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

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Thermodynamics of Continuous Media 93

  

gradgrad

grad

DVV

From the definition of mass fractions, we obtain:

0and1

2121

 

ZZZZ

gradgrad

and thus:

grad

VV  

[2.56]

Substituting [2.56] into [2.55] gives:

>@ >@

2211

ZUZU

gradDqgradDq

 

[2.57]

There are almost no static diffusion problems which lead to a motionless local

inertia center. Expression [2.57] will be used in the diffusion equation in the

presence of convection in Chapter 4.

NOTE – Using relations [2.39] and [2.40], diffusion velocities in the reference

frame

can be expressed, as before, as a function of the variables

and n

well as

with

and

grad

pgrad

DMgrad

  

[2.58]

grad

pgrad

DMgrad

  

[2.59]

With these concentration variables, diffusion velocities can no longer be

expressed by means of the same diffusion coefficient and so we have two different

diffusion coefficients, in which the molar average mass

is variable:

94 Fundamentals of Fluid Mechanics and Transport Phenomena

2.4.4.2.6. Unsteady evaporation

From a practical point of view, the consequences of the developments

outlined in the preceding sections are that the exact formulation of equations for

diffusion problems should only be performed after an explicit choice of

reference frame when possible; this allows us to obtain precise expressions for

[]V

JJG

and

[]V

JJG

. For example, in certain problems (steady evaporation, see

sections 2.4.1.2 and 2.4.4.2.4) one of the gases, say

, is stationary, and only

gas

moves during diffusion. Taking a reference frame R

linked with

, the

velocity

[]V

JJG

of gas

follows immediately from [2.41].

On the other hand, if the

evaporation is no longer steady

, then the air (species 1)

is no longer stationary. Consider the case where the tube in Figure 2.9b is filled only

with air at the initial instant; the total pressure remains constant; the rise of vapor

(species 2) in the tube moves the molar quantity corresponding to the air which is

now in movement and is thus no longer the favored reference frame.

Let us reconsider the evaporation problem for the unsteady 1D case, in an

atmosphere at constant temperature and pressure (Figure 2.9b). The total

volumetric number of moles

is constant. This leads to a situation where the

molar average velocity

and the total molar flux

nVq

are independent of

the abscissa x. However,

a priori

they do depend on time. All that remains is to

write the balance equation for the vapor in the reference frame of the tube:





[2.60]

The flux density of the vapor

is now a function of the two variables

and

It can be obtained by decomposing the velocity,

,VV V

¯



¡°

¢±

and by replacing

[]V



JJG

by its expression in [2.45]:

DVnq

222





[2.61]

By substituting into equation [2.60], we obtain the diffusion equation:





[2.62]

Thermodynamics of Continuous Media 95

As the velocity

of the air is zero at the abscissa

= 0 (impermeability of the

liquid surface), the total molar flux

nVq

is equal to the molar flux

of the vapor at the origin. This can be obtained by means of expression [2.44] of

Fick’s law and the condition

(0, )



ntn

VnnVq



[2.63]

The molar concentration

satisfies equation [2.62] and condition [2.63] with

which we can associate the conditions used in section 2.4.4.2.4 for positions 0 and A:

 

ntnntn

222

,,0

in addition to the concentration distribution

(x,0) at the initial time.

We will go no further with these calculations. Let us just

recover the steady

regime

studied earlier (section 2.4.4.2.4) by observing that in this case, gas 1 is

stationary; we therefore have, for all

VnnVq

Substituting this expression into [2.61] gives us:

DVn



Equation [2.62] is thus identical to [2.53]:



w

If evaporation occurs

at constant volume

in a container in which initially no

vapor is present, the pressure will increase during evaporation, which also produces

a displacement of the mixture. We will leave aside such static problems where the

equations of motion are to be solved.

96 Fundamentals of Fluid Mechanics and Transport Phenomena

2.4.4.2.7. Case of weak concentrations

When the mixture contains a weak proportion of species 2:

....or;or;

121212

 ppnn

it is then possible to confuse the different units of concentration (section 2.4.1.3).

By comparing relations [2.28] and [2.29], we can see that the three reference

frames

, R

and R

become identical to the first order. We can note that

expressions [2.47], or [2.58] and [2.59], show that the diffusion velocities

of component 1, which is excessively large, are negligible compared with the

diffusion velocity

of component 2.

It results in the preceding approximations that:

– the molecular diffusion velocity

of species 2 is thus the same in any one of

the three reference frames

, R

and R

;



any variable amongst those defined in section 2.4.1.3 (number of moles per

unit volume

, molar or mass concentration

or ǹ

, partial pressure

, molar or

mass fraction



or ȁ

, etc.) can be used to represent the concentration

component

to the second order excepted.

