Heat Transfer Modeling between the Mold and the Ingot for Convex Concave IngotSurface+1

Toshio Sakamoto1,+2, Akira Matsushita1,+3, Yasuhiro Oda1, Yuichi Motoyama2, Hitoshi Tokunaga2 andToshimitsu Okane2

1Central Research Institute, Mitsubishi Materials Co., Kitamoto 364-0028, Japan2Advanced Manufacturing Research Institute, National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8564, Japan

In this paper, a heat transfer model between the mold and the ingot with convex and concave surfaces for continuous casting (CC) processof copper and copper alloys is proposed and discussed. Conventionally, during CC process, vibration of the mold leads to ingot with convex andconcave surfaces known as oscillation marks. These marks may cause heat resistance between the mold and the ingot. In the model, three areaswhere heat resistances occurs were considered: (¦R1) non-contacting area, (¦R2) concave area derived from decrease of thermal conductivity,and (¦R3) area where non-effective heat flow exists in solid phase. The heat resistance values were obtained either analytically or by numericalmethods for conditions typically observed in the CC process. Quantitative analyses and comparison of heat resistance values indicated that ¦R3

was the most significant factor and that ¦R1 and¦R2 was negligible. Furthermore, it was found that slight changes in contact condition results ina large change in heat resistance. [doi:10.2320/matertrans.MT-M2020162]

(Received May 25, 2020; Accepted July 1, 2020; Published August 21, 2020)

Keywords: heat transfer model, continuous casting (CC) process, copper and copper alloy, mold, thermal resistance, numerical method

1. Introduction

In the continuous casting (CC) process of copper andcopper alloys, how the heat transfer coefficient between themold and ingot (the reciprocal of the thermal resistance) isdetermined has been an important research theme for a longtime. This heat transfer coefficient is thought to fluctuate dueto various factors. For example, it depends on the alloy type,the size of the ingot and the casting conditions. In somecases, even under the same casting conditions, it can alsofluctuate over time due to some reasons. Fluctuations in theheat transfer coefficient have a large influence on the qualityof the ingot, for instance, they can cause internal defects.However, the mechanism of determination and fluctuationof the value of the heat transfer coefficient has not beenclarified.

In general, on the surface of an ingot produced by theCC process of copper or copper alloys, wave-like periodicconvex and concave areas (hereinafter referred to as unevenareas or unevenness) are formed as the result of oscillation,that is, the vibration of the mold.1) The wavelength andamplitude of the unevenness is in the order of 10¹1 to 101mmand the formation of the unevenness may affect the heattransfer coefficient between the mold and ingot. Similarunevenness has been observed in continuous casting of steel,which has been reported to cause local delay of solidificationand cracking of the ingot surface.2)

To date, several reports have been made on heat transfermodels between the mold and ingot with an uneven surface,mainly in the steel industry. Ya Meng et al.3) assumed thatthe flux would enter the concave areas, and calculated thethermal resistance using the thermal conductivity of the flux.

Shibata et al.4) assumed that gas is filled in the concaveareas and proposed a model where the thermal resistanceis calculated by using both thermal conductivity of air andradiation. However, there are not many studies that haveexamined the thermal resistance in detail by paying attentionto the shape of unevenness.

The purpose of this study is to quantify the increase inthermal resistance of the ingot with an uneven surfaceaccording to the shape of the unevenness with respect tothe thermal resistance of the ingot with a flat surface. Inparticular, we expected that some areas in solid phaseclose to concave areas become so-called thermal dead spacesthat do not contribute to heat extraction, and the thermalresistance is increased by this. We named this thermal deadspace “stagnation of heat” and carried out analysis in detail.Including this “stagnation of heat”, we decomposed theincreased thermal resistance into three parts, and formulasthat determine these have been established, using a total ofeight parameters that mainly characterize the shape of theunevenness. Using these formulas, we clarified the dominantfactors of the increase in thermal resistance, and discussed thecauses of the thermal resistance fluctuations observed in theactual CC process.

