John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 89
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For both liquids and gases,the diffusion coefficient rises with increasing temperature. Typical diffusion coefficients in solids (not listed) may range from about 10−20 toabout 10−9 m2 /s, depending upon what substances are involved and thetemperature [11.6].The diffusion of water vapor through air is of particular technicalimportance, and it is therefore useful to have an empirical correlationspecifically for that mixture:2.072−10 Tfor 282 K ≤ T ≤ 450 K (11.35)DH2 O,air = 1.87 × 10pwhere DH2 O,air is in m2 /s, T is in kelvin, and p is in atm [11.7]. The scatterof the available data around this equation is about 10%.Coupled diffusion phenomenaMass diffusion can be driven by factors other than concentration gradients, although the latter are of primary importance.
For example, temperature gradients can induce mass diffusion in a process known as thermal diffusion or the Soret effect. The diffusional mass flux resulting fromboth temperature and concentration gradients in a binary gas mixture isthen [11.2]M1 M2k∇ln(T)(11.36)ji = −ρD12 ∇m1 +TM2Diffusion fluxes and Fick’s law§11.3Table 11.1 Typical diffusion coefficients for binary gas mixtures at 1 atm and dilute liquid solutions [11.4].Gas mixtureair-carbon dioxideair-ethanolair-heliumair-napthaleneair-waterT (K)276313276303313D12 (m2/s)1.42×10−51.456.240.862.88argon-helium29562810688.332.181.0(dilute solute, 1)-(liquid solvent, 2)T (K)D12 (m2/s)ethanol-benzenebenzene-ethanolwater-ethanolcarbon dioxide-waterethanol-water2882982982982882.25×10−91.811.242.001.00methane-water2753330.853.55pyridene-water2880.58where kT is called the thermal diffusion ratio and is generally quite small.Thermal diffusion is occasionally used in chemical separation processes.Pressure gradients and body forces acting unequally on the differentspecies can also cause diffusion.
Again, these effects are normally small.A related phenomenon is the generation of a heat flux by a concentrationgradient (as distinct from heat convected by diffusing mass), called thediffusion-thermo or Dufour effect.In this chapter, we deal only with mass transfer produced by concentration gradients.613An introduction to mass transfer61411.4§11.4Transport properties of mixtures6Direct measurements of mixture transport properties are not always available for the temperature, pressure, or composition of interest. Thus, wemust often rely upon theoretical predictions or experimental correlationsfor estimating mixture properties. In this section, we discuss methodsfor computing Dim , k, and µ in gas mixtures using equations from kinetic theory—particularly the Chapman-Enskog theory [11.2, 11.8, 11.9].We also consider some methods for computing D12 in dilute liquid solutions.The diffusion coefficient for binary gas mixturesAs a starting point, we return to our simple model for the self-diffusioncoefficient of a dilute gas, eqn.
(11.32). We can approximate the averagemolecular speed, C, by Maxwell’s equilibrium formula (see, e.g., [11.9]):C=8kB NA TπM1/2(11.37)where kB = R ◦ /NA is Boltzmann’s constant. If we assume the moleculesto be rigid and spherical, then the mean free path turns out to be=kB T1√= √ 22π 2N dπ 2d p(11.38)where d is the effective molecular diameter. Substituting these valuesof C and in eqn. (11.32) and applying a kinetic theory calculation thatshows 2ηa = 1/2, we findDAA = (2ηa)C=(kB /π )3/2d2NAM1/2T 3/2p(11.39)The diffusion coefficient varies as p −1 and T 3/2 , based on the simplemodel for self-diffusion.To get a more accurate result, we must take account of the fact thatmolecules are not really hard spheres.
We also have to allow for differences in the molecular sizes of different species and for nonuniformities6This section may be omitted without loss of continuity. The property predictionsof this section are used only in Examples 11.11, 11.14, and 11.16, and in some of theend-of-chapter problems.Transport properties of mixtures§11.4Figure 11.6 The Lennard-Jonespotential.in the bulk properties of the gas. The Chapman-Enskog kinetic theorytakes all these factors into account [11.8], resulting in the following formula for DAB :DAB =(1.8583 × 10−7 )T 3/2ABpΩD(T )211+MAMBwhere the units of p, T , and DAB are atm, K, and m2/s, respectively.
TheAB(T ) describes the collisions between molecules of A and B.function ΩDIt depends, in general, on the specific type of molecules involved and thetemperature.The type of molecule matters because of the intermolecular forcesof attraction and repulsion that arise when molecules collide. A goodapproximation to those forces is given by the Lennard-Jones intermolecular potential (see Fig. 11.6.) This potential is based on two parameters,a molecular diameter, σ , and a potential well depth, ε.
The potential welldepth is the energy required to separate two molecules from one another.Both constants can be inferred from physical property data. Some valuesare given in Table 11.2 together with the associated molecular weights(from [11.10], with values for calculating the diffusion coefficients of water from [11.11]).615An introduction to mass transfer616§11.4Table 11.2 Lennard-Jones constants and molecular weights ofselected species.Speciesσ (Å)ε/kB (K)AlAirArBr2CCCl2 F2CCl4CH3 OHCH4CNCOCO2C2 H6C2 H5 OHCH3 COCH3C6 H6Cl2F22.6553.7113.5424.2963.3855.255.9473.6263.7583.8563.6903.9414.4434.5304.6005.3494.2173.357275078.693.3507.930.6253322.7481.8148.675.091.7195.2215.7362.6560.2412.3316.0112.6abMkgkmol26.9828.9639.95159.812.01120.9153.832.0416.0426.0228.0144.0130.0746.0758.0878.1170.9138.00Speciesσ (Å)ε/kB (K)H2H2 OH2 OH2 O2H2 SHeHgI2KrMgNH3N2N2 ONeO2SO2Xe2.8272.655a2.641b4.1963.6232.5512.9695.1603.6552.9262.9003.7983.8282.8203.4674.1124.04759.7363a809.1b289.3301.110.22750474.2178.91614558.371.4232.432.8106.7335.4231.0Mkgkmol2.01618.0234.0134.084.003200.6253.883.8024.3117.0328.0144.0120.1832.0064.06131.3Based on mass diffusion data.Based on viscosity and thermal conductivity data.ABAn accurate approximation to ΩD(T ) can be obtained using the Lennard-Jones potential function.
