Entropy GMS IrrevThermo Presentacion Haase

7/21/2019 Entropy GMS IrrevThermo Presentacion Haase

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GMS Equations FromIrreversibleThermodynamics

ChEn 6603

References

• E. N. Lightfoot, Transport Phenomena and Living Systems, McGraw-Hill, New York 1978.

• R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena 2nd ed., Chapter 24

McGraw-Hill, New York 2007.

• D. Jou, J. Casas-Vazquez, Extended Irreversible Thermodynamics, Springer-Verlag, Berlin 1996.

• R. Taylor, R. Krishna Multicomponent Mass Transfer, John Wiley & Sons, 1993.

• R. Haase, Thermodynamics of Irreversible Processes, Addison-Wesley, London, 1969.

1Monday, February 27, 12



Outline

Entropy, Entropy transport

Entropy production: “forces” & “fluxes”

• Species diffusive fluxes & the Generalized Maxwell-Stefan Equations

• Heat flux

• Thermodynamic nonidealities & the “Thermodynamic Factor”

Example: the ultracentrifuge

Fick’s law (the full version)

Review




A Perspective

Reference velocities• Allows us to separate a species

flux into convective anddiffusive components.

Governing equations

• Describe conservation of mass,momentum, energy at thecontinuum scale.

GMS equations

• Provide a general relationshipbetween species diffusion fluxesand diffusion driving force(s).

• So far, we’ve assumed:

! Ideal mixtures (inelastic collisions)

!“small” pressure gradients

Goal: obtain a more generalform of the GMS equationsthat represents morephysics

• Body forces acting differentlyon different species (e.g.electromagnetic fields)

• Nonideal mixtures

• Large pressure gradients

(centrifugal separations)




Entropy

Entropy differential:

Total (substantial/material) derivative:Specific volumev

ρ = 1/v

Dρ

Dt= −ρ∇ · v ρ

ωi

Dt= −∇ · ji + σi

D

Dt≡

∂

∂ t+ v ·∇

Chemical potentialper unit mass

µi = µi/M iT ds = de + pdv −

nX

i=1

µidωi

Internal energye

T ρDs

Dt= ρ

De

Dt+ pρD

v

Dt−

nX

i=1

µiρDωi

Dt

T ρDs

Dt= ρ

De

Dt−

p

ρ

Dρ

Dt−

nX

i=1

µiρDωi

Dt

ρDe

Dt= −∇ · q− τ : ∇v− p∇ · v +

X

i=1

f i · ji




Entropy Transport

c h a i n

r ul e . . .

∇(αβ ) = α∇β + β ∇α

T ρDs

Dt= −∇ · q − τ : ∇v − p∇ · v +

n

i=1

f i · ji + p

ρρ∇ · v −

n

i=1

µi (−∇ · ji + σi) ,

= −∇ · q − τ : ∇v +n

i=1

f i · ji +n

i=1

µi∇ · ji −

n

i=1

µiσi,

ρDs

Dt = −∇·

1T q−

ni=1

µi ji Transport of s

+q ·∇ 1

T −

ni=1

ji ·∇ µi

T −

1

T τ : ∇v +

1

T

ni=1

f i · ji −

1

T

ni=1

µiσi Production of s




ρDs

Dt= −∇ · js + σsNow let’s write this in the form:

Look at this term(entropy production due to species diffusion)

