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Journal of Mechanical Science and Technology 26 (11) (2012) 3625~3629
www.springerlink.com/content/1738-494x
DOI 10.1007/s12206-012-0854-0
Mathematical modeling for surface hardness in investment casting applications†
Rupinder Singh*
Department of Production Engineering, Guru Nanak Dev Engineering College, Ludhiana, 141006, India
(Manuscript Received February 11, 2012; Revised May 13, 2012; Accepted June 10, 2012)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract
Investment casting (IC) has many potential engineering applications. Not much work hitherto has been reported for modeling the sur-
face hardness (SH) in IC of industrial components. In the present study, outcome of Taguchi based macro-model has been used for de-
veloping a mathematical model for SH; using Buckingham’s π-theorem. Three input parameters namely volume/surface-area (V/A) ratio
of cast components, slurry layer’s combination (LC) and molten metal pouring temperature were selected to give output in form of SH.
This study will provide main effects of these variables on SH and will shed light on the SH mechanism in IC. The comparison with ex-
perimental results will also serve as further validation of model.
Keywords: Investment casting; Surface hardness; Buckingham’s π-theorem; Volume/surface area; Pouring temperature
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction
IC is widely used technique for modern metal castings
which provides an economical means of mass producing
shaped metal parts containing complex features [1]. The
mould is made by surrounding a wax or plastic replica of the
part with ceramic material [2]. After the ceramic material so-
lidifies, the wax replica is melted out, and metal is poured into
the resulting cavity [3]. IC is used for the production of nu-
merous equipments like: dental tools, electrical/electronic
equipments, radar, guns, hand-tools, jewellery, M/C tools,
material handling equipments, metal working equipments,
agricultural equipments, cameras, pneumatics/hydraulics
components etc.
The literature review reveals that lot of work has been re-
ported on optimization of IC process [4-7]. Various process
parameters (like wax properties, number of slurry layers, size
of component and mould thermal conductivity etc.) for the
sound casting produced by IC process have been reported. But
hitherto very less has been reported for modeling the SH in IC
of industrial components. So, the present investigation has
been focused to develop mathematical model for SH in IC.
For IC of commercially used metals and alloys (like: Al,
M.S and S.S) an approach to model SH was proposed and
applied [8]. This model was an attempt for predicting the SH
as macro model in IC and is based upon robust design concept
of Taguchi technique. The model was mechanistic in sense
that parameters can be observed experimentally from a few
experiments for a particular material and then used in predic-
tion of SH over a wide range of process parameters. This was
demonstrated for IC of commercially used metals and alloys,
where very good predictions were obtained using an estimate
of multi parameters at a time. In that study, effects of three
process parameters (namely V/A ratio of cast components, LC
and molten metal pouring temperature) were revealed. Table 1
shows various input and output parameters used in experimen-
tal study.
2. Methodology
The levels of molten metal pouring temperature (as: 600°C,
1550°C, 1600°C), component V/A ratio (as 2.74, 3.78, 4.09
mm) were judicially selected (while pilot experimentation) for
IC of spherical discs of (Al, M.S and S.S) of three different
commercially used sizes (corresponding to diameter: 2”, 3”
and 4”) based upon field application as a case study of ball
valve.
LC of 1 + 1 + 2 + 4 represents one layer of zircon paint
*Corresponding author. Tel.: +91 9872257575, Fax.: +91 1612490339
E-mail address: [email protected] † Recommended by Associate Editor Dae-Cheol Ko
© KSME & Springer 2012
Table 1. Various input and output parameters.
Input parameters Output parameter
Three levels of component V/A ratio
(2.74, 3.78, 4.09 mm)
Three levels of LC
(1 + 1 + 2 + 4, 1 + 1 + 3 + 3 and 1 + 1 + 4 + 2)
Three levels of molten metal pouring temperature
(600°C, 1550°C, 1600°C)
SH
3626 R. Singh / Journal of Mechanical Science and Technology 26 (11) (2012) 3625~3629
(one, 1° layer), one layer of silica slurry of 80-100 mesh (one,
2° layer), two layers of silica slurry of 50-80 mesh (two, 3°
layers), and four layers of silica slurry of 30-50 mesh (four, 4°
layers). Similarly 1 + 1 + 3 + 3 represents one layer of zircon
paint (one, 1° layer), one layer of silica slurry of 80-100 mesh
(one, 2° layer), three layers of silica slurry of 50-80 mesh
(three, 3° layers), and three layers of silica slurry of 30-50
mesh (three, 4° layers) and (1 + 1 + 4 + 2) represents one layer
of zircon paint (one, 1° layer), one layer of silica slurry of 80-
100 mesh (one, 2° layer), four layers of silica slurry of 50-80
mesh (four, 3° layers), and two layers of silica slurry of 30-50
mesh (two, 4° layers). The total number of 1° + 2° + 3° + 4°
layers has been kept fixed equal to 8 based upon pilot experi-
mentation, as because during the process of shell formation, it
was observed from pilot experimentation that the shell with
less than 8 layers cracks while de-waxing. The drying condi-
tions were 27°C temperature and humidity 60%. The relation-
ships were studied by considering interaction between these
variables. Table 2 shows control log of experimentation (based
upon Taguchi L9 O.A) and experimental observations for SH.
