thermal analysis of piston casting using 3-d finite element method

*Corresponding author.E-mail address: [email protected] (R. Palaninathan).

Finite Elements in Analysis and Design 37 (2001) 85}95

Thermal analysis of piston casting using 3-D "nite elementmethod

D. Robinson, R. Palaninathan*

Department of Applied Mechanics, Indian Institute of Technology, Chennai-600036, India

Abstract

This paper presents a "nite element modelling of metal solidi"cation, applied to an aluminum}silicon alloypiston casting. Earlier studies on solidi"cation have been mostly con"ned to one- and two-dimensionalproblems whereas the present one deals with 3-D castings. It is assumed that the heat transfer within the meltis by conduction and that from the melt/casting to the mold is by the interfacial heat transfer process. The airgap formed between the casting and the mold at interface a!ects the heat transfer rate. A "nite elementformulation for the heat transfer through the air gap is presented for the 3-D problems. The same is appliedto the speci"c case of piston solidi"cation. Temperature pro"les at di!erent time steps are presented. Thetime taken for completion of solidi"cation with respect to the process parameters are discussed. ( 2001Elsevier Science B.V. All rights reserved.

1. Introduction

Casting is an important process in metal forming. Complex shapes are made using the castingprocesses for several industrial applications. Optimum mold design is a must for good qualitycastings. Development of mold through experimental study is a di$cult and costly route. Alterna-tively, modelling of the solidi"cation process, taking into account various factors close to thereality will help to achieve an optimum mold design. A mold design evolved through simulationstudies can be subjected to experimental veri"cation, with reduced cost and time. The processparameters a!ecting the quality of the casting are: pouring temperature (¹

1), mold temperature

(¹.), shape of the casting, rate of cooling etc. Basically, the metal solidi"cation process is a heat

transfer problem with phase change. Initially, the mold is "lled with liquid metal. As the heat isconducted through the mold to the atmosphere, the temperature reduces and solidi"cation starts

0168-874X/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 8 7 4 X ( 0 0 ) 0 0 0 2 0 - 2

from the mold wall and progresses inward. During this phase change, latent heat is released at thesolidi"cation front in the case of pure metal and in the mushy zone in the case of alloys. Themodelling of solidi"cation process has remained a topic of active interest for several decades.Earlier studies were con"ned mainly to one-dimensional problems with severe approximationssuch as properties remaining constant with temperature. Recent developments in analyticalmethods and advanced computational facilities have enabled improved modelling, taking intoaccount various parameters which in#uence the quality of castings.

The "nite element method (FEM) involves a physical approximation wherein the given region isdivided into small domains (elements). The "eld variable inside the element is approximated interms of nodal values. Element matrices are obtained using variational principles and are assem-bled in the same way as the elements constitute the given region. This procedure results in a set ofsimultaneous equations. The solution of the set gives the "eld variable at the nodes. Several textsare available on FEM [1,2]. Very few texts are available on heat transfer including solidi"cationusing FEM [3]. In addition, several references on thermal modelling are available in the literature[4,5]. In FEM solution, in addition to the formulations of element matrices, two aspectsconcerning solidi"cation to be considered are: (i) latent heat release [6}9] and (ii) metal}moldinterface heat transfer [10}12]. As may be seen from the literature, most of the studies have beenconcerned with 2-D modelling of solidi"cation. The present paper is concerned with the 3-Dsolidi"cation modelling. The derivation of element matrices are not presented here. The contribu-tions of this paper are the formulation for interfacial heat transfer (gap element matrix) in thecontext of 3-D problems and a numerical study on the solidi"cation of aluminium}silicon alloy pistons.

2. Finite element formulation

It is assumed that the mold is "lled with liquid metal with uniform initial temperature. The heattransfer from the molten metal to the mold by convection is neglected. The thermal behavior ofeither liquid or solid is governed by the same equation, namely the classical heat conductionequation, which is expressed in Cartesian co-ordinates as

LLx Ck

L¹Lx D#

LLy Ck

L¹Ly D#

LLz Ck

L¹Lz D"oC

p

L¹Lt

, (1)

where k is the thermal conductivity, o the density and Cp

the speci"c heat and ¹ the temperature.In matrix form the above equation can be written using weak form of representation [3] as

PV G

L¹Lx

L¹Ly

L¹LzHT

Ck 0 0

0 k 0

0 0 kD GLwLx

LwLy

LwLxH d<#P

V

oCp

L¹Lt

wd<#PA

h(¹!¹a )wdA"0. (2)

