183
Internal Forced Convection
the lower plane wall at uniform temperature T 0 and the upper
plane wall at T 1 . If the velocity profile is a parabolic profile,
find the temperature profile from the energy equation.
8.9. A 2-D channel flow is subjected to a uniform heat flux on one wall
and insulated on the other wall. Assume that the viscous dissipation is negligible and the properties are constant. Determine the
following.
a. The governing momentum and energy equations with the
appropriate BCs for the developing region. Do not solve the
equations; however, show the details of simplifying the governing equations. Sketch the temperature and velocity profiles
with respect to y at two x positions.
b. Repeat (a) for the fully developed region.
c. Sketch the temperature profile in the fully developed region
if both walls are insulated. Consider two cases: (1) viscous
dissipation is negligible and (2) viscous dissipation is not
negligible.
8.10. Consider liquid metal flow in a parallel-plate channel at a uniform
wall heat flux condition.
a. Using the momentum and energy differential equations and
making the appropriate assumptions, derive the surface Nusselt number if flow is laminar and the temperature profile is in
a fully developed condition.
b. Using the momentum and energy differential equations and
making the appropriate assumptions, outline the methods (no
need to solve) in order to determine the surface Nusselt number if the flow is laminar and the temperature profile is in a
developing condition. Describe that the surface Nusselt numbers for (b) will be higher or lower than those for (a). Explain
why.
c. Nu, T w , and T b versus x from the entrance to the fully
developed region.
8.11. 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
2
∂
∂
∂
∂ u
∂ u
u
∂
+ v
x
∂
= −
ν
y
ρ ∂ x
+
∂ x 2 +
∂ y 2
�
T
�
2 T
2
2
∂
∂ T
∂
∂ T
�
ν
p
�
∂ u
u
+ v
= α
+
+
∂ x
∂ y
∂ x 2
∂ y 2
c
∂ y
�
a. What is the physical meaning of the last term shown in the
above differential energy equation? Explain under what conditions the last term should be included in order to solve the
temperature distribution between two parallel plates.
b. Assume a low-speed, slug-flow velocity profile (i.e., a uniform velocity profile) between two parallel plates, and also
Internal Forced Convection
the lower plane wall at uniform temperature T 0 and the upper
plane wall at T 1 . If the velocity profile is a parabolic profile,
find the temperature profile from the energy equation.
8.9. A 2-D channel flow is subjected to a uniform heat flux on one wall
and insulated on the other wall. Assume that the viscous dissipation is negligible and the properties are constant. Determine the
following.
a. The governing momentum and energy equations with the
appropriate BCs for the developing region. Do not solve the
equations; however, show the details of simplifying the governing equations. Sketch the temperature and velocity profiles
with respect to y at two x positions.
b. Repeat (a) for the fully developed region.
c. Sketch the temperature profile in the fully developed region
if both walls are insulated. Consider two cases: (1) viscous
dissipation is negligible and (2) viscous dissipation is not
negligible.
8.10. Consider liquid metal flow in a parallel-plate channel at a uniform
wall heat flux condition.
a. Using the momentum and energy differential equations and
making the appropriate assumptions, derive the surface Nusselt number if flow is laminar and the temperature profile is in
a fully developed condition.
b. Using the momentum and energy differential equations and
making the appropriate assumptions, outline the methods (no
need to solve) in order to determine the surface Nusselt number if the flow is laminar and the temperature profile is in a
developing condition. Describe that the surface Nusselt numbers for (b) will be higher or lower than those for (a). Explain
why.
c. Nu, T w , and T b versus x from the entrance to the fully
developed region.
8.11. 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
2
∂
∂
∂
∂ u
∂ u
u
∂
+ v
x
∂
= −
ν
y
ρ ∂ x
+
∂ x 2 +
∂ y 2
�
T
�
2 T
2
2
∂
∂ T
∂
∂ T
�
ν
p
�
∂ u
u
+ v
= α
+
+
∂ x
∂ y
∂ x 2
∂ y 2
c
∂ y
�
a. What is the physical meaning of the last term shown in the
above differential energy equation? Explain under what conditions the last term should be included in order to solve the
temperature distribution between two parallel plates.
b. Assume a low-speed, slug-flow velocity profile (i.e., a uniform velocity profile) between two parallel plates, and also