A Dynamic Study of a Pilot Eckey Horizontal Fractionator

In order that the automatic control of a process system be optimal, the dynamics of the equipment involved must be throughly understood. Distillation is a complex non-linear process and as such, many of the dynamic system parameters are difficult to predict accurately and must be determined experimentally. The purpose of this study was to develop a series of dynamic models which characterize the response of the liquid temperature on the plates of a pilot Eckey Horizontal Fractionator, to variations in liquid feed rate, liquid feed composition, and vapor feed rate. This was done by experimentally determining the frequency response of the column indirectly, using the pulse technique. The eight inch, 32 plate column was run as a stripper using the binary methanol-water system. Liquid feeds were introduced at the top wlth raw steam being fed into the bottom of the column. The column was run at pressures of one atmosphere and about 200 mm Hg. Pulse-like variations were introduced into the liquid feed rate and composition and in the vapor feed rate and the time histories of these pulses and the column responses on several plates were recorded. From these data, values for the system frequency response were calculated numerically using the computerized TAFT routine. This data was then plotted on Bode diagrams. The Bode plots were analyzed to determine the form of the dynamie models of the column and their parameters. It was found that liquid rate and composition disturbances showed first order responses plus delays which increased with distance from the feed plate. Vapor rate responses also showed first order dynamics but without delays. The major first order time constants were relatively constant on all plates for the liquid and vapor rate responses but increased with distance from the feed plate for liquid composition responses. Of considerable interest was the presence of resonance peaks in the frequency response curves. These are thought to be the result of oscillating composition transients traveling down the column in the liquid and up in the vapor flow. A term to account for resonance was included in the models. It was found that the time delays and first order time constants could be related to the liquid residence time and scale-up equations are developed for applying the results of this study to other Eckey horizontal columns.

The eight inch, 32 plate column was run as a stripper using the binary methanol-water system. Liquid feeds were introduced at the top wlth raw steam being fed into the bottom of the column. The column was run at pressures of one atmosphere and about 200 mm Hg. Pulse-like variations were introduced into the liquid feed rate and composition and in the vapor feed rate and the time histories of these pulses and the column responses on several plates were recorded.
From these data, values for the system frequency response were calculated numerically using the computerized TAFT routine. This data was then plotted on Bode diagrams. The Bode plots were analyzed to determine the form of the dynamie models of the column and their parameters.
It was found that liquid rate and composition disturbances showed first order responses plus delays which increased with distance from the feed plate. Vapor rate responses also showed first order dynamics but without delays.
The major first order time constants were relatively constant on all plates for the liquid and vapor rate responses but increased with distance from the feed plate for liquid composition responses. Of considerable interest was the presence of resonance peaks in the frequency response curves.
These are thought to be the result of oscillating composition transients traveling down the column in the liquid and up in the vapor flow. A term to account for resonance was included in the models. It was found that the time delays and first order time constants could be related to the liquid residence time and scale-up equations are developed for applying the results of this study to other Eckey horizontal columns.
ACKNOWLEDGMENT At. this time, I would like to take the opportunity to thank those whose generous aid was given to the author -Dr. G. David Shilling whose advice, councel, and patience were greatly appreciated; Dr. Pasquale Marino who originated the concept of this study; and the Vulcan Manufacturing   In order to insure efficient and economical design and high quality performance of process equipment, there must be a thorough understanding of the unit's operating characteristics and behavior. This is especially true where adequate control of certain system parameters is to be achieved. In checking the performance of or pinpointing troubles in existing control systems, investigating the controllability of process equipment, or designing optimal control systems, it is necessary to have information concerning the dynamic behavior of t h e equipment involved.
This data is acquired by determining functional rela-. tionships between the independent and dependent variables of the system. A dynamic model may be rigorously derived if the physical laws describing the action and interaction of these variables are completely understood. Often however, the system is of such complexity that only an approximate model can be constructed and experimental information is required to determine the parameters occuring in this model and to verify its accuracy and reliability. The experimental testing usually consists of subjecting an input (~nde pendent variable to some type of disturbance and measuring the response of some output (dependent) variable to that disturbance. Appropriate analysis of the forcing and the system response will then yield the desired dynamic information.

