Peube J-L. Fundamentals of fluid mechanics and transport phenomena

Thermodynamics of Continuous Media 83

vertical position of this section (otherwise, we should have some accumulation of

vapor between two sections with different fluxes, which would increase with time).

2.4.3.5.

Diffusion in a moving medium

By virtue of the laws of dynamics, the characteristics of the inertia center of a

material system can be obtained through knowledge of the external forces acting on

the system. The same goes for fluid particles of a continuous medium if the external

forces acting on the fluid particle are known.

Diffusive phenomena amount to differences in velocity between different

components of a fluid. The separation of bulk movement and diffusive phenomena

can be achieved by identifying the

diffusive velocities in a reference frame

associated with the inertia center G of fluid particles

(mass or barycentric reference

frame) which move with velocity

G

V

G

(mass velocity) and we will denote simply as

V

G

, as obtained from the equations of motion which will be covered in Chapter 3.

From definition [2.29] we have, for the local inertia center:

¦

k

i

ii

VV

1

G

UU

[2.33]

Quantities measured in a reference frame

R

G

associated with this inertia center

are denoted

>@

G

(between brackets and with the superscript G). Relation [2.33] can

be written:

1

0, with:

k

GG

ii i i

i

 VVVV



¯ ¯



¡° ¡°

¢± ¢±



GGGG

[2.34]

This relation expresses a particular importance of the mass flux densities:

<>

1

,with: 0

k

G

ii mi

mi

i

q  Vq



¯



¡°

¢±



G

[2.35]

which describe the diffusive fluxes in this reference frame.

2.4.3.6.

Diffusion of components in weak concentration

The preceding difficulties disappear when a mixture is almost entirely comprised

of one of the components (for example species 1). Supposing that the number of

moles of the other components takes on a small value, of order

H, we have seen in

84 Fundamentals of Fluid Mechanics and Transport Phenomena

section 2.4.1.3 that all concentration definitions were equivalent for the first order. It

can be shown in the same way from expressions [2.28] and [2.29] that the

velocities

,V

G

*

V

G

and

1

V

G

are equal to order H, so that the predominant component

constitutes the natural reference frame.

In this case, the diffusion velocity of a component is its velocity with respect to

the predominant species, and any extensive quantities can be used to define a flux

density, as the reference frame is no longer described by a balance equation.

2.4.3.7

. General methodology: diffusion in an arbitrary reference frame

The preceding examples show that the velocity of a component is not the only

characteristic of its diffusion and we always have a global condition between the

ensemble of diffusion velocities which characterize the particular problem

considered. Diffusion itself is characterized by

diffusion velocities and flux densities

in the reference frame best adapted to the problem considered

. The flux density of

the extensive quantity associated with this velocity also depends on the choice of

reference frame.

The diffusion velocities and corresponding flux densities (molar, barycentric or

otherwise) of each component are summarized in Table 2.1 for different reference

frames, which are denoted by indices corresponding to the preceding quantities:

 the superscript * in the reference frame R

*

corresponding to the molar average

velocity

*

V

G

;

 the superscript

G

in the reference frame R

G

corresponding to the mass average

velocity

5

V

G

(G being the inertia center);

 the superscript

1

in the reference frame R

1

of fluid 1.

5

The mass speed

V

G

is the local speed of fluid medium intervening in the mechanical

equations (

G

VV

GG

).

Thermodynamics of Continuous Media 85

Reference frame Extensive

quantity

Diffusion

velocity

Diffusion flux densities

R

*

molar average

velocity

*

V

G

i

n

>@

*

VVV

ii

GGG



>@



>@

0

1

*



¦

k

i

ni

iiiini

q

VVnVnq

G

R

G

mass average

velocity

V

G

i

U

>@

VVV

i

G

i

GGG



>@



>@

0

1



¦

k

i

G

mi

ii

G

ii

G

mi

q

VVVq

G

UU

R

1

component 1

velocity

1

V

G

i

n

U

>@

0

1



V

VVV

ii

G

GGG

>@

>@>@

1

11

,1

0

mi i i

ni i i

mn

qVi

qnVi

qq

U

ªº

z

¬¼

ªº

z

¬¼

G

GG

weak concentrations

of components other

than 1

1*

RRR ##

G

1zic

i

>@ >@

>@

0

1

*

1

z

V

iV

VV

i

G

ii

G

>@



1

z



i

VVcVcq

iiiici

GGG

G

Table 2.1.

