Page 343 - Bird R.B. Transport phenomena

Problems 327
the temperatures at the inner and outer walls being T and T v The thermal conductivity of the
o
insulation varies linearly with temperature from k at T to k at T . }
o
x
0
(b) Estimate the rate of evaporation of liquid oxygen from a spherical container of 6 ft inside
diameter covered with a 1-ft-thick annular evacuated jacket filled with particulate insulation.
The following information is available:
Temperature at inner surface of insulation - 183°C
Temperature at outer surface of insulation 0°C
Boiling point of O 2 -183°C
Heat of vaporization of O 2 1636 cal/g-mol
4
Thermal conductivity of insulation at 0°C 9.0 X 1СГ Btu/hr • ft • F
4
Thermal conductivity of insulation at - 183°C 7.2 X 10" Btu/hr • ft • F
( 7 i 7 \ / T" 1 T \
-^y^Jb^T^J; (b) 0.198 kg/hr
10B.15, Radial temperature gradients in an annular chemical reactor. A catalytic reaction is being
carried out at constant pressure in a packed bed between coaxial cylindrical walls with inner
r
radius r 0 and outer radius . Such a configuration occurs when temperatures are measured
x
with a centered thermowell, and is in addition useful for controlling temperature gradients if
a thin annulus is used. The entire inner wall is at uniform temperature T , and it can be as-
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sumed that there is no heat transfer through this surface. The reaction releases heat at a uni-
form volumetric rate S c throughout the reactor. The effective thermal conductivity of the
reactor contents is to be treated as a constant throughout.
(a) By a shell energy balance, derive a second-order differential equation that describes the
temperature profiles, assuming that the temperature gradients in the axial direction can be
neglected. What boundary conditions must be used?
(b) Rewrite the differential equation and boundary conditions in terms of the dimensionless
radial coordinate and dimensionless temperature defined as

€ = у; © = J~J° (10B.15-1)

Explain why these are logical choices.
(c) Integrate the dimensionless differential equation to get the radial temperature profile. To
what viscous flow problem is this conduction problem analogous?
(d) Develop expressions for the temperature at the outer wall and for the volumetric average
temperature of the catalyst bed.
(e) Calculate the outer wall temperature when r 0 = 0.45 in., r 1 = 0.50 in., k eff = 0.3 Btu/hr • ft •
3
F, T = 900°F, and S = 4800 cal/hr • cm .
c
o
(f) How would the results of part (e) be affected if the inner and outer radii were doubled?
Answer: (e) 888°F
IOB.I60 Temperature distribution in a hot-wire anemometer. A hot-wire anemometer is essentially
a fine wire, usually made of platinum, which is heated electrically and exposed to a flowing
fluid. Its temperature, which is a function of the fluid temperature, fluid velocity, and the rate
of heating, may be determined by measuring its electrical resistance. It is used for measuring
velocities and velocity fluctuations in flow systems. In this problem we analyze the tempera-
ture distribution in the wire element.
We consider a wire of diameter D and length 2L supported at its ends (z = -L and z =
+ L) and mounted perpendicular to an air stream. An electric current of density / amp/cm 2
flows through the wire, and the heat thus generated is partially lost by convection to the air
stream (see Eq. 10.1-2) and partially by conduction toward the ends of the wire. Because of
their size and their high electrical and thermal conductivity, the supports are not appreciably
heated by the current, but remain at the temperature T , which is the same as that of the ap-
L
proaching air stream. Heat loss by radiation is to be neglected.