Document 1996

C H A P T E R2
Diffusion in Dilute Solutions
In this chapter,we considerthe basiclaw thatunderliesdiffusionandits application
to severalsimpleexamples.The examplesthat will be givenarerestrictedto dilute solutions.
Resultsfor concentratedsolutionsare deferreduntil Chapter3.
This focuson the specialcaseof dilute solutionsmay seemstrange.Surely,it would seem
more sensibleto treatthe generalcaseof all solutionsandthen seemathematicallywhat the
limit is like. Most booksusethis approach.Indeed,becauseconcentrated
dilute-solution
solutionsare complex,thesebooks often describeheattransf'eror fluid mechanicsfirst and
then teachdiffision by analogy.The complexityof concentrateddiffusion then becomesa
mathematicalcancergraftedonto equationsof energyand momentum'
I haverejectedthis approachfor two reasons.First,the mostcommondiffusionproblems
do take place in dilute solutions.For example,diffusion in living tissuealmost alwaysinvolvesthe transportof small amountsof soluteslike salts.antibodies,enzymes,or steroids'
Thus many who are interestedin diffusion neednot worry about the complexitiesof concentratedsolutions;they can work effectivelyand contentedlywith the simplerconceptsin
this chapter.
Secondand more important,diffusion in dilute solutionsis easierto understandin physical terms. A diffusion flux is the rate per unit areaat which massmoves. A concentration
profìle is simply the variation of the concentrationversustime and position. Theseideas
are much more easily graspedthan conceptslike momentumflux, which is the momentum
per areaper time. This seemsparticularlytrue for thosewhosebackgroundsarenot in engineering,thosewho needto know aboutdiftision but not aboutother transpol'tphenomena.
This emphasison dilute solutionsis found in the historicaldevelopmentof the basiclaws
sections2.2and2.3ofthischapterfocusontwosimple
rnvolved,asdescribedinSection2.L
diffusion into
diffusion acrossa thin film and unsteady-state
casesof {iffusion: steady-state
.rninfinite slab. This focus is a logical choicebecausethesetwo casesare so common. For
L'xample,diffusion acrossthin films is basic to membranetransport,and diffusion in slabs
rr importantin the strengthof welds and in the decayof teeth. Thesetwo casesare the two
r.xtretn€sin nature,and they bracketthe behaviorobservedexperimentally.In Sections2'4
.tnd2.5. theseideas are extendedto other examplesthat demonstratemathematicalideas
Lrsefulfbr other situations.
2.1 Pioneersin Diffusion
2.1.1 ThomasGraham
Our modern ideason diffusion are largely due to two men, ThomasGrahamand
\dolf Fick. Grahamwas the elder. Born on December20, 1805,Grahamwas the son of
i successfulmanufacturer.At 13 yearsof age he enteredthe University of Glasgowwith
:hc intention of becoming a minister, and there his interestin sciencewas stimulatedby
ThomasThomson.
l-l
2 / Di/Jusionin Dilute Solutions
l4
/ Pioneers irt Dir
t-r
\c.--=/
il-
D i f f u s i n gg o s
ti
( o)
F i g . 2 . l - 1C
. :
fiee diflìsri:
F i g . 2 . l - 1 . G r a h a m ' s d i f f u s i o n t u b e f o r g a s eTs h
. isapparatuswasusedinthebestearlystudyof
diftìsion. As a gas like hydrogen difTusesout through the plug, the tube is lowered to ensure
that there will be no Dressuredil'ference.
Graham'sresearchon the ditTusionof gases,largelyconductedduring the years 1828to
I 833, dependedstronglyon the apparatusshownin Fig. 2. 1-I (Graham,1829, | 833). This
apparatus,a "diffusion tube," consistsof a straightglasstube, one end of which is closed
with a densestuccoplug. The tubeis filled with hydrogen,and the end is sealedwith water,
as shown. Hydrogendiflises throughthe plug and out of the tube, while air diffusesback
throughthe plug and into the tube.
Becausethe diffusion of hydrogenis fasterthanthe diffusion of air, the waterlevelin this
tube will rise during the process.Grahamsaw that this changein water level would leadto
a pressuregradientthat in turn would alterthe diffusion. To avoid this pressuregradient,he
continuallyloweredthetubesothatthe waterlevelstayedconstant.His experimentalresults
then consistedof a volume-changecharacteristicof each gas originally held in the tube.
Becausethis volume changewas characteristicof diffusion, "the diffusion or spontaneous
intermixtureof two gasesin contactis effèctedby an interchangeof position of infinitely
minute volumes,being,in the caseof eachgas,inverselyproportionalto the squareroot of
the densityof the gas" (Graham,1833,p. 222). Gtaham'soriginal experimentwas unusual
becausethe diffusiontook placeat constantpressure,not at constantvolume(Mason, 1970).
Grahamalso perfbrmedimportantexperimentson liquid diffusion using the equipment
he workedwith dilutesolutions.
shownin Fig.2.l-2 (Graham,I 850);in theseexperiments,
In one seriesof experiments,he connectedtwo bottlesthat containedsolutionsat diffèrent
concentrations;he waiteil severaldays and then separatedthe bottles and analyzedtheir
contents.In anotherseriesofexperiments,he placeda small bottle containinga solutionof
kno'uvnconcentrationin a largerjar containingonly water. After waiting severaldays,he
removedthe bottleand analyzedits contents.
Graham'sresultswere sirnpleand definitive. He showedthat diffusion in liquids was at
leasrseveralthousandtimes slowerthandiflision in gases.He recognizedthat the diffusion
that "diffusion mustnecessarilyfollow
processgot still slowerastheexperimentprogressed,
Most important,he concludedfrom the resultsin Table2.1-1
a diminishingprogression."
that "rhe quanritiesdiftised appearto be closely in proportion . . . to the quantity of salt in
the ditTusionsolution" (Graham,1850,p. 6). In other words, the flux causedby difTusion
is proportionalto the concentrationdiffèrenceof the salt.
2.t.2.1d,
lE
r|lllr'
p
F l ' f r" r .
$sr
iW
i!ì-.]j
.--'
:,,lLttiotts
l5
2.1 / Pioneersin Dffision
t+
\/ ' /h
fi
Gloss
--
otot'
H
( o)
Fig.2.l-2. Graham's diffusion apparatusfbr liquids. The equipment in (a) is the ancestorof
fiee diffusion experiments; that in (b) is a forerunner of the capillary method.
:---'.tudr of
,
Table 2.1-l. Graham's resultsfor liquid difrusion
::1\ UI.e
Weight percentof
sodiumchloride
1
2
3
: , , : ' 'l S l E t o
\ :.ì I. ThiS
-:r r. Jlo\ed
r .,. :h \\ater,
...j. back
- . :: rnthiS
: .cld to
j ' ' . - t 3 n t .l ì e
: . , : .f - . U l t S
' ' . : ì at u b e .
-:- ....Lllel)US
:.:'.lr'ritely
- ---:l:' ltrL)t Of
.r.:' ,ìllllsUOl
I. . . . . I9 " 0 ) .
-:-.,.ìPlllÈnt
'- .LltrrnS.
.--r-::l'ierent
.- . ,:J their
. , . : t o nO f
-
-.,
. : . r r: . h g
.. .t. riiìS ot
J irl'iusion
..: ' itrÌlow
. , , ' 1 . l.. l - 1
: . , , r s a l ti n
. irtTusion
,1
Relativeflux
r.00
1.99
3.01
4.00
Source:Data from Graham(1850).
2.1.2 Adolf Fick
The next major advancein the theory of diffusion came from the work of Adolf
Eugen Fick. Fick was born on September3, 1829, the youngestof fìve children. His
ofbuildings. During his secondaryschooling,
father,a civil engineer,was a superintendent
Fick was delightedby mathematics,especiallythe work of Poisson.He intendedto make
mathematicshis career.However,an older brother,a professorof anatomyat the University
of Marlburg,persuadedhim to switch to medicine.
In the spring of 1847, Fick went to Marlburg, where he was occasionallytutored by
Carl Ludwig. Ludwig strongly believedthat medicine, and indeedlife itself, must have
a basis in mathematics,physics,and chemistry. This attitudemust have been especially
appealingto Fick, who saw the chanceto combine his real love, mathematics,with his
medicine.
ehosenprofession.
In the fall of 1849.Fick's educationcontinuedin Berlin. where he did a considerable
irmountof clinical work. In 185t he returnedto Marlburg,wherehe receivedhis degree.His
rhesisdealtwith the visualerrorscausedby astigmatism,againillustratinghis determination
t o c o m b i n e s c i e n c e a n d m e d i c i n e ( F1i c8k5,2 ) .I n t h e f a l l o f 1 8 5l , C a r l L u d w i g b e c a m e
professorof anatomy in Zurich, and in the spring of 1852 he brought Fick along as a
prosector.Ludwig movedto Menna in 1855,but Fick remainedin Zurich until 1868.
do not dependon diffuParadoxically,the majority of Fick's scientifìcaccomplishments
(Fick, 1903).He did
physiology
.ion studiesat all, but on his more generalinvestigationsof
outstandingwork in mechanics(particularlyas appliedto the functioning of muscles),in
hydrodynamicsand hemorheology,and in the visual and thermalfunctioningof the human
l6
2 / Diffusion in Dilute Solutions
body. He was an intriguing man. However,in this discussionwe are interestedonly in his
developmentof the fundamentallaws of difTusion.
In his first diffusion paper,Fick (1855a) codified Graham's experimentsthrough an
impressivecombinationof qualitativetheories,casualanalogies,and quantitativeexperiments. His paper,which is refreshinglystraightforward,deservesreadingtoday. Fick's
introductionof his basic idea is almost casual: "[T]he diffusion of the dissolvedmaterial
. . . is left completelyto the influenceof the molecularforcesbasic to the samelaw . . . for
the spreadingof warmth in a conductorand which hasalreadybeenappliedwith suchgreat
s u c c e s s t o t h e s p r e a d i n geol fe c t r i c i t y " ( F i c k1, 8 5 5 a , p . 6 5 ) .I n o t h e r w o r d s , d i f f u s i o n c a n
be describedon the samemathematicalbasisas Fourier'slaw for heatconductionor Ohm's
law for electricalconduction.This analogyremainsa usefulpedagogicaltool.
Fick seemedinitially nervousabout his hypothesis. He buttressedit with a variety of
argumentsbasedon kinetic theory. Although theseargumentsare now dated,they show
physicalinsightsthatwould be exceptionalin medicinetoday. For example,Fick recognized
that diffusion is a dynamic molecular process. He understoodthe differencebetweena
true equilibrium and a steadystate,possibly as a result of his studieswith muscles(Fick,
1856).Later,Fick becamemore confidentashe realizedhis hypothesiswas consistentwith
Graham'sresulrs(Fick, 1855b).
Using this basic hypothesis,Fick quickly developedthe laws of diffusion by meansof
analogieswith Fourier's work (Fourier, 1822). He defined a total one-dimensionalflux
./1 as
Jt: Ajr: -AD?
!
r
li
ù
a
;
ffiT:
rfr:l][email protected]
ffiù
@: I
F,út
lhr
(2.1-1)
d7.
whereA is the areaacrosswhich diffusionoccurs,/1 is the flux per unit a;tal.,
c1is concentration, and z is distance.This is the f,rst suggestionof what is now known as Fick's law. The
quantity D, which Fick called"the constantdependingof the natureof the substances,"
is,
of course,the diffusion coefficient.Fick alsoparalleledFourier'sdevelopmentto determine
the moregeneralconservation
equation
3r'r
/ d2c,
I AA At' \
a , : D [ * * A , ' à r )
(2.1-2)
When the area A is a constant,this becomesthe basic equation for one-dimensional
unsteady-state
diffusion, sometimescalled Fick's secondlaw.
Fick next had to prove his hypothesisthat diffusion and thermal conductioncan be
describedby the sameequations.He was by no meansimmediatelysuccessful.First, he
tried to integrateEq.2.l-2 for constantarea,but he becamediscouragedby the numerical
effort required. Second,he tried to measurethe secondderivativeexperimentally.Like
manyothers,he foundthatsecondderivativesaredifficult to measure:"the seconddifference
increasesexceptionallythe efTectof [experimental]errors."
