Fundamentals of Mass Transfer

When a system contains two or more components whose

concentrations vary from point to point, there is natural

tendency for mass to be transferred, minimizing the

concentration differences within the system.

Mechanisms of mass transfer

Molecular Transfer

Convective Transfer

Mass transfer

~ random molecular motion in quiescent fluid

Molecular transfer

Convective transfer

~ transferred from surface into a moving fluid

Analogy between heat and mass transfer

Heat transfer

Heat conduction

Heat convection

Fundamentals of Mass Transfer

Fick’s lawdz

dj A

ABAz

−= D

In vector form

Mass diffusivity

AABAj −= D

ABD

Prandtl number

=Pr

Schmidt number

Lewis number

ABAB DD

=

=Sc

ABABpC

ke

DD

=

=L

dz

dxCJ A

ABAz D−=

AABA xCJ −= D

Molecular Mass Transfer(Molecular Diffusion)

"Transport Phenomena" 2nd ed.,R.B. Bird, W.E. Stewart, E.N. Lightfoot

Mass Concentration

the mass of A per unit volume of mixture

( )Aa) mass concentration

=i

i

Molecule of

species A

dV in multicomponent

muxture

- Total mass concentration(density)

=

=

A

ii

AA

- mass fraction

=i

i 1

Molar Concentration

~ the number of moles of A present per unit volume

of mixture

( )ACb) molar concentration

=i

iCC- Total molar concentration

A

AA

MC

=

For ideal gas

=i

ix 1C

Cx A

A =

- molecular weight of AAM

RTnVP AA =

RT

P

V

nC AA

A == - partial pressure of AAP

- Mole fraction

for ideal gas mixture

for gasfor liquid

for ideal gasRT

P

V

nC total ==

C

Cy A

A = =i

iy 1

P

P

RTP

RTP

C

Cy AAA

A === Dalton’s law of gas mixture

Mass Average and Molar Average Velocity

a) mass average velocity

=

=

=

iii

iii

ii

iii

vvv

v - the absolute velocity of species i

relative to stationary coordinate axes

iv

b) molar average velocity

===i

iii

ii

ii

iii

vxC

vC

C

vCV

c) diffusion velocity - the velocity of a particular species relative to the mass

average or molar average velocity

vvi − and Vvi −

A species can have a diffusion velocity

only if gradients in the concentration existvvB −

vv A −

Bv

vAv

Mass Diffusivity

t

L

LLMtL

M

dzdC

J

A

AAB

2

32 1

1=

=

−=D

( )nCompositio,essurePr.,TempfAB =D

Gases 5x10-6 ~ 10-5 m2/s

Liquids 10-10 ~ 10-9

Solids 10-14 ~ 10-10

Molecular Mass and Molar Fluxes

ABD

( ) AABAAA xCVvCJ −=−= D

AJ

Molar flux

Mass flux

mass diffusivity(diffusion coefficient) for component A

diffusing through component B

the molar flux relative to the molar average velocity

( ) AABAAAvvj −=−= D

Convective Mass and Molar Fluxes

Molar flux

Mass flux relative to a fixed spatial coordinate system

+=+−=+−==i

iAAi

iAAABAAABAA njnvvn DD

( ) AABAAA xCVvCJ −=−= D

VCxCvC AAABAA +−= D

===i

iii

ii

ii

iii

vxC

vC

C

vCV

+−=i

iiAAABAA vCxxCvC D

define AAA vCN = the molar flux relative to a set of stationary axes

+=+−=i

iAA

i

iAAABA NxJNxxCN D

concentration

gradient

contribution

bulk motion

contribution

Related Types of Molecular Mass Transfer

Nernst-Einstein Relation

Au Mobility of component A, or the resultant velocity of

the molecule under a unit driving force

According to 2nd law of thermodynamics, system not equilibrium will tend

to move toward equilibrium with time

Driving force = chemical potential

cAB

cAART

uVv −==−D

( ) cAB

AAAART

CVvCJ −=−=D

in homogeneous ideal solution at constant T & P

Ac ClnRT+= 0

AABA CJ −= D Fick’s Equation

Mass Transfer by Other Physical Conditions

Thermal diffusion(Soret diffusion)

~ produce a chemical potential gradient

Mass transfer by applying a temperature gradient to a multicomponent system

Small relative to other diffusion effects in the separation of isotopes

Pressure diffusion

Mass fluxes being induced in a mixture subjected to an external force field

Separation by sedimentation under gravity

Electrolytic precipitation due to an electrostatic force field

Magnetic separation of mineral mixtures by magnetic force

Component separation in a liquid mixtures by centrifugal force

Knudsen diffusion

If the density of the gas is low, or if the pores through which the gas

traveling are quite small

the molecules will collide with the wall more frequently than each other

(wall collision effect increases)

the molecules are momentarily absorbed and than given off in the

random directions

gas flux is reduced by the wall collisions

Differential Equations of Mass Transfer

Conservation of Mass(overall)

( ) 0=

+ VA dV

tdAnv

Equation of Continuity for Component A

Conservation of Mass of Component A

0=−

+ A

AA r

tn

yy,Ayyy,Axx,Axxx,A xznxznzynzyn −+−++

0=−

+−+

+zyxrzyx

tyxnyxn A

A

zz,Azzz,A

0=−

+ A

AA R

t

CN

Differential

Equations of Mass

Transfer-1

Equation of Continuity for the Mixture

In binary mixture

0=−

+ A

AA r

tn

0=−

+ B

BB r

tn

( )( )

( ) 0=+−

+++ BA

BABA rr

tnn

vvvnn BBAABA =+=+

=+ BA

BA rr −=

0=

+

tv

Differential

Equations of Mass

Transfer-2

0=−

+ A

AA r

tn

vjn AAA +=

( )vjn AAA +=

( ) 0=−+

+ AA

A

Arv

tj

vvtt

AAAA ++

+

=0

0=−+

AA

A rjDt

D

Differential

Equations of Mass

Transfer-3

in terms of molar unit

0=−

+ A

AA R

t

CN

0=−

+ B

BB R

t

CN

But RA is not always –RB

for only stoichiometry reaction ( A B)

