Lewis Matheson 3x4rm

Studies in Distillation Design of Rectifying Columns for Natural and Refinery Gasolirw by. K.

LEITIS .4KD

G. L.

hfATHESON

Department of Chemical Engineering, Massachusetts Institute of Technology. Cambridge, Mas>.

T h e Hausbrand equations f o r binary mixtures are applied to the solution of problems irc continuous rectification of natural gasoline. Steps of the computation are outlined, the details of which are illustrated by a specijic problem

T

HE design of rectifying equipment for natural and

that the molal heat of vaporization of the componeiits is reasonably constant under ordinary conditions of rectification by assuming average constant values for Vn and On+ above the feed plate and for the corresponding values of V , and 0, - I below it, thus making it possible to use the graphical stepwise methods of design which have so generally replaced the former algebraic technic. I n using the above equation, Hausbrand calculated the vapor composition, y., from the liquid composition on the plate above, and then determined the liquid composition, xn,by the use of the equilibrium diagram, thus proceeding down the column. I n the case of natural gasoline the process can be simplified because of the fact that the components of the mixture follow Raoult's law, a t least within the accuracy of design. Thus, assuming 100 per cent plate efficiency, the partial pressure of any particular component above the nth plate is equal to XZ,, and this in turn is identical with its partial pressure in the vapor above tJhe plate, yn7r. Hence,

refinery gases is complicated by the presence in the mixtures of a large number of components. The problem can be solved by applying the ordinary Hausbrand equations for binary mixtures successively to all of the components whose propelties influence the separation. However, the multiplicity of the operations of computation is sufficient to confuse one unfamiliar with the technic of handling the equations. Because of the growing importance of rectification in the industry, it seems worth while to outline and illustrate the steps to be taken in solving a problem of this sort. The following nomenclature will be employed : V = moles of vapor rising from any plate per unit time 0 = moles of overflow descending from any plate per unit time z = mole fraction in liquid phase of any particular component under consideration y = mole fraction in vapor phaPe of any such component F = moles of feed to the column per unit time D = moles of final product (distillate), whether vapor or liquid, leaving apparatus per unit time W = moles of reqidue leaving bottom of apparatus per unit time P = vapor pressure of component under consideration in pure state at temperature in question T = total pressure on apparatus p = partial pressure of component under consideration

Count the plates up from the feed plate toward the top and down toward the bottom. Call any particular plate above the feed plate the nth plate and below, mth. Call the top plate the pth and the bottom plate-i. e., the stillthe wth plate. Designate the conditions referred to by means of a subscript indicating the location of the material in question or the point from which the material came. Thus, y,, is the composition (mole/fraction of the component in question) of the vapor rising from the nth plate, xf is the composition of the feed to the column, etc. Assuming continuous operation of such a rectifying column, Hausbrand equated input to output of a given component in a section of the apparatus above the nth plate, as follows: y,V,

= Tn+iOn+i

f

XdD

The left-hand side of the equation is the total amount of component entering the top of the column from the nth plate per unit time, while the first term of the right-hand side is the amount of this component flowing in the overflow from the top of the column down on the nth plate, and the second term is the amount in the distillate. Hausbrand employed weight units, but for reasons which will appear presently it is more convenient in this particular case to use moles. The expression X d D is determined by the conditions of the problem, and the amount of vapor and overflow can be calculated a t any temperature level from a heat balance. I n recent years it has become common practice in the case of binary mixtures to take advantage of the fact

Y" = znP,/n,

and consequently one may rewrite the Hausbrand equation as

Similarly, below the feed plate,

It should he clear that by means of these equations one can compute the change in concentration from plate to plate in any part of the column, once the conditions on any given plate are known. CHARACTEI~ISTIC PROBLEM IN DESIGN It will be worth while t o consider a characteristic problem in design. One will know the Composition and amount of the feed to the column; the temperature and pressure under which it exists; the temperature which it is practicable to maintain in the reflux condenser a t the top of the column with the cooling means available; the point a t which it is desired to effect the separation; and the allowable o v e r l a p i. e., the amount of high-boiling material which may be tolerated in the overhead distillate and of low-boiling material in the residue from the still a t the bottom of the column. With these data a t hand, the following steps of computation are necessary: (1) Calculate the amount and exact composition of the dietillate and residue. (2) Determine the pressure which must be maintained on the condenser in order to produce the reflux which is necessary for the functioning of the column. This is the operating pressure. 8, of the apparatus.

