Flux Boundary Research Articles

Restoring the discontinuous heat equation source using sparse boundary data and dynamic sensors

This study focuses on addressing the inverse source problem associated with the parabolic equation. We rely on sparse boundary flux data as our measurements, which are acquired from a restricted section of the boundary. While it has been established that utilizing sparse boundary flux data can enable source recovery, the presence of a limited number of observation sensors poses a challenge for accurately tracing the inverse quantity of interest. To overcome this limitation, we introduce a sampling algorithm grounded in Langevin dynamics that incorporates dynamic sensors to capture the flux information. Furthermore, we propose and discuss two distinct dynamic sensor migration strategies. Remarkably, our findings demonstrate that even with only two observation sensors at our disposal, it remains feasible to successfully reconstruct the high-dimensional unknown parameters.

Changes in core–mantle boundary heat flux patterns throughout the supercontinent cycle

SUMMARY The Earth’s magnetic field is generated by a dynamo in the outer core and is crucial for shielding our planet from harmful radiation. Despite the established importance of the core–mantle boundary (CMB) heat flux as driver for the dynamo, open questions remain about how heat flux heterogeneities affect the magnetic field. Here, we explore the distribution of the CMB heat flux on Earth and its changes over time using compressible global 3-D mantle convection models in the geodynamic modelling software ASPECT. We discuss the use of the consistent boundary flux method as a tool to more accurately compute boundary heat fluxes in finite element simulations and the workflow to provide the computed heat flux patterns as boundary conditions in geodynamo simulations. Our models use a plate reconstruction throughout the last 1 billion years—encompassing the complete supercontinent cycle—to determine the location and sinking speed of subducted plates. The results show how mantle upwellings and downwellings create localized heat flux anomalies at the CMB that can vary drastically over Earth’s history and depend on the properties and evolution of the lowermost mantle as well as the surface subduction zone configuration. The distribution of hot and cold structures at the CMB changes throughout the supercontinent cycle in terms of location, shape and number, indicating that these structures fluctuate and might have looked very differently in Earth’s past. We estimate the resulting amplitude of spatial heat flux variations, expressed by the ratio of peak-to-peak amplitude to average heat flux, q*, to be at least 2. However, depending on the material properties and the adiabatic heat flux out of the core, q* can easily reach values >30. For a given set of material properties, q* generally varies by 30–50 per cent over time. Our results have implications for understanding the Earth’s thermal evolution and the stability of its magnetic field over geological timescales. They provide insights into the potential effects of the mantle on the magnetic field and pave the way for further exploring questions about the nucleation of the inner core and the past state of the lowermost mantle.

Experimentally determining the thermophysical properties, heat transfer and friction factor Fe3O4-TiO2 magnetic hybrid nanofluids in a mini-heat sink under magnetic field: Proposing new correlations

The heat transfer and friction factor for the Fe3O4-TiO2/water nanofluids circulating in a mini-heat sink of comparable magnitude to personal computer or electronic gadgets with and without a magnetic field have been estimated experimentally. The Fe3O4-TiO2 nanomaterial was prepared using chemical coprecipitation and sol–gel technique. Synthesized Fe3O4-TiO2 nanoparticles were characterized by X-ray diffraction, and vibrating sample magnetometer. The water-based hybrid nanofluids were then used in a series of experiments for assessing heat transfer, and friction factor of a mini-heat sink assembly of five circular channels with 4 mm in diameter and 50 mm in length. The experiments were performed in the operating range: 238.66 < Re < 1873.36; 0 <ϕ < 2 %; and 0 < Gauss < 1000, respectively. Constant heat flux boundary condition was maintained at the bottom surface of the heat sink and an external magnetic field was provided with an electromagnet. Results indicate that, by using the Fe3O4-TiO2 hybrid nanofluid of 2 % vol. the Nusselt number, with no magnetic field, is enhanced over that of water by 38.16 %, and by 88.93 % when the magnetic field of 1000 gauss is applied. However, the hybrid nanofluid of 2 vol% imposed a penalty in increase in friction factor of 25.87 % without magnetic field, and 67.64 % when magnetic field of 1000 G is applied at a Reynolds number of 1873.33 over water data. The experimental data were used to develop generalized Nusselt number and friction factor correlations for a mini-heat sink subjected to an external magnetic field.

