C H A P T E R8
Fundamentalsof Mass Transfer
Diffusion is the processby which molecules,ions, or other small particlesspontaneouslymix, moving from regionsof relativelyhigh concentrationinto regionsof lower
concentration.This processcan be analyzedin two ways. First, it can be describedwith
Fick's law and a diffusion coeffìcient,a fundamentaland scientificdescriptionusedin the
fìrst two parts of this book. Second,it can be explainedin terms of a masstransfercoefficient, an approximateengineeringidea that often gives a simpler description. It is this
simpler idea that is emphasizedin this part of this book.
Analyzing diffusion with masstransfercoefficientsrequiresassumingthat changesin
concentrationare limited to that small part of the system'svolume nearits boundaries.For
example,in the absorptionof one gasinto a liquid, we assumethat all gasesand liquids are
well mixed, exceptnearthe gas-liquid interface.In the leachingof metal by pouring acid
exceptin a thin layer next to the solid ore
overore, we assumethat the acid is homogeneous,
particles.In studiesof digestion,we assumethat the contentsof the small intestinearewell
mixed, exceptnear the villi at the intestine'swall. Such an analysisis sometimescalled a
model" using
"lumped-parametermodel" to distinguishit from the "distributed-parameter
diffusion coefficients.Both models are much simpler for dilute solutions.
If you are beginning a study of diffusion, you may have troubìe deciding whether to
organizeyour resultsas masstransfercoefficientsor as diffusion coefficients.I have this
trouble too. The cliché is that you should use the mass transfercoeffìcientapproachif
the diffusion occurs acrossan interface,but this cliché has many exceptions. Insteadof
dependingon the cliché, I believeyou should always try both approachesto seewhich is
betterfor your own needs. In my own work, I have found that I often switch from one to
the other as the work proceedsand my objectivesevolve.
This chapterdiscussesmasstransfercoefficientsfor dilute solutions;extensionsto concentratedsolutionsare deferredto Section 13.5. In Section8.1, we give a basic definition
for a masstransfercoefficientand show how this coefficientcan be usedexperimentally.
In Section8.2, we presentother common definitionsthat representa thicket of prickly alternativesrivaledonly by standardstatesfor chemicalpotentials.Thesevariousdefinitions
are why masstransferoften has a reputationwith studentsof being a difficult subject. In
Section8.3, we list existing correlationsof masstransfercoefficients;and in Section8.4,
we explain how thesecorrelationscan be developedwith dimensionalanalysis.Finally, in
Section8.5, we discussprocessesinvolving diffusion acrossinterfaces,a topic that leadsto
overall masstransfercoeffìcientsfound as averagesof more local processes.This last idea
is commonly called masstransferresistancesin series.
8.1 A Definition of Mass Tlansfer Coefficients
The defìnition of masstransferis basedon empirical argumentslike thoseused
in developingFick's law in Chapter2. Imagine we are interestedin the transferof mass
211
lrl
8 / Fundamentalsof Mass Transfer
e.1/ A DeÍi
irom someinterfaceinto a well-mixed solution. We expectthat the amounttransferred
is
proportionalto the concentrationdifferenceand the interfacialarea:
(a)
/ rateol mass\ _ ,. ( inrcrfacial\ / concenrrarion
\
a r e a i \ d i f l e r e n c e)
\transferred/-"\
( 8 . 1 I-)
where the proportionality is summarizedby À, called a mass transfercoeffìcient. If we
divide both sidesof this equationby the area,we can write the equationin more familiar
symbols:
Nr:k(cri-ct)
(c
(8.r-2)
where N1 is the ffux at the interfaceand c1; and cl are the concentrationsat the interface
and in the bulk solution,respectively.The flux N1 includesboth diffusion and convection:
it is like the total flux nt exceptthat it is locatedat the interface.The concentrationc1; is
at the interfacebut in the samefluid as the bulk concentrationc1. It is often in equilibrium
with the concentrationacrossthe interfacein a second,adjacentfluid phase;we will defer
discussionof transportacrossthis interfaceuntil sectiong.5.
The flux equationin Eq.8.1-2 makespracticalsense.It saysthat if the concentration
differenceis doubled,the flux will double. It also suggeststhat if the areais doubled,the
total amountof masstransferredwill doublebut the flux per areawill not change.In
other
words, this definition suggestsan easy way of organizing our thinking around a simple
constant,the masstransfercoefîcient ft.
Unfbrtunately,this simple schemeconcealsa variety of approximationsand ambiguities. Before introducingthesecomplexities,we shall go over someeasyexamples.These
examplesare important. Study them carefully beforeyou go on to the hardermaterial
that
follows.
Example 8.1-l: Humidification Imaginethat wateris evaporatinginto initially dry air in
theclosedvesselshownschematically
in Fig. 8.1-I (a). The vesselis isothermalat 25"C,so
the water'svapor pressureis 23.8 mm Hg. This vesselhas 0.8 liter of water with 150 cm2
of surfacearea in a total volume of 19.2 liters. After 3 minutes, the air is five percent
saturated. What is the mass transfercoeffìcient? How long will it take to reach ninety
percentsaturation?
Solution The flux at 3 minutescan be found directly from the valuessiven:
vapor
(
I / air \
concentration
\
/
\ volume
/
Nr:
Fig
ma
airexc
is tt
zero. Thus,
4.
k
This value ii
The time
(
d
;
The air is in
r -
We usethì.
I
Reanangin
-
| mol \ /273 \
^ ^- /23.8\ /
U.U)t" tt
lt
,
6
0
/ \ 2 2 . 4t i r e r /s \ 2 9_8)/ t t g . 4 t i r e r s l
r
:4.4.10-E mol/cm2-sec
The concentrationdifferenceis that at the water's surfaceminus that in the bulk solution.
That at the water'ssurfàceis the valueat saturation;that in bulk at shorttimes is essentially
It takes trr ::
'ler
:r'J is
8.1 / A Defnition of Mass Transfer Coe.fficients
(a) Humidilication
t-E''ut
|
. l t )
i: rie
'-.rliar
-,iolll
IS
--Ltnl
i.'Ier
l-:iIilll
i. !he
. il c'f
' ' - -n l e
:::ui- ; - -\ c.
-
(b) Packod Bed
I
r fiàldìf'l
(c) LiquidDrops
(d) A Gas Bubble
rg1
| ? o"o.l
, i t)
: : : J ÙC
l'
"oà,ar
I
Fig. 8.1-1.Foureasyexamples.We analyzeeachof the physicalsituationsshownin termsof
masstransfercoefficients.In (a), we assumethatthe air is at constanthumidity,exceptnearthe
air-waterinterface.In (b), we assumethat waterflowingthroughthe packedbedis well mixed,
exceptvery closeto the solid spheres.In (c) and (d), we assumethatthe liquid solution,which
is thecontinuousphase,is at constantcomposition,
exceptnearthe dropletor bubblesurfaces.
zero. Thus, from Eq. 8.1-2, we have
t t:'
4 . 4 . l 0 - " m o l / c m 2 - s e:c k ( 2 : ' 8
. ?l -o\
\ 1 6 0 2 2 . 4 .l 0 ' c m ' 2 9 8
/
:
3
.
4
.
k
l0-2cm/sec
This value is lower than that commonly found for transferin gases.
The time requiredfor ninety percentsaturationcan be found from a massbalance:
,:i.tt
/ accumulation
\
I
l
gas
pnase
In
\
/
- : . 1l t l
a
;Vc1 :
: l
/ evaporation
\
I
rate
\
/
l1Y,
dt
-'..0
- tÌll
--:'llI
213
Ak[ct(sat)- c1]
:
The air is initially dry, so
/ :0,
c r: 0
We use this condition to integratethe massbalance:
Ll _1_"_(kAlv)t
c1(sat)
-
I
-
(
Rearrangingthe equationand insertingthe valuesgiven,we find
t:-LmúkA
\
''
\
cr (sat)/
1 8 . 4 .1 0 1c m 3
I n l| - 0 . 9 )
( 3 . 4 . l 0 - 2 c m / s e c ). ( 1 5 0c m 2 )
: 8 . 3 . 1 0 3s e c: 2 . 3 h r
It takesover 2 hoursto saturatethe air this much
I+
8 / Fundamentalsof Mass Transfer
Example 8.1-2: Mass transfer in a packed bed Imagine that 0.2 centimeter-diameter
spheresof benzoicacid arepackedinto a bed like that shownschematicallyin Fig. 8.1-1(b).
The sphereshave 23 cm2 surfaceper 1 cm3 of bed. Pure water flowing at a superficial
velocity of 5 cm/secinto the bed is sixty-two percentsaturatedwith benzoicacid after it
haspassedthrough 100centimetersof bed. What is the masstransfercoefficient?
Solution The answerto this problem dependson the concentrationdifference
usedin the definition of the masstransfercoefficient. In every definition, we choosethis
differenceas the value at the sphere'ssurfaceminus that in the solution. However, we
can definedifferent masstransfercoefficientsby choosingthe concentrationdifferenceat
variouspositionsin the bed. For example,we can choosethe concentrationdifferenceat
the bed's entranceand so obtain
Nr:klcr(sat)-01
0.62c1t sat){5 cm/sec)A
: (ct(sat)
( 2 3. r r / . * . x l o o . , , - ' ) A
where A is the bed's crosssection.Thus
k:1.3. 103cm/sec
This definition for the masstransfercoeffìcientis infrequentlyused.
Alternatively.we can chooseas our concentrationdifferencethat at a position e in the
bed and write a massbalanceon a differentialvolume AA,zat this position:
(accumurarior,
: (
"r,"lr"'fJ;"r, )
o : A ( c t r n l- r , r o l
\
l.-
amountof \
-, /
\dissolution/
8 . 1/ A D t
This valu
definition
A tang
dissolvin
in problen
that it is rh
so muchl
Benzo
is relatire
base.b1 L
carbon or
by one m
dissoluti
Third.and
intertace
aremuch
intellectu
"
can be a..
c h e m i c a.
Erample
\ \ a t e r .3 ì .
-ìminut3
5
) 11a1.;olrt
l:,1--l
where a is the spheresurfaceareaper bed volume. Substitutingfor N1 from Eq.8.l-2,
dividing by AAz, and taking the limit as Az goesto zero,we find
dc,
' :
47.
ka
-c1l
^lct(sat)
This is subjectto the initial conditionthat
z:0,
ct:0
Integrating,we obtain an exponentialof the sameform as in the fìrst example:
c1
c't(sat)
-_ r | - ( - - \ k a l u o ) z
Rearrangingthe equationand insertingthe valuesgiven, we find
* : ( ' n ) ' , l 'r -' 1' ('s a It ) /
\a:/
\
5 cm/sec
ln(l - 0.62)
(23 cmzlcm3)(100cm)
: 2 . 1 ' 1 0 - 3c m / s e c
5.::l::i
l
8.I / A Definition rf Mass Transfer Coefficients
.: ' l C f
. bt.
