1/40 Solidification of Metals and Alloys, Year 1999, Volume 1, Book No. 40 Krzepnięcie Metali i Stopów, Rok 1999, Rocznik 1, Nr 40 PAN Katowice PL ISSN 0208-9386 COMPUTER SIMULATION OF ROUND BAR FLUIDITY TEST BOROWIECKI Bogusław Institute of Materials Engineering, Technical University of Szczecin 70-350 Szczecin, ul. Aleja Piastów 19, POLAND ABSTRACT Computer simulation of filling with metal the vertical channels of various diameters in fluidity test has been concerned in the paper. Mathematical and numerical model of solidification process during liquid phase move in mould channels has been worked out. Numerical calculation results has been compared to the empirical results and obtained satisfactory compatible for low-overheated metal. Analysis of the numerical results confirmed hypothesis, that in the channels of the various diameters there creates boundary layer of the same thickness. The layer depends on length of channel, metal viscosity and thermal physical properties of mould material. 1. Introduction Explanation metal fluidity mechanisms in channel and its solidification many investigators were engaged in [2, 3, 4, 5]. From among varied fluidity tests of alloys metals it is worth to pay attention to the round bar fluidity test. The vertical channels of various diameters have been located equidistant from pouring gate, fig. 1. Metal was flowing with equal speed through pouring gate to vertical channels. The metal solidification in filling channel of fluidity test casting has been considered. System dimensions of fluidity test casting have been presented on figure 1. Superheated metal with various temperatures has been poured gravitational into moulds. The vertical spur has been placed in the middle of mould. Vertical channel with diameters: φ 12, φ 10, φ 8, φ 7, φ 6, φ 5, φ 4, φ 3, φ 2 millimeters were filling with metal from below. During the filling of channels metal was stopping at the time, when solidified growing boundary layer obtained a value of channel radius [2].
16 2. Mathematical problem expression Fig. 1. Fluidity test of cast iron Rys. 1. Prętowa próba lejności In this vertical channels metal fill and solidify at various heights, which are dependent on channel diameter. The thermal field and the velocity field have been defined in liquid phase for movable boundary value, which shapes solidification front and they have an influence on kinetic of solidification. The phenomena of flow ability and thermal that accompanying of solidification process as nonstationary states has been considered, then it permits to achieve an analysis of temporary system state. For this purpose of mathematical problem has been formulated and on this base of numerical model has been worked, put for example finite element method. This model has been used for construction of analytical program, which enabled an execution of numerical calculation. Mathematical model has been reduced to differential system of equation. A number of theirs has been limited through a selection evidently phenomena, that permitted to obtain an effective numerical solution. The simplified assumptions with reference to the medium properties. Taking above into consideration, has been determined liquid metal has characteristic of viscosity liquid and of thermal conducting and its flow has laminar it has been received.
17 The velocity fields with solution system of equations Navier s Stokes and an equation of continuity has been received. The thermal fields from solution Fourier- Kirchhoff s equation with convection term of an equation has been received. The liquid metal solidify in temperature interval liquidus/solidus [7, 9, 10]. Taking into consideration change of thermal-physical parameters of metal, that depends on temperature and volume fraction of the solid phase. Solidification process has been defined with differential system of equations [1, 11, 12]: Navier-Stokes equation and continuity equation dν 2 ρ = p + η ν + ρg, ν = 0 (1) dt where: ρ mass density [kg/m 3 ], η viscosity [Ns/m 2 ], ν flow velocity [m/s], p pressure [N/m 2 ], t time [s]; heat flow in the solidifying casting with convection element T k (2) t ( T ( x, t) ) ρ C ρ C T ν = 0 where: k coefficient of thermal conductivity [W/(m K), T temperature [K], L C = c + substitute thermal capacity where as in the semi-liquid [J/(kg K)], T L T S L heat of solidification [J/(kg K)], c specific heat [J/(kg K), T L liquidus temperature [K], T S solidus temperature [K], x geometrical co-ordinate [m]. equation of state reduced to density which changes with temperature f ( p ρ, T ) = ρ( T ), (3) Above-mentioned system of equations of uniqueness conditions has been completed. The initial conditions define the physical state of system in the initial moment related to velocity field and thermal field. This system was considering as problem of symmetrical axis and boundary conditions assumed, as on fig. 2. Above equations system adequate to uniqueness conditions has been completed. Initial conditions define physical state of system in the first moment with reference to velocity field and thermal field [12, 13].
