1

CH351 Mass TransferCH351 Mass Transfer

Sitaraman Krishnan9/26/07

2FickFick’’ss First Law: Steady State First Law: Steady State

Molecular DiffusionMolecular Diffusion

axis- along gradient ionconcentrat

/scm t,coefficien diffusion

s)mol/(cm axis,- along A component offlux

A

2AB

2A,z

AABA,z

zz

cD

zJz

cDJ

=∂∂

=

⋅=∂∂

−=

3

( ) ( ) ( )2

31

B3

1A

21

AB

75.1

AB00143.0

:equation Giddings and Schetler Fuller,

⎥⎦⎤

⎢⎣⎡ Σ+Σ

=

vvMP

TD

molecule a of mass molecules )(spherical of diameter

tcoefficien diffusion self 3

2

:GasesDensity Low orTheory Kinetic From

*AA

21

33

223*AA

==

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ κ

π=

mdD

mT

PdD

4

( )

B solvent in diffusing is A Species

104.7

:Liquids in Diffusion for Equation Chang-Wilke

6.0AB

21BB

8

VTMDAB

η

Φ×=

−

5

• pi = partial pressure of component i

• ci = molar conc. of component i

• P = total pressure (atm, bar, N/m2, mmHg, Torr)

• c = total molar concentration of gas (mol/cm3, mol/m3, …)

( ) ( )

cRTP

RTccpp

RTcpRTcp

RTVnpRTnVp

=

+=+

=

=

=⇒=

BABA

BB

AA

AAAA

Partial Pressures, Molar Partial Pressures, Molar ConcetrationsConcetrations, Total Pressure, Total Pressure

6 Concentrations, Partial Concentrations, Partial Pressures, and Mole FractionsPressures, and Mole Fractions

Pp

ccy

Pp

ccy

BBB

AAA

B of fraction Mole

A of fraction Mole

===

===

• At constant total pressure, P, and constant temperature, T, total molar concentration, c, is constant.

• For each species, mole fraction ∝concentration ∝ partial pressure

cRTPRTcpRTcp

==

=

BB

AA

7FickFick’’ss First Law in terms of Mole First Law in terms of Mole

FractionsFractions

zy

RTPD

zycD

zcDJ

∂∂

−=

∂∂

−=

∂∂

−=

AAB

AAB

AABA,z

zy

RTPD

zycD

zcDJ

∂∂

−=

∂∂

−=

∂∂

−=

BBA

BBA

BBAB,z

8

What is the net molar flux of A that a stationary observer would measure?

fluid element

direction of bulk flowdirection of bulk flow

direction of direction of diffusionaldiffusional

flowflowcA

position

observer stationary to relative A ofvelocity A,z =

=

vvcN A,zAA,z

9What is the net molar flux of A that an observer who moves with the fluid element would measure?

Moving observer will measure only flux due to diffusion.

( )

zcD

vvcJ A,zAA,z

∂∂

−=

−=

AAB

z *

10

Total Flux, Total Flux, NNA,zA,z = = JJA,zA,z + + ccAAvvzz**

( )

scmmol

scm

cmmol

**

23

A,zAA,z

B,zBA,zAz

zA,zAA,z

⋅=×

=

+=

−=

vcN

vyvyvvvcJ • vA,,z = velocity of A

measured by a stationary observer

• vB,z = velocity of B measured by a stationary observer

• vz* = molar average velocity

11

General Form of General Form of FickFick’’ss First LawFirst Law

( )

( )B,zA,zAA

ABA,z

BAAA,zA,z

zAA,zA,z

motion bulk to dueflux flux diffusive*

NNyz

ycDN

NNyJN

vcJN

++∂∂

−=

++=

+=

+=

12 EquimolarEquimolar Counter Diffusion Counter Diffusion (EMCD)(EMCD)

• A simplifying assumption that– allows easier

solution of the general form of Fick’s first law

– makes analytical solution possible

( )

zcDJN

NNyJN

NN

∂∂

−==

++=∴

−=

AABA,zA,z

B,zA,zAA,zA,z

B,zA,z

0

13

Problem 1 (EMCD assumption)Problem 1 (EMCD assumption)

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=

∂∂

−==

12

A,2A,1ABA,z

AABA,zA,z

zzyy

RTPDN

zy

RTPDJN

Mixture of two gases, A and B, in a tube with conc. gradient

z-axis

z = z1 = 0

yA,1

pA,1P = constant, T = constant

yA,2

pA,2

z = z2

( )112

A,2A,1A,1A zz

zzyy

yy −⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=

CONCENTRATION PROFILEFLUX

14 Problem 2 (EMCD assumption)Problem 2 (EMCD assumption)

sA

vpA

vpA

cRTpc

pp

RrA

RrA

==

=∴

=

=

0 bAA ≈=

∞→cc

r

Evaporation from surface of naphthalene ball

Air near surface is saturated with naphthalene

r

r+Δr

R

( )

