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Mathcad 11 CASA_06-27.mcd Appendix 1
γs 76972.857kg
m2
s2
= pcf 157.087464kg
m2
s2
=
ρs
γs
g:= ρs 7849.047053
kg
m3
=
A 8:= A 8=
m1
ρ0 a⋅
A:= m1 38.09371
kg
m2
=
ha
62.752:= h 0.004857 m=
m2 ρs h⋅:= m2 38.124515kg
m2
=m2 m1−
m1
0.000809=
m3
ρs π a h+( )2
⋅ π a2
⋅− ⋅
2 π⋅ a1
2h⋅+
⋅
:= m3 38.124515kg
m2
=m3 m1−
m1
0.000809=
R1 12.478:=E
K
ρ0
ρs 1 ν2
−( )⋅
⋅ 12.468436=
Skalak, R. (1956), Ref. 4"An extension of the theory of water hammer."Transactions of the ASME 78, 105-116.(PhD Thesis, Columbia University, New York, USA, 1954, Ref. 2)
Table 1
g 32.174049ft
s2
= g 9.80665m
s2
=
a 1 ft⋅:= a 0.3048m=
c 5000ft
s⋅:= c 1524
m
s=
ρ0 1.94slug
ft3
⋅:= ρ0 999.834908kg
m3
=
K ρ0 c2
⋅:= K 2.322193 109
× Pa=
E 30 106
⋅ psi⋅:= E 2.068427 1011
× Pa=
ν 0.3:= ν 0.3= pcflbf
ft3
:=
γs 490 pcf⋅:=
Mathcad 11 CASA_06-27.mcd Appendix 2
There is something wrong with Skalak's equations [74] and [76]: the -c02 term in
the last factor must be deleted in [74] and [76] misses a square root.
c2
c0
0.981061=c2
c3.464192=
c1
ce
0.999004=c1
c0.643647=
ce
c0.644288=
c0
c3.531067=
Check with Table 1
Eq. (76)c2 5279.429013m
s=
c2 c2 A⋅ R⋅ R+ R
21 ν
2−( )⋅+ 2 A⋅ R⋅ R+ R
21 ν
2−( )⋅+
2
4 R2
⋅ 1 ν2
−( )⋅ 2 A⋅ R+( )⋅−+
2 2 A⋅ R+( )⋅
0.5
⋅:=
Eq. (76)c1 980.917501m
s=
c1 c2 A⋅ R⋅ R+ R
21 ν
2−( )⋅+ 2 A⋅ R⋅ R+ R
21 ν
2−( )⋅+
2
4 R2
⋅ 1 ν2
−( )⋅ 2 A⋅ R+( )⋅−−
2 2 A⋅ R+( )⋅
0.5
⋅:=
Eq. (1)ce 981.895414m
s=ce
c
12 K⋅ a⋅
E h⋅+
:=
Korteweg - Joukowsky
A 7.993536=Aρ0 a⋅
m2
:=
R R2:=R2 12.468436=R2
c02
c2
:=
Nomenclaturec0 5381.346495m
s=c0
E h⋅
m2 1 ν2
−( )⋅
:=
Wave speeds
Mathcad 11 CASA_06-27.mcd Appendix 3
ct 5133.477378m
s=
ct
c0
0.953939=
With FSI:
γ2 c2F c2t+ ν2
ρf
ρt
⋅D
ee⋅ c2F⋅+:= γ2 2.883457 10
7×
m2
s2
=
λ211
2γ2 γ2
24 c2F⋅ c2t⋅−−
⋅:= λ21 9.622 10
5×
m2
s2
=
λ231
2γ2 γ2
24 c2F⋅ c2t⋅−+
⋅:= λ23 2.787 10
7×
m2
s2
=
λ1 λ21:= λ1 980.917501m
s= cF ct⋅ 5.17868431352064 10
6×
m2
s2
=
λ3 λ23:= λ3 5279.429013m
s= λ1 λ3⋅ 5.17868431352065 10
6×
m2
s2
=
λ3
ct
1.028431=Check with Table 1
λ1
c0.643647=
λ1
ce
0.999004=λ3
c3.464192=
λ3
c0
0.981061=
ρf ρ0:= ρf 999.834908kg
m3
= Tijsseling
D 2 a⋅:= D 0.6096 m=
ee h:= ee 0.004857m=
ρt ρs:= ρt 7849.047053kg
m3
=
Longitudinal wave speeds
Classical:
c2FK
ρf
1K D⋅
E ee⋅1 ν
2−( )⋅+
1−
⋅:= cF c2F:= cF 1008.806299m
s=
cF
ce
1.027407=
c2fK
ρf
1K D⋅
E ee⋅+
1−
⋅:= cf c2f:= cf 981.895414m
s=
cf
ce
1=
c2tE
ρt
:= ct c2t:=
Mathcad 11 CASA_06-27.mcd Appendix 4
d2 11.477473m
3
s=d1 2.925721
m3
s=
d2 c a2
⋅
A 4+( )c2
c
5c2
c
3
1 R+( )⋅−c2
c
R⋅+
16−c2
c
2
⋅ 2 A⋅ R+( )⋅ 8 R⋅ 2 A⋅ 1+( )⋅+ 8 R2
⋅ 1 ν2
−( )+
⋅:=Eq. (78)
d1 c a2
⋅
A 4+( )c1
c
5c1
c
3
1 R+( )⋅−c1
c
R⋅+
16−c1
c
2
⋅ 2 A⋅ R+( )⋅ 8 R⋅ 2 A⋅ 1+( )⋅+ 8 R2
⋅ 1 ν2
−( )+
⋅:=Eq. (78)
Nominal length of the wave front according to Tijsseling
d2old 11.477844m
3
s=d1old 2.925652
m3
s=
d2old c a2
⋅
A 4+( )c2
c
5c2
c
3
1 R+( )⋅−c2
c
R⋅+
c2
c
h2
12 a2
⋅
⋅ 1 2 R2
⋅ ν⋅+( )⋅−
16−c2
c
2
⋅ 2 A⋅ R+( )⋅ 8 R⋅ 2 A⋅ 1+( )⋅+ 8 R2
⋅ 1 ν2
−( )+
⋅:=Eq. (78)
d1old c a2
⋅
A 4+( )c1
c
5c1
c
3
1 R+( )⋅−c1
c
R⋅+
c1
c
h2
12 a2
⋅
⋅ 1 2 R2
⋅ ν⋅+( )⋅−
16−c1
c
2
⋅ 2 A⋅ R+( )⋅ 8 R⋅ 2 A⋅ 1+( )⋅+ 8 R2
⋅ 1 ν2
−( )+
⋅:=Eq. (78)
Nominal length of the wave front according to Skalak
λ3
c2
1=λ1
c1
1=
Check Skalak - Tijsseling
Mathcad 11 CASA_06-27.mcd Appendix 5
L2 5 s⋅( ) 15.669309 m=L2 5 s⋅( ) 51.408494 ft=
L2 1 s⋅( ) 9.163467 m=L2 1 s⋅( ) 30.06387 ft=
L1 5 s⋅( ) 9.935248 m=L1 5 s⋅( ) 32.595957 ft=
L1 1 s⋅( ) 5.810168 m=L1 1 s⋅( ) 19.062231 ft=
Check with Table 1
Eq. (64)L2 t( )3 π⋅
3d2 t⋅⋅
Γ1
3
sinπ
3
⋅
:=L1 t( )3 π⋅
3d1 t⋅⋅
Γ1
3
