Understanding Non-Equilibrium Thermodynamics - Springer 2008 Episode 3 pdf

The reciprocity relations imply that a temperaturegradient of 1◦C m−1 along the x-direction will give raise to a heat flux in the normal y-direction, which is the same as the heat flow gen

Box 2.2 Curie’s Law

The existence of spatial symmetries in a material system contributes to plify the scheme of the phenomenological relations Because of invariance ofthe phenomenological equations under special orthogonal transformations,some couplings between fluxes and forces are not authorized in isotropicsystems In the examples treated in this chapter, no fluxes and forces of ten-sorial order higher than the second will occur We shall therefore consider

sim-an isotropic material characterized by the three following phenomenologicalrelations

j = j(x , X), J = J (x , X), T = T(X), (2.2.1)

which contain fluxes and forces of tensorial order 0 (the scalar j), order

1 (the vectors J and x ) and order 2 (the symmetric tensors T and X).

Isotropy imposes that the above relations transform as follows under an

orthogonal transformation Q :

j(Q · x; Q · X · QT) = j(x , X), (2.2.2)

J (Q · x; Q · X · QT) = Q· J (x, X), (2.2.3)

T(Q· X · QT) = Q· T(X) · QT. (2.2.4)According to the theorems of representation of isotropic tensors (e.g

Spencer and Rivlin 1959), the functions j, J , and T are isotropic if and

only if

j(x , X) = j[I X , II X , III X , x · x, x · (X · x), x · (X · X · x)], (2.2.5)

J (x , X) = (A1 I + A2X + A3X· X) · x, (2.2.6)

T(X) = B1I + B2X + B3X· X, (2.2.7)

the flux j and the coefficients A i (i = 1, 2, 3) are isotropic scalar functions

of x and X and the B i are isotropic scalars of X alone; I X , II X , and III X

are the principal invariants of the tensor X, namely

I X = tr X, II X= 12[I X2 − tr(X · X)], III X = det X. (2.2.8)

By restricting the analysis to linear laws, (2.2.5)–(2.2.7) will take the simple

form

j = l(tr X), J = A1x , T = B1(tr X)I + B2 X, (2.2.9)

wherein l, A1, B1, and B2 are now scalars independent of x and X

Re-lations (2.2.9) exhibit the property that in linear constitutive equations,spatial symmetry allows exclusively the coupling between fluxes and forces

of the same tensorial order, as concluded from Curie’s law

Step 6 Restrictions on the sign of phenomenological coefficients

A direct restriction on the sign of the phenomenological coefficients arises as aconsequence of the second law Substitution of the linear flux–force relations(2.21) into (2.20) of the rate of entropy production yields the quadratic form

σ s=

αβ

According to standard results in algebra, the necessary and sufficient

condi-tions for σ s ≥ 0 are that the determinant |L αβ + L βα | and all its principal

minors are non-negative It follows that

con-Step 7 Restrictions on L αβ due to time reversal: Onsager–Casimir’s reciprocal relations

It was established by Onsager (1931) that, besides the restrictions on the sign,the phenomenological coefficients verify symmetry properties The latter werepresented by Onsager as a consequence of “microscopic reversibility”, which

is the invariance of the microscopic equations of motion with respect to time

reversal t → −t Accordingly, by reversing the time, the particles retrace their

former paths or, otherwise stated, there is a symmetry property betweenthe past and the future Invoking the principle of microscopic reversibilityand using the theory of fluctuations, Onsager was able to demonstrate thesymmetry property

In Chaps 4 and 11, we will present detailed derivations of (2.26)

It should, however, be stressed that the above result holds true only for

fluctuations a α (t) = A α (t) − Aeq

α of extensive state variables A αwith respect

to their equilibrium values, which are even functions of time a α (t) = a α(−t).