His third effort was more successful.He used a glass cylinder containingcrystalline
sodium chloride in the bottom and a large volume of water in the top, shown as the lower
apparatusin Fig. 2.1-3. By periodicallychangingthe water in the top volume,he was able
to establisha steady-state
concentrationgradientin the cylindrical cell. He found that this
gradientwas linear,as shownin Fig. 2. 1-3. Becausethis resultcan be predictedeitherfrom
Eq. 2.1-1or from Eq.2.l-2, this was a triumph.
GItlN
Jlil5
hIr
nafl,
IJJ
Nlffi
&iltuf
E
Àr,
[lm'
î,&
pmrrr
I| )11.t
) I / Pioneersin Diffusion
l t
nls
'--
,--:l .in
. -,.1
.
'.
.- ., .. ,À\
r
.-.::lfl
i-tl1
-
,
*::'.ìt
--
E
(!
(J
o
'6
o
CL
a
, ,ill
Distance,z
. ìt
.-ù
. - I
: l . t
-\ .
Fig. 2 I -3. Fick's experimental results. The crystals in the bottom of each apparatussaturatethe
adjacent solution, so that a fixed concentration gradient is establishedaìong the narrow, lower
partoftheapparatus. Fick'scalculationofthecurveforthefunnelwashisbestproofofFick's
law.
-h
Table 2.1-2. Fick's law for diffu,sionwithout conNection
For one-dimensionaldiffusion in
Cartesiancoordinates
- j t : D ,dc'
az
For radial diffusion in cylindrical
coordinates
- j r : D ,d c ,
ar
For radial diffusion in spherical
coordinates
- j t : D ,
dc,
ar
Nole..Moregeneralequations
aregivenin Table3.2_l.
:le
Ir
JI
- , - : :b e
: -. . he
But this successwas by no meanscomplete. After all, Graham'sdata for liquids anJlpatedFq.2.l-1. To try to strengthenthe analogywith thermal conduction,Fick used
- lower apparatusshown in Fig. 2.1-3. In this apparatus,he establishedthe steady-state
- 'ncentrationprofìle in the samemanneras before. He measuredthis profile and then tried
Dredicttheseresultsusing Eq. 2.1-2,in which the funnel areaA availablefor diffusion
,ned with the distance:. When Fick comparedhis calculationswith his experimental
-r'.ults,he found
the good agreementshownin Fig.2.l-3. Theseresultswere the initial
. r'rilìcationof Fick's law.
::ìJill
. Lrke
-
J:3ilCe
- - - ,l t n e
....,ble
':
, - .Ì h l s
-
i 1llÌl
2.1.3 Forms of Fick's Inw
useful forms of Fick's law in dilute solutionsare shown in Table 2.1-2. Each
r.lurìtioncloselyparallelsthat suggestedby Fick, that is, Eq. 2.1-I. Each involvesthe
'úrlle phenomenologicaldiffision coefficient.Each will be combinedwith mass
balances
, analyzethe problemscentralto the rest of this chapter.
One must rememberthat theseflux equationsimply no convectionin the samedirection
',' the one-dimensionaldiffusion. They are thus specialcasesof the generalequations
lr\en in Table3.2-1. This lack of convectionoften indicatesa dilute solution. In fàct.
t8
2 / Diffusíon in Dílute Solutions
Fig.2.2-1. Diîfusion across a thin fìlm. This is the simplest diffusion problem, basic to perhaps
80% of what follows. Note that the concentration profìle is independentof the diffusion
coelhcient.
the assumptionof a dilute solution is more restrictivethan necessary,for there are many
concentratedsolutionsfor which thesesimple equationscan be used without inaccuracy.
Nonetheless,for the novice, I suggestthinking of diffusion in a dilute solution.
2.2 SteadyDiffusion Acrossa Thin Film
In the previoussectionwe detailedthedevelopmentof Fick's law, thebasicrelation
for diffusion. Armed with this law,we cannow attackthe simplestexample:steadydiffusion
acrossa thin film. In this attack.we want to find both the diffusionflux andthe concentration
profile. In other words, we want to determinehow much solutemovesacrossthe film and
how the soluteconcentrationchangeswithin the film.
This problemis very important. It is one extremeof diffusion behavior,a counterpointto
diffusion in an infinite slab. Every reader,whethercasualor diligent, shouldtry to master
this problem now. Many will fail becausefilm diffusion is too simple mathematically.
Pleasedo not dismiss this important problem; it is mathematicallystraightforwardbut
physically subtle.Think aboutit carefully.
b,["s
2.2.1 The Physical Situation
Steadydiffusion acrossa thin film is illustratedschematicallyin Fig. 2.2-1. On
eachside of the film is a well-mixed solutionof one solute,speciesI . Both thesesolutions
are dilute. The solutediffusesfrom the fixed higher concentration,locatedat z < 0 on the
lefrhand side of the film, into the fixed, lessconcentratedsolution,locatedat z > / on the
right-handside.
We want to find the soluteconcentrationprofile and the flux acrossthis film. To do this,
we first write a massbalanceon a thin layer Az, locatedat somearbitrarypositionz within
the thin film. The massbalancein this layer is
/
I
solute
\ l:
/ r a t eo î d l f l u s t o n \ |
|
/ rate ol diffusion \
. . 1 . î . , _ . , _ _ . . . .ì
|| O U t o l t n e l a v e rI
\ a c c u m u l a t i o n /\ i n t o t h e l a y e r a t , / \ " " ; ; ; ' , ' i ? ' '
/
Becausethe processis in steadystate,the accumulationis zero. The diffusion rate is the
{N
:llu -
\r tlutictttr
- ) / SteadyDiffusion Across a Thin Film
19
.t'iusionflux times the film's areaA. Thus
0 : A ( j t l . - - / rr . + r . 7
(2.2-1)
):i iding this equationby the film's volume,AAz, andreaffanging,
-/rl')
o:-(rr '-r'
\(:+Ar)-zl
hen Az becomesvery small,this equationbecomesthe definitionof the derivative
"
d
0:_;jt
(2.2-3)
47.
'
.erhaps
,nrbiningthis equationwith Fick's law,
- j t : D
:r'I11&Iì!
r))-)t
dct
'
47.
() )-4\
.' find, for a constantdiffusion coefficientD,
, Juracy.
'1) "
0 : D';','
-
r? )-51
:r. differentialequationis subjectto two boundaryconditions:
z :0,
r-..1I1On
cr: clo
(2.2-6)
cl : ct!
(2.2-1)
. : ' r l\ i O n
z:
::rtion
.:il and
-
,u]t to
t:Jster
. : : . ,i t l l y .
I,
:.:.rrn.becausethis systemis in steadystate,the concentrations
c16and c17are independent
' lime. Physically,
this meansthat the volumes of the adjacentsolutionsmust be much
:'rter than the volume of the fìlm.
_::J but
2.2.2 Muthematical Results
The desiredconcentrationprofile and flux arenow easilyfound. First, we integrate
- ).2-5 twice to find
ct:albz
--r On
..ltlons
. , nt h e
, ì nt h e
itr this.
. \\ithin
--
;' constantsa andb canbe found from Eqs.2.2-6 and2.2-l, so the concentrationprofile is
'Z
c l : c l o + ( c l /_ c1o)
t
() ) _a\
:r' li1s41variationwas,of course,anticipated
by the sketchtnFig.2.2-1.
The flux is found by differentiatingthis profile:
jt:-D
dc,
D
, :;(cro-crr)
07.
: I: the
(2.2-8)
(2.2-10)
I
r3.'rìusethe systemis in steadystate,the flux is a constant.
.\s mentionedearlier,this caseis easymathematically.Although it is very important,it
. ,itten underemphasized
becauseit seemstrivial. Before you concludethis, try someof
r' e\amplesthat fbllow to make sureyou understandwhat is happening.
20
2 / Diffusion in Dilute Solutions
2 / SteadyDffisit
.:cidentsthatresult
'rembrane.
(a)
(b)
(c)
Fig.2.2-2. Concentration profiles across thin membranes. In (a). the solute is more soluble in
the membrane than in the adjacent solutions; in (b), it is less so. Both casesconespond to a
chemical potential gradient like that in (c).
This type of diff
. ,,lctly,in termsof :
-rossthe membra
,1.which drops .:
:ce responsiblefì.
npletely is Secti
The flux scroS:;
' . r ì l ew i t h F i c k ' :
i.
.
rI -
IDH
r
. is parallelto E.
Exampfe 2.2-l: Membrane diffusion Derive the concentrationprofile and the flux for
a single solutediffising acrossa thin membrane. As in the precedingcaseof a fìlm, the
membraneseparates
two well-stirredsolutions.Unlike thefilm, themembraneis chemically
differentfrom thesesolutions.
Solution As before,we first write a massbalanceon a thin layer Az:
-'rerhilitrr
on'l
.'
:r'nneabilitrPs
the diffusion;
- : 1 t è r e n c ei ns . ,
0 : A ( j t l . - , / rl : + r : )
This leadsto a differentialequationidenticalwith Eq. 2.2-5:
r)
O-Cr
0: Daz'
However,this new massbalanceis subjectto somewhatdifferentboundaryconditions:
z:0,
z,: I,
ct:HCrc
' , : m P l € 2 . 2 - 2 :P o r
, : l e a r ec h a n s :
Solution
, n s e ro n È - . 1
"' -:.:ne. Rath::
r ctreficre:
- - . r l È t h e n; *
ct: HCtt
where 11 is a partitioncoeffìcient,the concentrationin the membranedivided by that in the
adjacentsolution. This partitioncoeffìcientis a equilibriumproperty,so its useimplies that
equilibrium existsacrossthe membranesurface.
The concentrationprofìle that resultsfiom theserelationsis
l . l - . 1 :\ 1 .
: --'
^ . . : .'
ct:H('tl+HlC11 -C,r)ì
which is analogous
to Eq. 2.2-9.This resultlooksharmlessenough.However,it suggests
concentrationprofìleslikes thoseinFig.2.2-2, which containsuddendiscontinuitiesat the
interface.If the soluteis more solublein the membranethan in the surroundingsolutions,
then the concentrationincreases.If the solute is less soluble in the membrane.then its
concentrationdrops. Either caseproducesenigmas.For example,at the lefi-hand side of
the membranein Fig. 2.2-2(a),solutediffusesfiom the solution at r:y6into the membrane
at hi pher concentration.
This apparentquandaryis resolvedwhen we think carefully aboutthe solute'sdiffusion.
Diffusion often can occur fiom a region of low concentrationinto a region of high concentration;indeed,this is the basisof many liquid-liquid extractions.Thus the jumps in
concentrationinFig.2.2-2 arenot asbizarreasthey might appear;rather,they aregraphical
,!L-"'
,|[-
\olution.s
2.2 / Steudt'DiJJusionAcrcssa Thin Film
21
accidentsthatresultfrom usingthe samescaleto representconcentrations
insideandoutside
membrane.
This type of diffusion can also be describedin terms of the solute'senergy or, more
exactly,in termsof its chemicalpotential.The solute'schemicalpotentiaÌdoesnot change
acrossthe membrane'sinterface,becauseequilibrium existsthere. Moreover,this potential, which drops smoothly with concentrarion,as shown in Fig. 2.2-2(c), is the driving
fbrce responsiblefor the diffusion. The exact role of this driving force is discussedmore
completelyis Sections6.4 and1.2.
The flux acrossa thin membranecanbe found by combiningthe foregoingconcentration
profilewith Fick's law:
IDHl
-C11.)
,/t :--(C11y
\ fu)r
. .r h e
.-lllr
This is parallelto Eq. 2.2-10. The quantity in squarebrackersin this equationis calledthe
permeability,and it is ofien reportedexperimentally.Sometimesthis sameterm is called
the permeabilityper unit length. The partition coefficient11 is found to vary more widely
than the diffision coefficientD, so differencesin diffusion tend to be less importantthan
the differencesin solubilitv.
Example 2.2-2:Porous-membranediffusion Determinehow the resultsof the previous
exampleare changedif the homogeneousmembraneis replacedby a microporouslayer.
