A R C H I V E S

o f

F O U N D R Y E N G I N E E R I N G

Published quarterly as the organ of the Foundry Commission of the Polish Academy of Sciences

ISSN (1897-3310) Volume 10

Issue 4/2010

77 – 82

14/4

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Chvorinov’s rule and determination of coefficient of heat accumulation

of moulds with non-quartz base sands Dedicated to Ing. Nikolai Chvorinov (1903 – 1987) on the occasion of 70th anniversary of publishing

his solidification theory [3]

P. Jelínek*, T. Elbel

VŠB – Ostrava University of Technology, FMMI Foundry Department

*Corresponding author. E-mail address: petr.jelinek@vsb.cz

Received 16.06.2010; accepted in revised form 25.06.2010

Abstract Application of the „Chvorinov’s rule“ for calculation of the total time of casting solidification made also possible to determine chilling effect of foundry moulds (coefficient of heat accumulation of the mould, bf) with use of mixtures with new kinds of non-quartz base sands (Magnesite, Chromite, Olivine, Dunite, Kerphalit). Processes by several authors (G. Halbart, A. I. Vejnik, G. A. Anisovich) were used for mathematical treatment of measurement results and determination of bf. The highest values were achieved for magnesite moulds followed by chromite ones; the lowest values, approximately half-ones, represented the Dunite moulds. At the same time the results made possible to determine „the Chvorinov’s mean solidification constants“ (k) that are in direct proportional dependence on bf and indirect proportional to solidification time (τ1). Keywords: Coefficient of heat accumulation of moulds, Non-quartz base sands (Magnesite, Chromite, Olivine, Dunite, Kerphalit),

Chvorinov´s mean solidification constants

1. Introduction

A mould and its physical and chemical properties decide not only the surface quality of castings but their internal quality too. For simulating the foundry processes, casting crystallization and cooling in particular, it is necessary to know above all the chilling effect of the mould – thermal conductivity (λ), temperature diffusivity (a) and coefficient of heat accumulation of the mould (bf) (heat diffusivity coefficient). The bf coefficient is also necessary for determining the solidification constant when calculating the solidification time of castings (hot spots).

Therefore with introducing synthetic and natural non-quartz base sands it is necessary to know their physical characteristics [1] [2].

The term of solidification constant was introduced in foundry specialized literature by N. Chvorinov [3]. The contribution is engaged in experimental determination of the chilling effect expressed by the coefficient of heat accumulation of the mould and the mean solidification coefficient for non-quartz base sands bonded both with organic (furan self-setting, CO2 – resol) and inorganic binders (CO2 – process with sodium silicate, geopolymers).

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2. Heritage of Ing. Nikolai Chvorinov

Ing. Nikolai Chvorinov was an excellent specialist in the field of crystallization and solidification of castings and ingots, a person of versatile interests and artistic inclinations with an eventful fate. After complicated arrival in Czechoslovakia he graduated in 1928 from the Mining Academy in Příbram and from 1929 he worked in the research department of the Škoda plant in Plzeň. At first he was aimed at cast iron properties, latter on at questions of casting solidification. He gained the world fame with his cardinal work about the theory of casting solidification published in 1940 in German [3]. From that time his name is mentioned in all important works about casting solidification even in the beginning of 21st century. From the last ones it is a work by J. Campbell [5] who stated that convincing demonstrations of the Chvorinov’s rule accuracy have been made many times. 70 years passed this year from the first publication by N. Chvorinov [3] about control of casting solidification with the aid of calculation. It includes a relation for calculating the total time of casting solidification Z as follows: Z = (R/m)2 (1) in which the R is ratio of the volume V to the casting surface S and m is solidification constant. The mentioned article also contains a relation for calculation of solidification constant and graphical representation of the relation [3] in a double logarithmic scale in such a way as it is known from the most frequently cited publication by Chvorinov in German from the Giesserei magazine [3]. The relation (1) was latter on called „the Chvorinov’s rule“ or it is also called „ the Chvorinov’s law“. It is the most frequently cited relation in the field of metal solidification in castings. Chvorinov has called the R in the German original [3] „Grossenkoefizient“ – the size coefficient. Latter on in the Czech publication he used the term the thickness equivalent of the casting [6]. Due to the work by R. Wlodawer [7] the term Modul has been extended in German that was also adopted by other languages (Modulus in English). It is determined according to the relation [4] as follows:

