McCabe-Thiele Calculator: Stages and Minimum Reflux

Equilibrium and balances

The primary method is the McCabe–Thiele equilibrium-stage construction (McCabe and Thiele, 1925, DOI 10.1021/ie50186a023). Measured 101 kPa ethanol–water VLE comes from Susial Badajoz et al. (2026), Table 5, DOI 10.1021/acsomega.6c04827. The table and shape-preserving cubic interpolation (pchip) match the WiredWhite pot still calculator. Experimental points at x = 0.071 and 0.199 are withheld for checks. The pure-water endpoint is a dilute-limit approximation. Inverse equilibrium uses bisection on this same interpolated curve, rather than fitting an unrelated inverse.

Overall and light-component balances give $D/F=(z_F-x_B)/(x_D-x_B)$ and $B=F-D$. Internal flows are $L=RD$, $V=(R+1)D$, $L'=L+qF$, and $V'=V+(q-1)F$. The operating lines are $y_R=(Rx+x_D)/(R+1)$ and $y_S=(L'x-Bx_B)/V'$. Their intersection also satisfies $qx-(q-1)y=z_F$, which remains well defined at q = 0 and q = 1.

Full-curve minimum reflux

For each composition between the products, the reflux needed to bring either operating line onto equilibrium is calculated from these balances. The smaller of the two thresholds applies at that composition; its maximum is the controlling pinch. A 1001-point scan locates the highest candidate, followed by bounded scalar refinement. Positive stripping vapor flow is also required. This captures rectifying or stripping tangent pinches as well as the feed pinch.

Stage counting

Starting at $(x_D,x_D)$, move horizontally to equilibrium and vertically to the rectifying line, switching to the stripping line at the first equilibrium composition below the operating-line intersection. The suggested feed stage is that crossing stage, counted from the top. Stop at the first horizontal equilibrium point at or below xB. The fractional last stage uses the fraction of the last horizontal composition interval. Retain the final graphical point for counting and explicitly identify an overshoot below xB. The profile and stage table display construction coordinates, not a solved sequence of trays or a mass-balanced partial-reboiler state. The endpoint comparison separately shows the specified xB and its equilibrium vapor reference y*(xB). Round up for the whole equilibrium-stage count; subtract one for ideal trays because the partial reboiler is included. The total condenser is excluded. This follows the convention stated by LearnChemE. The total-reflux limit is stepped separately against y = x. A 500-stage or stalled construction is rejected.

Shortcut comparison

Fenske uses $N_{min}=\ln[x_D(1-x_B)/(x_B(1-x_D))]/\ln\bar\alpha$. Underwood solves $\bar\alpha z_F/(\bar\alpha-\theta)+(1-z_F)/(1-\theta)=1-q$ for $1<\theta<\bar\alpha$, then calculates $R_{min}+1$ from the distillate composition. Here $\bar\alpha$ is the geometric mean of relative volatility at the two product compositions. Gilliland uses the Molokanov fit documented by Difflow, with $X=(R-R_{min})/(R+1)$ and $Y=1-\exp[(1+54.4X)(X-1)/((11+117.2X)\sqrt X)]$, giving $N=(N_{min}+Y)/(1-Y)$. For stable inversion, evaluate $N=(N_{min}+1)/\exp(A)-1$, with $A=(1+54.4X)(X-1)/((11+117.2X)\sqrt X)$; this avoids subtracting a very small exponential from 1. Before exponentiation, test $\ln(N_{min}+1)-A>\ln(501)$. Values at or below that model's minimum reflux, or above the 500-stage reporting limit, are deliberately unavailable. A conditional message explains when to increase reflux; the main construction remains available when valid. The empirical shortcut is a comparison, not a correction to real ethanol–water stepping.

• Binary steady operation; one feed, two products, a total condenser, and an equilibrium partial reboiler. No side draws, reaction, entrainment, or noncondensables.
• Constant molar overflow within each section, with feed effects represented by q. Feed quality is an input, not calculated from temperature or an energy balance.
• All compositions are mole fractions. The ethanol model is fixed at 101 kPa on the water-rich side of its measured azeotrope; no pressure adjustment is implied.
• The constant-relative-volatility teaching model is idealized. User molar masses in that mode affect stream quantities only.
• Every theoretical tray reaches equilibrium. No tray-efficiency factor or packing-height conversion is applied.
• The one-hour mass basis represents continuous throughput, not a batch run. Metric/US display converts this input and product masses; molar masses and molar-flow results retain their declared chemical-engineering units.
• Mass and component balances and the full-curve reflux thresholds are derived here from conservation equations. Equilibrium values and the Gilliland correlation are sourced, not fitted to reference cases.

For preliminary binary separation studies and teaching. Ethanol–water compositions must satisfy 0 < xB < zF < xD < 0.899. The teaching model accepts constant α = 1.05–15. Feed quality is limited to 0–1.5 and operating reflux to 0.001–100, always above the computed minimum and with positive vapor flow in both sections. Designs requiring more than 500 stages or unable to advance are held back with an instruction to increase reflux or relax the separation.

The first measured nonzero liquid composition is 0.011; smaller compositions rely on the pure-water endpoint. The sparse measured curve has experimental uncertainty and is not a rigorous activity-coefficient model. Near the azeotrope, measured composition rounding matters greatly. Product fractions above 0.87 trigger a reminder to reduce purity or validate finer VLE data.

Fenske–Underwood–Gilliland assumes a representative constant relative volatility. It can substantially underpredict ethanol–water stages or minimum reflux and must not be used to bypass its azeotrope. Real-VLE total-reflux stepping is the primary minimum-stage estimate. Fractional stage interpolation is a graphical convention; whole stages are rounded up. The last plotted/table coordinate may be an overshoot below xB and is identified as such. It must not be interpreted as the actual partial-reboiler composition. The xB equilibrium vapor reference is also a target reference, not a separately solved column state. Gilliland estimates above 500 stages are omitted with an explanatory message, with a logarithmic guard preventing numerical overflow. Feed stage is a graphical suggestion for the specified equilibrium model, not a detailed optimization.

No mechanical sizing, column diameter, tray hydraulics, packing, heat duties, pressure containment, relief sizing, multicomponent separations, or azeotrope-breaking process is covered.

It does not assess fire protection, pressure relief or the safe operation of a column (ethanol vapor is flammable).