Solution The differencebetweenthis caseand the previousone is that diffusion
is no longer one-dimensional;it now wiggles along the tortuouspores that make up the
rnembrane. Rather than try to treat this problem exactly, you can assumean effective
diffusion coefîcient that encompasses
all ignoranceof the pore'sgeometry.All the earlier
answersare then adopted;for example,the flux is
lr:
:t the
. rhat
. - !ú\L\
'-.,ttne
- .i l o n s .
:l;'tl ltS
- r J eo f
îrane
,lrion.
: ùon-
I D-oHf
l-,LltCro-Crr)
t
t
l
rvhereD.s i, u n"*, "effective" diffusion coefficient.Sucha quantityis a flnction not only
of soluteand solventbut also ofthe local seometrv.
Example 2.2-3: Membrane diffusion with fast reaction Imagine that while a solute
:s diffusing steadily across a thin membrane,it can rapidly and reversibly react with
Ither immobile solutesfìxed within the membrane. Find how this fast reaction affects
:hesolute'sflux.
Solution The answeris surprising:The reactionhasno efTect.This is an excellent
:rample becauseit requirescareful thinking. Again, we begin by writing a massbalance
.rn a layer Az locatedwithin the membrane:
solute \
/
/ s o l u t ed i î f u s i o ni n \
/ a m o u n rp r o d u c e d \
*
\ u v c h e m i c a l r e a c r i /o n
\ a c c u m u l a t i o " i : \ m i n u s t h a r o u ri
Becausethe systemis in steadystate,this leadsto
rrt-ìs in
'rh ì,..1
0 : A( jrl: - ,/rl.:+r;)- rrAL.z
22
2 / Diftusion in Dilute Solutions
nùúuil!
||ùnu"tr I
Porous
Diaphragm
,UllL ),
ft":
fE rems '!,!r
fi"-:
tilHflraul riil*r
Fig. 2.2-3. A diaphragm cell for measuring diffusion coefîcients. Becausethe diaphragm has
a much smaller volume than the adjacent solutions. the concentration profile wìthin the
diaphragm has essentially the linear, steady-statevalue.
,Jili =
ír
ftrH,*,r'
ff
)
u
0:-..À--rr
dZ
where rl is the rate of disappearance
of the mobile speciesI in the membrane.A similar
massbalancefbr the immobile product2 gives
,
'
't-*#
d
U - - . . / 1 f / ' 1
47.
But becausethe productis immobile, j2 is zerc,and hence11is zero. As a result,the mass
balancefor speciesI is identicalwith Eq. 2.2-3,leavingthe flux and concentrationprofìle
unchanged.
This result is easierto appreciatein physicalterms. After the diffusion reachesa steady
state,the local concentrationis everywherein equilibrium with the appropriateamountof
do not changewith time, the
the fast reaction'sproduct. Becausetheselocal concentrations
amountsof the productdo not changeeither.Diffusion continuesunaltered.
This casein which a chemicalreactiondoesnot affect diffusion is unusual.For almost
any other situation,the reactioncan engenderdramaticallydifferent masstransfer. If the
reactionis irreversible,the flux can be increasedmany ordersof magnitude,as shown in
Section I 6.1. If the difTusionis not steady,the apparentdiffusion coefficientcan be much
greaterthan expected,as discussedin Example 2.3-3. However,in the casedescribedin
this example,the chemicalreactiondoesnot affect diffusion.
Example 2,2-4: Diaphragm-cell diffusion One easy way to measurediffusion coefîcients is the diaphragmcell, shown in Fig. 2.2-3. Thesecells consistof two well-stirred
volumesseparated
by a thin porousbarrieror diaphragm.In the more accurateexperiments,
the diaphragmis ofien a sinteredglassfrit; in many successfulexperiments.it is just a piece
of filter paper (seeSection5.5). To measurea diffusion coefficientwith this cell, we fill
the lower compartmentwith a solutionof known concentrationand the uppercompartment
with solvent. After a known time, we sampleboth upper and lower compartmentsand
measuretheir concentrations.
Find an equationthat usesthe known time and the measuredconcentrationsto calculate
the diffusion coefficient.
,un
&l
I
,F
\ttlurions
2.2 / SteadyDiJfusionAcross a Thin Film
L-')
Sucha
Solution An exactsolutionto this problemis elaborateand unnecessary.
solutionis known but neverused(Barnes,1934). The usefulapproximatesolutiondepends
value
on the assumptionthat the flux acrossthe diaphragmquickly reachesits steady-state
flux is approachedeventhoughthe concen(Robinsonand Stokes,1960).This steady-state
trationsin the upperand lower compartmentsare changingwith time. The approximations
introducedby this assumptionwill be consideredagainlater.
In this pseudosteadystate,the flux acrossthe diaphragmis that given for membrane
diffusion:
rrl,grn has
the
I DHI
, - l t C r . r o * . r - C lr p p . r )
/r :|
t
t
l
Here, the quantity H includes the fraction of the diaphragm'sarea that is availablefor
Jiffusion. We next write an overall massbalanceon the adjacentcompartments:
dCt.lu*",
Vtower#
dt
-Ajt
dCr.upp.,
Vuoocr---.::lAjt
"
d
Í
.' -\ similar
.vhereA is the diaphragm'sarea. If thesemassbalancesare divided bY Vru*",and yuppcr,
-espectively,and the equationsare subtracted,one can combine the result with the flux
tquationto obtain
te mass
d
- nrnlìle
,: .tead)
-.nLÌnt
of
: n t e .t h e
:: : rlntost
, :': If the
.'-:tr\\flilì
-3 nluch
,: -,:lbedin
.
, r coeffi:. l-stirred
,: rriments,
, . . ra p l e c e
-: .. rie fiÌl
ì.lrrlment
.-:r-nts and
: Jrlculate
^
;;{Cr
: Df(Ct,opp.,- Cl.lo*.,)
tu*.,- Ct.upp",)
r which
1 \
A H l I
YR- - - l - r - l
/ \vt"*t''v',,")
. a geometricalconstantcharacteristicof the particulardiaphragmcell being used. This
-.l-ferential
equationis subjectto the obviousinitial condition
: CÎ.to*..- Cf.uno.t
Cl.ror..- Ct.upp.,
/ : 0,
- rheuppercomparrmentis initially filled with solvent,then its initial soluteconcentration
:ll be zero.
Integratingthe differentialequationsubjectto this condition givesthe desiredresult:
Cl.lo*". -
Cl.upp.,
: ( ^-ftDr
Cî.,n*.. - C?.uoo"'
I
D:-ln|
pt
'tt
\
/ c P ''. ,' ,c. r - L l u P P c r I
\
C'.'u*",- Cr.upper
/
,-an measure the time r and the various concentrations directly. We can also determine
coeffìcient
Seometric factor B by calibration of the cell with a species whose diffusion
ro.uvn.Then we can determine the diffusion coefficients of unknown solutes.
) 1
2 / Diffusion in Dilute Solutions
Stead,tDiffusion At
There are two major ways in which this analysiscan be questioned.First, the diffusion
coefficientusedhereis an effèctivevalue alteredby the tortuosityin the diaphragm.Theoreticiansoccasionallyassertthat differentsoluteswill havedifferenttortuosities,so that the
diffusion coefficientsmeasuredwill apply only to that particulardiaphragmcell andwill not
be generallyusable.Experimentalistshavecheerfullyignoredtheseassertionsby writing
D-
|
/c9.
-cP
clo
\
- r n 1 ' i h ' u c r ' l u P PI e r
- Ct.,pr,,
B't
\C'.'"*",
/
wherep' is a new calibrationconstantthat includesany torluosityefTects.So far, the experimentalistshavegottenaway with this: Diffusion coefficientsmeasuredwith the diaphragm
cell do agreewith thosemeasuredby other methods.
The secondmajor questionaboutthis analysiscomesfrom the combinationof the steadystateflux equationwith an unsteady-state
massbalance.You may find this combinationto
be one ofthose areaswheresuperfìcialinspectionis reassuring,but wherecarefulreflection
is disquieting.I havebeentemptedto skip overthis point, but havedecidedthat I had better
not. Heregoes:
The adjacentcompartmentsare much larger than the diaphragm itself becausethey
containmuch more material. Their concentrationschangeslowly, ponderously,as a result
of the transfer of a lot of solute. In contrast, the diaphragm itself contains relatively
little material. Changesin its concentrationprofile occur quickly. Thus, even if this
profile is initially very diffèrent from steadystate,it will approacha steadysratebefore
the concentrationsin the adjacentcompartmentscan changemuch. As a result,the profìle
acrossthe diaphragmwill alwaysbe closeto its steadyvalue,eventhoughthe compartment
concentrationsare time dependent.
Theseideascan be placedon a more quantitativebasisby comparingthe relaxationtime
of the diaphragm,t21o, with that of the comparrmenrs,1l(Dp) The analysisusedhere
will be accuratewhen (Mills, Woolf, and Watts, 1968)
trr,t.l,/!:"
| /( p Dr1,
( 1
:vai"p,.ogn'
".r
+
\ Vt,,*.,
I
S m ol l
---diffusion
coefficient
trio ra-J C the diffu'ior. -
:r film. [ì rtc:ir
(.
i -
t l _
ci --
'j..ìlt.thei.-..
- -D.
)
Vuppr,
/
This type of "pseudosteady-state
approximation"is common and will be found to underlie
mostmasstransfercoefficients.
Example 2.2-5: concentration-dependent diffusion In all the examplesthus far, we
haveassumedthat the diffusion coefficientis constant.However.in somecasesthis is not
true; the diffusion coeffìcientcan suddenlydrop from a high value to a much lower one.
Suchchangescan occur for water difTusionacrossfìlms and in detergentsolutions.
Find the flux acrossa thin film in which diffision variessharply. To keep the problem
simple, assumethat below somecritical concentrationc1., diffusion is tast,but abovethis
concentrationit is suddenlymuch slower.
Solution This problem is best idealized as two films that are stuck together
(Fi$.2.2-4). The interfacebetweenthesefìlms occurswhen the concentrationequalsc1..
ll
= t )
D
[)í,iùsiorrAcrossa Thin Film
. Solutions
:.: :ic diffision
-:::,rgllì.
.
TheO-
J.. \(l that the
L0rge
d if f u s i o n
coefficient
: ' - : . . L n dw i l l n o t
-
' h1 rvriting
\ :.ii.theexpef: : : .: : - . d: i a p h r a g m
: :: I thesteady: , , : : b r n a t i o nt o
, . : r : r . i rl e f l e c t i o n
- : .::I hadbetter
: : 'iailusethey
: ,-. ., J\ a result
- ' . : : : ì -r e l a t i v e l y
. : 'e. n i f t h i s
: -., -:-,tr'betbre
- - . . : : t ep r o t ì l e
'_ - 'ìrt.Lrtment
.ri - Lrll tlllle
.
z=Q
z=zc
z=l
::g. 1.2-.1.Concentration-dependentdiffusion acrossa thin fiim. Above the concentration t 1,,
::e drffusion coeffìcient is small; below this critical value, it is larger.
'- j: tÌm. a steady-state
massbalanceleadsto the sameequation:
, d j t
dz
- -;.ult. the flux j1 is a constanteverywherein the film. However,in the leffhand film
- -': .oncentrationproducesa small diffusion
coefficient:
-.iJ here
dc,
/ t: - D ,
a?.
. ::.ult is easilyintegrated:
t.:
.rnderlie
f,
I ita, -D I
.ttt
dct
J,,u
- : t h er e s u l t
-
-.. îdr. wO
. . : . : : - . ìi.s n o t
. . \e r o n e .
: '.:J rroblem
- ,.: -:'!r\e this
:.-\ tlrgether
" 3 . l u a l cs t c .
- - (Dc r o - c r . )
,/r
z(.
--' risht-handfilm, the concentrationis small, and the diffusion coefficientis large:
t..
l r : - D ?az.
D
:,
L
_
-
aa
(clr_cu)
2 / Difrusionin Dilute Solutions
/(r
The unknownposition2,.can be found by recognizingthat the flux is the sameacrossboth
films:
I
t' : o!!r-r-t'
l(c19 -
c1,.)
The flux becomes
D ( c ' r o- c r , . )* D ( c r , - c t t )
.Il
-
If the critical concentrationequalsthe averageof c1sand c11,then the apparentdifTusion
coefficientwill be the arithmeticaverageof the two diffusion coefficients.
