| 1. |
- Löfgren, Hans B., et al.
(författare)
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Damping mechanisms of perturbations in electromagnetically braked horizontal film flows
- 2000
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Ingår i: Fluid Dynamics Research. - 0169-5983 .- 1873-7005. ; 26:1, s. 53-68
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Tidskriftsartikel (refereegranskat)abstract
- The stability of shear and surface modes in an electromagnetically braked free surface liquid metal flow over a horizontally moving boundary is considered. The main objective is to investigate the stabilizing effect of a transverse magnetic field of the film flow appearing in connection with the horizontal belt strip casting. The flow is two-dimensional and the upper boundary is a free surface. In the stability analysis the film flow is approximated as parallel and is therefore based on the MHD-modified Orr-Sommerfeld equation. Numerical calculations present the linear growth rate for shear and surface mode instabilities. A transverse magnetic field is found to have two damping mechanisms. For low Hartmann numbers the main mechanism is due to the braking of the flow, reducing the shear stress. The other mechanism is due to dissipation by the Joule effect.
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| 2. |
- Löfgren, Hans B., et al.
(författare)
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Electromagnetic braking of the flow of a liquid metal with a free surface
- 1998
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Ingår i: Fluid Dynamics Research. - 0169-5983 .- 1873-7005. ; 23:1, s. 1-25
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Tidskriftsartikel (refereegranskat)abstract
- The electromagnetic braking of liquid metal flowing over a moving boundary is considered. The flow is two dimensional and the upper boundary is a free surface. The main interest of this flow originates from the application of horizontal belt strip casting. In this process, the liquid metal is fed onto a single endless horizontal belt that runs between two rollers. The bottom of the belt is cooled by water and the liquid is fed onto the belt through a slit. This results in a flow that can be modelled as a flat free jet impinging onto a horizontally moving boundary. The flow after the impingement region is analysed using the boundary layer equations. Here, the flow depends on the Reynolds number R, the Froude number F, the Hartman number M, and the ratio of the belt velocity to jet velocity, β. For βF-2/3, there is no hydraulic jump and the film thickness grows smoothly into the final state with constant thickness and uniform velocity. The braking length for the case without hydraulic jump is shown to be of order O(F2/3Ra/2) without magnetic field and of order O(M-2Ra/2) with magnetic field, where a is the jet width. Thus, the magnetic field decreases the braking distance considerably. This result is favourable for both the stability of the flow and for the solidification process in the horizontal belt strip casting process.
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| 3. |
- Löfgren, Hans B., et al.
(författare)
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Initial solidification in liquid metal film flow over a moving bondary
- 2001
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Ingår i: International Journal of Heat and Mass Transfer. - 0017-9310 .- 1879-2189. ; 44:4, s. 837-842
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Tidskriftsartikel (refereegranskat)abstract
- The initial solidification problem of a two-dimensional liquid metal film flow over a heat extracting moving boundary is studied. Analytical solutions in the limit of large Peclet numbers are found. It is shown that the point of initial solidification depends on the Peclet number, the Biot number and the superheat. The initial growth of the solidified phase is found to have a quadratic dependence of the distance from the point of initial solidification. The results are applicable to continuous strip casters.
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| 4. |
- Löfgren, Hans B., et al.
(författare)
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The laminar stagnation-point flow against a solidifying moving shell
- 2000
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Ingår i: Acta Mechanica. - 0001-5970 .- 1619-6937. ; 142:1-4, s. 33-46
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Tidskriftsartikel (refereegranskat)abstract
- This paper considers the two-dimensional laminar stagnation-point flow due to a jet impinging onto a solidifying moving boundary. The flow is of interest in connection with the horizontal belt strip casting process. An exact solution to the Navier-Stokes equations is found that is shown to depend on a single ordinary differential equation. The solution is useful in the study of morphological and hydrodynamic instabilities within the impingement region. Solutions for the steady-state shape of the initial stages as well as the asymptotic behavior of the solidifying interface are also discussed in a perturbative manner.
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| 5. |
- Åkerstedt, Hans O., et al.
(författare)
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Free surface magnetohydrodynamic flow with solidification
- 2003
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Ingår i: European journal of mechanics. B, Fluids. - : Elsevier BV. - 0997-7546 .- 1873-7390. ; 22:6, s. 581-601
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Tidskriftsartikel (refereegranskat)abstract
- The paper considers an analysis of a liquid metal flow, occurring in the horizontal belt strip casting process. The liquid metal flows over a moving copper belt with a growing solidifying phase beneath the melt. The effect of applying a transverse magnetic field is investigated. A set of three-dimensional shallow water equations is derived. Supercritical flow is assumed and the shallow water equations are solved numerically using a shock-capturing method, which automatically takes care of the possibility of oblique hydraulic jumps.It is shown that non-uniform conditions introduced in the feeding region give a pattern of steady hydraulic jumps, which propagate downstream and are reflected at the sidewalls. The effect of the magnetic field is to brake the flow and damp the standing hydraulic jump pattern. Different feeding methods are compared and it is shown that the magnetic field erases the initial differences in liquid distribution using different feeding techniques.
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| 6. |
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| 7. |
- Löfgren, Hans B.