I!iE LITERATURE
Two main methods for determining dynamic properties, as discussed by Lees (58) are transient response testing and frequency response testing. Both of these methods yield sufficient information to construct or test a linearized model of the system. In transient response response testing, usually a ramp or step change in some system input is made and the time history of this and the system outputs of interest are recorded. These data are analyzed by standard methods to yield values for the time constants and other parameters oocuring in a dynamic model. However for higher order more complex systems, interpretation of transient data becomes difficult. Also, a permanent change in the system variables is produced which in practical situations may not be desirable.
Frequency response testing involves the technique of applying a steady sinusoidal variation to a system input and measuring the output response. This is a technique that has been long used in electrical engineering applications.
Theoretically the forcing can cover an infinite range of frequencies, but actually, data of interest usually lie in a more narrow range. By proper analysis of amplitude ratios and phase angle shifts between input forcing and the system's response over a range of input frequencies, it is possible to accurately determine values for the parameters occuring in a dynamic model of the system. From this information, one could then predict the response of the system to any disturbance in the range of linear operation. Frequency response data also yield information regarding stability of the system, reveal resonance effects, and are more useful for higher order systems than transient data.
The direct method of obtaining this data is to actually vary a system input sinusoidally and measure the response over a wide range of input frequencies. This however can be a very time consuming process and may require a large amount of expensive testing apparatus. Also if the equipment being tested is in use, the experimentation could cause dangerous side effects or result in the production of large quantities of off-specification product. Fortunately there are simpler indirect methods of obtaining frequency response data which are available. Gallier (37) and Lees (58) discuss methods of obtaining frequency response data by statistically analyzing random .variations in input and • utput variables using correlation techniques. However a more practical and widely applicable method is pulse testing.
It has been shown by Hougen (45,47), Clements and Schnelle (22), and Draper et al (28) that by subjecting a system to a single pulse-like disturbance, the same frequency response information can be acquired as that determined by forcing the system sinusoidally over a range of frequencies. The experimental testing time then is much shorter than that required for direct frequency response testing.
Since an exact sinusoidal variation in an input variable is not needed, the testing equipment can often be simpler. The pulse is made by varying the input from its steady-state value for some finite period of time and then returning it to this value. The time history of the input and the system response are recorded. The pulse must be large enough to generate a measurable response but should not be so large as to excite the system beyond linear operation. It may have a precise shape, such as triangular, rectangular, or half sine, but this is not a requirement.
Driefke (29) has made a det. ailed investigation into the effect of pulse height, width and shape on the accuracy of frequency response data derived from experimental pulse testing. Thie study involved analog testing of first, second, third, and higher order systems by forcing them with many different pulses. He coneluded that several different pulses gave good results, and in general, longer pulses gave more reliable low frequency data and shorter pulses more reliable high frequency data. It was found that half sine pulses gave better results than rectangular pulses and that the spectral content of the input pulse was very important. Accuracy was also affected by the data reduction process, confirming earlier work.
Some of the earliest applications of pulse testing were in the area of aircraft stability and control. Smith and Triplett (78) and Eggleston and Mathews (32) discuss several methods of obtaining frequency response data from pulse information gotten during flight tests. In recent years this technique has been applied in many areas of the chemioal process industry. Hougen and Lees (46) pulse tested a condenser to determine the response of the outlet liquid temperature to changes in the inlet liquid flow rate. The frequency response was then calculated, compared to the sinusoidal forced response, and fitted to a dynamic model of the process. Hougen and Walsh (47) discuss the application of the pulse testing method to ten afstems including servomechanisms, analog models, heat exchangers and mixing chambers.
The frequency response data obtained were found to be in excellent agreement with those acquired by direct sinusoidal forcing.
There have been numerous studies made of the dynamics of fractionating equipment. A distillation column consists of a complex sequence of interacting heat and mass transfer stages. A series of non-linear differential equations describing the distillation process can be written for each stage, but the simultaneous solution of these becomes quite difficult. Attempts to solve simplified versions of these have been made and the results compared "1th experimentally determined transient and frequency response data.
W illiams (83) and Archer and Rothfus (4) have completed extensive literature surveys of recent work done in the area of distillation dynamics. Rosenbrook (64) discusses a digital computer routine used to numerically solve the theoretical equati@ns.
A five plate benzene-carbon tetrachloride column was studied at total reflux by Armstrong and Wilkson (5,6)  that the responses at low frequency could be simply characterized by delays and first order lags.
In a series of papers, Gerster et al (10,11,36,38) conducted experimental transient response studies on a pilot plant distillation column.
Step changes in liquid feed rate, vapor rate, and reflux rate were made for five plate and ten plate, two foot diameter benzene-acetone columns. The results, when compared with analog solutions to the perturbation equations of Lamb and Pigford, showed good agreement.
Approximate first order responses were noted, with disturbances in vapor rates being felt almost immediately t hroughout the column.
More complex equations were developed by Franke et al (34), and their solutions were compared with experimental tests performed on a twelve plate methanol-tertiary butyl alcohol column. The response to step changes in reflux rate showed approximately equal first order lags on each plate.
Wahl and Harriot (81), in analyzing a eomputer-simulated column, felt changes in vapor rates showed first order responses on any plate while responses to liquid rate and concentration changes were nth order at a distance of n plates from the point of upset.
Wood and Armstrong (8,84) have developed and solved mathematical models of t heoretical columns to study the frequency response to sinusoidal changes in liquid feed com-pos1 tion and reflux rate. The presence of possi ble oscillatory effects is noted.
The frequency response of a 20 plate gasoline splitting column was studied by Aikman (1) in order to provide a high

