Generalized Thermodynamics of Global Irreversible Processes in a

J. Phys. Chem. B 1999, 103, 8583-8594

8583

Generalized Thermodynamics of Global Irreversible Processes in a Finite Macroscopic System Byung Chan Eu† Department of Chemistry, McGill UniVersity, 801 Sherbrooke Street West, Montreal, Quebec H3A 2K6, Canada ReceiVed: March 8, 1999; In Final Form: July 22, 1999

The second law of thermodynamics stated by Clausius and Kelvin gives rise to the notion of uncompensated heat, which identically vanishes if the processes are reversible. It is required to have a knowledge of the quantity and the associated compensated heat if a theory of irreversible processes is to be formulated. In this paper, to formulate a global theory of irreversible processes, we present a global form of uncompensated heat together with an attendant form of extended Gibbs relation. The formulas presented are then examined from the viewpoint of local irreversible processes by using local field equations for macroscopic variables, namely, the generalized hydrodynamic equations of fluids. The formulas obtained for global irreversible processes from the local field theory of macroscopic processes in space-time clarify the meanings of the aforementioned global expressions.

I. Introduction Within the current framework of natural sciences the laws of thermodynamics are believed to govern all natural macroscopic phenomena. These phenomena are either studied in such a detail with regards to their space-time behavior that their description requires a local theory of field variables, such as fluid velocity, pressure, temperature, and so on, which depend on local position and time, or studied on a gross scale so that only a global theory is required, in which close attention is not paid to the local behavior of the field variables but to their global behavior. Fluid dynamic description of natural macroscopic phenomena belongs to the former category of theories. The laws of thermodynamics, however, are phrased for macroscopic bodies (e.g., cycles and engines) without reference to the local behavior of field variables around points within the system; they are concerned with the global behavior in a gross scale comparable to the macroscopic size of the system. Therefore, making deductions directly from the laws of thermodynamics at the level of global processes, especially in the case of irreversible processes should be of considerable importance in thermodynamics. In this work we are interested in a global description of a macroscopic system undergoing irreversible processes. Henceforth the term global will be used to connote gross scale processes or phenomena in, or gross scale behaviors of, a macroscopic system, as opposed to the term local used in the sense of fluid dynamics or continuum field theory in which local field variables are of primary interest. It is now well recognized that the classical thermodynamics theory deals with an idealization of real processes and is incomplete because it is concerned with reversible processes only, although natural phenomena are generally irreversible. The classical thermodynamics is based on the notion of entropy which Clausius3 deduced from a mathematical representation of the second law of thermodynamics that holds for reversible † Also at the Centre for the Physics of Materials, McGill University and the Asia Pacific Center for Theoretical Physics, Seoul, Korea. Electronic mail address: [email protected].

processes only. It must be emphasized that the notion of entropy therefore is applicable only to reversible processes or equilibrium. Clausius discovered that there are two kinds of heat involved in thermal processes: compensated heat and uncompensated heat. The former is heat exchanged between the system and the surroundings, and the latter is the heat inherently generated within the system because of the processes being irreversible. He defined reversible processes as those in which the uncompensated heat identically vanishes. In the previous work1,2 on generalization of thermodynamics it has been shown that, even if the process involved is irreversible, the second law of thermodynamics as stated by Clausius and Kelvin3,4 can be mathematically represented in terms of calortropy (meaning heat evolution), compensated heat, and uncompensated heat, which are combined into a single exact differential form in an appropriate macroscopic variable space, and that a thermodynamically consistent theory of irreversible processes can be formulated on the basis of the notion of calortropy as a generalization of the Clausius entropy that holds only for reversible processes. Therefore, to determine either one of the aforementioned three quantities it is necessary to elucidate the other two quantities by some independent means and also specify the variable space spanned by experimental observables. It was in fact possible to deduce by thermodynamic correspondence some local forms for the compensated and uncompensated heats on the basis of local field theories derived from the kinetic theory of fluids such as the Boltzmann kinetic theory and the generalized Boltzmann equation for dense fluids.2 However, their global macroscopic forms for a system of a finite volume still remain to be investigated. The principal aim of this work is to elucidate the fundamental extended Gibbs relation, compensated heat, and uncompensated heat for global irreversible processes so as to erect a theoretical framework for generalized thermodynamics which can be applied for studying mean (volume-averaged) macroscopic irreversible processes in a system of a finite size. Often in macroscopic physics and thermal science in general it so happens that we are not interested in the details provided

10.1021/jp9908000 CCC: $18.00 © 1999 American Chemical Society Published on Web 09/23/1999

8584 J. Phys. Chem. B, Vol. 103, No. 40, 1999

Eu uB, diathermal, and permeable to matter. The direction of the outward normal to Bi is counted positive. The volume of the system is denoted by V, which may change with time. The surroundings may be regarded as adjoining systems or large heat or material reservoirs at equilibrium or, more precisely, at internal equilibrium. The component fluid velocities are denoted by ua and the barycentric velocity by u. The boundary conditions on fluid velocities are

Figure 1. Schematic representation of the system under consideration. The shaded rectangle is the system which is separated from the surroundings (e.g., heat or material reservoirs) by the boundaries B1 and B2, which may be movable, diathermal, and permeable to matter. The vertical side walls ϑ3 and ϑ4 are not shown in the figure. The boundaries ϑ1 and ϑ2 are rigid, adiabatic, and impermeable to matter. The boundaries ϑ1 and ϑ2, however, may move in parallel in opposite directions in the case where the system is sheared in plane Couette flow geometry.

by a local hydrodynamic theory of macroscopic irreversible phenomena but in a global description of irreversible processes which tells us about the phenomena occurring in the gross scale of the system of a finite volume. For example, many physicochemical measurements, such as flow rate of fluids through a tube or capillary, heat flow through a piece of a metal, flow of electric current in electric cells or electronic devices, and numerous other nonequilibrium effects, are performed without attention paid to the details of local processes within the system, the only quantities under our control being the boundary conditions on the variables involved or the values of the variables at the boundaries. Therefore a global formulation of irreversible processes is of great practical interest to our study of commonly occurring macroscopic phenomena which we encounter in a gross spatial scale. Nevertheless, the thermodynamic theory of such global processes and its molecular theory foundations are still obscure and remain to be studied seriously, if nothing else, for the information that can be extracted for material functions from experiments on global irreversible processes performed in the laboratory or phenomena observed in nature. The present work is a report on an aspect of our continuing efforts toward the goal of formulating such a global thermodynamic theory of irreversible processes on the basis of the mathematical representation that we have achieved for the second law of thermodynamics for irreversible processes in the work mentioned earlier.1,2 The notions of uncompensated and compensated heats accompanying global irreversible processes are important in such a theory and should be elucidated if a theory of global irreversible processes is to be formulated for a system of a finite volume. Studying the subject together with the differential form for calortropy will also give a more precise idea about the significance of the calortropy differential form itself in the case of global irreversible processes. To make the discussion more definite we consider a finite system (a mixture of reacting fluids) bounded by surfaces as described in Figure 1. The fluid mixture consists of r components which are denoted by Ca (a ) 1, 2, ..., r). There are m chemical reactions of compounds Ca (1 e a e r) r

