One-dimensional Heat Flux Research Articles

Study on dryout heat flux of axial stratified debris bed under top-flooding

The coolability of the debris bed with a simulant of solidified corium is experimentally studied, focusing on the effects of the structure of the axial stratified debris bed on the dryout heat flux (DHF). DHF was obtained for the four structures with different particle sizes for the axial stratified debris bed under top flooding. The experimental results show that the dryout position of the axial stratified debris bed is formed at the stratified interface indicated by the temperature rise, and the DHF of the axial stratified bed is much lower than that of the homogeneous bed packed with the upper small particles. To predict the dryout heat flux of the stratified debris beds, by considering the properties of the mixed area, a one-dimensional dryout heat flux model of the porous medium is derived from a water and vapor momentum equation for porous medium, two-phase permeability modifications, interfacial drag, and the correlation between capillary pressure and liquid saturation and verified with the experimental data. The modified model can give reasonable results under different structures.

Sea water freezing modes in a natural convection system

Sea ice is crucial in many natural processes and human activities. Understanding the dynamical couplings between the inception, growth and equilibrium of sea ice and the rich fluid mechanical processes occurring at its interface and interior is of relevance in many domains, ranging from geophysics to marine engineering. Here we investigate experimentally the complete freezing process of water with dissolved salt in a standard natural convection system, i.e. the prototypical Rayleigh–Bénard cell. Due to the presence of a mushy phase, the studied system is considerably more complex than the freezing of fresh water in the same conditions (Wanget al.,Proc. Natl Acad. Sci. USA, vol. 118, issue 10, 2021, e2012870118). We measure the ice thickness and porosity at the dynamical equilibrium state for different initial salinities of the solution and temperature gaps across the cell. These observables are non-trivially related to the controlling parameters of the system as they depend on the heat transport mode across the cell. We identify in the experiments five out of the six possible modes of heat transport. We highlight the occurrence of brine convection through the mushy ice and of penetrative convection in stably stratified liquid underlying the ice. A one-dimensional multi-layer heat flux model built on the known scaling relations of global heat transport in natural convection systems in liquids and porous media is proposed. Given the measured porosity of the ice, it allows us to predict the corresponding ice thickness, in a unified framework.

Experiments and COMSOL simulations: A comparative study of the heat flux plate method and the gradient method for soil heat flux measurements in barren sand

Soil heat flux (G) is important for studying the energy balance at the soil-atmosphere interface. In most cases, a reference flux (true value) is not available for soil heat flux. The heat flux plate method and the heat pulse sensor gradient method are two commonly used methods to measure soil heat flux. Both methods have inherent shortcomings that lead to unavoidable measurement errors. There is a need for a theoretical analysis to identify the origin of errors associated with the gradient method. In this study, by introducing a plane heater of known heating strength to generate a one-dimensional heat flux, we compared the measurements of soil heat flux by both methods at different soil moisture contents for laboratory (indoor) and field (outdoor) experiments, as well as computer simulations for a sandy soil. Differences between the COMSOL simulation results and the indoor measurements were less than 15%, indicating that the COMSOL simulated soil heat flux results were reliable. The indoor experiments provided close agreement between the heat flux plate method and the gradient method (Pearson's correlation coefficient r ≥ 0.993). The results of the field experiments showed that the correlation coefficient of the two methods decreased by at least 9.3%. Our field measurements indicated that the gradient method is not suitable for measuring G in the presence of alternating freezing-thawing soil conditions. Although heat flux plates have several short-comings, they may perform better than the heat pulse sensor gradient method in partially frozen soils. The finite difference approximation of nonlinear thermal gradients causes errors in the gradient method heat flux values. However, this gradient method error could be eliminated by altering the heat pulse probe positioning with depth to provide finer scale measurements of temperature with depth.

