Tutorial/HW Week #7

WRF Chapters 22-23; WWWR Chapters 24-25

ID Chapter 14

• Tutorial #7• WWWR# 24.1, 24.12, 24.13,

24.15(d), 24.22.

• To be discussed on March

10, 2020.

• By either volunteer or

class list.

Molecular Mass Transfer

• Molecular diffusion

• Mass transfer law components:

– Molecular concentration:

– Mole fraction:

(liquids,solids) , (gases)

c

cy

c

cx A

AA

A

RT

p

V

n

Mc AA

A

AA

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 AA

A

n

i

ii

n

i

i

n

i

ii

1

1

1

vv

v

c

cn

i

ii 1

v

V

mol

– 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

or, in the z-direction,

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

AABA cD J

dz

dcDJ A

ABzA ,

dz

dycDJ A

ABzA ,

– 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 A

ABzzAAzA )( ,,

zAA

ABzAA Vcdz

dycDvc ,

• As the total velocity,

• Or

• Which substituted, becomes

)(1

,, zBBzAAz vcvcc

V

)( ,, zBBzAAAzA vcvcyVc

)( ,,, zBBzAAAA

ABzAA vcvcydz

dycDvc

• 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

• Related molecular mass transfer

– Defined in terms of chemical potential:

– Nernst-Einstein relation

dz

d

RT

D

dz

duVv cABc

AzzA

,

dz

d

RT

DcVvcJ cAB

AzzAAzA

)( ,,

Diffusion Coefficient

• Fick’s law proportionality/constant

• Similar to kinematic viscosity, n, and

thermal diffusivity, a

t

L

LLMtL

M

dzdc

JD

A

zA

AB

2

32

,)

1

1)((

• Gas mass diffusivity

– Based on Kinetic Gas Theory

– l = mean free path length, u = mean speed

– Hirschfelder’s equation:

uDAA l3

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

– 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

Binary gas-phase Lennard-Jones

“collisional integral”

– With no reliable or e, we can use the Fuller

correlation,

– For binary gas with polar compounds, we

calculate by

23/13/1

2/1

75.13 1110

BA

BA

AB

vvP

MMT

D

*

2196.00 T

ABD

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

and

– For gas mixtures with several components,

– with

2/1

BAAB

3/1

23.11

585.1

bV

nn DyDyDyD

1

'

31

'

321

'

2

mixture1/...//

1

nyyy

yy

...32

2'

2

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 eq.B

ABr

TD

6

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 B,

589.014.151026.13 ABAB VD

3/1

A

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

– 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

ll2

• Pore diffusivity

– Diffusion of molecules within pores of porous

solids

– 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

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

A

pore

A

porepore

KAM

Td

M

NTdu

dD 4850

8

33

KAAB

A

Ae DD

y

D

111

a

A

B

N

N1a

AeAe DD 2' e

Example 6

Types of porous diffusion. Shaded areas represent nonporous solids

– Hindered diffusion for solute in solvent-filled

pores

• A general model is

• F1 and F2 are correction factors, function of pore

diameter,

• F1 is the stearic partition coefficient

)()( 21 FFDD o

ABAe

pore

s

d

d

2

2

1 2

( )( ) (1 )

pore s

pore

d dF

d

• F2 is the hydrodynamic hindrance factor, one

equation is by Renkin,

53

2 95.009.2104.21)( F

Example 7

Convective Mass Transfer

• Mass transfer between moving fluid with

surface or another fluid

• Forced convection

• Free/natural convection

• Rate equation analogy to Newton’s cooling

equation

AcA ckN

Example 8

Differential Equations

• Conservation of mass in a control volume:

• Or,

in – out + accumulation – reaction = 0

....0

vcscdV

tdA nv

• For in – out,

– in x-dir,

– in y-dir,

– in z-dir,

• For accumulation,

xxAxxxA zynzyn ,,

yyAyyyA zxnzxn ,,

zzAzzzA yxnyxn ,,

zyxt

A

• For reaction at rate rA,

• Summing the terms and divide by xyz,

– with control volume approaching 0,

zyxrA

, , , , , ,0

A x x x A x x A y y y A y y A z z z A z z AA

n n n n n nr

x y z t

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

x y z t

• 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 B

A B A Br rt

vvvnn BBAABA

BA rr

• So by conservation of mass,

• Written as substantial derivative,

– For species A,

0

t

v

0 v

Dt

D

0 AAA r

Dt

Dj

• In molar terms,

– For the mixture,

– And for stoichiometric reaction,

0

A

AA R

t

cN

0)(

BA

BABA RR

t

ccNN

0)(

BA RR

t

ccV