Lecture 2 Mass Transfer

8/11/2019 Lecture 2 Mass Transfer

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Combustion Technology

and Chemical ReactorsMASS TRANSFER2 N D L E C T U R E

A N D E R S C . O L E S E N

P O S T D O C , P H . D .


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Combustion Technology and

Chemical ReactorsMass Transfer2

Course Curriculum

Lecture

No:

Topic Literature

1 Introduction; Analogy between heat and mass

transfer; Mass diffusion; Boundary conditions; Steady

state mass diffusion through a wall

[1] 14.1-14.5

2 Water vapor migration in buildings; Transient mass

diffusion; Mass diffusion in a porous medium;

Diffusion in a moving medium

[1] 14.6-14.8

3 Mass convection; Simultaneous heat and mass

transfer

[1] 14.9-14.10

[1] Y. engel: Heat and Mass Transfer: Fundamentals andApplications, 3rded.


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Combustion Technology andChemical ReactorsMass Transfer

4

Todays lecture

Water vapor migration in buildings

Transient mass diffusion

Mass diffusion in a porous medium

Mass diffusion in a moving medium

Irreversible thermodynamics (i.e.

Maxwell-Stefan diffusion of a binarymixture)

Assignments


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Moisture transport in building materials (e.g.

wood) is important because of dimension

changes with moisture content

Excess moisture can also lead to:

Corrosionand rusting of metals

Rottingof woods

Peelingof paint

Moldsgrow on wood at a relative humidity of

higher than 85 %

Moisture content also affects the effective

conductivity of porous materialsheattransfer increases with moisture content


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Moisture migration can be controlled by either

vapor barriers(i.e. sheet metals impermeable to

moisture) or vapor retarders(commonly

reinforced plastics or metals, thin foils, plastic

films, )

Commonly vapor retarders with a permeance of57.410-9 kg/s-m2 are used in buildings

Chilled water lines and other cold surfaces must be

wrapped with a vapor barrier jacket


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The vapor permeabilityof the material is usually expressed for a giventhickness instead of per unit thickness, and this is called the permeance

The reciprocal of permeanceis called vapor resistance and is expressed as

1

Note that the amount of water vapor that enters or leaves the building by

diffusion is usually negligible compared to the amount that enters with

infiltrating air or leaves with extrafiltrating air


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Todays lecture





Irreversible thermodynamics (i.e.

Maxwell-Stefan diffusion of a binarymixture)

Assignments


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The differential equation describing transientmass diffusion can be derived by applying the

principle of conservation of mass speciesto

a elementary control volume

For a stationary fluid containing specie thisleads to the following differential equation:

Storage of species Net inflow of species

+

+

x x+ x

V = Ax

A|x A|x+ x


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Introducing Ficks Law of diffusionand substituting :

If we further assume that themixture density and binarydiffusion coefficient areconstant we obtain what is knownas Ficks second law of

diffusion:

For an semi-infinite mediumwe

can write the following initial and

boundary conditions:

, at , at , as


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The analogous case has been solved for heat

conduction: for mass transfer this means the

solution can be givens as:

, ,, , erfc

4

, ,, , , ,, ,

, ,, ,


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Just to clarifythe complementary error functionerfcis defined asfollows:

e r f c 1 e r f

2


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Transient mass diffusion (erfcfrom heat transfer)


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Transient mass diffusion (Penetration depth)

Penetration depthof a medium is proportionalto the square-root of diffusion coefficient andtime

, ,

= , ,

, ,

E.g. for diffusion coefficient of zinc in copper at

1000 C we have 5.0 x 10-13m2/s, sopenetration depth after 10 h will be 0.24 mm

Diffusion coefficients in solid are very low andhence only thin layer will be affected

Solid can conveniently be treated as semi-infinite medium


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Transient mass diffusion (Summary)

Transient mass diffusion in a stationary mediumis analogues to transient heat transfer providedthat

1. The solution is dilute and thus the density ofthe medium is constant.

2. The diffusion coefficient is constant. Thisis valid for an isothermal medium since DABvaries with temperature (corresponds toconstant thermal diffusivity).

