2 Prediction Of Diffusivities.pdf

1

PREDICTION OF DIFFUSIVITIES

Diffusivities are best determined by experimental

measurements.

The diffusivity of some common gases diffusing in

air at 0C and 1 atm (can be obtained from the text

book (McCabe, Smith and Harriott, Appendix 19).

The value of diffusivities can also be predicted using

various empirical equations.

1) Diffusion in gases

EKC 217: Mass Transfer - Prediction of Diffusivities

2

The equations developed by Chen and Othmer is:

------ (1.50)

where:

DAB = Diffusivity (cm2/s) T = Temperature (K) MA, Mb = Molecular weights of component A & B respectively TcA, TcB = Critical temperatures of A & B respectively (K) VcA, VcB = Critical molar volumes of A & B respectively (cm3/gmol) P = Pressure (atm)

24.04.01405.0

5.0

81.1

)()(

1101498.0

cBcAcBcA

BA

ABVVTTP

MMT

D


• Another rigorous equation called Chapman-Enskog equation is used to predict diffusivities:

5.0

2

5.1001858.0

BA

BA

DAB

ABMM

MM

P

TD

------ (1.51)

where:

DAB = Diffusivity (cm2/s)

T = Temperature (K)

MA, MB = Molecular weights of component A & respectively

P = Pressure (atm)

AB = average collision diameter

ΩD = collision integral based on the Lennard-Jones potential

= f (kT/AB )

k = Boltzmann’s constant

= Lennard-Jones force constant for common gases

BAAB

3 EKC 217: Mass Transfer - Prediction of Diffusivities

The collision integral, ΩD decreases with increasing T, that makes

DAB increase with more than the 1.5 power of the absolute T.

For diffusion in air at T = 300 – 1000 K, DAB varies with about T 1.7-

1.8, and T 1.75 can be used to extrapolate from room temperature

data.

Therefore, we can say that: DAB T 1.75 x 1/P

Hence, the diffusivity at T2 and P2 relative to the standard

temperature and pressure (STP) can be calculated as follows:

75.1

2

2

,2,

STP

STPSTPABAB

T

T

P

PDD ------ (1.52)


Eq. (1.51) is relatively complicated to use, and often some of the

constants such as AB are not available of difficult to estimate.

Hence, semi empirical method by Fuller et al. is often preferred:

23/13/1

5.0

75.17 )/1/1(10x00.1

BA

BAAB

P

MMTD

------ (1.53)

Eq. (1.53) was obtained by correlating many recent data and uses

atomic values from Table 6.2-2. This method can be used for

mixtures of nonpolar gases or for a polar-nonpolar mixture.

where: A = sum of structural volume increments



Example 5:

Predict the volumetric diffusivity for benzene in air at 100C and 2

atm by using the rigorous equation (1.51) and by extrapolating from

the published value for 0C and 1 atm.

Solution:

From Appendix 19 (McCabe, Smith and Harriott), the force

constants are as follows:

/k M

Benzene 412.3 5.349 78.1

Air 78.6 3.711 29


Thus,

53.42

711.3349.5

AB

1806.78x3.412k/ AB

072.2180

373k

AB

T

From Appendix 19, D = 1.062.

Substitute values into Eq. (1.51) gives:

/scm0668.0062.1x53.4x2

29x1.78

291.78373x001858.0

2

2

5.0

5.1

ABD


From Appendix 18 at standard temperature and pressure,

/scm0772.0/hft299.0 22 ABD

At 373 K and 2 atm,

/scm0666.0273

373

2

10772.0 2

75.1

ABD

Therefore, agreement with the value calculated from Eq. (1.51)

is very good.


Example 6:

In an oxygen-nitrogen gas mixture at 1 std atm, 25C, the

concentration of oxygen at two planes 2 mm apart are 10 and

20 vol %, respectively. Calculate the flux of diffusion of the

oxygen for the case where:

a) The nitrogen is non-diffusing

b) There is equimolecular counterdiffusion of the two gases.

DO2-N2 at 0C and standard atmospheric pressure

= 1.81 x 10-5 m2/s


Solution:

Component A = O2

Component B = N2

A + B A

2 mm

pA1

= 0.2PT

pA2

= 0.1PT

NB = 0

Given:

PT = 1 atm = 1.0133 x 105 N/m2

T = 25C = 298 K

DO2-N2 = 1.81 x 10-5 m2/s (0 C , 1 atm)

Diffusivity of O2-N2 mixture at 25C , 1

atm:

/sm 10 x 2.06

273

298x10x81.1

25-

5.1

5

298,22

KNOD


pA1 = 0.2 PT

pA2 = 0.1 PT

Then, pB1 = PT – 0.2PT = 0.8PT

pB2 = PT – 0.1PT = 0.9PT

)()(

21

12

AA

BM

TABA pp

pzzRT

PDN

)/ln( 12

12

BB

BBBM

pp

ppp

Calculate pBM :

and

T

TT

TBM P

PP

Pp 849.0

)8.0/9.0ln(

)8.09.0(

178.1

849.0

T

T

BM

T

P

P

p

P

a) For the case where nitrogen is non-diffusing, Eq. (1.37) is used:


sm

kmole 10 x 4.95

m

N 10 x 1.0133 x 1.02.0

)m002.0(K) )(298Kkmole

Nm (8314

178.1 s

m 10 x 2.06

)()(

2

5-

2

5

25-

21

122

AA

BM

TABO pp

pzzRT

PDN

Substitute the known values into Eq. (1.37):


b) When there is equimolecular counter diffusion of the two gases,

the following equations are applied:

)( 12

21

zzRT

ppDJ AAAB

A

)( 12

21

zzRT

ppDJ BBAB

B

Substitute the known values:

sm

O kmole 10 x 4.21

)m10 x 2)(K298(Kkmole

Nm8314

m

N10 x 1.0133 x 1.02.0

s

m10 x 2.06

2

25-

3

2

52

5

2

OJ


s/mN kmole 10 x 4.21-

m)10 x K)(2 (298Kkmole

Nm8314

m

N10 x 1.0133 x 0.90.8

s

m10x2.06

2

2

5-

3

52

5

2

NJ

For nitrogen:


16

Exercise: Estimation of Diffusivity of a Gas Mixture

Normal butanol (A) is diffusing through air (B) at 1 atm abs. Using

the Fuller et al. method, estimate the diffusivity DAB for the

following temperatures and compare with the experimental data:

a) For 0C

b) For 25.9C

c) For 0C and 2 atm abs.

Ans:

a) Deviation of +10% from the experimental value (in Table 6.2-1).

b) Deviation of +4% from the experimental value.

c) DAB = 3.865 x 10-6 m2/s


17

2) Diffusion in Small Pores

M

TrDK 9700

KABpore DDD

111

The diffusion process is called Knudsen diffusion.

Occurs when the pore size is much smaller than the normal mean free

path during gas diffusion in very small pores of a solid, for processes

like adsorption, drying of porous solids or membrane separation.

The diffusivity for a cylindrical pore is:

------ (1.54)

where: DK = Knudsen diffusivity, cm2/s

T = temperature, K

M = molecular weight

r = pore radius, cm

Diffusivity in the pore: ------ (1.55)


18

3) Diffusion in Liquids

V AB

BAB

TMD

6.0

5.016 )(

10x173.1

A widely used correlation for liquid diffusivity of small molecules,

called Wilke-Chang equation:

------ (1.56)

where: DAB = diffusivity, m2/s

T = absolute temperature, K

B = viscosity of B in Pa.s

VA = molar volume of solute as liquid at its normal

boiling point, m3/kgmole (taken from Table 6.3-2)

= association parameter of the solvent

MB = molecular weight of solvent B (kg/kgmole)


Recommended values for :

Water 2.6

Methanol 1.9

Ethanol 1.5

Benzene, heptane and other unassociated solvents 1.0

Eq. (1.56) is valid only at low solute concentrations and does not

apply when the solution has been thickened by addition of high-

molecular-weight polymers.


EKC 217: Mass Transfer - Prediction of Diffusivities 20

EKC 217: Mass Transfer - Prediction of Diffusivities 21

For water as solute, VA = 0.0756 m3/kgmole.

If other compound is the solute, e.g.: acetone (CH3COCH3) in

water, the value of VA can be calculated by considering the atomic

volumes as given in Table 6.3.2 – Atomic and Molar Volumes at the

Normal Boiling Point.

Acetone (CH3COCH3) has 3 carbons, 6 hydrogens and 1 oxygen.

Therefore,

VA = 3(14.8 x 10-3) + 6(3.7 x 10-3) + 1(7.4 x 10-3)

= 0.0740 m3/kgmole


Example 7:

Predict the diffusion coefficient of acetone (CH3COCH3) in water at

25C and 50C using the Wilke-Chang equation. The experimental

value is 1.28 x 10-9 m2/s at 25C (298 K).

Solution:

From Appendix 9 (McCabe and Thiele textbook), the viscosity of

water at 25C is 0.90x 10-3 Pa.s and at 50C, 0.55 x 10-3 Pa.s. From

Table 6.3-2, for CH3COCH3 with 3 carbons + 6 hydrogens + 1

oxygen,

VA = 3(0.0148) + 6(0.0037) + 1(0.0074) = 0.074 m3/kgmole


For water, the association parameter () is 2.6 and MB = 18.02 kg

mass/kgmole.

For 25C, T = 298 K, substituting known values into Eq. (1.56):

/sm 10 x 1.277

)(0.0740)10 x (0.9

(298)18.02) x )(2.610 x (1.173

)(10x173.1

29-

0.63-

1/216-

6.0

5.016

V AB

BAB

TMD


For 50C, T = 323 K:

/sm 10 x 2.25

)(0.0740)10 x (0.55

(323)18.02) x )(2.610 x (1.173

)(10x173.1

29-

0.63-

1/216-

6.0

5.016

V AB

BAB

TMD

Therefore, the predicted diffusivity is very close to the experimental

value.



26

Schmidt number

DDSc

k

cpPr

The ratio of the kinematic viscosity to the molecular diffusivity is

known as Schmidt number, Sc.

------ (1.57)

Similar to Prandtl number:

------ (1.58)

Sc for gases in air at 0C and 1 atm is given in Appendix 18, range

from 0.5 – 2.0.

Sc for liquids range from 102 to 105 for typical mixtures.

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