40
Analytical Heat Transfer
b. If during the air cooling, the annular fin has also received radiation energy from the surrounding environment, T sur , can you
sketch, compare, and comment on the fin temperature profile
T(r) with that in (a)?
2.7. Both sides of a very thin metal disk, as shown in Figure 2.9d, are
heated by convective hot air at temperature T ∞ . The convection
heat transfer coefficient h can be taken constant over the disk. The
periphery at r = R is maintained at a uniform temperature T R .
a. Derive the steady-state heat conduction equation of the disk.
b. Propose a solution method and the associated boundary conditions that can be used to determine the disk temperature
distributions. Sketch the disk temperature profile T(r). Do
you think the disk temperature is hotter at the center or the
periphery? Why?
c. If during the air heating, the disk has also emitted a net uniform
radiation flux q "" to the surrounding environment, derive the
rad
steady-state heat condition equation of the disk and propose
a solution method to determine the disk temperature distributions. Can you sketch, compare, and comment on the disk
temperature profile with that in (b)?
2.8. The front surface of a very thin metal disk is cooled by convective
air at temperature T ∞ while the back surface is perfectly insulated,
as shown in Figure 2.9d. The convection heat transfer coefficient h
can be taken to be constant over the front surface of the disk. The
periphery at r = R is maintained at a uniform temperature T R by
a heat source.
a. Derive the steady-state heat conduction equation of
disk.
b. Determine the disk temperature distributions with the associated boundary conditions. Sketch the disk temperature profile
T(r).
c. If during the air cooling, the disk has also emitted a net uniform
radiation flux q "" to the surrounding environment, derive
rad
the steady-state heat conduction equation of the disk and
determine the disk temperature distributions. Can you sketch,
compare, and comment on the disk temperature profile with
that in (b)?
2.9. A thin conical pin fin is shown in Figure 2.9a. Determine analytically the temperature profile in the pin fin. Also determine the
heat flux through the pin fin base.
Given:
Pin fin tip temperature: T R > T ∞ Pin fin height: l
Pin fin base diameter: d Pin fin base temperature: T b
Cooling air at T ∞ , h Hot wall at T b
2.10. The wall of a furnace has a height L = 1 m and is at a uniform temperature of 500 K. Three materials of equal thickness
t = 0.1 m, having the properties listed in the table attached, are
placed in the order shown in the Figure to insulated the furnace
Analytical Heat Transfer
b. If during the air cooling, the annular fin has also received radiation energy from the surrounding environment, T sur , can you
sketch, compare, and comment on the fin temperature profile
T(r) with that in (a)?
2.7. Both sides of a very thin metal disk, as shown in Figure 2.9d, are
heated by convective hot air at temperature T ∞ . The convection
heat transfer coefficient h can be taken constant over the disk. The
periphery at r = R is maintained at a uniform temperature T R .
a. Derive the steady-state heat conduction equation of the disk.
b. Propose a solution method and the associated boundary conditions that can be used to determine the disk temperature
distributions. Sketch the disk temperature profile T(r). Do
you think the disk temperature is hotter at the center or the
periphery? Why?
c. If during the air heating, the disk has also emitted a net uniform
radiation flux q "" to the surrounding environment, derive the
rad
steady-state heat condition equation of the disk and propose
a solution method to determine the disk temperature distributions. Can you sketch, compare, and comment on the disk
temperature profile with that in (b)?
2.8. The front surface of a very thin metal disk is cooled by convective
air at temperature T ∞ while the back surface is perfectly insulated,
as shown in Figure 2.9d. The convection heat transfer coefficient h
can be taken to be constant over the front surface of the disk. The
periphery at r = R is maintained at a uniform temperature T R by
a heat source.
a. Derive the steady-state heat conduction equation of
disk.
b. Determine the disk temperature distributions with the associated boundary conditions. Sketch the disk temperature profile
T(r).
c. If during the air cooling, the disk has also emitted a net uniform
radiation flux q "" to the surrounding environment, derive
rad
the steady-state heat conduction equation of the disk and
determine the disk temperature distributions. Can you sketch,
compare, and comment on the disk temperature profile with
that in (b)?
2.9. A thin conical pin fin is shown in Figure 2.9a. Determine analytically the temperature profile in the pin fin. Also determine the
heat flux through the pin fin base.
Given:
Pin fin tip temperature: T R > T ∞ Pin fin height: l
Pin fin base diameter: d Pin fin base temperature: T b
Cooling air at T ∞ , h Hot wall at T b
2.10. The wall of a furnace has a height L = 1 m and is at a uniform temperature of 500 K. Three materials of equal thickness
t = 0.1 m, having the properties listed in the table attached, are
placed in the order shown in the Figure to insulated the furnace