Diffusive Heat Transport Research Articles

New physics capabilities from the upgraded Thomson scattering diagnostic on MAST

The newly upgraded MAST Thomson scattering (TS) system provides excellent spatial resolution (∼1 cm) at over 130 radial locations across a full plasma diameter, and utilizes eight individual Nd: :YAG laser systems which can be fired sequentially, providing electron temperature and density profiles approximately every 4 ms throughout a plasma discharge. By operating the system in burst mode, whereby the laser separation can be adjusted to within a few microseconds of each other, it is possible to obtain detailed profiles of transient and periodic phenomena such as sawteeth crashes, massive gas injection for disruption mitigation and the temperature perturbations associated with neoclassical tearing mode (NTM) islands. Following Fitzpatrick et al (1995 Phys. Plasmas 2 825), we consider a simplified model in which finite parallel diffusive heat transport can provide a threshold for NTM island growth and demonstrate that the TS derived electron temperature profiles around an island can be used to obtain both the island width and the critical island width below which temperature gradients are maintained across the island, potentially removing the bootstrap current drive for the NTM. Initial results from high beta, neutral beam injection heated discharges on MAST show that the measured island width inferred from the TS data is in good agreement with magnetic estimates of the island width (considering both a cylindrical approximation and using a full field line tracing estimate). The temporal behaviour of the island width obtained from the magnetic diagnostics indicates that for the scenarios considered to date, finite parallel diffusion is likely to play an important role in NTM threshold physics in MAST.

Concurrent coupling of atomistic and continuum models at finite temperature

A concurrent multiscale method for coupling discrete (atomistic) and continuum models at finite temperatures is presented. Motion of atoms is governed by an inter-atomic potential and is represented by molecular dynamics. A thermo-mechanical continuum defined by standard differential equations of conservation of momentum and heat transport is used. The coupling is performed by an interface region where the two models overlap. The phonon spectrum of the discrete region is divided into a low frequency part which is transferred to the continuum model as mechanical waves, and a high frequency component which is modeled in the continuum as diffusive heat transport. Seamless mechanical coupling is ensured by imposing weak compatibility of displacements in the interface region. The method is implemented in 1D and full bi-directional thermal and mechanical coupling is demonstrated in thermodynamic equilibrium and in non-equilibrium.

Marangoni convection in molten salts

Marangoni convection is involved in many technological processes. The substances of industrial interest are often governed by diffusive heat transport and their physical modelling is limited with respect to the Prandtl number Pr. The present paper addresses this deficiency. Studies were made on molten salts having Pr values in an intermediate range well below that of the typically employed organics. Since some of the selected species have a relatively high melting point, a high-temperature facility which allows studying thermocapillary convection at temperatures in excess of 1,000°C was built. The results presented here were obtained in a cylindrical geometry, although the equipment that was built is not restricted to this configuration because of its modular construction. Modelled after some applications, the fluid was heated centrically on top. The bulk was embedded in a large thermostatically controlled reservoir so as to establish the lower ambient reference temperature. A characteristic size of the experimental cell was chosen such that, on the one hand, the dynamic Bond number Bo did not become too high; on the other hand, the liquid had to have a certain depth to allow particle image velocimetry. The complicated balance between body forces and thermocapillary forces in the case of intermediate Bo was found to result in a distinct local separation into a bulk motion governed by natural convection with a recirculating Marangoni flow on top. In contrast to low viscosity organics, the vapour pressure of which increases considerably with decreasing Pr, high values of the Marangoni number can be reached. Comparisons of the topology of Marangoni vortices between molten salts with 2.3 ⩽ Pr ⩽ 6.4 and a silicone oil with Pr typically one order of magnitude higher suggest that the regime of non-negligible heat diffusion is entered.

A Spectral Parameterization of Drag, Eddy Diffusion, and Wave Heating for a Three-Dimensional Flow Induced by Breaking Gravity Waves

AbstractThere are three distinct processes by which upward-propagating gravity waves influence the large-scale dynamics and energetics of the middle atmosphere: (i) nonlocalized transport of momentum through wave propagation in three dimensions that remotely redistributes atmospheric momentum in both zonal and meridional directions from wave generation to wave dissipation regions; (ii) localized diffusive transport of momentum, heat, and tracers due to mixing induced by wave breaking; and (iii) localized transport of heat by perturbing wave structures due to dissipation that redistributes the thermal energy within a finite domain. These effects become most significant for breaking waves when momentum drag, eddy diffusion, and wave heating— the “breaking trinity”—are all imposed on the background state. This paper develops a 3D parameterization scheme that self-consistently includes the breaking trinity in large-scale numerical models. The 3D parameterization scheme is developed based on the general relationship between the wave action flux and the subgrid-scale momentum and heat fluxes developed by Zhu in 1987 and a mapping approximation between the wave source spectrum and momentum deposition distribution developed by Alexander and Dunkerton in 1999. For a set of given input wind and temperature profiles at each model grid, the parameterization scheme outputs the vertical profiles of the subgrid-scale force terms together with the eddy diffusion coefficients in the momentum and energy equations for a 3D background flow.