S

VR = (2)

The above mentioned Chvorinov’s rule (1) was latter on

criticized by the author himself to the effect that it corresponds to one-dimensional heat transfer [6]. Heat transfer on concave mould surfaces curved inward (e.g. at a cylinder, a ball) is different and it can be faster than in case of a flat one-dimensional wall (a case of a plate). Difference of heat flows is caused by a fact that with the same value of relative thickness of a casting the sphere solidifies most quickly, the cylinder more slowly and the plate solidifies as a last one. In such a way a more general relation for calculating the solidification time has resulted as follows: τ = ε . k . R2 (3)

in which the ε is a shape coefficient and the k is a mean solidification coefficient corresponding to a reversed value of the Chvorinov’s solidification constant m2 from the equation (1). For a plate form casting the ε = 1 and solidification time τ1 is given by the equation as follows:

( )[ ]22c2

2st4

2st1t1cL

21

2

1S

1V1

λρ

−+ρπ=τ

(4)

In the equation the product of thermal and physical properties λ . c . ρ is expressed as products of heat accumulation 1 (diffusivity) of the mould bf [W . s1/2 . m-2 . K-1] by the relation as follows:

22c2fb λρ= (5)

If the time of casting solidification in moulds made from different moulding materials is compared then under the same conditions (the same metal, the same form and dimensions of the casting) the solidification time is indirect proportional to the coefficient bf. Heat behaviour of the foundry mould and its chilling effect can be also expressed with the aid of the mean solidification coefficient k from the equation (3) as follows: τ / R2 = ε . k (6) and the shape coefficient for the plate casting is ε = 1 and for other forms (cylinder, ball) it is ε < 1. Then the mean solidification coefficient for the same casting material is dependent on the coefficient of heat accumulation of the mould bf and it can be a criterion of chilling effect of the mould as already given by Chvorinov on several examples [3].

2.1 Non-quartz base sands and composition of mixtures

5 kinds of base sands were checked – Magnesite, Dunite

(Magnolite), Olivine, Chromite, Kerphalit KF and one mixture (Chromite/Olivine). Composition of individual mixtures (8 variants) is given in tab. I.

Sodium silicate binder (48/50 °Bé; m = 2.3; KP = 5.6 % Na2O) was hardened by CO2 – process. Resol resin with the Ecolotec 600 binder was also hardened by CO2. For hardening the mixture with furan resin (X 850 H.-A.) the hardener 100T3 (PTS – acid) was used. Geopol 515 geopolymer was hardened by the SA71 hardener.

2.2. Bulk density of the mould and a test casting

A mould of a plate test casting (fig. 1.) was formed by two

external cores for the reason of ensuring the defined and uniform

1 The authors of the paper prefere for bf the term “heat accumulation” because it better represents heat absorption in sand moulds.

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bulk density at ramming and the accurate position of thermocouples measuring the thermal field (tab. 2.). Table. 1. Composition of mixtures with non-quartz base sands

Mixture mark

Composition of mixtures [weight parts]

Moisture of mixtures

[%] No 1 100 Magnesite 6.4 sodium

silicate 0.2 water

3.2

No 2 100 Dunite 8.0 sodium silicate

0.2 water

3.9

No 3 100 Olivine 5.1 sodium silicate

0.2 water

2.2

No 4 100 Chromite 1.8 Resol resin (Ecolotec 600) Not

determined No 5 40 Chromite

60 Olivine 4.0 sodium

silicate 2.1

No 6 100 Chromite 1.2 furan resin 0.65 catalyst Not

determined No 7 100 Kerphalit 2 Geopol 515

0.28 hardener SA71

Not determined

No 8 100 Chromite 1.2 Geopol 515 0.17 hardener

SA71

Not determined

Table 2. Bulk density of cores

Mixture mark Bulk density of cores [kg/m3]

No 1 2694 No 2 1779 No 3 2011 No 4 3067 No 5 2555 No 6 2911

No 7 ** 1777 No 8 * 3131

* Chromite AFS 50 ** Kerphalit KF 200/400 µm, AFS 50 The test casting of dimensions 38 x 400 x 298 mm met a condition of one-dimensional thermal field (a semi-endless plate)

38

400

s

h= > 10 where the h is plate height and the s its relative

thickness (modulus R1 = 0.019 m). Test steel castings of gross weight 29.5 kg (ČSN 422650 standard) were cast from casting temperatures of 1615 – 1620 °C (measured dipping in the ladle). Dimensions of false plate-like cores were l 40 x 415 x 298 mm.

