Computer simulation of liquid flow patterns on distillation trays

This thesis describes work carried out to improve the fundamental modelling of liquid flows on distillation trays. A mathematical model is presented based on the principles of computerised fluid dynamics. It models the liquid flow in the horizontal directions allowing for the effects of the vapour through the use of an increased liquid turbulence, modelled by an eddy viscosity, and a resistance to liquid flow caused by the vapour being accelerated horizontally by the liquid. The resultant equations are similar to the Navier-Stokes equations with the addition of a resistance term.A mass-transfer model is used to calculate liquid concentration profiles and tray efficiencies. A heat and mass transfer analogy is used to compare theoretical concentration profiles to experimental water-cooling data obtained from a 2.44 metre diameter air-water distillation simulation rig. The ratios of air to water flow rates are varied in order to simulate three pressures: vacuum, atmospheric pressure and moderate pressure.For simulated atmospheric and moderate pressure distillation, the fluid mechanical model constantly over-predicts tray efficiencies with an accuracy of between +1.7% and +11.3%. This compares to -1.8% to -10.9% for the stagnant regions model (Porter et al. 1972) and +12.8% to +34.7% for the plug flow plus back-mixing model (Gerster et al. 1958). The model fails to predict the flow patterns and tray efficiencies for vacuum simulation due to the change in the mechanism of liquid transport, from a liquid continuous layer to a spray as the liquid flow-rate is reduced. This spray is not taken into account in the development of the fluid mechanical model. A sensitivity analysis carried out has shown that the fluid mechanical model is relatively insensitive to the prediction of the average height of clear liquid, and a reduction in the resistance term results in a slight loss of tray efficiency. But these effects are not great. The model is quite sensitive to the prediction of the eddy viscosity term. Variations can produce up to a 15% decrease in tray efficiency. The fluid mechanical model has been incorporated into a column model so that statistical optimisation techniques can be employed to fit a theoretical column concentration profile to experimental data. Through the use of this work mass-transfer data can be obtained.

