Heisler Charts

In the field of thermal sciences, solving transient heat conduction problems analytically often leads to daunting mathem

In the field of thermal sciences, solving transient heat conduction problems analytically often leads to daunting mathematical hurdles, such as infinite series or complex integral transforms. For engineers in the field, these solutions can be cumbersome to compute and difficult to interpret physically. To bridge the gap between rigorous mathematics and practical engineering, H.S. Heisler introduced a set of dimensionless graphical solutions in 1947, now universally known as Heisler Charts.

These charts provide a standardized method for solving one-dimensional transient heat conduction for three fundamental geometries: infinite plates, infinite cylinders, and spheres. By utilizing dimensionless parameters, Heisler Charts allow engineers to bypass tedious derivations and quickly determine temperature distributions at any internal point or the temperature change at the center of an object.
The utility of Heisler Charts relies on two critical dimensionless numbers. These parameters encapsulate the physical relationship between the material's internal thermal properties and the external environmental conditions.

1. The Fourier Number ($Fo$)

The Fourier number represents the relationship between the time scale of the process and the rate of thermal diffusion within the material. It can be thought of as a measure of "dimensionless time." It is defined as:

$$ Fo = \frac{\alpha t}{L_c^2} $$

Where:

  • $\alpha$ is the thermal diffusivity ($m^2/s$).
  • $t$ is the elapsed time ($s$).
  • $L_c$ is the characteristic length (the half-thickness for a plate, or the radius for a cylinder or sphere).

A higher $Fo$ indicates that heat has had sufficient time to diffuse deep into the body, leading to a more uniform temperature distribution.

2. The Biot Number ($Bi$)

The Biot number characterizes the ratio of internal conduction resistance to external convection resistance. It determines whether the temperature within the object is relatively uniform or if significant gradients exist. It is defined as:

$$ Bi = \frac{h L_c}{k} $$

Where:

  • $h$ is the convective heat transfer coefficient ($W/(m^2 \cdot K)$).
  • $k$ is the thermal conductivity of the material ($W/(m \cdot K)$).

The $Bi$ threshold is crucial for choosing a solution method:

  • If $Bi < 0.1$: The internal conduction resistance is negligible compared to the surface convection resistance. The temperature throughout the body is nearly uniform, and the Lumped Capacitance Method can be used.
  • If $Bi \ge 0.1$: Significant temperature gradients exist within the body, necessitating the use of Heisler Charts or analytical series solutions.

Structure and Interpretation of the Charts

Heisler Charts are typically organized into two distinct sets of graphs to provide a complete thermal profile.

1. Centerline Temperature Ratio Chart

This chart is used to find the dimensionless temperature at the very center of the object ($x=0$ or $r=0$).

  • Horizontal Axis: The Fourier number ($Fo$), often plotted on a logarithmic scale.
  • Vertical Axis: The dimensionless temperature ratio at the center, defined as:
    $$\theta^*_{\text{center}} = \frac{T(0,t) - T_\infty}{T_i - T_\infty}$$
  • Parameter Curves: A family of curves representing different values of $1/Bi$.

Workflow: Calculate $Bi$ and $Fo$, locate the curve corresponding to $1/Bi$, find the value on the $Fo$ axis, and read the corresponding $\theta^*_{\text{center}}$ on the vertical axis.

2. Position-Dependent Temperature Ratio Chart

Once the center temperature is known, this second chart allows you to find the temperature at any other specific location within the object.

  • Horizontal Axis: The dimensionless position, such as $x/L_c$ (for plates) or $r/R$ (for cylinders and spheres).
  • Vertical Axis: The ratio of the temperature at a specific position to the temperature at the center:
    $$\frac{\theta(x,t)}{\theta(0,t)} = \frac{T(x,t) - T_\infty}{T(0,t) - T_\infty}$$
  • Parameter Curves: Again, these are plotted as a function of $1/Bi$.

Final Temperature Calculation:
By combining the results from both charts, the actual temperature $T(x,t)$ at any position $x$ and time $t$ is calculated as:
$$ T(x,t) = T_\infty + \left[ \theta^*_{\text{center}} \times \left( \frac{\theta(x,t)}{\theta(0,t)} \right) \times (T_i - T_\infty) \right] $$

Practical Application Example

Consider a steel plate with a thickness of $2L = 0.1\text{ m}$ (meaning $L = 0.05\text{ m}$) at an initial temperature $T_i = 200^\circ\text{C}$. The plate is suddenly immersed in a cooling liquid at $T_\infty = 20^\circ\text{C}$ with a convection coefficient $h = 200\text{ W}/(m^2 \cdot K)$. The steel has a thermal conductivity $k = 50\text{ W}/(m \cdot K)$ and a thermal diffusivity $\alpha = 1.2 \times 10^{-5}\text{ m}^2/s$. We want to find the center temperature after $t = 300\text{ s}$.

  1. Calculate $Bi$:
    $$ Bi = \frac{200 \times 0.05}{50} = 0.2 \implies 1/Bi = 5 $$
  2. Calculate $Fo$:
    $$ Fo = \frac{1.2 \times 10^{-5} \times 300}{(0.05)^2} = 1.44 $$
  3. Consult the Charts:
    Using the Centerline Temperature Ratio chart, locate the curve for $1/Bi = 5$. At $Fo = 1.44$, we read a dimensionless ratio of approximately $\theta^*_{\text{center}} \approx 0.65$.
  4. Solve for $T$:
    $$ T(0,300) = 20 + 0.65 \times (200 - 20) = 20 + 117 = 137^\circ\text{C} $$

Scope, Limitations, and Advanced Usage

While Heisler Charts are incredibly powerful, they are not universal tools. Users must be mindful of their boundaries:

  • Applicability: The charts are strictly designed for one-dimensional problems involving constant material properties, no internal heat generation, and convective boundary conditions.
  • Precision: Because the method relies on visual interpolation, there is an inherent reading error (typically between 1% and 3%). For high-precision aerospace or nuclear applications, numerical simulations (such as FEA/CFD) or exact analytical series solutions are preferred.
  • The Product Solution for 3D Objects: For complex geometries like a rectangular block, the problem can be treated as a combination of three independent 1D problems (one for each dimension). By calculating the dimensionless temperature ratio for each dimension using the respective Heisler charts and multiplying them together, an accurate approximation for the 3D temperature distribution can be achieved.

Despite the rise of sophisticated computational software, Heisler Charts remain a cornerstone of thermal engineering education and a vital tool for rapid, intuitive engineering estimations.