UDK 621.74:519.68
Original scientific article/Izvirni znanstveni članek
ISSN 1580-2949 MTAEC9, 45(1)47(2011)
MODELLING OF THE DIRECTIONAL SOLIDIFICATION OF A LEADED RED BRASS FLANGE
MODELIRANJE USMERJENEGA STRJEVANJA PRIROBNICE IZ RDEČE SVINČEVE MEDENINE
Vladimir Grozdanic
University of Zagreb, Metallurgical Faculty, Aleja narodnih heroja 3, 44103 Sisak, Croatia
Prejem rokopisa - received: 2010-04-21; sprejem za objavo - accepted for publication: 2010-11-02
A mathematical model and a numerical simulation of the directional solidification of a leaded red brass flange are presented. The mathematical model is based on physically realistic assumptions and it is solved by the finite-difference method, implicit alternating directions method. This method has great accuracy and is of second order with regard to the approximation of time and space. The initial conditions are an analytical solution of the heat conduction equation in the case of the contact of two semi-infinite media. The latent heat of fusion incorporated into the equation for the specific heat capacity of metal, and temperature dependent thermophysical properties of all materials in the system mould-casting-core-chill, enables us to accurately numerically represent the flange solidification, which is casting and a common example in foundry practice. The simulation of the directional solidification is a modern and scientific way of pointing to casting points in which the defect occurrence is possible, and how to prevent them using the chill.
Keywords: mathematical model, directional solidification, leaded red brass flange, graphite chill
Predstavljena sta matematični model in numerična simulacija usmerjenega strjevanja prirobnice iz rdeče svinčeve medenine. Podlaga matematičnega modela so realne fizikalne predpostavke, model pa je razvit z metodo končnih razlik, implicitno metodo z alternativno smerjo. Ta metoda je zelo natančna in je drugega reda glede na približek časa in prostora. Začetni pogoji so analitične rešitve enačbe za toplotno prevodnost za primer kontakta dveh semineskončnih medijev. Latentna talilna toplota, kije uporabljena za specifično toplotno kapaciteto kovine in termofizikalne lastnosti vseh materialov v sistemu forma - litje - jedro -kokila omogočajo, da se numerično modelira strjevanje prirobnice, torej litje, kar je splošen primer dela v livarni. Simulacija usmerjenega strjevanja je moderna metoda, da se znanstveno opredeli mesta ulitka, kjer lahko nastajajo napake in kako te napake s kokilo preprečiti.
Ključne besede: matematični model, usmerjeno strjevanje, prirobnica, svinčeva rdeča medenina, grafitna kokila
1 INTRODUCTION
The directional solidification of castings of a given geometry may be obtained using risers or chills. Risers are reservoirs of molten metal and feed the casting in the liquid state. In the case of classical risers, only 17 pct. of the initial volume of the riser is available for the feeding of the casting 1. In contrast to risers, chills better lead away the heat and in this way speed up the solidification. Usually, they are used when the placement of risers is impossible. However, in the case of nonferrous metals
Figure 1: Flange and graphite chill with the corresponding dimensions Slika 1: Prirobnica in grafitna kokila z dimenzijami
this is not a rule. In this way, for example, in the investigated system of a leaded red brass flange, directional solidification is obtained using a graphite chill. With the placement of an appropriate chill, the hot spot is shifted in areas of the casting which are later machined. This is desirable because the hot spot solidifies last, and in their place the occurrence of casting defects (e.g., shrinkage cavity, porosity etc.) is possible. The investigated system is complex because it consists of a flange, a mould core and a chill, and it is shown in Figure 1. For a given system the mathematical model of the solidification is formulated and investigated.