3. There are no homogeneous reactionsinthe medium that generate or deplete thediffusing speciesA (corresponds to no heat

generation).4. Initially (t=0) the concentration of speciesA

is constantthroughout the medium(corresponds to uniform initial temperature).


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Todays lecture





Irreversible thermodynamics (i.e.Maxwell-Stefan diffusion of a binary

mixture)

Assignments


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Diffusion in a porous medium

For porous media the description of Ficks first law of diffusion is altered to

account for effective diffusion coefficient as follows:

, , Typical the effective diffusion coefficient accounts for effective ordinary

diffusionand Knudsen diffusionin the following approximate manner:

1,

1,F,

1,,

Where ,F,is the effective ordinary diffusion coefficient and ,,isthe effective Knudsen diffusion coefficient.


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Ficks and Knudsens effective diffusion coefficients for a porous medium is

respectively given by:

,F,

,, ,

The porosityor void fractionof a porous medium accounts for thereduction of the cross-sectional area for diffusion posed by the solid

material The tortuosityaccounts for the increased diffusion length due to tortuous

path of real pores, and for the effects of constrictions and dead-end pores


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Porosity Tortuosity


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Void volume over total

volume:

The simplest mathematical

quantification of tortuosity is

the arcchord ratio:

Representative

elementary volume,


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Example: Effectiveness of catalyst pellet (A. F. Mills, 1995)

When a chemical reaction takes

place within a porous pellet, a

concentration gradient arises, and

the surfaces on pores deep

within the pelletare exposed to a

lower concentration than surfacenear the pore opening.

Lets consider a fixed-bed reactor

filled with spherical catalyst

pellets, which promotes the

following reaction:

2 C O O 2CO


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The differential equation governing the concentration

distribution of speciesin an spherical pellet can befound by applying the principle of species

conservationto an elementary control volume:

Net inflow of species Rate of consumptionof species 0

4 4 + 4 0 0

r=R

r

s

r


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Assumptions: Isothermal pellet, constant total pressure and constant

effective diffusion coefficient.

Governing equations:

0

1

, 0

Boundary conditions:

0 : bounded : ,


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Lets simplifyour notationby defining , and introducinga new independent variable :

0

The previous equation has the following general solution:

sinh coshor

sinh cosh


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Applying the first boundary conditionat 0yields: sinh0= cosh0= 0

Applying the second boundaryconditionat yields:

,sinh Thus, the final solutionbecomes:

,

sinhsinh where

,

,

rR0

yA

yA,s

0


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The mass rate at which species is consumed within the catalyst pellet isgiven by:

, = 4= 4, =

4,, 1 tanh

We now define the pellet effectivenessas the ratio of actual consumption

rate divided by that of a pellet with infinite diffusion coefficient:

, ,,= 4

,, 1 tanh4 3 , 3

1tanh

1


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In chemical engineering practice the effectiveness is rewritten in terms of

the Thiele modulusgiven by:

3

Thus

11

tanh3 1

3


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Property Value

Pellet diameter 0.5cmPorosity 0 . 8Tortuosity 4Catalytic surface

area 7.1210 cm cm Average pore

radius

1mTemperature 800Ficks diffusion

coefficient for CO-

air, 1.0610 m

Reaction rate 4.1910 m

Lets calculate the

effectiveness of a

spherical

CuO-on-Alumina pellet

used in the catalyst bed

of a oxidation converterfor a car

Properties are as shown

in the table to right


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Effective diffusion coefficient:

,F, 0.84.0 1.0610 m 2.1210 m

,, 97 0.84.0 9 7 1 0m 800273K28 kg kmol 1.0510 m

, ,F,,,

,F,

,, 1.7610 m


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Thiele modulus:

3 3

, 0.25cm

34.1910 m 7 .1210 cm cm

1.7610 m

34.3

Pellet effectiveness:

1

1

tanh3 1

3

134.3 1tanh334.3 1334.3 0.0293 or 2.92%

Why so low?