The SH has been measured on the periphery of spherical disc
of the ball valve. Three measurements on different locations on
the periphery of spherical disc have been made and average
values are shown in Table 2. Further the experiment has been
repeated three times to reduce the experimental error.
On the basis of this model, Singh (2012) studied the rela-
tionships between SH and controllable process parameters [8].
These relationships agree well with the trends observed by
experimental observations made otherwise [3-7].
2.1 Description of the IC process
The IC process is shown in Fig. 1. It is a 12 step process,
which are: injecting wax into dies, ejection of patterns, pattern
assembly or tree making, slurry coating, stucco coating, mould
completion, pattern melt-out or de-waxing, mould baking,
pouring, shakeout, cutting of rise and at last the final product
produced [7-9]. The major IC process variables affecting SH
are shown as cause and effect diagram (Fig. 2).
The study presented in this paper is based on a previously
published macro model based on Taguchi robust design [8].
Now based upon geometric model, Buckingham’s π-
theorem has been used to study the relationships between SH
and controllable process parameters.
3. Mathematical modelling of SH
As per Taguchi design SH in IC was significantly depend-
ent on molten metal pouring temperature. Table 3 and 4 re-
spectively shows percentage contribution of input parameters
and geometric model for SH [8]. The case study under consid-
eration deals primarily with obtaining optimum system con-
figuration in terms of response parameters with minimum
expenditure of experimental resources. The best settings of
control factors have been determined through experiments.
The Buckingham’s π-theorem proves that, in a physical
problem including “n” quantities in which there are “m” di-
Fig. 1. IC process.
Fig. 2. Cause and effect diagram of SH.
Table 2. Control log of experimentation.
S.
No.
Ratio
(V/A)
LC ( Total
no. of layers
fixed to 8)
Type of
metal/pouring
temp. °C
SH
(HV)
1 2.74 1+1+3+3 Al (600°C) 4
2
4
3
4
1
2 2.74 1+1+2+4 S.S (1550°C)
2
3
9
2
3
7
2
3
7
3 2.74 1+1+4+2 M.S (1600°C)
1
8
5
1
8
4
1
8
2
4 3.78 1+1+3+3 S.S (1550°C)
2
4
6
2
4
8
2
4
7
5 3.78 1+1+2+4 M.S (1600°C)
1
6
3
1
6
7
1
6
0
6 3.78 1+1+4+2 Al (600°C) 5
0
5
4
5
1
7 4.09 1+1+3+3 M.S (1600°C)
1
8
0
1
8
3
1
7
8
8 4.09 1+1+2+4 Al (600°C) 4
2
4
3
4
5
9 4.09 1+1+4+2 S.S (1550°C)
2
4
8
2
5
0
2
5
1
R. Singh / Journal of Mechanical Science and Technology 26 (11) (2012) 3625~3629 3627
mensions, the quantities can be arranged in to “n-m” inde-
pendent dimensionless parameters. In this approach dimen-
sional analysis is used for developing the relations [10, 11].
Since SH, ‘H’ depends upon input parameters (namely: V/A
ratio, LC, molten metal pouring temperature, type of metal
(W/P hardness factor), mold thermal conductivity and solidifi-
cation time), therefore by selecting basic dimensions:
• M (mass);
• L (length);
• T (time); and
• θ (temperature).
The dimensions of foregoing quantities would then be:-
1. The SH “H” (kgf/mm2) Vickers hardness: M L
-1 T
-2
2. LC “N1” (mm): L
3. Component’s V/A ratio “R” (mm): L
4. Type of metal/ W/P hardness factor “F” (kgf/mm2) Vick-
ers hardness: M L-1 T
-2
5. Molten metal pouring temperature “θ” (°C): θ
6. Mold thermal conductivity “K” (Wm-1K
-1): M L T
-3 θ
-1
7. Solidification time “t” (min): T
Now, H = f (N1, R, F, θ, K, t). (1)
In this case n is 7 and m is 4. So, we can have (n-m = 3) π1,
π2 and π3 three dimensionless groups.
