Computational Fluid Mechanics and Heat Transfer Third Edition_17 pot

§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]. Gas mixture T (K) D 12 (m 2 /s) air-carbon dioxide 276 1.42×10 −5 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 (dilute solute, 1)-(liquid solvent, 2) T (K) D 12 (m 2 /s) ethanol-benzene 288 2.25×10 −9 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 where k T 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 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 11.4 Transport properties of mixtures 6 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 computing D im , k, and µ in gas mixtures using equations from ki- netic theory—particularly the Chapman-Enskog theory [11.2, 11.8, 11.9]. We also consider some methods for computing D 12 in dilute liquid solu- 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]): C =  8k B N A T πM  1/2 (11.37) where k B = R ◦ /N A is Boltzmann’s constant. If we assume the molecules to be rigid and spherical, then the mean free path turns out to be  = 1 π √ 2Nd 2 = k B T π √ 2d 2 p (11.38) where d is the effective molecular diameter. Substituting these values of C and  in eqn. (11.32) and applying a kinetic theory calculation that shows 2ηa = 1/2, we find D AA  = (2ηa)C = (k B /π) 3/2 d 2  N A M  1/2 T 3/2 p (11.39) The diffusion coefficient varies as p −1 and T 3/2 , 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. §11.4 Transport properties of mixtures 615 Figure 11.6 The Lennard-Jones potential. in the bulk properties of the gas. The Chapman-Enskog kinetic theory takes all these factors into account [11.8], resulting in the following for- mula for D AB : D AB = (1.8583 ×10 −7 )T 3/2 pΩ AB D (T )  1 M A + 1 M B where the units of p, T , and D AB are atm, K, and m 2 /s, respectively. The function Ω AB D (T ) describes the collisions between molecules of A and B. It depends, in general, on the specific type of molecules involved and the temperature. The type of molecule matters because of the intermolecular forces of attraction and repulsion that arise when molecules collide. A good approximation to those forces is given by the Lennard-Jones intermolec- ular potential (see Fig. 11.6.) This potential is based on two parameters, a molecular diameter, σ , and a potential well depth, ε. The potential well depth is the energy required to separate two molecules from one another. Both constants can be inferred from physical property data. Some values are given in Table 11.2 together with the associated molecular weights (from [11.10], with values for calculating the diffusion coefficients of wa- ter from [11.11]). 616 An introduction to mass transfer §11.4 Table 11.2 Lennard-Jones constants and molecular weights of selected species. Species σ(Å)ε/k B (K) M  kg kmol  Species σ(Å)ε/k B (K) M  kg kmol  Al 2.655 2750 26.98 H 2 2.827 59.72.016 Air 3.711 78.628.96 H 2 O2.655 a 363 a 18.02 Ar 3.542 93.339.95 H 2 O2.641 b 809.1 b Br 2 4.296 507.9 159.8H 2 O 2 4.196 289.334.01 C3.385 30.612.01 H 2 S3.623 301.134.08 CCl 2 F 2 5.25 253 120.9He2.551 10.22 4.003 CCl 4 5.947 322.7 153.8Hg2.969 750 200.6 CH 3 OH 3.626 481.832.04 I 2 5.160 474.2 253.8 CH 4 3.758 148.616.04 Kr 3.655 178.983.80 CN 3.856 75.026.02 Mg 2.926 1614 24.31 CO 3.690 91.728.01 NH 3 2.900 558.317.03 CO 2 3.941 195.244.01 N 2 3.798 71.428.01 C 2 H 6 4.443 215.730.07 N 2 O3.828 232.444.01 C 2 H 5 OH 4.530 362.646.07 Ne 2.820 32.820.18 CH 3 COCH 3 4.600 560.258.08 O 2 3.467 106.732.00 C 6 H 6 5.349 412.378.11 SO 2 4.112 335.464.06 Cl 2 4.217 316.070.91 Xe 4.047 231.0 131.3 F 2 3.357 112.638.00 a Based on mass diffusion data. b Based on viscosity and thermal conductivity data. An accurate approximation to Ω AB D (T ) can be obtained using the Len- nard-Jones potential function. The result is Ω AB D (T ) = σ 2 AB Ω D  k B T  ε AB  where, the collision diameter, σ AB , may be viewed as an effective molecu- lar diameter for collisions