ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Transients and Equilibration Time in
Continuous Distillation ARTHUR ROSE The Pennsylvania Sfafe University, University Park, Pa.
AND
CLARENCE
L. JOHNSON
AND
THEODORE J. WILLIAMS
U. S. Air Force lnsfifufe o f Technology, Wrighf-Pafferson Air Farce Base, Ohio
M
ATHEMATICAL or theoretical study of transient conditions in stagewise operations such as distillation, absorp tion, and ion exchange has been undertaken heretofore by only a few investigators (6). In addition, much of this work has been done only recently. Previously, most theoretical studies in these fields had been concerned with investigations of steady-state conditions. They discussed particularly the prediction of optimum design methods and optimum operation conditions for this state. Work done to date in determining the course of a transient operation has been concerned mainly with equilibration time and the effects of upsets in isotope separation plants (S), the equilibration time ( 1 , 5, 7 ) and operating curves (4, 10, 1 6 ) of batch distillation, and the effects of upsets in absorption and extraction equipment (6). Except for those investigations which were tied in very closely with experiments (4, 16) this early work has been handicapped by an important fact. If exact relations were derived they could not be solved by mathematical methods known at that time unless special assumptions were made as to holdup distribution (3,7 ) or in linearizing or otherwise simplifying the vapor-liquid equilibrium relations ( 1, 7 ) . However, this had the effect of severely limiting the application of the resulting solutions and only qualitative conclusions could then be drawn (6) The recent advent of readily available computing machines has offered a solution to this problem. Their great computing capacity compared t o previous hand methods makes practical more exact but much more tedious calculation methods. For example, the recent work of Rose and coworkers in the application of small digital computers to the batch distillation problem has been carried out without such simplifying assumptions and has succeeded in generally establishing the behavior of a batch distillation column under normal operating conditions ( 1 1 , 12). Other work in the application of these machines to adsorption (IS)and ion exchange (8) problems also shows promise of great success. The operation of continuous distillation has been essentially neglected in previous theoretical transients work on stagewise processes. Except for a brief mention of equilibration time in t h e books of Robinson and Gilliland (9) and Marshall and Pigford ( 7 ) little is available in published form. Recently, however, Smith and Polk (16)completed a thesis in which they made a preliminary study of this problem. Knowledge of the course of the transient state in continuous distillation is important principally in predicting the course and length of the start-up period of the column and also as an aid in the control of the column following a change in operating conditions if the column is not completely instrumented and automatically controlled. Work reported separately (14, 1 7 ) has shown that automatic control can handle any reasonable voluntary or involuntary July 1956
change in operating conditions. However, the instrumentation required is in many cases very complex and may not be available for the column in question. Thus this information would be very helpful to the operator in his work.
Mathematical model utilized to determine extent and course of transient state The transient condition calculations described were carried out both on the IBM card-programmed electronic calculator of The Pennsylvania State University and on the double unit Reeves electronic analog computer of the USAF Institute of Technology. The card-programmed calculator is a digital-type machine, while the Reeves electronic analog computer is an analog-type computer. The two machines were of comparable capacity and each could carry out the computations for the transient state for a continuous distillation column with as many as seven theoretical plates with the assumptions listed below. Results obtained on the two machines were identical. The object of this investigation was to determine the extent and course of the transient state in a distillation system and to define the effect of the common distillation variables on this condition. The calculations were accordingly kept as simple as possible-that is, all the usual simplifying assumptions common t o investigations of this type were made in this case, as follows: 1. A binary mixture with a constant relative volatility, a , will be separated. 2. The column is using perfect theoretical plates; thus the vapor stream leaving any one plate is in equilibrium with the liquid leaving that plate a t all times. Instantaneous mixing on all plates also is assumed. 3. All plates, including the still pot, have the same holdup, and this remains unchanged throughout all the calculations. This holdup is considered as liquid on the various plates. 4. Vapor holdu between plates is considered negligible. 5. Holdup, bo% vapor and liquid, in the condenser and associated lines is considered negligible. The condenser also operates as a total condenser. 6. Liquid and vapor rates are constant throughout the stripping or the enriching sections of the column, thus implying that the components of the mixture have equal heats of vaporization, that they show no heat effects upon mixing, and that the column is Operating adiabatically. 7 . No variations in feed rate are to be considered, only changes in feed composition. No controller is considered t o be present; therefore liquid and vapor hydraulic lag effects need not be considered, Ance there are no changes in column flow rates. Calculations were carried out for systems containing from one to seven plates, separating mixtures with relative volatilities, a, from 1 to 5 and for all possible combinations of initial compositions and feed compositions. Operation with the column a t some initial equilibrium state was compared to operation with the same composition of liquid on all plates. The effect of feed plate location was investigated.
