Temperature profiles along thin aluminium rods for

constant and sinusoidally varying base temperatures

Goh Wei Zhong

(Lab partner: Yuxin Yang)

9th May 2014

Abstract

The temperature profiles along long, thin aluminium rods were measured when its base

was maintained at a constant temperature of 65C and when its base temperature was sinus-

oidally varied around room temperature. For a constant temperature at the base, we found

an exponential decay constant of 5.60± 0.42 m−1 for its reduced temperature for a 1/2-inch

diameter aluminium rod, in agreement with theory. In addition, we found that foam insula-

tion decreased the heat dissipation along the lateral surface of the rod by 70%. For sinusoidal

temperature variations at the base, we found exponential decays in the amplitude, and linear

decreases in the phase, of temperature variation as we travel along the rod away from the

base, consistent with theory. From the exponential decay constant of the amplitude and the

gradient of the phase variations, we calculated the thermal conductivity along the rod to be

140±22 W m−1 K−1.

1

Electronics, which are becoming ever more prevalent, need to be cooled efficiently to oper-

ate optimally. In this lab, we measure the ability of an aluminium rod to conduct and dissipate

heat along its length.

1 Theory

Consider a thin, long cylindrical conducting rod, along which we orient the x-axis. Its base,

at x = 0, is maintained at a constant temperature T0, which is above the room temperature,

T∞. Heat flows along the rod and is dissipated at its lateral surface. The steady-state reduced

temperature at a point Θ(x) is given by

Θ(x) =T (x)−T∞

T0−T∞

= exp

−√

4hkd

x

= e−qlx,

where

• T (x) is the temperature at x,

• h is the average heat transfer coefficient, the amount of heat dissipated through a unit area

of the lateral surface of the rod, per unit time, per unit temperature difference between

the rod and the surroundings (aluminium to air: 25.0 W m−2 K−1),

• k is the thermal conductivity, the amount of heat conducted through a unit area of the

rod cross-section, per unit time, per unit temperature gradient along the rod (aluminium:

237 W m−1 K−1),

• d is the diameter of the cylindrical rod, and

• ql =

√4hkd is the decay constant.

Consider the same rod, whose base, at x = 0, is maintained at a time-varying temperature

T (0, t) = T∞ +∆T cosωt,

2

where ∆T is the amplitude of the sinusoidal oscillation in temperature and ω is the angular

frequency of oscillation. Heat waves flow along the rod, decaying in amplitude as heat is

dissipated at its lateral surface. The temperature at a point T (x, t) is given by

T (x, t) = T∞ +∆Te−qx cos(ωt−q′x+ ε

),

where

• q =

√ν+√

ν2+ω2

2κis the exponential decay constant of the amplitudes of temperature

variation along the rod,

• q′=√−ν+√

ν2+ω2

2κis the magnitude of the gradient of the phase shifts of the temperature

variation along the rod,

• ν = 4hcρd , κ = k

cρare constants that depend on the rod material and geometry,

• c is the specific heat capacity of the rod (aluminium: 900 J kg−1 K−1),

• ρ is the mass density of the rod (aluminium: 2700 kg m−3), and

• ε is an overall phase shift.

2 Setup

An aluminium rod was placed in a vertical slot in contact with a thermoelectric device, which

heats and cools the base of the rod. An EPCOS B57861S0202F040 thermistor was placed in

contact with the thermoelectric device, which measures the temperature at the device. Rings

were placed at various positions of the rod with holes for thermistors. EPCOS thermistors

were attached into the rings with thermal paste to facilitate thermal conduction and accurate

measurement of the rod temperature.

A Burr-Brown REF200 current source/sink supplied a constant current of 400 mA to the

thermistors, which were connected in series, and the thermistors were wired to a National

Instruments USB-6211 Data Acquisition Device. We wrote a LabVIEW program to read the

potential differences across each thermistor, from which we calculated the temperature at each

3

Position above first thermistor, x/cm 0.0 1.5 3.6 5.7 7.7 10.8 15.9 22.3Steady-state temperature, T/C 51.8 47.0 44.8 41.4 39.9 38.1 34.7 31.5

Steady-state reduced temperature, Θ 1.00 0.83 0.76 0.64 0.59 0.52 0.41 0.30

Table 1: Steady-state temperatures and corresponding reduced temperatures along the rod. Thereduced temperature is given by Θ(x) = T (x)−23C

51.8C−23C .

thermistor. We found that using the manufacturer’s value of 2 kΩ at 25C gave good results.

The LabVIEW program also uses a PI controller from the PID and Fuzzy Logic Toolkit to

control the temperature at the thermoelectric cooler. We used a proportional gain of 1 and an

integral time of 0.5 min.