of A and B.Ifσ A and σ B are the cross-sectional diameters of A and B,inÅ, 7 then σ AB = (σ A +σ B )  2 (11.40) The collision integral, Ω D is a result of kinetic theory calculations calcu- lations based on the Lennard-Jones potential. Table 11.3 gives values of 7 One Ångström (1 Å) is equal to 0.1 nm. §11.4 Transport properties of mixtures 617 Ω D from [11.12]. The effective potential well depth for collisions of A and B is ε AB = √ ε A ε B (11.41) Hence, we may calculate the binary diffusion coefficient from D AB = (1.8583 ×10 −7 )T 3/2 pσ 2 AB Ω D  1 M A + 1 M B (11.42) where, again, the units of p, T , and D AB are atm, K, and m 2 /s, respec- tively, and σ AB is in Å. Equation (11.42) indicates that the diffusivity varies as p −1 and is in- dependent of mixture concentrations, just as the simple model indicated that it should. The temperature dependence of Ω D , however, increases the overall temperature dependence of D AB from T 3/2 , as suggested by eqn. (11.39), to approximately T 7/4 . Air, by the way, can be treated as a single substance in Table 11.2 owing to the similarity of its two main constituents, N 2 and O 2 . Example 11.3 Compute D AB for the diffusion of hydrogen in air at 276 K and 1 atm. Solution. Let air be species A and H 2 be species B. Then we read from Table 11.2 σ A = 3.711 Å,σ B = 2.827 Å, ε A k B = 78.6K, ε B k B = 59.7K and calculate these values σ AB = (3.711 +2.827)/2 = 3.269 Å ε AB  k B =  (78.6)(59.7) = 68.5K Hence, k B T/ε AB = 4.029, and Ω D = 0.8822 from Table 11.3. Then D AB = (1.8583 ×10 −7 )(276) 3/2 (1)(3.269) 2 (0.8822)  1 2.016 + 1 28.97 m 2 /s = 6.58 ×10 −5 m 2 /s An experimental value from Table 11.1 is 6.34 × 10 −5 m 2 /s, so the prediction is high by 5%. Table 11.3 Collision integrals for diffusivity, viscosity, and thermal conductivity based on the Lennard-Jones potential. k B T/ε Ω D Ω µ = Ω k k B T/ε Ω D Ω µ = Ω k 0.30 2.662 2.785 2.70 0.9770 1.069 0.35 2.476 2.628 2.80 0.9672 1.058 0.40 2.318 2.492 2.90 0.9576 1.048 0.45 2.184 2.368 3.00 0.9490 1.039 0.50 2.066 2.257 3.10 0.9406 1.030 0.55 1.966 2.156 3.20 0.9328 1.022 0.60 1.877 2.065 3.30 0.9256 1.014 0.65 1.798 1.982 3.40 0.9186 1.007 0.70 1.729 1.908 3.50 0.9120 0.9999 0.75 1.667 1.841 3.60 0.9058 0.9932 0.80 1.612 1.780 3.70 0.8998 0.9870 0.85 1.562 1.725 3.80 0.8942 0.9811 0.90 1.517 1.675 3.90 0.8888 0.9755 0.95 1.476 1.629 4.00 0.8836 0.9700 1.00 1.439 1.587 4.10 0.8788 0.9649 1.05 1.406 1.549 4.20 0.8740 0.9600 1.10 1.375 1.514 4.30 0.8694 0.9553 1.15 1.346 1.482 4.40 0.8652 0.9507 1.20 1.320 1.452 4.50 0.8610 0.9464 1.25 1.296 1.424 4.60 0.8568 0.9422 1.30 1.273 1.399 4.70 0.8530 0.9382 1.35 1.253 1.375 4.80 0.8492 0.9343 1.40 1.233 1.353 4.90 0.8456 0.9305 1.45 1.215 1.333 5.00 0.8422 0.9269 1.50 1.198 1.314 6.00 0.8124 0.8963 1.55 1.182 1.296 7.00.7896 0.8727 1.60 1.167 1.279 8.00.7712 0.8538 1.65 1.153 1.264 9.00.7556 0.8379 1.70 1.140 1.248 10.00.7424 0.8242 1.75 1.128 1.234 20.00.6640 0.7432 1.80 1.116 1.221 30.00.6232 0.7005 1.85 1.105 1.209 40.00.5960 0.6718 1.90 1.094 1.197 50.00.5756 0.6504 1.95 1.084 1.186 60.00.5596 0.6335 2.00 1.075 1.175 70.00.5464 0.6194 2.10 1.057 1.156 80.00.5352 0.6076 2.20 1.041 1.138 90.00.5256 0.5973 2.30 1.026 1.122 100.00.5170 0.5882 2.40 1.012 1.107 200.00.4644 0.5320 2.50 0.9996 1.093 300.00.4360 0.5016 2.60 0.9878 1.081 400.00.4172 0.4811 618 §11.4 Transport properties of mixtures 619 Limitations of the diffusion coefficient prediction. Equation (11.42)is not valid for all gas mixtures. We have already noted that concentration gradients cannot be too steep; thus, it cannot be applied in, say, the interior of a shock wave when the Mach number is significantly greater than unity. Furthermore, the gas must be dilute, and its molecules should be, in theory, nonpolar and approximately spherically symmetric. Reid et al. [11.4] compared values of D 12 calculated using eqn. (11.42) with data for binary mixtures of