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Handout Gas-Liquid Diffusion

Handout Gas-Liquid Diffusion - 24.2 The Diffusion...

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Unformatted text preview: 24.2 The Diffusion Coefficient 431 greater than the free paths of gases at ordinary pressures. After the fast neutrons are slowed down through elastic-scattering collisions between the neutrons and the nuclei of the reactor‘s moderator, these slower moving neutrons, theme! neutrons, migrate from positions of higher concentration to positions of lower concentration. and their migration is described by Fick‘s law of diffusion. 24.2 THE DIFFUSION COEFFICIENT Gas Mass Dit'fusivity Fick’s law proportionality. DAB, is known as the diffusion coefficient. Its fundamental di- mensions. which may be obtained from equation (24-15) D = —JA‘: 2 M —.—l 2 Lil—2 “8'3 deg/dz L2! MILLNL I are identical to the fundamental dimensions of the other transport properties: kinematic viscosity, v, and thermal diffusivity. a, or its equivalent ratio. kfpcp. The mass diffusivity has been reported in cmzls; the SI units are mzls which is a factor 10'4 smaller. In the Eng- lish system ftzi'hr is commonly used. Conversion between these systems involves the sim- ple relations DAg(cm2!'s) _ 104 Emmi!” (2426) D 2 ———"“(ft in”) = 3.8? DA 311cm Is) The diffusion coefficient depends upon the pressure, temperature, and composition of the system. Experimental values for the diffusivities of gases, liquids, and solids are tabu- lated in Appendix Tables J. l, 1.2, and 1.3, respectively. As one might expect from consid- erat'ion of the mobility of the molecules, the diffusion coefficients are generally higher for gases (in the range of 5 X 10—6 to l X 10"5 mils}. than for liquids (in the range of 10" m to [0—9 mzfs) which are higher than the values reported for solids {in the range of 10' ‘4 to 10" '0 mzt's). in the absence of experimental data, semitheoretical expressions have been developed which give approximations, sometimes as valid as experimental values due to the difficul- ties encountered in their measurement. Theoretical expressions for the diffusion coefficient in low-density gaseous mixtures as a function of the system's molecular properties were derived by Sutherland.* Jeans“? and Chapman and Cowling,:|: based upon the Kinetic Theory of Gases. in the simplest model of gas dynamics, the molecules are regarded as rigid spheres that exert no inter— molecular forces. Collisions between these rigid molecules are considered to be com- pletely elastic. With these assumptions. a simplified model for an ideal gas mixture of * W. Sutherland. Phil. Mag, 36, 507; 38, l {1894). ' T J. Jeans. Dynamical Them)- quases. Cambridge University Press, London, 1921. i S. Chapman and T. G. Cowling, Mathematical Theory oann—Unifona Gases. Cambridge University Press. London. [959. 432 Chapter 24 Fundamentals of MaSs Transfer species A diffusing through its isotope 11* yields an equation for the self-difiusion coef- ficient, defined as D“. = %m actor} and )t is the mean free path of length of species A, given by T A = —K (24-28} Vin-7i.” where n is the mean speed of species A with respect to the molar average velocity 2 SKNT u i ”MA (2449) Insertion of equations (24-28) and (24-29) to equation {24-27) results in 2T3"2 K3N ”2 D + = .— — 24-30 M 37rmaiP (Ms ( ] where M, is the molecular Weight of diffusing species A. (g/rnole), N is Avogadro’s num- ber (6.022 X 1023 moleculesf'mole), P is the system pressure, It" is the absolute temperature (K), K is the Boltzmann‘s constant (1.38 'X 10—15 ergst), and 0,, is the lennard-Jones di- ameter of the Spherical molecule. Unlike the other two molecular transport coefficients for gases, viscosity and thermal conductivity. the gas-phase diffusion coefficient is dependent on both pressure and temperature. Specifically the gas-phase diffusion coefficient is an in- verse function of total system pressure 0M. oc§ [24-30 and a 3!?! power-law function of absolute temperature 0,”. a: TM (2442) Modern versions of the kinetic theory have been attempted to account for forces of attraction and repulsion between the molecules. Hirschfelder er al. (1949),* using the Lennard-Jones potential to evaluate the influence of the molecular forces, presented an equation for the diffusion coefficient for gas pairs of nonpolar, nonreacting molecules: 3:2 _1_ "L 1.!2 0.001858?" [MA+MH:I DAB P013013 (24-33) where DAB is the mass diffusivity of A through B, in cuffs; Tis the absolute temperature, in K: MA, M 3 are the molecular weights of A and B, respectively; P is the absolute pres- sure, in atmospheres; 0",”; is the “collision diameter," a Lennard-Jones parameter. in A: and (1-0 is the “collision integral" for molecular diffusion, a dimensionless function of the temperature and of the intermolecular potential-field for one molecule of A and one mole cule of 3. Appendix Table K.l lists [ID as a function of Kin-9,1,3; K is the Boltzmann con- stant, which is 1.38 X 10‘ '5 ergsJ’K; and a” is the energy of molecular interaction for the * J. O. Hirschfelder. R. B. Bird. and E. L. Spotz, Chem Rent. 44, 205 (i949). 24.2 The Diffusion Coefficient 433 binary system A and B, a Lennard-J ones parameter, in ergs; see equation (24-40). Unlike the other two molecular transport coefficients, viscosity and thermal conductivity, the dif- fusion coefficient is dependent on pressure as well as on a higher order of the absolute temperature. When the transport process in a single component phase was examined. we did not find any Composition dependency in equation (24-30) or in the similar equations for viscosity and thermal conductivity. The Lennard-Jones parameters, 0' and EA. are usually obtained from viscosity data. Unfortunately. this information is available for only a very few pure gases. Appendix Table K2 tabulates these values. In the absence of experimental data, the values for pure components may be estimated from the following empirical relations: or = 1.18 V3,” (24-34) a- = 0.841 Vi” (24-35) 1" 1n 0* = 2.446;) (24-36) EAI'K = 0.77 Tr (246?) and EA/K = 1.157}, (24-38) where Vb is the molecular volume at the normal boiling point, in (emf/g mole (this is evaluated by using Table 24.3); Vt. is the critical molecular volume, in (cmflg mole; 31. is the critical temperature. in K: T}, is the normal boiling temperature, in K; and PE is the crit- ical pressure, in atmospheres. For a binary system composed of nonpolar molecular pairs, the Lennard-Jones para- meters of the pure component may be combined empirically by the following relations: + on = 0" 2 a” (24—39) and em = files (2440) These relations must be modified for polar»polar and polar-nonpolar molecular pairs; the proposed modifications are discussed by Hirschfelder, Curtiss, and Bird.* The Hirschfelder equation (24-33} is often used to extrapolate experimental data. For moderate ranges of pressure, up to 25 atmospheres, the diffusion coefficient varies in- versely as the pressure. Higher pressures apparently require dense gas corrections; unfor- tunately, no satisfactory correlation is available for high pressures. Equation (24-33) also states that the diffusion coefficient varies with the temperature as TEE/(ID varies. Simplify- ing equation (24-33), we can predict the diffusion coefficient at any temperature and at any pressure below 25 atmospheres from a known experimental value by _ Pl T2 3:: inr. Dish. - Dis... (133(3) Dali. (24—41} In Appendix Table 1.], experimental values of the product DABP are listed for several gas pairs at a particular temperature. Using equation (24-41), we may extend these values to other temperatures. * J. O. Hirschfeldcr. C. F. Curtiss. and R. B. Bird. Molecular Theory of Gases and liquids, John Wiley 6}. Sons. lnc.,_ New York, 1954. 434 Chapter 24 Fundamentals of Mass Transfer g Evaluate the diffusion coefficient of carbon dioxide in air at 20°C and atmospheric pressure. Compare this value with the experimental value reported in Appendix Table 1. 