Heat Flux Boundary Research Articles

Staircase formation by resonant and non-resonant transport of potential vorticity

The E × B staircase is a quasi-periodic pattern of pressure profile corrugations. In this work, we present a new mechanism for E × B staircase formation that involves resonant transport versus non-resonant transport. We start from a potential vorticity evolution system and use quasi-linear theory, a model dispersion relation, and a bi-Lorentzian spectrum approximation, to construct the relation between the fluxes and the profiles. With these fluxes, we close the profile evolution equations and the extended turbulence intensity evolution equation, which together constitute a turbulence-profile evolution system. In this system, the Doppler effect from the E × B mean flow can cause resonance between trapped ion precession motion and the trapped ion mode, which drives a resonant transport contribution to the fluxes. The profiles will be flattened where the resonant transport is switched on. In contrast, for the regions of non-resonant transport, profiles are steeper. A quasi-periodic pattern of profile corrugation (the E × B staircase) spontaneously emerges in this system, which is the two states mentioned above, arranged as alternating layers in space. The feedback processes during the staircase pattern formation are identified. An estimate of the critical value of the boundary heat flux is obtained, above which the staircase formation will be triggered. An estimate scaling of the step size in the staircase pattern is obtained. The resonant turbulent transport is also a mechanism for collisionless saturation of zonal flow. This work is related to internal transport barrier formation and suggests some new scenarios, such as an enhanced confined L mode.

Narrow, Fast, and “Cool” Mantle Plumes Caused by Strain‐Weakening Rheology in Earth's Lower Mantle

AbstractThe rheological properties of Earth's lower mantle are key for mantle dynamics and planetary evolution. The main rock‐forming minerals in the lower mantle are bridgmanite (Br) and smaller amounts of ferropericlase (Fp). Previous work has suggested that the large differences in viscosity between these minerals greatly affect the bulk rock rheology. The resulting effective rheology becomes highly strain‐dependent as weaker Fp minerals become elongated and eventually interconnected. This implies that strain localization may occur in Earth's lower mantle. So far, there have been no studies on global‐scale mantle convection in the presence of such strain‐weakening (SW) rheology. Here, we present 2D numerical models of thermo‐chemical convection in spherical annulus geometry including a new strain‐dependent rheology formulation for lower mantle materials, combining rheological weakening and healing terms. We find that SW rheology has several direct and indirect effects on mantle convection. The most notable direct effect is the changing dynamics of weakened plume channels as well as the formation of larger thermochemical piles at the base of the mantle. The weakened plume conduits act as lubrication channels in the mantle and exhibit a lower thermal anomaly. SW rheology also reduces the overall viscosity, notable in terms of increasing convective vigor and core‐mantle boundary heat flux. Finally, we put our results into context with existing hypotheses on the style of mantle convection and mixing. Most importantly, we suggest that the new kind of plume dynamics may explain the discrepancy between expected and observed thermal anomalies of deep‐seated mantle plumes on Earth.

Mixed convective EMHD flow of bioconvectionin generalized viscoelastic nanofluid through a convectively heated Riga surface

The current investigation is considered to determine the impact of activation energy and bioconvection on the modified second-grade nanofluid along the Riga pattern. And also, the effects of convective boundary and nonlinear heat flux are included. The significant importance of the present study is the impact of the electrical field generated by the Riga plate. With the help of suitable transformations, partial differential equations of the flow are reduced to a nonlinear system of differential equations. The reduced model is solved by the most powerful method: the finite element method (FEM). From this study, an important observation is that heat source is the most relative technique considering improvement in thermal act based on coolants. Every outcome is validated by comparing it with available results in the literature. The parameter terms are varied and also observations are noticed in the form of numerical solutions and graphical representations. Therefore, if a nanofluid is desired to turn down the shear thickening situation, it is suggested to increase the thermal production of nano-structures . Using regression analysis, the Brownian movement and thermophoresis have negative impacts, while the radioactive heat factor and thermal Biot number have positive impacts on the estimate of heat transfer.

