Academic literature on the topic 'Break-even coefficient'

The aim of the research is to selection general coefficient of operating leverage of enterprise and to great factor model of this.The subject of the research is operating risk of enterprise. Accordingly had study coefficients operating leverage, the fixed and variable costs, the conception of marginal profit and break-even points.Had mace comparative analysis of well-known coefficients of enterprise operating leverage in the article. Here are discussed relationship between coefficients of Operating Leverage, Marginal Profit, Break-even point, the production Margin of Safety and structure of costs.Generally accepted methods of analysis are used in the paper, such as analysis and synthesis, induction and deduction, quantitative and qualitative analysis methods, traditional analysis methods.Results of research had proofed that fixed costs and operating profit ratio can be recognition with General Coefficient of Operating Leverage of enterprise. Had great four-factors model of this indicator too, whose practical use will help the management of enterprises to explore positive and negative factors through the traditional analytical methods and to the adequate decisions will making.

This study aimed to analyze the break even point and the risk of laying ducks farm income in Sugih Waras Village, Belitang Mulya Subdistrict, Ogan Komering Ulu (OKU) Timur District. The research location is determined on purposive and make all laying duck farmers using cages 15 x 5 m and 15 x 6 m or raising ± 150 laying ducks in Sugih Waras Village as the respondents of this study (census methods). The results showed that laying ducks farmers in Sugih Waras Village reach the break even point at the time of production by 740 egg per month with a price is Rp 1.111.675,13 per egg. Laying ducks farm in Sugih Waras Village likely losses indicated by the variation coefficient of 1,026 and the lower limit revenue of -1.380.102,9.Keywords : break even point, the risk of income, laying ducks

AbstractIn this paper a 600 MW oxy-type coal unit with a pulverized bed boiler and a membrane-cryogenic oxygen separator and carbon capture installation was analyzed. A membrane-cryogenic oxygen separation installation consists of a membrane module and two cryogenic distillation columns. In this system oxygen is produced with the purity equal to 95%. Installation of carbon capture was based on the physical separation method and allows to reduce the CO2emission by 90%. In this work the influence of the main parameter of the membrane process – the selectivity coefficient, on the efficiency of the coal unit was presented. The economic analysis with the use of the break-even point method was carried out. The economic calculations were realized in view of the break-even price of electricity depending on a coal unit availability.

AbstractThe usual way of solving a solvable quintic equation has been to establish more equations than unknowns, so that some relation among the coefficients comes up, leading to the solutions. In this paper, a relation among the coefficients of a principal quintic equation is established by effecting a change of variable and inserting a root to the quintic equation, and then equating odd-powers of the resulting sextic equation to zero. This leads to an even-powered sextic equation, or equivalently a cubic equation; thus one needs to solve the cubic equation.We break from this tradition, rather factor the even-powered sextic equation in a novel fashion, such that the inserted root is identified quickly along with one root of the quintic equation in a quadratic factor of the form, u2− g2 = (u + g)(u − g). Thus there is no need to solve any cubic equation. As an extra benefit, this root is a function of only one coefficient of the given quintic equation.

The article examines the interrelation of market and liquidation value of the appraisal subject. It was established that the approach of “break-even” sale of a subject at liquidation value, which is predominating in the literature sources, shows the results essentially different from the ratio of the prices of free and accelerated sales that can be observed on the market. The article offers an alternative approach to definition of market value coefficient, which considers switching to liquidation value. This approach is based on functional dependence of the coefficient on the ratio of limited and unlimited period of exposition.

The new azo benzene polyurethane (CAPU) was synthesized by diazo coupling reaction with 1,4-Diaminobenzene, sodium nitrite, and R(-)-3-chloro-1,2-propanediol. The solubility was measured with several organic solvents. Some mechanical properties of CAPU, such as tensile strength and ratio of elongation at break, were investigated. The good thermal stability of the CAPU was shown by its absorption spectrum characterized by UV-Vis spectroscopy and its melting point of 113.5°C. Moreover, the refractive index of CAPU was obtained by the attenuated total reflection (ATR) method at different temperature and wavelength, by which the materials thermo-optic coefficient was calculated. It is shown that the material has a large thermo-optic coefficient and an obvious birefringence phenomena. And the dispersion coefficient was also calculated by the Sellmeyer equation. Because of thermo-optical properties mentioned above, the CAPU could be used to design and generate the thermo-optic switch and even all-optical switch.

