10.3 Distillation and McCabe-Thiele Method

10.3 Distillation and McCabe-Thiele Method

Distillation is the most common industrial separation process, separating components based on differences in vapor pressure (volatility). Binary distillation involves a feed containing two components. The FE Chemical exam extensively tests the McCabe-Thiele graphical method, including operating lines, feed conditions, reflux ratios, minimum stages (Fenske equation), and efficiency calculations.

Vapor-Liquid Equilibrium (VLE)

The ease of separation by distillation is characterized by the relative volatility $\alpha_{AB}$, defined as the ratio of the distribution coefficients of the two components:

$\alpha_{AB} = \frac{y_A / x_A}{y_B / x_B} = \frac{y (1 - x)}{x (1 - y)}$

Where $y$ and $x$ represent the mole fractions of the more volatile component (light key) in the vapor and liquid phases, respectively. Solving for $y$ yields the equilibrium curve equation:

$y = \frac{\alpha x}{1 + (\alpha - 1) x}$

A larger relative volatility ($\alpha \gg 1$) results in a wider gap between the equilibrium curve and the $y = x$ diagonal line, indicating that fewer stages are required for separation. If $\alpha = 1$, no separation is possible (azeotrope).

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The McCabe-Thiele Graphical Method

The McCabe-Thiele method simplifies column design by assuming Constant Molar Overflow (CMO). Under CMO:

• The molar latent heats of vaporization of both components are approximately equal.
• Sensible heat changes, heat of mixing, and heat losses (adiabatic column) are negligible.
• As a result, the molar flow rate of liquid ($L$) and vapor ($V$) are constant in the rectifying section, and the liquid ($\bar{L}$) and vapor ($\bar{V}$) flow rates are constant in the stripping section. This allows operating lines to be plotted as straight lines on a $y\text{-}x$ diagram.

Rectifying Section Operating Line (ROL)

The rectifying section is the portion of the column above the feed tray. A mass balance around the condenser and a stage in this section gives:

$y = \frac{R}{R+1} x + \frac{x_D}{R+1}$

Where:

• $R = L/D$ is the reflux ratio (ratio of reflux liquid to distillate product).
• $x_D$ is the distillate mole fraction of the light key.
• The slope of the ROL is $\frac{R}{R+1}$ and its $y$-intercept is $\frac{x_D}{R+1}$. The ROL intersects the $y = x$ diagonal line at the point $(x_D, x_D)$.

Stripping Section Operating Line (SOL)

The stripping section is the portion of the column below the feed tray. A mass balance around the reboiler and a stage in this section gives:

$y = \frac{\bar{L}}{\bar{V}} x - \frac{B}{\bar{V}} x_B$

Where:

• $B$ is the bottoms molar flow rate.
• $x_B$ is the bottoms mole fraction of the light key.
• The SOL passes through the point $(x_B, x_B)$ on the $y = x$ diagonal and has a slope of $\frac{\bar{L}}{\bar{V}}$, which is always greater than 1.0.

Feed Line (q-line) and Thermal Conditions

The feed line (or $q$-line) represents the locus of the intersection points of the ROL and SOL. The parameter $q$ is defined as the mole fraction of liquid in the feed, or the heat required to vaporize 1 mole of feed divided by the molar latent heat of vaporization:

$y = \frac{q}{q-1} x - \frac{x_F}{q-1}$

The $q$-line intersects the $y = x$ diagonal at the feed composition point $(x_F, x_F)$ and has a slope of $\frac{q}{q-1}$. The thermal condition of the feed determines the slope:

• Subcooled Liquid ($q > 1$): Slope is positive ($> 1$).
• Saturated Liquid ($q = 1$): Slope is infinite (vertical line).
• Liquid-Vapor Mixture ($0 < q < 1$): Slope is negative.
• Saturated Vapor ($q = 0$): Slope is zero (horizontal line).
• Superheated Vapor ($q < 0$): Slope is positive ($< 1$).

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Operating Limits: Reflux Ratio

Minimum Reflux Ratio ($R_{\min}$)

Decreasing the reflux ratio $R$ decreases the slope of the ROL, moving it closer to the equilibrium curve. At $R_{\min}$, the intersection of the ROL, SOL, and $q$-line touches the equilibrium curve, creating a pinch point. At this point, the driving force is zero, and an infinite number of stages is required.

