Heat Transfer Intensity Research Articles

Experimental Investigation and Calculation of Convective Heat Transfer in Two-Component Gas–Liquid Flow Through Channels Packed with Metal Foams

This paper presents a study on heat transfer in two-phase mixtures (air–water and air–oil) flowing through heated horizontal channels filled with open-cell aluminum foams characterized by porosities of 92.9–94.3% and pore densities of 20, 30, and 40 PPI. The research included mass flux densities ranging from 2.82 to 284.7 kg/(m2·s) and heat flux densities from 5.3 to 35.7 kW/m2. The analysis examined the effects of flow conditions, fluid properties, and foam geometry on the intensity of heat transfer from the heated walls of the channel to the fluid. Results indicate that the heat transfer coefficient in two-component non-boiling flow exceeds that of single-phase flow, primarily due to fluid properties and velocities, with minimal impact from flow structures or foam geometry. An assessment of existing methods for predicting heat transfer coefficients in gas–liquid and boiling flows revealed significant discrepancies—up to several hundred percent—between measured and predicted values. To address these issues, a novel computational method was developed to accurately predict heat transfer coefficients for two-component non-boiling flow through metal foams.

Managing Heat Transfer Intensity in a Fluidized Particle-in-Tube Solar Receiver

Particle flow structure and associated heat transfer coefficient are examined as a function of temperature in a single-tube fluidized bed solar receiver operating in upward particle flow mode. It is found that temperature has a strong effect on both fluidization regimes and wall-to-bed heat transfer coefficient that varies in the range 800-1200 W/(m2.K). Turbulent fluidization regime results in the most intense heat transfer between the irradiated wall and the fluidized particle.

Numerical analysis of sloshing effects in cryogenic liquefied-hydrogen storage tanks for trains under various vibration conditions

This study examines the effects of hydrogen sloshing on internal pressure, temperature, and fluid behavior liquefied-hydrogen storage tanks designed for train usage by applying the natural frequency and frequency conditions from train vibration test standards. Notably, it investigates the impact on BOG generation using a transient volume-of-fluid phase change model. Here, simulations were conducted at vibrations of 0, 0.53, 1.53, and 3 Hz, which were established using sine wave acceleration. The results demonstrated that sloshing increased with higher frequencies, thereby resulting in a more intense heat transfer between the wall of the tanks and free surface of hydrogen and an increase in the BOG generation. Compared to the 0 Hz baseline, BOG generation increased by 13, 44, and 66 % at 0.53, 1.53, and 3 Hz, respectively.

Modeling of bed-to-wall heat transfer coefficient in fluidized adsorption bed by gene expression programming approach

Adsorption cooling and desalination methods with adsorption chillers (AC) are promising in energy technologies. However, the low-performance coefficient and bulkiness of traditional packed-bed ACs, primarily due to the high voidage of the sorbent beds leading to low heat transfer coefficients, pose significant challenges. Despite numerous attempts, a practical solution to this problem is yet to be found.In response to this challenge, we propose a novel approach: a fluidized adsorbent bed instead of the traditional packed bed. We also introduce gene expression programming (GEP) as an innovative artificial intelligence (AI) method for modeling the bed-to-wall heat transfer coefficient in the adsorption bed. Our study includes calculations and model validation for a heat transfer adsorption bed reactor designed for low-pressure adsorption processes. The fluidizing agent of the adsorbent bed was water vapor generated in the evaporator. Silica gel was used as the parent adsorption material in our tests. The heat transfer coefficient was successfully validated and determined through experiments and estimated using the formulated (b-t-wHTc) meta-model. The data evaluated by the model aligns well with the experimental results. Our calculations demonstrate that the GEP-based model accurately predicts the heat transfer coefficient and is suitable for analyzing the fluidized adsorption bed reactor.The outlined studies serve as a benchmark for subsequent simulations of the intensified heat transfer adsorption bed reactor, as they are integral to project No. 2018/29/B/ST8/00442, titled “Research on sorption process intensification methods in modified construction of adsorbent beds,” supported by the National Science Center in Poland.

