Non-uniform Heat Flux Boundary Conditions Research Articles

Analysis on the Effects of Different Receiver Structures and Porous Parameters on the Volumetric Effects and Heat Transfer Performance of Porous Volumetric Solar Receiver

The volumetric solar receiver is an important heat transfer component of the concentrated solar power (CSP) system. Moreover, in order to improve the absorption of concentrated solar radiation, the porous media are widely used in volumetric solar receiver. In recent years, many studies were concerned with the effects of porosity, pore number density and size, Reynolds number, and Darcy number on heat and flow performance in volumetric solar receiver. However, there are few studies on the effects of structure type and geometric parameters on the volumetric effects and the radiation characteristics of a porous volumetric solar receiver with nonuniform heat flux boundary condition. In this contribution, in order to analyze the effects of structures on the volumetric effects and heat transfer performance, we design six types of porous volumetric receiver structures, whereas the volume of different structures keeps consistent. Then, we apply the local non-thermal-equilibrium model and also consider the effect of structure shape on the concentrating solar radiation transport characteristic. Furthermore, we apply the Gaussian distribution model (GDM) to simulate the actual nonuniform heat flux boundary condition. The result shows that drum-type structure (RT-III) has the best heat transfer performance among the six structures; e.g., the outlet average temperature of fluid is up to 851 K when the pore size and porosity of porous media are 2 mm and 0.7, respectively, which is 41 K increased than the scaled cone-type structure. In addition, the pore diameter has less influence on the outlet temperature of fluid, but it has greater influence on the pressure drop. The contribution can provide a reference for this type of solar receiver design and reconstruction.

A novel method to realize a non-uniform heat flux distribution through the variable-speed scanning of an electron beam

Quartz lamp heaters and hypersonic wind tunnel are currently applied in thermal assessment of heat resistant materials and surface of aircraft. However, it is difficult to achieve precise heat flux distribution by quartz lamp heaters, while enormous energy is required by a large scale hypersonic wind tunnel. Electron beam can be focused into a beam spot of millimeter scale by an electromagnetic lens and electron-magnetically deflected to achieve a rapid scanning over a workpiece. Moreover, it is of high energy utilization efficiency when applying an electron beam to heat a metal workpiece. Therefore, we propose to apply an electron beam with a variable speed to establish a novel method to realize various non-uniform heat flux boundary conditions. Besides, an electron beam thermal assessment equipment is devised. To analyze the feasibility of this method, an approach to calculate the heat flux distribution formed by an electron beam with variable-speed scanning is constructed with beam power, diameter of the beam spot and dwell duration of the electron beam at various locations as the key parameters. To realize a desired non-uniform heat flux distribution of the maximum gradient of 1.1 MW/m3, a variable-speed scanning strategy is constructed on basis of the conservation of energy. Compared with the desired heat flux, the maximum deviation of the scanned heat flux is 4.5% and the deviation in the main thermal assessment area is less than 3%. To verify the method, taking the time-average scanned heat flux as the boundary condition, a heat transfer model is constructed and temperature results are calculated. The experiment of variable-speed scanning of an electron beam according to the scanning strategy has been carried out. The measured temperatures are in good agreement with the predicted results at various locations. Temperature fluctuation during the scanning process is analyzed, and it is found to be proportional to the scanned heat flux divided by volumetric heat capacity, which is applicable for different materials up to 3.35 MW/m2. This study provides a novel and effective method for precise realization of various non-uniform heat flux boundary conditions.

Numerical investigation of transient phase-change transpiration cooling based on variable properties of coolant

Aimed at promoting the practical application of transpiration cooling technology in the field of active thermal protective system of hypersonic vehicles, this study presents a comprehensive numerical approach combining two-phase mixture model (TPMM), local thermal non-equilibrium (LTNE) assumption and variable properties of coolant to investigate the transient transpiration cooling problems with phase change. The real variable physical properties of coolant as functions of temperature and pressure are taken into consideration based on the data from the National Institute of Standards and Technology (NIST). The model is verified by comparing with the numerical and experimental data in previous researches. Results suggest that a new phenomenon of “reverse temperature distribution” in two-phase region is observed in the transient simulation considering variable properties. Comparison between the constant-property model and variable-property model indicates the necessity of the real variable physical properties, especially at low-pressure conditions with large differential pressure gradient. Meanwhile, the simulation with a non-uniform heat flux boundary condition reveals the mechanism of heat transfer deterioration. A gap of liquid outflow is formed after vapor generation, next moves along the outlet, then disappears and finally regenerates. The non-uniform heat flux will certainly lead to an uneven pressure distribution, resulting in the deterioration of local heat transfer in the porous matrix.

