Reference ID: MET-B165 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Feed Condition, commonly referred to as the q-line, is a fundamental concept in the design and analysis of continuous binary distillation columns. In Process Engineering, the q-line represents the locus of points where the rectifying and stripping operating lines intersect on a McCabe-Thiele diagram. It defines the thermal state of the feed as it enters the column, which dictates the internal liquid and vapor flow rates within the stripping and rectifying sections.
Understanding the q-line is critical for determining the minimum reflux ratio and the number of theoretical stages required for a specific separation. It is used extensively in chemical plant design to ensure that the column operates within stable hydraulic limits while achieving the desired product purity.
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The calculation of the q-line is based on the molar energy balance around the feed stage. The parameter q is defined as the moles of saturated liquid produced per mole of feed introduced to the column. The following formulas define the relationship between the feed thermal state and the operating line geometry:
First, the liquid fraction q is determined by the vapor fraction φ of the feed:
\[ q = 1 - \phi \]
The operating line for the feed condition is expressed as a linear equation in the form y = mx + c, where the slope m and the intercept c are derived from the feed composition zF and the liquid fraction q:
\[ m = \frac{q}{q - 1} \]
\[ c = -\frac{z_{F}}{q - 1} \]
The resulting equation for the q-line is:
\[ y = \left( \frac{q}{q - 1} \right) \cdot x - \left( \frac{z_{F}}{q - 1} \right) \]
The following table summarizes the physical regimes of the feed based on the value of q:
Feed Condition
Liquid Fraction (q)
Slope (m)
Subcooled Liquid
q > 1
Positive (m > 1)
Saturated Liquid
q = 1
Undefined (Vertical Line)
Two-Phase Mixture
0 < q < 1
Negative (m < 0)
Saturated Vapor
q = 0
Zero (Horizontal Line)
Superheated Vapor
q < 0
Positive (0 < m < 1)
The q-value represents the fraction of liquid in the feed. You can calculate it based on the thermal state of the feed using the following criteria:
Subcooled liquid: q > 1
Saturated liquid: q = 1
Two-phase mixture: 0 < q < 1
Saturated vapor: q = 0
Superheated vapor: q < 0
The slope of the q-line is defined as q / (q - 1). This line represents the locus of points where the operating lines of the stripping and rectifying sections intersect. It effectively dictates how the feed condition shifts the internal liquid and vapor traffic within the column.
The feed condition directly impacts the slope of the q-line, which in turn changes the intersection point with the equilibrium curve. As the feed becomes more vaporous (lower q-value), the q-line becomes flatter, which generally leads to:
A change in the slope of the rectifying section operating line.
An increase in the minimum reflux ratio required for a given separation.
A reduction in the stripping section vapor load.
Worked Example: Feed Condition (q-line) Calculation
Scenario: A continuous binary distillation column operating at 1 atm separates an ethanol–water mixture. The feed enters the column with a mole fraction of ethanol of \(z_F = 0.4\). The feed is partially vaporized, with a measured vapor fraction of 0.5. Using the constant molar overflow assumption, we determine the q-line that will be used to locate the intersection of the rectifying and stripping operating sections on a McCabe–Thiele diagram.
Knowns:
Feed mole fraction, \(z_F = 0.4\) (dimensionless)
Vapor fraction of feed = 0.5 (dimensionless)
Step-by-step calculation:
Determine the q-factor. The q-factor is defined as the fraction of feed that is liquid on a molar basis. Since the vapor fraction is given:
\[
q = 1 - (\text{vapor fraction}) = 1.0 - 0.5 = 0.5
\]
A value of \(q = 0.5\) confirms the feed is a two-phase mixture (50% liquid, 50% vapor).
Compute the slope of the q-line. The q-line equation is:
\[
y = \frac{q}{q-1}\,x - \frac{z_F}{q-1}
\]
The slope is therefore:
\[
\text{Slope} = \frac{q}{q-1} = \frac{0.5}{0.5 - 1.0} = \frac{0.5}{-0.5} = -1.0
\]
A slope of \(-1.0\) is characteristic of a feed that is exactly half vaporized; the q-line will run parallel to the line \(y = -x\) shifted by the intercept.
Compute the intercept of the q-line. The intercept term (the constant added to the slope term) is:
\[
\text{Intercept} = -\frac{z_F}{q-1} = -\frac{0.4}{0.5 - 1.0} = -\frac{0.4}{-0.5} = 0.8
\]
Thus the full q-line equation is:
\[
y = -1.0\,x + 0.8
\]
Verify the q-line passes through the diagonal point. A fundamental property of the q-line is that it must intersect the 45° line \(y = x\) at \(x = z_F\). Substituting \(x = z_F = 0.4\):
\[
y = (-1.0 \times 0.4) + 0.8 = -0.4 + 0.8 = 0.4
\]
Since the computed \(y\) equals \(z_F = 0.4\), the line indeed passes through the point \((z_F, z_F) = (0.4, 0.4)\) on the diagonal, confirming mathematical consistency.
Final Answer:
q-factor: \(q = 0.5\) (dimensionless)
q-line slope: \(-1.0\) (dimensionless)
q-line intercept: \(0.8\) (dimensionless)
q-line equation: \(y = -1.0\,x + 0.8\)
Verification point: \((x = 0.4,\; y = 0.4)\) lies on the line.
Interpretation: With \(q = 0.5\), the feed enters as a 50% vapor, 50% liquid mixture. On the McCabe–Thiele diagram, the q-line will slope downward from the point (0.4, 0.4) and intersect the equilibrium curve, providing the junction point for the rectifying and stripping operating lines.
