Summary
The displacement of a liquid or gas in a capillary by another fluid of equal density and viscosity depends upon the hydrodynamic behaviour in the tube as well as on the molecular diffusivity. The case of Poiseuille flow is discussed in the light of the theories of Taylor, Bos worth and of an approach by the authors. The theoretical curves are compared with experimental results.
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Abbreviations
- a :
-
tube radius
- c :
-
concentration
- c 0 :
-
initial concentration
- D :
-
molecular diffusion coefficient
- E T :
-
Taylor diffusivity
- H :
-
holdback
- L :
-
tube length
- r :
-
radial distance
- Re=2au/v :
-
Reynolds group
- Sc=v/D :
-
Schmidt group
- t=L/u :
-
mean residence time
- u :
-
mean fluid velocity
- X :
-
axial distance
- α=Dt/a 2 :
-
radial diffusion group
- v :
-
kinematic viscosity
- τ:
-
dimensionless residence time
- ϕ:
-
\(\int\limits_0^\infty {cdx} \),
- ω n :
-
zero of J1(ω)
- J0, J1 :
-
Bessel functions
- I0, I1 :
-
modified Bessel functions
- Erf:
-
error function
- In erfc:
-
repeated integrals of the complementary error function
References
Bosworth, R. C. L., Phil. Mag.39 (1948) 847.
Taylor, Sir Geoffry, Proc. Roy. Soc. (London)A 219 (1953) 186.
Taylor, Sir Geoffry, Proc. Roy. Soc. (London)A 225 (1954) 473.
Danckwerts, P. V., Chem. Engng. Sci.2 (1953) 1.
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Van Deemter, J.J., Broeder, J.J. & Lauwerier, H.A. Fluid displacement in capillaries. Appl. sci. Res. 5, 374–388 (1955). https://doi.org/10.1007/BF03184600
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DOI: https://doi.org/10.1007/BF03184600