Metal–mold heat transfer coefficients during horizontal and vertical Unsteady-State solidification of Al–Cu and Sn–Pb Alloys

Abstract

In this work, metal–mold heat transfer coefficients (h) are determined during unidirectional solidification of Al–Cu and Sn–Pb alloys. The effects of casting assembly (horizontal and vertical), alloy composition, material and thickness of the mold and melt superheat are investigated. By using measured temperatures in both casting and metal, together with numerical solutions of the solidification problem, metal–mold heat transfer coefficients are quantified based on solution of the inverse heat conduction problem. Experimental temperatures are compared with simulations furnished by an explicit finite difference numerical model, and an automatic search selects the best theoretical–experimental fitting from a range of values of metal–mold heat transfer coefficients. Experiments were conducted to analyze the evolution of h during solidification of Al–2, 4.5, 5, 8, 10, 15, 33 wt% Cu alloys and Sn–5, 10, 15, 20, 30, 39 wt% Pb in horizontal and vertical steel chills. The results permitted the establishment of expressions as a power function of time, for different alloy compositions, casting assembly material and thickness of the mold and melt superheat.

Introduction

For the purpose of accurate mathematical modeling of solidification processes, it is essential that correct boundary conditions be established. The heat transfer at the metal–mold interface is one of these boundary conditions, which is of central importance when considering the magnitude of heat transfer during the early stages of solidification. The way heat flows across the casting and the mold surfaces directly affects the evolution of solidification and plays a notable role in determining the freezing conditions within the casting, mainly in foundry systems of high thermal diffusivity like chill casting. Gravity or pressure die casting, continuous casting and squeeze castings are some of the processes where product soundness is more directly affected by heat transfer at the metal–mold interface. The loss of heat when a metal first comes into contact with the mold is regulated not only by the heat storage capacity of the mold material, but also by the heat transfer conditions within the metal itself and particularly at the metal–mold interface. The solid bodies are only in contact at isolated points and the actual area of contact is only a small fraction of the nominal area, as shown in Fig. 1.

FIGURE 1 Heat flow across metal–mold interface and thermal profile in the volume element during solidification (massive mold).

Part of the heat flow follows the paths of actual contact, but the remainder must pass through the gaseous and nongaseous interstitial media between the surface peaks. The interstices are limited in size, so that convection can be neglected. If temperature differences are not extreme, radiation does not play a significant role and most of the energy passes by conduction across the areas of actual physical contact.

The heat flow across a casting–massive mold interface, as shown by the schematic representation of Fig. 1, can be characterized by a macroscopic average metal–mold interfacial heat transfer coefficient (h _i ), given by:

where q is the average heat flux across the interface [W], T _IC and T _IM are, respectively casting and mold surface temperatures [K] and A is area [m²]. In water-cooled molds the overall heat flow is affected by a series of thermal resistances, as shown in Fig. 2.

FIGURE 2 Thermal resistances and thermal profile in the volume element during solidification (cooled mold).

The interfacial resistance between the casting and the mold surface is generally the largest, and the overall thermal resistance (R _g = 1/h _g ) is given by:

where h _g is the overall heat transfer coefficient between casting surface and the coolant fluid [W/m² K], e is the wall thickness of the bottom of the mold [m] and h _w is the mold–coolant heat transfer coefficient. The average heat flux from casting surface to the cooling water is then given by:

where T _o is the temperature of the cooling water [K].

The present study describes a method for obtaining interfacial heat transfer coefficients as a function of time, from experimental data concerning the solidification of Al–Cu and Sn–Pb alloys in massive and cooled molds. Experimental temperatures in the casting and in the mold are compared with simulations furnished by a numerical model, and an automatic search selects the best fitting from a range of values of interfacial heat transfer coefficients. The effects of alloy composition, material and thickness of the mold and melt superheat are also investigated.

Heat Transfer Coefficient

Several researches have attempted to quantify the transient interfacial heat transfer during solidification in terms of a heat transfer coefficient [1–5]. These studies have highlighted the different factors affecting heat flow across an interface during solidification. These factors include the thermophysical properties of the contacting materials, the casting and mold geometry, the roughness of the mold contacting surface, mold coatings, contact pressure, melt superheat, initial temperature of the mold, etc. The heat transfer coefficient shows a high value in the initial stage of solidification and then decline to a low steady value because the casting contracts from the mold surface, creating an interfacial gap. Most of the methods of calculation of interfacial heat transfer (h) existing in the literature are based on temperature histories at interior points of the casting or mold together with mathematical models of heat transfer during solidification. Among these methods, those based on the solution of the inverse conduction problem have been widely used in the quantification of the transient interfacial heat transfer [6–9].

