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In the current study, a numerical investigation of three-dimensional combined convection-radiation heat transfer over an inclined forward facing step (FFS) in a horizontal rectangular duct is presented. The fluid is treated as a gray, absorbing, emitting and scattering medium. To simulate the incline surface of FFS, the blocked-off method is employed in this study. The set of governing equations for gas flow are solved numerically using the CFD technique to obtain the temperature and velocity fields. Since the gas is considered as a radiating medium, all of the convection, conduction and radiation heat transfer mechanisms are presented in the energy equation. For computation of radiative term in energy equation, the radiative transfer equation (RTE) is solved numerically by the discrete ordinates method (DOM) to find the divergence of radiative heat flux distribution inside the radiating medium. The effects of optical thickness, radiation-conduction parameter and albedo coefficient on heat transfer behavior of the system are carried out.

Forced convection flow over a backward or forward facing step in a three-dimensional channel is widely encountered in engineering applications. These geometries are important in industrial studies. In these types of convection flow, separating and reattaching regions exist because of the sudden changes in flow geometry. Separation flows accompanied with heat transfer are frequently encountered in many systems, such as cooling of electronic systems, power generating equipments, gas turbine blades, heat exchangers, combustion chamber and ducts flows used in industrial applications. In some of the mentioned advices, specially, when soot particles exist in the combustion product, the radiation effect may be important. Besides, the trend toward increasing temperature in modern technological systems has promoted concerted effort to develop more comprehensive and accurate theoretical methods to treat radiation. Therefore, for having more accurate and reliable results in the analysis of these types of flow, the flowing gas must be considered as a radiating medium and all of the heat transfer mechanisms including convection, conduction and radiation, must be taken into account. The flow over backward facing step (BFS) or forward facing step (FFS) has the most features of separated flows. Although the geometry of BFS or FFS flow is very simple, the heat transfer and fluid flow over these types of step contain most of complexities. Consequently, it has been used in the benchmark investigations. There are many studies about laminar convection flow over BFS in a 2-D duct by several investigators [

There are many engineering applications, in which the forward or backward-facing step is inclined. Simulations of three-dimensional laminar forced convection adjacent to inclined backward-facing step in rectangular duct were presented by Chen et al. [

Bouali and Mezrhab [

Simulations of laminar forced convection flow of a radiating gas adjacent to backward and forward facing steps in two-dimensional ducts under different conditions were presented in several studies [

The study of mixed convection heat transfer in 3-D horizontal and inclined ducts with considering radiation effects has been numerically examined in detail by Chiu et al. [

Although there are limited studies about laminar forced convection flow of radiating gas over a BFS and FFS, but base on the author’s knowledge, 3-D combined convection and radiation heat transfer over an inclined forward-facing step in a duct, is still not studied by DOM and block-off method. Therefore, the present research work deals the three-dimensional simulations of incompressible laminar forced convection flow of a radiating gas over an inclined forward facing step in rectangular duct, while the well known DOM and block-off method are employed to solve the radiation problem.

Three-dimensional combined convection and radiation heat transfer in a horizontal heated rectangular duct with an inclined forward facing step is numerically simulated. Schematic of the computational domain is shown in _{in}. All of the duct’s walls are kept at a constant temperature _{1} = 5H and the rest of the channel length is equal to L_{2} = 15H. This is made to ensure that the flow at the inlet and outlet sections is not affected significantly by the sudden changes in the geometry and flow at exit section becomes fully developed. Also, the step inclination angle indicated by

For incompressible, steady and three-dimensional laminar flow, the governing equations are the conservations of mass, momentum and energy that can be written as follows:

In the above equations, u, v and w are the velocity components in x-, y- and z-directions, respectively; _{p} the specific heat;

The boundary conditions are treated as no slip condition at the solid walls (zero velocity) and constant temperature of at all boundary surfaces. At the inlet duct section, the flow has uniform velocity of with uniform temperature of T_{in}, which is assumed to be lower than. At the outlet section, zero axial gradients for velocity components and gas temperature are employed.

In the energy equation, besides the convective and conductive terms, the radiative term as the divergence of the radiative heat flux, i.e.

In the above equation,

tion of

where

in which

boundary surface and

In the DOM, Equation (7) is solved for a set of n different directions,

where

subjected to the boundary conditions:

Also

At any arbitrary surface, heat flux may also be determined from the surface energy balance as:

In 3-D Cartesian coordinate system, Equation (10) becomes as follows [

where

In fact

in which

The details of the numerical solution of RTE by DOM were also described in the previous work by the second author, in which the thermal characteristics of porous radiant burners were investigated [

For the radiative boundary conditions, the walls are assumed to emit and reflect diffusely with constant wall emissivity,

In numerical solution of the set of governing equations including the continuity, momentum and energy, the following dimensionless parameters are used to obtain the non dimensional forms of these equations:

The non-dimensional forms of the governing equations are as follows:

The main physical quantities of interest in heat transfer study are the mean bulk temperature and Nusselt number. The mean bulk temperature along the channel was calculated using the following equation:

In the combined convection-radiation heat transfer, the energy transport from the duct wall to the gas flow depends on two related factors:

1. Fluid temperature gradient on the wall;

2. Rate of radiative heat exchange on boundary surface.

Therefore, total heat flux on the wall is the sum of convective and radiative heat fluxes such that

Therefore, the function of total Nusselt number

Total Nusselt numbers is given as follows [

where

Equation (24) contains two parts. The first term on the right hand side represents the convective Nusselt number, whereas the second term is the radiative Nusselt number. It should be considered that for pure convective hat transfer, total Nusselt number is equal to the convective one.

