Initial Temperature Profile Research Articles

Incorporating multi-source uncertainties in fast building wall thermal resistance estimation through physics-based and multi-fidelity statistical learning models

Meta-modelling and Bayesian inversion technique are proposed for fast and accurate in-situ estimation of the total thermal resistance (RTot) of walls using non-intrusive wall instrumentation. Various sources of uncertainties are taken into account to provide an enhanced credible interval for the thermal resistance estimate. In the considered protocol, a small zone of the internal surface of the wall is excited by a prototype to get faster in situ estimation and to limit the influence of external weather conditions. To be independent of the heat transfer coefficients and of the misknown layer thicknesses, which are difficult to estimate, we use herein the measurements of internal and external surface temperatures as boundary conditions in a thermal direct problem reformulated. A meta-model of the thermal model is created based on a statistical multi-fidelity approach with two levels of fidelity, Resistance-Capacitance (RC) and 1D models, to achieve the Bayesian estimation of the total thermal resistance in a reasonable computation time. The RTot estimation method is applied to realistic internal insulated walls (IIW), from poorly to highly insulated, under different weather conditions. Several numerical tests are carried out and credible intervals are provided to study the importance of the wall initial condition, the excitation time and the instrumentation. Finally, an experimental application is conducted using real measurements on an internal insulated wall (IIW) in Nancy (France). The obtained experimental results show that, in a short excitation time (10h) with a reduced instrumentation, the proposed method accurately estimates the minimum wall thermal resistance. For a standard wall of about 4 m2 K/W, a relevant estimate of RTot with a relative deviation less than 10% can be achieved by using a polynomial initial temperature profile proposed in this study, sufficient measurements and an excitation time of 3 days. This study thus offers prospects for improved energy assessment of buildings before and after renovation.

Alternative thermal histories of Earth-like planets: Influence of key parameters

The thermal evolution of any planet can be influenced by many factors: the initial temperature profile, the distribution of specific materials within the planet, the existence or lack of a gaseous atmosphere, the effects of early and “late” collision events. Insights concerning the influence of those factors can be obtained by examining solutions to the heat equation, applied to spherical bodies. General trends identified in this work include: 1) Moderate conductive materials contribute to efficiently flatten the temperature gradients, whereas insulating materials promote the preservation of steep temperature gradients. 2) It is not necessary to invoke convection to achieve a relatively flat temperature gradient; moderately conductive materials might achieve the same result without any advective motion involved. 3) Heat transport can take place both outwards and inwards, depending on the initial distribution of temperatures. 4) If the initial temperatures near the center of a planet are low, they will tend to remain low even if heat production takes place at its middle or upper parts. 5) Gradients of temperature near the surface of a planet may not reflect temperature variations at its middle or central parts. 6) Changes of phase exert a strong influence on the evolution of temperature profiles within a planet. 7) Highly insulating atmospheric layers can be important in determining the time of solidification of the upper layer of a magma ocean but not all atmospheres are equally efficient in that respect. As a result, models that give for granted the existence of deep mantle convection on Earth are justified only in the context of models of planet formation that require high initial temperatures; the standard model of a cold solar nebula is not consistent with such deep mantle convective movements.

Initial temperature profile recovery under a mixed boundary condition

Initial temperature profile recovery under a mixed boundary condition

Microwave antenna focusing for spatially resolved modulation of burn rate

This study explores the ability to dynamically modulate the burn rate of nanothermites through microwave generated heat zones. These heat zones were generated by embedding a microwave receiving antenna in 3D printed Al/CuO samples. These thermal hot spots in the nanothermites are generated by the induced high electric fields at the terminus of an embedded receiving antenna and are monitored by an IR camera. The burn rates were investigated for different MW induced hot spot temperatures, geometries and receiving antenna materials. The local burn rate increase closely tracks the hot spot spatial distribution, and for our experimental condition enabled an amplification of burn rate of ∼6×. An effective burn rate activation energy was calculated to be ∼18 kJ/mol from the Arrhenius plot of local burn rate and initial temperature. Three color pyrometry confirms that the main mechanism of the local burn rate increase is related to the initial temperature profile in the sample. This study demonstrates the sensitivity of the local burn rate of nanothermites to initial temperature and provides a new approach to modulate the burn rate by localizing MW energy.

