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

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Thermodynamics of Discrete Systems 43



[1.41]

and, generally, by using intensive energy variables

TZY  in the place of

intensive entropic variables



  

[1.42]

that is, to second order excepted, by linearizing in the vicinity of (

YT ):



;

iiii

 



 

[1.43]

By substituting expression [1.43] into expression [1.39] for the thermodynamic

flux in which the temperature appears, we obtain a linearized expression for the flux

as a function of the intensive energy variables (including the temperature):



2100

int1

jjiij

YYYTL



in which the matrix

can be easily deduced from the matrix

. It is easy to see

that the matrix

is not symmetric, as noted earlier.

In addition, the

terms of the matrix diagonal

are negative: the flux of the

extensive quantities is in the same direction as the decaying intensive energy

quantities.

Letting

i = j = 1, the heat flux

2,1

and the temperature T, we define the

thermal resistance R

which is the inverse of the term

of the first diagonal term

of the matrix

2,1



The entropy source at the interface of the two sub-systems can be written using

the energy variables (accurate to second order excepted):

44 Fundamentals of Fluid Mechanics and Transport Phenomena



IjiYYYYYL

iijij

,....2,1,'

2121

int



¦¦

1.4.2.6.3.

Conditions for the application of irreversible linear thermodynamics

We have already noted that it is possible to linearize the phenomenological law

which gives the flux of extensive variables if the imbalances are “small”. This idea

can be stated in the following usual way: if the imbalance corresponds to a weak

variation of a “regular” process, then the linearization may be satisfactory. This

leads us to invoke irreversible extensive quantity transfer mechanisms. Let us

examine the case of thermal transfers, which are primarily due to two mechanisms.

1) Intermolecular action within a material

Molecular agitation, the intensity of which increases with temperature, results in

the transmission of extensive properties via collisions between molecules, ions, etc.

This is a statistical mechanism, which tends to cause a uniform distribution of the

properties of a body. For example, the mechanical energy of molecules in hot

regions is transmitted to molecules in cold regions via collisions between the

molecules (gases), and/or by the action of intermolecular forces (liquids, solids).

Within the context of kinetic theory in traditional mechanics, molecules are

animated with a velocity in the order of the speed of sound; they cover a distance

called the mean free path between successive collisions. Under ordinary conditions

of pressure and temperature, this distance is in the order of 10

-7

meters. Thermal

energy is due to kinetic and potential energy of molecules. Let us take an example to

evaluate the imbalance due to a temperature gradient. If we admit that statistically

molecules lose one-thousandth of their energy with each collision, we can conclude

that about 100 collisions are necessary in order for gas molecules to lose one-tenth

of their energy. This loss corresponds to a temperature drop of about 30 Kelvin

which will be produced over a distance of the order of 10

-5

meters (10 Pm). This

corresponds to a considerable thermal imbalance. However, these collisions, which

correspond to a tiny mean energy loss of 1/1,000, are clearly very small processes in

comparison to two microscopic fluxes of mechanical energy due to the molecules

going through any plane in one direction and also in the opposite direction. These

opposite fluxes have nearly the same absolute value. The macroscopic mechanism

for irreversible transfer of extensive quantities by molecular collision is thus

statistically a tiny perturbation amongst the mechanisms of thermal agitation, and it

is thus not surprising that the macroscopic processes are linear. Our experience

verifies the arguments proposed by this rather simplistic reasoning.

However, we must realize that the collision properties can vary with temperature,

even if the mean free path is not very temperature sensitive. This implies that the

properties of thermal resistance can depend on the temperature chosen T

in relation

[1.43], in order to evaluate the thermal resistance (the inverse of the first diagonal

Thermodynamics of Discrete Systems 45

term of the matrix

). In fact, for large temperature differences, it is necessary to

consider thermal resistance as a succession of isothermic thermal resistances in

series and to perform an integration which takes the temperature variation into

account:



2,1

As a first approximation, we can often take the temperature T

as being equal to

the mean temperature

TT 

2) Thermal radiation

Thermal radiation is in fact an electromagnetic radiation, whose emitted power

per unit surface (thermal flux density) of a blackbody can be represented by

Stefan’s law:

(ı: Stefan constant;T Kelvin)

q  T

Consider two parallel planes, face to face and respectively heated to

temperatures of T

and T

< T

). The net heat flux density q

received by plane 1

is equal to:





TT q



The preceding relation will quickly deviate from a linear heat transfer law valid

for small temperature differences. The reader can verify that in this case the

preceding law can be represented by a thermal resistance R

equal to

31 T

which varies strongly with temperature.

