CHAPTER
r8
The E quati ons of C onti nui ty
for a B i nr
to consider some of their speciala1
combined heat and mass transfer, tt
and three-component ordinary diffu
provide the basis for dimensionalana
with the determination of the functior
correlations.
O F CO N
$r8.t TH E EQ UATTO NS
A BI NARY M I XTURE
In this section we apply the law of
volume element L,r L,y Az fixed in sp
A and.B is flowing. (SeeFig. 3.1-1
duced by chemical reaction at a ral
tributions to the mass balanceare
The Equationsof Change
for MulticomponentSystems
In chapter l7 a number of problemsin ordinary diffusionwereformulated
by making massbalanceson one of the diffusingspecies.In this chapterwe
begin by making a mass balanceover an arbitrary differentialfluid element
to establishthe equations of continuity for the two chemical speciesin a
binary fluid mixture. Then the insertionof the expressions
for the massflux
givesthe diffusionequationsin a variety of forms. Thesediffusionequations
can be used to set up any of the problemsin chapter 17 and more complicat edonesas wel l .
After this introduction to the equationsof ordinary diffusion for a binary
mixture, we proceedto the expositionof the equationsof changefor a multicomponent mixture with chemicalreactionsand heat effects,which include
an equation of continuity for eachchemicalspeciespresent,the equation of
motion, and the equation of energybalance. These relations supply a full
descriptionof multicomponentflow systemsand include the equaiionsgiven
in chapters 3 and l0 as specialcases. They are not normally used in their
completeform given here; usually one usesthem by discardingterms that
are identically zero or physicallynegligibleand by this procedureobtaining
simpler equationsfor the specialsituation under consideration.
After the presentationof thesegeneralequations,we are then in a position
554
ti me rate of c hange
of mass of A in
volume element
a^
-"i4 A
ot
input of .4 across
face at r
fiatl,
outirut of ,4 across
faceatrl L,z
nt , lr -
rate of production
of A by chemical
reaction
ra L,t
There are also input and output term
entire mass balance is written down a
obtains, after letting the size of the vc
opo (4,
ra - --\a r -i
This is the equation of continuityfot
describesthe change of mass concen
fixed point in space, this changeresu
reactions producing l. The quanti
components of the mass flux vector r
vector notation Eq. 18.1-5may be rer
?* o
ot
T h e E quati ons of C onti nui ty
CHA P T ER
r8
l or a B i nary Mi xture
to consider some of their special applications,such as the description of
combined heat and mass transfer, thermal, pressure,and forced diffusion,
and three-componentordinary diffusion. The equations of change also
provide the basisfor dimensionalanalysis,which is used later in connection
with the determination of the functional form of the mass-transfercoefficient
correlations.
0 r8 .r TH E E QU A TION SOF C ON TIN U ITY FOR
A B IN A R Y MIX TU R E
In this section we apply the law of conservationof mass of speciesA to a
volume elementLx Ly Lzfixed in space,through which a binary mixture of
A and,B is flowing. (SeeFig. 3.1-1.) Within this element,A may be produced by chemical reaction at a rate r, (g cm a sec-l). The various contributions to the mass balance are
tnge
Systems
n ordinarydiffusionwere formulated
diffusingspecies.In this chapterwe
n arbitrary differential fluid element
I f or t h e two c hem ic als pec ie si n a
for the massflux
r of the expressions
of forms. Thesediffusion equations
ms in Chapter 17 and more complins of ordinary diffusion for a binary
'the equationsof changefor a multi.ionsand heat effects,which include
rical speciespresent,the equation of
lance. These relations supply a full
:e m sand i n cl u det he equat ion sg i v e n
They are not normally used in their
usesthem by discardingterms that
ible and by this procedureobtaining
on underconsideration.
I equations,we are then in a position
time rate of change
of mass of ,4 in
volume element
Y!:! Lx Lu Lz
^,
ot
(18.1-1)
input of ,4 across
face at r
n1,1,L,YL,z
(18.1-2)
output of ,4 across
fa c e atrl L,r
nnrl r* 6, LY Lz
(18.1-3)
rate of production
of A by chemical
reaction
ro Lr Ly L,z
(18.1-4)
There are also input and output terms in the gr-and z-directions. When the
entire massbalanceis written down and divided through by Ar Ay Az, one
obtains. after lettine the size of the volume element decreaseto zero,
: ,,
**ou *':""\
*ol * (L?.
\or
oz/
( 18.1-5)
This is the equationof continuityfor componentA in a binary mixture. It
describesthe changeof mass concentrationof A with respectto time at a
fixed point in space,this change resulting from motion of I and chemical
reactions producing ,4. The quantities n1,, nlyt nAz are the rectangular
componentsof the mass flux vector fl,t: p,to,t defined in Eq. 16.1-5. In
vector notation Eq. l8.l-5 may be rewritten
oPn
+ (V' ne) : ra
0t
(18.1-6)
55 6
T h e Eq u a tio n s o f Ch a n g e for Mul ti component
S ystems
Similarly the equationof continuityfor componentB is
A^
" YB + (V' n -) :
0t
r-
(18.1-7)
Addition of thesetwo equationsgives
3
*(v'Pu):o
OT
( 18.1-8)
which is the equationof continuityfor the ntirture. It is the same as that for
a pur e f luid giv en i n E q .3 .1 -4 . In o b ta i n i n gE q . 18.1-8,w e have made use
o f t he r elat ion n1 * n n : p u (Eq . J i n T a b l e 16.1-3)and al so the l aw of
conservationof mass in the form 11 | ro: Q. Finally, we note that Eq.
1 8 . 1- 8bec om es
(18 . 1 -e )
(V'u):0
for a fluid of constantmass density p.
The foregoing developmentcould equally well have been made in terms of
molar units. If R, is the molar rate of production of I per unit volume,then
the molar analog of Eq. 18.1-6is
*
dI
+ p.N7):
R,,
(18.1-10)
The E quati ons of C onti nui ty
for a B i r
diffusion, we replacethe fluxesn_'
involving thc concentrationgradie
substi tutedint o Eq. 18. 1 6 and when
E q. 18.1-10,we get t he f ollowing
equati ons:
*OT * ( v ' p - , u )=
e"
? + tv'c-,ur; =
dt
Either one of theseequationsdescr
diffusingsystem. The only restrictio
forceddi ffu sion. Equat ions18. 1 14r
total density (p or c) and variabledj
B ecause
Eqs. 18. 1- 14and l5 ar ef a
In the anal ysisol dif f usingsyst emr
constant mass density or constant
simplification:
A ssumptionof Const antp and91
For thi s assum pt ion,Eq. l8. l- 14
Similarly,for componentB, we have
?+(V.N3):R3
0t
(1 8 . r-1 1 )
B ut accord ingt o Eq. 18. 1- 9,( V. , )
Mo we get
gives
Additionof thesetwo equations
(Rr R')
fi* f,'ca*): +
0r'
(18.1-12)
for the €quationof continuity for the mixture. Here we haveusedthe relation
N/ + Na : ct)*. However, since moles are not in general conserved,we
cannot set R,a f R, equal to zero unlessone mole of ,Bis produced for every
mole of I disappearing(or uice uersa). Finally, we note that Eq. l8.l-12
becomes
(V.u * ):
j1(R r
0p,, ,
; r r P- t ( V'u) f ( u
ot
* R r:)
(r8 . 1 -1 3 )
c
for a fluid of con.stantmolar density c.
Equations l8.l-6 and l0 are not in usefulform for obtainingconcentration
profiles. In order to get the equationsgenerallyused for describingbinary
+ e.yc,)
U:!
This equationis usuallyusedfor diff
temperatureand pressure.The left s
18.1 l Ti sof t hesam ef or m asEq.l0. l
for the analogiesthat are frequentl
in flowing fluids with constantp.
A ssumptionof Const antc and. Q .
For thi s assum pt ion,Eq. 18. 1- 1
0r,n
i c_r ( V' u*) * ( u
0t
C h a nge f or Mult i c o m P o n e n t
S y ste m s
tmponent B rs
B ):
rB
(18.1-7)
T h e E quati ons of C onti nui ty
for a B i nary Mi xture
S 57
diffusion,we replacethe fluxes n.. and N., by the appropriate expressions
involving the concentrationgradients. When Eq. ,4 of Table 16.2-1 is
substitutedinto Eq. l8.l-6 and wllen Eq. .Bof Table 16.2-l is substitutedinto
Eq. 18.1-10, we get the following completely equivalent binary diffusion
e q u a ti ons:
*
(l 8.1-8)
P u) : 0
e mixture. It is the sameas that for
ining Eq. 18.1-8,we have made use
L Table 16.l-3) and also the law of
Fi n a lly , we not e t ha t Eq .
ra :0.
( 18.1-e)
=0
ally well havebeenmade in terms of
roductionof I per unit volume,then
vr) : R'r
( 18.1-10)
* (V ' p,ru) : (V ' p?,nuV at,r)* r,1
(v'
R,,
'fi * Co'c,p*'): c9,,,flr,)1
( 18.l- 14)
( 18. 1- 15)
Either one of theseequationsdescribesthe concentrationprofilesin a binary
of thermal,pressurc,and
diffusingsystem. The only restrictionis the abser]ce
wit h var iable
fo rc eddi ffusi on. E quati ons18.1-14and l 5 are val i dfor systems
total density (p or c) and variable diffusivity !-t,t,tEqs. l8.l-14 and 15 are fairly general,they are alsofairly unwieldy.
Because
In the analysisof diffusingsystemsone can often legitimatelyassumeeither
constant mass density or constant molar density and thereby effect some
simplification:
Assumpti onof C onstantp andI,o6
For thi s assumpti on,E q. 18.1 14 becomes
Vr):
Rn
( 18.1-1r)
op,t
* r.a
+ pj(v' u) * (u'Vp-r) : (y',.,,fl)p1
0t
(18.116)
Bu t accordi ngto E q. 18.1-9,(V ' ,) i s zero. W hen E q. 18.1 l 6 i s dividedby
Mowe get
(Rj + R/J)
( I 8.1 -l 2 )
rture. Here we haveised the relation
es are not in generalconserved,we
s one mole of ,Bis produced for everY
Finall y,w e n ot e t hat E q. l8 .l -1 2
R1+ Rr)
(18.1-13)
*
* R,1
* rr.Vc,) : !y',,,P2c,n
(18.1-17)
This equationis usually usedfor diffusionin cliluteliquidsolutionsat constant
tempeiatureand pressure.The left sidecan be written as Dc.rlDt. Equation
1 8 '1-l Ti softhesameformasE q'10' l 25i r R .r : 0; thi ssi mi l arit yist hebasis
for the analogiesthat arc frequentlydrawn betwcenheat and masstransport
in flowing fluids with constantp.
Assumptionof Constantc and.Ci,.r,,
For thi s assumpti on,E q. 18.1-15becomes
ieful form for obtaining concentration
generallyused for describing binary
vr - t
At
* c-,(V' u*) + (u*'Vc.l) : 9,uF"cr * Rt
(18'1-18)
T h e Eq u a tio n s o f Ch a n g e for Mul ti component
Bu t ac c or dingt o Eq . 1 8 .1 1 3 , (V ' r* )
Henc eE q. 18. 1-1 8b e c o m e s l
u;*t*.vcr):91pv2c,1
S ystems
c a n b e repl acedby (l /c)(R " * R r).
This equation is usually used for low-densitygasesat constant temperature
and pressure. The left side of this equation cannotbe written as Dc^f Dt
becauseof the appearanceof u* insteadof u.
A s s um pt ionof Z e ro V e l o c i ty
Rectangular coor dinat es :
+.(+.ry
Cy Iindr ical coordinates :
There is one more simplifiedform of Eqs. 18.1-14and 15 that must be
mentioned. If there are no chemical changesoccurring, then ry, r6, Ra,
a nd R, ar e all z e ro . l l , i n a d d i ti o n ,u i s z e ro i n E q. l 8.l -17 or i f u* i szeroi n
Eq. 18. 1- 18,t hen w e g c t
a,
TAE
(
THE EQUATIONOF CONTINUITY
(Eq
* Rr - ! (n, * Rr) (18.1*19)
c
" .4 :
The Equation of Continuity in Curvi
q ,,,v z c ,
(18.1-20)
OT
which is called Fick's secondlaw of dtfusion or sometimessimply the diflusion
equation. This equation is usually used for diffusion in solids or stationory
in gasesz
in Eq. 18.1-17)and for equimolar,counter-dWsion
liquids (u:0
(u *: 0
in E q. 1 8 .1 -1 9 ). N o te th a t E q . 1 8 . 1-20i s si mi l ar to the heatconductionequationgiven in Eq. l0.l-26; this similarity is the basisfor the
analogous treatment of many heat-conductionand diffusion problems in
solids. Keep in mind that many hundredsof problemsdescribedby Fick's
second law have been solved; the solutions may be fbund in the monographs of Crank3 and of Carslaw and Jaeger'.4
+ . (:l{'*u,t
Spher ical coordinates :
+ . (i,'!,{"*n
^,*
TI
THE EQUATIONOF CONTINU
{t
Rectangular coordinates.
o
dc.t
1c,t
0c,
- . + fI u .- :j
+L 'z* u-,- :
oA
d
t
at
\*
OF A
$r 8. 2 T HE E QU AT ION OF C O N T T N U T T Y
I N CUR V IL IN E ARC OOR D IN AT E S
in this sectionthe most important equationsgivenin $18.1are summarized
in rectangular,cylindrical, and sphericalcoordinates. They are tabulated
for ready referencein setting up problems. In Table 18.2-l we give the
equati<inof continuity in terms of N;, and in Table 18.2-2, the diffusion
C ylindr ical coor dinates :
*,
* * (,,#*,,'r#
lla
= r t ; n\ ,
1 I n t h e a b s e n c eof ch e m ica l r e a ctio n s, Eq . l8 .t- 1 9 c an be w ri tten i n terms of u rather
than o* by using a different type of concentration, namely the logarithm of the mean
molecularweight:
A
; ln M + ( u. V ln M ) : 9noYz
ln M
(18.1-l9a)
0,
by C. H.
in which M : reMt * tnM* This relationseemsto have been first suggested
Bedingfield,
Jr., and T. B. Drew, Ind. Eng. Chem.,42,1164(1950).
'?By equimolar counter-diffusion,we mean that the total molar flux with resPectto
stationarycoordinatesis zero.
3 J. Crank, The Mathematicsof Difusion, Oxford UniversityPress(1956).
aH. S. Carslawand J. C. Jaeger,Heat Conductionin Solids,Oxford UniversityPress
(1959),SecondEdition.
a
Spherical coordinates :
*,,i**
T *(,,,*
:
" ^o( !,,* ( ," * )
.,
Cha nge f or Mult i c o m p o n e n t
S yste m s
T h e E quati on of C onti nui ty
i n C urvi l i near
s59
C oordi nates
TABLEI8.2-I
canbe replaced
by (llc)(R" * Rr).
TH E E Q U A T I o N o F C o N T I N U I T Y o F A | N V A R | o U s c o o R D | N A T E S Y S TEM S
(Eq. l8.l-10)
r *R-r -!(n-,
* R,u) (18.1-19)
c
'nsttj gasesat constant temperature
atlon cannot be written as Dc,,f Dt
of u.
Rectangular coordinates :
lt *(u*n,
at'\a'
(A)
+dN,t, *'Tr'\:n,
arl
oy
Cylindrical coordinates:
E qs. 18 .1 -1 4a nd 15 t hat m u s t b e
hangesoccurring, then r_r, rp, Ra,
z er oin Eq . l 8 .l - 17 or if u* is z e roi n
u r ^*( ,- 1 ,,' ^ ,_ ,+ 1 3 +
: n,
1 49Als\
t
'
'" '4 "
d0
At
r
\r ar
(B)
I
Spherical coordinat es :
,Y'r-o
(18.1-20)
'or or sometimessimply the difusion
for diffusion in solids or stationary
quimolar,counter-dffislon in gases2
q. 18.1-20is similar to the heat; this similarity is the basisfor the
Cuctionand diffusion problems in
ds of problemsdescribedby Fick's
tions may be fbund in the monoeger.4
: R.r(c)
0)*
]r*u,sin
?)
+ . $-|r**u,t.
