Mass transfer in multi-component mixturesusers.abo.fi/rzevenho/MTiMM-OHs.pdf · Multi-component...
Transcript of Mass transfer in multi-component mixturesusers.abo.fi/rzevenho/MTiMM-OHs.pdf · Multi-component...
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 1/76
Mass transfer in multi-component mixtures
Ron ZevenhovenÅbo Akademi University
Thermal and Flow Engineering Laboratorytel. 3223 ; [email protected]
See also Krishna & WesselinghChem. Eng. Sci.52(6) 1997 861-911
Chapters 1-10 ex. 7, of 25 of book
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 2/76
1 ”Old-school” mass transfer
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
J Ddcdzi
i ci
ci
diffusivity
kDzii
mass transfer coefficient
Fick’s law
Ji : flux of species i with respect to the mixture
J Dcz
k ci ii
i i
z
c c ci i i
Mass Transfer as you have learned it
phases α, β
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
N Ddcdz
Nxi ii
i differential equation
N k c Nxi i i i difference equation
Ni : flux with respect to an interface
diffusion flux
drift flux
Stefan or driftcorrection
N Nii
flux of mixture
Diffusion with Drift
k s ci i i
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
gas: c c c constant1 2
fluxes with respect to mixture
J Ddcdz1 1
1
J Ddcdz2 2
2
J J D
d c c
dz1 21 20
+
0 1
D
x1
only one binary D, which is independent
of composition
Classic - in gases
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft 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)
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
0.6
0.5
0.4
0.6
0.4
0.2
0.0
0 10 20
N2A
B
H2
CO2
mol
e fr
actio
n x i
reverse diffusion
timeh
Three gases (2)
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Two cations
H+
Cl-
Na+
Cl-
cation permeable membrane
highconcentration
lowconcentration
excess +charge
and electrical fieldso Na+ can move against its concentration gradient!
H+ moves rapidly H+
Na+
1
3
2
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft 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
Two gases in a porous plug (1)
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
He
298 K
100 kPa
Ar
298 K
101 kPa(for example)
main reason:viscous flow
retards He, accelerates Ar
p
N NHe Ar
Two gases in a porous plug (2)
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 11/76
2 Driving forces
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
1kg
1 m
the potential difference is the work required to change the condition of the weight
here: mg z 981 981. .J ( Nm)
or, per mole i iM g z
1kg
Fi
the driving force is the negative potential gradient:
Fddz
M gii
i
the force is downwards
Gravity - a simple potential
Δz
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Chemical potentialxi i
mixture ii aRTTpconst ln,
chemical potential
activity
pure i (one mole)
i const p T ,
iiii aRT ln
activity coefficientwork required: change in the chemical potential
iii xa
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 14/76
chemical potential in an ideal solution ii xRTTpconst ln),(
in an ideal gas
pp
RTTpconst ii ln),(
partial pressure
in an ideal solution
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 15/76
momentum ‘in’ momentum ‘out’
forces
change of momentum Fvmvm
dtmvd
outin )()()(
F
Momentum balance
Σ(ṁv)in Σ(ṁv)out
(ṁv)out(ṁv)in
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 16/76
(2)(1)
1u2u
zH2 CO2
dzz
Moving through each other
species velocities
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 17/76
dzzz
area A
volume Adz
zAp1 dzz
Ap1
)( 1221 uuppforce
friction
dzdp
forcedriving 1
Forces on hydrogen (1)
,,,
,
,1
11
ζD
RT
D
RTζt coefficienfriction with
)uu(xζ)uu(RTp F force Driving
moleper force )uu(RTpdz
dp
p
RT gives
RT
pcwith
per volume force )uu(ppdz
dp :balance Force Note:
often x2 ≈ 1and u2 = 0
Gases: u ~ 10-2 m/sLiquids:u ~10-4 m/s
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
dzda
aRT
dzad
RTdzd
F i
i
iii
ln
RTx
dxdzi
i
in ideal solutionsfor a given T and p
Driving force (per mole of i)
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
F x u ui i j j i jj i
,driving force
on i
friction coefficient between i and j
mole fractionof j
(diffusive) species velocities
Maxwell-Stefan equation
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
the driving force on a species i in
a mixture
the sum of the friction forces between i and the
other species j
the friction exerted by j on i is proportional - to the fraction of j in the mixture and- to the difference in velocity between i and j.
