#### Interactive Self-Study Module: Transient Conduction in a Sphere with Spatial Effects

##### Overview:

This module uses screencasts and interactive simulations to explain how you can calculate the temperature of a sphere that warms up or cools down nonuniformly over time. The sphere is surrounded by a warm or cool fluid, such as air or water. The fluid causes the temperature at the surface of the sphere to change first. Because it takes time for heat to diffuse from the surface to the center of the sphere, the temperature near the object’s center changes more slowly. Unlike lumped capacitance, temperature is now a function of time and location within the sphere, and the analysis becomes more complicated.

Your retention of material in this module will increase if you write down reasons for your answers to ConcepTests, questions in screencasts, and questions to answer before using interactive simulations, and you try to solve the example problems before watching the screencast solutions.

We suggest using the learning resources in the following order:

- Attempt to answer the multiple choice ConcepTest and solve the example problem before watching the screencasts or working with the simulations.
- Watch the screencasts that describe how you can solve for the temperature in a sphere as a function of location in the sphere and time.
- Review the important equations for nonuniform analysis.
- Use the interactive simulation to further understand how you can predict the temperature over time and location.
- Try to solve the example problems before watching the solutions in the screencasts.
- Answer the ConcepTests.
- Look at the list of key points, but only after you try to list the key points yourself.

##### Motivation:

- Engineers sometimes need to estimate the temperature of a sphere as a function of time. If the sphere is relatively small and has a relatively high thermal conductivity, you can assume that the temperature throughout the sphere is relatively uniform. However, this is usually not the case. In general, the temperature at the sphere’s surface changes more rapidly than the temperature at its center. This is commonly observed when cooking food. When boiling or baking a potato, for example, the potato will cook from the outside in. The surface of the potato heats up first. It takes longer for the center of the potato to heat up. You can use the topics in this module to estimate how long it takes to fully cook the potato.
- This module is primarily intended for a Heat Transfer course.

##### Before studying this module, you should be able to:

- Work with lumped capacitance problems.
- Explain the physical meaning of the Biot number and how to use it.
- Write a transient energy balance equation for a control volume.
- Explain the meaning of a heat transfer coefficient.
- Use partial differential equations.
- Specify an initial condition and boundary conditions and know how to use them.

##### After studying this module, you should be able to:

- DiscussÂ how spheres heat up or cool down over time.
- Calculate the temperature of a sphere as a function of position and time.
- Sketch qualitatively-accurate graphs of the temperature throughout the sphere at a particular time.
- Predict when a point in a sphere will reach a specified temperature.