-h
-,*
TABLE I8.2-2
T H E E Q U A T I O NO F C O N T I N U I T YO F A F O R C O N S T A N T p A ND 9 1 '
(Eq. l8.l-17)
Rectangular coordinates:
* *(,.**,,**,"+):'^"(#.# . +) * oio,
J I T Y OF A
rEs
t ionsgiv e ni n $ 18. 1ar e s um m a ri z e d
.l coordinates. They are tabulated
ems. In Table 18.2-l we give the
and in Table 18.2-| the diffusion
C vlindrical coor dinates :
.,"+)
* *(,,#*,,!,+
: '^"('l*'(:#
).'}#.#)*^ ^ (B )
. l- 19 can be writ t e n i n t e r m s o f u r a th e r
tion, namely the logarithm of the mean
t : 9 t f l zl n M
( 1 8 . 1 _ l9 a )
ns to have been first suggested by C. H.
r . ,42, 1164(1950).
hat the total molar flux with respect to
rrd University Press (1956).
tclion in .Solr'ds,Oxford University
press
Spherical coordinates:
dc,
;
dc,,
ldcs
,* "rink)
I
* l,'. a, + ''o; au
^ ll
: _rtt\7
a/,ac-1\
* \,' E
)
.#**(""'+).^k
#).t,
560
T h e Eq u a tio n s o f Ch a n g e for Mul ti component
S ystems
e quat ion in t he fo rm o f E q . 1 8 .1 -1 7 . o th e r equati onsof
$l g.l may be
written down by analogy. The notation for curvilinearcoordinatesystemsis
given in Fig. A.6-1. Note that the diffusion equationsfor solids can be
obtained by settingthe velocity componentsin Table lg.2-2 equal to zero.
T
AT T ON S
$ r 8. 3 T HE M U L T T C O MP O N ENEQU
OF C H A N GE
I N T E RM SO F T H E F L U X ES
In chapters 3 and l0 the equationsof changewere given for a pure fluid.
Here we extend these discussionsand give the equationsof change for a
nonisothermalmulticomponentfluid of n chemicalspecies.
(i) An equationof continuily for each chemical speciespresent in the fluid:
*0, :
i:1,2,"',n
Dt
..a
(conti nui t y)i p, :
ot
(18.3-1)
Addition of all z equationsof this kind givesthe equation of continuity for
the mixture, describedin Eq. l8.l-8. Any one of the n equationsabovemay
be replacedby the equationof continuityfor the mixtureinany givenproblem.
(ii) The equationof motion for the mixture:
n..
generallyneedscxplicit expressio
gradients and the transport coeffi
cientsalso needto be known as fu
It has been pointed out in prece
changetha t t heseequat ionsm ay as
ing on (a) whether DlDt or 0l0t is
for the fluxes,(c) whether masso
various forms of energy are broke
For example, we may rewrite E<
(moti on)
-p i (Y.u ) - (V .i o ) * rn
p+:
'
of Change in Terms of th
Equations
n
-[V
.* ] * Zp, E ,
- ' ilr
(18.3-2)
Here we have introduced for the sake of brevity the pressure tensor zr :
r * p6' in which r is the viscous part of the momentum flux (or shear
stresstensor),p is the static pressure,and 6 is the unit tensor. Note that
Eq. 18.3 2 differs from the equation of motion for a pure fluid (Eq.3.2-10)
only in the last term, where p! has been replaced,by Dprgr. In ihat term
accountis taken of the fact that eachchemicalspeciespresentmay be acted
on by a differentexternalforce per unit massg,.
(iii) The equationof energyfor the mixture:
p+ { 0 + i u " } :- ( v . s ) - o . [ ' . r ])+ t@ ,.il ( 1 8 .3 - 3 )
- ' Dt '
,= r
Note that this equation differs in appearancefrom the energy equation for a
pure fluid (Eq. 10.1-l l) only in the last term, where(pu . g tris ueenreplaced
by Xr(n, 'gf,). Here g is the multicomponentenergyflux relativeto the mass
averagevelocity u, defined in $18.4. Emission and absorption of radiant
energyare neglected.(See914.6.)
The complete descriptionof mass, momentum, and energy transport in
multicomponentsystemsis containedin theseequations. oni also needsthe
thermal equation of statep : pb, T, rr), the caloric equation of state 0 :
0(p, T, r,), and information about the chemical kinetics. In addition, one
- ( V'
{p, u*
fr0,:-[v'{pu
a^
(energy) ! p{0 * lu'} : -('
ot
+ i @,' l ' n
t=l
From Table 16.1 3 we know tha
just n,, the mass flux with resp
fluxesmay be introducedfor mom
define the following mass,mome
coordinate systemf xed in space:
ni:
P@iu+,
g: puu*t
e:
p{0 I
Equations 18.3-4,5, and 6 may b
(continuity)
1
lp,:-(!
0t
.a
(moti on)
"pu: - [ \
0t
(energy.)
+;,
!o t p1o
For one-dimensionalsteady-sta
external forces the fluxes n,, S, at
particularlyconvenientfor setting
masstransfer. (SeeExample18.5
7
The Equations of Change for Mutticomponent Systems
TABLE I8.3-I
THE EQU ATI O NO F ENERG YFoR M ULTIC o M P o N E N Ts Y s T E M s o
InTerrnsof
E:
p
O + fr +a:
(exactonlyfor
a6/ar:O)
DEn
: -(v .q) - (v .[n . u])+
Dt
.Z7r..g)
(A)
InTerrnsofO+fr:O+$u2:
Dr,
e .(U
+ K) : -(v.c) - (v'[n. u])+)
(ni.E)
(B)
In Termso1fr : !u2:
DK
:
' Dt
We now concludethis discuss
the equations of motion and en
equationof motion as it is usuall
forced-convection
problemsbut th
for describingthe limiting caseof
1
ture inequalitiesin the system.In
well as temperatureinequalities
10.3-2and use an equationofsta
of p in T and xu (for a two-comp
p: p+?rl^.r-
n
-(u . [v .rr]) * |
Fluxesin Terms of Transport pro;
ci@ . Ei)
(c)
:p- ofl tr - 11-
In Termsof 0:
n
DO
P Dt : -(V .el - Gc:Vu) * >(jt.C)
(D)
i:7
in which ( : -0ld(0pl1x),
is
indicateshow much the density
motion then becomes,for gravity
In Terms of fu:
DH
: -(v.s)
' Dt
Dp,
+ * - (t: vu) +ljt.EJ
UT
(E)
In Termsof Co:
^DT
_
'oc^
' Dt
n
/:r^
ii
\d
I'/pn
: -(v .q) - (c: vu)+ i<jo.ro)+ e::\
ln
n !,
n
2
Dt
.J,) - Ril
+ ) -tr,t<v
(r)b
In Terntsof e u:
^DT
_ : -(v .q) 'oc^.
" Dt
(rczY a+) i ri , .et + ( r - r g*\
j:t
\
In Terms of Ei:
\" r /e,r ,/
a p \ -f
* ilo,* lp/\ - r "'/
4lttv . J i) - n , 1
^ -f _)
) ,o .,
(c )
* ("'i*a) :o'kvr)
*(,2,"'n')
. ' # - ( r : vu )+ i{ in .s,)
Da_
_:
:
p
'Dt
_[ V. "]
_
This equation reducesto Eq. 10.
enceswithin the system. The las
force" resulting from deviationsfi
Next we consider the equationo.
pure fluids in $10.1that the eners
ways-in terms of 0, U, or f. ih
we have given the energyequation
of the energy equation are prese
necessaryto add a term ,S, to de
geneouschemical reactions. Thi
and C, and appears explicitly as Rememberthat in calculatins,p o
of the various speciesmust b-einc
$t8.4 THE M ULTTCO M PO N
OF THE TRANSPO RT
P
(H)"
" See $18.4 for the definition ofthe heat fluxes in these equations.
o L. B. Rothfcld,
personal communication.
c The usually unimportant Dufour
energy flux g(r) has been neglected here
In $18.3the equationsof chang
in terms of the fluxes of mass,mc
expressionsfor the profiles, we I
expressionsthat contain the tran
concentration, velocity, and teml
554
T h e Eq u a tio n s o f C hange for Mul ti compon€nt
S ystems
Fluxesin Terms of TransportP
before: in chapter 3 the equation of motion was rewritten by inserting
the
expressionfor mome'tum flux in terms of velocitygradients; in
chapter r0
the energyequationwas rewrittenby insertingthe expressionfor
energyflux
in terms of the temperaturegradient; and in
$ta.t the equationof coniinuity
was rewritten by inserting an expressionfor the mass flux in terms
of the
concentrationgradient.
Actually, the discussionswe have had so far regarding mass fluxes
and
concentrationgradientshave been somewhatoversimplifiid. certainly
the
most lmportant contribution to the mass flux is that resulting
from the
concentrationgradient. It is known, however,that even in an isothermal
system there are actually three "mechanical driving forces" that
tend to
produce the movement of a specieswith respectto the mean fluid
motion:
(a) the concentrationgradient, (b) the pressuregradient, and (c)
external
f or c esac t i n g u n e q u a l l yo n th e v a ri o u sc h emi calsfeci es. In
$r6.) and $l g.l
the secondand third of these"mechanicaldriving fbrces',hauebeenneglected
in order to simplify the discussions
there. In a multicomponentsystem,then,
we have fluxes of momentum, energy, and mass, each resulting
from an
associateddriving force as indicated by the main diagonal in Flg.
lg.4_1.
But the story is not quite that simplc. According to th"ethermodyiamics
of
irreversibleprocesses,
there will be a contribution to eachflux owing to each
driving force in the
This "coupling" can occur, however, onty
.system.
betweenflux-force pairs that are tensorsof equal order or which
differ in
order_bytwo. Consequently,in a multicomponentsystem(c) the momentum
flux. dependsonly upon the velocity gradients,(b) ihe .n..gy flux
depends
both on the temperaturegradient (heat conduction)and on-ih. mechanical
driving forces(the "diffusion-thermoeffect" or ..Dufour effect").
and (c) the
m as sf lux d e p e n d sb o th o n th e me c h a n i caldri vi ng forces(ordi nury.
pr.rrrr..
and forceddiffusion)and on the temperaturegraclient(the ..thermal-diffusion
effect" or "Soret effect"). Furthermore,the onsager reciprocal
relationsof
the thermodynamicsof irreversibleprocessesgive information
as to the
interrelationof the two coupledeffecti, the Dufour and the Soret.
In order
to describethe Soreteffect,an additionaltransportpropcrty (i.e.,in
addition
tq viscosity, thermal conductivity, and diffuiivlty) had to be introduced,
namely the "thermal diffusion ratio" or the "Soret coeflrcient,"depending
upon the exact definition. Becauseof the interconnectionof the
Soret and
Dufour effects,as describedby the onsager relations,this one
additionar
transport property wiil take care of the quantitative description
of both
phenom en a .(Se eth e n o n d i a g o n ael n tri e sl n fi g. l g.4_1.)
It is hoped that theseintroductoryremarkswiil give the beginner
a glimpse
of the insight that thermodynamicargumentscan offer in connection
with
coupled phenomena. AIso, perhaps, the reader will be somewhar
more
appreciativeof the set of generalflux relationsthat we are about to give
for
multicomponentsystems.Thosedesirousof exploringthe connection
between
thermodynamics and the trans
enoesin the literature.r
The expressionsfor the mo
mixtures as well as for pure sut
order tensor given by
a :
_p(yu
in which Vo is a dyadic produ
and 6 is the unit tensor. The cr
forces
Driving
gradi
Velocity
Fluxes
Momentum
(second
order
tensor)
Newton's
law
Itt,xl
Energy
(vector)
Mass
(vector)
Fi g, 18.4-1. S c hemati cdi agrams hov
forces i n a bi nary s y s tem. The as s oc
3.4-5,6 , and 7. Equat ion 18.
to the velocity gradients at an
18.4-l are the instantaneousl c
mixture. Expressions for t in
The expressionfor the energ
heat eonduction in pure substa
the conductive flux, contributi
r Two articlesdealingspecifical
energyfluxesare J. G. Kirkwoodar
(1952)andR. B. Bird,C. F. Curtis
. (
S ez' es,
No. 16, 51, 6985( 1955)M
are S. R. de Groot, Thermodynamics
Co., Amsterdam (1 95 1), pp. 9 4-123 fo
of Irreuersible Processes, Thomas, Spr
dynamics of the Steady Stale, Methur
are based upon the original develop
of L. Onsager, Physical Reuiew,37, L
-5--
C h ange f or Mult i c o m p o n e n t
S yste m s
F luxes i n Terms of Transport
rotionwas rewritten by insertingthe
of velocitygradients; in Chapter l0
iertingthe expressionfor energyflux
rd in $18.1the equationof continuity
n for the mass flux in terms of the
d so far regardingmass fluxes and
ewhat oversimplified. Certainly the
rassflux is that resulting from the
owever,that even in an isothermal
anical driving forces" that tend to
h respectto the mean fluid motion:
pressuregradient, and (c) external
:h e mic asp
l e ci e s .I n $16. 2and $ 1 8 .1
Lldrivingforces"havebeenneglected
. In a multicomponentsystem,then,
and mass, each resulting from an
y t h e ma i n d i a gonalin F ig. 18 .4 -1 .
\ccordingto the thermodynamicsof
rtributionto eachflux owing to each
lupling" can occur, however, only
:s of equal order or which differ in
omponentsystem(a) the momentum
adients,(b) the energy flux depends
conduction)and on the mechanical
ect" or "Dufour effect"),and (c) the
al driving forces(ordinary, pressure,
ture gradient(the "thermal-diffusion
, the Onsagerreciprocalrelationsof
ocessesgive information as to the
the Dufour and the Soret. In order
I transportproperty (i.e.,in addition
diffusivity) had to -be introduced,
r the "Soret coefficient,"depending
he interconnectionof the Soret and
rsagerrelations,this one additional
he quantitativedescription of both
e s in F ig . 1 8 .4 - 1. )
rarkswill give the beginnera glimpse
lmentscan offer in connectionwith
:he reader will be somewhat more
lations that we are about to give for
s of exploringthe connectionbetween
P roperti es
565
thermodynamicsand the transport processeswill find several suitable refereno€sin the literature.l
The expressionsfor the momentumfux r in Chapter 3 are valid for
mixtures as well as for pure substances.For Newtonian fluids r is a second
order tensor given bv
's : -p,(Vu + (Vu)t) + (?p - r)(V . u)6
(18.,t-1)
in which Vu is a dyadicproduct,(Vu)r is the transpose
of the dyadic Vu
and 6 is the unit tensor.Thecomponents
of c for r : 0 are givenin Tables
forces
Driving
gradients
Velocity
gradient
Temperature
Fluxes
Momentum
(second
ordertensor)
gradient
Concentration
gradient
Pressure
External
forcedifferences
Newton's
law
l P , xl
Energy
(vector)
Fourier's
law
Mass
(vector)
Soretetfect
lD,{rl
Ikl
Dufour
effect
r atun
I r
I
Fick's
law
lDAE l
F ig. !8.G1. S chemati cdi agram show i ng roughl y the rel ati ons betw een fl uxes and dri v i ng
fo r ces i n a bi nary system. The associ atedtransport coeffi ci entsare show n wi thi n brac k ets .
3 .4 -5,6, and7. E quati on 18.4-l show show the momentumflux is r elat ed
to the velocity gradientsat any point in the system. The pr and r in Eq.
18.4-l are the instantaneouslocal viscosityand bulk viscosityof the fluid
mixture. Expressionsfor r in non-Newtonianfluids were discussedin $3.6.
The expressionfor the energyfur g given in Eq. 8.1 6 is valid only for
heat conductionin pure substances.For mixtures there are, in addition to
the conductiveflux, contributions resulting from the interdiffusion of the
l Two articles dealing specifically with the expressions
for the mass, momentum, and
e n ergy fl uxes are J. G. K i rkw ood and B . L. C raw ford, Jr.,J. P hys. C hetn. ,56, 1048-1051
(1952) and R. B. Bird, C. F. Curtiss, and J. O. Hirschfelder, Chem. Eng. prog. Syntpostum
Se ri es,N o. 16,51,69-85 (1955). More general referencesto i rreversi bl ethermody nami c s
are S. R. de Groot, Thermodynamics of lrreuersible Processes,North Holland Publishing
Co .,A msterdam(1951),pp.9+l 23fortransportphenomena;
I.P ri gogi ne,Thermody nami c s
o flr reuersi bl eP rocesses,Thomas,S pri ngfi el d,
Il l i noi s(1955); K .G.D enbi gh,TheThermodynamics of the Steady Srale, Methuen, London (1951), pp. 78-86. All of these references
are based upon the original development of the concepts of irreversible thermodynamics
of L. Onsager, Physical Reuiew, 37, l, 405-426 (1911) ; ibid., 38, lI, 2265-2279 (193 l).