The dissociative recombination (DR) of is a key process in interstellar chemistry. More than 30 experimental studies of the DR process have been published in the literature. The DR rate coefficient results obtained from these measurements, however, have not always been consistent. The outcome seems to depend on the experimental method, on the exact measurement procedure and sometimes even on the interpretation of the experimental data. In the past two decades, heavy-ion storage rings have become the working horse for DR measurements, as they provide a direct measurement of the DR products. Furthermore, storage ring measurements yield energy-resolved rate coefficients with unprecedented resolution that allow for detailed comparison with theory. DR results from different storage ring facilities have shown a remarkable consistency throughout the years and they provide additional information on break-up dynamics and internal excitation. In this study, we will review the storage ring DR measurements that have been carried out for .

This paper considers estimating a panel data simultaneous equations model under both coefficient and covariance matrix restrictions in a scenario where one or the other set of identifying restrictions may be invalid or may hold only weakly. We study the limiting properties of various estimators in an asymptotic framework, which takes both the cross-sectional dimensionNand the time dimensionTto infinity. In this setting as in the pure cross-sectional setup, the performance of the 2SLS estimator depends on the strength of the identifying conditions imposed on the coefficients of the model, and it fails to be consistent once these conditions break down sufficiently resulting in instruments that are too weakly correlated with the endogenous regressors. On the other hand, the between-group (BG) estimator is consistent and asymptotically normal even when coefficient restrictions fail, but it has the shortcoming that its precision depends only on variations in the cross-sectional dimension; and, hence, it is less efficient and has slower rate of convergence than alternatives, which make better use of the large time dimension. A GMM estimator, which combines the moment conditions of theBGestimator with that of the within-group IV estimator, is more robust to instrument weakness than 2SLS and is more efficient than theBGestimator, but it has a second-order bias even under strong instruments if the assumed covariance restrictions do not hold. To remedy the deficiency of the aforementioned estimators, we propose in this paper two new model averaging estimators, which are weighted averages of the GMM estimator and a bias-corrected GMM estimator. The two proposed estimators have weighting functions that depend on alternative transformations of the Bayesian Information Criterion (BIC), which is employed here to assess the validity of the covariance restrictions. We show that these new estimators have some nice robustness properties against possible failure of either the coefficient restrictions or the covariance restrictions.

Background. The purpose of this study was to identify whether the learning effect, fatigue, motivation, effort and/or sex-specific neural, physiological and morphological factors influenced the results of the test–retest reliability of tests to assess cognitive function. Methods. The sample included ten men (age 21.2 ± 0.4 years; body mass 79.5 ± 8.3 kg) and ten women (age 22.0 ± 1 years; body mass 60.0 ± 10.0 kg). Participants accomplished six tests (three for memory and three for attention) four times, i.e. two times (with 24 hours’ break) on successive days (teaching) and two times (with 48 hours’ break) on the third and fifth days (re-testing to assess the reliability). The reliability was assessed by calculating the average of the population, standard deviation, and intraclass correlation coefficient (ICC). Results. In males and females, measurements of attention function were highly reliable over time (ICC > .84). The ICCs for volume of spatial memory were above .79, for memory of even number recognition above .57 for both genders and for memory of figure recognition .00 for males and .79 for females. Conclusion. In young healthy males and females, measurements of attention function were highly reliable over time. Meanwhile, reliability for volume of spatial memory was good/high for both sexes, but reliability of memory for even number recognition was insufficient for both sexes and results from memory of figure recognition showed good reliability for women and insufficient reliability for men.

