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Aondona Paul Ihom and Aniekan Offiong

University of Uyo, Uyo, Department of Mechanical Engineering, PMB 1017 Uyo, Akwa Ibom State- Nigeria.

Coauthor’s Email: irimanagement @yahoo.com, Tel: +2347032254829

Modeling of the effect of moulding variable parameters on some properties of clay-bonded sand has been carried out. The data used for the work was generated at the National Metallurgical Development Centre, Jos sand testing laboratory. The data was used for modeling the effect of moulding variables on the green shear strength and bulk density of clay-bonded sand. Individual regression equations were developed, multiple regression equations were developed alongside with their coefficient of correlation and coefficient of determination and multiple determinations. These were sufficient for the explanation of the effect of moulding variable parameters on the two properties of the clay-bonded sand. The green shear strength had a multiple coefficient of determination of 0.5 and bulk density had a multiple coefficient of determination of 0.47.

Moulding variable parameters are numerous and their control is very important in attaining acceptable moulding properties of clay-bonded sands. The mulling station is the central point where most of these variables are controlled. These variables include mulling time, clay content, moisture content, ramming and other additives. These variable parameters have varying effect and influence on the properties of clay- bonded sands. Ihom et al [1] and several other authors have clearly shown in their individual work, how clay and moisture content variation affect properties like green permeability, green compression strength, dry compression strength, dry shear strength, bulk density and shatter index [2-4]. Ihom [5] in his work

‘foundry raw materials for sand casting and testing procedures’ clearly explains the interactions between clay, sand and moisture and the effect these interactions have on properties of clay-bonded sands. According to Ihom [5] ‘clay acquires its bonding action in the presence of moisture. When water is added to clay it penetrates the mixture and forms a surface film which coats the surface of each flake as more water is added the thickness of the film increases up to a certain limit after which the excess water remain in the fluid state. The thickness of this water film varies with the clay mineral and the quantity of clay. The bonding of clay depends on the mixing and the thickness of water film it can contain. This has effect on properties like the strengths, permeability, mouldability, shatter index, compactability and flowability.’

When sand is rammed in the mould the sand grains are forced together the clay coating on each coating acts in such a way that it not only locks the grains in position, but also makes them retain that position, if the water is added in the exact quantity required, to form the films, bonding action is best. if water is in

excess the strength is reduced and the mould gets weakened. Rundman in Ihom [5] have explained how

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water is involved with the sand and clay in a moulding mixture using a low moisture model involving polarized water molecules on both clay and sand and a high moisture model involving ionic bonds between the Na ions and the water molecules. The polar bonding at low moisture is referred to as rigid water bonding and the ion water bonding at high moisture is referred to as bridge bonding. Surface defects on castings are, therefore more prevalent when the warm wet strength of the sand is low. This condition occurs when the clay binder has insufficient quantity of the adsorbed species which account for ionic bridge bonding, that portion of the bonding mechanism which is so important at high moistures and temperatures.

Modeling of the effect of moulding variable parameters on some properties of clay-bonded sand particularly the variation in clay and moisture content is important. The interrelations involved have been explained above. The mathematical models will further bring out the relationships and also make the relationship available for utilization for practical purposes. Using the developed models, a known variation in clay and moisture content can be used to predict the properties of the clay-bonded sand [6-

10].

The objective of this work is to undertake the modeling of the effect of moulding variable parameters on some properties of clay-bonded sand.

The materials that were used for this work included; foundry sand, clay binder from Baroid of Nig. Ltd., Port-Harcourt-Nigeria. Water was from the sand testing laboratory water system. The equipment used for the work included; sand rammer equipped with bulk density tester, universal strength testing machine, digital weighing balance, mixer, electric permmeters and sieve shaker among others.

The tests to determine the effect of moulding variable parameters on some properties of clay-bonded sand was carried out at National Metallurgical Development Centre, Jos, sand testing laboratory. During the mulling of the moulding mixture, the two variables were clay and moisture content. The mulling time, ramming, sand type, and other variables were kept constant. All the tests carried out conformed to AFS standard for testing moulding mixtures in the foundry. The results generated are presented in Table 1.

