Effects of finite ion size on transport of neutral solute across porous wall of a nanotube
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O R I G I NA L A RT I C L E
Saikat Bhattacharjee · Morteza Dejam · Sirshendu De
Effects of finite ion size on transport of neutral solute across porous wall of a nanotube
Received: 26 December 2019 / Accepted: 18 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Effect of finite ion size on the transport of a neutral solute across the porous wall of a nanotube is presented in this study. Modified Poisson–Boltzmann equation without the Debye–Huckel approximation is used to determine the potential distribution within the tube. Power law fluid is selected for the study, as its rheology resembles closely to the real-life physiological fluids. The flow within the tube is actuated by the combined effects of pressure and electroosmotic forces. Steady-state solute balance equation is solved by the similarity technique in order to track the solute transport across the tube. The effects of ionic radius, ionic concentration, and flow behavioral index on the length-averaged Sherwood number, permeate flux, and permeate concentration are analyzed. This study will be extremely helpful in predicting the transport characteristics of a neutral solute in real physiological systems and also to fine-tune the performance of microfluidic devices having porous wall. Keywords Finite ion size · Porous wall · Nanotube · Power law fluid · Solute transport · Sherwood number
List of symbols A
n Dimensionless parameter defined as G n n+1 + n
A1 a
Dimensionless parameter defined as A1 = Radius of ions (m)
B B1 B2 B3 c c0 cp cw
(κ R)n−1 n d(n) 1 4G ReSc L A
1/n (2B3 + B2 )
∗ 1/3 Dimensionless parameter defined as B = Pe2w Ax 1 Constant in ψ ∗ in Eq. (8) Constant in ψ ∗ in Eq. (8) Constant in ψ ∗ in Eq. (8) Concentration of solute (kg/m3 ) Initial concentration of neutral solute (kg/m3 ) Permeate concentration (kg/m3 ) Wall concentration (kg/m3 )
Communicated by Oleg Zikanov. S. Bhattacharjee · S. De (B) Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India E-mail: [email protected] M. Dejam Department of Petroleum Engineering, College of Engineering and Applied Science, University of Wyoming, 1000 E. University Avenue, Laramie, WY 82071-2000, USA
S. Bhattacharjee et al. ∗ cw c y=0 D d Ex e G k kb L Lp n n± n+ n− n0 Px Pew Pew R Re
Rr r r∗ Sc Sh Sh S1 T u u∗ up u HS v vw x x∗ y y∗ z
Dimensionless wall concentration ( ccw0 ) Concentration at the wall (kg/m3 ) Diffusivity of neutral solute (10−12 m2 /s) Diameter of tube (m) Applied electric field (V/m) Charge of an electron (1.6 × 10−19 ) up ) Ratio between the velocity of pressure-driven flow and electroosmotic velocity ( u HS Mass transfer coefficient (m/s) Boltzmann constant (1.38 × 1023 J/K) Length of the tube (m) Permeability of tube wall (m/Pa s) Flow behavioral index Number concentration of ions (m−3 ) Number concentration of cation (m−3 ) Number concentration of anion (m−3 ) Concentration of ions (mol/m3 ) Applied pressure (Pa) Dimensionless permeate flux ( vwdD ) Length-averaged per
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