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Cooldown curves, inflection points etc.

  1. Mar 31, 2008 #1
    I am curious to know whether in a real physical situation a cooldown curve (temperature vs. time plot for a given point) can exhibit inflection. Why or why not?

    Let me point out that there is no phase change involved during the cooling process.


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    Last edited by a moderator: Apr 23, 2017
  2. jcsd
  3. Mar 31, 2008 #2
    Short answer: yes. (You have experimental evidence that says so!)

    In the simple case known as "Newtonian cooling", where a system at some elevated temperature [tex]T_{system}[/tex] is suddenly placed in surroundings at a temperature [tex]T_{surroundings}[/tex], then if we define

    [tex]\Delta T = (T_{system} - T_{surroundings})[/tex]

    we should expect

    [tex]\Delta T = \Delta T_0 e^{-\alpha t}[/tex]

    where [tex]\alpha[/tex] is some time constant. This "decaying exponential" curve does not have a point of inflexion.

    However, real cooling can be nothing like this! Heat transfer is a diffusive process, so the "temperature" of the system can vary in space as well as in time. If you are trying to observe a close approximation to Newtonian cooling, then you might want to consider some of these factors:

    • Is the 'goodness' of the thermal contact between the system and the surroundings constant?
    • Where is your temperature sensor placed? What part of the system is it observing?
    • How quickly does your sensor respond to changes in temperature? (This is less important for you, as your experiment seems to be conducted on a timescale of many hours)
    Last edited: Mar 31, 2008
  4. Mar 31, 2008 #3
    Thank you for your response, pterid.

    Here is some more information on the data:

    My data is obtained from 3D FEA of a subsea oil production system. The system consists of parts made out of metal and some surfaces are insulated. The cooling occurs due to forced convection from seawater currents. The convection coefficients are calculated and modeled as a boundary conditions. The cooldown data points are queried for within the fluid volume.

    I don't think that FEA simulates non-Newtonian cooling. Can you explain the inflection in this curve considering this information?

  5. Mar 31, 2008 #4
    I don't know the exact geometry of the situation - and I'm not an expert on convective cooling processes! - but the system certainly seems complicated enough that non-Newtonian cooling shouldn't be something to worry about. Are you concerned that it might be indicating a problem with the simulation?
  6. Apr 1, 2008 #5
    What I was trying to say was that I am sure that FEA code only computes Newtonian solutions to the heat transfer problem. I don't think there is a problem with that.

    I am concerned about the results from FEA. What I am trying to understand is how the "cooling rate" will decrease and then increase as evident by the inflection point in the curve?

    Is it due to the interplay of this location with the rest of the system? I am querying 20 locations for nodal temperature data, out of which only 1 has a cooldown curve with inflection in it.
    Last edited: Apr 1, 2008
  7. Apr 1, 2008 #6


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    I would presume it to be due to some initial fluid temperature that cools at a varying rate due to the temperature of the ocean current varying. For example, if the ambient temperature around the pipeline where initially close (or equal) to the fluid in the pipeline, and then became cooler, the plot of the Newtonian cooling would look just like what you have attached.

    Hope that helps.

  8. Apr 1, 2008 #7
    The ocean currents are simulated by applying constant forced convection coefficients on the exposed surfaces. These coefficients do not vary with time or temperature. Natural convection within the seawater is neglected. Natural convection within the trapped system fluids is simulated by 'equivalent thermal conductivity'. Also, the material properties (density, Cp, K, viscosity) do not vary with time.

    In the face of all of this, the cooling curve behavior is puzzling.

    Here is a possible explanation I thought while composing this reply, but it is not very convincing to me ...

    The fact that the convection coefficients do not vary as the system cools is causing the inflection. In reality the convection coefficients will vary to a small extent as the temperature falls and will result in a 'smooth' cooldown curve.

    Can anyone offer a more convincing explanation?

    On a side note, I had posted this in the engineering forum also, but I think an admin has removed it from that section. Can this thread be cross-linked to the engineering forum as well, so that I can have some engineers ponder over this question? Thanks.
  9. Apr 1, 2008 #8


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    A negative value of [itex]\partial T/\partial t[/itex] (which is observed on the left side of your graph) would never occur for a lone system experiencing time- and temperature-independent boundary conditions and starting from a uniform initial temperature. I'm basing this on the model transient heat equation

    [tex]c\rho\frac{\partial T}{\partial t}=k\frac{\partial^2T}{\partial x^2}-\frac{h}{d}(T-T_\infty)[/tex]

    where the two terms on the right represent conduction and convection. This equation predicts only positive values of [itex]\partial T/\partial t[/itex].

    My guess is that conduction (or convection, but I think you said there is no actual convection) within your system is bringing a non-constant stream of heat to T15 and causing the negative curvature. Imagine, for example, thermal diffusion from a localized hot spot with no heat generation. The surrounding areas get hotter before they get colder. Could a similar case apply here?

    EDIT: I meant to put [itex]\partial^2 T/\partial t^2[/itex] in the text above.
    Last edited: Apr 1, 2008
  10. Apr 1, 2008 #9
    Mapes, thank you for putting the problem in this perspective. The FEA simulates a 3D system which has different heat fluxes at different locations. So what you are suggesting may happen, but I am not sure.

    Can you please explain how to conclude that [tex]\partial[/tex][tex]\Large T[/tex][tex]/[/tex][tex]\partial[/tex][tex]\Large t[/tex] is always positive?
  11. Apr 1, 2008 #10


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    Whoops, I meant [itex]\partial^2 T/\partial t^2[/itex] in those two spots. Sorry about that.

    The magnitude of the driving force to change the temperature at a certain node is itself dependent on the difference between the node temperature and the surrounding temperature. If the surrounding temperature and all material properties are constant, then the driving force can only decrease with time as the node temperature converges to the surrounding temperature. Thus [itex]\partial^2 T/\partial t^2<0[/itex] when the node is cooling.

    I don't know if that's the best way to describe it, but I hope it helps. And you can see how this argument breaks down in a 3-D object, where the "surrounding" temperature is just another node whose temperature is not constant at all.
  12. Apr 2, 2008 #11
    So, I should not be alarmed upon observing cooldown curves with inflection?

    I wonder if I can devise a simpler test model and generate such cooldown curves. The model I am working with is quite complex, and I have to explain the results to other people in my group. A test model will be easier to explain.
  13. Apr 2, 2008 #12


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    I think you probably can. Try a 2-D domain with 9 nodes, with the center node at a higher initial temperature than the others. Now impose a convective cooling BC at the perimeter. If the center node is hot enough, the edge nodes will get hotter before they get colder. I'll bet by playing around with the thermal conductivity, the convection coefficient, and the temperatures, you can demonstrate a case where the edge nodes are decreasing in temperature with negative curvature, then with positive curvature. Good luck!
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