12

CHAPTER 2

LITERATURE REVIEW

2.1 METAL – MOLD INTERFACIAL HEAT TRANSFER

During the solidification of metal castings, an interfacial heat transfer

resistance exits at the boundary between the metal and the mould. This heat

transfer resistance usually varies with time even if the cast metal remains in

contact with the mold, due to the time dependence of plasticity of the freezing

metal and oxide growth on the surface. This resistance to heat flow at the metal –

mold interface has a marked influence on the solidification rate of metal castings,

especially in permanent mould or die castings and in sand castings involving

chills.

The rate of heat transfer from the molten metal to the substrate is often

limited by the thermal contact resistance. Ruhl (1967) and Wang et al (1991)

have confirmed that the interfacial heat transfer coefficient between the casting

and the mould was among the most important variables that controlled the melt

cooling and solidification. In addition, it is possible to promote the accuracy of

numerical solutions for different solidification models done by Wang et al

(1992), Trapiga et al (1992), Bennett and Poulikakos (1994), Tong and Holt

(1996) and Rangel and Bian (1998), if information regarding interfacial heat

transfer coefficient is known with sufficient accuracy. It is therefore essential to

13

investigate interfacial thermal conductance during rapid contact solidification

process.

In order to determine this thermal resistance, it is necessary to develop

a mathematical method which enables the calculation of thermal resistances at

the interface from measurable quantities such as thermal histories at various

thermocouple locations.

Among the mathematical methods described in the literature, three

main groups may be identified including

i. Purely analytical.

ii. Serial – analytical and empirical

iii. Numerical techniques based on FDM and FEM.

In purely analytical techniques for analyzing temperature data,

Roberstone and Fascetta (1977), Garcia et.al. (1979) and Clyne and Garcia

(1980) have assumed a constant interfacial heat transfer coefficient, in order to

obtain an analytical solution for the Fourier equation of heat conduction.

In contrast, the use of semi – analytical or empirical techniques does

not attempt to solve rigorously the Fourier heat equation, but involves analyzing

the experimental data by means of semi-analytical formulae, done by Prates and

Biloni (1972) and Levy et.al. (1969), and on curve fitting done by Tillman and

Berry (1972).

14

In numerical technique, Ho and Pehlke (1983 and 1984) characterized

the heat transfer at the metal – mold interface in terms of physical mechanisms.

2.2 METHODS TO DETERMINE ‘h’ (HEAT TRANSFER

COEFFICIENT)

Attempts are done to calculate the heat transfer coefficient based upon

matching experimental solidification rates with analytical solutions. Flemings et

al (1961) and Prates et al (1972) conducted linear fluidity tests and Robertson et

al (1977) conducted experiments with metal at its melting point poured into a

semi-infinity mould. In the case of castings solidifying in permanent moulds it is

rather difficult to obtain exact analytical solutions for solidification rates in view

of the finite size of the mould and it has been recognized that numerical method

are better suited in these cases despite the approximate nature of the solutions

obtained as referred in Ruddle (1957).

The air-gap resulting from physical separation of the solidifying skin

results in a significant lowering of heat flux at the interface, and it is the most

important phenomena controlling the solidification of the casting in the metallic

mould. Many research workers like Ho and Pehlke (1984 and 1985), Sully

(1976), Issac et al (1984), Nishida et al (1986) Veinic (1968), Durham and Berry

(1974), Morgan et al (1981), Gozalan et al (1987) and Bamberger et al (1987)

have contributed to the understanding of heat flow across interface, the formation

air-gap, its spread over the entire casting surface, etc. The interfacial heat transfer

coefficient, h for various casting conditions have been reported by them.

The interfacial heat transfer coefficient, ‘h’ is measured by two methods viz.

15

(i) Inverse method and

(ii) Measurement of air-gap width.

2.2.1 Inverse Method

The basic idea of the inverse method is as follows:

Here an appropriate set of equations is required to describe the heat

transfer behavior in a thermal analysis. With the boundary condition, initial

condition and the thermo physical property of the material known, the

temperature at any point within the system can be obtained. Although, one of the

thermo physical properties of the material being unknown, the temperature

distribution can be obtained by the reverse method.

One of the first efforts to solve the inverse heat conduction problem

was reported by Stolz (1960). Sparrow et al (1964) adopted the use of operational

mathematics to solve the inverse heat conduction problem in one-dimensional

geometries.

