ke-42 3200 mass transfer

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Mass transfer 1

KE-42.3200 Fundamentals of separation processes

Mass transfer

1 Unified treatment of mass, heat and momentum transfer ...................... 2

1.1 Basic equations ................................................................................................................. 2

2 Mass transfer as a term in the material balance ..................................... 4

3 Mechanisms of mass transfer ................................................................ 4

3.1

Molecular diffusion and convective mass transfer ............................................................. 5

3.2 Fick’s law ......................................................................................................................... 5

4 Diffusion coefficients ........................................................................... 6

4.1

Gas mixtures ..................................................................................................................... 6

4.1.1

Example..................................................................................................................... 6

4.2 Liquids.............................................................................................................................. 7

5 Mass transfer in binary mixtures ........................................................... 7

5.1

Equimiolar mass transfer ................................................................................................... 8

5.2 Mass transfer through a stagnant phase ............................................................................. 9

5.2.1

Example................................................................................................................... 11

5.2.2

Example................................................................................................................... 12

5.3

Mass transfer when the flux ratios are known .................................................................. 13

6 Diffusion in solid materials ................................................................. 13

6.1

Diffusion in solid materials according to the Fick’s law .................................................. 14

6.1.1 Example................................................................................................................... 14

6.2

Diffusion in porous solid materials when structure is important....................................... 15

6.3

Knudsen diffusion ........................................................................................................... 15

7 Mass transfer in separation processes ................................................. 16

8 Two-film theory .................................................................................. 16

9 Mass transfer coefficients ................................................................... 18

9.1 Mass transfer coefficients according to the film model .................................................... 18

9.2

Overall mass transfer coefficient ..................................................................................... 20

10 Penetration and surface renewal theories ............................................ 22

11 Temperature effect to mass transfer .................................................... 23

12 Determination of mass transfer coefficients ........................................ 24

12.1

Mass transfer for a liquid film falling inside at tube ..................................................... 25

12.2 Flow around spherical particles ................................................................................... 25

12.3

Mass transfer in packed beds ....................................................................................... 25

12.4 Volumetric mass transfer coefficient ........................................................................... 26

12.5

Analogy between heat and mass transfer...................................................................... 26

13 Stage efficiencies ................................................................................27

14 Number of transfer units ..................................................................... 29

15 Symbols .............................................................................................. 30



Mass transfer 2

1 Unified treatment of mass, heat and momentum transfer

The most important transport phenomena in chemical engineering are mass, heat and momentum

transfer.

Momentum transfer is important e.g. when liquid, gases or various suspension flows are analyzed.

Analyzed equipment could be pumps, compressors, pipes, mixers, settling vessels, filters etc. Inmany practical applications, the actual momentum balance is not written, but the flow is analyzed

based on mechanical energy balances and pressure drop correlations. Momentum balances are needed

when velocity profiles need to be calculated.

Heat transfer mechanisms are conduction, convection and radiation. Depending on the process, any

one or several of these may be significant. Examples of separation processes where heat transfer is

important are distillation, evaporation and drying.

In separation processes, components that are initially mixed are separated into more concentratedstreams. Then components must be physically separated e.g. into different phases or different sides of

a membrane. Mass transfer is therefore essential in all separation processes, such as in distillation,absorption, drying, extraction, membrane operations etc. It has often significant role also in chemical

reactions, where the observed rate may be limited by mass transfer, not intrinsic chemical kinetics.

1.1 Basic equations

All three transport phenomena can be described with the same basic equation. Flux is the transferredmomentum, mass or energy divided by the cross-sectional area and time. Driving force is the gradient

of the transferring entity, and the resistance is the factor that hinders transfer.

cetanresis

forcedrivingFlux

Usually this is presented in a slightly different form:

Flux = transfer coefficient · driving force (1.)

so that the transfer coefficient is inverse of the resistance. Next the basic equations are shown for

momentum, heat and mass transfer.

Molecular momentum transfer can be described by the Newton’s law of viscosity

dz

dv xzx , (2.)

where

zx = momentum transfer, transfer of x-directional momentum into z-direction (N/m2)

= dynamic viscosity (Pas)

vx = flow velocity in x-direction (m/s)z = distance (m)

Here the transfer coefficient is viscosity, and driving force is velocity gradient (shear rate)



Mass transfer 3

For molecular heat transfer, the Fourier’s law of heat conductivity is used

dz

dTq (3.)

q = heat flux (W/m

2

) = heat conductivity (W/(mK))

In heat transfer, the transfer coefficient is thermal conductivity, and the driving force is temperature

gradient.

For molecular mass transfer, the Fick’s law is often used

dz

dcDJ A

ABA , (4.)

whereJA = component A diffusion flux (mol/(m

2s))

DAB = diffusion coefficient of A in B (or a mixture of A and B) (m2/s)

c = concentration (mol/m3)

For diffusion, the transfer coefficient is the diffusion coefficient, and the driving force is

concentration gradient. The diffusion equation can also be written so, that the total concentration is

included in the transfer coefficient so that mole fraction gradient remains as the driving force (c A =

ctot·xA). Sometimes the driving force is expressed in terms of chemical potential gradient. This way is

more tedious to use than the Fick’s law, and not considered in this course.

In film theories, the film thickness is included in the transfer coefficient (mass or heat transfer

coefficient), so that only temperature or concentration difference remains as the driving force (not

their gradient). Film theory is discussed in more detail later in this text.

