Formula Sheets

This formula sheet contains basic equations for heat and mass transfer.

This math help sheet inlcudes solutions to common derivatives and integral rules.

The Heisler Chart

These charts describe the relationship between variables of temperature, position, and time in a transient or unsteady state system. They plot the 1st term of a series solution for heat transfer, which is a sufficient approximation after “long times” (e.g., the Fourier number is greater than 0.2).

You can find images of the Heisler Charts here for slab, cylinder, or sphere geometries. In order to determine find fractional temperature change (y-axis) in Cartesian coordinates, for example, follow these steps:

Step 1. Calculate \( m \) to identify which set of lines to use on the chart:

where

• \( h \) = convective heat transfer coeffiecent \( [\frac{W}{m^{2} K}] \)
• \( L \) = characteristic length \( [m] \)
• \( k \) = thermal conductivity of the medium \( [\frac{W}{mK}] \)

Step 2. Calculate \( n \), the non-dimensional length variable, to identify which specific line to use within in the groupings of lines:

where

• \( x \) = location of interest within the solid \( [m] \)
• \( L \) = characteristic length \( [m] \)

Step 3. Calculate the Fourier Number, the non-dimensional time variable (on the x-axis):

where

• \( \alpha = \frac{k}{\rho c_{p}} \) = thermal diffusivity \( [\frac{m^{2}}{s}] \)
• \( t \) = time \( [s] \)
• \( L \) = characteristic length \( [m] \)

Step 4. Trace the intersection of those three variables to the fractional temperature change:

where

• \( T \) = surface temperature of the solid \( [K] \)
• \( T_{i} \) = initial temperature of the solid \( [K] \)
• \( T_{\infty} \) = temperature of bulk fluid or air \( [K] \)