Low Effective Thermal Conductivity Research Articles

Nano-Enhanced Graphite/Phase Change Material/Graphene Composite for Sustainable and Efficient Passive Thermal Management.

Passive battery thermal management systems (BTMSs) are critical for mitigation of battery thermal runaway (TR). Phase change materials (PCMs) have shown promise for mitigating transient thermal challenges. Fluid leakage and low effective thermal conductivity limit PCM adoption. Furthermore, the thermal capacitance of PCMs diminishes as their latent load is exhausted, creating an unsustainable cooling effect that is transitory. Here, an expanded graphite/PCM/graphene composite that solves these challenges is proposed. The expanded graphite/PCM phase change composite eliminates leakage and increases effective thermal conductivity while the graphene coating enables radiative cooling for PCM regeneration. The composite demonstrates excellent thermal performance in a real BTMS and shows a 26% decrease in temperature when compared to conventional BTMS materials. The composite exhibits thermal control performance comparable with active cooling, resulting in reduced cost and increased simplicity. In addition to BTMSs, this material is anticipated to have application in a plethora of engineered systems requiring stringent thermal management.

Modelling of hollow-fiber doping in silica aerogel composites for radiative and conductive insulation under high temperatures

Doping opacifier and fiber has been developed to improve the thermal insulation performance of silica aerogels at high tempearture. However, this will inevitably increase the density and enhance heat conduction in practical applications. In the present work, to explore a way of not only avoiding the thermal insulation compromise but also reducing the density, hollow fibers as dopants are investigated. Their radiative properties and the overall thermal insulation performance of hollow fiber-doped silica aerogel composites are numerically evaluated. In addition, the dependence of the optimal hollow ratio on the fiber diameter at different temperatures is analyzed, and the temperature-dependent optimal diameter of fibers for different hollow ratios is obtained. It is shown that the hollow-fiber-doped aerogel composite with a hollow ratio of 0.6 can present not only a low effective thermal conductivity (compared with the conventional solid fiber doped ones), but also a decrease in density by up to 11.45%, between 300 and 1300 K. The present work can illustrate the potential of hollow fiber doping towards an optimal trade-off between the density and thermal insulations of silica aerogels for high temperature circumstances and can inspire more explorations of hollow structures in silica aerogels.

Gas-solid coupling in a randomly distributed ceramic nanofibrous aerogel

Ceramic nanofibrous aerogel (CNA) has received tremendous interest recently due to its advantageous properties and potential applications. However, modeling its gas-contributed thermal conductivity is quite difficult due to the intricate gas-solid coupling heat transfer mechanism and complex structure. Herein, based on an equivalent method used for a gas in free space, a model for predicting gas-contributed thermal conductivity in its nanopores is built to provide a straightforward solution to the above-mentioned issues. We study both gas and gas-solid coupling effects on the effective thermal conductivity of the randomly distributed Al2O3-carbon core-shell nanofiber aerogel. The coupled term demonstrates low gas-contributed and effective thermal conductivity of 0.022 and 0.065 W m−1 K−1 at 1500 K, respectively. It is shown that the coupling effect of gas-solid interaction is noticeable and should not be ignored when the gas pressure pg ≤ 0.1 bar. The predicted effective thermal conductivity demonstrates good consistency with available experimental data, which could provide direction for quick and accurate design and optimization of high-temperature thermal insulation materials.

Lightweight aerogel-like silicon oxycarbide ceramics with directionally honeycomb-like structure for high temperature environments

Ceramic aerogels are considered promising thermal protection materials for their excellent thermal insulation performance and high-temperature stability. Lightweight aerogel-like silicon oxycarbide (SiOC) was prepared from polyhydromethylsiloxane (PHMS) as precursor and divinylbenzene (DVB) as crosslinker through solvothermal, freeze casting/drying and pyrolysis. The aerogel-like SiOC ceramic can achieve a low density of 0.087 g·cm−3 under the optimized mass ratio of DVB:PHMS of 1.5:1, with a directionally honeycomb-like structure induced by cyclohexane ice-crystals during the freeze casting. The unique porous structure endows the aerogel-like SiOC with low effective thermal conductivity (0.051–0.057 W·m−1·K−1) and good compressive strength (0.30–0.34 MPa). Moreover, the aerogel-like SiOC maintains structural and property stability after heat treatment (1400 °C in argon or 800 °C in air) for good thermal stability and oxidation resistance, which sheds light on the preparation of high-performance SiOC thermal insulation materials for high-temperature environments.