Influence of inhomogeneous magnetic field source location on the intensity of thermomagnetic convection in a closed loop

Purpose. Obtaining information on the influence of the location of the inhomogeneous magnetic field source relative to the heated section of a vertical hydrodynamic loop filled with magnetic fluid on the intensity of convective heat transfer along the loop.Methods. Experiments were carried out using a hydrodynamic loop made of a thin tube of circular cross-section and placed in a vertical plane. Heat source was carried out by a heater on a short vertical section of the loop, and heat removal was implement out by blowing the entire surface of the tube with thermostated air. The source of the magnetic field was the flat pole tips of the ferrite magnetic core, in the gap between which the heater was located. The position of the pole tips relative to the heater varied vertically in the experiments. In the control experiments, the source of the magnetic field was deleted. The circuit was filled with medium concentrated magnetic liquid of the type "magnetite - kerosene - oleic acid". The intensity of steady-state convective heat flow along the tube was calculated from the results of measuring the tube surface temperature by copper-constantane thermocouples. The measurement results were presented in dimensionless form - the relationship between the Nusselt number and Rayleigh number.Results. The unfluctuating mixed, thermomagnetic and gravitational, convection of the magnetic fluid in the loop was observed at any location of the pole tips of the magnetic core relative to the heater. At location of pole tips above the heater, competition of gravitational and thermomagnetic convection was observed, and the heat flux was weak. When the pole tips were placed below the heater, the Nusselt number was 2 - 4 times higher than in the control tests (only gravitational convection) with equal Rayleigh numbers. The highest Nusselt numbers were obtained when the field source was placed in center of the heater.Conclusion. Information on the influence of the relative location of the magnetic field source and the heater on convective heat transfer by magnetic fluid in a hydrodynamic loop is obtained experimentally. The optimal concerning of heat transfer intensity position of the field source was found.

Mechanistic impact of Stern bilayer-based electrolysis on the enhancement of pool boiling heat transfer

Pool boiling is a heat transfer method that utilizes phase change for efficient heat transfer, and enhancing its heat transfer intensity and understanding the mechanism is of great practical significance for refrigeration and microelectronics thermal management. This study employs experimental methods to innovatively use ionic surfactant modification on heating surfaces, building on the basis of strengthening heat transfer by generating hydrogen bubbles through water electrolysis to increase nucleation sites on the heating surface. The Stern double layer formed on the heating surface is utilized to control the growth rate and quantity of hydrogen bubbles, thus achieving controllable nucleation density on the heating surface. The study examines its impact on heat transfer efficiency and the onset of nucleate boiling (ONB). Results indicate that electrolysis can increase nucleation sites at low heat fluxes, thereby enhancing heat transfer. Using a CTAB solution at a concentration of 3200 ppm with an electrolytic current of 0.08A, under the Stern potential at saturated adsorption, the heat transfer coefficient increased by up to 3.16 times. Additionally, the superheat at ONB decreased from 12.6 K to 4.4 K under boiling heat flux. Therefore, utilizing electrolysis with the addition of surfactants to enhance rapid cooling of high-temperature surfaces provides a novel engineering application approach.

An experimental study of knock analysis of HCNG fueled SI engine by different methods and prediction of knock intensity by particle swarm optimization-support vector machine

Hydrogen and its careers have a lot of potential in transportation industry due to lesser emissions and better performance. Performance of hydrogen enriched compressed natural gas (HCNG) fueled engine is limited by knock. The purpose of this study is to evaluate and predict knock at different operating conditions by varying (0 %–40 %) hydrogen amount in HCNG, by varying (25 %–100 %) load on engine and by varying (700 rpm–1700 rpm) speed of engine. Knock is evaluated by knock ratio, knock intensity, exhaust temperature and in-cylinder heat transfer rate. Quasi-dimensional combustion model (QDCM) MATLAB program is used to calculate in-cylinder pressure theoretically. By increasing (0 %–40 %) hydrogen in HCNG 32.7 %, 7.6 %, 33.5 %, 21.5 % & 6.4 % increment observed in, in-cylinder pressure, knock intensity, knock ratio, in-cylinder heat transfer rate and exhaust temperature respectively. By increasing (25 %–100 %) load on engine 75.8 %, 69.8 %, 88.2 %, 77.8 % & 46.7 % increment observed in, in-cylinder pressure, knock intensity, knock ratio, in-cylinder heat transfer rate & exhaust temperature respectively. By increasing speed (1100–1700) rpm of engine aforementioned parameters decreases by 12.19 %, 38.9 %, 36.06 %, 47.2 % & 14.1 % respectively. Knock intensity is predicted by particle swarm optimization-support vector machine (PSO-SVM) algorithm effectively. To minimize the prediction error, 31 combination of different (1–5) inputs applied to predict knock intensity and found combination of 5 input variable is 67.2 % & 71.9 % more accurate than 1 input variable in-terms of mean squared error and mean absolute error respectively. Findings of this study can be used to train electronic control unit of engine and in the development of HCNG fueled engine.