monatomic, polyatomic, nonpolar, and polar gases of the sort appearing in Table 11.2. They reported an average absolute error of 7.3 percent. Better results can be obtained by using values of σ AB and ε AB that have been fit specifically to the pair of gases involved, rather than using eqns. (11.40) and (11.41), or by constructing a mixture-specific equation for Ω AB D (T ) [11.13, Chap. 11]. The density of the gas also affects the accuracy of kinetic theory pre- dictions, which require the gas to be dilute in the sense that its molecules interact with one another only during brief two-molecule collisions. Childs and Hanley [11.14] have suggested that the transport properties of gases are within 1% of the dilute values if the gas densities do not exceed the following limiting value ρ max = 22.93M  (σ 3 Ω µ ) (11.43) Here, σ (the collision diameter of the gas) and ρ are expressed in Å and kg/m 3 , and Ω µ —a second collision integral for viscosity—is included in Table 11.3. Equation (11.43) normally gives ρ max values that correspond to pressures substantially above 1 atm. At higher gas densities, transport properties can be estimated by a variety of techniques, such as corresponding states theories, absolute reaction-rate theories, or modified Enskog theories [11.13, Chap. 6] (also see [11.4, 11.8]). Conversely, if the gas density is so very low that the mean free path is on the order of the dimensions of the system, we have what is called free molecule flow, and the present kinetic models are again invalid (see, e.g., [11.15]). Diffusion coefficients for multicomponent gases We have already noted that an effective binary diffusivity, D im , can be used to represent the diffusion of species i into a mixture m. The pre- ceding equations for the diffusion coefficient, however, are strictly appli- cable only when one pure substance diffuses through another. Different equations are needed when there are three or more species present. 620 An introduction to mass transfer §11.4 If a low concentration of species i diffuses into a homogeneous mix- ture of n species, then  J j ∗  0 for j ≠ i, and one may show (Prob- lem 11.14) that D −1 im = n  j=1 j≠i x j D ij (11.44) where D ij 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 includes only small traces of several other species, then the diffusion coefficient of each trace gas is approximately the same as it would be if the other trace gases were not present. In other words, for any particular trace species i, D im D iA (11.45) Finally, if the binary diffusion coefficient has the same value for each pair of species in a mixture, then one may show (Problem 11.14) that D im =D ij , as one might expect. Diffusion coefficients for binary liquid mixtures Each molecule in a liquid is always in contact with several neighboring molecules, and a kinetic theory like that used in gases, which relies on detailed descriptions of two-molecule collisions, is no longer feasible. Instead, a less precise theory can be developed and used to correlate experimental measurements. For a dilute solution of substance A in liquid B, the so-called hydro- dynamic model has met some success. Suppose that, when a force per molecule of F A is applied to molecules of A, they reach an average steady speed of v A relative to the liquid B. The ratio v A /F A is called the mobil- ity of A. If there is no applied force, then the molecules of A diffuse as a result of random molecular motions (which we call Brownian mo- tion). Kinetic and thermodynamic arguments, such as those given by Einstein [11.16] and Sutherland [11.17], lead to an expression for the dif- fusion coefficient of A in B as a result of Brownian motion: D AB = k B T ( v A /F A ) (11.46) Equation (11.46) is usually called the Nernst-Einstein equation. §11.4 Transport properties of mixtures 621 To evaluate the mobility of a molecular (or particulate) solute, we may 