Solving the PDE for a Sphere in Vacuum: Temperature over Time

In summary: The boundary condition for the temperature is given by the Stefan-Boltzmann law which is a non-linear function of temperature to the fourth power. There are books and online resources available for reference on this topic. Numerical methods can also be used to solve this problem.
  • #1
MarkL
34
2
Is there a PDE describing temperature as a function of time for a sphere in a vacuum (a moon in space)?

Since it can only radiate, the boundary would be a function of temperature to the fourth power (nonlinear), right?

Is there a book or thread out there I can reference?

Maybe this can be handled easily numerically.

Thank you

Mark
 
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  • #2
MarkL said:
Is there a PDE describing temperature as a function of time for a sphere in a vacuum (a moon in space)?

Since it can only radiate, the boundary would be a function of temperature to the fourth power (nonlinear), right?

Is there a book or thread out there I can reference?

Maybe this can be handled easily numerically.

Thank you

Mark

You will need some assumptions about the energy coming in, and about the heat capacity of the surface. The latter tells you how quickly temperature drops as energy is radiated. It can be more complicated if the planet is water covered, or otherwise has some heat capacity below the surface.

Cheers -- sylas
 
  • #3
Apply the http://en.wikipedia.org/wiki/Heat_equation" . Due to the symmetry of a sphere it can be done as a one dimensional problem with the radius of the sphere as the variable.
 
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FAQ: Solving the PDE for a Sphere in Vacuum: Temperature over Time

What is a PDE?

A PDE (Partial Differential Equation) is a mathematical equation that involves partial derivatives of an unknown function. It is commonly used to model physical phenomena in science and engineering, such as heat transfer, fluid dynamics, and quantum mechanics.

Why is the PDE for a sphere in vacuum important?

The PDE for a sphere in vacuum is important because it allows us to model the temperature distribution over time for a spherical object in a vacuum environment. This can be useful in a variety of applications, such as studying the behavior of planets or stars, designing thermal insulation for spacecraft, or analyzing the efficiency of thermodynamic systems.

What are the boundary conditions for solving the PDE for a sphere in vacuum?

The boundary conditions for solving the PDE for a sphere in vacuum typically include the initial temperature distribution on the surface of the sphere, the thermal conductivity of the material, and the surrounding temperature of the vacuum. These boundary conditions help determine the temperature distribution over time and provide a realistic model for the physical system.

What methods can be used to solve the PDE for a sphere in vacuum?

There are several methods that can be used to solve the PDE for a sphere in vacuum, including analytical solutions, numerical methods, and computer simulations. Analytical solutions involve solving the PDE using mathematical techniques, while numerical methods use algorithms to approximate the solution. Computer simulations involve using specialized software to solve the PDE and visualize the results. The choice of method depends on the complexity of the problem and the desired level of accuracy.

How can the solution to the PDE for a sphere in vacuum be verified?

The solution to the PDE for a sphere in vacuum can be verified by comparing it to known analytical solutions, performing sensitivity analyses, or conducting experiments in a controlled environment. Additionally, the solution can be checked for physical plausibility and compared to real-world data to ensure its accuracy.

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