For turbulent flows, so called turbulent viscosity, heat transfer coefficient and diffusion coefficientcan be used similarly than the molecular transfer coefficients above. Molecular transfer is based on

the random movement of individual molecules, causing leveling out of any systematic differences inmolecule states. Some examples of these differences are molecules moving faster to a certain

direction (momentum transfer), generally more rapidly moving molecules (heat transfer), orstructurally different molecules (mass transfer, diffusion). In turbulent flow, also turbulent eddies

move randomly besides individual molecules, carrying material to different locations and thusleveling out local differences. Turbulent transfer coefficients are not material properties, but depend

on the turbulence level of the bulk flow. Mass, heat and momentum transfer by turbulence then occur

with the same mechanism.

dz

dv)( x

Tzx (5.)

dz

dT)(q T (6.)

dz

dc)DD(J A

TAB (7.)



Mass transfer 4

Turbulent contribution to the transfer rate is often substantially higher than molecular. This is not true

in the vicinity of various surfaces or interfaces, such as surfaces of pipes, particles, bubbles or

droplets. These surfaces dampen turbulence so that the molecular mechanism remains the only

significant factor.

2 Mass transfer as a term in the material balance

In material balance, the rate of change of a certain component is affected by the flow rates in and out

from the control volume, mass transfer, and reaction. Actually mass transfer is just one type of flow, but it is usually expressed as a separate term in the material balances. In the following, a time

dependent one phase material balance is shown for a component A and control volume V. Thiscontrol volume may be a part of a separation process.

Vr V Nnndt

dnAAout,Ain,A

A a (8.)

The mass transfer direction is here chosen so, that it is positive if material is transferred to the phase

for which the balance was written. For the other phase, term NaV would be negative. Mass transfer

rate is consisted of two terms, mass transfer flux N and mass transfer area, which is further divided

into specific surface area a and the control volume V.

Reaction rate (production rate of the component) r is written here relative to the volume of the

considered phase, mol/m3s. Volume is a product of volume fraction and total control volume V.

This course deals mainly with nonreactive systems, so for now the last term can be neglected.

Mass transfer is therefore a product of two terms, mass transfer flux and mass transfer area. Masstransfer area is typically obtained from the geometry of the system of from various correlations. Mass

transfer flux is obtained from the phase equilibrium, material balances and the mass transfer models

discussed in this hand-out.

3 Mechanisms of mass transfer

Mass transfer occurs whenever there are composition differences in the system. It is important in

various technical applications and also around us in the nature. For example water evaporates into

stagnant air since the water molecules in the liquid move to the air above the surface. If there is less

water in the air further away from the surface, i.e. air is less moist, the water molecules aretransferred from the vicinity of the surface to further away. Then more molecules are transferred

from the liquid water to the air above the surface than from the air back to the liquid, resulting in a

net evaporation of the water. A sugar cube in a cup of coffee or tea first dissolves to the liquid near

the surface of the sugar crystal and then diffuses further away. Several materials, e.g. wood, dry since

water diffuses from the core of the wood logs to the surface, and then diffuses as water vapor. In

fermenters, nutrients and oxygen first dissolve and then diffuse to the micro-organisms. In catalytic

reactions the reagents first diffuse to the active sites of the catalyst and the products back to the

surface of the catalyst and then further to the surrounding fluid.

In almost all separation processes, the components move with diffusive mechanisms. In separation

processes, the phases between which the separation occurs are brought into contact in such a way that



Mass transfer 5

the desired component moves with diffusion to the desired direction. The better is the contact, the

faster is the separation.

3.1 Molecular diffusion and convective mass transfer

The molecular diffusion is considered to occur due to random movement and collisions of molecules.Molecules move in such a way that the average concentration differences tend to level out.Convective mass transfer is the movement of molecules with the surrounding fluid.

If a droplet of dye is put into stagnant water, it starts to diffuse slowly, and the color changes in the

water tend to level out. This process can be speeded up with mechanical stirring, in which case also

convective mass transfer occurs.

Mass transfer is then a sum of the two terms described earlier: convective transfer along with the

flow, and diffusive movement, i.e. random movement that tends to level out differences. Mass

transfer flux is therefore movement with respect to the chosen coordinate system (often fixed to the

equipment or to an interface), and diffusion is movement with respect to the average flow.

Mass transfer is written as the sum of these two terms. Diffusion flux is described by the symbol J A

and convective flux by xA N.

NxJ N AAA (9.)

The convective part, i.e. flow that carries material, is the sum of individual component mass transfer

fluxes:

BA N N N (10.)

Mass transfer due to convection and diffusion are independent of each other. Both may occur

simultaneously, or just one of them.

Mass transfer flux NA can be determined only by an additional condition that determines the total

flux. This can be e.g. total amount condensed or evaporated, and calculated with the help of energy

balance and phase equilibrium.

3.2 Fick’s law

Let us first focus on diffusion, i.e. movement of molecules with respect of the average flow.

Diffusion flux occurs due to component concentration differences. The general form of the Fick’s

law is the following

dz

dxcDJ A

ABAB (11.)

If the total molar density (total concentration) c is constant, equation (5) follows immediately. To be

precise, the previous equation where the driving force is the mole fraction gradient describes

diffusion with respect to the average molar flow, and the “traditional” form described earlier(concentration gradient as the driving force), describes diffusion with respect to the average volumeflow. If the total concentration is constant, these two are the same.



Mass transfer 6

Other driving forces for diffusion are temperature, pressure, electrical potential and chemical

potential. In mass transfer context, the thermodynamic state is usually described with chemical

potentials, not fugacities. Chemical potential is more convenient in describing mass transfer since

diffusion is actually directly proportional to the gradient of chemical potential.

In practice, accounting for chemical non-idealities in mass transfer calculations of separation

processes is not very common. If the non-ideality is neglected in diffusion, the driving force reduces

to the mole fraction gradient as shown above. In this course, the non-idalities are neglected in mass

transfer calculations, although these must be accounted in the phase equilibrium calculations.

