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128
Analytical Heat Transfer
plate as shown in Figure 6.1, due to conductivity of fluid and velocity distribution, the temperature gradually decreases from its free-stream maximum
value to that at the flat plate. The hot fluid particle conducts heat from the
free-stream into the cold surface through the velocity boundary layer. Therefore, a thermal boundary-layer thickness is developed over a solid surface
due to fluid flow.
The temperature profile and associated thermal boundary-layer thickness
over a flat plate is solved in Chapter 7. In an ideal case (assume Pr = 1),
the thermal boundary layer is identical to hydrodynamic boundary layer as
shown in Figure 6.1 or 6.3. In this ideal case, the temperature profile is the
same as the velocity profile through the entire boundary layer over the flat
plate. Once we determine the temperature profile over a flat plate, T(y) at a
given distance x, the thermal boundary-layer thickness, the heat flux on the
surface, and the heat transfer coefficient (or Nusselt number) can be obtained
as follows:
√
Thermal boundary-layer thickness δ T (x) ∼ x
If Pr = 1, then δ(x) = δ T (x).
At the body surface, the heat flux is
∂T �
∂T �
q = −k
= −k f
≡ h(T w − T ∞ )
w
�
�
∂y
∂y
y=0
y=0
The heat transfer coefficient h with the unit of W/m 2 k can be expressed as
−k f (∂T/∂y) y=0
−k f ((T ∞ − T w )/δ T )
k f
k f
h =
∼
∼
∼
∼ k f U ∞
T w − T ∞
T w − T ∞
δ T
δ
In the laminar boundary layer, the temperature profile gradually changes
from the free-stream value to the surface as a parabolic shape, but, in the turbulent boundary layer, the temperature profile remains fairly uniform from
the free-stream to near the surface and then suddenly changes to the surface value. This is due to turbulent mixing (particle moves up and down,
back and forth); the hot (or cold) free-stream particle is able to move next to
the cold (or heated) surface due to random motion. From application point
of view, the heat flux (fluid conductivity × temperature gradient at the surface) and heat transfer coefficient (heat flux/temperature difference between
the free-stream and the surface) decrease with decreasing temperature gradient and fluid conductivity. Since temperature gradient decreases (because
thermal boundary-layer thickness increases) with increasing distance due to
fluid thermal conductivity, the heat flux (related to heat transfer rate) and heat
transfer coefficient decrease with increasing distance from the leading edge
of the flat plate. However, when flow transitions into the turbulent boundary
layer, the heat flux (and heat transfer coefficient) is much greater than the
laminar flow portion. This is because a major portion of heat transfer is due
to turbulent random motion in the turbulent boundary layer.
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128
Analytical Heat Transfer
plate as shown in Figure 6.1, due to conductivity of fluid and velocity distribution, the temperature gradually decreases from its free-stream maximum
value to that at the flat plate. The hot fluid particle conducts heat from the
free-stream into the cold surface through the velocity boundary layer. Therefore, a thermal boundary-layer thickness is developed over a solid surface
due to fluid flow.
The temperature profile and associated thermal boundary-layer thickness
over a flat plate is solved in Chapter 7. In an ideal case (assume Pr = 1),
the thermal boundary layer is identical to hydrodynamic boundary layer as
shown in Figure 6.1 or 6.3. In this ideal case, the temperature profile is the
same as the velocity profile through the entire boundary layer over the flat
plate. Once we determine the temperature profile over a flat plate, T(y) at a
given distance x, the thermal boundary-layer thickness, the heat flux on the
surface, and the heat transfer coefficient (or Nusselt number) can be obtained
as follows:
√
Thermal boundary-layer thickness δ T (x) ∼ x
If Pr = 1, then δ(x) = δ T (x).
At the body surface, the heat flux is
∂T �
∂T �
q = −k
= −k f
≡ h(T w − T ∞ )
w
�
�
∂y
∂y
y=0
y=0
The heat transfer coefficient h with the unit of W/m 2 k can be expressed as
−k f (∂T/∂y) y=0
−k f ((T ∞ − T w )/δ T )
k f
k f
h =
∼
∼
∼
∼ k f U ∞
T w − T ∞
T w − T ∞
δ T
δ
In the laminar boundary layer, the temperature profile gradually changes
from the free-stream value to the surface as a parabolic shape, but, in the turbulent boundary layer, the temperature profile remains fairly uniform from
the free-stream to near the surface and then suddenly changes to the surface value. This is due to turbulent mixing (particle moves up and down,
back and forth); the hot (or cold) free-stream particle is able to move next to
the cold (or heated) surface due to random motion. From application point
of view, the heat flux (fluid conductivity × temperature gradient at the surface) and heat transfer coefficient (heat flux/temperature difference between
the free-stream and the surface) decrease with decreasing temperature gradient and fluid conductivity. Since temperature gradient decreases (because
thermal boundary-layer thickness increases) with increasing distance due to
fluid thermal conductivity, the heat flux (related to heat transfer rate) and heat
transfer coefficient decrease with increasing distance from the leading edge
of the flat plate. However, when flow transitions into the turbulent boundary
layer, the heat flux (and heat transfer coefficient) is much greater than the
laminar flow portion. This is because a major portion of heat transfer is due
to turbulent random motion in the turbulent boundary layer.