Steady-state Heat Transport Research Articles

Gaussian Process Regression with Soft Equality Constraints

This study introduces a novel Gaussian process (GP) regression framework that probabilistically enforces physical constraints, with a particular focus on equality conditions. The GP model is trained using the quantum-inspired Hamiltonian Monte Carlo (QHMC) algorithm, which efficiently samples from a wide range of distributions by allowing a particle’s mass matrix to vary according to a probability distribution. By integrating QHMC into the GP regression with probabilistic handling of the constraints, this approach balances the computational cost and accuracy in the resulting GP model, as the probabilistic nature of the method contributes to shorter execution times compared with existing GP-based approaches. Additionally, we introduce an adaptive learning algorithm to optimize the selection of constraint locations to further enhance the flexibility of the method. We demonstrate the effectiveness and robustness of our algorithm on synthetic examples, including 2-dimensional and 10-dimensional GP models under noisy conditions, as well as a practical application involving the reconstruction of a sparsely observed steady-state heat transport problem. The proposed approach reduces the posterior variance in the resulting model, achieving stable and accurate sampling results across all test cases while maintaining computational efficiency.

Modulation of Steady-State Heat Transport in a Dissipative Multi-Mode Qubit-Photon System

Quantum heat transport is considered as an indispensable branch of quantum thermodynamics to potentially improve performance of thermodynamic devices. We theoretically propose a dissipative qubit-photon system composed of multiple coupled resonators interacting with a single two-level qubit, to explore the steady-state heat transport by tuning both the inter-resonator photon hopping and the qubit-photon coupling. Specifically in the three-mode case, the dramatic enhancement and suppression of the heat current into the central resonator can be modulated by the corresponding frequency, compared to the currents into two edge resonators. Moreover, fruitful cycle current components are unraveled at weak qubit-photon coupling, which are crucial to exhibit the nonmonotonic feature with increase of the reservoir temperature bias. In the one-dimensional case under the mean-field framework, the influence of the photon hopping on heat transport is analyzed. The steady-state heat current is comparatively enhanced to the single-mode limit at weak qubit-photon coupling, stemming from the nonvanishing mean-field photon excitation parameter and the additional cycle current component. We hope these obtained results may have possible applications in quantum thermodynamic manipulation and energy harvesting.

Environment-Assisted Modulation of Heat Flux in a Bio-Inspired System Based on Collision Model.

The high energy transfer efficiency of photosynthetic complexes has been a topic of research across many disciplines. Several attempts have been made in order to explain this energy transfer enhancement in terms of quantum mechanical resources such as energetic and vibration coherence and constructive effects of environmental noise. The developments in this line of research have inspired various biomimetic works aiming to use the underlying mechanisms in biological light harvesting complexes for the improvement of synthetic systems. In this article, we explore the effect of an auxiliary hierarchically structured environment interacting with a system on the steady-state heat transport across the system. The cold and hot baths are modeled by a series of identically prepared qubits in their respective thermal states, and we use a collision model to simulate the open quantum dynamics of the system. We investigate the effects of system-environment, inter-environment couplings and coherence of the structured environment on the steady state heat flux and find that such a coupling enhances the energy transfer. Our calculations reveal that there exists a non-monotonic and non-trivial relationship between the steady-state heat flux and the mentioned parameters.

Steady State Heat Transport by Microbubble Dispersions Mediating Convection With Phase Change Dynamics

A new theory for additional heat transfer convected by a dispersed phase of microbubbles was posited recently. An additional convection term in the heat transport equation reflects the latent heat of vapor of the liquid carried by the microbubbles from hot zones that vaporize more liquid to cold zones where condensation releases the latent heat. This theory was shown to be consistent with analysis of observations of freezing times measured by in the original Mpemba effect study, by inferring heat transfer coefficients fitted by Newton’s law of cooling. In this paper, the scaling analysis, leading to the proposition that the additional heat flux is proportional to the phase fraction of microbubbles, is tested by steady state solutions of the canonical hot wall / cold wall buoyant convection problem. For phase fractions 0.02 and 0.1, the maximum ratio of additional Nusselt number emergent is five, occurring in the microfluidic regime. Increasing the characteristic length of the domain maintains the monotonicity of the increase in additional Nusselt number ratio over the case of no microbubbles present. The additional heat transfer due to the microbubble dispersion, ranging from 5-50%, is found to be nearly proportional to the microbubble phase fraction for the range of 0.02 to 0.2. However, larger characteristic lengths introduce insufficient heat flux from the hot wall to maintain a “driven cavity” flow structure, so that the steady state structure that emerges is a stable stratification with thin boundary layers near the hot and cold walls, with weak shear flow convection. The stable stratification resultant at higher characteristic lengths suppresses the additional heat flux due to microbubble mediation, but only moderately deviating from proportionality.

