Experimental Study on Removing Chemical Oxygen Demand from Distillery Spent Wash

2nFactorial Experiment 4 Factor used to Remove Chemical Oxygen demand from Distillery Spent Wash R.K. Prasad and S.N. Srivastava (2009). Electrochemical degradation of distillery spent wash using catalytic anode: Factorial design of experiments, Chemical Engineering Journal, Vol. 146, pp. 22-29.

Data Description Response: Y = % Chemical Oxygen Demand Removed from Distillery Spent wash Factors and Levels: A: Current Density (mA/cm2) 14.285, 42.857 B: Dilution (%) 10, 30 C: Time (hrs) 2, 5 D: pH 4, 9 Experimental Runs: 16 All 24Combinations of levels of A,B,C,D

Data Normal Order Run 10 9 4 15 3 5 8 6 12 1 7 14 2 11 16 13 CurrDens Dilution 14.285 42.857 14.285 42.857 14.285 42.857 14.285 42.857 14.285 42.857 14.285 42.857 14.285 42.857 14.285 42.857 Time 2 2 2 2 5 5 5 5 2 2 2 2 5 5 5 5 pH 4 4 4 4 4 4 4 4 9 9 9 9 9 9 9 9 Label (1) a b ab c ac bc abc d ad bd abd cd acd bcd abcd y 10 10 30 30 10 10 30 30 10 10 30 30 10 10 30 30 28.265 32.520 32.230 36.210 28.265 38.560 34.562 40.230 65.230 56.680 62.135 64.430 71.210 68.720 72.270 68.260 For the Label, any factor at its high level appears in lower case form. (1) Corresponds to the case when all factors are at their low levels.

Table of Contrasts - I Create a Column for the intercept (I), one for each Main Effect and each Interaction (A, ,D, AB, ,CD, ABC, ,BCD, ABCD), and one for the response (y). If there were multiple replicates per treatment, replace y with the mean of those r replicates Create a row for each experimental run (treatment), using the Labels from the previous slide. For the Intercept Column, put +1 in each row For all Main Effects, Put +1 if that factor was at its high level, -1 if at its low level (Note: Books use +/-)

Table of Contrasts - II Trt (1) a b ab c ac bc abc d ad bd abd cd acd bcd abcd I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 B -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 C -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 D -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 AB AC AD BC BD CD ABC ABD ACD BCD ABCD y 28.265 32.520 32.230 36.210 28.265 38.560 34.562 40.230 65.230 56.680 62.135 64.430 71.210 68.720 72.270 68.260 For Interactions, multiply the coefficients in each row for the Main Effects that make up that Interaction. For Row 1 and Column AB: A has coefficient -1, B has -1, so AB has (-1)(-1) = +1 For Row 1 and Column ABC: (-1)(-1)(-1) = -1 For Row 1 and Column ABCD: (-1)(-1)(-1)(-1) = +1 An Interaction will have a coefficient of +1 if it has an even number of its Main Effects at their low levels, -1 if an odd number.

Table of Contrasts - III Trt (1) a b ab c ac bc abc d ad bd abd cd acd bcd abcd I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 B -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 C -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 D -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 AB 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 AC 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 AD 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 BC 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 BD 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 CD 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 ABC -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 ABD -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 ACD -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 BCD -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 ABCD 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 y 28.265 32.52 32.23 36.21 28.265 38.56 34.562 40.23 65.23 56.68 62.135 64.43 71.21 68.72 72.27 68.26 Create 4 Rows below this matrix : Contrast, Divisor, Effect, Sum of Squares n 2 = 1 In EXCEL, you can take the SUMPRODUCT of each Column with = Contrast k y k y i i i = 1 i n 2 for the Intercept (I) Column 2 for all other columns Contrast Divisor r SS = = = Divisor 1 n = Effect ( ) 2 Sum of Squar es Contrast Sum of Squares is not typically computed for Intercept 2n

