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PREDICTION OF DIFFUSIVITIES
Diffusivities are best determined by experimental
measurements.
The diffusivity of some common gases diffusing in
air at 0C and 1 atm (can be obtained from the text
book (McCabe, Smith and Harriott, Appendix 19).
The value of diffusivities can also be predicted using
various empirical equations.
1) Diffusion in gases
EKC 217: Mass Transfer - Prediction of Diffusivities
2
The equations developed by Chen and Othmer is:
------ (1.50)
where:
DAB = Diffusivity (cm2/s) T = Temperature (K) MA, Mb = Molecular weights of component A & B respectively TcA, TcB = Critical temperatures of A & B respectively (K) VcA, VcB = Critical molar volumes of A & B respectively (cm3/gmol) P = Pressure (atm)
24.04.01405.0
5.0
81.1
)()(
1101498.0
cBcAcBcA
BA
ABVVTTP
MMT
D
EKC 217: Mass Transfer - Prediction of Diffusivities
• Another rigorous equation called Chapman-Enskog equation is used to predict diffusivities:
5.0
2
5.1001858.0
BA
BA
DAB
ABMM
MM
P
TD
------ (1.51)
where:
DAB = Diffusivity (cm2/s)
T = Temperature (K)
MA, MB = Molecular weights of component A & respectively
P = Pressure (atm)
AB = average collision diameter
ΩD = collision integral based on the Lennard-Jones potential
= f (kT/AB )
k = Boltzmann’s constant
= Lennard-Jones force constant for common gases
BAAB
3 EKC 217: Mass Transfer - Prediction of Diffusivities
The collision integral, ΩD decreases with increasing T, that makes
DAB increase with more than the 1.5 power of the absolute T.
For diffusion in air at T = 300 – 1000 K, DAB varies with about T 1.7-
1.8, and T 1.75 can be used to extrapolate from room temperature
data.
Therefore, we can say that: DAB T 1.75 x 1/P
Hence, the diffusivity at T2 and P2 relative to the standard
temperature and pressure (STP) can be calculated as follows:
75.1
2
2
,2,
STP
STPSTPABAB
T
T
P
PDD ------ (1.52)
4 EKC 217: Mass Transfer - Prediction of Diffusivities
Eq. (1.51) is relatively complicated to use, and often some of the
constants such as AB are not available of difficult to estimate.
Hence, semi empirical method by Fuller et al. is often preferred:
23/13/1
5.0
75.17 )/1/1(10x00.1
BA
BAAB
P
MMTD
------ (1.53)
Eq. (1.53) was obtained by correlating many recent data and uses
atomic values from Table 6.2-2. This method can be used for
mixtures of nonpolar gases or for a polar-nonpolar mixture.
where: A = sum of structural volume increments
5 EKC 217: Mass Transfer - Prediction of Diffusivities
6 EKC 217: Mass Transfer - Prediction of Diffusivities
Example 5:
Predict the volumetric diffusivity for benzene in air at 100C and 2
atm by using the rigorous equation (1.51) and by extrapolating from
the published value for 0C and 1 atm.
Solution:
From Appendix 19 (McCabe, Smith and Harriott), the force
constants are as follows:
/k M
Benzene 412.3 5.349 78.1
Air 78.6 3.711 29
7 EKC 217: Mass Transfer - Prediction of Diffusivities
Thus,
53.42
711.3349.5
AB
1806.78x3.412k/ AB
072.2180
373k
AB
T
From Appendix 19, D = 1.062.
Substitute values into Eq. (1.51) gives:
/scm0668.0062.1x53.4x2
29x1.78
291.78373x001858.0
2
2
5.0
5.1
ABD
8 EKC 217: Mass Transfer - Prediction of Diffusivities
From Appendix 18 at standard temperature and pressure,
/scm0772.0/hft299.0 22 ABD
At 373 K and 2 atm,
/scm0666.0273
373
2
10772.0 2
75.1
ABD
Therefore, agreement with the value calculated from Eq. (1.51)
is very good.
