Thermal Penetration Depth Research Articles

The impact of stack geometry and mean pressure on cold end temperature of stack in thermoacoustic refrigeration systems

This paper reports on the experimental and simulation studies of the influence of stack geometries and different mean pressures on the cold end temperature of the stack in the thermoacoustic refrigeration system. The stack geometry was tested, including spiral stack, circular pore stack and pin array stack. The results of this study show that the mean pressure of the gas in the system has a significant impact on the cold end temperature of the stack. The mean pressure of the gas in the system corresponds to thermal penetration depth, which results in a better cold end temperature of the stack. The results also show that the cold end temperature of the pin array stack decreases more than that of the spiral stack and circular pore stack geometry by approximately 63% and 70%, respectively. In addition, the thermal area and viscous area of the stack are analyzed to explain the results of such temperatures of thermoacoustic stacks.

Thermodiffusion-Stipulated Anomalous Response of a Stratified Fluid to Mechanical Forcing

Steady perturbations introduced into a temperature-stratified fluid binary mixture, such as saline water, by inhomogeneous tangent stresses on its surface have been studied in a linear approximation. It has been shown that thermodiffusion may greatly increase the depth of thermal perturbation penetration into the medium despite the stable background density stratification.

Characterisation of high thermal conductivity thin-film substrate systems and their interface thermal resistance

This paper characterises the high thermal conductivity thin-film substrate systems and the interface thermal resistances between the films. First, three-omega (3ω) method was proposed and verified, obtaining a reliable thermal conductivity measurement at shallow thermal-penetration depths. The method was then applied to a thin-film/substrate system to identify the individual thermal conductivities. The interface thermal resistance between the thin films was then successfully characterised with the aid of theoretical modelling and experimental measurements. As an example of the method application in the IC industry, the AlN/Si systems were investigated. The study identified that the thermal conductivity of the 2μm-thick AlN film in an AlN/Si system is 172.1W/mK and the AlN/Si interface thermal resistance is 1.796×10−9m2K/W.

Thermal Penetration Depth Analysis and Impact of the BEOL Metals on the Thermal Impedance of SiGe HBTs

In this letter, we present a detailed investigation on how dynamic thermal phenomena take place in state-of-the-art SiGe HBTs when excited by sinusoidal power dissipation. To give a better insight into the mechanisms leading to the thermal impedance ( $\text{Z}_{{\textsf {th}}})$ decay, we introduce the concept of thermal penetration depth; then, with the help of 3-D thermal simulations, we illustrate its effect on the spatial distribution of the temperature variations within the transistor structure, according to the frequency of operation. In order to experimentally analyze the impact on a real device, dedicated HBT structures are designed; they consist of multi-finger SiGe HBTs realized in B55 technology from STMicroelectronics, for which modifications are made in the back-end-of-line (BEOL) metallization or in the transistor layout, increasing its deep trench isolation enclosed area. For these transistors, $\text{Z}_{{\textsf {th}}}$ measurements are carried out in the frequency range 10kHz–1GHz; the results show that the metal connections configuration in the BEOL or layout modifications can considerably impact the $\text{Z}_{{\textsf {th}}}$ decay at low frequencies. An identical $\text{Z}_{{\textsf {th}}}$ trend is instead measured above 1–2 MHz, demonstrating that at higher frequencies just the region close to the heat source is concerned by dynamic thermal phenomena.

Thermal Conductivity Measurement of Liquids by Using a Suspended Microheater

In this paper, the traditional $$3\omega $$ method is modified in order to measure the thermal conductivity of a droplet of liquid. The $$3\omega $$ sensor is microfabricated using bulk silicon etching on a silicon wafer to form a microheater on a suspended bridge structure. The Si substrate of over 400 $$\upmu \hbox {m}$$ thickness beneath the microheater is etched away so that the sample liquid can fill the gap created between the heater and the bottom boundary of the sensor. The frequency of the sinusoidal heating pulses that are generated from the heater is controlled such that the thermal penetration depth is much smaller than the thickness of the liquid layer. The temperature oscillation of the sample fluid is measured at the thin-film heater to calculate the thermal conductivity of the surrounding fluid. The thermal conductivity and measured values of the de-ionized water and ethanol show a good agreement with the theoretical values at room temperature.