( )( )

( ) 0=+−

+++ BA

BABA RR

t

CCNN

VCvCvCNN BBAABA =+=+

CCC BA =+

BA RR −=

( ) 0=+−

+ BA RR

t

CVC

Differential Equations

of Mass Transfer-4

AAABAA rvt

+=

+

2D

vCn AAABA +−= D

VCyCN AAABA +−= D

( )BAAAABA NNyyCN ++−= D

( )BAAAABA nnn ++−= D

or

If and are constant ABD

AAABAAAA rvv

tt+=++

+

2D

AAABAA RCCV

t

C+=+

2D

or

0=−+

AA

A rjDt

D

AAABAA rv

t+=

+

2D

AAA rnt

+−=

gvt

+−=

( )gvevUt

+−=

+

2

21

AAA vjn +=

vv +=

+++=

2

21 vUvvqe

( ) ( ) ( )AAgTTggpDt

vD −−−−−+−=

( ) ( ) ( )

( ) ( )AA

AA

,TA,T

A

TT

TTT

,T

AA

−−−−

+−

+−

+=

Boundary Conditions

Initial Conditions 00 or 0 AAAA CC,tat ===

Case 1) The concentration of the surface may be specified

Boundary Conditions

, , , , 10111 AAAAAAAAAA ,xxyyCC =====

Case 2) The mass flux at the surface may be specified

0 11

,0 , ,=

−====

z

AABsurfacez,AAAAAA dz

dDj)eimpermeabl(jNNjj

Case 3) The rate of heterogeneous chemical reaction may be specified

Case 4) The species may be lost from the phase of interest by convective mass

transfer

surface

n

Ansurfacez,A CkN =

( )

−= ,Asurface,Acsurfacez,A CCkN

Steady State Diffusion of A through Stagnant B

0=−+ zz,Azzz,A SNSN

At z=z1, gas B is insoluble in liquid A

Mass Balance Equation

( )z,Bz,AAA

ABz,A NNydz

dyCN ++−= D

z

Liquid A

zAN

zzAN+

S

Flowing of gas B

1zz =

2zz =

0=dz

dN z,A 0=dz

dN z,B

0= z,BN

~ Component B is stagnant

0=z,BN

dz

dy

y

CN A

A

ABz,A

−−=

1

D

22

11

AA,

AA,

yyzzat

yyzzat

==

==

Steady State

Diffusion of A

through Stagnant B-1

−

−= 2

1

2

1 1 A

A

y

y

A

AAB

z

zz,Ay

dyCdzN D

( )( )21

ln,12ln,

21

121

2

12

,1

1ln AA

B

AB

B

AAAB

A

AABzA yy

yzz

C

y

yy

zz

C

y

y

zz

CN −

−=

−

−=

−

−

−=

DDD

( ) ( )( ) ( )12

12

12

12

11

11

AA

AA

BB

BBln,B

yyln

yy

yyln

yyy

−−

−−−=

−=

Steady state diffusion of one gas through a second gas

~ absorption, humidification

P

Pyand

RT

P

V

nC A

A ===

( ) ln,

21

12

,

B

AAABzA

P

PP

zzRT

PN

−

−=

D

For ideal gas

Film Theory

~ a model in which the entire resistance to diffusion from the liquid surface

to the main gas stream is assumed to occur in a stagnant or laminar film of

constant thickness d.

zAN ,

Flowing of gas B

0=z

d=z

Liquid A

Slowly moving gas film

ln,

21,

B

AAABzA

P

PP

RT

PN

−

d=D

( ) ( )2121, AAc

AAczA PPRT

kCCkN −=−=

d=

ln,B

ABc

P

Pk

D

Convective Mass Transfer

1.0~0.5

ABck Dactual

Main fluid stream

in turbulent flow

Concentration Profile

0=dz

dN z,A

dz

dy

y

CN A

A

ABz,A

−−=

1

D

22

11

AA,

AA,

yyzzat

yyzzat

==

==

01

=

− dz

dy

y

C

dz

d A

A

ABD

If C and are constant under isothermal and isobaric conditionsABD

01

1=

− dz

dy

ydz

d A

A

( ) 211ln CzCyA +=−−

( ) ( )121

1

2

1 1

1

1

1zzzz

A

A

A

A

y

y

y

y−−

−

−=

−

−

ln,

12

12

lnB

BB

BBBB y

yy

yy

dz

dzyy =

−==

Pseudo Steady State Diffusion through a

Stagnant Gas Film

dt

dz

MN

A

zA

LA,

,

=

tzzttat

zztat

==

==

,

,0 0

the length of the diffusion path changes a small amount over a long

period of time ~ pseudo steady state model

ln,B

AAAB

A

AABz,A

y

yy

zz

C

y

yln

zz

CN 21

121

2

12 1

1 −

−=

−

−

−=

DD

Molar density of A in the liquid phase

ln,

21,

B

AAAB

A

LA

y

yy

z

C

dt

dz

M

−=

D

( )

−

−

=

2

2

0

2

21

ln,, zz

yyC

Myt t

AAAB

ABLA

D

( )

−

−

=

2

2

0

2

21

ln,, zz

tyyC

Myt

AA

ABLA

ABD

zAN ,

S

Flowing of gas B

0zz =

tzz =

Liquid A

Diffusion through a Isothermal Spherical Film

0=r,BN

( ) 0,

2 =rANrdr

d

01

2 =

− dr

dx

x

Cr

dr

d A

A

ABD

( ) ( )( ) ( )

21

1

11

11

1

2

1 1

1

1

1 rr

rr

A

A

A

A

x

x

x

x −

−

−

−=

−

−

for constant temperature is constantABCD

Temperature T1

r1

r2

Gas film

Temperature 12 TT =

( ) ( )

−

−

−

==

=1

2

21

,

2

11

1ln

11

44

1

A

AAB

rrrAAx

x

rr

CNrW

D

( )r,Br,AAA

ABr,A NNxdr

dxCN ++−= D

Since B is insoluble in liquid A

Diffusion through a Nonisothermal Spherical Film23

1

n

AB,1

AB

r

r

=

D

D

01

23

1

12 =

− dr

dx

r

r

x

RTPr

dr

d A

n

A

AB,1D

( ) ( )