Numerical analysis of the bistable propagation of lean premixed hydrogen flames

Premixed hydrogen-air flames that propagate in slender channels have been recently found to be bistable in the limit of fuel-lean mixtures and non-negligible heat losses. Specifically, two stable configurations conformed by either a circular or a double-cell flame front arise for the same combination of controlling parameters (fuel mixture, heat transfer). Nevertheless, this multiplicity of solutions lacks a satisfactory explanation that predicts the formation of either one or another structure given a particular ignition transient. In this work, the analysis of a set of numerical simulations is offered to unveil the underlying physics controlling the bistable behavior. The formulation employed here proves that simplified one-step Arrhenius reaction and quasi-two-dimensional equations suffice to reproduce the complex dynamics, rendering three-dimensional effects and detailed chemical kinetics as second-order contributions that may be important, however, to provide exact quantitative results. Specifically, the initial temperature profiles and subsequent expansion of the flow field prescribe the growth of the flame front leading to different curvatures and sizes of the kernel that control the evolution into one of the canonical structures. Moreover, the achievement of double-cell flames require splitting and reorientation processes prior to pairing of isolated flames, which call for specific initialization conditions that enable these transient dynamics.

Assessing the Impact of a Shallow Geothermal System Operation through Multi-Layer Temperature Monitoring in a Mediterranean Climate

Shallow Geothermal Energy (SGE) exchanges heat with the ground. In continuous, long-term operation, the initial temperature field can be disturbed, and subsurface thermal changes can be developed. In this paper, the thermal impact of a SGE system under a Mediterranean climate is handled. Temperature monitoring was conducted on 15 investigation boreholes equipped with a total of 92 thermal sensors placed at specific depths. Investigation boreholes were drilled 1–2 m from SGE system borehole heat exchangers installed in a university building. The analysis handles a one-year monitoring period of SGE system operation. Temperature depth profiles, reaching up to 140 m depth, were registered with a 10 min time step, resulting in a large amount of data. Ground thermal conductivity was estimated experimentally and semi-empirically, allowing us to obtain, using a numerical model, the initial undisturbed ground temperature profiles and compare them with the monitored values. Climate data were recorded by the university meteorological station. Globally, the measured and computed data were coherent, and a non-negligible impact of the SGE system operation in the first year was observed. The building orientation as well as the nearby departments had significant impacts on the shallow ground temperature. Maximum ground temperature changes observed at depths higher than 10–20 m, ranging from 2 to 3 °C as observed in different boreholes, indicate that the system is operating efficiently.

Pathway dynamics to double-cell premixed flames in lean hydrogen–air mixtures

The propagation of premixed flames over lean hydrogen–air mixtures have been found to be bistable in recent studies, considering slender channels with non-negligible conductive heat losses. In particular, two stable configurations conformed by either a circular or a double-cell flame front arise for the same combination of controlling parameters (fuel mixture, channel size and thermal conductivity). Nevertheless, this multiplicity of solutions lacks a satisfactory explanation to predict the formation of either one structure or the other during a particular ignition transient. In this work, detailed analyses are performed over the unsteady evolution of a set of numerical simulations. Specifically, the initial temperature profiles (distribution and peak value) and subsequent expansion of the flow field prescribe the early growth of the flame front leading to different curvatures and sizes of the kernel that control the evolution into each of the canonical structures. This study explores the underlying physics controlling the final-stage bistable behavior. In particular, the local convective effects and interactions between fronts have been identified as the key causes to produce each of the two stably propagating flames. Frequently found division of kernels cannot ensure the formation of double cells, which require additional reorientation dynamics and merging events due to asymmetric seeding of flame fragments or to interaction with neighboring structures.