sinπ
3
⋅
:=
3 π⋅
Γ1
3
sinπ
3
⋅
4.062354=Γ1
3
2
3π
3
1
2
Γ2
3
⋅⋅→sinπ
3
1
23
1
2⋅→
c2
c
h2
12 a2
⋅
⋅ 1 2 R2
⋅ ν⋅+( )⋅
A 4+( )c2
c
5c2
c
3
1 R+( )⋅−c2
c
R⋅+
0.000032−=
c1
c
h2
12 a2
⋅
⋅ 1 2 R2
⋅ ν⋅+( )⋅
A 4+( )c1
c
5c1
c
3
1 R+( )⋅−c1
c
R⋅+
0.000024=
Error because of "wrong" (h/a)2 term in Eq. (78)
h
a
2
0.000254=R 12.468436=A 7.993536=
Mathcad 11 CASA_06-27.mcd Appendix 6
0 1 2 3 4 5 6 7 8 9 100
5
10
15
20
L1 t( )
L2 t( )
t
TOL 1012−
:=even function to be integrated
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51
0
1
2
sin η η3
+( )η
η
Mathcad 11 CASA_06-27.mcd Appendix 7
0
9.3
ηsin η β η
3⋅+( )
η
⌠⌡
d 0.840919869378856=β 10:=
0
20
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.11015525927976=β 1:=
0
43
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.42315940650369=β1
10:=
0
93
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.56376138708481=β1
100:=
0
201
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.57083144591109=β1
1000:=
0
430
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.57077652953821=β1
10000:=
0
907
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.57076569930658=β1
100000:=
0
1843
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.57082809089641=β1
1000000:=
π
21.570796=
Eq. (49)
0
6600
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.57093027532399=β 0:=
Determination of the maximum possible upper boundaries of integrals:
Mathcad 11 CASA_06-27.mcd Appendix 8
0
20
ηsin η β η
3⋅+( )
η
⌠⌡
d 0.432399540407618=β 1−:=
0
43
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.73656497765972=β1−
10:=
0
94
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.71751324086010=β1−
100:=
0
202
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.18218831662671=β1−
1000:=
0
441
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.56865597358518=β1−
10000:=
0
962
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.70193923364200=β1−
100000:=
0
2115
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.62570169385300=β1−
1000000:=
π
21.570796=
0
6600
ηsin η β η
3⋅+( )
η
⌠⌡
d 1.57093027532399=β 0:=
0
2.0
ηsin η β η
3⋅+( )
η
⌠⌡
d 0.599018563818040=β 1000:=
0
4.3
ηsin η β η
3⋅+( )
η
⌠⌡
d 0.681123567409668=β 100:=
Mathcad 11 CASA_06-27.mcd Appendix 9
β 10−:=
0
9.3
ηsin η β η
3⋅+( )
η
⌠⌡
d 0.123095673983503−=
β 100−:=
0
4.3
ηsin η β η
3⋅+( )
η
⌠⌡
d 0.347949976012880−=
β 1000−:=
0
2.0
ηsin η β η
3⋅+( )
η
⌠⌡
d 0.444274408294637−=
Maximum possible upper boundaries b of integrals as function of β:
b β( ) 1843 β 0.000001≤if
907 0.000001 β< 0.00001≤if
430 0.00001 β< 0.0001≤if
201 0.0001 β< 0.001≤if
93 0.001 β< 0.01≤if
43 0.01 β< 0.1≤if
20 0.1 β< 1≤if
9.3 1 β< 10≤if
4.3 10 β< 100≤if
2.0 100 β< 1000≤if
0 otherwise
:=
Mathcad 11 CASA_06-27.mcd Appendix 10
Define range of n values for β:
n 1000:=
i 0 n..:= j i( )10
ni⋅ 7−:= β i( ) 10
j i( ):= β i( )
110 -7
1.02310 -7
1.04710 -7
1.07210 -7
1.09610 -7
1.12210 -7
1.14810 -7
1.17510 -7
1.20210 -7
1.2310 -7
1.25910 -7
1.28810 -7
1.31810 -7
1.34910 -7
1.3810 -7
1.41310 -7
= b β i( )( )1843
1843
1843
1843
1843
1843
1843
1843
1843
1843
1843
1843
1843
1843
1843
1843
=
i
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
= j i( )
-7
-6.99
-6.98
-6.97
-6.96
-6.95
-6.94
-6.93
-6.92
-6.91
-6.9
-6.89
-6.88
-6.87
-6.86
-6.85
=
Maximum possible upper boundaries b of integrals as function of β:
1 .107
1 .106
1 .105
1 .104
1 .103
0.01 0.1 1 10 100 1 .103
0
500
1000
1500
2000
b β i( )( )
β i( )
Mathcad 11 CASA_06-27.mcd Appendix 11
Eq. (49) for negative (f=wn) and positive (g=wp) values of β:
