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Mass Transfer Principles

1. Introduction

2. Mass Transfer Principles

3. Equilibrium Stage Operations

4. Distillation

5. Absorption6. Extraction

7. Leaching

The objective is:

Recognize and be able to use equilibria and material

and energy balances to carry out process calculations

Recognize the mass transfer concept in separation

process and their estimation

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Introduction

The transport of one constituent from a region

of higher concentration to that of a lower

concentration is called mass transfer.

Rate of a transfer process = driving force

resistance

Possible driving force for mass transfer

• Concentration different

• Pressure different

• Electrical gradient

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Examples of mass transfer

• Evaporation of water in the open pail to

atmosphere

• Coffee dissolves in water

• O2 dissolves in the solution to the

microorganism in the fermentation process

• Reaction occurs when reactants diffuse

from the surrounding medium to thecatalyst surface

• The mechanism of mass transfer involves

both molecular diffusion and convection.

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Convective mass transfer

• Using mechanical force or action to

increase rate of molecular diffusion

• e.g- stirred the water to dissolve coffee

during coffee making

Molecular Diffusion

Transfer of individual molecules

through a fluid by random, individual

movements of the molecules

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Molecular Diffusion

The basic of diffusion(Fick’s Law) wasenunciated by Adolf Eugen Fick, a

physiologist in 1885

“the molar flux of a species relative to an

observer moving with molar average

velocity is proportional to the conc.

gradient of the species”

Molecular Diffusion

Diffusion of molecules when the bulk fluid isstationary given by Fick’s Law :

dz

dxcD J A

AB A *

Molar flux of component A (kgmol A/s.m2)

Molecular diffusivity of the molecule A in B (m2 /s)

Total conc. of A and B (kgmol A+B/m3)

Mole fraction of A

A J *

AB D

c

A x

(1)

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Molecular DiffusionFick’s Law for molecular diffusion of mass at constant

total concentration c A = x Ac:

dz

dc D J A

AB Az *

Molar flux of component A in the z direction due tomolecular 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

AB D

Ac z

(2)

If c is varies, an average value is often used with equation (2).

Other driving forces (besides conc.) for diffusion also occur because of T,

P, electrical potential and other gradients. (transport phenomena TB)

Example

Molecular Diffusion of Helium in Nitrogen. Amixture of He and N2 gas is contained in apipe at 298 K and 1 atm total pressure whichis constant throughout. At one end of thepipe at point 1 the partial pressure p A1of He

is 0.60 atm and at the other end 0.2 m (20cm) p A2= 0.20 atm. Calculate the flux of Heat steady state if D AB of the He-N2 mixture is0.687 x 10-4m2/s (0.687 cm2/s).

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• Integrate the equation:

•From ideal gas law, p AV = n A RT ,

12

21 )(

z z

cc D J A A AB

A

)()(

12

21

z z RT p p D J A A AB

A

V

n

RT

pc A A A

11

)020.0)(298(8314

)10027.21008.6)(10887.0( 444

x x x J A

= 5.63 x 10-6 kgmolA/s.m2

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Think of the last time that you washed the dishes.

You placed your first greasy plate into the water,

and the dishwater got a thin film of oil on the top of

it, didn’t it? Find the flux, J, of oil droplets through

the water to the top surface. The sink is 18 cm

deep, and the concentration of oil on the plate is 0.1

mol/cm3. Assume that there is no oil at the top of

the sink yet.

Exercise

• Answer:• To solve this problem, we will need to apply the mass transfer equation we

just learned.

where: D AB = 7 x 10-7cm2/s

dc A= concentration at the top of the sink – the concentration of oil on the plate.

The concentration at the top of the sink = 0

The concentration of oil on the plate = 0.1 mol/cm3

dc A = 0 – 0.1 = -0.1 mol/cm3

dz = the depth of the sink = 18 cm

Since we know all of the numbers needed, we can calculate the flux.

