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.



Adobe Systems


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:




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 p2 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.



Adobe Systems



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.


Adobe Systems


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:



Adobe Systems


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. 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.



Adobe Systems


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). 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.



Adobe Systems


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 to give a diameter of 1824mm and a length of 78km. Elton to Towton (parallel pipes with offtakes on both branches)



Adobe Systems


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.


Adobe Systems


Figure 4.15

 Horndon – SE1 (parallel pipeline branches with differing pipe diameters in each branch)




Adobe Systems


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. 



[1] This technique is discussed in “Gas pipeline hydraulics”, Menon, E.S