mass transfer lecture no 5 estimation of diffusivities · self-diffusivity versus mutual...

Mass Transfer

Lecture No 5

Estimation of Diffusivities

1

Mass Transfer, Fall 2019

Bibliography

Material taken from:

1. E.L. Cussler, Diffusion – Mass transfer in Fluid Systems, Cambridge

Univ. Press, 3rd Ed., 2007 – Chapter 5.

2. Chapter 5: Values of diffusion coefficients

The Diffusion Coefficient D

• So far, we have treated D as a proportionality constant in Fick’s law.

• To calculate fluxes and concentration profiles, we need to know D.

• We must rely on experimental measurements because there exists no

universal theory to predict D.

• Unfortunately, the experimental measurements are unusually difficult to

perform.

• Thus, we also need to know how we evaluate these measurements.

3

11

dcj D

dz



Values of the Diffusion Coefficient D

• Diffusion coefficients in gases:

- around 10-1 cm2 / s

- can be estimated theoretically

• Diffusion coefficients in liquids:


- not as readily estimated as for gases

• Diffusion coefficients in solids:


- strongly dependent on T and material

• Diffusion coefficients in polymers and glasses:


- strongly dependent on T and solute concentration

4


5.1 Diffusion Coefficient D (diffusivity) of Gases

Background:

Kinetic theory of gases (proposed by Maxwell, Boltzmann, Clausius)

Assumptions of the theory

• A gas consists of molecules of diameter d [m], mass m [kg] and number

concentration c [#/cm3], that are in random motion.

• The size of the molecules is negligible. That is, their diameter d is much

smaller than the average distance between collisions (d << ).

• The molecules only interact through perfectly elastic collisions (no energy

transferred).

• Molecules have a distribution of speeds (Maxwell distribution of speeds).

5

Elements of the Kinetic Theory of Gases

2

2

3/2

14 exp

22

BB

mvf v v

k Tk T

m

2 2 2

3/2

1, , exp

22

x y z

x y z

BB

m v v vf v v v

k Tk T

m

Elements of the Kinetic Theory of Gases

Maxwell-Boltzmann distribution of molecular velocities:

Average velocity where

Molecular Collisions: Two molecules will collide when their centers come

within a distance (collision diameter) of each other; is approximately

equal to the diameter d of the molecule ( = d)

Let us calculate the frequency

of such collisions:

- We first imagine that the

positions of all molecules

except one are frozen.

- Then, we assume that this

molecule travels for a time

interval t.

8

Bk T

vm

Av

Mm

N

Molecular Collisions

• Molecule will collide with all

molecules that are in this collision

tube with cross section

• The # of molecules with centers

inside the collision tube is given by

the overall number concentration c

(= N/V) times the tube volume:

• Thus, collision frequency:

• Since molecules are not stationary,

the average relative velocity must

be used

• Thus,

* N c v t

*2

1 1

A A

NZ Z c v

t

2relv v

2

1 2 AZ c v 8

2 2 d

Molecular Collisions (cont.)

• Using the ideal gas law

• If a molecule travels with mean speed and collides with

frequency Z1A, then:

– it spends time 1/Z1A between collisions and

– travels the mean free path:

or

2

1 2 A

B

PZ v

k T

v

21 2

B

A

v k T

Z p2

1

2

c

9



• Mean free path of air:

• The average molecular separation is about 10 times the atomic diameter

• The mean free path is about 310 times the nominal atomic dimeter and

about 28 times the average molecular separation

air 1 atm, 298 K 65 nm

10



11


Diffusion coefficient from the kinetic theory of gases

• Let us determine the diffusive flux of molecules from a region of high

concentration to a region of low concentration through the area A.

• Assuming 1/3 of the molecules have motion in z-direction, then 1/6 of

the molecules have motion in positive z direction.

12



• Let cA be the concentration of molecules at the

plane A

• Then, number concentration c+ of molecules

moving towards plane A at a point one mean free

path away from plane A will be

• Similar, number concentration of molecules that

cross in the negative z-direction per unit area is:

• Then, net flux per unit area (molecules / (cm2 s)):

• Comparing with Fick’s law:

1

6

A

dcc c

dz

1

6

A

dcc c

dz

1

3

dcj j j v c c v

dz

1

3D v

13



• D from kinetic theory of gases:

• Substituting for :

• Equivalently:

• Describes dependence of D on T and P!