Three-equation local thermal non-equilibrium model for transient heat transfer in porous media: The internal thermal conduction effect in the solid phase

The present work developed a three-equation local thermal non-equilibrium (LTNE) model to model the initial effect on the internal thermal conduction in the solid phase of porous medium. The two energy equations in the original LTNE model were extended with a third equation to govern the dimensionless thermal penetration depth for the internal thermal conduction in the solid phase, I, which can describe the initial effect. The model was validated by comparisons between an analytical solution of the I equation and exact solutions of thermal conduction problems and between the macro-scale and the pore-scale numerical simulations. The comparisons demonstrate that the three-equation LTNE model is more accurate than the two-equation LTNE model with a constant I during the initial period. An important similarity number, η, is obtained for the enhanced geothermal system (EGS) heat extraction problem from the dimensionless three-equation LTNE model, which measures the internal thermal conduction effect in the solid phase of a porous medium. The initial effect in the EGS heat extraction can be neglected for η<0.5. Criteria for degeneration of the three-equation LTNE model are proposed.

A non-Fourier heat flux approach to model MHD Oldroyd-B fluid flow due to bidirectional stretching surface

Prime focus of this research is to address the MHD Oldroyd-B fluid flow and heat transfer due to stretching of plane elastic surface in two lateral directions. The phenomenon of heat transfer is modeled under a non-Fourier heat flux theory featuring thermal relaxation effects. Flow field is subjected to uniform vertical magnetic field. Furthermore, fluid with variable thermal conductivity is considered. Appropriate transformations are applied to yield the similar form of governing equations . A powerful analytical approach namely Homotopy Analysis Method (HAM) is opted to develop series solutions involving exponentially decaying functions. The results show that flow fields are appreciably affected with the variation in embedded parameters, especially with fluid relaxation and retardation times. It is found that the structure of solutions depends on a parameter measuring the ratio of stretching rates. Thermal penetration depth decreases for increasing values of thermal relaxation time. Thermal relaxation time has an effective role in improving the cooling process of the stretching surface which is vital in some industrial processes.

Thermal penetration depth enhancement in latent heat thermal energy storage system in the presence of heat pipe based on both charging and discharging processes

Latent Heat Thermal Energy Storage Systems (LHTESS) have attracted increasing attentions in recent years. In these systems, energy storage-retrieval are achieved during solid-liquid phase change of Phase Change Materials (PCM). Because of low thermal conductivity of PCMs, energy storage and retrieval cannot be achieved in desired time duration. In the present study, a novel fin configuration is optimized based on both charging and discharging processes using Response Surface Method (RSM). Following this, the optimum fin array is attached to heat pipe to improve its ability in increasing thermal penetration depth into the PCM. In the subsequent stage, the system with optimum finned heat pipe is compared to other systems in the presence of common fin arrays. The parameter of maximum energy storage capacity of the system is implemented quantitatively in the optimization procedure, and the efficiency of using this parameter as one of the objectives of LHTESS design is examined, which is proposed as the novelty here. Results indicate that attaching fin of suitable configuration to the heat pipe enhances its performance in increasing thermal penetration depth into the PCM. Moreover, it is indicated that choosing maximum energy storage capacity as one of the objectives of optimization procedure for LHTESS performance evaluation leads to efficient design.

Thermal analysis of high frequency electromagnetic heating of lossy porous media

Industrial applications of the electromagnetic wave as a source of energy have been increasing in recent years. One of the growing areas of application of electromagnetic heating (EMH), especially using high frequency waves, is the heating of geological media for extraction of hydrocarbons. In this work, electromagnetic field propagation in a lossy, electrically isotropic, and homogenous medium was studied to determine the effectiveness of EMH for heavy oil recovery. The electromagnetic heat source is modeled using Beer-Lambert’s law coupled with heat conduction. New analytical solutions are developed to find temperature distribution in linear and radial geometries subject to various boundary conditions. The effect of electromagnetic heat generation on thermal penetration depth is examined. One of the challenges in electromagnetic heating of geomaterials has been reflection of the incident wave due to impedance mismatch and thus loss of the applied power. The analytical solutions developed allow determination of absorbed power by the geological media and thus can be used as a tool to predict efficiency of electromagnetic heating. In addition, the aforementioned solutions can be used as forward model for design of experiments and to estimate the electromagnetic absorption coefficient of geomaterials.