( ) ( ) ( ) ( )

−

−

−

+==

++=1

2

2

1

21

2

21

1

,

2

11

1ln

11

2144

1

A

A

nnn

AB,1

rrrAAx

x

rrr

nRTPNrW

D

01

2 =

− dr

dx

x

Cr

dr

d A

A

ABD

2 −nif

Temperature T1

r1

r2

Gas film

Temperature

n

r

rTT

=

1

212

Diffusion with Heterogeneous Chemical Reaction

01

1

21

=

− dz

dx

xdz

d A

A

−d=

021,

1

1ln

2

A

ABzA

x

CN

D

Gas A Gas A & B

Sphere with coating of catalytic material

BA →2

Edge of hypotheticalStagnant gas film

Catalytic surface

A

B

Z=0

Z=d

Z xA

xB

xA0

zAzB NN ,21

, −=

dz

dx

x

CN A

A

ABzA

21,

1−−=

D

( ) ( ) ( )212121 ln2ln21ln2 KzKCzCxA −−=+=−−

0 ,

,0 0

=d=

==

A

AA

xzat

xxzat

( ) ( ) ( )d−−=−

z

AA xx1

021

21 11

Irreversible and instantaneous rxn

Solid catalyzed dimerization of CH3CH=CH2

Diffusion with Slow Heterogeneous Chemical Reaction

01

1

21

=

− dz

dx

xdz

d A

A

( )large 1

1ln

1

21

021

1

, kxk

CN

AAB

ABzA

−d+

d=

D

D

BA →2

zAzB NN ,21

, −=dz

dx

x

CN A

A

ABzA

21,

1−−=

D

( ) 21211ln2 CzCxA +=−−

( )AzA

zA

A

AA

CkNCk

Nxzat

xxzat

1,

1

,

0

,

,0

==d=

==

assume pseudo 1st order rxn

( ) ( ) ( )d−

d

−

−=−

z

A

z

AzA x

Ck

Nx

1

021

1

21 1

2

111

ABkDa Dd= 1 Damkohler No.: the effect of the surface reaction on the

overall diffusion reaction process

Diffusion with Homogeneous Chemical Reaction

ABBA →+

01,, =−−+

zSCkSNSN AzzzAzzA

dz

dCN A

ABzA D−=,

( )0 0 ,

,0

,

0

===

==

dzdCorNLzat

CCzat

AzA

AA

First order reaction

01

,=+ A

zACk

dz

dN

012

2

=− AA

AB Ckdz

CdD

02

2

2

=−

d

d

0AA CC= Lzζ = ABLk D2

1= Thiele modulus

0 ,1

1 ,0

==

==

ddat

at

Diffusion with

Homogeneous

Chemical Reaction-1

=−=

==

tanh0

00,

L

C

dz

dCN ABA

z

AABzzA

DD

ζsinhcosh 21 += CC

( )

−=

−=

cosh

1cosh

cosh

ζsinhsinhcoshcosh

( )( ) AB

AB

A

A

Lk

LzLk

C

C

D

D2

1

2

1

0 cosh

1cosh −=

( )

==

tanh

dz

dzL

0

L

0 0

0

, AA

A

avgA CC

C

C

Gas Absorption with Chemical Reaction

in an Agitated Tank

effect of chemical reaction rate on the

rate of gas absorption in an agitated

tank

Example) The absorption of SO2 or

H2S in aqueous NaOH solutions

Semiquantative understanding can

be obtained by the analysis of a

relatively simple model

Gas A in

Surface area

of the bubble

is S

Volume of liquid

phase is V

Liquid B

Gas Absorption with

Chemical Reaction

in an Agitated Tank-1

Assumptions:

1) Each gas bubble is surrounded by a stagnant liquid film of thickness d, which is

small relative to the bubble diameter

2) A Quasi-steady state concentration profile quickly established in the liquid film

after the bubble is formed

3) The gas A is only sparingly soluble gas in the liquid, so that we can neglect the

convection term.

4) The liquid outside the stagnant film is at a concentration CAd, which changes so

slowly with respect to time that it can be considered constant.

Z=0 Z=d

CAd

Liquid film

Liquid-gas

interface

Gas in

bubble

Main body of liquid

Without reaction

With reaction

Gas Absorption with

Chemical Reaction


012

2

=− AA

AB Ckdz

CdD

d=d=

==

AA

AA

CC,zat

CC,zat

0 0

( )

−+=

sinh

sinhcoshBcoshsinh

C

C

A

A ζ

0

0AA CC= dzζ =ABk D2

1d =

d

d=

= A

z

AAB CVk

dz

dC1SD-

( ) d+=

sinhSVcoshB

1

( )

d+−

=

d= =

sinhSVcoshcosh

sinhC

NN

ABA

zz,A 1

0

0

D

The total rate of absorption with chemical reaction

ζsinhcosh 21 += CC

Assumption (d)

Gas Absorption with

Chemical Reaction


for large value of , dimensionless surface

mass flux increases rapidly with and

becomes nearly independent of V/Sd.

for very slow reactions,

the liquid is nearly

saturated with dissolved gas

Chemical rxn is fast enough to keep

the bulk of the solution almost solute

free, but slow enough to have little

effect on solute transport in the film


Diffusion and Chemical Reaction inside a Porous

CatalystWe make no attempt to describe the diffusion inside the

tortuous void passages in the pellet. Instead, we describe the

“averaged diffusion” of the reactant ~ effective diffusivity

( ) 0444 22

,

2

, =++−+

rrRrrNrN ArrrArrA

( ) ( )A

rr,A

rrr,A

r

Rrr

NrNr

lim2

22

0

=−

+

→

( ) ArA RrNrdr

d 2

,

2 =


Diffusion and Chemical

Reaction inside a

Porous Catalyst-1

First order chemical reaction,

a is the available catalytic surface per unit volume

AA

A Rdr

dCr

dr

d

r−=

2

2

1D

dr

dCN A

AA -D=

AA

A aCkdr

dCr

dr

d

r1

2

2

1=

D

finiteC,rat

CC,Rrat

A

ARA

==

==

0

( )rfrC

C

AR

A 1= f

ak

dr

fd

A

=D

1

2

2

rak

sinhr

Cr

akcosh

r

C

C

C

AAAR

A

DD1211 +=

Raksinh

raksinh

r

R

C

C

A

A

AR

A

D

D

1

1

=

effective diffusivity

depends on P, T

and pore structure


Reaction inside a

Porous Catalyst-2

If the catalytically active surface were all exposed to the stream, then the

species A would not have to diffuse through the pores to a reaction site.