Steady-State Heat Transport and Work With a Single Artificial Atom Coupled to a Waveguide: Emission Without External Driving

We observe the continuous emission of photons into a waveguide from a superconducting qubit without the application of an external drive. To explain this observation, we build a two-bath model where the qubit couples simultaneously to a cold bath (the waveguide) and a hot bath (a secondary environment). Our results show that the thermal-photon occupation of the hot bath is up to 0.14 photons, 35 times larger than the cold waveguide, leading to nonequilibrium heat transport with a power of up to 132 zW, as estimated from the qubit emission spectrum. By adding more isolation between the sample output and the first cold amplifier in the output line, the heat transport is strongly suppressed. Our interpretation is that the hot bath may arise from active two-level systems being excited by noise from the output line. We also apply a coherent drive, and use the waveguide to measure thermodynamic work and heat, suggesting waveguide spectroscopy is a useful means to study quantum heat engines and refrigerators. Finally, based on the theoretical model, we propose how a similar setup can be used as a noise spectrometer which provides a new solution for calibrating the background noise of hybrid quantum systems.

Harmonic chains and the thermal diode effect.

Harmonic oscillator chains connecting two harmonic reservoirs at different constant temperatures cannot act as thermal diodes, irrespective of structural asymmetry. However, here we prove that perfectly harmonic junctions can rectify heat once the reservoirs (described by white Langevin noise) are placed under temperature gradients, which are asymmetric at the two sides, an effect that we term "temperature-gradient harmonic oscillator diodes." This nonlinear diode effect results from the additional constraint-the imposed thermal gradient at the boundaries. We demonstrate the rectification behavior based on the exact analytical formulation of steady-state heat transport in harmonic systems coupled to Langevin baths, which can describe quantum and classical transport, both regimes realizing the diode effect under the involved boundary conditions. Our study shows that asymmetric harmonic systems, such as room-temperature hydrocarbon molecules with varying side groups and end groups, or a linear lattice of trapped ions may rectify heat by going beyond simple boundary conditions.

Heat conduction in a hard disc system with non-conserved momentum

We describe results of computer simulations of steady state heat transport in a fluid of hard discs undergoing both elastic interparticle collisions and ‘pseudo collisions’ which do not conserve momentum. The latter are done by picking particles at random and randomizing the directions of their velocities. The system consists of N discs of radius r in a unit square, periodic in the y-direction and having thermal walls with different temperatures at x = 0 and at x = 1. We extrapolate results from different N, to N → ∞, r → 0, such that πr2N = 1/2. We find that in the (extrapolated) hydrodynamic limit N → ∞, the systems’ local density and temperature profiles are those of local thermodynamic equilibrium (LTE), the corresponding pressure is constant independent of position and the heat flux obeys Fourier’s law. The variance of global quantities, such as the total energy, deviates from its local equilibrium value in a form consistent with macroscopic fluctuation theory.

Advances in the greedy optimization algorithm for nodes and collocation points using the method of fundamental solutions

The Method of Fundamental Solutions and Boundary Element Method commonly involve development of approximation functions featuring linear combinations of basis functions dened over the problem domain and boundary. These basis functions are selected with respect to the governing partial differential equation (PDE) being approximated, and are customarily fundamental solutions of the PDE operator. Points where singularities occur are known as source points and are traditionally uniformly distributed outside the problem domain. In this paper, basis functions with singularities at source points are examined with interest in optimizing the placement of the corresponding nodes to reduce computational error while also reducing the number of nodes involved. A motivation for such an optimization is the reduction in matrix solver requirements in solving large dense matrix systems. An algorithm is explored that has been shown to provide such an optimization capability with a 3D case study in steady state heat transport. It is shown that by including the nodal position coordinates as additional variables to be optimized, the resulting approximation function is improved in computational accuracy by increasing the number of degrees of freedom to be optimized.

