computer simulation of round bar fluidity test

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
1/40
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].
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Fig. 1. Fluidity test of cast iron
Rys. 1. Prętowa próba lejności
2. Mathematical problem expression
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.
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The velocity fields with solution system of equations Navier’s – Stokes and an
equation of continuity has been received. The thermal fields from solution FourierKirchhoff’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
ρ
dν
= −∇p + η∇ 2ν + ρg ,
dt
∇ ⋅ν = 0
(1)
where:
ρ – mass density [kg/m3],
η – viscosity [Ns/m2],
ν – flow velocity [m/s],
p – pressure [N/m2],
t – time [s];
– heat flow in the solidifying casting with convection element
∇ ⋅ (k∇ ⋅ T (x, t )) − ρ ⋅ C
∂T
− ρ ⋅ C ⋅ ∇T ⋅ν = 0
∂t
(2)
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 − TS
L – heat of solidification [J/(kg·K)],
c – specific heat [J/(kg·K),
TL – liquidus temperature [K],
TS – 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].
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y
x
δ
l
∂v
=0
∂x
q=0
SAL
kA
∂T
=0
∂x
2r
k
∂T1
∂n
= kL
S AL
∂TL
∂n
S AL
∂T
= −α k (T − T A )
∂x
q=0
V = Vp
T = Tp
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 Tz. In calculation of initial
velocity vp 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: Tz = 1530 [K], TL = 1483 [K], Ts = 1428 [K].
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Temperature of environment Tw and initial temperature of sand mould were received
as equivalent Tf = Tw =303 [K].
The surface film conductance from sand mould to environment averages
αk = 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/m3]
k [W/(m·K)]
T ≤ TS
560
7100
40.5
TS ≤ T ≤ TL
724
7100 ÷ 6920
40 ÷ 24
T ≥ TL
851
6920
24
Coefficient of absolute viscosity varied with temperature Tz-TL 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/m3]
k [W/(m·K)]
η [N·s/m2]
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
178
1490 K
1040 K
744 K
445 K
295 K
x
φ 10
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.
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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. 821831.
[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.
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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