**M a s s T r a n s f e r P r i n c i p l e s**

**DONALD J. KIRVVAN**

*Department of Chemical Engineering*

*University of Virginia*

*Charlottesville, Virginia*

**2.1 INTRODUCTION**

**Mass transfer can be defined simply as the movement of any identifiable species from one spatial location**
**to another. The mechanism of movement can be macroscopic as in the flow of a fluid in a pipe (convection)****or in the mechanical transport of solids by a conveyor belt. In addition, the transport of a particular species**
**may be the result of random molecular motion (molecular diffusion) or random microscopic fluid motion****(eddy or turbulent diffusion) in the presence of a composition gradient within a phase. This chapter is**
**concerned primarily with mass transfer owing to molecular or microscopic processes.**

**Chemical processes produce complex mixtures of compounds from various feedstocks. Proper operation**
**of chemical reactors often requires that the feed contain only certain species in specified ratios. Thus,**
**separation and purification of species from feedstocks, whether petroleum, coal, mineral ores, or biomass,**
**must be accomplished. Similarly, a mixture leaving a reactor must be separated into purified products, **
**by-products, unreacted feed, and waste materials. Separation processes are also of importance where no reaction**
**is involved as in seawater desalination by reverse osmosis, crystallization, or evaporation; in the **
**fraction-ation of crude petroleum; or in the drying of solids or devolatizfraction-ation of polymers where diffusion within a**
**porous solid is of importance.**

**Since separation processes are based on the creation of composition differences within and between**
**phases, a consideration of mass transfer principles is necessary to the analysis and design of such processes.**
**It is the combination of mass transfer principles with phase-equilibrium relationships (Chapter 1), energy**
**considerations, and system geometry or configuration that permits the proper description of the separation**
**processes discussed in the remainder of this book. It is the goal of this chapter to provide quantitative**
**descriptions of the rate of species transport within phases under the influence of composition differences.**
**An understanding of the importance of physical properties, geometry, and hydrodynamics to mass transfer**
**rates is sought.**

**2.1-1 Factors Influencing Mass Transfer**

**To set the stage for the subject matter of this chapter, consider a familiar mass transfer situation—sweetening**
**a cup of coffee or tea. A series of experiments at our kitchen table can provide insight into a number of**
**characteristics of mass transfer. It could be quickly determined that an equal amount of granulated sugar**
**will dissolve more rapidly than a cube, that an increase in the amount of granulated sugar will increase the**
**dissolution rate, that stirring greatly increases the rate, that an upper limit to the amount that can be dissolved**
**is reached and this limit increases with temperature as does the rate. Careful sampling would also reveal**
**a greater sugar concentration near the sugar-solution interface which would be very nearly the final **
**satu-ration value that could be obtained.**

**The results of these simple experiments on mass transfer can be stated as follows:**

**1. Mass transfer occurs owing to a concentration gradient or difference within a phase.**

**2. The mass transfer rate between two phases is proportional to their interfacial area and not the**
**volumes of the phases present.**

**3. Transfer owing to microscopic fluid motion or mixing is much more rapid than that due to molecular**
**motion (diffusion).**

**4. Thermodynamics provides a limit to the concentration of a species within a phase and often governs**
**the interfacial compositions during transfer.**

**5. Temperature has only a modest influence on mass transfer rates under a given concentration driving**
**force.**

**This example also illustrates the use of the three basic concepts on which the analysis of more complex**
**mass transfer problems is based; namely, conservation laws, rate expressions\ and equilibrium ****thermo-dynamics. The conservation of mass principle was implicitly employed to relate a measured rate of ****accu-mulation of sugar in the solution or decrease in undissolved sugar to the mass transfer rate from the crystals.**
**The dependence of the rate expression for mass transfer on various variables (area, stirring, concentration,**
**etc.) was explored experimentally. Phase-equilibrium thermodynamics was involved in setting limits to the**
**final sugar concentration in solution as well as providing the value of the sugar concentration in solution**
**at the solution-crystal interface.**

**2.1-2 Scope of Chapter**

**The various sections of this chapter develop and distinguish the conservation laws and various rate **
**expres-sions for mass transfer. The laws of conservation of mass, energy, and momentum, which are taken as**
**universal principles, are formulated in both macroscopic and differential forms in Section 2.2.**

**To make use of the conservation laws it is necessary to have specific rate expressions or constitutive**
**relationships connecting generation and transport rates to appropriate variables of the system of interest.**
**In contrast to the universality of the conservation laws, both the form and the parameter values of rate**
**expressions can be dependent on the specific material under consideration. Fourier's Law of heat conduction**
**relating the heat flux to a temperature gradient, q = -k dT/dz, is applicable to both gases and metals yet****the magnitude and the temperature dependence of the thermal conductivity differ greatly between gases and**
**metals. Expressions for the dependence of the rate of a chemical reaction on the concentrations of **
**partic-ipating species vary widely for different reactions. In complex geometries and flow patterns, useful **
**expres-sions for the mass transfer flux are dependent on geometry, hydrodynamics, and physical properties in**
**ways not completely predictable from theory. In any case, the conservation equations cannot be solved for**
**any particular problem until appropriate rate expressions, whether obtained from theory or from experiment**
**and correlation, are provided. In Section 2.3 theoretical foundations, correlations, and experimental **
**mea-surements of molecular diffusion are treated along with problems of diffusion in static systems and in**
**laminar flow fields. Theories, experimental observations, and engineering correlations for mass transfer in**
**turbulent flow and in complex geometries and flow fields are presented in Section 2.4 with some examples**
**illustrating their use in analyzing separation phenomena and processes. A further important use of examples**
**in this chapter is to give the reader a "feel" for the order of magnitude of various mass transfer parameters**
**and the length and time scales over which mass transfer effects are of importance.**

**2.2 CONSERVATION LAWS**

**Expressions of the conservation of mass, a particular chemical species, momentum, and energy are **
**fun-damental principles which are used in the analysis and design of any separation device. It is appropriate**
**to formulate these laws first without specific rate expressions so that a clear distinction between conservation**
**laws and rate expressions is made. Some of these laws contain a source or generation term, for example,**
**for a particular chemical species, so that the particular quantity is not actually conserved. A conservation**
**law for entropy can also be formulated which contributes to a useful framework for a generalized transport**
**theory.1'2 Such a discussion is beyond the scope of this chapter. The conservation expressions are first**
**presented in their macroscopic forms, which are applicable to overall balances on energy, mass, and so**
**on, within a system. However, such macroscopic formulations do not provide the information required to**
**size equipment. Such analyses usually depend on a differential formulation of the conservation laws which**
**permits consideration of spatial variations of composition, temperature, and so on within a system.**

**2.2-1 Macroscopic Formulation of Conservation Laws**

**FIGURE 2.2-1* Mass balances on arbitrary volume of space.**

**Since, in non-nuclear events, total mass is neither created nor destroyed, the conservation law for total**
**mass in a system having a number of discrete entry and exit points may be written**

**(2.2-1)**

**The total mass of the system is m, n is the total mass flux (mass flow per unit area) relative to the system****boundary at any point, and S is the cross-sectional area normal to flow at that same location. The summations****extend over all the mass entry and exit locations in the system. The mass flux at any point is equal to pv,****where p is the mass density and i; is the velocity relative to the boundary at that point. Equation (2.2-1)****can be applied equally well to a countercurrent gas absorption column or to a lake with input and output**
**streams such as rainfall, evaporation, streams flowing to or from the lake, deposition of sediment on the**
**lake bottom, or dissolution of minerals from the sides and bottom of the lake. The steady-state version of**
**Eq. (2.2-1) (dm/dt = 0) is of use in chemical process analysis because it permits calculation of various****flow rates once some have been specified.**

