### 2.3 Practical Example: Oil Blending Model

 In order to appreciate the various approaches to practical MP, we shall now develop a simple MP model and see how it is implemented using different types of MP software. Our example is drawn from the downstream oil industry, the largest user of MP, and is concerned with the blending of oil products. [Note that the example has deliberately been simplified and ignores the distinction between those qualities which "blend by volume" (e.g. specific gravity) and those which "blend by weight" (e.g. sulphur content, Viscosity Blend Index). In building an MP model it is vital to be clear and precise in defining units and to ensure consistency. Here we shall fall below those requirements in order to make the example easier to understand.] Blending of Heavy Fuel Oil Heavy fuel oil (HFO) is blended from five components: light gas oil (LGO); heavy gas oil (HGO); waxy distillate (WAXD); atmospheric residue (ARES); vacuum residue (VRES). Three physical properties, or qualities are of interest: density; viscosity; and sulphur content. We have to produce HFO to a quality specification and want to know in what proportions to blend the components. It is quite deliberate (and typical of reality) that we are not given a precise statement of what the objective is: we must work it out for ourselves. Here are the qualities of the components: ` Density Viscosity SulphurLGO 0.83 15.0 1.0HGO 0.88 26.0 2.2WAXD 0.92 30.0 2.8ARES 0.97 40.0 4.1VRES 1.03 48.0 5.0` and the quality specification for the fuel oil: `HFO: max 0.98 37.0 3.7 min` Although some qualities do not blend linearly, we shall assume here that they do (for instance, viscosity does not blend linearly, but a function of viscosity, the Viscosity Blend Index, does). By "blending linearly" we mean that the value of quality q of a blend is simply the arithmetic mean of the values of quality q for the components weighted by the quantity of the component in the blend. Two-Component Blends Returning to our data, we can see immediately that ARES is just outside the quality specification but that the addition of a relatively small quantity of any of the lighter materials will produce an acceptable HFO. Thus the qualities of various acceptable blends are: ` Density Viscosity Sulphur12.9% LGO87.1% ARES 0.9519 36.78 3.70021.4% HGO78.6% ARES 0.9507 37.00 3.69330.8% WAXD69.2% ARES 0.9546 36.92 3.700` Note what I have done here. I have calculated just how much of the lighter material is needed to bring the quality of the ARES down so that the blend lies within the quality specification of the HFO. The specific gravity of all the materials (with the exception of VRES) lies within the quality specification. So there are no difficulties with this quality. The problem areas are Viscosity Blend Index (VBI) and Sulphur. It so happens, when you do the calculations, that the Sulphur Constraint is the tighter one when adding LGO or WAXD whereas VBI of the ARES - LGO and ARES - WAXD blends is less than 37.0 (viz 36.78 and 36.92) whereas the Sulphur content of the ARES - HGO blend is less than the maximum permitted (viz 3.693 instead of 3.7). We say that the Sulphur constraint is binding for the ARES - LGO and ARES - WAXD blends whereas the VBI constraint is binding for the ARES - HGO blend. Three-Component Blends Since we are able to make blends with maximum VBI by using HGO and with maximum Sulphur using LGO or WAXD you might very well ask whether we could not make a blend with max VBI and max Sulphur by using HGO and LGO or WAXD. This is indeed the case. ` Density Viscosity Sulphur 2.4% LGO17.1% HGO80.5% ARES 0.9512 37.00 3.70012.5% HGO12.5% WAXD75.0% ARES 0.9525 37.00 3.700` If our objective were to minimise quality giveaway and we ignore VRES we can now see that a blend of 75% ARES, 12.5% HGO and 12.5% WAXD would be the optimum. Both the viscosity and the Sulphur constraints are binding and the specific gravity of this blend is higher than that of the ARES - HGO - LGO blend. [Highly technical point: We have actually shown only that this is the best blend that we have yet considered, if indeed we prefer it to the ARES - WAXD blend which has a higher specific gravity although a lower VBI. It can however be proved rigorously that the optimum lies at one of these 'extreme' points, so if this is the preferred extreme point it is the optimum.] `previous contents next`