5 2 2 THE JOURiVAL OF INDUSTRIAL AND EHGIAVEERING CHEMISTRY. Aug., 1909

The methods for tin and antimony are in all essential respects well-known methods, but we have found in the laboratory that they yield good re- sults as the following indicate:

Antimony by alka-

method, with final mow- with final weigh- Tin-volu- weighing as Sb& volumetric. iug as SnOs. metric.

Alloy 17.28 17.76 7.82 7.98 4874 17.33 ,7,46 17.90 8.10 7.92

.... 17.68 . . . . . .... ....

Tin by alkaline line sulphide Anti- sulphide method

.... 17.5,) ..... .... ....

Antimony by Anti- Tin by alkaline alkaline sulpbide mony- rulphide method

Alloy. weighing as SbzSa. metric. weighing as SnOp metric. .... 8.02 1 8.45 73.98 73.87 4873

method, with final volu- with final Tin--volu-

.... 7.91 I 8.38 74.03 75.47 1 . . . . . . . . . 74.33 1

. ... 7.91 8.03 . . . . . . . . . 74.42 } 74.16

. . . . . . . . . . . . . . . . . 73.31 I

. . . . . . . . I . . . . ..... 74.98 I

. . . . . . . . . . . . ..... 73.51 J 828 I

J We wish to express our thanks to Mr. F. W.

Smither who made the analyses of the two samples of Babbitt metal by the alkaline sulphide method. He used the method described in N. W. Lords I Notes on Metallurgical Analysis, 2nd Ed. Also to Dr. H. C. McNeil for suggestions on the various methods tried.

[CONTRIBUTION N O . 6 FROM THE RESEARCH LABORATORY OF APPLIED CHEMISTRY OF THE MASSACHUSRTTS INSTITUTE OF TECHNOLOGY.]

THE THEORY OF FRACTIONAL DISTILLATION. BY WARREN R. LEWlS. Received May 4, 1909.

The theory of the separation of a binary liquid mixture into its components coming up for dis- cussion in a seminar conducted by the author in the Research Laboratory of Applied Chemistry of the Massachusetts Institute of Technology, i t was discovered that but very little work has been done on the subject, and that the results already obtained are incomplete and difficult of access. While qualitatively the phenomena are well under- stood, methods of calculation of the quantitative efficiency of separation by simple distillation have never been developed, and although separation by rectification has been ably treated mathematically by Hausbrand, still the developments in physical chemistry since the time of his publication enable us to grasp the concepts involved more clearly than was then possible, and thus to develop the formulae more simply, and to carry them somewhat further than he. The general importance of the subject, the entire absence of quantitative treat-

1 Hausbrand, Rectificir- und Distilir-Apparate, Julius Springer, Berlin, 1893.

ment of its most important phase, as well as the inaccessibility of Hausbrands and other pertinent articles,2 especially to American readers, justify the publication of the methods herein developed.

In the separation by distillation of a binary mixture of miscible liquids, we have, in the ter- minology of the phase rule, a divariant system. Throughout this discussion we shall make the assumption that the pressure remains constant, inasmuch as most distillations are carried on under atmospheric pressure, and very closely approximate this condition. All our formulae will hold equally well whatever this constant pressure may be, whether below or above atmospheric, and it must also be mentioned that these formulae apply like- wise to distillation a t constant temperature, the pressure being varied as may be necessary to main- tain the equilibrium between the gaseous and liquid phases. If, now, this one external con- dition (pressure or temperature) imposed upon the system, be kept constant throughout, the effect is the same as reducing i t to a mono-variant one. The phase rule, then, tells us that, which- ever of the remaining variables we may choose as the independent one, all others are functions

of i t alone. Since our problem is the separation of a binary mixture into its components, we naturally choose the compositions of the liquid and the gas above it as our variables. A t con- stant pressure (or tcmperature) the composition of the gas in equilibrium with a given liquid mix-

Zeatschrijt jur S#zralusznd, 1884. 1885 1896

LEWIS O N THE THEORY OF FRACTIONAL DISTILLATIOS. 523

ture is fixed, and we assume that the composition of the vapor evolved from a given liquid has been experimentally determined for the case to be studied. The methods of making these measure- ments are to be found in all the ordinary text books on the subject.

Let us define as the composition of a liquid or vapor mixture the fractional part by weight of the more volatile component; thus a liquid or vapor consisting of 40 lbs. of ether and 60 Ibs. of carbon tetrachloride would have a composition of 0.40. We will designate the composition of the liquid by c, that of the vapor by C. C is then a function of c alone, and we have assumed this function experi- mentally measured. These data will be points determining some curve such as A B C (Fig. I) . This curve can, by definition, not start a t A below the line d C. It may cross this line farther up, as shown dotted in A B D C. This case we will consider later. If this curve coincides with the line A C, we have

c = c ; the composition of the vapor is under all con- ditions the same as that of the liquid and separa- tion by distillation is impossible. This is the case in mixtures of optical isomers.

SIMPLE DISTILLATION.

