6.3.2.1 Objective Function

The objective function should reflect the net value added or profitability of the process. Table 6.9 indicates the typical value of each component in the feed, distillate and bottoms product, and also the value of energy. The values shown (in dollars per tonne) were assumed for this study and do not necessarily reflect current prices although care was taken to make the scenario realistic. With respect to the optimisation, the units and magnitudes are arbitrary (only the relative values affect the location of the optimum).

The high negative value of ethanol in the distillate reflects the potential to excessively age or poison downstream catalysts. The reduced value of C4 components in the bottoms product reflect the effect on the gasoline pool volatility and the need to balance C4 with higher value components to meet overall volatility specifications. The fixed operating costs do not need to be considered as these do not affect the results of the optimisation (only the magnitude of the profitability).

The overall objective function can be expressed as:

Profitability (P) = Product Value - Feed Cost - Energy Cost (6.9)

Feed and Product Values |
Feed |
Distillate |
Bottoms | |

Ethanol ($/tonne) |
- |
-250 |
450 | |

Isobutene ($/tonne) |
- |
150 |
120 | |

ETBE ($/tonne) |
- |
0 |
800 | |

Non-Reactive C4 ($/tonne) |
- |
150 |
120 | |

Overall ($/tonne) |
250 |
- |
- | |

Heating ($/kW) |
0.03 | |||

Cooling (S/kW) |
0.02 |

Adopting the nomenclature described below and the values from Table 6.9:

P = (450 x, + 120 x2 + 800 x3 + 120 x4) B + (-250 y, + 150 y2 + 150 y4) D

The net value added per unit of feed is also of interest since the production rate is likely to be determined by other factors (i.e. demand):

P* = (450 x, + 120 x2 + 800 x3 + 120 x4) B/F + (-250 y, + 150 y2 + 150 y4) D/F - 250 - 30 Q/F - 20 Q/F (6. il)

The reactive distillation column can only be operated within the constraints imposed by the equipment design. These constraints include the maximum duties of the reboiler and the condenser, the column capacity (i.e. flooding point) and the restrictions on the composition of products due to downstream processing or blending requirements. The following constraints were assumed for this example, using a consistent nomenclature:

Vapour Flooding Factor < 80% (6.14)

Downcomer Flooding Factor < 80% (6.15)

A concentration of greater than 2.5% ethanol in the distillate was assumed to cause irreversible catalyst damage to a downstream unit. A total C4 concentration of greater than 2.0% in the bottoms was assumed to violate an intermediate (or final) volatility specification. A minimum ETBE purity of 90% was specified as a blending requirement.

Either equation (6.10) or (6.11) can be used to optimise the column's operating conditions. The primary manipulated variables (i.e. the condenser duty for the column pressure and the reboiler duty and reflux rate for the product compositions) could be optimised directly but it is preferable to find the optimal control set-points instead. These set-points can then be used within the existing control structure to reject minor disturbances in the feed rate or composition, etc. The situation is analogous to the justification for simple cascade controllers.

The multi-variable optimisation was undertaken within SpeedUpâ„˘ using a feasible-path successive quadratic programming routine (Aspen Technology, 1993). The global system of equation includes several local maxima so that the selection of the starting point was important. A starting point that corresponded to a high ETBE purity in the bottoms product was found to converge to the global optimum in all the cases examined.

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