Methods for estimating the height of the active section of counterflow differential contactors such as packed towers, spray towers, and falling-film absorbers are based on rate expressions representing mass transfer at a point on the gas-liquid interface and on material balances representing the changes in bulk composition in the two phases that flow past each other. The rate expressions are based on the interphase mass-transfer principles described in Sec. 5. Combination of such expressions leads to an integral expression for the number of transfer units or to equations related closely to the number of theoretical stages. The paragraphs which follow set forth convenient methods for using such equations, first in a general case and then for cases in which simplifying assumptions are valid.

Use of Mass-Transfer-Rate Expression Figure 14-5 shows a section of a packed absorption tower together with the nomenclature that will be used in developing the equations that follow. In a differential section dh, we can equate the rate at which solute is lost from the gas phase to the rate at which it is transferred through the gas phase to the interface as follows:

In Eq. (14-5), GM is the gas-phase molar velocity [kmol/(s m2)], NA is the mass-transfer flux [kmol/(s m2)], and a is the effective interfacial area (m2/m3).

When only one component is transferred, dGM = -NAadh (14-6)

Substitution of this relation into Eq. (14-5) and rearranging yield dh = — -

For this derivation we use the gas-phase rate expression

Na = kG(y — y,) and integrate over the tower to obtain hT =

Multiplying and dividing by yBM place Eq. (14-8) into the HGNG format hT =

The general expression given by Eq. (14-8) is more complex than normally is required, but it must be used when the mass-transfer coefficient varies from point to point, as may be the case when the gas is not dilute or when the gas velocity varies as the gas dissolves. The values of y, to be used in Eq. (14-8) depend on the local liquid compositionx, and on the temperature. This dependency is best represented by using the operating and equilibrium lines as discussed later.

Example 2 illustrates the use of Eq. (14-8) for scrubbing chlorine from air with aqueous caustic solution. For this case one can make the simplifying assumption that y,, the interfacial partial pressure of chlorine over the caustic solution, is zero due to the rapid and complete reaction of the chlorine after it dissolves. We note that the feed gas is not dilute.

Example 2: Packed Height Requirement Let us compute the height of packing needed to reduce the chlorine concentration of 0.537 kg/(s^m2), or 396 lb/(h'ft2), of a chlorine-air mixture containing 0.503 mole-fraction chlorine to 0.0403 mole fraction. On the basis of test data described by Sherwood and Pig-ford (Absorption and Extraction, McGraw-Hill, 1952, p. 121) the value of kGayBM at a gas velocity equal to that at the bottom of the packing is equal to 0.1175 kmol/(s'm3), or 26.4 lb'mol/(hft3). The equilibrium back pressure y, can be assumed to be negligible.

Solution. By assuming that the mass-transfer coefficient varies as the 0.8 power of the local gas mass velocity, we can derive the following relation:

71y + 29(1 — y) t 1 — y -1y + 29(1 — y J { 1 — y where 71 and 29 are the molecular weights of chlorine and air respectively. Noting that the inert-gas (air) mass velocity is given by GM = Gm(1 — y) = 5.34 x 10-3 kmol/(s>m2), or 3.94 lb'mol/(h'ft2), and introducing these expressions into the integral gives hT = 1.82.

This definite integral can be evaluated numerically by the use of Simpson's rule to obtain hT = 0.305 m (1 ft).

Use of Operating Curve Frequently, it is not possible to assume that y, = 0 as in Example 2, due to diffusional resistance in the liquid phase or to the accumulation of solute in the liquid stream. When the backpressure cannot be neglected, it is necessary to supplement the equations with a material balance representing the operating line or curve. In view of the countercurrent flows into and from the differential section of packing shown in Fig. 14-5, a steady-state material balance leads to the following equivalent relations:

Operating curve, |

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