4 Diffusion coefficients

4.1 Gas mixtures

Binary diffusion coefficients of gaseous mixtures are measured widely, so values for any design

problem may be found in the literature. If this is not the case, or if the reported diffusion coefficientsare measured at very different temperatures than needed in the separation process model, the

coefficients need to be estimated. There are several correlations available for estimating gaseous

mixture diffusion coefficients. Typical gaseous diffusion coefficients at atmospheric pressures and

typical measurement temperatures 0 – 100 oC are around 0.05 – 1 cm2/s.

Note that the diffusion coefficient is not a property of certain chemical, but always a property of a

pair of components. Diffusion coefficient can be considered to be a certain kind of inverse friction

factor between the components. The larger is the diffusion coefficient, the smaller is the “friction”.

Diffusion coefficients in gaseous mixtures depend only slightly on mixture composition, but morestrongly on pressure. The diffusion coefficients also depend rather heavily on temperature.

4.1.1 Example

Mixture of helium and nitrogen is in a 20 cm long tube. Temperature is 25 C and pressure 1 atm. At

one end of the tube the partial pressure of helium is 0.6 atm and at the other end 0.2 atm. Estimate

helium flux. Diffusion coefficient of helium in the mixture is 0.69 ·10-4 m2/s.

The mixture can be assumed to be ideal gas. Concentration is then

RT

p

V

nc ,

where n is the total amount of moles, V is volume, T is temperature and R is the gas constant.

since p and T are constants, also the mixture total concentration is constant. Integrating the Fick’s

law we get

2

1

2A

1A

z

z

c

cAAB dcDdzJ



Mass transfer 7

12

2A1AAB

zz

)cc(DJ

From the ideal gas law it follows

RT

p

V

n

c1AA

1A

2

3524

12

2A1AAB

ms

mol106.5

m2.0K 298J314.8s

K molPa10)2.06.0(m1069.0

)zz(RT

) p p(DJ

4.2 Liquids

Typically diffusion coefficients in gases are approximately 105 times higher than diffusion

coefficients in liquids. Liquid diffusion coefficients also depend more strongly on composition.Typical liquid diffusion coefficients at most common measurement temperatures are around 0.1 –

310-9 m2/s.

One out of several estimation methods for liquid diffusion coefficients is the Wilke-Chang equation

6.0

3

AB

BB

8

2

AB

mol/cm

V

cP

K

T

mol/g

M104.7

s

cm

D

(12.)

where B is the solvent association factor, e.g. for water it is 2.6. Subscript A refers to the materialthat diffuses (diluted) and B into solvent (concentrated).

This equation is typical in the sense, that some pure component physical properties are needed (here

solvent viscosity and molar volume for the diffusing component). The equation also predicts typicalviscosity dependency, i.e. diffusion coefficient reduces if viscosity increases. Often mass transfer

efficiency can be qualitatively estimated based on liquid viscosity; in a certain equipment, e.g. in a

distillation column tray, mass transfer is more efficient if viscosity is low and the efficiency is

reduced if viscosity increases. This is also directly reflected into tray efficiency.

5 Mass transfer in binary mixtures

Previously diffusion flux was described with the Fick’s law. The proportionality coefficient is usually

written as DAB, which describes diffusion of component A in component B when there is lots of B

and only traces of A. Also the tabulated diffusion coefficients and the estimation methods usually predict these infinite dilution values. In practice, mixtures are not always infinitely dilute, so that the

diffusion coefficient must describe diffusion rate in the mixture of A and B. One possibility tocalculate this effective diffusion coefficient when there is a substantial fraction of both components

present is to use Wignes correction. For binary systems it is



Mass transfer 8

AB x0

BA

x0

ABAB DDD (13.)

Here the superscript0 refers to the infinite dilution value, and the symbol D AB without the superscript

to the coefficient where the true mixture composition is accounted. This correction is usually needed

only in liquid phase.

By summing up mass transfer fluxes for both components in a two-component mixture

N)xx(JJ N N N BABABA (14.)

we see, that

dz

dxcD

dz

dxcDJJ B

ABA

ABBA (15.)

Since the mole fractions sum up to unity, the mole fraction gradients have to be equal but with

different sign

BA dxdx

From this it results that in binary mixtures the diffusion coefficients have to be equal.

BAAB DD (16.)

In multicomponent mixtures, this is not generally valid. With this analysis it can be found that the

Fick diffusion coefficients must also be functions of compositions. When the Wignes correction is

used, the above condition is valid in binary systems.

5.1 Equimiolar mass transfer

In equimolar mass transfer, the total flux is zero

BA JJ0 N (17.)

In this case the diffusion fluxes are of equal magnitude and of different sign, and there is no

contribution from convection

BA JJ0 N (18.)



Mass transfer 10

At the bottom of the tube, on the surface of the water, air is practically insoluble to the water

(compared to other fluxes), so it cannot move downwards in the tube. Total mass transfer flux of air

is then zero. At the top of the tube, air is blown so that the diffusing water composition is practically

zero (air is assumed to be completely dry here just to make the example simpler). The tube is so

narrow, that the air blow at the top does not mix its contents.

Note that in the figure the arrow indicating mass transfer is the assumed mass transfer direction,

which is in the negative z-direction. Then as water is evaporating, its flux has negative values due to

the chosen z coordinate direction.

Let’s examine the situation in the tube at steady state. The mass transfer flux of the water vapor in the

tube (component A) can be expressed as

Nydz

dycD N A

A

ABA (20.)

Correspondingly for air, component B

Nydz

dycD N B

BBAB (21.)

In a binary mixture, these are not independent, so that only one equation is needed to solve the fluxes.

Total flux N is the sum of the individual component fluxes

BA N N N (22.)

In order to solve the problem, an additional condition for the total flux is needed. In this case the

required condition can be found from the fact that air is practically insoluble to the water, so that itsmass transfer flux is zero. Air may diffuse and be convected, but the sum of these two terms is zero

0 N B (23.)

Using equations (23), (22) and (20) we obtain

AAA

ABA Nydz

dycD N (24.)

