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Diffusion

051333 Unit operation in Agro-Industry IIIDepartment of Biotechnology, Faculty of Agro-Industry

Kasetsart UniversityLecturer: Kittipong Rattanaporn

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Outline

Definition and mechanismTheory of diffusionMolecular diffusion in gasesMolecular diffusion in liquidMass transfer

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Definition and mechanism

Diffusion is the movement of an individual component through a mixture. (the influence of a physical stimulus)Driving force is a concentration gradient of the diffusing component.A concentration gradient tends to move the component in the direction to equalize concentrations and destroy the gradient.

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Consider 2 gases B and E in a chamber, initially separated by a partition. At some instant in time, the partition is removed, and B and E diffuse in opposite direction as a result of the concentration gradients.

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Diffusion is the characteristic of many mass transfer operations.Diffusion can be causes by an activity gradient such as by a pressure gradient, by a temperature gradient, or by the application of an external force field.Role of diffusion in mass transfer.

DistillationLeachingCrystallizationHumidificationMembrane separation

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Theory of Diffusion

AssumptionDiffusion occurs in a direction perpendicular to the interface between the phases and at a definite location.Steady state. (The concentrations at any point do not change with time)Binary mixtures.

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Comparison of diffusion and heat transfer

Net flux

Driving force

The rate of temperature change.

The rate of change in concentration

Temperature gradientConcentration gradient

Heat transfer (conduction)

Diffusion

From the similarity, the equation of heat conduction can be adapted to problems of diffusion in solids or fluids.

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Comparison of diffusion and heat transfer

Differences between heat transfer and mass transfer

Heat transfer is an energy transition but diffusion is the physical flow of material.Heat transfer in a given direction is based on one temperature gradient and the average thermal conductivity.Mass transfer, there are different concentration

gradients for each component and often different diffusivities.

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Diffusion quantities

Velocity, u (length/time)Flux across a plane, N (mol/area time)Flux relative to a plane of zero velocity, J (mol/area time)Concentration, c and molar density ρM (mol/volume)Concentration gradient, dc/dx

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Molal flow rate, velocity, and flux

The total molal flux (mole/time area)N = ρMu0 (1)

Where ρM is the molar density of the mixture and u0 is the volumetric average velocity.

For component A and B, the molal fluxes areNA= uAcA (2)NB= uBcB (3)

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The molar flux of component A and B JA = cAuA – cAu0 = cA(uA – u0) (4)JB = cBuB – cBu0 = cB(uB – u0) (5)

The diffusion flux J is assumed to be proportional to the concentration gradient and the diffusivity of component (D)

(6)

(7)

dxdcDJ A

ABA −=

dxdcDJ B

BAB −=

12

Equation 6 and 7 are statements of Fick’sfirst law of diffusion for a binary mixture.The law is based on:

The flux is in moles per unit time.The diffusion velocity is relative to the volume-average velocity.The driving potential is in the molar concentration.

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The Fick’s law is similar to Fourier ‘s law of heat conduction (8) and Newton’s equation for shear-stress-strain relationship(9).

(8)

(9)

xTk

Aq

∂∂

−=

yu∂∂

−= µσ

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Diffusivity

DAB is mass diffusivity of component A on component B through the mixture.The dimensions of mass diffusivity are length squared divided by time, usually given as m2/s or cm2/s

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The magnitude of mass diffusivities for liquids or gases in solid are less than the mass diffusivities for gases in liquids.In solids, the mass diffusivities range from 10-9 to 10-1 cm2/s, in liquids, the mass diffusivities range from 10-6 to 10-5 cm2/s and for gases, the mass diffusivities range from 5x10-1 to 10-1 cm2/sThe mass diffusivity magnitudes are a function of temperature and concentrations; in the case of gases the mass diffusivity is substantially influenced by pressure.

