Z ˇ ivorad R. Lazic ´ Design of Experiments in Chemical Engineering Design of Experiments in Chemical Engineering.Z ˇ ivorad R. Lazic ´ Copyright  2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31142-4 Further Titles of Interest: Wiley-VCH (Ed.) Ullmann’s Chemical Engineering and Plant Design 2 Volumes 2004, ISBN 3-527-31111-4 Wiley-VCH (Ed.) Ullmann’s Processes and Process Engineering 3 Volumes 2004, ISBN 3-527-31096-7 R. Sundmacher, A. Kienle (Eds.) Reactive Destillation Status and Future Directions 2003, ISBN 3-527-30579-3 A. R. Paschedag CFD in der Verfahrenstechnik Allgemeine Grundlagen und mehrphasige Anwendungen 2004, ISBN 3-527-30994-2 Z ˇ ivorad R. Lazic ´ Design of Experiments in Chemical Engineering A Practical Guide Z ˇ ivorad R. Lazic ´ Lenzing Fibers Corporation 1240 Catalonia Ave Morristown, TN 37814 USA & All books published by Wiley-VCH are carefully produced. 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KGaA, Weinheim ISBN: 3-527-31142-4 Preface IX I Introduction to Statistics for Engineers 1 1.1 The Simplest Discrete and Continuous Distributions 7 1.1.1 Discrete Distributions 10 1.1.2 Continuous Distribution 13 1.1.3 Normal Distributions 16 1.2 Statistical Inference 22 1.2.1 Statistical Hypotheses 23 1.3 Statistical Estimation 30 1.3.1 Point Estimates 31 1.3.2 Interval Estimates 33 1.3.3 Control Charts 42 1.3.4 Control of Type II error-b 44 1.3.5 Sequential Tests 46 1.4 Tests and Estimates on Statistical Variance 52 1.5 Analysis of Variance 63 1.6 Regression analysis 120 1.6.1 Simple Linear Regression 121 1.6.2 Multiple Regression 136 1.6.3 Polynomial Regression 140 1.6.4 Nonlinear Regression 144 1.7 Correlation Analysis 146 1.7.1 Correlation in Linear Regression 148 1.7.2 Correlation in Multiple Linear Regression 152 II Design and Analysis of Experiments 157 2.0 Introduction to Design of Experiments (DOE) 157 2.1 Preliminary Examination of Subject of Research 166 2.1.1 Defining Research Problem 166 2.1.2 Selection of the Responses 170 2.1.3 Selection of Factors, Levels and Basic Level 185 2.1.4 Measuring Errors of Factors and Responses 191 VII Contents Design of Experiments in Chemical Engineering.Z ˇ ivorad R. Lazic ´ Copyright  2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31142-4 2.2 Screening Experiments 196 2.2.1 Preliminary Ranking of the Factors 196 2.2.2 Active Screening Experiment-Method of Random Balance 203 2.2.3 Active Screening Experiment Plackett-Burman Designs 225 2.2.3 Completely Randomized Block Design 227 2.2.4 Latin Squares 238 2.2.5 Graeco-Latin Square 247 2.2.6 Youdens Squares 252 2.3 Basic Experiment-Mathematical Modeling 262 2.3.1 Full Factorial Experiments and Fractional Factorial Experiments 267 2.3.2 Second-order Rotatable Design (Box-Wilson Design) 323 2.3.3 Orthogonal Second-order Design (Box-Benken Design) 349 2.3.4 D-optimality, B k -designs and Hartleys Second-order Designs 363 2.3.5 Conclusion after Obtaining Second-order Model 366 2.4 Statistical Analysis 367 2.4.1 Determination of Experimental Error 367 2.4.2 Significance of the Regression Coefficients 374 2.4.3 Lack of Fit of Regression Models 377 2.5 Experimental Optimization of Research Subject 385 2.5.1 Problem of Optimization 385 2.5.2 Gradient Optimization Methods 386 2.5.3 Nongradient Methods of Optimization 414 2.5.4 Simplex Sum Rotatable Design 431 2.6 Canonical Analysis of the Response surface 438 2.7 Examples of Complex Optimizations 443 III Mixture Design “Composition-Property” 465 3.1 Screening Design “Composition-Property” 465 3.1.1 Simplex Lattice Screening Designs 469 3.1.2 Extreme Vertices Screening Designs 473 3.2 Simplex Lattice Design 481 3.3 Scheffe Simplex Lattice Design 484 3.4 Simplex Centroid Design 502 3.5 Extreme Vertices Designs 506 3.6 D-optimal Designs 521 3.7 Draper-Lawrence Design 529 3.8 Factorial Experiments with Mixture 540 3.9 Full Factorial Combined with Mixture Design-Crossed Design 543 Appendix 567 A.1 Answers to Selected Problems 567 A.2 Tables of Statistical Functions 589 Index 607 ContentsVIII IX The last twenty years of the last millennium are characterized by complex automati- zation of industrial plants. Complex automatization of industrial plants means a switch to factories, automatons, robots and self adaptive optimization systems. The mentioned processes can be intensified by introducing mathematical methods into all physical and chemical processes. By being acquainted with the mathematical model of a process it is possible to control it, maintain it at an optimal level, provide maximal yield of the product, and obtain the product at a minimal cost. Statistical methods in mathematical modeling of a process should not be opposed to tradi- tional theoretical methods of complete theoretical studies of a phenomenon. The higher the theoretical level of knowledge the more efficient is the application of sta- tistical methods like design of experiment (DOE). To design an experiment means to choose the optimal experiment design to be used simultaneously for varying all the analyzed factors. By designing an experi- ment one gets more precise data and more complete information on a studied phe- nomenon with a minimal number of experiments and the lowest possible material costs. The development