A thermal conductivity measurement experiment investigates how easily heat is transferred through a material. Metals, glass, resins, and insulating materials transmit different amounts of heat even when subjected to the same temperature difference.
Materials with high thermal conductivity transfer heat more rapidly, while materials with low thermal conductivity resist heat flow. For example, metals such as copper and aluminum tend to have high thermal conductivity, whereas foamed plastics and materials containing large amounts of air tend to have low thermal conductivity.
A representative measurement method is the steady-state method, in which one end of a rod-shaped sample is heated and the thermal conductivity is determined from the temperature distribution and heat flow rate after steady state is reached. Ideally, the heat flow through the sample is assumed to be constant and to occur only along the length of the sample.
This article assumes an experiment in which a temperature difference is established between the ends of a rod-shaped or plate-shaped sample and the thermal conductivity is determined from Fourier’s law. It covers the introduction, precautions, items to record, sample experimental values, calculations, sources of error, example results, and discussion examples.
Note:
This article is educational reference material for discussing results obtained in physics experiments at schools, universities, and similar institutions. The conditions and measured values shown here are illustrative examples and are not actual experimental results.
When handling heaters, heated samples, hot water, or steam, take precautions against burns. Do not touch energized heater terminals or exposed wiring, and always turn off the power before changing connections. Because the sample may remain hot, allow it to cool sufficiently before touching it after measurement. Follow the laboratory manual and the instructions of your instructor or teaching assistant for actual procedures.
What Is Thermal Conductivity?
Thermal conductivity is a material property that indicates how readily heat is transferred through a material when a temperature difference exists within it. It is generally represented by the symbol k or λ.
Its SI unit is:
W/(m·K)
The greater the thermal conductivity, the more heat is transferred under the same cross-sectional area, length, and temperature difference.
Fourier’s Law
For one-dimensional steady-state heat conduction, the amount of heat transferred per unit time, or heat flow rate Q̇, is expressed by Fourier’s law as follows.
Q̇ = −kA(dT/dx)
Here, k is the thermal conductivity, A is the cross-sectional area through which heat flows, and dT/dx is the temperature gradient. The negative sign indicates that heat flows from the higher-temperature side to the lower-temperature side.
If the temperature is assumed to vary linearly over a sample length L from the hot-side temperature Th to the cold-side temperature Tc, the magnitude of the heat flow rate is:
Q̇ = kA(Th − Tc)/L
Therefore, the thermal conductivity can be determined from:
k = Q̇L/{A(Th − Tc)}
When Using an Electric Heater
If all electrical power supplied to the heater is treated as the heat flow through the sample, the power can be calculated from voltage V and current I.
P = VI
If heat loss is neglected, Q̇ = P, and the thermal conductivity becomes:
k = VIL/{A(Th − Tc)}
In practice, however, some heat escapes from the heater to the air, supports, wiring, and other surroundings, so not all input power necessarily passes through the sample.