f(x) is the integral from 0 to minus b(x) OR f(x) is minus the integral from 0 to b(x)
f x( )
0
b x( )
ηsin η x η
3⋅−( )
η
⌠⌡
d−:= g x( )
0
b x( )
ηsin η x η
3⋅+( )
η
⌠⌡
d:=
wni f β i( )( ):= wpi g β i( )( ):=
wp
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1.571
1.571
1.571
1.571
1.571
1.571
1.571
1.571
1.571
1.571
1.571
1.571
1.571
1.571
1.571
1.571
=wn
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-1.550
-1.591
-1.616
-1.570
-1.544
-1.525
-1.537
-1.547
-1.556
-1.564
-1.565
-1.559
-1.547
-1.534
-1.526
-1.536
=
ββi β i( ):= zni1−
3β i( )
:= zpi1
3β i( )
:=
ββ
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
110 -7
1.02310 -7
1.04710 -7
1.07210 -7
1.09610 -7
1.12210 -7
1.14810 -7
1.17510 -7
1.20210 -7
1.2310 -7
1.25910 -7
1.28810 -7
1.31810 -7
1.34910 -7
1.3810 -7
1.41310 -7
= zn
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-215.443
-213.796
-212.162
-210.539
-208.93
-207.332
-205.747
-204.174
-202.613
-201.064
-199.526
-198.001
-196.487
-194.984
-193.494
-192.014
= zp
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
215.443
213.796
212.162
210.539
208.93
207.332
205.747
204.174
202.613
201.064
199.526
198.001
196.487
194.984
193.494
192.014
=
Mathcad 11 CASA_06-27.mcd Appendix 12
Skalak, R. (1956), Ref. 4"An extension of the theory of water hammer."Transactions of the ASME 78, 105-116.
Figure 4, not scaled
30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 2 4 62
1
0
1
2
3
wni−
wpi−
π
2
π
2−
π
6−
zni zpi, zni, zpi, zpi,
Figure 4, scaled and vertically shifted
30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 2 4 60.4
0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1
π− wni⋅
1
2+
1
π− wpi⋅
1
2+
1
0
1
3
zni zpi, zni, zpi, zpi,
Mathcad 11 CASA_06-27.mcd Appendix 13
1000 800 600 400 200 0 200 400 600 800 10003
2
1
0
1
2
wni
wpi
π
2−
π
2
π
6
π
6
ββ i− ββ i, 0, 0, ββ i, ββ i−,
1 .107
1 .106
1 .105
1 .104
1 .103
0.01 0.1 1 10 100 1 .103
3
2
1
0
1
2
wni
wpi
π
2−
π
2
π
6
ββ i
Mathcad 11 CASA_06-27.mcd Appendix 14
Eq. (1-8), thesisw0 9.247031 106−
× m=w0
p0 a2
⋅
E h⋅:=
Eq. (3-24), thesiswI0 1.268253− 106−
× m=wI0
p0
6 m2⋅c1
c1−
⋅ D1⋅
:=
Eq. (3-22), thesiswIpinf 3.804758− 106−
× m=wIpinf
p0
2 m2⋅c1
c1−
⋅ D1⋅
:=
Eq. (3-22), thesiswIninf 3.804758 106−
× m=wIninf
p0−
2 m2⋅c1
c1−
⋅ D1⋅
:=
displacements
Eq. (56)CP 1.077752 1010
×Pa
m=CP
2 ρ0⋅ c12
⋅
ac1
2
c2
1−
⋅
−:=
Eqs. (47), (49)C 2.422184− 106−
× m=Cp0
π m2⋅c1
c1−
⋅ D1⋅
:=
p0 100000 Pa⋅:=h
2
12 a2
⋅
0.000021=D1 9.672924 108
×1
s2
=
Eq. (45),Eq. (3-14), thesis
D1
4 ρ0⋅ c14
⋅
m2 c2
⋅ a⋅c1
2
c2
1−
2
⋅
2 c02
⋅
a2
ν2
1c0
2
c12
−
21+ ν
2−
h2
12 a2
⋅
+
⋅+:=
Constants for "positive" waterhammer wave
Mathcad 11 CASA_06-27.mcd Appendix 15
Scaling
cPn CP C⋅ wn⋅1
2p0⋅+:= cPp CP C⋅ wp⋅
1
2p0⋅+:= Eqs. (47), (49), (56)
PIninf CP wIninf⋅1
2p0⋅+:= PIninf 91005.873668Pa= NOT equal to p0
because .....
PIpinf CP wIpinf⋅1
2p0⋅+:= PIpinf 8994.126332Pa= NOT equal to 0
because .....
PI0 CP wI0⋅1
2p0⋅+:= PI0 36331.375444Pa=
Dimensional pressure for p0 100000 Pa=
30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 2 4 60
1 .104
2 .104
3 .104
4 .104
5 .104
6 .104
7 .104
8 .104
9 .104
1 .105
1.1 .105
1.2 .105
cPni
cPpi
PIninf
PIpinf
PI0
zni zpi, zni, zpi, zpi,
Mathcad 11 CASA_06-27.mcd Appendix 16
Dimensionless pressure for a
h62.752=
30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 2 4 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
cPni
p0
cPpi
p0
PIninf
p0
PIpinf
p0
PI0
p0
zni zpi, zni, zpi, zpi,
PIninf
p0
0.910059=PIpinf
p0
0.089941=
0.089941 0.0046− 0.085341= off set add precursor, see Figure 5.