( )

( ) ⎟⎠⎞

⎜⎝⎛−π=

−×π=

⎟⎠⎞

⎜⎝⎛−

=

=−

−⇒=⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

∂∂

−=

=

smol4

4 rate nEvaporatio

s cmmol

0

state,steady At

bA

sAAB

bA

sA

AB2

2

bA

sA

ABA

bA

sA

bAAA2

AABA

ccDR

RccDR

RccDN

rR

cccc

rcr

r

rcDN

Rr

rr

15

UnimolecularUnimolecular DiffusionDiffusion( )

zy

yRTPD

yJ

N

NyJN

N

NNyJN

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎠⎞

⎜⎝⎛−=

−=

+=∴

=

++=

A

A

AB

A

A,zA,z

A,zAA,zA,z

B,z

B,zA,zAA,zA,z

11

1

0

16

Solvent Evaporation (UMD)Solvent Evaporation (UMD)

Liquid, A

z = z1 = 0 pA1, yA1

pA2, yA2

z

z = z2

• At z = z1, air is saturated with liquid: pA1 = vap. pressure

• At z = z2, all A carried away by air: pA2 ~ 0

17

Solvent EvaporationSolvent Evaporation

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−=

B,1

B,2

B,1B,2lmB,

12

A,2A,1

lmB,

ABA

A

B

ABA,z

ln

1

1

yy

yyy

zzyy

yRTPDN

zy

yRTPDN

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

−− 12

1

A,1

A,2

A,1

A

11

11 zz

zz

yy

yy

FLUX

CONC. PROFILE

18 Evaporation of Benzene Evaporation of Benzene ((bpbp 80 80 °°C) at 25 C) at 25 °°C and 70 C and 70 °°CC

25 25 °°CC 70 70 °°CC

19

Flux Profiles (UMD)Flux Profiles (UMD)

20 Evaporation of Dichloromethane Evaporation of Dichloromethane ((bpbp 40 40 °°C) at 25 C) at 25 °°CC

21Diffusion Into a Falling Liquid FilmDiffusion Into a Falling Liquid Film

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛δ

−μδρ

=22

y 12

zgzv

z

yLiquid

Gasvy(z)

cA(z)cA,i

W

x

OBJECTIVE:• To determine molar flux

(mol cm−1 s−1) at the gas−liquid interface.

• The overall rate of mass transfer (mol/s).

( )entrance top the from position at

)0( interface atflux molar local0zA

y

zyN ===

22

Liquid Element

y

y+Δy

zz+Δz

( ) ( )

( ) ( )

0

0

0

A,zyA,

zA,zzzA,zyyA,yyyA,

zzA,zzA,z

yyyA,yyA,

=∂

∂+

∂

∂

=Δ

−+

Δ

−

=Δ⋅−Δ⋅+

Δ⋅−Δ⋅

Δ+Δ+

Δ+

Δ+

zN

yN

z

NN

y

NN

yWNyWN

zWNzWN

*

*

zAA

ABA,z

yAA

AByA,

vcz

cDN

vcycDN

+∂∂

−=

+∂∂

−=

2A

2

ABA

y zcD

ycv

∂∂

=∂∂

yyA,N

zA,zN

ΔyyyA, +N

ΔzzA,z +N Δz Δy

W

23

z

yLiquid

Gasvy(z)

cA(z)cA,i

W

x

2A

2

ABA

y zcD

ycv

∂∂

=∂∂

( )

yvD

cz

cDN

vyD

zcy,zc

zz π

=∂∂

−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

==

maxy,ABiA,

0

AAB0A,z

maxy,AB

iA,A

4erfc

:solution eApproximat

24

( ) ( )

( ) ( )

( )yWy

vDc

yWN

y

yvD

cz

cDN

z

zz

d

d

area flux position at second per dtransferre A of Moles

maxy,ABiA,

0A,z

maxy,ABiA,

0

AAB0A,z

⋅×⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

π=

⋅×=

×=

π=

∂∂

−=

=

==

25

( )

gas and element liquid between contact of time where

12

2

2

d1

d

tower the of length over second per dtransferre A of moles total

ABiA,

maxy,ABiA,

maxy,ABiA,

0

maxy,ABiA,

0

maxy,ABiA,

=τ

⎟⎠⎞

⎜⎝⎛τ

⎟⎠⎞

⎜⎝⎛

π×=

π×=

π=

π=

⋅×⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

π=

∴

∫

∫

DcWL

LvD

cWL

LvDWc

yy

vDWc

yWy

vDc

L

L

L

26

Mass Transfer CoefficientMass Transfer Coefficient

• Fick’s law: flux ∝ conc. gradient– const. of proportionality = diffusion coefficient