The modeling of meso-structure of fiber/particle composites and its influence on ETC under different inner pressure: A comprehensive study by RNMG-LBM

Fiber/Particle composites (FPCs) have garnered increasing attention as promising insulation materials in the energy field. Establishing a suitable composite model that can be used to investigate each component is essential. In this paper, the random non-contact multi-phase generation (RNMG) method was proposed for generating the meso-structure of FPCs. Furthermore, a comprehensive investigation was conducted using the Lattice Boltzmann Method (LBM) to quantify the influence mechanisms of various physical parameters, including inner pressure, aspect ratio, and particle diameter, as well as orientation angle, on the effective thermal conductivity (ETC) of FPCs. The validity of the RNMG-LBM was examined by comparing its results with experimental data. The simulation results demonstrated that reducing the particle diameters from 5 nm to 2 nm in FPCs leads to decreased pore sizes. Consequently, this reduces the sensitivity of FPCs to pressure fluctuations in the range of 10−1 Pa–105 Pa. Moreover, a physical parameter, called the orientation angle, was introduced to describe and explore the relationship between heat flux orientation and fiber orientation. The findings revealed that when the fiber was oriented perpendicular to the heat flux direction, it exhibited the lowest ETC among the orientation angles ranging from 0° to 90°. In addition, both the diameter and aspect ratio of fiber impacted its thermal properties. It's suggested controlling the aspect ratio below 20 and reducing fiber diameter as much as possible will significantly decrease the ETC of FPCs. The conclusions are valuable for optimizing the manufacturing process of FPCs and further improving their insulation capacity.

Prediction of thermal conduction in a four-phase composite with lowly-conducting interfaces based on variational asymptotic method

Due to the complex physical and chemical reactions, surface contamination and debonding, the interfaces within the four-phase composite, represented by hollow-glass-bead reinforced cement based material (HGB-CBM), cannot be assumed to be perfect. In order to consider the effect of imperfect interfaces, a novel thermal conduct model is developed. An imperfect interface is used to replace the pre-existing transition layer in the composite material, and the discontinuous temperature field proportional to the normal heat flow is preserved by using Hadamard’s relation and Taylor’s expansion. Based on the principle of minimum functional and variational asymptotic method, the governing variational statement for thermal conduct model is derived, and validated through comparison with RVE-based FEM and experimental results. It can be concluded that HGB-CBM with lowly-conducting imperfect interfaces has lower effective thermal conductivity, smaller heat flux within HGB and more significant changes at imperfect interfaces, which may be due to longer and more complex heat flux paths along the imperfect interfaces. HGB-CBM with larger volume fraction of HGB and smaller thermal conductivity of bead wall can provide better thermal insulation effect, which could provide reference for using HGB-CBM as thermal-insulating material in building engineering.

Experimental study on evaporation heat transfer characteristics of multi-composite porous wick with three different powders

Based on our previous research, the composite porous wick with spherical-dendritic powders owns biporous structure and uneven local thermal conductivity. In this paper, a multi-composite porous wick sintered with three different materials and structures of spherical copper, spherical nickel and dendritic nickel is tested. The spherical copper is beneficial to expansion of the evaporation heat transfer area, the dendritic nickel can provide sufficient liquid for evaporation, and the spherical nickel is used to regulate the distribution of spherical-dendritic powders and large pores. Through the evaporation heat transfer experiment, it is found that there are five different types of operating temperature and evaporation mass curves, including continuous and intermittent overshoot, single overshoot, stable operation and temperature losing control. The occurrence of the overshoot is affected by the bubble behavior in the bottom of the porous wick. And two types of the vapor–liquid interface behavior occur in the overshoot operation, in which the porous wick is penetrated by a single bubble or the evaporating interface, respectively. The multi-composite porous wick can reach a maximum heat load of 520 W (the corresponding heat flux is 43.5 W/cm2), and the highest heat transfer coefficient of the open capillary evaporator can reach 2776.1 W/(m2·K), with the lowest total thermal resistance is 0.3089 K/W at 500 W heat load. As a reference using a composite porous wick prepared with spherical copper powder and dendritic nickel powder of the same volume ratio, its maximum heat load is only 210 W (the corresponding heat flux is 17.6 W/cm2). The multi-composite porous wick shows better evaporation heat transfer performance. After the evaporation heat transfer experiment, the oxidized multi-composite porous wick still shows good wettability. It is inferred that a mixed wetting structure is formed inside it due to the composite structure sintered by three kinds of metal powders, which may be beneficial for improving the evaporation heat transfer performance of the multi-composite porous wick. In addition, the conditions such as lower effective thermal conductivity, more uneven local thermal conductivity, and large pores provided by the unchanged spherical nickel powder may also play a role.