PIninf
p0
0.085341− 0.824718=PIpinf
p0
0.085341− 0.0046=
contributions of constant and other three solutions
PngIpinf Png2Ipinf+ Pg2Ininf+
p0
0+ 0.085332−= off set
osw
PngIpinf Png2Ipinf+ Pg2Ininf+
p0
0+:= osw 0.085332−=
Mathcad 11 CASA_06-27.mcd Appendix 17
t 1 s⋅:=
zzni
c1 t⋅ zni3
d1 t⋅⋅+
c1 t⋅:= zzpi
c1 t⋅ zpi3
d1 t⋅⋅+
c1 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
1t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
cPni
p0
cPpi
p0
PIninf
p0
PIpinf
p0
PI0
p0
zzni zzpi, zzni, zzpi, zzpi,
L1 t( ) 5.810168m= 0.005 c1⋅ t⋅ 4.904588 m=
Mathcad 11 CASA_06-27.mcd Appendix 18
t 5 s⋅:=
zzni
c1 t⋅ zni3
d1 t⋅⋅+
c1 t⋅:= zzpi
c1 t⋅ zpi3
d1 t⋅⋅+
c1 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
1t)
0.97 0.975 0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 1.020
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
cPni
p0
cPpi
p0
PIninf
p0
PIpinf
p0
PI0
p0
zzni zzpi, zzni, zzpi, zzpi,
L1 t( ) 9.935248m= 0.002 c1⋅ t⋅ 9.809175 m=
Mathcad 11 CASA_06-27.mcd Appendix 19
t 10 s⋅:=
zzni
c1 t⋅ zni3
d1 t⋅⋅+
c1 t⋅:= zzpi
c1 t⋅ zpi3
d1 t⋅⋅+
c1 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
1t)
0.99 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 1.010
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
cPni
p0
cPpi
p0
PIninf
p0
PIpinf
p0
PI0
p0
zzni zzpi, zzni, zzpi, zzpi,
L1 t( ) 12.517628m= 0.001 c1⋅ t⋅ 9.809175 m=
Mathcad 11 CASA_06-27.mcd Appendix 20
Eqs. (47), (49), (56)cPp2 CP2 C2⋅ wp⋅:=cPn2 CP2 C2⋅ wn⋅:=
Scaling
Eq. (3-24), thesis,Eq. (56)
P2I0 76.712762− Pa=P2I0
p0
6 m2⋅c2
c1−
⋅ D2⋅
CP2⋅:=
NOT equal to 0
because .....
Eq. (3-22), thesis,Eq. (56)
P2Ipinf 230.138286− Pa=P2Ipinf
p0
2 m2⋅c2
c1−
⋅ D2⋅
CP2⋅:=
Eq. (3-22), thesis,Eq. (56)
P2Ininf 230.138286Pa=P2Ininf
p0−
2 m2⋅c2
c1−
⋅ D2⋅
CP2⋅:=
Eq. (56)CP2 1.662263− 1010
×Pa
m=CP2
2 ρ0⋅ c22
⋅
ac2
2
c2
1−
⋅
−:=
Eqs. (47), (49)C2 8.813923 109−
× m=C2
p0
π m2⋅c2
c1−
⋅ D2⋅
:=
p0 100000 Pa=h
2
12 a2
⋅
0.000021=D2 3.844165 1010
×1
s2
=
Eq. (45),Eq. (3-14), thesis
D2
4 ρ0⋅ c24
⋅
m2 c2
⋅ a⋅c2
2
c2
1−
2
⋅
2 c02
⋅
a2
ν2
1c0
2
c22
−
21+ ν
2−
h2
12 a2
⋅
+
⋅+:=
Constants for "positive" precursor wave
Mathcad 11 CASA_06-27.mcd Appendix 21
Dimensional pressure for p0 100000 Pa=
30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 2 4 6300
200
100
0
100
200
300
400
cPn2i
cPp2i
P2Ininf
P2Ipinf
P2I0
zni zpi, zni, zpi, zpi,
Mathcad 11 CASA_06-27.mcd Appendix 22
Dimensionless pressure for a
h62.752=
30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 2 4 60.003
0.002
0.001
0
0.001
0.002
0.003
0.004
cPn2i
p0
cPp2i
p0
P2Ininf
p0
P2Ipinf
p0
P2I0
p0
zni zpi, zni, zpi, zpi,
P2Ininf
p0
0.002301=P2Ipinf
p0
0.002301−= off set
P2Ininf
p0
0.002301+ 0.004602=P2Ipinf
p0
0.002301+ 3.828584− 107−
×=
contributions of constant and other three solutions
PgIpinf PngIpinf+ Png2Ipinf+
p0
1
2+ 0.002308= off set
osp
PgIpinf PngIpinf+ Png2Ipinf+
p0
1
2+:= osp 0.002308=
Mathcad 11 CASA_06-27.mcd Appendix 23
t 1 s⋅:=
zzn2i
c2 t⋅ zni3
d2 t⋅⋅+
c2 t⋅:= zzp2i
c2 t⋅ zpi3
d2 t⋅⋅+
c2 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
2t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.003
0.002
0.001
0
0.001
0.002
0.003
0.004
cPn2i
p0
cPp2i
p0
P2Ininf
p0
P2Ipinf
p0
P2I0
p0
zzn2i zzp2i, zzn2i, zzp2i, zzp2i,
L2 t( ) 9.163467m= 0.002 c2⋅ t⋅ 10.558858 m=
Mathcad 11 CASA_06-27.mcd Appendix 24
t 5 s⋅:=
zzn2i
c2 t⋅ zni3
d2 t⋅⋅+
c2 t⋅:= zzp2i
c2 t⋅ zpi3
d2 t⋅⋅+
c2 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure displacement for a
h62.752= as function of z \ (c
2t)
0.995 0.996 0.997 0.998 0.999 1 1.001 1.002 1.003 1.004 1.0050.003
0.002
0.001
0
0.001
0.002
0.003
0.004
cPn2i
p0
cPp2i
p0
P2Ininf
p0
