where g is the acceleration due to gravity, β is the thermal coefficient of expan-
sion, is dynamic fluid viscosity, υ is the kinematic fluid viscosity, hc is the heat
transfer conductance, α is the thermal diffusivity, and k is the thermal conductiv-
ity of the fluid. Pr is the Prandtl number and Gr is the Grashof number. The two
remaining undefined terms, ∆T and L, are dependent on the boundary conditions
and geometry of the problem. In the simplest case, ∆T will be the temperature dif-
ference between a warm surface and a cold surface. The variable L is a character-
width is often used; other examples are discussed below. Correlations for Nu are
found in the form of
Nu = ARaB
(10)
when a specific material, such as air, is specified, or
Nu = AGr B
(11)
for the general case of natural convection in fluids or gases.
Heat transfer by radiation between two surfaces can have a large effect on the
heat transfer correlations. Experiments and analytical or numerical analysis can
include these effects or they can be removed. Radiation is primarily reflected in
the heat transfer conductance h. Generally, h should be considered to be the sum
vection. It is not always clear when examining heat transfer correlations if this is
the case, or if h represents merely hc.
A vertical rectangular cavity (enclosure) is defined as an enclosure bounded by
two vertical surfaces held at different temperatures. The other two parallel sur-
faces, top and bottom, are taken as insulated (Gebhart et al. 1988). Heat transfer
occurs only at the vertical surfaces. The characteristic length L for this geometry is
the distance between the hot and cold walls, and the characteristic temperature
∆T is the difference between the vertical wall temperatures. For an air-filled square
enclosure, Ostrach (1972) summarized the following numerical results for the
average Nusselt number in the form of eq 11.
Reference
A
B
eq
Newell and Schmidt (1969)
0.0547
0.397
(12)
Han (1967)
0.0782
0.3594
(13)
Elder (1965)
0.231
0.25
(14)
He reported tolerable agreement between these correlations and experiments.
Recently, de Vahl Davis and Jones (1983) presented a numerical benchmark
solution for air in a square vertical enclosure at Ra values from 103 to 106. Fitting
an equation to their Nusselt number data at the cold surface yields
Nu = 0.14162Gr 0.2996 .
(15)
Equations 1215 are drawn on Figure 2.
Correlations have also been developed for vertical enclosures with aspect ra-
tios (height/width) other than one. Gebhart et al. (1988) present several correla-
tions of the form
4
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