If T s ≈ T s u r in Equation 1.9, the radiation heat transfer coefficient may be approximated as h r , a = 4 ∈ σ T ¯ 3 where T ¯ ≡ ( T s + T s u r ) / 2 . We wish to assess the validity of this approximation by comparing values of h r and h r , a for the following conditions. In each case, represent your results graphically and comment on the validity of the approximation. (a) Consider a surface of either polished aluminum ( ∈ = 0.05 ) or black paint ( ∈ = 0.9 ) , whose temperature may exceed that of the surroundings ( T s u r = 25 ° C ) by 10 to 100°C. Also compare your results with values of the coefficient associated with flee convection in air ( T ∞ = T s u r ) , where h ( W/m 2 ⋅ K ) = 0.98 Δ T 1 / 3 . (b) Consider initial conditions associated with placing a workpiece at T s = 25 ° C in a large furnace whose wall temperature may be varied over the range 100 ≤ T s u r ≤ 1000 ° C . According to the surface finish or coating, its emissivity may assume values of 0.05, 0.2, and 0.9. For each emissivity, plot the relative error, ( h r − h r , a ) / h r , as a function of the furnace temperature.
If T s ≈ T s u r in Equation 1.9, the radiation heat transfer coefficient may be approximated as h r , a = 4 ∈ σ T ¯ 3 where T ¯ ≡ ( T s + T s u r ) / 2 . We wish to assess the validity of this approximation by comparing values of h r and h r , a for the following conditions. In each case, represent your results graphically and comment on the validity of the approximation. (a) Consider a surface of either polished aluminum ( ∈ = 0.05 ) or black paint ( ∈ = 0.9 ) , whose temperature may exceed that of the surroundings ( T s u r = 25 ° C ) by 10 to 100°C. Also compare your results with values of the coefficient associated with flee convection in air ( T ∞ = T s u r ) , where h ( W/m 2 ⋅ K ) = 0.98 Δ T 1 / 3 . (b) Consider initial conditions associated with placing a workpiece at T s = 25 ° C in a large furnace whose wall temperature may be varied over the range 100 ≤ T s u r ≤ 1000 ° C . According to the surface finish or coating, its emissivity may assume values of 0.05, 0.2, and 0.9. For each emissivity, plot the relative error, ( h r − h r , a ) / h r , as a function of the furnace temperature.
Solution Summary: The author compares polished aluminum and black paint to exceed the surrounding temperature and the relation of free convection coefficient. The expression for the approximated values of linearized radiation coefficients is valid within 2% of these conditions.
If
T
s
≈
T
s
u
r
in Equation 1.9, the radiation heat transfer coefficient may be approximated as
h
r
,
a
=
4
∈
σ
T
¯
3
where
T
¯
≡
(
T
s
+
T
s
u
r
)
/
2
. We wish to assess the validity of this approximation by comparing values of
h
r
and
h
r
,
a
for the following conditions. In each case, represent your results graphically and comment on the validity of the approximation. (a) Consider a surface of either polished aluminum
(
∈
=
0.05
)
or black paint
(
∈
=
0.9
)
, whose temperature may exceed that of the surroundings
(
T
s
u
r
=
25
°
C
)
by 10 to 100°C. Also compare your results with values of the coefficient associated with flee convection in air
(
T
∞
=
T
s
u
r
)
, where
h
(
W/m
2
⋅
K
)
=
0.98
Δ
T
1
/
3
. (b) Consider initial conditions associated with placing a workpiece at
T
s
=
25
°
C
in a large furnace whose wall temperature may be varied over the range
100
≤
T
s
u
r
≤
1000
°
C
. According to the surface finish or coating, its emissivity may assume values of 0.05, 0.2, and 0.9. For each emissivity, plot the relative error,
(
h
r
−
h
r
,
a
)
/
h
r
,
as a function of the furnace temperature.
Figure 3 shows cross section of a duct. If each wall is 100 cm, surface C is insulated, and duct is
very long. Evaluate;
a) net radiation per meter of duct for surface A,
b) temperature of surface C, and
c) effect of changing in the value of ɛ, on the results.
C,T3, ɛ3 = 0.8
Figure 3
B,T2 = 700 K, ɛ2 = 0.5
A,T, = 1000 K,ɛ, = 0.33
An electrical cable with 20 mm in diameter and emissivity equal to 0.85 is installed inside a conduit whose inner surface and air in the its interior is at 30oC. The electrical resistivity of the cable, ρe (µΩ.m), is a function of its temperature, given by ρe=a[1+b(T-T0)], where a=0.0171 µΩ.m, b=0.00396 K-1 and T0= 25oC.
The natural convection heat transfer coefficient is expressed by the relation h=cD-0.25(T-Tꚙ)0.25where c=1.21 W/(m1.75.K1.25) and D is the cable diameter.
Electrical resistance per unit of cable length is R’e=ρe/Ac (Ac is the cross-sectional area).
(a) For steady state operating conditions, estimate the maximum current that can be dissipated in the wire so that its temperature does not exceed 65oC;
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