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Analytical Heat Transfer
11.6. The glass of the skylight of a house as shown in Figure 11.12
has a spectral emissivity, ε λ (λ) [or absorptivity, α λ (λ)] distribution as shown: ε λ = 0.9 for 0 ≤ λ ≤ 0.3 μm, ε λ = 0 for 0.3 ≤
λ ≤ 2.0 μm, ε λ = 0.9 for λ ≥ 2.0 μm. During an afternoon when
the solar flux is 900 W/m 2 , the temperature of the glass is
27 ◦ C. The interior surfaces of the walls of the house and the
air in the house are at 22 ◦ C, and the heat transfer coefficient
between the glass of the skylight and the air in the house is
5 W/(m 2 K).
a. What is the overall emissivity, ε, of the skylight?
b. What is the overall absorptivity, α, of the skylight for solar
irradiation? You may assume that the sun emits radiation as a
blackbody at 6000 K.
c. What is the convective heat flux on the outer surface of the
""
skylight, q
, in W/m 2 ? Is the temperature
convection outside air
of the outside air higher or lower than the temperature of the
skylight? Please assume that the sky is at 0 ◦ C.
11.7. Consider a typical setup for a solar collector as shown in Figure
11.11. A special glass is used as a cover for the collector and a
specialized coating is used on the collector plate and tubes, where
the solar energy is collected, to maximize the performance of the
collector.
a. If you had to specify the value of the glass transmissivity, τ λ , as a function of λ to maximize the performance of
the collector, what would you choose and why? Explain.
Use illustrations or sketches if needed to help explain your
answer.
b. If you had to specify the value of the collector plate and tube
absorptivity, α λ , as a function of λ to maximize the performance of the collector, what would you choose and why?
Explain.
c. A manufacturing process calls for heating a long aluminum
rod that is coated with a thin film with an emissivity of ε.
The rod is placed in a large convection oven whose surface is
maintained at T w (K). Air at T ∞ (K) circulates in the oven at a
velocity of u (m/s) across the surface of the rod and produces
a convective heat transfer coefficient of h [W/(m 2 K)]. The rod
has a small diameter of d (m) and has an initial temperature
of T i (K). Here, T i < T ∞ < T w . What is the rate of change of
the rod temperature (K/s) when the rod is first placed in the
oven?
11.8. Asolar collector consists of an insulating back layer, a fluid conduit
through which a water–glycol solution flows to remove heat, an
absorber plate and a glass cover plate. The external temperatures
""
T air , T sky , and T ground are known. Solar radiation of intensity q s
""
(W/m 2 ) is incident on the collector and collected heat q (W/m 2 )
c
is removed by the fluid. The absorber plate is painted black with
Analytical Heat Transfer
11.6. The glass of the skylight of a house as shown in Figure 11.12
has a spectral emissivity, ε λ (λ) [or absorptivity, α λ (λ)] distribution as shown: ε λ = 0.9 for 0 ≤ λ ≤ 0.3 μm, ε λ = 0 for 0.3 ≤
λ ≤ 2.0 μm, ε λ = 0.9 for λ ≥ 2.0 μm. During an afternoon when
the solar flux is 900 W/m 2 , the temperature of the glass is
27 ◦ C. The interior surfaces of the walls of the house and the
air in the house are at 22 ◦ C, and the heat transfer coefficient
between the glass of the skylight and the air in the house is
5 W/(m 2 K).
a. What is the overall emissivity, ε, of the skylight?
b. What is the overall absorptivity, α, of the skylight for solar
irradiation? You may assume that the sun emits radiation as a
blackbody at 6000 K.
c. What is the convective heat flux on the outer surface of the
""
skylight, q
, in W/m 2 ? Is the temperature
convection outside air
of the outside air higher or lower than the temperature of the
skylight? Please assume that the sky is at 0 ◦ C.
11.7. Consider a typical setup for a solar collector as shown in Figure
11.11. A special glass is used as a cover for the collector and a
specialized coating is used on the collector plate and tubes, where
the solar energy is collected, to maximize the performance of the
collector.
a. If you had to specify the value of the glass transmissivity, τ λ , as a function of λ to maximize the performance of
the collector, what would you choose and why? Explain.
Use illustrations or sketches if needed to help explain your
answer.
b. If you had to specify the value of the collector plate and tube
absorptivity, α λ , as a function of λ to maximize the performance of the collector, what would you choose and why?
Explain.
c. A manufacturing process calls for heating a long aluminum
rod that is coated with a thin film with an emissivity of ε.
The rod is placed in a large convection oven whose surface is
maintained at T w (K). Air at T ∞ (K) circulates in the oven at a
velocity of u (m/s) across the surface of the rod and produces
a convective heat transfer coefficient of h [W/(m 2 K)]. The rod
has a small diameter of d (m) and has an initial temperature
of T i (K). Here, T i < T ∞ < T w . What is the rate of change of
the rod temperature (K/s) when the rod is first placed in the
oven?
11.8. Asolar collector consists of an insulating back layer, a fluid conduit
through which a water–glycol solution flows to remove heat, an
absorber plate and a glass cover plate. The external temperatures
""
T air , T sky , and T ground are known. Solar radiation of intensity q s
""
(W/m 2 ) is incident on the collector and collected heat q (W/m 2 )
c
is removed by the fluid. The absorber plate is painted black with