An analytical solution for the unidirectional solidification problem

An analytical solution for the unidirectional solidification problem K.Davey Department

of Mechanical

Engineering,

UMIST,

Manchester,

UK

The extraction of heat from a molten casting is resisted by an imperfect thermal contact at the mold-casting interface. The nature of the contact varies throughout the casting process and has the effect of increasing the thermal resistance at the interface. This can be modelled by incorporating a gaseous gap at the mold-casting interface that grows with increasing time. This paper is concerned with an analytical solution of the unidirectional solidiJication problem, which incorporates movement of the casting at the interface. The derivation of the analytical solution requires the simultaneous solution of the transient heat equations, for the mold, gaseous gap, and solid and liquid parts of the melt. The analytical solution is extended so that contamination layers on the mold and casting can be incorporated as well as an initial gap. This is achieved by introducing virtual layers of mold, gas, and casting. Using the extended solution, the efsects of interfacial resistance, air conductivity, and gap variation on solidiJication rates are examined.

Keywords:unidirectional

solidification, analytical solution, interfacial thermal resistance

1. Introduction Knowledge of solidification rates is of paramount importance in the field of casting technology. It has been established that resistance to heat transfer at the moldcasting interface has a major effect on casting times. This is especially true in the case of permanent mold, low- and high-pressure die casting, and in sand castings where chills are incorporated. The mechanisms of heat transfer across the metal-mold interface have been discussed by Ho and Pehlke.le3 The heat transfer at the interface has contributions from conduction through contact points and spaces formed between these points as well as radiation and convection. However, for relatively low interfacial temperatures and small gap widths, both radiative and convective effects can be ignored. Also in certain casting processes such as pressure die casting, die lubricants are sprayed directly onto the mold, forming a thin separating film that reduces metal-mold contact, that is, provided the lubricant does not evaporate prior to injection of the molten material. The main mechanism for heat transfer for this case is by conduction through a gaseous/liquid film. For sand casting the positioning of the chill relative to the melt determines the relative importance of the transport mechanisms. Positioning the chill below the melt ensures that contact is maintained, Address reprint requests to Dr. Davey at the Dept. Engineering, UMIST, Manchester M60 lQD, UK. Received 1993

658

20 July 1992; revised

1.5 February

Appl. Math. Modelling,

of Mechanical

1993; accepted

4 March

1993, Vol. 17, December

whereas a gap will form if the chill is placed above the melt.lp3 Analytical solutions have an important role to play as far as casting is concerned. They may be used to develop rule of thumb design rules and generally incorporate the essential physics of the process. They also aid in the validation of more comprehensive numerical models. Many attempts have been made to derive exact and approximate mathematical models to simulate the solidification process. Obtaining an analytical solution is extremely difficult, so simplifying assumptions are made. Schwarz,4v5 for example, assumes that the metal and mold interfacial temperatures are equal and constant. This means that no interfacial resistance is included, hence cooling rates can be expected to be high. An alternative model proposed by Lyubov allows for variations of the interfacial temperatures.4 In this model superheat cannot be included and the effects of the mold can only be considered via variation in the heat transfer coefficient. A solution proposed by Garcia et aL6,* represents the interfacial thermal resistance by virtual layers of solid metal and mold. This approach, which is called the virtual adjunct method, requires the interfacial heat transfer coefficient to be constant. In this paper a model is proposed that involves solving the transient heat equations for four separate domains: the mold, the gaseous gap, the solid part of the casting, and the liquid part. These domains are then linked via compatibility conditions, i.e., temperature and rates of heat transfer. It can be shown that the resulting system can be solved analytically if the interfacial gap between

0 1993 Butterworth-Heinemann

Analytical solution for the unidirectional solidification casting and mold grows parabolically with time. This solution can be extended further to include the effects of mold coatings and initial gaseous gap widths. Similar to the virtual adjunct method, virtual layers of mold, gas, and solid melt can be incorporated. It is well established that the solid-liquid interface moves parabolically with time.4~5,9~10 With this in mind it is possible to justify, especially for die casting, that the gap growth is proportional to the solidified distance between the two interfaces, hence varying parabolically with time. However, this is certainly not the case for sand casting problems where melt and mold contact may be maintained throughout cooling. Even for this case though, it has been shown’-3 that the heat transfer coefficient decreases with time, which can be simulated with a growing gap though of very small order, typically a few micrometers.

