Fractional flow

Pulp exiting the conditioners is diluted by usiag process brine to a soHds content of 30—35 wt % for use as feed for the flotation cells. In some plants, the coarse- and fine-fraction flows are floated separately. In most plants, the two fractions are recombiaed and the flotation is conducted ia a common operation. 

FIGURE 16.1 Preparative SEC of short-chain (scb) branched glucans of small" (<3S /u.m) starch granules of potato species Ostara separated on Sephacryl S-1000 (88 X 2.6 cm) eluent 0.005 M NaOH the normalized chromatogram (area = 1.0) was constructed from an off-line determined carbohydrate content of succeeding 5-ml fractions flow rate 0.67 ml/mln V d = 185 ml, V, = 460 ml fraction I high dp fraction fraction 2 low dp fraction. 

FIGURE 16.6 Waxymaize ( ) and amylomaize ( ) separated on Sephacryl S-IOOO (95 x 1.6 cm) pooled in 3-ml fractions for further analysis normalized (area = 1.0) eluogram profiles (ev) constructed from an off-line determined mass of carbohydrates for each of the 3 ml fractions flow rate 0.41 ml/ min = 75 ml, V , = 162 ml eluent 0.005 M NaOH. 

For the purpose of illustration, let us consider the degradation behavior of a hypothetical monodisperse polymer fraction flowing along the central streamline. 

The fractional flow rate, (1 - f) Lu,n> is then also the volumetric flow passing through the well-mixed region of settler phase volume, Vmix. The flows leaving the plug-flow and well-mixed regions, Xsp and Xsmn. respectively, then combine to give the actual exit concentration from the settler, X . 

Zhou et al. [55], The most effective method to assess the capacity is the flow simulation which includes volumetric formulas and more reservoir parameters rather than other methods [56], Mass balance and constitutive relations are accounted in mathematical models to capacity assessment and dimensional analysis consists of fractional flow formulation with dimensionless assessment and analytical approaches [33], From the formulations demonstrated by Okwen and Stewart for analytical investigation, it can be deduced that the C02 buoyancy and injection rate have affected the storage capacity [57], Zheng et al. have indicated the equations employed in Japanese and Chinese methodology and have noted that some parameters in Japanese relation can be compared to the CSLF and DOE techniques [58]. 

In order to evaluate pump flow rate reproducibility and pulsation, one method is commonly used to assess gradient formation capability. A certain amount of an analyte with adequate molar absorptivity at the wavelength employed for detection is introduced into one of the mobile phases employed to create the gradient. In the case described, 5% acetone was introduced into the mobile phase, distributed to the system by pump B. No UV-absorbing analyte was introduced into mobile phase A. The fractional flow rate of pump B relative to the total flow rate of the system (mandated by the sum of the flow rates of pumps A and B) was increased in individual steps to account for 0, 3,6,12.5,25, 50, and 100% fractional rates. The total flow for the system was maintained at 300 /jL/ min (for 24 columns), resulting in a per column flow rate of 12.5 /iL/min/column. 

In the model described in this work every effort has been made to ensure that it embodies physically meaningful parameters. It is inevitcible, however, that some simplistic idealizations of the physical processes involved in GPC must be made in order to arrive at a system of equations which lends itself to mathematical solution. The parameters considered are, the axial dispersion, interstitial volume fraction, flow rate, gel particle size, column length, intra-particle diffusivity, accessible pore volume fraction and mass transfer between the bulk solution eind the gel particles. 

The higher boiling aqueous product fraction flows downwards through the lower distillation section, 10, to a reboiler, 15, where it is heated by an electrical heater. A portion of this higher-boiling aqueous product is withdrawn via an exit line, 15, as shown, and the remainder of the aqueous distillation reaction product is returned to the reactive distillation column, 10, by a reboiler return line. 

By coupling flow field-flow fractionation (flow FFF) to ICP-MS it is possible to investigate trace metals bound to various size fractions of colloidal and particulate materials.55 This technique is employed for environmental applications,55-57 for example to study trace metals associated with sediments. FFF-ICP-MS is an ideal technique for obtaining information on particle size distribution and depth profiles in sediment cores in addition to the metal concentrations (e.g., of Cu, Fe, Mn, Pb, Sr, Ti and Zn with core depths ranging from 0-40 cm).55 Contaminated river sediments at various depths have been investigated by a combination of selective extraction and FFF-ICP-MS as described by Siripinyanond et al,55... 

It is necessary to note that the matrix K does not deal with the hold-ups at all. The same rule for its construction was suggested in [8,9] on the basis of particle fractions flows balance only, without referring to Markov chains models. Here it was obtained as a particular case of the developed model, which is presented by the matrix P and allows describing also the evolution of hold-ups in general case provided the smaller matrices of P are known. 

For small values of A, the return flow fraction flowing across the flight tip increases, thus increasing the potential for dispersion. This is in agreement with experimental results, e. g., Fig. 9.11. The return flow fraction also increases with a larger gap width s. However, the absolute shear and extension rates are then lower, which can have an adverse effect on dispersion. 

Volume fraction Flow resistance parameter Obstruction factor (rate equation)... 

The constants k are called the fractional flow rates. They have the dimension of time-1 and they are defined as follows ... 

In contrast to the clearance, the fractional flow rates indicate the direction of the flow, i.e., kji kij, the first subscript denoting the start compartment, and the second one, the ending compartment. The fractional flow rates and the volumes of distribution are usually called microconstants. 

When the volume of the compartment being cleared is constant, the assumption that the fractional flow rate is constant is equivalent to assuming that the clearance is constant. But in the general case, in which the volume of distribution cannot be assumed constant, the use of the fractional flow rates k is unsuitable, because the magnitude of k depends as much upon the volume of the compartment as it does upon the effectiveness of the process of removal. In contrast, the clearance depends only upon the overall effectiveness of removal, and can be used to characterize any process of removal whether it be constant or changing, capacity-limited or supply-limited [308]. 

For each compartment in the configuration, apply the mass-balance law to obtain the differential equation expressing the variation of amount per unit of time. In these expressions, constant or variable fractional flow rates k can be used. 

In the special case of a mammillary compartmental configuration, the above relation allows one to express the volume of distribution in peripheral compartments as functions of the fractional flow rates and the volume of distribution of the central compartment Vj = [kij/kji] V) for j = 2,m. Substituting this relationship in (8.6), we obtain... 

The deterministic model with random fractional flow rates may be conceived on the basis of a deterministic transfer mechanism. In this formulation, a given replicate of the experiment is based on a particular realization of the random fractional flow rates and/or initial amounts 0. Once the realization is determined, the behavior of the system is deterministic. In principle, to obtain from the assumed distribution of 0 the distribution of common approach is to use the classical procedures for transformation of variables. When the model is expressed by a system of differential equations, the solution can be obtained through the theory of random differential equations [312-314]. 

Besides the deterministic context, the predicted amount of material is subjected now to a variability expressed by the second equation. This expresses the random character of the fractional flow rate, and it is known as process uncertainty. Extensive discussion of these aspects will be given in Chapter 9. 

Figure 8.2 shows E[q(t) and E[q(t) yJVar [q (f)] with q0 = 1 and k -Cani(2, 2). Noteworthy is that confidence intervals are present due to the variability of the fractional flow elimination rate k. This variability is inherent to the process and completely different from that introduced by the measurement devices. ... 

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