A steady-state inverse heat conduction-convection conjugated problem in determining unknown spatially dependent surface heat flux

The unknown spatially dependent surface heat flux is estimated for a three-dimensional steady-state inverse heat conduction-convection conjugated problem (IHCCCP) in this study. Air with a specified velocity flows over a flat plate which was subjected to an unknown heat flux on the bottom surface and the simulated temperature measurements on the top surface were used for the estimations. As the functional form of the heat flux is considered to be initially unknown, this inverse problem falls within the category of function estimation. Optimization is performed using the conjugate gradient method (CGM) because this method does not require a priori information regarding the functional form of the unknown functions, enabling a large number of unknowns to be corrected and estimated in each iteration to always yield good estimates. This efficient algorithm has never been applied to the IHCCCP. The results of the inverse solutions are verified using numerical simulations with various inlet air velocities and plate thicknesses. The results show that using exact measurements always produces accurate boundary heat fluxes under thin plate conditions since the maximum error for the estimated heat flux using error measurements was found to be only 6.43%, and the air velocity does not affect the estimates. The measurement errors and their influence on the inverse solutions are analyzed. Finally, it is concluded that because the inverse problem is ill-posed, the estimated heat flux becomes less accurate as the plate thickness increases.

The influence of heterogeneous seafloor heat flux on the cooling patterns of Ganymede’s and Titan’s subsurface oceans

Several icy moons of Jupiter and Saturn are known to possess deep water oceans. Heating in the rocky mantle underneath often produces heterogeneous heat flux patterns at the ocean’s seafloor. How this internal ocean dynamically relates the seafloor to the surface ice shell is a crucial question to understand the long term evolution of icy moons. Here we investigate how a heterogeneous seafloor heat flux pattern affects the convection and heat transfer in the subsurface ocean of large icy worlds involving a high pressure ice layer beneath the seafloor such as Titan or Ganymede. We perform rotating convection simulations in a thin 3D spherical shell with a prescribed heterogeneous bottom heat flux inferred from 3D convection simulations of the underlying mantle (Choblet et al., 2017b). In our simulations, although the amplitude of imposed inner boundary heat flux heterogeneity is rather moderate, preferred longitudes of intense outer boundary heat flux are highly correlated with longitudes of intense inner boundary heat flux. In addition, a small imposed inner boundary large-scale order 2 pattern is amplified at the outer boundary heat flux by the convection in the thin shell. Lastly, deviations from axisymmetry and equatorial symmetry in the outer boundary heat flux increase with the main convection vigor and the amplitude of the inner boundary heterogeneity. In our models polar vs. equatorial cooling is mostly controlled by inertial effects, as was found by Amit et al. (2020) for homogeneous boundary conditions, with the latitudinally equilibrated inner boundary heterogeneity acting to reduce the amplitude of this effect. Our results support polar cooling for Titan’s sub-surface ocean.

Experimental arrangement for estimation of metal-mold boundary heat flux during gravity chill casting

The cooling rate of the castings is determined by the rate of heat transfer into the mould during solidification, which has a considerable impact on their characteristics. The thermo-physical properties of the casting and mould materials determine the heat flow rate/cooling rate. Heat flow on the inner and external surfaces of the mould can be calculated using the Inverse Heat Conduction Problem, often known as the Inverse Approach (IHCP). On one side of the rectangular mould chamber, cold materials such as cast iron, stainless steel, and copper are used. For the experimentation, a simple gravity die casting in a rectangle mould is explored. At various points throughout the mould, mineral insulated K-type thermocouples are put. The thermocouples are wired into a data collecting system that connects to a computer. During pouring and subsequent solidification, the mold's time–temperature history is recorded. These recorded transient temperatures are supplied into the inverse method as input (IHCP). The surface heat flux is calculated as a function of time by the solver.

