Diffusive Heat Transport Research Articles

Direct current (DC) resistivity and induced polarization (IP) monitoring of active layer dynamics at high temporal resolution

With permafrost thawing and changes in active layer dynamics induced by climate change, interactions between biogeochemical and thermal processes in the ground are of great importance. Here, active layer dynamics have been monitored using direct current (DC) resistivity and induced polarization (IP) measurements at high temporal resolution and at a relatively large scale at a heath tundra site on Disko Island on the west coast of Greenland (69°N). At the field site, the active layer is disconnected from the deeper permafrost, due to isothermal springs in the region. Borehole sediment characteristics and subsurface temperatures supplemented the DC-IP measurements. A time-lapse DC-IP monitoring system has been acquiring at least six datasets per day on a 42-electrode profile with 0.5m electrode spacing since July 2013. Remote control of the data acquisition system enables interactive adaptation of the measurement schedule, which is critically important to acquire data in the winter months, where extremely high contact resistances increase the demands on the resistivity meter. Data acquired during the freezing period of October 2013 to February 2014 clearly image the soil freezing as a strong increase in resistivity. While the freezing horizon generally moves deeper with time, some variations in the freezing depth are observed along the profile. Comparison with depth-specific soil temperature indicates an exponential relationship between resistivity and below-freezing temperature. Time-lapse inversions of the full-decay IP data indicate a decrease of normalized chargeability with freezing of the ground, which is the result of a decrease in the total unfrozen water and of the higher ion concentration in the pore-water. We conclude that DC-IP time-lapse measurements can non-intrusively and reliably image freezing patterns and their lateral variation on a 10–100m scale that is difficult to sample by point measurements. In combination with laboratory experiments, the different patterns in resistivity and chargeability changes will enable the disentanglement of processes (e.g., fluid migration and freezing, advective and diffusive heat transport) occurring during freezing of the ground. The technology can be expanded to three dimensions and also to larger scale.

A simple Boltzmann transport equation for ballistic to diffusive transient heat transport

Developing simplified, but accurate, theoretical approaches to treat heat transport on all length and time scales is needed to further enable scientific insight and technology innovation. Using a simplified form of the Boltzmann transport equation (BTE), originally developed for electron transport, we demonstrate how ballistic phonon effects and finite-velocity propagation are easily and naturally captured. We show how this approach compares well to the phonon BTE, and readily handles a full phonon dispersion and energy-dependent mean-free-path. This study of transient heat transport shows (i) how fundamental temperature jumps at the contacts depend simply on the ballistic thermal resistance, (ii) that phonon transport at early times approach the ballistic limit in samples of any length, and (iii) perceived reductions in heat conduction, when ballistic effects are present, originate from reductions in temperature gradient. Importantly, this framework can be recast exactly as the Cattaneo and hyperbolic heat equations, and we discuss how the key to capturing ballistic heat effects is to use the correct physical boundary conditions.

The Graetz–Nusselt problem extended to continuum flows with finite slip

AbstractGraetz and Nusselt studied heat transfer between a developed laminar fluid flow and a tube at constant wall temperature. Here, we extend the Graetz–Nusselt problem to dense fluid flows with partial wall slip. Its limits correspond to the classical problems for no-slip and no-shear flow. The amount of heat transfer is expressed by the local Nusselt number $\mathit{Nu}_{x}$, which is defined as the ratio of convective to conductive radial heat transfer. In the thermally developing regime, $\mathit{Nu}_{x}$ scales with the ratio of position $\tilde{x}=x/L$ to Graetz number $\mathit{Gz}$, i.e. $\mathit{Nu}_{x}\propto (\tilde{x}/\mathit{Gz})^{-{\it\beta}}$. Here, $L$ is the length of the heated or cooled tube section. The Graetz number $\mathit{Gz}$ corresponds to the ratio of axial advective to radial diffusive heat transport. In the case of no slip, the scaling exponent ${\it\beta}$ equals $1/3$. For no-shear flow, ${\it\beta}=1/2$. The results show that for partial slip, where the ratio of slip length $b$ to tube radius $R$ ranges from zero to infinity, ${\it\beta}$ transitions from $1/3$ to $1/2$ when $10^{-4}<b/R<10^{0}$. For partial slip, ${\it\beta}$ is a function of both position and slip length. The developed Nusselt number $\mathit{Nu}_{\infty }$ for $\tilde{x}/\mathit{Gz}>0.1$ transitions from 3.66 to 5.78, the classical limits, when $10^{-2}<b/R<10^{2}$. A mathematical and physical explanation is provided for the distinct transition points for ${\it\beta}$ and $\mathit{Nu}_{\infty }$.

