Determination of the Heat Transfer Coefficient at the Metal-Mold Interface During Centrifugal Casting SANTIAGO VACCA, MARCELO A. MARTORANO, ROMULO HERINGER, and MA´RIO BOCCALINI Jr. The heat transfer coefficient at the metal-mold interface (hMM) has been determined for the first time during the centrifugal casting of a Fe-C alloy tube using the inverse solution method. To apply this method, a centrifugal casting experiment was carried out to measure cooling curves within the tube wall under a mold rotation speed of 900 rpm, imposing a centrifugal force 106 times as large as the gravity force (106 G). As part of the solution method, a comprehensive heat transfer model of the centrifugal casting was also developed and coupled to an optimization algorithm. Finally, the evolution of hMM with time that gives the minimum squared error between measured and calculated cooling curves was obtained. The determined hMM is approximately 870 W m2 K1 immediately after melt pouring, decreasing to about 50 W m2 K1 when the average temperature of the tube is ~973 K (700 C), after the end of solidification. Despite the existence of a centrifugal force that could enhance the metal-mold contact, these values are lower than those generally reported for static molds with or without an insulating coating at the mold inner surface. The implemented model shows that the heat loss by radiation is dominant over that by convection at the tube inner surface, causing the formation of a solidification front that meets another front coming from the outer surface of the tube. DOI: 10.1007/s11661-015-2770-2  The Minerals, Metals & Materials Society and ASM International 2015

I.

INTRODUCTION

HORIZONTAL centrifugal casting is currently one of the most important processes in the industrial production of steel and cast iron tubes. In this process, the molten alloy is poured into a rotating cylindrical mold and is thrown against the internal wall by a centrifugal inertial force that is much larger than the gravitational force.[1] This force causes a pressure gradient in the radial direction, parallel to the heat flux, helping flotation of impurities, and feeding of solidification contraction, eventually improving the metallurgical quality of the casting.[2] Because of rotation, experimental thermal analysis of the centrifugal casting process is more difficult than that of static processes, increasing the importance of the mathematical modeling of heat transfer as an alternative tool. However, to construct useful and realistic heat transfer models, the heat flux at the contact interface between the cast tubes and the cylindrical molds must be known accurately. This heat flux can be defined in terms

SANTIAGO VACCA, Mechanical Engineer, and MARCELO A. MARTORANO, Associate Professor, are with the Department of Metallurgical and Materials Engineering, University of Sa˜o Paulo, Av. Prof. Mello Moraes, 2463, Sa˜o Paulo, SP 05508-900 Brazil. Contact e-mail: [email protected] ROMULO HERINGER, Professor, is with the Department of Materials Engineering, Federal University of Paraiba, Campus Universitario, Joa˜o Pessoa, PB 58051-900 Brazil. MA`RIO BOCCALINI Jr., Researcher, is with the Center for Metallurgical and Materials Technologies, Institute for Technological Research – IPT, Av. Prof. Almeida Prado, 532, Sa˜o Paulo, SP 05508901, Brazil. Manuscript submitted July 12, 2014. METALLURGICAL AND MATERIALS TRANSACTIONS A

of a heat transfer coefficient, hMM, which depends on several parameters, such as the thickness and the material of the coating on the internal mold wall, and the thickness of the air gap that forms at the metal-mold interface owing to the tube contraction. Values of hMM adopted in centrifugal casting simulations are unreliable and usually arbitrary. Ebisu[3] and Yang et al.[4] assumed an exponential decay for the radiation heat flux at the metal-mold interface. Martinez et al.,[5] Gao and Wang,[6] Kang et al.,[7] Kang and Rohatgi,[8] and Chang et al.[9] adopted constant arbitrary values of hMM, while Raju and Mehrotra,[10] Drenchev et al.,[11] Nastac et al.,[12] and Panda et al.[13] assumed that hMM decreased with time according to an arbitrary equation. Xu et al.[14] assumed perfect contact at the metal-mold interface and Humphreys et al.[15] calculated the heat flux at this interface using thermal resistances, but did not provide the parameters to calculate them. In Table I, the values of hMM adopted by these authors are summarized. In addition to unreliable hMM values, important effects were neglected in some of these models. As examples, in the centrifugal casting of tubes, calculation of the heat loss from the tube inner surface was neglected or oversimplified[3,5–8,12] and the mushy zone was approximated by a planar solid–liquid interface.[6–8] The objective of the present work is to obtain the heat transfer coefficient at the metal-mold interface as a function of time during the centrifugal casting of an FeC tube using the method of inverse solution to the heat conduction equation. To apply the inverse solution method, measurement of cooling curves within the metal and mold were carried out during casting of a tube

Table I.