A tool to minimize the need of Monte Carlo ray tracing code for 3D finite volume modelling of a standard parabolic trough collector receiver under a realistic solar flux profile

AbstractThe energy collection element of a parabolic trough collector includes a selective coated metallic receiver tube inside an evacuated glass tube. Perpendicularly incident sun light on the parabolic trough mirror aperture is concentrated on the receiver tube highly nonuniformly along its circular direction. This solar energy is collected as thermal energy circulating a suitable heat transfer fluid (HTF) through the tube. This conjugate heat transfer phenomenon under nonuniform heat flux boundary condition is computationally studied applying 3D finite volume (FV) modelling technique of computational fluid dynamics coupled with Monte Carlo ray tracing (MCRT) optical data. The MCRT model simulates the actual flux profile around the receiver tube. Apart from a FV model, this coupled study requires expertise in, and access to, a suitable MCRT code. A combination of polynomial correlations and user‐defined function (UDF) is introduced in this article in order to minimize the need of MCRT codes from subsequent FV modelling of the receiver tube of the Luz Solar 2 (LS2) collector. The correlations are developed from a verified 3D MCRT model, which is equivalent to the local irradiation data as a function of receiver circular location. The UDF includes two algorithms: one to develop solar flux profile from the correlations around the receiver, and the other to calculate heat loss from the receiver. Interpreting the UDF into ANSYS Fluent, a 3D FV model of the LS2 receiver is developed and validated with experimental results. The effectiveness of the UDF as an alternative to MCRT code is verified. The FV model is capable to investigate the heat transfer characteristics of the LS2 collector receiver at different solar irradiation level, optical properties of the collector components, glass tube conditions, HTFs, inserts or swirl generators, collector length, and internal diameter of the tube.

Effect of hybrid nanofluid on heat transfer performance of parabolic trough solar collector receiver

In this study, three-dimensional heat transfer and flow characteristics of hybrid nanofluids under turbulent flow condition in a parabolic trough solar collector (PTC) receiver has been investigated. Ag–ZnO/Syltherm 800, Ag–TiO2/Syltherm 800, and Ag–MgO/Syltherm 800 hybrid nanofluids with 1.0%, 2.0%, 3.0%, and 4.0% nanoparticle volume fractions are used as working fluids. Reynolds number is between 10,000 and 80,000. The temperature of the fluid is taken as 500 K. The C++ homemade code has been written for the nonuniform heat flux boundary condition for the outer surface of the receiver. Variations of thermal efficiency, heat transfer coefficient, friction factor, PEC number, Nusselt number, and temperature distribution are presented for three different types of hybrid nanofluids and four different nanoparticle volume fractions with different Reynolds numbers. Also, the graphs of the average percent increase according to Syltherm 800 are given for the working parameters. According to the results of the study, all hybrid nanofluids are found to provide superiority over the base fluid (Syltherm 800) with respect to heat transfer and flow features. Heat transfer augments with the growth of Reynolds number and nanoparticle volume fraction. Thermal efficiency, which is one of the important parameters for PTC, decreases with increasing Reynolds number and increases with the increasing volume fraction of nanoparticle. It is obtained that the most efficient working fluid for the PTC receiver is the Ag–MgO/Syltherm 800 hybrid nanofluid with 4.0% nanoparticle volume fraction.

Two-phase flow pattern and heat transfer coefficients of R245fa/R134a under non-uniform heat flux

