Linear and non-linear statistical modelling of green

Linear and non-linear statistical modelling of green sand mould system

Published by Maney Publishing (c) W. S. Maney & Son Limited

M. B. Parappagoudar, D. K. Pratihar* and G. L. Datta In the present work, design of experiments (DOE) technique with response surface methodology was used to develop both linear and non-linear models, to establish the input–output relationships in green sand mould system. Grain fineness number (GFN), amount of clay, amount of water and number of strokes (degree of ramming) were considered as the input variables (parameters), which control the outputs (i.e. mould properties). Full factorial DOE was utilised for the linear model, whereas central composite design and Box-Behnken designs were used to develop the non-linear models. Experiments were conducted to measure the green sand mould properties, such as permeability, green compression strength, mould hardness and bulk density. The adequacy of all the developed models was checked through statistical analysis. Twenty random test cases were considered, to validate the models and compare their performances. A model that is statistically adequate and gives minimum percentage of deviation in prediction was adjudged as the best model for a particular response. Keywords: Green sand mould, Statistical models, DOE, CCD, Box-Behnken, ANOVA

Introduction Green sand mould casting is one of the most popular casting methods. In order to maintain consistent quality of the castings, it is necessary to control the process efficiently.1 The quality of a green sand mould casting is largely influenced by the properties of mould, such as green compression strength, permeability, mould hardness and others, which depend on many input parameters. The relationship of these properties with input parameters, such as sand grain size and shape, amount of binder and water, etc., is complex in nature. The major source of defects in sand casting is the moulding sand mixture itself. The defects can be minimised by a proper control of the moulding sand properties, such as green compression strength, permeability, hardness and bulk density. Hence, it is important to establish their relationships with the input variables (moulding sand ingredients), namely sand grain fineness number (GFN), amount of binder and moisture for a given ramming condition. Statistical design of experiments (DOE) refers to the process of planning the experiment, so that an appropriate set of data can be collected, which will be analysed by using the statistical methods. Statistical analysis of the experimental data is necessary, to draw meaningful conclusions from the data.2 Design of experiments combined with response surface methodology is a powerful statistical tool to develop the input–output relationships. Full factorial DOE with the parameters set at their respective two levels, can be used to develop linear Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, Kharagpur 721302, India *Corresponding author, email [email protected]

ß 2007 W. S. Maney & Son Ltd. Received 15 December 2005; accepted 19 February 2007 DOI 10.1179/136404607X184952

relationships between the input–output parameters. The advantage of full factorial DOE is that the number of experiments required is less though the analysis provides complete information on the main and interaction effects of input parameters on the response. However, the main problem with this linear model lies in the fact that the curvature effect (if any) in the response function will not be recognised. Out of the different classes of response surface methodology (RSM) designs, the central composite design (CCD) and Box-Behnken design, which require the input parameters to be set at three levels were used to develop nonlinear models. The availability of several varieties of CCD enables their use under different experimental regions of interest. In the present work, non-rotatable CCD was used. The Box-Behnken designs are formed by combining 2k factorials with incomplete block designs, where k is the number of input variables. The resulting designs are usually very efficient in terms of the number of required runs, and they are either rotatable or nearly rotatable.3 Some responses may have linear relationships and others may have non-linear relationships with the input parameters. As the interdependency of the output responses cannot be considered in the statistical models, a separate model was developed for each response. In the present work, the above discussed three techniques were used to develop the models. The performances of the models were tested on the test cases and the best model was chosen for a particular response. The statistical models developed through DOE and response surface methodology are found to be more accurate and can be used to predict the responses. A lot of research was carried out on moulding sand and its mechanics during 1960s and 1970s around the

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Their model showed an average of 8% deviation in prediction for the test cases. In the present work, the following things have been attempted: (i) experiments were conducted as per the requirements of full factorial design. Linear models were developed for the responses such as permeability, green compression strength, mould hardness and bulk density. Adequacies of the developed models were checked by statistical analysis and their performances were tested on 20 random test cases (ii) non-rotatable CCD was used to develop nonlinear models. Statistical analysis was performed and adequacy of the models was checked. Twenty test cases were used to test the practical suitability of the developed models (iii) another set of non-linear relations were developed by using the Box-Behnken DOE and the performances of the models were tested statistically on the same 20 randomly generated cases (iv) the performances of all the three models were compared with the help of some randomly developed test cases and the best model was selected based on the minimum percentage deviation in prediction, for each response. The rest of the text is organised as follows: the section ‘Methodology’ explains the methodology. The methods of data collection are described in the section on ‘Experimental data collection’. Results are discussed in the section on ‘Results and discussion’. Some concluding remarks are made in the section on ‘Concluding remarks’.

