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

−= D

In vector form

Mass diffusivity

AABAj −= D

Prandtl number

Schmidt number

Lewis number

ABAB DD

ABABpC

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

Molecule of

species A

dV in multicomponent

muxture

- Total mass concentration(density)

- mass fraction

Molar Concentration

~ the number of moles of A present per unit volume

of mixture

( )ACb) molar concentration

iCC- Total molar concentration

For ideal gas

- molecular weight of AAM

RTnVP AA =

A == - partial pressure of AAP

- Mole fraction

for ideal gas mixture

for gasfor liquid

for ideal gasRT

nC total ==

A = =i

Cy AAA

A === Dalton’s law of gas mixture

Mass Average and Molar Average Velocity

a) mass average velocity

v - the absolute velocity of species i

relative to stationary coordinate axes

b) molar average velocity

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 −

Mass Diffusivity

( )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

( ) AABAAA xCVvCJ −=−= D

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

iAAABAAABAA njnvvn DD

( ) AABAAA xCVvCJ −=−= D

VCxCvC AAABAA +−= D

+−=i

iiAAABAA vCxxCvC D

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

+=+−=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

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

Equation of Continuity for Component A

Conservation of Mass of Component A

yy,Ayyy,Axx,Axxx,A xznxznzynzyn −+−++

+zyxrzyx

tyxnyxn A

zz,Azzz,A

Differential

Equations of Mass

Transfer-1

Equation of Continuity for the Mixture

In binary mixture

( )( )

( ) 0=+−

+++ BA

BABA rr

vvvnn BBAABA =+=+

BA rr −=

Differential

Equations of Mass

Transfer-2

vjn AAA +=

( )vjn AAA +=

( ) 0=−+

AAAA ++

0=−+

A rjDt

Differential

Equations of Mass

Transfer-3

in terms of molar unit

But RA is not always –RB

for only stoichiometry reaction ( A B)

( )( )

( ) 0=+−

+++ BA

BABA RR

VCvCvCNN BBAABA =+=+

CCC BA =+

BA RR −=

( ) 0=+−

+ BA RR

Differential Equations

of Mass Transfer-4

AAABAA rvt

vCn AAABA +−= D

VCyCN AAABA +−= D

( )BAAAABA NNyyCN ++−= D

( )BAAAABA nnn ++−= D

If and are constant ABD

AAABAAAA rvv

tt+=++

AAABAA RCCV

0=−+

A rjDt

AAABAA rv

AAA rnt

( )gvevUt

AAA vjn +=

21 vUvvqe

( ) ( ) ( )AAgTTggpDt

vD −−−−−+−=

( ) ( ) ( )

( ) ( )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 , ,=

−====

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

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

Liquid A

Flowing of gas B

dN z,A 0=dz

dN z,B

0= z,BN

~ Component B is stagnant

0=z,BN

−−=

yyzzat

Steady State

Diffusion of A

through Stagnant B-1

−= 2

dyCdzN D

( )( )21

ln,12ln,

1ln AA

AABzA yy

CN −

( ) ( )( ) ( )12

BBln,B

−−

−−−=

Steady state diffusion of one gas through a second gas

~ absorption, humidification

( ) ln,

AAABzA

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.

Flowing of gas B

Liquid A

Slowly moving gas film

AAABzA

( ) ( )2121, AAc

AAczA PPRT

kCCkN −=−=

Convective Mass Transfer

1.0~0.5

ABck Dactual

Main fluid stream

in turbulent flow

Concentration Profile

dN z,A

−−=

yyzzat

− dz

If C and are constant under isothermal and isobaric conditionsABD

− dz

( ) 211ln CzCyA +=−−

( ) ( )121

y−−

BBBB y

dzyy =

Pseudo Steady State Diffusion through a

Stagnant Gas Film

tzzttat

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

period of time ~ pseudo steady state model

AABz,A

Molar density of A in the liquid phase

ln,, zz

Flowing of gas B

Liquid A

Diffusion through a Isothermal Spherical Film

0=r,BN

( ) 0,

2 =rANrdr

− dr

( ) ( )( ) ( )

for constant temperature is constantABCD

Temperature T1

Gas film

Temperature 12 TT =

( ) ( )

rrrAAx

( )r,Br,AAA

ABr,A NNxdr

dxCN ++−= D

Since B is insoluble in liquid A

Diffusion through a Nonisothermal Spherical Film23

− dr

( ) ( )