If a binary mixture of composition c, be put into a still and brought to a boil, i t gives off a vapor of composition, C. This vapor is removed as rapidly as formed, and thereby the composition of the liquid, and in consequence that of the vapor as well, are lowered. Let the weight of liquid in the still equal L, and its composition c. The weight of volatile component is cL. If now an amount of liquid d L , of composition C , be boiled off, the composition of the liquid has changed to c + dc, where dc is a negative quantity, and it is evident that

cL + C d L c + & = ----, L + d L

Multiplying out,

CL + L d c + c d L + d c d L = CL + C d L . Discarding differentials of the second order, and

solving

or (C - C ) d L = Ldc,

d L - dc - _ - L c - c c

1 See Young, Fractional Distillation Macmillan & Co.

In - = - c. L, i,-

This equation gives us the ratios of the weights of the liquid in the still for a given composition change. A consideration of the limiting cases involved is interesting. Thus if all the liquid be boiled out of the still, i. e. , if L, = 0, it is evident that since dc cannot give an infinite value to the integral, C - c must become zero; that is to say, all the liquid cannot be boiled out of the still until the compositions of the liquid and vapor phases have become equal; the final composition of the liquid in the still will be represented by the point A in Fig. I (by the point C if the original mixture followed the dotted curve and lay between D and C ) . This equation is not applicable to condensa- tion, owing to our original assumption that the gaseous phase is removed from the system as soon as formed. This is indicated by the fact that the equation becomes indeterminate for L, = 0. Since we can give finite values to L, and L2, i t is clear that dc and C --. c can only have zero values simul- taneously, i. e . , if there be no change in composi- tion of the liquid during distillation, the com- positions of liquid and vapor are equal, and con- versely. Such would be the point D of Fig. I . That this is a coniposition of constant boiling point is proven by the fact already referred to, that the temperature is a function of the concentration alone.

If C can be expressed as a known function of C , this equation is capable of immediate integration and consequent solution. This is possible in the case of certain mixtures, and this solution would be of interest to the mathematical physicist; but inasmuch as these cases are without technical importance, a general graphical solution will be given instead, applicable in all cases where the c - C curves have been experimentally deter- mined, and accurate well within the experimental error of the data. If c be made the abscissae, and

the corresponding values of ~ - the ordinates

of a plot, the area under the curve obtained, be- tween the ordinates c2 and c1 is equal to the integral desired. This curve is shown for the case of alco- hol-water mixtures in Fig. 2. The data are from Gronings tables.

The usefulness of this curve can best be shown

I

c - c

1 Handb d Sfizratusfab I Maercker, p. 626

524 T H E JOURNAL OF I N D U S T R I A L A N D EIVGISEERIIVG C H E 3 1 I S T R Y . hug. , 1909

by a concrete example. Let us assume that we have a 50 per cent. alcohol (by weight), and we wish to know what percentage distillate we shall obtain if we distil this mixture, in a simple still,

C

I C-.'

T 4

till the liquid residue in the still is j per cent. The area ARCD, measured with a planimeter, is found equal to 1.075. To this a niiiius sign must be prefixed as the increment in c is negative. Hence

Lz - In - - -1.075. Ll If we start with one pound, I,, = I , and

ln I,, = -1.075 L, = 0.3413 lb.

We have, then, left in our still 0.3413 Ib. of mixture containing 0.0171 lb. of alcohol and 0.3242 lb. water. Since we started with 0.5 lb. of each, our distillate must contain the difference, i. e . , 0.4829 lb. alcohol and 0.1758 lb. water or 0.6587 lb. of a 73.3 per cent. alcohol. In this way we can find the separation obtainable in this or any other case, starting and ending with any desired composition in the still, if the C - c curve be known.

While this method suffices both theoretically and practically to solve all problenis as to the separation obtainable by simple distillation, still if these problems arise frequently and must be

solved rapidly, the use of the planimeter is some- what disadvantageous. In this case, however, a plot may be made embodying the results of pre- liminary calculations and suitable for future ref- erence. Assume that we start with a 50 per cent. alcohol in the still. The vapor arising from this is 81.7 per cent., but only an infinitesimal amount of this vapor can be collected, the residue remaining practically 50 per cent. We have al- ready shown that if the residue be boiled down to 5 per cent., the distillate is 73.3 per cent., while if the alcohol be entirely expelled, there is no residue in the still, and the distillate is j o per cent. Between these extremes still other points may be calculated, and a curve drawn as is done in Fig. 3. Such curves have been indicated for an original charge of IO, 30, 50, and 70 per cent. Thus for a charge of IO per cent., distilled till the residue in the still is 2.5 per cent., we read directly from the curve that the composition of the distillate is 0.432, i. e . , 43.2 per cent. alcohol. These curves may be calculated and plotted as closely together as neces- sary, and interpolated for intermediate values.

If the amount of distillate be likewise desired, i t may be obtained from the readings of these curves with great ease. Thus the curve tells us that a IO per cent. alcohol distilled to a residue of 2 . j per cent. yields a distillate of 43.2 per cent, Assume the charge one unit by weight, say one pound. The distillate is x lbs. The alcohol in the charge must be equal to that in the residue plus that in the distillate, i. e . ,

0.10 = O.O2j(I - X) f 0.432X x = 0.184 lb.

A IO per cent. alcohol separates on distillation giving 0.184 lb. of 43.2 per cent. distillate and 0.816 lb. of 2 . j residue per pound of charge.

