(BQ) Part 2 book Design of experiments in chemical engineering has contents: Defining research problem, selection of the responses, screening experiments, youdens squares, statistical analysis, gradient optimization methods, simplex lattice design, extreme vertices designs,...and other contents.
157 II Design and Analysis of Experiments 2.0 Introduction to Design of Experiments (DOE) Design of experiments, like any other scientific discipline, has its own terminology, methodology and subject of research The title of this scientific discipline itself clearly indicates that it deals with experimental methods A large number of experiments is done in research, development and optimization of the system This research is done in labs, pilot plants, full-scale plants, agricultural lots, clinics, etc An experiment may be physical, psychological or model based It may be performed directly on the subject or on its model The model usually differs from the subject in its dimensions and sometimes in its nature The experiment may also be done on an abstract mathematical model When a model describes the subject precisely enough, the experiment on the subject is generally replaced by an experiment on the model Lately, due to a rapid development of computer technology, physical models are more frequently replaced by abstract mathematical ones An experiment takes a central place in science, particularly nowadays, due to the complexity of problems science deals with The question of efficiency of using an experiment is therefore imposed J Bernal has made an estimation that scientific research is organized and done fairly chaotically so that the coefficient of its usability is about 2% To increase research efficiency, it is necessary to introduce something completely new into classical experimental research One kind of innovation could be, to apply statistical mathematical methods or to develop design of experiments-DOE DOE is a planned approach for determiniing cause and effect relationships Hereby, the following is essential: reduction or minimization of total number of trials; simultaneous varying of all factors that formalizes experimenter’s activities; choice of a clear strategy that enables reliable solutions to be obtained after each sequence of experiments The methodology of design of experiments has in developed countries made a special expansion in solving very complex problems in all fields of human activities It should be pointed out that an important place in this expansion was the developDesign of Experiments in Chemical Engineering Zˇivorad R Lazic´ Copyright 2004 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 3-527-31142-4 158 II Design and Analysis of Experiments ment of electronic computers, for they greatly accelerated and alleviated statistical calculations Chemical and engineering studies, as for those in other fields, are based on complex, long-term and relatively expensive experiments Experimental work is included in: physical and chemical studies for establishing constants and properties of elements, chemical compounds and materials; routine analyses of raw materials, intermediates and final products; lab studies for designing and developing technological processes; optimization of technological procedures in the lab, pilot-plant and full-scale plant systems; optimization of mixture or “composition-properties”; mathematical modeling of a system; selection of factors by the significance of their effects on a measured valueresponse; estimates and definitions of theoretic model constants, etc Hence, wherever experiments exist there should be new scientific disciplines dealing with their designing and analysis The efficiency of experimental research is determined by the degree of precision and completeness of data and information about the system that is being tested This degree results from applying the methodology of design on the experiments and on the way the obtained experimental data are analyzed It is important at this point to consider the manner in which the experimental data were collected as this greatly influences the choice of the proper technique for data analysis Before going any further it is well to point out that the person performing the data analysis should be fully aware of several things: What is the objective of the research? What is considered a significant research finding? How are the data to be collected and what are the factors that effect the responses? If an experiment has been properly designed or planned, the data will be collected in the most efficient form for the problem being considered Experimental design is the sequence of steps initially taken to insure that the data will be obtained in such a way that its analysis will lead immediately to valid statistical inferences Before a design can be chosen, the following questions must be answered: How to measure the response and the factor’s effect? How many of the factors will affect the response? How many of the factors will be considered simultaneously? How many replications (repetitions) of the experiment will be required? What type of data analysis is required (regression, ANOVA, etc.)