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.
The thermal conductivity of a sheet of rigid, extruded
insulation is reported to be k = 0.029 W/mK. The
measured temperature difference across a 20-mm-thick
sheet of the material is T₁ T₂ = 10°C.
-
(a) What is the heat flux through a 2 m x 2 m sheet of
the insulation?
(b) What is the rate of heat transfer through the sheet
of insulation?
A 158 cm long aluminum rod (thermal conductivity = 205 W/m-K) has a cross-section area of 1 cm2. If one end is kept in a steam bath and the other end in an ice-water mixture, what is the quantity of heat (in Joules) conducted by the rod in 4 days? Input only the numerical value of the answer.
Chapter 17 Solutions
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