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)

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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

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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: rupindersingh78@yahoo.com † 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.

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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%