IRREVERSIBLE THERMODYNAMlCS IN ENGINEERING J A M E S WE1
Is irreversible thermodynamics of practical value to the chemical engineer, or is it an interesting game f o r the ccpure’J researcher? Attitudes and answers to this question diverge shnrpb. In this review, a well known engineer expresses his views after careful consideration of the suyect. H i s views will help f o r m a clearer appreciation of thepresent andpotential vale of irreversible thermodynamics f o r engineers and researchers alike have been following irreversible thermodynamics
I (IT) for more than 10 years and have even published
a few papers on the subject. During that time, it has been said that the motto of irreversible thermodynamicists is “entropy is our most important product.” With this in mind, the question has naturally arisen, “Has this field produced anything other than journal papers?” For me, as a chemical engineer employed by industry, it is reasonable to question my return on the time I have invested. This paper offers a summary of an examination of the content and utility of IT. Perhaps it will stimulate others to undertake similar critical looks a t all of our tools. After a brief look at I T as a physical theory, we shall be concerned with the postulates and theorems on which IT is based. From there, we shall move to a n examination of the physical meaning of IT, and then to its applications-how it has been used and misused. Irreversible thermodynamics has also been called nonequilibrium thermodynamics, and the thermodynamics of the steady state. There is a movement to rename classical equilibrium thermodynamics as “thermo-statics” and to designate I T as “thermo-dynamics.” This movement is, I believe, created through an etymological mistake. The word “thermodynamics” was coined by the founding fathers by using the Greek words meaning “hot” and “power” (74a). They chose it because Sadi Carnot and other engineers were trying to convert heat into power. I n this context, “dynamics” does not mean time and motion and is not the antonym of “statics.” For the purposes of this discussion, we shall use IT to mean the description of kinetic systems using thermodynamic variables that are macroscopic and observable, such as temperature, pressure, concentration, or chemical potentials. We do not mean descriptions of states by keeping track of loz3molecules, each with six or more degrees of freedom. We do not mean the Roltzmann
equation, Chapman-Enskog, the kinetic theory of Maxwell, etc. Many people have sought to justify IT by using these molecular descriptions, and many have shown that IT is consistent with the laws of quantum fluids. But the laws and applications of IT itself are strictly observable and thermodynamic ; they contain no molecular description whatsoever. Irreversible Thermodynamics a s Physical Theory
We may grade our understanding of nature according to its depth of penetration, from the most superficial (barely adequate for some engineering processes) to the most fundamental (derived from quantum mechanical principles). A first-level theory is phenomenological in nature and seeks merely to describe and correlate raw data. Call it curve fitting, if you like. Fugacity charts in thermodynamics are a n example, as are Henry’s law and the half-order law in kinetics. At the second level, we look at molecular structure and interaction mechanisms and seek to derive properties of material from a knowledge of intermolecular potentials, statistical mechanics, kinetic theory, etc. An example would be the derivation of the second and third virial coefficients from the Lennard-Jones potential, or a study of molecular mechanism in chemical reactions. I n statistical thermodynamics, second-level understanding can be obtained for dilute gases and simple molecules. I n kinetics, it has been accomplished for very complicated systems, such as the free radical mechanism of Hinshelwood and Semenov, the Ssz substitution of Hughes and Ingold, and the Krebs cycle in glucose burning. At this level, we say that we really understand the system. A second-level understanding is not always available; but if it is, it always supersedes a firstlevel understanding. At the third level, we seek to derive everything a priori-including the Lennard- Jones potential and the strength of the hydrogen bond-from quantum mechanical principles. This level is in pure science and is not often attained. IT seeks to describe a system by macroscopic observables and is supposed to be true for any molecular structure and mechanism; it is certainly a first-level theory. No a priori estimates of the valid region of its laws can be given. I t is really what engineers use for a firstorder description, with no claims to having deep physical AUTHOR James Wei is on the staff of the Research Department, Central Research Division Laboratory, Mobil Oil Corp. He is recognized for many contributions to theJeld of kinetics and thermodynamics. This paper will be part of the Fourth Annual IMEC Summer Symposium, June 1967. VOL. 5 8
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understanding. I t is tentative and acceptable until someone does a thorough study on level two. If you do not really need a good understanding, or do not have time and money to do a thorough study, irreversible thermodynamics provides a notation and format for your system. That is where one would expect the maximum usefulness of irreversible thermodynamics. The linear equation of forces and fluxes is really a constitutive equation, defining a class of idealized material in a narrow range of conditions. I t is therefore especially unfortunate that the notation and format are quite complicated and are difficult for practical engineers to use.