Four different plastics were tested: potato starch based plastic (TPS-P)–BIOPLAST GF 106/02; corn starch based plastic (TPS-C)–BioComp BF 01HP; polylactic acid (polylactide) plastic (PLA)—BioComp BF 7210 and low density polyethylene, trade name Malen E FABS 23-D022; as a petrochemical reference sample. Using the blown film extrusion method and various screw rotational speeds, films were obtained and tested, as a result of which the following were determined: breaking stress, strain at break, static and dynamic friction coefficient of film in longitudinal and transverse direction, puncture resistance and strain at break, color, brightness and gloss of film, surface roughness, barrier properties and microstructure. The biodegradable plastics tested are characterized by comparable or even better mechanical strength than petrochemical polyethylene for the range of film blowing processing parameters used here. The effect of the screw rotational speed on the mechanical characteristics of the films obtained was also demonstrated. With the increase in the screw rotational speed, the decrease of barrier properties was also observed. No correlation between roughness and permeability of gases and water vapor was shown. It was indicated that biodegradable plastics might be competitive for conventional petrochemical materials used in film blowing niche applications where cost, recyclability, optical and water vapor barrier properties are not critical.

Bones are multifunctional passive organs of movement that supports soft tissue and directly attached muscles. They also protect internal organs and are a reserve of calcium, phosphorus and magnesium. Each bone is covered with periosteum, and the adjacent bone surfaces are covered by articular cartilage. Histologically, the bone is an organ composed of many different tissues. The main component is bone tissue (cortical and spongy) composed of a set of bone cells and intercellular substance (mineral and organic), it also contains fat, hematopoietic (bone marrow) and cartilaginous tissue. Bones are a tissue that even in adult life retains the ability to change shape and structure depending on changes in their mechanical and hormonal environment, as well as self-renewal and repair capabilities. This process is called bone turnover. The basic processes of bone turnover are: • bone modeling (incessantly changes in bone shape during individual growth) following resorption and tissue formation at various locations (e.g. bone marrow formation) to increase mass and skeletal morphology. This process occurs in the bones of growing individuals and stops after reaching puberty • bone remodeling (processes involve in maintaining bone tissue by resorbing and replacing old bone tissue with new tissue in the same place, e.g. repairing micro fractures). It is a process involving the removal and internal remodeling of existing bone and is responsible for maintaining tissue mass and architecture of mature bones. Bone turnover is regulated by two types of transformation: • osteoclastogenesis, i.e. formation of cells responsible for bone resorption • osteoblastogenesis, i.e. formation of cells responsible for bone formation (bone matrix synthesis and mineralization) Bone maturity can be defined as the completion of basic structural development and mineralization leading to maximum mass and optimal mechanical strength. The highest rate of increase in pig bone mass is observed in the first twelve weeks after birth. This period of growth is considered crucial for optimizing the growth of the skeleton of pigs, because the degree of bone mineralization in later life stages (adulthood) depends largely on the amount of bone minerals accumulated in the early stages of their growth. The development of the technique allows to determine the condition of the skeletal system (or individual bones) in living animals by methods used in human medicine, or after their slaughter. For in vivo determination of bone properties, Abstract 10 double energy X-ray absorptiometry or computed tomography scanning techniques are used. Both methods allow the quantification of mineral content and bone mineral density. The most important property from a practical point of view is the bone’s bending strength, which is directly determined by the maximum bending force. The most important factors affecting bone strength are: • age (growth period), • gender and the associated hormonal balance, • genotype and modification of genes responsible for bone growth • chemical composition of the body (protein and fat content, and the proportion between these components), • physical activity and related bone load, • nutritional factors: – protein intake influencing synthesis of organic matrix of bone, – content of minerals in the feed (CA, P, Zn, Ca/P, Mg, Mn, Na, Cl, K, Cu ratio) influencing synthesis of the inorganic matrix of bone, – mineral/protein ratio in the diet (Ca/protein, P/protein, Zn/protein) – feed energy concentration, – energy source (content of saturated fatty acids - SFA, content of polyun saturated fatty acids - PUFA, in particular ALA, EPA, DPA, DHA), – feed additives, in particular: enzymes (e.g. phytase releasing of minerals bounded in phytin complexes), probiotics and prebiotics (e.g. inulin improving the function of the digestive tract by increasing absorption of nutrients), – vitamin content that regulate metabolism and biochemical changes occurring in bone tissue (e.g. vitamin D3, B6, C and K). This study was based on the results of research experiments from available literature, and studies on growing pigs carried out at the Kielanowski Institute of Animal Physiology and Nutrition, Polish Academy of Sciences. The tests were performed in total on 300 pigs of Duroc, Pietrain, Puławska breeds, line 990 and hybrids (Great White × Duroc, Great White × Landrace), PIC pigs, slaughtered at different body weight during the growth period from 15 to 130 kg. Bones for biomechanical tests were collected after slaughter from each pig. Their length, mass and volume were determined. Based on these measurements, the specific weight (density, g/cm3) was calculated. Then each bone was cut in the middle of the shaft and the outer and inner diameters were measured both horizontally and vertically. Based on these measurements, the following indicators were calculated: • cortical thickness, • cortical surface, • cortical index. Abstract 11 Bone strength was tested by a three-point bending test. The obtained data enabled the determination of: • bending force (the magnitude of the maximum force at which disintegration and disruption of bone structure occurs), • strength (the amount of maximum force needed to break/crack of bone), • stiffness (quotient of the force acting on the bone and the amount of displacement occurring under the influence of this force). Investigation of changes in physical and biomechanical features of bones during growth was performed on pigs of the synthetic 990 line growing from 15 to 130 kg body weight. The animals were slaughtered successively at a body weight of 15, 30, 40, 50, 70, 90, 110 and 130 kg. After slaughter, the