Purification of plasmid DNA for use in human gene therapy

Two key issues defined the focus of this research in manufacturing plasmid DNA for use In human gene therapy. First, the processing of E.coli bacterial cells to effect the separation of therapeutic plasmid DNA from cellular debris and adventitious material. Second, the affinity purification of the plasmid DNA in a Simple one-stage process. The need arises when considering the concerns that have been recently voiced by the FDA concerning the scalability and reproducibility of the current manufacturing processes in meeting the quality criteria of purity, potency, efficacy, and safety for a recombinant drug substance for use in humans. To develop a preliminary purification procedure, an EFD cross-flow micro-filtration module was assessed for its ability to effect the 20-fold concentration, 6-time diafiltration, and final clarification of the plasmid DNA from the subsequent cell lysate that is derived from a 1 liter E.coli bacterial cell culture. Historically, the employment of cross-flow filtration modules within procedures for harvesting cells from bacterial cultures have failed to reach the required standards dictated by existing continuous centrifuge technologies, frequently resulting in the rapid blinding of the membrane with bacterial cells that substantially reduces the permeate flux. By challenging the EFD module, containing six helical wound tubular membranes promoting centrifugal instabilities known as Dean vortices, with distilled water between the Dean number's of 187Dn and 818Dn,and the transmembrane pressures (TMP) of 0 to 5 psi. The data demonstrated that the fluid dynamics significantly influenced the permeation rate, displaying a maximum at 227Dn (312 Imh) and minimum at 818Dn (130 Imh) for a transmembrane pressure of 1 psi. Numerical studies indicated that the initial increase and subsequent decrease resulted from a competition between the centrifugal and viscous forces that create the Dean vortices. At Dean numbers between 187Dn and 227Dn , the forces combine constructively to increase the apparent strength and influence of the Dean vortices. However, as the Dean number in increases above 227 On the centrifugal force dominates the viscous forces, compressing the Dean vortices into the membrane walls and reducing their influence on the radial transmembrane pressure i.e. the permeate flux reduced. When investigating the action of the Dean vortices in controlling tile fouling rate of E.coli bacterial cells, it was demonstrated that the optimum cross-flow rate at which to effect the concentration of a bacterial cell culture was 579Dn and 3 psi TMP, processing in excess of 400 Imh for 20 minutes (i.e., concentrating a 1L culture to 50 ml in 10 minutes at an average of 450 Imh). The data demonstrated that there was a conflict between the Dean number at which the shear rate could control the cell fouling, and the Dean number at which tile optimum flux enhancement was found. Hence, the internal geometry of the EFD module was shown to sub-optimal for this application. At 579Dn and 3 psi TMP, the 6-fold diafiltration was shown to occupy 3.6 minutes of process time, processing at an average flux of 400 Imh. Again, at 579Dn and 3 psi TMP the clarification of the plasmid from tile resulting freeze-thaw cell lysate was achieved at 120 Iml1, passing 83% (2,5 mg) of the plasmid DNA (6,3 ng μ-1 10.8 mg of genomic DNA (∼23,00 Obp, 36 ng μ-1 ), and 7.2 mg of cellular proteins (5-100 kDa, 21.4 ngμ-1 ) into the post-EFD process stream. Hence the EFD module was shown to be effective, achieving the desired objectives in approximately 25 minutes. On the basis of its ability to intercalate into low molecular weight dsDNA present in dilute cell lysates, and be electrophoresed through agarose, the fluorophore PicoGreen was selected for the development of a suitable dsDNA assay. It was assesseel for its accuracy, and reliability, In determining the concentration and identity of DNA present in samples that were eleclrophoresed through agarose gels. The signal emitted by intercalated PicoGreen was shown to be constant and linear, and that the mobility of the PicaGreen-DNA complex was not affected by the intercalation. Concerning the secondary purification procedure, various anion-exchange membranes were assessed for their ability to capture plasmid DNA from the post-EFD process stream. For a commercially available Sartorius Sartobind Q15 membrane, the reduction in the equilibriumbinding capacity for  ctDNA in buffer of increasing ionic demonstrated that DNA was being.adsorbed by electrostatic  interactions only. However, the problems associated with fluid distribution across the membrane demonstrated that the membrane housing was the predominant cause of the .erratic breakthrough curves. Consequently, this would need to be rectified before such a membrane could be integrated into the current system, or indeed be scaled beyond laboratory scale. However, when challenged with the process material, the data showed that considerable quantities of protein (1150 μg) were adsorbed preferentially to the plasmid DNA (44 μg). This was also shown for derived Pall Gelman UltraBind US450 membranes that had been functionalised by varying molecular weight poly-L~lysine and polyethyleneimine ligands. Hence the anion-exchange membranes were shown to be ineffective in capturing plasmid DNA from the process stream. Finally, work was performed to integrate a sequence-specific DNA·binding protein into a single-stage DNA chromatography, isolating plasmid DNA from E.coli cells whilst minimising the contamination from genomic DNA and cellular protein. Preliminary work demonstrated that the fusion protein was capable of isolating pUC19 DNA into which the recognition sequence for the fusion-protein had been inserted (pTS DNA) when in the presence of the conditioned process material. Althougth the pTS recognition sequence differs from native pUC19 sequences by only 2 bp, the fusion protein was shown to act as a highly selective affinity ligand for pTS DNA alone. Subsequently, the scale of the process was scaled 25-fold and positioned directly following the EFD system. In conclusion, the integration of the EFD micro-filtration system and zinc-finger affinity purification technique resulted in the capture of approximately 1 mg of plasmid DNA was purified from 1L of E.coli  culture in a simple two stage process, resulting in the complete removal of genomic DNA and 96.7% of cellular protein in less than 1 hour of process time.

Hydrodynamic modeling of mineral wool fiber suspensions in a two-dimensional channel flow

A consequence of a loss of coolant accident is the damage of adjacent insulation materials (IM). IM may then be transported to the containment sump strainers where water is drawn into the ECCS (emergency core cooling system). Blockage of the strainers by IM lead to an increased pressure drop acting on the operating ECCS pumps. IM can also penetrate the strainers, enter the reactor coolant system and then accumulate in the reactor pressure vessel. An experimental and theoretical study that concentrates on mineral wool fiber transport in the containment sump and the ECCS is being performed. The study entails fiber generation and the assessment of fiber transport in single and multi-effect experiments. The experiments include measurement of the terminal settling velocity, the strainer pressure drop, fiber sedimentation and resuspension in a channel flow and jet flow in a rectangular tank. An integrated test facility is also operated to assess the compounded effects. Each experimental facility is used to provide data for the validation of equivalent computational fluid dynamic models. The channel flow facility allows the determination of the steady state distribution of the fibers at different flow velocities. The fibers are modeled in the Eulerian-Eulerian reference frame as spherical wetted agglomerates. The fiber agglomerate size, density, the relative viscosity of the fluid-fiber mixture and the turbulent dispersion of the fibers all affect the steady state accumulation of fibers at the channel base. In the current simulations, two fiber phases are separately considered. The particle size is kept constant while the density is modified, which affects both the terminal velocity and volume fraction. The relative viscosity is only significant at higher concentrations. The numerical model finds that the fibers accumulate at the channel base even at high velocities; therefore, modifications to the drag and turbulent dispersion forces can be made to reduce fiber accumulation.