Forces and Fluxes. The rate of change of a degree of freedom is called a n “Onsager flux”
J , = da,/dt
(2)
For. each flux, J t , we define a “conjugate force”
x,= b S / d a ,
(3)
There is another school that defines Xias T .bS/bai, so that the sum of X , J , is T.dS/dtor 2 G, the Rayleigh dissipation function. For a continuous system, the rate of local entropy production at a point in space is given by dS/dt = C J i X i
(4)
i
Postulates of Irreversible Thermodynamics
First we shall discuss the assumptions or postulates of IT, above and beyond the laws of classical thermodynamics. Entropy Production. The second law of thermodynamics tells us that any spontaneous change in a closed system is accompanied by a rise in entropy. This statement implies that spatial variations of temperature, pressure, and chemical potentials tend to equalize with the passage of time. Let us postulate that the rate of increase of entropy be the potential for any spontaneous irreversible change ( 3 6 ) . We shall discuss the formulation in two idealized systems called the “discontinuous system” and the ‘(continuous system.” The discontinuous system consists of a finite number of regions, each with uniform values for the state intensive variables. These state variables have different values in different regions, and are discontinuous across boundaries. For example, we may consider two homogeneous fluid phases that are separated by a porous membrane. The state of the entire system is described by a number of Onsager coordinates, al, a2. . .a, (such as the temperature and pressure in each region), When equilibrium is established, these Onsager coordinates are taken to be zero. We then proceed to investigate the entropy of the entire system as a function of these deviation parameters :
.
S(a1, a 2 , . .a,>
(11
For the continuous system, where the state intensive variables are taken to be continuous functions of position in space, a complete description of the system requires the complete temperature profile, instead of merely the temperature in a finite number of regions. In an effort to reduce the complexity of the problem, it is customary to abandon the global formulation in favor of a local formulation. We shall describe each point in space by a set of Onsager coordinates, and we shall pretend that even for systems away from equilibrium we can investigate the entropy of each point as a function of these Onsager coordinates. Here we follow the development of Prigogine. The older developments of Onsager (36) and Casimir (4) are based on global properties. For instance: I n a problem concerning heat flow in a n isolated crystal, the Onsager coordinate ai is the x-moment of the distribution of energy. 56
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
and the overall rate of entropy production for an entire system is
b / d t . dV
(5)
For a discontinuous system, the overall rate of entropy production is given by Equation 4. I t is difficult and tedious to find the conjugate forces from Equation 3, so that in many important systems no one knows what are the right conjugate forces (17). Shortcuts are available by using the Gibbs equation TdS = dE PdV &&,--if we believe that it is applicable for each point in space while irreversible processes are taking place. A list of generally accepted forces and fluxes is given in Table I. The reader will probably notice that the usual concept of heat flux involves the quantity of heat flowing from one region to another, measured in calories/(time) (area). But an Onsager flux J must be the time derivative of an Onsager coordinate, such as bT(r, t)/bt, in degrees/time. The problem of reconciling the difference was attempted by Casimir ( 4 ) and was continued by Prigogine (38) and deGroot and Mazur (70), They took the divergence of the heat flux vector and made a Fourier transform of the temperature field. Unfor-
+
TABLE 1.