Erratum: Ballistic and diffusive transport of energy and heat in metals [Phys. Rev. B79, 184303 (2009)

Erratum: Ballistic and diffusive transport of energy and heat in metals [Phys. Rev. B<b>79</b>, 184303 (2009)

Book Reviews

Book Reviews

Thickness of the diffusive sublayer in turbulent convection

In this paper we address the following question: how thick is the diffusive fraction of the thermal boundary layer in highly turbulent thermal convection? We have studied this problem in a large-scale Rayleigh-Bénard experiment at fixed aspect ratio Gamma=1.13 and variable Rayleigh number 5.2x10(10)<Ra<9.6x10(11) using air as the working fluid. By measuring profiles of the mean temperature in the vicinity of the heated bottom plate with a higher spatial resolution than in any other previous experiment and by complementing them with simultaneous independent measurements of the local heat flux, we have determined the shares of diffusive and convective vertical heat fluxes inside the thermal boundary layer. Our measurements show that the thickness of the sublayer where heat is exclusively transported by diffusion is only about 1/25 of the nominal thickness of the thermal boundary layer and depends only weakly on Ra in the parameter domain investigated here. This result implies that phenomenological theories which assume a diffusive heat transport in the whole thermal boundary layer are unlikely to be correct at very high Rayleigh numbers.

Uncertainty analysis of thermo-hydro-mechanical coupled processes in heterogeneous porous media

In this paper we present an uncertainty analysis of thermo-hydro-mechanical (THM) coupled processes in a typical geothermal reservoir in crystalline rock. Fracture and matrix are treated conceptually as an equivalent porous medium, and the model is applied to available data from the Urach Spa and Falkenberg sites (Germany). The finite element method (FEM) is used for the numerical analysis of fully coupled THM processes, including thermal water flow, advective–diffusive heat transport, and thermoelasticity. Non-linearity in system behavior is introduced via temperature and pressure dependent fluid properties. Reservoir parameters are considered as spatially random variables and their realizations are generated using conditional Gaussian simulation. The related Monte-Carlo analysis of the coupled THM problem is computationally very expensive. To enhance computational efficiency, the parallel FEM based on domain decomposition technology using message passing interface (MPI) is utilized to conduct the numerous simulations. In the numerical analysis we considered two reservoir modes: undisturbed and stimulated. The uncertainty analysis we apply captures both the effects of heterogeneity and hydraulic stimulation near the injection borehole. The results show the influence of parameter ranges on reservoir evolution during long-term heat extraction, taking into account fully coupled thermo-hydro-mechanical processes. We found that the most significant factors in the analysis are permeability and heat capacity. The study demonstrates the importance of taking parameter uncertainties into account for geothermal reservoir evaluation in order to assess the viability of numerical modeling.

Visualization of diffusion and convection heat transport in a square cavity with natural convection

In this study, the total heatfunction equation which includes diffusion and convection transport is divided into the corresponding heatfunction equations. The superposition rule is used to obtain the mathematical definitions of diffusion and convection heatfunctions and corresponding boundary conditions. It is observed that the separate visualization of diffusion and convection heatlines provides significant information on understanding of the mechanism of heat transfer in a convective flow. The direction of the diffusion and convection heat transport as well as the strength of convection compared to the conduction in entire or in a portion of a domain can be visualized. The diffusion heatlines demonstrate a potential flow like behavior while convective heat flow rotates due to the source term of the convection heatfunction equation, similar to the rotation of fluid flow generated by fluid flow vorticity. The similarity between the streamfunction and the total heatfunction yields a concept of heat flow vorticity, Ω t . The obtained results show that the maximum absolute value of the convection heatfunction may be an appropriate parameter for determination of the convection strength. The diffusion and convection heatfunction equations for natural convection in a differentially heated square cavity for four different length of the heated surface on the right vertical wall as s p = L/4, L/2, 3 L/4 and L and a fixed length of the cooled surface on the right vertical wall as L/4 are obtained and corresponding heatlines are drawn. The values of the conduction heatfunction are positive while the sign of convection heatfunction values is negative for the studied cases. Based on the distribution of total heatlines, two regions are detected in the cavity, an active region with the positive values of heatlines signifying dominant conduction heat transfer and a passive region with the negative heatfunction values in where convection heat flow is dominant and heat only rotates in a closed contour pattern. The variations of average Nusselt number, average of heat flow vorticity, maximum absolute values of convection heatfunction and streamfunction at different Rayleigh numbers and lengths of the heated surface are presented.