86 D. Robinson, R. Palaninathan / Finite Elements in Analysis and Design 37 (2001) 85}95

The last term in Eq. (2) will be applicable for elements on the boundary with convective heattransfer. ¹a is the temperature of the medium surrounding body. A three-dimensional isoparamet-ric element with 8}20 variable nodes is employed. The #exibility in the choice of mid-side nodes inthree directions helps to model the linear or quadratic variations to the temperature pro"lesselectively. Following the established steps of FEM, Eq. (2) is transformed to [1]

[K]%M¹iN#[C]%M¹Q

iN"MFN%, (3)

where ¹iare the nodal temperatures and ¹Q

iare the time derivatives. [K]% is the conductivity matrix

and [C]% is the capacitance matrix. In general, the conductivity matrix is expressed in two parts as

[K]%"[K#]%#[K

)]%"P

V

[B]T[k][B]d<#PA

hMNNSNTdA. (4)

The second term in the above equation will be applicable only for those elements lying on theboundary and having speci"ed convective heat-transfer coe$cient. This term is employed to developthe gap element in the next section. The capacitance matrix and the thermal load vector are

[C]%"PV

oCpMNNSNTd<, MFN%"¹aP

A

hMNNdA. (5)

2.1. Casting}mould interface heat transfer

For modelling the heat transfer from the casting to the mold, coincident node technique has beenemployed. In this technique, the interfacial heat transfer coe$cient (IHTC) is used to account forthe heat transfer from casting to mold. The computation of the interfacial heat transfer coe$cient isa very complicated procedure. This is mainly controlled by the air gap formation between the moldand the casting. It is more appropriate to use the values obtained from experiments conducted onspeci"c systems of metal and the casting process. In general, in FEM context, for an element havingconvective heat transfer across its face, the conductivity matrix acquires an additional term (thesecond part on the right-hand side of Eq. (4)). Following the same procedure, convective part of theconductivity matrices of elements of the casting and the mold, lying on either side of the interfacescan be introduced. In the present analysis, these additional convective terms are not added to theconductive matrices of respective elements, but are grouped to give a conductivity matrix for thevirtual gap element. Hence, along with the casting elements and mold elements, a third type ofelement group, namely, gap elements are used in the FE modelling of metal solidi"cation process.The gap element located in between casting and mold elements may be visualized as shown inFig. 1. The formulation of the conductivity matrix of a gap element (8 noded) is as follows. The rateof heat transfer is expressed as

q"APA

h'(¹

#!¹

.) dA (6)

where ¹#

and ¹.

are the temperatures on the casting side and the mold side, respectively, h', the

interfacial heat transfer coe$cient, accounts for heat #ex from casting to the mold by convection,conduction and radiation. In Fig. 1, nodes c1 and ml, c2 and m2, c3 and m3, c4 and m4 are thecoincident nodes on the interface. Even though a gap may exist between them, same spatial

D. Robinson, R. Palaninathan / Finite Elements in Analysis and Design 37 (2001) 85}95 87

Fig. 1. The gap element arrangement.

coordinates are given for the pairs of nodes and hence the name `coincident node techniquea. Onedoes not have to bother about the actual air gap, which will be varying with time and locations onthe casting surface. The overall e!ect of gap is speci"ed through the heat transfer coe$cients.Typical variations of the interfacial heat transfer coe$cient with time can be seen in [12]. Fordeveloping the conductivity matrix for gap element, Eq. (2) is made use of. For an element lying onthe casting side with one face exposed to the gap, Eq. (2) can be rewritten as

PV G

L¹Lx

L¹Ly

L¹LzHT

Ck 0 0

0 k 0

0 0 kD GLwLx

LwLy

LwLzHd<#P

V

oCp

L¹Lt

w d<#PA

h(¹#!¹

.)w dA"0. (7)

It is to be noted that ¹#

and ¹.

are unknowns which can be expressed in terms of nodaltemperatures of the corresponding faces of the casting and the mold as follows:

¹#"

n+i/1

Ni(#1, s, t)¹#

i, ¹

."

n+i/1

Ni(!1, s, t)¹.

i. (8)

Substituting Eqs. (8) into (7) and following the usual FEM procedures we have for an element onthe casting side of the interface

[K]%M¹

iN#[C]M¹Q

iN#AP

A

hMNi(#1, s, t)NSN

i(#1, s, t)TM¹#

iN dA

!PA

hMNi(#1, s, t)NSN

i(!1, s, t)TM¹.

iNdAB"M0N. (9)


Similarly, the equation for the element on the mold side, with side (r"!1) facing the casting isexpressed as