The non-uniform distribution of the heat flux commonly occurs in systems where vapor-liquid phase changes, such as nuclear reactors, and aerospace thermal control systems. This poses a great challenge on the study of pressure drop, heat transfer coefficient, and flow pattern prediction. In addition, flow patterns and heat transfer characteristics are usually affected by differences in fluid properties (viscosity, density, and surface tension), thermodynamic conditions (pressure, temperature), and geometric dimensions. At present, most of the researches on non-uniform heat flux are concentrated in systems with pure working fluids. However, compared to the pure working fluid, the mixed working fluid, particularly the zeotropic working fluid can improve the cycle efficiency and has great application potential, chiefly due to the temperature glide. There are few studies on the two-phase flow pattern and heat transfer characteristics of the zeotropic working fluid under non-uniform heat flux. Aiming at the problem of vapor-liquid phase change of zeotropic working fluid under non-uniform heat flux boundary conditions, a two-phase flow boiling experimental system in a horizontal tube with a 10 mm inner diameter under non-uniform heat flow was established. The studied zeotropic working fluid is R245fa/R134a with a mass fraction of 0.7:0.3. Experiments were performed at conditions where the mass flux ranges from 175 to 373 kg/(m2 s), while the heat flux ranges from 9.95 to 47.57 kW/m2. Using a high-speed camera and image processing technology, the gray values of the observed bubble flow, plug flow, stratified flow, and annular flow were analyzed, and the flow patterns under both non-uniform and uniform heat flux were demonstrated. Effects of the vapor quality, mass flux, heat flux and saturation temperature on the two-phase heat transfer coefficient were analyzed. And on the basis of experimental data, the prediction accuracy of four commonly used correlations was evaluated. The results show that the gray value of bubble flow and plug flow fluctuate widely, while the gray value fluctuation range of stratified flow and annular flow is relatively small. The typical flow pattern under non-uniform heat flux is the same as flow pattern under uniform heat flux. With the same amount of the additional heat, under the condition of non-uniform heat flux, the initial vapor quality of annular flow is advanced, and the proportion of the annular flow is larger than the non-annular flow. The main reason for this phenomenon is that the flow instability caused by non-uniform heat flux accelerates the transition of intermittent flow to annular flow. In addition, the heat transfer coefficient changes little at the lower vapor quality, and the heat transfer coefficient of the intermittent flow region is independent of vapor quality. In the annular flow region with higher vapor quality, the heat transfer coefficient increases with the increase of vapor quality. The heat transfer coefficient increases with the increase of mass flux and heat flux. The saturation temperature has little effect on the heat transfer coefficient in the convection boiling region and has a certain influence in the nucleate boiling region. Furthermore, the prediction ability of the four correlations is poor in the non-annular flow region. Relatively, the prediction accuracy of the Sun & Mishima correlation is the highest in the annular flow region.

Geometry and orientation effects in non-uniformly heated microchannel heat exchangers using supercritical carbon dioxide

There is growing interest to use supercritical carbon dioxide (sCO2) as a working fluid in thermal management applications. This study investigates the thermal-hydraulic performance of microchannel heat sinks as a function of flow channel geometry and orientation at operating conditions representative of electronics cooling applications. Three different experimental test sections, subject to non-uniform heat flux boundary conditions, were investigated. Two of the test sections contained parallel arrays of rectangular microchannels with a hydraulic diameter of 750 μm and aspect ratios of 1 and 2, respectively. The third test section had a staggered array of diamond shaped micro-pins with a hydraulic diameter, based on the minimum flow area, of 525.2 μm. Data were collected for varying inlet temperature (16⩽Tin⩽50 °C), mass flux (315⩽G⩽1000 kg m−2 s−1), and heat flux (20⩽q″⩽50 W cm−2) at a fixed reduced pressure (PR) of 1.1. A data analysis method using 2-D and 3-D heat transfer models of the test sections was used to calculate the average heat transfer coefficients for each experimental condition. Additionally, a pressure drop model was developed to resolve the total measured pressure drop into its components. The results of this study indicate that the turbulent convective heat transfer was independent of orientation (top versus bottom heating) for square microchannel (aspect ratio = 1) for the conditions investigated. Increasing the channel aspect ratio from 1 to 2 led to an enhancement in thermal transport. Finally, the heat transfer performance of the staggered pin array flow geometry was superior to the rectangular channels, but this enhancement in heat transfer was commensurate with the increase in pressure drop. Based on these results, this paper concludes with general design recommendations for those considering the early adoption of supercritical carbon dioxide for thermal management applications.

