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 during the week 9-13 March, 2015.

• By either volunteer or class list.

• Homework #7

• (self practice)

• WWWR #24.2

• ID # 14.25.

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 AAA

n

iii

n

ii

n

iii

1

1

1

vvv

c

cn

iii

1

vV

– 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 relationdz

d

RT

D

dz

duVv cABc

AzzA

,

dz

d

RT

DcVvcJ cABAzzAAzA

)( ,,

Diffusion Coefficient

• Fick’s law proportionality/constant

• Similar to kinematic viscosity, , and thermal diffusivity,

t

L

LLMtL

M

dzdc

JD

A

zAAB

2

32

, )1

1)((

• Gas mass diffusivity– Based on Kinetic Gas Theory

– = mean free path length, u = mean speed

– Hirschfelder’s equation:

uDAA 3

1*

2/13

22/3

2/3

* )(3

2

AAAA M

N

P

TD

DAB

BAAB P

MMT

D

2

2/1

2/3 11001858.0

– Lennard-Jones parameters and from tables, or from empirical relations

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

– Extrapolation of diffusivity up to 25 atmospheres

2BA

AB

BAAB

2

1

1,12,2

2/3

1

2

2

1

TD

TD

ABAB T

T

P

PDD

PTPT

Binary gas-phase Lennard-Jones “collisional integral”

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

– For binary gas with polar compounds, we calculate by

23/13/1

2/1

75.13 1110

BA

BAAB

vvP

MMT

D

*

2196.00 T

ABD

where

bb

PBAAB TV

232/1 1094.1,

ABTT /* 2/1

BAAB

bT23.1118.1/

)exp()exp()exp( ****0 HT

G

FT

E

DT

C

T

ABD

and

– For gas mixtures with several components,

– with

2/1BAAB

3/1

23.11

585.1

bV

nn DyDyDyD

1

'31

'321

'2

mixture1 /...//

1

nyyy

yy

...32

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.

BAB r

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

3/1A

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

• 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

AAA M

NTuD

8

33*

• But if Kn >1, then

• If both Knudsen and molecular diffusion exist, then

• with

• For non-cylindrical pores, we estimate

Apore

A

poreporeKA M

Td

M

NTdu

dD 4850

8

33

KAAB

A

Ae DD

y

D

111

A

B

N

N1

AeAe DD 2'

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 oABAe

pore

s

d

d

22

1 2

( )( ) (1 )pore s

pore

d dF

d

• F2 is the hydrodynamic hindrance factor, one equation is by Renkin,

532 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

....

0vcsc

dVt

dA nv

• For in – out,– in x-dir,

– in y-dir,

– in z-dir,

• For accumulation,

xxAxxxA zynzyn ,,

yyAyyyA zxnzxn ,,

zzAzzzA yxnyxn ,,

zyxtA

• 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 BA B A Br r

t

vvvnn BBAABA

BA rr

• So by conservation of mass,

• Written as substantial derivative,

– For species A,

0

t

v

0 vDt

D

0 AAA r

Dt

Dj

• In molar terms,

– For the mixture,

– And for stoichiometric reaction,

0

A

AA R

t

cN

0)(

BABA

BA RRt

ccNN

0)(

BA RRt

ccV