Diffusive and phase change instabilities in a ternary mushy layer

AbstractWe analyse the stability of a mushy layer during the directional solidification of a ternary alloy. Our model includes diffusive and convective transport of heat and solutes, coupled by an equilibrium thermodynamic constraint of the ternary phase diagram. The model contains phase change effects due to latent-heat release, solute rejection and background solidification. We identify novel convective instabilities, both direct and oscillatory, which are present under statically stable conditions. We examine these instabilities asymptotically by simplifying to a thin mushy layer with small growth rates. We also discuss numerical results for the full problem, confirming the asymptotic predictions and providing the stability characteristics outside the small-growth-rate approximation. A physical explanation for these instabilities in terms of parcel arguments is proposed, indicating that the instability mechanisms generally involve different rates of solute diffusion, different rates of solute rejection and different background solute distributions induced by the initial alloy composition.

Importance of thermal diffusion in the gravomagnetic limit cycle

We consider the role of thermal diffusion due to turbulence and radiation on accretion bursts that occur in protoplanetary discs which contain dead zones. Using 1D viscous disc models we show that diffusive radial transport of heat is important during the gravo-magnetic limit cycle, and can strongly modify the duration and frequency of accretion outbursts. When the Prandtl number is large - such that turbulent diffusion of heat is unimportant - radial radiative diffusion reduces the burst duration compared to models with no diffusive transport of heat. When the Prandtl number is small ($\lesssim 25$) we find that turbulent diffusion dominates the radial transport of heat, reducing the burst duration to $\lesssim 10^3$ years as well as increasing the burst frequency. Furthermore, inclusion of radial transport of heat extends the range of infall rates under which the disc undergoes accretion bursts from $10^{-8}$ to $10^{-6}$ M$_\odot$ yr$^{-1}$ with no diffusion, to $10^{-8}$ to $\gtrsim10^{-4}$ M$_\odot$ yr$^{-1}$ with radiative and strong turbulent diffusion. The relative roles of radiative and turbulent thermal diffusion are likely to vary during an accretion burst, but simple estimates suggest that the expected Prandtl numbers are of the order of 10 in protoplanetary discs, and hence that turbulent diffusion is likely to be an important process during accretion outbursts due to the gravo-magnetic limit cycle.

Finite element analysis of transient ballistic–diffusive phonon heat transport in two-dimensional domains

While sub-continuum heat conduction becomes more important as the size of micro/nanodevices keeps shrinking under the mean free path of heat carriers, its computation still remains challenging to the general engineering community due to the lack of easily accessible numerical simulation tools. To address this challenge, this article reports the finite element analysis (FEA) of transient ballistic-diffusive phonon heat transport in a two-dimensional domain using a commercial package (COMSOL Multiphysics). The Boltzmann transport equation under the gray relaxation-time approximation was numerically solved by discretizing the angular domain with the discrete ordinate method (DOM) and the spatial domain with the FEA. The DOM-FEA method was validated by comparing the results with different benchmark studies, such as the equation of phonon radiative transfer, the ballistic-diffusive equation, and the finite difference method of the phonon Boltzmann transport equation. The calculation of phonon heat transport for a 2-D square slab reveals that heat conduction becomes more ballistic with temperature jumps at boundaries as Knudsen number (Kn) increases. The ballistic nature also significantly affects transient thermal behaviors at high Kn numbers. The obtained results clearly demonstrate the capability of the DOM-FEA as a promising engineering tool for calculating sub-continuum phonon heat transport. (C) 2014 Elsevier Ltd. All rights reserved. (Less)

How does the Redi parameter for mesoscale mixing impact global climate in an Earth System Model?

