CBE 3233 MassTransfer(2)

7/29/2019 CBE 3233 MassTransfer(2)

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CBE 3233 Chemical &Biomolecular Separation

Qing Song Spring 2013

Lecture 3 Mass Transfer

1


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Mass Transfer and Diffusion

Mass transfer by ordinary molecular diffusionoccurs because of a concentration gradient

Net movement of a species in a mixture from onelocation to another.2


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Applications of Mass TransferOperation Extract

Phase

RaffinatePhase

Distillation

Gas

adsorption/dehumidification

Membrane separationsAdsorption

Liquid extraction

Leaching

Crystallization

Drying

Vapor

Gas

Gas or liquid

Gas or liquidExtract

Liquid

Mother liquor

Gas (usually

air)

Liquid

Liquid

Gas or liquid

SolidRaffinate

Solid

Crystals

Wet solid

Transfer of material from one homogeneous phase to another

Driving force for transfer is a concentration difference 3


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Mass TransferAbsorption: solute transfer through the gas to the gas-liquid interface, across the interface, and into the liquid

Distillation: separate a liquid mixture by vaporization into

individual components.

A solute gas is absorbed from an inert gas into a liquid in

which the solute is more or less soluble.

Washing of ammonia from mixture of NH3 and air,

Remove of CO2 and H2S from natural gas.

The low boiler (more volatile) diffuses through the liquid

phase to the interface and away from the interface into

the vapor. The high boiler (less volatile) diffuses in the

reverse direction.

Example: ethanol and water, crude oil into gasoline,

kerosene, fuel oil, and lubricating stock. 4


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Mechanisms of Mass TransferMolecular diffusion: random, spontaneous,

Eddy (turbulent) diffusion by random,

macroscopic fluid motion

Both molecular and eddy diffusion may

involve the movement of different species in

opposing directions.

Bulk flow of the mixture in a direction

parallel to the direction of diffusion.

5


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Molar concentration:Mole fraction:

(liquids, solid (gases)

RT

p

V

n

Mc AA

A

AA

Mass Transfer Fundamentals

6

RT

p

V

n

Mc ii

i

ii

c

cx i

i

c

cx A

A

c

cy i

i

c

cy A

A

molar average velocity,c

cn

i

ii 1

M

v

v

velocity of a particular species relative to molar averagevelocity is the diffusion velocity

MiiD v


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Flux DefinitionsMolar flux iii cN With respect to a fixed reference frame

Molar fluxWith respect to a molar average velocity vM

Miii cJ

7


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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 Ficks First Law, for basic

molecular diffusion

Molar flux

or, in the z-direction,

For a general relation in a non-isothermal, isobaric

system,

AABA cD J

dz

dcDJ AABzA ,

dz

dycDJ AABzA ,

Theory of Diffusion

dz

dwDj A

ABA

Mass flux 8

ii cDJ

dz

dcDJ izi ,

dz

dycDJ izi ,

dz

dwDj i

i


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Units of Above ParametersJi: molar flux of component i, kgmol/m

2.h (kmol/m2.h) or lb

mol/ft2.h

D: diffusivity m2/h

Ci: mole concentration kgmol/m3, kmol/m3, or lbmol/ft3

Z: distance in direction of diffusion, m or ft

i: mass density of component i, kg/m3, or lb/ft3

xi: mole fraction of i

wi: mass fraction of i

9


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10

Units of Above Parameters for Binary SystemJA: molar flux of component i, kgmol/m

2.h (kmol/m2.h) or lb

mol/ft2.h

DAB: diffusivity m2/h

CA: mole concentration kgmol/m3, kmol/m3, or lbmol/ft3

Z: distance in direction of diffusion, m or ft

A: mass density of component A, kg/m3, or lb/ft3

xA: mole fraction of A

wA: mass fraction of A


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Similarities between Mass , Heat and Momentum Transfer

Mass

Heat

Momentum

dzdcDJ AABzA ,

Ficks first law

dz

dTkq Fouriers law

dz

vdx

zx

Newtowns law

11


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Since mass is transferred by two means: concentration differences and convection differences from densitydifferences

Flux Relationships

12

MiiMiiiMiii cNcccJ

Mii

Miii cdz

dcDcJN earranging:

Diffusion flux Convective flux

n

i

i

n

i

iiiin

i

i

iin

i

i

n

i

ii

iii Nc

cJc

c

cJ

c

c

cJN11

11

1

n

iiiii

NxJN1


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13

Flux Relationships for Binary System

dzdxcDJ AABA

BAAA

ABBAAAA NNxdzdxcDNNxJN


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Summary of Molar and Mass fluxes

14

JA, and NA are equivalent statement of theFick rate equation. Any one of theseequations is adequate to describe moleculardiffusion.

