Diffusion

Diffusivities in Gases: D AB =

Reid & Sherwood(1966)

0.0018583T 3 / 2 1/ M A + 1/ M B P(σ AB ) 2 Ω D , AB

Binary air-hydrocarbon or non-hydrocarbon gas mixtures at low pressure. 0.00100T 7 / 4 D AB = 1/ M A + 1/ M B Fuller et al. (1966) P[(∑ν )1A/ 3 + (∑ν )1B/ 3 ]2

0.0150T 1.81 1/ M A + 1/ M B 0.4 0.4 2 P (TCATCB ) 0.1405 (VCA + VCA )

Chen & Othmer (1962)

D AB =

Chen & Othmer (1962)

⎡ 1/ M A + 1/ M B ⎤ 2.74 D AB = (2.52 × 10 7 ) µ air ⎢ 0.4 0.4 2 ⎥ ⎣⎢ (VCA + VCB ) ⎥⎦

Gas diffusivity of binary hydrocarbon-hydrocarbon gas systems at low pressure 0.5 0.1014T 1.5 (1 / M A + 1 / M B ) Gilliland D AB = 2 P V A1 / 3 + VB1 / 3

(

)

Binary mixture low pressure-non polar: 0.0027 − 0.0005M 1AB/ 2 T 3 / 2 M 1AB/ 2 D AB = 2 Pδ AB ΩD

(

)

Diffusivities of multi-component gas mixtures: ⎡ ⎤ ⎛ Nc ⎞⎤ Nc ⎡⎛ xN ⎞ Stefan-Maxwell, Smith & Taylor Dim = ⎢1 − xi ⎜⎜ ∑ N j / N i ⎟⎟⎥ / ∑ ⎢⎜⎜ x j − i i ⎟⎟ / Dij ⎥ Ni ⎠ ⎢⎣ ⎝ j =1 ⎠⎥⎦ j =1 ⎣⎝ ⎦ −1

Blanc

⎛ Nc x ⎞ Dim = ⎜ ∑ j ⎟ ⎜ j =1 D ⎟ ij ⎠ ⎝

−1

Wilke

⎛ Nc ⎞ xj ⎟ ⎜ Dim = ⎜ ∑ ⎟ ⎜ ij≠=1j Dij ⎟ ⎝ ⎠

Diffusivities in Liquids: ° D AB µ AB 7.4 × 10 −8 (ξM B )1 / 2 = T VbA0.6

Wilk & Chang (1955)

(Excluded water as solute) For unassociated solvents ξ=1.0; for water ξ=2.6; for methanol ξ=2.6; for ethanol ξ=1.5.

Scheibel (1954)

D

° AB

⎡ ⎛ 3V ⎤ ⎡ T = 8.2 × 10 −8 ⎢1 + ⎜⎜ bB = K⎢ 1/ 3 ⎥ ⎢⎣ ⎝ VbA ⎣ µ ABVbA ⎦

⎞ ⎟⎟ ⎠

2/3

⎤⎡ T ⎤ ⎥⎢ 1/ 3 ⎥ ⎥⎦ ⎣ µ ABVbA ⎦

(Excluded water as solute) Special solvent cases:water VbA
Othmer & Thakar (1953)

D

° AB

14 × 10−5 = 0.6 1.1∆H BT ∆H WT VbA µ B′ µWT

(Excluded water as solute)

Othmer & Thakar (1953)

° DAB =

14 × 10−5 1.1 0.6 µWT VbA

(Aqueous solutions only)

Olander (1961)

D°Water as solute=(D°Wilk & Chang) / 2.3 ° AB

⎛ M 1 / 2 ∆H 1 / 3T ⎞ = 5.4 × 10 ⎜⎜ B 0.5 B 0.3 ⎟⎟ ⎝ µ BVbA ∆H A ⎠

0.93

−8

Sitaraman et al (1963)

D

King et al (1963)

° ⎛V ⎞ µB DAB = 4.4 × 10−8 ⎜⎜ bB ⎟⎟ T ⎝ VbA ⎠

Reddy & Doraiswamy (1967)

