## E l

and when we substitute for 0 and simplify, we obtain King's expression: LTmin/F = (rL,D — arH,D)/(a — 1). Another interesting case is minimum energy operation when we consider sharp split only between the most heavy and most light components, while all the intermediates are distributed to both products. This case is also denoted the preferred split, and in this case there will be a pinch region on both sides of the feed stage. The procedure is:

1. Compute all the Nc — 1 common roots (0) from the feed equation.

2. Set r1D = 1 and rNc D = 0 and solve the following linear equation set (Nc — 1 equations) with respect to [VT, r2,D, r3,D (Nc — 1 variables):

airi dzI

Note that, in this case, when we regard the most heavy and light components as the keys, and all the intermediates are distributed to both products, King's very simple expression will also give the correct

minimum reflux for a multicomponent mixture (for q = 1 or q = 0). The reason is that the pinch then occurs at the feed stage. In general, the values computed by King's expression give a (conservative) upper bound when applied directly to multicomponent mixtures. An interesting result which can be seen from King's formula is that the minimum reflux at the preferred split (for q = 1) is independent of the feed composition and also independent of the relative volatilities of the intermediates.

However, with the Underwood method, we also obtain the distribution of the intermediates, and it is easy to handle any liquid fraction (q) in the feed.

The procedure for an arbitrary feasible product recovery specification is similar to the preferred split case, but then we must only apply the Underwood roots (and corresponding equations) with values between the relative volatilities of the distributing components at the limit of being distributed. In cases where not all components distribute, King's minimum reflux expression cannot be trusted directly, but it gives a (conservative) upper bound.

Figure 9 shows an example of how the components are distributed to the products for a ternary (ABC) mixture. We chose the overhead vapour flow (V = VT) and the distillate product flow (D = V — L) as the two degrees of freedom. The straight lines, which are at the boundaries when a component is at the limit of appearing/disappearing (distribute/not distribute) in one of the products, can be computed directly by Underwood's method. Note that the two peaks (PAB and PBC) give us the minimum vapour flow for a sharp split between A/B and B/C. The point PAC, however, is at the minimum vapour flow for a sharp A/C split and this occurs for a specific distribution of the intermediate B, known as the preferred split.

King's minimum reflux expression is only valid in the triangle below the preferred split, while the Underwood equations can reveal all component recoveries for all possible operating points. (The shaded area is not feasible since reflux and vapour flow rates have to be positive (V > D, V > (1 — q) F.)

## 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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