The 3-omega method for thermal conductivity measurement using the hyperbolic heat conduction equation is presented. Mathematical expressions representing the conditions when non-Fourier effects cannot be neglected are formulated. Results obtained …
The 3-omega method for thermal conductivity measurement using the hyperbolic heat conduction equation is presented. Mathematical expressions representing the conditions when non-Fourier effects cannot be neglected are formulated. Results obtained from Fourier and hyperbolic heat conduction equations based on parameterized dielectric thermophysical parameters are compared. The analysis is applied to a thin Chemical vapor deposited (CVD) diamond film and bologna meat and it is shown in these cases that by neglecting the non-Fourier effects one can underestimate the thermal conductivity by 50%.
We present a 3ω method for simultaneously measuring the specific heat and thermal conductivity of a rod- or filament-like specimen using a way similar to a four-probe resistance measurement. The …
We present a 3ω method for simultaneously measuring the specific heat and thermal conductivity of a rod- or filament-like specimen using a way similar to a four-probe resistance measurement. The specimen in this method needs to be electrically conductive and with a temperature-dependent resistance, for acting both as a heater to create a temperature fluctuation and as a sensor to measure its thermal response. With this method, we have successfully measured the specific heat and thermal conductivity of platinum wire specimens at cryogenic temperatures, and measured those thermal quantities of tiny carbon nanotube bundles some of which are only ∼10−9 g in mass.
The 3ω method is widely employed for measuring the thermal conductivity of bulk, powder, liquid, and fibrous materials owing to its excellent versatility and accuracy. However, for sheet materials, the …
The 3ω method is widely employed for measuring the thermal conductivity of bulk, powder, liquid, and fibrous materials owing to its excellent versatility and accuracy. However, for sheet materials, the 3ω method usually requires depositing a metal electrode on the material, and sheet materials with rough surfaces, therefore, are difficult to meet the testing requirements of the 3ω method. In this study, a two-dimensional heat conduction model of the 3ω method suitable for thin sheet materials with/without rough surfaces was proposed, and it was used to measure the stainless-steel sheet and aluminum foil materials. The samples were suspended by four metal testing electrodes, and their thermal conductivity was measured and compared with standard values. The results indicate that this method can accurately measure the thermal conductivity of two-dimensional thin sheet conducting materials, thus extending the practical application scope of the 3ω method.
Generally, the 3ω method for the measurement of fluid thermal properties is based on an approximate solution to heat conduction model assuming vanishing heater-thickness, no dielectric layer and infinite heater-length. …
Generally, the 3ω method for the measurement of fluid thermal properties is based on an approximate solution to heat conduction model assuming vanishing heater-thickness, no dielectric layer and infinite heater-length. In this study, a novel three-dimensional model and partial differential equations of the dimensionless heating conduction in frequency domain were established, which took into account the finite thicknesses of the heater and dielectric layer as well as a finite heater length to investigate thermal conductivity and diffusivity of fluid. Through the numerical studies, it was found that we could reduce the ratio of thickness to width of heater or the thickness and thermal conductivity of the dielectric layer to minimize discrepancy between the numerical results and the approximate solutions. Additionally, the edge effects of a finite heater length can be ignored at low measurement frequencies and high ratio of length to width of the heater.
In the longitudinal heat flow method for the measurement of thermal conductivity, the surface losses present a very serious source of error, specially at high temperatures. A mathematical procedure has …
In the longitudinal heat flow method for the measurement of thermal conductivity, the surface losses present a very serious source of error, specially at high temperatures. A mathematical procedure has been proposed to eliminate this source of error. It is shown that if the temperature is measured at three points instead of two as is usual, the surface losses can be accounted for.
As an important factor affecting the accuracy of thermal conductivity measurement, systematic (bias) error in the guarded comparative axial heat flow (cut-bar) method was mostly neglected by previous researches. This …
As an important factor affecting the accuracy of thermal conductivity measurement, systematic (bias) error in the guarded comparative axial heat flow (cut-bar) method was mostly neglected by previous researches. This bias is primarily due to the thermal conductivity mismatch between sample and meter bars (reference), which is common for a sample of unknown thermal conductivity. A correction scheme, based on finite element simulation of the measurement system, was proposed to reduce the magnitude of the overall measurement uncertainty. This scheme was experimentally validated by applying corrections on four types of sample measurements in which the specimen thermal conductivity is much smaller, slightly smaller, equal and much larger than that of the meter bar. As an alternative to the optimum guarding technique proposed before, the correction scheme can be used to minimize the uncertainty contribution from the measurement system with non-optimal guarding conditions. It is especially necessary for large thermal conductivity mismatches between sample and meter bars.