Effect of Liquid and Vapor Flow Disturbances
The propagation of flow and concentration disturbances within a · column is a very complicated process. to construct simpler linearized models from frequency response data which characterize well the response of the sys• tem. This is common practice in dynamic analysis, (1,38,39,46,53,59,62,65).

System Transfer Function
These models are often convenient~ written in the form of a transfer function, which is the ratio of the La.Place transforms of the system response and the input forcing.
Consistent with results from dynamic response tests perfermed on other distillation units, the modeling equation for the Eckey Fractionator will be of the form: where G(s) defines the transfer function as the ratio of La.Place transforms and s is the LaPlacian operator. The response of a system to any input forcing can be found simply by multiplying the system transfer function by the La.Place transform of the forcing and taking the inverse transformation.
When a change is made in the liquid feed rate or concentration to a column, there is usually a period Gf time before a response is noted on the other plates in the column.
This time L is represented in the transfer function by the -~ expression for delay, e • It should be possible to relate L to the time it takes the liquid to flow from the point of disturbance to the plate at which the response is measured.
The quantity "t-' represents the major time constant associated with a first order model of the response. If more accuracy were warranted, a second order model could be used.

THEORY OF PULSE TESTING
The frequency response of a system is a set of forced responses to sinusoidal variations in some input covering a wide range of frequencies. If the relationships between the dependent and independent variables of the system can be defined by a set of linear differential equations with eonstant coefficients, the system is said to be linear. When an input to such a system is varied sinusoidally at some frequency, the forced response of the system will also be a sinusoid and of the same frequency. Figure 2 shows the forced response, y(t), of a typical linear system to a sinusoidal variation in input, x(t). The frequency response information deduced from a set of plots like this are the phase shift, ; , between the input and output, and the ratio of their amplitudes, A 1 /Ax. A different set of values of the phase angle and amplitude ratio will be obtained for each value of the frequency of the input forcing.
A convenient and useful way of presenting frequency response data is through the use of Bode diagrams. In this method a log-log plot of the amplitude ratios versus frequency is made accompanied by a semi-log plot of the phase angles versus the log of the frequency. Often the plot of amplitude ratios is normatized with respect to the amplitude 16 Amplitude ¢ Time Fi gure 2-Sinusoidal forced response of typical linear system ratio at zero frequency, i.e. the steady-state gain. From t hese plots it is possible to determine the maximum gain for stable operation, if the system is put Un.der feedback control, and the critical frequency of the system. This information is useful in determining optimum controller settings, judging the speed of response and comparing the contrGllabili ty of the system under various proposed control schemes.
When the frequency response of the components of a cascade of non-interacting subsystems are known it is a relatively simple matter to combine these to get the response of the entire system. Although frequency response analysis is theoretically accurate only when dealing with linear systems, it can provide information which is extremely useful when studying the dynamics of a system whose response can be assumed linear over a certain range of operation. By proper graphical analysis it is possible to determine values for the time constants occurring in linearized dynamic models of a system or validate the models. · The presence of resonance effects can also be determined.

