- Research article
- Open Access
Optimizing Bi_{2}O_{3} and TiO_{2}to achieve the maximum non-linear electrical property of ZnO low voltage varistor
- Yadollah Abdollahi^{1}Email author,
- Azmi Zakaria^{1}Email author,
- Raba’ah Syahidah Aziz^{2},
- Siti Norazilah Ahmad Tamili^{2},
- Khamirul Amin Matori^{1},
- Nuraine Mariana Mohd Shahrani^{2},
- Nurhidayati Mohd Sidek^{2},
- Masoumeh Dorraj^{1} and
- Seyedehmaryam Moosavi^{2}
https://doi.org/10.1186/1752-153X-7-137
© Abdollahi et al.; licensee Chemistry Central Ltd. 2013
- Received: 19 May 2013
- Accepted: 6 August 2013
- Published: 10 August 2013
Abstract
Background
In fabrication of ZnO-based low voltage varistor, Bi_{2}O_{3} and TiO_{2} have been used as former and grain growth enhancer factors respectively. Therefore, the molar ratio of the factors is quit important in the fabrication. In this paper, modeling and optimization of Bi_{2}O_{3} and TiO_{2} was carried out by response surface methodology to achieve maximized electrical properties. The fabrication was planned by central composite design using two variables and one response. To obtain actual responses, the design was performed in laboratory by the conventional methods of ceramics fabrication. The actual responses were fitted into a valid second order algebraic polynomial equation. Then the quadratic model was suggested by response surface methodology. The model was validated by analysis of variance which provided several evidences such as high F-value (153.6), very low P-value (<0.0001), adjusted R-squared (0.985) and predicted R-squared (0.947). Moreover, the lack of fit was not significant which means the model was significant.
Results
The model tracked the optimum of the additives in the design by using three dimension surface plots. In the optimum condition, the molars ratio of Bi_{2}O_{3} and TiO_{2} were obtained in a surface area around 1.25 point that maximized the nonlinear coefficient around 20 point. Moreover, the model predicted the optimum amount of the additives in desirable condition. In this case, the condition included minimum standard error (0.35) and maximum nonlinearity (20.03), while molar ratio of Bi_{2}O_{3} (1.24 mol%) and TiO_{2} (1.27 mol%) was in range. The condition as a solution was tested by further experiments for confirmation. As the experimental results showed, the obtained value of the non-linearity, 21.6, was quite close to the predicted model.
Conclusion
Response surface methodology has been successful for modeling and optimizing the additives such as Bi_{2}O_{3} and TiO_{2} of ZnO-based low voltage varistor to achieve maximized non-linearity properties.
Keywords
- Optimization
- ZnO-varistor
- Modeling
- RSM
- Bi_{2}O_{3}
- TiO_{2}
Background
Varistors are nonlinear electro-devices with a ceramics microstructure that are used as protectors in distribution and energy transmission lines against voltage surge [1]. In the past four decades, varistors based on ZnO and SnO_{2} have attracted attention because of their excellent non-ohmic behavior and low leakage currents [2, 3]. However, ZnO-based varistor has been demanded along with the development of very-large-scale integration electronics because it exhibits high nonlinear current–voltage (I-V) characteristics in lower voltage ranges [4, 5]. The non-linearity is expressed by I = KV^{α} where K is a constant, and 'α’ is nonlinear coefficient (alpha) [6]. The alpha originates from microstructure of the varistor ceramics which is composed of conductive n-type ZnO grains and small amount of a few metal oxide additives such as Bi_{2}O_{3}, TiO_{2}, Co_{3}O_{4}, Mn_{2}O_{3}, Sb_{2}O_{3} and Al_{2}O_{3}. The microstructure is made of ZnO grain surrounded by the melted additives as boundaries [4]. The boundaries contain of Bi-rich intergranular, metal oxide and secondary spinel phase and strictly influences on the alpha [4, 6–11]. The role of Bi_{2}O_{3}, as a former, is