? What level of difference in effects is considered significant? 2.0 Introduction to Design of Experiments (DOE) The purpose of statistically designing an experiment is to collect the maximum amount of relevant information with a minimum expenditure of time and resources It is important to remember also that the design of experiment should be as simple as possible and consistent with the requirements of the problem Hence, design of experiments requires a new approach to research, which is far from the traditional (classical) methods of empirical research The traditional approach demands considerable material expense and is more time consuming, for the effect of each factor experiment may be designed to investigate one factor at a time so that all other independent variables (factors) are held constant This is the so-called classical experimental design and is the one that has been favored almost exclusively among scientists and engineers At the same time, the factors have no more than or different values (levels of variation) as the total number of trials is particularly big If, for instance, the effect of five factors is to be tested where each of them may be varied at five levels, then for the complete testing of the research subject it is necessary to realize 55=3125 different combinations of factors-trials with no trial replications meant to reduce experimental errors The plotted number of classical experimental design points is hard to realize, so that in practice their number is reduced at the expense of either reducing the investigated factor space-domain or the number of factor levels In both cases, the confidence of conclusions, based on experimental results, is reduced Besides, a significant part of information obtained in a similar way is of no practical use for it refers to the region of factor space-domain, which is far from its optimum Even more drastic errors are possible if all the necessary trials are done However, due to the huge time consumption, uncontrolled changes in the quality of inlet raw materials or in the experimental plant are not accounted for The first and final trial results of an experimental program are not comparable from the accuracy point of view As an important drawback of classical experimenting, there also appears the fact that it is impossible to single out the effects of interactions between the analyzed factors This has a great influence on the errors in estimating the responses as functions of observed factors An additional difficulty also arises in an estimate on the lack of fit of the obtained mathematical model since the experimental error is usually missing Finally, interpreting the results of a classical experiment becomes difficult, because a simultaneous analysis is impossible due to a large number of tables and graphs Most of these problems can be avoided by applying the design of experiments and a simultaneous increase in efficiency of empirical research The consumption of research time may be reduced ten or more times Referring to the example where five factors are analyzed, it is possible to the designed experiment with 32 trials only by using rotatable design of second order Cases are known when, by applying the design of experiments, an optimal solution has been reached and where a classical experiment had no solution in a reasonable time period By using the design of experiments, a researcher’s intuition is developed and his way of thinking changed It may therefore be said, that the design and analysis of an experiment is a scientific method in elaborating experimental results, in finding optimal solutions and in research that has the experiment as their subject Design 159 160 II Design and Analysis of Experiments of experiments also uses a traditional approach in the research, namely the use of experimental data to obtain a mathematical model of a system In a general case and from a mathematical point of view, the used mathematical models may end up being complicated mathematical functions Response, aim function or optimization criterion may have the form: y=f(xi, zi, wi ) (2.1) where: y is response, aim function, optimization criterion; xi are the controllable independent variables, factors; zi,wi are variables and constants that affect y but are uncontrollable; f is the function that defines y, xi, zi, wi relationships Besides, one should also keep in mind the equations and non-equations that define the constraints of controllable factors Equation (2.1) defines the constraints of a research subject Research solutions may be considered optimal if they are the maximum and minimum of the response function for the given constraints It has to be remembered that each model is an approximate solution and generally is not a correct description of the research subject Optimal solution of a model is therefore considered an approximate optimum of the real system This assertion is both good and bad The good side is