V. N. SCHRODT
UNSTEADY-STATE PROCESSING Benejts can be obtained by deliberately operating a process in the unsteady state.
Increases in
eficiency and capacity of columns and jattening
o f velocity profiles in packed beds are some o f the aduantages ouer conuentional steady-state processing.
Examples are given from work in distil-
lation, extraction, crystallization, ion exchange, and reaction engineering nsteady-state processing promises to be an interesting of investigation in the future. Sufficient studies, both experimental and mathematical, have now been done to show that benefits can accrue from imposing a repetitive discontinuity on an otherwise continuous operation. I t is a fact that continuous operation has been viewed as the ultimate in processing concepts, that it is somehow technically better than batch processing. I n the schools, continuous processing is taught as though it were a goal toward which every good chemical engineer should strive, and in our chemical factories the operators are taught to try to keep all the lines on their strip charts straight. These viewpoints are unfortunate since there can be some definite technical disadvantages with continuous operation as compared to batch processing. The big advantage of continuous processing is that in certain instances, most notable in the petroleum industry, there is economic gain. The main technical disadvantage is the decreased reaction yield which results in a need to recycle, thus imposing a requirement for separation processes that might not otherwise be needed. Continuous processes, in short, may become more complex. Unsteady-state processing can combine the economic advantage of continuous operation, if such advantage exists, with the technical advantages of batch operation, and unsteady-state processing has some inherent advantages of its own. I t is not to be construed as any kind of panacea. The use of the unsteady state needs to be evaluated in the laboratory and pilot plant for each
U area
58
INDUSTRIAL A N D ENGINEERING CHEMISTRY
specific application Then an economic evaluation has to be made. The point here is that the unsteady-state concept is another processing tool which can be added to those already available. There is a very real advantage to thinking about a process in this way because it is so unconventional. The operation is generally conducted over a range of variables instead of taking place at some fixed set of conditions and this often will reveal aspects of the process which are useful. It is illuminating to look at many of our surroundings and observe just what things are really unsteady-state operations. Virtually all the phenomena of nature involve repetitive sequences of operations. A person’s bodily processes are intermittent. A jet of vapor issuing into the liquid from a hole in a distillation tray can be shown, by high speed photography, to consist of a series of discrete bubbles. Your automobile is driven by a controlled, repetitive sequence of explosions acting through an engine. This list, o€ course, could be continued, but the important item to observe is that very many of our commonplace surroundings have the unsteady state as an integral part of their operation. Fundamental Concept-Change
with l i m e
I t is necessary for the purpose of this article to define exactly what is meant and what is excluded by the term “unsteady-state processing.” Unsteady-state processing is defined here as any operation which is disturbed in time by a controlled repeating series of upsets. A simple way to visualize the process is by means of the ubiquitous “black box.” Any process, either batch or continuous, can be characterized by a set of inputs and outputs and by a definition of the process along with its limits.
INPUTS ___.*
-
OUTPUTS
The process may be a distillation, and for this case the black box is the column, the inputs are the feed stream and heat load, and the outputs are the overhead and bottoms streams. The process could be a chemical reaction. For this operation the box is a reactor, the input is the feed or the thermal load, and the output is the product yield. It is not necessary to have two phases in a reacting system.
