Power Corrections in the Transient Plane Source Method

The transient plane source (TPS) is a well-known method for measuring the thermal conductivity, thermal diffusivity, and specific heat capacity of solids, liquids, pastes, and powders. The method makes use of a resistive element acting as a heat source and a temperature sensor. This resistive element, also known as the TPS sensor, is shaped into a bifilar spiral and is enclosed in an electrically insulating sheet. At the start of the measurement, the TPS sensor is placed in between two large, identical, homogeneous samples in thermal equilibrium. The TPS sensor undergoes Joule heating upon the application of electrical current. At the same time, the TPS sensor records its own temperature. The thermophysical properties of the samples can be obtained from the temperature transient recorded by the TPS sensor with Gustafsson's mathematical model [1,2].

It was pointed out by Gustavsson and Gustafsson [3] that the power delivered by the TPS is not constant due to factors such as power loss in the electrical leads, the temperature coefficient of resistance (TCR) of the TPS sensor, the heat capacity of the sensor, etc. In this paper, we present a semi-analytical expression for the average temperatures of the TPS sensor with a first-order correction to the delivered power. The first-order correction includes the effects of the sensor heat capacity and the TCR of the sensor. To verify the correction, TPS measurements were made on Pyroceram 9606. Pyroceram 9606 was selected because it can be obtained from the market easily, is made with high quality, and has a measured thermal conductivity value reported in literature [4]. The measured properties of Pyroceram 9606 using Gustafsson's model and the corrected model were then compared over different initial powers.

where I₀ is the modified Bessel function of the first kind of order zero.

where $P0$ is the initial power delivered to the TPS sensor and $C$ is the sensor heat capacity in Joules per Kelvin. The minus sign indicates that a nonzero heat capacity reduces the power delivered to the TPS sensor to heat up the samples.

where $I1(x)$ is the modified Bessel function of the first order, and $L1(x)$ is the modified Struve function of the first order.

The average temperature rise of the sensor $ΔTave(t)$ can be obtained from Eq. (23) by taking its numerical inverse Laplace transform. In this paper, de-Hoog's algorithm is used, which treats the inverse Laplace transform as a Fourier transform. The convergence of the Fourier transform is then accelerated by a nonlinear double acceleration with a Padé approximation [6,7].

where $β$ and $C$ are fixed parameters determined before the start of the measurement, $t0$ is the time offset parameter introduced in Sec. 1, and $Tf$ is the temperature offset added in the model to account for contact resistance [1]. To minimize Eq. (25), algorithms such as the Levenberg-Marquardt algorithm can be used to find the best-fit parameters $t̂0$⁠, $T̂f$⁠, $Λ̂$⁠, and $κ̂$ [8].

To verify the new model, TPS measurements were performed on Pyroceram 9606 with Thermtest Instruments TPS 3317 sensor, a heat source made up of 16 concentric rings and a radius of 6.4 mm (Fig. 2). Pyroceram 9606 is used because it has a known thermal conductivity of 3.913 Wm⁻¹K⁻¹ at room temperature (23 °C) [4]. The measured TPS sensor temperatures were collected using Thermtest Instruments Measurement Platform-1. Measurements on Pyroceram 9606 were performed at multiple initial powers, from 200 to 1200 mW, with increments of 100 mW, all performed at room temperature (23 °C). Each measurement was 10 s long. The sample dimensions were large enough that the probing depth of the measurement at 10 s does not exceed the sample boundaries [1]. The thermal properties were then obtained from the sensor temperatures using Gustafsson's model [1] and the improved model in Eq. (24). The disk approximation used in Eq. (24) is valid since the TPS sensor has greater than 10 rings [1].