Multicomponent Diffusion - in Words
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
in a solid particle
zd10
z
d
gases
liquids
z 10 4 m
z 10 5 m
membrane
z 10 107 4 m
Film 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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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 22/76
Difference form of force
za
aRT
za
RTz
F i
i
iii
ln
zx
xRT i
i
in ideal solutions
for a given T and p
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
1
RT+1
0
-1
0 2
approximate
exact
‘approximate’ works out better in difference equations
1
1lna
a
11
11
1
1
5.0 aa
aa
aa
1
1
a
a
-2
Approximation
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
z 10 5 m
mole fractionof CO2
x1 0003 .
FRTx
dxdz
RTx
xz1
1
1
1
1
x1 0001 .
Forces in a glass of beer
growing bubble of CO2
Note: u2 = 0
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 25/76
Example (3.1 from book)
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 26/76
Example (3.1 from book): answer
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 27/76
3 Friction
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Friction coefficients of spheres
12 2 112 1 1
3 3 10, = A d Nmol ms1
F1
A 6 1023
molecules mol-1
coefficient of a single sphere
spheres ‘1’
liquid ‘2’
using:Stokes Law
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Maxwell-Stefan diffusivity of large molecules in dilute liquids (not gases)
sm
10104.01010106
300314.8 29
9323
ÐRT
1212
,,
ÐRT
d122 13, A
Diffusion and friction coefficients
2,12,1 Ð
RT
each others ‘inverse’
we use both
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
2 components: 1 relative velocity
1 independent equation
3 components: 2 relative velocities
2 independent equations
n components: n - 1 relative velocities
n - 1 independent equations
One equation missing
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
only relative
velocities
ij
jijjii uuxF )(,bootstrap
‘floating’ transport relations: have to be ‘tied’ to surroundings
Bootstrap (1)
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 32/76
N2
CO2
N2
H2
H+
Cl-
Na+
Cl-
HeAr
no net volume flow plug does not move
membrane does not move
(almost) no charge transfer
Bootstraps (2)
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 33/76
j
jijijiii uuxcxcxF )(,
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:
Fluxes
Fi / V ; xi = ci / c
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
x1
1xx1
positivedirection
binary:
infinitesimallayer
finite layer(approximate)
u1
212,1
21
1
uuÐRT
xdzda
aRT
dzÐ
uux
ada
2,1
212
1
1
z
Ðk
kuu
xaa
2,1
2,12,1
212
1
1
From differential to difference
1a
1a
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 35/76
iu
0iu
0
icaverage concentration ic
species velocity(depends on position in film)
species velocity at the average composition
positive velocity
Average velocityfilm
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 36/76
)(1
2122,11
1 uuxka
a
cxc 1
)(1
21122,11
11 NxNx
ckaa
x
Differences with fluxes
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 37/76
ij ji
jij
i
i
k
uux
aa
,
)(
using velocities using fluxes
ij ji
jiij
i
ii ck
NxNx
aa
x,
)(
for ideal solutions
ix
Multicomponent equations
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
100
10-2
10-4
10-6
ki j,1m s
kÐ
zi ji j
,,
gases 10 1 m s 1
liquids 10 4 m s 1
Transfer coefficients
in pores
in pores
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 39/76
changes are not very important
Temperature effectsMS-equation
terms) diffusion (thermal)u(uxζFj
jijji,i small
driving force
dzdx
xRT
dzd
F i
iT
ii
difference form:
zx
xTR
F i
ii
at constant temperature
average film temperature
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 40/76
4 Binary examples
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
x1
x1
0
drops on a tray
gas: trace of NH3 (1)
bulk of N2 (2)
..as you already knew..
Stripping - dilute
12 x
transport relation
bootstrap
flux
ckNxNx
x2,1
21121
02 N
12,11 xckN
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
0
x1 05 .x1
x1
Stripping - concentrated
12,112
2,11 2 xckx
x
ckN
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
heat
benzene (1), volatile
toluene (2)
x1
x2
x2
x1
y K x1 1 1
y K x2 2 2
vapour removed by convection
bootstrap:NN
yy
1
2
1
2
Vaporising droplet
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
NN
1
2
xx N x N
k c12 1 1 2
12,
xx N x N
k c21 2 2 1
12,
Nk c
x xx1
12
2 11
,
N
k c
x xx2
12
1 22
,
example 2 x x1 2 0 5 .
N k c x1 12 14 , N k c x2 12 22 ,
Fluxes from vaporising droplet
Stefan (drift) correctionsΔx1 < 0Δx2 > 0
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft 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 gasification
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
xx N x N
k c
x x N
k c12 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
,
. ..
