Chemical & Metallurgical Engineering, Vol. 49, No. 5

CHEMICAL

Ł m e t a l l u r g i c a l

ENGINEERING

ESTABLISHED 1902 S. D. KIRKPATRICK. Editor

MAY. 1942

“Acres of D i a m o n d s ” T

is the story which was so often told by

the late Russell H . Conwell in the famous lecture he called “ Acres of D iam onds.” I t was about an ancient P ersian who lived in contentm ent u n til a B uddhist p riest told him of m ythical mines of diam onds where the white sands of a riv er were satu ra te d w ith gems of great value. Then A1 H afed became discontented, sold his homestead and spent the rem ainder of his life in a futile search fo r this source of unbounded wealth. Finally, discouraged and disappointed, he took his own life when he found th a t the stranger to whom he had sold his farm had later discovered in the sands of one of its garden streams, literally acres of diamonds. F rom th a t old fa rm came gems th at have since decorated the heads of many monarchs.

F rom there came the famous Kohinoor diamond, now am ong E n g la n d ’s crown jewels.

One does not have to ponder long to find the moral of such a tale. We go to great extremes, to faraw ay places, to search fo r our fortunes only to find too often th a t they lie buried in our own back­

y ard s !

. Today in d u strial diam onds are alm ost priceless, b u t so, too, in a different sense, are the strategic m etals and m aterials so sorely needed to keep our w ar machine functioning ever faster and faster.

M ilitary uses will take all the magnesium, alum i­

num, nickel, chromium and copper, and m ost of the zinc and tin fo r the next two years. A new technique in war-time th r if t is therefore becom­

ing a prerequisite to the privilege of staying in business. I t is a job th a t chemical engineers in in ­ d u stry m ust face and do som ething about.

W illiam L. B att, D irector of M aterials, W ar P roduction Board, recently said, “ The biggest single source of strategic and scarce m etals lies above ground,—i.e., in the m aterials we are w ast­

in g !” H ere then is a challenge and an oppor­

tu n ity for every resourceful engineer. The better

the man, the better the job he will do. No one in the average industrial organization is better equipped to tackle the problem of salvaging indus­

trial scrap than the engineer who knows w hat unused equipm ent is obsolete and can be discarded, who knows w hat buildings can be demolished and where to tu rn for other sources of scrap and salvageable m aterials.

One fairly large company in New Jersey is salvag­

ing more than two tons a day of w7aste paper, c a rd ­ board and old rags. In addition it saves and u til­

izes old p a in t pails, 5-gal. tins, em pty drum s, b u r­

lap bags, old grinding wheels and files, barrels and nail kegs. Each of these items goes into its own special container and is collected, carefully checked and sorted. The old adage about “ w hat is worth doing, is worth doing w ell” certainly applies here. To save a critical metal only to render it useless by mixing it with other m aterials now falls into the category of crime.

Even though we would like to share in Jesse Jo n es’ optimism about the prospective achieve­

ments of our fellow chemical engineers in the syn­

thetic rubber business, we cannot honestly dodge the fact th a t we are approaching a ru bber crisis in this country. We need to salvage every possible pound of old rubber. Not only tires and tubes and hot w ater bottles and bath m ats, b u t rubber belting, bum pers and gaskets, fire and in d u strial hose, and mechanical rubber goods of all kinds, must somehow find th eir way p rom ptly back to th e reclaiming plants.

Each in d u stry nowadays has its own peculiar

shortages of raw m aterials. Some of these th rea ten

production more seriously th an all the wage-hour

and p aten t and profit controversies wre have been

making so m uch noise about. B u t hanging over

all our heads is the vital need fo r metals, not only

to make guns and ships, but, equally im portant, to

make the equipm ent and m achinery required fo r

the new synthetic rubber and other essential plants.

Perhaps, it is not exaggerating too much to think of these once common m etals as jewels of great value. I f so there are “ Acres of D iam onds” under our feet.

THIS MONTH'S REPORT

Ch e m i c a l e n g i n e e r s

have long recognized the fact th a t the basis for equipm ent design and operation was well grounded so fa r as stoichiometric and equilibrium relations were concerned. In most cases, however, the same certain ty has not been attainable in the phase of design which requires a knowledge of the tra n sfe r of m aterial between phases. A nd since m aterial tran sfer is often the factor which sets the time needed to accomplish a given operation or reaction, choice of equipment size has rem ained largely empirical. Reasoning has usually stemmed from the results of sim ilar operations or from small-scale testing.

It is only in the last decade th a t much effort has been applied to rig h t this situation. A relatively small group of men have devoted th eir efforts to the stu d y of diffusion. Some have developed the m athem atical theory. O thers have tested the theory and have attem pted to set u p correlations which can be used in practice. Much rem ains to be done arid readily useable results are still not very num er­

ous. B u t the work is progressing and chemical engineering will u ltim ately be benefited consider­

ably.

To in te rp re t the present state of knowledge in several of the m ajor diffusional operations is the purpose of the group of articles th a t follows in this issue. They are presented here in a special 48-page report— as one of

i&

long series of reports and review's of the u n it operations of chem­

ical engineering. W e commend the re p o rt to your attention, either for a quick review of im prove­

ments, new developments, and design m ethods; or as an up-to-the-m inute basis for thorough study.

W A S H I N G T O N H I G H L I G H T S

PLANNING even farth er ahead is now necessary. Prom pt delivery of many raw m aterials cannot be ex­

pected. M aterials o f construction and fo r maintenance and rep air are difficult to get and advanced ordering lielps but little. Im provisation is the keynote. M akeshift engineering be­

comes a new branch o f the profession.

SHORTAGES of material now limit the extent o f the war effort. W ash­

ington still makes more noise about wages, working hours, profits, p at­

ents and other economic factors. B ut a few people in high places are be­

ginning to realize that none o f these, probably not all combined, so seri­

ously lim it the magnitude of' actual delivery o f fighting equipment.

PATENTS will have little value to anyone if radical members o f the Senate have their way. Fortunately, it is doubtful whether Congress will go as f a r as these extremists would like. B u t it is no longer safe to use patents as a sole basis fo r industrial security. W hat they will be worth a fter the w ar is any man’s guess.

FOREIGN patent deals are all under fire in Washington. Those who would break down the whole patent system are arguing that every such arrangement has given foreign corpo­

rations or cartels control o f American enterprise. The fact that such ex­

changes have brought a tremendous amount o f useful information and

opportunity fo r aggressive develop­

ment in the United States is deliber­

ately Concealed. The campaign on this subject has a few ardent but very vocal supporters. They have no desire to seek the facts other than to distort them. Their real motive is to attack the patent system. Of course, the campaign is conducted in a costume o f patriotic zeal.

RUBBER plans were mismanaged by Je sse Jon es, so official Washington thinks. In any event, he is in the dog-house and much o f his power is being stripped from him. Thus the New Deal re-establishes its tradition that the Secretary o f Commerce is a nice figure-head who is not even p er­

mitted to speak influentially fo r busi­

ness.