that the models are not complicated, since, to be close to the real system, they would have to be very complex On the other hand, insufficient reality of a model reduces the solution confidence In classical research methods, the main objective is to define the rule/law, which has the property of an absolute category, at a given level of knowledge The law is either unconditionally correct or not Such an approach makes studying a complex system difficult, for when many factors have complex effects it is difficult to find the correct mathematical system in accord with the laws Also, approximate solutions are senseless for we cannot talk about “bad” and “good” laws In the new approach to solving problems, or in design of experiments, the mathematical model is not absolute It only offers an approximate idea on the research subject and one may speak of “good” and “bad” mathematical models The essence of design of experiments is that it enables optimal solutions to be obtained even when it is really impossible to get a functional (deterministic) mathematical model and define a rule precisely It is characteristic for design of experiments that it uses polynomial models since the quality of approximation may be improved by increasing a polynomial degree Such models are especially suitable for solving optimization problems as it makes it possible to take into account the effects of interaction and a large number of factors Besides, it makes it easy to estimate the degree of lack of fit of polynomial models of different orders A designed or active experiment is based on using general methodological concepts such as regression and correlation analysis, analysis of variance, randomization, optimal use of factor space, successive experimenting, replication, compactness of information, statistical estimates, etc The regression analysis mathematical apparatus is used in the design of experiments It is therefore suggested to take into account assumptions of regression anal- 2.0 Introduction to Design of Experiments (DOE) ysis when performing an experiment This means that the trial results are independently and normally distributed random values of equal variances In other words, the experimental results in each trial are obtained with certain probability so that the distribution of such values in each trial is subject to the normal distribution law, and variances typical for them are practically equal The law on the distribution of experiment results is observed because, the random value is defined if its distribution law is known The stress is on the normal distribution for then the used mathematical model is the most efficient The law on normal distribution of data is most frequently met in practice The fact that some experimental results not submit to this law is not upsetting as by mathematical transformations, given in section 1.5, such results may be brought down to the normal distribution law Equality of random-value variances is of particular importance in experiments with a minimal number of runs or design of experiments due to their confidence level This condition is fulfilled if the variance of one trial is equal to the same variance of any other trial This variance equality is checked by tests from section 1.5 In the case of inequality, it is solved by identical transformations, same as for the normality of data distribution These checks may be easily performed since replication of trials is available and replicated trials are a principle of design of experiments One assumption of regression analysis is the increased precision of measuring or fixing a factor When measuring or fixing a factor, such conditions are recommended where a factor measurement error is incomparably smaller when compared to an error in determining a response Randomization is also an important idea in the design of experiments It has to with the random sequence of doing trials so as to annul the influence of systematic factors, which are difficult to stabilize and control In this way one of the main concepts of classical experiment, having to with the necessity of fixing disturbance factors, is disrupted Randomization is the means used to eliminate any bias in the experimental units and/or treatment of combinations-trials If the data are random it is safe to assume that they are independently distributed Errors associated with experimental units, which are adjacent in time or space will tend to be correlated, thus violating the assumption of independence Randomization helps to make this correlation as small as possible so that the analyses can be carried out as though the assumption of independence were true The idea of the concept of successiveness in doing an experiment is as follows Empirical research should consist of separate successive stages or series of trials and not of designing a complete experimental research in advance An active experiment should have the property of successiveness, or, each next stage is projected and designed based on the results of previous trials Optimality of using