gwf Gas/Erdgas Gasnetze sind fit für die Energiewände (Vorschau)

Rohrnetz
Fachberichte
Im Rahmen der Durchführung einer konventionellen
Druckverlustberechnung ist der Leitungsenddruck
gemäß Gl. (1) zu ermitteln:
L 16 Tm
pn
p p
D T p V 2
K
2
=
1⋅ 1−λ⋅ ⋅ ⋅
5 2 n
⋅ ⋅ ⋅
2 n
⋅
m
π
ρ & (1)
n 1
L 16 Tm
pn
Will p man p den Druckverlauf D T entlang p V 2
des K
2
=
1⋅ 1−λ⋅ ⋅ ⋅
n
⋅ ⋅ ⋅
n
⋅
5 2
2 m
π
ρ &
Strömungsweges
x 16
( xauftragen, ) = p1 ⋅ 1−λgilt ⋅ ⋅
n 1
5
D π
ρ Tm
pn
2
L Gl. 16 (2): ⋅ T⋅ ⋅ ⋅ ⋅
2 n 2 n m
Tn
p V & K
m
pn
p p
1
D x 16
p( x) = p1 ⋅ 1−λ⋅ ⋅
5
D π
ρ
T T p
m
p
V 2
K
2
=
1⋅ 1−λ⋅ ⋅ ⋅
n
⋅ ⋅ ⋅
n
⋅
5 2
2 m
π
ρ &
n 1 n 2
⋅
2 n
⋅ ⋅ ⋅
2 n
⋅
m
T
2
x ⎡
2
1− ⎛ n
p V & K (2)
L 16 Tm
pn
p p
1
D p ⎞ ⎤
⎢
x 16
Folgender p( x) = p Zusammenhang
⎝ ⎜ 1 ⎠
⎟
⎥ kann ebenfalls Verwendung
finden:
1⋅ 1−λ⋅ ⋅
L
5
⎣⎢
p 2⎦⎥
D⎡
2
1
⋅ 1− ⎛ π
ρ
T T p
m
p
V 2
K
2
=
1⋅ 1−λ⋅ ⋅ ⋅
n
⋅ ⋅ ⋅
n
⋅
5 2
2 m
π
ρ &
n 1 n 2
⋅
2 n
⋅ ⋅ ⋅
2 n
⋅
m
⎞ ⎤Tn
p V & K
1
x p
⎢x
16
⎝ ⎜ 1 ⎠
⎟
⎥
p( x) = p1 ⋅ 1−λL
⋅
⎡
= − ⎛ 2
⎣⎢
⋅
5 p
⎦⎥
D
2
p ⎞ ⎤ ⎡ 5 2
2
&V ⎢
⎝ ⎜ 2
⎠
⎟ ⎥
⎣⎢
p
⎦⎥ ⋅ ⋅ D 2
L
⋅ π
⋅ Tn
T
⋅ p1
p
⋅ 1 ⋅ 1
( ) =
n
11⋅ 1− ⋅ 1− ⎛ π
ρ Tm
pn
2
⋅
2 n
⋅ ⋅ ⋅
2 n
⋅
m
⎞ ⎤Tn
p V & K
1
x p
p x p ⎢
1
λ 16
m n
ρn Km
⎡
= − ⎛ 2
⎝ ⎜ 1 ⎠
⎟
⎥ (3)
L
p ⎞ ⎤ ⎣⎢
p
5 ⎦⎥
2
2
&V ⎢
⎝ ⎜ 2
⎠
⎟ ⎥
⎣⎢
p
wm
⋅D⎦
⎥ ⋅ 1
⋅ D
L
⋅ π
⋅ Tn
T
⋅ p1
p
⋅ 1 ⋅ 1
n
1 x ⎡
2
Die ( ) Transportkapazität =
1⋅ 1− ⋅ 1
1
λ
− ⎛ p ⎞ ⎤
p x p ⎢
16 der Rohrleitung
m n
ρn K ist dann
m
wm⋅D⋅ρm
Re
Rem
= =
= − ⎛ 2
⎝ ⎜ 1 ⎠
⎟
⎥
L
⎡ ⎞ ⎤ 5 2
2
gemäß Gl. (4) bestimmbar: p
⎣⎢
p
νm
ηm
⎝ ⎜ 2
⎢
w ⎠
⎟
⎥
m
⋅D wm⋅D⋅ρm
Re = Re
⎣⎢
m
=
⎦⎥ ⋅ 1
⋅ D
⋅ π
⎦⎥
⋅ Tn
⋅ p1
⋅ 1 ⋅ 1
&V
n
1
p1
λ L 16 Tm pn ρn Km
=
4
= ⋅ V−&
⎛ 2
⎡
ν
⎞ ⎤ 5 2
2
p
m
ηm
m
w ⎝ ⎜ 2
⎢
m 2 w ⎠
⎟ ⎥
Re = Re π⎣
⎢
m
⋅
⋅D
4
⋅ =
⎦⎥ ⋅ ⋅ D
⋅ ⋅ Tn
