ChE 333 : Mass transfer

Textbook: Fundamentals of Momentum, Heat and Mass transfer. J.R.Welty, R.E.Wilson and C.E.Wicks. 5 th Edition , John Wiley (2007).

Reference: Fundamentals of Heat and Mass transfer. Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera and David P. DeWitt.

7 th Edition , John Wiley (2011)

Dr. Sharif Fakhruz ZamanDepartment of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University,

Jeddah, KSA

About myself

Dr. Sharif Fakhruz ZamanAssistant Professor at KAU.

PhD in Chemical and Biochemical Engineering, University of British Columbia, Vancouver Canada (2010).MS in Chemical Engineering, KFUPM, KSA .

BSc in Chemical Engineering, BUET, Bangladesh.

Field of interest : Heterogeneous catalysis and molecular modeling of catalyst and catalytic reactions, reaction kinetics, reactor design, diffusion in zeolites etc.

Office address :Room 216, Building 45, Department of Chemical and Materials Engineering

Faculty of Engineering, King Abdulaziz university, Jeddah, KSA.

E-mail : zfsharif@gmail.com or sfzaman@kau.edu.saPhone : cell : 0563063594, office : 6402000-ext-68044

Topic 1 : Fundamentals principles of mass transfer

Introduction to mass transfer and its industrial applications. Week 1

Topic 2 : Diffusion coefficients; mass transfer coefficients

Molecular mass transfer, Fick's rate equation, Diffusion coefficient, Gas mass

diffusivity, Liquid mass diffusivity.Week 2

Pore diffusivity, Knudsen diffusion, Solid mass diffusivity, Convective mass

transfer.Week 3

Topic 3: Differential equations of mass transfer

Modeling mass transfer phenomena, Special form of mass transfer equation, Fick's

second law. Week 4

Commonly encountered boundary conditions, steps for modeling process

involving molecular diffusion. Steps to solve mass transfer problems.Week 5

Major exam 1

Course syllabus

Topic 4 : Steady state molecular diffusion

One dimensional mass transfer independent of chemical reaction, Pseudo

steady state diffusion.Week 6

One dimensional system associated with reaction (heterogeneous system) Week 7

One dimensional system associated with reaction (homogeneous system),

film theory and penetration theory.Week 8

Major exam 2

Course syllabus

Topic 5 : Unsteady state molecular diffusion

Unsteady state diffusion and Fick's law, transient diffusion in a semi-

infinite medium.Week 9

Transient diffusion in a finite medium,

Concentration time chart for simple geometric shapes Week 10

Topic 6 : Convective mass transfer

Fundamental consideration of convective mass transfer, Significant

parameters in convective mass transfer, dimensionless analysis for

convective mass transfer, exact analysis of the laminar concentration

boundary layer.

Week 11

Approximate analysis of the laminar concentration boundary layer.

Mass transfer and momentum transfer analogy, Chilton Colburn

analogy.

Week 12

Major exam 3

Course syllabus

Topic 7 : Convective mass transfer correlations

Mass transfer for plates sphere and cylinders Week 13

Mass transfer through pipes, wetted wall columns,

mass transfer in packed and fluidized bedWeek 14

Final exam Week 15

Course syllabus

Class Schedule

Lectures : Sunday – Tuesday 8:00 – 9:30 am

Tutorial : Sunday : 2:30 -5:20 pm

Class Room : 220 , Building 45

Grading

My goal is that you to learn the material and make a high grade in the course!

Homeworks ----------------------------------------------------------- 5%

Midterm I and II and III ------------------------------------------ 40%

Weekly in-lecture quizzes + projects--------------------------- 15%

{ Based on class content or core homework problems +

Diffusion lab experiment report}

Written final exam --------------------------------------------------- 40%

Mass Transfer

When a system consists of two or more components whose concentration vary from point to point , there is a natural tendency of mass transfer minimizing the concentrationdifference within the system.

The transport of one constituent from one region of higher concentration to that of a lower concentration is called mass transfer.

What happens if a lump of sugar added to a cup of black coffee.

Sugar will eventually dissolvediffuse uniformly throughout the coffee.