Upper limit to the thermal penetration depth during modulated heating of multilayer thin films with pulsed and continuous wave lasers: A numerical study

In this study, we present a method to calculate the temperature and heat flux profiles as a function of depth and radius for bulk, homogeneous materials and samples with layered thin-film structures, including geometries supporting bidirectional heat fluxes, during pulsed and continuous wave (CW) laser heating. We calculate the temperature profiles for both modulated and unmodulated heating events to reveal that the thermal penetration depth (defined as the depth at which temperature decays to 1/e of the surface temperature) for a pulsed laser is highly dependent on time and repetition rate. In the high repetition rate limit, the temperature profile relaxes to that of a CW source profile, while in the opposite extreme, a single pulse response is observed such that the concept of the thermal penetration depth loses any practical meaning. For modulated heating events such as those used in time- and frequency-domain thermoreflectance, we show that there is a limit to the thermal penetration depth obtainable in an experiment, such that simple analytical expressions commonly used to determine thermal penetration depth break down. This effect is further compounded in samples with multiple layers, including the case when a ∼100 nm metallic transducer is deposited onto a bulk substrate, revealing that many recent studies relying on this estimation significantly over-predict the thermal penetration depth. Considering a bidirectional heat flow geometry (e.g., substrate/metal film/liquid), we find that heating from an unmodulated source results in an asymmetric heat flux about the plane of laser absorption to preserve a symmetric temperature profile when interfacial thermal resistance is negligible. However, the modulated case reveals a temperature asymmetry such that the thermal penetration depths in each side fall in line with those resulting from an insulated boundary condition.

Experimental Demonstration of the Dependence of Temperature Decrease on the Hydraulic Radius of Regenerator in a Traveling-Wave Thermoacoustic Refrigerator

Thermoacoustic refrigerators are cooling devices that employ sound wave in heat removal process. They mostly use air or other inert gas as working fluid and hence they are environmentally benign. The cooling process takes place in a porous medium (regenerator or stack). This paper presents an experiment on a loudspeaker-driven traveling-wave thermoacoustic refrigerator to demonstrate the dependence of temperature decrease on the hydraulic radius (rh) of regenerator. The refrigerator has a looped tube and a resonator tube, with the length of 260 cm and 150 cm, respectively. Six different regenerators are used, each of which is made of a tight stack of stainless-steel wire-mesh screens with different mesh sizes, providing rh from 0.07 mm until 0.28 mm. The regenerator length is 60 mm. Atmospheric air is used as the working gas, driven at operating frequency of 30 Hz, giving thermal penetration depth (δκ) of 0.49 mm. The experimental results show that there is an optimum rh which gives the largest temperature decrease at the cooling point. It is found, in this case, that the optimum rh is 0.10 mm (that is, rh/δκ = 0.20) and the corresponding temperature decrease is 18.7°C (i.e. from 28.0 °C down to 9.3 °C).

Design Analysis of Thermoacoustic Refrigerator Using Air and Helium as Working Substances

In this paper, thermoacoustic refrigerator design strategy with parameters normalization and literature review covering the recent development in the modification of the resonator shape and size is discussed. The design of a 10 W cooling power thermoacoustic refrigerator using air as working substance and the effect of operating frequency on viscous and thermal penetration depths, and on stack sheet thickness and spacing are discussed. The promising 10 W cooling power TDH (Taper and Divergent section with Hemispherical end) resonator design operating with air and helium gases as working substances is analyzed using DeltaEC software. The analysis results show that the TDH resonator design using helium as working substance operates at lower drive ratio (14%) compared to air (25%). In comparison, DeltaEC predicts a decent low temperature of -35.4 o C at cold heat exchanger with a COP of 0.5294 when operated using helium gas, and for air is -9 oC and 0.8113 respectively, and the results are discussed.

Numerical study of heat transfer in parallel-plate heat exchangers in thermoacoustic engine

For designing thermoacoustic devices, heat transfer problem of heat exchangers is a key point and is not solved completely yet. In the present study, both computational fluid dynamics (CFD) simulations and numerical analysis based on linear thermoacoustic theory are performed, and heat transfer characteristics, heat fluxes, and Nusselt numbers are investigated from both numerical results. Especially, through parametric studies by linear analysis, the effect of displacement amplitude and thermal penetration depth on heat transfer is discussed. In this simulation, a regenerator is located between hot and cold heat exchangers, and the effect of each channel size is also investigated. Furthermore, simulated results are compared with the conventional heat transfer models for oscillatory flow, and the verifications of those models or simulations are executed. On the other hand, by comparing the CFD results with linear analysis, non-linearity of heat transfer included in thermoacoustic phenomena is also discusse...