From where

dz

dycD)y1( N A

ABAA (25.)

A

AABA

y1

dycDdz N

(26.)

By integrating this from the top of the tube to the liquid surface we get

1

0

1 y

y A

AAB

z

0

Ay1

dycDdz N (27.)



Mass transfer 11

0

1

1

ABA

y1

y1ln

z

cD N

(28.)

In gaseous mixtures, compositions are often expressed in terms of partial pressures pA and total

pressure p. Total concentration c in gas mixtures is obtained from the equation of state. Here the ideal

gas equation is used

p

py A

A (29.)

RT

pc (30.)

Inserting these to (28) we get

0A

1A

1

ABA

p p p pln

RTz pD N

(31.)

5.2.1 Example

Water diffuses from the bottom of a narrow tube as described above. Temperature is constant 20 C

and pressure is atmospheric. Water surface is 15 cm below the top of the tube. Diffusion coefficient

of water in air in these conditions is 0.25·10-4 m2/s. Estimate water flux.

Density of liquid water is so high that it can be assumed that the water surface level does not changesignificantly during the time where stationary state is reached in the tube.

Water vapor pressure pA1 at temperature 20 C can be found from the steam tables or calculated with

a suitable correlation (e.g. Antoine)

atm0231.0 p 1A

0231.0 p

py 1A

1A

0y 0A

33 m

mol6.41

K 293

1

atmcm057.82

K molatm

RT

pc

0

1

1

ABA

y1

y1ln

z

cD N



Mass transfer 12

01

0231.01ln

m15.0

1

s

m1025.0

m

mol6.41 N

24

3A

2

4

Asm

mol106.1 N

5.2.2 Example

In the previous example, the liquid surface level slowly decreases in the tube. Derive an equation for

this.

Let us assume again that there is a stationary state in the gas volume in the tube, i.e. that the surface

level decreases slowly. Let z be the liquid surface distance from the top of the tube and y water

composition in the air. From the previous example, air above the tube is dry and the water flux is

y1lnz

cD N ABA

Amount of moles in the liquid water phase, n, can be expressed in terms of its density , tube cross-

sectional area A and surface level z. A dynamic material balance for water is

dt

dz

M

A

dt

M

Vd

dt

dnANA

so that

y1lnz

AcD

Mdt

dzA AB

dty1lncDM

zdzAB

Liquid level changes from z1 to z2 when time changes from 0 to t. Integration gives us

y1lnMcD2)zz(t

AB

2

2

2

1

Note that as water evaporates, z1 < z2, due to the chosen direction of coordinate z. If the liquid level is

measured from the bottom of the tube, indices 1 and 2 can be changed.

This equation can be used to estimate diffusion coefficients in gases. Liquid level in the tube isfollowed, so that the diffusion coefficient can be estimated from the equation above.



Mass transfer 13

5.3 Mass transfer when the flux ratios are known

In mass transfer through constant cross-sectional area, each component mass transfer flux N A = JA +xA N is constant throughout the diffusion path. However, compositions change. Then the convective

term xA N and correspondingly molecular diffusion JA must change.

Let us define the component A mass transfer ratio as

BA

AAA

N N

N

N

NZ

(32.)

Note that if the fluxes are constant, then also this ratio is constant.

Starting from the general mass transfer expression and inserting mass transfer ratio definition above

we get

A

AAAAAA

Z NxJ NxJ N (33.)

A

A

AA J

Z

x1 N

(34.)

dz

dxcD

xZ

ZJ

xZ

Z N A

AB

AA

AA

AA

AA

(35.)

If the physical properties (c and D) are constant, this can be integrated analytically over the diffusion path

1AA

2AA

12

A

AxZ

xZln

zz

ZDc N AB

, if AZ (36.)

)xx(zz

Dc N 1A2A

12

AAB

, if AZ (equimolar mass transfer) (37.)

6 Diffusion in solid materials

Diffusion of liquids, gases, and solids also occur in solid materials, but diffusion rates are usuallymuch lower than in liquids or gases. Diffusion in solids is nevertheless an important factor in several

industrial processes. Examples of such processes are leaching of soy beans or metals, drying ofwood, salts and food, diffusion of reactants in solid catalyst particles, and diffusion of gases through

packing material.

Diffusion in solid can be divided into two categories. The first follows Fick’s law, and the diffusing

material is basically dissolved into the solid matrix. The material structure does not significantly

affect the diffusion. The second category is such where the material structure does affect significantlyto the diffusion. In this case, diffusion occurs typically in small pores of the material.



Mass transfer 14

6.1 Diffusion in solid materials according to the Fick’s law

Mass transfer flux with Fick’s law for diffusion is

Nxdz

dxcD N A

AABA (38.)

Diffusion in solid materials often resembles diffusion through stagnant phase. Usually the solubility

of the diffusing component into the solid matrix is small ( )0x A so that the convective mass

transfer, i.e. xA N is small compared to the diffusive termdz

dxcD A

AB and can be left out from the

analysis. The mass transfer flux is then

dz

dxcD N A

ABA (39.)

This is actually the same than equimolar mass transfer equation. It can be integrated (assumingconstant physical properties) to

12

2A1AABA

zz

ccD N

(40.)

6.1.1 Example

Hydrogen temperature is 17 C and partial pressure within a tube 0.01 atm. It diffuses through 0.5

mm thick neoprene rubber. Estimate hydrogen flux.

Hydrogen diffusion coefficient in neoprene at the given conditions is 1.03 ·10-10 m2/s.