How long it will take to have uniform concentration of sugar in the coffee cup??

-It depends up on the process- - 1) Quiescent (being at rest/ quite / still)- -2) Mechanically agitated by a spoon.

Mechanism of mass transferDepends on the dynamics of the system in which it occurs

Mechanism of mass transferDepends on the dynamics of the system in which it

occurs.

Random molecular motion in quiescent fluid.

Transferred from a surface into a moving fluid aided by the dynamic characteristics of the flow.

Mass transfer

Molecular mass transfer. Convective mass transfer.

Molecular mass transfer examples

(1)Biological process(a)oxygenation of blood(b)Transportation of ions across membranes

within kidney

(2) Chemical processes(c) Chemical vapor deposition(d)Aeration of waste water(e)Purification of ores and isotopes

(3) Chemical separation processes(f) Adsorption(g)Crystallization(h)Absorption(i) Liquid liquid extraction

Component remains at the interface

Component penetrates to the interface and the transfer to the bulk of the 2nd phase.

We will talk little about interface mass transfer : Chapter 29

Molecular mass transfer

First observed by Parrot 1815.

A gas mixture contains two or more molecular species whose relative concentration varies from point to point, an apparently natural process which tends to diminish any inequalities of composition. This microscopic transport of mass, independent of any convection within the system is called molecular mass transfer.

Kinetic theory of gases can explain mass transfer in gaseous mixture in specific case.

At temperature above absolute zero, individual molecules are in a state of continual yet random motion.

Within dilute gas mixture each solute molecule behaves independently of the other solute molecule, since it seldom encounters them. Collision between solute and solvent molecules are continually occurring. As a result of collision the solute molecules move along a zigzag path sometime towards a region of higher concentration sometime towards a lower concentration.

Mass transfer refers to mass in transit due to a species concentration gradient in a mixture.

Must have a mixture of two or more species for mass transfer to occur.

The species concentration gradient is the driving potential for transfer.

Mass transfer by diffusion is analogous to heat transfer by conduction.

• Physical Origins of Diffusion:

Transfer is due to random molecular motion.

Consider two species A and B at the same T and p, but initially separated by a partition.

– Diffusion in the direction of decreasing concentration dictates net transport of A molecules to the right and B molecules to the left.

– In time, uniform concentrations of A and B are achieved.

HomeworkChapter 24 (WWWR) : 1,8,12,13,15,22

TutorialChapter 24 (WWWR) : 3,4,11,13,17,21,22

Example Chapter 24 (WWWR) : 1,2,3,4,5,6,7,8

Definitions:iCMolar concentration of species i. 3kmol/m

:iMass density (kg/m3) of species i.

:iMMolecular weight (kg/kmol) of species i.

i i iC M* :iJMolar flux of species i due to diffusion. 2kmol/s m

Transport of i relative to molar average velocity (v*) of mixture.

:iN Absolute molar flux of species i. 2kmol/s m Transport of i relative to a fixed reference frame.

:ijMass flux of species i due to diffusion. 2kg/s m Transport of i relative to mass-average velocity (v) of mixture.

Transport of i relative to a fixed reference frame.

:ix Mole fraction of species i / .i ix C C

:im Mass fraction of species i / .i im

Absolute mass flux of species i. 2kg/s m:in

Concentration :

In multi component mixture , the concentration of a molecular species can be expressed in many ways.

Mass concentration also known as density (gm/cm3)

Mass fraction , ωA:

Mass transfer occurs in mixtures, so it is important to evaluate the effect of each component in the transfer process. To explain the role of a component in the mixture we will use the following definitions.

Molecular concentration:

Mole fraction: (liquids,solids) ,

(gases)

RT

p

V

n

m

mol

molkg

mkg

Mc AA

A

AA

3

3

/

/

c

cy

c

cx A

AA

A For gases,

Velocity: mass average velocity,

molar average velocity,

Velocity of a particular species relative to mass/molar average is the diffusion velocity.

P

p

RTP

RTpy AAA

n

iii

n

ii

n

iii

1

1

1

vvv

c

cn

iii

1

vV

For GAS only

mol

Example # 1

Flux: A vector quantity denoting amount of a particular species that passes per

given time through a unit area normal to the vector, given by Fick’s First Law, for basic molecular diffusion.

