## G VgTZAi

In the above problem, the variables u and A£ were included in the vector A£ and the inequality constraint is the same as that of the original problem formulation. (ZTBZ is the Hessian matrix of the objective function $ and it is also updated during the quadratic programming step using the BFGS (Broyden-Fletcher-Goldfarb-Shanno) formula). The reduced gradients for the objective and constraint functions appearing in eqns [69] and [70] above are calculated in the iteration t during the integration step according to the formula:

Si Szf

NE K

In the equation above, the partial derivatives of the state variables at the rightmost exterior collocation point (zk + i), in relation to the optimized vector is calculated, via chainruling, by the formula:

The new set of control variables and element lengths calculated in the optimization step replaces the old one and the integration step is performed once again. The Kuhn-Tucker conditions, which determine the attainment of constrained minimum, are then checked and the calculations are stopped if these conditions have been reached. Otherwise, the optimization step is performed again and the whole procedure is repeated.

The preceding development typifies the complexity involved in rigorous optimization of batch distillation. The gains of reduced costs or shorter process times that can be achieved by such an optimization would not probably be worth the effort for routine operation. None the less, it is possible to utilize sophisticated laboratory-scale distillation units to test alternative control strategies indicated by computational approaches.

Subscripts i, j = tray number, component i = initial f = final R = reboiler w = material in the still tot = total

## Solar Panel Basics

Global warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products.

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