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Moshe  Matalon                                                                                  University  of  Illinois  at  Urbana-­‐Champaign   1  

Lecture 2 Constitutive Relations

Moshe  Matalon  

⇢De

Dt= �pr · v + ��r · q

⇢Dh

Dt=

Dp

Dt+ ��r · q

⇢Dv

Dt= �rp+r ·⌃+ ⇢g

⇢DYi

Dt+r · (⇢YiVi) = !i i = 1, 2, . . . , N

@⇢

@t+r · ⇢v = 0

h =NX

i=1

Yi

ho

i

+

ZT

T

o

cp

dT

We already have expressions for the viscous stress tensor ⌃ and the dissipa-

tion function �. We need constitutive relations for the heat flux vector q, thedi↵usion velocities Vi and the rates of consumption/production !i.

Conservation equations

p = ⇢RTNX

i=1

Yi

Wi

{

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q = ��rT + ⇢NX

i=1

YihiVi + qR

The total energy flux q is the sum of the separate energy fluxes, including the

fluxes due to di↵usion and the radiant flux qR , but neglecting the Dufour heat

flux (due to di↵erence in di↵usion velocities)

Heat flux vector

⇢Dh

Dt=

Dp

Dt+ �+r · (�rT )�r ·

NX

i=1

⇢YiVihi

Radiative heat transfer will also be removed and added only when necessary.

⇢Dh

Dt=

Dp

Dt+ �+r · (�rT )�r ·

NX

i=1

⇢YiVihi

| {z }the energy transfer is not

just by conduction, but also

through di↵usion fluxes

of the di↵erent species

�r · qR

Moshe  Matalon  

h =NX

i=1

Yi

ho

i

+

ZT

T

o

cp

dT

⇢Dh

Dt=

Dp

Dt+ �+r · (�rT )�r ·

NX

i=1

⇢YiVihi

⇢D

Dt

ZT

T

o

cp

dT +NX

i=1

⇢DY

i

Dtho

i

r ·NX

i=1

⇢Yi

Vi

[ho

i

+

ZT

T

o

cp

i

dT ]

r ·NX

i=1

⇢Yi

Vi

ho

i

+r ·NX

i=1

⇢Yi

Vi

ZT

T

o

cp

i

dT

⇢D

Dt

ZT

T

o

cp

dT =Dp

Dt+ �+r · (�rT )�

NX

i=1

✓⇢DY

i

Dt+r·⇢Y

i

Vi

◆

| {z }!

i

ho

i

�r ·NX

i=1

⇢Yi

Vi

ZT

T

o

cp

i

dT

⇢D

Dt

ZT

T

o

cp

dT

!=

Dp

Dt+�+r · (�rT )�r ·

NX

i=1

⇢Yi

Vi

ZT

T

o

cp

i

dT �NX

i=1

!i

ho

i

(I)

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⇢Dh

Dt=

Dp

Dt+ �+r · (�rT )�r ·

NX

i=1

⇢YiVihi

⇢cpDT

Dt+

NX

i=1

hi⇢DYi

Dt=

Dp

Dt+ �+r · (�rT )�r ·

NX

i=1

⇢YiVihi

⇢cpDT

Dt=

Dp

Dt+�+r·(�rT )�

NX

i=1

⇢YiVi·rhi�NX

i=1

hi

⇢DYi

Dt+r · (⇢YiVi)

�

| {z }!i

h =NX

i=1

hiYi ) dh =NX

i=1

dhiYi +NX

i=1

hidYi

=NX

i=1

cpiYidT +NX

i=1

hidYi = cpdT +NX

i=1

hidYi

) Dh

Dt= cp

DT

Dt+

NX

i=1

hiDYi

Dt

Alternative form of the energy equation

Moshe  Matalon  

hi

= ho

i

+

ZT

T

o

cp

i

dT ) rhi

= cp

i

rT

⇢cpDT

Dt=

Dp

Dt+ �+r · (�rT )�

NX

i=1

⇢YiVi ·rhi �NX

i=1

hi!i

(II)

and it can be shown that the two versions are equivalent. Moreover, when all

the cpi are the same, i.e., cpi = cp for all i, the energy equation reduces to

⇢cp

DT

Dt=

Dp

Dt+ �+r · (�rT )�

NX

i=1

ho

i

!i

⇢cpDT

Dt=

Dp

Dt+ �+r · (�rT )�

NX

i=1

⇢cpiYiVi ·rT �

NX

i=1

hi!i

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Di↵usion

In a binary mixture Fick’s law of di↵usion states that the di↵usion mass flux of

a given species is proportional to its concentration gradient

⇢Y1V1 = �⇢D12 rY1

⇢Y2V2 = �⇢D21 rY2

⇢DYi

Dt+r · (⇢YiVi) = !i i = 1, 2

⇢YiVi = �⇢DrYi

⇢DYi

Dt+r · (⇢DrYi) = !i i = 1, 2

For ⇢D = const. )

