Reference ID: MET-E57B | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The stripping-section operating line describes the material-balance relationship between the liquid and vapor phases inside the stripping section of a continuous distillation column. It is used to determine the local composition of vapor rising from a tray when the liquid composition on that tray is known, and vice-versa. Plotting this line on a McCabe-Thiele diagram allows engineers to step off the theoretical stages required to achieve the desired bottoms purity, making it a cornerstone of binary distillation design and troubleshooting.
Methodology & Formulas
Overall mole balance around the stripping section
\[
L' = V' + B
\]
where
L' is the molar liquid flow rate descending the column (kmol h-1)
V' is the molar vapor flow rate ascending the column (kmol h-1)
B is the molar bottoms product flow rate (kmol h-1)
Component mole balance for the more volatile component
\[
L' x = V' y + B x_B
\]
Rearranging gives the operating-line equation:
\[
y = \frac{L'}{V'} x - \frac{B x_B}{V'}
\]
x is the mole fraction of the more volatile component in the liquid
y is the mole fraction of the more volatile component in the vapor
xB is the mole fraction of the more volatile component in the bottoms product
Dimensionless parameters
Slope:
\[
m = \frac{L'}{V'}
\]
Intercept:
\[
c = -\frac{B x_B}{V'}
\]
Hence the operating line is compactly written as
\[
y = m x + c
\]
Typical range and validity criteria
Parameter
Condition
Remarks
xB
\(0 \le x_B \le 1\)
Mole fraction must be physically meaningful
B, L', V'
> 0
All flow rates must be positive
m = L'/V'
\(0 < m < 1\)
Slope outside this range suggests inconsistent input data or infeasible column operation
The stripping section operating line represents the mass-balance relationship between the liquid and vapor phases in the stripping column. It defines the trajectory that the operating point follows as solvent is removed from the feed, allowing engineers to predict product composition and column performance.
Derivation follows a material balance around the stripping section:
Write the total component balance: L – V = F(zF – yp)
Express liquid flow (L) and vapor flow (V) in terms of reflux ratio and boil-up ratio.
Rearrange to obtain \( y = \frac{L}{V}x + \frac{F}{V}(z_F - y_p) \).
Identify the slope (\( \frac{L}{V} \)) and intercept (\( \frac{F}{V}(z_F - y_p) \)).
This linear equation is the operating line used on an x-y diagram.
Key influences include:
Liquid-to-vapor ratio (L/V) – set by reflux and boil-up rates.
Feed composition (zF) and flow rate (F) – determine the intercept.
Desired product purity (yp) – shifts the line upward or downward.
Column pressure and temperature – affect vapor-liquid equilibrium and thus the effective L/V.
To meet a specific purity:
Increase the reflux ratio – raises the slope (L/V) and moves the line closer to the equilibrium curve.
Modify the boil-up ratio – changes the vapor flow, also affecting the slope.
Adjust feed rate or composition – alters the intercept, shifting the line vertically.
Re-evaluate the column’s stage count – a steeper line may require additional stages to reach the target.
Iterate these changes in a simulation or by hand calculations until the intersection of the operating line with the equilibrium curve yields the desired yp.
Worked Example: Stripping Section Operating Line
Scenario: A distillation column is used to separate a binary mixture. The stripping section operates with a bottom product flow rate of 100 kmol/h and a mole fraction of the more volatile component in the bottom product of 0.1. The liquid and vapor flow rates in the stripping section are 150 kmol/h and 200 kmol/h, respectively. Determine the equation of the stripping section operating line.
B = 100.0 kmol/h (bottom product flow rate)
xB = 0.1 (mole fraction of more volatile component in bottom product)
L' = 150.0 kmol/h (liquid flow rate in stripping section)
V' = 200.0 kmol/h (vapor flow rate in stripping section)
Calculate the slope of the operating line:
\[
\text{slope} = \frac{L'}{V'} = \frac{150.0}{200.0} = 0.750
\]
Determine the intercept of the operating line using the equation:
\[
\text{intercept} = -\frac{B}{V'} \cdot x_B = -\frac{100.0}{200.0} \cdot 0.1 = -0.050
\]
Write the operating line equation in the form \( y = mx + b \):
\[
y = 0.750x - 0.050
\]
Final Answer: The stripping section operating line is described by the equation \( y = 0.750x - 0.050 \).
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
