ARCHIVES
of
FOUNDRY ENGINEERING
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
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).
ARCHIVES of FOUNDRY ENGINEERING Volume 10, Issue 4/2010, 77-82
77
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:
R =
V
(2)
S
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
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:
τ1 =
 V1 
 
 S1 
2
[
(
2
π ρ1 L + c1 t 1 − t s
2
4 ts ρ2 c2 λ2
)]2
(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:
bf =
ρ2 c2 λ2
(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
(3)
1
The authors of the paper prefere for bf the term “heat accumulation”
because it better represents heat absorption in sand moulds.
78
ARCHIVES of FOUNDRY ENGINEERING Volume 10, Issue 4/2010, 77-82
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
Composition of mixtures
Moisture of
mark
[weight parts]
mixtures
[%]
No 1
100 Magnesite
6.4 sodium
3.2
silicate
0.2 water
No 2
100 Dunite
8.0 sodium
3.9
silicate
0.2 water
No 3
100 Olivine
5.1 sodium
2.2
silicate
0.2 water
No 4
100 Chromite
1.8 Resol resin
Not
(Ecolotec 600)
determined
No 5
40 Chromite
4.0 sodium
2.1
60 Olivine
silicate
No 6
100 Chromite
1.2 furan resin
Not
0.65 catalyst
determined
No 7
100 Kerphalit
2 Geopol 515
Not
0.28 hardener
determined
SA71
No 8
100 Chromite
1.2 Geopol 515
Not
0.17 hardener
determined
SA71
Table 2.
Bulk density of cores
Mixture mark
No 1
No 2
No 3
No 4
No 5
No 6
No 7 **
No 8 *
[b]
Bulk density of cores
[kg/m3]
2694
1779
2011
3067
2555
2911
1777
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)
h
s
=
400
[a]
> 10 where the h is plate height and the s its relative
38
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
ARCHIVES of FOUNDRY ENGINEERING Volume 10, Issue 4/2010, 77-82
79
the (ti) comes closer to the (ts) with lowering solidification
temperature of measured metal (alloy):
bf
=
R1
[
(
π . ρ1 L + c1 t1 − t s
R1
ρ1
L
=
=
=
c1
=
t1
τ1
=
=
(7)
τ1
2 ti
where:
)]
relative casting thickness (modulus)
bulk density of metal
latent crystallization temperature
(L = 272.1 . 103 [J . kg-1])
specific heat of metal
(c1 = 753.1 [J . kg-1 . K-1])
metal casting temperature
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:
bf =
R1
2
where:

 ρ t

π
τ1
c1 ln


tL 
t1
2
+
 ρ1 [L


(
+ c1 t L − t s
ts
)]


