fluxes in multicomponent systems - sutherland · reference velocities denote the velocity of...
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Fluxes in Multicomponent Systems
ChEn 6603
Taylor & Krishna §1.1-1.2
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Outline
Reference velocitiesTypes of fluxes (total, convective, diffusive)• Total fluxes
• Diffusive Fluxes
ExampleConversion between diffusive fluxes
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Reference VelocitiesDenote the velocity of component i as ui.
v =n�
i=1
�iui“Mass-Averaged” velocity ωi 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
�iui“Volume-Averaged” velocity�i � ciV̄i is the volume fraction of component i.
ci = xi c = ρi/Mi - molar concentration of i.
- partial molar volume of component i (EOS).V̄i
ua =n�
i=1
aiuiArbitrary reference velocityn�
i=1
ai = 1 ai is an arbitrary weighting factor.
Why ∑ai = 1?
T&K Table 1.2
What assumptions have 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 due
to 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 Flux of component i:ni � ⇥i�ui
= �iui
Total Mass Flux:
n �n�
i=1
ni
=n�
i=1
�iui,
= �v
Molar Flux of component i: Total Molar Flux:
Ni � xicui
= ciui
Nt =n�
i=1
Ni
=n�
i=1
ciui
= cu
ρ - mixture mass density
Total Fluxes may be written in terms of a component’s velocity and specific density
Note: there is a typo in T&K eq. (1.2.5)
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Diffusive FluxesConcepts:
• The total flux is partitioned into a convective and diffusive flux.
• The convective flux is defined in terms of an average velocity.
• The diffusive flux is defined as the difference between the total and convective fluxes. It is only defined once we have chosen a convective flux!
ji = ni � �iv= �i(ui � v)= ni � ⇥int
Total mass flux of component i
Mass diffusive flux of component i relative to a mass-averaged velocity.
Convective flux of component i due to a mass-averaged velocity
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Diffusion Flux
Total Flux
Convective Flux
Comments
Diffusive FluxesDiffusion fluxes are defined as motion relative to some reference velocity.
(Diffusion flux) = (Total Flux) - (convective flux)
ji ni �iv Mass diffusive flux relative to a mass-averaged velocity
jui �iuniMass diffusive flux relative to
a molar-averaged velocity
Ji Ni ciu Molar diffusive flux relative to a molar-averaged velocity
Jvi Ni civ Molar diffusive flux relative
to 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 FluxesT&K Table 1.3
Note: typo in table 1.3 (missing v superscript)
n�
i=1
�i
xiJv
i = 0
n�
i=1
ji =n�
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)
Spec
ies
Mol
e Fr
actio
ns
Acetone(1)Methanol(2)Air(3)
Example - Stefan TubeAt steady state (1D),Air
LiquidMixture
z = �
z = 0
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
1.6 x 10−4
z (m)
velo
city
(m/s
)
Molar AverageMass Average
0 0.05 0.1 0.15 0.2 0.250
0.5
1
1.5
2
2.5
3 x 10−3
z (m)
Spec
ies
Velo
citie
s (m
/s)
AcetoneMethanolAir
We will show this later...
0 0.05 0.1 0.15 0.2 0.250
0.5
1
1.5
x 10−3
z (m)
Diff
usiv
e Fl
uxes
for A
ceto
ne
jju
JJv
Molar, mass averaged velocities Acetone diffusive fluxes ji = ⇢i(ui � v)Species velocities ui =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 mass diffusive fluxes relative to mass and molar reference velocities.
To form [Buo]-1 this must be an n-1 dimensional 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)
Let’s try to get a jiu on the RHS ⇒ add & subtract ρiu.
ji = �i(ui � v),= �i(ui � u)⇤ ⇥� ⌅
jui
+�i(u� v)
ji = �i(ui � v)
ji = jui � !i
nX
j=1
jujThis 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] (cont’d)
Eliminate from the above equation...jun
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
n�1⇧
j=1
⇤�⇥n
xn
xj
⇥jjuj + juj
⌅,
=n�1⇧
j=1
⇤�ij � ⇥i
�1� ⇥n
xn
xj
⇥j
⇥⌅juj
ji = jui � !i
nX
j=1
juj
(j) = [Bou](ju)
juneliminate
separate out jun
gather terms on jun
jun = ��n
xn
n�1�
i=1
xi
�ijui�
nX
i=1
xi
�ijui = 0
ji = jui � !i
2
4jun +n�1X
j=1
juj
3
5
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