A boundary layer based integral analysis has been performed to investigate laminar natural convection heat transfer characteristics for fluids with arbitrary Prandtl number over a semi-infinite horizontal plate subjected either to a variable wall temperature or variable heat flux. The wall temperature is assumed to vary in the form $T¯w(x¯)-T¯∞=ax¯n$ whereas the heat flux is assumed to vary according to $qw(x¯)=bx¯m$. Analytical closed-form solutions for local and average Nusselt number valid for arbitrary values of Prandtl number and nonuniform heating conditions are mathematically derived here. The effects of various values of Prandtl number and the index n or m on the heat transfer coefficients are presented. The results of the integral analysis compare well with that of previously published similarity theory, numerical computations and experiments. A study is presented on how the choice for velocity and temperature profiles affects the results of the integral theory. The theory has been generalized for arbitrary orders of the polynomials representing the velocity and temperature profiles. The subtle role of Prandtl number in determining the relative thicknesses of the velocity and temperature boundary layers for natural convection is elucidated and contrasted with that in forced convection. It is found that, in natural convection, the two boundary layers are of comparable thickness if Pr ≤ 1 or Pr ≈ 1. It is only when the Prandtl number is large (Pr > 1) that the velocity boundary layer is thicker than the thermal boundary layer.

## References

References
1.
Burmeister
,
L. C.
,
1983
,
Convective Heat Transfer
,
John Wiley
,
New York
.
2.
Martynenko
,
O. G.
, and
Khramtsov
,
P. P.
,
2005
,
Free Convective Heat Transfer: With Many Photographs of Flows and Heat Exchange
,
Springer
,
New York
.
3.
Kays
,
W. M.
, and
Crawford
,
M. E.
,
1993
,
Convective Heat and Mass Transfer
,
3rd ed.
,
McGraw-Hill
,
New York
.
4.
Rajan
,
V. S. V.
, and
Picot
,
J. J. C.
,
1971
, “
Experimental Study of the Laminar Free Convection From a Vertical Plate
,”
Ind. Eng. Chem. Fundam.
,
10
(
1
), pp.
132
134
.10.1021/i160037a021
5.
Martynenko
,
O. G.
,
Berezovsky
,
A. A.
, and
Sokovishin
,
Yu. A.
,
1984
, “
Laminar Free Convection From a Vertical Plate
,”
Int. J. Heat Mass Transfer
,
27
(
6
), pp.
869
881
.10.1016/0017-9310(84)90008-5
6.
Aydin
,
O.
, and
Guessous
,
L.
,
2001
, “
Fundamental Correlations for Laminar and Turbulent Free Convection From a Uniformly Heated Vertical Plate
,”
Int. J. Heat Mass Transfer
,
44
(
24
), pp.
4605
4611
.10.1016/S0017-9310(01)00107-7
7.
Fujii
,
T.
, and
Imura
,
H.
,
1972
, “
Natural-Convection Heat Transfer From a Plate With Arbitrary Inclination
,”
Int. J. Heat Mass Transfer
,
15
(
4
), pp.
755
767
.10.1016/0017-9310(72)90118-4
8.
Goldstein
,
R. J.
, and
Lau
,
K. S.
,
1983
, “
Laminar Natural Convection From a Horizontal Plate and the Influence of Plate–Edge Extensions
,”
J. Fluid Mech.
,
129
, pp.
55
75
.10.1017/S0022112083000646
9.
Clifton
,
J. V.
, and
Chapman
,
A. J.
,
1984
, “
Natural Convection on a Finite Size Horizontal Plate
,”
Int. J. Heat Mass Transfer
,
12
(
12
), pp.
1573
1584
.10.1016/0017-9310(69)90092-1
10.
Yu
,
W.-S.
, and
Lin
,
H.-T.
,
1988
,
Free Convection Heat Transfer From an Isothermal Plate With Arbitrary Inclination
,”
Wärme-und Stoffübertragung
,
23
, pp.
203
211
.10.1007/BF01807322
11.
Lin
,
H.-T.
,
Yu
,
W.-S.
, and
Yang
,
S.-L.
,
1989
, “
Free Convection on an Arbitrarily Inclined Plate With Uniform Surface Heat Flux
,”
Wärme-und Stoffübertragung
,
24
, pp.
183
190
.10.1007/BF01590018
12.
Pretot
,
S.
,
Zeghmati
,
B.
, and
Le Palec
,
G.
,
2000
, “
Theoretical and Experimental Study of Natural Convection on a Horizontal Plate
,”
Appl. Therm. Eng.
