APPLICATIONS OF STATISTICAL EXPERIMENTAL DESIGN METHODS TO CHEMICAL ENGINEERING

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1 APPLICATIONS OF STATISTICAL EXPERIMENTAL DESIGN METHODS TO CHEMICAL ENGINEERING B.AKAY, S.ERTNC, N.BRSALI *, M.ALPBAZ, H.HAPOĞL Ankara nversty, Engneerng Faculty, Chem. Engneerng Department, 6 Tandoğan Ankara * Yenmahalle Muncpalty Health Department, Yenmahalle, Ankara SMMARY Desgn of Experments (DOE) s an effcent expermentaton method. There are a lot of DOE methods such as full twolevel factoral, fractonal factoral smplex desgn and Doptmal desgn. These are selected based on engneer objectves and examned factor number.doe conssts of two steps one of them s the plannng the experments the other s the statstcal analyss of expermental data. Statstcal analyss of data s made both quanttatvely (ANOVA) and qualtatvely. DOE methods have been used n a large area such as process and product mprovements and optmzaton In ths study a bref nformaton about DOE objectves, steps and analyss was gven Also basc concepts of Full twolevel, Fractonal factoral Desgns and Central Composte Desgn were explaned. At the end of the study a smple example was gven. Key words: Desgn of experments; Factoral desgns, Process mprovement; Statstcal expermental desgn.introdction When a scentst or an engneer wants to mprove current product and process or develop a new process or product he or she needs to perform experment. In an experment one or more process varables are changed delberately to observe the effects of these varables on the response(s) of the process. There are two dfferent expermentaton technques: one of them s the tradtonal method called onefactor at a tme method the other one s the Statstcal Expermental Desgn. Planng of Experments and analysng of expermental data by usng statstcal methods to yeld vald and objectve conclusons can be called Desgn of Experments or Statstcal Expermental Desgn and symbolsed wth DOE[].Desgn of Experments was frst ntroduced by an Englsh scentst Sr Ronald Fsher n 9. p to now DOE technques have been developed very much wth the contrbuton of other scentsts studes on DOE technques such as Yule, Box, Bll Hunter, Cox and Taguch. DOE technques have been used n large area for many years such as engneerng, medcal scences, drug and cosmetc ndustry. DOE technques s really very mportant n the development of new processes and products and mprovement of the exstng processes or products wth respect to chemcal engneerng[]. Wth the use of the DOE technques t s possble to acheve mproved yeld, reduced varablty and costs. There are a lot of Statstcal Expermentaton methods. But full twolevel factoral, fractonal factoral, PlackettBurmann and Central Composte Desgns are the well known and mostly used of all them. In ths study, a bref nformaton about objectves, analyss and applcaton areas of DOE methods were gven. At the end of the study a smple example llustratng the applcaton and analyss of twolevel factoral Expermentaton method was gven.

2 .OBJECTIVES OF DOE Statstcal Expermental Desgn methods can be used at dfferent stuatons based on the engneer objectves. These objectves were gven below[]; a.comparatve Experments: If an Engneer wants to make a decson about two or more current alternatves, he or she can use Desgn of Experments methods. For example, a new catalyst can be compared wth the exstng one or two currently used machnes must be replaced wth the new one but whch of them? b.screenng Experments: In order to determne the most mportant nput varables and the least mportant nput varables havng effect on the process response, t s possble to make screenng experments. c.response Surface Modellng: In ths stuaton an engneer wants to determne the best settng values of the nput varables to optmse the process response or wants to operate a robust process. Generally a secondorder or hgher order mathematcal model s created between process nputs and response. d.regresson Modellng: A mathematcal model s created between process nput(s) and output(s) and wth the use of DOE technques uncertanty connected wth each coeffcent s made as small as possble. Selecton of Expermental Desgn methods depends on both objectve of an engneer and examned factor number In Table Selecton of Expermental Desgn was gven. 3.STEPS OF DOE[] Settng of Objectves Selectng of Process Varables 3 Selectng of Expermental Desgn Executng of Expermental study and collectng data Checkng of expermental assumpton 6 Analysng and nterpretng of the results 7 Presentng and usng of the results (may be necessary further runs or a new DOE selecton) Table. Selecton of Expermental Desgn[] Num.of Factors Comparatve Objectve Screenng Objectve Response Surface Objectve or more factor completely randomzed Desgn Randomzed Block Desgn Randomzed Block Desgn Full or Fractonal Factoral Desgn Fractonal Factoral or PlackettBurmann Desgn Central Composte or Box Behnken Screenng frst to reduce number of experments.fll TWOLEVEL FACTORIAL EXPERIMENTAL DESIGN Twolevel Factoral expermental desgns are the most well known expermental desgns. All factors levels are studed at ther selected two levels[]. These levels are generally maxmum and mnmum and codfed by and respectvely. Factor levels may be quanttatve, such as