Two-phase investigation of water-Al2O3 nanofluid in a micro concentric annulus under non-uniform heat flux boundary conditions

PurposeThis is a 3D numerical study of convective heat transfer through a micro concentric annulus governing non-uniform heat flux boundary conditions employing water-Al2O3 nanofluid. The nanofluid is modeled using two-phase mixture model, as it has a good agreement to experimental results.Design/methodology/approachHalf of the inner pipe surface area of the annulus section of a double pipe heat exchanger is exposed to a constant heat flux which two models are considered to divide the exposing surface area to smaller ones considering the fact that in all cases half of the inner pipe surface area has to be exposed to the heat flux: in model (A), the exposing surface area is divided radially to two parts (A1), four parts (A2) and eight parts (A3) by covering the whole length of the annulus and in model (B) the exposing surface area is divided axially to two parts (B1), four parts (B2) and eight parts (B3) by covering half of the annulus radially.FindingsThe results reveal that model (B) leads to higher Nusselt numbers compared to model (A); however, at Reynolds number 10, model (A3) exceeds model (B3). The average Nusselt number is increased up to 142 and 83 per cent at models (A3) with Reynolds number 10 and model (B3) with Reynolds number 1000, respectively.Originality/valueThis paper is a two-phase investigation of water-Al2O3 nanofluid in a micro concentric annulus under non-uniform heat flux boundary conditions.

Three-objective shape optimization and parametric study of a micro-channel heat sink with discrete non-uniform heat flux boundary conditions

A water-cooled multi-die heat sink with parallel rectangular micro-channels was designed to satisfy the operational requirements of a multi-die processor. A shape optimization strategy based on the RSM (response surface method) was used to minimize pressure drop and die maximum case temperatures. The effects of the thermal interface materials and heat spreader between the dies and heat sink were captured by the numerical simulation. The optimization was performed for constant values of coolant flow rate and inlet temperature, as well as the power, location, and surface area of the dies. The influence of channel hydraulic diameter, Reynolds number, thermal entrance length, and total heat transfer surface area on the hydraulic and thermal performance of the heat sink was determined using CFD (computational fluid dynamics) simulations at RSM design points. A sensitivity analysis was performed to evaluate the effect of the design parameters on the response parameters. The optimum designs were achieved by minimizing a weighted objective function defined based on response parameters using JAYA algorithm. The results of weighted sum method were compared with Pareto based three-objective optimization with a NSGA-II (non-dominated sorting GENETIC algorithm). Finally, a parametric study was performed to see the effect of the design parameters on the response parameters.

Experimental investigation of supercritical carbon dioxide in horizontal microchannels with non-uniform heat flux boundary conditions

Supercritical carbon dioxide (sCO2) experiences a drastic change in its thermophysical properties near the thermodynamic critical point. A non-linear thermophysical property variation can influence the heat transfer behavior of sCO2 which is not predicted well by conventional single phase heat transfer theory. This can become a major hindrance in the effective design of heat exchangers using sCO2 as a heat transfer fluid and operating in the vicinity of the critical point. Previous investigations of sCO2 heating have been primarily focused on macroscale, circular and uniformly heated channels at relatively low heat fluxes. It is unclear if models and correlations developed from large circular tube data can be scaled down to the microscale, non-circular channels subject to non-uniform heating. The present study experimentally investigates the turbulent heat transfer performance of sCO2 in a microchannel heat exchanger operating in a horizontal configuration with a single wall non-uniform heat flux boundary condition. The test section has five parallel channels with a 0.75 mm hydraulic diameter and an aspect ratio of 1. The channels are fabricated using computer numerical control machining and the test section sealed using a diffusion bonding approach. Data analysis techniques which employ 2-D and 3-D heat transfer models of the experimental test section are developed to calculate the average heat transfer coefficients for a given set of experimental conditions. Data are obtained over a wide range of experimental parameters including test section applied heat flux (20⩽q″⩽40 W cm−2), mass flux (500⩽G⩽1000 kg m−2 s−1), reduced pressure (1.03⩽PR⩽1.1), and inlet temperatures (16⩽Tin⩽50 °C). The heat transfer data were screened for the presence of buoyancy and flow acceleration effects and then compared against correlations developed for turbulent subcritical and supercritical fluid flows.