Flux can be expressed in different ways , three different expression(1) Flux reference to a coordinate that are fixed in space (Total/absolute flux)

nA = Mass flux and NA = molar flux.

(2) Flux reference to a coordinate that are moving with the mass average velocity (jA).

(3) Flux reference to a coordinate that are moving with the molar average velocity (JA).

Fick’s first law defines the molar flux relative to molar average velocity.

or, in the z-direction,

JA,z = Molar flux of component A in z direction relative to molar average velocity.

DAB = Proportionality factor, Mass diffusivity, diffusion coefficient for component A diffusing through component B .

AABA cD J

dz

dcDJ AABzA ,

Isothermal, Isobaric system

For a general relation in a non-isothermal, isobaric system,

dz

dycDJ A

ABzA ,

Mass flux relative to Mass average velocity, jA,z.

dz

dDJ AABzA

,Concentration gradient in terms of mass fraction

Initial experimental investigation were unable to verify Fick’s 1st law

Why??

Since mass is transferred by two means:(1) concentration differences (concentration gradient)

and (2) convection differences from density differences (bulk motion)

For binary system with constant average velocity in z direction Vz,

Thus,

Rearranging to

)( ,, zzAAzA VvcJ

dz

dycDVvcJ A

ABzzAAzA )( ,,

zAA

ABzAA Vcdz

dycDvc ,

)(1

,, zBBzAAz vcvcc

V

)( ,, zBBzAAAzA vcvcyVc

Multiply by cA and rearrange

Molar average velocityc

cn

iii

1

vV

Which substituted, becomes

)( ,,, zBBzAAAA

ABzAA vcvcydz

dycDvc

Defining molar flux, N as flux relative to a fixed coordinate,

AAA c vN

)( ,,, zBzAAA

ABzA NNydz

dycDN

)( BAAAABA yycD NNN Generalized Eqn.

Finally

Bulk motion contributionConcentration gradient contribution

Average velocity

Stairs : Fixed coordinate

Escalator : Moving coordinate

Related molecular mass transfer

Defined in terms of chemical potential, molar diffusion velocity:

Nernst-Einstein relation

dz

d

RT

D

dz

duVv cABcAzzA

,

dz

d

RT

DcVvcJ cABAzzAAzA

)( ,,

Mobility of component A

dz

dcDJ AABzA ,

DIFFUSION COEFFICIENT

Fick’s law proportionality/constant

Similar to kinematic viscosity, n (momentum transfer) m2/s

and thermal diffusivity, (a heat transfer) m2/s

t

L

LLMtL

M

dzdc

JD

A

zAAB

2

32, )

1

1)((

Gas mass diffusivity {Sutherland – Jeans – Chapman - Cowling}

Theoretical expression for gas mass diffusivity for low density gas mixture Based on Kinetic Gas Theory Assumptions (i) Rigid sphere(ii) No intermolecular forces(iii) Elastic collision l = mean free path length, u = mean speed

uDAA l3

1* 2/1

3

22/3

2/3

* )(3

2

AAAA M

N

P

TD

MA = molecular weight ( gm/mol)N = Avogadro’s number = 6.022 x 10 23 molecules/moleP = system pressure (atm)T = absolute temperature (K)k = Boltzmann constant ( 1.38 x 10-16 erg/mol)σ AB = Lennard Jones diameter of the spherical molecule

Species ‘A’ diffusing through its isotopes ‘A*’

Hirschfelder’s equation: For Non polar and Non reacting molecule.

Molecular weight (periodic Table)

DAB

BAAB P

MMT

D

2

2/1

2/3 11001858.0

P in atm

Diffusivity in cm2/s.

