Lecture notes Mass transfer

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Part V Mass Transfer 59511. An introduction to mass transfer The edge of a colossal jungle, so dark-green as to be almost black, fringed with white surf, ran straight, like a ruled line, far, far away along a blue sea whose glitter was blurred by a creeping mist. The sun was fierce, the land seemed to glisten and drip with steam. Heart of Darkness, Joseph Conrad, 1902 11.1 Introduction We have, so far, dealt with heat transfer by convection, radiation, and diffusion (which we have been calling conduction). We have dealt only with situations in which heat passes through, or is carried by, a single medium. Many heat transfer processes, however, occur in mixtures of more than one substance. A wall exposed to a hot air stream may be cooled evaporatively by bleeding water through its surface. Water vapor may condense out of damp air onto cool surfaces. Heat will flow through an air-water mixture in these situations, but water vapor will diffuse or convect through air as well. This sort of transport of one substance relative to another is called mass transfer; it did not occur in the single-component processes of the preceding chapters. In this chapter, we study mass transfer phenomena with an eye toward predicting heat and mass transfer rates in situations like those just mentioned. During mass transfer processes, an individual chemical species trav- els from regions where it has a high concentration to regions where it has a low concentration. When liquid water is exposed to a dry air stream, its vapor pressure may produce a comparatively high concentration of wa- ter vapor in the air near the water surface. The concentration difference between the water vapor near the surface and that in the air stream will drive the diffusion of vapor into the air stream. We call this evaporation. 597598 An introduction to mass transfer §11.1 Figure 11.1 Schematic diagram of a natural-draft cooling tower at the Rancho Seco nuclear power plant. (From 11.1, courtesy of W. C. Reynolds.) In this and other respects, mass transfer is analogous to heat trans- fer. Just as thermal energy diffuses from regions of high temperature to regions of low temperature (following the temperature gradient), the mass of one species diffuses from regions high concentration to regions of low concentration (following its concentration gradient.) Just as the diffusional (or conductive) heat flux is directly proportional to a temper- ature gradient, so the diffusional mass flux of a species is often directly proportional to its concentration gradient; this is called Fick’s law of dif- fusion. Just as conservation of energy and Fourier’s law lead to equations for the convection and diffusion of heat, conservation of mass and Fick’s law lead to equations for the convection and diffusion of species in a mixture. The great similarity of the equations of heat convection and diffusion to those of mass convection and diffusion extends to the use of con- vective mass transfer coefficients, which, like heat transfer coefficients, relate convective fluxes to concentration differences. In fact, with sim- ple modifications, the heat transfer coefficients of previous chapters may often be applied to mass transfer calculations.§11.1 Introduction 599 Figure 11.2 A mechanical-draft cooling tower. The fans are located within the cylindrical housings at the top. Air is drawn in through the louvres on the side. Mass transfer, by its very nature, is intimately concerned with mix- tures of chemical species. We begin this chapter by learning how to quan- tify the concentration of chemical species and by defining rates of move- ment of species. We make frequent reference to an arbitrary “speciesi,” theith component of a mixture ofN different species. These definitions may remind you of your first course in chemistry. We also spend some time, in Section 11.4, discussing how to calculate the transport properties of mixtures, such as diffusion coefficients and