56 6
T h e Eq u a tio n s o f Ch a n g e for Mul ti component
S ystems
various speciespresent and the Dufour or diffusion-thermo effect. We may
then write for the total energy flux relative to the mass averagevelocity
q _ q (" )a q { a )q a 1 { a )
(18.4-2)
Here qt't : -kVT is the conductiveenergyflux, as definedin $8.1,and k is
the instantaneouslocal thermal conductivity of the mixture. The energy
flux q{atcausedby interdiffusionis definedfor a fluid containingn speciesby
the expression
nltn
s(d ':) # j ,:2 r7 ,J,
t=7 lVIi
r=l
(18.4-3)
Here ,9, is the partial molal enthalpy of the ith species.The Dufour energy
flux qt ' t is quit e c o m p l e xi n n a tu rea n d i s u s u a ll yof mi nor i mportance;hence
it is not further discussedhere.2The radiant energyflux gl'r may be handled
s epar at elyas de s c ri b e di n 8 1 4 .6 .
Frequently it is desirableto usethe energyflux with respectto stationary
c oor dinat ese,
, ra th e r th a n q . By u s i n g th e d e fi ni ti onof e, E q. 18.3-9,and
the foregoingexpressionfor q, we may write
e : q(c)+ g(,')+ g(")+ [*. ,] + p{0 + tur}a
of the Trans P ort
The expressionfor the massf
accordance with our brief Preli
butions associatedwith the me
contribution associatedwith the
i, : i!') -
Here we have written the massfl
(concentration)diffusion i!'', P
and thermal diffusion ilt). ttt.
are
n
e : _kV T +>F,t, * p u * p Uu
I:f
: -kvr +iT,t, * pftu
?2n
ii"):'
tM,M,
pRT Fr
il,)-'
\M,M,
PRTi=t
^2n
jlo':-4i*,
pRT Ft
(1g.4-4)
Wher rq( ' ) , [ r ' u] . a n d (!rp u ' )ua re o f n e g l i g i b lei mportance,w e may approxi mate e as
: -kvT+iE,t, +ir,fr,,
Fl uxes i n Terms
i: ') :
- Dir v
hT
In theseequationsG, and V, ar
energy) and volume, respectiv
coefficients,and the Drr are mu
The D,, and Dor have the follo'
Du:
(18.4-5)
iItI
i{*,*
i=1
W it h t he help of T a b l e l 6 .l -3 , w e re w ri teEq . 18.4 5 to get
e : _kvT +: N, 4
(18.4-6)
.:l
This approximateexpressionis the usualstartingpoint for engineeringstudies
on heat t r ans f erw i th s i m u l ta n e o u ma
s s stra n s fer.3
' ? T h e e x p l i c i t f o r m o f th e Du fo u r - e ffe ct te r m in m u lti component gas mi xtures has been
discusscd by J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular 7-heoryof Gases
an d L i q u i d s , W i l e y , N cw Yo r k ( 1 9 5 4 ) , p p .5 2 2 ,7 0 5 . T h e separati onof g i nto the component
pa r t s a b o v c i s n o t e n tir e ly cle a n - cu t b e ca u seo fa co n venti on adopted i n the defi ni ti on ofk
(s e e R . B . B i r d , C . F . Cu r tiss, a n d J. O. Hir sch fe ld e r , C hent.E ne. P roz. S yntposi um S eri es,
N o . 1 6 , 5 1 , 6 9 8 5 ( 1 9 5 5 ) , p . 7 7 , Eq s. 2 .1 5 a n d 2 .1 6 ) .
3 See, for examplc, T. K. Sherwood and R. L. Pigford, Absorption and Extraction,
M c G r a w - H i l l , N c w Yo r k ( 1 9 5 2 ) , Se co n d Ed itio n , p .9 6, E q. 128-112. S ee al so E xampl e
18 . 5 - l a n d ! 2 1 . 5 .
For n ) 2 the quantitiesD,, an
The ordinary difusion contrib
complicated way on the concen
The pressuredifusion term indic
ith speciesin a mixture if therei
the tendencyfor a mixtureto se
but use is made of this effectin
pressuregradientsmay be estab
in'
i mportanccin ionic syst em s,
the product of the ionic charg
ionic speciesmay thus be unde
is the only externalforce,thena
cally. The thermal diflusiontern
under the influenceof a tempe
r
C h ange f or M ulti c o m p o n e n t
S yste m s
or diffusion-thermoeffect. We may
tive to the massaveragevelocity
1{a)q q&\
(r8.4-2)
:rgy flux, as definedin $8.1,and k is
;tivity of the mixture. The energy
sd for a fluid containing n speciesby
n
:28, J ,
(18.4-3)
i=l
the ith species.The Dufour energy
usuallyof minor importance; hence
iant energyflux gt't may be handled
tergyflux with respectto stationary
the definitionof e, Eq. 18.3-9,and
rrite
n.ul * p { C + tr u,}a
( 18.4-4)
ligible
importancc,
we mayapproxi-
F luxes i n Terms
of the Transport
S O7
The expressionfor the mass.flu* j, in a multicomponent system will, in
accordancewith our brief preliminary discussion,consist of three contributions associatedwith the mechanicaldriving forces and an additional
contribution associatedwith the thermal drivins force
i,: il') + i:') + il")+ i!')
(r8.4-7)
Here we have written the massflux as the sum of terms describingordinarv
(concentration)diffusion ./j'), pressurediffusion jfe), forced didusion jjoi
and thermal diffusionjl'). ll'te formulas for thesemass flux contributions
are
v*-l
i:',: +
f ,,*,r,,lr,i (?9)
p,.t i=1
toru,T:;:r,
(18.4-8)
t\": # t,r,*,o,,1",*,(#,:)"r]
(r8.4-e)
L f;j
I
- 2,f t,)f
i:": - # 2,r,r,o,,12,r4,(t,
(r8.4-ro)
i:') : - Dirv l\ T
(18.4-11)
In theseequations G, and v, are the partial molal free enthalpy (Gibbs free
energy) and volume, respectively. The D, are multicomponent diffusion
coefficients,and the Drr are multicomponentthermal diffuiion coelicients.
The Dn, and Drr have the following pioperties:
,J,{ pu * pUu
.ri J p nu
Dii: o;
n
J, + l c , H , u
P roperti es
(18.+5)
n
2 r n' : o
(18.4-r2)
i:l
> {M iMhDih - MiM 1,D,,,}: o
(18.4-13)
i =7
e Eq. 18.4 -5to get
n
\NIJ
(18.4-6)
tartingpoint for engineeringstudies
s
transfer.
n mu l ti co m ponen t g a s m i x t u r e s h a s b e e n
td R. B. Bird, Molecular Theory of Gases
i . T he separat iono f q i n t o t h e c o m p on e n t
c on v entionadopt e d i n t h e d c f i n i t i o n o f k
lder, Chem. Eng. Proz. Syntposiunt Serics,
I6 ) .
L. Pigford, Absorption and Extraction,
n , p. 9 6, E q. 128-l / 2 . S e e a l s o E x a m p le
For n ) 2 the quantitiesD,, and D,, are not in generalequal.
The ordinary difusion contribution to the mass flux is seento depend in a
complicatedway on the concentrationgradientsof all the substances
present.
The pressuredffision term indicatesthat there may be a net movement of the
ith speciesin a mixture if thereis a pressuregradientimposedon the system;
the tendencyfor a mixture to separateunder a pressuregradientis very small,
but use is made of this effectin centrifugeseparationsin which tremendous
pressuregradientsmay be established.Theforceddifusion term is of pnmary
importancein ionic systems,in which the externalforce on an ion is equal to
the product of the ionic charge and the local electricfield strength; each
ionic speciesmay thus be undel the influenceof a differentforce. If gravity
i s the onl y externalforce,then al l the g, are the sameandT(f)va nishesident ically. The thermal difusion term describesthe tendencyfor speciesto diffuse
under the influenceof a temperaturegradient; this effectis quite small, but
5 68
T h e Eq u a tio n s o f Ch a n g e fo r M u lti component
S ystems
devicescan be arrangedto produce very steeptemperaturegradientsso that
separations
of mixtures are effected.4
To give the readera better feelingfor the physicalaspectsof thesegeneral
relations,we discusstwo important limiting cases.
Fluxes in Terms
of the Transport
Prope
and thermal diffusion in binary nonide
and 4.) Some sample values of k, fot
1 8 . 4 -r.
TABLE
E X PER IM EN TAL
TH ER M ALD IF
Bi n a rySy s t em s
AND LOW.DE
For a two-componentsystemof I and ,B the quantities Dro and DB,l
are e q u al,and E qs . 18. 4 -7th ro u g h 1 l b e c o me 5
r/ A o .r
I cz \
je:
-j n:'
- l- - li+ t , o 2 i v tn D o r.r.
" -t I V r,
l
lL
"
\ pRT l
L \1 r., M s t 7 .,
_t)orl' l _ D-1rv
- p!:G, - sa)
tn r
" * e
' M .t
(18.4_14)
P
Pt
T hi se q u at ionm ay be w ri tte n i n a n a l te rn a tefo rm b y usi ng (dGr)r,r:
RTdln a l and by dcfining6a "thermal diffusion ratio" hr:
(D
D.1B ) :
"rf
/^2,
f/:
r-
(pfc2M,aM) x
-l
Mro t,t,
M , r , lV,
l\ t E . t - E t,) + --_ " 1 :-._ ' _ _ l v p + AzV l n T | (18.4 15)
pRT
R T ' M.,
p'
I
This equationis the startingpoint for the study of ordinary,forced,pressure,
a Separations by thermal diffusion may be
considerably enhanced by combining the
eff ec t wit h f r e e c o n v e c t i o n i n a systcm sim ila r to th a t sh o wn in Fi g.9.9-1. A col umn
that makcs use of thesc two effects is callcd a Clusius-Dickel column. Such columns have
b een us c d f o r i s o t o p e s e p a r atio n s a n d fo r se p a r a tin g co m p le x mi xtures of very si mi l ar
organic c om p o u n d s . A v e r y r e a d a b le in tr o d u ctio n to th e r m a l d iffusi on i s the smal l book
by K. E. Grew and T. L. Ibbs, Thernnl DilJilsion in Gases, Cambridge University Prcss
(1 952). F or a p p l i c a t i o n s t o l iq u id s, se e A. L . Jo n e s a n d R. W. F oreman, Ind. E ng. C hcnt.,
44, 2249-22 5 3 ,( 1 9 5 2 ) , a n d A . L.Jo n e sa n d E.C.M ilb e r g cr ,
In d .Eryg.C hen.,45,2689-2696
(1953). The m a t h e m a t i c a l t h e o r y o f th e th e r m a l d iffu sio n co lu m n i s descri bedi n a cl assi c
article by R. C. Jones and W. H. Furry, Reu. Mod. Phys.,18, 151-224(1946).
5 Not e t h a t t h e q u a n t i t y i n th e sq u a r e b r a cke ts h a s
th e d im cnsi ons of force pcr uni t
mas s ; hence l c t r l s d e s i g n a t eitb y - F ,4 . Wh e n th e r e la tio n s in T abl e 16.1-3 arc used,one
can show that Eq. 18.4-14 (omitting the thermal diffusion tcrm) becomcs
D^"!l:F^
Components
A -B
T( "K)
C2H2Cln-z-CuHra
CrHnBrr-CrHoCl,
c2H2cl4-cc14
CBrn-CCln
ccl4-cH3oH
cH3oH-H2O
cyclo-CuHrr-CuH.
298
298
298
298
313
313
313
kr
no
0.5
1.0
0.5
0.22
0.5
0.06
0.09 0.12
0. 5
t . 23
0. 5 - 0. 13
0. 5
0.10
\
^
j . , : _jn : _ l, t ,l M . rMu D .,,,.
,
IL\ld::"ln ". r,. otI r . ,vr_
p
,
tt - u* :
^Rr
Liquids"
: ntnsFu
(18.4-14a)
w herernu, : D n l R T i s t h e "m o b ility." T h isr e la tio n wa su se d in E q. 16.5 l i nconnecri on
with the hydrodynamic theory of diffusion.
6 Other defincd quantities for binary systcms are the thernal
dilrttsionfactor z and the
Soret coellicient o
kr :
ttor o:
or AxBT
(l 8.4-l 5a)
Th equant ity a i s i L l m o s t i n d c p en d e n to fco n ce n tr a tio n fo r g a se s; t hequanti tyoi sgencral l y
us ed f or liq u i d s . N o t e t h a t Du ' :
- Du ' .
Be ca u se k" is d efi ned i n terms of D or,
wh en k , is p o s i t i v e , c o m p o ne n t,4 m o ve s to th e co ld e r r e g io n ; w hen k, i s negati ve,
component 14 moves to the warmer rcgion.
'Abstracted from R. L. Saxton. E. L. D
Phys.,22, ll66-1168 (195a); R. L. Saxto
L. J. Tichacek, W. S. Kmak, and H. G.
(1e56).
bAbstractedfrom tablesgivenby J. O
Bird, Molecular Theoryof GasesandLiqu
In consideringordinary diffusion onl
I
^2\
j e: _l | l uouu
\ p/
This should be compared with Eq. E ir
/.2\
i ", t : - l : - l U "\ p/
Comparisonof thesetwo equationssho
ideal solutions (i.e., activity proport
systemsone may use either Duu or !
above. Generally, experimentalists
r
since this requires no activity-concen
however, that Dup is less concentrat
phase. (SeeFig. 18.4-2.)
Ordi nary D if f usionin M ult icom pon
For an ideal gas mixture, Eq. 18.4^-n
jr:"_ lMrnt,nnpr
P i=r
C ha ng e for M ult i c o m p o n e n t
S y s te m s
steeptemperaturegradientsso that
:hephysicalaspectsof thesegeneral
ing cases.
rd B the quantities D,no and Do"
3come5
le #,),,
L \0r.,,
- 1)orl
-J p/
Vr',
D4rv \n T
(18.4-14)
ernateform by using (dG.)t,,,:
ffusionratio" kr:
(plczM.rM6) x
"" I v.r,
*A' T,P
,
t\
I
- -lvp * k7v In T |
o'
(18.415)
I
studyof ordinary,forced,pressure,
:onsiderably enhanced by combining the
t o t hat s hown in F i g . 9 . 9 - 1 . A c olu m n
lusius-Dickel coluntn. Such columns have
L r at i ngcom plc x m i x t u r e s o f v e r y s im ila r
o n t o th erm al dif f u s i o n i s t h e s m a l l b o o k
on in Gases, Cambridge Univcrsity Press
nes and R. W. Foreman, Ind. Eng. Chcnt.,
Milberger, Ind. Eng. Chenr.,45,2689 2696
diffusion column is dcscribed in a classic
o d . Phys. , 18, l5l-2 2 4 ( 1 9 4 6 ) .
)ts has the dimensions of force per unit
t h e r el a ti o nsin T ab l e l 6 . l - 3 a r e u s e d , o n e
I diffusion term) becomcs
--3 : iltnFt
Q8.4 14a)
a l ion wa s us c d in t g . 1 6 . 5 I i n c o n n ectio n
are the thernnl dilfusion factor a and the
or nzuT
(1 8.4 l5a)
a t io n fo r g as es ; t hc q u a n t i t y o i s g e n e r a lly
B eca usel, is det i n e d i n l e r m s o f Dr r ,
he colder region; when k" is ncgative,
Fluxes in Terms
of the Transport
Properties
569
and thermal diffusion in binary nonideal mixtures. (SeeExamples 18.5-2, 3,
and 4.) Some sample values of k, for gasesand liquids are given in Table
1 8.4-1.