The Pulse Method
Frequency response information can be extrauted from the time response of a system to a pulse-like disturbance in some system input. A periodic function such as a sequence of pulses can be written as a Fourier series of sinusoidal waves of different frequencies and amplitudes. Figure 3a shows a sequence of rectangular pulses of period P together with a line spectrum of the Fourier series coefficients associated with the pulse train. The ordinate represents the amplitude of the sine waves occurring in the series (the Fourier series coefficients) and the abcissa indicates frequency. If the pulse occurs at a frequency<>J , the frequencies of the sinusoids in the Fourier series will be integral multiples of w • As shown in Figure 3b, when the shape of the pulse remains the same, but its frequency of repetion decreases, more lines appear in the line spectrum.
where x(t) and y(t) are the input pulse and output response time histories respectively and Tx and Ty are the duration of x(t) and y(t).  (29), Marino (62) and Hougen (45).
In the development of the TAFT routine operating on a typical time function which differs from zero for a period of time T, (see Figure 4 for notation) it is first necessary to write the equation for the straight line connecting the ordinates fi and f'i + 1 over the interval A T: The transform of the function can now be evaluated 1n the 1th interval from the definition.
To determine the Fourier transform of the entire function, the integl;'als for the trapezoidal intervals , mu.st now Fi gure 4 --Notation for Trapez oida l ~pp roxirnati on to four ier'-'.!'._ransform-( TAFT routine ) be summed over the time period from 0 to T (i.e. i = 1 to N). lf TAFT ( jW) = \ TAFT ( j'-'>) T L. 1 1 When this is done, the Trapezoidal Approximation t0 the Fourier Transform, separated into its real and imaginary parts reads: The program used the TAFT routine to compute, for each value of w, the real and imaginary parts of the Fourier transform of the pulse forcing and system response (x and y) and the spectral content of the input pulse. The frequency response was then determined from the complex function F(y)/F(x), as values for each frequency; of the phase angle (arctan Im(F(y)/F(x) J /Re(F(y)/F(x)) ) and magnitude 2 2 (square root (Im(F(y)/F(x))) + (Re(F(y)/F(x))) ). The magnitude was then normalized by dividing each value by the magnitude at zero frequency. The frequency response data was then in a form suitable for plotting on a Bode diagram.
A copy of the computer program used and simplified data processing flow sheet are found in Appendix B.

EQUIPMENT
The overall equipment setup for the experimental work is shown in Figure 5.

0\
The column was run as a stripper using methanol-water for several reasons. In a distillation system which includes reboiler and reflux setups, the response of the column to liquid and vapor upsets will be affected by the response of the reboiler and reflux systems, since they are an integral part of the overall system. Using a reflux setup weuld complicate the system greatly. II.

Liquid Feed System
Liquid feed entered the column on plate 30 from either the main or auxiliary feed system. In both systems, the feed was pumped from stainless-steel storage tanks to ten gallon constant-head tanks mounted approximately ten feet above the column. To maintain a constant head, overflow pipes were mounted in the overhead tanks to carry excess feed back to storage. The feeds from both head tanks were filtered and passed through rotometers for flow rate measurement as shown in Figure 7. The capacities for the main and auxiliary feed systems were 100 and 60 gallons respectively. Flow was controlled in the auxiliary system by means of a hand operated needle valve, and switching between the feeds was accomplished using a 3-way solenoid valve.
Liquid flow rate disturbances were introduced into the system in the main feed line. A pulse in liquid rate resulted from a similar pulse in the air pressure signal to a Foxboro pnuematic control valve. The pulsed liquid flow rate was recorded using a Foxboro magnetic turbine flow transmitter. This instrument generates an oscillating voltage signal whose frequency is converted to an output voltage directly proportional to volumetric flow rate.
Rectangular pulses in feed liquid composition were made by switching from the main feed to the auxiliary feed of a different composition but the same flow rate for a fixed period of time. The duration of these pulses was recorded using an electric timer.

