Parallel Heat Conduction Research Articles

Thermographic validation of a novel, laminate body, analytical heat conduction model

The two-region fin model captures the heat spreading behaviour in multilayered composite bodies (i.e., laminates), heated only over a small part of their domains (finite heat source), where there is an inner layer that has a substantial capacity for heat conduction parallel to the heat exchange surface (convection cooling). This resulting heat conduction behaviour improves the overall heat transfer process when compared to heat conduction in homogeneous bodies. Long-term heat storage using supercooling salt hydrate phase change materials, stovetop cookware, and electronics cooling applications could all benefit from this kind of heat-spreading in laminates. Experiments using laminate films reclaimed from post-consumer Tetra Brik cartons were conducted with thin rectangular and circular heaters to confirm the laminate body, steady-state, heat conduction behaviour predicted by the two-region fin model. Medium to high accuracy experimental validation of the two-region fin model was achieved in Cartesian and cylindrical coordinates for forced external convection and natural convection, the latter for Cartesian only. These were conducted using constant heat flux finite heat source temperature profiles that were measured by infrared thermography. This validation is also deemed valid for constant temperature heat sources.

An Analytical Unit Cell Model for the Effective Thermal Conductivity of High Porosity Open-Cell Metal Foams

We present an analytical model for the effective thermal conductivity of fluid-saturated metal foams with open cells. For high porosity ranges $$(\varepsilon \ge 0.9)$$ , the model is derived based on a realistic representative unit cell (a tetrakaidecahedron with cuboid node) under the assumption of parallel heat conduction along the highly tortuous cell ligaments and the saturating fluid. Good agreement with existing experimental data as well as the present measurements of open-cell aluminum foams saturated with either air or water validates the present model. More realistic and reasonable node size is estimated and compared with other model predictions. Ligament shape and pore size (PPI) are found to have little influence upon the effective thermal conductivity of the bulk porous media. Further, the influence of the fluid phase as well as the interactive heat conduction between solid and fluid phase on the overall effective thermal conductivity is quantified.

Analyzing guard-heating to enable accurate hot-film wall shear stress measurements for turbulent flows

In this paper we examine guard-heated hot-film sensors for the measurement of wall shear stress fluctuations in turbulent flows, using analytical and numerical techniques. The new thermal sensor designs proposed here, seek to eliminate the most severe source of error in conventional single-element hot-film sensors – which is indirect heat transfer to the fluid flow through the solid substrate. Published studies of the single-element hot-film show that a significant portion of the heat generated in the hot-film travels through the substrate before reaching the fluid, which causes spectral and phase errors in the wall shear stress signal and can drastically reduce the spatial resolution of the sensor. In addition, heat conduction parallel to the surface, within the fluid near the sensor edges, can produce significant errors when sensors are made smaller to improve spatial resolution. Here we examine the effectiveness of forcing zero temperature gradient at the edges of the hot-film in eliminating these errors. Air and water flow over such guard-heated sensors are studied numerically to investigate performance and signal strength of the guard-heated sensors. Our results show, particularly for low-conductivity fluids such as air, that edge guard-heating needs to be supplemented by a sub-surface guard-heater. With this two-plane guard-heating, a strong non-linearity in the standard single-element designs can be corrected, and substrate conduction errors contributing to spectral distortion and phase errors can be eliminated.

