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McGraw-Hill Science & Technology Dictionary:

radiative transfer

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(′rād·ē′ād·iv ′tranz·fər)

(physics) The propagation of energy by radiative processes, involving emission, absorption, and scattering of electromagnetic radiation. Also known as radiative transport.

McGraw-Hill Science & Technology Encyclopedia:

Radiative transfer

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The study of the propagation of energy by radiative processes; it is also called radiation transport. Radiation is one of the three mechanisms by which energy moves from one place to another, the other two being conduction and convection. See also Electromagnetic radiation; Heat transfer.

The kinds of problems requiring an understanding of radiative transfer can be characterized by looking at meteorology, astronomy, and nuclear reactor design. In meteorology, the energy budget of the atmosphere is determined in large part by energy gained and lost by radiation. In astronomy, almost all that is known about the abundance of elements in space and the structure of stars comes from modeling radiative transfer processes. Since neutrons moving in a reactor obey the same laws as radiation being scattered by atmospheric particles, radiative transfer plays an important part in nuclear reactor design.

Each of these three fields—meteorology, astronomy, and nuclear engineering—concentrates on a different aspect of radiative transfer. In meteorology, situations are studied in which scattering dominates the interaction between radiation and matter; in astronomy, there is more interest in the ways in which radiation and the distribution of electrons in atoms affect each other; and in nuclear engineering, problems relate to complicated, three-dimensional geometry.

Radiative transfer is a complicated process because matter interacts with the radiation. This interaction occurs when the photons that make up radiation exchange energy with matter. These processes can be understood by considering the transfer of visible light through a gas made up of atoms. Similar processes occur when radiation interacts with solid dust particles or when it is transmitted through solids or liquids. See also Photon.

If a gas is hot, collisions between atoms can convert the kinetic energy of motion to potential energy by raising atoms to an excited state. Emission is the process which releases this energy in the form of photons and cools the gas by converting the kinetic energy of atoms to energy in the form of radiation. The reverse process, absorption, occurs when a photon raises an atom to an excited state, and the energy is converted to kinetic energy in a collision with another atom. Absorption heats the gas by converting energy from radiation to kinetic energy. Occasionally an atom will absorb a photon and reemit another photon of the same energy in a random direction. If the photon is reradiated before the atom undergoes a collision, the photon is said to be scattered. Scattering has no net effect on the temperature of the gas. See also Absorption; Atomic structure and spectra; Scattering of electromagnetic radiation.

Wikipedia on Answers.com:

Radiative transfer

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It has been suggested that radiative transfer equation and diffusion theory for photon transport in biological tissue be merged into this article or section. (Discuss) Proposed since December 2009.

Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission, and scattering processes. The equation of radiative transfer describes these interactions mathematically. Equations of radiative transfer have application in wide variety of subjects including optics, astrophysics, atmospheric science, and remote sensing. Analytic solutions to the radiative transfer equation (RTE) exist for simple cases but for more realistic media with complex multiple scattering effects numerical methods are required.

The present article is largely focused on the condition of radiative equilibrium.[1] [2]

Contents

Definitions

The fundamental quantity which describes a field of radiation is nowadays called the spectral radiance, traditionally called the specific intensity. If we think of a very small area element in the radiation field, there will be radiation energy flowing through that area element. The flow can be completely characterized by the amount of energy flowing per unit time per unit solid angle, the direction of the flow, and the wavelength interval being considered (polarization will be ignored for the moment).

In terms of the spectral radiance, I_\nu, the energy flowing across an area element of area da\, located at \mathbf{r} in time dt\, in the solid angle d\Omega about the direction \hat{\mathbf{n}} in the frequency interval \nu\, to \nu+d\nu\, is

dE_\nu = I_\nu(\mathbf{r},\hat{\mathbf{n}},t) \cos\theta \ d\nu \, da \, d\Omega \, dt

where \theta is the angle that the unit direction vector \hat{\mathbf{n}} makes with a normal to the area element. The units of the spectral radiance are seen to be energy/time/area/solid angle/frequency. In MKS units this would be W·m-2·sr-1·Hz-1 (watts per square-metre-steradian-hertz).

The equation of radiative transfer

The equation of radiative transfer simply says that as a beam of radiation travels, it loses energy to absorption, gains energy by emission, and redistributes energy by scattering. The differential form of the equation for radiative transfer is:

\frac{1}{c}\frac{\partial}{\partial t}I_\nu + \hat{\Omega} \cdot \nabla I_\nu + (k_{\nu, s}+k_{\nu, a}) I_\nu = j_\nu + \frac{1}{4\pi c}k_{\nu, s} \int_\Omega I_\nu d\Omega

where j_\nu is the emission coefficient, k_{\nu, s} is the scattering cross section, and k_{\nu, a} is the absorption cross section.