3 temperature, pressure or tme; or qualtatve such as two machnes, two operator. For a full twolevel factoral desgn wth k factor, number of expermental runs (N) s determned; N k Advantages of full twolevel factoral desgn are: Sutable for screenng experments especally n the early stage of the expermental study Economcal and smple Much nformaton n a short tme The relaton between real and codfed values of a factor s gven below; X () Table Expermental Desgn Matrx for full twolevel factoral desgn [] Treatment Combnaton a b ab Factoral effect X X X X X Desgn matrx has the followng propertes; t s orthogonal: It s n Standard Order :The frst column X conssts of successve mnus and plus sgns. The second column X consst of successve pars of mnus and plus sgns. In general, kth column conssts of k mnus sgns followed by k plus sgns. X X s called nteracton effect of factor X and X. Ths column s constructed by multplyng correspondng sgns of factor X and X n the same lne. X s called dummy varable and ts value s always Factor effects are determned from the followng equatons; A a ab b () ya ya [ a ab b () ] () n n B b ab a () yb yb [ b ab a () ] (3) n n ab () a b AB ) n n [ ab ( a b] () The expressons take place n the above brackets are called contrasts]]. Factor effects can be calculated smply by usng Yates s Algorthm and Least Square Method.

4 .TWOLEVEL FRACTIONAL FACTORIAL DESIGN The number of experments requred by full k factoral desgn ncreases geometrcally as the number of factor k s ncreased. At ths pont only a fracton of full factoral desgn s performed. In ths desgn fewer experments are performed than full factoral but man and nteracton effects of the factors are confounded because one of the man effect s dentfed beng the nteracton effect of the other examned factors. There are two mportant terms n a fractonal factoral desgn[3] a.defnng Relaton: IABCD (Identty column, I s always ) Defnng Relaton s used to determne the alased terms. b.generator of the desgn: DABC (A,B,C, and D are nvestgated factors) 6.CENTRAL COMPOSITE DESIGN If an engneer wants to dentfy a second order regresson model as gven below: n n y b b Χ b Χ Χ b Χ j j j uj u j jj u, j j u j n j at three dfferent levels of factors are studed.. But at ths tme the number of expermental run s ncreased. To reduce expermental run number BoxWlson Central Composte desgn can be used[]. Wth ths desgn factoral desgns are augmented wth a group of star ponts α that allow estmaton of curvature. Table 3. α Values for Central Composte Desgn[] () Number of Factork α Table. Central Composte Expermental Desgn Matrx for Factors[] run X X X 3 α 6 α 7 α 8 α 9 Number of experment: k< N n n n (6) k > N n n n (7)