world. Most of the research work performed during that period was based on experimental and theoretical approaches. Chang et al.4 developed a model to evaluate the performance of green sand mould. They tried to establish relative difference in density by varying percentage of water as the index of flowability. However, their analysis was based on one-dimensional (1D) model and developed two empirical relations for relative compression strength and density. Moreover, the percentage of water based on relative density may not be the only criterion to decide optimum level of water. Dietert5 established straight line relationships for the water requirement of additives through experiments. The work also describes the amount of water requirement for different additives. The experiments were conducted by keeping mouldability index constant. While developing the equations, the amount of free water required for plasticising the mixture and water evaporations were however not considered. Thus, the equations may not hold good with the change in mouldability index. Terashima et al.6 developed a dynamic control in real time to keep moldability value constant under certain quantity of sodium based bentonite, where water quantity was appropriately adjusted by using an adaptive– predictive control method. However, this model was 1D. Terashima et al.7 tried to control moldability value by applying fuzzy reasoning for variation of bentonite ratio, sand temperature, humidity of atmosphere, etc. However, permeability and bulk density were not considered in the analysis. Philips1 established the effect of chemicals and mulling on green compression strength and baked shear strength. From his work, it was showed that sand grain size distributions would be an important criterion, to determine clay and water requirements of the moulding sand system. However, this part of his work was not supported by experiments. Design of experiments had been used by various investigators in the past, to study the effects of different variables on the green sand mould properties. Rose et al.8 used the DOE technique to study the effects of amount of binder, geological origin of the base sand, degree of ramming and amount of water on green compression strength, rammed density, dry compression strength and dry shear strength. However, they did not develop any model to predict the responses. Parappagoudar et al.9 developed linear models for green compression strength and permeability. The model was not suitable for accurate prediction of permeability. Moreover, mould hardness and bulk densities were not considered in their experiments. An attempt was also made by Parappagoudar et al.10 following the same direction, but they did not consider the effect of degree of ramming (number of strokes) on mould properties. In another work carried out by Dhindaw et al.,11 the statistical DOE technique was applied to study and control the properties of different sand systems. They extended the DOE application to resin based sand systems, in addition to green sand. In the green sand system they considered bulk density of green sand along with the green compression strength. However, the effect of sand grain size was not considered. Permeability, which is an important mould property, was also left out in their analysis. Casalino et al.12 used the Taguchi technique to establish third order model for compression strength and permeability in laser sintered sand moulds.


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Methodology In the present work, the following steps are adopted.

Step 1: selecting parameters and their levels The principal properties of green sand mould are governed by the type of sand used and the amount of clay and water present in it. It is to be noted that the properties also depend on mulling time and in the present study, it has been kept constant at 3?0 min. Degree of ramming has its effect on mould strength, permeability, hardness and bulk density. In this work, the following parameters are considered for experimentation: (i) AFS grain fineness number of silica sand (ii) percentage of (bentonite) clay content (iii) clay/water ratio (iv) number of strokes. With a high amount of clay and a low amount of water, the water present may not be sufficient to activate (plasticise) the whole amount of clay added to sand. Similarly, a high level of water with a low level of clay will result in excess of water. In both the cases, green compression strength will be very less. Since water is added to activate clay, instead of water, clay/water ratio was selected as one of the variables.10 The levels of the variables were set in consultation with the foundry industries and a detailed literature survey on the sand casting of aluminium alloys.13–17 The parameters and their levels are listed in Table 1.

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Step 2: conducting experiments

Gelling index test

The following tests/experiments were carried out one after another: (i) sieve analysis test to determine the grain fineness number and size distributions of the silica sands (ii) gelling index test to know the strength of clay (iii) conducting experiments with different combinations of parameters as per the full factorial design, CCD and Box-Behnken design planning (iv) conducting experiments with randomly generated 20 combinations of the process parameters for testing the performances of the developed models.

Gelling index test was conducted to know the suitability of the binder used in the experiments. Gelling index is expressed as the ratio of the volume of water to the minimum weight of clay used.18 Ten test tubes with 10 mL of water in each, were labelled from 1 to 10 and 0?1 gm of clay was placed in the first test tube. The weight of clay was incremented by 0?1 gm for the successive test tubes. The water–clay mixture was kept for 24 h and observed for the formation of gel. The gel was found to form in the sixth test tube. Thus, the gelling index was calculated as follows:

Step 3: developing linear and non-linear models In the present paper, full factorial design was used to develop linear models for the responses such as permeability, green compression strength, mould hardness and bulk density. Moreover, CCD and Box-Bhenken designs were utilised to develop the non-linear models.



Step 4: performing statistical analysis of data and comparison of models Analysis of variance (ANOVA) was performed for each of the responses separately, to test the adequacy of the models. The models were compared response-wise for 20 test cases and the best model was identified.

