Question 6 of 6: Benzene–Toluene Distillation (McCabe–Thiele)
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2019 — 16-Chem-A3 Heat and Mass Transfer. Three-hour, open-book exam (one textbook of the candidate’s choice; any non-communicating calculator). Format: two parts — Part A (Q1–Q3) Heat Transfer and Part B (Q1–Q3) Mass Transfer; at least two questions must be attempted from each part and only the first two in each part are marked, so four questions (each 25 points) constitute a complete paper. All six questions are solved below for completeness. Property values not printed on the paper (molar masses, water latent heat, the dimensionless free-convection peak velocity $f'_{max}$, benzene/toluene physical properties) are stated explicitly in each Given block as open-book look-ups.
Reference texts: Coulson & Richardson (Backhurst, Harker & Richardson), Chemical Engineering, Vol. 1 — Fluid Flow, Heat Transfer and Mass Transfer (6th ed., Butterworth-Heinemann) — the source family for the crystalliser, tube-condenser, Stefan-tube and distillation problems; Incropera & DeWitt, Fundamentals of Heat and Mass Transfer (free- and forced-convection correlations, the Ostrach similarity solution); Treybal, Mass-Transfer Operations (3rd ed.) and McCabe, Smith & Harriott, Unit Operations of Chemical Engineering (7th ed.) — Stefan diffusion, Chilton–Colburn analogy, McCabe–Thiele; supporting property data from Perry’s Chemical Engineers’ Handbook (9th ed.) and the NIST Chemistry WebBook.
The obscured verb in Part B Q1 is taken as “calculate”, and the Part B Q2 water viscosity at 27 °C is $8.76\times10^{-4}$ Pa·s. One column of the Part A Q2 air-property table, the buoyancy group $g\beta/\nu^{2}$, is internally inconsistent: its printed values are about 40% below what $\rho$, $\mu$ and $\beta=1/T_f$ give. That column is not used; the Grashof number is built from the $\rho$ and $\mu$ columns and $\beta=1/T_f$.
Question 6: Benzene–Toluene Distillation (McCabe–Thiele) (Part B — 25 pts)
Reading the specification: the paper reads “waste of 4% mass toluene”, which would make the bottoms 96% benzene — impossible for a benzene-rich distillate. It is taken as the intended 4% mass benzene in the bottoms (96% toluene), the only physically consistent reading.
Find. (a) plate-2 composition; (b) plates + feed stage at $q=1$; (c) $R_{min}$; (d) $N_{min}$; (e) plates for subcooled (15 °C) feed at $R=4$.
Figure 6 — McCabe–Thiele construction ($q=1$, $R=4$): equilibrium curve (blue), rectifying line $y=0.8x+0.193$ (red), stripping line (green) and the vertical saturated-liquid $q$-line (purple). Stepping from $x_D$ to $x_W$ gives ten stages including the reboiler; the feed enters at stage 5.
Approach. Fix the rectifying line from $R$ and $x_D$, the $q$-line from the feed state, and the stripping line through $x_W$; then step between the operating lines and the equilibrium curve for (a),(b),(e); use the feed pinch for (c) and total reflux for (d).
(a) Second plate from the top. With a total condenser $y_1=x_D=0.966$; the equilibrium curve gives $x_1=0.915$. The rectifying line $y=\tfrac{4}{5}x+\tfrac{x_D}{5}=0.8x+0.193$ then gives $y_2=0.8(0.915)+0.193=0.925$, and equilibrium gives$$\boxed{x_2=0.83\ \text{(liquid on plate 2)},\quad y_2=0.925\ \text{(vapour from plate 2)}}.$$
(b) Plates and feed stage ($q=1$). The saturated-liquid $q$-line is vertical at $x=z_F=0.481$; it meets the rectifying line at $y=0.578$. The stripping line runs from $(x_W,x_W)=(0.047,0.047)$ to $(0.481,0.578)$. Stepping from $x_D$ down to $x_W$ (Figure 6) gives$$\boxed{10\ \text{theoretical stages incl. the reboiler}=9\ \text{plates + reboiler},\ \text{feed on plate 5}}.$$
(c) Minimum reflux. At $q=1$ the pinch is where the $q$-line meets the equilibrium curve: at $x=0.481$, $y^\*=0.705$. The rectifying line through $(x_D,x_D)$ and the pinch has slope $R_{min}/(R_{min}+1)$:$$\frac{R_{min}}{R_{min}+1}=\frac{x_D-y^\*}{x_D-z_F}=\frac{0.966-0.705}{0.966-0.481}=0.538\;\Rightarrow\;\boxed{R_{min}=1.17}.$$The operating $R=4$ is comfortably above this.
(d) Minimum plates (total reflux). At total reflux the operating line is the 45° diagonal; stepping between it and the equilibrium curve from $x_D$ to $x_W$ gives$$\boxed{N_{min}\approx8\ \text{stages incl. reboiler}\;(\approx7\ \text{plates})},$$consistent with Fenske ($N_{min}=\ln[(x_D/(1-x_D))((1-x_W)/x_W)]/\ln\alpha\approx6.9$ with $\alpha\approx2.5$).
(e) Subcooled feed at 15 °C. A cold feed condenses some vapour, so $q>1$: $q=1+c_{pL}(T_b-T_F)/\lambda$. Using open-book benzene–toluene values $c_{pL}\approx150$ J/mol·K and $\lambda\approx32$ kJ/mol with $T_b-T_F=95-15=80$ K gives $q=1.38$. The $q$-line (slope $q/(q-1)=3.6$) shifts the operating-line intersection up, and re-stepping at the same $R=4$ gives$$\boxed{\approx9\ \text{stages incl. reboiler}\;(\approx8\ \text{plates})}\ \text{— roughly one fewer than the saturated-liquid case}.$$
Check (part e data): the feed heat capacity and latent heat are not printed on the paper; $c_{pL}\approx150$ J/mol·K and $\lambda\approx32$ kJ/mol are open-book benzene–toluene estimates. The plate count is only mildly sensitive to $q$ here (a subcooled feed steepens the stripping section, saving about one stage). Stage counts come from linear interpolation of the tabulated equilibrium data (graphical accuracy ±1 stage).
