Abstract
A regular perturbation series method provides a solution to the diffusion equation when the boundary condition is a non-linear adsorption isotherm. For adsorption at the interface the Freundlich and Langmuir isotherms yield power series in the square root of time. Convergence of the power series solutions is improved by applying the Shanks transformation. The solutions are compared to limiting cases and to published numerical solutions. The results are most accurate for small time where the numerical finite difference method is least reliable.
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Abbreviations
- c :
-
concentration
- c i :
-
initial, or bulk, concentration
- D :
-
diffusion coefficient
- k :
-
q i/ci
- p :
-
parameter of Freundlich isotherm
- q :
-
surface concentration, adsorption
- q i :
-
surface concentration in equilibrium withc i
- t :
-
time
- x :
-
distance from the surface
- C :
-
c/c i
- F :
-
dimensionless isotherm
- Q:
-
q/q i
- s :
-
Laplace transform parameter
- X :
-
x/k
- θ :
-
Dt/k 2
- ε :
-
expansion parameter
- δ on :
-
unity ifn=0, zero otherwise
- γ :
-
gamma function
References
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Bender, C. M., S. A. Orszag, “Advanced Mathematical Methods for Scientists and Engineers”, McGraw-Hill, Chap. 8 (1978).
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McCoy, B.J. Analytical solutions for diffusion-controlled adsorption kinetics with non-linear adsorption isotherms. Colloid & Polymer Sci 261, 535–539 (1983). https://doi.org/10.1007/BF01419838
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DOI: https://doi.org/10.1007/BF01419838