Comparison of volumes

The basic equations for the reactors have already been seen as  

Analyzing the equations of PFR, CSTR, and batch reactors represented by Equations 16.1-16.3, one notes that the area under the curve AC represents the integral in the PFR equation, whereas the rectangle area ABCD represents the CSTR equation. The rate related to the outlet concentration in the CSTR is equal to the average concentration in the tank, because perfect mixture is considered. 

For a given constant temperature and reactions taking place at constant volume, these areas are directly proportional to the space time and also to the average residence time in the reactors, assuming the same inlet volumetric flow and the same final conversion. Therefore, the areas represent the reactors volume. 

If the reaction takes place at variable volume, then the average residence times are different from the space times and as a consequence, the volumes are not proportional. 

Let us consider some cases and compare CSTR and PFR volumes in order to verify the most significant comparison criteria. 

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FIG. 21 Effective diffusion coefficients from Refs. 337 and 193 showing comparison of volume average results (Ryan) with models of Maxwell, Weisberg, Wakao, and Smith for isotropic systems (a), and volume averaging calculations (solid lines) and comparison with data for anisotropic systems (b). (Reproduced with kind permission of Kluwer Academic Publishers from Ref. 193, Fig. 3 and 12, Copyright Kluwer Academic Publishers.)... 

Figure 21 Comparison of volume map for HRV-14 generated from x-ray data (left) and small-molecule energy-minimized structures (right). 

Table 2. Comparison of volume fraction and inter pore distance, IPD, for macroporous epoxies prepared with 6 and 7.5 wt % hexane via the CIPS technique caicuiated either with the classical method from SEM micrographs or based on density measurements...

Figure 21.2 Comparison of volume fraction transformed, , in the interior of a semi-...

Figure 6. Comparison of volume change for potassium in liquid ammonia (

The V—F—V angle of 150° in the vanadium pentafluoride structure lies between those for the niobium and rhodium compounds. Thus, the structure cannot be described in terms of either of the close-packed arrangements. Although in the structure some approach toward close-packed planes of fluorine atoms can be seen, a comparison of volume per formula unit for the pentafluorides shows that for the second and... 

Figure 8.3 Comparison of volumes irradiated by an electron beam (left) and emission volumes from the corresponding lines (right) for different incident electron energies and target materials (taken from F. Maurice.

Figure 22 Comparison of Volume, Surface Area, and AreaAtolume Ratio Using Eixperimental and Computational Techniques...

Fig. 5. Comparison of volume normal distribution function with data for = 0.001 0.005 [after Chen and Middleman (C8)].

Figure 7.30 Comparison of volume specific productivity at 99.9% purity.

Figure 7.32 Comparison of volume-specific productivity (VSP) between two batch modes.

Tip Calculations with the aid of the initial and final volume lead to the equilibrium volume of 40 ml C02 and 20ml CO, the comparison of volume... 

Generally, comparisons of volume of distribution are made by use of a parameter designated as the volume of distribution at steady state (Vss), which reflects the sum of the volumes of all the pools into which a drug may distribute. Vgg can be calculated from the area under the moment curve (AUMC) and the area under the curve (AUC) as defined by Benet and Galeazzi (11) ... 

Fig. 6.5-1S Comparison of volume of compacts (left) produced with the equipment shown in Fig. 6.5-17 and a conventional bale (right) from the same material [6.5.2.3]...

Fig. 6.5-17 Mobile hay collection equipment with high pressure ram press [6.5.2.3]. The cylindrical compacts are transferred into an open car by pushing them through the transport pipe on top of the machine Fig. 6.5-18 Comparison of volume of compacts (left) produced with the equipment shown in Fig. 6.5-17 and a conventional bale (right) from the same material [6.5.2.3]...

Table III. Comparison of Volume Fractions of PS Blocks Determined from PS Contents and Electron Micrographs...

When specifying the volume of a gas, the pressure and temperature must be specified because the volume of a gas depends strongly on both temperature and pressure (assuming that it is in an expandable container). In order to avoid specifying a different set of conditions for each measurement, an engineer can convert to standard temperature and pressure (typically 1 atm and OoC) which is common to all measurements. This allows direct comparisons of volume measurements, but also requires that one convert back to the actual conditions present in the system before the value can be used. 

Lee, M. S., and Olson, M. A. (2013). Comparison of volume and surface area nonpolar solvation free energy terms for implicit solvent simulations,/. Chem. Phys. 139, pp. 044119 1-6. 

Table 1. Comparison of volumes and histological characteristics of the vomeronasal epithelium between human fetal and adult samples...

Where R is the rate matrix that describes the NOE interactions across the system, and are the two NOE mixing and A(0) is the initial magnetization. To simplify this equation, a Taylor series expansion of the exponential can be made. Usually, only the first few terms in the expansion are kept for the approximation. The first term approximation is equivalent to the two-spin approximation (5,14). At realistic mixing times (50 ms or more), the Taylor series approximation also yields systematic error in determining the inter-proton distances (9). Figure 1 shows comparison of volumes simulated from the two-term Taylor series approximation and an exact rate-matrix calculation for the Dickerson dodecamer... 

Byun JH, Kim TK, Lee SS et al (2003) Evaluation ofthe hepatic artery in potential donors for living donor liver transplantation by computed tomography angiography using multidetector-row computed tomography comparison of volume rendering and maximum intensity projection techniques. J Comput Assist Tomogr 27 125-131... 

In this equation, the volume of the individual atom (i)is Vf and k indicates the atom type. The mean volume of that atom type is V, and is the standard deviation of that atom type. When an atom has a larger than average volume, it has a positive Z-score. The opposite is true for a smaller than average atom that would have a negative Z-score the ideal Z-score is zero. The second Z-score, called the Zrmsdj involves the comparison of volumes for all atoms or for groups of atoms (in residues). The Zrmsd is calculated as in Eq. 

Comparisons of volumes of activation in Table 4.2 from the work of Lawrance and Stranks provide some further insights. The authors suggest that the small values of AV for Ni(II) and Cr(III) imply an intramolecular... 

Edwards AJ, Wells IP, Roobottom CA (2005) Multidetector row CT angiography of the lower limb arteries a prospective comparison of volume-rendered techniques and intra-arterial digital subtraction angiography. Clin Radiol... 

Other investigations primarily in the linear range have been made by Rpbinet and Buvet and, extending to pressures where nonlinear effects would be expected, by Martynyuk and Findley. However, there appear to have been no comparisons of volume creep data with the Kovacs nonlinear theory. 

FIGURE 38.35 Comparison of volume (VLD) vs. anode (ALD) limited discharges of aluminum/air reserve power unit. (Courtesy of Alupower, Inc.)... 

Table 8 presents a series of calculated and experimental values for partial volumes of small molecules and polymers, together with a quantitative comparison of volumes. The table covers the most important classes of organic compounds and their derivatives, with special respect to biochemical model compounds. 

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