Application of Full Factorial Experimental Design and Response Surface Methodology for Chromite Beneficiation by Knelson Concentrator

The present work is undertaken to determine the effect of operational variables, namely: feed rate, centrifugal force and fluidization water flow rate on the efficiency of Knelson concentrator for chromite ore beneficiation. A full factorial design with three factors at three levels and response surface methodology (RSM) were applied for this purpose. The quadratic models were developed to predict the concentrate Cr2O3 grade and recovery as the process responses. The results suggest that all the variables affect the grade and recovery of the Cr2O3 concentrate to some degree. However, the fluidization water rate was found as the most effective parameter.


Introduction
Chromite ore (Mg,Fe)(Cr,Al,Fe) 2 O 4 is an oxide mineral which is the most important source of chromium.It is used for various purposes in metallurgical, refractory and chemical industries.In 2013, South Africa was the world's dominant chromite producer, followed by Kazakhstan, India and Turkey [1].
New generation gravity processing devices such as Mozley Multi Gravity Separator (MGS), Falcon Separator and Knelson Concentrator are promising technologies for the recovery of chromite at fine sizes.These gravity concentrators utilize centrifugal force to enhance the relative settling velocity between particles with different size and density [18][19][20][21][22][23].
The efficiency of centrifugal separators can be improved by adjusting the levels of operational parameters to provide the best conditions for separation [24,25].For this reason, the effects of different operational parameters are crucial to understanding and controlling the separation process.However, it is very time consuming to test the whole set of parameters at all levels.For this reason, a well-designed experimental test program is needed to determine the response of the separation process to each factor.As of today, factorial designs have been used successfully in many different areas for planning of experiments to develop empirical models [26][27][28][29].In this study, the effects of the operating variables such as feed rate, centrifugal force and fluidization water flow rate on the separation efficiency of Knelson concentrator were investigated using response surface methodology (RSM) and three-level and three-factor full factorial experimental design.Full factorial design contains all possible combinations of a set of factors.The influences of all factors and interaction effects on the responses are investigated systematically.

Materials
The chromite ore used in the study was supplied from Adana, Turkey.Microscopic studies indicated harzburgite and dunite which are serpentinized to different degrees as the host rocks.The specific gravity of the ore sample was determined as 3.2 g/cm 3 using Micrometrics, Accupyc II 1340 Gas Pycnometer (Micromeritics Instrument Corporation, Norcross, GA, USA).The specific gravity of the chromite mineral is about 4.6 g/cm 3 , whereas associated gangue minerals have a specific gravity of about 2.6 g/cm 3 .These specific gravity values indicate that separation of these minerals from each other using a gravity concentration technique is possible.
The ore sample was stage-crushed to 100% below 3.35 mm and then ground to 100% below 500 µm using a laboratory rod mill.Wet sieve analysis was carried out to determine the particle size distribution of the ground material using a vibratory sieve shaker and micro precision sieves.The results of size-wise chemical analysis are presented in Table 1.

Methods
A laboratory scale Knelson separator "KC-MD3" (FLSmidth USA Inc., Midvale, UT, USA) was used for the experiments.The concentrator has 45 kg/h dry and 8 L/min volumetric throughput.This manual discharge-laboratory scale separator has a 3 inches diameter truncated-cone-shaped separation bowl.The concentrate capacity of the separation bowl varies between 80 to 150 g depending on the fluidization conditions and heavy mineral properties.The amount of the feed material to be used in the study was determined as 220 g considering the chromite mineral content of the ore and results of the trial experiments.At the end of the study, the standard deviation of the weight of the concentrates was calculated as 1.8 g.The solid ratio of the feed slurry was kept constant 23% (w/w) during each test.The volumetric pulp flow rates corresponding to 12, 24 and 36 kg/h feed rates were 0.7, 1.4 and 2.2 L/min, respectively.The required fluidization water (FWFR) was supplied using a centrifugal pump with 4, 8 and 12 L/min flow rates.
The concentrate and tailing of the separation tests were filtered, dried and weighed prior to chemical analysis.Titration method (ASTM-E342-04, 1999) [30] was employed for the determination of the Cr 2 O 3 content of different products.
The effect of different operating parameters on the separation efficiency of Knelson separator was investigated using a three-level three-factor full factorial experimental design approach.The list of the independent variables (A, B, C) with their coded and actual levels are presented in Table 2. ´1: factor at low level; 0: factor at medium level; +1: factor at high level.
A second order polynomial equation was chosen to fit the experimental results.This model represents the effects of process variables (A, B, C) and their interactions on the response variables (Cr 2 O 3 grade and Cr 2 O 3 recovery).The general form of the model chosen is represented as following: where Stat-Ease Design Expert 9.0.3.1 software (Stat-Ease Inc., Minneapolis, MN, USA) was used to establish the validity of the models on the basis of analysis of variance (ANOVA).