• Another approach to quantify flux:– Flux ∝ conc. driving force, ΔcA

– const. of proportionality = mass transfer coefficient, k

– Similar to Newton’s “law” of cooling

27

⎟⎠⎞

⎜⎝⎛Δ⋅⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

⋅

=

Δ⋅=

ΔΔ

=

3A2A

AA

AABA

cmmol

scm

scmmol

scm of units has

tcoefficien transfer mass :approach gEngineerin

:law sFick'

ckN

k

kckN

zcDN

28 Evaporation from surface of Evaporation from surface of naphthalene ballnaphthalene ball

bA

sA

vpA

cc

cRTpc

rA

RrA

=

==

∞→

=

( )

cNk

ccc

Δ=

−=Δ=

A

bA

sA

t,coefficien transfer Mass

transfer mass for force driving Conc.

R

( )

( )( ) R

DccR

ccD

cNk

RccDN

Rr

ABb

As

A

bA

sA

AB

A

A

bA

sA

ABA

:us given hadlaw sFick' of Solution

=−

−

=Δ

=∴

−=

=

29

Sherwood NumberSherwood Number• For evaporation from surface of a sphere, we obtained:

• kD/DAB is a dimensionless quantity that often appears in mass transfer calculations.

• It is called the Sherwood number and is denoted by Sh.

• For diffusion from surface of a sphere into a stagnant fluid, Sh = 2

2

diameter) sphere (2

AB

ABAB

=⋅

∴

===

DDk

DD

DR

Dk

30

Convective Mass TransferConvective Mass Transfer• Material is transported between

– a solid surface and a moving fluid (gas or liquid)

– two relatively immiscible moving fluids (gas and liquid, or liquid and liquid)

• Examples– Mass-transfer of gas into a falling liquid film– Mass-transfer of naphthalene into a flowing

gas stream– Mixing in a stirred vessel

31

Turbulent FlowTurbulent Flow

• Laminar flow is characterized by streamlines

• Turbulent flow is characterized by chaotic flow of packets of fluid called “eddies”

• Rate of mass transfer in laminar flow is determined by molecular diffusivity, DAB

• Rate of mass transfer in turbulent flow is determined by both molecular diffusivity, DAB

and turbulent diffusivity, Dt

32 Mass Transfer Correlations for Mass Transfer Correlations for Convective FlowsConvective Flows

•• Dimensional AnalysisDimensional Analysis• Example

– The inner wall of a circular tube is coated with species A (e.g., naphthalene)

– Mass transfer occurs to a fluid flowing through the tube (e.g., air)

cAs cA

b D (cm)ρ (g cm−3)μ cm−1 s−1)DAB (cm2 s−1)

v (cm s−1)kc (cm s−1)

33

edcba

−1−1−100−1s21−1−311cm001100gm

DABvμρDkc

equations 3 variables, 51

1230

−=−−−=++−−

=+

edcedcba

cb

cedcaedb

cb

+−=−−+−=++−

−=

1123

( )

[ ]( ) ( ) ( ) ( ) ( )eABdcba

c

c

bA

sAcA

constant

tcoefficien transfer mass convective

DvDk

kcckN

μρ=

=

−⋅=

34

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−+−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

==⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−+−

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−+−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−−

caa

c

edb

cca

c

edb

cca

c

edb

1

11

113213

001

11

110213001

BAxAxA

BAx

11

35 [ ]( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ( )

[ ] ( ) ( ) ( )

[ ]

[ ]

( )ydiffusivit mass

ydiffusivit momentumnumber Schmidt

constant

constant

1constant

constant

constant

AB

1ca

AB

1a

AB

c

caAB

1ca11a

caAB

1acca

eAB

dcbac

=ρμ

==

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ

⎟⎟⎠

⎞⎜⎜⎝

⎛μρ

=⎟⎟⎠

⎞⎜⎜⎝

⎛

μρ⎟⎟⎠

⎞⎜⎜⎝

⎛μρ

=

μρ=

μρ=

βα

+++

−−++−−−+

−−+−

DSc

ScReSh

DDv

DDk

DDDv

DvD

DvDk

ca

36 [ ]( ) ( ) ( ) ( ) ( )

[ ] ( ) ( ) ( )

[ ]

[ ]