Collision diameterA Lennard Jones parameter (Å)

Collision integral

Temperature in Kelvin

For binary gas mixture

Diffusion coefficient for gases: DAB = DBA

Gas phase diffusion coefficient , DAB = f(P,T)

Lennard-Jones parameters and e from tables, or from empirical relations

for binary systems, (non-polar, non-reacting)

ΩD = f(T, intermolecular potential field for one molecule of A and one molecule of B). From Table : K-1;

Extrapolation of diffusivity up to 25 atmospheres

2BA

AB

BAAB eee

2

1

1,12,2

2/3

1

2

2

1

TD

TD

ABAB T

T

P

PDD

PTPT

2/3

1

2

2

1

1,12,2

T

T

P

PDD

PTPT ABAB

Quick estimation

Temperature dependency of collision integral is very nominal/small.

Temperature andPressure correction

of diffusivity

kkBA

AB

eee

BINARY GAS-PHASE LENNARD-JONES “COLLISION INTEGRAL”

If Lennard Jones parameter values are not reported :

Empirical relation to estimate Lennard Jones parameter for PURE COMPONENT.

Vb = Molecular volume at normal boiling point (cm3/g mol); Table 24.4

Tc = Critical temperature(K)

Pc = Critical pressure (atm)

Vc = Critical volume (cm3/g mol)

Tb = Normal boiling point temperature (K)

Gas constant R or Rg :

0.08205746(14) L atm K−1 mol−1

1.9858775(34) cal K−1 mol−1

1.9858775(34)×10−3 kcal K−1 mol−1

8.3144621(75)×107 erg K−1 mol−1

8.3144621(75) L kPa K−1 mol−1

8.3144621(75) m3 Pa K−1 mol−1

8.3144621(75) cm3 MPa K−1 mol−1

8.3144621(75)×10−5 m3 bar K−1 mol−1

8.205746×10−5 m3 atm K−1 mol−1

82.05746 cm3 atm K−1 mol−1

Unit conversion of pressure

Convert from these to pascals (Pa) multiply by

standard atmosphere (atm) 101 325

bar (bar) 100 000kilopascal (kPa) 1 000

megapascal (MPa) 1 000 000millibar (mbar) 100

std centimeter of mercury (cmHg) 1 333.224

millimetre of mercury (mmHg) 133.322 4

Unit conversion of viscosity (μ)

1 poise = dyne·s/cm² = g/cm·s = 1/10 Pa·s1 Pa·s = 1 N·s/m² = 1 kg/m·s

1 cP = 1 mPa·s = Pa·s/1000 = poise/100

Unit of work or energy1 erg = 10-7 J = 10-7 N-m

Interpolation

If you know the atomic wt. you should be able to calculate the molecular wt if you know the correct chemical formula. Periodic Table – keep one with you.

With no reliable s or e, we can use the Fuller-Schettler and Giddings correlation,

Careful about addition of structural correction; i.e. aromatic ring, for calculation of diffusion volume from atomic volume.

Calculation of volume of benzene (C6H6) [vapor/gas phase] from table 24.3.

Atomic diffusion volume : C = 16.5; H = 1.98 Diffusion volume = 16.5*6 +1.98*6 +aromatic ring correction

= 99 + 11.88- 20.2 = 90.68

23/13/1

2/1

75.13 1110

BA

BAAB

vvP

MMT

D

Gas Diffusivityin cm2/s

Pressure in atm

Atomic diffusion volume (Table 24.3) cm3/g mol

For binary gas mixture

Calculation of volume of benzene (C6H6) [vapor/gas phase] from table 24.3.Atomic diffusion volume : C = 16.5; H = 1.98 Diffusion volume = 16.5*6 +1.98*6 +aromatic ring correction

= 99 + 11.88- 20.2 = 90.68

where

bb

PBAAB TV

232/1 1094.1,

ABTT e /* 2/1

e

e

e BAAB

bT23.1118.1/ e

)exp()exp()exp( ****0 HT

G

FT

E

DT

C

T

ABD

For binary gas with polar compounds, we calculate by

*

2196.00 T

ABD

2/1BAAB

3/1

23.11

585.1

bV

Example of polar gases:NH3 , SO2, H2S, PH3

For binary gas mixture containing POLAR component

Suggested by Brokaw (1969)Hirschfelder equation is valid but need to estimate the collision integral (ΩD) in a

different way.