The efficiency of using simple heat pipes with a relatively low thermal conductivity for cooling transmit/receive modules

In this paper, we solved the problem of increasing the cooling efficiency of the basic version of the transmit/receive (T/R) module with a heat sink tilted in its working position while increasing its thermal power by a factor of 2 (from 112 W to 224 W). The purpose of the work is to confirm the correctness of the proposed strategy of cheap modernization of the T/R module through the use of inexpensive flat gravity heat pipes (HP) with a threaded evaporator and a relatively low effective thermal conductivity (3000 W/(m•°C)). The novelty of the work is the substantiation by the results of comparative numerical simulation of the possibility and efficiency of using such HPs in the basic version of the cooling system for T/R module and the determination of the critical geometric parameters and relationships necessary for this. It is shown that the integration of 16 flat HPs of a simplified design into the base of the radiator made it possible to reduce the maximum temperature non-uniformity of the mounting surface at an air speed of 1 m/s by more than two times and reduce the maximum temperature of the hottest transistor at 20.5 °C. The temperature non-uniformity of 8 transistors decreased from 10.5 °C to 1.6 °C. The minimum thermal impedance of the HP cooling system (0.06 °C/W) is half that of the base case. The correctness of the proposed strategy of cheap modernization of T/R modules is confirmed and the prospects of using HPs of a simplified design with working fluids with a freezing temperature below 0 °C are shown, which allows expanding the temperature range of operating conditions. The originality of the modernized cooling system and its world novelty are confirmed by the patent for the invention.

Novel prediction of thermal conductivities for nano-aerogel and its composites as vacuum insulation panel core

For its advanced adiabatic performance and low effective thermal conductivity (ETC), vacuum insulation panel (VIP), as one of newly developed thermal insulation materials, has already been widely employed in many energy saving fields, especially in buildings industry. Nano-porous aerogel is commonly selected as optimal core materials of VIP. Vacuum is a vital parameter affecting on ETC. A novel theoretical model was proposed and the accuracy was verified by experiments to analyze the ETC. A device was designed and verified in this work to accurately collect the VIP internal pressure. The density, temperature, and vacuum effects on ETC were investigated. The result implied that density had a remarkable effect on ETC. A minimum density of 120 kg/m3 at a temperature 300 K gained the lowest ETC. The increase in temperature caused an increase in the radiative thermal conductivity. At ρa = 100 kg/m3and P > 3 kPa, the gas conduction was predominant part influencing ETC. And at ρa = 200 kg/m3and P > 50 kPa, gas conduction became the major infector. As pure aerogels are fragile, the ETC of aerogel composite core with additive fibers was investigated. It was indicated that the optimal fiber content is about 5% when T < 550 K and about 25% when T > 550 K. At T > 400 K, the total ETC increases significantly versus fiber diameter.

Testing the thermal properties of modern ventilated facade fastening systems

The study reported in this paper investigated a set of building fasteners used in ventilated facades. For the building fasteners actually present in the industrial market the values of the effective thermal conductivity were measured experimentally. These values were used next in numerical simulations run with COMSOL Multiphysics software application. The validation of the simulation model was done in specific additional experimental test. The paper presents a method of determining the effective thermal conductivity coefficient for fasteners with a novel design. Temperature distributions and heat fluxes were determined for different variants of multilayer walls with the fasteners. The calculation of the effective thermal conductivity coefficient for a structural profile is based on the heat balance of the measuring stand. The performed tests show not only an expected reduction in the coefficient value for structures in which stainless steel is used. The results also demonstrate that the fasteners with holes cut out in their structures have significantly lower effective thermal conductivity coefficients than those with solid walls. This effect can be justified by the formation of labyrinth-like narrowings extending the conductive heat flow path in the fastener. As a final result of the experimental tests and the COMSOL simulations the application of the effective thermal conductivity as the new indicator of a thermal effectiveness of building fasteners is proposed in industrial practice. Consequently the design of the building fasteners with various shapes of holes is recommended for improving their insulation features.