(författare)
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Ideal solidification of a liquid metal boundary layer flow over a conveying substrate
- 2001
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Ingår i: Journal of Fluid Mechanics. - 0022-1120 .- 1469-7645. ; 446, s. 121-131
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Tidskriftsartikel (refereegranskat)abstract
- The ideal solidification problem of the two-dimensional boundary layer flow of a superheated liquid-metal binary alloy over a conveying substrate is examined. Analytical results for the velocities, the solute concentration and the temperature field are found in the asymptotic limits of small prandtl numbers and large schmidt numbers. The growth of the solidifying front is shown to follow the square-root law.
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| 8. |
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| 9. |
- Löfgren, Hans B.
(författare)
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Theoretical magnetohydrodynamic and solidification studies of continuous strip casting
- 2000
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The continuous casting of steel strips has been under investigation and development for several decades. The aim of this development is to be able to cast steel strips in a thickness range from 1 to 10 mm. Today, there are basically two such methods: the single-belt caster and the twin-roll caster. The advantage of these methods in comparison with the more conventionally produced strips, produced from slab-casting and hot-rolling, is that there is considerable energy savings if a continuous strip casting process can be successfully developed. In this thesis we consider the fluid dynamics and solidification process of the single-belt caster. This technique is used in a pilot-plant at MEFOS in Luleå, and in laboratory scale casters at the Clausthal University and at the Division of Fluid Mechanics at Luleå University of Technology. In this process the liquid metal is fed onto a single endless horizontal belt that runs between two rollers. The underside of the belt is cooled by water. The demands of the strip casting process regarding the shape and surface conditions of the strip are higher the smaller the thickness of the strip. Critical factors for the surface and the shape of the strip are the flow and the solidification conditions of the liquid metal at the feeding point onto the conveyor belt. One condition to be satisfied in the design of the feeding system, is that the average relative velocity between the liquid and the belt should not be too large, in order to avoid strong hydraulic jumps and turbulence. There are many different techniques for feeding the liquid metal onto the conveyor belt. The feeding systems in use are basically driven by gravity or pressure. In our study we consider the feeding through a slit or down an inclined plane. This generates approximately flat jets, falling vertically or obliquely onto the belt, with a nearly two-dimensional spreading in the casting direction. An interesting concept in the control of many metallurgical flows is to use electromagnetic forces. It is well known that magnetic fields may brake and stabilize liquid metal flow. In this thesis we analyse how the fluid dynamic conditions of the single-belt casting process can be improved with the application of a transverse DC magnetic field. The main complexity in the analysis of this kind of flow is the unknown position of the free surface and the occurrence of a hydraulic jump. It is found that with a sufficiently large ratio of the belt velocity to jet velocity the hydraulic jump can be eliminated. In this case, however, the characteristic distance it takes to decelerate the liquid metal smoothly to its final state, of constant thickness and uniform velocity, is very long. With the application of the magnetic field, this braking distance can be decreased considerably. As an example, a braking length of 0.25 m is achieved with a magnetic field of about 0.2 Tesla. A uniform velocity profile is favourable for flow stability and for the solidification process. From the solidification point of view, a homogenous velocity distribution is favourable since this causes minimum macrosegregation of the solidified strip. In the final state of constant thickness and uniform velocity distribution, which can be reached in a short distance with the application of a magnetic field, one therefore has good control of the solidification and the solidification process can easily be analysed. In the stability analysis we find two types of instabilities associated with this kind of flow. These are the surface mode and the shear modes. The stabilization of these modes can be accomplished with a moderate magnetic field of the order of 0.1 Tesla. From the stability point of view the transverse magnetic field acts in two ways. First, the field modifies the velocity profile due to the braking action into a more favourable "relaxed" profile. This is the most important stabilizing effect for weak magnetic fields of the order of 0.1 Tesla. Secondly, there is a resistive damping effect, which is small for weak magnetic fields. For the case of a uniform velocity distribution this is the only damping mechanism of the field. Therefore, in order to get an appreciable damping of ordinary gravity waves a stronger magnetic field of the order 1 Tesla is necessary. If the braking action of such a strong transverse magnetic field is too violent, a longitudinal magnetic field may be of interest. A longitudinal field causes no braking of the flow, but the resistive damping is still operative. The quality of the solidified steel depends to a large extent upon the degree of control during solidification. The solidification issues are therefore of great importance. Single-belt castings are characterized by an asymmetrical solidification structure, where the dendrites grow from the heat-extracting boundary towards the free surface of the casting. The resulting structure is finer on the contact side, due to the higher heat transfer rate, and coarsens across the thickness of the strip. The solidification process is primarily controlled by the heat transfer. Heat flows from the liquid metal to the heat-extracting conveyor belt. When the liquid metal reaches fusion temperature it solidifies onto a growing shell of steady state shape that is continuously conveyed downstream. The main difficulty in the determination of the growth of the solidification front is the coupling between heat and fluid flow. For the case of constant material properties, the heat flow is coupled to the fluid flow in terms of advection and the fluid flow is coupled to the heat flow by the boundary conditions at the solidification front. In this study we derive analytical results for the initial solidification process of a pure liquid metal over a moving boundary with a finite heat transfer coefficient. For the case of zero superheat in the melt, the initial growth of the solidified metal is immediate and linear. On the other hand, by contrast, for a weakly superheated melt the initial solidification is delayed somewhat with an initially quadratic growth of the solidification front. Further downstream the growth of the solidifying interface tends to the ideal case with a square root growth. In this thesis we also present the fluid flow at the stagnation point of the feeding jet as well as the boundary layer flow over an ideal solidification front. These solutions are useful in the determination of the morphological instabilities of the solidifying interface.
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