Transient Response Characteristics and Performance Assessment of a Calorimetric Surface Junction Probe Under Impulsive Thermal Loading

Abstract The measurement of surface temperatures in highly unsteady flow environments is a challenging task pertaining to the need for high-frequency sensors. This paper contains the sequential activities related to a fast response thermal probe (coaxial surface junction probe—CSJP). This probe is prepared in the laboratory, and its salient features are explored for short-duration (∼20 ms) experiments. The surface junction morphology and inside texture of the probe are examined under field emission scanning electron microscope (FESEM). The study confirms the plastic deformation of thermo-elements with an average junction thickness of 21 μm. The static calibration of the probe using a glycerin bath shows a linear trend between voltage and temperature from which the sensitivity value is calculated as 59 μV/°C. The “thermal product” of the probe is also measured experimentally through the “water droplet technique,” and its value is found to be 8677 J/m2 s0.5 K. The concept of one-dimensional heat flux modeling is followed to infer surface heat flux from transient temperatures. For assuring prediction of heat flux, the probe is calibrated experimentally by exposing to a laser source of known wattage (2 W and 3 W). These findings are also supported by numerical simulation of the probe with accuracy in prediction for surface temperature and heat flux as ± 2% and ±3%, respectively. The performance capability of the probe is demonstrated through shock-tube flow experiments to measure instantaneous heat flux. The comparison of the transient response behavior (9500 K/s) with pressure transducer justifies the utility of the probe under hostile flow environment.

Reply to short communication ‘Comment on “Numerical approach to define a thermodynamically equivalent material for the conjugate heat transfer simulation of very thin coating layers” by P. Olmeda, X. Margot, P. Quintero, J. Escalona, International Journal of Heat and Mass Transfer, Vol. 162(2020) 120377 ” by Jaal Ghandhi and Georgios Koutsakis

In the recent paper “Surface temperature of a multi-layer thermal barrier coated wall subject to an unsteady heat flux” by G. Koutsakis, G. F. Nellis, J. B. Ghandhi, Int. J. Heat Mass Transfer 155(2020), 119645 the authors have proposed an analytical solution to determine wall temperature of surfaces coated with multi-layer coatings. The solutions are obtained based on the assumptions of one-dimensional heat flux using the matrix method in conjunction with the 1-D Laplace transformed heat diffusion equation. The authors suggest that using this analytical approach a thicker pseudo-material could be found in order to address the 3D conjugate heat transfer problem. This approach is compared with another published methodology that employs a DoE (Design of experiments) in combination with a multiple regression analysis to define a thicker equivalent coating layer.

Evaluation of Grad's Second Problem Using Different Higher Order Continuum Theories

Abstract In our earlier work (Jadhav, and Agrawal, 2020, “Grad's second problem and its solution within the framework of Burnett hydrodynamics,” ASME J. Heat Transfer, 142(10), p. 102105), we proposed Grad's second problem (examination of steady-state solution for a gas at rest upon application of a one-dimensional heat flux) as a potential benchmark problem for testing the accuracy of different higher order continuum theories and solved the problem within the framework of Burnett hydrodynamics. In this work, we solve this problem within the moment framework and also examine two variants, Bhatnagar–Gross–Krook (BGK)–Burnett and regularized 13 moment equations, for this problem. It is observed that only the conventional form of Burnett equations which are derived retaining the full nonlinear collision integral are able to capture nonuniform pressure profile observed in case of hard-sphere molecules. On the other hand, BGK–Burnett equations derived using BGK-kinetic model predict uniform pressure profile in both the cases. It seems that the variants based on BGK-kinetic model do not distinguish between hard-sphere and Maxwell molecules at least for the problem considered. With respect to moment equations, Grad 13 and regularized 13 moment equations predict consistent results for Maxwell molecules. However, for hard-sphere molecules, since the exact closed form of moment equations is not known, it is difficult to comment upon the results of moment equations for hard-sphere molecules. The present results for this relatively simple problem provide valuable insights into the nature of the equations and important remarks are made in this context.

Novel inverse heat transfer methodology for estimation of unknown interfacial heat flux of a continuous casting mould: A complete three-dimensional thermal analysis of an industrial slab mould

Accurate quantification of interfacial heat flux between the continuous casting mould and slab/billet is essential for the design and development of the products as well as operation and quality control of the casting process. Apart from empirical correlations, inverse heat transfer methods are typically employed to predict the heat flux variation by assuming a one-dimensional variation of heat flux along the casting direction. The assumption of one-dimensional heat flux often leads to significant error especially in determining the corner heat flux which affects the shrinkage behaviour and thereby, the quality of the products of casting. Previous studies have recognized this error and assumed this error of heat flux as a constant deviation to account for the longitudinal and transverse variation of heat flux. In this work, a novel methodology has been presented to predict the variable interfacial heat flux of a three-dimensional mould domain by estimating the parameters of a characteristic heat flux profile using an inverse heat transfer technique based on Salp Swarm Algorithm (SSA). This methodology uses limited experimental data or on-site plant recorded data. Representative cases of billet and slab moulds have been considered based on the available plant data. Results indicate that the mould corner heat flux is considerably different from that of the typically assumed value of 67% of the centre heat flux as reported in the literature with deviation reaching as high as 56% near the meniscus for a slab casting mould. The proposed methodology is simple to implement and is recommended for studies aimed towards realistic estimation of unknown boundary heat flux in a continuous casting process.

