7. Use of Transient Temperature Charts

Slab

Dimensionless Equations: \[\begin{aligned} {3} \frac{\partial^2\theta}{\partial X^2} &= \frac{\partial \theta}{\partial \tau} \qquad & & \text{ in } 0 < X < 1, \text{ for } \tau > 0 \\ \frac{\partial \theta}{\partial X} &= 0 & & \text{ at } X = 0, \text{ for } \tau > 0 \\ \frac{\partial \theta}{\partial X} + \text{Bi}\; \theta &= 0 & & \text{ at } X = 1, \text{ for } \tau > 0 \\ \theta &= 1 & & \text{ in } 0 \le X \le 1, \text{ for } \tau = 0 \end{aligned}\] For plane wall, the solution involves several parameters: \[T = T (x, L, k, h, \alpha, T_i, T_\infty, t)\] By using dimensional groups, we can reduce the number of parameters. \[\theta = \theta(X, \text{Bi}, \text{Fo})\]

The solution for temperature will now be function of the dimensionless quantities: \[\theta = \theta(X,\text{Bi}, \text{Fo})\] The transient temperature charts shown in next slides for a large plane wall (also available for long cylinder, and sphere) were presented by M. P. Heisler in 1947 and are called Heisler charts.

There are three charts associated with each geometry:

  1. The first chart is to determine the temperature \(T_0\) at the center of the geometry at a given time \(t\).

  2. The second chart is to determine the temperature at other locations at the same time in terms of \(T_0\).

  3. The third chart is to determine the total amount of heat transfer up to the time \(t\).

These plots are valid for \(\text{Fo} > 0.2\).