Subtractive calculations

Sometimes the properties and yields of a process unit depend on the final optimization solution. Therefore they cannot be calculated ahead of time as a standard Base + Delta model, but must be calculated during optimization.

This approach is often helpful when trying to balance the properties of a process unit. For example, imagine a hydrotreater that always hydrotreats its product to 10ppm sulfur. The amount of hydrogen consumed, and the amounts of product and hydrogen sulfide produced, depend on the sulfur of the incoming feed. This is not known ahead of time, so a standard linear Base + Delta relationship between the hydrogen sulfide yield and incoming sulfur cannot be established, as the amount of H2S produced vary non-linearly with respect to the incoming sulfur content. Instead, it is useful to calculate the yield of H2S by balancing the sulfur across the unit, that is, calculating the yield of H2S (and DS product) based on the required amount of hydrotreating.

This can be achieved using subtractive property calculations, used to balance a property across a unit depending on the performance of the unit.

Property balances

Perfect separation

Imagine a separation unit that is fed with an impure benzene and paraffin mixture and completely removes benzene to a product stream, leaving raffinate as the product. The density of the feed is the weighted average (in volume terms) of the amount of benzene and other components, while the density of the pure benzene product is known. The density of the raffinate depends on the amount of benzene in the feedstock and the feedstock's density.

The feedstock of the product might be configured as shown:

An operating parameter to represent the raffinate density is required:

A calculation is required to determine the value of the operating parameter. The value of this parameter is then assigned to the raffinate density to close the density balance and ensure the final product density is equal to the blended feed density.

The incoming feed is 0.75 proportion benzene and has a density of 0.821 g/cc. Out of 100bbl, 75 bbl of are pure benzene, with a density of 0.8765 g/cc. As such, 0.75 × 0.8765 = 0.67375 g/cc of the density originates from benzene. By subtraction, 0.821 - 0.6735 = 0.163625 g/cc of density originates from raffinate, which makes up 0.25 proportion of the feed. Therefore the raffinate density must be 0.163625/0.25 = 0.6545 g/cc.

Another way of representing this problem is to use a property x yield prediction for the benzene, as the density could in theory change with respect to the final benzene yield prediction. This can be written as shown, and gives the same results as the simpler model above.

Imperfect separation

What happens if the separation is not perfect? Suppose the separation is 90% selective for benzene, so 90% of the feed benzene ends up in the benzene product, and 10% remains in the raffinate. Likewise, 10% of the raffinate leaves in the benzene product, so the benzene product is 90% benzene, and 10% other material.

The following table shows how the yields of the aromatic and raffinate fraction would change with respect to different feed proportions of benzene. Also shown is the composition of each final product in terms of its aromatic content and other materials. From the chart it is possible to see that the yield of the aromatic stream varies proportionally with the incoming benzene concentration.

Shown below are the calculations used to determine these values.

The gradient for the chart can be used to configure the Base + Delta model, and the reverse implies the yield of raffinate. The benzene concentration of the aromatic fraction can be calculated using a property x yield calculation, where the final benzene content of the aromatic product varies with respect to the incoming feed benzene content.

The aromatic stream density is determined by its composition in terms of benzene and other material. The density contribution from benzene is known and entered as 0.7889. This is 0.9 × 0.8765 (the density of pure benzene) and takes into account the fact that only a proportion of the incoming benzene contributes to the final stream density. The aromatics density also requires a component due to the other material in the feed, but the density of this is unknown. Therefore, an estimate of the contribution of this is made.

The final raffinate density can be used to close the material balance using the same calculation as above now that the feed density and other product density is known. The resultant calculation assigns the value of the operating parameter to the raffinate density, and this ensures that the unit is kept in density balance even where the other stream yields and densities are varying.