1

I3. MASS TRANSFER AND DIFFUSION

I3.1. INTRODUCTION

The movement of one type of molecules through other types of molecules is influenced by the

concentration gradient, the physical and molecular properties of the participating species and

the external forces. These factors affect the rate of transfer of the molecules. This molecular

interaction is the basis of determining the rate of mass transfer, which is important in the design

of mass transfer equipment such as gas absorbers, humidifiers, distillation columns, and others.

To simplify the discussion, only binary system will be considered in this presentation.

There are two types of diffusion that will be considered. One is molecular diffusion, which is

highly influenced by concentration gradient, and the other is eddy or turbulent diffusion, which

is influenced not only by concentration gradient but also by the movement or mixing of the

material due to some external applied force.

I3.2. MOLECULAR DIFFUSION

Consider a binary system where a certain species A is moving at an average velocity uA in a

bulk of material containing species B moving at an average velocity of uB. Let us assume that

the mixture is moving at a bulk velocity uo referred to a stationary observer. Then the molar

fluxes of A and B may be determined by the Fick’s Law of diffusion

AOA A A O AB

dcI c u u D

dZ (I3 – 1)

BOB B B O BA

dcI c u u D

dZ (I3 – 2)

where IoA is the molar flux of A through a plane moving at Ou and cA and cB are the

concentrations of species A and B while DAB is the diffusivity of A relative to B and DBA is the

diffusivity of B relative to A. The diffusivities are transport properties which may be

determined experimentally or estimated from empirical equations in terms of the physical and

molecular properties of the diffusing components.

In design calculations, what is more important is the diffusion flux, not relative to the

movement of the bulk but relative to a stationary observer. These diffusion fluxes, NA and NB

are given by

NA = cAuA (I3 – 3)

NB = cB uB (I3 – 4)

2

while the total flux of the entire bulk, N is given by

N = NA + NB = m uo (I3 – 5)

If uA, uB and uo are eliminated from Eqs. (I3 – 1) and (I3 – 2), we get the equations

( )A AOA A A B AB

m

c dcI N N N D

dZ (I3 – 6)

( )B BOB B A B BA

m

c dcI N N N D

dz (I3 – 7)

If we add Eqs. (I3 – 6) and (I3 – 7), it can easily be seen that

IOA + IOB = 0 (I3 – 8)

and

DAB = DBA = Dv (I3 – 9)

That is, the sum of the molar fluxes relative to the movement of the bulk is zero and, for binary

system, the diffusivity of A relative to B is the same as the diffusivity of B relative to A. Here,

we will just refer to this as the volumetric or mass diffusivity, Dv with units of m2/s or ft

2/hr. It

is important to note that the diffusivity is based on the movement of the entire bulk and not on a

stationary position. For gases, the diffusivity can also be expressed in terms of molar units, Dm

defined by

v Tm v m

D PD D

RT (I3 – 10)

where the units of Dm is in moles/time-length and m is in moles per unit volume.

Solving for the molar flux relative to a stationary observer, NA from Eq. (I3 – 6), we get

( )A AA v A B

m

dc cN D N N

dz (I3 – 11)

It is seen that the diffusion flux, NA is composed of two terms, the molecular diffusion flux as

given by Fick’s Law and another type of flux which we can consider here as convective flux or

phase drift.

The differential equation presented in Eq (I3 – 11) may be solved by considering two ideal

steady state diffusion models. These are Equimolar Counter Diffusion and Unicomponent

Diffusion. An example of the former is encountered in the rectification of volatile components

where both can co-exist in both phases such as ethanol-water system. An example of the latter

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is in the absorption of a soluble component from an inert gas that is insoluble in the solvent

where the soluble component is able to penetrate the solid-liquid interface while the inert gas

becomes stagnant since it cannot diffuse to the liquid phase.

I3.3. EQUIMOLAR COUNTER DIFFUSION

When the molar flux of A and B are moving at equal rates and in opposite direction,

NA = - NB or NA + NB = 0 (I3 – 12)

Equation (I3 – 11) reduces to

AA v

dcN D

dz (I3 – 13)

This equation may be integrated for the total molar rate of diffusion, NTA, if the diffusion area,

A, perpendicular to the direction of motion is constant,

1 2

2 1

TA A Av

N c cD

A z z

(I3 – 14)

It is to be noted that for constant area, the concentration profile is linear across the direction of

diffusion. If the diffusion area is not a constant, it must be expressed in terms of z and the

differential equation solved applying the limits from z1 to z2..