[ ] ( )

factor-J Colburn and Chilton

constant

constant

constant

constant

constant

D3/2c

323/2c

c

1ca

AB

ac

caAB

caa

caAB

1accac

==

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ

⎟⎟⎠

⎞⎜⎜⎝

⎛μρ

=⎟⎠

⎞⎜⎝

⎛

μρ⎟⎟⎠

⎞⎜⎜⎝

⎛μρ

=

μρ=

+δγ

δγ

++

−−+−−

−−+−

JScvk

ScReScvk

ScRevk

DDv

vk

vDDv

DvDk

ca

37

Turbulent Flow Inside PipesTurbulent Flow Inside Pipes

33.083.0

ABAB

c

Re023.0

:300060 and 2100 When

ScSh

Sc . Re

DScDvRe

DDkSh

=

<<>

ρμ

=μρ

==

In general, Sc for gases is in the range 0.5−3, and for liquids, Sc > 100.

38

Flow Past Single SpheresFlow Past Single Spheres

3/162.0

3/15.0

3/153.0

ABAB

c

Re347.0 :70001Re2000 liquids, For

Re95.02 :2000Re2 whenliquids, For

552.02

2.70.6 and 48000,1 whengases, For

ScSh

ScSh

ScReSh

ScRe

DScDvRe

DDkSh

=<<

+=<<

+=

<<<<

ρμ

=μρ

==

39

Naphthalene EvaporationNaphthalene Evaporation

• A sphere of naphthalene having a diameter of 1 cm is suspended in air at 1 atm, and flowing at a velocity of 0.3 m/s. The diffusion coefficient of naphthalene in air at this temperature is 6.92×10−6 m2/s. The molar density of naphthalene solid is 8.86×10−3 mol/cm3. Calculate the initial rate of evaporation of naphthalene from the surface. Estimate the time required to reduce the diameter from 1 to 0.5 cm.

40 Mass Transfer in Packed BedsMass Transfer in Packed Beds

tube of area sectional-crossrateflow volumetric

packing withouttubeempty the invelocity average gas of '

spheres the of diameter0.5. and 0.3 betweenusually is of value The

solid plus space void of volume totalspace void of volumebed the in fraction void

''

4548.0,0000110 whenspheres, of bed packed a in gases For

p

AB

p32cD

4069.0D

=

==

=ε

==ε

ρμ

=μ

ρ==

ε=

<<

−

velocity lsuperficiav

D

DSc

vDReSc

vkJ

ReJ

Re

41

Mass Transfer in Packed BedsMass Transfer in Packed Beds

( )

particles. spherical-non the of area surface the use flux, calculate Toparticle solid given the as area surface same the withsphere a of dia.

particles, spherical-non For

Re250.006901165 and ,150055 whenliquids, For

Geankoplis and WilsonRe09.170000165 and ,550.0016 whenliquids, For

p

31.0D

32D

=

ε=

<<<<ε

=

<<<<

−

−

D

J

ScRe

J

ScRe

42 Packed Bed of Benzoic Acid Packed Bed of Benzoic Acid SpheresSpheres

• Water at 26.1 °C flows at a rate of 5.514×10−7

m3/s through a packed bed of benzoic acid spheres having a diameter of 6.375 mm. The void fraction of the bed is 0.436. The tower diameter is 0.0667 m and the tower height is 0.1 m. The solubility of benzoic acid in water is 2.948 ×10−2 kg mol/m3. Predict the mass transfer coefficient kc and the outlet concentration of benzoic acid in water.

43

ε = 0.436Dp = 6.375×10−3 m, Ap = 0.01198 m2

DAB = 1.25×10−9 m2/s (Wilke-Chang)

( )

( )

( )( )( ) 150.1108718.0

7.99610578.110375.6'

sm10578.1

0667.04

10514.5

4

'

6.70210245.17.996

108718.0

3

43p

4

2

7

2

9

3

AB

=×××

=μ

ρ=

×=π

×=

π=

=×

×=

ρμ

=

−

−−

−−

−

−

vDRe

T

Qv

DSc

Water flow rate, Q = 5.514×10−7 m3/sμ = 0.8718×10−3 Pa·s, ρ = 996.7 kg/m3

cA,2 = ?

