1
I
Enai- neerin1Realigning Practice with Fundamentals P. LE GOFF he development of chemical engineering during the Tpast 40 is characterized by a constant effort
years a t ahtraction and by an elucidation and organization of a number of unitary concepts associated with various parts of the chemical industry. Unit operations constituted the first classification of chemical engineering. These operations were organized into several groups such as transfer of m a s between phases, mechanical operations on solids, etc., obeying a few general principles of conservation, equilibrium, and optimization. We have sought to extend the analysis of chemical engineering by further examination of how the basic laws of physics, chemistry, and economics are utilized.
The "ConslrailHs"
of a System
In general, the operational regime of a chemical reactor is governed by a combination of a very large number of characteristic factors which depend on the chemical transformation itself or on the processes for heat, mass, and momentum transport, etc. The constraints of the system are expressed by the mathematical relations between the different factors. When one mathematically describes the function of such a reactor, he finds it difficult to be certain that he has taken into account all possible constraints and, conversely, it is difficult to be sure that the numerous mathematical relations that are written are really independent of one another. Enumeration of these relations indicates that they fall into six groups: PRINCIPLES OF CONSERVATION of extensive quantities (mass, energy, electric charge, momentum, etc.). LAWSOF STATIC EQUILIBRIUM expressing mechanical, physical, or chemical equilibrium, which are applications of different forms of the first law of thermodynamics. For example, in order to express the mechanical equilibrium of a system, we can write the resultant moment of forces applied to the system as zero. The physico18
INDUSTRIAL A N D ENGINEERING CHEMISTRY
physicol prmssing me met by constructing plonis. Before o plant can be built the engineer must combine mony diverse and fiepenfly >competingfoctors into a ’ hmmonious whole. In proceedingfrom conception .tofin&he8 piont, the and
~
L
consisknt and highly ookptoblc mgonizotian O f knowledge is Dtlc of the most ooluable tools
chemical equilibrium of a system is expressed by the equality of the chemical potential of the constituents in each of the phases in contact. KINETIC LAWS express the direction and velocity of the changes in the system. In fact the direction of a change for any isolated system is fixed by a unique law, Le., the second law of thermodynamics which states that the entropy of this system can only increase. The expression for the rate of entropy production constitutes a unique and very general way of measuring the velocity of change of a system but this method is too abstract for practical use a t the present time. Furthermore, this general kinetic law will only inform us about the change in the total energy content of the system, even when there is interest in the evolution and transfer of the different forms of energy between constituents of the system. This justifies the practical importance of the numerous laws which mark the rates of the different elementary processes of chemical reaction, rate of mass transport (Fick’s law), of electricity (Ohm’s law), heat transport (Fourier’s law), momentum transport (Newton’s law), etc. The kinetic laws will shortly be considered in greater detail. CAPACITIVE RELATIONS. These are relations which express the amount per unit volume of each of the extensive quantities (mass, heat, electricity, momentum) and take, in each particular case, a familiar form. Such a relation would be, e.g., the equation of state for fluids. THEPROFIT FORMULAS. T o establish the economic balance it is necessary to know all the factors which affect the profit of the operation and which are expressed in comparable form. The profit formulas are all relations permitting conversion to dollars of
After having treated the material properties, consider now a continuous medium at the human level in the search for a physico-chemical interpretation of the properties of a discontinuous material structure on the molecular level. I t is convenient to consider material discontinuities on four levels : -the nuclear level is that in which the particles have dimensions much less than one Angstrom. -the molecular level is that in which the particle dimensions are between one and one thousand Angstroms. -the granular level is a level of aggregates and is now of great importance in industrial activities. It is the leuel of grains of dust, droplets of mists and emulsions, bubbles, etc., and also of eddies and This leuel covers, roughly, vortices in the turbulent Jow of &ids. the dimensional range between 0.7 micron and 7 centimeter. -the human level is that level of objects with which we are most familiar and includes chemical reactors.
