Session 16

Chairman: G. Z. Gershuni

COMPUTATION OF TWO-PHASE FLOW WITH HEAT TRANSFER IN A SCALE-MODEL ELECTRIC ARC FURNACE 

Processing steel in an electric arc furnace involves a glassy slag layer floating on top of the molten steel. The slag layer decreases heat transfer between the steel and the freeboard gasses above and is an important site for chemical reactions in the process. In current designs, fuel lances are introduced through the furnace side wall to supply reactant materials to the process but also to maintain high temperature near the liquid steel surface. For heat transfer purposes, one would like to position the lances so that the motion induced in the slag layer gives the highest net heat transfer rate to the steel.

The present work forms part of a project that aims to provide basic data and understanding needed to optimise the positioning and conditions of the lances. At the centre of the study is a scale model designed to simulate the major processes affecting the convective heat transfer across the slag layer. This model captures the impingement of the lance flow onto the slag surface and the resulting surface depression and motion of the slag layer, both of which can be expected to affect heat transfer through the slag. The heat transfer rates to the simulated liquid steel and to the cooled furnace ceiling will be measured as a function of jet position, orientation and momentum in the experiment. A second part of the work, and that which will be reported, is the computation of the 3-dimensional two-phase flow with heat transfer occurring in the physical model. It is felt that the computation will provide a sufficiently close representation of the flow processes affecting convective heat transfer to be helpful in determining optimum lance conditions. In any case, this second part of the project will allow a firm assessment of the suitability of the computational modelling.

In the physical model, the furnace enclosure is a vertical cylinder with the bottom end made of solid aluminium and held at 300 K to represent the dense molten steel and the top end cooled by liquid nitrogen to 170 K to simulate the water-cooled furnace ceiling. The sides of the model furnace are well insulated. Four symmetrically-placed air jets at 360 K simulate the lance jets and, to preserve the 90 sector angle symmetry, two regularly-spaced holes provide an outlet vent. A layer of liquid oil above the aluminium bottom plate represents the slag layer. Thus, a 3-dimensional flow computation with an account of the free surface and heat transfer in both phases is required. The computations will be made using the commercial computer code FLUENT¹ which incorporates a range of models that should allow a reasonable representation of all the major processes occurring in the model flows. The ‘volume of fluid’ technique will be used to compute flow with free surface. This involves the time dependent solution of a single set of flow equations for the entire domain with appropriate effective properties used for numerical cells containing the free surface and hence both phases. Perhaps most challenging will be satisfactorily representing turbulence effects in the vicinity of the slag/gas interface. The turbulence should be damped by the liquid ‘wall’ and yet this will not be captured by a standard turbulence model without special introduction of the dependence on distance from the free surface. It may be possible to capture such effects using a low Reynolds number turbulence model. Such a model will be tested.

The computational results will be compared with the experimental heat transfer rates and also with the slag surface depression of the slag layer found in the experiment. The initial stage, and that to be reported, examines the reliability of the computation model under the conditions of the experiment. Computations will be made with the aim of identifying shortcomings and strengths of the computational model. Preliminary conclusions regarding optimum lance practice will be made as appropriate.

1. FLUENT User’s Guide, Version 4.3, Fluent Incorporated, 1995.

Physical Vapour Transport (PVT) and Chemical Vapour Transport (CVT) are two common and versatile ways of growing crystals¹. Both techniques involve the sublimation of the solute material, which is then transported by an inert gas and then condensed on the cooled end plate. The difference between the two techniques is that CVT involves chemical reactions to generate the solute.

Both processes are such that significant variations occur in fluid properties due to large temperature and/or concentration variations, thus making it impossible to invoke the Boussinesq approximation.