2. Details of Heat Transfer Model

2.1 Hypothesis on the shape of unevennessThe aim of this heat transfer model is to calculate the

average thermal resistance in the direction perpendicular tothe surface of the ingot (hereafter such a direction is referredto as heat extraction direction or y-direction) when two-dimensional, periodical and uniform unevenness is formed onthe surface of the ingot in the casting direction (x-direction)during the CC process, as shown in Fig. 1(a). Here, two-dimensional means that, for example, in the case of acylindrical ingot, the unevenness is formed in rotationalsymmetry with respect to the central axis of the cylinder.Similarly, in the case of a prismatic ingot, it was assumed that

+1This Paper was Originally Published in Japanese in J. Japan Inst. Copper58 (2019) 109­115.

+2Corresponding author, E-mail: sakamoto@mmc.co.jp+3Present address: Corporate Products Research and Development Depart-ment Laboratories, Mitsubishi Aluminum Co., Ltd., Susono 410-1127,Japan

Materials Transactions, Vol. 61, No. 10 (2020) pp. 1974 to 1980©2020 Journal of Japan Institute of Copper

the shape of the unevenness does not change in thecircumferential direction.

In this study, the ingot size (radius or thickness of the ingotcross-section) was assumed to be more than one order largerthan the wavelength and amplitude of the unevenness.Therefore, as the surface can be regarded as substantiallyplanar macroscopically even in the case of a cylindrical ingot,it was assumed that the unevenness is formed on a two-dimensional flat plate that extends infinitely in z-direction ofthe Cartesian coordinate system as shown in Fig. 1(b). Theshape of the unevenness was expressed by a two-dimensionalcross section, and the thermal resistance and heat transfercoefficient were evaluated by the value per unit thicknessunless otherwise specified.

Furthermore, the following assumptions were used forsimplicity.

(1) The shape of the convex portion in uneven areasconstituted by the solid phase is regarded as trapezoidal orrectangular, as shown in Fig. 2.

(2) The uneven areas have already been formed at thebeginning of solidification, and the shape does not changethereafter.

(3) The interface between the solid phase and the melt isflat. Therefore, the solid-phase is divided into two parts, thoseare, the convex part and the flat plate part, as shown in Fig. 2.

(4) The concave areas outside the ingot are consideredto be adiabatic, and radiation and thermal conduction areignored for simplicity, since the purpose of this study is toanalyze the effect of the shape of unevenness. Generally, inthe CC process, concave areas are filled with lubricant orgas, so radiation and/or thermal conduction are not alwaysnegligible. However, the melting point of copper is lowerthan that of steel and the solidification of copper is aphenomenon that occurs in a temperature range where the

effect of radiation is relatively small. And the thermalconductivity of the copper solid phase is more than one orderhigher than the flux used for lubricants and the thermalconductivity of the gas. Therefore, this assumption is notconsidered to be significantly different from the actualsituation.

(5) The mold and solid phase are uniformly contacted withthe upper side of the trapezoid. As shown in Fig. 3, the localheat transfer coefficient h‘[W/m2K] between the mold andingot is set to h0 in the contacting positions (hereafter h0 isreferred to as the heat transfer coefficient of the contactingpositions), and is set to 0 in non-contacting positions, becausethe concave areas are adiabatic. In case of real CC process,the heat transfer coefficient might not be constant even withina contacting position but it might change according to theposition as shown by the arrow and dotted line at the rightend of Fig. 3. On the other hand, the depth of the unevennessstudied in this model is in the order of mm, resulting in alarge difference in the thermal resistance between thecontacting and non-contacting positions. For this reason,we concluded that it is unnecessary to model the slightchanges in heat transfer inside the contacting position indetail, and used the simple model. Moreover, the h0 valuemay vary with the progress of the solidification of the ingot,because it may change depending on, for example, the airgap formed between the mold and ingot by deformationof solidification shell. However, since the main purpose ofthis model is to analyze the effects of the shape ofunevenness, fluctuations in the h0 value were not considered.Consequently, the value of h0 was assumed to be constantregardless of the shape of the unevenness and the timeelapsed from the start of solidification.

(6) There is no thermal resistance between the melt andsolid phase.

(7) The thermal conductivity k[W/mK] of the solid phaseis constant regardless of temperature and position.