494

May, 1932

I N D U S T R I A L A S D E N G I NE E R I NG C H E A I I S T R Y

(3) Assuming for the moment a suitable value of the reflux, the top of the column from the reflux condenser, calculate

0 , to

,

the composition of the reflux. This will, of course, depend on the type and method of operation of the condenser employed. (4) Calculate the composition of the vapor from the top plate, and, from this, that of the liquid on the top plate and its temperature. ( 5 ) Calculate the temperature of the still and the composition of the vapors rising from it. (6). From a heat balance on the column, determine the variation in the reflux through the column. One can immediately determine by such a balance the reflux from the bottom of the column to the still, and it is usually sufficiently accurate to wsume the change in reflux per degree of temperature rise up the column constant, except at the feed plate where the reflux is increased by the amount of liquid in the feed. (7) By use of the Hausbrand equation applied from plate to plate calculate the concentrations and temperatures on the plates, working down from the top and up from the bottom. (8) Tnspection of the results of the preceding operations as the temperatures of the plates approach each other towards the middle of the column will enable one to judge with reasonable precision whether the amount of reflux assumed for the operation of the column is satisfactory. If not, readjust it to a satisfactory value and repeat steps 3 to 6, inclusive. (9) By study of the compositions determined by working up from the bottom, pick a suitable feed plate. Estimate the concentration on t h s plate of that component of highest boiling point which does not appear in the residue in appreciable amount. If necessary, recalculate the composition on the plate below and correct for the concentration of t,his component on it. Then proceed with the computation of the concentrations up the column until the concentration of the lowest-boiling constituent of the residue which does not appear in appreciable amount in the distillate has become negligible. If the concentrations thus determined correspond substantially with those calculated by starting down the column from the top, the problem is solved. If not, the concentration on the feed plate must be reeatimated, and this last operation repeated. The number of plates thus determined, corrected for the plate efficiency, gives the number of plates required in the column when operated under these conditions. (IO) To determine t,he influence of reflux ratio in column size and heat consumption, these operations must be repeated for a number of different values of reflux. The plotted results will make it easy to determine the best operating conditions. ILLUS'FRATIOX

OF

DETAILSOF

PROBLEM

The detail of these steps will be best understood from a specific illustration. Assume as a feed stock a material containing 26 mole per cent of methane and permanent gases, 9 per cent ethane, 25 per cent propane, 17 per cent butane, 11 per cent pentane, and 12 per cent of hexane and higher. Assume that the amount of isomers of the normal hydrocarbons may be neglected and that the hexane and higher hydrocarbons average heptane. Assume that conditions are such that one can maintain a n effective top temperature-i. e., a temperature of the gas and reflux, within the condenser itself, of 70" F. (294" K.). The feed is pumped in at 100" F. (311" K.). It is desired to take all propane and lighter overhead, and all butane and heavier as residue, but it is allowable to have 1 per cent of butane in the overhead distillate and 0.1 per cent propane in the residue. The reflux condenser is of a type in which the vapors~travel with the liquid through the condenser and are separated after having been brought to the final condenser temperature of 70" F. Consequently, the final gas is in equilibrium with the reflux liquid. STEP1. Were the separation complete, there would be 60 moles of distillate and 40 of residue for each 100 moles of feed. Consequently, in the actual column the butane In the distillate will be approximately 0.6 mole and the propane in the residue 0.04. On the basis of this, the following table, showing the distribution of the components of 100 moles of feed between distillate and residue, should be selfexplanatory:

HYDROCARBOS CHI+ CzHe C3H8 CIHK CSHIP CEHl4 +

Total

DISTILL~TE Moles M o l e yo 26.0 42.9 9.0 14.9 24.96 41.2 0.61 1.0 ~

...

Moles

RESIDCE .Vole %

...

...

...

0.04 16.39 11.0

100.0

39,43

0: 1 41.6 27.9 30.4 100.0

...

60.57

195

~

...

12.0

...

