2. Principles of Mass Transfer

CHG 3111

Unit Operation

Principles of Mass Transfer

Text Book: Chapters 6 & 7

CHG 3111/B. Kruczek 2

Introduction Basic definitions:

Mass transfer is a rate process that involves net movement of a component in a mixture from one location to another driven by a concentration gradient.

d riv in g fo rce

ra te o f a tran sfer p ro cessresistan ce

Mass transfer versus bulk fluid motion (convective flow)

• Bulk fluid motion (e.g. the flow of a fluid through a pipe, or a motion of air induced by a fan) is not considered as mass transfer

• However, mass transfer can be superimposed with bulk fluid flow

Analogy between mass and heat transfer

• Mass transfer by diffusion is analogous to conduction heat transfer

• Mass transfer by convection is analogous to heat transfer by forced convection


Steady State Diffusion Fick’s Law of diffusion:

Formulated in 1855 by Fick as an extension of Fourier’s law of conduction from 1822.

z

dTq kdz

• Fourier’s law: where: qz, k, T, z are heat flux, thermal conductivity,

temperature, direction of the transport

and * *A BAz AB Bz BA

dc dcJ D J D

dz dz

• Fick’s law (for a binary mixture of A and B):

• Molecular diffusion: random-walk process, which

yields a mean-square distance of travel for a given

time interval, but not the direction of travel

• Diffusion in the direction of decreasing

concentration (B to left, A to right), but the number

of molecules crossing a given plane in both

direction is the same


Steady State Diffusion Equimolar Counterdiffusion in Gasses

Consider two large, well-stirred reservoirs containing ideal mixtures of gases A and B at constant total pressure connected by a tube where steady state molecular diffusion occurs.

• Since the total pressure (P = pA + pB) is constant: * *A BJ J and where: A B

Pc c c cRT

• Also, since c is constant: thus: * *A B AA B Az AB B BA BA

dc dc dcdc dc J D J D D

dz dz dz

Consequently: AB BAD D

• In addition, partial pressures pA and pB

change linearly with z, thus:

1 2 1 2= = * A A A AAB

A AB

c c p pDJ D

z RT z

where: L is the length of tube


Steady State Diffusion General Case: Diffusion of Gases A and B Plus Convection

Consider the case when the whole fluid is moving in bulk (convective flow), but in addition there is a concentration gradient within the moving fluid

NB 1: Do not confuse

NB 2: The expression for diffusion of gases A and B plus convection holds for diffusion of liquid or solid.

• In general,

is the velocity of A relative to a stationary point

bulk velocity

diffusive velocity of A relative to the moving fluid

A

M

Ad

.A Ad M Multiplying both sides by cA: *A A A Ad A M A A A Mc c c N J c

• Since:

A BM A B M

N NN c N N

c

• Consequently: A A AA AB A B A AB A A B

dc c dxN D N N N cD x N N

dz c dz

• To solve the above ordinary differential equation (ODE), the relation between flux NA and

NB must be known.

• For the special case of equimolar counterdiffusion, NA = -NB, thus: *A AN J

with *A AN J


Steady State Diffusion Special Case: Diffusion of A through Stagnant, Nondiffusing B

Consider the following physical situation

• Evaporation of a pure liquid (A) at the bottom of a narrow tube,

where large amount of inert air (B) is passed over the top of

the tube, i.e., pA2 = 0

• Assuming that air is insoluble in benzene, the liquid surface at

point 1 is impermeable to air; consequently, air is considered

to be stagnant and nondiffusing, i.e., NB = 0

• Consequently: 0 A AB A AA AB A A A A

dx D dp pN cD x N N N

dz RT dz P• Rearranging and integrating:

2

1

2

1

11

z pAA AB A AB A

A Az pA A

p D dp D dpN N dz

P RT dz RT p P 2

2 1 1

lnAB A

A

A

D P P pN

RT z z P p

• Alternatively,

1 2

2 1

AB

A A A

BM

D PN p p

RT z z p

NB: pMB is the log mean pressure of inert component B in the tube.

Question: What is the partial pressure profile in the tube, i.e. pA = f(z)? How can you determine it?