∑νalCa ) 0

(m g l g 1)

a)1

where νal denotes the stoichiometric coefficients times the mass of species a, which are counted positive for the products, and negative for the reactants. It is assumed that the surfaces ϑi (i ) 1, ..., 4) are adiabatic, rigid, and impermeable to matter, whereas the surfaces Bi ) (i ) 1, 2) are movable at velocity

ua ) u ) 0 at ϑi (i ) 1, ..., 4) Therefore the boundaries ϑi do not play a role in the present consideration since it is assumed that they are not only immobile but also adiabatic and impermeable to matter. The configuration of the system and the boundary conditions considered in this work, albeit looking special, is representative of the configurations of wide-ranging global irreversible phenomena studied in the laboratory. Consequently, the theory developed for the system is sufficiently general; it is formulated in such a form that it can be readily adopted to different configurations. In section II some relevant aspects are discussed of the calortropy differential that arises from the second law of thermodynamics, and an expression for the uncompensated heat is obtained. Physical arguments are given for a pair of principal components in the expression for the uncompensated heat and, on the basis thereof, a proposition is made for the pair. This proposition basically represents what we understand about the phenomenological form of the uncompensated heat associated with global irreversible processes. The proposition is examined by calculating the various contributing factors on the basis of their local formulas in the subsequent sections. In section III the local form of the first law of thermodynamics is averaged over the volume of the system to identify the internal work. The local form of the compensated heat is also averaged over the volume to identify the heat transfer in excess of the heat transfer given by the first law. With results obtained in section III the global uncompensated heat is calculated from its local form and elucidated in section IV. The calculations performed for various quantities involved indicate the limitations, from the viewpoint of local continuum field theory, of the global extended Gibbs relation, compensated heat, and uncompensated heat, which have been presented on the phenomenological grounds in section II. Some examples are presented in section V, and the conclusion is presented in section VI. II. Global Calortropy and Related Quantities for a Finite System A. Calortropy Differential. Clausius3 introduced the notion of entropy in the case of reversible processes only, for which the uncompensated heat vanishes everywhere in the path of processes. If the reversible heat transfer in a reversible process is denoted by dQrev, then the entropy S introduced by Clausius for the reversible process is defined by

dS )

dQrev T

(1)

In equilibrium thermodynamics5 based on eq 1, the compensated heat dQrev is identified with the reversible heat change in the expression for the first law of thermodynamics, and the uncompensated heat, denoted by dN, vanishes identically. Therefore the process in question should be such that there is no energy dissipation. If the process involved is irreversible, the differential form 1 must be suitably generalized, and the

Thermodynamics of Global Irreversible Processes

J. Phys. Chem. B, Vol. 103, No. 40, 1999 8585

thermodynamics of irreversible processes crucially hinges on how the notion of entropy is generalized to nonequilibrium phenomena. It was previously shown1,2,10-12 that if there is an irreversible process which is accompanied by uncompensated heat, there exists an exact differential dΨ in an appropriate thermodynamic space necessary for description of the process, and the second law of thermodynamics can be mathematically represented by the differential form

dΨ )

dQ + dN T

(2)

This differential form dΨ is called the calortropy differential and follows from the Clausius inequality if the compensated and uncompensated heats are regarded as two physically independent quantities:

N)-I

dQ T

(3)

where dQ is the “heat” transfer between the body and the heat reservoir, which is called the compensated heat, and N is the uncompensated heat. The differential form 2 is applicable to an irreversible cycle operating between heat reservoirs of an infinitesimal temperature difference, material reservoirs of an infinitesimal concentration difference, and similarly for a pressure difference, and thus performing an infinitesimal amount of work. By the term infinitesimal we do not mean a cycle which is infinitesimally small in its physical size, but we mean a cycle which has an infinitesimally small difference in the boundary conditions that drive the cycle to produce an infinitesimal work. Here T is the temperature of a heat reservoir, and the way the Clausius inequality3,5 was obtained suggests that the temperature T may be taken as a mean quantity over the volume of the infinitesimal cycle involved. This temperature is made a local quantity in the local field theory of macroscopic variables, which underlies the global expression 2 for a system of a finite volume. The meaning of T for an infinitesimal cyclic process (e.g., an infinitesimal Carnot cycle) is discussed in detail in ref 2. The dN is the differential of the uncompensated heat N accompanying the irreversible process. To satisfy the second law of thermodynamics the inequality dN/dt g 0 should be satisfied everywhere on the cycle; otherwise, it is possible to construct a cycle that violates the second law of thermodynamics. Whereas dQ and dN are not exact differentials in the thermodynamic space, dΨ is an exact differential in the space by the demand of the second law of thermodynamics. This aspect of dΨ is similar to the case of the first law of thermodynamics where the internal energy differential dE is an exact differential, although the work and heat differentials making up dE are not exact differentials. Therefore, Ψ, called the calortropy,2,10 is a state function in the space of macroscopic variables characterizing the thermodynamic processes in the body of a finite volume; it is a global quantity for the whole macroscopic system. The calortropy is a quantity that generalizes to irreversible processes or nonequilibrium the notion of Clausius (equilibrium) entropy, which holds only for reversible processes or equilibrium. The physical meanings of compensated and uncompensated heats are not obvious from the expression 2 as it stands, and irreversible thermodynamics may be considered to be a subject of thermal science in which the meanings of compensated and uncompensated heats are sought in such a manner that the irreversible process of interest is explained in accordance with the laws of thermodynamics. We posit this as the general goal of irreversible thermodynamics.

If there is an irreversible process present in the system, then there is an internal work besides the work by the cycle designed to perform a particular mechanical task, and hence the meaning of dQ is not necessarily the same as the heat change simply calculated as the complementary component to the work in the expression for the first law of thermodynamics in the case of reversible processes. In other words, when the work of the task for a cycle (e.g., pressure-volume work in the case of a Carnot cycle which involves compression and dilatation of a gas in a piston) in terms of heat is calculated by using the first law of thermodynamics, dQ inevitably contains a contribution corresponding to the internal work that is irrelevant to the given task of mechanical work for the cycle, and dQ or the aforementioned contribution, together with the meaning of dN, remains to be elucidated. This point will be elaborated on later. From the mathematical viewpoint the nonequilibrium thermodynamic space characterizing the thermodynamic state of the system and therefore the calortropy Ψ is larger than that for the Clausius entropy S for a reversible process, since the former must include additional nonequilibrium variables among the variables spanning the thermodynamic space if the nonequilibrium process is to be adequately described in spacetime. Nevertheless, just as the integral of dS vanishes over a closed contour characterizing a reversible cyclic process in the equilibrium thermodynamic space, does the integral of dΨ identically vanish over a closed contour characterizing an irreversible cyclic process in the nonequilibrium thermodynamic space, since dΨ is an exact differential in the space mentioned:

I dΨ )

)0 ∫0τdtdΨ dt

(4)

Here τ is the period of the cycle. We remark that the circular integral here is in the thermodynamic space, but not over configuration (position) space. Therefore, written in the present form13 for an irreversible cyclic process returning to the initial state on compensation from the surroundings, the second law of thermodynamics may be expressed as a vanishing closed contour integral of the calortropy differential for the irreversible process of interest. It must be emphasized that dN g 0 everywhere along the path of integration. Since for a cyclic process the first law of thermodynamics is also expressible by a vanishing contour integral

I dE ) 0

(5)

the two thermodynamic laws can be mathematically represented by a pair of vanishing contour integrals. The differential form 2 is a direct consequence of eq 4. It is convenient to express the uncompensated heat differential

dN )

dN T

(6)

where by virtue of the second law

dN g0 dt

(7)

everywhere in the irreversible process. The equality holds only for reversible processes or at equilibrium. Therefore we may write eq 2 in the form

TdΨ ) dQ + dN

(8)

This is the fundamental differential form which we are going to study as a general mathematical representation of the second