In the case of odd parity of one of the variables, α or β, for which a α (t) =

−a α(−t), the coefficients L αβ are skew-symmetric instead of symmetric, as

shown by Casimir (1945), i.e

In a reference frame rotating with angular velocityω and in presence of an

external magnetic field H, Onsager–Casimir’s reciprocal relations take the

form

L αβ(ω, H) = ±L βα(−ω, −H), (2.28)

as it will be shown on dynamical bases in Chap 11 The validity of theOnsager–Casimir’s reciprocal relations is not limited to phenomenologicaltransport coefficients that are scalar quantities as discussed earlier Considerfor example an irreversible process taking place in an anisotropic crystal, suchthat

J α=

β

where fluxes and forces are vectors and Lαβ is a tensor of order 2 The

Onsager–Casimir’s reciprocal relations write now as

in an anisotropic crystal The reciprocity relations imply that a temperaturegradient of 1◦C m−1 along the x-direction will give raise to a heat flux in the normal y-direction, which is the same as the heat flow generated along the x-axis by a temperature gradient of 1 ◦C m−1 along y Another advan-

tage of the Onsager–Casimir’s reciprocal relations is that the measurement

(or the calculation) of a coefficient L αβ alleviates the repetition of the sameoperation for the reciprocal coefficient L βα; this is important in practice as

the cross-coefficients are usually much smaller (of the order of 10−3 to 10−4)

than the direct coupling coefficients, and therefore difficult to measure oreven to detect

Although the proof of the Onsager–Casimir’s reciprocal relations wasachieved at the microscopic level of description and for small deviations offluctuations from equilibrium, these symmetry properties have been widelyapplied in the treatment of coupled irreversible processes taking place at themacroscopic scale even very far from equilibrium It should also be kept inmind that the validity of the reciprocity properties is secured as far as theflux–force relations are linear, but that they are not of application in thenon-linear regime

2.5 Stationary States

Stationary states play an important role in continuum physics; they are fined by the property that the state variables, including the velocity, remainunchanged in the course of time For instance, if heat is supplied at one end

de-of a system and removed at the other end at the same rate, the temperature

at each point will not vary in time but will change from one position to the

other Such a state cannot be confused with an equilibrium state, which ischaracterized by a uniform temperature field, no heat flow, and a zero entropyproduction It is to be emphasized that the evolution of a system towards anequilibrium state or a steady state is conditioned by the nature of the bound-ary conditions Since in a stationary state, entropy does not change in thecourse of time, we can write in virtue of (2.7) that

Stationary states are also characterized by interesting extremum principles

as demonstrated by Prigogine (1961): the most important is the principle ofminimum entropy production, which is discussed further The importance ofvariational principles has been recognized since the formulation of Hamilton’sleast action principle in mechanics stating that the average kinetic energyless the average potential energy is minimum along the path of a particlemoving from one point to another Quoting Euler, “since the construction

of the universe is the most perfect possible, being the handy work of an wise Maker, nothing can be met in the world in which some minimum ormaximum property is not displayed” It is indeed very attractive to believethat a whole class of processes is governed by a single law of minimum ormaximum However, Euler’s enthusiasm has to be moderated, as most of thephysical phenomena cannot be interpreted in terms of minima or maxima Inequilibrium thermodynamics, maximization of entropy for isolated systems orminimization of Gibbs’ free energy for systems at constant temperature andpressure, for instance, provide important examples of variational principles.Out of equilibrium, such variational formulations are much more limited.For this reason, we pay here a special attention to the minimum entropyproduction theorem, which is the best known among the few examples ofvariational principles in non-equilibrium thermodynamics

all-2.5.1 Minimum Entropy Production Principle

Consider a non-equilibrium process, for instance heat conduction or

thermod-iffusion taking place in a volume V at rest subjected to time-independent constraints at its surface The state variables a1, a2, , a n are assumed to

obey conservation laws of the form

ρ ∂a α

∂t =−∇ · J α (α = 1, 2, , n), (2.32)

where ∂/∂t is the partial or Eulerian time derivative; processes of this kind,

characterized by absence of global velocity, are called purely dissipative We

have seen in Sect 2.4 that the total entropy P produced inside the system

and since the ∂α α /∂Γ β terms (which represent, for instance, minus the heat

capacity or minus the coefficient of isothermal compressibility) are negativequantities because of stability of equilibrium, one may conclude that

dP

This result proves that the total entropy production P decreases in the course

of time and that it reaches its minimum value in the stationary state Animportant aside result is that stationary states with a minimum entropy pro-duction are unconditionally stable Indeed, after application of an arbitrarydisturbance in the stationary state, the system will move towards a transitoryregime with a greater entropy production But as the latter can only decrease,the system will go back to its stationary state, which is therefore referred to

as stable It is also worth to mention that P, a positive definite functional

with a negative time derivative, provides an example of Lyapounov’s function(Lyapounov 1966), whose occurrence is synonymous of stability, as discussed

in Chap 6

It should, however, be emphasized that the above conclusions are farfrom being general, as their validity is subordinated to the observance ofthe following requirements:

1 Time-independent boundary conditions

2 Linear phenomenological laws

3 Constant phenomenological coefficients

4 Symmetry of the phenomenological coefficients

In practical situations, it is frequent that at least one of the above restrictions

is not satisfied, so that the criterion of minimum entropy production is of weakbearing It follows also that most of the stationary states met in the natureare not necessarily stable as confirmed by our everyday experience

The result (2.38) can still be cast in the form of a variational principle

δP = δ



σ s (Γ

α , ∇Γ α , ) dV = 0, (2.41)where the time derivative symbol d/dt has been replaced by the variational symbol δ Since the corresponding Euler–Lagrange equations are shown to

be the stationary balance relations, it turns out that the stationary state ischaracterized by an extremum of the entropy production, truly a minimum,

as it can be proved that the second variation is positive definite δ2P > 0.

It should also be realized that the minimum entropy principle is not anextra law coming in complement of the classical balance equations of mass,momentum, and energy, but nothing else than a reformulation of these laws

in a condensed form, just like in classical mechanics, Hamilton’s principle is

a reformulation of Newton’s equations

The search for variational principles in continuum physics has been a ject of continuous and intense activity (Glansdorff and Prigogine 1964, 1971;Finlayson 1972; Lebon 1980) A wide spectrum of applications in macroscopicphysics, chemistry, engineering, ecology, and econophysics is discussed inSieniutycz and Farkas (2004) It should, however, be stressed that it is only inexceptional cases that there exists a “true” variational principle for processesthat dissipate energy Most of the principles that have been proposed refer

sub-either to equilibrium situations, as the maximum entropy principle in rium thermodynamics, the principle of virtual work in statics, the minimumenergy principle in elasticity, or to ideal reversible motions as the principle

equilib-of least action in rational mechanics or the minimum energy principle forEulerian fluids

2.6 Applications to Heat Conduction, Mass Transport, and Fluid Flows

To better understand and illustrate the general theory, we shall deal withsome applications, like heat conduction in a rigid body and matter diffusioninvolving no coupling of different thermodynamic forces and fluid flow Morecomplex processes involving coupling, like thermoelectricity, thermodiffusionand diffusion through membranes are treated in Chap 3 The selection ofthese problems has been motivated by the desire to propose a pedagogicalapproach and to cover situations frequently met in practical problems byphysicists, chemists and engineers Chemistry will receive a special treatment

in Chap 4 where we deal at length with chemical reactions and their couplingwith mass transport, a subject of utmost importance in biology Despite itssuccess, CIT has been the subject of several limitations and criticisms, whichare discussed in Sect 2.7 of the present chapter

2.6.1 Heat Conduction in a Rigid Body

The problem consists in finding the temperature distribution in a rigid body

at rest, subject to arbitrary time-dependent boundary conditions on perature, or on the heat flux Depending on the geometry and the physicalproperties of the system and on the nature of the boundary conditions, awide variety of situations may arise, some of them being submitted as prob-lems at the end of the chapter For the sake of pedagogy, we follow the samepresentation as in Sect 2.4

tem-Step 1 State variable(s)

Here we may select indifferently the specific internal energy u(r , t) or the temperature field T (r , t), which should be preferred in practical applications.