J = -(7 x 10-7 cm2/s) * (-0.1 mol/cm3) / (18 cm)

J = 4 x 10-4 mol / (cm2s)

dz

dc D J A

AB Az *

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Convective Mass Transfer

• When fluid flowing outside a solid surfacein forced convection motion, rate ofconvective mass transfer is given by:

)cc( k N Li Lc A 1

Mass-transfer coefficient (m/s)Bulk fluid concentration (kgmol A/m3)Concentration in the fluid next to the surface of thesolid

ck

1 Lc

Lic

(2)

k c depends on >>>>>system geometry, fluid properties and flow velocity

The objective is:

Recognize and be able to use equilibria and material

and energy balances to carry out process calculations

Recognize the mass transfer concept in separation

process and their estimation

Learning Outcomes

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Diffusion in Gases p 414

• Outlines:1. Equimolar counter diffusion in gases

2. General case for diffusion of gases A and Bplus convection

3. Special case for A diffusing throughstagnant, non-diffusing B

4. Diffusion through varying cross-sectionalarea

5. Diffusion coefficients for gases

1. Equimolar counter-diffusion

• Consider:

– 2 gases A and B

– At constant total

pressure P

– Molecular diffusion at

steady-state – Partial pressures:

21 A A p p

12 B B p p

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Exercise

Check example 6.2-1

Equimolar counter diffusion is occurring at steady

state in a tube 0.11 m long containing N2 and CO

gases at the total pressure of 1.0 atm abs. The

partial pressure of N2 is 80mm Hg at one end and

10 mmHg at the other end. Given the D AB at 298K

is 2.05 x 10-5 m2/s

a) Calculate the flux in kg mol/s.m2 at 298 K for N2

b) Repeat at 473 K. Given that D AB at 493K is 4.60

x 10-5m2/s.

Exercise

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2. General diffusion & convection

Multiplying by c A,

M A Ad A A A vcvcvc

Hence,

M A A*

A vc J N

If N = total convective flux of the whole stream relative to the

stationary point, then

B A M N N cv N

c

N N v B A M

(11)

(12)

(10)

2. General diffusion & convectionSubstituting equation (11) and Fick’s law into (12),

B A A A

AB A N N c

c

dz

dxcD N

Note:For Equimolar counter-diffusion,Hence,

B A N N

dz

dxcD N A

AB A

Convectionterm

Diffusionterm

(13)

This is the general equation describing mass transfer of component-A by

diffusion through moving bulk fluid. It allows one to calculate the mass

transfer rate (molar flux, e.g. in kg-mole/m2.s) between 2 points.

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3. Stagnant, non-diffusing B


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

0 A A A

AB A N c

c

dz

dxcD N

If total pressure is kept constant P

(14)

RT

P c P x p A A

P

p

c

c A A

Substituting into (14)

A

A A AB

A N P

p

dz

dp

RT

D N (15)

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Re-arranging and integrating

1

2

12 A

A AB A

p P

p P ln

) z z ( RT

P D N

(16)

Or another form P =p A1 + p B1 =p A2 + p B2 ,

p B1 =P-p A1 and p B2 = P-p A2

21

12

A A

BM

AB A p p

p ) z z ( RT

P D N

(17)

dz

dp

RT

D

P

p N A AB A

A

1

p

p

dp

RT

Ddz N

A

A

p

AB

z

z

A

p

1

2

1

2

1

)/()(ln)/ln( 12

21

12

12

A A

A A

BM B B

B B BM

p P p P p p

p p p p p p

Water

vapor

example

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M Azmi Bustam

• 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

N N

A

A

Where

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

Cross-sectional area through which the diffusionoccurs

A N

A

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

A N A

(18)

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1. Diffusion from a sphere

BM

A A AB A

A

p

P p

RTr

P D N

r

N 21

1

12

14

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

(19)

1 A p P P p BM

112 Dr RT / pc A A 11

21

1

1

2 A A

AB A cc

D

D N (20)

24 r A


2. Diffusion through a conduit of non-uniform csa

dz P / p

dp

RT

D

r

N N

A

A AB A A

12

Defining1

12

12 r z z z

r r r

2

1

2

112

1

12

12

z

z

p

p A

A AB A A

A P / p

dp

RT

D

r z z z

r r

dz N

(21)

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a) Experimental determination

)(2 21

22

A A AB A

BM o F F

p p P D M RTp z z t


a) Experimental determination

t

V V A / L

V V Dexp

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.