TABLEI8.4_I
EXPERIMENTAL
THERMAL
DIFFUSION
RATIOS
FORLIQUIDS
AND LOW-DENSITY
GASES
Liquids"
Gasesb
I
Components
A -B
nn
T(K )
C2H2Cla-n-CuH1a
CrHrBrr-CrHnCl,
c2H2cl4-ccl4
CBrn-CClo
ccl4-cH3oH
cHsoH-H2O
cyclo-CoH12-CsHo
298
298
298
298
313
3l 3
313
Components
A - B
r(' K )
kr
0.5
1.08
0.5
0.225
0.5
0.060
0.09
0.t29
0.5
t.2f
0.5 -0.1 37
0.5
0.100
Ne-He
Nz-Hz
Dr-H,
kr
oa
330 0.20
0.60
2@ o.294
0.775
327 0. 10
0.50
0.90
0.0531
0.1004
0.0548
0.0663
0. 0145
0.0432
0.0166
u Abstractedfrom R. L. Saxton,E. L. Dougherty,and H. G. Drickamer,J.Chem.
Phys.,22,I166-1168(1954);R. L. Saxtonand H. G. Drickamer,ibid.,1287-1288;
L. J. Tichacek,W. S. Kmak, and H. G. Drickamer,J. Phys. Chem.,60,660-665
(1 956).
b Abstractedfrom tablesgiven by J. O. Hirschfelder,C. F. Curtiss,and R. B.
Bird, MolecularTheoryof GasesandLiquids,Wiley, New York (1954),$8.4.
In consideringordinary diffusiononly, we seethat Eq. 18.4-15simplifiesto
u
ie : - (!\
\ p/
\d ln x^/r.,
^,,t"r,'(+llo)
Y,n
(18.4-16)
This should be compared with Eq. E in Table 16.2-l:
i,t :
-
/ ^2\
\:- )tvt
(18.4-17)
eu eO aBV rA
Comparison of thesetwo equationsshows that DnuandgAB are identical for
ideal solutions (i.e., activity proportional to mole fraction). In nonideal
systemsone may use either Duu or 9uo and the corresponding equation
above. Generally, experimentalists report diffusion coemcients a.s 9aB,
since this requires no activity-concentrationdata. Available data indicate,
however, that Dnu is less concentration dependent than Ouo in the liquid
phase. (SeeFig. 18.42.)
Ordi nary D i ffusi oni n Mul ti componentGasesat Low D ensi ty
For an ideal gas mixture, Eq. l8.rt-8 becomes
jr :"
i*n*,r ,pr n
P i =L
i:1,2,"
,n
(18.4-18)
Y
T h e Eq u a tio n s o f Ch a n g e fo r Mul ti component
S ystems
For an n-componentideal-gasmixture the relation is known betweenthe
D,, (the diffusivity of the pair i-j in a multicomponent mixture) and the
9,, (the diffusivity of the pair i-j in a binary mixture).7 Because the Dn,
are concentrationdependent,Eq. 18.+18 is inconvenientto use. It has been
shown by Curtiss and Hirschfelder that Eqs. l8.rt-18 mav be ,.turned
wro ng- s ideout " t o o b ta i n
Fl uxes i n T erms of the Trans port p
point for the calculation of ord
mixtures. (SeeExample 18.5-5.)
For somecalculationsand for us
defineean efectiue binary dffisiuil
Recall that 9,ro was defined by
N't:
V ', - 2 : + ( u , - u , ) : i - t
and defne gr*by
rhese
equari.",
",..-i""T'
^,*"'|ri;t,.f;,':.:r:";^:^":.;", ,l
-c?tn\
this analogousrr
Ni:
- cQ
2. 5
By solving Eq. 18.4-21for Vr, and
Maxwell equations,we get immed
n
p Da B
2.0
I
--=cUr,,
trl
In general the 9r^ are dependentc
dependenceis slight, we may gene
mass-transfercoefficientcorrelation
some special kinds of diffusing s
9 r^ becomesparticularly simple:
1. 5
rtDtn
a. For trace components2, 3, . .
lnL
--0
u
1.0
9,,
Mole fractionether
Fig. 18 . # 2 . E f f e c to f a c tivity o n th e p r o d u ct o f visco sitya n d di ffusi vi tyforl i qui d mi xturesof
c hlorof o r m a n d e t h e r . [ F r o m R. E. Po we ll,W. L . Ro se ve a r ea, n d H enry E yri ng,tnd.E ng.chem.
3 3 ,4 30- 43s
( t e4t ) . 1
b. For systemsin which all theg
9,,
is the 9 n, tfrat appear here rather than the Do, and that the g
r, are virtually
independentof composition. (See Eq. 16.4-13.) This is the usual starting
? C. F . Cur t is s andJ .O.H i rs c h fe l d e r,J .C h e m .p h ys.,17,550-555(1949).
Forathree_
component system, the relations are of the form
o,,:
'-
c,,[t
' "1
+
xsl( M xtM ) 2 !3
- 2 n)\
r r ? zt * r z? r t * t"?rr|
(18.4-18a)
wit h s im i l a r r e l a t i o n s f o r Dr r , Dr ", D"", D,., a n d Dr r .
EThe analogous set of equations,
including pressure, thermal, and forced diffusion,
has also been derived by curtiss and Hirschfelder (toc. cit.); see also J. o. HirschfelJer.
c. F. curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, wiley, New york
(1954), E q . t t . 2 s 4 , p . 7 1 8.
c. For syst em sin which 2, 3, . . .
stationary),
l- x
%
In systemsin which the variationr
l i near vari at ion wit h com posit ion
eThe systematicuse of an effective
bina
by O. A. Hougen and K. M. Watson,Ci
York (1947),pp.977-979.Methodsofeva
by C. R. Wllke, Chem.Eng. Prog.,46,9
Note 3208(1954).
roH. W. Hsu and R. B. Bird, A.LC|.E.
i han ge fo r M ult ic o m p o n e n t
Systems
he relation is known between the
nulticomponentmixture) and the
inary mixture).7 Becausethe Dro
is inconvenientto use. It has been
t E qs . 1 8 .4 -18 m ay be " t ur n e d
F l uxes i n Terms of the Transport
. t c y/
t r,l g , - r ; N, )
( 18. 4 -1 9 )
ij
-Maru'ell equations.s Note that it
571
point for the calculation of ordinary diffusion in multicomponent gas
mixtures. (SeeExample 18.5-5.)
For somecalculationsand for use in somecorrelations,it is convenientto
defineean efectiue binary difusiuity 9,- for the diffusion of i in a mixture.
Recall that 9"u was defined by
Nt:
j.
:
.-,
P roperti es
and defne I,^
-c9,trrY*t
(18.4-20)
* tl(N,r * Nrr)
by this analogousrelation:
(r8.4-21)
Nt: -c7i,,Vr, * t, I N,
By solving Eq. 18.4-2lforVz, and equating;-i..r" It to Vr, in the StefanMaxwellequations,
we getimmediately
for collinearVt'
1
1
CQi,n
_
2r(11c9)(,Ii-,,N
rl
':/
n
N, -
r
(r8.4-22)
tL - - )
'N
In generalthe 9r^ are dependenton position. For situationsin which this
dependenceis slight, we may generalizethe binary diffusion formulas and
mass-transfercoemcientcorrelationsby simply replacing0,rtrby 9r*. For
some special kinds of diffusing systems,this formula (Eq. 18.4-22) for
I o- becomesparticularly simple:
a. For tracecomponents2, 3,. . .n i n nearl ypure speci esl ,
9rrr:
ether
; cosi tyanddif f us iv i t yf o r l i q u i d m i x t u r eso f
s eve are,
a nd Henry E y r i n g ,l n d . E n g .Ch e m .
: D, and that the I o, are virtually
t.4-13.) This is the usual starting
(1949).For a threePhys.,17,550-555
vr)gn - grrl\
rr9r" + q%l
(18.4-l8a)
D "r '
)ressure, thermal, and forced diffusion,
r (loc. cit.); see also J. O. HirschfelJer,
of Gases and Liquids, Wiley, New York
9i t
(18.4-23)
b. For systemsin which all the 9,, are the same
9' * :
9"
(18.4-24)
c. For systemsi n w hi ch 2,3,... r?move w i th the samevelocity(or are
stationary),
r,
I-" r:a
A rn,
tsz',/ r,
( 18.4-2s)
the assumptionof
ln systemsin which the variation of I i* is considerable,
has
proven
useful.r0 The
distance
linear variation with composition or
e The systcmatic use of an effective binary diffusivity seems first to have been suggested
by O. A. Hougen and K. M. Watson, Chentical Process Principles, Vol. III, Wiley, New
York(1947),pp.917-979. Methodsofeval uati ng9;-forspeci al caseshavebeendev el oped
b y C . R . Wrl ke, C hem. E ng. P rog.,46,95-104 (1950) and W. E . S tew a rt, N A C -A Tec h.
Note 3208 (1954.
10H. W. Hsu and R. B. Bird, A.I.C'h.E. Journal (in press).
s72
T h e Eq u a tio n s o f Ch a n g e for Mul ti component
S ystems
9 r- approach to solving multicomponent problems seemsto give pretty
good resultsfor calculatingmass-transfer
ratesbut a lesssatisfactoryquantitative descriptionof concentrationprofiles.
Substitutionof the expressions
for the fluxesgiven in this sectioninto the
equationsofchangeof $l 8.3producesthe generalpartial differentialequations
describingthe flow of a multicomponentfluid mixturewith heattransfer,mass
transfer,and chemicalreactionsoccurring. The word "general," of course,
always has to be used with some caution, for one can frequcntly think up
"more general" situations. In this case,the field of magnetohvdrodynantics
comes to mind; the equations describingmulticomponent fluid mixtures
with electromagneticeffects are the equations of change and Maxwell's
equationsof electromagnetictheory. This field is of interestin connection
with astrophysicalphenomena,ionized gas behavior,and plasmajcts.u,rz,rr
Another field not covered by our equations is that of relatit:isticfuid
mechanics; this subjcct includes the relativisticeffectsthat are important
when the fluid velocity is near the velocity of light.l4
U se of E quati ons of C hange
continuity and energyfor this sys
and 12 as
(continuityof l)
(energy)
Thereforcboth N,r, and e, arecon
$18. 5 US E O F EQU AT T ON SO F C H AN G E TO S E T U p
DI F F US IO NPR O B L EMS
All of the problems of Chapter 17, and more difficult ones as well, may
be set up directly by means of the differential equationsin this chapter.
As exampleswe considercombinedheatand masstransfer,thermal,pressure,
and forced diffusion,and three-componentordinary diffusion.
Exa mple
18. 5- 1. Sim ult aneous
o
o
l
6
Heat a n d M a s s T r a n s f e r
Develop expressionsfor the concentration profile r...r(z)and temperature profile
T(z) f or the systempictured in Fig. l 8.5-l , given the concentrationsand temperatures
at both film boundarics(z : 0, z : 6;. Here a hot condensablevapor, l, is diffusing
at steady state through a stagnant film of noncondensablegas, B, to a cold surface
at z :0 where I condenses. Assume that the gas behavior is ideal and that the
and the physical properties of the mixture are constant.l Neglect radiative
pressure_
heat transfer and thermal diffusion.
Solution. To dcterminethe desiredquantities,we must solve the equations of
11SeeT. G. Cowling, Magnetohydrodynamics,Inrerscience,
New York (1957).
P L. Spitzer,lr., Physicsof Fully lonizedGases,Interscicnce,
New York (1956).
r3B. T. Chu, Physicsof Fluids,2, 473-484(1959); in this paper the dcrivationsof the
equationsof changeare given for a pure, electricallyconductingffuid. In the cncrgy
equationtherc are termswhich accountfor the temperatureriseresultingfrom both viscous
(S" and S" in Chapter9).
dissipationand electricaldissipation
1{ L. D. Landau and E. M. Lifshitz, Myekhanika SploshnikhSred, Moscow (1954),
ChapterXV, pp. 606-616.
1 The simplesystemdescribedhereis often usedas a modelin psychromctriccalculations.
More generalmodelsare discussed
in $$21.5,
6, 7.
d-
Fig. 18.5-1. Condensation
of hot vap
d e n s a b lgea s8 .
To determine thc concentrationpr
through stagnant ,B:
N s" =
Insertion of Eq. 18.5-3 into Eq. I
profiles
( t- 'o
\r - ,r,
This resultwas obtainedin 917.2for r
flux Nr, is
"Az
-
C h a nge f or M ult ic o m p o n e n t
S y ste m s
)nt p robl e msse em slo giv e p re tty
ratesbut a lesssatisfactoryquantils.
fluxesgiven in this sectioninto the
general
partial differentialequations
uid mixturewith heattransfer,mass
g. The word "general," of course,
n, for one can frequently think up
the field of magnetoht,drodynantics
ng multicomponentffuid n.rixtures
uations of change and Maxwell's
risf ield i s o f i n ter es tin c onne c ti o n
a sbehavi o r,a n d plas m ajet s . rt,rz ,tr
Lationsis that of relatitistic /uid
ativisticeffectsthat arc important
v o f light.1 4
Use of Equations of Change
573
continuity and energy for this system. Thesc may be written from Eqs. 18. 3l0
an d 1 2 a s
(continuity
of .4)
: t
+
(energy)
( 18. 5- 1)
*dz :,
Therefore both Nr.
(l 8.5-2)
and c, are constant through
the film.
.-A
--n
T
=7 .
--0
Jr = rr^
/
7 T =T o
t o =, o t
B oundaryof
gas fi l m
.NGE T O SET UP
Directionof movement
A
e
of condensablvapor
rd more difficult ones as well, may
'erentialequations in this chapter.
nd masstransfer,thermal,pressure,
nt ordinarydiffusion.
o
o
l
!
a
Heat and Mass Transfer
t profilee',r(z)and ten.rperature
profile
entheconcentrations
andtemperatures
r hotcondensable
vapor,,4,is diffusing
Incondensable
gas,.8,to a cold surface
the gasbehavioris idcal and that the
ixtureareconstant.lNeglectradiativc
J
Fig. 18.5-1. Condensation
of hot vaporA on a cold surfacein the presence
of a nonconde n s a b lgea s8 .
To determine the concentrationprofilc, we need the mass flux for diffusion of I
through stagnantB:
ntities,we mustsolvcthc cquationsof
I n t ersci e nc e,New Y o r k ( 1 9 5 7 ) .
s , I n terscienc e,New Y o r k ( 1 9 5 6 ) .
9 59 ); i n t his pape r t h e d c r i v a t i o n s o f th e
) t ric al l y c onduc t in g f l u i d . l n t h e c n e r g y
mperature rise resulting from both viscous
i n Ch apter 9).
"
tanika Sploshnikh Sred, Moscow (1954),
I as a model in psychrometric calculations.
7.
j^,
v 'tt
:
-
C ?,1 6
| -
. |
d.r - 1
{ t d ' )-J )
,t,
Insertionof Eq. 18.5-3into Eq. 18.5-1and intcgrationgivesthe concentration
profiles
,l
-
l'
-l
\r -
r.
\
/,10./
/l
: l;
-
\t -
,'
\?/d
r,,_
r u,. /, !
r r 8 .5 ,4 )
This resultwas obtainedin $17.2for the isothermalproblem. As before,the constant
flux Nr, is
* n, : ' #r "*
( 18. 5- 5)
574
T h e Eq u a tio n s o f Ch a n g e f or Mul ti component
S ystems
Note that N.1, is negativein Fig. 18.5 1 becausez1 is condensing. The last two
exprcssionsmay be combined to put the concentrationprofilesin an alternateform:
t l' . 1rjrd-
t 1.
1o,
ex pl( Nt r lc o t t ) z l
_1:r_{o 1 - exp [(.N,rrlclu)6]
(18.56)
This mass flux will tend to establis
result in an opposed flux:
jy) : -:
I
When steady state is reached,there !
To determine lhc tctnperatureproJilc, we use the energy flux from Eq. 18.4-6
for an ideal gas:
-k
Use of Equations of Change
_l
o:j'n+jtl):
dT
,
I
+ lHjN4 ,+ HBNRz\
Thi s bul b mai ntai ned
ut
-k _ * N _ ,rc ,-r{
T-
To)
(18.5-7)
Here rve havc chosen 7o as the rcfcrence temperature for the enthalpy. Insertion
of this exprcssionfor c, into Ec1.18.5 2 and integration betweenthe limits T : To
at : : 0 and I : T6 at z : r) gives
T - 7'o_1c ,x p l (N k c t ,,tl k)zl
Ta- To l - expl(N/,C),''lk),Jl
(r 8.5-8)
It can be seenthat the temperatureprofile is not linear for this systcmcxccpt in the
linrit as N,1,C,,,a lk 0. Not c t hc s inr ilar it ybet weenE q s . 1 8 . 5 6 a n d 8 .
