CHAPTER 9
9.1
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s
R
qsol
A satellite shaped like a sphere (R=1 m) has a gray-diffuse surface coating
with ǫs=0.3 and is fitted with a long, thin, cylindrical antenna, as shown
in the adjacent sketch. The antenna is a specular reflector with ǫa=0.1,
ka=100 W/m K, and d=1 cm. Satellite and antenna are exposed to solar
radiation of strength qsol =1300 W/m2from a direction normal to the antenna.
Assuming that the satellite produces heat at a rate of 4 kW and—due to a
high-conductivity shell—is essentially isothermal, determine the equilibrium
temperature distribution along the antenna. (Hint: Use the fact that d≪Rnot only for conduction
calculations, but also for the calculation of view factors.)
T=Tsat x=R,dT
dx =0 at x→ ∞ (9.1-C)
or, from an energy balance for x→ ∞
qr→0 or T= 4qsol
,
ξ2
θ∞=1.19421/4=1.0454 →T∞=413.3K
In finite difference form this becomes
θi−1−(2 +c∆ξ2θ3
i)θi+θi+1=−c∆ξ2θ4
eff,ii=1,I−1
θ0=1i=0
θI+1=θI−1i=I
The above system can be solved, after linearization, by a simple tridiagonal scheme. Note that, because of
the strong nonlinearity, underrelaxation is necessary.