In this paper a highly accurate numerical method is presented for the solution of such problems. The growth of Iodine crystals (I₂) using PVT, for which data are available in the literature², has been used to investigate the accuracy and efficiency of the code. We have chosen two situations, in which very different flows result from the competition between solutal and thermal convections,. Two different inert gas are considered: H₂ and C₄F₈, typically used in such process. In the first case, the ratio between the molar masses of the inert gas and the solute material is very high: M(H₂)=2 kg/kmol while M(I₂)=254 kg/kmol. This mixture is mainly governed by solutal convection and exhibits high density gradients. The second pair involves C₄F₈ as inert gas, giving a closer molar mass with the solute, M(C₄F₈)=200 kg/kmol. The solutal convection is thus strongly reduced and the thermal convection becomes dominant.

Due to the large temperature gradient and different molecular masses of the species, there are significant variations of viscosity, thermal conductivity, specific heat and diffusion coefficient Therefore all properties are made variable. The compressible Navier-Stokes, energy and species equations are solved under the Low Mach Number flow hypothesis³. A pseudospectral collocation-Chebyshev method is used for the spatial resolution. The temporal integration is done with a second-order accurate semi-implicit scheme combining the second order Backward Euler and Adams-Bashforth schemes. Thus, at each time step, the different transport equations are reduced to simple Helmholtz problems, but with variable coefficients, solved directly using a complete diagonalization of operators in both space directions.

For the treatment of variable properties, each variable is decomposed in an “average” part and a “fluctuating” part. This allows the resolution of the compressible flows to be brought back to that of incompressible flows, i.e. Helmholtz equations with constant coefficients. All the “fluctuating” terms are treated as source terms. However, some internal iterations are required to make the resolution of the whole system more implicit. A time-splitting algorithm is used for the coupling of the velocity and the pressure, ensuring mass conservation at each time step.

An ideal gas equation of state is used with the molecular mass determined from the local species concentration. The other thermodynamic and transport properties are similarly determined from the local temperature and species concentration. Overall mass conservation is ensured numerically by requiring that the volume integral of the density remains constant and equal to the initial mass of the system. In compressible flow calculations this initial mass (i.e. the initial density field) plays a very important role and is sensitive to the choice of initial mass fractions. We have used the solution proposed by Greenwell et al. ⁴ for a one dimensional problem, as the initial condition.

The domain considered corresponds to a rectangular cavity with source material at one end and crystal at the other end, respectively kept at constant temperatures T_source and T_crystal, where T_s > T_c. We assume complete rejection of the inert gas and equilibrium of solute at both interfaces. The two other walls are impermeable to both components and have constant temperature profiles imposed.

Figure 1. Streamlines (top) and Iso-mass Fractions contours (bottom) for (a) H₂ – I₂ and (b) C₄ F₈ – I₂ . T_source= 368K and T_crystal= 343K. Dashed lines indicate a counter-clockwise rotation.

Our computed results show a good agreement with data available in the literature. Characteristic flow structures are displayed with iso-mass fraction contours of the solute for the two configurations of PVT processes (see Figure 1), indicating clearly the respective dominance of solutal (H₂-I₂) and thermal (C₄F₈-I₂) convections.

1. Rosenberger, F., Fundamentals of Crystal Growth. Macroscopic Equilibrium and Transport Concepts, Springer-Verlag, 1979.
2. Markham, B. L. and Rosenberger, F., Diffusive-Convective Vapor Transport across Horizontal and Inclined Rectangular Enclosures, J. Crystal Growth, Vol. 67, pp 241-254, 1984.
3. Paoluci, S., On the Filtering of Sound from the Navier-Stokes Equations, Sandia National Laboratory Report SAND82-8257, pp 5-52, 1982.
4. Greenwell, D. W., Markham, B. L. and Rosenberger, F., Numerical Modeling of Diffusive Physical Vapor Transport in Cylindrical Ampoules, J. Crystal Growth, Vol. 51, pp 413-426, 1981.