Porous-fin microchannel heat sinks for future micro-electronics cooling

Porous-fin microchannel heat sinks offering high solid-fluid interfacial areas have the potential to be an integral part of future microelectronic cooling systems. Replacing solid with porous fins reduces the pressure drop penalty of the straight plate-fin microchannel heat sinks. However, prior research has shown that in some cases porous fins adversely affect the thermal performance of the microchannel heat sinks. The reduced effective thermal conductivity of the porous fins compared to the solid ones has been considered the main reason for the reduced thermal performance of porous-fin microchannels. In this work, a detailed study is conducted to assess the feasibility of using porous fins to simultaneously improve the thermal and hydraulic performances of the straight plate-fin microchannel heat sinks. Results revealed that the porous-fin microchannels outperform the solid-fin microchannels at small channel heights. However, at high channel heights, the low effective thermal conductivity of porous fins results in a weak vertical thermal diffusion. Consequently, replacing solid with porous fins degrades the thermal performance of the microchannel at high channel heights. Additionally, the porous-fin microchannels exhibit better thermal performance than the solid-fin microchannels at fin to fluid width ratios above 0.25. The increased coolant mass flow penetrated the porous fin section is the main reason for the enhanced heat transfer rate of the porous-fin microchannels at higher width ratios. Moreover, the porous-fin microchannels offer a lower pressure drop penalty than the solid-fin microchannels under all examined operating conditions. This research confirms the potential of substituting solid with porous fins to simultaneously improve the thermal and hydraulic performances of the straight plate-fin microchannel heat sinks.

Impact of Various Thermistors on the Properties of Resistive Microbolometers Fabricated by CMOS Process.

Microbolometers based on the CMOS process has the important advantage of being automatically merged with circuits in the fabrication of larger arrays, but they typically suffer from low detectivity due to the difficulty in realizing high-sensitivity thermistors in the CMOS process. In this paper, two resistive microbolometers based on polysilicon and metal Al thermistors, respectively, are designed and fabricated by the standard CMOS process. Experimental results show that the detectivity of the two resistive microbolometers can reach a maximum of 1.78 ´ 109 cmHz1/2/W at 25 μA and a maximum of 6.2 ´ 108 cmHz1/2/W at 267 μA. The polysilicon microbolometer exhibits better detectivity at lower bias current due to its lower effective thermal conductivity and larger resistance. Even though the thermal time constant of the polysilicon thermistor is three times slower than that of the metal Al thermistor, the former is more suitable for designing a thermal imaging system with sensitive and low power consumption.

Development of a technology for manufacturing a heat-shielding structure on nitrogen cryocontainers, excluding heat transfer through gas

One of the important stages in the creation of the scientific and technical foundations for the calculation, design and manufacturing technology of the lowest heat-conductivity thermal protection from screen-vacuum thermal insulation (SVTI) is the development of a process for achieving the optimal vacuum P0 ≤ 10−3 Pa in the SVTI layers, since at this pressure, thermal conductivity (λeff) through the SVTI is carried out only due to the radiant (λrad) and contact-conductive (λk,k) components. It is proposed to obtain such a pressure in thermal insulation by using cushioning material in it, which was previously degassed in a separate vacuum chamber at 370−380 K for 12 h in order to remove water molecules from its structure and then replace them with nitrogen molecules. These molecules have 3−4 times less heat of adsorption; therefore they are pumped out faster. As a result, it becomes possible to accelerate (by ∼20 h) to achieve optimal vacuum in thermal insulation, as well as 11% lower effective thermal conductivity [equal to (14.1−14.3)⋅10−5 W/(m⋅K)]. The analysis carried out (according to the developed methodology) showed that the achieved optimal effective thermal conductivity of thermal insulation in a cryocontainers is determined by 33% of radiant thermal conductivity [4.7⋅10−5 W/(m⋅K)] and 67% of the contact-conductive component [9.4⋅10−5 W/(m⋅K)].