At this point our description is for any conventional process. To transfer from conventional to unsteadyprocess operation within the framework of our terminology, it is necessary to superimpose a repeating series of upets on the box. To do this a controller is installed on the input side so that the process inputs are alternately turned on and off for definite time intervals. The outputs may or may not be similarly turned on and off. Using this control scheme, we get the following operation :
The process is now operating in the unsteady state and the outputs vary with time during the on-time interval If the overall operation is a batch process then the inputs will vary with time, which would not be the case if the process were continuous in the overall sense. The important concept is time and the use that is made of it. The success of unsteady-state processing depends on overloading the process for short time periods-these being the on-time intervals. Almost any device or process can be overloaded without breakdown if the overload is only applied for a short time. I t is convenient to remember that a large overload can be applied for a short time, a more modest overload for a somewhat longer time, and no overload for an infinite time. The analogy here is that when a distillation column becomes floodedit does not do so instantaneously hut rather over a finite time period. In this article only those operations involving an onand-off sequence will be considered. The variance with time of an input which is not a square wave will not be considered. Sine wave-type variance has been applied to extraction columns and has been studied in detail. Applicolions lo Various Opemtions-Elhcts
Unsteady-state processing has been applied to a variety of chemical operations and to some nonchemical ones as well. Virtually all of the chemical engineering unit operations have been investigated, both experimentally and mathematically. The same is true for a number of reactor systems. Most of the work has been on what could be called a pilot plant scale. Cannon, in 1956, presented the principle of cycled
operation as it applies to staged separation processes and guided much of the early experimental work (2). Kiessling, apparently independently, published similar work along with an analysis of the operation in 1961 (70). In his studies he deduced all of the effects of stage efficiency, equilibrium relations, and operating parameters with which we are familiar at present (77, 72). Since then several investigators have discussed the effects of periodic operation on the efficiency of a staged operation. Chien, Sommerfeld, Schrodt, and Parisot presented an analytical solution of the equations for periodic
Figurc 1. Dirtillation column mrrmgcnantr
operation and derived asymptotic expressions for stage efficiencies (4). Horn assumed a column with an infinite number of stages and derived asymptotic efficiency relations similar to those obtained by Chien et al. (8). The various efforts concerned with simulation will be discussed later in this paper. The effects of unsteady-state processing are of primary concern. Those process parameters changed by unsteady operation are the capacity and the efficiency, while in packed beds an important change occurs in the flow pattern. In general, the capacity and efficiency of any operation may be improved by unsteady operation. In practical terms, more pounds of a better quality product may be produced by a given piece of equipment over a given time period. Conversely, a smaller operating unit will suffice to deliver the same quality and quantity of material that a larger unit will deliver when operating conventionally. The thermodynamics of an operation are not affected-Le., heat loads will be the same per unit of material produced except as process changes dictate. VOL 59
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It is difficult to talk about unsteady-state operation without describing specific systems. There are a number of distinct similarities and differences that can only be brought out by detailed discussion of just how a piece of equipment or a process functions in the unsteady state. Some of the processes studied in the on-off time domain are distillation, extraction, absorption, adsorption, ion exchange, crystallization, electrolysis, heat transfer, and chemical reaction. Distillation is the most common and widely used of the available separation methods and probably has been m a t completely studied in the unsteady state. Figure 1 illustrates the arrangement of Patch and continuous distillation columns for unsteadykitate operations. The construction of the trays for these columns is simple; there are no downcomen, wein, or other devices. They are just flat sheets of metal with holes which amount to about 20% of the cross-sectional area. The only column modification necessary for unsteady-state operation is the provision of on-off valves in the vapor lines to the column. When the valve is open, vapor flows up the column and there is no net downward flow of liquid. The liquid during this vapor flow period merely sits on the tray as the vapor blows through it. The vapor flow period time is that time necessary to accumulate the desired amount of product and enough liquid to replenish the top tray at the flow rate being used. When the valve is turned off, there is no vapor flow so that no force holds the liquid on the tray and the liquid then drops to the tray below. The optimum time for this liquid flow period is that time necessary for the liquid to drop one tray unit. One observed effect is an increase in the overall column efficiency which can be as large as 100%. This means a 10-plate column might have the separating power of a 20-plate column. This improvement in efficiency (not capacity) is for an ideal set of conditions which will not always be realized. The improvement depends on the tray efficiency, reflux ratio, equilibrium relation, and the amount of tray liquid transferred during the liquid flow period (78, 22). Simultaneously, and in addition to the efficiency improvement, there can be an increase in the capacity of the column. In some instances three-fold increases in capacity have been observed (75). These observed effects can be used for purposes other than just decreasing column size. Often the operating costs are more important than the capital cost and, since the efficiency has been increased, it would be possible to decrease the reflux ratio instead of cutting down the number of plates, thus lessening the cooling load on the condenser and the heat load on the reboiler. Packed distillation columns can be similarly operated but no efficiency increase is observed (79). The same kind of capacity increases are observed, however. There is one problem, as yet unsolved, in the distillation area, and that problem is derived from the existence of a transient pressure pulse up the column at the initiation of the liquid flow period. This pressure pulse interferes with the liquid flow in plate columns so that the lower trays drain to dryness before the upper trays begin to 60
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Figura 2. Extroctim column
+
INlERMlllENl FLOW
ROW
S M D Y SINE
Figure 3. Packed bcd vcIm't~pojlcs
r" ,............
