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Laminar forced convection to Bingham plastic flowing through a circular tube with internal energy generation

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Summary

Heat transfer to a Bingham plastic in laminar tube flow with transient internal heat generation is investigated. Numerical results are obtained for the temperature response of the fluid and tube wall when subjected to step and sinusoidal variations in the heat generation rate. A resonance phenomenon is observed for the frequency response, and the Bingham plastic responds faster than a Newtonian fluid following a step in the internal heat generation.

Experiments with both Newtonian fluids and slurries of aluminum in sulfuric acid show agreement with the theoretical predictions. However the effects of combined free and forced convection, as well as nonuniform dispersion of solids in a slurry, result in an axially asymmetric temperature distribution. Heat exchangers whose design is influenced by these gravity dependent phenomena include the homogeneous nuclear reactor and chemical reactors.

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Abbreviations

a :

Bingham plastic constant or “plug” radius, r*/R i

A :

amplitude ratio; A f=T f/T sf; A w=T w/T sw, dimensionless

C f :

fluid specific heat

C w :

wall specific heat

D i :

inside diameter of tube

g :

local acceleration of gravity

k :

thermal conductivity

N Gr :

Grashof number, ρ 2 gβ(T− T ∞) D 3i /μ 2, dimensionless

N Gz :

Graetz number, Re Pr D i/x*, dimensionless

Pr :

Prandtl number, μC f/k f, dimensionless

q″ :

volumetric heat generation rate, dimensionless

q0:

mean volumetric heat generation rate

R :

radius; R i, inside tube radius; R o, outside tube radius

Re :

Reynolds number, ρ u*D i/μ, dimensionless

r :

dimensionless radius, r*/R i

r*:

radial distance from the centerline of the tube

T*:

temperature

T :

dimensionless temperature, (T−T 0) k f/q0R 2i

t :

time

u :

dimensionless fluid velocity, u*/2u *m

u*:

fluid velocity

x :

dimensionless axial distance from inlet of test section, x*/R

x*:

axial distance from inlet of test section

α f :

fluid thermal diffusivity

α w :

wall thermal diffusivity

β :

dimensionless parameter, ρ f C f/ρ w C w

Γ :

dimensionless frequency, ω R 2i /α f

δ :

dimensionless wall thickness, R 0/R i−1

θ :

dimensionless time, α f t/R 2i

Λ :

dimensionless parameter, α w/α f

μ :

viscosity

μ B :

Bingham plastic viscosity

ρ :

density

τ :

shear stress, τ y, yield shear stress of Bingham plastic

ø :

phase lag of temperature with respect to the heat generation function, dimensionless

ω :

frequency

f:

fluid

i:

initial

m:

mean

o:

entrance

sf:

steady fluid

sw:

steady wall

w:

wall

References

  1. Schechter, R. S. and E. H. Wissler, Nucl. Sci. and Eng. 6 (1959) 371.

    Google Scholar 

  2. Poppendiek, H. F., Chem. Eng. Sympos. Ser. 50 (1954) 93.

    Google Scholar 

  3. Grigull, U., Chemie-Ing. Techn. 28 (1956) 553.

    Google Scholar 

  4. Sparrow, E. M. and R. Siegel, Nucl. Sci. and Eng. 4 (1958) 239.

    Google Scholar 

  5. Tachibana, F. and T. Morishita, Industrial Research, Tokyo University, 11 (1959) 45.

    Google Scholar 

  6. Michiyoshi, I., Bulletin of Japan Soc. Mech. Eng. 5 (1962) 315.

    Google Scholar 

  7. Michiyoshi, I., et al, Trans. Japan Soc. Mech. Eng. 28 (1962) 1533.

    Google Scholar 

  8. Ayers, D. L., The Asymptotic Behavior of Periodic Heat Transfer to Laminar Flow in Tubes, Ph.D. thesis, Department of Mechanical Engineering, University of Michigan 1963.

  9. Metzner, A. B., Non-Newtonian Technology: Fluid Mechanics, Mixing, and Heat Transfer, Advances in Chemical Engineering, Vol. 1, Academic Press Inc., New York 1956.

    Google Scholar 

  10. Thomas, D. G., E. L. Compere, and J. P. McBride, Nucleonics 18 (1960) 104.

    Google Scholar 

  11. Henning, C. D., Transient Heat Transfer to a Bingham Plastic with Internal Energy Generation, Ph.D. thesis, Department of Mechanical Engineering, University of Michigan 1965.

  12. Thomas, D. G., Homogeneous Reactor Project, Quarterly Progress Report for the Period Ending July 31, 1954, USAEC report ORNL-1813, Oak Ridge National Laboratory 1954.

  13. Lane, J. A., H. G. MacPherson, and Frank Maslon, Fluid Fuel Reactors, Addison-Wesley Publ. Co., Reading, Mass. 1958.

    Google Scholar 

  14. Metais, B. and E. R. G. Eckert, Trans. ASME J. of Heat Transfer 86 (1964) 295.

    Google Scholar 

  15. Bird, R. B., et al, Transport Phenomena, John Wiley and Sons, New York 1960.

    Google Scholar 

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Authors and Affiliations

  1. Mech. Eng. Dept., Lawrence Rad. Lab., Un. of California, Livermore, Cal., USA

    Carl D. Henning

  2. Mech. Eng. Dept., Un. of Michigan, Ann Arbor, Mich., USA

    Wen-Jei Yang

Authors
  1. Carl D. Henning
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  2. Wen-Jei Yang
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Henning, C.D., Yang, WJ. Laminar forced convection to Bingham plastic flowing through a circular tube with internal energy generation. Appl. Sci. Res. 18, 336–352 (1968). https://doi.org/10.1007/BF00382358

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  • DOI: https://doi.org/10.1007/BF00382358

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