For ideal gases, the diffusion equation may be expressed in terms of partial pressure, pA, that is,

AA

pc

RT (I3 – 15)

or

vTA ADN dp

A RT dz (I3 – 16)

I3.4. UNICOMPONENT DIFFUSION

For unicomponent diffusion of A through a stagnant component B, then NB = 0. Equation (I3-

11) becomes,

A AA v A

m

dc cN D N

dz (I3 – 17)

since cA + cB = m, the above equation may be converted to

4

1TA A Av

B

N c dcD

A c dz

(I3 – 18)

If this is expressed in terms of the mole fractions of A and B, that is xA and xB, the above

equation can be integrated in the form of

1 2

2 1 ln

vTA A A

B

DN c c

A z z x

(I3 – 19)

where xBln is the logarithmic mean of the mole fraction of B at point 2 and point 1.

For ideal gases, Eq. (I3 – 18) may be expressed in terms of pA,, that is,

v TTA A

B

D PN dp

A RTp dz (I3 – 20)

if the diffusion area is constant, with pB = PT – pA, the above equation can be integrated to give

2

1

lnv TTA T A

T A

D PN P p

A RT z P p

(I3 – 21)

It is noted that the concentration profile for this case is non-linear but logarithmic.

I3.5. EVALUATION OF DIFFUSIVITIES

The volumetric diffusivity, Dv for gases and liquids may be determined experimentally or from

empirical correlations based on the kinetic theory of gases. Some of the more important

equations are presented here.

I3.5.1. From Empirical Equations

1. For gases, Chen and Othmer Equation. (McCabe and Smith, 1976)

D

TM M

p T T V Vv

A B

CA CB CA CB

0 014981 11 81

0 5

0 1405 0 4 0 42

. .

.

. . . (I3 – 22)

2. For gases, Gilliland Equation. (Brown, et al., 1950))

5

DT

P V VM M

G

A BA B

0 0166 1 1

32

13

13

2

. (I3 – 23)

3. For gases, Chapman and Engskog Equation (Geankoplis, 1997))

1/ 2

7 3/ 2

2

,

1.8583 10 1 1AB

AB D AB A B

x TD

P M M

(I3 – 24)

4. For liquids, Stokes-Einstein Equation (Geankoplis, 1997))

16

1/3

9.96 10AB

A

x TD

V

(I3 – 25)

3. For Liquids, Wilke and Chang (Treybal, 1968)

D

M T

VAB

B

A

7 4 10 8 0 5

0 6

..

' .

(I3 – 26)

Other empirical equations maybe found from literature. The nomenclature used in these

equations is found in the Appendix.

I3.5.2. From Experimental Data

Sources of diffusivity data can be found in Perry and Green (1984), Green, et al. (1997),

McCabe, et al. (2001), Geankoplis (1995) and other textbooks.

If the diffusivity is given at a particular reference temperature, say 273K and 1 atm, it is

possible to estimate the diffusivity at a desired temperature and pressure by making use of the

empirical equations as the basis. If the calculation is based on Chen and Othmer correlation, the

equation becomes

D fT

pv

1 81.

(I3 – 27)

or

1

1.81

273,1

1

273v vT P atm

TD D

p

(I3 – 28)

I3.6. TURBULENT DIFFUSION

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The equation for molecular diffusion may be modified and applied to turbulent diffusion by

introducing a correction M referred to as the turbulent or eddy mass diffusivity. Thus, Eq. (I3 -

1) may now be written as

AoA AB M

dcI D

dz (I3 – 29)

I3.7. MASS TRANSFER COEFFICIENTS

For equimolal counter diffusion, IoA = NA. The above equation can therefore be integrated

across a film thickness of (z2 – z1), to give

1 2

2 1

( )AB MA A A

DN c c

z z

(I3 – 30)

This equation is then simplified by expressing it in terms of a convective mass transfer

coefficient, kc’ based on the movement of the entire bulk phase.