Fick’s law of binary di↵usion

The constrains Y1V1 + Y2V2 = 0, and Y1 + Y2 = 1, imply that D12 = D21 ⌘ D

DYi

Dt+Dr2Yi = !i i = 1, 2

Moshe  Matalon  

(Curtiss & Hirschfelder 1949)

rXi =NX

j=1

XiXj

Dij(Vj �Vi)

| {z }di↵erence in

di↵usion velocities

+(Yi �Xi)rp

p| {z }pressure

gradient

+NX

j=1

XiXj

⇢Dij

DT

j

Yj� DT

i

Yi

!rT

T| {z }

temperature gradient

Soret e↵ect

+⇢

p

NX

j=1

YiYj(fi � fj)

| {z }di↵erence in

body force

Stefan-Maxwell equations

rXi =NX

j=1

XiXj

Dij(Vj �Vi)

For multicomponent di↵usion in gases

The e↵ects of pressure gradients, and temperature gradients (known as Soret

e↵ect) on di↵usion are typically small. If, in addition, there are no significant

unequal forces acting on the species (often relevant for ions and electrons where

f is due to an electrical field), then

where Dij are the binary di↵usivities, which have been proven to be sym-

metric, i.e., Dij = Dji. For a dilute ideal gas mixture they are independent of

composition, and determined by a simple two-components experiment where a

species in low concentration di↵uses through a second species that is in abun-

dance.

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rX1 =X1X2

D12(V2 �V1)

rX2 =X2X1

D21(V1 �V2)

)

and the Stefan-Maxwell equations reduce to Fick’s law of di↵usion

r(X1 +X2)

X1X2= (V2 �V1)

✓1

D12� 1

D21

◆= 0

Y1V1 = �DrY1

Y2V2 = �DrY2

D12 = D21 ⌘ D

More generally, Fick’s law is recovered if one assumes that the binary di↵usion

coe�cients are all equal, i.e., Dij = D.

YiVi = �DrYi for i = 1, . . . , N

For a binary mixture

Moshe  Matalon  

rXi =NX

j=1

XiXj

Dij(Vj �Vi) =

Xi

D

NX

j=1

Xj(Vj �Vi) =Xi

D

NX

j=1

XjVj �Xi

D Vi

DrXi

Xi=

NX

j=1

XjVj �Vi

D✓rYi

Yi+

rW

W

◆= DrW

W�Vi

YiVi = �DrYi

Xi = WYi/Wi ) rXi

Xi=

rYi

Yi+

rW

W

If the binary di↵usion coe�cients are all equal, i.e., Dij = D, we again cover

Fick’s law.

DrXi

Xi= DrW

W�Vi

it can be shown, after some algebraic manipulations that

PNj=1 XjVj = D(rW/W )

and, since

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The use of the Stefan-Maxwell equations is quite complicated because the dif-

fusion velocities Vi are not expressed explicitly in terms of the concentration

gradients.

NX

i=1

YiDij = 0 for all j

Vi =NX

j=1

Dij rXj

Generalized Fick equations

The coe�cients Dij , which don’t have to be necessarily positive, satisfy

Dij = Dji, for all i, j

would be more convenient for the evaluation of the di↵usion term in the species

equations, but the ”Fick di↵usivities” Dij are not simply related to the binary

di↵usivities Dij , and they are concentration-dependent.

Note that there are many definitions for multicomponent di↵usivities; the one

quoted here (with the Dij symmetric) follow Curtiss & Bird (1999).

Moshe  Matalon  

Stefan-Maxwell

V1 = D11rX1 +D12rX2

V2 = D21rX1 +D22rX2

Multi-component di↵usion

D12 = D21D11Y1 +D12Y2 = 0

D12Y1 +D22Y2 = 0

D12 = D21

One can find the relation between the multi-component di↵usivities Dij and thebinary di↵usivities Dij . For a binary mixture

rX1 =X1X2

D12(V2 �V1)

rX2 =X1X2

D21(V1 �V2)

D11 = � Y22

X1X2D12

D22 = � Y12

X1X2D12

D12 = D21 =Y1Y2

X1X2D12

)

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Expressions for ternary and quaternary mixtures are available, but are more

complicated.