The Model Equations and Statistical Evaluation
A total of 32 tests including five control experiments (center points) were conducted during the experimental study.The actual data collected from the tests were used to construct the empirical models representing concentrate grade and recovery as process responds to the variables.The results obtained from the tests are presented in Table 3.The models fit the data well and the differences between the observed values and the model's predicted values are relatively small and unbiased.The predicted values are in consistent with the experimental results.The coefficients of determination (R 2 ) obtained for the grade and recovery are 0.9852 and 0.9830 respectively, indicating that the regression is significant as illustrated in Figure 1.Table 3 also shows the predicted responses of the process to the variables considered.The actual model equation for grade (Y1) and recovery (Y2) of the chromite concentrates is given in Equations ( 2) and ( 3) respectively.
The models fit the data well and the differences between the observed values and the model's predicted values are relatively small and unbiased.The predicted values are in consistent with the experimental results.The coefficients of determination (R 2 ) obtained for the grade and recovery are 0.9852 and 0.9830 respectively, indicating that the regression is significant as illustrated in Figure 1.The significance test of model fit for grade and recovery of the concentrates were performed using Design Expert software version 9.0.3.1 based on analysis of variance (ANOVA).The results showed that the models are significant as the F value is high and the Prob > F value is less than 0.05 (Table 4).The standard deviations of the predicted models were found 0.62 and 1.0 for grade and recovery respectively.Lack of Fit F-values were found as 1.23 for grade and 0.36 for recovery models implying that the Lack of Fit is not significant relative to the pure error for both models.The predicted R 2 values of grade and recovery models (0.9714 and 0.9678 respectively) are in reasonable consistency with the adjusted R 2 values (0.9791 and 0.9761), the difference between these two values are less than 0.2 for both models.
The significance level of the parameters, the coefficient estimates and parameter interactions of the empirical models are tabulated in Table 5.Values of "Prob > F" less than 0.05 for the models suggest that the model terms are significant.The significance test of model fit for grade and recovery of the concentrates were performed using Design Expert software version 9.0.3.1 based on analysis of variance (ANOVA).The results showed that the models are significant as the F value is high and the Prob > F value is less than 0.05 (Table 4).The standard deviations of the predicted models were found 0.62 and 1.0 for grade and recovery respectively.Lack of Fit F-values were found as 1.23 for grade and 0.36 for recovery models implying that the Lack of Fit is not significant relative to the pure error for both models.The predicted R 2 values of grade and recovery models (0.9714 and 0.9678 respectively) are in reasonable consistency with the adjusted R 2 values (0.9791 and 0.9761), the difference between these two values are less than 0.2 for both models.
The significance level of the parameters, the coefficient estimates and parameter interactions of the empirical models are tabulated in Table 5.Values of "Prob > F" less than 0.05 for the models suggest that the model terms are significant.

The Effect of Process Variables on Grade of the Knelson Separator Concentrate
The estimated coefficient values calculated for main effects of the process variables from the Equation 2 (Table 5) indicate that fluidization water flow rate (A) has a positive effect and G force (B) has a negative effect on the Cr2O3 content of the concentrate.It was also observed that feed rate (C) has no significant effect on the grade of the concentrate in the range of the parameter levels tested.
The Equation 2 dictates that the interaction of variables, fluidization water flow rate (A)-G force (B) has a negative effect on the Cr2O3 content of the concentrate whereas interactions of fluidization water flow rate (A)-feed rate (C) and G force (B)-feed rate (C) have a positive effect.On the other hand, fluidization water flow rate (A) has the highest quadratic effect on the concentrate Cr2O3 grade.