'

1 2( )A c A AN k c c (I3 – 31)

For mass transfer of A in a non-diffusing B, Equation (I3-19) may be modified to give

1 2

2 1 ln

( )AB M A AA

B

D c cN

z z x

(I3 – 32)

which may be simplified to '

1 2 1 2

ln

( ) ( )cA A A c A A

B

kN c c k c c

x (I3 – 33)

The mass transfer coefficients kc’ and kc have a unit of m/s or ft/hr. It is possible to express

these coefficients in terms of other units depending on the driving forces used in the defining

mass transfer equation. Examples are

1 2 1 2 1 2( ) ( ) ( )A G A A y A A x A AN k p p k y y k x x (I3 – 34)

I3.8. EVALUATION OF MASS TRANSFER COEFFICIENTS

I3.8.1. Dimensionless Numbers

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The dimensionless numbers obtained by the usual procedure of dimensional analysis that are

important in mass transfer operations are the following:

Reynolds Number, Re

inertia forces

viscous forces

DuN

(I3 – 35)

Schmidt Number, momentum

mass diffusivitySc

v

ND

(I3 – 36)

Sherwood Number, ' turbulent diffusion

molecular diffusion

cSh

AB

k LN

D (I3 – 37)

The mass transfer coefficient is correlated as a dimensionless, JD factor given by

' '

2 /3

1/3

Re

( )c c T ShD Sc

m Sc

k k P NJ N

v v N N (I3 – 38)

I3.8.2. Mass, Heat and Momentum Transfer Analogies

The transport mechanism of mass, heat and momentum have similarities that could be used to

relate the three mechanisms especially in determining approximate values of the transfer

coefficients in the absence of a more reliable experimental data. The more common analogies

are presented here.

Reynolds Analogy (NSc = NPr = 1.0)

'

2

c

p av

kf h

c G u (I3 – 39)

Chilton-Colburn Analogy

'2 /3 2/3

Pr( ) ( )2

cH D Sc

p av

kf hJ N J N

c G u (I3 – 40)

I3.8.3. Mass Transfer Coefficients

A. For Flow Inside Pipes

For Laminar flow, refer to Fig. 7.3 –2 (Geankoplis, 1995)

For Turbulent Flow, for NSc of 0.6 to 3000

8

' 0.83 0.33

Re0.023Sh c Sc

AB

DN k N N

D (I3 – 41)

B. For Flow Outside Solid Surfaces

1. Parallel Flat Plates

' 0.5 1/3

Re,0.664Sh c L Sc

AB

LN k N N

D (I3 – 42)

2. Flow Past Single Spheres

For gases, NSc = 0.6 to 2.7 and NRe = 1 to 48,000

0.53 1/3

Re2 0.552Sh ScN N N (I3 – 43)

For liquids, NRe = 2 to 2000

0.5 1/3

Re2 0.95Sh ScN N N (I3 – 44)

NRe = 2000 to 17,000

0.62 1/3

Re0.347Sh ScN N N (I3 – 45)

C. For Packed Beds

For Gases through spheres with NRe = 10 to 10,000

0.4069

Re

0.4545D HJ J N

(I3 – 46)

For Liquids with NRe = 0.0016 to 55 and NSc = 165 to 70,600

2/3

Re

1.09DJ N

(I3 – 47)

For Liquids with NRe = 55 to 1500 and NSc=165 to 10,690

0.31

Re

0.250DJ N

(I3 – 48)

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The representative equations given above are obtained from Geankoplis(1995). Many more

correlations are available in Green, et al. (Perry’s Handbook, 1997) and other references.

D. Penetration Theory of Mass Transfer

For cases where surface renewal rather than film theory applies, for equimolal diffusion, the

individual mass transfer coefficient is given by

L

vM

t

Dk

2 (I3 – 49)

where tL is the average time the fluid elements remain at the interface. This is dependent on the

fluid velocity, fluid properties and the geometry of the system.

THE WETTED WALL COLUMN

The wetted wall column is the most popular apparatus used in experimentally determining the

mass transfer coefficient of a system since the mass transfer area can be determined with

reasonable accuracy. Correlations on the behaviour of the dimensionless numbers such as the

Sherwood number, Reynolds number and Schmidt number under turbulent diffusion have been

derived using this apparatus.