18 y x δ q = 0 l v = 0 x T = 0 x S AL 2r k T 1 TL A = k L n S n AL S AL T k = α x ( T ) k T A q = 0 V = V p T = T p Fig. 2. Considered system and boundary conditions assumed in the problem under study Rys.2. Rozważany układ i warunki brzegowe przyjęte w rozwiązywanym zadaniu It has been assumed that at the inlet to channel there are boundary conditions of first type (Dirychlete). On the symmetry axis and on limiting lanes of considering field has been assumed of II type boundary conditions (Neuman). Heat outlet from casting mould surface was according to III type conditions (Newton). Above formulated problem with finite element method has been solved [1, 6, 14]. Considered subdomain of metal casting mould surroundings system, gird of finite element has been digitized. In the work for numerical solution of the problem the finite element method has been employed. In the work it has been solved numerical conjugate of equations system. 3. Numerical calculation Numerical calculation has been made for round bar fluidity test that is shown in fig. 1. Phenomena of thermal and flowing occurring in sphere of channel were analysed with effect from initial moment of pouring sand mould to stoppage stream flow in vertical channel throttling by growth solidifying boundary layer. The sand moulds were poured with superheated grey cast iron of temperature T z. In calculation of initial velocity v p at entries to testing channels it was considered velocity of flow from hand ladle and resistance of flow in pouring gate. It was received following real temperature values for liquid metal: T z = 1530 [K], T L = 1483 [K], T s = 1428 [K].
19 Temperature of environment T w and initial temperature of sand mould were received as equivalent T f = T w =303 [K]. The surface film conductance from sand mould to environment averages α k = 40 [W/(m.K)]. Thermal-physical properties for grey cast iron were received from literature [8]. Their values are presented in table 1. Table 1. Thermal-physical properties of pouring cast iron Tablica 1. Własności termofizyczne odlewanego żeliwa c [J/(kg K)] ρ [kg/m 3 ] k [W/(m K)] T T S 560 7100 40.5 T T 724 7100 6920 40 24 S T L T T L 851 6920 24 Coefficient of absolute viscosity varied with temperature T z -T L it is received from literature [7]. Similarly the heat of solidification (L = 290 000 J/kg) it was received. Thermal-physical parameters (c specific heat, ρ mass density, k thermal conductivity, η absolute viscosity) for remaining subdomains of considered system, presented in table 2. Table 2. Thermal physical parameters for sand mould and air Tablica 2. Stałe termofizyczne dla formy i powietrza c [J/(kg K)] ρ [kg/m 3 ] k [W/(m K)] η [N s/m 2 ] Material of sand mould 1135 1590 1.6 - Air 1000 1.1 0.027 0.000019
20 Fig. 3. Illustration of numerical calculation results of velocity field of metal (for diameters: φ 4 mm, φ 7 mm, φ 10 mm) in the fluidity casting test Rys. 3. Ilustracja wyników obliczeń numerycznych rozkładu prędkości podczas wypełniania pionowych kanałów (o średnicach: φ 4 mm, φ 7 mm, φ 10 mm) prętowej próby lejności
21 Inlet velocity to channels with various diameters has been determined with the Bernoulli equation for the real liquid [15]. Velocity distribution in vertical channels with various diameters has been shown on fig. 3. There are characteristic of flow: turbulent in zone of entry and laminar at farther. In entry to channels velocity is maximal and equal for all channels with various diameters. Velocity is decreased during filling channels, because metalostatic pressure diminishes. On cross-section of metal stream for various heights it has been confirmed characteristic parabolic velocity distribution. As you can see on figures (1, 2, 3), the length of round bar depends on solidification of alloys process. Metals that flow in cold channel of fluidity test give in supervision. Growing viscosity caused essential growth of the boundary layer thickness that created on channel wall. The thickness of boundary layer grows first of all with growth of length channel and as a result it decreases active cross section of channel. In the middle of channel section the speed of stream filament is maximal [2, 3]. y 1490 K 178 1040 K φ 10 744 K 445 K 295 K x Fig. 4. Temperature field in vertical channel (for diameter φ 10 mm) after time 1.47 s. Rys. 4. Pole temperatury w kanale pionowym (o średnicy φ 10 mm) po czasie 1,47 s. The isotherm distribution depicts of profile thermal boundary layer surface. Metal stream in entry zone of middle channel showed maximum temperature. Temperature of laminar boundary layer was a value of freezing temperature. Distribution of velocity field and of thermal field in vertical various diameters channels, filling metal from bellow, confirms the hypothesis, that in the channels of the various diameters there creates boundary layer of the same thickness.