' :, l l l
::': lt
:-:ì.'e
:irir
. ,'\e
,:
.lt
-:' ;.it
215
This value is typical of those found in liquids. This type of mass transfer coefficient
definition is preferableto that usedfirst, a point exploredfurther in Section8.2.
A tangentialpoint worth discussingis the specific chemical system of benzoic acid
dissolvingin water. This systemis academicallyubiquitous,showing up again and again
in problemsof masstransfer.Indeed,if you readthe literature,you can get the impression
that it is the only systemwheremasstransferis important,which is not true. Why is it used
so much?
Benzoic acid is studiedthoroughly for three distinct reasons. First, its concentration
is relatively easily measured,for the amount presentcan be determinedby titration with
base,by UV spectrophotometry
of the benzenering, or by radioactivelytaggingeither the
carbon or the hydrogen. Second,the dissolutionof benzoic acid is accuratelydescribed
by one mass transfercoefficient. This is not true of all dissolutions. For example,the
dissolutionof aspirin is essentiallyindependentof eventsin solution (see Section 15.3).
Third, andmostsubtle,benzoicacid is solid,so masstransfertakesplaceacrossa solid-fluid
interface.Suchinterfacesarean exceptionin masstransferproblems;fluid-fluid interfaces
are much more common. However,solid-fluid interfacesare the rule for heattransfer,the
intellectualprecursorof masstransfer.Experimentswith benzoicacid dissolvingin water
can be compareddirectly with heat transferexperiments.Thesethree reasonsmake this
chemicalsystempopular.
Example 8.1-3: Mass transfer in an emulsion Bromine is being rapidly dissolvedin
water,as shown schematicallyin Fig. 8.1-1(c). Its concentrationis about half saturatedin
3 minutes. What is the masstransfercoefficient?
Solution Again, we begin with a massbalance:
d
--Vct
A k l c t( s a t )- c r l
ANr:
dt
dct
,:kafct(sat)-crl
at
where a(: AIV) is the surfaceíìreaof the bromine droplets divided by the volume of
aqueoussolution. If the water initially containsno bromine,
f :0,
c r: 0
Using this in our integration,we find
ct
-1*r*kat
c1(sat)
Rearranging,
k o :- ! n ( r _ - 1 )
/
--*
\
I
-t mtn
c1$at) /
ln(l-0.5)
: 3 . 9 . 1 0 - 3s e c I
216
8 / Fundamentalst{ Mass TransJer
8.2/ OtherD
This is as far as we can go; we cannotfind the masstransfèrcoefficient,only its product
with a.
Such a product occurs often and is a fixture of many mass transferconelations. The
quantity fta is very similar to the rate constantof a flrst-orderreversiblereactionwith an
equilibrium constantequal to unity. This particularproblem is similar to the calculationof
a half-life for radioactivedecay.Sucha parallelis worth thinking throughand will become
a very usefulconceptin Chapter 15.
T.
Example 8.1-4: Mass transfer from an oxygen bubble A bubbleof oxygen originally
0.1 centimeterin diameteris injectedinto excessstirredwater,as shown schematicallyin
Fig. 8. 1-1(d). After 7 minutes,the bubbleis 0.054centimeterin diameter.What is the mass
transfercoefficient?
Solution This time. we write a massbalancenot on the surroundinssolutionbut
on the bubbleitself:
d / 4
d r\ ' 3
- l c r - Í r - l
. \ ANr
/
: -4trr2klcr(sat)- 0l
This equationis tricky; cl refersto the oxygenconcentrationin the bubble,1 mol/22.4liters
at standardconditions.but t 1(sat)rel'ersto the oxygenconcentration
al saturationin water.
about1.5 . l0-3 molesoer liter undersimilarconditions.Thus
dr
Effect
Mass transtèr
Diffusion
Dispersion
Homogenetru.
chemical
reaction
H e t e r o _ 9 e n. ,e"
chemicll
re acti(rn
, c1(sat)
dj
E.:
cl
: -0.034k
\\:
i:Iìnitrrrn,:
This is subiectto the condition
t :0,
r : 0 . 0 5c m
i:nnitrtrn.i
'r\' r" n" rt n" i' '-.
r
t
;
*
,
il)
....,
In ir;ì. :.-
sointegration
gives
. .
..\
L
t'.:\-,
.1.:L.l3l ,
-i-:
r : 0.0-5
cm - 0.034kr
Insertingthenumericalvaluesgiven,we fìnd
0.021 cm:
k:
0.05 cm - 0.034ft(420sec)
1 . 6 .l 0 I c m / s e c
Rememberthat this coefficientis definedin termsof the concentrationin the liquid; it would
be numericallydifferent if it were definedin termsof the gas-phaseconcentration.
.
:
-l
-'31
-_ -
. : . ' . a ._
. . . - - - .
t:iler
Table 8.2- | . Mass transJèrcofficient compared with other raîe coe.lf(ienîs
rJuct
The
:h an
. ì n0 f
Effect
Basic
equation
Mass transfer
Nr : ftAcr
Diffusion
-ii : DVcr
Dispersion
-c'rv't : PY7,
- rrllle
iaÌ11
i\ lll
l l . ì ss
r but
: ..i!-rs
'',rtùf.
211
8.2 / Other Defnitions of Mass Transfer Cofficients
Homogeneous rt:
chemical
reaction
Ktcl
H e t e r o g e n e o u sr t :
chemical
feaction
Ktcl
Rate
Force
Coefficient
Flux per area
relative to
an interface
Flux per area
relative to
the volume
average
velocity
Flux per area
relative to
the mass
average
velocity
Rate per
volume
Diffèrence of
The masstransfèrcoetficoncentration c i e n tf t ( [ : ] l / r ) i s a
function of flow
The diffusion coefficient
Gradientof
D (I:lL2 lt) is a
concentratlon
physical property
independentof flow
Flux per
interfacial
area
Concentration
The dispersioncoefficient
Gradientof
Lll:lL' /f ) depends
time average
concentratron on the flow
Concentration
The rate constant
rr ([:] 1/r) is
a physical property
independentof flow
The rate constant
rc1(:lL lt) is a surface
property often defined
in terms of a bulk
concentration
8.2 Other Definitions of Mass Thansfer Coefficients
llú
We now want to return to some of the problemswe glossedover in the simple
definition of a masstransfercoefficientgiven in the previoussection. We introducedthis
defìnitionwith the implicationthat it providesa simpleway of analyzingcomplexproblems.
We implied that the mass transfercoefficient will be like the density or the viscosity,a
physicalquantitythat is well defìnedfor a specifìcsituation.
In fact, the masstransfercoefficientis often an ambiguousconcept,reflectingnuances
of its basicdefinition. To begin our discussionof thesenuances,we first comparethe mass
transfercoefficientwith the other rate constantsgiven in Tàble 8.2-1. The masstransfer
coefficientseemsa curious contrast,a combinationof diffusion and dispersion.Because
it involves a concentrationdifference,it has different dimensionsthan the diffusion and
dispersioncoefficients.It is a rateconstantfor an interfacialphysicalreaction,most similar
to the rate constantof an interfàcialchemicalreaction.
Unfortunately,the definitionof the masstransfercoeffìcientin Table8.2-I is not so well
acceptedthat the coefficient'sdimensionsare always the same. This is not true for the
other processesin this table. For example,the dimensionsof the diffusion coefficientare
alwaystakenas L2 lt. Iî the concentrationis expressedin terms of mole fraction or partial
pressure,then appropriateunit conversionsaremadeto ensurethat the diffusion coefficient
keepsthe samedimensions.
This is not the case fbr mass transfercoefficients,where a variety of definitions are
accepted. Four of the more common of theseare shown in Table 8.2-2. This variety is
218
8 / Fundamental.sof Mass Transfer
8.2 / Other Dr
Air w
'iroc
Table 8.2-2. Common definitions of mass transfer cofficientso
omm
Basic equation
Typical units of /<b
Remarks
Nr : tAcr
cm/sec
Nt :
mol/cm2-sec-atm
Common in the older literature:used
herebecauseof its simple physical
signifìcance(Treybal, 1980)
Common for a gas absorption;equivalent
fbrms occur in biological problems
(McCabe Smith, and Harriot, 1985;
Sherwood,Pigford, and Wilke, 1975)
Preferred for practical calculations,
especiallyin gases(Bennett
and Myers, 1974)
Used in an effort to include diffusioninducedconvection(cf. fr in Eq. 13.5-2
et seq.) (Bird, Stewart,and Lightfoot, 1960)
kpLpt
Nr : fr.,4-rr
mol/cm2-sec
Nr :ÀAcr f cruo
cm/sec
Notes:oIn this table,Nt is defìnedas molesperL2t, andct as molesper L3. Paralleldefinitions
whereNr is in termsof M lL2t andcr is M f L3t areeasilydeveloped.
Definitions
mixingmolesand
massareinfrequentlyused.
bFora gasof constantmolarconcentration :
c, k
RTkp : k,lc. For a diluteliquid solution
k : 6421ùk,, whereúz is ttremolecular
weightof thesolvent,antlp is thesolutiondensity.
largely an experimentalartifact,arising becausethe concentrationcan be measuredin so
many differentunits, including partialpressure,mole and massfractions,and molarity.
In this book, we will most frequentlyusethe first definitionin Table8.2-2,implying that
masstransfercoefficientshave dimensionsof length per time. If the flux is expressedin
molesper areaper time we will usually expressthe concentrationin molesper volume. If
the flux is expressedin massper areaper time, we will give the concentrationin massper
volume. This choiceis the simplestfor correlationsof masstransfercoefficientsreviewedin
this chapterandfor predictionsof thesecoeffìcientsgivenin ChaptersI 3- I 4. Expressingthe
masstransfercoeffìcientin dimensionsof velocity is also simplestin the casesof chemical
reactionand simultaneousheatand masstransferdescribedin Chapters15, 16, and20.
However,in someother cases,alternativeforms of the masstransfercoefficientslead to
simplerfinal equations.This is especiallytrue in the designof equipmentfor gasadsorption,
distillation, and extractiondescribedin Chaptersg-l l. There, we will frequentlyuse k'
the third form in Table8.2.2, which expressesconcentrationsin mole fractions. In some
casesof gasadsorption,we will find it convenientto respectseventyyearsof tradition and
use kp, with concentrationsexpressedas partial pressures. In the membrane separations
in Chapter 17, we will mention forms like k, but will calry out our discussionin terms of
forms equivalentto k.