Eu

law of thermodynamics for irreversible processes in a system of a finite size. Since in the case of reversible processes dN ) 0, identically, the differential form 8 reduces to eq 1 for reversible processes. Therefore the present theory includes the classical thermodynamics as a special case. B. Thermodynamic Space. Having ascertained that according to the second law of thermodynamics the calortropy is a state function in thermodynamic space, it is now possible to deduce some consequences of this property. To make progress in this investigation the notion of thermodynamic space, in which the global calortropy Ψ lives, should be made explicit and specific. Thermodynamics is a theory in which heat, work, and energy are interrelated and various thermal phenomena are correlated and interpreted in a mutually and thermodynamically consistent manner. It therefore is reasonable to think that the thermodynamic state of the whole system should be described by energy E, volume V, and masses Ma of species (1 e a e r), as well as nonconserved variables which change in time and characterize the irreversible behavior of the system; consequently, Ψ should be a function of the same set of variables. The nonconserved variables will be denoted by Γka where the subscript k denotes the kind of nonconserved variable. For example, Γka may be the volume integrals of fluxes such as diffusion fluxes, heat fluxes, stresses, and so on, which are necessary to properly describe the global irreversible processes of interest. The thermodynamic space (manifold) is then assumed to be spanned by the global variables for the whole system, which include E, V, Ma, Γka; l g k g 1, r g a g 1. There are 2 + (l + 1)r variables. This set will be abbreviated by B. The union of B with Ψ will be called the Gibbs space BG. C. Extended Gibbs Relation and Uncompensated Heat for a Finite System. Since the second law of thermodynamics represented by the vanishing contour integral 4 suggests that the calortropy Ψ is a state function in the sapce of macroscopic variables, which we take to be space B or a surface in space BG, by using the chain rule it is possible to obtain the time derivative of Ψ in the form

dΨ dt

)

( ) ∂Ψ ∂E

( ) ∑( ) dE

+

V,M,Γ dt r

∂V

∂Ma

E,M,Γ

dMa

∂Ψ

a)1

dV

∂Ψ

E,M′,Γ

dt

+

dt r

+

l

( ) ∂Ψ

∑∑ a)1 kg1 ∂Γ

ka E,M,Γ′

dΓ dt

(9)

where the prime on the subscripts means the exclusion of the variable of differentiation. Here the time derivatives represent the changes in the entire system which arise from the influence from the surroundings and the processes inherent to the system itself. Therefore, for example, it is necessary to distinguish the mass change within the system that arises from chemical reactions and the mass change incurred by the mass transfer between the body and its surroundings. The total mass change of species a can thus be written in two parts:

deMa )dt

(10)

where diMa/dt denotes the mass change arising from the chemical reactions within the system and deMa/dt represents the mass change arising from the mass transfer between the system and its surroundings. If species a of mass density Fa flows out of the system at the velocity (ua - uB) across the boundaries then the transfer part is given by the expression

(11)

where the integral is over the surface of the boundaries whose outward normal vector is taken positive. The mass density can change owing to chemical reactions within the system. If the reaction rate of reaction l is denoted by Rl then the rate of mass change arising from the chemical reactions is given by the expression

diMa

m

)

dt

∫Vdr∑νalRl

(12)

l)1

If chemical reactions occur uniformly in volume V as is the case in the absence of localized reaction sites, this rate may be written as

diMa

m

)

dt

νalRlV ∑ l)1

(13)

In the case of volume V there is no decomposition possible as in eq 10. It is also reasonable to decompose the rate of change, (dΓka/dt), in global nonconserved variable Γka into the part related to the transfer of Γka between the system and the surroundings, and the intrinsic part that has to do with energy dissipation within the system

dΓka deΓka diΓka ) + dt dt dt

(14)

where the first term on the right represents the transfer part, and the second the intrinsic part. These decompositions of dMa/ dt and dΓka/dt will be later used in eq 9. The subscripts e and i on the time derivative operator are reserved to denote the contributions to the rates of change in global macroscopic quantities in the aforementioned sense. The de/dt will be called the transfer derivative, and di/dt the intrinsic derivative. The transfer and intrinsic parts in eq 14 have physical meanings similar to those of the rates of mass change in eqs 11 and 13. Their more precise mathematical meanings from the local theory viewpoint will be given as the theory is developed in detail. It is now necessary to consider the first law of thermodynamics if further progress is to be made. In ref 2 careful consideration was made with regards to the operational meanings, and thus measurements, of temperature, pressure, chemical potentials, and generalized potentials associated with Γka when the system is away from equilibrium. The work term in the first law of thermodynamics should consist of pressure-volume work and works arising from mass transfer and the transfer of nonconserved variables. If these works are incorporated into the work term, the global form of the first law of thermodynamics may be expressed by the differential form for the internal energy

dE dMa deMa diMa ) + dt dt dt

∫BdB‚Fa(ua - uB)

dt

)

dQE dt

dV

-p

dt

r

+

deMa

∑µˆ a a)1

dt

r

-

l

∑ ∑Xka a)1 kg1

deΓka dt

+

diW dt (15)

Here p denotes the hydrostatic (phenomenological) pressure, µˆ a the (phenomenological) chemical potential of species a, Xka the (phenomenological) generalized potential conjugate to Γka, and the last term on the right represents the internal work per unit time not accounted for by other works already taken into consideration. The second, third, and fourth term on the right



represent, respectively, the powers related to pressure-volume work, work arising from mass transfer, and work associated with transfer of nonconserved variable Γka. The signs of these terms are assigned in such a way as to be consistent with the sign convention2 on work. The diW/dt is the internal work whose meanings will be elucidated as the theory is developed. For the moment it is sufficient to define it through the relation r r l deMa deΓka dV Xka ) - p + µˆ a (16) dt dt dt a)1 dt dt a)1kg1

diW

dW

∑

∑∑

Therefore diW is what is left in dW beyond the last three kinds of work in eq 16. The dQE/dt is the heat transfer rate between the body and the surroundings. It must be remarked that since at this point in development of the theory the compensated heat dQ cannot be presumed14 to be necessarily the same as the heat transfer dQE in eq 15; we distinguish them by attaching the subscript E to the heat transfer in the expression for the first law of thermodynamics. This point has been mentioned in the previous section, and it is important to elaborate on it. Let us call the work of the task the particular kind of work to accomplish a given mechanical task which a cycle performs. In the case of a Carnot cycle, pressure-volume work is the work of the task. On the other hand, if generating heat by rubbing a substance or shearing a liquid is the aim of the operation, then the mechanical work associated with viscous heating is the work of the task, whereas the pressure-volume work always present in such an operation represents a mode of energy loss to the given mechanical task. Suppose the pressure-volume work is the work of the task for the cycle in question, as is often the case for an internal engine. Then the work of the task during a cyclic period is

Wtask ) - I dV p

(17)

The efficiency of the cycle is calculated with this Wtask. The first law of thermodynamics implies that

Wtask ) -I

(

dQE dt

r

+

∑µˆ a a)1

deMa dt

r

-

l

deΓka

∑∑Xka a)1kg1

dt

+

)

where dQn/dt, together with diW/dt, is a quantity that must be elucidated by some means. It will be found that dQn/dt is a source of uncompensated heat. The signs of these terms are assigned in such a way as to be consistent with the sign convention2 on work. Upon making use of eqs 15 and 8 in eq 9, we obtain

0)

[( ) ∂Ψ

-

T dt

V,M,Γ

∑

dQ dQE dQn ) + dt dt dt

]

p dV

]

E,M,Γ

[( ) ] ∑( ) ∑ ∑( ) ( )

Not only are the global variables E, V, Ma, and Γka independent, but also their variations are arbitrary, and moreover, Ψ is a state function in space B as a consequence of the second law of thermodynamics. On the other hand, since the intrinsic variations of Ma and Γka as well as the internal work are inherent to the system and the irreversible processes in question, and just as the uncompensated heat dN is intrinsic to the system and the processes, their variations cannot be arbitrarily manipulated by the observer. Therefore the intrinsic parts of the variations in macroscopic variables and the uncompensated heat cannot be made to vanish individually and independently of each other; they must therefore vanish collectively. Consequently it can be concluded that the coefficients to the transfer derivatives on the right in eq 21 should be equal to zero and the intrinsic derivatives should collectively add up to zero. Hence not only the derivatives of Ψ are given by the relations

