Central Composite Experimental Designs Applied to Chemical Systems John A. Paiasota and Stanley N. ~ e m i n g ' University of Houston, 4800 Calhoun Road, Houston, TX 77204-5641 MultifactorSystems in Chemistry In science there has always been a strong belief that "if you want to find out how one fador influences a response, you must hold all other factors constant and vary only that one factor." As a result, much research is still carried out using the single-fador-at-a-time method ( 1 ) . For chemical systems that have more than one independent variable or factor (multifactor systems), the singlefactor-at-a-time method often leads to an incomplete understanding of the behavior of the system, resulting in codusion and a lack of predictive ability. Some of this confusion may be avoided with the application of properly designed experiments and adequate multifactor models. Response Surfaces Figures 1-3 show typical response surfaces that are frequently encountered in chemistry A response surface is a plot of the system response (e.g., percent yield of a reaction) versus each of the factors or variables that have an influence on the response (e.g., temperature and pressure)
(2, 3). A Maximum For Figure 1, suppose that the following factors are represented by the following variables. The response yl is the percent yield. The fador xlis the reaction temperature. The fador xz is the reaction pressure.
Figure 2. Response surfaceforthe full No-factor second-order polynomial model. y2= 96.65 - 1 . 6 8 3 ~ -~ 2.183~ + ~0.01250(x~)~ + O.O1750(x~)~ +0.008660~~~~ Then at a reaction temperature of 50 'C and a reaction pressure of 50 atmospheres, the percent yield is 100%. At any other temperature and pressure, the percent yield is less than 100%.
A Minimum Chemical systems often have multiple responses. For example, in the reaction described above, both high percent yield and low percent impurity might be desirable. Figure 2 might show a response surface of percent impurity (response y2) versus reaction temperature (factor x , ) and reaction pressure (factor xz). At a reaction temperature of 50 'C and a reaction pressure of 50 atmospheres in Figure 2, the percent impurity is a minimum. At any other temperature and pressure, the percent impurity is greater than the minimum. Thus, if the reaction is run a t 50 'C and 50 atmospheres, the percent yield (Fig. 1)will be a maximum, while the percent impurity (Fig. 2) will be a minimum.
A Saddle Surface Figure 1. Response sulface for the full two-factor second-order polynomial model. yl = 3.349 + 1 . 6 8 3 ~ + ~2 . 1 8 3 -~ 0.01250(x~)~ ~ - O.O1750(x~)~ -0.008660~~~~
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Figure 3 illustrates a third response surface often encountered in chemistry: a saddle surface (4). Let the re'Authorto whom correspondence should be addressed. (713) 7432809.
0
I
I 5
0
15
10
20
25
50
Drops HZ02
Figure 4. Atwo-factorcentral composite design.
Figure 3. Response surfacefor the full two-factorsewnd-order polynomial model.
The Central Composite Design
yl = 93.30 - 0 . 3 6 6 0 ~-~1 . 3 6 6 ~- ~0.005000(x,)~
+ 0.005000(x~~ +0 . 0 1 7 3 2 ~ ~ ~ ~ sponse yl be the percent yield for a different reaction as a function of temperature (factor xl) and pressure (factor XZ). As the experimental wnditions change from a temperature of 50 'C and a pressure of 50 atmospheres, the percent yield either increases or decreases depending on the direction. Increasing or decreasing both temperature and pressure causes the percent yield to increase. Increasing one factor while decreasing the other factor causes the percent yield to decrease. The Polynomial Model
A full second-order polynomial empirical model is often adequate for describing a wide variety of multifactor chemical systems such as those shown inFigures 1-3. For a twofactor system the model is Yli =
Po + P+li + P&i
+ ~lloCl$~ + P Z ( ~ d+ 2 Pli(l82i
A significant interaction between factor xl and fador xz can be seen in Figure 3. When xl = 0, as xz increases, the response decreases. However, when xl = 100, a s x2 increases, the response increases. Thus, the response surface has the appearance of a twisted plane.