ρDs

Dt= −∇ ·

1

T

q−

ni=1

µi ji

Transport of s

+q ·∇

1

T

−

ni=1

ji · ∇

µi

T

− 1

T τ : ∇v +

1

T

ni=1

f i · ji − 1

T

ni=1

µiσi Production of s

js =1

T

q−

ni=1

µi ji

σs = q · ∇

1

T

−

ni=1

ji · ∇

µi

T

− 1

T τ : ∇v +

1

T

ni=1

f i · ji − 1

T

ni=1

µiσi,

= −

q

T ·∇ lnT −

ni=1

ji ·

∇ µi

T −

1

T f i−

1

T τ : ∇v −

1

T

ni=1

µiσi

T σs = −q · ∇ ln T −

ni=1

ji ·

∇T,pµi +

V i

M i∇ p− f i

Λi

−τ : ∇v−

ni=1

µiσi

diffusive transport of entropy

pr od u c t

i on

of en t r o

p y

∇

µi

T =

∂ µi

∂ T ∇

T

T +

1

T

∂ µi

∂ p ∇ p+

1

T ∇T,pµi,

=1

T

1

M i

∂ µi

∂ p ∇ p+∇T,pµi

,

=1

T

V i

M i∇ p+∇T,p µi

Note that wehaven’t “completed”the chain rule here.We will apply it to

species later...




Part of the Entropy Source Term…Why can we add this “arbitrary” term?What does this term represent?

From physical reasoning (recall di represents force per unit volumedriving diffusion) or the Gibbs-

Duhem equation,

n

i=1

di = 0

ωi

M i=

xi

M

φi = ciV i

µi =µi

M i

ji = ρωi (ui − v)

V i Partial molarvolume.

cRT di = ci∇T,pµi + (φi − ωi)∇ p− ωiρf i −

n

k=1

ωkf k

ni=1

ji ·

∇T,pµi +

V i

M i∇ p− f i Λi

=

ni=1

ji ·

Λi −

1

ρ∇ p+

nk=1

ωkf knX

i=1

ji · Λi =nX

i=1

ρωi(ui − v) ·

"∇T,p µi +

V iM i

− 1

ρ

∇ p− f i +

nXk=1

ωkf k

#!,

=nX

i=1

(ui − v) ·

ci∇T,pµi + (φi − ωi)∇ p− ρωi

f i −

nXk=1

ωkf k

! | z

cRT di

,

= cRT

n

Xi=1 di · (ui − v),

= cRT

nXi=1

1

ρωidi · ji




The Entropy Source Term - Summary

Interpretation of each term???

ρDs

Dt= −∇

· js + σs

From the previous slide:

i=1

ji · Λi = cRT

i=1

di · ji

ρi

js =1

T

q−

ni=1

µi ji

cRT di = ci∇T,pµi + (φi − ωi)∇ p− ωiρ

f i −

nk=1

ωkf k

T σs = −q · ∇ lnT −

ni=1

ji ·∇T,pµi +

V i

M i∇ p− f i

Λi

−τ : ∇v−

ni=1

µiσi

= −q · ∇ lnT 1

−

n

i=1cRT

ρi

di · ji

2

− τ : ∇v 3

−

n

i=1µiσi

4




!s ! Forces " Fluxes

Fundamentalprinciple ofirreversible

thermodynamics:

σs =

α

J αF α

T σs = −q·∇ lnT −

n

i=1

cRT

ρi di · ji − τ : ∇v −

n

i=1

µiσi

Flux, J α Force, F α

q −∇ lnT

ji −cRT

ρi

di

τ −∇v

L!" - Onsager (phenomenological) coefficients

J α = J α(F 1, F 2, . . . , F β ; T, p, ωi)

Lαβ ≡∂ J α

∂ F β

J α =β

∂ J α

∂ F β

F β +O (F βF γ )

≈

β

LαβF β

Lαβ = Lβα




Species Diffusive Fluxes

Tensorial order of “1” anyvector force may contribute.

From irreversible thermo:

Fick’s Law:

Generalized Maxwell-Stefan Equations:

Index form: n-1 dimensional matrix form

DT

i - Thermal Diffusivity

Flux: J α

q ji τ

Force: F α −∇ lnT −cRT

ρidi −∇v

ρ(d) = −[Bon]( j)−∇ lnT [Υ](D )di = −

nXj6=i

xixj

ρÐ ij

ji

ωi

− jj

ωj

−∇ lnT

nXj6=i

xixjαT ij

αT ij =

1

Ð ij

D

T i

ρi−

DT i

ρj

Dij - Fickian diffusivity

( j) = −ρ [L] (d) +∇ ln T (β q) ji = −

n−1X

j=1

LijcRT ρj

dj − Liq ∇ lnT

ji = −ρ

n−1X

j=1

D

ijdj −DT i ∇ lnT

( j) = −ρ [D] (d)−D

T ∇ lnT




Constitutive Law: Heat Flux

Choose Lqq= $T toobtain “Fourier’s Law”

“Dufuor” effect - mass drivingforce can cause heat flux!