2. The unidirectional solidification The thermal

problem

problem

to be considered is depicted in semi-infinite domains are considered. For convenience the origin of the x-axis is placed on the mold side of the metal-mold interface. It is assumed that material properties are independent of temperature and that the main heat transfer mechanism in the liquid melt is conduction. In addition, the casting material is assumed to have a unique melting temperature, i.e., the material is pure or of eutectic composition. Note that this is not a restriction as it is possible to incorporate a mushy region using an approach described in Ref. 5. The governing equation for each domain is of the form

Figure I, where four separate

1c~a~TJa2 = a7yat

in

Rixz

where i is either m, g, s, or 1. The conditions that completely as follows:

x

=

define the problem

then T,-+T,

Asx-r-cc At

(1)

(2)

0 T, = Tg and k, aTJax = k, aT,/ax

I

are

X(t)

I

d(t)

(3)

At

x

d(t)T, = q and

=

problem:

K. Davey

k,aT,/ax= k,aT,/ax (4)

At x = X(t)T, = T = Tf and

k,

aiyax -

k, a7yax= pLdX(t)/dt

As x + cc then T, ---) TP

(6)

It will be shown that the governing equations given in (1) can be solved exactly satisfying conditions (2)-(6), provided the interfacial gap d(t) is proportional to the square root of time, that is,

d(t) = q(4rcg t)1’2

(7)

where q is some scaling factor. The scaling factor q is either specified apriori or related to the solidification distance X(t) and determined in the course of the solution. Before going on to consider the derivation of the analytical solution, it is worth discussing the transfer mechanisms across the interfacial gap in greater detail.

3. Heat transfer across the interfacial gap The interfacial gap can be characterized in terms of the type of contact that exists between the mold and the melt.’ The type of contact can be classified as either conforming, nonconforming, or a clearance gap. In either case a macroscopic average heat transfer coefficient can be defined. h = q/K

(8)

- T’J

For a conforming contact, the temperatures T, and T, are not the actual measured temperatures, but can be considered as some local average. As the interface changes from a conforming to a nonconforming and then to a clearance gap, the temperatures will approach the actual measured values. Experimental work carried out illustrated that even when by Ho and Pehlkele3 metal-mold contact was maintained throughout the casting process the interfacial heat transfer coefficient decreased with time, which is a direct result of a decrease in the number of contact points. The heat transfer coefficient defined in (8) can also be represented in terms of macroscopically defined thermal conductivity and interfacial gap width, which approaches the measured values with formation of a clearance gap. The heat transfer coefficient is defined as h = kg/d(t)

MOULO

Figure

1.

Semi-infinite

SOLID

domains.

MELT

.IQUID

MELT

(5)

(9)

An increase in the interfacial distance d(t) with time will have the desired effect of reducing the heat transfer coefficient. The exact relationship between the gap distance d(t) and time t is not known. However, for pressure die casting where in general the metal does not come into contact with the mold, it is arguable that the main contribution to an increase in gap size is from thermal contraction rather than volumetric shrinkage, which results during a phase change. This is because, upon cooling a solidified shell surrounds the cast component, which resists contraction. Further shrinkage will only result as the internal energy of the already solidified part of the melt decreases. In this case an approximate re-

Appl.

Math.

Modelling,

1993,

Vol. 17, December

659

Analytical solution for the unidirectional

solidification problem:

lationship can be derived, so that the increasing gap size can be predicted with solidification of the melt. Assuming a linear temperature variation over the solidified part of the melt, the interfacial gap size can be calculated to be

K. Davey

where 2 is the root of the following

equation

(Tf - T,)exp[: -~‘l/(G + erfC4) - (Wb,P x exp[ - n2(lc,/~,)]/erfc[~(rcdrc,)‘12] = rF2L1/c, (22)

(10) where ATa_ = (Ts - TJ2 and TZ is the temperature of the casting surface at the interfacial gap. This relationship, although only approximate, gives some indication of the order of the gap size that occurs with thermal contraction. A more exact relationship than (10) is d(t) = aX(t)AT, where ii is specified in the light of experimental results.