(8) The shape of the solid phase is represented by thefollowing six parameters as shown in Fig. 4.(a) A half of the wavelength of the unevenness L[m].(b) The depth (amplitude) of the unevenness D[m].(c) The thickness of the flat plate part, which is the solid

phase thickness ‘[m].(d) Contact ratio A[-], which is the ratio of contact between

the mold and the solid phase, and is defined as A =

Mold

Solid (Convex/Concave)

(Plate Area)

Liquid

Direction of Heat Flux

y

x

z

Convex/Concave

(a) Ingot (b) Convex/Concave

Fig. 1 Schematic drawing of the ingot and convex/concave.

Liquid

Solid (Plate Area)

Solid (Convex/Concave)

Plate Area ℓ

D

2LMold

Convex/Concave

Fig. 2 Schematic drawing of the model of convex/concave.

0

h0

hav

hℓ

Fig. 3 The model of heat transfer coefficient.

Heat Transfer Modeling between the Mold and the Ingot for Convex Concave Ingot Surface 1975

a/2L, where a[m] is the length of the upper side of thetrapezoid of the convex part.

(e) Filling ratio B[-], which is the volume fraction of theconvex part to the whole unevenness, and is defined asB = b/2L, where b[m] is the average value of the widthof the trapezoid.

(f ) Connection ratio C[-], which is the ratio of the solidphase connection between the flat plate part and theuneven part, and is defined as C = c/2L, where c[m]is the length of bottom side of the trapezoid.

(9) Since the unevenness is repeated periodically in thecasting direction (x-direction), it is sufficient for the modelingarea to have only the length L in x-direction, as shown inFig. 4. As the value of ‘ is equal to the solidificationthickness, it increases with the progress of solidification, butit was considered to be uniform within the area of the model.

In this model, the increase in the resistance was expressedusing eight parameters in total, consisting of six parametersrelated to the shape (L, D, ‘, A, B and C), a physical propertyk and an engineering parameter h0.

2.2 Assumptions about increase in resistance andformula derivation method

In this model, as shown in Fig. 5, it was assumed that thefollowing three types of thermal resistances (unit is m2K/W)are generated when unevenness is formed, compared to thecase of the ingot with a flat surface.(a) Increase in contact thermal resistance ¦R1, which is

caused by the presence of non-contacting parts betweenthe mold and the solid phase.

(b) Increase in thermal resistance in uneven areas ¦R2,which is caused by the decrease in apparent thermalconductivity in uneven areas because concave areasare adiabatic.

(c) Increase in thermal resistance due to “stagnation ofheat” ¦R3.

Then, the increase in total thermal resistance ¦R isexpressed as ¦R = ¦R1 + ¦R2 + ¦R3.

Here, “stagnation of heat” means the solid phase portionsthat become dead spaces from the viewpoint of the heat flowfrom the molten metal to the mold and do not sufficientlycontribute to the heat extraction. It is thought to be generatedmainly in the place near the concave area shown by the circlein Fig. 5(c).

The equations that determine ¦R1, ¦R2 and ¦R3 wereestablished using the 8 parameters defined in the previoussection. The equations for ¦R1 and ¦R2 were expressedusing the basic formulas for heat transfer. Regarding ¦R3, itwas difficult to express it using an analytical solution, soan approximate expression was obtained by performing anumerical analysis using the two-dimensional finite differ-ence method.

3. Formulation of Increase in Resistance Value

3.1 Increase in contact thermal resistance: ¦R1

When the surface of the ingot is uneven, the average heattransfer coefficient between the mold and the solid phase bycontact is represented by hav = A · h0. Therefore, the thermalresistance R1 per unit cross section is expressed as R1 =1/(A · h0). On the other hand, since the thermal resistanceR1p in case of the ingot with a flat surface is 1/h0, increasedthermal resistance ¦R1 can be expressed as follows.