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Molecular diffusion in gas

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Using Ideal gas law(10)

the gas constant RB for gas B can be written in terms of the Universal gas constant Ru

(11)Ru is the universal gas constant 8.314 (m3 Pa)/(g-mol K) and MB is molecular weight of gas B.Thus,

(12)

or(13)

TRp BBB ρ=

B

uB M

RR =

TRMp

TRp

u

BBB

B

BB

=

=

ρ

ρ

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Since ρB is mass concentration, substitute equation 13in equation 6

(14)

(15)

The mass diffusivity DBE refers to diffusivity of gas B in gas E. Similarly, we can express diffusion of gas E in gas B

(16)

xp

TRMDJ

TRMp

xDJ

B

u

BBEB

u

BBBEB

∂∂

−=

∂∂

−=

xp

TRMDJ E

u

EEBE ∂

∂−=

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Steady-State Diffusion of gases

Assume the mass diffusivity does not depend on concentration, from equation 6, we obtain

(17)

By separating variables and integrating:

(18)

(19)

dxdcDJ A

ABA −=

)xx()cc(DJ

dcDdxJ

AAABA

c

c A

x

x ABAA

A

12

21

2

1

2

1

−−

=

−= ∫∫

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For the condition of steady state diffusion, the concentration at the boundaries must be constant with time and diffusion is limited to molecular motion with in the solid being described.The mass diffusivities are not influenced by magnitude of concentration and no temperature gradient.

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Example 1 Molecular diffusion of Helium in Nitrogen

A mixture of He and H2 gas is contained in a pipe at 298 K and 1 atm total pressure which is constant throughout. At one end of the pipe at point 1 the partial pressure pA1 of He is 0.60 atm and at the other end 0.2 m pA2 of is 0.20 atm. Calculate the flux of He at steady state if DAB of He-N2 mixture is 0.687 x 10-4 m2/s

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Equimolar Counter diffusion in Gases

Molecules of A diffuse to the right and B to the left. The total pressure is constant, the net moles of A diffusing to the right must equal the net moles of B to the left.

JAz = -JBz (20)

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Fick’s law for B for constant c,(21)

Now P = pA + pB = constant thenc = cA + cB (22)

Differentiating both sidesdcA = - dcB (23)

dcDJ B−=dzBAB

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Equating Fick’s law to (21)(24)

Substitute (23) into (24) and canceling like termDAB = DBA (25)

For a binary mixture of A and B the diffusivity coeff. DAB for A diffusing in B in the same as DBA for B diffusing into A.

dzdc)D(J

dzdcDJ B

BABA

ABA −−=−=−=

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Example 2 Equimolar Counterdiffusion

Ammonia gas (A) is diffusing through a uniform tube 0.10 m long containing N2 gas (B) at 1.0132 x 105 Pa and 298 K. At one end of the pipe at point 1 the partial pressure pA1 is 1.013 x 104 Pa and at the other end pA2 of is 0.507 x 104 Pa. The diffusivity DAB = 0.230 x 10-4 m2/s. Calculate the flux JA and JB at steady state.

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General case for Gases A and B Plus Convection

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 uM (m/s)Component A is still diffusing to the right, its diffusion velocity uAd is measured relative to the moving fluid.The velocity of A relative to the stationary point is the sum of the diffusion velocity and the average or convective velocity

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uA

uAd uM

uA = uAd + uM (26)Mutiply by cA

cAuA = cA uAd + cAuM (27)Rewrite (27) in terms of flux

NA = JA + cAuA (28)

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Let N is the total convective flux of the whole stream relative to the stationary point

N = cuM = NA + NB (29)

(30)Substitute (30) to (28)

(31)

cNNu BA

M+

=

( )BAA

AA NNc

cJN ++=

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When JA is Fick’s law.(32)

Equation 32 is the final generation for diffusion plus convection .For equimolar counter diffusion , NA = NB and the convective term in (32) become zero. Then NA = JA = -NB = -JB

( )BAAA

ABA NNc

cdz

dcDN ++−=

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Special case for A diffusing through stagnant, Nondiffusing B

In this case one boundary at the end of the diffusion path is impermeable to component B, so it cannot pass through.For example, the evaporation of organic solvent into air, Gas adsorption into liquid.

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To derive the case for A diffusing in stagnant, nondiffusing B, NB = 0 is substitute into the general equation (32)

(33)

Keeping the total pressure P constant,(34)

AAA

ABA Nc

cdz

dcDN +−=

AAAAB

A NPp

dzdp

RTDN +−=

32

Rearranging and integrating,(34)

(35)

(36)

dzdp

RTD

PpN AABA

A −=

−1

∫∫

−

−=2

1

2

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pA

pA A

AABz

zA

Pp

dpRTDdzN

( ) 1

2

12 A

AABA pP

pPlnzzRT

PDN−−

−=

33

Eq. 36 can write in a form of an inert B log mean value.