Grad's Second Problem and Its Solution Within the Framework of Burnett Hydrodynamics

Abstract In his seminal work, Grad not only derived 13 moment equations but also suggested two problems to check his derived equations. These problems are highly instructive as they bring out the character of the equations by examining their solutions to these problems. In this work, we propose Grad's second problem as the potential benchmark problem for checking the accuracy of different sets of higher-order transport equations. The problem definition can be stated as: examination of steady-state solution for a gas at rest in infinite domain upon application of a one-dimensional heat flux. With gas at rest (no bulk velocity), the interest lies in obtaining the solution for pressure and temperature. The problem is particularly interesting with respect to the solution for pressure when Maxwell and hard-sphere molecules are considered. For Maxwell molecules, it is well known that the exact normal solution of Boltzmann equation gives uniform pressure with no stresses in the flow domain. In the case of hard-sphere molecules, direct simulation Monte Carlo (DSMC) results predict nonuniform pressure field giving rise to stresses in the flow domain. The simplistic nature of the problem and interesting results for pressure for different interaction potentials makes it an ideal test problem for examining the accuracy of higher-order transport equations. The proposed problem is solved within the framework of Burnett hydrodynamics for hard-sphere and Maxwell molecules. For hard-sphere molecules, it is observed that the Burnett order stresses do not become zero; they rather give rise to a pressure gradient in a direction opposite to that of temperature gradient, consistent with the DSMC results. For Maxwell molecules, the numerical solution of Burnett equations predicts uniform pressure field and one-dimensional temperature field, consistent with the exact normal solution of the Boltzmann equation.

A Method for Evaluation of Thermal Characteristics of Complex Heat-Shield Assembles at Extremely High Energy Fluxes

A suitable laboratory test procedure is developed where a small fragment of the relatively thick multilayer heat shield is exposed to the extreme heat flux of up to several kW/cm2 in order to evaluate its performance. A nearly one-dimensional test condition has been achieved, where thermal behavior of a small fragment of a large shield became representative for the large heat-shield panel. A cylindrical test sample made of carbon-carbon composite has been surrounded by high-temperature thermal isolation made of zirconia concrete. The high heat flux has been provided by a power CW-laser. The conditions of one-dimensional axial heat flux through the cylindrical fragment of the heat shield are confirmed by the mathematical modeling of the experiment using experimentally measured temperature excursions along the sample axis.

One-dimensional heat flux estimation by sequential method using feedback control system: AFSMC approach

In this article, a novel sequential method for estimation of heat flux in inverse heat conduction problems is proposed. This technique is based on the feedback control system theory in which adaptive fuzzy sliding mode controller (AFSMC) is utilized. The objective is to estimate the input heat flux associated with a flat plate according to the time history of measured output temperature. In this closed-loop control system, at each time instant, the AFSMC calculates the proper heat flux value to track the measured temperature with the desired input temperature. Some experiments are designed to investigate the ability and accuracy of proposed method in estimating the heat flux of a one-dimensional flat plate. The effects of noise in data and sensor location on the exactness of proposed method are also assessed. Moreover, the distribution of heat flux is considered to be in three forms: constant, step and triangular. The influential parameters of suggested method are tuned once and then used in all subsequent numerical tests. This method does not rely on scanning the entire time history of measured temperatures, and successfully operates in an online manner for a wide range of heat flux distribution functions and their associated abrupt changes.