**The macroscopic version of the conservation law for a particular chemical species i that is produced**
**by chemical reaction at a rate per unit volume^ of r,- is written**

**(2.2-2)**

**Again it is assumed that there are a number of discrete entry and exit locations where species i crosses the**
**system boundary with a flux n,. The flux /I****1**** is also equal to p****i****v****i**** where p, is the mass concentration of /**
**(mass of iVvolume) and u, is the velocity of species i relative to the system boundary. In a multicomponent****mixture the velocities of each species may differ from one another and from the average velocity of the**
**mixture. The mass average velocity of a mixture is defined by**

**(2.2-3)**

**where the summation is over all species. The last term of Eq. (2.2-2) represents the total production**
**(consumption) of species i owing to chemical reaction within the control volume. Since the total mass of**

**$The term r, would actually represent the sum of the rates of all reactions producing or consuming**
**species i.**

**Mass flux**
**Species ffux**

**, Generation**

**the system is the sum of the mass of each species, Eq. (2.2-1) can be obtained by summing Eq. (2.2-2)**
**written for each species. The term E,- r, is identically zero since mass is neither created nor destroyed by**
**chemical reaction.**

**Conservation of species can also be expressed on a molar basis as shown by Eq. (2.2-4):**

**(2.2-4)**

**where moles of i in the system is Af,. The molar flux of species i (moles //time • area) relative to the system****boundary at any location is N****1**** m C****1U1****, where C****1**** is the molar concentration of species i. Summation of Eq.**
**(2.2-4) written for all species would yield a conservation expression for total moles which is of limited use**
**since a generation term representing net moles created or destroyed by reaction is present. A molar average**
**velocity of a mixture v* can be defined analogous to the mass average velocity**

**(2.2-5)**

**The total molar flux N = E1**** N****1**** is Cu*, where C is the total molar concentration E, C,.**

**The conservation of species equations, whether expressed in terms of mass [Eq. (2.2-2)] or moles [Eq.**
**2.2-4)], are entirely equivalent; the choice of which to use is made on the basis of convenience. The**
**steady-state forms of these material balances are universally employed in the design and analysis of chemical**
**processes including separation devices.3'4 Many examples will be encountered in later chapters of this**
**book. The primary purpose here is to set the stage for the differential balance formulation to follow shortly.**
**Since the macroscopic formulation of the momentum balance has little use in mass transfer theory it is**
**not presented here. The differential form is of importance because many mass transfer problems involve**
**the interaction of velocity and diffusion fields.**

**Similar to the mass conservation laws, the macroscopic formulation of the First Law of **
**Thermody-namics—that is, energy is conserved—is applicable to a wide range of chemical processes and process**
**elements. The energy conservation equation for a multicomponent mixture can be written in a bewildering**
**array of equivalent forms.5'6**** For the volume of space shown in Fig. 2.2-1 containing i species, one form****of the conservation equation for the energy (internal + kinetic -I- potential) content of the system is**

**(2.2-6)**

**Equation (2.2-6) states that the change with time of the total energy content of a volume is due to the net**
**difference between energy flowing into and out of the system with mass flow plus the shaft power being**
**delivered to the system plus the rate of energy delivered to the system as heat (both by conduction and**
**radiation). The partial mass enthalpy, specific kinetic energy, and specific potential energy of each species**
**i in the flow terms are evaluated at the entry or exit point conditions. The potential energy term, a result**
**of so-called body forces, can differ for different species, for example, the force of an applied electrical**
**field on ions in solution, but most commonly it is only that resulting from gravity.**

**The macroscopic energy balances, with appropriate simplifying assumptions, are used in many different**
**process systems. For example, under steady state conditions with unimportant mechanical energy and shaft**
**work terms, it is the basis of the so-called process "heat" balance which may be applied (along with mass**
**balances) to heat exchangers, chemical reactors, distillation columns, and so on to determine quantities**
**such as heat duties or exit temperatures from a unit. Application of the macroscopic balances generally**
**does not permit a determination of the composition or temperature profiles within a unit such as a packed**
**column gas absorber or tubular reactor. A differential formulation of the conservation laws is therefore**
**required.**

**2.2-2 Differential Formulation of Conservation Laws**

**FIGURE 2.2-2 One-dimensional shell balances: (a) rectangular and (b) cylindrical coordinates.**

**Although general and rather elegant derivations of the differential balances are available,6'7 it is more**
**instructive to use the simple "shell" balance approach. Figure 2.2-2 depicts situations where changes in**
**density, velocity, composition, and temperature occur continuously and in only one spatial direction, either**
**x or r. The change with time of the total mass within the volume element of Fig. 2.2-2a, owing to the****difference in mass flow into and out of the volume, is**

**where n****x**** denotes the mass flux in the x direction. In this rectangular coordinate system the cross-sectional****area is independent of position x and as Ax -> 0**

**(2.2-7)**

**Equation (2.2-7), known as the one-dimensional continuity equation, expresses the fact that the density**
**can only change at a particular location if there is a change in the mass flux with position. Conversely, if**
**the density does not change with time, the mass flux is independent of position.**

**Equation (2.2-7) can be generalized readily to three dimensions as shown in Table 2.2-1. The mass**
**flux in a particular coordinate direction (e.g., n****x****) is simply equal to the product of the density and the****component of the mass average velocity in that direction, pv****x****. These mass fluxes and velocities are with****reference to a stationary or laboratory coordinate system.**

**This same shell balance can be applied in curvilinear coordinate systems^ as illustrated in Fig. 2.2-26.**
**In this case the change of area with radial position must be taken into account, that is,**

**The three dimensional analogue of this equation and that for spherical coordinates are also listed in Table**
**2.2-1. The continuity equation can also be more compactly expressed in vector or tensor notation.**

**$It is preferable, however, to derive the equations in rectangular coordinates and then transform to another**
**system.**

**(a)**

**The continuity equation for total moles can also be written, but it must include a term representing the**
**net production of moles by chemical reaction. In rectangular coordinates the equation would be**

(2.2-8)

**where N****x****, N****yy**** N, are the components of the molar flux vector relative to stationary coordinates.****A shell balance applied to the mass of any particular species / would yield, in rectangular coordinates,**

**(2.2-9)**

**The notation n****itX**** denotes the x component of the mass flux of species i relative to laboratory coordinates.****Equation (2.2-9) expresses the fact that the mass concentration of a species will change with time at a**
**particular point if there is a change in the flux with position at that point or if a chemical reaction is**
**occurring there.**

**For analysis of mass transfer problems it is convenient to rewrite Eq. (2.2-9) in terms of a diffusive**
**flux or diffusion velocity. The mass diffusive flux in the x direction {j****iJC****) is defined relative to the mass****average velocity in that direction :$**

**(2.2-10)**

**The ratio pjp is the mass fraction of species /.**

**The conservation equation for species / can then be written in a form that explicitly recognizes transport**
**by convection at the mass average velocity and by diffusive transport relative to the average velocity:**

**$ln three directions the diffusive flux vector is defined as the difference between the total flux of / and that**
**carried at the mass average velocity.**

*Spherical Coordinates*
*Cylindrical Coordinates*
*Rectangular Coordinates*

## (2.2-11)

**Equation (2.2-11) can also be written in the form of a substantial derivative of the mass fraction****(w, = pilp) by use of the continuity equation for total mass (Table 2.2-1)**

(2.2-12)

**The substantial derivative can be inteipreted as the variation of w****(**** with time as seen by an observer moving**
**with the mass average velocity of the system:**

**(2.2-13)**

**The species conservation equations in cylindrical and spherical coordinates are shown in Table 2.2-2.**
**In rectangular coordinates the corresponding conservation of species equations in molar units are**

