Question $5$

Petroleum fuels are transported by pumping through an underground pipeline from the New Zealand Oil Refinery at Marsden Point to a tank farm on the Auckland waterfront. The pipeline is $200km$ long. Multiple fuels are transported in the same pipeline. Petrol might be pumped, and then after some time, the feed is switched to diesel.

When the fuel types are switched there will be an interface where petrol and diesel will mix. This material is recovered at the tank farm and returned for reprocessing by road transport.

A model is required to predict the volume and timing for this contaminated product.

a$)$ Formulate a model for this system. Assume the pipeline is of uniform diameter and there are no other adjoining flows into or out of the pipeline $($just the feed from a tank, pumped into a receiving tank$).$ State all assumptions made and define all the nomenclature you use.

$[12$ marks$]$

b$)$ The model is to be solved using finite differences. Define a grid and derive the finite difference approximations required to solve the model.

$[8$ marks$]$

c$)$ A tracer study was carried out to determine the dispersion coefficient. This provided the data in Table $5\mathrm{.}1.$ Use this data to determine the dispersion coefficient.

$[5$ marks$]$

Table $5\mathrm{.}1$ Tracer study data

$\backslash$table$[[$Time $($hours$),$Tracer measurement signal$],[15\mathrm{.}0,0\mathrm{.}0],[15\mathrm{.}5,1\mathrm{.}3],[16\mathrm{.}0,4\mathrm{.}2],[16\mathrm{.}5,5\mathrm{.}0],[17\mathrm{.}0,4\mathrm{.}5],[17\mathrm{.}5,1\mathrm{.}5],[18\mathrm{.}0,0\mathrm{.}3],[18\mathrm{.}5,0\mathrm{.}0]]$

$[$Total $25$ marks$]$