�R1 ¼ R1 � R1p ¼ ð1� AÞA � h0

ð1Þ

3.2 Increase in thermal resistance in uneven areas: ¦R2

When the shape of the convex portion is rectangularinstead of trapezoidal, the solid phase cross-sectional areain the uneven areas effective for thermal conduction in theheat extraction direction is thought to decrease from 1 toB because concave areas are assumed to be adiabatic.Therefore, the apparent thermal conductivity k2 of the unevenareas can be expressed as follows.

k2 ¼ B � k ðB < 1Þ ð2ÞThe thermal resistance R2 is obtained by multiplying thereciprocal of k2 by the depth of the unevenness D.

R2 ¼D

B � k ð3Þ

The thermal resistance R2p in the absence of unevennessis expressed as R2p = D/k. Consequently, the increase in theresistance ¦R2 can be expressed as follows.

�R2 ¼ R2 � R2p ¼ Dð1� BÞB � k ð4Þ

In the case of a trapezoid, since the width changes in theheight direction (y-direction), the thermal resistance also

L

ModelArea

a=A 2L

b=B 2L

c=C 2L

2L ℓ

D

Fig. 4 Six parameters describing the shape of convex/concave.

Contact non-Contact

Adiabatic

AdiabaticAdiabatic

Low Temp. Adiabatic

High Temperature

ΔR3

ΔR2

ΔR1

Heat flow

Less Heat flow

Adiabatic

B

1 C

A

C

(c)

(b)

(a)

(b')

1 A

1 B

A

C

B

Fig. 5 Schematic drawing of three kinds of heat resistance.

T. Sakamoto et al.1976

changes continuously in the height direction in proportion tothe reciprocal of the width. The thermal resistance betweenthe upper and lower sides of the trapezoid can beapproximately obtained by integrating the reciprocal of thewidth in the height direction, assuming that continuouslychanging thermal resistors are connected in series. Here, theratio of the trapezoidal thermal resistance value to that ofthe rectangle with the same area and height, r is determined.As shown in Fig. 6(a), the shape of the trapezoid is definedas base 1 + d, top 1 ¹ d and height 1. The width of thistrapezoid w at an arbitrary height position v is expressed bythe following equation.

w ¼ �2d � vþ ð1þ dÞWhen the thermal conductivity is 1, the relationship

between d and r can be expressed by eq. (5) and Fig. 6(b).

r ¼Z1

0

dv

w¼

Z1

0

dv

�2d � vþ ð1þ dÞ

¼ � 1

2dlnð�2d � vþ 1þ dÞ

� �10

r ¼ lnfð1þ dÞ=ð1� dÞg2d

ð5Þ

As can be seen from eq. (5) and Fig. 6(b), even for a verysharp trapezoid such as d = 0.9 (the ratio of the upper to thelower sides is 1:19), the value of r is at most 1.6, and thedifference in the thermal resistance between the trapezoidand rectangle is not very large. In other words, as shown inFig. 5(bA), the difference between the calculated value and thetrue value is not so large even if calculations are performedusing rectangles instead of trapezoids. For this reason, ¦R2

was evaluated using eq. (4) thereafter. In addition, the valueof B can be changed formally independently of A and Cso that this formula can be applied to shapes other thantrapezoids.

3.3 Increase in thermal resistance due to stagnation ofheat: ¦R3

3.3.1 Numerical analysis modelAs shown in Fig. 5(c), the area of numerical analysis for

obtaining ¦R3 was limited to the rectangular area in the flatplate part of solid phase. For the boundary conditions, it wasassumed that heat flows in from the entire lower side of therectangular area which is contacted to the molten metal, andheat flows out from a part of the upper side that is connected

to the convex part. The temperature of the lower side of therectangular was set to 1000°C, the part of the upper side,which is connected to the convex part was set to 0°C, and thepart of the upper side which is not connected to the convexpart was set to adiabatic. Since the temperature distributionin x-direction repeats periodically, the vertical sides of therectangular were set as the adiabatic boundary condition.Under these boundary conditions, two-dimensional thermalanalysis was performed to obtain the internal temperaturedistribution. In addition, using these results, the heat fluxat each position was calculated and the thermal resistancebetween the top and bottom sides of the trapezoid wereobtained. The values of four parameters related to this areawere changed in the series of calculation, those were, solidphase thickness (‘), solid phase thermal conductivity (k), onehalf of the wavelength (L) and connection rate (C).

