Statistical Experimental Design for the Extraction of Ti(IV) in the Ti(IV)-

The titled system has been investigated from the modelling point of view. The equilibration time for the system is 45 min, and 5% (v/v) heptane-1-ol in the organic phase is used as a de-emulsifier. The factors affecting the extent of extractions are [Ti(IV)], pH(eq), [Cyanex 301], [SO4 ] and temperature (T). The selected levels are high (+) and low (-) for these factors in the present investigation. Model equation for the extraction of Ti(IV) by Cyanex 301 is determined from 2 full factorial design. However, the success of the factorial design depends on the linear relationship between yield and factor. Plots of log D vs log [Ti(IV)], pH(eq), log [Cyanex 301] and log [SO4 ] are curves. Logistic functions involving these factors are considered in designing. While [Ti(IV)] < 1.00 g/L and [HA](o) > 0.10 mol/L, considered logistic functions viz. -log(1+316.2 ([Ti(IV)], mol/L)), -log (10 + 229 10 ), log ([HA](o), mol/L), -log (1+0.79 ([SO4 ], mol/L)) and absolute temperature are abbreviated as M, P, E, S and T, respectively. Model is log D = 5.847 + 0.964 M + 0.909 P + 2 E + 0.995 S – (1437.5/T). The experimental model illustrates that there is no interaction effect between the factors under investigation.


Introduction
In processing ilmenite (available in beach sand of south-eastern Bangladesh) for manufacturing pigment grade TiO 2 , it is required to extract Ti(IV) from the leach solution. In a review [1,2] it is concluded that the extraction of Ti(IV) from concentrated sulfuric acid medium by several organophosphorous extractants is the best achieved by using tri-octylphosphine oxide. However, the slow extraction kinetics, stripping difficulty and partial (~50% in a single stage) extraction are the disadvantages.
On the other hand, the extraction of Ti(IV) from Cl -/SO 4 2medium by organophosphorous extractants have been investigated by several workers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. These are mostly reviewed by Reddy and Saji [22] and Zhu et al. [23]. In last review, Zhu et al. [23] have also included a number of new methods to purify titanium in chloride solution by solvent extraction established by Lakshmanan et al. [24][25][26][27][28]; Willem et al. [29]; Duyvesteyn et al. [30]; Verhulst et al. [31] and Wang et al. [32]. The equilibrium of the liquid-liquid extraction of Ti(IV) from acidic sulfate medium by Cyanex 301 has been reported by Biswas and Karmakar [33]; where the extraction isotherms are provided and reported the value of log K ex as 1.117. In this paper, the same system is investigated from a modelling point of view. To model the system by factorial design, five logistic functions of the factors ([Ti(IV)], pH, [HA] (o) , [SO 4 2-] and T (absolute temperature)) have been considered. The investigated factor levels are at high (+) and low (-). This statistical design is used to verify the extraction data obtained from the thermodynamic view point [33]. This has been done to cross-check the extraction data obtained from thermodynamic view point [33] and also to optimize the factors.

Extraction procedure
The extraction process is given elsewhere [33,35]. Two phases at specified experimental parameters are stoppered in a quick fit reagent bottle. It is then agitated at O/A = 1 (O = 20 mL) for a predetermined time of 45 min. The phase separation after extraction is found to be quick; and on phase disengagement, the aqueous phase is analyzed for its equilibrium pH and Ti(IV)-content. Then the value of extraction or distribution ratio, D is calculated as follows:

Data treatment
The extraction isotherms, i.e. log D vs. pH and log D vs. log [HA] (o) plots, at a constant temperature, are supposed to be valid at a constant free [HA] (o) and a constant pH, respectively. In solvent extraction system, if the aqueous phase is not properly buffered (as usually the case; addition of foreign ions in the system not being practiced normally), the pH (eq)value is very much changed from the pH (eq) -value (usually decreased for liberation of H + due to extraction reaction), particularly when the latter value exceeds 1.
where the concentration terms are in mol/L, y = 2 for this case [33] and the last term within the second bracket represents [HA] (o, eq) . Similarly, the valid extraction isotherm in extractant dependence plot would be obtained when the experimental D-values are corrected to C D values at a constant pH (eq) or pH (ini) value on taking into consideration -the effect of pH variation (initial and equilibrium) on the value of experimental D value. Such a correction can be made by the equation given below (for value at constant pH (ini) ): where, x = 2 (pH dependence) as reported by Biswas and Karmakar [33], D is the experimental D-value for a pH (ini) -value which is changed to a pH (eq) -value and C D is the corrected D-value at a constant pH (ini) . When the correction for both parameters are required (as in the cases of [SO 4 2-] and temperature dependencies), the following combined equation can be used to get the corrected datum at a constant pH (ini) and [HA] (o,
The investigated system has been modelled by 2 5 factorial design which contains 32 trials. Subsequently, the 2 5 experimental model includes 32 trials and every trial run in twice. Hence, there will be 64 tests. At the middle point level of each factor, an extra test is repeated for four times. This additional trial is executed to analyze the inadequacy of fit due to curvature. The alteration of the average middle point value and the overall average value of the design points specify the severity of curvature.
Eqs. (8)(9)(10)(11)(12) [39,40] The estimation of variances for respective trial is then used in the calculation of a weighted average, i.e. the pooled variance of the individual variances for each trial.
Standard deviation pooled = √ [MIN] = t.s√ (11) [MINC] = t.s√ The student"s "t" table is used to find the "t" value (2.03) at 95% confidence level and 35 df (resulting from thirty-two (32) trials with two replicates and one trial with four replicates as df = 32(2-1)+1(4-1) = 35).  The studied experimental parameters of variables in the present system under investigation are displayed in Table 1. The coded form of the factors (2 5 experimental design) is given in the 3 rd to 7 th columns of  Table 2 represents the average values. The variance of two evaluations for individual trial is recorded in the end column of Table 2. Table 2 also stands for the mathematical investigation of the present experiment. The model matrix is accompanied with a computation matrix in this research. This process is used to observe any interaction result between the factors under investigation. A definite arithmetical multiplication of the coded factor levels utilized for the development of the computation matrix. In the test run 1, a and b are positive, respectively, therefore ab is positive. Similarly, in the test run 2, a is minus and b is plus, therefore ab is minus. The   (3 rd factor and interaction results are found in Table 2, as follows and given in Table 3. The sum of positive column (2 nd column of Table 3) accomplished by adding the column of Table 3) is achieved. The sum of these two columns value should be identical to the sum of all the average responses. It can be used as a check on arithmetic. The difference between the response values of the factor is at a high level (16 trails), and low level (16 trails), are displayed in the 4 th column of Table 3. The difference value is divided by the number of plus signs in the column to achieve the effect of the factors under investigation.
The system under investigation shows only the single factor effects with no other interaction effects. A first-order polynomial is used to express all the results as a numerical model. Table 3 displayed one-half values of the coefficient for the factor effects. Since the factor effects are created upon coded levels +1 and -1 that contrasted by two units. However, only factor effects are found in this investigation with no interaction effect. Therefore, the polynomial is: gives log K ex value of 1.072 at M = 0, P = 0, E = 0 and S = 0 which matches well with that (1.117) obtained from the factor-dependence studies [33]. The model can competently guess investigational log C D value which is displayed in Table 5, within a deviation of 0.20, on any set of observational parameters. The optimization of the factors to acquire more than 95% Ex of Ti(IV) are shown in Table 6. The % Ex of >95% Ex-values on five optimized settings have been tested by the shakeout investigation at the optimized situations. These resulting values are found to be similar.

Conclusion
In represents the equilibrium constant of the system at 303 K. Several conditions have been optimized for more than 95% extraction and at these conditions, the shake-out experiments yield % extractions which are very close to those predicted from the model.

Acknowledgment
Authors are grateful to Cytec Canada Inc. for providing Cyanex 301 as a gift. Organic phase (ini) Initial (eq) Equilibrium