Inverse estimation of heat flux under forced convection conjugate heat transfer in a vertical channel fully filled with metal foam

In this work, for the first time, a heat flux at the boundary is estimated for a conjugate heat transfer under forced convection in the presence of high porosity metal foams. For the forward problem a vertical channel experimental set up reported in the literature is considered. The metal foam placed in the vertical channel is subjected to constant heat flux through aluminum plate and airflow of various velocities is passed through vertical channel for removal of heat from the high porosity metal foam placed in the vertical channel. Six different velocities are considered and the required temperature distribution of the aluminum plate is obtained by solving Darcy extended Forchheimer and Local Thermal non-equilibrium models for metal foams. The forward problem, created using computational fluid dynamics in ANSYS-FLUENT, is substituted with Neural Network for faster computation of the forward problem. The maximum errors between the computational fluid dynamics and Artificial Neural Network models for the heat flux values of 466.66, 666.66 and 1133.3 W/m 2 are found to be 0.086, 0.043, 0.092 respectively. The heat flux to the forward problem is treated as unknown and the same is estimated using an inverse method that couples Particle Swarm Optimization with Bayesian framework. The result of inverse estimation of exact temperature data shows that for a heat flux of 1266.64 W/m 2 the error is found to be 1.6e−4%. Similarly, for the noise added temperature data, the absolute % error in heat flux of 599.985, 733.315 and 1266.635 W/m 2 is 4.80e−2%, 2.20e−2%, 2.30e−2% respectively. • Inverse estimation of boundary heat flux using particle swarm optimization method. • Forward problem is a conjugate heat transfer problem in a vertical channel. • Darcy–Forchheimer, Local Thermal Non-equilibrium models for modelling porous medium. • Different Artificial Neural Network models to facilitate faster computations. • Particle Swarm Optimization combined Bayesian framework to quantify modeling error.

Effects of the Conjugate Heat Transfer and Heat Flux Strength on the Thermal Characteristics of Impinging Jets

A numerical study using the conjugate heat transfer approach has been performed to investigate the effects of boundary heat flux, conduction effect, and working fluid on the thermal characteristics due to the jet impingement process. Air and water are used in this study as working fluids. For the water jet, the volume of fluid method is used to capture and track the interface in the multiphase flow. It is found that the wall conduction may change the fluid-solid interfacial thermal characteristics compared with no conduction or pure convection process. The amount of influence depends on the working fluid, nozzle size, metal thermal conductivity, metal thickness, and boundary heat flux. The conduction inside the solid wall tends to reorganize the uniform heat flux distribution at the boundary to a non-uniform heat flux distribution at the fluid-solid interface. This is mainly attributed to the conjugate effect of the solid. For a given jet Reynolds number and boundary heat flux, the conjugate heat transfer results divulge that the convective heat flux removed from the stagnation point is higher for the air jet than for the water jet. Contrary to the air jet, the effect of thermal boundary on the stagnation Nusselt number profile is negligible for the water jet. The disc material and thickness have no obvious effect on the stagnation Nusselt number profile for both air and water fluids.

Enhanced Conjugate Gradient Method for Heat Flux Estimation in Laminar and T-Junction Turbulent Mixing Flow

In this paper, the Jaya algorithm is coupled with the traditional conjugate gradient method (CGM) for solving the inverse heat transfer problem. The coupling of two algorithms is termed as enhanced conjugate gradient method (ECGM). Two different applications have been considered to determine the accuracy and stability of ECGM. First, the algorithm is applied in a steady-state two-dimensional laminar flow through a duct for the boundary heat flux estimation. The accuracy of estimation has been compared with traditional CGM by calculating the root mean square (RMS) error. The ECGM algorithm (RMS =0.142 kW/m2) is found to be more accurate than conventional CGM (RMS = 0.198 kW/m2). ECGM is then applied to determine the peripheral variation of heat flux at the fluid-solid interface of a tube at a section downstream to the T-junction. The experimental temperature readings at the outer periphery of the tube required in the inverse algorithm are simulated by solving the direct heat conduction problem. Here, the heat flux at the fluid-solid interface is obtained by solving the governing equations of the turbulent flow by using commercial computational fluid dynamics software. ECGM is found to be a stable and accurate inverse heat transfer algorithm.