P2Ipinf
p0
P2I0
p0
zzn2i zzp2i, zzn2i, zzp2i, zzp2i,
L2 t( ) 15.669309m= 0.0005 c2⋅ t⋅ 13.198573 m=
Mathcad 11 CASA_06-27.mcd Appendix 25
t 10 s⋅:=
zzn2i
c2 t⋅ zni3
d2 t⋅⋅+
c2 t⋅:= zzp2i
c2 t⋅ zpi3
d2 t⋅⋅+
c2 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
2t)
0.997 0.9975 0.998 0.9985 0.999 0.9995 1 1.0005 1.001 1.0015 1.0020.003
0.002
0.001
0
0.001
0.002
0.003
0.004
cPn2i
p0
cPp2i
p0
P2Ininf
p0
P2Ipinf
p0
P2I0
p0
zzn2i zzp2i, zzn2i, zzp2i, zzp2i,
L2 t( ) 19.742092m= 0.0004 c2⋅ t⋅ 21.117716 m=
Mathcad 11 CASA_06-27.mcd Appendix 26
Eq. (1-8), thesisw0 9.247031 106−
× m=w0
p0 a2
⋅
E h⋅:=
Eq. (3-24), thesiswnI0 2.749655− 107−
× m=wnI0
p0−
6 m2⋅c1
c1+
⋅ D1⋅
:=
Eq. (3-22), thesiswnIpinf 8.248965 107−
× m=wnIpinf
p0
2 m2⋅c1
c1+
⋅ D1⋅
:=
Eq. (3-22), thesiswnIninf 8.248965− 107−
× m=wnIninf
p0−
2 m2⋅c1
c1+
⋅ D1⋅
:=
displacements
Eqs. (47), (49)Cn 5.251454 107−
× m=Cnp0
π m2⋅c1
c1+
⋅ D1⋅
:=
p0 100000 Pa=h
2
12 a2
⋅
0.000021=D1 9.672924 108
×1
s2
=
Eq. (45),Eq. (3-14), thesis
D1
4 ρ0⋅ c14
⋅
m2 c2
⋅ a⋅c1
2
c2
1−
2
⋅
2 c02
⋅
a2
ν2
1c0
2
c12
−
21+ ν
2−
h2
12 a2
⋅
+
⋅+:=
Constants for "negative" waterhammer wave
Mathcad 11 CASA_06-27.mcd Appendix 27
6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 303.5 .10
4
4 .104
4.5 .104
5 .104
5.5 .104
6 .104
cPnni
cPnpi
PnIninf
PnIpinf
PnI0
zni zpi, zni, zpi, zni,
Dimensional pressure for p0 100000 Pa=
Eq. (56)PnI0 52963.447212Pa=PnI0 CP wnI0−⋅1
2p0⋅+:=
Eq. (56)PnIpinf 41109.658365Pa=PnIpinf CP wnIpinf−⋅1
2p0⋅+:=
NOT equal to 0
because .....PnIninf 58890.341635Pa=PnIninf CP wnIninf−⋅
1
2p0⋅+:=
Eqs. (47), (49), (56)cPnp CP Cn−⋅ wnp⋅1
2p0⋅+:=cPnn CP Cn−⋅ wnn⋅
1
2p0⋅+:=
wnp wn−:=wnn wp−:=
Scaling
Mathcad 11 CASA_06-27.mcd Appendix 28
Dimensionless pressure for a
h62.752=
6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300.35
0.4
0.45
0.5
0.55
0.6
cPnni
p0
cPnpi
p0
PnIninf
p0
PnIpinf
p0
PnI0
p0
zni zpi, zni, zpi, zni,
PnIninf
p0
0.588903=PnIpinf
p0
0.411097=
0.588903− 1.0025+ 0.413597= off set add precursor, see Figure 5.
PnIninf
p0
0.413597+ 1.0025=PnIpinf
p0
0.413597+ 0.824694=
contributions of constant and other three solutions
PgIninf Pg2Ininf+ Png2Ipinf+
p0
0+ 0.41363= off set
oswn
PgIninf Pg2Ininf+ Png2Ipinf+
p0
0+:= oswn 0.41363=
Mathcad 11 CASA_06-27.mcd Appendix 29
t 1 s⋅:=
zznni
c1− t⋅ zni3
d1 t⋅⋅+
c1 t⋅:= zznpi
c1− t⋅ zpi3
d1 t⋅⋅+
c1 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
1t)
1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92 0.90.35
0.4
0.45
0.5
0.55
0.6
cPnni
p0
cPnpi
p0
PnIninf
p0
PnIpinf
p0
PnI0
p0
zznni zznpi, zznni, zznpi, zznni,
L1 t( ) 5.810168m= 0.005 c1⋅ t⋅ 4.904588 m=
Mathcad 11 CASA_06-27.mcd Appendix 30
t 5 s⋅:=
zznni
c1− t⋅ zni3
d1 t⋅⋅+
c1 t⋅:= zznpi
c1− t⋅ zpi3
d1 t⋅⋅+
c1 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
1t)
1.02 1.015 1.01 1.005 1 0.995 0.99 0.985 0.98 0.975 0.970.35
0.4
0.45
0.5
0.55
0.6
cPnni
p0
cPnpi
p0
PnIninf
p0
PnIpinf
p0
PnI0
p0
zznni zznpi, zznni, zznpi, zznni,
L1 t( ) 9.935248m= 0.002 c1⋅ t⋅ 9.809175 m=
Mathcad 11 CASA_06-27.mcd Appendix 31
t 10 s⋅:=
zznni
c1− t⋅ zni3
d1 t⋅⋅+
c1 t⋅:= zznpi
c1− t⋅ zpi3
d1 t⋅⋅+
c1 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
1t)
1.01 1.008 1.006 1.004 1.002 1 0.998 0.996 0.994 0.992 0.990.35
0.4
0.45
0.5
0.55
0.6
cPnni
p0
cPnpi
p0
PnIninf
p0
PnIpinf
p0
PnI0
p0
zznni zznpi, zznni, zznpi, zznni,
L1 t( ) 12.517628m= 0.001 c1⋅ t⋅ 9.809175 m=
Mathcad 11 CASA_06-27.mcd Appendix 32
Pn2I0 42.344725Pa= Eq. (3-24), thesis
Scaling
wnn wp−:= wnp wn−:=
cPnn2 CP2 Cn2−⋅ wnn⋅:= cPnp2 CP2 Cn2−⋅ wnp⋅:= Eqs. (47), (49), (56)
Pn2Ininf Pn2Ininf−:= Pn2Ininf 127.034176− Pa= NOT equal to 0
because .....