Rr

AAR,AA

dr

dCRNRW

=

−== D22 44

−= R

akcothR

akCRW

AA

ARAADD

D 1114

( )( )( )AR,AR CkRW 13

34

0 −= a

( )132

0

−

== cothW

W

,AR

ARA

Aak D1= Thiele modulus

Effective factor

For nonspherical catalyst particles

AARpAR CakVW 1

=

p

p

nonsphS

VR 3 ( )133

3

12

−

= cothA

( )ppA SVak D1= generalized

modulus


Reaction inside a

Porous Catalyst-3


Diffusion in a Multi-Component System

( ) ( )

NxNxC

vvxx

xNN

−−=−−= == 11

1

DD

Liquid water(species 1) is evaporating into air, regarded as a binary mixture of

nitrogen(2) and oxygen(3) at a given T & P.

3 2 1 0 ,,dz

dN z,==

Since species 2 and 3 are not moving, 032 == z,z, NN

xC

N

dz

dx ;x

C

N

dz

dx z,z,

3

13

13

2

12

12

DD==

Maxwell Stefan equation for multicomponent diffusion in gases at low density

N,...,,, 321=

Textbook p567

;13

13

3 3

3

12

12

2 2

2 ==

L

z

z,Lx

x

L

z

z,Lx

xdz

C

N

x

dxdz

C

N

x

dx

DD

( ) ( )

−−=

−−=

13

1

3

3

12

1

2

2 ;DD C

zLNexp

x

x

C

zLNexp

x

x z,

L

z,

L

( ) ( )

−−−

−−−=

13

13

12

121 1

DD C

zLNexpx

C

zLNexpxx

z,L

z,L

At z=0

−−

−−=

13

13

12

1210 1

DD C

LNexpx

C

LNexpxx

z,L

z,L

Transient 1-D Diffusion of a Finite Slab~ negligible surface resistance

2

2

z

C

t

C AAB

A

=

D

( )

0 ,

0 ,0

Lz0 ,0 0

==

==

==

tLzatCC

tzatCC

tatzCC

sAA

sAA

AA

sAA

sAA

CC

CCYLet

−

−=

0

2

2

z

Y

t

YAB

=

D

0 , 0

0 ,0 0

Lz0 ,0 1

==

==

==

tLzatY

tzatY

tatY

by Separation variables method

( ) ( ) ( )tGzZtzY =,

z=0 z=L

( )zC A0

AsCtAsC

Textbook p612

2005. 05. 30

Transient 1-D

Diffusion of a

Finite Slab -1

( ) ( )

=

−

=

10 0

2sinsin

2 2

n

LXndz

L

znzYe

L

zn

LY D

( ) , 00 YzYif =

( )

=

−=

=

−

−

1

2

0

531 sin14 2

n

Xn

sAA

sAA ,,neL

zn

nCC

CCD

( ) ( )

=

−=

−=−=

1

2

0, 531 cos4 2

n

Xn

AsAABA

ABzA ,,neL

znCC

Ldz

dCN D

DD

( )22LtX ABD D=

Transient Diffusion in a Semi-Infinite Medium

0 ,

0 ,0

,0

0

0

→=

==

==

tzasCC

tzatCC

all ztatCC

AA

sAA

AA

~ Similarity Solution Technique

=

=

−

−22

110

0 ,z

tf

z

t

z

zf

CC

CC

AsA

AA

( ) ( )00

2 AsAAA CCCCY ,tz Let −−==

2

2

z

C

t

C AAB

A

=

D

( )

( ) ( ) 1 ,00

2

0

2

==

=

−

erferf

deerf

−=

−

−

t

zerf

CC

CC

ABAsA

AA

D21

0

0or

=

−

−

t

zerf

CC

CC

ABAAs

AAs

D20

Unsteady State Evaporation of a Liquid

0

0

0 0

A

A0A

A

==

==

==

x,zat

xx,zat

x,tat

001 =

−−=

z

A

A

ABz

z

x

xV

D

Continuity equation

zAN ,

S

Flowing of gas B

0=z

Liquid A

0=

z

Vz

( )tVV zz 0=

In the binary systemC

NNV

z,Bz,Az

00 +=

00 =z,BN Gas B is insoluble in Liquid A

2

2

001 z

x

z

x

z

x

xt

x AAB

A

z

A

A

ABA

=

−−−

=

DD

Textbook p613

Unsteady State

Evaporation of a

Liquid-1

0

1 0

==

==

X,Zat

X,Zat

Dimensionless molar

average velocity

( ) 022

2

=−+dZ

dXZ

dZ

Xd

tz Z,Xxx ABAA D4 0 ==

( )00

00

12

1

=−−==

ZA

AABzA

dZ

dX

x

xtVx D

( ) 21 −−= ZexpC

dZ

dXY

( ) 20

2

1 CZdZexpCXZ

+

−−=

( )( )

( )

dWWexp

dWWexp

ZdZexp

ZdZexp

ZX

ZZ

−

−

−

−

−−=

−−

−−

−=

2

2

0

2

0

2

11

( )( ) ( )

+

−−=

+

+−−=

erf

Zerf

erferf

erfZerfZX

1

11 Textbook p616

Fig. 20.1-1

Unsteady State

Evaporation of a

Liquid-2

tS

C

SN

dt

dV ABAA D== 0

( )( )