Fig. 1. Mould of the plate test casting with a false core (a) and a

core box with a compacted core (b) with premoulded channels for thermocouples.

3. Experimental methods of determination of the coefficient of heat accumulation of the mould (thermal diffusivity coefficient)

Many authors when calculating the coefficient of heat accumulation of the mould (bf) result from determination of solidification time of test castings of the semi-endless plate form with linear thermal field [8 – 12]. Solidification time of the plate was determined by direct measurement of temperatures in the casting thermal axis and it was checked by calculation from chemical composition of metal with use of a method designed by L. Kuchař and L. Repická [13]. G. Halbart [8] assumes that temperature of the mould – metal interface (ti) is equal to solidification temperature (ts). Generally

[a]

[b]

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the (ti) comes closer to the (ts) with lowering solidification temperature of measured metal (alloy):

( )[ ]1it2

st1t1cL1.1Rfb

τ

−+ρπ= (7)

where: R1 = relative casting thickness (modulus) ρ1 = bulk density of metal L = latent crystallization temperature (L = 272.1 . 103 [J . kg-1]) c1 = specific heat of metal (c1 = 753.1 [J . kg-1 . K-1]) t1 = metal casting temperature τ1 = casting solidification time A. I. Vejnik [11] divides the all solidification and cooling process in 3 time phases: Phase 1 - time of overheating heat removal (τp) 2 - solidification time (τt) 3 - cooling time (τch) Then he determines the bf from the sum of two first phases (τp + τt) as follows:

( )[ ]

−+ρ+ρ

τ

π=

st

stLt1cL1

2

Lt

1tln1ct

12

1Rfb

(8)

where: τ1 = τp + τt ρt = mass density of solidifying metal (within t1 – tL) tL = liquidus temperature of metal G. A. Anisovich [9] [10] has suggested quite original method of calculating the bf. Temperature field of the mould can be described with a parabola of the nth order. The amount of heat accumulated by the mould he

consider proportional to the *1S surface under the parabola.

Surface under the curve equals to 12n

1

+, to the S* +

*1S

quadrangle surface in which the parabola is situated (fig. 2.):

12n

*1S

*S*

1S+

+= (9)

If the S* and *1S surfaces are determined from the experiment

then it is possible to determine the parabola degree as follows:

*1S

*S

2n = (10)

Fig. 2. Scheme of the temperature field parabola

1.12n

2n2

it

L11R

fb

τ+

ρ=

(11)

3.1. Results of measurements of solidification time and temperature fields of moulds

Table 3 gives results of experiments for moulds from different moulding mixtures. The shortest solidification time showed the magnesite mould with sodium silicate binder, the longest one was observed in case of mould from Dunite bonded with sodium silicate too (CO2 – process).

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Table 3. Table of measured values

Mould

Solidus temperature

[°C]

Solidification time

τ1 [s]

Temperatures achieved in the moment of

solidification ending Thermocouple No Temperature

[°C]

1 -Magnesite

1479.2

127

1 – 40 mm from the face 2 – 30.5 3 – 21.5 4 – 13.5 5 ti mould/metal interface

99.5

99.5 99.0 492

1178.6

2 – Dunite

1473.1

368

1 2 3 4

5 ti

98.5 97.5 231

552.5 1401

3 – Olivine

1473.5

340

1 2 3 4

5 ti

98.5 99.5 184.5 626.5 1388.6

4 – Chromite

1474.1

215

1 2 3 4

5 ti

87.5 98.5 100

329.5 1329.9

5 –

Chromite Olivine

1470.5

294

1 2 3 4

5 ti

57.5 92.5 208

509.5 1408.2

6 – Chromite

1475.3

173

1 2 3 4

5 ti

22 30

81.5 274 1308

7 – Kerphalit

1473.9

345

1 2 3 4

5 ti

92.5 106

302.5 676.5 1370

8 – Chromite

1476.6

198

1 2 3 4

5 ti

27.5 64 102

364.5 1282

3.2. Determination of the coefficient of heat accumulation of the mould according to different authors

Table 4. bf results according to G. HALBART (equation 7)