(∂Ψ ∂E ) (∂Ψ ∂V ) ∂Ψ ∂Γka

)

1 T

(22)

)

p T

(23)

V,M,Γ

E,M,Γ

( ) ( ) ∂Ψ ∂Ma

(19)

(20)

-

∑∑

(18)

This or, more precisely, the difference of input Qin and output Qout was used in the original derivation of the Clausius inequality and is traditionally used in the textbooks of thermodynamics. This was the point we mentioned earlier; it is an important point of departure from the role we assign to dQE in the linear theory of irreversible processes,7-9 where the local equilibrium assumption is made for the entropy and the compensated heat dQ is not distinguished from the heat transfer dQE in the first law of thermodynamics. This subtle difference is generally overlooked in the case of irreversible processes, but it is important to keep it in mind when irreversible thermodynamics is formulated. To make the distinction between dQ and dQE we set

∂Ψ ∂V

[( )

di W dt

+

[( )

+ T dt r µˆ a deMa ∂Ψ + + T dt a)1 ∂Ma E,M′,Γ r l r Xka deΓka diMa ∂Ψ ∂Ψ + + T dt a)1kg1 ∂Γka E,M,Γ′ a)1 ∂Ma E,M′,Γ dt r l diΓka 1 diW dQn dN ∂Ψ (21) + T dt dt dt a)1kg1 ∂Γka E,M,Γ′ dt ∂E

but when the efficiency of the cycle is calculated, the right hand side of this equation is denoted by a contour integral of say, dQ; that is,

Wtask ) - I dQ ) -Q

]

1 dE

)-

E,M′,Γ

)

E,M,Γ′

µˆ a T

(24)

Xka T

(25)

but also the uncompensated heat change dN is determined by the equation

dN dt

)

diW dt

-

dQn dt

m

-

∑ l)1

r

AlRlV +

diΓka

l

∑∑

a)1kg1

Xka

dt

g0

(26)

where we have used eq 13 for diMa/dt in eq 21 and Al is the chemical affinity defined by r

Al )

∑µˆ aνal

(27)

a)1

This form of uncompensated heat generalizes the form by De Donder15 and Defay,16 who obtained the uncompensated heat for chemical reactions only. On use of relations 22-25 we


Eu

obtain the extended Gibbs relation for Ψ

dΨ ) T-1(dE + pdV -

r

∑

r

µˆ adMa +

a)1

l

∑ ∑XkadΓka)

(28)

a)1 kg1

This relation is for a global irreversible process and must be appended by the inequality 26 imposing a constraint on diΓka/ dt and other terms therein. A more detailed investigation is necessary for diW/dt and dQn/dt before the uncompensated heat is fully elucidated. Equations 22-26 in fact can be shown to give rise to eq 8, given the first law of thermodynamics in the form of eq 15 and Ψ ) Ψ (E, V, {Ma}, {Γka}). Therefore it can be concluded that, given eq 15 and eq 8, eqs 22-26 are the necessary and sufficient conditions for the global extended Gibbs relation 28 to hold in the thermodynamic space B. This result implies that the extended Gibbs relation 28 alone is not sufficient for a theory of irreversible thermodynamics, unless the theory is accompanied by the proposition for dN/dt, for example, as in eq 26 and, of course, the constitutive relations for the coefficients to the differentials in eq 28. Let us introduce mean thermodynamic forces, namely, volume averages of the thermodynamic forces conjugate to the nonconserved variables Γka and the fluxes Υka of Γka. The mean thermodynamic forces for Γka will be denoted by ωka and those for Υka by $ka. These mean thermodynamic forces drive the system in volume V to undergo energy dissipating irreversible processes. If the present theory is a generalization of the linear theory, it ought to be inclusive of the linear theory form of energy dissipation as a special case. Our experience with the linear theory of irreversible processes suggests that the energy dissipation should be given by a bilinear form of Γka and ωka and also of Υka and $ka. On the basis of this reasoning we propose the following. Proposition. The dissipation function ∑l ≡ T-1 (diW/dt dQndt) is a bilinear form of mean fluxes Γka and Υka and their mean conjugate thermodynamic forces ωka and $ka: r

Σl ) T-1

l

∑∑

(Γkaωka + Υka$ka)

(29)

a)1 kg1

T

-1

dN dt

-1

) Σl - T

III. Volume Averages of Local Forms In this and the following sections we delineate the proposition for dN/dt in the case of the configuration depicted in Figure 1. The discussion presented below not only elucidates various quantities involved but also demonstrates the importance of local continuum field theory in modeling global irreversible processes. The main aim is to see how and under what conditions the proposition arise from the continuum field theory and consequently to understand the limitations of the global thermodynamic theory of irreversible processes. A. Volume Average of the Internal Energy Balance Equation. For the aim we have in mind it is useful to note a general formula for the rate of change in a global variable M defined by its local density m (r, t)

M(t) )

r

l

AlRlV + T ∑ ∑Xka ∑ l)1 a)1 kg1 -1

dM ) dt

∫V(t) dr∂t∂ m(r, t) + ∫BdB‚uBm(r, t)

∂Fa ∂t

(32)

m

) -∇‚Faua +

νajRj ∑ j)1

where the last term on the right accounts for the contribution from the chemical reactions, eq 32 yields m

)-

dt g0

(31)

where the second integral on the right is a surface integral over surface B ) B1 + B2 (see Figure 1). 1. Rate of Mass Change. If m ) Fa, the mass density of species a, then since in the continuum theory the mass density balance equation for reacting species a is given by the equation

∫BdB Fa (ua - uB) + ∫Vdr∑νajRj j)1

diΓka dt

∫V(t)dr m (r, t)

It is well-known that the rate of change in M(t) is given by the formula17

dMa

Therefore the uncompensated heat dN is giVen by m

and diffusion. In order to ascertain the validity and limitations of this proposition and the global extended Gibbs relation it is necessary to consider the first and second laws of thermodynamics a little more in depth by using their local forms.

)

deMa diMa + dt dt

(33)

(30) A more detailed form for diΓka/dt must be taken such that this inequality is satisfied according to the requirement of the second law of thermodynamics. It must be noted here that the choice of the set {Γka: l g k g 1, r g a g 1} is not unique since it can depend on the kind of experiment considered. Therefore our understanding of what is meant by the uncompensated heat can change as the experiment is improved and the number of experiments to be considered increases. In this sense thermodynamics is anthropomorphic, but it enables us to comprehend in a logical manner what is observed of macroscopic processes in nature and in the laboratory in a coherent form and relate their behaviors to each other under the universal principles of thermodynamics. Examples will be considered for diΓka/dt in the next section. Examples for the mean thermodynamic forces will be also given in a later section where mean thermodynamic forces are explicitly worked out for shearing, bulk dilatation, heat flow,

This verifies eq 10. If the fluid sticks with the boundaries and moves with them so that ua ) uB, then deMa/dt ) 0 and the change in mass Ma in the system arises solely from the chemical reactions. 2. Rate of NonconserVed Variable Change. The stress tensors, heat fluxes, diffusion fluxes are typical examples of nonconserved variables appearing in the continuum theory of macroscopic processes. There are other nonconserved variables that may have to be included in an appropriate description of irreversible processes. Their evolution is described by partial differential equations which will be written in the form

∂Φka ) -∇‚(ψka + uΦka) + Zka + Λka ∂t

(34)

where Φka stands for a nonconserved variable, ψka is the flux of Φka, Zka is called the kinematic term which includes the driving force for Φka and is generally nonlinear, and Λka is called



the dissipation term which is the seat of energy dissipation associated with Φka. The dissipation term is generally nonlinear in Φka. The details of the evolution eq 34 can be found in refs 2 and 17. The volume integral of Φka gives the global nonconserved variables Γka:

Γka )