This study concerns the evaporation of a finite liquid film flowing on one of the walls of a vertical canal, 2m high. The canal consists of two parallel plates separated by the distance e. The wall on which the liquid flows is submitted to a constant heat flux, the other wall is thermally isolated (an adiabatic wall). To the forced convection due to the flow of air which enters the canal at uniform speed, is added the force of natural convection by the flow of heat imposed on the heated wall. The influences of inlet liquid temperature and imposed wall heat flux on liquid Nusselt number at the liquid-wall interface were examined in detail. The direct application of this study is the cooling of the walls and especially, electronic components¹. The latter being subjected to considerable heat flows, their efficiency depends essentially on the evacuation of this heat.

In the physical domain, we write equations that govern the transfer of movement and heat in the two phases and mass transfer in the gaseous phase. The reference chosen is at the upper extremity of the plate on which the liquid film flows. Abscissas are counted positively in the direction of the flow of the liquid (from the upper extremity to the lower), and the ordinates, from the upper extremity to the adiabatic plate. In order to constitute the equations, we must work within the framework of the hypothesis of the bidimensionnal layer limits and we study the physical properties of fluids in variable flows. The pressure gradient due to the depression of the flow in the canal is taken into account, as well as the convective terms in all the equations of movement and heat transfer.

A dimensionless equation and an appropriate variable change allow a generalization of the results and a precise location of the liquid-gas interface. Linear Equations and boundary conditions are obtained by the utilization of the finite difference implicit method. The coupled equations are solved by a successive repetitions based on an overall mass assessment in the two phases and of the overall thermal assessment at the liquid-gas interface. To do this, the governing equations are expressed in terms of finite difference approximations by employing the upstream difference in the axial convection terms and the central difference in the transverse convection and diffusion terms. The sampling of equations of movement in the two phases, liquid and vapour, gives a system of linear equations resolved by the Gauss method. The other equations are solved by using the Thomas Algorithm. The stages followed up for the resolution of the equations are :

1. At the entry of the canal, the thickness of the liquid and the profile of speeds in this phase are calculated by ignoring convective terms in the equation of the movement,
2. For a given temperature at the liquid-gas interface and a given thickness of the liquid film and a given pressure in the canal starting from the second calculation point, one proceeds by successive repetitions :
3. we give profiles of speeds, temperatures and concentrations,
4. Resolution of the equations of mass transfer in the gas, the equation of coupled heat transfer (liquid-gas) and the equation of coupled movement transfer (liquid-gas) associated with equations of the mass assessment in each phase,
5. utilisation of the mass, liquid and gas assessments to calculate the thickness of the liquid film with k as the number of repeats to the point of calculation under consideration,
6. resolution of the equation of continuity,
7. verification of the convergence of speeds, temperatures and gaseous mass concentrations by under-relaxation method with a coefficient of 0.5. If the test is not verified, one returns to stage 3 otherwise, one passes to the next stage,
8. comparison of the deduced interface temperature of stage 4 with that given or found in the iteration (k-1), if the difference between these two temperatures is inferior to 10-2, one passes to the next stage, otherwise one corrects this temperature and returns to stage 2,
9. comparison of the pressure calculated from the equation of the movement with that given in stage 2 or found in the iteration (k - 1),
10. passage to the next calculation section,
11. end of calculations at the exit of the canal.

Among results of this study, given in dimensionless abscissas and ordinates, we show that the direction of the flow in the gaseous phase can invert near the liquid-gas interface (figure 1). This inversion depends essentially of the imposed wall heat flux and the inlet liquid temperature.

We show, furthermore, that in practice, the Nusselt liquid number (at the wall-liquid interface), does not depend on the imposed wall heat flux (figure 2), while it increases with the inlet liquid temperature (figure 3).