quite important since it provides the medium for liquid-phase sintering, enhances the growth of ZnO grains, and finally stables the nonlinear current–voltage characteristics of the varistor [12]. High sintering temperature is necessary for ZnO grain growth despite the fact that at this condition Bi_{2}O_{3} tends to evaporate [13]. The melting point of Bi_{2}O_{3} is 825°C, and the eutectic temperature of ZnO-Bi_{2}O_{3} is only 740°C, thus a liquid phase is formed in the ZnO-Bi_{2}O_{3} specimens below 800°C. As soon as the eutectic liquid is formed, the mass loss starts to increase which indicates the vaporization of Bi_{2}O_{3}[14]. The sharp lost weight was reported above 1100°C since there was no reported peaks of β-Bi_{2}O_{3} at 1300°C [15, 16]. On the other hand, TiO_{2} increases reactivity of the Bi_{2}O_{3}-rich liquid phase with the solid ZnO during sintering process which prevents Bi_{2}O_{3} vaporization [13, 17–19]. The phase equilibrium and the temperature of liquid-phase formation are defined by the TiO_{2}/Bi_{2}O_{3} ratio [20]. According to the reports, the effect of TiO_{2} depends on Bi_{2}O_{3} that means the additives interact in low-voltage varistor ceramics fabrication. To determine the effect of the interactions on the varistors’ electrical properties, the molar ratios of the additives must simultaneously be considered. To the best of our knowledge, there is no study on the interactions which optimize the ratio of the additives and maximize the alpha. Recently, response surface methodology (RSM) has been accepted for modeling and optimizing of input intractable variables to achieve maximum yield product as output for productive process [21]. RSM is known as a semi-empirical method because the process could be optimized by using experimental results, and a group of mathematical and statistical techniques [22]. In this work, RSM was used for modeling and optimizing of molar ratio of Bi_{2}O_{3} and TiO_{2} as additives to achieve the maximum value of the alpha for low voltage varistor. The experiments were designed by central composite design (CCD) to obtain the empirical results (actual). The results were used for regression and fitting process to fine an appropriate model. The model was verified by several statistical techniques such as residual analysis, scaling residuals and prediction error sum of squares (PRESS). The model optimized the input additives and then maximized the alpha as output. In addition, the model predicted the desirable condition including minimum standard error and the maximum alpha which are validated by further experiments. The predicted samples were characterized by X-ray diffractometer (XRD), scanning electron microscope (SEM), variable pressure scanning electron microscope (VPSEM) and Energy-dispersive X-ray (EDX).
Experimental
Materials and methods
Experimental-design contain of the actual variables, and actual response and model predicted values of the alpha
Run | ZnO | TiO_{2} | Bi_{2}O_{3} | Co_{3}O_{4} | Mn_{2}O_{3} | Sb_{2}O_{3} | Al(NO_{3})_{3} | Alpha (Actual) | Alpha (Predicted) |
---|---|---|---|---|---|---|---|---|---|
1 | 96.50 | 1 | 1 | 0.5 | 0.5 | 0.5 | 0.00094 | 9.3 | 10.1 |
2 | 96.00 | 1.5 | 1 | 0.5 | 0.5 | 0.5 | 0.00094 | 3.9 | 3.9 |
3 | 96.00 | 1 | 1.5 | 0.5 | 0.5 | 0.5 | 0.00094 | 6.4 | 7.4 |
4 | 95.50 | 1.5 | 1.5 | 0.5 | 0.5 | 0.5 | 0.00094 | 10.5 | 10.6 |
5 | 96.35 | 0.896 | 1.25 | 0.5 | 0.5 | 0.5 | 0.00094 | 9 | 7.9 |
6 | 95.65 | 1.604 | 1.25 | 0.5 | 0.5 | 0.5 | 0.00094 | 5.6 | 5.8 |
7 | 96.35 | 1.25 | 0.896 | 0.5 | 0.5 | 0.5 | 0.00094 | 8.2 | 7.7 |
8 | 95.65 | 1.25 | 1.604 | 0.5 | 0.5 | 0.5 | 0.00094 | 11 | 10.5 |
9 | 96.00 | 1.25 | 1.25 | 0.5 | 0.5 | 0.5 | 0.00094 | 20 | 20 |
10 | 96.00 | 1.25 | 1.25 | 0.5 | 0.5 | 0.5 | 0.00094 | 19.6 | 20 |
11 | 96.00 | 1.25 | 1.25 | 0.5 | 0.5 | 0.5 | 0.00094 | 20.2 | 20 |
12 | 96.00 | 1.25 | 1.25 | 0.5 | 0.5 | 0.5 | 0.00094 | 20.7 | 20 |