A modified Levenberg–Marquardt method (LMM) for the identification of temperature-dependent thermal conductivity is proposed; the experiment and structure of the specimen for identification are also designed. The temperature-dependent thermal conductivities …
A modified Levenberg–Marquardt method (LMM) for the identification of temperature-dependent thermal conductivity is proposed; the experiment and structure of the specimen for identification are also designed. The temperature-dependent thermal conductivities of copper C10200 and brass C28000 are identified to verify the effectiveness of the proposed identification method. The comparison between identified results and the measured data of laser flash diffusivity apparatus indicates the fine consistency and potential usage of the proposed method.
The so-called "3ω method" is a well established method to measure heat conductivity of solids. It is a frequency-based method, in which the ratio between the first and third harmonics …
The so-called "3ω method" is a well established method to measure heat conductivity of solids. It is a frequency-based method, in which the ratio between the first and third harmonics of an induced voltage in an electric heater element can be shown to be (inversely) proportional to the thermal conductivity of a solid that the heater is in direct contact with. Commonly, the method utilizes Discrete Fourier analysis in an off-line setting, which is of course perfectly valid when measuring material properties in a static setting.In this paper we propose to make use of the measurement principle in a dynamic setting. We propose a novel timedomain approach to 3ω measurement, which can easily be implemented in a cheap micro-controller due to its modest memory and sampling rate requirements, and therefore likely to be useful for feedback in control loops or similar applications.The approach comprises two main elements, a discrete-time signal generator, which provides a steady-state sinusoidal current output, and a standard Luenberger-style state observer designed to estimate the associated third harmonic in the presence of noisy voltage measurements. We prove that the signal generator is robust to numerical inaccuracies. The approach is tested in simulation and on actual laboratory data, showing good agreement with traditional off-line analysis.
The problem of determining the temperature-dependent thermal conductivity coefficient of a substance is considered and investigated. This problem belongs to the class of problems of identification of model parameters. The …
The problem of determining the temperature-dependent thermal conductivity coefficient of a substance is considered and investigated. This problem belongs to the class of problems of identification of model parameters. The consideration is carried out on the basis of the first boundary value problem for the nonstationary heat equation. An algorithm for the numerical solution of the considered inverse problem is proposed. The identification problem was considered in one-dimensional, two-dimensional and three-dimensional formulations. The corresponding inverse coefficient problems were reduced to variational problems, which were solved numerically using gradient methods for minimizing cost functionals. The effective method is proposed for evaluation of the cost functional gradient. It is based on the fast automatic differentiation technique and produces the exact gradient for the chosen approximation of the optimal control problem.
Anisotropic thermal transport plays a key role in both theoretical study and engineering practice of heat transfer, but accurately measuring anisotropic thermal conductivity remains a significant challenge. To address this …
Anisotropic thermal transport plays a key role in both theoretical study and engineering practice of heat transfer, but accurately measuring anisotropic thermal conductivity remains a significant challenge. To address this issue, we propose the three-sensor 2ω method in this study, which is capable of accurately measuring the isotropic or anisotropic thermal conductivity of solid materials. In this method, several three-sensor groups following the design guidelines are fabricated upon the sample along different characteristic directions, and each group consists of three parallel metal sensors with unequal widths and distances optimally designed based on sensitivity analysis. Among the three sensors, the outer two serve as AC heaters and the middle one as a DC detector. The 2ω voltage signals across the detector in each three-sensor group are measured, and then the data are processed by the proposed Intersection Method to derive the thermal conductivities along directions of interest. The application of the detector's 2ω instead of the heater's 3ω voltage signals eliminates the errors introduced by the uncertainties of thermal resistance in superficial structures (metal layer, insulation layer, interface, etc.). Meanwhile, by replacing the fitting algorithm with the Intersection Method, the local optimum trap of multivariate fitting is avoided. To verify the accuracy and reliability, four typical monocrystalline semiconductors, i.e., Si, GaN, AlN, and {β-Ga _2 O _3}, are measured, and the results are consistent with the literature. This method will provide a comprehensive and versatile solution for the thermal conductivity measurements of solid materials.