This numerical analysis is not solidification analysis, butnumerical analysis for calculating the thermal resistance of acertain shape.3.3.2 Numerical analysis results

Figure 7(a) shows the temperature distribution of the flatplate part when C = 0.5 and ‘ ¼ 2L, as an example ofnumerical analysis results. The width of 4L is shownrepeatedly in x-direction in Fig. 7(a). For comparison,Fig. 7(b) shows the temperature distribution in the casewithout unevenness and the heat flux is completely uniformand one-directional. The isothermal line means that the widerthe interval, the smaller the temperature gradient and thesmaller the heat flux. In Fig. 7(a), the area shown by a blackcircle has a wider isotherm interval and smaller heat flux thanthe area shown in Fig. 7(b), and it indicates that there is a“stagnation of heat” there. On the contrary, the interval of theisothermal lines is narrow around the area shown by the blackarrow, and due to the influence of this, the interval of theisothermal lines is wide and the heat flux is small around thearea shown by the white arrow. Around the area shown bythe white arrow, the isothermal line is perpendicular to theheat extraction direction and almost flat, so the heat fluxaround this area as equal to the average heat flux of thisshape. This indicates that the total thermal resistance betweenthe upper and lower sides is increased due to the unevenness.

As described above, by comparing Fig. 7(a) and Fig. 7(b)it is clearly shown that the stagnation of heat is formed in the

1+d

1 d d

1

d

0.9

1.6

(a)Definition of d (b) The Relationship between r and d

Fig. 6 Heat resistance of trapezoid.

0 1000

Mold

L

ℓ=2L

Fig. 7 Temperature distribution of the plate area.

Heat Transfer Modeling between the Mold and the Ingot for Convex Concave Ingot Surface 1977

flat plate part, and the mechanism by which this phenomenonoccurs was clarified intuitively and visually.

Figure 8 shows the relationship between the thickness ofthe flat plate part ‘ and ¦R3 when the values of the thermalconductivity of the solid phase k and the wavelength L arechanged. The value of C is fixed at 0.5. Here, as shown inthe Fig. 8, the value of ‘ and ¦R3 were made dimensionlessand are expressed as ‘� ¼ ‘=L (‘� is the dimensionlesssolid phase thickness) and ¦R3

³ = k/¦R3/L (¦R3³ is the

dimensionless increase in resistance), respectively. It can beseen that all values of ¦R3

³ is almost independent of Land k.

From Fig. 8, it can be seen that the ¦R3³ value increases

with the increase of ‘� when ‘� is small, and reaches themaximum value (denoted as ¦R3³+) at ‘� ; 1 (‘ ; L), withno increase thereafter. This is because, as shown in Fig. 7(a),“stagnation of heat” exists only in the vicinity of unevenareas, and does not exist at a distance L or more from unevenareas.

As described above, since ¦R3³+ does not depend on L,

k and ‘, ¦R3³+ is considered to be a function of only the

connection rate C. Then, when the relationship between Cand ¦R3

³+ was obtained from the results of numericalanalysis, as shown in Fig. 9, It turned out that ¦R3

³+/(1 ¹ C) and ¹ln (C) have a nearly linear relationship.Consequently, it was found that ¦R3

+ can be expressed by thefollowing approximate equation.

�R�3 ¼ � 0:6071 � Lð1� CÞ � lnC

kð6Þ

As described above, the total thermal resistance increasedue to the formation of the unevenness ¦R can be calculatedusing eqs. (1), (4) and (6).

4. Heat Transfer Coefficient in Actual CC Process

In order to analyze fluctuations in the thermal resistanceoccurring in the CC process using this model, the moldtemperature in the actual CC process was measured and thechanges in the thermal resistance depending on the castingconditions were calculated.

4.1 The method and the results of temperature measure-ment

The measurement of the mold temperature was carried outon a low-alloy cylindrical ingot under conditions 1 and 2.Under conditions 1 and 2, the casting speed, molten metaltemperature (i.e. total amount of heat input), and the moldused were not changed while other conditions were changed.The temperature was measured at three locations, those are,upper, middle and lower points of the mold by changingthe mold height in the casting direction. Measurement wasperformed using a K type thermocouple at 1-second intervalsfor 20 minutes, and the average value was taken as thetemperature at each position.

