A mole balance on the solute impurity for stages from 1 to \(n\) (where \(n\) is any
integer from 1 to \(N\), and \(N\) is the total number of stages) is:
$$
x_0L + y_{n+1}V = x_nL + y_1V.
$$
Rearranging this yields the operating line for the column:
$$
y_{n+1} = \frac{L}{V}(x_n-x_0) + y_1,
$$
where \(L\) is the liquid solvent molar flow rate (Mmol/h), \(V\) is the gas molar flow rate
(Mmol/h),
\(x_0\) is the
solute mole ratio in the liquid feed to the column (stage 1), \(y_{n+1}\) is the solute mole
ratio in the gas stream entering stage \(n\), \(x_n\) is the solute mole ratio in the liquid
stream leaving stage \(n\), and \(y_1\)
is the solute mole ratio in the gas stream leaving the column from stage 1.
The equilibrium line is calculated using Henry's law:
$$
y_{n} = \frac{H}{P}x_n,
$$
where \(x_n\) and \(y_n\) are the solute mole ratios in the liquid and gas streams,
respectively, leaving stage \(n\) (these streams are in equilibrium), H(atm) is Henry's constant
at temperature T(K), and P is pressure (atm). The Henry's constant is related to temperature by:
$$
H = H^0e^{-\frac{E}{R}(\frac{1}{T}-\frac{1}{T_0})},
$$
where \(H^0\) is Henry's constant at \(T_0\) = 298 K, \(E\) is the activation energy (J/mol),
and \(R\) is the ideal gas constant (J/(mol K)).
A solute mole balance around stage \(n\) is:
$$
x_{n-1}L + y_{n+1}V = x_nL + y_nV
$$
where \(x_{n-1}\) is the solute mole ratio in the liquid stream entering stage \(n\).
Stages are counted by locating \(y_{n+1}\) on the operating line, where \(N\) is the last stage.
The corresponding \(x_N\) value on the operating line is from the mole balance:
$$
x_N = x_0 + \frac{y_{N+1} - y_1}{L/V}.
$$
A vertical line is then drawn from (\(x_N, y_{N+1}\)), to the equilibrium line. The values
(\(x_N, y_N\)) are the equilibrium mole fractions leaving stage \(N\). Next draw a horizontal
line from (\(x_N, y_N\)) to the operating line. From this point (\(x_{N-1}, y_N\)) on the
operating line, a vertical line is drawn to the equilibrium line, and this process is repeated
until \(x_1\) is reached.
References:
-
P.C. Wankat, Separation Process Engineering: Includes Mass Transfer Analysis , 3rd
ed., Upper Saddle River, NJ: Prentice Hall, 2012.
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