Finite difference forms of the continuity, momentum and energy equations ((18) to (22)) were obtained by integrating over an elemental cell volume with staggered control volumes for the x-, y- and z- velocity components. Other variables of interest were computed at the grid nodes.

The discretized forms of the governing equations were numerically solved by the SIMPLE algorithm of Patankar and Spalding [_{4} approximation has been used in this study.

Grid size | Value of the maximum convective Nusselt number | Value of the maximum radiative Nusselt number |
---|---|---|

100 × 15 × 15 | 22.351 | - |

160 × 20 × 20 | 26.354 | 16.738 |

240 × 25 × 25 | 29.172 | 18.957 |

300 × 30 × 30 | 31.259 | 20.554 |

360 × 36 × 36 | 32.183 | 21.275 |

420 × 40 × 40 | 32.301 | 21.352 |

In many cases, a computer program written for a regular grid can be improved to handle an irregularly shaped computational domain using the blocked-off method [

According to the blocked-off technique, known values of the dependent variables must be established in all inactive control volumes. If the inactive region represents a stationary solid boundary as in the case, the velocity components in that region must be equal to zero, and if the region is regarded as isothermal boundary, the known temperature must be established in the inactive control volumes.

As it was mentioned before, there are limited theoretical and experimental research works about heat transfer in laminar forced convection flow over FFS. First, the present numerical implementation was validated by reproducing the results of Iwai et al. [

To validate the present calculations in solving the RTE by DOM using the block-off method, there is not any theoretical result in combined convection and radiation heat transfer over an inclined forward facing step (FFS)

in a three-dimensional duct. But there is a research work about two-dimensional laminar forced convection of radiating gas flow over an inclined FFS by Ansari and Gandjalikhan Nassab [

Therefore, second validation is done based on the results of Ref. [

In this study, numerical results are presented for combined convection and radiation heat transfer over an inclined FFS in a three -dimensional duct with a contraction ratio of CR = 0.5 and aspect ratio of AR = 4 at different conditions. The fixed inclination angle of

In order to illustrate the radiation effect in thermal behavior of a convection flow, contours of total Nusselt number on the bottom wall with and without considering the radiation term in energy equation are presented in

The effect of radiation in Nu distribution can be found if one compares

the distribution of total Nusselt numbers on the bottom wall in the mid-plane of the duct

also from

In the convection flow of a radiating fluid, the optical thickness (t), the radiation-conduction parameter (RC) and the albedo coefficient (ω) are the main parameters that affect the thermal behavior of the radiation-convec- tion system. In the next sections, an attempt is made to study the effects of these parameters on thermal behavior of the thermal system.

A well-known radiation property and one of the important parameter in participating medium is the optical thickness that affects the temperature distribution inside the participating medium. High optical thickness means that the medium has great ability to absorb and emit radiant energy. For the convective flow with radiating heat transfer in the channel including an inclined 3-D forward facing step, as shown in

It is seen that optical thickness has a greater influence on the radiative Nusselt number than the convective

Nusselt number. For more study about the effect of optical thickness on the thermal behavior of the convection-radiation system, the counters of total Nusselt number on the bottom wall for two different values of the optical thickness are plotted in

In order to have another form of figures about the influence of optical thickness on the Nusselt number in convection-radiation system, the distributions of convective, radiative and total Nusselt numbers along the centerline on the bottom wall are plotted in Figures 10(a)-(c) for different value of the optical thickness. These figures show the same trend for variations of all types of Nusselt number as it was seen before in Figures 7-9.

Radiation-conduction parameter (RC) is another one of the main parameters in the combined radiation-conduc- tion systems, which show the relative importance of the radiation mechanism compared with its conduction counterpart. High value of RC parameter shows the radiation dominance in a thermal system. The effect of RC parameter on the contours of total Nusselt number is showed in

This figure illustrates that the total Nusselt number increases by increasing in RC parameter. This is due to this fact that under the effective presence of radiation mechanism at high values of RC, the radiative Nusselt number gets increase that consequently leads to an increase in total Nusselt. Distribution of total Nusselt number along the bottom wall on the centerline is represented in

To study the effect of radiation-conduction parameter (RC) on the mean bulk temperature distribution along the duct

The scattering albedo is an important parameter in radiating systems that can show the ability of participating medium to scatter thermal radiation. As it was mentioned before, scattering albedo, ω, is defined as

The extreme values of scattering albedo, i.e., ω = 1.0 and ω = 0.0 correspond to pure scattering and non-scat- tering cases, respectively. Therefore, the medium changes from pure absorption to pure scattering by increasing ω from 0 to 1.

The effect of scattering albedo coefficient on the mean bulk temperature is presented in

the case of no-radiation problems, the convective system has the same trend and behavior as pure scattering case with ω = 1.0.

Simulation of combined convection and radiation heat transfer over an inclined forward facing step (FFS) in a rectangular duct with constant step inclination angle in a 3-D horizontal duct is studied in this research work. The set of governing equations consisting of the continuity, momentum and energy is solved numerically by the CFD techniques in the Cartesian coordinate system. The blocked-off method is used in this study to simulate the incline surface of FFS. For calculating the radiative term in the energy equation, the RTE is solved by the DOM to obtain the distribution of radiant intensity inside the radiating medium. The effects of optical thickness, the radiation-conduction parameter and albedo coefficient on thermal behavior of the convective system are thoroughly explored by plotting the variations of Nusselt number (total, radiative and convective), and mean bulk temperature under different conditions.

A. Dehghani Rayeni,S. A. Gandjalikhan Nassab, (2016) Analysis of Combined Radiation and Forced Convection Heat Transfer in 3D Laminar Flow over an Inclined Forward Facing Step. Journal of Electronics Cooling and Thermal Control,06,1-18. doi: 10.4236/jectc.2016.61001