Values of hMM Adopted by Different Authors

Authors Ebisu[3] Yang et al.[4] Martinez et al.[5] Gao and Wang[6] Kang et al.[7] Kang and Rohatgi[8] Chang et al.[9] Raju and Mehrotra[10] Drenchev et al.[11] Nastac et al.[12] Panda et al.[13] Xu et al.[14] Humphreys et al.[15]

Type of hMM

Value (Wm2 K1)

variable variable constant constant constant constant constant variable variable variable variable — —

not available not available 700 to 500 2000 to 5000 100 to 1500 100 to 1500 103 to 2.6 9 103 102 to 103 102 to 103 3 9 102 to 2 9 104 50 to 104 perfect contact not available

under rotation and a comprehensive heat transfer model of centrifugal casting was proposed and implemented. The present work is the first attempt to obtain realistic values of the heat transfer coefficient at the metal-mold interface during the centrifugal casting of tubes. In nearly all previously proposed models, a wide range of arbitrarily chosen values was adopted, which might prevent realistic predictions, because the heat transfer at the metal-mold interface is one of the largest heat transfer barriers in this process.

II.

MEASUREMENTS OF COOLING CURVES DURING CENTRIFUGAL CASTING

A 100 kg charge of composition Fe-2.88 wt pct C-0.19 pct Si-0.40 pct Mn was melted in an induction furnace and poured at 1713 K (1440 C) during 21 seconds into the cylindrical gray cast-iron mold of a horizontal centrifugal casting machine. The mold was 30 cm in both length and inner diameter and rotated at 900 rpm (Figure 1). The inner surface and the lid in the front side of the mold were coated with a zircon-based insulating porous layer with about 2 and 5 mm thickness, respectively. The backside of the mold was covered with a MgO-based plate with 30 mm thickness. Before pouring, the mold was preheated to about 673 K (400 C) at the surface of the insulating layer and the backside of the mold was attached to a water jacket to protect the system from heat. The molten metal solidified as a tube with 30 cm length, 30 cm outer diameter, 17 cm inner diameter, and 6.5 cm wall thickness. The centrifugal force acting in the half thickness of the tube wall was approximately 106 times as large as the gravity force (106 G). Two tantalum sheathed R thermocouples (Pt/Pt-13 pct Rh) were inserted into the cast tube wall at 20 and 35 mm from the metal-mold interface to measure cooling curves[16] (Figure 2). Each thermocouple wire had 75 mm length and 0.254 mm diameter, was electrically insulated by a layer of hafnium oxide, and had a tantalum sheath of 1.6 mm outside diameter, mounted on a support made of niobium to resist the high temperatures. A third thermocouple, of type K

Fig. 1—Schematic view (not to scale) of the longitudinal section of the centrifugal casting mold and tube system: (a) mold and tube dimensions (mm); (b) details of the cast tube internal cavity showing five surface patches: (1) differential element on inner surface; (2) complete inner surface; (3) front lid; (4) back side of mold; (5) opening for pouring.

(chromel-alumel) with mineral insulation and a stainless steel sheath, was installed within the mold wall. All three thermocouples were fixed in the middle of the tube length and rotated with the mold. The thermocouple signals were amplified with current transmitters connected to copper rings and mounted on the rotating mold shaft. The rings were in electric contact with carbon brushes, attached to the motionless structure of the mold and connected to a system to acquire signals at a rate of 1 s1 for each thermocouple.

III.

CALCULATION OF THE HEAT TRANSFER COEFFICIENT

The inverse solution technique was used to solve the heat transfer model and obtain its unknown boundary conditions using measured cooling curves. The whole domain method[17] was the inverse solution technique used in the present work, requiring: (1) a mathematical model of the heat transfer in the system with an unknown heat transfer coefficient at the metal-mold interface (Section III–A); (2) measured cooling curves (Section II); and (3) an optimization algorithm that searches for the heat transfer coefficient that minimizes METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 2—Schematic view of the device used for temperature measurement in the cast tube wall: two thermocouples, fixing system including a stainless steel hollow rod embedded in the mold wall and a niobium support to keep the thermocouple junctions in their positions within the cast tube wall. A ceramic cement seal is also used to prevent liquid metal penetration into the device.