Time scale analysis for fluidizedbedmeltgranulation-II: binder spreading rate

The spreading time of liquid binder droplet on the surface a primary particle is analyzed for Fluidized Bed Melt Granulation (FBMG). As discussed in the first paper of this series (Chua et al., in press) the droplet spreading rate has been identified as one of the important parameters affecting the probability of particles aggregation in FBMG. In this paper, the binder droplet spreading time has been estimated using Computational Fluid Dynamic modeling (CFD) based on Volume of Fluid approach (VOF). A simplified analytical solution has been developed and tested to explore its validity for predicting the spreading time. For the purpose of models validation, the droplet spreading evolution was recorded using a high speed video camera. Based on the validated model, a generalized correlative equation for binder spreading time is proposed. For the operating conditions considered here, the spreading time for Polyethylene Glycol (PEG1500) binder was found to fall within the range of 10-2 to 10-5 s. The study also included a number of other common binders used in FBMG. The results obtained here will be further used in paper III, where the binder solidification rate is discussed.

Thin film bearings with phase change of lubricant

The possible evaporation of lubricant in fluid film bearings has been investigated theoretically and by experiment using a radial flow hydrostatic bearing supplied with liquid refrigerant R114. Good correlation between measured and theoretical values was obtained using a bespoke computational fluid dynamic model in which the flow was assumed to be laminar and adiabatic. The effects of viscous dissipation and vapour generation within the fluid film are fully accounted for by applying a fourth order Runge-Kutta routine to satisfy the radial and filmwise transverse constraints of momentum, energy and mass conservation. The results indicate that the radial velocity profile remains parabolic while the flow remains in the liquid phase and that the radial rate of enthalpy generation is then constant across the film at a given radius. The results also show that evaporation will commence at a radial location determined by geometry and flow conditions and in fluid layers adjacent to the solid boundaries. Evaporation is shown to progress in the radial direction and the load carrying capacity of such a bearing is reduced significantly. Expressions for the viscosity of the liquid/vapour mixture found in the literature survey have not been tested against experimental data. A new formulation is proposed in which the suitable choice of a characteristic constant yields close representation to any of these expressions. Operating constraints imposed by the design of the experimental apparatus limited the extent of the surface over which evaporation could be obtained, and prevented clear identification of the most suitable relationship for the viscosity of the liquid/vapour mixture. The theoretical model was extended to examine the development of two phase flow in a rotating shaft face seal of uniform thickness. Previous theoretical analyses have been based on the assumption that the radial velocity profile of the flow is always parabolic, and that the tangential component of velocity varies linearly from the value at the rotating surface, to zero at the stationary surface. The computational fluid dynamic analysis shows that viscous shear and dissipation in the fluid adjacent to the rotating surface leads to developing evaporation with a consequent reduction in tangential shear forces. The tangential velocity profile is predicted to decay rapidly through the film, exhibiting a profile entirely different to that assumed by previous investigators. Progressive evaporation takes place close to the moving wall and does not occur completely at a single radial location, as has been claimed in earlier work.