Burggraf (1964) solved the inverse heat conduction problem by

assuming a solution of the surface temperature in the form of an infinite series,

whose terms involve higher order derivatives of the temperature and heat flux

histories at an interior point multiplied by some unknown spatial functions.

Beck (1965, 1969, 1967, 1968, 1979, 1970 & 1980) has studied the

inverse heat conduction problem from the stand point of effective treatment of

experimental data.

16

Analytical and numerical methods are available for solving the heat

flow equations by Jones (1969), Carslaw and Jaeger (1959), Croft and Lilley

(1977) and Segerlind (1976). For solving the inverse heat conduction problems

(IHCP), exact solutions are available. These are analytical procedures which

develop expressions for the boundary condition for a given temperature history

in the casting. A few analytical solutions to the IHCP in which a temperature

sensor is placed at an arbitrary location in the conducting body are available in

literature Burggraf (1964), Grysa (1980), Stolz (1960), Tikhonov and Arsebub

(1977) and Imber and Khan (1972).

A combined method of function specification and regularization was

reported by Tu (1988). The method of dynamic programming and its use for the

formulation and solution of the IHCP are presented in matrix form Nicholas et al

(1988 & 1989) investigated the two dimensional linear IHCP. The non linear

estimation procedure was used by Beck (1985) for numerical solution of the

IHCP. This procedure has the advantage over the other numerical procedures

that it takes into account inaccuracies in measuring the location of the

thermocouples, statistical errors in temperature measurement and uncertainty in

material properties Beck (1970).

Woodbury et al (1998) used the inverse method to measure the IHTC

between a resin-bonded mould and cast aluminum plates. Shivkumar (1993)

made some preliminary work on the heat transfer in Lost Foam Casting (LFC).

Venkataramani and Ravindran (1996) have analyzed the effects of coating

thickness and pouring temperature on thermal response in LFC.

17

In this method, the temperature at some locations near the cast – mould

interface is measured and using that data, the interfacial heat transfer coefficient

is determined. Ho and Pehlke (1985) and Beck (1991) have used this technique to

find ‘h’, Hwang et al (1994) have employed both the techniques to measure ‘h’.

Mahallawy (1984) Assar (1992) and Taha et al (1992) have determined ‘h’ of the

chills. Nishida et al (1986) have estimated it for Al and Al alloy for plate and

cylindrical geometries. Zavarise et al (1992) used a thermo mechanical approach

while Zu Zanak (1991) gave criterion for the neglect of ‘h’ in metallic and sand

moulds.

2.2.2 Measurement of air-gap width

The air-gap, forms during casting solidification between the cast and

mould, has always been a matter of importance to researchers for simulating the

casting solidification. It is a cumulative movement of cast, which shrinks, and the

mould, which expands. This air-gap among several other factors resists heat

transfer between the cast and mould. It influences the solidification rate and

solidification time substantially. Interfacial heat transfer coefficient (IHTC)

decreases when the air-gap increases and there by increases the solidification

time of any casting. Hence to estimate the change in IHTC, the transient variation

of the size of air-gap is very much required.

Many researchers have used various techniques either to measure the

air-gap or to measure the IHTC directly. Henzel and Keverian (1960) have

reviewed in detail the literature on air gap formation. Veinik (1968), Mackenzie

and Donald (1950) have suggested that the total air gap formed is the sum of the

18

contraction of the casting and expansion of the mould. Based on their

experimental investigation, Mackenzie and Donald (1950) have found that the

time of start of formation of air gap is different from bottom to the top.

Lewis et al (1956) and Fredrickson et al (1979) have reported that the

air gap causes a substantial reduction in the rate of heat transfer from casting to

mould, which in turn may affect the size of the air-gap.

The rate of heat transfer may also be affected by the type of the contact

between mould surface and the liquid /thin shell of the casting as has been

discussed by Prates and Biloni (1972) and Davies (1980).

However, in most of the cases as suggested by other investigators,

Pehlke (1971) and Durham (1974) also mentioned that the heat transfer through

the air gap may be due to the combined conduction and radiation.

Nelson et al (1970), Roth (1933), Bishop et al (1951), Srinivasan et al

(1970), Nehru (1974) and Thamban (1978) have reported the influence of

volume ratio on solidification time. All the investigators have assumed certain

value for the thermal resistance offered by the air-gap, since no information was

available on the size, distribution and formation of air-gap.

Isotani et al (1969) and Nehru (1974) have studied the influence of

coating thickness on the solidification time.

Determination of the air-gap during solidification requires

simultaneous measurements of both the mould wall and casting movements.

