John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 88
Текст из файла (страница 88)
(11.25) into eqn. (11.23) gives$$$$+i = N=xi +NJi∗ = NJi∗Niiiiso that$Ji∗ = 0.(11.26)iThus, both the Ji∗ ’s and the ji ’s sum to zero.Example 11.2At low temperatures, carbon oxidizes (burns) in air through the surface reaction: C + O2 → CO2 . Figure 11.4 shows the carbon-air interface in a coordinate system that moves into the stationary carbonat the same speed that the carbon burns away—as though the observer were seated on the moving interface.
Oxygen flows towardthe carbon surface and carbon dioxide flows away, with a net flowof carbon through the interface. If the system is at steady state and,if a separate analysis shows that carbon is consumed at the rate of0.00241 kg/m2 ·s, find the mass and mole fluxes through an imaginary surface, s, that stays close to the gas side of the interface.
Forthis case, concentrations at the s-surface turn out to be mO2 ,s = 0.20,mCO2 ,s = 0.052, and ρs = 0.29 kg/m3 .Solution. The mass balance for the reaction is12.0 kg C + 32.0 kg O2 → 44.0 kg CO2Since carbon flows through a second imaginary surface, u, movingthrough the stationary carbon just below the interface, the mass fluxesare related bynC,u = −1212nO2 ,s =nCO2 ,s3244The minus sign arises because the O2 flow is opposite the C and CO2flows, as shown in Figure 11.4. In steady state, if we apply massMixture compositions and species fluxes§11.2607Figure 11.4 Low-temperature carbonoxidation.conservation to the control volume between the u and s surfaces, wefind that the total mass flux entering the u-surface equals that leavingthe s-surfacenC,u = nCO2 ,s + nO2 ,s = 0.00241 kg/m2 ·sHence,nO2 ,s = −nCO2 ,s =32(0.00241 kg/m2 ·s) = −0.00643 kg/m2 ·s1244(0.00241 kg/m2 ·s) = 0.00884 kg/m2 ·s12To get the diffusional mass flux, we need species and mass averagespeeds from eqns.
(11.18) and (11.19):vO2 ,s=nO2 ,sρO2 ,s=−0.00643 kg/m2 ·s0.2 (0.29 kg/m3 )= −0.111 m/snCO2 ,s0.00884 kg/m2 ·s==ρCO2 ,s0.052 (0.29 kg/m3 )1 $(0.00884 − 0.00643) kg/m2 ·sni ==vs =ρs i0.29 kg/m3vCO2 ,s =Thus, from eqn. (11.20), ji,s = ρi,s vi,s − vs⎧⎨−0.00691 kg/m2 ·s for O2=⎩ 0.00871 kg/m2 ·s for CO20.586 m/s0.00831 m/sAn introduction to mass transfer608§11.3The diffusional mass fluxes, ji,s , are very nearly equal to the speciesmass fluxes, ni,s . That is because the mass-average speed, vs , is muchless than the species speeds, vi,s , in this case. Thus, the convectivecontribution to ni,s is much smaller than the diffusive contribution,and mass transfer occurs primarily by diffusion.
Note that jO2 ,s andjCO2 ,s do not sum to zero because the other, nonreacting species inair must diffuse against the small convective velocity, vs (see Section 11.7).One mole of carbon surface reacts with one mole of O2 to formone mole of CO2 . Thus, the mole fluxes of each species have the samemagnitude at the interface:NCO2 ,s = −NO2 ,s = NC,u =nC,u= 0.000201 kmol/m2 ·sMCThe mole average velocity at the s-surface, vs∗ , is identically zero byeqn. (11.23), since NCO2 ,s + NO2 ,s = 0. The diffusional mole fluxes are⎧⎨−0.000201 kmol/m2 ·s for O2 ∗= ci,s vi,s − vs∗ = Ni,s =Ji,s⎩ 0.000201 kmol/m2 ·s for CO2=0These two diffusional mole fluxes sum to zero themselves becausethere is no convective mole flux for other species to diffuse against∗= 0).(i.e., for the other species Ji,sThe reader may calculate the velocity of the interface from nc,u .That calculation would show the interface to be receding so slowlythat the velocities we calculate are almost equal to those that wouldbe seen by a stationary observer.11.3Diffusion fluxes and Fick’s lawWhen the composition of a mixture is nonuniform, the concentrationgradient in any species, i, of the mixture provides a driving potential forthe diffusion of that species.
It flows from regions of high concentrationto regions of low concentration—similar to the diffusion of heat fromregions of high temperature to regions of low temperature. We havealready noted in Section 2.1 that mass diffusion obeys Fick’s lawji = −ρDim ∇mi(11.27)Diffusion fluxes and Fick’s law§11.3609which is analogous to Fourier’s law.The constant of proportionality, ρDim , between the local diffusivemass flux of species i and the local concentration gradient of i involvesa physical property called the diffusion coefficient, Dim , for species i diffusing in the mixture m.
Like the thermal diffusivity, α, or the kinematicviscosity (a momentum diffusivity), ν, the mass diffusivity Dim has theunits of m2/s. These three diffusivities can form three dimensionlessgroups, among which is the Prandtl number:The Prandtl number, Pr ≡ ν/αThe Schmidt number,3 Sc ≡ ν/DimThe Lewisnumber,4(11.28)Le ≡ α/Dim = Sc/PrEach of these groups compares the relative strength of two different diffusive processes.