the error between the measured cooling curves and those calculated with the model (Section III–B). A. Heat Transfer Model for the Centrifugal Casting of Tubes 1. Heat conduction model The following assumptions were adopted: (1) at the initial time, the molten alloy had the shape of a tube inside the mold; (2) heat conduction occurs only in the radial direction; (3) heat transfer by convection in the liquid was taken into account by multiplying the liquid thermal conductivity by 2.0;[18] (4) the alloy solidifies as a hypoeutectic binary Fe-2.9 wt pct C alloy according to the equilibrium lever-rule of the Fe-Fe3 C phase diagram, that is, solidification starts with primary austenite and ends with the austenite-carbide eutectic reaction. Considering these assumptions, the following heat conduction equation was written in a cylindrical reference system at the tube axis (Figure 1(b))     @T 1 @ @T @es @ee ¼ j þ DHe þ q DHf qcp @t r @r @r @t @t for Ri  r  Re

½1

All symbols are defined in the Nomenclature. The thermal conductivity of the alloy, j, is an average of the conductivities of the solid (js ) and liquid (jl ) weighted by their local volume fractions. The inner (Ri) and outer (Re) radius of the tube are 0.085 and 0.15 m, respectively. The volume fraction of primary austenite, es , is obtained from the equilibrium lever-rule as follows es ¼ 0  1  T es ¼ TTL  T 1k f  1  TE es ¼ TTL  TE 1k f

for T> TL for TE  T  TL

½2

for T
The corrected temperature and corrected solute partition coefficient are calculated as given below after linearizing the liquidus and solidus lines of the Fe-Fe3C phase diagram METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 3—Schematic curve of the linear piecewise heat transfer coefficient assumed at the metal-mold interface (hMM) as a function of time for N = 5.

Tf ¼ TE 

TE  TL ðCE  DCÞ CE  C0

k ¼

DC ¼

CSE þ DC CE þ DC

CE ðCSE  CS0 Þ  CSE ðCE  C0 Þ ; ðCE  C0 Þ  ðCSE  CS0 Þ

½3

½4

½5

where the concentrations CS0 and CSE are obtained from the Fe-Fe3 C phase diagram.[19] When the temperature at a point reaches TE, the solidification of primary austenite stops at this point, and the fraction of eutectic, ee , is calculated from Eq. [1] imposing T ¼ TE . When es þ ee ¼ 1, T is again calculated from Eq. [1].

Fig. 5—Cooling curves measured during the centrifugal casting of the cast iron tube by two thermocouples located within the tube wall at 20 mm (TM20) and 35 mm (TM35) from the metal-mold interface and by a thermocouple in the mold wall (Tm), at 3 mm from the metal-mold interface. The liquidus (TL) and eutectic (TE) temperatures are also shown.

In the simulations Ta = 25 C and Tm is assumed equal to the temperature measured as a function of time by the thermocouple positioned within the mold wall (Section II). Note that hMM is the unknown determined by the inverse solution. At t = 0, the liquid metal temperature is assumed uniform and equal to Ti (1598 K (1325 C)), defined as explained in Section IV–B. Equations [1] to [7] were solved using the implicit finite volume method.

Fig. 4—Transverse section of the tube wall: (a) Indication of inner (Ri ) and external radius (Re) showing porosity near Ri ; microstructures at (b) 20 mm and (c) 60 mm from the mold wall (Re), showing what was originally the austenite dendrites (dark) and the skeleton of ledeburite eutectic (gray).

The boundary conditions are j

@T ¼ hMM ðT  Tm Þ for @r

@T ¼ hRC ðTRi  Ta Þ for j @r

r ¼ Re

½6

2. Heat transfer at the inner surface of the cast tube The heat transfer coefficient hRC is used in Eq. [7] to calculate the heat flux by radiation and convection at the inner surface of the tube. In this case, hRC ¼ hR þ hC , where hR and hC are the radiation and convection components. The air flow causing convection within the tube cavity might be driven by its rotating inner surface (forced convection) and by buoyancy forces (natural convection) resulting from temperature differences between this inner surface and the air inside and outside (environment) the cavity. The following relations obtained by Seghir-Ouali et al.[20] for the steady-state heat transfer by convection inside a rotating tube without forced axial flow were adopted to calculate hC Nui ¼ 8:5101  106 Re1:4513 r for 1:6  103
r ¼ Ri