Convective heat transfer in porous channel with multi-layer fractures under Local thermal non-equilibrium condition

Based on Brinkman-extended-Darcy model and Local thermal non-equilibrium (LTNE) model, analytical solutions of velocity and temperature fields, pressure drop and Nusselt number of porous channels containing different number of fracture layers are obtained. The characteristics of fluid flow and heat transfer in multi-layer parallel fractured channels are further studied. Results show that for a single-layer or multi-layer fractured channel, the pressure drop changes sensitively at small hollow ratio and increases as the number of fracture layers increases with decreasing increase degree, but the pressure drop tends to be consistent regardless of the number of fracture layers when Darcy number is high. For small ratio of effective thermal conductivity, LTE model is valid regardless of Biot number, and the heat transfer intensity in multi-layer fractured channels is always higher than that in single-layer fractured channel at any hollow ratio, but the heat transfer intensity is still low. For large ratio of effective thermal conductivity, increasing Biot number or fracture layer number makes the Nusselt number increase under very small hollow ratio for both single and multiple fracture cases, and when both the ratio of effective thermal conductivity and Biot number are high, there is an intersection point occurs at hollow ratio near 0.1, which makes same heat transfer effect under different fracture numbers.

The Influence of Gas-Dynamic Non-Stationarity of Air Flow on the Heat Transfer Coefficient in Round and Triangular Straight Pipes with Different Turbulence Intensities

Unsteady gas-dynamic phenomena in pipelines of complex configuration are widespread in heat exchange and power equipment. Therefore, studying the heat transfer level of pulsating air flows in round and triangular pipes with different turbulence intensities is a relevant and significant task for the development of science and technology. The studies were conducted on a laboratory stand based on the thermal anemometry method and an automated system for collecting and processing experimental data. Rectilinear round and triangular pipes with identical cross-sectional areas were used in the work. Flow pulsations from 3 to 15.8 Hz were generated by means of a rotating flap. The turbulence intensity (TI) of the pulsating flows varied from 0.03 to 0.15 by installing stationary flat turbulators. The working medium was air with a temperature of 22 ± 1 °C moving at a speed from 5 to 75 m/s. It was established that the presence of gas-dynamic unsteadiness leads to an increase in the TI by 47–72% in a round pipe and by 36–86% in a triangular pipe. The presence of gas-dynamic unsteadiness causes a heat transfer intensification in a round pipe by 26–35.5% and by 24–36% in a triangular pipe. It was shown that a significant increase in the TI of pulsating flows leads to an increase in the heat transfer coefficient by 11–16% in a round pipe and a decrease in the heat transfer coefficient by 7–24% in a triangular pipe. The obtained results can be used in the design of heat exchangers and gas exchange systems in power machines, as well as in the creation of devices and apparatuses of pulse action.

Thermal conductivity impact on MHD convective heat transfer over moving wedge with surface heat flux and high magnetic Prandtl number

The present evaluation focused on heat transfer and magnetic flux along the moving non-conducting wedge shape with thermal conductivity and surface heat flux effects. A suitable and well-known similarity transformation for integration is used for the coupled non-linear partial differential equations with stream formulations. The Keller Box technique is utilized to integrate the final non-similar equations numerically. The discretized algebraic equations are displayed graphically and numerically using the MATLAB software. The physical characteristics like temperature, magnetic field, velocity profiles, skin friction, heat transfer and magnetic intensity are investigated with various factors. The influence of relevant characteristics like thermal conductivity ξ, Prandtl number Pr, wedge parameter β, variable viscosity ε, and Biot number Bi is interpreted graphically and numerically. It is noted that velocity graph effects are declined at lower value of Biot number and enhanced value is obtained at the higher value of Biot number. The fluid temperature with highest thermal slip effects is displayed with minimum value of thermal conductivity but minimum value is obtained at large value of thermal conductivity. The maximum heat flux value is obtained at large value of moving parameter.