1

3D v

2

1 8 1

3 2 Bk T

Dm c

3 3/2

3 1/2 22

1 8 2

3 32 B B Bk T k T k T

Dm m pp

14


Self-diffusivity versus mutual diffusivity

• Molecules A diffuse in a gas of

other molecules A, with with mass

mA and collision diameter σA:

• However, often molecules A diffuse

in a gas containing both molecules

A and B. Then, collision diameter

• Then, coefficient of mutual

diffusivity, DAB:

3

3

3 2

1

2

3

AB Av

AA

A

TMk N

Dp

1

2 AB A B

3

3

23

1 1

2 22

3

2

A BB AvAB

A B

TM Mk N

D

p

15Mass Transfer, Fall 2019


However the standard equation is that of Chapman and Enskog:

Here:

• T (temperature in K), p (pressure in atm), M (molecular weight in g/mol)

• AB (collision diameter in Å) and AB (collision integral, dimensionless) are

molecular properties obtained best from the book by Poling et al.: The

properties of gases and liquids.

References:

• B.E. Poling, J.M. Prausnitz, J.P. O’Connell, “The properties of gases and

liquids”, McGraw-Hill, 5th ed., 2000.

• Earlier editions by R.C. Reid, J.M. Prausnitz and B.E. Poling

3

3 2

2

1 1

1.858 10 in cm /s

A B

AB

AB AB

TM M

Dp

16


• Collision integral Ω can be obtained from tables!

• need to know the energy of interaction εAB (described by the Lennard-Jones

potential, also tabulated).

• typically we use:

• equation applies best to non-polar gases (not to H2O and NH3) and low

pressures (p < 10 atm).

• for higher pressures, polar gases and concentration-dependent diffusivity,

check the book by Poling et al.

AB A B

17



Source

E.L. Cussler, Diffusion – Mass transfer in Fluid

Systems, Cambridge Univ. Press, 3rd Ed., 2007.

18

Diffusion coefficients from empirical correlations

• The Chapman-Enskog theory requires the knowledge of Lennard-Jones

potential parameters which are not always known and assumes non-polar

molecules.

• Other estimates of diffusion coefficients are based on empirical correlations,

like the one of Fuller et al. (1966):

• T in K, p in atm, in g/mol

• Vij: volumes of parts of molecules i and j (according to the next Table)

1/2

1.75

3 2

21/3 1/3

1 2

1 1

10 in cm /s

A B

AB

i i

i i

TM M

D

p V V

Fuller, E.N., Schettler, P.D., Giddings, J.C. (1966), Ind. Eng. Chem. 58, 19.

M

19

Diffusion coefficients from empirical correlations

Fuller, E.N., Schettler, P.D., Giddings, J.C. (1966), Ind. Eng. Chem. 58, 19.

20


Diffusion coefficients in Gases at 1 atm: Some data

Source: E.L. Cussler, Diffusion – Mass transfer in Fluid

Systems, Cambridge Univ. Press, 3rd Ed., 2007.

5.2 Diffusion Coefficient D in Liquids

Background:

• The estimation of diffusivity in liquids is far more complex and relies heavily

on correlations.

• We will describe it here in the frame of the Stokes-Einstein equation (which

is also the basic framework for particle diffusivity in gases).

• This equation describes the diffusion of a spherical particle undergoing

Brownian motion in a quiescent fluid at uniform temperature.

A. Einstein (1905), Ann. d. Physik 17, 549. 23

t 1

t 0

t 2

t 3

x=0

x=0

t 0

Elements of Brownian theory

• Consider particle transport

in one dimension x across

a surface A

Elements of Brownian Motion

rate of accumulation rate of diffusion rate of diffusion =

in the volume into the layer at out of the layer at

A x x x x

1 1

1 1 1

, ,= =

c x t j x tA xc A j x A j x x

t t x

11

cj D

x24


t 1

t 0

t 2

t 3

x=0

x=0

t 0 2

1 1

2

, ,

c x t c x tD

t x

Governing equation:

Initial and Boundary conditions:

01 0

1

1

0,

, 0

, 0

Nt c x x

A

x c t

x c t

Solution:

Probability of molecule to be at

position x within dx after time t:

2

01 , exp

42

N xc x t

DtA Dt

21

, exp44

xp x t

DtDt25


t 1

t 0

t 2

t 3

x=0

x=0

t 0

Let us calculate the mean square

displacement (msd) in x after time t:

2 2

22

22

msd ,

1exp

44

1exp

44

x t x t p x t dx

xx dx

DtDt

xx dx

DtDt

If we use:

We find:

2 2 1 1exp , 4

2

x ax dx Dta a a

2 1 4msd 4 2

24

Dtx t Dt Dt

Dt 26

Measurement of Diffusion Coefficient D

Strategy:

• We can measure D by monitoring how particles are displaced in time.

• From the slope of the mean-square displacement (msd) in time, we can get

D!

Experiment (!):

• The French physicist and 1926-Nobel laureate Jean Baptiste Perrin studied

the motion of an emulsion.

• He monitored the msd of the droplets!

