4. TECHNICALITIES OF MODEL
4.3 Pressure/Flow
Relations
4.3.1 Equivalent Pipe Method for Parallel Pipes[1]
A pipe with a large diameter causes
less resistance to the flow of gas than a pipe with a smaller diameter would,
and therefore, the pressure drop across a large diameter pipe is smaller. This fact is evident in the General Flow equation,
equation (1). Installing pipes in
parallel has the same effect as installing one pipe with a larger diameter. For this reason, parallel pipes are often
used in the gas network to reduce pressure drops between nodes. Lower pressure drops across pipelines reduces
the number of compressor stations required across the network as a whole. It is often more economical to install an
additional pipeline in parallel with existing pipelines, especially when the
operating costs of compressor stations are considered.
Figure 4.8 Parallel pipes
Figure 4.8 shows an example
of parallel pipes. The pressure at node
1 is fixed. To maintain flow rates, and
to ensure gas continues to flow in the correct direction, the pressure of the
gas at the end of each pipe as it reaches node 2 must be equal. By examining the General Flow equation, it is
evident that the only variable affecting the pressure drop across each pipe is
the flow rate, assuming all other parameters to be constant. Therefore, the total input flow must be
distributed in such a way as to achieve an equal pressure drop across each pipe,
and hence equal pressures at the end of each pipe at node 2.
To avoid having to employ an
iterative approach, which can become quite lengthy for more than two pipes in
parallel, an equivalent pipe may be substituted in place of the parallel
pipes. This allows the common pressure
at node 2 to be calculated using only two equations.
The General Flow equation for
gas flow through pipes is:
_{}
An equivalent pipe will have
a flow rate equal to the sum off the flows through the original parallel
pipes. Therefore:
_{}
Cancelling constants in the
flow equation gives:
_{}
Since the pressure drop
between nodes 1 and 2 is constant, this equation becomes:
_{}
This leaves two unknowns for
the equivalent pipe, namely length and diameter. Arbitrarily setting the length to a
particular value allows the equivalent diameter to be calculated. For the model, the length of the longest
parallel pipe is used as the equivalent length.
Rearranging the above
equation to solve for the diameter of the equivalent pipe gives:
_{}
This technique can be
extended for any number of pipes in parallel.
The equation used in the model, for n
pipes in parallel, is:
_{} |
(3) |
Once the dimensions of the
equivalent pipe are known, the pressure at Node 2 can be calculated using the
rearranged General Flow formula. The
actual flow rates through each original pipe can then be calculated using this
value of p_{2} in the flow
equation.
This method is known as the
Hazen Williams technique.
4.3.2 Iterative Flow Distribution Method for Parallel
Pipes with Offtakes
The equivalent pipe method
described above cannot be used if there are any offtakes from the parallel pipelines,
due to the fact that some flow is removed at each offtake. When this occurs, the flow through the
equivalent pipe is no longer equal to the sum of the inlet flows to each of the
parallel pipes, which is the fundamental relationship on which the equivalent
pipe method is based.
Figure 4.9 shows an example
of parallel pipes with offtakes. A
branch consists of a series of pipes, each of which may have differing lengths
and diameters. For example, Pipe 2.1
refers to the first pipe on Branch 2. A
branch may have offtakes between pipes, causing the flow rate through each pipe
to be different.
Figure 4.9: Parallel pipes with offtakes
As is the case with the
equivalent pipe method described above, the pressure drop across each branch
must be equal so that the pressure of the gas at the end of each branch is
equal as it reaches the outlet node.
Again, the only variable affecting the pressure drop is the flow of gas
through the branch. The flow from the
inlet node must be distributed between the two branches in such a way as to
achieve equal pressure drops.
To achieve this, the model
uses an iterative approach that initially sends a small amount of flow, dQ, through Branch 2 and sends the
remaining flow through Branch 1. The
pressure at the end of each branch is checked and compared, using the
rearranged General Flow equation. If the
two values are not equal (within a set error margin), an additional increment is
taken from the flow to Branch 1 and added to Branch 2. The pressures are then checked again. This process is repeated until the pressures
are equal. This process is similar to
the Newton-Raphson and Gauss-Seidel techniques used for electrical load-flow
problems (Wood et al, 1996).