Recovery of an initial temperature of a one-dimensional body from finite time-observations

Under the Dirichlet boundary setting, Aryal and Karki (2022) studied an inverse problem of recovering an initial temperature profile from known temperature measurements at a fixed location of a one-dimensional body and at linearly growing finitely many later times within a bounded interval. This paper studies the problem under the Neumann boundary conditions. That is, under this boundary setting, we suitably select a fixed location x0 on the body of length π and construct finitely many times tk, k = 1, 2, 3, . . . , n that grow linearly with k and are in [0, T] such that from the temperature measurements taken at x0 and at these n times, we recover the initial temperature profile f(x) with a desired accuracy, provided f is in a suitable subset of L2[0, π].

The effect of temperature-dependent material properties on simple thermal models of subduction zones

Abstract. To a large extent, the thermal structure of a subduction zone determines where seismicity occurs through controls on the transition from brittle to ductile deformation and the depth of dehydration reactions. Thermal models of subduction zones can help understand the distribution of seismicity by accurately modelling the thermal structure of the subduction zone. Here, we assess a common simplification in thermal models of subduction zones, i.e. constant values for the thermal parameters. We use temperature-dependent parameterisations, constrained by lab data, for the thermal conductivity, heat capacity, and density to systematically test their effect on the resulting thermal structure of the slab. To isolate this effect, we use the well-defined, thoroughly studied, and highly simplified model setup of the subduction community benchmark by van Keken et al. (2008) in a 2D finite-element code. To ensure a self-consistent and realistic initial temperature profile for the slab, we implement a 1D plate model for cooling of the oceanic lithosphere with an age of 50 Myr instead of the previously used half-space model. Our results show that using temperature-dependent thermal parameters in thermal models of subduction zones affects the thermal structure of the slab with changes on the order of tens of degrees and hence tens of kilometres. More specifically, using temperature-dependent thermal parameters results in a slightly cooler slab with e.g. the 600 ∘C isotherm reaching almost 30 km deeper. From this, we infer that these models would predict a larger estimated seismogenic zone and a larger depth at which dehydration reactions responsible for intermediate-depth seismicity occur. We therefore recommend that thermo(-mechanical) models of subduction zones take temperature-dependent thermal parameters into account, especially when inferences of seismicity are made.

A numerical study on performance efficiency of a low-temperature horizontal ground-source heat pump system

Exploitation of shallow ground and its low-grade heat potential is fundamental to designing 5th generation district heating and cooling (DHC) networks. Horizontal ground-source heat pump (HGSHP) systems are a common way to utilize shallow geothermal energy. Realistic estimation and prediction of performance of a HGSHP system and shallow ground thermal behaviour should consider the whole system including building heating and cooling load, heat pump and ground heat exchanger, and the ground. This should be accompanied by realistic atmospheric and ground conditions. In this paper, a three-dimensional coupled thermal–hydraulic model with realistic boundary conditions adopting a whole system approach is presented. Dynamic heat pump coefficient of performance (COP) that depends on seasonal variation of heating/cooling demand and ground conditions are also considered. Model validations are conducted against experimental and analytical results in literatures. The model is applied for evaluating a HGSHP system to support development of a 5th generation DHC network on a potential site in the UK. Several influencing factors, such as ground moisture transfer, building thermal load mode, buried depth of ground loops, and initial ground temperature profile are studied to assess performance efficiency of the HGSHP system and evolution of ground thermal behaviour in response to heat extraction or rejection into the ground. The results show that 5% of the monthly total heat demand of the site could be met by the designed HGSHP system, consisting of 200 U-shaped ground loops buried at the depth of 3 m and pure water as the heat carrier. Overlooking the ground moisture transfer or hyperbolizing the ground saturation would overestimate the load-carrying capacity of the HGSHP system. The HGSHP system is more efficient with a higher heat pump COP under the heating and cooling mode than under the heating-only mode. Predicted performance of the HGSHP system improves with buried depth of the ground loops. The results also show that a 1 ℃ increase in the undisturbed ground temperature could suffice up to 8% of the monthly total heat demand of the site.