Chemically reactive aspects of stagnation-point boundary layer flow of second-grade nanofluid over an exponentially stretching surface

Assessing the stagnation-point flow of a second-grade nanofluid with an induced magnetic field and the effects of Joule heating toward an exponentially extending surface is the leading goal of the current effort. The modeling of the thermal and solute energy equations has taken into account nonlinear thermal radiation as well as the impact of activation energy. The boundary of the sheet is subjected to the zero-mass flux and thermal slip boundary conditions. The simulated flow equations are converted into coupled nonlinear ODEs (ordinary differential equations) using similarity variables. The BVP4C MATLAB technique is used to numerically resolve these ODEs. Graphs and tabular data are used to perform the physical debate. The importance of a greater estimation of the magnetic Prandtl number and magnetic parameter is emphasized in order to obtain a better induced magnetic field profile. Additionally, the greater estimations of the second-order fluid parameter increase the rates of friction and heat transmission.

Vorticity dynamics at partial-slip boundaries

In this paper we discuss the dynamics of vorticity at partial-slip boundaries. We consider the total vector circulation, which includes both the total vorticity of the fluid and the slip velocity at the boundary (the interface vortex sheet). The generation of vector circulation is an inviscid process, which does not depend on either viscosity or the slip length at the boundary. Vector circulation is generated by the inviscid relative acceleration between the fluid and the solid, due to either tangential pressure gradients or tangential acceleration of the partial-slip wall. While the slip length does not affect the creation of vector circulation, it governs how vector circulation is distributed between the total vorticity of the fluid and the interface vortex sheet. Specifically, the partial-slip boundary condition prescribes the ratio between boundary vorticity and the strength of the interface vortex sheet, and the viscous boundary flux transfers vector circulation between the interface vortex sheet and the fluid interior to maintain this condition. The interaction between a vortex ring and a partial-slip wall is examined to highlight various aspects of this formulation. For the head-on collision, the quantity of vector circulation diffused into the fluid as secondary vorticity increases as the slip length is decreased, resulting in a stronger secondary vortex and increased rebound of the vortex ring. For the oblique interaction, the extent to which the vortex ring connects to the boundary increases as the slip length is increased.

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

In the paper, we develop three new methods for estimating unknown initial temperature in a backward time-fractional diffusion problem, which is transformed to a space-dependent inverse source problem for a new variable in the first method. Then, the initial temperature can be recovered by solving a second-order boundary value problem. The boundary functions and a unique zero element constitute a group symmetry. We derive energetic boundary functions in the symmetry group as the bases to retrieve the source term as an unknown function of space and time. In the second method, the solution bases are energetic boundary functions, and then by collocating the governing equation we obtain the expansion coefficients for retrieving the entire solution and initial temperature. For the first two methods, boundary fluxes are over-specified to retrieve the initial condition. In the third method, we give two boundary conditions and a final time temperature to construct the bases in another symmetry group; the governing equation is collocated to a linear system to obtain the whole solution (initial temperature involved). These three methods are assessed and compared by numerical experiments.

Lattice Boltzmann Method Simulation of Nanofluid Natural Convection Heat Transfer in a Square Cavity with Constant Heat Flux at Walls

In this study, for the first time, a nanofluid's natural convection heat transfer in a two-dimensional square cavity has been numerically investigated by use of the lattice Boltzmann method with the constant heat flux boundary condition. The horizontal walls of the cavity are insulated, and the vertical walls are kept at a constant heat flux. The diameters of the nanoparticles inside the cavity are the same and have a homogeneous distribution, and there is no chemical reaction between the particles. The flow is also assumed to be the steady state and two-dimensional. Constant temperature, streamlines, velocity, and average Nusselt have been investigated for different nanoparticle volume fractions and Rayleigh numbers. The results showed that the lattice Boltzmann method efficiently analyzes the natural heat transfer of nanofluids; moreover, by use of nanofluid in the cavity increases the heat transfer rate. With the increase in the nanoparticle volume fraction, the average Nusselt number on the right wall of the cavity increased. For a volume fraction of 20% with Grashof number 105, the average Nusselt number increased by almost 50% compared to the base fluid at the same Grashof number. It has been observed that as the volume fraction of nanoparticles in the fluid increases, the fluid’s viscosity also increases; consequently, the velocity of the fluid is found to decrease.