22 4. Summary Mathematical and numerical model of solidification process during liquid phase move in mould channels has been worked out. The results of numerical calculation have been compared to the empirical results and have been obtained satisfactory compatible for low-overheated metal. Analysis of the numerical results confirmed hypothesis, that in the channels of the various diameters there creates boundary layer of the same thickness. Velocity distribution in vertical channels with various diameter shows the flow characteristic: turbulent in zone of entry and laminar farther. The isotherm distribution depicts surface profile of thermal boundary layer. Metal stream in entry zone of middle channel showed maximum temperature. Temperature of laminar boundary layer was the value of freezing temperature. The thickness of boundary layer depends on length of channel, metal viscosity and thermal physical properties of mould material. ACKNOWLEDGEMENTS Author is grateful to Dr Ing. L. Sowa for valuable help concerning the solution of numerical problem as mentioned in the text. Presented here in experiments have been financially supported by KBN-RKH 1998 funds. LITERATURE [1] Bokota A., Sowa L.: Modelowanie narastania fazy stałej w cylindrycznym kanale formy. Krzepnięcie metali i stopów, 1993, vol. 18, s. 29-36. [2] Borowiecki B.: Boundary layer distortion factors in fluidity tests. Archives of mechanical engineering and automatization, No. 12, KTBM O/PAN, Poznań, 1993, pp. 77-84. [3] Borowiecki B.: Flow curve of liquid grey cast iron. Solidification of metals and alloys, No. 37, Foundry Commission of Polish Academy of Sciences, Katowice 1998, pp. 25-30. [4] Bydałek A.: About mechanism of filling a casting mould with a liquid metal. Solidification of metals and alloys, No. 37, Katowice 1998, pp. 31-40. [5] Dhatt G., Gao D.M., A Finie element simulation of metal flow in moulds. International Journal for Numerical Method in Engineering, 1990, vol. 30, s. 821-831. [6] Engler S., Lee Z. H., Fliessvermorgen und Warmrissverhalten von Blei-Animon Legierungen, Gießerei Forsch., 1978, Nr. 3, s. 107-116. [7] Gawroński J., Jura S., Cholewa M., Szajnar J., and other: Computer aided technology of massive steel castings. Solidification of metals and alloys, No 30.