Variables characterizing diffusion for different reference frames

The situations described in Table 2.1 correspond to those most frequently

encountered. Other more complex situations may arise (unsteady evaporation,

macroscopic mixing of numerous components with walls which are permeable to

only some of these, etc.). In all cases, a global analysis is necessary for an

appropriate choice of representation for the diffusion equations.

2.4.4. Binary isothermal mixture

2.4.4.1. Expressions for diffusion velocities and flux densities

We have reviewed the different definitions which are possible for extensive

quantities in different reference frames. For n flux-density vectors, diffusion

phenomena are characterized by n-1 independent equations for the flux-densities or

the diffusion velocities, and by an equation which describes a global condition. The

n-1 differences in diffusion velocities, two by two are obviously independent of the

reference frame used, and they can be linked to causes of imbalance, such as

86 Fundamentals of Fluid Mechanics and Transport Phenomena

concentration gradients, by phenomenological relations. We here limit our treatment

of the problem to a binary mixture; the reader should consult texts treating the

problem of irreversible thermodynamics for developments associated with diffusion

in mixtures with more than two components.

For a binary mixture, the only quantity which is independent of the reference

frame and diffusion characteristic is the velocity difference

21

VV

GG



, of species 1

and 2 with respect to an arbitrary reference frame. Diffusion velocities in the

reference frames

R* or R

G

can then be easily expressed as a function of

21

VV

GG



.

In effect, in the reference frame corresponding to the mean molar velocity

*

,

V

G

we obtain relation [2.28]:

** *

*

11 2 2

0 with:

ii

nV n V V V V

¯ ¯ ¯

 

¡° ¡° ¡°

¢± ¢± ¢±

GG GGG

which can be written as:

>@ >@



21

2

*

2

1

*

1

1111

VV

n

nn

VV

n

V

n

V

GG

GGGG







[2.36]

or:

>@



21

*

22

*

2

*

11

*

1

VV

n

nn

VnqVnq

nn

GGG

G

  

[2.37]

In the reference frame

R

*

associated with the molar velocity, the diffusion

velocities and molar flux densities of each component can be written as a function of

the velocity difference





21

VV

GG



.

Similarly, using relation [2.34] between the mass flux densities in reference

frame

R

G

we obtain:

>@



21

222111

VVVqVq

G

m

G

m

GGG

G

  

U

UU

[2.38]

Comparing [2.36] and [2.38] gives:

Thermodynamics of Continuous Media 87

>@ >@



>@ >@

*

1

2

*

1

2211

221

*

1

2

1

V

M

V

MnMn

Mnn

V

n

V

G

GGGG



U

[2.39]

>@ >@

*

2

1

2

V

M

V

G

[2.40]

where



21

2211

nn

MnMn

M



is the local average molar mass of the mixture.

In summary, the diffusion velocity in a given reference frame can be expressed

as a function of the velocity difference between the two components by means of the

law which defines the reference frame. It is thus

velocity differences which

characterize binary diffusion phenomena independently of the reference frame

which is chosen.

2.4.4.2. Isothermal diffusion

2.4.4.2.1. Fick’s law

Isothermal diffusion is caused by the existence of concentration gradients which

we will characterize by a volumic number of moles.

In order to define a diffusion coefficient we will use the relation:

2

1

2

1

21

n

grad

DVV  

GG

[2.41]

the ratio

2

1

n

being dimensionless. It is equal to the concentrations ratio of two

species, expressed using a proportional definition for the two species

6

:

 

21

pp

ppgrad

D

grad

D

grad

D

grad

DVV

 

  

ZZ

UU

JJ

GG

[2.42]

6

Using partial pressures is only correct for perfect gases and ideal solutions.

88 Fundamentals of Fluid Mechanics and Transport Phenomena

Relation [2.41] shows that the diffusion coefficient can be expressed in

m

2

.s

-1

.

For a gas at atmospheric pressure, it is more or less independent of concentration.

