9711

First-Order Solution to Glass-Ice Stefan Problem

A Stefan problem is a type of boundary value problem in which a phase boundary can move with time. A first-order approximation of the Stefan problem is explored for a system consisting of a block of ice (0° C) attached to a 6 mm thick piece of glass pulled from boiling water at 100° C. Here the Stefan problem is simplified by assuming that the temperature of the glass decreases linearly across the glass from 100° C at the left edge to 0° C at the glass-ice interface, as indicated by the color gradient. As time progresses, heat flows from the glass into the water composite at a rate proportional to the temperature gradient of the glass. The heat entering the frozen ice melts a water layer. The thickness of the water is plotted as a function of time.
  • Contributed by: Sam Shames
  • (for Jeffrey C. Grossman; assisted by W. Craig Carter)

SNAPSHOTS

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

DETAILS

A first-order approximation is used, assuming the temperature of the glass decreases linearly with distance. This assumption reduces the problem to the following first-order differential equation
,
where is the width of the glass and is the temperature of the left edge of the glass at time . The constants depend on the heat capacity, thermal conductivity, and density of the glass and ice.
    • 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.









 
RELATED RESOURCES
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.
Computerbasedmath.org »
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 © 2014 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+