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Planejamento e Otimização
de Experimentos
Introdução
Prof. Dr. Anselmo E de Oliveira
anselmo.quimica.ufg.br
Apresentação
Aulas
Téoricas e práticas
Plano de ensino anselmo.quimica.ufg.br
Prova
03/11
Projeto
24/11 Aplicação do curso
Cálculos
Artigo Bibliografia
2011 a 2015
Qualis: A1... B4
The Analytical Process
Tools: Exploratory data analysis
Data mining
Calibration
Information/control theory
Optimization
Experimental design
Sampling theory
Luck
Information: chemical concentrations...
Measurements: voltages, currents, volumes...
Samples
System
Knowledge of properties of system
Fonte: M.A. Sharaf; D.L. Illman; B.R. Kowalski, Chemical Analysis: Chemometrics
Mechanistic and Empirical Models
Mechanistic models
Scientific phenomena are so well understood that useful results including mathematical models can be developed directly by applying these well-understood principles Ex: Perfect gas law: 𝑃𝑉 = 𝑛𝑅𝑇
Empirical models
Observation of the system at work and experimentation are required to elucidate information about why and how it works
Well-designed experiments can often lead to a model of system performance
General Model of a Process
Inputs Output
Uncontrollable factors
. . .
𝑧1 𝑧2 𝑧𝑞
𝒚
. . .
𝑥1 𝑥2 𝑥𝑝
Process
Controllable factors
Strategy of Experimentation
The objectives of the experiment may include the following
Determining which variables are most influential on the response y
Determining where to set the influential x’s so that y is almost near the desired nominal value
Determining where to set the influential x’s so that the variability in y is small
Determining where to set the influential y’s so that the effects of the uncontrollable variables z1, z2, ..., zq are minimized
Strategy of Experimentation
Usually, an objective of the experimenter is to determine the influence that these factors have on the output response of the system
Strategy of experimentation Analytical measurements
sampling
number of replicates
pH
solvent
GC, MS, HPLC
...
Strategy of Experimentation
Best-guess approach Selecting an arbitrary combination of the factors, test them, and
see what happens
One-factor-at-a-time (OFAT) Selecting a starting point, or baseline set of levels, for each factor,
and then successively varying each factor over its range with the other factors held constant at the baseline level
Factorial experiment All factors are varied togheter
Modeling
All models are approximations Mechanistic
Empirical
Sometimes an empirical model can suggest a mechanism 𝑦 = 𝑓 𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥𝑘
or: 𝐗 = 𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑘
𝑦 = 𝑓 𝐗
Example: reaction inside two chemical reactors
Yielding
170 oC < T < 190 oC
Suposition #1
reactor 1
reactor 2 170 190 Temperature /oC
Yie
ldin
g
Reactor 1
Reactor 2 𝒚 = 𝜶𝟏 + 𝜷𝟏𝒙
𝒚 = 𝜶𝟐 + 𝜷𝟐𝒙
Example: reaction inside two chemical reactors
Suposition #2: quadractic model
suposition #1
parallel curves
identical results for both reactors
𝜸𝟏 e 𝜸𝟐 = 𝟎
𝜷𝟏 = 𝜷𝟐; 𝜸𝟏 = 𝜸𝟐; 𝜶𝟏 𝜶𝟐
𝜶𝟏 = 𝜶𝟐; 𝜷𝟏 = 𝜷𝟐; 𝜸𝟏 = 𝜸𝟐
𝒚 = 𝜶𝟏 + 𝜷𝟏𝒙 + 𝜸𝟏𝒙𝟐
𝒚 = 𝜶𝟐 + 𝜷𝟐𝒙 + 𝜸𝟐𝒙𝟐
Graphical Representation: 2D and 3D Plots
𝒚 = 𝒇 𝒙𝟏
𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐
Graphical Representation: Contourn Plots
𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐
Graphical Representation: Contourn Plots
𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐
Graphical Representation: Surface Plots
𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐
Graphical Representation: 4D Plots
𝒚 = 𝒇 𝒙𝟏, 𝒙𝟐, 𝒙𝟑
Graphical Representation: 4D Plots
Graphical Representation: 4D Plots
Some Applications of Experimental Design
Evaluation and comparison of basic design configurations
Evaluation of material alternatives
Selection of design parameters so that the product will work well under a wide variety of field conditions, that is, so that the product is robust
Determination of key product design parameters that impact product performance
Formulation of new products
Fonte: D.C. Montgomery, Design and Analysis of Experiments, 8th ed.
Some Results of Experimental Design
The application of experimental design techniques early in process development can result in
Improved process yields
Reduced variability and closer conformance to nominal or target requirements
Reduced development time
Reduced overall costs
Guidelines for Designing Experiments
1. Recognition of and statement of the problem (a team approach to designing experiments is recommended)
Factor screening or characterization Which factors have the most influence on the response(s) of interest?
Optimization Find the settings or levels of the important factors that result in desirable values of the response
Confirmation
Discovery New materials, new factors, or new ranges for factors
Robustness Under what conditions do the response variables of interest seriously degrade?
What conditions would lead to unacceptable variability in the response variables?
Guidelines for Designing Experiments
2. Selection of the response variable Average or standard deviation (or both)
Decide how each response will be measured
The gauge or measurement system capability (or measurement error)
Identify issues related to defining the responses of interest and how they are to be measured before conducting the experiment
3. Choice of factors, levels, and range Potential design factors
Design (selected), Held-constant, and Allowed-to-vary factors
Nuisance factors Controllable, Uncontrollable (analysis of variance), and Noise factors
Choose the ranges over which these factors will be varied and the specific levels at which runs will be made (process knowledge)
Factor screening or process characterization: keep the number of factor levels low
Guidelines for Designing Experiments
4. Choice of experimental design Sample size (number of replicates)
Selection of a suitable run order for the experimental trials
Determination of whether or not blocking or other randomization restrictions are involved
Empirical model First-order model: 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + 𝜀
Interaction term: 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + 𝛽12𝑥1𝑥2 + 𝜀
second-order model: 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + 𝛽12𝑥1𝑥2 + 𝛽11𝑥112 + 𝛽22𝑥22
2 + 𝜀
Some of the factor levels will result in different values for the response. Identify which factors cause this difference and estimate the magnitude of the response change
Guidelines for Designing Experiments
5. Performing the experiment Prior to conducting the experiments a few trial runs or pilot runs
are often helpful
6. Statistical analysis of the data Results and conclusions must be objective
Graphical methods
Empirical model
7. Conclusions and recommendations Follow-up runs and confirmation testing
Experimentation is interative and we usually experiment sequentially
A good rule of thumb
It is usually a major mistake to design a single, large, comprehensive experiment at the start of a study. As a general rule, no more than 25 percent of the available resources should be invested in the first experiment
Mude,
mas começe devagar,
porque a direção é mais importante
do que a velocidade.
Clarice Lispector
