Abstract
The problem of natural convection over a semi-infinite flat plate with non-uniform wall temperature is studied by using a numerical method. The local rates of heat transfer as a function of the distance along the plate are tabulated for a range of Prandtl numbers (0.01 to 100) and for a few cases of wall temperature distributions. Such tabulations serve as a reference against which other approximate solutions can be compared in the future.
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Abbreviations
- a i, bi, ci :
-
constants and coefficients in Eq. (I-5) in Appendix
- A :
-
constant
- f :
-
dimensionless dependent variable, defined in (5)
- F :
-
representation of the differential equations in Appendix
- g :
-
dimensionless dependent variable, defined in (5)
- Gr :
-
Grashof number, gβ(T r−T ∞)L 3/v 2
- h :
-
heat transfer coefficient
- h j :
-
step size
- k :
-
heat conductivity
- K :
-
ratio of successive step sizes
- L :
-
length of the plate
- m :
-
constant
- M(ξ):
-
dimensionless surface mass transfer parameter
- n :
-
constant
- Nu :
-
Nusselt number, hx/k
- P(ξ):
-
wall temperature function, defined in Eq. (9)
- Pr :
-
Prandtl number
- r :
-
a function of x in Eq. (16)
- R :
-
a function of x in Appendix
- Re :
-
Reynold number
- S :
-
a function of x in Appendix
- S w :
-
dimensionless wall temperature
- t :
-
g′ in Appendix
- T :
-
temperature
- u :
-
velocity component in x-direction and f′ in Appendix
- v :
-
velocity component in y-direction and f″ in Appendix
- x :
-
coordinate along the plate
- y :
-
coordinate perpendicular to the plate
- θ :
-
(T−T ∞)/(T w−T ∞)
- ξ :
-
dimensionless coordinate along the plate
- η :
-
similar variable, defined in (5)
- ψ :
-
stream function
- α :
-
constant and a function of ξ in Appendix
- β :
-
bulk modulus and a function of P(ξ) in Appendix
- φ :
-
a function of x
- ε :
-
(T w1−T ∞)/(T w2−T ∞)
- o:
-
reference condition
- r:
-
reference condition
- w:
-
wall condition
- x:
-
local condition
- ∝:
-
mainstream condition
References
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Keller, H. B., A New Difference Scheme for Parabolic Problems, Numerical Solution of Partial Differential Equations, Edited by J. Bramble, Academic Press, New York, 1970.
Bellman, R. and R. E. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, American Elsevier, New York, 1965.
Kuiken, H. K., Appl. Sci. Res. 20 (1969) 205.
Kao, T. T., G. A. Domoto and H. G. Elrod, Jr., ASME paper 75-WA/HT-15, presented at 1975 Winter Annual Meeting, Houston, Texas, November 30-December 4, 1975.
Parikh, P. G., R. J. Moffat, W. M. Kays and O. Bershader, Int. J. Heat Mass Transfer 17, No. 12 (1974) 1465.
Sparrow, E. M. and J. L. Gregg, ASME Trans. 80 (1958) 379.
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Kao, T. T., Letters in Heat Mass Transf. 2 (1975) 419.
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Na, T.Y. Numerical solution of natural convection flow past a non-isothermal vertical flat plate. Appl. Sci. Res. 33, 519–543 (1977). https://doi.org/10.1007/BF00411829
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DOI: https://doi.org/10.1007/BF00411829