The determination of the root ,? enables the temperature profiles in the four domains to be evaluated via equations (15)-(18). More importantly, from a casting point of view, solidification times can be estimated by utilizing relationship (21). It is worth noting here, that as y + 0 the root equation (22) approaches that given by Schwarz.s The temperature at the interface on the mold side, written in terms of the constants defined above, is G = C,(b,lb,)(T’ and on the casting

4. Analytical solution

T2 = Cl(T,

The analytical solution to the solidification problem depicted in Figure 1, which satisfies the governing equations (1) and the boundary conditions (2t(6), is relatively straightforward. However, this would not be the case if boundary conditions (4) were not specified on a moving boundary that varied parabolically with time. The solutions that satisfy the governing differential equations (1) are T, = A(1 + erf [x/(4rc, t)“‘]) + T,

(11)

q = B + C erf [x/(4rcg t)“2]

(12)

T, = D + E erf [x/(4rc, t)“2]

(13)

IT;= T, - F erfc[x/(4rc, t)‘j2]

(14)

where A, B, C, D, E, and F are arbitrary constants to be determined by conditions (3)-(5). Note that equations (11) and (14) in their present form automatically satisfy conditions (2) and (6), respectively. The temperature profiles that satisfy the boundary conditions and the governing equations (1) are as follows: 7; = T, - S erfc[J.x/(rc,/lc,)“2X]/erfc[~/(rc,/rc,)’i2] T, = T, + (Tf - TJ(C,

+ erf[Lx/X])/(C,

T, = T, + C,(T, - T,J(b,/b,)

(15)

+ erf[A]) (16)

+ erf [I(K-,/rc,)“2x/X1) (17)

T, = T, + C,(b$b,)( Tf - T,)(l + erf [(~Jq,J”2Wxl) (18) where erfc is the complementary error function (see Ref. 11) and 2 is an unknown root. The constants C, and Cl are Co =

Wb,)erf Crl + (b,lb,))expC-12((KdK,) - erf CrW~P”l

Cl = (Co + erfC~l(K,/Kg)1’21)/(C0 + erfC4) x WCrll + @$U)

- 111 (19) (3-Y

The boundary condition (5) is satisfied if the solid-liquid interface moves parabolically with time, i.e., x(t) = 1(4Jc,ty

660

Appl. Math. Modelling,

(21)

1993, Vol. 17, December

- T,) + T,

(23)

side

- T,)(b,lb,)

+ erflvl) + T,

(24)

It is important to recognize that C,, in equation (22), is a function of r) and, if contraction of the solidified melt is used to determine gap movement, it is a function of 13. Substitution of equation (21) into (10) and equating with (7) gives (25) Note that r] depends on the casting surface temperature, T2, whose value is dependent on II as well as y. This means that the solution of equation (22) requires an iterative approach. One possible procedure is to initially set r~ = ‘lo where y10 is a guessed approximation. This allows the root equation (22) to be solved using a standard iterative procedure such as the Newton-Raphson method. Once the root is determined the mold surface temperature can be calculated using equation (24). This can then be used in equation (25) to give a new prediction for yl. The procedure can then be repeated until convergence is achieved. This method was successfully used in conjunction with the material properties depicted in the Appendix.

5. Effects of superheat It can be seen on examination of equation (22) that superheat has the effect of reducing 2 and therefore decreasing the rate at which the solid-liquid interface moves. However, in the majority of casting processes the superheat will be extracted away by the mold before the solid-liquid interface progresses into the liquid melt. In the casting of pure metals some supercooling is usually required to initiate the solidification process, which may result in the removal of superheat before nucleation proceeds. Other mechanisms such as convection also play a part in the removal of superheat. The convection may be natural, arising from density variations, or forced, resulting during the injection/pouring stage of casting. If it is the case that the superheat is extracted before the solidification process proceeds, then the rate at which the solid-liquid interface moves should not be affected. The superheat will in effect only delay the onset of the

Analytical solution for the unidirectional solidification problem: K. Davey solidification process. Prates et a1.9 suggested that constant (e.g., G) should be added to (21) to give X(t) = 1(4i~,t)“~ - G

a

(26)

The constant G, as explained by Prates et al. is dependent on the value of the superheat. However, it has been shown that G is not equal to zero if no superheat is present.’ This is because the solid-liquid interface does not move parabolically at the onset of solidification, and hence equation (21) is not an accurate representation over the whole time domain. Also it is clear that equation (26) does not describe the position of the solidification front for all t. The relationship gives a negative X(r) for small t, which has no physical interpretation. The main reason for the solidification front behaving differently at the onset of solidification is due to the influence of the thermal resistance at the interface between the casting and the mold. Although the analytical solution discussed above cannot account for loss of superheat through mechanisms other than conduction, it can be extended to take into account initial interfacial resistance. This will result in an equation, similar to (26), that is representative over the whole time domain.