The masstransfercoefîcients definedin Table8.2-2 arealso complicatedby the choice
of a concentrationdifference, by the interfacial areafor masstransfer, and by the treatment
of convection.The basic definitionsgiven in Eq. 8.1-2 or Table 8.2-l areambiguous,for
the concentrationdiffèrenceinvolved is incompletelydefined. To explore the ambiguity
more carefully,considerthe packedtower shown schematicallyin Fig. 8.2-1. This tower
is basically a piece of pipe standingon its end and filled with crushedinert material like
broken gìass. Air containingammoniaflows upward through the column. Water trickles
down through the column and absorbsthe ammonia: Ammonia is scrubbedout of the gas
Wcî€
s ot L. :
w l lt
om-
Fr,s :
mirtdenr.
diffe:
Irqu:
mixture \\ ith
concentratlon
The proponi.'
betr"'eenintei\\'hich \ alue ,
In this bt'o
n r r r ìr i o n i n t h .
r u i r h i t f r o n :.
'.\
L .( l l t e n n l u i :
.'t]lOr€ flC;r.
::nnrtion $;.
., '* .t: Ured :::
.ìn(rther :
-.:'ìnterll-t;t.:
- --:i.1!e iifa; ::
. - i r ' . :Ì ' U f ì k L , ,
a - r l r ' t . t lt . j , -r:J::Ìllnini::
:_ .-3nl l:\.
--,....
-
_ . : - . \ . . . _ j
8.2 / Other Definitions of Mass Transfer Cofficients
Air wilh only
<_
2t9
Pure
w0rer
The concentrolion
difference of
ommonio here...
...moybe very
ferent thon
lhol here
îîî
Fig.8.2-l . Ammonia
scrubbing.
In thisexample,
ammonia
is separated
by washing
a gas
mixturewithwater.As explained
in thetext,theexample
illustrates
ambiguities
in the
definition
of masstransfer
coefficients.
Theambiguities
occurbecause
theconcentration
difference
causingthemasstransfèrchanges
andbecause
gasand
theinterfacial
areabetween
liquidis unknown.
mixture with water. The flux of ammonia into the water is proportionalto the ammonia
concentrationat the air-water interfaceminus the ammoniaconcentrationin the bulk water.
The proportionalityconstantis the masstransfercoefficient.The concentrationdifference
betweeninterfaceand bulk is not constantbut can vary along the height of the column.
Which value of concentrationdifferenceshouldwe use?
In this book, we alwayschooseto use the local concentrationdifferenceat a particular
position in the column. Such a choiceimplies a "local masstransfercoefficient"to distinguish it from an "averagemasstransfercoefficient." Use of a local coefficientmeansthat
we often must make a few extra mathematicalcalculations.However.the local coefficient
is more nearly constant,a smooth function of changesin other processvariables. This
definitionwas implicitly usedin Examples8.1-1,8.1-3,and 8.1-4in the previoussection.
It was usedin parallelwith a type of averagecoefficientin Example 8. | -2.
Another potentialsourceof ambiguityin the definitionof the masstransfercoefficientis
the interfacialarea.As an example,we againconsiderthe packedtower in Fig. 8.2-1. The
surfaceareabetweenwater and gasis usually experimentallyunknown,so that the flux per
areais unknown as well. Thus the masstransfercoefficientcannotbe easily found. This
problemis dodgedby lumping the areainto the masstransfercoefficientandexperimentally
determiningthe product of the two. We just measurethe flux per column volume. This
may seemlike cheating,but it works like a charm.
Finally, masstransfercoefficientscan be complicatedby diffusion-inducedconvection
normalto the interface.This complicationdoesnot existin dilute solution,just asit doesnot
exist for the dilute diffusion describedin Chapter2. For concentratedsolutions,theremay
be a larger convectiveflux normal to the interfacethat disruptsthe concentrationprofiles
near the interface. The consequenceof this flux, which is like the concentrateddiffusion
problemsin Section3.3, is that the flux will not double when the concentrationdifference
is doubled. This diffusion-inducedconvectionis responsiblefor the last definitionin Table
8.2-2,where the interfacialvelocity is explicitly included. Fortunately,most solutionsare
dilute,so we can successfully
deferdiscussingthis problemuntil Section13.5.
I fìnd thesepoints difficult, hard to understandwithout careful thought. To spur this
thought,try solving the examplesthat follow.
220
8 / Fundamentalsof Mass Transfer
Example 8.2-1: The mass transfer coefficient in a blood oxygenator Blood oxygenators areusedto replacethe humanlungs during open-heartsurgery.To improveoxygenator
design,you arestudyingmasstransferof oxygeninto waterin onespecificblood oxygenator.
From publishedcorrelationsof masstransfercoefficients,you expectthat the masstransfer
coefficientbasedon the oxygenconcentrationdifferencein the water is 3.3 ' 10-3 centimeters per second. You want to use this coefficientin an equationgiven by the oxygenator
manufacturer
t ) / Other
Sr
correspon
N1:kp(po,-pó,)
wherep0, is the actualoxygenpartialpressurein the gas,and pfi. is the hypotheticaloxygen
partial pressure(the "oxygen tension") that would be in equilibrium with water under the
experimentalconditions. The manufacturerexpressedboth pressuresin millimeters of
02. You also know the Henry's law constantof oxygen in water at your experimental
conditions:
For the gas
k
Po' : 44,000 atm xo,
where xe, is the mole fraction of the total oxygen in the water.
Find the masstransfercoefficientko.
Solution Becausethe correlationsare basedon the concentrationsin the liquid.
the flux equationmust be
N r : 3 . 1 ' l 0 I c m / s e c( t ' e , ,- t ' 6 r )
whereca., andcs, refer to concentrations
at the interfaceand the bulk solution,respectively.
We can converttheseconcentrationsto the oxvsen tensionsas follows:
LUì -
L^Ur
-
( o no.\
Thesecon
Example I
porescon
placedtog
-\
By compa
\iln1
wherec is thetotalconcentration
in theliquidwater,p is theliquid'sdensity,andM is its
average
molecularweight.Because
thesolutionis dilute,
1 g/cmr
I
atm I
po'
t0' : |
4A. 104atn760.rn Hcl
118g/r.,-rot
Note that r
R'e u ar
expenme
\
Combiningwith theearlierdefinition,we see
. c m I 1 . 6 7 l. 0 - e m o l l,
- no,)
Nr : 3.3 l0-'|
sec L .-: o,'nHg )(n0.,
mol
12
: 5.5'10
cm2 secmm Hg
\Pot,
* here .\'
s
Poz)
The new coefficientko equals5.5 ' l0-12 in the unitsgiven.
Thus
Example 8.2-2: Converting units of ammonia mass transfer coefficients A packed
tower is being used to study ammonia scrubbingwith 25"C water. The mass transfer
coefficientsreportedfor this tower are I .18 lb-mol NHrftrr-ft2for the liquid and 1.09lb-mol
NHr/hr-ft2-atmfor the gas. What are thesecoefficientsin centimetersper second?
'.'.hich i. :
::lil3rrth3
'..,;;ter
, .:ena-
221
8.2 / Other Definitions of Mass Transfer Cofficients
Solution FromTable8.2-2,we seethattheunitsof theliquid-phase
coefficient
correspond
to k,. Thus
- : -l . l t O f
úr,,
,,
i :-iì,ttOf.
t : - ^ ,
l:,rll \lèf
:--:lme-
p
/ 1 8l b / m o l\ / t . I 8 l b - m o lN H r \ l t / 3 0 . 5c m \ l t / _ l h r
\
frr-hr
fr
3.ó00
sec
62.4
rb/fr'
\
/ \
/ \
/ \
/
: : l.ltof
: t _ _ _ _ _ _ _ _ _ _ _ _ 1t t _
: 2 . 9 . 1 0 - 3c m r s e c
'\\ gen
: rrr-the
: .ii:\
Forthe gasphase,we seefrom Table8.2-2thatthe coefficienthasthe unitsof kr. Thus
k:
Of
RTkp
-:::ntal
m )o l \ / 3 0 . 5 c m \ /
hr
\
_ t /_l . l3tl 4
-a
_ r| m_ -lf tt r_\ /[ l2. 0998l b" - K
ft
\ lb-mol-K/\hr-ft'-atm/\
/\3.600sec/
: 3.6 cm/sec
Theseconversionstake time and thousht.but are not difficult.
.1u
id.
.:: elY
"
Example 8.2-3:Averaging a masstransfer coefficient Imaginetwo poroussolidswhose
pores contain different concentrationsof a particular dilute solution. If these solids are
placedtogether,the flux N1 from one to the other will be (seeSection2.3)
N,:
JDlntLr,
By comparisonwith Eq. 8.1-2,we seethat the local masstransfercoefficientis
k: tfDlrt
U t. its
Note that this coefficientis initially infinite.
We want to correlateour resultsnot in terms of this local value but in terms of a total
experimentaltime /e. This implies an averagecoefficient.[. definedby
Nt : [at'
where N1 is the total solutetransferredper areadivided by 16.How is I relatedto ft?
Solution From the problem statement,we seethat
..
N l' :
' [,""
! ^ t -N1,tr
'
d,
.[,1,"
fi" JD/n nrlat : 2 V
rs
DltrtoLct
Thus
* : zvTl"n
:.,;ked
::n Slèr
.-ntol
which is twice the value of k evaluatedat /e. Note that "local" refershere to a particular
time ratherthan a particularposition.
8 / Fundamentalsof Mass Transfer
222
againthe packedbed of
Example 8.2-42Logmean mass transfer coefficients Consider
8. I -2' Masstransfer
benzoicacid spheresshownin Fig. 8.1-1(b)thatwasbasicto Example
a log mean driving force:
coeffìcientsin a bed like this are sometimesreportedin tetms of
N1
by the total surface
For this specificcase,N1 is the total benzoicacid leavingthe beddivided
concentrationat the
areain the bed. The bed is fed with pure water,and the benzoicacid
spheresurfacesis at saturation;that is, it equalsc1(sat)'Thus
. lcr(sat) 0] [c1(sat) cr(out)]
/ V 1: / < ' u t @
8.3/ Correi
and
À,i..
