As mentioned before, aeration is absorption. Thus, the discussions that follow apply equally to aeration (and air stripping). More specifically, the following discussions address the sizing of absorption and stripping towers.

Absorption and stripping are reverse processes to each. Thus, discussing one is the same as discussing the other. Two design parameters required for the design of absorption towers are the cross section and the height. The cross section is a function of the mass velocity through the section. If the mass or volume flow rate is known, the cross section can be found using the equation of continuity p1 A{V j = p2 A2V 2 or AjV 1 = A2V2, if the mass density p is constant. Indices 1 and 2 refer to points in the elevation of the tower; A is the superficial area of the tower; and V is the average superficial velocity through the tower.

The plot between the concentration of the solute in the liquid phase and that in the gas phase is called the operating line. Consider an absorption operation in a tower and let G be the mole flow rate of solute-free gas phase (carrier gas) carrying solute at a concentration [Y ] mole units per unit mole of the gas phase solute-free carrier gas. The corresponding quantities for the liquid phase are L and [X], where L is the mole flow rate of solute-free liquid phase (carrier liquid) and [X] is the mole of solute per unit mole of the solute-free liquid carrier.

The mole flow rate of solute out of the gas phase must be equal to the mole flow rate of the solute into the liquid phase. Thus, if d[Y ] and d[X ] are the respective differential changes of the concentrations [Y ] and [X ],

[Y] and [X] are also respectively equal to [Y] = [y]H-[y] and [X] = [x]H-[x], where [y] is the mole fraction of solute in the gas phase and [x] is the mole fraction of the solute in the liquid phase. Thus, d[ Y] = d[ y/]/(1-[y/])2 and d[X] = d[Xf]/(1-[Xf])2. Substituting all into the equation and integrating from [x] = [x^] to [x] = [x] and from [yf] = [y^] to [y] = [yf] yield

[x^] and [y^] are the concentrations at the bottom of the tower. Note that G and L are constants. Equation (9.44) yields the concentration at any elevation in the tower; thus, it is called the equation of the operating line. Solving for [y],

Lf G( 1 - [Xf!])(1 - [xf]) + L{ 1 - [yf!])(1 - [xf 1 ]) - (1 - [yf,])(1 - [xf])}

When plots of the operating line and the equilibrium line are far apart, the driving force will be large and the rate of mass transfer will also be large. When the two lines are close to each other, the driving force will be small, and accordingly, the rate of mass transfer will also be small. The limiting condition is reached when the two lines touch or intersect each other. The liquid flow rate (or the gas flow rate) corresponding to this condition is the minimum flow rate. In absorption, a multiplying factor of 1.5 to 2 is applied to the minimum liquid flow rate to get the actual liquid flow rate in the column; in stripping, this range is applied to the minimum gas flow rate to get the actual gas flow rate in the column.

Let us first derive the units of the overall mass transfer coefficients when the concentration units used are in mole fractions. Let the overall mass transfer coefficient for the gas side be Kyf and that for the liquid side be Kxf. Gd[Y ] is mole of solute flowing per unit time. Mass transfer is a process where mass crosses an area perpendicular to the direction of motion of the solute particles. This area is the contact area for mass transfer. Let the differential area be designated as dA. Thus, in terms of mass transfer, Gd[Y ] is equal to Kyf([ yf] - [ y f ])dA. From this expression, the dimensions of Kyf are mole per unit time per unit mole fraction per unit square area or M/t/mole fraction • L2. In an analogous manner, Ld[X] is equal to Kxf ([ xf ] -[Xf])dA; the dimensions of Kxf are similarly M/t /mole fraction • L2. The asterisks used as "exponent" denotes equilibrium concentrations.

Because the objective is to design the height of the tower, express dA in terms of height, along with other parameters. Call the superficial area as S, the height as Z, and the interfacial contact area per unit bulk volume of tower (the destination medium) as a. Thus, dA = aSdZ. Also, calling G' as the gas flow rate (mixture of carrier gas and solute), G is equal to G'(1 - [yf]); calling L' as the liquid flow rate (mixture of carrier liquid and solute), L is equal to L'(1 - [Xf]). As found before [Y] = [yf]/1 - [Xf] and [X] = [Xf]/1 - [x]. Performing the necessary substitutions, the mass transfer expressions become, for the gas side and liquid side, respectively,

where VMy is called the gas side molar mass velocity and VMx is called the liquid side molar velocity. Integrating the equations and simplifying produce the formulas for tower height based on the gas and liquid sides, respectively:

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