The iterative flow distribution
technique can be extended to calculate the flow distribution across parallel
branches consisting of pipes with differing lengths and diameters but with no
offtakes. This problem cannot be
overcome using the equivalent pipe method as it is limited to single pipe
branches only.
The Iterative Flow
Distribution technique could be extended to deal with 3 or more branches in
parallel with offtakes, but this is unnecessary as this situation does not
occur in the network.
4.3.3 Combination of Equivalent Pipe Method with
Iterative Flow Distribution Method
The techniques described in
the previous two sections can be combined to calculate the flow distribution
through a number of pipelines in parallel, with one branch including offtakes
as shown in figure 4.10 below. These
three techniques cover all parallel pipe situations that occur in the National
Transmission System.
Figure 4.10 shows three pipes
in parallel to a branch with two offtakes.
Figure 4.10: Three pipes in parallel
with a branch with offtakes
Pipes a, b and c are in parallel with no offtakes and
can therefore be replaced by a single equivalent pipe, Branch 2, as shown in
figure 4.11:
Figure 4.11: Equivalent pipe in parallel
with a branch with offtakes
The flow distribution between
Branch 1 and Branch 2 can now be calculated using the Iterative Flow
Distribution method. Since the inlet
pressure and flow through Branch 1 is now known, the outlet pressure can be
calculated, allowing the flow through pipes a,
b, and c to be calculated.
4.3.4 Descriptions of Complex Pressure/Flow Relations
in the Model
This section gives
descriptions of complex pressure/flow relations that are contained in the
model. All the examples use one of the
three techniques described in the previous sections.
4.3.4.1 Easington Gas
Terminal to Paull (parallel pipes)
Figure 4.12 show the parallel
pipes transporting gas from Easington terminal to the node at Paull. This is a simple example of parallel pipes in
the NTS that can be represented by the equivalent pipe method. The notation adjacent to the pipes gives the
diameter in millimetres and the length in kilometres, shown in brackets. The three pipes therefore have a diameter of
900mm with two having a length of 24km.
The remaining pipe is 23km long.
Figure 4.12: Parallel pipes
Using formula (3) and setting
the length of the equivalent pipe to 34km, the equivalent diameter is
calculated as follows.
_{}
Therefore the three parallel
pipes between Easington and Paull are equivalent to one pipe of 1401mm
diameter.
Using equation (1b), the
pressure at the Paull end of the equivalent pipe can be calculated, which then
allows the flow through each parallel pipe to be calculated using the General
Flow equation, equation (1).
4.3.4.2 St.Fergus Gas
Terminal to Aberdeen Compressor Station (parallel pipes with offtakes)
This is an example of pipes
in parallel with a branch of offtakes and is shown in figure 4.13.
Figure 4.13
The dimensions of the
equivalent pipe for the 4 pipes a, b, c
and d is calculated in the same way
as shown in section 4.3.4.1 to give a diameter of 1824mm and a length of 78km.
4.3.4.3 Elton to Towton
(parallel pipes with offtakes on both branches)
Figure 4.14
When drawn slightly
differently, figure 4.14 becomes as shown in figure 4.15. To overcome the problem of flow entering at
Little Burdon, a negative offtake is entered into the VBA function. This causes
flow to be added to the branch at this point rather than being removed as is
the usual case.
Figure 4.15
4.3.4.4 Horndon – SE1 (parallel
pipeline branches with differing pipe diameters in each branch)
Figure 4.16
The equivalent pipe method cannot
be used as the pipeline diameter is not constant for each branch between
Horndon and SE1. Therefore, the iterative
flow distribution technique must be used.
The function calculates the pressure at the end of each pipe within each
branch and can hence calculate the common pressure at the outlet node SE1.