Static current profile control and RFP confinement

Static current profile control (CPC) is shown numerically to substantially enhance plasma confinement in the reversed-field pinch (RFP). By suitable application of an auxiliary electric field and adjustment of its internal location, width and amplitude, strongly decreased levels of dynamo fluctuations are obtained. The simulations are performed using a fully non-linear, resistive magnetohydrodynamic model, including the effects of ohmic heating as well as parallel and perpendicular heat conduction along stochastic field lines. The importance of controlling the parallel current profile in the core plasma to minimize the effects of tearing modes on confinement is thus confirmed. A near three-fold increase in energy confinement is found and poloidal plasma beta increases by 30% from 0.20 to 0.27. The edge heat flux is reduced to a third of that of the conventional RFP. The high-confinement phase is interrupted here by a crash, characterized by a rapid decrease in confinement. A detailed study of the crash phase is carried out by the standard Δ′ theory and a fully resistive linearized time-spectral method; the generalized weighted residual method. The analysis suggests that the instability is caused by pressure-driven, resistive g-modes. Inclusion of anisotropic thermal conduction reduces the linear growth rates. As compared with our earlier numerical studies of CPC in the RFP, employing feedback control, the present static control scheme should be more easily implemented experimentally.

Effect of Phosphor Encapsulant on the Thermal Resistance of a High-Power COB LED Module

With their many advantages, such as small size, energy efficiency, and long lifetime, light emitting diodes (LEDs) are conquering the lighting world. Blue LEDs, because of their high efficiency, are commonly used and a phosphor is used to convert blue light into white light. The remote phosphor concept has gained attention since it promises to deliver better efficacy than solutions in which the phosphor is applied directly on the LEDs. In this paper, the effect of phosphor packaging on the thermal performance of a high-power chip-on-board LED module is studied. Both simulations and measurements show that, despite the added thermal load caused by white light conversion losses in the phosphor, the average temperature of the phosphor-coated LEDs matches with that of noncoated LEDs. The phosphor encapsulant generates a parallel heat conduction path which reduces the thermal resistance from the LED chips to ambient and compensates the thermal power increase.

The effect of anisotropic heat transport on magnetic islands in 3-D configurations

An analytic theory of nonlinear pressure-induced magnetic island formation using a boundary layer analysis is presented. This theory extends previous work by including the effects of finite parallel heat transport and is applicable to general three dimensional magnetic configurations. In this work, particular attention is paid to the role of finite parallel heat conduction in the context of pressure-induced island physics. It is found that localized currents that require self-consistent deformation of the pressure profile, such as resistive interchange and bootstrap currents, are attenuated by finite parallel heat conduction when the magnetic islands are sufficiently small. However, these anisotropic effects do not change saturated island widths caused by Pfirsch-Schlüter current effects. Implications for finite pressure-induced island healing are discussed.

Magnetohydrodynamic study of shock waves in the current sheet of a solar coronal magnetic loop

A Lagrangian Remap (LareXd) Code is employed to investigate the shock wave formation in the current sheet of a solar coronal magnetic loop and its effect on the magnetic reconnection. We constructed the slow shock structure in the presence of viscosity and heat conduction parallel and perpendicular to the magnetic field and pairs of slow shocks propagate away from the central current sheet, the so-called diffusion region. Significant jumps in plasma density, pressure, velocity and magnetic field occur across the main shock while the temperature appears in the foreshock. In the presence of dissipative effects, the distinct jumps disappear and the shock profiles show smooth transition between the downstream and the upstream regions while the plasma density and the pressure show very narrow and a sharp decrease with time. These results can be applied to the heating of the solar corona, the structure of the magnetic reconnection and the solar wind.

Effect of neutrals on the power decay length at the divertor target

EDGE2D modelling of JET divertor plasmas with wide variation of main plasma parameters was aimed at identification of operational domains of the applicability of the 2/7×λTe scaling for predicting the power deposition profile width at the outer divertor target plate in high recycling conditions. Contrary to expectations, under no conditions, including lowest density plasmas just entering the conduction regime, was this scaling capable of correctly predicting the width and peak value of target power deposition profiles. Poorer ion parallel heat conduction, compared to the electron channel, in combination with reduced energy convection and volumetric power losses in divertor, of which charge exchange was found to play the dominant role, resulted in large deviations from the predictions of the 2/7×λTe scaling.