[K]%M¹

iN#[C]M¹Q

iN!AP

A

hMNi(!1, s, t)N SN

i(#1, s, t)TM¹#

iNdA

!PA

hMNi(!1, s, t)N SN

i(!1, s, t)TM¹.

iNdAB"0. (10)

The terms within the larger parentheses of Eqs. (9) and (10) constitute the gap element conductivitymatrix which may be expressed as

[K']"P

A

h CN1N1 N1N4 N1N8 N1N5 !N1N1 !N1N4 !N1N8 !N1N5N4N1 N4N4 N4N8 N4N5 !N4N1 !N4N4 !N4N8 !N4N5N8N1 N8N4 N8N8 N8N5 !N8N1 !N8N4 !N8N8 !N8N5N5N1 N5N4 N5N8 N5N5 !N5N1 !N5N4 !N5N8 !N5N5

!N1N1 !N1N4 !N1N8 !N1N5 N1N1 N1N4 N1N8 N1N5!N4N1 !N4N4 !N4N8 !N4N5 N4N1 N4N4 N4N8 N4N5!N8N1 !N8N4 !N8N8 !N8N5 N8N1 N8N4 N8N8 N8N5!N5N1 !N5N4 !N5N8 !N5N5 N5N1 N5N4 N5N8 N5N5

DDJDdsdtG

M¹#iN

M¹.iNH. (11)

It is to be noted that N*in the above equation are functions of s and t only. The gap element does

not possess the capacitance matrix and the load vector. In the above matrix, the size 8]8corresponds to the 8 nodes in the interface (4 from casting side and another 4 from mold side). Thesubscripts of shape functions, N

icorrespond to the corner nodes, which follows the general 3-D

element description [1]. Similarly, for elements with mid-side nodes also, the gap element conduct-ivity matrix has been obtained, the maximum size of which is 16]16. The software programdeveloped has the capability for variable node gap element (8}16 nodes) formulation [13].

2.2. Non-linear transient solution procedure

In solidi"cation problems, the "eld variable, namely the temperature at all nodal points varieswith time. Moreover, since the thermal properties like, conductivity, density and speci"c heat alsovary with temperature and hence with time, a non-linear transient solution technique has to beused to solve this problem. There are several methods available in the literature for solvingnon-linear transient problems. A two-level time-stepping numerical scheme [13] is used in thepresent analysis. The Eq. (3) is transformed to

[KHH]M*¹N5#MRN"0, (12)


Table 1Process parameter details for piston casting

Case Pouring temperature¹

1(3C)

Initial temperature of mold andpunch ¹

.(3C)

1 700 1002 700 2003 700 4004 800 1005 800 2006 800 4007 800 Pro"le at the end of case 48 800 Pro"le at the end of case 7

Fig. 2. (a) Top view of piston casting}mold}punch arrangement. (b) Sectional views of Fig. 2a, showing locations fortemperature pro"les.

where M*¹N5

refers to the incremental nodal temperatures during a time step and [KHH] is thetransformed matrix. An iterative scheme is employed to solve Eq. (12). The detailed solutionprocedure is available in [13]. The software developed under this work has been validated againstseveral example problems available in the literature.

3. Thermal analysis of diesel piston casting

The solidi"cation of diesel engine piston is considered for analysis. Fig. 2a shows die-punch-casting in as-cast condition. The internal con"gurations have somewhat complicated shapes due to


Fig. 3. Temperature pro"les along A1 * case 2 (¹*"7003C, ¹

."2003C).

the existence of gudgeon pin boss. The aluminium !13.2% Silicon alloy is used. The metallic dieset consisted of four pieces (two halves of the mold), and one centre punch (core) and a bottom pad.The properties of the mold and the punch materials are: thermal conductivity k"33 w/m 3C;density o"7500 kg/m3 and speci"c heat C

p"624 J/kg 3C.

The FE idealization consisted of three groups of elements. The "rst group was that of castingwith 554 elements and the second group was that of punch and mold with 1141 elements. Thinnerelements (2-mm thick) with mid-side nodes in the radial direction were used on either side of theinterface, in view of the steep thermal gradient. The largest size elements were used at the centre ofcrown of the casting and also on the outer side of the mold. The third group represented theinterfacial gap, between the casting and the mold and between the casting and the punch (numberof elements being equal to 174). The variable interfacial heat transfer coe$cients have been used forthe air gap between the mold and casting [12], whereas a constant value of 20,000 W/m2 was usedfor the interface between the casting and the punch. This is because the casting makes a goodcontact with the punch due to shrinkage and hence there was no air gap formation. The totalnumber of nodes in the whole FE model was 5536, which is equal to the number of equations. Thelowest time-step used was 0.001 s which was gradually increased to 1 s. The thermal analysis of thecasting was carried out for eight cases of process parameter combinations as given in Table 1. Thepouring temperature, ¹