POWER decides the choice between aluminum and p ap er pulp as offered to Uncle Sam by Canada. There is not enough power to make as mueh o f both as we would like to buy.

This is typical o f the situation in regard to m any things on which diffi­

cult and strange decisions are re­

quired o f both public officials and company executives and engineers.

NO MORATORIUM has been declared on anti-trust actions o f the govern­

ment. W ar and N a vy Departm ent officials can request that actual prose­

cution o f anti-trust cases be delayed by the Departm ent o f Ju stice i f it appears that immediate court action

might interfere with war effort. But the laws regarding agreements in re­

straint o f trade are still in force and the Departm ent o f Ju stice is very much on the job enforcing them.

CONTRACTS w ill continue to be re­

adjusted to save the government money. W herever there is the slight­

est evidence o f an over-generous p ro ­ fit being made 011 w ar contracts, the government is going to ask fo r changes in contractual terms. There is no evidence that comparable read­

justments are being authorized when a contractor is losing money. A s usual, the business of contracting with Uncle Sam is a one-way street.

TARIFF rates are to be lowered on many goods as the State Departm ent negotiates more trade agreements with Latin Am erican countries. The cam paign which Secretary o f State H ull has waged so successfully fo r the past eight years is continuing in discussions with Chile, U ruguay, Peru, B olivia, and p articu larly with Mexico.

CONTAINERS to package im portant chemicals generally take the same preference ratin g as do other “ operat­

ing supplies” which are required in the m anufacture o f the chemicals themselves. This means that A -3, or sometimes much better, p rio rity can be had fo r drums and other contain­

ers needed in essential w ar enter­

prises.

7 6— 0 M A Y 19',2 • C H E M IC A L & M E T A L L U R G IC A L E N G IN E E R IN G

Diffusional Operations in Chemical Engineering

TO CHEMICAL ENGINEERS, PLANT EXECUTIVES AND PRODUCTION MEN IN PROCESS PLANTS

A lthough from its title this report m ay ap p ear to b e lim ited to the diffusional a sp ec ts of chem ical en gin eerin g, it m ight better b e d escrib ed a s a series of briefed-dow n Chem. & Met. reports on sev era l of the m ost im portant of the unit operations of chem ical en gin eerin g, including rectification, drying, absorption, extraction, crystallization, le a ch in g and d issolvin g. T hese operations are all cla ssed a s diffusional, sin ce in all m olecular diffusion p la y s an important part a s the m ean s of transfer of m aterial b e tw ee n p h ases. In som e the rate of m a ss transfer is the controlling factor in the sp ee d w ith w hich the operation c a n b e carried out; in others it is not controlling, but still of interest. E ach article attem pts to sh o w the extent to w hich diffusion d o es control, an d h o w far present k n o w le d g e permits rational d esign . More than this, ea ch article is a short but com prehensive rev iew , stressing both principles an d n e w developm ents.

**/>

Introduction to Diffusional Operations In Chemical Engineering

H. F. JOHNSTONE

A s is often p oin ted out, ch em ica l p r o c e sse s an d operations are lim ited not o n ly b y the equilibrium conditions w h ich determ ine how far the action ca n proceed , but a lso b y the rate-determ ining factors w h ich control to w h a t extent the equilibrium ca n b e ap p roach ed in a g iv en am ount of time. O n e of the m ost im portant of the rate- determ inants is d iffu sion al transfer of m aterials b e tw e e n p h a se s, and sin ce diffusion is a com m on characteristic of m a n y unit op era­

tions, it is often the controlling resista n ce of the action. W h en diffusional transfer controls, it fix e s the n e c e ssa r y retention tim e in equipm ent, and h en ce the equipm ent c a p a c ity an d size.

D espite the fact that diffusion's im portance h a s b e e n recogn ized for a g o o d m a n y y e a r s, it is o n ly quite recen tly that the stu d y of unit operations from the d iffusional standpoint h a s b e g u n to p a y divi­

d en d s in better u n d erstan d in g of the op eration s a n d in n e w tools for c h e m ica l en g in eerin g d e sig n . Even n o w our inform ation is far from com p lete, but the v a lu e of the diffusional ap p roach h a s b e e n dem onstrated, an d its understan d in g h a s facilitated d esig n , ev en w h en the n e c e ssa r y coefficien ts h a v e not b e e n o b tain ed . Som etim es the v a lu e is n e g a tiv e , in sh o w in g that diffusion is not the lim iting resista n ce. In a n y even t, this sy m p o siu m w ill clarify the situation for a cross-section al group of unit operations* an d w ill a ssist m ateri­

a lly in bringing together the d iverse p h a s e s of recent d ev elo p m en ts.

Not o n ly diffusion but other a sp e c ts a lso are d isc u sse d

I

The reaso n fo r th is is obvious. In chem ical en g in eerin g , we a re u s u a lly concerned w ith d y n a m ic r a th e r th a n w ith s ta tic co n d itio n s. In c o n tin u o u s o p e ratio n s , a chem ical re a c tio n , o r a p hy sical change, m u s t be accom plished w hile th e m a te ria l is a c tu a lly m oving th ro u g h th e eq u ip m en t. E ven in b a tc h o p e ratio n s , i t is g e n e ra lly d e sirab le to m a in ta in a g ita tio n o r c o n ta c t b e­

tw een th e phases a t a m o v in g b o u n d a ry in o rd e r to a cc ele rate th e in te ra c tio n . D iffusional o p e ra tio n s , in g e n era l, a re re la tiv e ly slow. T he tr a n s f e r of m a ­ te r ia l from one p h ase to a n o th e r, th e r e ­ fore. re q u ire s tim e. T h e slow er th e diffusion th e m ore im p o r ta n t i t becomes in e sta b lish in g th e size of th e eq u ip ­ m ent, a s well as th e co st o f th e o p e ra ­ tion.

A co n sid era tio n of w h ere th is b o ttle ­ neck of d iffu sio n m ay be m e t a n d how i t m ay be broken is one of th e o b jects o f th is series of p ap ers.

D u rin g th e p a s t decade, th e re h a s been co n sid erab le in crease in th e scien ­ tific know ledge of th e d iffu sio n al o p e ratio n s . T he re a liz a tio n of th e a n alo g y e x is tin g betw een th e v a rio u s o p e ra tio n s h a s re s u lte d in c o rre la tio n s w hich h av e been fa r-re a ch in g . T hese a r e b ased on a th e o ry w hich h a s r e ­ q u ired th e u se of a n u m b er of new te rm s, n o t a lw ay s fa m ilia r. F u r t h e r ­ m ore, som e of th e in d u s trie s based 011

d iffusion a re am ong th e o ld est in existence. I t is n o t s u rp ris in g , th e r e ­ fore, to find w o rd s a n d te rm s u sed w hich a re in d efin ite and o v erlap in m ean in g . In o rd e r to av o id confusion, a few of th e m ore common te rm s used

in d iffu sio n al o p e ra tio n s a re defined

011 p ag e 80.