**(2.2-14)**

**(2.2-15)**

**The molar diffusion flux J****1**** is defined with respect to the molar average velocity,**

**(2.2-16)**

**were CJC is simply the mole fraction of i in the system. Other definitions of the diffusion flux exist in the****literature5 but they are not necessary to the analysis of mass transfer problems.**

**If we examine the continuity equations of total mass and any species /, it is quite clear that the equations**
**cannot be solved for density and composition as a function of time and position until the velocity field is**
**known. In addition, appropriate rate expressions (Sections 2.3 and 2.4) must be available for mass transport**
**as well as for chemical reaction. These rates will depend on temperature so that the temperature field must**

**TABLE 2.2-2. Differential Forms of the Conservation Equation**
**for a Species (Mass Units)'**

*Rectangular Coordinates*

*Cylindrical Coordinates*

*Spherical Coordinates*

**also be specified. Although many practical mass transfer problems are solved by independently specifying**
**the velocity and temperature—for example, diffusion in a stagnant, isothermal system—the general problem**
**requires that the velocity and temperature be determined simultaneously by using appropriate expressions**
**of conservation of momentum and energy in differential form.**

**The differential linear momentum balance for a multicomponent mixture can be derived from a shell**
**balance approach or from a more sophisticated analysis using continuum mechanics. Such formulations are**
**tabulated in many texts, for example, that by Bird et al.,5 and will not be presented here in any detail. By**
**way of illustration the x component of the differential momentum balance in rectangular coordinates would****be**

**(2.2-17)**

**where p is the hydrostatic pressure, g****x**** is the x component of gravity, and r represents the stress tensor. It****is important to note that v****xf**** uv****, and v. in a multicomponent system are the components of the mass average****velocity of the system. If the fluid is Newtonian, then the stress tensor, which may also be thought of as**
**the momentum flux vector, is related to the velocity gradients in the system by Newton's Law of viscosity,**
**and the well-known Navier-Stokes equations result. Equation (2.2-17) and its analogues are the starting**
**point for the solution of problems in fluid mechanics and may be used simultaneously with the mass**
**conservation equations to solve many problems in mass transfer.**

**The differential form of the energy balance for a multicomponent mixture can be written in a variety**
**of forms.5'6 It would contain terms representing heat conduction and radiation, body forces, viscous **
**dis-sipation, reversible work, kinetic energy, and the substantial derivative of the enthalpy of the mixture. Its**
**formulation is beyond the scope of this chapter. Certain simplified forms will be used in later chapters in**
**problems such as simultaneous heat and mass transfer in air-water operations or thermal effects in gas**
**absorbers.**

**As shown in the following examples, the conservation laws are employed in specific instances by**
**making appropriate simplifying assumptions to reduce the general differential balances to a tractable form.**

**EXAMPLE 2.2-1 A DIFFUSION EXPERIMENT**

**Two pure fluids A and B are initially contained in vertical cylinders that are brought over one another at**
**time zero (Fig. 2.2-3). Write down the appropriate equations to determine the composition profile at any**
**later time if the fluids are (a) perfect gases and (b) liquids.**

**Solution: We make the following assumptions:**

**1. Variations of composition occur only in the z direction.****2. The system is at constant pressure and temperature.**
**3. No chemical reaction occurs.**

**The continuity and species equations in either mass or molar units may be used at the convenience of the**
**analyst.**

**For gases it is convenient to use molar units because for an ideal gas mixture the molar concentration****C = pi****1****RTis independent of composition. The one-dimensional continuity equation is**

**Since C does not change, dvf/dz = 0 and vf is a constant. Furthermore, since there is no flux through the****ends of the cylinder, vf = 0.**

**The species balance for A reduces to**

**This equation can be solved provided an appropriate rate expression is available for JA, the diffusive flux**
**relative to the molar average velocity.**

**If the conservation equations in mass units had been employed,**

**(A)**

**(B)**

**For an ideal gas mixture the mass density does depend on position since p = MpIRT, where M is the mean****molecular weight. Substitution for the density into Eqs. (A) and (B) would yield equations for v****z**** and p****A****after an appropriate flux relationship forjA was supplied. The z-component of the mass average velocity,**
**y., is not zero.**

**For liquids the equations in mass units are convenient because liquid mass densities are generally weak****functions of composition, so that the approximation dp/dt = dp/dz = 0 is made and v****z**** = 0 by Eq. (A).****Equation (B) is then**

**Solutions to the above equations will be given in Section 2.3. In this problem of free diffusion the**
**continuity equation combined with thermodynamic relationships allows the specification of the velocity**
**field.**

EXAMPLE 2.2-2

**Pure water can be obtained from brackish water by permeation through a reverse osmosis membrane.**
**Consider the steady laminar flow of a salt solution in a thin channel between two walls composed of a**
**membrane that rejects salt. Derive the governing equations for the salt distribution in the transverse direction**
**for a given water permeation flux (see Fig. 2.2-4).**

**Solution. We make the following assumptions:**

**1. The system exists in an isothermal steady state.**

**2. The density is independent of composition because it is a liquid.**

**3. The effect of H2O permeation through the walls on the velocity profile is neglected.**

**The continuity equation can be written**

**FIGURE 2.2-4 Laminar flow membrane channel.**

**The species balance for salt is**

**(B)**

**and the z-component of momentum balance is**

**(C)**

**To obtain the velocity profile assumption 3 is used, whereby we take v****v**** = 0 for low permeation rates4**
**From Eq. (A) dvJdz = 0 and Eq. (C) becomes**

**whose solution is the well-known parabolic velocity profile**

**An approximate solution for the salt distribution under low permeation rates can be obtained by **
**ne-glecting diffusion in the z direction with respect to convection (n****StZ**** « p****s****v****z****) and neglecting v****y****. The species****balance then becomes**

**where v****z****(y) is known. This equation can be solved provided an appropriate rate expression for the diffusive****flux is known as well as an expression relating the flux of water through the membrane to the salt **
**concen-tration at the membrane surface and the pressure difference across the membrane. A discussion of membrane**
**transport can be found in Chapter 10 while rate expressions for diffusion are the subject of Section 2.3.**

**The primary purpose of this section was to state the framework in which all mass transfer problems**
**should be formulated, that is, the conservation equations for mass, species, momentum, and energy. As**
**illustrated by the above examples complete solutions of problems in mass transfer require a knowledge of**

**rate expressions for species transport under various conditions. These rate expressions and their use are the**
**subject of the remainder of this chapter.**

**2.3 MOLECULAR DIFFUSION**

**Ordinary diffusion can be defined as the transport of a particular species relative to an appropriate reference**
**plane owing to the random motion of molecules in a region of space in which a composition gradient exists.**
**Although the mechanisms by which the molecular motion occurs may vary greatly from those in a gas to**
**those in a crystalline solid, the essential features of a random molecular motion in a composition gradient**
**are the same, as will be seen in the simplified derivations discussed here.**

**Consider a perfect gas mixture at constant temperature and pressure (therefore the molar concentration**
**C is constant and there is no bulk motion of the gas) in which a mole fraction gradient exists in one**
**direction as shown in Fig. 2.3-1. Molecules within some distance ±a, which is proportional to the mean**
**free path of the gas (X), can cross the z — 0 plane. The net molar flux from left to right across the z — 0****plane would be proportional to the product of the mean molecular speed v, and the difference between the****average concentration on the left and on the right.**

**The difference in the average mole fractions can be approximated by -a(dy****A****/dz)****z****^o- The diffusive flux J****A****is, therefore**