N exact1 0 046 0 047 . : . mol m s2 1
122 smmol094.0:092.0 exactN
Fluxes in gasification
almost the same
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Binary distillation
0
x1
N1
x1transport relation
N2
x2
x2
hexane (1)
heptane (2)
x
x N x N
k c12 1 1 2
12,
bootstrap N N1 2
(equimolar exchange)
xx x N
k c11 2 1
12,
N k c x1 12 1 , N k c x2 12 2 ,
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft 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 bootstraps
N N1
1
2
2
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 49/76
5 Ternary examples
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
ddz
x u u x u u
ddz
x u u x u u
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 - per mole of i
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
xddz
x x u u x x u u
xddz
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
, ,
, ,
forces per mole of mixture
these should cancel:
2 1 12 2 1 12 2 1 12, , , , , , Ð Ð k k
Ternary - per mole of mixture
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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
binary
ternary
quaternary
ckNxNx
ckNxNx
x3,1
3113
2,1
21121
ckNxNx
ckNxNx
x3,2
3223
1,2
12212
More components
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 52/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Condensor 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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Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Condenser (2)transport (MS) relations:
NH3 :
H2O :
bootstrap
three linear equations, three unknowns
exact solutions:
03 N
30)103(4.03.0
30)101(3.04.0
2.0 331
321
NNNN
30)103(3.03.0
30)101(3.04.0
4.0 332
312
NNNN
015.01 N12
2 smmol045.0 N
013.01 N 122 smmol049.0 N
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 54/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
H O2
NH3
H2
mixture velocity
H2O moves down its gradient
NH3 dragged against its gradient
H2 does not move at all
Condenser (3)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 55/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
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 water
2 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
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 56/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
0.018
0.020
0.0220.0214
0.018
0.020
0.0186
0.022
y2 y2
no motion
u22375 10 .
u22158 10 .
u22081 10 .
Butanol - which direction?
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 57/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Ammonia reaction
N2 (1)
H2 (2)
NH3(3)
catalytic surface
N H NH2 2 33 2
xx N x N
k cx N x N
k c12 1 1 2
12
3 1 1 3
13, ,
xx N x N
k cx N x N
k c21 2 2 1
12
3 2 2 3
2 3, ,
xx N x N
k cx N x N
k c31 3 3 1
13
2 3 3 2
2 3, ,
transport relations:
bootstrap:N N N1 2 3
1 3 2
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 58/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
x
xx
u uk
xu u
k1
12
1 2
123
1 3
13, ,
xu u
keffeff
eff
1
1,
a ternary can be approximated as a binary when
When is: 3 = 2?
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 59/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
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
xu u
k2
12
3
133
1 3
13, , ,
u uxk
xk
u u2 32
12
3
131 2
, ,
k k
x 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
ux u x u
x xeff
2 2 3 3
2 3
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 60/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft 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 , 1 3 2
1
2 1
12
3 1
13kx x
kx x
keff, , ,
with
similarly for H2 and NH3
Effective binary in a reactive system
If all fluxes Ni are related via the same reaction stoichiometry → pseudo - binary
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 61/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Water from a saltwater reservoir penetrates through a membrane, see the figure. One result is that the salt concentration rises near the membrane.a. Give the Maxwell-Stefan (MS) the equation for the salt, in differential form, as a function of the fluxes Ni.b. For a mass transfer coefficient k1,2 = 0.01 m/s, the mean substance concentration ĉ = 60000 mol / m3 and flux N1 = 300 mol / (m2· s) for the water, calculate the gradient Δx2 if x2α = 0.018. Give the answer with 2 decimals accuracy.
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 62/76
Example: membrane
membrane
water (1) + salt (2) water
water
z
z=α z=β
x2α
x2β
x2
in boundary layer
Δx2 = x2β - x2α
x2 = x2α + ½ Δx2
u1
u2 = 0, x2 << x1membrane
water (1) + salt (2) water
water
z
z=α z=β
x2α
x2β
x2
in boundary layer
Δx2 = x2β - x2α
x2 = x2α + ½ Δx2
u1
u2 = 0, x2 << x1
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo TRP RZ 63/76
Example: membrane
membrane
water (1) + salt (2) water
water
z
z=α z=β
x2α
x2β
x2
in boundary layer
Δx2 = x2β - x2α
x2 = x2α + ½ Δx2
u1
u2 = 0, x2 << x1membrane
water (1) + salt (2) water
water
z
z=α z=β
x2α
x2β
x2
in boundary layer
Δx2 = x2β - x2α
x2 = x2α + ½ Δx2
u1
u2 = 0, x2 << x1
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
ideal solution:
non-ideal solution:
difference equation
chemical potential
8 Non-ideality
i
iiii a
az
RTdz
adRT
dzd
F
ln
ii aRTconstant ln
i
iiii x
xz
RTdz
xdRT
dzd
F
ln
ii xa
iii xa
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 64/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
molecules are little (hard) spheres moving around with their thermal velocity and occasionally bumping into each other
dd d
1 21 2
2,
ÐRT
pd M M1 2 3
3 2
1 22
1 2
1 22 1 1
,
/
,
/
A
d1
d2
Ð
T
p M M1 28
175
11 3
21 3 2
1 2
1 2
316 101 1
,
.
