mass transfer estimation of diffusivities - eth z · mass transfer – diffusivities. 9-5. for...

8. November 2017

Estimation of DiffusivitiesLecture 8, 08.11.2017, Dr. K. Wegner

Mass Transfer

Mass Transfer – Diffusivities 9-2

8. ESTIMATION OF DIFFUSIVITIESDiffusion coefficients in gases: - Around 10-1 cm2/s- Can be estimated theoretically

Diffusion coefficients in liquids:- Around 10-5 cm2/s- Not as reliably estimated as for gases

Diffusion in solids:- Around 10-8 – 10-10 cm2/s- Strongly dependent on temperature- Strongly dependent on material (metals, glasses, polymers, etc.)


8.1 Diffusion Coefficient (Diffusivity) of GasesBackground:Kinetic theory of gases, proposed by Maxwell, Boltzmann, Clausius

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 (diameters much smaller than the average distance between collisions)

The molecules only interact through perfectly elastic collisions (no energy transferred).


Molecules have a distribution of speeds (“Maxwell distribution of speeds”) with an average molecular velocity of:

They collide when their centers come within a distance σ of each other where σ, the collision diameter, is of the order of the actual diameters d of the molecules.

mTk8v B

⋅π= where

ANMm = M = molar mass

σd

hit

miss

v

2A πσ=collision cross section

m = molecular mass


For calculating the frequency of such collisions, first imagine that the positions of all molecules but one are frozen.

The number of molecules with centers inside the collision tube is then given by the number concentration c and the tube volume:

( )tvc*z 2 ∆⋅⋅πσ⋅=

Since the molecules are not stationary, the average relative velocity must be used, which is

v2vrel ⋅=

v2cz 2 ⋅⋅πσ⋅=

The collision frequency is vct*zz 2 ⋅πσ⋅=

∆=

Thus,


Using the ideal gas law this can be expressed as:

v2TR

Npz2A ⋅⋅πσ⋅

⋅⋅

= (c= (# of molecules) / V)

Boltzmann’s constantA

B NRk =

If a molecule travels with mean speed and collides with frequency z,it spends time 1/z between collisions and travels the mean free path:

v

p2Tk

zv

2B

⋅πσ⋅⋅

==λ orc2

12 ⋅πσ⋅

=λ


Mean free path of air:

λair (1atm, 298K) ≈ 65 nm


A

z

c high c low

λ λ

Diffusion coefficient from the kinetic theory of gases

Determine the diffusive flux of molecules from a region of high concentration to 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.


If the concentration of molecules at the plane A is cA, the number concentration (c+) of molecules moving towards plane A at a point one mean free path away from plane A will then be 1/6 of the total number concentration.

λ−=+

dzdcc

61c A

The number concentration of molecules that cross in the negative z-direction per unit area is:

λ+=−

dzdcc

61c A


So the net flux per unit area (molecules / (cm2 s)) can be calculated by multiplying with the average molecular velocity :v

( )dzdcv

31ccvjjj ⋅λ⋅−=−⋅=−= −+−+

Comparing with Fick’s 1. law gives: λ⋅= v31D

or:c2

1mTk8

31D

2B

⋅σ⋅π⋅

⋅π=

pmTk

32

p2Tk

mTk8

31D 221

23

3

3B

2BB

⋅σ⋅⋅

π=

⋅σ⋅π⋅

⋅π=or:

“self-diffusivity”


For species “A” with mA=MA/NA and σA:

For diffusion of gas A and gas B:

Then the collision diameter is the arithmetic average of the collision diameters of the two species present:

)(21

BAAB σ+σ=σ

2A

A3

3A

3B

AA pM/1TNk

32D

σ⋅⋅

⋅π⋅

=

( ) ( )( )2

BA

BA3

3A

3B

AB

2p

M21M2/1TNk32D

σ+σ⋅

+⋅⋅

π⋅

=


However the standard equation is that of Chapman and Enskog:

in cm2/s

with T (temperature in K), p (pressure in atm), M (molecular weight in g/mol). σAB (collision diameter in Å) and ΩAB (collision integral, dimensionless) are molecular properties obtained best from the book by Poling et al. “The properties of gases and liquids”.

( )AB

2AB

BA3

3AB p

M1M1T10858.1D

Ω⋅σ⋅+⋅

⋅= −

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


The collision integral Ω can be obtained from tables when the energy of interaction εAB (described by the Lennard-Jones potential, also tabulated) is known.

BAAB εε=ε

This equation applies best to non-polar gases (not to H2O and NH3) and low pressures < 10 atm. For higher pressures, polar gases and concentration-dependent diffusivity, check the book by Poling et al..


Source: Cussler “Diffusion”, 3rd edition


Diffusion Coefficients from Empirical Correlations

[ ]21/3i2i

1/3i1i

1/221

1.75-3

) V( ) V( p)M~1/ M~(1/ T 10 D

Σ+Σ

+=

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, g/mol in M~

Vij: Volumes of parts of the molecule j according to the Table

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


Diffusion Coefficients in Gases

Experimental values of diffusion coefficients in gases at 1 atm. Source: Cussler “Diffusion”


8.2 Diffusivity in LiquidsThe estimation of diffusivity in liquids is far more complex and relies heavily on correlations. We will articulate 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.