LIST OF GENERALLY ACCEPTED FORCES AND FLUXES Forces (X)
I
Continuous Fluxes ( J )
Heat flow Diffusion Electrical conduction Volume Viscous tum flux
1
~
I
Grad ( 1 / T ) or - l / T 2 grad T
- Grad
(d T)
Discontinuous System
1/T2AT A (dT )
Jb
I
- I / T grad
JV
-Grad ( p / T )
cp
- l / T grad u
1 / TAP
A($/ T )
tunately, the heat flux vector contains much more information than its divergence-for instance, whether the rate of temperature rise a t a point is mainly due to heat conduction in the x or they direction. Given the heat flux vector, one can always construct the divergence uniquely ; but given the divergence, one cannot uniquely reconstruct the heat flux vector. Therefore, their approach is not completely successful, and it is not clear that we are justified in using heat flux as an Onsager flux. There was a time when people thought there was another shortcut: Any set of forces and fluxes that satisfies Equation 4 without explicitly satisfying Equation 3 still qualifies as a set of conjugate forces and fluxes. I n 1960, this idea was exploded by Coleman and Truesdell (7), whose discussion is also contained in Fitts (77) and in deGroot and Mazur (77). Not all the recent books have recognized this fact, for a new one by Katchalsky and Curran (24) has repeated the erroneous shortcut. Linear Laws. Each flux is assumed to be a linear function of all the forces k
Each of the coupling coefficients, Lik,may be a tensor and possess different components in the x - , y-, and z-directions. This law is a generalization of the well known laws of Fourier, Fick, Poiseuille, Ohm, D’Arcy, as well as of the old engineering film concepts (46). There is certainly a region about the equilibrium point where the linear laws are sufficiently good approximations. However, the size of this linear region must be determined experimentally, and could be very small. The formalism of Equation 6 brings our attention to the phenomenon of cross-coupling-i.e., a force X I may directly give rise to another flux, J z . I t seems that if we have only a single flux, or if we have several fluxes that are not coupled together, we are quite justified in ignoring the whole subject of irreversible thermodynamics-the engineers are quite capable of handling themselves in these circumstances. I t is only when the forces and fluxes are coupled-Le., when terms such as L I Z 0-that irreversible thermodynamics may have something useful to say.
+
Theorems of Irreversible Thermodynamics
The principal glory of thermodynamics arises because we have erected a stupendous superstructure of theorems (relations that correlate a lot of our knowledge, such as chemical equilibrium and the pressure increase of vapor pressure) from a few postulates (the three laws). Our awe of thermodynamics chiefly rests on the realization that we put in very little and came out with a lot. Unfortunately, IT provides few useful theorems. Here is the principal theorem in irreversible thermodynamics : T h e Onsager Reciprocity Relations (ORR): L,, = Lkt. I t is a great piece of scientific information derived with the help of the statistical mechanical principle called
“microscopic reversibility” (36, 48). ORR tells us there is a symmetry of cause and effect in nature. Properly speaking, ORR consists of two separate statements: of ‘(reciprocity of spatial coordinates” and of “reciprocity between processes.” The former refers to heat conduction in a n anisotropic crystal where the heat conductivity tensor has the spatial symmetry, L,, = LUz. The latter refers to a system where mass and heat are coupled together, so that the two coupling tensors, La, and L,,, are equal; it also refers to a multicomponent diffusion system where the diffusional velocities of species i and j are coupled, and Li, = Lji. The former refers to the equality of components within the same tensor; the latter refers to the equality of two separate tensors. There is a disagreement concerning the significance of the “reciprocity of spatial coordinates.” Onsager cited the works of Soret and Voigt to show that an experiment could be devised to prove or to disprove the symmetry of the conductivity tensor. But Casimir, as well as deGroot and Mazur, considered that an antisymmetric component of tensor L has no observable physical consequences whatever, and that it is permissible to put it equal to zero. A consequence of their argument is that ORR has no content as a reciprocity of spatial coordinates. I t cannot be overemphasized that ORR is the heart and soul of IT. But is ORR true? There are many plausible theoretical derivations, and there are many experiments to show that the agreement is not really so bad (34). Some authors think it should be considered as an independent postulate in IT. Since most of science rests on foundations that are no firmer than ORR, we can accept its validity unless proved otherwise. T h r e e Other Theorems Often Considered Basic to IT. CURIETHEOREM. Truesdell once remarked that even a statement of the Curie theorem cannot be found, to say nothing of a proof (45). I t is indeed difficult to find the essence of the Curie theorem from the various verbal statements in the major texts (9, 17, 23, 38). Before we use this theorem, let us at least attempt to define it. Let us follow Hirschfelder, Curtiss, and Bird (22): “ I n a n isotropic system, forces and fluxes that correspond to a coupling of tensors whose orders differ by an odd number do not occur.’’ The first important word is “isotropic,” for it causes a problem. I t is often not easy to tell whether a given system is isotropic or not-i.e., the system must have the same property in all directions. If you rotate the cause, or driving force, X , then the effect, J , will rotate in the same manner. Thus, the Curie theorem is not concerned with heat conduction in anisotropic crystals, diffusion across cell membranes, or the flow of oil through layered rocks. I t is a negative theorem since it denies some couplings in isotropic systems. Since the restrictions of Curie’s theorem apply only in isotropic systems, IT has a greater impact on anisotropic systems where there are more possibilities for cross-coupling. The second important word is “coupling.” I n Equation 6, force Xz is not coupled to flow J1 if the VOL. 5 8
