## Batch Rectification

A typical batch distillation column (a batch rectifier) is shown in Figure 3. The feed mixture is charged to a large reboiler, to which a heating medium (e.g. steam) is also fed. The vapour reaching the top of the column is condensed, and partially returned as reflux. A distillate stream is withdrawn, which sequentially feeds a series of tank receivers. The column may be a staged or a packed one.

Since in practice it is much more frequently requested to operate an existing column rather than to design a new one, only the issues of modelling and operating an existing rectifier will be addressed in the following sections.

Regardless of the operating mode, the column-modelling approaches can be grouped into three main classes: short-cut modelling, approximate modelling and rigorous modelling.

Short-cut models Short-cut models usually assume that the column dynamic behaviour can be approximated as a sequence of steady states (pseudo-steady-state approximation), which in turn is true when the column and reflux drum hold-ups can be neglected compared to that of the reboiler. This assumption allows the short-cut models that have been devised for the design of continuous columns to be employed at every time step in the simulation of batch columns. When binary mixtures are considered, a graphical representation of the process can be obtained by means of classical McCabe-Thiele diagrams. For multicomponent mixtures, a modification of the Fenske-Underwood-Gilliland design approach is often employed for simulating the batch column operation.

The main advantage of these models is their speed of computation; the main limits are their poor accuracy, and the fact that most of them are limited to constant relative volatility mixtures. They are mostly employed in preliminary process design and

Figure 3 Sketch of a batch rectifier.

optimization studies, when a large number of cases need to be evaluated quickly, and a result of limited accuracy is acceptable. Rigorous results can be obtained by the subsequent use of more accurate models.

Approximate models Approximate models are used whenever a more realistic representation of the column dynamics is required. In this case, the tray and reflux drum liquid hold-ups are explicitly taken into account, even though they are usually assumed to be constant throughout the whole operation. Other usual assumptions are: boiling feed, total condensation without subcooling, perfect mixing in all parts, negligible vapour hold-ups, constant boil-up rate, constant molar overflows, ideal tray behaviour and negligible pressure drops and heat losses.

The set of equations usually employed to model a column with a separating capacity of N ideal trays is as follows:

Reboiler dH dt

Tray i dx

H -dp = L(xi +1, j — Xi, j) — V(yi, j — yi-i, j)

Top tray (tray N)

N —dt^ _ l(xD, j — XN, j) — V(yN, j — yN— 1, j)

Reflux drum dHd dt

Hd dx

Thermodynamic equilibrium in stage i y j = K, jXi; j j = 1, 2, 2, Nc — 1

Summation constraints

Bottom tray (tray 1)

dx1,

In the above equations, H, HD and HB are the tray, reflux drum and reboiler hold-ups respectively; xi, j is the mole fraction of component j in stage i; L, V and D are the liquid, vapour and distillate molar flow rates (respectively); Ki; j is the vapour/liquid equilibrium ratio of component j in stage i; and t is the time. Any suitable thermodynamic model can be used for the representation of the equilibrium ratio; deviations from the ideal tray behaviour can be taken into account using the Murphree tray efficiency concept. Due to possible wide ranges in the relative volatilities and/or large differences in tray and reboiler hold-ups, the system of differential equations is frequently stiff. As a starting point for the integration, it is often assumed that the reboiler, reflux drum and all trays are filled with the boiling liquid feed at the beginning of the operation.

Approximate models provide a fairly accurate description of the column dynamics, and are frequently used in process design and process optimization applications.

Rigorous models When a still more accurate description of the process dynamics is needed (for example, when it is required to model accurately the operation of an existing column), a rigorous model should be employed. In this case, almost all of the previous assumptions are relaxed, which considerably increases the computational load. The deter mination of the system and equipment parameters (like the Murphree tray efficiency, for example) is done by fitting some of the modelled profiles to actual plant data.

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