The coeffìc
Many ar
usedrnostl\
closelyrela
critics asse
Why bother
This ar-e
approxlmal
ily accomp
this need.\
\t1(sat)-cr(out)/
Showhowklu*isrelatedtothelocalcoefficientftusedintheearlierproblem.
bed,we showed
Soluiion By integratinga massbalanceon a differentiallengthof
in Example8.1-2thatfor a bed of lengthL,
cr(out)
:1_
cr (sat)
n-(kaluo)L
Reananging,we fìnd
c 1 ( s a t -) r , ( o 9
, (t"t) - 0
_ o_(ka lu\r
Taking the logarithm of both sidesand rearranging'
kaL
, I
Multiplyingboth sidesby c1(out)'
cr (out)uo: kaL
By definition,
c r( o u t ) u u A
where A is the
In
cientsand h
a method ftr
convenlen
Experimen
Howerer
do not uant
This avoida
coefficient
someoneel,
(r(sat)-0 \
ln|-----__-_.I
( s a t ) - c 1( o u t ) /
\ c1
N r' : _ -
8.
a(AL)
r ( s a t )- 0 l [ c r ( s a t ) c r ( o u l
r ' r( s a t -) 0 \
/
lnl-'l
\ r'r(sat)- cl (out)/
8'-
In
sible. The.
dimension
engineersu
so scientiîì
The char
correlation
transfercoe
of diffusion
Damkohle
usedfor dif
A ke1'p,
p h y s i c asl y
will be the
dissolvings
of thirty pe
8.3 / Correlations of Mass Transfer Cofficients
: -:.ì Of
/-z-)
ano
. : - : : l\ f C f
ftros: ft
:Ì ece
:t the
',red
The coefficientsare identical.
Many arguethat the log mean mass transfercoefficientis superiorto the local value
usedmostly in this book. Their reasonsare that the coefficientsare the sameor (at worst)
closely relatedand that ft1o*is macroscopicand henceeasierto measure.After all, these
critics assert,you implicitly repeatthis derivationevery time you make a massbalance.
Why bother?Why not useft1o*and be done with it?
This argumenthasmerit, but it makesme uneasy.I fìnd that I needto think throughthe
approximationsof masstransfercoefficients
everytime I usethemandthatthis reviewis easiÌy accomplishedby making a massbalanceand integrating.I fìnd that most studentsshare
this need. My adviceis to avoid log meancoefficientsuntil your calculationsare routine.
8.3 Correlations
of Mass Tlansfer
Coefficients
In the previoustwo sectionswe havepresenteddefinitionsof masstransfercoefficientsand haveshownhow thesecoefficientscan be found from experiment.Thus we have
a methodfor analyzingthe resultsof masstransfèrexperiments.This methodcan be more
convenientthan diffusion when the experimentsinvolve mass transferacrossinterfaces.
Experimentsof this sort include liquid-liquid extraction,gas absorption,and distillation.
However,we often want to predicthow one of thesecomplexsituationswill behave.We
do not want to correlateexperiments;we wanf to avoid experimentsas much as possible.
This avoidanceis like that in our studiesof diffusion, where we often looked up diffusion
coefficientsso that we could calculatea flux or a concentrationprofile. We wantedto use
someoneelse'smeasurements
ratherthan painfullv make our own.
8.3.I Dimensionless Numbers
In the same way, we want to look up mass transfercoefficientswheneverpossible. These coefficientsare rarely reported as individual values,but as correlationsof
dimensionlessnumbers.Thesenumbersare often named,and they aremajor weaponsthat
engineersuse to confusescientists.Theseweaponsare effectivebecausethe namessound
so scientific,like closerelativesof nineteenth-century
organicchemists.
The characteristics
of thecommondimensionless
groupsfrequentlyusedin masstransfer
conelations are given in Table 8.3-1. Sherwoodand Stantonnumbersinvolve the mass
transfercoefficientitself. The Schmidt,Lewis, and Prandtlnumbersinvolve differentkinds
of diffusion, and the Reynolds,Grashof,and Pecletnumbersdescribeflow. The second
Damkohler number, which certainly is the most imposing name, is one of many groups
usedfor diffusion with chemicalreaction.
A key point about each of these groups is that its exact defìnition implies a specific
physical system. For example,the characteristiclength / in the Sherwood numberkllD
will be the membranethicknessfor membranetransport,but the spherediameterfor a
dissolvingsphere.A good analogyis the dimensionlessgroup "efficiency." An efficiency
of thirty percent has very different implications for a turbine and for a running deer. In
11A
8 / Fundamentalsof Mass Transfer
groups
Table 8.3-1. Significanr:eof r:ommondimensionLess
Group'
Physicalmeaning
KI
Sherwoodnumber D
masstransfèrvelocity
k
Stantonnumber ^
masstransfèrvelocity
flow velocity
t)u
diffusivity of momentum
Schmidtnumbe. I
D
Lewis number
diffusion velocity
diffusivity of mass
a
diffusivity of energy
D
diffusivity of mass
Prandtl number.
I
diffusivity of momentum
a
diffusivity of energy
luo
Reynoldsnumber v
inertial forces
-;-or
^
vlscouslorces
Used in
Usual dependentvariable
Occasional dependentvariable
Correlationsofgas or liquid data
Simultaneousheat and masstransfer
Heat transfer;includedhere for
compìeteness
Forcedconvection
flow velocity
"momentum velocity"
Grashóf
numb
rr'1#1t
Péclet number
uol
I
SecondDamkóhlernumber
buoyancy forces
viscousforces
flow velocity
diffusion velocity
reactionvelocity
diffusion velocity
Free convection
8.3 / Corr
Some i
The accur
elTors are
&Iì)
SOft tr
physicalp
forced con
I fbr gase
be reliable
plants.the
c h e c k so n
Manr r
involvea S
i n t e r e s tT. l
The \ arial
diffèrenrpi
Frrre\anlp
r . p u m p e J:
: h r o u g hr i ,
l r t r t h e r, .
- , : L l r e \J - : :
: intem".
Correlations of gas or liquid data
involvingreactions
Correlations
(seeChapters15-16)
:,:lutinl.
The i ::
-..
.. -. :::;:.
^ rc12
o r ( T h i e l em o d u l u s ) ' D
Note: u The symbolsand their dimensionsare as fbllows:
D diffusioncoefficient(L2lt)
g a c c e l e r a t i odnu et o g r a v i t y( L / r r )
k masstransfercoefficient(l, /r)
/ characteristiclength (L)
u" fluid velociry (Llt)
o thermaldiffusivity (L2ll)
r first-orderreactionrateconstant(t r)
u kinematicviscosity(L2/r)
Ap/p fractionaldensitychange
the same way, a Sherwood number of 2 means different things for a membrane and for a
dissoh'ing sphere. This flexibility is central to the correlations that follow.
8.3.2 Frequently Used Correlations
Correlationsof masstransfercoefficientsare convenientlydivided into thosefor
fluid-fluid interfacesand thosefor fluid-solid interlaces.The correlationsfor fluid-fluid
interfacesare by far the more important,for they are basicto gas adsorption,liquid-liquid
extraction.and nonideal distillation. Correlationsof thesemass transfercoefficientsare
also important for aerationand water cooling. Thesecorrelationsusually have no known
parallelcorrelationsin heattransfer,wherefluid-fluid interfacesare not common.
I
'r -
- . . .
I :,
. . , .\ .l e r
l:ir
:: i ll \tef
..-.ì\e for
- :-fluid
:-liquid
a--,tr Of€
r ÌlOWfl
8.3 / Correlations of Mass Transfer Coefficients
225
Someof the more useful correlationsfor fluid-fluid interfacesare given in Table8.3-2.
The accuracyof these correlationsis typically of the order of thirty percent,but larger
elTorsare not uncommon. Raw datacan look more like the result of a shotgunblast than
any sort of coherentexperimentbecausethe data include wide rangesof chemical and
physical properties. For example,the Reynoldsnumber,that characteristicparameterof
forced convection,can vary 10,000times. The Schmidtnumber,the ratio (vlD), is about
I for gasesbut about 1000for liquids. Over a more moderaterange,experimentaldatacan
be reliable' Still, while the correlationsare usefulfor the preliminary designof small pilot
plants,they shouldnot be usedfor the designof full-scaleequipmentwithout experimental
checkson the specificchemicalsystemsinvolved.
Many of the correlationsin Table 8.3-2 have the same generalform. They typically
involve a Sherwoodnumber,which containsthe masstransfercoefficient,the quantity of
interest.This Sherwoodnumbervarieswith Schmidtnumber,a characteristicof diffusion.
The variation of Sherwoodnumber with flow is more complex becausethe flow has two
differentphysicalorigins. In most cases,the flow is causedby extemalstirringor pumping.
For example,the liquids usedin extractionarerapidly stirred;the gasin ammoniascrubbing
is pumpedthroughthe packedtower;the blood in the artificialkidney is pumpedby the heart
throughthe dialysisunit. This type ofexternally driven flow is called "forced convection."
In other cases,the fluid velocity is a result of the masstransferitself. The masstransfer
causesdensity gradientsin the surroundingsolution; thesein turn causeflow. This type
of internally generatedflow is called "free convection." For example, the dispersalof
pollutantsand the dissolutionofdrugs are often acceleratedby free convection.
The dimensionlessform of the correlationsfor fluid-fluid interfacesmay disguisethe
very real quantitativesimilaritiesbetweenthem. To explorethesesimilarities,we consider
the variationsof the masstransfercoeffìcientwith fluid velocity and with diffusion coeffìcient. Thesevariationsare surprisinglyuniform. The masstransfercoefficientvarieswith
the 0.7 power of the fluid velocity in four of the five correlationsfor packedtowersin Thble
8'3-2. It varieswith the diffusion coefficientto the 0.5 to 0.7 power in every one of the
correlations.Thus any theory that we derivefor masstransferacrossfluid-fluid interfaces
shouldimpÌy variationswith velocity and diffusion coeffìcientlike thoseshownhere.
Somefrequentlyquotedcorrelationsfor fluid-solid interfacesare given in Table 8.3-3.
Thesecorrelationsare rarely important in common separationprocesseslike absorption
and extractions.They are importantin leaching,in membraneseparations,and in electrochemistry.However,the real reasonthat thesecorrelationsarequotedin undergraduate
and
graduatecoursesis that they are close analoguesto heat transfer.Heat transferis an older
subject,with a strongtheoreticalbasisand more familiar nuances.This analogygets lazy
lecturersmerely mumble, "Mass transferis just like heattransfer"and quickly comparethe
correlationsin Thble 8.3-2 with the heattransferparallels.
