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various thermal BCs such as variable surface heat flux as well as variable
surface temperature BCs; for flow in rectangular channels with various aspect
ratios and flow in annulus with various thermal BCs. They require more
complex mathematics and are beyond the intermediate-level heat transfer.
PROBLEMS
8.1. Consider a steady constant-property laminar flow between two
parallel plates at y = ±�. The plates are electrically heated to give a
uniform wall heat flux. The differential equations for momentum
and energy are listed here for reference:
∂ u
∂ u
1 ∂ P
∂ 2 u
∂ 2 u
u
+ v
= −
+ ν
+
∂ x
∂ y
ρ ∂ x
∂ x 2
∂ y 2
∂ T
∂ T
∂ 2 T
∂ 2 T
ν ∂ u
2
u
+ v
= α
+
+
∂ x
∂ y
∂ x 2
∂ y 2
c p ∂ y
a. Assume a low-speed, slug flow velocity profile (i.e., a uniform velocity profile) between two parallel plates, and also
assume a thermally, fully developed condition, and write
down the simplified equations for momentum and energy and
the associated BCs that can be used for this problem.
b. Under the assumption in (a), determine the Nusselt number
on the plate.
c. Consider a fully developed velocity profile (i.e., a parabolic
velocity profile) between two parallel plates and a thermally,
fully developed condition, and comment on whether the Nusselt number on the plate will be higher, the same, or lower than
those of symmetry linear velocity profile (uniform velocity
profile). Explain why.
8.2. Consider a steady constant-property laminar flow between two
parallel plates at y = ±�. The plates are electrically heated to give a
uniform wall heat flux. The differential equations for momentum
and energy are listed here for reference:
∂ u
∂ u
1 ∂ P
∂ 2 u
∂ 2 u
u
+ v
= −
+ ν
+
∂ x
∂ y
ρ ∂ x
∂ x 2
∂ y 2
∂ T
∂ T
∂ 2 T
∂ 2 T
ν ∂ u
� 2
u
+ v
= α
+
+
∂ x
∂ y
∂ x 2
∂ y 2
c p ∂ y
a. Assume a low-speed, symmetry linear velocity profile (i.e., u =
a + by with maximum velocity at y = 0, zero velocity at y =
±�) between two parallel plates, and also assume a thermally,
fully developed condition, and write down the simplified
equations for momentum and energy and the associated BCs
that can be used for this problem.
180
Analytical Heat Transfer
�
�
�
� �
�
�
�
�
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various thermal BCs such as variable surface heat flux as well as variable
surface temperature BCs; for flow in rectangular channels with various aspect
ratios and flow in annulus with various thermal BCs. They require more
complex mathematics and are beyond the intermediate-level heat transfer.
PROBLEMS
8.1. Consider a steady constant-property laminar flow between two
parallel plates at y = ±�. The plates are electrically heated to give a
uniform wall heat flux. The differential equations for momentum
and energy are listed here for reference:
∂ u
∂ u
1 ∂ P
∂ 2 u
∂ 2 u
u
+ v
= −
+ ν
+
∂ x
∂ y
ρ ∂ x
∂ x 2
∂ y 2
∂ T
∂ T
∂ 2 T
∂ 2 T
ν ∂ u
2
u
+ v
= α
+
+
∂ x
∂ y
∂ x 2
∂ y 2
c p ∂ y
a. Assume a low-speed, slug flow velocity profile (i.e., a uniform velocity profile) between two parallel plates, and also
assume a thermally, fully developed condition, and write
down the simplified equations for momentum and energy and
the associated BCs that can be used for this problem.
b. Under the assumption in (a), determine the Nusselt number
on the plate.
c. Consider a fully developed velocity profile (i.e., a parabolic
velocity profile) between two parallel plates and a thermally,
fully developed condition, and comment on whether the Nusselt number on the plate will be higher, the same, or lower than
those of symmetry linear velocity profile (uniform velocity
profile). Explain why.
8.2. Consider a steady constant-property laminar flow between two
parallel plates at y = ±�. The plates are electrically heated to give a
uniform wall heat flux. The differential equations for momentum
and energy are listed here for reference:
∂ u
∂ u
1 ∂ P
∂ 2 u
∂ 2 u
u
+ v
= −
+ ν
+
∂ x
∂ y
ρ ∂ x
∂ x 2
∂ y 2
∂ T
∂ T
∂ 2 T
∂ 2 T
ν ∂ u
� 2
u
+ v
= α
+
+
∂ x
∂ y
∂ x 2
∂ y 2
c p ∂ y
a. Assume a low-speed, symmetry linear velocity profile (i.e., u =
a + by with maximum velocity at y = 0, zero velocity at y =
±�) between two parallel plates, and also assume a thermally,
fully developed condition, and write down the simplified
equations for momentum and energy and the associated BCs
that can be used for this problem.
180
Analytical Heat Transfer