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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