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Numerical simulation of natural convection in a vertical annulus with a localized heat source

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Abstract

The present numerical investigation deals with the size and location effects of a single isoflux discrete heater on the buoyancy induced convection in a cylindrical annulus. A discrete heater is placed at the inner wall, while the top and bottom walls as well as the unheated portions of the inner wall are kept adiabatic, and the outer wall is maintained at a lower temperature. The influence of location and size of the discrete heater on the convective flow and the corresponding heat transfer are obtained for a wide range of physical parameters. The predicted numerical results reveal that the placement of heater near the middle portion of inner wall yields a maximum heat transfer and minimum hot spots rather than placing the heater near the top and bottom portions of the inner wall. We found that the location of heater affects the rates of flow circulation and heat transfer in a complex fashion. The rate of heat transfer is an increasing function of radii ratio of the annulus. Further, we found that the rate of heat transfer and maximum temperature in the annular cavity are significantly modified by the heater length and location.

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Abbreviations

A :

aspect ratio

D :

annulus gap, m

g :

acceleration due to gravity, m/s2

h :

dimensional length of the heater, m

H :

height of the annulus, m

l :

distance between the bottom wall and centre of the heater, m

L :

dimensionless location of the heater

Nu :

Nusselt number

\(\overline{\mathit{Nu}}\) :

average Nusselt number of the heater

p∗:

effective pressure, Pa

Pr :

Prandtl number

q h :

heat flux, W/m2

Ra :

Rayleigh number for isothermal heating \(( \mathit{Ra} = \frac{g\beta(\theta_{h} - \theta_{c})D^{3}}{\upsilon\kappa} )\)

Ra ∗ :

modified Rayleigh number

T :

dimensionless temperature

T max :

maximum temperature of the heater

t :

dimensional time, s

(r i ,r 0):

radii of inner and outer cylinders, m

(r,x):

dimensional radial and axial co-ordinates, m

(u,w):

dimensional velocity components in (r,x) directions, m/s

(R,X):

dimensionless radial and axial co-ordinates

(U,W):

dimensionless velocity components in (R,X) direction

β :

coefficient of thermal expansion, 1/K

ε :

dimensionless size of the heater

ζ :

dimensionless vorticity

θ :

dimensional temperature, K

κ :

thermal diffusivity, m2/s

λ :

radii ratio

ν :

kinematic viscosity, m2/s

ρ :

fluid density, kg/m3

τ :

dimensionless time

ψ :

dimensional stream function, m2/s

Ψ :

dimensionless stream function

h :

conditions at the heater wall

c :

conditions at the cold wall

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Acknowledgements

This work was supported by WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (Grant No. R32-2009-000-20021-0). The author M. Sankar gratefully acknowledges the Chairman and Principal, East Point College of Engineering and Technology, Bangalore, and Visvesvaraya Technological University, Belgaum for the their support and encouragement.

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Authors and Affiliations

  1. Department of Mathematics, East Point College of Engineering and Technology, Bangalore, 560049, India

    M. Sankar

  2. Department of Mathematics, Kyungpook National University, 1370 Sangyeok-Dong, Buk-Gu, Daegu, 702-701, Republic of Korea

    Soojin Hong & Younghae Do

  3. Ulsan National Institute of Science and Technology (UNIST), Ulsan, 689-805, Republic of Korea

    Bongsoo Jang

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  1. M. Sankar
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  2. Soojin Hong
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  3. Younghae Do
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  4. Bongsoo Jang
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Correspondence to Younghae Do.

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Sankar, M., Hong, S., Do, Y. et al. Numerical simulation of natural convection in a vertical annulus with a localized heat source. Meccanica 47, 1869–1885 (2012). https://doi.org/10.1007/s11012-012-9560-3

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