Bayesian Calibration Points to Misconceptions in Three‐Dimensional Hydrodynamic Reservoir Modeling

AbstractThree‐dimensional (3d) numerical models are state‐of‐the‐art for investigating complex hydrodynamic flow patterns in reservoirs and lakes. Such full‐complexity models are computationally demanding and their calibration is challenging regarding time, subjective decision‐making, and measurement data availability. In addition, physically unrealistic model assumptions or combinations of calibration parameters may remain undetected and lead to overfitting. In this study, we investigate if and how so‐called Bayesian calibration aids in characterizing faulty model setups driven by measurement data and calibration parameter combinations. Bayesian calibration builds on recent developments in machine learning and uses a Gaussian process emulator as a surrogate model, which runs considerably faster than a 3d numerical model. We Bayesian‐calibrate a Delft3D‐FLOW model of a pump‐storage reservoir as a function of the background horizontal eddy viscosity and diffusivity, and initial water temperature profile. We consider three scenarios with varying degrees of faulty assumptions and different uses of flow velocity and water temperature measurements. One of the scenarios forces completely unrealistic, rapid lake stratification and still yields similarly good calibration accuracy as more correct scenarios regarding global statistics, such as the root‐mean‐square error. An uncertainty assessment resulting from the Bayesian calibration indicates that the completely unrealistic scenario forces fast lake stratification through highly uncertain mixing‐related model parameters. Thus, Bayesian calibration describes the quality of calibration and correctness of model assumptions through geometric characteristics of posterior distributions. For instance, most likely calibration parameter values (posterior distribution maxima) at the calibration range limit or with widespread uncertainty characterize poor model assumptions and calibration.

Pulsed Photothermal Radiometric Depth Profiling of Bruises by 532 nm and 1064 nm Lasers.

Optical techniques are often inadequate in estimating bruise age since they are not sensitive to the depth of chromophores at the location of the bruise. To address this shortcoming, we used pulsed photothermal radiometry (PPTR) for depth profiling of bruises with two wavelengths, 532 nm (KTP laser) and 1064 nm (Nd:YAG laser). Six volunteers with eight bruises of exactly known and documented times of injury were enrolled in the study. A homogeneous part of the bruise was irradiated first with a 5 ms pulse at 532 nm and then with a 5 ms pulse at 1064 nm. The resulting transient surface temperature change was collected with a fast IR camera. The initial temperature-depth profiles were reconstructed by solving the ill-posed inverse problem using a custom reconstruction algorithm. The PPTR signals and reconstructed initial temperature profiles showed that the 532 nm wavelength probed the shallow skin layers revealing moderate changes during bruise development, while the 1064 nm wavelength provided additional information for severe bruises, in which swelling was present. Our two-wavelength approach has the potential for an improved estimation of the bruise age, especially if combined with modeling of bruise dynamics.

Numerical simulation of thermal stratification in Lake Qiandaohu using an improved WRF-Lake model

