# 8.5: Batch Distillation

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Distillation is a common method of separating the components of a solution. The degree of separation that can be achieved depends on the vapor-liquid equilibrium relation and the manner in which the distillation takes place. Salt and water are easily separated in solar ponds in a process that is analogous to batch distillation. In that case the separation is essentially perfect since a negligible amount of salt is present in the vapor phase leaving the pond.

In this section we wish to analyze the unit illustrated in Figure \(\PageIndex{1}\) which is sometimes referred to as a *simple still*.

The process under consideration is obviously a transient one in which the unit is initially charged with \(M_{ o}\) moles of a binary mixture containing species \(A\) and \(B\). The initial mole fraction of species \(A\) is designated by \(x_{A}^{ o}\), and we will assume that the mole fraction of species \(A\) is small enough so that the ideal solution behavior discussed in Chapter 5 (see Equation \((5.4.14)\)) provides an equilibrium relation of the form

\[y_{A} =\alpha_{AB} x_{A} \label{59}\]

Here \(y_{A}\) represents the mole fraction in the vapor phase and \(x_{A}\) represents the mole fraction in the liquid phase. In our analysis, we would like to predict the composition of the liquid during the course of the distillation process. The control volume illustrated in Figure \(\PageIndex{1}\) is fixed in space and can be separated into the volume of the liquid (the \(\beta\)-phase) and the volume of the vapor (the \(\gamma\)-phase) according to

\[\mathscr{V}=V_{\beta } (t) + V_{\gamma } (t) \label{60}\]

Under normal circumstances there will be no chemical reactions in a distillation process, and we can express the macroscopic mole balance for species \(A\) as

\[\frac{d}{dt} \int_{\mathscr{V}}c_{A} dV + \int_{\mathscr{A}}c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =0 \label{61}\]

In addition to the mole balance for species \(A\), we will need either the mole balance for species \(B\) or the total mole balance. The latter is more convenient in this particular case, and we express it as (see Sec. 4.4)

\[\frac{d}{dt} \int_{\mathscr{V}}c dV + \int_{\mathscr{A}}c \mathbf{v}^{*} \cdot \mathbf{n} dA =0 \label{62}\]

For the control volume shown in Figure \(\PageIndex{1}\), the molar flux is zero everywhere except at the exit of the unit and Equation \ref{61} takes the form

\[\frac{d}{dt} \left[\int_{\mathscr{V}_{\beta } (t)}c_{A\beta } dV + \int_{\mathscr{V}_{\gamma } (t)}c_{A\gamma } dV \right] + \int_{\mathscr{A}_{exit} }c_{A\gamma } \mathbf{v}_{A\gamma } \cdot \mathbf{n} dA =0 \label{63}\]

Here we have explicitly identified the control volume as consisting of the volume of the liquid (\(\beta\)-phase) and the volume of the vapor (\(\gamma\)-phase) At the exit of the control volume, we can ignore diffusive effects and replace \(\mathbf{v}_{A\gamma } \cdot \mathbf{n}\) with \(\mathbf{v}_{\gamma } \cdot \mathbf{n}\), and the concentration in both the liquid and vapor phases can be represented in terms of mole fractions so that Equation \ref{63} takes the form

\[\frac{d}{dt} \left[\int_{\mathscr{V}_{\beta } (t)}x_{A} c_{\beta } dV + \int_{\mathscr{V}_{\gamma } (t)}y_{A} c_{\gamma } dV \right] + \int_{\mathscr{A}_{exit} }y_{A} c_{\gamma } \mathbf{v}_{\gamma } \cdot \mathbf{n} dA =0 \label{64}\]

If the total molar concentrations, \(c_{\beta }\) and \(c_{\gamma }\), can be treated as constants, this result can be expressed as

\[\frac{d}{dt} \left(\langle x_{A} \rangle M_{\beta } \right) + \frac{d}{dt} \left(\langle y_{A} \rangle M_{\gamma } \right) + \int_{\mathscr{A}_{exit} }y_{A} c_{\gamma } \mathbf{v}_{\gamma } \cdot \mathbf{n} dA =0 \label{65}\]

in which \(\langle x_{A} \rangle\) and \(\langle y_{A} \rangle\) are defined by

\[\langle x_{A} \rangle =\frac{1}{V_{\beta } (t)} \int_{\mathscr{V}_{\beta } (t)}x_{A} dV , \quad \langle y_{A} \rangle =\frac{1}{V_{\gamma } (t)} \int_{\mathscr{V}_{\gamma } (t)}y_{A} dV \label{66}\]