1, and the initial mold temperature, ¹

., are assumed uniform at t"0 in

the "rst six cases. In the latter two cases, while ¹1

is assumed uniform, ¹.

has nonuniform


Fig. 4. Temperature pro"les along A2 * case 2 (¹*"7003C, ¹

."2003C).

distribution at t"0. In Case-7, ¹.

at t"0 corresponds to the mold temperature at the end ofcasting time (t"120 s) from Case-4. Similarly, ¹

.at t"0 for Case-8 corresponds to that of

Case-7. The temperature pro"les for only one combination (Case-2) are presented in this paper.The results are presented at chosen points (A, B and C) and along chosen lines A1, A2, and AB asindicated in Figs. 2a and b. Fig. 3 gives the temperature pro"le along A1 under Case-2 at 7 di!erenttimes. The horizontal line shows the solidi"cation temperature, ¹

4"5773C. Initially, the temper-

ature of the casting remains at the pouring level (¹1) and temperature of the mold remains at the

initial level (¹.) with a step at the interface. As time increases the temperature of the casting

decreases and that of the mold increases. It takes about 60 s for the last point to solidify in this case.It may be observed that the outer temperature of the mold was not a!ected even after 100 s.Fig. 4 shows the temperature pro"le along A2 (middle height of the piston), where the castingthickness is very small. It may be seen that solidi"cation starts as early as 5 s. The temperaturepro"le of the punch increases fast compared to that of the mold. Fig. 5 shows the variation oftemperature with time at the points A, B and C. B and C are coincident nodes on either side of theinterface. The initial temperature of the point C was 2003C. It suddenly shoots up to 5003C anddrops down to 4003C in 10 s. Point A is at the centre of the crown, at which the temperature dropsdown from 700 to 6403C within 5 s and reduces gradually to the solidi"cation temperature, ¹

4at


Fig. 5. Variation of temperature with time at selected points* case 2 (refer Fig. 2b for locations of points A, B and C).

about 60 s. Then it suddenly drops down to 5003C in about 5 s and then decreases slowly. Thelatent heat e!ect can be attributed for this kind of change in the temperature pro"le. Initially, theheat released from the sensible heat capacity of the casting is conducted through the interface. Thishappens for about 5 s for the material system selected for the problem. Subsequently, additionalheat (latent heat) is released from the casting and solidi"cation starts from the interface. Thisreduces the rate of change of temperature at the points inside, which continues till the solidi"cationreaches the centre at about 60 s. Fig. 6 shows the temperature pro"le along the line, AB drawnvertically at the centre of the casting. The distance z"40 mm refers to the bottom of the castingand z"75 mm refers to the point on the casting, coincident with the bottom-most point of thepunch. As seen from the "gure, the region of the casting close to the punch is the last to solidify,which takes place at about 60 s. The micro-structural examination of piston castings made in thelaboratory revealed existence of smaller size shrinkage porosity only at this region, which con"rmsthe fact that the region just below the punch tip is the last to solidify. The temperature pro"les forthe remaining seven cases are available in [13]. The initial mold temperature has greater e!ect onthe solidi"cation time (time taken for the last point to solidify). For the material system considered,the lowest solidi"cation time of 44 s is obtained under Case-1 (¹

1"7003C, ¹

."1003C) and the


Fig. 6. Temperature pro"les along line AB (Fig. 2b) * case 2.

highest time of 290 s is obtained under Case-6 (¹1"8003C, ¹

."4003C). Case-8 is the repeated

castings of Case-4 and the solidi"cation time is found to increase from 60 to 87 s.

4. Concluding remarks

A gap element with 8}16 variable nodes to model the heat transfer at the interface between thecasting and the mold has been formulated. The new gap element has been subjected to validationchecks using sample problems from literature and is found to perform satisfactorily. The solidi"-cation of an automotive piston casting in metallic mold with central core/punch was studied usingthe present formulation. The two process variables, namely, the initial melt and the moldtemperatures, were varied and the temperature pro"les at di!erent locations were obtained. Fromthe numerical examples, it is found that the skirt region, the thinnest part, solidi"es "rst, as early asabout 1 s and the central region of the crown is the last to solidify, at about 60 s. A combination ofmold and melt temperatures, respectively, either 1003C and 8003C and 2003C or 7003C seems to bethe appropriate value, for the casting process of the material system considered.


References

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thermal analysis of piston casting using 3-d finite element method

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