The study of green compression strength of a green sand mould using statistical approach has been undertaken. Empirically generated data in National Metallurgical Development Centre, Jos Sand Testing Laboratory were used for the study. Coefficient of correlation, coefficients of determination and coefficient of multiple determinations were used to explain the relationship existing between the two independent variables of clay and moisture content and green compression strength, the dependent variable. The study showed that the coefficient of determination for Ys: X 1 was 0.88 while the coefficient of correlation was 0.94, coefficient of determination for Ys: X 2 was 0.90 while the coefficient of correlation was 0.95 and the coefficient of multiple determination was 0.72; these coefficients assisted tremendously in the study of green compression strength. A mathematical model was developed for the prediction of green compression strength; it was tested and proved to be a good estimation tool for estimating green compression strength values on the foundry shop floor. The study has clearly shown that statistical approach is a good tool for studying green compression strength of green sand moulds.
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Statistical approach of the study of this important property will give in-depth information on how to produce good moulds. This fact has been upheld by several authors [
The objective of the work is to study green compression strength of a green sand mould using statistical approach.
The materials used for this work were obtained from empirically generated data on various moulding sand systems at the sand testing laboratory of the National Development Centre, Jos. This particular data was generated from green compression strength tests on green moulds using Bacita foundry sand, clay binder and moisture (H2O). Universal strength testing machines, manual green compression strength tester, sand rammer, mixer, measuring cylinders were used during the work as equipment. The main variables in the work were clay (binder) which was varied from 1.5% to 7% and moisture (water) which was varied from 2% to 8%. Other variables like mixing time, number of rams, sand (grain size and shape) were all kept constant.
S/NO | %Clay Content Variation X1 | % Moisture content Variation X2 | Green Compression Strength, kN/m2 (YS) |
---|---|---|---|
1 | 1.5 | 2 | 11.72 |
2 | 2 | 3 | 17.24 |
3 | 3 | 4 | 21.38 |
4 | 4 | 5 | 22.75 |
5 | 5 | 6 | 24.13 |
6 | 6 | 7 | 24.82 |
7 | 7 | 8 | 33.74 |
The basic two variable models (one dependent and one independent variable) is
Which can be solved using the normal equations thus:
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)
Which can be solved by the normal equations for a three variable model, as follows:
The line of best fit gives way to a plane of best fit, b1 is the slope of the plane along the X1 axis, b2 is the slope along the X2 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. 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
The regression equation for the relationship of clay content and green compression strength value of the green mould is
The coefficient of correlation for this relationship is
Substituting the values in
The regression equation for the relationship of moisture content with green compression strength value of the green mould is
The coefficient of correlation for this relationship is
The multiple regression calculations are carried out using the three variable normal Equations (4)-(7) with substitution with values from
Solving these three equations simultaneously gave
Now substituting the model equation for three variables in Equation (4), this new model equation is obtained
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 compression strength of the green sand mould.
From Equation (9)
The various coefficients of determination can now be summarized
Equations (9), (11) and (15) are the developed regression models for simple and multiple linear regression relationships existing between green compression strength and the two variables of clay content and moisture content in the green mould.
The various coefficients are interpreted as follows:
n | X1 (%) | X2 (%) | Ys (Empirical values of Green compression strength) kN/m2 | Ys (Mathematical model values of Green compression strength) kN/m2 |
---|---|---|---|---|
1 | 1.5 | 2 | 11.72 | 14.00 |
2 | 2.0 | 3 | 17.24 | 15.57 |
3 | 3.0 | 4 | 21.38 | 18.67 |
4 | 4.0 | 5 | 22.75 | 21.77 |
5 | 5.0 | 6 | 24.13 | 24.87 |
6 | 6.0 | 7 | 24.82 | 27.97 |
7 | 7.0 | 8 | 33.74 | 31.07 |
8 | 8.0 | 9 | - | 34.17 |
9 | 9.0 | 10 | - | 37.27 |
R2: this shows the combined influence of the two variables of clay content and moisture content %. The influence is a major influence and it is positive. 72% of the changes in green compression strength values in green sand mould are brought about by the combined influence of clay content % and moisture content %.
The individual correlation between clay content and green compression strength is given by the coefficient of correlation
Moisture content has a very strong positive correlation with green compression strength of green sand moulds. The correlation is 0.95 which means that 95% of the increase in green compression strength will be caused by increase in moisture content of the moulding mixture if the moisture content is the only variable. The importance of moisture is also observed practically while preparing moulds. Moulds with moisture content below optimum do tend to be lacking in green compression strength. Literatures that have being reviewed for this work have also shown that moisture content increase do lead to increase in green compression strength until an optimum point is reached where further increase leads to a decrease in green compression strength [
The combined effect of clay content and moisture content as well as the influence on green compression strength has been given by the coefficient of multiple determination and interpreted above. The established relationship from the coefficient of multiple determination agrees with what is obtained practically and empirically generated results both clay content and moisture content increase green compression strength of moulding mixture making it mouldable [
The study of green compression strength of a green sand mould using statistical approach has been successfully carried out and the following conclusions are from the study:
1) Statistical approach has been found to be a good method of studying green compression strength of green sand mould.
2) The coefficients of correlation, coefficients of determination and coefficient of multiple determinations are great tools for studying the relationship between the variables (independent and dependent variables).
3) The study has shown the positive strong relationship that exists between green compression strength of the dependent variable and the two independent variables of clay content and moisture content. The study showed a coefficient of multiple determination of 0.72 which translates to 72% of the variation in green compression strength coming from the combined effect of clay content and moisture content.
4) The study developed a mathematical model which can be used for the prediction and estimation of green compression strength in green sand moulds. This clearly confirms the crucial nature of statistical approach in the study of green sand moulds.
The authors hereby extend their sincere appreciation to the staff of the Sand Testing Laboratory of NMDC Jos who assisted with the generation of this data. We also acknowledge your effort for the enormous amount of data bank that now exists in that Laboratory on Nigerian foundry sands.