2 1 2 1

2 1 2 1

and B B A ABM

B B A A

p p P p P pp

p p P p P p

ln ln


Steady State Diffusion Diffusion coefficients for gases

Kinetic theory of gases

• Two molecular species A and A* having the same mass and the same size and shape, at low

(near atmospheric) pressures.

• If the molecules are rigid and spherical particles (dA) and their collisions are completely elastic,

neglecting attractive and repulsion forces, the diffusion coefficient (self diffusion coefficient)

DAA* is given by:

• Substituting expressions for the molecular speed and mean free path and using ideal gas law to

express n leads to:

1 2 2

23

A

1 3

: = 8 is the mean molecular speed; 1 2 is the mean free path,

in which = N is Boltzmann constant ( =1.38 10 J/

AA

A A

D u

where u T M d n

R

*

/

: AK) and N is the Avogadro's number, and is number densityn

1 23 3 2

3 2

2

3AA

A A

TDM Pd

*

• This approach can be extended into the situation when the molecular species are different, i.e.

to a binary mixture of gases A and B



Kinetic theory of gases

• More rigorous approach – considering intermolecular forces of attraction and repulsion

approximated by Lennard-Jones function

• Use of distribution function instead of mean free path (Chapman and Enskog approach).

• For a pair of nonpolar molecules A and B:

1 27 3 2

2

1 8583 10 1 1AB

AB D AB A B

TDP M M

,

.

where: AB is an “average collision diameter”, D,AB is a collision integral based on the Lennard-Jones

potential, and DAB in [m2/s], T in [K], P in [atm], MA and MB are the molecular weight of A and B in [kg/kmol]

• The collision integral (D,AB) is a ratio showing deviation of a gas with interactions compared to a

gas of rigid, elastic spheres (value of 1.0 indicates a gas with no interactions)

• Effect of concentrations of A and B on DAB is rather weak (less than 4%), and thus can be

neglected.

• Prediction of DAB from first principles is difficult to apply in practice, because the constants AB

and D,AB are rarely available.



Semi-empirical approach – method of Fuller

• Applicable for mixtures of nonpolar gases or for a polar- nonpolar mixture

1 27 1 75

21 3 1 3

1 00 10 1 1A BAB

A B

T M MD

P

/.. / / and DAB in [m2/s], T in [K], P in [atm], MA and

MB in [kg/kmol]

where: 𝜐i is the sum of structural volume increments which can be evaluated from:


Steady State Diffusion Diffusion coefficients for gases – empirical values at 1 atm

• If DAB is required at T and P different from those in Table 6.2-2, it can be evaluated using the

correction arising from the Fuller equation, i.e.: 1 75 ABD T P .


Steady State Diffusion Diffusion coefficients in liquids

Empirical equation of Wilke-Chang

• There is no rigorous theory of diffusion in liquids

• Unlike the binary gas mixtures, in liquid solution DAB depends on concentration of solute in

solvent; in the limiting case of dilute solutions:

1 216

0 61 713AB B

B A

TD MV

.

.

where: DAB is in [m2/s], MB are the molecular weight of B in [kg/kmol], T is the absolute temperature in [K], B is

the viscosity of B in [Pa s], VA is the solute molar volume in [m3/kmol], and is an “association parameter” of the

solvent, where is 2.6 for water, 1.9 for methanol, 1.5 for ethanol, 1.0 benzene, ether, heptane and other

unassociated solvents.

• If water is the solute, the above equation must be multiplied by 0.435

• For high molecular weight solutes (VA > 0.5 m3/kmol) Stokes-Einstein equation should be used:

16

1 3

9 96 10AB

A

TDV

/

.

• For low molecular solutes (VA < 0.5 m3/kmol), the solute molar volume can be can be

determined based on atomic volumes of the atoms of which the molecule is made of, or directly

from a table shown on the next slide.


Steady State Diffusion Diffusion coefficients in liquids

Empirical equation of Wilke-Chang

• There is no rigorous theory of diffusion in liquids

• DAB predicted using Wilke-Change equation deviate up to 10 - 15% for aqueous solutions and

up to 25% for non-aqueous solutions


Steady State Diffusion Diffusion coefficients for dilute liquids – empirical values

• Diffusivities in liquids are 4-5 orders

of magnitude lower than viscosities

of gases, but the corresponding

diffusions rates in liquids are only 1-

2 orders of magnitude. Why?