the factor space for an adequate multifactor experiment means an increase in experiment efficiency proportional to the increase in the number of its factors The estimate precision of a polynomial model regression coefficients rises with an increase in the number of factors, because the diameter of the sphere of factor space, within which variation limits of each factor lie, also increases 161 162 II Design and Analysis of Experiments The concept of information compactness refers to the result analysis of a designed experiment This means that final results not require a large number of tables and graphs The concept of statistical estimates refers to the threshold or significance level where the estimate of a parameter, model or solution is either accepted or rejected Finally, it should be pointed out once again that obtaining as precise and complete information on a studied chemical or physical system as possible, with a minimal number of experiments and the lowest possible expenses, is the necessary condition for efficient research work Therefore, application of modern mathematical and statistical methods in designing and analyzing experimental results is a real necessity in all fields and phases of work, starting with purely theoretical considerations of a process, its research and development, all the way to designing equipment and studying optimal operational conditions of a plant All empirical research methodologies may be divided into two large groups: classical or passive, active or statistically designed Classical design of experiments-one factor at a time Experiments may be designed to investigate one factor at a time so that all other independent variable-factors are held constant This is the so-called classical experimental design A classical experiment means researching mutual relationships between variables of a system, under “specially adapted conditions” Let us observe an example of system research where the effects of k factors on p levels are to be determined As we mention above, the classical system of experimenting requires each factor to be tested at p levels while others are kept constant at chosen fixed values The total number of trials to be done by this scheme is: N=k(p-1)+1 (2.2) Assume we have the production in a chemical reactor whereby the product yield y is essentially affected by three factors: X1 reaction mixture temperature, X2 pressure in reactor and X3 time of reaction If all factors are changed at two levels (p=2) then the research program is encompassed by four trials (N=4) The lower level factor values are marked by the symbol “-” and the upper ones by “+” The conditions of doing each run are shown in Table 2.1 Table 2.1 Experimental combinations Number of trials Factor level combinations X1 X2 X3 y Remark Reference run – – – y1 + – – + – – y2 – – + y3 y4 2.0 Introduction to Design of Experiments (DOE) After realizing each trial, it is possible to determine factor effects on product yields: EX1=y2-y1; temperature effect on yield; EX2=y3-y1; pressure effect on yield; (2.3) EX3=y4-y1; time effect on yield; Based on data analysis one can conclude that: lack of experimental error; lack of interaction effects; the result of referential trial (y1) is overestimated for it is used three times in determining the effects Based on this kind of analysis the researcher may decide to check the precision of the results by repeating the trials Precision is the repeatability of the results of a particular experiment However, apart from the possibility of determining experimental error, the trial repeating does not offer new information Statistical design of experiments-DOE The mentioned deficiencies of the classical design of an experiment may efficiently be removed and overcome by statistical design and calculation of obtained results by means of methods of statistical analysis If for the studied example, instead of repetition, the experimental program is expanded by additional combinations of factor levels-trials, as shown in Table 2.2, we get an experiment with eight trials Table 2.2 Additional experimental combinations Number of trials Factor level combinations y X1 X2 + + – y5 + – + y6 – + + y7 + + + y8 Remark X3 A complete design of experimental research, which includes all eight design points, is one of the best-known statistical experimental designs, the so-called full factorial design Factorial design of experiments, combined with statistical methods of data analysis, offers wider and more differentiated information on the system, while conclusions are of greater usability The results of all the eight runs in the analyzed example serve for determining the factor effects, with seven trials being independent possibilities of testing the effects and one serving for their comparison with the chosen fixed values Three out of seven independently determined factor effects serve for 163 164 II Design and Analysis of Experiments finding its basic effect: EX1; EX2 and EX3 and the other four to determine their mutual interactions: EX1X2 EX1X3 EX2X3 and