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Exercises 
1. The inner and outer surfaces a spherical shell with radii of 1 cm and 2 cm are 300°C and 
100°C, respectively. If the thermal conductivity of the shell is 2 W/m·K, determine the heat 
transfer through the shell. 
Ans 100.5 W 
2. The inner and outer surfaces of a cylindrical shell with radii of 1 cm and 2 cm are 300oC 
and 100oC, respectively. If the thermal conductivity of the shell is 2 W/m·K, determine the 
heat transfer through the shell per unit length of the cylinder. 
Ans 3626 W/m 
3. What would your answers to problems 1 and 2 be if the thermal conductivity were given 
by the equation k = a – b/T where a = 2 W/m·K and b = 100 W/m. Note that you must use T 
in kelvins when computing the thermal conductivity. 
Ans: 89.72 W for problem 1 and 3237 W/m for problem 2. 
4. Consider a homogeneous spherical piece of radioactive material of radius,R = 0.04 m 
that is generating heat at a constant rate of gen = 4 x 107 W/m3. The heat generated is 
dissipated to the environment steadily. The outer surface of the sphere is maintained at a 
uniform temperature of 80°C and the thermal conductivity of the sphere is k = 15 W/m·°C. 
Find: (a) the temperature at the center of the sphere, (b) the heat flux at the outer surface 
of the sphere. 
Ans: The temperature is found by plugging the data into the equation for heat generation in a 
solid sphere and setting r = 0 to get the temperature at the center of the sphere. Applying the 
given data to the equation for the heat flux in a solid sphere with heat conduction and setting r = 
0.04 m to get the heat flux at the outer surface gives. 
Note that the formula for the heat flux at the outer surface satisfies a simple energy balance. The 
heat flux leaving the surface equals the total heat generated divided by the outer surface area. 
5. What would your answer to problem 4 be if you did not know the outer-surface 
temperature of the sphere, but you were told that there was a convection between the 
outer surface of the sphere and a coolant that was at 700F with a heat transfer coefficient 
of 5000 W/m2·oC? 
Ans 
6. Steam at 320°C flows in a stainless steel pipe (k = 15 W/m·°C) whose inner and outer 
diameters are 5 cm and 5.5 cm, respectively. The pipe is covered with 3-cm-thick glass 
wool insulation (k = 0.038 W/m·°C). Heat is lost to the surroundings at 5°C by natural 
convection and radiation, with a combined natural convection and radiation heat transfer 
coefficient of 15 W/m2·°C. Taking the heat transfer coefficient inside the pipe to be 80 W/
m2·°C, determine the rate of heat loss from the steam per unit length of the pipe. Also 
determine the temperature drops across the pipe shell and the insulation. 
Ans 96.9 W/m 
7. The boiling temperature of nitrogen at atmospheric pressure at sea level (atmospheric 
pressure) is –196°C. Therefore, nitrogen is commonly used in low-temperature scientific 
studies since the temperature of liquid nitrogen in a tank 
open to the atmosphere will remain constant at –196°C until 
it is depleted. Any heat transfer to the tank will result in the 
evaporation of some liquid nitrogen, which has a heat of 
vaporization of 198 kJ/kg and a density of 810 kg/m3 at 
atmospheric pressure. Consider a 3-m-diameter spherical 
tank that is initially filled with liquid nitrogen at 
atmospheric pressure and –196°C. The tank is exposed to 
ambient air at 15°C, with a combined convection and 
radiation heat transfer coefficient of 35 W/m·°C. The 
temperature of the thin-shelled spherical tank is observed 
to be almost the same as the temperature of the nitrogen 
inside. Determine the rate of evaporation of the liquid 
nitrogen in the tank as a result of the heat transfer from the 
ambient air if the tank is (a) not insulated, (b) insulated with 
5-cm-thick fiberglass insulation (k = 0.035 W/m·°C), and (c) insulated with 2-cm-thick 
superinsulation which has an effective thermal conductivity of 0.00005 W/m·°C. 