Fouling of reverse osmosis membrane in sugar mill condensate purification under sub- and super-boundary flux conditions

The sugar production process, inherently water-intensive, is characterized by pronounced condensate formation during the evaporation and boiling phases. Given the environmental and resource implications, there is a scholarly emphasis on the potable reuse of such condensate by reverse osmosis (RO) in sugar mills. However, membrane fouling presents a significant challenge to the consistent operation of the RO process. This research examines the influence of operating parameters, namely transmembrane pressure and crossflow velocity, on the performance and fouling tendencies of an RO membrane during the purification of sugar mill condensate. The boundary flux theory was employed to model the permeate flux throughout this process. A boundary flux (5.92 L h−1m−2) and its corresponding operating pressure (10 bar) were determined. Moreover, a comparative analysis of the fouling mechanisms and the composition of the foulant layer under both sub-boundary and super-boundary conditions was undertaken. Under sub-boundary conditions, the fouling mechanism was identified as cake-intermediate blocking, while the super-boundary condition exhibited intermediate blocking. Under both conditions, organic fouling emerged as the primary fouling type. The TOC of surface contaminants ranged from 1.07 to 1.86 g/m2. Within this, proteins (484.0 to 850.3 mg m−2) and polysaccharides (201.5 to 379.2 mg m−2) were the dominant components. However, when operating conditions surpassed the boundary, there was a noticeable rise in inorganic fouling such as Ca (from 44.95 to 149.43 mg m−2) and Fe (from 4.40 to 112.09 mg m−2), which is likely due to the co-deposition of organic materials with inorganic ions. This research contributes to a more nuanced understanding of how operating conditions influence fouling mechanisms and provides foundational insights for addressing fouling challenges in sugar mill condensate purification.

Seasonal Variations in Eddy-Induced Atmospheric Perturbations in the South China Sea

Abstract This study investigates the impact of eddy-induced sea surface temperature (SST) anomalies on the overlying atmosphere in the South China Sea, utilizing observational and reanalysis datasets. The results reveal that SST anomalies caused by anticyclonic or cyclonic eddies have a significant impact on the acceleration or deceleration of surface winds, with a stronger response in summer compared to winter. Moreover, atmospheric responses, such as heat flux, precipitation, and marine atmospheric boundary layer (MABL) depth above anticyclonic or cyclonic eddies, exhibit in-phase seasonal variations as surface winds. The study also explores the mechanisms leading to atmospheric response, pointing toward the vertical mixing mechanism as the dominant cause, supported by both the in-phase relationship between SST and surface wind anomalies and the linear relationship between the wind stress divergence anomaly and downwind SST gradient anomaly. Seasonal variations in coupling intensity are attributed to varying background atmospheric conditions, with more effective vertical turbulence mixing and stronger coupling intensity caused by more unstable MABL and enhanced large-scale vertical wind shear during summer than winter. Besides, the atmosphere above eddies is under a quasi-equilibrium condition, in which the surface wind stress increases monotonically with the depth of MABL. Given that the MABL depth response to eddy-induced SST anomaly is stronger in summer than in winter, it is reasonable to expect a more intense wind response during this season. Thus, the MABL depth coupling works together with the vertical mixing mechanism to explain the proportional relationship between SST and wind anomalies, and why the atmospheric response is stronger in summer than in winter. Significance Statement The atmosphere exerts a significant influence on the ocean, with strong winds cooling the sea surface temperature on a large scale (more than 1000 km). Conversely, the ocean tends to drive the atmosphere at a mesoscale level (10–1000 km), whereby a warm sea surface temperature causes surface wind to accelerate. This study aims to better understand how mesoscale eddies in the South China Sea influence the overlying atmosphere in different seasons. Our research explores factors that control mesoscale air–sea coupling, and highlights the importance of stability and depth change of marine atmospheric boundary layer. These results will establish a foundation for future research examining how the atmospheric perturbations caused by mesoscale eddies can in turn impact both ocean circulation and eddies.

