1. Thermal conductivity of an Aluminum alloy
Neal Doolin

2. Overview
Thermal conductivity (Λ) is the rate at which energy (heat) is transferred through a medium.
Λ= 𝐻 𝐴 Δ𝑇 Δ𝑋
The objective of the experiment is to measure the thermal conductivity of an aluminum alloy rod.

3. Overview
This experiment involves a rod with a heater at one end while the other end is held at a constant temperature and ideally there is no heat loss from the sides.

4. Device

5. Device

6. Device
An aluminum rod with a diameter of 1.60cm, attached to heat sink: a large block of the same aluminum alloy via a screw.
A heater (transistor) is attached at one end.
Along the rod, thermisters (thermal resistors) have been placed at x = {1, 5, 9, 13, 32, 36, 40}cm
Where X is the distance from the thermister to the heat sink

7. Ideally…
Once heat in = heat out, the rod is in a thermal steady state (ie there is no temperature change across the rod). We can then describe the temperature at any point (x) with
𝑇 𝑥 = 𝐻𝑥 Λ𝐴 + 𝑇 0
Where H = the rate that heat is introduced, A is the cross sectional area of the rod, and T₀ is the temperature of the heat sink.

8. Calculation Recall 𝑇 𝑥 = 𝐻𝑥 Λ𝐴 + 𝑇 0 And Λ= 𝐻 𝐴 Δ𝑇 Δ𝑋
So to find thermal conductivity, we only need to find the slope of temperature vs position in steady state, the heat applied, and the cross-sectional area of the rod.

9. Experimental Setup
To keep the heat sink at a constant temperature, it was submerged in ice water (0 degrees).
To prevent heat loss, the rod was wrapped in fiberglass insulation. This encompassed the rod from the heater to the end, but not the heat sink.
The fiberglass was clipped in multiple places to prevent leakage.

10. Experimental Setup

11. Thermister Calibration
Thermal resistors’ resistance varies with temperature. Thus, they need to be calibrated while the rod is in steady-state so that different voltage readings across the thermisters result in reading the same temperature.

12. Thermister Calibration
First, the apparatus was switched on. Thermisters warm slightly when first used, so the rod was left to cool to room temperature.
An hour’s worth of thermister voltages were recorded, averaged, and defined as the room temperature as read by a mercury thermometer.

13. Measurement Example
Raw data. X-value is related directly to temperature (Red is the farthest from the heat sink)
The steady state condition of the raw data. The standard deviation of each is less than 0.04 degrees.

14. Data Collection
Four trials were done, identified steady state conditions for each.
Averaged the temperatures for each position inside the steady state condition.
Plotted the mean temperature (kelvin) against the distance from the heat sink

15. Data Collection

16. Heat Absorption
Power dissipated ( 𝑃 𝑑 ) by the transistor had to be indirectly measured.
Could only measure 𝑉 𝑐𝑏
𝑉 𝑐𝑒 = 𝑉 𝑐𝑏 + 0.5
𝑃 𝑑 = 𝐼 𝑐 * 𝑉 𝑐𝑒
𝑉 𝑐𝑏 was measured to be 13.5V, 𝐼 𝑐 to be 329mA. 𝑃 𝑑 was then 4.61W.

17. Results Trial Slope Thermal Conductivity (#) 𝐾 𝑚 𝑊 𝑚∗𝑘
(#) 𝐾 𝑚 𝑊 𝑚∗𝑘
± ± 10
± ± 10
± ± 9
± ± 9

18. 𝜎Λ components 𝜎𝑉 = 5% of 14V 𝜎I = 1% of .329A 𝜎H = 3% = 0.14 W
𝜎A ≈ 0% (5 measurements of the diameter with a caliper)
𝜎Λ propagated using these values along with the slope values.
Some simple statistical analysis was applied (number of trials, mean values, error in means, etc)

19. Final Result
Λ = 323 ± 5 𝑊 𝑚∗𝐾

20. Uncertainty Discussion
How did I manage to pull 14 Volts from a wall outlet (12V)?
Larger error than estimated in 𝜎V
Heat sink was not fully submerged in ice water. This area was not insulated. (Notice the graphs had intercepts around 10 degrees C.)
Irrelevant as long as the end held a constant temperature, but there was not a way to check for this

21. Acknowledgements Dr. Han Jessica Stockham Joseph Randall
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