Continuous distillation economics: the typical “optimum” reflux ratio is selected as a multiple of the minimum reflux ratio. Over common industrial conditions, this optimum is usually in the range of __________ times the minimum value.

Introduction / Context:
In continuous binary (or multicomponent) distillation, designers rarely operate exactly at minimum reflux (which would require an infinite number of stages) or at excessively high reflux (which would demand large condenser/reboiler duties). Instead, an economic optimum is selected as a modest multiple of the minimum reflux ratio to balance capital and energy costs.

Given Data / Assumptions:

• Constant molal overflow approximation is acceptable for preliminary design.
• Minimum reflux ratio is obtained from total/partial condenser methods (Underwood, McCabe–Thiele pinch at the top).
• Equipment and utility costs have typical industrial weightings (no extreme energy pricing).

Concept / Approach:
The “optimum” reflux ratio minimizes the total annualized cost. As reflux increases, stage count decreases (smaller column height but larger utilities). As reflux decreases toward minimum, stage count and column height grow rapidly. The shallow minimum in total cost typically occurs near 1.1–1.5 times the minimum reflux ratio for many separations and relative volatilities.

Step-by-Step Solution:
Identify Rmin from design methods (pinch analysis/Underwood).Evaluate total cost vs. R: capital roughly decreases with R, utilities increase with R.Select the economic minimum, commonly at R ≈ 1.1–1.5 Rmin.

Verification / Alternative check:
Sensitivity studies on tray count versus reflux show diminishing returns beyond about 1.3–1.5 Rmin; the drop in stages is modest while energy use escalates.

Why Other Options Are Wrong:
1.6 to 2 / 2.2 to 2.6 / 2.7 to 3: These are typically above the economic sweet spot and imply unnecessary energy costs for little reduction in stages under ordinary utility and capital cost ratios.

Common Pitfalls:
Confusing “optimum” with “minimum”; assuming a single universal value without checking utility pricing and tray/packing costs.

Final Answer:
1.1 to 1.5