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Green function for boundary value problems in heat conduction equation
The geometry of sunspots has been used to suggest a problem in heat flow. The equation of heat transport is solved for the case of a cylinder with a given thermal conductivity imbedded in an otherwise uniform medium with different conductivity. The surface of this region radiates heat with flux proportional to temperature. At a lower surface, either in heat flux or temperature is held constant. The cylinder can have an anisotropic thermal conductivity. The variations in temperature along the radiating surface have been determined. A simple approximation is noted which has been found to give a general solution with acceptable accuracy. This method may be of some use in other situations requiring the solution of Laplace's equation with a free surface. The analysis is used to set limits on the ratio of diameter to depth for cases which preserve the sharp surface temperature transition across the cylinder.
Numerical integration of steady state heat conduction equations representing diurnal variations in upper atmosphere
Heat Stress Indices are equations that integrate some or all variables (e.g. temperature, relative humidity, wind speed), directly or indirectly, to produce a number for thermal stress on humans for a particular environment. There are a large number of equations that have been developed which range from simple equations that may ignore basic factors (e.g. wind effects on thermal loading, fixed contribution from solar heating) to complex equations that attempt to incorporate all variables. Each equation is evaluated for a particular use, as well as considering the ease of use and reliability of the results. The meteorology group at the Dryden Flight Research Center has utilized and enhanced the American College of Sports Medicine equation to represent the specific environment of the Mojave Desert. The Dryden WBGT Heat Stress equation has been vetted and implemented as an automated notification to the entire facility for the safety of all personnel and visitors.
Contact problem for generalized system of differential equations of heat and mass transfer
Dynamic properties of stellar coronas & stellar winds and integration of heat-flow equation
A new method is derived for solving parabolic partial differential equations arising in transient heat conduction or in boundary-layer flows. The method is based on a combination of the modified differential quadrature (MDQ) method with the rational Runge-Kutta time-integration scheme. It is fully explicit, requires no matrix inversion, and is stable for any time-step for the heat equations. Burgers equation and the one- and two-dimensional heat equations are solved to demonstrate the accuracy and efficiency of the proposed algorithm. The present method is found to be very accurate and efficient when results are compared with analytic solutions.
We formulate a new second-order closure turbulence model by employing a recent closure for the pressure-temperature correlation at the equation level. As a result, we obtain new heat flux equations that avoid the long-standing issue of a finite critical Richardson number. The new, structurally simpler model improves on the Mellor-Yamada 1982 and Galperin et al. 1988 models; key feature includes enhanced mixing under stable conditions facilitating agreement with observational, experimental and high-resolution numerical data sets. The model predicts a planetary boundary layer height deeper than predicted by models with low critical Richardson numbers, as demonstrated in single column model runs of the GISS ModelE general circulation model.
Dynamical properties of stellar coronas and stellar winds - integration of heat flow equation - thermoconductivity
Variable property heat transfer equations and their applicability to nozzle of nuclear rocket Phoebus 2
Thermodynamic Property Generation program produces tables for any fluid for which state equation, vapor pressure equation, specific heat equation, and critical constants are known. Program calculates thermodynamic properties at any state providing way of obtaining properties at extreme environmental conditions.
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It is pointed out that sunspots represent a disruption in the uniform flow of heat through the convective zone. The basic sunspot structure is, therefore, determined by the energy transport equation. The solutions of this equation for the case of stochastic heat transport are examined. It is concluded that a sunspot is basically a region of enhanced, rather than inhibited, energy transport and emissivity. The heat flow equations are discussed and attention is given to the shallow depth of the sunspot phenomenon. The sunspot is seen as a heat engine of high efficiency which converts most of the heat flux into hydromagnetic waves.
Computer program to solve heat conduction equation in lunar surface for temperature-dependent thermal properties
The time-dependent, turbulent mean-flow, Reynolds stress, and heat flux equations in mass-averaged dependent variables are presented. These equations are given in conservative form for both generalized orthogonal and axisymmetric coordinates. For the case of small viscosity and thermal conductivity fluctuations, these equations are considerably simpler than the general Reynolds system of dependent variables for a compressible fluid and permit a more direct extension of low speed turbulence modeling to computer codes describing high speed turbulence fields.
In this paper, we extend the operator-split asymptotic-preserving, semi-Lagrangian algorithm for time dependent anisotropic heat transport equation proposed in Chacón et al. (2014) [18] to use a fully implicit time integration with backward differentiation formulas. The proposed implicit method can deal with arbitrary heat-transport anisotropy ratios $\mathcal{X}$∥ /$ \mathcal{X}$⟂ $\ggg$ 1 (with $\mathcal{X}$∥, $ \mathcal{X}$⟂ the parallel and perpendicular heat diffusivities, respectively) in complicated magnetic field topologies in an accurate and efficient manner. Further, the implicit algorithm is second-order accurate temporally and demonstrates an accurate treatment at boundary layers (e.g., island separatrices), which was not ensured by the operator-split implementation. The condition number of the resulting algebraic system is independent of the anisotropy ratio, and is inverted with preconditioned GMRES. We propose a simple preconditioner that renders the finite-dimensional linear operator compact, resulting in mesh-independent convergence rates for topologically simple magnetic fields, and convergence rates scaling as ~ (NΔt) 1/4 (with N the total mesh size and Δt the timestep) in topologically complex magnetic-field configurations. We demonstrate the accuracy and performance of the approach with test problems of varying complexity, including an analytically tractable boundary-layer problem in a straight magnetic field, and a topologically complex magnetic field featuring magnetic islands with extreme anisotropy ratios $\mathcal{X}$∥ /$ \mathcal{X}$⟂ = 10 10 ) .
Here, we extend the recently proposed semi-Lagrangian algorithm for the extremely anisotropic heat transport equation [Chacón et al., J. Comput. Phys ., 272 (2014)] to deal with arbitrary magnetic field topologies. The original scheme (which showed remarkable numerical properties) was valid for the so-called tokamak-ordering regime, in which the magnetic field magnitude was not allowed to vary much along field lines. The proposed extension maintains the attractive features of the original scheme (including the analytical Green's function, which is critical for tractability) with minor modifications, while allowing for completely general magnetic fields. The accuracy and generality of the approach are demonstrated by numerical experiment with an analytical manufactured solution.
A computer program (Program SPHERE) solving the inhomogeneous equation of heat conduction with radiation boundary condition on a thermally homogeneous sphere is described. The source terms are taken to be exponential functions of the time. Thermal properties are independent of temperature. The solutions are appropriate to studying certain classes of planetary thermal history. Special application to the moon is discussed.