## YHn1 LTVT1HHyHn DVtXhD ayHn 1 LtVTXHD

where a = (Lt/Vt)/Hh > 1 is the absorption factor. The corresponding Kremser formula for the heavy component in the vapour phase at stage NT counted from the top of the column (the accumulator in stage zero) is then:

(assuming we are in the region where a is constant, i.e. xH + 0).

For hand calculations one may use the McCabe-Thiele diagram for the intermediate composition region, and the Kremser formulas at the column ends where the use of the McCabe-Thiele diagram is inaccurate.

Example We consider a column with N = 40, Nf = 21, a = 1.5, Zl = 0.5, F = 1, D = 0.5, VB = 3.206. The feed is saturated liquid and exact calculations give the product compositions xH,D = xL,B = 0.01. We now want to have a bottom product with only 1 p.p.m. heavy product, i.e. xL,B = 1.e — 6. We can use the Kremser formulas to estimate easily the additional stages needed when we have the same energy usage, VB = 3.206. (Note that with the increased purity in the bottom we actually get D = 0.505.) At the bottom of the column Hl = a = 1.5 and the stripping factor is s = (Vb/Lb)Hl = (3.206/3.711)1.5 = 1.296.

With xL,B = 1.e — 6 (new purity) and xl,Nb = 0.01 (old purity) we find by solving the Kremser equation [31] for the top with respect to NB that NB = 34.1, and we conclude that we need about 34 additional stages in the bottom (this is not quite enough since the operating line is slightly moved and thus affects the rest of the column; using 36 rather than 34 additional stages compensates for this).

The above Kremser formulas are valid at the column ends, but the linear approximation resulting from the Henry's law approximation lies above the real VLE curve (it is optimistic), and thus gives too few stages in the middle of the column. However, if there is no pinch at the feed stage (i.e. the feed is optimally located), then most of the states in the column will be located at the column ends where the above Kremser formulas apply.

Approximate Formula with Constant Relative Volatility

We will now use the Kremser formulas to derive an approximation for the separation factor S. First note that for cases with high purity products we have S + 1/(xl, bxh,d). That is, the separation factor is the inverse of the product of the key component product impurities.

We now assume that the feed stage is optimally located such that the composition at the feed stage is the same as that in the feed, i.e. yH,NT = yH,F and xl,Nb = xL,F. Assuming constant relative volatility and using Hl = a, Hb = 1/a, a = ^lf/xlfV^hf/xhf) and N = Nt + Nb + 1 (including total reboiler) then gives:

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