Pn2Ipinf Pn2Ipinf−:= Pn2Ipinf 127.034176Pa= Eq. (56)
Pn2I0 Pn2I0−:= Pn2I0 42.344725− Pa= Eq. (56)
Constants for "negative" precursor wave
D2
4 ρ0⋅ c24
⋅
m2 c2
⋅ a⋅c2
2
c2
1−
2
⋅
2 c02
⋅
a2
ν2
1c0
2
c22
−
21+ ν
2−
h2
12 a2
⋅
+
⋅+:= Eq. (45),Eq. (3-14), thesis
D2 3.844165 1010
×1
s2
=h
2
12 a2
⋅
0.000021= p0 100000 Pa=
Cn2
p0
π m2⋅c2
c1+
⋅ D2⋅
:= Cn2 4.865203 109−
× m= Eqs. (47), (49)
Pn2Ininf
p0−
2 m2⋅c2
c1+
⋅ D2⋅
CP2⋅:= Pn2Ininf 127.034176Pa= Eq. (3-22), thesis
Pn2Ipinf
p0
2 m2⋅c2
c1+
⋅ D2⋅
CP2⋅:= Pn2Ipinf 127.034176− Pa= Eq. (3-22), thesis
NOT equal to 0
because .....
Pn2I0
p0−
6 m2⋅c2
c1+
⋅ D2⋅
CP2⋅:=
Mathcad 11 CASA_06-27.mcd Appendix 33
Dimensional pressure for p0 100000 Pa=
6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30150
100
50
0
50
100
150
200
250
cPnn2i
cPnp2i
Pn2Ininf
Pn2Ipinf
Pn2I0
zni zpi, zni, zpi, zni,
Mathcad 11 CASA_06-27.mcd Appendix 34
Dimensionless pressure for a
h62.752=
6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300.0015
0.001
5 .104
0
5 .104
0.001
0.0015
0.002
0.0025
cPnn2i
p0
cPnp2i
p0
Pn2Ininf
p0
Pn2Ipinf
p0
Pn2I0
p0
zni zpi, zni, zpi, zni,
Pn2Ininf
p0
0.00127034−=Pn2Ipinf
p0
0.00127034=
0.00127034 1+ 1.00127= off set see Figure 5
Pn2Ininf
p0
1.00127+ 1=Pn2Ipinf
p0
1.00127+ 1.00254=
contributions of constant and other three solutions
PgIninf PngIninf+ Pg2Ininf+
p0
1
2+ 1.001264= off set
ospn
PgIninf PngIninf+ Pg2Ininf+
p0
1
2+:= ospn 1.001264=
Mathcad 11 CASA_06-27.mcd Appendix 35
t 1 s⋅:=
zznn2i
c2− t⋅ zni3
d2 t⋅⋅+
c2 t⋅:= zznp2i
c2− t⋅ zpi3
d2 t⋅⋅+
c2 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
2t)
1.02 1.015 1.01 1.005 1 0.995 0.99 0.985 0.98 0.975 0.970.0015
0.001
5 .104
0
5 .104
0.001
0.0015
0.002
0.0025
cPnn2i
p0
cPnp2i
p0
Pn2Ininf
p0
Pn2Ipinf
p0
Pn2I0
p0
zznn2i zznp2i, zznn2i, zznp2i, zznn2i,
L2 t( ) 9.163467m= 0.002 c2⋅ t⋅ 10.558858 m=
Mathcad 11 CASA_06-27.mcd Appendix 36
t 5 s⋅:=
zznn2i
c2− t⋅ zni3
d2 t⋅⋅+
c2 t⋅:= zznp2i
c2− t⋅ zpi3
d2 t⋅⋅+
c2 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
2t)
1.005 1.004 1.003 1.002 1.001 1 0.999 0.998 0.997 0.996 0.9950.0015
0.001
5 .104
0
5 .104
0.001
0.0015
0.002
0.0025
cPnn2i
p0
cPnp2i
p0
Pn2Ininf
p0
Pn2Ipinf
p0
Pn2I0
p0
zznn2i zznp2i, zznn2i, zznp2i, zznn2i,
L2 t( ) 15.669309m= 0.0005 c2⋅ t⋅ 13.198573 m=
Mathcad 11 CASA_06-27.mcd Appendix 37
t 10 s⋅:=
zznn2i
c2− t⋅ zni3
d2 t⋅⋅+
c2 t⋅:= zznp2i
c2− t⋅ zpi3
d2 t⋅⋅+
c2 t⋅:= Eqs. (40), (48), (51)
Dimensionless pressure for a
h62.752= as function of z \ (c
2t)
1.002 1.0015 1.001 1.0005 1 0.9995 0.999 0.9985 0.998 0.9975 0.9970.0015
0.001
5 .104
0
5 .104
0.001
0.0015
0.002
0.0025
cPnn2i
p0
cPnp2i
p0
Pn2Ininf
p0
Pn2Ipinf
p0
Pn2I0
p0
zznn2i zznp2i, zznn2i, zznp2i, zznn2i,
L2 t( ) 19.742092m= 0.0004 c2⋅ t⋅ 21.117716 m=
Mathcad 11 CASA_06-27.mcd Appendix 38
H2(∞)Pn2Ipinf 127.034176Pa=
Png2Ipinf 127.034176Pa≡Png2Ininf 127.034176− Pa≡
H1(∞)PnIpinf1
2p0⋅− 8890.341635− Pa=
PngIpinf 8890.341635− Pa≡PngIninf 8890.341635Pa≡
I2(∞)P2Ipinf 230.138286− Pa=
Pg2Ipinf 230.138286− Pa≡Pg2Ininf 230.138286Pa≡
I1(∞)PIpinf1
2p0⋅− 41005.873668− Pa=
