Separable systems definition

We need to complicate these definitions further by noting that the above definitions are the per pass yield of a reactor. If unreacted A can be recovered and recycled back into the feed, then the overall yield of the reactor plus the separation system becomes the single-pass selectivity of the reactor because no unreacted A leaves the reactor system. 

This chapter introduces the reader to elementary concepts of modeling, generic formulations for nonlinear and mixed integer optimization models, and provides some illustrative applications. Section 1.1 presents the definition and key elements of mathematical models and discusses the characteristics of optimization models. Section 1.2 outlines the mathematical structure of nonlinear and mixed integer optimization problems which represent the primary focus in this book. Section 1.3 illustrates applications of nonlinear and mixed integer optimization that arise in chemical process design of separation systems, batch process operations, and facility location/allocation problems of operations research. Finally, section 1.4 provides an outline of the three main parts of this book. 

To illustrate the nature of the distillation-based separation system synthesis problem, let us consider its generic definition shown in Figure 9.1, which is as follows ... 

This chapter discusses mixed-integer nonlinear optimization applications in the area of distillation-based separation system synthesis. Section 9.1 introduces the reader to the generic definition of separation system synthesis and provides a classification of the different types of separations based on distillation column sequencing. For further reading on separation system synthesis the reader is referred to the reviews by Westerberg (1985) and Nishida et al. (1981)... 

A phase is a chemically homogeneous part of a system, separated by definite bounding surfaces from other similar phases. Within a phase, concentration gradient i.e., variation of concentration from point to point, may sometimes exist, but it would always tend to homogenise itself... 

The higher the Henry coefficient for a substance the stronger is its adsorption and thus the longer its retention time. This definition shows that for two components to be separated their Henry coefficients have to differ. According to Eqs. 2.3 and 2.37 the quotient in the Henry coefficients expresses the selectivity of a separation system. 

NMR-SIM uses the predefined command molecule and endmol to set up the individual spin systems or molecules within the spin system definition file. All statements relating to nuclei, interactions and initial spin states are defined in a block that begins with the command molecule and ends with the command endmol. All the statements that occur in this block only relate to that particular spin system. Each spin system defined in this way has to be named, the name appearing on the same line as the molecule command separated by a space. Finally the relative concentration of the individual spin systems is added, separated from the name by a space. 

Suppose the chemical potential is not equalized then there are two points, ri and r2, with /r(r i) < p(r2). Consider these two points to be separate systems with N = p r )dr and N2 = p r2)dr electrons, respectively. From the definition of the local chemical potential [Eq. (105)] and the assumption that p(ri) < p(r2), it follows that the state with Ni + dN= [p(r0 + 8p(r)]dr and 7V2—dTV = [/9(r2)—8/9(r)]dr electrons has lower energy than the initial state. Electronic states, then, are stable only if the chemical potential is constant throughout the system. 

Aerosol particles in the atmosphere usually carry with them some moisture. The amount of water associated with the aerosol depends on the relative humidity. Increasing the relative humidity condenses more water onto the particles, until finally, when the vapor pressure of water exceeds the saturation point, a certain number of particles grows into fog or cloud droplets. Meterologists call these particles condensation nuclei, or simply nuclei. Fogs and clouds are treated as separate systems and are not included in the normal definition of the atmospheric aerosol, even though they represent an assembly of particles suspended in air and thus constitute an atmospheric colloid. The smoothness of the transition from an assembly of aerosol particles to one of cloud elements makes it difficult to define a boundary line between both colloids. Due to the overlap of size ranges of the particles in both systems, any division will be rather arbitrary. 

A typical high-performance liquid chromatography (HPLC) system consists of two pumps, a mixer, an injector, a guard colunm, a separation column, a detector, an electrospray nozzle, and a mass spectrometer. The two pumps and the mixer allow establishment of desired solvent gradients. The injector is used to inject a small amount of sample into the column. In all separation techniques, definition of a small volume of injected sample is cmcial in preventing adverse broadening of peaks and a consequent loss in compmient resolution. The injected compounds are separated in the separation colunm and then detected via a simple detector or a mass spectrometer. All of the parts of the HPLC system have been miniaturized, and several research groups have been able to demonstrate on-chip HPLC separations. 

The dual-phase (DP) membrane used in analytical separation usually consists of a polymer, or in some cases a ceramic solid-phase support impregnated with a fluid (i.e., gaseous or liquid phase). If the fluid is air the DP membrane is known as a gas-diffusion membrane. DP membranes incorporating a liquid phase can be considered in a broader sense as liquid membranes. The liquid phase in a liquid DP membrane can be identical to the feed and/or receiver solution (e.g., dialysis membranes, membrane-assisted LLE (MALLE)) or it can form a third immiscible liquid phase in the membrane separation system (e.g., supported and polymer liquid membranes). Membranes incorporating a liquid phase immiscible with the feed and receiver solutions are usually referred to as liquid membranes. This narrower definition of liquid membranes, currently accepted in the literature, will be used in subsequent discussions. The... 

The retention and the selectivity of separation in RP and NP chromatography depend primarily on the chemistry of the stationary phase and the mobile phase, which control the polarity of the separation systems. There is no generally accepted definition of polarity, but it is agreed that it includes various selective contributions of dipole-dipole, proton-donor, proton-acceptor, tt-tt electron, or electrostatic interactions. Linear Free-Energy Relationships (LFER) widely used to charactaize chemical and biochemical processes were successfiiUy apphed in liquid chromatography to describe quantitative structure-retention relationships (QSRR) and to characterize the stmctural contributions to the retention and selectivity, using multiple linear correlation, such as Eq. 

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