1.4.3. Application to heat engines

A heat engine is a device in which a fluid is made to evolve according to a cycle

C: the material follows an evolution parameterized by the time t, after which the

final state is identical to the initial state. Diverse extensive quantities are exchanged

by this material with external sources of work and of heat. The objective of this

machine is to produce certain desired quantities (work for a heat engine, heat for a

heat pump, etc.) from external sources of other quantities.

46 Fundamentals of Fluid Mechanics and Transport Phenomena

The fluid which circulates in the engine passes through successive devices which

either furnish or extract work and heat. Consider a mass m of this fluid whose

quantities and in particular the temperature T(t) evolve in a quasi-static manner as a

time function t. Let GQ be an amount of heat which it receives between the instants t

and t+

t. During the cycle C, the variations of the fluid’s entropy and of its internal

energy are zero:



0;0  

³³³

CCC

dSQWQWdE

[1.44]

Thus, the evolution process of the fluid over a cycle, as for all processes, creates

entropy (see section 1.4.2.4). As the entropy variation of the fluid is zero over the

cycle, a generalized version of relation [1.33] can be written, T being the

temperature of the fluid:

On the other hand, the heat transfer from the sources at temperature T

(t) is also

accompanied by a creation of entropy, the reasoning of relation shows that we have

the Clausius inequalities:

³³³

CCC

The preceding reasoning has the advantage of highlighting the entropy sources

associated with the Clausius inequality; it also shows that the difference between the

inequality terms is greater in proportion to the level of irreversibility.

We will not get into a detailed discussion of such cycles and the efficiency of

heat engines, all of which can be deduced from the aforementioned inequalities. The

reader will find such discussions in texts on applied thermodynamics.

Chapter 2

Thermodynamics of Continuous Media

The properties of continuous media can be obtained by a limiting process on

variables of discrete systems. The exchange of extensive quantities is modeled by

means of flux densities. Irreversible thermodynamics can be transposed in the same

way, being represented by diffusion equations expressed in terms of intensive

quantities (heat and diffusion equations of chemical species). The principal results of

fluid statics are presented. The diffusion of material leads to the existence of several

macroscopic reference frames, which brings with it specific difficulties which are

discussed.

2.1. Thermostatics of continuous media

2.1.1. Reduced extensive quantities

The continuous medium is defined according to the usual method of passing

from the discrete to the continuous by letting the elementary sub-systems tend

towards zero, their number thereby increasing indefinitely. If the geometric

dimensions of a system tend toward zero, the extensive quantities of that system also

tend toward zero, whereas the intensive quantities do not change. A limiting process

is therefore necessary for the study of continuous media.

We define reduced extensive quantities, i.e. extensive quantities per unit of mass,

volume, or number of moles:

– a quantity per unit mass

48 Fundamentals of Fluid Mechanics and Transport Phenomena

– a volume quantity

– a molar quantity

The amount of G which is contained in the domain D is:

³³³

DDD

dvgndvggdvG

The concept of a local quantity corresponds to an average over a volume which

is large on the microscopic scale, but small on the macroscopic scale. If we imagine

that it is possible to measure an average quantity for a set of particles (molecules, for

example) of a continuous medium, contained within a sphere of radius r and

centered on a point M, the values obtained will only tend towards the value of the

reduced extensive quantity if r is sufficiently large. For small values of r, noticeable

fluctuations would be observed. Figure 2.1 shows the result of a measurement of the

average density which would be obtained for a sphere whose radius r is of the order

of the inter-molecular distances. On a larger scale we would of course observe the

gradient (macroscopic) of the reduced extensive quantity.