The correlationsfor solid-fluid interfacesin Table8.3-3aremuch like their heattransfer
equivalents.More significantly,theselessimportant,fluid-solid correlationsareanalogous
but more accuratethan the importantfluid-fluid correlationsin Table8.3-2. Accuraciesfor
solid-fluid interfacesaretypically averageIl07o; for somecorrelationslike laminarflow in
a singletube,accuraciescan be *l%o. Suchprecision,which is truly rare for masstransfèr
measurements,
reflectsthe simpler geometryand stableflows in thesecases.Laminar flow
of one fluid in a tube is much betterunderstoodthan turbulentflow of gas and liquid in a
packedtower.
The correlationsfor fluid-solid interfacesoften show mathematicalforms like thosefor
fluid-fluid interfaces. The masstransfercoefficientis most often written as a Sherwood
o
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8 / Fundamenîalsof Mass Transfer
228
E.-ì/ Cttrr,
number,though occasionallyas a Stantonnumber. The effect of diffusion coefficientis
most often expressedas a Schmidtnumber.The effect of flow is most often expressedas a
Reynoldsnumberfor forced convection,and as a Grashófnumberfor free convection.
These fluid-solid dimensionlesscorrelationscan concealhow the masstransfercoefficient varies with fluid flow u and diffusion coeffìcient D, just as those for fluid-fluid
interfacesobscuredthesevariations. Basically,ft often varies with the squareroot of u.
The variation is lower for some laminar flows and higher for turbulent flows. It usualÌy
varieswith D2l3, thoughthis variationis rarely checkedcarefully by thosewho developthe
correlations.Variationof k with D2l3 doeshave some theoreticalbasis,a point explored
further in Chapter13.
:
Example 8.3-1: Dissolution rate of a spinning disc A soÌid disc of benzoic acid
2.5 centimetersin diameter is spinning at 20 rpm and 25'C. How fast will it dissolve
in a largevolume of water? How fast will it dissolvein a largevolume of air? The diffusion
coefficientsare 1.00. l0-s cm2/secin waterand 0.233cm2Tsecin air. The solubilityof
benzoicacid in wateris 0.003g/cmr: its equilibriumvaporpressureis 0.30 mm Hg.
Solution Before startingthis problem, try to guessthe answer. Will the mass
transferbe higher in water or in air?
In eachcase,the dissolutionrate is
Nr : ftcr(sat)
wherec1(sat)is the concentrationat equilibrium. We can find k from Table 8.3-2:
ttutl 2 r U rl I
k:0.62D1-l
\r,,/
f- ì
\D/
For water.the masstransfercoefficientis
! r:mplr
r)
20160)(2rlsec
: 0 . 6 2 ( 1 . 0 0 .l 0 5 c m 2 7 s e c ,
( 10'01.tr'/r*
t12
0. 0 1cm2 lsec
( , .00I 0 *
sec) " '
a
'
l
: 0.90.l0-3 cm/sec
Thus the flux is
N 1 : ( 0 . 9 0 . l 0 I c m l s e c ) ( 0 . 0 0g3/ c m 3 )
:2.7 .10 6g/cm2-sec
For air, the valuesare very different:
: 0 . 1 1c m / s e c
t,t
r)
20160)(2r
/sec
0.l-5.t"? sec
(1
:0.62(0.233cm2lsec)
(
0.l 5 cm2/sec
0 2;-)-);"1/*. ) " '
l
- - '
8.3 / Correlations of Mass Transfer Coffit:ients
' - : 3 Ì ] tt s
. . r ' J a sa
A l m o s lp u r e
oir
:1.
î
:
229
Pure woler
rO€f-
l::
- i-fluid
.:0fu.
: ..ìtliìlly
. .. .rp the
:\:lofed
A i r & w o t e rs o l u b l ev o p o r
| ì. acid
- r..olve
lt:lusion
--:Lit\ of
r'
Gos dissolved
in wofer
Fig. 8.3- l. Gas scrubbing in a wetted-wall column. A water-soluble gas
is being dissolve6 in
a falìing fi lm of water. The problem is to calculate the length of the c'olumn
necessaryro
reach a liquid concentration equal to ten percenl saturatlon.
ll ldòù
which is much larger than before. However,the flux is
mmHs (
Imot
N 1 : ( 0 . 4 7 c m l s e fc()o ' :
\
\ (213\ /122s\l
L\760mmHe/ \22.4.t}t.r,/ \2e8l \ not)]
: 0 . 9 . l 0 - ó g / c ms2e c
The flux in air is aboutone-thirdof that in water,eventhoughthe masstransf-er
coefficient
in air is about 500 times largerthan that in water. Did you suessthis?
'.)'"
Example 8.3-2: Gas scrubbing with a wetted-wall column Air containing
a warersolublevapor is flowing up and water is flowing down in the experimental
column shown
in Fig' 8.3-l ' The water flow in the 0.07-centimeter-thick
film is 3 centimetersper second,
the columndiameteris l0 centimeters,
and the air is essentially
well mixed right up to the
interface' The diffusion coefficientin water of the absorbedvapor is 1.g .
10"5 cà27sec.
How long a column is neededto reach a gas concentrationin water that is ten percent
of
saturation?
Solution The first step is to write a massbalanceon the water in a differential
columnheightA: :
(accumulation): (flow in minus flow out) * (absorption)
g:[ndluucfl, _frdlu0ci
] : + a : * t d L z k l c t ( s a t )_ c 1 l
in which d is the column diameter,/ is the film thickness,u is the flow, and
c1 is the vapor
concentrationin the water. This balanceleadsto
, 1. . .
0 : - / r 0 " , ' - . 1k l c r ( s a -Ur . , l
dz
230
8 / Fundamentalsof Mass Transfer
From Table 8.3-2, we have
rrherer/ rr
flt-r* . Ctrn:i
I Prrtl't12
k:0.69( _ I
\ z /
We also know that the enteringwater is pure; that is, when
z:0,
cl:0
Combining theseresultsand integrating,we find
(r
c1(sat)
-_ || - ( - - l . 3 g ( D z / I 2 u o , t t / 2
Insertingthe numericalvaluesgiven,
_ _ t/ _ l| 2l un \f l \1.90DlL
/
\
l _
c,
\.l2
t l
.,(sat)/l
m')'2' .( 3
^c"mr /"s' e
, _c )- .\ '
/ 0 . 0 1 c- =
I'"'"'
I l t n t l_ 0 . t ) 1 2
.
t
\ t . o o rl . 8 l o - 5 c m 2 l s e /c "
:
: 4 . 8c m
This approximatecalculationhasbeenimprovedelaborately,eventhoughits practicalvalue
is small.
Example 8.3-3: Measuring stomach flow Imaginewe want to estimatethe averageflow
in the stomachby measuringthe dissolutionrateof a nonabsorbingsolutepresentas a large
sphericalpill. From in vitro experiments,we know that this pill's dissolutionis accurately
describedwith a masstransfercoefficient.How can we do this?
Solution We first calculatethe concentrationc1 of the dissolving solutein the
stomachand then show how this is relatedto the flow. From a massbalance.
dc,
V --; : rd'k[c1 (sat)- c1]
clt
where V is the stomach'svolume, nd2 is the pill's area,k is the masstransfercoefficient,
and c1(sat)is the solute'ssolubility.Becauseno soluteis initially present,
cr :0
when / :0
Inte-rrating,
, V / r ' 1 ( s a t ) \
A : - l n l - l
ndtt
\ c ' 1 ( s a t ) r ' 1/
If rl e assume that stomach flow is essentially forced convection, we find, from Table 8.3-3,
kl
\.-t/ C,,rr.
- : 2 + 0 . 6 {/ d- ul \ t " r lr -. ,ì ' , ,
\ D /
D
\ u /
!,,itts.fer
8.3 / Correlations of Mass Transfer Cofficients
zJl
whered is the pill diameter,D is the diffusion coefficient,and u is the unknown
stomach
flow. Combiningand rearranging.
t ' : 2e5(/ v t : ap : ' t \ f V / c r ( s a r )\ - ' l1 2
/ L " r * ' " ( ' . n " . l -);
which is the desiredresult. Note the assumptionsin this problem: The flow is due
to forced
convection,the pill diameteris constant,and flow in and out of the stomachis negligible.
-.,1value
Example 8.3-4: Gtucoseuptake by red blood cells The uptakeof glucoseacross
the red
blood cell membranehasa maximum raterangingfrom 0. I to 5 p*ov.rr-hr. Apparently,
thesedifferencesresult from differencesin experimentalconditions.Using the
correlation
for liquid dropsin Table8.3-2,estimatethe effectof masstransferin the bulk
to seewhen it
could have affectedtheseuptakerates. To make the estimationmore quantitative,
assume
thata typicalexperimentis madein a beakercontaining100cm3of redbl,oodcells
suspended
in 1 liter of plasma.The beakeris stirredwith a l/50-hp motor. The cellsoriginally
contain
little glucose.At time zero,radioactivelytaggedglucoseis addedand its uptake
measured.
The diffusion coefficientof glucoseis about 6 . 10-6 cm2
1sec,and the plasmaviscosityis
approximatelythat of water.
Solution We are interestedin the case in which glucose uptake is dominated
by mass transfer. In this case, glucose will diffuse to the membraneand
then almost
instantaneously
be takeninto the cell. Thus
Nt : kct
::ie flOW
-:. ,1large
,,urately
-:r' in the
where cl is the bulk concentration.If we can calculatek, then we can estimate
N1, the
desiredquantity.we seefrom Tàble 8.3-2 that for a suspensionof liquid drops,
(i)"'
ft: 0.13
: 0 . r 3(
p-t/1v-s/t2 D213
t l s o h p 1.45. l}e g-cm2\ t/a1 1* y-tn
\ l . 0 0 0c m r
hp-secr /
\.r, /
-5lr2 ,
t , 2 / 3
| . ,^-r-cm'\
/ 0 . 0 t c m 2\
to.lu
\ * . /
sec/
\
3:nClent,
: 5 . 7 . l 0 3 c m / s e c: 2I cmlhr
The flux is
Nt : (21 cmlhr)c1
- . ! 1 _ 1
Whetheror not diffusionoutsidethe cellsis significantdependson c1,theamount glucose
of
used. The flux equals5 pmol/cmz-hrwhen c1 is about 0.3 mmol/liter. If c1 far exceeds
0.3 mmol/liter, then the flux due to diffusion will be much fasterthan that due to the
cell
membrane. The measurements
will then truly representmembraneproperties. However,
if the glucose concentrationis less than 0.3 mmol/liter, then the measurementswill
be
z)z
8 / Fundamentalsof Mass Transfer
8 . 1/ D r n
functions both of the membraneand of mass transfer in plasma. Such restrictions
can
compromisemeasurements
in biological svstems.
