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Find Thermal Diffusivity from Kelvin Functions

  1. Feb 24, 2014 #1
    1. The problem statement, all variables and given/known data
    I performed an experiment where we have to find the thermal diffusivity of rubber. This is done using the phase difference of the axial and external temperatures of the rubber. The teacher's guide says to use the Kelvin functions, plot them against x. Where x = r(√(2pi/TD)) Where T is the period of the immersion, and D is the diffusivity. From the plot find the diffusivity, I'm not sure how I'm to do this from the plot.


    2. Relevant equations
    Bessel function: J(x) = 1 - [itex]\frac{x^{2}}{4}[/itex] + [itex]\frac{x^{4}}{64}[/itex] - [itex]\frac{x^{6}}{2304}[/itex]

    Ber(x) = 1 - [itex]\frac{x^{4}}{64}[/itex]
    Bei(x) = [itex]\frac{x^{2}}{4}[/itex] - [itex]\frac{x^{6}}{2304}[/itex]
    tan([itex]\Phi[/itex]) = bei(x)/ber(x)

    3. The attempt at a solution
    The data I have is that of the phase difference between the axial and the external temperature graphs. From this I calcualate the x values by setting cos([itex]\Phi[/itex]) = Ber(x)
    and sin([itex]\Phi[/itex]) = Bei(x). Then I plot those in terms of x. But I'm not sure if this is the way to go or how I can get diffusivity from the graph. Thanks all.
     
  2. jcsd
  3. Feb 24, 2014 #2
    What is the highest value of phi that you get from your data?

    Chet
     
  4. Feb 24, 2014 #3
    Hi the largest value of phi I get is 0.912, I'm not sure if this is reasonable though.
     
  5. Feb 24, 2014 #4
    That's radians, correct?
     
  6. Feb 24, 2014 #5
    yes in radians
     
  7. Feb 25, 2014 #6
    Here's what I would do. I would first plot a graph of the function x vs [itex]\phi[/itex], independent of the experimental data. I would use this as a "master plot." If the data points are [itex]\phi_i[/itex] vs Ti, I would use the master plot to determine an xi value for every [itex]\phi_i[/itex] value. I would then make another plot of Ti versus 1/xi. This should be a straight line with a slope that can be used to determine the best value of D for that set of data.

    Chet
     
  8. Feb 26, 2014 #7
    I'm not sure I understand as, I get phi from the experimental data. The data I measured was the temperature change of the rubber in ice, and boiling water over time. From the Temp vs time graph I measured the phase shift between the internal surface (the rubber) and the external surface temperature. So my main question is how I would get the values of x, since I only have the data of the phase shift, and the kelvin functions. I'm very confused. What I think your saying is to graph the phase shifts against the temperature values, but I'm not sure how this will give thermal diffusivity, could you explain why this could? Thanks
     
  9. Feb 26, 2014 #8
    You get the values of x from your theoretical equation involving ber and bei. You make a graph of phase shift vs x, as given by this equation. For each data point, there is a unique combination of phase shift, x, and period T. For each data point, you determine x from the phase shift. You then plot the x's as a function of the T's on a graph. The result should be a straight line, and you can get the diffusivity from the slope.
     
  10. Feb 26, 2014 #9
    I solve for x using the equation below?
    tan(Φ ) = bei(x)/ber(x)
    Isolating x is difficult from this. I used wolfram but it gives unusual stuff (things I don't know). I could use python I guess.

    Also why would graphing the x with period give the diffusivity. I know x = r√([itex]\frac{2\pi}{TD}[/itex]). Where r is the radius, maybe graphing [itex]\frac{1}{x2}[/itex] with T will give the diffusivity? Thanks again for your time.
     
    Last edited: Feb 26, 2014
  11. Feb 26, 2014 #10
    Plot a master graph of tan(Φ ) = bei(x)/ber(x) vs x. Rather than specifying ø and determining x, specify x and determine ø for plotting on the graph. Then, for any value of ø, you can pluck off a value of x from the graph.

    Yes. This is what I had in mind.

    Chet
     
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