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Separation of species of a binary fluid mixture confined between two concentric rotating circular cylinders in presence of a strong radial magnetic field

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Abstract

The effect of a radial magnetic field on separation of a binary mixture of incompressible viscous thermally and electrically conducting fluids confined between two concentric rotating circular cylinders with different angular velocity is examined. The equations governing the motion, temperature and concentration in cylindrical polar coordinate are solved analytically. The solution obtained in closed form for concentration distribution is plotted against the radial distances from the surface of the inner circular cylinder for various values of non-dimensional parameters. It is found that the non-dimensional parameters viz. the Hartmann number, thermal diffusion number, baro diffusion number, rotational Reynolds number, the product of Prandtl number and Eckert number, magnetic Prandtl number and the ratio of the angular velocities of inner and outer cylinders affects the species separation of rarer and lighter component significantly. The problem discussed here derives its application in the basic fluid dynamics separation processes to separate the rarer component of the different isotopes of heavier molecules where electromagnetic method of separation does not work.

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Abbreviations

a, b:

Radii of inner and outer cylinder respectively

B :

Magnetic flux density vector

B d :

Baro diffusion number, B d = Aρνω 2

c :

Concentration

c 1 :

Concentration of the first component of the binary mixture

c 2 :

Concentration of the second component of the binary mixture

c p :

Specific heat at constant pressure

c 0 :

Concentration of lighter and rarer component of the binary fluid mixture in contact with inner cylinder

\( {\mathcal{D}} \) :

Diffusion coefficient

E :

Electric field vector

E c :

Eckert number, \( E_{\text{c}} = {\frac{{\omega_{2}^{2} b^{2} }}{{c_{\text{p}} \left( {T_{2} - T_{1} } \right)}}} \)

F :

Body force per unit mass

H :

Magnetic field vector

H 0 :

Uniform magnetic field

i :

Diffusion flux density vector

J :

Current density vector

k p :

Baro-diffusion ratio

k T :

Thermal-diffusion ratio

M :

Hartmann number, \( M = aH_{0} \left( \frac{\upsigma}{\rho \nu } \right)^{1/2} \)

m 1 :

Mass of the first kind of the particle

m 2 :

Mass of the second kind of the particle

n :

Unit vector drawn at the solid surface of the boundary directed outwards

p :

Pressure

P m :

Magnetic Prandtl number, P m = μ e σν

p*:

Non-dimensionalised pressure

p ∞ :

Working pressure of the medium

P r :

Prandtl number, \( P_{\text{r}} = {\frac{{{{\upmu}}c_{\text{p}} }}{\kappa }} \)

r :

Co-ordinate measuring radial distance

\( R_{\text{e}}^{\prime} \) :

Rotational Reynolds number, \( R_{\text{e}}^{\prime} = {\frac{{b^{2} {{\upomega}}_{2} }}{\nu }} \)

R m :

Magnetic Reynolds number, \( R_{\text{m}} = R_{\text{e}}^{\prime} P_{\text{m}} \)

s T :

Soret coefficient

T :

Temperature, T 1, temperature of binary fluid mixture at the inner cylinder, T 2, temperature of binary fluid mixture at the outer cylinder, T*, non-dimensionalised temperature

t d :

Soret number, t d = s T(T 2 − T 1)

v :

Velocity component in r-direction

v :

Velocity vector

v*:

Non-dimensionalised velocity

v 1 :

Velocity of the first component of the binary mixture

v 2 :

Velocity of the second component of the binary mixture

z :

Co-ordinate measuring the axial distance

ϕ:

Heat due to viscous dissipation

η :

Non-dimensional variable measuring the radial distance

η 1 :

Ratio of inner radius to outer radius of the cylinders i.e. \( {{\eta}}_{1} = \frac{a}{b} \)

η m :

Coefficient of magnetic viscosity or magnetic diffusivity, \( \eta_{\text{m}} = {1 \mathord{\left/ {\vphantom {1 {\left( {\mu_{\text{e}} \sigma } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\mu_{\text{e}} \sigma } \right)}} \)

κ :

Thermal conductivity

μ e :

Magnetic permeability

ν :

Coefficient of kinematic viscosity

θ :

Co-ordinate measuring the azimuthal direction

ρ :

Density

ρ 1 :

Density of the first component of the binary mixture

ρ 2 :

Density of the first component of the binary mixture

σ :

Electrical conductivity

ω 1 :

Angular velocity of the inner cylinder

ω 2 :

Angular velocity of the outer cylinder

Ω:

Ratio of the angular velocities of inner cylinder to the outer cylinder i.e. \( \Upomega = {\frac{{{{\upomega}}_{1} }}{{{{\upomega}}_{2} }}} \)

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Acknowledgments

Authors are extremely thankful to the referees of this manuscript for their valuable comments and suggestions, which contributes in improving the work.

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Authors and Affiliations

  1. Department of Mathematics, Dibrugarh University, Dibrugarh, 786004, Assam, India

    B. R. Sharma

  2. Marwari Hindi High School, Dibrugarh, 786001, India

    R. N. Singh

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Sharma, B.R., Singh, R.N. Separation of species of a binary fluid mixture confined between two concentric rotating circular cylinders in presence of a strong radial magnetic field. Heat Mass Transfer 46, 769–777 (2010). https://doi.org/10.1007/s00231-010-0609-3

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