Mass Transfer 1 by Dr. M A Rashid

8/10/2019 Mass Transfer 1 by Dr. M A Rashid

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3. Molecular mass transport

3.1 Introduction to mass transfer 3.2 Properties of mixtures

3.2.1 Concentration of species

3.2.2 Mass Averaged velocity3.3 Diffusion flux3.3.1 Picks !a"3.3.2 #elation among molar fluxes

3.$ Diffusivity3.$.1 Diffusivity in gases3.$.2 Diffusivity in li%uids3.$.3 Diffusivity in solids

3.& 'teady state diffusion3.&.1 Diffusion t(roug( a stagnant gas film

3.&.2 Pseudo ) steady ) state diffusion t(roug( a stagnant gas film.3.&.3 *%uimolar counter diffusion.3.&.$ Diffusion into an infinite stagnant medium.3.&.& Diffusion in li%uids3.&.+ Mass diffusion "it( (omogeneous c(emical reaction.3.&., Diffusion in solids

3.+ -ransient Diffusion.

3.1 Introduction of Mass Transfer

(en a system contains t"o or more components "(ose concentrations varyfrom point to point/ t(ere is a natural tendency for mass to 0e transferred/minimiing t(e concentration differences "it(in a system. -(e transport of oneconstituent from a region of (ig(er concentration to t(at of a lo"er concentrationis called mass transfer.

-(e transfer of mass "it(in a fluid mixture or across a p(ase 0oundary is aprocess t(at plays a maor role in many industrial processes. *xamples of suc(processes are

4i5 Dispersion of gases from stacks4ii5 #emoval of pollutants from plant disc(arge streams 0y a0sorption4iii5 'tripping of gases from "aste "ater 4iv5 6eutron diffusion "it(in nuclear reactors4v5 Air conditioning

Many of air day70y7day experiences also involve mass transfer/ for example

1



4i5 A lump of sugar added to a cup of coffee eventually dissolves and t(eneventually diffuses to make t(e concentration uniform.

4ii5 ater evaporates from ponds to increase t(e (umidity of passing7air7stream

4iii5 Perfumes presents a pleasant fragrance "(ic( is imparted t(roug(out

t(e surrounding atmosp(ere.

-(e mec(anism of mass transfer involves 0ot( molecular diffusion andconvection.

3.2 Properties of Mixtures

Mass transfer al"ays involves mixtures. Conse%uently/ "e must account for t(evariation of p(ysical properties "(ic( normally exist in a given system. (en asystem contains t(ree or more components/ as many industrial fluid streams do/t(e pro0lem 0ecomes un"idely very %uickly. -(e conventional engineering

approac( to pro0lems of multicomponent system is to attempt to reduce t(em torepresentative 0inary 4i.e./ t"o component5 systems.

In order to understand t(e future discussions/ let us first consider definitions andrelations "(ic( are often used to explain t(e role of components "it(in a mixture.

3.2.1 Concentration of Species:

Concentration of species in multicomponent mixture can 0e expressed in many

"ays. 8or species A/ mass concentration denoted 0y ρ A is defined as t(e mass of

A/m A per unit volume of t(e mixture.

V

m A A = ρ 777777777777777777777777777777777777 415

-(e total mass concentration density ρ is t(e sum of t(e total mass of t(e mixture

in unit volume

∑=i

i ρ ρ

"(ere ρ i is t(e concentration of species i in t(e mixture.

Molar concentration of/ A/ C A is defined as t(e num0er of moles of A present perunit volume of t(e mixture.

9y definition/

Aof weight molecular

Aof massmolesof Number =

2



A

A A

M

mn = 77777777777777777777777777777 425

-(erefore from 415 : 425

A

A A A

M V

nC

ρ ==

8or ideal gas mixtures/

T R

V pn

A A = ; from Ideal gas la" P< = n#->

T R p

V nC A A A ==

"(ere p A is t(e partial pressure of species A in t(e mixture. < is t(e volume ofgas/ - is t(e a0solute temperature/ and # is t(e universal gas constant.

-(e total molar concentration or molar density of t(e mixture is given 0y

∑=i

i C C

3.2.2 Velocities

In a multicomponent system t(e various species "ill normally move at differentvelocities? and evaluation of velocity of mixture re%uires t(e averaging of t(evelocities of eac( species present.

If ν I is t(e velocity of species i "it( respect to stationary fixed coordinates/ t(en

mass7average velocity for a multicomponent mixture defined in terms of massconcentration is/

ρ

ν ρ

ρ

ν ρ

ν i

i i

i i

i i

i ∑

∑

∑==

9y similar "ay/ molar7average velocity of t(e mixture ν @ is

C

V C i

i i ∑=@ν

3



8or most engineering pro0lems/ t(ere "ill 0e title difference in ν @ and ν and so

t(e mass average velocity/ ν/ "ill 0e used in all furt(er discussions.

-(e velocity of a particular species relative to t(e mass7average or molar

average velocity is termed as diffusion velocity

4i.e.5 Diffusion velocity = ν i 7 ν

-(e mole fraction for li%uid and solid mixture/ x A /and for gaseous mixtures/ y A/are t(e molar concentration of species A divided 0y t(e molar density of t(emixtures.

C

C x A

A = 4li%uids and solids5

C

C y A A = 4gases5.

-(e sum of t(e mole fractions/ 0y definition must e%ual 1?

4i.e.5 ∑ =i

i x 1

∑ =i

i y 1

0y similar "ay/ mass fraction of A in mixture is?

ρ

ρ A Aw =

1. -(e molar composition of a gas mixture at 2,3 B and 1.& @ 1 & Pa is

2 ,C 1C 2 1&6 2 +E

Determine

a5 t(e composition in "eig(t percent05 average molecular "eig(t of t(e gas mixturec5 density of gas mixtured5 partial pressure of 2.

4



Calculations:!et t(e gas mixture constitutes 1 mole. -(en

2 = ., molC = .1 mol

C 2 = .1& mol6 2 = .+E mol

Molecular "eig(t of t(e constituents are

2 = 2 @ 1+ = 32 gFmolC = 12 G 1+ = 2E gFmolC 2 = 12 G 2 @ 1+ = $$ gFmol6 2 = 2 @ 1$ = 2E gFmol

eig(t of t(e constituents are 41 mol of gas mixture5

2 = ., @ 32 = 2.2$ gC = .1 @ 2E = 2.E gC 2 = .1& @ $$ = +.+ g6 2 = .+E @ 2E = 1H.$ g

-otal "eig(t of gas mixture = 2.2$ G 2.E G +.+ G 1H.$= 3.+E g

Composition in "eig(t percent

D3A.,1AA@

+E.3A

2$.22 ==O

D13.H1AA@+E.3A

EA.2==CO

D&1.211AA@+E.3A

+A.+2 ==CO

DA+.+21AA@+E.3A

A$.1H2 ==N

Average molecular "eig(t of t(e gas mixturemolesof 6um0er

mixturegasof eig(t=M

mol g M +E.3A

1

+E.3A==

Assuming t(at t(e gas o0eys ideal gas la"/P< = n#-

RT

P

V

n=

5



mV

n ρ == densitymolar

-(erefore/ density 4or mass density5 = ρ mM

(ere M is t(e molecular "eig(t of t(e gas.

