the blend using the following equation (known as a Refutas equation): (1) VBN = 14.534 × ln[ln(v + 0.8)] + 10.975 where v is the viscosity in square millimeters per second (mm²/s) or centistokes (cSt) and ln is the natural logarithm (loge). It is important that the viscosity of each component of the blend be obtained at the same temperature. The next step is to calculate the VBN of the blend, using this equation: (2) VBNBlend = [wA × VBNA] + [wB × VBNB] + ... + [wX × VBNX] where w is the weight fraction (i.e., % ÷ 100) of each component of the blend. Once the viscosity blending number of a blend has been calculated using equation (2), the final step is to determine the viscosity of the blend by using the invert of equation (1): (3) v = ee(VBN - 10.975) ÷ 14.534 − 0.8 where VBN is the viscosity blending number of the blend and e is the transcendental number 2.71828, also known as Euler's number. Example economics for a two-component blend A marketable fuel oil, such as for fueling a power station, might be required to have a viscosity of 40 centistokes at 100 °C. It might be prepared using either the virgin or visbroken residue described above combined with a distillate diluent ("cutter stock"). Such a cutter stock could typically have a viscosity at 100 °C of 1.3 centistokes. Rearranging equation (2) above for a simple two component blend shows that the percentage of cutterstock required in the blend is found by: (4) %cutter stock = [VBN40 − VBNresidue] ÷ [VBNcutter stock − VBNresidue] Using the viscosities quoted in the tables above for the residues from Arab Light crude oil and calculating VBNs according to equation (1) gives: For virgin residue (i.e., the unconverted feed to the visbreaker): 27.5% cutter stock in the blend For visbroken residue: 13.3% cutter stock in the blend. As middle distillates have a far higher value in the market place than fuel oils, it can be seen that the use of a visbreaker will considerably improve the economics of fuel oil manufacture. For example, if the cutter stock is taken to have a value of $300 per tonne and fuel oil $150 per ton (oil prices naturally change quickly, but these prices, and more importantly the differences between them, are not unrealistic), it is a simple matter to calculate