Practical considerations for the adoption of Anderson acceleration in nonlinear diffusion accelerated transport

The use of the Anderson Acceleration (AA) method within the context of the coarse mesh finite difference (CMFD)-accelerated neutron transport k-eigenvalue calculation is explored for practical problems. Current application of CMFD only has a linear convergence rate. The AA method is similar to a quasi-Newton method and can improve the convergence rate of fixed-point iterations. We therefore implement AA for the 2D calculation in MPACT and study its performance in pressurized water reactor (PWR) problems. The initial results show that the traditional AA method can improve the convergence rate in a trivial problem and improve the stability of the CMFD method for some problems. However, the success of AA in the practical problems, where simulations with symmetry or domain decomposition, relies on one essential factor–the treatment of iterative angular flux boundary conditions in the AA. Additionally, the efficiency of AA is also affected by the computational cost for solving the optimization problem, and the computational cost of the transport solver and the CMFD solver. To address these issues, we propose a new implementation of AA that considers only the mutligroup scalar flux in the optimization problem, but the angular flux and the scalar flux in the mixing. The new approach achieves better stability and efficiency in practical simulations, and can be explained by the theory of Randomized Numerical Linear Algebra (RandNLA).

Time-periodic natural convection in a pentagonal house with pitched roof during one day

A numerical investigation of unsteady natural convective flow and heat transfer in a pentagonal house is performed. We set a sinusoidal periodic heat flux boundary condition for the pitched roof. Local radial basis function method and Fourier spectral method are adopted to solve the governing equations. The Rayleigh number varies from 103 to 105, the period of the roof heat flux ranges from 10 to 1,000, and the amplitude is set as 0.1, 0.3, and 0.5, respectively. It is found that all the control parameters significantly impact the fluid flow and heat transfer. The temperature and flow fields exhibit periodic behavior due to the sinusoidal heat source. The increasing Rayleigh number leads to a larger periodic average Nusselt number of the top and right walls and a smaller value of the bottom wall. Moreover, an obvious hysteresis phenomenon of heat transfer occurs especially for high Rayleigh number and large period.

Effects of different cross-sections of Body Centered Cubic cells on pressure drop and heat transfer of additively manufactured heat sinks

In many industrial applications, heat loads management requires the design and production of compact heat exchangers which are expected to handle high thermal loads with acceptable pressure losses, while assuring good mechanical performances. These challenging targets can be achieved by filling the cavities where the cool/hot fluid circulates with lattice structures promoting the heat exchange between the fluid and the cavity boundaries. Such lattice structures can be only produced through Additive Manufacturing due to their high geometric complexity. Recent experimental investigations proved the effectiveness of some kinds of lattice structures having a circular cross section. Here the aerothermal behaviour of Body-Centred Cubic (BCC) lattice stagger arrays in a rectangular channel was experimentally investigated by considerably extending the previous studies to higher Reynolds numbers (up to 30′000) and to new types of lattice structures. Specifically, three new BCC structures having a cam-like, drop-like and elliptical cross section were explored in this work and compared against those having circular cross section. All the samples were manufactured by means of Laser Powder Bed Fusion and made from AlSi10Mg. At first, the heat exchangers were comprehensively characterized by means of optical non-destructive methods. Successively they were tested in a dedicated rig by imposing constant heat flux boundary conditions. The characteristics of the transitional or fully turbulent approaching flow to the test section are also reported thanks to dedicated flow field measurements performed by Particle Image Velocimetry. According to the obtained results, the BCC structure with the circular cross section of larger diameter is the most effective in terms of heat transfer, although it is largely penalized by the pressure losses. Similar heat transfer performances were achieved by the tapered cross section of elliptical shape with the advantage of a considerably lower friction factor. Pressure losses resulted almost identical for all the tapered cross sections but lower than those of the circular one having an equal frontal dimension. When considering the thermal performance factor the circular shape becomes unfavourable for Re>20′000, while the elliptical cross section is the best choice to efficiently promote heat transfer up to Re=30′000.

