LRadd
4. Radiation
4.5 Radiative Exchange Between a Number of Grey Surfaces
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Since (reciprocity rule) Aj Fji = Ai Fij then the net radiation exchange between surface i and surface j is:
Qij = Qi-j - Qj-i = Ai Fij ı (Ti4 - Tj4) (4.12)
This can be easily generalised to the case of N black surfaces, which form an enclosure, as:
) (
4 41
j i N
j
i ij
i
F A T T
Q ¦ V
(4.13)
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j i N
j ij i
iG F A J
A
¦
1 (4.18) Substituting (4.18) into (4.14), for G, gives:
,
1 i N
j ij i
i J F J
Q
¦
) (
1
j i N
j ij i
i A F J J
Q
¦
(4.19)
Finally, combining Equations 4.19 and 4.17, which both express the heat flow in terms of radiosity gives a result that can be used directly:
) / (
) 1 (
) (
1 ,
j i N
j ij i i i
i i b i
i A E J A F J J
Q
H ¦
H
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How do we use this relationship ?
Apply Equation 4.20 to each surface in turn, e.g for a total of N = 3 surfaces: i = 1, j = 1, 2, 3; i = 2, j = 1, 2, 3; i = 3, j = 1, 2, 3. This will result in N (i.e. 3) simultaneous equations for the
radiosities J1, J2 and J3, which can be eliminated and providing some temperatures are known, the other unknown temperatures may be found as may be the various heat flows. The following example will illustrate the practical application of this technique as well as how to deal with convective heat transfer in addition to radiation.
Example 4.2
High temperature gas flows through the inside of a pipe of outer radius r2 = 30 mm. To reduce the thermal radiation from the pipe to an electrical control panel mounted nearby, a semi-circular radiation shield of radius r1 = 100 mm is placed concentrically around the pipe. Thermal
radiation from the pipe is radiated to bothe the shield and the surroundings which are at 310 K.
The radiation view factor from the shield to itself is F11 = 0.3345.
a) Obtain values for the remaining view factors
b) Consider a radiation balance on the system. The pipe temperature is 900 K; the shield temperature is T1; the emissivity of the inner surface of the shield is H1 = 0.8 and the emissivity of the outer surface of the pipe is H2 = 0.5. Assume the surroundings can be approximated by a black body at 310 K and neglecting any heat loss by convection, estimate the surface temperature of the shield when the emissivity of the outer surface of the radiation shield is H0 = 0.1
Figure 4.4 Radiation shield - Example 4.2
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Figure 4.4 shows a cross-section through the pipe and radiation shield. A third, imaginary surface, Surface 3 is drawn to form the enclosure. The pipe and radiation shield are assumed to be long in relation to their diameters so it is a reasonable assumption to neglect radiation from the open ends.
Solution
a) Since F21 + F22 + F23 = 1, and F22 = 0, then by symmetry F21 = F23 = 0.5 From Example 4.1(a) F11 = 0.35
F12 = F21 A2 / A1= 0.15 F13 = 1 – F12 – F11 = 0.5 F32 = F23 A2 / A3 = 0.15 F33 = F11 (symmetry) = 0.35 F31 = 1 – F32 – F33 = 0.5
b) Using Equation (4.20), applied to Surface 1
) (
) / (
1 1 1 12 1 2 13 1 3
1 1
, J F J J F J J
Eb
H H
Note the first term in the expansion of the series F11(J1 – J1) has been omitted because it is clearly zero.
Since H1= 0.8, then 1 - H1/ H1 = 0.25, also Eb,1 = ıT14 and if the surroundings, represented by Surface 3, can be approximated as a black body then J3 = ıT33. After doing the algebra to isolate J1 and substituting the above quantities this gives
) (
25 . 0 1
) (
25 . 0
13 12
4 3 13 2
12 4
1
1
F F
T F J
F J T
V
V
Substituting F12 = 0.15 and F13 = 0.5 and with T3 = 310 K, ı = 56.7 x 10-9 W / m2 K4, this finally gives:
J1 = 4.877 x 10-8 T14 + 0.03226 J2 + 56.3 (W/m2)
Obviously this alone is insufficient to solve the problem, because we have three unknows J1, J2 and T1. A similar analysis must be carried out on Surface 2 (i.e. i = 2, j = 1,2,3).
This gives
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) (
) / (
1
2
3 2 23 1 2 21 2 2 2
, J F J J F J J
Eb
H H
where 1 - H2/ H2 = 1, also Eb,2 = ıT24 and J3 = ıT33.
23 21
4 3 23 1
21 4 2
2
1
) F
F
T F J
F J T
V
V
Substituting F21 + F23 = 1, T2 = 900 K and ı = 56.7 x 10-9 W / m2 K4, gives J2 = 18731 + 0.25 J1 (W/m2)
Combining the above two equations to eliminate J2 gives J1 = 4.9 x 10-8 T14 + 666 (W/m2)
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However, we have one equation with two unknowns so we need to use a further relationship.
This comes from applying a heat balance on the inner and outer surfaces of the radiation shield itself. For the inside of the radiation shield
1 1
1 1 ,
1 1
H
/H
J q Eb
and in the absence of losses by convection this heat flux leaves the outside surface of the radiation shield, so:
-q1 = H0 ı(T14 – T34)
Combining these two expressions for q1 and substituting J1 = 4.9 x 10-8 T14 + 666, results in an equation with a single unknown quantity T1.
8 4
4 3
1
( 1 0 . 025 ) 4 . 9 10 666 025
. 0
u
V
V T T
From which, with T3 = 310 K, gives the final result that T1 = 522 K.
It is instructive to continue with this example to examine the effect on the surface temperature of the radiation shield if we increase the emissivity of its outer surface and decrease the emissivity of its inner surface. Qualitatively we would expect the surface temperature to be reduced because less radiation from the hot gas pipe is being absorbed by the shield. If we now make H0 = 0.8 and H1 = 0.1, then we obtain:
J1 = 8.112 x 10-9 T14 + 0.193 J2 + 347.6 (W/m2)
J2 = 18731 + 0.25 J1 (W/m2) which is the same as before because H1 is not involved in this equation.
When combined this gives
J1 = 8.534 x 10-9 T14 + 4163.5 (W/m2)
From then energy balance on the radiation shield T1 = 465 K.
It is also useful to look at the effect of convective heat transfer, h, from the outer surface of the radiation shield. The energy balance on the outer surface of the shield becomes:
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) (
) / (
1 1 3
4 3 4 1 0 1
1 1 1 ,
1 E J T T h T T
q b
H V
H H
Taking h = 10 W/m2 K, T3 = 310 K, H0 = 0.1 and H1 = 0.8, gives 9.1175 x 10.9 T14 + 2.5T1 – 1453 = 0
which is satisfied for T1 = 442 K
In determining what constitutes a surface it is necessary to recognise that these should be
approximately isothermal. So, for example, in applying this technique to say a turbine disc where the temperature can vary significantly over the radius of the disc, it may be necessary to
subdivide the disc into 10 or more sectors. This will of course result in a system of simultaneous equations that is no longer amenable to solving by hand and a computer matrix method will need to be used.