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?
Suppose a person is covered head to foot by wool clothing with an average thickness of d = 1.95 cm and is transferring energy by conduction through the clothing at the rate of Q / Δt = 45 W.
What is the temperature difference, in terms of the quantities given in the problem statement, across the clothing? Denote the surface area of the wool by A and the thermal conductivity by k.
On a multi-layered square wall, the thermal resistance of the first layer is 0.005 ° C / W, the resistance of the second layer is 0.2 ° C / W, and the third layer is 0.1 ° C / W. The overall temperature gradient in the wall is multilayered from one side. to the other side is 60 ° C.
a. Determine the heat flux through the walls. = Answer
watts / m2.
b. If the thermal resistance of the second layer is changed to 0.3 ° C / W, what is the effect in% on heat flux, assuming the temperature gradient remains the same? = Answer
Answer
%.
Consider a flat-plate solar collector placed on the roof of a house. The temperatures at the inner and outer surfaces of the glass cover are measured to be 33°C and 31°C, respectively. The glass cover has a surface area of 2.5 m2, a thickness of 0.6 cm, and a thermal conductivity of 0.7 W/m·K. Heat is lost from the outer surface of the cover by convection and radiation with a convection heat transfer coefficient of 10 W/m2·K and an ambient temperature of 15°C. Determine the fraction of heat lost from the glass cover by radiation.
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