Tower dia. T = 0.0667 m; Height H = 1 m

cA,1 = 0

44

( )

( )( )( )

( ) ( )

( )

( )

( )( ) ( )( ) 2

3

3

p

b

2pp

b3

pp

p

b

3322b

63/243/2Dc

3/23/2D

4

m 1855.010375.6

436.0110494.3616

spheres of area surface Total

16

bed the in spheres of number Let1 bed the in spheres of volume

fraction Void

m 10494.31.00667.044

bed of volume Total

m/s 10447.46.70210578.1277.2'

277.2150.10436.009.109.1

m/s 10578.1'150.16.702

=×

−×=

ε−⋅=

π⋅==

ε−⋅=⎟⎠⎞

⎜⎝⎛ π⋅∴

=

ε−⋅=∴ε=

×=π

=π

==

×=×=⋅⋅=

==ε

=

×===

−

−

−

−−−−

−−

−

DV

DNA

VDN

NV

HTV

ScvJk

ReJ

vReSc

45

( ) ( )

( ) ( ) ( )( )( )

( )( ) A,22

bA

3A,11

bA

32sA

2b

As

A

1b

As

A

2b

As

A1b

As

Alm

bA

sA

3

3

32

A,1A,2lmb

As

Ac

c outlet at acidbenzoic of conc.

mol/m kg 0c inlet at acidbenzoic of conc.

mol/m kg 10948.2

ln

mmol kg

sm

mmol kgm

sm

:balance Mole

==

===

×=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−

−

−−−=−

⋅⋅⋅

−⋅=−⋅⋅

−

c

c

c

cccc

cccccc

ccQccAk

46

( ) ( )

( )

( )32

7

622

cA,1

sA

sAA,2

A,1A,2

A,2s

A

A,1s

A

A,1A,2c

mol/m kg 10287.2

10514.51855.010447.4exp*010948.210948.2

exp

ln

−

−

−−−

×=

⎟⎟⎠

⎞⎜⎜⎝

⎛

×××

−−×−×=

⎟⎠

⎞⎜⎝

⎛−−−=∴

−⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−

−⋅⋅

QAkcccc

ccQ

cccc

ccAk

47 Mass Transfer to Bubbles and Mass Transfer to Bubbles and ParticlesParticles

• Small particles in suspension– Mass transfer from small gas bubbles to liquid

phase– Mass transfer from liquid phase to the surface

of catalyst particles, microorganisms, liquid drops, etc.

– Mass transfer coefficient depends on free fall or rise of particles due to gravitational forces

48• Mass transfer coefficient, kc, in mass transfer to small particles is

affected by natural convection. Natural convection occurs when there is significant density difference between the particles of the dispersed phase (gas bubbles or liquid drops) and the fluid (continuous phase). In such cases, kc is expected to depend on: – diameter Dp of the particles of dispersed phase (e.g., O2 bubble)

– density ρc of the continuous phase (e.g., water)

– viscosity μc of the continuous phase (e.g., water)

– the buoyant force gΔρ where Δρ is the density difference between the dispersed and continuous phases, and g is the acceleration due to gravity (9.81 m/s2)

– the diffusion coefficient of the molecules of dispersed phase (oxygen) in the continuous phase (water).

• Using dimensional analysis, derive an expression for the mass transfer coefficient in terms of the dimensionless numbers Sh and Sc. The dimensionless group with the buoyant force variable, gΔρ, is called the Grashof number, Gr. Derive an expression for Gr.

49

( )

positive) (always phases continuous and dispersed the between differencedensity the of magnitude

m/s 9.81

solution the ofviscosity

particle solid or bubble gas the of diameter solution in A solute the ofy diffusivit

tcoefficien transfer mass side'-liquid'

31.02

2c

p

AB

L

3/1

2c

c3/2

p

ABL

=ρΔ=

=μ

ρμ=

==

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

ρ

ρμΔ+= −

g

DSc

DDk

gScDDk

ABcc

Mass Transfer to Small Particles Mass Transfer to Small Particles (< 0.6 mm)(< 0.6 mm)

LOW DENSITY SOLIDS OR SMALL GAS BUBBLES IN AGITATED SYSTEMS

50 Mass Transfer to Large Gas Mass Transfer to Large Gas Bubbles or Liquid Drops (> 2.5 mm)Bubbles or Liquid Drops (> 2.5 mm)

positive) (always phases continuous and dispersed the between differencedensity the of magnitude

m/s 9.81

solution the ofviscosity

tcoefficien transfer mass side'-liquid'

42.0

2c

ABc

c

L

3/1

2c

c5.0L

=ρΔ=

=μ

ρμ

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

ρ

ρμΔ= −

g

DSc

k

gSckAERATION OF PURE LIQUIDS IN MIXING VESSELS; SIEVE-PLATE COLUMNS

51 Mass Transfer to Particles in Mass Transfer to Particles in Highly Turbulent MixersHighly Turbulent Mixers

• Turbulent forces become larger than gravitational forces. Mass transfer coefficient is determined primarily by agitation power-input and not by buoyant forces (natural convection).