(A) Inverse Solution

In the present work, a similar procedure determines the values of interfacial heat transfer coefficient by solving the IHCP (Inverse Heat Conduction Problem) by a numerical method. According to Beck, the IHCP is the estimation of the time or spatial profile of surface heat flux given one or more measured temperatures inside a body [6]. Among the variety of proposed methods for solving the IHCP, the iterative gradient technique is a procedure that minimizes a least squares function of the differences between the temperatures measured by thermocouples and temperatures predicted by a direct model of the problem.

A one-dimensional solidification model of the metal–mold system that predicts the thermal field in a longitudinal section through the system was used to provide a forward problem solution for the inverse problem. This model is based on a previous model developed by Spim, Santos, Quaresma and Garcia [10–12]. A schematic representation of the longitudinal section of the system is shown in Fig. 3.

FIGURE 3 Schematic representation of a longitudinal section of the metal–mold system.

The inverse problem consists of estimating the boundary heat transfer coefficient at the metal–mold interface from measured mold, water and metal temperatures. The inverse problem can be stated as follows:

a.	given M measured temperatures T j (j = 1, 2, 3,…, N),
b.	estimating the heat transfer coefficient given by its components h i (i = 1, 2, 3,…N). Fig. 4 shows a schematic of the stated problem.

FIGURE 4 Diagram showing domain for inverse heat conduction problem.

To solve the problem, the estimated temperature computed from the solution of the direct problem using the estimated values of the heat transfer coefficient components h _i (i = 1, 2, 3,…,N), should match the measured temperatures as close as possible. This matching can be done by minimizing the standard least squares norm with respect to each of the unknown heat transfer coefficient components.

In the present work, a similar procedure determines the value of h _i which minimizes an objective function [10] defined by the equation:

where T _est and T _exp are respectively the estimated and the experimentally measured temperatures at various thermocouples locations and times, and n is the iteration stage. A suitable initial value of h _i is assumed and with this value, the temperature of each reference location in casting and mold at the end of each time interval Δt is simulated by using an explicit finite difference technique. The correction in h _i at each iteration step is made by a value Δh _i , and new temperatures are estimated [T _est(h _i + Δh _i ) or T _est(h _i − Δh _i )]. With these values, sensitivity coefficients (ϕ) are calculated for each iteration, given by:

The sequence of the solution involves the calculation of the sensitivity coefficients for measured temperatures. The assumed value of h _i is corrected using the relation:

The above indicated procedure is repeated for a new value of h _i , and is continued until:

(B) Heat Flow Model [10–12]

The differential equation for heat transfer is known as “General Equation of Heat Conduction in an Unsteady State”, and by considering constant thermal conductivity along heat flux directions and with internal heat generation, it is given for three-dimensional heat flux by:

where ρ is material density [kg/m³]; c is specific heat [J/kg K]; k is thermal conductivity [W/m K], t is time and is the heat source term. In order to model the mathematical treatment, it was assumed that the heat flux is unidirectional from the centre to the surface and can be considered negligible along the vertical direction. Then, Eq. (8) becomes:

In this study, a fixed grid methodology is used with a heat source term due to phase change, which is given by an explicit solid fraction–temperature relationship as , where the solid fraction depends on a number of parameters and L is the latent heat of fusion [J/kg]. However, it is quite reasonable to assume that fs varies only with temperature and the fs can be obtained from:

where T _L is the liquidus temperature, T _s is the solidus temperature, T _f is the fusion temperature and k _o is the partition coefficient; and by using the pseudo-specific heat concept, the following is obtained:

The model permits the insertion of physical properties as a function of temperature, considering the amount of fs, as: k = (k _s − k _l )·f _s + k _l ; ρ = (ρ _s − ρ₁)·f _s + ρ _l and c = (c _s − c _l )·f _s + c _l − (L*df _s ), where indices s and l indicate solid and liquid respectively.

Analogy between thermal and electrical systems:

Approximating Eq. (9) by finite-difference terms, and by multiplying heat flux area (At) and Δx, which is equal to the volume, the following is obtained:

and by making an analogy between thermal system and the passive elements of an electrical circuit, where the thermal capacity is given by:

The thermal resistance at the heat flux line from point i + 1 or i − 1 to point i is given by:

This term is given by the sum of thermal resistance inside element i (from the centre to the surface) according to the following equations:

then:

and

This equation can be written as:

where

Equation (18) or (19) are generic and can be applied to any geometry, by varying only area and volume to be considered.