Figure 2.1. Local value of the average specific mass in a

sphere of radius r and fluctuations at a molecular scale

2.1.2. Local thermodynamic equilibrium

The description of a given continuous medium can be performed by means of a

field of reduced extensive quantities. We use the same methodology used for finite

discrete systems, as the reduced extensive quantities chosen only allow for a

description of the continuous medium if a hypothesis of local thermodynamic

equilibrium is made: any very small volume obeys the general equation of state for a

system in equilibrium, with all its consequences. We obtain the thermostatic

Thermodynamics of Continuous Media 49

relations for continuous media by a limiting process when the characteristic

dimension of the discrete sub-systems tends to zero.

General equilibrium equation [1.6] for a discrete system is first degree

homogenous; it can be applied immediately for reduced extensive quantities. By

performing a trivial transposition of the reduced notation (X

x, E

e, S

s, etc.),

the notation S which here designates the mathematical function of formula [1.6], we

have:

(,)or: (,)or: (,)s Sxe s Sxe s Sxe



[2.1]

The consequences of the hypothesis of local thermostatic equilibrium have

already been described in Chapter 1; the diverse relations obtained being first degree

homogenous relations, they are entirely transposable via a replacement of the

extensive quantities by the corresponding reduced extensive quantities. We note

however that the variance has decreased by one unit. We will often consider a

divariant fluid (two independent state variables) which will often be a perfect gas

with constant C

, for which the equations of state become:

const



TCerT

The local thermodynamic equilibrium hypothesis translates the existence of two

rapid dynamic processes which are opposed, such that locally there is a balanced

exchange between them: in effect, the energy mechanism and momentum

transmission on the molecular scale involves molecular collisions during which the

exchange of extensive quantities occurs in both directions. The macroscopic transfer

is no more than the residuum of these exchanges in opposing directions. Molecular

displacements occur with a velocity in the order of the molecular agitation velocity

(in the order of the speed of sound). The corresponding fluxes are individually very

high, but they are in opposing directions and it is their very weak net outcome that

we observe at the macroscopic scale for irreversible phenomena. These amount to a

very weak perturbation of the local equilibrium of the continuous medium, which

obeys thermostatic relations.

The local thermodynamic equilibrium hypothesis is based on the fact that the

time required for the local gas equilibrium to be achieved (relaxation time) is small

compared with the times associated with the macroscopic gas evolution. For the

molecules of mean free path

and molecular velocity c, the relaxation time is in the

order of

/c, if we consider that each molecular collision is efficient in the transfer of

extensive molecular quantities (quantum conditions).

50 Fundamentals of Fluid Mechanics and Transport Phenomena

The molecular interactions which are at the heart of the aforementioned

processes have a very short radius of action, whether we consider the intermolecular

forces or the collisions characterized by their mean free paths. These actions are in

fact volumetric at the scale of the given distances, and at the macroscopic scale they

appear as contacts, i.e. flux densities of extensive quantities, which are determined

by local conditions, i.e. surface forces.

The condition of local thermodynamic equilibrium is not always satisfied in the

case of certain gases undergoing rapid changes (for example, tri-atomic gases during

the passage of a series of shock waves, or in supersonic nozzles) and plasmas. We

are therefore led to separate different populations (ions and electrons) or different

forms of molecular energy (translation and vibration), for which we must introduce

a further extensive quantity and supplementary hypotheses (relaxation law, etc.).

Such separations into sub-systems are not only spatial. The interested reader can

refer to [BAS 98] or texts dealing with ultrasound.