Dependence of the boundary heat flux width on core and edge profiles in Alcator C-Mod

This work presents new evidence that the heat flux width, λ q , in the Alcator C-Mod tokamak scales with the edge electron pressure, as observed in the ASDEX Upgrade (AUG) tokamak (Silvagni et al 2020 Plasma Phys. Control. Fusion 62 045015), but the scaling with volume-averaged pressure, , from the plasma stored energy, found by Brunner et al (2018 Nucl. Fusion 58 094002), is a better predictor of λ q in Alcator C-Mod than the edge electron pressure. These previous studies, which find that λ q decreases with increasing plasma pressure, imply that a high performance core at high pressure will lead to challenging heat and particle exhaust due to very small λ q . This concern has led to our significant enlargement of the C-Mod database with the electron density, temperature, and pressure profile data from the Thomson scattering and electron cyclotron emission diagnostics. Using the C-Mod database augmented with new profile data, we find that λ q decreases with increasing edge electron pressure as , similar to results from AUG, and showing the strength of cross-machine comparisons. We also find that , consistent with the original finding from C-Mod that the heat flux width scales as (Brunner et al 2018 Nucl. Fusion 58 094002). The scalings of λ q with separatrix pressure and gradient scale length are found to match the AUG results qualitatively. The C-Mod scalings with edge plasma quantities have more scatter than the scaling, and, importantly, show different trends for H-modes relative to L- and I-mode. Investigating the source of this discrepancy presents an opportunity for further study that may improve our ability to predict the heat flux width in different confinement scenarios in the pursuit of optimizing core-edge performance in future reactors.

Heat transfer behaviour of funnel mould copper plates with high casting speed during different water supply processes

ABSTRACT In the process of high casting speed, the study of funnel mould (FM) cooling process is particularly important. Based on the copper plate temperature measured by thermocouple, a mathematical model of the inverse heat transfer problem of the thin slab/copper plate was developed to calculate the boundary heat flux, then a positive mathematical model of thin slab/copper plate/cooling water coupled heat transfer has been developed, and the influence of the cooling water flows from top to bottom in the water channels on the heat transfer behaviour of the FM copper plate was analysed. The results show that when the cooling water flows from top to bottom in the water channels was adopted, the peak temperature of the copper plate hot face decreased 117°C, the maximum temperature of the cooling water decreased 24°C. Cooling water velocity and temperature have little influence on heat transfer of copper plate.

Theoretical Research on Flow and Heat Transfer Characteristics of Hydrostatic Oil Film in Flat Microfluidic Boundary Layer

The hydrostatic bearing is the core component of ultra-precision computer numerical control (CNC) machine tools. Because the temperature rise in the oil film of hydrostatic bearings seriously affects the working accuracy of the bearings, it is important to study the flow and heat transfer characteristics of the oil film. Based on the physical model of an incompressible viscous fluid flowing in a flat microfluidic boundary layer, velocity, temperature and heat flux distribution equations of oil film are constructed by theories of heat transfer and hydrodynamics. Then, the effects of several parameters on velocity distribution, temperature distribution and heat flux distribution are analyzed, such as the upper plate velocity, the channel length, and so on. The results show that the dimensionless velocity of the oil film decreases with the increase in the upper plate velocity and the channel length. The oil film temperature distribution can be divided into three zones: the increasing zone, stabilizing zone and decreasing zone. The heat flux decreases linearly with the increase in the plate thickness, and increases linearly with the increase in the temperature difference.

Real-time reconstruction of thermal boundary condition of porous media via temperature sequence

Porous media are widely applied in numerous technological field due to its excellent thermal, mechanical and exchange properties. Because of involving multi-scale and multi-effect coupling, only a few studies are dedicated to heat flux estimation of porous media. In this study, on the basis of the finite volume method (FVM) and discrete ordinate method (DOM), a novel method is proposed to introduce the geometric analytical formulas of thermal properties into the multi-scale heat transfer problem of porous materials. Then, the temperature sequence in a short time are used as measurement information for inverse analysis. The time-varying heat flux of NiCrAl porous materials, copper and aluminum are reconstructed based on Kalman filter coupled with the recursive least square estimator (KF-RLSE). In addition, the sensitivity of the photothermal properties to temperature response information is analyzed. The effects of porosity, scattering albedo and specularity parameter on the accuracy and stability of the reconstructed heat flux are investigated. The numerical results with the noise level 0–0.3 show that the predicted boundary heat flux is less than 0.2 for NRMSE and more than 0.99 for the correlation coefficient, proving the validity of the proposed method.

