mass transfer & separation tectransfer & separation
TRANSCRIPT
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MassMass transfer & separation technology 424302 2008 transfer & separation technology 424302 2008 -- APPENDIXAPPENDIX
Åbo Akademi - TkF - Värmeteknik - Biskopsgatan 8, 20500 Åbo MÖF-ST RZ 2008 1/42
MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
MassMass transfer and transfer and separation technologyseparation technologyMassMassööverfverfööring och separationsteknik ring och separationsteknik ((””MMÖÖFF--STST””))
Appendix. Appendix. MassMass transfer in transfer in multimulti--componentcomponent mixturesmixtures
Ron ZevenhovenÅbo Akademi University
Heat Engineering Laboratorytel. 3223 ; [email protected]
See alsoKrishna & WesselinghChem. Eng. Sci.52(6) 1997 861-911
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
A.1 A.1 ””OldOld--schoolschool”” massmass transfer transfer
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
J Ddcdzi
i= − ciα
ciβ
βdiffusivity
k Dzii=
Δ
mass transfer coefficient
Fick’slaw
Ji : flux of i with respect to the mixture
J D cz
k ci ii
i i= − = −ΔΔ
Δ
Δz
Δc c ci i i= −β α
Mass TransferMass Transfer as you have learned it
α
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
N D dcdz
Nxi ii
i= − +differential equation
N k c Nxi i i i= − +Δdifference equation123 {
Ni : flux with respect to an interface
diffusion flux
drift flux
Stefan or drift correction
N Nii
=∑flux of mixture
Diffusion with DriftDiffusion with Drift
−k s ci i iΔ
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
gas: c c c constant1 2+ = =
fluxes with respect to mixture
J D dcdz1 1
1= −
J D dcdz2 2
2= −
( )J J Dd c c
dz1 21 20+ = = −+
+
0 1
D
x1
only one binary D, which is independent
of composition
Classic Classic -- in Gasesin Gases
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
N2, CO2N2, H2
ideal gases, 100 kPa, 298 K
A B
beginning: xN2 = 0 46.xH2 = 0 54.
xN2 = 0 52.xCO2 = 0 48.
Question: Does N2 transfer (a) from A to B? (b) from B to A?(c) not at all? (d) or does it do (a), (b) and (c)?
Three Gases (1)Three Gases (1)
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0.6
0.5
0.4
0.6
0.4
0.2
0.00 10 20
N2 A
B
H2
CO2
A
B
B
A
mol
e fra
ctio
n x i
reverse diffusion
timeh
Three Gases Three Gases (2)(2)
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Two Two CationsCations
H+
Cl-
Na+
Cl-
cationpermeable membrane
highconcentration
lowconcentration
excess +chargeand electrical fieldso Na+ can move
against its concentration gradient!
H+ moves rapidly H+
Na+
1
3
2
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
He
298 K
100 kPa
Ar
298 K
100 kPa
!friction (He / plug) < friction (Ar / plug)
the plug, matrix or membrane is a
(pseudo)component
MM
Ar
He
N NHe Ar≈ −3
2 Gases in a Porous Plug (1)2 Gases in a Porous Plug (1)
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
He
298 K
100 kPa
Ar
298 K
101 kPa(for example)
main reason:viscous flow
retards He, accelerates Ar
Δp
N NHe Ar= −
2 Gases in a Porous Plug (2)2 Gases in a Porous Plug (2)
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A.2 Driving A.2 Driving forcesforces