9 EKC 217: Mass Transfer - Prediction of Diffusivities
Example 6:
In an oxygen-nitrogen gas mixture at 1 std atm, 25C, the
concentration of oxygen at two planes 2 mm apart are 10 and
20 vol %, respectively. Calculate the flux of diffusion of the
oxygen for the case where:
a) The nitrogen is non-diffusing
b) There is equimolecular counterdiffusion of the two gases.
DO2-N2 at 0C and standard atmospheric pressure
= 1.81 x 10-5 m2/s
10 EKC 217: Mass Transfer - Prediction of Diffusivities
Solution:
Component A = O2
Component B = N2
A + B A
2 mm
pA1
= 0.2PT
pA2
= 0.1PT
NB = 0
Given:
PT = 1 atm = 1.0133 x 105 N/m2
T = 25C = 298 K
DO2-N2 = 1.81 x 10-5 m2/s (0 C , 1 atm)
Diffusivity of O2-N2 mixture at 25C , 1
atm:
/sm 10 x 2.06
273
298x10x81.1
25-
5.1
5
298,22
KNOD
11 EKC 217: Mass Transfer - Prediction of Diffusivities
pA1 = 0.2 PT
pA2 = 0.1 PT
Then, pB1 = PT – 0.2PT = 0.8PT
pB2 = PT – 0.1PT = 0.9PT
)()(
21
12
AA
BM
TABA pp
pzzRT
PDN
)/ln( 12
12
BB
BBBM
pp
ppp
Calculate pBM :
and
T
TT
TBM P
PP
Pp 849.0
)8.0/9.0ln(
)8.09.0(
178.1
849.0
T
T
BM
T
P
P
p
P
a) For the case where nitrogen is non-diffusing, Eq. (1.37) is used:
12 EKC 217: Mass Transfer - Prediction of Diffusivities
sm
kmole 10 x 4.95
m
N 10 x 1.0133 x 1.02.0
)m002.0(K) )(298Kkmole
Nm (8314
178.1 s
m 10 x 2.06
)()(
2
5-
2
5
25-
21
122
AA
BM
TABO pp
pzzRT
PDN
Substitute the known values into Eq. (1.37):
13 EKC 217: Mass Transfer - Prediction of Diffusivities
b) When there is equimolecular counter diffusion of the two gases,
the following equations are applied:
)( 12
21
zzRT
ppDJ AAAB
A
)( 12
21
zzRT
ppDJ BBAB
B
Substitute the known values:
sm
O kmole 10 x 4.21
)m10 x 2)(K298(Kkmole
Nm8314
m
N10 x 1.0133 x 1.02.0
s
m10 x 2.06
2
25-
3
2
52
5
2
OJ
14 EKC 217: Mass Transfer - Prediction of Diffusivities
s/mN kmole 10 x 4.21-
m)10 x K)(2 (298Kkmole
Nm8314
m
N10 x 1.0133 x 0.90.8
s
m10x2.06
2
2
5-
3
52
5
2
NJ
For nitrogen:
15 EKC 217: Mass Transfer - Prediction of Diffusivities
16
Exercise: Estimation of Diffusivity of a Gas Mixture
Normal butanol (A) is diffusing through air (B) at 1 atm abs. Using
the Fuller et al. method, estimate the diffusivity DAB for the
following temperatures and compare with the experimental data:
a) For 0C
b) For 25.9C
c) For 0C and 2 atm abs.
Ans:
a) Deviation of +10% from the experimental value (in Table 6.2-1).
b) Deviation of +4% from the experimental value.
c) DAB = 3.865 x 10-6 m2/s
EKC 217: Mass Transfer - Prediction of Diffusivities
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2) Diffusion in Small Pores
M
TrDK 9700
KABpore DDD
111
The diffusion process is called Knudsen diffusion.