3 Results

3.1 Constant temperature at rod base

3.1.1 1/2-inch rod, uninsulated

The thermoelectric device, to which the base of the rod was attached, was kept at 65C and

the 1/2-inch aluminium rod was allowed to heat to steady state. The steady-state temperat-

ures and corresponding reduced temperatures along the rod are tabulated in Table 1. In the

formula for the reduced temperature, Θ(x) = T (x)−T∞

T0−T∞, T∞ = 23C and T0 = 51.8C. The re-

duced temperatures at various positions along the rod are shown in Fig 1, and fit well to a

decaying exponential Θ = (0.939±0.026)e−(5.60±0.42)x. This decay constant corresponds to

a decay length of 0.178±0.013 m. The coefficient on the exponent is close to the theoretical

value of 1. The decay constant agrees with the theoretical value of√

4hkd = 5.73 m−1, where

d = 12.9 mm for the 1/2-inch rod is the average of five measurements taken close to the base

of the rod using vernier callipers.

3.1.2 1/2-inch rod, insulated

The experiment was repeated with a 1/2-inch rod with foam insulation wrapped around its

lateral surface. The reduced temperatures at various positions along the rod are shown in Fig 2,

and fit well to a decaying exponential Θ = (0.944±0.019)e−(2.99±0.18)x. This decay constant

4

Q = 0.939 e-5.60 x

0.00 0.05 0.10 0.15 0.20 0.25x m

0.2

0.4

0.6

0.8

1.0Q

Reduced temperatures alonga 12-inch uninsulated rod

Figure 1: Reduced temperature at each position along a 1/2-inch uninsulated rod

Q = 0.944 e-2.99 x

0.0 0.1 0.2 0.3 0.4x m

0.2

0.4

0.6

0.8

1.0Q

Reduced temperatures alonga 12-inch insulated rod

Figure 2: Reduced temperature at each position along a 1/2-inch insulated rod

5

Q = 0.946 e-23.7 x

0.00 0.05 0.10 0.15 0.20 0.25x m

0.2

0.4

0.6

0.8

1.0Q

Reduced temperatures alonga 18-inch uninsulated rod

Figure 3: Reduced temperature at each position along a 1/8-inch uninsulated rod

corresponds to a decay length of 0.334± 0.020 m—almost twice that of the uninsulated rod.

The thermal conductivity within the rod, k, stays the same, but the heat transfer coefficient out

of the lateral surface, h, decreases due to the insulation. Using the value of k for aluminium

and assuming that the diameter is the same as that of the uninsulated rod, we calculate from

the decay constant that with foam insulation,

h = 6.81 W m−2 K−1.

3.1.3 1/8-inch rod, uninsulated

The experiment was repeated for an 1/8-inch aluminium rod. The reduced temperatures at

various positions along the rod are shown in Fig 3, and fits reasonably to a decaying expo-

nential Θ = (0.946±0.039)e−(23.7±2.1)x. This decay constant corresponds to a decay length

of 0.0422± 0.0037 m. It is likely that the slightly higher fractional uncertainty in the decay

parameters, 8.7%, as compared to 7.5% for the 1/2-inch rod, reflects the greater fractional

uncertainty in the position measurements.

Our decay constant is half of the theoretical value of√

4hkd = 11.5 m−1, where d = 3.16 mm

6

for the 1/8-inch rod is the average of five measurements taken close to the base of the rod using

vernier callipers. It is possible that due to the curvature of the lateral surface of the 1/8-inch

rod, it was more difficult for the thermistors to be in good thermal contact with the surface of

the 1/8-inch rod as compared to the 1/2-inch rod, causing the temperature to be underestimated.

In addition, the ratio of the size of the rings to which the thermistors were attached, to the size

of the rod itself, was higher for the 1/8-inch rod as compared to the 1/2-inch rod. It is possible

that heat losses to the rings affected the 1/8-inch rod disproportionately.

3.2 Sinusoidally varying temperature at rod base

The setpoint temperature of the thermoelectric device was sinusoidally varied 10C around

the room temperature at periods of 100 s, 150 s, 200 s and 250 s in separate trials, and left

until steady variations were observed at each position along the 1/2-inch aluminium rod. The

variation with time of the temperature in the case of 150 s period is shown in Fig 4. The

dark blue curve corresponds to the measured position closest to the base, and the orange curve

corresponds to the measured position furthest from the base. As we move away from the

base, the sinusoidal variation in temperature decreases in amplitude and increases in phase lag.

To study this system quantitatively, we fit four to five periods of the sinusoidal variation in

temperature to a cosine curve,

T (x, t,ω) = T∞ +B(x,ω)cos(ωt + εi (x,ω)) ,

with fit parameters T∞, B(x,ω), ω and εi (x,ω). Although we took data every 0.1 s in Lab-

VIEW, we only used data points at 5 s intervals for the fit procedure to succeed in Mathematica.

An example of a fit is shown in Fig 5. The fits were very good across the board. No further use

was made of the parameters T∞ and ω , other than to check that they were constant throughout

the rod and corresponded to the actual room temperatures and angular frequencies of variation.

For each period, a plot of the amplitude of variation B(x,ω) against position along the rod

is shown in Fig 6. The position x = 0 now corresponds to the base of the rod, so that the first

7

23

24

25

26

27

28

29

30

600 700 800 900 1000 1100

T /

°C

t / s

Period 150 s

2

3

4

5

6

7

Figure 4: Variation with time of temperature for 150 s period when the base temperature issinusoidally varied.