° µ B 10 × 10−8 M B1 / 2 VbB DAB = ; ≤ 1.5 T VbA1 / 3VbB1 / 3 VbA

Reddy & Doraiswamy (1967)

° D AB µ B 8.5 × 10 −8 M B1 / 2 VbB = ; > 1.5 T VbA VbA1 / 3VbB1 / 3

Lusis & Ratcliff (1968)

1/ 3 ° ⎡ ⎛ VbB ⎞ DAB V ⎤ µB −8 −1 / 3 ⎟⎟ + bB ⎥ = 8.52 × 10 VbB ⎢1.40⎜⎜ T VbA ⎥ ⎢⎣ ⎝ VbA ⎠ ⎦

1/ 6

(Organic solvents)

1/ 2

⎛ ∆HmB ⎞ ⎜⎜ ⎟⎟ ∆ Hm A ⎠ ⎝

D AB =

Sun & Chen

1.23 ×10 −10 T µ 0.799VCA0.49

(ρ D AB = 5.152 DC Tr

Catchpole & King

⎛ M ⎞ − 0.4510 ⎜⎜1 + A ⎟⎟ R ⎝ MB ⎠

)

−0.667

(1 + (V

General mixtures: ° AB

=

Tyn-Claus

D

Umesi-Danner

° D AB =

Siddiqi-Lucas

D

° AB

(

8.93 ×10 −8 V A / VB2

0.333 2

) (Ψ 1/ 6

B

µB

(

)

/ VCA )

CB

/ ΨA ) T 0.6

)

2.75 × 10 −8 RB / R A2 / 3 T

µ

−8

9.89 × 10 VB0.265T = µ B0.907V A0.45

Gases in low viscosity liquids: Sridhar-Potter

D

° AB

D

° AB

⎛V ⎞ = DBB ⎜⎜ CB ⎟⎟ ⎝ VCA ⎠

2/3

⎛ VB ⎜⎜ ⎝ VmlB

⎞ ⎟⎟ ⎠

(βVCB )2 / 3 (RTCB )1/ 2 (V − 1)⎛⎜ T ⎞⎟ r 1/ 3 ⎜T ⎟ M 1A/ 6 (M BVCA ) ⎝ CB ⎠

1/ 2

Chen-Chen

= 2.018 ×10

Aqueous Solution:

−9

13.16 × 10 − 5 µW1.14V A0.589

Hayduk-Laudie

° = D AW

Siddiqi-Lucas

° D AB = 2.98 ×10 −7 V A−0.5473 µW−1.026T

Hydrocarbon Mixtures: Hayduk-Mihas

° D AB = 13.3 ×10 −8 T 1.47µ B(10.2 / VA −0.791)V A−0.71

Matthewes-Akgerman

° D AB = 32.88M A−0.61VD−1.04T 0.5 (VB − VD )

Riazi-Whiston

D AB

( ρD AB )° ⎛ µ ⎜ = 1.07 ρ

⎞ ⎜ µ ° ⎟⎟ ⎝ ⎠

−0.27 −0.38ω + ( −0.05+ 0.1ω ) Pr

Diffusion of dilute species A in a mixture of two solvents: Cullinan & Cusick (1967)

⎡ ⎤ xC xB lim D = ⎢ + ⎥ ° )xB (α ABC DBC° )xC (DAC° )xC (α ACB DCB° )xB ⎥⎦ ⎢⎣ (D AB xA → 0 ° A

α ABC =

° ° / DBC ) VC (1 − D AC D° , 1 − AC > 0.25 ° V A − VB DBC

α ABC =

VC D° , 1 − AC < 0.25 ° VA DBC

α ACB =

° ° / DCB ) VB (1 − D AB D° , 1 − AB > 0.25 ° V A − VC DCB

α ACB

° VB D AB = , 1 − ° < 0.25 VA DCB

Simpler and somewhat more effective form:

(lim D )µ ° A

xA → 0

ABC

(

° = D AB µB

)

xB

(

° + D AC µC

)

xC

−1

Concentrated Solutions of Non-Electrolyte: Gordon (1937) James et al. (1939) Aqueous solutions:

d ln γ A ° ⎟⎟ ⎜⎜1 + (D A )Conc. = D AB

⎞ µB

⎛

d ln x A ⎠ µ AB

⎝

Powell et al. (1941) Wilke (1949) ⎛ D° µ D° µ ⎛ D A µ AB ⎞ Ideal solutions: = ⎜⎜ BA A − AB B ⎟ ⎜ T ⎝ T ⎠ Conc. ⎝ T

⎞ D° µ ⎟⎟ x A + AB B T ⎠

Powell et al. (1941) Wilke (1949) ⎡⎛ D ° µ D° µ ⎛ D A µ AB ⎞ = ⎢⎜⎜ BA A − AB B Non-ideal solutions: ⎜ ⎟ T ⎝ T ⎠ Conc. ⎣⎝ T

(

)

(

)

Caldwell-Babb

° ° DAB = xB DBA + xB DAB βA

Rathbun-Babb

° ° DAB = x A DBA + xB DAB β An

DAB

⎡ K ⎛ ∂ ln xA ⎞⎤ = D° ⎢1 + − 1⎟⎥ ⎜ ⎣ x A x B ⎝ ∂ ln aA ⎠⎦

DAB

KT = 1/ 3 2πµmix (V / A)

Ausfour-Dullien

DAB

⎛ D° ⎞ = ⎜⎜ AB ⎟⎟ ⎝ µB ⎠

Siddiqi-Lucas

° ° DAB = C BVB DAB + C AVA DBA βA

Cussler

Cullinan

xB

⎞ D ° µ ⎤⎛ d ln γ A ⎞ ⎟⎟ x A + AB B ⎥⎜⎜1 + ⎟ T ⎦⎝ d ln x A ⎟⎠ ⎠

−1 / 2

⎡ ⎤ 2πx A xB β A ⎢1 + β (2πx x − 1) ⎥ A A B ⎣ ⎦

1/ 2

xA

° ⎛ DBA ⎞ ⎜⎜ ⎟⎟ ξµβ A µ ⎝ A ⎠

(

)

Non-associated solutions, ideal and non-ideal; also associated solutions if degree of association is constant. Poor for binaries of n-alkanes. Vignes (1966)

° (DA )Conc. = (DAB )x

Leffler & Cullinan (1970)

x ° (D A µ AB )Conc. = (D AB µB )

B

(D )

° xA BA

B

⎛ d ln γ A ⎞ ⎟⎟ ⎜⎜1 + ⎝ d ln x A ⎠

(D

° BA

⎛

µ A ) ⎜⎜1 + xA

⎝

d ln γ A ⎞ ⎟ d ln x A ⎟⎠

Electrolytes: Dilute solution of a single salt ⎛ ι 0ι 0 ⎞⎛ Z + Z − ⎞ ⎟⎟ DA° = 8.931 × 10 −10 T ⎜⎜ 0 + − 0 ⎟⎟⎜⎜ + ⎝ ι+ + ι− ⎠⎝ Z + Z − ⎠ ι : Cationic / anionic conductance at infinite dilution, mho/equivalent

Nernst-Haskell

Self Diffusivity, High Pressure: Mathur-Thodos

D AA =

10.7 ×10 −7 Tr ; ρ r ≤ 1.5 β .ρ r

Lee-Thodos

D AA =

0.77 ×10 −5 Tr ; ρr ≤ 1 δ .ρ r

Lee-Thodos

D AA =

(0.007094G + 0.001916)2.5 Tr ; ρ δ

r

> 1, G < 1

Pore Diffusivity, Gas phase diffusion in small pores at low pressure: 1/ 2 1 ⎡ 3 ⎛ πM ri ⎞ 1⎤ DPi = ⎟ + ⎥ ⎜ ⎢ Di ⎥⎦ τ P ⎣⎢ 4rPore ⎝ 2 RT ⎠

Satterfield (1970)

For liquid-phase diffusion of large adsorbate molecules: DPi =

λm =

Di

τP

rm rPore

(1 − λm )−2 ⎡⎢1 + 9 λm ln λm − 1.239λm ⎤⎥ ⎣

8

⎦

rm: Stokes-Einstein radii of the solute

−1