where‡

þ G R xt dx

xb   xð1—xÞ ln½aðxÞ]

  aðxÞ

ð23Þ

kx  ¼ KxðvL; qL; lL; cL; DL; deÞ (30)

In each case, we have assumed that there are seven quan- tities with physical dimensions。 Further,  we  note  that there are four units of dimension  in the relationships above    (mass,

mðxÞ ¼ 

ð1 þ ½aðxÞ— 1]xÞ

(24)

length, time, and mole number)。 By the Buckingham P theo- rem, a relationship exists among three dimensionless group- ings。25 It is straightforward to show in each case that the Sherwood number can be taken to depend on the Reynolds

*It is less disturbing to do this with a packed column where it is easier to imag-

ine the composition being continuous with packed   depth。

†See Appendix A。

‡We have ignored terms involving da/dx in the derivation of Eq。 24 from Eq。

16。 For the standard distillation systems studied in most laboratories, the variation

number and the Schmidt  number。

kyde m     n

of the relative volatility with composition is small, and the neglect of these terms has negligible impact。 See Appendix  B。

ShV ¼

V    V

¼ AVReV ScV (31)

fractional mass-transfer area will be a function of six dimen- sionless groupings。 In one method for finding the dimension- less groupings, we assume that the functional form for am/ad is a power law。 Doing so allows us to write down a set of linear equations relating the power-law exponents on the var- ious physical quantities。

am b v d

e / c g i

ad /

ðqVÞaðlVÞ  ðvVÞ ðqLÞ ðlLÞ ðvLÞ  ðrÞ ðdeÞ ðgÞ 

(34)

a b。 。v d

e。 。/

c 。 。i

m0l0t0 ¼ 。 。 。 m 。

l 。 m。 。  m  。   l

。 m。 l

ðlÞ

l3

Thus

l — t

t l3

l — t t t2

t2

(35)

0 ¼ a þ b þ d þ e þ c ðmassÞ 0 ¼ —3a — b þ v — 3d — e þ / þ g þ i ðlengthÞ 0 ¼ —b — v — e — / — 2c — 2i ðtimeÞ

(36)

Some manipulation allows us to express c, g, and i in terms of a, b, v, d, e,  and  /。

c ¼ —a — b — d — e