Molar fluxes, JA and NA, are used to describemass-transfer operations in which chemicalreactions are involved. They are often used todescribe the mass transfer in diffusion cellsfor measuring diffusion coefficients.


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Molecular Diffusion

15

Unit area

NA NB

NA is positive NB is negativeNet flux=N=NA+NB

A(Water) B(Ethanol)


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16

Equimolar Counter Diffusion (EMD)NxJN AAA

BA

AA

ABA NNc

c

dz

dcDN

BABBBAB NNcc

dz

dcDN

dzdcD

dzdcD

cccNNNN BBA

A

AB

BA

BABA

similarly,

Adding them together,

dz

dcD

dz

dcD B

BA

A

AB

BA JJ 0 BA NNN


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17

In the Absence of Bulk flowdz

dxcDJN AABAA

dz

dxcDJN B

ABBB

(3-16)(3-17)

Total concentration c, T, P are constants,z=z1, xA=xA1, z=z2, xA=xA2

A

AB

A dxdzcDN

A

A

x

x

A

z

z AB

A dxdzcDN

11


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18

11 AAAB

A xxzzcD

N

Integrate between z1 and z

Equimolar Counter Diffusion (EMD)

11

AA

AB

A xxzz

cDJ

BB

AB

Bxx

zz

cDJ

1

1

(3-18)

(3-19)


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19

Unimolecular Diffusion (UMD)N=NA, NB=0

Only one component such as A is being transferred,

Example: ammonia (NH3) (A) & air (B) absorbed by water

AA

A

ABA Nxdz

dxcDN (3-26)

dz

dxcDNx AABAA )1(

dzdx

x

cD

NA

A

AB

A 1 (3-27)

Integrating eq.(3-27)

At quasi-steady-state conditions and constant molar activity

)1ln(

1

1

11

1

1

1A

x

x A

AB

A

Ax

xA

AB

A

A

x

x A

ABz

zxd

N

cD

x

xd

N

cD

x

dx

N

cDdz

A

A

A

A

A

A

1

11

1ln)1(ln

1

A

A

A

ABx

xA

A

AB

x

x

N

cDx

N

cDzz

A

A(3-31)


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20

AB

A

A

A

cD

zzN

x

x 1

11

1ln

AB

AcD

zzN

A

A ex

x 1

11

1

AB

A

cD

zzN

AA exx

1

111

Unimolecular Diffusion (UMD)

(3-32)

1

2

12 1

1

lnA

AAB

A x

x

zz

cD

N

1

2

21

1

2

12

1

1ln

1

1ln

11

1

A

A

AA

A

A

AA

LMA

x

x

xx

x

x

xx

x

z=z2, eq.(3-21) becomes

LMA

AAAB

Ax

xx

zz

cDN

1

21

12


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Mass Transfer Coefficients

21

AcA ckN

AC

Solute A

CA1

Rate equation

Steady state diffusion of solute A

through a membrane as shown in

figure. After the solute diffuses

through the membrane, it is swept

from the external surface by a gas

stream. A mass transfer coefficient

for transfer of component A into the

free stream is defined in terms of

diffusion at the interface by

AA

z

A

AA

Ac

cc

z

cD

cc

Jk

1

0

1

kc: mass transfer coefficients


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22

Equimolar Counter Diffusion (EMD)

AA

ABA

ccD

J 1JA=-JB 11

AA

AB

A xxzz

cDJ

AB

c

D

k

'

NA=-NB 11

AAAB

A xxzz

cDN

AA

ABA

ccDN 1

AAcA cckN 1'

For liquids AALAAcA cckcckN 1'

1

'

AAxAALA xxkxxckN 1'

1

'And

For gases AAGAAc

A ppkppRT

kN 1'

1

'

RTP

Vnc t

tAAPyp

Dalton's law )( 1'

1

'

AAyAAGtA yykyykPN

nRTVPt


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23

Unimolecular Diffusion (UMD)NA=constant, NB=0

LM

A

AAAB

LMA

AAAB

A

c

c

ccD

x

xx

zz

cDN

11

11

1

cc

cc

c

c

c

c

c

c

A

A

AA

LM

A

1

1

1

1ln

11

1

LMA

AB

ccc

Dk

1

AALMA

c

AAcAcc

cc

kcckN

1

'

11


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24

AALAAcA cckcckN 11 AAxLAALA xxkxxckN 11

For liquids

or

Unimolecular Diffusion (UMD)