Gas solubility is often expressed in dissolved volume of gas in NTP (0 C and 101.325 kPa). A mole

of gas occupies 22.4 dm3 in NTP, which can be used to transform volume into moles. Gas solubility

is usually directly proportional to the gas partial pressure. Hydrogen solubility in neoprene is

atm

p051.0

rubber m

H NTPm/c A

3

2

3

A

From this, the hydrogen composition at the rubber surface can be calculated

rubber m

molH0228.0

atm

atm01.0

NTPm4.22

kmol

rubber m

H NTPm051.0c

3

2

33

2

3

A

Hydrogen composition outside the tube is 0

rubber mm5.0

1

rubber m

kmolH1028,2

s

m1003.1

zz

ccD N

3

252

10

12

2A1AABA



Mass transfer 15

rubber sm

molH107.4 N

2

29

A

Hydrogen flux in this case is rather small. If the partial pressure is higher, the flux is correspondingly

higher. Temperature also has an effect; diffusivity and solubility increase with temperature. Often

when hydrogen is processed, temperatures and partial pressures are high.

6.2 Diffusion in porous solid materials when structure is important

In porous solids, material needs to diffuse in tortuous path which makes the path longer. This is

usually accounted in the diffusion equation by adding a tortuosity term in the nominator, and

usually it assumes values around 1.5 – 4. This term describes the length of the diffusion pathcompared to the shortest possible path.

Another term, void fraction , is added to take account that the diffusion occurs only in pores (in this

case).

)zz(

ccD N

12

2A1AABA

(41.)

The effective diffusion coefficient is thus obtained by multiplying the free fluid diffusion coefficient

by void fraction and dividing by tortuosity. Both terms slow down observed diffusion rate, since void

fraction is always lower and tortuosity always higher than 1.

6.3 Knudsen diffusion

Knudsen diffusion refers to a situation, where molecule collisions with the walls is significant

compared to the intermolecular collisions. This occurs in such cases, where the pores are very narrow

so that the mean free path of molecules in the pores is significant compared to the pore diameter.

Knudsen diffusion coefficient can be estimated from

A

pKAM

Td5.48D (42.)

where d p is the pore diameter (m), T is temperature (K), and MA is the diffusing molecule molar mass

(g/mol). Effective diffusion coefficient for case where both wall and intermolecular collisions aresignificant, can be calculated similarly to resistances in series, i.e.

ABKA

A

D

1

D

1

1D (43.)

Also void fraction and tortuosity effects are considered in this equation. Diffusion in porous material

is slower than in free fluid, since pore walls slow down diffusion.



Mass transfer 16

Knudsen diffusion model is applied mainly for gases. For liquids, a parameter describing pore size

affect to the diffusion can be correspondingly estimated. The effect is significant mainly when the

pore diameter is smaller than 100 times the molecular diameter. Molecular diameter is slightly vague

term in this context, since long and narrow molecule can fit into a small pore in lengthwise direction,

but not otherwise. In practice, these correlations can be used to estimate effect of small pores in

diffusion, but it would be better to measure diffusion coefficient in the actual solid material if

possible.

7 Mass transfer in separation processes

In separation processes, mass is usually transferred between phases. The phases are brought into

contact in such a way, that the desired components move towards lower chemical potential. Usuallythis means lower molar fraction. At the phase interface, lower chemical potential (or fugacity) could

mean higher mole fraction (i.e. higher solubility).

The mass transfer operations can be classified according to the next table, depending on the phases

present

Processed Processing Operation

material material

gas gas rare

gas liquid absorption, scrubbing

gas solid adsorption, membrane separation

liquid gas distillation, absorption, stripping, desorption

liquid liquid extractionliquid solid adsorption, membrane separation

solid gas dryingsolid liquid leaching, washing

solid solid rare

As previously explained, there are two mass transfer mechanisms; diffusion and convection. Despite

this these operations are sometimes called as diffusion processes.

8 Two-film theory

Two-film theory is often used to describe mass transfer between phases. In the two-film theory it isassumed that there is an infinitely thin interface between the two phases, and on the both sides of this

interface there are boundary layers where all mass transfer resistance is located. Outside these layers,

in so called bulk phase, turbulence is so high that all concentration differences level out. According

to the model, compositions outside the films are thus constant. In the following figure, the model is

shown schematically for gas-liquid mass transfer. It can, however, be used for mass transfer between

any phases.



Mass transfer 17

Gas liquidinterface

Eddies, turbulent

mass transfer Eddies, turbulent

mass transfer A film on both sides of

the interface. Mass

transfer resistance is

in these films

In the next figure, schematic composition profiles are shown for the film model and for the actualsituation. The real profile may naturally change depending on the situation. The film model profile

may be slightly curved if the mass transfer is not equimolar, but diffusion coefficient is neverthelessassumed constant in the film. In reality, the profiles are curved significantly due to turbulent diffusion

which is more pronounced further away from the interface.

location

c o m p o s i t i o n

True profile

Profile according to the film model

Interface "bulk"

phase

As mass transfers from one phase to another, the resistance is assumed to be completely in these two

films. At the interface, there is no resistance. The interface is in physical and chemical equilibrium.

Temperature and velocity is the same immediately on both sides of the interface. All equilibriumconditions discussed in the phase equilibrium part of this course apply.

Chemical equilibrium means that the component chemical potentials (or fugacities) are equal on both

sides of the interface, but compositions are generally not. As the equilibrium is described at theinterface, all nonidealities should be accounted, although nonidealities are often neglected when mass

transfer fluxes in the films are estimated.

Although the interface is in equilibrium, the bulk phases are not. Each component is diffusingtowards lower chemical potential or fugacity. Mass transfer slows down only when the composition

differences between the interface and the bulk phases disappear.



