Short Answer:

The Heisler chart is a graphical tool used to find the temperature distribution and heat transfer in solids during transient (time-dependent) heat conduction. It provides quick solutions for temperature changes in geometries like plane walls, cylinders, and spheres without solving complex equations.

In simple words, the Heisler chart helps engineers easily estimate how temperature varies with time and position inside an object when it is heated or cooled. It is mainly used when the lumped capacitance method cannot be applied, that is, when the Biot number is greater than 0.1.

Detailed Explanation:

Heisler chart

The Heisler chart is a graphical representation developed by M. P. Heisler to simplify the analysis of transient heat conduction in solids. It provides ready-to-use solutions for temperature variation and heat transfer in materials over time. In many practical heat transfer problems, such as cooling of machine parts, quenching of metals, or heating of walls, the temperature changes with time and position. Solving these problems mathematically using Fourier’s law often becomes complex. Therefore, the Heisler chart serves as an effective shortcut method to find accurate solutions quickly.

Purpose of Heisler chart

The main purpose of the Heisler chart is to estimate the transient temperature distribution in objects of regular shape such as plane walls, long cylinders, and spheres. It relates the dimensionless temperature, time, and position for different Biot and Fourier numbers. Engineers use these charts to predict how a solid material reacts to heating or cooling over a certain period without performing complex calculations.

The chart is particularly helpful when:

• The Biot number (Bi) is greater than 0.1, meaning the temperature inside the body is not uniform.
• The process is transient, meaning temperature varies with time.
• The object has simple geometry like a slab, cylinder, or sphere.

Governing Parameters

The Heisler chart uses two important dimensionless numbers:

1. Biot Number (Bi):

It represents the ratio of internal conduction resistance to external convection resistance.

1. Fourier Number (Fo):

It indicates the ratio of heat conduction rate to the rate of thermal storage and is directly related to time.

Here,

•  = heat transfer coefficient (W/m²·K)
•  = characteristic length (m)
•  = thermal conductivity (W/m·K)
•  = thermal diffusivity (m²/s)
•  = time (s)

These two numbers are essential for reading and interpreting the Heisler chart.

Types of Heisler charts

The Heisler chart is actually a set of three separate charts, each representing a different relationship:

1. Chart 1: Dimensionless temperature at the center of the object versus Fourier number for various Biot numbers.
2. Chart 2: Dimensionless temperature distribution within the object (from center to surface) for different Biot numbers.
3. Chart 3: Dimensionless heat transfer from the object versus Fourier number for various Biot numbers.

Each of these charts helps find specific information about temperature and heat transfer depending on what is needed.

Use of Heisler chart

The steps to use a Heisler chart are simple and systematic:

1. Determine the Biot number (Bi) using .
2. Calculate the Fourier number (Fo) using .
3. Locate the Bi and Fo values on the chart corresponding to the shape of the object.
4. Find the dimensionless temperature (θ/θi) from the chart.
5. Convert the dimensionless temperature back to actual temperature using:

where  = temperature at a certain point,  = initial temperature, and  = surrounding temperature.

1. If required, the heat transfer rate can also be found using the third Heisler chart.

Applications of Heisler chart

The Heisler chart is used in various engineering situations, including:

• Cooling of metal rods or plates after hot working.
• Design of heat exchangers, where tube walls undergo transient heating or cooling.
• Thermal treatment of materials, such as annealing or quenching.
• Estimation of temperature in machine components, where heat transfer varies with time.
• Thermal analysis in electronics, to predict temperature changes in semiconductor materials.

The charts save time and effort by providing direct solutions that would otherwise require solving complicated differential equations.

Advantages of Heisler chart

1. Quick and reliable: Eliminates the need for complex mathematical computation.
2. Visual representation: Makes it easy to interpret heat transfer behavior.
3. Applicable to different geometries: Works for slabs, cylinders, and spheres.
4. Widely used in engineering design: Especially when the Biot number is greater than 0.1.
5. Accurate for practical use: Provides results very close to exact analytical solutions.

Limitations of Heisler chart

1. Valid only for regular shapes (plane wall, long cylinder, or sphere).
2. Assumes constant properties of the material, which may not hold true in all cases.
3. Limited Biot number range: Charts are reliable only for certain ranges of Bi and Fo.
4. Steady environment: Surrounding temperature and heat transfer coefficient are assumed constant.
5. Cannot handle complex geometries or multi-dimensional problems.

Example

Suppose a long steel rod (diameter = 0.02 m) initially at 200°C is suddenly exposed to air at 25°C with . To find its center temperature after 3 minutes, we calculate Bi and Fo, then use the Heisler chart for a cylinder. This gives the dimensionless temperature, which helps estimate the actual temperature directly.

Conclusion

The Heisler chart is an essential engineering tool that simplifies the analysis of transient heat conduction problems. It provides quick graphical solutions for temperature and heat transfer in common geometries such as slabs, cylinders, and spheres. Although it has certain limitations, it remains one of the most useful methods for understanding and solving transient heat conduction in engineering applications, especially when the Biot number is significant.