The nonlinear Poisson equation is considered, in which the thermal conductivity is a function of temperature λ(T ) = p1T+p2, where p1, p2 are the unknown parameters. To solve the …
The nonlinear Poisson equation is considered, in which the thermal conductivity is a function of temperature λ(T ) = p1T+p2, where p1, p2 are the unknown parameters. To solve the inverse problem consisting in the identification of p1 and p2 the additional information connected with the knowledge of temperature T at the set of points (sensors) selected from the domain considered is necessary. The fundamental problem is the selection of sensors location and here the algorithm assuring the optimal sensors location is proposed. In the final part of the paper the results of computations are shown. 1. Formulation of the problem The following 2D problem is considered [ ] : λ ( ) ( ) ( ) 0 : ( ) ( ) b x T T x + Q x = x T x = T x ∈Ω ∇ ∇ ∈Γ (1) where T is the temperature, x = (x1, x2) are the spatial coordinates, λ(T ) is the thermal conductivity, Q (x) is the source function, Tb (x) is known boundary temperature. We assume that 1 2 λ ( ) T = T + p p (2) where p1, p2 are the coefficients. When the direct problem is considered then all geometrical and thermophysical parameters appearing in the mathematical model (1) are known. In the paper the inverse parametric problem is discussed in which it is assumed that the coefficients p1, p2 are unknown. To solve the inverse problem the additional information is necessary. So, we assume that the temperatures at the selected points x ∈Ω are given 1 2 ( ) 1 , 2 , . . . , i i i d d T = T , , i = N x x (3) where N is the number of sensors. Please cite this article as: Ewa Majchrzak, Katarzyna Freus, Sebastian Freus, Experiment design for estimation of temperature dependent thermal conductivity, Scientific Research of the Institute of Mathematics and Computer Science, 2010, Volume 9, Issue 1, pages 83-88. The website: http://www.amcm.pcz.pl/ E. Majchrzak, K. Freus, S. Freus 84 The accuracy of identification depends significantly on the choice of sensors location and this problem is here discussed. 2. Algorithm of optimal sensors location Let X = {x, x,...,x } denotes the set of spatial points at which measurements may be taken. The practical design problem consists in selection of corresponding weights w1, w2,...,wM which define the best experimental conditions [1]. To solve this problem the following iterative algorithm under the assumption that number of unknown parameters equals 2 and number of sensors equals N can be applied [2]. At first, the sensitivity matrix is constructed
Abstract : A quantitative analysis of the Cut-Bar method of measuring the thermal conductivity of solids is performed. The mathematical model, which corrects for the difference in heat flux in …
Abstract : A quantitative analysis of the Cut-Bar method of measuring the thermal conductivity of solids is performed. The mathematical model, which corrects for the difference in heat flux in the specimen and reference standard, is that of the two dimensional steady heat conduction equation applied to an annulus of insulation. The solution is presented in detail and found to be comprised of two physically distinct parts, a conductivity factor and a geometrical factor. A number of charts and graphs are presented for clarification as to the nature and magnitude the relative sizes of the various components will have on the accuracy over different conductivity ranges. The complexity of the geometrical factor required digital computer programs which are included. Reference is made to a similar study, performed by researchers at the National Bureau of Standards. It is found that the differences in the guard temperature distribution results in a substantial change in the geometrical factor.
“3ω” experiments aim at measuring thermal conductivities and diffusivities. Data analysis relies on integral expressions of the temperature. In this paper, we derive new explicit analytical formulations of the solution …
“3ω” experiments aim at measuring thermal conductivities and diffusivities. Data analysis relies on integral expressions of the temperature. In this paper, we derive new explicit analytical formulations of the solution of the heat diffusion equation, using Bessel, Struve, and Meijer-G functions, in the 3ω geometry for bulk solids. These functions are available in major computational tools. Therefore numerical integrations can be avoided in data analysis. Moreover, these expressions enable rigorous derivations of the asymptotic behaviors. We also underline that the diffusivity can be extracted from the phase data without any calibration while the conductivity measurement requires a careful one.
