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Combined thermal convective magneto-hydrodynamic flow

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Summary

The role of combined natural and forced thermal convection in the flow of an electrically conducting fluid through a magnetic field is analyzed. The conditions under which the magnetic field renders the free convective contribution to the total thermal exchange negligible are discussed. Effects of wall electrical conductivity, magnetic field strength, internal energy generation, free convection, and thermal Prandtl number on the flow and heat transfer are presented and discussed.

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Abbreviations

a :

half-width of channel

a m :

Fourier coefficient

A :

temperature gradient, defined in equation (21)

b :

half-breath of channel

B :

magnetic flux density vector

B :

dimensionless magnetic flux density,B = B x/B 0 C

c :

heat capacity

C :

parameter defined as\((a^3 c/vk)/\left( { - \frac{{\partial p}}{{\partial x}} - \rho g} \right)\)

C*:

parameter defined as\((a^3 c/vk)/\left( { - \frac{{\partial p}}{{\partial x}} - \rho g + \sigma B_0 E_0 } \right)\)

D :

dielectric displacement vector

E :

electric field intensity vector

F :

internal energy generation index,F = Qa/kAC

F*:

internal energy generation index,F* = Qa/kAC*

g,g :

gravitational acceleration magnitude, vector

G∞G*:

mean flow rate-pressure gradient ratio, defined in equations (42), (62)

h :

thickness of channel walls

H :

magnetic field strength vector

J :

current density vector

k :

thermal conductivity

M :

Hartmann number,aB 0(σ/μ)1/2

m, n :

summation indices

Pr :

thermal Prandtl number,ρνc/k

Pr m :

magnetic Prandtl number,σμ 0 ν

p :

pressure

p 1(y, z):

pressure function defined in equation (24)

Q :

internal volumetric energy generation rate

Ra :

Rayleigh number,Ra = gβρCa 4 A/vk

Rm :

magnetic Reynolds' number,Rm = μ0σaū

t :

time

T, T B :

temperature, bulk fluid temperature

u :

local velocity inx-direction

ū :

mean velocity inx-direction

U :

dimensionless form ofu,U = apcu/kC

V :

velocity field vector

x, y, z :

rectangular coordinates

α i :

integration constants defined in Appendix B

i :

1 to 6

β :

thermal expansion coefficient

γ :

aspect ratio,b/a

ε 0 :

inductive capacity

ζ :

dimensionless form ofz, ζ=z/a

η :

dimensionless form ofy, η=y/a

θ :

dimensionless temperature difference,θ =(T − T) w)/aAC

κ, τ′, κ i :

integration constants

λ i :

parameters defined in Appendix A

μ 0 :

magnetic permeability

ν :

kinematic viscosity

ρ :

mass density

ρe :

charge density

σ :

electrical conductivity

Φ :

viscous dissipation term

φ, φ 1,φ 2 :

wall electrical conductance ratios

mn :

double Fourier transform

m orn :

single Fourier transform

w :

wall conditions

x, y, z :

scalar component inx, y, z direction

0:

applied value

∞:

parallel-plate case

*:

small magnetic Prandtl number case

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Authors and Affiliations

  1. Engineering Research Section, Reactor Engineering Division, Argonne National Laboratory, Argonne, Illinois, USA

    Ralph M. Singer

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  1. Ralph M. Singer
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Work performed under the auspices of the U.S. Atomic Energy Commission.

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Singer, R.M. Combined thermal convective magneto-hydrodynamic flow. Appl. Sci. Res. 12, 375–404 (1965). https://doi.org/10.1007/BF00382134

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  • DOI: https://doi.org/10.1007/BF00382134

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