Test your level of competency for GATE Mechanical Engineering and UPSC ESE Mechanical Engineering exam for the topic of Steady State Heat Conduction from Heat Transfer.

Q1) The wall of an industrial furnace is constructed from 0.15-m-thick fireclay brick having a thermal conductivity of 1.7 W/mK. Measurements made during steady-state operation reveal temperatures of 1400 and 1150 K at the inner and outer surfaces, respectively. What is the rate of heat loss through a wall that is 0.5 m x 1.2 m on a side?

Q2) An uninsulated steam pipe passes through a room in which the air and walls are at 25oC. The outside diameter of the pipe is 70 mm, and its surface temperature and emissivity are 200oC and 0.8, respectively. What are the surface emissive power and irradiation? If the coefficient associated with free convection heat transfer from the surface to the air is 15 W/m2 K, what is the rate of heat loss from the surface per unit length of pipe?

Q3) The blades of a wind turbine turn a large shaft at a relatively slow speed. The rotational speed is increased by a gearbox that has an efficiency of . In turn, the gearbox output shaft drives an electric generator with an efficiency of . The cylindrical nacelle, which houses the gearbox, generator, and associated equipment, is of length L = 6 m and diameter D = 3 m. If the turbine produces P = 2.5 MW of electrical power, and the air and surroundings temperatures are  and , respectively, determine the minimum possible operating temperature inside the nacelle. The emissivity of the nacelle is , and the convective heat transfer is . The surface of the nacelle that is adjacent to the blade hub can be considered to be adiabatic, and solar irradiation may be neglected.

Q4) The thermal conductivity of a sheet of rigid, extruded insulation is reported to be k = 0.029 W/m.K. The measured temperature difference across a 20 mm thick sheet of the material is

1. What is the heat flux through a 2 m x  2 m sheet of the insulation?
2. What is the rate of heat transfer through the sheet of insulation?

Q5) A thermodynamic analysis of a proposed Brayton cycle gas turbine yields P = 5 MW of net power production. The compressor, at an average temperature of , is driven by the turbine at an average temperature of  by way of an L = 1-m-long, d = 70-mm-diameter shaft of thermal conductivity .

1. Compare the steady-state conduction rate through the shaft connecting the hot turbine to the warm compressor to the net power predicted by the thermodynamics-based analysis.
2. A research team proposes to scale down the gas turbine of part (a), keeping all dimensions in the same proportions. The team assumes that the same hot and cold temperatures exist as in part (a) and that the net power output of the gas turbine is proportional to the overall volume of the device. Plot the ratio of the conduction through the shaft to the net power output of the turbine over the range . Is a scaled-down device with L = 0.005 m feasible?

Q6) What is the thickness required of a masonry wall having thermal conductivity 0.75 W/mK if the heat rate is to be 80% of the heat rate through a composite structural wall having a thermal conductivity of 0.25 W/mK and a thickness of 100 mm? Both walls are subjected to the same surface temperature difference.

Q7) A wall is made from an inhomogeneous (nonuniform) material for which the thermal conductivity varies through the thickness according to k = ax + b, where a and b are constants. The heat flux is known to be constant. Determine expressions for the temperature gradient and the temperature distribution when the surface at x = 0 is at temperature T1.

Q8) The 5-mm-thick bottom of a 200-mm-diameter pan may be made from aluminum (k = 240 W/m K) or copper (k = 390 W/mK). When used to boil water, the surface of the bottom exposed to the water is nominally at 110oC. If heat is transferred from the stove to the pan at a rate of 600 W, what is the temperature of the surface in contact with the stove for each of the two materials?

Q9) A square silicon chip (k = 150 W/mK) is of width w = 5 mm on a side and of thickness t = 1 mm. The chip is mounted in a substrate such that its side and back surfaces are insulated, while the front surface is exposed to a coolant. If 4 W are being dissipated in circuits mounted to the back surface of the chip, what is the steady-state temperature difference between back and front surfaces?

Q10) Humans are able to control their heat production rate and heat loss rate to maintain a nearly constant core temperature of  under a wide range of environmental conditions. This process is called thermoregulation. From the perspective of calculating heat transfer between a human body and its surroundings, we focus on a layer of skin and fat, with its outer surface exposed to the environment and its inner surface at a temperature slightly less than the core temperature, . Consider a person with a skin/fat layer of thickness L = 3 mm and effective thermal conductivity k = 0.3 W/mK. The person has a surface area A = 1.8 m2 and is dressed in a bathing suit. The emissivity of the skin is .

1. When the person is in still air at , what is the skin surface temperature and rate of heat loss to the environment? Convection heat transfer to the air is characterized by a free convection coefficient of h = 2 W/m2 K.
2. When the person is in water at , what is the skin surface temperature and heat loss rate? Heat transfer to the water is characterized by a convection coefficient of h = 200 W/m2K.

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