bblee@UniMAP

1

Part I

bblee@UniMAP 2

1. Introduction to Mass Transfer and Diffusion

2. Molecular Diffusion in Gasses3. Molecular Diffusion in Liquids

4. Molecular Diffusion in Biological Solutions and Gels

5. Molecular Diffusion in Solids6. Unsteady State Diffusion

Part I

Part II

bblee@UniMAP 3

7. Convection Mass Transfer Coefficients

8. Mass Transfer Coefficients for various geometries

9. Mass Transfer to Suspensions of Small Particles

Part III

Part II

bblee@UniMAP 4

1. Introduction to Mass Transfer and Diffusion1.1 Fick’s law for molecular diffusion

2. Molecular Diffusion in Gasses2.1 Equimolar counterdiffusion in gases2.2 General case for diffusion of gases A

& B plus convection2.3 Special case for A diffusing through

stagnant, nondiffusing B

bblee@UniMAP 5

2.4 Diffusion through a varying cross-sectional area

2.5 Diffusion coefficient for gases3. Molecular Diffusion in Liquids

3.1 Introduction3.2 Equation for diffusion in liquids3.3 Diffusion coefficients for liquids3.4 Prediction of diffusivities in liquids

bblee@UniMAP 6

Mass transfer occurs:

Water evaporates into still air.

Sugar dissolves & diffuses to the

surrounding solution.

bblee@UniMAP7

Mass transfer occurs:

Distillation

Drying

Liquid-liquid extraction

Crystallization

Adsorption

bblee@UniMAP 8

Molecular diffusion (transport): the transfer or movement of

individual molecules through a fluid by means of the random, individual movements of the molecules.

bblee@UniMAP 9

Figure 6.1-1: Schematic diagram of molecular diffusion process.High

concentration

Low concentration

bblee@UniMAP 10

Consider: the diffusion of molecules when the

whole bulk fluid is not moving but is stationary.

due to a concentration gradient. The Fick’s law equation:

dz

dxcDJ A

AB

*

AZ

Total concentration of A &B (kg mol/m3)

The mole fraction of A in mixture of A & B

Distance, m

The molecular diffusivity of the molecule A in B, m2/s

bblee@UniMAP 11

If c is constant, then cA=cxA,

cdxA = d(cxA)=dcA

For constant total concentration:

dz

dcDJ A

AB

*

AZ

bblee@UniMAP12

Example 6.1-1

bblee@UniMAP13

Example 6.1-1

bblee@UniMAP 14

Example 6.1-1

bblee@UniMAP15

2.1 Equimolar Counterdiffusion in Gases Two gases A & B at constant total

pressure P in two large chambers connected by a tube where molecular diffusion at steady state is occurring.

Figure 6.2-1: Equimolarcounterdiffusionof gases A and B

bblee@UniMAP 16

Stirring in each chamber keeps the concentrations in each chamber uniform.

The partial pressure pA1>pA2 and pB2>pB1. Molecules of A diffuse to the right and

B to the left. Since the total pressure P is constant

throughout, the net moles of A diffusing to the right must equal to the net moles of B to the left.

bblee@UniMAP17

This means that

Writing Fick’s law for B for constant c,

Now since P=pA+pB = constant, then

Differentiating both sides,

*

B

*

A JJ

dz

dcDJ B

BA

*

B

BA ccc

BA dcdcdc

BA dcdcdc=0,

bblee@UniMAP 18

Equating the equations,

Finally,

This shows that for a binary gas mixture of A & B, the diffusivity coefficient DAB for A

diffusing into B is the same as DBA

for B diffusing into A.

dz

dcD)(J

dz

dcDJ B

BA

*

BA

AB

*

A

BAAB DD Molecular diffusivity

bblee@UniMAP19

Example 6.2-1

bblee@UniMAP 20

Example 6.2-1

bblee@UniMAP 21

The diffusion flux J*A occurred because of the concentration gradient. The rate at which moles of A passed a

fixed point to the right (positive flux). This flux can be converted:

where vAD is the diffusion velocity of A (m/s).

3

2

m

Akgmol

s

mcv)m.s/Amolkg(J AAD

*

A

bblee@UniMAP 22

Consider when the whole fluid is moving in bulk or convective flow to the right.