Green Shear Strength KN/m2 (Y1 ) | 0 | 8.96 | 10.34 | 7.59 | 8.96 | 9.65 | 11.03 |

Bulk Density g/cm3 (Y2 ) | 1.48 | 1.48 | 1.51 | 1.58 | 1.58 | 1.60 | 1.62 |

Clay Content % (X1 ) | 1.5 | 2 | 3 | 4 | 5 | 6 | 7 |

Moisture Content % (X2 ) | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

The basic two variable models (one dependent and one independent variable) is

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Y = 𝑎 + 𝑏𝑥 (1)

Which can be solved using the normal equations thus:

∑ 𝑌 = 𝑎𝑛 + 𝑏 ∑ 𝑥 (2)

∑ 𝑋𝑌 = 𝑎 ∑ 𝑋 + 𝑏 ∑ 𝑋2 (3)

From this can be developed models with more than two variables and this is illustrated below using a 3

variable model (one dependent and two independent variables, Y, X1 , and X2 )

Y = 𝑎 + 𝑏1𝑋1 + 𝑏2 𝑋2 (4)

This can be solved by the normal equations for a three variable model, as follows:

∑ 𝑌 = 𝑎𝑛 + 𝑏1 ∑ 𝑋1 + 𝑏2 ∑ 𝑋2 (5)

∑ 𝑋1 𝑌 = 𝑎 ∑ 𝑋1 + 𝑏1 ∑ 𝑋2

+ 𝑏2 ∑ 𝑋1 𝑋2 (6)

∑ 𝑋2 𝑌 = 𝑎 ∑ 𝑋2 + 𝑏1 ∑ 𝑋1 𝑋2 + 𝑏2 ∑ 𝑋2

(7)

The line of best fit gives way to a plane of best fit, b1 is the slope of the plane along the X1 axis, b 2 is the slope along the X 2 axis and the plane cuts the Y axis at `a’. The aim of adding to the simple two variable models is to improve the fit of the data.

bx =

𝑛 ∑ 𝑋𝑌 − ∑ 𝑋 ∑ 𝑌

𝑛 ∑ 𝑋 2 − (∑ 𝑋)2

(8)

ax =

∑ 𝑌 −

𝑛

𝑏𝑋∑ 𝑋

𝑛

(9)

The coefficient of correlation between the variables is given by

𝑟𝑋 = __ __

𝑛 ∑ 𝑋 𝑌1 − ∑ 𝑋 ∑ 𝑌1

2

(10)

�𝑛 ∑ 𝑋 2− (∑ 𝑋)2 × �𝑛 ∑ 𝑌1

− (∑ 𝑌)2

𝑟2 𝑖𝑠 Coefficient of determination for Y: X

The closeness of fit is measured by the coefficient of multiple determination R2 for which the general

formula and a useful computational formula is given below

R2 =

𝑎 ∑ 𝑌 + 𝑏1 ∑ 𝑋1𝑌 + 𝑏2 ∑ 𝑋2𝑌 −

∑ 𝑌 2 − (∑ 𝑌)2

𝑛

(∑ 𝑌)2

𝑛

(11)

Table 2 gives the calculation of separate regressions with green shear strength as Y1, clay content as X1

and moisture content as X2.