of statistical methods for data analysis, combined with de- velopment of computers, has revolutionized the research and development work in all domains of human activities. Due to the fact that statistical methods are abstract and insufficiently known to all researchers, the first chapter offers the basics of statistical analysis with actual exam- ples, physical interpretations and solutions to problems. Basic probability distribu- tions with statistical estimations and with testings of null hypotheses are demon- strated. A detailed analysis of variance (ANOVA) has been done for screening of fac- tors according to the significances of their effects on system responses. For statisti- cal modeling of significant factors by linear and nonlinear regressions a sufficient time has been dedicated to regression analysis. Introduction to design of experiments (DOE) offers an original comparison be- tween so-called classical experimental design (one factor at a time-OFAT) and statis- tically designed experiments (DOE). Depending on the research objective and sub- ject, screening experiments (preliminary ranking of the factors, method of random balance, completely randomized block design, Latin squares, Graeco-Latin squares, Youdens squares) then basic experiments (full factorial experiments, fractional fac- Preface Design of Experiments in Chemical Engineering.Z ˇ ivorad R. Lazic ´ Copyright  2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31142-4 Preface torial experiments) and designs of second order (rotatable, D-optimality, orthogonal, B-designs, Hartleys designs) have been analyzed. For studies with objectives of reaching optima, of particular importance are the chapters dealing with experimental attaining of an optimum by the gradient method of steepest ascent and the nongradient simplex method. In the optimum zone up to the response surface, i.e. response function, one can reach it by applying second- order designs. By elaborating results of second-order design one can obtain square regression models the analysis of which is shown in the chapter on canonical analy- sis of the response surface. The third section of the book has been dedicated to studies in the mixture design field. The methodology of approaching studies has been kept in this field too. One begins with screening experiments (simplex lattice screening designs, extreme ver- tices designs of mixture experiments as screening designs) through simplex lattice design, Scheffe's simplex lattice design, simplex centroid design, extreme vertices design, D-optimal design, Draper-Lawrence design, full factorial mixture design, and one ends with factorial designs of process factors that are combined with mix- ture design so-called "crossed" designs. The significance of mixture design for developing new materials should be partic- ularly stressed. The book is meant for all experts who are engaged in research, devel- opment and process control. Apart from theoretical bases, the book contains a large number of practical exam- ples and problems with solutions. This book has come into being as a product of many years of research activities in the Military Technical Institute in Belgrade. The author is especially pleased to offer his gratitude to Prof. Dragoljub V. Vukovic ´ , Ph.D., Branislav Djukic ´ , M.Sc. and Paratha Sarathy, B.Sc. For technical editing of the manuscript I express my special gratitude to Predrag Jovanic ´ , Ph.D., Drago Jaukovic ´ , B.Sc., Vesna Lazarevic ´ , B.Sc., Stevan Rakovic ´ , machine technician, Dus ˇ anka Glavac ˇ , chemical technician and Ljiljana Borkovic. Morristown, February 2004 Z ˇ ivorad Lazic ´ X 1 Natural processes and phenomena are conditioned by interaction of various factors. By dealing with studies of cause-factor and phenomenon-response relationships, science to varying degrees, has succeeded in penetrating into the essence of phe- nomena and processes. Exact sciences can, by the quality of their knowledge, be ranked into three levels. The top level is the one where all factors that are part of an observed phenomenon are known, as well as the natural law as the model by which they interact and thus realize the observed phenomenon. The relationship of all fac- tors in natural-law phenomenon is given by a formula-mathematical model. To give an example, the following generally known natural laws can be cited: E ¼ mw 2 2 ; F ¼ ma ; S ¼ vt ; U ¼ IR ; Q ¼ FW The second group, i.e. at a slightly lower level, is the one where all factors that are part of an observed phenomenon are known, but we know or are only partly aware of their interrelationships, i.e. influences. This is usually the case when we are faced with a complex phenomenon consisting of numerous factors. Sometimes we can link these factors as a system of simultaneous differential equations but with no so- lutions to them. As an example we can cite the Navier-Stokes’ simultaneous system of differential equations, used to define the flow of an ideal fluid: r DW X @X ¼ @p @x þ l r 2 W X þ 1 3 @Q f @X  r DW y @y ¼ @p @y þ l r 2 W y þ 1 3 @Q f @y  r DW z @z ¼ @p @z þ l r 2 W z þ 1 3 @Q f @z  8 > > > > > > > < > > > > > > > : An an even lower level of knowledge of a phenomenon is the case when only a certain number of factors that are part of a phenomenon are known to us, i.e. there exists a large number of factors and we are not certain of having noticed all the vari- ables. At this level we do not know the natural law, i.e. the mathematical model by which these factors act. In this case we use experiment (empirical research) in order to reach the noticed natural law. I Introduction to Statistics for Engineers Design of Experiments in Chemical Engineering.Z ˇ ivorad R. Lazic ´ Copyright  2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-31142-4 [...]