__

STEP 2. At the temperature of the condenser the pressure must Le such that the overhead ga.j or distillate is a t its dew point, since it was separated from a liquid condensed from it by cooling at constant pressure. Therefore, the partial pressure of each component in the gas must of necessity equal the partial pressure of that same component in the liquid from which the gas was last separated, and with which it was in equilibrium. Kow a t the effective condenser temperature, 294" K., the pressure of pure ethane is 38 atmospheres, of propane 8.8, and of butane 2.2. Consequently, one can write the following equations: pz = 3822 = 0 . 1 4 9 ~ p , = 8 . 8 ~ 3= 0 . 4 1 2 ~ pa = 2.224 = 0.01 T

Furthermore, neglecting the solubility of methane and permanent gases in the liquid, the sum of the mole fractions must obviously be unity-i. e., x~ x3 z4 = 1. Solving these equations, one obtains s = 18.1 atmospheres, which is the necessary operating pressure on the condenser, and which, neglecting pressure drop through the colunin, is the pressure throughout the apparatus. From the above calculations, 100 x2 = 7.1 mole per cent ethane in the liquid condensate, 100 z3 = 84.7 per cent propane, and 100 x4 = 8.2 per cent butane. STEP3. Since, from the condenser on this equipment, the final overhead gas leaves in equilibrium with the total condensate, the figures just given also represent the composition of the reflux to the column. STEP 4. Assume a reflux equal to double the amount of overhead vapor. On the basis of 100 moles of distillate, there will therefore be 200 moles of reflux. Consequently, the vapor rising into the partial condenser from the top plate will have the composition shown by the following table :

+ +

CHI = 4 2 . 9 C?.Ho = 1 4 . 9 CSHO = 4 1 . 2 CiHio = 1 . 0

Total moles 42.9 29.1 210.6 17.4 300.0

+ 14.2 = ++ 169.4 = 16.4 =

?& 14.3 9.7 70.2

5.8 -

100.0

Since this vapor must be in equilibrium with the liquid on the top plate, one can write the following equations: 1 = 320' P z = 0 . 0 9 7 ( 1 8 . 1 ) = 1 . 7 5 5 = zzPz P3 = 0 . 7 0 2 ( 1 8 . 1 ) = 1 2 . 7 = Zap3 PI = 0 . 0 5 8 ( 1 8 . 1 ) = 1 . 0 5 = zip,

K

P 64 17 4.65

7

0.027 0.747

0.226

1 ,000

These equations cannot be solved directly because they contain six unknowns for three equations, although there is the fourth relationship, Zx = 1. However, the values of Pz,Pa, and P4 are determined by the unknown temperature of the top plate. This temperature must be such that i t will satisfy the above relation. It can easily be determined by successive approximation. As a matter of fact, in this case the top-plate temperature is 320" K., a t which temperature the pressure of each pure component and the corresponding mole fraction in the liquid phase on the top plate are shown in the two columns following the equation. It will be found that at no other temperature will the equations be satisfied. While solution by successive approximation is always in a certain sense unsatisfactory, in the

I

I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R I'

496

case of equations of this type a little experience soon makes it easy. STEP5. The composition of the boiling liquid in the still is known. Furthermore, it is boiling under a pressure of 18.1 atmospheres. Consequently, the temperature of the

VOI.

24,

NO.

5

When one endeavors to make a heat balance on this column to determine the reflux a t the bottom, one finds that the data available on heat of vaporization a t these high pressures (1) are unsatisfactory, particularly in the case of the high-boiling constituents (i. e., the liquid boiling in the still). Csing the highest values which the available data would indicate as possible, the overflow decreases somewhat as one goes down the column. Using lower values gives a n opposite result. I n order to eliminate confusing complications, it will therefore be assumed that, in the case of this column, the change in the overflow and consequently in the vapor ri the column is negligible from plate to plate except a t the feed plate. Hence, on the basis of 100 parts of feed, the overflow in the upper part of the column is 2 (60.57) = 121.1, and the vapor, 3 (60.57) = 181.7. The corresponding values below the feed plate are 176.3 and 136.9. STEP 7. The Hausbrand equation of the operating line above the feed plate is

However, in this equation the term P , is unknown because one does not know the temperature of the plate below. This must be assumed, the value of P read off for each component, the corresponding values of z computed, and the process repeated until a value of t is found which makes 22 = 1. 1 =

Pmz

= Pax3 = Pmr =

FIGURE1. CONCENTRATION GRADIENT THROUGH

CaHs C4HlO CaHiz CBHl4'

0.1 41.6 27.9 30.4

t =

413.5' K.