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N | Y1 | Y21 | X1 | X21 | X2 | X22 | X1Y1 | X2Y1 | X1X2 |

1 | 0 | 0 | 1.5 | 2.25 | 2 | 4 | 0 | 0 | 3.0 |

2 | 8.96 | 80.28 | 2 | 4 | 3 | 9 | 17.92 | 26.88 | 6.0 |

3 | 10.34 | 106.92 | 3 | 9 | 4 | 16 | 31.02 | 41.36 | 12.0 |

4 | 7.59 | 57.61 | 4 | 16 | 5 | 25 | 30.36 | 37.95 | 20.0 |

5 | 8.96 | 80.28 | 5 | 25 | 6 | 36 | 44.80 | 56.76 | 30.0 |

6 | 9.65 | 93.12 | 6 | 36 | 7 | 49 | 57.9 | 67.55 | 42.0 |

7 | 11.03 | 121.66 | 7 | 49 | 8 | 64 | 77.21 | 88.24 | 56.0 |

∑ | 56.53 | 539.87 | 28.5 | 141.25 | 35 | 203 | 259.21 | 318.74 | 169.0 |

The regression equation for the relationship of % clay content and green shear strength of the mould is

𝑌1𝑋1 = 3.39 + 1.15𝑋1 (12)

The coefficient of correlation for this relationship in equation 12 (from equation 10) is

𝑟𝑋1 = 0.63

The coefficient of determination r2 for regression Y1X1: X1 is

𝑟2 = 0.40

The regression equation for the relationship of green shear strength of the moulding mixture with %

moisture content variation (from equation 1) is

𝑌1𝑋2 = 1.63 + 1.29𝑋2 (13)

The coefficient of correlation for this relationship in equation 13 (from equation 10) is

rX2 = 0.75

The coefficient of determination r2 for regression Y1X2 : X2 is

2

𝑋2

= 0.56

The multiple regression calculations are carried out using the three variable normal equations (equations

4-7) and this gave rise to a multiple regression equation with best fit with X1 and X2 considered in the relationship. The developed mathematical model based on the two variables of clay and moisture content is

𝑌1 = −22.68 − 36.79𝑋1 + 36.11X2 (14)

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This mathematical model is derived with the combined influence of clay content and moisture content all taken into account and therefore can be used to predict the green shear strength of the sand mould. Table

3 shows empirical values of Green shear strength alongside with calculated values using mathematical

model of equation 14

𝒏 | X1 | X2 | Empirical values of Green Shear Strength kN/m2 (Y1) | Calculated values of Green Shear Strength kN/m2 (Y1) |

1 | 1.5 | 2 | 0 | -5.65 |

2 | 2 | 3 | 8.96 | 12.07 |

3 | 3 | 4 | 10.34 | 11.39 |

4 | 4 | 5 | 7.59 | 10.71 |

5 | 5 | 6 | 8.96 | 10.03 |

6 | 6 | 7 | 9.65 | 9.35 |

7 | 7 | 8 | 11.03 | 8.67 |

Coefficient of Multiple Determination, R2

The coefficient of multiple determination R2 for the developed model equation for the relationship between green shear strength Y1 and the two variables of clay and moisture content is

R2 = 0.5

𝒏 | Y2 | Y2 | X1 | 2 X1 | X2 | 2 X 2 | X1Y2 | X2Y2 | X1X2 |

1 | 1.48 | 2.19 | 1.5 | 2.25 | 2 | 4 | 2.22 | 2.96 | 3 |

2 | 1.48 | 2.19 | 2.0 | 4.00 | 3 | 9 | 2.96 | 4.44 | 6 |

3 | 1.51 | 2.28 | 3.0 | 9.00 | 4 | 16 | 4.53 | 6.04 | 12 |

4 | 1.58 | 2.50 | 4.0 | 16.00 | 5 | 25 | 6.32 | 7.90 | 20 |

5 | 1.58 | 2.50 | 5.0 | 25.00 | 6 | 36 | 7.90 | 9.48 | 30 |

6 | 1.60 | 2.56 | 6.0 | 36.00 | 7 | 49 | 7.60 | 11.20 | 42 |

7 | 1.62 | 2.62 | 7.0 | 49.00 | 8 | 64 | 11.34 | 12.96 | 56 |

∑ | 10.85 | 16.89 | 28.5 | 141.25 | 35 | 203 | 44.87 | 54.98 | 169 |

The regression equation for the relationship of bulk density of the moulding mixture with % clay content variation from equation 1 is

