The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 52
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It is exact for a binary gas mixture, for which thekinetic theory of gases givesj1 = - rD12 Ñm1 kg m 2 sec© 1999 by CRC Press LLC(4.7.24a)4-212Section 4on a mass basis, andJ1* = - cD12 Ñx1 kg m 2 sec(4.7.24b)on a molar basis; D12 (m2/sec) is the binary diffusion coefficient (or mass diffusivity), and D21 = D21.Equations (4.7.24a) and (4.7.24b) are mathematically equivalent; however, notice that it is incorrect towrite(4.7.25)ji = -D12 Ñr1since ,r1 ¹ r , m1 in general.
Fick’s law in the form of Equations (4.7.24a) and (4.7.24b) is also validfor dilute liquid and solid solutions, for which it is often possible to assume r (or c) constant, and thenEquation (4.7.25) or its molar equivalent are good approximations.Ordinary diffusion in multicomponent systems is described by the Stefan–Maxwell equations (Hirschfelder et al., 1954). These equations are difficult to use for engineering analysis. In gas mixturescontaining species that do not have widely varying molecular weights, it is possible to model approximately the diffusion process by using an effective binary diffusion coefficient in Fick’s law. Thiscoefficient is a suitable average over the species in the mixture, and may be calculated fromD 1m =(1 - x1 )nå(xi;x1 ! 1(4.7.26)D1i )i =2This equation works well for most mixtures of combustion gases (except those containing appreciableconcentrations of H or H2).Binary diffusion coefficients at 300 K are of the order of 10–5 m2/sec in gases at 1 atm, 10–9 m2/secin aqueous solutions, and 10–10 to 10–13 m2/sec in solids.
However, the product rD or (cD) is, at most,one order of magnitude different for gases and liquids. Data for diffusion coefficients may be found inTables 4.7.2 through 4.7.5.Molecules in a gas mixture, and in a liquid or solid solution, can diffuse by mechanisms other thanordinary diffusion governed by Fick’s law. Thermal diffusion is diffusion due to a temperature gradientand is often called the Soret effect. Thermal diffusion is usually negligible compared with ordinarydiffusion, unless the temperature gradient is very large.
However, there are some important processesthat depend on thermal diffusion, the most well known being the large-scale separation of uraniumisotopes. Pressure diffusion is diffusion due to a pressure gradient and is also usually negligible unlessthe pressure gradient is very large. Pressure diffusion is the principle underlying the operation of acentrifuge. Centrifuges are used to separate liquid solutions and are increasingly being used to separategaseous isotopes as well. Forced diffusion results from an external force field acting on a molecule.Gravitational force fields do not cause separation since the force per unit mass of a molecule is constant.Forced diffusion occurs when an electrical field is imposed on an electrolyte (for example, in chargingan automobile battery), on a semiconductor, or on an ionized gas (for example, in a neon tube or metalion laser).
Depending on the strength of the electric field, rates of forced diffusion can be very large.Some interesting diffusion phenomena occur in porous solids. When a gas mixture is in a poroussolid, such as a catalyst pellet or silica–gel particle, the pores can be smaller than the mean free pathof the molecules.
Then, the molecules collide with the wall more often than with other molecules. Inthe limit of negligible molecule collisions we have Knudsen diffusion, also called free molecule flow inthe fluid mechanics literature. If the pore size approaches the size of a molecule, then Knudsen diffusionbecomes negligible and surface diffusion, in which adsorbed molecules move along the pore walls,becomes the dominant diffusion mechanism.© 1999 by CRC Press LLC4-213Heat and Mass TransferTABLE 4.7.2Diffusion Coefficients in Air at 1 atm (1.013 ´ 105 Pa)aBinary Diffusion Coefficient (m2/sec ´ 104)aT(K)O2CO2COC7H6H2NOSO2He2003004005006007008009001000120014001600180020000.0950.1880.3250.4750.6460.8381.051.261.522.062.663.324.034.800.0740.1570.2630.3850.5370.6840.8571.051.241.692.172.753.283.940.0980.2020.3320.4850.6590.8541.061.281.542.092.703.374.104.870.0360.0750.1280.1940.2700.3640.4420.5380.6410.8811.131.411.722.060.3750.7771.251.712.443.173.934.775.697.779.9012.515.218.00.0880.1800.3030.4430.6030.7820.9781.181.411.922.453.043.704.480.0580.1260.2140.3260.4400.5760.7240.8871.0601.4401.8702.3402.8503.3600.3630.7131.141.662.262.913.644.425.267.129.2011.513.916.6Owing to the practical importance of water vapor-air mixtures, engineers have used convenientempirical formulas for D H 2O air .
A formula that has been widely used for many years isæP öæ T öD H 2O air = 1.97 ´ 10 -5 ç 0 ÷ ç ÷è P ø è T0 ø1.685m 2 / sec; 273 K < T < 373 Kwhere P0 = 1 atm; T0 = 256 K. More recently, the following formula has found increasing use.(Marrero, T.R. and Mason, E.A. 1992. Gaseous diffusion coefficients, J. Phys. Chem. Ref. Data,1, 3–118):D H 2O air = 1.87 ´ 10 -10T 2.072; 280 K < T < 450 KPT 1.632; 450 K < T < 1070 KPfor P in atmospheres and T in kelvins.
