Batch Distillation

Batch Distillation Column In batch distillation, a tank is charged with feed and then heated. Vapor flows overhead, is condensed and collected in a receiver. The liquid remaining in the tank is generally called the residue. The composition of the material collected in the receiver varies with time, so the composition of the product is an average of all the material collected. Often, the receiver will be emptied or switched several times during a distillation to collect separate cuts of product.

A batch process is inherently dynamic -- it cannot be modeled steady state.

Batch distillation can be conducted with or without reflux. When reflux is used, any of several different operating policies may be used -- you might use a constant reflux rate, you might vary it, etc.

Batch distillation is most common:

We want to consider two main variants -- with and without reflux.

"Differential" Distillation

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Batch distillation without reflux is often called differential distillation. Because there is no reflux, the vapor product is in equilibrium with the liquid residue in the tank at any given time.

There is one product stream, D, leaving the system, so the total material balance is:

'deriv(n,t)=_D
where n(t) is the total moles of material in the distillation vessel. For a binary system, the component balance is:
'deriv(n*x,t)=_D*x;D
This may be expanded via the chain rule
n*'deriv(x,t)+x*'deriv(n,t)=_D*x;D
combined with the total material balance
n*'deriv(x,t)+x*'deriv(n,t)=x;D*'deriv(n,t)
multiplied through by dt
n*dx+x*dn=x;d*dn
and rearranged to obtain
'quot(dx,x;D-x)='quot(dn,n)
Integrating yields
'integ('quot(1,n),n,n;0,n;1) = ln('quot(n;1,n;0) = 'integ('quot(1,y;D-x),x, x;0, x;1)
This is the Rayleigh Equation (MSH6 Eq. 21.75) which relates the amount of residue to the composition. Any of the four variables n0, n1, x0, x1 can be found, provided the others are known.

If there is no reflux, there is a single equilibrium stage, and yD is in equilibrium with xD (yD=yequilibrium(xD)).

The right hand side integral of the Rayleigh equation can usually only be evaluated for constant pressure systems. Often, it is necessary to solve the integral numerically or graphically. The latter is done by making a plot of 'quot(1, y(x)-x) vs. x and finding the area under the curve between the initial and final concentrations.

The Rayleigh equation can also be used for any two components of a multicomponent system.

Determination of the residue amounts or compositions using the Rayleigh Equation is one of the more common batch distillation calculations. Another is to determine the average overhead product composition. The composition of the overhead varies with time, so the average composition is that of a mixing tank that collects all of the distillate. It can be found from material balance to be:

x;Dmean = 'quot(n;0*x;0-n;1*x;1,n;0-n;1)

Special Case Applications of the Rayleigh Equation

Consider the case where equilibrium constants are available and the K values are relatively constant (as for close-boiling mixtures with limited delta T). Then, the equilibrium relationship is y=Kx, and

ln('quot(n;1,n;0))='quot(1,K-1)*ln('quot(x;1,x;0))

Another case of interest is that where the relative volatility is constant. Then, for a binary system:

y(x)='quot(?alpha?*x,1+x*(?alpha?-1))
which can be substituted into the Rayleigh equation and integrated to get
ln('quot(n;0,n;1))='quot(1,?alpha?-1)*(ln('quot(x;0,x;1))+?alpha?*ln('quot(1-x;1,1-x;0))
?alpha?*ln('quot(n;0,n;1))-ln('quot(n;0,n;1))=ln('quot(x;0,x;1))+?alpha?*ln('quot(1-x;1,1-x;0))
?alpha?*(ln('quot(n;0,n;1))-ln('quot(1-x;1,1-x;0)))=ln('quot(n;0,n;1)+ln('quot(x;0,x;1))
?alpha?*(ln('quot(n;0*(1-x;0),n;1*(1-x;1)))=ln('quot(n;0*x;0,n;1*x;1))
which can be useful for some calculations. Your text prefers to do a further rearrangement to get the equation in terms of the two components, A and B.
ln('quot(n;B0,n;B1)='quot(1,?alpha?)*ln('quot(n;A0,n;A1))
'quot(n;B0,n;B1)='exp('quot(n;A0,n;A1),'quot(1,?alpha))
(MSH6 Eq. 21.87).

Batch Distillation with Reflux

To use the Rayleigh equation when a column is refluxed, you generally must do one or more iterative calculations. Typically, a McCabe-Thiele diagram is used to obtain some of the information needed in the calculation.

A key factor in how you approach the problem is the operating policy chosen for the system. We will consider two such policies:

Variable Reflux Policy (Constant xD)

Variable reflux is generally more expensive to implement, since it requires composition measurements. Typically, a calculation follows

  1. For a known charge (n0, x0), fix the desired product composition xD and the heat input (vapor rate).
  2. Determine the number of ideal stages in the system -- this can be done from a total reflux construction
  3. Use McCabe-Thiele plots, and trial and error to determine the L/V ratio for the system. As the residue composition changes, the operating line will have to change -- it will rotate through the equilibrium envelope with its top end fixed at xD. You will get a fan shaped array of operating lines.
  4. The L/V ratio can be used in subsequent calculations.
Two formulas of interest (they will not be derived here) are:
n;t=n;0*'quot(x;D-x;0,x;D-x;t)
which relates the amount of residue and the various compositions at any time t during the run (xt is the composition and nt the amount of residue at time t), and (SH eq 13.16)
t='quot(n;0*(x;D-x;0),V)*'integ('quot(1,(1-'quot(L,V))*(x;D-x;R)^2,x,x;0,x;Rfinal)
to calculate run length.

Variable Composition Policy (Constant Reflux)

In other circumstances, it makes sense to operate with a constant reflux rate. In this situation, you fix the molar vapor rate to avoid flooding. The operating curves shift downward in parallel to keep the slope and number of stages constant. The batch time is given by (SH eq 13.11):

t='quot(n;0-n;1,V*(1-'quot(L,V)))='quot(R;D+1,V)*(n;0-n;1)

Examples

A set of five examples of batch distillation calculations may be downloaded as a Mathcad 6.0 file.

References:

  1. McCabe, W.L., J.C. Smith, P. Harriott, Unit Operations of Chemical Engineering, 5th Edition, McGraw-Hill, 1993, pp. 576-580.
  2. Seader, J.D. and E.J. Henley, Separation Process Principles, John Wiley, 1998, pp. 681-691.
  3. Treybal, R.E., Mass-Transfer Operations, 3rd Edition (Reissue), McGraw-Hill, 1987, p. 367-371.

R.M. Price
Original: 2/9/98
Revised: 3/18/98; 3/25/99; 2/20/2003

Copyright 1998, 1999, 2003 by R.M. Price -- All Rights Reserved