+

−

−=

erf

erf

x

xx

A

AA

11

1 2

0

00

( ) 120

11

1−

++=

experfxA

Rate of production of vapor from surface of area S

VA: the volume of A evaporation up to time t

tSV ABA D4=

2

2

z

x

t

x AAB

A

=

D

=

tSxV AB

AFick

A

D40

=t

SxV ABAA

D40 0Ax=

Deviation from the Fick’s second law caused

by the nonzero molar average velocity

( )00

00

12

1

=−−=

ZA

AA

dZ

dX

x

xx

Gas Absorption with Rapid Reaction

( )

( )

==

−=

−=

===

==

==

BB

BBS

AASR

BAR

AA

BB

CC, at z

z

C

bz

C

a, tzat z

CC, tzat z

CC, at z

for zCC, at t

11

0

0

0 0

0

DD

Gas A is absorbed by a stationary liquid solvent S containing solute B

aA + bB products

( )

( )

=

=

zt for zz

C

t

C

tzz for z

C

t

C

RB

BSB

RA

ASA

0

2

2

2

2

D

D

Gas A

Solute B

Liquid-Vapor Interface

ZR

Textbook p617

Gas Absorption with

Rapid Reaction-1

( )

( ) +=

+=

zt for zt

CCC

C

tzz for t

CCC

C

R

BSB

B

R

ASA

A

0

43

21

0

4D

zerf

4D

zerf

( )( )

( )

( )( )

( ) −

−−=

−=

zt for zt

t

C

C

tzz for t

t

C

C

R

BS

BS

B

B

R

AS

AS

A

A

1

11

0 10

4Dzerf

4Dzerf

4Dzerf

4Dzerf

R

R

−

=

−

BSASASAS

BS

A

B

BS

experfbC

aCerf

DDDD

D

D 0

1

ttanconstzR == 42 ZR increases as t

( )z

C

bz

C

a, tzat z C. B. of instead CC , at z C. B. from B

BSA

ASRBB

−=

−=== DD

11

Gas Absorption

with Rapid

Reaction-2

( ) 0

0

0terf

C

z

CN AS

AS

A

z

AASz,A

=

−=

=

D

DD

The average rate of absorption up to t

( ) 2

1 0

000

terf

CdtN

tN AS

AS

At

z,Aavg,z,A

== D

D

Gas Absorption with Rapid Reaction-2


Diffusion into a Falling Liquid Film(Gas Absorption)

x=0 x=d

xy

z

x

z

CA0

CA0

V(x)

0=−+−++

zWNzWNxWNxWNxxx,Axx,Azzz,Azz,A

( )

d−=

2

1x

vxv maxz

0=

+

x

N

z

N x,Az,A

( ) ( )xvCNNxz

CN zAz,Bz,AA

AABz,A ++

= -D

( )x

CNNx

x

CN A

ABx,Bx,AAA

ABx,A

++

= D-D

2

2

x

C

z

Cv A

ABA

z

=

D

Textbook p.558

Diffusion into a

Falling Liquid

Film(Gas

Absorption)-1

2

22

1x

C

z

Cxv A

ABA

max

=

d− D

0 x

0

0 0

0

=

d=

==

==

A

AA

A

C,xat

CC,xat

C,zat

2

2

x

C

z

Cv A

ABA

max

=

D

0

0

0 0

0

==

==

==

A

AA

A

C,xat

CC,xat

C,zat

The substance A has penetrated only a short distance into the film~penetration model

( ) −

−=maxvzABx

A

A dexpC

C D4

0

2

0

21

maxABmaxABA

A

vz

xerfc

vz

xerf

C

C

DD 441

0

=−=

z

vC

x

CN maxAB

A

x

AABxx,A

=

−=

==

DD 0

00

L

vWCdz

z

vWCdzdyNW AB

A

LABAx

W L

xAA

=

== =

max00

max000 0 ,0

1 4DD

Rybczynski-Hadamard circulation

Gas absorption from rising bubbles

Gas bubbles rising in liquids free surface –active

agents undergo a toroidal circulation

0

00 A

tAB

x

AABxx,A C

D

v

x

CN

4DD =

−=

==

D

vt t

osureexp =

For creeping flow00 3

AtAB

xx,A CD

vN

4D=

=

Trace of surface-active agents cause a marked decrease in absorption rates from small bubbles

By preventing internal circulation31

0

/

ABxx,AN D=


Diffusion into a Falling Liquid Film(Solid Dissolution)

d

y

z

CA=0

CA0

d−

d

d=

d−−

d=

2222

22

112

yygygvz

( ) ( ) 310 9 zay η,fCC ABAA D==

Slightly soluble

Wall made of A

=saturation

concentration

Parabolic

Velocity

Profile of

Fluid B

At end adjacent to the wall

( ) ayygvz d=

( ) ( )dd yy2

2

2

y

C

z

Cay A

ABA

=

D

0

0

0 0

0

==

==

==

A

AA

A

C,yat

CC,yat

C,zat

Similarity Solution Technique

03 2

2

2

=

+ d

df

d

fd

0

1 0

==

==

f,at

f,at

Textbook p.562

( )( ) ( )

31

34

0

0

31

34

3

0

00

0

00

99

=

−−−=

−=

−=

=

===

z

aC

z

aexpC

yC

C

d

dC

y

CN

AB

AAB

yAB

AAB

yA

AAAB

y

AAByy,A

D

D

DD

DD

20

3

1 CdexpCf +

−=

( )34

3

0

3

3

0

−

=

−

−

=

dexp

dexp

dexp

C

C

A

A

( )

31

37

0

0 0 0,09

2

== = L

aWLCdzdxNW

AB

AABW L

yyAAD

D

( ) ( ) 1907134

34

37 .==

( ) 32

0 LW ABA D

( ) 8930034 .=

Diffusion, Convection and Chemical Reaction

z

Liquid B with

small amounts

of A and C

Liquid B

Porous plug A

(slightly soluble

in B)

AA

ABA Ck

dz

Cd

dz

dCv 12

2

0 −=D

0

0 0

==

==

A

AA

C,zat

CC,zat

A C

by first order

reaction

( )( )( ) ABAB

A

A zvvkexpC

CDD 2141 0

201

0

−+−=

Textbook p.585

Analytical Expressions for Mass Transfer Coefficients

( )( ) AmcAAB

A CAkCWLL

vW −

= 0

,0max

0 04D

( ) 210

128144

ScRe.LvLvLk

ShAB

max

AB

max

AB

m,cm =

=

==

DDD

Mass Transfer in Falling Film on Plane surfaces

The dissolution of a slightly soluble solid into a falling liquid film

( )( )( ) Am,cA

AB

ABA CAkCWL

L

aW =−

= 0

0

31

370 0

9

2

D

D

( )( )

( )( ) 31

33

37

3

2

37

0

0719

161

9

22ScRe

L.

