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Homework Help: Newton's Law of Cooling

  1. Apr 11, 2006 #1
    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. jcsd
  3. Apr 11, 2006 #2

    siddharth

    User Avatar
    Homework Helper
    Gold Member

    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
  4. Apr 11, 2006 #3
    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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