Abstract
Despite single-phase parallel flows for which analytical solutions are available for species conservation equation in its most general form, little progress has been made in analytical modeling of cross-stream diffusion in stratified multiphase flows. The main reason is that solute concentration and fluid properties are discontinuous at the interface of phases. In the present study, a new method of solution is presented for the analytical treatment of solute transport in stratified multiphase flows. The solution methodology starts with developing separate species conservation equations for the phases, which, upon non-dimensionalization, are replaced with a single equation for which series solutions are obtained utilizing the variational calculus. Using the method proposed, analytical solutions are obtained for mass transport in microfluidic two-phase extraction by taking the influences of the non-uniform velocity and the axial diffusion effects into account. The inclusion of the non-uniform velocity effects, which leads to 3D solutions, enables us to capture the heterogeneous transport of solutes, a phenomenon that is ignored by the available simple 2D solutions. The results indicate that axial diffusion in stratified multiphase flows is significantly more important than in single-phase flow and is not characterized solely by the Péclet number. It is also found that the only parameter controlling the length required for a complete extraction is the solvent-to-solution viscosity ratio.
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Abbreviations
- \(A\) :
-
Coefficient
- \({\mathbb{A}}\) :
-
Channel cross-sectional area \(\left({\mathrm{m}}^{2}\right)\)
- \(c\) :
-
Number concentration \(\left({\mathrm{m}}^{-3}\right)\)
- \({c}_{\mathrm{inf}}\) :
-
Concentration in the limit \(x\to \infty\) \(\left({\mathrm{m}}^{-3}\right)\)
- \({c}_{m}\) :
-
Mean concentration \(\left({\mathrm{m}}^{-3}\right)\)
- \({c}_{0}\) :
-
Inlet concentration \(\left({\mathrm{m}}^{-3}\right)\)
- \(D\) :
-
Diffusivity \(\left({\mathrm{m}}^{2} {\mathrm{s}}^{-1}\right)\)
- \({E}_{e}\) :
-
Extraction efficiency \(\left[={c}_{m,2}/{c}_{\mathrm{inf},2}\right]\)
- \({E}_{r}\) :
-
Extraction ratio \(\left[={Q}_{2}{c}_{m,2}/{Q}_{1}{c}_{0}\right]\)
- \(f,g\) :
-
Basis function
- \(F\) :
-
Flow rate ratio \(\left[={Q}_{2}/{Q}_{1}\right]\)
- \(G\) :
-
Eigen function
- \(H\) :
-
Channel height \(\left(\mathrm{m}\right)\)
- \({I}_{G},{I}_{u}\) :
-
Functional
- \(K\) :
-
Partition coefficient function
- \({K}_{P}\) :
-
Partition coefficient
- \(L\) :
-
Channel length \(\left(\mathrm{m}\right)\)
- \(p\) :
-
Pressure \(\left(\mathrm{Pa}\right)\)
- \(Pe\) :
-
Péclet number \(\left[={\mathrm{Wu}}_{m}/{D}_{1}\right]\)
- \(Q\) :
-
Volumetric flow rate \(\left({\mathrm{m}}^{3} {\mathrm{s}}^{-1}\right)\)
- \({t}_{r}\) :
-
Residence time \(\left(\mathrm{s}\right)\)
- \(u\) :
-
Axial velocity \(\left(\mathrm{m }{\mathrm{s}}^{-1}\right)\)
- \({u}_{m}\) :
-
Mean velocity \(\left(\mathrm{m }{\mathrm{s}}^{-1}\right)\)
- \(W\) :
-
Channel width \(\left(\mathrm{m}\right)\)
- \(x,y,z\) :
-
Coordinates \(\left(\mathrm{m}\right)\)
- \(\alpha\) :
-
Channel aspect ratio \(\left[=H/W\right]\)
- \(\beta\) :
-
Eigenvalue
- \(\Gamma\) :
-
Mean velocity ratio \(\left[={u}_{m,2}/{u}_{m,1}\right]\)
- \(\delta\) :
-
Kronecker delta
- \({\eta }_{D}\) :
-
Diffusivity ratio \(\left[={D}_{2}/{D}_{1}\right]\)
- \({\eta }_{\mu }\) :
-
Viscosity ratio \(\left[={\mu }_{2}/{\mu }_{1}\right]\)
- \(\mu\) :
-
Dynamic viscosity \(\left(\mathrm{Pa s}\right)\)
- \(\omega\) :
-
Holdup
- \(1\) :
-
Phase 1
- \(2\) :
-
Phase 2
- \(*,\sim\) :
-
Dimensionless parameter
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The author sincerely thanks the Iran National Science Foundation (INSF) for their financial supports during the course of this work.
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Sadeghi, A. A new method for analytical modeling of microfluidic extraction. Microfluid Nanofluid 25, 45 (2021). https://doi.org/10.1007/s10404-021-02444-9
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DOI: https://doi.org/10.1007/s10404-021-02444-9