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NPTEL  – Chemical  – Mass Transfer Operation 1 

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MODULE 3: MASS TRANSFER COEFFICIENTS

LECTURE NO. 5

3.5 THEORIES OF MASS TRANSFER

Many theories are available to explain the behavior of mass transfer coefficient

such as the film, penetration, surface renewal and others. The theories are basedon both molecular and turbulent diffusion. The theories are explained in details in

the following sections.

3.5.1 Film theory

In case of one dimensional convective mass transfer, the concentration

distribution is linear. The coefficient of mass transfer in this special case is

proportional to the diffusion coefficient and inversely proportional to the film. This

phenomenon is commonly known as the film theory. The schematic diagram of

concentration profile with stagnant film is shown in Figure 3.3.

The expression of mass transfer coefficient is developed theoretically by

visualizing the mechanism. This is done by film theory. Film theory is based on

the following assumptions:

a) Mass transfer occurs by molecular diffusion through a fluid layer at phase

boundary (solid wall). Beyond this film, concentration is homogeneous and

is C  Ab.

b) Mass transfer through the film occurs at steady state.

c) Flux is low and mass transfer occurs at low concentration.

Hence,dZ 

dC  D N 

  A AB

 A

  (3.51)

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Figure 3.3: Schematic diagram of concentration profile with stagnant film.

Steady-state mass balance is done over an elementary volume of thickness Z.

Rate of input of solute at Z=N  A|Z

Rate of output of solute at Z+Z=N  A|Z+Z 

Rate of accumulation=0= (rate of input-rate of output)

Therefore, at steady state, N  A|Z-N  A|Z+Z=0

00

  Z 

 N  N 

 Lim  Z  Z  A Z  A

 Z    (3.52)

0dZ 

dN  A   (3.53)

0 

  

  

dZ 

dC  D

dZ 

d  A AB   (3.54)

02

2

dZ 

C d  D A AB   (3.55)

02

2

dZ C d   A   (3.56)

Integrating Equation (3.55) for the following boundary conditions:

C  A=C  Ai  when Z =0

C  A=C  Ab when Z=  

We have now,

Z

C  A 

Actual concentration profile

Concentration profile in stagnant filmC  Ai 

C  Ab 

0

C  Ab C  Ai 

 N  A|Z    N  A|Z+ Z  

 Z  

Z

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 Z C C C C  Ab Ai Ai A

  )(     (3.57)

Hence, according to film theory, concentration profile in stagnant film is linear.

Molar flux through film, N  A:

0

 Z 

 A AB

 A

dZ 

dC  D N   

 

)( Ab Ai AB

 A

C C  D N 

    (3.58)

Now comparing Equation (3.58) with Equation (3.8),

 

 AB

 L

 Dk      (3.59)

Film theory is useful in the analysis of mass transfer with chemical reaction. Forturbulent flow, the mass transfer coefficient has much smaller dependency

compared to laminar flow. In the turbulent flow, the mass transfer coefficient is

proportional to n

 AB D  where n may be zero to 0.9, depending upon the operating

conditions. Although the film theory offers some explanation of the mechanism of

mass transfer in fluid media, it does not explain the estimation thickness of the

film. Due to this disadvantage, application of the model is restricted to mass

transfer in a diffusion cell.