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4.2 Design of Adsorption and Ion-Exchange Processes 319

Langmuir Freundlich

X Set ( ) Assume (Xi)
NO

Calculate (Xi) Calculate (X )
Is the desirable (X) ?

Calculate θτ -X τ YES

Calculate θτ -X τ
Figure 4.26 The use of Miura’ s models.

The limiting cases of the analytical solutions for external fluid-film mass transfer con-
trolling ( 0) and solid diffusion controlling ( ) are the follo wing:
Langmuir

0
1 La (4.181)
C ln X ln( 1 ) 1 X

1 La 1 La
and

1  La 1 
X  X ln ln 1 X 1  (4.182)

1  La 1 La 

Freundlich

0
Fr ( 1 Fr ) / Fr 2 (4.183)
Fr
X 1ln X ln 1 X I A

1 Fr 1 Fr
and

1  1 1 
X  ln 1 X 1 Fr I B  (4.184)

Fr  1 Fr 1 