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CP302 Separation Process PrinciplesMass Transfer / Set 1

(Worked) Examples in Mass Transfer ! "iff#sion in $inar! S!stems

Example 1: Diffusion through a stagnant gas filmOxygen is diffusing in a mixture of oxygen-nitrogen at 1 std atm, 25°C. Concentration of oxygen

at planes 2 mm apart are 1 and 2 !olume " respecti!ely. #itrogen is non-diffusing.

$a% Deri!e the appropriate expression to calculate the flux oxygen. Define units of eachterm clearly.

$&% Calculate the flux of oxygen. Diffusi!ity of oxygen in nitrogen ' 1.()x1 *5 m 2+sec.

Solution: et us denote oxygen as and nitrogen as . /lux of $' #   % is made up of t0ocomponents, namely that resulting from the &ul motion of $' #x % and that resulting frommolecular diffusion  :

 A A A   J  Nx N    +=   $1.1%

/rom /ic3s la0 of diffusion, z d 

C d  D J 

  A

 AB A   −=   $1.2%

4u&stituting $1.2% in $1.1%, 0e get  z d 

C d 

 D Nx N   A

 AB A A   −=   $1.%

4ince # ' #  6 #  and x  ' C  + C, $1.% &ecomes   ( ) z d 

C d  D

C 

C  N  N  N 

  A

 AB

 A

 B A A   −+=  

7earranging the terms and integrating &et0een the planes &et0een 1 and 2,

( )∫ ∫    +−−=   2

1

 A

 A

C 

C  B A A A

 A

 AB   N  N C C  N 

dC 

CD

 z d   $1.8%

4ince is non-diffusing #  ' . lso, the total concentration C remains constant. 9herefore,$1.8% &ecomes

∫    −−=   2

1

 A

 A

C 

C  A A A

 A

 AB   C  N C  N 

dC 

CD

 z  

1

2ln

1

 A

 A

 A   C C 

C C 

 N    −

−

=

9herefore,

1

2ln

 A

 A AB

 AC C 

C C 

 z 

CD N 

−

−=   $1.5%

7eplacing concentration in terms of pressures using deal ;as la0, $1.5% &ecomes

1

2ln

 At 

 At t  AB

 A p P 

 p P 

 RTz 

 P  D N 

−−

=  

$1.<% 0here

D AB ' molecular diffusi!ity of in P t  ' total pressure of systemR  ' uni!ersal gas constantT  ' temperature of system in a&solute scalez  ' distance &et0een t0o planes across the direction of diffusion p A1 ' partial pressure of at plane 1, and p A2 ' partial pressure of at plane 2

;i!en are D AB ' 1.()x1 *5 m2+sec= P t  ' 1 atm ' 1.125x15 #+m2= T  ' 25°C ' 2)( >=

z  ' 2 mm ' .2 m= p A1 ' .2x1 ' .2 atm= p A2 ' .1x1 ' .1 atm

4u&stituting these in $1.<%, 0e get,

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( ) ( )( ) ( ) ( )

     

  

−−

=−

2.01

1.01ln

002.02988314

10*01325.110*89.1   55

 A N   ' 8.55x1 *5 mol+m 2.sec

Example 2: E?uimolar counter diffusionO&tain the expression descri&ing the molar flux for steady-state e?uimolar counter diffusion and

its concentration profile.

Solution: 9he molar flux # , for a &inary system at constant temperature and pressure isdescri&ed &y

  ( ) z d 

C d  D

C 

C  N  N  N 

  A

 AB

 A

 B A A   −+=   $2.1%

E?uimolar counter diffusion is gi!en &y #  ' - #, 0hich reduces $2.1% to

 z d 

C d  D N    A

 AB A   −=  

$2.2%/or steady state diffusion, $2.2% may &e integrated, using the &oundary conditions as

  ∫ ∫    −=2

1

2

1

 A

 A

C 

C 

 A AB

 Z 

 Z 

 A   C d  D z d  N 

0hich gi!es   )( 21

12

 A A AB

 A   C C  z  z 

 D N    −

−=   $2.%

/or ideal gases,T  R

 p

V 

nC    A A

 A   ==  

9herefore, $2.% &ecomes )()(

  21

12

 A A AB

 A   P  P  z  z T  R

 D N    −

−=   $2.8%

$2.% and $2.8% are e?uations descri&ing the molar flux for steady-state e?uimolar counterdiffusion.

Concentration profile in the e?uimolar counter diffusion may &e o&tained from

0)(   = A N  z d 

d   $4ince #  is constant o!er the diffusion path%.

