Homework Help: Newton's Law of Cooling

1. Apr 11, 2006

Chubbly

Hello,

I am an undergrad student studying Physics.

We are currently studying thermodynamics. The lecturer has set us a bunch of problems relating to Thermodynamics. While I can do most of them, I am getting completely thrown by one question:

An object with a surface area of 0.2 m², an emissivity of 0.8 and a heat capacity of 1320.0 J/K has a temperature of 59.0 °C. It is then place in an environment at 38.0 °C and it eventually cools until it is in thermal equilibrium with the new environment.

Stefan constant: σ = 5.68 x 10-8 W m-2 K-4
Coefficient of convective heat transfer = 6.0 W m-2 K-1

Approximately how long does the object take to cool to a temperature 14.0 K below its initial temperature?

The lecturer has said that Newton's Law of Cooling can be assumed to be valid. So I dug up the approximation of Newton's Law of Cooling which is, as I understand it:

dT/dt=(A/C)(q+εσ(Tave)^3) ∆T

or, in words

rate of change in temperature with respect to time = (surface area/heat capacity)(Coefficient of convective heat transfer + emissivity*Stefan's constant*average temperature cubed)*change in temperature.

The working
I'm sorry it's so messy, its hard to put a complicated formula in word characters.

dt=dT/((A/C)(q+εσ(Tave)^3) ∆T)

dT=59+273.16
A=0.2
C=1320
q=6
ε=.8
σ=5.68 x 10-8
(Tave)^3=((59+(59-14))/2+273.16)^3=34378849.96
∆T=14

It seems like a basic number plugger but every time I put the values in, it's wrong.

I tried unit cancellation to check, I tried all manner of different forms of the values but nothing works.

Help me, please!

(the answer is given, and it's 220s)

2. Apr 11, 2006

siddharth

These sites covers it pretty well
http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html" [Broken]
http://www.math.wpi.edu/Course_Materials/MA1022A96/lab2/node5.html" [Broken]

Last edited by a moderator: May 2, 2017
3. Apr 11, 2006

Chubbly

Thanks for the help but those notes use information I don't have and doesn't use information that I think should be relevant: emissivity, surface area etc.

If someone could confirm that dT is what I am saying it is, that could be helpful.

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