# Transient Conduction through a Plane Wall

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A plane wall of thickness is initially at uniform temperature. At time , its surface is exposed to a liquid at a different temperature. The centerline temperature (at ) is plotted as a function of time on the left, and the temperature distribution as a function of location is plotted on the right; it varies in time.

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Contributed by: Rachel Saker and Rachael L. Baumann (February 2017)

Additional contributions by: John L. Falconer

(University of Colorado Boulder, Department of Chemical and Biological Engineering)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Lumped capacitance is a quick method for determining the temperature within an object, such as the plane wall considered here. This method assumes a uniform temperature at any point within the solid (regardless of proximity to the surface) and therefore is only reliable for Biot number . The Biot number is

,

where is the convective heat transfer coefficient (), is the thermal conductivity () of the material, and is the cross-sectional length of the wall (m). The lumped capacitance method, which assumes a uniform temperature distribution through the wall, is only a function of the wall thickness and not a function of position within the wall. The temperature as a function of time is calculated using the lumped capacitance method from the ratio of the temperature differences:

,

,

where is temperature (K), the subscripts and denote the initial and ambient temperatures, and is the Fourier number (dimensionless):

,

,

where is thermal diffusivity (), is time (), is density () and is heat capacity ().

When , uniform temperature in the solid is not a valid assumption, and thus the lumped capacitance method is not accurate. Instead, the one-term approximation models the temperature distribution with better accuracy, and is valid for :

,

,

where is the position within the solid (), is dimensionless, and and are constants that can be found in Table 5.1 on p. 301 of [1].

Reference

[1] T. L. Bergman, A. S. Lavine, F. P. Incropera and D. P. DeWitt, *Introduction to Heat Transfer*, 6th ed., Hoboken, NJ: John Wiley and Sons, 2011.

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