Water-Jet Impingement Cooling of Hot Moving Steel Sheets
Ever-increasing demand from the automotive and building industries on sheet steel quality require steel strip to be tough, ductile, fatigue resistant, weldable and corrosion resistant. These properties are achieved, in addition to other initiatives, by tight control of the finishing and coiling temperatures, the latter of which is achieved by precise control of the cooling at the run-out-table (ROT). Design of the ROT is governed by the considerations of high cooling efficiency and achieving the desired steel properties. Planar jets (water curtains) provide very high cooling efficiency with minimum splashing, but they produce non-uniform cooling on the top and bottom surface as well as over the length of the cooling zone. Spray cooling has a low specific cooling performance, and incurs high maintenance costs. A compromise between high specific cooling performance and uniform strip cooling is provided by an array of round laminar jets impinging on the steel sheet. Cooling of the strip, typically in excess of 800oC, leads to boiling heat transfer characterised by forced convection, nucleate boiling, transition boiling and film-boiling regimes. In stationary strip jet-impingement (non-moving sheet), these boiling regimes are present simultaneously at differing distances as measured from the jet impingement centre-line. The thermal zones for jet impingement may be delineated into a free-jet region, where the velocity and temperature distributions of the jet are not affected by the presence of the strip surface; a jet-impingement or stagnation region, where single-phase forced convection takes place over several jet widths and the cooling effectiveness is high; and the wall-jet region, separated into a small region of transition and nucleate boiling before entering the parallel film-boiling region separating the strip surface from the jet by a vapour layer which reduces cooling effectiveness considerably. Due to both fluid flow and strip movement, the water-jet impingement cooling of a hot strip involves the solution of transport equations under certain initial and boundary/interface conditions. Limitations in the analytical solutions of such equations demand a numerical approach. Traditional Eulerian finite difference methods for the solution of such advection equations incur severe Courant number stability restrictions and Peclet number induced spurious oscillations which are not improved by upwinding methods. Purely Lagrangian methods which deal admirably with advection problems are not so successful when diffusion is present. A combined hybrid technique called the Eulerian-Lagrangian method (ELM) has been shown to be highly effective in the solution of the transport equation even for very large Peclet numbers and Courant numbers in excess of one. The impact of a water-jet can be divided into two regions: the first being the jet domain and the second the moving steel sheet. In addition to the difficulties presented by the numerical solution of transport equations, both in the jet and the steel domain, the interface between the two is governed by strong temperature gradient discontinuities as well as discontinuities in velocity and thermophysical properties. In that case numerical techniques which combine both the ability to accurately model the influence of diffusion and convection as well as discontinuities in temperature, velocity and material properties is required. This can be done through a combination of the Eulerian-Lagrangian method for the solution of transport equations and the Immersed Interface Method of LeVeque for the solution of PDE's with immersed interfaces to be called the Eulerian-Lagrangian Immersed Interface Method (ELIIM). In the jet domain a realistic fluid dynamical description is needed for the velocity fields of the jet itself. In the case of steady, inviscid, axisymmeric flow the shape of the flow field may be solved semi-analytically through a separation of variables approach in combination with an underrelaxed predictor-corrector numerical method of Phares. Other Useful ReferencesBooks or papers which are of use in the construction of ELIIM include:
- The book Numerical Analysis for Applied Mathematics, Science, and Engineering by Greenspan and Casulli.
- The papers and thesis of Anabela de Oliveira.
- In conjunction with the research of Antonio Baptista.
- For an introduction and error to ELM methods refer to Vincenzo Casulli
- For the development and important papers on the Immersed Interface Method refer to Randy LeVeque
- The velocity field (2D or axisymmetric) of an inviscid impinging jet see Phares, Smedley and Flagan