following bones were separated from the right half-carcass: humerus, 3rd and 4th metatarsal bone, femur, tibia and fibula as well as 3rd and 4th metatarsal bone. The features of bones were determined using methods described in the methodology. Describing bone growth with the Gompertz equation, it was found that the earliest slowdown of bone growth curve was observed for metacarpal and metatarsal bones. This means that these bones matured the most quickly. The established data also indicate that the rib is the slowest maturing bone. The femur, humerus, tibia and fibula were between the values of these features for the metatarsal, metacarpal and rib bones. The rate of increase in bone mass and length differed significantly between the examined bones, but in all cases it was lower (coefficient b <1) than the growth rate of the whole body of the animal. The fastest growth rate was estimated for the rib mass (coefficient b = 0.93). Among the long bones, the humerus (coefficient b = 0.81) was characterized by the fastest rate of weight gain, however femur the smallest (coefficient b = 0.71). The lowest rate of bone mass increase was observed in the foot bones, with the metacarpal bones having a slightly higher value of coefficient b than the metatarsal bones (0.67 vs 0.62). The third bone had a lower growth rate than the fourth bone, regardless of whether they were metatarsal or metacarpal. The value of the bending force increased as the animals grew. Regardless of the growth point tested, the highest values were observed for the humerus, tibia and femur, smaller for the metatarsal and metacarpal bone, and the lowest for the fibula and rib. The rate of change in the value of this indicator increased at a similar rate as the body weight changes of the animals in the case of the fibula and the fourth metacarpal bone (b value = 0.98), and more slowly in the case of the metatarsal bone, the third metacarpal bone, and the tibia bone (values of the b ratio 0.81–0.85), and the slowest femur, humerus and rib (value of b = 0.60–0.66). Bone stiffness increased as animals grew. Regardless of the growth point tested, the highest values were observed for the humerus, tibia and femur, smaller for the metatarsal and metacarpal bone, and the lowest for the fibula and rib. Abstract 12 The rate of change in the value of this indicator changed at a faster rate than the increase in weight of pigs in the case of metacarpal and metatarsal bones (coefficient b = 1.01–1.22), slightly slower in the case of fibula (coefficient b = 0.92), definitely slower in the case of the tibia (b = 0.73), ribs (b = 0.66), femur (b = 0.59) and humerus (b = 0.50). Bone strength increased as animals grew. Regardless of the growth point tested, bone strength was as follows femur > tibia > humerus > 4 metacarpal> 3 metacarpal> 3 metatarsal > 4 metatarsal > rib> fibula. The rate of increase in strength of all examined bones was greater than the rate of weight gain of pigs (value of the coefficient b = 2.04–3.26). As the animals grew, the bone density increased. However, the growth rate of this indicator for the majority of bones was slower than the rate of weight gain (the value of the coefficient b ranged from 0.37 – humerus to 0.84 – fibula). The exception was the rib, whose density increased at a similar pace increasing the body weight of animals (value of the coefficient b = 0.97). The study on the influence of the breed and the feeding intensity on bone characteristics (physical and biomechanical) was performed on pigs of the breeds Duroc, Pietrain, and synthetic 990 during a growth period of 15 to 70 kg body weight. Animals were fed ad libitum or dosed system. After slaughter at a body weight of 70 kg, three bones were taken from the right half-carcass: femur, three metatarsal, and three metacarpal and subjected to the determinations described in the methodology. The weight of bones of animals fed aa libitum was significantly lower than in pigs fed restrictively All bones of Duroc breed were significantly heavier and longer than Pietrain and 990 pig bones. The average values of bending force for the examined bones took the following order: III metatarsal bone (63.5 kg) <III metacarpal bone (77.9 kg) <femur (271.5 kg). The feeding system and breed of pigs had no significant effect on the value of this indicator. The average values of the bones strength took the following order: III metatarsal bone (92.6 kg) <III metacarpal (107.2 kg) <femur (353.1 kg). Feeding intensity and breed of animals had no significant effect on the value of this feature of the bones tested. The average bone density took the following order: femur (1.23 g/cm3) <III metatarsal bone (1.26 g/cm3) <III metacarpal bone (1.34 g / cm3). The density of bones of animals fed aa libitum was higher (P<0.01) than in animals fed with a dosing system. The density of examined bones within the breeds took the following order: Pietrain race> line 990> Duroc race. The differences between the “extreme” breeds were: 7.2% (III metatarsal bone), 8.3% (III metacarpal bone), 8.4% (femur). Abstract 13 The average bone stiffness took the following order: III metatarsal bone (35.1 kg/mm) <III metacarpus (41.5 kg/mm) <femur (60.5 kg/mm). This indicator did not differ between the groups of pigs fed at different intensity, except for the metacarpal bone, which was more stiffer in pigs fed aa libitum (P<0.05). The femur of animals fed ad libitum showed a tendency (P<0.09) to be more stiffer and a force of 4.5 kg required for its displacement by 1 mm. Breed differences in stiffness were found for the femur (P <0.05) and III metacarpal bone (P <0.05). For femur, the highest value of this indicator was found in Pietrain pigs (64.5 kg/mm), lower in pigs of 990 line (61.6 kg/mm) and the lowest in Duroc pigs (55.3 kg/mm). In turn, the 3rd metacarpal bone of Duroc and Pietrain pigs had similar stiffness (39.0 and 40.0 kg/mm respectively) and was smaller than that of line 990 pigs (45.4 kg/mm). The thickness of the cortical bone layer took the following order: III metatarsal bone (2.25 mm) <III metacarpal bone (2.41 mm) <femur (5.12 mm). The feeding system did not affect this indicator. Breed differences (P <0.05) for this trait were found only for the femur bone: Duroc (5.42 mm)> line 990 (5.13 mm)> Pietrain (4.81 mm). The cross sectional area of the examined bones was arranged in the following order: III metatarsal bone (84 mm2) <III metacarpal bone (90 mm2) <femur (286 mm2). The feeding system had no effect on the value of this bone trait, with the exception of the femur, which in animals fed the dosing system was 4.7% higher (P<0.05) than in pigs fed ad libitum. Breed differences (P<0.01) in the coross sectional area were found only in femur and III metatarsal bone. The value of this indicator was the highest in Duroc pigs, lower in 990 animals and the lowest in Pietrain pigs. The cortical index of individual bones was in the following order: III metatarsal bone (31.86) <III metacarpal bone (33.86) <femur (44.75). However, its value did not significantly depend on the intensity of feeding or the breed of pigs.