Numerical and experimental investigations for insulation particle transport phenomena in water flow

The investigation of insulation debris generation, transport and sedimentation becomes more important with regard to reactor safety research for pressurized and boiling water reactors, when considering the long-term behaviour of emergency core coolant systems during all types of loss of coolant accidents (LOCA). The insulation debris released near the break during a LOCA incident consists of a mixture of a disparate particle population that varies with size, shape, consistency and other properties. Some fractions of the released insulation debris can be transported into the reactor sump, where it may perturb or impinge on the emergency core cooling systems. Open questions of generic interest are for example the particle load on strainers and corresponding pressure-drop, the sedimentation of the insulation debris in a water pool, its possible re-suspension and transport in the sump water flow. A joint research project on such questions is being performed in cooperation with the University of Applied Science Zittau/Görlitz and the Forschungszentrum Dresden-Rossendorf. The project deals with the experimental investigation and the development of computational fluid dynamic (CFD) models for the description of particle transport phenomena in coolant flow. While the experiments are performed at the University Zittau/Görlitz, the theoretical work is concentrated at Forschungszentrum Dresden-Rossendorf. In the present paper, the basic concepts for computational fluid dynamic (CFD) modelling are described and experimental results are presented. Further experiments are designed and feasibility studies were performed.

Computationsl modelling of dimethyl ether separation and steam reforming in fluidized bed reactors

This study presents a computational fluid dynamic (CFD) study of Dimethyl Ether (DME) gas adsorptive separation and steam reforming (DME-SR) in a large scale Circulating Fluidized Bed (CFB) reactor. The CFD model is based on Eulerian-Eulerian dispersed flow and solved using commercial software (ANSYS FLUENT). Hydrogen is currently receiving increasing interest as an alternative source of clean energy and has high potential applications, including the transportation sector and power generation. Computational fluid dynamic (CFD) modelling has attracted considerable recognition in the engineering sector consequently leading to using it as a tool for process design and optimisation in many industrial processes. In most cases, these processes are difficult or expensive to conduct in lab scale experiments. The CFD provides a cost effective methodology to gain detailed information up to the microscopic level. The main objectives in this project are to: (i) develop a predictive model using ANSYS FLUENT (CFD) commercial code to simulate the flow hydrodynamics, mass transfer, reactions and heat transfer in a large scale dual fluidized bed system for combined gas separation and steam reforming processes (ii) implement a suitable adsorption models in the CFD code, through a user defined function, to predict selective separation of a gas from a mixture (iii) develop a model for dimethyl ether steam reforming (DME-SR) to predict hydrogen production (iv) carry out detailed parametric analysis in order to establish ideal operating conditions for future industrial application. The project has originated from a real industrial case problem in collaboration with the industrial partner Dow Corning (UK) and jointly funded by the Engineering and Physical Research Council (UK) and Dow Corning. The research examined gas separation by adsorption in a bubbling bed, as part of a dual fluidized bed system. The adsorption process was simulated based on the kinetics derived from the experimental data produced as part of a separate PhD project completed under the same fund. The kinetic model was incorporated in FLUENT CFD tool as a pseudo-first order rate equation; some of the parameters for the pseudo-first order kinetics were obtained using MATLAB. The modelling of the DME adsorption in the designed bubbling bed was performed for the first time in this project and highlights the novelty in the investigations. The simulation results were analysed to provide understanding of the flow hydrodynamic, reactor design and optimum operating condition for efficient separation. Bubbling bed validation by estimation of bed expansion and the solid and gas distribution from simulation agreed well with trends seen in the literatures. Parametric analysis on the adsorption process demonstrated that increasing fluidizing velocity reduced adsorption of DME. This is as a result of reduction in the gas residence time which appears to have much effect compared to the solid residence time. The removal efficiency of DME from the bed was found to be more than 88%. Simulation of the DME-SR in FLUENT CFD was conducted using selected kinetics from literature and implemented in the model using an in-house developed user defined function. The validation of the kinetics was achieved by simulating a case to replicate an experimental study of a laboratory scale bubbling bed by Vicente et al [1]. Good agreement was achieved for the validation of the models, which was then applied in the DME-SR in the large scale riser section of the dual fluidized bed system. This is the first study to use the selected DME-SR kinetics in a circulating fluidized bed (CFB) system and for the geometry size proposed for the project. As a result, the simulation produced the first detailed data on the spatial variation and final gas product in such an industrial scale fluidized bed system. The simulation results provided insight in the flow hydrodynamic, reactor design and optimum operating condition. The solid and gas distribution in the CFB was observed to show good agreement with literatures. The parametric analysis showed that the increase in temperature and steam to DME molar ratio increased the production of hydrogen due to the increased DME conversions, whereas the increase in the space velocity has been found to have an adverse effect. Increasing temperature between 200 oC to 350 oC increased DME conversion from 47% to 99% while hydrogen yield increased substantially from 11% to 100%. The CO2 selectivity decreased from 100% to 91% due to the water gas shift reaction favouring CO at higher temperatures. The higher conversions observed as the temperature increased was reflected on the quantity of unreacted DME and methanol concentrations in the product gas, where both decreased to very low values of 0.27 mol% and 0.46 mol% respectively at 350 °C. Increasing the steam to DME molar ratio from 4 to 7.68 increased the DME conversion from 69% to 87%, while the hydrogen yield increased from 40% to 59%. The CO2 selectivity decreased from 100% to 97%. The decrease in the space velocity from 37104 ml/g/h to 15394 ml/g/h increased the DME conversion from 87% to 100% while increasing the hydrogen yield from 59% to 87%. The parametric analysis suggests an operating condition for maximum hydrogen yield is in the region of 300 oC temperatures and Steam/DME molar ratio of 5. The analysis of the industrial sponsor’s case for the given flow and composition of the gas to be treated suggests that 88% of DME can be adsorbed from the bubbling and consequently producing 224.4t/y of hydrogen in the riser section of the dual fluidized bed system. The process also produces 1458.4t/y of CO2 and 127.9t/y of CO as part of the product gas. The developed models and parametric analysis carried out in this study provided essential guideline for future design of DME-SR at industrial level and in particular this work has been of tremendous importance for the industrial collaborator in order to draw conclusions and plan for future potential implementation of the process at an industrial scale.

