## DIFFUSION APPROXIMATION IN MULTIDIMENSIONAL RADIATIVE TRANSFER PROBLEMS

## Diffusion Approximation in Multidimensional Radiative Transfer Problems

### Leonid A. Dombrovsky

* Following from: *Differential approximations; P

_{1}approximation of spherical harmonics method; Solutions for one-dimensional radiative transfer problems

* Leading to: *Radiation of isothermal volumes of scattering medium: An error of the diffusion model; Finite-element method for radiation diffusion in nonisothermal and nonhomogeneous media; Radiative transfer in multidimensional problems: A combined computational model

It is known that the diffusion approximation is not as good for complex multidimensional problems as for the one-dimensional problems considered in the previous section. Even in the case of large optical thickness of a medium in the computational region, one can expect large errors of the diffusion approximation in the regions of sharp spatial variation of the medium properties and temperature, as well as near the corners of the complex region. Nevertheless, the diffusion approximation is usually a good approach for estimating the radiation energy density in scattering media, and the calculated field of this quantity can be used at the second step of the combined iterative solution (Dombrovsky, 1996a) (also see the article, Radiative transfer in multidimensional problems: A combined computational model). The diffusion approximation was directly employed (without additional iterations) in early calculations of the infrared radiation transfer in atmospheric clouds (Davies, 1978; Liou and Ou, 1979; Harshvardhan et al., 1981) and in supersonic nozzles of solid-propellant rocket engines (Vafin et al., 1981; Dombrovsky and Barkova, 1986; Dombrovsky, 1996b).

With the use of the diffusion approximation, the two- and three-dimensional radiation transfer problems are formulated as boundary-value problems for the nonhomogeneous modified Helmholtz equation (6) from the article Differential approximations [in addition, see papers by Modest (1989) and Cheng (1979)]. This type of problem is traditional for mathematical physics, and the problems mentioned can be solved by known methods. Analytical solutions can be obtained in the case of computational regions of simple form at constant equation coefficients and boundary condition parameters. Such solutions are applied directly or for numerical algorithm testing, as well as for initial approximation, by obtaining more accurate solutions to the radiative transfer equation (RTE). The majority of the realistic problems have no analytical solutions; therefore, the numerical algorithms are certainly necessary. The numerical methods have obvious preferences, since they are more general and can be used for nonhomogeneous media in complex computational regions. In our calculations we use the finite-element method (FEM), which has found numerous applications in engineering problems (Zienkiewicz et al., 2005). One of the first applications of the FEM to the radiative-transfer problem in diffusion approximation has been reported by Hänisch (1982). This numerical method was also employed in several studies by Dombrovsky et al. (1986, 1991, 2001, 2007). One should also remember early papers by Fernandes and Francis (1982), Razzaque et al. (1983, 1984), and Chung and Kim (1984) where the FEM was used to solve some combined radiative-conductive heat-transfer problems.

A general variational finite-element–spherical harmonics method for multigroup equations of neutron transport with anisotropic scattering and sources has been developed by Oliveira (1986). One can find the details of FEM applications for particle transport in reactor and radiation physics in the book by Ackroyd (1997). The FEM has also become well established in biomedical optics (Yamada, 1995). One should refer to papers by Arridge et al. (1993), Schweiger et al. (1995), and Schweiger and Arridge (1997), where this method was developed for solving the steady-state, time-dependent, and frequency-domain versions of the diffusion approximation, respectively. It is known that the diffusion approximation is a good approach for radiative transfer in optically thick media with small absorption and relatively high scattering. The biological tissues obey these conditions in the near-infrared spectral range. Therefore, the diffusion approximation (P_{1}) is employed in many medical applications such as disease diagnostics and medical imaging using optical techniques. Although media such as skin, bone, brain matter, and breast tissue satisfy the conditions of the diffusion approximation applicability, there also exist low-scattering, almost clear regions in a human body. The diffusion approximation fails to accurately describe the radiation propagation in these regions (Arridge et al., 2000). Moreover, diffusion theory can give inaccurate descriptions of transition zones between materials with significantly different absorption and scattering properties. The alternative combined techniques to overcome these difficulties have been developed in recent papers by Arridge et al. (2000), Ripoll et al. (2000), and Aydin et al. (2002, 2004). Similar problems take place in the description of thermal radiation of disperse systems. A typical situation when the diffusion approximation error increases is illustrated in the article Radiation of isothermal volumes of scattering medium: An error of the diffusion model.