DIFFUSIONAL OPERATIONS As show n in th e tab le , w ith th e e xception of th e sim p le m ix in g of gases a n d liq u id s, a n d th e d isp e rsio n of sm okes a n d fum es, th e d iffu sio n al o p e ra tio n s c o n sist in th e tr a n s f e r of m a te ria l a cro ss a p h a se b o u n d a ry . A t th e a c tu a l in te rfa c e , m o le c u la r m o tio n is so r a p id t h a t e q u ilib riu m is in s ta n ­ ta n e o u sly e stab lish e d . T he e n tir e r e ­ sista n c e to th e p a ssa g e of m a te ria l fro m one p h ase to th e o th e r, th ere fo re, m u s t lie in th e s lo w n e s s , w ith w hich th e m olecules a re b ro u g h t u p to th e in te rfa c e on one sid e a n d a re rem oved from th e in te rfa c e on th e o th e r side.

The s itu a tio n is s im ila r to th e tr a n s f e r of h e a t th ro u g h tw o so lid s w hich a re in close c o n ta c t w ith each o th e r. T h ere is no re s is ta n c e to h e a t tr a n s f e r a t th e a c tu a l in te rfa c e betw een th e tw o solids,

t h a t is, th e te m p e ra tu r e of one su rfa ce is e x a c tly eq u al to th e te m p e ra tu re of th e o th e r, p ro v id ed , of course, t h a t c o n ta c t is p e rfe c t. T h ere m u s t be a te m p e ra tu re difference, how ever, from th e o u tsid e of one solid u p to th e in ­ te rfa c e a n d from th e in te rfa c e to th e o u ts id e of th e o th e r so lid in o rd e r for h e a t to flow.

T h ere is one im p o r ta n t difference betw een h e a t tr a n s f e r a n d m ass t r a n s ­ fer. W hen tw o bodies a r e in th e rm a l e q u ilib riu m , th e te m p e ra tu re of each is id e n tic a l. On th e o th e r h a n d , w hen tw o ph ases a r e in e q u ilib riu m , th e con­

c e n tra tio n of m olecules is seldom th e sam e in both p h ases. T h u s, th e d e n s ity of s a tu r a te d ste am is fa r d iffe re n t from t h a t of w a te r w ith w hich i t is in

* A notable omission from tile opera­

tions dealt with is humidlllcation which, however, Is being treated a t length in Edw ard Simons’ cu rren t series of articles on cooling tower design.—Editor.

7S— 5 M A Y 19.’,2 • C H E M IC A L & M E T A L L U R G IC A L E N G IN E E R IN G

e q u ilib riu m . L ikew ise a so lu te d is ­ trib u te d betw een tw o liq u id solv en ts is u s u a lly found to h av e e n tire ly d if ­ fe re n t c o n c e n tra tio n s in each liquid.

T h ere is p e rh a p s one im p o r ta n t e x ception to th e ru le of in s ta n ta n e o u s e s ta b lis h m e n t of e q u ilib riu m a t the in te rfa c e . T h is is in th e c ry s ta lliz a tio n of so lid s from s a tu r a te d s o lu tio n s . T he f a c t t h a t m olecules m u s t fit in to d efin ite p o s itio n s in th e c ry s ta l la ttic e seem s to re q u ire a d d itio n a l tim e so t h a t d iffusion u p to th e in te rfa c e re p re ­ s e n ts only a p a r t of th e to ta l re sistan c e in th is process.

F ilm C oncept— A t a m oving b o u n d ­ a r y , th e m ech an ism of tr a n s f e r of m a te ria l up to th e in te rfa c e is n o t one of sim p le diffu sio n . Im m e d iately a d ja c e n t to th e in te rfac e , th e fluid is com pletely s ta tio n a r y b u t th e v elo city in cre as es ra p id ly w ith d is ta n c e aw ay fro m th e b o u n d a ry . I f th e flow of th e b u lk of th e fluid is tu r b u le n t, th e re is 110 s h a rp lin e of d e m a rc a tio n s e p a r a t­

ing th e la m in a r film from th e tu r b u le n t core. F u rth e rm o re , m a te ria l m u s t be tr a n s p o r te d from th e core to th e film an d th ro u g h th e film to th e in te rfa c e . I n o rd e r to avoid h a v in g to ta k e in to c o n sid e ra tio n a ll of th e c o m p licated m ech an ism s of t r a n s p o r t a n d v a r ia ­ tio n s of v elocity, i t is sim p le r to use th e co n cep t of “ effective film th ic k ­ ness,” as if th e m ech an ism w ere one of tr u e d iffusion th ro u g h a s tr ic tly s ta g n a n t lay e r. S ince the. th ic k n e ss of th e film is h y p o th e tic a l, i t is n o t stir p ris in g to find t h a t i t v a rie s w ith q u a n titie s w hich o th erw ise sh o u ld h av e no effect. F o r ex am p le, i t h a s been fou n d to depend on th e d iffu s iv itv of th e s u b s ta n c e a c tu a lly d iffu sin g .

I n th o s e o p e ra tio n s in w hich m a te ria l is p a ssin g from one fluid p h a se to a n o th e r, a n effective film th ic k n e ss m u s t He co n sid ered 011 each side of th e in te rfa c e . T he re s is ta n c e s offered by th e tw o film s a re seldom th e sam e.

N o t only m ay th e effective th ic k n e ss of th e film s d iffer, h u t also th e con­

c e n tra tio n d ifference w hich causes th e tr a n s f e r from th e h u lk of th e fluid to th e in te rfac e , o r v ic e versa.

W e h av e a lre a d y seen t h a t th e con­

c e n tra tio n s in th e tw o p h ases a t th e in te rfa c e a re seldom th e sam e. C on­

se q u e n tly , th e re m ay be a v e ry larg e diffu sio n p o te n tia l (i.e., c o n c e n tra tio n d ifferen ce) 011 one side of th e in te rfa c e a n d only a sm all p o te n tia l on th e o th er side. I n th is case, o th e r th in g s bein g equal, th e re la tiv e im p o rta n c e of th e film on th e sid e of th e sm a lle r p o te n tia l, in c o n tro llin g th e tr a n s f e r ra te , w ill be g r e a te r th a n t h a t on th e sid e of the la rg e r p o te n tia l. I t is for th is re aso n t h a t we speak of one film o r th e o th er, o r b o th , as “ c o n tro llin g ” in a d if ­ fu sio n a l o p e ra tio n . In o rd e r to a c ­ co m plish r a p id o v e rall tr a n s f e r , i t is o b viously d e sira b le to decrease th e effective film th ic k n e ss on th e side of th e c o n tro llin g resistan c e.

