Discrete bubble method

The class of DPMs and DEMs has been extended to discrete bubble methods (DBMs) for bubbly flows where in addition to bubble coalescence and breakup also the Hft force and added mass have to be taken into account. 

Sha et al [130, 131] developed a similar multifluid model for the simulation of gas-liquid bubbly flow. To guarantee the conservation of mass the population balance part of the model was solved by the discrete solution method presented by Hagesaether et al [52]. The 3D transient simulations of a rectangular column with dimensions 150 x 30 x 2000 (mm) and the gas evenly distributed at the bottom were run using the commercial software CFX4.4. For the same bubble size distribution and feed rate at the inlet, the simulations were carried out as two, three, six and eleven phase flows. The number of population balance equations solved was 10 in all the simulations. It was stated that the higher the number of phases used, the more accurate are the results. 

Hagesffither et al [28, 29, 30] extended the model by Luo and Svendsen [74], but the resulting breakage model was still not completely conservative. To ensure number and mass conservation they thus adopted a numerical procedure redistributing the bubbles on pivot points in accordance with the discrete solution method [94]. 

The main premise of these methods is that one in practice is not necessarily interested in the number density probability /i, but rather in the number density Ni (i.e., the number of bubbles of a particular size or size interval per unit volume). The methods within this category either divide the size domain into finite regions on which the number density is integrated to provide balances between the number of particles in each bin, or solve the balance equations accurately for a limited number of discretization points on the size domain. 

The above expressions contain the continuous bubble number probability density, f d,t). The source terms must be expressed entirely in terms of the dependent variable IVj. This can be achieved by using the mean value theorem [151]. Note, as mentioned before, at this point the discrete method of Ramkrishna [151] may deviate slightly from the multi-group method. 

An overview of the different types of discontinuity used in automatic methods and their characteristics is presented in Table 7.1. The most common discontinuity in discrete and robotic methods is the absence of flow, which involves keeping the samples in separate vessels for measurement. On the other hand, automatic continuous methods use very different kinds of discontinuity or do not use one at all. The discrete nature of segmented methods is determined by the presence of bubbles and wash cycles as a means of avoiding carryover, whereas that of unsegmented methods is dictated by the manner in which the sample —and reagent— is introduced into the system. There is only a single type of method using no discontinuity completely continuous flow analysis (CCFA). 

The problem of dynamic strength of a liquid was considered in a number of works, where pulse methods [l-6] were used. The amplitude of maximum tensile stresses achieved in liquid depends, as it shown in [2,3], on the parameters of rarefaction wave (RW) and on the parameters of initial gas-containing of liquid. The fast growth of cavitation nuclei in RW leading to the relaxation of tensile stresses in liquid in a time of order of 10 s [2-4]. Further two ways of the cavitation process development are possible (i) the bubble damped oscillations occur and (ii) the irreversible development of cavitation zone take place [5,6], which leads to the formation of fosuning structure and the liquid fracture into discret particles. The main principle structure peculiarities of process in the second case remain vague. The structure of flows which forms an axial explo-... 

A technique has been developed for the continuous measurement of emulsion surface tension based on the pressure necessary to form a bubble in liquid. Details of the method may be found in Schork and Ray [24]. With a laboratory prototype of the bubble tensiometer, it has been possible to measure surface tensions continuously to within 1 to 2% [24]. A commercial instrument based on these principles is now available. Figures 5.5 and 5.7 demonstrate the use of the bubble tensiometer to monitor the surface tension of methyl methacrylate emulsion during continuous and batch polymerization. It will be noted that during conversion oscillation the surface tension oscillated as well, in accordance with the discrete initiation mechanism often postulated to explain this phenomenon. 

A hierarchy of computational models is available to simulate dispersed gas-liquid-solid flows in three-phase slurry and fluidized bed reactors [84] continuum (Euler-Euler) method, discrete particle/bubble (Euler-Lagrange) method, or front tracking/capturing methods. While every method has its own... 

The heat exchanger is calculated by discretizing it. In the discretization points, either the bubble-dew point method or the equal enthalpy methods can be used to compute the heat transfer. The first one reduces to the second if there is no condensation or evaporation within the heat exchanger. Thus, we use the default method and obtain Figure 8.32. Related to the wet or dry wall, it is related to the condensation or not of the streams when hitting the tubes. 

The discrete phase simulation method described in Secs. 4.1 through 4.4 is capable of predicting the flow behavior in gas-liquid-solid three-phase flows. In this section, several simulation examples are given to demonstrate the capability of the computational model. First, the behavior of a bubble rising in a liquid-solid suspension at ambient pressure is simulated and compared to experimental observations. Then the effect of pressure on the bubble rise behavior is discussed, along with the bubble-particle interaction. Finally, a more complicated case, that is, multibubble formation dynamics with bubble bubble interactions, is illustrated. 

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