#### REFERENCES

Ackroyd, R. T., *Finite Element Method for Particle Transport: Application to Reactor and Radiation Physics*, *Research Studies in Particle and Nuclear Technology*, London: Research Studies Press, 1997.

Arridge, S. R., Schweiger, M., Hiraoka, M., and Delpy, D. T., A finite element approach for modeling photon transport in tissue, *Med. Phys*., vol. **20**, no. 2, pp. 299–309, 1993.

Arridge, S. R., Dehghani, H., Schweiger, M., and Okada, E., The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions, *Med. Phys*., vol. **27**, no. 1, pp. 252–264, 2000.

Aydin, E. D., de Oliveira, C. R. E., and Goddard, A. H., A comparison between transport and diffusion calculations using a finite element—Spherical harmonics radiation transport method, *Med. Phys*., vol. **29**, no. 9, pp. 2013–2023, 2002.

Aydin, E. D., de Oliveira, C. R. E., and Goddard, A. H., A finite element-spherical harmonics radiation transport model for photon migration in turbid media, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **84**, no. 3, pp. 247–260, 2004.

Cheng, P., Exact Solutions and Differential Approximation for Multi-Dimensional Radiative Transfer in Cartesian Coordinate Configurations, in *Progress in Astronautics and Aeronautics,* vol. **31**, *Thermal Control and Radiation*, 1979.

Chung, T. J. and Kim, J. Y., Two-dimensional, combined-modes heat transfer by conduction, convection, and radiation in emitting, absorbing, and scattering media–Solution by finite elements, *ASME J. Heat Transfer*, vol. **106**, no. 2, pp. 448–452, 1984.

Davies, R., The effect of finite geometry on the three-dimensional transfer of solar irradiance in clouds, *J. Atmos. Sci*., vol. **35**, no. 9, pp. 1712–1725, 1978.

Dombrovsky, L. A. and Barkova, L. G., Solving the two-dimensional problem of thermal-radiation transfer in an anisotropically scattering medium using the finite element method, *High Temp*., vol. **24**, no. 4, pp. 585–592, 1986.

Dombrovsky, L. A., *Radiation Heat Transfer in Disperse Systems*, New York: Begell House, 1996a.

Dombrovsky, L. A., A theoretical investigation of heat transfer by radiation under conditions of two-phase flow in a supersonic nozzle, *High Temp*., vol. **34**, no. 2, pp. 255–262, 1996b.

Dombrovsky, L. A., Kolpakov, A. V., and Surzhikov, S. T., Transport approximation in calculating the directed-radiation transfer in an anisotropically scattering erosional flare, *High Temp*., vol. **29**, no. 6, pp. 954–960, 1991.

Dombrovsky, L. A., Calculation of radiation heat transfer in a volume above the surface of a corium pool, *Therm. Eng*., vol. **48**, no. 1, pp. 42–49, 2001.

Dombrovsky, L. A., Lipiński, W., and Steinfeld, A., A diffusion-based approximate model for radiation heat transfer in a solar thermochemical reactor, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **103**, no. 3, pp. 601–610, 2007.

Fernandes, R. L. and Francis, J. E., Combined radiative and conductive heat transfer in a planar medium with flux boundary conditions using finite elements, AIAA Paper no. 0910, 1982.

Hänisch, M., Berechnung des strahlungsaustausches in umschlossenen räumen mit hilfe der Methode der finiten Elemente (MFE), *Wiss. Z. Tech. Univ. Dresden*, vol. **31**, no. 4, pp. 125–131, 1982.

Harshvardhan, Weinman, J.A., and Davies, R., Transport of infrared radiation in cuboudal clouds, *J. Atmos. Sci*., vol. **38**, no. 11, pp. 2500–2513, 1981.

Liou, K.-N. and Ou, S.-Ch. S., Infrared radiative transfer in finite cloud layers, *J. Atmos. Sci*., vol. **36**, no. 10, pp. 1985–1996, 1979.

Modest, M. F., Modified differential approximation for radiative transfer in general three-dimensional media, *J. Thermophys. Heat Transfer*, vol. **3**, no. 3, pp. 283–288, 1989.

de Oliveira, C. R. E., An arbitrary geometry finite element method for multi-group neutron transport with anisotropic scattering, *Prog. Nucl. Energy*, vol. **18**, no. 1-2, pp. 227–236, 1986.