In passing,we shouldrecognizethat the concentrationprofile shown in Fig' 2.2-4 im'
-
. efluxacrossthefilmisconstantand
p l i c i t l y g i v e s t h e r a t i o otfh e d i f f i s i o n c o e f f l c i e n tTs h
is proportionalto the concentrationgradient.Becausethe gradientis largeron the left, the
difTusioncoeffìcientis smaller. Becausethe gradientis smalleron the right, the diffusion
coefficientis larger. To test your understandingof this point, you should considerwhat
the concentrationprofìle will look like if the diffusion coeffìcientsuddenlydecreasesas
will help you understandthe next and final
the concentrationdrops. Such consi<lerations
+
:rt(,
examplein this section.
ui
Example 2.2-6: Skin diffusion The diffusion of inert gasesthrough the skin can cause
itching,burning rashes,which in turn can lead to vertigo and nausea.Thesesymptomsare
believedto occur becausegaspermeabilityand diffision in skin are variable.Indeed,skin
behavesas if it consistsof two layers,eachof which has a different permeability(Idicula
et at., 1976).Explain how thesetwo layerscan lead to the rashesobservedclinically.
Solution This problemis similarto Examples2.2-l and2.2-5,butthe solution
is very complex in terms of concentration.We can reducethis complexity by defining a
new variable: the gas pressurethat would be in equilibrium w'ith the locctlconcentration.
The "concentrationprofìles" acrossskin are much simpler in terms of this pressure'even
though it may not exist physically. To make theseideasmore specific,we label the two
layersof skin A and B. For layer A,
!!u e]l-:u
MÍ i
,lliirr
.,uÍil.
lttfir! [r r, ]|]:,ù,
!Îlr]l]rrt[|r i'ì
trtrlnr{u gî:L'r
|lnùtlmF[]L,
fi
ilÍlÍ0tclúl'h{r
lr"*l
!l|r,[. le.
u u-s J
úuftrv:r nr
îîìr
!Ín{rXT
P t : P t . g a+s
- Pt go')
i(ttti
3,
5!:i:
@lg
rùifîrn\E
Pil
muu:1
and for layer B,
f
pt : Pti +
z
f{rt.ti..ue
- Pl;)
-Ìc
The interfacialpressure
I ' l t
-
r
ulf
iilrhlt(xl.l.;.
/ D,qHa\
/ DnHa\
,^
r,
\
f
)Ptc"*\
h
-
/Pt
ilfullilÌ$hì
ri"ue
k
can be found fiom the fàct that the flux throughlayer A equalsthat throughlayer B.
li:t:t1
M'urii:'-r'
{m:!
2l
2.2/ Stead,-DiJJusionAc'rossa Thin FiLm
tn
I P1iîP?i
p1îp?
Loyer
P1,qos
P2, tissue
. , n
:ll-
-,:'lJ
.
G o so u t s i d e
the body
:|lr'
't]
::.1t
':\
'L.11
z =O
. ifc'
z= lA
z=lA + lA
Fig. 2.2-5. Gas difiusion across skin. The gas pressuresshown are those in equilibriurn with
the actual concentrations. In the specifìc case considered here, gas 2 is more permeable in
layer B, and gas I is more permeable in layer A. The resulting total pressurecan have major
physiologic effècts.
- rìll
-,|a
. . , .J I ì
- i I
'
, t l,
J.-lì
:',\o
Theseprofiles,which are shownin Fig. 2.2-5,imply why rashesform in the skin. In
particular,thesegraphsillustratethe transportofgas I from the suffoundingsinto the tissue
and the simultaneousdiffusion of gas 2 acrossthe skin in the oppositedirection. Gas I
is more permeablein layer A than in layer B; as a result, its pressureand concentration
gradientsfall lesssharplyin layer A thanin layer B. The reverseis true fbr gas2; it is more
permeable
in layer B thanin A.
Thesedifferent permeabilitieslead to a total pressurethat will have a maximum at the
rnterfacebetweenthe two skin layers. This total pressure,shown by the dotted line in
Fig. 2.2-5, may exceedthe surroundingpressureoutsidethe skin and within the body. If
it doesso, gasbubbleswill form aroundthe interfacebetweenthe two skin layers. These
bubblesproducethe medicallyobservedsymptoms.Thusthis conditionis a consequence
of
unequaldifTusion(or, more exactly,unequalpermeabilities)acrossdiffèrentlayersof skin.
The examplesin this section show that diffusion acrossthin films can be diffìcult to
understand.The difficulty doesnot derivefrom mathematicalcomplexity; the calculation
rs easy and essentiallyunchanged.The simplicity of the mathematicsis the reasonwhy
Jrffusion acrossthin films tends to be discussedsuperfìciallyin mathematicallyoriented
books. The difîculty in thin-film diffusion comes from adaptingthe same mathematics
ttr widely varying situationswith different chemical and physical effects. This is what is
Jifîcult to understandaboutthin film diffusion. It is an understandingthat you must gain
befbreyou can do creativework on hardermasstransferproblems.
2 / Diffusion in Dilute Solutions
28
2.3 UnsteadyDiffusionin a SemiinfiniteSlab
We now turn to a discussionof diffusion in a semiinfìniteslab. We considera
volume of solutionthat startsat an interfaceand extendsa very long way' Such a solution
can be a gas,liquid, or solid. We want to find how the concentrationvariesin this solution
as a result of a concentrationchangeat its interface. ln mathematicalterms, we want to
fìnclthe concentrationand flux as functionsof position and time'
This type of masstransferis often calledfiee diffusion (Gosting, 1956)simply because
this is briefer than "unsteadydiffusion in a semiinfiniteslab."At first glance,this situation
may seemrare becauseno solutioncan extendan infinite distance.The previousthin-film
examplemademore sensebecausewe can think of many more thin films than semiinfinite
slabs.Thus we might concludethat this semiinfinitecaseis not common. That conclusion
would be a seriouserrol.
The important caseof an infinite slab is common becauseany diffusion problem will
behaveas if the slabis infinitely thick at shorlenoughtimes. For example,imaginethat one
of the thin membranesdiscussedin the previoussectionseparatestwo identical solutions,
so that it initially containsa solute at constantconcentration.Everything is quiescent,at
equilibrium. Suddenlythe concentrationon the leffhand interfaceof the membraneis
raìsed,as shown in Fig. 2.3-1. Just after this suddenincrease,the concentrationnear this
left interf'acerisesrapidly on its way [o a new steadystate. In thesefirst few seconds,the
concentrationat the right interfaceremainsunaltered,ignorantof the turmoil on the left'
The left might as well be infinitely far away; the membrane,for thesefirst few seconds,
might as *"ll b. infinitely thick. Of course,at largertimes,the systemwill slitherinto the
steidy-statelimit in Fig. 2.3-l(c). But in those first seconds,the membranedoesbehave
like a semiinfìniteslab.
This example points to an important corollary, which statesthat casesinvolving an
infinite slab an<la thin membrane will bracket the observed behavior. At short times,
dif1ision will proceedas if the slab is infìnite; at long times, it will occur as if the slab
is thin. By focusing on these limits, we can bracket the possiblephysical responsesto
different diflision Problems.
2.3.1 The PhYsical Situation
The diffusion in a semiinfiniteslab is schematicallysketchedin Fig. 2'3-2' The
as
slab initially contains a uniform concentrationof solute clÉ. At Sometime, chosen
although
increased,
abruptly
and
is
suddenly
interface
the
time zero, the concentrationat
the solute is always presentat high dilution. The increaseproducesthe time-dependent
concentrationprofile that developsas solutepenetratesinto the slab.
We want to fìnd the concentrationprofile and the flux in this situation,and so again we
needa massbalancewritten on the thin layer of volume A Az:
/ soluteaccumulation\ -in volumeAAz /
\
rate of diffusion /
\
\into thelaYeratz )
f rateof diffusion\
/
[*:::'l'i1"'
(2.3-r)
ln mathematicalterms,this is
!o +u rt1
: A (.i tl-. ,/rl.+r .)
12.3-2)
2.3 / UnsteadyDffision in a SemiinfiniteSlab
C o n c e n î r o t i o np r o f i l e i n
o m e m b r o n eo l e q u i l i b r i u m
!l
-4 .
29
.i-Lrll
.i1'rll
'':
Ir)
..,1Ìl
: .L n l
C o n c e n t r oi fo n p r o fi l e s l i g h îl y
o f î e r î h e c o n c e n t r o f i o no n
îhe lefî is roised
.ilg
..'rll
I ncreose
lll
'ne
Il \.
rl
..'ìi is
,.: :his
Limiîing concenlrolion
p r o fi l e o f l o r g e f i m e
J -ait.
, - I
-
.,uì.
"j an
Fig. 2.3- I . Unsteady- versus steady-statediffusion. At small times, difTusion will occur only
near the lefrhand side of the membrane. As a result. at these small times. the diffusion will be
the same as if the membrane was infinitely thick. At large times, the results become those in
the thin fìim.
..:ì1es.
' j
.ìiìb
' .1.
tO
- The
.ifl
AS
::10ugh
: sndent
:rln we
1 . 3I-)
').3-2)
Fig.2.3-2. Free diffusion. ln this case,the concentration at the left is suddenly increasedto
a higher constant value. Diflìsion occurs in the region to the right This case and that in
Fig.2.2-1 are basic to most diffusion problems.
30
2 / Dilt'usionin Dilute Solutions
tr'î 'î : "".
We divide by AA: to find
Dcr
at
:
-
l
-
( j t l , + t , - , r rl . \
l
(2.3-3)
\(:+a:)-:/
We then let A: go to zero and usethe definitionof the derivative
àcr _
à/r
_
3t
3:
(2.3-4)
Combining this equation with Fick's law, and assumingthat the diffusion coefficient is
independeo
n lt 'c o n c e n t r a l i o w
n .e g e t
^
d C r'
- D
^1
d'Ct
(2.3-s)
' ò ::2
At
This equationis sometimescalled Fick's secondlaw, and it is often referred to as one
exampleof a "diffusion equation."In this case,it is subjectto the following conditions:
/:0,
a l l: ,
cl:cre
/>0.
z:0.
cl:cto
(2.3-6)
(2.3-1)
i:@,
cl:clx
(2.3-8)
Notethatbothcl6ÀDdct0aretakenasconsta
Tnhtesc. o n c e n t r a t i o n c l - i s c o n s t a n t b e c a u s e
it is so fàr fiom the interfàceas to be unaffèctedby eventsthere; the concentrationc1eis
kept constantby adding materialat the interface.
2.3.2 Mathematical Solution
The solutionof this problem is easiestusing the methodof "combinationof variables."This method is easy to fbllow, but it must have been difficult to invent. Fourier,
Graham,and Fick failed in the attempt;it requiredBoltzman'storturedimagination(Boltzm a n ,1 8 9 4 ) .
The trick to solving this problem is to definea new variable
'rrom r*'
:IIlr :".
q Lr U
(2.-r-9)
'/4Dt
The differential equation can then be w ntten as
dt, /à(\
dl,, /;t. 1:
. I
ti1
l : D d, (-- ; t . ,I
\àt /
\azl
(2.3r0)
nu
or
,,
d ' c r'
Lt<'
,
| 1r
d r:t
-
n
d<
( 2 . 3r -l )
In otherwords,the partialdiff'erentialequationhasbeenalmostmagicallytransformedinto
an ordinarydifferentialequation.The magic also works for the boundaryconditions;from
F,q.2.3-7,
(:0'
cl:cro
(2.3-12)
'ffius**
"*
) -l / Unstead) DrJJusionin a Semiinfinite Slab
a t
J 1
nd fiom Eqs.2.3-6and 2.3-8,
(:oo,
' l_11
(2.3-13)
cl:ctn
'\ ith the methodof combinationof variables,the transformationof the initial andboundary
,nditionsis often more critical than the transformationof the differentialequation.