Experimental data collection High silica sand and sodium based bentonite clay were used in the experiments. The sand used was collected from the costal Karnataka, India. The following tests/ experiments were conducted as discussed below.

Sieve analysis test Sieve analysis test was carried out and a standard procedure was used to determine grain fineness number of the sand. The distribution of different grain sizes present in sand is a more significant index. For good compaction of sand, the amounts retained on three or four consecutive sieves should be in the range of 75– 80%. In addition, the sieve distribution (percentage of sand retained on various sieves) should not show a double peak in the relationship between sieve size and percentage of sand retained. The distribution should show a normal curve with a single peak.14,15 The sieve analysis test was performed for both fine (94 AFS) and medium (52 AFS) sands using a standard AFS sieve set.15 The sieve analysis test shows 80% of sand was retained on three consecutive sieves for medium sand and 90% of the sand was retained on three consecutive sieves for the fine sand. Both the sands had a single peak. The sieve analysis test indicates that both the sands are suitable for foundry applications.9

gelling index~volume of water=weight of clay ~10=0:6~16:67 Thus, in the present work, sodium based clay having the gelling index of 16?67 is used. This clay will provide good bonding between the sand particles.

Determination of permeability, green compression strength, mould hardness and bulk density Experiments were conducted to measure green compression strength, permeability number, mould hardness and bulk density of the green sand mould. The standard test specimens were prepared with different combinations of the input variables and their levels as per the DOE, which is explained below. The height of the test specimen was measured with the help of height gauge and was maintained between 5?0 and 5?1 cm. The time required for 2000 cm3 of air to pass through the specimen in permeability meter, was noted by using a stop watch. The air pressure (cm of water) was recorded from permeability meter. The weight of the specimen was determined by using a digital balance. The following relationship was used to determine the permeability number P~

V |H p|A|T

(1)

where H is the height of specimen in cm, V indicates the volume of air in cm3, p is the pressure of air in gm cm22, A represents cross-sectional area of the specimen in cm2 and T is the time in minutes. Green compression strength and mould hardness were measured by using a strength measuring unit and mould hardness tester respectively. The green compression strength, measured in psi was converted to kilo Pascal (SI unit). Bulk density expressed in gm cm23 was determined by utilising the following relationship bulk density~ weight of the specimen=volume of the specimen

Table 1 Input parameters and their levels Parameters

Levels

Sample no.

Description

Notation

High (z1)

Middle (0)

Low (21)

1 2 3 4

AFS grain fineness no. Percentage of clay Clay/water ratio No. of strokes

A B C D

94 12 3 5

73 10 2.25 4

52 8 1.5 3


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Table 2 Design matrix for full factorial design


Input parameters

4

Responses

Sample no.

A

B

C

D

Permeability

Green compression strength

Hardness

Bulk density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

21 z1 21 z1 21 z1 21 z1 21 z1 21 z1 21 z1 21 z1

21 21 z1 z1 21 21 z1 z1 21 21 z1 z1 21 21 z1 z1

21 21 21 21 z1 z1 z1 z1 21 21 21 21 z1 z1 z1 z1

21 21 21 21 21 21 21 21 z1 z1 z1 z1 z1 z1 z1 z1

– – – – – – – – – – – – – – – –

– – – – – – – – – – – – – – – –

– – – – – – – – – – – – – – – –

– – – – – – – – – – – – – – – –

The experiments were conducted as per the planning required for the full factorial design, central composite design and Box-Behnken design and the responses such as permeability, green compression strength, mould hardness and bulk density were determined for each set of the input parameters. The design matrices are shown in Tables 2–4 for the full factorial design, central composite design and BoxBehnken design respectively. It is to be noted that five replicates for each response have been considered for each set of input parameters, so that the adequacy of the models could be checked statistically. As there are four input parameters (refer to Table 1) and two levels are considered for each input parameter, there are 24516 combinations of input parameters (refer to Table 2) as per the full factorial DOE.

A design is called rotatable, if the accuracy in prediction of a response is the same on a sphere around the centre of the design. Rotatable designs require the input parameters to be set at their five levels, whereas non-rotatable designs require the parameters to be fixed at three levels. The choice of a rotatable or nonrotatable design depends on the geometric nature of the practical constraints on the design region. Unless some practical considerations dictate the choice of a rotatable design, it should not be strictly followed. In the present work, due to some practical constraints, non-rotatable design is considered. Table 3 shows the design matrix of a non-rotatable central composite design for four input parameters, each set at their three levels. The data were collected according to this table and used to develop non-linear models for the responses such as

Table 3 Design matrix for central composite design Input parameters

Responses

Sample no.