viscosities. Consider a typical technology that is dominated by mass transfer pro- cesses. Figure 11.1 shows a huge cooling tower used to cool the water leaving power plant condensers or other large heat exchangers. It is es- sentially an empty shell, at the bottom of which are arrays of cement boards or plastic louvres over which is sprayed the hot water to be cooled. The hot water runs down this packing, and a small portion of it evapo- rates into cool air that enters the tower from below. The remaining water, having been cooled by the evaporation, falls to the bottom, where it is collected and recirculated. The temperature of the air rises as it absorbs the warm vapor and, in600 An introduction to mass transfer §11.2 the natural-draft form of cooling tower shown, the upper portion of the tower acts as an enormous chimney through which the warm, moist air buoys, pulling in cool air at the base. In a mechanical-draft cooling tower, fans are used to pull air through the packing. Mechanical-draft towers are much shorter and can sometimes be seen on the roofs of buildings (Fig. 11.2). The working mass transfer process in a cooling tower is the evapora- tion of water into air. The rate of evaporation depends on the tempera- ture and humidity of the incoming air, the feed-water temperature, and the air-flow characteristics of the tower and the packing. When the air flow is buoyancy-driven, the flow rates are directly coupled. Thus, mass transfer lies at the core of the complex design of a cooling tower. 11.2 Mixture compositions and species fluxes The composition of mixtures A mixture of various chemical species displays its own density, molecular weight, and other overall thermodynamic properties. These properties depend on the types and relative amounts of the component substances, which may vary from point to point in the mixture. To determine the local properties of a mixture, we must identify the local proportion of each species composing the mixture. One way to describe the amount of a particular species in a mixture is by the mass of that species per unit volume, known as the partial density. The mass of speciesi in a small volume of mixture, in kg, divided by that 3 volume, in m , is the partial density,ρ , for that species, in kg ofi per i 3 m . The composition of the mixture may be describe by stating the partial density of each of its components. The mass density of the mixture itself, ρ, is the total mass of all species per unit volume; therefore, ρ= ρ (11.1) i i The relative amount of speciesi in the mixture may be described by the mass ofi per unit mass of the mixture, which is simplyρ/ρ. This i ratio is called the mass fraction,m : i ρ mass of speciesi i m ≡ = (11.2) i ρ mass of mixture§11.2 Mixture compositions and species fluxes 601 This definition leads to the following two results: m = ρ/ρ= 1 and 0m  1 (11.3) i i i i i 3 The molar concentration of speciesi in kmol/m ,c , expresses con- i centration in terms of moles rather than mass. If M is the molecular i weight of speciesi in kg/kmol, then ρ moles ofi i c ≡ = (11.4) i M volume i The molar concentration of the mixture,c, is the total number of moles for all species per unit volume; thus, c= c. (11.5) i i The mole fraction of speciesi,x , is the number of moles ofi per mole i of mixture: c moles ofi i x ≡ = (11.6) i c mole of mixture Just as for the mass fraction, it follows for mole fraction that x = c/c= 1 and 0x  1 (11.7) i i i i i The molecular weight of the mixture is the number of kg of mixture per kmol of mixture: M ≡ρ/c. By using eqns. (11.1), (11.4), and (11.6) and (11.5), (11.4), and (11.2), respectively,M may be written in terms of either mole or mass fraction 1 m i M= xM or = (11.8) i i M M i i i Mole fraction may be converted to mass fraction using the following re- lations (derived in Problem 11.1): xM xM Mm m/M i i i i i i i m = = and x = = (11.9) i i M x M M m /M k k i k k k k In some circumstances, such as kinetic theory calculations, one works directly with the number of molecules ofi per unit volume. This