Taking H, V and P as the quantities which directly go in π1,
π2 and π3 respectively, it can be written as:
π1= H. (K)α1. (F)
β1. (N1)
γ1. (t)
δ1 (2)
π2= R (K)α2. (F)
β2. (N1)
γ2. (t)
δ2 (3)
π3= θ. (K)α3. (F)
β3. (N1)
γ3. (t)
δ3. (4)
Substituting the dimensions of each quantity and equating
to zero, the ultimate exponent of each basic dimension has
been achieved, since the “πis” are dimensionless groups. Thus
αi, βi, γi, δi, (where i = 1, 2, 3...) can be solved.
Solving for π1, we get
π1 = (M L-1T
-2). (M L T
-3 θ
-1) α1. (M L
-1T
-2) β1. (L)
γ1. (T)
δ1. (5)
Here,
M: 1 + α1 + β1= 0
L: -1 + α1 - β1 + γ1= 0
T: -2-3α 1- 2β1 + δ1= 0
θ: α 1 = 0.
Solving, we get:
α1 = 0, β1 = -1, γ1 = 0, δ1 = 0.
Thus
π1= H/F. (6)
Similarly we get:
π2 = ( L). (M L T-3 θ
-1) α2. (M L
-1T
-2) β2. (L)
γ2. (T)
δ2 (7)
M: α2 + β2 = 0
L: 1 + α2 – β2 + γ2 = 0
T: -3α2-2β2 + δ2 = 0
θ: α2 = 0.
Solving, we get:
α2= 0, β2= 0, γ2= -1, δ2= 0.
π2= R/N1. (8)
Similarly:
π3 = (θ). (M L T-3 θ
-1) α3. (M L
-1 T
-2) β3. (L)
γ3. (T)
δ3. (9)
Here,
M: α3+β3= 0
L: α3 - β3 + γ3= 0
T: -3α3-2β3 + δ3= 0
θ: 1-α3 = 0.
Solving, we get:
α3 = 1, β3 = -1, γ3 = -2, δ3 = 1.
Thus
π3 = θ. (K). (F)-1. (N1)
-2. (t). (10)
The ultimate relationship can be assumed to be of the form
πi = f (πj , πk). (11)
Let’s assume i = 1, j = 2, k = 3 Then functional relationship
is of the form
π1 = f (π2, π3)
or
H/F = f (R/N1, θ. (K). (F)-1. (N1)
-2. (t)).
It has been experimentally found that H directly goes with θ
[8]. This means metal pouring temperature significantly af-
fects the SH. Therefore metal pouring temperature has been
Table 3. Percentage contribution for SH.
Parameters Sum of square Percentage contribution
V/A 0.203193 0.0559208
LC 1.5739795 0.4331754
Pouring temp. 360.75569 99.28369
Error 0.203193 0.2272136
Table 4. Geometric model for SH [8].
Optimized SH
conditions Al S.S M.S
V/A 3.78 mm 4.09 mm 2.74 mm
LC 1+1+4+2 1+1+4+2 1+1+4+2
Pouring temp. 600°C 1550°C 1600°C
3628 R. Singh / Journal of Mechanical Science and Technology 26 (11) (2012) 3625~3629
taken as representative for development of mathematical
equation.
Thus the equation becomes
H = f {θ. K. t. R. 1/(N1)3} (12)
H = C. {θ. K. t. R/ (N1)3}. (13)
Here ‘C’ represents constant of proportionality.
Now by keeping {K. t. R/ (N1)3} fixed, experiments were
performed for different values of θ, to find out ‘H’ and ‘C’ in
Eq. (13).
The actual experimental data for metal pouring temperature
have been collected and plotted in Fig. 3.
The data collected has been further used for finding best fit-
ting curve. The second degree polynomial equation comes out
to be best fitted curve with coefficient of co-relation equals to
“1”. Thus Eq. (13) of SH for this case becomes:
(For LC = 1 + 1 + 2 + 4)
SH = [(-0.0022)θ2 + (5.4993)θ - 2931.3] [K. t. R/ (N1)
3]
(14)
(For LC = 1 + 1 + 3 + 3 )
SH = [(-0.002)θ2 + (5.0713)θ - 2707.3] [K. t. R/ (N1)
3] (15)
(For LC = 1 + 1 + 4 + 2)
SH = [(-0.002)θ2 + (4.9915)θ - 2656.8] [K. t. R/ (N1)
3]. (16)
4. Results and discussion
As per predicted values the SH values are maximum around
1100~1300°C The variation in SH may be explained on the
basis of different cooling rates obtained with different number
of layer combinations and pouring temperature. This may be
critical range for re-crystallization temperature. Since this
model is based upon Taguchi based model of SH, in which
component’s V/A ratio and LC are already optimized [8].
Therefore these parameters have not been varied while devel-
oping mathematical model. The second degree polynomial
equation has been used only to find best fir curve with coeffi-
cient of co-relation close to 1. Here the mechanism of SH in
IC involves primarily the effect of pouring temperature (which
depends upon the material to be casted).This model is useful
in deciding the range of process parameters in field applica-
tion of IC for getting desired SH without pilot experimentation.