D AB , range from 0.05 x 10-4 m2/s, to about 1.0 x 10-4 m2/s(H2)


c) Prediction of diffusivity for gases

Semi-empirical method of Fuller et al.

23131

2175171110

/

B

/

A

/

B A

.

AB

vv P

M / M / T D

WhereSum of structural volume increments Av

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a. Equimolar counter diffusion

12

21

12

21 )()(

z z

x xc D

z z

cc D N A Aav AB A A AB

A

Diffusion in Liquids

Where,

2/21

21

M M M

cav

av

c A1 – conc. A at

point 1

x A2 – mole frac.

A at point 1

cav – average total

conc. A + B in

kgmol/m3

b. Diffusion of A through nondiffusing B,

in gas

Rewrite in terms of conc. by substituting

Where pBM is “log mean partial conc. of B” between the location z 2 and z 1

BM

A Aav AB A

x z z

x xc D N )(

)(

12

21


Where,

)/ln( 12

12

B B

B B BM

x x

x x x

21

12

A A

BM

AB A p p

p ) z z ( RT

P D N

P

p x

RT

pc

RT

P c BM

BM A

A AV ,, 11

12

21 )(

z z

cc D N A A AB

A

Very dilute soln. x BM close to 1

and c constant

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3.Diffusion coefficients for liquids

(several methods)- unsteady state diffusion in a

long capillary tube- conc. profile

- quasi-steady state diffusion


cc D N AB A

Where,

the fraction of area of the

glass open to diffusion

c conc. in the lower chamber

c’ conc. in the upper chamber

effective diffusion length

t DV

A

cc

cc AB

o

oo

2ln

'

'

Where,

V

A

2 - cell constant

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4. Prediction of diffusivities in liquids

(theory for diffusion in liquid is not well established)

Modifying from the Stokes-Einstien equation:


3

1

161096.9

A

AB

V

T D

By assuming all molecules are alike and arrange in a cubic lattice and

expressing the molecular radius in term of molar volume

Where:

D AB : Diffusivity in m2/s

T : Temp.

: viscosity in Pa.s or kg/m.s

V A : solute molar volume at its normal boiling point.

4. Prediction of diffusivities in liquids

Modifying from the Stokes-Einstien by Wilke-

Chang:


6.0

2/116 )(10173.1 A B

B AB

V

T M D

Where:

D AB : Diffusivity in m2/s

T : Temp.

B : viscosity of B in Pa.s or kg/m.s

V A : solute molar volume at its normal boiling point.

M B : Molecular weight of solvent B

: an “associate parameter” of the solvent

Refer example 6.3-2 for your exercise. (p432)

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Mass transfer

Molecular diffusionConvective mass

transfer

• Gases1. Equimolar counter diffusion 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 area5. Diffusion coefficients for gases

• Liquid

• Solid

dz dc D J A

AB Az *

Molar flux of component

A in the z direction:

Bz *

Az * J J

Equimolar counter diffusion:

)cc( k N Li Lc A 1

Convective mass transfer:

B A

A A

AB A N N c

c

dz

dxcD N

General diffusion & convection

0 A A A

AB A N c

c

dz

dxcD N

Stagnant, non-diffusing B

1

2

12 A

A AB A

p P

p P ln

) z z ( RT

P D N

21

12

A A

BM

AB A p p

p ) z z ( RT

P D N

Stagnant non-diffusion B,

another form….