**(2.3-1)**

**The molecular diffusivity (or diffusion coefficient) D, defined as the proportionality constant between**
**the diffusive flux and the negative of the composition gradient, is therefore proportional to the product of**
**the mean molecular speed and the mean distance between collisions. If the simple kinetic theory expressions**
**for the mean molecular speed and the mean free path of like molecules are used, one finds a modest**
**temperature and pressure dependence of the diffusivity.**

**(2.3-2)**

**where d****A**** is the collision diameter and AfA is the molecular weight of species A. Detailed kinetic theory**
**analysis or the more exact results of Chapman and Cowling1 taking into account intermolecular forces and**

**differing sizes and masses of molecules are quoted in Section 2.3-2. The purpose here is simply to illustrate**
**that a diffusive flux originates from the random motion of molecules in a composition gradient.**

**For diffusion in condensed phases whether solid or liquid, Eq. (2.3-1) could still hold except it must**
**be recognized that not all molecules are free to move. Thus, the expression for the diffusive flux must be**
**multiplied by a probability that the molecule is available to move or jump. This probability is generally**
**related to an energy barrier which must be overcome. The nature of the barrier is dependent on the**
**mechanism of the diffusive process, for example, the motion of atoms and vacancies in a metal crystal**
**would be characterized by a different energy barrier than that for the movement of molecules in the liquid**
**state. In any event, the diffusivity in the condensed phase would be given by an expression of the form2**
**v****o****b****2**** exp (—EIRT). In this expression the mean free path has been replaced by a characteristic intermolecular****spacing b, and the mean speed by the product of a vibration frequency and this spacing. Again, the diffusion****coefficient contains parameters characteristic of random molecular motion within a phase.**

**Appropriate definitions of diffusive fluxes and a phenomenological theory for the relationship of the**
**fluxes to the driving forces or potentials for diffusion are presented first. Methods of predicting and **
**cor-relating diffusion coefficients in gases, liquids, and solids are discussed in Section 2.3-2, along with some**
**typical experimental values of diffusion coefficients. Section 2.3-3 describes a number of experimental**
**methods for measuring diffusion coefficients along with the appropriate mathematical analysis for each**
**technique. Finally, some sample problems of diffusion in stagnant phases and in laminar flow fields are**
**considered.**

**2.3-1 Phenomenological Theory of Diffusion**

**A general theory of transport phenomena3 suggests that the diffusive flux would be made up of terms**
**associated with gradients in composition, temperature, pressure and other potential fields. Only ordinary**
**diffusion, the diffusive motion of a species owing to a composition gradient, is considered in detail here.**
**The assumed linear relationship between diffusive flux and concentration gradient is traditionally known as****Fick's First Law of Diffusion. In one dimension,**

**(2.3-3)**

**Equation (2.3-3) can be generalized to relate the diffusive flux vector to the gradient of composition in**
**three coordinate directions. Cussler4 has recently summarized the historical background to the development**
**of Fick's Law. In fact, there have been a number of ambiguities or inconsistencies in the use of rate**
**expressions for diffusion primarily arising from an unclear view of the definitions used in a particular**
**analysis.**

**In Section 2.2 both the mass diffusive flux relative to the mass average velocity and the molar diffusive**
**flux relative to the molar average velocity were defined:**

**(2.3-4)**

**(2.3-4a)**

**Other definitions of the diffusive flux such as the mass flux relative to the molar average velocity could be**
**employed but they have no particular value. Equation (2.3-4) or (2.3-4a) can be used in conjunction with**
**the definitions of i; and v* to show immediately that**

**(2.3-5)**

**and**

**(2.3-5a)**

**These relationships among the diffusive fluxes of various species provide restrictions on the diffusion**
**coefficients appearing in the rate equations for the diffusive flux.**

**BINARY SYSTEMS**

**Consider a binary isotropic$ system composed of species A and B at uniform temperature and pressure and**
**ask what the appropriate rate expression for ordinary diffusion should be. The thermodynamics of **

**sible processes suggests that the driving force for diffusion would be the isothermal, isobaric gradient in**
**chemical potential, that is, the gradient in chemical potential owing to composition changes. It further**
**postulates a linear relationship between flux and driving force. There is nothing to say whether this **
**com-position should be expressed in terms of a mass or mole fraction or a volume concentration; indeed, all**
**are quite proper measures of composition.5 In mass units, for one-dimensional diffusion, either Eq.**
**(2.3-6) or (2.3-6a) can be used.**

**(2.3-6)**

**(2.3-6a)**

**The expressions are different, of course, unless the density is independent of position (composition).**
**If Eq. (2.3-6) is chosen as the definition, then it immediately follows by use of Eq. (2.3-5) that**
**A \ B — ^BA; that is, there is a single mutual diffusion coefficient in a binary system. If Eq. (2.3-6a) is**
**chosen, D****BA**** qfc D****AB, but there would be a relationship between them so that there still is only a single****characteristic diffusivity for a binary system. The choice of Eq. (2.3-6) is preferred because of its simplicity**
**although it certainly is not intrinsically better then Eq. (2.3-6a).**

**The analogous rate expressions for the molar diffusive flux relative to the molar average velocity can**
**be used. The preferred definition relates the flux to the mole fraction gradient**

(2.3-7)

**It can be shown that D****AB**** in this definition is identical to that in Eq. (2.2-6). The particular choice of**
**reference velocity and corresponding flux expression can be made for the convenience of the analyst as**
**illustrated by the extension of the analysis carried out in Example 2.2-1 on the interdiffusion of two gases**
**or liquids.**

**For the case of a perfect gas mixture whose molar concentration is independent of composition, the**
**continuity and species balance equations reduce to the statements**

**Since C is constant /A>, can be written — DAB dCAfdz, and the final form of the species balance becomes**

**(2.3-8)**

**Equation (2.3-8) is known as Fick '5 Second Law of Diffusion, but recognize that it is the result of assuming****constant molar concentration, determining v* = 0 from the continuity equation, and substituting Fick's****First Law in the species conservation equation. For a binary liquid solution whose mass density is **
**approx-imately independent of composition, the species balance and continuity equation would also reduce to the**
**form of Fick's Second Law**

**(2.3-9)**

**As illustrated in the following example, it is preferable to start with the conservation equations, rate**
**expression, and thermodynamic relationships, rather than immediately applying Fick's Second Law to a**
**diffusion problem.**

EXAMPLE 2.3-1

**Solution, The continuity equation and species balances in molar units are**

**(A)**

**(B)**

**By means of Eq. (A) the species balance can be written in the alternative form**

**(C)**

**Application of the chain rule for the variation of the molar density allows the continuity equation to be**
**written in the form**

**Elimination of dxjdt by Eq. (C) provides an expression for the velocity gradient**

**(D)**

**or**

**The experiment can usually be arranged so that v*(-oo,t) = 0. This general result for the average velocity****in terms of the diffusive flux in a free diffusion experiment was published by Duda and Vrentas.6**

**The density is related to the partial molar volumes by**

**For our case V****A**** and V****B**** are equal to their pure-component values, V% and Kg, so that the molar average****velocity is**

**Substitution for v* in Eq. (B) and use of the relationship**

**yield after some algebra**

**Thus, Fick's Second Law can be rigorously applied for the case of the interdiffusion of ideal solutions.**
**For nonideal solutions, the simultaneous solution of Eqs. (D) and (B) can be carried out if density versus**
**composition data are available for the system.6**

**It should be emphasized that in the phenomenological theory the diffiisivity is presumed independent**
**of composition gradient but still may be strongly dependent on composition as well as temperature.**

MULTICOMPONENT SYSYEMS

**effects may be seen in systems that are thermodynamically highly nonideal, in membrane diffusion, when**
**electrostatic effects occur, or when there are strong associations among particular species. Cussler has**
**written an excellent book4 on the experimental and theoretical aspects of multicomponent diffusion to which**
**the reader is referred for an extensive discussion.**

