in Curved Nozzle Pipes – Discussions
• Appendix
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Bend Combinations With Nozzle
Geometry of bends with varying anglesNondimensionalized by pipe diameter (p.d.);φ range: 0⁰, 30⁰, 60⁰, 90⁰.
Mesh at the cross-section (half model)nr × nθ =152×64 (the 1st grid center at y+≈1)
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Mesh for bends(a) 0⁰/0⁰ ; (b) 30⁰/30⁰; (c) 60⁰/60⁰; (d) 90⁰/90⁰ nx =40 +nφ+20+ nφ +60+70where nφ depends on the bend angle φ
(a)
(b) (c)
(d)
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Mercury Flow in Curved Nozzle Pipes (1)
Longitudinal distribution of static pressure for 90⁰/90⁰ CombinationVelocity inlet condition
Pin 30bar;uin fully developed velocity profile;
Outlet flow condition
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Mercury Flow in Curved Nozzle Pipes (2)
• Main Loss
Assume smooth pipe, thus Pmain=0
• Minor Loss– Elbow Loss
– Contraction Loss
• Total Loss
2/)/(2/)/( 22ininoutoutmain uDLudlP ρλρλ +=
mguh inelbow 2618.028.9136.43.02/ 22 =÷÷×== ξ
mgKuh outcontr 9694.028.9200475.02/ 22 =÷÷×==
Pa 198196944 )2( =++= contrelbowmainloss hhgPP ρ
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Mercury Flow in Curved Nozzle Pipes (3)
• Bernoulli’s Law
• Comparison
Pa 3534.208465944.1981965.013546205.013546136.410 3
2/5.02/5.0226
12
12
=⇒+××+=××+×
+++=++
out
out
lossoutoutinin
PP
PghuPghuP ρρρρ
%0898.9%1003534.208465
3534.2084653.189516
%100
=×−
=
×−
=ana
ananum
PPPError
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Mercury Flow in Curved Nozzle Pipes (4)
• Turbulence Level (1)
Pipes Imean(Nozzle)
Imean(No nozzle)
90⁰/90⁰ 5.143341% 5.204039%
60⁰/60⁰ 2.858747% 5.323943%
30⁰/30⁰ 2.701366% 5.368109%
0⁰/0⁰ 2.182027% 4.649036%
Fig. Two shown planes at the pipe outlet Table. Mean turbulence level at pipe outlet Mean Iin=4.599195%
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Mercury Flow in Curved Nozzle Pipes (5)• Turbulence Level (2)
Turbulence intensity distribution at the pipe exit(a) Horizontal plane ; (b) Vertical plane
(a) (b)
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Mercury Flow in Curved Pipes (6)
• Momentum Thickness
Momentum thickness distributionalong the wall at the pipe exit(a) 0⁰/0 ⁰ bend (b)30⁰/30 ⁰ bend (c) 60⁰/60 ⁰ bend (d) 90⁰/90 ⁰ bend
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Discussions (1)
Turbulence intensity distribution at the pipe exit(a) Horizontal plane (b)Vertical plane
(a) (b)
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Discussions (2)
The comparison of momentum thickness distribution at the pipe exit between pipe with/without nozzle(a) 0⁰/0 ⁰ bend (b)30⁰/30 ⁰ bend (c) 60⁰/60 ⁰ bend (d) 90⁰/90 ⁰ bendRed: pipe without nozzle;Blue: pipe with nozzle.
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Discussions (3)
• Nozzle Effects– Nozzle and turbulence level
• Nozzle reduces the turbulence level;– The 90⁰/90⁰ bend doesn’t change too much;
• I of the 90⁰/90⁰ bend is far different from other bends
– Nozzle and θt
• Nozzle decreases θt
• Very uniform and similar θt distribution at the nozzle exit for all bend pipes
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Outlines
• Region of interests• Bend Combinations without nozzle
– Bend Combinations Without Nozzle– Turbulence Model Comparisons– Mercury Flow in Curved Pipes – Discussions
• Bend Combinations with Nozzle– Bend Combinations With Nozzle– Mercury Flow in Curved Nozzle Pipes – Discussions
• Appendix
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Appendix(1)
• Continuity EquationThe two-phase model considers mixture comprising of liquid, vapor and non-condensable gas (NCG). Gas is compressible, the liquid and vapor are impressible. The mixture is modeled as incompressible.