... probability of 0.5 to either event Generally the probabilities of all possible events are chosen to total 1.0 If we toss two coins, we note that the fall of each coin is independent of the other The probability of either coin landing heads is thus still 0.5 The probability of both coins falling heads is the product of the probabilities of the single events, since the single events are independent:... outcomes of one kind Heads or tails with a coin nÂMðNÀMÞðNÀnÞ M objects of one kind, 2 N ðNÀ1Þ N objects of another kind k objects of kind M found in a drawing of n objects The n objects are drawn from the population without replacement after each drawing 1Àp Number of failures 2 p before the first success in a sequence of Bernoulli trials kt Random occurrence with time Probability of k occurrences in interval... specific values in a continuous distribution is: b P ða ð bÞ ¼ f ðx Þdx X (1.28) a where f(x) is the probability density function of the underlying population model Since all values of X lie between minus infinity and plus infinity ½À1; þ1Š, the probability of finding X within these limits is 1 Hence for all continuous distributions: þ1 ð f ðX Þdx ¼ 1 (1.29) À1 The expected value of a continuous distribution... usually an infinite gathering of elements-units For example, we can take each hundredth sample from a steady process and expose it to chemical analysis or some other treatment in order to establish a certain property (taking a sample from a chemical reactor with the idea of establishing the yield of chemical reaction, taking a sample out of a rocket propellant with the idea of establishing mechanical... by the middle of the nineteenth century Probably the first instance of applied statistics came in the application of probability theory to games of chance Even today, probability theorists frequently choose I Introduction to Statistics for Engineers a coin or a deck of cards as their experimental model Application of statistics in biology developed in England in the latter half of the nineteenth century... value in a definite range Examples of continuous random variables: waiting time for a bus, time between emission of particles in radioactive decay, etc The simplest probability model Probability theory was originally developed to predict outcomes of games of chance Hence we might start with the simplest game of chance: a single coin We intuitively conclude that the chance of the coin coming up heads... theory are being applied in all fields of engineering With the development of electronic computers, statistical methods began to thrive and take an ever more important role in empirical researches and system optimization Statistical methods of researching phenomena can be divided into two basic groups The first one includes methods of recording and processing-description of variables of observed phenomena... expected value of X; By appropriate manipulation, it is possible to determine the expected value of various functions of X, which is the subject of probability theory For example, the expected value of X is simply the sum of squares of the values, each weighted by the probability of obtaining the value The population variance of the random variable X is defined as the expected value of the square of the difference... strictly determined, i.e random and deterministic phenomena are the left and right limits of stochastic phenomena In order to find stochastic relationships the present-day engineering practice uses, apart from others, experiment and statistical calculation of obtained results Statistics, the science of description and interpretation of numerical data, began in its most rudimentary form in the census... first important application of statistics in the chemical industry also occurred in a factory in Dublin, Ireland, at the turn of the century Out of the need to approach solving some technological problems scientifically, several graduate mathematicians from Oxford and Cambridge, including W S Gosset, were engaged Having accepted the job in l899, Gosset applied his knowledge in mathematics and chemistry . R. Lazic ´ Design of Experiments in Chemical Engineering Design of Experiments in Chemical Engineering. Z ˇ ivorad R. Lazic ´ Copyright  2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN:. Correlation in Linear Regression 148 1.7.2 Correlation in Multiple Linear Regression 152 II Design and Analysis of Experiments 157 2.0 Introduction to Design of Experiments (DOE) 157 2.1 Preliminary. of statistics in biology developed in England in the latter half of the nineteenth century. The first important application of statistics in the chemical industry also occurred in a factory in