.......
~~
~~
.....
0
14-
drain at all. It is possible to operate 10-plate columns quite satisfactorily but as more plates are added the operation becomes marginal (20). This phenomenon is observed only in staged operations with a compressible phase. It may be possible to construct a manifold so that the pressure wave could be destroyed, thus allowing liquid to drop uniformly from each tray. This has been done by Robinson for a small 5-plate column with low freearea trays and was quite successful (77). The pressure wave problem increases in magnitude with the number of trays so that the problem cannot be considered solved. This same problem does not exist for extraction and an extractor G inches in diameter with 40 trays has been built and operated (7). Extraction is an operation where there are other advantages to unsteady-state operation besides the capacity and efficiency increases. Figure 2 shows the assembly of an extractor for unsteady-state operation. The plates in this extractor are again perforated sheets of metal, the hole size is 0.040 inch, typically, and the per cent hole area is 7.5. It has been shown that the extractors scale up directly. Capacity increases linearly with increasing cross-sectional area and the stage efficiency remains constant (7). In the operation of this extractor there are ordinarily four distinct flow periods in place of the usual two for distillation. These periods are a heavy-phase flow period, a coalescence period, a light-phase flow period, and a coalescence period, followed by repetition of the cycle. The feed stream may be added during either flow period or it may be added independently as an additional part of the cycle. Observed capacity increases have ranged from two to 10 times over that obtainable in conventional extraction units. In addition efficiency increases of the same order of magnitude, up to loo%, may be observed as was the case for distillation. There is an additional advantage in employing the unsteady state in extraction processesthe large degree of control that can be extended over the mixing (flow) and coalescence periods. The flow periods may involve short times and large flows, resulting in vigorous agitation which is especially appropriate for phases which separate easily. For systems which have a tendency to form emulsions, a longer time and a lower flow rate promotes more gentle mixing and reduces the tendency to emulsify the system. Obviously the coalescence periods must be sufficiently long to allow the phases to separate. Probably the main disadvantage with unsteady-state extraction is the need to use trays with small holes to keep two phases between plates. If larger holes are used and significant axial mixing takes place, most of the improvement due to cycled operation will be lost. In many, if not most, of the larger scale industrial appli-
AUTHOR V. N . Schrodt is a Research Specialist in the Agricultural Division of the Monsanto Co., St. Louis, Mo. H e presided at the 1@EC Division Symposium on Cyclic Processing at the 151st ACS Meeting held in Pittsburgh in March 1966.
