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UNSTEADY FRICTION IN TRANSIENT FLOW
DR. WALTER F. SILVA-ARAYA
ERDC – VICKSBURG JULY, 2014
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UNSTEADY PIPE FLOW EQUATIONS
• G!"#$%$& "'()*%$+ $"-%"$+%$)/ ($+*") %" 3
C$*%$(%*
5"$*( "'()*%$
Js )$ Ju )#" *6" +*") )$ ($+*") #%7*%$ *"#+.
T6" +*") #%7*%$ 7)$ 8" "9#"++" 8 *6" D)#7-W"%+8)76 #"/ :several equations available to estimate f ;
S/(*%$ 8 *6" "*6 76)#)7*"#%+*%7+ :5OC; # H )$ V :"
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UNSTEADY FRICTION 5ODELS
1. I$+*)$*)$"(+ A77"/"#)*%$ 5"/+ :B#($$" "* )/. 1R)+ "* )/. 200, V%*!+ "* )/. 200;
2. W"%&6*" F($7*%$ B)+" "/+ :%"/", 1@, V)#>3)$&, 1@@1, V)# )$ B#3$ 2004, 200;
. T3 %"$+%$)/ "/+ :S%/!)-A#)) )$ C6)(6# P"??%$&), 1@@@;
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INSTANTANEOUS ACCELERATION BASED 5
• T6" IAB "/+ )++(" *6)* *6" 3)// +6")# +*#"++ %+ ) (
*6" %$+*)$*)$"(+ /7)/ )77"/"#)*%$+ )$ )$ "%#%7)/ 736%76 ) 8" ) ($7*%$ *6" (% #"#*%"+.
• O$" 7"
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INSTANTANEOUS ACCELERATION BASED 5
• %+ *6" +%&$ *6" 7$!"7*%!" )77"/"#)*%$ *"# )$ 3)+
%$*#(7" * #(7" )%$& (#%$& )77"/"#)*%$ )$ 6
"H"7* (#%$& "7"/"#)*%$+.
• K ut )$ K ux )#" *3 decay coefcients #"/)*" * *6" /7
7$!"7*%!" )77"/"#)*%$+ #"+"7*%!"/.
• I* 6)+ 8""$ !"#%" $("#%7)// *6)* *6" *"# K ut ∂V/∂t )H
6)+" +6%* *#)$+%"$* #"++(#" 3)!"+ )$ K ux ∂V/∂x *6"
:R)+ "* )/. 200;.
•
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ESTI5ATION OF DECAY COEFFICIENTS
• T6" $(8"# )#)"*"#+ 7(/ 8" $" # *3 ""$%$& $ *6" 76+"
• T6" "%#%7)/ 7"
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GENETIC ALGORIT>5S
• A &"$"*%7 )/&#%*6 :GA; %+ ) +")#76 *"76$%'(" (+" %$ 7(*%
"9)7* # )#9%)*" +/(*%$+ * *%%?)*%$ )$ +")#76 #8
T6" GA %+ ) #)$ +")#76 )/&#%*6 8)+" $ *6" 7$7"* $)+"/"7*%$ %$6"#"$* %$ $)*(#)/ &"$"*%7+, #"+"$*+ ) #8(+* "*6+")#76 # *6" *%( +/(*%$ * *6" 7/"9 #8/"+.
• T6" )#*%7%)/ +(#!%!)/ 8"**"# +/(*%$ %$ GA +")#76 *"76$%'(" %)76%"!" 3%*6 &"$"*%7 "#)*#+ %$6"#%*)$7", (*)*%$, +"/"7*%
7#++!"# 8)+" $ "!/(*%$)# 8%/&.
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GA OBJECTIVE FUNCTION
• T6" /")+* +'()#"+ :LS; )#" 7$/ (+" )+ *6" 8"7*%!" ($7*%$ %$ *%%?)
#8/"+.
• T6" $#)/%?" #* ")$-+'()#" "!%)*%$ # "### :NR5SE; %+ ) +7)/" !)/("8"7*%!" ($7*%$.