19

Gittus (1954) observed both motions by measuring the movements of silica

probes in sand and metal by means of dial gauges. Engler et al (1973) improved

measurement accuracy by employing inductive movement sensors.

Malur N.Srinivasan (1982) has done experiments with plate and

cylindrical castings by pouring the metal at different temperatures into the

moulds of different wall thickness and pre heated to different temperatures.

Solidification time of these castings was matched with the experimental

solidification times and the heat transfer coefficients at the cast-mould interface

in the experimental castings were computed.

Pehlke et al (1982) have conducted experiments to find gap formation

of vertically cast cylinders of iron, copper and aluminum based alloys in dry and

green sand. The advantage of a vertical configuration is that the gap should be

radially symmetric about the perimeter of the cylinder.

Chiesa (1990) measured the temperature variation of molten metal

during the casting. By comparing the theoretical model, he obtained the ‘h’ under

various coating conditions. His results also demonstrated that interfacial heat

transfer resistance existed even in the filling period when the metal was still in

liquid state. Kumar and Prabhu (1991) found that the heat flux was actually an

exponential function of time.

Ho and Pehlke (1983) have used two methods to measure the

interfacial heat transfer coefficient, viz.

20

(i) computer simulation of the inverse heat conduction problem

using thermocouple measurements at selected locations in the

casting and the mould and

(ii) measurement of the variation of the interfacial gap size with

time using a differential displacement between two transducers

which continuously record the mould and metal movements

respectively in the vicinity of the interface.

They explained three different mechanisms which may affect the

transition of a metal and mould in solid contact to an interfacial gap viz.

(i) surface interaction of the metal and mould,

(ii) transformation of metal and mould materials and

(iii) effects of geometry.

They studied the interfacial heat transfer on two related types of

castings. The influence of air-gap on solidification time with three mould

materials is compared by a numerical example, and criteria for utilizing

chvorinov’s rule are discussed.

Isaac et al (1985) have done experimental investigations and found

that the time of start of air-gap formation, its growth, total size and solidification

time depend on the different combinations of volume ratio and coating thickness.

Hao et al (1987) have indicated, based on experiments with ductile iron and

calculations that there is no air-gap between the ductile iron casting and the

mould due to the expansion of graphite precipitated during the solidification

period. Hou and Pehlke (1988) have done experiments with Al 13% Si casting

21

and measured the air-gap size and from that they determined the interfacial heat

transfer coefficients at the interface. Lukens, Hou and Pehlke (1990) have

measured the air-gap formation in cylindrical castings of Al alloy 356

horizontally in dry and green sand. They found that more heat transfer occurred

in the drag and the final solidification occurred in the top of the mould cavity.

Chiesa (1990) has measured the thermal conductance at the interface in

permanent moulds and studied the influence of insulation coating thickness on

the thermal conductance. Prasanna Kumar and Narayan Prabhu (1991) have done

experiments with Al13.2%Si and Al3%Cu 4.5%Si alloys of casting square bars

for different metal / chill combinations with and without coatings. Hwang et al

(1994) have found, by the measurement of air-gap size, the value of h varies with

time/temperature during casting. ‘h’ is very high as the air-gap starts forming and

keeps dropping by as the air-gap size increases. It approaches constant when the

air-gap fully developed. By inverse method, they found that the value of ‘h’

increases in the beginning and reaches a peak value. Then it drops rapidly at the

eutectic temperature and rises again until the end of solidification. After that ‘h’

keeps dropping until the end of casting.

Gerber et al (1995) has done numerical investigation on the influence

of air-gaps upon the solidification in a rotary caster. Mythily Krishnan and

Sharma (1996) have determined ‘h’ in steady state unidirectional heat flow using

Beck’s non-linear estimation procedure. Lau et al (1998) has used inverse

heat-conduction analysis to find ‘h’. They suggested a three stage segmented

linear equation for ‘h’. Khan et al (2000) have used inverse heat conduction

method on LFC and found that the value of the IHTC initially as 120-150

W/m2Kand gradually decreased to approximately 100W/m2K. IHTC was not

significantly affected by the amount of super heat over the range studied.

22

Shenefelt et al (2000) have used a new method for solving linear inverse heat

conduction problems using temperature data. The method employs a pulse

sensitivity matrix and singular value decomposition to convert the heat transfer

problem into the frequency domain. This method reduced material size, robust

treatment of noisy temperature data and lack of ad hoc parameters.