Simulation of continuous casting of steel by a meshless technique

A recently developed local radial basis function collocation method is used for the solution of the transient convective–diffusive heat transport in continuous casting of steel. The solution of the thermal field with moving boundaries due to phase-change and the growing computational domain is based on the mixture continuum formulation. The growth of the domain and the movement of the starting block are described by activation of additional nodes and by the movement of the boundary nodes through the computational domain, respectively. Time-stepping is performed in an explicit way by a simple characteristic procedure. A two-dimensional transient test case solution is shown at different times and its accuracy is verified by comparison with the reference finite volume method results. The method is very attractive in the present context due to its trivial implementation of curved geometry for two and three dimensions, accuracy and stability of the results.

Ballistic and diffusive transport of energy and heat in metals

We exploit a recently developed formalism to study the temporal behavior of the energy and heat transport in metals. This formalism shows the transition between ballistic and diffusive regimes and it highlights some interesting effects such as oscillations in the energy transport at very short-time scales. The energy relaxation of the conduction-band electrons in metals is considered to be governed by electron-phonon scattering, and the scattering time is taken to be averaged over the Fermi surface. The analysis separates the diffusive and the nondiffusive contributions to the heat transport. While the diffusive contribution shows an almost exponentially decaying behavior with time, the nondiffusive part shows a damped oscillating behavior. The origin of this oscillation will be discussed as well as the effect of the ambient temperature on the dynamics of the energy modes transport.

Absence of thermal rectification in asymmetric harmonic chains with self-consistent reservoirs

An exact analysis using the Langevin equation approach proves that an asymmetric-inhomogeneous harmonic chain with self-consistent inner reservoirs does not rectify heat, while it may show diffusive heat transport properties. Thus, we argue that the onset of a formal Fourier's law in an asymmetric system does not guarantee an asymmetric heat flow.

Shape optimization for 2D diffusive scalar transport

A new theoretical formulation is presented for the shape optimization problem associated with maximizing or minimizing the diffusive scalar transport from a two-dimensional body. In particular, we consider the diffusive transport of heat from an isothermal body into a medium with constant temperature at the far-field. The formulation also applies to mass and momentum transport. The diffusion problem, which is governed by the Laplace equation, is addressed using conformal mapping techniques where the two-dimensional domain is mapped onto a simpler domain where an analytical solution can be readily obtained. The objective function of the optimization problem is the length of the object in the transformed domain and the variables of the optimization are the parameters of the Schwarz-Christoffel transformation. The length of the object in the transformed domain is related to the scalar displacement, which corresponds to a far-field temperature drop or rise (slip velocity in case of momentum transport), that depends on the shape of the body and it quantifies the enhancement or reduction in transport rate. The mathematical formulation is validated by addressing two fundamental shape optimization problems associated with maximizing or minimizing the transport rate (drag in case of momentum transport) from a two-dimensional body of unit span: i) for a given surface area to obtain the shape that maximizes the transport rate from a body, ii) for a given volume to obtain the shape that minimizes the transport rate from a body. For both cases we compute numerically that the cylinder is the optimal shape. The versatility of the formulation is further demonstrated by including constraints with respect to the length of the body.