**Early theories for multicomponent diffusion in gases were obtained from kinetic theory approaches and**
**culminated in the Stefan-Maxwell equations7*8 for dilute gas mixtures of constant molar density. In one**
**dimension,**

**(2.3-10)**

**The Dy are the binary diffusion coefficients for an / — j mixture; thus, no additional information is required****for computations of multicomponent diffusion in dilute gas mixtures although one might prefer a form in**
**which the fluxes appeared explicitly. Generalization of the Stefan-Maxwell form to dense gases and to**
**liquids has been suggested,9 but in these cases there is no rigorous relationship to the binary diffusivities.**
**Furthermore, the form of Eq. (2.3-10) in which the fluxes do not appear explicitly has little to **
**recom-mend it.**

**A phenomenological theory of ordinary diffusion in multicomponent systems based on irreversible**
**thermodynamics3 suggests that the ordinary diffusive flux of any species is a linear function of all the**
**independent composition gradients. In one dimension,**

**(2.3-11)**

**In a system containing c species there are c - 1 independent fluxes (since Zj****1**** = 0) and c - 1 independent****composition gradients. There would appear to be (c — 1) x (c — 1) diffusion coefficients, but use of the**
**Onsager reciprocity relationships of irreversible thermodynamics shows that there are actually only**
**c(c - l)/2 independent coefficients. Because of the complexity of the relationships among the Z>,****y of Eq.**
**(2.3-9) (requiring knowledge of the thermodynamic properties of the system), Cussler4 suggests that as a**
**practical matter the D****0**** be treated as independent coefficients in many cases. The key aspect of Eq.**
**(2.3-11) is the understanding that in a multicomponent system the diffusion rate of a particular species can**
**be affected by the composition gradients of other species as well as by its own. The D****u****, of course, are****composition dependent.**

**A striking example of the effects of a third component on the diffusive rate of a species is shown in**
**Fig. 2.3-2 in which the acceleration of the flux of sodium sulfate in aqueous solution owing to the presence**
**of acetone is shown. Some of the most dramatic examples of multicomponent effects in diffusion occur in**
**both natural and synthetic membranes.**

**In many practical instances involving multicomponent diffusion the diffusive flux is simply modeled in**
**terms of an effective diffusivity of i in the mixture (D****im****) which relates the flux of i only to its own****composition gradient.**

**FIGURE 2.3-2 Enhancement of Na2SO4 flux in aqueous**
**solution owing to acetone addition. Reprinted with **
**per-mission from E. L. Cussler, Multicomponent Diffusion,****Elsevier, Amsterdam, 1976.**

**Weight fraction acetone**
**Water**

**Na2SO4**
**Acetone,**
**V Water/**

Relativ

e

flu

x sodiu

## (2.3-12)

**The quantity D****im**** would not only be a function of the composition of the mixture but, in fact, could be**
**dependent on the gradients of various other species and therefore could depend on time and/or position**
**during the diffusion process. Nevertheless, Eq. (2.3-12) is often employed with reasonable success unless**
**some highly specific interactions exist. D****im**** should certainly be measured under the mixture composition**
**of interest, yet it is often estimated as the binary diffusivity of i in the solvent.**

**2.3-2 Diffusion Coefficients in Cases, Liquids and Solids**

**This section provides some typical experimental values of interdiffusion coefficients in gases, in solutions**
**of nonelectrolytes, electrolytes, and macromolecules, in solids, and for gases in porous solids. Theoretical**
**and empirical correlations for predicting diffusivities will also be discussed for use in those cases where**
**estimates must be made because experimental data are unavailable. A critical discussion of predicting**
**diffusivities in a fluid phase can be found in the book by Reid et al.10**

**CASES**

**At ambient temperature and pressure, gas-phase diffusion coefficients are of the order of 10"4-10~5 ft2/s**
**(10~5-10~6 m2/s). Table 2.3-1 presents some representative values for binary gas mixtures at 1 atm.**
**Marrero and Mason11 have provided an extensive review of experimental values of gas phase binary**
**diffusivities.**

**Kinetic theory predicts the self-diffusion coefficient, that is, the diffusion coefficient for a gas mixture**
**in which all molecules have identical molecular weights and collision diameters, to be12**

**while that for mutual diffusion of A and B is**

**(2.3-13)**

**TABLE 2.3-1 Experimental Values of Binary Gaseous Diffusion**
**Coefficients at 1 atm and Various Temperatures**

**Gas Pair T(K) D^****3X 105**** (m2/s)a**

**Air-CO2 317 1.77**

**Air-H2O 313 2.88**

**Ak-H-C6H14 328 0.93**

**Ar-O2 293 2.0**

**CO2-N2 273 1.44**

**CO2-N2 298 1.66**

**CO2-O2 293 1.53**

**CO2-CO 273 1.39**

**CO2-H2O 307 1.98**

**CO2-H2O 352 2.45**

**C2H4-H2O 328 2.33**

**H2-CH4 298 7.26**

**H2-NH3 298 7.83**

**H2-NH4 473 18.6**

**H2-C6H6 288 3.19**

**H2-SO2 473 12.3**

**N2-NH3 298 2.30**

**N2-NH3 358 3.28**

**N2-CO 423 6.10**

**O2-C6H6 311 1.01**

**O2-H2O 352 3.52**

**"Diffusion coefficients in ftVs are obtained by multiplying table entries by**
**10.76.**

**Gas Pair**

**Air-CO2**
**Air-H2O**
**Ak-H-C6Hi4**
**Ar-O2**
**CO2-N2**
**CO2-N2**
**CO2-O2**
**CO2-CO**
**CO2-H2O**
**CO2-H2O**
**C2H4-H2O**
**H2-CH4**
**H2-NH3**
**H2-NH4**
**H2-C6H6**
**H2-SO2**
**N2-NH3**
**N2-NH3**
**N2-CO**
**O2-C6H6**
**O2-H2O**

**T(K)**
**317**
**313**
**328**
**293**
**273**
**298**
**293**
**273**
**307**
**352**
**328**
**298**
**298**
**473**
**288**
**473**
**298**
**358**
**423**
**311**
**352**

**TABLE 2.3-2 Values of the Collision**
**Integral for Diffusion**

**kT/e****AB**** Q0**

**0.30 2.662**
**0.50 2.066**
**0.75 1.667**
**1.00 1.439**
**1.50 1.198**
**2.00 1.075**
**3.00 0.9490**
**4.00 0.8836**
**5.0 0.8422**
**10 0.7424**
**20 0.6640**
**50 0.5756**
**100 0.5170**

**where d****{**** is the molecular diameter of / and M****1**** is its molecular weight. The pressure dependence of Eq.**
**(2.3-13) appears to be correct for low pressures; but gas phase diffiisivities vary somewhat more strongly**
**with temperature than is indicated by Eq. (2.3-13).**

**By taking into account forces between molecules, Chapman and Enskog as described by Chapman and**
**Cowling1 derived the following expression for the binary diflfusivity at low pressures:**

**(2.3-14)**

**In Eq. (2.3-14) DA B is in m2****/s, 7\ is in K, p is in atm, and a****AB is in A . The dimensionless collision**
**integral for diffusion Q****D**** is a function of the intermolecular potential chosen to represent the force field**
**between molecules as well as of temperature. '•l3 The Lennard-Jones 6-12 potential is most commonly used**
**for calculation of Q****D**** and extensive tabulations as a function kT/e****AB**** are available.t3< u A small tabulation**
**is included in Table 2.3-2, while Table 2.3-3 provides values of intermolecular diameters a,- and energy****parameters €, for some common gases. Empirical methods for estimating a, and €/ from other physical****property data are available.10**** The combining rules for the Lennard-Jones potential are a****AB**** = (aA + <JB)/2**
**and eAB**** = (e****A****e****B****)****t/2****. Equation (2.3-14) represents well the diffiisivities of binary gas pairs that have ****ap-proximately spherical force fields and are only slightly polar. Use of a different intermolecular potential**
**function for polar gases has been suggested.>4 There also are no satisfactory theories and little data available**
**for the diffiisivities of gases at pressures greater than about 10 atm.**