Lake thermal stratification is important for regulating lake environments and ecosystems and is sensitive to climate change and human activity. However, numerical simulation of coupled hydrodynamics and heat transfer processes in deep lakes using one-dimensional lake models remains challenging because of the insufficient representation of key parameters. In this study, Lake Qiandaohu, a deep and warm monomictic reservoir, was used as an example to investigate thermal stratification via an improved parameterization scheme of the Weather Research and Forecast (WRF)-Lake. A comparison with in situ observations demonstrated that the default WRF-Lake model was able to simulate well the seasonal variation of the lake thermal structure. However, the simulations exhibited cold biases in lake surface water temperature (LSWT) throughout the year while generating weaker stratification in summer, thereby leading to an earlier cooling period in autumn. With an improved parameterization (i.e., via determination of initial lake water temperature profiles, light extinction coefficients, eddy diffusion coefficients and surface roughness lengths), the modified WRF-Lake model was able to better simulate LSWT and thermal stratification. Critically, employing realistic initial conditions for lake water temperature is essential for producing realistic hypolimnetic water temperatures. The use of time-dependent light extinction coefficients resulted in a deep thermocline and warm LSWT. Enlarging eddy diffusivity led to stronger mixing in summer and further influenced autumn cooling. The parameterized surface roughness lengths mitigated the excessive turbulent heat loss at the lake surface, improved the model performance in simulating LSWT, and generated a warm mixed layer. This study provides guidance on model parameterization for simulating the thermal structure of deep lakes and advances our understanding of the strength and revolution of lake thermal stratification under seasonal changes.

Unsteady thermal transport in an instantly heated semi-infinite free end Hooke chain

We consider unsteady ballistic heat transport in a semi-infinite Hooke chain with free end and arbitrary initial temperature profile. An analytical description of the evolution of the kinetic temperature is proposed in both discrete (exact) and continuum (approximate) formulations. By comparison of the discrete and continuum descriptions of kinetic temperature field, we reveal some restrictions to the latter. Specifically, the far-field kinetic temperature is well described by the continuum solution, which, however, deviates near and at the free end (boundary). We show analytically that, after thermal wave reflects from the boundary, the discrete solution for the kinetic temperature undergoes a jump near the free end. A comparison of the descriptions of heat propagation in the semi-infinite and infinite Hooke chains is presented. Results of the current paper are expected to provide insight into non-stationary heat transport in the semi-infinite lattices.

Unsteady two-temperature heat transport in mass-in-mass chains.

We investigate the unsteady heat (energy) transport in an infinite mass-in-mass chain with a given initial temperature profile. The chain consists of two sublattices: the β-Fermi-Pasta-Ulam-Tsingou (FPUT) chain and oscillators (of a different mass) connected to each FPUT particle. Initial conditions are such that initial kinetic temperatures of the FPUT particles and the oscillators are equal. Using the harmonic theory, we analytically describe evolution of these two temperatures in the ballistic regime. In particular, we derive a closed-form fundamental solution and solution for a sinusoidal initial temperature profile in the case when the oscillators are significantly lighter than the FPUT particles. The harmonic theory predicts that during the heat transfer the temperatures of sublattices are significantly different, while initially and finally (at large times) they are equal. This may look like an artifact of the harmonic approximation, but we show that it is not the case. Two distinct temperatures are also observed in the anharmonic case, even when the heat transport regime is no longer quasiballistic. We show that the value of the nonlinearity coefficient required to equalize the temperatures strongly depends on the particle mass ratio. If the oscillators are much lighter than the FPUT particles, then a fairly strong nonlinearity is required for the equalization.

An approach for recovering initial temperature via a bounded linear time sampling

We have improved the results of DeVore and Zuazua's work in [6] and addressed some questions posed in their paper. By constructing a linearly growing finitely many future times in a bounded interval, we have improved their result of recovering initial temperature profile from known temperature measurements at a fixed location of a body and at finitely many later times. More explicitly, we have constructed a finitely many times with a linear growth in a bounded interval such that the temperature measurements taken at a fixed location of a thin uniform homogeneous rod of length π and at these finite time instances lead to a recovery of the initial temperature profile with a desired accuracy provided the initial temperature is in a suitable closed subset of L2[0,π].