Then we have, in the three reference frames previously chosen to characterize

the diffusion the relation:

cgrad

DV 

[2.64]

In these conditions, we can define the flux density of species 2 by means of the

variable

2222

cgradDVcq



[2.65]

The flux

of the variable

across the surface (

) related to any one of the

three mentioned reference frames is:

³³



dscgradDdsq ..

222



[2.66]



Thermodynamics of Continuous Media 97

This approximation of law [2.41] in balance equation [2.8] allows the diffusion

equation to be obtained:







cgradDdiv

[2.67]

The volume source

of species 2 expresses, with the units of concentration

the quantity per unit of time of this species created by a homogenous chemical

reaction.

For weak concentrations, we can usually assume that the coefficient of diffusion

is constant; the diffusion equation is then reduced to the heat equation:

'



[2.68]

This approximation for weak concentrations is often used, either on account of

experimental conditions or because it allows for a simplified approximate treatment

of the problem, particularly in situations where it is associated and coupled with a

heat release and with chemical reactions. On the other hand, diffusion often occurs

in flows which present difficulties, and thus the preceding approximation is

necessary.

2.4.5. Coupled phenomena with diffusion

2.4.5.1

. Binary non-isothermal mixtures

The elementary discussion of the last section is not entirely rigorous, even in the

absence of movement in the gas, because all phenomena characterized by scalar

quantities are coupled (sections 2.1.5.2 and 2.1.5.3.4): the diffusion of a species in

another species is accompanied by a thermal flux and a temperature gradient

(Dufour effect). However, this is very often negligible in the absence of thermal

conditions leading to a temperature gradient. Similarly, a temperature gradient

(differences in the energy of molecular agitation of diluted gases) will induce a

concentration gradient (i.e. a difference in the ways two different species of

molecule will move in a gaseous mixture, some moving to the cold regions, others

moving in the opposite direction).

The action of a mass force field (gravity, inertial forces, etc.) on the two species

is manifest in a static equilibrium comprising a pressure gradient (or of the volumic

number of moles), as we saw in section 2.2.1.5 (atmospheric equilibrium). In such a

98 Fundamentals of Fluid Mechanics and Transport Phenomena

situation, the lighter molecules tend to rise. The thermodynamic force responsible

for this effect is the pressure gradient. In practice, the diffusion of pressure is

entirely negligible in the atmosphere, where the quantity

ngrad

is nearly always

negligible compared with

ngrad

; diffusion therefore occurs as if there

was no force field. The diffusion of pressure must be accounted for in mixtures

subjected to high accelerations, as in centrifuges.

Another cause for the molecular migration of a chemical species with respect to

molecules of another species is a difference in the forces exerted between the two,

for example between ions with different charges placed in an electric field. This

thermodynamic force is a vector field, in other words of the same rank as the

gradient of a scalar quantity. These phenomena are encountered, for example, during

the electrolysis of solutions or in an ionized gas in the presence of an electric field.

Kinetic gas theory allows us to construct the theory of diffusive phenomena. The

principle involves seeking distribution functions for neighboring velocities of the

Maxwell-Boltzman distribution (Chapman-Enskog method). We find that first order

deviations from this equilibrium distribution lead to the irreversible processes

previously discussed. This theory also shows a decoupling between imbalances of

different tensor orders such as thermal, mass or viscous fluxes. We will not develop

the corresponding equations. For velocity differences in a gas ([PEN 55]) we obtain:

>@>@



Tgrad

pnn

pgrad

MMDn

grad

DVV





 

[2.69]

is the thermal diffusion coefficient; we usually define the thermal diffusion

ratio

by the relation:

DkD

The thermal diffusion ratio

is more or less independent of temperature, but

varies strongly with concentration: in particular, it tends to zero with each of the

concentrations. Its maximal value, attained when the concentrations of components

are in the same order of magnitude, is of the order of 0.1-0.2.

The preceding expression shows that a temperature gradient leads to

additional diffusion phenomena (thermal diffusion) which can be used to

separate two components of a gaseous mixture when chemical or other physical

Thermodynamics of Continuous Media 99

methods are not practicable (for example, in gaseous mixtures comprising two

isotopes).

2.4.5.2.

Mixtures with several components

Once a mixture is comprised of many components, the number of reduced

extensive variables increases, each component requiring a concentration variable.

There thus exist as many mass or molar flux densities as there are species present.

For each species, we have a balance equation [2.24].

Following the principle outlined in sections 1.4.2.6 and 2.1.5.2, all diffusive fluxes

depend on all of the thermodynamic forces of the same tensor rank. We thus have a

matrix of diffusion coefficients. The general discussion concerning the choice of

reference frames which characterize the diffusion processes is identical to that outlined

earlier.