Our considerations, so far, have been purely theoretical, Let us now consider the disturbances that enter into the operation. These will fall under three heads, superheating, chilling of the vapor, and priming. Local superheating in the liquid can introduce no serious disturbances. The mixture is so agitated by the ebullition that the composition differences a t various points cannot be great and cannot long persist. Even though ebullition be excessive a t isolated points, the vapor evolved must of necessity have the equilibrium composition, required by our equations. Super- heating of the upper part of the still cannot change the composition of the escaping vapors, and hence

LEWIS 0-V THE THEORY OF FR4CTIOSAL DISTILLATIOS. 5 2 5

cannot affect the separation, though it would involve a waste of heat.

If the vapors be cooled they partially condense, depositing their equilibrium liquid, thereby enriching themselves. The condensate runs back into the still, to be boiled over again. This phenomenon will be treated under condensation, but i t may be noted here that devices intended to increase the efficiency of separation by preventing superheating of the

Xot so with chilling of the vapors.

A+. 3. upper part of the still always depend on chilling of the yapors to attain their end, ignoring the increased heat consumption involved in the re- evaporation of the partial condensate.

X vapor escaping from a liquid always carries a certain amount of that liquid with it in mechanical suspension. This suspended liquid is known as priming, and the amount of it depends on the rate of ebullition and the nature (viscosity, etc.) of the liquid. Borrowing a term from the steam engineer, let us call the fractional part of the total vapor which is true vapor the condition of the same; thus if a mixture is 95 per cent. vapor and j per cent. suspended liquid, its condition is 0.95. I t is evident that the suspended liquid must be of the equilibrium cornposition, since it was swept out of such a liquid and has no tendency to change. Our original equation for the change of composition with the loss in weight on distillation becomes

c L + x C d L + ( I --X)cdL L + dL c + dc = -___ f

1 See Jour. Am. Chem. Soc , SO, 1282.

or CL + Ldc + d c d L + d c L = CL + x C d L + dcL -

x c d L , whence,

Ldc = (C - c ) x d L

d L - L X ( C - C )

dc - -

In general this equation cannot be integrated, graphically or otherwise, unless x be known as a function of c. In a mixture like glycerine and water, where the one component is exceedingly viscous, x would perhaps be roughly proportional to the glycerine concentration, but in a mixture where the viscosities of the two liquids are not widely different, such as benzol-carbontetrachloride, alcohol-ether, alcohol-water, etc., i t is probably safe to assume that the condition is a function of the rate of ebullition alone, and would be prac- tically constant during a given distillation. In this case our equation becomes

The graphical evaluation of the integral, by means of the curve of Fig. 2 , makes the solution of this equation a t once possible, enabling one to calculate the decrease in efficiency due to priming.

If we have the curves of Fig. 3, vie have already

shown how - can be calculated by their use for any desired c1 and c. The logarithm of this ratio is, however, the value of the integral required, between the indicated limits, for the case x = 1.0. In this way the integral can be determined, and introduced into the above equation, thus solving it without the use of a planimeter.

A word as to the experimental determination of the condition will not be out of place. If the tem- perature and pressure be measured, the composi- tion of both phases in the mixture can be obtained directly from the tables. The measurement of one other factor dependent for its value on the relative amounts of these phases suffices to deter- mine x . Thus the heat content may be measured by the use of a throttling calorimeter, expanding the vapor adiabatically till superheated, and finding the resultant condition. Another method

L Ll

526 T H E JOURNAL OF INDUSTRIAL

would be to condense the whole mixture and find If a mixture

of liquid and vapor is a t a temperature, t , and pressure, e , empirical tables give us the com- positions of the phases, c and C. Per unit of weight there are x parts of vapor and I - x of liquid.

* the composition of the condensate.

The composition of the whole is then,

x c + (I - x ) c = Y . If Y be measured, we have

Y - c x = __- c- c To determine Y , and thereby x, a small portion

only of the vapor need be tapped off, condensed, and analyzed.

The equations we have developed suggest a new method of a t least theoretical interest for deter- mining the C - c curve. If a measured amount, I,,, of a liquid mixture of known composition, c,, be distilled, the final weight, L,, and composition, c2, be found, then the natural logarithm of is

the value of the

L L,

[ dc I C l c=c

Let

In 5 = A. L,

If various values of L,, and the corresponding c, be determined, and the A - c2 curve be drawn, then the tangent to this curve a t any point, c.

is the value of __ at that point. The tangent c -- c may be determined graphically or empirically and the value of C calculated. The usual precautions to prevent partial condensation, etc., would, of course, be necessary.

I

PARTIAL CONDENSATION.

I n the ultimate analysis the only method of increasing the composition of a vapor, that is, enriching it in the more volatile component, is partial condensation. This may be direct, with or without the aid of a rectifying column, or total, followed by a partial revaporization (redistilla- tion). Redistillation evidently involves the greater heat consumption, and hence the importance of the other process.

There are theoretically two different ways of 1 If the composition, c, of the liquid in the still be known, C may be

read directly from the curve. Fig. 1, or the corresponding tables.

