181
Internal Forced Convection
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 the symmetry linear velocity profile (u = a + by)?
Explain why.
8.3. Internal flow, fully developed laminar forced convection: Consider a low-speed, constant-property, fully developed laminar
flow between two parallel plates, with one plate insulated and
the other uniformly heated. Determine the Nusselt number.
8.4. Consider a concentric circular-tube annulus with a radius ratio
of r i /r o = 0.6. Let the inner tube wall be heated at a constant
rate and the outer tube wall remains insulated. Let the fluid be
a low-Prandtl number fluid, and assume a slug flow inside of the
annulus. Develop an expression for the Nusselt number at the
inner surface by means of the following steps:
a. Discuss how temperature and velocity profiles develop in the
system by considering Pr and the type of flow present. Indicate the system conditions. (Is the flow and or temperature
developed?)
b. Let T m be the mass average fluid temperature, and T s,r i the surface temperature at the inner surface with radius r i . Form the
thermal BCs; what can be said about
(
T m − T s,r
∂ /∂
i
x
c.
)
and T
?
Draw a diagram, indicating BCs for both temperature and
velocity and indicate the assumptions used to simplify the
energy equation.
d. Solve for T using the simplified velocity profile, BCs, and the
energy equation.
e. Find the heat transfer coefficient and the Nusselt number by
calculating T m , the mass average fluid temperature, and heat
flux at the inner surface.
Remember that the energy equation in cylindrical coordinates is
given by
�
∂T
∂T
v θ ∂T
∂T
ρc
v r
v x
∂t
+
∂r
+ r ∂θ
+
∂x
�
1 ∂
=
�
∂T
rk
�
1 ∂
∂T
∂
∂T
+
r ∂r
∂r
r 2 ∂θ
�
k
�
+
�
k
�
+ q ˙,
∂θ
∂x
∂x
where x represents the axial direction of the cylinder.
8.5. Consider a low-speed, constant-property fluid, fully developed
laminar flow between two parallel plates located at y = ±b. The
plates are electrically heated to give a uniform heat flux.
a. Determine the velocity profile and define the bulk velocity (u b )
in terms of the pressure gradient driving the flow.
b. Using the hydraulic diameter D h = 4A c /P, show that the
friction factor is given by f = 96/Re D h .
Internal Forced Convection
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 the symmetry linear velocity profile (u = a + by)?
Explain why.
8.3. Internal flow, fully developed laminar forced convection: Consider a low-speed, constant-property, fully developed laminar
flow between two parallel plates, with one plate insulated and
the other uniformly heated. Determine the Nusselt number.
8.4. Consider a concentric circular-tube annulus with a radius ratio
of r i /r o = 0.6. Let the inner tube wall be heated at a constant
rate and the outer tube wall remains insulated. Let the fluid be
a low-Prandtl number fluid, and assume a slug flow inside of the
annulus. Develop an expression for the Nusselt number at the
inner surface by means of the following steps:
a. Discuss how temperature and velocity profiles develop in the
system by considering Pr and the type of flow present. Indicate the system conditions. (Is the flow and or temperature
developed?)
b. Let T m be the mass average fluid temperature, and T s,r i the surface temperature at the inner surface with radius r i . Form the
thermal BCs; what can be said about
(
T m − T s,r
∂ /∂
i
x
c.
)
and T
?
Draw a diagram, indicating BCs for both temperature and
velocity and indicate the assumptions used to simplify the
energy equation.
d. Solve for T using the simplified velocity profile, BCs, and the
energy equation.
e. Find the heat transfer coefficient and the Nusselt number by
calculating T m , the mass average fluid temperature, and heat
flux at the inner surface.
Remember that the energy equation in cylindrical coordinates is
given by
�
∂T
∂T
v θ ∂T
∂T
ρc
v r
v x
∂t
+
∂r
+ r ∂θ
+
∂x
�
1 ∂
=
�
∂T
rk
�
1 ∂
∂T
∂
∂T
+
r ∂r
∂r
r 2 ∂θ
�
k
�
+
�
k
�
+ q ˙,
∂θ
∂x
∂x
where x represents the axial direction of the cylinder.
8.5. Consider a low-speed, constant-property fluid, fully developed
laminar flow between two parallel plates located at y = ±b. The
plates are electrically heated to give a uniform heat flux.
a. Determine the velocity profile and define the bulk velocity (u b )
in terms of the pressure gradient driving the flow.
b. Using the hydraulic diameter D h = 4A c /P, show that the
friction factor is given by f = 96/Re D h .