Empirical models, response surface designs

A complete list of the reaction conditions tested for this response surface design can be found in [76], The center point reaction condition was repeated six times. This was done to measure the variability of the reaction system. Also, the space velocity is kept constant, as it was the least important factor predicted by screening design, for all the reaction conditions. The purpose of this nested response surface design was to develop an empirical model in the form of Eqn (5) to relate the five reaction condition variables and the three catalyst composition variables to the observed catalytic performance. 

The response surface designs require at least three levels for each variable, in order to be able to detect and model curvature in the response. The model is very often an empirical second-order or quadratic one (see Eq. (6.2)). The coefficients in the second-order model are estimated using multiple regression and they allow to predict... 

The response surface design results are analysed by building, interpreting and validating an empirical model describing the relationship between responses and the studied factors. A second-order polynomial model is selected usually because frequently only two or three important factors are optimised, but for more factors the models would be similar. Equations (3.11) and (3.12) present the models for two x and x ) and three factors (xi, X2 and x ), respectively. 

To develop an empirical model for a response surface, it is necessary to collect the right data using an appropriate experimental design. Two popular experimental designs are considered in the following sections. 

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. 

The following set of experiments provides practical examples of the optimization of experimental conditions. Examples include simplex optimization, factorial designs used to develop empirical models of response surfaces, and the fitting of experimental data to theoretical models of the response surface. 

The resulting data of the Box-Behnken design were used to formulate a statistically significant empirical model capable of relating the extent of sugar 3deld to the four factors. A commonly used empirical model for response surface analysis is a quadratic polynomial of the type... 

A researcher is therefore recommended to use the design of experiments or to achieve an optimum in an experimental way. A researcher who designs an experiment does not know beforehand where in the studied response surface the optimum is located and what the shape of the surface is. Therefore he uses two approaches to reach the optimum. By one approach, he approximates in the given experimental region his experimental data by an assumed empirical model, or fits the response surface to the degree of the needed polynomial accuracy. Based on such an analytical model, he performs analytical optimization. Reaching an optimum in this case is more efficient if the obtained analytical model is adequate. By another approach, the researcher does not form an analytical model, but he does his experiments iteratively by prior established rules until he reaches the optimum. 

Response Surface Methodology (RSM) is a statistical method which uses quantitative data from appropriately designed experiments to determine and simultaneously solve multi-variate equations (3). In this technique regression analysis is performed on the data to provide an equation or mathematical model. Mathematical models are empirically derived equations which best express the changes in measured response to the planned systematic... 

A potential concern in the use of a two-level factorial design is the implicit assumption of linearity in the true response function. Perfect linearity is not necessary, as the purpose of a screening experiment is to identify effects and interactions that are potentially important, not to produce an accurate prediction equation or empirical model for the response. Even if the linear approximation is only very approximate, usually sufficient information will be generated to identify important effects. In fact, the two-factor interaction terms in equation (1) do model some curvature in the response function, as the interaction terms twist the plane generated by the main effects. However, because the factor levels in screening experiments are usually aggressively spaced, there can be situations where the curvature in the response surface will not be adequately modeled by the two-factor interaction... 

TJ apid entrainment carbonization of powdered coal under pressure in a partial hydrogen atmosphere was investigated as a means of producing low sulfur char for use as a power plant fuel. Specific objectives of the research were to determine if an acceptable product could be made and to establish the relationship between yields and chemical properties of the char, with special emphasis on type and amount of sulfur compound in the product. The experiments were conducted with a 4-inch diameter by 18-inch high carbonizer according to a composite factorial design (1, 2). Results of the experiments are expressed by empirical mathematical models and are illustrated by the application of response surface analysis. 

Selected blends of styrene-acrylonitrile copolymer (30 to 55%), a styrene-butadiene copolymer grafted with styrene and acrylonitrile (45 to 70%), and a coal-tar pitch (0 to 25%), were prepared. Physical properties of the experimental blends were determined and statistical techniques were used to develop empirical equations relating these properties to blend composition. Scheff canonical polynominal models and response surfaces provided a thorough understanding of the mixture system. These models were used to determine the amount of coal-tar pitch that could be incorporated into ABS compounds that would still meet ASTM requirements for various pipe-material designations. ... 

Response Surface Methodology (RSM) is a well-known statistical technique (1-3) used to define the relationships of one or more process output variables (responses) to one or more process input variables (factors) when the mechanism underlying the process is either not well understood or is too complicated to allow an exact predictive model to be formulated from theory. This is a necessity in process validation, where limits must be set on the input variables of a process to assure that the product will meet predetermined specifications and quality characteristics. Response data are collected from the process under designed operating conditions, or specified settings of one or more factors, and an empirical mathematical function (model) is fitted to the data to define the relationships between process inputs and outputs. This empirical model is then used to predict the optimum ranges of the response variables and to determine the set of operating conditions which will attain that optimum. Several examples listed in Table 1 exhibit the applications of RSM to processes, factors, and responses in process validation situations. 

Subsequently, we need to understand how the critical inputs affect the critical outputs (item 4). A second type of designed experiment, called response surface methodology (RSM), is used to accomplish this task. Sometimes we are fortunate and know the equation in advance. For example, the equation for dosage above is D = V x C. However, when the equation is not known, a DOE can be used to empirically fit a model. A response surface study is also presented as part of this case study. 

The most common nonlinear empirical model is a second order polynomial of the design variables, often called a quadratic response surface model, or simply, a quadratic model. It is a linear plus pairwise interactions model added with quadratic terms, i.e. design variables raised to power 2. For example, a quadratic model for two variables is y = 0 + b-yX-y +12X2+ bi2XiX2 + + 22XI. In general, we use the notation that b,- is the... 

To overcome the problem of strict sequentiality. Scheffe (26) proposed a modih-cation of his simplex lattice designs. He proposed a reduced empirical model containing only product terms. Like all of the other response surface models, this model is only an approximation and the product terms of increasing order can be added one after the other. The model proposed, called centered simplex model, is as follows ... 

Often one is not able to describe the response function (or response surface) from theory. It sometimes permits one to derive what type of function (linear, quadratic, etc.) should be obtained, but it is rarely able to give the coefficients in that function. In this case one uses empirical models to determine the relationship and experimental designs to develop the models. 

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