EX1X2X3, following these expressions: Table 2.3 k Full factorial design2 Number of trials Factor level combinations X1 Response y Remark Reference trial X2 X3 – – – + – – y1 y2 – + – y3 + + – y4 – – + y5 + – + y6 – + + + + + y7 EX1=(y2+y4+y6+y8)/4-(y1+y3+y5+y7)/4 EX2=(y3+y4+y7+y8)/4-(y1+y2+y5+y6)/4 EX3=(y5+y6+y7+y8)/4-(y1+y2+y3+y4)/4 EX1X2=(y1+y4+y5+y8)/4-(y2+y3+y6+y7)/4 EX2X3=(y1+y2+y7+y8)/4-(y3+y4+y5+y6)/4 EX1X3=(y1+y3+y6+y8)/4-(y2+y4+y5+y7)/4 EX1X2X3=(y2+y3+y5+y8)/4-(y1+y4+y6+y7)/4 y8 (2.4) As has been said before, in the case of the classical experiment, which with replication has N=24=8 trials, the results of three trials are used for establishing basic factorial effects, one as a referential value and the remaining four for determining experimental error The advantages of factorial design are evident and they prove to be the best in experiments with a larger number of factors Basic advantages of design of experiment when compared to the one factor at a time classical one, are as follows: it makes possible asserting lawfulness of phenomena in the experimental space-domain as a whole, and hence drawing conclusion on results is of wider usability value; it offers wider possibilities of testing, the effects of factor varying on final result, since results of all trials are used for calculation of the effects; it enables establishing the size of factor interactions, moreover, this is the only way such interactions may be determined; data accuracy from an active experiment is reached through considerably fewer statistically designed trials, i.e at the same number of trials an active experiment offers more complete and precise information; the final research objective set up is achieved in a systematic, well thought out and organized way in a short time with considerably fewer runs and the lowest possible material costs; 2.0 Introduction to Design of Experiments (DOE) in a classical experiment one is usually unable to take into account uncontrolled changes, errors resulting from material variation, bias errors and errors resulting from the sequence of testing; a classical experiment has a lack of information about experimental error, which serves as an estimate of the lack of fit for the obtained mathematical model; when doing a classical experiment one obtains clumsy tables and graphs that are difficult for a simultaneous analysis; an active experiment eliminates one of the main assumptions of classical experimentation having to with the necessity of fixing disturbance factors A researcher is consciously suggested to make random situations-randomizations so that hard to stabilize and uncontrolled factors could have a random character; an active experiment has a successive property, or, each next stage is projected and designed based on results of a previous series of trials; an active experiment changes the way of the experimenter’s reasoning, increases his intuition and makes him active in projecting further stages of an experiment, requires use of empirical and scientific background; a classical experiment is a special case of an active statistical design of experiment where the individual effect of certain factors on system response is tested From a mathematical point of view, a classical experiment offers partial effect, while the active one gives the total effect, for with it all factors are simultaneously varying in the experiment Table 2.4 shows basic statistical designs for all kinds of quantitative and categorical/qualitative factors Table 2.4 Basic DOE Designs Experimental design Factors Application Simple comparative designs Categorical/qualitative and quantitative Differences between batches, treatments, samples Check of method, testing of single factor effect Calculation of effects with elimination of inequality of experimental conditions Screening of factors Random blocks and Latin squares Fractional replicate designs Random balance design Full factorial designs Central composite rotatable designs Central composite orthogonal designs Categorical/qualitative and quantitative Categorical/qualitative and quantitative Categorical/qualitative, quantitative and combined Quantitative Quantitative Screening of factors Choice of factors, calculation of main effects and interactions Regression models of second order Regression models of second order 165 166 II Design and Analysis of Experiments Table 2.4 Continued Experimental design Factors Application Simplex lattice design Quantitative Extreme vertex design Quantitative with constraints Hartley's, Kono’s, Kifer’s, D-Optimal Higher-order designs Quantitative Mixture problems, regression models of second and higher orders Mixture problems, regression models of second and higher orders Regression models Quantitative Regression models of higher order 2.1 Preliminary Examination of Subject of Research 2.1.1 Defining Research Problem Experimental research of the system must be preceded by preliminary examination of the subject of research aimed at obtaining information necessary for defining the research objective The