• Momentum Equations
0=∂∂
j
j
xu
j
ji
ij
ji
xxP
xuu
∂∂
+∂∂
−=∂∂ τ
ρ
)( and )( where2
t ερµµµτ µ
kCxu
xu
i
j
j
itij =
∂∂
+∂∂
+=
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Appendix(2)
• Spalart-Allmaras Model
νν
ν
ν
νρννρµσ
νρ~
22
~
])~
(~
)~([1~SY
xC
xxG
xu
j
b
jji
i +−∂∂
+∂∂
+∂∂
+=∂
∂
)/(~~ and ~~ termproduction The 222~1 dfvSSvSCG vb κρν +≡=
21 )/~( n termdestructio The dvfCY ww ρν =
3.0,1.7 2/3, ,4187.0,622.0,1355.0 21~21 ====== wvbb CCCC νσκ
2/)(,2 and )1/(1 where ,,1~2~ ijjiijijijvv uuSff −=ΩΩΩ≡+−= χχ
)~/(~ ),( ,)]/()1[( where 2262
6/163
663 dSvrrrCrgCgCgf wwww κ≡−+=++=
vbbww CCCC ~22
113 /)1(/2.0, σκ ++==
stresses Reynolds theestimating ignored , termsource defined-User vS
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Appendix(3)
• Turbulent Viscosity (SA)
vvC
fv
v /~ and function damping visouswhere31~
3
3
1~ ≡+
= χχχ
1~~
vt fνρµ =
. viscositykinematicmolecular theis v
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Appendix (4)
• Standard K-ε model
εεεε
ε
ερεεσµµ
ερρε Sk
CGCGk
Cxxx
ut bk
j
t
jj
j +−++∂∂
+∂∂
=∂∂
+∂∂ 2
231 )(])[(
kMbk
jk
t
jj
j SYGGxk
xxku
tk
+−−++∂∂
+∂∂
=∂∂
+∂∂ ρε
σµµ
ρρ ])[(
RTkMMY
xTgG
xu
uuG ttM
it
tib
i
jjik γ
ρµβρ ==∂∂
=∂∂′′−= ,2,
Pr, 2
3.1,0.1,09.0,92.1,44.1 21 ===== εµεε σσ kCCC
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Appendix (5)
• Turbulent Viscosity (SKε)
ερµ µ
2kCt =
constant. a is µC
gradients;ity mean veloc toduek of generation :kG
buoyance; toduek of generation :bG
rate;n dissipatio overall the toturbulence lecompressibin dilatation gfluctuatin theofon contributi :MY
; and for numbers Prandtl turbulent:, εσσ ε kk
terms;source defined-user :, εSSk
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Appendix (6)
• Realizable K-ε model
kMbk
jk
t
jj
j SYGGxk
xxku
tk
+−−++∂∂
+∂∂
=∂∂
+∂∂ ρε
σµµ
ρρ ])[(
εεε
ε
εε
ερερεσµµ
ερρε SCCk
Cvk
CSCxxx
ut b
j
t
jj
j +++
−+∂∂
+∂∂
=∂∂
+∂∂
31
2
21])[(
ijij SSSkSC 2,,]5
,43.0max[1 ==+
=ε
ηηη
RTkMMY
xTgG
xu
uuG ttM
it
tib
i
jjik γ
ρµβρ ==∂∂
=∂∂′′−= ,2,
Pr, 2
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Appendix (7)
• Turbulent Viscosity (RKε)
ερµ µ
2kCt =
kijkijijkijkijijijijijij
s
SSUkUAAC ωεωε
ε
µ −Ω=Ω−Ω=ΩΩΩ+≡+
= ,2~,~~,1 **
0
jiijkijkij
s SSSS
SSSWWAA =====
− ~,~,3
)6(cos,cos6,04.43
1
0 φφ
2.1,0.1,9.1,44.1,2 21 ====+
= εε σσ kjiij
ij CCuu
S
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Appendix(8)Models Pros Cons
Spalart-Allmaras(SA)
Low-cost;Aerospace applications involving wall-bounded flows and with mild separation;
Can not predict the decay of homogeneous, isotropic turbulence; No claim is made regarding its applicability to all types of complex engineering flows;
Standard K-ε(Skε)
Robust and reasonably accurate;Most widely-used engineering model industrial applications;Contains submodels for buoyancy, compressibility, combustion, etc;
Must use wall function for near wall calculation;Performs poorly for flows with strong separation, large streamline curvature, and large pressure gradient;
Realizable K-ε(Rkε)
superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation
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Appendix (9)
• Properties of fluid
• Steady incompressible turbulent flow
Variable ValuesPipe ID 0.884 inch
Nozzle exit ID 0.402 inch
Driving pressure 30 bar
Frequency for Hg supply 12 s
Maximum Cycles 100
Jet Velocity 20 m/s
Jet Diameter 1 cm
Jet Flow rate 1.6 L/sec
Environment 1 atm air/vacuum
Mercury Properties (25 ⁰C )
Density 13.546 kg/L
Sound Speed 1451 m/s
Bulk Modulus 2.67×1010Pa
Dynamic Viscosity 1.128×10-7 m2/s
Thermal Conductivity
8.69 W/m⋅K
Electrical Conductivity
106 Siemens/m
Specific Heat 0.139 J/kg⋅KPrandtl Number 0.025
Surface Tension 465 dyne/cm