conduction questions

1. 1.Consider a shielding wall for a nuclear reactor. The wall
receives a gamma-ray flux such that heat is generated within
the wall according to the relation 𝑞 = 𝑞oe−ax where 𝑞o is the
heat generation at the inner face of the wall exposed to the
gamma-ray flux and a is a constant. Using this relation for heat
generation, derive an expression for the temperature
distribution in a wall of thickness L, where the inside and
outside temperatures are maintained at Ti and To, respectively.
Also obtain an expression for the maximum temperature in the
wall.

3. 2. The steady-state temperature distribution in a one–dimensional wall of
thermal conductivity 50W/mK and thickness 50 mm is observed to be T(℃)
= a+bx2, where a=200℃, b = - 2000 ℃/m2, and x in meters.
(a) What is the heat generation rate in the wall?
(b) Determine the heat fluxes at the two wall faces.
In what manner are these heat fluxes related to the heat generation rate?

5. 3.Consider a large plane wall of thickness L. The wall surface at
x = 0 is insulated, while the surface at x = L is maintained at a
temperature of T2. The thermal conductivity of the wall is k
(W/m°C), and heat is generated in the wall at a rate of 𝑞 =
𝑞 oe−0.5x/L W/m3. Assuming steady one-dimensional heat
transfer, (a) express the differential equation and the boundary
conditions for heat conduction through the wall, (b) obtain a
relation for the variation of temperature in the wall by solving
the differential equation.

7. 4. A heat flux meter attached to the inner surface of a 3 cm thick
refrigerator door indicates a heat flux of 25 W/m2 through the door. Also,
the temperatures of the inner and the outer surfaces of the door are
measured to be 7℃ and 15℃, respectively. Determine the average
thermal conductivity of the refrigerator door under steady state
condition.

9. 5. A truncated cone 30 cm high is constructed of aluminium [k =204
W/m.℃]. The diameter at the top is 7.5 cm, and the diameter at the
bottom is 12.5 cm. The lower surface is maintained at 93℃; the upper
surface, at 540℃. The other surface is insulated. Assuming one
dimensional heat flow, what is the rate of heat transfer in watts?

11. 6. Consider a steam pipe of length L= 20 m, inner radius r1=6 cm, outer radius r2=8 cm,
and thermal conductivity k = 20 W/m°C, as shown in Figure. The inner and outer
surfaces of the pipe are maintained at average temperatures of T1= 150°C and T2=
60°C, respectively. Obtain a general relation for the temperature distribution inside
the pipe under steady conditions, and determine the rate of heat loss from the steam
through the pipe.

13. 7. A stainless-steel sphere [k =16 W/m.℃] having a diameter of 4 cm is
exposed to a convection environment at 20℃, h=15 W/m2℃. Heat is
generated uniformly in the sphere at the rate of 1.0 MW/m3. Calculate
the steady-state temperature for the center of the sphere.

15. 8. An electric heater with a capacity P = 1500 W is used to heat air in a spherical chamber. The
inside radius is ri = 10 cm and outside radius ro = 14 cm and the conductivity is k = 2.4 W/m K.
At the inside surface heat is exchanged by convection. The inside heat transfer coefficient is hi =
10 W/m2 K. Heat loss from the outside surface is by radiation. The surroundings temperature is
Tsur = 15℃ and the surface emissivity ϵ = 0.81. Assuming one-dimensional steady state
conduction, determine:
a. The temperatures of the inner and outer surface of the spherical chamber and the heater’s
filament temperature. Assume that the filament is as hot as the air inside the spherical chamber.
b. The temperature distribution in the spherical chamber’s wall.
c. Draw the thermal resistance circuit and find the values of all the resistances involved due to
different modes of heat transfer.

17. 9. For a solid and hollow sphere, it is given that qG is the heat
generation term. Derive the temperature distribution assuming 1-D
steady state conditions.

19. 10. A shaft of radius 𝑅𝑠 rotates inside a sleeve of inner radius 𝑅𝑠 and outer radius 𝑅o.
Frictional heat is generated at the interface at flux 𝑞𝑖′′. The outside surface of the sleeve
is cooled by convection with ambient fluid at 𝑇∞. The heat transfer coefficient is h.
Consider one-dimensional steady-state conduction in the radial direction, determine the
temperature distribution in shaft and sleeve.