+ 'li
(1)
where yli is the single response in the ith experiment (e.g., absorbance):XI and xl are the factors or ex~erimentalvariables (e.g., G e n t r a t i o n of reactant A k d wncentration of reactant B); p, is the intercept term; pl and pz are slopes with respect to each of the two factors; pll and bz are curvature terms; and plz is the interaction term. The interaction term is a measure of how much the slope, with respect to one factor, changes as the other factor increases or decreases (56).The full second-orderpolynomial model usually provides a good approximation of the true behavior of a given system over a modest factor domain (2, 5, 7). Interpretation of Terns in the Model
Assuming that the model of eq 1will adequately describe the behavior of a two-factor system, it is necessary to choose an experimental design that will provide sufficient data to estimate the !3's Box and Wilson (8)introduced the central composite design in 1951. A central wmposite design (Fig. 4) wnsists of a two-level full factorial design (triangles) superimposed on-a star design (dots). The centers of the two designs coincide (9,10). This design allows the estimation of the intercept, slope, curvature, and interaction terms in the model of eq 1.In a coded factor space, in which the factorial points are f1 from the center, the star points are usually located a distance a = zki4
from the center, where k is the number of fadors (2). A wded central wmposite design (11) for a system with two factors (k = 2) is listed in the table. The two factors (XI and x2)might be pH and viscosity, temperature and pressure, or volume of reactant A and volume of reactant B. The coded values in the design matrix represent the following values that the two factors take for this particular design in a specified region of factor space.
..
the lowest value: the low value: -1 the middle value: 0 the high value: +1 the highest value: +a
Coded Central Composite Design for a system with two factors ( k = 2).
Design Point
Factor XI
Factor rn
Figures 1-3 show that changes in the Po term cause changes in the intercept (the response at zero xl and zero XZ). Likewise, as the magnitude of the curvature terms (Pll and &2) increases and decreases, the parabolic shape of the response surface increases and decreases. When the signs of pll and pzz are both negative, the parabola opens downward (Fig. 1); when the signs of Pl1 and Pzz are both positive, the parabola opens upward (Fig. 2). Volume 69 Number 7 July 1992
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The uncoded values may be calculated from the following relationship.
where xl, is the uncoded value, xi, is the coded value, cl is the center point value expressed in real units, and d l is the length expressed in real units between the center point and the +1 value ofthe factor. The values of el and dl must be such that the experimental design lies within any real boundaries. Although the central composite design specifies only nine factor combinations, three additional center-point replicates are usually performed to determine the reproducibility of the system. Regression-analysis computer programs using matrix least squares are available for calculating the P values (the parameters). Details on matrix least squares can be foundin the literature (2,3,6, 12-15). To demonstrate the use of a central composite design, the absorbance resvonse from a constant volume of a solution of vanadyl $ul& (VOS04)can be investigated as a function of the volume in drops of hydrogen peroxide (HzOz) and sulfuric acid (HzS04)that are added (16,171. The first factor xlis the number of drops of 1%Hz02 The second factor xz is the number of drops of 20% HzS04. The measured responseyl is the absorbance. The experimental factor space can be bounded between 0 drops and 30 drops of Hz02 and HzSOa. When using cl = 15, dl = 5, and eq 2, the following equations give the real (uncoded)values chosen for both factors. lowest (4= 8
Figure 5. Graph of the fitted full two-factor second-order pOlynOmial model, eq 4.
The following is the fitted model in uncoded factor space in real units.