cations, there are quantities of dirt that would probably eause the trays nearest the feed lines to plug so that a filtering system would be necessary to prevent this. There is also a problem with just how to support the thin perforated trays in the larger diameters but this is less serious than the fouling problem. One problem which can be nicely solved by the use of interface controls at the ends of the column is a change in the phase volumes along the column. This change must be compensated or it will badly upset the column operation. These controls are not shown on Figure 2. One important item in connection with distillation and extraction is the kind of valves to use. Ordinary ball valves are good for the continual on-off service. They hold up well and can be equipped with various kinds of actuators for automatic operation. One type of actuator that is satisfactory is an air cylinder, which should be oversized to give quick open and shut action. It was stated previously that the use of unsteady-state operation in packed beds causes a change in the flow pattern and others have capitalized on this flow phenomenon in the areas of crystallization and ion exchange, The velocity profile that results from unsteady operation is compared with the conventional steady flow velocity profile in Figure 3. This is a rather surprising development but one which has some definite uses. The very flat profile has been used in crystallizer beds as a means of removing the motor liquor adhering to crystal surfaces (14). I t has been used in resin beds to decrease greatly the amount of resin required for a given ion exchange process (23). In the application to crystallization, it was demonstrated that this flat velocity profile could be used to increase p-xylene purities from 85 to >99% by simply using pure liquor to wash off the impure mother liquor. A schematic diagram of the crystallizer unit is shown in Figure 4 and it has been demonstrated that laboratory and pilot plant units can be directly scaled up. In this device part of the crystal bed is melted and, as the piston descends, a flow of liquid in the upward direction takes place. Because of the flat velocity profile the mother liquor is washed off the crystals by pure liquor and no product is lost. On the return cycle when slurry is added to the bed, the mother liquor moves downward pushing pure melt out of the crystallizer and the process is reversed. The application of this flow phenomenon to the purification of seawater is an immediate end use. One of the serious problems with seawater purification by freezing is the fact that ice crystals have a high surface-to-volume ratio which causes a large amount of mother liquor to adhere to each crystal relative to the amount of pure water in the crystal. A further application of the flat velocity profile obtained by unsteady-state operation has been made in the ion exchange field. The application of the unsteady state has been shown to reduce resin volume requirements to 0.5 to 10% of the amount needed for conventional exchanger operation. Figure 5 shows an unsteady-state exchanger compared with a conventional exchanger. The conventional multicolumn ion exVOL. 5 9
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change plant gives way to a single pancake unit which needs 0.5 to 10% of the resin used in the multicolumn arrangement. The ion exchange application capitalizes on the flat velocity profile to prevent mixing of the various streams and, in addition, takes advantage of the high m a s transfer rates that exist at the beginning of the transfer periods. The exchange resin is only loaded to the extent of 5% by the incoming feed so that the transfer essentially takes place at the maximum rate possible. Then the product is removed and regeneration takes place. There is no reason why this same phenomenon could not be capitalized on in such areas as adsorption and leaching since the same general principles apply to these operations. I n fact, the arrangement could be set up so that a countercurrent staged process would be operated in the unsteady state thus taking maximum advantage of the overall operation from both a capacity and a c i e n c y standpoint. There are no immediate obvious disadvantages in the use of the unsteady state to gain the flat velocity profile. It is a situation with a definite advantage. The possibilities for application of the unsteady state are limited only by the imagination of the user. The application generally leads to improvements and a better understanding of the operation which is being conducted. One side-light for which there is spme limited p m f along with some intuition is the fact that transfer rata must at some early stage be very much higher than the averaged rates which are ordinarily measured. In extraction, for instance, some 80% or more of the transfer