-the quantities of mass consumed and produced (both the materials of reaction and of construction) -the quantities of energy produced and consumed -the labor furnished -the quantities of information consumed and produced (the purchase and sale of patents and augmentation of “intellectual potential”) -the amortization of the capital investment
I t is noted that an entity in a certain level generally contains a large number of smaller entities from a lower level and is a part of a super entity of a higher level. Interest in this classification of four levels is largely due to the similarities and differences between the many physico-chemical laws which are operative depending upon the levels a t which they are applied. One can, e g . , see that at the granular level as well as at the human level matter is considered as a continuous medium but the essential difference between the levels resides in the fact that surface phenomena (Le., phenomena involving the properties of the surface separating phases) at the human level are negligible compared with the volumetric phenomena, while, on the contrary, they play a preponderant role at the granular level.
THE PRINCIPLE
Molecular Structure and Granular Texture
There is really only one equation of optimization. It is that which consists of whatever is necessary to yield maximum profit for the operator by giving the most general sense to the word profit. But, just as the second law of thermodynamics is replaced by certain kinetic laws. so the principle of optimization is often replaced by certain laws which are easier to apply in the search f c r optimal conditions and which assume either maximum capacity of production or the basic hourIy profit. OF
OPTIhIIZATION.
Thermodynamics and Material Discontinuity
The physico-chemical study of continuous media formerly was divided into several domains, which are now regrouped in a single science, viz., the thermodynamics of irreuersible phenomena which is composed of several major subjects: 20
-Classical thermodynamics is the study of systems in equilibrium (or nearly so); it would be more appropriate to call it thermostatics. It consists principally of the knowledge of the chemical potential of substances in their pure state and in either nonreactive mixtures (physical thermodynamics) or in reactive mixtures (chemical thermodynamics). -Kinetics is the study of systems undergoing changes and is further divided into physical kinetics (transport of mass, heat, and momentum and transfer of these properties between phases) and chemical kinetics (transformation of chemical species).
INDUSTRIAL A N D ENGINEERING CHEMISTRY
For an interpretation of the properties of apparently continuous media with discontinuous structures it is necessary to employ statistical mechanics, Le., the science of movements of systems composed of a great number of objects. Owing to the very large value of Avogadro’s number, efforts have been primarily devoted to systems composed of a very large number of particles in the interpretation of their macroscopic properties (pressure, temperature, entropy, etc.) . Under such conditions, fluctuations of the macroscopic properties are negligible. Such would not be the case if Avogadro’s number were only lo3 or lo4. Little work has been devoted to the study of systems in which the number of constituents is large but not very large. Gibbs, Defay, and others have studied the influence of drop or bubble size on the vapor pressure of the fluids from which they are composed. The gap between small and wry large numbers of particles has
been partially filled, however, by the recent original work of Hill (4). This work will certainly find numerous applications in chemical engineering. Moreover, it is of interest to note that it can be used at two different levels : -on the one hand to study chemical potential and, more generally, all of the thermodynamic properties of matter contained i n small grains composed of a moderately large number of atcms (colloids, aerosols, alloys, microporous catalysts with a high spec@ surface) -on the other hand to study the macroscopic properties of objects composed of an assembly of a moderately large number of grains (calculation of the entropy of a disordered, random packing of spheres, study of apparent viscosity and surface tension of juidized beds, etc.)