• NR5SE %+ *6" #* ")$-+'()#" "!%)*%$ :R5SD; %!%" 8 *6" #)$&" ")
!)/("+.
• NR5SD #!%"+ ) ")+(#" *6" &$"++ * 8"*3""$ *6" "9"#%"$*)/ )$
)*) )//3%$& 7)#%+$ 8"*3""$ %H"#"$* )*) +"*+ 8"7)(+" %+ $$ %"$+%
• D%H"#"$7"+ 8"*3""$ *6" !)/("+ #"%7*" 8 "/ )$ *6" ")+(#" !)/("+ )*6%+ )#)"*"# 36%76 %+ %$%%?" 8 GA
• Complete time-history of pressure measurements at the valve location are used to estimate the IAB
this research using NRMSE as the objective function
N ! total number of time levels of data
J = time level H j* " H j ! measured and computed pre
H max* & H min* ! ma#imum and minimum values of mea
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IAB ONE COEFFICIENT
P+%*%!" 7)*%8%/%*
P+%*%!" 76)#)7*"#%+*%7
N"&)*%!" 7)*%8%/%*
N"&)*%!" 76)#)7*"#%+*%7
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IAB ONE COEFFICIENT
R")##)$&%$& +%*%!" 76)#)7*"#%+*%7"'()*%$
R")##)$&%$& $"&)*%!" 76)#)7*"#%+*%7
"'()*%$
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IAB TWO COEFFICIENT 5ODEL
P+%*%!" 7)*%8%/%*
P+%*%!" 76)#)7*"#%+*%7
N"&)*%!" 7)*%8%/%*
N"&)*%!" 76)#)7*"#%+*%7
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DATA BASE OF EPERI5ENTAL DATACase L (m) D (m) Vo (m/s) a
(m/s)
Pipe
material
Valve
location Re
[email protected] 0.01 0.0 12@ C"# 1112 0.0 D3$+*#")
A)3+% )$"#C! 1M.00 0.02M 0.0M 110 C"# 1424 0. D3$+*#")
S)**)# ""#C$ 1M.00 0.02M 0.0@4 101 C"# 24 0.4 D3$+*#")
S)**)# "
IMP-PA$
@.11 0.01 0.40 12@ C"# M2@ 0.1 D3$+*#")
A)3+% )$%& .20 0.022 0.00 120 C"# M@ 0.M D3$+*#")
B"#&)$* )$
IMP-PA'
@.11 0.01 0.1 12@ C"# 102 0.0 D3$+*#")
A)3+% )$
IMP-PA&
@.11 0.01
0.@40
12@ C"# 1M 0.0 D3$+*#")
A)3+% )$R! 2.00 0.04 0.@2 PVC 100 0.22 D3$+*#")
R)+ "R$ 14.00 0.04 0.@0 M0 PVC 414@ 0.2 D3$+*#")
R)+ "P! 14.0 0.0M 0.01 1 S*""/ @00 .4 U+*#") P"??%$&) )$P$ 14.0 0.0M 0.100 1 S*""/ M2@M 1.2 U+*#") P"??%$&) )$%' .20 0.022 0.00 120 C"# M@ 0.M U+*#") B"#&)$* )$P' 14.0 0.0M 0.14 1 S*""/ @42 0.@ U+*#") P"??%$&) )$P& 14.0 0.0M 0.2 1 S*""/ 1M24@ 0.44 U+*#") P"??%$&) )$
Case L (m) D (m) Vo (m/s) a
(m/s)
Pipe
material
Valve
location Re
[email protected] 0.01 0.0 12@ C"# 1112 0.0 D3$+*#")
A)3+% )$"#C! 1M.00 0.02M 0.0M 110 C"# 1424 0. D3$+*#")
S)**)# ""#C$ 1M.00 0.02M 0.0@4 101 C"# 24 0.4 D3$+*#")
