Question 5 of 7: Benzene / toluene distillation (McCabe–Thiele)
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. EGBC 04-CHEM-A3 Mass Transfer Operations, May 2016 — three-hour open-book exam. Seven questions in three parts: Part A (Q1–2, molecular diffusion), Part B (Q3–4, film mass transfer), Part C (Q5–7, staged and equilibrium separations). The candidate answers one of Q1–2, one of Q3–4 and two of Q5–7 (four questions of equal value). All seven are worked in full below.
Reference texts. J.M. Coulson & J.F. Richardson, Chemical Engineering Vol. 1 (Fluid Flow, Heat and Mass Transfer) and Vol. 2 (Particle Technology & Separation Processes) — the source of all seven problems; C.J. Geankoplis, Transport Processes and Separation Process Principles; R.E. Treybal, Mass-Transfer Operations; Bird, Stewart & Lightfoot, Transport Phenomena; Perry’s Chemical Engineers’ Handbook.
The waste is taken as $x_W=0.05$ benzene (95 mol% toluene). The printed “5% mole toluene” would make the bottoms 95% benzene, which is not a waste stream; the parallel with the 95% benzene product confirms $x_W=0.05$ benzene is intended.
Find. (a) $x$ on plate 2; (b) $N$ and feed plate; (c) $R_{\min}$; (d) $N_{\min}$; (e) $N$ for feed at 288 K.
$x$ (benzene, liq.)
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.95
$y$ (benzene, vap.)
0.21
0.37
0.51
0.64
0.72
0.79
0.86
0.91
0.96
0.98
Approach. Build the rectifying and stripping operating lines from $R$, $x_D$, $x_W$ and the $q=1$ feed line; step off equilibrium stages between the curve and the operating lines; obtain $R_{\min}$ from the feed-point pinch and $N_{\min}$ at total reflux.
Operating lines. Rectifying $y=\dfrac{R}{R+1}x+\dfrac{x_D}{R+1}=0.8x+0.19$. The $q=1$ feed line is vertical at $x=0.40$; it meets the rectifying line at $(0.40,\,0.51)$, so the stripping line runs from $(0.05,0.05)$ to $(0.40,0.51)$.
Second plate from the top (part a). A total condenser gives $y_1=x_D=0.95$; equilibrium $\Rightarrow x_1=0.88$; the rectifying line gives $y_2=0.8(0.88)+0.19=0.894$; equilibrium again gives
$$\boxed{x_2\approx0.77\ \text{(benzene mole fraction on plate 2).}}$$
Number of plates and feed location (part b). Stepping from $x_D$ down to $x_W$ across the two operating lines (figure):
$$\boxed{N\approx9\ \text{equilibrium stages}=8\ \text{plates}+\text{reboiler, feed on plate }\approx5.}$$
Minimum reflux (part c). For $q=1$ the pinch sits at $x_F$, where $y^{*}(0.40)=0.64$. The limiting rectifying line through $(x_D,x_D)$ and $(0.40,0.64)$ has slope $\dfrac{R_{\min}}{R_{\min}+1}=\dfrac{0.95-0.64}{0.95-0.40}=0.564$, so
$$\boxed{R_{\min}=1.29.}$$
Minimum plates (part d). At total reflux the operating line is the diagonal; stepping between the diagonal and the curve from $x_D$ to $x_W$ gives
$$\boxed{N_{\min}=7\ \text{stages (incl. reboiler).}}$$
Sub-cooled feed at 288 K (part e). A liquid entering 80 K below its 368 K bubble point has $q=1+\dfrac{c_{p}\,(T_b-T_F)}{\lambda}=1+\dfrac{159(80)}{32\,000}\approx1.40$. The feed line now rises with slope $q/(q-1)=3.5$, shifting the intersection to $(0.44,0.54)$. Re-stepping gives
$$\boxed{N\approx9\ \text{stages}}$$
— essentially unchanged, because the extra internal liquid from the sub-cooled feed only marginally re-positions the feed line here.
McCabe–Thiele construction: equilibrium curve (green), rectifying and stripping operating lines, the $q=1$ feed line and the stepped stages — about 9 equilibrium stages with the feed near plate 5.
Quantity
Value
(a) $x$ on plate 2
≈ 0.77
(b) Stages / feed plate
9 (8 plates + reboiler), feed ≈ plate 5
(c) $R_{\min}$
1.29
(d) $N_{\min}$ (total reflux)
7 stages
(e) Stages, feed at 288 K ($q=1.40$)
≈ 9
Check — part (e) physical data
The exam does not tabulate the feed heat capacity or latent heat, so $c_p\approx159\text{ kJ kmol}^{-1}\text{K}^{-1}$ and $\lambda\approx32\,000\text{ kJ kmol}^{-1}$ (typical benzene–toluene values) are assumed to obtain $q\approx1.40$; the plate count is insensitive to modest changes in these.