Application of the thermoelectric phenomena to study the unsteady-state thermal conductivity

We present an experimental set-up designed to investigate the unsteady-state thermal conductivity. A sine-shaped thermal wave is produced by a thermoelectric device and the change in temperature at two points in a metal rod is measured. The investigation is carried out for seven thermal wave frequencies. The thermal wave penetration depth and the thermal conductivity are determined by two methods: from the wave amplitude ratio and from the wave phase shift at two locations. The presented system also offers a determination of the thermal wave propagation velocity and the thermal diffusivity coefficient of the medium. The obtained measurement results are discussed. The specification of the measurement system is preceded by a theoretical and comprehensive description of the phenomena taking part in the experiment. With regard to the role of thermoelectric phenomena in contemporary science and technology the presented experiment is suitable for students in university laboratories studying metrology, electronics, space technology, energy harvesting, energo-mechanics, renewable energy science, chemical technology, bio-engineering and other similar courses.

Effect of quantum correction on nonlinear thermal wave of electrons driven by laser heating

In thermal interaction of laser pulse with a deuterium-tritium (DT) plane, the thermal waves of electrons are generated instantly. Since the thermal conductivity of electron is a nonlinear function of temperature, a nonlinear heat conduction equation is used to investigate the propagation of waves in solid DT. This paper presents a self-similar analytic solution for the nonlinear heat conduction equation in a planar geometry. The thickness of the target material is finite in numerical computation, and it is assumed that the laser energy is deposited at a finite initial thickness at the initial time which results in a finite temperature for electrons at initial time. Since the required temperature range for solid DT ignition is higher than the critical temperature which equals 35.9 eV, the effects of quantum correction in thermal conductivity should be considered. This letter investigates the effects of quantum correction on characteristic features of nonlinear thermal wave, including temperature, penetration depth, velocity, heat flux, and heating and cooling domains. Although this effect increases electron temperature and thermal flux, penetration depth and propagation velocity are smaller. This effect is also applied to re-evaluate the side-on laser ignition of uncompressed DT.

Thermal depth profiling of materials for defect detection using hot disk technique

A novel application of the hot disk transient plane source technique is described. The new application yields the thermal conductivity of materials as a function of the thermal penetration depth which opens up opportunities in nondestructive testing of inhomogeneous materials. The system uses the hot disk sensor placed on the material surface to create a time varying temperature field. The thermal conductivity is then deduced from temperature evolution of the sensor, whereas the probing depth (the distance the heat front advanced away from the source) is related to the product of measurement time and thermal diffusivity. The presence of inhomogeneity in the structure is manifested in thermal conductivity versus probing depth plot. Such a plot for homogeneous materials provides fairly constant value. The deviation from the homogeneous curve caused by defects in the structure is used for inhomogeneity detection. The size and location of the defect in the structure determines the sensitivity and possibility of detection. In addition, a complementary finite element numerical simulation through COMSOL Multiphysics is employed to solve the heat transfer equation. Temperature field profile of a model material is obtained from these simulations. The average rise in temperature of the heat source is calculated and used to demonstrate the effect of the presence of inhomogeneity in the system.

Experimental study of the influence of cold heat exchanger geometry on the performance of a co-axial pulse tube cooler

Improving the performance of the pulse tube cooler is one of the important objectives of the current studies. Besides the phase shifters and regenerators, heat exchangers also play an important role in determining the system efficiency and cooling capacity. A series of experiments on a 10W @ 77K class co-axial type pulse tube cooler with different cold heat exchanger geometries are presented in this paper. The cold heat exchangers are made from a copper block with radial slots, cut through using electrical discharge machining. Different slot widths varying from 0.12mm to 0.4mm and different slot numbers varying from around 20–60 are investigated, while the length of cold heat exchangers are kept the same. The cold heat exchanger geometry is classified into three groups, namely, constant heat transfer area, constant porosity and constant slot width. The study reveals that a large channel width of 0.4mm (about ten times the thermal penetration depth of helium gas at 77K, 100Hz and 3.5MPa) shows poor performance, the other results show complicated interaction effects between slot width and slot number. These systematic comparison experiments provide a useful reference for selecting a cold heat exchanger geometry in a practical cooler.