A

B

C

D

Permeability


Hardness

Bulk density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

21 z1 21 z1 21 z1 21 z1 21 z1 21 z1 21 z1 21 z1 0 21 z1 0 0 0 0 0 0 0 0

21 21 z1 z1 21 21 z1 z1 21 21 z1 z1 21 21 z1 z1 0 0 0 21 z1 0 0 0 0 0 0

21 21 21 21 z1 z1 z1 z1 21 21 21 21 z1 z1 z1 z1 0 0 0 0 0 0 21 z1 21 0 0

21 21 21 21 21 21 21 21 z1 z1 z1 z1 z1 z1 z1 z1 0 0 0 0 0 0 0 0 0 z1 0

– – – – – – – – – – – – – – – – – – – – – – – – – – –

– – – – – – – – – – – – – – – – – – – – – – – – – – –

– – – – – – – – – – – – – – – – – – – – – – – – – – –

– – – – – – – – – – – – – – – – – – – – – – – – – – –


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permeability, green compression strength, mould hardness and bulk density. Five replicates for each response were tried for each of the above 27 combinations of input parameters. In the Box-Behnken design involving four input parameters, the design can be arranged in three blocks. Each block consists of nine distinct points, including a centre point. Five replicates for each response are determined corresponding to a set of input parameters, as per the planning of Box-Behenken DOE. The nonlinear models were developed for all four responses mentioned in the table. Experiments were also conducted to generate 20 test cases at random, which are required to compare the performances of the regression analysis, carried out based on the above three designs of experiments (refer to Appendix). These experiments were conducted under the conditions similar to those of the experiments conducted for collecting the data required for model building.


Response – permeability

The following linear and non-linear models had been developed for the response – permeability: (i) full factorial design (linear model) Pfact ~2398:35{23:216A{185:165B{ 769:43C{451:795Dz1:79201ABz 8:44065ACz4:49427ADz75:2983BCz 38:83BDz175:023CD{0:799412ABC{ 0:38335ABD{1:90059ACD{17:362BCDz 0:183894ABCD (ii) central composite design (non-linear model) Pccd ~1192:51{15:98A{35:66Bz9:51C{ 105:66Dz0:07A2 z0:45B2 {4:13C 2 z4:22D2 z 0:11ABz0:2ACz0:52ADz1:19BCz

Results and discussion

1:99BD{3:1CD

Experimental data were collected as per full factorial design (2-level), CCD and Box-Behnken DOE. The collected data were analysed separately using the regression analysis/response surface methodology. Significance test, model adequacy checking and finally testing of the model on some random test cases were carried out for each model. Moreover, the performances of the models were compared among themselves.

(iii) Box-Behnken design (non-linear model) Pbox ~1149:93{19:1A{9:01B{1:36C{ 106:42Dz0:08A2 {0:29B2 z9:29C 2 z 5:62D2 z0:22AB{0:1ACz0:81AD{ 0:68BC{0:8BD{3:35CD

Model developments and statistical analysis

Significant tests were carried out for the different models separately and the significant terms were identified. Moreover, the coefficients of multiple correlations were determined to test the accuracy of the model. Table 5 shows the coefficient of multiple correlation values and

The experimental data were used to develop the linear and non-linear models for the different responses. Response-wise models and the results of their statistical analysis are given below. Table 4 Design matrix for Box-Behnken design Input parameters

Responses

Sample no.

A

B

C

D

Permeability


Hardness

Bulk density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

21 z1 21 z1 0 0 0 0 0 21 z1 21 z1 0 0 0 0 0 0 0 0 0 21 z1 21 z1 0

21 21 z1 z1 0 0 0 0 0 0 0 0 0 21 z1 21 z1 0 21 z1 21 z1 0 0 0 0 0

0 0 0 0 21 z1 21 z1 0 0 0 0 0 21 21 z1 z1 0 0 0 0 0 21 21 z1 z1 0

0 0 0 0 21 21 z1 z1 0 21 21 z1 z1 0 0 0 0 0 21 21 z1 z1 0 0 0 0 0

– – – – – – – – – – – – – – – – – – – – – – – – – – –

– – – – – – – – – – – – – – – – – – – – – – – – – – –

– – – – – – – – – – – – – – – – – – – – – – – – – – –

– – – – – – – – – – – – – – – – – – – – – – – – – – –


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Table 5 Coefficient of multiple correlations and insignificant terms of different models for the response – permeability Model

Coefficient of correlation with all terms R

Coefficient of correlation without insignificant terms

Insignificant terms

Linear Non-linear (CCD) Non-linear (Box-Behnken)

0.9989 0.9818 0.9747

0.9989 0.9793 0.9716

Nil B, C, BB, CC, DD, BC, CD BB, DD, AC, BC, BD, CD

the insignificant terms of different models for the response – permeability. From the table, it is seen that all terms of linear regression were significant and its coefficient of multiple correlation was more than that of non-linear models. Some terms were identified as the insignificant ones in both the non-linear models (refer to Table 5). ANOVA test was conducted to check the adequacy of the model as shown below. Table 6 shows that in the linear model, ‘P’ values for main effects, 2-way interactions, 3-way interactions and 4way interactions are less than 0?05 (corresponding to 95% confidence level). It indicates that the above mentioned terms of linear model were found to be significant. Table 7 shows the results of ANOVA test for the nonlinear models developed for the response – permeability. It has been observed from this table that the lack of fit becomes significant for both the non-linear models, if the insignificant terms are excluded from the models.