S)**)# "
IMP-PA$
@.11 0.01 0.40 12@ C"# M2@ 0.1 D3$+*#")
A)3+% )$%& .20 0.022 0.00 120 C"# M@ 0.M D3$+*#")
B"#&)$* )$
IMP-PA'
@.11 0.01 0.1 12@ C"# 102 0.0 D3$+*#")
A)3+% )$
IMP-PA&
@.11 0.01
0.@40
12@ C"# 1M 0.0 D3$+*#")
A)3+% )$R! 2.00 0.04 0.@2 PVC 100 0.22 D3$+*#")
R)+ "R$ 14.00 0.04 0.@0 M0 PVC 414@ 0.2 D3$+*#")
R)+ "P! 14.0 0.0M 0.01 1 S*""/ @00 .4 U+*#") P"??%$&) )$P$ 14.0 0.0M 0.100 1 S*""/ M2@M 1.2 U+*#") P"??%$&) )$%' .20 0.022 0.00 120 C"# M@ 0.M U+*#") B"#&)$* )$P' 14.0 0.0M 0.14 1 S*""/ @42 0.@ U+*#") P"??%$&) )$P& 14.0 0.0M 0.2 1 S*""/ 1M24@ 0.44 U+*#") P"??%$&) )$
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IAB ONE COEFFICIENT
•
V)#->3)$& :1@@M;
• V)#->3)$& :200;
• R" "* )/. :2012;
$%&'' ( R e ( )'%'''
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DECAY COEFFICIENTS :USVC AND DSV
$'' $''' $'''' $'''
'
''$
''*
''+
''&
'')
'',
''. /A 0S12 . /A 3S12 . 1B *''+ . 1B $44)
Reynolds Number
IAB 0ecay 2oefficients
1B - $44)
5aminar
6lo7
812 8ipe
1B- *''+
R-S-2
-*'$*
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IAB TWO DECAY COEFFICIENTS 5ODE• W%*6%$ *6" #)$&" R"$/+ $(8"#+ %$7/(" %$ *6%+ )$)/+%+, *6" "7)
# *6" *3-7"
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WEIG>TING FUNCTION BASED 5ODEL
• %"/" :1@; #"+"$*" ) 3"%&6*%$& ($7*%$ "/
*#)$+%"$* /)%$)# %" #%7*%$.
• T6%+ "/ )77($*+ # *6" %$("$7" *6" /7)/ !"/
6%+*# )* )$ %$* )/$& ) %" 36"$ "(7%$& *6"
%$+*)$*)$"(+ )&$%*(" *6" /7)/ +%$ #%7*%$.
• T6" $"9* +*)&" %$ 7/"9%* %+ * %$*#(7" #)*"+ 7
)77"/"#)*%$ )$ 6%&6"# #"# "#%!)*%!"+, /")%$& *
#(/)" *6" *"
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WEIG>TING FUNCTION BASED 5ODEL• F//3%$& %"/"+ %") *6" ($+*") +6")# +*#"++ 7)$ 8"
"9#"++" )+ ) 3"%&6*" %$*")/ )77"/"#)*%$+ )* )// #*%"+
•F# 7(*"# )/%7)*%$+t * ela+se time since calculations be)ant, * - . t is measure bac0ars in time from t1e current instant# t-1e 0ei)1tin) function 2 is a unique function of t,U/t is a function of t(
i ** N T* * %+ *6" *%" %$*"#!)/ 8"*3""$ +(77"++%!" 7)/7(/)*%$I$ #)7*%7", *6" 3"%&6*%$& ($7*%$ %+ +)// 36"$ *%+ /)#&" + *6" $("#%7)/ %$7)$ 8" *#($7)*". >3"!"#, *6" $(8"# *"#+ * 8" #"*)%$" %+ %$!"#+"/
* *6" )&$%*(" *6" %$*")*%$ *%" +*" )$ %+ +"*%"+ !"# /)#&".