Figure 10 shows the measurement results. Under con-ditions 1 and 2, the mold temperature changed by several tensof K. This is considered to be due to the change in the heattransfer coefficient (thermal resistance) between the mold andingot.

4.2 Calculation method and results of thermal resist-ance

In order to calculate the difference in thermal resistancebetween condition 1 and condition 2 from the measuredtemperature data, a two-dimensional axisymmetric thermalconductivity analysis of the CC process was performed usingcasting conditions as input conditions. First, in order todetermine the overall heat transfer coefficient (heat transfercoefficient between the cooling water and ingot) undercondition 1, numerical calculations were performed so thatthe mold temperature measured under condition 1 and themold temperature calculated under condition 1 match.

As the overall heat transfer coefficient includes twounknown values; one is the heat transfer coefficient betweenthe mold and ingot and the other is the heat transfercoefficient between the mold and cooling water (the watercooling transfer coefficient), in order to determine the bothheat transfer coefficients respectively, more information wasneeded. Consequently, calculation was also performed so

Fig. 8 The relationship between ¦R3³ and ‘�.

Fig. 9 The relationship between ¦R3³+ and C.

Fig. 10 The result of mold temperature measurement.

T. Sakamoto et al.1978

that the depth of the molten pool (distance from the moltenmetal surface position to the deepest point of the pool)measured under condition 1 and the depth of the moltenpool calculated under condition 1 become almost same.Next, in order to obtain the heat transfer coefficient betweenthe mold and ingot under the condition 2, calculation wasperformed so that the mold temperature measured under thecondition 2 and the mold temperature calculated under thecondition 2 were almost the same value, without changingthe water cooling transfer coefficient that was obtainedbefore.

As the result, the values of the overall heat transfercoefficient were as follows.(1) Condition 1: 1500W/m2K(2) Condition 2: 1320W/m2KFrom these results, the difference in thermal resistance

¦Rx between condition 1 and condition 2 was calculated as¦Rx = 0.90 © 10¹4 [m2K/W]. According to the simulationresults, the depth of the molten pool under condition 2 wasestimated to be about 8.2% deeper than that undercondition 1. Although detailed description is omitted, theingot produced under the condition 1 had internal defects,and the ingot produced under the condition 2 was sound. Asexplained so far, in the actual continuous casting process, theheat transfer coefficient of about 10% easily fluctuates, whichmay cause a great change in the quality of the ingot.

5. Discussion

5.1 Dominant factors of increase in thermal resistancedue to unevenness

Using eqs. (1), (4) and (6), the eight parameters werevaried to calculate the values ¦R1, ¦R2 and ¦R3

+, and themagnitude of the values was compared. For reference, thethermal resistance of the solid phase Rs ¼ ‘=k was alsocalculated and compared. The range of variation of theparameters was as follows, which is usually seen for copperand copper alloys.

Since a low-concentration copper alloy with a conductivityof 20% IACS or higher was assumed, k was set to 50 to400W/mK. L was set to 10mm at maximum because it isalmost the same as the wavelength of the unevenness. D wasalso set to 10mm at maximum because it is the depth of theunevenness, and ‘ was set to 0 to 200mm because it is theradius of the ingot or one half of the thickness of the ingot. Inthe range of the general shape of unevenness, the value of A,B and C varied from 0.1 to 0.9, and A ¼ B ¼ C, because theshape of the convex part is trapezoidal and it becomes thinnerfrom the bottom to the top. Since the h0 value is equivalentto the average heat transfer coefficient between the mold andingot without unevenness, it was estimated to be approx-imately 1000 to 2000W/m2K from actual measurementexperience. It is unknown whether the value is the same forthe ingot with an uneven surface, but in this analysis, themaximum was set to 2000W/m2K.

First, as a common example of copper, the combination ofparameter values shown in set 1 of Table 1 is called thedefault condition. As a result of this combination, ¦R2 and¦R3

+ were extremely small and negligible compared to ¦R1

and Rs.