13 | 96.00 | 1.25 | 1.25 | 0.5 | 0.5 | 0.5 | 0.00094 | 19.4 | 20 |
The breakdown voltage (E_{b}) was determined by measuring E at J = 0.75 mA/cm^{2} and the leakage current (J_{L}) was determined evaluating J at 0.8E_{b} where J (mA/cm^{2}) is the current density and E is the electrical field (V/mm). To characterize the microstructure, the both surfaces of samples were polished by aluminum oxide powder. Then, they were etched at 160°C under sintering time with heating and cooling rate, 10°C/min. Phase analysis was conducted using XRD (PANalytical, Philips-X’pert Pro PW3040/60) with CuKα source. The sample were radiated with Ni-filtered CuKα radiation (λ = 1.5428) within the 2θ scan range of 20–80°. Surface morphology and elemental analyses of sintered samples were studied under SEM (JEOL JSM 6400) and VPSEM (LEO 1455) which attached to EDX. The samples was mounted on Al stub using carbon paint and coated by gold layer. Average grains size of the ZnO in the varistor was evaluated by measuring 100 grains in SEMs images.
Experimental design
The variables and employed levels in the CCD for ZnO low voltage varistor fabrication
Level of variables | ||||
---|---|---|---|---|
Symbol | Unit (%) | Low (-1) | Middle (0) | High (+1) |
Bi_{2}O_{3} (x_{1}) | mol | 1.0 | 1.25 | 1.5 |
TiO_{2} (x_{2}) | mol | 1.0 | 1.25 | 1.5 |
The RSM description
where Y_{1i} is the experimental single response, x_{1} and x_{2} are the coded factors (Table 2), β_{0} is the intercept term, β_{1} and β_{2} are slopes with respect to each of the two factors, β_{11} and β_{22} are curvature terms, and β_{12} is the interaction term. To estimate the β’s, the fitting process provides the sufficient data by regression tools [33, 34]. In the process, the actual responses are fitted to the polynomial models by sequential model sum of squares (SMSS) [33, 34]. The SMSS compares the linear, two-factor interaction (2FI), quadratic and cubic models by using the statistical significance of adding new model terms, step by step in increasing order [35]. To select the provisional model, the lack of fit of those models is compared by minimum p-values and PRESS. The other assessments to select the best provisional model are maximum adjusted R-squared (R_{Adj}) and predicted R-squared (R_{Pred}) [36]. The p-value is one of the most important evidences which was used to study significant effect of the parameters [33]. In addition, the lack-of-fit test diagnoses how well each terms of the full model fit the data that pillared by statistical parameters such as R_{Adj}, R_{Pred} and PRESS [36, 37]. Therefore, the provisional model with minimum p-value and PRESS and also maximum R_{Adj}, R_{Pred} is selected to investigate in details. The details are provided by using analysis of variance (ANOVA) which contains a collection of terms statistical evidences. The ANOVA determines the significance of intercept, linear, interaction and square terms of the provisional model by using minimum p-value. For more evaluation, the normality of residuals, constant error and residual outlier is checked by various diagnostic plots [38]. The validated model, the relationship between variables and response, is created in coded and actual variables. The model indicates the effect of linear, quadratic and the parameters interactions on the interested response. The effects are presented by estimated coefficients and the related positive and negative signs (+, -). The coefficients are specific weight of the parameters in the model while the signs (+) and (-) operate as synergistic and antagonistic effects on the response [39]. Then the optimization process investigates combination of variables levels that produces the maximum response to a surface area simultaneously. Moreover, the model predicts the yield of product in specific condition such as individual standard error, the range of variables and responses. The prediction could be performed by further experiments.