J. Perrin (1909), "Mouvement Brownian et Réalité Moléculaire", Annales de

Chimie et de Physique 18, 5-114.

2

2msd 2 2

x t

x t Dt Dt

2 2 2

6

x t y t z tD

t

27

Measurement of Diffusion Coefficient D

Remark: The fluid will play a major role in

the msd of the particles!

Question: Can we relate D to the properties

of the fluid?

28



Force balance on moving particle:

Assumption: Particle size >> fluid particle size

Properties: m = mass of particle, = friction coefficient, v = velocity

Properties of the random force:

Our goal: Let us try to calculate the msd for this Brownian particle!

r

dvm v F

dt

Frictional resistance

(proportional to v)

(Random) fluctuating

force arising from the

thermal motion of fluid

molecules

Mass times

Acceleration

0 0

0

0 for

r

r r

F t

F t F t t t

29


2 2

2

r

r r

r

d xv FdY dx dvv x vv x v

dt dt dt dt m m

F FdY dYv x xv v x Y

dt m m dt m m

FdYY v x

dt m m

We formulate first an equation for the quantity: Y xv

The latter is of the form:

It can be solved for Y(t):

2, .,

rF tdYP t Y Q t P t const Q t v

dt m m

e e

P t dt P t dt

Y t Q t dt C

30



Substituting for P(t) and Q(t), and

doing the integrations, we get:

: a new integration variable.

Now, we should take the average of

this quantity over many particle

configurations or particle

trajectories:

e e

P t dt P t dt

Y t Q t dt C

2

0 0

e

t tt t t t

rm m m mF t

xv v e dt e xe dtm

t

2

0 0

e

t t

t t t trm m m m

Fxv v e dt e x t e dt

m

31



But the mean value of the random force

over many different particle positions

vanishes. Thus:

Also (equi-partition theorem):

Thus, overall:

But,

2

0

e

t

t tm mxv v e dt

2 21 1

2 2 B

B

k Tm v k T v

m

0

e

1

t

t tB m m

tB m

k Txv e dt

m

k Txv e

221 1

2 2

d xdx dxxv x

dt dt dt32



Combining the last two equations, we get:

This can be integrated for <x2> to give:

For sufficiently long times:

This can be compared with Einstein’s theory:

2

11

2

tB m

d x k Te

dt

2

0

2

21

21

t

tB m

tB m

k Tx e dt

k T mx t e

2 22 2

B Bk T k Tmx t x t

2 2x Dt 33


Bk T

DFinal equation for D:

We also know that:

Thus, we arrive at the Stokes-

Einstein expression for D in terms

of the properties of the fluid and the

particle through the friction

coefficient!!!

6 HR

6 B

H

k TD

R

34



Source: E.L. Cussler, Diffusion – Mass

transfer in Fluid Systems, Cambridge Univ.

Press, 3rd Ed., 2007.

35



Perrin’s experiments allowed him to determine Avogadro's constant as

R: the gas constant

Thus, Perrin gave an experimental proof of the kinetic theory by measuring the net

displacement.

Modern methods show that

J. Perrin’s experiment

36

2

2 3 3 B

p Av p

x k T R TD

t d N d

23

2

27x10 molecules/mole

3 Av

p

RT tN

d x

236.023x10 molecules/moleAvN

J. Perrin (1909), "Mouvement Brownian et Réalité Moléculaire",

Annales de Chimie et de Physique 18, 5-114.

• The Stokes-Einstein equation is limited to cases in which the solute is larger

than the solvent.

• Correlations for cases in which solute and solvent are similar in size, e.g.:


5.3 Diffusion Coefficient D in Solids

• Diffusion in solids is too slow (except for hydrogen – also known as the

hydrogen diffusion)

• However, these values increase quickly with the temperature T

• Typical expression for D:

• R0 = spacing between atoms, N = fraction of vacant sites in the crystal, =

jump frequency (number of jumps from one position to the other per unit time)

• Dependence on T:

• H = enthalpy of activation (quite large, often above 100 kJ/mole)

2

0 D R N

/

0 e H RTD D

38


5.4 Experimental measurement of diffusion coefficients

1. Diaphragm Cell

2. Infinite Couple

3. Taylor Dispersion Lecture No 9

4. Spin Echo NMR

5. Dynamic Light Scattering

…

…

…

…

1, bottom 1, top initial

1, bottom 1, top at time t

1ln

c cD

t c c

1 1

1, 1

erf4

c c z

c c Dt

39


40

Lecture No 5 – Summary

1. Kinetic Theory of Gases

2. Brownian motion

3. Measurement of diffusivities

End of Lecture No 5!

Thank you!!!


mass transfer lecture no 5 estimation of diffusivities · self-diffusivity versus mutual...

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