Solutions to the equation of radiative transfer

Solutions to the equation of radiative transfer form an enormous body of work. The differences however, are essentially due to the various forms for the emission and absorption coefficients. If scattering is ignored, then a general solution in terms of the emission and absorption coefficients may be written:

I_\nu(s)=I_\nu(s_0)e^{-\tau_\nu(s_0,s)}+\int_{s_0}^s j_\nu(s') e^{-\tau_\nu(s',s)}\,ds'

where \tau_\nu(s_1,s_2) is the optical depth of the medium between positions s_1 and s_2:

\tau_\nu(s_1,s_2) \ \stackrel{\mathrm{def}}{=}\ \int_{s_1}^{s_2} \alpha_\nu(s)\,ds

Local thermodynamic equilibrium

A particularly useful simplification of the equation of radiative transfer occurs under the conditions of local thermodynamic equilibrium (LTE). In this situation, the absorbing/emitting medium consists of massive particles which are in equilibrium with each other, and therefore have a definable temperature. The radiation field is not, however in equilibrium and is being entirely driven by the presence of the massive particles. For a medium in LTE, the emission coefficient and absorption coefficient are functions of temperature and density only, and are related by:

\frac{j_\nu}{\alpha_\nu}=B_\nu(T)

where B_\nu(T) is the black body spectral radiance at temperature T. The solution to the equation of radiative transfer is then:

I_\nu(s)=I_\nu(s_0)e^{-\tau_\nu(s_0,s)}+\int_{s_0}^s B_\nu(T(s'))\alpha_\nu(s') e^{-\tau_\nu(s',s)}\,ds'

Knowing the temperature profile and the density profile of the medium is sufficient to calculate a solution to the equation of radiative transfer.

The Eddington approximation

The Eddington approximation is a special case of the two stream approximation. It can be used to obtain the spectral radiance in a "plane-parallel" medium (one in which properties only vary in the perpendicular direction) with isotropic frequency-independent scattering. It assumes that the intensity is a linear function of \mu=\cos\theta. i.e.

I_\nu(\mu,z)=a(z)+\mu b(z)

where z is the normal direction to the slab-like medium. Note that expressing angular integrals in terms of \mu simplifies things because d\mu=-\sin\theta d\theta appears in the Jacobian of integrals in spherical coordinates.

Extracting the first few moments of the spectral radiance with respect to \mu yields

J_\nu=\frac{1}{2}\int^1_{-1}I_\nu d\mu = a
H_\nu=\frac{1}{2}\int^1_{-1}\mu I_\nu d\mu = \frac{b}{3}
K_\nu=\frac{1}{2}\int^1_{-1}\mu^2 I_\nu d\mu = \frac{a}{3}

Thus the Eddington approximation is equivalent to setting K_\nu=1/3J_\nu. Higher order versions of the Eddington approximation also exist, and consist of more complicated linear relations of the intensity moments. This extra equation can be used as a closure relation for the truncated system of moments.

Note that the first two moments have simple physical meanings. J_\nu is the isotropic intensity at a point, and H_\nu is the flux through that point in the z direction.

The radiative transfer through an isotropically scattering medium at local thermodynamic equilibrium is given by

\mu \frac{dI_\nu}{dz}=- \alpha_\nu (I_\nu-B_\nu) + \sigma_{\nu}(J_\nu -I_\nu)

Integrating over all angles yields

\frac{dH_\nu}{dz}=\alpha_\nu (B_\nu-J_\nu)

Premultiplying by \mu, and then integrating over all angles gives

\frac{dK_\nu}{dz}=-(\alpha_\nu+\sigma_\nu)H_\nu

Substituting in the closure relation, and differentiating with respect to z allows the two above equations to be combined to form the radiative diffusion equation

\frac{d^2J_\nu}{dz^2}=3\alpha_\nu(\alpha_\nu+\sigma_\nu)(J_\nu-B_\nu)

This equation shows how the effective optical depth in scattering-dominated systems may be significantly different from that given by the scattering opacity if the absorptive opacity is small.

See also

References

  1. ^ S. Chandrasekhar (1960). Radiative Transfer. Dover Publications Inc.. p. 393. ISBN 0-486-60590-6. 
  2. ^ Jacqueline Lenoble (1985). Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. A. Deepak Publishing. p. 583. ISBN 0-12-451451-0. 

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