**A number of empirical correlations for gaseous diffusion coefficients at low pressures based **
**approxi-mately on the ideal gas form have been proposed.15"19 According to Reid et al.10 the equation of Fuller,**
**Schettler, and Giddings19 appears to be the best, generally predicting diffusivities of nonpolar gases within**
**10%, although the temperature dependence may be underestimated.**

**TABLE 2.3-3 Lennard-Jones Parameters for Some**
**Common Gases**
***77eAB**
**0.30**
**0.50**
**0.75**
**1.00**
**1.50**
**2.00**
**3.00**
**4.00**
**5.0**
**10**
**20**
**50**
**100**
**Q0**
2.662
2.066
1.667
1.439
1.198
1.075
0.9490
0.8836
0.8422
0.7424
0.6640
0.5756
0.5170
Gas
Air
CO
CO2
CH4

C2H4

C3H8

[image:17.396.107.289.476.612.2]**(2.3-15)**

**Equation (2.3-15) is a dimensional equation with D****AB**** in m2****/s, Tin K, and p in atm. This equation must****make use of the diffusion volumes that are tabulated for simple molecules and computed from atomic**
**volumes for more complex molecules. Table 2.3-4 provides some of these values. In summary, it may be**
**said that gaseous diffusion coefficients at low or modest pressures can be readily estimated from either Eq.**
**(2.3-14) or (2.3-15) if experimental data are not available.**

EXAMPLE 2.3-2

**Estimate the diffusivity of a CO2-O2 mixture at 200C and 1 atm and compare it to the experimental value**
**in Table 2.3-1. Repeat the calculation for 1500 K where the experimental value is 2.4 x 10"4 m2/s (2.6**
**x 10~3 ft2/s).**

**Solution. Using Eq. (2.3-14) and data from Table 2.3-3,**

**At 293 K (from Table 2.3-2),**

**At 1500 K,**

**Using Eq. (2.3-15) and data from Table 2.3-4**

**TABLE 2.3-4 Atomic Volumes for Use in Predicting Diffusion Coefficients by the Method of**
**Fuller et al.**

**Atomic and Structural Diffusion Volume Increments**

**C**
**H**
**O**
**N**

**16.5**
**1.98**
**5.48**
**5.69**

**Cl**
**S**

**Aromatic ring**
**Heterocyclic ring**

**19.5**
**17.0**
**- 2 0 . 2**
**- 2 0 . 2**

**Diffusion Volumes for Simple Molecules**

**H2**
**He**
**N2**
**O2**
**Air**
**Ar**
**CO**

**7.07**
**2.88**
**17.9**
**16.6**
**20.1**
**16.1**
**18.9**

**CO2**
**N2O**
**NH3**
**H2O**
**SF6**
**Cl2**
**SO2**

**26.9**
**35.9**
**14.9**
**12.7**
**69.7**
**37.7**
**41.1**

**At 293 K,**

**At 1500 K,**

**LIQUIDS**

**Diffusion coefficients in liquids are of the order of 10~8 ft2/s (IO"9 m2/s) unless the solution is highly**
**viscous or the solute has a very high molecular weight. Table 2.3-5 presents a few experimental diffusion**
**coefficients for liquids at room temperature in dilute solution. Ertl and Dullien,20 Johnson and Babb,21 and**
**Himmelblau22 provide extensive tabulations of diffusion coefficients in liquids. It is probably safe to say**
**that most of the reported experimental diffiisivities were computed based on Fick's Second Law without**
**consideration of whether or not the system was thermodynamically ideal. Since the binary diffusion **
**coef-ficient in liquids may vary strongly with composition, tabulations and predictive equations usually deal**
**with the diffusivity of A at infinite dilution in B, D£B, and the diffusivity of B at infinite dilution in A,**
**D |A. Separate consideration is then given to the variation of the diffusivity with composition.**

**Hydrodynamic theories for prediction of liquid-phase diffusion coefficients at infinite dilution are **
**rep-resented by the Stokes-Einstein equation23 which views the diffusion process as the motion of a spherical**
**solute molecule A through a continuum made up of solvent molecules B.**

**TABLE 2.3-5 Diffusion Coefficients in Solutions at Low Solute Concentrations Near Room**
**Temperature**

**Solute**

**CH3OH**
**C2H5OH**
**C2H4O**
**Ethylene glycol**
**Glycerol**
**Ethylene**
**Toluene**
**O2**
**H2**
**CO2**
**NH3**
**H2O**
**H2O**
**H2O**
**CH3OH**
**n-Hexyl alcohol**
**H2O**

**Glycerol**
**C2H4O**
**C6H12**
**C2H5OH**
**CO2**
**C3H8**
**CO2**
**O2**
**C2H4O2**
**H2O**

**Ribonuclease (MW = 13,683)**
**Albumin (MW = 65,000)**
**Myosin (MW = 493,000)**
**Polystyrene (MW = 414,000)**
**Polystyrene (MW = 50,000)**
**Polystyrene (MW = 180,000)**
**Polystyrene (MW = 402,000)**
**Polyacrylonitrile (MW = 127,000)**

**Solvent**

**H2O**
**H2O**
**H2O**
**H2O**
**H2O**
**H2O**
**H2O**
**H2O**
**H2O**
**H2O**
**H2O**
**C2H5OH**
**Ethylene glycol**
**Glycerol**
**Glycerol**
**Glycerol**
**C2H4O**
**C2H5OH**
**CHCl3**
**C6H6**
**C6H6**
**«-C6H14**
**W-C6H14**
**C2H5OH**
**C2H5OH**
**Ethyl acetate**
**Ethyl acetate**
**Aqueous buffer**
**Aqueous buffer**
**Aqueous buffer**
**Cyclohexane**
**Cyclohexanone**
**Cyclohexanone**
**Cyclohexanone**
**Dimethylformamide**

**T(0K)**

**283**
**283**
**293**
**293**
**293**
**298**
**296**
**298**
**293**
**298**
**285**
**298**
**298**
**293**
**294**
**298**
**298**
**293**
**298**
**298**
**288**
**298**
**298**
**290**
**303**
**293**
**298**
**293**
**293**
**293**
**308**
**298**
**298**
**298**
**298**

**D°****AB**** x IO9 (m2/sec)**

**(2.3-16)**

**where fi****B**** is the viscosity of the solvent and d****A**** is the diameter of the solute molecule. The most widely**
**used correlation for DAB is that of Wilke and Chang24 which represents an empirical modification of Eq.**
**(2.3-16) to fit available experimental data:**

**(2.3-17)**

**where [VX)****0 6**** is the molar volume of pure A at its normal boiling point (cm****3/mol) and <£ is an association**
**factor for the solvent. The constant in Eq. (2.3-17) is not dimensionless so the viscosity must be expressed**
**in centipoise (cP) and the temperature in K to obtain D in units of m****2/s. Wilke and Chang gave empirical**
**values of <f> ranging from 2.6 for water and 1.9 for methanol to 1.0 for nonassociated solvents. The****temperature dependence of the diflfusivity in Eq. (2.3-17) is primarily represented by the temperature**
**dependence of the viscosity. Thus, the group D****AB****fi****B****/T may be assumed constant over modest temperature****ranges. Over wider temperature ranges or when viscosity data are unavailable, the diffiisivity is conveniently**
**correlated by an equation of the Arrhenius form:**