I n som e cases, only one film e x is ts th ro u g h w hich m a te ria l m u s t be tra n s -

p o rted by diffusion. T h is is tr u e w hen one of th e p hases is a solid, a s in leaching, o r in a d so rp tio n . L ikew ise, in a d ia b a tic h u m id ifica tio n , only th e gas film is effective, since th e te m p e ra ­ tu re of th e in te rfa c e is th e sam e as th a t of th e bulk of th e liq u id an d th ere is n e ith e r m ass n o r h e a t tr a n s f e r 011

th e liquid side. On th e o th e r h a n d , both films m u s t be considered in de- h u m id ificatio n , because h e a t m u s t be tra n s fe rre d th ro u g h th e liq u id aw ay from th e in te rfac e . In porous solids, th e re is, in g e n era l, m uch m ore r e ­ sista n ce to th e tr a n s f e r of m a te ria l th ro u g h th e c a p illa rie s an d in te rs tic e s to th e su rfa ce of th e p a rtic le th a n th ro u g h the re la tiv e ly th in film o u tsid e of th e p a rticle . A c tu ally , th e tra n s fe r inside of the pores is n o t a lw ay s by diffusion. In th e d ry in g of porous

solids, for exam ple, i t h a s been show n t h a t liq u id is g e n e ra lly tra n s p o rte d to th e s u rfa c e by c a p illa ry a c tio n .1

R a te o f D iffu sio n -—I n a ll cases in w hich d iffusion is th e m eans of t r a n s ­ fe rrin g m a te ria l, th e r a te of tr a n s f e r m ay be s ta te d by th e sim p le e q u a tio n :

d W /d t = K A A C (1) H ere d W / d t is th e r a te of m ass t r a n s ­ fer, expressed a s a d iffe re n tia l q u a n tity since th e o th e r te rm s m ay v a ry fro m p o in t to p o in t; K is the tr a n s f e r co­

efficient a s defined in th e ta b le of d efin itio n s 011 page 80; t is th e a re a th ro u g h w hich th e d iffusion is ta k in g place; a n d AC is th e c o n ce n tra tio n difference c a u s in g th e diffu sio n . S ince th e c o n c e n tra tio n a t th e in te rfa c e is unknow n, i t is comm on to u se a n o v erall c o n c e n tra tio n difference and

C lassification of D iffusional O perations

Proccss

Class

Diffusing From

Diffusing To

Type of Diffusion

Relative Importance of Diffusion in Controlling Rate

Remarks

Unsteady state mixing of gases in closed containers Dispersion of smokes and fumes

Absorption of gases, Dehumidification

Adsorption of gases

Contact catalysis.

Reactions between solids and gases

Vaporization, Humidification

Desorption, Steam dis­

tillation, Stripping

Distillation

Solvent extraction

Crystallization

Drying

Sublimation Extraction, Leaching, Dissolving, Washing

H eat treatm ent of metals and o th er solids.

Devitrification

Gas

Gas

Gas

Gas

Gas

Liquid

Liquid

Liquid <

Liquid

Liquid

Solid

Solid Solid

Solid Gas

Gas

Liquid

Solid

Solid

Gas

Gas

Gas

Liquid

Solid

Gas

Gas Liquid

Solid

Molecular diffusion

Eddy diffusion

Eddy diffusion in core and mostly molecular diffusion through lay­

ers at interface Mostly molecular dif­

fusion outsideof solid entirely molecular in­

side pores Same as above

Eddy diffłision in tur­

bulent core; mostly molerular diffusion in gas film

Same as above

Same as above

Probably always molecular

Mostly molecular

Usually capillary ac­

tion inside solids;

sometimes molecular diffusion of vapor in pores

Molecular diffusion Dialysis through cell walls, molecular dif­

fusion in pores Molecular diffusion throughcrystal lattice

Eutirely controls

May be controlling, depends on atmos­

pheric and physio­

graphic conditions Overall diffusion rate controls the rate of the entire process

Diffusion may control if activity of adsor­

bent is high Diffusion controls in some cases (eg., oxi­

dation of SCh), but not in others Controls if adequate heat transfer to sup­

ply latent heat of va­

porization Overall diffusion la te controls

Controls entirely

Entirely controls in continuous equip­

ment; with mixers and settlers, rate of sepa­

ration may control Sometimes diffusion controls, but in gen­

eral rate of crystal growth is im portant Entirely controls if heat transfer is suf­

ficient

Same as above Diffusion may control entirely if material is tightly held by inert solid

Diffusion may control eutirely

Very slow unless thermal currents or other agitation Scale of eddies varies tre­

mendously

For absorption, diffusion in either gas or liquid films, or both, may be im­

portant depending on solu- bility'of gas

Most adsorbents show selectivity for specific gases

Influence of temperature on overall reaction rate is usually ■ small when dif­

fusion controls In general, only diffusion through the gas is im port­

an t

Diffusion in gas usually th e most im portant, ex­

cept when dealing with very low concentrations of gases, as in deaeration Resistance to transfer in vapor phase is probably always major factor.

Counter-diffusion appar­

ently does not cause any greater resistance than dif­

fusion through stagnant inert gas

B oth films are im portant

Crystal habit is im portant

Very few solids are suf­

ficiently homogeneous for diffusion laws to apply in falling rate period

Porosity of solid is import­

ant ; for small porous par­

ticles equilibrium appears to be reached rapidly Very slow process

C H E M IC A L & M E T A L L U R G IC A L E N G IN E E R IN G • M A Y 19Ą2 5— 79

th e n becomes th e d ifference betw een th e a c tu a l c o n c e n tra tio n of th e so lu te in one phase, an d t h a t w h ich w ould be in e q u ilib riu m w ith th e s o lu te con­

c e n tra tio n in th e o th e r phase. The c o n c e n tra tio n m ay be ex p ressed in te rm s of w e ig h t (o r m oles) of so lu te p e r u n it volum e of e ith e r p h ase, an d th e a b so lu te v a lu e of th e c o n c e n tra ­ tio n difference, o r d iffu sio n p o te n tia l, dep en d s on w hich p h ase is chosen as th e b a sis of th e ex p ressio n . Since th e a c tu a l r a te of d iffu sio n m u s t be th e sam e re g a rd le s s of how th e c o n c e n tra ­ tio n is ex p ressed , th e v a lu e of th e o v e rall tr a n s f e r coefficient dep en d s on th e m an n e r in w hich th e c o n c e n tra tio n difference is expressed. T h u s, we find i t comm on to use a s u b s c rip t on th e o v e rall tra n s fe r coefficient, su c h a s K , a n d K„ ( o r i l, a n d K 2), in d ic a tin g th e p h ase in w hich th e c o n c e n tra tio n s a re expressed.

