Find Thermal Diffusivity from Kelvin Functions

[gothloli]

Homework Statement

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.

Homework Equations

Bessel function: J(x) = 1 - $\frac{x^{2}}{4}$ + $\frac{x^{4}}{64}$ - $\frac{x^{6}}{2304}$

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

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($\Phi$) = Ber(x)
and sin($\Phi$) = 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.

 

[Chestermiller]

What is the highest value of phi that you get from your data?

Chet

 

[gothloli]

Hi the largest value of phi I get is 0.912, I'm not sure if this is reasonable though.

 

[Chestermiller]

Hi the largest value of phi I get is 0.912, I'm not sure if this is reasonable though.

That's radians, correct?

 

[gothloli]

yes in radians

 

[Chestermiller]

Here's what I would do. I would first plot a graph of the function x vs $\phi$, independent of the experimental data. I would use this as a "master plot." If the data points are $\phi_i$ vs T_i, I would use the master plot to determine an x_i value for every $\phi_i$ value. I would then make another plot of T_i versus 1/x_i. 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

 

[gothloli]

Here's what I would do. I would first plot a graph of the function x vs $\phi$, independent of the experimental data. I would use this as a "master plot." If the data points are $\phi_i$ vs T_i, I would use the master plot to determine an x_i value for every $\phi_i$ value. I would then make another plot of T_i versus 1/x_i. 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

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

 

[Chestermiller]

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

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.

 

[gothloli]

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√($\frac{2\pi}{TD}$). Where r is the radius, maybe graphing [itex]\frac{1}{x²}[/itex] with T will give the diffusivity? Thanks again for your time.

 

[Chestermiller]

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.

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.

Also why would graphing the x with period give the diffusivity. I know x = r√($\frac{2\pi}{TD}$). Where r is the radius, maybe graphing [itex]\frac{1}{x²}[/itex] with T will give the diffusivity? Thanks again for your time.

Yes. This is what I had in mind.

Chet