∫Vdr Φka ≡ 〈Φka〉V

(35)

Here

〈A〉V )

∫Vdr A

(36)

nonconserved variables Πa, ∆a, Q′a, Ja, etc., correspond to the set {Γka} in which the leading elements will be arranged in the following order:

Γ1a ) 〈Πa〉V, Γ2a ) 〈∆a〉V )

∫BdB‚ψka + 〈(Zka + Λka)〉V

The first term on the right of the equation above gives the transfer of Φka, and the second term the intrinsic change in Γka associated with the energy dissipation owing to Φka. Thus with the definitions

deΓka )dt

∫BdB‚ψka

diΓka ) 〈(Zka + Λka)〉V dt

∂FE

1 ∆a ) TrPa - pja 3

dE d ) dt dt

Q′a ) Qa - hˆ aJa where δ is the unit second rank tensor, pja and hˆ a are the partial pressure and the enthalpy per unit mass of species a, respectively, and the superscript t denotes the transpose of the tensor. The Qa is the energy flux, which is often called the heat flux, Pa is the stress tensor of species a, and Ja is the diffusion flux of species a. When the species components of the variables are r summed over all species, we obtain, for example, P ) ∑a)1 r Pa, Q ) ∑a)1 Qa, and so on. The volume integrals of

(42)

∫V(t)dr FE (r,t)

dQE dWp dW′int + + dt dt dt

(43)

where various derivatives are defined by

(39)

(40)

∑Ja‚Fa

where F is the mass density, E is the internal energy density, and Fa is the external force per unit mass of species a. The potential of force Fa per unit mass of species a will be denoted by φa, namely Fa ) -∇φa. By using the general formulas 31 and 32 we then obtain dE/ dt from the local internal energy balance eq 42:

dQE )dt

1 1 Πa ) (Pa + Pta) - δTrPa 2 3

(41)

a)1

)

which elucidates eq 14 from the viewpoint of the continuum field theory. It is a global form of the evolution equation for a nonconserved variable which must be modeled such that the second law of thermodynamics (eq 30) is satisfied. By using molecular theory models available for Zka and Λka, the righthand side of eq 38 will be calculated for some special cases in the next section. The scalar products or the contractions of ψka, Φka, and its conjugate variable Yka, which are either vectors or tensors, will henceforth be denoted by simple products without the contraction symbol, such as a dot, a double dot, or a multiple dots, for the sake of brevity of notation. For example, the product YkaΦka means the scalar product Yka .l Φka where .l denotes contraction of rank l tensors Yka and Φka. The following symbols are useful for performing calculations:

V

r

) -∇‚(Q + uFE) - P:∇u +

∂t

we obtain the rate of change in Γka in the form

dΓka deΓka diΓka ) + dt dt dt

a

3. Rate of Internal Energy Change. If the continuum mechanics method is applied to derive the internal energy balance equation in local form, it is given by the equation2,17

(37) (38)

a

Γ3a ) 〈Q′a〉V ) 〈Qa - hˆ aJa〉V, Γ4a ) 〈Ja〉V, etc.

Integrating eq 34 over the volume of the system under the boundary condition ua ) uB at the boundaries we obtain

dΓka )dt

〈31TrP - pj 〉

∫BdB‚[Q + (u - uB)FE]

dWp )dt dW′int dt

)

(44)

∫V(t)drpj∇‚u

(45) r

∫

dr[- (P - pjδ):∇u + V

∑Ja‚Fa]

(46)

a)1

Here pj denotes the local hydrostatic pressure of the system in a local elementary volume. This hydrostatic pressure is quantifiable by inserting a minute pressure gauge into the elementary volume in question. We define the mean pressure by the volume average

〈pj〉 ≡ V-1〈pj〉V ) V-1

∫V(t)dr pj

(47)

Henceforth the angular brackets 〈...〉 will denote the volume average. In the case of a finite system the mean values of the gradients of intensive quantities and the fluid velocity, namely, the thermodynamic forces, appear as finite differences of the intensive quantities, including the velocity, at the boundaries. It is useful to define the following symbols:

χAh ) -

() A

∑i V-1 ∫B dBi T (r, t) ) i

-V-1

∫V(t)dr ∇

() A T

〈 ( )〉

(r, t) ≡ - ∇

A T

(48)

with A denoting 1 or the local chemical potential µ j a or the local generalized potential Yka. In the geometry of the boundaries depicted in Figures 1 χAh is given in terms of the differences of surface averages of A/T at the boundaries B1 and B2, or the


Eu

χu ) -V-1〈∇u〉V

values of A/T at B1 and B2 if A/T is uniform at the boundaries. For example, if A is a vector and A/T are uniform over the boundaries,

[( )

χV ) -V-1〈∇‚u〉V

() ]

A A χAh ) - (ΩB/V) B ˆ + B ˆ T B1 1 T B2 2

χfa ) V-1〈Fa〉V

where ΩB is the surface of the boundaries, B ˆ 1 and B ˆ 2 are unit normal vectors, and (A/T)B1 and (A/T)B2 are the values of |A/T| at the boundaries 1 and 2, respectively. Since B ˆ 2 ) -B ˆ 1 by the definition of the positive direction of the normal vector, it follows that

[( ) ( ) ]

()

A A A B ˆ ≡ -L-1∆ B ˆ T B1 T B2 1 T 1

χAh ) - (ΩB/V)

Since V/ΩB ) L, where L is the gap between boundaries B1 and B2, χAh in this case is seen to be a finite difference of A/T over the distance L. Some examples of χA for scalar, vectorial, and tensorial A will be explicitly worked out to indicate their mathematical structures and more transparent physical meanings later at a more appropriate stage. The second term in the surface integral of eq 44 accounts for the internal energy flow accompanied by mass flow relative to the boundary velocity uB. Because u ) uB, the term does not contribute to the heat transfer. Thus we obtain

dQE )dt

∫BdB‚Q

dV ) 〈∇‚uB〉V dt

(50)

(51)

Since the pressure in a nonequilibrium system is generally a point function depending on position, the volume average 〈pj〉 - pj does not vanish everywhere in V. Therefore ∆p ) pj - 〈pj〉 varies from point to point; that is, there are fluctuations in pj if the system is mechanically in nonequilibrium. If the fluctuations in pressure vanishes, namely, ∆p ) pj - 〈pj〉 ) 0, the integral in eq 51 vanishes under the boundary condition u ) uB at boundary B, and if the mean pressure 〈pj〉 is identified with the phenomenological pressure p; namely, if 〈pj〉 ) p, then eq 43 may be written in the global form of the first law of thermodynamics as given in eq 15:

dV dW′int dE dQE ) -p + dt dt dt dt

(52)

dt where

r

)

∑[Γ1a:χu + Γ2aχV + Γ4aχfa]

a)1

dt

r

)

deMa

∑µˆ a a)1

r

-

dt

l

∑ ∑Xka a)1 kg1

deΓka

+

dt

diW (55)

dt

On use of eq 53 under the conditions stated, we find

diW

r

)

∑[Γa1:χu + Γa2χV + Γa4‚χfa] -

a)1

r

∑

a)1

µˆ a

deMa

r

+

dt

deΓka

l

∑∑

Xka

a)1 kg1

dt

(56)

This elucidates the physical meaning of diW/dt. This result will be used later to calculate the uncompensated heat. If the boundary condition u * uB at boundary B is taken, then the second term in the surface integral in eq 44 contributes, and it accounts for the energy carried by the matter relative to the boundary velocity uB and the meaning of dQE/dt is accordingly altered. This case is excluded in the present study. We also remark that Q in eq 42 is the flux of internal energy in excess of the convective flow of internal energy. B. Volume Average of the Compensated Heat. The previous work2,10-12 on the local theory of irreversible processes and kinetic theory suggests that the compensated heat dQ is related to the local calortropy flux Jc defined by

