## THERMAL SCIENCE

International Scientific Journal

### INVERSE BOUNDARY DESIGN PROBLEM OF COMBINED RADIATION CONVECTION HEAT TRANSFER IN A DUCT WITH DIFFUSE-SPECTRAL DESIGN SURFACE

**ABSTRACT**

In the present work, an optimization technique is applied for inverse boundary design problem of radiative convective heat transfer of laminar duct flow by numerical method. The main goal is to verify how the solution of inverse problem is affected by the spectral behavior of the boundary surfaces. The conjugate gradient method is used to find the unknown temperature distribution over the heater surface to satisfy the prescribed temperature and heat flux distributions over the design surface. The bottom boundary surface (including design surface) is diffuse-spectral, while the top wall (heater surface) behaves as gray one. The variation of emissivity with respect to the wavelength is approximated by considering a set of spectral bands with constant emissivity and then the radiative transfer equation is solved by the discrete ordinates method for each band. The performance of the present method is evaluated by comparing the results with those obtained by considering a diffuse-gray design surface. Finally an attempt is made to investigate the spectral behavior of the design surface on the calculated temperature distribution over the heater surface.

**KEYWORDS**

PAPER SUBMITTED: 2016-10-11

PAPER REVISED: 2017-04-01

PAPER ACCEPTED: 2017-04-07

PUBLISHED ONLINE: 2017-05-06

**THERMAL SCIENCE** YEAR

**2019**, VOLUME

**23**, ISSUE

**Issue 1**, PAGES [319 - 330]

- Rousse, D. R., Numerical predictions of two-dimensional conduction, convection, and radiation heat transfer— I. Formulation, International Journal of Thermal Sciences, 39 (2000), pp. 315-331
- Chandrasekhar, S., Radiative transfer, Clarendon Press, Oxford, 1950
- Carlson, B. G., Lathrop, K. D., Transport theory —the method of discrete ordinates in: Computing methods of reactor physics, Gordon & Breach, New York, 1968
- Fiveland, W.A., Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures, Journal of Heat Transfer, 106 (1984), 4, pp.699-706
- Truelove, J. S., Three-dimensional radiation in absorbing-emitting-scattering in using the discrete ordinates approximation, Journal of Quantitative Spectroscopy &Radiative Transfer, 39 (1998), pp. 27-31
- Razzaque, M. M., Howell, J. R., Klein, D. E., Coupled radiative and conductive heat transfer in a two dimensional rectangular with gray participating media using finite elements, Journal of Heat Transfer, 106 ( 1984), pp. 613-619
- Tan, Z., Combined radiative and conductive transfer in two-dimensional emitting, absorbing, and anisotropic scattering square media, International Communications in Heat and Mass Transfer, 16 (1989), pp. 391-401
- Kim, T. Y., Baek, S. W., Analysis of combined conductive and radiative heat transfer in a two dimensional rectangular enclosure using the discrete ordinates method, International Journal of Heat and Mass Transfer, 34 (1991), pp. 2265-2273
- Rousse, D. R., Gautier, G., Sacadura, J. F., Numerical predictions of two-dimensional conduction, convection and radiation heat transfer - II. Validation, International Journal of Thermal Sciences,39 ( 2000), pp. 332-353
- Talukdar, P., Mishra, S. C., Transient conduction and radiation heat transfer with heat generation in a participating medium using the collapsed dimension method, Numerical Heat Transfer—Part A, 39 (2001), pp. 79-100
- Mahapatra, S. K., Nanda, P., Sarkar, A., Analysis of coupled conduction and radiation heat transfer in presence of participating medium using a hybrid method, Heat Mass Transfer, 41 (2005), pp. 890-898
- Amiri, H., Mansouri, S. H., Safavinejad, A., Combined conductive and radiative heat transfer in an anisotropic scattering participating medium with irregular geometries, International Journal of Thermal Sciences, 49(2010), pp. 492-503