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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Derivation of Equations. A continuous distillation column, whether of the bubble plate variety or a packed column, can be represented as divided into separate stages equivalent to each theoretical plate of the column, and the liquid hoIdup can be considered t o be concentrated a t these stages or plates. The column investigated and described by the assumptions can thus be represented as in Figure 1. A material balance of the more volatile component taken about each plate during the transient pcriod gives a system of simultaneous, first-order differential equations (11), The resulting set of equations, when modified according to the assumptions above, is as follows for a seven-plate column:
The problem, therefore, resolves itself t,o the solution of this resulting group of equations. However, because of the assumed vapor-liquid equilibrium relation (Equation l l ) , the equations are nonlinear, and thus their solution by the classical methods is extremely difficult if not impossible. Solution is possible, however, by means of automatic computers. As many as seven theoretical plates could be handled on either of the machines which were utilized for these calculations-the card-programmed calculator rrith one No. 941 storage unit, or the two-unit Reeves electronic analog computer. Definition of Equilibration Time. The computer presents its results as a graph of the compositions on each of the various plates versus a n arbitrary time scale. The important quantity t o be derived from the compute1 results is the “equilibration” time, or the time a t which all of the column compositions finally have attained their steady-state values. The final value is approached asymptotically, however, so that the exact determination of this quantity is very difficult if not impossible. Therefore, some other representative quantity, which can more easily be determined, must be defined and made the basis of future correlations. A “pseudo” equilibration time can be defined as that time at which all of the plate compositions finally have attained a value within 1 0 . 0 0 1 of their final values. This is the term which is meant whenever equilibration time is discussed in this article. It is realized, of course, t h a t the choice of a different degree of approach to final value or of the attainment of equilibrium on a particular plate rather than the whole column would result in a different set of equilibration times. The equilibration times which have been determined are expressed in arbitrary time units. This arbitrary time unit is defined as the amount of time required t o take off an amount of distillate product equal t o I / I O of the holdup in any one plate. This was kept fixed throughout all the calculations, since holdup remained constant a t all times
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The equilibration times derived from the computer results of this study have been plotted against each of the pertinent distillation variables in turn in order to correlate the effect of each of these variables on the distillation system-Le., the vaporliquid interactions on the various plates.
Effects of distillation variables o n equilibration time correlated
Feed Composition. Figure 2 shows the equilibration time, as defined, for several computations carried out for a series of values of feed composition. Column parameters included a constant p of 0.5 for all feed compositions. Distillate and bottoms take-off rates were maintained equal a t all times and feed was maintained on plate 4 in all cases. A mixture with a relative volatility, CY, of 5.0 was utilized. Therefore, the column was actually giving its best separation only for a feed composition of 0.5 mole fraction of more volatile component. Computations were carried out for the complete range of initial compositions of the liquid in the column betwen 0.0 and 1.0 mole fraction more volatile component and for a complete range of feed compositions between 0.0 and 1.0 mole fraction more volatile component. It was assumed in each rase that all the plates of the column had been charged with liquid of the same composition and that no separation had taken place prior to initiation of feed input and the start of the calculation. The very definite effect of feed composition on equilibration time can only be suggested by graphs of the type shown in Figure 2. However, the definite dependence of the eqiiilibration time on the proximity of the feed composition to that composition which permits the column to give its best separation is definitely estab-
Figure 1.
Diagram of column model
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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT lished. The feed composition which permits this best separation will be referred to as the critical feed composition, z j C , Initial Composition. The effect of the initial composition of the liquid in the column on the equilibration time is shown by Figure 3. This graph is essentially a replotting of the same results shown by Figure 2, except that much more information is presented in each graph. There is a radical change in the slope of the curves, depending on whether the feed composition is greater or less than the critical feed composition. Relative Volatility. Graphs for relative volatilities of 3.0, 2.0, and 1.5, are shown in Figures 4,5, and 6. The degree of the effect of feed composition on equilibration time is a very decided function of the relative volatility of the mixture being separated. For the case of a relative volatility of 1.0, equilibration time reduces to the mixing time for the liquid on the plates. The effect of relative volatility also can be shown by different types of graphs. For example, Figure 7 shows a family of curves of different feed compositions plotted onto a plane of relative volatility versus equilibration time. Similar graphs could be drawn for each of the intermediate initial compositions between 0.0 and 1.0, and they would closely resemble Figure 7 . Initial Equilibrium in Column. All the computations reported have been for t,he case in which all of the plates of the column are a t the same composition prior to initiation of feed. A study of the effects of a feed composition upset on normal column operation, however, requires that the column be a t some equilibrium state prior to the initiation of the upset. A set of computations has been carried out for this condition. It will, therefore, be as-
EFFECT OF FEEO COMPOSITION ON EQUILIBRATION TIME " 7 , R=4/l,or:5.0, F R . 4 I
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sumed that the column is at some original equilibrium such as would be caused by a feed composition equal to one of those compositions originally used to charge the plates to constant composition in the previously described calculations. A comparison of Figure 8 with Figure 3 summarizes the overall effect of the presence or the lack of initial equilibrium in the column. While the over-all trends in the two graphs are essentially the same, there is a decided difference in the actual shapes
EFFECT OF I N I T I A L AND F E E D COMPOSITION ON EQUILIBRATION TIME
EFFECT O F I N I T I A L AND F E E D COMPOSITION ON EQUILIBRATION T I M E R E L A T I V E VOLATILITY, u ,EOUALS 5 0
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EFFECT O F I N I T I A L AND FEEO COMPOSITION ON EQUILIBRATION T I M E R E L A T I V E VOLATILITY. a ,EQUALS 3 0
EFFECT O F I N I T I A L AND F E E D COMPOSITION ON EOUILIBRATION TIME R E L A T I V E VOLATILITY,W , EQUALS 3 0
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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT EFFECT O F INITIAL A N D F E E D COMPOSITION O N EQUILIBRATION TIME RELATIVE VOLATILITY, d , E Q U A L S 2 0
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