Parker and Piwonka (2000) have used eddy-current proximity sensor

for non-contact measurement of the gap formation between a solidifying

aluminum casting and a resin bonded sand mould. Woodburry et al (2000) have

investigated the interfacial heat transfer coefficients for resin bonded sand

casting of plate aluminum castings. Lewis and Ransing (2000) have done a

thermo-mechanical analysis of solidification process to predict the air-gap. They

used Lewis-Ransing correlation to link the stress model with an optimization

model for the optimal feeding design. Krishna et al (2001) have done the

evaluation of ‘h’ in LPPM (Low Pressure Permanent Mould) on Aluminum

casting process. They observed that the air-gap does not influence ‘h’ after a

lapse of about 300sec. and it approaches a constant value of about 400W/m2K for

any size of the air-gap. Santos et al (2001) have determined the transient ‘h’

values in chill mould castings. Using FDM technique, air-gap determined in

Al-Cu and Sn-Pb alloys were used for their study. Robinson and Palaninathan

(2001) have done 3D FE modeling of solidification of piston castings.

Wang and Qiu (2002) have studied the interfacial thermal conductance

in rapid contact solidification process. They used a temperature sensor of 1µm

thick to measure the rapid temperature changes. Wang and Mathys (2002) have

done experiments to determine ‘h’ during cooling and solidification of molten

23

metal droplets on a metallic substrate. The effect of melt super heat and surface

roughness were studied.

Gafur (2003) has investigated the effect of chill thickness and super

heat on ‘h’ during solidification of commercially pure aluminum. They have

done inverse analysis of the non-linear one-dimensional Fourier heat conduction

equation. Narayan Prabhu and Ravishankar (2003) have investigated the effect of

modification melt treatment on ‘h’ and electrical conductivity of Al 13%Si alloy.

Kayikci (2003) has used ultrasonic flaw detection techniques for the

determination of casting-chill contact during solidification. They used Al alloy

on to a Cu chill and observed a peak in ultrasound transmission, in the first two

seconds correlating to a maximum in the area of casting-chill contact. This was

followed by a decrease in the ultrasound transmission that corresponded to actual

contact areas between the casting and the chill in the region of 5 to 10%. Heinrich

and Poirier (2004) examined the effect of volume change during directional

solidification of binary alloys through numerical simulations of hypoeutectic Pb

– Sn alloys.

24

2.3 TECHNIQUES TO MEASURE THE AIR-GAP WIDTH

The interfacial heat transfer coefficient (IHTC), ‘h’ can be calculated

either by measuring the width of air-gap and using the formula,

h = k/a (2.1)

where

k – Thermal conductivity of air or gases in the air-gap and

a – width of the air-gap,

or by measuring the temperature at certain locations of the metal and mould using

the inverse method. Many researchers measured the air-gap width using various

techniques. Ho and Pehlke (1984) have explained the mechanisms of heat

transfer at the metal-mould interface. They explained three different mechanisms

which may affect the transition of a metal and mould in solid contact to an

interfacial gap viz.

(i) surface interaction of the metal and mould,

(ii) transformation of metal and mould materials and

(iii) effects of geometry.

Their work has drawn the importance of knowing what is in the air-gap

and the size of the gap width. Although the gap is universally called as the

‘air-gap’, this term may be somewhat misleading. In fact in many instances there

won’t be any air in it. For example, iron and steel castings in sand moulds, the

main gas present is H2. Campbell (1991) has estimated the size of the air-gap as

25

a = R [ αc (Tf – T) + αm ( Ti – To )] (2.2)

where

a – width of air-gap,

R – Radius of the casting,

αc & αm – thermal coefficients of the casting and mould,

Tf & T – freezing and final temperature of casting,

Ti – To – instantaneous and original temperature of the mould interface.

Kesavan and Seethramu (1994) have presented an entirely numerically

simulated program, which needs only the properties of the cast metal and mould

material as input. Ho and Pehlke (1983) have measured the magnitude of the

dynamic gap using the differential displacement between two transducers, which

continuously record the mould and metal movements respectively in the vicinity

of the interface. They used an experimental setup with a casting geometry of a

bottom gater vertical cylinder of 12.7cm diameter and 15.0cm height as shown in

Figure 2.1.

Using an insulating riser sleeve whose outer wall is covered by a layer

of fibrous kaowool minimizes heat flow in the radial direction. A Cu chill placed

on the top opening of the riser sleeve is forced cooled by a water cooled cap

which also supports linear transducers for measuring chill and metal movements.

