Errors dispersion phasing

For most purposes, the integration can be extended from 0 to oo without introducing any significant error. Therefore, the following quantities can be defined for the dispersed phase ... 

In this equation. Act is taken as the maximum possible surface tension lowering. Hence for a solute-free continuous phase, Aa is the difference between the interfacial tension for the solvent-free system and the equilibrium interfacial tension corresponding to the solute concentration in the dispersed phase. Equation (10-6) indicates a strong effect of the viscosity ratio k on the mass transfer coefficient as found experimentally (LI 1). For the few systems in which measurements are reported (Bll, Lll, 04), estimates from Eq. (10-6) have an average error of about 30% for the first 5-10 seconds of transfer when interfacial turbulence is strongest. 

We now consider a 40% silicone oil premixed emulsion dispersed in an aqueous phase. In Fig. 9 the evolution of mean diameter is plotted as a function of the applied shear rate. The dispersed phase volume fraction is kept constant at 75%, while the emulsifier concentration in the continuous medium is varied from 15 wt % to 45 wt %. The error bars show the distribution width deduced from the measured uniformity. At a given shear rate, smaller droplets with lower uniformity are produced (see Fig. 9) when surfactant concentration increases. For example at 45% of Ifralan 205 the uniformity never exceeds 15% whatever the applied shear rate, whereas it is of the order of 25% for 15% of Ifralan 205. Some microscope pictures of the emulsions obtained are given in Fig. 10. To understand the evolution, we may argue that the continuous phase viscosity increases... 

This second edition is vastly improved over the first. Numerous small errors present in the original were rooted out and banished, both in the text and the figures. The clarity of numerous figures was improved, and new tables were added and mathematical nomenclature made more uniform. This second edition includes two new chapters, one on macromolecular crystallization, and a second on X-ray diffraction data collection. Refinement and anomalous dispersion phasing are treated somewhat more extensively. More than 35 new figures have been incorporated. 

In an early attempt to calculate the phase fractions in an approximate implicit volume fraction-velocity-pressure correction procedure, Spalding [176, 177, 178, 180] calculated the phase fractions from the respective phase continuity equations. However, experience did show that it was difficult to conserve mass simultaneously for both phases when the algorithm mentioned above was used. For this reason, Spalding [179] suggested that the volume fraction of the dispersed phase may rather be calculated from a discrete equation that is derived from a combination of the two continuity equations. An alternative form of the latter volume fraction equation, particularly designed for fluids with large density differences, was later proposed by Carver [26]. In this method the continuity equations for each phase were normalized by a reference mass density to balance the weight of the error for each phase. 

No Dispersed phase Flow type Solution method Error, % Sources... 

For a simultaneous determination of fluid and particle velocity by LDA the fluid flow has to be additionally seeded by small tracer particles which are able to follow the turbulent fluctuations. The remaining task is the separation of the Doppler signals resulting from tracer particles and the dispersed phase particles. In most cases this discrimination is based on the scattering intensity combined with some other method in order to reduce the error due to the Gaussian beam effect The discrimination procedure introduced by Durst (1982) for example, was based on the use of two receiving optical systems and two photodiodes... 

A comparison of (162) with the solution of (161) leads to the following definition of the dispersion or phase error or phase-lag and the dissipative error ... 

Several studies have attempted to correlate the characteristics of the final products to the initial structures prior to polymerization. It must be reminded that the accurate determination of a microemulsion structure is rather difficult. In particular, when performing scattering experiments, which in principle provide the droplet size, the system must be diluted. However, the dilution procedure is not trivial because of the partitioning of the components of the microemulsion between continuous and dispersed phases. Experiments performed at finite concentration can suffer by a large error, in particular in the vicinity of a critical point where the radiation scattering probes critical fluctuations with a characteristic length much larger than the droplet radius [4]. 