In conclusion, the complete, physico-chemical study of the operation of a chemical reactor, therefore, implies a knowledge of the macroscopic thermodynamic and kinetic properties at the human level as well as the interpretation of these properties through the molecular structure and granular texture of the constituents. A resume of the program outlined above is given in Figure 1. Elementary Actions
Three Material “Actions.” It has been previously stated that the kinetic laws are all those which describe the velocities of the elementary processes. All phenomena of change in matter can be represented as the combination of three elementary processes that will be called “actions” : -the transport of matter from one point to another without a change in phase -the transfer of matter from one phase to another -the transformation of matter without displacement
Each of these three processes is applicable at different levels. The transformations, e.g., may be carried out at the molecular level (chemical reactions), the granular level (grinding, fritting, emulsification, coalescence), or at the nuclear level (nuclear reactions). I t is important to note that the modes of reasoning are often the same in these different cases. The theory of chain processes of fission propagated by neutrons, e.g., is analogous to chain polymerization propagated by free radicals. Another example; one finds that the kinetics of grinding is similar to the kinetics of first and second order reactions. Similarly, the processes of mass transport within a phase may be effected in either of two ways, whatever the level: by bulk flow or by diffusion. I t is noted that the general theory of diffusion is applied almost indistinguishably to the diffusion of neutrons in a nuclear reactor, to the diffusion of molecules in a chemical reactor, to the diffusion of globules in an emulsion or even to the diffusion of vortices in an agitated fluid. In conclusion phase changes of the usual kind, i.e., vaporization, sublimation, fusion, etc., are noted. These changes consist of a transfer of molecular entities, from one super-entity (the domain of the phase at the human or granular level) to another. This concept can be applied to the nuclear level where radioactivity is represented as a vaporization from a nucleus considered as a liquid drop. In the same way the sifting of fine grains from a mechanically agitated bed of grains and
the elutriation and pneumatic entrainment of grains from a fluidized bed are analogous to vaporization. T h e Three Energy “Actions.” I n addition to the three, elementary material actions, we can define the same for energy, Le., three elementary actions of transport, transfer, and transformation. -the transport of internal energy occurs at the same time that transport of the material entity itself occurs -the transfer of energy from one material system to another system can be accomplished i n two ways: either the two systems came into contact and the energy is transferred by collision (molecular collision or collision of aggregates on the human level) or the transfer of energy is done by electromagnetic radiation without the systems coming into contact -the transformation of one diverse form of energy into another is often associated ruith the simultaneous material transformations. Such are, e.g., the transformations of chemical energy into thermal energy or into electrical energy
Informational “Actions.” It is easy to consider that these six elementary processes of transport, transfer, and transformation of mass and energy constitute the fundamentals of all physico-chemical kinetics as a consequence of the scientific study of industrial operations. It is necessary to consider, in addition, a third quantity (other than mass and energy), Le., information. For information, one defines yet three more elementary processes of transport, transfer, and transformation which are necessary to describe the operation of an industrial installation connected by automatic control loops. The Notions of Extensity and Potential
The thermodynamics of irreversible phenomena permits the quantitative expression, in unique form, of all types of quantities, apparently without connection. This discipline expresses, notably, that every irreversible phenomenon which manifests itself in the form of the evolution in a certain quantity of position (or extensity) under the influence of a potential difference is accompanied by a diminution of available energy. Rigorous development of this discipline may be found in several works (2, 7,S), but sufficient for present purposes will be Table I which summarizes the principles of couples composed of potentials and extensities. I n fact, matter is always displaced in a transport process (molecule, grain, object) but this matter constitutes the carrier of an extensity which is displaced in a corresponding field. I t is, e.g., mass in a gravitational field, the quantity of heat in a temperature field, or a chemical specie diffusing in a concentration field. This presentation of irreversible phenomena calls upon numerous conventions for which one may refer to authors already cited. I t is important to draw attention to the danger of a too rapid generalization because the analogies between the different forms of energy are limited. I t is noted, in particular, that because of convenient calculation, certain quantities have been coupled in pairs in order to preserve the roles of extensity and potential, although the product of the two quantities is not always an energy variation. Such is the case with temperature and the quantity of heat in heat conduction and the case with molar concentration and the number of moles VOL. 5 6
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NDUSTRIAL A N D ENGINEERING CHEMISTRY
TABLE I .
POTENTIALS AND EXTENSITIES
Product
~
Elementary Processes A.
B.
(Potentzal)(Extenszty) Nature or Sign$cance of the energy
Z’otentznl
TRANSPORT PROCESSES Transport of a single material en tity in a force field Example : an object in the (A1titude)g earth’s gravitational field Transport of a single material entity in a collection of simi, lar entities Example : molecular diffusion First Approximation Chemical potential Second Approximation Molar activity
Mass
Number of moles Number of moles
p
hgdm
Gra\,itational energy
. daV
Chemical diffusion energy ?(diffusion in a nonideal medium)
?