½7

Nui ¼ 2:85  10

4

Re1:19 r

for Rer  2:77  10

½8 5

METALLURGICAL AND MATERIALS TRANSACTIONS A

Table II. Material Tube (white iron)

Properties of Materials Used in the Simulations

Property e1,e2 (oxide) q (kg m3) DHf (J kg1) DHe (J kg1) cp (kJ kg1 K1)

js (W m1 K1)

Front lid and back of mold (gray iron) Zircon coating (front lid) Magnesium oxide coating (back of mold) Air

jl (W m1 K1) TL (C) TE (C) C0 (pct) CS0 (pct) CE (pct) CSE (pct) T*f (C) k* () j3m, j4m (W m1 K1) e3 j3c (W m1 K1) e4 j4c (W m1 K1) ja (W m1 K1) ma 9 106 (m2 s1)

Value

Reference

5.71 9 10 T  0.821 7860 263 9 103 205 9 103 0.49704 + 0.13343 9 103 T T £ 935 K 0.06993 + 0.59 9 103 T 935 < T £ 1076 K 0.55623 + 0.138 9 103 T 1076 473 K 14.312 + 0.015075 T T £ 633 K 47.45  0.037319 T 633 < T £ 766 K 17.042 + 0.0023883 T T > 766 K 2.0 9 40.3 1304 1148 2.9 1.25 4.3 2.11 1781 0.614 87.541  0.075714 T T £ 623 K 53.312  0.020858 T 623 < T £ 999 K 69.681  0.037237 T T > 999 K 4.62 9 104 T + 1.03 T £ 773 K 8.48 9 104 T + 1.22 T > 773 K 6.91  0.00504 T + 1.42 9 106 T2 6.36 9 104 T + 72.5 T £ 1098 K 0.20 T > 1098 K 65.8  0.12 7 T + 1.02 9 104 T2  2.96 9 108 T3 1.26 9 102 + 5.30 9105 T 7.26 + 0.0579 T + 6.68 9105 T2

[24] [25] [26] [27] [28]

5

[28] [29] [19] [19] [19] [19] [19] [19] — — [28] [24] [30] [24] [30] [23] [23]

Temperature should be given in K in property calculations.

The radiation component at the inner surface of the tube, hR, was calculated by the model detailed in the appendix. For the same casting conditions, Ri , w, and Ta are given constants, implying that hC is only a function of the film temperature, which is the average of TRi and the temperature of the air within the cavity, Tc , assumed as ðTRi þ Ta Þ=2. Consequently, hC ¼ hC ðTRi Þ. As explained in the appendix, hR ¼ hR ðTRi Þ, and thus hR þ hC ¼ hRC ðTRi Þ for the same cast conditions. B. Inverse Solution Method The whole domain method was used to obtain hMM, which was assumed to change linearly within N predefined time intervals.[17] In the last time interval, hMM was constant (Figure 3) to prevent unrealistic negative values. To describe hMM as a function of time, N values of the hMM components and N  1 values of time had to be determined. Although the inverse solution process is simpler when the function hMM ¼ hMM ðtÞ has fewer degrees of freedom, a more general function was preferred in the present work to avoid limiting the possible behaviors of hMM. A more convenient METALLURGICAL AND MATERIALS TRANSACTIONS A

power-law function of time,[21] hMM ¼ h0 sn ðt þ sÞn , was also tested, where h0, s, and n are the unknowns. For the linear piecewise hMM function, the direct solution of the heat transfer model (Section III–A) is where T ¼ Tðt2 ; t3 ; . . . ; tN ; h1 ; h2 ; h3 ; . . . ; hN Þ, t2 ; t3 ; . . . ; tN ; h1 ; h2 ; h3 ; . . . ; hN are the 2N  1 unknowns of the inverse problem. Using the optimization algorithm described by Martorano and Capocchi,[17] these unknowns were calculated to give the minimum average squared error (SE) between the cooling curves obtained from the direct solution of the heat transfer model and those measured experimentally (TM) by the two thermocouples within the cast tube wall.

IV.