1. From Appendix Table K2 the values of 0' and six are obtained a; in A EAIK, in K Carbon dioxide 3.996 190 Air 3.617 9? The various parameters for equation (24-33) may be evaluated as follows: can 2 2 Elm/K = V (EA/KXEBJ'K) = V'(l90)(9?) = 136 T: 20 + 273 = 293K P = 1 atm E _ lie = KT — 293 0.463 KT _ a; — 2.16 QB (Table KJ) = 1.047 Mcoi = 44 and Man = 29 Substituting these values into equation (24—33), we obtain D = 0.0018'58T3’EUIMA + 19.13)"; AB Faisal!) _ (0.001858)(293)3”(1I44 + 1129)”2 (1)(3.806)2(1.O47} = 0.147 cmzls From Appendix Table J .1 for CO2 in air at 273- K, 1 atmosphere, we have DAB = 0.136 cuffs Equation (24-41) will be used to Correct for the differences in temperature Dam. = :13 3” flair: Dam": T2 nolr. Values for no may be evaluated as follows: at T2 = 273 5,1ng = % = 0.498 Quin = 1.074 at T; = 293 DE.”} = 1.04? (previous calculations) The corrected value for the diffusion coefficient at 20°C is 2713 (1.04?) M 1. 4 , DWI = (393) ( 07 )(0.136) 20.155cm21's (1.55 x 10—5m‘is) 24.2 The Difi‘usion Coefficient 435 We readily see that the temperature dependency of the “collision integral” is very small. Accordingly, most scaling of diffusivities relative to temperature only include the ratio 1mg)”. Equation (24-33) was developed for dilute gases consisting of nonpolar, spherical monatomic molecules. However, this equation gives good results for most nonpolar bi- nary gas systems over a wide range of temperatures.* Other empirical equations have been propo‘sedt for estimating the diffusion coefficient for nonpolar. binary gas systems at low pressures. The empirical correlation recommended by Fuller, Schettler. and Giddings permits the evaluation of the diffusivity when reliable Lennard-Jones parameters, a; and a}, are unavailable. The Fuller correlation is (24-42) where DAB is in cm‘ils. Tis in K. and P is in atmospheres. To determine the 1) terms. the authors recbtmnend the addition of the atomic and structural diffusion-volume increments 0 reported in Table 24.3. Table 24.3 Atomic Diffusion Volumes for use in Estimating 1D,, 3 by Method of Fuller, Schenler, and Giddings -————————______________ Atomic and Structure Diffusion-Volume Increments, v C 16.5 Cl 1 9.5 H 1.98 S 17.0 0 5 .48 Aromatic ring- - 20.2 N 5.69 Heterocyclie ring —20.2 Diffusion Volumes for Simple Molecules. u H2 7.07 Ar 16.1 H20 12.7 132 6.70 Kr 22.8 CCIF2 1 14.8 He 2.88 co 18.9 SF. 69.7 1v1 17.9 co2 26.9 (:12 37.7 02 16.6 No 35.9 Br. 67.2 Air 2.0.1 Nit. 14.9 so2 41.1 -————————______________ Banner and Dauberti: have recommend increments for C be corrected to 15.9 and f be corrected to 6.12 and for air to 19.7. * R. C. Reid. J. M. Prausnitz. and T. K. Sherwood. McGraw-Hill Book Company. New York. l977. Chapter 1 1. T J_. H. Arnold. J. Am. Chem. Soc, 52, Slattery and R. B. Bird. A.I.Ch.E. 1.. 4 3937 (1930). E. R. Gilliland, Ind. Eng. Chem, 26, 581 (1934). J. C. . 137 (1958). D. F. 0thmer and H. T. Chen. Ind. Eng. Chem. Prat-m Des. Den. 1, 2749 0962). R. G. Bailey, Chem. Engrc. 82(6). 36, (1975). E. N. Fuller. P. D. Schettler. and .1. C. GiddingS. ind. Eng. Chem. 58(5). 18 (1966). i R. P. Banner. and 'I‘. E. Daubert, Manualfor Predicting. Chemical Process Design Dara, A.l.Ch.E.. (1983). ed the atomic and structure diffusion-volume or H to 2.31 and the diffusion volumes for H2 The Properties afGaaer and Liquids, Third Edition. 436 Chapter 24 Fundamentals of Mass Transfer Reevaluate the diffusion coefficient of carbon dioxide in air at 20°C and atmospheric pres- sure using the Fuller, Schettler, and Giddings equation and compare the new value with the 'one reported in example 2. 10-3;MS L+i ”2 Ma Ms D“ = Home + (Eu)? 12 0—3 29 1.?5 l I ”2 1 ( 3) (E + E) = (1)[(26.9)”3 + (20.1)”3]2 = 0.152 cmzi‘s This value compares very favorably to the value evaluated with Hirschfelder equation, 0.155 cmzfs, and its detennination was easily accomplished. Brokaw* has suggested a method for estimating diffusion coefficient for binary gas mixtures containing polar compounds. The Hirschfelder equation (24-33} is- still used; however, the collision integral is evaluated by 0.19651 oD = (1.,“ + T“ (24-43} where 5.43 = (5.463)!” . . 