LvLLvLkSh

AB

max

AB

max

AB

m,cm

d=

d=

d

==

DDD

d=d= maxvga 2

The absorption of a slightly soluble gas into a falling liquid film

Analytical Expressions for

Mass Transfer

Coefficients-1

( ) AcmAAB

avg,A CkCD

vN −

= 0

0 03

4 D

( ) 210

641503

4

3

4ScRe.

DvDvDkSh

ABABAB

m,cm =

=

==

DDD

Mass Transfer for Flow Around Spheres

Creeping flow around a solid sphere with slightly soluble coating

( )( )

( ) Am,cAABAB

avg,A CkCD

vN =−

= 03

2

2

3737

32

0 02

23 DD

( )( )

( )( )

( ) 313

3737

32

3

3737

320

99102

3

2

3ScRe.

DvDvDkSh

AB

max

ABAB

m,cm =

=

==

DDD

The gas absorption from a gas bubble surrounded by liquid in creeping flow

( ) 21641502 ScRe.Shm +=20 == mSh Re

20 == mSh Re ( ) 3199102 ScRe.Shm +=

Blasius’s Solution

2

2

y

v

y

vv

x

vv xx

yx

x

=

+

0 ,0 ,0 ==== AsA

AsAyx

-CC

-CC

v

v

v

vyat

2

2

y

C

y

Cv

x

Cv A

ABA

yA

x

=

+

D

0=

+

y

v

x

v yx

Boundary Conditions

Governing Equations

1c =

=AB

SifD

(thermal boundary layer = hydrodynamic boundary layer)

AsA

AsAx

CC

CC

v

vf

−

−==

22

xRe,x

yxv

x

y

x

vy

222=

=

=

1 ,1 , ==→ AsA

AsAx

-CC

-CC

v

vyas

Blasius’s Solution-1

( )( )

( ) ( ) ( )

( ) 328.1

Re,2

2

Re,2

20

000

=−−

===

=

=

= x

AsAAsA

x

x

xyd

CCCCd

xyd

vvdf

d

df

( )

−=

=

21

0

Re,332.0

xsAA

y

A

xCC

dy

dC

( )0=

−=−=

y

AABAsAcAy

y

CCCkN D

( )21

0

Re,332.0

xAB

y

A

AsA

ABc

xy

C

CCk

DD=

−−=

=

21Re,332.0 x

AB

cAB

xkNu ==

D

31ScM

=d

d 3121

, Re,332.0 ScNu xxAB =- Pohlhausen

( )3121

3 43

0

Re,1

332.0Sc

xxNu xAB

−=

sAAAA C, Cx and xC, Cxif x == 00

Mean Nusselt Number

( )

−=

=

3121

0

3320ScRe,

x

.CC

y

CxsAA

y

A

31213320 ScRe,.Nuxk

xAB,x

AB

c ==D

( ) ( )

( ) ( )

−=−=

−=−=

A

xABAsAAsAc

AAsAcAsAcA

x

dAScRe,.CCCCWLk

dACCkCCAkW

31213320

D

31216640 ScRe,.Lk

L

AB

c=

D

Mass, Energy and Momentum Transfer Analogy

00 ==

−

−=

ys,A,A

s,AA

y

x

CC

CC

dy

d

v

v

dy

d

( )( )

0=

−−=−=

y

s,AAAB,As,Acy,A

y

CCCCkN D

0=

=

y

xc

dy

dv

vk

1c =

=AB

SifD

At steady state

0

22

2

2=

=

=

dy

dv

vvC xw

f

2

f

p

C

Cv

h=

Reynolds Analogy (1) Sc=1, and (2) no form drag ~ only skin drag

00 ==

−

−=

ys,A,A

s,AAAB

y

x

CC

CC

dy

d

v

v

dy

dD

2

fcC

v

k=

Chilton-Colburn Analogy

The Schmidt number is other than unity

Chilton-Colburn Analogy (1) 0.6<Sc<2500, and (2) laminar flow

3121Re,332.0 Scxk

Nu x

AB

cAB ==

D2131 Re,

332.0

Re, xx

AB

Sc

Nu=

2Re,Re,

32

31

f

x

AB

x

ABC

ScSc

Nu

Sc

Nu==

2

32 fcD

CSc

v

kj =

Colburn j factor for mass transfer

2

f

HD

Cjj == 3232Pr Sc

v

k

Cv

h c

p

=

2

3232 fcAB

AB

c C

v

SckSc

xv

xk==

D

D

(1) 0.6<Pr<2500

(2) 0.6<Sc<100

Mass Transfer to Non-Newtonian Fluids

C*

C(x,z)

fully developed flow

incompressible power-law fluid,

mass transfer rate is small

no chemical reaction

the solid surface consist of

a soluble material of length L

z

cxV

x

cD

zA

=

)(

2

2

oc, cat z == 0B.C. 1

*0 c, cat x ==B.C. 2

0=

=

x

c, at x dB.C. 3

no mass transfer

solubility

Mass Transfer to a Power-Law Fluid

Flowing on an Inclined Plate


The stress distribution in the film

d cos)( −= xgxz

The mass transfer is confined to a region near the solid surface

d cosgxz

−

0=

=

x

c, at x dB.C. 3 o

c, cat x =→B.C. 3’

for power-law fluids

d cosgdx

dVm

n

z or xm

gV

n

z

1

cos

=

d

the shear rate at the plane surface

n

wm

g1

cos

=

d

Mass Transfer to a Power-Law Fluid Flowing on an Inclined Plate-1

z

cx

x

cD

wA

=

2

2

By using Laplacce transform

dzccszsco))(exp()( 0

−−=

)()(

2

2

scxsx

scD

wA=

a form of the Bessel equation

−

−=

−

−

)()(3

2)3)(()(

31

31

61

31

*tItIx

s

Dccsc

w

A

oo

612

1

3

2x

D

st

A

w

=

)( ),(31

31

tItI−

the modified Bessel functions of the first kind

of order -1/3 and 1/3, respectively


the average(over the length L) mass transfer coefficient

( )( )