Conserved method for specified heat flux boundary in UGKS simulation of microscale gas flow

The unified gas kinetic scheme (UGKS) is a powerful method to simulate microscale gas flows. However, due to the lack of corresponding specified heat flux boundary (SHFB) treatment, there is great difficulty for UGKS to simulate problem with SHFB. In this paper, a SHFB treatment for UGKS is proposed based on gas kinetic theory. In the proposed boundary treatment, the heat flux term is directly added into the gas kinetic conservation equation of heat flux on boundary. In this conservation equation, the heat flux transferring from gas to the boundary is calculated using an extrapolated distribution function of inside gas, and the heat flux transferring form boundary to the gas is calculated using an equilibrium rebounding distribution function. Using the conservation equation of mass on boundary to close conservation equations, the gas density and temperature terms for rebounding gas distribution function can be calculated simultaneously. Distribution function on SHFB is reconstructed using rebounding gas distribution function and extrapolated distribution function of inside gas. Fluxes of gas distribution function and macro variables on SHFB is calculated with the reconstructed distribution function. Typical benchmark cases including steady and unsteady Fourier flow, steady Couette flow, pressure-driven flow, cavity flow and supersonic flow around square cylinder are tested to validate the proposed boundary treatment. The validations prove the accuracy, reliability and universality of the proposed method.

Numerical and experimental verification of the single neural adaptive PID real-time inverse method for solving inverse heat conduction problems

For the problems of difficult on-line tuning of PID parameters, weak anti-interference ability and poor self-adaptive ability in the classical proportional-integral-derivative (PID) inverse method, combining single artificial neuron with PID inverse algorithm, a single neural adaptive PID (SNA-PID) inverse method is established to estimate the thermal boundary condition. Compared with the classical PID inverse method, the main difference is that SNA-PID can adaptively adjust the weights of PID parameters. In the SNA-PID inverse method, firstly, the deviation signal between the output of the direct heat conduction model and the measured temperature and its first-order difference and second-order difference are taken as three inputs of a single neuron (dendritic input signal of neuron), and the connection strength of each input signal is determined according to the updated weight of the adaptive learning algorithm, The excitation value of the boundary heat flux increment of the heat conduction system is generated according to the linear truncation function. Finally, the real-time estimation of boundary heat flux of heat conduction system is realized. The unsteady boundary heat flux of one-dimensional heat conduction object is estimated through numerical and experimental tests, which verifies the effectiveness of the proposed SNA-PID inverse method.

Interfacial thermal resistance in phonon hydrodynamic heat conduction

Two-dimensional materials are usually predicted to have ultrahigh thermal conductivity because of the numerous phonon normal scatterings, which might cause hydrodynamic heat conduction. In addition, boundary and interface are significant in the polycrystalline structure and material contacts. Therefore, this article investigates the thermal behaviors at the boundary and interface in phonon hydrodynamics. Monte Carlo simulation is adopted to study the heat conduction phenomena in Poiseuille hydrodynamics and Ziman hydrodynamics. The concept of a boundary temperature step is defined to depict the temperature decline behaviors at the boundary in steady hydrodynamic heat conduction. Interfacial thermal behaviors can be treated as a combination of the boundary effects and phonon transmission effects, where the interface properties can be described by the interface transmissivity and the specular reflectivity. Moreover, the inverse temperature difference at the interface is observed, which means that the heat is transported from low temperature to high temperature, implying that the definition of temperature in phonon hydrodynamic heat conduction ought to be further investigated. Then, two theoretical models are proposed to describe these phenomena, namely, the particle propagation model and the dual boundary flux model. The particle propagation model tries to trace the propagation and evolution of phonons with simpler rules, and it finds that the heat flux reduction originates from the backward phonons that are scattered by the normal scattering process. The dual boundary flux model divides the whole boundary heat flux into the hydrodynamic heat flux and the diffusive heat flux, and the boundary temperature step appears in the transition between these two fluxes. These two models are compared with the results obtained by Monte Carlo simulations.