The molar average velocity of the whole fluid relative to a stationary point is vM

m/s. Component A is still diffusing to the

right, but now its diffusion velocity, vAd is measured relative to the moving fluid.

bblee@UniMAP23

To a stationary observer A is moving faster than the bulk of the phase, since its diffusion velocity vAd is added to that of the bulk phase vM.

The velocity of A relative to the stationary point is the sum of the diffusion velocity & the average or convective velocity:

Where vA – velocity of A relative to a stationary point.

MAdA vvv

bblee@UniMAP24

Multiplying by cA,

Let N be the total convective flux:

vA

vAd vM

MAAdAAA vcvcvcNA

(kgmol A/s,m2) J*A

BAM NNcvN

bblee@UniMAP25

So,

Since J*A is Fick’s law,

BAA*

AA NNc

cJN

BAAA

ABA NNc

c

dz

dxcDN

BABB

BAB NNc

c

dz

dxcDN

Equimolarcounterdiffusion

NA = -NB

bblee@UniMAP26

In the evaporation of a pure liquid (e.g. benzene (A) at the bottom of a narrow tube, where a large amount of inert or nondiffusing air (B) is passed over the top.

Figure 6.2-2a: Diffusion of A

through stagnant, nondiffusing B: (a)

benzene evaporating into air

bblee@UniMAP 27

The benzene vapor (A) diffuses through the air (B) in the tube.

The boundary at the liquid surface at point 1 is impermeable to air, since air is insoluble in benzene liquid.

Air (B) cannot diffuse into or away from the surface.

At point 2 the partial pressure pA2=0, since a large volume of air is passing by.

bblee@UniMAP28

In the absorption of NH3 (A) vapor which is in air (B) by water. The water surface is impermeable to

the air, since air is only very slightly soluble in water.

Figure 6.2-2: Diffusion of A through stagnant, nondiffusingB. (b) ammonia in air being absorbed into water.

bblee@UniMAP29

thus, since B cannot diffuse, NB = 0.

Keeping the total pressure P constant, substituting

0AAA

ABA Nc

c

dz

dxcDN

Convective flux of A

RT

Pc Pxp AA

P

p

c

c AA

bblee@UniMAP30

Then,A

AAABA N

P

p

dz

dp

RT

DN

dz

dp

RT

D

P

pN AABA

A 1

2

11

2

1

A

A

p

p A

AAB

z

z

APp

dp

RT

DdzN

1

2

12 A

AABA

pP

pPln

zzRT

PDN

bblee@UniMAP31

A log mean value of the inert B is defined:

Then,

2211 BABA ppppP

11 AB pPp 22 AB pPp

12

21

12

12

AA

AA

BB

BBBM

pPpPln

pp

ppln

ppP

21

12

AA

BM

ABA pp

pzzRT

PDN

bblee@UniMAP32

EXAMPLE 6.2-2

bblee@UniMAP33

EXAMPLE 6.2-2

bblee@UniMAP34

EXAMPLE 6.2-3

bblee@UniMAP35

EXAMPLE 6.2-3

bblee@UniMAP36

So far, the cross-sectional area A m2

through which the diffusion occurs has been constant with varying distance z. In some situations the area A may vary.

At steady state, will be constant but not A for a varying area.

A

NN A

A

kg moles of A / s

AN

bblee@UniMAP37

2.4.1 Diffusion from a sphereThe evaporation of a drop of liquid, the

evaporation of a ball of naphthalene, and the diffusion of nutrients to a sphere-like microorganism in a liquid.

Figure 6.2-3a: A sphere of fixed

radius, r1 (m) in an infinite gas

medium.

bblee@UniMAP 38

Since this is a case of A diffusing through stagnant, nondiffusing B:

Note that dr was substituted for dz. Integrating between r1 and some point r2

a large distance away:

24 rπ

N

A

NN AA

A

drPp

dp

RT

D

rπ

NN

A

AABAA

14 2

bblee@UniMAP39

r2>>r1, 1/r2≈0.

drPp

dp

RT

D

r

dr

π

N A

A

p

p A

AAB

r

r

A2

114

2

1

2

1

2

21

11

4 A

AABA

pP

pPln

RT

PD

rrπ

N

BM

AAABA

A

p

pp

RTr

DN

rπ

N 21

1

12

14

bblee@UniMAP 40

If pA1 is small compared to P (a dilute gas phase), pBM≈P,

2r1=D1 (diameter), cA1=pA1/RT

This equation can be used for liquids, where DAB is the diffusivity of A in the liquid.