Table of Contrasts - IV Trt (1) a b ab c ac bc abc d ad bd abd cd acd bcd abcd I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 B -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 C -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 D -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 AB 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 AC 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 AD 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 BC 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 BD 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 CD 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 ABC -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 ABD -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 ACD -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 BCD -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 ABCD 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 y 28.265 32.52 32.23 36.21 28.265 38.56 34.562 40.23 65.23 56.68 62.135 64.43 71.21 68.72 72.27 68.26 Divisor Contrast Effect SumSq 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 799.777 49.986 11.443 1.430 8.184 20.877 2.610 27.241 44.377 5.547 123.082 4163.250 258.093 32.262 4.423 0.553 1.223 7.483 0.935 3.500 -36.953 -4.619 85.345 -3.743 -0.468 0.876 -10.367 -1.296 6.717 19.593 2.449 23.993 -16.717 -2.090 17.466 14.227 1.778 12.650 -7.973 -0.997 3.973 -4.367 -0.546 1.192 -8.013 -1.002 4.013 = Effect : A y + y A,High A,Low + + + + + + + + + + + + + 32.52 36.21 38.56 40.23 56.68 64.43 68.72 68.26 8 Effect : AB y y = + 28.265 32.23 28.265 34.562 65.23 62.135 71.21 72.27 8 405.61 394.167 8 = = 1.430 AB,High + AB,Low + + + + + + + + + + + + 28.265 36.21 28.265 40.2 3 65.23 64.43 71.21 68.26 8 32.52 32.23 38.56 34.562 56.68 62.135 68.72 72.27 8 402.1 397.677 8 = = 0.553 Alternative Approach: + + + + + + 36.21 40.23 64.43 68.26 4 32.52 38.56 56.68 68.72 4 32.23 34.562 62.135 72.27 4 28.265 28.265 65.23 71.21 4 209.13 201.197 4 196.48 192.97 4 = + = = Effect @ 1: 1.98325 A B + + + + + + = = = Effect @ 1: 0.8775 A B 1 2 1.98325 0.8775 = Effect : 0.553 AB

Analysis of Variance = = 2 2 + + y y y y ) ( ) ( ) ( ( ) 2 2 A,Low A,High A,Low A,High = + + = 1 1 n n 2 2 SSA r y y y y r y y A,Low A,High A,Low A,High 2 2 ( ) ( ) 2 4 2 2 1 n n 2 2 (2) r r y y y y = 1 n ( ) ( ) 2 A,Low A,High A,High A,Low = + = 1 n 2 Effect Contrast r A A 1 2 2 4 2 ( ) ( 2 n 2 r 1 r r )( 1 ) ( ) ( ) ( ) 2 2 2 2 = = = = Contrast Contrast 11.43 8.184 Contrast A A SSA A n n n 2 16 2 2 2 2 Notes: Factor D (pH) has by far the largest effect on the outcome. With all mean effects and interactions, there are no error degrees of freedom, and no tests can be conducted Consider dropping interactions with small sums of squares to obtain an error term (Authors dropped: AB, AC, BC, and BCD) Source A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD df SS 1 1 1 1 4163.250 1 1 1 85.345 1 1 1 23.993 1 17.466 1 12.650 1 1 1 8.184 27.241 123.082 Source A B C D AD BD CD ABC ABD ACD ABCD Error df SS MS F_obs F(.05) P-value 1.223 3.500 1 1 1 1 4163.250 4163.250 2452.601 1 85.345 85.345 1 6.717 1 23.993 23.993 1 17.466 17.466 1 12.650 12.650 1 3.973 1 4.013 4 6.790 8.184 27.241 123.082 8.184 27.241 123.082 4.821 16.048 72.509 7.709 7.709 7.709 7.709 7.709 7.709 7.709 7.709 7.709 7.709 7.709 0.093 0.016 0.001 0.000 0.002 0.118 0.020 0.033 0.052 0.201 0.199 0.876 6.717 50.278 3.957 14.134 10.289 7.452 2.341 2.364 6.717 3.973 1.192 4.013 3.973 4.013 1.697