In this paper, the effect of seal clearance on the efficiency of a turbine with a shrouded rotor is compared with the effect of the tip clearance when the same turbine has an unshrouded rotor. The shrouded versus unshrouded comparison was undertaken for two turbine stage designs one having 50% reaction the other having 24% reaction. Measurements for a range of clearances, including very small clearances, showed three important phenomena. Firstly, as the clearance is reduced, there is a “break-even clearance” at which both the shrouded turbine and the unshrouded turbine have the same efficiency. If the clearance is reduced further, the unshrouded turbine performs better than the shrouded turbine, with the difference at zero clearance termed the “offset loss”. This is contrary to the traditional assumption that both shrouded and unshrouded turbines have the same efficiency at zero clearance. The physics of the break-even clearance and the offset loss are discussed. Secondly, the use of a lower reaction had the effect of reducing the tip leakage efficiency penalty for both the shrouded and the unshrouded turbines. In order to understand the effect of reaction on the tip leakage, an analytical model was used and it was found that the tip leakage efficiency penalty should be understood as the dissipated kinetic energy rather than either the tip leakage mass flow rate or the tip leakage loss coefficient. Thirdly, it was also observed that, at a fixed flow coefficient, the fractional change in the output power with clearance was approximately twice the fractional change in efficiency with clearance. This was explained by using an analytical model.

