### What is the modified transient plane source method?

The modified transient plane source (MTPS) method uses a modified design of the transient plane source method to directly measure thermal effusivity and thermal conductivity of a given material. The standard transient plane source method typically performs double sided measurements with the sensor sandwiched between two identical samples. The modified transient plane source requires only one sample material to conduct its measurements, but this is an available testing option for the standard transient plane source method, as well. Additionally, the mathematical approach to measuring the temperature of the plane significantly varies between the two methods.

### Mathematical considerations of the modified transient plane source method

The mathematical calculations for the modified transient plane source method treats the sensor as a simple plane. This contrasts the original transient plane source method, where the sensor is approximated by a series of concentric rings. Both methods are estimates of real situations, however the modified method yields simpler mathematical expressions at the expense of experimental accuracy.

With continuous power flux, \(G\), applied to the sensor, resulting in a continuous heat flow density \(G\prime\), the equation of state is

\[\rho{c}_{p}\frac{\partial{T}}{\partial{t}}= \lambda \frac{\partial^{2}T}{\partial{x}^{2}}+{G\prime}\]

Where λ is equivalent to the thermal conductivity in W/mK, ρ is the density in Kg/m^{3}, and *C _{p }*is the specific heat capacity in J/kgK. The equation can then be solved as follows:

\[\Delta{T}(x,t)=\frac{2G \sqrt{t}}{e_{1}+e_{2}}ierfc\frac{\mid{x}\mid}{2 \sqrt{a{t}}}\]

Where ΔT is the change in temperature of the sensor surface in °C, *G *is the power flux applied to the sensor in W/m^{2}, *t *is the time measured from the start of the process in seconds, \(e_{1}\) and \(e_{2}\) are the effusivities of the sensor and material, respectively, and α, whose value may vary with position, is the diffusivity at that point. However, this becomes unimportant, as at *x*=0 (the contact point between the material and the sensor) the expression becomes:

\[\Delta{T}(0,t)=\frac{1.1284G\sqrt{t}}{e_{1}+e_{2}}\]

This expression for the change in temperature is simpler than the transient plane source technique. For the measurement of temperature, the voltage is measured and temperature is calculated by using the expression of resistance

\[R(t)=R_{o}+AT(0,t)\]

Where \(R_{o}\) is the initial resistance. The change in resistance is then \(A\Delta{T}\), and if the current, \(I\), is kept constant, the change in voltage is

\[\Delta{V}(t)=\frac{1.1284IAG\sqrt{t}}{e_{1}+e_{2}}\]

### Calibrating the modified transient plane source

From here, calibration is used to obtain the effusivity of the sensor, \(e_{1}\). The modified transient plane source system is calibrated through computer software. In the first step, the user must select multiple calibration materials that most closely resembles the test sample. Then, at least three calibration materials with thermal conductivity values within the range of the sample material must be inputted into the software along with at least one temperature. These values will then be applied to the calibration tests that are conducted over a range of temperatures. Next, the user must input the known thermal conductivity range of the sample material into the software. Contact agent type, measurement time, and cooling times, as well as the power level, are selected depending on the characteristics of the sample material. Once the parameters have been decided, the contact agent is applied to the surface of the sensor, which is then placed under the calibration material and a small weight. The graph that the computer software generates is a representation of the linearity of the calibration curve; R^{2} values greater than 0.995 are acceptable. If the calibration results are incorrect, the measurements can be performed again, or different parameters can be selected.

### Measurements with the modified transient plane source

The modified transient plane source method can be used for a wide variety of materials across a range of temperatures. With proper calibration, it is claimed that the method can accurately measure liquids, powders, insulators and conductors up to 500 W/mK, from -50 °C to 500 °C. This method is fairly new, and its accuracy has not undergone significant testing, but conductivity and effusivity measurements claim to have an accuracy of better than 5%. However, it is unlikely that this level of accuracy is valid, due to the extremely calibrated nature of the instrument.

### Internationally recognized standards

This instrument was developed in accordance with the ASTM standard D7984, which is the standard test method for measuring the thermal effusivity of textiles and fabrics with the modified transient plane source.

### Additional literature

The following papers outline experimental procedures with the MTPS instrument that exemplify real-world applications of the instrument.

- TCi Principles of Operation. Author: Michael Emmanuel (2006).
*C-Therm Application Note,*1-7. - Measuring the Thermal Conductivity of Heat Transfer Fluids via the Modified Transient Plane Source (MTPS). Authors: A. Harris, S. Kazachenko, R. Bateman, J. Nickerson, and M. Emanuel (2014).
*Journal of Thermal Analysis and Calorimetry*116(3). - Application of the Modified Transient Plane Source Technique in Testing the Thermal Conductivity of Concrete. Authors: A. Harris, D. Kuvandykova, and R. Bateman (2011). In:
__Thermal Conductivity 31/Thermal Expansion 19__.