# Shortcut Distillation Column Calculator Documentation

# 1. Objective

The Shortcut Distillation Column Calculator solves preliminary design and rating problems for binary distillation columns. It utilizes the classical Fenske-Underwood-Gilliland (FUG) shortcut framework combined with Kirkbride's feed tray location formula and Souders-Brown vapor capacity hydraulics to size columns on a mass basis ( $\text{kg/h}$ ) and molar basis ( $\text{kmol/h}$ ).

# 2. Design Basis & Mass-Molar Conversion Methodology

# 2.1 Feed Mass Flow Rate & Composition Conversion

Plant distillation columns are typically operated on a mass flow basis ( $F_{\text{mass}}$ in $\text{kg/h}$ ).

Given:

Feed mass flow rate: $F_{\text{mass}}$ ( $\text{kg/h}$ )
Light Key component Molecular Weight: $M_L$ ( $\text{g/mol}$ or $\text{kg/kmol}$ )
Heavy Key component Molecular Weight: $M_H$ ( $\text{g/mol}$ or $\text{kg/kmol}$ )

# Converting Mass Fractions ( $w$ ) to Mole Fractions ( $z, x$ ):

z_F = \frac{\frac{w_F}{M_L}}{\frac{w_F}{M_L} + \frac{1 - w_F}{M_H}}

x_D = \frac{\frac{w_D}{M_L}}{\frac{w_D}{M_L} + \frac{1 - w_D}{M_H}}

x_B = \frac{\frac{w_B}{M_L}}{\frac{w_B}{M_L} + \frac{1 - w_B}{M_H}}

# Average Stream Molecular Weights:

M_{F, \text{avg}} = z_F M_L + (1 - z_F) M_H \quad [\text{kg/kmol}]

M_{D, \text{avg}} = x_D M_L + (1 - x_D) M_H \quad [\text{kg/kmol}]

M_{B, \text{avg}} = x_B M_L + (1 - x_B) M_H \quad [\text{kg/kmol}]

# Molar & Mass Overall Material Balances:

F_{\text{molar}} = \frac{F_{\text{mass}}}{M_{F, \text{avg}}} \quad [\text{kmol/h}]

D_{\text{molar}} = F_{\text{molar}} \times \left(\frac{z_F - x_B}{x_D - x_B}\right) \quad [\text{kmol/h}]

B_{\text{molar}} = F_{\text{molar}} - D_{\text{molar}} \quad [\text{kmol/h}]

D_{\text{mass}} = D_{\text{molar}} \times M_{D, \text{avg}} \quad [\text{kg/h}]

B_{\text{mass}} = B_{\text{molar}} \times M_{B, \text{avg}} \quad [\text{kg/h}]

# 3. FUG Shortcut Method Equations

Fenske Minimum Stages ( $N_{\text{min}}$ at Total Reflux):

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

Underwood Minimum Reflux Ratio ( $R_{\text{min}}$ ):
For saturated liquid feed ( $q = 1.0$ ):

R_{\text{min}} = \frac{1}{\alpha - 1} \left[ \frac{x_D}{z_F} - \frac{\alpha (1 - x_D)}{1 - z_F} \right]

Operating Reflux Ratio ( $R_{\text{actual}}$ ):

R_{\text{actual}} = R_{\text{min}} \times f_R \quad (\text{typically } f_R = 1.20 \text{ to } 1.50)

Gilliland Stage Correlation ( $N_{\text{theoretical}}$ ):

X_G = \frac{R_{\text{actual}} - R_{\text{min}}}{R_{\text{actual}} + 1}

Y_G = 0.75 \times (1 - X_G^{0.5668})

N_{\text{theoretical}} = \frac{N_{\text{min}} + Y_G}{1 - Y_G}

Actual Physical Trays:

N_{\text{actual}} = \left\lceil \frac{N_{\text{theoretical}}}{\eta_{\text{stage}} / 100} \right\rceil

Kirkbride Optimum Feed Stage ( $N_{\text{feed}}$ from top):

\log\left(\frac{N_{\text{rect}}}{N_{\text{strip}}}\right) = 0.206 \log\left[ \left(\frac{B_{\text{molar}}}{D_{\text{molar}}}\right) \left(\frac{1 - z_F}{z_F}\right) \left(\frac{x_B}{1 - x_D}\right)^2 \right]

# 4. Column Hydraulic Sizing (Souders-Brown)

Overhead Vapor Flow:

V_{\text{top, molar}} = D_{\text{molar}} \times (R_{\text{actual}} + 1) \quad [\text{kmol/h}]

V_{\text{top, mass}} = V_{\text{top, molar}} \times M_{D, \text{avg}} \quad [\text{kg/h}]

Souders-Brown Maximum Flooding Velocity ( $u_{\text{flood}}$ ):

u_{\text{flood}} = C_{\text{SB}} \sqrt{\frac{\rho_L - \rho_V}{\rho_V}} \quad [\text{m/s}]

u_{\text{design}} = 0.75 \times u_{\text{flood}}

Column Diameter ( $D_{\text{col}}$ ):

A_{\text{col}} = \frac{Q_{\text{vapor}}}{u_{\text{design}}} \quad \implies \quad D_{\text{col}} = \sqrt{\frac{4 A_{\text{col}}}{\pi}} \quad [\text{meters} \implies \text{mm}]

# 5. Industrial Binary Solvent Systems Included

Methanol / Water ( $\alpha = 2.80$ )
Ethanol / Water ( $\alpha = 2.20$ )
Acetone / Water ( $\alpha = 3.50$ )
Isopropanol (IPA) / Water ( $\alpha = 2.10$ )
Benzene / Toluene ( $\alpha = 2.40$ )
Ethyl Acetate / Ethanol ( $\alpha = 1.85$ )