AbstractA coupled climate model is used to examine the impact of an increase in the mixing due to mesoscale eddies on the global climate system. A sixfold increase in the Redi mixing coefficient ARedi, which is within the admissible range of variation, has the overall effect of warming the global‐mean surface air and sea surface temperatures by more than 1°C. Locally, sea surface temperatures increase by up to 7°C in the North Pacific and by up to 4°C in the Southern Ocean, with corresponding impacts on the ice concentration and ice extent in polar regions. However, it is not clear that the changes in heat transport from tropics to poles associated with changing this coefficient are primarily responsible for these changes. We found that the changes in the transport of heat are often much smaller than changes in long‐wave trapping and short‐wave absorption. Additionally, changes in the advective and diffusive transport of heat toward the poles often oppose each other. However, we note that the poleward transport of salt increases near the surface as ARedi increases. We suggest a causal chain in which enhanced eddy stirring leads to increased high‐latitude surface salinity reducing salt stratification and water column stability and enhancing convection, triggering two feedback loops. In one, deeper convection prevents sea ice formation, which decreases albedo, which increases SW absorption, further increasing SST and decreasing sea ice formation. In the other, increased SST and reduced sea ice allow for more water vapor in the atmosphere, trapping long‐wave radiation. Destratifying the polar regions is thus a potential way in which changes in ocean circulation might warm the planet.

Goldstone modes in the neutron star core

We studied the effect of the diverse Goldstone boson modes in the transport properties of dense neutron matter. The two Goldstone bosons associated with density oscillations of the neutron and electron + proton fluids, called superfluid phonons, mix and couple strongly to electrons reducing their mean free time and suppressing their contribution to transport coefficients. For typical neutron star temperatures in the range T=106–109 K, the Goldstone modes associated with rotational symmetry, called angulons, couple weakly to each other and to electrons and, consequently, have anomalously large mean free paths and can contribute to both diffusive and ballistic transport of heat and momentum. Still, their contribution is smaller that coming from the electrons on account of their smaller density. We speculate that long-wavelength superfluid phonons and angulons can play a role in neutron star seismology, and lead to interesting phenomenology especially since angulons couple to magnetic fields and have anisotropic dispersion relations.

Optimal performance and entropy generation transition from micro to nanoscaled thermoelectric layers

In this paper we address the heat and electric charge transport in a thermoelectric thin layer. The analysis is based on hyperbolic type equations describing the time evolution of dissipative flows including size effects on the thermal and electric conductivities. We explore the effects of the layer width on the time evolution of the system and, particularly, on the thermal figure of merit and the entropy generation. We find a transition between two dynamic regimes controlled by the system’s size when going from the micro to the nanometric scale of lengths. The transition is featured by a marked increasing of the thermal figure of merit and the total entropy generation. The low entropy generation regime is dominated by the diffusive transport of heat and the regime with high entropy generation values by wave energy transfer (this last corresponding to the nanometric scale). We use here the spectral methods of solution which assure a well representation of wave behaviour of heat given their spectral convergence.

