The following diagram shows a staged absorption column in which \(n\) -hexane
(H) is absorbed from a gas into a heavy oil.A gas feed stream containing 5.0
mole \(\%\) hexane vapor and the balance nitrogen enters at the bottom of an
absorption column at a basis rate of \(100 \mathrm{mol} / \mathrm{s}\), and a
nonvolatile oil enters the top of the column in a ratio 2 mol oil fed/mol gas
fed. The absorber consists of a series of ideal stages (see Problem 6.66),
arranged so that gas flows upward and liquid flows downward. The liquid and
gas streams leaving each stage are in equilibrium with each other (by the
definition of an ideal stage), with compositions related by Raoult's law. The
absorber operates at an approximately constant temperature \(T\left(^{\circ}
\mathrm{C}\right)\) and \(760 \mathrm{mm}\) Hg. Of the hexane entering the
column, \(99.5 \%\) is absorbed and leaves in the liquid column effluent. At the
given conditions it may be assumed that \(\mathrm{N}_{2}\) is insoluble in the
oil and that none of the oil vaporizes.(a) Calculate the molar flow rates and
mole fractions of hexane in the gas and liquid streams leaving the column.
Then calculate the average values of the liquid and gas molar flow rates in
the column, \(\dot{n}_{\mathrm{L}}(\mathrm{mol} / \mathrm{s})\) and
\(\dot{n}_{\mathrm{G}}(\mathrm{mol} / \mathrm{s}) .\) For simplicity, in
subsequent calculations use the average values for liquid and gas molar flow
rates within the column, but the actual values for the corresponding flow
rates entering and leaving the column.(b) Considering the bottom stage to be
ideal, estimate the mole fractions of hexane in the gas leaving that stage
\(\left(y_{N}\right)\) and in the liquid entering it \(\left(x_{N-1}\right)\) if
the column temperature is \(50^{\circ} \mathrm{C}\).
(c) Suppose that \(x_{i}\) and \(y_{i}\) are the mole fractions of hexane in the
liquid and gas streams leaving stage \(i\) Derive the following equations from
an equilibrium relationship and a mass balance around a section of the column
encompassing stage \(i\) and the bottom of the
column:$$\begin{array}{c}y_{i}=x_{i} p_{i}^{*}(T) / P \\
x_{i-1}=\left(x_{N} n_{L, N}+y_{i} \dot{n}_{\mathrm{G}}-y_{N+1}
\dot{n}_{\mathrm{G}+1}\right) / \dot{n}_{\mathrm{L}}\end{array}$$Verify that
these equations yield the answers you calculated in Part (b). (d) Examine the
effect of operating temperature on the column by estimating the number of
ideal stages necessary to achieve the desired separation. In the calculations,
which will be done using a spreadsheet, take the pressure in the column to be
constant at 760 torr, but consider three different operating temperatures:
\(30^{\circ} \mathrm{C}\) \(50^{\circ} \mathrm{C},\) and \(70^{\circ} \mathrm{C} .\)
The calculations will follow a stage-to-stage strategy beginning at the bottom
of the column and repeatedly applying Equations (1) and (2) until the mole
fraction of hexane in the vapor leaving the column is less than or equal to
that calculated in Part (a). You may use APEx or the Antoine equation and
Table B.4 to estimate the hexane vapor pressure. The calculations for the case
of \(T=30^{\circ} \mathrm{C}\) illustrate how to proceed; for this case,
\(y_{N-1} < y_{1}=0.00263\) after only two stages.(e) You can see that the
number of stages required increases as the column temperature increases. In
fact, there is a maximum temperature beyond which the required separation
cannot be achieved.
At that temperature, the entering gas and leaving liquid are approximately in
equilibrium, so that \(x_{N} p^{*}(T)=y_{N+1} P .\) Use either APEx or the
Antoine equation to estimate the maximum temperature at which the separation
can be achieved.
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