## INVERSE DESIGN OF ENCLOSURES WITH PARTICIPATING MEDIA AND MULTIMODE HEAT TRANSFER

*
Following from:
*
Ill posedness of inverse problems

*
Leading to:
*
Numerical methods for inverse radiation problems

Recently, inverse design problems have been attracting interest in the field of radiation heat transfer in participating media and multimode heat transfer including radiation. Inverse design problems are classified as (i) inverse boundary design problems and (ii) inverse heat source design problems (Fig. 1). In the former, the objective is to find a set of heaters over some part of the boundary, namely, the “heater surface,” in such a way that the desired temperature and heat flux distributions are recovered over other part of the boundary surface, namely, the “design surface” (Fig. 1a).

**Figure 1. Geometry of a radiant enclosure filled with participating media: (a) inverse boundary design problem; (b) inverse source design problem **

Regularization techniques were widely used for inverse boundary design of enclosures with participating media. Morales et al. (1996) first used the modified truncated singular value decomposition (MTSVD) regularization technique for inverse boundary design of radiating enclosures with an isothermal participating medium. Inverse radiation boundary design within enclosures with nonisothermal participating media based on the truncated singular value decomposition (TSVD) regularization approach was investigated by Franca et al. (1998a). The procedure was extended by Franca et al. (1998b) for inverse boundary design of radiant enclosures with nongray participating media. The TSVD regularization technique was applied by Franca et al. (2007) for inverse boundary design combining radiation and convection heat transfer through a channel with an absorbing-emitting medium. The application of variational methods of regularization to boundary design of radiant enclosures with absorbing-emitting media was investigated by Rukolaine (2007). Recently, Mossi et al. (2008) solved an inverse boundary design problem involving combined radiative and turbulent convective heat transfer in a rectangular cavity.

The application of optimization techniques for inverse boundary design of enclosures containing participating media were investigated by many researchers. Sarvari et al. (2003a) first used the conjugate gradient method (CGM) for boundary design of 2D radiant enclosures containing absorbing-emitting media with arbitrary geometries. The procedure was extended by Sarvari et al. (2003b) to solve boundary design problems involving combined conduction and radiation heat transfer. The application of the Levenberg-Marquart optimization technique for inverse boundary design of 3D radiant enclosures with absorbing-emitting media was investigated by Sarvari et al. (2003c). Pourshaghaghy et al. (2006) used the CGM for inverse boundary design of enclosures filled with absorbing-emitting-scattering media. The regularized variable metric method (VMM) and the CGM in solution of radiative boundary design problem were compared by Kowsary et al. (2007). Kim and Baek (2006, 2007) used the Levenberg-Marquart method for solving the inverse radiation and radiation-conduction design problem in a participating concentric medium. The second type of design problems consists of estimating a suitable heat source distribution throughout some part of the medium, namely, the “HS region,” to achieve a uniform distribution of temperature and heat flux over the “design surface” (Fig. 1b).

Franca et al. (1999a) used the TSVD regularization method for determination of heat source distribution in radiative systems with participating media. They extended the approach to solve the inverse heat source design problem combining radiation and conduction heat transfer (Franca et al., 1999b). The application of CGM for determination of heat source distribution in participating media was investigated by Sarvari and Mansouri (2004). Sarvari (2005) extended the application of the optimization procedure based on the CGM to solve the inverse heat source design problem in conductive-radiative media.

#### REFERENCES

Franca, F., Morales, J. C., Oguma, M., and Howell, J., Inverse radiation heat transfer within enclosures with nonisothermal participating media, *Proc. of 11th Int. Heat Transfer Conference*, Korea, vol. **1**, pp. 433–438, 1998a.

Franca, F., Oguma, M., and Howell, J. R., Inverse radiation heat transfer within enclosures with non-isothermal, non-gray participating media, *Proc. of ASME 1998 Int. Mechanical Engineering Congress and Exposition*, Anaheim, vol. **5**, pp. 145–151, 1998b.