21

1

1

2AA

ABA cc

D

DN

bblee@UniMAP41

Example 6.2-4

bblee@UniMAP 42

Example 6.2-4

bblee@UniMAP 43

If the sphere is evaporating, the radius r of the sphere decreases slowly with time.

The time it takes for the sphere to evaporate completely can be derived by assuming pseudo-steady state and by equating the diffusion flux equation, where r is now a variable, to the moles of solid A evaporated per dt time and per unit area as calculated from a material balance.

bblee@UniMAP 44

21

2

1

2 AAABA

BMAF

ppPDM

RTprpt

Density of the sphere

Original sphere radius

Molecular weight

bblee@UniMAP 45

2.5.1 Experimental determination of diffusion coefficients

To evaporate a pure liquid un a narrow tube with a gas passed over the top (see Fig 6.2-2a).

The fall in liquid level is measured with time and the diffusivity calculated:

21

2

1

2 AAFA

BMAAB

ppPtM

RTprpD

bblee@UniMAP 46

The common method is the two-bulb method (N1).

Pure gas A is added to V1 and pure B to V2

at the same pressures.

Figure 6.2-4: Diffusivity

measurement of gases by the

two-bulb method.

bblee@UniMAP47

The valve is opened, diffusion proceeds for a given time, and then the valve is closed and the mixed contents of each chamber are sampled separately.

Assumptions:Neglecting the capillary volume &

assuming each bulb is always of a uniform concentration.

bblee@UniMAP48

Quasi-steady-state diffusion in the capillary,

The rate of diffusion of A going to V2 is equal to the rate of accumulation in V2:

L

ccD

dz

dcDJ AB

AB

*

A12

Concentration of A in V2 at

time t

Concentration of A in V1 at

time t

dt

dcV

L

AccDAJ AB*

A2

212

bblee@UniMAP49

The average value cav at equilibrium can be calculated by a material balance from the starting composition c1

0 and c20 at

t=0:

A similar balance at time t gives,

0

22

0

1121 cVcVcVV av

221121 cVcVcVV av

bblee@UniMAP 50

Rearranging & integrating between t=0 & t=t,

if c2 is obtained by sampling at t, DAB

can be calculated.

12

21

0

2

2

VVAL

VVDexp

cc

cc AB

av

av

bblee@UniMAP51

Some typical data are given in Table 6.2-1, Perry & Green (1984) and Reid et al., (1938).

Table 6.2-1: Diffusivity coefficient of gases at 101.32 kPa Pressure

bblee@UniMAP52

The diffusivity of a binary gas mixture in the dilute gas region, that is, at low pressure near atmospheric, can be predicted using the kinetics theory of collision with another molecule, which implies that momentum is conserved.

bblee@UniMAP53

The final relation for predicting the diffusivity of a binary gas pair of A & B molecules is:

21

2

237 111085831/

BAAB,DAB

/

ABMMζP

Tx.D

Ω

Average collision diameter

A collision integral based on the Lennard-Jones

potential

bblee@UniMAP 54

The above equation is relatively complicated to use (σAB is not available or are difficult to estimate).

23131

217517 1110001/

B

/

A

/

BA

.

ABννp

MMTx.D

ΣΣ

Sum of structural volume increments

Note: DAB α 1/P, DAB α T1.75, DAB α T1.75/P

bblee@UniMAP55

Table 6.2-2: Atomic diffusion volumes for use with Fuller et al., method.

bblee@UniMAP56

Example 6.2-5

bblee@UniMAP57

Example 6.2-5

bblee@UniMAP58

3.1 IntroductionDiffusion of solutes in liquids is

important in many industrial processes:

Solvent extraction

Gas absorption Distillation

bblee@UniMAP 59

Diffusion in liquids also occurs in nature:

Oxygeneration of rivers and lakes by

the air

Diffusion of water in blood

bblee@UniMAP 60

It should be apparent that the rate of molecular diffusion in liquids is considerably slower than in gases. The molecules of the diffusing solute

A will collide with molecules of liquid B more often and diffuse more slowly than in gasses.