The solutionis now straightforward.One integrationof Eq. 2.3-1I gives
'l-:ll
dc:1
- :ae
\'
() 7-14\
d<
. rJnt ls
:erea is an integrationconstant.A secondintegrationand useofthe boundarycondition
.- \
't-5r
cr - clo
clc
-
(2.3l s)
- È r t r
clu
,t. Olì9
:'ft
. , \ l l\ :
'ì-6r
ì ì-Rì
1i i a u s e
... !
:,-
1 r )r ò
e r f1 : l :
) r q
I e "ds
t/Í
(2.3-t6)
.lo
- ir is the errorfunctionof {. This is the desiredconcentrationprofile giving the variation
- rìcentrationwith position and time.
nranypracticalproblems,the flux in the slabis ofgreaterinterestthantheconcentration
' .: itself.This flux
can againbe foundby combiningFick's law with Eq. 2.3-15:
j, :
Ar,
-O'i)
t11
: 1/OJ"t,
:'l4Dt(crc- cr-)
(2.3-t7)
:.rrticularly useful limit is the flux acrossthe interfaceat z - 0:
'I Van-
F',urier,
Boltz-
I l-g)
t -r-10)
r 3 - l1 )
:nlrd into
, ' n \ :f r o m
(2.3-18)
./rl::o: f Dlrt(c11'-cr!)
- ' .'
lr is the valueat the particulartime / andnot that averagedovertime. This distinction
, -,' ulÌportantin Chapter13.
- . ::rispoint, I havethe samepedagogicalproblem I had in the previoussection:I must
'.e vou that the appzrentlysimple resultsin Eqs. 2.3-15and 2.3-18 are valuable.
-'" .
. , :r'\ultsare exceededin importanceonly by Eqs.2.2-9 and2.2-10. Fortunately,
the
î.- - :ìr.rticsmay be difficult enoughto sparkthought and reflection;if not, the examples
r,.,rrl shclulddo so.
[,r,--nple 2.3-1: Diffusion across an interface The picture of the processin Fig. 2.3-2
nr-- -. ihat the concentrationat z : 0 is continuous.This would be true, for example,if
* " . - ' ( ) t h e r e w aas s w o l l e ng e l ,a n d w h e n z< 0 t h e r ew a sa h i g h l y d i l u t es o l u t i o n .
-- : r\ er. a much more common caseoccurswhen thereis a gas-liquid interfàceat
- =
t lrtlinarily,the gas at ; < 0 will be well mixed. but the liquid will not. How will
î
i::.rcerffect the resultsgivenearlier?
Solution Basically,it will haveno effect. The only changewill be a newboundary
Pro
'1 ì-l?\
tl
r
2 / Dffision ín Dilute Solutions
-)z
wherec I is the concentrationof solutein the liquid, x I is its mole fraction,p ro is its partial
pressurein the gas phase,H is the solute'sHenry's law constant,and c is the total molar
concentrationin the liquid.
The difficulties causedby a gas-liquid interfaceare anotherresult of the plethoraof
units in which concentrationcan be expressed.Thesediffìculties require concernabout
units, but they do not demandnew mathematicalweapons. The changesrequired for a
liquid-liquid interfacecan be similarly subtle.
Example 2.3-2: Free diffusion into a porous slab How would the foregoingresultsbe
changedif the semiinfiniteslab was a poroussolid? The diffusion in the gas-filledporesis
much fasterthan in the solid.
Solution This problem involves diffusion in all three directionsas the solute
moves through the tortuouspores. The common method of handling this is to define an
effectivediffision coefficientD.6 and treatthe problem as one-dimensional.The concentration profile is then
('t - cto
- : e
('r\-cru
z.
" îJ 4 D , r l
and the interfacial flux is
--*:,lrlr:o: 1/D.xlnt(crc - c1r)
This type of approximationoften works well if the distancesover which diffusion occurs
are largecomparedwith the size of the pores.
Example 2.3-3: Free diffusion with fast chemical reaction In many problems, the
diffusing solutesreactrapidly and reversiblywith sunoundingmaterial. The surrounding
materialis stationaryand cannotdiffuse. For example,in the dyeing of wool, the dye can
react quickly with the wool as it diffusesinto the fiber. How does such a rapid chemical
reactionchangethe resultsobtainedearlier?
Solution In this case,the chemical reaction can radically changethe process
by reducingthe apparentdiffusion coeflìcientand increasingthe interfacialflux of solute.
Theseradical changesstandin stark contrastto the steady-state
result,wherethe chemical
reactionproducesno elTect.
To solve this example,we first recognizethat the solute is effectively presentin two
forms: (l) free solutethat can diffuse and (2) reactedsolutefixed at the point of reaction.
If this reactionis reversibleand fasterthan diffusion.
c2: Kct
where c2 is the concentrationof the solutethat has alreadyreacted,c1 is the concentration
of the unreactedsolutethat can diffuse,and K is the equilibriumconstantof the reaction.If
the reactionis minor, K will be small; as the reactionbecomesirreversible,K will become
verv larse.
l{lllnri;n,.'*
ltb
nu"^,
nÌ
urt-
- .1/ UnsteadyDffision in a Semiinfinite Slah
:
r -
. :.rfiial
- :.nrllar
, -:'ì rbout
\
IL)f a
_ . : : 3 \ 1 5
, .,,lute
-
/ a c c u m u l a t i o\ n
I
- _ - - . . - . 1b, e
'
With thesedefinitions,we now write a massbalancefbr eachsoluteform. Thesemass
--,lancesshouldhavethe form
- : l.,rfî Of
-:-
. i : : 1 3a n
: ì .e n -
J-')
-
inAA:
l -
/
/
|
d i f f u s i o ni n \ |
\ m i n u st h a to u r)
/ a m o u n to r o d u c e d
bv\
J' |
\
r e a c t i oi n A A :,
" l
/
,r thediffusingsolute,thisis
À
-d t [ A A z c r l: A ( j t l . - , t rl . + r . )+ r l A L z
. :irre r; is therateof productionper volumeof speciesI, the diffusingsolute.By arguments
- r.rÌogous
to Eqs.2.3-2to 2.3-5,this becomes
d(r
rl'('t
: D
*
dI
u
d7'
t i ' t
:r' term on the left-handsideis the accumulation;the first term on the right is the diffusion
" nrinusthe diffusion out; the term 11is the effect of chemicalreaction.
\\'hen we write a similar massbalanceon the secondspecies,we find
À
-[AAec2] :
ot
-rrALz
ocz
At
-. the
'
. .r' ,.i"ì .n,o. è
,,3 aan
::.nical
'1-' do not get a diffusion term becausethe reactedsolutecannotdiffuse. We get a reaction
:::l that has a different sign but the samemagnitude,becauseany solutethat disappears
-. 'peciesI reappears
as species2.
To solvethesequestions,we first add them to eliminatethe reactionterm:
ò
à2c,
I t ' - l -r ' r ) : D -
dt
dz'
,r i now usethe fàct that the chemicalreactionis at equilibrium:
I lLrieSS
. ,r l u t e .
Jn l l C a l
:l îwo
:-::tiOn.
.:,ttlon
t , , n .I f
:J()mg
3
i Jl c ,
-(triKcll:Ddt
dz.'
dct
D
ò 2 ,t
Ar
lfK
0:2
.:.ri\ result is subjectto the sameinitial and boundaryconditionsas before in Eqs. 2.3-6,
- -ì-7. and 2.3-8. As a result, the only differencebetweenthis example and the earlier
:oblemis that D/(1 f K) replacesD.
This is intriguing. The chemicalreactionhas left the mathematicalform of the answer
-:r;hanged,but it has alteredthe diffusion coefficient.The concentrationprofile now is
ct -cto
(r\-(ro
_ - .-,'-J q ù i + z- R Í l l
__2 / Dilfusion in Dilute Solutions
34
and the interfacialflux is
r K l l " t ( ( r o- ( r à )
,/rl::o: V6i
The flux hasbeenincreasedby the chemicalreaction.
These effects of chemical reaction can easily be severalorders of magnitude. As
will be detailedin Chaprer5, diffusion coeffìcientstend to fall in fairly narrow ranges.
Thosecoefficientsfor gasesare around0.3 cm2/sec;thosein ordinaryliquids clusterabout
l0-5cm2/sec.Deviationsfromthesevaluesofmorethananorderofmagnitudeareunusual.
However,differencesin the equilibrium constantK of a million or more occur frequently.
Thus a fast chemicalreactioncan tremendouslyinfluencethe unsteadydiffusion process.
Example 2.3-4: Determining diffusion coefficients from free diffusion experiments
Diffusion into a semiinfiniteslab is the geometryusedfor the most accuratemeasurement
determinethe concentration
of diffusion coefficients.Thesemost accuratemeasurements
interferometer,uses
Rayleigh
the
profile by interferometry.One relativelysimple method,
index (Dunlop
in
refractive
function
a rectangularcell in which there is an initial step
shining
collinated
by
profile
is
followed
et al. 1912). The decay of this refiactive index
refractive
index
record
the
These
fringes
light throughthe cell to give interferencefringes.
v e r s u sc a m e r ap o s i t i o na n dt i m e .
Find equationsthat allow this informationto be usedto calculatediffusion coefficients.
Solution The concentrationprofìlesestablishedin the diffision cell closely approachtheprofilescalculatedearlierfor a semiinfiniteslab. The cell now effectivelycontains
two semiinfiniteslabsjoined togetherat z : 0. The concentrationprofile is unalteredfrom
Eq.2.3-15
ct-clg
ctx
-
7,
,-
-
:efl
Threr t
cto
"/4Dt
wherecle[: (cr-*cr--)/2] istheaverageconcentrationbetweenthetwoendsofthecell.
How accuratethis equationis dependson how exactly the initial changein concentration
can be realízed in practicethis changecan routinely be within 10 secondsof a true step
function.
We must convert the concentrationand cell position into the experimentalmeasured
refractiveindex and cameraposition. The refractiveindex n is linearly proportionalto the
concentratron:
p:4.o1u"n,f/rr.1
is the refractiveindex of the solvent. Each position in the camerais proporwhere nro1u.n1
tional to a positionin the diffusion cell:
Z:az,
wherea is the magnificationof the apparatus.It is experimentallyconvenientnot to measure
the positionof one fiinge but ratherto measurethe intensityminima of many fringes. These
minima occur when
n-no
l
ttx-tto
J12
iffi*
\ , 1 i l I 1 (r fl , t
35
).4 / ThreeOther Examoles
\\'herer?{ and r?0are the refiactiveindicesat z - oo and z - 0, respectively;./ is the total
numberof interferencefringes,and j is an integercalledthe fringe number. This number
r\ most convenientlydefìnedas zero at í - 0, the center of the cell. Combining these
iquations,
:--
-..Ì3. As
:-
i
. - . .. : 3 r l b o u t
l: -:l: -,llUrUOl.
, - :'J,lì.lentlY.
.
j
l:\rCeSS.
',""
érl
iilfl9e S.
J1 2
zi
-
uJ4a
.thereZlis the intensityminimum associated
with the 7th tiinge. Becausea and t are
-'\perimentallyaccessible,measurementsof 210. "/) can be used to find the diffusion
- Lrefficient
D. While the accuracyof interferometricexperimentslike this remainsunrivaled,
ic useofthese methodshas declinedbecausethey are tedious.
r\prriments
--
::'-ifÈl]l€lÌt
-.'nîration
' i : 3 r .u s e s
:-
'- - llinated
: - - . . - :. ' i n d e x
- - ::lìcients.
: , .eÌr'aP'
,
: .. ,'.r1113i15
,. :::r'J frOm
,- ::ht'cell.
'' - r':'rtration
- - ::!re slep
.
-
:-tJit\Ufed
:r.11
to the
2.4 ThreeOther Examples
The two previoussectionsdescribedifTusionacrossthin films and in semiinfinite
..lbs. In this section,we turn to discussingmathematicalvariationsof diffusion problems.
. his mathematicalemphasischangesboth the pace and the tone of this book. Up to now,
. e haveconsistentlystressedthe physicalorigins of the problems,constantlyharping on
-ituraleffectslike changingliquid to gas or replacinga homogeneousfluid wìth a porous
. 'lid. Now we shift to the more common textbook composition,a sequenceof equations
. ,metimesasjarring as a twelve-toneconcerto.
In theseexamples,we havethreeprincipal goals:
( I ) We want to show how the diff-erentialequationsdescribingdiffusion are derived.
