Page 529 - Bird R.B. Transport phenomena

Problems 509
radiation and reflection from the planets. The solar con- 16B.5. Cooling of a black body in vacuo. A thin black
stant at Earth is given in Example 16.4-1. body of very high thermal conductivity has a volume V,
(b) Extend part (a) to give the lunar surface temperature as surface area A, density p, and heat capacity C . At time t =
p
a function of angular displacement from the hottest point. 0, this body at temperature Tj is placed in a black enclo-
sure, the walls of which are maintained permanently at
16B.1. Reference temperature for effective emissivity. temperature T (with T <T ). Derive an expression for the
}
2
2
Show that, if the emissivity increases linearly with the tem- temperature T of the black body as a function of time.
perature, Eq. 16.5-3 may be written as
16B.6. Heat loss from an insulated pipe. A standard
Schedule 40 two-inch steel pipe (inside diameter 2.067 in.,
in which e° is the emissivity of surface 1 evaluated at a ref- wall thickness 0.154 in.) carrying steam is lagged (i.e., insu-
erence temperature T° given by lated) with 2 in. of 85% magnesia and tightly wrapped
T5 _ T-5 with a single outer layer of clean aluminum foil (e = 0.05).
T° = — 2 - (16B.1-2)
л The inner surface of the pipe is at 250°F, and the pipe is
horizontal, surrounded by air at 1 atm and 80°F.
16B.2. Radiation across an annular gap» Develop an ex-
Compute
pression for the radiant heat transfer between two long, (a) (cond) the conductive heat flow per unit length,
gray coaxial cylinders 1 and 2. Show that Q /L, through the pipe wall and insulation for as-
sumed temperatures, T , of 100°F and 250°F at the outer
o
(16B.2-1) surface of the aluminum foil.
Q12 =
1 (b) Compute the radiative and free-convective heat losses,
A2V2 Q (rad) /L and Q (conv) /L, for the same assumed outer surface
temperatures T .
where A is the surface area of the inner cylinder. o
}
(c) Plot or interpolate the foregoing results to obtain the
16B.3. Multiple radiation shields, steady-state values of T and Q (cond) /L = Q {rad) /L + Q (conv) /L
o
(a) Develop an equation for the rate of radiant heat transfer 16C1. Integration of the view-factor integral for a pair
through a series of n very thin, flat, parallel metal sheets, of disks (Fig. 16C.1). Two parallel, perfectly black disks of
each having a different emissivity e, when the first sheet is radius R are placed a distance H apart. Evaluate the view-
at temperature T } and the nth. sheet is at temperature T .
n
Give your result in terms of the radiation resistances factor integrals for this case and show that
2
1 + 2B - Vl + 4B 2 (16.1-1) 2
_ (16B.3-D = F , , = • IB 2
in which B = R/H.
for the successive pairs of planes. Edge effects and conduc-
tion across the air gaps between the sheets are to be ne-
glected.
(a) Determine the ratio of the radiant heat transfer rate for
n identical sheets to that for two identical sheets.
(c) Compare your results for three sheets with that ob- H
tained in Example 16.5-1.
The marked reduction in heat transfer rates produced
by a number of radiation shields in series has led to the use
of multiple layers of metal foils for high-temperature insu- Fig. 16.C-1. Two perfectly
lation. black disks.
16B.4. Radiation and conduction through absorbing 16D.1. Heat loss from a wire carrying an electric cur-
media. A glass slab, bounded by planes 2 = 0 and 2 = 5, is rent. 3 An electrically heated wire of length L loses heat to
of infinite extent in the x and у directions. The tempera- the surroundings by radiative heat transfer. If the ends of
tures of the surfaces at 2 = 0 and 2 = 5 are maintained at T o the wire are maintained at a constant temperature T , ob-
o
and T , respectively. A uniform monochromatic radiant tain an expression for the axial variation in wire tempera-
g
beam of intensity q^ in the 2 direction impinges on the face ture. The wire can be considered to be radiating into a
at 2 = 0. Emission within the slab, reflection, and incident black enclosure at temperature T .
o
radiation in the negative 2 direction can be neglected.
(a) Determine the temperature distribution in the slab, as- 2 C. Christiansen, Wiedemann's Ann. d. Physik, 19, 267-283
suming m and к to be constants. (1883); see also M. Jakob, Heat Transfer, Vol. II, Wiley, New York
a (1957), p. 14.
(b) How does the distribution of the conductive heat flux 3 H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids,
q depend on m l 2nd edition, Oxford University Press (1959), pp. 154-156.
a
z