The so-called "3ω method" is a well established method to measure heat conductivity of solids. It is a frequency-based method, in which the ratio between the first and third harmonics …
The so-called "3ω method" is a well established method to measure heat conductivity of solids. It is a frequency-based method, in which the ratio between the first and third harmonics of an induced voltage in an electric heater element can be shown to be (inversely) proportional to the thermal conductivity of a solid that the heater is in direct contact with. Commonly, the method utilizes Discrete Fourier analysis in an off-line setting, which is of course perfectly valid when measuring material properties in a static setting.In this paper we propose to make use of the measurement principle in a dynamic setting. We propose a novel timedomain approach to 3ω measurement, which can easily be implemented in a cheap micro-controller due to its modest memory and sampling rate requirements, and therefore likely to be useful for feedback in control loops or similar applications.The approach comprises two main elements, a discrete-time signal generator, which provides a steady-state sinusoidal current output, and a standard Luenberger-style state observer designed to estimate the associated third harmonic in the presence of noisy voltage measurements. We prove that the signal generator is robust to numerical inaccuracies. The approach is tested in simulation and on actual laboratory data, showing good agreement with traditional off-line analysis.
This work is devoted to analytical and numerical studies of diffusive heat conduction in configurations considered in 3ω experiments, which aim at measuring thermal conductivity of materials. The widespread 2D …
This work is devoted to analytical and numerical studies of diffusive heat conduction in configurations considered in 3ω experiments, which aim at measuring thermal conductivity of materials. The widespread 2D analytical model considers infinite media and translational invariance, a situation which cannot be met in practice in numerous cases due to the constraints in low-dimensional materials and systems. We investigate how thermal boundary resistance between heating wire and sample, native oxide and heating wire shape affect the temperature fields. 3D finite element modelling is also performed to account for the effect of the bonding pads and the 3D heat spreading down to a typical package. Emphasis is given on the low-frequency regime, which is less known than the so-called slope regime. These results will serve as guides for the design of ideal experiments where the 2D model can be applied and for the analyses of non-ideal ones.
Purpose Prompted by the reliability and robustness of the previously proposed method of non-destructive measurement of thermal conductivity (TC) for anisotropic materials, the enhanced approach is presented in this study. …
Purpose Prompted by the reliability and robustness of the previously proposed method of non-destructive measurement of thermal conductivity (TC) for anisotropic materials, the enhanced approach is presented in this study. The main improvement lies in the substitution of the analytic solution of direct problem solver with a numerical one. This solver, used during the inverse procedure that fits measurement data into simulated ones, is proposed to be a numerical one (finite volume method). Moreover, the purpose of this study is to show the applicability of the reduce order model for retrieving thermal conductivity of solid body. Design/methodology/approach In the proposed methodology, both the laser heat source and temperature measurements are performed on the same side of the sample material, which is the main difference with respect to the classic Parker flash method. To speed up the computational time, the full numerical model used in the course of inverse solution is replaced by the proper orthogonal decomposition (POD)-radial basis function (RBF) reduced order model, which is fast and accurate. Findings The TCs measured using the proposed methodology are in good agreement with the well established (but destructive) measurement methods. The advantage of the proposed approach lies in the optimal approximation properties of the POD approximation basis used in reduced order model, as well as in its regularization properties. Practical implications The proposed technique has high application potential in the design of novel apparatus for non-destructive measurement of TCs for both isotropic and anisotropic materials. Originality/value This is the first time when the POD-RBF reduced order model is used in the procedure of non-destructive TC measurement for anisotropic bodies.
In this work, we report the fabrication of an experimental setup for high temperature thermal conductivity (κ) measurement. It can characterize samples with various dimensions and shapes. Steady state based …
In this work, we report the fabrication of an experimental setup for high temperature thermal conductivity (κ) measurement. It can characterize samples with various dimensions and shapes. Steady state based axial heat flow technique is used for κ measurement. Heat loss is measured using parallel thermal conductance technique. Simple design, lightweight, and small size sample holder is developed by using a thin heater and limited components. Low heat loss value is achieved by using very low thermal conductive insulator block with small cross-sectional area. Power delivered to the heater is measured accurately by using 4-wire technique and for this, the heater is developed with 4 wires. This setup is validated by using Bi0.36Sb1.45Te3, polycrystalline bismuth, gadolinium, and alumina samples. The data obtained for these samples are found to be in good agreement with the reported data. The maximum deviation of 6% in the value κ is observed. This maximum deviation is observed with the gadolinium sample. We also report the thermal conductivity of polycrystalline tellurium from 320 K to 550 K and the nonmonotonous behavior of κ with temperature is observed.