Anomalous intensification of vortex heat transfer in the case of separated air flow over an inclined groove in a hot isothermal region of a flat plate

Anomalous heat transfer intensification in turbulent separated air flow over a long groove of moderate depth made in a plate inclined at an angle of 45°to the freestream is revealed both experimentally and numerically. The region under investigation includes a rectangle heated to 100°C by saturated water vapor. The Reynolds number varied from 103to 3×104. Using the gradient heatmetry the twofold increase, as compared with the case of a flat plate, of the heat transfer coefficient on the groove bottom is established at the Reynolds number Re = 3×104. The relative Nusselt number in different regions of the groove is determined both in the physical experiment and in the RANS calculations with the application of multiblock computational technologies and the SST model in the VP2/3 software package. The results are in good agreement in the turbulent flow regime at Re = (5, 10, and 30) ×103.

Experimental research of convective heat transfer between nanofluids and high-temperature dense granite in deep geothermal reservoirs

Global warming is increasing as CO2 emissions from the use of conventional fossil fuels continue to rise. The development of geothermal energy is gradually becoming the key to the sustainable development of human society in the future. Enhancing the heat transfer performance of circulating heat transfer fluids is one of the effective means to efficiently exploit deep geothermal resources. In this study, a self-developed circulating heat exchange experimental setup was established. The convection heat transfer characteristics of nanofluids in high-temperature rocks were experimentally investigated. Al2O3 nanofluid and CuO nanofluid were used as the working medium for heat transfer in high temperature granite samples. The effects of the Reynolds number (1000–6000) and parameters such as nanoparticle type (CuO & Al2O3), size (20 nm & 40 nm) and mass fraction (0 wt%-2wt%) on the heat transfer intensity were analyzed. The effect of nanoparticle accumulation on the rock surface on the local heat transfer strength was also investigated. The results indicate that the increase of Reynolds number enhances the heat transfer strength of nanofluids in rocks. Nanofluid containing 20 nm Al2O3 nanoparticles has better heat transfer performance in rock than nanofluid containing 20 nm CuO nanoparticles. Meanwhile, the heat transfer performance of nanofluid containing 20 nm Al2O3 nanoparticles in rock is better than that of nanofluid containing 40 nm Al2O3 nanoparticles. The heat transfer intensity increases and then decreases as the nanoparticle mass fraction increases. The heat transfer coefficient at Reynolds number 6000 is 52.2 % higher than that of water. For nanofluids and water, the local heat transfer intensity decreases with increasing distance from the inlet. Meanwhile, accumulation of nanoparticles on rock surfaces during convective heat transfer enhances heat transfer intensity. The results show that nanofluids have the potential to be applied to deep geothermal resource extraction. This paper provides data and technical support for the application of nanomaterials in medium and deep geothermal systems to enhance heat transfer efficiency.

Study of the Dynamics of a Single Bubble

The behaviour of bubbles in cavitation and boiling processes is determined by the thermodynamic parameters of the two-phase medium and the intensity of heat and mass transfer, which affect the final dynamic effects. In this review, we analyse the influences of these factors on bubble behaviour, as described in existing mathematical models. In particular, we analyse the physical processes that govern bubble behaviour, the influence of mass transfer, vapor and liquid temperature, vapour, and liquid pressure on the inertial and dynamic stages of development. In conclusion, we summarize the problems associated with modelling, the accuracy of numerical predictions, and propose directions for further research.

New correlations for focusing effect evaluation of the light metal layer in the lower head of a nuclear reactor in case of severe accident

In case of a severe accident (SA) in a nuclear reactor, the core heats up and materials melt and relocate to the lower head of the vessel. In such configuration, stratification occurs between the oxide and metal phases present in the melt. When the metal phase is lighter than the oxide phase, the maximum heat flux applied to the vessel wall is concentrated at the location of this top metal layer, creating the so-called “focusing effect”. The modelling of the focusing effect is essential for SA codes to properly evaluate the vessel failure time or to demonstrate the vessel integrity, when In Vessel Retention (IVR) strategy is implemented. However, existing correlations used in SA codes for focusing effect modelling have some limitations: if the thickness of the metal layer becomes very small, they predict an extremely high heat flux. Moreover, they were validated with tests using water, whereas the Prandtl number was found to play a significant role in this heat transfer.In this paper, the focusing effect is studied based on 3D Direct Numerical Simulations (DNS) of a cylindrical metal layer, covering all possible metal layer characteristics in terms of geometry and boundary conditions, especially for its top surface where the intensity of radiative heat transfer may vary. The analysis of the fluid behavior and the quantitative results obtained allow to derive new correlations and a new model based on the radial profile of the mean temperature of the system and on the radial profile at the top surface. Results at reactor scale confirm the overestimation of existing 0D model used in SA codes for a thin metal layer and highlight the relevance of the proposed model for focusing effect evaluation during transient situations.

