#### Interactive Self-Study Module: Transient Conduction into a Semi-infinite Solid

##### Overview:

This module uses screencasts and interactive simulations to explain how you can calculate the temperature change of a very thick solid. The temperature changes over time because of conduction. It is a function of time and depth within the solid. If the surface of the solid is warmed up, the thermal energy slowly diffuses deep into the solid. Mathematically the solid is infinitely thick. Strictly speaking this does not apply to any real parts, which have a finite thickness. However, at relatively short times the temperature change never reaches the center of the part, and it is reasonable to assume it is infinitely thick.

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.
- 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 variation near the surface of a thick part. If you zoom into the part, the local surface begins to look like a plane wall with a very thick depth. If you place the part into a furnace, its surface will warm up first, followed by the material below the surface. If it is not placed in the furnace for a long period of time, the material deep inside the part hasn’t had time to change temperature yet. This scenario is well characterized as a semi-infinite solid. Another example is placing your palm on a very cold surface. It takes time to first sense the cold, and a while longer to get frostbite. This is because it takes a finite amount of time to cool your skin deep enough for your nerves to sense it, and a while longer for your skin to freeze. Using the math outlined in this module, you can estimate the severity of the frostbite. You can do so by calculating the depth of your skin that reaches freezing over a given amount of time.
- This module is primarily intended for a Heat Transfer course.

##### Before studying this module, you should:

- Know the meaning of a heat transfer coefficient.
- Know basic heat transfer parameters, such as the thermal diffusivity, \(\alpha = \frac{k}{\rho c_p}\)
- Be able to use partial differential equations.
- Be able to specify an initial condition and boundary conditions and know how to use them.

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

- Explain how thick solids heat up or cool down over time.
- Sketch qualitatively-accurate graphs of the temperature of the solid at various points in time, demonstrating that the temperature of the solid deep beneath the surface does not change as rapidly.
- Calculate the amount of time it takes for the temperature at a given depth to reach a particular value.
- At a given point in time, calculate the temperature at a particular depth.