Third Approximation
Molar concentration
Number of moles
Other Examples: Neutron diffusion Globule migration Transport of a single extensive property carried by matter Translational momentum Rotational momentum Heat Exact Expression Approximate Expression Electricity Magnetism
Translational velocity Rotational velocity
Translational momentum Rotational momentum
- ud[ mu) urd( mwr)
Kinetic energy of translation Kinetic energy of rotation
Absolute temperature Temperature e.m.f. m.m.f.
Entropy Quantity of heat Charge Pole Strength
TdS
Quantity of heat
TRANSFORMATION PROCES5 O F MATERIAL ENTITIES Change of the size and shape of a n object Change in fluid volume Change in the amount of surface limiting a fluid Change in solid shape Elongation or compression Torsion Change of the nature of the constituents of the material entity Example : molecular transformation (reaction) First form Second Form Other examples : Nuclear transformation Globule transformation _
Pressure
-Volume change
?
?(diffusion in a n ideal medium)
?
3
EdQ 3? .dQ
Electrical energy Magnetic energy
- PdV
Flow or expansion energy of fluids
Surface tension
Interfacial area
rdA
Mechanical surface energy
Normal force
Length
FdL
Mechanical cnergy of elongation
Couple
Angle
Cda
Mechanical energy of torsion
Chemical potential Chemical affinity
Number of moles Degree of advancement of reaction
pdn‘
Chemical reaction energy Chemical reaction energy
_
Ad:
~
in molecular diffusion (in an ideal system). The analogy between the two examples is only that of their mathematical formulation. I t is further noted that the extensities are conserved in the course of the transformations in a large number of phenomena but that such need not necessarily be true : the entropy, interfacial area and degree of advancement are not conservative quantities. Linear Laws of Molecular Transport. It is now convenient to pose the important problem of knowing the velocity, or, as well as possible, the flow7 with which an 24
=
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
extensity is displaced under the influence of an imposed potential difference. The response is that, in a first approximation, t h e j u x of the extensity at some point is proportional to the driving force, i.e., to the gradient of the potential at this point. Let (flux of extensity) = (conductivity) X (potential gradient) + or p = --X grad V __f
This constitutes a definition of the generalized conductivity of the medium.
TABLE II.
Extensity
Law
LINEAR TRANSPORT LAWS
I
Number of moles of a constituent in a mixture Quantity of heat Quantity of electricity Momentum Mass of fluid
Fick Fourier Ohm Newton Darcy
Among the numerous elementary processes of transport, the simplest are those related to movements which are, at the molecular level, in immobile media (without bulk flow) and in quiet media (without eddies or turbulence). Thermal agitation of molecules, of ions or of free electrons in an ensemble results in macroscopic evolutions of the systems which, at the human level, are described by the simple, basic, linear transport laws of chemical engineering (Table 11). The Linear Laws of Capacity. Consider a domain of space into which flows one of the five preceding extensities, G. At each point of the domain there exists, at the stationary state, a certain concentration of this extensity, g, Le., a volumetric density of the extensity which is a function of the local value of the potential V. I n the first approximation, the relation between these two quantities is linear, i.e.,
g - go
C(V - VO) The potential having a n arbitrary origin permits go = 0 and V , = 0, and thus g = cv or (volumetric density of extensity) = (capacity) X (potential) =
The proportionality factor thus defined is the capacity of the medium at the point considered : -for transport of electricity, it is the electrical ca,bacit>j expressed in farads -for the transport of heat, i t is the thermal capacity expressed as the product of the sepecaj5c heat and the volumetric density -for the transport of a constituent o f a mixture by molecular daffusion, the capacity is equal to one; indeed the concentration of extensity and the potential are one and the same quantity (in the particular case of an ideal solution) -for laminar flow of a perfect gas, the relation between volumetric mass andpressure is
P =
[&I
P
T h e capacity o j the medium is, thus, C = ( m / R T ) . On the contrary the capacity of an incompessible fluid is 0 since p is a constant.