The charge effect on the hindrance factors for diffusion and convection of a solute in pores II

The diffusion and convection of a solute suspended in a fluid across porous membranes are known to be reduced compared to those in a bulk solution, owing to the fluid mechanical interaction between the solute and the pore wall as well as steric restriction. If the solute and the pore wall are electrically charged, the electrostatic interaction between them could affect the hindrance to diffusion and convection. In this study, the transport of charged spherical solutes through charged circular cylindrical pores filled with an electrolyte solution containing small ions was studied numerically by using a fluid mechanical and electrostatic model. Based on a mean field theory, the electrostatic interaction energy between the solute and the pore wall was estimated from the Poisson-Boltzmann equation, and the charge effect on the solute transport was examined for the solute and pore wall of like charge. The results were compared with those obtained from the linearized form of the Poisson-Boltzmann equation, i.e.the Debye-Hückel equation. © 2012 The Japan Society of Fluid Mechanics and IOP Publishing Ltd.

Deflection of jets discharged into a reservoir with a free surface

Deflections of jets discharged into a reservoir with a free surface are investigated numerically. The jets are known to deflect towards either side of the free surface or the bottom, whose direction is not determined uniquely in some experimental conditions, i.e. there are multiple stable states realizable in the same condition. The origin of the multiple stable states is explored by utilizing homotopy transformations in which the top boundary of the reservoir is transformed from a rigid to a free boundary and also the location of the outlet throat is continuously moved from mid-height to the top. We depicted bifurcation diagrams of the flow compiling the data of numerical simulations, from which we identified the origin as an imperfect pitchfork bifurcation, and obtained an insight into the mechanism for the direction to be determined. The parameter region where such multiple stable states are possible is also delimited. © 2011 The Japan Society of Fluid Mechanics and IOP Publishing Ltd.

Charge effects on hindrance factors for diffusion and convection of solute in pores I

The transport of a spherical solute through a long circular cylindrical pore filled with an electrolyte solution is studied numerically, in the presence of constant surface charge on the solute and the pore wall. Fluid dynamic analyses were carried out to calculate the flow field around the solute in the pore to evaluate the drag coefficients exerted on the solute. Electrical potentials around the solute in the electrolyte solution were computed based on a mean-field theory to provide the interaction energy between the charged solute and the pore wall. Combining the results of the fluid dynamic and electrostatic analyses, we estimated the rate of the diffusive and convective transport of the solute across the pore. Although the present estimates of the drag coefficients on the solute suggest more than 10% difference from existing studies, depending on the radius ratio of the solute relative to the pore and the radial position of the solute center in the pore, this difference leads to a minor effect on the hindrance factors. It was found that even at rather large ion concentrations, the repulsive electrostatic interaction between the charged solute and the pore wall of like charge could significantly reduce the transport rate of the solute.

