Effect of Thermal Radiation on the Conjugate Heat Transfer from a Circular Cylinder with an Internal Heat Source in Laminar Flow

The effect of thermal radiation on the two – dimensional, steady-state, conjugate heat transfer from a circular cylinder with an internal heat source in steady laminar crossflow is investigated in this work. P0 (Rosseland) and P1 approximations were used to model the radiative transfer. The mathematical model equations were solved numerically. Qualitatively, P0 and P1 approximations show the same effect of thermal radiation on conjugate heat transfer; the increase in the radiation – conduction parameter decreases the cylinder surface temperature and increases the heat transfer rate. Quantitatively, there are significant differences between the results provided by the two approximations.


Introduction
All materials with the temperature in the range of 30 to 30,000 K emit and absorb thermal radiation. The emission of thermal radiation is due to the conversion of the internal energy into energy transported by electromagnetic waves or photons. For heat transfer applications wavelengths between 10 -7 m and 10 -3 m are important.
The radiative transfer equation (RTE) is an integro-differential equation very difficult to solve. Exact analytical solutions exist only for simple situations such as onedimensional plane parallel media without scattering. Therefore, approximate mathematically less complicated but accurate models for the RTE have been developed. Examples are zeroth order diffusion or Rosseland approximation [1], high order diffusion approximations like PN [2] and SPN [3], and the moments method [4], [5] (and the references quoted herein). The numerical methods proven to be effective for solving RTE are the zonal method [1], the discrete ordinates method [6] (and the references quoted herein), the finite volume method [6] (and the references quoted herein) and the finite element method [7,8]. The Monte Carlo [9] and the lattice Boltzmann [10] methods were also used to solve RTE. An extensive presentation of these approximate models is outside the aims of the present work. Reviews can be viewed in [1] and [11].
The RTE solving necessitates the knowledge of the temperature profiles. In almost all the articles that investigate the RTE solving, the energy balance equation considered is the one phase, transient heat conduction equation. The influence of thermal radiation on more complex heat transfer problems was investigated by (the citation is restricted to the cases of forced / mixed convectionradiation heat transfer in external flows) Hossain and Takhar [12], Andrienko et al.
The effect of thermal radiation on conjugate, forced convection heat transfer was analysed only for an internal flow problem by Nia and Nassab [21,22]. The aim of the present work is to investigate the effect of thermal radiation on the conjugate, forced convection heat transfer for the external flow case. To the best of our knowledge, this problem is reported for the first time here.
The test problem models the steadystate conjugate heat transfer from a circular cylinder with an internal heat source in steady laminar crossflow. The P0 (Rosseland) and P1 approximations were used to model the radiative transfer.
This paper is organized as follows. In Sect.
where qr,r and qr,θ are the dimensionless normal and tangential components of the radiative heat flux vector.
The boundary conditions to be satisfied by the dimensionless temperature are: -Symmetry axis (θ = 0, π);

Rosseland approximation
The radial and tangential components of the dimensionless radiative heat flux vector given by Rosseland approximation [1] read as: where Rd0 is the Rosseland radiationconduction parameter, 0 = 4 , equation (2) can be rewritten as

P1 approximation
For P1 approximation, the dimensionless radiative heat flux vector satisfies the equation, [1], where G is the dimensionless directedintegrated intensity of the radiation, . Substituting equation (6) into equation (2), it results: Note that some elementary algebraic manipulations were made in order to obtain for Rd1 an expression similar to that for Rd0. The dimensionless directedintegrated intensity of radiation G verifies the equation [1]: The boundary conditions to be satisfied by G are, [1]: -Symmetry axis (θ = 0, π); where ℰ = 2 ( 2− ) . Two boundary conditions were proposed and tested for G at free stream. The boundary condition (9c) considers the free stream as an inflow / outflow boundary with null intensity of radiation. The boundary condition (9d) assumes radiative equilibrium at free stream.
It must be mentioned that for the P1 approximation, the dimensionless radiative heat flux is given by [1], The physical quantities of interest are the cylinder surface dimensionless average temperature ̄, , the local Nusselt number, Nu (θ), and the average Nusselt number, Nu.
Considering as driving force the difference (T0 -T∞), the local Nusselt number based on the diameter of the cylinder is given by (for Φ ≥ 1): The average Nu number is given by the relation: The cylinder surface dimensionless average temperature ̄, , was computed with the relations:

Method of solution
The energy balance equations (1,2) belong to the class called interface problem, [23]. The spatial derivatives (equations (1b, 2) were rewritten as a single equation with discontinuous coefficients) were discretized with the upwind and centered finite difference schemes (a double discretization required by the defectcorrection iteration) on a vertex-centered grid, [23]. The spatial derivatives of the radiative transfer equation (8)

Results and discussions
The dimensionless groups of the present problem can be divided into the following two classes: (1) conjugate convectiondiffusion heat transfer dimensionless groups, Pr, Re, Φ and ζ; (2) radiative dimensionless groups, , ℰ, and Rd0(1).
The The quantities used to quantify the influence of the thermal radiation on the conjugate heat transfer are the ratios: In the next paragraphs, the ratios ηS and ηN will be called surface ratio and flux ratio, respectively.
The effect of thermal radiation on the conjugate heat transfer is considered significant when The first aspect analysed is the influence of the boundary conditions (9c) and (9d) on the numerical solution of the P1 model. The P0 approximation reduces the radiationconvectionconduction problem to a standard convectionconduction problem with strongly temperature dependent thermal conductivity. The increase in the thermal conductivity of the fluid decreases the temperature gradient at the interface but amplifies the heat flux. The global result is the enhancement of the heat transfer rate even for small values of Rd0. It must be also mentioned that, for the Rosseland approximation, the same results were obtained neglecting the radiation transfer in the tangential direction.
In a first approximation, one can consider the P1 model similar to the model of mass transfer accompanied by a reversible chemical reaction with an unusual reaction rate and equilibrium constant equal to unity (see for example [26]). The dimensionless temperature is the reactant of the reversible chemical reaction while the dimensionless directedintegrated intensity of the radiation is the product of the reversible chemical reaction. However, there are differences between the present mathematical model and the mathematical model for the mass transfer accompanied by a reversible chemical reaction. In the case of the mass transfer accompanied by a reversible chemical reaction all the species involved in process obey the same mass transfer mechanism, convectiondiffusionreaction. For the present mathematical model, equation (7) is a convectiondiffusionreaction equation while equation (8)  The effect of the order of approximation of spherical harmonics model on the solution of the present problem is the last issue discussed in this section. The results presented in [1] and [27] for a thick medium show that the differences between the P1 approximation, high order spherical harmonics approximations and the solution of the full radiative transfer equation are small.
Significant differences exist between the P0 approximation, high order spherical harmonics approximations and the solution of the full radiative transfer equation. Thus, one can consider the P1 approximation used in this work an efficient and sufficiently accurate solution for the present radiative heat transfer problem (the optical thickness for the present medium is very large).

Conclusions
The effect of thermal radiation on the steady-state, conjugate heat transfer from a circular cylinder with an internal heat source in steady laminar crossflow was analysed in this work. The