Coefficient of heat accumulation Mould composition bf [W.s0.5.m-2.K-1] [%]

1 – Magnesite 3510 100 6 – Chromite 2765 78.8 4 – Chromite 2391 68.1 8 – Chromite 2319 66.0 5 – Chromite Olivine 1930 55.0 3 – Olivine 1821 51.9 2 – Dunite 1734 49.4 7 – Kerphalit 1665 47.4

Table 5. bf results according to A. I. VEJNIK (equation 8)

Coefficient of heat accumulation Mould composition bf [W.s0.5.m-2.K-1] [%]

1 – Magnesite 2339 100 6 – Chromite 2044 87.4 8 – Chromite 1903 81.4 4 – Chromite 1834 78.4 5 – Chromite Olivine 1587 67.8 7 – Kerphalit 1453 62.1 3 – Olivine 1453 62.0 2 – Dunite 1397 59.7

Table 6. bf results according to G. A. ANISOVICH (equation 11)

Coefficient of heat accumulation Mould composition bf [W.s0.5.m-2.K-1] [%]

1 – Magnesite 2212 100 6 – Chromite 1738 78.5 8 – Chromite 1725 78.0 4 – Chromite 1472 66.6 7 – Kerphalit 1273 57.5 5 – Chromite Olivine 1222 55.3 3 – Olivine 1152 52.1 2 – Dunite 1093 49.2

3.3. Discussion of results The highest chilling effect had a mould from magnesite mixture bonded with sodium silicate (CO2 – process). Besides high bulk density of the mould the chilling effect is further on enforced by high water concentration (green mould). Chromite moulds (compositions of moulds No 4, 6 and 8) hold the second place in sequence and they represent 66 – 78 % (according to G. Halbart), 78 – 87 % (according to A. I. Vejnik) and 66 – 78 % (according to G. A. Anisovich) of the chilling effect of the magnesite mould (100 %). Generally absolutely the highest values for bf are achieved by calculation according to the relation by G. Halbart, the lowest ones according to G. A. Anisovich.

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Due to high chromite price there is an endeavour for partial substituting it with cheaper base sands (Olivine, Kerphalit). With 60 % part of Olivine in chromite (mixture No 5) a mould with 55 – 68 % chilling effect of the magnesite mould is obtained, i.e. such one between the „pure“ chromite and olivine moulds. Mutual combinations of non-quartz base sands in mixtures must result from their chemical nature (in relation to the binder) and a kind of cast alloy. The chromite mould with Geopol geopolymer binder has higher chilling effect by 26 – 28 % in comparison with synthetic aluminosilicate base sand Kerphalit. The lowest bf values showed the mould with Dunite (Magnolite), 50 – 60 % of chilling effect of magnesite only. Results of bf enable to determine also mean solidification coefficient (k) according to the Chvorinov’s relation for the plate casting as follows:

1

1Rk

τ= (12)

Results (tab. 7.) fully follow the sequence of values of chilling effect of moulds (bf) determined according to equations by all three used authors. Table 7. Mean solidification coefficient

Mould composition

−

⋅ 2

1

smk

1 – Magnesite 0.00168 6 – Chromite 0.00144 8 – Chromite 0.00135 4 – Chromite 0.00129 5 – Chromite Olivine 0.00110 3 – Olivine 0.00103 7 – Kerphalit 0.00102 8 – Dunite 0.00099

4. Conclusion Chvorinov with his rule (law) has become a term acknowledged all over the world. With use of knowledge of solidification time of simple casting forms (a plate) with the aid of determining the bf it is possible to test the chilling effect of moulds with use of different types of non-quartz base sands (moulding mixtures). The bf is not a constant but a coefficient characterizing the thermal and physical properties of the mould under particular conditions of casting solidification (metal temperature, casting form, used moulding mixtures). Knowledge of bf of different moulds with non-quartz base sands gives to technologists in particular the extensive possibilities of influencing the solidification time and quality of produced castings.

Acknowledgements The work has been worked out with a financial support from The Grant Agency of the Czech Republic in the framework of the project „Graphite nucleation and possibilities of control its morphology in ferrous alloys” registration No 106/08/0789

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[10] G.A.Anisovich, N.P.Žmakin, Ochlaždenije otlivki v kombinovanoj forme.Mašinostrojenije, Moskva,1969, 134.

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