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Question- Newton's Law of Cooling

  1. Jun 10, 2013 #1
    1. The problem statement, all variables and given/known data

    A hot body placed in the surrounding of temperature θo obeys Newton's law of cooling dθ/dt = -k(θ-θo), where k is any constant. Its temperature is θ1 at t=0. t is time. The specific heat capacity of body is "s" and its mass is "m". Find the time starting from t=0 in which it will lose 90% of the maximum heat.

    2. Relevant equations


    http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html


    3. The attempt at a solution

    I solved it via integration and got the correct answer as well. As far as I know, it can be solved without integration:

    The maximum heat body can lose has to be ms(θ1o). If only 90 percent of this maximum heat is lost, we have, Heat lost = 9ms(θ1o)/10. As ms is constant we have, Temperature difference = 9(θ1o)/10

    Initial temperature of body at time t=0 : θ1
    Final temperature of body when it cools as per condition : θ1 - 9(θ1o)/10 = (θ1 + 9θo)/10

    Average temperature of body = (Initial temperature + Finial temperature)/2 = (11θ1 + 9θo)/20

    In magnitude only, we can write law as, Δθ/Δt= k(θ-θo)

    Now change in temperature Δθ = Final temperature of body when it cools as per condition - Initial temperature of body at time t=0 = 9(θ1o)/10

    Now Δt = t-0=t
    θ-θo = Average temperature of body - θo = 11(θ1o)/20

    Now putting all these values in Δθ/Δt= k(θ-θo), we get

    9/t = 11k/2 => t = 18/11k

    But the answer is t= ln(10)/k = 2.303/k.

    Where did I go wrong ?

    Please help !!

    Thanks in advance.... :smile:
     
  2. jcsd
  3. Jun 10, 2013 #2

    Mute

    User Avatar
    Homework Helper

    Your problem is that the equation Δθ/Δt= k(θ-θo) is only valid when Δθ and Δt are very small, but they're not small here. You're basically treating the problem as though the heat flow is constant over the total duration of the problem, but you know that's not true: as the temperature difference between the body and the environment decreases, the rate of heat flow also decreases. That's why you need to solve the problem by integration.
     
  4. Jun 10, 2013 #3
    But my textbook and reference books say that Newton's law of cooling in only valid when temperature difference between body and surrounding is not very large that is small. So why cannot I use the method in "Attempt at solution" here ?

    In other words the books say that Newton's law of cooling itself is true for only small temperature difference. Question says that the body obeys Newton;'s law of cooling. So temperature difference between body and surrounding is also small. Am I correct ?
     
  5. Jun 10, 2013 #4

    Mute

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    Homework Helper

    Small compared to what? You need to be clear on what "small" and "large" are in reference to when using approximation techniques.

    In this case, using your "attempt at solution" method requires that Δθ = θ(t + Δt) -θ(t) is small in the sense that Δθ = θ(t + Δt) -θ(t) ≈ θ'(t)Δt is a valid approximation, which requires that Δt is small compared to the timescale 1/k in the problem. Your method finds that the time between the initial and final condition is ~ 1.64/k, which is of course not small compared to the typical 1/k timescale. Hence, the conditions under which you used your approximation are violated, as indicated by your solution.

    Note that these conditions of "smallness" are different from the conditions under which even the differential form of Newton's law, θ'(t) = k(θ-θo), holds. There, the "small" temperature difference is likely referring to the difference between the body and the environment. Probably, the "smallness" is determined by requiring that the heat-transfer coefficient between the two bodies is approximately constant (temperature independent). This temperature-scale could be much larger than the Δθ = θ(t + Δt) -θ(t), and so has no bearing on whether you can approximate Δθ = θ(t + Δt) -θ(t) ≈ θ'(t)Δt (except to say that the whole law is invalid if the heat-transfer coefficient is not constant).
     
  6. Jun 11, 2013 #5
    Great Explanation Mute !! I get the concept now !! Thanks a lot !!! :)
    :smile:
     
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