In Equation \ref{65} we have used \(M_{\beta }\) and \(M_{\gamma }\) to represents the total number of moles in the liquid and vapor phases respectively. We can simplify Equation \ref{65} by imposing the restriction

\[\frac{d}{dt} \left(\langle y_{A} \rangle M_{\gamma } \right)<<\frac{d}{dt} \left(\langle x_{A} \rangle M_{\beta } \right) \label{67}\]

since \(c_{\gamma }\) is generally much, much less than \(c_{\beta }\). Given this restriction, Equation \ref{65} takes the form

\[\frac{d}{dt} \left(\langle x_{A} \rangle M_{\beta } \right) + \int_{\mathscr{A}_{exit} }y_{A} c_{\gamma } \mathbf{v}_{\gamma } \cdot \mathbf{n} dA =0 \label{68}\]

and we can express the flux at the exit in the traditional form to obtain

\[\frac{d}{dt} \left(\langle x_{A} \rangle M_{\beta } \right) + \langle y_{A} \rangle_{exit} c_{\gamma } Q_{\gamma } =0 \label{69}\]

This represents the governing equation for \(\langle x_{A} \rangle\) and it is restricted to cases for which \(c_{\gamma } <<c_{\beta }\).

In addition to \(\langle x_{A} \rangle\), there are other unknown terms in Equation \ref{69}, and the *total mole balance* will provide information about one of these. Returning to Equation \ref{62} we apply that result to the control volume illustrated in Figure \(\PageIndex{1}\) to obtain

\[\frac{d}{dt} \left[\int_{\mathscr{V}_{\beta } (t)}c_{\beta } dV + \int_{\mathscr{V}_{\gamma } (t)}c_{\gamma } dV \right] + \int_{\mathscr{A}_{exit} }c_{\gamma } \mathbf{v}_{\gamma }^{*} \cdot \mathbf{n} dA =0 \label{70}\]

At the exit of the control volume, we again ignore diffusive effects and replace \(\mathbf{v}_{\gamma }^{*} \cdot \mathbf{n}\) with \(\mathbf{v}_{\gamma } \cdot \mathbf{n}\) so that this result takes the form

\[\frac{d}{dt} \left(M_{\beta } +M_{\gamma } \right) + c_{\gamma } Q_{\gamma } =0 \label{71}\]

At this point, we again impose the restriction that \(c_{\gamma } <<c_{\beta }\) which allows us to simplify this result to the form

\[\frac{d M_{\beta } }{dt} + c_{\gamma } Q_{\gamma } =0 \label{72}\]

We can use this result to eliminate \(c_{\gamma } Q_{\gamma }\) from Equation \ref{69} so that the mole balance for species \(A\) takes the form

\[\frac{d}{dt} \left(\langle x_{A} \rangle M_{\beta } \right) - \langle y_{A} \rangle_{exit} \frac{d M_{\beta } }{dt} =0 \label{73}\]

At this point we have a single equation and three unknowns: \(\langle x_{A} \rangle\), \(M_{\beta }\) and \(\langle y_{A} \rangle_{exit}\), and our analysis has been only moderately restricted by the condition that \(c_{\gamma } <<c_{\beta }\). We have yet to make use of the equilibrium relation indicated by Equation \ref{59}, and to be very precise in the next step in our analysis we repeat that equilibrium relation according to

Equilibrium relation: \[y_{A} =\alpha_{AB} x_{A} , \quad \text{ at the vapor-liquid interface} \label{74}\]

In our macroscopic balance analysis, we are confronted with the mole fractions indicated by \(\langle x_{A} \rangle\) and \(\langle y_{A} \rangle_{exit}\), and the values of these mole fractions *at the vapor-liquid interface* illustrated in Figure \(\PageIndex{1}\) are *not available to us.* Knowledge of \(x_{A}\) and \(y_{A}\) at the \(\beta - \gamma\) interface can only be obtained by a detailed analysis of the *diffusive transport *^{9} that is responsible for the separation that occurs in batch distillation. In order to proceed with an *approximate solution* to the batch distillation process, we replace Equation \ref{74} with

Process equilibrium relation: \[\langle y_{A} \rangle_{exit} =\alpha_{eff} \langle x_{A} \rangle \label{75}\]

Here we note that Eqs. \ref{74} and \ref{75} are analogous to Eqs. \((5.6.8)\) and \((5.6.9)\) if the approximation \(\alpha_{eff} =\alpha_{AB}\) is valid. The *process equilibrium relation* suggested by Equation \ref{75} may be acceptable if the batch distillation process is *slow enough*, but we do not know what is meant by *slow enough* without a more detailed theoretical analysis or an experimental study in which theory can be compared with experiment.