Investigation of process-structure-property relationships in polymer extrusion based additive manufacturing through in situ high speed imaging and thermal conductivity measurements

Additive manufacturing has gained significant research attention due to multiple advantages over traditional manufacturing technologies. A fundamental understanding of the relationships between process parameters, microstructure and functional properties of built parts is critical for optimizing the additive manufacturing process and building parts with desired properties. This is also critical for a multi-functional part where the process needs to be optimized with respect to disparate performance requirements such as mechanical strength and thermal conductivity. This paper presents in situ high speed imaging and build-direction thermal conductivity measurements of polymer extrusion based additively manufactured parts in order to understand the effect of process parameters such as raster speed, infill percentage and layer height on build-direction thermal conductivity. Measurements of thermal conductivity using a one-dimensional heat flux method are correlated with in situ process images obtained from a high speed camera as well as cross section images of the built part. Results indicate strong dependence of build-direction thermal conductivity on raster speed, layer thickness and infill percentage, which is corroborated by high speed imaging of the printing process at different values of these process parameters. Key trade-offs between process throughput and thermal properties are also identified. In addition to enhancing our fundamental understanding of polymer extrusion based additive manufacturing and its influence on thermal properties of built parts, results presented here may facilitate process optimization towards parts with desired thermal and multi-functional properties.

Effective interface thermal resistance and thermal conductivity of dielectric nanolayers

Analytical expressions for the effective interface thermal resistance (ITR) and thermal conductivity k of dielectric nanolayers are derived and analyzed, based on the analytical solution of the phonon Boltzmann transport equation under the gray relaxation time approximation. This is achieved by using accurate expressions for the temperature and one-dimensional heat flux propagating across nanolayers supporting a diffusive phonon scattering at their interfaces. It is shown that the effective ITR between two layers can be symmetric on their thermal properties, such that its asymptotic value in the ballistic regime is higher than that in the diffusive one. In the ballistic-diffusive regime, the effective ITR depends strongly on the ratio λ=L/l, between the layer thickness L and mean free path l of phonons. Our predictions for the effective ITR in the ballistic regime are in good agreement with those of the diffuse mismatch model, while they differ by about 16% in the diffusive regime. On the other hand, k increases with λ until reaching saturation for bulk layers and agrees rather well with previous predictions reported in the literature. The obtained results could be useful for analytically describing the heat transport in dielectric nanothin films and superlattices, in which the gray approximation is valid.

Multi-site calibration and validation of a net ecosystem carbon exchange model for croplands

Croplands play an important role in the carbon budget of many regions. However, the estimation of their carbon balance remains difficult due to diversity and complexity of the processes involved. We report the coupling of a one-dimensional soil water, heat, and CO2 flux model (SOILCO2), a pool concept of soil carbon turnover (RothC), and a crop growth module (SUCROS) to predict the net ecosystem exchange (NEE) of carbon. The coupled model, further referred to as AgroC, was extended with routines for managed grassland as well as for root exudation and root decay. In a first step, the coupled model was applied to two winter wheat sites and one upland grassland site in Germany. The model was calibrated based on soil water content, soil temperature, biometric, and soil respiration measurements for each site, and validated in terms of hourly NEE measured with the eddy covariance technique. The overall model performance of AgroC was sufficient with a model efficiency above 0.78 and a correlation coefficient above 0.91 for NEE. In a second step, AgroC was optimized with eddy covariance NEE measurements to examine the effect of different objective functions, constraints, and data-transformations on estimated NEE. It was found that NEE showed a distinct sensitivity to the choice of objective function and the inclusion of soil respiration data in the optimization process. In particular, both positive and negative day- and nighttime fluxes were found to be sensitive to the selected optimization strategy. Additional consideration of soil respiration measurements improved the simulation of small positive fluxes remarkably. Even though the model performance of the selected optimization strategies did not diverge substantially, the resulting cumulative NEE over simulation time period differed substantially. Therefore, it is concluded that data-transformations, definitions of objective functions, and data sources have to be considered cautiously when a terrestrial ecosystem model is used to determine NEE by means of eddy covariance measurements.