(37)

Substitute 37 into 36

(38)

)pP/pPln(pp

)p/pln(ppP

AA

AA

BB

BBBM

12

21

12

12

−−−

=−

=

( ) ( )2112

AABM

ABA pp

PzzRTPDN −

−=

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Example 3 Diffusion of water through stagnant, Nondiffusing Air

Water in the bottom of a narrow tube is held at a constant temperature of 293 K. The total pressure of air is 1.01325x105 Pa (assumed dry air) and the temperature is 293 K. Water evaporates and diffuses through the air in tube and diffusion path is 0.1524 m long.

Calculate the rate of evaporation at steady state The diffusivity of water vapor at 293 K and 1 atm is 0.25 x 10-4 m2/sThe vapor pressure of water at 293 K is 0.0231 atm

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Molecular diffusion in liquids

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Diffusion of solutes in liquids is very important in many industrial process, especially in separation process.The rate of molecular diffusion in liquids is slower than in gases because the molecules are very close together compared to gas.In generally, the diffusion coefficient in a gas will be greater than in a liquid but the flux in a gas is not that much greater because the concentrations in liquids are considerably higher than in gases.

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In diffusion in liquids an important difference from diffusion from gases is that the diffusivities are often quite dependent on the concentration of the diffusing component

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Calculation of diffusion in liquids

Equimolar counterdiffusion(39)

(40)

(41)

BA NN −=

( ) ( )12

21

12

21

zzxxcD

zzccDN AAavABAAAB

A −−

=−−

=

22

2

1

1

+=

=

MMMc

avav

ρρρ

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Diffusion of A through nondiffusing BAn example is a dilute solution of propionic acid in a water solution being contacted with toluene

(42)( )avAB xxcDN −= ( ) 2112

AABM

A xzz −

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Where(43)

For dilute solution xBM is close to 1.0 and c is constant

(44)

)x/xln(xxx

BB

BBBM

12

12 −=

( )( )12

21

zzxxDN AAAB

A −−

=

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Example4 : Diffusion of Ethanol through Water

An ethanol-water solution in the form of stagnant film 2.0 mm thick at 293 K is in contact at one surface with an organic solvent in which ethanol is soluble and water is insoluble. Hence, NB= 0. At point 1 the concentration of ethanol is 16.8 wt% and the solution density is 978.2 kg/m3. The diffusivity of ethanol is 0.740 x 10-9 m2/s. Calculate the steady state flux NA.

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Molecular diffusion in solid

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Rate of diffusion in solids is generally slower than rates in liquids and gases.For example

Leaching of fooddrying of foodsDiffusion and catalytic reaction in solid catalystSeparation of fluids by membrane

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We can classify transport in solids into two types of diffusion

Diffusion in solids following Fick’s law.Diffusion in porous solids that depends on structure.

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Diffusion in solids following Fick’s law

This type of diffusion does not depend on the actual structure of solid,The diffusion occurs when the the fluid or solute diffusing is actually dissolve in the solid to form a more or less homogenous solution.

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Using the general equation for binary diffusion (Eq. 32)

The bulk flow term is usually small, so it is neglected.

(45)dcDN A−=

( )BAAA

ABA NNc

cdz

dcDN ++−=

dzABA

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Integration of equation 45 for a solid slab at steady state

(46)

For the case of diffusion radially through a cylinder wall inner radius r1 and outer r2 and length L,

(47)

( )( )12

21

zzccDN AA

ABA −−

=

drdcD

rLN A

ABA −=

π2

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(48)

The diffusion coefficient is not dependent upon the pressure of gas or liquid on the outside of the solid.The solubility of gas or liquid is directly proportional to pA by Henry ‘s law.For gas,

(49)

( ) ( )12212

rrlnLccDN AAABA

π−=

solidmkgmolA

.Spc A

A 3422=

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Diffusion in porous solid that depends on structure

The porous solid that have pores or interconnected voids in the solid would effect the diffusion.For the situation where the voids are filled with liquid or gas, the concentration of solute at boundary is diffusing through the solvent in the void volume takes a tortuous path which is greater than z2 – z1 by a factor, τ, called tortuous.

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For a dilute solution,

(50)

Where ε is the open void fractionτ is the correction factor of the path longer than z.