III. Vapor Feed and Bottoms Liquid Systems
Pure steam at 90-100 psig was available as vapor feed to the bottom of the column from the University high pressure steam line shown in Figure 8. This circuit is shown in Figure 10.   In order for dynamic testing to be successful, a change in some system input must produce a .measurable change in some system output. Prior to the pulse runs a series of steady state runs at various liquid and vapor feed rates and liquid feed compositions were made at both atmospheric and reduced pressure. This permitted optimum operating conditions for the dynamic tests to be determined and insured that pulse-like changes in vapor and liquid flows would produce measurable changes in the plate liquid temperatures in the column.
It was decided to make atmospheric steady state runs using feed liquid concentrations of about 30, 40 and 50 mole percent methanol. It was felt that the optimum feed liquid concentration for the pulse runs to follow would fall somewhere in this range. From inspection of the methanol-water vapor liquid equilibrium diagram, it can be seen that several equilibrium stages will be needed to strip the alcohol from liquid feeds of the above compositions. Using feeds of lower methanol concentrations would result in methanol removal occurring in only a few stages at the top of the column.
Using a highly concentrated feed would not appreciably increase the number of equilibrium stages required for stripping. Highly concentrated solutions could also be extremely dangerous to handle and would require large amounts of the limited supply of methanol available. Actual liquid feed composition used during these runs were 29.9, 42.l and 50.3 mole percent methanol. During each run, several liquid to vapor ratios were used.
To conserve feed, the column was first heated until a thermal equilibrium had been reached by feeding steam in the bottom and using pure water as liquid feed. Equilibrium was reached when the plate liquid reached the saturation temperature at column pressure. Following this, the methanol-water feed solution was introduced into the column. Liquid bottoms product and condensate samples, together with plate liquid temperature readings, were taken at various time intervals until steady state conditions were reached. At this point the liquid to vapor ratio was changed and product samples and temperature readings were again taken until the next steady state was reached. This~procedure was followed for each run using a different liquid feed composition.
Another series of steady state runs was made, prior to the reduced pressure pulse tests, at a column pressure of about 200 mm Hg. Liquid feed concentrations used were 39.5 and 47.5 mole percent methanol. For each feed, various liquid to vapor ratios were tried with the sampling procedure identical to that for atmospheric steady state runs.
The compositions of the distillate and bottoms products were determined from specific gravity measurements made on a Christian Becker balance. The plate liquid compositions were found from their liquid temperatures assuming saturation at column pressure. This is justified on the basis that when pure water was used as the liquid feed, the plate liquids did indeed reach and maintain their saturation temperatures. Using these measurements, plate liquid temperature and composition profiles throughout the column were constructed. The operating conditions and results of the steady state runs are shown in Table 1  Measurement of the actual value of the liquid and vapor rate to more than two figure accuracy was not necessary. It was also possible to estimate the maximum column response which might be expected when changes were made in vapor rate and liquid rate and composition.
It was found that the feed liquid was stripped of its alcohol in the top two thirds of the column and was essentially pure water at this point, regardless of the feed used.
Therefore no column response would be felt by the thermocouples in the liquid on plates 11 and 2. During the pulse runs the liquid temperatures on these plates were not recorded. A variety of triangular, half-sine and rectangular pulses were tried during both the high and low pressure runs.
Because of non-linearities inherent in the pulse circuit, transducer and control valve, the resulting pulses in liquid rate became somewhat deformed (see Figure 11) and the triangular and .half-sine pulses appeared quite similar in shape.
This was also true for the pulses made in the vapor flow rate to the bottom of the column. The effect is to partially smooth out any discontinuities in the pulse (e.g. to round off the point of the triangular pulses). It was possible to make more vapor pulses than liquid pulses during equal testing periods because the column responded more rapidly to vapor disturbances.  To determine the response of the column to pulses in liquid feed composition, the unit was first brought to steady state using the main feed system. The column was then switched to the auxiliary feed of a different composition for a short period of time and back to the main feed.

Liquid Composition Pulses
The pulse duration was measured using an electric timer accurate to .1 seconds. During this period, the volumetric flow rate was maintained constant. The response of the column to this disturbance was recorded until it had returned to steady state. Identical procedures were followed for both atmospheric and low pressure runs with pulses of several different amplitudes (compositions) being used. Pulses made in this way would be considered rectangular in shape.

Response Data
The thermocouple emf recordings representing the plate liquid responses to liquid and vapo~ pulses were hand smoothed prior to data processing. The pulses, to three significant figures, were read off the smoothed response curves for time intervals of eight seconds. From the unsmoothed input pulse curves values were taken for two second intervals.
These intervals were decided upon after some experimentation indicated that using smaller time intervals (more data points), while requiring more processing time, would not improve the accuracy of the results. Using much larger time intervals would have unduly distorted the functions being approximated. When pure time delays were apparent in the response curves, these were recorded, and the zero point was shifted to the point of first deviation from the steady state. The rectangular concentration pulses were all assigned an amplitude of unity for their duration. The input-response data were then processed by the TAFT routine which calculates the input and output Fourier transforms and from these, standard frequency response information. While producing a printed output, the program also utilized the computer plotting system to automatically construct and plot the results as Bode diagrams.