Razzaque, M. M., Klein, D. E., and Howell, J. R., Finite element solution of radiative heat transfer in a two-dimensional rectangular enclosure with gray participating media, *ASME J. Heat Transfer*, vol. **105**, no. 4, pp. 933–936, 1983.

Razzaque, M. M., Howell, J. R., and Klein, D. E., Coupled radiative and conductive heat transfer in a two-dimensional rectangular enclosure with gray participating media using finite elements, *ASME J. Heat Transfer*, vol. **106**, no. 3, pp. 613–619, 1984.

Ripoll, J., Nieto-Vesperinas, M., Arridge, S. R., and Dehghani, H., Boundary conditions for light propagation in diffusive media with nonscattering regions, *J. Opt. Soc. Am. A*, vol. **17**, no. 9, pp. 1671–1681, 2000.

Schweiger, M., Arridge, S. R., Hiraoka, M., and Delpy, D. T., The finite element model for the propagation of light in scattering media: Boundary and source conditions, *Med. Phys*., vol. **22**, no. 11, pp. 1779–1792, 1995.

Schweiger, M. and Arridge, S. R., The finite-element model for the propagation of light in scattering media: Frequency domain case, *Med. Phys*., vol. **24**, no. 6, pp. 895–902, 1997.

Vafin, D. B., Dregalin, A. F., and Shigapov, A. B., Radiation of two-phase flows in laval nozzles, *J. Eng. Phys. Thermophys*., vol. **41**, no. 1, pp. 702–706, 1981.

Yamada, Y., Light-tissue interaction and optical imaging in biomedicine, *Annu. Rev. Heat Transfer*, vol. **6,** pp. 1–59, 1995.

Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z., *The Finite Element: Its Basis and Fundamentals*, 6th ed., Oxford: Elsevier, 2005 (see also the first edition: Zienkiewicz, O. C., *The Finite Element Method in Engineering Science*, London: McGraw-Hill, 1967).