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1kg
1 m
the potential difference is the work required to change the condition of the weight
here: Δψ Δ= = ≡mg z 981 981. .J ( Nm)
or, per mole Δψ Δi iM g z=
1kg
Fi
the driving force is the negative potential gradient:F d
dzM gi
ii= − = −
ψ
the force is downwards
Gravity Gravity -- a simple Potentiala simple Potential
Δz
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Chemical PotentialChemical Potentialxi γ i
mixture ( ) ii aRTTpconst ln, +=μ
chemical potential
activity
pure i (one mole)
( )μi const p T∗ = ,
iiii aRT ln−=μ−μ=μΔ ∗
activity coefficientwork required: change in the chemical potential
iii xa γ=
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chemical potential in an ideal solution ii xRTTpconst ln),( +=μ
in an ideal gas ⎟⎠⎞
⎜⎝⎛+=μ
ppRTTpconst i
i ln),(
partial pressure
μμ in an in an Ideal SolutionIdeal Solution
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momentum ‘in’ momentum ‘out’
forces
∑ invm )( &
change of momentum ∑∑∑ +−= Fvmvm
dtmvd
outin )()()(&&
∑ outvm )( &
∑F
Momentum BalanceMomentum Balance
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
(2)(1)
1u2u
zH2 CO2
dzz+
Moving Through Each OtherMoving Through Each Other
species velocities
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
dzz+z
area A
volume Adz
zAp1 dzzAp+1
)( 1221 uuppforce
friction−∝⎟
⎠⎞
⎜⎝⎛
dzdp
forcedriving 1−=⎟
⎠⎞
⎜⎝⎛
Forces on Hydrogen (1)Forces on Hydrogen (1)
121221
21
21221212
2121
1
21211
===
−=−∝
−∝−=
−∝−
,,,
,
,1
11
ζDRT
DRTζt coefficienfriction with
)uu(xζ)uu(RTp F force Driving
moleper force )uu(RTpdzdp
pRT gives
RTpcwith
per volume force )uu(ppdzdp :balance Force Note:
often x2 ≈ 1and u2 = 0
Gases: u ~ 10-2 m/sLiquids:u ~10-4 m/s
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
dzda
aRT
dzadRT
dzdF i
i
iii −=−=
μ−=
ln−
RTx
dxdzi
i
in ideal solutionsfor a given T and p
Driving Force (per Mole of Driving Force (per Mole of ii))
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
( )F x u ui i j j i jj i
= −≠∑ζ ,driving force
on i
friction coefficient between i and j
mole fractionof j
(diffusive) species velocities
MaxwellMaxwell--Stefan EquationStefan Equation
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in a solid particle
Δz d≈10
Δzd
gases
liquids
Δz ≈ −10 4 m
Δz ≈ −10 5 m
membrane
Δz = − −10 107 4K m
Film Theory, Thickness of FilmsFilm Theory, Thickness of Films
eddies & large scale convection
‘film’: no eddies
phase boundary
two thin, one dimensional ‘films’ next to the phase boundary
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Difference Form of ForceDifference Form of Force
za
aRT
zaRT
zF i
i
iii Δ
Δ−=
ΔΔ
−=ΔμΔ
−=ln
zx
xRT i
i ΔΔ
−
in ideal solutions
for a given T and p
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Δμ1
RT+1
0
-1
0 2
approximate
exact
‘approximate’ works out better in difference equations
⎟⎟⎠
⎞⎜⎜⎝
⎛
α
β
1
1lnaa
( )αβ
αβ
+−
=Δ
11
11
1
1
5.0 aaaa
aa
α
β
1
1
aa
−∞
-2
ApproximationApproximation
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Δz ≈ −10 5 m
mole fractionof CO2
x1 0003α = .
F RTx
dxdz
RTx
xz1
1
1
1
1= − ≈ −ΔΔ
x1 0001β = .