Physics-informed CoKriging: A Gaussian-process-regression-based multifidelity method for data-model convergence

In this work, we propose a new Gaussian process regression (GPR)-based multifidelity method: physics-informed CoKriging (CoPhIK). In CoKriging-based multifidelity methods, the quantities of interest are modeled as linear combinations of multiple parameterized stationary Gaussian processes (GPs), and the hyperparameters of these GPs are estimated from data via optimization. In CoPhIK, we construct a GP representing low-fidelity data using physics-informed Kriging (PhIK), and model the discrepancy between low- and high-fidelity data using a parameterized GP with hyperparameters identified via optimization. We prove that the physical constraints in the form of deterministic linear operators are satisfied up to an error bound. Furthermore, we combine CoPhIK with a greedy active learning algorithm for guiding the selection of additional observation locations. The efficiency and accuracy of CoPhIK are demonstrated for reconstructing the partially observed modified Branin function, reconstructing the sparsely observed state of a steady state heat transport problem, and learning a conservative tracer distribution from sparse tracer concentration measurements.

Thermal rectification and heat amplification in a nonequilibrium V-type three-level system.

Thermal rectification and heat amplification are investigated in a nonequilibrium V-type three-level system with quantum interference. By applying the Redfield master equation combined with full counting statistics, we analyze the steady-state heat transport. The noise-induced interference is found to be able to rectify the heat current, which paves a new way to design quantum thermal rectifier. Within the three-reservoir setup, the heat amplification is clearly identified far from equilibrium, which is in absence of the negative differential thermal conductance.

Measurement of in-plane sheet thermal conductance of single-walled carbon nanotube thin films by steady-state infrared thermography

A useful and rapid method of measuring the in-plane sheet thermal conductance of single-walled carbon nanotube (SWNT) thin films by steady-state infrared thermography is proposed in this study. The SWNT thin films are suspended between the free ends of two cantilevered silicon thin plates, and the two anchored ends are kept at different but constant temperatures to achieve steady-state heat transport through this silicon–SWNT–silicon bridge in vacuum. The temperature gradient along the bridge is captured with an infrared camera. A silicon plate with known thermal conductivity serves as a reference for calculating the heat flux, so that the sheet thermal conductance of SWNT thin films can be calculated using Fourier’s law after deducting the influence of thermal radiation. The thermal conductivity of the 50-nm-thick SWNT thin film is 68.1 W m−1 K−1.

Transient and steady state heat transport in layered materials from molecular dynamics simulation

In this paper, both transient and steady state heat transfer in graphite are investigated with molecular dynamics (MD) simulations. The simulation results demonstrate that elastic anisotropy controls heat transfer in the transient state, which makes the outermost isothermal surface of temperature distribution similar to the phonon group velocity surface that has a shape of very flat ellipse in layered materials. In steady state, with the help of phonons engaging in sufficient scattering, the basal plane phonon modes determine the thermal conductivity along all directions except those very close to the cross plane. Our simulation results confirm that the classical theoretical model can accurately predict the thermal conductivity along arbitrary directions in layered materials.

Prospects for Assessing Enhanced Geothermal System (EGS) Basement Rock Flow Stimulation by Wellbore Temperature Data

We use Matlab 3D finite element fluid flow/transport modelling to simulate localized wellbore temperature events of order 0.05–0.1 °C logged in Fennoscandia basement rock at ~1.5 km depths. The temperature events are approximated as steady-state heat transport due to fluid draining from the crust into the wellbore via naturally occurring fracture-connectivity structures. Flow simulation is based on the empirics of spatially-correlated fracture-connectivity fluid flow widely attested by well-log, well-core, and well-production data. Matching model wellbore-centric radial temperature profiles to a 2D analytic expression for steady-state radial heat transport with Peclet number Pe ≡ r0φv0/D (r0 = wellbore radius, v0 = Darcy velocity at r0, φ = ambient porosity, D = rock-water thermal diffusivity), gives Pe ~ 10–15 for fracture-connectivity flow intersecting the well, and Pe ~ 0 for ambient crust. Darcy flow for model Pe ~ 10 at radius ~10 m from the wellbore gives permeability estimate κ ~ 0.02 Darcy for flow driven by differential fluid pressure between least principal crustal stress pore pressure and hydrostatic wellbore pressure. Model temperature event flow permeability κm ~ 0.02 Darcy is related to well-core ambient permeability κ ~ 1 µDarcy by empirical poroperm relation κm ~ κ exp(αmφ) for φ ~ 0.01 and αm ~ 1000. Our modelling of OTN1 wellbore temperature events helps assess the prospect of reactivating fossilized fracture-connectivity flow for EGS permeability stimulation of basement rock.