𝑌2𝑋1 = 1.44 + 0.028𝑋1 (15)

The coefficient of correlation for this relationship in equation 15 (from equation 10) is

𝑟𝑋1 = 0.51

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The coefficient of determination r2 for regression Y2X1 : X1 is

𝑟2 = 0.26

The regression equation for the relationship of bulk density of the moulding mixture with % moisture content variation from equation 1 is

𝑌2𝑋2 = 1.42 + 0.026𝑋2 (16)

The coefficient of correlation r for this relationship in equation 16 (from equation 10) is

rX2 = 0.51

The coefficient of determination r2 for regression Y2X2: X2 is

2

𝑋2

= 0.26

The multiple regression calculations are carried out using the three variable normal equations (equations

4-7) and this gave rise to a multiple regression equation with best fit with X1 and X2 considered in the relationship. The developed mathematical model for the relationship between bulk density and the two variables of clay and moisture content is

𝑌2 = 1.48 + 0.09𝑋1 − 0.059X2 (17)

Table 5 shows empirical values of bulk density alongside with calculated values using mathematical

model of equation 17

𝒏 | X1 | X2 | Empirical values of Bulk Density g/cm3 (Y2) | Calculated values of Bulk Density g/cm3 (Y2) |

1 | 1.5 | 2 | 1.48 | 1.497 |

2 | 2 | 3 | 1.48 | 1.483 |

3 | 3 | 4 | 1.51 | 1.514 |

4 | 4 | 5 | 1.58 | 1.545 |

5 | 5 | 6 | 1.58 | 1.576 |

6 | 6 | 7 | 1.60 | 1.607 |

7 | 7 | 8 | 1.62 | 1.638 |

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From equation 11 the coefficient of Multiple Determination R2 for the regression equation of equation 17 is

R2 = 0.47

The result of the regression analysis is here presented

Regression of green shear strength Y1 on % clay content variation X1

𝑌1𝑋1 = 3.39 + 1.15𝑋1 (12)

The coefficient of correlation r and determination r2 for Y1X1 : X1 are

𝑟𝑋 = 0.63 and 𝑟2 = 0.40

Regression of green shear strength Y1 on % moisture content variation X2 (Y1X1 : X2)

𝑌1𝑋2 = 1.63 + 1.29𝑋2 (13)

The coefficients of correlation r and determination r2 for Y1X2: X2 are 𝑟𝑋

= 0.75 and 𝑟2 = 0.56

The multiple regression of green shear strength on X1 and X2 (Y1: X1 and X2 ) is found to be

𝑌1 = −22.68 − 36.79𝑋1 + 36.11X2 (14)

The coefficient of multiple determination, R2 for green shear strength Y1 is R2 = 0.50

Table 3 gives the empirical values of green shear strength alongside calculated values using equation 14. The results for regression study of bulk density are as follows:

Regression Y2 : X1 is

𝑌2𝑋1 = 1.44 + 0.028𝑋1 (15)

The coefficients of correlation r and determination r2 for equation 15 are

𝑟𝑋 = 0.51 and 𝑟2 = 0.26

Regression Y2 : X2 is

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𝑌2𝑋2 = 1.42 + 0.026𝑋2 (16)

The coefficients of correlation r and determination, r2 for equation 16 are

𝑟𝑋2 = 0.51 and 𝑟2 = 0.26

The multiple regression (Y2: X1 and X2) for bulk density on X1 and X2 is found to be

𝑌2 = 1.48 + 0.09𝑋1 − 0.059X2 (17)

Table 5shows empirical values of bulk density alongside calculated values using equation 17