Regression Approach 1 if is at High Level 1 if is at Low Level A + A = Let: Similarly defined ( ), ( ), ( ) X X B X C X D 1 2 3 4 E Y + = + + + + + + + + + X X X X + + + + X X X X X X X X + X X X X X X X X 0 1 1 2 2 3 3 4 4 12 1 2 13 1 3 14 1 4 23 2 3 24 2 4 34 3 4 X X X X X X X X X X X X 123 1 2 3 124 1 2 4 134 1 3 4 234 2 3 4 1234 1 2 3 4 1 2 ^ ^ ^ ( ) = Note: Effect Effect since Effect 2 and so on... I A A y y 0 1 1 A,High A,Low Full Model: ^ = + + + + + + 49.986 0.715 2.310 X X 1.045 X X X 1.305 2.774 X X 16.131 X X 0.276 0.468 Y X X X + X X X X X 1 X X 2 3 1.225 X X X 4 + 1 2 1 3 0.234 0.889 + 0.648 1 4 2 3 2 4 3 4 0.498 0.273 0.501 X X X X X X X X X X 1 2 3 1 2 4 1 3 4 2 3 4 1 2 3 4 Reduced Model (Coefficients do not change due to orthogonal design): ^ 49.986 0.715 + 2.310 1.045 X X X = + + + 1.305 2.774 X X 16.131 Y X X X + X 1 X X 2 3 4 + 0.648 0.889 + 1.225 X X 1 4 2 4 3 4 0.498 0.501 X X X X X X X X X X 1 2 3 1 2 4 1 3 4 1 2 3 4

Further Model Reduction (Simplification) When testing the effects after removing the Interactions with the smallest effects, we find BD, ACD, and ABCD all have P-values that are > 0.10. Now we remove them for a simpler model. Source A B C D AD CD ABC ABD Error df SS MS F_obs F(.05) P-value 1 1 1 1 4163.250 4163.250 1355.908 1 85.345 85.345 1 23.993 23.993 1 17.466 17.466 1 12.650 12.650 7 21.493 8.184 27.241 123.082 8.184 27.241 123.082 2.665 8.872 40.086 5.591 5.591 5.591 5.591 5.591 5.591 5.591 5.591 0.147 0.021 0.000 0.000 0.001 0.027 0.049 0.082 This model has: + + + 8.184 + 12.650 + = = 2 0.9952 R 8.184 12.650 21.493 27.796 7.814 5.688 4.120 3.070 ^ = 49.986 0.715 + + + + + + 1.305 2.774 16.131 2.310 1.225 1.045 0.889 Y X X X X X X X X X X X X X X 1 2 3 4 1 4 3 4 1 2 3 1 2 4

Normal Probability Plot of Factor & Interaction Effects Under hypothesis of no main effects or interactions, estimated effects should be approximately normally distributed with mean 0. Construct a normal probability plot of estimated effects Trt AD ABC BD ABCD ACD BCD BC AB AC A ABD CD B C D Effect std.norm -1.739 -1.245 -0.946 -0.714 -0.515 -0.335 -0.165 0.000 0.165 0.335 0.515 0.714 0.946 1.245 1.739 Normal Probability Plot of Estimated Factor Effects and Interactions -4.619 -2.090 -1.296 -1.002 -0.997 -0.546 -0.468 0.553 0.935 1.430 1.778 2.449 2.610 5.547 32.262 2.000 1.500 1.000 Std. Normal Quantiles 0.500 0.000 -0.500 -1.000 -1.500 Clearly, several effects fall well away from central line -2.000 -10.000 -5.000 0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000 Effect

A Simple Test for Effects & Interactions Method described by Lenth (1989): Obtain s0 = 1.5*median(|Effects|) Compute: pseudo standard error: PSE = median(|Effects|*Indicator(|Effect| < 2.5*s0)) Compute Simultaneous Margin of Error: SME = t(.05/(2*Cm),d)*PSE where m = # of Effects, Cm=m(m-1)/2, d=m/3 Consider effect significant if |Effect| > SME s0 2.5xs0 PSE d t(.975;d) ME gamma SME 2.145563 5.363906 1.943813 5 Based on this criteria, only pH main effect is significant. When not making adjustment for multiple tests (ME), 3 effects are significant or very close 2.570582 4.996729 0.998293 10.14408

Estimates of Contrasts Simultaneous Margin of Error = 10.14 35 30 25 20 15 10 5 0 A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD -5 -10