Main Items to Record in the Results
- Sample material
- Sample shape
- Sample length
- Sample diameter or width and thickness
- Cross-sectional area perpendicular to heat flow
- Heater voltage
- Heater current
- Input power
- Hot-side temperature
- Cold-side temperature
- Position and temperature of each measurement point
- Time required to reach steady state
- Temperature difference
- Temperature gradient
- Heat flow rate
- Thermal conductivity
- Reference or literature value
- Relative error
- Room temperature
- Mean and variation of repeated measurements
Example Experimental Conditions
| Item | Example Condition |
|---|---|
| Sample | Aluminum rod |
| Sample length L | 0.200 m |
| Sample diameter d | 0.0200 m |
| Cross-sectional area A | 3.14 × 10−4 m2 |
| Heater voltage V | 12.0 V |
| Heater current I | 0.500 A |
| Input power P | 6.00 W |
| Hot-side temperature Th | 78.0 ℃ |
| Cold-side temperature Tc | 61.0 ℃ |
| Room temperature | 23.0 ℃ |
| Time to steady state | Approx. 25 min |
Example Experimental Values
Temperature Distribution Along an Aluminum Rod
| Distance x from hot end (m) | Temperature T (℃) |
|---|---|
| 0.000 | 78.0 |
| 0.040 | 74.6 |
| 0.080 | 71.2 |
| 0.120 | 67.8 |
| 0.160 | 64.4 |
| 0.200 | 61.0 |
When the Input Power Is Varied
| Input power P (W) | Hot-side temperature (℃) | Cold-side temperature (℃) | Temperature difference ΔT (K) | Apparent thermal conductivity (W/(m·K)) |
|---|---|---|---|---|
| 3.00 | 52.1 | 43.5 | 8.6 | 222 |
| 4.00 | 61.0 | 49.6 | 11.4 | 223 |
| 5.00 | 69.6 | 55.4 | 14.2 | 224 |
| 6.00 | 78.0 | 61.0 | 17.0 | 225 |
Reference Thermal Conductivities of Different Materials
| Material | Reference thermal conductivity (W/(m·K)) |
|---|---|
| Copper | Approx. 400 |
| Aluminum | Approx. 230 |
| Iron | Approx. 80 |
| Stainless steel | Approx. 15 |
| Glass | Approx. 1 |
| Water | Approx. 0.6 |
| Wood | Approx. 0.1–0.2 |
| Air | Approx. 0.026 |
Temperature Change Until Steady State Is Reached
| Time after heating begins (min) | Hot-side temperature (℃) | Cold-side temperature (℃) |
|---|---|---|
| 0 | 23.0 | 23.0 |
| 5 | 48.0 | 28.5 |
| 10 | 63.5 | 40.2 |
| 15 | 72.4 | 51.0 |
| 20 | 76.8 | 58.0 |
| 25 | 78.0 | 61.0 |
| 30 | 78.1 | 61.1 |
Calculation Method
Cross-Sectional Area of a Cylindrical Sample
For a cylindrical sample with a diameter of 0.0200 m, the radius is 0.0100 m.
A = πr2
= π × (0.0100)2
= 3.14 × 10−4 m2
Heater Input Power
For a voltage of 12.0 V and a current of 0.500 A:
P = VI
= 12.0 × 0.500
= 6.00 W
Temperature Gradient
For a hot-side temperature of 78.0 ℃, a cold-side temperature of 61.0 ℃, and a measurement interval of 0.200 m, the magnitude of the temperature gradient is:
|dT/dx| ≈ (78.0 − 61.0)/0.200
= 85.0 K/m
Thermal Conductivity
Assuming that all input power passes through the sample:
k = PL/(AΔT)
= 6.00 × 0.200/{(3.14 × 10−4) × 17.0}
= 225 W/(m·K)
Heat Flux
The heat flow rate per unit area is called the heat flux q.
q = Q̇/A
= 6.00/(3.14 × 10−4)
= 1.91 × 104 W/m2
Relative Error
For an experimental value of 225 W/(m·K) and a reference value of 230 W/(m·K):
Relative error (%) = |experimental value − reference value|/reference value × 100
= |225 − 230|/230 × 100
= 2.17%
Thermal Resistance
The resistance of a sample to heat flow can also be expressed as thermal resistance Rth.
Rth = L/(kA)
Under the present conditions:
Rth = 0.200/{225 × 3.14 × 10−4}
= 2.83 K/W
The temperature difference can be expressed as the product of heat flow rate and thermal resistance.