The coefficient of multiple determination, R2 for bulk density Y2 is found to be

R2 = 0.47

The result of the study has shown the various stages adopted in the modeling of the effects of the two variables of clay and moisture content on the two properties of the clay-bonded sand. Equation 12 is a modeled equation showing the relationship between green shear strength and clay content only. The relationship has a coefficient of correlation r, of 0.63 and a coefficient of determination r2x1 of 0.4. The correlation is positive and above average, with the coefficient of determination of 0.4 it means 40% of the increase in green shear strength is caused by increase in clay content. The developed model equation in equation 13 shows the relationship between green shear strength and moisture content. The relationship has a coefficient of correlation of 0.75 and a coefficient of determination r2x2 of 0.56. The coefficient of correlation is strongly positive which means any change in moisture content will bring about a positive change in green shear strength. The 0.56 value of coefficient of determination means that 56% change or increase in green shear strength is brought about by changes in the value of moisture content of the moulding moisture. The above relationships are based on the individual relationship or influence of clay and moisture content on green shear strength. In real world of moulding mixture the two variables of clay and moisture content have a resultant effect and the resultant effect is shown in equation 14. This equation can be used in predicting the green shear strength of the moulding mixture. This is because it takes into account the combined effect of the two variables on the properties of the moulding mixture. Table 3 shows the empirical values of green shear strength alongside the calculated values using the developed mathematical model of equation 14. Looking at Table 3, at 1.5% clay and 2% moisture content the empirical value is 0. The moulding mixture was not mouldable, the calculated or predicted value is -5.65

KN/m2, this value confirms why the moulding mixture could not be moulded. The value of the green

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shear strength is too small for the moulding mixture to be moulded. Comparing the empirical values and the calculated values using the model equation the values are close but not exact, but can be used for estimation purposes, more so that the equation is the best fit for the empirical result. The coefficient of multiple determination of green shear strength on clay and moisture content is 0.5. It is the combined effect of the clay and moisture content on the green shear strength, and it is moderately strong relationship. 50% of the changes in green shear strength are due to the combined effect of clay and moisture content. The modeled effects of the two independent variables on the dependent variable, (green shear strength) agrees with observations made by several authors [9-15]

The effect of the two variables (of clay and moisture content) on bulk density has equally been examined. Equation 15 shows the relationship between bulk density and clay content. The model equation has a correlation coefficient of 0.51 and a coefficient of determination of 0.26. The correlation is moderately strong but the coefficient of determination shows that only 26% of the changes in bulk density are due to changes in clay content. Equation 16 is the developed model for the relationship between bulk density and moisture content. The relationship has a correlation coefficient of 0.51 and a coefficient of determination of 0.26. The relationship has a moderately strong positive correlation but the coefficient of determination indicates that only 26% of the changes in bulk density are due to changes in moisture content of the moulding mixture. The multiple regression takes into account the combined effect of the two independent variables of clay and moisture content on the properties of the moulding mixture. The developed model equation in equation 17 shows the relationship between bulk density and clay and moisture content. Their combined effect is given by the coefficient of multiple determination which is 0.47 which means that 47% of the changes or increase in bulk density is due to the combined effect of clay and moisture content. Table 3 presents the empirical values of the bulk density alongside the calculated or predicted values using the developed model equation of equation 17. Comparing the values on both sides it can be seen that they are quite close such that the developed model equation in equation 17 can be used to predict bulk density values on the shop floor. Appraising the developed models and the defined effects of the two moulding mixture variables on the properties of moulding mixtures with respect to green shear strength and bulk density; the models and their explanation are in agreement with observations and existing theories made by other authors in their individual work [13-16]. Apart from clay and moisture content other variables too contribute to green shear strength and bulk density of moulding mixtures. The contributions made by the two variables have been shown in this work.

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The study ‘modeling of the effect of moulding variable parameters on some properties of clay-bonded sand’ has been carried out. The study has carefully drawn up the following conclusions:

1. Two models have been developed, one for the relationship between green shear strength and the two variables of clay and moisture content and the other one between bulk density and the two variables of clay and moisture content.

2. The two models can be used to predict green shear strength and bulk density

3. The work has established the effect of the independent variables on shear strength and bulk density through the models developed in the study.

