Relativistic heat conduction

Last updated

Relativistic heat conduction refers to the modelling of heat conduction (and similar diffusion processes) in a way compatible with special relativity. This article discusses models using a wave equation with a dissipative term.

Contents

Heat conduction in a Newtonian context is modelled by the Fourier equation: [1]

where θ is temperature, [2] t is time, α = k/(ρc) is thermal diffusivity, k is thermal conductivity, ρ is density, and c is specific heat capacity. The Laplace operator,, is defined in Cartesian coordinates as

This Fourier equation can be derived by substituting Fourier’s linear approximation of the heat flux vector, q, as a function of temperature gradient,

into the first law of thermodynamics

where the del operator, , is defined in 3D as

It can be shown that this definition of the heat flux vector also satisfies the second law of thermodynamics, [3]

where s is specific entropy and σ is entropy production.

Hyperbolic model

It is well known that the Fourier equation (and the more general Fick's law of diffusion) is incompatible with the theory of relativity [4] for at least one reason: it admits infinite speed of propagation of heat signals within the continuum field. For example, consider a pulse of heat at the origin; then according to Fourier equation, it is felt (i.e. temperature changes) at any distant point, instantaneously. The speed of information propagation is faster than the speed of light in vacuum, which is inadmissible within the framework of relativity.

To overcome this contradiction, workers such as Cattaneo, [5] Vernotte, [6] Chester, [7] and others [8] proposed that Fourier equation should be upgraded from the parabolic to a hyperbolic form,

.

In this equation, C is called the speed of second sound (i.e. the fictitious quantum particles, phonons). The equation is known as the hyperbolic heat conduction (HCC) equation.[ citation needed ] Mathematically, it is the same as the telegrapher's equation, which is derived from Maxwell’s equations of electrodynamics.

For the HHC equation to remain compatible with the first law of thermodynamics, it is necessary to modify the definition of heat flux vector, q, to

where is a relaxation time, such that

The most important implication of the hyperbolic equation is that by switching from a parabolic (dissipative) to a hyperbolic (includes a conservative term) partial differential equation, there is the possibility of phenomena such as thermal resonance [9] [10] [11] and thermal shock waves. [12]


Notes

  1. Carslaw, H. S.; Jaeger, J. C. (1959). Conduction of Heat in Solids (Second ed.). Oxford: University Press.
  2. Some authors also use T, φ,...
  3. Barletta, A.; Zanchini, E. (1997). "Hyperbolic heat conduction and local equilibrium: a second law analysis". International Journal of Heat and Mass Transfer. 40 (5): 1007–1016. doi:10.1016/0017-9310(96)00211-6.
  4. Eckert, E. R. G.; Drake, R. M. (1972). Analysis of Heat and Mass Transfer. Tokyo: McGraw-Hill, Kogakusha.
  5. Cattaneo, C. R. (1958). "Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée". Comptes Rendus. 247 (4): 431.
  6. Vernotte, P. (1958). "Les paradoxes de la theorie continue de l'équation de la chaleur". Comptes Rendus. 246 (22): 3154.
  7. Chester, M. (1963). "Second sound in solids". Physical Review . 131 (15): 2013–2015. Bibcode:1963PhRv..131.2013C. doi:10.1103/PhysRev.131.2013.
  8. Morse, P. M.; Feshbach, H. (1953). Methods of Theoretical Physics. New York: McGraw-Hill.
  9. Mandrusiak, G. D. (1997). "Analysis of non-Fourier conduction waves from a reciprocating heat source". Journal of Thermophysics and Heat Transfer. 11 (1): 82–89. doi:10.2514/2.6204.
  10. Xu, M.; Wang, L. (2002). "Thermal oscillation and resonance in dual-phase-lagging heat conduction". International Journal of Heat and Mass Transfer. 45 (5): 1055–1061. doi:10.1016/S0017-9310(01)00199-5.
  11. Barletta, A.; Zanchini, E. (1996). "Hyperbolic heat conduction and thermal resonances in a cylindrical solid carrying a steady periodic electric field". International Journal of Heat and Mass Transfer. 39 (6): 1307–1315. doi:10.1016/0017-9310(95)00202-2.
  12. Tzou, D. Y. (1989). "Shock wave formation around a moving heat source in a solid with finite speed of heat propagation". International Journal of Heat and Mass Transfer. 32 (10): 1979–1987. doi:10.1016/0017-9310(89)90166-X.