ΔT = Q̇Rth
Main Sources of Error
| Source of error | Effect on measurement | Typical tendency in results | Improvement |
|---|---|---|---|
| Heat loss from the sides | Part of the input heat escapes to the air | Heat flow through the sample is overestimated, tending to make thermal conductivity too large | Insulate the sides of the sample |
| Radiative heat loss | Heat is emitted from hot surfaces to the surroundings | Error increases at higher temperatures | Use a radiation shield |
| Thermal contact resistance | Temperature differences occur at interfaces between heater, sample, and cooling section | The internal sample temperature difference may be overestimated, tending to make thermal conductivity too small | Smooth contact surfaces and use thermal grease |
| Temperature sensor position error | Temperature is measured at a position different from the assumed position | The temperature gradient is incorrect | Measure sensor positions accurately |
| Poor temperature sensor contact | The sensor does not accurately reflect sample temperature | Measurements become unstable | Secure the sensor firmly against the sample |
| Steady state not reached | Heat is still accumulating within the sample | Calculated thermal conductivity changes with time | Wait until temperatures at all points become constant |
| Input power fluctuation | Heat flow rate is not constant | Temperature difference fluctuates | Use a regulated power supply |
| Assuming 100% heater efficiency | Heat entering the sample is overestimated | Thermal conductivity is overestimated | Correct for losses or perform comparative measurements |
| Error in sample cross-sectional area | Heat flux calculation is incorrect | Underestimating area makes thermal conductivity too large | Measure diameter or thickness at multiple points |
| Error in sample length | Temperature gradient calculation is incorrect | Overestimating measurement distance makes thermal conductivity too large | Directly measure the distance between temperature sensors |
| Material nonuniformity | Thermal conductivity varies locally | Temperature distribution deviates from a straight line | Use a homogeneous sample |
| Temperature dependence | Thermal conductivity changes with temperature | A single constant value does not apply across the hot and cold sides | Compare with a reference value corresponding to the mean temperature |
| Natural convection | Surrounding air carries heat away | Results vary with apparatus orientation and airflow | Avoid drafts and use an insulating cover |
| Heat conduction through sensor wires | Heat escapes through the wiring | Local temperature changes | Use thin wires and arrange them to minimize heat flow |
Why the Temperature Distribution Becomes Linear
When a uniform sample is used, thermal conductivity is constant, there is no heat loss from the sides, and steady state has been reached, the heat flow rate through the sample is the same at every position.
According to Fourier’s law, a constant heat flow rate produces a constant temperature gradient, so the relationship between temperature and position becomes linear.
Why Increasing Input Power Increases the Temperature Difference
If the sample material, length, and cross-sectional area are constant, Fourier’s law shows that the temperature difference is proportional to the heat flow rate. Therefore, increasing heater power increases the temperature difference between the hot and cold sides.
Why a Larger Cross-Sectional Area Transfers More Heat
Heat is conducted through the entire cross section of the sample. A larger cross-sectional area provides a wider path for heat flow, increasing the heat flow rate for the same temperature difference.
Why a Longer Sample Transfers Heat Less Easily
A longer sample has a smaller temperature gradient for the same temperature difference. Therefore, for the same material and cross-sectional area, a longer sample transfers less heat per unit time.
Example of How to Write the Results
One end of an aluminum rod 0.200 m long and 0.0200 m in diameter was heated with an electric heater, and the hot-side and cold-side temperatures were measured under steady-state conditions.
The heater voltage was 12.0 V and the current was 0.500 A, giving an input power of 6.00 W. The hot-side temperature was 78.0 ℃ and the cold-side temperature was 61.0 ℃, giving a temperature difference of 17.0 K.
Using Fourier’s law with a sample cross-sectional area of 3.14 × 10−4 m2, the thermal conductivity was calculated to be 225 W/(m·K). The relative error compared with the reference value for aluminum, 230 W/(m·K), was 2.17%.
Points to Connect to the Discussion
- Did the temperature at each measurement point become constant over time?
- Was the relationship between temperature and position linear?
- Were input power and temperature difference proportional?
- Was the measured thermal conductivity close to the reference value?
- Could heat loss from the sides be neglected?
- Was it valid to treat all heater input power as the heat flow rate?
- How did thermal contact resistance affect the result?
- Were the temperature sensor positions and contact conditions appropriate?
- Were the sample cross-sectional area and distance between sensors measured accurately?
- Could the temperature dependence of thermal conductivity be neglected?
- Was there heat loss due to radiation or natural convection?
- Was the reproducibility of the measurements sufficient?
Discussion Example
In this experiment, one end of an aluminum rod was heated with an electric heater, and the thermal conductivity was determined from the steady-state temperature distribution and input power.
Immediately after heating began, the temperatures on both the hot and cold sides increased with time, but after approximately 25 minutes the changes at each measurement point became small. At this stage, the heat entering and leaving the sample were approximately equal, indicating that steady state had been reached.
The relationship between distance from the hot end and temperature was approximately linear. For a uniform sample with constant thermal conductivity and heat flow only along its length, Fourier’s law predicts a constant temperature gradient.