(8)
τ1 = τp + τt
ρt = mass density of solidifying metal (within t1 – tL)
tL = liquidus temperature of metal
bf
1
*
, to the S* + S1
n2 + 1
quadrangle surface in which the parabola is situated (fig. 2.):
*
*
S + S1
*
S1 =
n2 + 1
R 1 ρ1 L
=
ti
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 S1 surface under the parabola.
Surface under the curve equals to
Fig. 2. Scheme of the temperature field parabola
2n 2
n2 + 1
. τ1
(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).
(9)
*
If the S* and S1 surfaces are determined from the experiment
then it is possible to determine the parabola degree as follows:
n2 =
80
S
*
*
S1
(10)
ARCHIVES of FOUNDRY ENGINEERING Volume 10, Issue 4/2010, 77-82
Table 3.
Table of measured values
Mould
Solidus
temperature
Solidification
time
[°C]
τ1 [s]
1 -Magnesite
1479.2
127
2 – Dunite
1473.1
368
3 – Olivine
1473.5
340
4–
Chromite
1474.1
215
1470.5
294
5–
Chromite
Olivine
6–
Chromite
1475.3
173
7–
Kerphalit
1473.9
345
8–
Chromite
1476.6
198
Temperatures achieved
in the moment of
solidification ending
Thermocouple No
Temperature
[°C]
99.5
1 – 40 mm
from the
face
99.5
2 – 30.5
99.0
3 – 21.5
492
4 – 13.5
5 ti
1178.6
mould/metal
interface
1
98.5
2
97.5
3
231
4
552.5
5 ti
1401
1
98.5
2
99.5
3
184.5
4
626.5
5 ti
1388.6
1
87.5
2
98.5
3
100
4
329.5
5 ti
1329.9
1
57.5
2
92.5
3
208
4
509.5
5 ti
1408.2
1
22
2
30
3
81.5
4
274
5 ti
1308
1
92.5
2
106
3
302.5
4
676.5
5 ti
1370
1
27.5
2
64
3
102
4
364.5
5 ti
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
6 – Chromite
2765
4 – Chromite
2391
8 – Chromite
2319
5 – Chromite Olivine
1930
3 – Olivine
1821
2 – Dunite
1734
7 – Kerphalit
1665
[%]
100
78.8
68.1
66.0
55.0
51.9
49.4
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
6 – Chromite
2044
8 – Chromite
1903
4 – Chromite
1834
5 – Chromite Olivine
1587
7 – Kerphalit
1453
3 – Olivine
1453
2 – Dunite
1397
[%]
100
87.4
81.4
78.4
67.8
62.1
62.0
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.
ARCHIVES of FOUNDRY ENGINEERING Volume 10, Issue 4/2010, 77-82
81
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:
k =
References
[1]
(12)
τ1
[3]
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
1 – Magnesite
6 – Chromite
8 – Chromite
4 – Chromite
5 – Chromite Olivine
3 – Olivine
7 – Kerphalit
8 – Dunite
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
[2]
R1
Mould composition
Acknowledgements
[4]
[5]
[6]
k
 m ⋅ s − 12 


0.00168
0.00144
0.00135
0.00129
0.00110
0.00103
0.00102
0.00099
4. Conclusion
[7]
[8]
[9]
[10]
[11]
[12]
[13]
W. Tilch, et al., Einfluss alternativer Formgrundstoffe auf
die Eigenschaften von Formstoff und Gussteil. Giesserei,
93, No 8 (2006) 12 – 24.
U. Recknagel, M. Dahlmann, Spezialsande Formgrundstoffe
für die moderne Kern – und Formherstellung. Giesserei
Praxis, Special, 11, 2008, 401 – 406.
N. Chvorinov, Theorie der Erstarrung von Gussstücken.
Giesserei, 27, 1940, Heft 10, pp. 177 – 188, Heft 11, pp.
201 – 208, Heft 12, pp. 222 – 225.
J. Hučka, Hutnické listy, vol. 2009.
J. CAMPBELL, Castings. Elsevier Butterwiörth
Heinemann, Oxford 2003.
N. Chvorinov, Studium tuhnutí oceli. Teoretická řešení.
Hutnické listy, vol. VI, No 11 (1951) 594 – 598.
R. Wlodawer, Die gelengte Erstarrung von Stahlguss.
Giesserei-Verlag G.mb.H. Düsseldorf, 1067.
G. Halbart, Éléments dune théorie mathématique de la
Fonderie. Liége, 1945.
G. A. Anisovich, R. N. Grinkevich, Metod opredelenija
termofyzičeskich svojstv formovočnych materialov. In
Problemy teploobmena pri litje. Minsk, 1960.
G.A.Anisovich, N.P.Žmakin, Ochlaždenije otlivki v
kombinovanoj forme.Mašinostrojenije, Moskva,1969, 134.
A.
I.
Vejnik,
Termodinamika
litejnoj
formy.
Mašinostrojenije, Moskva, 1968, 333.
X. Virolle, R. Chevriot, M. Jeancolas, Étude expérimentale
de la diffusivité thermique des matériaux de moulage.
Fonderie, 241, mars 1966.
T. Elbel, Výpočet intervalu teplot tuhnutí, Slévárenství,
No 8 (1980) 318
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.
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