Experimental investigation on heat dissipation of partitioned and non-partitioned rectangular enclosures under no-wind condition

Under no-wind condition, heat dissipation in partitioned and non-partitioned rectangular enclosures formed by quartz glass cover and base plate made of composite stone was experimentally investigated. Internal and external combined heat transfer coefficients hin and hex of enclosure, total thermal resistance Rtot of heat dissipation and correlations of average radiative and convective heat loss Nusselt numbers outside the glass cover were obtained and discussed. Also, the impacts of the gradient parameter k and magnitude of heat flux and tilt angle were compared. Results show that the local temperature differences (tl − ta) on the base plate and glass cover change in the same trend, while the impacts of partitions on average temperature on glass cover is greater than that on base plate. Compared with non-partitioned condition, hex is decreased and hin is increased, meanwhile, Rtot is higher when tilt angle θ < 60° for partitioned condition. Rtot for non-uniform heat flux boundary condition is higher than that for uniform heat flux boundary condition. Both for partitioned and non-partitioned conditions, thermal resistance Rex outside the glass cover is the main limited factor to control entire heat transfer process under θ < 60°. For non-partitioned condition, average convective heat loss Nusselt number Nucm increases with the augment of tilt angle, while for partitioned enclosure, Nucm for θ = 60° shows no relative tendency when k = 0 and k = 0.1. As for Nurm, the influence of tilt angle is less pronounced when compared with other parameters.

Numerical analysis of inserts configurations in a cavity receiver tube of a solar power tower plant with non-uniform heat flux

In order to analyse the effects of inserts configurations on heat transfer performance of solar cavity receiver tube (RT), a three-dimensional numerical model with four different inserts under non-uniform heat flux boundary condition was developed. Based on this model, the effects of inserts configurations, the effects of non-uniform heat flux boundary condition, the effects of thickness and positions of twisted-tape were analyzed. The numerical results show that RT-I with the twisted-tapes has the better comprehensive performances than those of other three kinds of RTs, and the RT-I with twisted-tapes is recommended to the solar cavity tower receiver. The twisted-tape RT-I has the highest Nu and the lowest circumferential temperature difference ΔT and average temperature of outer surface of receiver tube Tm, which are 21.74, 43 K, and 586 K, respectively. In addition, the thickness of twisted-tape has the significant effect on heat transfer performances of RTs. The twisted-tape with much more thickness will enhance the heat transfer and reduce the ΔT and Tm. Moreover, under the non-uniform boundary heat fluxes, RT-I with twisted-tapes placed in the back-sunlight domain is favorable than that of twisted-tapes placed in the sunlight concentrated domain. The contribution can provide a reference for this type of receiver design and reconstruction.

Experimental study of natural convection heat transfer from a nonuniformly heated flat plate simulating PV panel

In view of the characteristic of Photovoltaic (PV) conversion, an experimental study has been conducted to investigate the natural convection heat transfer from a flat plate. In order to simulate the real PV cells, three electrical heating circuits were employed to achieve a linear nonuniform heat flux boundary condition. The major parameters, such as the gradient parameter of heat flux k, tilt angle θ and heat flux qw were introduced to analyze their influence on heat transfer ability. Both the local temperature distribution and the overall trend of Nusselt number have been presented. Comparing with the uniform thermal boundary condition, the results show that the local convection heat transfer coefficient is fluctuated at the positions where heat flux has changed. When gradient parameter of heat flux increases, the difference of local convection heat transfer coefficient at the top of the plate between tilt angle θ = 0° and 90° becomes smaller. Moreover, it is observed that the gradient parameter of heat flux has a promotion effect on average convection Nusselt number at larger tilt angle, but the effect becomes complex as tilt angle is smaller. Finally, a correlation combining all significant factors has been put forward to estimate the natural convection heat transfer.

Infrared thermography with non-uniform heat flux boundary conditions on the rotor endwall of an axial turbine

It is technically challenging to measure heat fluxes on the rotating components of gas turbines, yet accurate knowledge of local heat loads under engine-representative conditions is crucial for ensuring the reliability of the designs. In this work, quantitative image processing tools were developed to perform fast and accurate infrared thermography measurements on 3D-shaped film-heaters directly deposited on the turbine endwalls. The newly developed image processing method and instrumentation were used to measure the heat load on the rotor endwalls of an axial turbine. A step-transient heat flux calibration technique is applied to measure the heat flux generated locally by the film heater, thus eliminating the need for a rigorously iso-energetic boundary condition. On-board electronics installed on the rotor record the temperature readings of RTDs installed in the substrate below the heaters in order to evaluate the conductive losses in the solid. Full maps of heat transfer coefficient and adiabatic wall temperature are produced for two different operating conditions, demonstrating the sensitivity of the technique to local flow features and variations in heat transfer due to Reynolds number effect.