(ii) central composite design (non-linear model) CSccd ~17:2527{1:7384A{2:7463Bz 32:3203Cz6:575Dz0:014A2 z0:0945B2 { 7:7857C 2 {1:2079D2 z0:0468AB{ 0:1215AC{0:0451ADz0:5516BCz 0:6378BDz2:689CD (iii) Box-Behnken design (non-linear model) CSbox ~33:0795{0:6358A{9:0986Bz 14:0303Cz3:4178Dz0:001928A2 z 0:663625B2 {0:2758C 2 {0:8274D2 z 0:0451AB{0:0088AC{0:0016AD{ 1:1261BCz0:7412BDz0:12411CD

Response – green compression strength

The coefficients of correlations and insignificant terms of the linear and non-linear models for the response – green compression strength, are shown in Table 8. To test the adequacy of the models, ANOVA tests were carried out both on the linear and non-linear models. The results of ANOVA test for the linear model of the response – green compression strength are shown in Table 9. All the main factors, combined 2-way interaction terms and combined 3-way interaction terms (although ABC, BCD are seen to be insignificant) are found to be significant. The ‘P’ value for four factor interaction term was found to be greater than 0?05, which indicates that it is statistically insignificant. The results of ANOVA test for the non-linear models of green compression strength are shown in Table 10. Table 8 indicates that there are a few insignificant terms in the non-linear models, but the ANOVA table (refer to Table 10) shows that the lack of fit becomes significant, if these insignificant terms are removed from the models.

The following linear and non-linear regression equations were obtained by using the MINITAB software for the response – green compression strength: (i) full factorial design (linear model) CSfact ~29:4603z0:14282A{12:8365Bz 8:7071C{17:2927Dz0:1736AB{ 0:420251ACz0:060739ADz3:0988BCz 4:07036BDz2:20632CD{0:0191521ABC{ 0:0381673ABDz0:045965ACD{ 0:846521BCDz0:0076608ABCD Table 6 Results of ANOVA – permeability (linear model) Source

DF Seq SS Adj SS Adj MS

F

P

Main effects 4 233 407 233 407 58 351.9 7133.8 0 2-way interactions 6 14 222 14 222 2370.4 289.79 0 3-way interactions 4 3321 3321 830.1 101.49 0 4-way interactions 1 2684 2684 2684.4 328.18 0 Residual error 64 523 523 8.2 Pure error 64 523 523 8.2 Total 79 254 158

Response – mould hardness

Linear and non-linear relations for the response – mould hardness had been established with the help of a response surface methodology. The ANOVA test was performed and the coefficients of correlations were

Table 7 Results of ANOVA – permeability (non-linear models) Design

6

Central composite

Box-Behnken

Source

DF

Seq SS

Adj SS

Adj MS

F

P

Seq SS

Adj SS

Adj MS

F

P

Regression Linear Square Interaction Residual error Lack of fit Pure error Total

14 4 4 6 120 10 110 134

308 261 32 14 11 10

308 48 32 14 11 10

22 042 12 157 8168 2370 97.4 1082 7.9

226 125 84 24

0 0 0 0

15 681 43 428 9533.4 1279.8 96.2 1099.1 5

0 0 0 0

0

219 526 173 714 38 134 7679 11 539 10 991 548

163.07 451.64 99.14 13.31

137

219 526 173 714 38 134 7679 11 539 10 991 548 231 065

220.56

0

592 699 671 222 683 815 868 32 027

592 630 671 222 683 815 867.7


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determined to test statistical adequacy of the models. The response equations as found out by using the different models are given below: (i) full factorial design (linear model)


(iii) Box-Behnken design (non-linear model) Hbox ~34:8682z0:3149Az1:7369Bz 6:8984Cz4:8952D{0:0014A2 z0:0563B2 {

Hfact ~36:9524z0:304762Az2:78095Bz

1:111C 2 {0:175D2 {0:0012AB{0:0032AC{ 0:0095ADz0:1BCz0:0BD{0:4CD

6:7873Cz4:8381D{0:0130952AB{ 0:0920635AC{0:0238095AD{0:435714BC{

Table 11 shows the coefficients of correlation for the different models and their insignificant terms. The coefficients of multiple correlations were found to be close to 1?0, which indicate that all models fit well to the assumed response equations. Table 12 shows the results of ANOVA test conducted on the linear model, for the response – mould hardness. The P values shown in this table indicate that all the terms (except the four factor interaction terms) are significant. Results of the ANOVA tests carried out for the non-linear models have been tabulated in Table 13. It is interesting to note that the lack of fit becomes significant with the removal of insignificant terms from the models.