3&

2,3@E31$

+E.3@1@&.1mkg

RT

PM M Density m === ρ

= 2.3 kgFm 3

Partial pressure of 2 = ;mole fraction of 2> @ total pressure

( )&1A@&.1@1AA

,=

= ., @ 1.& @ 1 &

= .1& @ 1 &

Pa

3.3 Diffusion flux

ust as momentum and energy 4(eat5 transfer (ave t"o mec(anisms fortransport7molecular and convective/ so does mass transfer. Jo"ever/ t(ere areconvective fluxes in mass transfer/ even on a molecular level. -(e reason for t(isis t(at in mass transfer/ "(enever t(ere is a driving force/ t(ere is al"ays a netmovement of t(e mass of a particular species "(ic( results in a 0ulk motion ofmolecules. f course/ t(ere can also 0e convective mass transport due to

macroscopic fluid motion. In t(is c(apter t(e focus is on molecular mass transfer.

-(e mass 4or molar5 flux of a given species is a vector %uantity denoting t(eamount of t(e particular species/ in eit(er mass or molar units/ t(at passes pergiven increment of time t(roug( a unit area normal to t(e vector. -(e flux ofspecies defined "it( reference to fixed spatial coordinates/ 6 A is

A A A C N ν = 7777777777777777777777 415

-(is could 0e "ritten interms of diffusion velocity of A/ 4i.e./ ν A - ν) and average

velocity of mixture/ ν/ as

ν ν ν A A A A C C N +−= 54 777777777777777 425

9y definition

C

C i

i i ∑==

ν

ν ν @

6



-(erefore/ e%uation 425 0ecomes

∑+−=i

i i A

A A A C C

C C N ν ν ν 54

∑+−=i

i i A A A C y C ν ν ν 54

8or systems containing t"o components A and 9/

5454 A A A A A A C C y C N ν ν ν ν ++−= 5454 A A A A N N y C ++−= ν ν

N y C N A A A A +−= 54 ν ν 77777777777 435

-(e first term on t(e rig(t (and side of t(is e%uation is diffusional molar flux of A/and t(e second term is flux due to 0ulk motion.

3.3.1 Fic!s la":

An empirical relation for t(e diffusional molar flux/ first postulated 0y 8ick and/accordingly/ often referred to as 8icks first la"/ defines t(e diffusion ofcomponent A in an isot(ermal/ iso0aric system. 8or diffusion in only t(e Kdirection/ t(e 8icks rate e%uation is

! "

C " D# A

A A −=

"(ere D AB is diffusivity or diffusion coefficient for component A diffusing t(roug(component 9/ and dC A F dK is t(e concentration gradient in t(e K7direction.

A more general flux relation "(ic( is not restricted to isot(ermal/ iso0asic systemcould 0e "ritten as

! "

y " DC # A

A A −= 77777777777777777 4$5

using t(is expression/ *%uation 435 could 0e "ritten as

N y ! "

y " DC N A

A A A +−= 777777777777777 4&5

3.3.2 #elation amon$ molar fluxes:

8or a 0inary system containing A and 9/ from *%uation 4&5/

7



N y # N A A A +=

or N y N # A A A += 77777777777777777777777 4+5

'imilarly/

N y N # += 77777777777777777777 4,5

Addition of *%uation 4+5 : 4,5 gives/

N y y N N # # A A A 54 +−+=+ 7777777777 4E5

9y definition 6 = 6 A G 6 9 and y A G y 9 = 1.-(erefore e%uation 4E5 0ecomes/ A G 9 = A = 7 9

! "

y " DC

$ "

y " DC

A A

A −= 777777777777777 4H5

8rom y A G y 9 = 1 dy A = 7 dy 9

-(erefore *%uation 4H5 0ecomes/

D A9 = D 9A 77777777777777777777777777777777777 415

-(is leads to t(e conclusion t(at diffusivity of A in 9 is e%ual to diffusivity of 9 in A.

3.% Diffusi&it'

8icks la" proportionality/ D A9/ is kno"n as mass diffusivity 4simply as diffusivity5or as t(e diffusion coefficient. D A9 (as t(e dimension of ! 2 F t/ identical to t(e

fundamental dimensions of t(e ot(er transport properties Binematic viscosity/ νη= 4µ F ρ5 in momentum transfer/ and t(ermal diffusivity/ α 4= k F ρ C ρ 5 in (eat

transfer.

Diffusivity is normally reported in cm2 F sec? t(e 'I unit 0eing m2 F sec.

Diffusivity depends on pressure/ temperature/ and composition of t(e system.

In ta0le/ some values of D A9 are given for a fe" gas/ li%uid/ and solid systems.

8



Diffusivities of gases at lo" density are almost composition independent/ incease"it( t(e temperature and vary inversely "it( pressure. !i%uid and soliddiffusivities are strongly concentration dependent and increase "it( temperature.

Leneral range of values of diffusivity

Lases & 1 )+ 7777777777777 1 17& m2 F sec.!i%uids 1 )+ 7777777777777 17H m2 F sec.'olids & 1 )1$ 7777777777777 1 171 m2 F sec. In t(e a0sence of experimental data/ semit(eoretical expressions (ave 0eendeveloped "(ic( give approximation/ sometimes as valid as experimental values/due to t(e difficulties encountered in experimental measurements.

3.%.1 Diffusi&it' in (ases:

Pressure dependence of diffusivity is given 0y

p

D A1

∝ 4for moderate ranges of pressures/ upto 2& atm5.

And temperature dependency is according to

23

T D A ∝

Diffusivity of a component in a mixture of components can 0e calculated usingt(e diffusivities for t(e various 0inary pairs involved in t(e mixture. -(e relationgiven 0y ilke is

n

nmixture

D

y

D

y

D

y D

−−−

− ′++

′+

′=

131

3

21

21

...........

1

(ere D 17mixture is t(e diffusivity for component 1 in t(e gas mixture? D 17n is t(ediffusivity for t(e 0inary pair/ component 1 diffusing t(roug( component n? and

ny ′ is t(e mole fraction of component n in t(e gas mixture evaluated on a

component )1 ) free 0asis/ t(at is

ny y y

y y

.......32

22

++

=′

H. Determine t(e diffusivity of Co 2 415/ 2 425 and 6 2 435 in a gas mixture (avingt(e composition

Co2 2E.& / 2 1&/ 6 2 &+.&/

-(e gas mixture is at 2,3 k and 1.2 @ 1 & Pa. -(e 0inary diffusivity values aregiven as 4at 2,3 B5

9



D 12 P = 1.E,$ m 2 PaFsecD 13 P = 1.H$& m 2 PaFsecD 23 P = 1.E3$ m 2 PaFsec

Calculations:Diffusivity of Co 2 in mixture

13

3

12

2

11

D

y

D

y D m ′

+′=

"(ere 21.A&+&.A1&.A

1&.A

32

22 =

+=

+=′

y y

y y

,H.A&+&.A1&.A&+&.A

32

33 =

+=

+=′

y y y y

-(erefore

H$&.1

,H.A

E,$.1

21.A

11

+=P D m

= 1.H3 m 2.PaFsec

'ince P = 1.2 @ 1 & Pa/

sec1A@+1.1

[email protected]

H3.1 2&

&1 mD m−==

Diffusivity of 2 in t(e mixture/

23

3

21

1

21

D

y

D

y D m ′

+′=

(ere 33&.A&+&.A2E&.A

2E&.A

31

11 =

+=

+=′

y y

y y

4mole fraction on72 free 0ans5.

and

++&.A&+&.A2E&.A

&+&.A

31

33 =

+=

+=′

y y

y y

andD 21 P = D 12 P = 1.E,$ m 2.PaFsec

-(erefore

10



E3$.1

++&.A

E,$.1

33&.A

12

+=P D m

= 1.E$, m 2.PaFsec

sec1A@&[email protected]

E$,.1 2&&2 mmD −==

9y 'imilar calculations Diffusivity of 6 2 in t(e mixture can 0e calculated/ and isfound to 0e/ D 3m = 1.&EE @ 1 )& m 2Fsec.