Franca, F., Ezekoye, O. A., and Howell, J. R., Inverse determination of heat source distribution in radiative systems with participating media, *Proc. of 33rd National Heat Transfer Conference*, Albuquerque, pp. 1–8, 1999a.

Franca, F., Ezekoye, O. A., Howell, J. R., Inverse heat source design combining radiation and conduction heat transfer, *Proc. of ASME–IMECE*, Nashville, pp. 45–52, 1999b.

Franca, F., Ezekoye, O. A., Howell, J. R., Inverse boundary design combining radiation and convection heat transfer, *J. Heat Transfer*, vol. **123**, pp. 884–891, 2007.

Kim, K. W. and Baek, S. W., Inverse radiation design problem in a two-dimensional radiatively active cylindrical medium using automatic differentiation and Broyden combined update, *Numer. Heat Transfer, Part A*, vol. **50**, no, 6, pp. 525–543, 2006.

Kim, K. W. and Baek, S. W., Inverse radiation-conduction design problem in a participating concentric cylindrical medium, *Int. J. Heat Mass Transfer*, vol. **50**, pp. 2828–2837, 2007.

Kowsary, F., Pooladvand, K., and Pourshaghaghy, A., Regularized variable metric method versus the conjugate gradient method in solution of radiative boundary design problem, *J. Quant. Spectrosc. Radiat. Transfe*, vol. **108**, pp. 277–294, 2007.

Morales, J. C., Harutunian, V., Oguma, M., and Howell, J. R., Inverse design of radiating enclosures with an isothermal participating medium, *Radiative Transfer I: Proc. of First Int. Symp. on Radiative Heat Transfer*, M. Pinar Mengüç (ed.), Begell House, New York, and Redding, CT, pp. 579–593, 1996.

Mossi, A. C., Vielmo, H. A., França, F. H. R., and Howell, J. R., Inverse design involving combined radiative and turbulent convective heat transfer, *Int. J. Heat Mass Transfer*, vol. **51**, pp. 3217–3226, 2008.

Pourshaghaghy, A., Pooladvand, K., Kowsary, F., and Karimi-Zand, K., An inverse radiation boundary design problem for an enclosure filled with an emitting, absorbing, and scattering media, *Int. Commun. Heat Mass Transfer*, vol. **33**, pp. 381–390, 2006.

Rukolaine, S. A., Regularization of inverse boundary design radiative heat transfer problems, *J. Quantitative Spectroscopy and Radiative Transfer*, vol. **104**, pp. 171–195, 2007.

Sarvari, S. M. H., Inverse determination of heat source distribution in conductive-radiative media with irregular geometry, *J. Quantitat. Spectrosc. Radiat. Transfer*, vol. **93**, pp. 383–395, 2005.

Sarvari, S. M. H., Howell, J. R., and Mansouri, S. H., Inverse boundary design conduction-radiation problem in irregular two-dimensional domains, *Numer. Heat Transfer, Part B*, vol. **44**, pp. 209–224, 2003b.

Sarvari, S. M. H., Mansouri, S. H., and Howell, J. R., Inverse boundary design radiation problem in absorbing-emitting media with irregular geometry, *Numer. Heat Transfer, Part A*, vol. **43**, pp. 565–584 2003a.

Sarvari, S. M. H., Mansouri, S. H., and Howell, J. R., Inverse design of three-dimensional enclosures with transparent and absorbing-emitting media using an optimization technique, *Int. Commun. Heat Mass Transfer*, **vol. 30**, pp. 149–162, 2003c.

Sarvari, S. M. H., Mansouri, S. H., Inverse design for radiative heat source in an irregular 2-D participating media, *Numer. Heat Transfer, Part B*, vol. **46**, pp. 283–300, 2004.