( ) ( ) ( ) ( )

rrrAAx

nRTPNrW

− dr

2 −nif

Temperature T1

Gas film

Temperature

Diffusion with Heterogeneous Chemical Reaction

− dz

Gas A Gas A & B

Sphere with coating of catalytic material

BA →2

Edge of hypotheticalStagnant gas film

Catalytic surface

zAzB NN ,21

, −=

1−−=

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

( ) ( ) ( )d−−=−

AA xx1

Irreversible and instantaneous rxn

Solid catalyzed dimerization of CH3CH=CH2

Diffusion with Slow Heterogeneous Chemical Reaction

− dz

( )large 1

BA →2

zAzB NN ,21

, −=dz

1−−=

( ) 21211ln2 CzCxA +=−−

( )AzA

assume pseudo 1st order rxn

( ) ( ) ( )d−

−=−

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

ABzA D−=,

( )0 0 ,

dzdCorNLzat

First order reaction

=− AA

AB Ckdz

0AA CC= Lzζ = ABLk D2

1= Thiele modulus

Diffusion with

Homogeneous

Chemical Reaction-1

dCN ABA

AABzzA

ζsinhcosh 21 += CC

ζsinhsinhcoshcosh

( )( ) AB

0 cosh

1cosh −=

avgA CC

Gas Absorption with Chemical Reaction

in an Agitated Tank

effect of chemical reaction rate on the

rate of gas absorption in an agitated

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

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

Liquid film

Liquid-gas

interface

Gas in

bubble

Main body of liquid

Without reaction

With reaction

Gas Absorption with

Chemical Reaction

=− AA

AB Ckdz

CC,zat

sinhcoshBcoshsinh

0AA CC= dzζ =ABk D2

AAB CVk

dC1SD-

( ) d+=

sinhSVcoshB

sinhSVcoshcosh

zz,A 1

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

, =++−+

rrRrrNrN ArrrArrA

( ) ( )A

( ) ArA RrNrdr

Diffusion and Chemical

Reaction inside a

Porous Catalyst-1

First order chemical reaction,

a is the available catalytic surface per unit volume

AA -D=

A aCkdr

finiteC,rat

CC,Rrat

( )rfrC

A 1= f

akcosh

DD1211 +=

Raksinh

raksinh

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.

AAR,AA

dCRNRW

−== D22 44

−= R

akcothR

ARAADD

D 1114

( )( )( )AR,AR CkRW 13

0 −= a

( )132

== cothW

Aak D1= Thiele modulus

Effective factor

For nonspherical catalyst particles

AARpAR CakVW 1

nonsphS

VR 3 ( )133

= cothA

( )ppA SVak D1= generalized

modulus

Reaction inside a

Porous Catalyst-3

Diffusion in a Multi-Component System

( ) ( )

−−=−−= == 11

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

dx z,z,

Maxwell Stefan equation for multicomponent diffusion in gases at low density

N,...,,, 321=

Textbook p567

( ) ( )

−−=

2 ;DD C

zLNexp

( ) ( )

−−−

−−−=

zLNexpx

zLNexpxx

At z=0

−−

−−=

1210 1

LNexpx

LNexpxx

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

Lz0 ,0 0

tLzatCC

tzatCC

tatzCC

CCYLet

0 ,0 0

Lz0 ,0 1

tLzatY

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

( ) ( )

2sinsin

( ) , 00 YzYif =

531 sin14 2

sAA ,,neL

( ) ( )

−=−=

0, 531 cos4 2

AsAABA

ABzA ,,neL

( )22LtX ABD D=

Transient Diffusion in a Semi-Infinite Medium

tzasCC

tzatCC

all ztatCC

~ Similarity Solution Technique

( ) ( )00

2 AsAAA CCCCY ,tz Let −−==

( ) ( ) 1 ,00

erferf

Unsteady State Evaporation of a Liquid

xx,zat

−−=

Continuity equation

Flowing of gas B

Liquid A

( )tVV zz 0=

In the binary systemC

z,Bz,Az

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

−−−

Textbook p613

Unsteady State

Evaporation of a

Liquid-1

Dimensionless molar

average velocity

( ) 022

=−+dZ

tz Z,Xxx ABAA D4 0 ==

=−−==

xtVx D

( ) 21 −−= ZexpC

( ) 20

1 CZdZexpCXZ

−−=

( )( )

dWWexp

ZdZexp

−−=

−−

( )( ) ( )

