For the special case of absorption from lean gases with relatively large amounts of solvent, the equilibrium lines are defined for each component by the relation
Thus, the equilibrium line for each component passes through the origin with slope K', where
and K = y°/x. When the system is sufficiently dilute, K = K
The liquid-to-gas ratio is chosen on the basis of the least soluble component in the feed gas that must be absorbed completely. Each component will then have its own operating line with slope equal to LM/GM (i.e., the operating lines for the various components will be parallel).
A typical diagram for the complete absorption of pentane and heavier components is shown in Fig. 14-11. The oil used as solvent is assumed to be solute-free (i.e., X2 = 0), and the "key component," butane, was identified as that component absorbed in appreciable amounts whose equilibrium line is most nearly parallel to the operat-in^ lines (i.e., the K value for butane is approximately equal to lm/gm).
In Fig. 14-11, the composition of the gas with respect to components more volatile than butane will approach equilibrium with the liquid phase at the bottom of the tower. The gas compositions of the components less volatile (heavier) than butane will approach equilibrium with the oil entering the tower, and since X2 = 0, the components heavier than butane will be completely absorbed.
Four theoretical trays have been stepped off for the key component (butane) on Fig. 14-11, and are seen to give a recovery of 75 percent of the butane. The operating lines for the other components have been drawn with the same slope and placed so as to give approximately the same number of theoretical trays. Figure 14-11 shows that equilibrium is easily achieved in fewer than four theoretical trays and that for the heavier components nearly complete recovery is obtained in four theoretical trays. The diagram also shows that absorption of the light components takes place in the upper part of the tower, and the final recovery of the heavier components takes place in the lower section of the tower.
Algebraic Design Method for Dilute Systems The design method described above can be performed algebraically by employing the following modified version of the Kremser formula:
Yt -Y2 _ (A0)N+1 - A0 Yx - mXj _ (A0)N+ 1 - 1 (14-55)
where for dilute gas absorption A0 _ LM/mGM and m ~ K _ y"/x.
The left side of Eq. (14-55) represents the efficiency of absorption of any one component of the feed gas mixture. If the solvent is solutefree so that X2 = 0, the left side is equal to the fractional absorption of the component from the rich feed gas. When the number of theoretical trays N and the liquid and gas feed rates LM and GM have been fixed, the fractional absorption of each component may be computed directly, and the operating lines need not be placed by trial and error as in the graphical method described above.
According to Eq. (14-55), when A0 is less than unity and N is large,
Equation (14-56) may be used to estimate the fractional absorption of more volatile components when A0 of the component is greater than A0 of the key component by a factor of 3 or more.
When A0 is much larger than unity and N is large, the right side of Eq. (14-55) becomes equal to unity. This signifies that the gas will leave the top of the tower in equilibrium with the incoming oil, and when X2 _ 0, it corresponds to complete absorption of the component in question. Thus, the least volatile components may be assumed to be at equilibrium with the lean oil at the top of the tower.
When A0 = 1, the right side of Eq. (14-56) simplifies as follows:
For systems in which the absorption factor A0 for each component is not constant throughout the tower, an effective absorption factor for use in the equations just presented can be estimated by the Edmister formula
This procedure is a reasonable approximation only when no pinch points exist within the tower and when the absorption factors vary in a regular manner between the bottom and the top of the tower.
Example 8: Multicomponent Absorption, Concentrated Case
A hydrocarbon feed gas is to be treated in an existing four-theoretical-tray absorber to remove butane and heavier components. The recovery specification for the key component, butane, is 75 percent. The composition of the exit gas from the absorber and the required liquid-to-gas ratio are to be estimated. The feed-gas composition and the equilibrium K values for each component at the temperature of the (solute-free) lean oil are presented in the following table:
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