View
173
Download
6
Category
Preview:
Citation preview
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
Introducción a la Ecuación de Maxwell-Stefan para transferencia de masa
Wesselingh, J.A. and Krishna, R., Mass Transfer in Multicomponent Mixtures,
VSSD, Delft (2000)
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
J Ddcdzi
i= − ciα
ciβ
βαdiffusivity
kDzii=
∆
mass transfer coefficient
Fick’slaw
Ji : flux of i with respect to the mixture
J Dcz
k ci ii
i i= − = −∆∆
∆
∆z
∆c c ci i i= −β α
Mass Transfer as you have learned it
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
N Ddcdz
Nxi ii
i= − +differential equation
N k c Nxi i i i= − +∆difference equation123 {
Ni : flux with respect to an interface
diffusion flux
drift flux
Stefan or drift correction
N Nii
=∑flux of mixture
with Drift
− k s ci i i∆
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
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
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
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)
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
0.6
0.5
0.4
0.6
0.4
0.2
0.00 10 20
N2A
B
H2
CO2
A
B
B
A
mole fraction xi
reverse diffusion
timeh
Three Gases (2)
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
Two Cations
H+
Cl -
Na+
Cl-
cationpermeable membrane
highconcentration
lowconcentration
excess +chargeand electrical field
so Na+ can move against its concentration gradient!
H+ moves rapidly H+
Na+
1
3
2
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
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)
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
He
298 K
100 kPa
Ar
298 K
101 kPa(for example)
main reason:viscous flow
retards He, accelerates Ar
∆p
N NHe Ar= −
2 Gases in a Porous Plug (2)
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
1kg
1 m
the potential difference is the work required to change the condition of the weighthere: ∆ψ ∆= = ≡mg z 981 9 81. .J ( Nm)
or, per mole ∆ψ ∆i iM g z=
1kg
Fi
the driving force is the negative potential gradient:
Fddz
M gii
i= − = −ψ
the force is downwards
Gravity - a simple Potential
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
Chemical Potential
xi γ i
mixture ( ) ii aRTTpconst ln, +=µchemical potential
activity
pure i (one mole)
( )µi const p T∗ = ,
iiii aRT ln−=µ−µ=µ∆ ∗
activity coefficientwork required: change in the chemical potential
iii xa γ=
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
chemical potential in an ideal solution ii xRTTpconst ln),( +=µ
in an ideal gas
+=µpp
RTTpconst ii ln),(
partial pressure
µ in an Ideal SolutionIntroduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
momentum ‘in’ momentum ‘out’
forces
∑ invm )( &
change of momentum ∑∑∑ +−= Fvmvm
dtmvd
outin )()()(
&&
∑ outvm )( &
∑F
Momentum Balance
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
(2)(1)
1u2u
zH2 CO2
dzz +
Moving Through Each Other
species velocities
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
dzz +z
area A
volume Adz
zAp1 dzz
Ap +1
)( 1221 uuppforce
friction −∝
dzdp
forcedriving 1−=
Forces on Hydrogen (1)
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
dzda
aRT
dzad
RTdzd
F i
i
iii −=−=µ−= ln − RT
xdxdzi
i
in ideal solutionsfor a given T and p
Driving Force (per Mole of i)
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
( )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
Maxwell-Stefan Equation
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
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
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
engineer
scientist
great scientist
one day
one week
one month
∑≠
−ζ=∆ψ∆−
ijjijji uux
z)(,
( )∑≠
−ζ=µ−ij
jijjii uux
dzd
,
( )∑≠
−ζ=µ∇−ij
jijjii uuxrr
,
Why Simplify?
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
eddies & large scale convection
‘film ’: no eddies
phase boundary
two thin, one dimensional ‘films’next to the phase boundary
the other phase
Film Theory
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
gases
liquids
membrane
in a solid particle
∆zd≈10
∆z ≈ −10 4 m
∆z ≈ −10 5 m
∆z = − −10 107 4K m
∆zd
Thickness of Films
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
Difference Form of Force
za
aRT
za
RTz
F i
i
iii ∆
∆−=∆
∆−=∆µ∆−= ln
zx
xRT i
i ∆∆−
in ideal solutions
for a given T and p
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
∆µ1
RT+1
0
-1
0 2
approximate
exact
‘approximate’ works out better in difference equations
α
β
1
1lna
a
( )αβ
αβ
+−
=∆
11
11
1
1
5.0 aa
aa
aa
α
β
1
1
a
a
− ∞
-2
Approximation
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
growing bubble of CO2 ∆z ≈ −10 5 m
mole fractionof CO2
x1 0003α = .
FRTx
dxdz
RTx
xz1
1
1
1
1= − ≈ − ∆∆
x1 0 001β = .
Forces in a Glass of Beer
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
1 kg of CO2 FRT
zx
x11
1
5
8
9
8314 30010
0 0020 002
24 10
5 10
≈ −
= − × × −+
= ×
≈ ×
−
∆∆
. ..