Applying the material balance and the rate of mass transfer of component A around the

differential area dA yields

dNA = V’dY = ky(Yi - Y) dA (I3 – 50)

Since, V’ = V(1-y) and

Y

ydY

dy

y

1

1 12

and

Substituting in Equation (I3 – 50)), we get

A

o

y

y i

y

iy

yyy

dy

V

dAk

dAyyky

dyyV

2

1

1

11

2

(I3 – 51)

Under adiabatic conditions, the temperature of the liquid remains constant, thus the interfacial

concentration, yi may be taken also as constant. Integrating Equation (I3 - 51), we get

........ 1

1ln

1

1

12

21

yyy

yyy

yV

Ak

i

i

i

y

(I3 – 52)

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With the temperature, flow rate and concentrations measured experimentally, together with the

surface area of contact between the gas and the liquid, the mass transfer coefficient of the

diffusing component maybe determined.

Several correlations have been derived for wetted-wall columns. An example is the Gilliland-

Sherwood Equation (McCabe and Smith, 1976) given by

44.081.0

Re023.0 ScSh NNN (I3 – 53)

which is very similar to Eq. (I3 – 41). The equation applies for NRe between 2,000 to 35,000;

NSc from 0.6 to 2.5; and over a pressure range of 0.1 to 3 atm.

A second correlation for wetted-wall columns, which shows the general analogy for

momentum, heat and mass transfer, although less precise than the above equation, can be

written as

2.0

Re023.02

Nf

jj HM (I3 – 54)

where f is the Fanning friction factor for flow in smooth pipes. The above equation is not

applicable if form drag exists.

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NOMENCLATURE

Symbol Description Units

A Area perpendicular to the moving species m2

cA Concentration of species A kg-mole/m3

cp heat capacity J/kg-K

DAB Diffusivity of A relative to B m2/s

Dm molal diffusivity kg-mole/s-m

Dv volumetric diffusivity m2/s

f Fanning friction factor [ - ]

G mass velocity kg/m2-s

h heat transfer coefficient W/m2-K

IoA Molar flux of A relative to bulk motion kg-mole/s-m2

kc mass transfer coefficient for unicomponent

diffusion

m/s

kc’ mass transfer coefficient for equimolar

diffusion

m/s

M molecular weight kg/kg-mols

NA Molar flux of A kg-mol/m2-s

NTA Total moles of A diffusing kg-mol/m2-s

NRe Reynolds Number

NSc Schmidt Number

NSh Sherwood Number

P, PT total pressure atm or Pa

pA partial pressure of A mm Hg or Pa

R universal gas constant=8314.34 J/kg-mol-K

T temperature K

Tc critical temperature K

u linear velocity m/s

VA solute molar volume at normal boiling point m3/kg-mol

Vc critical volume m3

xA Mole fraction of species A in liquid phase [ - ]

y mole fraction in the gas phase [ - ]

z Distance in the direction of moving species M

porosity of bed [ - ]

viscosity Pa-s

association parameter of the solvent [ - ]

AB average collision diameter M

D collision integral [ - ]

m Molal density of mixture kg-mols/s-m

M eddy or turbulent mass diffusivity m2/s

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References:

Brown, George G., D. Katz, A.L. Foust and R. Schneidewind. (1950). "Unit Operations", John

Wiley and Sons, New York

Foust, A.S., L.A. Wenzel, C.W. Clump, L. Maus and L.B. Andersen. (1960) "Principles of Unit

Operations", John Wiley and Sons, New York.

Geankoplis, Christie J. (1995) “Transport Processes and Unit Operations”, 3rd

edition.

Printice-Hall International ed.,

Green, Don W.(ed) and James O. Maloney (asoc. ed), (1997) “Perry's Chemical Engineers'

Handbook, 7th

edition", McGraw-Hill Book, New York

McCabe, Warren L., Julian C. Smith and Peter Harriott,(2001) Unit Operations of Chemical

Engineering, 6th

edition, McGraw-Hill International.

Perry, Robert H. and D. Green. (1984). "Perry's Chemical Engineers' Handbook, 6th

edition",

McGraw-Hill Book, New York.

Treybal, Robert E., (1968), “Mass Transfer Operations”, 2nd

edition, McGraw-Hill Kogakusha,

Ltd., Tokyo

TABLES NEEDED:

Diffusion Coefficients of Combination of Gases at 1 atm

Diffusion Coefficients of a Gas in Air at 1 atm and 273K

Atomic Diffusion Volumes

Diffusion coefficients for Dilute Liquid Solutions

Atomic and Molar Volumes at Normal Boiling Point

Diffusion Coefficients for Dilute Solutions of Gases in Water at 20oC