0:25BD{0:06349CDz0:00357143ABCz 0:0063492ACD{0:0039683BCDz 0:00039683ABCD (ii) central composite design (non-linear model) Hccd ~38:2843{0:0494Az2:4746Bz


7:8434Cz7:774Dz0:001A2 {0:00389B2 { 1:6988C 2 {0:6556D2 {0:00015AB{ 0:0151AC{0:0006AD{0:075BC{ 0:1938BDz0:65CD

Table 8 Coefficient of multiple correlations and insignificant terms of different models for the response – green compression strength Model

Coefficient of correlation with all terms

Coefficient of correlation without insignificant terms

Linear Non-linear (CCD) Non-linear (Box- Behnken)

0.9852 0.9638 0.9220

0.9817 0.9592 0.9121

Insignificant terms ABC, BCD, ABCD B, D, BB, DD, AD, AC A, AA, BB, CC, DD, AB, AC, AD, BC, BD, CD

Table 9 Results of ANOVA – green compression strength (linear model) Source

DF

Seq SS

Adj SS

Adj MS

F

P

Main effects 2-way interactions 3-way interactions 4-way interactions Residual error Pure error Total

4 6 4 1 64 64 79

20 429.5 1183.9 385.3 4.7 665.3 665.3 22 668.7

20 429.5 1183.9 385.3 4.7 665.3 665.3

107.439 197.32 96.33 4.66 10.40 10.40

5107.36 18.98 9.27 0.45

0 0 0 0.506

Table 10 Results of ANOVA – green compression strength (non-linear models) Design

Central composite

Box-Benkhen

Source

DF

Seq SS

Adj SS

Adj MS

F

P

Seq SS

Adj SS

Adj MS

F

P


14 4 4 6 120 10 110 134

24 727.9 22 937.1 606.9 1183.9 1900.0 963.7 936.3 26 627.9

24 727.9 22 937.1 606.9 1183.9 1900.0 963.7 936.3

1766.281 5734.27 151.720 197.324 15.833 96.368 8.512

111.56 362.17 9.58 12.46

0 0 0 0

1399.364 4775.99 74.127 31.775 28.837 284.552 5.590

0 0 0.041 0.365

0

19 591.1 19 103.9 296.5 190.6 3460.5 2845.5 614.9

48.53 165.62 2.57 1.10

11.32

19 591.1 19 103.9 296.5 190.6 3460.5 2845.5 614.9 23 051.6

50.90

0

Table 11 Coefficient of multiple correlations and insignificant terms of different models for the response – mould hardness Model

Coefficient of correlation with all factors

Coefficient of correlation without insignificant factors

Insignificant terms

Linear Non-linear (CCD) Non-linear (Box-Behnken)

0.9931 0.9839 0.9628

0.9914 0.9818 0.9581

AB, AD, ABD, BCD, ABCD A, BB, AB, AD, BC BB, DD, AB, AC, AD, BC, BD, CD


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(iii) Box-Behnken design (non-linear model)

Response – bulk density

Response equations developed for bulk density by using different models, such as full factorial design, CCD and Box-Behnken, are shown below: (i) full factorial design (linear model)

BDbox ~{0:26183z0:022882Az0:080797Bz 0:051977Cz0:221722D{0:000092A2 { 0:0034B2 {0:00058C 2 {0:00836D2 { 0:000076AB{0:00019AC{0:00179AD{

BDfact ~2:40669{0:0111753A{ 0:0908045B{0:512701C{0:284517Dz

0:00262BCz0:0013BD{0:00789CD

0:00110279ABz0:00588003ACz 0:00332374ADz0:0490025BCz

Table 14 indicates the insignificant terms identified in both linear and non-linear models. Moreover, this table includes the values of coefficient of correlations separately for the different models obtained by considering all the factors and the significant factors only. All the main factors, combined 2-way interaction terms , combined 3-way interaction terms and combined 4-way interaction terms were found to be significant through the ANOVA tests (refer to Table 15). Table 16 shows the results of ANOVA tests carried out on both the non-linear models. The non-linear models, developed based on both CCD and BoxBehnken designs of experiments have some insignificant terms (refer to Table 14). It is important to mention that the lack of fit becomes significant after removing the insignificant terms from the models. From the above discussion, it is clear that all the models developed for predicting the responses, such as

