Diffusion Coefficients in Gases and Liquids

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 1

Diffusion Coefficients in Gases and Liquids (Chapter 5 of the Text)

Measurement: Diffusion coefficients in liquids and gases may be measured from both steady state and unsteady state experiments:

♦ Steady state: design an experiment to measure concentration of diffusing species as

a function of distance. ♦ Unsteady state: design an experiment to measure concentration as a function of time

at one or more fixed locations. ♦ The experiment should be designed in such a way so that the diffusion coefficient is

the only unknown in the mathematical model for the system. ♦ The accuracy/quality of the measured diffusion coefficient largely depends on the

extent to which the model assumptions could be satisfied in the experiments. ♦ Common sources of error in the diffusion measurement are:

I) Unrecognized intrusion of other rate processes II) Inadequate estimate of the secondary rate process, if any

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 2

Diffusion Coefficients in Gases

Molecular diffusion in binary gas systems has been the subject of theoretical and experimental investigation for nearly a century.

Experimental values are available for many gas pairs. Published values should be used whenever available.

As a first approximation, the diffusion coefficients are:

♦ Inversely proportional to pressure ♦ Varies by 1.5 to 1.8 power of the temperature

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 3

Experimental values of diffusivity in gases at 1 atm

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 4

Experimental values of diffusivity in gases at 1 atm

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 5

Estimation of Gas Phase Diffusivity - Chapman-Enskog Theory ♦ Based on molecular motion in dilute gases, subject to the following additional

assumptions: 1. Non-polar gases 2. Molecular interaction involves collision between only two molecules at a time

♦ ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛

Ωσ

⎟⎟⎠

⎞⎜⎜⎝

⎛+×

=

−

scm

P

M1

M1T1086.1

D2

212

21

21

233

12

D12 binary molecular diffusivity of gas 2 in gas 1

σ1 = ½ (σ1 + σ2) (Å)

σ1 and σ2 are collision diameters of gases 1 and 2 calculated from Lennard-Jones potential,

Ω collision integral, ⎟⎠⎞

⎜⎝⎛ εkT

f where 21εε=ε and kkk21 εε

=ε

ε1 and ε2 are the Lennard-Jones force constants for 1 and 2; k is the Boltzman constant

T temperature (K); P total pressure (atm); M1, M2 molecular weights of gases 1 and 2;

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 6

Lennard-Jones potential parameters from viscosity

Average deviation of Chapman-Enskog theory is ~8%

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 7

Approximate estimation of σ and (ε/k) ♦ The following approximate rules are useful for species for which tabulated values are

not available

cT75.0k=

ε

( ) 31cV

65

=σ

Tc and Vc are the critical temperature and volume, respectively.

♦ Tc and Vc are important pure component physical constants and are available for many species in various physical property handbooks.

Appendix A of the following book is one such source:

R.C. Reid, J.M. Prausnitz and B.E. Poling, "The properties of gases and liquids", McGraw-Hill, 4th Edition, Singapore, 1998.

♦ Group contribution methods are available for estimation of these critical constants when measured values are not available. (See chapter 2 of the above book)

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 8

Gas phase diffusivity from empirical correlations ♦ Reliable values may be estimated from a number of semi-empirical corrrelations

developed by fitting limited experimental data. ♦ The following correlation is given by Fuller, Schettler and Giddings (Industrial

Engineering Chemistry, 58 (5), 1966, p.19):

( )

( ) ( )[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=∑∑ s

cm

VVP

M1

M1T00100.0

D2

312

311

21

21

75.1

12

♦ The quantities (ΣV)1 and (ΣV)2 are obtained by summing atomic diffusion volumes for

each constituent of the binary. Other symbols have the same meanings as before. ♦ The constant and the exponents were obtained by fitting 340 data points. 4-7%

deviation at low pressure.

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 9

Values of V and (ΣV) for some simple molecules

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 10

Gas phase diffusion at high pressure ♦ Chapman-Enskog theory and the semi-empirical correlations work well at relatively

low pressures. ♦ Their direct application at high pressure is not very successful. ♦ The following empirical suggestion given by Reid, Sherwood and Prausnitz (Properties

of Gases and Liquids, 3rd Edition, 1977)

ρD = ρoDo Here ρ is density. Subscript o refers to value at low pressure but at the same temperature as the high- pressure data.