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 (Re D < 1) F A = 6πµ B v A R A  1 +2µ B /βR A 1 +3µ B /βR A  (11.47) Here, R A is the radius of sphere A and β is a coefficient of “sliding” friction, for a friction force proportional to the velocity. Substituting eqn. (11.47) in eqn. (11.46), we get D AB µ B T = k B 6πR A  1 +3µ B /βR A 1 +2µ B /βR A  (11.48) This model is valid if the concentration of solute A is so low that the molecules of A do not interact with one another. For viscous liquids one usually assumes that no slip occurs between the liquid and a solid surface that it touches; but, for particles whose size is on the order of the molecular spacing of the solvent molecules, some slip may very well occur. This is the reason for the unfamiliar factor in parentheses on the right side of eqn. (11.47). For large solute particles, there should be no slip, so β →∞and the factor in parentheses tends to one, as expected. Equation (11.48) then reduces to 8 D AB µ B T = k B 6πR A (11.49a) For smaller molecules—close in size to those of the solvent—we expect that β → 0, leading to [11.19] D AB µ B T = k B 4πR A (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 the group Dµ  T is the size of the diffusing species, with a secondary depen- dence on intermolecular forces (i.e., on β). More complex theories, such 8 Equation (11.49a) was first presented by Einstein in May 1905. The more general form, eqn. (11.48), was presented independently by Sutherland in June 1905. Equa- tions (11.48) and (11.49a) are commonly called the Stokes-Einstein equation, although Stokes had no hand in applying eqn. (11.47) to diffusion. It might therefore be argued that eqn. (11.48) should be called the Sutherland-Einstein equation. 622 An introduction to mass transfer §11.4 Table 11.4 Molal specific volumes and latent heats of vapor- ization for selected substances at their normal boiling points. Substance V m (m 3 /kmol)h fg (MJ/kmol) Methanol 0.042 35.53 Ethanol 0.064 39.33 n-Propanol 0.081 41.97 Isopropanol 0.072 40.71 n-Butanol 0.103 43.76 tert-Butanol 0.103 40.63 n-Pentane 0.118 25.61 Cyclopentane 0.100 27.32 Isopentane 0.118 24.73 Neopentane 0.118 22.72 n-Hexane 0.141 28.85 Cyclohexane 0.117 33.03 n-Heptane 0.163 31.69 n-Octane 0.185 34.14 n-Nonane 0.207 36.53 n-Decane 0.229 39.33 Acetone 0.074 28.90 Benzene 0.096 30.76 Carbon tetrachloride 0.102 29.93 Ethyl bromide 0.075 27.41 Nitromethane 0.056 25.44 Water 0.0187 40.62 as the absolute reaction-rate theory of Eyring [11.20], lead to the same dependence. Moreover, experimental studies of dilute solutions verify that the group Dµ/T is essentially temperature-independent for a given solute-solvent pair, with the only exception occuring in very high viscos- ity solutions. Thus, most correlations of experimental data have used some form of eqn. (11.48) as a starting point. Many such correlations have been developed. One fairly successful correlation is due to King et al. [11.21]. They expressed the molecular size in terms of molal volumes at the normal boiling point, V m,A and V m,B , and accounted for intermolecular association forces using the latent heats of [...]... heat transfer problem as GrL = g∆ρL3 ρν 2 and RaL = g∆ρL3 g∆ρL3 = ραν µα (11.79) although ∆ρ would still have had to have been evaluated from ∆T With Gr and Pr expressed in terms of density differences instead of temperature differences, the analogy between heat transfer and low-rate mass transfer may be used directly to adapt natural convection heat transfer predictions to natural convection mass transfer. .. variation of density Rather than solving eqn (8.3) and 646 An introduction to mass transfer §11.6 the species equation for specific mass transfer problems, we again turn to the analogy between heat and mass transfer In analyzing natural convection heat transfer, we eliminated ρ from eqn (8.3) using (1 − ρ∞ /ρ) = β(T − T∞ ), and the resulting Grashof and Rayleigh numbers came