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Mathematical model for the unidirectional solidification of metals: II. Massive molds

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Abstract

In this paper, a new mathematical model recently outlined by the present authors 1 is applied to the case of unidirectional solidification via heat extraction through massive uncooled molds of effectively semi-infinite thickness. The model permits measurement of the Newtonian heat transfer coefficient at the metal/mold interface and a complete description of the kinetics and thermal characteristics of solidification is subsequently possible. Experimental results are compared with predictions for the case of lead and the effect of mold thickness in the effectively finite regime is also investigated.

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Abbreviations

a m :

thermal diffusivity of mold material (= km/cmdm), m2/s,

a s :

thermal diffusivity of solid metal (= ks/csds) m2/s,

A m :

first integration constant of thermal profile in mold, K,

A s :

first integration constant of thermal profile in metal, K,

B m :

second integration constant of thermal pro-file in mold, K,

B s :

second integration constant of thermal pro-file in metal, K,

c m :

specific heat of mold material, J/kg·K,

c s :

specific heat of solid metal, J/kg ·K,

d m :

density of mold material, kg/m3,

d s :

density of solid metal, kg/m3,

E o :

thickness of “pre-existing” adjunct to mold in virtual system, m,

h i :

Newtonian heat transfer coefficient of metal/mold interface, J/m2·K· s,

h im :

heat transfer coefficient on mold side of metal/mold interface, J/m2 ·K·s,

h is :

heat transfer coefficient on metal side of metal/mold interface,

H :

latent heat of fusion of metal, J/kg,

k m :

thermal conductivity of mold material, J/m·K·s,

k s :

thermal conductivity of solid metal, J/m ·K·s,

S :

thickness of solidified metal in real system, m,

S′ :

thickness of solidified metal in virtual systems, m,

S o :

thickness of “pre-existing” adjunct to metal in virtual system, m,

t :

time from zero point in real system, s,

t′ :

time from zero point in virtual systems, s,

t o :

time to produce “pre-existing” adjuncts in virtual systems, s,

T :

absolute temperature in real and virtual systems, K,

T f :

freezing temperature of metal, K,

T i :

invariant temperature of hypothetical plane at metal/mold interface, K,

T im :

temperature of mold at metal/mold interface, K,

T is :

temperature of metal at metal/mold interface, K,

T m :

temperature at any point in the mold, K,

T o :

initial temperature of massive mold (ambient temperature), K,

T s :

temperature at any point in the solidified metal, K,

V :

velocity of liquid/solid interface in real system, m/s,

x :

distance from metalJ.mold interface in real system, m,

x′ :

distance from metalJ.mold interface in virtual systems, m,

α :

first constant of Eq. [6] (= 1/4asθ2), s/m2,

β :

second constant of Eq. [6] (= So/2asθ2), s/m.

E *o :

dimensionless thickness of “pre-existing” adjunct to mold, Eohi/km,

H* :

dimensionless latent heat of fusion of metal,H/c s(Tf-T o)

M :

ratio of heat diffusivities of solid metal and mold material,k sdscs/kmdmcm)1/2,

N :

square root of ratio of thermal diffusivities of solid metal and mold material,a s/am)1/2,

S* :

dimensionless thickness of solidified metal in real system,Sh i/ks,

S *o :

dimensionless thickness of “pre-existing” adjunct to metal,S ohi/ks

t* :

dimensionless time from zero point in real system,thi2k sdsCs,

T *m :

dimensionless temperature at any point in the mold, (T m -To)/(Tf -To)

T *s :

dimensionless temperature at any point in the metal, (T s-To)/(Tf - To),

x*(x > 0):

dimensionless distance into metal from metalJ.mold interface,xh i/ks,

*(x \lt 0):

dimensionless distance into mold from metalJ.mold interface,xh i/km ,

θ :

dimensionless solidification constant, Eq. [20].

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Authors and Affiliations

  1. Department of Mechanical Engineering, State University of Campinas (UNICAMP), Campinas, SP, Brazil

    A. Garcia (Professor), T. W. Clyne (Professor) & M. Prates (Professor)

Authors
  1. A. Garcia
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  2. T. W. Clyne
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  3. M. Prates
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Garcia, A., Clyne, T.W. & Prates, M. Mathematical model for the unidirectional solidification of metals: II. Massive molds. Metall Trans B 10, 85–92 (1979). https://doi.org/10.1007/BF02653977

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