Transient Heat Transfer Overview

I first saw a figure like this in Professor Adrian Bejan’s book Heat Transfer  and I really liked the succinct way that it illustrates the ballpark relationship between various transient conduction heat transfer approximations that we commonly use.

In log-log coordinates, we have the Biot number (Bi), which is a dimensionless heat transfer coefficient plotted on the vertical axis, and the Fourier number (Fo), which is dimensionless time, plotted on the horizontal axis.

In a solid which is initially at a uniform temperature and is subjected to a sudden change in the boundary condition, the temperature changes will initially occur (Fo <<1) mostly very close to the surface, and the temperature field can be approximated by the solution for a semi-infinite body.  Professor Bejan calls this the “early” regime, and it is represented by blue shading on the left half of the figure.  If the heat transfer coefficient is high enough (large Bi) or for some reason the surface temperature is specified,  then the solution for a fixed surface temperature (still semi-infinite) is appropriate as shown in a different shade of blue in the upper part of the left side of the figure.

For cases where the conduction within the solid is rapid relative to the heat transfer out, the temperature variation across the body can be neglected.  The entire body can be characterized by a single temperature at any given time and the temperature variation of the body with time can be approximated with an approach called “lumped capacitance”.  The Biot number serves as a measure of when the spatial temperature variation is expected to be small relative to other temperature differences.  Lumped capacitance solutions are appropriate for Bi << 1, or a common rule of thumb that is sometimes cited is Bi < 0.1.  Professor Bejan terms this the “late” regime, and it is represented by the reddish shading in the lower right side of the figure.

For cases where neither the Fo nor the Bi number is very small, the exact solutions must be applied.  Actually, the exact solutions could be applied anywhere, but since they are in the form of infinite series, it is most convenient to use one of the simpler approximations, when appropriate.  These infinite series converge rapidly for large values of time, so a for Fo > 1, it is an excellent approximation to use only the first term of the series.  A common rule of thumb for using only the first term of these series is Fo > 0.2.  A graphical presentation of the temperature distribution based on the first term of these series has become known as the “Heisler Charts”.  The area covered by the first term of the exact solutions, or by the Heisler Charts, is shown in the upper right quadrant of the figure shaded green.

Definitions:

 L_c= a characteristic length of the solid

h = convective heat transfer coefficient at the surface of the solid

k = thermal conductivity of the solid

a = thermal diffusivity of the solid

t = time

Reference
Bejan, Adrian.  Heat Transfer.  New York:  Wiley, 1993.