−−=

+−−=

erferf

erfZerfZX

11 Textbook p616

Fig. 20.1-1

Unsteady State

Evaporation of a

Liquid-2

dV ABAA D== 0

( )( )

( ) 120

experfxA

Rate of production of vapor from surface of area S

VA: the volume of A evaporation up to time t

tSV ABA D4=

tSxV AB

SxV ABAA

D40 0Ax=

Deviation from the Fick’s second law caused

by the nonzero molar average velocity

=−−=

Gas Absorption with Rapid Reaction

CC, at z

a, tzat z

CC, tzat z

CC, at z

for zCC, at t

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

aA + bB products

zt for zz

tzz for z

Solute B

Liquid-Vapor Interface

Textbook p617

Gas Absorption with

Rapid Reaction-1

( ) +=

zt for zt

tzz for t

( )( )

( ) −

−−=

zt for zt

tzz for t

4Dzerf

BSASASAS

experfbC

ttanconstzR == 42 ZR increases as t

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

−=== DD

Gas Absorption

with Rapid

Reaction-2

AASz,A

The average rate of absorption up to t

z,Aavg,z,A

Gas Absorption with Rapid Reaction-2

Diffusion into a Falling Liquid Film(Gas Absorption)

x=0 x=d

0=−+−++

zWNzWNxWNxWNxxx,Axx,Azzz,Azz,A

vxv maxz

N x,Az,A

( ) ( )xvCNNxz

CN zAz,Bz,AA

AABz,A ++

ABx,Bx,AAA

Textbook p.558

Diffusion into a

Falling Liquid

Film(Gas

Absorption)-1

d− D

CC,xat

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

( ) −

−=maxvzABx

A dexpC

maxABmaxABA

DD 441

CN maxAB

AABxx,A

vWCdzdyNW AB

max000 0 ,0

Rybczynski-Hadamard circulation

Gas absorption from rising bubbles

Gas bubbles rising in liquids free surface –active

agents undergo a toroidal circulation

AABxx,A C

osureexp =

For creeping flow00 3

xx,A CD

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

By preventing internal circulation31

ABxx,AN D=

Diffusion into a Falling Liquid Film(Solid Dissolution)

d−−

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

CC,yat

Similarity Solution Technique

Textbook p.562

( )( ) ( )

−−−=

AAByy,A

1 CdexpCf +

0 0 0,09

== = L

aWLCdzdxNW

AABW L

( ) ( ) 1907134

37 .==

( ) 32

0 LW ABA D

( ) 8930034 .=

Diffusion, Convection and Chemical Reaction

Liquid B with

small amounts

of A and C

Liquid B

Porous plug A

(slightly soluble

ABA Ck

dCv 12

0 −=D

CC,zat

by first order

reaction

( )( )( ) ABAB

A zvvkexpC

CDD 2141 0

−+−=

Textbook p.585

Analytical Expressions for Mass Transfer Coefficients

( )( ) AmcAAB

A CAkCWLL

vW −

( ) 210

128144

ScRe.LvLvLk

m,cm =

Mass Transfer in Falling Film on Plane surfaces

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

( )( )( ) Am,cA

ABA CAkCWL

aW =−

( )( )

( )( ) 31

22ScRe

LvLLvLkSh

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 −

( ) 210

641503

4ScRe.

DvDvDkSh

ABABAB

m,cm =

Mass Transfer for Flow Around Spheres

Creeping flow around a solid sphere with slightly soluble coating

( )( )

( ) Am,cAABAB

avg,A CkCD

vN =−

( )( )

( ) 313

3ScRe.

DvDvDkSh

m,cm =

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

0 ,0 ,0 ==== AsA

Boundary Conditions

Governing Equations

(thermal boundary layer = hydrodynamic boundary layer)

1 ,1 , ==→ AsA

Blasius’s Solution-1

( )( )

( ) ( ) ( )

( ) 328.1

=−−

AsAAsA

Re,332.0

−=−=

AABAsAcAy

CCCkN D

Re,332.0

−−=

21Re,332.0 x

xkNu ==

d 3121

, Re,332.0 ScNu xxAB =- Pohlhausen

( )3121

332.0Sc

xxNu xAB

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

Mean Nusselt Number

3320ScRe,

31213320 ScRe,.Nuxk

( ) ( )

−=−=

xABAsAAsAc

AAsAcAsAcA

dAScRe,.CCCCWLk

dACCkCCAkW

31213320

31216640 ScRe,.Lk

Mass, Energy and Momentum Transfer Analogy

ys,A,A

( )( )