0:0263778BDz0:13375CD{ 0:000562454ABC{0:000279432ABD{ 0:00148235ACD{0:0127726BCDz 0:000137433ABCD (ii) central composite design (non-linear model)


BDccd ~1:02616z0:01316A{0:00052B{ 0:06845Cz0:0083D{0:00008A2 z0:0009B2 z 0:0239C 2 {0:00107D2 {0:00004AB{ 0:00018ACz0:00029AD{0:00302BC{ 0:00019BD{0:00186CD

Table 12 Results of ANOVA – hardness (linear model) Source

DF

Seq SS

Adj SS

Adj MS

F

P


4 6 4 1 64 64 79

586.25 36.875 4.45 0.012 8.8 8.8 636.387

586.25 36.875 4.45 0.012 8.8 8.8

146.562 6.146 1.112 0.012 0.137 0.137

1065.91 44.7 8.09 0.09

0 0 0 0.764

Table 13 Results of ANOVA – hardness (non-linear models) Design

Central composite

Box-Benkhen

Source

DF

Seq SS

Adj SS

Adj SS

F

P

Seq SS

Adj SS

Adj MS

F

P


14 4 4 6 120 10 110 134

732.564 645.078 50.611 36.875 24.369 8.369 16.000 756.933

732.56 48.982 50.611 36.875 24.369 8.369 16

52.33 12.25 12.65 6.146 0.203 0.837 0.146

258 60 62 30

0 0 0 0

477.767 458.267 16.35 3.15 37.833 20.633 17.2 515.6

477.767 458.267 16.35 3.15 37.833 20.633 17.2

34.126 114.57 4.088 0.525 0.315 2.063 0.156

108.24 363.38 12.96 1.67

0 0 0 0.14

5.8

0

13.2

0

Table 14 Coefficient of multiple correlations and insignificant terms of different models for response – bulk density

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Model

Coefficient of correlation with all factors

Coefficient of correlation without insignificant factors

Insignificant terms

Linear Non-linear (CCD) Non-linear (Box- Behnken)

0.9883 0.9571 0.9127

0.9856 0.9518 0.9017

BD, CD, ABC, ABD B, C, BB, CC, DD, BC, CD CC, DD, AB, AC, BC, BD, CD


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a full factorial design; b CCD; c Box-Behnken design 1 Comparison of model predicted values of permeability with experimental results for 20 test cases

that permeability might have non-linear relationship with the input variables. Figure 2 shows the values of percentage deviation in prediction of permeability for all the three models. It is seen from the above figure that the values of percentage deviation in prediction of permeability are lying between 250 and z17% for linear model, 210 and z18% for CCD, and 212 and z12% for Box-Behnken design. Both the non-linear models have outperformed the linear model in most of the test cases. It could be due to the reason that the input–output relationship for permeability is so much non-linear that they cannot be modelled accurately using a linear model. It is important to note that CCD has yielded a slightly better performance compared to the Box-Behnken DOE.


permeability, green compression strength, mould hardness and bulk density, are statistically adequate.

Testing and comparison of models In the previous section, the models have been validated statistically. Now the performances of the developed models have been compared response-wise, as discussed below. Response – permeability

Twenty test cases were generated at random and real experiments were conducted to record the different responses for the above test cases. The predicted values of permeability obtained by using different models based on full factorial design, CCD and Box-Behnken design were compared with their respective target (i.e. experimental) values, as shown in Fig. 1. In the case of linear model shown Fig. 1a, the best fit line of the above 20 points, was found to deviate from the ideal y5x line. Moreover, some of the points were seen to lie away from the best fit line. On the other hand, both the non-linear models, as shown in Fig. 1b and c, had given better predictions compared to the linear model. It indicates

Response – green compression strength

Model predicted values of green compression strength had been compared to their respective target values, for 20 test cases (refer to Fig. 3). Thus, 20 points were obtained on each of the plots shown in Fig. 3 and the best fit line was determined for each case. It is interesting to note that non-linear models had shown a slightly better performance compared to the linear model.