A N D ELVGINEERING CHEMISTRY. Aug., I909

producing a partial condensation, that form two limiting cases between which all actual condensa- tions lie. The first, which we have chosen to call simple condensation, is characterized by keeping the condensate in contact and equilibrium .with the residual gas, till the partial condensation is complete. The second, differential condensation, consists in removing the condensate from the system, as soon as formed. Continuous simple condensation may be perfectly and easily realized by maintaining a certain amount of liquid con- densate in the condenser, bubbling the incoming vapors through that liquid, securing the partial condensation by cooling the liquid itself, and re- moving the excess of liquid formed by a con- tinuous overflow at the surface where the vapors escape. We have no mechanical device to accom- plish a perfect differential separation, though this is the end a t which condenser design must aim.

If we have a weight of gas, G,, of composition C,, and wish to enrich i t by simple condensation to a composition, C,, we know that the condensate must have a conlposition, c2, connected with C, by the curve of Fig. I . Let the residual weight of gas be G,. Equating the original and final amounts of volatile component,

C1G1 = C,G2 + cZ(G, - GZ), or

Since neither G, nor C, - c2 can be negative, this equation loses its physical significance if c2 be greater than C,; in other words the limit of enrich- ment of a vapor by simple condensation is that vapor which is in equilibrium with a liquid of the composition of the original vapor, while the yield of enriched vapor is in this limiting case infini- tesimal.

If the condensation be a differential one, we know that the new composition of the gas is equal to the new amount of volatile component in the same, divided by its total weight, i. e.,

CG + cdG G + d G C+dC=---

whence,

CG + GdC + CdG + dCdG = CG + cdG. Discarding the differential of the second order and

solving, dG dC G -c=C _ -

LEMIS O X T H E THEORY OF FRACTIONAL DISTILLATIOA. 527

c, In G2 - - --\c,c.l dC

GI -r c dC - - C G, = GI@ C,

Since C -- c is always positive, G, can become infinitesimal only when C - c becomes zero, i. e . , when the composition of the vapor has reached the point C (or D, as the case may be) of Fig. I .

A differential condensation may be shown to be more efficient than a simple between the same limits, as follows: From the nature of the case we know,

except for limiting cases when they may become equal. Hence,

c < c , < c

c---c> c-cc, ,

c-c C-C2 I I < - --

P C , PC,

P C ,

PC.,

dC c -c2 >L c - c C , - C c ? :L, ___

Therefore,

c, - c2 C, - c2 >G,sim.) = G ~

G2(dif.) = G,e 3 C Hence a differential condensation is more efficient

than one, or any number of successive finite, simple, condensations between the same limits, represents the theoretical maximum of efficiency of. separa- tion, and may be considered an infinite series of infinitesimal simple condensations. The perform- ance of a condenser should always be given in per cent. of this theoretical maximum, which may be determined by measuring the area under the curve of Fig. 4 (for alcohol-water).

Let us suppose that we require an 85 per cent. distillate from a 2 0 per cent. alcohol. The still yields LIS a vapor of 67.7 per cent. We have, then,

C , = 0.677, C , = 0.85 and c, = 0.657 (from the c - C curve). A simple condensation yields us then 10.4 per cent. of the original weight of vapor. To calculate the yield of a differential separation, we find the area under the curve of Fig. 4, between the ordinates 0.85 and 0.677 to be 0.458. Hence,

i. e . , the yield is 62.3 per cent. of the original vapor, obtained as 85 per cent. mixture. A simple con- densation in this case has an efficiency of only

16.7 per cent. of the best achievable. If instead of one simple condensation we use three, enriching the vapor successively to 74, 80, and 85 per cent., we obtain a yield of 8 j per cent. alcohol equal to 5 2 . 2 per cent. of the original weight of vapor, i. e . , an efficiency of 84 per cent. One readily sees where the efficiency of the ordinary laboratory types of still head has its origin, and how the efficiency of a given type can be very closely

528 T H E JOURLVAL OF I S D U S T R I A L A S D EAVGIAVEERI+VG C H E M I S T R Y . Aug., 1909

estimated. But of infinitely more importance is the application of this method of calculation to the study of the performance of technical con- densers, and to the problem of their improvement and perfection, which is being taken up by this laboratory. The thing to strive for in condenser design is the removal by suitable mechanical devices of the condensate from the vapor as rapidly as possible after formation, and the success with which this ideal is approached is measured by the efficiency of the condenser, calculated as shown.

INTERMITTENT RECTIFICATION.

We assume a still containing the charge, sur- mounted by a plate tower of the usual type, above which is a partial condenser, the condensate re- turning to the tower, and the residual vapor pass- ing out to a total condenser to form the required product. For our theoretical discussion we make the following entirely allowable assumptions: ( I ) that the liquid in the still is great enough in amount so that for a finite time the whole apparatus may be considered as acting as though a vapor of constant composition were indefinitely fed into the bottom of the tower; ( 2 ) that the rate of flow a t all points in the apparatus is constant, thus pro- ducing a stationary condition; and (3) that there is no priming of the vapors.

We shall designate by V the weight of vapor passing through a given section of the apparatus in a unit of time; by C the composition of this vapor; by 0 the weight of liquid passing the sec- tion in the same time; and by c the composition of this liquid. The plate from which this liquid or vapor comes will be indicated by subscripts; thus through a section above the liquid in the still and below the lowest plate passes a rising vapor, V,, of composition C,, coming from the charge in the still of composition c,; and a re- turning overflow, o,, composition c l , from the first plate. Through a section above the n-th plate rises a vapor, V,, composition C,, and falls a liquid, 0, TI, composition cn + I , from the plate directly above, There are p plates in the tower, and on to the fi-th plate flows the con- densate, Oc, c,, from the condenser; and from the condenser passes the distillate V,, C,. The heat required to form the vapor, V, from its equilibrium liquid (of indefinite amount) at the boiling point, is K.