3.%.2 Diffusi&it' in li)uids:

Diffusivity in li%uid are exemplified 0y t(e values given in ta0le N Most of t(esevalues are nearer to 17& cm2 F sec/ and a0out ten t(ousand times s(o"er t(an

t(ose in dilute gases. -(is c(aracteristic of li%uid diffusion often limits t(e overallrate of processes accruing in li%uids 4suc( as reaction 0et"een t"o componentsin li%uids5.

In c(emistry/ diffusivity limits t(e rate of acid70ase reactions? in t(e c(emicalindustry/ diffusion is responsi0le for t(e rates of li%uid7li%uid extraction. Diffusionin li%uids is important 0ecause it is slo".

Certain molecules diffuse as molecules/ "(ile ot(ers "(ic( are designated aselectrolytes ionie in solutions and diffuse as ions. 8or example/ sodium c(loride46aCl5/ diffuses in "ater as ions 6a G and Cl7. -(oug( eac( ions (as a different

mo0ility/ t(e electrical neutrality of t(e solution indicates t(e ions must diffuse att(e same rate? accordingly it is possi0le to speak of a diffusion coefficient formolecular electrolytes suc( as 6aCl. Jo"ever/ if several ions are present/ t(ediffusion rates of t(e individual cations and anions must 0e considered/ andmolecular diffusion coefficients (ave no meaning.

Diffusivity varies inversely "it( viscosity "(en t(e ratio of solute to solvent ratioexceeds five. In extremely (ig( viscosity materials/ diffusion 0ecomesindependent of viscosity.

3.%.3 Diffusi&it' in solids:

-ypical values for diffusivity in solids are s(o"n in ta0le. ne outstandingc(aracteristic of t(ese values is t(eir small sie/ usually t(ousands of time lesst(an t(ose in a li%uid/ "(ic( are inturn 1/ times less t(an t(ose in a gas.

Diffusion plays a maor role in catalysis and is important to t(e c(emicalengineer. 8or metallurgists/ diffusion of atoms "it(in t(e solids is of moreimportance.

11



3.* Stead' State Diffusion

In t(is section/ steady7state molecular mass transfer t(roug( simple systems in"(ic( t(e concentration and molar flux are functions of a single space coordinate

"ill 0e considered.

In a 0inary system/ containing A and 9/ t(is molar flux in t(e direction of / asgiven 0y *%n 4&5 is ;section 3.3.1>

54 A A A

A A N N y $ "

y " DC N ++−= 777 415

3.*.1 Diffusion t+rou$+ a sta$nant $as film

-(e diffusivity or diffusion coefficient for a gas can 0e measured/ experimentallyusing Arnold diffusion cell. -(is cell is illustrated sc(ematically in figure.

fi$ure

-(e narro" tu0e of uniform cross section "(ic( is partially filled "it( pure li%uid A/ is maintained at a constant temperature and pressure. Las 9 "(ic( flo"sacross t(e open end of t(e tu0/ (as a negligi0le solu0ility in li%uid A/ and is alsoc(emically inert to A. 4i.e. no reaction 0et"een A : 95.

Component A vapories and diffuses into t(e gas p(ase? t(e rate of vaporiationmay 0e p(ysically measured and may also 0e mat(ematically expressed intermsof t(e molar flux.

Consider t(e control volume ' ∆ / "(ere ' is t(e cross sectional area of t(e

tu0e. Mass 0alance on A over t(is control volume for a steady7state operationyields

;Moles of A leaving at G ∆> ) ;Moles of A entering at > = .

4i.e.5 .A=−∆+ $

A$ $

A N % N % 77777777777777 415

Dividing t(roug( 0y t(e volume/ '∆K/ and evaluating in t(e limit as ∆K

approac(es ero/ "e o0tain t(e differential e%uation

A=$ "

N " A 7777777777777777777777777 425

-(is relation stipulates a constant molar flux of A t(roug(out t(e gas p(ase fromK1 to K2.

A similar differential e%uation could also 0e "ritten for component 9 as/

12



/A=! "

N "

and accordingly/ t(e molar flux of 9 is also constant over t(e entire diffusion pat(

from 1 and 2.

Considering only at plane 1/ and since t(e gas 9 is insolu0le is li%uid A/ "erealie t(at 69/ t(e net flux of 9/ is ero t(roug(out t(e diffusion pat(? accordingly9 is a stagnant gas.

8rom e%uation 415 4of section 3.&5

54 A A A

A A N N y $ "

y " DC N ++−=

'ince 6 9 = /

A A A

A A N y $ "

y " DC N +−=

#earranging/

$ "

y "

y

DC N A

A

A A −

−=1

777777777777 435

-(is e%uation may 0e integrated 0et"een t(e t"o 0oundary conditions at = 1 O A = O A1

And at = 2 O A = y A2

Assuming t(e diffusivity is to 0e independent of concentration/ and realiing t(at6 A is constant along t(e diffusion pat(/ 0y integrating e%uation 435 "e o0tain

∫ ∫ −−

= 2

1

2

11

A

A

y

y A

A A

!

! A

y

y " DC $ " N

−−

−=

1

2

12 1

1ln

A

A A A

y

y

! !

DC N 777777777777774$5

-(e log mean average concentration of component 9 is defined as

13



−=

1

2

12/

ln

lm

y y

y y y

'ince Ay y

−=1

/

−=

−−−=

1

2

21

1

2

12/

lnln

514514

A

A

A A

A

A

A Alm

y y

y y

y y

y y y

7777777 4&5

'u0stituting from *%n 4&5 in *%n 4$5/

lm

A A A Ay

y y $ !

DC N /21

1254 −−= 77777777777777777777 4+5

8or an ideal gasT R

p

V

nC == / and

for mixture of ideal gasesP

py A

A =

-(erefore/ for an ideal gas mixture e%uation. 4+5 0ecomes

lm

A A A A

p

p p

$ $ RT

DN

/

21

12

54

54−

−=

-(is is t(e e%uation of molar flux for steady state diffusion of one gas t(roug( asecond stagnant gas.

Many mass7transfer operations involve t(e diffusion of one gas componentt(roug( anot(er non7diffusing component? a0sorption and (umidification aretypical operations defined 0y t(ese e%uation.

-(e concentration profile 4y A vs. 5 for t(is type of diffusion is s(o"n in figure

Fi$ure

12. xygen is diffusing in a mixture of oxygen7nitrogen at 1 std atm/ 2& °C.