The Effect of Process Variables on Grade of the Knelson Separator Concentrate
The estimated coefficient values calculated for main effects of the process variables from the Equation 2 (Table 5) indicate that fluidization water flow rate (A) has a positive effect and G force (B) has a negative effect on the Cr 2 O 3 content of the concentrate.It was also observed that feed rate (C) has no significant effect on the grade of the concentrate in the range of the parameter levels tested.
The Equation 2 The predicted response (Y 1 ) of the process to the independent variables are presented graphically in three dimensional (3D) space to help to visualize the shape of the response surface in Figures 3-5.
The effect of the G force and fluidization water flow rate on grade (Cr 2 O 3 %) of the concentrate at the center level of feed rate is shown in Figure 3.As can be seen from Figure 3, a higher grade concentrate is obtained at higher fluidization water flow rate and at lower G force.The grade of the concentrate tends to decrease as the applied G force increases.This can be explained by the fact that the cut-density of the separation is reduced with increasing G force which in turn allows the lower grade composite particles to enter the concentrate product and decrease the Cr 2 O 3 grade.The predicted response (Y1) of the process to the independent variables are presented graphically in three dimensional (3D) space to help to visualize the shape of the response surface in Figures 3-5.
The effect of the G force and fluidization water flow rate on grade (Cr2O3 %) of the concentrate at the center level of feed rate is shown in Figure 3.As can be seen from Figure 3, a higher grade concentrate is obtained at higher fluidization water flow rate and at lower G force.The grade of the concentrate tends to decrease as the applied G force increases.This can be explained by the fact that the cut-density of the separation is reduced with increasing G force which in turn allows the lower grade composite particles to enter the concentrate product and decrease the Cr2O3 grade.Figure 4 illustrates the effect of feed rate and fluidization water flow rate on grade (Cr2O3 %) of the concentrate at the center level of G force.It is seen from Figure 4 that there is not any significant effect of feed rate on the grade of the concentrate at a constant water flow rate in the range of the parameter levels tested.However, increasing the fluidization water flow rate leads to obtaining higher grade concentrate due to the increased number of light particles transported to the outside of separation cone.4 that there is not any significant effect of feed rate on the grade of the concentrate at a constant water flow rate in the range of the parameter levels tested.However, increasing the fluidization water flow rate leads to obtaining higher grade concentrate due to the increased number of light particles transported to the outside of separation cone.The predicted response (Y1) of the process to the independent variables are presented graphically in three dimensional (3D) space to help to visualize the shape of the response surface in Figures 3-5.
The effect of the G force and fluidization water flow rate on grade (Cr2O3 %) of the concentrate at the center level of feed rate is shown in Figure 3.As can be seen from Figure 3, a higher grade concentrate is obtained at higher fluidization water flow rate and at lower G force.The grade of the concentrate tends to decrease as the applied G force increases.This can be explained by the fact that the cut-density of the separation is reduced with increasing G force which in turn allows the lower grade composite particles to enter the concentrate product and decrease the Cr2O3 grade.Figure 4 illustrates the effect of feed rate and fluidization water flow rate on grade (Cr2O3 %) of the concentrate at the center level of G force.It is seen from Figure 4 that there is not any significant effect of feed rate on the grade of the concentrate at a constant water flow rate in the range of the parameter levels tested.However, increasing the fluidization water flow rate leads to obtaining higher grade concentrate due to the increased number of light particles transported to the outside of separation cone.5 that changing feed rate has no significant effect on the concentrate grade in the range of the parameter levels tested.On the other hand, the use of higher G force at a constant feed rate results in a decrease in concentrate Cr2O3 grade.