Unsteady ballistic heat transport in two-dimensional harmonic graphene lattice

We study the evolution of initial temperature profiles in a two-dimensional isolated harmonic graphene lattice. Two heat transfer problems are solved analytically and numerically. In the first problem, the evolution of a spatially sinusoidal initial temperature profile is considered. This profile is usually generated in real experiments based on the transient thermal grating technique. It is shown that at short times the amplitude of the profile decreases by an order magnitude and then it performs small decaying oscillations. A closed-form solution, describing the decay of the amplitude at short times is derived. It shows that due to symmetry of the lattice, the anisotropy of the ballistic heat transfer is negligible at short times, while at large times it is significant. In the second problem, a uniform spatial distribution of the initial temperature in a circle is specified. The profile is the simplest model of graphene heating by an ultrashort localized laser pulse. The corresponding solution has the symmetry of the lattice and many local maxima. Additionally, we show that each atom has two distinct temperatures corresponding to motions in zigzag and armchair directions. Presented results may serve for proper statement and interpretation of laboratory experiments and molecular dynamics simulations of unsteady heat transfer in graphene.

Initialization of thermal models in cold and warm permafrost

Equilibrium modelling, also known as spin-up, is a technique for initializing a stable thermal regime in ground temperature models for permafrost regions. The results act as a baseline for subsequent transient analyses of ground temperature response to climate change or infrastructure. In practice, spin-up procedures are often loosely described or neglected, and the criteria by which a stable thermal regime is evaluated are rarely defined or presented explicitly. In this paper, model results show that no single criterion based on thresholds of inter-cycle temperature change can be used to identify a stable thermal regime in all spin-up scenarios. Results from simulations using a wide range of initialization temperatures and conditions show the number of spin-up cycles can range between 10 and 10 000, and a spin-up criterion as fine as 0.00001 °C/cycle is required to achieve a stable thermal regime suitable for deeper warm permafrost models. The implications of selected threshold criteria are examined in follow-up transient analyses and show that warm permafrost models can be highly sensitive to initial temperature profiles based on the criterion used. The results alert scientists and engineers to the importance of initialization on site-specific and regional permafrost models for transient ground temperature analyses.

Numerical simulation and production prediction assessment of Takigami geothermal reservoir

A numerical model was developed for the Takigami geothermal reservoir. A conceptual model of the field was constructed, initial and boundary conditions were defined according to available data. For the optimum model, permeability values of assigned rock types, mass flow rates, enthalpies, and locations of recharge zones were estimated according to matching between computed temperature for wells and their temperature profiles before the exploitation. Observed and calculated temperature profiles confirmed the validity of the conceptual model. The best model could successfully reproduce the initial temperature profiles of 13 wells located mainly in the production area. A developed model was used as an initial model for future prediction of the reservoir performance. The prediction simulation was conducted by assuming two different development scenarios for the Takigami geothermal power plant. Scenario I was continuing the current power production. Scenario II was to investigate producing 8.6 MWe more electricity by employing bottoming binary cycle to the currently under operation single flash plant. Effects of production and reinjection temperatures under proposed development scenarios were evaluated. Simulation results indicated that most probably there is no direct interaction between reinjection and production zones in the Takigami reservoir, and installing a binary plant will not have any severe impact on reservoir performance.

Ventilation of a Mid-Size City under Stable Boundary Layer Conditions: A Simulation Using the LES Model PALM

City centers have to cope with an increasing amount of air pollution. The supply of fresh air is crucial yet difficult to ensure, especially under stable conditions of the atmospheric boundary layer. This case study used the PArallelized Large eddy simulation (LES) Model PALM to investigate the wind field over an urban lake that had once been built as a designated fresh air corridor for the city center of Münster, northwest, Germany. The model initialization was performed using the main wind direction and stable boundary layer conditions as input. The initial wind and temperature profiles included a weak nocturnal low-level jet. By emitting a passive scalar at one point on top of a bridge, the dispersion of fresh air could be traced over the lake’s surface, within street canyons leading to the city center and within the urban boundary layer above. The concept of city ventilation was confirmed in principle, but the air took a direct route from the shore of the lake to the city center above a former river bed and its adjoining streets rather than through the street canyons. According to the dispersion of the passive scalar, half of the city center was supplied with fresh air originating from the lake. PALM proved to be a useful tool to study fresh air corridors under stable boundary layer conditions.