Forces in a Forces in a Glass of BeerGlass of Beer
growing bubble of CO2
Note: u2 = 0
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ExampleExample (3.1 from (3.1 from bookbook))
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ExampleExample (3.1 from (3.1 from bookbook):): answeranswer
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A.3 A.3 FrictionFriction
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Friction Coefficients of SpheresFriction Coefficients of Spheres
( )ζ πη12 2 112 1 1
3 3 10, = A× ≈ × − − −d Nmol ms1
F1F1
A = ×6 1023
molecules mol-1
coefficient of a single sphere
spheres ‘1’
liquid ‘2’
SeePTG § 6.7:Stokes Law
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Maxwell-Stefan diffusivity of large molecules in dilute liquids (not gases)
( ) sm10
104.01010106300314.8 2
99323
−−− ≈
××××××
→
Ð RT12
12,
,
≡ζ
Ð RTd12
2 13, = A πη
Diffusion and Friction CoefficientsDiffusion and Friction Coefficients
2,12,1 Ð
RT≡ζ
each others ‘inverse’
we use both
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2 components: 1 relative velocity1 independent equation
3 components: 2 relative velocities2 independent equations
n components: n - 1 relative velocitiesn - 1 independent equations
One Equation MissingOne Equation Missing
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only relative
velocities
∑≠
−ζ=ij
jijjii uuxF )(,bootstrap
‘floating’ transport relations: have to be ‘tied’ to surroundings
Bootstrap Bootstrap (1)(1)
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N2CO2
N2H2
H+
Cl-Na+
Cl-
HeAr
no net volume flow plug does not move
membrane does not move
(almost) no charge transfer
Bootstraps (2)Bootstraps (2)
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∑ −ζ=j
jijijiii uuxcxcxF )(,
∑ −ζ=j
jiijjii NxNxf )(,
force on i per unit volume of mixture
in practical problems we use fluxes:
N u c u cxi i i i i= =
flux form of MS-equation:
FluxesFluxes
Fi / V ; xi = ci / c
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x1α
β1xx1
positivedirection
α β
binary:
infinitesimallayer
finite layer(approximate)
u1( )21
2,12
1
1
uuÐRTx
dzda
aRT
−=−
( )( )dzÐ
uuxa
da2,1
212
1
1 −=−
( )z
Ðk
kuux
aa
Δ=
−=
Δ− 2,1
2,12,1
212
1
1
From Differential to DifferenceFrom Differential to Difference
α1a
β1a
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876film
iu
0iu
α β0
icaverage concentration ic
species velocity(depends on position in film)
species velocity at the average composition
positive velocity
Average VelocityAverage Velocity
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)(1212
2,11
1 uuxka
a−=
Δ−
cxc 1×
)(12112
2,11
11 NxNx
ckaax −=
Δ−
Differences with FluxesDifferences with Fluxes
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
∑≠
−=
Δ−
ij ji
jij
i
i
kuu
xaa
,
)(
using velocities using fluxes
∑≠
−=
Δ−
ij ji
jiij
i
ii ck
NxNxaax
,
)(
for ideal solutions
ixΔ−
Multicomponent EquationsMulticomponent Equations
APPENDIX chapter to course material for ÅA TkFcourse 424302 / 2008
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
100
10-2
10-4
10-6
ki j,1m s−
kÐ
zi ji j
,,=
Δ
gases ≈ − −10 1 m s 1
liquids ≈ − −10 4 m s 1
Transfer Transfer CoefficientsCoefficients
in pores
in pores
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
changes are not very important
changes are not very important
Temperature EffectsTemperature EffectsMS-equation
terms) diffusion (thermal)u(uxζFj
jijji,i +−=∑small
driving force
dzdx
xRT
dzdF i
iT
ii −=⎟
⎠⎞
⎜⎝⎛ μ−=
difference form:
zx
xTRF i
ii Δ
Δ−=
at constant temperature
average film temperature
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
A.4 A.4 BinaryBinary examplesexamples
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
x1α
x1
0α β
drops on a tray
gas: trace of NH3 (1)
bulk of N2 (2)
..as you already knew..
Stripping Stripping -- dilutedilute
12 ≈x
transport relation
bootstrap
flux
ckNxNxx
2,1
21121
−=Δ−
02 =N
12,11 xckN Δ−=
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
0β
x1 05= .x1β
α
x1α
Stripping Stripping -- concentratedconcentrated
12,112
2,11 2 xckx
xck
N Δ−=Δ−=
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
heat
benzene (1), volatile
toluene (2)
x1α
x2α
x2β
x1β
y K x1 1 1= β
y K x2 2 2= β
vapour removed by convection
bootstrap:NN
yy
1
2
1
2
= = ν
Vaporising DropletVaporising Droplet
APPENDIX chapter to course material for ÅA TkFcourse 424302 / 2008
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
NN
1
2
= ν
− =−
Δx x N x Nk c1
2 1 1 2
12,
− =−
Δx x N x Nk c2
1 2 2 1
12,
Nk c
x xx1
12
2 11= −
−,
νΔ N
k cx x
x212
1 22= −
−,
νΔ
example ν = 2 x x1 2 0 5= = .