Step 2 Evolution equation

In absence of source term, the evolution equation (2.18) for u(r , t) is simply

ρ du

here d/dt reduces to the partial time derivative ∂/∂t as v = 0 Equation (2.42) contains two unknown quantities, the heat flux q to be given by a constitutive

relation and the internal energy u(T ) to be expressed by means of an equation

of state

Step 3 Entropy production and second law

According to the second law, the rate of entropy production defined by

pression cannot contain a flux term like∇ · (J s − T −1 q ), which describes the

rate of exchange with the outside, as a consequence this term must be setequal to zero so that

whereas (2.45) of σ sreduces to

This illustrates the general statement (2.20) that the entropy production is

a bilinear form in the force∇T −1 (the cause) and the flux of energy q (the

effect)

Step 4 Linear flux–force relation

The simplest way to ensure that σ s ≥ 0 is to assume a linear relationship

between the heat flux and the temperature gradient; for isotropic media,

which is nothing else than the Fourier’s law stating that the heat flux is

proportional to the temperature gradient We observe in passing that Curie’sprinciple is satisfied as (2.49) is a relationship between flux and force of thesame tensor character, namely vectors In an anisotropic crystal, Fourier’srelation reads as

where the heat conductivityλ is now a tensor of order 2

Step 5 Restriction on the sign of the transport coefficients

Substitution of (2.49) in (2.48) of the rate of entropy production yields

σ s= 1

and from the requirement that σ s ≥ 0, it is inferred that λ ≥ 0 Roughly

speaking this means that in an isotropic medium, the heat flux takes place

in a direction opposite to the temperature gradient; therefore, heat will flowspontaneously from high to low temperature, in agreement with our everydayexperience In an anisotropic system, flux and force will generally be oriented

in different directions but the positiveness of tensorλ requires that the angle

between them cannot be smaller than π/2.

Step 6 Reciprocal relations

In the general case of an anisotropic medium, the flux–force relation is theFourier’s law expressed in the form (2.50) According to Onsager’s recipro-cal relations, the second-order tensor λ is symmetric so that, in Cartesiancoordinates,

a result found to be experimentally satisfied in crystals wherein, however,spatial symmetry may impose further symmetry relations For instance, incrystals pertaining to the hexagonal or the tetragonal class, spatial symme-try requires that the conductivity tensor is skew-symmetric By combiningthis result with the symmetry property (2.52), it turns out that the elements

λ ij (i = j) are zero; it follows that for these classes of crystal, the application

of a temperature gradient in the x-direction cannot produce a heat flow in the perpendicular y-direction Very old experiences by Soret (1893) and Voigt

(1903) confirmed this result, which is presented as one of the confirmations

of the Onsager–Casimir’s reciprocal relations

Step 7 The temperature equation

We now wish to calculate the temperature distribution in an isotropic rigidsolid as a function of time and space The corresponding differential equation

is easily obtained by introducing Fourier’s law (2.49) in the energy balanceequation (2.42) and the result is

where use is made of the definition of the heat capacity c v = ∂u/∂T

In the case of constant heat conductivity and heat capacity, and

introduc-ing the heat diffusivity defined by χ = λ/ρc v, (2.53) reads as

equation under given typical initial and boundary conditions for an infiniteone-dimensional rod It is the same kind of equation that governs matterdiffusion, as shown in Sect 2.6.2.

Box 2.3 Method of Solution of the Heat Diffusion Equation

A convenient method to solve (2.54) is to work in the Fourier space (k, t) with k designating the wave number It is interesting to recall that Fourier

devised originally the transform bearing his name to solve the heat diffusion

equation Let us write T (x, t) as a Fourier integral of the form

When the initial temperature dependence corresponds to a local heating at

one particular point x0 of the solid, namely

where g0 is an arbitrary constant proportional to the energy input at the

initial time, and δ(x − x0) the Dirac function, (2.3.6) will be given by

T (x, t) = g0

(4πχt)1/2 exp[−(x − x0)2/4χt]. (2.3.8)

We deduce from (2.3.8) that the temperature diffuses in the whole rod from

the point x0over a distance proportional to (χt)1/2; as a consequence, large

values of the thermal diffusivity imply a rapid diffusion of temperature tion (2.3.8) exhibits also the property that, after application of a temperaturedisturbance at a given point of the system, it will be experienced instanta-neously everywhere in the whole body Such a result is in contradiction withthe principle of causality, which demands that response will be felt after theapplication of a cause In the above example, cause and effect occur simul-taneously This property is typical of classical irreversible thermodynamicsbut is not conceptually acceptable This failure was one of the main motiva-tions to propose another thermodynamic formalism currently known underthe name of Extended Irreversible Thermodynamics (see Chap 7)