Utilization of the Crank-Nicolson technique to investigate thermal enhancement in 3D convective Walter-B fluid by inserting tiny nanoparticles on a circular cylinder

&lt;abstract&gt; &lt;p&gt;The current study is based on the mechanism of mixed convection and solar thermal radiation in Walters'-B fluid considering tera-hybrid nano-structures using convective boundary constraints (CBC) and (CHF) constant heat flux. The heat transmission phenomenon of the current study is taken into account under the influence of triple-suspended nanoparticles. The current problem has several potential applications, including improvements in solar thermal energy systems, nanofluids, aerospace, cooling processes, automotive engineering, and numerical modeling methods. A numerical approach, namely Crank-Nicolson, is utilized in the modeling of 3D Walter's B fluid past over a 3D circular cylinder whose radius varies sinusoidally for evaluation of velocity and temperature distributions. For mathematical modeling, the Cartesian coordinate system was used for the current study. Comparative analysis between constant heat flux (CHF) and convective boundary constraints (CBC) was demonstrated graphically against multifarious parameters towards the temperature profile and velocity profiles along the x-axis and in the y-axis. Moreover, comparative analysis for dissimilar parameters was manifested for Nusselt number through tables, and graphically for skin friction co-efficient and Nusselt number and has shown excellent accuracy. It was estimated that by enhancing values of &lt;italic&gt;Q&lt;sub&gt;sr&lt;/sub&gt;&lt;/italic&gt;, &lt;italic&gt;C&lt;/italic&gt;, &lt;italic&gt;H&lt;sub&gt;s&lt;/sub&gt;&lt;/italic&gt; and &lt;italic&gt;E&lt;sub&gt;c&lt;/sub&gt;&lt;/italic&gt;, it was addressed that temperature curve increases for CHF and CBC cases.&lt;/p&gt; &lt;/abstract&gt;

CONVECTIVE HEAT TRANSFER INSIDE A ROTATING HELICAL PIPE FILLED WITH SATURATED POROUS MEDIA

In this paper, we study the hydrothermal characteristics of flow inside a rotating helical pipe filled with saturated homogeneous porous medium. The analysis is being carried out for the case of small curvature and torsion. Using the perturbation approach, velocity and temperature fields are solved for both uniform wall heat flux and uniform wall temperature boundary conditions. Perturbation expansion up to the third order is carried out to investigate the effect of rotation on the flow. The influence of rotation on velocity is noticed as early as the first order, and on temperature solution, it has an effect in the third order. The influence of rotation on the Nusselt number does not appear till third order, and it is discovered that the Nusselt number grows as dimensionless curvature increases. Moreover, the theoretical results have been verified against experimental data from existing literature for the special case of zero rotation and curvature of the pipe. The available experimental data align well with and support the theoretical results in this limiting case.

IMPACT OF MEASUREMENTS TECHNIQUES ON HEAT TRANSFER CHARACTERISTICS IN AIR JET ARRAYS

Air impact processes have diverse applications in engineering, including backflow welding, textile drying, and gas turbine blade and combustion liner cooling. This research examines the influence of experimental methodologies and measurement tools on convective heat transfer in adjustable air jet assemblies. The experiment involves the use of heated targets made of thin stainless steel foil with constant heat flux boundary conditions. Thermography measures target surface temperatures by analyzing how internal passage cross-flow affects convective heat transfer via outflow adjustments. The experiments involve two arrays of jet nozzles: inline and staggering, each comprising 44 impingement jet nozzles arranged in 4 rows with 11 jet holes in each row. The study presents unsteady time average local and spatial Nusselt numbers as functions of jet Reynolds number (4630-14000) and explores their dependence on the jet nozzle diameter. Cross-flow levels significantly affect spatial and local Nusselt numbers in both local and span-wise averaged values, regardless of the Reynolds number. Strong cross-flow (single configuration) distributes flow causing turbulence and uneven heat distribution, reducing Nusselt numbers. In contrast, moderate cross-flow (double configuration) improves heat transfer and increases Nusselt numbers. The study emphasizes the crucial role of experimental techniques in heat transfer evaluation and demonstrates agreement with prior studies within a standard error below 5%.