Jc )

Qc

r

Qca ∑ a)1

(57)

∑Yka(r, t) ψka

(58)

) T(r, t)-1

T(r, t)

where l

j a(r, t)Ja + Qca ) Qa - µ

kg1

As to the term dW′int/dt, if the fluctuations of Πa, ∆a, and Ja from the volume averages Γ1aV-1, Γ2aV-1, and Γ4aV-1, respectively, are negligible; namely, ∆Πa ) Πa - Γ1aV-1 ) 0, etc., then the internal work term may be written in the form

dW′int

dW′int

(49)

we may write dWp/dt in the form

∫V(t) dr (pj∇‚u - 〈pj〉 ∇‚uB)

The derivation of the first law from the local energy balance equation presented here shows that from the local field theory viewpoint the phenomenological expression for the first law tacitly assumes a mean pressure over the volume in the pressure-volume work term as well as mean stresses and diffusion fluxes. We emphasize that p here must be understood as 〈pj〉 from the viewpoint of the local field theory of hydrodynamic processes under the conditions, such as vanishing fluctuations, mentioned earlier. The term dW′int/dt defined by eq 46 accounts for an internal work and must be related to the internal work term in eq 15 as follows:

dt

This gives the rate of energy transfer across the boundaries between the system and the surroundings. Since the rate of volume change is given by the formula

dWp dV ) - 〈pj〉 dt dt

(54)

(53)

In this expression Qa is a species component of heat flux Q already introduced in eq 42, µ j a is the local chemical potential per mass of species a, and Yka is the intensive variable (generalized potential) conjugate to the nonequilibrium variable j a and generalized potentials Yka as Φka. Chemical potentials µ well as other variables in eq 57 all depend on position and time in local field theory. Since Yka and ψka are either vectors or tensors, their products Ykaψka must be understood as scalar products of vectors or tensors or contractions of the latter. It was shown in the previous work1,2,10-12 that the calortropy flux is directly related to the compensated heat in the manner


1 dQ )T dt


∫BdB‚Jc

(59)

Comparing this expression with eq 20 we find

dQn

Using the formula for Jc in eq 57 the surface integral can be recast into a more suitable form as follows:

1 dQ

)-

T dt

∫Vdr

r

1

l

Yka∇‚ψka) ∑(∇‚Qa - µj a∇‚Ja + ∑ T(r, t)a)1 kg1

r

[ () ( ) 1

∫Vdr∑ Qa∇‚ T a)1

- Ja‚∇

µ ja

l

∑

+

T

ψka∇‚

kg1

( )]

(60)

Under the assumption that the fluctuations of intensive variables from their volume averages are negligible in the first group of integrals, namely,

∆

()

〈〉 ( )

() 〈 〉

〈〉

µ ja µ ja µ ja 1 1 1 ) ) 0, ∆ ) )0 T T T T T T ∆

Yka Yka Yka ) )0 T T T

and that the fluctuations of Qa, Ja, and ψka from V-1 〈Qa〉V, etc., as defined earlier, in the second group of integrals in eq 60 are also negligible, we find that the compensated heat in eq 60 can be expressed in the form

1 dQ T dt

)

〈〉

1 dQE

r

∑

-

T dt

a)1

〈〉

µ j a deMa T

r

+

dt

l

∑∑

a)1 kg1

〈〉

Yka deΓka T

+

dt

1 diQ

r

)-

T dt

[ 〈〉 1

∑ 〈Qca〉V‚ ∇T a)1

〈 〉

1 - 〈Ja〉V‚ ∇µ ja + T l

〈〉

〈〉

pj p ) , T T

〈〉

µˆ a µ ja ) , T T

〈 〉] 1

diW dt

dt

)

dQE dt

r

-

deMa

∑µˆ a a)1

dt

r

l

dQn

deΓka dt

(65)

dt

r

)

∑[Γ1a:χu + Γ2aχV + Γca‚χlnT + a)1

) TΣl

∑ΥkaχX ] kg1 ka

(66)

where from the local theory viewpoint

j aJa + Γca ) 〈Qa - µ ) 〈Qa〉V -

(67)

〈〉 µˆ a T

∑XkaΥka〉V

kg1

1

〈T〉 〈Ja〉V +

∑ kg1

〈〉 Yka T

〈T〉 〈ψka〉V

l

) 〈Qa〉V - µˆ a〈Ja〉V +

〈 〉

∑Xka〈ψka〉V

kg1 l

) 〈Qa〉V - µˆ aΓ4a +

∑XkaΥka

(68)

kg1

χlnT ) -〈∇ ln T〉

(69)

µˆ φa ) µˆ a + φa (Fa ) -∇φa)

(70)

and

ka

∑∑Xka a)1kg1

dt

l

Xka Yka (63) ) T T

a

+

-

Γ4a‚χµˆ φa -

(62)

Thus the compensated heat is given by

dQ

dt

di Q

l

〈∇T1〉 ) - 〈T1∇ ln T〉 ) - T1〈∇ ln T〉 〈T1∇µj 〉 ) T1〈∇µj 〉 〈T1∇X 〉 ) T1〈∇X 〉 ka

+

A. Phenomenological Form for the Uncompensated Heat. The terms in the expression for dQn/dt are amenable to phenomenological interpretations since they represent energy or matter transfer between the system and the surroundings across the boundaries in the course of irreversible processes. Similarly, diW/dt represents energy transfer accompanying stresses applied at the boundaries or body forces. Analysis made so far on the basis of a continuum theory model indicates that we may phrase these quantities in terms of such effects on purely phenomenological grounds, but it would have been difficult to guess them without the kind of analysis made so far. On combining the expression 65 with the expression for diW/ dt in eq 55 with the help of eq 53 which holds true in the limit of vanishing fluctuations from the mean values of macroscopic variables, we find

In connection with these identifications the following approximations are used in the limit of vanishing fluctuations of T-1:

a

deΓka

All the quantities in this expression can now be calculated in terms of observables.

Note that 〈Qa〉V can be given in terms of Γ3a and 〈Ja〉V is equal to Γ4a. Replacing the mean values of the intensive variables with their phenomenological counterparts in accordance with the thermodynamic correspondence principle we identify

1 1 ) , T T

dt

l

∑∑Xka a)1kg1

Υka ) 〈ψka〉V

∑〈ψka〉V T∇Yka

kg1

r

+

T dt (61)

if the boundary condition u ) uB holds at B. The last term on the right in eq 61 is defined by

1 diQ

dt

∑µˆ a a)1

IV. Global Uncompensated Heat for a Finite System

Yka T

de M a

r

)-

+

d iQ dt

(64)

The second line in eq 68 is obtained under the assumption of vanishing fluctuations in the intensive variables and the third line follows on identification of the mean values of the intensive variables with their phenomenological counterparts as in eq 63. As in the case of mean values of other products of local macroscopic quantities which have appeared in the present work, replacing them with products of mean quantities entails an approximation in which fluctuations are neglected, and such an approximation limits the range of validity of the global theory


Eu

from the viewpoint of local continuum theory; it requires care when the global theory is applied. Comparing the first line of eq 66 with the proposition made for ∑l we identify ωka and $ka:

ω1a ) χu, ω2a ) χV, ω3a ) χlnT, ω4a ) χµˆ φa, $ka ) -χXka (71) These results provide a complete form for ∑l for global irreversible processes, which appear in eq 29 for the proposition presented earlier. B. Uncompensated Heat from a Local Model. Since diΓka/ dt in the expression 30 for the uncompensated heat represents a constitutive equation for Γka it may be suitably postulated so that experiment can be phenomenologically explained. Its precise global theory form, however, is not a priori known on the phenomenological grounds. To gain an idea about diΓka/dt and ultimately the uncompensated heat we consider it from the viewpoint of local field equations for macroscopic variables. For this we rely on the kinetic theory results. The local form of uncompensated heat is identified by the formula2,10-12