Particle diffusivity in gases is taken as a model system for molecular diffusion in liquids.

D = f (particle size and gas properties)A. Einstein (1905), Ann. d. Physik 17, 549.

8.2.1 Particle Diffusivity in Gases


t 1

t 0

t 2

t 3

x=0

x=0

t 0

Consider particle transport in one dimension, x

Release N0 equally sized particles at t=0 and observe the distribution of n in space and time

∂∂

∂

∂

nt

D nx

=2

2 (1)

For the boundary conditions at x = 0, x = ∞ and t = 0, the particleconcentration n(x,t) is (see Chapter on Fick’s 2nd law):

( )n x t NDt

xDt

, exp= −

02

2 4π(2)


The mean square displacement of the particles from x=0 at time t is:

( )∫∞+

∞−

= dxt,xnxN1x 2

0

2

Noting that

(3)

x e dxa a

ax2 2 12

−

−∞

+∞=∫

π(see math tables)

eq. (3) becomes using eq. (2):

∫+∞

∞−

−

π= dx

Dt4xexpx

Dt2N

N1x

220

0

2

Dt2

Dt41

Dt412

1Dt2

1=

π

π

= (4)


We can measure by putting spheres in a liquid and follow their motion through a chequered glass.

2x

The goal is to relate the mean square displacement of a particle with the energy required for this “job”.

Force balance on a particle in Brownian motion:

)t(F uf- dtdum +⋅= (5)

Force actingon a particle

Frictional resistance(proportional to u)

(Random) fluctuatingforce arising from thethermal motion of fluid

molecules

where m is the particle mass, u is the particle velocity, t is time and f the friction coefficient. For spheres (Stokes law): f = 3π µ dp with thedynamic viscosity μ of the fluid.


Now multiply both sides of eq. (5) by the displacement x and divide by m. For a single particle:

( ) xm

tFxumf

dtudx +−= (6)

define as β = f/m and A = F(t)/m and remember that:

( ) 2d ux du dx dux u x udt dt dt dt

= + = +

Using these expressions eq. (6) becomes

( ) 2d uxux u Ax

dt+ β = +


Consider ux as the variable, say, ydydt

y u Ax+ = +β 2

Apply the standard formula for ordinary differential equations and integrate from t=0 to t

and obtain:

ux e u e dt e A xe dtt tt

t tt

=′

′ + ′− − ′∫ ∫β β β β2

0 0(7)

where t´ is a variable of integration representing time.

( ) ( ) Pdx Pdxdy P x y Q x y e Qe dx C dt

− ∫ ∫+ = ⇒ = + ∫


Average over all particles:

( ) ( )ux uxN

ux1 2

0

+ +=

Since the mean value of F(t) over a large number of particles vanishes at any given time, A= F(t)/m → 0 and the second term of eq. (7) vanishes:

( ) [ ] =β

=′ββ

= ′ββ−′ββ− ∫t0

t2

tt

0

t2t euetde1ueux

[ ] [ ]= − = −− −e u e u et t tβ β β

β β

2 21 1 (8)

You can also write: ux xdxdt

dxdt

dxdt

= = =2 2

212

(9)


Because the derivative of the mean over particles with respect to time is equal to the mean of the derivative:

From eq. (8) & (9): [ ]dxdt

u e t2 2

21= − −

ββ

Integrate over time from t = 0 to t

( )1eutueutu2x t

2

22t

0

t2

222

−β

+β

=β

+β

= β−β−

( )= + −

−u t e t2 1 1β β

β


For t >> 1/β (or βt >> 1):

( ) tu1tu101tu2x 2222

β≈

β

−β

=

−

β+

β≈ (10)

Invoke the equipartition of energy, meaning that the kinetic energy of particles is equal to that of the surrounding gas molecules:

2Tk

2umW B

2

kin == (11)

Considering (10) and (11) eq. (4) becomes:fTk

fmu

ttu

t2xD B

222==

β==

The Stokes-Einstein expression for D relates D to the properties of the fluid and the particle through the friction coefficient.

(1 dimensional space!)


The Stokes-Einstein equation is limited to cases in which the solute is larger than the solvent. Thus investigators have developed correlations for cases in which solute and solvent are similar in size, e.g.:


The French physicist and 1926-Nobel laureate Jean Baptiste Perrin verified Einstein's theoretical explanation of Brownian motion by studying the motion of an emulsion.

J. Perrin (1909), "Mouvement Brownian et Réalité Moléculaire", Annales de Chimie et de Physique 18, 5-114.


PAP

B2

d3T

NR

d3Tk

t2xD

µπ⋅=

µπ==

232

PA 107

xd3t2TRN ×=

µπ=

His experiments allowed Perrin to determine Avogadro's constant as

where R is the gas constant

Thus, he gave an experimental proof of the kinetic theory by measuring the net displacement. Modern methods show that

23A 10023.6N ×= molecules/mol


Diffusion Coefficients in Liquids

Source: Cussler “Diffusion”

Diffusion coefficients at infinite dilution in non-aqueous liquidsat 25°C, unless noted

Diffusion coefficients at infinite dilution in water at 25°C

mass transfer estimation of diffusivities - eth z · mass transfer – diffusivities. 9-5. for...

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