Applied Scientific Research

, Volume 22, Issue 1, pp 223–238 | Cite as

The effect of radial diffusion on the performance of a liquid-liquid displacement process

  • Chang Dae Han
Article
Received:
Revised:

Abstract

The effect of radial diffusion on the performance of a liquid-liquid displacement process is considered in fluid flow between porous parallel plates and through a porous tube, as examples of a two-zone problem in unsteady-state mass transfer. The double Laplace transformation is applied to the system equations. In obtaining the inversion of the Laplace transformed equations the first inversion (with respect to the transformed dimensionless axial distance) is performed by use of the residue method, and then the second inversion (with respect to the transformed dimensionless time) is performed by use of the numerical Laplace transform technique advanced by Bellman et al. A numerical example is shown and discussed.

Keywords

Radial Diffusion Porous Plate Residence Time Distribution Dimensionless Distance System Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

C1

concentration of solute in zone 1

C2

concentration of solute in zone 2

C0

initial concentration of solute in zone 2

C1

Laplace transformed variable of C 1 with respect to ξ 1

\(\tilde C_1 \)

double Laplace transformed variable of C 1

\(\tilde \bar C_1 \)

average value of C1

D

molecular diffusivity

De

effective diffusivity

r

coordinate in the radial direction in a tube

R

inner radius of a tube

t

time

x

coordinate perpendicular to the flow direction between the parallel plates

X

distance between the center of the two parallel plates and the inner wall of a plate

v

average fluid velocity in zone 1

wn

roots of Eq. (59)

y

coordinate in the flow direction between the parallel plates

z1

dimensionless distance, x/X

z2

dimensionless distance, r/R

βn

roots of Eq. (55)

λ

\(\sqrt {D/D_e } \)

ξ1

dimensionless time, (t−y/v) D/X 2

ξ2

dimensionless time, (t−y/v) D/R 2

η1

dimensionless axial distance, yD/vX 2

η2

dimensionless distance, yD/vR 2

σ1

dimensionless wall thickness of a porous plate, δ/X

σ2

dimensionless wall thickness of a porous tube, δ/R

units

cm, g, s

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References

  1. [1]
    Carslaw, H. S. and J. C. Jaeger, Conduction of Heat in Solids, Oxford Un. Press, 1947.Google Scholar
  2. [2]
    Crank, J., The Mathematics of Diffusion, Oxford Un. Press, 1956.Google Scholar
  3. [3]
    Taylor, Sir Geoffrey, Proc. Roy. Soc. (London) A 219 (1953) 186.ADSGoogle Scholar
  4. [4]
    Coste, J., D. Rudd, and N. R. Amundson, Can. J. Chem. Eng. 39 (1961) 149.Google Scholar
  5. [5]
    Farrel, M. A. and E. F. Leonard, AIChE J. 9 (1963) 191.Google Scholar
  6. [6]
    Advonin, N. A., Soviet Phys. — Doklady 8 (1964) 782.Google Scholar
  7. [7]
    Bellman, R., H. Kagiwada, R. Kalaba, and M. Prestruc, Invariant Imbedding and Time-Dependent Transport Process, American Elsevier, New York 1964.Google Scholar

Copyright information

© Martinus Nijhoff 1970

Authors and Affiliations

  • Chang Dae Han
    • 1
  1. 1.Dept. of Chem. Eng.Polytechnic Institute of BrooklynBrooklynUSA

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