the initial virtual gaseous layers d, are assumed for convenience to be the same. The solutions that satisfy the governing differential equations (1) in the extended virtual domains are T, = A(1 + erf [x’/(4ic, t’)“2]) + T,

(27)

T, = B + C erf [x’/(4ic9 r’)“‘]

(28)

T, = D + E erf [x’/(4ic, t’)“2]

(29)

T1 = T, - F erfc[x’/(4ic, t’)1/2]

(30)

where the constants are the same as those specified in equations (11)-(14), which are defined implicitly by equations (15)--(18). The x’-axis is indicated in Figure 2 and the time t’ equals t + t,, where t, is the time taken for the virtual layer S, to solidify. Alternatively t, can be considered to be the time required to form the virtual gap 2d,, i.e., t, = d$(q2 rcg).To obtain an analytical solution it is assumed that the gaseous gap moves parabolically with time, i.e., d’(t’) = n(4@)“2 and to satisfy the constant the solid-liquid interface

The analytical solution discussed above can be extended to obtain a more generalized solution that incorporates an initial interfacial resistance. An initial resistance to heat transfer is present in all casting processes. The resistance may arise from an initial gap, the use of mold coatings, imperfect contact due to rugosity of the mold surface, contamination layers, i.e., oxidization films, etc. To extend the analytical solution a procedure similar to that adopted by Garcia et a1.6-8 is used. However, a slight complication arises due to the fact that both the mold and metal sides of the interface must be considered together. To account for the initial resistance four virtual components are introduced-one on the mold, two in the gas, and a presolidified solid melt. The four components are illustrated in Figure 2 where it can be seen that

I

do

d(t)

I

do

I

I

I

So

,

X(t) = (4K,12

t + o;)1’2

-

D,,

(34)

This relationship can be compared with that of (21) where the effect of the initial thermal resistance can be seen, with the inclusion of the constant D,. It is worth noting here that 1 is unaffected by the initial resistance. However, this does not mean that the solidification rate is not affected because

d0’

D(t)

(33)

where D, = S, + 2d, and X(t) = D(t) + d(t), the following relationship is obtained

Equations coordinate

1

= 2~,1~/(4rc,A~ t + D;)1’2 < 2K,~2/(‘tK,~2 t)"'

(35)

(27H30) can be written in terms of the original system and the three unknowns E,, D,, and

T, = A(1 + erfC&cJK,)1’2(x - E,)/(X + Do)] + r,

I I

(36)

I I

Tg = B + C er~@c$lc,)“2(x

I

I

MOULD

GAS

GAS

I &

1

2.

SOLID

I

I Figure

I I I I I I I I I I

x’

Semi-infinite

domains

at (32)

X’(t,) = &, = A(4K,t0)1’2

dX(t)/dt

I

requirement

where d’ is the gaseous gap thickness and X’ is the distance of the solid-liquid interface in the virtual coordinate system. Substituting t + t, for t’ in (32) and X(t) + D, for X’ and given that

x,(t,)

Eo

temperature

X’(t’) = 12(4K,t’)1’2

6. Extension of the analytical solution

d’(t’)

(31)

for virtual system.

LIOUID

+ d,)/(X

+ Do)]

K = D + E erfCi(x + D,)/(X + D,)] 7; = T, - F erfc[+&cJ”2(x The virtual equation

2d, =

gap 2d, is related

(37) (38)

+ D,)/(X + Do)] (39) to Do by the following (40)

?fDo(~,/Ks)"2/,?

which is obtained after algebraic manipulation of (31) and (32) with the time t set equal to zero. The unknowns

Appl. Math. Modelling,

1993, Vol.

17, December

661

Analytical solution for the unidirectional

Eo

,

do

,

I

do

solidification problem:

heat is not included, then as the mold conductivity, k,, tends to a high value, Lyubov’s model is approached. Thus it is reasonable to conclude the solidification times predicted by the above analytical solution will compare favorably with the results of Prates et al. It is of interest to observe the effect that initial interfacial resistance and gas conductivity have on solidification rates and gap growth. The equations that describe the motion of the solid-liquid interface and the casting surface can be conveniently written as

so

, I

I

I I

SOLID

TF

I Til

Figure

3.