out in terms of an appropriate... the heat flux from a surface, q, as the product of a heat transfer coefficient, h, and a driving force for heat transfer, ∆T Thus, in the notation of Fig 11.14, qs = h (Ts − Te ) (1.17) In convective mass transfer problems, we would therefore like to express the diffusional mass flux from a surface, ji,s , as the product of a mass transfer coefficient and the concentration difference between the s-surface and. .. Napthalene sublimation can be used to infer heat transfer coefficients by measuring the loss of napthalene from a model over some length of time Experiments are run at several Reynolds numbers The lost mass fixes the sublimation rate and the mass transfer coefficient The mass transfer coefficient is then substituted in the analogy to heat transfer to determine a heat transfer Nusselt number at each Reynolds... the Prandtl number under the conditions of interest, some assumption about the dependence of the Nusselt number on the Prandtl number must usually be introduced Boundary conditions When we apply the analogy between heat transfer and mass transfer to calculate gm,i , we must consider the boundary condition at the wall We have dealt with two common types of wall condition in the study of heat transfer: ... liquid-phase correlations and provide an assessment of their accuracies An introduction to mass transfer 624 §11.4 The thermal conductivity and viscosity of dilute gases In any convective mass transfer problem, we must know the viscosity of the fluid and, if heat is also being transferred, we must also know its thermal conductivity Accordingly, we now consider the calculation of µ and k for mixtures of... 0.01787 The predictions thus agree with the data to within about 2% for µ and within about 4% for k To compute µm and km , we use eqns (11.54) and (11.55) and the tabulated values of µ and k Identifying N2 , O2 , and Ar as species 1, 2, and 3, we get φ12 = 0.9894, φ21 = 1.010 φ13 = 1.043, φ31 = 0.9445 φ23 = 1.058, φ32 = 0.9391 and φii = 1 The sums appearing in the denominators are  0.9978 for i =... the Lewis number is far from unity, the analogy between heat and mass transfer breaks down in those natural convection problems that involve both heat and mass transfer, because the concentration and thermal boundary layers may take on very different thicknesses, complicating the density distributions that drive the velocity field 11.7 Steady mass transfer with counterdiffusion In 1874, Josef Stefan presented... that gm,i is analogous to h cp (11.76e) Mass transfer at low rates §11.6 643 From these comparisons, we conclude that the solution of a heat convection problem becomes the solution of a low-rate mass convection problem upon replacing the variables in the heat transfer problem with the analogous mass transfer variables given by eqns (11.76) Convective heat transfer coefficients are usually expressed in... mass flux is principally carried by diffusion In this section, we examine diffusive and convective mass transfer of dilute species at low rates These problems have a direct correspondence to the heat transfer problems that we considered Chapters 1 through 8 We refer to this correspondence as the analogy between heat and mass transfer We will focus our attention on nonreacting systems, for which ˙ ri = 0 . 0.02615 Ar 521.5 0. 0178 2 0. 0178 7 The predictions thus agree with the data to within about 2% for µ and within about 4% for k. To compute µ m and k m , we use eqns. (11.54) and (11.55) and the tabulated. diffuse as a result of random molecular motions (which we call Brownian mo- tion). Kinetic and thermodynamic arguments, such as those given by Einstein [11.16] and Sutherland [11 .17] , lead to an expression. Lennard-Jones intermolec- ular potential (see Fig. 11.6.) This potential is based on two parameters, a molecular diameter, σ , and a potential well depth, ε. The potential well depth is the energy