( )12 zzA −τ( )21 ccDN AAAB −

=ε

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Combined into an effective diffusivity

(51)For diffusion of gases in porous solids

(52)

ABAeff DDτε

=

( )( )12

21

zzRTppDN AAAB

A −−

=τε

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Mass transfer

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Mass transfer is the phrase commonly used in engineering for physical processes which involve molecular and convective transport of atom and molecule within physical systems.

ref: http://en.wikipedia.org/wiki/Mass_transfer

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There are many difference of mass transfer but the most cases of mass transfer is treating using the same type of equations, which feature a mass transfer coefficient, km.The mass transfer coefficient km is defined as the rate of mass transfer per unit area per unit concentration difference (m3/m2 s, m/s).

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(53)

The coefficient represents the volume (m3) of component B transported across a boundary of one square meter per second

( ) ( )2121 BB

B

BB

Am ccA

mcc

Jk−

=−

=•

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For the relationship of the ideal gas law, the mass transport due to convection become

(54)

When the specific application of mass transport is water vapor in air, the equation 54 is used for computing the convective transport of water vapor in air.

(55)

( )21 BBAu

BmB pp

TRAMkm −=

•

( )21622.0WW

TRpAMkm

Au

BmB −=

•

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The Dimensional analysis for predicting the mass transfer coefficient.

The variables are grouped in the dimensionless number.(56)

(57)

(58)

(59)ABpLe

c

ABSc

AB

cmSh

DckN

udN

DN

DdkN

ρ

µρρµ

=

=

=

=

Re

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Consider a fluid flow over a flat plate. for the boundary layer from the leading edge of the plate, we can write the equation for concentration.

(60)2

2

ycD

ycu

xcu A

ABA

yA

x ∂∂

=∂∂

+∂∂

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From the equation 60, cA represents concentration of component A at location within the boundary layer.

(61)The Prandtl number provides the link between velocity and temperature profile.If (62)

then velocity and concentration profile have the same shape.

(63)

kìc

á

ìNrantl numbePr pPr ===

ρ

1=ABDρ

µ

ABSc D

NmberSchmidt nuρµ

==

60

The concentration and temperature profile will have the same shape if

(64)The ratio

(65)The functional relationship that correlated these dimensional number for forced convection are :

(66)

(67)

(68)

1=ABDα

ABLe D

NerLewis numb α==

( )Scsh NNfN ,Re=

diffusion molecular by ed transferrmass totaled transferrmass total

=ShN

mass ofdiffusion molecular momentum ofdiffusion molecular

=ScN

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These correlations are based on the assumption:constant physical propertiesno chemical reaction in the fluidsmall bulk flow at the interfaceno viscous dissipationno interchange of radiant energyno pressure, thermal, or forced diffusion.

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Laminar Flow Past a Flat PlateWhen Nre < 5x105

(69)

Km,x is the mass transfer coefficient at a fixed location. The dc used in the Sherwood and Reynolds number is the distance from the leading edge of the plate.

(70)when flow is laminar over the entire length of the plate. The dc is the total length of the plate.The mass transfer coefficient is the average value for the entire plate .

6.0 322.0 3/12/1Re

,, ≥== ScScL

AB

xmxsh NNN

Dxk

N

6.0 664.0 3/12/1Re

,, ≥== ScScL

AB

LmLsh NNN

Dxk

N

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Turbulent Flow Past a Flat Plate

When Nre > 5x105

(71)The characteristic dimension (dc) used in the Sherwood and Reynolds number is the distance from the leading edge of the plate.

(72). The characteristic dimension (dc) is the total length of the plate.

30000.6 0296.0 3/15/4Re,

,, <<== ScScx

AB

xmxsh NNN

Dxk

N

036.0 33.08.0Re

,, Sc

AB

LmLsh NN

Dxk

N ==

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Laminar Flow in a Pipe

(73)

Where the characteristic dimension, dc, is the diameter of the pipe.

(74)Where the characteristic dimension, dc, is the diameter of the pipe.

000,10N /

86.1 Re

3/1Re,

, <

==

c

Scd

AB

cmdSh

dLNN

DdkN

Turbulent Flow in a Pipe

000,10N N023.0 Re1/3Sc

8.0Re,, >== d

AB

cmdSh N

DdkN

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Mass Transfer for Flow over Spherical objects

(75)

For mass transfer from a freely falling liquid droplet

(76)

)N06.04.0(0.2 0.4Sc

3/2Re,

2/1Re,, dddSh NNN ++=

)N6.00.2 1/3Sc

2/1Re,, ddSh NN +=

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The end.