## VrmJF

Note that the recoveries

We call this expression the feed equation since only the feed properties (q and z) appear. Note that this is not the equation which defines the Underwood roots and the solutions (0) apply as roots of the defining equations only for minimum reflux conditions (N = r). The feed equation has Nc roots (but one of these is not a common root) and the Nc!1 common roots obey: a1 > 01 > a2 > 02 > 2 > 0Nc — 1 > aNc. Solution of the feed equation gives us the possible common roots, but all pairs of roots (^ and ^ + 1) for the top and bottom section do not necessarily coincide for an arbitrary operating condition. We illustrate this with the following example.

Example Assume we start with a given product split (D/F) and a large vapour flow (V/F). Then only one component i (with relative volatility ai) can be distributed to both products. No roots are common. Then we gradually reduce V/F until a second component j (this has to be a component j = i + 1 or j = i — 1) becomes distributed, e.g. for j = i + 1 one set of roots will coincide: ^ = +1 = 0i, while the others do not. As we reduce V/F further, more components become distributed and the corresponding roots will coincide, until all components are distributed to both products, and then all the Nc!1 roots from the feed equation also are roots for the top and bottom sections.

An important property of the Underwood roots is that the value of a pair of roots which coincide (e.g. when ^ = + 1 = 0i) will not change, even if only one, two or all pairs coincide. Thus all the possible common roots are found by solving the feed equation once.

### Minimum Energy - Infinite Number of Stages

When we go to the limiting case of infinite number of stages, Underwood's equations become very useful. The equations can be used to compute the minimum energy requirement for any feasible multicomponent separation.

Let us consider two cases: first we want to compute the minimum energy for a sharp split between two adjacent key components j and j + 1 (rj D = 1 and r;n =

## Solar Panel Basics

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