5 7.DOE ANALYSIS STEPS The basc DOE analyss steps are gven below[].expermental data are examned wth respect to response dstrbuton, possble tme and factor effects.a theoretcal mathematcal model s created. 3.Mathematcal model s dentfed by usng the data..resdual graphs are examned to test the model assumpton. Based on these result, ANOVA s examned and data transformaton, model reducton are consdered. f t s necessary at ths stage return to Step 3 to bult a new mathematcal model.results are presented and used.for example: sgnfcant factor effects, optmum set of operatng parameters. 7. Quanttatve Statstcal Analyze Technque Analyss of Varance(ANOVA) Statstcal procedure for analyss of sgnfcance of varous factors can be called Analyss of Varance.The ANOVA uses the Fsher s F test[]. MSeffect F (8) MS error SSeffect MSeffect (9) df SSerror MSerror df () SS ( y y) error centerpo nt () Sum of Square of an effect can be easly calculated from The Yates s Algorthm. To examne the curvature propertes of the nvestgated area[]: nfnc ( yf yc ) SSCurvature () nf nc If calculated F values of factor effects greater than theoretcal (tabulated) Fvalue then factor s statstcally sgnfcant 7. Qualtatve Statstcal AnalyzeTechnques[] There are many graphcal technque to examne factor and ther effects on the response. Some of them: Dex(Desgn of Experments)Mean : Detect Important Factors wth Respect to Locaton Box Plot : Check locaton and varaton shfts Dex Scatter Plot : Determne Important Factors wth Respect to Locaton and Scale HalfNormal Probablty Plot : Ths plot can be used to choose sgnfcant effects. Resduals vs Predcted: It tests the assumpton of constant varance. The plot should be a random scatter (constant range of resduals across the graph.) Expandng varance ("megaphone pattern <") n ths plot ndcates the need for a transformaton. Resduals vs Factor: Ths s a plot of the resduals versus any factor of selected. The plot should exhbt a random scatter

6 8.AN EXAMPLE:APPLICATION OF FLL TWOLEVEL FACTORIAL DESIGN[] Examne the effect of culture temperature and ph on the volumetrc productvty (gcells/l.h).full twolevel one replcated factoral expermental desgn was selected. Three replcated experments at center ponts of factors to estmate expermental error and curvature executed. Expermental Desgn Matrx was gven n Table. Factor effects and sum of squares of them were estmated from Yates Algorthm as gven n Table 6.Varance Analyze was gven n Table 7. At α sgnfcance level % (α.) Tabulated F value F. (,) 8.. The analyss of varance ndcates that only man effect of Temperature has sgnfcant effect on the volumetrc productvty under the selected factor levels. As can be seen from Normal plot of Factors[] (Fgure.) man effect of temperature s sgnfcant. Ths confrm varance analyss. Table Expermental Desgn Matrx Run X Ph Temp X X V(g/l.h) Table 6 Applcaton of Yates Algorthm V(g/l.h) () () Effect Average A B AB Estmate of effect Sum of Squares () k () n k Table 7 Analyss of Varance Source of varaton A (ph) B (Temperature) AB Curvature Error Total Sum of Squares Degrees of freedom 6 Mean Square F

7 DESIGNEXPERT Plot verm Normal plot A: ph B: Temperature 99 Normal % probablty A AB B Effect Fgure. Normal Plot of Factor Effects[] NOMENCLATRE df :degree of freedom for an effect, df : degree of freedom for error. SS effect :Sum of square of an effect SS error :sum of squares of error. k : factor number n : number of experments at center pont MS effect : mean sum of square of an effect MS error : mean sum of square of error REFERENCES. Engneerng Statstcs Handbook (NIST), [ July. Montgomery D.C., Desgn and Analyss of Experments, 3 rd ed, New York, Wley, Box G.E. P., Hunter W.G. and Hunter J.S., Statstcs for Expermenters, New York, Wley, Bursalı, N., Keskl Br Byoreaktöre Optmum Adaptf Genelleştrlmş Predktf Kontrol Metodunun ygulanması, Doktora Tez,Ankara Ünverstes Fen Blmler Ensttüsü,997.