( )

solution the ofdenstity solution the ofviscosity

tcoefficien transfer mass side'-liquid'

/13.0

c

c

ABc

c

L

3/1

2c

c3/2L

=ρ

=μ

ρμ

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

ρ

μ= −

DSc

k

VPSck

52 InterInter--phase Mass Transferphase Mass Transfer• Mass transfer at gas−liquid or liquid−liquid

interfaces• Consider mass transfer from naphthalene to air,

or from benzoic acid to water– Mass transfer rate is determined by concentration

gradients in air or water

• In mass transfer at interface of two immiscible liquids A and B (or gas A and liquid B)

• Mass transfer rate is determined by concentration gradients on both sides of the interface (in phase A and phase B)

53

Concentration Profile at InterfaceConcentration Profile at Interface

• Consider evaporation of napthalene from surface of a sphere (1 cm dia.) at 45 °C and 1 atm total pressure.

bAA

sAA

A2 0

cc

cc

rcr

r

r

Rr

=

=

=⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

∞→

=

rR

cccc

=−

−∴ b

As

A

bAA

cm 5.0mol/cm 0

mol/cm 10798.23b

A

38sA

=

≈

×= −

Rc

c

54

55 ‘‘TwoTwo--FilmFilm’’ Theory & Overall Theory & Overall Mass Transfer CoefficientMass Transfer Coefficient

gas−liquid interface

Gas phase Liquid phase

Mass transfer from gas to liquid

pAb

pAi

cAi

cAb

56 ( )

( )

( ) ( )bA

iAl

iA

bAgA

l

2b

Ai

Al

2g

2i

Ab

Ag

equal. are interface the across fluxes two Thes

cmtcoefficien transfer mass side-liquid

scmmol side'-liquid' onFlux

scmatmmoltcoefficien transfer mass side-gas

scmmol side'-gas' onFlux

cckppkN

k

cck

k

ppk

−=−=⇒

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛

⋅−=

⎟⎠⎞

⎜⎝⎛

⋅⋅=

⎟⎠⎞

⎜⎝⎛

⋅−=

57

⎟⎠⎞

⎜⎝⎛

⋅

⋅=∴

3

A

iAA

iA

cmatmmol Aspecies for

constant) sHenry' (e.g., constant mequilibriu the is

solution liquid the and mixture gas the between mequilibriu is there interface, the at that, Assume

H

pHc

58

. is phase gas the in A of pressure partial

the whenliquid the in A of conc. saturation ** ionconcentrat Define

1: for Solving

and Now,

bA

A

bAAA

lg

A

bA

bAA

AA

l

AbA

g

AbAA

g

AbA

iA

l

AbA

iA

p

cpHc

kkH

cpHNN

kNc

kNpH

kNpp

kNcc

=

=

+

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟

⎟⎠

⎞⎜⎜⎝

⎛−∴

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

59

additive. are interface an across sresistance transfer mass Thus,

'resistance transfer mass'tcoefficien transfer mass

1

11

11

* t,coefficien transfer mass all'-Over'

1*

lg

A

L

lg

Ab

AA

AL

lg

A

bAA

A

=

+=∴

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=−

=

+

−=∴

kkH

K

kkHcc

NK

kkH

ccN

60 Laminar Boundary Layer on a Laminar Boundary Layer on a Flat Plate (Flat Plate (ReRexx < 5< 5××101055))

x

y

x = 0

v0

Flat plate

0.99v0

δxvx

0.99v0

Velocity Velocity boundary boundary

layerlayer

3

xx0

x5.0

x

x0x 5.05.196.4

⎟⎟⎠

⎞⎜⎜⎝

⎛δ

−⎟⎟⎠

⎞⎜⎜⎝

⎛δ

==δ

μρ

=yy

vv

RexxvRe

61 Laminar Boundary Layer on a Laminar Boundary Layer on a Flat PlateFlat Plate

( )μρ

=⎟⎠⎞

⎜⎝⎛−=

∂∂

=

0x

3/12/1x

sA

bA

0

A where332.0 xvReScRex

ccyc

y

Conc. Conc. boundary boundary

layerlayer

x

y

x = 0

v0

Flat plate

δc

cA

cAs

cAb

62

( ) ( )