As a result an approximately vertical heat flow pattern develops in the cast metal

as solidification progresses.

26

Figure 2.1 Schematic diagram of the apparatus (Ho and Pehlke, 1983)

.

The measurement of metal-chill interfacial gap width is carried out using two

linear transducers, which continuously record the movements of the metal and

chill independently in the vicinity of the interface. These linear transducers

thereby show the interfacial gap width. As solidification commences, a dual-pen

recorder continuously records outputs from these two linear transducers.

Isaac et al (1985) revealed their investigations that the gap starts

forming at different times on the surface of the casting and that the gap sizes on

the surface vary from point to point during solidification. The influence of

coating thickness and mould volume ratio on behavior of the air-gap has also

been studied. They conducted experiments and found that the air-gap formation

starting time, its growth, total size and solidification time were depend on the

different combinations of volume ratio and coating thickness. In their

experiment, aluminum was poured into a cast iron mould with a cavity size of

7x7x28cm as shown in Figure 2.2 (a). The casting was fed from the sides (two

27

risers) and only the central section was investigated. Their mechanical air-gap

measuring device is shown Figure 2.2 (b).

(a) (b)

Figure 2.2 (a-b) Mechanical Air-Gap Measuring Device (Isaac et al, 1985).

(a) Half Mould, (b) Measuring Device.

The principle of the method adopted for measuring the gap size is to

follow the movement of the casting surface by a very light spring-loaded plunger.

To prevent the plunger from getting in to the casting a small thin aluminum plate

is kept as a facing plate. The thickness is sufficient to retain its shape and support

the plunger during the initial; period before the casting shell was formed. This

setup takes care of mould thermal expansion. The movement of this plunger is

transferred to the LVDT plunger. UV recorder reads the output of the LVDTs.

Two LVDTs are used simultaneously for measurement of the air-gaps at the

corner and middle of the casting face. They found that the gap starts to form at the

corner first and then at the middle of the cast. Initially for about 25 Seconds, the

growth of gap is faster at the corner than at the middle and during the remaining

28

period the gap at the middle grows faster. At the end of solidification, the

maximum values of gap at both the corner and the middle come out to be about

85 microns.

Lukens et al (1990) conducted an experiment to measure the

mold/metal gap formation. They used two linear transducers connected to a

regulated power supply and a two-channel strip-chart recorder for reading the

movements of mould and metal. The schematic drawings of the mould

configuration for the horizontal cylinder are shown in Figure 2.3 (a). Figure 2.3

(b) shows the vycor tubes which were used to eliminate mould/probe friction and

as protection sleeves for the tubes.

(a) (b)

Figure 2.3 Air-Gap Measuring Device (Lukens, Hou and Pehlke, 1990).

(a) Mould Configuration for the Horizontal Cylinder, (b)

Probe Configuration used to Independently Measure the

Mould and Cast behavior.

29

They conducted the experiment for a horizontally cast cylinder of

aluminum alloy 356 in both dry and green silica sand. Results of these

experiments showed that mould wall movement was greatest at the bottom of the

mould cavity and least at the top. Dry sand showed movements of 0.22mm,

0.15mm and 0.07mm at the end of solidification for the bottom, side and top of

the mould cavity respectively. Green sand showed greater movements with

0.40mm at bottom, 0.30-0.35mm at the side and 0.14-0.18mm at the top. This

difference in wall movements between the two sands may be attributed to

moisture evolution at the mould-metal interface in green sand and greater mould

rigidity in dry sand. The results of gap measurements showed that more heat

transfer occurred in the drag or mould bottom.

Hwang et al (1994) have measured the air-gap using ceramic tubes for

displacement measurements. Optical displacement gauges are placed as shown in

Figures 2.4 (a), to reflect the expansion and shrinkage of the casting and mould.

As seen from Figure 2.4 (b), when molten metal was poured into the

cavity, both metal and mould expand outward. At approximately 340 seconds,

the movement of metal started to slow down while the mould still moves at the

same pace. That was the instant when gap starts to form. At approximately 1240

seconds, the outward movement of metal stopped and started to move inward

while the mould kept moving outward. At that point, the size of the gap abruptly

increased. The rate of increase of the gap slowed down when the mould stopped

expanding at approximately 1675 seconds. However the metal kept on shrinking

until the end of recording.

30

(a)

(b)

Figure 2.4 Air-Gap Measuring Device based on Optical Measuring

Device. (a) Apparatus setup for gap size measurement,

(b) Amounts of movements of casting and mould, and air-gap

size during whole casting period. (Hwang et al, 1994).