4. The developed models were used in explaining the relationship between the dependent variable

and the independent variables. These explanations were in agreement with existing theories as presented by other authors.

5. The developed models have clearly presented the effect of moulding variable parameters on the

two properties of clay-bonded sand (green shear strength and bulk density)

I wish to acknowledge the contributions of the staff of the sand testing laboratory of National Metallurgical Development Centre, Jos where the data for this work was generated. Their assistance greatly contributed to the success story of this work.

[1] Ihom, A.P., Yaro, S.A. and Aigbodion, V.S. Application of Multiple Regression-Model to the Study of Foundry Clay- Bonded Sand Mixtures, JICCOTECH Vol.2, 2006, 161-168

[2] Ihom, A.P., Agunsoye, J., Anbua, E.E. and Bam, A. The use of Statistical Approach for Modeling and Studying the Effect of Ramming on the Mould Parameters of Yola Natural Sand, Nigerian Journal of Engineering vol. 16.1., 2009, 186-192

[3] Zrimsek, A.H. and Vingas, G.J. Effect of Moulding Variables Parameters on the Properties of Clay

Bonded Sands, Modern Casting, 1961, 39, 85

[4] Chakoraborty, M and Dhindaw, B.K., Utilization of Regression Equations in Optimization of Properties of Clay Bonded Sands, Proceedings of the 41st International Foundry Congress, Liege, Belgium, 1979, 8

[5] Ihom, A.P. (2012) Foundry Raw Materials for Sand Casting and Testing Procedures, 1st Edition, A2P2

Transcendent Publishers, Jos-Nigeria pp71-92.

[6] Lucey, T.(2000) Quantitative Techniques 7th Edition Low Priced Edition Funded by the British

Government, pp99-150

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[7] Chakoraborty, M and Dhindaw, B.K. Application of the Statistical Design of Experiments in the Study of Clay-bonded Sand Mixtures, the British Foundryman, Journal of the Institute of the British Foundrymen, 1985, 114-118

[8] Eckey, D.C. and Goldress, J.E., The Design of Experiments in the Sand Casting Foundry, Transactions of the American Foundryman Society, 62, 1989, 605

[9] Ihom, A.P., Allen, M. A., Nwonye, I. E., Ilochionwu, C., and Ogbodo, N. J., Investigation of the Accuracy of Non-linear Model in the Study of Green Permeability of Green Sand Moulds, African Journal of Engineering Research, Vol. 2(1), 14-20, 2014

[10] Ihom, A.P., Ogbodo, J.N., Allen, A. M., Nwonye, E.I. and Ilochionwu, C., Analysis and Prediction of Green Permeability Values in Sand moulds using Multiple Linear Regression Model, African Journal of Engineering Research, Vol.2(1), 8-13, 2014

[11] Ihom, A.P., Agunsoye, J, Anbua, E.E., Ogbodo, J., Effect of Moisture Content on the Foundry

Properties of Yola Natural Sand, LEJPT, ISSN 1583-1078,19., 2011, 85-96

[12] Khanna, O.P. (2008) A Text Book of Material Science and Metallurgy, 8th Reprint Dhanpat Rai

Publications, India, pp 51-8-10

[13] Brown, J.R. (1994) Foseco Foundryman’s Handbook, 10th Edition Pergamon press PLC, pp10-68 [14] Heine, R.W.,Loper, Jr. C.R. and Rosenthal, P.C.(1967) Principles of Metal Casting, 2nd Edition, Tata

Mcgraw-Hill Publishing Company Ltd, New Delhi, pp100-150

[15] Ihom,A.P., Suleiman, U.M, and Nyior, G.B., Impact of Swelling Indices of Sokoto Clays on the Moulding Properties of the Clays in Sand Mixtures, J. of Min. and Matls. Characterisation and Engineering, 11, 2012, 1050-1054

[16] Jain,R.K.(2009) Production Technology, 16th Edtion, Khana Publishers, Nai Sarak-Delhi-India, pp153-158

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