3/12/13/1

AB

2/10

AB

c

0xc,c

3/12/1xx

3/1

AB

2/10

AB

xc,

bA

sA

0

AAB

bA

sA

xyA,xc,

664.0664.0

d1 : length and widthof plate flat a

overflow laminar for tcoefficien transfer-mass Mean

332.0332.0

n)(definitio

:by given is tcoefficien transfer mass convection Local

ScReShD

LvD

Lk

xkL

kLW

ScReShD

xvD

xk

cc

ycD

cc

Nk

LL

L

y

=⇒⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ

⎟⎟⎠

⎞⎜⎜⎝

⎛μρ

=⎟⎟⎠

⎞⎜⎜⎝

⎛

=

=⇔⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ

⎟⎟⎠

⎞⎜⎜⎝

⎛μρ

=

−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−

=−

≡

∫

=

63

edcba

−1−2−100−1s2−2−1−311cm011100gm

DABgΔρμρDkc

equations 3 variables, 512

12230

−=−−−=+−−−

=++

edcedcba

dcb

dcedceba

dcb

212123

−−=++=+−

−−=

[ ]( ) ( ) ( ) ( ) ( )eABdcba

c constant DgDk ρΔμρ=

PROBLEM 18

64

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

==⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−++−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−++−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−−

dcdc

d

eba

dcdc

dc

eba

dcdc

dc

eba

21

13

2121

100001213

2121

100231010

BAxAxA

BAx

11

65

[ ]( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ( )

[ ]( ) ( ) ( ) ( ) ( )

[ ] ( ) ( ) ( )

( ) ( ) 2dc2dcAB

d

2dcc2dcdc3d2dc

AB

2dcAB

dcdc3d

AB

c

2dc1AB

dcdc13d

eAB

dcbac

constant

constant

constant

constant

++−−

−−++−−+

−−−−

−−−−−

ρΔ

μρ⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ

=

ρΔμρ=∴

ρΔμρ=

ρΔμρ=

Dg

DD

DgDD

DkDgD

DgDk

66

[ ] ( ) ( ) ( ) ( ) ( )

[ ]

[ ]

2

3

2

32dc

ABAB

c

0AB

d2dd3d2dc

ABAB

c

number Grashof

constant

constant

constant

μρΔρ

==

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

μρΔρ

⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ

=⎟⎟⎠

⎞⎜⎜⎝

⎛

ρΔμρ⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ

=⎟⎟⎠

⎞⎜⎜⎝

⎛

βα

+

−+

gDGr

GrScSh

gDDD

Dk

DgDDD

Dk

d

67 PROBLEM 17

( ) ( )

( )

38

3s

A

sAc

sAc

bA

sAc

cmmol 10797.2

K 4515.273molK

atm cm 06.82

1atm760555.0

0 surface atFlux

−×=

+××⎟

⎠⎞

⎜⎝⎛=

=−≈−=

c

ckckcck

68

503.13552.02

2.70.6 and 48000,1 whengases, For:SPHERE SINGLE A PAST FLOW

514.2

scm 0692.0

scm 0.174

41.172

scm 0.174

scm 30 cm 1

cm 1 Initially,

3/153.0

2

2

ABAB

2

=+=

<<<<

==ν

=ρμ

=

=×

=ν

=μρ

=

=

ScReSh

ScRe

DDSc

DvDvRe

D

69

( ) ( )

smol 10211.8

cm 0.1cmmol 10797.2

scm934.0

s

mol nevaporatio of rate Initial

scm 934.0

cm 1

cm 0692.0503.13

503.13

8

223

8

2sAc

2

ABc

AB

c

−

−

×=

×π×××=

π×=⎟⎠⎞

⎜⎝⎛

=×

=⋅

=∴

==

Dck

sDDShk

DDkSh

70

scm 10267.9

cmmol 1086.8

smol 10211.8

s

cm nevaporatio of rate Initial

36

33

8

m

2sAc

3

−

−

−

×=

×

×=

ρπ×

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

Dck

71

m

sAc

m

2sAc2

m

2sAc

m

2sAc

2

3

dd

4dd4

4dd

dd4

dd

34

ρ−=⇒

ρπ×

−=π∴

ρπ×

−=ρ

π×−=−=

π=

π=

cktR

RcktRR

RckDckQtV

tRR

tV

RV

72

R

DRv

RcD

t

DRv

RcD

tR

DRv

RDSh

RD

DDShk

cktR

d2552.02

2d

2552.0221

dd

2552.0222

dd

33.0

AB

53.0sAAB

m

33.0

AB

53.0

m

sAAB

33.0

AB

53.0ABABAB

c

m

sAc

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ ν⎟⎠⎞

⎜⎝⎛

ν+

⎟⎟⎠

⎞⎜⎜⎝

⎛ ρ−=∴

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ ν⎟⎠⎞

⎜⎝⎛

ν+⋅⎟

⎠⎞

⎜⎝⎛⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛

ρ−=∴

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ ν⎟⎠⎞

⎜⎝⎛

ν+==

⋅=

ρ−=

73

cm 0.5 and cm 1

d2

d2552.02

2d

21

sAAB

m

33.0

AB

53.0sAAB

m

0

1

2

2

1

==

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛ ρ=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ ν⎟⎠⎞