8.4 DimensionalAnalysis:The route to Correlations
The correlationsin the previous section provide a useful and compact way of
presentingexperimentalinformation. Use of thesecorrelationsquickly gives reasonable
estimatesof masstransfercoefficients.However,when we find the correlationsinadequate,
we will be forced to make our own experimentsand developour own correlations.How
can we do this?
The basicform of masstranstèrcorrelationsis easily developedusing a methodcalled
dimensionalanalysis(Bridgeman,1922;Becker,l9j6). This methodis easily learnedvia
the two specifìcexamplesthat follow. Before embarkingon this description,I want to
emphasizethat most people go through three mental statesconcerningthis method. At
fìrst they believe it is a route to all knowledge,a simple techniqueby which any set of
experimentaldata can be greatly simplified. Next they becomedisillusionedwhen they
havedifficultiesin the useof the technique.Thesedifficultiescommonly resultfrom efforts
to be too complete.Finally, they learn to use the methodwith skill and caution,benefiting
both from their pastsuccesses
and from their frequentfailures. I mentionthesethreestages
becauseI am afraidmanymay give up at the secondstageandmisstherealbenefitsinvolved.
We now turn to the examples.
8.4.1 Aeration
Aerationis a commonindustrialprocessand yet one in which thereis often serious
disagreementabout correlations.This is especiallytrue for deep-bedfermentorsand for
sewagetreatment,where the rising bubblescan be the chief meansof stirring. We want
to study this processusing the equipmentshown schematicallyin Fig. g.4-I . we plan to
inject oxygeninto a variety of aqueoussolutionsand measurethe oxygen concentrationin
the bulk with oxygen selectiveelectrodes.We expectto vary the averagebubble velocity
u, the solution'sdensity p and viscosity Ér,the enteringbubble diameterd, and the depth
o f t h eh e dL .
We measurethe steady-state
oxygen concentrationas a function of position in the bed.
Thesedatacan be summarizedas a masstransfercoefficientin the following wav. From a
massbalance.we seethat
dc,
0 : -ùE
r k u l c l( s a t )- r ' 1|
(8.4-l)
u'herezzis the total bubbleareaper column volume. This equation,a closeparallel to the
nlanyrîassbalancesin Section8.1, is subiectto the initial condition
: :0,
c r: 0
(8.4-2)
Thus
t.., _
u
/
c1(sat)
-ln{
. : \\ cr (sat)- cr (z)
-:
(8.4-3)
' ! È .
j
\ .
J l :
r
1,.;rt.sle
8.4 / DimensionalAnalysis: The route to Correlations
233
, :l\ Can
: ..Ar Of
:.' rn&ble
&t9,
: J r't.|U
.l iìlled
r:iad via
. .\ Jnt to
:-.,d. At
.: ' .et Of
.:3n they
::r 3lforts
Fig. 8.4-I . An experimentalapparatusfor the studyof aeration.Oxygenbubblesfrom the
spargerat the bottom of the tower partially dissolvein the aqueoussolution. The concentration
in this solutionis measured
with electrodes
thatarespecificfor dissolvedoxygen.The
concentrationsfound in this way tlle interpretedin terrnsofmass transfercoefîcients;this
interpretation
assumes
thatthe solutionis well mixed,exceptvery nearthe bubblewalls.
-.--:rrting
'.',.'...È
1':' .tà89S
-r rrlved.
Ideally, we would like to measure ft and a independently, separating the effects of mass
transfer and geometry. This would be difficult here, so we report only the product ftd.
Our experimental results now consist of the following:
ka : ka(u,O,p, d, z')
(8.4-4)
We assumethat this function has the form
:':ì .ÈflOUS
-. .,ndfor
\\ l rvant
, \ : p l a nt o
- , : . 1 t i o inn
: . r'loctty
: .:ì.' depth
- :hebed.
. . F r o ma
1)
S.,1.- :l to the
r 8.;1-2)
/s4 :
[constant)uqpfl Ut 4t re
(8.4-5)
where both the constantin the squarebracketsand the exponentsare dimensionless.Now
the dimensionsor units on the left-handside of this equationmust equalthe dimensionsor
units on the right-handside. We cannothavecentimetersper secondon the left-handside
equaÌto gramson the right. Becauseka has dimensionsof the reciprocalof time (1 lt), u
has dimensionsof length/ttme(L lt), p has dimensionsof massper lengthcubed (M I Lr,
and so forth, we fìnd
1
/L\" /M\fl
-r:r
(;i (z.j ( # ) ' ( L t ò ( L t '
(8.4-6)
The only way this equationcanbe dimensionallyconsistentis if the exponenton time on the
left-handsideofthe equationequalsthe sumofthe exponentson time on the right-handside:
-l:-d-y
(8.4-1)
Similar equationshold for the mass:
0:f_ty
(8.4-8)
andfor thelength:
0:a-3fl*y*6*e
s.-r-3)
Equations
8.4-1to 8.4-9givethreeequations
for thefiveunknownexponents.
(8.4-e)
8 / Fundamentalsof Mass Transfer
l-'i+
We can solve these equationsin terms of the two key exponentsand thus simplify
Eq. 8.4-5. We choosethe two key exponentsarbitrarily. For example,if we choosethe
exponenton the viscosity y andthat on column height e , we obtain
a : l - y
(8.4r0)
F:-y
( 8 . 4 -ll)
(8.4-12)
A - _ " ,
-e-1
(8.4-13)
(8.4-14)
Insertingtheseresultsinto Eq. 8.4-5 and rearranging,we find
/ kad\
l;/
: [constant]
(+)'(;)'
(8.4-15)
The left-handsideof this equationis a type of Stantonnumber.The first term in parentheses
on the right-handside is the Reynoldsnumber,and the secondsuch term is a measureof
the tank's depth.
This analysissuggestshow we should plan our experiments. We expect to plot our
measurements
of Stantonnumberversustwo independentvariables:Reynoldsnumberand
zld. We want to cover the widest possiblerangeof independentvariables.Our resulting
correlationwill be a convenientand compactway of presentingour results,and everyone
will live happily ever after.
Unfortunately,it is not alwaysthatsimplefor a varietyof reasons.First,we hadto assume
that the bulk liquid was well mixed, and it may not be. If it is not, we shallbe averagingour
valuesin someunknownfashion,and we may find that our correlationextrapolatesunreliably.Second,wemayfindthatourdatadonotfitanexponential
formlikeEq.8.4-5.Thiscan
happenif theoxygentransferredis consumedin somesortof chemicalreaction,which is true
in aeration.Third, we do not know which independentvariablesare important. We might
suspectthatka varieswith tank diameter,or spargershape,or surfacetension,or the phases
of the moon. Suchvariationscan be includedin our analysis,but they make it complex.
Still, this strategyhasproduceda simplemethodof correlatingourresults.The foregoing
objectionsareimportantonly if theyareshownto be so by experiment.Until then,we should
use this easystrategy.
8.4.2 The Artificial Kidney
The secondexampleto be discussedin this sectionis the masstn.,rsferout of the
tube shownschematicallyin Fig. 8.4-2. Suchtubesarebasicto the artificial kidney. There,
blood flowing in a tubularmembraneis dialyzedagainstwell-stirredsalinesolution. Toxins
in the blood diffuse acrossthe membraneinto the saline,thus purifying the blood. This
dialysisis often slow; it can takemore than40 hoursper week. Increasingthe masstransfer
in this systemwould greatlyimproveits clinicalvalue.
The first stepin increasingthis rate is to stir the surroundingsalinerapidly. This mixing
increasesthe rate of masstransferon the salineside of the membrane,so that only a small
part of the concentrationdifferenceis there,as shownin Fig. 8.4-2. In otherwords,we have
decreasedthe resistanceto masstransferon the salineside. The secondstepin increasing
8.1 / Dittttr
1,.;,t.s.ler
8.4 / DimensionalAnalysis: The route to Correlations
:nplify
,.. the
235
Blood fiow
. - 1 l-0 )
. l-l l)
r J,l?l
r +-13)
.1-1rl)
. _ l _l 5 )
' : ' - - l h se e s
::'tlf€
Of
: ot our
--r.r and
:i.ulting
:-. afvone
,rt:UlÎe
:ìiO!
OUf
:-.unleliT h i sc a n
Fig. 8.4-2. Mass transf'erin an artificial kidney. Arterial blood flows through a dialysis tube that
is immersed in saline. Toxins in the blood diffuse acrossthe tube wall and into the saline. If the
sallne is well stirred and if the tube wall is thin, then the rate of toxin removal dependson the
concentration gradient in the blood. Experiments'in this situation are easily correlated using
d i m e n s i o n aul n a l y s i s .
the rate is to make the membraneas thin as possible.Although too thin a membranewould
rupture,existing membranesare alreadyso thin that the membranethicknesshas only a
minor effect. The resultis that the concentrationdifferenceacrossthe membraneis not the
largestpart of the overall concentrationdifference,againas shownin Fig. 8.4-2.
The rate of toxin removalnow dependsonly on what happensin the blood. We want to
correlateour measurements
of toxin removalas a function of blood flow. tube size.and so
forth. To do this, we find the flux for eachcase:
flux N1 -
.-:,1\tfUe
'*: night
.-:'phases
:ì.llex.
: :egoing
.,.;'.hould
amounttransferred
(area)(time)
By definition,
Nt:k(ct-cn)
-
^L I
: . nllxlng
' .L\mall
, '.J have
" -:rrsing
(8.4-17)
Becausewe know N1 and c1,we carìfind the masstransfercoefficientft.
As befbre, we recognizethat the masstransfercoefficientof a particulartoxin varies
with the system'sproperties:
k : k ( u ,p , I , r D
, ,d)
-:i of the
:. There,
- Toxins
.r This
-. :riìnsfer
(8.4r6)
(8.4-18)
where u, p, and p are the velocity, density,and viscosity of the blood, D is the diffusion
coeffìcientof the toxin in blood. and d is the diameterof the tube. We assumethat this
relationhasthe form
ft - [constant)u"pfruv PdO'
(8.4-l9)
wherethe constantis dimensionless.The dimensionsor units on the left-handside of this
equationmust equal the dimensionsor units on the right-handside; so
(+)'"
i,'(i)"(#)'(X)'
(8.4-20)
236
8 / Fundamentalsof Mass Transfer
This equationwill be dimensionallyconsistentonly if the exponenton the length on the
left-handside ofthe equationequalsthe sum ofthe exponentson the right-handside:
(R 4-) l\
l:a-3F-V*23+e
8.5 / Mass T
The reasc
beginning.in
the densitr 1
kd
Similar equationsmust hold for mass:
D
(8.4-22)
0 :f*y
and fbr time:
-l : _o _ y - 3
( 9 ,L _ ) \ \
which is the
haveno rea:
and make int
sharpens.