number density,N , is given by i N =N c (11.10) i A i 26 whereN is Avogadro’s number, 6.02214× 10 molecules/kmol. A602 An introduction to mass transfer §11.2 Ideal gases The relations we have developed so far involve densities and concentra- tions that vary in as yet unknown ways with temperature or pressure. To get a more useful, though more restrictive, set of results, we now com- bine the preceding relations with the ideal gas law. For any individual component,i, we may write the partial pressure,p , exterted byi as: i p =ρRT (11.11) i i i In eqn. (11.11),R is the ideal gas constant for speciesi: i ◦ R R ≡ (11.12) i M i ◦ whereR is the universal gas constant, 8314.472 J/kmol· K. Equation (11.11) may alternatively be written in terms ofc : i   ◦ R p =ρRT =(Mc) T i i i i i M i ◦ =c R T (11.13) i Equations (11.5) and (11.13) can be used to relatec top andT p p i c= c = = (11.14) i ◦ ◦ R T R T i i ◦ Multiplying the last two parts of eqn. (11.14)byR T yields Dalton’s law 1 of partial pressures, p= p (11.15) i i Finally, we combine eqns. (11.6), (11.13), and (11.15) to obtain a very useful relationship betweenx andp : i i c p p i i i x = = = (11.16) i ◦ c cR T p in which the last two equalities are restricted to ideal gases. 1 Dalton’s law (1801) is an empirical principle (not a deduced result) in classical thermodynamics. It can be deduced from molecular principles, however. We built the appropriate molecular principles into our development when we assumed eqn. (11.11) to be true. The reason that eqn. (11.11) is true is that ideal gas molecules occupy a mixture without influencing one another.§11.2 Mixture compositions and species fluxes 603 Example 11.1 The most important mixture that we deal with is air. It has the fol- lowing composition: Species Mass Fraction N 0.7556 2 O 0.2315 2 Ar 0.01289 trace gases 0.01 Determinex ,p ,c , andρ for air at 1 atm. O O O O 2 2 2 2 Solution. To make these calcuations, we need the molecular weights, which are given in Table 11.2 on page 616. We can start by checking the value ofM , using the second of eqns. (11.8): air   −1 m m m N O Ar 2 2 M = + + air M M M N O Ar 2 2   −1 0.7556 0.2315 0.01289 = + + 28.02 kg/kmol 32.00 kg/kmol 39.95 kg/kmol = 28.97 kg/kmol We may calculate the mole fraction using the second of eqns. (11.9) m M (0.2315)(28.97 kg/kmol) O 2 x = = = 0.2095 O 2 M 32.00 kg/kmol O 2 The partial pressure of oxygen in air at 1 atm is eqn. (11.16) 4 p =x p=(0.2095)(101, 325 Pa)= 2.123× 10 Pa O O 2 2 We may now obtainc from eqn. (11.13): O 2 p O 2 c = O 2 ◦ R T 4 =(2.123× 10 Pa) (300 K)(8314.5J/kmol·K) 3 = 0.008510 kmol/m Finally, eqn. (11.4) gives the partial density 3 ρ =c M =(0.008510 kmol/m )(32.00 kg/kmol) O O O 2 2 2 3 = 0.2723 kg/m604 An introduction to mass transfer §11.2 Velocities and fluxes Each species in a mixture undergoing a mass transfer process will have  an species-average velocity,v , which can be different for each species in i the mixture, as suggested by Fig. 11.3. We may obtain the mass-average 2 velocity, v, for the entire mixture from the species average velocities using the formula ρv= ρv. (11.17) i i i This equation is essentially a local calculation of the mixture’s net mo- mentum per unit volume. We refer toρv as the mixture’s mass flux,n,  2 ˙ and we call its scalar magnitudem ; each has units of kg/m ·s. Likewise, the mass flux of speciesi is n =ρv (11.18) i i i and, from eqn. (11.17), we see that the mixture’s mass flux equals the sum of all species’ mass fluxes n= n =ρv (11.19) i i Since each species diffusing through a mixture has some velocity rel-  ative to the mixture’s mass-average velocity, the diffusional mass flux,j , i of a species relative to the mixture’s mean flow may be identified:   j =ρ v −v . (11.20) i i i The total mass flux of theith species,n , includes both this diffusional i mass flux and bulk convection by the mean flow, as is easily shown:  n =ρv =ρv+ρ v −v i i i i i i  =ρv+j i i (11.21)  = mn + j i i       convection diffusion 2 The mass average velocity,v, given by eqn. (11.17) is identical to the fluid velocity, u, used in previous chapters. This is apparent if one applies eqn. (11.17) to a “mix- ture” composed of only one species. We use the symbolv here becausev is the more common notation in the mass transfer literature.