Experimental Procedure

The casting assemblies used in both horizontal and vertical solidification experiments are shown in Fig. 5. Horizontal solidification experiments were performed with Sn, Pb, Al and Al–Cu alloys (4.5; 15; 33 wt% Cu) and Sn–Pb alloys (5; 10; 15 and 39 wt% Pb), including short and long freezing range alloys, as well as eutectic compositions. The casting and chill materials selected for experimentation, and the employed thermophysical properties are summarized in Table I.

FIGURE 5 Casting arrangement and position of the thermocouples in mold wall and metal.

TABLE I Casting and chill materials used for experimentation and the corresponding thermophysical properties [12–15]

	Al	Al–2%Cu	Al–4.5%Cu	Al–5%Cu	Al–8%Cu	Al–15%Cu	Al–33%Cu	Steel SAE 1010	Pb	Sn–39%Pb	Sn–30%Pb	Sn–20%Pb	Sn–15%Pb	Sn–10%Pb	Sn–5%Pb	Sn	Copper
k s	222	209	193	192.4	188.4	179	155	46	34.7	54.7	57.3	59	62.2	63	64	67	372
k L	92	90.7	85	89	87	80	141		29.7	31.7	32.0	32	32.5	33	33	33
c s	1123	1109	1092	1090	1088	1080	1070	527	129.8	186.2	193.6	200	207.3	209	221	221	419
c L	1086	1074	1059	1057	1039	999	895		138.2	212.9	222.8	231	240.9	243	259	259
ρ s	2550	2588	2650	2667	2746	2910	3410	7860	11340	8840	8512.3	8250	7906.0	7840	7720	7300	8960
ρ L	2380	2423	2480	2501	2580	2760	3240		10678	8400	8103.4	7860	7551.1	7480	7380	7000
α s (10^−5)	7.75	7.28	6.57	6.62	6.31	5.67	4.25		2.37	3.35	3.48	3.58	3.79	3.84	3.91	4.15
α L (10^−5)	3.36	3.49	3.24	3.36	3.24	2.90	2.45		2.04	1.79	1.77	1.76	1.79	1.81	1.82	1.82
L	385 000	383 566	381 900	381 415	379 264	274 270	350 000		26 205	47 560	50 359	52 580	55 534	56 140	57 120	60 710
T F	660	660	660	660	660	660	660		327	232	232	232	232	232	232	232
T E		548	548	548	548	548	548			183	183	183	183	183	183
T L		654	645	643	633	618					198	202	210	216	220
ε								0.8									0.023
k o		0.17	0.17	0.17	0.17	0.17					0.0656	0.0656	0.0656	0.0656	0.0656

The main design criteria were to ensure a dominant unidirectional heat flow during solidification. This objective was achieved by adequate insulation of the chill and casting chamber. Copper and low carbon steel chills were used, with the heat-extracting surfaces being polished. In order to investigate the influence of chill thickness on heat flow, five different thicknesses of chills were used (X = 6, 17, 28, 39 and 60 mm). Two chromel–alumel thermocouples were introduced in the chill; one near the chill–casting interface and the other at the outer surface, and a third one was placed in the casting and located at 20 mm from the interface, as indicated in Fig. 5. Each alloy was melted in an electric resistance-type furnace until the molten metal reached a predetermined temperature. It was then stirred until the temperature was brought to a specified value and poured into the casting chamber. The aluminum alloys were degassed with hexachloroethane tablets before pouring. The effect of liquid metal superheat on heat transfer coefficient was also investigated, by using a Sn–10 wt% Pb alloy, a 60 mm thick carbon steel chill and different degrees of superheat: 20, 40, 70 and 100°C above liquidus temperature. The vertical solidification apparatus was designed in such a way that the heat was extracted only through the water-cooled bottom, promoting upward directional solidification. The use of such experimental configuration permits natural convection to be minimized, as well as solute convection due to buoyancy forces if the rejected solute has a higher density than the alloy melt. A stainless steel mold was used having an internal diameter of 50 mm, height 110 mm and a wall thickness of 5 mm. The inner vertical surface was covered with a layer of insulating alumina to minimize radial heat losses, and a top cover made of an insulating material was used to reduce heat losses from the metal–air surface. The bottom part of the mold was closed with a thin (3 mm) disc of carbon steel. The alloys were melted in situ and the lateral electric heaters had their power controlled in order to permit a desired superheat to be achieved. To begin solidification, the electric heaters were disconnected and at the same time the water flow was initiated. Experiments were carried out with Al–Cu alloys (2; 5; 8 and 10 wt% Cu) and Sn–Pb alloys (5, 10, 15, 20 and 30 wt% Pb) at various superheats. Temperatures in the casting were monitored during solidification via the output of a bank of type K thermocouples (1.6 mm diameter) accurately positioned with respect to the heat-extracting surface.