We solve theseequationsin termsof the exponentsa and 6:
(8.4-24)
A : A
Y : 7 * a - 3
6 : ó
(8.4-25)
(8.4-26)
(8.4-21)
€ : a - l
(8.4-28)
B : q l 3 - l
8.5
We combinetheseresultswith Eq. 8.4-19and collect tems:
kdo
t t
/duo\"/ u \
lconstantì{ } lr^l
\ u /
\ p D /
ó
(8.4-2e)
Byconvention,wemultiplybothsidesofthisequationbythedimensionlessquarftity
ttlpD
to obtain
kct
/ctuo\"/ u \'o
: lconstant]('
,
s- ) \A
)
n-
(8.4-30)
This equationis the desiredcorrelation.As in the first example,a key stepin the analysis
is the arbitrary choice of the two exponentscvand 6. Any other pair of exponentscould
have been chosenand would have given a completely equivalentcorrelation. However,
the particular manipulationshere are made so that the dimensionlessgroups found are
consistentwith traditionalpatterns.Such traditionalpatternssometimesreflectexperience
and sometimesmerely mirror convention. Multiplying both sidesof Eq. 8.4-29plpD
involvesthesefactors.
The trouble with this analysisis that it is not the whole story. From experimentsat low
flow, we would f,nd that the masstransfercoeffìcientk doesvary with the cube root of the
velocity u and with the two thirds power of the diffusion coefficientD. This suggeststhat
Eo. 8.4-30can be rewritten
kd
/,1u\r'l
- : l c o n s t a n t-l \l -Dl /
D
(8.4-31)
t h a ti s . t h a t a : ( l - ó ) : l 1 3 . H o w e v e ri,f w e t h e n m e a s u r e d fats a f u n c t i o n o ft h e t u b e
d i a m e t e r d . w e w o u l d f i n d i t p r o p o r t i o n adl t- ot / 3 , n o t d - 2 / 3a s s u g g e s t e d b y E q . 8 . 41-.3
Afier a good start,our dimensionalanalysisis failing.
r :
231
r
;t1.\re
8.5 / Mass Transfer AcrossIntetfaces
ìn the
The reasonfbr this failure is that we did not chooseall the relevantvariablesat the very
beginning,in Eq. 8.4-l8. We shouldhaveincludedthetubelengthL; we couldhaveomitted
the density p and viscosity tr. Had we done so, we would haveobtained:
r_ll)
( r12u\ttl
kd
D
\
' l
J -
Iconstant]
\DLl
(8.,1-32
)
which is the result quoted in Table 8.3-3. However,from dimensionalanalysisalone,w'e
have no reasonto be critical of our original result in Eq. 8.4-30. We can only be critical
and make improvementsas our experimentalexperiencegrows and as our physicalinsight
' _r-t3)
sharpens.
8.5 Mass T[ansfer Across Interfaces
r J- )-ll
\ J-l5l
. J_r6l
' -+-17)
) +-28)
. l_29)
In the previous sections,we used mass transfercoeffìcientsas an easy way of
describingdiffusion occurring from an interfaceinto a relatively homogeneoussolution.
Thesecoefficientsinvolvedmany approximationsand sparkedthe explosionof defìnitions
exemplifiedby Table8.2-2. Still, they area very easyway to correlateexperimentalresults
or to make estimatesusing the publishedrelationssummarizedin Tables8.3-2 and 8.3-3.
In this section.we extend these definitions to transfer acrossan interfàce.from one
well-mixed bulk phaseinto anotherdifferent one. This caseoccursmuch more frequently
than doestransfèrfrom an interfaceinto one bulk phase;indeed,I had troubledreamingup
examplesearlierin this chapter.Transferacrossan interfaceagainsparkspotentiallymajor
problemsof unit conversion,but theseproblemsare often simplifìedin specialcases.
' .- . ;, , - p' D
8.5.1 The Basic Flux Equation
\ +-30)
- .rrìalysis
' j'-:. could
'-i
..\
-r\eî
..-:
- 'r
-.'L)i
Dl
ìtl
llli
-' .'.'\ (.t'r/1Ar
\ -1-31)
: :hetube
r - . r . J - 31.
Presumably,we can describemasstransferacrossan interf'acein termsof the same
type of flux equationas before:
NI : KAcr
( 8 . s1- )
..,,hereNr is the soluteflux relative to the interface,K is called an "overall masstransfer
qrr - : -: ":.-iJ Jcl is some appropriateconcentraftondifference. Rut what is Act ?
CTr, - -:nxppropriatevalueof Ac1 tums out to be difficult. To illustratethis, consider
"
,.:ijirrlsshownin Fig. 8.5-1. In the f,rst term, hot benzeneis placedon top of
LÙr',-'
r r : - i-,' benzenecools and the water warrnsuntil they reachthe sametemperature.
fl
'-,,:i.:irre is the criterion for equilibrium, and the amount of energy transferred
itf
-i
rrlD
;rroportionalto the temperaturedifferencebetweenthe liquids. Everything
rcCIllS SOCUf9.
As a secondexample,shownin Fig. 8.5-1(b),imaginethata benzenesolutionof bromine
is placedon top of water containingthe sameconcentrationof bromine. After a while, we
find that the initially equalconcentrationshavechanged,that the bromine concentrationin
the benzeneis much higher than that in water. This is becausethe bromine is more soluble
in benzene,so that its concentrationin the final solutionis higher.
This result suggestswhich concentrationdifferencewe can usein Eq. 8.5-I . Vy'eshould
not usethe concentrationin benzeneminus the concentrationin water;that is initially zero,
and yet thereis a flux. Instead,we can usethe concentrationactuallyin benzeneminus the
238
8 / FundamentalsoJ'MassTransfer
8.5 / Mass Tr,
(o)Heot tronsfer
lì-l-,.n,.n"
trF;l
ww.",",w
Iniliol
( b ) B r o m i n ee x t r o c t i o n
t---l
lEquot//
p,a
_
f---r
Benzene .l- aon"
lEquol -
7777-2
Finol
)
|,/,/,/)
V7-7V
, I
woter----+ lDitute)
lnllioI
/,/
,/)
Finol
(c) Bromine voporizotion
['';]-^' F-.;-l
F;1F*,*
Wú
Frg::
diîìu.::
c h a ng r '
Finol
Fig. 8.5-1.Driving fbrcesacrossinterfaces.In heattransfer,îhe amountofheat transfèrred
dependson the temperature
differencebetweenthe two l'iquids,as shownin (a). In mass
transfèr,the amountof solutethatditfusesdependson the solute's"solubility"or, moreexactly,
on its chemicalpotential.Two casesareshown.In (b), brominediffusesfrom waterinto
benzenebecause
it is muchmoresolublein benzene;in (c), bromineevaporates
until its
chemicalpotentialsin the solutionsareequal.This behaviorcompìicates
analysisof mass
transtèr.
concentration that would be in benzene that was in equilibrium with the actual concentration
in water. Svmbolicallv.
Nr : Klcr(in benzene)- Hc(inwater)]
coetl.:
îe\t.
:he lett is bein-l
.\-. :
.,.here,('-.ì: rhe
: : r - b u l kp r e \ \ u
.reflur acro..
Thu..
(8.5-2)
where rY is a partition coefficient,the ratio at equilibrium of the concentrationin benzene
to that in water. Note that this doespredict a zero flux at equilibrium.
A better understandingof this phenomenonmay come from the third case,shown in
Fig. 8.5-1(c)' Here, bromine is vaporizedfrom water into air. Initially, the bromine's
concentrationin water is higher than that in air; afterward, it is lower. Of course,this
reversalof the concentrationin the liquid might be expressedin molesper liter and that in
gasas a partial pressurein atmospheres,
so it is not surprisingthat strangethings happen.
As you think about this more carefilly, you will realizethat the units of pressureor
concentrationcloud a deepertruth: Mass transfercan be describedin termsof more fundamentalchemicalpotentials.If this were done,the peculiarconcentrationdifferenceswould
disappear.However,chemicalpotentialsturn out to be very difficult to use in practice,and
so the concentrationdifferencesfor masstransferacrossinterfaceswill remaincomplicated
by units.
8.5.2 The Overall Mass Transfer Cofficient
'We
want to include thesequalitativeobservationsin more exactequations.To do
this, we considerthe exampleof the gas-liquid interfacein Fig. 8.5-2. In this case,gason
r. .r r- rr r. -!
t( hr r: r I i . , "gu.r
-::J r and r
\\'e norr nee_
. : . r l n t o . ta j l . . .
|
11.
- )
-
. , :. C r e H r r . r l r :
i . : . S . - r - -tìh r r *
(
-i theflur
l' .,,t.rter
- -.'J
'r 3\ilctly,
8.5 / Mass TransferAcross Interfaces
239
Fig. 8.5-2. Mass transfer across a gas-liquid interface. In this example, a solute vapor is
diffusing from the gas on the left into the liquid on the right. Because the solute concentration
changesboth in the gas and in the liquid, the solute's flux must depend on a mass transfer
coefficient in each phase. These coefficients are combined into an overall flux equation in the
text.
the left is being transferredinto the liquid on the right. The flux in the gas is
(8.5-3)
Nt:kp(prc-pn)
- r llratlOn
whereftr, is the gas-phasemasstransfercoefficient(typically in mol/cmr-sec-atm),p1e is
the bulkpressure,and p11is the interfacialpressure.Becausethe interfacialregion is thin,
the flux acrossit will be in steadystate,and the flux in the gaswill equalthat in the liquid.
Thus.
Nr : kr (cri - cro)
- -inzene
. :1,r\\ n ln
-:.rnline's
.-:.r-. this
.-J thatin
. ::ppen.
l:'-.LlfC Of
-: iunda- r. \\ OUld
where the liquid-phasemasstransfercoefficientk1 is typically in centimetersper second
and c1; and c10are the interfacialand bulk concentrations,
respectively.
We now needto eliminatethe unknowninterfacialconcentrationsfrom theseequations.
In almostall cases,equilibrium existsacrossthe interface:
Pti :
"
Pri
H
krPn * ktcrc
knHlkl
(8.s-6)
and the flux
*t :,It,
To do
!its on
(8.5-5)
Hcti
where 11 is a type of Henry's law or partition constant(herein cmr-atm/mol). Combining
Eqs. 8.5-3 through8.5-5,we can find the interfacialconcentrations
:,..-t-. OIìd
:'riicated
(8.5-4)
I
* ul,rr(prc
- Hcrc)
(8.s-7)
You shouldcheckthe derivationsofthese results.
