### What is the transient plane source?

The transient plane source (TPS) method is a rapid approach to determine thermal conductivity, thermal diffusivity, thermal effusivity and heat capacity of solids, liquids, pastes and powders, between thermal conductivities of 0.005 and 1800 W/mK, and temperature ranges from -160 °C to 1000 °C. The transient plane source is capable of making double- or single-sided measurements, with no calibration or contact agent required.

### Mathematical considerations of the transient plane source

The transient plane source sensor is a small, coiled circuit that is placed between identical sample pieces. The circuit is composed of a bifilar nickel spiral, which transfers heat to the material, and ultimately monitors the resulting temperature variation through a change in resistance. The difference in resistance due to temperature (and the resulting altered voltage) is measured directly with a voltmeter. From this, the change in temperature can be calculated from,

\[ R(t)=R_{0}(1+\alpha \overline{\Delta T(t)}) \]

where \( R_{o} \) is the initial resistance, \( \alpha \) is the temperature coefficient of resistance of the nickel spiral, and \(\overline{\Delta T(t)} \) is the average temperature change. From the average temperature change, thermal conductivity and thermal diffusivity can be found by fitting,

\[ \overline{\Delta T(t)} = P_{o}\Big( \pi^{\frac{3}{2}}aK \Big)^{-1}D_{s}(\tau) \]

to experimental measurements. \( P_{o} \) is the total output of power, and \( a \) is the radius of nickel spiral. \( K \) is the thermal conductivity, \( \tau \) is the characteristic time, \( \tau = \sqrt{\frac{t\kappa}{a^{2}}} \), \( \kappa \) is the thermal diffusivity, and \( D_{s}(\tau)\) is the theoretical expression that determines the time dependent increase, created by solving the equations of state for the system.

### Measurements with the transient plane source

The transient plane source method, as described in the name, is a transient method. The double spiral sensor configuration is most often formulated as a time dependent consideration, where the total time of experimentation is not enough for the thermal wave to extend past the edges of the material. The probing depth is the distance travelled by 1% of the wave, and it is determined by

\[ d = \sqrt{4{\kappa}t} \]

Where *d *is equal to the diffusion depth, *κ *is the thermal diffusivity, and *t *represents time. Any sample to be measured with the transient plane source method should have a depth greater than what is determined by this equation, and should extend beyond the edges of the double spiral sensor by at least that amount.

### Instrument calibration

The transient plane source method is absolute and does not require calibration. The software associated with the transient plane source allows users to manually eliminate the effect of contact resistance from the beginning of each measurement, negating the need for calibration. The transient plane source method offers high accuracy and repeatability across a wide range of materials. Thermal conductivity measurements have an accuracy of 5% or better, and thermal diffusivity measurements are 10% or better.

### Internationally recognized standards

The transient plane source follows the ISO Standard 22007-2:2015, which explains the procedure for determining the thermal conductivity and thermal diffusivity of plastics with the transient plane source (hot disc) method.

### Additional literature

The following papers expand on the theory of the transient plane source method and provide a more in depth look at its application in experimental procedures.

- Transient Plane Source Techniques for Thermal Conductivity and Thermal Diffusivity Measurements of Solid Materials. Author: Silas E. Gustafsson (1991).
*Review of Scientific Instruments* - Transient Plane Source (TPS) Technique for Measuring Thermal Transport Properties of Building Materials. Authors: Torgrim Log and Silas E. Gustafsson (1995).
*Fire and Materials* - Use of the Transient Plane Source Technique for Rapid Multiple Thermal Property Measurements. Authors: T. Boumaza and John Redgrove (2003).
*International Journal of Thermophysics*24(2). - On the Use of Transient Plane Source Sensors for Studying Materials with Direction Dependent Properties. Authors: Mattias Gustavsson and Silas E. Gustafsson (2005). In:
__Thermal Conductivity 26: Thermal Expansion 14__. - Rapid Thermal Conductivity Measurement with a Hot Disk Sensor: Part 1. Theoretical Considerations. Author: Yi He (2005).
*Thermochimica Acta*436(1-2):122-129. - Rapid Thermal Conductivity Measurement with a Hot Disk Sensor: Part 2. Characterization of Thermal Greases. Author: Yi He (2005).
*Thermochimica Acta*436(1-2):130-134