Results and discussion
Modeling
The sequential model fitting summary for the actual responses which shows statistics conformation of the regression process, DF is degree of freedom
Source | Sum of squares | DF | Mean square | F-Value | p-value | Remark |
---|---|---|---|---|---|---|
Sequential model sum of squares | ||||||
Mean vs Total | 2064.18 | 1 | 2064.18 | - | - | |
Linear vs Mean | 12.07 | 2 | 6.04 | 0.13 | 0.8818 | |
2FI vs Linear | 22.31 | 1 | 22.31 | 0.44 | 0.5216 | |
Quadratic vs 2FI | 447.11 | 2 | 223.55 | 356.58 | < 0.0001 | Suggested |
Cubic vs Quadratic | 1.56 | 2 | 0.78 | 1.38 | 0.3325 | Aliased |
Residual | 2.83 | 5 | 0.57 | - | - | |
Total | 2550.06 | 13 | 196.16 | - | - | |
Source | Sum of squares | DF | Mean square | F-Value | p-value | Remark |
Lack of fit tests | ||||||
Linear | 472.78 | 6 | 78.80 | 307.73 | < 0.0001 | |
2FI | 450.47 | 5 | 90.09 | 351.85 | < 0.0001 | |
Quadratic | 3.36 | 3 | 1.12 | 4.38 | 0.0938 | Suggested |
Cubic | 1.80 | 1 | 1.80 | 7.03 | 0.0569 | Aliased |
Pure Error | 1.02 | 4 | 0.26 | - | - | |
Source | Std.Dev. | R _{ Adj } | R _{ Pred } | R | PRESS | Remark |
Model summary statistics | ||||||
Linear | 6.88 | 0.025 | -0.170 | -0.571 | 763.55 | |
2FI | 7.08 | 0.071 | -0.239 | -1.195 | 1066.45 | |
Quadratic | 0.79 | 0.991 | 0.985 | 0.947 | 25.53 | Suggested |
Cubic | 0.75 | 0.994 | 0.986 | 0.760 | 116.85 | Aliased |
Analysis of variance (ANOVA) for response surface quadratic model, MS is mean Square, DF is degree of freedom and SS is sum of squares while x_{ 1 } and x_{ 2 } introduce in Table 2
Source | SS | DF | MS | F-Value | p-value (Prob > F) | Suggestion |
---|---|---|---|---|---|---|
Model | 481.5 | 5 | 96.3 | 153.6 | < 0.0001 | significant |
x_{1} | 4.6 | 1 | 4.6 | 7.4 | 0.0299 | |
x_{2} | 7.4 | 1 | 7.4 | 11.9 | 0.0108 | |
x_{1}x_{2} | 22.3 | 1 | 22.3 | 35.6 | 0.0006 | |
x_{1}^{2} | 298.7 | 1 | 298.7 | 476.4 | < 0.0001 | |
x_{2}^{2} | 205.5 | 1 | 205.5 | 327.7 | < 0.0001 | |
Residual | 4.4 | 7 | 0.6 | |||
Lack of Fit | 3.4 | 3 | 1.1 | 4.4 | 0.0938 | not significant |
Pure Error | 1.0 | 4 | 0.3 | < 0.0001 | ||
Cor Total | 485.9 | 12 | 0.0299 | |||
Std.Dev. | R _{ Adj } | R _{ Pred } | R ^{ 2 } | PRESS | C.V.% | Adeq Precision |
0.792 | 0.985 | 0.947 | 0.991 | 25.5 | 6.284 | 29.9 |
The model presentation
where the actual values of the variables x_{1} and x_{2} were shown in Table 2 and Y is the alpha. As shown, the parameters including linear (x_{1}, x_{2}), quadratic (x_{1}^{2}, x_{2}^{2}) and interaction (x_{1}x_{2}) affected on the interested response. The effects are presented by the individual coefficients and the related signs (+, -) in the model. The coefficients indicate the specific weight of the parameters in the model while the signs are synergistic (+) and antagonistic (-) effects of variables on the response (Y). The weights determine the importance of the parameters roles in the modeling. The model is able to optimize input variables and also approximately predict the response inside of the actual experimental region that confirmed by minimum standard error (Figure 1) [33]. Therefore, the model was used to optimize the molar ratio of Bi_{2}O_{3} and TiO_{2} to achieve maximized alpha.