Effect of KC-MD3 Knelson Concentrator Variables on Concentrate Recovery
The estimated coefficient and Prob > F values obtained for main effects of the independent variables from Equation 3indicate that water flow rate (A) has a positive effect on the Cr2O3 recovery of the concentrate, whereas, G force (B) and feed rate (C) negatively affect the Cr2O3 recovery.
The interaction of variables fluidization water flow rate (A)-G force (B) has a significant positive effect on the Cr2O3 recovery of the concentrate, while interactions of fluidization water flow rate (A)feed rate (C) and G force (B)-feed rate (C) have a negative effect.It is also important to note that fluidization water flow rate (A) has a significant quadratic effect on the concentrate Cr2O3 recovery compared to the other independent variables.
Figure 6 illustrates the effect of G force and fluidization water flow rate on the recovery of Cr2O3 in the Knelson concentrate.Feed rate was kept constant at the center level during this test.At low level of fluidization water rate, as the G force increases, recovery of Cr2O3 decreases significantly due to inefficient fluidization of the separation layer inside the bowl.The highest recovery value was obtained by increasing fluidization water level to maximum for providing the appropriate conditions of fluidization.
Figure 7 shows the effect of fluidization water flow rate and feed rate on recovery of Cr2O3 in the concentrate of Knelson separator at the center level of G force.As can be seen from Figure 7, increasing fluidization water rate provides better fluidization conditions in the separation layer and helps to increase Cr2O3 recovery of concentrate.It should be noted that there is a significant decrease in the recovery of Cr2O3 at high feed rate when the fluidization water rate is low.Figure 8 demonstrates the effect of feed rate and G force on Cr2O3 recovery of the concentrate at the center level of fluidization water flow rate.As the G force decreases the transport of the lighter particles to the tailing fraction degreases which in increases recovery of Cr2O3 to the concentrate fraction.
Figure 8 was plotted representing the recovery of Cr2O3 concentrate vs. feed rate and G force.The Cr2O3 recovery of the concentrate increases with decrease in feed rate and G force.The lowest level of both independent variables appears to be providing favorable conditions for an efficient separation at this level of fluidization water flow.Increasing any of these variables to a higher level seems to change the fluidization conditions and lower the Cr2O3 recovery due to the disruption of the separation layer.Figure 7 shows the effect of fluidization water flow rate and feed rate on recovery of Cr 2 O 3 in the concentrate of Knelson separator at the center level of G force.As can be seen from Figure 7, increasing fluidization water rate provides better fluidization conditions in the separation layer and helps to increase Cr 2 O 3 recovery of concentrate.It should be noted that there is a significant decrease in the recovery of Cr 2 O 3 at high feed rate when the fluidization water rate is low.

Validation of the Models
For validation of the obtained models for predicting the grade and recovery of Cr 2 O 3 content in concentrate, nine additional tests were conducted using different levels of independent variables.The test conditions and results are presented in Table 6.The observed results and the predicted results for both responses are illustrated in Figure 9. Figure 9 shows that the predicted values are consistent with the observed values, with R 2 of 0.9254 percent for grade and R 2 of 0.9635 percent for recovery of chromite concentrates.

Validation of the Models
For validation of the obtained models for predicting the grade and recovery of Cr2O3 content in concentrate, nine additional tests were conducted using different levels of independent variables.The test conditions and results are presented in Table 6.The observed results and the predicted results for both responses are illustrated in Figure 9. Figure 9 shows that the predicted values are consistent with the observed values, with R 2 of 0.9254 percent for grade and R 2 of 0.9635 percent for recovery of chromite concentrates.

Optimization
The optimized independent variable levels for making the grade and recovery values maximum were also determined using the optimization module of Design Expert version 9.0.3.1.The numerical optimization option of the module was utilized by employing the developed models to search the factor space for the best trade-offs to achieve the highest grade and recovery in the range of the experimental data.The results showed that a concentrate assaying 43% Cr2O3 with 72% Cr2O3 recovery can be obtained by adjusting the variables to the following levels; fluidization water rate of 11 L/min, G force of 60 G, feed rate of 12 kg/h.

Conclusions
In this study, the effect of feed rate (kg/h), G force and fluidization water flow rate (L/min) on the chromite beneficiation process by KC-MD3 Knelson concentrator was investigated.Full factorial experimental design and response surface methodology were used to develop mathematical models for both grade and recovery of Cr2O3 concentrate.Mathematical model equations were derived using experimental data and mathematical software package Design Expert 9.0.3.1.It was found that the

Optimization
The optimized independent variable levels for making the grade and recovery values maximum were also determined using the optimization module of Design Expert version 9.0.3.1.The numerical optimization option of the module was utilized by employing the developed models to search the factor space for the best trade-offs to achieve the highest grade and recovery in the range of the experimental data.The results showed that a concentrate assaying 43% Cr 2 O 3 with 72% Cr 2 O 3 recovery can be obtained by adjusting the variables to the following levels; fluidization water rate of 11 L/min, G force of 60 G, feed rate of 12 kg/h.

Conclusions
In this study, the effect of feed rate (kg/h), G force and fluidization water flow rate (L/min) on the chromite beneficiation process by KC-MD3 Knelson concentrator was investigated.Full factorial experimental design and response surface methodology were used to develop mathematical models for both grade and recovery of Cr 2 O 3 concentrate.Mathematical model equations were derived using experimental data and mathematical software package Design Expert 9.0.3.1.It was found that the predicted values were in reasonable consistency with the experimental data which were also confirmed by the validation experiments.The ANOVA results indicated that fluidization water flow rate was the dominant independent variable affecting both grade and recovery of the concentrate.The results of the numerical optimization in the range of the experimental data showed that it is possible to produce a concentrate assaying 43% Cr 2 O 3 with 72% Cr 2 O 3 recovery by operating Knelson concentrator at the fluidization water rate of 11 L/min, G force of 60 G and feed rate of 12 kg/h.
, Y is the predicted response, b 0 is model constant; b 1 , b 2 and b 3 are linear coefficients; b 12 , b 13 and b 23 are cross product coefficients and b 11 , b 22 and b 33 are the quadratic coefficients.Statistical

