OCR Text |
Show NUMERICAL MODELING Solving a set of 3-D equations for the entire furnace domain which includes more than 100 flame jets impinging on the metal load is a rather costly and' time consuming problem, even for the most powerful computers. Experience in furnace design shows that it may be necessary to ~ake hundreds of calculations to choose the optimum set of parameters. It is, therefore, desrrable for practical applications to create a relatively simple numerical simulation that 1) gives acceptable accuracy for engineering purposes, and 2) allows personal computers to reach the solution within a few minutes. A simple 2-D axi-symmetric scheme of a single flame jet together with the key geometrical parameters is shown in Figure 2. The model includes 2-D momentum and transport equations for scalar variables: temperature, turbulent kinetic energy and its rate of dissipation, and concentrations of the species. To solve for radiant heat transfer, the zonal method was used. Radiation and convection energy transfer were incorporated using the procedure described in Reference [10]. The model was simplified by assuming: • Cross flow influence is negligible (for the experimental furnace, the ratio of the maximum cross flow momentum to the main flame jet flow momentum was less than 0.0005) • Interaction between neighboring flame jets is negligible. • Temperature and velocity fields are similar for all flame jets (measurements have shown the maximum deviation between the different jets to be 5%). The schematic diagram of the calculation domain and the boundary conditions is presented in Figure 2. The flame jet, emitted from a nozzle of diameter do, impinges on a plane surface of radius Rm placed at a distance x=H from the nozzle edge. The boundary conditions on the solid walls and on the jet axis are standard for such a case and need not to be discussed. The problem was to set them on a 'free' boundary (1 or 2) . Several tests were carried out at the often used condition of dF/dn = 0 ( F is a stream function, and n is the normal direction) for all variables including mass flows. An orthogonal grid of 33 by 25 (in x, y-directions) was used in all calculations. The grid's steps near the solid surfaces were about 0.05. The two momentum equations were written in the pressure-velocity form and were solved by means of the algorithm, described in references [11,12]. The main feature of the method consists of integrating the momentum equations along a closed contour, thus eliminating the pressure variable from the equations. U sing mass conservation equation to express U velocity as a function of V velocity, one can derive the resulting equation which contains only V velocity and has a five diagonal matrix form which can be easily solved with the standard Gauss linear five diagonal matrix algorithm. This method provides a convergence rate that is comparable and sometimes faster than the well known SIMPLEC code with optimal relaxation parameters. But in contrast to SIMPLEC, it has fewer problems with relaxation parameters especially for sharp, non unifonn grids and complex geometries and flows. 6 |