The diffusion coefficient in a gas will be on the order of magnitude of about 105 times greater than in a liquid.

bblee@UniMAP 61

The flux in a gas is being only 100 times faster, since the concentrations in liquids are

considerably higher than in gases.Since the molecules in a liquid are

packed together much more closely than in gases, the density & resistance to diffusion in a liquid are much greater. the attractive forces between

molecules play important role in diffusion.

bblee@UniMAP62

Diffusion of liquids: the diffusivities are often dependent on the concentration of the diffusing components.

3.2.1 Equimolar counterdiffusion An equation similar to gases at steady

state [NA=-NB]:

12

21

12

21

zz

xxcD

zz

ccDN AAavABAAAB

A

Kg mol A/s.m2 m2/s

Concentration of A (kg mol A/m3)

at point 1.

Mole fraction of A at point 1.

bblee@UniMAP63

cav defined by

The average value of DAB may vary some with concentration, &

the average value of c may vary with concentration Linear average of c is usually used.

22

2

1

1

M

ρ

M

ρ

M

ρc

av

av

Average total concentration of A+B (kg mol/m3)

Average molecular weight of the solution at point 1

(kg mass / kg mol)

Average density of

the solution(kg/m3)

bblee@UniMAP64

A dilute solution of propionic acid (A) in a water (B) solution being contacted with toulene.Only the propionic acid (A) diffuses

through the water phase (B), to the boundary & then into the toluene phase.The toulene-water interface is a

barrier to diffusion of B and NB = 0. Substituting

RT

Pcav

RT

ρc A

A1

1P

px BM

BM

bblee@UniMAP65

For liquids at steady state:

Note xA1+xB1=xA2+xB2=1.0 For dilute solution, xBM is close to 1.0 c is essentially constant.

21

12

AA

BM

avABA xx

xzz

cDN

)xxln(

xxx

BB

BBBM

12

12

12

21

zz

ccDN AAAB

A

bblee@UniMAP66

Example 6.3-1

bblee@UniMAP67

Example 6.3-1

bblee@UniMAP68

3.2.3.1 Experimental determination of diffusivities

A relatively dilute solution & a slightly more concentrated solution are placed in chambers on opposite sides of a porous membrane of sintered glass (see Fig6.3-1).

Figure 6.3-1: Diffusion cell for determination of diffusivity in a

liquid

bblee@UniMAP69

Quasi-steady-state diffusion in the membrane is assumed:

ηδ

'ccDεN ABA

Concentration in the lower chamber at a time, t

Concentration in the upper chamber

Fraction of area of the glass open to diffusion

bblee@UniMAP 70

Combining & integrating:

tDVηδ

Aεln AB'cc

cc ' 200

Initial concentrations

Final concentrationsCell constant

Note: For liquids, unlike gases, the diffusivity DAB does not equal DBA.

bblee@UniMAP71

Table 6.3-1: experimental diffusivity data for binary mixtures in the liquid phase are given.

The diffusivity values are quite small and in the range of about 0.5x10-9 to 5x10-9

m2/s for relatively nonviscous liquids. Diffusivities in gases are larger by a

factor of 104-105.

bblee@UniMAP72

Table 6.3-1: experimental

diffusivity data for binary

mixtures in the liquid phase are

given.

bblee@UniMAP73

The Stokes-Einstein equation was derived for a very large spherical molecule (A) diffusing in a liquid solvent (B) of small molecules.Stokes’ law was used to describe the

drag on the moving solute molecule.

31

1610969/

A

ABVμ

Tx.D

Viscosity, Pa.s or kg/m.s

Solute molar volume at its normal boiling

point, m3/kgmol

KDiffusivity, m2/s

bblee@UniMAP74

The Wilke-Chang correlation can be used for most general purposes, where the solute (A) is dilute in the solvent (B).

60

2116101731.

AB

/

BABVμ

TMθx.D

Molecular weight of solvent B

Viscosity of B, Pa.s or kg/m.s

Solute molar volume at the boiling point (Table 6.3-2)

Association parameter

of the solvent

bblee@UniMAP75

The association parameter (φ):

φ φ

Water 2.6 Benzene 1.0

Methanol 1.9 Ether 1.0

Ethanol 1.5 Heptane 1.0

Unassociated solvents

1.0

bblee@UniMAP76

Example 6.3-2

bblee@UniMAP77

Example 6.3-2