The conduction energyflux at the wall is greaterhere than in the absenceof mass
transfer. Thus, using a superscriptzero to indicate conditions in the absenceof
mass transfer, wc may write
- k(dTldz), o
- kktwtt-,
(N.l,e
D.tlk),5
( 18.5-9)
We seethcn that the rate of heat transfer is directly affectedby simultaneousmass
transfer, whcreas (if *'e neglect thermal diffusion) the mass flux is not directly
affcctcdby simultaneousheat transfer. In many applications,for example in most
psy ch rome tricpro bl em s , N. 1, ( - - , , 11k
is s m all. and t h e r i g h t s i d e o f E q . 1 8 . 5 9 i s
very nearly unity. (Scc Problem 18.A.) The rclationshipsbctweenheat and mass
transfer arc further discussedin Chaptcr 21.
Fig. 18.5-2, Steady-state
binarytherma
gasesA and B tendsto separate
underth
When kr is positive, component I n
I moves to the hot region. We may
dr'
E
In general, the degree of separation
is small. We may therefore ignore t
E q . 1 8 . 5 - 13to o b ta i n
xs2- fia
Becausethe dependenceof k7 on 7is
k? constant at some mean temperat
approximately
r A2 - n
The recommendedzmean temperatur
Ex am ple 18. 5- 2. Ther m al
Di f f u s i o n
Considcr two bulbs joined by an insulatcdtube of small diametcr and filled with
a mixture of ideal gasesI and B as shown in Fig. 18.5 2. The bulbs are maintained
at constanttemperaturesof f, and ?"r,rcspectively,
and the diameterof thc insulated
tubc is snrall enough to eliminate convection currents substantially. Develop an
expressionfor the steady-statediffercncein composition of the two bulbs.
So lutio n. Acco rding t o Eq. 18. 4 15, t he t em per at u r eg r a d i e n ti n t h e s y s t c mw i l l
causea mass flux given by
(18.5-r0)
Tm -Equation 18.5-15 is useful for est
diffusion effects.
Exa m p l e 1 8 .5
A binary liquid solution is mount
centrifuge, as shown in Fig. 18.5-3.
2H. Brown, Phys.Rer:.,58, 661-662(l
Changefor Multicomponent Systems
Use of Equations of Change
)causeI is condensing. The last two
:entrationprofiles in an alternate form:
This mass flux will tend to establisha concentrationgradient, which in turn will
result in an opposedflux:
s'
c2
)l (NA , l c a])zl
t[(N,a,lct111)d)
(18'5-6)
575
i9 : - 7 *or o"onT
( 18.5- 1r )
When steady state is reached, there will be no net mass flux and we may write
use the energy ffux from Eq. 18.4-6
o : i',t)
+j'I) : -! u ,rr" ,r\* .+n
u85 12)
,{A, + frBNR,)
Thi s bul b mai ntarned
:rA(T-
To)
(r8.5-7)
nperaturefor the enthalpy. lnsertion
integrationbetweenthe limits T : To
(r8.5-8)
not line;rr for this system except ir.rthe
be t w e e nE qs. 18.5-6 an d 8 .
reaterherethan in the absenceof mass
indicateconditions in the absenceof
(NA , et , Al k)i
tf-ry,,,f'",/t),ll
"p
( r 8.s9)
Jirectlyaffectedby simultaneousmass
ffusion)the mass flux is not directly
any applications,for exanrplcin most
, a n d t he right sid c of Eq. 18 .5- 9 is
: relationshipsbct'nveenheat and mass
Fig. 18.5-2. Steady-state
binarythermaldiffusionin a two-bulbapparatus.The mixtureof
gasesA and I tendsto separate
underthe influenceof the thermalgradient.
When k, is positive,component I moves to the cold region; when it is negative,
I moves to the hot region. We may write Eq. 18.5 12 as
dz,
kr dr
( I 8 .5 * 1 3 )
' t":- TE
In general, the degree of separation in an apparatus of the kind being considered
is small. We may therefore ignore the effect of composition on k7 and integrate
Eq . 1 8 . 5 1 3 t o o b t a i n
fr, k, ,_
xA2-xAt:-|,".
( r 8.5-1
4)
io,
!
r7
Becausethe dependenceof k7 on 7 is rather complicated, it is customary to assume
k" constant at some mean temperature?",n.Integration of Eq. 18.5-14then gives
approximately
rAz- rAt: - kThT
( l 8. 5-15)
The recommended2mean temperature ?n- is
rma l D iffusion
tube of small diameter and llllcd with
Fig. 1 8 . 5 2. T he b ulb s a re ma inta ined
ively,and the diamctcr of the insulated
n currentssubstantially. Dcvelop an
compositionof the two bulbs.
emperaturegradient in the systcmwill
k rd r
.
R - , 1R T E
( 18. 5 -1 0 )
T,^:
TrT,
,:jrln
.
T2
7,
( 18. 5l6)
Equation 18.5-15 is useful for estimating the order of magnitude of thermal
diffusion effects.
Example 18.5-3. Pressure Diffusion
A binary liquid solution is mounted in a cylindrical cell in a very high-speed
centrifuge, as shown in Fig. 18.5 3. The length of the cell L may be considercd
2H. Brown, Phys.Reu.,58,661-662(1940).
576
The Equations of Change for Multicomponent Systems
short with respect to the radius of rotation Ro, and the solution density may be
considereda function of composition only. Detcrmine the distribution of the two
components at stcady state in terms of their partial molal volumes,position in the
cell, and the pressuregradient dpldz : - pgt> -pO2R0 in which O is the angular
velocity of the centrifuge. Neglect changes in the partial molal volumes with
composition and assumethat the activity coefficientsare constantover the range of
compositionsexisting in the cell.
of Change
Exam ple 18
A quiescentaqueoussolutionof
plates of metallic M, as shownin F
between the plates under such co
dissolutionof the anodeand deDos
tration profileof MX in the solut
Cathodeof
metal l i cM
(M- + e--M)
\\"0/
\\/
\
\
Use of Equations
\/\,,
ll)
I
I
F
I
tl
rl
Mixtureof A and B
in d iffusi oncel l
Fig. 18.5-3. Steady-state
pressure
diffusion
in a centrifuge.The mixtureofA and8 tendsto
separateby virtue of the pressuregradientproducedin the centrifuge.
Solution.
18 .4 -15a s
At steady state the net mass flux j4, is zero, and we may write Eq.
d
_
,
t-
rrr4J
-
-
-(#)
di stri buti on
|
L
0
( r 8.5-l7)
- /"M)H)
(#) "(#)'-': "*o(rzo,'rn
1 8 . 5 - 1 8 d e s c r i b e s th e ste a d y- sta te co n ce n tr a tion
,n/
Fig. 18.5-{. {
(M,a-pv,1)
This expression,:rwhich may also be obtained by thermodynamic arguments,
can be integrated to give
E q.
xu*
I
trsr-,sl
for
a bi nary
system in a constant centrifugal field.
3 S e c P r o b l e m l 8 . I fo r th e d e ta ils o f th is in te g r a tio n . A devel opment of E q. 18.5-17
by thermodynamic arguments is given by E. A. Guggenheim, Thermodynanics, North
Hol l a n d P u b l i s h i n g C o ., Am ste r d a m ( 1 9 5 0 ) , p p . 3 5 6 - 3 60. N ote, how ever, that Guggenhei m ' s f i n a l r e s u l t i s i n co r r e ct b e ca u seo f a n a lg e b r a ice r ror i n the thi rd l i ne ofE q.l 1.10-3.
Assume the solutions to be quite r
physical properties.
Solution. We may considerthe
of water and the two ionic speciesl
reactions that Ny., the magnitude,
current density in the cell and that
further seen that the local molar c
equal by the requirement of elect
dilute, we may consider that the flu
We may immediately write
r:N,r
0:Nr
Here I is the current density in the r
Changefor Multicomponent Systems
r Ro, and the solution density may be
Determinethe distribution of the two
partial molal volumes, position in the
) -- - pO2R0in which () is the angular
:s in the partial molal volumes with
:fficientsare constant over the range of
Use of Equations
sTl
of Change
Example 18.5-{. Forced Difrusion
A quiescentaqueoussolution of a salt MX is situated betweentwo flat parallel
platesof metallicM, as shownin Fig. 18.5-4. A constantdirect currentis passed
between the plates under such conditions that the only electrode reactions are
dissolutionof the anodeand depositionof M on the cathode. Estimatethe concentration profile o[ MX in the solution and the madmum possiblecurrent density.
I
I Motion
I
Cathodeof
metal l i cM
Anodeof
metal l i cM
(MM'+
(Mr +e--M)
r+
tz l
oIe
e-)
|
ll
ll
!<-
r,
+l
Mixtureof A and B
e n t r i fuge. The m ix t u r e o f A a n d I t e nd s to
r ce di n the c ent rif u g e .
l ux T_ a,i s z ero, an d w e m a y w r i t e Eq .
ozL
Fig. 18.5'{. Concentration
polarization.
(Mt
-
( 18. 5t7 )
p/,)
rined by thermodyna4ric arguments,
' oM o -
9n;
\
V "M .'
' '1RTI
";:=l
"
(18 .5 l8)
centration distribution for a binary
:gration.A development
of Eq. 18.5-17
\. Guggenheim,Thermodlnoniics,North
35G360. Note, however,that Guggenbraicerrorin the third line of Eq.1I . l0 3.
Assume the solutions to be quite dilute and ignore changes of temperature and
physical properties.
Solution. We may consider the solution in this apparatus as a ternary mixture
of water and the two ionic speciesM+ and X-. lt can be seen from the electrode
reactions that Nly*, the magnitude of the molar flux of M+ , is proportional to the
current density in the cell and that the fluxes of X- and water are zero. It can be
further seen that the local molar concentrations of M+ and X- must always be
eqrtal by the requirement of electrical neutrality. Finally, since the solution is
dilute, we may consider that the fluxes N,. and Jr* are equal.
We mav immediatelv write
r : Ny+
t!|ft
=
Ji
bt
0:N x-
+ J#l\
(r8.5-r
e)
(r8.5-20)
* rl l or
as equivalentsper unit area
Here I is the currentdensityin the solution,expressed
s76
T h e Eq u a tio n s o f Ch a n g e for Mul ti component
S ystems
short with respect to the radius of rotation Ro, and the solution density may be
considereda function of composition only. Determine the distribution of the two
components at steadystate in terms of their partial molal volumes, position in the
-pO2R0 in which O is the angular
cell, and the pressuregradient dpldz : - pgg
velocity of the centrifuge. Neglect changes in the partial molal volumes with
composition and assumethat the activity coefficientsare constantover the range of
compositionsexisting in the cell.
fto
r\/
.,
\/t-
L
of Change
Exam ple I
A quiescentaqueoussolutionol
platesof metallicM, as shownin
betweenthe plates under suchc(
dissolutionof the anodeanddepo
tration profile of MX in the solu
\
(M-
V',
If---12=
Use of Equations
Cathodeof
metal l i cM
+ e--M)
i-r'
I
F
I
,il
lVixtureof A and B
ll
rl
Fig. 18.5-3. Steady-state
pressure
diffusion
in a centrifuge.The mixtureofA andI tendsto
separateby virtue of the pressure
gradientproducedin the centrifuge.
Solution.
18.4-1 5as
-l
L
0
At steady state the net mass flux7.1" is zero, and we may write Eq.
d
; ln. r r :
(#) (M,t- pv.t)
(r8.5r7)
This e4pression,3 which may also be obtained by thermodynamic arguments,
can be integrated to give
(#)"(#)": .*o(,r,,ro- z"M)H)
Eq.
ru+ fI
xn 7
1 8 . 5 - 1 8 d e s c rib e s th e ste a d y- sta te co n ce n tr a ti on
di stri buti on
t,u,-',;
for
a bi nary
system in a constant centrifugal field.
3 S e e P r o b l e m l 8 .I fo r th e d e ta ils o fth is in te g r a tio n. A devel opment ofE q. 18.5 17
by thermodynamic arguments is given by E. A. Guggenheim, Thermodynanics, North
Ho l l a n d P u b l i s h i n g Co ., Am ste r d a m ( 1 9 5 0 ) , p p .3 5 6 - 3 60. N ote, how ever, that Guggenhe i m ' s f i n a l r e s u l t i s i n co r r e ctb e ca u se o f a n a lg e b r a ice r rori nthethi rdl i neofE q.l l .l 0
3.
Fig. 18.5-{.
Assume the solutions to be quite
physical properties.
Solution. We may consider the
of water and the two ionic species
reactions that N.y., the magnitude
current density in the cell and that
further seen that the local molar (
equal by the requirement of elect
dilute, we may consider that the flu
We may immediately write
1 :N l
0 :N :
Here f is the current density in the
' Cha n g efor Multico mpo ne nt Sys t em s
)n R0, and the solution density may be
. Determinethe distribution of the two
ir partial molal volumes, position in the
To : - pO2Roin which O is thc angular
ges in the partial molal volumes with
remcientsare constant over the range of
Use of Equations
s77
of Change
Example 18.5-rl.Forced Difrusion
A quiescentaqueoussolutionof a salt MX is situatedbetweentwo flat parallel
platesof metallicM, as shownin Fig. 18.5-4. A constantdirect current is passed
betweenthe plates under such conditions that the only electrode reactions are
dissolutionof the anodeand depositionof M on the cathode. Estimatethe concentration profile of MX in the solution and the maximumpossiblecurrent density.
Cathodeof
metal l i cM
(M- + e--M)
I c entri fu ge.The m i x t u r e o f A a n d 8 t e n d s to
d u c ed i n t he c ent ri f u g e .
flux/_'z is zero, and we may write Eq.
ozL
Fig. 18.5-1. Concentrationpolarization.
iltu.,-
pl.,l
o8.5 17)
rtained by thermodynirmic arguments,
ileMn - v"M^)Yr)
( I 8 .5 -t8 )
)ncentration distribution for a binary
r t eg ra ti o n. A dev e l o p m e n t o f E q . 18 .5 - 1 7
A. Guggcnheim, Thermodynanrcs, North
rp. 35G360. Note, however, that Guggeng e bra i cerror in t he t h i r d l i n e o f E q . l l.l0 3 .
Assume the solutions to be quite dilute and ignore changes of temperature and
physical properties.
Solution. We may consider the solution in this apparatus as a ternary mixture
of water and the two ionic species M+ ar,d X-. lt can be seen from the electrode
reactions that Nr11',the magnitude of the molar flux of M+, is proportional to the
current density in the cell and that the fluxes of X- and water are zero. It can be
further seen that the local molar concentrations of M+ and X- must always be
equal by the requirement of electrical neutrality. Finally, since the solution is
dilute, we may consider that the fluxes N; and Jr* are equal.
We mav immediatelv write
1 : N , 1 1 +=J i ; i t +J i ; t l
0:Nx-
'Jtr?t *tfit:t
(18.s-19)
(l 8.5-20)
Here I is the currentdensityin the solution,expressed
as equivalentsper unit area
fr-'
578
T h e Eq u a tio n s o f Ch a n g e for Mul ti component
S ystems
per unit time. To expressEqs. 18.5-19 and 20 in terms of the transport coefficients,
we must make some further assumptions. We assume that the M+ and X- ions
are in such small concentration as to have no appreciable effect on one another.
Then we considereach ion as diffusing in a binary systemwith water as the second
component. We further assume the activity coefficients of the ions to be unity.
Then dGi : RT dlne, and for the ionsErz : Gilm)( -d6ldz). Here { is the local
electrostaticpotential, er is the ionic charge, and mi is the ionic mass. We may then
write Eq. 18.4-15 for ionic speciesI diffusing through water (w) as
Ji"*: -rr*(9*.##)
Use of Equations of Change
At the catalyst surface I is removed r
N t':
ko "
Here ko" and /r are constants assumed
reaction rate in terms of the gas comp
ture and pressure and ideal-gas behav
(18.5-2
l)
By substituting
Eq. 18.5-21into Eqs. 18.5-19and 20,we obtain
ru:_ce1**(ry
.+#)
o:_csx_w(+-+#)
(r8.5-22)
( 18.5-23)
We now take advantage of the fact that the mole fractions of the two ions are the
same to eliminate the potential gradient between these two equations:
[r:
dr n",
-2 c ? u * r-i -
(18.5-24)
rAd =
Since for dilute isothermal solutions the quantity cQM*, is nearly constant, we may
integrateEq. 18.5-24to obtain
rM+-ro__
I"
z
2 c9 tr 1+,
(r8.5-2s)
Here ro is the mole fraction of M+ at the cathode.