1.94 x 10311.3, (2444) up = dipole moment, debyes Vb = liquid molar volume of the specific compound at its boiling point. cm3fg mol Tb = normal boiling point. K and I“: = KTdé‘AE where E _ £12 ”2 K K K (2445) eh: = 1.18(1 + 1.3 82)}?J 6 is evaluated with (24-44). And 0”“ _ W)” + mm”) + export) + exp(HT*) {2446) * R. S. Brokaw. Ind. East: Chem. Process Des. Den, 8. 240 (I969). 24.2 The Diffusion Coefficient 437 with A = 1.060 36 E = 1.035 87 B=0.15610 F= 1.52996 C=0.l9300 G=l.76474 D = 0.476 35 H = 3.894 11 The collision diameter, or”, is evaluated with “as = may)” (2447} with each component’s characteristic length evaluated by 1.585 Vb "'3 = —— 24-48 J (t + 1.3 52) l ) Reid, Prausnitz, and Sherwood“ noted that the Brokaw equation is fairly reliable, permit- ting the evaluation of the diffusion coefficients for gases involving polar compounds with errors less than 15%. Mass transfer in gas mixtures of several components can be described by theoretical equations involving the diffusion coefficients for the various binary pairs involved in the mixture. Hirschfelder, Curtiss, and Bird? present an expression in its most general form. Willie: has simplified the theory and has shown that a close approximation to the correct form is given by the relation DI -mixtttre = I (24-49] J’iIDi—z + Yi’Dl—s + ' ' ' + yit'rDi—n where Dhmim is the mass diffusivity for component 1 in the gas mixture; DH, is the mass diffusivity for the binary pair, component 1 diffusing through component a; and y; is the mole fraction of component a in the gas mixture evaluated on a component-l-free basis. that is t_ y; )2 y2+_l’3+"‘+}’n In the chemical vapor deposition of silane (SiHi) on a silicon wafer, a process gas stream rich in an inert nitrogen (N2) carrier gas has the following composition: ySiHi = 0.00?5. y“: : 0.0151yN2 = 0.97?5 The gas mixture is maintained at 900 K and 100 Pa total system pressure. Determine the diffusivity of silane through the gas mixture. The Lennard-Jones constants for silane are 5,41»: = 207.6 K and 0-,, = 4.03 A. The binary diffusion coefficients at 900 K and 100 Pa total system pressure estimated by the Hirschfelder equation (24-33) are DSiH4—N: : 1.09 X 103 szfs and Dst‘_H3 = 4.06 X 103 szi’S * R._C. Reid, J. M. Prausnitz. and T. K. Sherwood, The Properties of Gases and Liquids. Third Edition, McGraw—Hill Book Company, New York, I977. Chapter 1]. 1' J. 0. Hirsehfelder, C. F. Curtiss. and R. B. Bird, Molecular Theory ofGases and Liquids. Wiley, New York, p. 718. i C. R. Wilke. Chem. Engn Ping, 4a, 954 04 (I950). 438 Chapter 24 Fundamentals of Mass Transfer The binary diffusion coefficients are relatively high because the temperature is high and the total system pressure is low. The composition of nitrogen and hydrogen on a silane» free basis are . _ fl 2 , = 0.015 = yN; ‘— 1 _ 000;!5 09849 and YH, 1 _ 000.15 00151 Upon substituting these values into the Wilke equation (24-49), we obtain D = + + Sim—mixture y 1:13 y a: 0.9849 + 0.0151 + 3 3 DEW“: DSiIL—H3 1.09x10 4.06x10 |[ =1.10>< 1035155 This example verifies that for a dilute multicomponent gas mixture, the diffitsion coeffi- cient of the diffusing species in the gas mixture is approximated by the binary diffusion coefficient of the diffusing species in the carrier gas. Liquid-Mass Difiusivity In contrast to the case for gases, where we have available an advanced kinetic theory for explaining molecular motion, theories of the structure of liquids and their transport char- acteristics are still inadequate to permit a rigorous treatment. Inspection of published ex- perimental values for liquid diffusion coefficients in Appendix 1.2 reveals that they are several orders of magnitude smaller than gas diffusion coefficients and that they depend on concentration due to the changes in viscosity with concentration and changes in the de- gree of ideality of the solution. Certain molecules diffuse as molecules, while others which are designated as elec- trolytes