DAB

BAAB P

MMT

D

2

2/1

2/3 11001858.0

μb = dipole moment, Debye.Vb = Liquid molar volume of the specific compound at its boiling point, cm3/g mol, Table 24-4 and 24-5.Tb = Normal boiling point in K.

Problem Exercise 24-11

For gas mixtures with several components (multiple components),

nnmixture DyDyDy

D

1'

31'321

'2

1 /...//

1

1

2

32

2'2 1... y

y

yyy

yy

n

1

3

32

3'3 1... y

y

yyy

yy

n

Calculate D1-2, D1-3 --- using any empirical equation or you can get it from literature i.e values reported in Appendix J-1. You may need to perform temperature correction of the diffusivity value if you use the value from the App. J.1.

In appendix J.1 : diffusivity values are reported in the form of => DAB.P [cm2 .atm/s]

you need to divide the value with the system pressure to get the actual value at the reported temperature.

2/3

1

2

1,12,2

T

TDD

PTPT ABAB

132

'

1... y

y

yyy

yy n

n

nn

yn’= Mole fraction of component ‘n’ in the gas mixture evaluated on a component -1-free basis.

2

Liquid mass diffusivity No rigorous theories Diffusion as molecules or ions Eyring theory Hydrodynamic theory

Stokes-Einstein equation

Equating both theories, we get Wilke-Chang equation

BAB r

TD

6

6.0

2/18104.7

A

BBBAB

V

M

T

D

Viscosity in centipoises (cP) unit 1 cP = 10−2 P = 10−3 Pa·s = 1 mPa·s

Non electrolyte solute in low concentration solution

Association parameter for solvent, B

Molar volume atNormal boiling point

A = soluteB = solvent

If data for computing the molar volume of solute at its normal boiling point, VA is not available, Tyne and Calus (1975) recommended the following correlation :

Vc = critical volume of species A in cm3/g. mol. Values are tabulated in literature {Reid, Prausnitz and Sherwood, 1977}.

Molar volume of ethanol

6.0

2/18104.7

A

BBBAB

V

M

T

D

For infinite dilution of non-electrolytes in water, W-C is simplified to Hayduk-Laudie eq.

Scheibel’s equation eliminates FB, simplified form of W-C

Exceptions:1) for benzene as a solvent, if VA < 2VB , use K = 18.9x10-8.

2) For other organic solvents, if VA < 2.5 VB, use K = 17.5x 10-8.

Liquid diffusion coefficient in concentrated solution:

Combine the infinite dilution coefficient DAB and DBA =>

Where, DAB = infinitely dilute diffusion coefficient of A in solvent B.

For associating compound , i.e alcohols,

589.014.151026.13 ABAB VD

3/1A

BAB

V

K

T

D

3/2

8 31)102.8(

A

B

V

VK

As diffusivity changes with temperature, extrapolation of DAB is by

Tc = Critical temperature of solvent B in Kelvin. Temperatures are in Kelvin unit.

n = Exponent related to the heat of vaporization of solvent(B), ΔHv, at its normal boiling point temperature {Text book page 419 - table for the value of n}.

For diffusion of univalent salt in dilute solution, we use the Nernst equation:

DAB = Diffusion coefficient based on the molecular concentration of A in cm/s2.

R = Gas constant 8.316 Joules/K/g mol. F = faraday’s constant 96500 coulombs/g equivalent

= Limiting ionic conductance in (amp/cm2 )(volt/cm)(g equivalent/cm3)

n

c

c

ABT

ABT

TT

TT

D

D

1

2

)(

)(

2

1

F

RTDAB )/1/1(

200

ll

00 /1/1 ll and

polyvalent salt solution

Replace the constant 2 in the univalent salt diffusion equation with where n+ and n- are the valances of the cation and anion of the polyvalent salt.

nn

11

Pore diffusivityDiffusion of molecules within pores of porous solids

example Heterogeneous catalysts are porous solids containing active

material inside the pore wall. So reactants need to diffuse through the pore and reach the active metal surface to convert into products.

Separation of solute from dilute solution by the process of adsorption.

Knudsen diffusion for gases in cylindrical pores(1)Pore diameter smaller than mean free path, and (2) density of gas is low Knudsen number

If Kn>>>1 then Knudsen diffusion is important.