Ξc ) Ξl + T-1

r

V. Some Examples for a Finite System If eq 38 is taken for the derivative diΓka/dt in eq 30, the uncompensated heat is identified a little more closely with the molecular expression, since Λka can be endowed with a molecular expression. The explicit form for diΓka/dt depends on the system under consideration since it contains the constitutive information. Some examples will be given after the global thermodynamic forces have been calculated in the following. We now examine the global derivatives χ1h, χuj, etc. appearing in eq 48. They will be put into more transparent forms. A. Temperature Gradient. We calculate the global temperature gradient in detail:

χ1h ) -V-1

∑ ∑Yka (Zka + Λka) g 0

(72)

where r

m

r

dr ) dBdx

l

µ j aνalRl) + T-1∑ ∑ψka∇Yka ∑ l)1 a)1 kg1

(73)

If it is assumed that temperature is uniform over the area ΩB of the cross section of the boundary B we find

l

j aJa + Qca ) Qa - µ

∑Ykaφka kg1

〈Ξc〉V ) 〈Ξl〉V +

∑ ∑〈T

-1

Yka(Zka + Λka)〉V

a)1 kg1

) Σl + T-1

r

l

∑ ∑Xka〈(Zka + Λka)〉V

(75)

a)1 kg1

where in the second line 〈Ξl〉V corresponds to Σl given in eq 29 and the correspondence is also made that 〈T-1Yka〉 w T-1Xka in the limit of vanishing fluctuations in T-1Yka; namely, T-1Yka 〈T-1Yka〉 ) 0. By using the definitions of deΓka/dt and diΓka/dt given earlier and comparing eq 75 with eq 30 under the correspondence

〈Ξc〉 ) T-1

χ1h ) -

(74)

and Zka and Λka are, respectively, the kinematic and dissipation terms in the evolution equation for nonconserved variables Φka considered earlier. Integrating Ξc over the volume and neglecting the fluctuations of the nonconserved variables so that Φka may be replaced by Γka we find the volume integral of Ξc l

(77)

The configuration of the system depicted in Figure 1 is as follows: the horizontal direction is taken parallel to the x axis, the horizontal length of the system is L, and the origin of the coordinate system is at the lower left corner of the system. Then it is appropriate to express the volume element in the form

(Πa:∇u + Qca‚∇ ln T + Ja‚∇µ j φa + ∑ a)1

r

∫Vdr ∇ (T1)

l

a)1 kg1

Ξl ) -T-1

It must be noted that in the phenomenological theory the righthand side of eq 76 does not have to be given in terms of the volume average of (Zka + Λka) as long as diΓka/dt does not violate the inequality 30 and gives rise to results for transport processes which are consistent with experiment.

(78)

1 T ) (T0 + TL) 2 ∆T ) TL - T0

(79)

Therefore it is clear that χ1h is the global thermodynamic force corresponding to the temperature difference across the horizontal dimension L of the system. B. Chemical Potential Gradient. The volume averages of other vectorial thermodynamic gradients can be similarly calculated. Since the boundaries ϑ1 and ϑ2 play no role in transport of energy and matter, by using the same method as for χ1h we obtain

( ) [( ) ( ) ]

χµj φa ) -V-1 )

we find a continuum theory model for diΓka/dt:

) (76)

[( ) ]

)

where δx is the unit vector along the x axis, TL and T0 are, respectively, the temperatures at x ) L and 0, and

dN dt

diΓka ) 〈(Zka + Λka)〉V dt

(

ΩB 1 ∆T 2 1 ∆T δ +O δ ) V T L T 0 x T 2L x T

where

µˆ φa T

∫Vdr ∇ -

L

∆µ j φa δ L x

µˆ φa T

µˆ φa δ T x 0

L-1δx (80)


∆µ j φa )


( ) ( ) () () µˆ φa µˆ φa LT T

µˆ a ) T

where

0

µˆ a + T0-1∆φa + O(φa∆T/T0) L T 0

(81)

Therefore χµj φa is easily seen to describe mass and energy transport arising from the differences in T and µ j φa across the horizontal dimension L of the system. One can calculate ∆µ j φa a little more explicitly by using the extended Gibbs relation 28. However, we will not pursue it here, leaving it to the future occasion of application of the theory developed here. C. Tensorial Thermodynamic Gradients. The gradients ∇ (u/T) and ∇ (Xka/T) belong to this class of thermodynamic forces. To calculate these tensorial thermodynamic gradients it is necessary to have some knowledge of vector u or tensor Xka. Since the tensor Xka, and its local behavior in particular, is not well known at present, one can only perform formal calculation which would not yield a more insightful result as for the velocity gradient and such a formal calculation can be performed in the same manner as for the latter, we will consider only ∇(u/T) here, leaving the study of ∇(Xka/T) to the future work which will become feasible when more details of the flow behavior of Xka will be known. Since we are better informed of the behavior of flow velocity u we will closely examine the tensor ∇(u/T) in the case of plane Couette flow geometry appropriate for the configuration of the system depicted in Figure 1. In this case it is assumed that the boundaries ϑ1 and ϑ2 move in opposite directions parallel to the x axis at velocity (u/2, whereas the side walls ϑ3 and ϑ4 are stationary and boundaries B1 and B2, perpendicular to the x axis, move parallel to the x axis at velocities (u/2. Therefore as will be seen, there is no role played by boundaries B1 and B2 either. The gap between the boundaries ϑ1 and ϑ2 is denoted by D. The vertical side walls are assumed located at a sufficiently large distance apart. The area of the upper and lower boundaries (ϑ1 and ϑ2) is Ω. In this case of the system configuration we may first write χuj in the form -1

χuj ) -V

∫

∂ u ∂ u ∂ u + δy + δz dr δx V ∂x T ∂y T ∂z T

[

]

(82)

where δx, δy, and δz are unit vectors in the x, y, and z directions. If the flow is laminar in the x direction and the fluid is incompressible, there is no velocity components in the x and z directions and the x component of the velocity ux is a function of y. Therefore there are only two terms contributing in eq 43, which then may be written in the form

χuj ) -V-1

u

since there is a temperature gradient along the x axis. Thus we find

[( ) ( ) ]

ΩB ux V T

-

D

ux T

0

δx δy -

(

)

∫BdB ux(y)

1 1 -1 δxδx V TL T0 u ∆T ) -T-1 δxδy - 2 ujxδxδx D TL

∫BdB ux(y) ) D-1∫0Ddy ux(y)

(85)

This is simply the flow rate through unit area of the boundary B. If L is infinite under the assumption of infinite channel length in the present flow geometry, then ∆T/L ) 0 and we find

u χuj ) -T-1 δxδy D

(86)

The result obtained here indicates that χuj is related to the shear rate γ ≡ (u/D) whereas there is no normal component of χuj in the case of flow under consideration. The treatment of ∇Xka can be made in the same manner as for ∇ (u/T), but since there are many components in the tensor some more detailed knowledge of Xka is necessary for explicit and useful analysis. For this reason we will not pursue it here, but leave it to future studies of phenomena requiring the gradients of Xka. D. Examples for Global Constitutive Equations. We close this section with two examples for diΓka/dt in the cases of processes corresponding to the global thermodynamic forces χuj and χ1h calculated earlier. For simplicity we consider a single component fluid. Therefore the subscript a for species will be omitted. If the system is near equilibrium, Zka and Λka may be taken in linear forms2 and in the case of a single component system it is possible to obtain their volume integrals in the forms

diΓ1 p ) 2pTV χuj - Γ1 dt η0 pTC ˆp diΓ3 ) pTC ˆ pVχ1h Γ dt λ0 3

(87)

where η0 and λ0 are the viscosity and the thermal conductivity of the fluid. Note that for this we have used approximate constitutive relations for Xk, which are linear with respect to Γ1 and Γ3, respectively. The global constitutive equations in eq 87 are the global forms of the Maxwell-Cattaneo-Vernotte equations often used in extended irreversible thermodynamics.21,22 At the steady state where

diΓk )0 dt

u

∫ϑdϑ∫0Ddy δxδy∂y∂ Tx - ∫BdB∫0Ldxδxδx∂x∂ Tx (83)