31

34

32

61

0

)3)(*(

−−=

−

= A

w

o

xD

scc

x

c

( )

31

31

31

0

3)*(

−−=

= zDcc

x

c

A

w

o

x

( )

31

31

32

34

0

0

2

3

)*(

1

=

−

−

= =

L

D

cc

dzx

cD

Lk wAL

o

x

A

a


2

2

y

cD

z

c

R

yV

Ao

=

oc, cat z == 0B.C. 1

*0 c, cat y ==B.C. 2

oc, cat y ==B.C. 3

solubility

completely analogous to convective heat transfer in Poiseuille flow

inlet concentration

B.C. 3 is valid only for the case of short contact time for which the mass transfer, or diffusion, proceeds in the vicinity of the wall

Dimensionless variables

2 , ,

*

*

RV

zD

R

y

cc

ccC A

o

==−

−=

Mass Transfer to a Power-Law Fluid in Poiseuille Flow


The velocity profile for power-law fluids

10 == , Cat B.C. 1

00 == , Cat B.C. 2

1== , Cat B.C. 3

2

21

3

=

+

CC

n

Similarity solution technique

)( =31

)1(3

9

+=

n

( )

−

=

de3

34

1

032

2

2

=+

d

dC

d

Cd100 == ) and C()C(

−−

+

+==

+n

n

n

R

Q

zzV

V

V 11

1

1

2

)1(11

3

3

0

13

R

Q

ndy

dV

y

z

w

+−=

−=

=

2

13

R

Q

nRV

wo

+−=−=

Mass Transfer to a Power-Law Fluid in Poiseuille Flow-1

The local mass transfer coefficient

*)(0

ccky

cD

bloc

y

A−=

−

=

For short contact time, cbco

cb is the bulk solute concentration in the fluid

0

02

*)(

22

=

=

=

−

==

C

cc

y

cR

D

RkSh

o

y

A

loc

loc

( )3

2

34

09

13

22

zD

RVnC

ShA

loc

+

=

=

=

local Sherwood number(Nusselt number for mass transfer)


average Sherwood number

( )3

2

340

9

13

312

LD

RVn

dzShLD

RkSh

A

L

loc

A

a

a

+

===

31

31

4

13

75.1 GznSha

+

=LD

VRGz

A

=

2

=

=

=

L

RScRe

L

R

D

VR

LD

VRGz

AA2

2

2

2

Shear thinning(as n decreases) enhances the mass transfer rate


Concentration Distributions in Turbulent Flows

( ) ( )t,z,y,xvz,y,xvv xxx+=

nAnAABA

A CkCDCvt

C−=+

2

( ) ( )t,z,y,xCz,y,xCC AAA+=

+

−+=+

22

2

12

AA

A

AAABAA

CCk

Ck

CvCCvt

CD

( )A

t

A CvJ =

( ) ( )( ) gpDt

vD tv+++−=

0=+

vDt

D

( ) ( )

+

−

+−= 22

2

1

AA

At

A

v

AA

CCk

Ck

JJDt

CD

Concentration

Distributions in

Turbulent Flows-1

Prandtl’s mixing length ~ normal to the direction of bulk flow

( )

dy

Cd

dy

vdlCvJ

Ax

Ay

t

A2−=−=

in the y direction

( ) ( )

dy

CdJ

A

AB

t

AtD−=

( )( )

( ) 1==tDAB

tt

Sc

Turbulent Prandtl Number

Fig. 22.1-1. Example of mass transfer across aplane boundary: 포화 평판의 건조

Fig. 22.1-2. Two rather typical kinds of membrane separators, classified here according to a Peclet number, or the flow through the membrane. The heavy line represents the membrane, and the arrows represent the flow along or through the membrane.

the rates of mass transfer across phase boundaries to the

relevant concentration differences, mainly for binary systems

TRANSFER COEFFICIENTS IN ONE PHASE

진짜 상 경계를 가짐

선택적 투과막


Fig. 22.1-3. Example of mass transfer through a porous wall: transpiration cooling.

Fig. 22.1-4. Example of a gas-liquid contactingdevice: the wetted-wall column. Two chemical species A and B are moving from the downward-flowing liquid stream into the upward-flowing gasstream in a cylindrical tube

유체역학적 물성의 급격한 변화


N=

+−=N

HTk1

e

Partial molar enthalpy

미분 면적에 정의되는 국부 전달계수

BA

*

B

*

A xx ,JJ −=−= 00 loc,xBloc,xA kk =

if the heat of mixing is zero (as in ideal gas mixtures)

( )o

,pAA TTC~

H −= 00 : reference temperatureoT

Partial molar enthalpy

n,P,Tn

HH

=

( ) ( ) ,n,n,nkH,kn,kn,knH 321321 =

=

HnHBy Euler Theorem

Enthalpy ~ Extensive property

Homogeneous of degree 1

nB

BAx

H~

xH~

H

−=

nB

ABx

H~

xH~

H

+= ( ) n/Hnn/HH

~BA =+=

nB

BA

nA x

H

n

x

n

HH

n

H

B

−==

( ) nxn ;nxn BBBA =−= 1

( ) ( )nHnHx

n

n

H

x

n

n

H

x

HBA

nB

B

nBnB

A

nAnBAB

++−=

+

=

( )BBAA HnHnHnH +==

"apparent" mass transfer coefficient

The superscript 0 indicates that these quantities are applicable only for small mass-transferrates and small mole fractions of species A.

mass transfer coefficient

Interphase Mass Transfer

000 AliquidAgasA NNN ==

( ) ( )AbAloc,xAAbloc,yA xxkyykN −=−= 0

0

0

0

0

Assuming equilibrium across the interface

( )00 AA xfy =

( ) ( )AbAeloc,xAeAbloc,yA xxKyyKN −=−= 00

0

Gas phase composition in equilibrium with a liquid at composition

TRANSFER COEFFICIENTS IN TWO PHASES

평형곡선

Aey

Liquid phase composition in equilibrium with a gas at compositionAex

Abx

Aby

Overall concentration difference


Fig. 22.4-2. Relations among gas- and

liquid phase compositions, and the

graphical interpretations of mx, and my.