For a saturated liquid feed ($q=1$), the pinch point occurs at $x = x_F$. The minimum reflux ratio can be calculated using the equilibrium vapor concentration in contact with the feed ($y_F^*$):

$R_{\min} = \frac{x_D - y_F^*}{y_F^* - x_F}$

Total Reflux ($R = \infty$)

At total reflux, all overhead vapor is condensed and returned as reflux ($D=0$, $L=V$). The ROL and SOL lie exactly on the $y = x$ diagonal line. This represents the minimum number of theoretical stages ($N_{\min}$) required for the separation.

The Fenske equation analytically calculates $N_{\min}$ for a system with constant relative volatility:

$N_{\min} = \frac{\ln\left[ \left(\frac{x_D}{1-x_D}\right) \left(\frac{1-x_B}{x_B}\right) \right]}{\ln \alpha} - 1$

Where the $-1$ term accounts for the partial reboiler (which acts as one equilibrium stage). If a total reboiler is used, the minimum stages inside the column is $N_{\min} + 1$, or simply $N_{\text{stages}} = N_{\min} + 1$.

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Stage and Column Efficiency

The theoretical stage calculation assumes that the vapor and liquid leaving each tray are in perfect thermodynamic equilibrium. In real columns, contact is incomplete, and trays are less than $100%$ efficient.

Murphree Vapor Efficiency ($E_{MV}$)

The efficiency of a specific tray $n$ is defined by:

$E_{MV} = \frac{y_n - y_{n+1}}{y_n^* - y_{n+1}}$

Where $y_n$ is the actual vapor composition leaving stage $n$, $y_{n+1}$ is the actual vapor composition entering stage $n$, and $y_n^*$ is the vapor composition in equilibrium with the liquid composition ($x_n$) leaving stage $n$.

Overall Column Efficiency ($E_O$)

The actual number of trays needed is related to the ideal (theoretical) stages by:

$E_O = \frac{N_{\text{ideal}}}{N_{\text{actual}}}$

Where $N_{\text{ideal}}$ is the number of equilibrium stages determined from the McCabe-Thiele diagram (excluding the reboiler if it is a partial reboiler).

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Worked Example: Fenske and Minimum Reflux

Problem: A binary distillation column is separating benzene and toluene. The feed is a saturated liquid ($q = 1.0$) with $x_F = 0.45$. The distillate must contain $95\text{ mol}%$ benzene ($x_D = 0.95$) and the bottoms must contain $5\text{ mol}%$ benzene ($x_B = 0.05$). The average relative volatility is $\alpha = 2.4$.

1. Calculate the minimum number of theoretical stages inside the column using the Fenske equation.
2. Determine the minimum reflux ratio ($R_{\min}$).

Solution:

1. Apply the Fenske equation: $N_{\min} = \frac{\ln\left[ \left(\frac{0.95}{1 - 0.95}\right) \left(\frac{1 - 0.05}{0.05}\right) \right]}{\ln 2.4} - 1$ $N_{\min} = \frac{\ln\left[ (19) (19) \right]}{\ln 2.4} - 1 = \frac{\ln(361)}{0.8755} - 1 = \frac{5.8889}{0.8755} - 1 = 6.73 - 1 = 5.73\text{ stages}$ Thus, the minimum number of theoretical stages inside the column is approximately $5.7$ stages.

2. Since the feed is a saturated liquid ($q = 1.0$), the pinch point occurs at $x = x_F = 0.45$. Find the equilibrium vapor concentration ($y_F^*$): $y_F^* = \frac{\alpha x_F}{1 + (\alpha - 1) x_F} = \frac{2.4 \times 0.45}{1 + (2.4 - 1) \times 0.45} = \frac{1.08}{1 + 1.4 \times 0.45} = \frac{1.08}{1.63} = 0.6626$

3. Apply the minimum reflux ratio equation: $R_{\min} = \frac{x_D - y_F^*}{y_F^* - x_F} = \frac{0.95 - 0.6626}{0.6626 - 0.45} = \frac{0.2874}{0.2126} \approx 1.35$

The minimum reflux ratio is $1.35$.