Keeping in mind the uncertainty associated with Equation \ref{75}, we use Equation \ref{75} in Equation \ref{73} to obtain

\[M_{\beta } \frac{d \langle x_{A} \rangle }{dt} + (1-\alpha_{eff} ) \langle x_{A} \rangle \frac{dM_{\beta } }{dt} =0 \label{76}\]

The initial conditions for the mole fraction, \(\langle x_{A} \rangle\), and the number of moles in the still, \(M_{\beta } (t)\), are given by

I.C.1 \[\langle x_{A} \rangle =x_{A}^{ o} , \quad t=0 \label{77}\]

I.C.2 \[M_{\beta } =M_{\beta }^{ o} , \quad t=0 \label{78}\]

At this point we have a *single* differential equation and *two* unknowns, \(\langle x_{A} \rangle\) and \(M_{\beta } (t)\). Obviously we cannot determine both of these quantities as a function of time unless some additional information is given. For example, if \(M_{\beta } (t)\) were specified as a function of time we could use Equation \ref{76} to determine \(\langle x_{A} \rangle\) as a function of time; however, without the knowledge of how \(M_{\beta } (t)\) changes with time we can only determine \(\langle x_{A} \rangle\) as a function of \(M_{\beta }\). This represents a classic situation in many batch processes where one can only determine the *changes that take place between one state and another.* In this analysis the state of the system is characterized by \(\langle x_{A} \rangle\) and \(M_{\beta }\).

Returning to Equation \ref{76} we divide by \(\langle x_{A} \rangle M_{\beta }\) and multiply by \(dt\) in order to obtain

\[\frac{d\langle x_{A} \rangle }{\langle x_{A} \rangle } =- \left(1-\alpha_{eff} \right)\frac{dM_{\beta } }{M_{\beta } } \label{79}\]

Since \(\alpha_{eff}\) will generally depend on the temperature, and the temperature at which the solution boils will depend on \(\langle x_{A} \rangle\), we need to determine how \(\alpha_{eff}\) depends upon \(\langle x_{A} \rangle\) before the variables in Equation \ref{79} can be completely separated. Here we will avoid this complication and treat \(\alpha_{eff}\) as a constant so that Equation \ref{79} can be integrated leading to

\[\int_{\eta =x_{A}^{ o} }^{\eta =\langle x_{A} \rangle }\frac{d\eta }{\eta } =- \left(1-\alpha_{eff} \right) \int_{\xi =M_{\beta }^{ o} }^{\xi =M_{\beta } (t)}\frac{d\xi }{\xi } \label{80}\]

Evaluation of the integrals allows one to obtain a solution for \(\langle x_{A} \rangle\) given by

\[\langle x_{A} \rangle =x_{A}^{ o} \left[\frac{M_{\beta } (t)}{M_{\beta }^{ o} } \right]^{ \alpha_{eff} - 1} \label{81}\]

Once again, we must remember that \(\alpha_{eff}\) is a *process equilibrium relation* that will generally depend on the temperature which will change during the course of a batch distillation. Nevertheless, we can use Equation \ref{81} to provide a *qualitative indication* of how the mole fraction of the liquid phase changes during the course of a batch distillation process.

When \(\alpha_{eff}\) is greater than one (\(\alpha_{eff} >1\)) we can see from Equation \ref{75} that the vapor phase is *richer* in species \(A\) than the liquid, and Equation \ref{81} predicts a *decreasing* value of \(\langle x_{A} \rangle\) as the number of moles of liquid in the still decreases. For the case where \(\alpha_{eff}\) takes on a variety of values, we have indicated the normalized mole fraction \(\langle x_{A} \rangle /x_{A}^{ o}\) as a function of \(M_{\beta } (t)/M_{\beta }^{ o}\) in Figure \(\PageIndex{2}\). There we can see that a significant separation takes place when \(\alpha_{eff}\) is either large or small compared to one. The results presented in Figure \(\PageIndex{2}\) are certainly quite plausible; however, one must keep in mind that they are based on the *process equilibrium relation* represented by Equation \ref{75}. Whenever one is confronted with an assumption of uncertain validity, experiments should be performed, or a more comprehensive theory should be developed, or both.