#### References

- Ackroyd, R. T.,
*Finite Element Method for Particle Transport: Application to Reactor and Radiation Physics*,*Research Studies in Particle and Nuclear Technology*, London: Research Studies Press, 1997. - Arridge, S. R., Schweiger, M., Hiraoka, M., and Delpy, D. T., A finite element approach for modeling photon transport in tissue,
*Med. Phys*., vol.**20**, no. 2, pp. 299–309, 1993. - Arridge, S. R., Dehghani, H., Schweiger, M., and Okada, E., The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions,
*Med. Phys*., vol.**27**, no. 1, pp. 252–264, 2000. - Aydin, E. D., de Oliveira, C. R. E., and Goddard, A. H., A comparison between transport and diffusion calculations using a finite element—Spherical harmonics radiation transport method,
*Med. Phys*., vol.**29**, no. 9, pp. 2013–2023, 2002. - Aydin, E. D., de Oliveira, C. R. E., and Goddard, A. H., A finite element-spherical harmonics radiation transport model for photon migration in turbid media,
*J. Quant. Spectrosc. Radiat. Transf.*, vol.**84**, no. 3, pp. 247–260, 2004. - Cheng, P., Exact Solutions and Differential Approximation for Multi-Dimensional Radiative Transfer in Cartesian Coordinate Configurations, in
*Progress in Astronautics and Aeronautics,*vol.**31**,*Thermal Control and Radiation*, 1979. - Chung, T. J. and Kim, J. Y., Two-dimensional, combined-modes heat transfer by conduction, convection, and radiation in emitting, absorbing, and scattering media–Solution by finite elements,
*ASME J. Heat Transfer*, vol.**106**, no. 2, pp. 448–452, 1984. - Davies, R., The effect of finite geometry on the three-dimensional transfer of solar irradiance in clouds,
*J. Atmos. Sci*., vol.**35**, no. 9, pp. 1712–1725, 1978. - Dombrovsky, L. A. and Barkova, L. G., Solving the two-dimensional problem of thermal-radiation transfer in an anisotropically scattering medium using the finite element method,
*High Temp*., vol.**24**, no. 4, pp. 585–592, 1986. - Dombrovsky, L. A.,
*Radiation Heat Transfer in Disperse Systems*, New York: Begell House, 1996a. - Dombrovsky, L. A., A theoretical investigation of heat transfer by radiation under conditions of two-phase flow in a supersonic nozzle,
*High Temp*., vol.**34**, no. 2, pp. 255–262, 1996b. - Dombrovsky, L. A., Kolpakov, A. V., and Surzhikov, S. T., Transport approximation in calculating the directed-radiation transfer in an anisotropically scattering erosional flare,
*High Temp*., vol.**29**, no. 6, pp. 954–960, 1991. - Dombrovsky, L. A., Calculation of radiation heat transfer in a volume above the surface of a corium pool,
*Therm. Eng*., vol.**48**, no. 1, pp. 42–49, 2001. - Dombrovsky, L. A., Lipiński, W., and Steinfeld, A., A diffusion-based approximate model for radiation heat transfer in a solar thermochemical reactor,
*J. Quant. Spectrosc. Radiat. Transf.*, vol.**103**, no. 3, pp. 601–610, 2007. - Fernandes, R. L. and Francis, J. E., Combined radiative and conductive heat transfer in a planar medium with flux boundary conditions using finite elements, AIAA Paper no. 0910, 1982.
- Hänisch, M., Berechnung des strahlungsaustausches in umschlossenen räumen mit hilfe der Methode der finiten Elemente (MFE),
*Wiss. Z. Tech. Univ. Dresden*, vol.**31**, no. 4, pp. 125–131, 1982. - Harshvardhan, Weinman, J.A., and Davies, R., Transport of infrared radiation in cuboudal clouds,
*J. Atmos. Sci*., vol.**38**, no. 11, pp. 2500–2513, 1981. - Liou, K.-N. and Ou, S.-Ch. S., Infrared radiative transfer in finite cloud layers,
*J. Atmos. Sci*., vol.**36**, no. 10, pp. 1985–1996, 1979. - Modest, M. F., Modified differential approximation for radiative transfer in general three-dimensional media,
*J. Thermophys. Heat Transfer*, vol.**3**, no. 3, pp. 283–288, 1989. - de Oliveira, C. R. E., An arbitrary geometry finite element method for multi-group neutron transport with anisotropic scattering,
*Prog. Nucl. Energy*, vol.**18**, no. 1-2, pp. 227–236, 1986. - Razzaque, M. M., Klein, D. E., and Howell, J. R., Finite element solution of radiative heat transfer in a two-dimensional rectangular enclosure with gray participating media,
*ASME J. Heat Transfer*, vol.**105**, no. 4, pp. 933–936, 1983. - Razzaque, M. M., Howell, J. R., and Klein, D. E., Coupled radiative and conductive heat transfer in a two-dimensional rectangular enclosure with gray participating media using finite elements,
*ASME J. Heat Transfer*, vol.**106**, no. 3, pp. 613–619, 1984. - Ripoll, J., Nieto-Vesperinas, M., Arridge, S. R., and Dehghani, H., Boundary conditions for light propagation in diffusive media with nonscattering regions,
*J. Opt. Soc. Am. A*, vol.**17**, no. 9, pp. 1671–1681, 2000. - Schweiger, M., Arridge, S. R., Hiraoka, M., and Delpy, D. T., The finite element model for the propagation of light in scattering media: Boundary and source conditions,
*Med. Phys*., vol.**22**, no. 11, pp. 1779–1792, 1995. - Schweiger, M. and Arridge, S. R., The finite-element model for the propagation of light in scattering media: Frequency domain case,
*Med. Phys*., vol.**24**, no. 6, pp. 895–902, 1997. - Vafin, D. B., Dregalin, A. F., and Shigapov, A. B., Radiation of two-phase flows in laval nozzles,
*J. Eng. Phys. Thermophys*., vol.**41**, no. 1, pp. 702–706, 1981. - Yamada, Y., Light-tissue interaction and optical imaging in biomedicine,
*Annu. Rev. Heat Transfer*, vol.**6,**pp. 1–59, 1995. - Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z.,
*The Finite Element: Its Basis and Fundamentals*, 6th ed., Oxford: Elsevier, 2005 (see also the first edition: Zienkiewicz, O. C.,*The Finite Element Method in Engineering Science*, London: McGraw-Hill, 1967).

#### Following from:

Solutions for one-dimensional problemsP1 approximation of spherical harmonics method

Differential approximations

#### Leading to:

RADIATIVE TRANSFER IN MULTIDIMENSIONAL PROBLEMS: A COMBINED COMPUTATIONAL MODELRadiation of isothermal volumes of scattering medium: An error of the diffusion model

Finite-element method for radiation diffusion in nonisothermal and nonhomogeneous media

#### This article belongs to the following areas:

Differential approximations in FundamentalsD in A-Z Index

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