As show n by E q u a tio n ( 1 ) , th e fac­

to rs w hich influence th e r a te of t r a n s ­ fer a r e : (1 ) d iffusion p o te n tia l, o r c o n c e n tra tio n difference, (2 ) th e v a lu e of th e o v e rall tr a n s f e r coefficient its e lf, and (3 ) th e in te rfa c ia l a re a . I t is now in o rd er to co n sid er, in a g e n era l w ay, how th ese fa c to rs a r e influenced by v a rio u s o p e ra tin g c o n d itio n s a n d how th ey m ay be e stim a te d so t h a t th e eq u atio n can be u se d fo r d esig n purposes.

D iffu sio n P o te n tia l— One of th e te rm s t h a t d e te rm in e th e d iffu sio n p o te n tia l is re a d ily know n, viz., th e a c tu a l con­

c e n tra tio n of th e m a te ria l in th e b u lk of th e fluid. T he o th e r te rm is th e so lu te c o n c e n tra tio n w hich w o u ld be in eq u ilib riu m w ith th e b u lk of th e o th e r phase. W hen th e tw o a re equal, no d iffusion can ta k e place. I n a c o n tin u o u s o p e ra tio n , th e re m ay be p o in ts in th e a p p a r a tu s in w hich th e p o te n tia l is la rg e b u t, if a t a n y one p o in t i t bccom cs sm a ll, o r v a n ish es because th e tw o p h ases a p p ro ac h eq u ilib riu m , th e size o f th e e q u ip m e n t re q u ire d becom es in fin itely larg e . T h is is th e b asis of th e r u le t h a t e q u ilib riu m c an n o t be reached a t a n y p o in t in a c o n tin u o u s o p e ra tio n d ep en d in g on d if­

fusion. T h e e q u ilib riu m re la tio n s h ip , th ere fo re, is of g ro a t im p o rta n c e in d e te rm in in g th e size of th e e q u ip m e n t re q u ire d .

T h ere a re m an y ty p es of e q u ilib riu m re la tio n s h ip s a n d i t is n o t a lw ay s p os­

sib le to fo rete ll w h a t ty p e a p a r tic u la r sy ste m fits w ith o u t a c tu a lly h a v in g th e e x p e rim e n ta l d a ta . F o rtu n a te ly , some of th e law’s a v ailab le fo r id eal so lu ­ tio n s a r e a p p lic ab le over a co nsiderable ra n g e of c o n ce n tra tio n . T h e m o st im p o r ta n t of th ese is the la w of L in e a r D is trib u tio n , i.e., th e c o n c e n tra tio n in o n e p h ase is p ro p o rtio n a l to t h a t in th e o th e r. T h ere a re m an y exam ples o f th is w hich m ay be fou n d fo r th e d is tr ib u tio n of so lu tes betw een a ll ty p e s of phases. T he m o st f a m ilia r is H e n r y ’s law fo r gases. T h is is a p p lic ­ a b le to a b s o rp tio n c a lc u la tio n s a n d h a s been found to hold for th e so lu b ility

Definitions

A bsorption—Rem oval of one or more soluble com ponents of a g a se o u s mix­

ture from a n inert g a s by dissolving in a liquid solvent. The term is distin­

guished from adsorption w hich refers to the rem oval of a component from a g as or liquid m ixture b y m eans of a porous solid with w hich it combines chem ically or physically.

Continuous Countercurrent Decantation

— The w ash in g of a solid b y a series of mixers an d thickeners a rra n g e d for counterflow of solids a n d solution.

Diffusion— The sp o ntaneous tran sfer of a component of a fluid m ixture from one point in the fluid to another, w ithout re ­ spect to fluid flow. M olecular diffusion refers to the net tran sfer of m aterials cau sed b y the norm al Brow nian m ove­

m ent of the m olecules, w hile e d d y dif­

fusion refers to the mixing of fluids cau sed b y eddies or turbulence. In both types, the net transfer of m aterial is p ro ­ portional to the difference in concentra­

tion a t the tw o points. The term "dif­

fusion" h a s also b e en u se d in the sen se of rem oving a soluble m aterial from an insoluble resid u e, such a s s u g a r from beet pulp, or tan n in g extract from wood.

U ndoubtedly, the m echanism in this case is b y d ialysis through the cell w a lls an d diffusion through the cap illaries. The operation is a n exam ple of the g e n era l­

ized type of extraction.

DU iusirity or Diffusion Coefficient—

A num erical coeflicient rep resen tin g the ten d en cy of a com ponent of a system to diffuse, a n alo g o u s to h e a t conductivity.

It rep resen ts the q u an tity w hich w ill dif­

fuse acro ss unit a r e a in unit time with unit volum e concentration g rad ien t. The dim ensions, therefore, a re (length)Ytime.

D issolving— The process of dissolution ol a com pletely soluble m aterial, either liquid or solid, in a solvent.

Effective Film Thickness— The thick­

ness of a film h av in g the sam e resistance a s the com bined resistan ces of the core an d lam in ar la y e r a t a m oving interface to the tran sfer of m ass or heat.

Extraction—The g e n e ra l term for tra n s­

ferring m aterial from a n in ert solid or liquid to a solvent.

a t low p re s s u re s of m a n y g a se s w hich do n o t fo rm com p o u n d s w ith th e s o l­

v en t. A sp e c ia l case of H e n ry ’s law w hich is o fte n a p p lic a b le to a d iffe re n t ra n g e of c o n c e n tra tio n is R a o u lt’s law . The law o f L in e a r D is tr ib u tio n also hold s fo r m a n y s o lu te s d is tr ib u te d b e­

tw een tw o im m iscib le liq u id s.

I t is in te r e s tin g to n o te t h a t in th e e x tr a c tin g of so lid s, su ch a s by lea ch ­ in g a n d w a sh in g , th e r e is fre q u e n tly lin e a r d is tr ib u tio n b etw een th e a m o u n t of s o lu te h eld in th e p o res of th e so lid a n d th e a m o u n t p re s e n t in th e c le ar so lu tio n . T he re a so n fo r th is is t h a t th e a m o u n t of s o lu tio n le f t in th e so lid a f te r d r a in in g is o fte n in d e p e n d e n t of th e c o n c e n tra tio n of th e so lu tio n . Con-

H.T.U.— The h eight of pack in g req u ired for one tran sfer unit. This q u an tity is proportional to the m ass velocity divided b y the m ass tran sfer coefficient.

Leaching—The rem oval of a re ad ily soluble m aterial from a n inert solid:

u su a lly refers to extraction w ith w ater.

L ixiriation—A n old term m ean in g es­

sen tially the sam e a s leaching. Accord­

ing to g e n era l u sa g e , leach in g a n d lixivl- ation re fe r to the rem oval of m aterials w hich a re v e ry soluble, w hile extraction refers to the rem oval of a m aterial w hich is quite firmly h eld b y the insoluble residue.