Over the temperature range 290 to 330 K, the discrepancybetween the two formulas is less than 2.5%. For small concentrations of water vapor in air, theolder formula gives a constant value of Sc H 2O air = 0.61 over the temperature range 273 to 373K. On the other hand, the Marrero and Mason formula gives values of Sc H 2O air that vary from0.63 at 280 K to 0.57 at 373 K.= 2.75 ´ 10 -9Very small particles of 10–3 to 10–1 mm size — for example, smoke, soot, and mist — behave muchlike large molecules. Ordinary diffusion of such particles is called Brownian motion and is described inmost elementary physics texts. Diffusion of particles due to a temperature gradient is called thermophoresis and plays an important role for larger particles, typically in the size range 10–1 to 1 mm.
Diffusionof particles in a gas mixture due to concentration gradients of molecular species is called diffusiophoresis.Forced diffusion of a charged particle in an electrical field is similar to that for an ionized molecularspecies. Thermal and electrostatic precipitators are used to remove particles from power plant andincinerator stack gases, and depend on thermophoresis and forced diffusion, respectively, for theiroperation.
Diffusion phenomena are unimportant for particles of size greater than about 1 mm in air at1 atm; the motion of such particles is governed by the laws of Newtonian mechanics. Transport ofparticles is dealt with in the aerosol science literature.Species Conservation EquationThe principle of conservation of a chemical species is used to derive the species conservation equation.On a mass basis this equation is© 1999 by CRC Press LLC4-214Section 4TABLE 4.7.3 Schmidt Number for Vapors in Dilute Mixture in Air atNormal Temperature, Enthalpy of Vaporization, and Boiling Point at 1 atmaVaporAcetoneAmmoniaBenzeneCarbon dioxideCarbon monoxideChlorineEthanolHeliumHeptaneHydrogenHydrogen sulfideMethanolNaphthaleneNitric oxideOctaneOxygenPentaneSulfur dioxideWater vaporabcChemical FormulaScbhfg , J/kg ´ 10–6TBP , KCH3COCH3NH3C6H6CO2COCl2CH3CH2OHHeC7H16H2H2SCH3OHC10H8NOC8H18O2C5H12SO2H2O1.420.611.791.000.771.421.320.222.00.200.940.982.35c0.872.660.831.491.240.610.5271.3700.3950.3980.2170.2880.854—0.3400.4540.5481.110—0.4650.3030.2140.3570.3982.257329240354194812383524.337220.321333849112139990.6309263373With the Clausius-Clapeyron relation, one may estimate vapor pressure as1 ö ïüïì Mh fg æ 1_ expíPsat ~ç ÷ ý atm for T ~ TBPïî R è T TBP ø ïþThe Schmidt number is defined as Sc = m/rD = n/D.
Since the vapors are in smallconcentrations, values for m, r, and n can be taken as pure air values.From a recent study by Cho, C., Irvine, T.F., Jr., and Karni, J. 1992. Measurementof the diffusion coefficient of naphthalene into air, Int. J. Heat Mass Transfer, 35,957–966.
Also, hsg = 0.567 ´ 106 J/kg at 300 K.¶ri+ Ñ × ni = r˙i¢¢¢¶t(4.7.27)¶ci+ Ñ × Ni = R˙ i¢¢¢¶t(4.7.28)and on a molar basiswhere r˙i¢¢¢ and R˙ i¢¢¢ are the mass and molar rates of production of species i due to chemical reactions.Summing Equation 4.7.27 over all species gives the mass conservation or continuity equation,¶r+ Ñ × rv = 0¶t(4.7.29)The molar form is¶c+ Ñ × cv * =¶tå R˙ ¢¢¢i(4.7.30)isince, in general, moles are not conserved in chemical reactions. A useful alternative form to Equation4.7.27 can be obtained using Equations (4.7.23a) and (4.7.29) and is© 1999 by CRC Press LLC4-215Heat and Mass TransferTABLE 4.7.4Schmidt Numbers for Dilute Solution in Water at 300 KaSoluteHeliumHydrogenNitrogenWaterNitric oxideCarbon monoxideOxygenAmmoniaCarbon dioxideHydrogen sulfideEthyleneMethaneNitrous oxideSulfur dioxideSodium chlorideSodium hydroxideAcetic acidAcetoneMethanolEthanolChlorineBenzeneEthylene glycoln-Propanoli-PropanolPropaneAnilineBenzoic acidGlycerolSucroseaScM120190280340350360400410420430450490490520540490620630640640670720720730730750800830104016704.0032.01628.0218.01630.0128.0132.0017.0344.0134.0828.0516.0444.0264.0658.4540.0060.0558.0832.0446.0770.9078.1162.0760.0960.0944.0993.13122.1292.09342.3Schmidt number Sc = m/rD; since the solutions are dilute, m and r can be taken as pure watervalues.
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