John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 90
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Childsand Hanley [11.14] have suggested that the transport properties of gasesare within 1% of the dilute values if the gas densities do not exceed thefollowing limiting value(11.43)ρmax = 22.93M (σ 3 Ωµ )Here, σ (the collision diameter of the gas) and ρ are expressed in Å andkg/m3 , and Ωµ —a second collision integral for viscosity—is included inTable 11.3. Equation (11.43) normally gives ρmax values that correspondto pressures substantially above 1 atm.At higher gas densities, transport properties can be estimated by avariety of techniques, such as corresponding states theories, absolutereaction-rate theories, or modified Enskog theories [11.13, Chap. 6] (alsosee [11.4, 11.8]). Conversely, if the gas density is so very low that themean free path is on the order of the dimensions of the system, we havewhat is called free molecule flow, and the present kinetic models are againinvalid (see, e.g., [11.15]).Diffusion coefficients for multicomponent gasesWe have already noted that an effective binary diffusivity, Dim , can beused to represent the diffusion of species i into a mixture m.
The preceding equations for the diffusion coefficient, however, are strictly applicable only when one pure substance diffuses through another. Differentequations are needed when there are three or more species present.619620An introduction to mass transfer§11.4If a low concentration of species i diffuses into a homogeneous mix∗ture of n species, then Jj 0 for j ≠ i, and one may show (Problem 11.14) thatD−1im =n$xjj=1j≠iDij(11.44)where Dij is the binary diffusion coefficient for species i and j alone.This rule is sometimes called Blanc’s law [11.4].If a mixture is dominantly composed of one species, A, and includesonly small traces of several other species, then the diffusion coefficientof each trace gas is approximately the same as it would be if the othertrace gases were not present.
In other words, for any particular tracespecies i,Dim DiA(11.45)Finally, if the binary diffusion coefficient has the same value for eachpair of species in a mixture, then one may show (Problem 11.14) thatDim = Dij , as one might expect.Diffusion coefficients for binary liquid mixturesEach molecule in a liquid is always in contact with several neighboringmolecules, and a kinetic theory like that used in gases, which relies ondetailed descriptions of two-molecule collisions, is no longer feasible.Instead, a less precise theory can be developed and used to correlateexperimental measurements.For a dilute solution of substance A in liquid B, the so-called hydrodynamic model has met some success. Suppose that, when a force permolecule of FA is applied to molecules of A, they reach an average steadyspeed of vA relative to the liquid B.
The ratio vA /FA is called the mobility of A. If there is no applied force, then the molecules of A diffuseas a result of random molecular motions (which we call Brownian motion). Kinetic and thermodynamic arguments, such as those given byEinstein [11.16] and Sutherland [11.17], lead to an expression for the diffusion coefficient of A in B as a result of Brownian motion:DAB = kB T (vA /FA )Equation (11.46) is usually called the Nernst-Einstein equation.(11.46)§11.4Transport properties of mixturesTo evaluate the mobility of a molecular (or particulate) solute, wemay make the rather bold approximation that Stokes’ law [11.18] applies,even though it is really a drag law for spheres at low Reynolds number(ReD < 1)1 + 2µB /βRAFA = 6π µB vA RA(11.47)1 + 3µB /βRAHere, RA is the radius of sphere A and β is a coefficient of “sliding”friction, for a friction force proportional to the velocity.
Substitutingeqn. (11.47) in eqn. (11.46), we getkB1 + 3µB /βRADAB µB=(11.48)T6π RA 1 + 2µB /βRAThis model is valid if the concentration of solute A is so low that themolecules of A do not interact with one another.For viscous liquids one usually assumes that no slip occurs betweenthe liquid and a solid surface that it touches; but, for particles whose sizeis on the order of the molecular spacing of the solvent molecules, someslip may very well occur. This is the reason for the unfamiliar factor inparentheses on the right side of eqn.
(11.47). For large solute particles,there should be no slip, so β → ∞ and the factor in parentheses tendsto one, as expected. Equation (11.48) then reduces to8kBDAB µB=T6π RA(11.49a)For smaller molecules—close in size to those of the solvent—we expectthat β → 0, leading to [11.19]DAB µBkB=T4π RA(11.49b)The most important feature of eqns. (11.48), (11.49a), and (11.49b)is that, so long as the solute is dilute, the primary determinant of thegroup Dµ T is the size of the diffusing species, with a secondary dependence on intermolecular forces (i.e., on β). More complex theories, such8Equation (11.49a) was first presented by Einstein in May 1905.