Figure 2
Figure2shows graphically the residuals for predicted values of grade and recovery.As can be seen from these figures that the residual values are uniformly distributed.The significance test of model fit for grade and recovery of the concentrates were performed using Design Expert software version 9.0.3.1 based on analysis of variance (ANOVA).The results showed that the models are significant as the F value is high and the Prob > F value is less than 0.05 (Table4).The standard deviations of the predicted models were found 0.62 and 1.0 for grade and recovery respectively.Lack of Fit F-values were found as 1.23 for grade and 0.36 for recovery models implying that the Lack of Fit is not significant relative to the pure error for both models.The predicted R 2 values of grade and recovery models (0.9714 and 0.9678 respectively) are in reasonable consistency with the adjusted R 2 values (0.9791 and 0.9761), the difference between these two values are less than 0.2 for both models.The significance level of the parameters, the coefficient estimates and parameter interactions of the empirical models are tabulated in Table5.Values of "Prob > F" less than 0.05 for the models suggest that the model terms are significant.

Figure 2
Figure2shows graphically the residuals for predicted values of grade and recovery.As can be seen from these figures that the residual values are uniformly distributed.The significance test of model fit for grade and recovery of the concentrates were performed using Design Expert software version 9.0.3.1 based on analysis of variance (ANOVA).The results showed that the models are significant as the F value is high and the Prob > F value is less than 0.05 (Table4).The standard deviations of the predicted models were found 0.62 and 1.0 for grade and recovery respectively.Lack of Fit F-values were found as 1.23 for grade and 0.36 for recovery models implying that the Lack of Fit is not significant relative to the pure error for both models.The predicted R 2 values of grade and recovery models (0.9714 and 0.9678 respectively) are in reasonable consistency with the
dictates that the interaction of variables, fluidization water flow rate (A)-G force (B) has a negative effect on the Cr 2 O 3 content of the concentrate whereas interactions of fluidization water flow rate (A)-feed rate (C) and G force (B)-feed rate (C) have a positive effect.On the other hand, fluidization water flow rate (A) has the highest quadratic effect on the concentrate Cr 2 O 3 grade.

Figure 3 .
Figure 3.The response surface plot representing the effect of fluidization water flow rate and applied G force on the Cr2O3 content of the concentrate.

Figure 4 .
Figure 4.The response surface plot representing the effect of fluidization water flow rate and feed rate on the Cr2O3 content of the concentrate.

Figure 3 .
Figure 3.The response surface plot representing the effect of fluidization water flow rate and applied G force on the Cr 2 O 3 content of the concentrate.

Figure 4
Figure4illustrates the effect of feed rate and fluidization water flow rate on grade (Cr 2 O 3 %) of the concentrate at the center level of G force.It is seen from Figure4that there is not any significant effect of feed rate on the grade of the concentrate at a constant water flow rate in the range of the parameter levels tested.However, increasing the fluidization water flow rate leads to obtaining higher grade concentrate due to the increased number of light particles transported to the outside of separation cone.

Figure 3 .
Figure 3.The response surface plot representing the effect of fluidization water flow rate and applied G force on the Cr2O3 content of the concentrate.

Figure 4 .
Figure 4.The response surface plot representing the effect of fluidization water flow rate and feed rate on the Cr2O3 content of the concentrate.

Figure 4 .
Figure 4.The response surface plot representing the effect of fluidization water flow rate and feed rate on the Cr 2 O 3 content of the concentrate.

Figure 5
Figure 5 represents the effect of G force and feed rate on grade (Cr 2 O 3 %) of the Knelson concentrate at the center level of fluidization water flow rate.It is observed from Figure5that changing feed rate has no significant effect on the concentrate grade in the range of the parameter levels tested.On the other hand, the use of higher G force at a constant feed rate results in a decrease in concentrate Cr 2 O 3 grade.