Related Research Articles

Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.

Divergence Vector operator producing the scalar quantity of a flow at a point

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

Navier–Stokes equations Equations describing the motion of viscous fluid substances

In physics, the Navier–Stokes equations, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes, describe the motion of viscous fluid substances.

The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by , , or .

Thermal conduction is the transfer of internal energy by microscopic collisions of particles and movement of electrons within a body. The colliding particles, which include molecules, atoms and electrons, transfer disorganized microscopic kinetic and potential energy, jointly known as internal energy. Conduction takes place in all phases: solid, liquid, and gas. The rate at which energy is conducted as heat between two bodies depends on the temperature difference between the two bodies and the properties of the conductive interface through which the heat is transferred.

Heat equation partial differential equation for distribution of heat in a given region over time

In physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, 2 or Δ. The Laplacian ∇·∇f(p) of a function f at a point p is the rate at which the average value of f over spheres centered at p deviates from f(p) as the radius of the sphere shrinks towards 0. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form.

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

Onsager reciprocal relations

In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.

In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named after William Rowan Hamilton and Carl Gustav Jacob Jacobi.

Helmholtz equation

In mathematics, the eigenvalue problem for the laplace operator is called Helmholtz equation. It corresponds to the linear partial differential equation:

In physics and engineering, the Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes transient heat conduction. Conceptually, it is the ratio of diffusive or conductive transport rate to the quantity storage rate, where the quantity may be either heat or matter (particles). The number derives from non-dimensionalization of the heat equation or Fick's second law and is used along with the Biot number to analyze time dependent transport phenomena.

In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.

In fluid mechanics and mathematics, a capillary surface is a surface that represents the interface between two different fluids. As a consequence of being a surface, a capillary surface has no thickness in slight contrast with most real fluid interfaces.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

Miniaturizing components has always been a primary goal in the semiconductor industry because it cuts production cost and lets companies build smaller computers and other devices. Miniaturization, however, has increased dissipated power per unit area and made it a key limiting factor in integrated circuit performance. Temperature increase becomes relevant for relatively small-cross-sections wires, where it may affect normal semiconductor behavior. Besides, since the generation of heat is proportional to the frequency of operation for switching circuits, fast computers have larger heat generation than slow ones, an undesired effect for chips manufacturers. This article summaries physical concepts that describe the generation and conduction of heat in an integrated circuit, and presents numerical methods that model heat transfer from a macroscopic point of view.

The multiphase particle-in-cell method (MP-PIC) is a numerical method for modeling particle-fluid and particle-particle interactions in a computational fluid dynamics (CFD) calculation. The MP-PIC method achieves greater stability than its particle-in-cell predecessor by simultaneously treating the solid particles as computational particles and as a continuum. In the MP-PIC approach, the particle properties are mapped from the Lagrangian coordinates to an Eulerian grid through the use of interpolation functions. After evaluation of the continuum derivative terms, the particle properties are mapped back to the individual particles. This method has proven to be stable in dense particle flows, computationally efficient, and physically accurate. This has allowed the MP-PIC method to be used as particle-flow solver for the simulation of industrial-scale chemical processes involving particle-fluid flows.

Multipole radiation is a theoretical framework for the description of electromagnetic or gravitational radiation from time-dependent distributions of distant sources. These tools are applied to physical phenomena which occur at a variety of length scales - from gravitational waves due to galaxy collisions to gamma radiation resulting from nuclear decay. Multipole radiation is analyzed using similar multipole expansion techniques that describe fields from static sources, however there are important differences in the details of the analysis because multipole radiation fields behave quite differently from static fields. This article is primarily concerned with electromagnetic multipole radiation, although the treatment of gravitational waves is similar.

References