Multi-scale numerical analysis of flow and heat transfer for a parabolic trough collector

This paper numerically investigated the coupled flow and heat transfer of a parabolic trough collector (PTC), with the non-uniform heat flux boundary condition on the absorber wall and the rarefied gas effects in the annular vacuum gap being taken into consideration. A fully coupled cross-sectional heat transfer model is established with Direct Simulation Monte Carlo (DSMC) method for the rarefied gas flow and heat transfer in the vacuum annual gap. The PTC tube efficiency can be obtained from the above simulation for a given HTF temperature. Such simulation is conducted for several specified HTF temperature and different efficiency data are obtained. These data are fitted by an equation. This equation is then used to advance the HTF temperature in the axial direction. In such a way a simplified 3D model for the design of a PTC receiver is obtained. Cross-sectional simulation results show that when the gas pressure is less than 0.1Pa further decrease in pressure makes no further contribution to reduce the heat loss. The effects of periphery non-uniform distribution of heat flux, coating material emissivity, envelope diameter and HTF inlet velocity on the PTC efficiency are discussed. An operation variant is proposed by using the 3D model by which the total PTC tube length can be reduced for a given thermal load.

Numerical optimization of catalyst configurations in a solar parabolic trough receiver–reactor with non-uniform heat flux

A three-dimensional numerical model of a solar parabolic trough receiver–reactor (SPTRR) with methanol steam reforming reaction was developed. Based on this model, the characteristics of hydrodynamics, heat transfer and chemical reaction of SPTRR were studied. Also, the effects of non-uniform heat flux boundary condition, catalyst layout and mass flow rate on the performances of SPTRR were analyzed. The results showed that SPTRRs partially filled with catalyst have better comprehensive performance than those fully filled with catalyst, and the SPTRR with horizontal cylindrical segment shaped catalyst in the lower part of receiver tube is recommended to substitute the fully catalyst-filled SPTRR. The chemical energy conversion per unit pump power of the proposed partially catalyst-filled SPTRR is one order of magnitude larger than that of the fully catalyst-filled SPTRR, and the mean outer surface temperature and maximum outer surface temperature difference of the proposed partially catalyst-filled SPTRR can be as low as that of the fully catalyst-filled SPTRR, when the mass of the catalyst in the partially catalyst-filled SPTRR is 0.4 times as much as that in the fully catalyst-filled SPTRR.

On moving heat sources

The two-dimensional thermal problem due to relative motion of a medium and a suddenly activated circular heat source is solved for several boundary conditions. The solutions can be interpreted as for a moving heat source in a stationary medium or a medium moving past a stationary heat source. Uniform and non-uniform temperature, and uniform and non-uniform heat flux boundary conditions are considered. The effect of velocity and radial direction on the temperature distribution is examined. Average, steady-state Nusselt numbers are derived. The transient response of a continuous line source is obtained as a limiting case of the prescribed heat flux solution.

Modeling studies of heat transfer and phase distribution in two-phase geothermal reservoirs

Phase distribution as well as mass flow and heat transfer behavior in two-phase geothermal systems have been studied by numerical modeling. A two-dimensional porous-slab model was used with a non-uniform heat flux boundary condition at the bottom. Steady-state solutions are obtained for the phase distribution and heat transfer behavior for cases with different mass of fluid (gas saturation) in place, permeabilities, and capillary pressures. The results obtained show very efficient heat transfer in the vapor-dominated zone due to the development of heat pipes and near-uniform saturations. The phase distribution below the vapor-dominated zone depends on permeability. For relatively high-permeability systems, single-phase liquid zones prevail, with convection providing the energy throughput. For lower permeability systems, a two-phase liquid-dominated zone develops, because single-phase liquid convection is not sufficient to dissipate heat released from the source. These results are consistent with observations from the field, where most high-temperature liquid-dominated two-phase systems have relatively low permeabilities (e.g. Krafla, Iceland; Olkaria, Kenya; Baca, New Mexico). The numerical results obtained also show that for high heat flow a high-temperature single-phase vapor zone can develop below a typical (240°C) vapor-dominated zone, as has recently been found at The Geysers, California, and Larderello, Italy.