RESULTS
A system subjected to a pulse forcing will be excited at all frequencies expect those at which the frequency con-

Determination of First Order Time Constants
Although it was possible to characterize the column responses by first, second or higher order approximations, after careful examination of the experimental Bode diagrams, it was decided that determination of more than a major first order time constant would be unwarranted. There were several reasons for this. The response of distillation units to changes in liquid flows (rate and composition) is often estimated to be first order on each plate in the column, making the total response nth orde~ at a distance n plates from the point of disturbance. It is common practice in cases such as this for the resulting response to be estimated by a time delay and a first order time constant (1,7,9,10,36,57). In this study, as previously mentioned, time delays were extracted from the experimental time histories of the responses to liquid rate and composition pulses before data processing by the TAFT routine.
The approximate first order time constants were determined by superimposing on the amplitude ratio curves of each Bode diagram, transparencies on which were plotted similar curves for known first order systems. Thus an observable best fit was obtained which would be considered accurate enough for engineering applications. The experimental amplitude ratio curves showed good agreement with first order dynamics at low frequencies but the . true higher order nature of the responses became evident because of the more rapid attenuation of the curves at high frequencies. The phase lags also showed lower values at the corner frequencies 0 (reciprocal of the first order time constants) than the -45 predicted from first order dynamics. It was felt however that any attempt to calculate a second time constant would produce values highly inaccurate and of little significance.
Because of the extremely low pressure drop in the column, a vapor pulse is felt almost immediately on each plate.
The response then contains no time delay. In this case a first order approximation to the response is more viable, although again at high frequencies, the higher order nature of the response is evident.

Characterization of Resonance Peaks
Perhaps the most distinct characteristic of the experimentally determined frequency response curves is presence of resonance peaks. Lamb and Pigford (57) and Wood (84)  In theory, these peaks should occur on all plates at the same frequencies. The experimental curves bear this out.
As expected, the resonance frequencies were not noticeably affected by the shape, height or duration of the input pulse.
In addition, resonance peaks were evident in the frequency response curves for all liquid and vapor forcings at both atmospheric and reduced pressure.
The quantity B in the experim~ntal transfer functions was calculated from the period of the resonance peaks occurring in the amplitude ratio curves of the frequency response plots. The quantity A is related to the amplitude of the resonance peaks and was calculated, where possible from the phase angle curves. Details of the methods of calculation of these parameters are shown below with reference to Figure 12.

Determination of Parameters A and B
Let the transfer function for a system which shows resonance peaks in the frequency response be of the form: where G(s) = total system transfer function ::.n(jw) [(1 -AcosBw)+ j(AsinBw)J

Calculation of B
Obtain magnitude of G(jw) Oscillations in a plot of AR( G) versus log w as shown in Figure 12 are caused by the term, -2AcosB~. This oscillates with a "period", P= From Figure 12, the "period" is the difference between successive peaks or troughs. Therefore B is determined from the relation: This term is oscillatory as seen from the table below. Table I.

Reliability and Accuracy of Results
The reliability and hence the usefulness of the dynamic response models is a function of the accuracy of the experimental input pulse -output response time histories, the point reading process, the computerized TAFT routine used to calculate and plot the frequency response data, and the interpretation of the Bode diagrarn~, to determine model para- It is often convient to truncate the output pulse responses to facilitate data processing. This is done when the response decays at a very slow rate or does not return to zero, as was often the case when very strong input forcings were used. When this is done, the low frequency part of the amplitude ratio curve is flattened while the high frequency part is attenuated too sharply. However Driefke (29) found that moderate truncation (up to 25-50 percent of the time history) had negligible deterimental effect.
It has been shown that calculation of frequency response data numerically, using the TAFT routine, is extremely accurate as long as a sufficient number of points are used (22, 29, 47). Hougen (47) recommends at least ten data points to approximate the input pulse while Clements and Schnelle (22) feel it is rarely necessary to approximate the output by more than 50 points. In this study it was found that a maximum 5 minutes of response time, with points taken at 8 second intervals, sufficiently described the output response. It is felt that the data processing routine con-

Further work can be suggested in several areas:
The effect on the column dynamics of (1) using different binary or multicomponent systems, (2) running at high vacuum, (3) using reboiler and reflux setups should be investigated, Further investigation into the resonance effects using direct sinusoidal forcings or carefully selected pulses would be interesting.
An attempt should be made to mathematically derive and solve a set of modeling equations for the col-umn. The validity of these could then be checked using the experimental results.