31

Parker and Piwonka (2000) used eddy current technique to measure

the gap formation in resin-bonded moulds.

Figure 2.5 Eddy current sensor technique of air-gap formation

measurement. (Parker and Piwonka, 2000).

As shown in Figure 2.5, an eddy current sensor was buried in the sand

mould and the molten alloy was poured into the mould. The sensor initially

registered the presence of the metal and gave an output of a particular frequency

as a function of distance D1. As the metal cooled, the gap was formed between

the metal and mould. The metal moved away from the sensor, denoted D2 and the

output of the sensor changed accordingly. The output of the sensor and time were

recorded continuously. The distance between the D1 and D2 showed the gap

formed.

Kayikci et al (2003) have used ultrasonic technique for the

determination of casting-chill contact during solidification. They used an

D2 – D1

32

experimental apparatus shown in Figure 2.6, to measure the contact area up on

the first contact of the molten alloy with a cold surface.

Figure 2.6 Experimental apparatus used to investigate the ultrasonic

reflection from the casting-mould interface. (a) For liquid Hg

cast onto a Cu mould. (b) For Al4.5wt%Cu alloy cast onto a

Cu mould. (Kayikci et al, 2003).

33

Figure 2.7 Schematic of the output from the ultrasonic flaw detector. It

shows H0, the amplitude of the echo from the surface of the

mould and H1, the amplitude of the echo from the upper

surface of the mould after pouring. (Kayikci et al, 2003).

Cu chill of 25mm diameter and 100mm length was inserted into the

bottom of a 300mm long refractory tube. The base of the Cu chill was surrounded

by a water jacket and the transducer of an ultrasonic flaw detector was applied to

the base of the chill. The water-cooling prevented the base of the chill becoming

hot, which might cause degradation of the coupling gel applied between the

transducer and the baser of the chill and a consequent loss in efficiency in the

ultrasonic signal transmission.

With the ultrasonic transducer placed at the base of the chill, the

monitor showed a strong echo from the upper surface of the chill. The amplitude

of this echo, H0 corresponded to a condition of complete reflection of the

ultrasonic signal from the upper surface of the chill. Upon contact of the cast

34

liquid metal with the chill surface some of the ultrasonic signal would be

transmitted through the areas of contact between them, resulting in a reduction in

the amplitude of the initial echo to new amplitude, H1 as shown in Figure 2.7.

They concluded that the results could only be obtained for the first few seconds

of casting and that suggested a peak in contact area, which could be as great as

70%, followed by a decline to values of around 5 to 10%.

2.4 FE SIMULATION OF CASTING SOLIDIFICATION

The finite element method involves a physical approximation of the

domain, wherein the given domain is divided into small domains, called as

elements. The field variable inside the element is approximated using its value at

the nodes. Elemental matrices are obtained using variational principles and are

assembled in the same way, as the elements constitute the domain. This

procedure results in a set of simultaneous equations. The solution of the set gives

the field’s variable at the nodes. Several texts are available on FEM like Bathe

(1996), Zienkiewicz (1988) and S.S.Rao (2001). The use of FEM enables

thermal modeling of solidification close to reality, taking into account

(i) Variation of properties with temperature ,

(ii) Effect of latent heat and

(iii) Casting – mould interface heat transfer.

Lewis et al (1996) dealt in detail with the FEM in heat transfer

analysis. References on thermal modeling are available in Comini et al (1974),

Morgan et al (1978 and 1981) and Rolph et al (1982). Many researchers like

Zienkiewicz and Parekh (1970), Bruch and zyvoloski (1974) and Tamma and

35

Namburu (1988) have used this method to solve non– linear transient heat

transfer problem.

The advantage of using FEM is the ability to handle complex

boundaries, the ease in implementing boundary conditions. But this method

requires much effort for formulation of the problem, data preparation and need

long processing time.

2.4.1 Incorporation of latent heat

The material properties undergo a drastic change during phase change.

In pure metal/eutectic alloys, phase change occurs at a particular temperature,

called as solidification temperature, Ts, and the total latent heat is released at this

temperature. In alloys, where solidification occurs over a range of temperature

(Tl – Ts) and the total latent heat is released over this range of temperature.