1. Richard, C. Chu, "Heat transfer in electronic system", 8th Int. Heat Conf., San Francosco, pp 293-305, 1986.
2. Dukler, A. E., "Fluid mechanics and heat transfer in vertical falling film system", Chem. Eng. Prog. Symp. Ser., Vol. 56, pp 1-10, 1960.
3. Smith, R. A., "Vaporizers", John Wiley and Sons, Inc., New York, NY, 1986.
4. Chun, K. R. and Seban, R. A., "Heat transfer to evaporating liquid films", J. Heat Transfer, Vol. 93, pp 391-396, 1971.
5. Yan, W. M., "Effects of film evaporation on laminar mixed convection heat and mass transfer in a vertical channel", Int. J. Heat Mass Transfer, Vol. 35, pp 3419-34

AN ADAPTIVE METHOD FOR HEAT TRANSFER IN ANNULAR LIQUID JETS

J. I. Ramos
Departamento de Lenguajes y Ciencias de la Computación
E. T. S. Ingenieros Industriales, Universidad de Málaga
Plaza El Ejido, s/n, 29013-Málaga, Spain

Annular liquid jets are thin sheets of liquid falling under gravity which, under certain conditions, merge in the centerline to become round liquidjets (Figure 1). The volume enclosed by annular liquid jets has been proposed as a chemical reactor for scrubbing of radioactive and non-radioactive materials, burning of toxic wastes, reduction of zirconium from zirconium tetrachloride and sodium, etc., because the liquid jet contains the reaction and the gases generated in, for example, the combustion of toxic wastes may be absorbed by the liquid which may be collected, purified and recycled so that the combustion-generated gases do not flow through the jet's outer interface and, therefore, do not pollute the environment.

Figure 1. Schematic of an annular liquid jet

Most previous studies of annular liquid jets have been concerned with the analysis of their fluid dynamics and heat/mass transfer under steady state conditions by assuming that the axial velocity component of the liquid is governed by Torricelli's free-fall formula since the geometry of the jet is curvilinear. Moreover, studies on mass transfer in annular liquid jets have taken profit of the fact that the binary diffusion coefficient of gases in liquids is much smaller than that of gases; as a consequence, mass absorption by annular liquid jets is a rather slow process, and the decrease or growth that the volume enclosed by the annular liquid jet experiences due to the mass absorbed by the liquid is mainly controlled by the mass of the gases generated by the combustion of toxic wastes.

When the volume enclosed by annular liquid jets is used to burn toxic wastes, in addition to the absorption of the gaseous combustion products by the liquid, one must account for the heat exchanges between this volume and the jet. However, since the heat diffusion coefficient in the liquid is much larger than the mass diffusion coefficient of the gases absorbed by the liquid, it is to be expected that liquid jet absorb the heat at a faster rate than the gaseous combustion products and, as a consequence, the combustion of toxic wastes in the volume enclosed by annular liquid jets is characterized by four time scales: the characteristic residence time which depends on the liquid volumetric flow rate, the characteristic reaction time, and the characteristic times for heat and mass absorption by the liquid. Since the latter is much larger than the other three, one may assume that mass absorption is a quasi-steady phenomenon. Moreover, if the liquid does not absorb the heat at a sufficiently large rate, the heating of the gases enclosed by the annular jet will increase their temperature, pressure and volume; therefore, under these conditions, the curvilinear geometry of the annular liquid jet will be a function of time. We are thus faced with a free-surface problem, i.e., the annular liquid jet, which affects and is affected by the combustion of the toxic wastes that it encloses.

In this paper, a set of one-dimensional equations for the fluid dynamics of slender, inviscid, annular liquid jets subject to gravity, surface tension and pressure differences across the jet obtained from the two-dimensional Euler equations subject to kinematic and normal stress conditions at the jet interfaces, and the two-dimensional heat transfer equation for the liquid have been solved numerically by means of an adaptive finite difference method which maps the unknown, time-dependent, curvilinear geometry of the annular liquid jet into a unit square, and conservative, second-order accurate finite difference formulae. The fluid dynamics equations have been obtained by means of perturbation techniques by using a long wavelength approximation and a high Reynolds number aproximation. Since the axial location at which the annular liquid jet merges on the symmetry axis to become a round one is not known and must be determined from the solution, an ordinary differential equation for this location has been obtained from the liquid's continuity and linear momentum equations by imposing the condition that the jet's inner radius at this location is zero.