Uncertainties in vertical groundwater fluxes from 1‐D steady state heat transport analyses caused by heterogeneity, multidimensional flow, and climate change

AbstractSteady state 1‐D analytical solutions to estimate groundwater fluxes from temperature profiles are an attractive option because they are simple to apply, with no complex boundary or initial conditions. Steady state solutions have been applied to estimate both aquifer scale fluxes as well as to estimate groundwater discharge to streams. This study explores the sources of uncertainty in flux estimates from regional scale aquifers caused by sensor precision, aquifer heterogeneity, multidimensional flow and variations in surface temperature due to climate change. Synthetic temperature profiles were generated using 2‐D groundwater flow and heat transport models with homogeneous and heterogeneous hydraulic and thermal properties. Temperature profiles were analyzed assuming temperature can be determined with a precision between 0.1°C and 0.001°C. Analysis of synthetic temperature profiles show that the Bredehoeft and Papadopulos (1965) method can provide good estimates of the mean vertical Darcy flux over the length of the temperature profile. Reliable flux estimates were obtained when the ratio of vertical to horizontal flux was as low as 0.1, and in heterogeneous media, providing that temperature at the upper boundary was constant in time. However, temporal increases in surface temperature led to over‐estimation of fluxes. Overestimates increased with time since the onset of, and with the rate of surface warming. Overall, the Bredehoeft and Papadopulos (1965) method may be more robust for the conditions with constant temperature distributions than previously thought, but that transient methods that account for surface warming should be used to determine fluxes in shallow aquifers.

Study of heterogeneous vertical hyporheic flux via streambed temperature at different depths

Abstract. The hyporheic flux can be characterized using the heat-tracing method. Based on the analytical solution of the one-dimensional steady-state heat transport equation under vertical groundwater discharge conditions, hyporheic flux was obtained via a curve fitting method. The temperature data used was obtained from monitoring three different sections of the DaWen River, Shandong Province. The distribution of the depth of the hyporheic zone was analysed by a curve relating groundwater temperature and the depth of the hyporheic zone. The study results showed that the vertical hyporheic flux was significantly heterogeneous along the three sections. The hyporheic flux ranged from 99.61 to 356.25 L/m2 per day. In the summer, the low temperature area on streambed profile was in accordance with the high value areas of hyporheic flux. There were several strong discharge zones within the same section and these flux values were normally distributed. The depth of the hyporheic zone was inversely proportional to the hyporheic flux and the hyporheic zone depth, also, presented great spatial heterogeneity.

Combined heat, air, moisture modelling: A look back, how, of help?

Trials to model combined heat, air, moisture transfer in and through building assemblies started in the 1930-ties, when the first methodologies surfaced that coupled steady state vapour diffusion to steady state heat transport. Thanks to H. Glaser and his papers published end of the 1950-ties, that diffusion/conduction approach gained physical correctness. Some 13 years later, capillary suction was added as transport mechanism. At that time, computer software already helped solving models that linked transient heat transport to moisture transfer by diffusion and suction in composite assemblies. Later, air got included as carrier for heat and vapour while increased computer power allowed analyzing two- and three-dimensional geometries. After 2000, the turn from the assembly to the whole building level gained attention.Although the theory looks well established and the computer software, actually available, quite complete, still it does not always help explaining and curing the damage cases, encountered in practice. As built complicates things and physics related pitfalls remain: simulations base on too simple drawings, inability to correctly include airflow, overlooking pressure and gravity driven water flow, uncertainty in material properties, difficulties to grasp the real initial and boundary conditions, the complexity of the envelope/building interactions, etc.