PgIpinf 41005.873668− Pa≡PgIninf 41005.873668Pa≡
Global definition of values at infinity
p0 CP⋅
2 m2⋅c1
c1−
⋅ D1⋅
p0 CP⋅
2 m2⋅c1
c1+
⋅ D1⋅
−p0 CP2⋅
2 m2⋅c2
c1−
⋅ D2⋅
+p0 CP2⋅
2 m2⋅c2
c1+
⋅ D2⋅
−
p0
0.499993−=
PIninf PnIninf+ P2Ininf+ Pn2Ininf+
p0
1.499993=Solution for z = negative infinity,see Eq. (56)
PIpinf PnIpinf+ P2Ipinf+ Pn2Ipinf+
p0
0.500007=Solution for z = positive infinity,see Eq. (56)
Mathcad 11 CASA_06-27.mcd Appendix 39
PngIpinf Png2Ipinf+ Pg2Ininf+
p0
0+ 0.085332−= p. 16
PgIpinf PngIpinf+ Png2Ipinf+
p0
1
2+ 0.002308= p. 22
PgIninf Pg2Ininf+ Png2Ipinf+
p0
0+ 0.41363= p. 28
PgIninf PngIninf+ Pg2Ininf+
p0
1
2+ 1.001264= p. 34
Solution for z = positive infinity,see Eq. (56)
PgIpinf PngIpinf+ Pg2Ipinf+ Png2Ipinf+
p0
0.499993−=
Solution for z = negative infinity,see Eq. (56)
PgIninf PngIninf+ Pg2Ininf+ Png2Ininf+
p0
0.499993=
Mathcad 11 CASA_06-27.mcd Appendix 40
Dimensional pressure for p0 100000 Pa=
zzJ0 PIninf:= zzJ1 PIpinf:= Joukowsky
30 26 22 18 14 10 6 2 2 60
2 .104
4 .104
6 .104
8 .104
1 .105
1.2 .105
scaled axial distance (-)
pre
ssu
re (
Pa)
Factors for conversion from radial wall displacement to pressure
E h⋅
a2
1.081428 1010
×Pa
m=
E h⋅
a2
w0
zzJ0
⋅ 1.09883=
CP 1.077752 1010
×Pa
m= CP
w0
zzJ0
⋅ 1.095095=
Mathcad 11 CASA_06-27.mcd Appendix 41
zzznp2i c2− t⋅ zpi3
d2 t⋅⋅+:= Eqs. (40), (48), (51)
off sets
osw osw p0⋅:= osp osp p0⋅:=
PIpinf osw+ 460.957159 Pa= P2Ininf osp+ 460.957159 Pa=
oswn oswn p0⋅:= ospn ospn p0⋅:=
PnIninf oswn+ 100253.387765 Pa= Pn2Ipinf ospn+ 100253.387765 Pa=
axial coordinates of end points of connecting horizontal lines
z0w0
700
m⋅:= zwp1200
3000
m⋅:= zpw3000
4800
m⋅:= zinfp5800
6000
m⋅:=
Positive and negative waterhammer and precursor wavesin one figure
t 1 s⋅:= c1 t⋅ 980.917501 m= zn1 213.796209−= zp1 213.796209=
3d1 t⋅ 1.430247 m= znn 0.1−= zpn 0.1=
zzzni c1 t⋅ zni3
d1 t⋅⋅+:= zzzpi c1 t⋅ zpi3
d1 t⋅⋅+:= Eqs. (40), (48), (51)
zzzn2i c2 t⋅ zni3
d2 t⋅⋅+:= zzzp2i c2 t⋅ zpi3
d2 t⋅⋅+:= Eqs. (40), (48), (51)
zzznni c1− t⋅ zni3
d1 t⋅⋅+:= zzznpi c1− t⋅ zpi3
d1 t⋅⋅+:= Eqs. (40), (48), (51)
zzznn2i c2− t⋅ zni3
d2 t⋅⋅+:=
Mathcad 11 CASA_06-27.mcd Appendix 42
Dimensional pressure for a
h62.752= and p0 100000Pa=
as function of z, WATERHAMMER + precursor, upstream
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 60002 .10
4
0
2 .104
4 .104
6 .104
8 .104
1 .105
1.2 .105
cPni osw+( )
cPpi osw+( )
PIninf osw+( )
PIpinf osw+( )
cPn2i osp+( )
cPp2i osp+( )
P2Ininf osp+( )
P2Ipinf osp+( )
zzzni zzzpi, z0wi, zwpi, zzzn2i, zzzp2i, zpwi, zinfpi,
L1 t( ) 5.810168m= 0.005 c1⋅ t⋅ 4.904588 m=
Mathcad 11 CASA_06-27.mcd Appendix 43
Dimensional pressure for a
h62.752= and p0 100000Pa=
as function of z, PRECURSOR, upstream
3000 3500 4000 4500 5000 5500 6000100
0
100
200
300
400
500
600
cPn2i osp+( )
cPp2i osp+( )
P2Ininf osp+( )
P2Ipinf osp+( )
zzzn2i zzzp2i, zpwi, zinfpi,
L2 t( ) 9.163467m= 0.002 c2⋅ t⋅ 10.558858 m=
Mathcad 11 CASA_06-27.mcd Appendix 44
Dimensionless pressure for a
h62.752= and p0 100000Pa=
as function of z, WATERHAMMER + precursor, downstream
