Larson equation

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

These fundamental equations apply to many systems involving diserete entities aerosols, moleeules, and partieles, even people. A full review of their derivation of these equations is to be found in Randolph and Larson (1988), who have pioneered their applieation to industrial erystallizers in partieular. 

For solution of the population balanee equation, many forms exist for the partiele disruption terms Ba and Da respeetively (Randolph and Larson, 1988 Petanate and Glatz, 1983) but a partieularly simple form, whieh requires no integration of a fragment distribution, is the two-body equal-volume breakage funetion. It is assumed that eaeh partiele breaks into two smaller pieees, eaeh of half the original volume from whieh it follows that... 

The following equation of moments (Randolph and Larson, 1988) for an ideal MSMPR erystallizer... 

The population balanee eoneept enables the ealeulation of CSD to be made from basie kinetie data of erystal growth and nueleation and the development of this has been expounded by Randolph and Larson (1988), as summarized in Chapters 2 and 3. Bateh operation is, of eourse, inherently in the unsteady-state so the dynamie form of the equations must be used. For a well-mixed bateh erystallizer in whieh erystal breakage and agglomeration may be negleeted, applieation of the population balanee leads to the partial differential equation (Bransom and Dunning, 1949)... 

Larson and Garside (1973) extended the eontrolled eooling approaeh to provide simplified equations to prediet programmed evaporative erystallization polieies based on the population balanee. 

R. G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston (1988). 

Larson RG (1988) Constitutive equations for polymer melts and solutions, Butter-worths, Boston, p 256... 

Pioneering work on the desulphonylation of jS-ketosulphones was carried out by Corey and Chaykovsky - . This reaction was part of a sequence which could be used in the synthesis of ketones, as shown in equation (53). The main thrust of this work was in the use of sulphoxides, but Corey did stress the merits of both sulphones and sulphonamides for different applications of this type of reaction. The method soon found application by Stetter and Hesse for the synthesis of 3-methyl-2,4-dioxa-adamantane , and by House and Larson in an ingenious synthesis of intermediates directed towards the gibberellin skeleton, and also for more standard applications . Other applications of the method have also been madealthough it does suffer from certain limitations in that further alkylation of an a-alkyl- -ketosulphone is a very sluggish, inefficient process. Kurth and O Brien have proposed an alternative, one-pot sequence of reactions (equation 54), carried out at — 78 to — 50°, with yields better than 50%. The major difference between the two routes is that the one-pot process uses the desulphonylation step to generate the enolate anion, whereas in the Corey-House procedure, the desulphonylation with aluminium amalgam is a separate, non-productive step. 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

Larson, R. G. Constitutive Equations for Polymeric Melts, Butterworth, Stoneham, MA, 1988, pp. 46-53. 

The MRL was based on a hepatic NOAEL of 3 ppm chloroform administered for 6 hours a day for 7 consecutive days to mice (Larson et al. 1994c). Female mice exposed to 100 or 300 ppm exhibited centrilobular hepatocyte necrosis and severe diffuse vacuolar degeneration of midzonal and periportal hepatocytes, while exposure to 10 or 30 ppm resulted in mild-to-moderate vacuolar changes in centrilobular hepatocytes. Decreased eosinophilia of the centrilobular and midzonal hepatocyte cytoplasm relative to periportal hepatocytes was observed at 30 ppm. Livers of mice in the 1 and 3 ppm groups did not differ significantly from control animals and were considered to be NOAELs for liver effects. The NOAEL of 3 ppm was converted to the Human Equivalent Concentration (HEC) as described in Equation 4-10 in Interim Methods for Development of Inhalation Reference Concentrations (ERA 1990b). This calculation resulted in a NOAEL hec] of 3 ppm. An uncertainty factor of 30 (3 for extrapolation from animals to humans and 10 for human variability) was applied to the NOAEL hec] value, which resulted in an MRL of 0.1 ppm. 

Another approximate evaluation of Equation 19 has been given by Larson and Garside (9)... 

The dynamic model used in predicting the transient behavior of isothermal batch crystallizers is well developed. Randolph and Larson (5) and Hulburt and Katz (6) offer a complete discussion of the theoretical development of the population balance approach. A summary of the set of equations used in this analysis is given below. 

In some areas, e.g. aerosol physics and crystallisation, population balance models are used in situations when a number balance equation is required as well as conventional mass and energy balances. Randolph and Larson review this theory as it applies specifically to particulate systems [15], whilst Froment and Bischoff [16] present population balance equations in the context of an extension of classical RTD theory. 

For the calculations, averages of the results of the two 5. -equilibrium models of Ca2+ = 35 p.p.m., Mg2+ = 7 p.p.m., and alkalinity = 1.55 X 10 3 equiv./liter are used. Solubility data of Larson and Buswell (11), carbon dioxide solubility data of Hamed and Davies (2), and the carbonate ionization data of Hamed and Hammer (3) and Hamed and Scholes (4) are used. Linear interpolations are made for dolomite between pK(soly) = 16.3(5°C.) and 17.0(25°C.). Equations outlining the calcite and dolomite calculations are ... 

Using Equations 3.3a and b, Englezos et al. (1987a) calculated the critical radius of methane hydrate to be 30-170 A. In comparison, critical cluster sizes using classical nucleation theory are estimated at around 32 A (Larson and Garside, 1986), while computer simulations predict critical sizes to be around 14.5 A (Baez and Clancy, 1994 Westacott and Rodger, 1998 Radhakrishnan and Trout, 2002). 

Temperature-dependent equilibrium constants for carbonate are shown in table 13.1 (Larson and Buswell, 1942). Equations 13.4 and 13.5 can result in more alkaline waters due to the generation of OH- this is typical of what may be found in lakes and streams due to high carbonates in the drainage basin. The percolation of H2O through soils results in the enrichment of CO2 from plant and microbial decay processes forming H2CO3 which can... 

Temkin and Pyzhev also checked Equation (1), using a = 0.5 on the experimental data published by Larson and Tour (68), which cover a range of pressures up to 100 atm., and later Emmett and Kummer (66)... 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

Larson RG, "Constitutive Equations for Polymer Melts and Solutions", Butterworth, London, 1988. 

The development of equations that successfully predict multicomponent phase equilibrium data from binary data with remarkable accuracy for engineering purposes not only improves the accuracy of tray-to-tray calculations but also lessens the amount of experimentation required to establish the phase equilibrium data. Such equations are the Wilson equation (13), the non-random two-liquid (NRTL) equation (14), and the local effective mole fractions (LEMF) equation (15, 16), a two-parameter version of the basically three-parameter NRTL equation. Larson and Tassios (17) showed that the Wilson and NRTL equations predict accurately ternary activity coefficients from binary data Hankin-son et al. (18) demonstrated that the Wilson equation predicts accurately... 

Many improvements or modifications to the UCM model can be found in the literature. These csm lead to various classes of constitutive equations keeping the differential nature of the equation [2, 3, 35]. As pointed out by Larson [43], a systematic classification of these can be performed by rewritting the UCM model as ... 

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