From Kinetic Theory of Gases,

diameterpore

speciesgdiffuofpathfreemean

dKn

pore

sin

l

AAA M

NTuD

ll 8

33*

But if Kn >1, then

If both Knudsen and molecular diffusion exist, then

with

For non-cylindrical pores, we estimate

Apore

A

poreporeKA M

Td

M

NTdu

dD 4850

8

33

KAAB

A

Ae DD

y

D

111

A

B

N

N1

AeAe DD 2' e

dpore in cm

Diffusivity value in cm2/s

Void fraction in the solid, ε = void volume /(total volume of solid + void)

Types of porous diffusion. Shaded areas represent nonporous solids

EXAMPLE 6

Updating the diffusivity value to a different temperature and pressure

Hindered diffusion for solute in solvent-filled pores

Diffusion of a solute molecule through tiny capillary pore filled with liquid solvent. A general model is

DAB0 = Liquid liquid diffusivity.

F1 and F2 are correction factors, function of pore diameter, values between 0 and 1.

If φ >1, solute molecule is greater than pore diameter, This phenomena is known as solute exclusion. It is used to separate large biomolecules such as proteins from dilute aqueous mixture.

)()( 21 FFDD oABAe

pore

s

d

d

ds = diameter of solute molecule

F2 is the hydrodynamic hindrance factor, one equation is by Renkin, range in between 0<=φ=>0.6

532 95.009.2104.21)( F

Assumptions for the model:

-Spherical rigid solute in straight cylindrical pore.

-Ignore electrostatic or other energetic solute, solvent and pore wall interaction; polydispersity of solute diameter and noncircular pore diameter.

o F1 is the stearic partition coefficient: geometric hindrance

22

1 2

( )( ) (1 )pore s

pore

d dF

d

EXAMPLE 7

CONVECTIVE MASS TRANSFER

Mass transfer between moving fluid with surface or another fluid

Forced convection Free/natural convectionRate equation analogy to Newton’s cooling equation

AAScAcA CCkckN

Surface concentration (mol/m3)

Bulk fluid concentration

Mass transfer coefficient (m/s)

Film mass transfer and film mass transfer co-efficient:

Fluid flowing past a surface, there is a thin layer close to the surface where the fluid is laminar and the fluid particles next to the solid boundary are at rest.

Laminar boundary layer

No slip condition

Mechanism of mass transfer between the surface and stagnant and laminar layer is by molecular mass transfer.

Controlling resistance to the convective mass transfer sometime is from this laminar “film” due to molecular diffusion.

Momentum transfer/ Fluid Mechanics

h

wL

W

L

EXAMPLE 8

Diffusion in solids

Diffusion of atoms within solids. Mainly covered in the Materials Engineering course.

Examples:Semiconductor manufacturing process :Impurity atoms : DOPANTS are introduced to the solid silicon to control the conductivity in a semiconductor device.

Hardening of still (i.e. Carburization) : Diffusing carbon(C) atom through iron (Fe).

Solid diffusion mechanism :

(1)Vacancy Diffusion(2)Interstitial diffusion

Vacancy diffusion:

Transported atoms JUMPS from a lattice position of the solid into the neighboring unoccupied solid site or vacancy.

Atoms continues to diffuse by a series of jumps into the neighboring vacancy.

This normally requires a distortion of lattice or lattice defect sites.

Interstitial Diffusion

the diffusing atom is not on a lattice site but on an interstice. The diffusing atom is free to move to any adjacent interstice, unless it is already occupied.

The rate of diffusion is therefore controlled only by the ease with which a diffusing atom can move into an interstice. 

Appendix J-3 : Values of binary diffusion coefficient in solids

Do = proportionality constantQ = activation energy , kcal/mol, J/mol etc.R = Gas constant = 8.314 J/mol/K

Effect of temperature :

Diffusion coefficient value increases with temperature according to Arrhenius equation

There is an energy barrier to change the places of atom.Eyring “unimolecular rate theory” concept explains the mechanism of the

diffusion.

Problem : 24.19

Linear Interpolation formula

X1 Y1

X ( @ known x value) Y (unknown)

X2 Y2