⋅ p
&V
⋅ ⋅
n
1
L 116
πT
1 1
m
pn
1
p p
(4)
p1
D λw L
m⋅D
16 ⋅Tρ
Tp p
V 2
m n
ρ
K
2
=
1⋅ 1−λ⋅ ⋅ ⋅
n
⋅ ⋅ ⋅
n
⋅
5 2
2 m
π
ρ &
K
nm
1 n m
m
=
V&
ν L
16
m m
η
m
Außerdem w geht
m
π⋅Dn
V&
m
V&
n
p 2
wT
⋅
aus der Herleitung der Druckverlustgleichung
Re p( x= ) Re = pklar = ⋅ = hervor = [1], dass sowohl die Rohrreibungs-
m
⋅ ⋅
4
m
p
T
K
x w16
m⋅D⋅
ρm
m1 V&
⋅ 1−λ⋅ ⋅
5
zahl n
m
n m
V&
m
= V&
n
⋅ p m
wλ als auch die
νm
Kompressibilitätszahl
D π η
ρ T
Tm
pn
p p
D T
p V 2
K
2
=
1⋅ 1−λ⋅ ⋅ ⋅
n
⋅ ⋅ ⋅
n
⋅
5 2
2 m
π
ρ &
n
1
m
pn
2
⋅
2 n
⋅ ⋅ ⋅
2 n
⋅
m
m Tn
p V & K
1 K für über den
2
π D T
⋅ ⋅
m
p 3 3
2
T
K
gesamten − pn
2
= ⋅ Strömungsweg x 16
gemittelte Drücke und Temperaturen
4 V&
2
m
n
m 2
V&
3
1
⋅ p
m
V&
⋅ p m
w zu bestimmen sind:
2
π D T
n 3 2⋅
3 m
2
p1
m −Tp
K
2
( ) =
1⋅ ⋅ 1− ⎛ p( x) = p
1⋅ 1−λ⋅ ⋅
5
D π
ρ Tm
pn
2
⋅
2 n
⋅ ⋅ ⋅
2 n
⋅
m
T
⎡ ⎞ ⎤
n
p V & K
1
xL
16 p T
l = l m = l (T m , p
n2
m ) ⎝ ⎜ m
pn
p
x p p ⎢
1 ⎠
⎟
⎥
DL
⎣⎢
p T
⎦⎥
p V 2
K
2
=
1⋅ 1−λ⋅ ⋅ ⋅
n
⋅ ⋅ ⋅
n
⋅
5 2
2 m
π
ρ &
m
= ⋅
2 2
3
1n
− pm
2 2
V&
V &
n⋅⎛
p T
K m = K (T m , p⋅ ⋅p
⎞
3 m )
3 2 m
m
2
+
⎝
⎜
p
1m
1−Tp
K
n 1
2
x ⎡
2
( ) = 1⋅ 1− ⋅ 1− ⎛ p ⎞ ⎤
p x p ⎢
n2
p + ⎠
⎟
m
= ⎡⋅
2 12
p
2 2
Für
= 23
⎛p1
− p2p
⎞
2
pm
=
die
⋅
Berechnung
− ⎛ 2
p ⎞ ⎤
⎝ ⎜ 1 ⎠
⎟
⎥
L
⎣⎢
x 16p
5 2⎦
⎥
p( x) = p
2
1⋅ 1−λ⋅ &V ⎢
+
3 2⎝
⎜ p
der mittleren Rohrreibungszahl
3⎝ ⎜ 2
⎠
⎟ ⎥
1 3
2⎣
⎢ p
p −
+ 2⎠
⎟
ist die 1
p ⎦⎥ ⋅ 1 ⋅
5
⋅ D
L
⋅ π
⋅ n
T
⋅ 1
p
⋅ 1 ⋅ 1
n
1 D π
ρ Tm
pn
2
⋅
2 n
⋅ ⋅ ⋅
2 n
⋅
m
Tn
p V & K
1
1
λ 16
m n
ρn Km
2
p mittlere 1
p2
m
p⋅
+ 2p Reynolds-Zahl
1⋅ 2
Re m zu verwenden. Es gilt:
2 2
m 23
⎛p
1
− p2p
⎞
= ⋅ 3
2
m 2
( p
+
1+
p2)
2
3p ⎝
⎜ p
1wm
⋅D = − ⎛ 2
⎡ ⎞ ⎤ 5 2
2
p
⎝ ⎜ 2
⎢
⎠
⎟ ⎥
⎣⎢
⎦⎥ ⋅ 1
⋅ D
⋅ π
⋅ Tn
⋅ p
⋅ 1 ⋅ 1
&V
n
1
2
1
p x ⎡
1
λ
w
L 2