Figure 5
Figure 5 represents the effect of G force and feed rate on grade (Cr2O3 %) of the Knelson concentrate at the center level of fluidization water flow rate.It is observed from Figure5that changing feed rate has no significant effect on the concentrate grade in the range of the parameter levels tested.On the other hand, the use of higher G force at a constant feed rate results in a decrease in concentrate Cr2O3 grade.

Figure 5 .
Figure 5.The response surface plot representing the effect of G force and feed rate on the Cr2O3 content of the concentrate.

Figure 5 .
Figure 5.The response surface plot representing the effect of G force and feed rate on the Cr 2 O 3 content of the concentrate.

3. 3 .
Effect of KC-MD3 Knelson Concentrator Variables on Concentrate Recovery The estimated coefficient and Prob > F values obtained for main effects of the independent variables from Equation 3 indicate that water flow rate (A) has a positive effect on the Cr 2 O 3 recovery of the concentrate, whereas, G force (B) and feed rate (C) negatively affect the Cr 2 O 3 recovery.The interaction of variables fluidization water flow rate (A)-G force (B) has a significant positive effect on the Cr 2 O 3 recovery of the concentrate, while interactions of fluidization water flow rate (A)-feed rate (C) and G force (B)-feed rate (C) have a negative effect.It is also important to note that fluidization water flow rate (A) has a significant quadratic effect on the concentrate Cr 2 O 3 recovery compared to the other independent variables.Figure 6 illustrates the effect of G force and fluidization water flow rate on the recovery of Cr 2 O 3 in the Knelson concentrate.Feed rate was kept constant at the center level during this test.At low level of fluidization water rate, as the G force increases, recovery of Cr 2 O 3 decreases significantly due to inefficient fluidization of the separation layer inside the bowl.The highest recovery value was obtained by increasing fluidization water level to maximum for providing the appropriate conditions of fluidization.

Figure 8
demonstrates the effect of feed rate and G force on Cr 2 O 3 recovery of the concentrate at the center level of fluidization water flow rate.As the G force decreases the transport of the lighter particles to the tailing fraction degreases which in increases recovery of Cr 2 O 3 to the concentrate fraction.

Figure 8 of 11 Figure 6 .
Figure 6.The response surface plot representing the effect of fluidization water flow rate and applied G force on the Cr2O3 recovery of the concentrate.

Figure 7 .
Figure 7.The response surface plot representing the effect of fluidization water flow rate and feed rate on the Cr2O3 recovery of the concentrate.

Figure 8 .
Figure 8.The response surface plot representing the effect of G force and feed rate on the Cr2O3 recovery of the concentrate.

Figure 6 .of 11 Figure 6 .
Figure 6.The response surface plot representing the effect of fluidization water flow rate and applied G force on the Cr 2 O 3 recovery of the concentrate.

Figure 7 .
Figure 7.The response surface plot representing the effect of fluidization water flow rate and feed rate on the Cr2O3 recovery of the concentrate.

Figure 8 .
Figure 8.The response surface plot representing the effect of G force and feed rate on the Cr2O3 recovery of the concentrate.

Figure 7 .of 11 Figure 6 .
Figure 7.The response surface plot representing the effect of fluidization water flow rate and feed rate on the Cr 2 O 3 recovery of the concentrate.

Figure 7 .
Figure 7.The response surface plot representing the effect of fluidization water flow rate and feed rate on the Cr2O3 recovery of the concentrate.

Figure 8 .
Figure 8.The response surface plot representing the effect of G force and feed rate on the Cr2O3 recovery of the concentrate.

Figure 8 .
Figure 8.The response surface plot representing the effect of G force and feed rate on the Cr 2 O 3 recovery of the concentrate.

Figure 9 .
Figure 9. Relationship between actual and predicted results of the validation tests.((a) Grade; (b) Recovery)

Figure 9 .
Figure 9. Relationship between actual and predicted results of the validation tests.((a) Grade; (b) Recovery)

Table 1 .
Size-wise chemical analysis results of chromite.

Table 2 .
Variables and levels for the three-level and three-factor full factorial design.

Table 3 .
Design matrix and the results of the three-level and three-factor factor full factorial design.

Table 4 .
Analysis of variance (ANOVA) table derived for the grade and recovery models.

Table 5 .
Estimated coefficient values for the parameter and parameter interaction effects.

Table 4 .
Analysis of variance (ANOVA) table derived for the grade and recovery models.

Table 5 .
Estimated coefficient values for the parameter and parameter interaction effects.

Table 6 .
The conditions and actual and predicted results of the validation tests.

Table 6 .
The conditions and actual and predicted results of the validation tests.