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KSME International Journal

, Volume 15, Issue 10, pp 1451–1460 | Cite as

The onset of natural convection and heat transfer correlation in horizontal fluid layer heated uniformly from below

  • Min Chan Kim
  • Sin Kim
Thermal Engineering · Fluid Engineering · Energy and Power Engineering

Abstract

The critical condition of the onset of buoyancy-driven convective motion of uniformly heated horizontal fluid layer was analysed by the propagation theory which transforms the disturbance quantities similarly. The dimensionless critical time,τ c , is obtained as a function of the Rayleigh number and the Prandtl number. Based on the stability criteria and the boundary-layer instability model, a new heat transfer correlation which can cover whole range of Rayleigh number was derived. Our theoretical results predict the experimental results quite reasonably.

Key words

Buoyancy Effect Stability Analysis Heat Transfer Correlation Propagation Theory 

Nomenclature

a

Horizontal wave number [−]

d

Fluid depth [m]

g

Gravitational acceleration [m/s2]

k

Thermal conductivity [J/mK]

Nu

Nusselt number (=qw/kΔT) [−]

P

Pressure [Pa]

Pr

Prandtl number (=ν/α) [−]

Ra

Rayleigh number (=gβΔTd3/αν) [−]

Raq

Rayleigh number based on the heat flux (=gβqwd4/kαν) [−]

qw

Wall heat flux [J/m2]

T

Temperature [K]

t

Time [s]

\(\overrightarrow U \)

Velocity vector [m/s]

w

Dimensionless vertical velocity [−]

X, Y, Z

Space in Cartesian coordinate [m]

Greeks

α

Thermal diffusivity [m2/s]

β

Thermal expansion coefficient [1/K]

ΔT

Temperature difference [K]

ζ

Similarity variable [−]

θ

Dimensionless temperature [−]

μ

Viscosity [Pa s]

ν

Kinematic viscosity [m2/s]

ρ

Density [kg/m3]

τ

Dimensionless time [−]

