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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.)
Mass Transfer – Diffusivities 9-3
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).
Mass Transfer – Diffusivities 9-4
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
Mass Transfer – Diffusivities 9-5
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,
Mass Transfer – Diffusivities 9-6
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
Mass Transfer – Diffusivities 9-7
Mean free path of air:
λair (1atm, 298K) ≈ 65 nm
Mass Transfer – Diffusivities 9-8
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.
Mass Transfer – Diffusivities 9-9
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
Mass Transfer – Diffusivities 9-10
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”
Mass Transfer – Diffusivities 9-11
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⋅
=
Mass Transfer – Diffusivities 9-12
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
Mass Transfer – Diffusivities 9-13
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..
Mass Transfer – Diffusivities 9-14
Source: Cussler “Diffusion”, 3rd edition
Mass Transfer – Diffusivities 9-15
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.
Mass Transfer – Diffusivities 9-16
Diffusion Coefficients in Gases
Experimental values of diffusion coefficients in gases at 1 atm. Source: Cussler “Diffusion”
Mass Transfer – Diffusivities 9-17
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
Mass Transfer – Diffusivities 9-18
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)
Mass Transfer – Diffusivities 9-19
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)
Mass Transfer – Diffusivities 9-20
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.
Mass Transfer – Diffusivities 9-21
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 = +
Mass Transfer – Diffusivities 9-22
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
− ∫ ∫+ = ⇒ = + ∫
Mass Transfer – Diffusivities 9-23
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)
Mass Transfer – Diffusivities 9-24
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
Mass Transfer – Diffusivities 9-25
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!)
Mass Transfer – Diffusivities 9-26
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.:
Mass Transfer – Diffusivities 9-27
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.
Mass Transfer – Diffusivities 9-28
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
Mass Transfer – Diffusivities 9-29
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