The thermocouples were calibrated at the melting points of aluminum (for Al–Cu alloys) and Tin (for Sn–Pb alloys), exhibiting fluctuations of about 1.0 and 0.4°C respectively. The experimental profiles plotted are the averages of three thermocouples readings at each location in chill and casting. Results from repeated experiments have shown differences not greater than 4°C. All of the thermocouples were connected by coaxial cables to a data logger interfaced with a computer, and the temperature data were acquired automatically. The temperature files were used in a finite-difference heat flow program to estimate the transient heat transfer coefficients.

Results and Discussion

Horizontal Casting

Effect of Alloy Composition

Solidification simulation of each test casting was performed by adopting two different approaches for the liberation of the latent heat of fusion. For eutectic alloys and pure metals, the latent heat (L) was transformed into equivalent number of degrees by considering a temperature accumulation factor (λ) related to L by the specific heat (λ = L/c). For short or long freezing range alloys, the latent heat evolution was taken into account by using Scheil's equation until the remaining liquid reached the eutectic composition. Temperature was experimentally measured in two locations: in the chill at 3 mm from the metal–mold interface and in the casting at 20 mm from this interface. In Figs. 6 and 7 typical experimental thermal responses are compared to those numerically simulated by using the transient h _i profile which provides the best curve fitting for Al–Cu and Sn–Pb alloys.

FIGURE 6 Experimental and simulated temperatures responses at two locations in casting and chill: 60 mm thick steel chill and a superheat ΔT = 0.1 T _L (10% of liquidus temperatures) Al–4.5 wt% Cu.

FIGURE 7 Experimental and simulated temperatures responses at two locations in casting and chill: 60 mm thick steel chill and a superheat ΔT = 0.1 T _L , Sn–5 wt% Pb.

Figure 8 shows the metal–mold heat transfer coefficients profiles as a function of time for the case of Al–Cu alloys solidifying against a 60 mm thick carbon steel chill.

FIGURE 8 Evolution of the metal–mold interfacial heat transfer coefficients as a function of alloy composition: Al–Cu system, 60 mm thick steel chill and ΔT = 0.1 T _L .

The observed differences in the h _i profiles between the pure metals and the other alloys examined can be explained by the total shrinkage accompanying solidification, the extent of the solidification range and the wetting of the mold by the melt. It can be seen that the h _i profile increases with increasing mushy zone length. For longer mushy zones, the interdendritic liquid can feed better the solidification contraction causing a continued presence of liquid at the interface, leading to higher values of h _i . This can be taken as a general trend, but care should be exercised when applying this conclusion to the early beginning of solidification. As can be seen in Fig. 8, the Al–15 wt% Cu alloy exhibits initial h _i values higher than those corresponding to the Al–4.5 wt% Cu alloy, which has a longer mushy zone. At the initial stage of solidification the wetting of the mold by the melt seems to be the dominant factor controlling heat transfer coefficient. In any case a more complex experimental setup would be necessary for an accurate characterization of initial heat transfer coefficients, as fluid flow in the cast alloys, with its associated heat transfer, would be at its strongest.

Effects of Chill Material and Chill Thickness

The effects of chill material and chill thickness on heat transfer coefficient are shown in Figs. 9 and 10, for the cases of a Sn–10 wt% Pb alloy solidifying respectively against a carbon steel and copper chills at a superheat of ΔT = 0.1 T _L .

FIGURE 9 Evolution of the metal–mold heat transfer coefficients as a function of chill thickness: Sn–10 wt% Pb alloy, steel chill and ΔT = 0.1 T _L .

FIGURE 10 Evolution of the metal–mold heat transfer coefficients as a function of chill thickness: Sn–10 wt% Pb alloy, copper chill and ΔT = 0.1 T _L .

As can be seen by comparing Figs. 9 and 10, the heat transfer coefficient profiles increase with increasing thermal diffusivity of the chill material. These results are in agreement with other studies in the literature [3,7]. It can also be seen that the h _i profiles increase with decreasing chill thickness. The chill temperature rises more rapidly from beginning of solidification with decreasing chill thickness. As a consequence, mold expansion favors the thermal contact between metal and chill surface and as the solidified shell is not so thick as for thicker chills, this translates to lower contraction away from the chill. Both factors will contribute to an increase in h _i values.