Concentration of oxygen at planes 2 mm apart are 1 and 2 volume respectively. 6itrogen is non7diffusing.

14



4a5 Derive t(e appropriate expression to calculate t(e flux oxygen. Defineunits of eac( term clearly.

405 Calculate t(e flux of oxygen. Diffusivity of oxygen in nitrogen = 1.EH @ 1 )&

m 2Fsec.

Solution:

!et us denote oxygen as A and nitrogen as 9. 8lux of A 4i.e.5 6 A is made up of t"o components/ namely t(at resulting from t(e 0ulk motion of A 4i.e.5/ 6x A andt(at resulting from molecular diffusion A

A A A # Nx N += 7777777777777777777777777777777777 415

8rom 8icks la" of diffusion/

$ "

C " D#

A A A −= 77777777777777777777777777777777777777777 425

'u0stituting t(is e%uation 415

$ "

C " DNx N

A A A A −= 77777777777777777777777777777 435

'ince 6 = 6 A G 6 9 and x A = C A F C e%uation 435 0ecomes

( )$ "

C " D

C

C N N N

A A

A A A −+=

#earranging t(e terms and integrating 0et"een t(e planes 0et"een 1 and 2/

( )∫ ∫ +−−= 2

1

A

A

C

C A A A

A

A N N C C N

"C

cD

$ " 77777777777777 4$5

'ince 9 is non diffusing 6 9 = . Also/ t(e total concentration C remains constant.-(erefore/ e%uation 4$5 0ecomes

∫ −−= 2

1

A

A

C

C A A A

A

A C N C N

"C

CD

$

1

2ln

1

A

A

A C C

C C

N −

−=

-(erefore/

15



1

2ln

A

A A A

C C

C C

$

CDN

−

−= 7777777777777777777777777777 4&5

#eplacing concentration in terms of pressures using Ideal gas la"/ e%uation 4&50ecomes

1

2ln

At

At t A A

P P

P P

RT$

P DN

−

−= 777777777777777777777777777 4+5

"(ereD A9 = molecular diffusivity of A in 9P - = total pressure of system# = universal gas constant- = temperature of system in a0solute scale = distance 0et"een t"o planes across t(e direction of diffusionP A1 = partial pressure of A at plane 1/ andP A2 = partial pressure of A at plane 2

(i&en:D A9 = 1.EH @ 1 )& m2FsecP t = 1 atm = 1.132& @ 1 & 6Fm 2

- = 2&°C = 2,3 G 2& = 2HE B

= 2 mm = .2 mP A1 = .2 @ 1 = .2 atm 48rom Ideal gas la" and additive pressure rule5P A2 = .1 @ 1 = .1 atm

'u0stituting t(ese in e%uation 4+5

( ) ( ) ( )

−−=

−

2.A1

1.A1ln

AA2.A2HEE31$

1A@A132&[email protected] &&

AN

= $.&& @ 1 )& kmolFm 2.sec

3.*.2 Psuedo stead' state diffusion t+rou$+ a sta$nant film:

In many mass transfer operations/ one of t(e 0oundaries may move "it( time. Ift(e lengt( of t(e diffusion pat( c(anges a small amount over a long period of

time/ a pseudo steady state diffusion model may 0e used. (en t(is conditionexists/ t(e e%uation of steady state diffusion t(roug( stagnant gas can 0e usedto find t(e flux.

fi$ure

If t(e difference in t(e level of li%uid A over t(e time interval considered is only asmall fraction of t(e total diffusion pat(/ and t ) t is relatively long period of time/

16



at any given instant in t(at period/ t(e molar flux in t(e gas p(ase may 0eevaluated 0y

lm

A A A A

$y

y y DC N

/

21 54 −= 777777777777777777 415

"(ere e%uals 2 ) 1/ t(e lengt( of t(e diffusion pat( at time t.

-(e molar flux 6 A is related to t(e amount of A leaving t(e li%uid 0y

t "

$ "

M N

A

& A A

/ ρ = 77777777777777777777777777 425

"(ere A

& A

M

/ ρ is t(e molar density of A in t(e li%uid p(ase

under Psuedo steady state conditions/ e%uations 415 : 425 can 0e e%uated to give

lm

A A A

A

& A

y $

y y DC

t "

$ "

M /

21/ 54 −=

ρ

777777777777777 435

*%uation. 435 may 0e integrated from t = to t and from = t to = t as

∫ ∫ −=

=

t

t

!

! A A A

Alm& At

t

"$ $ y y DC

M y "t

A54 21

//

A

ρ

yielding

−−

=254

2A

2

21

// t t

A A A

Alm& A $ $

y y DC

M y t

ρ

77777777 4$5

-(is s(all 0e rearranged to evaluate diffusivity D A9 as/

−−

=254

2A

2

21

// t t

A A A

lm& A A

$ $

t y y C M

y D

ρ

1. , &ertical $lass tu-e 3 mm in diameter is filled "it+ li)uid toluene to a

dept+ of 2mm from t+e top openend. ,fter 2/* +rs at 30.% C and a total

pressure of / mm $ t+e le&el +as dropped to mm from t+e top.Calculate t+e &alue of diffusi&it'.

Data

vapor pressure of toluene at 3H.$°C = ,.+$ k6 F m2/

17



density of li%uid toluene = E& kgFm3 Molecular "eig(t of toluene = H2

4C + J+ CJ35

( )

−

−= 2

2A

2

21

/ t t

A A A

lm& A

A

! !

t y y C M

y D

ρ

fi$ure

"(ere

−=

1

2

12/

ln

ml

y

y

y y y

y 92 = 1 ) y A2 y 91 = 1 ) y A1

3.1A1

+$.,11 ==

P

py

A A 4,+ mm Jg = 11.3 k6Fm25

= .,&$ y 91 = 1 ) .,&$ = .H2$+

y A2 = y 9 = 1 ) y A2 = 1

-(ereforeH+1E.A

H2$+.A

1ln

H2$+.A1/ =

−

=lmy

( )$.3H2,3@E31$

1A@A132&.1 &

+==

T R

P C

= .3H k mol Fm3

-(erefore

( )

−−= 2

A2.AAE.A

3+AA@2,&@AA,&[email protected]@H2

H+1E.A@E&A 22

AD

= 1.&2+2 @ 1 )3 4.E 2 ) .2 25= H.1&,2 @ 17+ m2Fsec.

3.*.3 4)uimolar counter diffusion:

A p(ysical situation "(ic( is encountered in t(e distillation of t"o constituents"(ose molar latent (eats of vaporiation are essentially e%ual/ stipulates t(at t(eflux of one gaseous component is e%ual to 0ut acting in t(e opposite direction

from t(e ot(er gaseous component? t(at is/ 6 A = 7 69.

-(e molar flux 6 A/ for a 0inary system at constant temperature and pressure isdescri0ed 0y

54 A A A

A A N N y $ "

y " DC N ++−=

18



or 54 A A A

A A N N y $ "

C " DN ++−= 7777777 415

"it( t(e su0stitution of 69 = 7 6 A/ *%uation 415 0ecomes/

$ "

C " DN A A A −= 77777777777777777 425

8or steady state diffusion *%uation. 425 may 0e integrated/ using t(e 0oundaryconditions at = 1 C A = C A1

and = 2 C A = C A2

Living/

∫ ∫ −=

2

1

2

1

A

A

C

C A A

!