The relation between upstream density and temperature widths in the scrape-off layer and the power width in an attached divertor

The target power width is one of the most critical practical quantities in the development of magnetic fusion energy. It is essential to know how to scale this quantity to future devices. At present the controlling physics is not adequately understood, making reliable prediction difficult. It seems likely that two important processes effecting are (a) cross-field transport, e.g. D⊥, and (b) volumetric power loss processes in the edge plasma, with the latter tending to occur mainly in the divertor for attached divertor conditions. It is hypothesized that a simple relation exists between the ‘upstream’ radial profiles of ne and Te in the main scrape-off layer, , , and the parallel power flux density at the divertor entrance, . Such a simple relation is found here in 2D SOLPS edge code simulations of attached divertor conditions, which contain a wide range of more or less complex edge physics effects. It is found that , as can be expected on the basis of flux-limited parallel heat conduction, rather than Spitzer–Harm conduction for which is expected. For the relatively open divertor configuration considered, and for attached divertor conditions, it is found that the flux-limited relationship also holds for the SOLPS power flux density deposited on the target , even including the radiation load; this despite the fact that up to half the power into the SOL is dissipated radiatively. Comparing with experimentally measured target power widths for H-mode discharges, better agreement is found assuming flux limited rather than Spitzer–Harm transport although definitive conclusions will require analysis of specific discharges in specific tokamaks. This study is a necessary preliminary work to an equivalent treatment of the case where volumetric losses in the divertor are stronger, including the detached, strongly radiating divertor case with momentum loss.

Electron thermal transport within magnetic islands in the reversed-field pinch

Tearing mode induced magnetic islands have a significant impact on the thermal characteristics of magnetically confined plasmas such as those in the reversed-field pinch (RFP). New Thomson scattering diagnostic capability on the Madison Symmetric Torus (MST) RFP has enabled measurement of the thermal transport characteristics of islands. Electron temperature (Te) profiles can now be acquired at 25 kHz, sufficient to measure the effect of an island on the profile as the island rotates by the measurement point. In standard MST plasmas with a spectrum of unstable tearing modes, remnant islands are present in the core between sawtoothlike reconnection events. Associated with these island remnants is flattening of the Te profile inside the island separatricies. This flattening is characteristic of rapid parallel heat conduction along helical magnetic field lines. In striking contrast, a temperature gradient within an m=1, n=5 island is observed in these same plasmas just after a sawtooth event when the m=1, n=5 mode may briefly come into resonance near the magnetic axis. This suggests local heating and relatively good confinement within the island. Local power balance calculations suggest reduced thermal transport within this island relative to the confinement properties of standard MST discharges between reconnection events. The magnetic field and island structure is modeled with three-dimensional nonlinear resistive magnetohydrodynamic simulations (DEBS code) with Lundquist numbers matching those in MST during standard discharges. During improved confinement plasmas with reduced tearing mode activity, temperature fluctuations correlated with magnetic signals are small with characteristic fluctuation amplitudes of order T̃e/Te∼2%.

Effective heat conduction in a configuration with nonoverlapped magnetic islands

The effective radial heat conduction κeff in a plasma configuration with nonoverlapped magnetic island chains is assessed by applying an “optimal path” method. This approach implies that heat is transported predominantly along paths rendering the minimum temperature variation and is related to the principle of minimum entropy production. Paths combined of up to three radial sections and two segments aligned along magnetic field lines are considered. It is demonstrated that the enhancement of κeff over the level of perpendicular heat conduction κ⊥ arising due to flows along magnetic field lines is controlled only by the Chirikov parameter and by the value 4br2κ∥∕κ⊥, where br is the relative amplitude of the radial field resonant harmonic and κ∥ is the parallel heat conduction.