We find then that the concentration gradient in the cell is linear. The maximum
current density is reached when the salt concentration at the cathode is zero;
that is,
,
'max
4 C? y+ *x^ u .
L
Fi g. 18 .5-5. D i ffus i on of A through
Solution. From the stoichiomet
equati
- nN,t^". The Stefan-Maxwell
**:.,"('-,
(18.5-26)
Equations 18.5-25 and26 provide a good qualitative description of "concentration
polarization," even though their quantitative application is limited to quite dilute
systems. For example,Eq. 18.5-26shows that in an electroplatingbath there is a
limit to the rate at which metal may be deposited. Similarly, in a battery the
diffusion ol electrolyte limits the rate at which current can be drawn.
I
dto
Y l ,,n
aL
---;
:("-;)
Here yAn, : N pblc9,r^o, rn : 9,t
These equations may be integrated wj
and ro : rBo 4t ( : 0. We first integ
r B:
Exa mple 18. 5- 5.Thr ee- Com ponent O r d i n a r y D i f f u s i o n w i t h
Heterogeneous Chemical Reactiona
A gas I is diffusing at steady state in the z-direction through a film of stagnant
B to a catalytic surface (see Fig. 18.5-5), where an irreversible reaction of the
following form takes place:
n A- An
4 H . W . H s u a n d R. B, Bir d , A.I.C| .E. Jo u r n a l6 ,5 1 6-524 (1960).
rn(
We now subs
Here R -- ro - lln).
integrate the resulting first-order linear
ra : (ras-
N r + M-1.us6)exp (N
(r8.s-27)
HereN:(1
-rr)and
M :1 - n
Y
C h an ge f or Mul t i c o m p o n e n t
S yste m s
20 in terms of the transport coemcients,
We assume that the M+ and, X- ions
I no appreciable effect on one another.
binary system with water as the second
ty coefficientsof the ions to be unity.
: (enlm)(-dgldz). Here / is the local
and m; is the ionic mass. We may then
g through water (w) as
aC1
1,
'z * "t.t
( l 8 .5 -2 1 )
19 and 20, we obtain
z,y-le,l d{\
-- ' ,
(r8.s-22)
xT dzf
E)
"r
_ '"-1.'i4\
Use of Equations of Change
At the catalyst surface I
579
is removed at a rate that may be expressed as
NA:
ktr"*rh
atz
( l 8. 5- 28)
:6
Here kn" and I are constants assumedto be known. Develop an expression for the
reaction rate in terms of the gas composition at z : 0. Assume constant temperature and pressure and ideal-gas behavior.
--l
( I 8.5-23)
KT d rl
mole fractionsof the tw, ions are the
weenthesetwo equationsl
dty+
'r-A;
(r8.s-24)
'a nn
K
'rtity cQya, is nearly constant, we rnay
-;;-
I,
LLZM+w
(l 8.5-25)
thode.
:nt in the cell is linear. The maximum
:oncentration at the cathode is zero:
W+wravg
L
Fi g. 18.5-5. D i ffusi on ofA through stagnant8 to form A " at a catal ytics url ac e.
of the problem,it is requiredthat Nr, :
Solution. From the stoichiometry
for I and -Bare
equations
- nNt_r. The Stefan-Maxwell
I dr,
ll
"',l "u ;:t-r"
"s
( I 8.5-26)
ralitativedescription of "concentration
e applicationis limited to quite dilute
rat in an electroplating bath there is a
teposited. Similarly, in a battery the
ch current can be drawn.
I
\J'
/
\
,./
dru
: /
,t(
\' o-
1\
il''o
z-direction through a film of stagnant
where an irreversible reaction of the
n
zl 6,51G524 (1960).
* (.ra,-
rs)rq- ra*
( l 8. 5- 2e)
( 18. 530)
2,t,nl 9,1t", and ( : zlr\.
Here a^^n : N,ar6lc?t^o, rB : 9_0,,n19s3, r,t,:
These equations may'be integrated with the boundary conditions that .r1 :.r
',,
We first integrateEq. 18.5 30 to obtain
and nr : rlo at ( :0.
rB :
nt O r d i nary D iffu sio n with
nical Reactiona
l\
-;l * t
rBoexp (Rrl.,iro
( 1 8 .5 3 1 )
Here R : rnOln). We now substitute this expressioninto Eq. 18.5 29 and
integrate the resulting flrst-order linear differential equation to get
r7 : (zAo -
N-1 + M-rrp1,) exp (Nr,r,r_t,,10
+ N 1-
( l 8.5-27)
Her e N : ( l - n - 1 )
and M:l-
nll(l-
M - 'r 3 r , e x p ( R r . r , r 0
rs,,)l?r-
( 1 8 .5 - 3 2 )
r 1 , , ) ) . F i n al l y, w e u se
580
The Equations of Change for Multicomponent Systems
Eqs. 18.5 28 and 32 to obtain the desiredrelations for the dimensionlessreaction
rate v 4^r] as the following transcendentalequation:
t"
^JaF:(r.ro-
N 1 + M lrps)exp(Nr,a,u.rp)
+ N-r - M-1r30exp(Rr,_r,.n)(19.5_33)
in whichK : kn"6lc?s,r. That is, Eq. 18.533 givesl,,r.r as an implicitfunction
of K, n, r3, rA^,r-{0,artd;rrr,,.
A N A L YS | SO F T H E E QU A TTON S
OF C H A N GE
$t 8. 6 DT M E N ST ON AL
F O R A B IN A R Y ISOT H ER M ALF L U ID MIX TU R E
Thus far in this chaptcrwe havediscussed
the equationsof changcfor a fluid
m ix t ur e, and in $ 1 8 .5w e i l l u s tra te dh o w to s e t up and sol vesomc probl ems
i n dif f us ion. Ne x t w e c o n s i d e rth e d i m e n s i o n alanal ysi sof the equati onsfor
an isothermal binary fluid mixture of constant viscosity p and constant
diffusivity!) ,tt. ln addition, we assumethe rangeof compositionto be small
e nought hat bot h ma s sd e n s i typ a n d m o l a r d e nsi tyc are cssenti al lconstant.
y
With theserestrictions,we write the equationsof changeas
(v'u ) : o
(continuity)
Dxj
(continuity of ,4)
DI
(r8 . 6 -1 )
: !1 ,,,v z r ,
/l R 6-)l
fo rcetl
(motion
p : uv'u*{_oJr**,,0t_
I
ff
rn)
(18.6-3)
(18.6-4)
d i m e n s i o n l c svse l o c i ty
Y
p*
' :
P 1.
: d i rn e n s i o n l c spsr essure
( r 8.6-5)
.P ;'
pv"
Vt
t * - i --: : d i me n s i o n l e stisme
D
r'r* :
(18.6-6)
dimensiolrless
concentration
(1s.6-'7)
tffrr:
5 F o r t h c s p e c i a l ca se s, n :1 ,
Bird, op. cit.
r 1 n :r r ,
a n d ( n r 3 -1):ru(n
A nal y s i s of the E quat
in which V, D, and rAr - r'Ao ar
linear dim ension,ar
characte r ist ic
in the system. The equationsof ,
form by multiplying Eq. 18.6-l b
E q. 18.6 - 3by DlpVz:
(v
(continuity)
(continuity of ,4)
DrA*
-
l ), see H su ancl
.
Dt*
(motion)
The dimensionlessgroups appea
the Reynolds number, Re : D
PlPQan : Y19tn' For isotherm
a role analogousto that of the Pr
In free conrection we again pr
S10.6. We define a reducedvelo
Then, by multiplying Eq. 18.6-l b
and Eq. 18.6-3by D3pf pP,we ge
(v
(continuity)
J ree
i n whic h r r o is s o m ere fe re n c e
c o n c e n tra ti o n .In E q. 18.6 3 w e have w ri tten
the equationof motion to includeeither the forced-or free-conrcctionterms.
we first considerforced conrectiotlfor which it is convenientto define
th e r educ edqua n ti ti e smu c h a s i n Q1 0 .6 :
o* : !.:
t/
D i mensi onal
2g
(continuityof .4)
(moti on)
Dl**
Du**
Dt**
i n w hi ch G r ut :
pz( gD3( r ut transfer. We see that Grrn diff
(@n, - rro), whereasGr contain
In C hapt er2l use is m adeof t l
c
the correlation of mass-transfer
ment exactly parallelsthat for he
the dissipationfunction, the corre
the same. We may thereforeexp
heat and masstransfer. Howeve
gies cannot be fully understoodw
onp,a, Z, and ro. Thestudentwi
I Changefor Multicomponent Systems
Dimensi onal
relationsfor the dimensionlessreaction
rquation:
in which V, D, and nAr - rAo are, respectively,a characteristic velocity, a
characteristiclinear dimension, and a characteristicconcentration difference
in the system. The equationsof changemay now be put into dimensionless
form by multiplying Eq. 18.6-l by DlV, Eq. 18.6-2by DIV(rnt- rno),and
E q . 18.6-3by D l pV z:
Nr7 1,p)
^u
N-t - M-lrrts exp (Rr,_1,7;)(18.5,33)
.5 33 g i vesr_r"r :ts a n implicit func t ion
TH E E Q UA T I ONS OF CHANG E
- F L U I D MIX T URE
sedthe equations ofchange for a fluid
i l to s e t up and solve some p rob lem s
:ns i o n a lanalysis of th c eq ua tions f or
constant viscosity p, and constant
the r a n ge of com p osition to b c s m all
rl a r de n sity c are cssen tially co ns t ant .
l at i o n s of chang e a s
:0
(r8.6-t)
,ttvzr,r
( 18.62)
-t pg
forced
( 18.6*3)
( 18.64)
velocity
nlesspressure
( 18.6s)
s time
( r 8.6-6)
rs i o nlessco n c ent r at ion ( 18 .6 -7 )
n d ( n ro-
l):
rn ^ Q t -
l), see Hsu and
D rA *
(continuity of ,4)
(motion)
( 18. 6- 8)
(v* .0* ):0
(continuity)
I
:
Dt*
Du*
Dt*
g* rr" *
(18.6-e)
Re Sc
1 g * t r*
v x^p * *
Re
+Lg
Ftg
(18.6-10)
The dimensionlessgroups appearingare the Froude number, Fr : V2lgD,
the Reynolds number, P.e: DVplp, and the Schmidt number, Sc:
plp9,cn : vl9en. For isothermal mass transfer, the Schmidt number plays
a role analogousto that of the Prandtl number in heat transfer.
In free conuectionwe again proceed analogously to the development in
$10.6. We define a reducedvelocity and time as in Eqs. 10.6-14 and 15.
Then, by multiplying Eq. 18.6 | by D2pl1t,Eq. 18.6-2by pD2lp@^ - r,to),
and Eq. 18.6-3by Dtplt"', we get the equationsof changein the form
(continuity)
(e_r-
Jree
":n)
rti on. I n Eq . 1 8. 6- 3wc hav e w ri tte n
t he f orce d -o r free- c onv ec t ion
te rms .
b r w hich i t i s c onv enientt o d e fi n e
581
A nal ysi s of the E quati ons of C hange
(continuity of l)
(motion)
(18.6-1
1)
(V * ' u* * ) : Q
D r,t*
Dt**
I g* zr" *
:
(18.6-12)
Sc
D'** :v*2u**
Dt**
6
-
xA* Gt,ru9
(18.6-13)
C'
o
in which Grtn:
p2(gD\(xnr- r',)l 1"" is the Grashof number for mass
transfer. We see that G1173differs from Gr only in that Gry6 contains
t@n - oro), whereasGr containsfr(T, - Tl.
In Chapter 2l use is made of the groups Sc and Grr' in connectionwith
the correlation of mass-transfercoefficients.Note that the foregoing treatment exactlyparallelsthat for heat transferand that, with the exceptionof
the dissipationfunction, the correspondingequationsin the two sectionsare
the same. We may thereforeexpect to find many close analogiesbetween
heat and masstransfer. However,the extent and limitations of theseanalogiescannot be fully understoodwithout examiningthe boundary conditions
on p, a, T, and rn The studentwill find it profitable to look for theseanalogies
582
T h e Eq u a tio n s o f Ch a n g e for Mul ti component
S ystems
-and
also for differences between these two transfer Drocesses-in
remaining chapters ol the text.
the
P robl ems
use, thi s si tu ati on oc c urs at R e grea
R eynol ds nu mber has a pronounc ed e
by a number of different investigators,
Ex am ple
18. 6- 1.Blending of M i s c i b l e F l u i d s
Develop by the methods of dimensionalanalysisthe generalform of a correlation
for the tin.rerequired to blend two misciblefluids in an agitatedtank. Assume that
the two fluids and their mixtures have essentiallythe same physicalproperties.
Solution, It will be assumedhere that the achievementof "equal degreesof
blending" in any two operations means the achievementof the same reduced
concentration pattern in cach; that is, for any component I the concentration
profile
t , r : r , qt * , 0, 2* )
is the same. Here r* and z* are reduced position variables,rlD and z/D, respectively, in which D is the diameter of the impeller. lt will also be convenientto define
reduced time, velocity, and pressure,as /N, alND, and plpN2Dz, respectively,in
which N is the rate of rotation of the impeller, revolutions per unit time, and
p is the fluid density.
Then from Eqs. 18.6-9 and 10 it can be seenthat r,a(r*,0, z*) dependsupon /*,
Rc, Sc, Fr, geonretry,and initial and boundary conditions. Here Re : D2Nplt,
Sc: plp9,rij, and Fr : pU2k. Important among the initial conditions are the
relative amounts of the two fluids, the manner in which thcy are introduced to the
tank, and the flow pattcrns at the start oi mixing.
Frequently the number of independent dimensionlessgroups can be further
reduced. For example, it has been experimentallyobserved that if the tank is
baflledt no vortices of importance occur; that is, the fluid surface is effectively
level. Undcr thesecircumstances,or in the absenceof a free fluid surface,gravitational forces, hence the Froude number, are unimportant. At a high Reynolds
number the effcctsof both Re and Sc can also be ncglccted2so that .r.n(r*,0, z*)
dependsonly upon l*: that is, the reducedmixing time is constant for any desired
degree of blending. In other words, the number of revolutions of the impeller
requiredto producc a given degreeof mixing is fixed. For most impellersin common
r A commonand effective
baffiingarrangement
for verticalcylindricaltankswith axially
mountedimpellcrsis to mount four o'enly spaccdstrips,with their flat surfaces
in planes
through.thetank axis,along thc tank wall, cxtendingfrorn the top to the bottom of the
tank and irt leasttwo tenthsof the distanccto the tirnk center.
! This indcpcndence
of mixingtime upon Reynoldsnumberscan be seenintuitir.clyfrom
smallin relationto the accelerathe fact that thc tcrr.n(V*!u+)rRein Eq. 18.6 l0 becomes
howevcr,and the cfTect
tion termsat high Re. Suchintuitivcargumentsare dangerous,
of solid surfaces.Here the
of Re is alwaysinrportantin the immediateneighborhood
irmountof mixing taking placc in thc immcdiatcncighborhoodof solid surfacesis small
of the effectof Reynoldsnumber,
and can bc ncglected.For a more completediscussion
seeH. Schlichting,BoundarvLayer Theory,McGrarv-Hill,New York (1955),Chapter4.
The indcpcndence
of n.rixingtimc on Schmidtnumberscan bcstbe sccnfrom the timeaveraged
equationof continuityin Chapter20. At high Reynoldsnumbersthe turbulent
massflux is much grcaterthan that due to moleculardiffusionexceptin the immediate
neighborhood
of the solidsurfaces.