modern approach to experimental research presupposes that to obtain the optimal solution it is necessary to define the research problem correctly It should be defined in such a way to enable the most efficient algorithms and methods of a designed experiment For a concrete definition of a research problem, it is necessary to formulate clearly its objective, choose the research subject model and analyze its preliminary information Special attention should be paid to the setup conditions in the problem with reference to the capability of the available experimental plant The next step is the choice of preliminary design of experiment When choosing it one must take into account all the singularities of the research problem and all known design of experiments must be analyzed in this respect The design or method that is most efficient in the particular analyzed case is chosen The methods and designs of experiments for further research stages will be considered after completing and analyzing the previous research As Fig 2.1 shows, the new approach to experimental research requires long prior preparation of the experiment aimed at increasing experimentation efficiency The research objective may be defined if the research subject or optimization subject is defined, if its requirements are known and if there exist interactions that change the quality of a research subject with the change of requirements The next step is choice of research subject model It has been said before that design of experiments rests on cybernetic concepts about the research subject A “black box› is therefore recommended as the research subject model, which will be affected by various controllable factors The defining principles of such a model cor- 558 III Mixture Design “Composition-Property” Table 3.60 (continued) N r g/cm3 rz~ 10-3 psi rs 10-3 psi rp 10-3 psi E 10-3 psi 28 1.047 1.170 3.550 9.325 29 1.070 1.018 2.074 2.096 5.458 5.961 9.936 14.849 31 32 1.599 1.701 6.842 5.523 113.891 92.042 4.75 46.18 1.408 2.061 5.786 6.140 151.090 233.562 2.75 45.63 33 1.390 1.488 4.923 7.655 69.508 127.688 11.75 52.90 34 1.413 2.648 6.893 8.795 188.910 238.930 2.75 51.54 35 1.361 2.683 6.832 8.820 177.011 228.251 3.83 47.51 36 1.218 2.230 4.952 7.444 125.650 167.862 4.75 45.09 37 1.200 1.221 1.742 2.967 5.723 7.502 8.605 11.180 65.118 168.391 118.473 209.106 15.67 3.25 52.35 51.00 39 40 1.170 2.837 7.476 12.713 147.005 187.590 6.58 46.96 1.181 1.376 4.225 13.302 35.650 47.539 28.10 59.62 41 1.204 2.465 6.813 16.235 106.444 121.308 9.83 58.26 42 1.152 2.156 7.116 10.277 93.620 127.054 24.30 54.23 43 1.266 3.671 8.489 20.086 183.699 233.242 3.00 56.90 44 1.174 3.725 8.810 17.021 184.266 188.491 5.17 52.87 45 1.122 1.259 3.467 2.222 8.203 5.554 17.576 8.330 163.933 138.991 205.205 81.023 7.50 4.50 48.84 50.25 1.259 2.101 5.449 9.419 130.262 78.496 5.50 50.25 1.259 2.089 5.560 9.391 133.316 78.339 5.25 50.25 30 38 46 47 48 Table 3.61 Es 10-3 psi e % Price (/l 36.676 22.903 20.00 54.64 98.425 101.368 117.795 121.118 6.10 11.70 53.28 49.25 Regression coefficients rs r e B0 B1 1.13249 -114.19733 0.07269 0.01064 0.01384 0.56932 0.4745 1.37649 95.98950 -0.02732 0.00078 -0.00283 0.14554 1.79251 B2 1.57042 167.01782 -0.05128 -0.00615 -0.01668 -5.75500 0.70308 B3 -1.02582 -14.61224 -0.04799 -0.00967 -0.020115 -1.22441 -0.29696 B4 -1.10084 109.44629 -0.03456 -0.00979 -0.02067 0.22113 -1.35077 B5 1.58401 160.08569 -0.06056 -0.00244 -0.01579 1.72925 1.58983 B6 0.27504 82.41577 -0.03130 -0.00383 0.00103 -0.63911 0.50197 B7 0.08755 0.34422 290.07031 256.43359 -0.12926 -0.22232 -0.01465 -0.00569 -0.00056 -0.02062 -0.33234 -1.35132 -0.64604 0.43842 B8 B11 rp rz~ Regression coefficients Es E 0.0 182.80933 -0.05150 -0.02704 -0.04074 -2.97189 -3.16558 B22 0.0 39.53931 0.00961 -0.00091 0.01035 8.39853 -0.30142 B33 0.0 84.78394 0.02581 0.00139 -0.00693 0.72176 -0.41278 B44 0.0 137.30420 -0.00937 -0.00172 -0.01476 1.06233 0.57664 3.9 Full Factorial Combined with Mixture Design-Crossed Design Table 3.61 (continued) Regression coefficients r e rp rz~ rs Es E B55 0.0 119.84521 0.05115 -0.01412 0.03872 -4.55200 -2.13765 B66 0.0 25.01221 -0.02219 -0.00174 -0.04392 0.89334 -0.21522 B77 0.0 -113.98730 0.07174 0.00431 -0.01337 -0.23637 0.16125 B88 0.0 0.0 37.80859 0.0 0.21372 0.0 0.00680 0.0 0.01492 0.0 3.09570 1.93091 -0.41095 0.0 B12 B13 B14 0.0 0.0 0.0 0.0 0.0 4.53545 0.0 0.0 0.0 0.0 0.0 0.0 -2.24657 0.0 B15 0.0 0.0 0.0 0.0 0.0 1.72719 0.0 B16 0.0 0.0 0.0 0.0 0.0 5.74762 0.0 B17 0.0 -270.89063 0.0 0.0 0.0 -0.56160 -2.24062 B18 0.0 0.0 0.0 0.0 0.0 2.11084 0.0 B23 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 8.43297 6.87164 0.0 0.0 B25 B26 0.0 0.0 0.0 0.0 0.0 0.37309 0.0 0.0 0.0 0.0 0.0 0.0 2.11751 0.0 B27 0.0 0.0 0.0 0.0 4.83197 -1.45135 B24 -342.40527 B28 0.0 0.0 0.0 0.0 0.0 10.49878 0.0 B34 0.0 0.0 0.0 0.0 0.0 -1.32475 0.0 B35 0.0 0.0 0.0 0.0 0.0 -5.29202 0.0 B36 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -3.41042 0.98099 0.0 0.0 B38 B45 0.0 0.0 0.0 0.0 0.0 3.12524 0.0 0.0 0.0 0.0 0.0 0.0 0.58264 0.0 B46 0.0 B47 0.0 B37 0.0 -196.06665 0.0 0.0 0.0 -3.37820 0.0 0.0 0.0 0.0 -1.92389 1.13045 0.0 B48 0.0 0.0 0.0 0.0 0.0 1.41870 B56 0.0 0.0 0.0 0.0 0.0 2.08801 0.0 B57 0.0 0.0 -372.80078 61.21167 0.0 0.00910 0.0 -0.01003 0.0 -0.00587 2.08035 2.39307 -2.25305 0.276081 B67 B68 0.0 -188.13574 0.0 0.0 0.0 0.07410 -1.06024 0.0 0.0 0.0 3.12915 0.0 B78 0.0 0.29166 2.00905 0.02934 0.20239 -0.40131 B58 0.0 0.0 -520.73828 559 560 III Mixture Design “Composition-Property” Example II [14] When changing technological parameters in the process of refining of raw benzene, composition of waste liquid is also changed Waste liquid is often used for obtaining expensive materials characterized by their density and viscosity Water content in such a liquid varies from to 15%, and the contents of ashes does not exceed 10% The research objective is to obtain a regression model that will adequately describe density and viscosity in the function of proportions of organic matter, water and ashes Since proportions of components are limited, the experiment has been done by extreme vertices design The local factor space is given in Table 3.62, and design matrix with outcomes of trials in Table 3.63 Table 3.62 Local factor space Components Proportions of components x2 x1 x3 X1 95 X2 80 15 X3 90 10 Table 3.63 Extreme vertices design Response Mark Density g/cm3 Design matrix Viscosity p X1 X2 X3 r1 r2 r m1 m2 m 16.85 y1 0 1.128 1.126 1.127 17.1 16.6 y2 1.070 1.072 1.071 4.0 5.0 4.50 y3 0 1.139 1.136 1.1375 21.0 19.1 20.05 12.70 y12 0.5 0.5 1.112 1.114 1.113 12.6 12.8 y13 0.5 0.5 1.135 1.130 1.1325 18.6 17.9 18.25 y23 1/3 0.5 1/3 0.5 1/3 1.118 1.122 1.120 1.123 1.119 1.1225 14.6 15.0 14.8 15.6 14.70 15.30 y123* The geometrical interpretation of local factor space is given in Fig 3.38 3.9 Full Factorial Combined with Mixture Design-Crossed Design 0.2 0.05 0.15 water ash 0.10 0.10 0.15 0.05 1.0 0.95 0.90 0.85 0.2 0.8 organic matter Figure 3.38 Local factor space Connections between real and coded factors are given by these relations: x1=0.95X1 + 0.8X2 + X3 x2=0.15X2 x3=0.05X1 + 0.05X2 +0.1X3 (3.157) By calculating regression coefficients, these second-order regression models have been obtained: _ r ¼ 1:127X1 þ 1:071X2 þ 1:138X3 þ 0:046X1 X2 À 0:002X1 X3 þ 0:058X2 X3 (3.158) _ m ¼ 16:85X1 þ 4:5X2 þ 20:05X3 þ 8:1X1 X2 þ 9:7X1 X3 À 0:8X2 X3 (3.159) To check lack of fit of the obtained regression models, errors of the experiment have been calculated: À6 Sr ¼ 3:6 Â 10 ; Sm ¼ 0:422; f ¼ 7: A check of lack of fit has been done in control point y123 Calculated values of density and viscosity in control points have by Eqs (3.158) and (3.159) these values, respectively are: r123=1.123 and m123=15.69 _ 123 À r 123 ¼ j1:1225 À 1:1230j ¼ 0:0005; Dr123 ¼ r _ Dm123 ¼ m123 À m 123 ¼ j15:3 À 15:69j ¼ 0:39 pffiffiffi pffiffiffi 0:0005 0:39 pffiffiffiffiffiffiffiffiffiffiffi ¼ 0:655: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:292; tm ¼ tr ¼ À6 1:276 0:422 1:276 3:6Â10 Since the tabular value is t(0.025;7)=2.3646, calculated values tR are smaller, so that regression models are adequate The geometric interpretation of regression models as contour lines and in coded factors is given in Figs 3.39 and 3.40 561 562 III Mixture Design “Composition-Property” X2 0.2 0.8 0.6 0.4 15 16 0.4 0.6 17 18 0.2 0.8 19 0.6 0.8 X1 Figure 3.39 0.2 0.4 X3 Viscosity contour lines X2 1.090 0.8 0.2 1.100 1.11 0.6 0.4 1.120 1.125 0.4 0.6 1.130 0.2 0.8 1.135 X1 Figure 3.40 0.8 0.4 X3 Density contour lines Example III [29] The research objective has been to define the durability of a coating depending on mixture composition Ni-Cr-B Besides, one had to determine the optimal composition of the given three-component mixture Since there is a linear correlation between resistance on wear-out and hardness of coating, Rockwell hardness (HRC) has been chosen as the system response Based on preliminary information, it is known that the response surface is smooth and continuous Hence, it may be 3.9 Full Factorial Combined with Mixture Design-Crossed Design approximated by a lower-order polynomial The design of experiment matrix with outcomes is given in Table 3.64 (trials to 6) The second-order regression model for hardness by Rockwell has this form: _ y ¼ 22X1 þ 35X2 þ 51X3 þ 38X1 X2 þ 34X1 X3 þ 52X2 X3 (3.160) Lack of fit of the obtained regression model has been checked in trials 7, 10, 11, 12 and 13 Control trials have been chosen under the assumption that they are in the optimum zone and may simultaneously be used for a model augmenting to a higher-order regression model The analyzed regression model has not been adequate in control design points To obtain an incomplete cube model, trial No has been used to calculate regression coefficients The regression model of incomplete third order has this form: _ y ¼ 22X1 þ 35X2 þ 51X3 þ 38X1 X2 þ 34X1 X3 þ 52X2 X3 þ 222X1 X2 X3 (3.161) Lack of fit of the obtained model has been checked by trials 8, 9, 10, 14 and 15 Regression model (3.161) has not been adequate in chosen control trials To calculate regression coefficients for a