Microfluidic platforms: a mainstream technology for the preparation of crystals

Microfluidics is a multidisciplinary field of science based on the manipulation of fluids in sub-millimeter dimensions where the fundamental fluid physics changes dramatically when compared to macroscale fluid physical phenomena. Therefore, the conditions that microfluidic technologies offer are completely different from those of bulk set-ups, and thus they are very interesting for the study of crystallisation because diffusion, mixing and mass and heat transport are all finely controlled, and are easily modulated. This tutorial review is intended to give a broad and up-to-date overview of the distinct microfluidic approaches that have been employed so far for crystallisation studies for the uninitiated in these techniques. Main emphasis will be given to microfluidic platforms operating under continuous flow regimes, droplet-based methods, valve-based approaches, well-based methods, and digital microfluidics. This tutorial does not intend to give detailed methodology, but rather provides illustrative examples which capture the attention of the reader and allow them to appreciate the unique features that microfluidic technologies can offer towards the study of different crystallisation processes. Indeed, crystallisation studies of different types of crystalline matter including organic, inorganic and metal-organic materials are presented.

A multiscale thermal modeling approach for ballistic and diffusive heat transport in two dimensional domains

The modeling of heat transport for small length scales requires accounting for the physics of heat carrier transport such as the propagation, scattering, and relaxation of phonons. However, the models used to describe such phenomena are often too computationally expensive to model large domains where relevant system boundaries are far removed from the region where ballistic-diffusive phonon transport is dominant. To address the response of such systems to its far field thermal boundary conditions, a multiscale thermal model is needed to efficiently account for transport phenomena in each domain. In this work, a multiscale thermal modeling approach is presented where the transport of phonons is treated with a Finite Volume Discrete Ordinates Model (FVDOM) solution to the phonon Boltzmann Transport Equation (BTE) which is embedded in a region treated by diffusive thermal transport. The approach shows that it is possible to create a coupled multiscale model where the boundary conditions of the FVDOM model are derived from the diffusive transport model. The limitations of the method and the key parameters for accurate modeling are investigated and discussed. The final model is used to explore the impact of the size of heat generation regions with respect to the overall domain size on the peak temperature. The results show that by using such a model it is possible to capture the ballistic-diffusive transport in pertinent areas of a domain that is mostly dominated by diffusive transport.

Size dependence of efficiency at maximum power of heat engine

We perform a molecular dynamics computer simulation of a heat engine model to study how the engine size difference affects its performance. Upon tactically increasing the size of the model anisotropically, we determine that there exists an optimum size at which the model attains the maximum power for the shortest working period. This optimum size locates between the ballistic heat transport region and the diffusive heat transport one. We also study the size dependence of the efficiency at the maximum power. Interestingly, we find that the efficiency at the maximum power around the optimum size attains a value that has been proposed as a universal upper bound, and it even begins to exceed the bound as the size further increases. We explain this behavior of the efficiency at maximum power by using a linear response theory for the heat engine operating under a finite working period, which naturally extends the low-dissipation Carnot cycle model [M. Esposito, R. Kawai, K. Lindenberg, C. Van den Broeck, Phys. Rev. Lett. 105, 150603 (2010)]. The theory also shows that the efficiency at the maximum power under an extreme condition may reach the Carnot efficiency in principle.

Ultrasonic Intensification of Grape Stalk Convective Drying: Kinetic and Energy Efficiency

Grape stalk is a by-product of the winemaking process with a high antioxidant content. Drying is a necessary stage before antioxidant extraction, which may affect not only kinetic and energy efficiency but also product quality. Coupling non-thermal technologies, such as power ultrasound, to convective drying is considered a strategy for process intensification in order to prevent certain drawbacks of conventional technologies. In this work, the use of power ultrasound in the convective drying of grape stalk was tested in order to estimate its influence on kinetic and energy efficiency. For this purpose, convective drying kinetics of grape stalk were carried out with and without power ultrasound application (21.8 kHz, at two ultrasonic power levels: 45 and 90 W). In addition, the inlet and outlet air temperatures of the drying chamber were monitored. The drying process was modeled considering heat and mass transfer phenomena jointly with the total energy consumption and the energy efficiency of the system. Power ultrasound application involved a shortening of grape stalk drying time, which was dependent on the drying air temperature (40 and 60°C) and the ultrasonic power applied (45 and 90 W). The modeling showed the increase in diffusion and convective heat transport phenomena produced by ultrasound application, despite grape stalk being a low-porosity product and, therefore, only slightly sensitive to ultrasonic effects. In addition, it was also highlighted that ultrasound application increased the energy efficiency during the drying of grape stalk.

