1. Heisler chart
2. Importance of numerical methods

1. Heisler chart

• Heisler charts are a graphical analysis tool for the evaluation of heat transfer in thermal engineering. They 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. Heisler charts permit evaluation of the central temperature for transient heat conduction through an infinitely long plane wall of thickness 2L, an infinitely long cylinder of radius ro, and a sphere of radius ro.
• These first Heisler-Gröber charts were based upon the first term of the exact Fourier Series solution for an infinite plane wall:

$$\frac{T(x, t) - T_∞}{T_i - T_∞} = \sum_{n = 0}^∞ \bigg[\frac{4 sin λ_n}{2λ_n + sin 2λ_n}e^{-λ_n^2 \frac{at}{L^2}} cos \frac{λ_n x}{L}\bigg]$$

where $T_i$ is the initial temperature of the slab, $T_∞$ is the constant temperature imposed at the boundary, x is the location in the plane wall, $λ_n$ is π(n+1/2), and α is thermal diffusivity. The position x=0 represents the center of the slab.

Limitations

Although Heisler-Gröber charts are a faster and simpler alternative to the exact solutions of these problems, 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.

1. Importance of numerical methods

• The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following:

  • Advanced numerical methods are essential in making numerical weather prediction feasible.
  • Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.
  • Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically.
  • Hedge funds (private investment funds) use tools from all fields of numerical analysis to attempt to calculate the value of stocks and derivatives more precisely than other market participants.
  • Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of operations research.
  • Insurance companies use numerical programs for actuarial analysis.