(2) We want to examinethe effectsof sphericaland cylindrical geometries.
(3) We want to supply a mathematicalprimer for solving thesedifferent diffusion
equations.
:: all threeexamples,we continueto assumedilute solutions.The threeproblemsexamined
'.j\t are physicallyimportantand will be referredto againin this book. However,they are
-rroducedlargely to achievethesemathematicalgoals.
\
2.4.1 Decay of a Pulse (Inplace Transforms)
propor-
:.r nlcasure
_r3\.These
As a first example,we considerthe diffusion away from a sharppulse of solute
rc that shownin Fig.2.4-l. The initially sharpconcentrationgradientrelaxesby diffusion
:r the z direction into the smoothcurvesshown (Crank, 1975). We want to calculatethe
. rape of thesecurves.This calculationillustratesthe developmentof a differentialequation
.-:riiits solutionusing Laplacetransforms.
As usual, our first step is to make a massbalanceon the differential volume AAz as
- 1rlwn:
/
solute
\
solute \
solute \
/
/
airiusion
outof I
aitfusion
inro
I
|
|
volume
volume
this
this
/
\
/
\
l"::r'f:"',)
:
Q.4-tl
/
2 / Diffusion in Dilute Solutions
36
'itr'r'e
Oîlt,
rble cond
I
.-.l:L'gfJII'
. . ' ,e t h r -
P o s i î i o nz
: 0 diffuses as
Fig. 2.4- l. Diffusion of a pulse. The concentratedsolute originally located at z
t
h
e
t
h
r
e
e
m
o
s
t
i
m
p
o
r
t
a
n
t
c
a
s
e
s
,
alongwiththose
T
h
i
s
i
s
t
h
e
t
h
i
r
d
o
f
theGaussianprofileshown.
in Figs.2.2-l and2.3-2.
In mathematicalterms,this is
a
;dlt A A : r ' 1l : A i r l .- A . llr,rr .
t) L-)\
Dividing by the volume and taking the limit as Au goesto zero gives
3r'r
}jt
àt
ò2.
r) 4-7t
Combining this relationwith Fick's law of diffusion,
0 c'' - D
At
à2c,
.'
àz.'
(2.4-4)
This is the samedifferent equationbasic to the free diffusion consideredin the previous
section.The boundarycon<litionson this equationare as follows' First, far from the pulse,
the soluteconcentrationis zero:
/ > 0.
: : oQ, cr :0
Q'4-5)
Second.becausediffusion occurs at the samespeedin both directions,the pulse in symmetric:
r>0,
':0,
dct
-o
(2.4-6)
oz
This is equivalentto sayingthat at z : 0, the flux has the samemagnitudein the positive
and negativedirections.
The initial condition for the pulse is more interestingin that all the solute is initially
locatedatz:0:
/ : 0,
M
cr : -ó(:)
lfilL
: -
tLa
t ''
(2.4-1)
[email protected]
where A is still the cross-sectionalareaover which diffusion is occurring, M is the total
amountof solutein the system,and 6(z) is the Dirac function. This can be shownto be a
*ttrr
ile*-
: 'Tltrer Otlter Erttrttltle.s
), '..lil( )/1.\
3l
,rronabl€
conditionb1,a massbalance:
/./ .
t,Att::./
f'M
, iu,.tAdz:
M
(2.4-8)
thisintegration,
we shouldremember
tható(z)hasdimensions
of (length)-r.
rotur this problem,we fìrsrtakerhet-uptu."
rransfbrm
of Eq.z.+r+withrespectro
,,ll
.."0
' t t '-t. ! . t ( t ' _0 , - o ?
a7'
':ere
c-1is the transformed concentration.
The boundary conditions are
d|t
z -('l
dz
.li tìuses as
: u ith those
a : oc,
_ _M/A
2D
Q'4-g\
(2.4_10)
f't :0
(2.4_r1)
-
;' first of thesereflectsthe properties
of the Dirac function, bul the secondis routine.
' -.ration2.4-9
canthen easily by integratedto give
r14)t
cr : aey'slDz r be-/4
nz
-:re
(2.4-3)
cl
-
| 2.4_4)
previous
hr'pulse,
rl.4-5)
in sym-
t2.4-6)
ìi positive
. initially
\2.4-7)
. rhe total
',ntobea
(2.4-t2)
a and ó areintegrationconstants.Clearly,
a is zeroby Et1.2.4_11.
UsingEq.2.4-10,
: fìndà andhencelr:
:
M/E
';;
T,
yro-1tr- 1/ s/ Dz.
(2.4-13)
.: inverscLaplacetransformof this function
gives
_
M/A,-z2141nt
lq;
Dt-
(2.4-11)
l:ch is a Gaussiancurve. You may wish
to integratethe concentrationover the entire
.rcmto checkthat the total
solutepresentisM.
This solutioncan be usedto solve many
unsteadydiffusion problemsthat haveunusual
' ral
conditions(crank, l9?5). More important,
it is often ur.à to corelate the dispersion
' rollutants,
especiallyin the air, as discussed
in Chapter4.
2'4'2 steady Dissorution of a sphere (sphericat
coordinates)
our secondexample,which is easiermathematicary,
is the steadydissorutionof
':herical particle,as shown
inFig.2.4-2. The sphereis of a sparingly
sorubremateriar,
hat the sphere'ssizedoesnot changemuch.
However,this materiarquickly dissorvesin
': rulroul.ìding
solvent'so that solute'sconcentrationat the
sphere'ssurfàceis saturated.
r"ausethe sphereis immersedin a very
largefluid volume,,l..on."nirltion far from
-:r-rels
the
zero.
The goal is to find both the dissolution
rate and the concentrationprofile around
-rere'
the
Again, the fìrst stepis a massbarance.
In contrastwith the pi.urou, exampres,
' nlassbalanceis most conveniently
made in sphericalcoordinateso'rigrnating
from the
2 / Dift'usionin Dilute Solutions
38
2.4/ ThreeOther
This basicdiffere
r:Ro.
S o l u t ef l u x
owoy
the sphere werr
rhanged, but tl
E q . 2 . 4 - l 8f i e
dct
dr
' -':e
a is an inte
c 1: f i .
D i s t o n c ef r o m
s D h e r es c e n i e r
\ t
diffusion
asphere.Thisproblemrepresentsanextensionof
Fig.2.4-2.steadydissolutionof
this dissolutioncanbe
thJoryto a sphericallysymmetricsiìuation.In actualphysicalsituations,
'12)
by fiee convectioncausedby diffision (seeChapter
complicated
shell of thickness ar
center of the sphere. Then we can make a mass balance on a spherical
shell is like the rubber
located at some arbitrary clistancer from the sphere. This spherical
of a balloon of surface area4T rz and thickness Ar'
earlier:
A mass balance on this shell has the same general fbrm as those used
ai[1sio1
d i f r u s i o\ n_ l
/ soluteaccumulation
\ _
.,)
- (
s
h
e
l
l
/
t
h
e
i
n
t
o
within the shell )
\
\ o u t o l t h es h e l l/
\
\2.4-t5)
In mathematicalterms,this is
a
"
i),*6,
tru) : o : (4trr2
i;, - (4trr2
fi6rrt
\2.4-r6)
is
The accumulationon the left-handside of this massbalanceis zero' becausediffusion
point
by
this
steady,not varying with time. Novices frequently make a seriousefror at
term r2;1 is
The
wrong.
is
This
side.
righlhand
on
the
terms
both
of
;";""li"g the ,i oit
at (r + Ar) in
evaluateclat r in the first term; that is,-it is r21;r l,). The term is evaluated
the secondtem; so it equals(r * Lr)'(irl'+a').
the Imit
If we divide both sidesof this equationby the sphericalshell'svolume and take
as Ar '+ 0. we find
t d
0:-,;(r'j1)
t2.4-17)
rar
is constant'
Combinins this with Fick's law and assumingthat the diffusion coefficient
o" -
D d .dct
--y'dr
12dr2
:thet$oL'
(2.4-18)
-
r
\oltrÍions
2.4 / Three Other Examples
39
This basicdifferentialequationis subjectto two boundaryconditions:
r :
Ro, ct : ct (sat)
r:@,
ct:0
(2.4-19)
() 4-)O\
If the spherewere dissolvingin a partiallysaturatedsolution,this secondcondition would
be changed,but the basicmathematicalstructurewould remain unaltered.One integration
of Eq. 2.4-18 yields
dct
a
dr
12
() 4-)1t
where a is an integration constant. A secondintegrationgives
() 4i)\
r
Use of the two boundaryconditionsgivesthe concentrationprofile
cl :
-l \ 1On
,ncanbe
. .R
tl(sat) -
dc,
ar
D R,,
:_jr.1(sat)
() 4-)4\
which, at the sphere'ssurface,is
lr :
r.-1-l5)
() 4-)11
The dissolutionflux can then be found from Fick's law:
jr:_D,
, \ n e S SA r
.:rerubber
0
r
D
-cr(sal)
R6
t) 4-)5\
ifthe sphereis twice as large,the dissolutionrate per unit areais only halfas large,though
the total dissolutionrate over the entire surfaceis doubled.
This examplesforms the basisfor suchvariedphenomenaas the growth of fog droplets
.rndthe dissolutionof drugs. It is included here to illustrate the derivationand solution
rf differential equationsdescribingdiffusion in sphericalcoordinatesystems. Different
.oordinatesystemsare also basicto the final examplein this section.
l.:1-I 6)
, : : i u s i o ni s
. point by
r:lll rlJrt is
- -1,r)rn
. . r t h el i m i t
t) . 1 - 1 7 )
li \tant,
rl.;1-18)
2.4.3 Unsteady Diffusinn Into Cylinders (Cylindrical Coordinates and
Separatio n of Variable s)
The final example,probably the hardestof the three,concernsthe diffusion of a
.oluteinto the cylindershownin Fig.2.4-3. The cylinderinitially containsno solute.At
:mezero,it is suddenlyimmersedin a well-stirredsolutionthatis of suchenormousvolume
iet its soluteconcentrationis constant.The solutediffusesinto the cylindersymmetrically.
ri-oblemslike this are importantin the chemicaltreatmentof wood.
We want to find the solute'sconcentrationin this cylinder as a function of time and
Jation. As in the previousexamples,the first stepis a massbalance;in contrast,this mass
.,ìanceis madeon a cylindrical shell locatedat r, of area2r Lr, andof volume 2n Lr Lr.
-re
basicbalance
. o l u t ea c c u m u l a t i o n _
solutediffusion - / solutediffusion \
\
/
\
n t h i sc y l i n d r i c asl h e l l/
\ into the shell /
\ out of the shell /
(2.4-26)
2 / Dffision in Dilute Solutions
40
( c)
z
tr
É.
F
z
trj
z
o
P O S I ft o N
Fig.2.4-3.Waterproofìngafencepost.Thisproblemismodeledasdiffusioninaninfinite
situation. In reality, the
.yiin.l.r, and so representsan extenslon to a cylindrically symmetrìc
the grain is faster than
with
diffusion
because
ends ofthe post must be considered,especially
acrossthe grain.
becomesin mathematicalterms
LPrrLLrc,l
dt
: (2trLj),
- (2rrL11),',6,
Q'4-27)
small:
we can now divide by the shell'svolume and take the limit as ^r becomes
l a
a
'òt": -i u'l'
t) 4-)?\
equation
combining this expressionwith Fick's law givesthe requireddifferential
(2.4-2e)
à c t_ D _ r , . d !
'òt
r òr òr
which is subjectto the fbllowing conditions:
1<0,
allr,
/ > 0,
r :
Ro,
r:0,
ct:0
ct : ct (surface)
Ac
=o lr : o
r) 4-30ì
(2.4-3r)
t) l-7)\
- : / Three Other Examples
4l
-. lheseequations,
c1(surface)is the concentrationat the cylinder's surfaceand Rs is the
-. :nder'sradius. The fìrst of the boundaryconditionsresultsfrom the large volume of
-l\)unding solution,and the secondreflectsthe symmetryof the concentrationprofiles.
Problemslike this are often algebraicallysimplified if they are written in terms of di- :rrionlesq variables.