that does occur can take place while a drop is being formed and this formation is a very rapid process. The use of the unsteady state limits the transfer to just such pericds and this is an area which deserves further study. The whole field of chemical reactor technology is probably the place where most of the significant gains will be made but as yet it has received little attention. Some researchers have investigated packed columns in the laboratory and others have made theoretical studies of stirred-tank and packed-bed reactors. Figure 6 illustrates the latter two types of reactor arranged for unsteady-state operation. Recent calculations using model systems have revealed that it is entirely p i b l e to increase the conversion of raw materials to products in stirred-tank reactors by intermittently feeding reactants or by raising and lowering the reactor temperature or by some combination of feed and temperature upsets. Mathematical analyses of periodic reactor operation have been published by Douglas for stirred-tank reactors and by Gore for fixed-bed reactors (5, 7). The results obtained depend significantly on the kinetics which are assumed in formulating the problem. The results so far indicate that the more nonlinear a system is the greater will be the gain from unsteady-state operation (6). Horn and Lin have made a significant contribution to the theory and optimization of periodic operation of reactors (9). They show the similarity between recycle and periodic 61
INDUSTRIAL A N D ENGINEERING CHEMISTRY
MOMITK ‘VU#
Figure 6. Raoctas
processes, present a numerical method for the solution of periodic problems, and develop a method to determine the optimum control policy for periodic operation. There has been no experimental work on this subject to date so no definitive conclusions may be drawn. Reaction kinetics and the behavior of chemical reactors are still very much applied science so this is a ripe field for a good experimentalist. The application of unsteady-state operation to fixedbed reactors is a promising area. Early laboratory wotk has been done using gas chromatographic equipment in which the column is constructed so that reactions take place and separation occurs simultaneously. A typical example is the use of such a scheme to produce butadiene by the dehydrogenation of butenes (27). The reactor is operated by pulsing reactant butenes into a diluent stream; separation of reactant and products occurs in the reactor to the point where conversion to butadiene far in excess of equilibrium is achieved. In fact, reasonable conversions can be achieved at temperatures so
low that under steady flow conditions no measurable amount of butadiene would be formed because of the equilibrium. The use of a chromatograph as a reactor has become more promising within the past 2 yean since at least two companies have claimed the ability to produce 4-inchdiameter chromatograph columns with good separating power. One of these companies claims to be developing a I-foot-diameter unit and has a 4-foot-diameter column in design (3). This would open up a new area of reactor technology since it would provide an ability to manufacture material that might otherwise be difficult or impossible to produce because of temperature or pressure restrictions and the existence of unfavorable equilibriums. A mathematical study has indicated that with favorable conditions it would even be feasible to apply the concept to the manufacture of commodity items because of their volume and relatively low cost (7). This area is one which has received considerable experimental study, much of which is reported in the foreign literature. It is a field where the existing technology for laboratory study is very good. Available gas chromatographs are excellent instruments and are reasonably cheap so that quite detailed and complete experimental studies can be done. The application awaits only the further development of the large scale chromatography units. Application of Uns-y
Slab-Practical
Aspects
Before addressing the theory of the unsteady state as it applies to the separations processes of chemical engineering, it is necessary to consider some of the more practical aspects that arise when the unsteady state is being considered at the plant design stage. In applying the unsteady state, particular attention must be paid to coordinating the various units that make up an operating plant. It is necessary to provide surge capacity so that the performance of an individual piece of equipment is not tied too tightly to the operation of another part of the plant. This is always done in conventional plants generally between major parts of a process. With unsteady operation it would probably be necessary to provide surge capacity between two distillation columns operating in seriesjust to avoid synchronization problems.