Experimental Study on a Standing Wave Thermoacoustic Prime Mover with Air Working Gas at Various Pressures

Thermoacoustic prime mover is an energy conversion device which converts thermal energy into acoustic work (sound wave). The advantages of this machine are that it can work with air as the working gas and does not produce any exhaust gases, so that it is environmentally friendly. This paper describes an experimental study on a standing wave thermoacoustic prime mover with air as the working gas at various pressures from 0.05 MPa to 0.6 MPa. We found that 0.2 MPa is the optimum pressure which gives the lowest onset temperature difference of 355 °C. This pressure value would be more preferable in harnessing low grade heat sources to power the thermoacoustic prime mover. In addition, we find that the lowest onset temperature difference is obtained when rh/δk ratio is 2.85, where rh is the hydraulic radius of the stack and δk is the thermal penetration depth of the gas. Moreover, the pressure amplitude of the sound wave is significantly getting larger from 2.0 kPa to 9.0 kPa as the charged pressure increases from 0.05 MPa up to 0.6 MPa.

Thermal dynamics of thermoelectric phenomena from frequency resolved methods

Understanding the dynamics of thermoelectric (TE) phenomena is important for the detailed knowledge of the operation of TE materials and devices. By analyzing the impedance response of both a single TE element and a TE device under suspended conditions, we provide new insights into the thermal dynamics of these systems. The analysis is performed employing parameters such as the thermal penetration depth, the characteristic thermal diffusion frequency and the thermal diffusion time. It is shown that in both systems the dynamics of the thermoelectric response is governed by how the Peltier heat production/absorption at the junctions evolves. In a single thermoelement, at high frequencies the thermal waves diffuse semi-infinitely from the junctions towards the half-length. When the frequency is reduced, the thermal waves can penetrate further and eventually reach the half-length where they start to cancel each other and further penetration is blocked. In the case of a TE module, semi-infinite thermal diffusion along the thickness of the ceramic layers occurs at the highest frequencies. As the frequency is decreased, heat storage in the ceramics becomes dominant and starts to compete with the diffusion of the thermal waves towards the half-length of the thermoelements. Finally, the cancellation of the waves occurs at the lowest frequencies. It is demonstrated that the analysis is able to identify and separate the different physical processes and to provide a detailed understanding of the dynamics of different thermoelectric effects.

Thermal wave imaging of microelectronics

Thermal waves can reveal thermal properties of different layers forming a multilayer structure. If the thickness of each layer is known, specific ranges of thermal wave frequencies can be implemented to study the thermal response of a specific number of layers and eventually extract the thermal properties of individual layers. As a first approach this idea can be simplified by means of the thermal penetration depth parameter, δ. The thermal penetration depth is defined as, δ=k/πCf, where k and C are respectively the thermal conductivity and volumetric heat capacity of the material carrying the thermal wave and f is the frequency of the thermal wave. From this expression it can be seen how it is possible to constrain the material thermal response to a desired depth by controlling the frequency. Thus, using high enough frequencies, the top layer properties would be measured first. Decreasing the thermal wave frequency by an appropriate amount would include the next layer in the thermal response. Since the properties of the first layer are now known, it would be possible to extract the properties of the current layer. The measurement would continue in a similar fashion for the remaining layers. Frequency domain thermoreflectance (FDTR) can be used to generate thermal waves. In this technique, a periodically modulated continuous wave laser (red pump beam) provides the periodic heat flux input into the material while a second laser (green probe beam) monitors the surface temperature through a proportional change of the surface reflectivity. The measured value is the phase lag (degrees) between the incoming thermal wave and the surface temperature response. In this study, an FDTR system was used in conjunction with a piezo stage to obtain thermal images of two different multilayer structures. The first one consisted of a CPU chip formed mainly by layers of SiO2 and Cu. The second case consisted of a TFT LCD screen from a mobile device. Regarding the CPU chip, the low frequency thermal wave travelled well past the second layer of Cu wires and provided thermal information about the bottom layers of the chip. In contrast, the high frequency wave could not penetrate through the second layer, which resulted in a more sensitive response upon the Cu wires close to the surface. A similar phenomenon occurred with the LCD screen. In this case the top layer was a glass layer used to sandwich the liquid crystal and the second layer is composed of the ITO electrodes that provide the electric field. It can be observed how the high frequency wave did not penetrate through the top glass layer providing no thermal information about the bottom layer as opposed to the low frequency wave, which clearly shows the ITO electrodes. The estimated thermal penetration depths displayed on top of each image were calculated using the equation provided before with known thermal properties of SiO2, Cu and ITO.