P

P

80 32.2 13.3

0.08 13,39 3.71 0.90 18.08

2.95

P

2

90 21.5 6.2

0.014 0.536 0.450

Y (as % ) 0.44 74.06 20,52 4.98 100.00

The composition of the vapor from the still is given in the fifth column. STEP6. It is first necessary to determine the condition in which the feed enters the column. This calculation is most easily made by reference to the following equation, derived from Raoult's law and a material balance. Call N the total moles of feed which enter in the liquid state per 100 moles of total feed, the total moles of a given component per 100 moles of feed, and n the moles of the component entering in the liquid in the feed. Equality of input and output gives n o = ' + ; ( + P) 100 n

The composition of the liquid on the second plate down is given in the last column. By repetition of this process, using the same equations but with the proper values of P and xn + 1, one can go down the column from plate to plate The Hausbrand equation below the feed plate is

x,-,

P,T'

= zm -

TO

+ zul

= 0.0429xmP,

However, for 1 This temperature IS above the crltical point of ethane. the purpose of romputing t h e small amount of ethane which wlll dlssolve i n the liquid under theae conditions, in the absence of exact d a t a as t o the solubility, it is allowable to extrapolate the vapor pressure curve to the temperature i n question. The figure is rough but undoubtedly sufficiently accurate for the purpose In hand

+ 0.2237~,

Since the temperature of the still is known, the corresponding values of P are determined, and one can calculate the composition of the overflow from the plate above directly as shown by the equations in the following table. The arithmetical accuracy of the operation is checked by the summation of the concentrations to unity. However, before one can proceed further up the column, it is necessary to know the temperature of the plate above. This must be such that the pressure will be 18.1 atmospheres. It is determined by successive approximation as before, as shown in the last two columns of the table: f =

-

23

=

24

=

25 27

I

0.0429zmPm 0.0429zmPm 0.0429zmPm 0.0429zmPm

+ + +

+

395' K.

0,000224 0,093 0,0624 0,068

= =

Z p 0.00365 62 0,6680 23.5 9.25 0.2214 0.1066 1.9 ~ ~

The equation is applied successively to each component and the condition imposed that Z n = N . At the feed temperature, 311" K., the pressure of ethane is 53 atmospheres,' of propane 13.7, butane 3.7, pentane 1.05, and heptane 0.11. Solving these equations, N equals 55.2. I n other words, the feed enters this column as 55.2 per cent liquid and the rest gas.

-

1,000

CoLu>rN

still must be determined by successive approximation as in the preceding case, and Zy must. equal unity. The technic is shown in the following table: %

331' K.

++ 0.899 = 1.23 = 11.52 + 2.485 0.0603 = 2.79

12.OGzn+i 12.06xn+i 12,06xn+i

0.99965

Pa

0.23 15.63 2.04 0.20

18.1

One can proceed from plate to plate up the column by repeating this step. STEP 8. Working down the column from the top and up the column from the bottom in this way, one finds that the propane and butane concentrations come together quite rapidly. I n other words, the reflux assumed is adequate. If a reflux ratio of one be assumed in this case, it is found that the concentration and temperature changes from plate to plate through the column are far less and are too small for satisfactory operation. I n other words, a reflux ratio

I N D U S T R I A L A N D E N G I N E E R I N G C H E RI I S T R Y

May, 1932

of 1: 1 is too small. Exactly what ratio to use must be decided by the engineer on the basis of this type of computation, interpreted in the light of the operating conditions in question. STEP9. The conditions of the various plates in the column thus computed are given in Figure 1. It will be noted that four plates below the top plate the propane has fallen to 13.9 mole per cent, n-hile five plates above the still it has risen t o substantially the same value. In other words, this fifth plate above the still may well be chosen as the feed plate. If, however, it is desired to introduce the feed on plates above or below this point, this may be tried out by the methods now to be discussed.

