Environmental Engineering Reference
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standard temperature and pressure (W/m-
◦
C); b
=
interaction coefficient;
λ
=
mean-
free-path between collisions of a molecule (cm);
a
=
accommodation coefficient;
γ
=
ratio of specific heats for the annulus gas (air); T
po-gi
=
average temperature
T
gi
)/2 (
◦
C); P
a
(T
po
+
=
annulus gas pressure (mmHg); and
δ =
molecular diameter of
annulus gas (cm).
This correlation slightly overestimates the heat transfer for very small pressures
(
∼
<
0.013 Pa). The molecular diameter of air,
δ
, is obtained from Marshal (1976) and
10
−
8
cm, the thermal conductivity of air is 0.02551 W/m-
◦
C, the
interaction coefficient is 1.571, the mean-free-path between collisions of a molecule
is 88.67 cm, and the ratio of specific heats for the annulus air is 1.39. These are for
an average fluid temperature of 300
◦
C and pressure equal to 0.013 Pa. Using these
values, the convection heat transfer coefficients (h
po-gi
) obtained from Equation 6.2.14
is equal to 0.0001115 W/m
2
-
◦
C.
is equal to 3.55
×
b)
Pressure in annulus
If the receiver is filled or partially filled with ambient air
(
pressure
>
0.013 Pa) or if the receiver annulus vacuum is lost, the convection heat
transfer between the receiver pipe and glass envelope occurs by natural convection.
For this purpose the Raithby and Holland's correlation for natural convection in an
annular space (enclosure) between horizontal concentric cylinders is used, given by
(Cengel, 2006):
∼
2
πk
eff
ln (
D
gi
/D
po
)
(
T
gi
−
q
po- gi,conv
=
T
po
)
10
2
10
7
For: 0
.
7
≤
Pr
po-gi
≥
6000
and
≤
F
cyl
Ra
po-gi
≥
(6.2.17)
0
.
386
1
/
4
k
eff
Pr
po-gi
0
.
861
(F
cyl
Ra
Dpo
)
1
/
4
k
ag
=
(6.2.18)
+
Pr
po-gi
[ln(D
gi
/
D
po
)]
4
L
c
(D
−
3
/
5
=
F
cyl
(6.2.19)
D
−
3
/
5
)
5
−
po
gi
(D
gi
−
D
po
)
In these equations the critical length is given by: L
c
=
2
thermal conductivity of annulus gas at T
po-gi
(W/m-
◦
C); T
po
where: k
ag
=
=
outside
receiver pipe surface temperature (
◦
C); T
gi
=
inside glass envelope surface temperature
(
◦
C); D
po
=
=
outside receiver pipe diameter (m); D
gi
inside glass envelope diameter
=
=
(m); Pr
po-gi
Prandtl number for gas properties evaluated at T
po-gi
;Ra
Dpo
Rayleigh
number evaluated at D
po
; and T
po-gi
T
gi
)/2 (
◦
C).
This correlation assumes long, horizontal, concentric cylinders at uniform tem-
peratures, which is perfectly applied for a PTC. All physical properties are evaluated
at the average temperature (T
po
+
=
average temperature, (T
po
+
T
gi
)/2.
6.2.3.2 Radiation heat transfer
Several assumptions were made in deriving an equation for the radiation heat transfer,
as follows:
•
The surfaces are gray,
•
Diffuse reflections and irradiation,
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