One habitually combines the two preceding linear laws by eliminating the potential between their expressions and writing + cp -
-
-D grad g with D
= X/c
D is the diffusivity of the medium which has always the T+), whatever extensity is transsame dimension (L2 ported. I t is noted that the preceding differential relations representing the local properties of the medium are frequently replaced by still more approximate linear relations which express a mean total property of the system.
I
Potential Molar concentration of the constituent Temperature Electrical potential Flow velocity Pressure
Conductivity Coefficient of diffusion Thermal conductivity Electrical conductivity Dynamic viscosity Permeability
One writes, therefore, that the specific flux of extensity is proportional to a mean difference of potential :
which defines the conductance h of the system; a n analogous equation defines the total capacitance of the system. One notes that the coefficient of diffusion of matter defined by Fick’s law is at the same time a conductivity and a diffusivity. Frequently difficulties in the reasoning concerning analogies between transport processes arise from the fact that the same quantities are involved in the diffusivities and conductivities of mass, heat, and momentum transport. Applications to Systematic Research of New Materials. The concepts introduced previously lead to a methodical investigation of new paths for research in physical kinetics. One knows, in effect, that simple, linear laws are only valuable in a limited domain of variation of their parameters. Thus, one may arrange the list of media possessing exceptional properties by extending the region of validity of the laws. This is done by successively testing materials having extreme values of conductivity or capacities or, alternatively, testing under extreme conditions of potential. Extension of the Domain of Available Conductance. The thermal conductivities of all the materials which we know cover only a small range, varying by a factor of a few magnitudes only. It is evident that research into better insulating or better heat conducting media would be of interest industrially. For the transport of electricity it doesn’t appear that we can expect to discover better insulators. O n the other hand, the superconductors which only up to now have been laboratory curiosities, have become industrially important. For momentum transport, it doesn’t appear that fluids of nearly zero viscosity (helium-2) have found industrial applications. On the other hand, industry has solved a number of problems involved in the flow of very viscous materials (glasses and plastics, generally non-Newtonian) . The research on the more rigid materials and on the materials with great plasticity has preoccupied the attention of many scientists. The builders of equipment have always been desirous of being able to prepare a material which offers all the mechanical properties of the metals and the optical properties of glass. Some progress has been recently made in this area. There is no doubt that new materials to be developed in the next few years to solve the problem of thermal shock on satellites will be VOL. 5 6
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AUGUST 1964
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perfected and become available for constructional use in the chemical industry. Extension of the Domain of Available Capacity. Materials with large thermal capacity are strongly desired. One knows that the effectiveness of many chemical reactors, as well as nuclear reactors, is limited by the problem of heat removal accompanying exothermic reactions. A means of reducing the problem consists of using fluids with much larger thermal capacities. Such fluids currently under investigation include liquid metals, emulsions, suspensions, and special reactive systems. Electrical supercapacity is also a highly prized property in view of the current interest in piles and accumulators in satellite engines. Fluid compressibility is receiving attention. The old distinction between compressible gases and liquids is no longer valid. I n numerous applications such as the hydraulic transmission of information and the hydraulic damping of vibrations, one seeks fluids which are either very compressible or slightly compressible. Investigating the Extended Domain of the Potential. Much work has been devoted to the investigation of materials destined for use under conditions of extreme potential or potential gradients listed in Table I. Familiar industrial temperatures seldom extend beyond a minimum of -195’ C. and a maximum of 3000’ C., but it is possible that these limits will be surpassed within 10 years. At very high temperatures the properties of plasmas are still unknown and should be determined. At very low temperatures (liquid H and He) the purely scientific information is already abundant but the same cannot be said for application techniques. In the domain of pressure