Transition between ballistic and diffusive heat transport regimes in silicon materials

We study the extent of ballistic and diffusive thermal transport and the range of application of the Casimir and Fourier theories in semiconductor materials by using a theoretical model based on the Boltzmann transport equation. We show that combined effects of length scale, temperature, and boundary roughness are responsible for thermal transport transitions in silicon nanowires and thin films. We also introduce a more accurate principle for ballistic transport that considers the balance between internal and surface scattering. Phonon quantum confinement effects as well as the conditions for phonon wave interference in nanoscale heat transport are discussed.

A numerical benchmark test for continuous casting of steel

There is a continuously developing need for benchmarking of solidification simulations - from the theoretical as well as from the applied points of view. The benchmarking is usually done in two parts. The verification part confirms the proper numerical solution (Are we solving the equations correctly?) and the validation part (Are we solving the right equations?) confirms the proper response of the simulations regarding the experimental evidence. The history of related benchmarking shows differences of the results between different numerical methods, and differences in comparison with the experiments when solving even quite simple solidification situations. The present benchmark test proposes the verification benchmark for continuous casting of steel. Since the simulations of the temperatures, velocities, pressures and concentrations in the continuous casting represent a multiscale and multiphysics problem of high complexity (far away from a closed form solution), the verification can be done only by comparing the results of different numerical methods for solving the same governing equations. This has been done in the context of continuous casting by breaking the considered two dimensional problem into several sub problems by increasing complexity of geometry (straight in vertical direction, curved), boundary conditions (linear, non-linear), material properties (only Fe and Fe - C), microscopic considerations (Lever rule and Scheil rule). The governing equations complexity first involves only convective - diffusive heat transport with a predetermined velocity field, and assumed laminar and turbulent velocity calculations afterwards, modelled by a low Reynolds number turbulence model. Further complications involve the presence of the electromagnetic forces. The paper represents guidelines for the presentation of the numerical method, discretisation and results, as well as some of the results, obtained by the commercial finite volume based code and our in-house meshless method based laboratory code.

Determination of electron thermal diffusivity at the WEGA stellarator

For determining the thermal diffusivity in WEGA the electron cyclotron resonance heating power from a 28 GHz gyrotron is square-wave modulated to provide a periodically varying energy source at the plasma centre causing a modulation mainly of the electron temperature Te. A fast Si-diode bolometer system, whose 16 channel view the entire plasma cross-section, is employed to monitor the heat propagation process. A fast Fourier transform analysis shows clear coherency of the line-integrated signals among different channels and monotonic radial increment of the phase delays of the fundamental components in the central channel signals, allowing the determination of the electron thermal diffusivity coefficient D. A purely diffusive heat transport is assumed and is simulated using a Monte Carlo method. The thermal diffusivity is determined by matching the simulated results to the measured ones. A typical value of D = 1.9 m2 s−1 is obtained. This value is then compared with the result based on a local power balance analysis. Within the error bars the local power balance calculation yields a similar diffusivity value. The limitations and conditions of using bolometric diagnostic for this purpose are discussed.

Simulating Interfacial Thermal Conductance at Metal-Solvent Interfaces: The Role of Chemical Capping Agents

With the non-isotropic velocity scaling (NIVS) approach to reverse non-equilibrium molecular dynamics (RNEMD), it is possible to impose an unphysical thermal flux between different regions of inhomogeneous systems such as solid/liquid interfaces. We have applied NIVS to compute the interfacial thermal conductance at a metal/organic solvent interface that has been chemically capped by butanethiol molecules. Our calculations suggest that coupling between the metal and liquid phases is enhanced by the capping agents, leading to a greatly enhanced conductivity at the interface. Specifically, the chemical bond between the metal and the capping agent introduces a vibrational overlap that is not present without the capping agent, and the overlap between the vibrational spectra (metal to cap, cap to solvent) provides a mechanism for rapid thermal transport across the interface. Our calculations also suggest that this is a nonmonotonic function of the fractional coverage of the surface, as moderate coverages allow diffusive heat transport of solvent molecules that have been in close contact with the capping agent.