Re = Re1
= 2+ = p1⋅ p +
2 ⎠
⎟ m⋅D
16
⋅ρ
Tm pn ρn Km
( ) =
1⋅ 1− ⋅ 1− ⎛ p ⎞ ⎤
p x p ⎢
m
m 1
p=
2 (5)
⎝
p
⎜ 1 ⎠
⎟
⎥
L
m ν 2⎣
⎢ p
2 ⎛
m
η ⎦⎥
3 p ⎞
m
2
pm
⋅ ⋅ ( p+
1+
2)
2
2
3p ⎝
⎜ p
1w Die mittlere 2
1
+ m
⋅D w
p1⋅ p +
2 ⎠
⎟
1
p2
pm = 4
= ⋅ Strömungsgeschwindigkeit m⋅D⋅ρm
Re = Rem
= =
folgt aus
V&
ν
der 3
m m
ηm
wKontinuitätsgleichung
m= π ⋅ 2( 1+
p2)
2⋅
2
D
− ⎛ 2
⎡ ⎞ ⎤ 5 2
2
p
2
p1
+ p1⋅ p2+
p2
pm = ⋅ ⎝
4 V&
⎜ 2
⎢
⎠
⎟
⎥
⎣⎢
⎦⎥ ⋅ 1
⋅ D
⋅ π
⋅ Tn
⋅ p1
⋅ 1 ⋅ 1
&V
n
1
p1
λ L 16 Tm pn ρn Km
m 3
⋅ n( p1m+
p2)
V&
m
= V&
n
2
p m
w
2
π⋅D
T
⋅ ⋅
m
p
w
m
T
K
m
⋅D wm⋅D⋅ρm
Re = Rem
n
n
= ⋅ m
V&
⋅ ⋅
3 3
2
1
−
m
V&
p = =
Tν
m
n
m
pm
Tp
K
ηm
n2
pm
= ⋅
2 2
3
4
Der = mittlere ⋅ p
V&
m
w
1
− p
m Druck 2 muss grundsätzlich mit Hilfe des
32
3
2π
⋅
pD
−
Mittelwertsatzes p
1
p2
m
= ⋅ der Integralrechnung berechnet wer-
2
2
2
2
3 ⎛
p
−
den [1]. Für den 1
phier 2p
⎞
2 behandelten Fall der horizontalen
pm
+
3 ⎝
⎜
n1
m
Gastransportleitung V&
p + ⎠
⎟
m
= V&
⋅ p T
n
⋅ ⋅
1 mpist 2
der mittlere Pipelinedruck wie
2
pm
T
K
n 2
folgt ⎛
p
⎞
analytisch = ⋅ 2
2
p bestimmbar:
2
pm
+
1
+ p1⋅ 2 3 ⎝
⎜ p1
p
+
⎠
⎟
1
p
3 3 2
m 2 p1
− p2
pm
= ⋅3
(6)
2 2
3
⋅ ( p1+
p
2p1
− p2
2)
2
p 2
+ ⋅ +
1
p1 p2 p2
p
m
=
Dieser Zusammenhang 3
kann formal weiter umgeformt
= ⋅ ⎞
2 ⎛
⋅ ( p
1+
2
)
2
2
p werden, p
m
+
3 ⎝
⎜ p womit man eine in der russischsprachigen
Literatur häufig 1
1
p + ⎠
⎟ p2verwendete Formulierung findet:
p
2
p1
+ p ⋅ p + p
=
⋅ ( p1+
p2)
2
m
3
2
1 2 2
p
L 16 Tm
pn
p
D T p V 2
= ⋅ 1−λ⋅ ⋅ ⋅ K
n
⋅ ⋅ ⋅
2 n
⋅
π
ρ &
2 1 5 2
n
x 16
( x) = p1 ⋅ 1−λ⋅ ⋅
5
D π
ρ Tm
pn
2
L 16 ⋅
2
Tn
⋅ ⋅ ⋅ ⋅
2 n m
Tn
p V & K
m
pn
p p
1
D T p V 2
K
2
=
1⋅ 1−λ⋅ ⋅ ⋅
n
⋅ ⋅ ⋅
n
⋅
5 2
2 m
π
ρ &
n 1
Bild 1. Situationsskizze horizontale Gastransportleitung.