3D Edge Transport Studies with EMC3‐EIRENE for the Dynamic Ergodic Divertor (DED) at TEXTOR

AbstractIn this paper we introduce some model extensions to the energy balances of the 3D fluid Monte‐Carlo code EMC3 and study their influence on the simulation results for TEXTOR‐DED. We implemented local kinetic corrections to the classical parallel heat conductivity of electrons (heat flux limit). Depending on the DED configuration, a cooling effect in the outermost SOL region, resulting from the heat flux limit, can be observed. In addition, also the adiabatic cooling term $1 \over 2$mnV2$\vec V$, which was neglected before, was implemented into the energy iterations. This term causes a localized cooling of the ions in front of the wall, where the plasma is accelerated towards the first wall. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Elucidation of the Heat-Flux Limit from Magnetic-Island Heating

Recent experiments on heating of magnetic islands in a tokamak are analyzed to assess plasma transport characteristics. By comparing with the experimental data, both perpendicular and parallel components of the electron heat conductivity in the island are determined. As a consequence, the so-called heat-flux limit factor xi, the ratio of the parallel heat conduction flux in a collisionless plasma to that transferred by free-streaming electrons, can be estimated. The found factor xi does not contradict that established earlier by interpreting laser plasma experiments.

A model for particle and heat losses by type I edge localized modes

A model to estimate the particle and energy losses caused in tokamaks by type I edge localized modes (ELMs) is proposed. This model is based on the assumption that the increase in transport by ELM is due to flows along magnetic field lines perturbed by ballooning–peeling MHD modes. The model reproduces well the experimentally found variation of losses with the plasma collisionality ν*, namely, the weak dependence of the particle loss and significant reduction of the energy loss with increasing ν*. It is argued that the electron parallel heat conductivity is dominating in the energy loss at not very large ν*.

Extending the collisional fluid equations into the long mean-free-path regime in toroidal plasmas. III. Parallel heat conduction

It is illustrated that plasma transport processes in the direction of the magnetic field are local in the vicinity of the magnetic island in the long mean-free-path regime where the collisionality parameter ν* is larger than 10−2, and the width of the island is about 3% of the minor radius or smaller. This is because the plasma temperature variation on the magnetic surface that results from the magnetic reconnection is gentle. Both the electron and the ion parallel transport fluxes including parallel heat flow in the banana regime where ν*<1 are calculated using a model Coulomb collision operator that conserves momentum.

Investigations of the electron temperature profiles at the WEGA stellarator

This paper presents a study of the hollow bulk electron temperature profiles measured in the low-temperature (2–15 eV) plasmas of the WEGA stellarator. For this the global power balance equation was solved for the bulk electrons. The observed hollow temperature profiles could be reproduced satisfactorily assuming that the bulk electrons are heated through energy transfer from fast electrons and that the parallel heat conduction in the scrape-off layer is determined by the potential drop in the sheath in the vicinity of the wall. These results imply that the cause of the hollow shape of bulk electron temperature profiles is the heating of them through collisions with fast electrons produced by the heating method.

Evidence for electromagnetic fluid drift turbulence controlling the edge plasma state in the Alcator C-Mod tokamak

Plasma profiles across the separatrix and scrape-off layer (SOL) in Alcator C-Mod are examined for a range of plasma densities, currents and magnetic fields in Ohmic L-mode discharges and for a subset of conditions in Ohmic H-mode discharges. In all plasmas, electron pressure gradient scale lengths () exhibit a minimum value just outside the separatrix (i.e. in the near SOL), forming the base of a weak (strong) pedestal in L-mode (H-mode) plasmas. Over a wide range of conditions in Ohmic L-mode discharges, at this location are found to track with a monotonic function of electron collision frequency, when this quantity is normalized according to the framework of electromagnetic fluid drift turbulence (EMFDT) theory. Moreover, at fixed values of normalized collisionality parameter (characterized as the ‘diamagnetic parameter’, αd), electron pressure gradients in the near SOL increase with plasma current squared, holding the MHD ballooning parameter, αMHD, unchanged. Thus, the state of the near SOL is restricted to a narrow region within this two-parameter phase-space. An implication is that cross-field heat and particle transport are strong functions of these parameters. Indeed, as αd is decreased below ∼0.3, cross-field heat convection increases sharply and competes with parallel heat conduction along open field lines, making high plasma density regions of αMHD–αd space energetically inaccessible. These observations are consistent with the idea that the operational space of the edge plasma, including boundaries associated with the tokamak density limit, is controlled by EMFDT.