! A C " D$ " N

from "(ic(

54 2112

A A A

A C C $ $

DN −

−= 7777777777777777777 435

8or ideal gases/T R

p

V

nC A A A == . -(erefore *%uation. 435 0ecomes

5454 21

12 A A

A A P P

$ $ T R

DN −

−= 7777777777 4$5

-(is is t(e e%uation of molar flux for steady7state e%uimolar counter diffusion.

Concentration profile in t(ese e%uimolar counter diffusion may 0e o0tained from/

A54 = AN $ "

" 4'ince 6 A is constant over t(e diffusion pat(5.

And from e%uation. 425

$ "

C " DN A A A −= .

-(erefore

19



A=

−

$ "

C " D

$ "

" A A .

or .A2

2

=$ "

C " A

-(is e%uation may 0e solved using t(e 0oundary conditions to give

21

1

2

1

1$ $

$ $

C C

C C

A

A

A

A

−−

=−

− 77777777777777 4&5

*%uation/ 4&5 indicates a linear concentration profile for e%uimolar counterdiffusion.

3. Met(ane diffuses at steady state t(roug( a tu0e containing (elium. At point 1t(e partial pressure of met(ane is p A1 = && kPa and at point 2/ .3 m apart P A2 =1& BPa. -(e total pressure is 11.32 kPa/ and t(e temperature is 2HE B. At t(ispressure and temperature/ t(e value of diffusivity is +.,& @ 1 )& m 2Fsec.

i5 calculate t(e flux of CJ $ at steady state for e%uimolar counter diffusion.

ii5 Calculate t(e partial pressure at a point .2 m apart from point 1.

Calculation:

8or steady state e%uimolar counter diffusion/ molar flux is given 0y

( )21 A A A

A p p$ T R

DN −= 777777777777777777777777777 415

-(erefore?

( )sec.

1&&&A3.A@2HE@31$.E

1A@,&.+2

&

m

kmol N A −=

−

sec1A@+33.3

2&

m

kmol −=

And from 415/ partial pressure at .2 m from point 1 is

( ) A p−=−

− &&A2.A@2HE@31$.E

1A@,&.+1A@+33.3

&&

p A = 2E.33 kPa

20



11. In a gas mixture of (ydrogen and oxygen/ steady state e%uimolar counter

diffusion is occurring at a total pressure of 1 kPa and temperature of 2°C. If

t(e partial pressures of oxygen at t"o planes .1 m apart/ and perpendicular tot(e direction of diffusion are 1& kPa and & kPa/ respectively and t(e massdiffusion flux of oxygen in t(e mixture is 1.+ @ 1 )& kmolFm 2.sec/ calculate t(e

molecular diffusivity for t(e system.

Solution:

8or e%uimolar counter current diffusion

( )21 A A A

A p pRT$

DN −= 777777777777777777777777 415

"(ere

6 A = molar flux of A 41.+ @ 1 )& kmolFm 2.sec5D A9 = molecular diffusivity of A in 9# = niversal gas constant 4E.31$ kFkmol.k5- = -emperature in a0solute scale 42,3 G 2 = 2H3 B5 = distance 0et"een t"o measurement planes 1 and 2 4.1 m5P A1 = partial pressure of A at plane 1 41& kPa5? andP A2 = partial pressure of A at plane 2 4& kPa5

'u0stituting t(ese in e%uation 415

( ) ( ) ( ) ( )&1&

A1.A2H331$.E1A@+.1 &

−=

− AD

-(erefore/ D A9 = 3.EHE @ 1 )& m 2Fsec

2. A tu0e 1 cm in inside diameter t(at is 2 cm long is filled "it( Co 2 and J2 at a

total pressure of 2 atm at °C. -(e diffusion coefficient of t(e Co2 ) J2 system

under t(ese conditions is .2,& cm2Fsec. If t(e partial pressure of Co2 is 1.& atmat one end of t(e tu0e and .& atm at t(e ot(er end/ find t(e rate of diffusion for

i5 steady state e%uimolar counter diffusion 46 A = 7 6 95ii5 steady state counter diffusion "(ere 6 9 = 7.,& 6 A/ andiii5 steady state diffusion of Co2 t(roug( stagnant J2 469 = 5

i5 ( ) A A A

A A N N y $ "

y " DC N ++−=

Liven6 9 = 7 6 A

21



-(erefore$ "

C " D

$ "

y " DC N

A A

A A A −=−=

48or ideal gas mixtureT R

pC A A = "(ere p A is t(e partial pressure of A? suc( t(at

p A G p 9 = P5

-(erefore( )

$ "

RT p" DN A

A A −=

8or isot(ermal system/ - is constant

-(erefore$ "

p"

RT

DN

A A A

−=

4i.e.5 ∫ ∫ −=

2

1

2

1

A

A

P

P A

A!

! A p"

RT

D$ " N

( )21 A A A

A p p$ RT

DN −= 7777777777777777777777777777777777 415

"(ere K = K 2 ) K 1

Liven D A9 = .2,& cm 2Fsec = .2,& @ 1 )$ m 2 Fsec ? - = °C = 2,3 k

( )&&$

1A@A132&.1@&.A1A@A132&.1@&.12.A@2,3@E31$

1A@2,&.A −=−

AN

sec1A@13E.+

2+

mmol k −=

#ate of diffusion = 6 A '

(ere ' is surface area

-(erefore rate of diffusion = +.13E @ 1 7+ @ π r 2

= +.13E @ 1 )+ @ π 4.& @ 1 )25 2

= $.E21 @ 1 )1 k molFsec= 1.,3& @ 1 )3 molF(r.

ii5 ( ) A A A

A A N N y $ "

y " DC N ++−=

given 6 9 = 7 .,& 6 A

-(erefore ( ) A A A A

A A N N y $ "

y " DC N ,&.A−+−=

22



A A A

A N y $ "

y " DC 2&.A+−=

$ "

y " DC N y N

A A A A A −=− 2&.A

A

A A A

y

y " DC $ " N

2&.A1−−=

for constant 6 A and C

∫ ∫ −−=

2

1

2

12&.A1

A

A

y

y A

A A

!

! A

y

y " CD$ " N

( )

+=

+∫ x bab x ba

x " ln

1

( ) ( )[ ] 2

1

2&.A1ln

2&.A

1 A

A

y

y

A A A y DC $ N −

−−=

−−

−=1

2

2&.A1

2&.A1ln

$

A

A A A

y

y

$

CDN 7777777777777777777777777777777777 425

(i&en:

3&

AEH3.A2,3@E31$

1A@A132&.1@2mmol '

T R

pC ===

,&.A2

&.111 ===

P

py

A A

2&.A2

&.A2

2 === P

p

y

A

A

'u0stituting t(ese in e%uation 425/

−−

=−

,&.A@2&.A1

2&.A@2&.A1ln

2.A

1A@2,&[email protected]@$ $

AN

sec1A@A2E.,

2+

m

kmol −=

#ate of diffusion = 6 A ' = ,.2E @ 1 )+ @ π @ 4.& @ 1 )25 2

= &.&2 @ 1 )1 kmolFsec = 1.HE, @ 1 )3 molF(r.

iii5 ( ) A A A

A A N N y $ "

y " CDN ++−=

(i&en: 6 9 =

-(erefore A A A

A A N y $ "

y " CDN +−=

23



∫ ∫ −−=

2

1

2

11

A

A

y

y A

A A

!