The more generalform, eqn. (11.48), was presented independently by Sutherland in June 1905. Equations (11.48) and (11.49a) are commonly called the Stokes-Einstein equation, althoughStokes had no hand in applying eqn. (11.47) to diffusion. It might therefore be arguedthat eqn. (11.48) should be called the Sutherland-Einstein equation.621622An introduction to mass transfer§11.4Table 11.4 Molal specific volumes and latent heats of vaporization for selected substances at their normal boiling points.SubstanceMethanolEthanoln-PropanolIsopropanoln-Butanoltert -Butanoln-PentaneCyclopentaneIsopentaneNeopentanen-HexaneCyclohexanen-Heptanen-Octanen-Nonanen-DecaneAcetoneBenzeneCarbon tetrachlorideEthyl bromideNitromethaneWaterVm (m3 /kmol)0.0420.0640.0810.0720.1030.1030.1180.1000.1180.1180.1410.1170.1630.1850.2070.2290.0740.0960.1020.0750.0560.0187hfg (MJ/kmol)35.5339.3341.9740.7143.7640.6325.6127.3224.7322.7228.8533.0331.6934.1436.5339.3328.9030.7629.9327.4125.4440.62as the absolute reaction-rate theory of Eyring [11.20], lead to the samedependence.
Moreover, experimental studies of dilute solutions verifythat the group Dµ/T is essentially temperature-independent for a givensolute-solvent pair, with the only exception occuring in very high viscosity solutions. Thus, most correlations of experimental data have usedsome form of eqn. (11.48) as a starting point.Many such correlations have been developed. One fairly successfulcorrelation is due to King et al.
[11.21]. They expressed the molecular sizein terms of molal volumes at the normal boiling point, Vm,A and Vm,B , andaccounted for intermolecular association forces using the latent heats of§11.4Transport properties of mixtures623Figure 11.7 Comparison of liquid diffusion coefficients predicted by eqn. (11.50) with experimental values for assortedsubstances from [11.4].vaporization at the normal boiling point, hfg,A and hfg,B . They obtainedDAB µB= (4.4 × 10−15 )TVm,BVm,A1/6 hfg,Bhfg,A1/2(11.50)which has an rms error of 19.5% and for which the units of DAB µB /T arekg·m/K·s2 .
Values of hfg and Vm are given for various substances in Table 11.4. Equation (11.50) is valid for nonelectrolytes at high dilution, andit appears to be satisfactory for both polar and nonpolar substances. Thedifficulties with polar solvents of high viscosity led the authors to limiteqn. (11.50) to values of Dµ/T < 1.5×10−14 kg·m/K·s2 . The predictionsof eqn. (11.50) are compared with experimental data in Fig. 11.7. Reid etal.
[11.4] review several other liquid-phase correlations and provide anassessment of their accuracies.An introduction to mass transfer624§11.4The thermal conductivity and viscosity of dilute gasesIn any convective mass transfer problem, we must know the viscosity ofthe fluid and, if heat is also being transferred, we must also know itsthermal conductivity. Accordingly, we now consider the calculation of µand k for mixtures of gases.Two of the most important results of the kinetic theory of gases arethe predictions of µ and k for a pure, monatomic gas of species A: 3M TA−6(11.51)µA = 2.6693 × 10σA2 Ωµand0.083228kA =σA2 Ωk2TMA(11.52)where Ωµ and Ωk are collision integrals for the viscosity and thermalconductivity.
In fact, Ωµ and Ωk are equal to one another, but they aredifferent from ΩD . In these equations µ is in kg/m·s, k is in W/m·K, T isin kelvin, and σA again has units of Å.The equation for µA applies equally well to polyatomic gases, butkA must be corrected to account for internal modes of energy storage—chiefly molecular rotation and vibration. Eucken (see, e.g., [11.9]) gave asimple analysis showing that this correction was9γ − 5µcp(11.53)k=4γfor an ideal gas, where γ ≡ cp /cv . You may recall from your thermodynamics courses that γ is 5/3 for monatomic gases, 7/5 for diatomicgases at modest temperatures, and approaches unity for very complexmolecules.
Equation (11.53) should be used with tabulated data for cp ;on average, it will underpredict k by perhaps 10 to 20% [11.4].An approximate formula for µ for multicomponent gas mixtures wasdeveloped by Wilke [11.22], based on the kinetic theory of gases.
He introduced certain simplifying assumptions and obtained, for the mixtureviscosity,µm =n$i=1xi µin#xj φijj=1(11.54)Transport properties of mixtures§11.4625whereφij1 + (µi /µj )1/2 (Mj /Mi )1/4=1/2√ 2 2 1 + (Mi /Mj )2The analogous equation for the thermal conductivity of mixtures wasdeveloped by Mason and Saxena [11.23]:km =n$i=1xi kin#xj φij(11.55)j=1(We have followed [11.4] in omitting a minor empirical correction factorproposed by Mason and Saxena.)Equation (11.54) is accurate to about 2 % and eqn. (11.55) to about 4%for mixtures of nonpolar gases. For higher accuracy or for mixtures withpolar components, refer to [11.4] and [11.13].Example 11.4Compute the transport properties of normal air at 300 K.Solution. The mass composition of air was given in Example 11.1.Using the methods of Example 11.1, we obtain the mole fractions asxN2 = 0.7808, xO2 = 0.2095, and xAr = 0.0093.We first compute µ and k for the three species to illustrate the useof eqns.
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