/rom $2.2%, z d 

C d  D N    A

 AB A   −=

9herefore,   0=   

  

 −

 z d 

C d  D

 z d 

d   A AB   or .0

2

2

= z d 

C d   A

9his e?uation may &e sol!ed using the &oundary conditions to gi!e

 21

1

21

1

 z  z 

 z  z 

C C 

C C 

 A A

 A A

−−

=−−

  $2.5%

$2.5% indicates a linear concentration profile for e?uimolar counter diffusion.

Example :@ethane diffuses at steady state through a tu&e containing helium. t point 1 the partialpressure of methane is  p 1  ' 55 Aa and at point 2, . m apart,  p 2 ' 15 >Aa. 9he totalpressure is 11.2 Aa, and the temperature is 2)( >. t this pressure and temperature, the!alue of diffusi!ity is <.B5x1 *5 m 2+sec.

$a% Calculate the flux of C 8 at steady state for e?uimolar counter diffusion.$&% Calculate the partial pressure at a point .2 m apart from point 1.

Solution: /or steady state e?uimolar counter diffusion, molar flux is gi!en &y

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( )21   A A

 AB

 A   p p z T  R

 D N    −=   $.1%

9herefore,

( )secm

kmol1555

03.0298314.8

1075.62

5

⋅−

×××

=−

 A N secm

kmol10633.3

2

5

⋅×=   −

 nd from $.1%, partial pressure at .2 m from point 1 is calculated as follo0s:

( ) A p−××

×=×−− 55

02.0298314.8

1075.610633.3

55

 p A ' 2(. Aa

Example 8:n a gas mixture of hydrogen and oxygen, steady state e?uimolar counter diffusion is occurring

at a total pressure of 1 Aa and temperature of 2°C. f the partial pressures of oxygen at t0o

planes .1 m apart, and perpendicular to the direction of diffusion are 15 Aa and 5 Aa,respecti!ely, and the mass diffusion flux of oxygen in the mixture is 1.<x1   *5  mol+m  2.sec,

calculate the molecular diffusi!ity for the system.

Solution: /or e?uimolar counter current diffusion, ( )21   A A

 AB

 A   p p RTz 

 D N    −=   $8.1%

0hereN  A ' molar flux of $1.<x1 *5 mol+m 2.sec%D AB ' molecular diffusi!ity of in R  ' ni!ersal gas constant $(.18 +mol.%T  ' 9emperature in a&solute scale $2B 6 2 ' 2) >%z  ' distance &et0een t0o measurement planes 1 and 2 $.1 m%P  A1 ' partial pressure of at plane 1 $15 Aa%= andP  A2 ' partial pressure of at plane 2 $5 Aa%

4u&stituting these in $8.1%, 0e get( ) ( ) ( )

  ( )51501.0293314.8

106.1   5 −=×   −   AB D

9herefore, D AB ' .()(x1 *5 m 2+sec

Example 5:  tu&e 1 cm in inside diameter that is 2 cm long is filled 0ith CO2 and 2 at a total pressure of 

2 atm at °C. 9he diffusion coefficient of the CO2 * 2 system under these conditions is .2B5

cm2+sec. f the partial pressure of CO2 is 1.5 atm at one end of the tu&e and .5 atm at the other end, find the rate of diffusion for the follo0ing cases:

i% steady state e?uimolar counter diffusion $#  ' - # %

ii% steady state counter diffusion 0here #  ' -.B5 # , andiii% steady state diffusion of CO2 through stagnant 2 $# ' %

 Answers: Denote CO2 &y and 2 &y .

i% = A N  <.1(x1-< mol+m2.sec= 7ate of diffusion of ' 1.B5x1 * mol+hr 

ii% = A N  B.2(x1-< mol+m2.sec= 7ate of diffusion of ' 1.)(Bx1 * mol+hr 

iii% = A N  1.8)x1-5 mol+m2.sec= 7ate of diffusion of ' .(18 mol+hr 

CP302 Separation Process PrinciplesMass Transfer / Set 1

(Worked) Examples in Mass Transfer ! "iff#sion in $inar! S!stems3

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contin#ed

Example <:ater in the &ottom of a narro0 metal tu&e is held at T  ' 2) >. /or air, P  ' 1.125x15 Aa $'1 atm% and T = 2) >. ater e!aporates and diffuses through the air in the tu&e and Fz   '.1528 m. Calculate the rate of e!aporation at steady state in mol+s.m2. 9he diffusi!ity of 0ater !apour in air at 2) > and 1 atm pressure is .25x1 -8  m2+s. ssume that the system is

isothermal. 