One-dimensional coolability and concrete ablation analysis of volumetrically heat-generating debris bed

We present a preliminary study on the feasibility of the pre-flooded cavity strategy for ex-vessel cooling of APR1400 NPPs. Fragmented particles form a porous debris bed, so understanding the heat transfer mechanism in a porous medium and its critical heat flux is essential to assess the coolability of the debris bed. An one-dimensional dryout heat flux model of porous medium is derived from a water and vapor momentum equation for porous medium, two-phase permeability modifications and the correlation between capillary pressure and liquid saturation. The developed model is validated by comparing dryout heat flux data of DEBRIS, STXY, POMECO and COOLOCE tests. STXY BED, POMECO BED-5 and COOLOCE-8 BED are made to reflect the particle size distribution results of the melt fragmentation tests such as FARO, CCM, MIRA and DEFOR. We analyze the ex-vessel coolability by assuming that these three beds represent debris beds in actual accidents. In addition, one-dimensional coolability analysis is conducted for three cases of cylinder shape debris bed at containment pressures of 1–5bar. Based on these, one-dimensional ablation concrete depth analysis for the three cases of debris shape and the three kinds of the debris beds of APR1400 NPPs is conducted. As a result, if fragmented particles are distributed evenly and the POMECO BED-5 or STXY BED like debris bed is formed in the cavity, the debris bed coolability is so good that ablation depth is negligible or less than the thickness of sacrificial concrete. Even if the COOLOCE-8 BED like debris bed is formed, the integrity of the containment is kept during ex-vessel cooling if the containment pressure is maintained 3bar or more. However, if the solidified particles spread only in a particular area, the leak tightness of containment building could not be maintained in the ex-vessel cooling process. Because solidified particles lose their fluidity, so it is questionable that all of them are distributed uniformly in the cavity. Therefore, a further research is required on how fragmented particles are distributed in pre-flooded cavities to validate the pre-flooding strategy of APR1400 NPPs.

Thermal Conductivity Measurements By Means of a New ‘Small Hot-Box’ Apparatus: Manufacturing, Calibration and Preliminary Experimental Tests on Different Materials

The evaluation of the thermal performance building components requires a high level of accuracy. Windows, doors and thermal bridges are not homogeneous, and their thermal transmittance can be evaluated by means of Hot-Box, used for full-scale elements. For homogeneous materials and one-dimensional heat flux, the thermal conductivity can be easily measured through other experimental apparatuses, such as the guarded hot plate and the heat flow meters. This study presents a new experimental apparatus named Small Hot-Box, built at the University of Perugia. No European standards are available for this innovative facility, but it takes into account some prescriptions of EN ISO 8990 and EN ISO 12567; it was built for the evaluation of the thermal properties of small specimens. The apparatus was designed, built, and calibrated by means of preliminary measurements. It is composed of a hot and a cold side, and the external walls are made of thick insulation. The thermal conductivity can be calculated by two different methodologies: the Hot-Box and the thermal flux meter method. Preliminary calibrations were carried out and different materials with known thermal transmittance were tested. The aim is the development of a new experimental apparatus; guidance documents could be defined for the measurements methodology requirements.

Optimization of Thermoelectric Components for Automobile Waste Heat Recovery Systems

For a typical spark ignition engine approximately 40% of available thermal energy is lost as hot exhaust gas. To improve fuel economy, researchers are currently evaluating technology which exploits exhaust stream thermal power by use of thermoelectric generators (TEGs) that operate on the basis of the Seebeck effect. A 5% improvement in fuel economy, achieved by use of TEG output power, is a stated objective for light-duty trucks and personal automobiles. System modeling of thermoelectric (TE) components requires solution of coupled thermal and electric fluxes through the n and p-type semiconductor legs, given appropriate thermal boundary conditions at the junctions. Such applications have large thermal gradients along the semiconductor legs, and material properties are highly dependent on spatially varying temperature profiles. In this work, one-dimensional heat flux and temperature variations across thermoelectric legs were solved by using an iterative numerical approach to optimize both TE module and TEG designs. Design traits were investigated by assuming use of skutterudite as a thermoelectric material with potential for automotive applications in which exhaust gas and heat exchanger temperatures typically vary from 100°C to over 600°C. Dependence of leg efficiency, thermal fluxes and electric power generation on leg geometry, fill fractions, electric current, thermal boundary conditions, etc., were studied in detail. Optimum leg geometries were computed for a variety of automotive exhaust conditions.