potentials one observes that the use of pressures of tens or thousands of atmospheres has been introduced into industry. With respect to low pressures, industrial technology is still at the high vacuum stage ( l o + mm. Hg) but space simulators have demonstrated the practicality of large enclosures wherein the residual pressure is of the order of 10-10 mm. Hg. Change in Level. As stated earlier, the five laws of linear transport listed in Table I1 are only the macroscopic manifestations of the random movements and collisions of molecular entities. The same statement may be integrally repeated by changing levels, Le., by considering the movements and collisions of grains of material. The development of industrial procedures for fluids handling has, consequently, treated suspensions of grains transported by pumps for circulation in conduits and through heat exchangers, The hydraulic or pneumatic transport of ground minerals, the technique of fluidized beds, mixing of powders, etc., treat dispersed media as fluids. These pseudo-fluids often have very particular properties ; they are not Newtonian and the apparent viscosity that one attributes to them is only a mathematical artifice. It is, however, necessary to represent these properties, in the first approximation, by linear laws of transport and capacity. The apparent large thermal conductivity of fluidized beds constitutes their chief advantage in industrial applications. The 26
INDUSTRIAL AND ENGINEERING CHEMISTRY
converted from one to another. Coupling simultaneous systems constitutes an important scientific step and a same is true for the thermal capacity of emulsions and suspensions. The study of the electrical conductivity of fluidized beds has been recently undertaken ( 3 ) . Analogous to ionic propulsion of rockets, a study has been undertaken to investigate propulsion by the ejection of charged colloidal particles. The Study of Nonideal Systems. We propose the name “ideal system’’ for those which obey one or two of the simple laws stated above. We note that each of these laws is characterized by a single parameter, viz., the conductance or capacity. The nonideal systemi, on the contrary, don’t obey the simple laws and require the employment of relations with many parameters. Nonideal systems may be divided into four classes: -nonlinear systems are those in which the conductance or cupacity are not constant; indeed they vary with the potential. Such is the case in the nonohmic conductance of electricity, Jlow of nonNewtonian fluids, etc. -anisotropic systems are those of which the conductance is a function of the direction of transport, e.g., the thermal conductance of crystals -nonhomogeneous systems are those in which the cabacity is not the same at two points of the medium, e.g., in t w o phase media -stochastic systems are (as contrasted with deterministic systems) those in which the conductance and the capacity are not dejned by dejnite relations but rather by laws of probability
The Study of Coupled Systems. Until now there has been consideration of nonideal systems in which only one form of energy was involved. However, observation of the real world reveals that most of the industrial operations result from the combination of several transport systems, transfer systems, and transformation systems simultaneously in which the forms of energy are TABLE I I I.
SEPARATION OF A M I X T U R E OF GRAINS BY SEDIMENTATION
1
Material Regime
~
Flux
~-
=
(Conductzvity)(Driving Force)
TABLE IV. FRACTIONATION O F M I X T U R E S BASED ON DIFFERENCES O F CONDUCTANCE I N T H E TRANSPORT OF EXTENSITIES _____ Applied Diytrence of
Potential
-
Electric potential Electric- potential Magnetic potential Velocity potential (translation) Velocity potential (rotation) Chemical potential Thermal potential
Operation
Sedimentation-decantation Electrolysis : separation of ions based on their migrational velocity Electrophoresis: separation of colloids Electrostatic precipitation Enrichment of ferrous materials Separation of qraiiis from gases by i t i c w i i i (baffled apparatus:, Centrifugation
Gravitational potential Electric potential
1
Diffusion of gases or liquids ~ l i r o i i g h porous barriws Thermal diffiision
1110
WLW
I I
Finally, in industrial heat exchangers, e.g., the general notion of coupling involves the losp of mechanical energy of fluid motion (transfer of momentum to the w a h ) which produces turbulence in the fluid and, conseheat transfer. Similarly, mam transfer quently, be* involves the dissipation of mechanical energy (such as in an absorption or distillation column) of the fluid which provokes miximg and consequently the transfer of material between phases. Condunion. There is evidence of a need to construct a list of fundamental phenomena which have resisted treatment. Perhaps it is useful to proceed as above, on a methodical investigation of research efforts with successive examinations of the three aspects of the problem:
V lESlDU1 D U U ~
2.
