masstransfer fundamentals

7/28/2019 Masstransfer Fundamentals

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Tutorial

Tutorial #7

WWWR# 24.1, 24.12, 24.13,24.15(d), 24.22.


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Molecular Mass Transfer

Molecular diffusion

Mass transfer law components:

Molecular concentration:

Mole fraction:

(liquids,solids) , (gases)

c

cy

c

cx AA

AA

RT

p

V

n

Mc AA

A

AA


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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

i

ii

n

i

i

n

i

ii

1

1

1

vv

v

c

cn

i

ii 1

v

V


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mol


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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

or, in the z-direction,

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

AABA cD J

dz

dcDJ AABzA ,

dz

dycDJ AABzA ,


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Since mass is transferred by two means:

concentration differences

and convection differences from density differences

For binary system with constant Vz,

Thus,

Rearranging to

)( ,, zzAAzA VvcJ

dz

dycDVvcJ AABzzAAzA )( ,,

zAA

ABzAA Vcdz

dycDvc ,


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As the total velocity,

Or

Which substituted, becomes

)(1 ,, zBBzAAz vcvcc

V

)( ,, zBBzAAAzA vcvcyVc

)( ,,, zBBzAAAA

ABzAA vcvcydz

dycDvc


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

And finally,

Or generalized,

AAA c vN

)( ,,, zBzAAA

ABzA NNydz

dycDN

)( BAAAABA yycD NNN


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Related molecular mass transferDefined in terms of chemical potential:

Nernst-Einstein relation dz

d

RT

D

dz

duVv cABcAzzA

,

dz

d

RT

D

cVvcJ

cAB

AzzAAzA

)( ,,


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Diffusion Coefficient

Ficks law proportionality/constant

Similar to kinematic viscosity, , and

thermal diffusivity, a

t

L

LLMtL

M

dzdc

J

D A

zA

AB

2

32

,

)1

1

)((


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Gas mass diffusivity

Based on Kinetic Gas Theory

= mean free path length, u = mean speed

Hirschfelders equation:

uDAA 3

1*

2/13

22/3

2/3

* )(3

2

AA

AAM

N

P

TD

DAB

BA

ABP

MMT

D

2

2/1

2/3 11001858.0


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Lennard-Jones parameters and e from tables,

or from empirical relations

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

Extrapolation of diffusivity up to 25

atmospheres

2

BAAB

BAAB eee

2

1

1,12,2

2/3

1

2

2

1

TD

TD

ABABT

T

P

PDD

PTPT


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Binary gas-phase Lennard-Jones

collisional integral


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With no reliable ore, we can use the Fuller

correlation,

For binary gas with polar compounds, we

calculate by

23/13/1

2/1

75.1311

10

BA

BA

AB

vvP

MMT

D

*

2

196.00 T

ABD


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where

bb

PBAAB

TV

232/1 1094.1,

ABTT e /*

2/1

e

e

e BAAB

bT23.1118.1/ e

)exp()exp()exp( ****0 HTG

FT

E

DT

C

T

ABD


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and

For gas mixtures with several components,

with

2/1BAAB 3/1

23.11585.1

bV

nn DyDyDyD

1

'

31

'

321

'

2

mixture1/...//

1

nyyy

yy

...32

2'

2


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2


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Liquid mass diffusivity

No rigorous theoriesDiffusion as molecules or ions

Eyring theory

Hydrodynamic theory Stokes-Einstein equation

Equating both theories, we get Wilke-Chang eq.B

ABr

TD

6

6.0

2/18104.7

A

BBBAB

V

M

T

D


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For infinite dilution of non-electrolytes in

water, W-C is simplified to Hayduk-Laudie eq.

Scheibels equation eliminates B,

589.014.151026.13 ABAB VD

3/1

A

BAB

V

K

T

D

3/2

8 31)102.8(A

B

VVK


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As diffusivity changes with temperature,

extrapolation of DAB

is by

For diffusion of univalent salt in dilute solution,

we use the Nernst equation

n

c

c

ABT

ABT

TT

TT

D

D

1

2

)(

)(

2

1

F

RTDAB )/1/1(

200


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Pore diffusivity

Diffusion of molecules within pores of poroussolids

Knudsen diffusion for gases in cylindrical pores

Pore diameter smaller than mean free path, and

density of gas is low

Knudsen number

From Kinetic Theory of Gases,

poredKn

AAA M

NTuD

8

33*


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But if Kn >1, then

If both Knudsen and molecular diffusion exist, then

with

For non-cylindrical pores, we estimate

A

pore

A

porepore

KAM

TdMNTdudD 48508

33

KAAB

A

Ae DDy

D111 a

A

B

N

N1a

AeAe DD2' e


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Example 6


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Types of porous diffusion. Shaded areas represent nonporous solids


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Hindered diffusion for solute in solvent-filled

pores

A general model is

F1 and F2 are correction factors, function of porediameter,

F1 is the stearic partition coefficient

)()( 21 FFDDo

ABAe

pore

s

d

d

2

2

1 2

( )( ) (1 )

pore s

pore

d dF

d


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F2 is the hydrodynamic hindrance factor, one

equation is by Renkin,

53

2 95.009.2104.21)( F


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Example 7


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Convective Mass Transfer

Mass transfer between moving fluid with

surface or another fluid

Forced convection

Free/natural convection

Rate equation analogy to Newtons cooling

equation

AcA ckN


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Example 8


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Differential Equations

Conservation of mass in a control volume:

Or,

inout + accumulationreaction = 0

.... 0vcsc dVtdA nv


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For inout,

in x-dir,

in y-dir,

in z-dir,

For accumulation,

xxAxxxAzynzyn

,,

yyAyyyA zxnzxn ,,

zzAzzzA yxnyxn ,,

zyxt

A


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For reaction at rate rA

,

Summing the terms and divide by xyz,

with control volume approaching 0,

zyxrA

, , , , , , 0A x x x A x x A y y y A y y A z z z A z z A An n n n n n

rx y z t

, , ,0AA x A y A z An n n r

x y z t


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We have the continuity equation for

component A, written as general form:

For binary system,

but and

0

A

AA r

t

n

n n 0A BA B A Br rt

vvvnn BBAABA

BA rr


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So by conservation of mass,

Written as substantial derivative,

For species A,

0

t

v

0 vDtD

0 AAA r

Dt

D j


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In molar terms,

For the mixture,

And for stoichiometric reaction,

0

A

AA Rt

cN

0)( BABA

BA RRt

ccNN

0)( BA RR

t

ccV

masstransfer fundamentals

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