Newton's Law of Cooling and other Models

[leehufford]

Hello,

In my Differential Equations class we are learning about modelling with first order differential equations. We learned that Newton's Law of Cooling breaks down when the temperature of the object is approaching the temperature of the room its in. You eventually get to a point where you have

0 = e^x

or some variation of that, where of course there is no solution. This leads me to a few questions.

1.) Are there mathematical models that don't break down, i.e maybe they aren't perfect but they are still a good approximation and

2.) Have we come up with a newer model of cooling that does not break down at the point were the object temperature reaches the ambient temperature?

Thanks for your time,

-Lee

 

[UltrafastPED]

See http://en.wikipedia.org/wiki/Heat_transfer
and http://en.wikipedia.org/wiki/Heat_equation

Heat transfer is often taught as a senior level course in mechanical engineering.

 

[leehufford]

See http://en.wikipedia.org/wiki/Heat_transfer
and http://en.wikipedia.org/wiki/Heat_equation

Heat transfer is often taught as a senior level course in mechanical engineering.

I was able to understand some of that material. It actually got me more excited about my differential equations course to know that I am working toward such cool stuff as the heat equation.

It seemed that the convergence toward equilibrium is still a problem in the heat equation but is dealt with by one-parameter semigroups theory? This is way over my head but is that a correct assessment? Thanks for the relpy,

-Lee

 

[UltrafastPED]

They usually don't go into applications in the beginning differential equations class because it takes too much time to properly motivate each problem ... thus they keep it to the mathematics, and just teach the methods.

You will start using differential equations in your upper level courses, especially physics and engineering.

If you are a math major they may offer a course on the theory of ordinary differential equations; you may enjoy this; it would be a senior level or beginning graduate level course.

 

[blue_raver22]

Hello Lee,

That's a great question and it's great to hear that you are learning about modelling with first order differential equations in your class. You are correct in noting that Newton's Law of Cooling may not always be applicable in situations where the object's temperature approaches the ambient temperature.

1.) To answer your first question, yes, there are mathematical models that do not break down in these situations. One example is the Stefan-Boltzmann law, which describes the rate of heat loss from a black body in terms of its temperature and the surrounding temperature. This model takes into account the emissivity of the object and the radiative heat transfer between the object and its surroundings. While this model may not be perfect, it is a good approximation in many situations and does not break down at the point where the object's temperature approaches the ambient temperature.

2.) As for your second question, there have been newer models developed to address the limitations of Newton's Law of Cooling. One example is the non-linear cooling model, which takes into account the non-linear relationship between the temperature difference and the rate of heat transfer. This model has been found to be more accurate in situations where the temperature difference is small.

It's important to remember that mathematical models are simplifications of real-world phenomena and may not always be applicable in all situations. As scientists, we are constantly working to improve and refine these models to better describe the natural world. I hope this helps to answer your questions and good luck with your studies!

Best,