T12

Semi-infinite

domains for virtual system at time f = 0.

and D, can be related to the initial thermal resistance h; r. Refer to Figure 3, which depicts the virtual system at time t’ = t,. Balancing the energy transfer across the virtual interface at x’ = -I& gives E,

hi(Tf - TJ = k, aT,jax’ Substituting

(41)

(27) in (41) yields D

W,,(T, - T,~IGJ~‘~

0

l”

z”2hiDo(Tr - T,)

E, =

(42)

~(~SIGI)1’2

The constant Do can be determined with knowledge of the initial gap size by using equation (40). However, it should be recognized that the resistance due to the initial virtual gap is already incorporated in hi. In fact it is possible to estimate the size of 2d, from 2d

K. Davey

k,(G - Ti,) ’ = h,(Tf - To)

(43)

Thus any specification of gap size will need to be consistent with this relationship.

X(t) = (D,t + Di)l’2 - Do

(44)

d(t) = (D3t + D$)1’2 - D,

(45)

where D, = 4u,L2, D, = 41cgy2 and D, = 2d, The constants Do, D,, D,, and D, are determined in Tables Z-3 for a particular casting operation. The casting material is zinc and the mold material is a high-speed steel normally used for pressure die casting. The thermal properties for these materials along with boundary conditions are given in the Appendix. It can be seen on examination of Table 1 that the initial interfacial resistance has a profound effect on the gap and solidification growth rates. Increasing the resistance correspondingly increases the constants Do and D,. This has the effect of significantly decreasing the growth rates especially at the onset of solidification. It can be seen on examination of Table 2 that increasing the thermal conductivity by several orders of magnitude does not give rise to similar changes in the growth rates. The main reason for this is that the mold and initial resistance restricts the rate of solidification. However, for relatively low values of gaseous thermal conductivity the growth rates are reduced as expected. It is of interest to note that k, affects both the solidification growth rate constants Do and D,, whereas changes in hi alter D, only. Table 2. Variation of growth rate constants with thermal conductivity of the gaseous interface Conductivity of gas

7. Discussion

and verification

The above analytical solution can be compared with the experimental results obtained by Prates et a1.9 Prates achieved reasonable accuracy on comparing his results against the analytical solutions of Schwarz and Lyubov.5 As already mentioned the root equation (22) approaches that obtained by Schwarz as q + 0. In addition, if super-

Constants for growth equations

k,/10-3 W/mm C

Do mm

DI mm*/s

D2/10-“ mm

D3/10-6 mm*/s

0.0034 0.034 0.34 34.0 3400.0

1.44 2.91 3.89 4.22 3.14

3.95 8.08 15.14 11.33 11.31

3.73 15.15 26.87 30.71 22.87

0.26 2.1 9 5.12 6.00 6.01

Table 1. Variation of growth rate constants with initial interfacial resistance

Table 3. Variation of growth parameter 7

Interfacial heat transfer coefficient

Gap scaling

hi

662

scaling

Constants for growth equations

Constants for growth equations

Do

DI

W/mm* C

mm

0.0004 0.004 0.04 0.4

29.10 2.91 0.291 0.0291

ADDI. Math.

rate constants with gap

mm*/s

D,/l O-4 mm

Da/l O-6 mm*/s

8.08 8.08 8.08 8.08

151.50 15.15 1.515 0.1515

2.19 2.19 2.19 2.19

Modelling,

1993,

Vol. 17, December

q/l o-s

Do mm

Dl mm*/s

D2/l O-4 mm

D3/1 O-6 mm*/s

0.002 0.004 0.04 0.02 0.40

4.03 3.96 2.91 1 .Ol 0.40

11.12 10.93 8.09 2.71 0.98

0.89 1.77 15.10 45.26 59.65

0.00544 0.0218 2.18 54.40 218.0

Analytical

solution

This indicates that a decrease in k, has a greater effect on solidification rates as time proceeds, whereas hi predominantly affects the solidification rate when the freezing front is near to the casting-mold interface. For the results in Tables I and 2 the gap scaling parameter v] was determined using equation (25). However, this was not the case for the results in Table 3, where it can be seen that an increase in the scaling parameter v] gives rise to a corresponding decrease in both D, and D,. Thus, as expected, increasing q increases the interfacial thermal resistance and correspondingly reduces the solidification rate.

