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Heat transfer in magnetohydrodynamic laminar radial channel flow

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Abstract

Heat transfer in the steady axisymetrical laminar source flow of an incompressible electrically conducting fluid between two parallel disks in the presence of a transverse applied magnetic field is analyzed. The energy equation is solved numerically for the temperature distribution, where both Joulean and viscous heating are included. Both local and average Nusselt numbers for the case of constant wall temperature are evaluated. For fluids of moderate and high Prandtl numbers, Nusselt number is seen to be a strong function of both Hartmann number and a heat generation parameter together with a modified Peclet number. However, for fluids of small Prandtl number, Joulean heating and viscous dissipation can be neglected without appreciable error.

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Abbreviations

A(n):

coefficient in the implicit finite difference equation

Bo :

applied magnetic induction

B(n):

coefficient in the implicit finite difference equation

C(n):

coefficient in the implicit finite difference equation

C p :

specific heat at constant pressure

D(n):

coefficient in the implicit finite difference equation

F 1 :

radial velocity function, M \(\left[ {\frac{{cosh My - cosh M}}{{sinh M - cosh M}}} \right]\)

Gz :

Graetz number, Eq. (14)

H 0 :

axial applied magnetic field

J :

total electric current density (J r, J θ, J z)

K :

thermal conductivity

M :

Hartmann number, μ 0 H 0 t(σ/μ)1/2

m :

integer designation for columns in the finite-difference grid

N :

number of grid points in the axial direction in the finite-difference grid

Nu(η):

local Nusselt number

004E-0304;u :

average Nusselt number

n :

integer designation for rows in the finite-difference grid

Pr :

Prandtl number

Q :

source discharge

q :

heat flux

r :

radial coordinate

r i :

inlet radius of the two disks

r o :

outlet radius of the two disks

Re :

channel Reynolds number, U 0 t/ν

T :

temperature

T 0 :

initial temperature

T w :

wall temperature

t :

half channel width

u :

radial velocity

U 0 :

reference velocity Q/4πt 2

ū :

non-dimensional radial velocity, u/U 0

W :

axial velocity

W:

non-dimensional axial velocity, W/U 0

x :

non-dimensional radial coordinate, r/t

y :

non-dimensional axial coordinate, z/t

z :

axial coordinate

β :

non-dimensional constant, Eq. (5)

γ :

heat generation number, β/Re

Δ :

finite change in the coordinate direction

η :

modified radial coordinate, 1/Gz

θ :

non-dimensional temperature

μ :

viscosity

η 0 :

magnetic permeability

ν :

kinematic viscosity

ρ :

density

σ :

electrical conductivity

φ :

viscous dissipation function, Eq. (2)

References

  1. Thomas, R. A. and M. M. Cobble, J. Heat Transfer ASME. 85 (1963) 189.

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  2. Kreith, F., Int. J. Heat and Mass Transfer 9 (1966) 265.

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  3. Khader, M. S. and J. S. Goodling, Appl. Sci. Res. 27 (1972) 93.

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  4. Kreith, F., Phy. of Fluids 8 (1965) 1189.

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  5. Khader, M. S., Ph.D. Diss. Auburn University, U.S.A. (1973).

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Author notes

  1. M. S. Khader

    Present address: Dept. of Mech. Eng. Cairo Univ., Cairo, Egypt

Authors and Affiliations

  1. Dept. of Mech. Eng., Auburn Univ., Auburn, Alabama, USA

    M. S. Khader & R. I. Vachon

Authors
  1. M. S. Khader
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  2. R. I. Vachon
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Khader, M.S., Vachon, R.I. Heat transfer in magnetohydrodynamic laminar radial channel flow. Appl. Sci. Res. 29, 321–341 (1974). https://doi.org/10.1007/BF00384156

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  • DOI: https://doi.org/10.1007/BF00384156

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