Mixture design chapter

The experimental designs discussed in Chapters 24-26 for optimization can be used also for finding the product composition or processing condition that is optimal in terms of sensory properties. In particular, central composite designs and mixture designs are much used. The analysis of the sensory response is usually in the form of a fully quadratic function of the experimental factors. The sensory response itself may be the mean score of a panel of trained panellists. One may consider such a trained panel as a sensitive instrument to measure the perceived intensity useful in describing the sensory characteristics of a food product. 

A section has been added to Chapter 1 on the distinction between analytic vs. enumerative studies. A section on mixture designs has been added to Chapter 9. A new chapter on the application of linear models and matrix least squares to observational data has been added (Chapter 10). Chapter 13 attempts to give a geometric feel to concepts such as uncertainty, information, orthogonality, rotatability, extrapolation, and rigidity of the design. Finally, Chapter 14 expands on some aspects of factorial-based designs. 

ICLS Example 1 The first ICLS example involves samples that contain three cemponents—A. B, and C. Tlie ranges of interest are 0.1 -0,5 mol fraction (A), 0.2-O.6 mol fraction (B), and 0.3-0.7 mol fraction (C). An extreme veniccs mixture design is used to formulate the concentration matrix. Table 5.7 and Figure 5.32 display the concentrations of the calibration samples that are used to estimate the pure spectra. Spectra comprised of 200 measurement -ariabies are obtained from these mixture samples. The pure spectra are estimated fftxn these mixtures following the "Six Habits of an Effective Chemometrican" which are detailed in Chapter 1. 

The California Air Resources Board has prepared risk assessments for a number of toxic airborne compounds and mixtures, designated as toxic air contaminants, TACs (Table 16.15). For example, risk assessments for individual compounds such as benzene, benzo[a]pyrene (see Chapter 10), formaldehyde, and vinyl chloride have been carried out, in addition to complex mixtures such as diesel exhaust (California Air Resources Board, 1997a) and environmental tobacco smoke (California Environmental Protection Agency, 1997). These risk assessment documents form the basis for controls imposed as part of the risk management process (e.g., see Seiber, 1996). 

Product formulations are complex, consisting of numerous additives designed to perform certain functions. The performance of these additives depends on other components of a mixture. Similarly, fillers are added to perform certain tasks, and their performance might be enhanced or retarded by other components of the mixture. This chapter reviews the current understanding of these interactions, in order to highlight potential improvements or potential risks related to the application of fillers in complex formulations, which contain components that may interact due to physical or chemical forces. 

Chapter 8 covers non-standard designs and the last two chapters are concerned with mixtures. In chapter 9, the standard mixture design and models are described and chapter 10 shows how the methods discussed in the rest of the book may be applied to the more usual pharmaceutical formulation problem where there are severe constraints on the proportions of the different components. In all of these chapters we indicate how to choose the optimal design according to the experimenter s objectives, as those most commonly used — according to the pharmaceutical literature — are by no means optimal in all situations. 

Validation can be considered to start with the conception of the formulation. Thus the presence of each component may be justified as a result of the experimental designs, carried out at the preformulation and formulation stages of the project, such as the screening and compatibility testing described in chapter 2. The quantitative composition is supported by the response surface and optimization studies carried out according to the methods of this and the previous chapter and, more specifically, for mixtures in chapters 9 and 10. 

In this section the control variables have been process variables, but we may also wish to adjust the proportions of drug substance and excipients so that the formulation is insensitive to noise factors. The control factors are thus studied in a mixture design of the kind that will be described in the final two chapters. 

There are some excellent review articles on various aspects of the toxicology of mixtures of chemicals (Calabrese, 1995 Kri.shnan and Brodeur, 1991), pesticides (lyaniwura, 1990 Murphy, 1980), or OP compounds (Cohen, 1984 DuBois. 1961 Murphy, 1969. 1980). Many of these reviews, although not recent, provide a comprehensive overview of the possible mechanisms underlying interactions of chemicals in a mixture and, specifically, interactions of OP compounds in a mixture. This chapter does not provide an in-depth discussion of the mechanisms underlying interactions of OP or CM pesticides in a mixture. Rather, this chapter is designed to (i) address issues of experiment design of mixture studies (ii) summarize the available literature on OP pesticide mixtures, CM... 

Olefin metatheses are equilibrium reactions among the two-reactant and two-product olefin molecules. If chemists design the reaction so that one product is ethylene, for example, they can shift the equilibrium by removing it from the reaction medium. Because of the statistical nature of the metathesis reaction, the equilibrium is essentially a function of the ratio of the reactants and the temperature. For an equimolar mixture of ethylene and 2-butene at 350°C, the maximum conversion to propylene is 63%. Higher conversions require recycling unreacted butenes after fractionation. This reaction was first used to produce 2-butene and ethylene from propylene (Chapter 8). The reverse reaction is used to prepare polymer-grade propylene form 2-butene and ethylene ... 

In our discussion of (vapor + liquid) phase equilibria to date, we have limited our description to near-ideal mixtures. As we saw in Chapter 6, positive and negative deviations from ideal solution behavior are common. Extreme deviations result in azeotropy, and sometimes to (liquid -I- liquid) phase equilibrium. A variety of critical loci can occur involving a combination of (vapor + liquid) and (liquid -I- liquid) phase equilibria, but we will limit further discussion in this chapter to an introduction to (liquid + liquid) phase equilibria and reserve more detailed discussion of what we designate as (fluid + fluid) equilibria to advanced texts. 

A gas-liquid mixture will have a lower density than the liquid alone. Therefore, if in a U-tube one limb contains liquid and the other a liquid-gas mixture, the equilibrium height in the second limb will be higher than in the first. If two-phase mixture is discharged at a height less than the equilibrium height, a continuous flow of liquid will take place from the first to the second limb, provided that a continuous feed of liquid and gas is maintained. This principle is used in the design of the air lift pump described in Chapter 8. 

The design equations for a CSTR do not require that the reacting mixture has constant physical properties or that operating conditions such as temperature and pressure be the same for the inlet and outlet environments. It is required, however, that these variables be known. Pressure in a CSTR is usually determined or controlled independently of the extent of reaction. Temperatures can also be set arbitrarily in small, laboratory equipment because of excellent heat transfer at the small scale. It is sometimes possible to predetermine the temperature in industrial-scale reactors for example, if the heat of reaction is small or if the contents are boiling. This chapter considers the case where both Pout and Tout are known. Density and Q ut wiU not be known if they depend on composition. A steady-state material balance gives... 

At this point we introduce the formal notation, which is commonly used in literature, and which is further used throughout this chapter. In the new notation we replace the parameter vector b in the calibration example by a vector x, which is called the state vector. In the multicomponent kinetic system the state vector x contains the concentrations of the compounds in the reaction mixture at a given time. Thus x is the vector which is estimated by the filter. The response of the measurement device, e.g., the absorbance at a given wavelength, is denoted by z. The absorbtivities at a given wavelength which relate the measured absorbance to the concentrations of the compounds in the mixture, or the design matrix in the calibration experiment (x in eq. (41.3)) are denoted by h. ... 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

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