Numerical analysis of thermal mechanism in downward flame spread over inverted surfaces of discrete inclined solid fuels

This investigation is propelled by engineering challenges associated with heat transfer during fire incidents on inclined surfaces, such as those observed in pitched roof fires, which hold significant implications for fire safety engineering and energy conservation. Drawing inspiration from real-world fire dynamics, this study delves into the downward flame spread over inverted solid fuel surfaces across a range of inclination angles, bridging the gap between theory and practice. Detailed analysis of temperature distribution, flame characteristics, and gas-phase reactions reveals critical insights into the complex interplay governing heat behavior under these conditions. The observed reduction in flame thickness and standoff distance with increasing inclination underscores the influence of buoyancy-induced plumes. This work developed a dimensionless formula that accounts for both buoyancy-induced plumes and diffusion rates perpendicular to the fuel, successfully capturing the variation in flame thickness with inclination angles. Furthermore, our findings show that the average flame spread rate escalates with higher inclination angles, attributed to enhanced gas-phase reactions near the flame front that lead to intensified heat transfer and a subsequent increase in temperature within the solid phase. The study also uncovers a competitive mechanism between flame “jumps” and heat transfer to the unburned zone, causing an initial increase and then a decrease in flame spread rates with increasing discrete gap size at larger inclination angles. Conversely, at smaller inclination angles, flame progression halts as the discrete gap size increases, due to inadequate heat transfer to the adjacent discrete fuel, resulting in the cessation of flame spread. This study not only advances our understanding of the thermal processes underpinning flame spread in inclined configurations but also offers crucial insights for developing more effective fire protection strategies and thermal protection systems in engineering applications.

Experimental and numerical investigation of heat sinks constructed by anisotropic 3-D Turing patterns

Heat dissipation is a critical issue for various engineering applications such as electronic devices, turbine blades and batteries. Existing heat sink structures contained inherent shortages in geometric flexibility which prevented further development of heat dissipation technologies. This study proposed a self-organized design method for heat sinks inspired by Turing Patterns (TP) in nature morphogenesis. The biomimetic structures were parametrically controlled with the aid of reaction-diffusion equations in terms of feature size, feature density, structural morphology and structural orientation. The metallic selective laser melting (SLM) technology was adopted to fabricate the three-dimensional (3-D) heat sinks. Experimental and numerical evaluations were conducted to compare TP structures with traditional pin-fin (PF) arrays baselines. Results showed substantial advantages of the TP structures in regulating flow field and enhancing height-wise fluid mixing. The thermal performance factor (TPF) of TP structures is approximately 45 % higher than traditional PF structures with similar feature sizes. The globally averaged magnitude of height-wise velocities of TP structure is around 2.4 times larger than that of PF structures due to the existence of the threshold-like and bridge-like structures. With the aid of three-dimensional structures, full utilization of the coolant was achieved to further increase the heat transfer intensity, while a high synergic level was formed within the flow field and temperature field to reduce the pressure drop. The design method proposed in this study demonstrated strong capability in tuning three-dimensional flow field and is expected to provide new insights to the heat transfer area that faces nonuniform thermal conditions and inhomogeneous flow conditions.

Mechanism chaotic movement of Leidenfrost droplets

One of the most effective methods for cooling overheated surfaces is drip irrigation. If the surface temperature exceeds the Leidenfrost temperature, then a vapor film is formed between the droplet and the surface, which leads not only to a decrease in heat transfer intensity but also causes droplet mobility. For a number of applications, the mobility of droplets is an undesirable phenomenon, so the analysis of the factors responsible for their movement is a relevant task. Here we analyze the movement mechanism of the Leidenfrost droplets with variations in the composition and volume of the droplets. The data obtained show that the droplet speed increases with an increase in the droplet volume. However, smaller droplets change direction of motion more often than larger droplets. To substantiate the experimental data, a hypothesis is proposed, according to which the mechanism of movement of Leidenfrost droplets is caused by the reactive force that arises due to the evaporation of liquid. A Leidenfrost droplet changes the direction of its movement due to the deformation of its surface under the influence of gravity and capillary force. To substantiate the experimental data a simple phenomenological model is proposed.