/ /
/
.
empirical modification:
diffusion volume liquid volume, m3 mol-1
9 Diffusivities in Gases
M = molar mass
kg/mol
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 65/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Gases, empirical equation
Ð
T
p M M1 28
175
11 3
21 3 2
1 2
1 2
316 101 1
,
.
/ /
/
.
H2 N2 CO2 NH3 H2O
7.07 17.9 26.9 14.9 12.7
i
10 6 m mol3 1
diffusion volume liquid volume, m3 mol-1
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 66/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Gases, diffusivity example
N2 (1) CO2 (2) T 300 K p 105 Pa
Ð1 2
8175
5 6 1 3 6 1 3 2
3 3 1 2
5
316 10300
10 17 9 10 26 9 10
10
28
10
44175 10
,
.
/ /
/
.. .
.
m s2 1
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 67/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Dilute Species in a Liquid
d11
1 3
A
/
u1 ÐRT
d1 22 13,
A
this ‘constant’ varies with the size ratio of the species:
d d1 2
d d1 2
3
2
1
0 2 4
dd
1
2
129
2212
2,1 sm102
dd
RTÐ
A
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 68/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Non-ideal Binary
ethanol (1) - water (2)
Ð D1 2 1 2, ,
interpolation roughly logarithmic for Ð
MS and Fick diffusivities differ
only at x x1 21 1 and isÐ D1 2 1 2, ,
25 oC
Ð1 2,
5
2
1
0.50 1x1
D1 2,
Ð D1 2 1 2
910, ,,
m s2 1
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 69/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
10 Dimensionless Groups
Shi ji j
i j
k d
Ð,,
,
Sc i ji jÐ,,
Re vd
Sherwood number
Reynolds number
Schmidt number
mass transfer
fluid flow
mixture property
diameterfilm thickness
Re 1 laminarRe 1 turbulent
Sc 1 in gasesSc 103 in liquids
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 70/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Mass transfer coefficientsoutside rigid interfaces
kg
Ði j i j, ,
/
.
0 3 2
1 3
, ,and Ði j are values in the fluid outside
for short times
for small particles
for large particles
boundary layer theory
kÐ
ti ji j
,,. 113
kÐ
di ji j
,, 2
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 71/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
surface active agentsswept to the back
these immobilise the drop or bubble when
df
g
1 2/
f is the fouling (fudge) factorf 0 2. in clean liquidsf 1 in dirty liquids
the transition is at d 1 mm
cause surface tension gradients
d
Mobile or Rigid?
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 72/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Mass transfer coefficientsBubbles with mobile interfaces
bubbles:
kg
Ði j i j, ,
/
.
0 4
2 23
1 6
and are for the liquid (outside)
outside
kg
Ði jd
i jd
, ,
/
.
0 4
2 23
1 6
inside (?)
Ði jd,
Ði j,
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 73/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Mass transfer coefficientsmobile drops in a liquid
Ði jd,
Ði j,
d
mobile drops in a liquid
calculate the rigid and mobile (bubble) coefficients separately
interpolate using d
k k ki j i jmobile
i jrigid
, , ,.
.
.
1
1 0 3
0 3
1 0 3
(these differ for the outer and inner coefficients)
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 74/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
Mass transfer coefficientsdrops in a gas
dg
2
1 2
/
dg
4
1 2
/
rigid sphere mobile oscillating drop drop shatters
februari 2018 Åbo Akademi - Biskopsgatan 8, 20500 Åbo RZ 75/76
Transport processes 424522
Multi-component mass transferusing transparancies that accompany ”Mass Transfer in Multicomponent mixtures”
by J.A. Wesselingh & R. Krishna, Delft University Press (2000)
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