This is standardpracticein many advancedtextbooks.I ofien find
- ' procedure
confusing,becausefor me it producesonly a small gain in algebraat the
- r 3 n s e o f a l a r g e l o s s i n p h y s i c a l i n s i gNh ot .n e t h e l e s s , w e s h a l l f o l l o w t h i s p r o c e d u r e h e r e
lustratethe simplificationpossible.we first definethreenew variables:
('l
dimensionlessconcentratio
: ln ' . -0
(2.4-33)
c I (surface)
r
dimensionless
position: f :
&
Dt
Ré
d i m e n s i o n l e s s t i m e : z :.-..-
(2.4-34)
(2.4-3s)
liflerential equationand boundaryconditionsnow become
a 0_ l a , l ) e
ò'-4afta6
(2.4-36)
r:0,
all{.
r>0, f :1,
:\. tlìe
,1.-+-28)
(1.4-30)
(2.4-40)
Si:t)J(€)
Eqs.2.4-36 and2.4-40are combined,the resultingtangleof termscan be separated
. r o nw i t h S G ) f ' ( € ' ) :
K- {. t. tq d . d f ( E- t
î , - . d R ( r ):
J ' 5 t - ,
.
d r
d €
€ d q
t ds(r)
e(r) dr
t).4-29)
(2.4-39)
oE
r'novice, this manipulationcan be more troublesomethan it looks.
'olve theseequations,we first assumethat the solutionis the productof two functions,
: time and one of radius:
p(r,€):
).1-21)
(2.4-37)
(2.4-38)
0:0
'Àe
:" - g
È:0.
'-:la
:': ihan
0:l
1
(t)
d
"df
d€'
d€
€.fG)
(2.4-4t)
: rrnefìxesf and changes,, ./(6) remainsconstantbut g(z) varies.As a result.
I dg(t)
((r) dt
)
t) 4-4) \
\2.4-31)
i\ a constant.Similarly, if we hold z constantand let f change,we realize
t2.4-32)
-,
-'r (€) riq' r/6
1,
n
u
, l f tr ts \
F u . ,
/
2
(2.4-43)
2 / Diffusion in Dilute Solutions
A )
Thusthepartialdifferentialequation2.4-36hasbeenconvertedinto two ordinarydifferential
equations2.4-42 and2.4-43.
The solutionof the time-dependentpart of this result is easy:
(2.4-44)
wherea'is an integrationconstant.The solutionfor /(6) is more complicated,but straightforward:
-r [email protected]
J G) : [email protected])
(2'4-45)
where -/sand Is areBesselfunctionsand a andb are two more constants.From F,q.2.4-39
we seethafb :0. From Eq. 2.4-38.we seethat
(2.4-46)
0: [email protected])
Becaused cannotbe zero,we recognizethat theremust be an entirefamily of solutionsfor
which
Je(a,) :
(2.4-41)
Q
ú..
.llllll",,
The most generalsolutionmust be the sum of all solutionsof this form found for different
integralvaluesof n:
ll,llf."! f
ù :
É t r . È ) : f1 - " ( . 1 , r ' . )J,s 1 a , , € , \ e ' l ' '
(2.4-48)
n:l
We now usethe initial conditionF,q.2.4-37to find the remainingintegrationconstant(aa'),,'.
!;
t-f
(2.4-4e)
We multiply both sidesof this equationby f Je(cv,,f
) and integratefrom f :0
flnd (aa'\. The total resultis then
o : itI-
t
to t :
ill r
Jll
1 to
- I
["-
lrn.,-.É)e-o,,'
(2.4-50)
L a , , J 1 ( a ], , )
nirrrl
Aú,i
or, in terms of our original variables,
I
ùu-u
-'-
@h'
( t(surface)
e-DoÎ[email protected],llno)
_ | _ )-{
,t-,
(2.4-5r)
u , , J 1 ( u , , t ' fR 1 1 l
i[],--. --
mtul-
This is the desiredresult,though the cy,must still be founclfrom 8q.2.4-41 .
This problem clearly involvesa lot of work. The seriousreadershouldcertainly work
one more problemof this type to get a feel for the idea of separationof variablesand for the
practiceof evaluatingintegrationconstants.Even the seriousreaderprobablywill embrace
the ways of avoidingthis work describedin the next chapter.
1lîr'
rem-i:
i'
, .\olutk)ns
,rtvection
and Dilute Dffision
+)
. Jrîferential
t2.1-44)
Stolic eleclrode
of which soluîe
c o n c e n l r o î i o ni s
r10
D i r e c l i o no f
diffusion
:, rut straighti I J-215'l
Moving electrode
ol which soluîe
c o n c e n l r o l i o ni s e 1 7
- : Eq.2.4-39
\).4-46)
' . rLrtions
for
r).1-47)
',
"o"'nrn,,rl'ou'o
Fig. 2.5- l. Steady diflision in a moving lìlm. This case is mathematically the same as diffusion
acrossa stagnant 1ìlm, shown in Fig. 2.2-1. It is basic to the film theory of mass tlanstèr
ciescribedin Section I l. L
,r Jifterent
2.5 Convection and Dilute Diffusion
(1..1-48)
ritl (aa'),r'.
t).4-49)
ri :l t o
In many practicaìproblems,both diffusion and convectiveflow occur. In some
--...speciallyinf-astmasstransferinconcentratedsolutions,thediffusionitselfcausesthe
:,tion. This type of masstransfer,the subjectof Chapter3, requiresmore complicated
r'- -.rl and mathematical
analyses.
:'reis anothergroup of importantproblemsin which diffusion and convectioncan be
: '. :rsily handled.Theseproblemsarisewhen diffusion and convectionoccur normal to
: ". - ther. In other words,diffusion occursin one direction,and convectiveflow occursin
. -':-:c'ndiculardirection. Two of theseproblemsare examinedin this section. The first,
- ..-:f)nacross
a thin fowing f1m, parallelsSection2.2;thesecond,diffusioninto a liquid
'
.\ a less obvious analogueto Section2.3. Thesetwo examplestend to bracketthe
- . :,.ed experimentalbehavior,and they are basic to theoriesrelatingdiffusion and mass
(seeChapterI 3).
:- .::r coefficients
(1.4-50)
2.5.1 Steady Diffusion Across a Falling Film
(2.4-51)
,:r.unly work
^ :. .rndfor the
.,.ilì embrace
The first of the problemsof concernhere,sketchedin Fig. 2.5-1, involvesdiffusion
- .- a thin, moving liquid film. The concentrations
on both sidesof this film are fixed by
, ::.rchemicalreactions,but the fìlm itself is moving steadily.I havechosenthis example
' îr'ccuSeit occursofien
but becauseit is simple. I hopethat readersorientedtowardthe
-:Lial will wait for later examplesfor resultsof greaterapplicability.
. , solvethis problem,we make threekey assumptions:
. r The liquid solutionis dilute. This assumptionis the axiom for this entire chapter.
I r The liquid is the only resistanceto masstransfer. This implies that the electrode
reactionsare fast.
2 / Dffision in Dilute Solutions
A A
++
(3)
Masstransportis by diffusionin thez directionandby convectionin thex direction'
Transportby the other mechanismsis negligible.
It is the lastof theseassumptionsthat is most critical. It impliesthat convectionis negligible
in the z direction. ln fact, diffusion in the z direction automaticallygeneratesconvection
in this direction.but this convectionis small in a dilute solution. The last assumptionalso
suggeststhat thereis no difTusionin the x direction. Theremay be suchdiffusion,but it is
assumedmuch slower and hencemuch lessimportantin the x directionfhan convectton.
This problem can be solved by writing a mass balance on the differential volume
W Lx Lz, where W is the width of the liquid film, normal to the plane of the paper:
/ s o l u t ea c c u m u l a t i o\n -in WA,r'A.:
\
i
/ s o l u t ed i f f u s i n gi n a t z m i n u s\
\ s o l u t ed i f f u s i n go u t a t . r L z .)
, / soluteflowing in at x minus \
r
\ sofute flowing out at r + Lr )
( 2 . s1- )
or, in mathematicalterms,
(l
^ (crl{ArA:):
(tt
[(lrWLx). - (71WAr).*o.l
* [ ( c r u .W
, A z ) - .- ( c ru . ,W A z ) , . + l ' l
(2.s-2)
The term on the left-handsideis zerobecauseof the steadystate.The secondterm in square
bracketson the right-handside is also zero,becauseneithercl nor ur changeswith x. The
concentrationc1 <loesnot changewith x becausethe fìlm is long, and thereis nothing that
will causethe concentrationto changein the x direction. The velocity u., certainlyvaries
with how far we are acrossthe film (i.e., with z), but it doesnot vary with how far we are
alongthe film (i.e.,with -r).
After dividing by lVA,rA: and taking the limit as this volume goes to zero, the mass
balancein Eq. 2.5-2 becomes
.I:
^
uJ1
r? 5-31
47.
This can be combinedwith Fick's law to give
0 - D
r)
O-Cl
,
1
(2.s-4)
a 7.-
This equationis subjectto the boundaryconditions
z:0,
('r:clo
r?5-5ì
z: I,
cr: clt
(2.s-6)
When theseresultsarecombinedwith Fick's law, we haveexactlythe sameproblemas that
in Section2.2. The answersare
cl :t'lnttCti-{lgl7
(2.s-7)
D
jr:-(.cn-ctt)
(2.s-8)
I
The flow has no effèct. Indeed,the answeris the sameas if the fluid was not flowing.
lb
uaunl,:
;
' - 1 . t ! ( ,S o l u t i o n s
,JlreCIlOn.
Convectionand Dilute Diffusion
45
L i qu i d
s o lv e nt
:r:uli_eible
- :l\ ection
::1\lnalSO
:r.but it is
' r. S a t l o n .
S o l uî e g o s
,,. r rllume
Li.
C o nv e c li o n
r 5_?\
:l \quare
:r .i..The
:nc that
.r Varies
r \\e are
hemass
Liquid with
dissolved
solulegos
F i g . 2 . 5 - 2u. n s t e a d y - s t a t e d i f f u s i o n i n t o a f a l r i n T
gh
f ìirsma.n a l y s i s t u r n s o u t t o b e
mathematically
equivalentto fiee diflusionlseeFig. 2.3-2). rtis basicto the penetratron
theory
of masstransfèrdescribedin SectionI 1.2.
.fhisansweristypical
ofmanyproblemsinvolvingdiffusionandflow.
Whenthesolutions
lilute' the diffusion and convection often
are perpendicular to each other and the solution
(1.5_4)
I
'
\ - \ ì
(2.s-6)
:- -.ent asthat
(2.5-7)
(2.5-8)
i ng.
":aightfbrward. You may armostfeergyppediyou girdedyourserftbr a diffìcurtprobrem
- iound an easyone. Restassuredthat more
diflìcutt problemsfollow.
2.5.2 Diffusion Into a Falling Fitm
The secondproblem of interestis iilustrated
schematicailyin Fig. 2.5-2 (Bird,
' 'irr' and Lightfoot, r 960).
A thin liquid film flows slowly and wirhour ripptes
down a
'urlàce. one side of this firm wets
the surfàce;the other side is in contactwith a gas,
'h is sparinglysolubrein the liquid. we want
to find out how muchgascrissorves
in the
J
' ' solvethis problem,we again go throughthe
increasingryfamiliar ritany; we write a
" balanceas a differentialequation,combine this with Fick's
law, and then integrate
'r fìnd the desiredresurt.we do this subject
to four key assumptions:
. r The solutionare alwaysdilute.
I
Mass transportis by 3 diffusion anc.lr convection.
2 / Dilfusion in Dilute Solutions
( 3 ) T h e g a si s P u r e .
(4) The contactbetweengas and liquid is short'
example' The third
The fìrst two assumptìonsare identicalwith thosegiven in the earlier
liquid. The final
in
the
only
phase,
gas
in
the
meansthat therels no resistanceto diffusion
assumptionsimplifiesthe analysis.
shownin the inset
we now make a massbalanceon the differentialvolume IV in width,
in Fis.2.5-2:
/
\
massaccumulation :
\
w i t h i nW A x A : I
/ mass<liffusingin at z minus \
diffusing out at ^z+ Lz.)
