**5. Conclusions**

154 Computational Simulations and Applications

particles. In fact, the numerically obtained profiles of particle mass concentration are highly appreciated because of the efficient operation of CFB units. On the contrary, in the flow domain the gradient profiles of mass concentration can cause retard of enhancement of the combustion process. Thus, an additional sand mass fraction brought to the flow domain

> **0 0.25 0.5 0.75 1 Radial distance**

Fig. 6. Distribution of particle mass concentration for ash and sand solid phases in different

Finally, the Fig. 7 shows distribution of turbulent the energy across the flow. All considered results for given three different regimes: mixture1, mixture2 and mixture3 are matched between each other and versus also turbulent energy of single phase flow. As a whole, the trend shows that the particles in all the observed regimes generate the turbulence, which stems from the vortex shedding phenomenon behind the particles which is input to the level of turbulence generated by the flow itself. This effect of turbulence modulation, namely, the turbulence enhancement due to the presence of coarse particles is explained and computed using the four-way coupling model by Crowe (2000). This amount of an additional turbulent energy is proportional to the square of velocity slip between the gaseous and the solid phases following the model by Crowe (2000) and it is substantial because of large velocity slip between the phases owing to high inertia of large particle size. Following to the model of Crowe (2000), this generation term is balanced by the introduced dissipation rate of turbulent energy and calculated via the hybrid turbulence length scale (last term in the right-hand side of Eq. 5). The given four-way coupling model by Crowe (2000) is based on the criteria of turbulence modulation by particles considering the ratio of particle size to the integral turbulence length scale. In accordance with this criterion for the considered cases of two-phase turbulent flow loaded by 500 and 1000µm particles, this scale ratio is far above 0.1 and therefore the particles enhanced the turbulence of the carrier gas-phase flow. In addition, the effect of increase of polydispersity grade, i.e. particle size variation from 500 up to 1000µm occurred for the mixture2 (only with ash particles) is less pronounced than that with increase of mass flow ratio up to 20kg/kg occurred in the case of the mixture3 (cf. bold dashed line in Fig. 7), on forming the shape and magnitude level of turbulent energy.

flow conditions shown in previous Figs. 4 and 5: for mixtures 1 and 3.

may contribute to the improvement of the combustion process in CFB cycles.

**mixture1:ash mixture1:sand mixture3:ash mixture3:sand**

**0.975**

**0.98**

**0.985**

**0.99**

**0.995**

**Particle mass concenration**

**1.005**

**1.01**

**1**

The numerical study of particulate turbulent flow modelled by 2D RANS (Euler/Euler) approach showed importance of addition of second solid fraction, characterized by heavy (sand) particles along with existence of first solid fraction of lighter (ash) particles in the mixing process taken place in freeboard CFB process. The main contribution to the flow formation stems from the inclusion of inter-particle collisions and four-way coupling turbulence modulation due to the presence of polydispersed solid particles with various physical properties. Other forces exerted on the motion of solids are: the gravitation, viscous drag and lift forces. On the basis of the performed calculations one can conclude:

a. variation of solids material properties results in the enhancement of flow turbulence in comparison with the turbulence level of the flow loaded by one particle fraction;

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**1. Introduction**

numerical solutions.

the direction normal to the plane.

two-dimensional steady state case.

Fig. (1).

Vai Kuong Sin and Chon Kit Chio

The full Navier-Stokes equations are difficult or impossible to obtain an exact solution in almost every real situation because of the analytic difficulties associated with the nonlinearity due to convective acceleration. The existence of exact solutions are fundamental not only in their own right as solutions of particular flows, but also are agreeable in accuracy checks for

**Computation of Non-Isothermal Reversed** 

**Stagnation-Point Flow over a Flat Plate** 

In some simplified cases, such as two-dimensional stagnation point flows, by introducing coordinate variable transformation, the number of independent variables is reduced by one or more. The governing equations can be simplified to the non-linear ordinary differential equations and are analytic solvable. The classic problems of two-dimensional stagnation-point flows can be analyzed exactly by Hiemenz Hiemenz (1911), one of Prandtl's first students. These are exact solutions for flow directed perpendicular to an infinite flat plate. Howarth Howarth (1951) and Davey Davey (1961) extended the two-dimensional and axisymmetric flows to three dimensions, which is based on boundary layer approximation in

The similarity solutions for the temperature field were studied by Eckert Eckert (1942). Case corresponding a step change in wall temperature or in wall heat flux in laminar steady flows at a stagnation point has been also investigated by several authors (see Chao et al. Chao & Jeng (1965), Sano Sano (1981) and Gorla Gorla (1988)). Further, Lok et al. Lok et al. (2006) investigated the mixed convection near non-orthogonal stagnation point flow on a vertical plate with uniform surface heat flux, where the results published are very good with present value of *θ*(0) for the constant wall temperature boundary condition. On the contrary, reversed stagnation-point flow over an infinite flat wall does not have analytic solution in

The aim of this study is to investigate the unsteady viscous reversed non-isothermal stagnation-point flow, which is started impulsively in motion with a constant velocity away from near the stagnation point. A similarity solution of full Navier-Stokes equations and energy equation are solved by applying numerical method. Studies of the reversed stagnation-point flow have been considered during the last few years, as this flow can be applied in different important applications that occur in oil recovery industry, as shown in

*University of Macau Taipa, Macao SAR, China*

**8**