ALLOWAKCESFOR HIGH- AKD LOW-BOILING RIATERTALS The difficulty with the computations so far conducted lies in the fact that they take no cognizance of pentane or heavier above the feed plate, or ethane below the feed plate. There is a certain amount of high-boiling constituents in the overhead product. Had this been exactly known, it could have been allowed for in the computations, and the amount of these constituents in the upper part of the column computed. Similarly, had the exact data on the trace of ethane in the residue from the still been known, its amount up the column could have been calculated. Furthermore, the composition of the liquid on the feed plate as determined by these two computation operations-i. e., from the top down and from the bottom up-would be the same. However, the analytical methods are incapable of determiiiing with precision these traces of the lowest-boiling, materials in the still product and the highest-boiling in the overhead even in the case of an operating unit. Still less is the designer in a position to predict these in advance. However. to design the unit with assurance, allowance must be made for these factors. Theoretically, one could make this allowance by assuming the amount of these contaminating traces and checking up by trial and error on the column calculations until the compositions on the feed plate come together. The following is a practical method of making suitable allowance for these corrections. The percentage of ethane on the feed plate must necessarily be less than the percentage of thiq component on the plate immediately above the feed. However, assume, for the moment, that these concentrations differ by a negligible amount and apply the Hausbrand equation for ethane over the feed, .cqL+l

0.083~J'~ - 0.0745

The temperature of the feed is 363" IC., a t which 1' (extrapolated) is 125 atmospheres. Using this value and the assumption that z,,+~ = xn, it is seen that the concentration of ethane on the feed plate must be a t least 0.00795. As will appear later, it must actually be higher than this by a small amount. Below the feed plate the operating equation for ethane isz,-, = 0 . 0 4 2 9 ~ 2 , . B y means of this, one can readily determine the ethane concentration. Thus, on the plate below the feed, the concentration is about 0.0015. On the next plate below it would be 0.0003. It is obvious that quantities as small as this may be neglected. Now using any desired assumed value for the ethane concentration on the feed plate, ing that this value must be somewhat but not much above 0.008, one can calculate concentrations up the column from the feed plate by using the Hausbrand operating equation for conditions above the feed already employed above. Doing this, one finds that the concentration of pentane and heavier above the feed fades

497

away rapidly and conditions then approximate those coniputed by figuring down from the top. If one assumes a wrong value of ethane on the feed plate, difficulties are encountered. Thus, if the value is too small, the equations will indicate a negative incrpment in ethane concentrations going up the column a t some plate, usually the feed plate or the one above it. Larger assumed values will avoid this difficulty, but, unless they are. correct, will indicate on the upper plates where the concentrations of pentane and higher have fallen to a negligible point-ratios of ethane:propane:butane which are out of line with those computed by working down the column and which, therefore, are incompatible with the column set-up. Thus, by successive approuimation, one can determine the proper ethane concentration on the feed plate. P L A T E NUMBER FROM TOP OF COLUMN

42

40

L

z 36 34

32 PLATE

NUMBER F R O M BOTTOM OF COLUMN

FIGIJRE2 . TEVPER 9TURE G R ~ D I E UTHROUGH T

COLUMU

This process may sound involved, but aboye the feed plate the pentane and heavier usually fade out of the picture so quickly that the estimation is reasonably rapid. At any rate, it is far shorter and more satisfactory than the more obvious method of estimating overhead and bottom concentrations of the materials present in traces only and checking them by refiguring the whole column. I n computing the ethane, it is desirable to express the amount on the feed plate to a precision far beyond that with which it can possibly be known, a precision unjustified, for example, by the uncertainty in the extrapolated value of Pz which is us3d in determining it. The reason for this is the extreme sensitiveness of the larger and important concentrations on the plates above to very minor changes in this value of the Concentration on the feed plate.

DISCLSSIOS OF REsuLw The concentrations and temperatures thus computed are plotted in Figures 1 and 2. The abscissas are plate numbers indicated by subscripts 1, 2, etc. Where the point was computed from the bottom of the column, the abscissas are given a t the bottom of the figures; where computed from the top the plate numbers are s h o m a t the top of the diagram. It will be noted that these abscissas do not correspond exactly. The concentrations working up meet the curve working down in between plates. This means that the column balance is not such that an integral number of theoretically perfect plates will function in this way. However, as in the case of design of columns for binary mixtures, using the next largest number of integral plates will give a column which will give a somewhat hetter separation. I n this case, as in the ordinary one, it is unnecessary to try to readjust top and bottom conditions so that the plate numbers come out exactly integral.