A non-volatile thermal switch for building energy savings

Compared with traditional static insulation, a thermally switchable building envelope could reduce annual heating and cooling loads by intermittently coupling to the outside environment when beneficial. Here, we demonstrate a voltage-actuated, contact/non-contact thermal switch that meets the unique challenges of this application. The switch is non-volatile, consuming electricity only briefly while switching and none to hold steady state. The switch ratio is 12, the off state has a low effective thermal conductivity of 0.045 W m - 1 K - 1 , comparable to fiberglass insulation, and the performance is stable over 1,000 switching cycles. Numerical simulations using real-world climate data show that combining this thermal switch with a thermal storage layer in a building envelope can yield annual energy savings of 9%–55% (heating) and 17%–76% (air conditioning), depending on the climate zone. The greatest benefits are realized when the exterior temperature crosses well above and below the desired interior temperature within a single 24 h period. • Bi-stable voltage-controlled thermal switch designed for building envelopes • Low off-state thermal conductivity of 0.045 W m - 1 K - 1 is close to fiberglass • Simulations show 9%–55% and 17%–76% annual energy savings for heating and cooling Like opening and closing windows, a thermally switchable building envelope could reduce heating and cooling loads by selectively coupling/decoupling to the exterior environment when beneficial. Miao et al. experimentally demonstrate a voltage-driven thermal switch suitable for this application. The bi-stable mechanism minimizes the energy required for device operation.

Theoretical and experimental investigation of a heat pipe heat exchanger for energy recovery of exhaust air

AbstractHeat pipes and two‐phase thermosyphon systems are passive heat transfer systems that employ a two‐phase cycle of a working fluid within a completely sealed system. Consequently, heat exchangers based on heat pipes have low thermal resistance and high effective thermal conductivity, which can reach up to the order of (105 W/(m K)). In energy recovery systems where the two streams should be unmixed, such as air‐conditioning systems of biological laboratories and operating rooms in hospitals, heat pipe heat exchangers (HPHEs) are recommended. In this study, an experimental and theoretical study was carried out on the thermal performance of an air‐to‐air HPHE filled with two refrigerants as working fluids, R22 and R407c. The heat pipe heat exchanger used was composed of two rows of copper heat pipes in a staggered manner, with 11 pipes per row. Tests were conducted at different airflow rates of 0.14, 0.18, and 0.22 m3/h, evaporator inlet‐air temperatures of 40, 44, and 50°C, filling ratios of 45%, 70%, and 100%, and ratios of heat capacity rate of the evaporator to condenser sections (Ce/Cc) of 1 and 1.5. For HPHE's steady‐state operation, a mathematical model for heat‐transfer performance was set and solved using MATLAB. Results illustrated that the heat transfer rate was in direct proportion with the evaporator inlet‐air temperature and flow rate. The highest HPHE's effectiveness was obtained at a 100% filling ratio and (Ce/Cc) of 1.5. The predicted and experimental values of condenser outlet‐air temperature were in good agreement, with a maximum difference of 3%. HPHE's effectiveness was found to increase with the increase in evaporator inlet‐air temperature and number of transfer units (NTU) and with the decrease in airflow rate, up to 33% and 20% for refrigerants R22 and R407c, respectively. Refrigerant R22 was the superior of the two refrigerants investigated.

Investigation of hydration/dehydration processes in a fluidized bed reactor using MgO/Mg(OH)2 thermochemical energy storage system

Thermochemical energy storage (TCES) is a promising energy storage method because of its high energy storage density and long-term storage ability. The reactor is the most critical part of the TCES system, and the fluidized bed reactor (FBR) is a novel reactor, which is not restricted by the low effective thermal conductivity of energy storage materials. However, there is little theoretical guidance about the design and operation of the FBR for a TCES system. In this study, we investigated the performance of an FBR using MgO/Mg(OH)2 as reaction materials by establishing a numerical model followed by experimental validation. The influence of operating parameters on the bed temperature was discussed based on numerical results. According to the sensitivity analysis, the reactor inner diameter, material mass, and inlet gas velocity were the main factors affecting the bed temperature. The optimal boundary surface temperature was analyzed, which influences the proportion of TCES in the total heat storage.