We make considerable use of the Schmidt number laterin this chapter.When diffusion occurs in mixtures of only two species—so-called binary mixtures—Dim reduces to the binary diffusion coefficient, D12 . Infact, the best-known kinetic models are for binary diffusion.5 In binarydiffusion, species 1 has the same diffusivity through species 2 as doesspecies 2 through species 1 (see Problem 11.5); in other words,D12 = D213(11.29)Ernst Schmidt (1892–1975) served successively as the professor of thermodynamics at the Technical Universities of Danzig, Braunschweig, and Munich (Chapter 6, footnote 3).
His many contributions to heat and mass transfer include the introduction ofaluminum foil as radiation shielding, the first measurements of velocity and temperature fields in a natural convection boundary layer, and a once widely-used graphicalprocedure for solving unsteady heat conduction problems. He was among the first todevelop the analogy between heat and mass transfer.4Warren K. Lewis (1882–1975) was a professor of chemical engineering at M.I.T. from1910 to 1975 and headed the department throughout the 1920s. He defined the originalparadigm of chemical engineering, that of “unit operations”, and, through his textbookwith Walker and McAdams, Principles of Chemical Engineering, he laid the foundationsof the discipline.
He was a prolific inventor in the area of industrial chemistry, holdingmore than 80 patents. He also did important early work on simultaneous heat andmass transfer in connection with evaporation problems.5Actually, Fick’s Law is strictly valid only for binary mixtures. It can, however, often be applied to multicomponent mixtures with an appropriate choice of Dim (seeSection 11.4).610An introduction to mass transfer§11.3A kinetic model of diffusionDiffusion coefficients depend upon composition, temperature, and pressure. Equations that predict D12 and Dim are given in Section 11.4. Fornow, let us see how Fick’s law arises from the same sort of elementarymolecular 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) inwhich the molecules A of one species are very similar to the molecules Aof a second species (as though some of the molecules of a pure gas hadmerely been labeled without changing their properties.) The resultingprocess is called self-diffusion.If we have a one-dimensional concentration distribution, as shown inFig.
11.5, molecules of A diffuse down their concentration gradient inthe x-direction. This process is entirely analogous to the transport ofenergy and momentum shown in Fig. 6.13. We take the temperature andpressure of the mixture (and thus its number density) to be uniform andthe mass-average velocity to be zero.Individual molecules move at a speed C, which varies randomly frommolecule to molecule and is called the thermal or peculiar speed.
Theaverage speed of the molecules is C. The average rate at which moleculescross the plane x = x0 in either direction is proportional to N C, whereN is the number density (molecules/m3 ). Prior to crossing the x0 -plane,the molecules travel a distance close to one mean free path, —call it a,where a is a number on the order of one.The molecular flux travelling rightward across x0 , from its plane oforigin at x0 − a, then has a fraction of molecules of A equal to the valueof NA /N at x0 − a. The leftward flux, from x0 + a, has a fractionequal to the value of NA /N at x0 + a.
Since the mass of a molecule ofA is MA /NA (where NA is Avogadro’s number), the net mass flux in thex-direction is thenjA M N AA= η NCNN x0ANA −N x0 +ax0 −a(11.30)where η is a constant of proportionality. Since NA /N changes little in adistance of two mean free paths (in most real situations), we can expandthe right side of eqn. (11.30) in a two-term Taylor series expansion aboutDiffusion fluxes and Fick’s law§11.3611Figure 11.5 One-dimensional diffusion.x0 and obtain Fick’s law:jA d(NA /N ) = η NC−2aNAdxx0dmA = −2ηa(C)ρdx x0 M A(11.31)x0(for details, see Problem 11.6). Thus, we identifyDAA = (2ηa)C(11.32)and Fick’s law takes the formjA = −ρDAAdmAdx(11.33)The constant, ηa, in eqn. (11.32) can be fixed only with the help of a moredetailed kinetic theory calculation [11.2], the result of which is given inSection 11.4.The choice of ji and mi for the description of diffusion is really somewhat arbitrary.
The molar diffusion flux, Ji∗ , and the mole fraction, xi ,are often used instead, in which case Fick’s law reads∗Ji = −cDim ∇xi(11.34)Obtaining eqn. (11.34) from eqn. (11.27) for a binary mixture is left as anexercise (Problem 11.4).612An introduction to mass transfer§11.3Typical values of the diffusion coefficientFick’s law works well in low density gases and in dilute liquid and solidsolutions, but for concetrated liquid and solid solutions the diffusion coefficient is found to vary with the concentration of the diffusing species.In part, the concentration dependence of those diffusion coefficients reflects the inadequacy of the concentration gradient in representing thedriving force for diffusion in nondilute solutions.
Gradients in the chemical potential actually drive diffusion. In concentrated liquid or solidsolutions, chemical potential gradients are not always equivalent to concentration gradients [11.3, 11.4, 11.5].Table 11.1 lists some experimental values of the diffusion coefficientin binary gas mixtures and dilute liquid solutions. For gases, the diffusion coefficient is typically on the order of 10−5 m2 /s near room temperature. For liquids, the diffusion coefficient is much smaller, on theorder of 10−9 m2 /s near room temperature.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