2 MATHEMATICAL MODEL
In the development of the mathematical model, the following partial differential equation of heat flow corresponding to the flange (Figure 1) 2 should be solved:
(d 2 T 1 dT d 2 T ^
dT dt
- = a


dr2 r dr dz2
(1)
Since for the horizontal axis of the flange symmetry r = 0, equation (1) should be modified according to L' Hospital's rule considering the equation:
ar dt
d 2 T d 2 T
2 a;^ "a^^
(2)
Initial conditions
The mould temperature and the temperature of its outer side are equal to Ts, whereas the temperature of the metal is equal to the casting temperature TL. The initial temperature at the mould/casting boundary interface can be obtained by solving Fourier's differential equation for heat flow through the contact area of two semi-infinite media:
T = T +-
TL - T s
1+
K.,
K
(3)
K
K
K
K
K
K
dn
dTm
dn
dTm
dn
dTK
dn dn dn
= K
=K
=K
=K
=K
dn 37j
dn
dn 37,
dn
dTk
dn
dr,
dn
(4)
(5)
(6)
(7)
(8)
(9)
Figure 2: Temperature-dependent thermal conductivity of the leaded red brass flange
Slika 2: Odvisnost toplotne prevodnosti od temperature za prirobnico iz svinčeve rdeče medenine
Thermophysical properties of the material
It has been assumed that the thermal properties of the mould, metal, core and chill are temperature dependent 4, which is as shown in Figures 2 to 9.
The derivation of eq. (3) is given in Appendix 2.
Boundary conditions
The outer mould surface maintains a constant temperature Ts. For the contacts mould/metal, metal/core, mould/core, metal/chill, mould/chill and chill/core area there are continuous heat flows for which the boundary condition of the fourth type holds 3: dTm dTK
= kk
Figure 3: Temperature-dependent specific heat capacity of the leaded red brass flange
Slika 3: Odvisnost specifične toplotne kapacitete od temperature za prirobnico iz svinčeve rdeče medenine
Figure 4: Temperature-dependent thermal conductivity of the graphite chill
Slika 4: Odvisnost toplotne prevodnosti od temperature za grafitno kokilo
Figure 5: Temperature-dependent specific heat capacity of the graphite chill
Slika 5: Odvisnost specifične toplotne kapacitete od temperature za grafitno kokilo
m
a
k
Figure 6: Temperature-dependent thermal conductivity of the mould Slika 6: Odvisnost toplotne prevodnosti od temperature za formo
Figure 7: Temperature-dependent temperature diffusivity of the mould Slika 7: Odvisnost toplotne difuzivnosti od temperature za formo
000 t; »C
Figure 8: Temperature-dependent thermal conductivity of the core Slika 8: Odvisnost toplotne prevodnosti od temperature za jedro
Figure 9: Temperature-dependent temperature diffusivity of the core. Slika 9: Odvisnost toplotne difuzivnosti od temperature za jedro
3 IMPLICIT ALTERNATING DIRECTION METHOD
Differential heat-flow equations (1) and (2) with the corresponding initial and boundary conditions have been numerically solved using the implicit alternating direction method 5,6. The method utilises the division of the time interval into two steps.
In the first half of the time interval the equation is solved implicitly for z and explicitly for the r direction. The procedure is reversed in the second half of the time interval.
Consequently, for the differential equation (1) and the first half of the time interval At/2 we have:
rri n	r\ rp n .rri n	rji n	rji n
T i,j-1 - 2T i,j + T i,j +1 T i,j +1 - T i,j-1 -+ —:-:-+
(Ar)2	■ 2rj -Ar
Ti-1 ^ - 2Ti: + t;+ 1 ^ + -
1
rrr * rp n
At/ 2
(Az)2
Whereas for the second At/2 we obtain:
(10)
rji n + 1 _ o 7""
T i,j-1 2T ij
(Ar)
n+1 + T n+1 T n+1 _t n+1
+ T i,j+1 T i,j+1 T i,j -1 - + - +
2r -Ar
Th J - 2T^j + Tl 1 ^ (Az)2
^ n + 1 rpj
i,j i,j
At/ 2
(11)
The numerical solution for the differential equation (2) of the heat flow for the first At/2 is:
- T1 Tl, .1 -2T:, + Tl 1.1 4-:— + —
(Ar)
(Az) 1
*n
T i.1 - T i.1
ai ,1, n
and for the second At/2:
At/ 2
(12)
4
TX' - T1+1 T:_1 .1 -2Tij + T:^ 1.1
(Ar)
(Az)
^ n + 1 rj-i* T i.1 — T i.1
At/ 2
(13)
The use of the implicit alternating direction method results in a system of simultaneous linear algebraic equations with the variables V1, Vn of tri-diagonal form:
b1 v1 + c 1 v2 = d1
a2 v1 + b2 v2 + c2 v3 = d2
a3 Vo + b3 v3 + c4 v4 = d3
ai Vi-1 + bi vi + ci Vi+1 = di (14)
aN-1 VN-2 bN-1 VN-1 + cN-1 VN = dN-1 aNVN-1 + bNVN= dN
The special efficient algorithm for solving the tri-diagonal system of equations is 7:
+
a
1
a
(a)
VM - /N	y'^-'j
cy.