Consider the whole apparatus above a section between the n-th and n + plate. There is

entering this part of the system only the vapor V,, from the plate below. There is leaving i t the distillate V,, and the overflow 0, + Accord- ing to assumption ( 2 ) these must be equal:

1)

The amount of volatile component entering and v, = V, + 0, + I.

leaving must likewise be the same, or

Applied to a section just below the first plate

(1a)

these equations become

and v, = v, + o,,

c,v, = cy, + clo,. I za ) Solving these last two equations simultaneously

gives us,

z b ) V, c1 = c, - - (C, - C,). 0,

I t is required of a rectifier to take a charge of a definite composition and yield a definite distillate, a t a specified rate. We may then consider C, and V, as known from the nature of the problem presented. In other words, the composition of the overflow back into the still is a function of the amount of that overflow alone. This composition, c,, cannot exceed the value C,, because the liquid formed by partial condensation is always lower in composition than the vapor from which i t is formed, and the two compositions can only be equal in the limiting case of total condensation, i. e . , 0, in- finitely great. c1 cannot be less than c, since then the vapor above the first plate would be lower in composition than that leaving the still, i. e . , our separation would be a negative one. We have then,

c, < c, < c,. Equation ( z b ) transformed gives us

v (C - C,) 0 -LC_Cp Co - ct 1 -

whence do, - - Vc(C,-C,~

- dc, (C, - c,)

Since our tower gives us a separation of the components, Cc > C,, and hence this differential coefficient is positive, increasing in value with increasing cy, it is evident that 0, is a mini- mum when c, has its lowest possible value, i. e . , when c1 = c,. Since from equation ( I ) the less the overflow, 0,, the greater V, for a given ITo, the maximum efficiency is obtained from a tower when c, = c,.

LELV-I.5 0.V T H E T H E O R Y OF FRA4CTI@.\rAL DISTILL.4TIO.V. 529

I t might be suggested that this is not of necessity have all the da ta necessary for calculating the a point of maximum heat efficiency if one com- actual working efficiency of the system. The ponent of high heat of vaporization should de- crease in the overflow, but be overbalanced by a

increase is shown, however, by the fact that

r ec t i~e r is yielding c --GI pounds of distillate per pound of vapor from the still, whereas a perfect

co - co pounds. The efficiency type would yield

per cent. is equal, then, to I O O

cc - C l greater increase in the second component of very low heat of vaporization. That both components

c, - co cco - Cl)(CC - co)

dC, ' (C, - C A Z ' (C, - c1) (C, - c, 1

VC(CC - C') v d(ClO1) dc1

d ( I ___ -- ,)O,= __ ( I - c ) VdCC -~ ___ - C0) and Equations ( I ) and ( 2 ) give us

c n -1 = c, --5- (CC--CtL). (C' - C,)Z ' o c - I

- c o

both positive quantities. These equations only say what is self-evident, that the only way to increase the overflow is to boil up more vapor out of the still and completely condense this increase.

This argument may be applied with equal validity to equations ( I ) and ( z ) , and so we may say that a tower is working a t its maximum effi- ciency only when the compositions of the liquids on succeeding plates do not differ more than differ- entially. But in this case the vapors rising from them and in equilibrium with them can likewise differ only differentially; to obtain a finite separa- tion a t maximum efficiency requires an infinite number of plates.

These equations enable us to calculate directly the performance of a theoretically perfect rectifier, for any desired separation, namely by making c, = c, in equation ( z b ) and solving for 0,. Sub- stituted in (IU) this gives us,

C' -- c c o - c o ---I Yo = s-,[I +

( I b )

If a rectifier receive a charge of composition c,, and yield a distillate, C,, the theoretical maxi-

cc - C' = v,-* co - co

mum of efficiency is '' - " pounds of distillate c c - c,

recovered per pound of vapor boiled out of the still. If I(, be the heat of vaporization of the vapor issuing from the still, the heat conslimption

per pound of distillate would be KO

E. T. U., not allowing for bringing the liquid up to the boiling point.

If we can measure the actual composition of the overflow into the still of an apparatus in operation, by the insertion of a thermometer into the liquid on the first plate, by removing a sample and analyzing it, or by any other suitable means, we

c -co c, --Go

I n a working column it is essential to have c n L x , for a given C,, as large as possible, and this is attained by increasing O,, + I. We have already shown that the greater O n + I , the greater c, ,+, and in consequence the greater the rate of separa- tion as we go up the tower. I t is desirable, then, to maintain O n T I as great as possible on every plate. If, due to poor insulation, we have partial condensation in the lower part of a column, the overflow thus formed does not pass the upper plates, and in consequence these plates are less efficient than they would be if this condensation took place above them and returned through the whole system. In other words, the more perfect the insulation of the plate tower of a rectifier, the better its performance. The condensate should be entirely formed in the condenser a t the top, and return through the entire length of the rectifier.