Introducing ideal gas law and make use of dalton's

law

AGAyAAGtAAGA yykyykPppkN 111

or

AALMB

Gt

AA

LMB

ct

A ppp

kPpp

pRT

kPN 1

'

1

'


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Units of Mass Transfer Coefficients

25

EMD UMD

Gases:

Units

AB

cDk '

RT

Dk ABG

'

RT

DPk ABty

'

Liquids:

AB

LDk '

AB

x

cDk '

LMB

ABtc

PDPk

LMB

ABt

G

pRT

DPk

MB

ABL

xDk

LMB

ABt

ypRT

DPk

2

mole(time)(area)(mol/vol)

mole(time)(area)(pressure

mole(time)(area)(mol fra.)

mole(time)(area)(mol/vol)

mole(time)(area)(mol fra.)

LMB

AB

x

x

cDk


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26

Conversion between Mass Transfer Coefficients

LMBLLMBxLLx xCkxkkM

Ckk '''

LMBLMB

xCC

P

kPkPkPk

RT

Pk yLMBGLMByG

c '''

Gases:

Liquids:


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Example 1

27


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28

Film Theory

Resistance to the diffusion is equivalent to that in astagnant film of a certain thickness

A thin boundary layernear the interface wherethe fluid is in laminar flowAll resistance in the filmTransport or Mass transferis by molecular diffusion


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29

Penetration Theory


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Diffusion of a short distance Unsteady-state molecular transport Mass flux at interface of gas and liquid is:

Surface elements will be renewed byeddies from the turbulent core Instantaneous mass transfer, with solutepenetrating into eddy after exposure tosurface

AbAiABA cctD

N

30

Penetration Theory


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Total solute transferred is

with average mass transfer rate

With distribution in element ages at thesurface, rate of surface renewal is constantand given a factor s, so mass transfer is

2/1

exp

0 2exp

tD

ccdtNAB

AbAi

t

A

2/1

exp

2

tDccN ABAbAiA

AbAiABA ccsDN 31

Penetration Theory


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32

For absorption, the solute may diffuse through a gasphase and then diffuse through and be absorbed in anadjacent and immiscible liquid phase.

The two phases are in direct contact with each other,and the interfacial area between the phases is usuallynot well defined. A concentration gradient must exist to cause this mass

transfer through the resistances in each phase.

Mass Transfer Between Phases

Ammonia (NH3) (A) +air (B) absorbed by water (C)


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33

distance from interface

NA

yAGyAi

interface

gas-phase mixtureof A in gas Gliquid-phase solutionof A in liquid L.

xAixAL

WhereyAG = concentration of A in the bulk gas phaseyAi = concentration of gas A at the interfacexAi =concentration of liquid A at the interfacexAL = concentration of A in the bulk liquid phase



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3434


ALAixAiAGyA xxkyykN ''

Equimolar counterdiffusion (EMD)

Whereky = Gas phase mass transfer coefficient (kg mol/s.m2.mol frac)

kx = Liquid phase mass transfer coefficient (kg mol/s.m2.mol frac)

Diffusion of A through stagnant or

nondiffusing B (UMD)

ALAixAiAGyA xxkyykN

LMA

y

y

y

kk

1

'

LMA

x

x

x

kk

1

'

Where


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35

The interface composition (xAi and yAi ) can be determined by drawing

he line PMwith the slope (-kx/ky) intersecting the equilibrium line


y*A xAL~ x*A~yAG

yAG

yAi

y*AxAL xAi x*A

M

D

E

P equilibriumline

slope = my

slope = mx

AiA

AiAG

yxx

yym

*

AGAi

AAi

x xx

yym

*

AiAL

AiAG

y

x

xxyy

kkslope

'

'

Fig.: Concentration driving force and interface concentrations in inter

phase mass transfer (equimolar counter diffusion EMD)


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36

yAG

yAi

y*AxAL xAi x*A

M

D

E

P equilibriumline

slope = my

slope = mx

AiA

AiAGy

xxyym *

AGAi

AAi

xxx

yym

*

ALAiAiAG

LMAy

LMAx

xx

yy

yk

xk

slope

1

1

'

'

Fig.: Concentration driving force and interface concentrations in inter

phase mass transfer (A diffusing through stagnant B).


y*A xAL~ x*A~yAG


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37

Overall mass-transfer coefficientSimilar to overall heat-transfer coefficient

Mass Transfer Between PhasesUnimolecular diffusion (UMD)

ALAxAAGyA xxKyyKN **

ALAixAiAGyA xxkyykN


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Overall Mass Transfer Coefficient

