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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 elasticscattering 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 (2415) 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 lowdensity 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 selfdiﬁusion coef
ﬁcient, deﬁned as D“. = %m actor}
and )t is the mean free path of length of species A, given by
T
A = —K (2428}
Vin7i.” 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 (2428) and (2429) to equation {2427) results in 2T3"2 K3N ”2
D + = .— — 2430
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 lennardJones di
ameter of the Spherical molecule. Unlike the other two molecular transport coefﬁcients for
gases, viscosity and thermal conductivity. the gasphase diffusion coefﬁcient is dependent
on both pressure and temperature. Speciﬁcally the gasphase diffusion coefﬁcient is an in
verse function of total system pressure 0M. oc§ [2430
and a 3!?! powerlaw 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
LennardJones 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 (2433)
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 LennardJones parameter. in A:
and (10 is the “collision integral" for molecular diffusion, a dimensionless function of the
temperature and of the intermolecular potentialﬁeld for one molecule of A and one mole
cule of 3. Appendix Table K.l lists [ID as a function of Kin9,1,3; 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 LennardJ ones parameter, in ergs; see equation (2440). 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 (2430) or in the similar equations
for viscosity and thermal conductivity. The LennardJones 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,” (2434)
a = 0.841 Vi” (2435)
1" 1n
0* = 2.446;) (2436)
EAI'K = 0.77 Tr (246?)
and
EA/K = 1.157}, (2438) 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 LennardJones 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 polarnonpolar molecular pairs; the
proposed modiﬁcations are discussed by Hirschfelder, Curtiss, and Bird.* The Hirschfelder equation (2433} 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 (2433) also
states that the diffusion coefﬁcient varies with the temperature as TEE/(ID varies. Simplify
ing equation (2433), 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 (2441), 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 (2433) 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 (2441) 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 (2433) 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 LennardJones parameters, a; and
a}, are unavailable. The Fuller correlation is (2442) 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 diffusionvolume 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 DiffusionVolume 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. McGrawHill 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. Pratm 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 diffusionvolume
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. 103;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 (2433} is still used;
however, the collision integral is evaluated by 0.19651
oD = (1.,“ + T“ (2443}
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 (2444). 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
= —— 2448
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 (2449] 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 componentlfree
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 LennardJones 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 (2433) 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 (2449), 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. LiquidMass 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 lowconcentration
solutions. In the Eyring concept the ideal liquid is treated as a quasicrystalline 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, McGrawHill 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
StokesEinstein equation KT DAB = 67177143 (2450) 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 DANLs
«T —ﬁV) (2451)
in which ﬂV) is a function of the molecular volume of the diffusing solute. Empirical cor
relations, using the general form of equation (2451), 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 _ 74 X 10 _s(q’BMn)m
T 1136 (2452) 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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