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.