Many micromachines use rotating shafts and other moving parts which carry a load and need fluid bearings for support. Most of them operate with air or water as the lubricating fluid. The present study analyzes air microbearing represented as an eccentric cylinder rotating in a stationary housing. The fluid mechanics and operating characteristics of microbearing are different from their larger cousins. The small length-scale may invalidate the continuum approximation in Navier-Stokes equations, and slip flow, rarefaction, compressibility and other unconventional effects may have to be taken into account. Surface effects dominate in small devices due to a high surface-volume ratio. In this study, two-dimensional eccentric-shaft journal microbearings with different eccentricities are simulated by direct simulation Monte Carlo (DSMC) code incorporated with a Volume-CAD software. The diffuse reflection model and Cercignani-Lampis-Lord (CLL) model are applied to model the molecule-surface interaction by considering the accommodation coefficients of shaft wall and housing wall separately. The distribution of mean free path in the flow field indicates that the continuum model may break down and it is necessary to carry our molecular modeling. Calculation results show that at high eccentricity and high accommodation coefficient on the housing wall (ACO) the flow may develop a recirculation region. However, the accommodation coefficient on shaft wall (ACI) does not have any effect on the occurrence of recirculation and the size of recirculation zone. There is antisymmetry of the pressure about a vertical axis, which produces a pressure force on the shaft wall. The influence of ACI to isobars is larger than that of ACO. The shear stress profile on shaft wall is big at low ACI. At the region of short clearance between the shaft wall and housing wall, it is also influenced by the surface condition of housing wall and may even change its direction at low ACO. The pressure profile is reduced in amplitude as the ACI increases, but it is enhanced a little with the increase of ACO. The ACO has great impact on the viscous force in the case of big eccentricity. With the increase of ACI, the viscous force decreases. The pressure force is high at large eccentricity. The influence of ACO to pressure force is insignificant, but the pressure force fall is enormous when ACI increases, especially for large eccentricity. The total force decreases markedly at high eccentricity when ACI increases. The ACO almost has no impact on the total force. The torque increases with ACO, but decreases with ACI. The eccentricity also has great impact on the torque, and the microbeaing may have large torque at high eccentricity. The method developed in this paper would be very useful for designing and evaluating journal microbearing.

The flow and heat transfer in microchannels has been of great interest for some years now due to the significantly higher heat transfer coefficients useful for enhancing the heat transfer in very small but high heat flux applications such as electronics cooling. Nucleate boiling heat transfer in microchannels is also of great interest for generating even higher heat transfer rates; however, numerous studies have shown that the bubble formation immediately fills the entire microchannel with vapor significantly reducing the heat transfer since the bubble size is normally of the same size as the microchannel. The bubble growth process is very fast and difficult to study experimentally, even with high speed cameras. This study numerically analyzes the flow and bubble growth in a microchannel for various conditions by solving the Navier-Stokes equations with the VOF model with an analytical microlayer model to provide the large amount of vapor produced by the curved region of the microlayer. As each bubble forms, the large pressure drop around the bubble causes the bubble to quickly break away from the nucleation site and move quickly downstream. The bubbles are quite small with the size depending on the bulk flow velocity, subcooling and the heating rate. The numerical results compare quite well with preliminary experimental observations of bubble growth on a microheater embedded in the channel wall for FC-72 flowing in a microchannel.

