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.
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Abbreviations
- C 1 :
-
concentration of solute in zone 1
- C 2 :
-
concentration of solute in zone 2
- C 0 :
-
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
- D e :
-
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
- w n :
-
roots of Eq. (59)
- y :
-
coordinate in the flow direction between the parallel plates
- z 1 :
-
dimensionless distance, x/X
- z 2 :
-
dimensionless distance, r/R
- β n :
-
roots of Eq. (55)
- λ :
-
\(\sqrt {D/D_e } \)
- ξ 1 :
-
dimensionless time, (t−y/v) D/X 2
- ξ 2 :
-
dimensionless time, (t−y/v) D/R 2
- η 1 :
-
dimensionless axial distance, yD/vX 2
- η 2 :
-
dimensionless distance, yD/vR 2
- σ 1 :
-
dimensionless wall thickness of a porous plate, δ/X
- σ 2 :
-
dimensionless wall thickness of a porous tube, δ/R
- units:
-
cm, g, s
References
Carslaw, H. S. and J. C. Jaeger, Conduction of Heat in Solids, Oxford Un. Press, 1947.
Crank, J., The Mathematics of Diffusion, Oxford Un. Press, 1956.
Taylor, Sir Geoffrey, Proc. Roy. Soc. (London) A 219 (1953) 186.
Coste, J., D. Rudd, and N. R. Amundson, Can. J. Chem. Eng. 39 (1961) 149.
Farrel, M. A. and E. F. Leonard, AIChE J. 9 (1963) 191.
Advonin, N. A., Soviet Phys. — Doklady 8 (1964) 782.
Bellman, R., H. Kagiwada, R. Kalaba, and M. Prestruc, Invariant Imbedding and Time-Dependent Transport Process, American Elsevier, New York 1964.
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Han, C.D. The effect of radial diffusion on the performance of a liquid-liquid displacement process. Appl. Sci. Res. 22, 223–238 (1970). https://doi.org/10.1007/BF00400529
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DOI: https://doi.org/10.1007/BF00400529