As we have already said (section 2.4.2.3), the

balance equations for each

component leads to a global mixing equation that describes the conditions under

which the mixture will evolve (in movement, at rest in a fixed container, during

evaporation). The

equations are generally replaced by this global equation and

–

1 equations characterizing the components of the mixture.

The interested reader should consult textbooks covering problems of

irreversible thermodynamics ([BIR 02], [BOC 92], [CHA 91], [DEG 62], [DOU

01], [EU 92], [GER 94], [LEV 62], [PRI 68]).

2.4.6. Boundary conditions

In the absence of chemical reactions, the boundary conditions can be identical to

those of the thermal problem (section 2.3.2). The existence of a heterogenous

reaction on the wall P leads to the production or absorption of the components.

Chemical kinetics provides the law for the reaction speed for the components

concerned. The flux density of a component at the wall must be equal to that

produced or absorbed by the chemical reaction, for example for a reaction of order



[2.70]

The coefficient

is often an expression of the form



RTUkk  exp

, where

U is the activation energy of the reaction,

designating the absolute temperature.

The form of this relation shows a strong coupling between the temperature and the

100 Fundamentals of Fluid Mechanics and Transport Phenomena

reaction speed. The heat release due to heterogenous chemical reactions must be

taken into account in the boundary conditions of the energy equation ([BOR 00],

[PEN 55], [WIL 65]).

Chapter 3

Physics of Energetic Systems in Flow

In the preceding chapters, we examined the physical and mechanical properties

of matter independent of the dynamic effects induced by motion. Before dealing in

the next chapter with the general equations of fluid dynamics and the transfer of

quantities in flows, we will first recall the basic laws of mechanics and their role in

thermodynamics; we will then outline the formalism used to describe the motion of

continuous media and finally we will examine the mechanical properties of moving

fluids.

3.1. Dynamics of a material point

3.1.1. Galilean reference frames in traditional mechanics

As geometric space is homogenous and isotropic, the translational motion of an

isolated material point is necessarily rectilinear and uniform. In effect, for any other

kind of trajectory, a favored direction could be defined and any non-uniform

movement of an isolated material point would imply an inhomogenous time. We

thus postulate the existence of Galilean reference frames in which the distance

traveled by an isolated material particle is a linear function of time. The laws of

physics should be the same in all Galilean the reference frames. We must now

change the reference frame to where the transformation matrix is a function of time

and where all uniform translational movement are required to have the same

properties in the new reference frame. For Cartesian reference frames this results in

new coordinate systems which are in uniform rectilinear translation with respect to

one another, and which form a group.

102 Fundamentals of Fluid Mechanics and Transport Phenomena

In traditional mechanics, changes in the reference frame which conserve

distances and time belong to the Galilean group. They have the form:

ttitVxx

iii

 '),3,2,1('

The presence of the time variable in these reference frame changes leads to

specific properties of temporal derivatives. As time is the same in all Galilean

reference frames, the values of extensive scalar quantities are independent of the

Galilean reference frame used. On the other hand, components of vector or tensor

quantities vary in reference frame changes according to the usual formulae

(covariant or contravariant according to the case considered). We say that these

quantities are invariant for (geometric) changes of the Galilean reference frame (see

texts on linear or tensor algebra). However, certain vectors (position or velocity of a

particle) are defined with respect to a given Galilean reference frame; the evaluation

of their temporal derivatives depends on the reference frame chosen for this

definition. For example, the components in reference frame R’ of the velocity of a

point defined in reference frame R are not equal to the components in reference

frame R’ of the velocity of the point defined in reference frame R’. In a general

manner, so-called cinematic operations (such as calculations of temporal derivatives

velocity, acceleration) lead to formulae of changing reference frames dependent on

their relative motion. We will assume that these ideas are known to the reader.

Let us recall the following elementary formulae, which will subsequently prove

useful:

rrearea

VVVV

 

ZJJJ

in which the indices a and r indicate that derivatives with respect to time are

calculated in a Galilean reference frame (a) or in some other reference frame (r)

which is in motion relative to the former. In the expression of acceleration the first

term e represents the drag term (centripetal acceleration in the case of a rotating frame)

and the third term is the Coriolis acceleration.

In the following, where not indicated otherwise, momentum, velocity and

acceleration will be calculated in a Galilean reference frame.

3.1.2. Isolated mechanical system and momentum

We have seen in Chapter 2 that we characterize motionless matter by convenient

extensive properties and that the assumption of thermodynamic equilibrium leads to

the existence of relations between thermodynamic properties.