ionize in solutions and diffuse as ions. For example, sodium chloride, NaCl, dif- fuses in water as the ions Na+ and CI‘. Though each ion has a different mobility, the electrical neutrality of the solution indicates the ions must diffuse at the same rate; ac- cordingly, it is possible to speak of a diffusion coefficient for molecular electrolytes such as NaCl. However, if several ions are present, the diffusion rates of the individual cations and anions must be considered, and molecular diffusion coefficients have no meaning. Needless to say, separate correlations for predicting the relation between the liquid mass diffusivities and the properties of the liquid solution will he required for electrolytes and nonclcctrolytes. Two theories, the Eyring “hole" theory and the hydrodynamical theory, have been pos~ tulated as possible explanations for diffusion of nonelectrolyte solutes in low-concentration solutions. In the Eyring concept the ideal liquid is treated as a quasi-crystalline lattice model interspersed with holes. The transport phenomenon is then described by a unimolec~ ular rate process involving the jumping of solute molecules into the holes within the lattice model. These jumps are empirically related to Eyring's theory of reaction rate.* The hydro- dynamical theory states that the liquid difiusion coefficient is related to the solute mole- cule’s mobility; that is. to the net velocity of the molecule while under the influence of a unit driving force. The laws of hydrodynamics provide relations between the force and the * S. Glasstone, K. .I. Laidler, and H. Eyring, Theory ofRare Processes, McGraw-Hill Book Company, New York. 194], Chap. IX. 24.2 The Diffusion Coefficient 439 velocity. An equation that has been developed from the hydrodynamical theory is the Stokes-Einstein equation KT DAB = 67177-143 (24-50) where D,” is the diffusivity ofA in dilute solution in B; K is the Boltzmann constant; Tis the absolute temperature; r is the solute particle radius; and its is the solvent viscosity. This equation has been fairly successful in describing the diffusion of colloidal particles or large round molecules through a solvent which behaves as a continuum relative to the diffusing species. The results of the two theories can be rearranged into the general form DAN-Ls «T —fiV) (24-51) in which flV) is a function of the molecular volume of the diffusing solute. Empirical cor- relations, using the general form of equation (24-51), have been developed which attempt to predict the liquid diffusion coefficient in terms of the solute and solvent properties. Wilke and Chan g* have proposed the following correlation for nonelectrolytes in an infi- nitely dilute solution: ' DABHH _ 7-4 X 10 _s(q’BMn)m T 113-6 (24-52) where Dan is the mass diffusivity ofA diffusing through liquid solvent B, in cmzls; lug is the viscosity of the solution, in centipoises; T is absolute temperature, in K: M3 is the mol- ecular weight of the solvent; VA is the molal volume of solute at normal boiling point. in cmjfg mol; and (1),:I is the “association" parameter for solvent 8. Molecular volumes at normal boiling points, VA. for some commonly encountered compounds. are tabulated in Table 24.4. For other compounds, the atomic volumes of each element present are added together as per the molecular formulae. Table 24.5 lists Table 24.4 Molecular Volumes at Norma] Boiling Point for Some Commonly Encountered Compounds ————————___—__ Molecular Molecular volume, volume. in Compound cm-‘fg mole Compound crui‘fg mole Hydrogen, H: 14.3 Nitric oxide, ND 23.6 Oxygen. 03 25.6 Nitrous oxide. N30 36.4 Nitrogen, N; 31.2 Ammonia. N H, 25.8 Air 29.9 Water, H30 18.9 Carbon monoxide, CO 30.? Hydrogen sulfide, HES 329 Carbon dioxide. CO: 34.0 Bromine, Br2 53.2 Carbonyl sulfide, COS 51.5 Chlorine. C12 48.4 Sulfur dioxide, SD; 44.8 Iodine, I2 71.5 _——————_—_—___ * C. R. Wilke and P. Chang. A.I.Ch.E..L, I. 264 (1955). 440 Chapter 24 Fundamentals of Mass Transfer Table ...
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