38

ALAixAiAGAAiAiAGAAG xxmyyyyyyyy **

mx

is the slope of ME

x

A

x

y

A

y

A

k

Nm

k

N

K

N

x

x

yy km

kK

111

ALAiAiAGy

ALAiAiAALA xxyym

xxxxxx 1**

x

A

yy

A

x

A

k

N

km

N

K

N

xyyx kkmK

111

Similarly

my is the slope of DM

(3-242)

(3-243)

AGAi

AAi

x xx

yym

*

AiA

AiAG

y

xx

yym

*


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39

Mass Transfer Resistance

39

Resistance in gas phase

Total resistance in both phasesy

y

K

k

1

1

Resistance in liquid phase

Total resistance in both phasesx

x

K

k

1

1

Solute A is very soluble, mx is small

Solute A is insoluble, my is large


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40

Two Resistance Theory

Three resistance: the gas phase, the liquid phase, andthe interface.The interface resistance is negligible and anyfluctuations in yAi, and xAi are small, yAi and xAi areequilibrium values given by the systems equilibrium-distribution curve


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Individual Mass Transfer Coefficient

41

iAgAGzA ppkN ,,, LAiALzA cckN ,,,

Combining both and rearrange to

iALA

iAGA

G

L

cc

pp

k

k

,,

,,


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42

Overall Mass Transfer Coefficient*

, AGAGA ppKN

iAiAmcp ,, LAA mcp ,

* *, AGA mcp

zA

LAiA

zA

iAGA

G Nccm

Npp

K ,

,,

,

,,1

At low concentrations, we have linear

equilibrium relations

LGGk

m

kK

11

Where : PAG the bulk composition in the gas phase

PA* is the partial pressure of A in equilibrium with the bulkconcentration of CALKG the overall mass transfer coefficient based on partial pressure

(time)(interfacial area)(pressure)


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4343

Overall Mass Transfer CoefficientLAALA ccKN ,*

iAiAmcp ,, LAA mcp ,

* *, AGA mcp At low concentrations, we have linear

equilibrium relations

LGL kmkK

111

Where : CA* the bulk composition in the liquid in equilibrium with

the bulk gas phaseCAL is the bulk concentration in the liquid

KL the overall mass transfer coefficient based on a liquid driving force

(time)(interfacial area)(mole A/volume)

zA

LAiA

zA

iAGA

zA

ALA

LN

cc

mN

pp

N

CC

K ,

,,

,

,,

,

*1

oncentration Driving Force for the Two-resistance Theory


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44

*

AP

*

AC

APtotal

AGP

ALP

AC

AGC

ALC

total


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45

Ratio of resistances in individual phase to

total resistance

Ratio of resistances in individual phase to

total resistance

L

L

totalA

filmliquidA

K

k

C

C

/1

/1

,

,

G

G

totalA

filmgasA

K

k

P

P

/1

/1

,

,


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Mass Transfer Resistance

46

Resistance in gas phase

Total resistance in both phases G

G

K

k

1

1

Resistance in liquid phase

Total resistance in both phasesL

L

K

k

1

1

Solute A is very soluble, mx is small

Solute A is insoluble, my is large


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47

Wetted Wall Column

Wetted Wall Column (glass tube)

Header

Header

Water

Moist air

Moist air

Experimental Setup

Blower

Thomas

Meter

Wet and Dry Bulb

Temperature

Thermometer

Hygrometer

Area for mass transfer is defined


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Film TheoryResistance to the diffusion is equivalent to that in astagnant film of a certain thickness

Aicc 0z

Abcc z

AbAiABA ccD

J

AB

c

Dk

atat

is not directly measureable contains all of the fluiddynamics in the system and willchange with the Re

0z z

cAb

Thin layer

(stagnant)

able to calculate (Re)from experimental data

48

Example : Interface Composition in Interphase Mass Transfer.


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49

The solute A is being absorbed from a gas mixture of A and B in a

wetted-wall tower with the liquid flowing as a film downward along the

wall. At certain point in the tower the bulk gas concentration

yAG = 0.380 molfraction and the bulk liquid concentration isxAL = 0.1. The tower is operating at298 K and 1.013 x 10

5 Paand the

equilibrium data are as follows:

xA yA

0 0

0.05 0.022

0.1 0.052

0.15 0.087

0.2 0.131

0.25 0.187

0.3 0.265

0.35 0.385


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50

The solute A diffuse through stagnant B in the gas

phase and then through a nondiffusing liquid.Using correlations for dilute solutions in wetted-wall

towers, the film mass-transfer coefficient for A in the

gas phase is predicted as

ky = 1.465 x 10-3 kg mol A/s.m2 mol frac. And for the

liquid phase as

kx = 1.967 x 10-3 kg mol A/s.m2 mol frac.