Breakage model development and application with CFD for predicting breakage of whey protein precipitate particles

Particle breakage due to fluid flow through various geometries can have a major influence on the performance of particle/fluid processes and on the product quality characteristics of particle/fluid products. In this study, whey protein precipitate dispersions were used as a case study to investigate the effect of flow intensity and exposure time on the breakage of these precipitate particles. Computational fluid dynamic (CFD) simulations were performed to evaluate the turbulent eddy dissipation rate (TED) and associated exposure time along various flow geometries. The focus of this work is on the predictive modelling of particle breakage in particle/fluid systems. A number of breakage models were developed to relate TED and exposure time to particle breakage. The suitability of these breakage models was evaluated for their ability to predict the experimentally determined breakage of the whey protein precipitate particles. A "power-law threshold" breakage model was found to provide a satisfactory capability for predicting the breakage of the whey protein precipitate particles. The whey protein precipitate dispersions were propelled through a number of different geometries such as bends, tees and elbows, and the model accurately predicted the mean particle size attained after flow through these geometries. © 2005 Elsevier Ltd. All rights reserved.

Vortex shedding from a row of square bars

Interactions of wakes in a flow past a row of square bars, which is placed across a uniform flow, are investigated by numerical simulations and experiments on the tassumption that the flow is two-dimensional and incompressible. At small Reynolds numbers the flow is steady and symmetric with respect not only to streamwise lines through the center of each square bar but also to streamwise centerlines between adjacent square bars. However, the steady symmetric flow becomes unstable at larger Reynolds numbers and make a transition to a steady asymmetric flow with respect to the centerlines between adjacent square bars in some cases or to an oscillatory flow in other cases. It is found that vortices are shed synchronously from adjacent square bars in the same phase or in anti-phase depending upon the distance between the bars when the flow is oscillatory. The origin of the transition to the steady asymmetric flow is identified as a pitchfork bifurcation, while the oscillatory flows with synchronous shedding of vortices are clarified to originate from a Hopf bifurcation. The critical Reynolds numbers of the transitions are evaluated numerically and the bifurcation diagram of the flow is obtained.

Modeling of Cylindrical Couette Flow of Rarefied Gas Between Rotating Cylinders

Петър Господинов, Добри Данков, Владимир Русинов, Стефан Стефанов - Изследвано е цилиндрично течение на Кует за разреден газ между два въртящи се цилиндъра. Получени са профилите на налягането, скоростта и температурата по метода на прякото статистическо моделиране (DSMC) и чрез числено решаване на уравненията на Навие-Стокс за свиваем флуид. Резултатите сочат много добро съвпадение за малки числа на Кнудсен Kn = 0.02. Показано е, че при различни кинематични гранични условия, газът изостава или избързва спрямо скоростта на стената, или има поведение на твърдо еластично тяло. Получените резултати са важни при решаването на неравнинни, задачи от микрофлуидиката с отчитане на ефектите на кривината.

Numerical Analysis of Chemical Reactions by Monte Carlo Simulation

Добри Данков, Владимир Русинов, Мария Велинова, Жасмина Петрова - Изследвана е химическа реакция чрез два начина за моделиране на вероятността за химическа реакция използвайки Direct Simulation Monte Carlo метод. Изследван е порядъка на разликите при температурите и концентрациите чрез тези начини. Когато активността на химическата реакция намалява, намаляват и разликите между концентрациите и температурите получени по двата начина. Ключови думи: Механика на флуидите, Кинетична теория, Разреден газ, DSMC

Comparative Analysis of the Opposite Rotating Cylinders Impact over Couette Flow for Rarefied Gas

Петър Господинов, Добри Данков, Владимир Русинов, Стефан Стефанов - Иследвано е цилиндрично течение на Кует на разреден газ в случая на въртене на два коаксиални цилиндъра с еднакви по големина скорости, но в различни посоки. Целта на изследването е да се установи влиянието на малки скорости на въртене върху макрохарактеристиките – ρ, V , . Числените резултати са получени чрез използване на DSMC и числено решение на уравненията на Навие-Стокс за относително малки (дозвукови) скорости на въртене. Установено е добро съвпадение на резултатите получени по двата метода за Kn = 0.02. Установено е, че съществува “стационарна” точка за плътността и скоростта. Получените резултати са важни при решаването на неравнини, задачи от микрофлуидиката с отчитане на ефектите на кривината. Ключови думи: Механика на флуидите, Кинетична теория, Разреден газ, DSMC