Research on the uniqueness and algorithm of multi-parameter identification in coupling thermal acoustic and solid problem

<p indent="0mm">In the coupled thermal-acoustic solid field, we consider the identification problem of initial boundary value from the boundary measurements of temperature and ultrasonic propagation time to reconstruct the boundary heat flux and thickness of a solid structure simultaneously. Through rigorous theoretical analysis, the uniqueness of the simultaneous identification is demonstrated, which provides theoretical support for engineering applications. By reformulating an inverse problem as an optimization problem and deriving the adjoint equation, we propose an efficient alternating iteration algorithm for solving the inverse problem. Rigorous convergence analysis of the proposed algorithm is performed. Finally, the reliability and feasibility of the algorithm are verified via numerical examples, and the algorithm’s accuracy is verified by comparing it with a single-parameter identification algorithm that only identifies heat flux.

Thermocouple response time estimation and temperature signal correction for an accurate heat flux calculation in inverse heat conduction problems

Many heat transfer processes do not allow temperature measurements with advanced instruments like IR cameras, and thus require the use of thermocouples – often inserted sheathed thermocouples – in order to measure the temperature in some key-regions. Nevertheless, thermocouples have an intrinsic response time that can dampen substantially temperature measurements and, when applicable, affect heat flux estimations by inverse methods. In this study, a thermocouple measurement correction method is proposed, especially for fast thermal transients like quenching, to avoid underestimation of the boundary heat fluxes due to delayed temperature responses. A simplified energy balance at the hot junction allowed modeling the temperature measurement by a thermocouple, in which heat losses and the response time were taken into account so they can be estimated using the least squares method on experimental data. A series of tests was performed to validate the present method with a heated jet impinging on a copper body instrumented with two thermocouples: one ungrounded and sheathed, then with high response time (”slow”); and another with exposed welded wires, hence with low response time (”fast”). After having estimated the slow thermocouple response time and heat loss parameters, the model allowed a good reconstruction of the fast thermocouple signal using the slow thermocouple measurements. Also, the heat flux estimation by the inverse method using the reconstructed signal resulted in practically the same results obtained using the fast thermocouple data. A parametric sensitivity analysis showed the heat loss parameters can be neglected for heat flux estimations, while simulations showed that temperature noises degrade the response time estimation but data filtering can mitigate this noise effect. Finally, the present method was applied to a large scale jet-cooling of a hot plate – near industrial conditions – and showed a substantially higher dissipated heat flux than the initially estimated. In fact, the results using reconstructed signals demonstrated that the heat flux dissipated by the jet has a higher dependence on the jet Reynolds number than the observed with the original thermocouple measurements, showing how important the temperature measurement correction is to evaluate fast thermal processes.

Stability analysis of functionals in variational data assimilation with respect to uncertainties of input data for a sea thermodynamics model

AbstractThe problem of stability and sensitivity of functionals of the optimal solution of the variational data assimilation of sea surface temperature for the model of sea thermodynamics is considered. The variational data assimilation problem is formulated as an optimal control problem to find the initial state and the boundary heat flux. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is studied. Computing the gradient of the response function reduces to the solution of a non-standard problem being a coupled system of direct and adjoint equations with mutually dependent initial and boundary values. The algorithm to compute the gradient of the response function is presented, based on the Hessian of the original cost functional. Stability analysis of the response function with respect to uncertainties of input data is given. Numerical examples are presented for the Black and Azov seas thermodynamics model.

Legendre spectral collocation technique for fractional inverse heat conduction problem

For fractional inverse heat conduction problem (FIHCP), this paper introduces a numerical study. For the proposed FIHCP, in addition to the unknown function of temperature, the boundary heat fluxes are also unknown. Related to the two independent variables, the proposed scheme uses a fully spectral collocation treatment. Our technique is determined to be more accurate, efficient and practicable. The obtained results confirmed the exponential convergence of the spectral scheme.