multicomponent mass transfer:fluxes & velocities
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Fluxes in
Multicomponent Systems
ChEn 6603
Taylor & Krishna 1.1-1.2
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Outline
Reference velocities
Types of fluxes (total, convective, diffusive)
Total fluxes
Diffusive Fluxes
Example
Conversion between diffusive fluxes
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Reference Velocities
Denote the velocity of component i as ui.
v =
n
i=1
iuiMass-Averaged velocity i is the mass fraction of component i.
Molar-Averaged velocity u =
n
i=1
xiui xi is the mole fraction of component i.
uV=
n
i=1
iuiVolume-Averaged velocityi ciVi is the volume fraction of component i.
ci =xi c = i/Mi - molar concentration ofi.
- partial molar volume of component i (EOS).Vi
ua=
n
i=1
aiuiArbitrary reference velocity
n
i=1
ai = 1 ai is an arbitrary weighting factor.
Why ai = 1?
What assumptionshave we made here?
How do we define ai such that ua = uj?
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Types of Fluxes
Total Flux
Amount of a quantity passing through a unit surface area per unit time.
Convective Flux: Amount of a quantity passing through a unit surface area per unit time that
is carried by some reference velocity.
Diffusive Flux:
Amount of a quantity passing through a unit surface area per unit time dueto diffusion.
The difference between the Total Flux and the Convective Flux.
Cannot be defined independently of the total & convective fluxes!
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Total Fluxes
Mass Fluxof component i:
ni iui
= iui
Total Mass Flux:
n
n
i=1
ni
=
n
i=1
iui,
= v
Molar Fluxof component i: Total Molar Flux:
Ni xicui
= ciui
Nt =
i=1
Ni
=
n
i=1
ciui
= cu
- mixture mass density
Note: there is a typoin T&K eq. (1.2.5)
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Diffusive Fluxes
Concepts: The totalflux is partitioned into a convective and diffusive flux.
The convective fluxis defined in terms of an average velocity.
The diffusive fluxis defined as the difference between the
totaland convective fluxes. It is only defined once we havechosen a convective flux!
i = ni iv
= i(ui v)
= ni int
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Diffusion
Flux
Total
Flux
Convective
Flux Comments
Diffusive Fluxes
Diffusion fluxes are defined as motion relative to some reference velocity.
(Diffusion flux) = (Total Flux) - (convective flux)
ji ni ivMass diffusive flux relative to
a mass-averaged velocity
ji
iuniMass diffusive flux relative to
a molar-averaged velocity
Ji Ni ciuMolar diffusive flux relativeto a molar-averaged velocity
v
i Ni civMolar diffusive flux relativeto a mass-averaged velocity
ji = ni iv
= i(ui v)
= ni int
Ji = Ni ciu
= ci(ui u)
= Ni xiNt
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Linear Dependence of Diffusive Fluxes
Note: typo in table 1.3
(missing v superscript)
n
i=1
i
xi
Jv
i= 0
i=1
ji =
i=1
(iui iv)
= n
i=1
(iui iv),
=
v +
iui
= v + v,
= 0.
n
i=1
ji = 0For mass diffusive flux relative
to mass-averaged velocity:Appropriately weighted
diffusive fluxes sum to zero.
Why?
Total fluxes are all
independent.
Convective fluxes are not all
independent.
From the previous slide: ji = i (ui v)
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0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1
z (m)
SpeciesMoleFractions
Acetone(1)
Methanol(2)
Air(3)
Example - Stefan TubeAt steady state (1D),Air
Liquid
Mixture
z =
z =
ni = i
Ni = i
0 0.05 0.1 0.15 0.2 0.25
1.1
1.2
1.3
1.4
1.5
1.6x 10
4
z (m)
velocity(m/s)
Molar Average
Mass Average
0 0.05 0.1 0.15 0.2 0.25
0
0.5
1
1.5
2
2.5
3x 10
3
z (m)
SpeciesVelocities(m/s)
Acetone
Methanol
Air
We will show this later...
0 0.05 0.1 0.15 0.2 0.250
0.5
1
1.5
x 103
z (m)
DiffusiveFluxesforAcetone
j
ju
J
Jv
Molar, mass averaged velocities Acetone diffusive fluxes ji = i(ui v)Species velocitiesui =
ni
i
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Conversion Between Fluxes
(ju) = [Buo](j)
(j) = [Bou](ju)
We can convert between various diffusive fluxes via linear transformations.
Conversion between massdiffusive fluxes relative to massand molar reference velocities.
To form [Buo]-1 this must be an n-1dimensional system of equations!
(why?)
Derivation of[Bou]...
n
i=1
jui
=
n
i=1
iui
n
i=1
iu,
= v u,
1
n
i=1
jui
= v u.
jui= i(ui u)
Lets try to get a jiu on the RHS add & subtract iu.
ji = i(ui
v),
= i(ui u)
jui
+i(u v)
ji = i(ui v)
ji = jui !iX
j=1
juj
This is an n-dimensional set of
equations. If we derive [Bou]from this, it will not be full-rank.
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Derivation of[Bou] (contd)
Eliminate from the above equation...ju
n
we have n-1 of these
equations (i=1...n-1)
Bouij = ij i
1
n
xn
xj
j
The inverse [Buo]=[Bou]-1 can be obtained
from equations A.3.21-A.3.23 in T&K(note the typo in A.3.22).
[B]1 = [A]
1
1
[A]
1 (u) (v)T [A]
1,
= 1 + (v)T [A]1 (u)
ji = jui i
n1
j=1
n
xn
xj
jjuj +j
uj
,
=
n1j=1
ij i
1
n
xn
xj
j
juj
ji=
ju
i
!iXj=1
ju
j
(j) = [Bou](ju)
ju
n
eliminate
separateout j
n
gather
termson j
n
jun = n
xn
n1
i=1
xi
i
juiX
i=1
xi
i
jui = 0
ji = jui !i
24jun +
n1Xj=1
juj
35
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