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169
Internal Forced Convection
For the laminar flow, the thermal entrance length to tube diameter ratio is
about 5% of Reynolds number (based on the tube diameter) times Prandtl
number. This implies that the thermal entrance length increases with increasing Reynolds number (because a thinner boundary layer requires longer
distance for the boundary layer to merge) and Prandtl number (because lower
thermal conductivity requires longer distance to merge) [1–4].
Figure 8.2 also shows that the heat transfer coefficient decreases from the
entrance along the tube and becomes a constant value when thermal boundary
layer reaches the fully developed condition, and the heat transfer coefficient
increases with Reynolds number (because of a thinner boundary layer from
the entrance and the longer entrance length). It is noted that the thermal
entrance length is identical to the hydrodynamic entrance length if Pr = 1.
For the turbulent flow, the thermal entrance length is harder to determine;
just like the hydrodynamic entrance length, the thermal entrance length is
around 10–20 tube diameter. It is hard to distinguish whether the turbulent flow is thermally fully developed or not from 10 to 20 tube diameter
downstream.
8.2 Fully Developed Laminar Flow and Heat Transfer
in a Circular Tube or between Parallel Plates
For fluid flow in a circular tube, the Reynolds number is defined as
ρVD
VD
4m ˙
Re D =
=
=
(8.1)
μ
ν
πDμ
Laminar flow is observed if Re D ≤ 2300.
At a certain distance from the entrance, the velocity profile u(r) remains
unchanging along the tube (if the fluid properties remain constant). Correspondingly, there will be no velocity component in the radial direction. Also,
the axial pressure gradient required to sustain the flow against the viscous
forces will be constant along the tube (no momentum change). The flow is
hydrodynamically fully developed. The differential governing equations for
the flow inside a circular tube are [1–4]
∂u 1 ∂(vr)
+
= 0
(8.2)
∂x r ∂r
∂u
∂u
1 ∂P
1 ∂
∂u
u
+ v
= −
+ υ
r
(8.3)
∂x
∂y
ρ ∂x
r ∂r
∂r
∂ (uT) ∂ (vT)
1 ∂
∂T
+
= α
r
(8.4)
∂x
∂y
r ∂r
∂r
�
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169
Internal Forced Convection
For the laminar flow, the thermal entrance length to tube diameter ratio is
about 5% of Reynolds number (based on the tube diameter) times Prandtl
number. This implies that the thermal entrance length increases with increasing Reynolds number (because a thinner boundary layer requires longer
distance for the boundary layer to merge) and Prandtl number (because lower
thermal conductivity requires longer distance to merge) [1–4].
Figure 8.2 also shows that the heat transfer coefficient decreases from the
entrance along the tube and becomes a constant value when thermal boundary
layer reaches the fully developed condition, and the heat transfer coefficient
increases with Reynolds number (because of a thinner boundary layer from
the entrance and the longer entrance length). It is noted that the thermal
entrance length is identical to the hydrodynamic entrance length if Pr = 1.
For the turbulent flow, the thermal entrance length is harder to determine;
just like the hydrodynamic entrance length, the thermal entrance length is
around 10–20 tube diameter. It is hard to distinguish whether the turbulent flow is thermally fully developed or not from 10 to 20 tube diameter
downstream.
8.2 Fully Developed Laminar Flow and Heat Transfer
in a Circular Tube or between Parallel Plates
For fluid flow in a circular tube, the Reynolds number is defined as
ρVD
VD
4m ˙
Re D =
=
=
(8.1)
μ
ν
πDμ
Laminar flow is observed if Re D ≤ 2300.
At a certain distance from the entrance, the velocity profile u(r) remains
unchanging along the tube (if the fluid properties remain constant). Correspondingly, there will be no velocity component in the radial direction. Also,
the axial pressure gradient required to sustain the flow against the viscous
forces will be constant along the tube (no momentum change). The flow is
hydrodynamically fully developed. The differential governing equations for
the flow inside a circular tube are [1–4]
∂u 1 ∂(vr)
+
= 0
(8.2)
∂x r ∂r
∂u
∂u
1 ∂P
1 ∂
∂u
u
+ v
= −
+ υ
r
(8.3)
∂x
∂y
ρ ∂x
r ∂r
∂r
∂ (uT) ∂ (vT)
1 ∂
∂T
+
= α
r
(8.4)
∂x
∂y
r ∂r
∂r