Diffusion

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Mass Transfer Principles

Dr.T.Murugesan

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Mass Transfer

Transport of one component from the region of Higher Concentration to lower Concentration

Migration of one substance through another under the

influence of concentration gradient

Involves the Diffusional Transport of some component

within a single phase or between two immiscible phases which are in contact.

Components may migrate from the bulk of one phase to

interphase between phases and remains there

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Examples of Diffusion Processes

Sugar added to a cup of tea/ water

Evaporation of water from ponds

Room refreshener

Oxygen into blood

Function of kidney membrane

Controlled release of drugs

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Separation Process involving Mass Transfer

Distillation

Absorption & Desorption

Liquid Extraction

Leaching

Adsorption

Crystallization

Humidification & Dehumidification

Drying

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Classification of Mass Transfer process

Molecular Diffusion in stagnant media

Molecular diffusion in fluids in Laminar

flow

Eddy diffusion or Diffusion in turbulent

stream

Mass transfer between Phases

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Diffusion

Diffusion (considered in this chapter)

The movement of a substance from anarea of high concentration to an area oflow concentration

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Example:

A = blue liquid dye

B = water

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Basic definitions

Total mass of the mixture: = i

Mass fraction:

For gases: CA = nA/ V = PA / RT

Mole fraction XA =

For multi component systems

Mass average velocity:

n

i i

i

A

A

Mw

Mw

1

n

i

i

n

i

iiv

v

1

1

Molar average velocity:

n

i

i

n

i

ii

v

vC

V

1

1

n

i

i

AAw

1

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Fick’s Law

Fick’s Law for molecular diffusion of mass for constant total concentration:

dz

dcDJ A

ABAz*

Molar flux of component A in the z direction due to molecular diffusion (kgmol A/s.m2)Molecular diffusivity of the molecule A in B (m2/s)Concentration of A (kgmol/m3)Distance of diffusion (m)

Az*J

ABD

Acz

(1)

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Mass Transfer Coefficient

The rate of convective mass transfer for a fluidflowing outside a solid surface in forced convectionfrom the surface to the solid (vice versa):

)cc(kN LiLcA 1

Mass-transfer coefficient (m/s)Bulk fluid concentration (kgmol A/m3Concentration in the fluid next to the surface of the solid

ck

1Lc

Lic

(2)

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Diffusion in Gases

Outlines:1. Equimolar counterdiffusion in gases

2. General case for diffusion of gases A and B plus convection

3. Special case for A diffusing through stagnant, non-diffusing B

4. Diffusion through varying cross-sectional area

5. Diffusion coefficients for gases

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1. Equimolar counter-diffusion

Consider:

2 gases A and B

At constant total pressure P

Molecular diffusion at steady-state

Partial pressures:

21 AA pp

12 BB pp

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Net moles A and B diffusing right to left (and vice versa) are equal since total pressure P is constant,

Bz*

Az* JJ

Fick’s law for B for constant total concentration c,

Since total pressure P is constant, then

BA ccc BA dcdc

dz

dcDJ

B

BAB*

(3)

(4)

(5)


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Combining (1), (3) and (4)

dz

dcD

dz

dcD B

BAA

AB (6)

Substituting (5) into (6) and canceling

BAAB DD (7)

i.e. for binary gas mixture of A and B, the diffusivity coefficient DAB

for A diffusing into B is the same as DBA for B diffusing into A


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2. General diffusion & convection

For diffusion in stationary fluid, the diffusion flux J*A passing a fixed

point from left (high concn.) to right (low concn.) can be expressed in terms of velocity of diffusion of A,

AAdA* cvJ

(m/s)(kgmol A/m3)(kgmol A/sm2)

Diffusion velocity of A

If convective flow (to the right) presents, A is diffusing at velocity vAd plus the convective velocity of the bulk fluid. Hence,

MAdA vvv Convective velocity of the

bulk fluid

Velocity of A relative to a

stationary point

(8)

(9)

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Multiplying by cA,

MAAdAAA vcvcvc

Hence,

MAA*

A vcJN

If N = total convective flux of the whole stream relative to the stationary point, then

BAM NNcvN

c

NNv BA

M

(11)

(12)

(10)

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Substituting equation (12) and Fick’s law into (11),

BAAA

ABA NNc

c

dz

dxcDN

Note:For Equimolar counter-diffusion,Hence,

BA NN

dz

dxcDN A

ABA

Convection term

Diffusion term

(13)

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3. Stagnant, non-diffusing B

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For A diffusing in stagnant, non-diffusing B, in equation (12) set 0BN

0 AAA

ABA Nc

c

dz

dxcDN

If total pressure is kept constant P

(14)

RT

Pc

P

xp A

A P

p

c

c AA

Substituting into (14)

AAAAB

A NP

p

dz

dp

RT

DN (15)

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Re-arranging and integrating

1

2

12 A

AABA

pP

pPln

)zz(RT

PDN

(16)

Or another form

21

12

AA

BM

ABA pp

p)zz(RT

PDN

(17)

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Sphere to surrounding medium

Evaporation of a drop of liquid

Evaporation of a ball of naphthalene

Diffusion of nutrients to a sphere-like micro-organism in a liquid

Conduit of non-uniform csa

4. Varying cross-sectional area

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Define

A

NN

A

A

WhereKg moles of A diffusing per second (kgmol/s)

Cross-sectional area through which the diffusion occurs

AN

A

At steady-state, will be constant but not for varying area.

AN A

(18)

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1. Diffusion from a sphere

BM

AAABA

A

p

Pp

RTr

PDN

r

N 21

1

12

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If is small compared to (a dilute phase), Also, setting , diameter, and

(19)

1Ap P PpBM

112 Dr RT/pc AA 11

21

1

1

2AA

ABA cc

D

DN (20)

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2. Diffusion through a conduit of non-uniform csa

dzP/p

dp

RT

D

r

NN

A

AABAA

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Defining 1

12

12 rzzz

rrr

2

1

2

112

1

12

12

z

z

p

p A

AABAA

AP/p

dp

RT

D

rzzz

rr

dzN

(21)

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a) Experimental determination

t

VVA/L

VVDexp

cc

cc AB

av

av

12

21

0

2

2

Where is the average concentration value at equilibriumavc

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b) Experimental diffusivity data

Available in Perry and Green or Reid at al.Typical data as in Geankoplis pg 424.

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c) Prediction of diffusivity for gases

Semi-empirical method of Fuller et al.

23131

217517 1110

/

B

/

A

/

BA

.

AB

vvP

M/M/TD

WhereSum of structural volume increments Av

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d) Schmidt number of gases

AB

ScpD

N

The Schmidt number of a gas mixture of dilute A in B is defined by:

WhereViscosity of the gas mixture (B for dilute) (Pas or kg/ms)Diffusivity (m2/s)Density of the mixture (kg/m3)

ABD

p

It is dimensionless. For gases, values range from 0.5 – 2.0.

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Diffusion in Liquids

Outlines:

1. Equations for diffusion in liquids

2. Diffusion coefficients for liquids

3. Prediction of diffusivities in liquids

4. Prediction of diffusivities of electrolytes in liquids

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Diffusion in solids

Outlines:

1. Types of diffusion in solids

2. Diffusion in solids following Fick’s Law

3. Diffusion in porous solids that depend on structure

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Diffusion in biological solutions

and gels

Outlines:

1. Diffusion of biological solutes in liquids

2. Diffusion in biological gels

Diffusion

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