wFiguru 7. I*lorion bcfcucnload and time
Present control and instrumentation layouts have two deficiencies which render them inadequate for control and sensing of unsteady-state operation. Most control schemes are programmed to damp out fluctuations, and the response of measuring units is too slow. For the operation of an unsteady-state unit two readouts are ‘necessary, one to indicate the instantaneous values and another to give an averaged reading. The control mechanism must operate so as to maintain both the repeating pattern of the instantaneous values and the average thereof. Link bdweon Labomlory and Applicdion-Theory
It has not been possible to treat mathematically the increased capacity which is observed during unsteadystate operation of staged equipment. The increase can be rationalized by observing the overload-time relationship as shown by Figure 7. What has been observed is that very high rates are possible for short time periods and the reset period is quite short, so that the average rate is greater than for conventional operation. The parameters must be established experimentally. It has been possible to ascertain the reasons for the increased column efficiency. Simulations of the unsteady-state operation of distillation columns, both packed and plate types, and of plate-type absorption columns have revealed that the stage efficienciesof platetype units are considerably higher in the unsteady-state mode of operation. Although there is also a capacity increase for paeked columns there is no efficiency increase. This agreed with experiment (79). The separating ability of an unsteady-state column may be related to that of a conventional column by the construction of an analogy between unsteady-state distillation and conventional distillation with transvene concentration gradients across the plates of the column. The analogy involves replacement of distance as the independent variable in the conventional case with time in the unsteady-state case. The beneficial effects of transvene concentration gradients on the separating ability of a distillation column were shown by Lewis (73). A significant benefit of this analogy is that the theoretical performance of a controlled cycling column may be accurately predicted from that of its conventional counterpart. Because of the existence of this analogy it is p s i b l e to make conventional equilibrium and efficiency determinations and then convert the results to unsteady-state operation to determine stage requirements. This will be explained later. The following will present a numerical technique for the simulation of the operation of plate-type separations equipment in the unsteady state. Although presented for distillation it is applicable to any separation process when mass transfer may be neglected during one flow period (75, 78,22). For operations where there is transfer during both flow periods additional equations are necessary. Dynamic material-balance equations for the liquidflow and vapor-flow periods of the unsteady-state distillation of a binary mixture in a rectification s t i l l and the assumptions underlying them are given. As is the V O L 5 9 NO. 6 JUNE 1 9 6 7
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case for conventional distillation, the various assumptions may be removed by appropriate changes. For the ith plate of the column (reboiler = stage 1) during the vapor-flow portion of the operation we have:
where Vis the constant molar boilup rate, H i s the liquidphase molar holdup on the plate (assumed to be the same and constant for all plates in the column), xi is the liquid-phase mole fraction of the more volatile component on the ith plate, and y t is the mole fraction of the more volatile component in the vapor leaving the ith plate. The definition of the Murphree (vapor) point efficiency is :
Optimum performance is obtained when $I = 1.0. It will be noted that there is no change in the condenser composition during the liquid-flow period. McWhirter was able to show (75, 76) that the condition of plug flow during the liquid-flow period was obtained in columns as large as 6 inches in diameter. His work was done using packed plate columns and it has been shown that columns with plates of large free area perform similarly to packed plate columns (20). The length of the vapor-flow period, r , is determined by the variables, plate holdup H, the reflux ratio R, fraction of a plate holdup dropped during the liquid-flow period 4, and the boilup rate, V. In conventional distillation the reflux ratio is L R=-=-
D
where yt* is the composition of the vapor that would be in equilibrium with the liquid on the ith plate. I t is calculated from the particular vapor-liquid relationship assumed-e.g., the van Laar relationship. The Murphree point efficiency, E, is assumed to be the same and equal for all plates in the column and is equated to the overall plate efficiency at any instant of time. Equations 3 and 4 are for the special cases of the reboiler and condenser, respectively.