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 11

Estimation of Liquid Phase Diffusivity

Molecular theory of the liquid state, as it stands today, does not allow prediction of diffusion coefficient with the same degree of confidence possible for dilute gas systems.

Experimental values should be used as much as possible.

Stokes-Einstein Equation ♦ The system is idealised as liquid solute sphere moving slowly through a continuum of

solvent liquid (1: solute, 2: solvent)

♦ Net velocity of the solute is expected to be proportional to the force acting on it: 1vfForce ×=

where f is the friction coefficient, v1 is microscopic velocity

♦ Stokes law [which applies for Re ≤ 0.1 based on particle (solute in this case) diameter] is valid here:

ooo R4R2R6f πμ+πμ=πμ=

where Ro is the radius of the solute sphere (=½σ) and μ is viscosity of the solvent. Form drag Friction drag

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 12

♦ Einstein took chemical potential gradient as the force acting on the solute molecule:

( ) 1o1 vR6

dzd

πμ=μ

−

Assuming that the solution is ideal: 11o1 xlnkT+μ=μ−

21

110 cc

clnkT+

+μ=

( )21o1 clnclnkT −+μ= k Boltzmann's constant x solute mole fraction c1, c2 concentration of solute, solvent T temperature (K) μ1(μ1o) chemical potential, at reference state

Since dilute solution, c1 + c2 ≈ c2 ≈ constant

dzdc

c1kT

dzdc

dcclndkT

dzclndkT

dzd 1

1

1

1

111 ===μ

∴

μ is viscosity and μ1 is chemical potential

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 13

Flux = c1v1 = -dz

dcD 112

or, dz

dcDdz

dR6

1c 112

1

o1 −=⎟⎟

⎠

⎞⎜⎜⎝

⎛ μπμ

−

or,dz

dcDdz

dcckT

R61c 1

121

1o1 −=⎟⎟

⎠

⎞⎜⎜⎝

⎛πμ

−

or, o

12 R6kTDπμ

=

How far can it capture experimental trends?

Diffusivity is approximately proportional to T

Inversely proportional to μ when

The prediction breaks down with decreasing solute size

In high viscosity solvents, the diffusivity seems to vary with (-2/3) power of viscosity

In very high viscosity medium, diffusivity becomes independent of viscosity

Solute radius Solvent radius

>5

D12 will be in (cm2/s) if T in (K), Ro in (cm), μ in (g/cm-s) and k=1.38×10-16 g-cm2/s2-K) are used.

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 14

Liquid phase diffusivity from empirical correlations ♦ A number of empirical correlations have been proposed in the literature for

estimating diffusion coefficients. ♦ It is interesting to note that all the correlations in the attached table show the same

temperature and viscosity dependence as the Stoke-Einstein equation. ♦ The Wilke & Chang correlation is more popular. It accounts for solute-solvent

interaction through the factor φ. ♦ All the correlations require molal volume of the solute ( 1V ) or both solute and

solvent ( 21 V,V ) , which are obtained from LeBas atomic and group volumes listed below:

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 15

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 16

Table: Alternatives to Stoke-Einstein equation for diffusion in liquids

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 17

Plot of Wilke-Chang correlation A = 1 (i.e., solute); B = 2 (i.e., solvent)

♦ The correlations are valid for dilute solutions and work better for when solute and solvent are similar in size.

♦ Based on experimental

results for solvent viscosity effect, it should be clear that these correlations will not work for diffusion in high viscosity solvents.

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 18

Experimental data on liquid phase diffusivity ♦ Gives idea about order of

magnitude ♦ Will serve as useful reference

source

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 19

S Farooq/Advanced Separation Processes – Review – Estimation of Diffusion Coefficients 20

Diffusion in Polymer Solutions

Two different categories are of interest here: ♦ Diffusion of high molecular weight substance in a liquid solvent.

♦ Diffusion of gases and other low molecular weight solutes in a solution of

polymer.

For large unhydrated molecules at low concentration in water, the following simple correlation has been reported:

315M1074.2D −−×= (cm2/s)

M is the molecular weight of the large molecule. For a modest range of polymer concentration, the diffusion coefficient of small, nonreacting solutes are not much smaller in polymer solutions than in the pure solvent.

Solvent viscosity should be that of the polymer solution.