User:Jonbarzel

Heisler Charts
Heisler Charts are a set of two charts per included geometry introduced in 1947 by M. P. Heisler which were supplemented by a third chart per geometry in 1961 by H. Gröber. They are used to provide a graphical analysis tool for the evaluation the central temperature for transient heat conduction through an infinitely long plane wall, an infinitely long cylinder of radius ro, and a sphere of radius ro.

Though the Heisler-Gröber Charts are faster and simpler alternative to the exact solutions of these problems, it should be noted that there are some limitations. First, the body must be at uniform temperature initially. Additionally, the temperature of the surroundings and the convective heat transfer coefficient must remain constant and uniform. Also, there must be no heat generation from the body itself.

Infinitely Long Plane Wall (of thickness 2L)
These first Heisler-Gröber Charts were based upon the first term of the exact Fourier Series solution for an infinite plane wall:

x is the location in the plane wall

The first chart for the plane wall is plotted using 3 different variables. Plotted along the vertical axis of the chart is θ o*, where θ o* $$= \frac{T_o-T_\infty}{T_i-T_\infty}$$. Plotted along the horizontal axis is the Fourier Number, Fo=αt/L2. The curves within the graph are a selection of values for the inverse of the Biot Number, where "Bi = hL/k. k is the thermal conductivity of the material and h is the heat transfer coefficient.

The second chart is used to determine the variation of temperature within the plane wall for different Biot Numbers. The vertical axis is the ration of a given temperature to that at the centerline  θ / θ o$$= \frac{T-T_\infty}{T_i-T_\infty}$$ where the x/L curve is the position at which T is taken. The horizontal axis is the value of Bi-1.

The third chart in each set was supplemented by Gröber in 1961 and this particular one shows the dimensionless heat transferred from the wall as a function of a dimensionless time variable. The vertical axis is a plot of Q/Qo, the ratio of actual heat transfer to the amount of total possible heat transfer before T=T∞. On the horizontal axis is the plot of (Bi2)(Fo), a dimensionless time variable.



Infinitely Long Cylinder
For the infinitely long cylinder, the Heisler chart is based on the first term in an exact solution to a Bessel Function.

Each chart plots similar curves to the previous examples, and on each axis is plotted a similar variable.





Sphere
The Heisler Chart for a sphere is based on the first term in the exact solution:

These charts can be used similar to the first two sets and are plots of similar variables.





Modern Alternatives
Currently there are programs that provide numerical solutions to the same problems, without using transcendental functions or infinite series. Examples of these programs can be found here or here.