The result is AB2(T ) = σABΩD kB T εABΩDwhere, the collision diameter, σAB , may be viewed as an effective molecular diameter for collisions of A and B. If σA and σB are the cross-sectionaldiameters of A and B, in Å,7 then(11.40)σAB = (σA + σB ) 2The collision integral, ΩD is a result of kinetic theory calculations calculations based on the Lennard-Jones potential. Table 11.3 gives values of7One Ångström (1 Å) is equal to 0.1 nm.Transport properties of mixtures§11.4617ΩD from [11.12]. The effective potential well depth for collisions of Aand B is√(11.41)εAB = εA εBHence, we may calculate the binary diffusion coefficient fromDAB(1.8583 × 10−7 )T 3/2=2pσABΩD211+MAMB(11.42)where, again, the units of p, T , and DAB are atm, K, and m2/s, respectively, and σAB is in Å.Equation (11.42) indicates that the diffusivity varies as p −1 and is independent of mixture concentrations, just as the simple model indicatedthat it should.
The temperature dependence of ΩD , however, increasesthe overall temperature dependence of DAB from T 3/2 , as suggested byeqn. (11.39), to approximately T 7/4 .Air, by the way, can be treated as a single substance in Table 11.2owing to the similarity of its two main constituents, N2 and O2 .Example 11.3Compute DAB for the diffusion of hydrogen in air at 276 K and 1 atm.Solution. Let air be species A and H2 be species B. Then we readfrom Table 11.2εAεB= 78.6 K,= 59.7 KσA = 3.711 Å, σB = 2.827 Å,kBkBand calculate these valuesσAB = (3.711 + 2.827)/2 = 3.269 Å4εAB kB = (78.6)(59.7) = 68.5 KHence, kB T /εAB = 4.029, and ΩD = 0.8822 from Table 11.3. Then2(1.8583 × 10−7 )(276)3/211+m2 /sDAB =2(1)(3.269) (0.8822)2.016 28.97= 6.58 × 10−5 m2 /sAn experimental value from Table 11.1 is 6.34 × 10−5 m2 /s, so theprediction is high by 5%.Table 11.3 Collision integrals for diffusivity, viscosity, andthermal conductivity based on the Lennard-Jones potential.kB T /ε0.300.350.400.450.500.550.600.650.700.750.800.850.900.951.001.051.101.151.201.251.301.351.401.451.501.551.601.651.701.751.801.851.901.952.002.102.202.302.402.502.60618ΩD2.6622.4762.3182.1842.0661.9661.8771.7981.7291.6671.6121.5621.5171.4761.4391.4061.3751.3461.3201.2961.2731.2531.2331.2151.1981.1821.1671.1531.1401.1281.1161.1051.0941.0841.0751.0571.0411.0261.0120.99960.9878Ωµ = Ωk2.7852.6282.4922.3682.2572.1562.0651.9821.9081.8411.7801.7251.6751.6291.5871.5491.5141.4821.4521.4241.3991.3751.3531.3331.3141.2961.2791.2641.2481.2341.2211.2091.1971.1861.1751.1561.1381.1221.1071.0931.081kB T /ε2.702.802.903.003.103.203.303.403.503.603.703.803.904.004.104.204.304.404.504.604.704.804.905.006.007.08.09.010.020.030.040.050.060.070.080.090.0100.0200.0300.0400.0ΩDΩµ = Ωk0.97700.96720.95760.94900.94060.93280.92560.91860.91200.90580.89980.89420.88880.88360.87880.87400.86940.86520.86100.85680.85300.84920.84560.84220.81240.78960.77120.75560.74240.66400.62320.59600.57560.55960.54640.53520.52560.51700.46440.43600.41721.0691.0581.0481.0391.0301.0221.0141.0070.99990.99320.98700.98110.97550.97000.96490.96000.95530.95070.94640.94220.93820.93430.93050.92690.89630.87270.85380.83790.82420.74320.70050.67180.65040.63350.61940.60760.59730.58820.53200.50160.4811§11.4Transport properties of mixturesLimitations of the diffusion coefficient prediction.
Equation (11.42) isnot valid for all gas mixtures. We have already noted that concentrationgradients cannot be too steep; thus, it cannot be applied in, say, theinterior of a shock wave when the Mach number is significantly greaterthan unity. Furthermore, the gas must be dilute, and its molecules shouldbe, in theory, nonpolar and approximately spherically symmetric.Reid et al.
[11.4] compared values of D12 calculated using eqn. (11.42)with data for binary mixtures of monatomic, polyatomic, nonpolar, andpolar gases of the sort appearing in Table 11.2. They reported an averageabsolute error of 7.3 percent. Better results can be obtained by usingvalues of σAB and εAB that have been fit specifically to the pair of gasesinvolved, rather than using eqns. (11.40) and (11.41), or by constructingAB(T ) [11.13, Chap. 11].a mixture-specific equation for ΩDThe density of the gas also affects the accuracy of kinetic theory predictions, which require the gas to be dilute in the sense that its moleculesinteract with one another only during brief two-molecule collisions.
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