The model optimization
The summary of optimized input variables and obtained maximized alpha% by canonical, graphical, numerical methods and validation value of the photodegradation
Method | Bi_{2}O_{3} (%mol) | TiO_{2} (%mol) | Alpha |
---|---|---|---|
Canonical (point) | 1.195 | 1.025 | 15.03 |
Graphical (area) | Area around 1.25 | Area around 1.25 | 20.03 |
Numerical (prediction) | 1.24 | 1.27 | 20.03 |
Validated sample | 1.24 | 1.27 | 21.6 |
The validated varistor
Conclusions
This work reports modeling and optimization of the molar ratio of Bi_{2}O_{3} and TiO_{2} by RSM. The fabrication was designed by CCD using two variables and a response. To obtain actual responses, the design was performed in laboratory by conventional fabrication methods. The actual responses were fitted into a quadratic model. The model was validated by ANOVA which provided evidences such as high F-value (153.6), very low p-value (<0.0001), R_{adj} (0.985) and R_{Pred} (0.947). The results of the validation showed the model was significant. The model tracked the optimum of the designed additives by using 3D plots. In the optimum condition, the molars ratio of Bi_{2}O_{3} and TiO_{2} were around 1.25 that maximized the alpha value at 20. Moreover, the model suggested a solution to predict the optimum amount of the additives. In this case, the condition of the solution included standard error of 0.35, Bi_{2}O_{3} of 1.24, TiO_{2} of 1.27 and alpha of 20.03. The solution was tested by further experiments. As the validation test showed, the obtained value of the alpha (21.6) was very close to the predicted value (20.03). Therefore, RSM was succeeded in modeling of the additives in fabrication of zinc oxide based low voltage varistor to achieve maximum alpha.
Declarations
Acknowledgements
The author would like to express acknowledgement to Ministry of Higher Education Malaysia for granting this project under Research University Grant Scheme (RUGS) of Project No. 05-02-12-1878. The authors wish to thank Dr Pedram Lalbakhsh for polishing and proofreading the manuscript.
Authors’ Affiliations
References
- Ramírez M, Bueno P, Ribeiro W, Varela J, Bonett D, Villa J, Márquez M, Rojo C: The failure analyses on ZnO varistors used in high tension devices. J Mater Sci. 2005, 40: 5591-5596. 10.1007/s10853-005-1366-4.View ArticleGoogle Scholar
- Ramirez M, Bassi W, Bueno P, Longo E, Varela J: Comparative degradation of ZnO-and SnO2-based polycrystalline non-ohmic devices by current pulse stress. J Phys D Appl Phys. 2008, 41: 122002-122002. 10.1088/0022-3727/41/12/122002.View ArticleGoogle Scholar
- Ramírez MA, Bassi W, Parra R, Bueno PR, Longo E, Varela JA: Comparative electrical behavior at low and high current of SnO_{2} and ZnO based varistors. J Am Ceram Soc. 2008, 91: 2402-2404. 10.1111/j.1551-2916.2008.02436.x.View ArticleGoogle Scholar
- Levinson L, Philipp H: Zinc oxide varistors–a review. Am Ceram Soc Bull. 1986, 65: 639-Google Scholar
- Eda K: Zinc oxide varistors. IEEE Electr Insul Mag. 2002, 5: 28-30.View ArticleGoogle Scholar