Thermal evolution of cratonic roots

Cratons stabilized in the Archean because their mantle roots remained strong despite temperatures in the convecting mantle and amounts of radiogenic elements in the crust and in the lithospheric mantle that were larger than today. Thermal evolutionary models are developed using constraints from heat flow and heat production data in Archean provinces. The large time-scale of diffusive heat transport implies that the lithospheric mantle can remain thermally decoupled from the crust for as long as 1 Gyr. Heat production in the lithospheric mantle is a key variable in determining thermal conditions that permit stabilization of the crust and the preservation of a thick cratonic mantle root. Archean provinces are currently characterized by low heat flow, with an average of 41 mW m − 2 less than the global continental average (56 mW m − 2 ). The range of regionally averaged heat flow values in Archean provinces (18–54 mW m − 2 ) is narrower than in Proterozoic and Paleozoic terranes. However, at the end of the Archean, when crustal heat production was double the present value, surface heat flow would have varied over a range (≈ 45–90 mW m − 2 ) at least as wide as that presently observed in Paleozoic provinces. The high crustal heat production during the Archean is not sufficient to account for elevated lower crustal temperatures recorded in some metamorphic assemblages without some additional heat input, or without the crust being thicker, or the vertical distribution of radioelements being different from today's. Thermal models and the observed correlation between the degree of upper crustal enrichment and heat production values indicate that crustal melting and differentiation were largely driven by heating due to in-situ radiogenic heat production. Even for values of the surface heat flow higher than average in cratons, the crust can be stabilized before 2.5 Ga if the radioelements are confined to shallow crustal layers. For present surface heat flow of 40–45 mW m − 2 , calculations indicate that, prior to differentiation, the lowermost crust was near the solidus. Present heat production in the mantle root is constrained by estimates of the mantle heat flow. Further constraints can be obtained by modeling the past thermal regime of the root when heat production was higher. If heat generation is high and/or if the root is thick, the lower lithosphere must have cooled more rapidly than the convecting mantle, which implies that the temperature gradient was inverted at the base of the lithosphere and a weak mechanical layer formed in the middle of the root, precluding its survival. A low temperature gradient at the base of the root leads to the development of convective instabilities and possible removal of the lowermost part. Following its isolation from the convecting mantle and stabilization, thick continental lithosphere adjusts slowly to in-situ heat production and basal heat supply. We show that, due to in-situ heat production, temperatures in the lithospheric mantle may rise above their initial values inherited from the process of root formation due to in-situ radiogenic heat production.

Modeling the Merensky Reef, Bushveld Complex, Republic of South Africa

The Merensky pegmatoid (normal reef) in the western Bushveld Complex is commonly characterized as a pyroxene-rich pegmatoidal unit with a base that is enriched in chromite and platinum-group element-bearing sulfides overlying a leuconorite footwall. Models for its formation have ranged from those that view it as entirely a magmatic cumulate succession to those that have suggested that it is a zone of volatile-induced remelting. The consequences of the latter interpretation are investigated using the numerical modeling program IRIDIUM, which links diffusive and advective mass and heat transport with a phase equilibration routine based on the MELTS program. The initial system consists of a simple stratigraphic succession of a partially molten leuconorite overlain by a partially molten pyroxenite, both initially at 1,190°C and 2 kbar. 2 wt% of a volatile fluid composed of 75 mol% H2O, 20 mol% CO2 and 5 mol% H2S is then added to the lower 20 cm of the pyroxenite. The system is then allowed to evolve under conditions of chemical diffusion in the liquid. The addition of the volatile components results in a modest increase in the amount of melt in the pyroxenite. However, chemical diffusion across the leuconorite–pyroxenite boundary leads to more extensive melting at and below the boundary with preferential loss of opx from the underlying leuconorite, preferential re-precipitation of sulfide and chromite and concentration of the PGE at this boundary. These results mimic actual mineral and compositional profiles across the Merensky pegmatoid and illustrate that long-term diffusion process can effectively produce mineralogical and compositional layering not present in the original assemblage.

THE MECHANISM OF UNSTEADY DOWNSTREAM INTERACTIONS OF PREMIXED HYDROGEN-AIR FLAMES

The process of flame annihilation resulting from downstream interactions of premixed hydrogen-air flames is studied using Direct Numerical Simulations (DNS). The process is investigated during unsteady interaction between a vortex pair and a premixed flame kernel in 2D. The annihilation process results from the interactions of the premixed flames on their products' side. The simulations show that the mechanism of extinction during downstream interaction is different from an upstream interaction, which is governed by the sequence of the interactions of the different preheat and reaction layers on both processes and the diffusive transport of heat and mass. In contrast to observations related to upstream interactions, downstream interactions lead to a shift in the equivalence ratio towards the richer conditions with a steady decrease in chemical activity and no radical pool build-up during the stages of extinction. The effect of vortex sizes is qualitatively the same between the different cases considered.