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Mass-averaged velocity:

or

Mass flux of species :

Total mass flux:

Diffusional mass flux of species is defined relative to the mass-

averaged velocity:


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The total mass flux of species includes both the diffusional massfluxand the bulk fluid mass flux:

vv

Dv

wi

i i

0


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Mole-averaged velocity:

or

Molar flux of species :

Total molar flux:

Diffusional molar flux of species is defined relative to the mole-

averaged velocity:

Notice that I have

changed the notation of

the book slightly!


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The total molar flux of species includes both the diffusional molarfluxand the bulk fluid molar flux:

vv

Dv

yi

i i

0


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We can now more precisely define Ficks first law of diffusion as the mass

flux or molar flux of species relative tothe mass-average or molaraveraged velocity, respectively:

These two equation are mathematical equivalent. However, this does not

in general imply that , actually:


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The mathematical equivalence for a binary mixturecan be shown as

follows:

1


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For a binary mixture in a moving mediumwe can now state the following:

The mass and molar flux are related as follows:


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For a binary mixture we can further show that the binary diffusion coefficient

ofand must be equal to each other: 1

0


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Mass diffusion in a moving medium (Stefan flow)

Stefan flowis the diffusion of vapor through a

stationary gas.

Assumptions: Constant pressure , temperature and thus concentration , the gas is unsoluble inthe liquid, the vapor pressure is equal to thesaturation pressure at 0(equilibrium), thevapor pressure at is lower than at 0, andsteady state has been reached.

The diffusion of species occurs towards theopening, creating a bulk fluid transportthat

opposes the diffusion of species

0 and 0


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The molar flux of the binary mixture then becomes:

1

Applying the principle of conservation of mass species yields the governing

equation:

1 0


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The solution of the previous equation is known as Stefans Law:

ln1 ,1 ,

The molar fraction profilesare given by the following relations:

1 1 , 1 ,1 ,

and ,

,

The described induced convective flow that enhances mass diffusion is

called Stafan flow


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Mass diffusion in a moving medium (Equimolar)

Consider two large reservoirs connected

by a channel of length and containinga binary mixture of gases and at auniform pressure and temperature.

For ideal gases ( ) and thespecified assumptions the totalconcentration is constant:

or

0


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Interestingly, the mixture is stationaryon a molar basis:

0 0 However, it is not stationaryon a mass basis:

It is

incorrect in

engel


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For one-dimensional flow through a channel of uniform cross sectional areawith no homogeneous chemical reaction the molar fractions are:

, , , ,

, ,

, , , , , ,


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Combustion Technology andChemical ReactorsMass Transfer 48

Todays lecture






mixture)

Assignments


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Mass diffusion in an moving medium

(Irreversible thermodynamics)

The force acting per mole of species

A is balanced by friction between the

diffusing species A and B:

,Dv F

Where is the chemical potentialof species, is the Maxwell-Stefan diffusion coefficient and

is the relative velocitybetween species and .

A

B

B

BB

uA

uB


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Chemical potential is defined as follows:

,,

The change in chemical potential due to pressure is given by:

l n The activity is an effective mole fraction and for a real fluid it is defined

as:

Where is the chemical potential at standard state and is the activity

coefficient.


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The previous equation is now multiplied by the mole fraction andrewritten in terms of molar fluxes:

,

The diffusive molar flux can be rewritten as follows:

,


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The chemical potential of a real-fluid is rewritten in terms of molar fractionas follows:

, 1

The binary diffusion is now a function of the binary Maxwell-Stefandiffusion coefficient:

1 . This now implies that:

,


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Todays lecture






mixture)

Assignments


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Assignments

Problem numbering is based on the file ch14.pdf!

P14-56C14-62C

P14-70C14-72C

P14-77C14-81C

P14-64

P14-76

P14-84


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Lecture 2 Mass Transfer