**(2.3-18)**

**Figure 2.3-3 illustrates this correlation for a few systems. The activation energy E****a**** for a diffusion process**
**is typically a few kilocalories per mole.**

**After comparing a number of correlations for liquid-phase diffusivities, Reid et al.10 conclude that the**
**Wilke-Chang correlation is to be preferred for estimating infinite-dilution coefficients of **
**low-molecular-weight solutes in nonpolar and nonviscous solvents. For highly viscous solvents the Wilke-Chang **

**come-Curve Solute Solvent**

**(A) (B) SN**

**1 Benzene n-Heptane**
**2 n- Heptane Benzene****3 Methane Water**
**4 Vinyl chloride Water**
**5 n-Butane Water**

**lation may underestimate the diffusivity by as much as an order of magnitude. The correlation of Gainer**
**and Metzner,25 based on Absolute Rate Theory, is quite successful at predicting diffusion coefficients in**
**viscous liquids.**

**The application of the Wilke-Chang correlation to the diffusion of solutes that would be gases if they**
**existed in a pure state at the temperature of the solution has yielded conflicting results. Wise and Houghton26**
**and Shrier27 found the diffusivities to be significantly underestimated while Himmelblau22 found good**
**agreement between the correlation and literature values. Witherspoon and Bonoli28 found agreement between**
**their measurements of the diffusivity of hydrocarbon gases in water and the Wilke-Chang equation. **
**Ak-german and Gainer29 presented a method based on Absolute Reaction Rate Theory to predict the diffusion**
**coefficients of gases in solution.**

**Composition Dependence of Z)****AB. The diffusion coefficient of a binary solution is a function of**
**composition. If the true driving force for ordinary diffusion is the isothermal, isobaric gradient of chemical**
**potential, then the binary difrusivity can be written in the form**

**(2.3-19)**

**where 7A**** is the activity coefficient at mole fraction x****A****. This result, which suggests that the diffusivity can****be corrected by a thermodynamic activity term, has been successful, in some cases, in accounting for much**
**of the composition dependence of the diffusion coefficient (Fig. 2.3-4).**

**Vignes30 correlated the composition dependence of binary diffusion coefficients in terms of their **
**infinite-dilution values and this thermodynamic correction factor.**

**(2.3-20)**

**Some representative curves from his paper are also shown in Fig. 2.3-4. The correlation appears to be**
**quite successful unless the mixtures are strongly associating (e.g., CH3Cl-acetone). Leffler and Cullinan31**
**have provided a derivation of Eq. (2.3-20) based on Absolute Rate Theory while Gainer32 has used Absolute**
**Rate Theory to provide a different correlation for the composition dependence of binary diffusion ****coeffi-cients.**

**Multicomponent Effects. Only limited experimental data are available for multicomponent diffusion****in liquids. The binary correlations are sometimes employed for the case of a solute diffusing through a**
**mixed solvent of uniform composition.533 It is clear that thermodynamic nonidealities in multicomponent**
**systems can cause significant effects. The reader is referred to Cussler's book4 for a discussion of available**
**experimental information on diffusion in multicomponent systems.**

**Ionic Solutions. Since salts are partially or completely dissociated into ions in aqueous solution, the****diffusion coefficient of the salt must be related to those of its component ions. Comprehensive discussions**
**of electrolytic solutions may be found in a number of references.34****" For a single salt in solution a molecular****diffusivity can be assigned to the salt and is calculable at infinite dilution from the ionic conductances of**
**the ions:37**

**(2.3-21)**

**where Z+1_ is the valence and X+ _ the ionic conductance of positive or negative ions. No adequate theory**
**exists for the effect of increasing salt concentration on the diffusion coefficient, although it is generally**
**found that the diffusion coefficient first decreases with increasing salt concentration and then increases.**
**Diffusion of ions in mixed electrolyte solutions is a complex phenomenon involving transport owing to**
**both composition and electrical potential gradients. Even without an applied potential the differences of**
**mobility among the ions in solution give rise to an electrical potential. Mixed electrolytic solutions are in**
**fact multicomponent solutions so that a number of diffusion coefficients are required to describe the system**
**property. Indeed, some interesting multicomponent effects are the result of electrostatic interactions.4**

**FIGURE 2.3-4 Composition dependence of binary diffusion coefficients in liquids, (in units of cm2****/s): (a)****acetone (A)-CCl4****, (b) methanol (A)-H****2O (c) ethanol (A)-H2****O, (d) acetone (A)-CH****3Cl. Reprinted with**
**permission from A. Vignes, lnd. Eng. Chem. Fundam., 5, 198 (1966). Copyright 1966 American Chemical****Society.**

**Macromolecular Solutions. In these systems two diffusion phenomena could be of importance: the****mutual diffusion coefficient of a macromolecule and its solvent and the influence of a macromolecule in a**
**solution on the diffusion rate of smaller solutes. As can be seen in Table 2.3-5 the mutual diffusion**
**coefficients of proteins in aqueous solution or of a synthetic polymer in a solvent are in the range of 10~l 0**
**-10"!l m2/s at room temperature. Experimental methods and theoretical concepts dealing with the diffusion**
**coefficients of macromolecules can be found in Tanford.41 Representative data are those of Paul42 and**
**Osmers and Metzner.43 Duda and Vrentas44 have reviewed the theory and experiments dealing with the**
**mutual diffusion coefficient of amorphous polymers and low molecular weight solvents. The difftisivities**
**of macromolecules appear to vary with the square root of the molecular weight of the solute, although**
**solute shape also has an effect.**

**The difrusivities of low molecular weight solutes in polymer solutions do not appear to follow a**
**predictable pattern as a function of polymer concentration. It is clear that the results cannot be predicted**
**with the Wilke-Chang equation by merely using the increased viscosity of the solution. Diffusivities have**
**been reported to increase, decrease, or go through a maximum with increasing polymer concentration**
**despite solution viscosity increases.43"47 Li and Gainer47 measured the diffusivities of four solutes in seven**

**(a)****(b)**

**(C)**

**aData from Ref. 38-40.**

**polymer solutions and developed a predictive theory that could account for both increases and decreases**
**in diffusion coefficients. Navari et al.48 examined the diffusion coefficients of small solutes in mixed protein**
**solutions, for example, O2 or glucose in solutions containing various amounts of albumin and gamma**
**globin, and concluded that the ratio of diffusivity in the solution to that in the pure solvent was independent**
**of the solute. They also observed some rather dramatic decreases in diffusivity at particular protein **
**com-positions. At the present time, it is advisable to obtain experimental data for the system of interest because**
**an understanding of all the interactions that occur in such systems does not exist. Although these are**
**multicomponent solutions, studies to date have only considered the situation where there is no gradient in**
**polymer concentration.**

**EXAMPLE 2.3-3**

**It is desired to estimate the diffusivity of a thymol (C,oH,4)-salol (C13H10O3) organic melt at 27°C. The**
**measured viscosities of pure thymol and salol at 270C are 11 and 29 cP, respectively.**

**Solution Although the liquids are somewhat viscous, first estimate the infinite-dilution diffusivities****using the Wilke-Chang correlation, Eq. (2.3-17). The molar volumes at the boiling point are estimated by**
**Shroeder's method10 as**

**It is not clear what the association factor <i> should be for these mixtures. Aromatic mixtures have****10 0 «**
**0.7 while hydrogen bonded systems have <f> « 1.0. Use <f> « 1.0**