N k c x1 12 14= − , Δ ( )N k c x2 12 22= − − , Δ
Fluxes from Vaporising DropletFluxes from Vaporising Droplet
Stefan (drift) correctionsΔx1 < 0Δx2 > 0
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
O C CO2 2 2+ →
C
O2 1( )
CO( )2
1.0
0.60.4
0.0
both components are moving and have a high concentration
k12210, = − −ms 1
bootstrap:N N2 12= −
calculate N1 and N2
c = −10 mol m 3
Carbon GasificationCarbon Gasification
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
( )− =−
=+
Δx x N x Nk c
x x Nk c1
2 1 1 2
1 2
2 1 1
1 2
2
, ,
N N2 12= −
( )Nk c
x xx1
1 2
2 11
2
210 10
0 7 2 0 30 6= −
+= −
×+ ×
−−
,
. ..Δ
( )N exact1 0 046 0 047= − −. : . mol m s2 1
( ) 122 smmol094.0:092.0 −−−−= exactN
Fluxes in GasificationFluxes in Gasification
almost the same
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
Binary DistillationBinary Distillation
0β
x1β
N1
α
x1αtransport relation
N2
x2α
x2β
hexane (1)
heptane (2)− =
−Δx x N x N
k c12 1 1 2
12,
bootstrap N N1 2= −
(equimolar exchange)
( )→ − =+
Δxx x N
k c11 2 1
12,
N k c x1 12 1= − , Δ N k c x2 12 2= − , Δ
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
1
2
3
4
5
6
membrane stagnant
bulk stagnant (absorption)
trace stagnant (polarisation)
equimolar exchange (distillation)
interface determined (vaporisation)
reaction stoichiometry
uM = 0
02 =N
u1 = 0
N N1 2 0+ =
NN
yy
1
2
1
2
=
Some BootstrapsSome Bootstraps
N N1
1
2
2ν ν=
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
A.5 A.5 TernaryTernary examplesexamples
APPENDIX chapter to course material for ÅA TkFcourse 424302 / 2008
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by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
( ) ( )
( ) ( )
− = − + −
− = − + −
ddz
x u u x u u
ddz
x u u x u u
μζ ζ
μζ ζ
112 2 1 2 13 3 1 3
22 1 1 2 1 2 3 3 2 3
, ,
, ,
forces per mole of ‘1’
forcesper mole of ‘2’
Ternary Ternary -- per mole of per mole of ii
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
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( ) ( )
( ) ( )
− = − + −
− = − + −
⎫
⎬⎪
⎭⎪
x ddz
x x u u x x u u
x ddz
x x u u x x u u
11
12 1 2 1 2 13 1 3 1 3
22
2 1 1 2 2 1 2 3 2 3 2 3
μζ ζ
μζ ζ
, ,
, ,
forces per mole of mixture
these should cancel:
ζ ζ2 1 12 2 1 12 2 1 12, , , , , ,≡ ≡ ≡Ð Ð k k
Ternary Ternary -- per mole of Mixtureper mole of Mixture
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binary
ternary
quaternary
ckNxNx
ckNxNxx
3,1
3113
2,1
21121
−+
−=Δ−
ckNxNx
ckNxNxx
3,2
3223
1,2
12212
−+
−=Δ−
More ComponentsMore Components
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
CondensorCondensor vapour
liquid
cooling water
H O2
NH3
H2
0.6
0.4
0.2
0.0
NH3 (1) and H2O (2) condense on a tube
find the velocities in the gas film
H2 (3) does not condense
k k13 2 333 10, ,= = × − −ms 1
k1231 10, = × − −ms 1
Mix: NH3+H2O + H2
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Condenser (2)Condenser (2)transport (MS) relations:
NH3 :
H2O :
bootstrapthree linear equations, three unknowns
exact solutions:
03 =N
30)103(4.03.0
30)101(3.04.02.0 3
313
21−− ×
−+
×−
=−NNNN
30)103(3.03.0
30)101(3.04.04.0 3
323
12−− ×
−+
×−
=NNNN
015.01 =N 122 smmol045.0 −−=N
013.01 =N 122 smmol049.0 −−=N
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H O2
NH3
H2
mixture velocity
H2O moves down its gradient
NH3 dragged against its gradient
H2 does not move at all
Condenser (3)Condenser (3)
APPENDIX chapter to course material for ÅA TkFcourse 424302 / 2008
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Ternary Distillation (1)Ternary Distillation (1)
in which direction does 2 move?