.N
mol CO
kgfkgCO
2
2500 000 tons
1025
particles
Supertanker!
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
J. C. MaxwellJ. C. Maxwell J. StefanJ. Stefan
The great-great-GrandfathersIntroduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
A. FickA. Fick
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
Friction 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’
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
Maxwell-Stefan diffusivity of large molecules in dilute liquids (not gases)
( ) sm
10104.01010106
300314.8 29
9323−
−− ≈×××××
×→
ÐRT
1212
,,
≡ζ
ÐRT
d122 13, =
A πη
Diffusion and Friction Coefficients
2,12,1 Ð
RT≡ζ
each others ‘inverse’we use both
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
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
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
only relative
velocities
∑≠
−ζ=ij
jijjii uuxF )(,
bootstrap
‘floating’ transport relations: have to be ‘tied’ to surroundings
Bootstrap (1)
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
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)
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
∑ −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:
Fluxes
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
x1α
β1xx1
positivedirection
α β
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
α1a
β1a
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
876film
iu
0iu
α β0
icaverage concentration ic
species velocity(depends on position in
film)species velocity at the average composition
positive velocity
Average Velocity
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
)(1
2122,11
1 uuxka
a −=∆−cxc 1×
)(1
21122,11
11 NxNx
ckaa
x −=∆−
Differences with Fluxes
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
∑≠
−=∆−
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
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
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 Coefficient
s
in pores
in pores
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
x1α
x1
0α β
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 ∆−=
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
0β
x1 05= .x1β
α
x1α
Stripping - concentrated
12,112
2,11 2 xckx
x
ckN ∆−=∆−=
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
Polarisationwater (1) permeates
u1
x2α
x2 ∆x2
salt (2) does not:
x xx
2 22
2= +α
∆
transport:
also
so
∆xx
2
2α
3
2
1
0 0 1 2
exact
− = −∆xx
uk
2
2
1
1 2
10
,
x u2 2 0small, =
∆xx
uk
uk
2
2
1
12
1
12
2
2α
=−
,
,uk
1
12, exact solution:∆xx
uk
2
2
1
12α
=
exp
,
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
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 Drop
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
NN
1
2
= ν
− = −∆xx N x N
k c12 1 1 2
12,
− = −∆xx N x N
k c21 2 2 1
1 2,
Nk c
x xx1
12
2 11= −
−,
ν∆ N
k c
x xx2
12
1 22= −
−,
ν∆
example ν = 2 x x1 2 0 5= = .
N k c x1 12 14= − , ∆ ( )N k c x2 12 22= − − , ∆
Fluxes from Vaporising Drop
Stefan (drift) corrections
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
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
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
( )− = − =+
∆xx N x N
k c
x x N
k c12 1 1 2
1 2
2 1 1
12
2
, ,
N N2 12= −
( )Nk c
x xx1
12
2 11
2
210 10
07 2 0306= −
+= − ×
+ ×−
−,
. ..∆
( )N exact1 0046 0047= − −. : . mol m s2 1
( ) 122 smmol094.0:092.0 −−−−= exactN
Fluxes in Gasification
almost the same
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
Binary Distillation
0β
x1β
N1
α
x1αtransport relation
N2
x2α
x2β
hexane (1)
heptane (2)− = −∆x
x N x Nk c1
2 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= − , ∆
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
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
2ν ν=
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
( ) ( )
( ) ( )
− = − + −
− = − + −
ddz
x u u x u u
ddz
x u u x u u
µ ζ ζ
µ ζ ζ
112 2 1 2 13 3 1 3
22 1 1 2 1 2 3 3 2 3
, ,
, ,
forces per mole of ‘1’
forces per mole of ‘2’
Ternary - per mole of i
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Introduccion a la Ecuación de Maxwell-Stefan para Transferencia de Masa
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
( ) ( )
( ) ( )
− = − + −
− = − + −
xddz
x x u u x x u u
xddz
x x u u x x u u
11
1 2 1 2 1 2 1 3 1 3 1 3
22
2 1 1 2 2 1 2 3 2 3 2 3
µ ζ ζ
µ ζ ζ
, ,
, ,
forces per mole of mixture
these should cancel:
ζ ζ2 1 12 2 1 12 2 1 12, , , , , ,≡ ≡ ≡Ð Ð k k
Ternary - per mole of Mixture
Introduccion a la Ecuación de Maxwell-Stefan
Dr. Ing. Bernardo Yépez Silva Santisteban. EPG-UNALM 2006
binary
ternary
quaternary
ckNxNx
ckNxNx
x3,1
3113
2,1
21121
−+−=∆−
ckNxNx
ckNxNx
x3,2
3223
1,2
12212
−+−=∆−
More Components
Introduccion a la Ecuación de Maxwell-Stefan
PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Recommended