L v-L
where D and L are the distillate and reflux rates, respectively, expressed as amounts of material per unit time. In unsteady-state distillation, the reflux ratio is given in terms of one complete cycle. The total boilup in one period is Vr, and the amount of liquid reflux at the end of the period is 4H. Therefore, the reflux'ratio is:
R=
+H VT - q5H
(9)
so :
(3)
(4) where P and C are liquid-phase holdups of the reboiler and condenser at time t, and I is the number of actual plates in the column. These equations describe the dynamics during the vapor-flow period. In Equation 3 the reboiler is assumed to be equivalent to one theoretical stage. I t is assumed that no mass transfer occurs during the liquid-flow period of the cycle and that constant and equal amounts of liquid flow from each stage in plugflow fashion. Including these assumptions, the material-balance equations for the liquid-flow period are :
where the superscripts I and v refer to conditions at the ends of the liquid-flow and vapor-flow parts of the cycle, and 4, expressed as a fraction of a plate holdup, is the amount of liquid that flows during the liquid-flow period. For the reboiler and condenser during the liquid-flow period
x,z = x,' (7) where Pzis the holdup in the reboiler at the end of the liquid-flow period. The conditions denoted by the superscript 1 are also the initial conditions for the vaporflow part of the next cycle. Equations 5 and 6 are valid only for 0 5 4 5 1. For larger values of 4, decreased efficiencies are observed. 64
INDUSTRIAL A N D ENGINEERING CHEMISTRY
If values of V, H, and r are given, then specification of 4 defines R. At total reflux ( R = m ) :
The duration of the liquid-flow period is determined by the hydrodynamics of a real column, and this period is not relevant from a purely mathematical point of view. To show the effect of mixing during the liquid-flow period on the separating ability of an unsteady-state column, Equations 5 and 6 are replaced by differential equations which describe the dynamics of the plates and the reboiler for complete mixing during the liquid-flow period. These new equations are
H
dx
-'dt
=:
L x , + ~-
where L is the liquid molar flow rate during the liquidflow period, which at total reflux is equal to V if T ~ the , duration of the liquid-flow period, is equal to the duration of the vapor-flow period, r . I t was observed that complete mixing during this period negates the separation advantages to be gained by unsteady-state operation and results in essentially conventional operation. Close approach to liquid plug flow down the column is necessary in unsteady-state distillation. The equations describing the column operation are
conveniently solved by rewriting them in finite difference form and using standard numerical techniques on a large, fast digital computer. The analogy mentioned previously is of value in the design of distillation columns to operate in the unsteady state. I n conventional design, some estimate of either individual plate efficiencies by the use of an Oldershaw column or of overall column efficiency has to be made. Which efficiency to use is immaterial as the two can always be related to one another. I n the case of a binary separation, one may use a McCabe-Thiele diagram to determine the required number of theoretical stages and divide by the overall column efficiency to obtain the necessary number of actual plates. If we assume that some reasonable estimate of the overall column efficiency under conventional conditions can be made, this quantity can then be related to individual plate efficiencies. By use of the analogy, expected values for plate efficiencies under unsteady-state conditions may be computed and a new overall column efficiency calculated. The new overall plate efficiency, E,, is calculated from : =
[-+-]ln+ 1 1
E where
+-I
NOMENCLATURE
c = liquid phase holdup in condenser, moles D = distillate rate, moles/unit time
E = Murphree vapor point efficiency, defined by Equation 2 E, = effective plate efficiency H = liquid phase holdup on a plate, moles
z =
number of plates in column plate number L = reflux rate, moles/unit time P = liquid phase holdup in still-pot, moles R = reflux ratio, moles liquid/moles vapor t = time At = time interval boilup rate, moles/unit time x = mole fraction of more volatile component in liquid phase Y = mole fraction of more volatile component in vapor phase y * = equilibrium vapor composition x = ratio of slopes of equilibrium and operating lines + = flow during liquid-flow period as fraction of plate holdup i - = length of vapor-flow period i-L, = length of liquid-flow period $ = defined by Equation 15 t
=
v =
(14)
+=l+EE,(X-l)
and X is the ratio of the slopes of the equilibrium and operating lines. This relation is for the case where 4 = 1.0, which is for the preferred method of operation. Complex multicomponent distillations are normally handled today by means of generalized digital computer programs. Most of these have provisions for plate efficiencies other than unity and, hence, values of these under unsteady-state conditions, as estimated using the analogy, may be used directly (78, 22). I n addition, this analogy is valid for many other types of countercurrent operation-e.g., absorption, stripping, and extraction, which are carried out in a plate column with transfer during one flow period. Often these latter operations are characterized by poor efficiencies under conventional conditions where, because of the strong dependence of E , on E, only small improvements in efficiency may be realized through unsteadystate operation. Nonetheless, the principle remains valid. I t is possible to write out the equations governing virtually any process, to convert them to finite difference equations, and then to solve these by numerical methods. The wide availability of digital computers makes this a relatively easy task. Furthermore, the use of hybrid units should be mentioned since these, by their very nature, are quite useful for unsteady-state processing problems. One serious drawback at present is the lack of knowledge of the hydrodynamics which makes it impossible to predict capacity in the unsteady state. State of the Art-More
tions as regards process efficiency is quite well worked out and has been treated by different techniqws all leading to the same result, namely that 100% efficiency increases may be expected for ideal cases. The mathematical techniques necessary are available and the agreement with experiment is good. The same may be. said for the reactor work except that little experimental work is available for comparison with the theory. I n none of the areas where unsteady-state operation may be applied is it possible to make predictions governing flow behavior. This is an area which can benefit from some research effort.
