Aluminum Sphere Temperature Distribution for Heat Transfer by Radiation

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In summary, the temperature distribution for an aluminum sphere can be found by solving the equation dU/dt = 4πr2σT4(r)dr, where dU/dt is the rate of change of internal energy and σ is the Stefan-Boltzmann constant. By integrating this equation and using the conservation of energy, we can determine that the temperature at a distance r from the center of the sphere is given by T(r) = [(H/4πσ)ln(r/R) + C]1/4, where H is the heat radiated from the sphere, R is the radius of the sphere, and C is an integration constant.
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Homework Statement


I need to find the temperature distribution for an aluminum sphere given that the heat radiated from the sphere is equal to the rate of change of internal energy.

Homework Equations


I couldn't understand the solution of the following integral.
 

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The attempted solutionThe solution is as follows:Let T(r) be the temperature at a distance r from the center of the sphere. Then the rate of change of internal energy with respect to time is given by:dU/dt = 4πr2σT4(r)drwhere σ is the Stefan-Boltzmann constant.Integrating this equation, we get:U(t) = 4π∫r2σT4(r)drBy the conservation of energy, this is equal to the heat radiated from the sphere. Thus,U(t) = Hwhere H is the heat radiated from the sphere.Substituting for U(t) and rearranging, we get:4πr2σT4(r)dr = HorT4(r)dr = H/(4πr2σ)Integrating both sides, we get:T4(r) = (H/4πσ)ln(r/R) + Cwhere R is the radius of the sphere and C is an integration constant.Finally, solving for T(r):T(r) = [(H/4πσ)ln(r/R) + C]1/4
 

FAQ: Aluminum Sphere Temperature Distribution for Heat Transfer by Radiation

1. What is the purpose of studying aluminum sphere temperature distribution for heat transfer by radiation?

The purpose of studying aluminum sphere temperature distribution for heat transfer by radiation is to better understand how heat is transferred through radiation in various materials. This knowledge is important in many industries, such as aerospace and manufacturing, where controlling temperature is crucial for maintaining the integrity and functionality of materials.

2. How does the temperature distribution of an aluminum sphere affect heat transfer by radiation?

The temperature distribution of an aluminum sphere plays a significant role in heat transfer by radiation. It determines the amount of heat that is radiated from the sphere and the direction in which it is radiated. A higher temperature leads to a greater amount of heat being radiated, and a more even temperature distribution results in a more uniform heat transfer.

3. What factors influence the temperature distribution of an aluminum sphere for heat transfer by radiation?

Several factors can influence the temperature distribution of an aluminum sphere for heat transfer by radiation. These include the material properties of the sphere, such as its thermal conductivity and emissivity, as well as the ambient temperature, the intensity of the radiation source, and the size and shape of the sphere.

4. How can the temperature distribution of an aluminum sphere be calculated for heat transfer by radiation?

The temperature distribution of an aluminum sphere for heat transfer by radiation can be calculated using mathematical models and equations, such as the Stefan-Boltzmann law and the heat transfer coefficient equation. These equations take into account the factors that influence temperature distribution and can provide accurate predictions of the temperature distribution at different points on the sphere.

5. What are the practical applications of understanding aluminum sphere temperature distribution for heat transfer by radiation?

The understanding of aluminum sphere temperature distribution for heat transfer by radiation has many practical applications. It can be used in the design of thermal insulators and heat exchangers, as well as in the development of new materials with improved thermal properties. This knowledge is also essential for the development of more efficient and cost-effective heating and cooling systems in various industries.

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