• Diffusivities in liquids are practically

independent on the liquid pressure.

• Dependence on temperature is

stronger than a direct proportionality,

because the viscosity of liquids is

greatly affected by temperature.


Steady State Diffusion Example 1 – Equimolar counterdiffusion

Two bulbs A and B of the same volume are connected by a tube of length 5 cm and the internal diameter of 0.5 cm. Initially, when bulb A contains pure helium while bulb B pure methane, the molar flow rate of helium from bulb A to bulb B is 4.33 x 10-6 mol/s. The temperature and the total pressure in both bulbs are the same and equal to 25oC and 1 atm., respectively.

a) What is the initial molar flux of methane from bulb B to bulb A?

b) What is the diffusivity coefficient of helium in methane at the above conditions?

c) What will be the partial pressure of helium in the connecting tube 4 cm from bulb A when the when the partial pressure of helium in bulb A is 0.8 atm?

d) If the volume of each bulb is 0.001 m3, how long will it take for the partial pressure of helium in bulb A to drop from 1 atm to 0.8 atm?


Convective Mass Transfer Analogy with convective heat transfer

Transfer of a component from a moving fluid to a surface of solid or a surface of another fluid which is immiscible with the moving fluid.

Question: Is it physically possible for ci to be greater than cLi? (case in Fig (c) above)?

Moving fluid

• Rate equation in convective mass transfer equivalent to Newton’s cooling law:

1 where: is a convective mass transfer coefficient A c L Li cN k c c k

Equilibrium distribution coefficient (K) between moving fluid and a surface:

1

Li

L

cK

c

• In heat transfer, unless there was a contact resistance, the temperature profile in the system

would be continues

• In mass transfer, unless K = 1, the concentration profile have discontinuity


Convective Mass Transfer Types of Mass Transfer Coefficients

Mass transfer coefficient for equimolar counterdiffusion (NA = -NB)

Mass transfer coefficient for A diffusing through stagnant, nondiffusing B (NB = 0)

1 2 'A AA AB M A A B AB M A c A A

dx dxN c D x N N c D N k c c

dz dz

• Since the driving force can be expressed not only in terms of concentration difference, the rate

equation can be written in alternative forms:

1 2 1 2 1 2Gases: ' ' 'A c A A G A A y A AN k c c k p p k y y

1 2 1 2 1 2Liquids: ' ' 'A c A A L A A x A AN k c c k c c k x x

1 2 1 2 1 2

2 1

'AB M c

A A A A A c A A

BM BM

D kN c c c c k c c

z z x x 2 1

2 1

where: and

ln

B BBM

B B

x xx

x x

'c

c

BM

kk

x

1 2 1 2 1 2Gases: A c A A G A A y A AN k c c k p p k y y

1 2 1 2 1 2Liquids: A c A A L A A x A AN k c c k c c k x x

NB: Do not confuse with 'k k


Convective Mass Transfer Summary of Flux Equations and Mass Transfer Coefficients


Convective Mass Transfer Coefficient Correlations for mass transfer coefficients

Dimensionless groups to correlated data

Chilton and Colburn J-factor analogy

NB: The properties in dimensionless groups (viscosity and density) are of the flowing mixture, which for

diluted solutions are those of pure fluid B.

Reynolds number: where: is the characteristic length

Re

LN L

Question: What are the heat transfer equivalents of Schmidt and Sherwood numbers?