! A

y

y " CD$ " N

−

−=

1

2

1

1ln

A

A A

y

y

!

CD

−−

=−

,&.A1

2&.A1ln

2.A

1A@2,&[email protected] $

sec.1A@3$H.1

2

&

m

kmol −=

#ate of diffusion = 1.3$H E 1 )& @ π @ 4.& @ 1 )25 2

= 1.&H Bmol F sec = 3.E1$ molF(r

3.*.% Diffusion into an infinite standard medium :Jere "e "ill discuss pro0lems involving diffusion from a sp(erical particle

into an infinite 0ody of stagnant gas. -(e purpose in doing t(is is to demonstrate(o" to set up differential e%uations t(at descri0e t(e diffusion in t(ese processes.-(e solutions/ o0tained are only of academic interest 0ecause a large 0ody of gas in "(ic( t(ere are no convection currents is unlikely to 0e found in practice.Jo"ever/ t(e solutions developed (ere for t(ese pro0lems actually represent aspecial case of t(e more common situation involving 0ot( molecular diffusion andconvective mass transfer.

a5 4&aporation of a sp+erical Droplet:

As an example of suc( pro0lems/ "e s(all consider t(e evaporation of sp(erical droplet suc( as a raindrop or su0limation of nap(t(alene 0all. -(evapor formed at t(e surface of t(e droplet is assumed to diffuse 0y molecular motions into t(e large 0ody of stagnant gas t(at surrounds t(e droplet.

Consider a raindrop/ as s(o"n in figure. At any moment/ "(en t(e radius

Fi$ure4&aporation of a raindrop

of t(e drop is r / t(e flux of "ater vapor at any distance r from t(e center is given0y

( ) A A A

A A N N y r " y " DC N ++−=

Jere 6 9 = 4since air is assumed to 0e stagnant5-(erefore/

A A A

A A N y r "

y " DC N +−=

#earranging/

24



r "

y "

y

DC N

A

A

A A −

−=

1 QQQQQQQQQQ 415

-(e flux 6 A is not constant/ 0ecause of t(e sp(erical geometry? decreasesas t(e distance from t(e center of sp(ere increases. 9ut t(e molar flo" rate at r

and r G δr are t(e same.

-(is could 0e "ritten as/

r r A

r A N AN A

δ +

= QQQQQQQQQQ 425

"(ere A = surface area of sp(ere at r or r G δr.'u0stituting for A = $ π r 2 in e%uation 425/

A$$ 22

=−

+ r A

r r A N r N r π π

δ

or

Alim

22

A=

−+

→ r

N r N r r

Ar r

A

r δ

δ

δ

ss ( ) A2 = AN r "r

" QQQQQQQQQQ 435

Integrating/

constant2 = AN r QQQQQQQQQQ 4$5

8rom e%uation 4$5/ A

2A

2 A A N r N r =

'u0stituting for 6 A from e%uation 415/

A

2A

2

1 A

A

A

AN r

r "

y "

y

DC r =

−−

∫ ∫ −−=

A

A A A

y

y " DC

r

r " N r

12

2A A QQQQQQQQQQ 4&5

9oundary condition At r = r y A = y A'

And

At r = ∞ y A = y A∞

-(erefore e%uation 4&5 0ecomes/

( )[ ] ∞−=

−

∞ A

A%

y

y A Ar

A y DC r

N r 1ln1

A

A

2A

'implifying/

−−

= ∞

% A

A A A

y

y

r

DC N

1

1ln

AA

QQQQQQQQQQ 4+5

-ime re%uired for complete evaporation of t(e droplet may 0e evaluated frommaking mass 0alance.

timeunit

droplett(eleaving"ater of moles

timeunit

diffusing"ater of Moles=

−=

A A

M r

"t

" N r &

ρ π π

3AA

2A 3

$$

25



t "

r "

M r

A

& A2A$

ρ π −= QQQQQQQQQQ 4,5

'u0stituting for 6 A from e%uation 4+5 in e%uation 4,5/

t "

r "

M y

y

r

DC

A A%

A A & A

A 1

1ln

ρ −=

−

− QQQQQQQQQQ 4E5

Initial condition (en t = r = r 1

Integrating e%uation 4E5 "it( t(ese initial condition/

∫ ∫

−

−

−=

∞

A

AAA 1

1

1ln

11

r

% A

A A A

t

r " r

y

y DC M t "

& ρ

−

−=

∞

% A

A A A

y

y

r

DC M t

&

1

1ln

2

121

ρ

QQQQQQQQQQ 4H5

*%uation 4H5 gives t(e total time t re%uired for complete evaporation of sp(ericaldroplet of initial radius r 1.

-5 Com-ustion of a coal particle:-(e pro0lem of com0ustion of sp(erical coal particle is similar to

evaporation of a drop "it( t(e exception t(at c(emical reaction 4com0ustions5occurs at t(e surface of t(e particle. During com0ustion of coal/ t(e reaction

C G 2 → C 2

cccurs. According to t(is reaction for every mole of oxygen t(at diffuses to t(esurface of coal 4maximum of car0on5/ react "it( 1 mole of car0on/ releases 1

mole of car0on dioxide/ "(ic( must diffuse a"ay from t(is surface. -(is is a caseof e%uimolar counter diffusion of C 2 and 2. 6ormally air 4a mixture of 6 2 and 25 is used for com0ustion/ and in t(is case 6 2 does not takes part in t(e

reaction/ and its flux is ero. Ai.e.2 =N N .

-(e molar flux of 2 could 0e "ritten as

( )2222

2

22 N COOO

O

gasOO N N N y r "

y " DC N +++−= −

QQQQQQQQQQ 415

(ere gasOD −2 is t(e diffusivity of 2 in t(e gas mixture.

'ince A2

=N N / and from stoic(iometry 22 COO N N −= / e%uation 415

0ecomes

r "

y " DC N

O

gasOO2

22 −−= QQQQQQQQQQ 425

8or steady state conditions/

( ) A2

2 =ON r r "

" QQQQQQQQQQ 435

Integrating/

26



sOO N r N r 22

2A

2 constant == QQQQQQQQQQ 4$5

(ere r is t(e radius of coal particle at any instant/ and sON 2 is t(e flux of 2

at t(e surface of t(e particle.

'u0stituting for 2ON from e%uation 425 in e%uation 4$5/

sOO

gasO N r r "

y " DC r 2

2

2

2A

2 =− − QQQQQQQQQQ 4&5

9oundary condition

At sOO y y r r 22A ==

And

At ∞=∞=22 OO y y r

it( t(ese 0oundary condition/ e%uation 4&5 0ecomes

∫ ∫ ∞

−

∞−=

2

2

22

A

A 2

2A

O

sO

y

y OgasO

r A y " DC

r

r " N r

"(ic( yields

( )∞−

−=22

2

2A

OsOgasO

sO y y r

DC N QQQQQQQQQQ 4+5

8or fast reaction of 2 "it( coal/ t(e mole fraction of 2 at t(e surface of particle

i ero. 4i.e./5 A2

=sOy .