E U E S TION S
l . D i scuss the appl i c abi l i ty of the equa
l 0 to the desc ri pti on of s y s temsof more t
2. A s the c onc entrati on of c omponent.
p and c beco me nearl y c ons tant; both E r
of thesc tw o e quati ons s ugges tsthat r' : r
3. When i s (V . r' * ) equal to z ero?
4. Is the de fi ni ti on of q gi v en i n C hapt
transfcr i n s y s tems of i nterdi ffus i ng pai
isomers ?
5. What do es the term )i ,E ,J r mean?
6. What ar e Fi c k ' s fi rs t and s ec ondl aw s
7. Show that forced diffusion does no
mass acts on al l s pec i es .
8. Define thc Soret coefficient. What a
9. Why sh oul d the c onnec ti ng tube i n I
i nsul ated'l
10. The ratio of partial specifc volume
spondi ng rati o for egg al bumi n and w ater
ul tracentri fugi ng than the l atter. W hy ?
11. What happens to the w ater i n E x an
12. In gaseous di ffus i on s y s temsai r i s
reasonabl e?
13. A gas i i s di ffus i ng through a mi x tur
system bc con s i dc red as a bi nary one?
14. C ompare the effec ts of mas s trans
heat transfer on s i mul taneousmas s trans f
15. C ompare the !)uu and D i ; us ed i n tl
16. What are the mas s trans fc r anal ogs
17. In rvhat w ay s are heat and mas strar
18. Why i s the ri ght s i de of E q. 18.5-9
'19.
IJow do forced and pressure diffusi
PR(
l 8.A r
D ehumi di fi c ati on
of A i r
C onsi der a s y s tem s uc h as that pi c ture
the stagnant gas i s ai r. A s s ume the fol l o
condi ti oni ng) : (i ) at , : r), i " : 80' F, an
3 E . A . Fox and V . E . Gex , A .l .C h.E . J
B aars, and W. H . K nol l , C hen. E ng. S c i .,
S ci .,4, 178 2 00,209 220 (tgs s ).
Y
f C hange f or M u f t i c o m p o n e n t
Syste m s
hese two transfer processes-in
the
Problems
583
use, this situation occurs at Re greater than lOa to 2 x 104. At lower Re the
Reynolds number has a pronounced effect. This behavior has been substantiated
by a number of different investigators.3
ng o f M iscible Fluid s
lnalysis the general form of a correlation
fluids in an agitatedtank. Assume that
ntially the same physical properties.
t the achievementof "equal degreesof
the achievement of the same reduced
)r any component A the concentration
*r 0, " * )
rosition variables,rlD and:/D, respeceller. It will also be convenientto define
V, alND, and plpN2D2, respectively,in
rpeller, revolutions per unit time, and
ieen that .t,q(r*,0, z*) dependsupon /*,
ndary conditions. Here Re : D2Nplp,
lt among the initial conditions are the
ner in which they are introduced to the
nixing.
dimensionlessgroups can be further
imentally observedthat if the tank is
that is, the fluid surface is effectively
absenceof a free fluid surface, gravitarre tnimportant. At a high Rcynolds
also be neglected2
so that :r1(r*, 0, z*)
mixing time is constantfor any desired
number of revolutions of the impellcr
is fixed. For mosl impellersin common
nt for verticalcylindricaltankswith axially
:ed strips,with their flat surfaccsin planes
endingfrorn the top to the bottom of the
he tank center.
noldsnumbers
can bc seenintuitivelyfrom
) becomcs
small in relzrtionto thc accclcrartsare dangcrous,however,and thc effect
neighborhood
of solid surfaces.Herc the
te neighborhood
of solid surfaccsis small
rcussion
of the ellcct of Reynoldsnuntber,
:Graw-Hill,Ncw York (1955),Chaptcr4.
. numberscan bestbe secnfrom thc timeAt high Reynoldsnunrbersthe turbulent
rleculardiffusionexceptin the immediate
QU E S TION S
FOR D IS C U S S ION
1 D i scuss the appl i cabi l i ty of the equati ons of change as devcl oped i n Chapters 3 and
l0 t o the descri pti on of systemsof more than one chemi cal speci es.
2 . A s the concentrati on of component I of a bi nary mi xture becomesvery s mal l , both
p a nd c become nearl y constant; both E qs. l 8.l -17 and l 9 are then val i d. C ompari s on
o f these tw o equati ons suggeststhat u : r'*. Is thi s true ?
3 . When i s (V . r'*) equal to zero?
4 . Is the defi ni ti on of q gi ven i n C hapter 8 sui tabl e to descri becombi ned heat and mas s
tr a nsfer i n systents of i nterdi ffusi ng pai rs of i sotopcs? Gcometri c i som ers ? Opti c al
isomers ?
5 . What does the term X ;F;J1 mean?
6. What are Fick's first and second laws ? Under what circumstances are they applicable ?
7. Show that forccd diffusion does not occur when the same extcrnal force oer unit
m a s s acts on al l speci es.
8 . D efi ne the S oret coe{fi ci ent. What are i ts uni ts?
9 . Why shoul d theconnecti ng tube i n Fi gure 18.5-2 be smal l i n di ametc r? N eed i t be
in sul ated?
10. The ratio of parlial specific volumes of PbNO3 and water is greater than the corresp o ndi ng rati o for egg al bumi n and w atcr. H ow ever, i t i s harder to scparatcthe former by
u ltr acentri fugi ng than the l atter. Why?
1 1. What happens to the w ater i n E xampl e 18.5-4?
1 2. In gaseous di ffusi on systems ai r i s usual l y treated as a si ngl e compo nent. Is thi s
r e a sonabl e?
1 3. A gas i i s di ffusi ng through a mi xture of tw o heavy i sotopesTandT*. W hen c an thi s
system be consi dercdas a bi nary onc')
1 4. C omparc the effects of mass transfer on si mul taneous heat transfer w i th thos e of
h ca t transfer on si mul taneousmass transfcr.
1 5. C ornpare the !]tn and D ,; used i n thi s chapter.
1 6. What are the mass transfer anal ogs of the Grashof and P randtl numb ers ?
1 7. In w hat w ays are hcat and mass transfer anal ogous and i n w hat w ays d o they di ffer?
1 8. Why i s the ri ght si de of E q. 18.5-9 grcater than uni ry?
1 9. H ow do forced and oressuredi ffusi on di ffer'l
P R OB LE MS
l8 .Ar
D ehumi di fi cati on
of A i r
Consi dcr a system such as that pi ctured i n Fi g. 18.5 1 i n w hi ch thc vapor i s H ,O and
th e stagnant gas i s ai r. A ssume the fol l ow i ng condi ti ons (w hi ch are repres c ntati rc i n ai r
co n di ti oni ng): (i ) at : : d, 7 : 80' F, and r",,, : 0.018; (i i ) at z : 0, f : 50' F.
3 E. A . Fox and V . E . C ex, A .l .C h.E . Journal ,2, 539 544 (1956); H . K ramers , G. M.
Ba a rs,andW.H .K nol l ,
C hen.E ns.S ci .,2,35 42(19.55); J.G.vandcV usse, C hc nt.E nq.
Sci.,4, 178 200,209,220 (t955).
58,f
T h e Eq u a tio n s o f Ch a nge for Mul ti component
S yrtems
Problems
a. Calculatethe ratio
(r8.A-r)
I - exp (NA,e,alk)6
in Eq. 18.5-9,forp : 1 n1r.
6. Compareg(') and Qtotat z : 0. What is the significance
of your answer?
Answer:a' 1'004
f g.Br Thermal Diffusion
Fstimatethe steady-state
separationof H, and D, achievedin the simplethermal
..a.
diffusion apparatusshown in Fig. 18.5-2under the following condirions: rj is zoo.x,
7, is 600" K, the mole fraction of deuteriumin the chargeis 0.10,and the effectiveaverage
k, is 0.0166.
b. At what temperatureshould this averagekrhave beenevaluatecl?
Answerb:a. r"ris higherby 0.0183in hot bulb
b. 330"K
l8.Cr Ultracentrifuging of Proteins
Estimate the steady-stateconcentrationprofile when a typical albumin solution is
subjectedto a centrifugalfield of 50,000 tjmes rhe force of giavity under the following
conditions:
Cell length: 1.0cm
Molecularweight of albumin : 45,000
Apparentdensityof albuminin solution: M,tl V^: 1.34g/cm3
Mole fractionof albuminat z : O,rro : 5 x l0-c
Apparentdensityof waterin the solution: 1.00g/cmr
: 75'F
Temperature
Answ er: rt:
5 x l o-t exp (-22.72);
18.F, Setting Up DiffusionProblem
Show how the generalequationsof (
problemsof Chapter17: (i) l7.D; (ii)
(vii) 17.K; (viii) 17.L.
I8.G, Alternative Forms of the Equ
Showthat Eq. 18.1-6may be written
d^
a.+
dt
+ ( v. p, r ')* ( v .j) : 11
b. -i i + ( v. cr u*)+ ( v. Jr *) : R/
dt
Discussthe significanceof eachterm in I
I8.H,
Simplification of the Multico
Binary System
Showthat Eqs. 18.4 7 through1l ma
caseof a binary mixture. (It will proba
j e *i o : 0 . )
18,1, PressureDiffusion
a. CombineEq. 18.5-17with theana
{grtovo-
z i s i n centi meten
l8.Dr Electrode Polarization
Rr'
v1
The effectivediffusivityof Ag+ in diluteaqueoussolutionsat 20. c is about l0-5 cm2sec-r.
Estimatethe limiting currentdensityin amperes/cm,for the apparatusshownin Fig. lg.5-4
if the solution is tenth-normalin the absenceof any appliedvoltageand the thicknessof
the solutionis 0.10cm.
Answer:3.g6 x l0-r amperes/cmr
l8.Er EffectiveBinary Diffusivitiesin a Multicomponent Gaseoul Mixture
Integrate this expression to obtain Eq. 1
b. E xtend E x ampl e 18.5-3 to the c as ei
to R0. Note that for this case dpldr : p
of rotation of the centrifuge.
c. S i mpl ify E q. 18.5-18 for the c as e
negl i gi bl e. (S ee P robl em 18.C .)
compute cg;^ for each speciesin the surface-catalyzed
gas-phasehydrogenationof
benzene,assumingthat only the following reactionoccurs:
18.J, Mobi l i ty
CrH. * 3H2+cysls-6.H,,
( | 8 . E - t)
The calculationis to be madefor one point in the reactor,near the catalystsurface.where
con ditio nsare T:5 00" K, p: lO at m , r , ( C. Hr ) : 0 . 1 0 , r 2 ( H r ) : 0 . g 0 , : r . . ( C . H , r ; :
0.05, r.(CH.) : 0.05. CH. doesnot reactin this system.
The binary cg;t in E-molecm-r sec-r are
c? tz:c9 2 t:) 4 .2 r 1 9 - '
c2 s:
c? t:
1 .9 5 x
cQla : cg l:
6 .2 1 x
cg zN: cg st:2 0 .7
x
c9 2 1 :cOp :4 1 .3
x
co sa : cQs:
5 .4 5 x
l 0-'
l 0-.
l 0-c
l 0-'
l0-o
Answer: Component:
cQi^, g-mole cm-r sec-r x 10..
a. Estimate the total force required I
sol uti on at I c m s ec 1 at 25' C i f the di f
b. Comparable forces are clearly reqr
the motive force for such materials in or
18.K 3 B i na ry and Ternary D i ffus i on
Air is conventionally treated as a singl
Thus low-temperature viscosity data for i
gas vi scosi t y formul a (E q. 1.4-18) w i th t
o :3 .6 1 'l
whereas, for Or,
o :3 .4 3 3
C.Ho
6.63
H2 CoH.
20.8 8.71
CH.
60.
and, for Nr,
o : 3 .6 8
Ch ange f or Mult i c o m p o n e n t
S yste m r
585
Pr o bl ems
(r8.A-r)
ofyour answer?
the significance
Answer:a. 1.004
{, and D, achievedin the simple thermal
er the followingconditions: fr is 200'K,
thechargeis 0.10,and the effectiveaverag€
k. havebeenevaluated?
erl: a. xr, is higherby 0.0183in hot bulb
, . 3 30' K
ofile when a typical albumin solution is
s the force of gravity under the following
1 8 .F, S etti ng U p D i ffusi on P robl ems
Show how the general equations of Chapter l8 can be used to set up the following
p r o bl ems of C hapter 17: (i ) 17.D ; (i i ) l 7.E ; (i i i ) l 7.F; (i v) l 7.G; (v) l 7 .H ; (v i ) l 7.J ;
( vii) 17.K ; (vi i i ) 17.L.
18.G, Alternative Forms of the Equation of Continuity
Showthat Eq. 18.1-6may be writtenin theseforms:
a.+
OT
( 1 8 .G- l )
+ ( V . p , r ') *( Y . j ) : r t
t ? + (V .c,u*)+ (v.Jr*) : R,{
OT
(18.c-2)
Discuss the significance of each term in these equations.
l8 .H ?
S i mpl i fi cati on of the Mul ti comP onent
B i nary S ystem
Mass Fl ux E xpressi ons for U s e i n a
S how that E qs. 18.4-7 through I I may be combi ned to yi el d E q. 18.4-14 for the s pec i al
case of a binary mixture. (It will probably be convenient to take advantage of the relation
,000
ution : MAI VA : 1.34g/cm3
t/ o : 5 x 10-6
rlution: 1.00g/cm3
Answer:rt: 5 x l0-c exp (-22.12);
z is in centimeters
i n * in:O.)
Diffusion
18.1, Pressure
B to obtain
for species
equation
withtheanalogous
Eq. 18.5-17
a. Combine
dz: -r^ff + v"ft
#rr"r^- Mov)
( 18. r - l)
at 20' C is aboutl0-s cmr sec-r.
ussolutions
shownin Fig. 18.5-4
/cmtfor theapparatus
rf any appliedvoltageand the thicknessof
Answer:3.86 x l0 3 amperes/cmz
In tegrate thi s expressi onto obtai n E q. 18.5-18.
b. Extend Example 18.5-3 to the case in which L cannot be considered short with respect
to R0. Note that for this case dpldr : pC)2r,in which r is distance measured from the axis
of rotation of the centrifuge.
c. S i mpl i fy E q. 18.5-18 fbr the case i n w hi ch the mol e fracti on of one c omP onent i s
,rlticomponentGaseousMixture
n e gl i gi bl e. (S ee P robl cm 18.C .)
gas-phasehydrogenationof
bce-catalyzeci
non occurs:
1 8 .J, Mobi l i ty
cyclo-CrH,,
( | 8. E- l)
lhereactor,near the cata.lystsurface,where
:
:
z.(C'H'r) :
"-rHr) 0.10,zr(Hr) 0.80,
his system.
24.2 x
1 . 9 5x
6.21x
20.7x
41 . 3x
5.45x
I8 .K3
B i nary and Ternary D i ffusi on i n A i r
Air is conventionally treated as a single component in evaluation of transPort properties.
Thus low-tempcrature viscosity data for air have been nicely fitted by using the monatomicgas viscosity formula (Eq. l.a-l8) with the following Lennard-Jones Parameters:
l0-c
10-6
l0-t
l0-'
l 0-'
l0-'
Hr C6Hrt
C.H.
-rsec-rx 10.. 6.63 2 0 . 8 8 . 7 1
rz. Estimate the total force required to move one equivalent of silver ions through a
1solution at I cm sec 1 at 25" C if the diffusivity of silver in the solution is 10-5 cmz sec
b. Comparable forces are clearly required to move uncharged species. What provides
the motive force for such materials in ordinary diffusion?
Answer: a. 2.53 x lOe kilograms force
o : 3 . 6 1 7A"€
,
- : 9 7 . 0 'K
o : 3.433
A,
1 : tt:.0" r
o:3.681 A,
- : 9 1 . 5 'K
K
whereas, for Or,
CH{
60.
K
a n d, for N r,
K
s86
T h e Eq u a tio n s o f Ch a n g e for Mul ti component
S ystemr
a. Compute cQABfor the diffusion of methane in air at 300" K, treating air as one
component with the Lennard-Jones parameters given above.
b. Compute cQ A^for methane in air at 300" K, taking air to be 2l rnole per cent oxygen
and 79 mole per cent nitrogen. Assume that the nitrogen and oxygen move at the same
velocity.
Note that the two methods are in good agreement, as is usually to be expected. In cases
in which the relative concentrations or ffuxes of nitrogen and oxygen vary appreciably
they should be treated as separate components.