2 3 2 3

1 3

' '/ /

/

Re

....c G ShD Sc Sc

M Sc

k k P NJ N N

G N N

Sherwood number: '

' ' .....xSh c c BM

AB AB

kL LN k k yD c D

Schmidt number:

Sc

AB

ND

Stanton number: where:

'' '

.......yc G

St M

M M av

kk k PN G c

G G M

• Mass transfer coefficient is often correlated as a dimensionless JD factor


Convective Mass Transfer Coefficient

Mass transfer in flows inside pipes

Driving force for mass transfer in internal flows

Laminar flow, i.e.,

0

0

ln

Ai AL Ai AAi A LM

Ai AL Ai A

c c c cc c

c c c c

• Concentration of species A, which being transferred to, or from the moving fluid, can be

assumed constant at the inner surface of the pipe (cAi)

• Since the concentration of A in the fluid entering the pipe (cA0) is different from cAi, the

concentration of A in the fluid leaving the pipe (cAL) must also be different from cAi

• It can be shown that average driving force for mass transfer of A is given by the log-mean-

concentration difference (cAi – cA)LM

2100

Re, :D

DN

2 3

0 06683 66

1 0 04

'Re

Re

..

.

SccSh

AB Sc

N N x Dk DN

D N N x D

• For fully developed flow, i.e. as x/D 3 66 .ShN :

Turbulent flow:

2100 and 0.5 < < 3000:Re,D ScN N 0 83 0 33

0 023. .

Re.Sh ScN N N• For


Convective Mass Transfer Coefficient Correlations for external mass transfer

Mass transfer in flow parallel to flat plates

• For 15 000: Re, ,LN 0 5 0 50 0 33

0 664 0 664

'

. . .

Re, Re,. .cD L Sh L Sc

AB

k LJ N N N N

D

• For 15 000 300 000: Re,, ,LN 0 2 0 80 0 33

0 036 0 036

'

. . .

Re, Re,. .cD L Sh L Sc

AB

k LJ N N N N

D

Mass transfer for flow past single spheres

• For gases with 1 48 000 and 0 6 2 7 Re, , . . :D ScN N 0 53 0 33

2 0 552 . .

, Re,.Sh D D ScN N N

• For liquids with 2 2 000: Re, ,DN 0 50 0 33

2 0 95 . .


• For liquids with 2 000 17 000: Re,, ,DN 0 62 0 33

2 0 95 . .


0 6 2 6 . .ScN

Mass transfer for flow past single cylinders

• For gases with and liquids with 1 000 3 000 , , :ScN 0 487

0 600

.

Re,.D DJ N


Convective Mass Transfer Coefficient Mass transfer in packed beds

Basic terms an definitions

• Void fraction or bed porosity:

Correlations for mass transfer coefficient

’, cA1

’, cA2

, Dp, a

cAi = const

Re 'pN D

• Diameter of particles in the bed (Dp). If bed is made of non-spherical particles,

Dp is the diameter of sphere having the same surface area as the particle

• Specific surface (a) is the total surface area of the total bed volume in [m2/m3]

and: 6 1 pa D

• Superficial velocity (’) is the empty bed velocity, which is less than the

actual velocity in the bed (),

• Flow regime is determined by Reynolds number:

'

total solids totalV V V

• The average driving for mass transfer is the

log mean concentration difference:

0

0

ln

Ai AL Ai AAi A LM

Ai AL Ai A

c c c cc c

c c c c

• For liquids in static bed with 00016 55 and 165 70 000 Re, , :D ScN N 2 31 09 /Re.DJ N

• For liquids in static bed with 55 1 500 and 55 10 690 Re, , , :D ScN N 0 310 25 .Re.DJ N

• For static bed with and fluidized bed with 10 10000 ReN 0 40690 4548 .Re.DJ N10 4000: ReN

• For liquids in fluidized bed with 1 10 Re, :DN 0 721 1068 .Re.DJ N


Convective Mass Transfer Coefficient Example 2 – Laminar and turbulent flow in circular tube

Water at 25oC flows at 2 m/s through a straight circular tube cast from benzonic

acid of 1 cm inside diameter. The solubility and diffusivity of benzonic acid in water

are 0.0034 g/cm3, and 9.18 x 10-6 cm2/s, respectively, while the water viscosity is

8.9 10-4 kg/m s.

a) If the water entering the tube has the concentration of benzonic acid of

0.00002 g/cm3, what should be the length of the tube for the concentration

at the tube exit of 0.0002 g/cm3? What is the corresponding average mass

transfer coefficient?

b) Recalculate the required length of the tube to and the average mass transfer

coefficient if the velocity of water were 0.02 m/s.

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2. Principles of Mass Transfer