vi = 7 i -
ßi
i = n - 1, n - 2, ..., 1
(15)
(16)
where ß and 7 are calculated from recursive formulas
ßi = bi -
ßi = bi 7i = di/ßi
a.. c..
7 i = -
ßi-i '
- a. 7 i-i ßi
i = 2,3,...,n
i = 2,3,...,n
(17)
(18)
(19)
(20)
The tri-diagonal coefficients that give the algorithm of the flange solidification in a sand mould are presented in Appendix 3. Based on the presented algorithm a computer program was written in FORTRAN 77 and solved on an Athlon AMD 64 computer.
4	FLOW DIAGRAM
A detailed flow diagram is shown in Figure 10a and 10b.
The main feature of the program is the use of two temperature matrixes, namely T and V. The first matrix contains the temperatures at the start and the end of the time step, and the second contains the temperature at the end of first A?/2. Initial values are assigned to the program variables and constants. The program module TYP provides for the initial values of the temperatures, of particular net points as well as for the standardization of all the points in the mould, the casting, the chill core and their boundary interfaces. The module PRINT i prints out the initial temperature distribution. The system of tri-diagonal equations is then solved firstly, row by row (module ROWS), and then column by column (module COLS). The results are periodically printed over the whole geometry of the casting, the mould, the chill and the core (module PRINT i) or over the casting geometry only (module PRINT 2) until the prescribed
time tmax.
5	DISCUSSION
Leaded red brass (alloy C83600) has the following composition: 85 % Cu, 5 % Sn, 5 % Pb and 5 % Zn 8. The alloy C83600 is used not only for flanges, but also for valves, pipe fittings, water pumps' meter housings and impellers, small gears, high-quality plumbing goods, statuary and plaques. These applications are possible because of its corrosion resistance, machinability, strength, bearing properties, colour and the excellent castability of the alloy.
The simulation of the solidification of the leaded red brass flange in a sand mould is carried out by the space
Figure 10: Flow diagram Slika 10: Flow diagram
steps Az = i5 mm and Ar = 7 mm and the time step At = 5 s to tmax = 600 s.
The casting temperature was ii80 °C and the initial temperature of the mould, the core and the chill was 25 °C. The initial temperature at the sand/casting interface was i057 °C, on the casting/core interface it was i0i3 °C, and at the casting/chill interface it was 577 °C. On the basis of successive temperature print outs for particular net points, a solidification time of 367 s was obtained.
Figure 11: Progress of the isosolidus (854 °C) after (150, 240 and 300) s
Slika 11: Napredovanje izotr. (854 °C) po (150, 240 in 300) s
Using the isosolidus (854 °C) shift, as seen in Figure 11, it can be concluded that the potential defect site is obviously the point of the final solidification of the heat centre, i.e. in the vicinity of the core.
The accuracy of the simulation is limited, depending on the assumptions used in the mathematical model, the method of the numerical analysis and the values of ther-mophysical material properties used. Several assumptions were used in the elaboration of the mathematical model. The most important are as follows: the complete heat-transfer rate, a casting temperature equal to the initial temperature of the metal in the mould, and a mould/casting interface with perfect thermal contact. The first assumption restrains the analysis to the mould-casting-core-chill system with heat conduction only, i.e., partial heat flows associated with the mould and core moisture are not considered. The second assumption is the simplification introduced to avoid a complex consideration of the metal flow through the gate system and the mould cavity and the matching heat transfer. The assumption of a perfect thermal contact on the interface is acceptable because there is only a partial appearance of a gaseous clearance, and therefore in the mathematical formulation the boundary condition of the fourth kind is usually taken as valid. The partial differential equation for the heat flow is solved using a numerical method of finite difference - the implicit alternating direction method, which was chosen because of its high accuracy during the approximation of both time and space. Insufficient knowledge about the ther-mophysical properties of the material, especially at high temperatures, has a strong influence on the simulation of the solidification. It holds especially with respect to the thermal properties of the mould and the core material, which can be determined only experimentally. Moreover, the values for the thermal properties at higher temperatures show a considerable a dissipation.
