Heat Transfer Coefficients — External Natural Convection
Vertical Wall
Figure 4-19: Schematic representation of geometry and parameters for the heat transfer coefficient correlation applied to natural convection on a vertical wall.
The following correlations correspond to equations 9.26 and 9.27 in Ref. 18:
(4-151)
where the height of the wall, L, is a correlation input.
The Rayleigh number is defined as:
(4-152)
if the density depends on temperature only, or as:
(4-153)
if the density depends on other parameters.
All material properties are evaluated at (T + Text) ⁄ 2, except ρs, which is evaluated at the wall temperature, T, and g is the acceleration of gravity equal to 9.81 m/s2. This correlation is valid for 104≤ RaL ≤ 1013.
Inclined Wall
Figure 4-20: Schematic representation of geometry and parameters for the heat transfer coefficient correlation applied to natural convection on an inclined wall.
The following correlations correspond to equations 9.26 and 9.27 in Ref. 18 (the same as for a vertical wall):
(4-154)
where the length of the wall, L, is a correlation input and is the tilt angle (the angle between the wall and the vertical direction; for vertical walls). These correlations are valid for −60° <  < 60° and 104≤ RaL ≤ 1013.
The definition of the Raleigh number, RaL, is analogous to the one for vertical walls and is given by the following:
(4-155)
if the density depends on temperature only, or as:
(4-156)
if the density depends on other parameters.
g denotes the gravitational acceleration, equal to 9.81 m/s2.
For turbulent flow, 1 is used instead of cos  in the expression for h, because this gives better accuracy (see Ref. 38).
According to Ref. 18, correlations for inclined walls are only satisfactory for the top side of a cold plate or the down face of a hot plate. Hence, these correlations are not recommended for the bottom side of a cold face and for the top side of a hot plate.
The laminar-turbulent transition depends on (see Ref. 38). Unfortunately, little data is available about transition. There is some data available in Ref. 38 but this data is only approximate, according to the authors. In addition, data is only provided for water (Pr around 6). For this reason, the flow is defined as turbulent, independently of the value, when
All material properties are evaluated at (T + Text) ⁄ 2, except ρswhich is evaluated at the wall temperature, T.
Horizontal Plate, Upside
Figure 4-21: Schematic representation of geometry and parameters for the heat transfer coefficient correlation applied to natural convection on the top surface of an horizontal plate.
The following correlations correspond to equations 9.30–9.32 in Ref. 18 but can also be found as equations 7.77 and 7.78 in Ref. 38.
If ρ < ρext, then
(4-157)
while if ρ ≥ ρext, then
(4-158)
RaL is given by Equation 4-152 or Equation 4-153, and L, the characteristic length (defined as area/perimeter, see Ref. 38) is a correlation input. The material data are evaluated at (T + Text) ⁄ 2, except ρswhich is evaluated at the wall temperature, T.
When the density depends only on temperature, the conditions ρ < ρext and ρ ≥ ρext can be replaced by T > Text and T ≤ Text respectively.
Horizontal Plate, Downside
Figure 4-22: Schematic representation of geometry and parameters for the heat transfer coefficient correlation applied to natural convection on the bottom surface of an horizontal plate.
Equation 4-157 is used when ρ ≥ ρext (or T ≤ Text) and Equation 4-158 is used when ρ < ρext (or T > Text). Otherwise it is the same implementation as for Horizontal Plate, Upside.
Long horizontal cylinder
Figure 4-23: Schematic representation of geometry and parameters for the heat transfer coefficient correlation applied to natural convection on a long horizontal cylinder.
The following correlations correspond to equations 9.34 in Ref. 18. It is validated for RaD ≤ 1012.
(4-159)
Here D is the cylinder diameter and RaD is given by
if the density depends on temperature only, or as:
if the density depends on other parameters.
The material data are evaluated at (T + Text) ⁄ 2, except ρswhich is evaluated at the wall temperature, T.
Sphere
Figure 4-24: Schematic representation of geometry and parameters for the heat transfer coefficient correlation applied to natural convection on a sphere.
The following correlations correspond to equation 9.35 in Ref. 18. It is validated for RaD ≤ 1011 and Pr ≥ 0.7.
(4-160)
Here D is the cylinder diameter and RaD is given by
if the density depends on temperature only, or as:
if the density depends on other parameters.
The material data are evaluated at (T + Text) ⁄ 2, except ρswhich is evaluated at the wall temperature, T.
Vertical Thin Cylinder
Figure 4-25: Schematic representation of geometry and parameters for the heat transfer coefficient correlation applied to natural convection on a vertical thin cylinder.
The following correlation corresponds to equation 7.83 in Ref. 38. It is validated only for side walls of the thin cylinder (δT ≥ D), the horizontal disks (top and bottom) should be treated as horizontal plates. If the boundary thin layer is much smaller than D, vertical wall correlations should be used.
where D is the cylinder diameter, H is the cylinder height, and RaH is given by
if the density depends on temperature only, or as:
if the density depends on other parameters.
The material data are evaluated at (T + Text) ⁄ 2, except ρswhich is evaluated at the wall temperature, T.