In a plot of heat capacity vs temperature, there will be a sudden jump at

Ts for pure / eutectic / metal and Ts and Tl for binary alloys, due to the release /

absorption of latent heat as the metal solidifies / melts. This jump can be

categorized as a dirac – delta function in mathematical modeling of the process

.However in the conventional finite element procedure, it is difficult to capture

this effect. Mostly it will be slipped, as the solidification front generally lies away

from the nodes.

36

The Latent heat release can be considered in two ways viz.

(i) Front tracking method and

(ii) Fixed grid method.

The front tracking method was used by Husan (1987). The fixed grid

method include

(i) Enthalpy method,

(ii) Fictious heat flow method

(iii) Capacitance method ,

(iv) Temperature recovery method and

(v) Inverse finite element method.

Among them the enthalpy method was used by Comini et al (1974),

Morgan et al (1978), Hogge et al (1982), Karl (1985), Kumar et al (1990), Droux

(1991), Chow et al (1992) and Chidiac et al (1993).

Diego et al (1994) used the temperature recovery method and Nicholas

Zabaras (1990) used the inverse finite element method.

37

In the Front Tracking Method, the phase change problem is treated as a

moving boundary problem. Here the grid is deformed in such a way that certain

modal points coincide with the solidification front. The solidification front is

tracked continuously and the latent heat is treated as a convective load term in the

final set of equations.

Among the fixed grid methods, enthalpy method is used here. The

effect of phase change can be incorporated by considering a sudden variation in

the heat capacity of the material at Ts. To account for this, a new variable viz. the

enthalpy (H) is introduced which is the sum of the specific heat and the latent

heat added over a range of temperature. In case of problems where phase change

occurs at fixed temperature, Ts, the enthalpy can be defined as:

T

s psT0

H (T) C (T)dT for T <Ts (2.3)

and

T T

s ps l plT T

s

0 s

H (T) C (T)dT L (T) C (T)dT for T >Ts (2.4)

where

To - the reference temperature,

Ts - the solidification temperature,

Suffixes S and L refer solid end liquid respectively and

L is the latent heat.

38

Nicholas Zebras (1990) has calculated enthalpy for a binary alloy

where the phase change occurs over a range of temperature as follows:

T

s psT0

H (T) C (T)dT for T < Ts (2.5)

T T

ss ps m pm

T Tl s

s

0 s

T TH (T) C (T)dT L (T)C (T)dTT T

for Ts < T < Tl (2.6)

T T

s ps m l s l plT T

s

0 l

H (T) C (T)dT L C (T T ) (T)C (T)dT

for T > Tl (2.7)

where

Cpl, Cpm and Cps are specific heat of metal in liquid, mushy and solid

conditions respectively.

Cm is the specific heat at freezing range s lC C2

(2.8)

Using the above equations, the enthalpy, H vs Temperature, T

relationship for a given system of metal / alloy can be constructed for a

temperature of interest.

The effective heat capacity, C* is obtained by differentiating H w.r.t.

T, when is expressed as

C* = dH/ dT (2.9)

39

But in numerical procedures this form is not convenient.

Morgan et al (1978) expressed the effective heat as follows:

1/ 222 2

*22 2

H H Hx y z

CT T Tx y z

(2.10)

Here temperature and enthalpy are expressed as functions of space

variables within the domain, it is easier to compute.

Hence in the fixed grid method there is no necessity to search for the

location of the solidification front and deform the FE grid to make it to coincide

with the solidification front.

2.4.2 Incorporation of Air-Gap

For the simulation of casting solidification, the behavior of the

interface is very important as cooling rate is controlled by the interface.

To model the interface, two methods are used, viz.

(i) Thin element for air gap.

(ii) Coincident node technique.

40

(i) Thin element for air-gap

Morales et al (1979) incorporated the air gap during simulation by

introducing a virtual element of appropriate thickness for the air gap and giving

an appropriate value for the interfacial heat transfer coefficient, ’h’. This

arrangement of the thin element at the interface is shown as follows:

5 6 7 8

Cast

Air

gap

Mould

1 2 3 4

Figure 2.8 Liner quadrilateral elements with thin interface element.

(ii) Coincident node technique

In this technique, the nodes of cast and mould at the interface have the

same special coordinates. The heat transfer from the cast to the mould is

incorporated through the convective heat transfer coefficient. This method offers

significant savings in computer time and greater ease in modeling the mesh.