The models of van Baten and Krishna (2004) and Vandu et al. (2005), for gas-Uquid bubble flows, showed little or no agreement with the experimental results. Van Baten and Krishna (2004) developed their model (Eq. 7.1.1) over a wide range of parametric values (ID = 1.5-3 mm, Luc = 0.015-0.05 m). Their model underestimated the current mass transfer coefficients for all the channels. It is worth noting that in this work the length of the unit cells (Luc) and the velocity of the dispersed phase (Up) were one order of magnitude lower than those used by Van Baten and Krishna (2004). In the model by Vandu et al. (2005) (Eq. 7.1.2), which was evaluated for channel sizes from 1 to 3 mm ID and unit cell lengths from 5 to 60 mm, the only contribution on the mass transfer coefficient is by the film. The kuu obtained for 0.5 and 1 mm ID channel seem to fall within the predictions of their model (for C = 8.5), whilst mass transfer is underestimated in all cases for the 2 mm ID channel with a relative error from 40 to 60 %. The discrepancies between the experimental results and the gas-liquid models may be attributed to the more complex hydrodynamics in the liquid-liquid systems. In addition, there is less resistance to mass transfer by diflusion within a gas plug compared to a liquid one. 

In the dispersed phase, only the unsteady state method is applicable. The tracer must be introduced in the form of marked drops and its concentration recorded at two positions downstream from the point where the drops were introduced. Conventional techniques proved to be too inaccurate for this purpose and special probes were therefore developed which coalesced the dispersed phase and measured the light absorption directly in the column. It was found that any attempt to withdraw drops from the column and carry out time-dependent measurements on them led to unacceptable errors. 

Ideally, the substantial reduction in dispersed phase size morphology is evidence of improvements in interfacial adhesion and hence will translate into inprovements in mechanical properties. Tensile properties of the blends are displayed in Figures 1-2 inconsistency in the results as a function of compatibilizer fraction is much larger than the error bars wottld suggest. Error bars were calculated by running multiple samples cut from the same sheet, hence it is likely that molding inconsistencies are the cause of the disagreement. 

Errors in the molecular weight data from HPSEC are usually due to improperly prepared samples, column dispersity, or flow rate variations. The sample to be analyzed should be completely dissolved in the mobile phase and filtered prior to injection onto the column. A plugged column inlet frit will invalidate results. In addition, do not load the column with excess sample. Column overloading affects the accuracy of data by broadening peaks, reducing resolution, and increasing elution volume. For best results, the concentration of the injected sample should be as low as possible while still providing adequate... 

Thus either the penetration theory or the film theory (equation 10.144 or 10.145) respectively can be used to describe the mass transfer process. The error will not exceed some 9 per cent provided that the appropriate equation is used, equation 10.144 for L2 jDt > n and equation 10.145 for L2/Dt < n. Equation 10.145 will frequently apply quite closely in a wetted-wall column or in a packed tower with large packings. Equation 10.144 will apply when one of the phases is dispersed in the form of droplets, as in a spray tower, or in a packed tower with small packing elements. 

The main consequences are twice. First, it results in contrast degradations as a function of the differential dispersion. This feature can be calibrated in order to correct this bias. The only limit concerns the degradation of the signal to noise ratio associated with the fringe modulation decay. The second drawback is an error on the phase closure acquisition. It results from the superposition of the phasor corresponding to the spectral channels. The wrapping and the nonlinearity of this process lead to a phase shift that is not compensated in the phase closure process. This effect depends on the three differential dispersions and on the spectral distribution. These effects have been demonstrated for the first time in the ISTROG experiment (Huss et al., 2001) at IRCOM as shown in Fig. 14. 

The selection of a suitable emulsifying agent and its appropriate concentration are matters of experience and of trial and error. It is not necessary to use emulsifier amounts above the required quantities to produce complete interfacial films, unless an increase in the viscosity of the dispersion medium is intended. Reducing the interfacial tension makes emulsification easy but does not by itself prevent coalescence of the particles and resultant phase separation. Frequently, combinations of two or more emulsifying agents are used [2] to (a) adequately reduce the interfacial tension, (b) produce a sufficiently rigid interfacial film, and (c)... 

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