The input power calculated from a heater voltage of 12.0 V and current of 0.500 A was 6.00 W. Using a hot-side temperature of 78.0 ℃, a cold-side temperature of 61.0 ℃, a sample length of 0.200 m, and a cross-sectional area of 3.14 × 10−4 m2, the thermal conductivity was calculated to be 225 W/(m·K).
This value was close to the reference value for aluminum of 230 W/(m·K), with a relative error of 2.17%. The result therefore reflects the high thermal conductivity of aluminum.
However, the calculation assumed that all power supplied to the heater passed through the sample. In reality, heat escapes to the surroundings from the heater surface, sample sides, supports, wiring, and other parts.
If some input power is lost without passing through the sample but the full input power is used as the heat flow rate, the heat flow through the sample is overestimated and the calculated thermal conductivity tends to be higher than the true value.
Thermal contact resistance also exists at the interfaces between the heater and sample, the sample and cooling section, and the temperature sensors and sample. Small gaps or air layers at these interfaces can produce additional temperature differences.
If a contact temperature drop is included in the temperature difference attributed to the sample itself, the temperature gradient within the sample is overestimated and the calculated thermal conductivity may be too small.
Because side heat loss and thermal contact resistance can affect the thermal conductivity in opposite directions, an experimental value close to the reference value does not necessarily mean that both errors are negligible; they may partly cancel each other.
Temperature sensor position error is also important. The temperature-gradient calculation uses not only the temperature difference but also the distance between sensors, so even a displacement of several millimeters may produce a non-negligible error for a short sample.
If a temperature sensor is not in good thermal contact with the sample, it may indicate a temperature between the sample temperature and the surrounding air temperature. Thermal grease and suitable fixtures can be used to maintain consistent contact.
When the input power was varied from 3.00 to 6.00 W, the temperature difference increased approximately in proportion to input power. If thermal conductivity, sample length, and cross-sectional area remain constant, Fourier’s law predicts that temperature difference is proportional to heat flow rate, so this trend agrees with theory.
At higher temperatures, however, heat losses through natural convection and radiation increase, so the relationship may deviate from ideal proportionality as input power increases.
Measurement accuracy can be improved by insulating the sides of the sample, smoothing contact surfaces and applying thermal grease, waiting until all temperatures have sufficiently stabilized, and calibrating heater heat losses.
In summary, the thermal conductivity of the aluminum rod was determined from its temperature distribution and heat flow rate. The result was close to the reference value, while heat loss from the sides, thermal contact resistance, temperature sensor position and contact, and estimation of input power were identified as major sources of error.
Discussion Example When Thermal Conductivity Is Higher Than the Reference Value
One reason the measured thermal conductivity may be higher than the reference value is the assumption that all input power passes through the sample. In reality, heat escapes from the sides, heater surface, supports, and other parts, so the heat flow through the sample is smaller than the electrical input power. Using the full input power therefore overestimates the heat flow rate and produces a thermal conductivity that is too large.
Discussion Example When Thermal Conductivity Is Lower Than the Reference Value
A measured thermal conductivity lower than the reference value may be caused by thermal contact resistance between the heater and sample, the sample and cooling section, or the temperature sensor and sample. If the additional temperature differences at these interfaces are treated as part of the temperature difference within the sample, the temperature gradient is overestimated and the calculated thermal conductivity becomes too small.
Discussion Example When the Temperature Distribution Is Not Linear
A nonlinear temperature distribution may result from failure to reach steady state, position-dependent heat loss from the sides, a nonuniform sample, or inconsistent contact between the temperature sensors and the sample.
Discussion Example When Measurements Show Large Scatter
Large scatter in the measured values may result from fluctuations in input power, insufficient fixation of the temperature sensors, indoor airflow, or differences in the criterion used to judge steady state. It is necessary to confirm that the rate of temperature change at each measurement point has become sufficiently small and to perform repeated measurements under the same conditions.
Summary
In thermal conductivity measurements, Fourier’s law is used to determine thermal conductivity from the heat flow rate through the sample, sample length, cross-sectional area, and temperature difference.
In the steady-state method, it is important to confirm that the temperatures at all measurement points have become constant with time and that the temperature distribution is approximately linear.
In the discussion, heat loss from the sides, thermal contact resistance, temperature sensor position and contact, estimation of input power, and the determination of steady state should be related to whether the experimental value is higher or lower than the reference value.