Steady-state heat transport: Ballistic-to-diffusive with Fourier's law

It is generally understood that Fourier's law does not describe ballistic phonon transport, which is important when the length of a material is similar to the phonon mean-free-path. Using an approach adapted from electron transport, we demonstrate that Fourier's law and the heat equation do capture ballistic effects, including temperature jumps at ideal contacts, and are thus applicable on all length scales. Local thermal equilibrium is not assumed, because allowing the phonon distribution to be out-of-equilibrium is important for ballistic and quasi-ballistic transport. The key to including the non-equilibrium nature of the phonon population is to apply the proper boundary conditions to the heat equation. Simple analytical solutions are derived, showing that (i) the magnitude of the temperature jumps is simply related to the material properties and (ii) the observation of reduced apparent thermal conductivity physically stems from a reduction in the temperature gradient and not from a reduction in actual thermal conductivity. We demonstrate how our approach, equivalent to Fourier's law, easily reproduces results of the Boltzmann transport equation, in all transport regimes, even when using a full phonon dispersion and mean-free-path distribution.

Uncertainty Estimation in One‐Dimensional Heat Transport Model for Heterogeneous Porous Medium

In many practical applications, the rates for ground water recharge and discharge are determined based on the analytical solution developed by Bredehoeft and Papadopulos (1965) to the one-dimensional steady-state heat transport equation. Groundwater flow processes are affected by the heterogeneity of subsurface systems; yet, the details of which cannot be anticipated precisely. There exists a great deal of uncertainty (variability) associated with the application of Bredehoeft and Papadopulos' solution (1965) to the field-scale heat transport problems. However, the quantification of uncertainty involved in such application has so far not been addressed, which is the objective of this wok. In addition, the influence of the statistical properties of log hydraulic conductivity field on the variability in temperature field in a heterogeneous aquifer is also investigated. The results of the analysis demonstrate that the variability (or uncertainty) in the temperature field increases with the correlation scale of the log hydraulic conductivity covariance function and the variability of temperature field also depends positively on the position.

Characterizing streambed water fluxes using temperature and head data on multiple spatial scales in Munsan stream, South Korea

Summary Accurate characterization of water fluxes in the streambed where surface water–ground water (SW–GW) interacts is essential to evaluate the fate and transport of a contaminant plume and to investigate aquatic ecosystems in aquifers connected to rivers or streams. Vertical streambed water fluxes were characterized on multiple spatial scales using temperature-depth profiles and head data measured in a reach of Munsan stream, Paju-si, South Korea. This study applied an exponential model and type curves to determine water fluxes on a large scale of aquifer thickness including a streambed (=15 m) under the assumption of steady state heat transport. On a small scale of the streambed (=0.3 m) Darcy’s law was used to calculate water fluxes through the streambed. The results showed that regional ground water discharges occurred to the stream through the streambed over the reach considered in both August, 2008 and January, 2009 while significant spatial variability in vertical streambed fluxes was found for the streambed of 0.3 m in thickness due to heterogeneities in streambed hydraulic conductivity, variations in stream discharge, and hydraulic gradients in the streambed. Therefore, it is concluded that vertical water fluxes in the streambed of the site are affected by regional ground water flow moving from a deep aquifer to stream and by local small-scale upward or downward hyporheic flow simultaneously. From seepage flux estimates and electrical conductivity (EC) data of water samples, the origins of water in the streambed were inferred. It suggested that SW–GW mixing zones were present at 0.3 m deep or probably even deeper streambeds at S3–S5 located in mid- to downstream at the site. Results have significant implications on effective water resources managements and aquatic habitat investigations in ground water dependent ecosystems of the site.

Thermal Relaxation and Heat Transport in the Spin Ice Material Dy2Ti2O7

The thermal properties of single crystalline Dy2Ti2O7 have been studied in a temperature range from 0.3 K to 30 K and magnetic fields applied along [110] direction up to 1.5 T. Based on a thermodynamic field theory various heat relaxation and thermal transport measurements were analysed. So we were able to present not only the heat capacity of Dy2Ti2O7 in the whole temperature and magnetic field range, but also the different contributions of the magnetic excitations and their corresponding relaxation times in the spin ice phase. In addition, the thermal conductivity and the shortest relaxation time were determined by thermodynamic analysis of steady state heat transport measurements. Finally, we were able to reproduce the temperature profiles recorded in heat pulse experiments on the basis of the thermodynamic field theory using the previously determined heat capacity and thermal conductivity data without additional parameters. Thus, the thermodynamic field theory has been proved to be thermodynamically consistent in describing three thermal transport experiments on different time scales. The observed temperature and field dependencies of heat capacity contributions and relaxation times indicate the magnetic excitations in the spin ice material Dy2Ti2O7 as thermally activated monopole-antimonopole defects.