Effect of Superheat

The heat transfer coefficient increases with increasing values of superheat, as can be seen in Fig. 11 for a Sn–10 wt% Pb alloy solidifying against a 60 mm thick carbon steel chill. The fluidity of molten alloys increase with increasing superheat, favoring the wetting of the chill by the melt [9,16]. Some results reported in the literature indicate that the surface of solidified shell becomes smoother as the superheat increases for the same chill microgeometry, thus increasing the interfacial contact [17]. Some differences can be observed for values higher than 40°C (Fig. 11). This is not the case for Al, where fluidity plays a more significant role. The initial value of h _i rises from about 1500 to 6000 W/m² K if the superheat is increased from 10% of the melting point (T _f ) to 20% of T _f [12]. Table II summarizes all the values of h _i , expressed as a power function of time, determined during the present experimental investigation for Al–Cu and Sn–Pb alloys under different conditions.

FIGURE 11 Evolution of the metal–mold heat transfer coefficients as a function of superheat: Sn–10 wt% Pb alloy, 60 mm thick steel chill.

TABLE II Metal/mold heat transfer coefficients during the horizontal directional solidification

Alloy	Pouring Temperature (°C)	Mold condition		hi (W/m^2 K), t(s)
Al–Pure	726	1010 Steel mold		2000 (t)^−0.17
	792			12500 (t)^−0.17
Al–4.5 wt% Cu	645	1010 Steel mold		8650 (t)^−0.17
Al–15 wt% Cu	618	1010 Steel mold		17000 (t)^−0.54
Al–33 wt% Cu	548	1010 Steel mold		4900 (t)^−0.54
Sn–Pure	232	1010 Steel mold		6800 (t)^−0.47
Sn–5 wt% Pb	220	1010 Steel mold		18000 (t)^−0.47
Sn–10 wt% Pb	210	1010 Steel mold		9600 (t)^−0.47
			Thickness	–
			6 mm	20500 (t)^−0.47
			17 mm	18500 (t)^−0.47
			28 mm	15000 (t)^−0.47
Sn–10 wt% Pb	231	1010 Steel mold	39 mm	11500 (t)^−0.47
			60 mm	9600 (t)^−0.47
			6 mm	14000 (t)^−0.37
			17 mm	12800 (t)^−0.37
		Copper mold	28 mm	10200 (t)^−0.37
			39 mm	8800 (t)^−0.37
			60 mm	8000 (t)^−0.37
Sn–10 wt% Pb	230	1010 Steel mold		9600 (t)^−0.47
	250			9600 (t)^−0.47
	280			11500 (t)^−0.47
	310			14800 (t)^−0.47
Sn–20 wt% Pb	202	1010 Steel mold		8400 (t)^−0.47
Sn–39 wt% Pb	183	1010 Steel mold		7800 (t)^−0.47

Vertical Casting

Figure 12 shows typical examples of the evolution of thermal profiles during the course of different experiments of upward directional solidification of an Al–Cu alloy and a Sn–Pb alloy.

FIGURE 12 Typical experimental temperatures responses at three locations in casting: Al–Cu and Sn–Pb alloys (coated mold): (A) Al–2 wt% Cu, (B) Sn–30 wt% Pb.

Figure 13 shows a typical example of the evolution of the overall metal–mold heat transfer coefficient (h _g ) during the course of different experiments of upward directional solidification of an Al–5 wt%Cu alloy.

FIGURE 13 Evolution of the overall metal–mold heat transfer coefficient as a function of superheat: Al–5 wt% Cu, water-cooled steel mold.

Table III summarizes all the values of h _g , expressed as a power function of time, determined during the present experimental investigation for Al–Cu and Sn–Pb alloys under different conditions.