For mixtures with air, it usually lies between 1.0 × 10

-5

and 2.5 × 10

-5

m

2

.s

-1

, the

lowest values corresponding to heavy molecules (carbon dioxide, ethanol, benzene,

etc.). Very light molecules (hydrogen, helium) give larger values (up to 13.2 × 10

-5

m

2

.s

-1

for hydrogen-helium mixtures).

In liquid mixtures, the diffusion coefficient is in the order of 10

-9

m

2

.s

-1

.

Diffusion in solids results from different mechanisms, depending on whether we are

dealing with diffusion of impurities which move from a free position in one

crystalline structure to another, or with particles (atoms, etc.) capable of moving

around the structural grid. The diffusion coefficient in solids varies from 10

-12

to

10

-14

m

2

.s

-1

.

Expression [2.41] shows that the quantity

gradD

has the dimension of velocity

and that this gives an order of magnitude of

AD

for the diffusion velocities, where

A

is the distance over which the concentration gradient is extended. Taking for

example

A

= 0.1 meter, we can see that the diffusion velocity is in the order of 10

-4

m

.s

-1

in gases and 10

-8

m

.s

-1

in liquids. These velocities increase considerably if the

distance

A

is significantly diminished; as for momentum transfer (section 6.5.3),

convection effect can reduce this quantity to values comparable with the thickness of

a boundary layer, leading to a significant increase in diffusion velocity ([BIR 01]).

We define the

Lewis number

Le

as the dimensionless ratio between the diffusion

coefficient and thermal diffusivity

a

:

a

D

Le

Excluding instances of extremely strong force fields or accelerations, the total

number of moles

n

per unit volume is often nearly constant (

n

1

+

n

2

#

constant)

under standard conditions (in particular for ideal gases). Thus, we have:

0

21

 ngradngrad

[2.43]

Substituting [2.43] into Fick’s law [2.41], the velocity difference can be written

as a function of the concentration

n

1

alone:



1

11

1

21

ngrad

nnn

n

Dngrad

nn

n

DVV



  

GG

[2.44]

Thermodynamics of Continuous Media 89

2.4.4.2.2. Diffusion in the local reference frame of molar velocity

We use the local reference frame

R

*

of molar velocity for diffusion in a closed

fixed volume. From [2.41] and [2.44] we obtain an expression for the molar flux in

the reference frame

R

*

:

>@ >@

2

*

2

1

*

1

n

ngrad

DV

n

ngrad

DV  

GG

[2.45]

From this we can deduce the molar flux densities of the two species:

>@

21

*

22

*

2

*

11

*

1

ngradDngradDVnqVnq

nn

  

G

[2.46]

Instead of representing the concentrations by

n

1

and

n

2

, diffusion velocities in

the fixed reference frame can be expressed as a function of quantities of the two

gases which are proportional to these concentrations, such as the partial specific

masses

U

1

and

U

2

, the partial pressures

p

1

and

p

2

, and the molar fractions

J

1

and

J

2

:

>@

2

*

2

1

*

1

J

U

J

U

grad

D

p

pgrad

D

grad

DV

grad

D

p

pgrad

D

grad

DV

  

G

[2.47]

The mass fractions cannot be directly used as concentration variables, because

the specific mass of the mixture varies within the volume.

NOTE

–

Expression

[2.47]

,

using the molar fractions as variables

,

is exact

, even if

the volumic number of moles

n

has a non-zero gradient. This results from the

definition:

12 1 2

1 and: 0 grad grad  

JJJJJG JJJJJG

and from expression [2.42] of Fick’s law using molar fractions which can be written:

1

21

J

JJ

grad

D

VV  

G

[2.48]

90 Fundamentals of Fluid Mechanics and Transport Phenomena

From [2.47], we obtain the densities of molar flux:

>@ >@

2

*

21

*

1

JJ

gradDnqgradDnq

nn

 

G

[2.49]

2.4.4.2.3. Application to diffusion in a medium at macroscopic rest

Here we consider a medium at macroscopic rest, in other words a medium

whose very slight movements are related to diffusion phenomena. We assume

that the medium satisfies the equations of fluid statics and that differences of

specific mass do not lead to movements associated with natural convection.