D11 = �(Y2+Y3)

2

X1D23+ Y2

2

X2D13+ Y3

2

X3D12

X1D12D13

+ X2D12D23

+ X3D13D23

D12 =Y1(Y2+Y3)

X1D23+ Y2(Y1+Y3)

X2D13� Y3

2

X3D12

X1D12D13

+ X2D12D23

+ X3D13D23

with the additional entries generated by cyclic permutation of the indices.

For the general case, one can write a relation between the matrix of Fick di↵u-

sivities Dij and that of binary (Stefan-Maxwell) di↵usivities Dij ; see Curtiss &

Bird Ind. Eng. Chem. Res. 38:2515-2522 (1999).

For example, for a ternary mixture

Moshe  Matalon  

Vi =NX

j=1

D̂ij rYj

We are interested in rewriting the generalized Fick equations in terms of the

mass fraction

rXi = �W 2

Wi

NX

j=1j 6=i

1

W+ Yi

✓1

Wj� 1

Wi

◆�rYj .

This leads, in general to complicated expressions for the

ˆDij .

which requires expressing rXi in terms of rYi, or using the di↵erential relation

(Bird, Stewart & Lightfoot)

rX1 =X1X2

Y1Y2rY1 rX2 =

X2X1

Y2Y1rY2

For a binary mixture,

D̂11 = �Y2

Y1D D̂22 = �Y1

Y2D D̂12 = D̂21 = D

and we recover Fick’s law, as we should.

Y1V1 = �DY2rY1 +DY1rY2 = �D(Y2 + Y1)rY1 = �DrY1

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Yi ⌧ 1 for i = 1, · · · , N � 1, YN ⇠ 1

Xi ⌧ 1 for i = 1, · · · , N � 1, XN ⇠ 1

W =NX

i=1

XiWi ⇠ WN

NX

i=1

YiVi = 0, ) VN ⌧ 1

YiVi = �Di rYi for i = 1, 2, · · · , N � 1

Dilute mixture

There is an abundant species in the mixture, denoted by N , while all other

species may be treated as trace species (for combustion in air, nitrogen is abun-

dant)

rXi =

NX

j=1

XiXj

Di,j(Vj �Vi) ⇠ �XiVi

Di,Nfor i = 1, 2, · · · , N � 1

rYi ⇠ �YiVi

DiNfor i = 1, 2, · · · , N � 1

Fick’s law may be used as an approximation with the di↵usion coe�cient inter-

preted as the binary di↵usion coe�cient with respect to the abundant species,

i.e., Di = DiN . The equation for N is still complicated, but is not needed

because YN = 1�PN�1

i=1 Yi Moshe  Matalon  

note, however, that this can be only used forN�1 species, with the last obtained

from the constrain

PYiVi = 0.

YiVi = �Di,e↵rYi or XiVi = �Di,e↵rXi

rXi =NX

j=1

XiXj

Di,j(Vj �Vi) ⇡ �

NX

j=1

XiXj

Di,jVi = �XiVi (

Xi

Di,e↵+

NX

j=1j 6=i

Xj

Dij)

�XiVi

Di,e↵⇡ �XiVi (

Xi

Di,e↵+

NX

j=1j 6=i

Xj

Dij)

Di,e↵ =1�Xi

NPj=1j 6=i

Xj/Dij

A useful approximation is the e↵ective binary di↵usivity for the di↵usion of

species i in the mixture, obtained by assuming that Vj ⇡ 0 for all the species,

except for Vi 6= 0.

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⇢D

Dt

ZT

T

o

cp

dT

!=

Dp

Dt+�+r·(�rT )+r ·

NX

i=1

⇢Di

rYi

ZT

T

o

cp

i

dT

!

| {z }�

NX

i=1

!i

ho

i

⇢D

Dt

ZT

T

o

cp

dT

!=

Dp

Dt+�+r·(�rT )�r·

NX

i=1

⇢Yi

Vi

ZT

T

o

cp

i

dT

!�

NX

i=1

!i

ho

i

Introducing the di↵usion law YiVi = �Di rYi in the energy equation

If Di = D for all i

energy transfer due to di↵usion

fluxes caused by imbalances in the

di↵usivities and specific heats.

⇢D

Dt

ZT

T

o

cp

dT

!=

Dp

Dt+�+r·(�rT )+r·⇢D

NX

i=1

rYi

ZT

T

o

cp

i

dT

!

| {z }�

NX

i=1

!i

ho

i

NX

i=1

rYi

Z T

T o

cpidT

!= r

NX

i=1

Yi

Z T

T o

cpidT

!�

NX

i=1

Yir Z T

T o

cpidT

!