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parameter L12 = 0. Coupling is thus a direct, linear, causal relation. The Curie theorem cannot rule out indirect effects or nonlinear effects. If we keep in mind the valid, but restricted, meaning of these two words, we may state the consequences of the Curie theorem: Vectors (mass and heat flux) do not couple with scalars (chemical reaction) ; neither do vectors couple with second-order tensors (viscous forces). But do second-order tensors couple with scalars? I t is not ruled out by Curie’s theorem, and there are no experimental data to show whether such coupling exists. In IT, the word coupling is used in an exceedingly narrow sense ; a lack of direct coupling does not imply a lack of “interference” or interaction. Everybody must have seen a t least a hundred papers entitled “Heat and Mass Transfer with Chemical Reaction”-but this phenomenon is called interference, rather than coupling, in IT. I n IT, heat and mass transfer cannot be ‘(coupled” with chemical reaction, unless we believe that the rate of chemical reaction, at a point in space, is a function of temperature and concentration gradient as well as of local temperature and concentrations. O n the other hand, chemical reactions alter concentration gradients and give rise to mass transfer-this is called interference-whereas the concept of coupling refers to a chemical reaction that takes place without (miraculously) causing a change in concentration gradient, and yet gives rise to mass transfer (forbidden by Curie’s theorem). Two simultaneous chemical reactions, such as
A*B
Application of Irreversible Thermodynamics
A*C
Where is irreversible thermodynamics applicable and fruitful? Presumably, it never hurts to have another way of looking at the same object, to have another viewpoint toward a system. However, an engineer does not have unlimited time to study all possible viewpoints. I n the interest of economy, we shall therefore adopt the more restricted view of applicability: IT is applicable to a system if it improves our ability to describe, to measure, and to predict outcomes, and if there is no better way to handle the system. I n a field that is already well understood, we do not count it as an application when someone shows that IT is consistent with what we already know. Ask not what we can do for I T ; ask what IT can do for us. Let us examine some of the teachings of IT and see where they are applicable: ( u ) IT provides a linear model of forces and fluxesa property that is sometimes misused. It could be said that many processes have been reformulated, or shoehorned, into the framework of IT. Some were ancient and honorable long before IT was dreamed of: Fourier’s law of heat conduction, Fick’s law of diffusion, Ohm’s law of electric conduction, Poiseuille’s law of capillary flow, D’Arcy’s law of flow in porous solids, and the Whitman two-film theory (47). One may even force-fit those chemical kinetics that obey the mass action lawsomething done in great detail in many texts and papers. But in chemical kinetics, the linear region is vanishingly
certainly interfere since reaction one will alter the concentration of species A and affect reaction two. But, these two reactions do not “couple,” unless we believe that in a n equilibrium system, the addition of some B will directly cause reaction two to proceed even if A and C are in equilibrium with each other. Such intrinsic chemical couplings have never been found (32, 37). There exist fluids whose viscosities are markedly changed by the presence of an electric field, but electric field is not “coupled” to viscous flow unless fluids are set into motion by an electric field even in the absence of a pressure gradient. So, with a sigh of relief, we can abandon what we never believed in (coupling of mass transfer with chemical reaction) and return to a favorite old subject (interference of mass transfer and chemical reaction). This is the doctrine MINIMUM ENTROPY PRODUCTION. that a steady state under constraint is characterized by minimum entropy production. I n a more poetic version, we read that IT “ . . .sheds new light on ‘the wisdom of living organisms.’ Life is a constant struggle against the tendency to produce entropy by irreversible processes. But since there is no possibility of escaping the entropic doom imposed on all natural phenomena under the Second Law of Thermodynamics, living organisms choose the least evil-they produce entropy 58
at a minimal rate by maintaining a steady state” (29). This statement echoes the long tradition of the concept of “least action” in physics and theology, and is particularly pleasing esthetically. The doctrine is true in linear systems. The founders of irreversible thermodynamics hoped that it would be valid even in nonlinear regions, but it was found to be not true (26, 27). Therefore, we have no assurance that we can use this doctrine in biological systems, or in any other systems far from equilibrium. By use of local potentials, generalized minimum entropy production, and thermokinetic potentials, Glansdorff and Prigogine ( X I ) , Callen ( 3 ) , and Li (30) made attempts to carry the theory into nonlinear regions. As yet, we do not have any useful results and must wait for those restrictions and revisions we can surely expect. I n one instance, Finlayson and Scriven (76) showed that local potential is just one form of the Galerkin method. THERMODYNAMIC COUPLIXG.This is the concept that one process may depart from equilibrium and consume entropy, provided it is coupled with a second process that is producing entropy even faster. It is, of course, the basic principle of any pump, whether it moves water uphill or moves heat toward a higher temperature region. Thermodynamic coupling simply stems from our knowledge of the Second Law; a spontaneous change is possible if the entropy of the entire closed system is increased, and the entropy change of a nonisolated subsystem can be of either sign. I t is hardly a new principle.