0.530.45
0.63
0.35
0.02 0.02
1
3
2
1 ethanol 3 water2 a trace of butanol
large friction between 1 2andk12
28 10, = × − −ms 1
k k13 2 3220 10, ,= = × −
bootstrap: equimolar exchange
vapour liquid
m s-1
u y u y u y1 1 2 2 3 3 0+ + =
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0.018
0.020
0.0220.0214
0.018
0.020
0.0186
0.022y2α y2β
no motion
u22375 10= − × −.
u22158 10= − × −.
u220 81 10= + × −.
ButanolButanol -- which direction?which direction?
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Ammonia reactionAmmonia reaction
N2 (1)
H2 (2)
NH3(3)
α β
catalytic surface
N H NH2 2 33 2+ ⇔
− =−
+−
Δx x N x Nk c
x N x Nk c1
2 1 1 2
12
3 1 1 3
13, ,
− =−
+−
Δx x N x Nk c
x N x Nk c2
1 2 2 1
12
3 2 2 3
2 3, ,
− =−
+−
Δx x N x Nk c
x N x Nk c3
1 3 3 1
13
2 3 3 2
2 3, ,
transport relations:
bootstrap: N N N1 2 3
1 3 2= =
−
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− =−
+−Δx
xx u u
kx u u
k1
12
1 2
123
1 3
13, ,
L=−x u u
keffeff
eff
1
1,
a ternary can be approximated as a binary when
When is: 3 = 2?When is: 3 = 2?
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
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1
2
3
one friction term dominates:
(example: mobile species in many membranes)equal velocity of two species:
(example: Na+ and Cl- in water)equal diffusivities (‘1’ in m- and p-xylene)
xk
xk
x u uk
2
12
3
133
1 3
13, , ,
<< →−
( )u u xk
xk
u u2 32
12
3
131 2= → +
⎛
⎝⎜
⎞
⎠⎟ −
, ,
( ) ( )k kx x u x u x u
k12 132 3 1 2 2 3 3
12, ,
,
= →+ − +
x xeff = 3
k keff1 13, ,=
xk
xk
xk
eff
eff1
2
12
3
13, , ,
= +⎛
⎝⎜
⎞
⎠⎟
x x xeff = +2 3
u x u x ux xeff =++
2 2 3 3
2 3
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MultiMulti--componentcomponent massmass transfertransferusingusing transparanciestransparancies that that accompanyaccompany ””MassMass Transfer in Transfer in MulticomponentMulticomponent mixturesmixtures””
by J.A. by J.A. WesselinghWesselingh & R. Krishna, & R. Krishna, DelftDelft University Press (2000)University Press (2000)
simplifying transport equation of N2 in ammonia formation:eliminate N2 and N3 with N N N N2 1 3 13 2= = −
N k c xeff1 1 1= − , Δ1 3 21
2 1
12
3 1
13kx x
kx x
keff, , ,
=−
++
with
similarly for H2 and NH3
Effective Binary in Reactive Effective Binary in Reactive SystemSystem
If all fluxes Ni are related via the same reaction stoichiometry→ pseudo - binary
APPENDIX chapter to course material for ÅA TkFcourse 424302 / 2008
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