Generalscheme jor irothcmnl separ
-tha utiliurtion: what is th8 possible inciddnca of opplicotion of the exceptional properties to i&trid c h m i s ~ ? Application: Comparison of the Muhods of Fmdionation of Mixtures
Figwe 3 (left). BiMIy &tillation column. Th8 Q ~ O Q betmen solWnr and enthalpy is illustrotad by “re&ing” enthalpy from the cond a m to the balm ma Q had pMIp Figwe 4 (right). Separati&n o j “binary” mixhor of powdns in an elutrid01 arrociarad wifh o cy~lommhne momentum of the carrier gas Carupondr to the utrcrcting sdony
complete development requires the thermodynamics of irreversible processes. For example, the phenomena of electrolysis, electrophoresis, elecuudialysis, and electroosmosis correspond to a coupling of electrical energy with mechanical energy. Coupled transport of material and energy constitute the phenomenon of thermal diffusion (Soret effect and Dufour effect) which is being more frequently used for difficult organic chemical separations. It is known that coupling can be accomplished among quantities of the same r a d , viz.: quantifies (r& of mtoid tranrjonndim) q m A t i a s vI.r of trmpwi ojrcdur extmifies) --tarmr pmtihk (flux of frmpwi of mtor e m ’ h k )
-scdur
-vector
According to the Onsager relations, for each coupling of two extensive quantities there is a single coefficient of mutual interaction which is a generalization of the idea of conductlvity defined for each quantity.
AU methods of separation of mixturn may be shown to fall into two general families. The first family is basically concerned with differences in conductivity of the transport processes, whereas the second family deals with the differences in driving force in the transfer of material between phases. This clasilication is valid for molecular and granular mixtures. Fractionation in tramport processes. Consider the example of sedimentation in a gravitational field. As indicated in Table 111the flux of material is equal to the velocity of fall, D, multiplied by the number N of p i n s per unit volume. We write that the flux is proportional to the driving force g(pD - pc) (gradient of gravitational potemtial); the d c i e n t of proportionality is, therefore, the conductivity. In the Stokes regime, the system is linear because the conductivity is independent of the flux, and of the forces, but in Newton’s regime, the system is strongly nonlinear since the conductivity varies as l/E/gbD pc). Bosworth ( I ) has stated that the separation of the constituents of the mixture by transport phenomena only is pwible if the system is nodincur. One observes, on the contrary, that the separation is possible in so far as the constituents of the mixture have different conductivities (grains of the constituents are of different diameters) whether the system is linear or not. I n a vacuum all the grains fall with the same velocity and no fractionation occurs. The separation of the grains by sedimentation results from the drag, i.e., by the degradation of mechanical energy to heat energy, i.e., veation of entropy. Generally we note that in order to separate the constituents of a mixture of molecules, ions, grains, etc., we must assign to the particles an extensity which is the mas, the electric charge, the magnetic pole, the momentum, quantity of heat, etc. Subjecting the mixture to an appropriate potential
-
VOL $6
NO. 8 A U G U S T 1 9 6 4
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difference produces a corresponding separation which is proportional to the drag involved with the creation of entropy. Table IV presents a list of the industrial operations which illustrate these general principles. The table gives only the difference of potential applied to the exterior of the system. In general, in the interior of the equipment, several extensities and several potentials are simultaneously coupled (e.g., thermal diffusion, electrolysis). Fractionation in Transfer Processes. All the operations of isothermal transfer of material between phases (absorption, extraction, etc.) provide examples of the scheme of Figure 2. The feed is composed of a solute and an inert diluent which are contacted in a mass exchanger with an extracting phase (frequently a pure solvent). The extract is composed of a solvent enriched with solute. The mixture is next treated to remove the solute and the solvent is recycled to the exchanger. The flux of solute transfer from the bulk of one phase to the bulk of another is generally represented by the phenomenological law : p = Ko,(y - ,v*) = KOL(X* - X ) where Kon and KOL are two expressions of the overall conductance of transfer ; the overall conductance depends essentially on the partial conductance of transport (by molecular diffusion and turbulence) in both phases and to some extent on the conductance of the interfacial crossing which is frequently very great compared with the other two. When several