Calculate the interface concentrations yAi and xAi

and the flux NA.

SolutionFirst we plot the data


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51

yA

xA

00

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4


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52

Now we need to find the point P on the graphSo :Since the correlations are fordilute solutions, (1-yA)iM and (1-

xA)iM are approximately 1.0 and the coefficients are the sameas ky and kx .

Point P is plotted at yAG = 0.380 and xAL = 0.1.For the first trial (1-yA)iM and (1-xA)iM are assumed as 1.0 andthe slope of line PM is,

A line through point P with a slope of1.342 is plotted in thefigure intersecting the equilibrium line at M1, whereyAi = 0.183 and xAi = 0.247.

342.10.1/10465.1

0.1/10967.1

)1/(

)1/(3

3

'

'

LMAy

LMAx

yk

xkslope


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53

For the second trial we use yAi and xAi fromthe first trial to calculate the new slope.Substituting into corresponding Eqs.,

)]1/()1ln[(

)1()1()1(

AGAi

AGAiLMA

yy

yyy

715.0)]38.01/()183.01ln[()380.01()183.01(

)]1/()1ln[(

)1()1()1(

AiAL

AiALLMA

xx

xxx

825.0)]247.01/()1.01ln[(

)247.01()1.01(


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54

Substituting into Eq. (10.4-9) to obtain the new slope,

A line through point P with a slope of1.163 is

plotted and intersects the equilibrium line at M,

where yAi = 0.197 and xAi = 0.257. Using these newvalues for the third trial, the following values are

calculated:

163.1715.0/10465.1

825.0/10967.1

)1/(

)1/(

3

3

'

'

LMAy

LMAx

yk

xkslope

709.0)]38.01/()197.01ln[(

)380.01()197.01()1(

LMAy

820.0)]257.01/()1.01ln[(

)257.01()1.01()1(

LMAx

Please notice the valuesof xAi and yAi didntchange much from thefirst trial and that meanswe are on the right wayof solving the problem(refining the answer)

8200/10961)1/(3'k


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55

This slope of1.160 is essentially the same as theslope of1.163 for the second trial.Hence, the final values are yAi= 0.197 and xAi =

0.257 and are shown as point M. To calculate the

flux,

Note that the flux NA through each phase is the

same as in other phase, which should be the case at

steady state.

160.1709.0/10465.1

820.0/10967.1

)1/(

)1/(3

3

'

LMAy

LMAx

yk

xkslope

24

3'

./1078.3

)197.0380.0(709.0

10967.1)(

)1(

mskgmol

yyy

kN AiAG

LMA

y

A


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56

Fig.: Location of interface concentrationsfor example.

yAG

yAi

y*A

xAL xAi x*A

M

D

E

P

M1

00

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4


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Example 3

57


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Example 4

62


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Diffusion Coefficients

69

Diffusion can be considered by a mechanisticapproach in which a consideration of atommovement is important or by a continuum approachsuch as with Ficks first law, where no considerationis given to the actual mechanism by which atomtransfer occurs.Gases: molecules are far apart, intermolecular forcescan often be disregarded or considered only duringcollisions.A molecule is assumed to travel along a straight lineuntil it collides with another molecule, after which itsspeed and direction are altered.

Diffusion Coefficients for Gases


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70

The diffusion coefficient can be derived foran idea gas by using the simplified discussionpresented by Sherwood et al (1975).From kinetic theory, the diffusion coefficientis assumed to be directly proportional to themean molecular velocity , and the meanfree path U

UD

Diffusion Coefficients for Gases


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71

For an idea gas the motion of the molecule is assumedto be totally random, with the mean free path , beinginversely proportional to both the average cross-sectional area of the molecules, A, and the numberdensity, n, of all molecules in a specified volume.Thenumber density of an idea gas varies directly withpressure and inversely with the temperature,

PA

T

nA

1

Mean molecular velocity is related to the temperatureand molecular weight of the molecule by theexpression

PA

T

M

T

M

TD

BA

AB

21

21

23

' 11

BAavg

ABMMPA

TKD

Aavg: average cross-sectional areaof both types of moleculesK a costant


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Semi-empirical Equations

231312175.100143.0

BVAVAB

BAAB

PM

TDD

Equation of Fuller, Schettler, and Giddings

DAB is in cm2/sP: atmT: K V =summation of atomic and structural diffusionvolumes from Table 3.1, which includes diffusionvolumes of simple molecules

CBE 3233 MassTransfer(2)

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Transcript of CBE 3233 MassTransfer(2)