- Bernik S, Zupancic P, Kolar D: Influence of Bi_{2}O_{3}/TiO_{2}/Sb_{2}O_{3} and Cr_{2}O_{3} doping on low-voltage varistor ceramics. J Eur Ceram Soc. 1999, 19: 709-713. 10.1016/S0955-2219(98)00301-X.View ArticleGoogle Scholar
- Suzuki H, Bradt R: Grain growth of ZnO in ZnO-Bi_{2}O_{3} ceramics with TiO_{2} addition. J Am Ceram Soc. 1995, 78: 1354-1360. 10.1111/j.1151-2916.1995.tb08494.x.View ArticleGoogle Scholar
- Clarke DR: Varistor ceramics. J Am Ceram Soc. 1999, 82: 485-502.View ArticleGoogle Scholar
- Pillai SC, Kelly JM, McCormack DE, O'Brien P, Ramesh R: The effect of processing conditions on varistors prepared from nanocrystalline ZnO. J Mater Chem. 2003, 13: 2586-2590. 10.1039/b306280e.View ArticleGoogle Scholar
- Elfwing M, Österlund R, Olsson E: Differences in wetting characteristics of Bi_{2}O_{3} polymorphs in ZnO varistor materials. J Am Ceram Soc. 2000, 83: 2311-2314.View ArticleGoogle Scholar
- Lao Y-W, Kuo S-T, Tuan W-H: Effect of Bi_{2}O_{3} and Sb_{2}O_{3} on the grain size distribution of ZnO. J Electroceram. 2007, 19: 187-194. 10.1007/s10832-007-9187-2.View ArticleGoogle Scholar
- Peiteado M, Fernández Lozano JF, Caballero AC: Varistors based in the ZnO-Bi_{2}O_{3} system: microstructure control and properties. J Eur Ceram Soc. 2007, 27: 3867-3872. 10.1016/j.jeurceramsoc.2007.02.046.View ArticleGoogle Scholar
- Peiteado M, De la Rubia M, Velasco M, Valle F, Caballero A: Bi_{2}O_{3} vaporization from ZnO-based varistors. J Eur Ceram Soc. 2005, 25: 1675-1680. 10.1016/j.jeurceramsoc.2004.06.006.View ArticleGoogle Scholar
- Metz R, Delalu H, Vignalou J, Achard N, Elkhatib M: Electrical properties of varistors in relation to their true bismuth composition after sintering. Mater Chem Phys. 2000, 63: 157-162. 10.1016/S0254-0584(99)00227-8.View ArticleGoogle Scholar
- Wong J: Sintering and varistor characteristics of ZnO Bi_{2}O_{3} ceramics. J Appl Phys. 1980, 51: 4453-4459. 10.1063/1.328266.View ArticleGoogle Scholar
- Yilmaz S, Ercenk E, Toplan HO, Gunay V: Grain growth kinetic in x TiO 2–6 wt.% Bi 2 O 3–(94- x) ZnO (x = 0, 2, 4) ceramic system. J Mater Sci. 2007, 42: 5188-5195. 10.1007/s10853-006-0581-y.View ArticleGoogle Scholar
- Toplan HÖ, Karakaş Y: Processing and phase evolution in low voltage varistor prepared by chemical processing. Ceram Int. 2001, 27: 761-765. 10.1016/S0272-8842(01)00027-X.View ArticleGoogle Scholar
- Yan MF, Heuer AH: Additives and interfaces in electronic ceramics. 1983, Columbus, OH: Amer Ceramic SocietyGoogle Scholar
- Peigney A, Andrianjatovo H, Legros R, Rousset A: Influence of chemical composition on sintering of bismuth-titanium-doped zinc oxide. J Mater Sci. 1992, 27: 2397-2405. 10.1007/BF01105049.View ArticleGoogle Scholar
- Bernik S, Daneu N, Rečnik A: Inversion boundary induced grain growth in TiO_{2} or Sb_{2}O_{3} doped ZnO-based varistor ceramics. J Eur Ceram Soc. 2004, 24: 3703-3708. 10.1016/j.jeurceramsoc.2004.03.004.View ArticleGoogle Scholar
- Montgomery DC: Design and analysis of experiments. 2008, New York: John Wiley & Sons IncGoogle Scholar
- Khuri AI, Cornell JA: Response surfaces: designs and analyses. 1996, New York: CRCGoogle Scholar