6 rONCLUSTONS
dification was developed assuming thermal conduction as only heat flow in the system, which is considered as a physically real assumption. A differential equation for the heat flow suited to the flange's geometry was modified and numerically solved by the use of implicit alternating direction method. The temperature dependence of the thermophysical material's properties was taken into account. Based on the obtained algorithm a computer program written in FORTRAN 77 for an Athlon AMD 64 computer was used for the simulation of the solidification. It was determined that the complete solidification takes 367 seconds. The progress of the solidification as well as the hot spot, i.e., the site of any potential shrinkage cavity, can be determined by a shift of isosolidus.
7 REFERENCES
1	V. Grozdanic, Metalurgija 25 (1986) 2, 51
2	J. H. Leinhard IV, J. H. Leinhard V, A heat transfer textbook, 3rd ed., Phlogiston Press, Cambridge, 2006
3	V. P. Isachenko, V. A. Osipova, A. S. Sukomel, Heat transfer, 3 rd ed., Mir Publishers, Moscow 1980
4	R. D. PehIke, A. Jeyarajan, H. Wada, Summary of thermal properties for casting alloys and mold materials, University of Michigan, Ann Arbor, 1982
5	J. Douglas, H. H. Rachford, Trans. Amer. Math. Soc., 82 (1956), 421
6	V. Grozdanic, Materiali in tehnologije 43 (2009) 5, 233
7	B. Carnahan, H. A. Luther, J. O. Wilkes, Applied numerical methods, John Wiley, New York, 1969
8	AFS, Casting coppar-base alloys, American Foundrymen's Society, Des Plaines, 1984
9	M. R. Spiegel, Laplace transforms, McGraw-Hill, New York, 1965
10	M. Abramowitz, I. A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards, Washington, 1964
Appendix 1
Abbreviations used: a -- temperature conductivity
at, bi, Ci, di - coefficients adjoining unknowns in a tri-dia-
gonal system of algebraic equations Cp - specific heat capacity at constant pressure k - thermal conductivity n - vertical direction r - space coordinate t - time
T - temperature
vi - unknown in system of simultaneous equations 7 - space coordinate
A numerical simulation of the directional solidification of a leaded red brass flange was carried out on the basis of a suitable mathematical model. The complex model system is composed of four materials: the mould, the core, the chill and the casting of comparatively complicated geometry. The mathematical model of soli-
Appendix 2
The following partial differential equation of heat flow with the appropriate initial and boundary conditions must be solved to derive the equation for the temperature
distribution at the mould/metal interface/
dT d2 T
- = a-
dt
dx2
T(x,0) = Ts
T(0,t) = Ti	(22)
\T(x,t)\ < M
where M is a positive real constant.
The equation of heat flow should be solved for the case of the contact of two semi-infinite media. The partial differential equation (21) must be transformed into a common differential equation by the Laplace transform defined as 9:
Two semi-infinite media (mould and metal) are in (21) interfacial contact and the boundary condition of the fourth kind is valid:
l{t(x, t)}= e(x, s) = e-stT(x, t)dt
(23)
Equation (21) in the Laplace region has the from: d2 Q 1 1
—^ ^ sQ=--T^	(24)
dx 2 a	a s
The solution of (24) has the form:
___ T
Q(x,s) = Cj(s) exp(Ws / a) + c2(s) exp(-Ws / a) ^(25)
Because of temperature limitations, |T(x,t)| < M, Ci = 0 for x^^, i.e.
T
Q(x, s) = c2 exp(-^Vs7ä)(26) For the boundary condition T(0,t) = Ti, i.e.