Mass transfer 18

In the next figures, composition and chemical potential profiles are schematically shown for mass

transfer from gas to liquid.

gas liquidinterface

Composition profiles for sparingly soluble component. Mole fraction is not a continuous function at

the interface.

gas liquidinterface

Composition profiles for highly soluble component.

gas liquidinterface

Component chemical potentials. Chemical potential is a continuous function at the interface since the

interface itself was assumed to be in equilibrium.

The gradients in the films are opposite, if the component is transferring from liquid to gas.

The film thicknesses are often about 0.1 mm for gases and 0.01 mm for liquids. Note that the filmthickness is not a direct indication about the resistance, since also total concentration and diffusion

coefficient contribute to the mass transfer resistance.

9 Mass transfer coefficients

9.1 Mass transfer coefficients according to the film model

Let composition of component A in a binary mixture be x in liquid and y in vapor. Thesecompositions are constant outside the films, and they can be obtained from the process material

balances. Compositions in both films change linearly if mass transfer is equimolar. If the convective

mass transfer is significant, the composition profiles are slightly curved also according to the filmmodel.



Mass transfer 19

Interface compositions in liquid and vapor side are marked with xi and yi.

Mass transfer coefficients on the liquid (k x) and vapor (k y) sides are defined as follows, when the

convective flux is negligible:

)yy(k N iyA (44.)

)xx(k N ixA (45.)

Similar analysis applies also for mass transfer between any other two phases as well. These

coefficients are defined for each component; subscripts are left out for simplicity in this analysis.

Theoretically these coefficients are not independent for all components, since diffusion fluxes should

sum up to unity. This is unfortunately often forgotten in mass transfer analysis.

Dimension of mass transfer coefficient, according to the previous definition, is2ms

mol. There is also

another possible definition which is quite often applied

)yy(k c N iyyA (46.)

)xx(k c N ixxA (47.)

The only difference is that in the latter set of equations, the total concentration is separately shown.

With these definitions, the mass transfer coefficient dimension is m/s. This describes the maximum

velocity by which the material can move in the film due to diffusion. Equations must be used with

care, since the same symbols are used in the two different meanings.

Mass transfer coefficient is defined as a proportionality factor between the flux and the driving force,

when the convective flux is small (xA N = 0).

2A1A

12

AAxx

zz

DcJ N AB

(48.)

Proportionality coefficient is thus

12 zz

D

ck AB

(49.)

If the total concentration is not part of the mass transfer coefficient, the definition would be

12 zz

Dk AB

. Mass transfer coefficient according to the film model can therefore be obtained by

dividing diffusion coefficient by the diffusion path length.



Mass transfer 20

9.2 Overall mass transfer coefficient

When the mass transfer flux equations are written, it would be easier if the mass transfer could bedefined based on the bulk compositions that are available from the equipment material balances.

Then the interface compositions would not be needed. The most typical way to accomplish this is todefine overall mass transfer coefficients in the following way

)yy(K N *

yA (50.)

)xx(K N *

xA (51.)

Either of these can be used. Subscript * refers to the equilibrium composition. y* refers to such vapor

composition, which would be in equilibrium with the bulk liquid x. These equilibrium compositionscan be estimated if the equilibrium coefficient is known: y* = Kx* and x* = y*/K. The variables are

shown in the following figure for situation where mass transfer direction is from gas to liquid, andthe value of the equilibrium coefficient is above unity. y* and x* do not correspond to any real

composition in the system, but are calculated with the equilibrium coefficient.

gas liquidinterface

y

yi

xi

x

x*=y/K

y*=Kx

These definitions are analogous to any formulation for resistances in series, but due to the nature of

the chemical equilibrium (compositions are not the same), the same thing can be expressed in two

alternative ways. These definitions simplify the mass transfer model, but on the other hand bring a

new challenge, namely how to calculate the overall mass transfer coefficient.

The coefficients of relative velocity can be accounted analogically to the individual film mass

transfer coefficients earlier. This definition is theoretically not quite justified, and the coefficients of

relative velocity are almost exclusively left out when the overall mass transfer coefficients are

considered.

When the mass transfer is stationary, its magnitude can be expressed both with overall or individualfilm mass transfer coefficients

)xx(K )xx(k )yy(K )yy(k N *

xix

*

yiyA (52.)

from where

xx

yy

k

1

k

1

yy

yy

k

1

k

1

)yy(

)yy()yy(

k

1

)yy(k

)yy(

K

1

i

*

i

xyi

*

i

yyi

*

ii

yiy

*

y

(53.)

By denoting



Mass transfer 21

xx

yym

i

*

i

(54.)

the connection between the individual phases and overall mass transfer coefficient is obtained

xyy k

m

k

1

K

1

(55.)

x xix*

y

yi

y*

A

F

D

B

C

In the figure above, the equilibrium line is shown and the following points

x: liquid phase bulk composition, y* corresponding equilibrium vapor composition

y: vapor phase bulk composition, x* corresponding equilibrium liquid composition

xi: interface composition on the liquid sideyi: interface composition on the vapor side

xi and yi are on the equilibrium curve according to the assumption that the interface itself is in

equilibrium.

From the figure it can be seen that

CD

BC

xx

yym

i

*

i

which corresponds to the equilibrium line slope between points B and D.

If the mass transfer flux is expressed in terms of the liquid side overall mass transfer coefficient, we

get

yxx mk

1

k

1

K

1 , (56.)

wherei

*

i

xx

yym

Overall mass transfer coefficients K x and K y are constant in the modeled unit only if both film mass

transfer coefficients are constant and the equilibrium line is straight. Usually these assumptions aremade in order to simplify calculations.



Mass transfer 22

Since the mass transfer can be calculated with either individual phase (flux must be equal if there is

no accumulation at the interface)

)xx(k )yy(k N ixiyA (57.)

the slope AB in the previous figure can be calculated as

y

x

i

i

k

k

xx

yy

(58.)