Before proceedingfurther, we make a quick analogy. This result is often comparedto
an electriccircuit containinqtwo resistances
in series.The flux correspondsto the current.
240
8 / Fundamentalsof Mass Transfer
and the concentrationdifferenceprc - Hcrc conespondsto the voltage. The resistanceis
then I f k,, + H lkL, which is roughly a sum of two resistances
in series.This is a good way
of thinking about theseeffects. You must remember,however,that the resistancesl/kn
and 1I kl are not directly added,but alwaysweightedby partition coefficientslike t1.
We now want to write Eq. 8.5-7 in the form of Eq. 8.5-1. We can do this in two wavs.
First. we can write
Nr :Kz(r:ì-cro)
(8.s-8)
Kr:
(8.5-e)
where
l lkt -l I lkpH
and
.
Dtrr
'
H
(8.5r0)
K1 is called an "overall liquid-side masstransfercoefficient,"and ci is the hypotheticat
liquid concentrationthat would be in equilibrium with the bulk gas. Alternatively,
Ny: Kp(pn - pi)
( 8 . 5l -r )
I
K '.' - _
llkr+ H/kL
(8.sl2)
Pî : Hcut
( 8 . sl -3 )
where
and
K2 is an "overall gas-sidemasstransfercoefficient,"and pi is the hypotheticalgas-phase
concentrationthat would be in equilibrium with the bulk liquid.
We now turn to a varietyof examplesillustratingmasstransferacrossan interface.These
exampleshavethe annoyingcharacteristicthat they are initially diffìcult to do, but they are
trivial after you understandthem. Rememberthat most of the difficulty comesfrom that
a n c i e nbt u t c o m m o nc u r s e :u n i t c o n v e r s i o n .
Example 8.5-1: Oxygen mass transfer Estimatethe overall liquid-side mass transfer
coeffìcientat 25'C for oxygen from water into air. In this estimate,assumethat each
individual masstransfercoefficientis
k :
D
0 . 0 1c m
T h i s r e l a t i o ni s j u s t i f i e di n S e c t i o n1 3 . 1 .
Solution For oxygenin air, the diffusion coefficientis 0.23 cm2lsec;for oxygen
in water.the diffusioncoefficientis 2.1 . l0-5 cm2/sec.The Henry'slaw constantin this
caseis 4.,1. 101atmospheres.We needonly calculateft1-and k, andplug thesevaluesinto
r
T,,irts.fe
iJnce lS
Eq. 8.5-9.Findingft1 is easy
,,,,ìway
:, Ilkp
H.
, r r\ &YS.
s.5-8)
241
8.5 / Mass TransferAcross Interfaces
2 . 1. 1 0 - 5c m 2 / s e c
D,
"
0 . 0 1c m
0 . 0 1c m
: 2 . 1 . 1 0 3c m / s e c
Finding kn and H is harderbecauseof unit conversions.From Eq. 8.5-3 and Table 8.2-2
s.s -9 )
D6
k6
t_
K p :
RT
( 0 . 0 1c m ) ( R Z )
0.23 cm2lsec
:
(0.01cmX82 cm3-atm/g-mol-'K)(298"K)
: 9.4 . l0-a g-mol/cm2-sec-atm
r 5-10)
From the units of the Henry's law constant,we seethat the value given implies
,thetical
Ptr: H'rti
By comparisonwith Eq. 8.5-5,
r . 5 -I l )
p 1 ;: H c 1 ; : ( H c ) x 1 ;
Thus
s . 5 1- 2 )
H :
/ H'\
4 . 4 ' l O aa t m
:J.9.
| - I - -,
lg-mol/l8cmr
\, /
l0' atm-cm'/g-mol
Insertingtheseresultsinto Eq. 8.-5-9,we find
s . - 5l -3 )
;,r.-phase
K "r - _
I
llkt+llkpH
I
,:. These
.: theyare
: r-rìlnthat
',:.- transfer
tl
t
lol.rn/*.
-
l
e|1
: 2 . 1 . 1 0 - 3c m / s e c
The masstransferis dominatedby the liquid-sideresistance.This would also be true if we
calculatedthe overallgas-sidemasstransfercoeffìcient,a consequence
of the slow diffusion
in the liquid state.It is the usualcasefbr problemsof this sort.
: :hat each
- r oxygen
- . 1 . , : irnt t h i s
- . , , l u e si n t o
Example 8.5-2: Perfume extraction Jasmone(CllHl60) is a valuablematerial in the
perfume industry, used in many soapsand cosmetics. Supposewe are recoveringthis
materialfrom a water suspensionof jasmine flowers by an extractionwith benzene.The
phaseis continuouslthe masstransfercoefficientin the benzenedropsis 3.0' l0 +
aqueous
per second;the masstransfèrcoefficientin the aqueousphaseis 2.4. l0 l
centimeters
centimetersper second.Jasmoneis about 170 times more solublein benzenethan in the
suspension.What is the overall masstransfercoeffìcient?
''t ^"t
8 / Fundamentalsof Mass Transfer
Solution For convenience,we designateall concentrationsin the benzenephase
with a prime and all thosein the water without a prime. The flux is
8 . 6/ C o r
W h e nt h e
N r : k ( c r o- c r , ) : k ' ( c ' r ,- c 1 o )
The interfacialconcentrationsare in equilibrium:
c'li :
Hcli
l n p a s sni '
\\-e c;:
Eliminating theseinterfacialconcentrations,we find
t
N,:l
t
l
l ( H c ," n - t : , ^ l
L l l k ' +H l k j
The quantity in squarebracketsis the overall coefficientK' that we seek. This coeffìcient
is basedon a driving force in benzene.Insertingthe values,
K,:
3 . 0 . 1 0 - 1c m / s e c
+'
lz0
2 . 4 . 1 0 - 3c m / s e c
1 . 3 .1 0 ' c m / s e c
:
Similar resultsfor the overallcoefficientbasedon a driving force in water are easilyfound.
Two points about this problem deservemention. First, the result is a completeparallel
to Eq. 8.5-12, but for a liquid-liquid interfaceinsteadof a gas-liquid interface. Second,
masstransferin the water dominatesthe processeventhough the masstransfercoefficient
in water is largerbecausejasmoneis so much more solublein benzene.
'We
are studying
Example 8.5-3: Overall mass transfer coefficients in a packed tower
packed
tower
containin
a
pressure
total
2.2
atmospheres
gas absorptioninto water at
we
believe
that
methane,
and
with
ammonia
experiments
ing Berl saddles. From earlier
is
per
volume
packing
tower
area
times
the
for both gasesthe mass transfercoefficient
for
The
values
side.
the
liquid
for
l8 lb-mol/hr-ft3for the gas side and 530 lb-molftrr-ftr
thesetwo gasesmay be simiìar becausemethane,and ammonia have similar molecular
for ammonia
weights. However,their Henry's law constantsaredifferent: 9.6 atmospheres
mass
transfer
coefficient
gas-side
fbr methane.What is the overall
and 41,000atmospheres
for eachgas?
Solution This is essentiallya problem in unit conversion. Although you can
extractthe appropriateequationsfrom the text, I alwaysfeel more confidentif I repeatparts
of the derivation.
The quantity we seek,the overall gas-sidetransfercoefficientK., is definedby
N 1 a:
K , . rtu1 1 0- r ' i )
: k r a ( . y r o- y r , )
: k,a(xti - r'n)
where r'1 and 11 are the gasand liquid mole fractions.
The interfacialconcentrationsare relatedby Henry's Ìaw:
/)1,
pJ t, -
Hxti
Tre c.ì.
243
8.6 / Conclusirns
When theseinterfacialconcentrationsare eliminated,we find that
1
1
Hlo
- - - - L
Kro- kru' kra
In passing,we recognizethat yi mustequalHxrclp.
We can now fìnd the overall coefficientfbr each sas. For ammonia,
9.6 atmlZ.2 atm
-
["d
l
.
T
| 6 lb-mot/hr-It-
530 1b-mol/hr-ft3
K,.a - l6 1b-mol/hr-ft3
The gas-sideresistancecontrolsthe rate. For methane,
I
1
--]_
Kr"
l8 1b-mol/hr-ft'
41.000 atntl2.2 atm
530Ib-molftr-ft3
K,'a : 0.03 lb-mol/hr-ft3
: ,ìnd.
' . : - r1l e l
: -"nd.
The coefficientfor methaneis smaller and is dominatedby the liquid-side masstransf.er
coefficient.
--'lr'fll
8.6 Conclusions
- j\ lng
-.i.ìin: : thàt
-::lL-
1S
-:" for
:-;ular
: ì'rflnia
::-Jìent
. Jcan
l.:i f iìrtS
This chapterpresentsan alternativemodel for diffusion, one using masstransfer
coefficientsratherthandiffusion coefficients.The model is most usefulfor diffusion across
phaseboundaries.It assumesthat large changesin the concentrationoccur only very near
theseboundariesand that the solutionsfar from the boundariesare well mixed. Such a
model.
descriptionis called a lumped-parameter
resultscan be convertedinto mass
we
have
shown
how
experimental
In this chapter,
can be efficiently orgaWe
have
also
shown
how
these
coeffìcients
transfercoefficients.
published
correlationsthat are
and
we
have
cataloged
nized as dimensionlesscorrelations,
problems
with
units that come
by
commonly useful. Thesecorrelationsare compromised
out of a plethoraof closely relateddefinitions.
Masstransfercoeffìcientsprovideespeciallyusefuldescriptionsof diffusion in complex
niultiphasesystems. They are basic to the analysisand design of industrial processes
ìike absorption,extraction,and distillation. They should fìnd major applicationsin the
.tudy of physiologic processeslike membranediffusion. blood perfusion.and digestion;
physiologistsandphysiciansdo not often usethesemodelsbut would benefitfrom doing so.
Mass transfercoeffìcientsare not useful in chemistry when the focus is on chemical
kinetics or chemical change. They are not useful in studies of the solid state, where
modelsdo not help
;oncentrationsvary with both positionand time, and lumped-parameter
ntuch. However,mass transfercoeffìcientsare used in analyzingetching processes.like
thoseusedin makingsiliconchips.
All in all, the materialin this chapteris a solid alternativefor analyzingdiffusion near
rnterfaces.It is basic stuff for chemicalengineers,but it is an unexploredmethodfor many
,rthers.It reDavscareful studv.