McCabe-Thiele Feed Quality Analysis

This question relates the position of the intersection point between the q-line and the vapor-liquid equilibrium (VLE) curve on a McCabe-Thiele diagram to the condition of the feed stream.

Understanding the q-line Intersection

In the McCabe-Thiele method for distillation column design:

• The VLE curve represents the equilibrium relationship between the mole fraction of the more volatile component in the liquid phase ($x$) and the vapor phase ($y$). Let this be described by the function $y = f(x)$.
• The q-line represents the feed condition. It passes through the point $(x_f, x_f)$ on the $y=x$ diagonal, where $x_f$ is the mole fraction of the feed.
• The intersection point $(x_{int}, y_{int})$ is where the q-line meets the VLE curve.
• The question states a specific condition: the x-coordinate of the intersection point is greater than the x-coordinate of the feed point, i.e., $x_{int} > x_f$.

Relating Intersection to Feed Quality (q value)

The relationship between the compositions on the q-line and the feed can be mathematically expressed. For the intersection point $(x_{int}, y_{int})$ on the VLE curve ($y_{int} = f(x_{int})$), the condition derived from the q-line properties is:

Where '$q$' is the fraction of the feed stream that is in the liquid phase.

Analyzing the Condition $x_{int} > x_f$

We analyze the equation based on the value of $q$, assuming a typical VLE curve where $f(x) > x$ for $x < 1$ (thus $f(x_{int}) - x_{int} > 0$):

• If $q > 1$ (Liquid feed): Then $q-1 > 0$. For the equation $\frac{x_{int}-x_f}{q-1} > 0$ to hold, we must have $x_{int}-x_f > 0$, which means $x_{int} > x_f$. This matches the question's condition. A feed with $q>1$ is liquid below its bubble point.
• If $0 < q < 1$ (Mixed feed): Then $q-1 < 0$. For $\frac{x_{int}-x_f}{q-1} > 0$ to hold, we must have $x_{int}-x_f < 0$, meaning $x_{int} < x_f$. This contradicts the condition.
• If $q = 1$ (Saturated liquid feed): The q-line is a vertical line $x = x_f$. Thus, $x_{int} = x_f$. This contradicts the condition $x_{int} > x_f$.
• If $q = 0$ (Saturated vapor feed): The q-line is a horizontal line $y = x_f$. The intersection is $(x_{int}, x_f)$, so $f(x_{int}) = x_f$. Since $f(x_{int}) > x_{int}$ (for $x_{int}<1$), it follows that $x_f > x_{int}$. This contradicts the condition.
• If $q < 0$ (Superheated vapor feed): Then $q-1 < 0$. For $\frac{x_{int}-x_f}{q-1} > 0$ to hold, we must have $x_{int}-x_f < 0$, meaning $x_{int} < x_f$. This contradicts the condition.

Conclusion

The condition where the x-coordinate of the intersection point $(x_{int})$ is greater than the feed point x-coordinate $(x_f)$ occurs only when $q > 1$. This corresponds to a feed that is liquid below its bubble point.