In some cases, the interface composition, point B in the figure, needs to be calculated. This can be

done by starting from the known compositions x and y and by calculating the mass transfer

coefficients, after which the interface compositions can be estimated with the equilibrium line and theAB slope.

10 Penetration and surface renewal theories

Penetration and surface renewal theories are similar in structure and an alternative to the film model.In these theories it is assumed that there is a fluid element that arrives to the interface, stays there for

some time, during which the mass transfers, and after a while is replaced by a new element. Duringthis contact time the material is thus “penetrated” to the element at the surface. The difference in

penetration and surface renewal theories is mainly which kind of contact time distribution is assumedfor the fluid elements. In penetration theory it is assumed that each element stays the same time, and

in the surface renewal theory a residence time distribution is applied. The origin of these theories

comes from the observation, that in some cases contact times between the phases are so short that thematerial does not have time to reach a steady state in the film, assumed by the film theory.

Time dependent change of component concentration in the fluid element can be described by the

following diffusion equation

2

A

2

A

z

cD

t

c

(59.)

Initial value is a constant concentration 0AA cc everywhere in the fluid. One boundary value is

obtained from the equilibrium condition with the other phase AiA c)0z(c . The other boundarycondition is given by the assumption that far from the interface the composition remains in the bulk

phase value

0AA

AiA

0AA

c)z(c

0t,c)0z(c

0t,cc

(60.)

This equation can be solved with Laplace transformation or by using a combined variableDt

z. The

solution is a complementary error function



Mass transfer 23

)Dt2

z(erfc)cc(cc 0AAi0AA (61.)

xerf 1xerfc (62.)

Definition of the error function is

x

0

2d)exp(

2)x(erf (63.)

By taking a derivative from the concentration with respect to the location, the fluxes at the surfacecan be calculated at time t

)cc(t

D

dz

dcDtJ 0AAi

t

(64.)

Average flux in the time interval 0 - tT can be obtained from the previous

)cc(t

D2

t

dtD

t

ccJdt

t

1J 0AAi

T

t

0T

0AAi

t

0T

TT

(65.)

so that the average mass transfer coefficient is

TT t

D13,1

t

D2k

(66.)

In the penetration theory, each fluid element remains at the interface the same time. Another option is

to have a residence time distribution for the fluid elements. Both of these theories predict that the

mass transfer coefficient should depend on the square root of diffusion coefficient.

In penetration theory, the contact time is needed. Often it can be assumed that this contact time is the

ratio of bubble, drop or particle diameter and its slip velocity in the continuous phase. This is

equivalent to assuming that the contact time is such where the particle rises or settles a distance equal

to its own diameter.

11 Temperature effect to mass transfer

Temperature rise

- typically reduces gas solubility to liquids (there are important exceptions, such as hydrogen

solubility in hydrocarbons)

- typically increases liquid mutual solubilities

Liquid temperature rise

- reduces viscosity and makes mass transfer films thinner

- increases liquid diffusion coefficients Mass transfer is enhanced



Mass transfer 24

Gas temperature rise

- increases gas viscosity

- increases gas diffusion coefficients

- lowers total concentration at constant pressure

Effect to mass transfer is usually small

12 Determination of mass transfer coefficients

Mass transfer coefficient can be measured similarly than the diffusion coefficient. Based on these

measurements, there are several correlations available, where the mass transfer coefficient can be

estimated when physical properties and fluid flow characteristics are known. The functional formscan be taken from the boundary layer (film) or penetration theory, and the variables can be found

with the dimensional analysis.

The dimensional analysis proposes the following dependency

...

d

l,

d

l,Sc,Ref Sh 21 (67.)

Here the Sherwood number Sh is defined based on the mass transfer coefficient (here with the

dimension of m/s), characteristic length d, and diffusion coefficient DAB

ABD

kdSh (68.)

Reynolds number Re is defined based on velocity v, viscosity , density and characteristic length d

vd

Re (69.)

Schmidt number is defined based on the ratio of kinematic viscosity

and diffusion coefficient

ABDSc

(70.)

The correlations are sometimes written by using the Stanton number or mass transfer factor jM

v

k

ScRe

ShSt (71.)

3/2

M StSc j (72.)

In the following chapters, there are some examples of the mass transfer coefficient correlations in

various geometries



Mass transfer 25

12.1 Mass transfer for a liquid film falling inside at tube

Mass transfer coefficient in this case can be estimated from the following correlation

44.081.0 ScRe023.0Sh (73.)

This is valid when 00035Re0002 , 5.2Sc6.0 and kPa300 pkPa10 .

Another correlation suitable for higher Schmidt numbers (430 - 100 000), can be used for viscous

liquids

346.0913.0 ScRe0096.0Sh (74.)

12.2 Flow around spherical particles

As fluid flow around a spherical particle slows down, the Sherwood number approaches 2. Onetypical correlation for situation where there is some flow around a sphere (Re < 1000) is

3 ScRe6.02Sh (75.)

This is very useful correlation, since quite often fluid particles, i.e. bubbles and drops, can be

assumed to be spherical. This assumption is good at least for rather small bubbles and drops.

Similar equation can be used for other than spherical particles. Then the characteristic dimension for

the particle need to be estimated e.g. based on the particle surface area and volume.

12.3 Mass transfer in packed beds

In a bed of spherical particles, when void fraction is 40 – 45 %, the following correlation can be used

for mass transfer between gas and liquid

33.0585.0 ScRe17.1Sh (76.)

For cylindrical particles, the cylinder diameter is used when Sherwood and Reynolds numbers are

calculated.

In the modern packing material, the void fraction is typically very high, often above 90%. There arespecific correlations for these packings, e.g. the following by Onda

2

p p

33.0

V

7.0

V

V p

ydScRe23.5

D

k aa

(77.)