Ans: 1.055 kg/s; 0.0214 kg/s; 0.00076 kg/s 
8. A 10-in-thick, 30-ft-long, and 10-ft-high wall is 
to be constructed using 9-in-long solid bricks (k 
= 0.40 Btu/h·ft·°F) of cross section 7 in by 7 in, 
or identical size bricks with nine square air 
holes (k = 0.015 Btu/h·ft·°F) that are 9 in long 
and have a cross section of 1.5 in by 1.5 in. 
There is a 0.5-in-thick plaster layer (k = 0.10 Btu/
h·ft·°F) between two adjacent bricks on all four 
sides and on both sides of the wall. The house 
is maintained at 80°F and the ambient 
temperature outside is 30°F. Taking the heat 
transfer coefficients at the inner and outer 
surfaces of the wall to be 1.5 Btu/h·ft2·°F and 
4 Btu/h·ft2·°F, respectively, determine the rate of 
heat transfer through the wall constructed of (a) 
solid bricks and (b) bricks with air holes. 
(Problem 3-58E in text.) 
9. Consider a sphere of diameter 5 cm, a cube of side length 5 cm, and a rectangular prism 
of dimension 4 cm by 5 cm by 6 cm, all initially at 0oC and all made of silver (k = 429 W/
m2·oC, ρ = 10,500 kg/m3, cp = 0.235 kJ/kg·oC). Now all three of these geometries are 
exposed to ambient air at 33oC on all of their surfaces with a heat transfer coefficient of 12 
W/m2·oC. Determine how long it will take for the temperature of each geometry to rise to 
25oC. 
Ans 2428 s for sphere; 428 s for cube. 2363 s for prism. 
10. Cylindrical brass pellets (k = 64.1 Btu/h ft·oF, ρ = 532 lbm/ft3, and cp = 0.092 Btu/lbm·oF) 
that are 2-in in diameter and 1 in long initially at 250oF are quenched in a water bath at 
120oF for a period of 2 minutes. If the convection heat transfer coefficient is 42 Btu/h ft2·oF, 
determine the temperature of the balls after quenching 
Ans:153oF 
11. A 35-cm diameter cylindrical shaft made of stainless steel 304 (k = 14.9 W/m·oC, ρ = 
7900 kg/m3, cp = 477 kJ/kg·oC) comes out of an oven at a uniform temperature of 400oC. 
The shaft is then allowed to cool slowly in a chamber at 150oC with an average heat 
transfer coefficient of 60 W/m2·oC. Determine the temperature center of the shaft 20 min 
after the start of the cooling process. 
T = 380oC 
12. The soil temperature in the upper layers of the earth varies with the variations in the 
atmospheric conditions. Before a cold front moves in, the earth at a location is initially at 
a uniform temperature of 10oC. Then the area is subjected to a temperature of –10oC and 
high winds that resulted in a convection heat transfer coefficient of 40 W/m2·oC on the 
earth’s surface for a period of 10 h. Taking the properties of the soil at that location to be k 
= 0.9 W/m·oC and α = 1.6x10-5 m2/s, determine the soil temperature at distances 0, 10, 20, 
and 50 cm from the earth’s surface at the end of this 10-h period. 
13. A short brass cylinder (ρ = 8530 kg/m3, cp = 0.389 kJ/kg·oC, k _ 110 W/m·oC, and α = 
3.39x10-5 m2/s) of diameter D = 8 cm and height H = 15 cm is initially at a uniform 
temperature of Ti = 150oC. The cylinder is now placed in atmospheric air at 20oC, where 
heat transfer takes place by convection with a heat transfer coefficient of h = 40 W/m2·oC. 
Calculate (a) the center temperature of the cylinder; (b) the center temperature of the top 
surface of the cylinder; and (c) temperature of the top outer radius of the cylinder, and (d) 
the total heat transfer from the cylinder 15 min after the start of the cooling.