- Harutunian, V., Morales, J. C., Howell, J. R., Radiation exchange within an enclosure of diffuse-gray surfaces: the inverse problem (No. CONF-950828), American Society of Mechanical Engineers, New York, NY (United States)
- Howell, J., Ezekoye, O., Morales, J., Inverse design model for radiative heat transfer, Journal of heat transfer, 122 (2000), 3, pp. 492-502
- Morales, J. C., Harutunian, V., Oguma, M., Howell, J. R., Inverse design of radiating enclosures with an isothermal participating medium, ICHMT Digital Library Online: Begel House Inc., (1995), www.dl.begellhouse.com/references/1bb331655c289a0a,63045c9947e5cdf6,6d79ace923db0293.html
- Franca, F., Morales, J.C., Oguma, M., Howell, J.R., Inverse radiation heat transfer within enclosures with non isothermal participating media, In Heat Transfer Conference, (1998), Vol. 7, pp. 433-438
- Franca, F. H., Ezekoye, O. A., Howell, J. R., Inverse boundary design combining radiation and convection heat transfer, ASME Journal of heat transfer, 123 (2001), 5, pp. 884-891
- Fedorov, A. G., Lee, K. H., Viskanta, R., Inverse optimal design of the radiant heating in materials processing and manufacturing, Journal of Materials Engineering and Performance, 7 (1998), 6, pp. 719-726
- Daun, K. J., Howell, J. R., Morton, D. P., Design of radiant enclosures using inverse and non-linear programming techniques, Inverse Problems in Engng., 11 (2003), 6, pp. 541-560
- Hosseini Sarvari, S. M., Mansouri, S. H., Howell, J. R., Inverse boundary design radiation problem in absorbing-emitting media with irregular geometry, Numerical Heat Transfer: Part A: Applications, 43 (2003), 6, pp. 565-584
- Sarvari, S. H., Mansouri, S. H., Howell, J. R., Inverse Design of Three Dimensional Enclosures with Transparent and Absorbing-Emitting Medial using an Optimization Technique, International communications in heat and mass transfer, 30 (2003), 2, pp. 149-162
- Hosseini Sarvari, S. M., Howell, J. R., Mansouri, S. H., Inverse boundary conduction-radiation problem in irregular two-dimensional domains, Numerical Heat Transfer: Part B: Fundamentals, 44 (2003), 3, pp.209-224
- Sarvari, S. H., Mansouri, S. H., Inverse design for radiative heat source in two-dimensional participating media, Numerical Heat Transfer, Part B, 46 (2004), 3, pp. 283-300
- Sarvari, S. H., Inverse determination of heat source distribution in conductive-radiative media with irregular geometry, Journal of Quantitative Spectroscopy and Radiative Transfer, 93 (2005), 1, pp. 383-395
- Bayat, N., Mehraban, S., Sarvari, S. H., Inverse boundary design of a radiant furnace with diffuse-spectral design surface, International Communications in Heat and Mass Transfer, 37 (2010), 1, pp. 103-110
- Modest, M. F., Radiative heat transfer, Oxford: Academic press, (2013), pp. 498-529
- Siegel, R., Howell, J.R., Thermal Radiation Heat Transfer, Published by Taylor & Francis, New York, USA, Fourth Edition, (2002),pp. 44-46
- Atashafrooz, M. and Gandjalikhan Nassab, S. A., Numerical analysis of laminar forced convection recess flow with two inclined steps considering gas radiation effect, Computer & Fluids, 66(2012), pp. 167-176
- Ozisik, M. N., Orlande, H. R. B., Inverse heat transfer, Published by Taylor & Francis, New York, USA, (2000), pp. 58-66
- Bahraini, S., Gandjalikhan Nassab, S.A., Sarvari, S.M.H., Inverse Convection-Radiation Boundary Design Problem of Recess Flow with Different Conditions, Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 40 (2016),3, pp.215-222.
- Toulukain, Y. S. and DeWitt, P. D., Thermophysical Properties of Matter - The TPRC Data Series. Volume 7. Thermal Radiative Properties - Metallic Elements and Alloys,(1970)