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Transient Conduction through a Plane Wall

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.
Click the "start heating" or "start cooling" play button to initiate heating or cooling. For heating, the initial temperature is 300 K, and for cooling the initial temperature is 800 K. During heating, the fluid temperature is 1100 K (green line), and for cooling, the fluid temperature is 300 K.
Because the exact solution to the transient heat conduction equations gives an infinite series, the first term in the series was used to calculate temperature as a function of time and location; the black line in the left plot shows the one-term solution. The dashed blue line is the lumped capacitance solution, which is nearly the same as the one-term solution at low Biot numbers, but it deviates significantly at high Biot numbers. Set the Biot number with a slider. The lumped capacitance method assumes that the plane wall temperature is uniform at a given time. The Biot number is a dimensionless ratio of convection at the surface to conduction within the solid.

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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