RESULTS AND DISCUSSION

A. Microstructure of the Centrifugally Cast Tube The microstructures of samples extracted from a transverse section at the middle length of the tube were ground and polished using standard metallographic techniques and finally etched with the chemical reagent Nital 3pct (3 ml HNO3, 97 ml ethyl alcohol). The

Fig. 6—Measured (TM20, TM35) cooling curves at 20 and 35 mm from the metal-mold interface compared with calculated (T20, T35) cooling curves using a heat transfer coefficient at the metal-mold interface (hMM) defined with two different number of components, namely (a) N = 1 and (b) N = 5, and (d) using a power-law equation, with corresponding average squared errors (SE). Calculated temperature in the tube wall, at the interface with the mold (TRe) and at the inner surface (TRi) are also shown in (b). The heat flux at the metal-mold interface as a function of time for N = 5 is shown in (c).

transverse section of the tube wall is shown in Figure 4(a). The micrographs of the samples extracted at 20 and 60 mm from the mold wall (the latter being 5 mm from the tube internal surface) are shown in Figures 4(b) and (c), respectively. The microstructures are typical of white cast irons and consist of pearlite (dark areas), formed from primary austenite dendrites, and ledeburite (eutectic) in the interdendritic regions. Dendrites are columnar from the tube outer surface up to 25 mm into the tube wall (Figure 4(b)) and apparently coarser and equiaxed in the remaining region, up to the inner surface (Figure 4(c)). Features of this microstructure were used to verify the equilibrium lever-rule hypothesis assumed in Section III–A. The Fourier number (a) defined in models of solute distribution within the mushy zone[22] was calculated using Ds  6:3  1011 m2 s1 , tf  258 s, and k ¼ 41 lm,

measured at 20 mm from the tube outer surface, yielding a = 16.4 1, which suggests that the lever-rule hypothesis is reasonable. B. Heat Transfer Coefficient at Metal-Mold Interface Pouring of the molten alloy began at ~150 seconds and, thus, measured curves TM20 and TM35 for t < 150 seconds show the temperature of the air inside the mold cavity (near the inner surface of the mold), which was ~873 K (600 C) (Figure 5). At the beginning of pouring, these curves indicate an abrupt increase to ~1620 K (1347 C), which is 43 C above the liquidus temperature (TL = 1577 K (1304 C)). To define the initial condition, a uniform temperature of 1598 K (1325 C) (Ti ) was used in the heat transfer model, because it yielded good agreement between calculated and measured cooling curves. METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 7—Calculated evolution with time of the position of the mushy zone and eutectic fronts growing from the outer (Re) and inner (Ri) surfaces of the tube.

Properties of the tube and mold materials are given in Table II. The thermal conductivity for the liquid alloy was that of liquid iron multiplied by 2.0, which is a simple method to consider liquid convection in heat transport.[18] The liquidus and solidus temperatures and the equilibrium compositions of the liquid and solid were extracted from the Fe-Fe3C phase diagram for a Fe-2.9 pct C alloy.[19] After solving the radiation model described in the appendix, the approximation give below was adopted hRC ðWm2 K1 Þ ¼ 8  106 T2Ri þ 2:7  103 TRi þ 3:83; ½9 where TRi should be given in C. Since the number (N) and size of time intervals to define hMM are unknown, different N values were examined. The measured and calculated cooling curves and the hMM as a function of time are given in Figures 6(a) and (b) for N = 1 (constant hMM) and N = 5, respectively, showing better fitting and a decrease in SE when N increased. For N > 5 the fitting did not improve and hMM began to oscillate with time; therefore, N = 5 was adopted. The maximum in hMM (~870 Wm2 K1) occurred immediately after pouring, reflecting the closest contact between the surfaces of the tube and mold. Then hMM decreased with time to ~50 Wm2 K1, decreasing the heat flux at the metalmod interface (Figure 6(c)), because the conforming contact was lost as the solid shell grew and cooled, contracting, and probably causing the formation of a gap at the metal-mold interface. The cooling curve calculated at the outer surface of the tube (TRe ) shows reheating at t ~ 300 seconds, caused by the abrupt decrease in hMM. In Figure 6(d), the inverse solution for hMM as a power-law function of time is also given.