,
20
(
10
), pp.
873
891
.10.1016/S1359-4311(99)00067-8
13.
Mahajan
,
R. L.
, and
Gebhart
,
B.
,
1980
, “
Higher Order Boundary Layer Effects in Plane Horizontal Natural Convection Flows
,”
ASME J. Heat Transfer
,
102
(
2
), pp.
368
371
.10.1115/1.3244291
14.
Pera
,
L.
, and
Gebhart
,
B.
,
1973
, “
Natural Convection Boundary Layer Flow Over Horizontal and Slightly Inclined Surfaces
,”
Int. J. Heat Mass Transfer
,
16
(
10
), pp.
1131
1146
.10.1016/0017-9310(73)90126-9
15.
Afzal
,
N.
,
1985
, “
Higher Order Effects in Natural Convection Flow Over a Uniform Flux Horizontal Surface
,”
Wärme –und Stoffübertragung
,
19
(
3
), pp.
177
180
.10.1007/BF01403753
16.
Schlichting
,
H.
, and
Gersten
,
K.
,
2004
,
Boundary-Layer Theory
,
8th ed.
,
Springer
,
New Delhi
.
17.
Stewartson
,
K.
,
1958
, “
On the Free Convection From a Horizontal Plate
,”
Z. Angew. Math. Phys.
,
9
(
3
), pp.
276
282
.10.1007/BF02033031
18.
Gill
,
W. N.
,
Zeh
,
D. W.
, and
Del Casal
,
E.
,
1965
, “
Free Convection on a Horizontal Plate
,”
Z. Angew. Math. Phys.
,
16
(
4
), pp.
539
541
.10.1007/BF01593934
19.
Dayan
,
A.
,
Kushnir
,
R.
, and
Ullmann
,
A.
,
2002
, “
Laminar Free Convection Underneath a Hot Horizontal Infinite Flat Strip
,”
Int. J. Heat Mass Transfer
,
45
(
19
), pp.
4021
4031
.10.1016/S0017-9310(02)00116-3
20.
Rotem
,
Z.
, and
Claassen
,
L.
,
1969
, “
Natural Convection Above Unconfined Horizontal Surfaces
,”
J. Fluid Mech.
,
38
(pt
1
), pp.
173
192
.10.1017/S0022112069002102
21.
Gebhart
,
B.
,
Jaluria
,
Y.
,
Mahajan
,
R. L.
, and
Sammakia
,
B.
,
1988
,
Buoyancy-Induced Flows and Transport
,
Hemisphere Publishing Corporation
,
New York
.
22.
Chen
,
T. S.
,
Tien
,
H. C.
, and
Armaly
,
B. F.
,
1986
, “
Natural Convection on Horizontal, Inclined and Vertical Plates With Variable Surface Temperature or Heat Flux
,”
Int. J. Heat Mass Transfer
,
29
(
10
), pp.
1465
1478
.10.1016/0017-9310(86)90061-X
23.
Samanta
,
S.
, and
Guha
,
A.
,
2012
, “
A Similarity Theory for Natural Convection From a Horizontal Plate for Prescribed Heat Flux or Wall Temperature
,”
Int. J. Heat and Mass Transfer
,
55
(
13–14
), pp.
3857
3868
.10.1016/j.ijheatmasstransfer.2012.02.031
24.
Fishenden
,
M. W.
, and
Saunders
,
0. A.
,
1950
,
An Introduction to Heat Transfer
,
Oxford University Press
,
London
.
25.
Kihm
,
K. D.
, and
Cheeti
,
S. K. R.
,
1994
, “
Study of Thermal Flows From Two-Dimensional, Upward-Facing Isothermal Surfaces Using a Laser Speckle Photography Technique
,”
Exp. Fluids.
17
(
4
), pp.
246
252
.10.1007/BF00203043
26.
Levy
,
S.
, and
,
N. Y.
,
1955
, “
Integral Method in Natural Convection Flow
,”
J. Appl. Mech.
,
77
, pp.
515
522
.
27.
Samanta
,
S.
, and
Guha
,
A.
,
2013
, “
Similarity Theory for Forced Convection Over Horizontal Plates
,”
AIAA J. Thermophys. Heat Transfer
,
27
(
3
), pp.
506
514
.10.2514/1.T4033
28.
Ghiaasiaan
,
S. M.
,
2011
,
Convective Heat and Mass Transfer
,
Cambridge University Press
,
Cambridge, UK
.