Rotor bearing systems frequently utilise hydrodynamic bearings whose dynamic properties are generally influenced by the bearing reaction forces (which determine the bearing stiffness and damping coefficients). These reaction forces are frequently unknown and are generally determined from the solution of the Reynolds equation using rotor motion measurements as input. Of interest is the attainable accuracy of such bearing force determinations, and for experimental evaluation, a test rig was fabricated, the design specification of which required that the rotor system run stably over its operating speed range. This paper describes the commissioning of this rig for stability purposes with the aid of natural frequency analyses, noting the required design modifications to ensure stable operation. Stability was found to be significantly influenced by the extent of the continuous fluid film in the hydrodynamic circumferentially grooved bearings. It was concluded that the assumption of a 180 degree film extent was totally inappropriate even though the bearing ends were open to the atmosphere, whereas the assumption of fluid film break up at the lubricant saturation vapour pressure proved appropriate for stability predictions provided one ensured that the bearings were flooded. Preliminary bearing force evaluations proved inconclusive, primarily because the self aligning bearings nevertheless experienced angular misalignment; and because there was uncertainty as to how much air was entrained in the bearings, in spite of attempts to prevent air ingress.

Prediction of the mixing of multi-component fluids is important in many chemical process applications. Although laminar mixing is a complicated process per se (involving multi-component diffusion coefficients, for example), there is a far greater challenge in predicting mixing in turbulent flows because of their intrinsic, chaotic nature. In turbulent flows, large-scale eddies with coherent structures are mainly responsible for the mixing of passive scalars. The large-scale eddies embody themselves in the form of identifiable and organized distributions of vorticity. In addition, the mixing process involves all mechanisms typically found in vortex dynamics, such as stretching, break-up, concatenation, and self-induction of vortices. Experimental work suggests that large-scale, time-dependent structures, with periods much longer than the time of an impeller revolution, are involved in many of the fundamental hydrodynamic processes in stirred vessels. For example, local velocity data histograms may be bi-modal or tri-modal, even though they are being analyzed as having only one mode in most Laser-Doppler experiments. Digital particle image velocimetry experiments have shown that large-scale asymmetries with periods up to several minutes exist in stirred vessels equipped with axial flow impellers. These complex phenomena are not limited to single-phase systems. Many industrial vessels are operated with a multiphase flow. In such systems, the gas holdup distribution may be asymmetric and oscillating. In solids suspension processes, solids can be swept from one side of the vessel to the other in an oscillating pattern, even in dilute suspensions. The numerical modeling of these complicated mixing processes is a daunting task. Direct numerical simulation (DNS) provides the most exact approach in which the mechanism involved in turbulent mixing can be accurately represented. DNS requires resolving the smallest eddies, which makes the approach prohibitively expensive, even with the most powerful computers of the present day and foreseeable future as well. On the other hand, the popular approaches based on Reynolds-Averaged Navier-Stokes (RANS) equations amount to averaging out the large eddies that are primarily responsible for mixing. One is left to model the effects of large eddies by relying on empirical data and phenomenological reasoning and hypotheses, which are often questionable. The advantage of Large Eddy Simulation (LES) is that it explicitly resolves the large eddies, which are responsible for much of the mass, energy, and momentum transport, and only small eddies are modeled with a sub-grid model. In this lecture we will first briefly review the fundamentals of turbulent flows in stirred vessels, and how modeling these has evolved during the past decade. The focus will be on those aspects of turbulence that are relevant to mixing processes and the modeling thereof. We will continue with a discussion of the applicability of various turbulence models. For single-phase systems, we will then discuss the application of LES to the prediction of large-scale chaotic flow structures in stirred tanks. The focus of those studies is on systems with unsteady flows that are especially difficult to model with eddy-viscosity style models, namely those with strong swirl such as glass-lined mixing vessels (which usually have one baffle) and multiple impeller systems with strong interaction between the impeller flows. For multiphase systems, turbulence modeling is an even greater challenge. Interesting developments in this field include the use of LES models coupled with discrete particle simulations. More recently, full Reynolds stress models for use on unstructured finite volume meshes for Eulerian-Eulerian multiphase flow models have become available. Recent results with these models and expected future developments will be discussed.