§11.2 Mixture compositions and species fluxes 605 Figure 11.3 Molecules of different species in a mixture moving with different average velocities. The velocity v is the average over all molecules of i speciesi. Although the convective transport contribution is fully determined as soon as we know the velocity field and partial densities, the causes of diffusion need further discussion, which we defer to Section 11.3. Combining eqns. (11.19) and (11.21), we find that    n= n = ρv+ j =ρv+ j =n+ j i i i i i i i i i i Hence  j = 0 (11.22) i i Diffusional mass fluxes must sum to zero because they are each defined relative to the mean mass flux. Velocities may also be stated in molar terms. The mole flux of the 2   ith species, N,is cv , in kmol/m · s. The mixture’s mole flux, N,is i i i obtained by summing over all species ∗   N= N = cv =cv (11.23) i i i i i ∗ where we define the mole-average velocity,v , as shown. The last flux ∗  we define is the diffusional mole flux,J : i  ∗ ∗  J =c v −v (11.24) i i i606 An introduction to mass transfer §11.2 It may be shown, using these definitions, that ∗    N =xN+J (11.25) i i i Substitution of eqn. (11.25) into eqn. (11.23) gives ∗ ∗       N= N =N x + J =N+ J i i i i i i i i so that ∗  J = 0. (11.26) i i ∗   Thus, both theJ ’s and thej ’s sum to zero. i i Example 11.2 At low temperatures, carbon oxidizes (burns) in air through the sur- face reaction: C+ O → CO . Figure 11.4 shows the carbon-air in- 2 2 terface in a coordinate system that moves into the stationary carbon at the same speed that the carbon burns away—as though the ob- server were seated on the moving interface. Oxygen flows toward the carbon surface and carbon dioxide flows away, with a net flow of carbon through the interface. If the system is at steady state and, if a separate analysis shows that carbon is consumed at the rate of 2 0.00241 kg/m ·s, find the mass and mole fluxes through an imagi- nary surface,s, that stays close to the gas side of the interface. For this case, concentrations at thes-surface turn out to bem = 0.20, O ,s 2 3 m = 0.052, andρ = 0.29 kg/m . CO ,s s 2 Solution. The mass balance for the reaction is 12.0kg C+ 32.0kg O → 44.0kg CO 2 2 Since carbon flows through a second imaginary surface, u, moving through the stationary carbon just below the interface, the mass fluxes are related by 12 12 n =− n = n C,u O ,s CO ,s 2 2 32 44 The minus sign arises because the O flow is opposite the C and CO 2 2 flows, as shown in Figure 11.4. In steady state, if we apply mass§11.2 Mixture compositions and species fluxes 607 Figure 11.4 Low-temperature carbon oxidation. conservation to the control volume between theu ands surfaces, we find that the total mass flux entering theu-surface equals that leaving thes-surface 2 n =n +n = 0.00241 kg/m ·s C,u CO ,s O ,s 2 2 Hence, 32 2 2 n =− (0.00241 kg/m ·s)=−0.00643 kg/m ·s O ,s 2 12 44 2 2 n = (0.00241 kg/m ·s)= 0.00884 kg/m ·s CO ,s 2 12 To get the diffusional mass flux, we need species and mass average speeds from eqns. (11.18) and (11.19): 2 n −0.00643 kg/m ·s O ,s 2 v = = =−0.111 m/s O ,s 2 3 ρ 0.2(0.29 kg/m ) O ,s 2 2 n 0.00884 kg/m ·s CO ,s 2 v = = = 0.586 m/s CO ,s 2 3 ρ 0.052(0.29 kg/m ) CO ,s 2 2 1 (0.00884− 0.00643) kg/m ·s v = n = = 0.00831 m/s s i 3 ρ 0.29 kg/m s i Thus, from eqn. (11.20), ⎧ ⎨ 2  −0.00691 kg/m ·s for O 2 j =ρ v −v = i,s i,s i,s s 2 ⎩ 0.00871 kg/m ·s for CO 2608 An introduction to mass transfer §11.3 The diffusional mass fluxes,j , are very nearly equal to the species i,s mass fluxes,n . That is because the mass-average speed,v , is much i,s s less than the species speeds,v , in this case. Thus, the convective i,s contribution ton is much smaller than the diffusive contribution, i,s and mass transfer occurs primarily by diffusion. Note thatj and O ,s 2 j do not sum to zero because the other, nonreacting species in CO ,s 2 air must diffuse against the small convective velocity, v (see Sec- s tion 11.7). One mole of carbon surface reacts with one mole of O to form 2 one mole of CO . Thus, the mole fluxes of each species have the same 2 magnitude at the interface: n C,u 2 N =−N =N = = 0.000201 kmol/m ·s CO ,s O ,s C,u 2 2 M C ∗ The mole average velocity at thes-surface,v , is identically zero by s eqn. (11.23), sinceN +N = 0. The diffusional mole fluxes are CO ,s O ,s 2 2 ⎧ ⎨ 2  −0.000201 kmol/m ·s for O 2 ∗ ∗ J =c v −v =N = i,s i,s i,s s i,s 2 ⎩  0.000201 kmol/m ·s for CO 2 =0 These two diffusional mole fluxes sum to zero themselves because there is no convective mole flux for other species to diffuse against ∗ (i.e., for the other speciesJ = 0). i,s The reader may calculate the velocity of the interface fromn . c,u That calculation would show the interface to be receding so slowly that the velocities we calculate are almost equal to those that would be seen by a stationary observer. 11.3 Diffusion fluxes and Fick’s law When the composition of a mixture is nonuniform, the concentration gradient in any species,i, of the mixture provides a driving potential for the diffusion of that species. It flows from regions of high concentration to regions of low concentration—similar to the diffusion of heat from regions of high temperature to regions of low temperature. We have already noted in Section 2.1 that mass diffusion obeys Fick’s law  j =−ρD ∇m (11.27) i im i§11.3 Diffusion fluxes and Fick’s law 609 which is analogous to Fourier’s law. The constant of proportionality, ρD , between the local diffusive im mass flux of speciesi and the local concentration gradient ofi involves a physical property called the diffusion coefficient,D , for speciesi dif- im fusing in the mixturem. Like the thermal diffusivity,α, or the kinematic viscosity (a momentum diffusivity),ν, the mass diffusivityD has the im 2 units of m/s. These three diffusivities can form three dimensionless groups, among which is the Prandtl number: The Prandtl number, Pr≡ν/α 3 The Schmidt number, Sc≡ν/D (11.28) im 4 The Lewis number, Le≡α/D = Sc/Pr im Each of these groups compares the relative strength of two different dif- fusive processes. We make considerable use of the Schmidt number later in this chapter. When diffusion occurs in mixtures of only two species—so-called bi- nary mixtures—D reduces to the binary diffusion coefficient,D .In im 12 5 fact, the best-known kinetic models are for binary diffusion. In binary diffusion, species 1 has the same diffusivity through species 2 as does species 2 through species 1 (see Problem 11.5); in other words, D =D (11.29) 12 21 3 Ernst Schmidt (1892–1975) served successively as the professor of thermodynam- ics at the Technical Universities of Danzig, Braunschweig, and Munich (Chapter 6, foot- note 3). His many contributions to heat and mass transfer include the introduction of aluminum foil as radiation shielding, the first measurements of velocity and temper- ature fields in a natural convection boundary layer, and a once widely-used graphical procedure for solving unsteady heat conduction problems. He was among the first to develop the analogy between heat and mass transfer. 4 Warren K. Lewis (1882–1975) was a professor of chemical engineering at M.I.T. from 1910 to 1975 and headed the department throughout the 1920s. He defined the original paradigm of chemical engineering, that of “unit operations”, and, through his textbook with Walker and McAdams, Principles of Chemical Engineering, he laid the foundations of the discipline. He was a prolific inventor in the area of industrial chemistry, holding more than 80 patents. He also did important early work on simultaneous heat and mass transfer in connection with evaporation problems. 