And also at some distance a"ay from t(e surface of t(e particle21.A

22 == ∞OO y y 40ecause air is a mixture of 21 mole 2 and ,H mole 6

25

it( t(ese conditions/ e%uation 4+5 0ecomes/

A

2

2

21.A

r

DC N gasO

sO−= QQQQQQQQQQQQ 4,5

Fi$ureCom-ustion of a particle of coal

&. A sp(ere of nap(t(alene (aving a radius of 2mm is suspended in a largevolume of s(ell air at 31E B and 1 atm. -(e surface pressure of t(e nap(t(alenecan 0e assumed to 0e at 31E B is .&&& mm Jg. -(e D A9 of nap(t(alene in air at 31E B is +.H2 @ 1 )+

m 2Fsec. Calculate t(e rate of evaporation of nap(t(alene

from t(e surface.

Calculation

'teady state mass 0alance over a element of radius r and r G δr leads to

A=−+ r r

Ar

A N % N % δ 7777777777777777777777777777 415

"(ere ' is t(e surface are 4= $ π r 25

dividing 415 0y 'δr/ and taking t(e limit as δr approac(es ero/ gives

27



(A

2

=r "

N r " A

Integrating r 2 6 A = constant 4or5 $ π r 2 6 A = constant

e can assume t(at t(ere is a film of nap(t(alene ) vapor F air film aroundnap(t(alene t(roug( "(ic( molecular diffusion occurs.

Diffusion of nap(t(alene vapor across t(is film could 0e "ritten as/

( ) A A A

A A N N y r "

y " CDN ++−=

6 9 = 4since air is assumed to 0e stagnant in t(e film5

A A A

A A N y r "

y " CDN +−=

−−=

A

A A A

y

y

r "

" CDN

1

( )[ ]r "

y " CDN

A A A

−=

1ln

A = #ate of evaporation = $ π r 2 6 A # = constant.

( )( )r "

y " CDr (

A A A

−=

1ln$ 2π

( )∫ ∫ −= A A A y " C Dr

r " ( 1ln$

2 π

9oundary condition

At r = # $1A@3A3.,,+A

&&&.A −== Ay

ln 41 ) y A5 = 7 ,.3 @ 1 )$

At r = ∞ y A = ln 417y A5 =

-(erefore( )[ ]

∫ ∫ −−

∞−=

A

1A@3.,2 $

1ln$ A AR

A y " C D

r

r " ( π

( )[ ]A1A@3., $1ln$

1−−

∞−=

−

A AR

A y C Dr

( π

[ ]$1A@3.,A$1

A −+=

+ C D

R ( A A π

A = $ π # D A9 C @ ,.3 @ 1 )$

28



31E@E31$

1A@A132&.1

@tan

&

==T t cons)as

P C

= .3E3 kmolFm 3

Initial rate of evaporation

-(erefore A = $ @ 3.1$2 @ 2 @ 1 )3 @ +.H2 @ 1 )+ @ .3E3 @ ,.3 @ 1 )$ = $.E+3 @ 1 )12 kmolFsec= 1.,&1 @ 1 )& molF(r.

3.*.* Diffusion in 6i)uids:*%uation derived for diffusion in gases e%ually applies to diffusion in

li%uids "it( some modifications. Mole fraction in li%uid p(ases is normally "rittenas Rx 4in gases as y5. -(e concentration term RC is replaced 0y average molar

density/a* M

ρ

.

a5 8or steady ) state diffusion of A t(roug( non diffusivity 96 A = constant / 6 9 =

( )21 A Aa* M

A A x x

M x $

DN −

= ρ

"(ere K = K 2 ) K 1/ t(e lengt( of diffusion pat(? and

−=

1

2

12

ln

M

+

+

+ + +

05 8or steady ) state e%uimolar counter diffusion 6 A = 7 6 9 = const

( ) ( )2121 A Aa*

A A A

A A x x

M !

DC C

!

DN −

=−=

ρ

$. Calculate t(e rate of diffusion of 0utanol at 2°C under unidirectional steady

state conditions t(roug( a .1 cm t(ick film of "ater "(en t(e concentrations of 0utanol at t(e opposite sides of t(e film are/ respectively 1 and $ 0utanol 0y"eig(t. -(e diffusivity of 0utanol in "ater solution is &.H @ 1 )+ cm 2Fsec. -(e

densities of 1 and $ 0utanol solutions at 2°C may 0e taken as .H,1 and

.HH2 gFcc respectively. Molecular "eig(t of 9utanol 4C $ J H J5 is ,$/ and t(atof "ater 1E.

Calculations

8or steady state unidirectional diffusion/

lm

A A A A

x

x x C

$

DN

/

21 −=

"(ere C is t(e average molar density.

29



a*g M

= ρ

Conversion from "eig(t fraction t(e Mole fraction

( )

( ) A2+.A

1EH.A,$1.A

,$1.A

1 =+

= A

x

( )( )

A1A.A1EH+.A,$A$.A

,$A$.A2 =

+= A x

Average molecular "eig(t at 1 : 2

( )'mol kg M $,.1H

1EH.A,$1.A

11 =

+=

( )'mol kg M &+.1E

1EH+.A,$A$.A

12 =

+=

2

2211 M M

M a*g

ρ ρ ρ +=

2

&+.1EHH2.A$,.1HH,1.A +=

= .&1, gmol F cm 3 = &1., kmolFm 3

( )

−

−

−−−=

−=

1

2

12

12

12/

1

1ln

11

ln

A

A

A A

lm

x

x

x x

x x

x x x

4i.e.5

( ) ( )

−−

−−−=

A2+.A1

A1.A1ln

A2+.A1A1.A1/lm x

HE2.AA1+3.A

A1+.A ==

-(ereforelm

A A

a*g

A A

x

x x

M

DN

/

21

2

−

= ρ

( )

HE2.A

A1A.AA2+.A@

[email protected]

,.&1@1A@1A@H.&2

$+ −=

−

−−

sec1A@H,.$

2,

m

kmol −=

..,EH.1

2 hr m

gmol =

..,$@,EH.1

2 hr m

g =

30



..$.132

2 hr m

g =

Mass diffusion "it+ +omo$eneous c+emical reaction: A0sorption operations involves contact of a gas mixture "it( a li%uid and

preferential dissolution of a component in t(e contacting li%uid. Depending on t(ec(emical nature of t(e involved molecules/ t(e a0sorption may or may notinvolve c(emical reaction.

-(e follo"ing analysis illustrates t(e diffusion of a component from t(e gasp(ase into t(e li%uid p(ase accompanied 0y a c(emical reaction in t(e li%uidp(ase. Consider a layer of a0sor0ing medium 4li%uid5 as s(o"n in diagram.