Work Is Needed
I t is apparent that unsteady-state processing is in an early development state. The theory of staged opera-
SUBSCRIPTS 1 = value for plate 1 (reboiler) 2 = value for plate 2
i
= value for condenser = value for ith plate
Z
= value for Zth plate
c
SUPERSCRIPTS I = value for end of liquid-flow period v
= value for end of vapor-flow period
LITERATURE C I T E D (1) Belter, P. A., Speaker, S. M., IND.END. CHEM.PROC.DESIGNDEVELOP.6, 36 (1 967). ( 2 ) Cannon, M . R., Oil Gas J. 58, 68 (1956). (3) Chem. Eng. News, 52 (May 23, 1966). (4) Chien H H Sommerfeld, J. T., Schrodt, V. N., Pariaot, P. E., Separationr Sci. 1 anh 3j: 281 (1966). (5) Douglas, J. M., IND.ENC. CHEM.PROC.DESIGN DEVELOP. 6 , 43 (1967). ( 6 ) Douglas, J. M., Rippin, D. W. T., Chem. Eng. Sci. 21, 305 (1966). (7) Gore, F. E., IND.ENG.CHEM.PRCO.DESIGNDEVELOP. 6 , l l (1967). (8) Horn, F. J. M., Ibid., p. 30. ( 9 ) Horn, F. J. M., Lin, R. C., Zhid., p. 21. (10) Kiessling, R., dissertation, University of Leipzig, 1961. (11) Kiessling, R., Kernenergie 6 , 168 (1963). (12) Kiessling, R., Wetzel, K., Ibid.,5 , 28 (1962). (13) Lewis, W. K . , IND.ENG.CHEM.28, 399 (1936). 6 , 16 (14) McKay, D. L., Brown, B. T., IND.ENC.CHEM.PROO.DESIGNDEVELOP. (1967). (15) McWhirter, J. R., M.S. thesis, Pennsylvania State University, January 1961 ; Ph.D. thesis, December 1962. (16) McWhirter, J. R., Lloyd, W. A., Chem. Eng. Progr. 5 9 (G), 58 (1963). (17) Robinson, R. G., Ph.D. thesis, Pennsylvania State University, December 1964. (18) Robinson, R. G., Engel, A. J., TND. ENG.CHEM.59 (3), 22 (1967). (19) Schrodt, V. N., IND.ENC.CHEM.FUNDAMENTALS 4, 108 (1965). (20) Schrodt, V. N., Sommerfeld, J. T., Martin, 0. R., Parisot, P. E., Chien, H . H., Chem. Eng. Sci.,in preas. (21) Semenenko, E. I., Roginskii, S . Z . , Yanovskii, N. I., Kinetika i Kntalir 5 , 490 (1964). (22),Sommerfeld, J. T., Schrodt, V. N., Parisot, P. E., Chien, H . H., Separations Scr. 1 (2 and 31, 245 (1966). (23) Spinner, I. R., Simmons, P., paper presented a t 55th National Meeting, A.I.Ch.E., Houston, Tex., 1965.
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