TABLE III Overall heat transfer coefficients during the vertical directional solidification

Alloy	Pouring temperature Tp (°C) and mold surface condition	hg (W/m^2 K), t(s)
Al–2 wt% Cu	T p = 660°C – Polished mold	2600 (t)^−0.14
	T p = 674°C – Polished mold	2100 (t)^−0.14
	T p = 689°C – Polished mold	2000 (t)^−0.14
	T p = 709°C – Polished mold	1850 (t)^−0.14
	T p = 689°C –Coated mold	1900 (t)^−0.18
Al–5 wt% Cu	T p = 655°C – Polished mold	3500 (t)^−0.27
	T p = 663°C – Polished mold	3200 (t)^−0.27
	T p = 680°C – Polished mold	2450 (t)^−0.27
	T p = 709°C – Polished mold	2400 (t)^−0.27
	T p = 709°C –Coated mold	1500 (t)^−0.20
Al–8 wt% Cu	T p = 640°C – Polished mold	5100 (t)^−0.33
	T p = 652°C – Polished mold	5100 (t)^−0.33
	T p = 690°C – Polished mold	5100 (t)^−0.33
	T p = 652°C – Coated mold	1000 (t)^−0.065
Al–10 wt% Cu	T p = 700°C – Polished mold	5700 (t)^−0.33
	T p = 705°C – Polished mold	5700 (t)^−0.33
	T p = 705°C – Coated mold	1000 (t)^−0.075
Sn–10 wt% Pb	T p = 220°C – Coated mold	1200 (t)^−0.01
Sn–15 wt% Pb	T p = 234°C – Coated mold	1200 (t)^−0.1
Sn–30 wt% Pb	T p = 224°C – Coated mold	1100 (t)^−0.01
Sn–5 wt% Pb	T p = 225°C – Stainless steel mold	650 (t)^−0.07
Sn–10 wt% Pb	T p = 220°C – Stainless steel mold	1300 (t)^−0.07
Sn–20 wt% Pb	T p = 207°C – Stainless steel mold	900 (t)^−0.07
Sn–30 wt% Pb	T p = 197°C – Stainless steel mold	400 (t)^−0.07

In can be seen both in Table III and Fig. 13 that as expected, under the same condition of melt superheat, the heat transfer coefficient decreases when the mold surface is coated with an insulating layer of alumina. It can also be observed that the increase in melt superheat decreases the heat transfer coefficient. This is, apparently, in contradiction with results obtained in a previous article concerning the horizontal directional solidification of Al–Cu and Sn–Pb alloys [12] and can be explained by the differences on the physical configuration of the two experimental setups. In both cases the superheat delays the solidification evolution. In the horizontal solidification this will translate to a higher h profile for higher melt superheats, since the contraction of metal from the mold wall will also be delayed. In the upward solidification the casting weight will contribute to a better metal–mold thermal contact if lateral contraction is effective (permitting the ingot to be gradually detached from lateral walls). This will happen sooner for solidification without superheat, and as a consequence a higher h profile will be provided with decreasing melt superheat.

Conclusions

From the experimental work on horizontal and vertical directional solidification of Al–Cu and Sn–Pb alloys, the following conclusions can be drawn:

1.	for both the horizontal and vertical castings, a rapid drop in interfacial heat transfer coefficient occurs during the initial stages of solidification;
2.	the metal–mold (h i ) or the overall heat transfer coefficient (h g ) can be expressed as a power function of time, of the form h = C 1·(t)− m , where h [W/m2 K], t [s] and C 1 and m are constants, which depend on alloy composition, chill material and superheat.
3.	the transient heat transfer coefficients profiles increase with increasing melt superheat for horizontal directional solidification. A reverse situation has been observed for vertical upward directional solidification, where the h g profiles decrease with increasing melt superheat.

Nomenclature

Table

A	=	area (m^2)
c	=	specific heat (J/kg K)
e	=	mold wall thickness (m)
h i	=	metal–mold heat transfer coefficient (W/m^2 K)
h o	=	mold–environment heat transfer coefficient
h g	=	overall heat transfer coefficient
h w	=	mold–coolant heat transfer coefficient
k	=	thermal conductivity (W/m K)
k o	=	partition coefficient
L	=	latent heat of fusion (J/kg)
q	=	heat flux (W)
t	=	time (s)
T	=	temperature (°C)
T F	=	fusion temperature
T E	=	eutectic temperature
T S	=	solidus temperature
T L	=	liquidus temperature
T P	=	pouring temperature
T o	=	cooling water temperature

Greek Symbols

Table

ε	=	emissivity
ρ	=	density (kg/m^3)
α	=	thermal diffusivity (m^2/s)

Subscripts

Table

I	=	interface
S	=	solid
L	=	liquid
M	=	mold
IC	=	casting surface
IM	=	mold surface
w	=	water
m	=	metal

Acknowledgments

The authors would like to acknowledge financial support provided by FAPESP (The Scientific Research Foundation of the State of São Paulo, Brazil) and CNPq (The Brazilian Research Council).