N um ber of Transler Units—A m ea su re ­ m ent of the difficulty of a tran sfer o p e ra ­ tion e x p ressed m ath em atically a s a n in­

teg ra l of approxim ately the ratio of the differential q u an tity tran sferred , to the diffusion potential. For diffusion in one direction it is given b y the equation

f Vl

Percolation, or Percolation Filtering—

The extraction ol a m aterial b y a liquid solvent from a n inert solid (or adsorption irom solution b y a n inert solid) in w hich the solid is statio n ary a n d the solvent flow is continuous.

Rectification — The sep a ra tio n of a liquid m ixture b y distillation In w hich a p a rt of the con d en sate is retu rn ed a s reflux lor contact w ith the vap o rs.

Transfer Coefficient— A num erical co­

efficient indicating the ra te of tran sfer in diffusional p rocesses p e r unit interfaclal a r e a a n d unit diffusion p otential (concen­

tration or p ressu re). W hen the inter­

facial a r e a is not know n, a volum e tra n s ­ fer coeflicient m ay b e used, representing the ra te p e r unit volume of p ack in g an d unit concentration difference.

W a sh in g —In g e n era l, refers to the re ­ m oval of a sm all am ount of dissolved m aterial from a n insoluble liquid or solid b y displacem ent with iresh solvent.

se q u e n tly , th e a m o u n t of s o lu te in th e d ra in e d so lid s is ' p ro p o rtio n a l to th e a m o u n t in th e c le a r so lu tio n , if th e c o n c e n tra tio n of th e s o lu tio n is u n i­

form . T he r a tio of th e a m o u n t of so lu te le f t in th e so lid to t h a t in th e cle a r s o lu tio n w ould dep en d , of course, on th e m eth o d of s o lu tio n rem o v al u sed, su c h a s th ic k e n in g , filte rin g , o r cen­

tr ifu g in g . T h is im p o r ta n t p rin c ip le w ill em p h asize th e a n alo g y betw een e x tra c tio n a n d a b so rp tio n arid is th e b a s is of a com m on m eth o d of c a lc u la ­ tio n to be d escrib ed below w h ich is a p p lic ab le w h en t h e law of L in e a r D is ­ tr ib u tio n h olds.

O verall T r a n sfe r C oefficient — T he o v e rall tr a n s f e r coefficient i m ay -be

8 0 ^-5 i f A Y 1942 • C H E M IC A L & M E T A L L U R G IC A L E N G IN E E R IN G ?

con sid ered a s th e recip ro ca l of th e re s is ta n c e to m ass tr a n s f e r from th e body of one p h a se to t h a t of th e o th er.

I f tw o film s e x is t a t th e in te rfa c e an d th e re s is ta n c e of b o th is a p p reciab le, th e o v e ra ll re s is ta n c e is a c tu a lly com ­ posed o f tw o re s is ta n c e s in se ries an d is eq u al to th e sum of th e re sista n c e s of th e in d iv id u a l film s. I n te rm s of th e in d iv id u a l film tr a n s f e r coefficients, th e re fo re , th e o v e rall coefficient m ay be re p re se n te d by

1/JT i « 1 / h + m /k , (2) T h e c o n s ta n t m is n e ce ssa ry becausc th e u n its in w hich th e in d iv id u a l co ­ efficients, a n d a r e ex p ressed a re n o t th e sam e, ow ing to th e d ifference in th e m a n n e r of e x p ressin g c o n c e n tra ­ tio n in th e tw o film s, a s a lr e a d y ex ­ p lain ed . A c tu a lly , m re p re s e n ts th e r a ti o betw een th e e q u ilib riu m concen­

tr a tio n s in th e tw o p h ases. U n less th e la w of L in e a r D is tr ib u tio n h o ld s, m m ay n o t be a c o n s ta n t. T lic r e la tio n ­ sh ip betw een th e tw o o v e ra ll coefficients ex p ressed in th e te rm s of th e con­

c e n tr a tio n in th e tw o p h a se s is a s fo llo w s:

m K i = K i (3)

W e s h a ll now co n sid er th e fa c to rs w hich affect th e in d iv id u a l film co­

efficients, a s th e s e m u s t in t u r n affect th e o v e rall tr a n s f e r coefficient. T he film coefficient is a p p ro x im a te ly p ro ­ p o rtio n a l to th e d iffu s iv ity of th e s o lu te a n d in v e rs e ly p ro p o rtio n a l to th e effec­

tiv e film th ic k n e ss. W hen th e concen­

tr a t i o n of th e d iffu sin g m a te ria l is h ig h , i t is k n o w n t h a t a n o th e r fa c to r m u s t be co n sid ered , viz., th e a c tu a l c o n c e n tra tio n of th e so lv e n t, o r in e r t m a te ria l. F o r m o s t cases th is m a y be neg lected , b u t fo r c e rta in cases of a b ­ s o rp tio n a n d c o n d en s atio n i t m ay b e­

come im p o rta n t.

In fo rm a tio n on th e a b s o lu te v a lu e of th e d iffu s iv ity of so lu te s in g ases is in f a ir ly good sh a p e b u t th e re is m uch to be le a rn e d b efo re a n y g e n e ra l c o r­

re la tio n can be m ad e fo r d iffu siv itie s in liq u id s. F o r g ases, th e k in e tic th e o ry serv es a s a b a sis fo r m a n y th e o re tic a l c a lc u la tio n s a n d se v e ra l e q u a tio n s h av e been d e riv e d b y w hich th e d iffu siv ity m ay be c a lc u la te d fro m c o n s ta n ts w hich a re know n, o r c an be a p p ro x im a te d . T h ere a re co n sid erab le e x p e rim e n ta l d a ta a v a ila b le for th e d iffu s iv itie s in g a s sy stem s. A few y e a rs ago th e en ­ t ir e s u b je c t w as su rv ey ed by G illi­

la n d 10 w ho recom m ended th e follow ­ in g e q u a tio n a s th e m o s t s u ita b le fo r c a lc u la tin g d iffu siv itie s for sy stem s fo r w hich th e d a ta a re n o t a v a ila b le : D = 0.0043X

T 3* I 1 1

P i v . v * + r # y y j r m + w i W I n th is e q u a tio n , th e d iffu s iv ity /) is g iv en in ( c m .) 1 p er sec.; P is th e to ta l p re s s u re in a tm o s p h e re s ; T is th e a b ­ s o lu te te m p e ra tu re in deg. K . ; M . a n d M b a r e th e m o lecu lar w e ig h ts of th e tw o g a s e s ; a n d V m a n d V* a r e th e

m o lecu lar volum es. T he v a lu e s of l a a n d V b a re o b tain ed b y u s in g th e law of a d d itiv e volum es, w ith th e ru le s a n d v a lu e s given by Le B as. T hese a rc given in G illila n d ’s p a p e r a n d a r c also pu b lish ed in m an y te x t books. I t is im ­ p o r t a n t to n o te t h a t th e d iffu siv ity is in v ersely p ro p o rtio n a l to th e p re ssu re a n d d ire c tly p ro p o rtio n a l to th e sq u a re ro o t of th e su m of th e re cip ro ca ls of th e m o lecu lar w eig h ts of th e gases.