⎜⎝⎛

ν+

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛ ρ−=

∫

∫∫

RR

RShR

cDt

R

DRv

RcD

t

R

R

R

R

t

74 D R Re Sc Sh 2R/Shcm cm cm1.00 0.50 172.414 2.514 13.503 0.07410.98 0.49 168.966 2.514 13.380 0.07320.96 0.48 165.517 2.514 13.257 0.07240.94 0.47 162.069 2.514 13.132 0.07160.92 0.46 158.621 2.514 13.006 0.07070.90 0.45 155.172 2.514 12.878 0.06990.88 0.44 151.724 2.514 12.749 0.06900.86 0.43 148.276 2.514 12.619 0.06820.84 0.42 144.828 2.514 12.487 0.06730.82 0.41 141.379 2.514 12.354 0.06640.80 0.40 137.931 2.514 12.220 0.06550.78 0.39 134.483 2.514 12.084 0.06460.76 0.38 131.034 2.514 11.946 0.06360.74 0.37 127.586 2.514 11.806 0.06270.72 0.36 124.138 2.514 11.665 0.06170.70 0.35 120.690 2.514 11.521 0.06080.68 0.34 117.241 2.514 11.376 0.05980.66 0.33 113.793 2.514 11.229 0.05880.64 0.32 110.345 2.514 11.080 0.05780.62 0.31 106.897 2.514 10.928 0.05670.60 0.30 103.448 2.514 10.775 0.05570.58 0.29 100.000 2.514 10.618 0.05460.56 0.28 96.552 2.514 10.459 0.05350.54 0.27 93.103 2.514 10.298 0.05240.52 0.26 89.655 2.514 10.134 0.05130.50 0.25 86.207 2.514 9.966 0.0502

Integral 0.0157 cm

t 71875.03 s1197.92 min19.97 h

( ) ( ) ( )xyyyyyxy n

x

x

n

Δ⋅⎭⎬⎫

⎩⎨⎧ +++++= −∫ 132n12

1d1

L

75

Evaporation in Stagnant AirEvaporation in Stagnant Air

m

sAAB

m

sAAB2

m

sAAB

2

3

dd

4dd4

4dd

dd4

dd

34

ρ−=⇒

ρπ

−=π∴

=ρ

π−=−=

π=

π=

cDtRR

RcDtRR

RcDQtV

tRR

tV

RV

76

( )

( )

( )( )

days 97.4s 35.429147

cm25.05.0

cmmol10797.2

scm0692.02

cmmol1086.8

2

02

222

38

2

33

22

21s

AAB

m

m

sAAB

2 2

1

==

−

⎟⎠⎞

⎜⎝⎛××⎟⎟

⎠

⎞⎜⎜⎝

⎛×

⎟⎠⎞

⎜⎝⎛×

=

−ρ

=∴

−ρ

−=⎥⎦

⎤⎢⎣

⎡

−

−

RRcD

t

tcDRR

R

77

9/28/079/28/07

• Dimensional analysis– Application in developing correlations

• Correlations for convective mass transfer coefficients

• Mass transfer calculations for packed beds

78

10/3/0710/3/07

• Inter-phase mass transfer– Two-Film theory– Gas-side and liquid-side mass transfer

coefficients– Overall mass transfer coefficient

• Review– Dimensional analysis (HW Problem 18)– Mass transfer coefficient (HW Problem 17)

• Use of correlation

79

HW ProblemsHW Problems

• Mass transfer in packed beds (benzoic acid): Problem 19, HW # 5

• Kinematic viscosity (slide 35): – ν = μ / ρ (cm2/s)– Re = D v ρ/μ = D v / ν– Sc = μ / (ρ DAB) = ν / DAB

• Problem 14: “Write 3 sentences explaining the answers to part 3 of problems 11 and 13.”

80

10/5/0710/5/07

• Inter-phase mass transfer• Review of topics covered• Simple quiz on fundamental concepts?

– Definitions, units, < 5-min problems, etc.

81

ReviewReview

• Slides 42 to 46• Correlations

[ ]

33.083.0023.0:pipes throughflow Turbulent

constant

ScReSh

ScReSh

=

= βα

82

• Quiz–16 questions, < 40 min

–Questions at the beginning require less time than those at the end

• Course evaluation–CH351: Mass Transfer and

Stagewise Operations

–Instructor: Sitaraman Krishnan

83 Benzoic acid spheres in a Benzoic acid spheres in a packed bedpacked bed

cA,2

cA,1Saturation solubility of benzoic acid in water = cA

s

Mass transfer driving force at inlet = cA

s – cA,1

= (2.948×10−5 − 0) mol/cm3

Mass transfer driving force at outlet = cA

s – cA,2

= (2.948×10−5 − cA,2) mol/cm3