METALLURGICAL AND MATERIALS TRANSACTIONS A

The positions of the mushy zone fronts as a function of time are given in Figure 7. There are two fronts: one growing into the liquid from the outer surface (r ¼ Re ) and another growing from the inner surface (r ¼ Ri ) of the tube. Solidification started first at the outer surface (in contact with the mold) and later from the inner surface owing to heat losses by convection and radiation. These fronts met at ~7 mm from the inner surface of the tube. The solidification front growing from the inner surface would have not been predicted if this surface had been considered adiabatic, as assumed by Ebisu[3] and Martinez et al.[5] Also shown in Figure 7 are the positions of the two eutectic fronts growing into the mushy zone. These fronts meet at ~5 mm from the inner surface, indicating the last part of the tube to solidify, which is a region prone to the formation of shrinkage porosity. In Figure 4(a), porosity is observed near this region, being consistent with the model calculations. The present values of hMM, which lie between 870 and 50 Wm2 K1, are generally lower than those reported for static molds with or without thermal insulating coatings. For several alloy systems, the maximum reported hMM values were between 1500 and 8000 Wm2 K1[17,21,31–33] for uncoated mold surfaces, whereas for coated surfaces, they lie between[17,34–36] 350 and 2000 Wm2 K1 for the whole solidification and cooling periods. The present low values of hMM, especially those after the initial decay ~ (<100 Wm2 K1 ), are unexpected, since the centrifugal force was believed to enhance the contact between casting and mold surfaces, increasing the heat transfer at the metal-mold interface. This force, however, might have been insufficient to deform the solid shell and decrease the gap size at this interface. The insulating zircon coating applied to the internal mold surface in the present experiment may also have created a heat transfer barrier much larger than that at the contact between casting and coating, making hMM insensitive to any enhancement of the metal-mold contact due to the centrifugal force. Finally, the values of heat transfer coefficients reported for static molds were frequently obtained in experiments in which the metal-mold interface was below the casting and consequently had the casting weight on it, preventing the establishment of a continuous gap. C. Analysis of the Heat Transfer at the Inner Surface of the Tube Calculations with the radiation model for hRC, hR, and hC as a function of time and a polynomial approximation for hRC ¼ hRC ðTRi Þ are shown in Figure 8(a). The coefficient hRC increases with TRi , because hR increases. On the other hand, hC decreases owing to the lower density of air within the cavity, decreasing Rer in Eq. [8] as a result of the larger importance of viscous forces. The maximum value of hC (~3 Wm2 K1) is lower than those adopted by Kang and Rohatgi[8] and Nastac et al.[12], namely, 8.4 and 10 W m2 K1, respectively.

Fig. 8—Heat transfer coefficient calculated with the present model at the inner surface of the tube as a function of the surface temperature: (a) total heat transfer coefficient (hRC) and its polynomial approximation (hRC(poly)), the radiation (hR) and convective (hC) components; (b) hR for different ratios of tube length to internal cavity diameter (L=2Ri ); and (c) hR obtained from the model compared with that suggested by Kang et al.[7]

Figure 8(a) shows that radiation is dominant over convection heat transfer for TRi > 1073 K (800 C) and, therefore, also dominates the heat transfer at the tube inner surface during all solidification period and part of the subsequent cooling. The effect of the ratio of the tube length to its inner diameter ðL=2Ri Þ on hR is examined in Figure 8(b). As ðL=2Ri Þ increased from 1.8 (present experiment) to 10, the importance of radiation decreased, because a larger portion of the radiation flux ~ 4; hR is blocked by the tube inner surface. For L=2Ri > and hC are an order of magnitude lower than hMM, indicating that adiabatic conditions could be assumed at the tube inner surface, as done by Ebisu[3] and Martinez et al.[5] The calculation of the radiation heat flux implies that presented by Kang et al.[7]

  hR ¼ re T2Ri þ T2a ðTRi þ Ta Þ, which gives values nearly an order of magnitude larger than those from the present model, as illustrated in Figure 8(c). This expression is valid when the inner surface of the tube is completely exposed to the environment, which is hardly the case for cast tubes, even for L=2Ri as low as in the present experiment.

V.

CONCLUSIONS

The inverse solution method was used to determine the heat transfer coefficient at the metal-mold interface for the first time during the centrifugal casting of a cast iron tube. To apply this method, an experiment was METALLURGICAL AND MATERIALS TRANSACTIONS A

carried out to measure the cooling curves within the cast tube wall under rotation and a comprehensive heat conduction model was developed. The obtained heat transfer coefficient is ~870 Wm2 K1 immediately after pouring the melt, decreasing to ~ 50 Wm2 K1 after 2000 seconds, when the tube is completely solid and is at an average temperature of ~983 K (710 C). Although the centrifugal force is expected to enhance the contact at the metal-mold interface, the present heat transfer coefficient is lower than that reported by several authors for static molds with or without a thermal insulating coating on the mold surface. The model results also shows that the heat loss by radiation is dominant over that by convection at the tube inner surface, causing the formation of a solidification front that meets another front coming from the outer surface of the tube.