Usually neglected.

Tensorial order of “1”

anyvector force may contribute.

Flux: J α q ji τ

Force: F α −∇ lnT − cRT

ρidi −∇v

q = −Lqq∇ lnT −

n

i=1

Lqi

cRT

ρi

di

The “Species” term is typically included here, even though it does not come fromirreversible thermodynamics. Occasionally radiative terms are also included here...

here we have substitutedthe RHS of the GMS

equations for di.

q = −λ∇T | z Fourier

+

nXi=1

hi ji

| z Species

+

nXi=1

nXj6=i

cRTDT xixj

ρiÐ ij

ji

ρi− jj

ρj

| z

DufourNote: the Dufour effect

is usually neglected.




Observations on the GMS Equations

What have we gained?

• Thermal diffusion (Soret/Dufuor)& its origins.!Typically neglected.

• “Full” diffusion driving force!Chemical potential gradient (rather than

mole fraction). More later.

! Pressure driving force.

! When will %i" #i? More later.

! Body force term.

! Does gravity enter here?

Onsager coefficients themselvesnot too important from a“practical” point of view.

Still don’t know how to get the

binary diffusivities.

cRT di = ci∇T,pµi + (φi − ωi)∇ p− ωiρ

f i −

n

k=1ωkf k

di =

n

j=1

xiJj − x

jJi

cDij−∇ lnT

n

j=1

xixjαT ij




The Thermodynamic Factor,#

xi

RT ∇T,pµi =

xi

RT

n−1j=1

∂ µi

∂ xj

T,p,Σ

∇xj ,

= xi

RT

n−1j=1

RT

∂ ln γ ixi

∂ xjT,p,Σ

∇xj ,

= xi

n−1j=1

∂ lnxi

∂ xj+

∂ lnγ i

∂ xj

T,P,Σ

∇xj ,

=

n−1

j=1δ ij + xi

∂ lnγ i

∂ xj T,p,

Σ∇xj ,=

n−1j=1

Γij∇xj

& - Activity coefficientMany models available(see T&K Appendix D)

Γij ≡ δ ij + xi

∂ lnγ i

∂ xj

T,p,Σ

µi(T, p) = µi + RT ln γ ixi

µi = µi(T,p,xj)

∇T,pµi =n−1

j=1

∂ µi

∂ xj

T,p,P∇xj

di =

xi

RT ∇

T,pµi +

1

ctRT (φi−

ωi)∇ p−

ρi

ctRT f

i−

nk=1

ωkf k

di =n−1

j=1

Γij∇xj + 1

ctRT (φi − ωi)∇ p−

ρi

ctRT f i −

n

k=1

ωkf k Note: forideal gas,

p = ctRT




Example: The UltracentrifugeUsed for separating mixtures based on components’ molecular weight.

f = f i = Ω2r

Ω

depleted in

dense species

For a closed centrifuge (no flow) with a known

initial charge, what is the equilibrium species profile?

Consider a closed system...




∂ρi

∂ t= −∇ · ni + si

∂ ni

∂ r= 0

ni = ρivr + ji,r = 0 ji,r = J i,r = 0

Species equations:

GMS Equations:

The generalized diffusion driving force:

di =n−1

Xj=1

Γij∇xj + 1

ctRT

(φi − ωi)∇ p− ωiρ

ctRT f i −

n

Xk=1

ωkf k!

di =

nX

j=1

xiJj − xjJi

cDij

= 0

steady, 1D,no reaction

For an ideal gas mixture, %i = xi, and !ij = "ij.

We don’t know d p/ dr or xi0

(composition at r = 0).