This condition of local thermodynamic equilibrium does not allow the entire

medium considered to be in global thermodynamic equilibrium. It can be shown

using statistical methods taken from kinetic gas theory, for example, that the net

balance of the extensive molecular quantities gives, at first order, a distribution of

Maxwell-Boltzmann velocities which correspond to a local statistic mechanical

equilibrium ([CHA 91], [HIR 64]). At second order we have phenomena associated

with the irreversible transfer of extensive quantities ([BIR 02]); in section 3.4.1.3 we

will discuss the mechanism of this irreversible transfer in the case of momentum

transfer. The statistical equations describing turbulence are identical to the preceding

molecular statistical transfer equations, in which turbulent fluctuations play the role

of molecular fluctuations for the transfer of these extensive quantities.

Unfortunately, turbulent fluctuations do not verify general statistical laws and the

preceding analogy is only formal ([COU 89], [MAT 00], [TEN 72]).

2.1.3. Flux of extensive quantities

2.3.1.1. Flux density

An extensive quantity G can be transferred in a continuous medium by different

processes, the nature of which does not interest us here. The definition of a flux

of the quantity G across the surface S has already been given for the surface of

finite systems (the amount of the quantity G which crosses the surface S per unit

time).

We will show that in a continuous medium, the flux of an extensive scalar

quantity G is characterized by a flux density vector.

Thermodynamics of Continuous Media 51

Consider the tetrahedron OA

whose corners are the origin O and three

points lying on the axes (Figure 2.2). Suppose that the surface dimensions of this

tetrahedron are small, of order

. Let

be, respectively,

the fluxes of quantity G across the surfaces OA

, OA

and OA

(of

respective surface areas ds

, ds

and ds

, of order

) in the positive direction along

the coordinate axes.

Let

be the flux leaving the face A

of area ds of the tetrahedron.

Suppose that the (algebraic) sources of the quantity G are volumetric. The amount of

the quantity created in the tetrahedron is O(

). The balance of the extensive

quantity G leaving the tetrahedron can be written:

)(

332211

Odsqdsqdsqd

GGGG



Figure 2.2.

Quantities balance on an elementary tetrahedron

We obtain:

)3,2,1( idsds

where

designates the directional cosines of the normal

oriented towards the

exterior of the tetrahedron.

By replacing the surfaces ds

with their previous expressions and by letting

H

towards 0, the terms of order

should cancel out. By defining vector

or the flux

density of the quantity G, with components



321

GGG

qqq

, we obtain the

elementary flux of the quantity G across ds (the quantity of G crossing ds per unit

time in the direction normal

dsnqdsqd

GiGiG

[2.2]

52 Fundamentals of Fluid Mechanics and Transport Phenomena

The flux

of the quantity G across the surface S can be written:

dsnq

[2.3]

By a similar argument, it can be shown that the transfer of a vector quantity G

would be characterized by a tensor

Gij

whose flux across an elementary surface ds

would give the vector quantity G

which crossed ds per unit time:

dsnqdsnqd

jGijGi

[2.4]

where the tensor

is defined, as before, using the vector fluxes across the co-

ordinate planes.

2.1.3.2. Examples of flux of extensive quantities

The preceding property did not require any hypothesis regarding the transfer

mechanisms of the quantity G (action by contact or action at a distance): it results

solely from the notion of balance for an extensive quantity. As an example, we can

consider the flux of extensive quantities due either to propagative phenomena

(example 1) or phenomena involving action by direct contact (examples 2 and 4):

1) Energy transfer by electromagnetic radiation (energy-flux density vector

in lighting, infra-red heating, etc.) or by acoustic propagation (acoustic intensity

vector).

2) Heat transfer in a material (molecular agitation energy at the molecular

scale) is represented by the thermal flux density vector (section 2.1.5.3.1).

3) The diffusion of a chemical species in a material medium is represented

by the molar flux density vector of the species considered (n: the number of

moles/volume) (section 2.4.2).

4) For example, if the extensive quantity is a force density (or momentum

density at the microscopic scale), the corresponding flux density is the stress

tensor

. However, the convention for the orientation of the normal is here

reversed. The elementary force is the force exerted on the surface ds, by the

material situated on the side of the normal (side 1 in Figure 2.3a), towards the

other side:

dsndf

jiji

[2.5]