Structure of intermediate shocks and slow shocks in a magnetized plasma with heat conduction

The structure of slow shocks and intermediate shocks in the presence of a heat conduction parallel to the local magnetic field is simulated from the set of magnetohydrodynamic equations. This study is an extension of an earlier work [C. L. Tsai, R. H. Tsai, B. H. Wu, and L. C. Lee, Phys. Plasmas 9, 1185 (2002)], in which the effects of heat conduction are examined for the case that the tangential magnetic fields on the two side of initial current sheet are exactly antiparallel (By=0). For the By=0 case, a pair of slow shocks is formed as the result of evolution of the initial current sheet, and each slow shock consists of two parts: the isothermal main shock and the foreshock. In the present paper, cases with By≠0 are also considered, in which the evolution process leads to the presence of an additional pair of time-dependent intermediate shocks (TDISs). Across the main shock of the slow shock, jumps in plasma density, velocity, and magnetic field are significant, but the temperature is continuous. The plasma density downstream of the main shock decreases with time, while the downstream temperature increases with time, keeping the downstream pressure constant. The foreshock is featured by a smooth temperature variation and is formed due to the heat flow from downstream to upstream region. In contrast to the earlier study, the foreshock is found to reach a steady state with a constant width in the slow shock frame. In cases with By≠0, the plasma density and pressure increase and the magnetic field decreases across TDIS. The TDIS initially can be embedded in the slow shock’s foreshock structure, and then moves out of the foreshock region. With an increasing By, the propagation speed of foreshock leading edge tends to decrease and the foreshock reaches its steady state at an earlier time. Both the pressure and temperature downstreams of the main shock decrease with increasing By. The results can be applied to the shock heating in the solar corona and solar wind.

Modelling of heat transport in magnetised plasmas using non-aligned coordinates

We present discretisation schemes for the heat diffusion equation in strongly magnetised plasmas in 2-d Cartesian and 3-d cylindrical geometry. The algorithms described do not use field-aligned coordinates, but show, nevertheless, a strong reduction of the pollution of perpendicular heat flux caused by parallel heat conduction, compared to standard finite element or other finite difference formulations. Results for test examples show independence of the numerical error from χ∥ for values of χ∥/χ⊥ up to 109 in plane and 1012 in periodic cylinder geometry (simulating toroidal connection). The algorithms appear particularly well suited for dynamic MHD calculations, where the effort of using exactly aligned coordinates becomes prohibitive.

Nonlocal closures for plasma fluid simulations

The application of fluid models in studies of transport and macroscopic stability of magnetized, nearly collisionless plasmas requires closure relations that are inherently nonlocal. Such closures address the fact that particles are capable of carrying information over macroscopic parallel scale lengths. In this work, generalized closures that embody Landau, collisional and particle-trapping physics are derived and discussed. A gyro/bounce-averaged drift kinetic equation is solved via an expansion in eigenfunctions of the pitch-angle scattering operator and the resulting system of algebraic equations is solved by integrating along characteristics. The desired closure moments take the form of integral equations involving perturbations in the flow and temperature along magnetic field lines. Implementation of the closures in massively parallel plasma fluid simulation codes is also discussed. This implementation includes the use of a semi-implicit time advance of the fluid equations to stabilize the dominant closure terms which are introduced explicitly. Application of the nonlocal, parallel heat flow closure, q∥, in studies of temperature flattening across helical magnetic islands in toroidal geometry reveal a scaling of temperature versus critical island width for flattening of T∼wd−1.5. This result predicts more robust flattening at small island widths when compared to the diffusive scaling, T∼wd−1.7, which assumes a Braginskii-type parallel heat conductivity. Preliminary application of q∥ to tokamak disruption simulations shows qualitative agreement of wall heat loads with experimental observations, smooth distribution in toroidal angle, and striation in the poloidal direction along the wall.