 F12 þ F13 þ F14 þ F15 þ þ F14 J4 ¼

e1 r T4  F15 rT5a ð1  e1 Þ Ri ½10

 F23 þ F24 þ F25 þ ¼

 e2 J2 þ F23 J3 þ F24 J4 ð1  e2 Þ

e2 r T4  F25 rT5a ð1  e2 Þ Ri ½11  F32 J2 þ

F32 þ F34 þ

þ F34 J4 

F42 J2 þ F43 J3 F42 þ F43

METALLURGICAL AND MATERIALS TRANSACTIONS A

 e4 þ F45 þ J4 ð1  e4 Þ

e4 rT34 T4 ¼ F45 rT4a  ð1  e4 Þ e3 J3  ð1  e3 Þ

"

½12

½13

# e3 rT33 þh3c þ T3 ¼ ð1  e3 Þ   l3c l3m 1 1  h3c Tc  þ þ Ta j3c j3m h3ca

l3c l3m 1 þ þ j3c j3m h3ca

APPENDIX The radiation component of the heat transfer coefficient, hR, is required to calculate hRC and the total heat flux at the inner surface of the tube, defined by Eq. [7]. To construct a radiation model, the internal surface that forms the complete tube cavity was subdivided into 5 surface patches (Figure 1b): (1) differential surface element on the inner surface of the tube in the middle of its length; (2) complete inner surface; (3) front lid; (4) backside of the mold; and (5) opening in the front lid. The boundary at which Eq. [7] is applied coincides with surface patch (1) and the net radiation heat flux leaving this patch is exactly the radiation part of the total flux calculated by Eq. [7]. Each of the 5 patches was assumed to have uniform radiosity and uniform temperature, which vary from patch to patch. Surface patch (5), the lid opening, was assumed black and patches (1) to (4), opaque, diffuse, and gray. Patches (3) and (4) are actually the surfaces of the coatings on the front lid and back of the mold, respectively, and they exchange heat by radiation and convection due to the air flow within the cavity. The heat conducted through the coatings is also conducted through the walls of the front lid and back of mold and, eventually, transferred to the environment (front lid) and to the cooling water (back of mold). These heat transfer steps are included in the model using thermal resistances. Considering these assumptions, the following system of equations can be written[23]

 e3 J3 ð1  e3 Þ

e3 rT33 T3 ¼ 0 ð1  e3 Þ



ACKNOWLEDGMENTS The authors thank Conselho Nacional de Desenvolvimento Cientı´ fico e Tecnolo´gico (CNPq) for the financial support (grant 310923/2011-5) and CAPES (Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´ vel Superior) for the scholarship to S. Vacca. They also thank the Key Reader for the comprehensive list of suggestions and comments.

 e1 J1 þ F12 J2 þ F13 J3 ð1  e1 Þ

1

½14 e4 J4  ð1  e4 Þ

"

#  l4c l4m 1 1 e4 rT34 þ þ þh4c þ T4 ¼ j4c j4m h4ca ð1  e4 Þ   l4c l4m 1 1  h4c Tc  þ þ Ta j4c j4m h4ca ½15

The view factor between surface patches j and i, denoted as Fji , is calculated as explained by Modest.[24] Note that the temperature of surface patch (5) is the environment temperature, Ta, and the temperature of patches (1) and (2) are equal to TRi, obtained as part of the solution of Eq. [1], i.e., T at r = Ri. The convective heat transfer coefficients h3c , h4c , and h3ca were calculated using the following relation developed for the heat exchanged between a rotating disk and the surrounding fluid[37] Nuc ¼ 0:33 Re0:5 c

for Rec <2:5  105 ðlaminar flowÞ ½16

The heat transfer between the external surface of the mold back and the cooling water of the water jacket was approximated by the heat transfer between a static