= rNX

i=1

Yi

Z T

T o

cpidT �NX

i=1

YicpirT = rNX

i=1

Yi

Z T

T o

cpidT � cprT =

z }| {

rZ T

T o

cpdT � cprT

Moshe  Matalon  

r · ⇢D rZ T

T o

cpdT � cprT

!

⇢D

Dt

ZT

T

o

cp

dT

!=

Dp

Dt+�+r·(�rT )+r · ⇢D

NX

i=1

rYi

ZT

T

o

cp

i

dT

!

| {z }�

NX

i=1

!i

ho

i

⇢cp

DT

Dt�r · (�rT ) =

Dp

Dt+ ��

NX

i=1

!i

ho

i

⇢D

Dt

ZT

T

o

cp

dT

!�r ·

"⇢Dr

ZT

T

o

cp

dT + �⇣1� ⇢Dc

p

�

⌘

| {z }rT

#=

Dp

Dt+ ��

NX

i=1

!i

ho

i

= 0 whenD = �/⇢cp

When all the cpi are also the same, cpi=cp=cp(T ) for all i, the energy equation

reduces to

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Governing Equations

with Fickian di↵usion (dilute mixture) and equal cpi

@⇢

@t+r · ⇢v = 0

⇢Dv

Dt= �rp+r · µ[(rv)+(rv)T] +r(� 2

3µ)(r · v) + ⇢g

⇢DYi

Dt�r · (⇢DirYi) = !i, i = 1, 2, . . . , N � 1

⇢cp

DT

Dt�r · (�rT ) =

Dp

Dt+ ��

NX

i=1

!i

ho

i

Moshe  Matalon  

Transport Coe�cients

The bulk viscosity is negligible in combustion processes; ⇡ 0.

⌃ = µ[(rv) + (rv)T]� 2

3µ(r · v)I

� =1

2µ

✓@vi

@xj+

@vj

@xi

◆2

� 2

3µ(r · v)2

Although the transport coe�cients �, µ in a multicomponent mixture depend in

general on the concentrations, this dependence is generally ignored in theoreti-

cal studies and average quantities are used; they remain dependent on pressure

and temperature.

Similarly W is taken as constant, so that the equation of state reduces to

p = ⇢RT/W .

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The thermal conductivity depends mostly on temperature, and behaves as

� ⇠ T↵/p with 1/2 ↵ 1. More relevant, however, is the thermal di↵u-

sivity �/⇢cp which has the same units and the same temperature and pressure

dependence as the molecular di↵usivities and the kinematic viscosity. In partic-

ular

�/⇢cp ⇠ T↵/p 3/2 ↵ 2

Typical values at p = 1atm are in the range 0.1� 1 cm2/s.

The dependence of the binary di↵usivities Di on pressure and temperature is

Di ⇠ T↵/p 3/2 ↵ 2

Typical values at p = 1atm are in the range 0.01� 10 cm2/s.

µ/⇢ ⇠ T↵/p 3/2 ↵ 2

Typical values at p = 1atm are in the range 0.1� 1 cm2/s.

The kinematic viscosity µ/⇢ has the same units and the same pressure and tem-

perature dependence as the molecular di↵usivities

Moshe  Matalon  

Since �/⇥cp, µ/⇥,Di have the same dependence on temperature, their ratios are

nearly constant.

We can write ⇢Di = Le

�1i (�/cp) and µ = Pr(�/cp) with � = �(T ). A good

approximation (Smooke & Giovangigli, 1992) for the latter is

�

cp= 2.58 · 10�4

✓T

298K

◆0.7g

cm s

The Prandtl and Lewis numbers are nearly constant; Pr = 0.75 and for common

combustible mixtures Le varies in the range 0.2� 1.8.

�/⇥cpDi

= Lei

µ/⇥

�/⇥cp= Pr

µ/�

Di= Sci Schmidt number

Prandtl number

Lewis number

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p = �RT/W

�Dv

Dt= �⇥p+ µ

⇥⇥2v + 1

3⇥(⇥ · v)⇤+ �g

⇥�

⇥t+� · �v = 0

If the transport coe�cients µ,�, ⇢Di assume constant values (an average value,

say) the equations simplify to

⇢cp

DT

Dt� �r2T =

Dp

Dt+ ��

NX

i=1

!i

ho

i

⇢DYi

Dt� ⇢Dir2Yi = !i, i = 1 . . . N � 1

there are N+6 variables (�, p, T,v, Yi) and N+5 equations, supplemented with

the constrain

PNi=1 Yi = 1.

Moshe  Matalon