Example B: Diffusion in i?uidsCalculate the rate of diffusion of &utanol at 2°C under unidirectional steady state conditions

through a .1 cm thic film of 0ater 0hen the concentrations of &utanol at the opposite sides of the film are 1" and 8" &utanol &y 0eight, respecti!ely. 9he diffusi!ity of &utanol in 0ater 

solution is 5.)x1 *< cm2+sec. 9he densities of 1" and 8" &utanol solutions at 2°C may &e

taen as .)B1 and .))2 g+cc, respecti!ely. @olecular 0eight of &utanol $C 8)O% is B8, andthat of 0ater is 1(.

Solution:

E?uation deri!ed for diffusion in gases e?ually applies to diffusion in li?uids 0ith somemodifications. @ole fraction in li?uid phases is normally 0ritten as x   ' C +C and the

concentration term, C, is replaced &y a!erage molar density,av M 

    

    ρ 

.

a% /or steady state e?uimolar counter diffusion, #  ' - #  ' const

( ) ( )   ( )212121   A A

av

 AB A A

 AB

 A A AB

 A   x x M  z 

 D x xC 

 z 

 DC C 

 z 

 D N    − 

  

  =−=−=

  ρ 

$B.1%&% /or steady state diffusion of through non diffusi!e , #  ' constant and #  '

( )21

, A A

avlm B

 AB

 A

  x x M  x z 

 D N 

  −  

 

 

 

=

  ρ 

$B.2%

0here z  ' z 2 * z 1, the length of diffusion path= and   

  

−=

1

2

12

,

ln B

 B

 B B

lm B

 x

 x

 x x x

$B.%

9o calculate the rate of diffusion of $&utanol% under steady state unidirectional diffusion, use$B.2%.

Con!ersions from 0eight fraction the mole fraction gi!e the follo0ing:

( )( )

  026.0189.0741.0

741.01   =

+= A x   and

( )( )

  010.01896.07404.0

7404.02   =

+= A x

 !erage molecular 0eight at 1 G 2 are gi!en &y the follo0ing:

( )  kmolkg47.19

189.0741.0

11   =

+

= M  and

( )  kmolkg56.18

1896.07404.0

12   =

+= M 

(2

2211   M  M 

 M  av

 ρ  ρ  ρ    += 

  

  

2

56.18992.047.19971.0   +=

33 mmol7.51cmmol0517.0   k ==  

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$B.% gi!es,

( )

( ) ( )

   

  

 −−

−−−=

−=

1

2

12

12

12,

1

1ln

11

ln

 A

 A

 A A

 B B

 B Blm B

 x

 x

 x x

 x x

 x x x

 

( ) ( )

   

  

−−

−−−=

026.01

01.01ln

026.0101.01

 

982.00163.0016.0 ==

9herefore( )

lm B

 A A

avg 

 AB A

 x

 x x

 M 

 D N 

,

21

2

−   

  =

  ρ 

( )982.0

010.0026.0

101.0

7.5110109.52

46 −×

××××

= −

−−

secm

kmol1097.4

2

7

⋅×=   −

hr m

gmol789.1

2 ⋅=

hr m

g74789.1

2 ⋅×=

hr m

g4.132

2 ⋅=

Example (: Diffusion through a stagnant air film $past paper ?uestion%  circular tan < m in diameter contains &enHene at 22oC, 0hich is exposed the atmosphere for < hours per 0oring day in such manner that the li?uid is co!ered 0ith a stagnant air filmestimated to &e 5 mm thic. 9he concentration of &enHene &eyond the stagnant air film isnegligi&le. 9he !apour pressure of &enHene at 22oC is 1 mm g. if a litre of &enHene costs 7s15+', estimate the daily loss due to e!aporation of &enHene. 9he specific gra!ity anddiffusi!ity of &enHene at storage conditions are .(( and .2))x2.5(1x1 -5 m2+s, respecti!ely.@olecular 0eight of &enHene is B( g+mol. ni!ersal gas constant is (.18 +mol.>. 9he

atmospheric pressure may &e taen as 1.1 &ar. $1 &ar ' 1 #+m

2

 ' B5 mm g%

Example ): Diffusion through a stagnant air film $past paper ?uestion%  pool of 0ater 1 mm in depth lies on a horiHontal surface in contact 0ith dry air at 2 oC and 1atm. E!aporation taes place and can &e treated as molecular diffusion through a 5 mm thicfilm of moist air immediately a&o!e the 0ater surface. &o!e this gas film, 0ater !apour isassumed to &e perfectly mixed 0ith the surrounding $0ith the Hero partial pressure%. ssumingthat e!aporation does not affect the temperature of 0ater, calculate $i% the molar flux of 0ater !apour from the surface, and $ii% the time taen for complete e!aporation of the li?uid. Diffusi!ityof 0ater in air can &e taen as 2.<x1 -5 m2+s, saturated !apour pressure of 0ater at 2oC as28 #+m2 and gas constant as (18 +mol.>.

5