Shear heating-induced thermal pressurization during earthquake nucleation

[1] We model earthquake nucleation (in 2D) on narrow faults with coupled rate-state friction and shear heating-induced thermal pressurization, including diffusive transport of heat and pore pressure. Thermal pressurization increases pore pressure p, decreasing frictional resistance. Observed fault core permeability is generally too low to mitigate thermal pressurization at subseismic slip speeds. Under drained, isothermal conditions, nucleation with the aging law is crack like, with the interior of the slip zone always near maximum slip speed. When thermal pressurization is included, it can dominate weakening at speeds of 0.02–20 mm/s for hydraulic diffusivities chyd from 10−8 to 10−3 m2/s and nominal material parameters well before seismic radiation occurs. Dramatic along-strike localization of slip occurs due to feedback in which the area of maximum slip experiences the greatest weakening, which in turn favors more slip. With the slip law, however, nucleation is pulse like, with slip speed decaying behind the pulse tip. Thermal pressurization is diminished relative to the aging law case since most weakening occurs in locations with limited slip, yet we find that it can overwhelm frictional weakening at slip speeds in the range of 1–100 mm/s for chyd from 10−8 to 3 × 10−5 m2/s. At higher slip speeds, the finite thickness of the shear zone becomes significant, reducing thermal pressurization. Even if not the dominant weakening mechanism, thermal pressurization is likely to be significant at or before the onset of seismic radiation.

The Role of Nonuniformity in Convective Heat and Mass Transfer through Porous Media, Part 2

Nonuniformities in porous materials can play a significant role in the convective and diffusive transport of fluid, heat, and mass. This study provides experimental results and corresponding numerical simulation results. The experiments report continuous data in transient test runs with measurements of temperature and velocity at distributed locations in the domain immediately downstream of the porous materials. An increasing degree of nonuniformity was found to produce a lower drying rate as well as an earlier onset of falling rate drying. The numerical model provides results of the effects of different nonhomogeneities such as distributed holes or distributed regions of varying permeability and moisture content. Comparisons between numerical and experimental test results indicate general agreement with differences with regard to details of the drying curve features. This provides a tool for studying the role of nonuniformity in fluid flow and heat and mass transfer in porous media.

Self-consistent electrothermal analysis of nanotube network transistors

We develop an electrothermal transport model for nanocomposite thin films based on self-consistent solution of drift-diffusion and Poisson equations for electrons coupled with diffusive transport of heat. This model is used to analyze the performance of an electronic display the pixels of which are controlled by carbon nanotube (CNT) network thin-film transistors (TFTs). The effect of electrothermal coupling on device performance and steady state temperature rise is analyzed as a function of key device parameters such as channel length, network density, tube-to-substrate thermal conductance, and tube-to-substrate thermal conductivity ratio. Our analysis suggests that device on-current Ion may reduce by 30% for a 1 μm channel length devices due to self-heating. The temperature rise in such devices can be as high as 500 K in extreme cases due to the thermally insulating substrate and the low tube-to-substrate thermal conductance. These results suggest that an appropriate combination of network density, channel length and width should be selected for CNT-TFTs to avoid device temperature rise above acceptable limits. We analyze the effectiveness of active cooling in reducing the temperature and enhancing the performance of the device. We find that the high thermal spreading resistance between the CNT device and the electronic display reduces the effectiveness of forced convective cooling, necessitating the exploration of alternative designs for viable CNT-FET based display technology.