We see then that in an ideal rectifier the vapors from the still pass up against the counterflow of the partial condensate formed from them in the condenser. As the vapors enter each plate they condense, boiling off with their heat of condensa- tion a new and richer gas mixture from the liquid there, the composition of the vapor above a plate being connected by the c -C curve with the liquid on it. The function of the column is to remove as far as possible, from the condensate of the condenser above, the volatile constituent, returning as a final overflow to the still only the non-volatile plus what volatile cannot be boiled out. Since a differential condensation leaves least volatile component in the liquid i t leaves least for the e column to perform, and a given tower gives its best performance when the condenser over it is a differential one.

If we consider that part of a rectifying column between two sections, one just below, and one just above the n-th plate, equating receipts and deliveries,

5 30 THE JOURNAL OF INDUSTRIAL AND ENGINEERING CHEMISTRY. Aug., 1909

Vn-1 + On+I=Vn + O n , (2%-1 Vx-I+ c ~ + I On+I= CnVn + CnOn,

v, + 0, = v, + o,,

(3)

(4)

13a)

(4a)

and

which, applied to the first plate, give

and c,v, + c,o, = cy, + clo,.

Having shown its desirability we now introduce a new assumption, vuiz., (4) that the walls of the plate column are non-conducting. On to and from each plate are running overflows approximately equal in amount and temperature, and so the quantities of heat brought in and taken out by them are very nearly the same; this must then likewise be true of the heats of the vapors con- densing and evolving, i. e . ,

V,K, = V,Kn = Vn+,K, t l = V,K,=V,K,. (5) This equation is beyond doubt very closely

fulfilled because these heats of vaporization are so much larger than the corresponding specific heats of the liquids flowing through, that a rather large change in the heat of the liquid would vitiate the above equation but little.

Now given K n = F,C,), (6)

c, = f,C,\, ( 7 )

K, = W,) (6a)

c, = f(C11, (7a)

and

with their special cases,

and

we have in equations (~a), (za), (3a) , (4a), (5), (6a\, and (7a) the known quantities c,, C,, V,, C,, and KO, while unknown are V,, 0,, c,, 0,, VI, c,, C,, and K,. With the help of these seven equa- tions, however, we can solve for all these unknowns in terms of any one of them, let us say, V,. Ap- plying these equations successively to each plate as we go up the tower we can express all unknowns in terms of this same V,, getting finally,

c, = (p(V,), and V, = #(Vo). We know, however, that

the nature of the function depending on the method of condensation employed. These three equa- tions enable us to evaluate V,, and from the point of view of the mathematical physicist our problem is solved. Practically, however, we must proceed somewhat differently.

cc = W6,CC) 3

Applied to the top plate, equations ( I ) and ( 2 ) give

arid o,=v,-vv, ( I C )

(2c) ccoc = c,v, - c,v,. We know what the apparatus is to deliver, i. e . ,

we know V, and C,. Now assume either ( I ) a simple condensation, in which case c, is a t once known from the c - C curve; or (2) a condensation definitely known from the measured performance of the particular condenser used, giving C, empir- ically; or (3) for the ideal case a differential con- denser. I n either case assume a value for either C,, V, or 0,. Under the first two assumptions all these quantities are a t once known; under the third, knowing C, and C,, c, can be calculated a t once; if V, or 0, be given, readily by successive approximation.

Equations (3) , (4), and ( 5 ) g' ive us v*-I + 0, = v, + o,,

c,-Iv,-I + c,o, = cpvp + cpop, 3c)

(4C)

We must now introduce our last assumption, which is unfortunately not so valid as our previous ones, namely, that the heat of vaporization of the two components may be considered as constant over the temperature interval involved, and that the heat of vaporization of the mixture equals the sum of that of the components. While this assumption is only an approximation to the func- tion, K = F(C), i t is the best we can make a t present and in many cases will beyond doubt give very close results. It cannot be used when the heat of mixing of the liquids is large, as in the case of sulfuric acid and water, for example. A n exhaustive study of the heats of vaporization of important mixtures must be undertaken. Calling k, and k, the heat of vaporization of the com- ponents, we have

V,Kp = vp-,Kp-lI. (5C)

K = k,C + k , ' ~ - C). (6) Knowing C, we can calculate K, a t once, and

write : (5d) VpKp = Vp-,K,- I =

V~-x[kaCp-l + k,'I - c p - x ) ] . The equations ( 3 c ) , (4c), and (5d) contain only

They are solved as follows: three unknowns, Vp-,, Oj, and Cp--r.

. ( 7 ) c p - rvp - I - CY, 0, = vr-I - v, = CP

LEWIS ON THE THEORY OF FRACTIONAL DISTILLATION. 53

CgVc - ccv, - V S O __ denser, and is there enriched to 8 j per cent. Assume ~fi-Cfi-1 kb +Cg-x(ka-k,) ( 8 ) a delivery of IOO Ibs. 8 j per cent. alcohol per unit

Equation (8) is a simple algebraic containing of time. We have already shown under con- densation that in this case each IOO Ibs. of 67.7 per cent. vapor entering the condenser yields 63.3 Ibs. enriched product and a condensate of 36.7 lbs. of 37.9 per cent. alcohol. Hence we have

V, = 100, C, = 0.85; V p = 158, C, = 0.677;

- - V p 1 = ~

only cg-I undetermined. tained, (8 ) , being in its more general form

cfi-I being thus ob- Equation the c - c curve gives US cg-I.

