Page 308 - Process Equipment and Plant Design Principles and Practices by Subhabrata Ray Gargi Das

310 Chapter 11 Distillation

Minimum number of total trays in a column
The minimum number of trays in a column with total condenser under condition of total reflux is
given by Fenske’s equation (Eq. 11.13).

log x LK;D =x HK;D x HK;B =x LK;B
(11.13)
S m ¼ðN min þ 1Þ¼ avg
log a LK HK
where N min is the minimum number of trays in the column and S m is the minimum number including
the reboiler. x LK;D and x HK;D are the mole fraction of the light and heavy key in the distillate and x LK;B ,
avg
x HK;B denote the corresponding mole fractions in the bottoms product. a LK HK is the average relative
volatility between the key components and is given by Eq. 11.12.
Since vapoureliquid contacting on the feed tray is expected to be poor, one more tray is added to
give S m ¼ðN min þ2Þ.
Further, in case of a partial condenser due to the additional stage of vapoureliquid contacting, it
provides S m ¼ðN min þ3Þ.
Eq. 11.13 shows that the minimum number of equilibrium stages depends on the degree of sep-
aration of the key components and their relative volatility. It is independent of the feed phase and
composition. This is expected as the operating lines for rectification as well as the stripping section
under total reflux coincide with the 45 degree line on the x-y plane, making the location of feed
composition irrelevant. The non-key components influence N min only by their effect on a LK HK . The
Fenske equation needs to be used with caution when (i) the relative volatility varies appreciably over
the column and (ii) the mixture forms a nonideal liquid solution. Once N min is known, the split for all
non-key components is calculated using Eq. 11.13 by substituting ‘LK’ with component ‘i’.
Minimum Reflux e Algebraic expression for the minimum reflux ratio (R min ) given by the Un-
derwood’s equation is valid for ideal or near-ideal systems.
When feed is at its bubble point (q ¼ 1),

" #
avg
1 1 x LK;D
x LK;D
LK HK
a
R min ¼ avg (11.14a)
a LK HK 1 x LK;F 1 x LK;F
When feed at its dew point (q ¼ 0),
2 3
avg
1 4 LK HK x LK;D 1 x LK;D
a
R min ¼ avg 5 1 (11.14b)
a LK HK 1 y LK;F 1 y LK;F
In the general case 0 < q < 1, the explicit algebraic expression for R min does not exist and the
implicit relationship between the parameters is

R min x LK;F þ qx LK;D a ðR min þ 1Þy LK;F þðq 1Þx LK;D
(11.15)
¼
R min 1 x LK;F þ q 1 x LK;D ðR min þ 1Þ 1 x LK;F þðq 1Þ 1 x LK;D
Eqs. 11.14 and 11.15 show that R min depends mainly on the feed condition and relative volatility
and to a lesser extent on the degree of separation of the two key components.