low (-1) = 10 middle (0)= 15 high (+I)= 20 highest (+a) = 22
Procedure Add dropwise the amounts of 1%HzOzand 20% HzS04, in that order, that are specified by the experimental design to 40 drops of a stwk VOSOl solution (about 0.1 g dissolved in 250 mL of distilled water). Stir the resulting mixture, and allow it to equilibrate for 5 min after the addition of HzSOa. Then measure the percent transmittance a t 460 nm using a visible spectrophotometer (e.g., Bausch & Lomb Spectronic 20). Calculate the absorbance. The procedure is a modification of a vanadium spot test used in inorganic analysis (16, 17). Results Figure 4 shows the numerical results of the central composite design. This design is moderately large and lies in the center of the bounded factor domain. The absorbance obtained from each of these experiments (~1000)is indicated at each vertex. Four replicate measurements were carried out at the center ~oint. It is possible to "let the data speak to us". A strong interanion hetwcen the numberofdroos ofH,SOa and the number of drops of HzOzcan be seen in Fi&e 4. At low levels of HzOz, increasing the amount of HzS04 causes the absorbance to decrease. However, at high levels of HzOz, increasing the amount of HzS04 causes the absorbance response to increase. Regression analysis gives the fitted model in coded factor space. 562
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Agraph of the fitted full second-order polynomial model (Fig. 5) gives a geometric view of the shape of the estimated response surface. Addiiional Experiments Equation 4 (Fig. 5 ) adequately describes the observed data over the range of experimentation.However, students might speculate about the observed decrease in absorbance as the amounts of Hz02 and HzSOaare increased from 5 to 20 drops each. Is this a simple dilution effect caused by the increased total volume? While eq 4 might adequately describe the observed data over the range of experimentation,does it extrapolate very well into unexplored regions? Students should wonder, for example, why the fitted model (Fig. 5) predicts a high absorbance at 0 drops of Hz02 and 0 drops of HzSOa. Does this prediction make chemical sense? Would there really be a high absorbance if an experiment were carried out at 0 drops of HzOz and 0 drops of H2S04?An experiment should be carried out there to confirm or deny any hypothesis. Similarly, does the predicted increased absorbance at 30 drops of HzOzand 30 drops of HzS04make chemical sense? An additional experiment should be carried out at 30 drops of each reagent to confirm or deny any hypothesis. Conclusion The quality of information obtained from multifactor experimental designs, such as the central composite design, eives the researcher a broader understanding of the chem" ical system being investigated. It also dvesa richer basis for askine funhcr ouestions about the behavior of the system (181.-
-
Acknowledament This work was supported by Grant No. 003652.108 from the Advanced Technology Program, Texas Higher Education Coordinating Board, Division of Research Programs. Literature Cited 1. Bor, G . E. P Biometries 1 9 M , 1 0 , 1 ~ . 2. Darning, S. N.; MorgshS. L.ErperimntolI*sign:A C h m t r i c A p p m o c h ; E l a e u ier: Amaterdarn, 1987. 3. Daming, S. N. C H E M T E C H I W , 19,5258. 4. Khlmi, A. I.; cornell, J.A Respanap Surf0Crfoc:Des~andAw1yso~;Dekker: New York,1987. 5. Box, G. E. P;Hunteq W. G.;Hunter, J. 8.Slotiadurforfijxr~nfern:AnInlmdur tion toDadgn. Doto Anolysu, and MdelBuilding; W~ley:NewYork. 1918.
6. Draper, N. R.;Smith, H. Applld Ropesaion Andyeis, 2nd ed.; Wiley: New York, 19111
7. HarUey, H.0.Biomtrics 1959,15,611-624. 8. Box, G. E. P ; Wilaon, K B. J Royol Slot. Soc 1W1.13, 1-45. 9. Rubin, I. B.; Mitehell, T. . I . Goldstein, ; G.Annl. C k m . 1W1,43,711-724. 10. Read. D. R.Biomtrica 1954.10.1-13. 11. Bsyne,C. K:R"bin,I. B.Pm~dEzpronontolDe~ei*ondOpllmuotionMethads forchemlab; VCH:Florida, 1986. 12. Deming, S. N.:Morgan, 8. L.Clin. Chem 1979,25,840655. 13. Dermng, S.N. CHEMTECHlW,19,249-255. 14. Deming, S. N. CHEMTECX 1989,19,50P-511. 15. Deming,S.N. CHEMTECH1990,20,11&126. 6th ed.; Elsevier: New York, 16. Feigl. F;Anger, V. Spot D t s m inogonic A&&, lo"? 17. Shavers, C. L.;P m s , M. L.; Deming, 8. N. J. Chem Edue 1979,56.307309. 18. Strange, R 8. J. C k m . Edue. 1990,67,llb115.
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