Table 15 Results of ANOVA – bulk density (linear model) Source

DF

Seq SS

Adj SS

Adj MS

F

P


4 6 4 1 64 64 79

0.123193 0.005585 0.001721 0.001499 0.003134 0.003134 0.135132

0.123193 0.005585 0.001721 0.001499 0.003134 0.003134

0.0307983 0.0009308 0.0004303 0.0014993 0.000049 0.000049

628.97 19.01 8.79 30.62

0 0 0 0

Table 16 Results of ANOVA – bulk density (non-linear models) Design

Central composite

Box-Benkhen

Source

DF

Seq SS

Adj SS

Adj MS

F

P

Seq SS

Adj SS

Adj MS

F

P


14 4 4 6 120 10 110 134

0.17386 0.14562 0.02265 0.00559 0.01603 0.00961 0.00642 0.18989

0.1739 0.0207 0.0226 0.0056 0.016 0.0096 0.0064

0.012 0.005 0.006 9E-04 1E-04 1E-03 6E-05

93 39 42 7

0 0 0 0

0.0115 0.0204 0.0125 0.0049 0.0003 0.0026 6E-05

0 0 0 0

0

0.16115 0.08161 0.04988 0.02967 0.0324 0.02593 0.00647

42.64 75.57 46.19 18.31

16

0.16115 0.08161 0.04988 0.02967 0.0324 0.02593 0.00647 0.19355

44.08

0


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2 Values of percentage deviation in prediction of permeability for test cases

a full factorial design; b CCD; c Box-Behnken design 3 Comparison of model predicted values of green compression strength with experimental results for 20 test cases

4 Values of percentage deviation in prediction of green compression strength for test cases

Three models have been compared in Fig. 4, in terms of percentage deviation in prediction of green compression strength. The values of percentage deviation in prediction were seen to be within 210 to z12% for linear model,

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210 to z10% for CCD and 214 to z16% for BoxBenkhen model. Central composite design model showed the best performance in modelling green compression strength as the function of different input parameters.

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a full factorial design; b CCD; c Box-Behnken design 5 Comparison of model predicted values of mould hardness with experimental results for 20 test cases

6 Values of percentage deviation in prediction of mould hardness for test cases

a full factorial design; b CCD; c Box-Behnken design 7 Comparison of model predicted values of bulk density with experimental results for 20 test cases

Response – mould hardness

Figure 5 shows the comparisons of the predicted values of mould hardness obtained through different models with their respective target values. The values of percentage deviation in prediction of mould hardness are shown for 20 test cases, in Fig. 6. The values of percentage deviation in prediction of hardness were found to lie between 22?5 and z2?0% for linear model, 23?0 and z1?0% for CCD, and 23?5 and z2?0% for Box-Behnken model. For this response, the

linear model was found to perform better than the non-linear models. It could be due to the fact that the input–output relationship for this response is not so much non-linear. Response – bulk density

Model predicted values of bulk density were compared with their respective target values for the response – bulk density, as shown in Fig. 7. The linear model was seen to outperform the non-linear models. Figure 8 shows the


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8 Values of percentage deviation in prediction of bulk density for test cases

comparisons of the above models in terms of percentage deviation in prediction of bulk density, for 20 random test cases. The values of percentage deviation were found to lie in the ranges of 21?0 to 1?0%, 24?0 to 0?5% and 23?0 to 3?0% for the linear, CCD and Box-Behnken models respectively. It might have happened due to the reason that the relationship of bulk density with the input parameters is not so much non-linear. Thus, the performance of a model depends on the dataset indicating input–output relationships. The linear model was seen to outperform the non-linear models for this response.

For each response, the average value of absolute percentage deviations in prediction was calculated for the different models separately. Figure 9a shows the comparison of the linear model and two non-linear models (such as CCD and Box-Behnken), in terms of average of absolute percentage deviations in prediction of permeability. The model developed based on CCD had outperformed the other two models. Similar comparisons were also made for the other responses, namely green compression strength, mould hardness and bulk density, which are shown in Fig. 9b–d respectively. It

a permeability; b green compression strength; c mould hardness; d bulk density 9 Comparison of models in terms of average of absolute percentage deviation in prediction of responses

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is interesting to note that the non-linear model developed based on CCD showed the best performance in predicting permeability and green compression strength. On the other hand, the linear model built based on full factorial design had shown the best performance in predicting both mould hardness and bulk density.


Concluding remarks To determine input–output relationships in green sand mould system, full factorial design was adopted for the linear modelling and two techniques, namely central composite design and Box-Behnken design were used for developing the non-linear models. To validate the results of the developed models, 20 randomly generated test cases were examined and deviations in prediction were determined. For each response, the predicted values using three different models were compared separately with their respective target values. Moreover, three models were compared, in terms of percentage deviation in prediction of different responses. The average value of absolute percentage deviations in prediction was used as the criterion to select the best model, for a particular response. The central composite design based non-linear model has shown the best performance in predicting permeability and green compression strength, whereas linear model has outperformed the non-linear models for making predictions of mould hardness and bulk density. Thus, the selection of a suitable model for finding/establishing the input– output relationships of a process is dependent on the output variables.

Acknowledgement The first author gratefully acknowledges the help of the Quality Improvement Program, Ministry of Human Resource Development, Government of India for


providing the financial help for carrying out the research work.

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