χuj ) -

ujx ) Ω-1 B

(84)

the constitutive equations become

Γ1 ) -2η0TVχuj ) 2η0γVδxδy Γ3 ) λ0Vχ1h ) -λ0

|∆T|V δx T 2L

(88)

These equations are the global forms of Newton’s law of viscosity and Fourier’s law of heat conduction to which they reduce if the stress and the heat flux are uniform over the volume of the system. In summary for ∑l, with the mean thermodynamic forces calculated in this section the dissipation function is given by

8594 J. Phys. Chem. B, Vol. 103, No. 40, 1999 r

Σl ) -

|∆T|

u

]

∑T-1 Γ1a: D δδ + Γ2aL + Γca‚ TL δ a)1 r

-

[ () u

Eu

|∆µ j φa|

∑Γ4a‚ a)1

L

r

δ+

∑ ∑ΥkaχX a)1 kg1

ka

(89)

Here we have used V-1 (dV/dt) ) u/L in the case of a system depicted in Figure 1. On use of this result in the expression for the uncompensated heat in eq 30 we have a formula for the latter in the case of global irreversible processes we have set out to obtain. The energy dissipation ∑l has now acquired a clear physical meaning for global irreversible processes. VI. Discussion The form of uncompensated heat presented here, eq 30, generalizes the form which was obtained by Th. De Donder15,16 in connection with chemical reactions and which also forms the foundations of the theory of irreversible processes of chemical reactions formulated by Prigogine7 and his school,18,19 whereas the form for ∑l generalizes the formula for entropy production in the theory of linear irreversible processes to global processes occurring far removed from equilibrium. Here we have so generalized their interpretation of uncompensated heat as to include transport processes induced by shearing, heat flow, external forces, and so on. The last term in eq 66 for ∑l and the term involving diΓka/dt in eq 30 represent the aforementioned generalizations. The ∑l may be in fact regarded as a Rayleigh dissipation function for global irreversible processes. The global uncompensated heat elucidated here, especially, eq 30, is consistent with the local calortropy production in eq 72. In the literature2,17,20-22 the extended Gibbs relation appears usually in local forms, but they sometimes appear also in a global form holding for a system of a finite volume. The condition under which the local extended Gibbs relation can be directly transcribed into the global one and vice versa has not been made clear in the literature. In this work we have made an analysis of the condition. Consideration made in this work indicates that from the local field theory viewpoint there are some important restrictions; when a global extended Gibbs relation is used, that should be clearly understood, especially, with regard to the meanings of intensive variables and nonconserved variables. The same comment can be applied to the situation where the local extended Gibbs relation is interpreted from the global phenomenological theory viewpoint. The analysis made in the present work indicates that irreversible thermodynamics based on the global extended Gibbs relation for systems far removed from equilibrium is a rather subtle and delicate subject which requires careful treatments of various intensive variables, especially, with regards to their physical and operational meanings. We reiterate that eq 30 and related formulas are for a global system in the large, but not for a local irreversible process which contains much more information on the local behavior of the flow of interest. The global formulas enable us to study irreversible processes in a global scale without paying attention to the local behavior. In this approach to irreversible phenomena the demanding necessity of solving local field equationss hydrodynamic equationssis avoided at the cost of losing the knowledge of local flow behavior of the fluid, but such local information often is not required in understanding the global irreversible flow behavior of the system. The interpretation

presented here for the uncompensated heat provides us with a mathematical machinery to begin such studies in irreversible phenomena in a gross scale. Nevertheless, considerable care must be exercised in using the global thermodynamic formalism for systems of a finite volume, because finer details provided by the local continuum field theory are neglected, but they may be important for some phenomena. In summary for the system depicted in Figure 1, the fundamental thermodynamic equations for global irreversible processes are

dΨ dt

(

) T-1

dE dt

dV

+p

r

-

dt

∑ a)1

µˆ a

dMa dt

r

+

∑ ∑Xka a)1 kg1

)

dΓka dt

which must be accompanied by the formula for the uncompensated heat

dN dt

) T-1

dN dt

) Σl - T-1

m

r

AlRlV + T-1∑ ∑Xka ∑ l)1 a)1 kg1

diΓka dt

g0

with ∑l given by eq 89 and with dN/dt related to dΨ/dt through eq 8. Acknowledgment. This work was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada. References and Notes (1) Eu, B. C. Phys. ReV. E 1995, 51, 768. (2) Eu, B. C. Nonequilibrium Statistical Mechanics: Ensemble Method; Kluwer: Dordrecht, 1998. (3) Clausius, R. Philos. Mag. Ser. 4 1851, 2, 1, 102. Ann. Phys. (Leipzig) 1865, 125, 355. (4) Thomson, W. (Lord Kelvin) Mathematical and Physical Papers of William Thomson; Cambridge University Press: London, 1822; pp 174200. (Trans R. Soc. Edinburgh 1853, 20, 261. (5) Kirkwood, J. G.; Oppenheim, I. Chemical Thermodynamics; McGraw-Hill: New York, 1961. (6) For historical accounts of and discussions on the second law of thermodynamics, The Second Law of Thermodynamics; Dowden, Hutchinson & Ross: Stroudburg, PA, 1976. (7) Prigogine, I. Thermodynamics of IrreVersible Processes, 3rd ed.; Interscience, New York, 1967. (8) de Groot, S. R.; Mazur, P. Nonequilibrium Thermodynamics; NorthHolland, Amsterdam, 1962. (9) Meixner, J.; Reik, H. G. Thermodynamik der irreversiblen Prozesse. In Handbuch der Physik; Flugge, S., Ed.; Springer: Berlin, 1959; Vol. 3. (10) Eu, B. C. J. Chem. Phys. 1995, 103, 10652. (11) Eu, B. C. J. Chem. Phys. 1996, 104, 1105. (12) Eu, B. C. J. Chem. Phys. 1997, 107, 222. (13) It is useful to note that the second law of thermodynamics was originally stated for cyclic processes only. (14) This point has an indirect relevance to the assertion made by Mu¨ller that the entropy flux has a nonclassical contribution, see: Mu¨ller, I. Z. Phys. 1967, 198, 329. (15) De Doinder, Th. L’Affinite´ ; Gauthier-Villars: Paris, 1928. (16) Defay, R. Bull. Acad. R. Belg. (Cl. Sci.) 1938, 24, 347. (17) Eu, B. C. Kinetic Theory and IrreVersible Thermodynamics; Wiley: New York, 1992. (18) Glansdorff, P.; Prigogine, I. Thermodynamic Theory of Structure, Stability, and Fluctuations; Wiley: New York, 1967. (19) Nicolis, G.; Prigogine, I. Self-Organization in Nonequilibrium Systems; Wiley: New York, 1977. (20) Mu¨ller, I.; Ruggeri, T. Extended Thermodynamics; Springer: Heidelberg, 1993. (21) Jou, D.; Casas-Vazquez, J.; Lebon, G. Extended IrreVersible Thermodynamics; Springer: Heidelberg, 1993. (22) Garcia-Colin, L. S. ReV. Mex. Fis. 1988, 34, 344. Garcia-Colin, L. S.; Uribe, F. J. Non-Equilibr. Thermodyn. 1991, 16, 89. Garcia-Colin, L. S. Mol. Phys. 1995, 86, 69.

Generalized Thermodynamics of Global Irreversible Processes in a

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