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WEIG>TING FUNCTION BASED 5ODEL
• A *(#8(/"$7" "/ %+ #"'(%#" # *6" +6")# +*#"++
%+*#%8(*%$
• V)# )$ B#3$ :1@@M; (+" " !%+7+%* :+"!"#)/
!"#+%$+ 6)!" 8""$ #+";
#)*" 76)$&" %$")*%7 !%+7+%* # )* *6" 3)// * )* *6" 7#"+6")#-/
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EQUATIONS OF FLOW
• S6")# L)"#
• S/!" 8 L)/)7" T#)$+#. O8*)%$ *#)$+#" *6" 3"%&6*%$& (
F%$)// %$!"#+" *#)$+# * 8*)%$ . :V)# )$ B#3$, 1@@M;
• S"!"#)/ !)#%)*%$+ *6%+ "*6 6)!" 8""$ #+".
•
3
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CASE STUDY
0.000 0.M00 1.000 1.M00 2.000 2.M00
-20
-10
0
10
20
0
40
M0
0
0
Measred pressre istory at te valve
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CASE STUDY• S*") S*)*" F#%7*%$
0 0.2 0.4 0. 0. 1 1.2 1.4
-20
-10
0
10
20
0
40
M0
0
0
C(*" E9"#%"$*)/
*ime (s)
Pressre (m)
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CASE STUDY• +neCoe/M+C
0.0 0.2 0.4 0. 0. 1.0 1.2 1.4 1
-20
-10
0
10
20
0
40
M0
0
0
0E9"# %"$*)/ C(*"
*ime (s)
Pressre (m)
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CASE STUDY• *,oCoe/D
0.0 0.2 0.4 0. 0. 1.0 1.2 1.4
-20
-10
0
10
20
0
40
M0
0
0
0 E9"# %"$*)/ C(*"
*ime (s)
Pressre (m)
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CASE STUDY• *,oCoe/M+C
0.0 0.2 0.4 0. 0. 1.0 1.2 1.4 1.
-20
-10
0
10
20
0
40
M0
0
0
0 E9"# %"$*)/ C(*"
*ime (s)
Pressre (m)
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CASE STUDY
• Vardy and .ro,n ($$) 0.
0 0.2 0.4 0. 0. 1 1.2
-20
-10
0
10
20
0
40
M0
0
0 C(*" E9"# %"$*)/
*ime (s)
Pressre ead (m)
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CASE STUDY
• Vardy-.ro,n ($&) 0. model or ro1 pipes
0 0.2 0.4 0. 0. 1 1.2
-20
-10
0
10
20
0
40
M0
0
0 C(*" E9"#%"$*
*ime (s)
Pressre ead (m)
Valve Clos(re
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CASE STUDY• S%/!) C6)(6#+ "/
0 0.M 1 1.M 2
-20
-10
0
10
200
40
M0
0
0
C(*" )$ 5")+(#" P#"++(#" >") )* *6" V)/!"
C(*" P#"++(#" >") 5")+(#" P#"++(#" >")
T%" :+;
P#"++(#" >") :;
0 0.M0
0.2
0.4
0.
0.
*
A2is *itle
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CASE STUDY• S%/!) C6)(6#+ "/
0 0.01 0.02 0.0 0.04
Velocity Pro3les
0.M + 0. + 0.@ + 1.1
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CONCLUSIONS
• T3 %"$+%$)/ # '()+%-*3-%"$+%$)/ ($+*") #%7*%$
)#" 7(*)*%$)// *%"-%$*"$+%!" 8(* #(7" )$ )#9%
*6" !"/7%* #/"+ %$ *#)$+%"$* 3+, %$7/(%$& ) 3 #"!"#
*6" 3)//.
• I *6" "$"#& %++%)*%$ )++7%)*" 3%*6 *6"+" #/"+ %+ )77
#, *6"$ *6" "/ +6(/ #!%" & #"+(/*+ # +%(/)*%
/$&"# "#%+ *%"+.
• T(#8(/"$7" "/+ # *#)$+%"$* 3+ )#" ) )**"# #"+")#
63"!"#, *6" "!"/"$* "
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*4A5#