Next, we investigated whether the combinations ofparameters may or may not exist such that ¦R2 and ¦R3

+

equal to or greater than ¦R1. As shown in Set2, whenA < B < C with respect to the default condition, ¦R2, ¦R3

+

decreased significantly more than ¦R1. Moreover, as shownin Set3, when values A, B and C increased simultaneouslywhile maintaining A = B = C with respect to the defaultcondition, ¦R1 and ¦R2 decreased simultaneously, and ¦R3

+

decreased significantly. As shown in Set4 to Set7, the valueof ¦R1 was always maximum even when L, D, k and h0 wereappropriately varied from the default value. Furthermore,as shown in Set 8, ¦R1 was maximum even when all theparameters were selected so that the values of ¦R2 and ¦R3

+

became the largest relative to ¦R1.From the above results, for copper and copper alloys, the

increase in resistance when unevenness is formed on thesurface of the ingot is approximately ¦R1 º ¦R2 ¿ ¦R3

+,indicating that ¦R1 was dominant. The effect of “stagnationof heat” focused on in this study was not so great.

5.2 Estimation of factors of thermal resistance changedue to differences in casting conditions

The reasons for the change in the thermal resistance ofcasting conditions 1 and 2 calculated in Chapter 4 wereestimated using the results obtained so far. First, from theresults of the discussion in Section 5.1, the change in ¦Ris considered to be due to the fact that only ¦R1 changed.Consequently;

�Rx ¼ �R1ðCond: 2Þ ��R1ðcond: 1Þ ð7ÞHere, substituting the values of the contact rate A under

conditions 1 and 2 at A1 and A2, respectively, and the valuesof the respective heat transfer coefficients h0 to be h01 and h02,the following is established.

�Rx ¼1

A2 � h02� 1

A1 � h01¼ 0:90� 10�4 ð8Þ

Since eq. (8) included only two kinds of parameters A andh0, the change in the change in thermal resistance fromcondition 1 to condition 2, ¦Rx is thought to be the resultsof the change in A or h0. Therefore, the two cases in whicheither A or h0 has changed were reviewed as describedbelow.

First, in Case 1, the change in A was considered. Here,when A1 = 0.5 and h01 = h02 = 2000W/m2K, the value of

Table 1 Parameter sets used for evaluation of heat resistance.

Heat Transfer Modeling between the Mold and the Ingot for Convex Concave Ingot Surface 1979

A2 became 0.458, and the difference in the value of Abetween conditions 1 and 2 became about 10%. Even if thevalue of A1 was changed to a value other than 0.5, the changein A2 was almost the same. Although the shape of theunevenness of the ingot produced in this casting test was notbeen measured, such a change in the shape of unevennessmay occur generally in the continuous casting of copper andcopper alloys.

Therefore, it was clearly demonstrated that the cause ofthe fluctuation obtained by this actual CC process data can beexplained by the change in the shape of the unevenness.

Next, in Case 2, only h0 was changed. Here, when A1 =A2 = 0.5, and h01 = 2000W/m2K, the value of h02 became1833W/m2K, and the change was found to be about 10%.The value of h0 changes due to some cause other than theformation of unevenness, and is considered to changedepending on, for example, the air gap due to the detachmentof the ingot from the mold. The possibility that h0 changes byabout 10% due to these factors cannot be denied at present.Therefore, the fluctuations in thermal resistance observed inthe actual CC process this time could be due to the change inthe contact ratio A, but it was not confirmed that it was theonly cause.

6. Conclusion

The increase in thermal resistance when unevenness isformed on the surface of an ingot produced by the CCprocess was divided into three factors (¦R1, ¦R2 and ¦R3),and each thermal resistance increase value was expressedusing a simple equation consisting of eight parameters. Then,using these equations, the causes of the fluctuation in thethermal resistance observed in the actual CC process werediscussed.

As a result, it was found that the most dominant factorof the fluctuation in the thermal resistance was the contactratio, A.

In addition, it was found that if the cause of the fluctuationin thermal resistance observed in the actual CC process islimited only to the change of the shape of the unevenness, thecause can be explained by the change in A.

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T. Sakamoto et al.1980