5 Actually, Fick’s Law is strictly valid only for binary mixtures. It can, however, of- ten be applied to multicomponent mixtures with an appropriate choice of D (see im Section 11.4).610 An introduction to mass transfer §11.3 A kinetic model of diffusion Diffusion coefficients depend upon composition, temperature, and pres- sure. Equations that predictD andD are given in Section 11.4. For 12 im now, let us see how Fick’s law arises from the same sort of elementary molecular kinetics that gave Fourier’s and Newton’s laws in Section 6.4. We consider a two-component dilute gas (one with a low density) in  which the moleculesA of one species are very similar to the moleculesA of a second species (as though some of the molecules of a pure gas had merely been labeled without changing their properties.) The resulting process is called self-diffusion. If we have a one-dimensional concentration distribution, as shown in Fig. 11.5, molecules of A diffuse down their concentration gradient in the x-direction. This process is entirely analogous to the transport of energy and momentum shown in Fig. 6.13. We take the temperature and pressure of the mixture (and thus its number density) to be uniform and the mass-average velocity to be zero. Individual molecules move at a speedC, which varies randomly from molecule to molecule and is called the thermal or peculiar speed. The average speed of the molecules isC. The average rate at which molecules cross the planex=x in either direction is proportional toNC, where 0 3 N is the number density (molecules/m ). Prior to crossing thex -plane, 0 the molecules travel a distance close to one mean free path,—call ita, wherea is a number on the order of one. The molecular flux travelling rightward acrossx , from its plane of 0 origin atx −a, then has a fraction of molecules ofA equal to the value 0 ofN /N atx −a. The leftward flux, fromx +a, has a fraction A 0 0 equal to the value ofN /N atx +a. Since the mass of a molecule of A 0 A isM /N (whereN is Avogadro’s number), the net mass flux in the A A A x-direction is then          M N N  A A A   j  =η NC − (11.30) A   x 0 N N N A x −a x +a 0 0 whereη is a constant of proportionality. SinceN /N changes little in a A distance of two mean free paths (in most real situations), we can expand the right side of eqn. (11.30) in a two-term Taylor series expansion about§11.3 Diffusion fluxes and Fick’s law 611 Figure 11.5 One-dimensional diffusion. x and obtain Fick’s law: 0       M d(N /N)  A A   j =η NC −2a A  x 0 N dx x A 0   dm A  =−2ηa(C)ρ (11.31)  dx x 0 (for details, see Problem 11.6). Thus, we identify  D =(2ηa)C (11.32) AA and Fick’s law takes the form dm A  j =−ρD (11.33) A AA dx The constant,ηa, in eqn. (11.32) can be fixed only with the help of a more detailed kinetic theory calculation 11.2, the result of which is given in Section 11.4. The choice ofj andm for the description of diffusion is really some- i i ∗ what arbitrary. The molar diffusion flux,J , and the mole fraction,x , i i are often used instead, in which case Fick’s law reads ∗  J =−cD ∇x (11.34) i im i Obtaining eqn. (11.34) from eqn. (11.27) for a binary mixture is left as an exercise (Problem 11.4).612 An introduction to mass transfer §11.3 Typical values of the diffusion coefficient Fick’s law works well in low density gases and in dilute liquid and solid solutions, but for concetrated liquid and solid solutions the diffusion co- efficient is found to vary with the concentration of the diffusing species. In part, the concentration dependence of those diffusion coefficients re- flects the inadequacy of the concentration gradient in representing the driving force for diffusion in nondilute solutions. Gradients in the chem- ical potential actually drive diffusion. In concentrated liquid or solid solutions, chemical potential gradients are not always equivalent to con- centration gradients 11.3, 11.4, 11.5. Table 11.1 lists some experimental values of the diffusion coefficient in binary gas mixtures and dilute liquid solutions. For gases, the diffu- −5 2 sion coefficient is typically on the order of 10 m /s near room tem- perature. For liquids, the diffusion coefficient is much smaller, on the −9 2 order of 10 m /s near room temperature. For