Solu-Box 2.4 Stationary States and Minimum Entropy Production Principle

In the case of thermal conduction described by the phenomenological

equa-tion q = L qq ∇T −1 , the total entropy production P in the system, say a one-dimensional rod of length l, is according to (2.47),

c v ∂T /∂t with c v > 0, it is finally found that

Lya-that the stationary state is stable It is, however, important to realize Lya-that

the phenomenological coefficient L qq is related to the heat conductivity by

L qq = λT2 and therefore the validity of the minimum entropy production

principle is conditioned by the condition that λ varies like T −2; the

princi-ple is therefore not applicable to systems with a constant heat conductivity

or with an arbitrary dependence of λ with respect to the temperature This

is the reason why the criterion of minimum entropy production remains anexception

2.6.2 Matter Diffusion Under Isothermal and Isobaric

Conditions

In this section, we briefly discuss the problem of isothermal and isobaricdiffusion of two non-viscous isotropic non-reacting fluids, in the absence ofexternal forces There is no difficulty to generalize the forthcoming consid-

erations and results to the more general case of a n-component mixture of

viscous fluids

Let ρ1 and ρ2 denote the densities (mass per unit volume) and c1(=

ρ1/ρ), c2(= ρ2/ρ) the mass fractions of the two constituents (c1+ c2= 1),

ρ = ρ1+ ρ2is the total mass density Designating by v k (r , t) the local scopic velocity of substance k, the centre of mass or barycentric velocity is

and is expressed in kg m−2s−1; in some circumstances, it may be convenient

to replace in (2.56) the barycentric velocity by an arbitrary reference velocity(de Groot and Mazur 1962) In virtue of the definition of the barycentric

velocity, it is directly seen that J1+ J2= 0

The analysis carried out in Sect 2.6.1 for heat transport is easily repeatedfor matter diffusion In the present problem, the Gibbs’ equation takes theform

wherein ¯µ1 and ¯µ2 are the respective chemical potentials and where use has

been made of dc2 = −dc1 Combining (2.57) with the balance equations(2.16)–(2.18) written without source terms, it is easily shown (see Prob-lem 2.6) that the entropy production is given by

T σ s=−J1· ∇(¯µ1− ¯µ2)≥ 0. (2.58)Within the hypothesis of linear flux–force relations, one obtains

J1=−L∇(¯µ1− ¯µ2) (L > 0), (2.59)

or, in virtue of Gibbs–Duhem’s relation c1d¯µ1+c2d¯µ2= c1∇¯µ1+c2∇¯µ2= 0,

J1=− L

c2∇¯µ1=− L(∂ ¯ µ1/∂c1)T,p

wherein (∂ ¯ µ1/∂c1)T,p ≥ 0 to satisfy the requirement of stability of

equilib-rium After identifying the coefficient of ∇c1 with ρD, where D (in m2s−1)

is the positive diffusion coefficient, one finds back the celebrated Fick’s law

Elimination of J1 between Fick’s law and the balance equation ρ dc1/dt =

−∇ · J1leads to a parabolic differential equation in c1(r , t) similar to (2.54).

In the particular case of diffusion of N material particles located at a point

r = r0 at time t0, the distribution of the mass fraction c1(r , t) obeys the

same law as (2.3.8), namely

c1(r , t) = N

8(πDt)3/2 exp[−(r − r0)· (r − r0)/(4Dt)]. (2.62)

The characteristic displacement length is now given by l ∝ √ Dt, and this

result constitutes the main characteristic of diffusion phenomena, namelythat the mean distance of diffusion is proportional to the square root of thetime Note that it is also usual to express Fick’s law in terms of the molarconcentration rather than the mass fraction; in such a case, Fick’s law takesthe form (2.2) instead of (2.61)

differen-The set of basic variables giving a complete knowledge of the system in

space and time are the mass density ρ(r , t), the velocity v (r , t) and the temperature T (r , t) fields.