0 =

n−1Xj=1

Γij

dxjdr

+ 1

ctRT (φi − ωi)

d p

dr−

ωiρ

ctRT

Ω2r −

nXk=1

ωkΩ2r

!

n−1

Xj=1

Γijdxj

dr

= 1

ctRT

(ωi − φi)d p

dr

dxi

dr=

1

ctRT (ωi − xi)

d p

dr


d 1 d



For species i,

Species mole balance:

Z rL

0

p xi r dr = p∗

x∗

i

r2

L

2

Must know p(r) and

xi(r) to integrate this.

Momentum:

at steady state(no flow):

d p

dr = ρ

nX

i=1

ωif r,i = ρΩ2r

∂ρv

∂ t

= −∇ · (ρvv)−∇ · τ − ∇ p + ρ

nX

i=1

ωif i

We don’t know p0

(pressure at r = 0).

The momentum equationgives the pressure profile,

but is coupled to the species

equations through M .d p

dr

= ρΩ2r =

pM

RT

Ω2r

rL

0

cxi2πr dr =Z rL0

c

∗

x

∗

i 2πr dr

dxi

dr=

1

ctRT (ωi − xi)

d p

dr

* indicates the

initial condition(pure stream).




* indicates the initial condition (pure stream).

c =RT

Total mole balance (at equilibrium):

Substitute p(r) and solve this for p0...

Z V

cdV =

Z V

c∗dV

Z rL

0

cr dr = c∗r2

L

2Z rL

0

pr dr = p∗r2

L

2

dV = L2πrdr

dxi

dr =

M

RT (ωi − xi)Ω

2r

Solve theseequations:

With theseconstraints:

Z rL

0

pr dr = p∗r2

L

2

Z rL

0

p xi r dr = p∗

x∗

i

r2

L

2

d p

dr = ρΩ

2r = pM

RT Ω

2r Note: M couples all of theequations together andmakes them nonlinear.

Note: for tips on solving ODEs numerically in Matlab, see my wiki page.17Monday, February 27, 12

http://www.che.utah.edu/~sutherland/wiki/index.php/ODEs

http://www.che.utah.edu/~sutherland/wiki/index.php/ODEs



Example: separation of Air into N2, O2.




Fick’s Law (revisited)

Ignoring thermal diffusion,

How do we interpret each term?When is each term important?

Notes: [ D]=[ B]-1[#]

In the binary case: D11=#11 '12

For ideal mixtures: [#]=[ I ]

di = −

j=1

xixj

ρDij ji

ωi

− jj

ωj−∇ lnT

j=1

xixjα

T

ij

= −

nj=1

xjJi − xiJj

cDij

−∇ lnT

nj=1

xixjαT ij J = −c[B]−1(d)−∇ lnT (DT )

This is the same [ B] matrix asbefore (T&K eq. 2.1.21-2.1.22)

di =n−1

j=1

Γij∇xj + 1

ctRT

(φi − ωi)∇ p− ρi

ctRT f i −

n

k=1

ωkf k

(J) = −c[B]−1[Γ](∇x)

1

− ∇ p

RT [B]−1 ((φ) − (ω))

2

− ρ

RT [B]−1[ω] ((f ) − [ω](f + f n))

3


R i



Review:Where we are, where we’re going…

Accomplishments• Defined “reference velocities” and

“diffusion fluxes”

• Governing equations formulticomponent, reacting flow.

!mass-averaged velocity…

• Established a rigorous way tocompute the diffusive fluxes fromfirst principles.

!Can handle diffusion in systems ofarbitrary complexity, including:

! nonideal mixtures, EM fields, large pressure& temperature gradients, multiple species,chemical reaction, etc .

• Simplifications for ideal mixtures,negligible pressure gradients, etc .

• Solutions for “simple” problems.

Still Missing:• Models for binary diffusivities.

! Given a model, we are good to go!

Roadmap:

• Models for binary diffusivities.(T&K Chapter 4) - we won’tcover this...

• Simplified models formulticomponent diffusion

• Interphase mass transfer (surface

discontinuities)

• Turbulence - models for diffusionin turbulent flow.

• Combined heat, mass, momentumtransfer.

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Entropy GMS IrrevThermo Presentacion Haase