Heat Transfer Augmentation for Impingement of Steady Air Jet under Exponential Heat Flux Boundary Condition

Heat Transfer Augmentation for Impingement of Steady Air Jet under Exponential Heat Flux Boundary Condition

PARAMETRIC STUDY AND DEVELOPMENT OF A CORRELATION FOR NUSSELT NUMBER IN NANOFLUID JET IMPINGEMENT

Numerical simulations for nanofluid ( water with Al&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;3&lt;/sub&gt; nanoparticles) jet impinging perpendicularly on a flat circular heated p late have been performed. A heated p late is subjected to constant heat flux boundary condition. A two-phase modified mixture mo del was used for the prediction of heat transfer coefficient, and comparisons are made with standard mixture model. Present results for average Nusselt number are validated with experimental data available in the literature. Though a standard mixture model predicted heat transfer with accepted accuracy, it was found that accuracy of modified mixture model is better (around 5&amp;#37; improvement) compared to standard mixture model. Thereafter, parametric study was performed considering nozzle exit Reynolds number (Re), spacing ratio (&lt;i&gt;H/D&lt;/i&gt;), nanoparticle volume fraction (&amp;phi;), and nanoparticle diameter (dp) on heat transfer prediction. The results reveal that particle diameter in the range 10-100 nm has no effect on the Nusselt number, Furthermore, heat transfer increased with increasing Reynolds number and volume fraction. However, spacing ratio shows first increasing and then, decreasing trend (similar to a log-normal distribution curve) in the prediction of heat transfer. Finally, a new correlation was developed for Nusselt number using nonlinear regression analysis. In the correlation, a two-phase multiplier was used, which is the ratio between two-phase Nusselt number (Nu&lt;sub&gt;nf&lt;/sub&gt; ) and single-phase Nusselt number (Nu&lt;sub&gt;sp&lt;/sub&gt;). The simplified correlation is found to predict data with maximum error of 8.9&amp;#37;, average error of 2.74&amp;#37; and &lt;i&gt;R&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt; &amp;#61; 0.955.

The use of variable coil pitch of helical tube on the hydro-thermal performance improvement

The use of helical tubes in heat transfer appliances is desirable due to their better heat transfer characteristics, but the higher pressure drop decreases the overall performance. The variation of pitch design of a helical tube is proposed to alleviate this situation so that the pitch number does not remain constant in the total tube length. A total of six variable pitch numbers with three different diameters are proposed and investigated on the thermal and fluid characteristics. To better understand the helical tube efficiency, PEC (performance evaluation criteria) is selected as a performance indicator in the present work and simulations are performed in a laminar regime (100 ≤ Re≤ 1600) at a constant heat flux boundary condition using computational fluid dynamics. The numerical results show that variable radial pitch has higher effects on the overall performance than variable axial pitch, and it can intensify the helical tube performance by up to 10%. The results also indicate that increasing the tube diameter leads to heat transfer and friction factor increment while increasing the Reynolds number deteriorates the overall performance.

A source-to-surface model of heat and fluid transport in the Taupō Rift, New Zealand