These variables obey the classical balance equations of mass, momentum,and energy already given in Sect 2.4 but recalled here for the sake of clarity:

The quantity V is the symmetric part of the velocity gradient tensor; in

Cartesian coordinates, V ij = 12(∂v i /∂x j + ∂v j /∂x i) In fluid mechanics, the

symmetric pressure tensor P is usually split into a reversible hydrostatic

pressure pI and an irreversible viscous pressure Pv in order that P = pI +

Pv with I the identity tensor The symmetric viscous pressure tensor Pv

may further be decomposed into a bulk part pv(= 13 tr Pv) and a tracelessdeviatoric part

0

Pvso that P = pvI+

0

Pv The possibility and physical meaning

of an antisymmetric contribution to Pv will be analysed in Problem 2.9.Gathering all these results, the pressure tensor can be written as

P = (p + pv)I +

0

Instead of the pressure tensor P, some authors prefer to use the stress tensor

σ, which is equal to minus the pressure tensor All the results derived aboveand the forthcoming remain valid by working with the stress tensor Theevolution equations (2.63)–(2.65) constitute a set of five scalar relations with

16 unknown quantities namely, ρ, v, p, pv,

0

Pv, q , u, and T It is the aim

of the classical theory of irreversible processes to provide the eleven missingequations As usual, we start from Gibbs’ relation

wherein du/dt and dρ −1 /dt will be replaced, respectively, by expressions

(2.65) and (2.63) This yields the following balance equation for the specific

entropy s:

ρ ds

dt =−∇ · q

T + q · ∇T −1 − T −1 pv∇ · v − T −1P0v :V,0 (2.68)whereV is the traceless part of tensor V From (2.68), it is inferred that the0

expressions of the entropy flux and the entropy production are given by

Pv, and the forces ∇T −1,

T −1(∇ · v), T −1V; by assuming linear relations between them and invoking0

Curie’s law, one obtains the following set of phenomenological relations, alsocalled the transport equations:

We recognize in (2.71) the Fourier’s law by identifying the

phenomenolog-ical coefficient L qq /T2 with the heat conductivity λ; (2.72) is the Stokes’ relation if l vv /T is identified with the bulk viscosity ζ, and finally (2.73) is the Newton’s law of hydrodynamics when L vv /T is put equal to 2η, with η

the dynamic shear viscosity It follows from (2.71)–(2.73) that the fluid will beset instantaneously in motion as soon as it is submitted to a force, in contrastwith elastic solids, which may stay at rest even when subject to stresses Atequilibrium, both members of equations (2.71)–(2.73) vanish identically as it

should The phenomenological coefficients λ, ζ, and η depend generally on ρ and T but in most practical situations the dependence with respect to ρ is

negligible In the case of incompressible fluids for which∇·v = 0, the viscous

pressure pv is zero and the bulk viscosity does not play any role However,

in compressible fluids like dense gases and bubbly liquids, the bulk viscosity

is by no means negligible The bulk viscosity vanishes in the case of perfect

gases; in real gases, it is far from being negligible and the ratio ζ/η between

bulk and shear viscosity may even be of the order of hundred

After substitution of the transport equations (2.71)–(2.73) in (2.70) of theentropy production, one obtains

By combining the nine transport equations (2.71)–(2.73) with the fiveevolution equations (2.63)–(2.65), one is faced with a set of 14 equationswith 16 unknowns The missing relations are provided by Gibbs’ equation

(2.67) from which are supplemented the two equations of state T = T (u, ρ),

p = p(u, ρ), or solving with respect to u,

These equations, together with the evolution and transport relations describecompletely the behaviour of the one-component isotropic fluid after thatboundary and initial conditions have been specified The state equations(2.76) have a status completely different from the phenomenological equa-tions (2.71)–(2.73) because, in contrast with the latter, they do not vanishidentically at equilibrium By substitution of the phenomenological laws ofStokes (2.72) and Newton (2.73) in the momentum balance equation (2.64),one finds back the well-known Navier–Stokes’ equation when it is assumedthat the viscosity coefficients are constant:

ρ dv

dt =−∇p + 2η∇2v +

2

3η + ζ



∇(∇ · v) + ρF. (2.77)