−−=−=

s,AAAB,As,Acy,A

CCCCkN D

At steady state

vvC xw

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

ys,A,A

s,AAAB

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

cAB ==

D2131 Re,

Re, xx

2Re,Re,

32 fcD

Colburn j factor for mass transfer

Cjj == 3232Pr Sc

3232 fcAB

(1) 0.6<Pr<2500

(2) 0.6<Sc<100

Mass Transfer to Non-Newtonian Fluids

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

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

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

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

c, at x dB.C. 3 o

c, cat x =→B.C. 3’

for power-law fluids

d cosgdx

z or xm

the shear rate at the plane surface

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

By using Laplacce transform

dzccszsco))(exp()( 0

−−=

a form of the Bessel equation

2)3)(()(

*tItIx

)( ),(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

( )( )

)3)(*(

−−=

= zDcc

Lk wAL

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

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

Similarity solution technique

)( =31

Cd100 == ) and C()C(

−−

+−=−=

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

The local mass transfer coefficient

For short contact time, cbco

cb is the bulk solute concentration in the fluid

local Sherwood number(Nusselt number for mass transfer)

average Sherwood number

dzShLD

75.1 GznSha

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−=+

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

−+=+

AAABAA

CvCCvt

A CvJ =

( ) ( )( ) gpDt

vD tv+++−=

( ) ( )

+−= 22

Concentration

Distributions in

Turbulent Flows-1

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

vdlCvJ

A2−=−=

in the y direction

( ) ( )

AtD−=

( )( )

( ) 1==tDAB

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

Partial molar enthalpy

미분 면적에 정의되는 국부 전달계수

A xx ,JJ −=−= 00 loc,xBloc,xA kk =

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

,pAA TTC~

H −= 00 : reference temperatureoT

Partial molar enthalpy

n,P,Tn

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

HnHBy Euler Theorem

Enthalpy ~ Extensive property

Homogeneous of degree 1

+= ( ) n/Hnn/HH

~BA =+=

( ) nxn ;nxn BBBA =−= 1

( ) ( )nHnHx

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

Assuming equilibrium across the interface

( )00 AA xfy =

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

Gas phase composition in equilibrium with a liquid at composition

TRANSFER COEFFICIENTS IN TWO PHASES

평형곡선

Liquid phase composition in equilibrium with a gas at compositionAex

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

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

approx,xkmk

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

Penetration model holds in each phase

ABlloc,xloc,x

ACgloc,yloc,y

Total molar concentraion in each phase

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

0 1 −=

Atot,A wtm =( )tk

tkexp'''

1 −=

−−=1

QVkwVkK ''''''

11 == When the tank is full

wVt ( ) VkwwVdt

AAA 10 −−=

AFA ==1

( ) ( )

( )( )11

1 −+−=+−

( ) 01 AAA K

=++ ( )tVw =

( )wVkK '''

Binary Splitters

xWyPzF

( )xyz −+= 1

Separation factor XY =

1Mole ratio

( ) ( )

−−=

Gas-liquid Splitters(equilibrium distillation) Relative volatility

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

서로 다른 분리공정의 유효성을 비교하기 위한 기준 수립

( )( ) ( )y1-x

약간 농축되는 계

( )( )

( ) ( )xxx

xxy −−+

−+= 11x

( ) −1

Dirac 분리용량(separation capacity)

)()()( zFvxWvyPv −+=

xWyPzF

( ) ( )x-yx-z or 1 =−+= xyz

Binary Splitters

( ) )()(1)( zvxvyvF

−−+=

( ) ( ) ( ) ( ) +−+−+= xvxyxvxyxvyv2

21)()(

( ) ( ) ( ) ( ) +−+−+= xvxzxvxzxvzv2

21)()(

( )( ) )(12

21 xvxy

F−−=

( )( )

( ) ( )xxx

xxy −−+

−+= 11x

( )( ) ( ) )(111222

21 xvxx

F−−−=

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

( )( )221 11 −−=

( )222

−= ( )

( ) 2121

ln12)( CxCx

xxxv ++−

−−=

적분 상수를 구하기 위한 조건 선택 ( )( )x

−−=

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

nn rVQ

+−= =1

Hemodialysis

Initial condition

0202 and 0

−===

( ) GQdt

dV +−−= 21

( ) 2212

dV −−=

during dialysis period

122 t === 10Initial condition

For the recovery period

2132 Ct

VVQexpCcf, +

For the recovery period

during dialysis period

Fundamentals of Mass Transfer

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