Determination of the effective thermal conductivity of the porous media based on digital rock physics

The thermal properties of a porous medium have been emphasized for its essential value in rock thermal engineering. Traditional theoretical and experimental methods on thermal conductivity studies of porous media usually contain homogeneous and isotropic assumptions with subjective idealization. Starting from the Digital Rock Physics (DRP), this paper establishes a digital internal structure model of a homogeneous rock through the processes of noise reduction, threshold segmentation, algorithmic process and three-dimensional (3D) reconstruction based on a series of micro-CT (computer tomography) images of the rock. With high quality meshes compatible with the Computational Fluid Dynamics (CFD) software, the Effective Thermal Conductivity (ETC) values are numerically obtained in three Representative Element Volume (REV) blocks with consistent porosity property. The results show that there is a 5% variation between the highest and lowest ETC data along different heat conduction directions. In addition, the simulation results satisfactorily fit with the empirical self-consistent model for ETC prediction of porous media. It is expected the innovative method reported in the present work can provide a feasible and reliable alternative for analyzing the rock thermal and other physical properties based on the accurate description of the internal microscale structures of the porous media.

Numerical simulation on thermal performance of vacuum insulation panels with fiber /powder porous media based on CFD method

Up to now, the fiber/powder porous media is getting more and more attention as a new development direction in the field of vacuum insulation panels. In this paper, the random generation-growth method is used to generate the structure of the fiber/powder porous media. The numerical simulation method is used to simulate the thermal conductivity of the fiber/powder porous media based on Fourier's law to study the correlation between the effective thermal conductivity and the microstructure of the fiber/powder porous media under vacuum conditions. The model was verified with experimental and theoretical data. The effect of different microstructure parameters such as powder diameter and fiber orientation angle on the effective thermal conductivity of the fiber/powder porous media is introduced in detail. The results show that the finer the diameter of the powder leads to the smaller the pore size of the fiber/powder porous media and has the stronger the ability to maintain a lower effective thermal conductivity under higher pressure. In addition, the more inconsistent between the fiber axial direction and the heat transfer direction, the better the thermal insulation performance. The results are very important for the optimization of the core material of the vacuum insulation panel.

Mathematical model of bent-flattened sintered-grooved heat pipe with concept of receding-and-excessing fluid

A mathematical model that uses the concept of evaporation and condensation to explain the process of heat and mass transfer within the bent, flattened heat pipe has been established to predict its wall temperature and thermal resistance. When the heat pipe was bent and flattened in the application, the wick was damaged, resulting in less liquid being returned. It caused the wick to become unsaturated at the evaporator section. It can be determined that the liquid-vapor interface will gradually recede into the wick. This is defined as the “Receding” model (R). In the condenser section, the remaining condensate was collected as excess fluid. This is referred to as the “Excess Fluid” model (EF). It was observed that the low thermal conductivity of the unsaturated wick decreased the effective thermal conductivity at the evaporator section. The effective thermal conductivity at the condenser section also decreased due to the low thermal conductivity of the liquid film. The low effective thermal conductivity resulted in higher evaporator temperatures and lower condenser temperatures, resulting in higher thermal resistance. The established “REF” model has been validated in the same conditions by the experimental results and provides accurate predictions for the heat transfer performance of heat pipes.

A dual-role theory of the aspect ratio of the nanofillers for the thermal conductivity of graphene-polymer nanocomposites

The extremely low thickness-to-length ratio (aspect ratio) of graphene nanofillers has long been considered beneficial to the overall thermal conductivity of graphene-polymer nanocomposites as it creates large surface area and extra channels for phonon transport. From time to time, however, a lower aspect ratio of the nanofillers did not yield a higher thermal conductivity as expected. To explain such phenomena, we explored the dual role played by the aspect ratio of graphene nanofillers in the effective thermal conductivity via an effective-medium approximation based on Maxwell's far-field matching with consideration of imperfect graphene interfaces. On one hand, we found that the initiation of the graphene thermal networks relies solely on the aspect ratio. A lower aspect ratio tends to boost the establishment of the thermal channels. On the other, we discovered a persistent exponential law that relates the graphene interfacial thermal coefficient to the aspect ratio regardless of the matrix material. A lower aspect ratio causes a higher interfacial thermal resistance, thus a possible lower effective thermal conductivity. These two effects are competing against each other in the framework of this theory. The origin of the second effect was discussed and the effect cannot be neglected with an aspect ratio below 0.01. This dual-role theory was confirmed by a large number of experimental observations.