D iffu siv ities in liq u id s a r c m uch low er th a n those in gases. A p p lica tio n of th e k in e tic th eo ry to p re d ic tio n of th e coefficient h as o n ly been p a r tia lly successful d u e to th e a b n o rm a l effects caused by th e a sso c iatio n o r bonding of b o th so lv en t and so lu te m olecules.

D a ta a re av ailab le in th e In te r n a tio n a l C ritic a l T ab les fo r th e d iffu siv itie s of a n u m b er of dissolved gases a n d o th er so lu tes in w a ter, an d fo r a few m a te ­ ria ls in o rg an ic so lv en ts. I n th e a b ­ sence of f u r th e r in fo rm a tio n , i t is su g g ested t h a t d a ta for su b stan ces a s n e a rly as possible lik e th e so lu te a n d so lv e n t in q u e stio n be used. F o r tu ­ n a te ly , liq u id film d iffusion coefficients fo r sim p le so lu tes do n o t show th e g r e a t ra n g e in v a lu e s show n by gas d iffu sio n coefficients.

E v en if d iffu siv ities w ere k now n w ith g r e a t a c c u rac y th e in d iv id u a l film co­

efficients could n o t be calcu la te d w ith ­ o u t k n o w in g th e effective film th ic k ­ ness. T h is is a m uch m ore co m p li­

c ated p ro p o sitio n . The m o s t im p o r­

t a n t fa c to rs affectin g th e film th ic k ­

ness a re th e in te r f a c ia l v elo cities of th e fluids an d th e d eg ree of tu rb u le n c e caused by th e sh a p e of th e in te rfac e . F o r ilu id s in tu r b u le n t flow', th e th ic k ­ ness of th e film is a p p ro x im a te ly p ro ­ p o rtio n a l to V '0 *, w here V ,is th e in ­ te rfa c ia l v elocity. T he th ic k n e ss is also affected b y th e p ro p e rtie s of th e fluid a n d of th e m a te ria l d iffu sin g . I t is a p p ro x im a te ly p ro p o rtio n a l to th e sq u a re r o o t of th e v isc o sity of th e fluid a n d of th e d iffu s iv ity of th e so lu te, a n d in v ersely p ro p o rtio n a l to th e sq u a re ro o t of th e d e n s ity of th e fluid.

I n te r fa c ia l A rea — T he a r e a of th e in te rfa c e a cro ss w hich d iffu sio n ta k e s place is a n o th e r q u a n tity on w hich th e re is u s u a lly l it t le e x a c t in fo rm a ­ tio n av ailab le. I t is w ell k now n t h a t in a packed to w er, fo r in sta n c e , th e a re a of th e p a ck in g is by no m ean s th e sam e as th e a re a of th e in te rfa c e . T h is is because of th e c h an n e lin g of one o r b o th fluids, an d th e existence of pockets in w hich th e re c a n be no flow. C e rta in ty p es of s tre a m lin e p a c k ­ in g , of course, h av e m o re effective a re a p e r u n i t volum e th a n o th e rs . T here m u s t also be a b alan ce betw'een th e a re a a v a ila b le a n d o th e r c h a ra c ­ te ris tic s of th e p a ck in g , su ch a s th e p re ssu re d ro p a n d flooding v elo city . B ecause of lack of in fo rm a tio n on th e a b so lu te v alu e of A , i t is com m on to com bine th e a re a w ith K in to a volum e tr a n s f e r coefficient K a , wTliere a is th e a c tu a l in te rfa c ia l a re a per u n i t volum e of p ack in g . A few e x p erim e n tal v a lu e s

Diffusion is a controlling factor in rectification

C H E M IC A L <t M E T A L L U R G IC A L E N G IN E E R IN G • M A Y 101,2 ^{5 — 81}

m

Fig. 1— Perform ance efficiency of m ulti-stage countercurrent diffusional operation for case of lin ear distribution of solute

betw een p h a se s (after Sherwood) of K a fo r g a s a b so rp tio n on v a rio u s

ty p e s of p a ck in g a re a v a ila b le in th e l ite r a tu r e .“ In d iv id u a l film coefficients fo r v a rio u s p ack in g s h av e been d e te r ­ m ined for a few g a se s.17 O nly recen tly , in fo rm a tio n on film coefficients for liq u id -liq u id e x tra c tio n h av e begun to a p p e a r.1’ ’■T' 13

The T h eo retica l P la te or R ta p e — W hen th e r a te of tr a n s f e r o f a m a ­ te r ia l betw een tw o ph ases is e n tire ly u nknow n, i t is som etim es a d v an tag e o u s to p re d ic t th e difficulty of se p a ra tio n by a n e n tire ly d iffe re n t concept. In th is , %ve assu m e t h a t th e tw o ph ases a re b ro u g h t to a c tu a l e q u ilib riu m w ith each o th e r b y th o ro u g h m ix in g , a n d th en a re allow ed to s e p a ra te an d flow in th e ir re sp ec tiv e d ire c tio n s , e ith e r e o u n te rc u rre n tly o r in p a ra lle l. T he m o st f a m ilia r a p p lic a tio n of th is idea is in th e M cC abe-Thielc d ia g ra m for d is tilla tio n .“ H ere i t is po ssib le to p re ­ d ic t th e n u m b er of th e o re tic a l p la te s re q u ire d fo r a giv en s e p a ra tio n by a g ra p h ic a l m eth o d , u s in g th e e q u ilib ­ riu m betw een liq u id an d v a p o r a n d a specified o p e ra tin g r a ti o of v a p o r to liq u id . In o rd e r to kn o w th e a c tu a l size of th e d is tilla tio n co lu m n re q u ire d , it is n ecessary to know how efficient th e v a rio u s p la te s a r e in b rin g in g th e tw o stre a m s to e q u ilib riu m an d also to h av e in fo rm a tio n on w h a t v a p o r v elo city is allow able.

A p p lica tio n of th is co n cep t to a [d ate colum n is so m e w h a t m ore ev id en t th a n it is to a packed colum n. I f we conceive of a section of p a ck in g as being req u ired to b r in g a b o u t e q u i­

lib riu m betw een th e p h ases, we can s till use th e sam e concept of a th e o ­ re tic a l p la te an d d e te rm in e e x p e ri­

m e n ta lly th e h e ig h t e q u iv a le n t to th e th e o re tic a l p la te , o r H .E .T .P . for th e p acking.

