SIMULATION OF GAS PIPELINES LEAKAGE USING CHARACTERISTICS METHOD Author: Ehsan Nourollahi...
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Transcript of SIMULATION OF GAS PIPELINES LEAKAGE USING CHARACTERISTICS METHOD Author: Ehsan Nourollahi...
SIMULATION OF GAS
PIPELINES LEAKAGE USING
CHARACTERISTICS METHOD
Author: Ehsan Nourollahi
• Organization: NIGC (National Iranian Gas Company)
• Department of Mechanical Engineering , Ferdowsi University, Iran
Topics:
1. Introduction
2. Characteristics method
3. The numerical solution method & The implementation of the leakage effect
4. Results & Conclusions
Introduction
The pipe surface leakage or the pipe section dismissal can be created of some various reasons like as corrosion, earthquake or mechanical stroke which may be implemented in the pipe surface and also overload compressors.
Figure (1)
After the leakage creation, the flat expansion pressure waves are propagated in two converse sides
These waves have the sonic speed and after clashing to the upstream and downstream boundaries, return to the form of compression or expansion wave depending on the edge type
In the leak location, depending on ratio of pressure to ambient pressure be more or less than CPR quantity, the flow will be sonic and ultrasonic or subsonic respectively.
If the flow be sonic and ultra sonic, the sonic reporter wave don’t leak from out of the pipe to inter the pipe practically. Hence the changes of the flow field are accomplished due to the flat pressure waves and the real boundary conditions on the start and end of the pipe
mass flow outlet of the hole only depends on the stagnation pressure in the leak location and on area of the hole and is not related to the form of the orifice cross section
1
1 1
2
k
k
cr
out
kP
PCPR
Characteristics method
The continuity equation is:
The momentum equation is:
By extension of these equation, we have:
With attention to the definition of speed of the sound by:
For an ideal gas:
tx
u
)(
Dt
Du
x
P
kp
a 2
s
pa )(2
01
x
u
x
u
t
01
x
uu
t
u
x
Third condition of continuity for isentropic flow is:
or:
For isentropic flow are constant, then we have:
)1/(2)( k
Aref a
a
)1/(2)( kk
Aref a
a
p
p
t
a
akt
1
1
21
x
a
ak
k
x
p
p
1
1
21
refrefA pa ,,
By using of the relationship between the sonic speed and the pressure in an ideal gas, these equations are changed to the below forms after some steps of rewriting of the mass and momentum conservation equations:
These equations are set of quasi-linear hyperbolic partial differential equations.
0})({2
1})({
x
uau
t
uk
x
aau
t
a
0})({2
1})({
x
uau
t
uk
x
aau
t
a
Therefore a solution of the form: is
required. Except of special cases, there are no
analytical solution for these equations, then we should
study numerical solutions. In this paper we use
Characteristics method to achieve a numerical
solution.
Base of this method is transferring of two independent
equations as and to another group
as or
),(
),(
txuu
txaa
),( txaa ),( txuu
),( txcc ),( aucc
Figure 2. Graphical interpretation of the characteristics method
(a) Three-dimensional surface defining
(b) Projection of line on characteristics surface to plane at
The solution may be represented by the curved surface bounded by edges PQRS
0c
),( txcc
if in a special point on the surface of for a
reviewing special curve from that point, the slope of the
projected curve on the x-t plane be equal with quantity
of curve of that point, the passing direction of that point
is known as the characteristic direction. We have in
mathematical expression:
By using of this complete derivative definition, the sonic
speed and particle speed parameters are determined
with respect to the time of a characteristic length like as
the below :
),( txcc
Cdtdx char )/(
x
ac
t
a
dt
da
char
x
uc
t
u
dt
du
char
Therefore if defined as the form of:
In length of two characteristics, they are rewritten like as:
),( aucc auc
auc
2
1
02
1
02
1
22
11
cc
cc
dt
duk
dt
da
dt
duk
dt
da
The numerical solution method
At first the none-dimensional parameters of A and U are defined as below in the characteristics method:
In the above equation is sonic speed in the start point. Then Reimann non-dimensional characteristics are defined as following:
refref a
aA
a
uU ;
refa
Uk
AUk
A2
1;
2
1
An explicit equation between Reimann variables in
inner points of the solution field is presented below
which is for each step:
By distinction of the state equation in the boundary, a
mono-equation is created between Reimann variables.
So always in any boundary, one of these variables is
known and the other one is unknown then the
unknown Reimann variable can be calculated, so the
effect of the boundary transfers to the solution field is
obtained.
nini
ni
ni
ni
ni ab
x
t
1111
nini
ni
ni
ni
ni ab
x
t
1111
The implementation of the leakage effect
For implementation of the leakage effect on the flow field, the mesh is chosen in a way that the hole location would be stated between two nodes
When the hole is created in the pipe surface, as it’s said, the pressure ratio to the ambient pressure in below the hole which is inter the pipe, is more than the CPR in the later time steps.
Figure (3)
The leakage point in any time step act as a boundary
and two expansion waves depending to direction of
flow in the pipe, would reach to a and b points with a
little time difference and create the same change in
non-dimensional speed of U like the below form:
bb
bb
aa
aa
Um
QUU
Um
QUU
Therefore, the flow is checked in
the hole location and outflow of
the leak location, calculated by:
1
1
1
2...
k
k
llor k
kZRT
MPAQ
Then the unknown parameters and are
calculated like the below form:
Therefore, the state of two points in any time step with
considering to corrected leakage effect and hence by
notice to the equations that governed to the problem
are type of the hyperbolic equations, during the time of
the leakage effect is transferred permanently as a third
boundary addition to the upstream and downstream
boundaries to the solution field.
ba
bbb
aaa
Uk
Uk
)1(
)1(
Results and Conclusions
Consumptions:
Pipe Length : 250 meter
Hole Area : 1 cm
Number of grid system : 100 nodes
Initial gas pressure : 30 bar
Initial gas speed : 41 ft/s
Also temperature is constant and there are non viscose flow.
Boundary conditions:Upstream boundary condition is the reservoir with constant
pressure and the downstream boundary condition is stated
with three forms:
• The boundary with no changes with respect to the location
• The valve with constant coefficient of pressure drop
• close end
Figure 4. State (1) of the boundary conditions:4-a Pressure changes by increasing of the pipe length at primary
times4-b Changes of the exit mass flux by time
Figure 5. State (1) of the boundary conditions:5-a changes of the exit mass flux by increasing the hole area and
pipe length5-b changes of the exit mass flux by increasing the pipe pressure
and pipe length
Figure 6. State (2) of the boundary conditions:6-a. Pressure changes by increasing of the pipe length at primary
times6-b. Changes of the exit mass flux by time
Figure 7. State (2) of the boundary conditions:7-a Changes of the exit mass flux by increasing the hole area and
pipe length7-b Changes of the exit mass flux by increasing the pipe pressure
and pipe length
Figure 8. state (3) of the boundary conditions:8-a Pressure changes by increasing of the pipe length at primary
times8-b Changes of the exit mass flux by time
Figure 9. State (2) of the boundary conditions:9-a Changes of the exit mass flux by increasing the hole area and
pipe length9-b Changes of the exit mass flux by increasing the pipe pressure
and pipe length
With best wishes
of Iranian People