8

700 800 900 1000t s

25

26

27

28

29

T °C

Period 100 s, Position 0.022 m

Figure 5: Temperature variation at the thermistor closest to the base for a period of 100 s, andthe fit to a cosine curve.

thermistor on the rod is at a position x = 2.2 cm. B(x,ω) fit well to

B(x,ω) = 10e−q(ω)x+δ ,

with fit parameters q(ω) ,δ . Notice that the larger the period, the more sustained the heating

and cooling would be, leading to the larger amplitude of variation at each position.

A plot of the phase shift εi (x,ω) against position along the rod is shown in Fig 7. They

fit well to a linearly decreasing relation with the magnitude of the gradient, q′ (ω), of the

fits annotated on the graphs. The larger the period, the less the relative phase shift in the

temperature variations along the rod, in agreement with the fact that in the limit of a large

period relative to the time taken for the heat wave to propagate along the rod, the temperature

variations would be uniform along the rod.

The best fit values of q and q′, along with values expected from theory, are plotted against

ω in Figs 8 and 9. In accordance with theory, q and q′ increase with ω , but the experimental

values of q′, and especially q, both exceed theoretical predictions from published values of

aluminium.

9

q = 27.3 -1m

0.00 0.02 0.04 0.06 0.08 0.10 0.12x m

1

2

3

4

5B

Period 100 s

(a)

q = 22.9 -1m

0.00 0.02 0.04 0.06 0.08 0.10 0.12x m

1

2

3

4

5B

Period 150 s

(b)

q = 19.8 -1m

0.00 0.02 0.04 0.06 0.08 0.10 0.12x m

1

2

3

4

5B

Period 200 s

(c)

q = 17.2 -1m

0.00 0.02 0.04 0.06 0.08 0.10 0.12x m

1

2

3

4

5B

Period 250 s

(d)

Figure 6: Amplitudes of variation B(x,ω) against position along rod for various periods

10

q ' = 17.6

0.02 0.04 0.06 0.08 0.10 0.12x m

-1

1

2

3

4Εi

Period 100 s

(a)

q ' = 16.1

0.02 0.04 0.06 0.08 0.10 0.12x m

-1

1

2

3

4Εi

Period 150 s

(b)

q ' = 14.7

0.02 0.04 0.06 0.08 0.10 0.12x m

-1

1

2

3

4Εi

Period 200 s

(c)

q ' = 13.6

0.02 0.04 0.06 0.08 0.10 0.12x m

-1

1

2

3

4Εi

Period 250 s

(d)

Figure 7: Amplitudes of variation B(x,ω) against position along rod for various periods

11

0.02 0.03 0.04 0.05 0.06 0.07Ω s-1

15

20

25

q m-1

Decay constant for amplitudes of temp variation vs angular frequency

Theory

Figure 8: Best fit values of q against ω , along with values expected from theory. Error bars arestandard errors from Mathematica’s NonlinearModelFit.

0.02 0.03 0.04 0.05 0.06 0.07Ω s-110

12

14

16

18

20

q¢ m-1

Gradient of phase shifts of temp variation vs angular frequency

Theory

Figure 9: Best fit values of q′ against ω , along with values expected from theory. Error barsare standard errors from Mathematica’s LinearModelFit.

12

0.02 0.03 0.04 0.05 0.06 0.07Ω s-1100

120

140

160

180

200

220

240

k W m-1 K-1

Thermal conductivity vs angular frequency

Theory

Figure 10: Experimental values of thermal conductivity k against ω , along with the valueexpected from theory, 237 W m−1 K−1.

Using qq′ = ω

2κand κ = k

cρ, we use the product of q and q′ to calculate κ and the exper-

imental thermal conductivity k, as shown in Fig 10. Within uncertainties, the experimental

values of k are consistent with each other with a mean of

k = 140±22 W m−1K−1,

around half the value expected from theory. The discrepancy warrants further investigation,

but one possibility is that the rod is made up of an aluminium alloy with different h and k

values, but a similar value for hk , than published for pure aluminium. This might have allowed

experiment to agree with theory in the case of a constant high temperature but not in that of a

sinusoidally varying temperature.

4 Conclusion

In this lab, we affirmed several quantitative relations, such as the exponential decay in reduced

temperature when the base of a rod is kept at a high temperature. For sinusoidal temperature

variations at the base, we have also found exponential decays in the amplitude, and linear

13

decreases in the phase, of temperature variation as we travel along the rod away from the base,

consistent with theory.

For the 1/8-inch rod with the base at a constant high temperature, we might have found

better agreement with theory in the value of the decay constant if depressions were made

along the rod so that there is better thermal contact between the thermistor and the rod. In

addition, performing the experiment for sinusoidally varying base temperatures on a day in

spring, where there is no need for indoor heating or cooling, would help reduce the effects of

indoor air currents and temperature gradients, which taint the experimentally calculated values

of material constants.

14