This paper presents the development of instrumentation for the measurement of high-temperature thermal conductivity of bulk and coatings using a modulated photothermal radiometry (MPR) method, where a sample is heated …
This paper presents the development of instrumentation for the measurement of high-temperature thermal conductivity of bulk and coatings using a modulated photothermal radiometry (MPR) method, where a sample is heated by an intensity-modulated laser to probe into different layers of the sample. While MPR has been previously established, most of the previous studies only focus on the measurement at room temperature. The MPR has not been well studied for measurements of bulk and coating materials at high temperatures, which are increasingly important for a multitude of applications, such as materials used in the concentrating solar power (CSP) plants and the nuclear reactors. MPR is a non-contact technique that utilizes the intrinsic thermal emission from the specimens for thermometry, which is favorable for measurements at high temperatures in harsh environment. The authors designed and utilized a sample holder suitable for high temperature measurement up to 973 K with good temperature uniformity within the sample. The high-temperature MPR setup was validated by measuring bulk materials with known thermal conductivity. The setup and technique were then extended to the measurement of black solar-absorbing coatings of 10 to 50 {\mu}m thick on various substrates by modulating the frequency of the laser heating beam and the thermal penetration depth. The studies showed that thermal conductivities of typical solar-absorbing coatings are 0.4 ~ 0.8 W m-1 K-1, indicating a possibly large temperature drop within the coating under high solar irradiation flux, such as over 1000-sun for central solar towers in CSP plants.
Silicon Carbide (SiC) is a typical material for third-generation semiconductors. The thermal boundary resistance (TBR) of the 4H-SiC/SiO2 interface was investigated by both experimental measurements and theoretical calculations. The structure …
Silicon Carbide (SiC) is a typical material for third-generation semiconductors. The thermal boundary resistance (TBR) of the 4H-SiC/SiO2 interface was investigated by both experimental measurements and theoretical calculations. The structure of 4H-SiC/SiO2 was characterized by using transmission electron microscopy and X-ray diffraction. The TBR was found to be 8.11 × 10−8 m2K/W at 298 K by the 3ω method. Furthermore, the diffuse mismatch model was employed to predict the TBR of different interfaces, which is in good agreement with measurements. Heat transport behavior based on the phonon scattering perspective was also discussed to understand the variations of TBR across different interfaces. Besides, the intrinsic thermal conductivity of SiO2 thin films (200–1500 nm in thickness) on 4H-SiC substrates was measured by the 3ω procedure, to be 1.42 W/m K at 298 K. It is believed the presented results could provide useful insights into the thermal management and heat dissipation for SiC devices.
We present a 3ω method for simultaneously measuring the specific heat and thermal conductivity of a rod- or filament-like specimen using a way similar to a four-probe resistance measurement. The …
We present a 3ω method for simultaneously measuring the specific heat and thermal conductivity of a rod- or filament-like specimen using a way similar to a four-probe resistance measurement. The specimen in this method needs to be electrically conductive and with a temperature-dependent resistance, for acting both as a heater to create a temperature fluctuation and as a sensor to measure its thermal response. With this method, we have successfully measured the specific heat and thermal conductivity of platinum wire specimens at cryogenic temperatures, and measured those thermal quantities of tiny carbon nanotube bundles some of which are only ∼10−9 g in mass.
We have performed thermal conductance measurements on individual single crystalline silicon suspended nanowires. The nanowires (130 nm thick and 200 nm wide) are fabricated by e-beam lithography and suspended between …
We have performed thermal conductance measurements on individual single crystalline silicon suspended nanowires. The nanowires (130 nm thick and 200 nm wide) are fabricated by e-beam lithography and suspended between two separated pads on Silicon On Insulator (SOI) substrate. We measure the thermal conductance of the phonon wave guide by the 3 method. The cross-section of the nanowire approaches the dominant phonon wavelength in silicon which is of the order of 100 nm at 1K. Above 1.3K the conductance behaves as T3, but a deviation is measured at the lowest temperature which can be attributed to the reduced geometry.