0loc,yK the overall mass transfer coefficient "based on the gas phase’

( ) ( )AbAeloc,xAeAbloc,yA xxKyyKN −=−= 000

0loc,xK the overall mass transfer coefficient "based on the liquid phase"


If the mass-transport resistance of the gas phase has little effect,and it is said that the mass transfer is liquid-phase controlled. In practice, this means that the system design should favor liquid-phase mass transfer.

If then the mass transfer is gas-phase controlled. In a practicalsituation, this means that the system design should favor gas-phase mass transfer.

If , roughly, one must be careful to consider the interactionsof the two phases in calculating the two-phase transfer coefficients.

100 loc,yloc,x mkk

100 loc,yloc,x mkk

1010 00 loc,yloc,x mkk.

Mean two phase mass transfer coefficient

bulk concentrations in the two adjacent phase do not change significantly over the total mass transfer surface

( ) ( )00

0

11

1

ymxxm

approx,xkmk

K+

=

an oxygen stripper, in which oxygen(A) from the water(B) diffuses into the nitrogen gas(C) bubbles.

Penetration model holds in each phase

exp

ABlloc,xloc,x

tckk

D40 =

exp

ACgloc,yloc,y

tckk

D40 =

Total molar concentraion in each phase

11

0

0

=mc

c

mk

k

AC

AB

g

l

loc,y

loc,x

D

D

Only liquid phase resistance is significant

Absorption or desorption of sparingly soluble gases is almost always liquid phase controlled


the law of conservation of mass of chemical species in a multicomponent macroscopic flow system

the mass rate of addition of species to the system by mass transfer across the bounding surface

the instantaneous total mass of in the system

the net rate of productionof species by homogeneous and hetero-geneous reactions within the system

in molar units

Disposal of an unstable waste product

wVt tot,A

'''

Atot,A mkwmdt

d10 −= 00 == tot,Am t

( )( )tkexpk

wm '''

'''

Atot,A 1

1

0 1 −=

Atot,A wtm =( )tk

tkexp'''

'''

A

A

1

1

0

1 −=

K

e K

A

AF

−−=1

0

QVkwVkK ''''''

11 == When the tank is full


wVt ( ) VkwwVdt

d '''

AAA 10 −−=

AFA ==1

( ) ( )

( )( )11

0

0

1

1 −+−=+−

+−

K

AAF

AA eK

K

( ) 01 AAA K

d

d

=++ ( )tVw =

( )wVkK '''

AAA

1

00

11 +=

+=

Binary Splitters

WPF

xWyPzF

+=

+=

FP=

( )xyz −+= 1

Separation factor XY =

x

xX

y

yY

−=

−=

1 ,

1Mole ratio

( ) ( )

1x or

11 y

y

x

xy

−−=

−+=

Gas-liquid Splitters(equilibrium distillation) Relative volatility

Cut

Dirac 분리용량(separation capacity) 및 가치함수(value function)

서로 다른 분리공정의 유효성을 비교하기 위한 기준 수립

( )( ) ( )y1-x

x-y1

1

1+=

−

−=

xx

yy

약간 농축되는 계

( )( )

( ) ( )xxx

xxy −−+

−+

−+= 11x

11

1

( ) −1

Dirac 분리용량(separation capacity)

)()()( zFvxWvyPv −+=

WPF

xWyPzF

+=

+=

FP=

( ) ( )x-yx-z or 1 =−+= xyz

Cut

Binary Splitters

( ) )()(1)( zvxvyvF

−−+=

( ) ( ) ( ) ( ) +−+−+= xvxyxvxyxvyv2

21)()(

( ) ( ) ( ) ( ) +−+−+= xvxzxvxzxvzv2

21)()(

( )( ) )(12

21 xvxy

F−−=

( )( )

( ) ( )xxx

xxy −−+

−+

−+= 11x

11

1

( )( ) ( ) )(111222

21 xvxx

F−−−=

계의 분리용량은 실질적으로 농도에 무관하다고 가정 ( ) 1)(122 =− xvxx

( )( )221 11 −−=

F

( )222

2

1

1)(

xxdx

xvd

−= ( )

( ) 2121

ln12)( CxCx

xxxv ++−

−−=

적분 상수를 구하기 위한 조건 선택 ( )( )x

xxxv

−−=

1ln12)(

0)( ,0)(21

21 == vv

Compartmental Analysis

021 0 ===t

A complex system is treated as a network of perfect mixers, each constant volume, connected by ducts of negligible volume, with no dispersion occurring in the connecting ducts

The volumetric flow rate of solvent flow from unit m to unit n

( ) nnnm

N

mmn

nn rVQ

dt

dV

+−= =1

Hemodialysis

Initial condition

mnQ

21

2

21

2

2212

22

V

G

V

Q

V

D

V

Q

dt

d

V

D

V

Q

V

Q

dt

d=+

+++

2

0202 and 0

V

D

dt

dt

−===

( ) GQdt

dV +−−= 21

11

( ) 2212

2

DQdt

dV −−=

during dialysis period

122 t === 10Initial condition

21

2

21

2122

2

VV

QG

dt

d

VV

VVQ

td

d=

++

For the recovery period

4

21

2132 Ct

VV

VVQexpCcf, +

+=

21

2VV

tGpi,

+

=

2

222

V

D

dt

d

dt

d +=

For the recovery period

during dialysis period



Fundamentals of Mass Transfer

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