INDUSTRIAL A N D ENGINEERING CHEMISTRY
small, and the results are so unfruitful that practically no one is proud of them. Engineers are, of course, quite familiar with linear models and used them for years before the emergence of IT. They should use the linear model provided by IT only if there is something to be gained. They will continue to use the gradient of temperature instead of the gradient of (1/T) as a driving force unless there is a strong motivation to change. There may be fields that are exceedingly underdeveloped, where even linear models are unheard of; then the framework of IT may seem as manna from heaven. ( b ) IT says that if we pick the fluxes and forces in a conjugate manner, then LiIC= LICf. This is the alpha and omega of irreversible thermodynamics; it is the only reason anyone would like to put his problem in the framework of IT. In all systems where a level-two understanding is not available, O R R is always experimentally found to be true regardless of our lack of understanding of molecular structure and mechanism. O R R means a saving in experimental labor; perhaps as good engineers, we should use it as a redundancy check on the accuracy and consistency of our data. In application, it is sometimes difficult or inconvenient to put a problem in IT format. First, a knowledge of the conjugate force X presupposes a knowledge of S(a1, u2, . . , u s ) . The shortcuts have been destroyed by Coleman and Truesdell. If one cannot find the conjugate forces, one does not have O R R and there is no point to IT. Second, engineers untutored in IT will probably use the following, somewhat more natural, linear law : da,/dt = - ~ M , , U ’
(7)
j
The coefficients of the matrix M are related to the matrix L by
where G CI’
-
3% baICba5
Since L is symmetric, it can be shown that the matrix M is diagonalizable, and all of its eigenvalues are real and positive or zero. This is a useful result based on IT (43, 44) since it implies that irreversible systems in general do not oscillate indefinitely about the equilibrium point in a closed system (eigenvalues are not complex). Toor also considered oscillation in the partial differential equations of multicomponent diffusion with chemical reaction. By using the formalisms of IT, we will show that oscillations do not take place even in a nonideal system: When the system is very close to equilibrium, one may reformulate the field equations by using chemical potential as driving force for both diffusion and chemical reaction
The chemical potential can be approximated by
cm
where is the global equilibrium concentration of species m and is independent of spatial position, and
We have then
b
(Ct - CJ = C [-K\‘. E., Prober, R.,I N D . ESG. CHEM.FUNDAMENTALS 3, 224 (1964). (44) T o o r , H. L., Am. Inst. Chem. Eng. J. 10, 448 (1964); Chem, Eng. Sci. 20, 941 (1965). (45) Truesdell C., “ T h e Principles of Continuum Mechanics,” p. 328, “Mobil O i l Corporation’Lectures in Pure and Applied Science,” 1960. (46) Walker, W.H., Lewis, W K McAdams W.H Gilliland E. R., “Principles lib; McGraw:Hill, Gkw York, i937. of Chemical Engineering,” (47) Whitman, W. G . , Chem. M e t . Eng. 29 (1923). (48) Wigner, E. P., J . Chem. Phys. 11, 1912 (1954).