C n V c - CCVC - - - V I S , __ Cn-Cn-1 k6 + Cn-x(ku-kb)

can be applied a t once to find Cp-2, and so on down the still to c,, below the bottom plate. Equa- tion (8) gives us V, and (7) 0,. We have found then the theoretical performance of the given tower, with the assumed condensation. If this con- densation were differential, the performance calcu- lated is the theoretically maximum of which the given still is capable. Varying the overflow by assuming other values for the condensation pro- cess, we determine as many points as may be

curve, interpolating for necessary on the c, - -

any desired values. For a given column, with a given method of condensation, a series of such

curves may be drawn for varying dis- V O c, - - lc

tillates (i. e . , varying C,) and these curves would give a t once by substitution in equation (IU) and (za) the maximum efficiency for a required opera- tion, of which the apparatus is capable.

The heat consumption q, corresponding to a charge in the still of composition, c,, and a distil- late, Cc, requiring a vaporization from the still of vo pounds per pound of distillate obtained is

VO

IT,

v c

. T. U. per lb. distillate, disregarding = VOK - 0 B v, the heat necessary to raise the liquid in the still to the boiling point. Of this heat consumption

are extracted by the cooling water of the partial condenser, while the rest, K, B. T. U., must be absorbed in the final condenser, plus the amount necessary to cool the distillate to room temperature.

A numerical illustration will most readily make clear the use of these equations. A plate tower, with a differential condenser above, is yielding an 85 per cent. distillate. We assume that the liquor on the top plate is 20 per cent., thus giving off a vapor of 67.7 per cent., which enters the con-

0, = 58, C, = 0.379, cp = o . z o ; ~ , = 371, k, = 979. The equation connecting the composition of the

liquid on the plate with that of the vapor entering i t

V K CPVC - ccvc - - fi P-- cg- cg-1 JEb + C,--I(k, - b)

becomes then for this column under these con- di tions,

979 - 608 C n F 1 = 899700 (c, - cw-I). 100 cn - 8 j

Applied to the top plate, where c, = 0.20 , this equation gives us Cg-I = 0.633, as the com- position of the vapor rising from the p - l - f k plate. The liquid on this plate, c g b X , must be, from the c - C curve for alcohol, c + - - ~ = 0.158. Sub- stituting this value, and solvingfor the +-2-th plate, we obtain Cg-2 = 0.621; hence cP-* == 0.149. I n exactly the same manner we obtain, c p F 3 = 0.148 and cp,4 has practically the same value. In other words, further increasing the length of the tower mill not materially improve the separation. Assuming therefore four plates we have cb-4 = c,, the liquid in the still. c, is so nearly equal to c, that, barring inaccuracies involved in our assump- tions and the available data, this combination gives us an efficiency of 99 per cent. of the theoret- ically maximum. This is due to the fact that we assumed a differential condenser, and such figures as these show, more clearly than any algebraic equations can, how much the efficiency of a rectifier depends on the condensation. The perfection of the condenser is the problem of the still designer to- da y .

To obtain the heat consumption, vL(c~-c0) 100(0.85 - 0.619)

co -c1 0.619 - 0.148 = 49.1, - 0, = ~- -

Ir, = v, $- 0, = 149.1, VOKO - Q = - - 897 R.T.U. per pound of 85 per cent.

VC distillate produced from 14.8 per cent. beer.

Let us assume an original charge of composition cz and weight I, in our still, rectified to yield a t all

532 THE JOUR,VAL OF I,VDliSTRIAL A11D EA-GISEERI:YG CHELI.IISTRY. Aug., 1909

times a distillate, Cc. This is accomplished by gradually increasing the flow of cooling water through the condenser, thus enlarging the over- flow Oc, and in consequence the vapor consumption

%. If this process be carried on till the residue in the still is of a required purity, c:, what has been the total heat Consumption, Q, per pound of distillate ?

We have shown ( I ) how for a given still in opera-

curve may be exactly tion points on the c, - - measured; ( 2 ) how this same curve may be calcu- lated for a given still working a t its theoretical maximum efficiency; and (3) how this curve may be found for the theoretically perfect still. We shall now give the formulae, which, if the values read from the first curve be inserted, will give the actual heat consumption of the operation measured ; if from the second curve, the theoretically minimum heat consumption of the still studied; and if from the third, the theoretically minimum amount of heat to give the separation required. If we call

these heats Q1, Q2, and Q3 we may consider as

the theoretical efficiency of the still in question,

Q as the actual working efficiency of the separa-

tion, and - as the efficiency of operation of the still.

Calling D the weight of the total distillate ob- tained when the composition of the residue in the still has reached c,, we have

VC

VG VC

Q Q2

a, Q a,

c:L - C,D CG = ~ , or L - D

0 c -c, cc - c,

D = L A - - .

We know likewise that the heat consumption per pound of distillate a t this composition in the still c, is

curve we can By the use of the proper c, - ~-

thus calculate for varying values of D the corre- sponding values of q, and construct a curve with 131 as abscissae and q as ordinates. The total heat consumption Q is

IT, VC

Q = qdD, where sl co - c

D , = L U c,- c:, Q is measured by planimetering the area between

the D-q curve, the D axis, and the ordinate D,. Whether the Q obtained is theoretical or actual

depends on the c , - ~ curve used to calculatc

the D-q relationship, as was pointed out in the preceding paragraph.