This arrangement of the coincident nodes at the interface is shown as follows:

5 6&7 8

Cast

Mould

1 2&3 4

Figure 2.9 Linear Quadrilateral elements with thin interface element

2.4.3 Time Stepping Methods

41

Lewis et al (1996) used the equation for transient heat conduction

problem as

pTC div (k gradT) Qt

(2.11)

The FE equation for transient heat conduction is given by:

[K] {T} [C] {T} {F}

(2.12)

Although an analytical solution of this equation is possible, it is more

usual for the system of Ordinary Differential Equation to be discretised in time,

from which solution at various times can be obtained. The two commonly

employed techniques to fully discretise the system of equations are

(i) Finite difference method (FDM)

(ii) Finite Element method (FEM)

(iii) Weighted Residual Method

Finite difference method had been adopted by Alifanov (1985).

Alifanov (1985) assumed that the thermal histories at two designated interior

points in a one-dimensional body are known, and developed Finite Difference

Equations at each node, including nodes at the surface and the two designated

interior points. The surface temperature and heat flux can accordingly be

determined at each time interval by solving this set of equations. He employed a

modified explicit finite difference scheme at each time step to solve the

42

temperature distribution between two designated interior points with known

temperature histories.

In order to derive the various difference schemes, the partial difference

equation (PDE), which is a simplified version of equation (2.9) given by

2

p 2

u uCt x

(2.13)

If we let t

uut

and

2

xx 2

uux

, then we have

ut = uxx (2.14)

For approximating the time derivative Ut in terms of values at discrete

time interval t , such that

n nu u(t ) u (n t) (2.15)

It is essential to establish a rule where by the next value of ‘U’ is

dependent upon time steps.

n 1 n n 1 0u g (u , u , .... u ) (2.16)

43

The simplest representation of the derivative approximation is intuitively

n 1 n

t

u uut

(2.17)

and substituting this in equation (2.12) leads to the simplest time stepping

algorithm, the forward difference method or Euler’s scheme, given by

n 1 n nxxu u t (u ) (2.18)

The error involved in this approximation is proportional to the time

step length Δt, O(Δt). It can be seen from the equation (2.18) that the value of U

at the next time step, Un+1, is directly evaluated from the current value, Un. In the

same way, a system of equations describing all grid points or nodal values of the

variable leads to the direct solution of the matrix of the unknown nodal values.

The same scheme is therefore explicit.

In Finite Element Time stepping, In order to produce a finite element

algorithm, the temperature is discretised in time by the normal finite element

procedure, i.e.

i iT N (t) T (2.19)

The differential equation given by Equation (2.9) is first order w.r.t

time derivative. It is therefore only necessary to provide first –order (linear)

shape function, Ni (t) in time. The problem can now be solved by weighted

residual method.

44

In Weighted Residual method the discretisation of the equation (2.10)

produces

n n 1 n n 1n n 1 n n 1C(T N T N K(T N T N ) f

(2.20)

where

n 1 nn n 1N (1 ), N , N 1/ t, N 1/ t

Applying weighted residual method in equation (2.18), we get

1

n n 1 n n 1n n 1j n n 1

0W [C (T N T N ) K(T N T N ) f ] d 0

(2.21)

Substituting in the expressions for the shape function and their

derivatives, lead to

n 1 nC C ˆK T K(1 ) T ft t

(2.22)

where

1 1

j j0 01 1

j j0 0

w d w f dˆand f

w d w d

If the same spatial interpolation is used for both f and T, then f̂ is

given by

n n 1f̂ f (1 )f (2.23)

45

2.5 OBJECT ORIENTED PROGRAMMING

Procedural languages like C, Pascal, and Fortran consider data to be

passive and memory occupying elements. Only function and procedures

manipulate data. Since data and algorithms posses different structure, they

should therefore be declared and defined in different manner. In contrast to

procedural languages, an object includes data as well as functions, which are

called methods. An object can only operate on its own data with its own methods,

and external functions cannot change the data of this object. External functions

and other objects can communicate with an object only by sending messages. In

general, a message is a call by a method. So, if an object receives a message, it

interprets it and executes one of its procedures. Data encapsulation avoids

undesirable side effects and, during program verification, it is only necessary to

prove the methods of an object. Thus the program is independent of change of

methods.

C++ is an extension of C having object oriented concepts. This

language includes all the features of C and because of its efficiency, it is well

suited to solve numerical problems.

Masters et al, (1997) described the implementation of C++ for the

solution of transient heat conduction problem with phase change. Mackie (1992)

suggested an object oriented implementation of FEM and illustrated the

advantage of the approach. Scholz (1992) and Zeglinski et al, (1994) also dealt

with object oriented implementation in FEM programming.