4.0

p p

5.0

L

667.0

L

33.0

L

Lx dScRe0051.0

gk a

(78.)



Mass transfer 26

For structured packings, the correlation by Bravo has provided good results. The characteristic length

is the packing fold length s

33.0

G

8.0

GG ScRe0338.0Sh (79.)

s

uD

2k

eff ,LL

L (80.)

12.4 Volumetric mass transfer coefficient

Sometimes the correlation gives the product of mass transfer coefficient and specific mass transfer

area k a. This product is called volumetric mass transfer coefficient, and it is somewhat easier to use

than separate correlations for mass transfer coefficient and mass transfer area. Both of these terms are

affected by the flow conditions; turbulence enhances mass transfer by making the mass transfer film

thinner and also at the same time increases mass transfer area in case of mass transfer between two

fluid phases. These two terms are, however, fundamentally different, and they are explained withdifferent physical phenomena. The terms can also be measured separately, although they appear

together in the mass transfer equation. For example bubble or droplet specific surface area a can beestimated by taking photographs, and the mass transfer rate (which depends on the product k a) can be

estimated by measuring compositions.

12.5 Analogy between heat and mass transfer

Mass transfer coefficients are measured much less than heat transfer coefficients. Due to the analogy between the mass and heat transfer, correlations developed for heat transfer can be used also for

estimating mass transfer coefficient. According to the analogy, the dimensionless numbers in the heat

transfer correlations should be replaced by corresponding dimensionless numbers for mass transfer.

Heat transfer Mass transfer

Nusselt Nu Sherwood Sh

Stanton St Stanton St

Heat transfer j-factor Mass transfer j-factor

Prandtl Pr Schmidt Sc

In the following figure, mass and heat transfer factors are compared for flow perpendicularly outsidea cylinder. The analogy is surprisingly good



Mass transfer 27

The previous figure can be expressed as a correlation for mass transfer

3 ScRe61.0Sh (81.)

13 Stage efficiencies

By using the two-film theory, stage efficiencies for a mass transfer stage can be estimated. Here a

cross-flow stage is considered, where liquid flows horizontally and vapor vertically. Such situation is

typically on distillation column trays. Similar analysis can be carried out also for other flow

solutions.

As vapor flows through the liquid, mass transfers between them. Due to this, vapor compositionchanges in vertical direction. Liquid is assumed to be completely mixed in vertical direction. The

situation is schematically shown in the following figure

Variables are:

V : vapor molar flow per cross-sectional area Ai

y : mole fraction in vapor

a : specific surface area; m2 of interface area / m3

dispersion

K y : vapor side overall mass transfer coefficient(mol/m

2s)

From these we get:

Component molar flow: VyA (mol/s)

Control volume size: Adz (m3)Surface area within the control volume: Aadz (m

2)

Vz, yz

Vz+dz, yz+dz

dz

Cross-sectional area A



Mass transfer 28

Mass transfer direction is here defined positive if mass transfers from liquid to vapor. Total flux is

assumed to be small. The mass transfer flux expression in this case is

)yy(K N *

y (82.)

Let us write a steady state material balance for the vapor. The first term describes material flow in,the second term mass transfer from liquid to vapor and the term on the right hand side flow out from

the control volume

AyV)yy(K dzAAyV dzzdzz

*

yzz a (83.)

When the slice height dz approaches zero, this can be expressed in a differential form: dy = yz+dz - yz

dz)yy(K Vdy *

y a (84.)

In the previous equation, total vapor flow rate was assumed to be constant. Actually this assumption

is not necessary, if we define V = Vin, i.e. the vapor inflow.

By integrating over the liquid height on the tray from point 1 (at the bottom) to point 2 (at the

surface) we get

2

1

2

1

z

z

y

y

y

*dz

V

K

yy

dy a (85.)

V

ZK exp

yy

yy y

1

*

2

* a (86.)

where Z is the distance between points 1 and 2, i.e. z2-z1. From the previous it can seen that the term

V

K yawas assumed constant (it was taken outside the integral).

The point efficiency was defined as

0

1

*

12 Eyy

yy

, (87.)

from where

) Nexp(V

ZK exp

yy

yy

yy

yyyyE1 y0

y

1

*

2

*

1

*

121

*0

a (88.)



Mass transfer 29

TermV

ZK N

y

y0

a is called vapor side number of overall mass transfer units. It is directly

proportional to the mass transfer coefficient, as can be seen from the previous equation. Often thestage efficiency calculations are carried out by using the number of transfer units. Another symbol

that is used for this variable is NTU.

In distillation, the number of transfer units on a tray is typically 1.5-2 irrespective of the vapor flowrate. Then the point efficiency is about 78-86%. Stage efficiency is typically higher than that, if the

liquid flow path on the tray is so long that there are composition gradients in the liquid.

14 Number of transfer units

The following expressions can be written for the numbers of transfer units

y0

y N

V

ZK

a (89.)

Here V is the vapor flow per cross sectional area, a is the specific surface area, and Z is the height

over which the mass transfers. For example on a distillation tray Z is the height of the bubbling

liquid.

Similar expressions can be defined also based on individual film mass transfer coefficients. Then the

liquid side (Nx) and vapor side (Ny) mass transfer coefficients are obtained

y

y N

V

Zk

a (90.)

xx NL

Zk

a (91.)

Previously the connection between the overall mass transfer coefficient and individual phase

coefficients was derived (assuming small total flux, x and y = 1)

xyy k

m

k

1

K

1

By inserting the mass transfer coefficient expressions as functions of transfer units, the number of

overall mass transfer units is obtained

mL N

Z

V N

Z

V N

Z

xyy0

aaa (92.)

that is

L N

Vm

N

1

N

1

xyy0

(93.)

ke-42 3200 mass transfer

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