INDUSTRIAL AND ENGINEERING CHEMISTRY

498

It is worthy of note that the temperature gradient of Figure 2 is nearly uniform above and below the feed plate. This is quite different from the corresponding plot of a binary mixture, where the temperature gradient is large near the feed plate and small toward both ends of the column. This difference is due to the fact that, in complex hydrocarbon mixtures of several components, the components of intermediate boiling point accumulate in the middle of the column in a way that effectively flattens out the temperature curve. It should be clear that the low-temperature portion of the curves is calculated down from the top of the column, and the high-temperature part up from the bottom. These two portions blend in tangentially where they meet. However, if the upper portions are continued down, they give a too low value of temperature and a too high value of the low-boiling constituents, because in calculating downward from the top it is impossible to allow for the high-boiling constituents. This portion of the curve is shown dotted in Figure 1. The only point regarding the concentration curve which merits special attention is the accumulation of butane in the middle part of the column. This is the component of boiling point intermediate between the bottom and top temperatures of the column, and, as already indicated, this accumulation in the middle of the column is characteristic. The slight rise in butane concentration on the first plate above the feed is more striking. While the general trend of butane concentration above the feed plate is downward, a rise of this sort may occur where the temperature is sufficiently high to give a value of P, sufficient to reverse the slope of the Hausbrand line. Dependable data as to plate efficiency in natural gasoline columns are exceedingly meager, but all the indications are that the efficiencies are high. Thus, a sample of the liquid on the plate in the upper part of a column gave upon analysis 15.7 mole per cent propane and 70 mole per cent

Vol. 24, NO.,^

butane. Using Raoult’s law, the vapor in equilibrium with this liquid should contain 47 per cent propane and 49.3 per cent butane. A sample of the vapor rising from this plate showed upon analysis 49.3 per cent propane and 53.8 per cent butane. I n other words, the vapor rising from the plate was richer in butane and poorer in propane than it would have been had it left the plate in equilibrium with the liquid on it. This is another way of saying that the liquid on the plate had not completely dissolved the butane out of the vapor rising into the plate, down to the equilibrium value. However, the difference in composition is small. The vapor approached closely to equilibrium with the liquid, which means that the plate efficiency was reasonably high. It is very desirable that accurate determinations of actual plate efficiency and of the height of the equivalent theoretical plate for the different types of tower-filling used in the industry be made in order to serve as a suitable guide for deg and operating engineers. SUhlM.4RY

The Hausbrand equation for the calculation of plate-toplate concentration gradients in the rectification of binary mixtures can be applied directly to the calculation of gradients in the isopiestic rectification of mixtures however compIex, provided the composition of the feed and the point and sharpness of cut be known, and the components of the mixture follow Raoult’s law. These equations are, therefore, directly applicable in the design of rectification equipment for natural or refinery gasolines and, in general, for any mixture of hydrocarbons in which the concentrations of the individual components are known.

LITERATURECITED (1) McAdams, W.

H., and Morrell. J. C. I N D ENO

(’HEX..16. .<7.i

(1924).

RECEIVED March 18, 1932.

--

Studies in Distillation Graphical Method of Computation for Rectifying Complex Hydrocarbon Mixtures J. Q. COPE,JR., AND W. K. LEWIS Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.

The graphical method used in treating problems in the rectification of binary mixtures is developed to be applicable to hydrocarbon mixtures, however complicated, and is illustrated by a typical problem in the fractionation of natural gasoline

T

HE preceding article has shown how the classical

methods of design of rectifying columns for binary mixtures may be applied to the rectification of complex mixtures of hydrocarbons. During the last decade the algebraic technic of design has, in the case of binary mixtures, been replaced by graphical methods. It is the purpose of this article to develop suitable graphical procedure for the case of the complex mixtures in question.

REPRESENTATION OF MATERIAL BALANCE The graphical technic for computation of the continuous rectification of binary mixtures is based upon the facts that the relation between the composition of the vapor rising into any plate in a column and the liquid flowing down from that

plate on to the plate below is given by a simple equation representing a material balance, and that, assuming perfect between liquid and vapor on a plate, the composition of the vapor rising from a plate is a unique, known function of the composition of the liquid on it.* Both of these relations can be represented graphically on a vapor-liquid composition diagram, the former, through proper choice of basis, at least approximately by a straight line, and the latter by the equilibrium curve. The change in concentration from plate to plate is obtained by staircasing between the two. The line representing the materia1 balance is called the “operating line” of the column. 1 A minor modification of t h e method takes care of t h e effect of plate e5ciency, provided this is known.