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National Advisory Committee for Aeronautics, Technical Notes - Combined Natural and Forced Convection Laminar Flow and Heat Transfer of Fluids with and Without Heat Sources in Channels with Linearly Varying Wall Temperatures

The flow of fluids with and without heat sources and subject to
body forces between two plane parallel surfaces which are oriented in
the direction of the generating body force is analyzed under the condi-
tion that the temperature vary linearly along these surfaces. It is
found that a modified.Rayleigh number (product of the reciprocal of the
ratio of specific heats and the Prandtl number Pr and the modified
Grashof number GrA) as well as a parameter KA = Pr GrA is of
significance in this problenu where B is the volumetric egpansion
coefficient, fX is the negative of the X-component of body force per
unit mass, d is the characteristic length, and cP is the specific
heat at constant pressure. Solutions of this problem are obtained in
terms of "universal" functions which are tabulated for simple applica-
tion to specific cases. Representative velocity and temperature dis-
tributions from which detailed study of the heat transfer is made are
then computed. When the ratio of CKA (where C is related to the mass
flow) to the Rayleigh number is of unit order of magnitude, the effects
of aerodynamic or frictional heating can be appreciable. Asymptotic
solutions (for large values of the Rayleigh number) which render the
computations simple are also presented.

Comparison of the'results from the method given herein with those
obtained elsewhere in an approximate manner for a special case simu—
lating the natural—convection flow of fluids with heat sources in a com—
pletely enclosed region shows that the approximate method is suffi-
ciently accurate for prdblems in which the modified Rayleigh number is
less than 104.

In recent years the transfer of heat to and from enclosed or par-
tially enclosed regions by means of natural convection or by a combina-
tion of natural and forced convection has taken on new significance in
the fields of aeronautics, atomic power, electronics, and chemical en—
gineering. Most of the information on these modes of heat transfer under
such conditions is of a semiempirical or specialized nature; relatively
little detailed information exists for internal natural-convection flows.
In reference 1 there appears one of the few attempts to determine theo-
retically the velocity and temperature distributions in detail and hence
the heat transfer for an internal flow problem of this kind.