Fi$ure

At t(e surface of t(e li%uid/ t(e composition of A is CA . -(e t(ickness of t(e film/

δ is so defined/ t(at 0eyond t(is film t(e concentration of A is al"ays ero ? t(at is

C Aδ = . If t(ere is very little fluid motion "it(in t(e film/

( ) A A A

A A N N C

C

$ "

C " DN ++−= QQQQQQQQQQQQ 415

If concentration of A in t(e film/ C A is assumed small/ e%uation 415 0ecomes

$ "

C " DN

A A A −= QQQQQQQQQQQQ 425

-(e molar flux 6 A c(anges along t(e diffusion pat(. -(is c(ange is due tot(e reaction t(at takes place in t(e li%uid film. -(is c(anges could 0e "ritten as

( ) A=− A A r N $ "

" QQQQQQQQQQQQ 435

"(ere )r A is t(e rate of disappearance of A. 8or a first order reaction/ A

k →

A A C k r =− QQQQQQQQQQQQ 4$5

"it( t(e su0stitution from e%uation 4$5 and 425 in e%uation 435/

A=+

− A

A A C k

$ "

C " D

$ "

"

8or constant Diffusivity/

A2

2

=+− A A

A C k $ "

C " D QQQQQQQQQQQQ 4&5

"(ic( is a second order ordinary differential e%uation. -(e general solution to t(is

e%uation is

+

= $

Dk hC $

Dk hC C

A A A sincos 21 QQQQQQQ 4+5

-(e constants of t(is e%uation can 0e evaluated from t(e 0oundary conditionsat K = C A = C A

And at K = δ C A = .

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-(e constant C 1 is e%ual to C A / and C 2 is e%ual to

−δ

A

AD

k h

C tanA

"it( t(is su0stitution e%uation 4+5 0ecomes/

−

=

$ D

k h

$

D

k hC

$ D

k hC C

A

A

A

A A A

tan

sin

cos

A

A

QQQQQQQ 4,5-(is e%uation gives t(e variation of concentration of A "it( 4i.e

concentration profile of A in t(e li%uid5. -(e molar flux at t(e li%uid surface can 0edetermined 0y differentiating e%uation 4,5/ and evaluating t(e derivative

Aat =$ $ "

C " A

Differentiating C A "it( respect to /

−

=

δ A

A A A

A A A

A

Dk h

$ D

k hD

k C

$ D

k hD

k C $ "

C "

tan

cos

sin

A

A

QQQQQQQQQQQQ 4E5'u0stituting = in e%uation 4E5 and from e%uation 425/

=

=

δ

δ

δ

A

A A A

! A

Dk h

Dk

C DN

tan

A

A QQQQQQQQQQQQ 4H5

8or a0sorption "it( no c(emical reaction/ t(e flux of A is o0tained from e%uation425 as

δ

A A A

A

C DN = QQQQQQQQQQQQ 415

"(ic( is constant t(roug(out t(e film of li%uid. n comparison of e%uation 4H5 and

415/ it is apparent t(at t(e term

δ δ

A A Dk h

Dk tan s(o"s t(e

influence of t(e c(emical reactions. -(is terms a dimensionless %uantity/ is oftencalled as Jatta 6um0er.

3.*./ Diffusion in solidsIn certain unit operation of c(emical engineering suc( as in drying or in

a0sorption/ mass transfer takes place 0et"een a solid and a fluid p(ase. If t(etransferred species is distri0uted uniformly in t(e solid p(ase and forms a(omogeneous medium/ t(e diffusion of t(e species in t(e solid p(ase is said to

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0e structure independent. In t(is cases diffusivity or diffusion coefficient isdirection ) independent.

At steady state/ and for mass diffusion "(ic( is independent of t(e solidmatrix structure/ t(e molar flux in t(e direction is

constant=−=$ "

C " DN

A A A / as given 0y 8icks la".

Integrating t(e a0ove e%uation/

$

C C DN

A A A A

21 −=

"(ic( is similar to t(e expression o0tained for diffusion in a stagnant fluid "it( no0ulk motion 4i.e. 6 = 5.

Diffusion in process solids:In some c(emical operations/ suc( as (eterogeneous catalysis/ an

important factor/ affecting t(e rate of reaction is t(e diffusions of t(e gaseouscomponent t(roug( a porous solid. -(e effective diffusivity in t(e solid is reduced

0elo" "(at it could 0e in a free fluid/ for t"o reasons. 8irst/ t(e tortuous nature of t(e pat( increases t(e distance/ "(ic( a molecule must travel to advance a givendistance in t(e solid. 'econd/ t(e free cross ) sectional area is restricted. 8or many catalyst pellets/ t(e effective diffusivity of a gaseous component is of t(eorder of one tent( of its value in a free gas.

If t(e pressure is lo" enoug( and t(e pores are small enoug(/ t(e gasmolecules "ill collide "it( t(e "alls more fre%uently t(an "it( eac( ot(er. -(is iskno"n as Bnudsen flo" or Bnudsen diffusion. pon (itting t(e "all/ t(emolecules are momentarily a0sor0ed and t(en given off in random directions.-(e gas flux is reduced 0y t(e "all collisions.

9y use of t(e kinetic flux is t(e concentration gradient is independent of

pressure ? "(ereas t(e proportionality constant for molecular diffusion in gases4i.e. Diffusivity5 is inversely proportional to pressure.

Bnudsen diffusion occurs "(en t(e sie of t(e pore is of t(e order of t(emean free pat( of t(e diffusing molecule.

3. Transient Diffusion-ransient processes/ in "(ic( t(e concentration at a given point varies

"it( time/ are referred to as unsteady state processes or time ) dependentprocesses. -(is variation in concentration is associated "it( a variation in t(emass flux.

-(ese generally fall into t"o categoriesi5 t(e process "(ic( is in an unsteady state only during its initial

startup/ andii5 t(e process "(ic( is in a 0atc( operation t(roug(out its operation.In unsteady state processes t(ere are t(ree varia0les7concentration/ time/

and position. -(erefore t(e diffusion process must 0e descri0ed 0y partial rat(er t(an ordinary differential e%uations.

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Alt(oug( t(e differential e%uations for unsteady state diffusion are easy toesta0lis(/ most solutions to t(ese e%uations (ave 0een limited to situationsinvolving simple geometries and 0oundary conditions/ and a constant diffusioncoefficient.

Many solutions are for one7directional mass transfer as defined 0y 8icks

second la" of diffusion

2

2

$

C D

t

C A A

A

∂

∂=

∂

∂ QQQQQQQQQQ 415

-(is partial differential e%uation descri0es a p(ysical situation in "(ic( t(ere isno 0ulk)motion contri0ution/ and t(ere is no c(emical reaction. -(is situation isencountered "(en t(e diffusion takes place in solids/ in stationary li%uids/ or insystem (aving e%uimolar counter diffusion. Due to t(e extremely slo" rate of

diffusion "it(in li%uids/ t(e 0ulk motion contri0ution of flux e%uation 4i.e./ y A ∑ 6 i5

approac(es t(e value of ero for dilute solutions ? accordingly t(is system alsosatisfies 8icks second la" of diffusion.

-(e solution to 8icks second la" usually (as one of t(e t"o standard

forms. It may appear in t(e form of a trigonometric series "(ic( converges for large values of time/ or it may involve series of error functions or related integrals"(ic( are most suita0le for numerical evaluation at small values of time. -(esesolutions are commonly o0tained 0y using t(e mat(ematical tec(ni%ues of separation of varia0les or !aplace transforms.

Mass Transfer 1 by Dr. M A Rashid

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