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Theoretical investigation of the effect of an internal heat source on turbulent liquid metal heat transfer in a channel with uniform heat flux

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Summary

The effect of an internal heat source on the heat transfer characteristics for turbulent liquid metal flow between parallel plates is studied analytically. The analysis is carried out for the conditions of uniform internal heat generation, uniform wall heat flux, and fully established temperature and velocity profiles. Consideration is given both to the uniform or slug flow approximation and the power law approximation for the turbulent velocity profile. Allowance is made for turbulent eddying within the liquid metal through the use of an idealized eddy diffusivity function. It is found that the Nusselt number is unaffected by the heat source strength when the velocity profile is assumed to be uniform over the channel cross section. In the case of a 1/7-power velocity expression, the Nusselt numbers are lower than those in the absence of internal heat generation, and decrease with diminishing eddy conduction. Nusselt numbers, in the absence of an internal heat source, are compared with existing calculations, and indications are that the present results are adequate for preliminary design purposes.

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Abbreviations

A :

hydrodynamic parameter

a :

half height of channel

a 1 :

a constant, 1+0.01\(\bar \psi \) Pr Re 0.9

a 2 :

a constant, 0.01\(\bar \psi \) Pr Re 0.9

C p :

specific heat at constant pressure

D h :

hydraulic diameter of channel, 4a

h :

heat transfer coefficient, q w/(t w−t b)

I 1 :

integral defined by (17)

I 2 :

integral defined by (18)

k :

diffusivity parameter, (1+0.01 \(\bar \psi \) Pr Re 0.9)1/2

m :

exponent in power velocity expression

Nu :

Nusselt number, hD h/κ

Nu 0 :

Nusselt number in absence of internal heat generation

Pr :

Prandtl number, ν/α

Q :

heat generation rate per volume

q w :

wall heat flux

Re :

Reynolds number for channel, 2ūα/ν

s :

ratio of heat generation rate to wall heat flux, Qa/q w

T :

dimensionless temperature, (t w−t)/(t w−t b)

t :

fluid temperature, t w wall temperature, t b fluid bulk temperature

u :

fluid velocity in x direction, ū, fluid mean velocity

x :

longitudinal coordinate measured from channel entrance

x + :

dimensionless longitudinal coordinate, 2(x/a)/Pr Re

y :

transverse coordinate measured from channel centerline

z :

transverse coordinate measured from channel wall, a−y

α :

molecular diffusivity of heat, κ/ρC p

β :

dummy variable of integration

γ :

dummy variable of integration

ε H :

eddy diffusivity of heat

ε M :

eddy diffusivity of momentum

ζ :

dummy variable of integration

κ :

fluid thermal conductivity

λ T :

dimensionless diffusivity, Pr (ε H/ν)

ν :

fluid kinematic viscosity

ξ :

dummy variable of integration

ρ :

fluid density

τ :

dummy variable of integration

ψ :

ratio of eddy diffusivity for heat transfer to that for momentum transfer, ε H/ε M

\(\bar \psi \) :

average value of ψ

ω :

dimensionless velocity distribution, u/ū

References

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Authors and Affiliations

  1. Lewis Research Center National Aeronautics and Space Administration, Cleveland, Ohio, USA

    R. M. Inman

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  1. R. M. Inman
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Inman, R.M. Theoretical investigation of the effect of an internal heat source on turbulent liquid metal heat transfer in a channel with uniform heat flux. Appl. Sci. Res. 18, 460–477 (1968). https://doi.org/10.1007/BF00382367

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  • DOI: https://doi.org/10.1007/BF00382367

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