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This paper investigates suitable approximation for Calculating the thermal radiation flux divergence and effect of errors on performance evaluation of porous radiant burners (PRB).Thus, a single layer and a double layer of buried flame type of the porous radiant burners have been selected and numerically simulated. Due to the significant difference in the temperature of the solid matrix and the fluid passing the burner, the energy equations was considered as a non-local thermal equilibrium. Complete kinetics of methane air was used for combustion modeling. Since the effect of lateral walls should be neglected the problem was solved in 1D to present exact solution of RTE and compares the other approximations. Results show that discrete ordinate as well finite volume approximation of RTE show that eight directional spherical split is the best selection. Lower ordinates have substantial deviation and increasing the number of division enlarges computation cost without any considerable improvement on errors reductions. Furthermore, two flux method and Rosseland approximation are not valid for this kind of modeling.

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In this investigation, an exact solution is proposed for the nonlinear forced vibration analysis of nanobeams made of functionally graded materials (FGMs) in thermal environment with considering the effects of surface stress and nonlocal elasticity theory. The physical properties of FGM nanobeams are assumed to vary through the thickness direction on the basis of the power law distribution. The geometrically nonlinear equations of motion and corresponding boundary conditions are derived using Hamilton’s principle on the basis of the Euler-Bernoulli beam theory. Using the Gurtin-Murdoch and Eringen elasticity theories, the surface stress and nonlocal effects are taken into account in obtained equations, respectively. For the solution purpose, first, the Galerkin procedure is employed in order to reduce the nonlinear partial differential governing equation into a nonlinear ordinary differential equation. This new equation is solved analytically by the multiple scales perturbation method. In the results section, the influences of different parameters including power law index, surface stress, nonlocal parameter, boundary conditions and temperature changes on the nonlinear forced vibration response of nanobeams are investigated. Also, comparisons are made between the results obtained from the classical, Gurtin-Murdoch and Eringen elasticity theories. It is shown that as the thickness decreases, the surface stress effect moderates the hardening-type nonlinear behavior of nanobeams. This effect is more prominent at low magnitudes of thickness. Moreover, one can find that by increasing the nonlocal parameter, the hardening-type response of nanobeams is intensified.

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A three-dimensional (3D) Peano series solution is presented for the static analysis of functionally graded (FG) and layered magneto-electro-elastic (MEE) plates resting on elastic foundations with considering imperfect interfacial bonding. The interfacial imperfection is modeled using a generalized spring layer. Regardless of the number of layers, the equations of motion, Gauss’ equations for electrostatics and magnetostatics, the boundary and interface conditions are satisfied exactly. No assumption on deformations, stresses, magnetic and electric field along the thickness direction is introduced. The governing partial differential equations are finally solved using the state-space method. The present formulation has been validated through comparison with other similar works available in the open literature. Effects of two-parameter elastic foundation, gradient index, bonding imperfection, applied mechanical and electrical loads on the static and dynamic response of the functionally graded magneto-electro-elastic (FGMEE) plate are discussed. It is worthy to note that the present novel exact formulation includes all previous solutions, such as piezoelectric, piezomagnetic, purely elastic solution, elastic foundation and interlayer slip problems, as special cases. The obtained exact solution can be used to assess the accuracy of layered FGMEE plate theories and/or validating finite element codes.

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