both liquids and gases, the diffusion coefficient rises with increasing temperature. Typical dif- −20 fusion coefficients in solids (not listed) may range from about 10 to −9 2 about 10 m /s, depending upon what substances are involved and the temperature 11.6. The diffusion of water vapor through air is of particular technical importance, and it is therefore useful to have an empirical correlation specifically for that mixture:   2.072 T −10 D = 1.87× 10 for 282 K≤T ≤ 450 K (11.35) H O,air 2 p 2 whereD is in m /s,T is in kelvin, andp is in atm 11.7. The scatter H O,air 2 of the available data around this equation is about 10%. Coupled diffusion phenomena Mass diffusion can be driven by factors other than concentration gradi- ents, although the latter are of primary importance. For example, tem- perature gradients can induce mass diffusion in a process known as ther- mal diffusion or the Soret effect. The diffusional mass flux resulting from both temperature and concentration gradients in a binary gas mixture is then 11.2 M M 1 2  j =−ρD ∇m + k ∇ ln(T) (11.36) i 12 1 T 2 M§11.3 Diffusion fluxes and Fick’s law 613 Table 11.1 Typical diffusion coefficients for binary gas mix- tures at 1 atm and dilute liquid solutions 11.4. 2 Gas mixture T (K) D (m /s) 12 −5 air-carbon dioxide 276 1.42×10 air-ethanol 313 1.45 air-helium 276 6.24 air-napthalene 303 0.86 air-water 313 2.88 argon-helium 295 8.3 628 32.1 1068 81.0 2 (dilute solute, 1)-(liquid solvent, 2) T (K) D (m /s) 12 −9 ethanol-benzene 288 2.25×10 benzene-ethanol 298 1.81 water-ethanol 298 1.24 carbon dioxide-water 298 2.00 ethanol-water 288 1.00 methane-water 275 0.85 333 3.55 pyridene-water 288 0.58 wherek is called the thermal diffusion ratio and is generally quite small. T Thermal diffusion is occasionally used in chemical separation processes. Pressure gradients and body forces acting unequally on the different species can also cause diffusion. Again, these effects are normally small. A related phenomenon is the generation of a heat flux by a concentration gradient (as distinct from heat convected by diffusing mass), called the diffusion-thermo or Dufour effect. In this chapter, we deal only with mass transfer produced by concen- tration gradients.614 An introduction to mass transfer §11.4 6 11.4 Transport properties of mixtures Direct measurements of mixture transport properties are not always avail- able for the temperature, pressure, or composition of interest. Thus, we must often rely upon theoretical predictions or experimental correlations for estimating mixture properties. In this section, we discuss methods for computingD ,k, andµ in gas mixtures using equations from ki- im netic theory—particularly the Chapman-Enskog theory 11.2, 11.8, 11.9. We also consider some methods for computingD in dilute liquid solu- 12 tions. The diffusion coefficient for binary gas mixtures As a starting point, we return to our simple model for the self-diffusion coefficient of a dilute gas, eqn. (11.32). We can approximate the average molecular speed,C, by Maxwell’s equilibrium formula (see, e.g., 11.9):   1/2 8k N T B A C= (11.37) πM ◦ wherek =R /N is Boltzmann’s constant. If we assume the molecules B A to be rigid and spherical, then the mean free path turns out to be 1 k T B √ √ = = (11.38) 2 2 π 2Nd π 2d p where d is the effective molecular diameter. Substituting these values ofC and in eqn. (11.32) and applying a kinetic theory calculation that shows 2ηa= 1/2, we find  D =(2ηa)C AA   3/2 1/2 3/2 (k /π) N T B A = (11.39) 2 d M p −1 3/2 The diffusion coefficient varies as p and T , based on the simple model for self-diffusion. To get a more accurate result, we must take account of the fact that molecules are not really hard spheres. We also have to allow for differ- ences in the molecular sizes of different species and for nonuniformities 6 This section may be omitted without loss of continuity. The property predictions of this section are used only in Examples 11.11, 11.14, and 11.16, and in some of the end-of-chapter problems.