6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 02 .10
4
0
2 .104
4 .104
6 .104
8 .104
1 .105
1.2 .105
cPnni oswn+( )
cPnpi oswn+( )
PnIpinf oswn+( )
PnIninf oswn+( )
cPnn2i ospn+( )
cPnp2i ospn+( )
Pn2Ipinf ospn+( )
Pn2Ininf ospn+( )
zzznni zzznpi, z0w−( )i
, zwp−( )i
, zzznn2i, zzznp2i, zpw−( )i
, zinfp−( )i
,
Mathcad 11 CASA_06-27.mcd Appendix 45
Dimensionless pressure for a
h62.752= and p0 100000Pa=
as function of z, precursor, downstream
6000 5750 5500 5250 5000 4750 4500 4250 4000 3750 3500 3250 30009.99 .10
4
1 .105
1.001 .105
1.002 .105
1.003 .105
1.004 .105
cPnni oswn+( )
cPnpi oswn+( )
PnIpinf oswn+( )
PnIninf oswn+( )
cPnn2i ospn+( )
cPnp2i ospn+( )
Pn2Ipinf ospn+( )
Pn2Ininf ospn+( )
zzznni zzznpi, z0w−( )i
, zwp−( )i
, zzznn2i, zzznp2i, zpw−( )i
, zinfp−( )i
,
Mathcad 11 CASA_06-27.mcd Appendix 46
fstatic 5215.453765Hz=fstatic
c1
λstatic
:=
λstatic 188.079033 mm=λstatic 2 π⋅
4
a2
h2
⋅
3 1 ν2
−( )⋅
⋅:=
h 1.5 mm⋅:=
a 40 mm⋅:=
Asselman (1969), eq (2.2.6), Fig. 2.10, static solution,MSc Thesis, Dept. of Applied Physics, TU Eindhoven
1f3ring
f0ring
− 0.422494=
f3ring 1548.007982Hz=f3ring1
2 π⋅
4 E⋅
D2
ρs⋅ 1D
8 ee⋅
ρf
ρs
⋅+
⋅
⋅:=
Barez et al (1979), part II, eq (2), Ref. 16
1f2ring
f0ring
− 0.477614=
f2ring 1400.260208Hz=f2ring1
2 π⋅
4 E⋅
D2
ρs⋅ 1D
6 ee⋅
ρf
ρs
⋅+
⋅
⋅:=
Kellner and Schoenfelder (1982), Ref. 17
1f1ring
f0ring
− 0.552642=
f1ring 1199.146458Hz=f1ring1
2 π⋅
4 E⋅
D2
ρs⋅ 1D
4 ee⋅
ρf
ρs
⋅+
⋅
⋅:=
Walker and Phillips (1977), Ref. 18
f0ring 2680.506233Hz=f0ring1
π D⋅
E
ρs
⋅:=
Ring frequencies
Mathcad 11 CASA_06-27.mcd Appendix 47
flobar 6( ) 279.339293Hz=flobar 3( ) 50.508256Hz=
flobar 2( ) 16.070848Hz=flobar 5( ) 181.179919Hz=
flobar 1( ) 0Hz=flobar 4( ) 105.158248Hz=
Lobar modes
c0
ee1.108 10
6× Hz=flobar n( ) f0ring Ω2 n( )⋅:=
(2-32)Ω2 n( ) β2n
2n
21−( )2
⋅
1 n2
+ 2 n⋅ µ⋅+
⋅:=
c0
D8827.668135Hz=
f00ring
f0ring
1.048285=f00ring
c0
πD:=
Lobar frequencies
aee
2+
R0.024641m=β2 2.082909 10
5−×=β2
ee2
12 aee
2+
2
⋅
:=
1
4
ρf
ρt
⋅D
ee⋅ 3.996768=µ 3.965174=µ
ρf Af⋅
ρt At⋅:=
Constants
It 44253.676749 cm4
=At 93.762442 cm2
=Af 0.29186351 m2
=
Itπ
4a ee+( )
4a
4− ⋅:=At π a ee+( )
2a2
− ⋅:=Af π a2
⋅:=
Cross-sectional areas
Input data
Lobar frequenciesDe Jong (1994) Section 2.4PhD Thesis, Dept. of Mechanical Engineering, TU Eindhoven
Mathcad 11 CASA_06-27.mcd Appendix 48
Ovalizing frequency
flobar 2( ) 16.070847864527Hz=
foval2 ee⋅
Dρf
ρt
D
ee⋅ 5+⋅
f0ring⋅:= PS9, Time Scales, Eq. (14), wrong
foval 9.324237 Hz=
fcoval2 3⋅ ee⋅
Dρf
ρt
D
ee⋅ 5+⋅
f0ring⋅:= PS9, Time Scales, Eq. (14), corrected
fcoval 16.150053 Hz=
fcoval2 3⋅ ee⋅
D 4 µ⋅ 5+⋅f0ring⋅:= PS9, Time Scales, Eq. (14), corrected, more accurate
fcoval 16.1988983478942 Hz= fcovala
aee
2+
⋅ 16.070847864527Hz=
Mathcad 11 CASA_06-27.mcd Appendix 49
Dimensional pressure for a
h62.752= and p0 100000Pa=
as function of z, WATERHAMMER + precursor, upstream
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 60002 .10
4
0
2 .104
4 .104
6 .104
8 .104
1 .105
1.2 .105
connects to figure on (Ap)p. 44