- Bernik S: Microstructural and electrical characteristics of ZnO based varistor ceramics with varying TiO_{2}/Bi_{2}O_{3} ratio. Advances in science and technology. 1999, 151-158.Google Scholar
- Williges RC, Clark C: Illinois Univ Savoy Aviation Research Lab. Response surface methodology design variants useful in human performance research. 1971Google Scholar
- Myers RH, Anderson-Cook CM: Response surface methodology: process and product optimization using designed experiments. 2009, New Jersey: John Wiley and Sons, IncGoogle Scholar
- Clark C, Williges RC: DTIC Document, New York. Response surface methodology design variants useful in human performance research. 1971Google Scholar
- Deming S: Quality by design-part 5. Chemtech. 1990, 20: 118-120.Google Scholar
- Hader R, Park SH: Slope-rotatable central composite designs. Technometrics. 1978, 20: 413-417. 10.1080/00401706.1978.10489695.View ArticleGoogle Scholar
- Palasota JA, Deming SN: Central composite experimental designs: applied to chemical systems. J Chem Educ. 1992, 69: 560-10.1021/ed069p560.View ArticleGoogle Scholar
- Rubin IB, Mitchell TJ, Goldstein G: Program of statistical designs for optimizing specific transfer ribonucleic acid assay conditions. Anal Chem. 1971, 43: 717-721. 10.1021/ac60301a003.View ArticleGoogle Scholar
- Baş D, Boyacı İH: Modeling and optimization I: usability of response surface methodology. J Food Eng. 2007, 78: 836-845. 10.1016/j.jfoodeng.2005.11.024.View ArticleGoogle Scholar
- Khuri AI, Mukhopadhyay S: Response surface methodology. Wiley Interdiscip Rev Comput Stat. 2010, 2: 128-149. 10.1002/wics.73.View ArticleGoogle Scholar
- Anderson MJ, Whitcomb PJ: RSM simplified: optimizing processes using response surface methods for design of experiments. 2005, Virginia: Productivity PrGoogle Scholar
- Oehlert GW: A first course in design and analysis of experiments. 2000, New York: WH FreemanGoogle Scholar
- Deming S: Quality by design-part 3. Chemtech. 1989, 19: 249-255.Google Scholar
- Whitcomb PJ, Anderson MJ: RSM simplified: optimizing processes using response surface methods for design of experiments. 2004, New York: Productivity PressGoogle Scholar
- Allen DM: The relationship between variable selection and data agumentation and a method for prediction. Technometrics. 1974, 16: 125-127. 10.1080/00401706.1974.10489157.View ArticleGoogle Scholar
- Stuart A, Ord JK: Kendall’s advanced theory of statistics: vol 1. Distribution theory. 1994, London: ArnoldGoogle Scholar
- Abdollahi Y, Zakaria A, Abdullah AH, Masoumi HRF, Jahangirian H, Shameli K, Rezayi M, Banerjee S, Abdollahi T: Semi-empirical study of ortho-cresol photo degradation in manganese-doped zinc oxide nanoparticles suspensions. Chem Cent J. 2012, 6: 1-8. 10.1186/1752-153X-6-1.View ArticleGoogle Scholar
- Box GEP, Wilson K: On the experimental attainment of optimum conditions. J Roy Stat Soc B. 1951, 13: 1-45.Google Scholar
- Stuart A, Ord JK, Arnold S: Kendall’s advanced theory of statistics: vol 2A. Classical inference and the linear model. 1999, London: ArnoldGoogle Scholar
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