We have thus solved the problem we set before us, showing how the theoretical working of a rectifier may be calculated and the actual separa- tion measured and followed. To calculate the disturbances due to priming, to poor insulation and resultant local condensation, and to the im- perfect establishment of equilibrium would lead us a t this time too far, but the methods here out- lined can be applied to these problems and are being taken up in this laboratory. The still de- signer must likewise have data as to the influence of the rate of flow of the vapors through the tower on the rate with which equilibrium between the vapors and the liquids through which they pass is reached. The object of this paper was not to attack these problems, but to develop methods of calculation that would show what data are needed, and prove, we trust, an incentive to experimental work in this so important field of chemical engineer- ing.

v, VC

CONTINUOUS RECTIFICATION.

While continuous rectification is the most im- portant type of practical distillation, and while the equations for such columns should in a suitable place be published a t length, the method of calcula- tion is so entirely similar to that just given, that from our present point of view nothing would be gained by developing them here. We will only outline the scheme to be followed:

The theoretically minimum heat consumption of separation is the same, down to the delivery plate, whether the process be continuous or not and is given by calculating Q, as already shown. For a given still, of given delivery, we follow down through the tower exactly as before, except that when we reach the plate upon which the beer enters we must change our equations for the new overflow. Thus, we get the theoretical efficiency of the given still.

KELLE Y O S THE GROlVTH OF PISEAPPLES. 533

To calculate the working heat consumption, know- ing the composition of the beer and distillate we need measure only that of the slop, exactly as be ore we needed to know only that of the over- flow into the still. One point must be mentioned; if the steam be injected into the vapor space above the slop we have C, = 0 ; if, as should be done, the vapor for running the tower be obtained by boiling the slop, either with coils or direct flame, as the case may be, C, corresponds to the c, of the slop.

______-_

[CONTRIBIJTION FROM THE LABORATORY OB THE HAWAII EXPERIMENT STATION. I

MANGANESE IN SOME OF ITS RELATIONS TO THE GROWTH OF PINEAPPLES.

B y W. P. KELLEY. ReceivedMay 13, 1909.

For a number of years pineapples have been grown on the upland plains of the Hawaiian Is- lands. I n some of the fields devoted to this crop there are spots varying from one to sometimes more than twenty acres in size on which pine- apples do not grow well. On these areas the young plants usually make a fair growth for a few months; but then the leayes begin to show a reddish purple color, which soon gives place to a yellowish white appearance, which color persists throughout the remaining life of the plant. Many such plants never bear fruit ; that which is produced, how- ever, is always inferior in size and quality. The areas thus affecting this crop are quite definite and with the rapid expansion of the pineapple industry in recent years, such soil has been found to be much greater in extent than was formerly thought. All efforts of the best growers, including the applications of fertilizers and lime, good tillage, drainage, etc., have not effectually changed the growth of the pines on these lands.

Practically all the soil of the Hawaiian Islands is of volcanic origin and one of its chief character- istics is its high percentage of iron, which, in many instances, imparts to the soil a brick-red color. The soil on which the pineapples become yellow, however, is usually dark in color, some- times almost Mack. From this difference in color, i t is common in Hawaii to speak of pineapple soils as being red or black, meaning thereby good or poor pineapple land ; and i t is in this sense that the terms red and black soils are used in this paper.

See Laws and Soils of the Hawaiian Island by Maxwell.

In addition to this difference in color, the black soils appear to have a finer texture than the red soils. When reasonably dry and in good tilth, the red soil usually has a granular or shot ty texture, while the black soil, under similar con- ditions, is reduced to a finer state of division. Upon thorough wetting, however, either of these types may be crushed between the fingers to such a fine state of division that practically all of i t will pass through a one-hundred mesh sieve. There is, therefore, very little true grit in this section, a very complete disintegration of the lava having taken place, which, in some localities, extends to a depth of thirty feet or more.

In consideration of the extent of the black soil and its unusual character, the Hawaii Experi- ment Station has undertaken an extended in- vestigation of the question. This work is still in progress, but i t is thought that results of sufficient importance and scientific interest have been ob- tained to justify their publication a t this time, This paper should, therefore, be regarded as a preliminary report, rather than a complete dis- cussion of the subject.

I n September, 1908, two extensive series of fertilizer plot experiments were begun : one on red soil, and one on black, There was a twofold ob- ject in these experiments: first, to determine the fertilizer requirements of pineapples in Hawaii ; second, to ascertain the cause of pineapple yellow- ing on the back soil. This paper deals with the latter of these only.

-It the beginning of these experiments, a number of samples of soil and sub-soil were taken from various parts of the pineapple district. About one-half of these samples were from the black soil, where the pines were very yellow; and one- half from red soil, producing good pines. The samples of soil were taken to a depth of eight inches; and the sub-soil from eight to twenty inches below the surface. Each of these samples represents a composite, taken from not less than six different places; and, therefore, the average of the six samples of each type may be looked upon as representing not less than thirty-six differ- ent places in the red and black soils, respectively. I n the following table is given the average water- free composition of the two types as determined by the methods of The Association of Official Agricultural Chemists.

The word type is not used in this1paper In its usual sense, but rather for the sake of brevity.