F7. Absorption Formulas

In an absorption column or tower, a gas is fed from the bottom of an absorption tower. The tower is packed with packing to increase contact time and removal efficiency. For dilute gases, the equilibrium relationship can be described by Henry' law. A typical mass balance over a column is given as

where L, and G are molar flow rates per unit cross section of the tower for liquid, and gas respectively. x and y are concentrations of absorbing constituents expressed in mole fraction. Subscript 1 represents the bottom of the tower where gas enters the column. We can find minimum amount of liquid required to result the desired removal of solute from the gas as described below. Assume that the liquid exiting from the bottom of the tower is in complete equilibrium with entering gas then the solute absorbed by liquid is equal to the solute released from gas.

Where superscript represents equilibrium condition. The quantity x₁^* is given by y₁/m, where m is Henry's law constant. The quantity of liquid used in achieving the desired absorption level is 20 % to 30 % higher than the minimum value. We can find concentration of the solute in liquid exiting the tower as

The height of the absorption tower can be calculated by a number of ways. If the height of a gas- and liquid-phase units (H_G, H_L) are provided then height of packing required is given by

Where H_G is the height of a transfer unit based upon resistance in the gas phase, H_OG is the height of a transfer unit based upon overall resistance in the gas phase, and H_ETP is the height of a unit equivalent to a theoretical plate. H_OG is given as

And H_ETP is given as

The number of units based on the overall resistance in the gas phase can be calculated as

Where A is absorption factor and is defined as A = L/(mG).

The number of units based on the gas phase resistance can be calculated as

The number of theoretical plates N_p can be calculated as

Mass Transfer Coefficients: The transfer of a solute from water to air follows the two-film theory covering transfer from (1) bulk-liquid to liquid-film, (2) liquid-film to air-film, and (3) air-film to bulk-air. The rate at which a solute is transferred from water to air, for low solubility solutes, is represented by an overall transfer rate constant, K_La which is the product of two variables; K_L, the liquid mass transfer coefficient, and a, the area-to-volume ratio of the packing. For design purposes, K_La should be determined experimentally. However, for dilute solutions, K_La can be determined from the Sherwood and Holloway equation. Mathematical correlations developed by Onda also provide values of K_La that are valid over a range of solutes.

Where a_w = wetted packing area, m^-1

a_t = total packing area, m^-1

σ = surface tension of water, N/m

σ_c = critical surface tension of packing material, N/m

Re = Reynolds number, dimensionless = L/(a_tμ_L)

Fr = Froude number, dimensionless = L²a_t/(ρ_L²g)

We = Weber number, dimensionless = L²/(ρ_L²σa_t)

L = liquid molar loading rate, kmol/(s·m²)

ρ_L = density of water, kg/m³

ρ_G = density of air, kg/m³

g = gravity, m²/s

μ_L = liquid viscosity, Pa·s

Using this value of a_w and a_t, liquid phase mass transfer coefficient can be found as

where k_L = liquid-phase mass-transfer coefficient, m/s

L = liquid mass diffusion loading rate, kg/(m²·s)

ρ_L = density of liquid, kg/m³

D_L = liquid-diffusion coefficient, m²/s

where k_G = gas-phase mass-transfer coefficient, m/s

G = gas mass diffusion loading rate, kg/(m²·s)

ρ_G = density of gas, kg/m³

D_G = gas-diffusion coefficient, m²/s

Air-stripping design considerations: In practice, stripping towers have diameters of 0.5 m to 3 m and heights of 1 m to 15 m. The ratios of air- to liquid-flow rate ranges from as low as 5 to several hundred and is controlled primarily by flooding and pressure drop considerations. As the airflow in a tower is increased, it will ultimately hold back the free downward flow of water and cause flooding of the tower. The smaller size packings are more susceptible to flooding. Channeling occurs when water flows down the tower wall rather through the packing. Distribution plates should be placed approximately every 5 diameters to avoid this. Using smaller-size packing will also reduce the tendency of flow to channel. The pressure drop in a tower should be between 0.35 in to 0.5 in water per foot to avoid flooding. The pressure drop appears to be the parameter that most directly affects the operating cost of the tower. Capital cost is a function of the tower volume. Flooding and pressure drop calculations are simplified through the use of Figure 7.2.

Steam Stripping: Steam stripping is used for the removal of volatile and sometimes semi- volatile compounds from water. The process is capable of reducing volatile organic compounds in water to very low concentrations. Both steam and air strippers are based on the transfer of organics from the liquid phase to the gas. The higher concentration of organics in a steam stripper requires more complex process design techniques than for an air stripper. The functional differences between steam stripping and air stripping are:

(1) Steam, rather than air, is used as stripping gas,

(2) Stripping gas, steam, is infinitely soluble in the liquid phase, water,

(3) Steam strippers operate at much higher temperatures than air strippers,

(4) The organics in the water are recovered as a separate liquid phase.

The bottom portion of the steam stripper is stripping wastewater in a manner similar to an air stripper. This is known as the stripping section of the column. The top section of the column, above the feed plate is known as the rectifying section. This section of the column enriches the organic content of the steam to a point where a separate organic phase can be achieved in the overhead decanter.

Steam Stripping design theory and considerations: The Henry's constant is applicable to a limited temperature range (usually 0 °C to 35 °C), to simple binary systems (usually water and a single organic), and to very dilute systems. These simplifications are not valid within a steam stripper.

The feed water to a steam stripper generally is 100 mg/L or more, and the concentration of the organics in the liquid phase increases above the feed point. Consequently, the assumption of a very dilute system is not valid in a steam-stripping column. In a steam stripper, the stripping- gas is entirely soluble in water (steam is water vapor), and the concentration of the organics at the top (rectifying) section of a steam stripper can be several percent. The operating gas-liquid conditions do significantly approach liquid-vapor equilibrium conditions. As a result, a steam stripper is designed as an equilibrium stage process; mass transfer resistances are not directly considered in this design approach.

In an air stripper, the feed water concentration is generally less than 100 mg/L, and the concentration of organics decreases in the liquid phase as the water is processed through the air stripper.

Air is so sparingly soluble in water that air is treated as a water-insoluble gas for mass transfer purposes. In addition, the ratio of the air-flow to liquid-flow is so high that there is no significant approach to equilibrium conditions between the organic concentrations in the gas phase (air) and the liquid phase (water). The design of an air stripper is consequently based on an estimation of the resistances to the mass transfer (e.g. two-film theory).

The stripping of a solute from the absorbing solution is required to regenerate the solution and to make the solution available for absorption of the solute from the gas. In a typical steam-stripping unit, the rich solution is fed from the top of the column and the steam is injected from the bottom of the tower. The lean solution leaves the stripper from the bottom of the tower and desorbed solute goes along with the steam from the top. In a typical setting, the inlet concentration of the rich solution is known. It is required to reduce the concentration of the solute to a given value. In other word, we do know the amount of solute that needs to be removed in a stripping column. Stripping steam is void of the solute, and the outlet solute concentration in the steam depends upon the quantity of the steam used.