Radioactive wastes are packed in a thin-walled spherical container. The wastes generate thermal energy nonuniformly according to the relation ġ = 4, 1-(r/r.)* | where ġ is the local rate of energy generation per unit volume, ġ, is a constant, and r, is the radius of the container. Steady- state conditions are maintained by submerging the container in a liquid that is at T, and provides a uniform convection coefficient h. Coolant T, h - ġ = ¿, (1– (r/r,)²] Determine the temperature distribution, T(r), in the container. Express your result in terms of 4,, r,, T, , h, and the thermal conductivity k of the radioactive wastes.

Radioactive wastes are packed in a thin-walled spherical container. The wastes generate thermal energy nonuniformly according to the relation ġ = 4, 1-(r/r.)* | where ġ is the local rate of energy generation per unit volume, ġ, is a constant, and r, is the radius of the container. Steady- state conditions are maintained by submerging the container in a liquid that is at T, and provides a uniform convection coefficient h. Coolant T, h - ġ = ¿, (1– (r/r,)²] Determine the temperature distribution, T(r), in the container. Express your result in terms of 4,, r,, T, , h, and the thermal conductivity k of the radioactive wastes.

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)

8th Edition

ISBN:9781305387102

Author:Kreith, Frank; Manglik, Raj M.

Publisher:Kreith, Frank; Manglik, Raj M.

Chapter1: Basic Modes Of Heat Transfer

Section: Chapter Questions

Problem 1.66P

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3.5 In a ball-bearing production facility, steel balls that are each of 15 mm in diameter are annealed by first heating them to 870°C and then slowly cooling in air to 125°C. If the cooling air stream temperature is 60°C, and it has a convective heat transfer coefficient of , determine the time required for the cooling.
2.2 A small dam, which is idealized by a large slab 1.2 m thick, is to be completely poured in a short Period of time. The hydration of the concrete results in the equivalent of a distributed source of constant strength of 100 W/m3. If both dam surfaces are at 16°C, determine the maximum temperature to which the concrete will be subjected, assuming steady-state conditions. The thermal conductivity of the wet concrete can be taken as 0.84 W/m K.
1.15 A thermocouple (0.8-mm-diameter wire) used to measure the temperature of the quiescent gas in a furnace gives a reading of . It is known, however, that the rate of radiant heat flow per meter length from the hotter furnace walls to the thermocouple wire is 1.1 W/m and the convection heat transfer coefficient between the wire and the gas is K. With this information, estimate the true gas temperature. State your assumptions and indicate the equations used.
2.7 A very thin silicon chip is bonded to a 6-mm thick aluminum substrate by a 0.02-mm thick epoxy glue. Both surfaces of this chip-aluminum system are cooled by air at , where the convective heat transfer coefficient of air flow is . If the heat dissipation per unit area from the chip is under steady-state conditions, draw the thermal circuit for the system and determine the operating temperature of the chip.
2.15 Suppose that a pipe carrying a hot fluid with an external temperature of and outer radius is to be insulated with an insulation material of thermal conductivity k and outer radius . Show that if the convection heat transfer coefficient on the outside of the insulation is and the environmental temperature is , the addition of insulation actually increases the rate of heat loss if , and the maximum heat loss occurs when . This radius, is often called the critical radius.
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