\ r-t-tuts
massflowingin at x minus
_
- /
-l Lx )/
\ massflowingout at x
6/AFinalPerspecti
rs again reflectsthe
.ult, the solutecan di
the exact location o
Ì be infinitely far au
lhis problemis des
..rsionin a semiinfì
, t. Becausethe ml
p leis
t t
c flux at the inte:
tr., ,l
: Y D
This result is parallelto thosefound in earliersections:
l à
|
- (IVA/r).+r.]
| - t c 1 a , l l : l v ) I : [(WAjr/r):
tdt
ì
* [ ( l 4 z A z c r u ' ) '- ( W L z c t u . ) , . + 1 . , ]
( 2 . sl -0 )
the left-handside
when the systemrs at steadystate,the accumulationis zero. Therefore,
vary with both z andx'
of the equaiionis zero. No otherterms are zero,because71and c1
goesto zero,we find
tf we divide by the volume lv ax az and takethe limit as this volume
ol1
0:-;.
dz.
,l
-
(2.5-r r)
o
^ (lui
dx
law and set u' equal
We now maketwo further manipulations;we combinethis with Fick's
of short
assumption
the
reflects
change
second
This
to its maximum vaìue, a consùnt.
and ìt
interface'
the
cross
to
a
chance
has
barely
contacttimes. At such times, the solute
the
reaches
velocity
fluid
the
region'
interfacial
diffusesonly slightly into the fluid. In this
a
serious
not
probably
is
value
constant
of
a
maximum suggestedin Fig. 2.5-2, so the use
assumption.Thus the massbalanceis
,òct
.tetnative vier'
balance.ln t
. , h i c hl i q u i d
:. -statediffer
,ngwith the
. equationli
. .rer is that t
- nethod use
, : . c r i b e da :
': of watch
- : i d g e .u e
3. \\'erea
\ Fina
the soluteflow out minusthatin; theright-hand
The left-handsideof this equationrepresents
side is the diffusion in minus the diffusion out'
This massbalanceis subjectto the following conditions:
.r :0
. r: 0 .
all:,
r>0.
z:0,
cr:ct(sat)
--t
(
' 'I - - ( t \ ' ,
(2.5-l3r
(2.5-l4r
(2'5-15
with the gasitself, and /
where c1(sat)is the concentrationof dissolvedgasin equilibrium
threeboundaryconditions
is the thicknessof the falling film in Fig. 2.5-2. The last of these
is replacedwith
z:oo,
-"
'
,rrethe answef\
-r answersapP
l . T h o s es t u d r
J think about :
i masstransfe
now conside
(2.5-12l
: DA:r-:
dz'
0 (.r/u-"*)
x>0.
: l -
c1(sat)
cl:0
(2.5-t6
. .rnd:l
' ,,ndtll
:3
lllS:
-' Th
-.::er
:
ì--Ji
. - '::ì
, Solutions
The third
The final
t h ei n s e t
).6 / A Final Perspective
This again reflectsthe assumptionthat the film is exposed
only a very short time. As a
:esult,the solutecan diffuse only a short way into the film. Its
diffusion is then unaffected
îy the exact location of the other wall, which, from the standpoint
of diffusion, might as
n ell be infinitely far away.
This problem is describedby the samedifferentialequationand boundary
conditionsas
Jiffusionin a semiinfiniteslab.The soledifferenceis thatthe quantity
x /u^u"replacesthe
lime /' Becausethe mathematicsis the same,the solutionis the
same. The concentration
profileis
ct
c 1( s a )t
rI 5-9t
' : - r r i n ds i d e
. : r. a n dx .
::'. $e find
il.-5-ll)
-,'lr'Ì equal
n of short
" . . ! i e .a n d i t
:iitchesthe
I I SenOUS
(2.5-12)
: rishrhand
( 2 . 51- 3 )
(.2.5-14)
( 2 . 5l-s )
." itself,andI
.1r\conditions
(2.s-t6)
:
l-erf-è
J4Dx f u^r^
(2.s-11)
. r n dt h e f l u x a t t h e i n t e r l a c e
is
. t rl . - o :
r. 5r-0 )
41
V'Dr^^J,"-cr(sat)
( 2 . 5l 8
-)
Theseare the answersto this problem.
Theseanswersappearabruptly becausewe can adopt the mathematical
resultsof Secrion 2'2. Those studyingthis materialfor the first time often find
this abruptnessjarring.
Stop and think about this problem. It is an important problem,
basic to the penetration
theoryof masstransferdiscussedin Section 13.2.To supplya forum
for furtherdiscussion,
rveshall now considerthis problem from anotherviewpoint.
The alternativeviewpoint involveschangingthe differentialvolume
on which we make
the massbalance.In the foregoingproblem, we chosea volume
fixed in space,a volume
rhrough which liquid was flowing. This volume accumulated
no solute, so its use led
to a steady-state
differentialequation. Alternatively,we can choosea differentialvolume
f l o a t i n g a l o n g w i t h t h e f l u i d a t a s p e e d uT- "h*e. u s e o f t h i s v o l u m e l e a d s t o a n u n s t e a d y - s t a t e
differentialequarionlike Eq. 2.3-5. Which viewpoint is conect?
The answeris that both are correct;both eventuallylead to the same
answer.The fixedcoordinatemethodusedearlieris often dignified as "Eulerian,"
and the movrng-coordinate
plcture is describedas "Langrangian."The difference
betweenthem can be illustrateclby
the situationof watchingfish swimming upstreamin a fast-flowing
river. If we watch the
tìshfrom a bridge,we may seeonly slow movement,but if we
watch the fish from a freelv
floatingcanoe,we realizethat the fìsh are moving rapidly.
2.6 A Final Perspective
This chapteris very important,a keystoneof this book. It introduces
Fick's law for
dilute solutionsand showshow this law can be combinedwith
massbalancesto calculate
concentratlonsand fluxes. The massbalancesare madeon thin shells.
When theseshells
are very thin, the massbalancesbecomethe ditferentialequations
necessaryto solve the
Variousproblems.Thus the bricksfrom which this chapteris built
arelargelymathematical:
shellbalances,diffèrentialequations,and integrationsin different
coord-inate
systems.
However, we must also see a different and broader blueprint based
on physics, not
mathematics.This blueprint includesthe two limiting casesof'diffusion
acrossa thin fìlm
anddiffision in a semiinfiniteslab. Most diffusion problemsfall between
thesetwo limits.
The first, the thin fìlm, is a steady-stateproblem, mathematically
easy and sometimes
physicallysubtle.The second,the unsteady-state
problemof the thick slab,is a little harder
to calculatemathematically,and it is the limit at short times.
48
2 / Dffision in Dilute Solutions
In many cases,we can use a simple criterionto decidewhich of the two centrallimits is
more closely approached.This criterion hingeson the magnitudeof the Fourier number
(length)2
/ diffusion
\
I
;. .- . lltime)
\ coelnclent/
This variableis the argumentof the error function of the semiinfiniteslab, it determines
the standarddeviation of the decayingpulse, and it is central to the time dependenceof
diffusion into the cylinder. In other words, it is a key to all the foregoing unsteady-state
problems.Indeed,it can be easily isolatedby dimensionalanalysis.
This variablecan be usedto estimatewhere limiting caseis more relevant.If it is much
larger than unity, we can assumea semiinfinite slab. If it is much less than unity, we
should expect a steadystateor an equilibrium. If it is approximatelyunity, we may be
forced to make a fancier analysis. For example,imagine that we are testinga membrane
for an industrial separation. The membraneis 0.01 centimetersthick, and the diffusion
coefficient in it is l0 7cm2/sec. If our experimentstake only 10 seconds,we have an
unsteady-state
problem like the semiinfiniteslab; it they take three hours we approacha
steady-state
situation.
In unsteady-state
problems,this samevariablemay also be usedto estimatehow far or
how long masstransferhasoccurred.Basically,the processis significantlyadvancedwhen
this variableequalsunity. For example,imagine that we want to guesshow far gasoline
has evaporatedinto the stagnantair in a glass-fiberfilter. The evaporationhas beengoing
on about 10 minutes,and the diffusion coefficientis about0. lcm2/sec. Thus
( l e n g t h) 2
(0.I cm2/sec)(600sec)
- l;
length: 8cm
Alternatively,supposewe find thathydrogenhaspenetratedabout0. 1 centimeterinto nickel
Becausethe diffusion coefficientin this caseis about l0-8 cm2lsec.we can estimatehou
long this processhasbeengoing on:
(10-1cm2)
:
(10-8 cmzlsec)(time)
l:
t i m e:
lOdays
This sort ofheuristic argumentis often successful.
A secondimportantperspectivebetweenthesetwo limiting casesresultsfrom compering
their interfacialfluxesgivenin Eqs.2.2-10and 2.3-18:
jr :
D
7
tcr
(rhin fitm)
i, : f D 1nt Lc1 (rhickslab)
Although the quantitiesDll and (Dlrt;l/2 vary differently with diffusion coefficients.
they both have dimensionsof velocity; in f'act,in the lifè sciences,they sometimesare
called "the velocity of diffusion." In later chapters,we shall discoverthat thesequantities
are equivalentto the masstransfercoefficientsusedat the beginningofthis book.
Final PersJ
Further
s , C . ( 1 9 3 4 ) .f
R. B., Stewan
nann,L. ( 189
. l. (1975) T;
r . P J . ,S t e e l
e d s .G . \ \ ;
r . E . ( 1 8 5 1 r'
:. E. ( 185-5r
: E.( 185-ir
, E . ( 1 8 5 ó ,'
- E.( l90i
, rB
. .( l s : :
- L. J.(l!:-
T .r l E l "
T rlS-r-:
r ,l\:,
trr,l:; -
IJ
\ ..tnons
ll]ltS 1S
:ber
l:1: lllllfl9S
- -1.'nceOf
: .,-:r - statg
i
. llluCh
- 'i\' we
r.j '-'lùv be
a:-:lbrane
r i.:iusion
: i.r\ e iìfl
::::'rach a
- .\ tar or
. - iJ when
:: i,t\()line
-:r-n
-9ollì$
nickel.
rtehow
nparlng
r:llClents,
: Illes afe
-,r.tntities
1A Final Perspective
49
Further Reading
- .:nes.C. (1934).Pà,i'.sics,5.4.
- -r. R. 8., Stewart,W. E., and Lightfbot, E. N. (1960).
TransportPhenomena.New york: Wiley.
- irmann,L. ( 1894).Annalender Physikund Chemie,53,
959.
-.,nk.J. (1975). TheMathematics
of Dffision, 2nd ed. Oxford: ClarendonPress.
. :rìop,P J., Steele,B. J.. and Lane, J. E. (1972). ln.. PhysicalMethodso.fChemistry,-,
eds.G.Weissberger
and B. W. Rossiter.New York: Wiley.
,.. A. E. (1852). Zeítschríftfùr RationelleMedicin,2,83.
-.. A. E. (1855a). Poggendorff'sAnnelender Physik,94, 59.
-.. A. E. (1855b).PhilisophicalMagazine,l0,30.
,. A. E. (1856). MedizinischePlrysik Burnswick.
-.. A. E. (1903). GesammelteAbhandlungen.W'jrzburg.
.-rier,J. B. (1822). Théorieanal,-îiquede la chaleur. Pa,ris.
.ting, L. J. (1956). Advancesin Proîein Chemístry,ll, 429.
-.ham, T. (1829).
Quarterly Journal of Science,Literature and Art,27,74.
',ram, T. (1833).Phiktsophical
Magazine,2,175,222,351.
-'ram, T. (1850). P/zrlosophicalTransaction.s
of the Rov-alSociett:of London, l40,1.
, - ula. J., Graves.D. J., Quinn, J. A., and Lambersten,C. J. ('|976). In. IJndem-aterPhysiology,
Vol.5, ed.C. J. Lambersten,p. 355. New York: Academic.
'-.or. E. A. (1970).
Philosophit:al
Journal,7,99.
1 i.. R., Woolf, L. A., and Watts,R. O. ( I 968). American Instituteof ChemicatEngineersJoumal,
t 4 . 6 7t .
.rnson, R. .A.,and Stokes,R. H. (1960). ElectrolyteSolutions.London: Butterworth.
`