A hollow cylinder has length L , inner radius a , and outer radius b , and the temperatures at the inner and outer surfaces are T 2 and T 1 . (The cylinder could represent an insulated hot-water pipe.) The thermal conductivity of the material of which the cylinder is made is k . Derive an equation for (a) the total heat current through the walls of the cylinder; (b) the temperature variation inside the cylinder walls. (c) Show that the equation for the total heat current reduces to Eq. (17.21) for linear heat flow when the cylinder wall is very thin. (d) A steam pipe with a radius of 2.00 cm, carrying steam at 140°C, is surrounded by a cylindrical jacket with inner and outer radii 2.00 cm and 4.00 cm and made of a type of cork with thermal conductivity 4.00 × 10 −2 W/m · K. This in turn is surrounded by a cylindrical jacket made of a brand of Styrofoam with thermal conductivity 2.70 × 10 −2 W/m · K and having inner and outer radii 4.00 cm and 6.00 cm ( Fig. P17.115 ). The outer surface of the Styrofoam has a temperature of 15°C. What is the temperature at a radius of 4.00 cm, where the two insulating layers meet? (e) What is the total rate of transfer of heat out of a 2.00-m length of pipe? Figure P17.115
A hollow cylinder has length L , inner radius a , and outer radius b , and the temperatures at the inner and outer surfaces are T 2 and T 1 . (The cylinder could represent an insulated hot-water pipe.) The thermal conductivity of the material of which the cylinder is made is k . Derive an equation for (a) the total heat current through the walls of the cylinder; (b) the temperature variation inside the cylinder walls. (c) Show that the equation for the total heat current reduces to Eq. (17.21) for linear heat flow when the cylinder wall is very thin. (d) A steam pipe with a radius of 2.00 cm, carrying steam at 140°C, is surrounded by a cylindrical jacket with inner and outer radii 2.00 cm and 4.00 cm and made of a type of cork with thermal conductivity 4.00 × 10 −2 W/m · K. This in turn is surrounded by a cylindrical jacket made of a brand of Styrofoam with thermal conductivity 2.70 × 10 −2 W/m · K and having inner and outer radii 4.00 cm and 6.00 cm ( Fig. P17.115 ). The outer surface of the Styrofoam has a temperature of 15°C. What is the temperature at a radius of 4.00 cm, where the two insulating layers meet? (e) What is the total rate of transfer of heat out of a 2.00-m length of pipe? Figure P17.115
A hollow cylinder has length L, inner radius a, and outer radius b, and the temperatures at the inner and outer surfaces are T2 and T1. (The cylinder could represent an insulated hot-water pipe.) The thermal conductivity of the material of which the cylinder is made is k. Derive an equation for (a) the total heat current through the walls of the cylinder; (b) the temperature variation inside the cylinder walls. (c) Show that the equation for the total heat current reduces to Eq. (17.21) for linear heat flow when the cylinder wall is very thin. (d) A steam pipe with a radius of 2.00 cm, carrying steam at 140°C, is surrounded by a cylindrical jacket with inner and outer radii 2.00 cm and 4.00 cm and made of a type of cork with thermal conductivity 4.00 × 10−2 W/m · K. This in turn is surrounded by a cylindrical jacket made of a brand of Styrofoam with thermal conductivity 2.70 × 10−2 W/m · K and having inner and outer radii 4.00 cm and 6.00 cm (Fig. P17.115). The outer surface of the Styrofoam has a temperature of 15°C. What is the temperature at a radius of 4.00 cm, where the two insulating layers meet? (e) What is the total rate of transfer of heat out of a 2.00-m length of pipe?
A hollow cylinder has length L, inner radius R, and thickness d, and the temperatures at the inner and
outer surfaces are Tn and Ti, respectively. The thermal conductivity of the material of which the cylinder
is made is k.
Derive an equation for the rate of heat transfer P through the walls of the cylinder. Simplify this
equation assuming that the thickness of the cylinder d is much smaller than the its inner radius R.
Geologists measure conductive heat flow out of the earth by drilling holes (a few hundred meters deep) and measuring the temperature as a function of depth. Suppose that in a certain location the temperature increases by 20°C per kilometer of depth and the thermal conductivity of the rock is 2.5 W/m·K. What is the rate of heat conduction per square meter in this location? Assuming that this value is typical of other locations over all of earth's surface, at approximately what rate is the earth losing heat via conduction? (The radius of the earth is 6400 km.)
T (C)
50
4. As a physicist, you put heat into a 600-g solid
sample at the rate of 15 kJ/min while recording its
temperature as a function of time. You plot your
data and obtain the graph shown on the left. (a)
What is the latent heat of fusion of this solid? (b)
40
30
20
10
What are the specific heats of the liquid and solid
states of the material?
1 (min)
4
2
3
Chapter 17 Solutions
University Physics with Modern Physics (14th Edition)
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