Derivations of extended Navier-Stokes equations from upscaled molecular transport considerations for compressible ideal gas flows: Towards extended constitutive forms

This paper deals with the derivations of the extended forms of the continuum conservation equations for ideal gas flows with heat and mass transfer, from upscaled molecular transport considerations. It is shown that for strong local temperature and/or density gradients, diffusive transport of momentum, heat, and mass over subcontinuum length scales gives rise to additional terms in the corresponding continuum level constitutive relationships. An exact analogy with kinematically hypothesized extended rheological forms is established by casting the expression for momentum flux in an equivalent linear form, with the postulation of a gradient-based hypothesis. Possible considerations of extending the theory to more general flow conditions are also discussed.

A numerical model for estimating groundwater flux from subsurface temperature profiles

AbstractA simple numerical model is presented for estimating vertical groundwater flux from transient subsurface temperature profiles obtained from field measurements. The model developed utilizes the MacCormack scheme, which is based on the Finite Difference Method (FDM), for solving the governing partial differential equation of convection–diffusion heat transport with appropriate initial and boundary conditions within the subsurface. In order to validate the model, numerical solutions obtained for the study area located in the Nagoka plain, Japan are compared with the published measured data and results obtained by others. Results obtained show good agreement and fit the observed data with a correlation coefficient, R2, of 0·88. The estimated groundwater flux is 1·85 × 10−7 m s−1. Sensitivity analyses were also carried out to investigate the effect of variations in groundwater fluxes, thermal properties and the annual thermal variability due to climatic changes on the transient subsurface temperature profiles and to have a better understanding of the subsurface thermal dynamics. A substantial effect of annual climatic variability is observed on the temporal distributions of temperature depth profiles, and a better estimate of thermal parameters is required to estimate vertical groundwater flux. The largest change in subsurface temperature depth profiles due to groundwater flux over a year is within ± 4 °C. The influence of groundwater flux on subsurface temperature distributions in space and time may be more pronounced in areas where the top of the saturated layer fluctuates considerably. Variation in thermal diffusivity results in temperature change up to ± 1·5% and may cause change in groundwater flux estimate by ± 18%. The model presented has merits over analytical solutions (type curve matching techniques) in terms of suitability and applicability to real field problems, and can be a good asset to hydrological models as quantifying groundwater recharge or deducing it from other quantities, such as rainfall, evapotranspiration and runoff, is often complicated. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Microgravity experiments on columnar–equiaxed transition in Al based alloys

The investigation on solidification of metallic alloys under well defined experimental conditions is of fundamental interest for the understanding of the physics involved in microstructure formation. In technical casting processes, these microstructures play an important role in the properties and the quality of the final cast product. The authors have investigated the transition from columnar to equiaxed grain growth (CET) in the framework of the ESA map CETSOL in Al–Si hypoeutectic alloys under microgravity conditions on the sounding rocket flight MAXUS-7. In the directionally solidified rod-like samples, the bulk melt flow is suppressed and diffusive mass and heat transport without sedimentation of refiner particles or equiaxed grains is obtained. The analysis of the processed samples creates benchmark data for validation and development of physical models for the formation of equiaxed grains and the CET. Here the authors present preliminary results from the analysis of two AlSi7 samples, one with and the other without addition of grain refiners (Ti and B) under identical experimental conditions. Critical parameters at the CET for temperature gradient and solidification rate are obtained from the experiments, as well as some characteristics of the grain structure. In a first approach the critical parameters are compared to predictions from Hunt's mechanical blocking model.

Numerical modeling of diffusive heat transport across magnetic islands and highly stochastic layers

Diffusive heat transport across magnetic islands and highly stochastic layers is studied numerically for realistic values of χ‖∕χ⊥ in cylindrical geometry, where χ‖ denotes the heat diffusion coefficient parallel and χ⊥ the one perpendicular to the magnetic field lines. The computations are performed with a second-order finite difference scheme, for which the numerical errors are independent from the value of χ‖∕χ⊥ [S. Günter et al., J. Comput. Phys. 209, 354 (2005)]. Sufficient spatial resolution is ensured by using an unsheared helical coordinate system. The heat flux around magnetic islands as well as the effective radial heat diffusivity χr are examined and compared to analytical theory. The temperature perturbations caused by magnetic islands and the resulting bootstrap current perturbations essential for the stability of neoclassical tearing modes are analyzed and compared to analytical predictions [R. Fitzpatrick, Phys. Plasmas 2, 825 (1995)]. Agreement is found in the “small” and “large” island limits, but an enhanced NTM drive is observed in between. A correction factor that can reproduce the numerical results very well is presented. For a highly stochastic layer, produced by five strongly overlapping islands, the radial heat diffusivity χr is determined and compared to several analytical theories.