EXAMPLE $4-4.$ McCabe$-$Thiele analysis of open steam heating

A $60$mol$\%$ methanol and $40$mol$\%$ water feed is input as a two$-$phase mixture that flashes so that $\frac{V_F}{F}=0\mathrm{.}3.$ Feed flow rate is $350$kmo$\frac{l}{h}.$ The column is well insulated and has a total condenser. The reflux is returned to the column as a saturated liquid. An external reflux ratio of $\frac{L_0}{D}=3\mathrm{.}0$ is used. We desire a distillate concentration of $95$mol$\%$ methanol and a bottoms concentration of $8$mol$\%$ methanol. Instead of using a reboiler, saturated steam at $1$atm is sparged directly into the bottom of the column to provide boilup $($called direct or open steam$).$ Column pressure is $1$atm. Calculate the number of equilibrium stages and the optimum feed plate location.

Solution

A$.$ Define. It is helpful to draw a schematic diagram of the apparatus $($Figure $4-15),$ particularly given that a new type of distillation is involved. We wish to find the optimum feed plate location, $N_F,$ and the total number of equilibrium stages, $N.$ We could also calculate $Q_c,D,B,$ and the steam rate $S,$ but these are not asked for. We assume that the column is adiabatic because it is well insulated.

B$.$ Explore. The first thing we need is equilibrium data. Fortunately, these are readily available $($see Table $2-8$ in Problem $2.$D$1).$

Second, we would like to assume CMO so that we can use the McCabe$-$Thiele analysis procedure. An easy way to check this assumption is to compare the latent heats of vaporization per mole:

$H_{\text{v}\text{a}\text{p}}$ methanol $(at$ boiling point$)=8\mathrm{.}43$kca$\frac{l}{m}ol$

$H_{\text{v}\text{a}\text{p}}$ water $(at$ boiling point$)=9\mathrm{.}72$kca$\frac{l}{m}ol$

These values are not equal, and in fact, water's latent heat is $15\mathrm{.}3\%$ higher than methanol's. Thus, CMO is not valid; however, we will solve this problem assuming CMO and check our results with a process simulator.

Solve example $4-4$ with L$0/$D $=1\mathrm{.}5$ instead of $3\mathrm{.}0$ by hand with McCabe$-$Thiele analysis. In addition to calculating the optimum feed plate location and the total number of equilibrium stages, also calculate Qc$,$ D$,$ B$,$ and the stream rate S$.$