In the case of a closed impermeable container in which a fluid is at rest, the

molar average velocity is assumed to be zero at all points of the medium in

a

cartesian reference frame associated with the container.

The diffusion equation of the components can be obtained by applying [2.8]

for the number of moles

n

1

and

n

2

,

j

jn

G

x

q

t

n

w



w

1

V

which, on accounting for expression [2.46], gives:





¸

¹

·

¨

©

§



w



¸

¹

·

¨

©

§

w

GG

jj

ngradDdiv

t

n

x

n

D

xt

n

VV

1

111

[2.50]

The volume source

V

G

of species 1 expresses the number of moles per unit volume

of this species created by a homogenous chemical reaction inside the container. It is

fixed by chemical kinetics. It is zero in the absence of any chemical reaction.

In the common case where the diffusion coefficient is constant and where there

is no chemical reaction, equation [2.50] reduces to the heat equation:

1

nD

t

n

'

w

[2.51]

The boundary conditions for impermeable walls can be represented by a zero

normal component

1

ngrad

at the walls ( :

w

n

G

normal to the wall):

Thermodynamics of Continuous Media 91

0.

1

¸

¹

·

¨

©

§

w

n

ngradn

G

2.4.4.2.4. Diffusion with respect to a fixed component

The discussion of the steady evaporation problem given in section 2.4.3.4

(Figure 2.9b) showed us that gas 1 is fixed. In this case it suffices to directly apply

Fick’s law in the form [2.44]:

2

21

2

ngrad

nn

n

DV 

G

The flux density of species 2 can thus be written as:

2

222

nn

ngrad

nDVnq

n



G

The conservation equation of species 2 can be written:

0

2

¸

¹

·

¨

©

§

 nn

ngrad

nDdiv

[2.52]

For the 1D evaporation problem of Figure 2.9b, we obtain, by integrating [2.52]:

constant

2



dx

dn

nn

nD

q

n

[2.53]

We consider the following boundary conditions:

 at the surface of the liquid (

0 x

), the number of moles per unit volume n

S

, is

that of the saturated vapor at the experimental temperature;

 at the tube extremities (

A x

), we assume that the very slight movements of

atmospheric air suffice to eliminate all vapor, such that we can write that the vapor

concentration

n

2

is equal to the water vapor concentration n

2a

of the atmospheric air.

The distribution of the water vapor concentration can be obtained by

integrating equation [2.53]:



¸

¹

·

¨

©

§

 

nD

qx

nnnn

n

S

2

exp

92 Fundamentals of Fluid Mechanics and Transport Phenomena

Expressing, at the abscissa

A

, the concentration condition for the ambient air

we obtain a relation giving the flux density of the vapor:

¸

¹

·

¨

©

§



S

an

nn

nD

q

22

ln

A

For relatively weak concentrations we have:



A

aS

n

nnD

q

2



[2.54]

Numerical application

Take a tube of height

A

= 0.10 m. We base our reasoning on a density which is

proportional to the volumic number of moles for water vapor at 20ºC, assuming the

external air to be perfectly dry (n

2a

= 0) and using the following data: D=2.6 × 10

-5

m

2

.s

-1

:

U

s

= 0.018 kg.m

-3

. We find, for the flux density evaporated:

126

5

2

..1068.4

1.0

018.0106.2





u

uu

smkgq

m

This corresponds to a reduction in the level of the free surface of about 0.4 mm

per day. For acetone, whose saturated vapor pressure is 10 times greater, the

evaporation is 10 times more rapid.

Evaporation rate in static conditions is very weak. Diffusion in a stationary

medium is a very slow process, which only becomes efficient in the presence of

convection (section 2.4.4.2.1).

2.4.4.2.5. Diffusion in a reference frame related to the local inertia center

Equation [2.38] gives an expression for the mass flux in a reference frame

associated with the local inertia center:

>@



21

222111

VVVqVq

G

m

G

m

G

  

U

UU

[2.55]

The expression of Fick’s law [2.42] which is best adapted to this reference frame

uses mass fractions for the concentration variables:

Peube J-L. Fundamentals of fluid mechanics and transport phenomena

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