The plot of thermal conductivity (and thermal diffusivity) plotted against power is displayed in Fig. 3. The cross marks represent measurements using Gustafsson's model, while the gray circles represent measurements using the model in Eq. (23). In the thermal conductivity plot, the dashed line represents the reference value reported in Ref. [4]. Notice that there is little to no difference in the measured thermal properties with Gustafsson's model and Eq. (23) at the lowest power (200 mW). However, as the initial power is increased, the measured thermal properties exhibit a trend with respect to the power applied. To better quantify this trend, the percent deviation in measured thermal conductivity against the reference value of 3.913 Wm⁻¹K⁻¹ [4] is plotted in Fig. 4. For measurements performed at 200 mW, the percent deviation is low (below 0.5%), but as the initial power increases, the measured thermal conductivity becomes larger relative to the reference value. Without any power corrections, the percent deviation in thermal conductivity at 1200 mW is 3.15%, compared to 2.3% when a power correction is applied. The change in conductivity with power was measured to be 0.18 Wm⁻¹K⁻¹ per watt (4.7% per watt) without this correction, and 0.095 Wm⁻¹K⁻¹ per watt (2.4% per watt) with this correction, nearly a 50% improvement.

The power correction is a modest correction on the measured thermal conductivity. However, it does not improve the thermal diffusivity measurements, as seen in Fig. 3(b). At the lowest power (200 mW), the thermal diffusivity readings are similar. However, as the power is increased, the measured thermal diffusivity increases with Gustafsson's model, while it decreases with the power correction model. The measured thermal diffusivity at 1200 mW is 1.4% higher relative to the 200 mW measurement using Gustafsson's model, while the measured thermal diffusivity using the power correction model decreased by −1.8% relative to the 200 mW measurement.

A more accurate description of the TPS sensor is presented in this work, accounting for the change in power due to the TCR and the finite heat capacity of the sensor. The corrections were verified with TPS measurements on Pyroceram 9606. Using the power correction model made the thermal conductivity readings less sensitive to the initial power used relative to Gustafsson's model, where a constant power is assumed.

• National Research Council of Canada through their Industrial Research Assistance Program (IRAP), with project code NRC-IRAP (No. 928449; Funder ID: 10.13039/501100000046).

David Landry and Renzo Flores are employed by Thermtest Inc. and as such have beneficial interest in the work.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

a =

TPS sensor radius (m)

C =

sensor heat capacity (J/m²K)

I(t) =

electric current passing through the TPS sensor (A)

I₁(x) =

modified Bessel function of order 1

J₁(x) =

Bessel function of order 1

L₁(x) =

modified Struve function of order 1

m =

number of concentric rings on the TPS sensor

P₀ =

initial power delivered by the TPS sensor (W)

P(t) =

power delivered by the TPS sensor (W)

q =

$s/κ$ (m⁻¹)

Q(r, z, t) =

source term

r =

radial coordinate (m)

R_L =

total electrical resistance of the leads to the sensor (Ω)

R_REF =

electrical resistance of the reference resistor (Ω)

R₀ =

initial electrical resistance of the TPS sensor (Ω)

R(t) =

electrical resistance of the TPS sensor (Ω)

s =

Laplace variable (s⁻¹)

t =

time (s)

t_i =

ith timestamp reading (s)

$t̂0$ =

fitted time offset parameter (s)

T₀ =

initial temperature of the system (K)

$Tf̂$ =

fitted temperature offset parameter (K)

T_i =

ith temperature reading (K)

T(r, z, t) =

temperature of the system at coordinates (r, z) at time t (K)

u(t) =

Heaviside step function

U =

voltage across the bridge (V)

Greek Symbols

α =

temperature coefficient of resistance (K⁻¹)

β =

α (R_REF – R₀)/(R_REF + R₀) (K⁻¹)

δ(z) =

delta function (m⁻¹)

ΔP(t) =

P(t) – P₀ (W)

ΔR(t) =

R(t) – R₀ (Ω)

$Δθ¯ave$ =

Laplace transform of the TPS temperatures for a constant power P₀ (K s)

ΔT =

T(r,z,t) – T₀ (K)

ΔT_ave =

average temperature of the heat source (K)

$ΔT¯$ =

Laplace transform of ΔT(r,z,t) (K s)

κ =

thermal diffusivity (m²/s)

$κ̂$ =

fitted thermal diffusivity (m²/s)

Λ =

thermal conductivity (Wm⁻¹K⁻¹)

$Λ̂$ =

fitted thermal conductivity (Wm⁻¹K⁻¹)

σ =

Hankel variable (m⁻¹)