23 [8] Ignaszak Z.: Thermal-physical properties of mould materials in the aspect of casting solidification control. Rozprawy nr 211, Polit. Poznańska, Poznań 1989. [9] Ignaszak Z., Baranowski A.: Morfologia krzepnięcia a warunki zasilania odlewów. Krzepnięcie metali i stopów, z. 19, wyd. PAN, 1994, s. 19-24. [10] Longa W.: Teoretyczna formuła do obliczania czasów zalewania form piaskowych żeliwem. Krzepnięcie metali i stopów, nr 24, 1995, s.71-76. [11] Mishima S., Szekeley J., The modelling of fluid and heat transfer in mould filling. ISIJ Inernational, 1989, vol. 29, nr 4, s. 324-332. [12] Mochnacki B., Suchy J.: Modelling and simulation of casting solidification. WNT, Warszawa 1993. [13] Sowa L., Symulacja narastania fazy stałej podczas przepływu metalu w zakrzywionym kanale dopływowym. Krzepnięcie Metali i Stopów, 1995, vol. 23, s. 26-36. [14] Szczygiol N., Sowa L.: Numerische Simulation des Anwachsen der festen Phase im Anschnittsystem einer Gießform. ZAHM. 75, 1994, s. 507-508. [15] Prosnak W.J.: Mechanika płynów. Wyd. PWN, Warszawa 1970. SYMULACJA KOMPUTEROWA PRĘTOWEJ PRÓBY LEJNOŚCI STRESZCZENIE Badania symulacyjne procesu wypełniania pionowych kanałów formy o różnych średnicach z jednoczesnym uwzględnianiem zjawiska krzepnięcia metalu umożliwiają wytłumaczenie wielu zjawisk [6, 7, 8, 11]. Celem badań była weryfikacja przyjętej hipotezy w myśl której grubość warstwy przyściennej metalu w kanale pionowym formy nie zależy od średnicy kanału lecz od lepkości metalu i wymiany ciepła na granicy metal forma, która wpływa na rozkład pola prędkości i temperatur w tym kanale. Rozważaniom poddano proces krzepnięcie metalu w wypełnianym kanale pionowym próby lejności w formie piaskowej, rys. 2. Oprócz pól temperatur określano tu jednocześnie pola prędkości w fazie ciekłej w obszarze o ruchomych brzegach, kształtujące front krzepnięcia i wpływające na kinetykę krzepnięcia. Zjawiska cieplnoprzepływowe, towarzyszące procesowi krzepnięcia, rozpatrywano jako nieustalone, co pozwoliło dokonywać analizy chwilowych stanów układu. Aby osiągnąć ten cel sformułowano model matematyczny i na jego bazie opracowano model numeryczny w metodzie elementów skończonych, który posłużył do konstrukcji programu obliczeniowego umożliwiającego wykonanie symulacji numerycznych tak postawionego problemu. Dążąc do uwzględniania wszelkich zjawisk we wzajemnym wpływie na siebie, model matematyczny sprowadzono do rozbudowanego układu równań różniczkowych. Ich liczbę ograniczono, przez wybór najistotniejszych zjawisk dla analizy rozważanego problemu, tak aby uzyskać efektywne rozwiązanie numeryczne. Sformułowano również założenia upraszczające odnośnie właściwości rozważanego ośrodka i charakteru jego zachowania się pod wpływem oddziaływań zewnętrznych.
24 Mając powyższe na uwadze, przyjęto że ciekły metal ma cechy cieczy lepkiej nieściśliwej i przewodzącej ciepło, a jego przepływ jest laminarny. Pola prędkości otrzymano z rozwiązania układu równań Naviera-Stokesa i równania ciągłości strugi, natomiast pola temperatur z rozwiązania równania Fouriera-Kirchhoffa z członem konwekcyjnym [11, 13, 14, 15]. Założono, że front krzepnięcia jest rozmyty, tzn. ciekły metal krzepnie w przedziale temperatur likwidus-solidus [5, 8, 9]. W modelowaniu tego zjawiska równanie przewodnictwa ciepła uzupełniono o człon ciepła krzepnięcia przez wprowadzenie go do efektywnej pojemności cieplnej [6]. Uwzględniano zmianę parametrów termofizycznych od temperatury i od udziału fazy stałej w dwufazowej strefie przejściowej. Porównanie wyników obliczeń numerycznych z wynikami pomiarów długości powstałych prętów żeliwnych w formach eksperymentalnych (rys. 3) wykazuje zadowalającą zgodność dla żeliwa o niskim stopniu przegrzania. Rozkład pól prędkości oraz pól temperatury w kanałach pionowych wypełnianych ciekłym stopem potwierdzają przyjętą hipotezę o mechanizmie narastania warstwy przyściennej metalu w kanale pionowym. Recenzował Prof. dr hab. inż. Stanisław Jura