The concept of th e th e o re tic a l p la te , o r stag e, is a p p lic ab le to a ll ty p es of diffu sio n al o p e ratio n s . T h e l ite r a tu r e c o n ta in s a num ber of v a lu a b le p a p ers d escrib in g th e use of th e m eth o d fo r tlie g ra p h ic a l tre a tm e n t of problem s on a b so rp tio n .13 liu in id ific atio n ,10 liquid- liq u id e x tr a c tio n ," ',2' 11 a n d th e leach ­ ing a n d e x tra c tio n of solids.9 In these o p e ra tio n s th e re is one c o m p lica tio n w hich does n o t e x is t in d is tilla tio n c a l­

c u la tio n s , nam ely, i t is u s u a lly in c o r­

re c t to assu m e t h a t th e o p e ra tin g line is a s tr a ig h t line. T he re aso n for th is is t h a t in m o st d iffu sio n al o p e ra tio n s th e r a tio of th e tw o p hases is n o t con­

s t a n t fo r a ll sta g es, as it is in d is ­ tilla tio n . T h is difficulty is avoided, how ever, by th e use of tr ia n g u la r d ia ­ g ra m s, o r of r e c ta n g u la r d ia g ra m s of th e M erkel, o r P onelion ty p e.“ ’

W hen th e o p e ra tio n c o n sists of th e se p a ra tio n of tw o o r m ore so lu tes from a n in e r t so lv e n t by tr a n s f e r to a n o th e r p h ase, th es e m eth o d s of c a lc u la tio n , a t b est, a re o n ly a p p ro x im a tio n s. The a n alo g y w h ic h can be s e t u p betw een c e r ta in d iffu sio n al o p e ratio n s , su ch as e x tra c tio n a n d d is tilla tio n , how ever, h a s been e x tre m e ly h e lp fu l. T h u s, in so lv e n t e x tra c tio n , th e benefit to be

deriv ed fro m r e tu r n in g a reflu x stre a m in c o u n te rc u rre n t o p e ra tio n is im m e d i­

a te ly a p p a r e n t by a c a lc u la tio n of th is ty p e.”’ I2,11

T h e K re m se r F o rm u la — T he concept of th e th e o re tic a l sta g e , o r p la te , is esp ecially v a lu a b le w hen th e law of L in e a r D is trib u tio n h o ld s fo r th e e q u i­

lib riu m betw een th e tw o p h ases. F o r th is case, K re m se r h a s d e riv e d a n a lg e ­ b ra ic ex p ressio n r e la tin g th e com posi­

tio n of th e s tre a m s e n te r in g a n d lea v ­ ing th e e q u ip m e n t to th e n u m b er of sta g es.“ ' 18 F o r th e im p o r ta n t case of th e c o u n te r c u r r e n t a b so rp tio n of g a s, w here th e c o n c e n tra tio n s of th e s o lu te in th e g a s an d liq u id a re low so t h a t th e s tre a m s a r e e s s e n tia lly c o n s ta n t, th e fo rm u la becom es

( R _ \ n+l _ I±

F i - Yo = \ m ) m Yi - m X 0 / f i \ n+1 _ x w

H e re 1’, an d V0 a r e th e g a s c o n c e n tra ­ tio n s e n te r in g a n d le a v in g th e a b ­ so rb er ; A'0 is th e c o n c e n tra tio n of s o ­ lu te in th e so lv e n t e n te r in g th e a b s o rb e r; m is th e c o n s ta n t d i s t r ib u ­ tio n r a ti o of th e so lu te b etw een th e g as a n d liq u id a t e q u ilib riu m ; R is th e r a ti o of th e liq u id v e lo c ity to the g as v e lo c ity ; and » is th e n u m b er of th e o re tic a l p la te s . T h e le f t sid e of th e e q u a tio n re p re s e n ts th e r a tio of th e a c tu a l ch an g e in g a s c o m p o sitio n to th e ch an g e w hich w ould o ccu r if th e g a s lea v in g w ere in e q u ilib riu m w ith th e so lv e n t e n te r in g th e a b so rb er. I f fres h so lv e n t is u sed , c o n ta in in g no so lu te, X 0 is zero, a n d th e r a ti o b e­

comes th e a c tu a l fra c tio n of th e g as recovered, a n d re p re s e n ts th e efficiency of th e colum n. In F ig . 1 is show n a p lo t of th e K re m se r fo rm u la w hich in d ic a te s th e effect of th e r a t i o of liq u id to g a s, a n d th e n u m b er of th e o ­ re tic a l p la te s , on th e p e rce n ta g e r e ­ covery fo r a giv en v a lu e of th e d i s t r i ­ b u tio n coefficient m , a n d w hen fresh

so lv e n t, free from so lu te , is se n t to th e a b so rp tio n tow er.

E x a c tly th e sam e ty p e of fo rm u la m ay be d e riv e d fo r a c o u n te rc u rre n t e x tra c tio n , o r w a sh in g sy s te m , such as a se ries of m ix e rs a n d s e ttle r s . L e t Ji re p re s e n t th e r a tio of so lv e n t in th e overflow , to so lv e n t in th e underflow fro m each th ic k e n er. T hen, a ssu m in g t h a t th is r a tio re m a in s c o n s ta n t r e ­ g a rd le s s of co m p o sitio n of th e s o lu ­ tio n , a n d p u re so lv e n t is s e n t to th e la s t s ta g e, th e fra c tio n of so lu te r e ­ covered is giv en by a fo rm u la id en tica l w ith E q u a tio n ( 5 ) , ex cep t t h a t It r e ­ places R /m .

T h e T r a n s fe r U n it— A n id ea som e­

w h a t s im ila r to th e th e o re tic a l p la te , b u t m o re lo g ically a p p lie d to packed co lu m n s, w as su g g ested se v e ra l y e a rs ag o by C h ilto n a n d C olburn.” I n th is , th e difficulty of a tr a n s f e r o p e ra tio n is e x p ressed in te rm s of th e c o n c e n tra ­ tio n ch an g e to be effected in e ith e r fluid s tre a m , a n d of th e d riv in g force.

T h is q u a n tity is defined a s “ th e n u m ­ b e r of tr a n s f e r u n its .” T he e q u a tio n giv en in th e ta b le of d e fin itio n s on page SO defines th e q u a n tity m a th e m a tic a lly for d iffu sio n in one d ire c tio n , such as in a b s o rp tio n a n d e x tra c tio n . F or eq u al m o la r c o u n ter-d iffu sio n , a s in d is tilla tio n — o r w hen y is sm a ll com ­ p a re d to u n ity — th e n u m b er of tr a n s f e r u n its is given by th e fo rm u la

n = i f * - * , (6)

J y , y - y

H e re, y re p re s e n ts th e m ole fr a c tio n of th e so lu te b ein g s e p a ra te d , th e s u b ­ s c r ip ts 1 a n d 2 r e fe r r in g to th e t e r ­ m in a l c o n d itio n s. In th e cases w here H e n ry ’s la w o r o th e r s im ila r L in e a r D is tr ib u tio n law s h o ld , i t m a y be show n t h a t

y* = m (y — y* )/R (7) H ere m is a g a in th e e q u ilib riu m d is ­ tr ib u tio n c o n s ta n t a n d R is th e r a tio

82— 5 M A Y 1943 • C H E M IC A L & M E T A L L U R G IC A L E N G IN E E R IN G