4. Diffusion through varying cross-sectional area A

N N

A

A

BM

A A AB A

A

p

P p

RTr

P D N

r

N 21

1

12

14

Sphere

a conduit of non-uniform csa

dz P p

dp

RT

D

r

N N

A

A AB A

A/12

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5. Diffusion coefficients for gases )(2 21

22

A A AB A

BM F F F p p P D M

RTp z z t

t

V V A / L

V V Dexp

cc

cc AB

av

av

12

21

0

2

2

23/13/1

2/175.17 /1/110

B A

B A AB

vv P

M M T D

AB

Sc pD

N

Stagnant non-diffusion B

The two bulb method

Semi-empirical method of

Fuller et al.

Schmidt number of gases

• Liquid

• Solid

12

21

12

21 )()(

z z

x xc D

z z

cc D N A Aav AB A A AB

A

t DV

A

cc

cc AB

o

oo

2ln

'

'

Diffusion in liquid

Diffusion in coefficients liquid

Prediction of diffusion in liquid

6.0

2/116 )(10173.1 A B

B ABV T M D

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Diffusion in solids

Typical values for diffusivity in gases, liquids

and solids are shown in table.

General range of values of diffusivity:

Gases : 5 X 10 –6 ~ 1 X 10-5 m2 / sec.

Liquids : 10 –6 ~10-9 m2 / sec.

Solids : 5 X 10 –14~1 X 10-10 m2 / sec.

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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Introduction

- Molecular diffusion in solids slower than gasesand liquids;

- Very important in chemical and biological processe.g. leaching of food, drying thing (timber, salts andfoods), diffusion and catalytic reaction, treatment ofmetal in high temp. etc

Types of diffusion in solids

1. Follow Fick’s law 2. depend on actual structure and void channels

Diffusion in solids

Diffusion in solids1.Diffusion in solids following Fick’s Law p 441

B A

A A

AB A N N

c

c

dz

dxcD N Using general equation for binary diffusion;

Bulk-flow is small, it is neglected. Also c is assumed constant.

Giving diffusion in solids;

B A

A N N c

c

dz

dc D N A AB

A

12

21 )(

z z

cc D N A A AB

A

In case of diffusion through cylinder wall of inner

radius r 1 and outer r 2 and length of L;

)/ln(

2)(

12

21r r

Lcc D N A A AB A

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2. Diffusion in porous solids that depend on structure

Diffusion in solids

)(

)(

)(

)(

12

21

12

21

z z RT

p p D

z z

cc D

N A A AB A A AB

A

)(

)(

12

21

z z

cc D N A A AB

A

Diffusion of salt in water

at steady state;

Where;

: open void fraction

: tortuosity

Diffusion of gases in porous solids;

Mass transfer

Molecular diffusionConvective mass

transfer

• Gases1. Equimolar counter diffusion in gases

2. General case for diffusion of gases A

and B plus convection3. Special case for A diffusing through

stagnant, non-diffusing B

4. Diffusion through varying cross-

sectional area


dz

dc D J A

AB Az *

Molar flux of component

A in the z direction:

Bz *

Az * J J

Equimolar counter diffusion:

)cc( k N Li Lc A 1

Convective mass transfer:

B A A A

AB A N N c

c

dz

dxcD N

General diffusion & convection

0 A A A

AB A N c

c

dz

dxcD N

Stagnant, non-diffusing B

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• Liquid

• Solid

12

21

12

21 )()(

z z

x xc D

z z

cc D N A Aav AB A A AB A

t DV

A

cc

cc AB

o

oo

2ln

'

'

Diffusion in liquid

Diffusion in coefficients liquid

Prediction of diffusion in liquid

6.0

2/116 )(10173.1 A B

B ABV

T M D

12

21 )(

z z

cc D N A A AB

A

Diffusion in solids

Problem based learning

)(

)(

)(

)(

12

21

12

21

z z RT

p p D

z z

cc D N A A AB A A AB

A

Diffusion in porous solids

Diffusion of liquidDiffusion of gas

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