The gases enclosed by the annular liquid jet have been assumed to be homogeneous, isobaric and isothermal owing to the their large mass and energy diffusivities and the low Mach number of the flow, whereas the jet's outer interface was assumed to be isothermal. The energy boundary conditions at the jet's inner interface use continuity of temperature and heat flux, so that the mass and temperature of the gases enclosed by the annular liquid jet are governed by ordinary differential equations in time which are nonlinearly coupled to the fluid dynamics and energy equations for the liquid. The downstream boundary condition for the liquid's thermal energy was determined by assuming that, at the axial location where the annular liquid jet merges onto the symmetry axis to become a round one, the derivative of the temperature along the local streamlines is zero.

Due to the nonlinear coupling between the jet's velocity components and inner and outer radü, the one-dimensional fluid dynamics equations for the liquid were solved by means of a Newton- Kantorovich method which provides a set of linear equations for the jet's mass per unit length, mean radius, and axial and radial velocity components. The governing equations were also solved by means of a time-linearized method for both the fluid dynamics and gas concentration equations. This method is based on the time linearization of the Crank-Nicolson technique and, therefore, is implicit, and provides a system of block-tridiagonal, linear, algebraic equations for the fluid dynamics equations which can be solved by means of LU decomposition. The two-dimensional energy equation was factorized into one-dimensional operators which were also solved by means of LU decomposition. In order to accurately resolve the steep temperature gradients at the jet's inner interface due to the large thermal Péclet number, the grid was not distributed uniformly in the radial direction; in fact, the rid was concentrated near the et s inner interface. Furthermore, since the thermal conductivity of liquids is large, a small time step was used in the calculations in order to accurately resolve the initial transient phenomena which correspond to the rapid cooling of the liquid for the case that combustion takes place in the volume enclosed by the annular liquid jet.

The discretized equations were solved as follows. First, the geometry of the jet was determined under isothermal conditions, i.e., when the temperatures of the combustion products and the liquid are identical, for which the fluid dynamics and heat transfer phenomena are uncoupled. Once, a steady state was obtained, combustion within the volume enclosed by the annular liquid jet was started until a new equilibrium state (if any) was reached so that the temperature at the jet's inner interface is identical to that of the gases enclosed by the jet, and there is a temperature gradient in the liquid. Then, combustion was assumed to be shut off instantaneously, so that one could study the unsteady fluid dynamics of and heat transfer in annular liquid jets.

Some sample results for the conditions described in the previous section are presented in Figures 2 and 3 which show the pressure coefficient of the gases enclosed by the annular liquid jet, heat fluxes at the jet's inner and outer interfaces, volume of the gases enclosed by the jet and the temperature at the jet's inner interface as functions of time for three different Weber numbers. These figures clearly indicate that the volume enclosed by the jet, the pressure coefficient, the temperature at the jet's inner interface and the heat flux at the jet's inner interface decrease as a function of time due to the heat absorbed by the liquid. They also show the initial steep drop in the pressure and heat flux. Since the time was nondimensionalized with respect to the residence time, Figures 2 and 3 clearly indicate that a large amount of cooling occurs initially, and this requires the use of a small time step to be resolved accurately. Although not shown here, the axial location at which the annularjet becomes a round one and the liquid's axial velocity component there decrease, whereas the jet's thickness increases with time.

Figure 2. Pressure coefficient (top left), temperature at the jet's inner interface (top right), heat flux at the jet's inner interface (bottom left) and volume enclosed by the annular jet (bottom right) as functions of time and Weber number.

Figure 3. Pressure coefficient (top left), temperature at the jet's inner interface (top right), heat flux at the jet's inner interface (bottom left) and volume enclosed by the annular jet (bottom right) as functions of time and annular jet's thickness-to-radius ratio at the nozzle exit.

ACKNOWLEDGEMENTS

This work has been supported by Project No. PB94-1494 from the DGICYT of Spain.