Occurs when the pore size is much smaller than the normal mean free
path during gas diffusion in very small pores of a solid, for processes
like adsorption, drying of porous solids or membrane separation.
The diffusivity for a cylindrical pore is:
------ (1.54)
where: DK = Knudsen diffusivity, cm2/s
T = temperature, K
M = molecular weight
r = pore radius, cm
Diffusivity in the pore: ------ (1.55)
EKC 217: Mass Transfer - Prediction of Diffusivities
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3) Diffusion in Liquids
V AB
BAB
TMD
6.0
5.016 )(
10x173.1
A widely used correlation for liquid diffusivity of small molecules,
called Wilke-Chang equation:
------ (1.56)
where: DAB = diffusivity, m2/s
T = absolute temperature, K
B = viscosity of B in Pa.s
VA = molar volume of solute as liquid at its normal
boiling point, m3/kgmole (taken from Table 6.3-2)
= association parameter of the solvent
MB = molecular weight of solvent B (kg/kgmole)
EKC 217: Mass Transfer - Prediction of Diffusivities
Recommended values for :
Water 2.6
Methanol 1.9
Ethanol 1.5
Benzene, heptane and other unassociated solvents 1.0
Eq. (1.56) is valid only at low solute concentrations and does not
apply when the solution has been thickened by addition of high-
molecular-weight polymers.
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EKC 217: Mass Transfer - Prediction of Diffusivities 20
EKC 217: Mass Transfer - Prediction of Diffusivities 21
For water as solute, VA = 0.0756 m3/kgmole.
If other compound is the solute, e.g.: acetone (CH3COCH3) in
water, the value of VA can be calculated by considering the atomic
volumes as given in Table 6.3.2 – Atomic and Molar Volumes at the
Normal Boiling Point.
Acetone (CH3COCH3) has 3 carbons, 6 hydrogens and 1 oxygen.
Therefore,
VA = 3(14.8 x 10-3) + 6(3.7 x 10-3) + 1(7.4 x 10-3)
= 0.0740 m3/kgmole
22 EKC 217: Mass Transfer - Prediction of Diffusivities
Example 7:
Predict the diffusion coefficient of acetone (CH3COCH3) in water at
25C and 50C using the Wilke-Chang equation. The experimental
value is 1.28 x 10-9 m2/s at 25C (298 K).
Solution:
From Appendix 9 (McCabe and Thiele textbook), the viscosity of
water at 25C is 0.90x 10-3 Pa.s and at 50C, 0.55 x 10-3 Pa.s. From
Table 6.3-2, for CH3COCH3 with 3 carbons + 6 hydrogens + 1
oxygen,
VA = 3(0.0148) + 6(0.0037) + 1(0.0074) = 0.074 m3/kgmole
23 EKC 217: Mass Transfer - Prediction of Diffusivities
For water, the association parameter () is 2.6 and MB = 18.02 kg
mass/kgmole.
For 25C, T = 298 K, substituting known values into Eq. (1.56):
/sm 10 x 1.277
)(0.0740)10 x (0.9
(298)18.02) x )(2.610 x (1.173
)(10x173.1
29-
0.63-
1/216-
6.0
5.016
V AB
BAB
TMD
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For 50C, T = 323 K:
/sm 10 x 2.25
)(0.0740)10 x (0.55
(323)18.02) x )(2.610 x (1.173
)(10x173.1
29-
0.63-
1/216-
6.0
5.016
V AB
BAB
TMD
Therefore, the predicted diffusivity is very close to the experimental
value.
25 EKC 217: Mass Transfer - Prediction of Diffusivities
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Schmidt number
DDSc
k
cpPr
The ratio of the kinematic viscosity to the molecular diffusivity is
known as Schmidt number, Sc.
------ (1.57)
Similar to Prandtl number:
------ (1.58)
Sc for gases in air at 0C and 1 atm is given in Appendix 18, range
from 0.5 – 2.0.
Sc for liquids range from 102 to 105 for typical mixtures.