third-order model, it has been necessary to include these trials: 1, 2, and 7, 8, 9, 10, 11, 12 and 13 The third-order regression model is: _ y ¼ 22X1 þ 35X2 þ 51X3 þ 11:25X1 X2 þ 38:25X1 X3 þ 58:50X2 X3 þ15:75X1 X2 ðX1 À X2 Þ þ 51:75X1 X3 ðX1 À X3 Þ þ 22:50X2 X3 ðX2 À X3 Þ þ270X1 X2 X3 (3.162) Lack of fit of the obtained regression model has been checked in control trials 4, 5, and 14, 15, 16, 17 and 18 Three trials 16, 17 and 18 lie inside concentration triangle The obtained third-order regression model has been adequate in all control trials For a three-component mixture it is easiest to determine the optimum from geometric interpretation of the regression model The contour lines of regression model (3.162) are shown in Fig 3.41 563 564 III Mixture Design “Composition-Property” Design matrix for a three-component mixture Table 3.64 N Design matrix X2 X1 HRC X3 Augmenting of models II order model Incomplete III III order 1 0 22 + + + 35 + + + 0 51 + + + 0.5 0.5 33 + + check of lack 0.5 0.5 45 + + of fit 0.5 0.5 56 + + 0.333 0.333 0.333 58 check of lack + 0.666 0.333 30 10 0.333 0.666 32 0.666 0.333 55 11 0.333 0.666 57 12 0.666 0.333 44 + 13 0.333 0.666 46 + 14 0.75 0.25 29 check of lack check of lack 15 0.75 0.25 0.25 32 54 of fit of fit 16 0.25 0.50 17 18 0.25 0.50 0.25 58 0.25 0.25 0.50 56 + of fit of fit + of fit + 35 40 45 50 55 58 60 Figure 3.41 + + check of lack X2 X1 check of lack X3 Contour lines of regression model Eq (3.162) 3.11 Reference The figure clearly shows that the optimum is in this factor space: Ni˛(20–30 %); Cr˛(30–50 %); B˛(30–40 %) It should be noted that by including trials No 16, 17 and 18 one may calculate regression coefficients for a fourth order regression model 3.11 Reference Snee, D R., Technometrics, V 17., No 2, 1975 19 Kiefer, J., J Roy, Statist Soc., Ser br 30, No 2, 1968 Snee, D R., Marquardt, W D., Technometrics, V 18., No 1, 1976 20 Kiefer, J., Ann Math Statistics, 32, No 1, 1961 Cox, D R., Biometrika, 58, 1971 Marquardt, W D., Snee, D R., Technometrics, 21 Draper, N., Lawrence, C., J Roy, Statist Soc., 16 1974 Scheffe, H., J Roy, Statist Soc., B20 1958 Scheffe, H., J Roy, Statist Soc., B20, 1958 Kurotori, I S., Industrial Quality Control, 22, Ser B, 27, No 3, 1965 22 Kenworthy, O., Industr Quality Control, l9 No 12, 1963 23 Cornell, J A., Experiments with Mixtures: 1966 Designs, Models, and the Analysis of Mixture Data, John Wiley and Sons, New York, 1981 Elfving, G., Ann Math Statist., 23, 1952 Snee, D R., Marquardt, W D., Technometrics, 24 Gorman, J W., Cornell, J A., Technometrics, 17., 1975 10 Kennard, R W., Stone, L., Technometrics, 11, 24., 1982 25 Wagner, T O., Gorman, J W., Application of 1969 11 Kurnakov, N S., Vedenie v Fiziko-himicˇeskij Analiz, Izd.: AN SSSR, 1940 12 Ahnazarova, L S., Kafarov, V V., Optimizacija 13 14 15 16 17 18 26 Eksperimenta v Himii i Himicˇeskoj Tehnologii, VysÐaja kola, Moskva, 1978 ˇ , OreÐcˇanin, R., Naucˇno-tehnicˇki Lazic´, R Z 27 pregled, Vol XXX, Br 8, 1980 Vinarskij, S M., Lur,e, V M., Planirovanie Eksperimenta v Tehnologiceskih Issledovanijah, Tehnika, Kiev, 1975 Ilcˇenko, D K., Rozengat, I JU., Zlydina, P A., 28 Metalurgicˇeskie pecˇi, br 2, 1977 McLean, R., Anderson, V., Technometrics, 8, No 3, 1966 ˇ , Journal of Propulsion and Power, 29 Lazic´, R Z vol 6, br 5, sept.-oct.1990 ˇ , Vuga, S., Jaukovic´, D., NaucˇnoLazic´, R Z tehnicˇki pregled, Vol XXXII, br 3, 1982 Statistics and Computers to Fuel and Lubricant Research Problems, Proceedings of a Symposium, San Antonio, TX, 1962 John, P W., Gorman, J W., Experiments with Mixtures, Gordon Research Conference on Statistics in Chemistry and Chemical Engineering, 1961 Ziegel, E R., Discussion of the paper “Designs for Mixture Experiments Involving Process Variables”, Presented at the Annual Meeting of the American Statistical Association, Chicago, IL, 1977 Fricke, A L., Natarajan, K V., Caskey, A J., Kutat, L H., Journal of Applied Polymer Science, V 17, pp 3529-3544, 1973 Markova, E V., Zavodskaja Laboratorija, No 7, 1968 565 567 Appendix A.1 Answers to Selected Problems Chapter I 1.1 Sample 10 11 12 13 14 15 16 17 18 19 20 Values 3.4; 3.9; 3.9; 9.9; 3.5 5.5; 8.5; 6.7; 0.7; 4.5 1.5; 0.6; 3.3; 4.2; 5.7 2.2; 8.2; 9.4; 0.4; 9.9 8.4; 5.0; 0.8; 6.6; 3.1 9.9; 9.3; 5.1; 4.8; 9.2 6.5; 4.0; 9.2; 4.6; 2.1 1.8; 3.6; 5.2; 9.4; 2.9 7.5; 1.7; 5.6; 3.3; 8.0 4.2; 5.1; 2.3; 5.8; 2.6 6.8; 8.2; 3.6; 2.8; 8.3 6.1; 6.0; 2.8; 6.9; 9.1 3.5; 8.2; 6.5; 3.2; 2.9 3.6; 9.7; 8.6; 8.8; 2.3 5.2; 4.7; 8.8; 2.4; 7.4 0.9; 0.0; 4.7; 4.4; 5.9 7.5; 8.0; 6.8; 3.0; 0.0 6.8; 2.2; 0.2; 5.9; 7.0 9.4; 0.3; 4.0; 9.8; 0.9 8.6; 9.9; 1.4; 5.7; 6.4 Averages: Population parameters: Averages Variances 5.7 5.2 3.6 4.9 4.1 5.9 3.9 4.9 4.5 3.8 5.4 4.8 4.5 6.0 5.1 4.2 3.7 4.1 4.6 5.9 4.7 4.5 8.4 4.8 4.9 12.3 9.2 10.5 7.4 8.9 7.1 4.8 6.9 12.1 6.5 6.2 5.2 11.9 12.2 10.9 15.1 6.3 8.5 8.3 Design of Experiments in Chemical Engineering Zˇivorad R Lazic´ Copyright 2004 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 3-527-31142-4 568 Appendix 1.2 a) b) c) d) e) f) g) h) i) 1.3 a) b) P(Z£1.4)-P(Z