T, Ti 1 Q(0, s) = C2 = C2 = -(T1 - Ts)
The final result in the Laplace region is:
Ti - Ts	I- Ts
Q(x, s) ^ ^-exp(-^Vs/a)	(27)
ss
By passing from the Laplace region into real space, we have:
T(x, t) = (Tj - T^.)erfc

T
respectively
(T - T,)
= erf
x
^ 2401
(Ts - T,) ^ where the error function is defined as 10:
2
erf(t) = -
jn
e " dM
The temperature gradient along the x-axis is:
dT T^ - T,
respectively
dx -Jnat
dT
exp(-x2 / 4at)
= 0
Ts - T,
-Jnat
(28)
(29)
(30)
(31)
(32)
-K
(dT, ^
dx
/"\T-' \

dT
dx
By including the proper temperature gradients:

T - T
^ = k
T, - Ti
.jnäkt ^ na^t
(33)
(34)
Finally, the initial temperature distribution on the boundary mould/metal surface is obtained:
T, - Ts
T = T
1+
Kk la K
(35)
Appendix 3
The constants, which in tri-diagonal coefficients, are: aAt
Pi = P2 =
2(Ar)2 aAt 4rj - Ar
P3 =P1 -P2 P4 =P1 + P2
At( k A + k b)
P5 =
P6 =
2c(Ar)2 At( k A + k b)
4crj - Ar
kA k B
c = —+—
=
q 2 = q 3 =
aAt 2(Az)2 k A At c(Az)2 K At
q 4 ="
q5 =
q6 =
c(Az)2 At( k A + k B)
2c(Az)2 k A At c(Ar)2 k B At c(Ar)2
Tri-diagonal coefficients
1. Point (i,j) in the mould, metal core or chill - first At/2:
/x = 0
x = 0
aA aB
ai = Ci = -qi bi = 1 + 2q1 di = p 3 Tn-1 + (1-2P1) Tn + p 4 Tn+1
- second Af/2:
(36)
aj = -P3 bj= 1 + 2pi
Cj = -P4	(37)
dj = q 1 Ti-1, j +(1-2q i)Ti: j + q 1 Ti+ i.j
2.	Point (i',j) on the boundary surface parallel to the r axis separating the material A (left) and B (right)
-	first Af/2:
ai =-q 2 bi =1+q 2 + q 3
Ci =-q 3	(38)
= (P 5 - P 6)Ti"j -1 + (1-2P 5)Ti"j + (P 5 + P 6)Ti"j+1
-	second Af/2:
aj =-(P5 -P6)
bj =1+2p 5
Cj =-( P 5 - P 6)	(39)
dj = q 2 Ti-1, j +(1-q 2- q 3)T*j + q 3 Ti+1.j
3.	Point (i',j) on the boundary surface parallel to the z axis separating the material A (down) and B (up).
-	first Af/2:
a i = c, =-q4
b, = 1+ 2q 4	(40)
d, = (q5 -q6)Tin-1 +(1-2p.^t;. + (q5 + q1
-	second Af/2:
aj = P6- q 5
b. =1+p 5
Cj =-(q 5 - P 6)	(41)
dj = q 4 Ti*-1, j + (1-2q 4) Ti: j + q 4 Ti: 1,.
4.	Point (i',1) out of the boundary surface
-	first Af/2:
ai = Ci =-q 1
b, =1+2q1	(42)
d, = (1-4 p1)Ti"1 + 4 P1 Ti"2
-	second Af/2:
bj =1+4p1
C. = 4 P1	(43)
di = q 1 Ti*-1,1 + (1-2q1) Ti:1 + q 1 Ti: 1,1
5.	Point (i',1) on the boundary surface, which separates the material A (left) and B (right)
-	first Af/2:
a, =-q 2 b^ = 2q 4 +1 Ci =-q?3	(44)
di = (1-4 p,)Tn, + 4 P5 Ti"2
-	second Af/2:
bj = 4 p 5 +1
C. =-4 p 5	(45)
dj = q 2 Ti*-1,1 +(1-2q 4)Ti*1 + q 3 Ti*+1,1