Subscripts

0

Basic quantity

1

Disturbed quantity

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References

  1. Arpaci, V. S., 1997, “Microscales of Turbulent Heat and Mass Transfer Correlations,”Advances in Heat Transfer, Vol. 30, pp. 1–91.Google Scholar
  2. Busse, F. B., 1967, “On the Stability of Two-Dimensional Convection in a Layer Heated from Below,”J. Math. Phys., Vol. 46, pp. 140–150.MATHGoogle Scholar
  3. Carslaw, H. S. and Jaeger, J. C., 1959,Conduction of Heat in Solid, 2nd ed., Oxford.Google Scholar
  4. Chen, K. Chen, M. M. and Sohn, C. W., 1983, “Thermal Instability of Two-Dimensional Stagnation-Point Boundary Layers,”J. Fluid Mech., Vol. 132, pp. 49–63.MATHCrossRefGoogle Scholar
  5. Cheung, F. B., 1980, “Heat Source-Driven Thermal Convection at Arbitrary Prandtl Number,”J. Fluid Mech., Vol. 97, pp. 734–758.CrossRefMathSciNetGoogle Scholar
  6. Choi, C. K. and Kim, M. C., 1994, “Buoyancy Effects in Plane Couette Flow Heated from Below,”Proc. 10th Int. Heat Transfer Conf., Brighton, Vol. 5, pp. 453–458.Google Scholar
  7. Choi, C. K., Shin, C. B. and Hwang, S. T., 1986, “Thermal Instability in Thermal Entrance Region of Couette Flow Heated Uniformly from Below,”Proc. 8th Int. Heat Transfer Conf., San Francisco, Vol. 3, pp. 1389–1394.Google Scholar
  8. Chu, T. Y., 1990, “Developing Convection above a Finite Horizontal Surface,”Proc. 9th Int. Heat Transfer Conf., Jerusalem, Vol. 2, pp. 169–174.Google Scholar
  9. Chun, Y. H. and Choi, C. K., 1991, “Thermal Instability of Natural Convection over Inclined Isothermal Heated Plates,”Hwahak Konghak, Vol. 29, pp. 381–387 (in Korean).Google Scholar
  10. Currie, I. G., 1967, “The Effect of Heating Rate on the Stability of Stationary Fluid,”J. Fluid Mech., Vol. 29, pp. 337–347.CrossRefGoogle Scholar
  11. Foster, T. D., 1965, “Stability of Homogeneous Fluid Cooled Uniformly from Above,”Phys. Fluids, Vol. 8, pp. 1249–1257.CrossRefMathSciNetGoogle Scholar
  12. Foster, T. D., 1969, “Onset of Manifest Convection in a Layer of Fluid with a Time-Dependent Surface Temperature,”Phys. Fluids, Vol. 12, pp. 2482–2487.CrossRefGoogle Scholar
  13. Howard, L. N., 1964, “Convection at High Rayleigh Numbers,”Proc. 11th Int. Cong. Appl. Mech., Munich, pp. 1109–1115.Google Scholar
  14. Jhaveri, B. S. and Homsy, G. M., 1982, “The Onset of Convection in Fluid Layer Heated Rapidly in a Time-Dependent Manner,”J. Fluid Mech., Vol. 114, pp. 251–260.MATHCrossRefGoogle Scholar
  15. Kim, M. C., Baik, J. S., Hwang, I. G., Yoon, D. Y., Choi, C. K., 1999a, “Buoyancy-Driven Convection in Plane Poiseuille Flow,”Chem. Eng. Sci., Vol. 54, pp. 619–632.CrossRefGoogle Scholar
  16. Kim, M. C., Choi, K. H. and Choi, C. K., 1999b, “The Onset of Thermal Convection in Initially, Stably Stratified Fluid Layer,”Int. J. Heat Mass Transfer, Vol. 42, pp. 4253–4258.MATHCrossRefGoogle Scholar
  17. Kim, K.-H. and Kim, M.-U., 1986, “The Onset of Natural Convection in a Fluid Layer Suddenly Heated from Below,”Int. J. Heat Mass Transfer, Vol. 29, pp. 193–201.MATHCrossRefGoogle Scholar
  18. Lick, W., 1965, “The Instability of a Fluid Layer with Time Dependent Heating,”J. Fluid Mech., Vol. 21, pp. 565–576.MATHCrossRefMathSciNetGoogle Scholar
  19. Lee J. D., Choi, C. K. and Yoon, D. Y., 1988, “An Anlaysis of Thermal Convection in a Horizontal Fluid Layer Heated Uniformly from Below,”Proc. 1st KSME-JSME Thermal and Fluid Eng. Conf., Vol. 2, pp. 342–347.Google Scholar
  20. Long, R. R., 1976, “The Relation Between Nusselt Number and Rayleigh Number in Turbulent Thermal Convection,”J. Fluid Mech., Vol. 73, pp. 445–451.CrossRefGoogle Scholar
  21. Morton, B. R., 1957, “On the Equilibrium of a Stratified Layer of Fluid,”J. Mech. Appl. Math., Vol. 10, pp. 433–447.MATHCrossRefMathSciNetGoogle Scholar
  22. Nielsen, R. C. and Sabersky, R. H., 1973, “Transient Heat Transfer in Benard Convection,”Int. J. Heat Mass Transfer, Vol. 16, pp. 2407–2420.CrossRefGoogle Scholar
  23. Patrick, M. A. and Wragg, A. A., 1975, “Optical and Electrochemical Studies of Transient Free Convection Mass Transfer at Horizontal Surfaces,”Int. J. Heat Mass Transfer, Vol. 18. pp. 1397–1407.CrossRefGoogle Scholar
  24. Sparrow, E. M., Goldstein, R. J. and Jonsson, V. K., 1964, “Thermal Instability in a Horizontal Fluid Layer: Effect of Boundary Condtions and Nonlinear Temperature Profiles,”J. Fluid Mech., Vol. 18, pp. 513–528.MATHCrossRefMathSciNetGoogle Scholar
  25. Stuart, J. T., 1964, “On Cellular Patterns in Thermal Convection,”J. Fluid Mech., Vol. 18, pp. 481–498.MATHCrossRefMathSciNetGoogle Scholar
  26. Wankat, P. C. and Homsy, G. M., 1977, “Lower Bounds for the Onset Time of Instability in Heated Layers,”Phys. Fluids, Vol. 20, pp. 1200–1201.CrossRefGoogle Scholar
  27. Yoon, D. Y. and Choi, C. K., 1989, “Thermal Convection in a Staturated Porous Medium subjected to Isothermal Heating,”Korean J. Chem. Eng., Vol. 6, pp. 144–149.CrossRefGoogle Scholar

Copyright information

© The Korean Society of Mechanical Engineers (KSME) 2001

Authors and Affiliations

  1. 1.Department of Chemical EngineeringCheju National UniversityCheju-doKorea
  2. 2.Department of Nuclear and Energy EngineeringCheju National UniversityCheju-doKorea

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