solutes are transferred simultaneously, their separation results from the difference between their transfer fluxes. It is important to note that the transport conductances of the different constituents in a mixture are frequently nearly the same (they depend essentially on the turbulence of the medium). O n the contrary, the differences in the driving forces of the various constituents result in the separation. I n resume, the fractionation of a mixture of solutes depends more on the static quantities (notably the partition coefficient) than on the dynamic quantities. T h e Enthalpy-Solvent Analogy. Operations of fractional distillation and fractional crystallization may be shown to be included in the same classification, though apparently no extracting solvent is used in these systems. In reality the enthalpy functions very much like the solvent. Provided the analogy between the general idea of extraction in Figure 2 and the scheme of Figure 3 is accepted, then Figure 3 represents a column for the continuous fractionation of a binary mixture. This analogy between the solvent and the enthalpy isn’t just a convenience; it has practical utility and has been employed by Mole (5) for developing a general mathematical method. This analogy also permitted
Pratt (6) to uncover new processes of liquid-liquid extraction, inspired by the technique of continuous distillation. Fractionation of Mixtures of Grains. The analysis of a mixture is actually at the state of development of the analysis of molecular mixtures at the beginning of the century. The result of a granular analysis is the distribution curve for grains of equivalent diameters and is comparable to the curve of relative volatilities (ASTM distillation curve) which provided the principal experimental data for petroleum mixtures. By comparison with the methods effective in chromatography, mass and optical spectrometry analysis, one can hardly imagine the methods to be developed in the future for the determination of the form, dimension, and specific properties of granular species. One might ask how the efficiency of the fractionation equipment for powders compares with that of distillation columns (which contain hundreds of theoretical plates). Figure 4 illustrates the principle of the fractionation equipment for a binary mixture of powders by elutriation. It is the momentum of the fluid carrier which assumes the role of the solvent in extraction. The solute separator is represented by the cyclone which separates the grains from the fluid. The elutriation column contains a dense, pseudo-liquid phase surmounted by a dilute psuedo-gas phase. The overall conductance of transfer of fine grains from the bulk of the pseudo-liquid phase to the pseudo-gas phase is, again, a function of the partial conductance of granular diffusion within the pseudo-fluids and of the conductance of “evaporation.” The conductances have nearly the same values for various constituents and, as in the molecular level, it is the differences in the driving forces of the constituents which permits the possibility of separation. One might believe that the driving forces are nearly equal for the various constituents which, in fact, don‘t exhibit the attractive or repulsive forces analogous to molecular interactions (one regards a mixture of grains as comparable to an ideal mixture). In reality this isn’t quite true because there are interactions of the grains within the strong force field and fluid carrier velocitv field; one must, thus, treat free and hindered settling, respectivelv, as an ideal and a nonideal system. Furthermore, the general principle of countercurrent flow which is indispensable for attaining high efficiencies in molecular transfer is equally important in granular transfer where it has yet to be used. In conclusion, the improvements in fractionation equipment for powders will result from better applications of the countercurrent principle and in the profound study of the driving forces involved. LITERATURE CITED (1) Bosworth, R. C. L., “Transport Processes in Applied Chemistry,” Horwitz Publications, Ltd., Sydney, 1956. (2) De Groot, S. R., Mazur, P., “Non-equilibrium Thermodynamics,” North Holland Publishing Co., Amsterdam, 1962. (3) Goldschmidt, D., Le Gofl, P., Chcm. Ens. Sci. 18, 805 1963. (4) Hill, T. L., “Thermodynamics of Small Systems,’’ W. A. Benjamin Co., New York. 1963. ( 5 ) Mole, P. D. A , , Chcm. Ens. Sci. 7, 236 (1958). (6) Pratt, H. R . C., Ibid., 3, 189 (1954). (7) Prigogine, I., “Etude Thermodynamique des Phenomenes Irreversibles,” Masson, Paris, 1947. (8) Prigogine, I., ed. “Transport Processes in Statistical Mechanics,” Interscience Publishers, Brussels, 1956. ~
AUTHOR Pierre L e Gof is Professor of Chemical Engineering and Director of the Center for Physico-Chemical Kinetic Research at the University of Nancy (France). 28
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