And gas and liquid molar flow rates
§ 7. Non-linear Equilibrium and the McCabe-Thiele Construction
Learning Outcomes
A linear equilibrium relationship (Henry’s Law) may be used;
The material balance is linear because the total molar flow rates of gas and liquid remain constant from stage to stage
The graphical construction is equivalent to solving simultaneously the material balance and the equilibrium relationship.
§ 7.2. A cross-flow cascade system
The intersection of the operating line with the equilibrium curve gives the exit compositions (xn, yn) from stage n
Figure 7.2. The cross-flow cascade construction on any y-x diagram, showing four stages.
As N → ∞, yN → y∞, where, y∞ is in equilibrium with the solvent feed mole fraction, xF. Thus, the solute cannot be completely removed from the gas stream, unless pure liquid solvent is used.
§ 7.3. A countercurrent cascade system (design and rating case)
$y_{\text{n-1}} = \text{\ \ }\frac{L}{G}x_{n} + \left( \text{\ y}_{N} - \frac{L}{G}x_{F} \right)$ (7.3)
This operating line has a slope of L/G and connects all the points (xn, yn-1) to the same point (xF, yN).
The equilibrium relationship between the gas (y) and liquid (x) mole fractions is known at the operating pressure and temperature and may be plotted as shown in Figure 7.4.
Figure 7.4. A McCabe-Thiele construction for a counter-current cascade of equilibrium stages.
Starting at point B, (the bottom of the gas absorption column), draw a vertical construction line, along constant x1, to the equilibrium curve to find y1, the gas and liquid streams leaving stage 1 are in equilibrium
We can use the material balance (i.e. the operating line) to find the mole fraction in the liquid, x2 entering stage 1: this is equivalent to constructing a horizontal line from the point (x1, y1) back to the operating line.
The procedure described above is an example of a design calculation to find the number of equilibrium stages for a given L, G, xF, yF and a specification of yN, plus the provision of an equilibrium curve.
Although the example in Figure 7.4 gives an integer number of stages, non-integer number of stages may also result, as shown in Figure 7.5.
In this case, the gas absorption column already exists, so that the number of stages is fixed
The purpose of the calculation is to determine the gas outlet mole fraction
Conversely, if the number of stages calculated is too many, then the value of yN should be increased.
§ 7.4. The minimum L/G
Figure 7.6. An end pinch at the minimum liquid flow rate.
There is a range of possible liquid flow rates that gives satisfactory operation of the gas absorption column
$\left( \frac{L}{G} \right)_{\text{min}} = \frac{y_{F} - y_{N}}{x_{1} \ast - x_{F}}$ (7.5)
In practice, the column must operate with L > Lmin.
However, often, there are inert species (e.g. air in the ammonia-air gas stream) that are not transferred between the counter-flowing streams and the flow rates of these species must remain constant along the cascade.
This suggests, that we could work on a solute-free basis, using mole ratios in place of mole fractions.
$\text{G'\ } = \text{\ molar\ flow\ rate\ of\ inert\ gas} = G_{n}\left( 1 - y_{n} \right) = \frac{G_{n}}{\left( 1 + Y_{n} \right)}$ (7.10)
We shall use the prime to indicate a solute-free molar flow rate and a capital X or Y to indicate a mole ratio.
y = mx (7.12)
Or in terms of mole ratios:
