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

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Thermodynamics of Discrete Systems 13

1.1.4. Processes and systems

1.1.4.1. Definition of a process

Certain authors define a process (a,b) as a pair of states: initial (a) and final (b).

They are thereby led to distinguish between states which are possible and those

which are not. Insofar as we limit ourselves to only consider processes which are

truly observed (physical processes), the discussion of an axiomatization concerning

impossible processes, being ill-defined, is beyond the scope and objectives of this

book.

From a physical perspective, this means that a process can only be defined if the

initial and boundary conditions are entirely determined during the process. It makes

no sense to speak of a process which allows us to pass from a state (a) to state (b)

unless the external conditions which constrain that process are specified. This is no

longer a mathematical question, but rather a problem related to a determinism which

amounts to admitting that an initial state (a), well-defined and always subjected to

the same constraints, will always lead to the same final state (b). It will always be

possible to relate two given states, under the condition that, on the one hand,

exchanges with the exterior furnish the necessary physical quantities, and on the

other, that the internal system processes which redistribute these quantities allow the

desired distribution of these system quantities to be achieved. For example, it is not

possible to realize a state consisting of a given mass whose temperature distribution

comprises a central peak (Figure 1.1) by means of an action at the exterior walls.

The necessary energy must be directly supplied to the central zone, which must be

insulated from the adjacent regions.

By definition, a process is a series of states. This mathematical definition only

has physical relevance for processes representative of real evolutions. While not

precluding a choice of states with no link (a rabbit, a carrot, etc.), the obtention of

physical evolution laws for matter implies a “certain continuity of content” for this

ensemble of states. The same goes for all practical problems. A process is therefore

a succession of states which must be uniquely defined. Apart from some exceptions

(shocks), we will only consider processes comprising a continuous series of states,

described by variables which must be continuous functions of time. We will

however allow situations with discontinuities (shocks, shockwaves, deflagration)

which momentarily violate this continuity condition.

1.1.4.2.

The notion of a system

The notion of a system is a relatively vague one; it is in fact included in that of a

process: a system is an entity which we consider during a process. As our

considerations often take a differential form, the system is the principal part (zero

order) on which we perform differential balances. The notion of a system is not

14 Fundamentals of Fluid Mechanics and Transport Phenomena

clearly identified if we do not entirely state the conditions of the process under

study.

Take the following examples:

1) The matter studied remains enclosed in a fixed volume (a cylindrical

calorimetric bomb, for example). The observation domains D and D' at times t and t'

are contained in the interior of the cavity C (Figure 1.3a). The presence of the barrier

constituted by the rigid wall leads to the matter inside C being constrained to remain

within the system (Figure 1.3a). This is a closed system.

C=D=D’

D(t)

D’=D(t’)

(a) (b) (c)

Figure 1.3. (a) and (b): closed systems; (c) open system

2) Now consider the case where matter is caused to move at a velocity

with

respect to the used reference. The observation and description of this matter in

movement can be performed:

– either, by following the matter in its movement, in which case the observation

domain D(t) is displaced in time (Figure 1.3b); the matter contained within the

domain D(t) constitutes a closed system in the sense just defined;

– or, by considering a fixed domain D

in which the matter is continually

renewed; the ensemble of states contained within the fixed geometric domain is

qualified as an open system, in other words a system which exchanges matter with

the exterior (Figure 1.3c).

In Chapter 3 we will encounter these different ways of describing the movement

of matter in the form of substantial (Lagrangian variables) and spatial (Eulerian

variables) descriptions.

By definition, we will say that a process describing the evolution of material

elements which are identified, and remain unchanged, operates on a closed system,

in other words a system which does not exchange (does not provide or receive)

matter with the exterior of the system. We can also use the denomination material

system.

Thermodynamics of Discrete Systems 15

On the other hand, a process (a series of states) during which some matter passes

from the inside of the domain to its outside corresponds to an open system.

1.1.4.3.

Types of processes and states

We normally define the terminology of processes and states in the following

way:

– a natural process is undergone by an isolated system which is not subjected to

any external action;

– a reversible process is a process for which the direction of time can be

changed. It is of course a process in which no entropy is created. Such processes

occur over infinitely slow transformations;

– a quasi-static process is a succession of close infinitely equilibrium states;

– a possible process is a process where the constraints placed on entropy are

obeyed. In the opposite case we speak of an impossible process;

– a state of equilibrium is in fact the result of a succession of states whose

variables remain constant, any exchanges of its physical properties with the exterior

having ceased.

1.1.4.4.

Enclosures and walls

A diathermic wall is a wall which is permeable to heat and to external sources of

entropy.

An adiabatic wall is impermeable to heat and does not allow the passage of

entropy. It thermally insulates the system from the exterior.

1.2. Axioms of thermostatics

1.2.1. Introduction

The traditional presentation of thermodynamics usually begins with a direct

definition of the various quantities (force, pressure, etc.) which are then used in the

subsequent definition of elementary work and heat. The first and second principles

(conservation of energy and entropy respectively) are then stated, to which further

laws are then added (conservation of mass, chemical species, etc.). This all leads to

the differential form of energy being written as the differential of a function E,

energy. The result is a lack of coherence well known by students of

thermodynamics. This situation can be avoided by means of a more structured

presentation concerning the extensive quantities, among which energy plays a

particular role in physics, whereas entropy is the basis of all considerations

16 Fundamentals of Fluid Mechanics and Transport Phenomena

pertaining to irreversible thermodynamics. We will also more clearly outline the fact

that the more complex and irreversible the evolution considered, the greater the

number of variables required.

1.2.2. Extensive quantities

1.2.2.1. Definition of extensive quantities

Among the quantities used for the description of a state, we postulate (basic

principle) that for every system S there exists an ensemble of n extensive quantities

whose measure is proportional to the extension of the system, and which are always

defined regardless of the state considered.

Figure 1.4. Disjoint states

Consider two disjoint states, i.e. states which have no matter in common; the

extensive quantity X

associated with the ensemble of the two states (a) and (b) is

equal to the sum of the quantities corresponding to each state:

)()()( bXaXbaX

iii

 

If the sub-ensembles corresponding to (a) and (b) are not disjoint, we clearly

have:

)()()()( baXbXaXbaX

iiii

 

This definition only concerns the description of a state (a collection of matter) at

a given instant. Under no circumstances does it imply the same property for two

separated sub-systems (a) and (b) which we bring together in an externally applied

field (force field, electromagnetic field, etc.), as this would constitute a process

(series of states).

The index i takes on small integer values as these quantities are generally small

in number for a state. The following quantities are extensive: mass, volume, number

of moles, energy, entropy, force, etc.

Thermodynamics of Discrete Systems 17

1.2.2.2.

Natural processes

The usual (historic) presentation of thermodynamics begins with a statement of

the first law of thermodynamics, which sees energy play a central role compared

with the other extensive quantities. This is in fact only one of the conservation

principles among the axioms of thermostatics, which affirm the existence of

conservation laws all possessing the same structure. Consider a material system (i.e.

a collection of defined material elements) which evolves according to a natural

process (without exchange with the exterior). The system is described in a Galilean

(or inertial) reference frame in relation to which it is fixed (no dynamic effects in the

sense of mechanics are authorized), and only geometric changes of reference frame

are permitted.

The following principles can be established:

– The entropy S of the material system is non-decaying during the natural

evolution of a process.

– The other extensive quantities X

of a material system remain constant during

the natural evolution of a process.

The preceding axioms actually express the existence of physical conservation

laws, except for the entropy which can only increase during the spontaneous

evolution of an isolated system. The axiom of extensive quantities also implies that

the concept of variation in the extensive quantities of a system (a) is meaningful: it

is possible to conceive of the quantity

of which (a) has gained an amount X.

(a)

(b)

abba



Figure 1.5. Action and reaction in a system

1.2.2.3.

Action and reaction

The preceding definition implies a principle of action and reaction: let us divide

the considered system into two sub-systems (a) and (b), separated by a surface S

(Figure 1.5). The extensive quantity

gained by (a) to the detriment of (b) is

opposed to the extensive quantity

gained by (b) to the detriment of (a). This

amounts to saying that the action exercised by (a) on (b), the source of the transfer

between (a) and (b), is opposed to the action of (b) on (a):

18 Fundamentals of Fluid Mechanics and Transport Phenomena

0 

oo abba

XX [1.2]

To be more general, we can consider a process during which the sub-systems (a)

and (b) receive in addition the quantities X

a ext

and X

b ext

from the exterior of the

system S:

baextbbabextaa

XXXXXX

  ;

The conservation of the quantity X in the system implies that the total amount of

the quantity X is equal to the amount of this quantity which is received from the

exterior:

extbextaba

XXXX  

Equality [1.2] is once again the result.

The preceding reasoning is not applicable to entropy, for which we do not have

conservation, but only non-decay. This results in the principle of action and reaction,

which, as applied to the entropy, is written:

0t

oo abba

SS [1.3]

This inequality implies that one of the sub-systems has gained more entropy than

the other has lost.

The actions leading to a gain in the quantity X in system (a) can be classed into

two categories:

– volume actions related to fields which can exert their influence from a distance:

an electric field produces a displacement of electric charges, electromagnetic

radiation can lead to a the creation of heat by absorption in the material medium,

etc.;

– contact actions (on a surface) between (a) and (b) (Figure 1.5) essentially

results from the action between the molecules in the immediate vicinity of the

surface S (at distances of the order of the intermolecular distances or of the mean

free path in a gas). As this volume is based on the surface S and is of small

thickness, it can be modeled by the surface S, on which we can concentrate the

interaction between (a) and (b). The usual contact actions are pressure, frictional

force, thermal conduction, molecular diffusion, etc.

Thermodynamics of Discrete Systems 19

1.2.2.4.

States of equilibrium and extensive variables

Let us consider a material system in a given initial state. Suppose that excepting

its entropy, the values of its extensive quantities are known. We thus have a partial

characterization which does not allow a full description of the state of the system. It

is clear that a knowledge of the global values of mass, energy, etc., does not provide

any understanding of how these quantities are distributed within the system.

Let us now isolate the system and leave it to evolve according to a natural

process; the extensive quantities which it contains will naturally distribute

themselves in a balanced way throughout the system. It is easy to understand that the

final equilibrium state of the system is unique, as confirmed by experience. In other

words, the extensive quantities of a material system allow a description of the

equilibrium state achieved by a system at the end of all natural processes (provided

the system remains isolated). Following the axiom pertaining to the entropy (section

1.2.2.2), the entropy associated with such a material system increases in order to

attain a maximum value corresponding to this final state of equilibrium. We thereby

deduce that this entropy of the equilibrium state is a well-determined function of the

other extensive quantities for the system considered:

)(

XSS

[1.4]

Relation [1.4] constitutes the entropic representation of the system in

equilibrium. It is in fact a state equation (in the thermodynamic sense) for

equilibrium states which are only comprised of the extensive variables. In order not

to confuse it with the usual state equations, we will refer to it as the general

equation of state of a system. This relation is unique for a given material system,

although numerous state equations, more or less dependent on it, can be derived

from it. The variance of a system is the number of independent extensive quantities

necessary for its representation.

The existence of this general equation of state leads to the properties of

thermostatics.

1.2.2.5.

Homogenity

We have said that the extensive quantities of a material system are proportional

to its extension without stating exactly what this extension is. Let us suppose that

these extensive quantities X

and the entropy S are all associated with a given

volume of ordinary 3D space. Taking two systems S

and S with material contents

which are homothetic (or which are geometrically similar) with a ratio of O and

which are constituted from the same matter, under the same physical conditions, at

homologous points, we can say that the extensive quantities X

are proportional to a

cube with a reference dimension:

20 Fundamentals of Fluid Mechanics and Transport Phenomena

SSXX

Denoting our two states by (a) and p (a), with p = O

, state relation [1.4] can be

written:

)()(

iip

XpSpXSS

[1.5]

The function



XS is a first degree homogenous function.

1.2.2.6.

Note

This property no longer holds if the extensive quantities are associated with

spaces of different dimensions. As an example consider a drop of water whose total

energy is the sum of two terms, on the one hand an energy associated with its

weight, E

, proportional to its volume, and on the other hand an energy associated

with its surface tension, E

, proportional to its surface:

EEE 

The total energy of the system S

becomes, considering a homothetic ratio O

EEE



Other extensive quantities

(for example, mass or volume) of this drop remain

proportional to the homothetic ratio

and respect relation [1.4]. It is therefore clear

that relation [1.5] is no longer respected. We will not consider such cases in what

follows.

1.2.3. Energy, work and heat

1.2.3.1.

Energetic representation of a system

Among the extensive quantities, the energy, which we denote

is often

assigned a particular role. Depending on whether it is explicitly solved in

, or in

(energetic representation), the general equation of state [1.4] of a system can be

written:

),(or),(

XSEEXESS

[1.6]

Thermodynamics of Discrete Systems 21

In practical applications at usual temperatures, energy and entropy only appear in

terms of their variations. In these conditions, it is not necessary to define them in an

absolute sense and it is sufficient to evaluate them to the nearest constant.

For a system at rest in a Galilean reference frame, the energy of a system is

essentially comprised of its internal energy and the different forms of potential energy

of the system elements. Changing the Galilean reference frame would amount to adding

a constant amount of kinetic energy, which clearly does not change our description.

This will no longer be the case if elements of the system have different movements, in

which case their respective kinetic energies would need to be accounted for.

1.2.3.2.

Work and heat

By calculating the differential of relation [1.6]:





¦¦

we can define the

elementary work

and the

elementary heat

by the relations:

¦¦



dWdS

[1.7]

such that we obtain the classic definition

dQdWdE 

, in which the differential

forms

and

are not differentials of a function, i.e. they are not exact differentials.

1.3. Consequences of the axioms of thermostatics

1.3.1. Intensive variables

1.3.1.1.

Definition and properties

We will only study equilibrium states entirely characterized by the values of the

extensive variables and the relation of state [1.4] or [1.6] between them.

We define

entropic intensive variables

using the relations:

[1.8]

(resp. 1/

) being called the conjugated variable of

(resp.

) with respect to

22 Fundamentals of Fluid Mechanics and Transport Phenomena

The

energetic intensive variables

are similarly defined:

[1.9]

(resp.

) is the conjugated variable of

(resp.

) with respect to

The differentials

of the energy and

of the entropy can be written:



dXYTdSdE

[1.10]



dXZdE

[1.11]

From this, the relation between the energetic

and entropic intensive variables,

respectively

and

can be obtained:

TZY 

[1.12]

Consider a fluid defined by the extensive variables (volume

, number of moles

, energy

and entropy

), the energetic intensive variables are the pressure

(to

the nearest sign), the temperature

and the chemical potential



SVNVSN

dNTdSpdVdE

,,,

;; :with





1.3.1.2.

Consequences of homogenity

Differentiating relation [1.5] with respect to p, and letting p = 1, we obtain:

¦¦



iii

[1.13]

and:



XYTSE

[1.14]

We note that the general relation [1.13] or [1.14] does not allow the

characterization of a system. It constitutes a differential equation which is satisfied