CHEMICAL ENGINEERING

MASS TRANSFER 281

Fraction of surface of age 0 to

1

s

DK

∫ 1 /s

0

estdtDK

[

est

s

] 1 /s

0

D

K

s

 1 e^1

Fraction of surface of age

1

s

to1DK^0

∫ 1

1 /s

e^2 stdtDK^0

[

e^2 st

 2 s

] 1

1 /s

D

K^0

2 s

e^2

The total surface is unity or:

K

s

 1 e^1 C

K^0

2 s

e^2

AttD 1 /s

, both age distribution functions must apply, and:

Ke^1 DK^0 e^2 or:K^0 DKÐe

Thus, for the total surface: 1D

K

s

 1 e^1 C

Ke

2 s

e^2

D

K

s

(

1 e^1 C^12 e^1

)

D

K

s

(

1 ^12 e^1

)

Thus: KDs 1 ^12 e^1 ^1 K^0 Dse 1 ^12 e^1 ^1

Thus 0<t<

1

s

ft

Ds

(

1 ^12 e^1

) 1

est

1

s

<t< 1 ft

Dse

(

1 ^12 e^1

) 1

e^2 stDs

(

1 ^12 e^1

) 1

e^1 ^2 st

PROBLEM 10.46

Derive the partial differential equation for unsteady-state unidirectional diffusion accom-
panied by annth-order chemical reaction (rate constantk):

∂CA

∂t

DD

∂^2 CA

∂y^2

kCnA

whereCAis the molar concentration of reactant at positionyat timet.
Explain why, when applying the equation to reaction in a porous catalyst particle, it is
necessary to replace the molecular diffusivityDby an effective diffusivityDe.
Solve the above equation for a first-order reaction under steady-state conditions, and
obtain an expression for the mass transfer rate per unit area at the surface of a catalyst
particle which is in the form of a thin platelet of thickness 2L.
Explain what is meant by the effectiveness factor;for a catalyst particle and show
that it is equal to 1 /f tanhffor the platelet referred to previously wherefis the Thiele
modulusL

p
k/De.
For the case where there is a mass transfer resistance in the fluid external to the
particle (mass transfer coefficienthD), express the mass transfer rate in terms of the bulk
concentrationCAorather than the concentrationCASat the surface of the particle.
For a bed of catalyst particles in the form of flat platelets it is found that the mass
transfer rate is increased by a factor of 1.2 if the velocity of the external fluid is doubled.