Question 18.4: A mixture of 2 mole percent ethanol and 98 mole percent wate......
A mixture of 2 mole percent ethanol and 98 mole percent water is to be stripped in a plate column to a bottom product containing not more than 0.01 mole percent ethanol. Steam, admitted through an open coil in the liquid on the bottom plate, is to be used as a source of vapor. The feed is at its boiling point. The steam flow is to be 0.2 mol per mole of feed. For dilute ethanol-water solutions, the equilibrium line is straight and is given by y_{e}=9.0x_{e}. How many ideal plates are needed?
Since both equilibrium and operating lines are straight, Eq. (17.27) rather than a graphical construction may be used. The material-balance diagram is shown in Fig. 18.23. No reboiler is needed, as the steam enters as a vapor. Also, the liquid flow in the tower equals the feed entering the column. By conditions of the problem
F={\overline{{L}}}=1\qquad{\overline{{V}}}=0.2\qquad y_{b}=0\qquad x_{a}=0.02
x_{b}=0.0001\qquad m=9.0\qquad y_{a}^{*}=9.0\times0.02=0.18
y_{b}^{*}=9.0\times0.0001=0.0009
To use Eq. (17.27), y_{a}, the concentration of the vapor leaving the column, is needed. This is found by an overall ethanol balance
V={\overline{{V}}}+(1-q)F\qquad{\mathrm{and}}\qquad V-{\overline{{V}}}=(1-q)F (17.27)
\overline{{{V}}}(y_{a}-y_{b})=\overline{{{L}}}(x_{a}-x_{b})\qquad0.2(y_{a}-0)=1(0.02-0.0001)
from which y_{a}=0.0995. Substituting into Eq. (17-27) gives
N={\frac{\ln\left[(0.0995-0.18)/(0-0.0009)\right]}{\ln\left[(0.0009-0.18)/(0-0.0995)\right]}}
={\frac{\ln89.4}{\ln1.8}}=7.6{\mathrm{~ideal~plates}}