Transient Cooling of a Sphere

This Demonstration shows transient heat conduction in a sphere of radius . At time , the sphere is held at a uniform temperature . At time , the sphere is immersed in a well-mixed cooling bath at temperature . The sphere loses heat from its surface according to Newton's law of cooling: , where is a heat transfer coefficient. Assume that at any time in the cooling process, the temperature distribution within the sphere depends solely on the radial coordinate; in a spherical coordinate system, the temperature is symmetric with respect to the azimuthal and polar angles.
The Demonstration finds the 15 first roots of and displays the density plot of the sphere's temperature for user-set values of the Biot number, , and the dimensionless time . Larger values of or correspond to cooler temperatures of the sphere.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The mathematical problem involved is:
subject to the following initial conditions (IC) and boundary conditions (BC)
IC: ,
BC 1: for ,
BC 2: .
The material parameters of the sphere that affect heat transfer are , the thermal diffusivity and , the thermal conductivity .
The heat transfer coefficient for the sphere/bath system accounts for the resistance associated with heat loss from the sphere to the fluid and is a function of the local mixing properties of the bath as well as the fluid thermal properties.
To solve this problem it is convenient to introduce the following dimensionless variables:
, , and .
Thus the PDE and IC/BCs become:
for ,
IC : ,
BC1: for ,
BC2: ,
where is a dimensionless quantity called the Biot number, a measure of the relative importance of resistance, heat conduction within the sphere, and resistance of heat loss to the surrounding fluid.
Using the method of separation of variables, you obtain the following solution:
where are the roots of and .
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students. »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+