We construct 2-D numerical models of bulk heat and water transport from the mid-crust (10 km depth) to the surface in a 20 km-wide continental rift, based on observations from the Taupō Rift in the central Taupō Volcanic Zone (TVZ) of New Zealand.The model represents a simplified geological setting with two low-permeability basement-like rock-types, the first defining the margins of the rift and the second a distinct ‘rifted basement’. The rift is overlain with a 2 km-thick, 20 km-wide layer of shallow volcanic infill. At the base of model there is a 10 km-deep heat source (the ‘hotplate’) with a specified TVZ-like heat flux of 0.77 Wm−2 and a ‘target’ average hotplate temperature of 700o to 900 °C, required to maintain the low melt fractions at ∼10 km depth inferred from geophysics.Parameterising the permeability of the rifted basement is a key part of the paper as it enables both the heat flux boundary condition and a temperature constraint to be satisfied at the hotplate. More generally, it allows the flexibility to control the depth of temperature contours which represent proxies for geophysical measurements. It is described as a power-law for log10(horizontal component of permeability, kh) with specified values of kh at two depths and a power law exponent as free parameters. Viable models which satisfy all constraints for the hydrothermal system beneath the TVZ rift have a power-law exponent less than ∼0.4, shallow (2 km deep) kh's of 10–14.0 to 10–13.0 m2, and deep (10 km) kh‘s <10–16.5 m2. The vertical component of the rifted basement permeability, kv, is ten times higher than kh.A generic feature of the models is that irregular, unsteady convection occurs in the upper ∼5 km of the rift. This results in temporal and spatial fluctuations in heat and fluid flow which are interpreted as high-temperature TVZ-like geothermal systems which have temperatures of ca. 300 °C at 2 km depth. Over the nominal simulation time of 3 Myr, approximately 40% of the geothermal systems are clustered within ∼2 km of the rift margins, with the remainder distributed roughly uniformly across the rift. Between the geothermal systems cool recharge from the surface occurs, with temperatures as low as 50 °C occurring at 5 km depth. The models successfully match temperature proxies for the depths of maximum seismicity (450 °C) and shallowest partial melt (> 700 °C) inferred from geophysical observations.Heat transport within the rift is dominated by convection. This is true even at 10 km depth, at the base of the geothermal systems, where kh in the rifted basement approaches 10−17 m2. In this situation, the relatively large value of vertical permeability kv, ∼ 10−16 m2, allows convection to occur. Conductive heat transport dominates in regions ∼1 km in vertical extent immediately above the hotplate and between the geothermal systems.The vigour of the hydrothermal systems is controlled by the permeability of the shallow volcanics, through which all cold surface recharge and hot outflows of fluid occur. Models with low permeability for the shallow volcanics produce longer lasting and higher temperature geothermal systems, and those with high permeability produce fewer and cooler geothermal systems.

A Machine Learning Accelerated Hierarchical 3D+1D Model for Proton Exchange Membrane Fuel Cells

Physics-based continuum models for proton exchange membrane fuel cells (PEMFCs) are an essential tool for fuel cell design and management. To date, many continuum models, ranging from 1D to 3D, have been developed for PEMFCs. Although computationally efficient, 1D models do not account for heterogeneity in flow fields, which negatively impact their accuracy. In contrast, 2D and 3D models are usually more representative of actual operating conditions but computationally intensive due to the coupled partial differential equations and large number of mesh elements involved. To overcome these issues, a hierarchical approach that combines a 2D/3D description of flow fields, gas diffusion layers (GDLs) and a simplified microporous layer (MPL)/catalyst layer (CL)/membrane sub-model has been proposed in the literature. However, studies based on this method often use a simplified or 0D MPL/CL/membrane sub-model, whose results may deviate from a full 1D description due to the neglected nonlinearity, especially at higher loads.In this study, we present a computationally efficient 3D+1D hierarchical model for PEMFCs accelerated by machine learning. The 3D model, which captures the two-phase flow in the gas channels and GDLs, is coupled with a full 1D description of the MPLs, membrane, CLs, and CL agglomerates by exchanging boundary values and fluxes, as shown in the figure. To avoid the high computing cost increase associated with the full 1D description, we develop a physics-informed neural network to replace the 1D sub-model for coupling with the 3D model, while maintaining the full description of fuel cell internal states. Large synthetic datasets are generated using the 1D model for training the neural network, ensuring the accuracy of the model. The proposed 3D+1D model is validated against experimentally obtained polarization curves and high frequency resistances under different relative humidities. The proposed model is then used to study the nonlinear distribution of the internal states along the thickness direction of the membrane electrode assembly as well as in the 3D flow field. The model is also highly effective in elucidating the dominant voltage loss factor under different operating conditions. Our developed model offers high accuracy at low computing cost under a wide range of operating conditions, potentially aiding the rapid optimization of both the membrane electrode assembly and the gas channel geometry and advance the water and thermal management of existing fuel cell designs. Figure 1