In casting while solidification process is going on, there are
possibilities of gas holes and shrinkage cavities (having some
definite dimensions). Also the numbers of such type of cavities
can be counted. Further to check the internal defects of the
castings obtained the radiography analysis was done as per
ASTM E 155 standard for gas holes and shrinkages at opti-
mized conditions suggested by Taguchi design (Table 5). The
results obtained shows that the components prepared are ac-
ceptable in accordance with ASTM E155 standard (Fig. 4).
The present results are valid for 90-95% confidence interval.
For validation of this model, final observations were made
under both experimental conditions (based upon Taguchi de-
sign) and theoretically developed mathematical equations.
Corollary:
The data for experiment no. 8 (Table 2) has been used for
verification of mathematical equation. The experimental value
for SH is 43HV. Now by considering Eq. (14), for LC =
1 + 1 + 2 + 4 the input data is as under:
SH = | [(-0.0022) θ2 + (5.4993) θ - 2931.3] [K. t. R/ (N1)
3]|
θ = 600°C, K = 1.1 W/mK (or 0.01 W/cm°C), t = 4800 sec,
R = 4.09 mm, N1 = 27 mm
Fig. 3. SH Vs metal pouring temperature.
Table 5. Radiography analysis of castings.
S.
No.
Ratio
(V/A)
LC (Total
no. of
layers fixed
to 8)
Type of
metal/pouring
temp. °C
Gas
hole
level
Shrink
age
level
1 2.74 1+1+3+3 Al (600°C) 3 --
2 2.74 1+1+2+4 S.S (1550°C) 4 3
3 2.74 1+1+4+2 M.S(1600°C) - -
4 3.78 1+1+3+3 S.S (1550°C) 2 2
5 3.78 1+1+2+4 M.S(1600°C) -- 3
6 3.78 1+1+4+2 Al (600°C) - -
7 4.09 1+1+3+3 M.S(1600°C) 5 4
8 4.09 1+1+2+4 Al (600°C) -- 4
9 4.09 1+1+4+2 S.S (1550°C) -- -
Fig. 4. Radiographic analysis of casting prepared by IC at LC (1+1+4
+2).
R. Singh / Journal of Mechanical Science and Technology 26 (11) (2012) 3625~3629 3629
SH =| [(-0.0022) (600)2 + (5.4993)600 - 2931.3] [0.01.
(80*60). (0.409)/ (2.7)3]/ 9.8*|
= 43.081
Note: 9.8* unit conversion of kgf into N.
Comparison of SH result obtained experimentally agrees
very well with predictions through mathematical equations.
The verification experiment revealed that on an average there
is 7% improvement in SH (Table 6).
Therefore the equations developed for SH represent the mi-
cro-modeling which is based on an in-depth understanding of
the system. It begins by developing a mathematical model of
the system, which, in this case, is SH of IC. When systems are
complex, as in this case study, one must make assumption that
simplify the operation, as well as put forth considerable effort
to develop the model. Furthermore, the more simplifying we
do, the less realistic the model will be, and, hence, the less
adequate it will be for precise optimization. But once an ade-
quate model is constructed, a number of well-known optimi-
zation methods, can be used to find the best system configura-
tion. For developing a micro-model in this case under study,
initially a macro-model based upon concept of robust design
has been made and output of this macro-model has been used
for developing a micro-model.
5. Conclusions
The Buckingham’s π-theorem has been used for mathemati-
cal modeling of SH in IC process. The interactions among input
parameters have been considered for developing the model. The
following conclusions can be drawn from this study:
The molten metal pouring temperature contributes sig-
nificantly for SH of IC (99%). The mathematical equation
developed here sufficiently express all input parameters (Eq.
(14)-(16)) that contributes for SH of IC. As regard to
mathematical model second degree polynomial equation for
SH is giving best fitting curve with coefficient of co-
relation ≈ 1.
The verification experiment revealed that on an average
there is 7% improvement in surface hardness, for selected
workpiece. Also as regards to surface integrity of castings is
concerned no surface defects have been observed in radio-
graphic image at optimized conditions.
Acknowledgment
The authors would like to thank DST (Government of In-
dia) for financial support.
References
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Rupinder Singh is Associate Professor
(P.E) in Guru Nanak Dev Engg. College,
Ludhiana (India). He has published
around 200 research papers at Na-
tional/International level and supervised
55 M.Tech theses. His area of interest is
rapid casting, non-traditional machining,
and manufacturing processes.
Table 6. Percentage improvement in SH.
SH of castings
prepared in HV Type of
cast
metal At initial
settings
At final
settings
Percentage
improvement
in SH
Average
percentage
improvement
in SH
Al 46 50 8.7%
S.S 170 182 7.05%
M.S. 237 251 5.9%
7.2%