for the unidirectional

Do Eo

so

A, B, C, D, E, F

gap

1

liquid melt mold solid melt

m

k) h

hi k L

4 S t

to T

T, Tf G G T, X(t) o! K P r ! d0

(C

References

7

8

Nomenclature heat diffusivity (kcp)‘12 (J/mm2 Cs112) specific heat (J/g C) distance between mold and solid melt (interfacial gap) (mm) heat transfer coefficient (W/mm2 C) heat transfer coefficient at castingmold interface (W/mm2 C) thermal conductivity (W/mm C) latent heat of fusion (J/g) rate of heat transfer per unit area (Wlmm2) superheat T, - T, (C) time (s) time for gaseous gap, 2d,, to form (s) temperature (C) ambient temperature (C) melting temperature (C) mold surface temperature (C) casting surface temperature (C) pouring/injection temperature (C) distance between mold and solidliquid interfaces (mm) coefficient of thermal expansion (C-r) thermal diffusivity k/cp (mm2/s) density (g/mm3) time domain interfacial gap scaling factor root virtual gas layer (mm)

K. Davey

So + 2d, (mm) virtual mold layer (mm) virtual solidified metal layer (mm) constants used in analytical solutions

9

9

b

problem:

Subscripts

8. Conclusion This paper describes an analytical solution to the unidirectional solidification problem. The solution allows for incorporation of an initial and temporally varying thermal resistance at the casting-mold interface. The movement of the solidification front and the casting surface is described by equations (45) and (44). The solidification rates are significantly reduced with decreases in the initial heat transfer coefficient and gaseous conductivity as well as increases in the interfacial gap size. An increase in the initial interfacial resistance has the effect of reducing rates of heat transfer at the initial stages of solidification. This can be compared with a reduction in gaseous conductivity, which has a more pronounced effect on rates of heat transfer in the latter stages of solidification.

solidification

10 11

Ho, K. and Pehlke, R. D. Metal-mould interfacial heat transfer. Metallurgical Transactions 1985, 16B, 585-594 Ho, K. and Pehlke, R. D. Mechanisms of heat at a metal-mould interface. AFS Transactions 1984,92, 587-597 Ho, K. and Pehlke, R. D. Transient methods for determination of metal-mould interfacial heat transfer. AFS Transacfions 1983, 91, 685-698 Ruddle, R. W. The Solidification of Casting. Institute of Metals Monograph and Report Series No.“~, 2nd Ed., 1957 Carslaw, H. and Jaeger, J. C. Conduction of Heat in Solids, 2nd Ed., Oxford and Clarendon Press, London. 1959 Garcia, A. and Prates, M. Mathematical model for the unidirectional solidification of metals: I. Cooled molds. Metallurgical Transactions 1978,9B, 449 Clyne, T. W. and Garcia, A. Assessment of a new model for heat flow during unidirectional solidification of metals. J Heat Mass Transfer 1980, 23, 773-782 Garcia, A., Clyne, T. W., and Prates, M. Mathematical model for the unidirectional solidification of metals: I. Massive molds. Metallurgical Transactions 1979, lOB, 82 Prates, M., Fissolo, J., and Biloni, H. Heat flow parameters affecting unidirectional solidification of pure metals. Metallurgical Transactions 1972, 3, 1419-1425 Prates, M. and Biloni, H. Variables affecting the nature of the chill zone. Metallurgical Transactions 1972, 3, 1419-1425 Abramowitz, M. and Stegun, 1. Handbook of Mathematical Functions. Dover, New York, 1968

Appendix Die material (high-speed steel-H13) thermal conductivity specific heat c, density pm ambient temperature

k,

(at 150 C)

T,

28.6 lop3 W/mm C 0.46 J/g C 7.76 10m3 g/mm3 2oc

thermal conductivity k, specific heat cg density ps initial thermal resistance

34.0 10m6 W/mm C 1.0 J/g C 0.10 10e6 g/mm3 0.004 W/mm2 C

Air in gap

hi

Zinc thermal conductivity k, thermal conductivity k, specific heat c, specific heat c1 density ps density pI latent heat L coefficient of thermal expansion solidification temperature Tf pouring temperature T,

Appl. Math. Modelling,

c1

0.10886 W/mm C 0.05 W/mm C 0.419 J/g C 0.505 J/g C 6.70 10e3 g/mm” 6.50 10m3 g/mm3 126.0 J/g 27.4 1O-6 C-r 383 C 403 c

1993, Vol. 17, December

663