Convective Heat Transfer Analysis of Supercritical CO2 in Artificial Rock Fractures

Geothermal energy has emerged as a promising renewable and sustainable resource. This research article explores the utilization of supercritical CO2 and convective heat transfer in porous rock to enhance geothermal extraction. Artificial cement rock with a smooth and manmade slotted surface fracture was used as a working sample for heat transfer source in the test. Heat transfer characteristics under various flow rates, fluid inlet temperature and pressure, rock initial temperature are analyzed. The results indicate that the geometry characteristics of fracture surface impact the overall heat transfer intensity to a certain extent, whereas lithology has little influence on the heat transfer intensity. In addition, it is found that increasing injection pressure affects the operating temperature and density of the injected fluid, which helps to improve the production effect, to obtain more heat. However, limited by the heat transfer area, low injection flow is required. The maximum error of the heat transfer coefficient model is 7.05% compared with the empirical value and less than 10% compared with the experimental results. By examining the interaction between fluid and porous fractures, we aim to provide insights into the potential of these techniques for maximizing energy efficiency and tapping into deep geothermal resources.

Experimental investigation of the structure effects on the energy recovery and jet impinging process in a fast cooling Joule-Thomson cryocooler

Compared to other Joule-Thomson (J-T) refrigeration systems, open-cycle miniature J-T cryocoolers offer exceptional rapid cooling capabilities, making them ideal for applications such as infrared guidance in missiles. The energy recovery in the heat exchanger enables the refrigerant reach saturation temperature quickly and improve the jet liquefaction rate. The heat transfer intensity of the impinging jet determines the cooling rate of the target. Hence, heat recovery and the impact jet process are the primary factors behind this rapid cooling, with distinct roles that require separate consideration. The structural differences will directly affect the energy recovery and jet impact process. To investigate these affects, an experimental system for rapid cooling J-T cryocoolers was established and three distinct cryocoolers with substantial structural variations were designed. The important structures, including jet height, orifice diameter, enhanced heat transfer treatment of the cold plate, heat exchanger height, and heat exchanger cone angle, were closely studied. In the range of our experiments, it was found that larger heat exchanger cone angle leading better energy recovery performance, while the length of the heat exchanger is limited by the type of refrigerant. Longer heat exchanger actually introduce too much thermal mass for the refrigerant with better energy recovery performance. In the aspect of jet impingement, enhanced heat transfer treatment and larger jet height will improve the jet heat transfer intensity.

Effect of shape and thermal conductivity of microstructured capillary-porous coatings on heat transfer during evaporation and boiling in thin horizontal liquid layers

Investigation results on heat transfer during evaporation and boiling in thin horizontal liquid layers on 2D modulated capillary-porous coatings with a sinusoidal profile in a wide range of relative pressures and layer heights are presented in this paper. Coatings were made on a 3D laser printer using the technology of additive selective laser sintering (SLS). One coating was made of bronze with a modulation wavelength equal to the Laplace constant of the working fluid and two coatings were made of stainless steel and bronze with a doubled wavelength. The experimental data obtained on the coatings were compared with each other and with the values obtained by evaporation and boiling of n-dodecane on a smooth surface under the same conditions. It is shown that in the region of low relative pressures in liquid layers with a height less than the Laplace constant, heat transfer intensification achieved on a sample made of a material with low thermal conductivity (stainless steel) was five times higher as compared to a smooth surface. In the range of relative pressures, when nucleate boiling was observed in the layers, the temperature difference reached on the bronze coatings was approximately 3–4 times less than on a smooth surface. The temperature difference for the stainless steel coating changed little with an increase in the heat flux. In pre-crisis conditions, the temperature difference during boiling on the stainless steel coating was less than on the bronze coatings in the entire range of relative pressures and layer heights. The highest critical heat fluxes were obtained on a bronze coating with a modulation wavelength equal to the Laplace constant.