Problems
a. Show that the total molar rate of co
- ( N r .* N c'
['
Here the subscripts0 and d refer to condi
mole fraction of component I in the con
6. Show that the energy flux through tl
1 8 . L 3 D i f f u s i v i t y in a n Aq u e o u s So lu tio n o f a Sin gl e S al t
When a single salt (Me\"(Xa)D
diffuses through water in the absence of electric current,
the molar fluxes J.* of the two ionic species are related by the requirement of electrical
neutrality. Show by a procedure analogous to that used in Example 18.5-4 that, in dilute
solution, this requirement may be used to determine the following relation between the
di f f u s i v i t y o f t h e s alt in wa te r , 9 ,*,a n d th e d iffu sivitie sof the ani on and cati on, 9r,and
9r,,, respectively:
On _ ? *.( a * p)
-. (18.L-r)
'"p9r, * agx.
cg m '
,/:J-tnl;
- !,1,:o: ( T o-,;) ( +
\o
Here B :
-(N,a. * Nc,)eo,^''6lk^h an
This result was first obtained by Nernst in 1888.r
18 . M ,
T h e B e d i n gfie ld - Dr e w Eq u a tio n o f Co n tin u ity
S h o w h o w t o g e t Eq . 1 8 .1 l9 a fr o m Eq . l8 .l- 1 9 .
Nole: Solution is lengthy.
I8.Ns
T= Tc
C o n d e n s a t io n o f M ixe d Va p o r s2
Chloroform (C) and benzene (B) vapors are condensing on a cold surface from an
eq u i m o l a r m i x t u r e a t I a tm . ( Se e F ig . 1 8 .N.) T h e te m peratureT"of thecol dsurfacecan
be varied. The more volatile species (C) accumulates near the cold surface, so that the
condensation process is retarded by the presenceofa gas "film" in which both the temperature and the composition depend on the distance z from the liquid surface. The film thickness d is 0.1 mm, and the heat-transfer coefficient ft of the condensate film is 200 Btu hr-l
f t - ' ? o F - r . T h e r a t i o o fb e n ze n e to ch lo r o fo r m in th e condensate can be consi dered as
eq u a l t o t h e r a t i o of r a te s o f co n d e n sa tio na n d th e fo llow i ng data may be used:
7rc : 0.050cmz sec r;
Co c:
1 6 .5 Btu /lb - m o le ' F;
C on:22.8
B tu/l b-mol e'F
Vapor-liquid equilibrium data for this system at I atm are as follows:
uc (vaPor)
0
0
Saturation
temPerature,
"c
0 .1 0 0.20
0 .4 0 t0 .5 0 t0.60
0. 29 10. 36 1 0 . 4 4
1.00
0.54 | 0.66
77. t | 76. 4 | 7 5 . 3 | 1 4 . 0 | 7 t . 9
The molar heat of vaporizationimtx and thermalconductivityk-,* of chloroform-benzene
mixtures may be consideredconstantat 12,800Btu/lb-moleand 0.007 Btu/hrft'F,
respectively.The effectof temperatureon physicalpropertiesmay be neglected.
I W. Nernst,Z. physik.Chem.,2,613-637
(1888).
2 For a discussionof the condensationof mixed vapors,seeA. P. Colburn and T. B.
Drew, Trans.Am. Inst. Chem.Engrs.,33, 197-212(1937).
Fig. 18.N. Conde
c. Estimatethe ratesof condensation
an
condenserwall temperatures. Note that th
a condensatecomposition. (i) This cho
from-the result of part (a). (ii) The energ
calculated from the result of part (r). (iii
estimated by an energy balancethrough
balance,assumethat all energytransport-t
according to the expressionq, : h(Tc - Tc
Sample answer: If I" is g6oF, the rate I
and the condensate
will
d. When, if ever,will the condensate
be .
e. What is the'(approximate)minimun
condensate-vaporinterface and what will
under theselimiting conditions? Explaintl
18.O, Constant-EvaporatingMixtures
A mixture of toluene and ethanol is ev:
nitrogen film of thicknessd to a streamof 1
f Changefor Multicomponent Systemj
hanein air at 300" K, treating air as one
's givenabove.
'K, takingair to be 2l mole per c€nt oxygen
the nitrogen and oxygen move at the same
ement,as is usuallyto be expected.In cases
s of nitrogenand oxygen vary appreciably
s.
of a SingleSalt
587
Pr o bl ems
a. Show that the total molar rate of condensation is
-(N^a,
* Na:cff ne=#,\:'ff ne=#\ 08N-,)
0 and d referto conditionsalz:0
andz: d. The quantityz!)isthe
Here the subscripts
mole fraction of component ,{ in the condensate.
g ig
D. Show that the energy flux through the condensate
flm at z :
(--!
-i',,, 1N,,+ N",; (r8.N-2)
l\t-e-01-,\
HereB : -(N.e,* Nc)ee,^r"6lk^,*
andQ,-1. : a']'e,"l rti'e,".
-!,1,:o:(ra-rJ({+)
'-\6
oughwaterin the absenceof electriccurrent,
are relatedby the requirementof electrical
r that usedin Example 18.5-4that, in dilute
rterminethe following relation betweenthe
ffusivitiesof the anion and cation,9r, and
t x, ( a 4 p)
( 18. L- l)
o * a9r.
B oundaryof
stagnantgas film
rB d =rC 6 =0.50
i g. 1
C o n ti nu it y
t - t9.
T=Tc
T :
tre condensing on a cold surface from an
The temperature 7" of the cold surface can
umulatesnear the cold surface, so that the
:eof a gas "film" in which both the temperace z from the liquid surface. The film thickent , of the condensate film is 200 Btu hr-r
rr in the condensate can be considered as
I the following data may be used:
r - m ol e" F ;
e ,n:
2 2 . 8 B t u / l b - m ole " F
n at I atm are as follows:
lal conductivityk.,* of chloroform-benzene
,800 Btuilb-mole and 0.007 Btu/hr ft " F.
sicalpropertiesmay be neglected.
888).
lixed vapors,seeA. P. Colburn and T. B.
2120937).
T o : 7 6 . 4 'C .
Drrectionof movemenl
of vapor
of mixedvapors.
Fig. 18.N. Condensation
c. Estimatethe ratesof condensationand the compositionof the condensatefor several
condenserwall temperatures. Note that the calculation will be simplified by first choosing
a condensatecomposition. (i) This choice permits direct calculation of Nr, * Ns,
from the result of part (a). (ii) The energy flux through the condensatefilm may then be
calculated from the result of part (D). (iii) The required wall temperature may then be
estimatedby an energy balance through the condensatefilm. In making this energy
film occursby conduction
balance,assumethat all energytransportthroughthe condensate
accordingto the expressionqz : h(7" - T).
Sampleansvr'er:If I" is 86" F, the rate of condensationwill be 1.26lb-moleshr-r ft-2
and the condensatewill be 44 mole per cent chloroform.
d. When, if ever,will the condensatebe in equilibrium with the saturatedvaPor?
e. What is the'(approximate)minimum temPeraturethat can be maintained at the
interfaceand what will the (exact) compositionof the condensatebe
condensate-vapor
under theselimiting conditions? Explain the significanceof your answers.
18.O, Constant-Evaporating Mixtures
A mixture of toluene and ethanol is evaporating through a one-dimensionalstagrant
nitrogen film of thicknessd to a stream of pure nitrogen. The entire systemis maintained
T h e Eq u a tio n s o f € h a n ge for Mul ti comP onent
588
S ystems
diffusivities.of.toluene and
at 60" F and at constant pressure. The ratio of the binary
e tha no linn itrog en( 9y t "lgo, u) is about 0' 695, and t h e v a p o r - l i q u i d e q u i l i b r i u m d a t a f o r
systemat 60'F are
the toluene-ethanol
Mole fraction toluene in liquid
0.096
Mole fraction toluene in vaPor
0.14'7
388
Total pressure,mm Hg
0. 15 5
39'7
0.233
o.274
0.375
0.242
0.256
0.277
391
395
P robl ems
D. Show that the effect of the Potentia
the solution is negligible. The purpose
is to reduce the potential gradient so thi
diffusion. Systems of the type pictun
widely used for polarographic analysisar
=:v
taE
390
9!!x
R
Fb
;Y=
mixture of toluene
It is desired to determine the composition of the constant-evaporating
a n d e t h a n o , l ( i . e . 'th e liq u id co m p o sitio n fo r wh ich th e r ati oofevaporati onratesi sthe
r a t i o o f c o n c e n t r a tio n sin th e liq u id ) u n d e r a va r ie tyo fcondi ti ons.A ssumei deal gas
behavior.
r-;
a . S h o w f r o m t h e Sto ich io m e tr yo fth isp r o b le m th a tth erati ooftol uenetoethanol
in a constant-evaporating mixture is given by the expression
6)
"Em
6)
x
F
(oo) . t
-
It
v^
L-
c
08.o-l )
mmHg,760mm
of3e6
tofalpresslure
"t;::l,i::;:XT:i:Ji:1:11,H:':il're *rg
--
i liiliillli:[:::::;il.^;l.,,,.
Po la r iza tio n in th e Pr e se ntu o'on Indi fferent E l ectrol yte
18.5-4 contains both Ag+NO"- at
consider that the apparatus described in Example
6 N and K+NO3- at an average concentration of 0'1 N. A
l0
concentration
an average
at the cathode to drop essentially
voltagejrist sufficient to cause the silver ion concentration
that no electrode reactions
assumed
b-e
again
may
It
cell.
the
across
impressed
to..ioi.
that is, the transport
silver;
metallic
of
out
plating
take place except the dissolution and
unitY'
is
ion
the
silver
of
number
C o n c e n t r a tio n
spectes Present' assumlng'
a. Calculatethe concentration gradients of the three ionic
another and that bulk flow
one
of
independently
diffuse
they
that
18.5-4,
as in ExamPle
An swe r : S ee Fi g. 18.P .
is negligible.
All concentration gradients are linear'
3G. W. Bennettand w. A. wright, Ind. E4q. Chem.,28,646-648(1936)obtainedan
iompositionof 20 per centtoluene.However,becausc
constant-evaporating
experimental
betweenthe
; e*perimentattechrlque,iheir resultsshould be intermediate
;i;;;;.i;
composiconstant-evaporating
true
the
and
toluene,
cent
per
25
.o-fi.i,ion,
;;.;;;;;l;
reasonable'
quite
tion. The calculatedvalue of l7 per cent thus appears
{_
!q
:O
-l
H e r e t h e g ' ^ a n d 9 ,^ a r e th e e ffe ctive b in a r yd iffu sivitie s ofethanol andtol uenei nthe
in the gas at the
andar* are the mole fractions of ethanol and toluene
mixture unJyr*
i n t e r f a c e ' ( N o t e t h a ty,*a n d yE*a r e fu n ctio n so fto ta lPr essureasw el l astemperature
and liquid comPosition.)
grn"for
diffusivities 9"rrand
b. show that it is reasonable to substitute the binary
I p^ and 9 r^.
"".
mixtures for
results of parts (a) and (b), estimate the constant evaporating
uring'i;"
l8.Ps
r
Fi g. 18.P. C onc entrati on profi l es for
el ectrolyte.
c. It is frequently stated that in sys
the nondischarging species." Is this a
t8.Q3
Di ffus i on of A through a S tr
ComP os i ti on
Gas ,4 is diffusing in the positive z-c
of thickness d, which contains nonmovi
B in the film on an l-free basis is ra'.
c. Show that the Stefan-Maxwell eq
r'tsrn and (dx"ldo
C as (dtoldL):
through the film and r'r, : 6Nn"lc9^,.
b. Integrate these two differential t
rro, afl d aI z : 6,
that at z : O, rt:
-
4F
-
/t_( t - 'uJ
- \e xP
-
I
(0 -
c. We maywritcrr' :
'^o)
exP
rr
I
)or"Otl .
to c
and integrate
into this expression
r/
?tnt?^c
H ere r:
contai ne d i n rrr.
' o : l l -fl l g
r \e x
and Q:11
-t
590
T h e Eq u a tio n s o f Ch a n ge for Mul ti comP onent
S ystems
l8.Rn The Equations of Mechanical and Thermal Energy for MulticomPonent
Systems
a. obtain an expressionfor the rate of changeof kinetic energyPer unit massin multicomponentsystems,Eq. C of Table I 8.3-l, by taking the dot productof u and the equation
of motion,Eq. 18.3-2.
6. Su btra ctth er es ult of par t ( a) f r om Eq. ( B) of T a b l e l 8 . 3 - l t o o b t a i n t h e e q u a t i o n o f
internal energy,Eq. D of Table 18.3-1.
l8.S{ Diffusion-ControlledCatalytic Cracking with A * pP * CQ
An ideal gasI diffusesat steadystatein the positivez-directionthrough a flat gas film
18. S. At z : 6 t her e i s a s o l i d c a t a l y t i c s u r f a c e a t w h i c h l
of thickn essd ,as s howninFig.
P robl ems
c. Solve these equations simultaneou
rpos' * (Bpao -
s' - ( Ar '
d. Take the inversetransform ofi, I
r, :
-l
dr
rI
-
r^(cr*t
d-L
* (BPeo - A4t,
+ (BPCa_ CPA
Here aa : tr7-, + Ad + \/(A, - A,
e. Is this solution valid for the follc
a- :0.
xP
Catalyticsurface
a t wh ich
p P + qQ
A- - >
z =6
F ig . 1 8 .S. Diffu sio n - co n tr o lle dcatal yti c cracki ng.
undergoes an irreversible nth-order reaction: A*PP
Atz:0 ,r p :z^ a n d
e x p r e i e d a s N r l r :a :kn "n f.
fractions ate assumed to be known.
The reaction rate may be
*qQ.
a9:rgoi
both of these mol e
a. Show that the Stefan-Maxwell equations for components P and Q are
t*T:A p'+B pe*Cp
*r#:A {c*B enp* c e
( l 8.s-l)
(r 8.s-2)
Here vro: Nilcgpe, Ap: Q - p)rr - 1, Bp: (l - rp)P,Cr: prr' The quantities
Aa, Bc, and Ce may be obtained by everywherereplacingP by Q and p by q. In these
rp:9rrf 9^" and ro: 9ral9tl.
expressions
6. Take the LaplacetransformofEqs. 18.S-l and 2 with resPectto r"o( to obtain the
for the transformedmole fractions;"(s) and fq(s):
foltowing algebraicexpressions
r r o* Cps - r + Bp e
s- Ap
Z^:
'
reo *
-
Cos-'t * Boip
s- Ao
(l 8.s-3)
(r8.s-4)
Y
,f Change for Multicomponent Systems
Problems
I Thermal Energy for Multicomponent
591
c. Solvetheseequationssimultaneouslyto get
rp.'sz* (Brrqo - Aerro * Cr)s * (BpCr)- CpAd I
_
'P -s, - (Ap + Aals + UpAa - BpU
;
ngeof kineticenergyper unit massin multitakingthe dot productof u and the equation
(l 8.s-5)
d. Take the inversetransformofi, to obtain
.B)of Table 18.3-l to obtain the equation of
zr:
ckingvvithA+ PP+ qQ
re positivez-directionthrough a flat gas film
d there is a solid catalytic surfaceat which z{
lI
-l
d+ -
r"o(ured+urQl
d.-eo-'PQl)
-
d- L
+ (Bprao- A{^
}
Cp)(eq+'pe1
--ea-'pel)
_,,leq-,ecl
* (BpCc- C.lo)\
- |
,,
- ""-,.0{
,_
- l\l
(18.s-6)
) )
Herea1 : LlUp+ Ad + l1a - eola +aao1
e. Is this solution valid for the following special cases? (i) a. :
e - :0.
I
I
dic surface
I w h ich
: r olle d ca t aly t icc ra c k in g .
A -pP l qQ.
The reaction rate may be
p : ,ro nnd no: rgoi both of these mole
P and Q are
i for components
' IB p c *C r
* Bq p * Cq
(1 8 .s -1 )
(l 8.s-2)
Bp: 0 - rr)p, Cp * prr. The quantities
rerereplacingP by Q and p by q. In these
S-1 and 2 with respectto ?pe{ to obtain the
rrmedmole fractionser(s) and tro(s):
lot-r t
_ AP
B*q
.-es-L+ BaiP
-A^
(18.S-3)
(18.S-4)
a-, (ii) au * 0 and