Experimental investigation of radiative thermal rectifier using vanadium dioxide

Vanadium dioxide (VO2) exhibits a phase-change behavior from the insulating state to the metallic state around 340 K. By using this effect, we experimentally demonstrate a radiative thermal rectifier in the far-field regime with a thin film VO2 deposited on the silicon wafer. A rectification contrast ratio as large as two is accurately obtained by utilizing a one-dimensional steady-state heat flux measurement system. We develop a theoretical model of the thermal rectifier with optical responses of the materials retrieved from the measured mid-infrared reflection spectra, which is cross-checked with experimentally measured heat flux. Furthermore, we tune the operating temperatures by doping the VO2 film with tungsten (W). These results open up prospects in the fields of thermal management and thermal information processing.

Observations of vertical eddy diffusivities in a shallow tropical reservoir

Tropical lakes are of key importance as fresh water supply sources in tropical regions. One of the biggest threats to tropical lakes is nuisance algal blooms which at times can be harmful to human health. Research shows that thermal and flow dynamics, through their effects on the light and nutrient accessibility to phytoplankton, have a strong bearing on the growth of phytoplankton. We report herein results combining long-term and short-term observations of vertical diffusivities which impact the evolution of the surface mixed layer and the vertical transport of nutrients for a shallow tropical reservoir, Kranji Reservoir in Singapore.The one-dimensional heat flux method (Jassby and Powell, 1975; Henderson-Sellers, 1985) is applied to obtain estimates of the vertical mixing in Kranji Reservoir using water temperature and meteorological data collected during a two month field experiment conducted in 2007. To fit the assumptions of the method and particularly the heating duration, a time period of 11:00–15:00 is selected for heat flux method when the water body is strongly stratified. The vertical eddy diffusivity, Kz varies with depth and spans two orders of magnitude, from 10−5 to 10−3 m2/s. In addition, short-term observations from two 24hr turbulence microstructure measurements conducted in 2011 are considered. The heat flux method derived diffusivities are compared directly against those derived from the microstructure measurements with reasonable agreement with the latter when the system fits a one-dimensional assumption. The microstructure measurements further provides Kz observed values over the entire day and this is discussed in the context of the thermal forcing regimes in Kranji reservoir.

Carbonyl sulfide hydrolysis in Antarctic ice cores and an atmospheric history for the last 8000 years

©2014. American Geophysical Union. All Rights Reserved. Carbonyl sulfide (COS) was measured in Antarctic ice core samples from the Byrd, Siple Dome, Taylor Dome, and West Antarctic Ice Sheet Divide sites covering the last 8000 years of the Holocene. COS levels decrease downcore in most of these ice cores. The magnitude of the downcore trends varies among the different ice cores and is related to the thermal histories of the ice sheet at each site. We hypothesize that this is due to the temperature-dependent hydrolysis of COS that occurs in situ. We use a one-dimensional ice flow and heat flux model to infer temperature histories for the ice core samples from different sites and empirically determine the kinetic parameters for COS hydrolysis. We estimate e-folding lifetimes for COS hydrolysis ranging from 102 years to 106 years over a temperature range of 0°C to -50°C. The reaction kinetics are used to estimate and correct for the in situ COS loss, allowing us to reconstruct paleoatmospheric COS trends during the mid-to-late Holocene. The results suggest a slow, long-term increase in atmospheric COS that may have started as early as 5000 years ago. Given that the largest term in the COS budget is uptake by terrestrial plants, this could indicate a decline in terrestrial productivity during the late Holocene.

Experimental validation of a numerical code by thin film heat flux sensors for the resolution of thermal bridges in dynamic conditions

Thermal bridges are areas of risk of the building envelope, inducing uncontrolled increments of heat transfer, mainly in winter. This notwithstanding, thermal bridges are often neglected in energy audits, since many numerical codes for building energy simulations adopt heat transfer models based on one-dimensional heat flux: this can cause lack of reliability.This paper validates a numerical model, written in MatLab™, which solves thermal bridges – under bi-dimensional heat transfer conditions – in transient regime. More in detail, the model – developed by the same authors – allows estimation of thermal bridge effects under dynamic conditions (hourly and sub-hourly). The model is validated through a comparison with experimental data, measured by means of a proper apparatus, consisting in a set of heat flow meters, thin flux sensors, temperature probes, infrared thermographic camera. A typical L-shaped thermal bridge has been investigated and the numerical methodology – compared to the experimental study – reveals very satisfactory results in reliability and accuracy. The percentage gap between calculated and measured heat fluxes varies between −12% and +6%, with an average value close to zero. A satisfactory agreement has been evaluated also with reference to the thermal power through the investigated envelope element.