**A single experimental measurement49 of the average diffusivity over the whole composition range gave**
**a value of 7.4 X 10"!1 m2/s. These solutions do form nearly ideal liquid mixtures so the diffusivity of a**
**50/50 mixture could be estimated from Vignes' correlation:**

**The infinite dilution diffusivities can also be estimated using the Gainer-Metzner correlation25 with the**
**results: £>!r = 8.0 x 10"l ! m2****/s, Df****9**** = 6.7 x 10"!1 m2****/s and for the 50/50 mixture, D****ST**** = 7.3 x**
**10" u m2/s.**

**TABLE 2.3-6 Diffusion Coefficients of Gases in Aqueous Electrolyte Solutions' at 25 0C**

**Gas**
**O2**
**O2**
**O2**
**O2**
**O2**
**O2**
**H2**
**H2**
**H2**
**H2**
**H2**
**CH4**
**CH4**
**CH4**
**CO2**
**CO2**
**CO2**
**CO2**
**CO2**
**Electrolyte**
**None**
**NaCl**
**KCl**
**KCl**
**KCl**
**KCl**
**None**
**KCl**
**KCl**
**MgSO4**
**MgSO4**
**None**
**MgSO4**
**MgSO4**
**None**
**NaCl**
**NaCl**
**MgSO4**
**MgSO4**
**Concentration (mol/L)**
**0.15**
**0.01**
**0.10**
**1.0**
**4.0**
**1**
**3**
**0.05**
**2.0**
**0.5**
**2.0**
**1.04**
**3.77**
**0.20**
**0.97**

**D j8 X 109 (m2/s)**

**Both methods give values in reasonable agreement with experiment although the Wilke-Chang method**
**does have the adjustable parameter <f>. The Gainer-Metzner correlation is to be preferred as the viscosity****of a solution liquid increases.**

**SOLIDS**

**Diffusion rates of ions, atoms, or gases through crystalline solids often govern the rates of important **
**solid-state reactions and phase changes such as the formation of oxide films. Extensive discussions of these**
**phenomena are given by Barrer50 and Jost,2 including tabulations of diffusion coefficient data. Recent**
**developments can be found in Diffusion in Solids by Nowick and Burton.****51 Since the solid phase can exist**
**over a large temperature range, diffusivity values in solids can vary over an enormous range as can be seen**
**from the representative data shown in Table 2.3-7. An Arrhenius form for diffusion in metals (ZV"^*7)**
**has been shown to correlate diffusion data over many orders of magnitude of the diffusivity and hundreds**
**of kelvins. Some representative data taken from Birchenairs text52 for self-diffusion in various metals are**
**shown in Fig. 2.3-5.**

**In metal alloys forming solid solutions over significant composition ranges, the binary diffusivity varies**
**with composition. Figure 2.3-6 shows some typical data for diffusion in the Fe-Ni system at 1185°C.**
**Vignes and Birchenall54 have presented a correlation for the concentration dependence in binary alloys**
**similar to Eq. (2.3-18) for liquids but they required an additional factor involving the melting temperature**
**of the alloy to obtain a satisfactory correlation.**

**POROUS SOLIDS**

**The diffusion rates of fluids, particularly gases, through porous solid materials are of particular interest**
**because of the common practice of using porous solids as catalysts and adsorbents. Diffusional transport**
**rates within the pores may limit the rates by which such processes occur.**

**Satterfield's book, Mass Transfer in Heterogeneous Catalysis*****5**** is an excellent source of information**
**on diffusional rates in porous media. Transport in porous solids can occur by ordinary diffusion, by Knudsen**
**diffusion, and by surface diffusion of adsorbed species. Ordinary diffusion simply represents the transport**
**of material by diffusion in the gas phase within the pores. Since the whole cross-sectional area of the solid**
**is not available for diffusion and the pores do not run straight through the material but consist of a randomly**
**connected network, the effective diffusivity is given by**

**(2.3-22)**

**where Z>AB**** is the gas phase diffusion coefficient, € is the porosity of the solid, and r is the tortuosity. The****tortuosity factor, accounting for the random direction of pores, can vary widely but typical values are in**
**the range of 1 to 10.55**

**When the pressure is lowered or the pore radius is small, the mean free path of a gas molecule becomes**
**comparable to that of the pore radius. In this situation, called the Knudsen diffusion regime, collisions**
**between gas molecules and the wall are as important as those among molecules. The Knudsen diffusion**
**coefficient is given by the expression55**

**(2.3-23)**

**where the effective pore radius is represented by 2dp^****R**** and p»$****h**** is the total surface per volume of solid.**
**The Knudsen diffusion coefficient is independent of pressure and only a weak function of temperature,**

**TABLE 2.3-7 Some Binary Diffusion Coefficients in Solids**

**System**

**H2 in Fe**
**H2 in Fe**
**He in Pyrex**
**He in Pyrex**
**Al in Cu**
**Al in Cu**
**C in Fe**
**C in Fe**
**Ni (10%) in Fe**
**Ni (10%) in Fe**

**Temperature (0K)**

**283**
**373**
**293**
**773**
**293**
**1123**
**1073**
**1373**
**1293**
**1568**

**DAB x 1012 (m2/sec)**

**0.16**
**12.4**

**0.0045**
**2**

**1.3 x 10"26**
**0.22**

**1.5**
**45**

**FIGURE 2.3-5 Temperature dependence of diffusion coefficients in metals. Reprinted with permission**
**from Ref. 52.**

## Diffusio

## n coefficien

## t

t## (cm

2

## /$

## )

**No**

**Cu**
**;Ni**
**,Co**
**^Fe**

*Body*

* -centered*

* cubic*

**Pb**

*Face*

* - centered*

* cubic*

Ge

*Diamond*
*cubic*

**In in InSb**
T l

**FIGURE 2.3-6 Composition dependence of diffusion coefficient in iron-nickel alloys at 11850C.53**

**while the diffusion coefficient in a gas is inversely proportional to pressure and varies approximately as**
**T****1****'****75****. Whether ordinary or Knudsen diffusion governs the transport in the gas depends on the magnitudes****of DAB cff**** and D****K**** for a given situation. In the transition region where both mechanisms are of importance,**
**it has been shown that an overall diffusion coefficient is given approximately by55*56**

(2.3-24)

**Satterfield55 summarizes experimental data illustrating the transition from one regime to the other as a**
**function of pore size, pressure, and chemical composition of the gases involved.**

**A third transport mechanism operating in parallel with the above mechanisms in a porous solid occurs**
**when the gaseous species adsorbs and migrates along the surface of the pores in the direction of decreasing**
**surface concentration. For this to be a significant mechanism sufficient surface coverage must be achieved,**
**yet the absorbed species cannot be tightly bound or little motion can occur. The total flux of a species**
**owing to a concentration gradient in a porous solid can be represented by an expression of the form**

**(2.3-25)**

**if the relationship between adsorbed and gaseous concentrations can be represented by a linear isotherm,**
**CAS**** = KC****A****. The surface diffusion coefficient (D****s****) typically is in the range of 10~****7-10~n m2/s for physically**
**adsorbed species. Sladek57 has developed an extensive correlation for surface diffusion coefficients of both**
**physically and chemically adsorbed species.**

**Zeolites of controlled pore sizes of the order of a few angstroms have become popular catalysts and**
**adsorbents because of their ability to discriminate effectively the shape and size differences among **
**mole-cules. Weisz et al.58 describe applications of these shape-selective catalysts in oxidation of normal paraffins**
**and olefins in the presence of the branched isomers. Observed diffusion coefficients in these zeolites are**
**often found to be quite small (10~14 m2/s) but precise interpretations of the measurements and mechanisms**
**of transport in zeolites are not available as yet.**

POLYMERS