Minimum Liquid Rate in Gas Absorption: Reading the Operating Line

The Operating Line Is Just a Mass Balance

In a countercurrent absorber, gas carrying a solute enters at the bottom and rises, while solvent enters at the top and falls, picking up solute on the way down. Write a solute balance between the top of the column and any point inside it and you get a straight relationship between the gas composition and the liquid composition passing each other at that level. Plotted on y–x axes, this is the operating line, and its slope is the liquid-to-gas flow ratio, L/G.

Strictly, that line is only straight when compositions are expressed on a solute-free basis, as mole ratios Y and X with solute-free flows L_s and G_s, because the total flows change as solute transfers. For dilute gases the difference is small and plain mole fractions work fine. Either way, the operating line connects the two ends of the column: the lean solvent and clean gas at the top, the rich solvent and feed gas at the bottom.

The Pinch and the Minimum Liquid Rate

For absorption, the operating line sits above the equilibrium curve. The vertical gap between them is the driving force for mass transfer: how much more solute the gas carries than it would in equilibrium with the liquid beside it. A big gap means fast transfer and a short column. A small gap means slow transfer and a tall one.

Now reduce the solvent flow while keeping the same gas feed and the same required recovery. The top end of the line stays fixed, but the slope L/G falls, so the line swings down toward the equilibrium curve. At some solvent rate it touches. At that point the driving force is zero, and reaching it would take an infinitely tall column. That touching point is the pinch, and the slope at which it happens is (L/G)min. For a straight equilibrium line the pinch occurs at the rich end of the column; for a curved one it can occur as a tangent somewhere in the middle, so it is worth checking the whole curve rather than just the end point.

The Absorption Factor and the Kremser Equation

For dilute systems with a straight equilibrium line y = mx, the behaviour of the whole column collapses into one dimensionless group, the absorption factor A = L/(mG). It is the ratio of the operating-line slope to the equilibrium-line slope. Kremser showed in 1930 that the number of ideal stages needed for a given recovery can then be written in closed form, an expression now called the Kremser equation, with A and the recovery as its only inputs.

The absorption factor explains a lot at a glance. If A is less than one, the operating line converges with equilibrium at the rich end, and the fraction of solute you can ever remove is capped at roughly A, no matter how many stages you add. Above one, any recovery is possible with enough stages, and the number required falls rapidly as A rises.

Choosing the Actual Solvent Rate

Running at the minimum liquid rate is not an option, so real designs use a multiple of it, commonly around 1.2 to 2 times (L/G)min, with 1.5 a frequent starting point. The choice is an economic trade. More solvent shortens the column, but every extra kilogram of solvent has to be pumped and, usually, regenerated in a stripper, which costs energy.

Colburn's 1939 analysis framed this in terms of the absorption factor and suggested that the economic optimum usually falls between about 1.25 and 2. That range still appears in textbooks as a rule of thumb. It is a sensible place to begin, but a real design checks it against solvent cost, regeneration energy, and the column's flooding limits, since a higher liquid rate also moves the column closer to flooding.

References

1. Kremser, A. (1930). Theoretical analysis of absorption process. National Petroleum News, 22(21), 42–49.
2. Colburn, A. P. (1939). The simplified calculation of diffusional processes. Transactions of the American Institute of Chemical Engineers, 35, 211–236.
3. Seader, J. D., Henley, E. J., & Roper, D. K. (2011). Separation Process Principles (3rd ed.), Chapter 6. Hoboken, NJ: Wiley.
4. Treybal, R. E. (1980). Mass-Transfer Operations (3rd ed.), Chapter 8. New York: McGraw-Hill.