#### References

- Franca, F., Morales, J. C., Oguma, M., and Howell, J., Inverse radiation heat transfer within enclosures with nonisothermal participating media,
*Proc. of 11th Int. Heat Transfer Conference*, Korea, vol.**1**, pp. 433â€“438, 1998a. - Franca, F., Oguma, M., and Howell, J. R., Inverse radiation heat transfer within enclosures with non-isothermal, non-gray participating media,
*Proc. of ASME 1998 Int. Mechanical Engineering Congress and Exposition*, Anaheim, vol.**5**, pp. 145â€“151, 1998b. - Franca, F., Ezekoye, O. A., and Howell, J. R., Inverse determination of heat source distribution in radiative systems with participating media,
*Proc. of 33rd National Heat Transfer Conference*, Albuquerque, pp. 1â€“8, 1999a. - Franca, F., Ezekoye, O. A., Howell, J. R., Inverse heat source design combining radiation and conduction heat transfer,
*Proc. of ASMEâ€“IMECE*, Nashville, pp. 45â€“52, 1999b. - Franca, F., Ezekoye, O. A., Howell, J. R., Inverse boundary design combining radiation and convection heat transfer,
*J. Heat Transfer*, vol.**123**, pp. 884â€“891, 2007. - Kim, K. W. and Baek, S. W., Inverse radiation design problem in a two-dimensional radiatively active cylindrical medium using automatic differentiation and Broyden combined update,
*Numer. Heat Transfer, Part A*, vol.**50**, no, 6, pp. 525â€“543, 2006. - Kim, K. W. and Baek, S. W., Inverse radiation-conduction design problem in a participating concentric cylindrical medium,
*Int. J. Heat Mass Transfer*, vol.**50**, pp. 2828â€“2837, 2007. - Kowsary, F., Pooladvand, K., and Pourshaghaghy, A., Regularized variable metric method versus the conjugate gradient method in solution of radiative boundary design problem,
*J. Quant. Spectrosc. Radiat. Transfe*, vol.**108**, pp. 277â€“294, 2007. - Morales, J. C., Harutunian, V., Oguma, M., and Howell, J. R., Inverse design of radiating enclosures with an isothermal participating medium,
*Radiative Transfer I: Proc. of First Int. Symp. on Radiative Heat Transfer*, M. Pinar MengÃ¼Ã§ (ed.), Begell House, New York, and Redding, CT, pp. 579â€“593, 1996. - Mossi, A. C., Vielmo, H. A., FranÃ§a, F. H. R., and Howell, J. R., Inverse design involving combined radiative and turbulent convective heat transfer,
*Int. J. Heat Mass Transfer*, vol.**51**, pp. 3217â€“3226, 2008. - Pourshaghaghy, A., Pooladvand, K., Kowsary, F., and Karimi-Zand, K., An inverse radiation boundary design problem for an enclosure filled with an emitting, absorbing, and scattering media,
*Int. Commun. Heat Mass Transfer*, vol.**33**, pp. 381â€“390, 2006. - Rukolaine, S. A., Regularization of inverse boundary design radiative heat transfer problems,
*J. Quantitative Spectroscopy and Radiative Transfer*, vol.**104**, pp. 171â€“195, 2007. - Sarvari, S. M. H., Inverse determination of heat source distribution in conductive-radiative media with irregular geometry,
*J. Quantitat. Spectrosc. Radiat. Transfer*, vol.**93**, pp. 383â€“395, 2005. - Sarvari, S. M. H., Howell, J. R., and Mansouri, S. H., Inverse boundary design conduction-radiation problem in irregular two-dimensional domains,
*Numer. Heat Transfer, Part B*, vol.**44**, pp. 209â€“224, 2003b. - Sarvari, S. M. H., Mansouri, S. H., and Howell, J. R., Inverse boundary design radiation problem in absorbing-emitting media with irregular geometry,
*Numer. Heat Transfer, Part A*, vol.**43**, pp. 565â€“584 2003a. - Sarvari, S. M. H., Mansouri, S. H., and Howell, J. R., Inverse design of three-dimensional enclosures with transparent and absorbing-emitting media using an optimization technique,
*Int. Commun. Heat Mass Transfer*,**vol. 30**, pp. 149â€“162, 2003c. - Sarvari, S. M. H., Mansouri, S. H., Inverse design for radiative heat source in an irregular 2-D participating media,
*Numer. Heat Transfer, Part B*, vol.**46**, pp. 283â€“300, 2004.