surface and an impinging jet,[23] enabling the calculation of h4ca . In this calculation, the diameter of the water inlet was 5 cm, the distance between inlet exit and backside of mold was 10 cm, and the average velocity at the inlet was 3 ms1. The system of Eqs. [10] through [15] was solved by the iterative Gauss–Seidel method for six unknowns, namely J1 ; J2 ; J3 ; J4 ; T3 , and T4 , from which J1 is the necessary variable to finally calculated hR as follows  4  rTRi  J1 e1 : ½17 hR ¼ ð1  e1 Þ ðTRi  Ta Þ The temperature at surface patch (1), TRi , and the temperatures of the air in the environment, Ta, and in the cavity, Tc , should be given to enable a unique solution of the equation system, i.e., hR ¼ hR ðTRi ; Ta ; Tc Þ. In the present work, Ta = 25 C and, since the temperature of the air in the internal cavity, Tc, is unknown, the approximation Tc ¼ ðTRi þ Ta Þ=2 was considered. A few tests with the model revealed that changes in Tc in the range from TRi to Ta caused negligible variation in hR. Therefore, if TRi is given, hR can be obtained, i.e., hR ¼ hR ðTRi Þ. For the specific conditions of the present simulations, the complete radiation model was solved and this function was approximated by a polynomial of degree 2.

NOMENCLATURE CE C0 cp CSE CS0 Ds Fji hC hi hj hic hMM h0 hRC hR h3ca h4ca Ji

Eutectic composition (pct) Average alloy composition (pct) Specific heat (J kg1 K1) Composition of the solid in equilibrium with the liquid at TE (pct) Composition of the solid in equilibrium with the liquid at TL (pct) Diffusion coefficient of carbon in austenite (m2 s1) View factor between surface patches j and i () Convection component of hRC (W m2 K1) i component of hMM (W m2 K1) mean heat transfer coefficient between rotating disk and fluid for j = 3c, 4c, 3ca (W m2 K1) Convection heat transfer coefficient between surface patch i and air within tube cavity (W m2 K1) Heat transfer coefficient at the metal-mold interface (W m2 K1) Constant of power-law model (W m2 K1) Heat transfer coefficient at the inner surface of tube (W m2 K1) Radiation component of hRC (W m2 K1) Convection heat transfer coefficient between front lid and environment (W m2 K1) Convection heat transfer coefficient between backside of mold and cooling water (W m2 K1) Radiosity of surface patch i (W m2)

k* lic l3m l4m L n N Nuc Nui r Ri Re Rec Rer t tf ti T Ta Tc TE T*f Ti Ti TL Tm TM TRe TRi w

Corrected solute partition coefficient () Thickness of the coating on surface patch i (m) Thickness of the front lid of mold (m) Thickness of backside of mold (m) Length of cast tube (m) Constant of power-law model () Number of hi components of hMM Nusselt number at the surface of a rotating disk (Rihj/ja) Nusselt number at the inner surface of tube (2RihC/ja) Radial coordinate (m) Inner radius of cast tube (m) Outer radius of cast tube (m) Reynolds number for a rotating disk (R2i w/va) Rotational Reynolds number at the inner surface of tube (2R2i w/va) Time (s) Local solidification time (s) Time of index i to define components of hMM (s) Calculated temperature (K) Temperature of air outside the mold or of cooling water (K) Temperature of air inside the tube cavity (K) Eutectic temperature (K) Corrected temperature of the pure metal (K) Initial temperature of the molten metal (K) Temperature of surface patch i (K) Liquidus temperature of the alloy (K) Temperature measured in the mold (K) Temperature measured by thermocouple (K) Temperature of outer surface of tube (K) Temperature of inner surface of tube, also surface patch (1) (K) Angular velocity of mold (rad s1)

GREEK SYMBOLS a DC DHe DHf ee ei es j ja jic jim jl js k va

Fourier number (4Dstf/k2) Concentration factor Latent heat of eutectic solidification (J kg1) Latent heat of austenite solidification (J kg1) Volume fraction of eutectic () Emissivity of surface patch i () Volume fraction of austenite () Average thermal conductivity (W m1 K1) Thermal conductivity of air at the film temperature (W m1 K1) Thermal conductivity of the coating on surface patch i ( W m1 K1) Thermal conductivity of mold material (W m1 K1) Thermal conductivity of liquid melt (W m1 K1) Thermal conductivity of austenite (W m1 K1) Secondary dendrite arm spacing (m) Kinematic viscosity of air at the film temperature (m2 s1)

METALLURGICAL AND MATERIALS TRANSACTIONS A

q r s

Density (kg m3) Stefan–Boltzmann constant (W m2 K4) Constant of power-law model (s) REFERENCES

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