### CENTRE for INNOVATION inMATHEMATICS TEACHING

PIPELINES

 Esso uses its Fawley Refinery (near Southampton) to process most of its oil. 70% of its products are now distributed by a network of pipelines to a number of depots with storage facilities throughout the country. From there, the fuel is transported to customers, such as petrol stations.The pipeline is expensive to build, but once laid it is essentially invisible, environmentally friendly, safe and inexpensive to maintain.The current pipeline network and depots are shown on the map opposite.

The flow in the pipelines is continuous every hour of every day and different products follow one another down the pipeline. It has been found that these different products will not mix, so long as the speed is at least 6 km/hour. Consequently the small interface between different products is maintained. Speed is controlled by having pumping stations at regular intervals.

One of the many mathematical problems associated with pipelines occurs at a junction. The flow must be continuous and the speeds need to be adjusted according to the diameter of the pipes

 Problem 1By equating the flow of oil per hour into and out of the junction, find the speed, v km per hour, needed for the two outflow pipes.Problem 2If the outflow speed in each pipe is 6 km per hour, what must be the inflow speed?

 Problem 3 For the same junction, find the speed for one of the outlet flows, if the other is 6km/hour and the inflow is 8km/hour.

 Problem 4 Write down one basic equation that must be satisfied for the junction opposite.

 Extension At regular times, the state of the pipework is monitored by sending a robot 'pig' along the pipe, from Fawley to the junction near Birmingham.It is not motorised, but moves along with the product (at speed say 6 km/hour).If it leaves Fawley at 11.00am on Monday, when should it arrive at the junction, 220 kilometres away?