182
Analytical Heat Transfer
c. Obtain an expression for the heat convection coefficient and
the Nusselt number.
(Hint: Consider the fully developed condition to approximate
∂ /∂
� b
T x; and define Y(y) = T − T b , where T b = 2 uTdy/2bu b .)
0
8.6. 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
u
v
v
∂ x
+
∂ y
= ρ ∂x
+
�
2
∂ u
2
∂ u
+ 2
∂x 2
∂y
�
2
∂ T
∂ T
2
∂ T
2
∂ T
ν ∂ u
u
∂ x
+ v
= α
�
∂ x
+
∂ y
2
∂ y 2
�
+ c p
�
∂ y
�
a. Assume a low-speed, linear velocity profile (i.e., u = u m
(
1 − y/�
)
with maximum velocity at y = 0, and 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.
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 in (a)? Explain why.
8.7. Consider an incompressible laminar 2-D flow in a parallel plate
channel as shown below. The top plate is pulled at a constant
velocity U T . The top and bottom plates are maintained at constant
heat flux q s . Flow is both hydrodynamically and thermally fully
developed. Assume that the pressure gradient is zero in a parallel
plate channel.
a. Obtain differential equations governing the velocity U(Y) and
temperature T(Y) fields.
b. Use appropriate BCs to evaluate U(Y). Obtain T(Y). Do not
attempt to evaluate constants of integration for the temperature field.
8.8. Find the Nusselt number for the following problems.
a. Fully developed Couette flow (i.e., assume that velocity and
temperature profiles do not change along the channel) with
the lower plane wall at uniform wall temperature T 0 and the
upper plane wall at T 1 . If the velocity profile is a linear profile
(U = 0 at the lower plane wall, U = V at the upper plane wall),
find the temperature profile from the energy equation.
b. Fully developed Poiseuille flow (i.e., assume that velocity and
temperature profiles do not change along the channel) with
Analytical Heat Transfer
c. Obtain an expression for the heat convection coefficient and
the Nusselt number.
(Hint: Consider the fully developed condition to approximate
∂ /∂
� b
T x; and define Y(y) = T − T b , where T b = 2 uTdy/2bu b .)
0
8.6. 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
u
v
v
∂ x
+
∂ y
= ρ ∂x
+
�
2
∂ u
2
∂ u
+ 2
∂x 2
∂y
�
2
∂ T
∂ T
2
∂ T
2
∂ T
ν ∂ u
u
∂ x
+ v
= α
�
∂ x
+
∂ y
2
∂ y 2
�
+ c p
�
∂ y
�
a. Assume a low-speed, linear velocity profile (i.e., u = u m
(
1 − y/�
)
with maximum velocity at y = 0, and 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.
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 in (a)? Explain why.
8.7. Consider an incompressible laminar 2-D flow in a parallel plate
channel as shown below. The top plate is pulled at a constant
velocity U T . The top and bottom plates are maintained at constant
heat flux q s . Flow is both hydrodynamically and thermally fully
developed. Assume that the pressure gradient is zero in a parallel
plate channel.
a. Obtain differential equations governing the velocity U(Y) and
temperature T(Y) fields.
b. Use appropriate BCs to evaluate U(Y). Obtain T(Y). Do not
attempt to evaluate constants of integration for the temperature field.
8.8. Find the Nusselt number for the following problems.
a. Fully developed Couette flow (i.e., assume that velocity and
temperature profiles do not change along the channel) with
the lower plane wall at uniform wall temperature T 0 and the
upper plane wall at T 1 . If the velocity profile is a linear profile
(U = 0 at the lower plane wall, U = V at the upper plane wall),
find the temperature profile from the energy equation.
b. Fully developed Poiseuille flow (i.e., assume that velocity and
temperature profiles do not change along the channel) with