2
x ⎡
2
1− ⎛ p ⎞ ⎤
p x
p
⎢x
16
( ) =
⎝ ⎜ 1 ⎠
⎟
⎥
1⋅ 1−λ⋅ ⋅
L 5
D⎣
⎢ πp
ρ Tm
pn
2
⋅ ⋅ ⋅ ⋅ ⋅
2 n 2 n m
⎦⎥
Tn
p V & K
1
= − ⎛ 2
2
⎡ ⎞ ⎤ 5 2
2
p x ⎡
⎢
⎝ ⎜ 2
&V
⎠
⎟ ⎥
⎣⎢
p
⎦⎥ ⋅ ⋅ D 2
L
⋅ π
⋅ Tn
T
⋅ p1
p
⋅ 1 ⋅ 1
n( ) = 11⋅ 1− ⋅ 1− ⎛ p ⎞ ⎤
p x p ⎢
1
λ ⎝ ⎜ 1 16 ⎠
⎟
⎥
L
⎣⎢
p
⎦⎥
m n
ρn Km
wm
⋅D wm⋅D⋅ρm
Re
Rem
= =
= − ⎛ 2
⎡ ⎞ ⎤ 5 2
2
p
νm
ηm
⎝ ⎜ 2
⎢
⎠
⎟ ⎥
⎣⎢
⎦⎥ ⋅ 1
⋅ D
⋅ π
⋅ Tn
⋅ p1
&V
⋅ 1
p L
⋅ 1
n
1
1
λ 16 Tm pn ρn Km
4
= ⋅ V&
m
wm
2 wm
⋅D wm⋅D⋅ρm
Re = Re π⋅D
m
= =
ν η
m
1
n m
V&
m
= V&
n⋅ p T
⋅ ⋅
m
4 p
T
K
= ⋅ V&
m
w
n
m
Bild 2.
2
π⋅Situationsskizze D
Gastransportleitung mit Höhendifferenzen.
3 3
2 p1
− p2
pm
= ⋅
2 2
3
n
= ⋅ 1
−⋅ pm
V&
m
V&
p T
n
2⋅
p T
K
m
n
m
m
2
⎛ p ⎞
3 3 2
m
2
+
⎝
⎜pp1
+ ⎠
⎟ (7)
1
− p2
pm
= ⋅
1
p
2 2 2
3 p1
− p2
Nach dem Ausrechnen von Gl. (6) erhält man mit
2
2
Gl. (8) p1
+ p1⋅ p2+
p2
pm = eine weitere nutzbare 2
2 ⎛
Beziehung. Diese Schreibweise
wird
3
p ⎞
2
pm
= ⋅ in ⋅ ( p
+ DVGW-G
1+
p2)
2000 u. a. zur Berechnung des
3 ⎝
⎜ p1
Netzinhalts
2 p + ⎠
⎟
1
p2
verwendet.
p
2
p1
+ p ⋅ p + p
=
⋅ ( p1+
p2)
2
m
3
2
1 2 2
(8)
Alle oben zur Berechnung des mittleren Druckes
angegebenen Gleichungen führen zu identischen
Resultaten.
2.2 Gastransportleitungen mit Höhendifferenzen
Mit den Bezeichnungen gemäß Bild 2 lassen sich die
nachstehenden Gebrauchsgleichungen anschreiben
[1]. Es ist zu beachten, dass bei steigenden Leitungen für
ΔH das Vorzeichen „+“ und bei fallenden Leitungen „–“
zu wählen ist.
Verwendet man wieder die Normdichte und den
Normvolumenstrom, erhält man die sog. Ferguson-Gleichung
[1]:
2 2
p1
−p2
⋅e
ξ
e −1
ξ
2 gTn
n
= ⋅ ⋅ ρ
ξ ⋅ ⋅∆H
p K T
n
ξ
L 16 Tm
2
= λ⋅ ⋅ ⋅ ⋅p ⋅ρ
⋅V&
⋅K
5 2
D π T
⋅ m 1
n
m
n n n m
(9)
April 2012
gwf-Gas Erdgas 259